Properties

Label 1008.2.bt
Level 1008
Weight 2
Character orbit bt
Rep. character \(\chi_{1008}(17,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 32
Newform subspaces 4
Sturm bound 384
Trace bound 7

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Defining parameters

Level: \( N \) = \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1008.bt (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1008, [\chi])\).

Total New Old
Modular forms 432 32 400
Cusp forms 336 32 304
Eisenstein series 96 0 96

Trace form

\( 32q + 4q^{7} + O(q^{10}) \) \( 32q + 4q^{7} - 12q^{19} - 8q^{25} - 12q^{31} + 8q^{43} - 4q^{67} + 24q^{73} + 36q^{79} - 32q^{85} + 36q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1008, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1008.2.bt.a \(4\) \(8.049\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-2\) \(q-\beta _{2}q^{5}+(-2+3\beta _{1})q^{7}+(\beta _{2}+\beta _{3})q^{11}+\cdots\)
1008.2.bt.b \(4\) \(8.049\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(2\) \(q+(-\beta _{1}-\beta _{3})q^{5}+(2-3\beta _{2})q^{7}+\beta _{1}q^{11}+\cdots\)
1008.2.bt.c \(8\) \(8.049\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(-4\) \(q+(-\zeta_{24}^{5}-\zeta_{24}^{7})q^{5}+(-\zeta_{24}^{2}-\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
1008.2.bt.d \(16\) \(8.049\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(8\) \(q+(\beta _{4}-\beta _{10})q^{5}+(\beta _{3}+\beta _{9})q^{7}+(\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1008, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1008, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \))(\( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2}( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} ) \))(\( 1 - 26 T^{2} + 343 T^{4} - 3142 T^{6} + 22997 T^{8} - 145796 T^{10} + 840842 T^{12} - 4532272 T^{14} + 23186506 T^{16} - 113306800 T^{18} + 525526250 T^{20} - 2278062500 T^{22} + 8983203125 T^{24} - 30683593750 T^{26} + 83740234375 T^{28} - 158691406250 T^{30} + 152587890625 T^{32} \))
$7$ (\( ( 1 + T + 7 T^{2} )^{2} \))(\( ( 1 - T + 7 T^{2} )^{2} \))(\( ( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 4 T + 4 T^{3} + 29 T^{4} + 28 T^{5} - 1372 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} \))(\( 1 + 20 T^{2} + 279 T^{4} + 2420 T^{6} + 14641 T^{8} \))(\( ( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} )^{2} \))(\( 1 + 34 T^{2} + 631 T^{4} + 7742 T^{6} + 56309 T^{8} + 36244 T^{10} - 4803190 T^{12} - 80571856 T^{14} - 990277910 T^{16} - 9749194576 T^{18} - 70323504790 T^{20} + 64208456884 T^{22} + 12070334230229 T^{24} + 200807541260942 T^{26} + 1980348305710951 T^{28} + 12911494341830194 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 20 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 - 22 T^{2} + 597 T^{4} - 10874 T^{6} + 145448 T^{8} - 1837706 T^{10} + 17050917 T^{12} - 106189798 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} \))(\( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} \))(\( 1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 295936 T^{10} + 39923038 T^{12} - 772402208 T^{14} + 6975757441 T^{16} \))(\( 1 - 84 T^{2} + 3336 T^{4} - 99240 T^{6} + 2736514 T^{8} - 66944172 T^{10} + 1418515200 T^{12} - 27661974396 T^{14} + 497038747923 T^{16} - 7994310600444 T^{18} + 118475808019200 T^{20} - 1615869570797868 T^{22} + 19089257897900674 T^{24} - 200067234680558760 T^{26} + 1943627783398482696 T^{28} - 14143737430989678036 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 + T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \))(\( ( 1 + 3 T + 22 T^{2} + 57 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 - 12 T + 92 T^{2} - 528 T^{3} + 2487 T^{4} - 10032 T^{5} + 33212 T^{6} - 82308 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} + 174 T^{3} + 567 T^{4} - 1140 T^{5} - 10940 T^{6} - 109212 T^{7} - 514114 T^{8} - 2075028 T^{9} - 3949340 T^{10} - 7819260 T^{11} + 73892007 T^{12} + 430841226 T^{13} + 1928881121 T^{14} + 5363230434 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \))(\( 1 + 14 T^{2} - 333 T^{4} + 7406 T^{6} + 279841 T^{8} \))(\( ( 1 + 28 T^{2} + 255 T^{4} + 14812 T^{6} + 279841 T^{8} )^{2} \))(\( 1 + 84 T^{2} + 2840 T^{4} + 59304 T^{6} + 1536034 T^{8} + 51810444 T^{10} + 1287398720 T^{12} + 22322505660 T^{14} + 403780671955 T^{16} + 11808605494140 T^{18} + 360266945203520 T^{20} + 7669805137024716 T^{22} + 120288335965115554 T^{24} + 2456757821014240296 T^{26} + 62237533386937711640 T^{28} + \)\(97\!\cdots\!56\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 + 14 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 82 T^{2} + 3369 T^{4} - 137570 T^{6} + 4900772 T^{8} - 115696370 T^{10} + 2382829689 T^{12} - 48775512322 T^{14} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 - 7 T + 31 T^{2} )^{2}( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \))(\( ( 1 - 6 T + 23 T^{2} - 66 T^{3} - 468 T^{4} - 2046 T^{5} + 22103 T^{6} - 178746 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 4 T + 48 T^{2} + 404 T^{3} + 1502 T^{4} + 12524 T^{5} + 46128 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2}( 1 + 8 T + 72 T^{2} + 172 T^{3} + 1373 T^{4} + 5332 T^{5} + 69192 T^{6} + 238328 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 - 11 T + 37 T^{2} )^{2}( 1 + 10 T + 37 T^{2} )^{2} \))(\( ( 1 + 8 T - 8 T^{2} - 16 T^{3} + 1447 T^{4} - 592 T^{5} - 10952 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 2 T - 39 T^{2} + 686 T^{3} - 133 T^{4} - 26040 T^{5} + 186136 T^{6} + 542152 T^{7} - 8048970 T^{8} + 20059624 T^{9} + 254820184 T^{10} - 1319004120 T^{11} - 249263413 T^{12} + 47569954502 T^{13} - 100063329951 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 68 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 28 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 20 T^{2} - 1146 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 + 204 T^{2} + 20760 T^{4} + 1374660 T^{6} + 65555246 T^{8} + 2310803460 T^{10} + 58662798360 T^{12} + 969021265164 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 - 11 T + 43 T^{2} )^{4} \))(\( ( 1 - T + 43 T^{2} )^{4} \))(\( ( 1 + 8 T + 84 T^{2} + 344 T^{3} + 1849 T^{4} )^{4} \))(\( ( 1 + 2 T + 139 T^{2} + 134 T^{3} + 8158 T^{4} + 5762 T^{5} + 257011 T^{6} + 159014 T^{7} + 3418801 T^{8} )^{4} \))
$47$ (\( 1 - 88 T^{2} + 5535 T^{4} - 194392 T^{6} + 4879681 T^{8} \))(\( 1 + 56 T^{2} + 927 T^{4} + 123704 T^{6} + 4879681 T^{8} \))(\( 1 - 152 T^{2} + 13198 T^{4} - 834176 T^{6} + 42212419 T^{8} - 1842694784 T^{10} + 64402029838 T^{12} - 1638440730008 T^{14} + 23811286661761 T^{16} \))(\( 1 - 200 T^{2} + 20236 T^{4} - 1289200 T^{6} + 54604010 T^{8} - 1402967000 T^{10} + 1590108464 T^{12} + 2082887856200 T^{14} - 134280636932141 T^{16} + 4601099274345800 T^{18} + 7759222059719984 T^{20} - 15122883392481143000 T^{22} + \)\(13\!\cdots\!10\)\( T^{24} - \)\(67\!\cdots\!00\)\( T^{26} + \)\(23\!\cdots\!76\)\( T^{28} - \)\(51\!\cdots\!00\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 34 T^{2} - 1653 T^{4} + 95506 T^{6} + 7890481 T^{8} \))(\( 1 + 98 T^{2} + 6795 T^{4} + 275282 T^{6} + 7890481 T^{8} \))(\( 1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 2482296446 T^{10} + 108517785193 T^{12} + 3501969058382 T^{14} + 62259690411361 T^{16} \))(\( 1 + 82 T^{2} + 1819 T^{4} - 293986 T^{6} - 25475671 T^{8} - 757240268 T^{10} + 15438451790 T^{12} + 2705625303848 T^{14} + 146204406396370 T^{16} + 7600101478509032 T^{18} + 121816810518410990 T^{20} - 16783746761372742572 T^{22} - \)\(15\!\cdots\!31\)\( T^{24} - \)\(51\!\cdots\!14\)\( T^{26} + \)\(89\!\cdots\!79\)\( T^{28} + \)\(11\!\cdots\!58\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8} \))(\( 1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8} \))(\( 1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 104631898 T^{10} - 57278765447 T^{12} - 1602860278358 T^{14} + 146830437604321 T^{16} \))(\( 1 - 322 T^{2} + 52107 T^{4} - 6104798 T^{6} + 600323513 T^{8} - 51577574244 T^{10} + 3916097487742 T^{12} - 266299877019064 T^{14} + 16439030582633874 T^{16} - 926989871903361784 T^{18} + 47452766970162888862 T^{20} - \)\(21\!\cdots\!04\)\( T^{22} + \)\(88\!\cdots\!73\)\( T^{24} - \)\(31\!\cdots\!98\)\( T^{26} + \)\(92\!\cdots\!67\)\( T^{28} - \)\(19\!\cdots\!42\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4} )^{2} \))(\( ( 1 - 12 T + 176 T^{2} - 1536 T^{3} + 15591 T^{4} - 93696 T^{5} + 654896 T^{6} - 2723772 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 + 70 T^{2} + 4306 T^{4} + 9720 T^{5} - 90560 T^{6} + 641520 T^{7} - 8009105 T^{8} + 39132720 T^{9} - 336973760 T^{10} + 2206255320 T^{11} + 59620191346 T^{12} + 3606426205270 T^{14} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 + 7 T - 18 T^{2} + 469 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 5 T + 67 T^{2} )^{2} \))(\( ( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} )^{4} \))(\( ( 1 - 14 T - 125 T^{2} + 1238 T^{3} + 28877 T^{4} - 158888 T^{5} - 2519254 T^{6} + 1335236 T^{7} + 243583474 T^{8} + 89460812 T^{9} - 11308931206 T^{10} - 47787631544 T^{11} + 581903921117 T^{12} + 1671454882466 T^{13} - 11307297771125 T^{14} - 84849962474522 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 + 20 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 92 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2} \))(\( ( 1 - 392 T^{2} + 75892 T^{4} - 9334040 T^{6} + 790096294 T^{8} - 47052895640 T^{10} + 1928543294452 T^{12} - 50215311297032 T^{14} + 645753531245761 T^{16} )^{2} \))
$73$ (\( ( 1 - 17 T + 73 T^{2} )^{2}( 1 - 10 T + 73 T^{2} )^{2} \))(\( ( 1 - 10 T + 73 T^{2} )^{2}( 1 + 7 T + 73 T^{2} )^{2} \))(\( ( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 16131 T^{4} - 117384 T^{5} + 969878 T^{6} - 4668204 T^{7} + 28398241 T^{8} )^{2} \))(\( ( 1 + 30 T + 591 T^{2} + 8730 T^{3} + 105765 T^{4} + 1168044 T^{5} + 12012906 T^{6} + 115920888 T^{7} + 1041481850 T^{8} + 8462224824 T^{9} + 64016776074 T^{10} + 454388972748 T^{11} + 3003539959365 T^{12} + 18097915006890 T^{13} + 89438527736799 T^{14} + 331421955572910 T^{15} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 5 T - 54 T^{2} - 395 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 14 T + 7 T^{2} + 434 T^{3} + 13996 T^{4} + 34286 T^{5} + 43687 T^{6} + 6902546 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 16 T + 64 T^{2} - 56 T^{3} - 1765 T^{4} + 55364 T^{5} - 19696 T^{6} - 3181796 T^{7} + 14045608 T^{8} - 251361884 T^{9} - 122922736 T^{10} + 27296611196 T^{11} - 68746892965 T^{12} - 172315158344 T^{13} + 15557597153344 T^{14} - 307262543778544 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 + 16 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 112 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 + 278 T^{2} + 32811 T^{4} + 1915142 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 438 T^{2} + 94557 T^{4} + 13226610 T^{6} + 1298783528 T^{8} + 91118116290 T^{10} + 4487516458797 T^{12} + 143199883535622 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( 1 + 38 T^{2} - 6477 T^{4} + 300998 T^{6} + 62742241 T^{8} \))(\( 1 - 154 T^{2} + 15795 T^{4} - 1219834 T^{6} + 62742241 T^{8} \))(\( ( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 - 384 T^{2} + 64644 T^{4} - 8318208 T^{6} + 1128930634 T^{8} - 127988142720 T^{10} + 11460228290064 T^{12} - 1112004852903552 T^{14} + 109920677865950547 T^{16} - 8808190439849035392 T^{18} + \)\(71\!\cdots\!24\)\( T^{20} - \)\(63\!\cdots\!20\)\( T^{22} + \)\(44\!\cdots\!54\)\( T^{24} - \)\(25\!\cdots\!08\)\( T^{26} + \)\(15\!\cdots\!24\)\( T^{28} - \)\(75\!\cdots\!44\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 86 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 190 T^{2} + 20643 T^{4} - 1787710 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 - 234 T^{2} + 40305 T^{4} - 5045730 T^{6} + 535964996 T^{8} - 47475273570 T^{10} + 3568172670705 T^{12} - 194915449153386 T^{14} + 7837433594376961 T^{16} )^{2} \))
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