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Properties

Label 322.2.e
Level $322$
Weight $2$
Character orbit 322.e
Rep. character $\chi_{322}(93,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $32$
Newform subspaces $4$
Sturm bound $96$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(322, [\chi])\).

Total New Old
Modular forms 104 32 72
Cusp forms 88 32 56
Eisenstein series 16 0 16

Trace form

\( 32 q - 16 q^{4} - 8 q^{6} - 4 q^{7} - 12 q^{9} + O(q^{10}) \) \( 32 q - 16 q^{4} - 8 q^{6} - 4 q^{7} - 12 q^{9} - 4 q^{10} + 8 q^{11} + 4 q^{14} + 16 q^{15} - 16 q^{16} - 16 q^{17} + 20 q^{21} + 16 q^{22} + 4 q^{24} - 28 q^{25} + 12 q^{26} - 24 q^{27} + 8 q^{28} - 12 q^{30} + 4 q^{31} - 24 q^{33} - 12 q^{35} + 24 q^{36} - 4 q^{38} + 4 q^{39} - 4 q^{40} - 24 q^{41} - 4 q^{42} - 24 q^{43} + 8 q^{44} + 12 q^{45} + 16 q^{47} + 20 q^{49} + 4 q^{51} - 16 q^{53} + 16 q^{54} + 40 q^{55} + 4 q^{56} + 24 q^{57} + 24 q^{59} - 8 q^{60} - 28 q^{61} - 24 q^{62} + 28 q^{63} + 32 q^{64} + 28 q^{65} - 32 q^{66} + 8 q^{67} - 16 q^{68} - 16 q^{69} - 20 q^{70} + 40 q^{73} + 16 q^{74} - 24 q^{75} + 8 q^{77} + 16 q^{78} + 8 q^{79} + 40 q^{83} - 4 q^{84} - 88 q^{85} + 16 q^{86} - 8 q^{88} - 12 q^{89} - 8 q^{90} - 60 q^{91} + 48 q^{93} + 4 q^{94} + 12 q^{95} + 4 q^{96} + 72 q^{97} + 24 q^{98} - 144 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(322, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
322.2.e.a 322.e 7.c $8$ $2.571$ 8.0.310217769.2 None \(-4\) \(-3\) \(-7\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{5})q^{2}+(\beta _{1}+\beta _{2}-\beta _{5})q^{3}+\cdots\)
322.2.e.b 322.e 7.c $8$ $2.571$ 8.0.1767277521.3 None \(-4\) \(5\) \(5\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{5}q^{2}+(1+\beta _{2}-\beta _{5}+\beta _{6})q^{3}+(-1+\cdots)q^{4}+\cdots\)
322.2.e.c 322.e 7.c $8$ $2.571$ 8.0.1767277521.3 None \(4\) \(-1\) \(-3\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{6})q^{2}+(\beta _{4}+\beta _{5}-\beta _{6}-\beta _{7})q^{3}+\cdots\)
322.2.e.d 322.e 7.c $8$ $2.571$ 8.0.6498455769.2 None \(4\) \(-1\) \(5\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{5})q^{2}-\beta _{1}q^{3}-\beta _{5}q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(322, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(322, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(161, [\chi])\)\(^{\oplus 2}\)