Properties

Label 324.3.f
Level $324$
Weight $3$
Character orbit 324.f
Rep. character $\chi_{324}(55,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $92$
Newform subspaces $19$
Sturm bound $162$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 36 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 19 \)
Sturm bound: \(162\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

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

Total New Old
Modular forms 240 100 140
Cusp forms 192 92 100
Eisenstein series 48 8 40

Trace form

\( 92q + 2q^{4} + O(q^{10}) \) \( 92q + 2q^{4} - 20q^{10} + 4q^{13} + 2q^{16} - 6q^{22} - 186q^{25} + 60q^{28} + 40q^{34} - 8q^{37} - 98q^{40} + 432q^{46} + 242q^{49} + 52q^{52} - 80q^{58} + 4q^{61} - 412q^{64} - 54q^{70} - 56q^{73} - 420q^{76} - 416q^{82} + 104q^{85} - 186q^{88} + 60q^{94} + 124q^{97} + O(q^{100}) \)

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.f.a \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(-4\) \(0\) \(-2\) \(12\) \(q-2q^{2}+4q^{4}+(-2+2\zeta_{6})q^{5}+(8+\cdots)q^{7}+\cdots\)
324.3.f.b \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(-4\) \(0\) \(7\) \(-15\) \(q-2q^{2}+4q^{4}+(7-7\zeta_{6})q^{5}+(-10+\cdots)q^{7}+\cdots\)
324.3.f.c \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-7\) \(15\) \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-7+7\zeta_{6})q^{5}+\cdots\)
324.3.f.d \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(2\) \(-12\) \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(2-2\zeta_{6})q^{5}+\cdots\)
324.3.f.e \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-1}) \) \(-2\) \(0\) \(8\) \(0\) \(q-2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}+(8-8\zeta_{6})q^{5}+\cdots\)
324.3.f.f \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-1}) \) \(2\) \(0\) \(-8\) \(0\) \(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}+(-8+8\zeta_{6})q^{5}+\cdots\)
324.3.f.g \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-2\) \(-12\) \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-2+2\zeta_{6})q^{5}+\cdots\)
324.3.f.h \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(7\) \(15\) \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(7-7\zeta_{6})q^{5}+\cdots\)
324.3.f.i \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(-7\) \(-15\) \(q+2q^{2}+4q^{4}+(-7+7\zeta_{6})q^{5}+(-10+\cdots)q^{7}+\cdots\)
324.3.f.j \(2\) \(8.828\) \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(2\) \(12\) \(q+2q^{2}+4q^{4}+(2-2\zeta_{6})q^{5}+(8-4\zeta_{6})q^{7}+\cdots\)
324.3.f.k \(4\) \(8.828\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(-4\) \(0\) \(-8\) \(0\) \(q+(-2+2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+(-4\zeta_{12}+\cdots)q^{5}+\cdots\)
324.3.f.l \(4\) \(8.828\) \(\Q(\sqrt{-3}, \sqrt{13})\) None \(-3\) \(0\) \(0\) \(0\) \(q+(-1+\beta _{1})q^{2}+(-\beta _{1}+3\beta _{2}+\beta _{3})q^{4}+\cdots\)
324.3.f.m \(4\) \(8.828\) \(\Q(\sqrt{-3}, \sqrt{13})\) None \(3\) \(0\) \(0\) \(0\) \(q+(1-\beta _{1})q^{2}+(-\beta _{1}+3\beta _{2}+\beta _{3})q^{4}+\cdots\)
324.3.f.n \(4\) \(8.828\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(4\) \(0\) \(8\) \(0\) \(q+(2-2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+(4\zeta_{12}+\cdots)q^{5}+\cdots\)
324.3.f.o \(8\) \(8.828\) 8.0.207360000.1 None \(0\) \(0\) \(0\) \(-12\) \(q+(-\beta _{2}+\beta _{3})q^{2}+(-1+\beta _{1}+\beta _{7})q^{4}+\cdots\)
324.3.f.p \(8\) \(8.828\) 8.0.207360000.1 None \(0\) \(0\) \(0\) \(12\) \(q-\beta _{2}q^{2}+(1-\beta _{5}-\beta _{7})q^{4}+(\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
324.3.f.q \(12\) \(8.828\) 12.0.\(\cdots\).2 None \(-1\) \(0\) \(-2\) \(0\) \(q+(-\beta _{3}+\beta _{6})q^{2}+(1+\beta _{2}+\beta _{4})q^{4}+\cdots\)
324.3.f.r \(12\) \(8.828\) 12.0.\(\cdots\).2 None \(1\) \(0\) \(2\) \(0\) \(q+(\beta _{3}-\beta _{6})q^{2}+(1+\beta _{2}+\beta _{4})q^{4}+(1+\cdots)q^{5}+\cdots\)
324.3.f.s \(16\) \(8.828\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{5}q^{2}+(1-\beta _{2}-\beta _{3}-\beta _{7})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)

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

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