Properties

Label 2009.2.a
Level 2009
Weight 2
Character orbit a
Rep. character \(\chi_{2009}(1,\cdot)\)
Character field \(\Q\)
Dimension 136
Newforms 21
Sturm bound 392
Trace bound 3

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Defining parameters

Level: \( N \) = \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2009.a (trivial)
Character field: \(\Q\)
Newforms: \( 21 \)
Sturm bound: \(392\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2009))\).

Total New Old
Modular forms 204 136 68
Cusp forms 189 136 53
Eisenstein series 15 0 15

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

\(7\)\(41\)FrickeDim.
\(+\)\(+\)\(+\)\(29\)
\(+\)\(-\)\(-\)\(37\)
\(-\)\(+\)\(-\)\(41\)
\(-\)\(-\)\(+\)\(29\)
Plus space\(+\)\(58\)
Minus space\(-\)\(78\)

Trace form

\( 136q - 2q^{2} + 136q^{4} + 2q^{6} - 6q^{8} + 132q^{9} + O(q^{10}) \) \( 136q - 2q^{2} + 136q^{4} + 2q^{6} - 6q^{8} + 132q^{9} - 6q^{11} - 8q^{12} - 12q^{13} - 2q^{15} + 120q^{16} - 8q^{17} - 22q^{18} - 4q^{19} + 16q^{20} - 8q^{22} - 12q^{23} + 26q^{24} + 124q^{25} + 8q^{26} - 6q^{27} - 8q^{29} + 10q^{30} + 8q^{31} - 14q^{32} - 4q^{33} + 140q^{36} + 16q^{37} + 26q^{38} - 16q^{39} + 20q^{40} - 4q^{41} - 8q^{43} - 38q^{44} + 32q^{45} + 20q^{46} + 4q^{48} - 6q^{50} - 32q^{51} - 36q^{52} + 4q^{53} + 30q^{54} - 18q^{55} - 4q^{57} - 32q^{58} - 16q^{59} + 34q^{60} - 4q^{61} + 80q^{64} + 4q^{65} + 96q^{66} + 6q^{67} + 32q^{68} + 32q^{69} - 12q^{71} - 50q^{72} - 52q^{73} - 16q^{74} + 24q^{75} + 24q^{76} - 24q^{78} - 24q^{79} + 36q^{80} + 96q^{81} - 2q^{82} + 4q^{83} - 44q^{85} + 36q^{86} - 4q^{87} + 28q^{88} + 44q^{89} - 32q^{90} + 20q^{92} + 8q^{93} + 42q^{94} - 18q^{95} + 46q^{96} - 36q^{97} - 24q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2009))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 7 41
2009.2.a.a \(2\) \(16.042\) \(\Q(\sqrt{5}) \) None \(-1\) \(1\) \(-1\) \(0\) \(-\) \(-\) \(q-\beta q^{2}+(1-\beta )q^{3}+(-1+\beta )q^{4}+(-1+\cdots)q^{5}+\cdots\)
2009.2.a.b \(2\) \(16.042\) \(\Q(\sqrt{5}) \) None \(-1\) \(3\) \(1\) \(0\) \(-\) \(+\) \(q-\beta q^{2}+(1+\beta )q^{3}+(-1+\beta )q^{4}+(1+\cdots)q^{5}+\cdots\)
2009.2.a.c \(2\) \(16.042\) \(\Q(\sqrt{13}) \) None \(0\) \(-1\) \(1\) \(0\) \(-\) \(-\) \(q-\beta q^{3}-2q^{4}+(1-\beta )q^{5}+\beta q^{9}+(-2+\cdots)q^{11}+\cdots\)
2009.2.a.d \(2\) \(16.042\) \(\Q(\sqrt{13}) \) None \(0\) \(1\) \(-1\) \(0\) \(+\) \(+\) \(q+\beta q^{3}-2q^{4}+(-1+\beta )q^{5}+\beta q^{9}+\cdots\)
2009.2.a.e \(2\) \(16.042\) \(\Q(\sqrt{5}) \) None \(1\) \(-2\) \(2\) \(0\) \(-\) \(-\) \(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}+q^{5}-\beta q^{6}+\cdots\)
2009.2.a.f \(2\) \(16.042\) \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(-2\) \(0\) \(+\) \(+\) \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}-q^{5}+\beta q^{6}+\cdots\)
2009.2.a.g \(3\) \(16.042\) 3.3.148.1 None \(-1\) \(0\) \(2\) \(0\) \(-\) \(+\) \(q+(-\beta _{1}-\beta _{2})q^{2}-\beta _{2}q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)
2009.2.a.h \(3\) \(16.042\) 3.3.257.1 None \(0\) \(-3\) \(0\) \(0\) \(-\) \(+\) \(q-\beta _{2}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
2009.2.a.i \(3\) \(16.042\) 3.3.257.1 None \(0\) \(3\) \(0\) \(0\) \(-\) \(-\) \(q-\beta _{2}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
2009.2.a.j \(3\) \(16.042\) 3.3.257.1 None \(1\) \(1\) \(-6\) \(0\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
2009.2.a.k \(3\) \(16.042\) \(\Q(\zeta_{14})^+\) None \(4\) \(-5\) \(-2\) \(0\) \(-\) \(+\) \(q+(1+\beta _{1})q^{2}+(-2+\beta _{1})q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)
2009.2.a.l \(5\) \(16.042\) 5.5.233489.1 None \(-2\) \(-2\) \(2\) \(0\) \(+\) \(+\) \(q-\beta _{2}q^{2}-\beta _{3}q^{3}+(1+\beta _{1})q^{4}+\beta _{3}q^{5}+\cdots\)
2009.2.a.m \(5\) \(16.042\) 5.5.233489.1 None \(-2\) \(2\) \(-2\) \(0\) \(-\) \(-\) \(q-\beta _{2}q^{2}+\beta _{3}q^{3}+(1+\beta _{1})q^{4}-\beta _{3}q^{5}+\cdots\)
2009.2.a.n \(5\) \(16.042\) 5.5.633117.1 None \(-1\) \(-4\) \(5\) \(0\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(-1+\beta _{1})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
2009.2.a.o \(6\) \(16.042\) 6.6.185257757.1 None \(-1\) \(4\) \(1\) \(0\) \(-\) \(+\) \(q-\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(1+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
2009.2.a.p \(7\) \(16.042\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(-7\) \(-4\) \(0\) \(-\) \(-\) \(q+\beta _{2}q^{2}+(-1+\beta _{1})q^{3}+(1-\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots\)
2009.2.a.q \(7\) \(16.042\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(7\) \(4\) \(0\) \(-\) \(+\) \(q+\beta _{2}q^{2}+(1-\beta _{1})q^{3}+(1-\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots\)
2009.2.a.r \(17\) \(16.042\) \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(3\) \(-1\) \(1\) \(0\) \(-\) \(+\) \(q+\beta _{1}q^{2}+\beta _{12}q^{3}+(1+\beta _{2})q^{4}+\beta _{13}q^{5}+\cdots\)
2009.2.a.s \(17\) \(16.042\) \(\mathbb{Q}[x]/(x^{17} - \cdots)\) None \(3\) \(1\) \(-1\) \(0\) \(+\) \(-\) \(q+\beta _{1}q^{2}-\beta _{12}q^{3}+(1+\beta _{2})q^{4}-\beta _{13}q^{5}+\cdots\)
2009.2.a.t \(20\) \(16.042\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-2\) \(-8\) \(-8\) \(0\) \(+\) \(+\) \(q-\beta _{1}q^{2}-\beta _{9}q^{3}+(1+\beta _{2})q^{4}-\beta _{18}q^{5}+\cdots\)
2009.2.a.u \(20\) \(16.042\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-2\) \(8\) \(8\) \(0\) \(+\) \(-\) \(q-\beta _{1}q^{2}+\beta _{9}q^{3}+(1+\beta _{2})q^{4}+\beta _{18}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2009))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2009)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(287))\)\(^{\oplus 2}\)