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Properties

Label 490.4.a
Level $490$
Weight $4$
Character orbit 490.a
Rep. character $\chi_{490}(1,\cdot)$
Character field $\Q$
Dimension $41$
Newform subspaces $25$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 25 \)
Sturm bound: \(336\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(490))\).

Total New Old
Modular forms 268 41 227
Cusp forms 236 41 195
Eisenstein series 32 0 32

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(-\)$-$\(6\)
\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(-\)\(-\)$+$\(6\)
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(6\)
\(-\)\(-\)\(+\)$+$\(7\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(23\)
Minus space\(-\)\(18\)

Trace form

\( 41 q + 2 q^{2} + 4 q^{3} + 164 q^{4} - 5 q^{5} - 24 q^{6} + 8 q^{8} + 361 q^{9} + O(q^{10}) \) \( 41 q + 2 q^{2} + 4 q^{3} + 164 q^{4} - 5 q^{5} - 24 q^{6} + 8 q^{8} + 361 q^{9} - 10 q^{10} - 32 q^{11} + 16 q^{12} + 90 q^{13} + 40 q^{15} + 656 q^{16} - 78 q^{17} - 150 q^{18} + 360 q^{19} - 20 q^{20} + 136 q^{22} - 204 q^{23} - 96 q^{24} + 1025 q^{25} + 196 q^{26} + 304 q^{27} + 410 q^{29} + 200 q^{30} + 104 q^{31} + 32 q^{32} + 520 q^{33} - 348 q^{34} + 1444 q^{36} + 2038 q^{37} - 160 q^{38} + 1608 q^{39} - 40 q^{40} + 850 q^{41} + 640 q^{43} - 128 q^{44} + 135 q^{45} - 328 q^{46} - 1204 q^{47} + 64 q^{48} + 50 q^{50} - 312 q^{51} + 360 q^{52} - 570 q^{53} + 432 q^{54} - 420 q^{55} - 2968 q^{57} + 588 q^{58} - 696 q^{59} + 160 q^{60} - 886 q^{61} + 384 q^{62} + 2624 q^{64} + 1130 q^{65} + 288 q^{66} - 3240 q^{67} - 312 q^{68} - 248 q^{69} - 1216 q^{71} - 600 q^{72} - 1942 q^{73} + 756 q^{74} + 100 q^{75} + 1440 q^{76} - 832 q^{78} - 4672 q^{79} - 80 q^{80} + 5265 q^{81} - 444 q^{82} - 4396 q^{83} + 10 q^{85} - 1880 q^{86} + 3408 q^{87} + 544 q^{88} - 462 q^{89} - 530 q^{90} - 816 q^{92} - 2120 q^{93} + 1096 q^{94} - 1580 q^{95} - 384 q^{96} + 3074 q^{97} + 5488 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(490))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 7
490.4.a.a 490.a 1.a $1$ $28.911$ \(\Q\) None \(-2\) \(-10\) \(5\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-10q^{3}+4q^{4}+5q^{5}+20q^{6}+\cdots\)
490.4.a.b 490.a 1.a $1$ $28.911$ \(\Q\) None \(-2\) \(-4\) \(-5\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-4q^{3}+4q^{4}-5q^{5}+8q^{6}+\cdots\)
490.4.a.c 490.a 1.a $1$ $28.911$ \(\Q\) None \(-2\) \(-1\) \(-5\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+4q^{4}-5q^{5}+2q^{6}+\cdots\)
490.4.a.d 490.a 1.a $1$ $28.911$ \(\Q\) None \(-2\) \(1\) \(5\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
490.4.a.e 490.a 1.a $1$ $28.911$ \(\Q\) None \(-2\) \(1\) \(5\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
490.4.a.f 490.a 1.a $1$ $28.911$ \(\Q\) None \(-2\) \(3\) \(-5\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}-5q^{5}-6q^{6}+\cdots\)
490.4.a.g 490.a 1.a $1$ $28.911$ \(\Q\) None \(-2\) \(8\) \(5\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+8q^{3}+4q^{4}+5q^{5}-2^{4}q^{6}+\cdots\)
490.4.a.h 490.a 1.a $1$ $28.911$ \(\Q\) None \(-2\) \(10\) \(-5\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+10q^{3}+4q^{4}-5q^{5}-20q^{6}+\cdots\)
490.4.a.i 490.a 1.a $1$ $28.911$ \(\Q\) None \(2\) \(-7\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-7q^{3}+4q^{4}+5q^{5}-14q^{6}+\cdots\)
490.4.a.j 490.a 1.a $1$ $28.911$ \(\Q\) None \(2\) \(-5\) \(-5\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-5q^{3}+4q^{4}-5q^{5}-10q^{6}+\cdots\)
490.4.a.k 490.a 1.a $1$ $28.911$ \(\Q\) None \(2\) \(-1\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
490.4.a.l 490.a 1.a $1$ $28.911$ \(\Q\) None \(2\) \(-1\) \(5\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
490.4.a.m 490.a 1.a $1$ $28.911$ \(\Q\) None \(2\) \(1\) \(-5\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+4q^{4}-5q^{5}+2q^{6}+\cdots\)
490.4.a.n 490.a 1.a $1$ $28.911$ \(\Q\) None \(2\) \(1\) \(-5\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+4q^{4}-5q^{5}+2q^{6}+\cdots\)
490.4.a.o 490.a 1.a $1$ $28.911$ \(\Q\) None \(2\) \(8\) \(-5\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+8q^{3}+4q^{4}-5q^{5}+2^{4}q^{6}+\cdots\)
490.4.a.p 490.a 1.a $2$ $28.911$ \(\Q(\sqrt{177}) \) None \(-4\) \(-5\) \(-10\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-2-\beta )q^{3}+4q^{4}-5q^{5}+\cdots\)
490.4.a.q 490.a 1.a $2$ $28.911$ \(\Q(\sqrt{2}) \) None \(-4\) \(-2\) \(10\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+3\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
490.4.a.r 490.a 1.a $2$ $28.911$ \(\Q(\sqrt{46}) \) None \(-4\) \(-2\) \(10\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1+\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
490.4.a.s 490.a 1.a $2$ $28.911$ \(\Q(\sqrt{2}) \) None \(-4\) \(2\) \(-10\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+3\beta )q^{3}+4q^{4}-5q^{5}+\cdots\)
490.4.a.t 490.a 1.a $2$ $28.911$ \(\Q(\sqrt{46}) \) None \(-4\) \(2\) \(-10\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta )q^{3}+4q^{4}-5q^{5}+(-2+\cdots)q^{6}+\cdots\)
490.4.a.u 490.a 1.a $2$ $28.911$ \(\Q(\sqrt{177}) \) None \(-4\) \(5\) \(10\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(3-\beta )q^{3}+4q^{4}+5q^{5}+(-6+\cdots)q^{6}+\cdots\)
490.4.a.v 490.a 1.a $3$ $28.911$ 3.3.115880.1 None \(6\) \(-4\) \(-15\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1-\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
490.4.a.w 490.a 1.a $3$ $28.911$ 3.3.115880.1 None \(6\) \(4\) \(15\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
490.4.a.x 490.a 1.a $4$ $28.911$ \(\Q(\sqrt{2}, \sqrt{113})\) None \(8\) \(-10\) \(-20\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-3-\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
490.4.a.y 490.a 1.a $4$ $28.911$ \(\Q(\sqrt{2}, \sqrt{113})\) None \(8\) \(10\) \(20\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(2-\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(490))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(490)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)