Properties

Label 1728.2.c
Level 1728
Weight 2
Character orbit c
Rep. character \(\chi_{1728}(1727,\cdot)\)
Character field \(\Q\)
Dimension 32
Newform subspaces 7
Sturm bound 576
Trace bound 13

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Defining parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(576\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1728, [\chi])\).

Total New Old
Modular forms 324 32 292
Cusp forms 252 32 220
Eisenstein series 72 0 72

Trace form

\( 32q + O(q^{10}) \) \( 32q + 8q^{13} - 32q^{25} - 8q^{37} - 32q^{49} - 24q^{61} + 16q^{85} + 16q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1728, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1728.2.c.a \(2\) \(13.798\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{6}q^{7}-7q^{13}+5\zeta_{6}q^{19}+5q^{25}+\cdots\)
1728.2.c.b \(2\) \(13.798\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{6}q^{7}+5q^{13}+\zeta_{6}q^{19}+5q^{25}+\cdots\)
1728.2.c.c \(4\) \(13.798\) \(\Q(\sqrt{3}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{5}-\beta _{2}q^{7}-\beta _{3}q^{11}-2q^{13}+\cdots\)
1728.2.c.d \(4\) \(13.798\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}+\beta _{2}q^{7}+\beta _{3}q^{11}+q^{13}+\cdots\)
1728.2.c.e \(4\) \(13.798\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{5}-\zeta_{12}^{2}q^{7}+\zeta_{12}^{3}q^{11}+\cdots\)
1728.2.c.f \(8\) \(13.798\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{2}q^{5}+(\zeta_{24}+\zeta_{24}^{3})q^{7}-\zeta_{24}^{6}q^{11}+\cdots\)
1728.2.c.g \(8\) \(13.798\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{3}q^{5}-\zeta_{24}^{7}q^{7}+(\zeta_{24}+\zeta_{24}^{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1728, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 4 T^{2} + 6 T^{4} - 100 T^{6} + 625 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} + 51 T^{4} - 250 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( ( 1 - 5 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} ) \))(\( ( 1 - T + 7 T^{2} )( 1 + T + 7 T^{2} ) \))(\( ( 1 + T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2}( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 5 T + 7 T^{2} )^{2}( 1 + 5 T + 7 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 99 T^{4} - 686 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 - 14 T^{2} + 123 T^{4} - 686 T^{6} + 2401 T^{8} )^{2} \))
$11$ (\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 5 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 + 28 T^{2} + 390 T^{4} + 3388 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 + 22 T^{2} + 267 T^{4} + 2662 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{4} \))(\( ( 1 - T + 13 T^{2} )^{4} \))(\( ( 1 - 2 T + 13 T^{2} )^{4} \))(\( ( 1 + 2 T + 15 T^{2} + 26 T^{3} + 169 T^{4} )^{4} \))(\( ( 1 - 4 T + 6 T^{2} - 52 T^{3} + 169 T^{4} )^{4} \))
$17$ (\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 2 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} + 246 T^{4} - 5780 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 - 22 T^{2} + 289 T^{4} )^{4} \))
$19$ (\( ( 1 - T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 35 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 14 T^{2} + 361 T^{4} )^{4} \))
$23$ (\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 22 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 23 T^{2} )^{4} \))(\( ( 1 - 20 T^{2} + 726 T^{4} - 10580 T^{6} + 279841 T^{8} )^{2} \))(\( ( 1 + 38 T^{2} + 529 T^{4} )^{4} \))
$29$ (\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 38 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 26 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 52 T^{2} + 1590 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 28 T^{2} + 342 T^{4} - 23548 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( ( 1 - 47 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 - 68 T^{2} + 2310 T^{4} - 65348 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 - 62 T^{2} + 2283 T^{4} - 59582 T^{6} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - T + 37 T^{2} )^{2} \))(\( ( 1 + 11 T + 37 T^{2} )^{2} \))(\( ( 1 - 4 T + 37 T^{2} )^{4} \))(\( ( 1 - T + 37 T^{2} )^{4} \))(\( ( 1 + 8 T + 37 T^{2} )^{4} \))(\( ( 1 - 6 T + 71 T^{2} - 222 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 + 50 T^{2} + 1369 T^{4} )^{4} \))
$41$ (\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))(\( ( 1 - 2 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 - 100 T^{2} + 5094 T^{4} - 168100 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 52 T^{2} + 2502 T^{4} - 87412 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 26 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2}( 1 + 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - 116 T^{2} + 6678 T^{4} - 214484 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 82 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 14 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 + 76 T^{2} + 4662 T^{4} + 167884 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 + 100 T^{2} + 5382 T^{4} + 220900 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 101 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 74 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 25 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 20 T^{2} - 1194 T^{4} - 56180 T^{6} + 7890481 T^{8} )^{2} \))(\( ( 1 - 26 T^{2} + 387 T^{4} - 73034 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 94 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 + 28 T^{2} + 4806 T^{4} + 97468 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 148 T^{2} + 10902 T^{4} + 515188 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 13 T + 61 T^{2} )^{2} \))(\( ( 1 - T + 61 T^{2} )^{2} \))(\( ( 1 - 4 T + 61 T^{2} )^{4} \))(\( ( 1 + 11 T + 61 T^{2} )^{4} \))(\( ( 1 - 4 T + 61 T^{2} )^{4} \))(\( ( 1 + 2 T + 15 T^{2} + 122 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 4 T + 61 T^{2} )^{8} \))
$67$ (\( ( 1 - 11 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \))(\( ( 1 - 74 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 11 T + 67 T^{2} )^{2}( 1 + 11 T + 67 T^{2} )^{2} \))(\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \))(\( ( 1 - 134 T^{2} + 12699 T^{4} - 601526 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} - 522 T^{4} - 89780 T^{6} + 20151121 T^{8} )^{2} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 34 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 + 52 T^{2} + 7302 T^{4} + 262132 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 17 T + 73 T^{2} )^{2} \))(\( ( 1 - 7 T + 73 T^{2} )^{2} \))(\( ( 1 - 5 T + 73 T^{2} )^{4} \))(\( ( 1 + T + 73 T^{2} )^{4} \))(\( ( 1 - T + 73 T^{2} )^{4} \))(\( ( 1 - 6 T + 107 T^{2} - 438 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 + 6 T + 131 T^{2} + 438 T^{3} + 5329 T^{4} )^{4} \))
$79$ (\( ( 1 - 13 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( ( 1 - 16 T + 79 T^{2} )^{2}( 1 + 16 T + 79 T^{2} )^{2} \))(\( ( 1 - 155 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 146 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 62 T^{2} + 2643 T^{4} - 386942 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 154 T^{2} + 6241 T^{4} )^{4} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 19 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 70 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 139 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 268 T^{2} + 30966 T^{4} + 1846252 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 - 74 T^{2} + 12747 T^{4} - 509786 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 - 158 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 170 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 308 T^{2} + 39126 T^{4} - 2439668 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 188 T^{2} + 21222 T^{4} - 1489148 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 5 T + 97 T^{2} )^{2} \))(\( ( 1 - 19 T + 97 T^{2} )^{2} \))(\( ( 1 - 11 T + 97 T^{2} )^{4} \))(\( ( 1 + 13 T + 97 T^{2} )^{4} \))(\( ( 1 + 5 T + 97 T^{2} )^{4} \))(\( ( 1 - 7 T + 97 T^{2} )^{8} \))(\( ( 1 + 5 T + 97 T^{2} )^{8} \))
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