Properties

Label 108.3.d
Level 108
Weight 3
Character orbit d
Rep. character \(\chi_{108}(55,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 4
Sturm bound 54
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 108.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(54\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(108, [\chi])\).

Total New Old
Modular forms 42 16 26
Cusp forms 30 16 14
Eisenstein series 12 0 12

Trace form

\( 16q - 2q^{4} + O(q^{10}) \) \( 16q - 2q^{4} + 14q^{10} + 8q^{13} + 10q^{16} - 102q^{22} + 72q^{25} - 150q^{28} - 40q^{34} + 8q^{37} + 170q^{40} + 372q^{46} - 176q^{49} + 44q^{52} - 340q^{58} - 184q^{61} - 326q^{64} - 78q^{70} - 64q^{73} + 576q^{76} + 596q^{82} + 400q^{85} + 342q^{88} - 300q^{94} + 128q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(108, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
108.3.d.a \(2\) \(2.943\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(14\) \(0\) \(q+(-1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}+7q^{5}+\cdots\)
108.3.d.b \(2\) \(2.943\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-14\) \(0\) \(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}-7q^{5}+\cdots\)
108.3.d.c \(4\) \(2.943\) \(\Q(\sqrt{-3}, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}-\beta _{2})q^{2}+(3+\beta _{3})q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots\)
108.3.d.d \(8\) \(2.943\) 8.0.207360000.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+(-1+\beta _{5})q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(108, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(108, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 4 T^{2} \))(\( 1 - 2 T + 4 T^{2} \))(\( 1 - 5 T^{2} + 16 T^{4} \))(\( 1 + 2 T^{2} - 12 T^{4} + 32 T^{6} + 256 T^{8} \))
$3$ 1
$5$ (\( ( 1 - 7 T + 25 T^{2} )^{2} \))(\( ( 1 + 7 T + 25 T^{2} )^{2} \))(\( ( 1 + 37 T^{2} + 625 T^{4} )^{2} \))(\( ( 1 + 44 T^{2} + 1014 T^{4} + 27500 T^{6} + 390625 T^{8} )^{2} \))
$7$ (\( ( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} ) \))(\( ( 1 - 11 T + 49 T^{2} )( 1 + 11 T + 49 T^{2} ) \))(\( ( 1 - 59 T^{2} + 2401 T^{4} )^{2} \))(\( ( 1 - 70 T^{2} + 5307 T^{4} - 168070 T^{6} + 5764801 T^{8} )^{2} \))
$11$ (\( 1 - 167 T^{2} + 14641 T^{4} \))(\( 1 - 167 T^{2} + 14641 T^{4} \))(\( ( 1 - 95 T^{2} + 14641 T^{4} )^{2} \))(\( ( 1 - 124 T^{2} + 15126 T^{4} - 1815484 T^{6} + 214358881 T^{8} )^{2} \))
$13$ (\( ( 1 - 20 T + 169 T^{2} )^{2} \))(\( ( 1 - 20 T + 169 T^{2} )^{2} \))(\( ( 1 + 16 T + 169 T^{2} )^{4} \))(\( ( 1 + 2 T + 159 T^{2} + 338 T^{3} + 28561 T^{4} )^{4} \))
$17$ (\( ( 1 + 8 T + 289 T^{2} )^{2} \))(\( ( 1 - 8 T + 289 T^{2} )^{2} \))(\( ( 1 + 370 T^{2} + 83521 T^{4} )^{2} \))(\( ( 1 + 140 T^{2} + 136662 T^{4} + 11692940 T^{6} + 6975757441 T^{8} )^{2} \))
$19$ (\( 1 - 614 T^{2} + 130321 T^{4} \))(\( 1 - 614 T^{2} + 130321 T^{4} \))(\( ( 1 + 682 T^{2} + 130321 T^{4} )^{2} \))(\( ( 1 - 695 T^{2} + 130321 T^{4} )^{4} \))
$23$ (\( 1 - 1046 T^{2} + 279841 T^{4} \))(\( 1 - 1046 T^{2} + 279841 T^{4} \))(\( ( 1 - 758 T^{2} + 279841 T^{4} )^{2} \))(\( ( 1 - 604 T^{2} + 632886 T^{4} - 169023964 T^{6} + 78310985281 T^{8} )^{2} \))
$29$ (\( ( 1 - 10 T + 841 T^{2} )^{2} \))(\( ( 1 + 10 T + 841 T^{2} )^{2} \))(\( ( 1 - 866 T^{2} + 707281 T^{4} )^{2} \))(\( ( 1 + 2468 T^{2} + 2752998 T^{4} + 1745569508 T^{6} + 500246412961 T^{8} )^{2} \))
$31$ (\( ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) \))(\( ( 1 - 31 T + 961 T^{2} )( 1 + 31 T + 961 T^{2} ) \))(\( ( 1 - 1883 T^{2} + 923521 T^{4} )^{2} \))(\( ( 1 - 1540 T^{2} + 1702662 T^{4} - 1422222340 T^{6} + 852891037441 T^{8} )^{2} \))
$37$ (\( ( 1 + 10 T + 1369 T^{2} )^{2} \))(\( ( 1 + 10 T + 1369 T^{2} )^{2} \))(\( ( 1 - 26 T + 1369 T^{2} )^{4} \))(\( ( 1 + 14 T + 2607 T^{2} + 19166 T^{3} + 1874161 T^{4} )^{4} \))
$41$ (\( ( 1 + 50 T + 1681 T^{2} )^{2} \))(\( ( 1 - 50 T + 1681 T^{2} )^{2} \))(\( ( 1 + 3310 T^{2} + 2825761 T^{4} )^{2} \))(\( ( 1 + 3140 T^{2} + 5167302 T^{4} + 8872889540 T^{6} + 7984925229121 T^{8} )^{2} \))
$43$ (\( 1 - 3398 T^{2} + 3418801 T^{4} \))(\( 1 - 3398 T^{2} + 3418801 T^{4} \))(\( ( 1 - 3542 T^{2} + 3418801 T^{4} )^{2} \))(\( ( 1 - 4516 T^{2} + 10784166 T^{4} - 15439305316 T^{6} + 11688200277601 T^{8} )^{2} \))
$47$ (\( 1 + 3082 T^{2} + 4879681 T^{4} \))(\( 1 + 3082 T^{2} + 4879681 T^{4} \))(\( ( 1 - 4406 T^{2} + 4879681 T^{4} )^{2} \))(\( ( 1 - 4444 T^{2} + 14436726 T^{4} - 21685302364 T^{6} + 23811286661761 T^{8} )^{2} \))
$53$ (\( ( 1 + 47 T + 2809 T^{2} )^{2} \))(\( ( 1 - 47 T + 2809 T^{2} )^{2} \))(\( ( 1 + 925 T^{2} + 7890481 T^{4} )^{2} \))(\( ( 1 - 508 T^{2} + 14912358 T^{4} - 4008364348 T^{6} + 62259690411361 T^{8} )^{2} \))
$59$ (\( 1 - 5762 T^{2} + 12117361 T^{4} \))(\( 1 - 5762 T^{2} + 12117361 T^{4} \))(\( ( 1 - 1154 T^{2} + 12117361 T^{4} )^{2} \))(\( ( 1 - 6076 T^{2} + 23942166 T^{4} - 73625085436 T^{6} + 146830437604321 T^{8} )^{2} \))
$61$ (\( ( 1 + 64 T + 3721 T^{2} )^{2} \))(\( ( 1 + 64 T + 3721 T^{2} )^{2} \))(\( ( 1 - 8 T + 3721 T^{2} )^{4} \))(\( ( 1 - 10 T + 7287 T^{2} - 37210 T^{3} + 13845841 T^{4} )^{4} \))
$67$ (\( 1 - 1478 T^{2} + 20151121 T^{4} \))(\( 1 - 1478 T^{2} + 20151121 T^{4} \))(\( ( 1 - 5078 T^{2} + 20151121 T^{4} )^{2} \))(\( ( 1 - 8110 T^{2} + 40110387 T^{4} - 163425591310 T^{6} + 406067677556641 T^{8} )^{2} \))
$71$ (\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 6194 T^{2} + 25411681 T^{4} )^{2} \))(\( ( 1 - 292 T^{2} + 42446598 T^{4} - 7420210852 T^{6} + 645753531245761 T^{8} )^{2} \))
$73$ (\( ( 1 + 55 T + 5329 T^{2} )^{2} \))(\( ( 1 + 55 T + 5329 T^{2} )^{2} \))(\( ( 1 + 19 T + 5329 T^{2} )^{4} \))(\( ( 1 - 58 T + 10779 T^{2} - 309082 T^{3} + 28398241 T^{4} )^{4} \))
$79$ (\( 1 - 12434 T^{2} + 38950081 T^{4} \))(\( 1 - 12434 T^{2} + 38950081 T^{4} \))(\( ( 1 - 9986 T^{2} + 38950081 T^{4} )^{2} \))(\( ( 1 - 934 T^{2} - 28603749 T^{4} - 36379375654 T^{6} + 1517108809906561 T^{8} )^{2} \))
$83$ (\( 1 - 12911 T^{2} + 47458321 T^{4} \))(\( 1 - 12911 T^{2} + 47458321 T^{4} \))(\( ( 1 - 311 T^{2} + 47458321 T^{4} )^{2} \))(\( ( 1 - 26116 T^{2} + 265140006 T^{4} - 1239421511236 T^{6} + 2252292232139041 T^{8} )^{2} \))
$89$ (\( ( 1 - 10 T + 7921 T^{2} )^{2} \))(\( ( 1 + 10 T + 7921 T^{2} )^{2} \))(\( ( 1 + 9550 T^{2} + 62742241 T^{4} )^{2} \))(\( ( 1 + 30668 T^{2} + 360580758 T^{4} + 1924179046988 T^{6} + 3936588805702081 T^{8} )^{2} \))
$97$ (\( ( 1 + 25 T + 9409 T^{2} )^{2} \))(\( ( 1 + 25 T + 9409 T^{2} )^{2} \))(\( ( 1 - 119 T + 9409 T^{2} )^{4} \))(\( ( 1 + 62 T + 8259 T^{2} + 583358 T^{3} + 88529281 T^{4} )^{4} \))
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