Properties

Label 2002.2.a
Level $2002$
Weight $2$
Character orbit 2002.a
Rep. character $\chi_{2002}(1,\cdot)$
Character field $\Q$
Dimension $61$
Newform subspaces $19$
Sturm bound $672$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 2002 = 2 \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2002.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(672\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(5\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2002))\).

Total New Old
Modular forms 344 61 283
Cusp forms 329 61 268
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.

\(2\)\(7\)\(11\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(5\)
\(+\)\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(+\)\(-\)\(-\)$+$\(4\)
\(+\)\(-\)\(+\)\(+\)$-$\(3\)
\(+\)\(-\)\(+\)\(-\)$+$\(4\)
\(+\)\(-\)\(-\)\(+\)$+$\(5\)
\(+\)\(-\)\(-\)\(-\)$-$\(3\)
\(-\)\(+\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(+\)\(-\)$+$\(2\)
\(-\)\(+\)\(-\)\(+\)$+$\(2\)
\(-\)\(+\)\(-\)\(-\)$-$\(6\)
\(-\)\(-\)\(+\)\(+\)$+$\(3\)
\(-\)\(-\)\(+\)\(-\)$-$\(6\)
\(-\)\(-\)\(-\)\(+\)$-$\(5\)
\(-\)\(-\)\(-\)\(-\)$+$\(2\)
Plus space\(+\)\(27\)
Minus space\(-\)\(34\)

Trace form

\( 61 q + q^{2} - 4 q^{3} + 61 q^{4} - 2 q^{5} + 4 q^{6} + q^{7} + q^{8} + 65 q^{9} + O(q^{10}) \) \( 61 q + q^{2} - 4 q^{3} + 61 q^{4} - 2 q^{5} + 4 q^{6} + q^{7} + q^{8} + 65 q^{9} + 6 q^{10} - 3 q^{11} - 4 q^{12} + q^{13} + q^{14} + 8 q^{15} + 61 q^{16} + 2 q^{17} + 13 q^{18} + 4 q^{19} - 2 q^{20} + 4 q^{21} + q^{22} - 8 q^{23} + 4 q^{24} + 59 q^{25} + q^{26} + 8 q^{27} + q^{28} + 6 q^{29} + 24 q^{30} + 8 q^{31} + q^{32} - 4 q^{33} + 18 q^{34} - 2 q^{35} + 65 q^{36} - 10 q^{37} + 4 q^{38} + 4 q^{39} + 6 q^{40} + 10 q^{41} + 4 q^{42} + 20 q^{43} - 3 q^{44} + 38 q^{45} + 8 q^{46} + 8 q^{47} - 4 q^{48} + 61 q^{49} + 31 q^{50} + 40 q^{51} + q^{52} - 26 q^{53} + 40 q^{54} - 18 q^{55} + q^{56} + 48 q^{57} + 22 q^{58} + 20 q^{59} + 8 q^{60} + 46 q^{61} - 19 q^{63} + 61 q^{64} - 2 q^{65} + 4 q^{66} - 12 q^{67} + 2 q^{68} + 16 q^{69} - 2 q^{70} - 48 q^{71} + 13 q^{72} - 54 q^{73} + 6 q^{74} + 4 q^{75} + 4 q^{76} + q^{77} - 4 q^{78} - 56 q^{79} - 2 q^{80} + 21 q^{81} + 26 q^{82} - 76 q^{83} + 4 q^{84} - 4 q^{85} - 28 q^{86} - 72 q^{87} + q^{88} - 14 q^{89} - 2 q^{90} - 3 q^{91} - 8 q^{92} - 64 q^{93} - 16 q^{94} - 88 q^{95} + 4 q^{96} - 54 q^{97} + q^{98} - 39 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2002))\) 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 7 11 13
2002.2.a.a 2002.a 1.a $1$ $15.986$ \(\Q\) None \(-1\) \(0\) \(2\) \(-1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}-q^{7}-q^{8}-3q^{9}+\cdots\)
2002.2.a.b 2002.a 1.a $1$ $15.986$ \(\Q\) None \(-1\) \(0\) \(2\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
2002.2.a.c 2002.a 1.a $1$ $15.986$ \(\Q\) None \(1\) \(0\) \(4\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+4q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
2002.2.a.d 2002.a 1.a $2$ $15.986$ \(\Q(\sqrt{5}) \) None \(-2\) \(-1\) \(-1\) \(-2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}+(-1+\beta )q^{5}+\beta q^{6}+\cdots\)
2002.2.a.e 2002.a 1.a $2$ $15.986$ \(\Q(\sqrt{13}) \) None \(-2\) \(1\) \(-1\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+(-1+\beta )q^{5}-\beta q^{6}+\cdots\)
2002.2.a.f 2002.a 1.a $2$ $15.986$ \(\Q(\sqrt{5}) \) None \(2\) \(-3\) \(-5\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-2-\beta )q^{5}+\cdots\)
2002.2.a.g 2002.a 1.a $2$ $15.986$ \(\Q(\sqrt{5}) \) None \(2\) \(-1\) \(-1\) \(-2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}+(-1+\beta )q^{5}-\beta q^{6}+\cdots\)
2002.2.a.h 2002.a 1.a $2$ $15.986$ \(\Q(\sqrt{13}) \) None \(2\) \(-1\) \(-1\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta q^{3}+q^{4}+(-1+\beta )q^{5}-\beta q^{6}+\cdots\)
2002.2.a.i 2002.a 1.a $3$ $15.986$ 3.3.469.1 None \(-3\) \(1\) \(5\) \(3\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+(2+\beta _{2})q^{5}-\beta _{1}q^{6}+\cdots\)
2002.2.a.j 2002.a 1.a $3$ $15.986$ 3.3.733.1 None \(-3\) \(3\) \(-1\) \(-3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}-\beta _{1}q^{5}+(-1+\cdots)q^{6}+\cdots\)
2002.2.a.k 2002.a 1.a $3$ $15.986$ 3.3.229.1 None \(3\) \(-3\) \(-5\) \(3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-2-\beta _{2})q^{5}+\cdots\)
2002.2.a.l 2002.a 1.a $4$ $15.986$ 4.4.14272.1 None \(-4\) \(-2\) \(-2\) \(-4\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{3})q^{5}+\cdots\)
2002.2.a.m 2002.a 1.a $4$ $15.986$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(-4\) \(-2\) \(-2\) \(4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(-1+\beta _{3})q^{5}+\cdots\)
2002.2.a.n 2002.a 1.a $5$ $15.986$ 5.5.2055632.1 None \(-5\) \(-2\) \(-6\) \(5\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-2+\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
2002.2.a.o 2002.a 1.a $5$ $15.986$ 5.5.4179152.1 None \(-5\) \(-2\) \(0\) \(-5\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{3}q^{5}+\beta _{1}q^{6}+\cdots\)
2002.2.a.p 2002.a 1.a $5$ $15.986$ 5.5.9353072.1 None \(5\) \(0\) \(4\) \(-5\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+(1+\beta _{2})q^{5}+\beta _{1}q^{6}+\cdots\)
2002.2.a.q 2002.a 1.a $5$ $15.986$ 5.5.3430384.1 None \(5\) \(4\) \(2\) \(5\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{4})q^{3}+q^{4}+\beta _{1}q^{5}+(1+\cdots)q^{6}+\cdots\)
2002.2.a.r 2002.a 1.a $5$ $15.986$ 5.5.1590832.1 None \(5\) \(4\) \(4\) \(5\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1+\beta _{3})q^{5}+\cdots\)
2002.2.a.s 2002.a 1.a $6$ $15.986$ 6.6.1385718192.1 None \(6\) \(0\) \(0\) \(-6\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{4}q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2002))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2002)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(286))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\)\(^{\oplus 2}\)