Properties

Label 324.3.d
Level $324$
Weight $3$
Character orbit 324.d
Rep. character $\chi_{324}(163,\cdot)$
Character field $\Q$
Dimension $44$
Newform subspaces $9$
Sturm bound $162$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(162\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(324, [\chi])\).

Total New Old
Modular forms 120 52 68
Cusp forms 96 44 52
Eisenstein series 24 8 16

Trace form

\( 44 q + 2 q^{4} + O(q^{10}) \) \( 44 q + 2 q^{4} + 4 q^{10} + 4 q^{13} - 10 q^{16} + 78 q^{22} + 144 q^{25} + 48 q^{28} - 2 q^{34} + 16 q^{37} + 112 q^{40} - 60 q^{46} - 136 q^{49} - 44 q^{52} + 52 q^{58} + 124 q^{61} - 262 q^{64} - 168 q^{70} + 16 q^{73} - 54 q^{76} - 50 q^{82} - 64 q^{85} - 306 q^{88} - 336 q^{94} + 124 q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
324.3.d.a \(2\) \(8.828\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-1}) \) \(-4\) \(0\) \(-8\) \(0\) \(q-2q^{2}+4q^{4}+(-4+\beta )q^{5}-8q^{8}+\cdots\)
324.3.d.b \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(8\) \(0\) \(q+(-1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}+4q^{5}+\cdots\)
324.3.d.c \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-8\) \(0\) \(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}-4q^{5}+\cdots\)
324.3.d.d \(2\) \(8.828\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-1}) \) \(4\) \(0\) \(8\) \(0\) \(q+2q^{2}+4q^{4}+(4+\beta )q^{5}+8q^{8}+(8+\cdots)q^{10}+\cdots\)
324.3.d.e \(6\) \(8.828\) 6.0.3636603.2 None \(-1\) \(0\) \(-2\) \(0\) \(q-\beta _{3}q^{2}-\beta _{1}q^{4}+(-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
324.3.d.f \(6\) \(8.828\) 6.0.3636603.2 None \(1\) \(0\) \(2\) \(0\) \(q-\beta _{1}q^{2}-\beta _{2}q^{4}+(-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
324.3.d.g \(8\) \(8.828\) 8.0.\(\cdots\).1 None \(-3\) \(0\) \(6\) \(0\) \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{5}-\beta _{6})q^{4}+\cdots\)
324.3.d.h \(8\) \(8.828\) 8.0.\(\cdots\).4 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}+(-1-\beta _{6})q^{4}+(-2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
324.3.d.i \(8\) \(8.828\) 8.0.\(\cdots\).1 None \(3\) \(0\) \(-6\) \(0\) \(q+\beta _{6}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{4}+\beta _{6})q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(324, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)