Properties

Label 322.2.a
Level $322$
Weight $2$
Character orbit 322.a
Rep. character $\chi_{322}(1,\cdot)$
Character field $\Q$
Dimension $11$
Newform subspaces $7$
Sturm bound $96$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(322))\).

Total New Old
Modular forms 52 11 41
Cusp forms 45 11 34
Eisenstein series 7 0 7

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

\(2\)\(7\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(4\)
Minus space\(-\)\(7\)

Trace form

\( 11 q + 3 q^{2} + 11 q^{4} - 2 q^{5} - q^{7} + 3 q^{8} + 15 q^{9} + O(q^{10}) \) \( 11 q + 3 q^{2} + 11 q^{4} - 2 q^{5} - q^{7} + 3 q^{8} + 15 q^{9} + 6 q^{10} - 2 q^{13} - q^{14} - 16 q^{15} + 11 q^{16} - 10 q^{17} + 7 q^{18} - 8 q^{19} - 2 q^{20} + 4 q^{21} - q^{23} - 3 q^{25} - 2 q^{26} - 24 q^{27} - q^{28} + 2 q^{29} - 16 q^{31} + 3 q^{32} + 14 q^{34} - 2 q^{35} + 15 q^{36} - 10 q^{37} + 8 q^{38} - 40 q^{39} + 6 q^{40} - 18 q^{41} - 4 q^{42} - 8 q^{43} + 6 q^{45} + 3 q^{46} + 8 q^{47} + 11 q^{49} + 5 q^{50} + 32 q^{51} - 2 q^{52} - 18 q^{53} + 24 q^{54} - 8 q^{55} - q^{56} - 14 q^{58} - 16 q^{60} + 14 q^{61} - 16 q^{62} - 13 q^{63} + 11 q^{64} + 36 q^{65} - 16 q^{66} - 10 q^{68} - 4 q^{69} - 2 q^{70} - 24 q^{71} + 7 q^{72} - 10 q^{73} - 10 q^{74} - 24 q^{75} - 8 q^{76} + 12 q^{77} - 8 q^{78} - 32 q^{79} - 2 q^{80} + 3 q^{81} + 30 q^{82} + 16 q^{83} + 4 q^{84} + 44 q^{85} + 24 q^{86} + 24 q^{87} + 6 q^{89} - 34 q^{90} + 10 q^{91} - q^{92} + 24 q^{93} + 8 q^{94} - 8 q^{95} + 6 q^{97} + 3 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(322))\) 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 7 23
322.2.a.a 322.a 1.a $1$ $2.571$ \(\Q\) None \(-1\) \(0\) \(-2\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
322.2.a.b 322.a 1.a $1$ $2.571$ \(\Q\) None \(-1\) \(2\) \(0\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-2q^{6}+q^{7}-q^{8}+\cdots\)
322.2.a.c 322.a 1.a $1$ $2.571$ \(\Q\) None \(1\) \(-2\) \(-2\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}+q^{4}-2q^{5}-2q^{6}-q^{7}+\cdots\)
322.2.a.d 322.a 1.a $1$ $2.571$ \(\Q\) None \(1\) \(2\) \(-2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+q^{7}+\cdots\)
322.2.a.e 322.a 1.a $2$ $2.571$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(-2\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
322.2.a.f 322.a 1.a $2$ $2.571$ \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
322.2.a.g 322.a 1.a $3$ $2.571$ 3.3.316.1 None \(3\) \(2\) \(4\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+(1-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(322))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(322)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 2}\)