Properties

Label 1296.4.c
Level $1296$
Weight $4$
Character orbit 1296.c
Rep. character $\chi_{1296}(1295,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $8$
Sturm bound $864$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(864\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1296, [\chi])\).

Total New Old
Modular forms 684 72 612
Cusp forms 612 72 540
Eisenstein series 72 0 72

Trace form

\( 72 q + O(q^{10}) \) \( 72 q - 1800 q^{25} + 1188 q^{37} - 3528 q^{49} + 1404 q^{61} + 2484 q^{73} + 3996 q^{85} + 108 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1296, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1296.4.c.a 1296.c 12.b $2$ $76.466$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-7\zeta_{6}q^{5}+5\zeta_{6}q^{7}-39q^{11}+43q^{13}+\cdots\)
1296.4.c.b 1296.c 12.b $2$ $76.466$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-7\zeta_{6}q^{5}-5\zeta_{6}q^{7}+39q^{11}+43q^{13}+\cdots\)
1296.4.c.c 1296.c 12.b $4$ $76.466$ \(\Q(\sqrt{-2}, \sqrt{3})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(2\beta _{1}-\beta _{2})q^{5}+(46+\beta _{3})q^{13}+(-\beta _{1}+\cdots)q^{17}+\cdots\)
1296.4.c.d 1296.c 12.b $8$ $76.466$ 8.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(3\beta _{1}+\beta _{3})q^{5}+\beta _{4}q^{7}+\beta _{5}q^{11}+\cdots\)
1296.4.c.e 1296.c 12.b $10$ $76.466$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{2}-\beta _{3})q^{5}+(2\beta _{2}+\beta _{3}+\beta _{9})q^{7}+\cdots\)
1296.4.c.f 1296.c 12.b $10$ $76.466$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{2}-\beta _{3})q^{5}+(-2\beta _{2}-\beta _{3}-\beta _{9})q^{7}+\cdots\)
1296.4.c.g 1296.c 12.b $12$ $76.466$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{5}-\beta _{2}q^{7}+\beta _{8}q^{11}+(4+\beta _{4}+\cdots)q^{13}+\cdots\)
1296.4.c.h 1296.c 12.b $24$ $76.466$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{4}^{\mathrm{old}}(1296, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1296, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 2}\)