Properties

Label 8008.2.a
Level 8008
Weight 2
Character orbit a
Rep. character \(\chi_{8008}(1,\cdot)\)
Character field \(\Q\)
Dimension 180
Newforms 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: \(\Q\)
Newforms: \( 26 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7 11 13
8008.2.a.a \(1\) \(63.944\) \(\Q\) None \(0\) \(-2\) \(-1\) \(1\) \(+\) \(-\) \(+\) \(+\) \(q-2q^{3}-q^{5}+q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
8008.2.a.b \(1\) \(63.944\) \(\Q\) None \(0\) \(-2\) \(1\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q-2q^{3}+q^{5}-q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
8008.2.a.c \(1\) \(63.944\) \(\Q\) None \(0\) \(-1\) \(-3\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-3q^{5}-q^{7}-2q^{9}-q^{11}-q^{13}+\cdots\)
8008.2.a.d \(1\) \(63.944\) \(\Q\) None \(0\) \(0\) \(-2\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q-2q^{5}+q^{7}-3q^{9}+q^{11}-q^{13}+\cdots\)
8008.2.a.e \(1\) \(63.944\) \(\Q\) None \(0\) \(3\) \(1\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+3q^{3}+q^{5}+q^{7}+6q^{9}-q^{11}+q^{13}+\cdots\)
8008.2.a.f \(2\) \(63.944\) \(\Q(\sqrt{5}) \) None \(0\) \(1\) \(1\) \(-2\) \(+\) \(+\) \(-\) \(+\) \(q+\beta q^{3}+(1-\beta )q^{5}-q^{7}+(-2+\beta )q^{9}+\cdots\)
8008.2.a.g \(2\) \(63.944\) \(\Q(\sqrt{5}) \) None \(0\) \(1\) \(1\) \(2\) \(-\) \(-\) \(-\) \(-\) \(q+\beta q^{3}+(1-\beta )q^{5}+q^{7}+(-2+\beta )q^{9}+\cdots\)
8008.2.a.h \(2\) \(63.944\) \(\Q(\sqrt{13}) \) None \(0\) \(3\) \(5\) \(2\) \(+\) \(-\) \(+\) \(+\) \(q+(1+\beta )q^{3}+(2+\beta )q^{5}+q^{7}+(1+3\beta )q^{9}+\cdots\)
8008.2.a.i \(3\) \(63.944\) 3.3.229.1 None \(0\) \(2\) \(6\) \(-3\) \(-\) \(+\) \(+\) \(-\) \(q+(1+\beta _{2})q^{3}+(2-\beta _{1})q^{5}-q^{7}+(1+\cdots)q^{9}+\cdots\)
8008.2.a.j \(5\) \(63.944\) 5.5.668973.1 None \(0\) \(1\) \(-1\) \(5\) \(+\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{3}+\beta _{3}q^{5}+q^{7}+(1+\beta _{2}-\beta _{4})q^{9}+\cdots\)
8008.2.a.k \(6\) \(63.944\) 6.6.244558277.1 None \(0\) \(-1\) \(1\) \(6\) \(+\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{3}-\beta _{5}q^{5}+q^{7}+(2+\beta _{2})q^{9}+\cdots\)
8008.2.a.l \(8\) \(63.944\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-5\) \(-7\) \(8\) \(-\) \(-\) \(-\) \(-\) \(q+(-1+\beta _{3})q^{3}+(-1+\beta _{5})q^{5}+q^{7}+\cdots\)
8008.2.a.m \(9\) \(63.944\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-5\) \(-5\) \(-9\) \(-\) \(+\) \(+\) \(-\) \(q+(-1+\beta _{1})q^{3}+(-1+\beta _{6})q^{5}-q^{7}+\cdots\)
8008.2.a.n \(9\) \(63.944\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-3\) \(1\) \(-9\) \(+\) \(+\) \(-\) \(+\) \(q-\beta _{1}q^{3}-\beta _{6}q^{5}-q^{7}+(1+\beta _{4}-\beta _{5}+\cdots)q^{9}+\cdots\)
8008.2.a.o \(9\) \(63.944\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-2\) \(-4\) \(9\) \(+\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{3}+\beta _{4}q^{5}+q^{7}+\beta _{2}q^{9}-q^{11}+\cdots\)
8008.2.a.p \(9\) \(63.944\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(1\) \(-4\) \(-9\) \(+\) \(+\) \(-\) \(-\) \(q+\beta _{1}q^{3}+\beta _{3}q^{5}-q^{7}+(-\beta _{3}+\beta _{5}+\cdots)q^{9}+\cdots\)
8008.2.a.q \(9\) \(63.944\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(3\) \(8\) \(9\) \(-\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{3}+(1-\beta _{7})q^{5}+q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
8008.2.a.r \(9\) \(63.944\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(5\) \(3\) \(-9\) \(-\) \(+\) \(+\) \(+\) \(q+(1-\beta _{1})q^{3}+\beta _{6}q^{5}-q^{7}+(1+\beta _{3}+\cdots)q^{9}+\cdots\)
8008.2.a.s \(10\) \(63.944\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-3\) \(-4\) \(10\) \(+\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{3}+\beta _{9}q^{5}+q^{7}+(1+\beta _{5}-\beta _{6}+\cdots)q^{9}+\cdots\)
8008.2.a.t \(10\) \(63.944\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(1\) \(3\) \(10\) \(-\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{3}+\beta _{6}q^{5}+q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8008.2.a.u \(10\) \(63.944\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(2\) \(-4\) \(-10\) \(+\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{3}-\beta _{6}q^{5}-q^{7}+(\beta _{1}+\beta _{2})q^{9}+\cdots\)
8008.2.a.v \(11\) \(63.944\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(-2\) \(2\) \(-11\) \(-\) \(+\) \(-\) \(+\) \(q-\beta _{1}q^{3}+\beta _{3}q^{5}-q^{7}+(1+\beta _{3}+\beta _{9}+\cdots)q^{9}+\cdots\)
8008.2.a.w \(11\) \(63.944\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(0\) \(3\) \(-2\) \(-11\) \(-\) \(+\) \(-\) \(-\) \(q+\beta _{1}q^{3}+\beta _{6}q^{5}-q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8008.2.a.x \(12\) \(63.944\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(4\) \(6\) \(12\) \(+\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{3}+\beta _{4}q^{5}+q^{7}+(1+\beta _{2})q^{9}+\cdots\)
8008.2.a.y \(14\) \(63.944\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(-3\) \(-6\) \(14\) \(-\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{3}+\beta _{10}q^{5}+q^{7}+(2+\beta _{2})q^{9}+\cdots\)
8008.2.a.z \(15\) \(63.944\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(0\) \(-1\) \(4\) \(-15\) \(+\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{3}-\beta _{11}q^{5}-q^{7}+(2+\beta _{2})q^{9}+\cdots\)

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

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