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Properties

Label 490.4.e
Level $490$
Weight $4$
Character orbit 490.e
Rep. character $\chi_{490}(361,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $80$
Newform subspaces $27$
Sturm bound $336$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 27 \)
Sturm bound: \(336\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(490, [\chi])\).

Total New Old
Modular forms 536 80 456
Cusp forms 472 80 392
Eisenstein series 64 0 64

Trace form

\( 80 q + 12 q^{3} - 160 q^{4} + 10 q^{5} + 8 q^{6} - 282 q^{9} + O(q^{10}) \) \( 80 q + 12 q^{3} - 160 q^{4} + 10 q^{5} + 8 q^{6} - 282 q^{9} + 20 q^{10} + 2 q^{11} + 48 q^{12} - 16 q^{13} + 160 q^{15} - 640 q^{16} + 132 q^{17} - 112 q^{18} - 250 q^{19} - 80 q^{20} - 112 q^{22} - 168 q^{23} - 16 q^{24} - 1000 q^{25} + 156 q^{26} - 816 q^{27} + 748 q^{29} + 140 q^{30} - 96 q^{31} + 128 q^{33} + 480 q^{34} + 2256 q^{36} - 980 q^{37} + 704 q^{38} - 1816 q^{39} + 80 q^{40} - 1096 q^{41} - 464 q^{43} + 8 q^{44} + 380 q^{45} + 424 q^{46} - 500 q^{47} - 384 q^{48} + 1296 q^{51} + 32 q^{52} + 1716 q^{53} - 556 q^{54} + 480 q^{55} - 1992 q^{57} + 240 q^{58} - 1140 q^{59} - 320 q^{60} - 1146 q^{61} + 160 q^{62} + 5120 q^{64} - 1210 q^{65} - 2880 q^{66} + 1124 q^{67} + 528 q^{68} - 1684 q^{69} + 2752 q^{71} - 448 q^{72} + 972 q^{73} + 1108 q^{74} + 300 q^{75} + 2000 q^{76} - 2176 q^{78} + 6108 q^{79} + 160 q^{80} - 752 q^{81} - 1536 q^{82} - 968 q^{83} - 4000 q^{85} + 1692 q^{86} - 2580 q^{87} + 224 q^{88} + 162 q^{89} - 1800 q^{90} + 1344 q^{92} + 4684 q^{93} - 1084 q^{94} - 600 q^{95} - 64 q^{96} - 3504 q^{97} + 17396 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(490, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
490.4.e.a 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(-2\) \(-8\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.b 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(-2\) \(-7\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-7+7\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.c 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(-2\) \(-5\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-5+5\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.d 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(-2\) \(-1\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.e 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(-2\) \(1\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.f 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(-2\) \(1\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.g 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(-2\) \(5\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(5-5\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.h 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(-2\) \(7\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(7-7\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.i 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(-2\) \(8\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(8-8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.j 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(-8\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.k 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(-4\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.l 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(-3\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.m 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(-1\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.n 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(-1\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.o 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(1\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.p 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(3\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.q 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(4\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(4-4\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.r 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(8\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(8-8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.s 490.e 7.c $2$ $28.911$ \(\Q(\sqrt{-3}) \) None \(2\) \(10\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(10-10\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.t 490.e 7.c $4$ $28.911$ \(\Q(\sqrt{-3}, \sqrt{-59})\) None \(4\) \(-5\) \(-10\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{1}q^{2}+(-3-3\beta _{1}-\beta _{3})q^{3}+(-4+\cdots)q^{4}+\cdots\)
490.4.e.u 490.e 7.c $4$ $28.911$ \(\Q(\sqrt{-3}, \sqrt{46})\) None \(4\) \(-2\) \(10\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{2}q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+(-4+\cdots)q^{4}+\cdots\)
490.4.e.v 490.e 7.c $4$ $28.911$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(4\) \(-2\) \(10\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2+2\beta _{2})q^{2}+(3\beta _{1}+\beta _{2}+3\beta _{3})q^{3}+\cdots\)
490.4.e.w 490.e 7.c $4$ $28.911$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(4\) \(2\) \(-10\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2+2\beta _{2})q^{2}+(3\beta _{1}-\beta _{2}+3\beta _{3})q^{3}+\cdots\)
490.4.e.x 490.e 7.c $4$ $28.911$ \(\Q(\sqrt{-3}, \sqrt{-59})\) None \(4\) \(5\) \(10\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{1}q^{2}+(3+3\beta _{1}+\beta _{3})q^{3}+(-4+\cdots)q^{4}+\cdots\)
490.4.e.y 490.e 7.c $6$ $28.911$ 6.0.\(\cdots\).2 None \(-6\) \(4\) \(15\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{3})q^{2}+(\beta _{1}-\beta _{3})q^{3}+4\beta _{3}q^{4}+\cdots\)
490.4.e.z 490.e 7.c $8$ $28.911$ 8.0.\(\cdots\).19 None \(-8\) \(-10\) \(-20\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{2}+(-3-3\beta _{1}+\beta _{5})q^{3}+(-4+\cdots)q^{4}+\cdots\)
490.4.e.ba 490.e 7.c $8$ $28.911$ 8.0.\(\cdots\).19 None \(-8\) \(10\) \(20\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{2}+(3+3\beta _{1}-\beta _{5})q^{3}+(-4+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(490, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(490, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)