Properties

Label 4010.2.a
Level $4010$
Weight $2$
Character orbit 4010.a
Rep. character $\chi_{4010}(1,\cdot)$
Character field $\Q$
Dimension $135$
Newform subspaces $15$
Sturm bound $1206$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(1206\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4010))\).

Total New Old
Modular forms 606 135 471
Cusp forms 599 135 464
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(5\)\(401\)FrickeDim
\(+\)\(+\)\(+\)$+$\(16\)
\(+\)\(+\)\(-\)$-$\(18\)
\(+\)\(-\)\(+\)$-$\(21\)
\(+\)\(-\)\(-\)$+$\(13\)
\(-\)\(+\)\(+\)$-$\(22\)
\(-\)\(+\)\(-\)$+$\(12\)
\(-\)\(-\)\(+\)$+$\(9\)
\(-\)\(-\)\(-\)$-$\(24\)
Plus space\(+\)\(50\)
Minus space\(-\)\(85\)

Trace form

\( 135 q - q^{2} + 135 q^{4} - q^{5} - 4 q^{6} + 8 q^{7} - q^{8} + 131 q^{9} + O(q^{10}) \) \( 135 q - q^{2} + 135 q^{4} - q^{5} - 4 q^{6} + 8 q^{7} - q^{8} + 131 q^{9} - q^{10} - 12 q^{11} + 6 q^{13} - 4 q^{15} + 135 q^{16} + 6 q^{17} + 3 q^{18} + 4 q^{19} - q^{20} - 8 q^{21} - 4 q^{22} - 16 q^{23} - 4 q^{24} + 135 q^{25} + 2 q^{26} + 8 q^{28} + 18 q^{29} + 16 q^{31} - q^{32} + 16 q^{33} + 6 q^{34} + 131 q^{36} + 38 q^{37} + 8 q^{38} + 40 q^{39} - q^{40} - 2 q^{41} + 32 q^{42} + 28 q^{43} - 12 q^{44} + 3 q^{45} + 8 q^{46} + 32 q^{47} + 159 q^{49} - q^{50} - 24 q^{51} + 6 q^{52} + 14 q^{53} + 8 q^{54} - 4 q^{55} + 32 q^{57} + 2 q^{58} + 28 q^{59} - 4 q^{60} + 10 q^{61} + 24 q^{62} + 48 q^{63} + 135 q^{64} - 6 q^{65} + 24 q^{66} + 40 q^{67} + 6 q^{68} + 72 q^{69} - 8 q^{70} + 8 q^{71} + 3 q^{72} + 38 q^{73} + 2 q^{74} + 4 q^{76} + 16 q^{77} + 16 q^{78} - 24 q^{79} - q^{80} + 95 q^{81} - 26 q^{82} - 20 q^{83} - 8 q^{84} - 6 q^{85} - 28 q^{86} + 48 q^{87} - 4 q^{88} + 22 q^{89} + 3 q^{90} + 32 q^{91} - 16 q^{92} - 16 q^{93} - 24 q^{94} + 12 q^{95} - 4 q^{96} - 2 q^{97} + 7 q^{98} - 20 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\) 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 5 401
4010.2.a.a 4010.a 1.a $1$ $32.020$ \(\Q\) None \(-1\) \(-2\) \(1\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-4q^{7}+\cdots\)
4010.2.a.b 4010.a 1.a $1$ $32.020$ \(\Q\) None \(-1\) \(2\) \(-1\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}-q^{5}-2q^{6}-q^{8}+\cdots\)
4010.2.a.c 4010.a 1.a $1$ $32.020$ \(\Q\) None \(-1\) \(2\) \(1\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}-q^{8}+\cdots\)
4010.2.a.d 4010.a 1.a $1$ $32.020$ \(\Q\) None \(-1\) \(3\) \(-1\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}-q^{5}-3q^{6}+q^{7}+\cdots\)
4010.2.a.e 4010.a 1.a $1$ $32.020$ \(\Q\) None \(1\) \(-1\) \(1\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+3q^{7}+\cdots\)
4010.2.a.f 4010.a 1.a $1$ $32.020$ \(\Q\) None \(1\) \(2\) \(1\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+4q^{7}+\cdots\)
4010.2.a.g 4010.a 1.a $2$ $32.020$ \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}+q^{4}+q^{5}-\beta q^{6}+2\beta q^{7}+\cdots\)
4010.2.a.h 4010.a 1.a $9$ $32.020$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(-4\) \(9\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{7}q^{3}+q^{4}+q^{5}-\beta _{7}q^{6}+\cdots\)
4010.2.a.i 4010.a 1.a $10$ $32.020$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-10\) \(-4\) \(10\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{7}q^{3}+q^{4}+q^{5}+\beta _{7}q^{6}+\cdots\)
4010.2.a.j 4010.a 1.a $12$ $32.020$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(12\) \(-2\) \(-12\) \(-9\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
4010.2.a.k 4010.a 1.a $15$ $32.020$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-15\) \(-6\) \(-15\) \(-5\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{9}q^{3}+q^{4}-q^{5}+\beta _{9}q^{6}+\cdots\)
4010.2.a.l 4010.a 1.a $17$ $32.020$ \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(-17\) \(3\) \(-17\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
4010.2.a.m 4010.a 1.a $20$ $32.020$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-20\) \(4\) \(20\) \(11\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
4010.2.a.n 4010.a 1.a $22$ $32.020$ None \(22\) \(1\) \(22\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$
4010.2.a.o 4010.a 1.a $22$ $32.020$ None \(22\) \(2\) \(-22\) \(13\) $-$ $+$ $+$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4010))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4010)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(401))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(802))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2005))\)\(^{\oplus 2}\)