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Properties

Label 4004.2.a
Level $4004$
Weight $2$
Character orbit 4004.a
Rep. character $\chi_{4004}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $11$
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: \(\Q\)
Newform subspaces: \( 11 \)
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

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

Display 100 coefficients 10 coefficients

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

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

\( S_{2}^{\mathrm{old}}(\Gamma_0(4004)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(182))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(286))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(364))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(572))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2002))\)\(^{\oplus 2}\)