Properties

Label 4004.2.a
Level 4004
Weight 2
Character orbit a
Rep. character \(\chi_{4004}(1,\cdot)\)
Character field \(\Q\)
Dimension 60
Newforms 11
Sturm bound 1344
Trace bound 3

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Defining parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.a (trivial)
Character field: \(\Q\)
Newforms: \( 11 \)
Sturm bound: \(1344\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4004))\).

Total New Old
Modular forms 684 60 624
Cusp forms 661 60 601
Eisenstein series 23 0 23

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.
\(-\)\(+\)\(+\)\(+\)\(-\)\(10\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(10\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(10\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(10\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(20\)
Minus space\(-\)\(40\)

Trace form

\(60q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 64q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(60q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 64q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 8q^{17} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 76q^{25} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut +\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 12q^{37} \) \(\mathstrut -\mathstrut 8q^{41} \) \(\mathstrut +\mathstrut 12q^{43} \) \(\mathstrut +\mathstrut 8q^{45} \) \(\mathstrut +\mathstrut 8q^{47} \) \(\mathstrut +\mathstrut 60q^{49} \) \(\mathstrut +\mathstrut 16q^{51} \) \(\mathstrut +\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 12q^{55} \) \(\mathstrut +\mathstrut 16q^{57} \) \(\mathstrut -\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 8q^{61} \) \(\mathstrut +\mathstrut 16q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 28q^{69} \) \(\mathstrut +\mathstrut 12q^{71} \) \(\mathstrut +\mathstrut 40q^{73} \) \(\mathstrut +\mathstrut 48q^{75} \) \(\mathstrut +\mathstrut 20q^{79} \) \(\mathstrut +\mathstrut 140q^{81} \) \(\mathstrut +\mathstrut 32q^{83} \) \(\mathstrut +\mathstrut 56q^{85} \) \(\mathstrut -\mathstrut 48q^{87} \) \(\mathstrut +\mathstrut 52q^{89} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 60q^{93} \) \(\mathstrut -\mathstrut 52q^{95} \) \(\mathstrut +\mathstrut 28q^{97} \) \(\mathstrut +\mathstrut 8q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\) 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
4004.2.a.a \(1\) \(31.972\) \(\Q\) None \(0\) \(-2\) \(0\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q-2q^{3}+q^{7}+q^{9}-q^{11}+q^{13}+6q^{17}+\cdots\)
4004.2.a.b \(1\) \(31.972\) \(\Q\) None \(0\) \(-1\) \(-3\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-3q^{5}-q^{7}-2q^{9}-q^{11}+q^{13}+\cdots\)
4004.2.a.c \(1\) \(31.972\) \(\Q\) None \(0\) \(2\) \(4\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+2q^{3}+4q^{5}+q^{7}+q^{9}+q^{11}-q^{13}+\cdots\)
4004.2.a.d \(4\) \(31.972\) 4.4.3981.1 None \(0\) \(-3\) \(-1\) \(4\) \(-\) \(-\) \(-\) \(-\) \(q+(-1-\beta _{3})q^{3}+(\beta _{2}+\beta _{3})q^{5}+q^{7}+\cdots\)
4004.2.a.e \(4\) \(31.972\) \(\Q(\zeta_{15})^+\) None \(0\) \(1\) \(1\) \(-4\) \(-\) \(+\) \(-\) \(+\) \(q+(-\beta _{2}-\beta _{3})q^{3}+(1-\beta _{1}+\beta _{3})q^{5}+\cdots\)
4004.2.a.f \(5\) \(31.972\) 5.5.463341.1 None \(0\) \(3\) \(0\) \(-5\) \(-\) \(+\) \(+\) \(-\) \(q+(1-\beta _{1})q^{3}-\beta _{2}q^{5}-q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
4004.2.a.g \(6\) \(31.972\) 6.6.246302029.1 None \(0\) \(-2\) \(-3\) \(6\) \(-\) \(-\) \(+\) \(+\) \(q-\beta _{1}q^{3}+(-1-\beta _{4})q^{5}+q^{7}+(1+\beta _{3}+\cdots)q^{9}+\cdots\)
4004.2.a.h \(9\) \(31.972\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(3\) \(0\) \(9\) \(-\) \(-\) \(-\) \(+\) \(q+\beta _{1}q^{3}-\beta _{4}q^{5}+q^{7}+(2+\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
4004.2.a.i \(9\) \(31.972\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(4\) \(4\) \(9\) \(-\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{3}-\beta _{7}q^{5}+q^{7}+(1+\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
4004.2.a.j \(10\) \(31.972\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-2\) \(-2\) \(-10\) \(-\) \(+\) \(+\) \(+\) \(q-\beta _{1}q^{3}+\beta _{3}q^{5}-q^{7}+(1+\beta _{2})q^{9}+\cdots\)
4004.2.a.k \(10\) \(31.972\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(1\) \(4\) \(-10\) \(-\) \(+\) \(-\) \(-\) \(q+\beta _{1}q^{3}-\beta _{4}q^{5}-q^{7}+(2+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4004))\) into lower level spaces

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