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Let f be a newform for $\Gamma_0(4)$ with a trivial character.

I guess that the eigenvalue $\lambda(2)$ for $T_2$ is 0. I want to know that this is known result or not. If so, could you explain or give a reference?

Edit: 1)As Ramsey pointed out, here $T_2$ means $U_2$ operator. 2)I am doing some research for automorphic L-functions for $\Gamma_0(4)$ and I could see that automorphic forms with trivial character has zero 2nd coefficient..so I asked the above question..

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# 2nd eigenvalues for cusp forms for $\Gamma_0(4)$

Let f be a newform for $\Gamma_0(4)$ with a trivial character.

I guess that the eigenvalue $\lambda(2)$ for $T_2$ is 0. I want to know that this is known result or not. If so, could you explain or give a reference?