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Properties

Label 4030.2.a
Level $4030$
Weight $2$
Character orbit 4030.a
Rep. character $\chi_{4030}(1,\cdot)$
Character field $\Q$
Dimension $121$
Newform subspaces $18$
Sturm bound $1344$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4030.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(1344\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4030))\).

Total New Old
Modular forms 680 121 559
Cusp forms 665 121 544
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.

\(2\)\(5\)\(13\)\(31\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(7\)
\(+\)\(+\)\(+\)\(-\)$-$\(9\)
\(+\)\(+\)\(-\)\(+\)$-$\(10\)
\(+\)\(+\)\(-\)\(-\)$+$\(6\)
\(+\)\(-\)\(+\)\(+\)$-$\(8\)
\(+\)\(-\)\(+\)\(-\)$+$\(7\)
\(+\)\(-\)\(-\)\(+\)$+$\(6\)
\(+\)\(-\)\(-\)\(-\)$-$\(9\)
\(-\)\(+\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(+\)\(-\)$+$\(7\)
\(-\)\(+\)\(-\)\(+\)$+$\(6\)
\(-\)\(+\)\(-\)\(-\)$-$\(9\)
\(-\)\(-\)\(+\)\(+\)$+$\(6\)
\(-\)\(-\)\(+\)\(-\)$-$\(8\)
\(-\)\(-\)\(-\)\(+\)$-$\(9\)
\(-\)\(-\)\(-\)\(-\)$+$\(6\)
Plus space\(+\)\(51\)
Minus space\(-\)\(70\)

Trace form

\( 121 q - 3 q^{2} - 4 q^{3} + 121 q^{4} - 3 q^{5} - 12 q^{6} + 8 q^{7} - 3 q^{8} + 117 q^{9} + O(q^{10}) \) \( 121 q - 3 q^{2} - 4 q^{3} + 121 q^{4} - 3 q^{5} - 12 q^{6} + 8 q^{7} - 3 q^{8} + 117 q^{9} + q^{10} + 4 q^{11} - 4 q^{12} + q^{13} + 4 q^{15} + 121 q^{16} - 6 q^{17} - 7 q^{18} - 20 q^{19} - 3 q^{20} + 16 q^{21} - 12 q^{22} + 8 q^{23} - 12 q^{24} + 121 q^{25} + q^{26} + 8 q^{27} + 8 q^{28} + 6 q^{29} - 4 q^{30} + q^{31} - 3 q^{32} + 16 q^{33} - 38 q^{34} + 8 q^{35} + 117 q^{36} - 10 q^{37} + 12 q^{38} + 4 q^{39} + q^{40} + 10 q^{41} + 4 q^{43} + 4 q^{44} - 7 q^{45} - 16 q^{47} - 4 q^{48} + 73 q^{49} - 3 q^{50} + 40 q^{51} + q^{52} + 6 q^{53} - 24 q^{54} - 4 q^{55} - 16 q^{57} - 2 q^{58} - 76 q^{59} + 4 q^{60} + 6 q^{61} + q^{62} + 24 q^{63} + 121 q^{64} + q^{65} + 32 q^{66} - 28 q^{67} - 6 q^{68} - 80 q^{69} - 24 q^{70} + 24 q^{71} - 7 q^{72} + 2 q^{73} + 22 q^{74} - 4 q^{75} - 20 q^{76} - 64 q^{77} - 4 q^{78} - 48 q^{79} - 3 q^{80} + 177 q^{81} - 30 q^{82} - 4 q^{83} + 16 q^{84} + 18 q^{85} + 12 q^{86} - 8 q^{87} - 12 q^{88} - 22 q^{89} + 13 q^{90} - 8 q^{91} + 8 q^{92} - 4 q^{93} - 8 q^{94} + 4 q^{95} - 12 q^{96} - 86 q^{97} - 11 q^{98} - 28 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\) 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 13 31
4030.2.a.a 4030.a 1.a $1$ $32.180$ \(\Q\) None \(-1\) \(0\) \(1\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+q^{5}-q^{8}-3q^{9}-q^{10}+\cdots\)
4030.2.a.b 4030.a 1.a $2$ $32.180$ \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(-2\) \(-1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2\beta q^{3}+q^{4}-q^{5}-2\beta q^{6}+\cdots\)
4030.2.a.c 4030.a 1.a $6$ $32.180$ 6.6.3081125.1 None \(-6\) \(-1\) \(-6\) \(4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{3}q^{3}+q^{4}-q^{5}+\beta _{3}q^{6}+\cdots\)
4030.2.a.d 4030.a 1.a $6$ $32.180$ 6.6.3728437.1 None \(-6\) \(-1\) \(6\) \(-2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{3}q^{3}+q^{4}+q^{5}-\beta _{3}q^{6}+\cdots\)
4030.2.a.e 4030.a 1.a $6$ $32.180$ 6.6.6550837.1 None \(6\) \(-7\) \(6\) \(-8\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
4030.2.a.f 4030.a 1.a $6$ $32.180$ 6.6.4418197.1 None \(6\) \(-3\) \(-6\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1}+\beta _{3})q^{3}+q^{4}-q^{5}+\cdots\)
4030.2.a.g 4030.a 1.a $6$ $32.180$ 6.6.10369693.1 None \(6\) \(-3\) \(6\) \(-10\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
4030.2.a.h 4030.a 1.a $7$ $32.180$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-1\) \(7\) \(-4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
4030.2.a.i 4030.a 1.a $7$ $32.180$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(3\) \(-7\) \(2\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
4030.2.a.j 4030.a 1.a $7$ $32.180$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(7\) \(-3\) \(-7\) \(4\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{3}q^{3}+q^{4}-q^{5}-\beta _{3}q^{6}+\cdots\)
4030.2.a.k 4030.a 1.a $8$ $32.180$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(-1\) \(-8\) \(-8\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
4030.2.a.l 4030.a 1.a $8$ $32.180$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(1\) \(8\) \(11\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{6}q^{3}+q^{4}+q^{5}+\beta _{6}q^{6}+\cdots\)
4030.2.a.m 4030.a 1.a $8$ $32.180$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(5\) \(8\) \(5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
4030.2.a.n 4030.a 1.a $8$ $32.180$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-1\) \(-8\) \(1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
4030.2.a.o 4030.a 1.a $8$ $32.180$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(3\) \(8\) \(7\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
4030.2.a.p 4030.a 1.a $9$ $32.180$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(-3\) \(-9\) \(-3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
4030.2.a.q 4030.a 1.a $9$ $32.180$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(3\) \(-9\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
4030.2.a.r 4030.a 1.a $9$ $32.180$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(3\) \(9\) \(9\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{2}q^{3}+q^{4}+q^{5}-\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4030))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4030)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(403))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(806))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2015))\)\(^{\oplus 2}\)