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Properties

Label 294.8.e
Level $294$
Weight $8$
Character orbit 294.e
Rep. character $\chi_{294}(67,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $92$
Newform subspaces $29$
Sturm bound $448$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 294.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 29 \)
Sturm bound: \(448\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(294, [\chi])\).

Total New Old
Modular forms 816 92 724
Cusp forms 752 92 660
Eisenstein series 64 0 64

Trace form

\( 92 q - 2944 q^{4} + 4 q^{5} - 864 q^{6} - 33534 q^{9} + O(q^{10}) \) \( 92 q - 2944 q^{4} + 4 q^{5} - 864 q^{6} - 33534 q^{9} + 3792 q^{10} - 4696 q^{11} + 7904 q^{13} + 21276 q^{15} - 188416 q^{16} + 71732 q^{17} + 1796 q^{19} - 512 q^{20} + 57248 q^{22} + 198804 q^{23} + 27648 q^{24} - 935040 q^{25} - 113312 q^{26} + 1433912 q^{29} - 157248 q^{30} + 223610 q^{31} + 207954 q^{33} - 515136 q^{34} + 4292352 q^{36} - 421504 q^{37} - 40224 q^{38} - 41472 q^{39} + 242688 q^{40} + 180432 q^{41} - 7006448 q^{43} - 300544 q^{44} + 2916 q^{45} + 404128 q^{46} - 2099004 q^{47} + 2138624 q^{50} - 1259712 q^{51} - 252928 q^{52} + 5491820 q^{53} + 314928 q^{54} + 5440260 q^{55} - 8706528 q^{57} + 3296656 q^{58} - 6658856 q^{59} - 680832 q^{60} + 920792 q^{61} + 2067904 q^{62} + 24117248 q^{64} + 20147764 q^{65} + 2299968 q^{66} + 298572 q^{67} + 4590848 q^{68} - 6244344 q^{69} + 21337304 q^{71} + 5120456 q^{73} + 3753440 q^{74} + 2830248 q^{75} - 229888 q^{76} - 11197440 q^{78} - 7424042 q^{79} + 16384 q^{80} - 24446286 q^{81} + 19568640 q^{82} + 3986336 q^{83} + 53699224 q^{85} - 5515616 q^{86} + 1395198 q^{87} - 1831936 q^{88} - 17273088 q^{89} - 5528736 q^{90} - 25446912 q^{92} - 30360960 q^{93} + 23143680 q^{94} - 7273244 q^{95} + 1769472 q^{96} + 316556 q^{97} + 6846768 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(294, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
294.8.e.a 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(-8\) \(-27\) \(-470\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.b 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(-8\) \(-27\) \(30\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.c 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(-8\) \(-27\) \(114\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.d 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(-8\) \(27\) \(-114\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.e 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(-8\) \(27\) \(-30\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.f 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(-8\) \(27\) \(470\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.g 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(-27\) \(-410\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.h 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(-27\) \(-270\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.i 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(-27\) \(-18\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.j 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(-27\) \(122\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.k 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(-27\) \(220\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(-3^{3}+3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.l 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(27\) \(-290\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.m 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(27\) \(-220\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.n 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(27\) \(-122\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.o 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(27\) \(18\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.p 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(27\) \(165\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.q 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(27\) \(270\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.r 294.e 7.c $2$ $91.841$ \(\Q(\sqrt{-3}) \) None \(8\) \(27\) \(410\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\zeta_{6}q^{2}+(3^{3}-3^{3}\zeta_{6})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.s 294.e 7.c $4$ $91.841$ \(\Q(\sqrt{-3}, \sqrt{6001})\) None \(-16\) \(-54\) \(288\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{1}q^{2}+(-3^{3}+3^{3}\beta _{1})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.t 294.e 7.c $4$ $91.841$ \(\Q(\sqrt{-3}, \sqrt{2881})\) None \(-16\) \(-54\) \(309\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{1}q^{2}+(-3^{3}+3^{3}\beta _{1})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.u 294.e 7.c $4$ $91.841$ \(\Q(\sqrt{-3}, \sqrt{6001})\) None \(-16\) \(54\) \(-288\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{1}q^{2}+(3^{3}-3^{3}\beta _{1})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.v 294.e 7.c $4$ $91.841$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(16\) \(-54\) \(-540\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{2}q^{2}+(-3^{3}-3^{3}\beta _{2})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.w 294.e 7.c $4$ $91.841$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(16\) \(-54\) \(972\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{2}q^{2}+(-3^{3}-3^{3}\beta _{2})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.x 294.e 7.c $4$ $91.841$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(16\) \(54\) \(-972\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{2}q^{2}+(3^{3}+3^{3}\beta _{2})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.y 294.e 7.c $4$ $91.841$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(16\) \(54\) \(540\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{2}q^{2}+(3^{3}+3^{3}\beta _{2})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.z 294.e 7.c $6$ $91.841$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(-24\) \(81\) \(-70\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8-8\beta _{1})q^{2}-3^{3}\beta _{1}q^{3}+2^{6}\beta _{1}q^{4}+\cdots\)
294.8.e.ba 294.e 7.c $6$ $91.841$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(-81\) \(-110\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-8\beta _{1}q^{2}+(-3^{3}-3^{3}\beta _{1})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)
294.8.e.bb 294.e 7.c $8$ $91.841$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-32\) \(-108\) \(-432\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\beta _{1})q^{2}-3^{3}\beta _{1}q^{3}-2^{6}\beta _{1}q^{4}+\cdots\)
294.8.e.bc 294.e 7.c $8$ $91.841$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-32\) \(108\) \(432\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-8+8\beta _{1})q^{2}+3^{3}\beta _{1}q^{3}-2^{6}\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(294, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(294, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)