Properties

Label 9.6.a
Level 9
Weight 6
Character orbit a
Rep. character \(\chi_{9}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(9))\).

Total New Old
Modular forms 7 2 5
Cusp forms 3 1 2
Eisenstein series 4 1 3

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

\(3\)Dim.
\(-\)\(1\)

Trace form

\( q + 6q^{2} + 4q^{4} - 6q^{5} - 40q^{7} - 168q^{8} + O(q^{10}) \) \( q + 6q^{2} + 4q^{4} - 6q^{5} - 40q^{7} - 168q^{8} - 36q^{10} + 564q^{11} + 638q^{13} - 240q^{14} - 1136q^{16} - 882q^{17} - 556q^{19} - 24q^{20} + 3384q^{22} + 840q^{23} - 3089q^{25} + 3828q^{26} - 160q^{28} - 4638q^{29} + 4400q^{31} - 1440q^{32} - 5292q^{34} + 240q^{35} - 2410q^{37} - 3336q^{38} + 1008q^{40} + 6870q^{41} + 9644q^{43} + 2256q^{44} + 5040q^{46} + 18672q^{47} - 15207q^{49} - 18534q^{50} + 2552q^{52} - 33750q^{53} - 3384q^{55} + 6720q^{56} - 27828q^{58} + 18084q^{59} + 39758q^{61} + 26400q^{62} + 27712q^{64} - 3828q^{65} - 23068q^{67} - 3528q^{68} + 1440q^{70} + 4248q^{71} - 41110q^{73} - 14460q^{74} - 2224q^{76} - 22560q^{77} + 21920q^{79} + 6816q^{80} + 41220q^{82} - 82452q^{83} + 5292q^{85} + 57864q^{86} - 94752q^{88} + 94086q^{89} - 25520q^{91} + 3360q^{92} + 112032q^{94} + 3336q^{95} + 49442q^{97} - 91242q^{98} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
9.6.a.a \(1\) \(1.443\) \(\Q\) None \(6\) \(0\) \(-6\) \(-40\) \(-\) \(q+6q^{2}+4q^{4}-6q^{5}-40q^{7}-168q^{8}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T + 32 T^{2} \)
$3$ 1
$5$ \( 1 + 6 T + 3125 T^{2} \)
$7$ \( 1 + 40 T + 16807 T^{2} \)
$11$ \( 1 - 564 T + 161051 T^{2} \)
$13$ \( 1 - 638 T + 371293 T^{2} \)
$17$ \( 1 + 882 T + 1419857 T^{2} \)
$19$ \( 1 + 556 T + 2476099 T^{2} \)
$23$ \( 1 - 840 T + 6436343 T^{2} \)
$29$ \( 1 + 4638 T + 20511149 T^{2} \)
$31$ \( 1 - 4400 T + 28629151 T^{2} \)
$37$ \( 1 + 2410 T + 69343957 T^{2} \)
$41$ \( 1 - 6870 T + 115856201 T^{2} \)
$43$ \( 1 - 9644 T + 147008443 T^{2} \)
$47$ \( 1 - 18672 T + 229345007 T^{2} \)
$53$ \( 1 + 33750 T + 418195493 T^{2} \)
$59$ \( 1 - 18084 T + 714924299 T^{2} \)
$61$ \( 1 - 39758 T + 844596301 T^{2} \)
$67$ \( 1 + 23068 T + 1350125107 T^{2} \)
$71$ \( 1 - 4248 T + 1804229351 T^{2} \)
$73$ \( 1 + 41110 T + 2073071593 T^{2} \)
$79$ \( 1 - 21920 T + 3077056399 T^{2} \)
$83$ \( 1 + 82452 T + 3939040643 T^{2} \)
$89$ \( 1 - 94086 T + 5584059449 T^{2} \)
$97$ \( 1 - 49442 T + 8587340257 T^{2} \)
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