Properties

Label 304.2.a
Level $304$
Weight $2$
Character orbit 304.a
Rep. character $\chi_{304}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $7$
Sturm bound $80$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(80\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(304))\).

Total New Old
Modular forms 46 9 37
Cusp forms 35 9 26
Eisenstein series 11 0 11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(19\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(6\)

Trace form

\( 9 q - 2 q^{5} - 2 q^{7} + 9 q^{9} + O(q^{10}) \) \( 9 q - 2 q^{5} - 2 q^{7} + 9 q^{9} + 2 q^{11} - 2 q^{13} + 12 q^{15} + 2 q^{17} + 3 q^{19} - 8 q^{21} - 4 q^{23} + 3 q^{25} - 2 q^{29} - 8 q^{33} + 18 q^{35} - 2 q^{37} + 4 q^{39} + 2 q^{41} - 14 q^{43} - 18 q^{45} + 14 q^{47} + q^{49} + 4 q^{51} + 6 q^{53} - 18 q^{55} - 16 q^{59} - 10 q^{61} - 30 q^{63} + 12 q^{65} - 20 q^{67} - 24 q^{69} + 4 q^{71} + 2 q^{73} - 4 q^{75} - 8 q^{77} - 8 q^{79} + 17 q^{81} + 4 q^{83} - 12 q^{85} + 12 q^{87} + 18 q^{89} - 40 q^{91} + 8 q^{93} - 8 q^{95} + 10 q^{97} + 42 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(304))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 19
304.2.a.a 304.a 1.a $1$ $2.427$ \(\Q\) None \(0\) \(-2\) \(-1\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+3q^{7}+q^{9}-5q^{11}+\cdots\)
304.2.a.b 304.a 1.a $1$ $2.427$ \(\Q\) None \(0\) \(-1\) \(0\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{7}-2q^{9}-2q^{11}+q^{13}+\cdots\)
304.2.a.c 304.a 1.a $1$ $2.427$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{7}-2q^{9}+6q^{11}+5q^{13}+\cdots\)
304.2.a.d 304.a 1.a $1$ $2.427$ \(\Q\) None \(0\) \(1\) \(-4\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}-3q^{7}-2q^{9}-2q^{11}+\cdots\)
304.2.a.e 304.a 1.a $1$ $2.427$ \(\Q\) None \(0\) \(2\) \(-1\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+3q^{7}+q^{9}+3q^{11}+\cdots\)
304.2.a.f 304.a 1.a $1$ $2.427$ \(\Q\) None \(0\) \(2\) \(3\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+3q^{5}+q^{7}+q^{9}-3q^{11}+\cdots\)
304.2.a.g 304.a 1.a $3$ $2.427$ 3.3.961.1 None \(0\) \(-1\) \(1\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{2}q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(304))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(304)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 2}\)