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 $7$

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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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 23 29
4002.2.a.a \(1\) \(31.956\) \(\Q\) None \(-1\) \(-1\) \(-2\) \(-2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}-2q^{7}+\cdots\)
4002.2.a.b \(1\) \(31.956\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-4\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-4q^{7}-q^{8}+\cdots\)
4002.2.a.c \(1\) \(31.956\) \(\Q\) None \(-1\) \(-1\) \(1\) \(4\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+4q^{7}+\cdots\)
4002.2.a.d \(1\) \(31.956\) \(\Q\) None \(-1\) \(1\) \(-3\) \(-3\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}-3q^{7}+\cdots\)
4002.2.a.e \(1\) \(31.956\) \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
4002.2.a.f \(1\) \(31.956\) \(\Q\) None \(-1\) \(1\) \(2\) \(0\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+2q^{5}-q^{6}-q^{8}+\cdots\)
4002.2.a.g \(1\) \(31.956\) \(\Q\) None \(-1\) \(1\) \(4\) \(0\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}-q^{8}+\cdots\)
4002.2.a.h \(1\) \(31.956\) \(\Q\) None \(1\) \(-1\) \(-2\) \(-2\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-2q^{7}+\cdots\)
4002.2.a.i \(1\) \(31.956\) \(\Q\) None \(1\) \(-1\) \(-2\) \(2\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}+2q^{7}+\cdots\)
4002.2.a.j \(1\) \(31.956\) \(\Q\) None \(1\) \(-1\) \(1\) \(-4\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-4q^{7}+\cdots\)
4002.2.a.k \(1\) \(31.956\) \(\Q\) None \(1\) \(-1\) \(1\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
4002.2.a.l \(1\) \(31.956\) \(\Q\) None \(1\) \(-1\) \(2\) \(4\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+4q^{7}+\cdots\)
4002.2.a.m \(1\) \(31.956\) \(\Q\) None \(1\) \(1\) \(-3\) \(-4\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-4q^{7}+\cdots\)
4002.2.a.n \(1\) \(31.956\) \(\Q\) None \(1\) \(1\) \(-2\) \(-4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-2q^{5}+q^{6}-4q^{7}+\cdots\)
4002.2.a.o \(1\) \(31.956\) \(\Q\) None \(1\) \(1\) \(-1\) \(-3\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-3q^{7}+\cdots\)
4002.2.a.p \(1\) \(31.956\) \(\Q\) None \(1\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
4002.2.a.q \(1\) \(31.956\) \(\Q\) None \(1\) \(1\) \(0\) \(4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+4q^{7}+q^{8}+\cdots\)
4002.2.a.r \(1\) \(31.956\) \(\Q\) None \(1\) \(1\) \(3\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+q^{7}+\cdots\)
4002.2.a.s \(2\) \(31.956\) \(\Q(\sqrt{17}) \) None \(-2\) \(-2\) \(-3\) \(3\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}+q^{6}+\cdots\)
4002.2.a.t \(2\) \(31.956\) \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+2\beta q^{5}+q^{6}+\beta q^{7}+\cdots\)
4002.2.a.u \(2\) \(31.956\) \(\Q(\sqrt{17}) \) None \(-2\) \(2\) \(1\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}+\beta q^{5}-q^{6}+\beta q^{7}+\cdots\)
4002.2.a.v \(2\) \(31.956\) \(\Q(\sqrt{17}) \) None \(2\) \(-2\) \(-3\) \(0\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
4002.2.a.w \(2\) \(31.956\) \(\Q(\sqrt{3}) \) None \(2\) \(2\) \(6\) \(6\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
4002.2.a.x \(2\) \(31.956\) \(\Q(\sqrt{6}) \) None \(2\) \(2\) \(8\) \(4\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}+4q^{5}+q^{6}+(2-\beta )q^{7}+\cdots\)
4002.2.a.y \(3\) \(31.956\) 3.3.1772.1 None \(-3\) \(-3\) \(-1\) \(6\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}-\beta _{1}q^{5}+q^{6}+(2+\cdots)q^{7}+\cdots\)
4002.2.a.z \(3\) \(31.956\) 3.3.316.1 None \(3\) \(-3\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-\beta _{2}q^{5}-q^{6}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
4002.2.a.ba \(4\) \(31.956\) 4.4.16448.2 None \(-4\) \(-4\) \(2\) \(6\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}+q^{4}+\beta _{3}q^{5}+q^{6}+(1+\cdots)q^{7}+\cdots\)
4002.2.a.bb \(4\) \(31.956\) 4.4.23252.1 None \(-4\) \(4\) \(-3\) \(-2\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+(-1+\beta _{3})q^{5}-q^{6}+\cdots\)
4002.2.a.bc \(4\) \(31.956\) 4.4.11324.1 None \(4\) \(4\) \(-4\) \(-4\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+q^{3}+q^{4}+(-1+\beta _{1})q^{5}+q^{6}+\cdots\)
4002.2.a.bd \(4\) \(31.956\) 4.4.19796.1 None \(4\) \(4\) \(-1\) \(2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-\beta _{3}q^{5}+q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)
4002.2.a.be \(5\) \(31.956\) 5.5.2389280.1 None \(-5\) \(-5\) \(1\) \(-4\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+\beta _{2}q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
4002.2.a.bf \(6\) \(31.956\) 6.6.61157024.1 None \(-6\) \(6\) \(3\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+\beta _{4}q^{5}-q^{6}+(\beta _{4}+\cdots)q^{7}+\cdots\)
4002.2.a.bg \(7\) \(31.956\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-7\) \(-7\) \(2\) \(-5\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}-q^{3}+q^{4}-\beta _{5}q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
4002.2.a.bh \(7\) \(31.956\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(7\) \(7\) \(1\) \(-2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+\beta _{3}q^{5}+q^{6}-\beta _{6}q^{7}+\cdots\)
4002.2.a.bi \(8\) \(31.956\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-8\) \(8\) \(-3\) \(-6\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-\beta _{4}q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
4002.2.a.bj \(8\) \(31.956\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-8\) \(1\) \(0\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}-q^{6}-\beta _{4}q^{7}+\cdots\)
4002.2.a.bk \(8\) \(31.956\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(-8\) \(2\) \(5\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}-q^{6}+(1+\cdots)q^{7}+\cdots\)

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

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