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Properties

Label 5010.2.a
Level $5010$
Weight $2$
Character orbit 5010.a
Rep. character $\chi_{5010}(1,\cdot)$
Character field $\Q$
Dimension $109$
Newform subspaces $29$
Sturm bound $2016$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 5010 = 2 \cdot 3 \cdot 5 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5010.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 29 \)
Sturm bound: \(2016\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5010))\).

Total New Old
Modular forms 1016 109 907
Cusp forms 1001 109 892
Eisenstein series 15 0 15

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

\(2\)\(3\)\(5\)\(167\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(7\)
\(+\)\(+\)\(+\)\(-\)$-$\(7\)
\(+\)\(+\)\(-\)\(+\)$-$\(9\)
\(+\)\(+\)\(-\)\(-\)$+$\(5\)
\(+\)\(-\)\(+\)\(+\)$-$\(5\)
\(+\)\(-\)\(+\)\(-\)$+$\(8\)
\(+\)\(-\)\(-\)\(+\)$+$\(6\)
\(+\)\(-\)\(-\)\(-\)$-$\(7\)
\(-\)\(+\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(+\)\(-\)$+$\(6\)
\(-\)\(+\)\(-\)\(+\)$+$\(4\)
\(-\)\(+\)\(-\)\(-\)$-$\(10\)
\(-\)\(-\)\(+\)\(+\)$+$\(4\)
\(-\)\(-\)\(+\)\(-\)$-$\(9\)
\(-\)\(-\)\(-\)\(+\)$-$\(11\)
\(-\)\(-\)\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(43\)
Minus space\(-\)\(66\)

Trace form

\( 109 q + q^{2} - 3 q^{3} + 109 q^{4} + q^{5} + q^{6} + q^{8} + 109 q^{9} + O(q^{10}) \) \( 109 q + q^{2} - 3 q^{3} + 109 q^{4} + q^{5} + q^{6} + q^{8} + 109 q^{9} + q^{10} + 12 q^{11} - 3 q^{12} + 6 q^{13} + 8 q^{14} + q^{15} + 109 q^{16} + 18 q^{17} + q^{18} + 12 q^{19} + q^{20} + 4 q^{22} + 8 q^{23} + q^{24} + 109 q^{25} - 2 q^{26} - 3 q^{27} + 14 q^{29} + q^{30} - 8 q^{31} + q^{32} + 4 q^{33} + 2 q^{34} - 8 q^{35} + 109 q^{36} - 2 q^{37} + 4 q^{38} - 10 q^{39} + q^{40} + 26 q^{41} - 16 q^{42} + 4 q^{43} + 12 q^{44} + q^{45} + 8 q^{46} + 32 q^{47} - 3 q^{48} + 93 q^{49} + q^{50} - 14 q^{51} + 6 q^{52} + 54 q^{53} + q^{54} - 4 q^{55} + 8 q^{56} - 12 q^{57} - 42 q^{58} + 28 q^{59} + q^{60} + 22 q^{61} + 16 q^{62} + 109 q^{64} + 14 q^{65} - 4 q^{66} - 4 q^{67} + 18 q^{68} - 24 q^{69} + 8 q^{70} - 8 q^{71} + q^{72} - 14 q^{73} + 22 q^{74} - 3 q^{75} + 12 q^{76} + 48 q^{77} - 2 q^{78} + 24 q^{79} + q^{80} + 109 q^{81} + 10 q^{82} + 20 q^{83} - 6 q^{85} + 12 q^{86} + 14 q^{87} + 4 q^{88} + 26 q^{89} + q^{90} - 48 q^{91} + 8 q^{92} - 48 q^{93} - 12 q^{95} + q^{96} - 22 q^{97} + 25 q^{98} + 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5010))\) 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 3 5 167
5010.2.a.a 5010.a 1.a $1$ $40.005$ \(\Q\) None \(-1\) \(-1\) \(1\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+2q^{7}+\cdots\)
5010.2.a.b 5010.a 1.a $1$ $40.005$ \(\Q\) None \(-1\) \(-1\) \(1\) \(5\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+5q^{7}+\cdots\)
5010.2.a.c 5010.a 1.a $1$ $40.005$ \(\Q\) None \(-1\) \(1\) \(-1\) \(-4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)
5010.2.a.d 5010.a 1.a $1$ $40.005$ \(\Q\) None \(-1\) \(1\) \(-1\) \(2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
5010.2.a.e 5010.a 1.a $1$ $40.005$ \(\Q\) None \(1\) \(-1\) \(-1\) \(-4\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)
5010.2.a.f 5010.a 1.a $1$ $40.005$ \(\Q\) None \(1\) \(-1\) \(-1\) \(2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
5010.2.a.g 5010.a 1.a $1$ $40.005$ \(\Q\) None \(1\) \(1\) \(-1\) \(0\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
5010.2.a.h 5010.a 1.a $1$ $40.005$ \(\Q\) None \(1\) \(1\) \(1\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
5010.2.a.i 5010.a 1.a $2$ $40.005$ \(\Q(\sqrt{33}) \) None \(-2\) \(-2\) \(-2\) \(3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+(1+\beta )q^{7}+\cdots\)
5010.2.a.j 5010.a 1.a $2$ $40.005$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(2\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+\beta q^{7}+\cdots\)
5010.2.a.k 5010.a 1.a $2$ $40.005$ \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(-2\) \(3\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+3\beta q^{7}+\cdots\)
5010.2.a.l 5010.a 1.a $2$ $40.005$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(2\) \(6\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(3+\beta )q^{7}+\cdots\)
5010.2.a.m 5010.a 1.a $3$ $40.005$ 3.3.785.1 None \(-3\) \(-3\) \(3\) \(-5\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+(-2+\cdots)q^{7}+\cdots\)
5010.2.a.n 5010.a 1.a $3$ $40.005$ 3.3.316.1 None \(-3\) \(-3\) \(3\) \(1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+(1-2\beta _{1}+\cdots)q^{7}+\cdots\)
5010.2.a.o 5010.a 1.a $3$ $40.005$ 3.3.321.1 None \(3\) \(3\) \(-3\) \(-3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
5010.2.a.p 5010.a 1.a $3$ $40.005$ \(\Q(\zeta_{14})^+\) None \(3\) \(3\) \(3\) \(-7\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+(-2+\cdots)q^{7}+\cdots\)
5010.2.a.q 5010.a 1.a $4$ $40.005$ 4.4.314425.1 None \(-4\) \(-4\) \(4\) \(-7\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+(-2+\cdots)q^{7}+\cdots\)
5010.2.a.r 5010.a 1.a $4$ $40.005$ 4.4.27329.1 None \(-4\) \(4\) \(-4\) \(1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+\beta _{1}q^{7}+\cdots\)
5010.2.a.s 5010.a 1.a $4$ $40.005$ 4.4.2777.1 None \(4\) \(-4\) \(4\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
5010.2.a.t 5010.a 1.a $5$ $40.005$ 5.5.2931521.1 None \(-5\) \(-5\) \(-5\) \(6\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+(1+\beta _{4})q^{7}+\cdots\)
5010.2.a.u 5010.a 1.a $5$ $40.005$ 5.5.138136.1 None \(-5\) \(5\) \(-5\) \(3\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
5010.2.a.v 5010.a 1.a $5$ $40.005$ 5.5.1686952.1 None \(-5\) \(5\) \(5\) \(-3\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
5010.2.a.w 5010.a 1.a $5$ $40.005$ 5.5.1387436.1 None \(5\) \(-5\) \(-5\) \(-5\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
5010.2.a.x 5010.a 1.a $6$ $40.005$ 6.6.50061269.1 None \(-6\) \(6\) \(6\) \(-6\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
5010.2.a.y 5010.a 1.a $7$ $40.005$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-7\) \(-7\) \(-12\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+(-2+\cdots)q^{7}+\cdots\)
5010.2.a.z 5010.a 1.a $7$ $40.005$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(7\) \(-7\) \(-7\) \(10\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
5010.2.a.ba 5010.a 1.a $9$ $40.005$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(9\) \(-9\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+\beta _{6}q^{7}+\cdots\)
5010.2.a.bb 5010.a 1.a $10$ $40.005$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(10\) \(-10\) \(10\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+\beta _{1}q^{7}+\cdots\)
5010.2.a.bc 5010.a 1.a $10$ $40.005$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(10\) \(10\) \(10\) \(5\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5010))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5010)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(334))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(501))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(835))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1670))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2505))\)\(^{\oplus 2}\)