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Properties

Label 162.8.c
Level $162$
Weight $8$
Character orbit 162.c
Rep. character $\chi_{162}(55,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $56$
Newform subspaces $18$
Sturm bound $216$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 18 \)
Sturm bound: \(216\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(162, [\chi])\).

Total New Old
Modular forms 402 56 346
Cusp forms 354 56 298
Eisenstein series 48 0 48

Trace form

\( 56 q - 1792 q^{4} - 830 q^{7} + O(q^{10}) \) \( 56 q - 1792 q^{4} - 830 q^{7} - 18470 q^{13} - 114688 q^{16} + 227608 q^{19} - 70512 q^{22} - 293512 q^{25} + 106240 q^{28} - 128498 q^{31} + 177456 q^{34} + 482788 q^{37} + 840772 q^{43} - 1204032 q^{46} - 2481558 q^{49} - 1182080 q^{52} + 488124 q^{55} + 1730640 q^{58} - 2167046 q^{61} + 14680064 q^{64} + 7691980 q^{67} + 9850320 q^{70} - 1526792 q^{73} - 7283456 q^{76} + 34746808 q^{79} + 967680 q^{82} - 5422374 q^{85} - 4512768 q^{88} + 56885120 q^{91} + 11585184 q^{94} + 45244606 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(162, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
162.8.c.a 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 2.8.a.a \(-8\) \(0\) \(-210\) \(-1016\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}-210\zeta_{6}q^{5}+\cdots\)
162.8.c.b 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 54.8.a.a \(-8\) \(0\) \(-120\) \(-377\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}-120\zeta_{6}q^{5}+\cdots\)
162.8.c.c 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 54.8.a.b \(-8\) \(0\) \(105\) \(937\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}+105\zeta_{6}q^{5}+\cdots\)
162.8.c.d 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 6.8.a.a \(-8\) \(0\) \(114\) \(1576\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}+114\zeta_{6}q^{5}+\cdots\)
162.8.c.e 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 162.8.a.a \(-8\) \(0\) \(165\) \(508\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}+165\zeta_{6}q^{5}+\cdots\)
162.8.c.f 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 54.8.a.c \(-8\) \(0\) \(312\) \(-323\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}+312\zeta_{6}q^{5}+\cdots\)
162.8.c.g 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 54.8.a.c \(8\) \(0\) \(-312\) \(-323\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8-8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}-312\zeta_{6}q^{5}+\cdots\)
162.8.c.h 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 162.8.a.a \(8\) \(0\) \(-165\) \(508\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8-8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}-165\zeta_{6}q^{5}+\cdots\)
162.8.c.i 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 6.8.a.a \(8\) \(0\) \(-114\) \(1576\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8-8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}-114\zeta_{6}q^{5}+\cdots\)
162.8.c.j 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 54.8.a.b \(8\) \(0\) \(-105\) \(937\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8-8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}-105\zeta_{6}q^{5}+\cdots\)
162.8.c.k 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 54.8.a.a \(8\) \(0\) \(120\) \(-377\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8-8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}+120\zeta_{6}q^{5}+\cdots\)
162.8.c.l 162.c 9.c $2$ $50.606$ \(\Q(\sqrt{-3}) \) None 2.8.a.a \(8\) \(0\) \(210\) \(-1016\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8-8\zeta_{6})q^{2}-2^{6}\zeta_{6}q^{4}+210\zeta_{6}q^{5}+\cdots\)
162.8.c.m 162.c 9.c $4$ $50.606$ \(\Q(\sqrt{-3}, \sqrt{329})\) None 54.8.a.g \(-16\) \(0\) \(48\) \(-880\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\beta _{1})q^{2}-2^{6}\beta _{1}q^{4}+(24\beta _{1}+\cdots)q^{5}+\cdots\)
162.8.c.n 162.c 9.c $4$ $50.606$ \(\Q(\sqrt{-3}, \sqrt{-643})\) None 162.8.a.c \(-16\) \(0\) \(114\) \(-280\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\beta _{1}q^{2}+(-2^{6}-2^{6}\beta _{1})q^{4}+(57+\cdots)q^{5}+\cdots\)
162.8.c.o 162.c 9.c $4$ $50.606$ \(\Q(\sqrt{-3}, \sqrt{-643})\) None 162.8.a.c \(16\) \(0\) \(-114\) \(-280\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{1}q^{2}+(-2^{6}-2^{6}\beta _{1})q^{4}+(-57+\cdots)q^{5}+\cdots\)
162.8.c.p 162.c 9.c $4$ $50.606$ \(\Q(\sqrt{-3}, \sqrt{329})\) None 54.8.a.g \(16\) \(0\) \(-48\) \(-880\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\beta _{1}q^{2}+(-2^{6}+2^{6}\beta _{1})q^{4}+(-24+\cdots)q^{5}+\cdots\)
162.8.c.q 162.c 9.c $8$ $50.606$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 162.8.a.g \(-32\) \(0\) \(-528\) \(-560\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8-8\beta _{1})q^{2}+2^{6}\beta _{1}q^{4}+(132\beta _{1}+\cdots)q^{5}+\cdots\)
162.8.c.r 162.c 9.c $8$ $50.606$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 162.8.a.g \(32\) \(0\) \(528\) \(-560\) $\mathrm{SU}(2)[C_{3}]$ \(q+(8+8\beta _{1})q^{2}+2^{6}\beta _{1}q^{4}+(-132\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(162, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(162, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)