Properties

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

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 4009 = 19 \cdot 211 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4009.a (trivial)
Character field: \(\Q\)
Newforms: \( 6 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\) into irreducible Hecke orbits

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

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