Properties

Label 4009.2.a
Level $4009$
Weight $2$
Character orbit 4009.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

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

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 19 211
4009.2.a.a 4009.a 1.a $1$ $32.012$ \(\Q\) None \(-1\) \(2\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}-q^{4}+3q^{5}-2q^{6}+3q^{8}+\cdots\)
4009.2.a.b 4009.a 1.a $3$ $32.012$ \(\Q(\zeta_{14})^+\) None \(2\) \(-2\) \(-3\) \(3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(-1+\beta _{1})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
4009.2.a.c 4009.a 1.a $71$ $32.012$ None \(-15\) \(-8\) \(-18\) \(-19\) $-$ $-$ $\mathrm{SU}(2)$
4009.2.a.d 4009.a 1.a $75$ $32.012$ None \(-11\) \(-4\) \(-18\) \(-19\) $+$ $+$ $\mathrm{SU}(2)$
4009.2.a.e 4009.a 1.a $82$ $32.012$ None \(15\) \(12\) \(9\) \(14\) $-$ $+$ $\mathrm{SU}(2)$
4009.2.a.f 4009.a 1.a $83$ $32.012$ None \(11\) \(0\) \(15\) \(19\) $+$ $-$ $\mathrm{SU}(2)$

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

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