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Properties

Label 324.2.b
Level $324$
Weight $2$
Character orbit 324.b
Rep. character $\chi_{324}(323,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $3$
Sturm bound $108$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(108\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(324, [\chi])\).

Total New Old
Modular forms 66 28 38
Cusp forms 42 20 22
Eisenstein series 24 8 16

Trace form

\( 20 q + 2 q^{4} + O(q^{10}) \) \( 20 q + 2 q^{4} - 8 q^{10} + 4 q^{13} + 14 q^{16} + 6 q^{22} - 24 q^{28} + 10 q^{34} + 4 q^{37} + 4 q^{40} - 24 q^{46} + 8 q^{49} - 32 q^{52} - 20 q^{58} - 8 q^{61} + 2 q^{64} - 24 q^{70} + 4 q^{73} - 54 q^{76} + 46 q^{82} - 28 q^{85} - 18 q^{88} - 12 q^{94} - 8 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
324.2.b.a 324.b 12.b $4$ $2.587$ \(\Q(\sqrt{-2}, \sqrt{3})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{2}-2q^{4}-\beta _{2}q^{5}-2\beta _{1}q^{8}+\cdots\)
324.2.b.b 324.b 12.b $8$ $2.587$ 8.0.170772624.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}-\beta _{2}q^{4}+(\beta _{3}-\beta _{4}+\beta _{7})q^{5}+\cdots\)
324.2.b.c 324.b 12.b $8$ $2.587$ 8.0.5780865024.3 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{3}+\beta _{5})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(324, [\chi]) \cong \)