Properties

Label 4009.2.a
Level 4009
Weight 2
Character orbit a
Rep. character \(\chi_{4009}(1,\cdot)\)
Character field \(\Q\)
Dimension 315
Newform subspaces 6
Sturm bound 706
Trace bound 1

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Defining parameters

Level: \( N \) = \( 4009 = 19 \cdot 211 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4009.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(706\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4009))\).

Total New Old
Modular forms 354 315 39
Cusp forms 351 315 36
Eisenstein series 3 0 3

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

\(19\)\(211\)FrickeDim.
\(+\)\(+\)\(+\)\(75\)
\(+\)\(-\)\(-\)\(84\)
\(-\)\(+\)\(-\)\(82\)
\(-\)\(-\)\(+\)\(74\)
Plus space\(+\)\(149\)
Minus space\(-\)\(166\)

Trace form

\( 315q + q^{2} + 319q^{4} - 12q^{5} - 2q^{7} + 9q^{8} + 311q^{9} + O(q^{10}) \) \( 315q + q^{2} + 319q^{4} - 12q^{5} - 2q^{7} + 9q^{8} + 311q^{9} - 2q^{10} - 2q^{11} - 8q^{12} - 6q^{13} - 20q^{14} - 4q^{15} + 323q^{16} - 4q^{17} + 9q^{18} - 3q^{19} - 30q^{20} - 12q^{21} + 4q^{22} - 4q^{23} - 36q^{24} + 279q^{25} - 2q^{26} - 10q^{29} - 16q^{31} - 11q^{32} - 22q^{34} + 26q^{35} + 295q^{36} - 26q^{37} + 3q^{38} + 40q^{39} + 30q^{40} - 22q^{41} + 4q^{42} + 14q^{43} + 8q^{44} - 64q^{45} + 32q^{46} - 22q^{47} + 44q^{48} + 305q^{49} - 13q^{50} + 12q^{51} - 22q^{52} - 18q^{53} - 20q^{54} - 42q^{55} - 84q^{56} + 4q^{57} + 26q^{58} - 8q^{59} - 8q^{60} - 24q^{61} + 24q^{62} - 22q^{63} + 359q^{64} - 44q^{66} + 4q^{67} - 18q^{68} - 4q^{69} - 4q^{70} - 24q^{71} + 33q^{72} - 40q^{73} - 6q^{74} + 36q^{75} - 3q^{76} + 34q^{77} - 16q^{78} - 16q^{79} - 106q^{80} + 307q^{81} + 22q^{82} - 20q^{83} - 56q^{84} - 26q^{85} - 28q^{87} + 108q^{88} - 70q^{89} + 6q^{90} - 28q^{91} - 56q^{92} + 28q^{93} + 56q^{94} - 12q^{95} - 152q^{96} - 66q^{97} - 43q^{98} - 18q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 19 211
4009.2.a.a \(1\) \(32.012\) \(\Q\) None \(-1\) \(2\) \(3\) \(0\) \(+\) \(-\) \(q-q^{2}+2q^{3}-q^{4}+3q^{5}-2q^{6}+3q^{8}+\cdots\)
4009.2.a.b \(3\) \(32.012\) \(\Q(\zeta_{14})^+\) None \(2\) \(-2\) \(-3\) \(3\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+(-1+\beta _{1})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
4009.2.a.c \(71\) \(32.012\) None \(-15\) \(-8\) \(-18\) \(-19\) \(-\) \(-\)
4009.2.a.d \(75\) \(32.012\) None \(-11\) \(-4\) \(-18\) \(-19\) \(+\) \(+\)
4009.2.a.e \(82\) \(32.012\) None \(15\) \(12\) \(9\) \(14\) \(-\) \(+\)
4009.2.a.f \(83\) \(32.012\) None \(11\) \(0\) \(15\) \(19\) \(+\) \(-\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4009))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4009)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(211))\)\(^{\oplus 2}\)