Properties

Label 63.6.e
Level $63$
Weight $6$
Character orbit 63.e
Rep. character $\chi_{63}(37,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $32$
Newform subspaces $6$
Sturm bound $48$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(48\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(63, [\chi])\).

Total New Old
Modular forms 88 36 52
Cusp forms 72 32 40
Eisenstein series 16 4 12

Trace form

\( 32 q - 268 q^{4} - 60 q^{5} - 70 q^{7} + 72 q^{8} + O(q^{10}) \) \( 32 q - 268 q^{4} - 60 q^{5} - 70 q^{7} + 72 q^{8} + 282 q^{10} - 42 q^{11} - 236 q^{13} - 1884 q^{14} - 3892 q^{16} - 498 q^{17} + 3982 q^{19} + 1968 q^{20} - 9204 q^{22} + 5220 q^{23} - 6478 q^{25} - 15882 q^{26} + 22712 q^{28} + 12624 q^{29} + 16426 q^{31} + 3288 q^{32} - 41940 q^{34} - 8262 q^{35} - 7424 q^{37} - 27612 q^{38} + 50004 q^{40} + 43812 q^{41} - 30716 q^{43} + 32580 q^{44} + 2862 q^{46} - 40074 q^{47} + 9398 q^{49} + 1308 q^{50} + 27436 q^{52} - 44610 q^{53} + 41472 q^{55} - 22992 q^{56} - 76128 q^{58} + 1956 q^{59} + 13102 q^{61} + 282144 q^{62} + 157544 q^{64} + 37206 q^{65} + 147586 q^{67} - 61152 q^{68} - 150822 q^{70} - 301764 q^{71} - 37352 q^{73} - 287940 q^{74} - 586736 q^{76} + 139206 q^{77} + 51130 q^{79} - 28860 q^{80} + 2712 q^{82} + 639396 q^{83} + 363228 q^{85} + 224106 q^{86} + 440364 q^{88} - 189786 q^{89} - 193502 q^{91} - 812544 q^{92} - 212502 q^{94} - 482982 q^{95} - 803528 q^{97} - 58050 q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(63, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
63.6.e.a 63.e 7.c $2$ $10.104$ \(\Q(\sqrt{-3}) \) None 21.6.e.a \(-2\) \(0\) \(11\) \(259\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}+28\zeta_{6}q^{4}+(11-11\zeta_{6})q^{5}+\cdots\)
63.6.e.b 63.e 7.c $2$ $10.104$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 63.6.e.b \(0\) \(0\) \(0\) \(-211\) $\mathrm{U}(1)[D_{3}]$ \(q+2^{5}\zeta_{6}q^{4}+(-149+87\zeta_{6})q^{7}-427q^{13}+\cdots\)
63.6.e.c 63.e 7.c $4$ $10.104$ \(\Q(\sqrt{-3}, \sqrt{-83})\) None 21.6.e.b \(-3\) \(0\) \(-33\) \(-350\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{1}-\beta _{3})q^{2}+(34\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
63.6.e.d 63.e 7.c $4$ $10.104$ \(\Q(\sqrt{-3}, \sqrt{37})\) None 7.6.c.a \(2\) \(0\) \(-38\) \(-168\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{2})q^{2}+(-6+6\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
63.6.e.e 63.e 7.c $8$ $10.104$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 21.6.e.c \(3\) \(0\) \(0\) \(258\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{1}+\beta _{2})q^{2}+(-\beta _{1}+18\beta _{2}+\cdots)q^{4}+\cdots\)
63.6.e.f 63.e 7.c $12$ $10.104$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 63.6.e.f \(0\) \(0\) \(0\) \(142\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(30\beta _{2}-\beta _{5}-\beta _{8})q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(63, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(63, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)