Properties

Label 2100.2.x
Level 2100
Weight 2
Character orbit x
Rep. character \(\chi_{2100}(1357,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 48
Newform subspaces 4
Sturm bound 960
Trace bound 11

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Defining parameters

Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.x (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 35 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(960\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2100, [\chi])\).

Total New Old
Modular forms 1032 48 984
Cusp forms 888 48 840
Eisenstein series 144 0 144

Trace form

\( 48q + O(q^{10}) \) \( 48q - 32q^{11} + 4q^{21} - 8q^{23} - 16q^{37} - 48q^{43} + 32q^{51} + 40q^{53} + 8q^{57} + 48q^{67} + 64q^{71} + 24q^{77} - 48q^{81} - 44q^{91} + 8q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2100, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2100.2.x.a \(8\) \(16.769\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{3}+(2\zeta_{24}+\zeta_{24}^{7})q^{7}-\zeta_{24}^{3}q^{9}+\cdots\)
2100.2.x.b \(8\) \(16.769\) 8.0.157351936.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{3}+\beta _{7}q^{7}-\beta _{3}q^{9}-2q^{11}+\cdots\)
2100.2.x.c \(16\) \(16.769\) 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{3}+(\beta _{12}-\beta _{13}-\beta _{14})q^{7}-\beta _{11}q^{9}+\cdots\)
2100.2.x.d \(16\) \(16.769\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{8}q^{3}+(-\beta _{3}-\beta _{4}+\beta _{10})q^{7}-\beta _{5}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2100, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2100, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(700, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1050, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{4} \))(\( ( 1 + T^{4} )^{4} \))
$5$ 1
$7$ (\( 1 - 94 T^{4} + 2401 T^{8} \))(\( ( 1 + 49 T^{4} )^{2} \))(\( 1 + 73 T^{4} + 2928 T^{8} + 175273 T^{12} + 5764801 T^{16} \))(\( 1 - 32 T^{3} + 36 T^{4} - 96 T^{5} + 512 T^{6} - 1216 T^{7} + 3334 T^{8} - 8512 T^{9} + 25088 T^{10} - 32928 T^{11} + 86436 T^{12} - 537824 T^{13} + 5764801 T^{16} \))
$11$ (\( ( 1 + 3 T + 11 T^{2} )^{8} \))(\( ( 1 + 2 T + 11 T^{2} )^{8} \))(\( ( 1 + T + 8 T^{2} + 11 T^{3} + 121 T^{4} )^{8} \))(\( ( 1 - 4 T + 36 T^{2} - 116 T^{3} + 546 T^{4} - 1276 T^{5} + 4356 T^{6} - 5324 T^{7} + 14641 T^{8} )^{4} \))
$13$ (\( ( 1 + 146 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 + 146 T^{4} + 28561 T^{8} )^{2} \))(\( ( 1 - 383 T^{4} + 86256 T^{8} - 10938863 T^{12} + 815730721 T^{16} )^{2} \))(\( 1 - 536 T^{4} + 97308 T^{8} - 1310248 T^{12} - 1567451002 T^{16} - 37421993128 T^{20} + 79377124999068 T^{24} - 12487773625649816 T^{28} + 665416609183179841 T^{32} \))
$17$ (\( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 289 T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 254 T^{4} + 83521 T^{8} )^{2} \))(\( ( 1 - 16 T^{2} + 289 T^{4} )^{4}( 1 + 16 T^{2} + 289 T^{4} )^{4} \))(\( 1 - 904 T^{4} + 210332 T^{8} + 67692360 T^{12} - 44568692154 T^{16} + 5653733599560 T^{20} + 1467225014080412 T^{24} - 526690502455703944 T^{28} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 + 26 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 + 10 T^{2} + 361 T^{4} )^{4} \))(\( ( 1 + 29 T^{2} + 576 T^{4} + 10469 T^{6} + 130321 T^{8} )^{4} \))(\( ( 1 + 120 T^{2} + 6692 T^{4} + 228264 T^{6} + 5227942 T^{8} + 82403304 T^{10} + 872108132 T^{12} + 5645505720 T^{14} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 - 697 T^{4} + 279841 T^{8} )^{2} \))(\( ( 1 - 734 T^{4} + 279841 T^{8} )^{2} \))(\( ( 1 + 521 T^{4} + 16944 T^{8} + 145797161 T^{12} + 78310985281 T^{16} )^{2} \))(\( ( 1 + 4 T + 8 T^{2} - 4 T^{3} - 1548 T^{4} - 3956 T^{5} - 3432 T^{6} + 53556 T^{7} + 1130086 T^{8} + 1231788 T^{9} - 1815528 T^{10} - 48132652 T^{11} - 433193868 T^{12} - 25745372 T^{13} + 1184287112 T^{14} + 13619301788 T^{15} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 + 23 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 47 T^{2} + 1080 T^{4} - 39527 T^{6} + 707281 T^{8} )^{4} \))(\( ( 1 - 136 T^{2} + 8924 T^{4} - 381688 T^{6} + 12377958 T^{8} - 320999608 T^{10} + 6311775644 T^{12} - 80895971656 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 - 14 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 34 T^{2} + 961 T^{4} )^{4} \))(\( ( 1 - 113 T^{2} + 5100 T^{4} - 108593 T^{6} + 923521 T^{8} )^{4} \))(\( ( 1 - 136 T^{2} + 8612 T^{4} - 354840 T^{6} + 11712390 T^{8} - 341001240 T^{10} + 7953362852 T^{12} - 120700500616 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 - 529 T^{4} + 1874161 T^{8} )^{2} \))(\( ( 1 + 1369 T^{4} )^{4} \))(\( ( 1 + 1249 T^{4} + 2134416 T^{8} + 2340827089 T^{12} + 3512479453921 T^{16} )^{2} \))(\( ( 1 + 8 T + 32 T^{2} + 56 T^{3} - 2980 T^{4} - 15992 T^{5} - 31008 T^{6} + 263416 T^{7} + 5332518 T^{8} + 9746392 T^{9} - 42449952 T^{10} - 810042776 T^{11} - 5584999780 T^{12} + 3883261592 T^{13} + 82103245088 T^{14} + 759455017064 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 41 T^{2} )^{8} \))(\( ( 1 + 30 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 + 12 T^{2} - 250 T^{4} + 20172 T^{6} + 2825761 T^{8} )^{4} \))(\( ( 1 - 200 T^{2} + 20668 T^{4} - 1400376 T^{6} + 67532294 T^{8} - 2354032056 T^{10} + 58402828348 T^{12} - 950020848200 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 + 23 T^{4} + 3418801 T^{8} )^{2} \))(\( ( 1 - 334 T^{4} + 3418801 T^{8} )^{2} \))(\( ( 1 + 478 T^{4} - 4140477 T^{8} + 1634186878 T^{12} + 11688200277601 T^{16} )^{2} \))(\( ( 1 + 24 T + 288 T^{2} + 2504 T^{3} + 19620 T^{4} + 145992 T^{5} + 988256 T^{6} + 5831128 T^{7} + 34926502 T^{8} + 250738504 T^{9} + 1827285344 T^{10} + 11607385944 T^{11} + 67076875620 T^{12} + 368109141272 T^{13} + 1820552558112 T^{14} + 6523646666568 T^{15} + 11688200277601 T^{16} )^{2} \))
$47$ (\( ( 1 - 1054 T^{4} + 4879681 T^{8} )^{2} \))(\( ( 1 - 3518 T^{4} + 4879681 T^{8} )^{2} \))(\( ( 1 + 3682 T^{4} + 4879681 T^{8} )^{4} \))(\( 1 + 2312 T^{4} - 7152868 T^{8} - 8327985864 T^{12} + 35781048692550 T^{16} - 40637914388829384 T^{20} - \)\(17\!\cdots\!48\)\( T^{24} + \)\(26\!\cdots\!92\)\( T^{28} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( ( 1 + 2809 T^{4} )^{4} \))(\( ( 1 + 466 T^{4} + 7890481 T^{8} )^{2} \))(\( ( 1 - 3292 T^{4} + 13762470 T^{8} - 25975463452 T^{12} + 62259690411361 T^{16} )^{2} \))(\( ( 1 - 20 T + 200 T^{2} - 1284 T^{3} + 2996 T^{4} + 21028 T^{5} - 195432 T^{6} + 604308 T^{7} - 304634 T^{8} + 32028324 T^{9} - 548968488 T^{10} + 3130585556 T^{11} + 23639881076 T^{12} - 536963013012 T^{13} + 4432872225800 T^{14} - 23494222796740 T^{15} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 + 10 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 6 T^{2} + 3481 T^{4} )^{4} \))(\( ( 1 + 42 T^{2} + 3481 T^{4} )^{8} \))(\( ( 1 + 280 T^{2} + 33212 T^{4} + 2343784 T^{6} + 136126182 T^{8} + 8158712104 T^{10} + 402441793532 T^{12} + 11810549419480 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )^{4}( 1 + 14 T + 61 T^{2} )^{4} \))(\( ( 1 - 10 T^{2} + 3721 T^{4} )^{4} \))(\( ( 1 - 233 T^{2} + 21000 T^{4} - 866993 T^{6} + 13845841 T^{8} )^{4} \))(\( ( 1 - 312 T^{2} + 47324 T^{4} - 4644168 T^{6} + 328520806 T^{8} - 17280949128 T^{10} + 655240579484 T^{12} - 16074356800632 T^{14} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 - 8809 T^{4} + 20151121 T^{8} )^{2} \))(\( ( 1 + 2258 T^{4} + 20151121 T^{8} )^{2} \))(\( ( 1 + 4247 T^{4} + 20151121 T^{8} )^{4} \))(\( ( 1 - 24 T + 288 T^{2} - 2984 T^{3} + 22788 T^{4} - 56616 T^{5} - 752032 T^{6} + 14609960 T^{7} - 156156506 T^{8} + 978867320 T^{9} - 3375871648 T^{10} - 17027998008 T^{11} + 459203745348 T^{12} - 4028773319288 T^{13} + 26052014064672 T^{14} - 145457078527752 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 - 3 T + 71 T^{2} )^{8} \))(\( ( 1 + 2 T + 71 T^{2} )^{8} \))(\( ( 1 - 11 T + 158 T^{2} - 781 T^{3} + 5041 T^{4} )^{8} \))(\( ( 1 + 8 T + 140 T^{2} + 1024 T^{3} + 14962 T^{4} + 72704 T^{5} + 705740 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{4} \))
$73$ (\( ( 1 - 24 T + 288 T^{2} - 1752 T^{3} + 5329 T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 1752 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 8158 T^{4} + 28398241 T^{8} )^{2} \))(\( ( 1 - 8542 T^{4} + 28398241 T^{8} )^{4} \))(\( 1 - 9816 T^{4} + 48230108 T^{8} - 385344995944 T^{12} + 2907384584774982 T^{16} - 10943120062961734504 T^{20} + \)\(38\!\cdots\!48\)\( T^{24} - \)\(22\!\cdots\!36\)\( T^{28} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( ( 1 - 157 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{4} \))(\( ( 1 - 59 T^{2} + 13224 T^{4} - 368219 T^{6} + 38950081 T^{8} )^{4} \))(\( ( 1 - 248 T^{2} + 39836 T^{4} - 4744648 T^{6} + 419330758 T^{8} - 29611348168 T^{10} + 1551615426716 T^{12} - 60285688969208 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 - 13294 T^{4} + 47458321 T^{8} )^{2} \))(\( ( 1 + 6889 T^{4} )^{4} \))(\( ( 1 + 10948 T^{4} + 66965670 T^{8} + 519573698308 T^{12} + 2252292232139041 T^{16} )^{2} \))(\( 1 + 11592 T^{4} + 91463132 T^{8} + 885997919352 T^{12} + 6666213559678726 T^{16} + 42047973661939327992 T^{20} + \)\(20\!\cdots\!12\)\( T^{24} + \)\(12\!\cdots\!12\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 + 70 T^{2} + 7921 T^{4} )^{4} \))(\( ( 1 + 89 T^{2} )^{8} \))(\( ( 1 - 40 T^{2} - 2226 T^{4} - 316840 T^{6} + 62742241 T^{8} )^{4} \))(\( ( 1 + 328 T^{2} + 42716 T^{4} + 2964280 T^{6} + 188103750 T^{8} + 23480061880 T^{10} + 2680097566556 T^{12} + 163009863435208 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 + 12866 T^{4} + 88529281 T^{8} )^{2} \))(\( ( 1 + 17282 T^{4} + 88529281 T^{8} )^{2} \))(\( ( 1 + 4993 T^{4} + 1167168 T^{8} + 442026700033 T^{12} + 7837433594376961 T^{16} )^{2} \))(\( 1 - 1624 T^{4} + 150045276 T^{8} - 609733450088 T^{12} + 17981575892543942 T^{16} - 53979263937940026728 T^{20} + \)\(11\!\cdots\!36\)\( T^{24} - \)\(11\!\cdots\!84\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))
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