Properties

Label 216.2.f
Level 216
Weight 2
Character orbit f
Rep. character \(\chi_{216}(107,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 2
Sturm bound 72
Trace bound 4

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Defining parameters

Level: \( N \) = \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 216.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 24 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(72\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(216, [\chi])\).

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

Trace form

\( 16q - 2q^{4} + O(q^{10}) \) \( 16q - 2q^{4} + 6q^{10} - 14q^{16} + 8q^{19} + 14q^{22} + 16q^{25} - 30q^{28} - 16q^{34} - 18q^{40} - 16q^{43} - 48q^{46} - 8q^{49} + 12q^{52} + 12q^{58} + 34q^{64} - 64q^{67} + 66q^{70} + 8q^{73} + 44q^{76} + 56q^{82} - 34q^{88} - 24q^{91} + 48q^{94} - 16q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(216, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
216.2.f.a \(8\) \(1.725\) 8.0.\(\cdots\).3 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}+(-1-\beta _{1}-\beta _{4})q^{4}+\beta _{2}q^{5}+\cdots\)
216.2.f.b \(8\) \(1.725\) 8.0.170772624.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{2}-\beta _{3}q^{4}+\beta _{2}q^{5}+(\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(216, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(216, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} + 6 T^{4} + 8 T^{6} + 16 T^{8} \))(\( 1 - T^{2} - 4 T^{6} + 16 T^{8} \))
$3$ 1
$5$ (\( ( 1 + 8 T^{2} + 54 T^{4} + 200 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 + 8 T^{2} + 33 T^{4} + 200 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( ( 1 - 10 T^{2} + 75 T^{4} - 490 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 129 T^{4} - 784 T^{6} + 2401 T^{8} )^{2} \))
$11$ (\( ( 1 - 16 T^{2} + 198 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 11 T^{2} + 121 T^{4} )^{4} \))
$13$ (\( ( 1 - 22 T^{2} + 267 T^{4} - 3718 T^{6} + 28561 T^{8} )^{2} \))(\( ( 1 - 2 T - 6 T^{2} - 26 T^{3} + 169 T^{4} )^{2}( 1 + 2 T - 6 T^{2} + 26 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( ( 1 - 16 T^{2} + 54 T^{4} - 4624 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 40 T^{2} + 846 T^{4} - 11560 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 4 T + 39 T^{2} - 76 T^{3} + 361 T^{4} )^{4} \))(\( ( 1 + 2 T + 6 T^{2} + 38 T^{3} + 361 T^{4} )^{4} \))
$23$ (\( ( 1 + 56 T^{2} + 1542 T^{4} + 29624 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 32 T^{2} + 1182 T^{4} + 16928 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 + 20 T^{2} + 1014 T^{4} + 16820 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 + 80 T^{2} + 3150 T^{4} + 67280 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 28 T^{2} + 1926 T^{4} - 26908 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 64 T^{2} + 2913 T^{4} - 61504 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - 94 T^{2} + 4515 T^{4} - 128686 T^{6} + 1874161 T^{8} )^{2} \))(\( ( 1 + 32 T^{2} + 1806 T^{4} + 43808 T^{6} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 - 52 T^{2} + 2310 T^{4} - 87412 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 88 T^{2} + 4110 T^{4} - 147928 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 2 T + 43 T^{2} )^{8} \))(\( ( 1 + 4 T + 43 T^{2} )^{8} \))
$47$ (\( ( 1 + 56 T^{2} + 4902 T^{4} + 123704 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 44 T^{2} + 2790 T^{4} + 97196 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 + 68 T^{2} + 5046 T^{4} + 191012 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 + 176 T^{2} + 13065 T^{4} + 494384 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 - 208 T^{2} + 17670 T^{4} - 724048 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 74 T^{2} + 3481 T^{4} )^{4} \))
$61$ (\( ( 1 - 214 T^{2} + 18699 T^{4} - 796294 T^{6} + 13845841 T^{8} )^{2} \))(\( ( 1 - 100 T^{2} + 7830 T^{4} - 372100 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 + 8 T + 123 T^{2} + 536 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{8} \))
$71$ (\( ( 1 + 140 T^{2} + 13254 T^{4} + 705740 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 + 176 T^{2} + 16638 T^{4} + 887216 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 2 T + 99 T^{2} + 146 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 - 4 T + 117 T^{2} - 292 T^{3} + 5329 T^{4} )^{4} \))
$79$ (\( ( 1 - 130 T^{2} + 13635 T^{4} - 811330 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 256 T^{2} + 28734 T^{4} - 1597696 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 - 172 T^{2} + 20406 T^{4} - 1184908 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 214 T^{2} + 23115 T^{4} - 1474246 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 80 T^{2} + 16470 T^{4} + 633680 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 52 T^{2} - 2490 T^{4} - 411892 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 2 T + 87 T^{2} + 194 T^{3} + 9409 T^{4} )^{4} \))(\( ( 1 + T + 97 T^{2} )^{8} \))
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