You have a good answer; here are some computations about primitive representations by two forms, $g_0$ is the underlying $x^2 + y^2 + 3 z^2 + xy,$ while $g_1$ is your $x^2 + 3 y^2 + 3 z^2.$ I was a little surprised to find a distinction requiring mod 72, mod 36 was not enough. Compare $7 \pmod {72}$ to $43 \pmod {72}.$ Alright, it amounts to $\pmod {24}.$ When I have time I will compute the cases divisible by 2 or 3 or both.
Made a little summary: the ratios $(r_0/h(4M), r_1/h(4M))$ are split three ways when $\gcd(M,6) = 1.$ We get ratios $(12,4)$ when $M \equiv 1, 13,25,37,49,61 \pmod {72},$ which is all $M \equiv 1 \pmod {12}.$ We get ratios $(24,16)$ when $M \equiv 7,31,55 \pmod {72},$ which is all $M \equiv 7 \pmod {24}.$We get ratios $(16,8)$ when $M \equiv 19,43,67 \pmod {72},$ which is all $M \equiv 19 \pmod {24}.$
---------------------------------------
g0 : 9 : 1 1 3 0 0 1 auto 24
g1 : 36 : 1 3 3 0 0 0 auto 16
---------------------------------------
---------------------------------------
M r0 r1 h(4M) %72 r0/h r1/h
1 6 2 1 N 1 6 2 1 = 1
3 8 4 1 N 3 8 4 3 = 3
4 12 8 1 N 4 12 8 4 = 2^2
6 12 4 2 N 6 6 2 6 = 2 * 3
7 24 16 1 N 7 24 16 7 = 7
10 24 8 2 N10 12 4 10 = 2 * 5
12 12 8 2 N12 6 4 12 = 2^2 * 3
13 24 8 2 N13 12 4 13 = 13
15 24 16 2 N15 12 8 15 = 3 * 5
16 24 16 2 N16 12 8 16 = 2^4
19 48 24 3 N19 16 8 19 = 19
21 24 8 4 N21 6 2 21 = 3 * 7
22 24 8 2 N22 12 4 22 = 2 * 11
-------------------------------------------------------------------------
1 6 2 1 N 1 6 2 1 = 1
73 48 16 4 N 1 12 4 73 = 73
145 96 32 8 N 1 12 4 145 = 5 * 29
217 96 32 8 N 1 12 4 217 = 7 * 31
289 96 32 8 N 1 12 4 289 = 17^2
361 120 40 10 N 1 12 4 361 = 19^2
433 144 48 12 N 1 12 4 433 = 433
505 96 32 8 N 1 12 4 505 = 5 * 101
577 96 32 8 N 1 12 4 577 = 577
649 240 80 20 N 1 12 4 649 = 11 * 59
721 192 64 16 N 1 12 4 721 = 7 * 103
793 96 32 8 N 1 12 4 793 = 13 * 61
865 192 64 16 N 1 12 4 865 = 5 * 173
937 240 80 20 N 1 12 4 937 = 937
1009 240 80 20 N 1 12 4 1009 = 1009
1081 192 64 16 N 1 12 4 1081 = 23 * 47
1153 192 64 16 N 1 12 4 1153 = 1153
1225 192 64 16 N 1 12 4 1225 = 5^2 * 7^2
1297 144 48 12 N 1 12 4 1297 = 1297
1369 216 72 18 N 1 12 4 1369 = 37^2
------------------------------------------------------------------------
7 24 16 1 N 7 24 16 7 = 7
79 120 80 5 N 7 24 16 79 = 79
151 168 112 7 N 7 24 16 151 = 151
223 168 112 7 N 7 24 16 223 = 223
295 192 128 8 N 7 24 16 295 = 5 * 59
367 216 144 9 N 7 24 16 367 = 367
439 360 240 15 N 7 24 16 439 = 439
511 336 224 14 N 7 24 16 511 = 7 * 73
583 192 128 8 N 7 24 16 583 = 11 * 53
655 288 192 12 N 7 24 16 655 = 5 * 131
727 312 208 13 N 7 24 16 727 = 727
799 384 256 16 N 7 24 16 799 = 17 * 47
871 528 352 22 N 7 24 16 871 = 13 * 67
943 384 256 16 N 7 24 16 943 = 23 * 41
1015 384 256 16 N 7 24 16 1015 = 5 * 7 * 29
1087 216 144 9 N 7 24 16 1087 = 1087
1159 384 256 16 N 7 24 16 1159 = 19 * 61
1231 648 432 27 N 7 24 16 1231 = 1231
1303 264 176 11 N 7 24 16 1303 = 1303
1375 480 320 20 N 7 24 16 1375 = 5^3 * 11
------------------------------------------------------------------
13 24 8 2 N13 12 4 13 = 13
85 48 16 4 N13 12 4 85 = 5 * 17
157 72 24 6 N13 12 4 157 = 157
229 120 40 10 N13 12 4 229 = 229
301 96 32 8 N13 12 4 301 = 7 * 43
373 120 40 10 N13 12 4 373 = 373
445 96 32 8 N13 12 4 445 = 5 * 89
517 144 48 12 N13 12 4 517 = 11 * 47
589 192 64 16 N13 12 4 589 = 19 * 31
661 216 72 18 N13 12 4 661 = 661
733 168 56 14 N13 12 4 733 = 733
805 192 64 16 N13 12 4 805 = 5 * 7 * 23
877 120 40 10 N13 12 4 877 = 877
949 144 48 12 N13 12 4 949 = 13 * 73
1021 264 88 22 N13 12 4 1021 = 1021
1093 120 40 10 N13 12 4 1093 = 1093
1165 240 80 20 N13 12 4 1165 = 5 * 233
1237 168 56 14 N13 12 4 1237 = 1237
1309 288 96 24 N13 12 4 1309 = 7 * 11 * 17
1381 312 104 26 N13 12 4 1381 = 1381
------------------------------------------------------------------
19 48 24 3 N19 16 8 19 = 19
91 96 48 6 N19 16 8 91 = 7 * 13
163 48 24 3 N19 16 8 163 = 163
235 96 48 6 N19 16 8 235 = 5 * 47
307 144 72 9 N19 16 8 307 = 307
379 144 72 9 N19 16 8 379 = 379
451 288 144 18 N19 16 8 451 = 11 * 41
523 240 120 15 N19 16 8 523 = 523
595 192 96 12 N19 16 8 595 = 5 * 7 * 17
667 192 96 12 N19 16 8 667 = 23 * 29
739 240 120 15 N19 16 8 739 = 739
811 336 168 21 N19 16 8 811 = 811
883 144 72 9 N19 16 8 883 = 883
955 192 96 12 N19 16 8 955 = 5 * 191
1027 192 96 12 N19 16 8 1027 = 13 * 79
1099 288 144 18 N19 16 8 1099 = 7 * 157
1171 336 168 21 N19 16 8 1171 = 1171
1243 192 96 12 N19 16 8 1243 = 11 * 113
1315 288 144 18 N19 16 8 1315 = 5 * 263
1387 192 96 12 N19 16 8 1387 = 19 * 73
------------------------------------------------------------------
25 24 8 2 N25 12 4 25 = 5^2
97 48 16 4 N25 12 4 97 = 97
169 72 24 6 N25 12 4 169 = 13^2
241 144 48 12 N25 12 4 241 = 241
313 96 32 8 N25 12 4 313 = 313
385 96 32 8 N25 12 4 385 = 5 * 7 * 11
457 96 32 8 N25 12 4 457 = 457
529 144 48 12 N25 12 4 529 = 23^2
601 240 80 20 N25 12 4 601 = 601
673 144 48 12 N25 12 4 673 = 673
745 192 64 16 N25 12 4 745 = 5 * 149
817 144 48 12 N25 12 4 817 = 19 * 43
889 192 64 16 N25 12 4 889 = 7 * 127
961 192 64 16 N25 12 4 961 = 31^2
1033 144 48 12 N25 12 4 1033 = 1033
1105 192 64 16 N25 12 4 1105 = 5 * 13 * 17
1177 144 48 12 N25 12 4 1177 = 11 * 107
1249 384 128 32 N25 12 4 1249 = 1249
1321 288 96 24 N25 12 4 1321 = 1321
1393 192 64 16 N25 12 4 1393 = 7 * 199
-------------------------------------------------------------------
31 72 48 3 N31 24 16 31 = 31
103 120 80 5 N31 24 16 103 = 103
175 144 96 6 N31 24 16 175 = 5^2 * 7
247 144 96 6 N31 24 16 247 = 13 * 19
319 240 160 10 N31 24 16 319 = 11 * 29
391 336 224 14 N31 24 16 391 = 17 * 23
463 168 112 7 N31 24 16 463 = 463
535 336 224 14 N31 24 16 535 = 5 * 107
607 312 208 13 N31 24 16 607 = 607
679 432 288 18 N31 24 16 679 = 7 * 97
751 360 240 15 N31 24 16 751 = 751
823 216 144 9 N31 24 16 823 = 823
895 384 256 16 N31 24 16 895 = 5 * 179
967 264 176 11 N31 24 16 967 = 967
1039 552 368 23 N31 24 16 1039 = 1039
1111 528 352 22 N31 24 16 1111 = 11 * 101
1183 336 224 14 N31 24 16 1183 = 7 * 13^2
1255 288 192 12 N31 24 16 1255 = 5 * 251
1327 360 240 15 N31 24 16 1327 = 1327
1399 648 432 27 N31 24 16 1399 = 1399
---------------------------------------------------------------
37 24 8 2 N37 12 4 37 = 37
109 72 24 6 N37 12 4 109 = 109
181 120 40 10 N37 12 4 181 = 181
253 48 16 4 N37 12 4 253 = 11 * 23
325 144 48 12 N37 12 4 325 = 5^2 * 13
397 72 24 6 N37 12 4 397 = 397
469 192 64 16 N37 12 4 469 = 7 * 67
541 120 40 10 N37 12 4 541 = 541
613 120 40 10 N37 12 4 613 = 613
685 144 48 12 N37 12 4 685 = 5 * 137
757 120 40 10 N37 12 4 757 = 757
829 264 88 22 N37 12 4 829 = 829
901 288 96 24 N37 12 4 901 = 17 * 53
973 144 48 12 N37 12 4 973 = 7 * 139
1045 192 64 16 N37 12 4 1045 = 5 * 11 * 19
1117 168 56 14 N37 12 4 1117 = 1117
1189 240 80 20 N37 12 4 1189 = 29 * 41
1261 240 80 20 N37 12 4 1261 = 13 * 97
1333 240 80 20 N37 12 4 1333 = 31 * 43
1405 288 96 24 N37 12 4 1405 = 5 * 281
------------------------------------------------------------------
43 48 24 3 N43 16 8 43 = 43
115 96 48 6 N43 16 8 115 = 5 * 23
187 96 48 6 N43 16 8 187 = 11 * 17
259 192 96 12 N43 16 8 259 = 7 * 37
331 144 72 9 N43 16 8 331 = 331
403 96 48 6 N43 16 8 403 = 13 * 31
475 192 96 12 N43 16 8 475 = 5^2 * 19
547 144 72 9 N43 16 8 547 = 547
619 240 120 15 N43 16 8 619 = 619
691 240 120 15 N43 16 8 691 = 691
763 192 96 12 N43 16 8 763 = 7 * 109
835 288 144 18 N43 16 8 835 = 5 * 167
907 144 72 9 N43 16 8 907 = 907
979 384 192 24 N43 16 8 979 = 11 * 89
1051 240 120 15 N43 16 8 1051 = 1051
1123 240 120 15 N43 16 8 1123 = 1123
1195 384 192 24 N43 16 8 1195 = 5 * 239
1267 288 144 18 N43 16 8 1267 = 7 * 181
1339 384 192 24 N43 16 8 1339 = 13 * 103
1411 192 96 12 N43 16 8 1411 = 17 * 83
-------------------------------------------------------------------
49 48 16 4 N49 12 4 49 = 7^2
121 72 24 6 N49 12 4 121 = 11^2
193 48 16 4 N49 12 4 193 = 193
265 96 32 8 N49 12 4 265 = 5 * 53
337 96 32 8 N49 12 4 337 = 337
409 192 64 16 N49 12 4 409 = 409
481 192 64 16 N49 12 4 481 = 13 * 37
553 96 32 8 N49 12 4 553 = 7 * 79
625 120 40 10 N49 12 4 625 = 5^4
697 96 32 8 N49 12 4 697 = 17 * 41
769 240 80 20 N49 12 4 769 = 769
841 168 56 14 N49 12 4 841 = 29^2
913 144 48 12 N49 12 4 913 = 11 * 83
985 288 96 24 N49 12 4 985 = 5 * 197
1057 192 64 16 N49 12 4 1057 = 7 * 151
1129 192 64 16 N49 12 4 1129 = 1129
1201 192 64 16 N49 12 4 1201 = 1201
1273 240 80 20 N49 12 4 1273 = 19 * 67
1345 192 64 16 N49 12 4 1345 = 5 * 269
1417 192 64 16 N49 12 4 1417 = 13 * 109
------------------------------------------------------------------
55 96 64 4 N55 24 16 55 = 5 * 11
127 120 80 5 N55 24 16 127 = 127
199 216 144 9 N55 24 16 199 = 199
271 264 176 11 N55 24 16 271 = 271
343 168 112 7 N55 24 16 343 = 7^3
415 240 160 10 N55 24 16 415 = 5 * 83
487 168 112 7 N55 24 16 487 = 487
559 384 256 16 N55 24 16 559 = 13 * 43
631 312 208 13 N55 24 16 631 = 631
703 336 224 14 N55 24 16 703 = 19 * 37
775 288 192 12 N55 24 16 775 = 5^2 * 31
847 240 160 10 N55 24 16 847 = 7 * 11^2
919 456 304 19 N55 24 16 919 = 919
991 408 272 17 N55 24 16 991 = 991
1063 456 304 19 N55 24 16 1063 = 1063
1135 432 288 18 N55 24 16 1135 = 5 * 227
1207 432 288 18 N55 24 16 1207 = 17 * 71
1279 552 368 23 N55 24 16 1279 = 1279
1351 576 384 24 N55 24 16 1351 = 7 * 193
1423 216 144 9 N55 24 16 1423 = 1423
--------------------------------------------------------------
61 72 24 6 N61 12 4 61 = 61
133 48 16 4 N61 12 4 133 = 7 * 19
205 96 32 8 N61 12 4 205 = 5 * 41
277 72 24 6 N61 12 4 277 = 277
349 168 56 14 N61 12 4 349 = 349
421 120 40 10 N61 12 4 421 = 421
493 144 48 12 N61 12 4 493 = 17 * 29
565 144 48 12 N61 12 4 565 = 5 * 113
637 144 48 12 N61 12 4 637 = 7^2 * 13
709 120 40 10 N61 12 4 709 = 709
781 240 80 20 N61 12 4 781 = 11 * 71
853 120 40 10 N61 12 4 853 = 853
925 144 48 12 N61 12 4 925 = 5^2 * 37
997 168 56 14 N61 12 4 997 = 997
1069 360 120 30 N61 12 4 1069 = 1069
1141 288 96 24 N61 12 4 1141 = 7 * 163
1213 120 40 10 N61 12 4 1213 = 1213
1285 144 48 12 N61 12 4 1285 = 5 * 257
1357 192 64 16 N61 12 4 1357 = 23 * 59
1429 264 88 22 N61 12 4 1429 = 1429
----------------------------------------------------------------
67 48 24 3 N67 16 8 67 = 67
139 144 72 9 N67 16 8 139 = 139
211 144 72 9 N67 16 8 211 = 211
283 144 72 9 N67 16 8 283 = 283
355 192 96 12 N67 16 8 355 = 5 * 71
427 96 48 6 N67 16 8 427 = 7 * 61
499 144 72 9 N67 16 8 499 = 499
571 240 120 15 N67 16 8 571 = 571
643 144 72 9 N67 16 8 643 = 643
715 192 96 12 N67 16 8 715 = 5 * 11 * 13
787 240 120 15 N67 16 8 787 = 787
859 336 168 21 N67 16 8 859 = 859
931 288 144 18 N67 16 8 931 = 7^2 * 19
1003 192 96 12 N67 16 8 1003 = 17 * 59
1075 288 144 18 N67 16 8 1075 = 5^2 * 43
1147 288 144 18 N67 16 8 1147 = 31 * 37
1219 288 144 18 N67 16 8 1219 = 23 * 53
1291 432 216 27 N67 16 8 1291 = 1291
1363 288 144 18 N67 16 8 1363 = 29 * 47
1435 192 96 12 N67 16 8 1435 = 5 * 7 * 41
-----------------------------------------------------------------------
M r0 r1 h(4M) %72 r0/h r1/h
---------------------------------------
g0 : 9 : 1 1 3 0 0 1 auto 24
g1 : 36 : 1 3 3 0 0 0 auto 16
---------------------------------------
---------------------------------------
M r0 r1 h(4M) %72 r0/h r1/h