Properties

Label 4010.2.a
Level 4010
Weight 2
Character orbit a
Rep. character \(\chi_{4010}(1,\cdot)\)
Character field \(\Q\)
Dimension 135
Newforms 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: \(\Q\)
Newforms: \( 15 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 401
4010.2.a.a \(1\) \(32.020\) \(\Q\) None \(-1\) \(-2\) \(1\) \(-4\) \(+\) \(-\) \(+\) \(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-4q^{7}+\cdots\)
4010.2.a.b \(1\) \(32.020\) \(\Q\) None \(-1\) \(2\) \(-1\) \(0\) \(+\) \(+\) \(+\) \(q-q^{2}+2q^{3}+q^{4}-q^{5}-2q^{6}-q^{8}+\cdots\)
4010.2.a.c \(1\) \(32.020\) \(\Q\) None \(-1\) \(2\) \(1\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}-q^{8}+\cdots\)
4010.2.a.d \(1\) \(32.020\) \(\Q\) None \(-1\) \(3\) \(-1\) \(1\) \(+\) \(+\) \(-\) \(q-q^{2}+3q^{3}+q^{4}-q^{5}-3q^{6}+q^{7}+\cdots\)
4010.2.a.e \(1\) \(32.020\) \(\Q\) None \(1\) \(-1\) \(1\) \(3\) \(-\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+3q^{7}+\cdots\)
4010.2.a.f \(1\) \(32.020\) \(\Q\) None \(1\) \(2\) \(1\) \(4\) \(-\) \(-\) \(-\) \(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+4q^{7}+\cdots\)
4010.2.a.g \(2\) \(32.020\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(2\) \(0\) \(+\) \(-\) \(-\) \(q-q^{2}+\beta q^{3}+q^{4}+q^{5}-\beta q^{6}+2\beta q^{7}+\cdots\)
4010.2.a.h \(9\) \(32.020\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(9\) \(-4\) \(9\) \(-7\) \(-\) \(-\) \(+\) \(q+q^{2}-\beta _{7}q^{3}+q^{4}+q^{5}-\beta _{7}q^{6}+\cdots\)
4010.2.a.i \(10\) \(32.020\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-10\) \(-4\) \(10\) \(-3\) \(+\) \(-\) \(-\) \(q-q^{2}-\beta _{7}q^{3}+q^{4}+q^{5}+\beta _{7}q^{6}+\cdots\)
4010.2.a.j \(12\) \(32.020\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(12\) \(-2\) \(-12\) \(-9\) \(-\) \(+\) \(-\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
4010.2.a.k \(15\) \(32.020\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-15\) \(-6\) \(-15\) \(-5\) \(+\) \(+\) \(+\) \(q-q^{2}-\beta _{9}q^{3}+q^{4}-q^{5}+\beta _{9}q^{6}+\cdots\)
4010.2.a.l \(17\) \(32.020\) \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(-17\) \(3\) \(-17\) \(4\) \(+\) \(+\) \(-\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
4010.2.a.m \(20\) \(32.020\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-20\) \(4\) \(20\) \(11\) \(+\) \(-\) \(+\) \(q-q^{2}+\beta _{1}q^{3}+q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
4010.2.a.n \(22\) \(32.020\) None \(22\) \(1\) \(22\) \(0\) \(-\) \(-\) \(-\)
4010.2.a.o \(22\) \(32.020\) None \(22\) \(2\) \(-22\) \(13\) \(-\) \(+\) \(+\)

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

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