Properties

Label 1350.2.f
Level 1350
Weight 2
Character orbit f
Rep. character \(\chi_{1350}(107,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 48
Newform subspaces 6
Sturm bound 540
Trace bound 31

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Defining parameters

Level: \( N \) = \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1350.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(540\)
Trace bound: \(31\)
Distinguishing \(T_p\): \(7\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1350, [\chi])\).

Total New Old
Modular forms 612 48 564
Cusp forms 468 48 420
Eisenstein series 144 0 144

Trace form

\( 48q + 4q^{7} + O(q^{10}) \) \( 48q + 4q^{7} - 24q^{13} - 48q^{16} + 4q^{22} + 4q^{28} - 24q^{31} - 24q^{37} + 72q^{43} + 16q^{46} + 24q^{52} - 16q^{58} + 40q^{61} - 8q^{67} - 52q^{73} + 8q^{76} - 32q^{82} + 4q^{88} + 96q^{91} + 12q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1350, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1350.2.f.a \(8\) \(10.780\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{24}q^{2}+\zeta_{24}^{3}q^{4}+(-1+\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
1350.2.f.b \(8\) \(10.780\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{2}-\zeta_{24}^{3}q^{4}+(3\zeta_{24}-\zeta_{24}^{2}+\cdots)q^{7}+\cdots\)
1350.2.f.c \(8\) \(10.780\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}q^{2}+\zeta_{24}^{2}q^{4}+\zeta_{24}^{3}q^{8}+\zeta_{24}^{4}q^{11}+\cdots\)
1350.2.f.d \(8\) \(10.780\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}q^{2}+\zeta_{24}^{3}q^{4}+\zeta_{24}^{7}q^{7}+\zeta_{24}^{5}q^{8}+\cdots\)
1350.2.f.e \(8\) \(10.780\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{5}q^{2}-\zeta_{24}^{3}q^{4}+(3\zeta_{24}-\zeta_{24}^{2}+\cdots)q^{7}+\cdots\)
1350.2.f.f \(8\) \(10.780\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(8\) \(q-\zeta_{24}^{5}q^{2}-\zeta_{24}^{3}q^{4}+(1-2\zeta_{24}^{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(450, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(675, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))
$3$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$5$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$7$ (\( ( 1 + 2 T + 2 T^{2} + 12 T^{3} + 71 T^{4} + 84 T^{5} + 98 T^{6} + 686 T^{7} + 2401 T^{8} )^{2} \))(\( 1 + 28 T^{4} + 3270 T^{8} + 67228 T^{12} + 5764801 T^{16} \))(\( ( 1 + 49 T^{4} )^{4} \))(\( ( 1 + 23 T^{4} + 2401 T^{8} )^{2} \))(\( 1 + 28 T^{4} + 3270 T^{8} + 67228 T^{12} + 5764801 T^{16} \))(\( ( 1 - 2 T + 7 T^{2} )^{4}( 1 - 10 T^{2} + 49 T^{4} )^{2} \))
$11$ (\( ( 1 - 16 T^{2} + 231 T^{4} - 1936 T^{6} + 14641 T^{8} )^{2} \))(\( ( 1 - 13 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 + 5 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 11 T^{2} )^{8} \))(\( ( 1 - 13 T^{2} + 121 T^{4} )^{4} \))(\( ( 1 - 22 T^{2} + 267 T^{4} - 2662 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 - 334 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 - 49 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 - 337 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 + 169 T^{4} )^{4} \))(\( ( 1 - 49 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 + 12 T + 72 T^{2} + 336 T^{3} + 1343 T^{4} + 4368 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 - 124 T^{4} - 77946 T^{8} - 10356604 T^{12} + 6975757441 T^{16} \))(\( 1 + 796 T^{4} + 309894 T^{8} + 66482716 T^{12} + 6975757441 T^{16} \))(\( ( 1 + 47 T^{4} + 83521 T^{8} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \))(\( 1 + 796 T^{4} + 309894 T^{8} + 66482716 T^{12} + 6975757441 T^{16} \))(\( 1 - 802 T^{4} + 296739 T^{8} - 66983842 T^{12} + 6975757441 T^{16} \))
$19$ (\( ( 1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 20 T^{2} + 714 T^{4} - 7220 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 34 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 37 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 - 20 T^{2} + 714 T^{4} - 7220 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 56 T^{2} + 1410 T^{4} - 20216 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 + 706 T^{4} + 279841 T^{8} )^{2} \))(\( ( 1 + 311 T^{4} + 279841 T^{8} )^{2} \))(\( ( 1 + 311 T^{4} + 279841 T^{8} )^{2} \))(\( ( 1 - 958 T^{4} + 279841 T^{8} )^{2} \))(\( ( 1 + 311 T^{4} + 279841 T^{8} )^{2} \))(\( 1 - 1522 T^{4} + 1068819 T^{8} - 425918002 T^{12} + 78310985281 T^{16} \))
$29$ (\( ( 1 + 64 T^{2} + 2514 T^{4} + 53824 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 + 6 T + 40 T^{2} + 174 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 + 31 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 50 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 6 T + 40 T^{2} - 174 T^{3} + 841 T^{4} )^{4} \))(\( ( 1 + 106 T^{2} + 4467 T^{4} + 89146 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 + 4 T - 9 T^{2} + 124 T^{3} + 961 T^{4} )^{4} \))(\( ( 1 + 2 T + 31 T^{2} )^{8} \))(\( ( 1 + 5 T + 31 T^{2} )^{8} \))(\( ( 1 - 7 T + 31 T^{2} )^{8} \))(\( ( 1 + 2 T + 31 T^{2} )^{8} \))(\( ( 1 - 2 T + 57 T^{2} - 62 T^{3} + 961 T^{4} )^{4} \))
$37$ (\( ( 1 + 12 T + 72 T^{2} + 12 T^{3} - 1294 T^{4} + 444 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} )^{2} \))(\( 1 - 1442 T^{4} + 2270595 T^{8} - 2702540162 T^{12} + 3512479453921 T^{16} \))(\( ( 1 + 1369 T^{4} )^{4} \))(\( ( 1 - 2737 T^{4} + 1874161 T^{8} )^{2} \))(\( 1 - 1442 T^{4} + 2270595 T^{8} - 2702540162 T^{12} + 3512479453921 T^{16} \))(\( ( 1 + 1369 T^{4} )^{4} \))
$41$ (\( ( 1 - 52 T^{2} + 2838 T^{4} - 87412 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 46 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 26 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 - 41 T^{2} )^{8} \))(\( ( 1 - 46 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 - 136 T^{2} + 7890 T^{4} - 228616 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 12 T + 72 T^{2} - 660 T^{3} + 5906 T^{4} - 28380 T^{5} + 133128 T^{6} - 954084 T^{7} + 3418801 T^{8} )^{2} \))(\( 1 - 5444 T^{4} + 14231334 T^{8} - 18611952644 T^{12} + 11688200277601 T^{16} \))(\( ( 1 - 217 T^{4} + 3418801 T^{8} )^{2} \))(\( ( 1 + 3191 T^{4} + 3418801 T^{8} )^{2} \))(\( 1 - 5444 T^{4} + 14231334 T^{8} - 18611952644 T^{12} + 11688200277601 T^{16} \))(\( ( 1 - 24 T + 288 T^{2} - 2688 T^{3} + 20327 T^{4} - 115584 T^{5} + 532512 T^{6} - 1908168 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 - 1054 T^{4} + 4879681 T^{8} )^{2} \))(\( ( 1 - 4249 T^{4} + 4879681 T^{8} )^{2} \))(\( ( 1 - 4249 T^{4} + 4879681 T^{8} )^{2} \))(\( ( 1 + 2209 T^{4} )^{4} \))(\( ( 1 - 4249 T^{4} + 4879681 T^{8} )^{2} \))(\( 1 - 3026 T^{4} + 4575795 T^{8} - 14765914706 T^{12} + 23811286661761 T^{16} \))
$53$ (\( 1 + 878 T^{4} + 6253683 T^{8} + 6927842318 T^{12} + 62259690411361 T^{16} \))(\( 1 + 508 T^{4} - 3205722 T^{8} + 4008364348 T^{12} + 62259690411361 T^{16} \))(\( ( 1 - 4174 T^{4} + 7890481 T^{8} )^{2} \))(\( ( 1 - 4174 T^{4} + 7890481 T^{8} )^{2} \))(\( 1 + 508 T^{4} - 3205722 T^{8} + 4008364348 T^{12} + 62259690411361 T^{16} \))(\( ( 1 - 3854 T^{4} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 112 T^{2} + 8370 T^{4} + 389872 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 - 6 T + 19 T^{2} - 354 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 10 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 10 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 6 T + 19 T^{2} + 354 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 + 110 T^{2} + 3481 T^{4} )^{4} \))
$61$ (\( ( 1 - 12 T + 146 T^{2} - 732 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 8 T + 111 T^{2} + 488 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 - 8 T + 61 T^{2} )^{8} \))(\( ( 1 - 5 T + 61 T^{2} )^{8} \))(\( ( 1 + 8 T + 111 T^{2} + 488 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 12 T + 152 T^{2} + 732 T^{3} + 3721 T^{4} )^{4} \))
$67$ (\( ( 1 - 4 T + 8 T^{2} - 60 T^{3} - 2254 T^{4} - 4020 T^{5} + 35912 T^{6} - 1203052 T^{7} + 20151121 T^{8} )^{2} \))(\( 1 + 3196 T^{4} + 5515494 T^{8} + 64402982716 T^{12} + 406067677556641 T^{16} \))(\( ( 1 - 8302 T^{4} + 20151121 T^{8} )^{2} \))(\( ( 1 - 8302 T^{4} + 20151121 T^{8} )^{2} \))(\( 1 + 3196 T^{4} + 5515494 T^{8} + 64402982716 T^{12} + 406067677556641 T^{16} \))(\( ( 1 + 4 T + 8 T^{2} + 268 T^{3} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 - 28 T^{2} - 2010 T^{4} - 141148 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 158 T^{2} + 12435 T^{4} - 796478 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 34 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 34 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 158 T^{2} + 12435 T^{4} - 796478 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 88 T^{2} + 2418 T^{4} - 443608 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 17 T + 73 T^{2} )^{4}( 1 + 143 T^{2} + 5329 T^{4} )^{2} \))(\( 1 - 7292 T^{4} + 67324998 T^{8} - 207079973372 T^{12} + 806460091894081 T^{16} \))(\( ( 1 - 9214 T^{4} + 28398241 T^{8} )^{2} \))(\( ( 1 - 10657 T^{4} + 28398241 T^{8} )^{2} \))(\( 1 - 7292 T^{4} + 67324998 T^{8} - 207079973372 T^{12} + 806460091894081 T^{16} \))(\( ( 1 - 8 T + 32 T^{2} - 552 T^{3} + 9506 T^{4} - 40296 T^{5} + 170528 T^{6} - 3112136 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 - 146 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 212 T^{2} + 21018 T^{4} - 1323092 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 157 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 + 11 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 212 T^{2} + 21018 T^{4} - 1323092 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 - 286 T^{2} + 32715 T^{4} - 1784926 T^{6} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 + 6286 T^{4} + 19632339 T^{8} + 298323005806 T^{12} + 2252292232139041 T^{16} \))(\( 1 + 4516 T^{4} + 69906534 T^{8} + 214321777636 T^{12} + 2252292232139041 T^{16} \))(\( ( 1 - 13294 T^{4} + 47458321 T^{8} )^{2} \))(\( ( 1 + 3122 T^{4} + 47458321 T^{8} )^{2} \))(\( 1 + 4516 T^{4} + 69906534 T^{8} + 214321777636 T^{12} + 2252292232139041 T^{16} \))(\( 1 - 20132 T^{4} + 192702054 T^{8} - 955430918372 T^{12} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 + 244 T^{2} + 30294 T^{4} + 1932724 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 6 T + 160 T^{2} - 534 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 + 70 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 + 70 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 + 6 T + 160 T^{2} + 534 T^{3} + 7921 T^{4} )^{4} \))(\( ( 1 + 40 T^{2} - 5358 T^{4} + 316840 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 - 30 T + 450 T^{2} - 4080 T^{3} + 35471 T^{4} - 395760 T^{5} + 4234050 T^{6} - 27380190 T^{7} + 88529281 T^{8} )^{2} \))(\( 1 - 2258 T^{4} - 119271597 T^{8} - 199899116498 T^{12} + 7837433594376961 T^{16} \))(\( ( 1 - 11422 T^{4} + 88529281 T^{8} )^{2} \))(\( ( 1 - 4657 T^{4} + 88529281 T^{8} )^{2} \))(\( 1 - 2258 T^{4} - 119271597 T^{8} - 199899116498 T^{12} + 7837433594376961 T^{16} \))(\( ( 1 + 24 T + 288 T^{2} + 3768 T^{3} + 45698 T^{4} + 365496 T^{5} + 2709792 T^{6} + 21904152 T^{7} + 88529281 T^{8} )^{2} \))
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