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Properties

Label 4970.2.a
Level $4970$
Weight $2$
Character orbit 4970.a
Rep. character $\chi_{4970}(1,\cdot)$
Character field $\Q$
Dimension $141$
Newform subspaces $30$
Sturm bound $1728$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 4970 = 2 \cdot 5 \cdot 7 \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4970.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 30 \)
Sturm bound: \(1728\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4970))\).

Total New Old
Modular forms 872 141 731
Cusp forms 857 141 716
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.

\(2\)\(5\)\(7\)\(71\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(+\)\(-\)$-$\(12\)
\(+\)\(+\)\(-\)\(+\)$-$\(11\)
\(+\)\(+\)\(-\)\(-\)$+$\(7\)
\(+\)\(-\)\(+\)\(+\)$-$\(8\)
\(+\)\(-\)\(+\)\(-\)$+$\(10\)
\(+\)\(-\)\(-\)\(+\)$+$\(9\)
\(+\)\(-\)\(-\)\(-\)$-$\(9\)
\(-\)\(+\)\(+\)\(+\)$-$\(10\)
\(-\)\(+\)\(+\)\(-\)$+$\(7\)
\(-\)\(+\)\(-\)\(+\)$+$\(8\)
\(-\)\(+\)\(-\)\(-\)$-$\(9\)
\(-\)\(-\)\(+\)\(+\)$+$\(5\)
\(-\)\(-\)\(+\)\(-\)$-$\(12\)
\(-\)\(-\)\(-\)\(+\)$-$\(13\)
\(-\)\(-\)\(-\)\(-\)$+$\(5\)
Plus space\(+\)\(57\)
Minus space\(-\)\(84\)

Trace form

\( 141 q - 3 q^{2} - 4 q^{3} + 141 q^{4} + q^{5} - 4 q^{6} + q^{7} - 3 q^{8} + 145 q^{9} + O(q^{10}) \) \( 141 q - 3 q^{2} - 4 q^{3} + 141 q^{4} + q^{5} - 4 q^{6} + q^{7} - 3 q^{8} + 145 q^{9} + q^{10} - 12 q^{11} - 4 q^{12} - 2 q^{13} + q^{14} + 4 q^{15} + 141 q^{16} - 6 q^{17} - 7 q^{18} - 4 q^{19} + q^{20} + 4 q^{21} - 12 q^{22} + 8 q^{23} - 4 q^{24} + 141 q^{25} + 14 q^{26} - 40 q^{27} + q^{28} - 18 q^{29} - 4 q^{30} - 3 q^{32} - 16 q^{33} - 22 q^{34} + q^{35} + 145 q^{36} + 30 q^{37} - 4 q^{38} - 8 q^{39} + q^{40} + 2 q^{41} + 4 q^{42} + 12 q^{43} - 12 q^{44} + 13 q^{45} - 8 q^{46} - 16 q^{47} - 4 q^{48} + 141 q^{49} - 3 q^{50} + 56 q^{51} - 2 q^{52} + 22 q^{53} + 8 q^{54} - 4 q^{55} + q^{56} - 10 q^{58} + 20 q^{59} + 4 q^{60} + 30 q^{61} + 13 q^{63} + 141 q^{64} + 14 q^{65} - 16 q^{66} - 20 q^{67} - 6 q^{68} + 16 q^{69} + q^{70} + q^{71} - 7 q^{72} - 30 q^{73} + 22 q^{74} - 4 q^{75} - 4 q^{76} - 4 q^{77} + 24 q^{78} + 16 q^{79} + q^{80} + 197 q^{81} - 14 q^{82} - 4 q^{83} + 4 q^{84} + 18 q^{85} - 12 q^{86} + 88 q^{87} - 12 q^{88} + 82 q^{89} + 13 q^{90} + 14 q^{91} + 8 q^{92} + 64 q^{93} + 16 q^{94} + 20 q^{95} - 4 q^{96} - 38 q^{97} - 3 q^{98} - 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4970))\) 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 71
4970.2.a.a 4970.a 1.a $1$ $39.686$ \(\Q\) None \(-1\) \(-2\) \(-1\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}+q^{7}+\cdots\)
4970.2.a.b 4970.a 1.a $1$ $39.686$ \(\Q\) None \(-1\) \(-1\) \(1\) \(1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
4970.2.a.c 4970.a 1.a $1$ $39.686$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
4970.2.a.d 4970.a 1.a $1$ $39.686$ \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
4970.2.a.e 4970.a 1.a $1$ $39.686$ \(\Q\) None \(-1\) \(1\) \(1\) \(1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
4970.2.a.f 4970.a 1.a $1$ $39.686$ \(\Q\) None \(-1\) \(2\) \(1\) \(-1\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}-q^{7}+\cdots\)
4970.2.a.g 4970.a 1.a $1$ $39.686$ \(\Q\) None \(1\) \(0\) \(-1\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-3q^{9}+\cdots\)
4970.2.a.h 4970.a 1.a $1$ $39.686$ \(\Q\) None \(1\) \(0\) \(-1\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-3q^{9}+\cdots\)
4970.2.a.i 4970.a 1.a $1$ $39.686$ \(\Q\) None \(1\) \(0\) \(-1\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-3q^{9}+\cdots\)
4970.2.a.j 4970.a 1.a $1$ $39.686$ \(\Q\) None \(1\) \(0\) \(1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{5}-q^{7}+q^{8}-3q^{9}+\cdots\)
4970.2.a.k 4970.a 1.a $1$ $39.686$ \(\Q\) None \(1\) \(2\) \(1\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}-q^{7}+\cdots\)
4970.2.a.l 4970.a 1.a $2$ $39.686$ \(\Q(\sqrt{17}) \) None \(-2\) \(1\) \(2\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+q^{5}-\beta q^{6}-q^{7}+\cdots\)
4970.2.a.m 4970.a 1.a $3$ $39.686$ 3.3.316.1 None \(3\) \(-4\) \(-3\) \(3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{1})q^{3}+q^{4}-q^{5}+(-1+\cdots)q^{6}+\cdots\)
4970.2.a.n 4970.a 1.a $4$ $39.686$ 4.4.12357.1 None \(-4\) \(-1\) \(-4\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{2}q^{3}+q^{4}-q^{5}-\beta _{2}q^{6}+\cdots\)
4970.2.a.o 4970.a 1.a $4$ $39.686$ 4.4.133593.1 None \(-4\) \(1\) \(4\) \(4\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
4970.2.a.p 4970.a 1.a $4$ $39.686$ 4.4.3981.1 None \(4\) \(-3\) \(4\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{3})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
4970.2.a.q 4970.a 1.a $5$ $39.686$ 5.5.729621.1 None \(-5\) \(5\) \(5\) \(5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
4970.2.a.r 4970.a 1.a $5$ $39.686$ 5.5.81589.1 None \(5\) \(-5\) \(5\) \(5\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
4970.2.a.s 4970.a 1.a $5$ $39.686$ 5.5.320837.1 None \(5\) \(-1\) \(-5\) \(5\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
4970.2.a.t 4970.a 1.a $6$ $39.686$ 6.6.158745965.1 None \(-6\) \(-1\) \(-6\) \(6\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{3}q^{3}+q^{4}-q^{5}-\beta _{3}q^{6}+\cdots\)
4970.2.a.u 4970.a 1.a $7$ $39.686$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-6\) \(7\) \(-7\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
4970.2.a.v 4970.a 1.a $7$ $39.686$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-5\) \(7\) \(7\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
4970.2.a.w 4970.a 1.a $7$ $39.686$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(7\) \(-2\) \(-7\) \(-7\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-\beta _{1}+\beta _{2})q^{3}+q^{4}-q^{5}+\cdots\)
4970.2.a.x 4970.a 1.a $7$ $39.686$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(7\) \(1\) \(-7\) \(-7\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
4970.2.a.y 4970.a 1.a $8$ $39.686$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(4\) \(8\) \(-8\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
4970.2.a.z 4970.a 1.a $9$ $39.686$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(4\) \(-9\) \(9\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
4970.2.a.ba 4970.a 1.a $11$ $39.686$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-11\) \(2\) \(-11\) \(11\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
4970.2.a.bb 4970.a 1.a $11$ $39.686$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(11\) \(-2\) \(11\) \(-11\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
4970.2.a.bc 4970.a 1.a $12$ $39.686$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-12\) \(0\) \(-12\) \(-12\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
4970.2.a.bd 4970.a 1.a $13$ $39.686$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(13\) \(6\) \(13\) \(13\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4970))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4970)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(71))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(142))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(355))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(497))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(710))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(994))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2485))\)\(^{\oplus 2}\)