Properties

Label 3.9
Level 3
Weight 9
Dimension 2
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 4 4 0
Cusp forms 2 2 0
Eisenstein series 2 2 0

Trace form

\( 2q + 90q^{3} - 496q^{4} + 3024q^{6} - 3500q^{7} - 5022q^{9} + O(q^{10}) \) \( 2q + 90q^{3} - 496q^{4} + 3024q^{6} - 3500q^{7} - 5022q^{9} + 10080q^{10} - 22320q^{12} + 51460q^{13} - 30240q^{15} - 135040q^{16} + 272160q^{18} + 37876q^{19} - 157500q^{21} - 312480q^{22} + 24192q^{24} + 680450q^{25} - 1042470q^{27} + 868000q^{28} + 453600q^{30} - 702956q^{31} + 937440q^{33} - 3362688q^{34} + 1245456q^{36} + 2670340q^{37} + 2315700q^{39} + 80640q^{40} - 5292000q^{42} - 7052300q^{43} - 2721600q^{45} + 21123648q^{46} - 6076800q^{48} - 5404602q^{49} + 10088064q^{51} - 12762080q^{52} + 4653936q^{54} + 3124800q^{55} + 1704420q^{57} - 20694240q^{58} + 7499520q^{60} + 1507204q^{61} + 8788500q^{63} + 31425536q^{64} - 14061600q^{66} + 4537780q^{67} - 63370944q^{69} - 17640000q^{70} + 2177280q^{72} + 55345540q^{73} + 30620250q^{75} - 9393248q^{76} + 77807520q^{78} - 45961964q^{79} - 60872958q^{81} - 84208320q^{82} + 39060000q^{84} + 33626880q^{85} + 62082720q^{87} - 2499840q^{88} - 25310880q^{90} - 90055000q^{91} - 31633020q^{93} + 183238272q^{94} - 197987328q^{96} + 294542020q^{97} + 84369600q^{99} + O(q^{100}) \)

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

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
3.9.b \(\chi_{3}(2, \cdot)\) 3.9.b.a 2 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8 T^{2} + 65536 T^{4} \)
$3$ \( 1 - 90 T + 6561 T^{2} \)
$5$ \( 1 - 730850 T^{2} + 152587890625 T^{4} \)
$7$ \( ( 1 + 1750 T + 5764801 T^{2} )^{2} \)
$11$ \( 1 - 380283362 T^{2} + 45949729863572161 T^{4} \)
$13$ \( ( 1 - 25730 T + 815730721 T^{2} )^{2} \)
$17$ \( 1 - 8342551298 T^{2} + 48661191875666868481 T^{4} \)
$19$ \( ( 1 - 18938 T + 16983563041 T^{2} )^{2} \)
$23$ \( 1 + 64711613182 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} \)
$29$ \( 1 - 788066452322 T^{2} + \)\(25\!\cdots\!21\)\( T^{4} \)
$31$ \( ( 1 + 351478 T + 852891037441 T^{2} )^{2} \)
$37$ \( ( 1 - 1335170 T + 3512479453921 T^{2} )^{2} \)
$41$ \( 1 - 12452468931842 T^{2} + \)\(63\!\cdots\!41\)\( T^{4} \)
$43$ \( ( 1 + 3526150 T + 11688200277601 T^{2} )^{2} \)
$47$ \( 1 - 30967680304898 T^{2} + \)\(56\!\cdots\!21\)\( T^{4} \)
$53$ \( 1 - 80936075395298 T^{2} + \)\(38\!\cdots\!21\)\( T^{4} \)
$59$ \( 1 - 105562517046242 T^{2} + \)\(21\!\cdots\!41\)\( T^{4} \)
$61$ \( ( 1 - 753602 T + 191707312997281 T^{2} )^{2} \)
$67$ \( ( 1 - 2268890 T + 406067677556641 T^{2} )^{2} \)
$71$ \( 1 - 1001758688017922 T^{2} + \)\(41\!\cdots\!21\)\( T^{4} \)
$73$ \( ( 1 - 27672770 T + 806460091894081 T^{2} )^{2} \)
$79$ \( ( 1 + 22980982 T + 1517108809906561 T^{2} )^{2} \)
$83$ \( 1 - 2352070843223138 T^{2} + \)\(50\!\cdots\!81\)\( T^{4} \)
$89$ \( 1 - 2600204109557762 T^{2} + \)\(15\!\cdots\!61\)\( T^{4} \)
$97$ \( ( 1 - 147271010 T + 7837433594376961 T^{2} )^{2} \)
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