Properties

Label 27.3
Level 27
Weight 3
Dimension 35
Nonzero newspaces 3
Newform subspaces 4
Sturm bound 162
Trace bound 1

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Defining parameters

Level: \( N \) = \( 27\( 27 = 3^{3} \) \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 4 \)
Sturm bound: \(162\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(27))\).

Total New Old
Modular forms 69 51 18
Cusp forms 39 35 4
Eisenstein series 30 16 14

Trace form

\( 35q - 3q^{2} - 6q^{3} - 13q^{4} - 21q^{5} - 18q^{6} - 11q^{7} - 9q^{8} + O(q^{10}) \) \( 35q - 3q^{2} - 6q^{3} - 13q^{4} - 21q^{5} - 18q^{6} - 11q^{7} - 9q^{8} + 3q^{10} - 3q^{11} - 15q^{12} - 23q^{13} - 21q^{14} - 9q^{15} - 13q^{16} - 9q^{17} + 63q^{18} - 2q^{19} + 219q^{20} + 132q^{21} + 51q^{22} + 168q^{23} + 144q^{24} + 29q^{25} - 90q^{27} - 110q^{28} - 246q^{29} - 243q^{30} - 41q^{31} - 387q^{32} - 207q^{33} - 81q^{34} - 252q^{35} - 360q^{36} + 16q^{37} - 51q^{38} + 15q^{39} - 21q^{40} + 249q^{41} + 486q^{42} + 43q^{43} + 639q^{44} + 477q^{45} + 165q^{46} + 483q^{47} + 453q^{48} + 39q^{49} + 264q^{50} + 36q^{51} + 91q^{52} - 54q^{54} - 114q^{55} - 363q^{56} - 192q^{57} - 129q^{58} - 561q^{59} - 846q^{60} - 191q^{61} - 900q^{62} - 585q^{63} + 53q^{64} - 435q^{65} - 423q^{66} + 256q^{67} + 126q^{68} + 99q^{69} + 591q^{70} + 315q^{71} + 720q^{72} + 97q^{73} + 219q^{74} + 255q^{75} + 451q^{76} + 195q^{77} + 180q^{78} + 151q^{79} + 36q^{81} - 330q^{82} + 51q^{83} - 588q^{84} - 207q^{85} - 75q^{86} - 279q^{87} - 717q^{88} + 72q^{89} + 288q^{90} - 7q^{91} - 51q^{92} + 591q^{93} - 741q^{94} + 615q^{95} + 270q^{96} - 470q^{97} + 882q^{98} + 513q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(27))\)

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
27.3.b \(\chi_{27}(26, \cdot)\) 27.3.b.a 1 1
27.3.b.b 2
27.3.d \(\chi_{27}(8, \cdot)\) 27.3.d.a 2 2
27.3.f \(\chi_{27}(2, \cdot)\) 27.3.f.a 30 6

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(27))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(27)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T )( 1 + 2 T ) \))(\( 1 + T^{2} + 16 T^{4} \))(\( 1 - 3 T + 7 T^{2} - 12 T^{3} + 16 T^{4} \))
$3$ 1
$5$ (\( ( 1 - 5 T )( 1 + 5 T ) \))(\( 1 - 41 T^{2} + 625 T^{4} \))(\( 1 + 6 T + 37 T^{2} + 150 T^{3} + 625 T^{4} \))
$7$ (\( 1 + 13 T + 49 T^{2} \))(\( ( 1 - 5 T + 49 T^{2} )^{2} \))(\( ( 1 - 11 T + 49 T^{2} )( 1 + 13 T + 49 T^{2} ) \))
$11$ (\( ( 1 - 11 T )( 1 + 11 T ) \))(\( 1 - 17 T^{2} + 14641 T^{4} \))(\( 1 - 3 T + 124 T^{2} - 363 T^{3} + 14641 T^{4} \))
$13$ (\( 1 + T + 169 T^{2} \))(\( ( 1 + 10 T + 169 T^{2} )^{2} \))(\( 1 - 4 T - 153 T^{2} - 676 T^{3} + 28561 T^{4} \))
$17$ (\( ( 1 - 17 T )( 1 + 17 T ) \))(\( 1 - 254 T^{2} + 83521 T^{4} \))(\( 1 - 335 T^{2} + 83521 T^{4} \))
$19$ (\( 1 - 11 T + 361 T^{2} \))(\( ( 1 + 16 T + 361 T^{2} )^{2} \))(\( ( 1 - 11 T + 361 T^{2} )^{2} \))
$23$ (\( ( 1 - 23 T )( 1 + 23 T ) \))(\( 1 - 914 T^{2} + 279841 T^{4} \))(\( 1 - 48 T + 1297 T^{2} - 25392 T^{3} + 279841 T^{4} \))
$29$ (\( ( 1 - 29 T )( 1 + 29 T ) \))(\( 1 - 782 T^{2} + 707281 T^{4} \))(\( 1 + 78 T + 2869 T^{2} + 65598 T^{3} + 707281 T^{4} \))
$31$ (\( 1 + 46 T + 961 T^{2} \))(\( ( 1 + T + 961 T^{2} )^{2} \))(\( 1 + 32 T + 63 T^{2} + 30752 T^{3} + 923521 T^{4} \))
$37$ (\( 1 - 47 T + 1369 T^{2} \))(\( ( 1 - 20 T + 1369 T^{2} )^{2} \))(\( ( 1 + 34 T + 1369 T^{2} )^{2} \))
$41$ (\( ( 1 - 41 T )( 1 + 41 T ) \))(\( 1 + 238 T^{2} + 2825761 T^{4} \))(\( 1 - 21 T + 1828 T^{2} - 35301 T^{3} + 2825761 T^{4} \))
$43$ (\( 1 + 22 T + 1849 T^{2} \))(\( ( 1 - 50 T + 1849 T^{2} )^{2} \))(\( ( 1 - 83 T + 1849 T^{2} )( 1 + 22 T + 1849 T^{2} ) \))
$47$ (\( ( 1 - 47 T )( 1 + 47 T ) \))(\( 1 - 4382 T^{2} + 4879681 T^{4} \))(\( 1 - 84 T + 4561 T^{2} - 185556 T^{3} + 4879681 T^{4} \))
$53$ (\( ( 1 - 53 T )( 1 + 53 T ) \))(\( 1 - 4889 T^{2} + 7890481 T^{4} \))(\( ( 1 - 53 T )^{2}( 1 + 53 T )^{2} \))
$59$ (\( ( 1 - 59 T )( 1 + 59 T ) \))(\( 1 - 6062 T^{2} + 12117361 T^{4} \))(\( 1 + 87 T + 6004 T^{2} + 302847 T^{3} + 12117361 T^{4} \))
$61$ (\( 1 + 121 T + 3721 T^{2} \))(\( ( 1 + 76 T + 3721 T^{2} )^{2} \))(\( 1 + 56 T - 585 T^{2} + 208376 T^{3} + 13845841 T^{4} \))
$67$ (\( 1 + 109 T + 4489 T^{2} \))(\( ( 1 + 10 T + 4489 T^{2} )^{2} \))(\( 1 - 31 T - 3528 T^{2} - 139159 T^{3} + 20151121 T^{4} \))
$71$ (\( ( 1 - 71 T )( 1 + 71 T ) \))(\( 1 - 1982 T^{2} + 25411681 T^{4} \))(\( 1 - 9110 T^{2} + 25411681 T^{4} \))
$73$ (\( 1 + 97 T + 5329 T^{2} \))(\( ( 1 - 65 T + 5329 T^{2} )^{2} \))(\( ( 1 - 65 T + 5329 T^{2} )^{2} \))
$79$ (\( 1 - 131 T + 6241 T^{2} \))(\( ( 1 - 14 T + 6241 T^{2} )^{2} \))(\( 1 + 38 T - 4797 T^{2} + 237158 T^{3} + 38950081 T^{4} \))
$83$ (\( ( 1 - 83 T )( 1 + 83 T ) \))(\( 1 - 13769 T^{2} + 47458321 T^{4} \))(\( 1 - 84 T + 9241 T^{2} - 578676 T^{3} + 47458321 T^{4} \))
$89$ (\( ( 1 - 89 T )( 1 + 89 T ) \))(\( 1 - 7742 T^{2} + 62742241 T^{4} \))(\( 1 - 290 T^{2} + 62742241 T^{4} \))
$97$ (\( 1 - 167 T + 9409 T^{2} \))(\( ( 1 + 85 T + 9409 T^{2} )^{2} \))(\( 1 - 115 T + 3816 T^{2} - 1082035 T^{3} + 88529281 T^{4} \))
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