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Properties

Label 162.7.d
Level $162$
Weight $7$
Character orbit 162.d
Rep. character $\chi_{162}(53,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $48$
Newform subspaces $7$
Sturm bound $189$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 7 \)
Sturm bound: \(189\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(162, [\chi])\).

Total New Old
Modular forms 348 48 300
Cusp forms 300 48 252
Eisenstein series 48 0 48

Trace form

\( 48 q + 768 q^{4} - 1200 q^{7} + O(q^{10}) \) \( 48 q + 768 q^{4} - 1200 q^{7} - 8400 q^{13} - 24576 q^{16} - 14520 q^{19} + 7200 q^{22} + 96312 q^{25} - 76800 q^{28} - 70680 q^{31} + 54720 q^{34} - 59808 q^{37} - 208380 q^{43} + 270144 q^{46} - 539064 q^{49} + 268800 q^{52} - 1751976 q^{55} - 796320 q^{58} + 56112 q^{61} - 1572864 q^{64} - 528924 q^{67} - 185760 q^{70} + 241800 q^{73} - 232320 q^{76} + 3161292 q^{79} - 787104 q^{82} - 2051784 q^{85} - 230400 q^{88} - 4259328 q^{91} + 1473696 q^{94} - 3107676 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(162, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
162.7.d.a 162.d 9.d $4$ $37.269$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 54.7.b.b \(0\) \(0\) \(0\) \(-778\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2\beta _{1}-2\beta _{3})q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+\cdots\)
162.7.d.b 162.d 9.d $4$ $37.269$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 6.7.b.a \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}-\beta _{3})q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+30\beta _{1}q^{5}+\cdots\)
162.7.d.c 162.d 9.d $4$ $37.269$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 54.7.b.a \(0\) \(0\) \(0\) \(410\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{3})q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+\cdots\)
162.7.d.d 162.d 9.d $4$ $37.269$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 18.7.b.a \(0\) \(0\) \(0\) \(968\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-4\beta _{1}+4\beta _{3})q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+\cdots\)
162.7.d.e 162.d 9.d $8$ $37.269$ \(\Q(\zeta_{24})\) None 54.7.b.c \(0\) \(0\) \(0\) \(-836\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{24}^{5}q^{2}+2^{5}\zeta_{24}^{2}q^{4}+(5\zeta_{24}-5\zeta_{24}^{3}+\cdots)q^{5}+\cdots\)
162.7.d.f 162.d 9.d $8$ $37.269$ \(\Q(\zeta_{24})\) None 162.7.b.a \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q+4\zeta_{24}^{3}q^{2}+(2^{5}-2^{5}\zeta_{24})q^{4}+(95\zeta_{24}^{5}+\cdots)q^{5}+\cdots\)
162.7.d.g 162.d 9.d $16$ $37.269$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 162.7.b.b \(0\) \(0\) \(0\) \(-964\) $\mathrm{SU}(2)[C_{6}]$ \(q+4\beta _{8}q^{2}+(2^{5}-2^{5}\beta _{2})q^{4}+(-12\beta _{8}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(162, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(162, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)