↑

Properties

Label 4002.2.a
Level $4002$
Weight $2$
Character orbit 4002.a
Rep. character $\chi_{4002}(1,\cdot)$
Character field $\Q$
Dimension $101$
Newform subspaces $37$
Sturm bound $1440$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 37 \)
Sturm bound: \(1440\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4002))\).

Total New Old
Modular forms 728 101 627
Cusp forms 713 101 612
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.

\(2\)\(3\)\(23\)\(29\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(+\)\(-\)$-$\(5\)
\(+\)\(+\)\(-\)\(+\)$-$\(8\)
\(+\)\(+\)\(-\)\(-\)$+$\(7\)
\(+\)\(-\)\(+\)\(+\)$-$\(4\)
\(+\)\(-\)\(+\)\(-\)$+$\(8\)
\(+\)\(-\)\(-\)\(+\)$+$\(6\)
\(+\)\(-\)\(-\)\(-\)$-$\(6\)
\(-\)\(+\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(+\)\(-\)$+$\(5\)
\(-\)\(+\)\(-\)\(+\)$+$\(4\)
\(-\)\(+\)\(-\)\(-\)$-$\(9\)
\(-\)\(-\)\(+\)\(+\)$+$\(5\)
\(-\)\(-\)\(+\)\(-\)$-$\(9\)
\(-\)\(-\)\(-\)\(+\)$-$\(9\)
\(-\)\(-\)\(-\)\(-\)$+$\(2\)
Plus space\(+\)\(43\)
Minus space\(-\)\(58\)

Trace form

\( 101 q + q^{2} - 3 q^{3} + 101 q^{4} + 6 q^{5} + q^{6} + q^{8} + 101 q^{9} + O(q^{10}) \) \( 101 q + q^{2} - 3 q^{3} + 101 q^{4} + 6 q^{5} + q^{6} + q^{8} + 101 q^{9} + 6 q^{10} + 12 q^{11} - 3 q^{12} + 6 q^{13} + 8 q^{14} + 6 q^{15} + 101 q^{16} + 18 q^{17} + q^{18} - 4 q^{19} + 6 q^{20} - 16 q^{21} + 12 q^{22} + q^{23} + q^{24} + 115 q^{25} + 14 q^{26} - 3 q^{27} + q^{29} + 6 q^{30} + 8 q^{31} + q^{32} + 12 q^{33} + 18 q^{34} + 48 q^{35} + 101 q^{36} + 14 q^{37} + 20 q^{38} - 26 q^{39} + 6 q^{40} + 26 q^{41} + 8 q^{42} + 4 q^{43} + 12 q^{44} + 6 q^{45} - 7 q^{46} + 16 q^{47} - 3 q^{48} + 117 q^{49} + 15 q^{50} + 18 q^{51} + 6 q^{52} - 10 q^{53} + q^{54} + 8 q^{55} + 8 q^{56} - 4 q^{57} - 3 q^{58} + 4 q^{59} + 6 q^{60} + 22 q^{61} + 8 q^{62} + 101 q^{64} - 28 q^{65} + 12 q^{66} - 28 q^{67} + 18 q^{68} - 7 q^{69} + 16 q^{70} - 56 q^{71} + q^{72} - 14 q^{73} - 34 q^{74} + 3 q^{75} - 4 q^{76} - 48 q^{77} - 2 q^{78} - 40 q^{79} + 6 q^{80} + 101 q^{81} - 6 q^{82} - 84 q^{83} - 16 q^{84} - 36 q^{85} - 20 q^{86} + q^{87} + 12 q^{88} - 70 q^{89} + 6 q^{90} + 32 q^{91} + q^{92} - 24 q^{93} - 8 q^{94} - 88 q^{95} + q^{96} - 6 q^{97} + 25 q^{98} + 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\) 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 23 29
4002.2.a.a 4002.a 1.a $1$ $31.956$ \(\Q\) None \(-1\) \(-1\) \(-2\) \(-2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}-2q^{7}+\cdots\)
4002.2.a.b 4002.a 1.a $1$ $31.956$ \(\Q\) None \(-1\) \(-1\) \(0\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}-4q^{7}-q^{8}+\cdots\)
4002.2.a.c 4002.a 1.a $1$ $31.956$ \(\Q\) None \(-1\) \(-1\) \(1\) \(4\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+4q^{7}+\cdots\)
4002.2.a.d 4002.a 1.a $1$ $31.956$ \(\Q\) None \(-1\) \(1\) \(-3\) \(-3\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots\)
4002.2.a.e 4002.a 1.a $1$ $31.956$ \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
4002.2.a.f 4002.a 1.a $1$ $31.956$ \(\Q\) None \(-1\) \(1\) \(2\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+2q^{5}-q^{6}-q^{8}+\cdots\)
4002.2.a.g 4002.a 1.a $1$ $31.956$ \(\Q\) None \(-1\) \(1\) \(4\) \(0\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}-q^{8}+\cdots\)
4002.2.a.h 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(-1\) \(-2\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-2q^{7}+\cdots\)
4002.2.a.i 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(-1\) \(-2\) \(2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}+2q^{7}+\cdots\)
4002.2.a.j 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(-1\) \(1\) \(-4\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-4q^{7}+\cdots\)
4002.2.a.k 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
4002.2.a.l 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(-1\) \(2\) \(4\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+4q^{7}+\cdots\)
4002.2.a.m 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(1\) \(-3\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-4q^{7}+\cdots\)
4002.2.a.n 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(1\) \(-2\) \(-4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-2q^{5}+q^{6}-4q^{7}+\cdots\)
4002.2.a.o 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(1\) \(-1\) \(-3\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-3q^{7}+\cdots\)
4002.2.a.p 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(1\) \(-1\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
4002.2.a.q 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(1\) \(0\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+4q^{7}+q^{8}+\cdots\)
4002.2.a.r 4002.a 1.a $1$ $31.956$ \(\Q\) None \(1\) \(1\) \(3\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots\)
4002.2.a.s 4002.a 1.a $2$ $31.956$ \(\Q(\sqrt{17}) \) None \(-2\) \(-2\) \(-3\) \(3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}+q^{6}+\cdots\)
4002.2.a.t 4002.a 1.a $2$ $31.956$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+2\beta q^{5}+q^{6}+\beta q^{7}+\cdots\)
4002.2.a.u 4002.a 1.a $2$ $31.956$ \(\Q(\sqrt{17}) \) None \(-2\) \(2\) \(1\) \(1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+\beta q^{5}-q^{6}+\beta q^{7}+\cdots\)
4002.2.a.v 4002.a 1.a $2$ $31.956$ \(\Q(\sqrt{17}) \) None \(2\) \(-2\) \(-3\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
4002.2.a.w 4002.a 1.a $2$ $31.956$ \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(6\) \(6\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
4002.2.a.x 4002.a 1.a $2$ $31.956$ \(\Q(\sqrt{6}) \) None \(2\) \(2\) \(8\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+4q^{5}+q^{6}+(2-\beta )q^{7}+\cdots\)
4002.2.a.y 4002.a 1.a $3$ $31.956$ 3.3.1772.1 None \(-3\) \(-3\) \(-1\) \(6\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-\beta _{1}q^{5}+q^{6}+(2+\cdots)q^{7}+\cdots\)
4002.2.a.z 4002.a 1.a $3$ $31.956$ 3.3.316.1 None \(3\) \(-3\) \(0\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-\beta _{2}q^{5}-q^{6}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
4002.2.a.ba 4002.a 1.a $4$ $31.956$ 4.4.16448.2 None \(-4\) \(-4\) \(2\) \(6\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+\beta _{3}q^{5}+q^{6}+(1+\cdots)q^{7}+\cdots\)
4002.2.a.bb 4002.a 1.a $4$ $31.956$ 4.4.23252.1 None \(-4\) \(4\) \(-3\) \(-2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+(-1+\beta _{3})q^{5}-q^{6}+\cdots\)
4002.2.a.bc 4002.a 1.a $4$ $31.956$ 4.4.11324.1 None \(4\) \(4\) \(-4\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+(-1+\beta _{1})q^{5}+q^{6}+\cdots\)
4002.2.a.bd 4002.a 1.a $4$ $31.956$ 4.4.19796.1 None \(4\) \(4\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-\beta _{3}q^{5}+q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)
4002.2.a.be 4002.a 1.a $5$ $31.956$ 5.5.2389280.1 None \(-5\) \(-5\) \(1\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+\beta _{2}q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
4002.2.a.bf 4002.a 1.a $6$ $31.956$ 6.6.61157024.1 None \(-6\) \(6\) \(3\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+\beta _{4}q^{5}-q^{6}+(\beta _{4}+\cdots)q^{7}+\cdots\)
4002.2.a.bg 4002.a 1.a $7$ $31.956$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-7\) \(2\) \(-5\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-\beta _{5}q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
4002.2.a.bh 4002.a 1.a $7$ $31.956$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(7\) \(7\) \(1\) \(-2\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+\beta _{3}q^{5}+q^{6}-\beta _{6}q^{7}+\cdots\)
4002.2.a.bi 4002.a 1.a $8$ $31.956$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(8\) \(-3\) \(-6\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-\beta _{4}q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
4002.2.a.bj 4002.a 1.a $8$ $31.956$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-8\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}-q^{6}-\beta _{4}q^{7}+\cdots\)
4002.2.a.bk 4002.a 1.a $8$ $31.956$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-8\) \(2\) \(5\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}-q^{6}+(1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4002))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4002)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(667))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\)\(^{\oplus 2}\)