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Properties

Label 1176.3.f
Level $1176$
Weight $3$
Character orbit 1176.f
Rep. character $\chi_{1176}(97,\cdot)$
Character field $\Q$
Dimension $40$
Newform subspaces $4$
Sturm bound $672$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1176.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(672\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1176, [\chi])\).

Total New Old
Modular forms 480 40 440
Cusp forms 416 40 376
Eisenstein series 64 0 64

Trace form

\( 40 q - 120 q^{9} + O(q^{10}) \) \( 40 q - 120 q^{9} + 16 q^{11} - 24 q^{15} - 96 q^{23} - 256 q^{25} + 64 q^{29} + 8 q^{37} + 48 q^{39} - 144 q^{43} + 48 q^{51} + 48 q^{53} + 120 q^{57} + 528 q^{65} + 48 q^{67} - 32 q^{71} - 56 q^{79} + 360 q^{81} - 96 q^{85} - 216 q^{93} + 416 q^{95} - 48 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(1176, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1176.3.f.a 1176.f 7.b $8$ $32.044$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+(-\beta _{2}-\beta _{5})q^{5}-3q^{9}+(-3+\cdots)q^{11}+\cdots\)
1176.3.f.b 1176.f 7.b $8$ $32.044$ 8.0.339738624.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{3}+(2\beta _{2}-\beta _{5}-\beta _{7})q^{5}-3q^{9}+\cdots\)
1176.3.f.c 1176.f 7.b $8$ $32.044$ 8.0.\(\cdots\).9 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(\beta _{1}-\beta _{7})q^{5}-3q^{9}+(5+\beta _{2}+\cdots)q^{11}+\cdots\)
1176.3.f.d 1176.f 7.b $16$ $32.044$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(-\beta _{2}+\beta _{11})q^{5}-3q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1176, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1176, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)