Properties

Label 162.13.d
Level $162$
Weight $13$
Character orbit 162.d
Rep. character $\chi_{162}(53,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $96$
Newform subspaces $8$
Sturm bound $351$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 8 \)
Sturm bound: \(351\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(162, [\chi])\).

Total New Old
Modular forms 672 96 576
Cusp forms 624 96 528
Eisenstein series 48 0 48

Trace form

\( 96 q + 98304 q^{4} + 343200 q^{7} + O(q^{10}) \) \( 96 q + 98304 q^{4} + 343200 q^{7} - 4250400 q^{13} - 201326592 q^{16} + 308198640 q^{19} + 134668800 q^{22} + 1794454512 q^{25} + 1405747200 q^{28} - 3001349040 q^{31} + 755573760 q^{34} + 27365822400 q^{37} - 4733380680 q^{43} - 33186650112 q^{46} - 34079455152 q^{49} + 8704819200 q^{52} + 301214327184 q^{55} - 7326144000 q^{58} + 6311563104 q^{61} - 824633720832 q^{64} + 173066025720 q^{67} - 64914186240 q^{70} - 284072089200 q^{73} + 315595407360 q^{76} - 22027137144 q^{79} + 753291878400 q^{82} - 31677547824 q^{85} - 275801702400 q^{88} - 1979828714496 q^{91} - 311763336192 q^{94} + 522094640520 q^{97} + O(q^{100}) \)

Decomposition of \(S_{13}^{\mathrm{new}}(162, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
162.13.d.a 162.d 9.d $4$ $148.067$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-67144\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2^{5}\beta _{1}+2^{5}\beta _{3})q^{2}+(2^{11}-2^{11}\beta _{2}+\cdots)q^{4}+\cdots\)
162.13.d.b 162.d 9.d $4$ $148.067$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(98744\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2^{5}\beta _{1}+2^{5}\beta _{3})q^{2}+(2^{11}-2^{11}\beta _{2}+\cdots)q^{4}+\cdots\)
162.13.d.c 162.d 9.d $8$ $148.067$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-271484\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}-2^{11}\beta _{1}q^{4}+(\beta _{2}+139\beta _{3}+\cdots)q^{5}+\cdots\)
162.13.d.d 162.d 9.d $8$ $148.067$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(-153080\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{3}+\beta _{6})q^{2}+(2^{11}+2^{11}\beta _{1})q^{4}+\cdots\)
162.13.d.e 162.d 9.d $8$ $148.067$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(68284\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(2^{11}-2^{11}\beta _{1})q^{4}+(60\beta _{2}+\cdots)q^{5}+\cdots\)
162.13.d.f 162.d 9.d $16$ $148.067$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(393320\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{8}q^{2}-2^{11}\beta _{1}q^{4}+(-10^{2}\beta _{8}+10^{2}\beta _{9}+\cdots)q^{5}+\cdots\)
162.13.d.g 162.d 9.d $24$ $148.067$ None \(0\) \(0\) \(0\) \(54336\) $\mathrm{SU}(2)[C_{6}]$
162.13.d.h 162.d 9.d $24$ $148.067$ None \(0\) \(0\) \(0\) \(220224\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{13}^{\mathrm{old}}(162, [\chi])\) into lower level spaces

\( S_{13}^{\mathrm{old}}(162, [\chi]) \cong \) \(S_{13}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{13}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)