Properties

Label 306.2.a
Level $306$
Weight $2$
Character orbit 306.a
Rep. character $\chi_{306}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $6$
Sturm bound $108$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 306.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(108\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(306))\).

Total New Old
Modular forms 62 8 54
Cusp forms 47 8 39
Eisenstein series 15 0 15

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

\(2\)\(3\)\(17\)FrickeDim
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(1\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(7\)

Trace form

\( 8 q + 8 q^{4} + 6 q^{5} + 4 q^{7} + O(q^{10}) \) \( 8 q + 8 q^{4} + 6 q^{5} + 4 q^{7} + 2 q^{10} - 2 q^{11} + 4 q^{13} + 4 q^{14} + 8 q^{16} + 2 q^{17} + 8 q^{19} + 6 q^{20} + 2 q^{22} + 4 q^{25} - 4 q^{26} + 4 q^{28} + 14 q^{29} - 4 q^{31} - 2 q^{34} - 8 q^{35} - 18 q^{37} + 2 q^{40} - 12 q^{41} - 4 q^{43} - 2 q^{44} - 24 q^{46} - 16 q^{47} + 8 q^{49} + 12 q^{50} + 4 q^{52} - 4 q^{53} - 40 q^{55} + 4 q^{56} - 6 q^{58} - 12 q^{59} - 26 q^{61} - 20 q^{62} + 8 q^{64} - 28 q^{65} - 12 q^{67} + 2 q^{68} - 32 q^{70} - 24 q^{73} + 10 q^{74} + 8 q^{76} + 24 q^{77} + 8 q^{79} + 6 q^{80} - 4 q^{82} - 4 q^{83} + 2 q^{85} - 28 q^{86} + 2 q^{88} + 32 q^{89} - 24 q^{91} - 16 q^{94} + 24 q^{95} - 20 q^{97} - 8 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(306))\) 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 3 17
306.2.a.a 306.a 1.a $1$ $2.443$ \(\Q\) None \(-1\) \(0\) \(0\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-4q^{7}-q^{8}-6q^{11}+2q^{13}+\cdots\)
306.2.a.b 306.a 1.a $1$ $2.443$ \(\Q\) None \(-1\) \(0\) \(2\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}-q^{8}-2q^{10}+4q^{11}+\cdots\)
306.2.a.c 306.a 1.a $1$ $2.443$ \(\Q\) None \(1\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{7}+q^{8}+2q^{13}+2q^{14}+\cdots\)
306.2.a.d 306.a 1.a $1$ $2.443$ \(\Q\) None \(1\) \(0\) \(4\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+4q^{5}-2q^{7}+q^{8}+4q^{10}+\cdots\)
306.2.a.e 306.a 1.a $2$ $2.443$ \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+\beta q^{5}+(2+\beta )q^{7}-q^{8}+\cdots\)
306.2.a.f 306.a 1.a $2$ $2.443$ \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+\beta q^{5}+(2-\beta )q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(306))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(306)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 2}\)