Properties

Label 216.2.i
Level $216$
Weight $2$
Character orbit 216.i
Rep. character $\chi_{216}(73,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $6$
Newform subspaces $2$
Sturm bound $72$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(72\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 96 6 90
Cusp forms 48 6 42
Eisenstein series 48 0 48

Trace form

\( 6q - 2q^{5} + O(q^{10}) \) \( 6q - 2q^{5} + 7q^{11} + 14q^{17} + 6q^{19} + 4q^{23} - 3q^{25} - 12q^{29} - 6q^{31} - 36q^{35} - 9q^{41} - 9q^{43} - 9q^{49} + 16q^{53} - 12q^{55} + 25q^{59} - 6q^{61} - 14q^{65} - 3q^{67} + 8q^{71} - 18q^{73} - 12q^{77} + 6q^{79} + 26q^{83} + 12q^{85} + 12q^{89} + 12q^{91} - 16q^{95} + 21q^{97} + O(q^{100}) \)

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.i.a \(2\) \(1.725\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(3\) \(q-\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(5-5\zeta_{6})q^{11}+\cdots\)
216.2.i.b \(4\) \(1.725\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(-1\) \(-3\) \(q+(-\beta _{1}+\beta _{3})q^{5}+(-2+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(216, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 + T - T^{2} - 8 T^{3} - 26 T^{4} - 40 T^{5} - 25 T^{6} + 125 T^{7} + 625 T^{8} \))
$7$ (\( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} \))(\( 1 + 3 T + T^{2} - 18 T^{3} - 48 T^{4} - 126 T^{5} + 49 T^{6} + 1029 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( ( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} )^{2} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 130 T^{5} + 169 T^{6} + 10985 T^{7} + 28561 T^{8} \))
$17$ (\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 5 T + 32 T^{2} - 85 T^{3} + 289 T^{4} )^{2} \))
$19$ (\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 7 T + 42 T^{2} - 133 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + T - 22 T^{2} + 23 T^{3} + 529 T^{4} \))(\( 1 - 5 T - 19 T^{2} + 10 T^{3} + 832 T^{4} + 230 T^{5} - 10051 T^{6} - 60835 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 522 T^{5} - 36163 T^{6} + 73167 T^{7} + 707281 T^{8} \))
$31$ (\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 + 7 T - 17 T^{2} + 28 T^{3} + 1876 T^{4} + 868 T^{5} - 16337 T^{6} + 208537 T^{7} + 923521 T^{8} \))
$37$ (\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 - 6 T + 50 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 + 12 T + 59 T^{2} + 36 T^{3} - 360 T^{4} + 1476 T^{5} + 99179 T^{6} + 827052 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 + 8 T - 5 T^{2} - 136 T^{3} + 160 T^{4} - 5848 T^{5} - 9245 T^{6} + 636056 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 + 3 T - 38 T^{2} + 141 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 79 T^{2} + 18 T^{3} + 5112 T^{4} + 846 T^{5} - 174511 T^{6} - 311469 T^{7} + 4879681 T^{8} \))
$53$ (\( ( 1 + 2 T + 53 T^{2} )^{2} \))(\( ( 1 - 10 T + 98 T^{2} - 530 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 - 11 T + 62 T^{2} - 649 T^{3} + 3481 T^{4} \))(\( ( 1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4} )^{2} \))
$61$ (\( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \))(\( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 488 T^{5} - 420473 T^{6} - 226981 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - T - 66 T^{2} - 67 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 89 T^{2} - 116 T^{3} + 5464 T^{4} - 7772 T^{5} - 399521 T^{6} + 1203052 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 + 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 7 T + 84 T^{2} + 511 T^{3} + 5329 T^{4} )^{2} \))
$79$ (\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 - 7 T - 113 T^{2} - 28 T^{3} + 16132 T^{4} - 2212 T^{5} - 705233 T^{6} - 3451273 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - T - 82 T^{2} - 83 T^{3} + 6889 T^{4} \))(\( 1 - 25 T + 311 T^{2} - 3700 T^{3} + 39832 T^{4} - 307100 T^{5} + 2142479 T^{6} - 14294675 T^{7} + 47458321 T^{8} \))
$89$ (\( ( 1 - 18 T + 89 T^{2} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{4} \))
$97$ (\( 1 - 13 T + 72 T^{2} - 1261 T^{3} + 9409 T^{4} \))(\( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 13192 T^{5} - 1063217 T^{6} - 7301384 T^{7} + 88529281 T^{8} \))
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