↑

Properties

Label 90.6.e
Level $90$
Weight $6$
Character orbit 90.e
Rep. character $\chi_{90}(31,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $40$
Newform subspaces $4$
Sturm bound $108$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 90.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(108\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(90, [\chi])\).

Total New Old
Modular forms 188 40 148
Cusp forms 172 40 132
Eisenstein series 16 0 16

Trace form

\( 40 q + 8 q^{2} - 22 q^{3} - 320 q^{4} + 50 q^{5} + 88 q^{6} + 116 q^{7} - 256 q^{8} - 652 q^{9} + O(q^{10}) \) \( 40 q + 8 q^{2} - 22 q^{3} - 320 q^{4} + 50 q^{5} + 88 q^{6} + 116 q^{7} - 256 q^{8} - 652 q^{9} - 78 q^{11} + 704 q^{12} - 724 q^{13} + 40 q^{14} - 1100 q^{15} - 5120 q^{16} - 5628 q^{17} - 2864 q^{18} + 2036 q^{19} + 800 q^{20} + 9964 q^{21} + 1896 q^{22} + 132 q^{23} - 512 q^{24} - 12500 q^{25} - 13376 q^{26} - 34144 q^{27} - 3712 q^{28} + 3162 q^{29} + 3800 q^{30} - 1084 q^{31} + 2048 q^{32} + 13866 q^{33} + 7656 q^{34} - 9800 q^{35} + 9248 q^{36} - 20632 q^{37} + 5800 q^{38} + 11116 q^{39} - 53052 q^{41} - 6640 q^{42} - 2938 q^{43} + 2496 q^{44} + 100 q^{45} + 18960 q^{46} - 44772 q^{47} - 5632 q^{48} - 25374 q^{49} + 5000 q^{50} + 132486 q^{51} - 11584 q^{52} - 10344 q^{53} + 128272 q^{54} + 640 q^{56} - 17954 q^{57} - 104910 q^{59} - 17600 q^{60} + 122462 q^{61} - 160352 q^{62} + 128200 q^{63} + 163840 q^{64} - 10100 q^{65} - 51024 q^{66} - 18562 q^{67} + 45024 q^{68} + 23382 q^{69} + 36600 q^{70} + 279912 q^{71} + 46720 q^{72} + 221684 q^{73} + 78832 q^{74} - 13750 q^{75} - 16288 q^{76} - 200196 q^{77} - 329776 q^{78} + 13016 q^{79} - 25600 q^{80} - 329620 q^{81} + 111408 q^{82} + 113892 q^{83} + 67360 q^{84} - 17700 q^{85} + 142696 q^{86} + 392964 q^{87} + 30336 q^{88} - 223716 q^{89} + 142400 q^{90} + 470104 q^{91} + 2112 q^{92} + 166900 q^{93} + 48120 q^{94} + 35800 q^{95} - 14336 q^{96} + 196862 q^{97} - 1347984 q^{98} - 775560 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(90, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
90.6.e.a 90.e 9.c $8$ $14.435$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(-16\) \(24\) \(-100\) \(172\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\beta _{1})q^{2}+(1+3\beta _{1}+\beta _{3})q^{3}+\cdots\)
90.6.e.b 90.e 9.c $10$ $14.435$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-20\) \(-39\) \(125\) \(-109\) $\mathrm{SU}(2)[C_{3}]$ \(q-4\beta _{1}q^{2}+(-4+\beta _{1}+\beta _{2})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
90.6.e.c 90.e 9.c $10$ $14.435$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(20\) \(9\) \(-125\) \(-16\) $\mathrm{SU}(2)[C_{3}]$ \(q+4\beta _{1}q^{2}+(1+\beta _{2}-\beta _{4})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)
90.6.e.d 90.e 9.c $12$ $14.435$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(24\) \(-16\) \(150\) \(69\) $\mathrm{SU}(2)[C_{3}]$ \(q-4\beta _{1}q^{2}+(3\beta _{1}+\beta _{2}-\beta _{3})q^{3}+(-2^{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(90, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)