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Properties

Label 490.3.f
Level $490$
Weight $3$
Character orbit 490.f
Rep. character $\chi_{490}(197,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $82$
Newform subspaces $17$
Sturm bound $252$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 490.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 17 \)
Sturm bound: \(252\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(490, [\chi])\).

Total New Old
Modular forms 368 82 286
Cusp forms 304 82 222
Eisenstein series 64 0 64

Trace form

\( 82 q - 2 q^{2} - 4 q^{3} - 8 q^{5} - 8 q^{6} + 4 q^{8} + O(q^{10}) \) \( 82 q - 2 q^{2} - 4 q^{3} - 8 q^{5} - 8 q^{6} + 4 q^{8} + 22 q^{10} + 16 q^{11} - 8 q^{12} + 6 q^{13} - 52 q^{15} - 328 q^{16} - 2 q^{17} - 62 q^{18} + 20 q^{20} - 16 q^{22} + 76 q^{23} + 86 q^{25} + 52 q^{26} + 176 q^{27} - 72 q^{31} + 8 q^{32} + 120 q^{33} + 508 q^{36} - 206 q^{37} - 104 q^{38} - 20 q^{40} - 240 q^{41} - 92 q^{43} - 126 q^{45} - 136 q^{46} + 28 q^{47} + 16 q^{48} - 142 q^{50} + 376 q^{51} + 12 q^{52} + 18 q^{53} + 328 q^{55} - 264 q^{57} - 192 q^{58} + 24 q^{60} + 288 q^{61} + 232 q^{62} - 326 q^{65} - 64 q^{66} + 92 q^{67} + 4 q^{68} - 72 q^{71} - 124 q^{72} + 106 q^{73} + 180 q^{75} - 80 q^{76} + 472 q^{78} + 32 q^{80} - 1010 q^{81} - 240 q^{82} - 564 q^{83} + 6 q^{85} + 440 q^{86} + 408 q^{87} + 32 q^{88} + 54 q^{90} + 152 q^{92} + 376 q^{93} + 240 q^{95} + 32 q^{96} + 514 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(490, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
490.3.f.a 490.f 5.c $2$ $13.352$ \(\Q(\sqrt{-1}) \) None \(-2\) \(-8\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+(-4+4i)q^{3}+2iq^{4}+\cdots\)
490.3.f.b 490.f 5.c $2$ $13.352$ \(\Q(\sqrt{-1}) \) None \(-2\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+(2-2i)q^{3}+2iq^{4}+\cdots\)
490.3.f.c 490.f 5.c $2$ $13.352$ \(\Q(\sqrt{-1}) \) None \(-2\) \(8\) \(6\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+(4-4i)q^{3}+2iq^{4}+\cdots\)
490.3.f.d 490.f 5.c $4$ $13.352$ \(\Q(i, \sqrt{14})\) None \(-4\) \(-4\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{2})q^{2}+(-1+\beta _{2}+\beta _{3})q^{3}+\cdots\)
490.3.f.e 490.f 5.c $4$ $13.352$ \(\Q(i, \sqrt{30})\) None \(-4\) \(-4\) \(12\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{2})q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+\cdots\)
490.3.f.f 490.f 5.c $4$ $13.352$ \(\Q(i, \sqrt{11})\) None \(-4\) \(-2\) \(6\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{2})q^{2}+(-1+\beta _{1}-\beta _{3})q^{3}+\cdots\)
490.3.f.g 490.f 5.c $4$ $13.352$ \(\Q(\zeta_{8})\) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2})q^{2}+5\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.h 490.f 5.c $4$ $13.352$ \(\Q(\zeta_{8})\) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2})q^{2}+\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.i 490.f 5.c $4$ $13.352$ \(\Q(\zeta_{8})\) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2})q^{2}+4\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.j 490.f 5.c $4$ $13.352$ \(\Q(\zeta_{8})\) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}^{2})q^{2}+\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.k 490.f 5.c $4$ $13.352$ \(\Q(i, \sqrt{11})\) None \(-4\) \(2\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{2})q^{2}+(-\beta _{1}+\beta _{2}-\beta _{3})q^{3}+\cdots\)
490.3.f.l 490.f 5.c $4$ $13.352$ \(\Q(i, \sqrt{30})\) None \(-4\) \(4\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{2})q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+\cdots\)
490.3.f.m 490.f 5.c $4$ $13.352$ \(\Q(\zeta_{8})\) None \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\zeta_{8}^{2})q^{2}+\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.n 490.f 5.c $8$ $13.352$ 8.0.\(\cdots\).20 None \(8\) \(-4\) \(4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{3})q^{3}-2\beta _{1}q^{4}+\cdots\)
490.3.f.o 490.f 5.c $8$ $13.352$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+2\beta _{2}q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
490.3.f.p 490.f 5.c $8$ $13.352$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{2})q^{2}+\beta _{4}q^{3}-2\beta _{2}q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
490.3.f.q 490.f 5.c $12$ $13.352$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(12\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{4})q^{2}+\beta _{7}q^{3}-2\beta _{4}q^{4}+(\beta _{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(490, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(490, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)