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

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

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

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