Properties

Label 8008.2.a
Level $8008$
Weight $2$
Character orbit 8008.a
Rep. character $\chi_{8008}(1,\cdot)$
Character field $\Q$
Dimension $180$
Newform subspaces $26$
Sturm bound $2688$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 26 \)
Sturm bound: \(2688\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8008))\).

Total New Old
Modular forms 1360 180 1180
Cusp forms 1329 180 1149
Eisenstein series 31 0 31

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
\(+\)\(+\)\(+\)\(+\)$+$\(10\)
\(+\)\(+\)\(+\)\(-\)$-$\(15\)
\(+\)\(+\)\(-\)\(+\)$-$\(11\)
\(+\)\(+\)\(-\)\(-\)$+$\(9\)
\(+\)\(-\)\(+\)\(+\)$-$\(14\)
\(+\)\(-\)\(+\)\(-\)$+$\(9\)
\(+\)\(-\)\(-\)\(+\)$+$\(10\)
\(+\)\(-\)\(-\)\(-\)$-$\(12\)
\(-\)\(+\)\(+\)\(+\)$-$\(10\)
\(-\)\(+\)\(+\)\(-\)$+$\(13\)
\(-\)\(+\)\(-\)\(+\)$+$\(11\)
\(-\)\(+\)\(-\)\(-\)$-$\(11\)
\(-\)\(-\)\(+\)\(+\)$+$\(14\)
\(-\)\(-\)\(+\)\(-\)$-$\(11\)
\(-\)\(-\)\(-\)\(+\)$-$\(10\)
\(-\)\(-\)\(-\)\(-\)$+$\(10\)
Plus space\(+\)\(86\)
Minus space\(-\)\(94\)

Trace form

\( 180 q + 180 q^{9} + O(q^{10}) \) \( 180 q + 180 q^{9} - 12 q^{11} - 24 q^{15} + 180 q^{25} - 24 q^{27} + 8 q^{37} + 16 q^{41} - 24 q^{45} - 24 q^{47} + 180 q^{49} - 8 q^{53} + 16 q^{59} - 8 q^{67} - 24 q^{69} - 32 q^{71} + 16 q^{73} - 24 q^{75} + 32 q^{79} + 180 q^{81} - 32 q^{83} + 96 q^{87} - 12 q^{91} - 8 q^{93} + 72 q^{95} + 32 q^{97} - 36 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\) 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
8008.2.a.a 8008.a 1.a $1$ $63.944$ \(\Q\) None \(0\) \(-2\) \(-1\) \(1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
8008.2.a.b 8008.a 1.a $1$ $63.944$ \(\Q\) None \(0\) \(-2\) \(1\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}-q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
8008.2.a.c 8008.a 1.a $1$ $63.944$ \(\Q\) None \(0\) \(-1\) \(-3\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}-q^{7}-2q^{9}-q^{11}-q^{13}+\cdots\)
8008.2.a.d 8008.a 1.a $1$ $63.944$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}+q^{7}-3q^{9}+q^{11}-q^{13}+\cdots\)
8008.2.a.e 8008.a 1.a $1$ $63.944$ \(\Q\) None \(0\) \(3\) \(1\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+q^{7}+6q^{9}-q^{11}+q^{13}+\cdots\)
8008.2.a.f 8008.a 1.a $2$ $63.944$ \(\Q(\sqrt{5}) \) None \(0\) \(1\) \(1\) \(-2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1-\beta )q^{5}-q^{7}+(-2+\beta )q^{9}+\cdots\)
8008.2.a.g 8008.a 1.a $2$ $63.944$ \(\Q(\sqrt{5}) \) None \(0\) \(1\) \(1\) \(2\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1-\beta )q^{5}+q^{7}+(-2+\beta )q^{9}+\cdots\)
8008.2.a.h 8008.a 1.a $2$ $63.944$ \(\Q(\sqrt{13}) \) None \(0\) \(3\) \(5\) \(2\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(2+\beta )q^{5}+q^{7}+(1+3\beta )q^{9}+\cdots\)
8008.2.a.i 8008.a 1.a $3$ $63.944$ 3.3.229.1 None \(0\) \(2\) \(6\) \(-3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}+(2-\beta _{1})q^{5}-q^{7}+(1+\cdots)q^{9}+\cdots\)
8008.2.a.j 8008.a 1.a $5$ $63.944$ 5.5.668973.1 None \(0\) \(1\) \(-1\) \(5\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{3}q^{5}+q^{7}+(1+\beta _{2}-\beta _{4})q^{9}+\cdots\)
8008.2.a.k 8008.a 1.a $6$ $63.944$ 6.6.244558277.1 None \(0\) \(-1\) \(1\) \(6\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{5}q^{5}+q^{7}+(2+\beta _{2})q^{9}+\cdots\)
8008.2.a.l 8008.a 1.a $8$ $63.944$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-5\) \(-7\) \(8\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{3})q^{3}+(-1+\beta _{5})q^{5}+q^{7}+\cdots\)
8008.2.a.m 8008.a 1.a $9$ $63.944$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-5\) \(-5\) \(-9\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1+\beta _{6})q^{5}-q^{7}+\cdots\)
8008.2.a.n 8008.a 1.a $9$ $63.944$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-3\) \(1\) \(-9\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{6}q^{5}-q^{7}+(1+\beta _{4}-\beta _{5}+\cdots)q^{9}+\cdots\)
8008.2.a.o 8008.a 1.a $9$ $63.944$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-2\) \(-4\) \(9\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{4}q^{5}+q^{7}+\beta _{2}q^{9}-q^{11}+\cdots\)
8008.2.a.p 8008.a 1.a $9$ $63.944$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(1\) \(-4\) \(-9\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{3}q^{5}-q^{7}+(-\beta _{3}+\beta _{5}+\cdots)q^{9}+\cdots\)
8008.2.a.q 8008.a 1.a $9$ $63.944$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(3\) \(8\) \(9\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(1-\beta _{7})q^{5}+q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
8008.2.a.r 8008.a 1.a $9$ $63.944$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(5\) \(3\) \(-9\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+\beta _{6}q^{5}-q^{7}+(1+\beta _{3}+\cdots)q^{9}+\cdots\)
8008.2.a.s 8008.a 1.a $10$ $63.944$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-3\) \(-4\) \(10\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{9}q^{5}+q^{7}+(1+\beta _{5}-\beta _{6}+\cdots)q^{9}+\cdots\)
8008.2.a.t 8008.a 1.a $10$ $63.944$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(1\) \(3\) \(10\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{6}q^{5}+q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8008.2.a.u 8008.a 1.a $10$ $63.944$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(2\) \(-4\) \(-10\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-\beta _{6}q^{5}-q^{7}+(\beta _{1}+\beta _{2})q^{9}+\cdots\)
8008.2.a.v 8008.a 1.a $11$ $63.944$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(-2\) \(2\) \(-11\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{3}q^{5}-q^{7}+(1+\beta _{3}+\beta _{9}+\cdots)q^{9}+\cdots\)
8008.2.a.w 8008.a 1.a $11$ $63.944$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(3\) \(-2\) \(-11\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{6}q^{5}-q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8008.2.a.x 8008.a 1.a $12$ $63.944$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(4\) \(6\) \(12\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{4}q^{5}+q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8008.2.a.y 8008.a 1.a $14$ $63.944$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(-3\) \(-6\) \(14\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{10}q^{5}+q^{7}+(2+\beta _{2})q^{9}+\cdots\)
8008.2.a.z 8008.a 1.a $15$ $63.944$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(0\) \(-1\) \(4\) \(-15\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{11}q^{5}-q^{7}+(2+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8008))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8008)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(286))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(364))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(572))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(616))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(728))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2002))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\)\(^{\oplus 2}\)