Properties

Label 9.3
Level 9
Weight 3
Dimension 2
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 18
Trace bound 0

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(9))\).

Total New Old
Modular forms 10 6 4
Cusp forms 2 2 0
Eisenstein series 8 4 4

Trace form

\( 2q - 3q^{2} - 3q^{3} - q^{4} + 6q^{5} + 9q^{6} - 2q^{7} - 9q^{9} + O(q^{10}) \) \( 2q - 3q^{2} - 3q^{3} - q^{4} + 6q^{5} + 9q^{6} - 2q^{7} - 9q^{9} - 12q^{10} - 3q^{11} + 6q^{12} + 4q^{13} + 6q^{14} + 11q^{16} + 22q^{19} - 6q^{20} - 6q^{21} + 3q^{22} - 48q^{23} - 45q^{24} - 13q^{25} + 54q^{27} + 4q^{28} + 78q^{29} + 18q^{30} - 32q^{31} + 27q^{32} + 9q^{33} - 27q^{34} - 9q^{36} - 68q^{37} - 33q^{38} - 24q^{39} + 30q^{40} - 21q^{41} + 61q^{43} - 54q^{45} + 96q^{46} - 84q^{47} + 33q^{48} + 45q^{49} + 39q^{50} + 81q^{51} + 4q^{52} - 81q^{54} - 12q^{55} - 30q^{56} - 33q^{57} - 78q^{58} + 87q^{59} + 18q^{60} - 56q^{61} + 36q^{63} - 142q^{64} + 24q^{65} - 18q^{66} + 31q^{67} - 27q^{68} + 12q^{70} + 135q^{72} + 130q^{73} + 102q^{74} - 39q^{75} - 11q^{76} + 6q^{77} + 36q^{78} - 38q^{79} - 81q^{81} + 42q^{82} - 84q^{83} - 6q^{84} - 54q^{85} - 183q^{86} - 234q^{87} + 15q^{88} + 54q^{90} - 16q^{91} + 48q^{92} + 192q^{93} + 84q^{94} + 66q^{95} + 115q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(9))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
9.3.b \(\chi_{9}(8, \cdot)\) None 0 1
9.3.d \(\chi_{9}(2, \cdot)\) 9.3.d.a 2 2

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 7 T^{2} + 12 T^{3} + 16 T^{4} \)
$3$ \( 1 + 3 T + 9 T^{2} \)
$5$ \( 1 - 6 T + 37 T^{2} - 150 T^{3} + 625 T^{4} \)
$7$ \( ( 1 - 11 T + 49 T^{2} )( 1 + 13 T + 49 T^{2} ) \)
$11$ \( 1 + 3 T + 124 T^{2} + 363 T^{3} + 14641 T^{4} \)
$13$ \( 1 - 4 T - 153 T^{2} - 676 T^{3} + 28561 T^{4} \)
$17$ \( 1 - 335 T^{2} + 83521 T^{4} \)
$19$ \( ( 1 - 11 T + 361 T^{2} )^{2} \)
$23$ \( 1 + 48 T + 1297 T^{2} + 25392 T^{3} + 279841 T^{4} \)
$29$ \( 1 - 78 T + 2869 T^{2} - 65598 T^{3} + 707281 T^{4} \)
$31$ \( 1 + 32 T + 63 T^{2} + 30752 T^{3} + 923521 T^{4} \)
$37$ \( ( 1 + 34 T + 1369 T^{2} )^{2} \)
$41$ \( 1 + 21 T + 1828 T^{2} + 35301 T^{3} + 2825761 T^{4} \)
$43$ \( ( 1 - 83 T + 1849 T^{2} )( 1 + 22 T + 1849 T^{2} ) \)
$47$ \( 1 + 84 T + 4561 T^{2} + 185556 T^{3} + 4879681 T^{4} \)
$53$ \( ( 1 - 53 T )^{2}( 1 + 53 T )^{2} \)
$59$ \( 1 - 87 T + 6004 T^{2} - 302847 T^{3} + 12117361 T^{4} \)
$61$ \( 1 + 56 T - 585 T^{2} + 208376 T^{3} + 13845841 T^{4} \)
$67$ \( 1 - 31 T - 3528 T^{2} - 139159 T^{3} + 20151121 T^{4} \)
$71$ \( 1 - 9110 T^{2} + 25411681 T^{4} \)
$73$ \( ( 1 - 65 T + 5329 T^{2} )^{2} \)
$79$ \( 1 + 38 T - 4797 T^{2} + 237158 T^{3} + 38950081 T^{4} \)
$83$ \( 1 + 84 T + 9241 T^{2} + 578676 T^{3} + 47458321 T^{4} \)
$89$ \( 1 - 290 T^{2} + 62742241 T^{4} \)
$97$ \( 1 - 115 T + 3816 T^{2} - 1082035 T^{3} + 88529281 T^{4} \)
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