Properties

Label 1008.2.cq
Level 1008
Weight 2
Character orbit cq
Rep. character \(\chi_{1008}(431,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 32
Newform subspaces 3
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.cq (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 84 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(19\)

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

\( 32q + O(q^{10}) \) \( 32q - 16q^{13} - 8q^{25} - 8q^{37} - 16q^{49} + 16q^{61} - 40q^{73} + 96q^{85} + 80q^{97} + O(q^{100}) \)

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.cq.a \(8\) \(8.049\) 8.0.796594176.1 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{5}-\beta _{6}q^{7}+(-\beta _{1}-\beta _{4})q^{11}+\cdots\)
1008.2.cq.b \(12\) \(8.049\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(-2\) \(q+\beta _{2}q^{5}+\beta _{5}q^{7}+(-\beta _{1}+\beta _{8})q^{11}+\cdots\)
1008.2.cq.c \(12\) \(8.049\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(2\) \(q+\beta _{2}q^{5}-\beta _{5}q^{7}+(\beta _{1}-\beta _{8})q^{11}+(-1+\cdots)q^{13}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1008, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))
$3$ (\( \))(\( \))(\( \))
$5$ (\( ( 1 + 8 T^{2} + 39 T^{4} + 200 T^{6} + 625 T^{8} )^{2} \))(\( 1 + 14 T^{2} + 80 T^{4} + 240 T^{6} + 868 T^{8} + 7822 T^{10} + 51214 T^{12} + 195550 T^{14} + 542500 T^{16} + 3750000 T^{18} + 31250000 T^{20} + 136718750 T^{22} + 244140625 T^{24} \))(\( 1 + 14 T^{2} + 80 T^{4} + 240 T^{6} + 868 T^{8} + 7822 T^{10} + 51214 T^{12} + 195550 T^{14} + 542500 T^{16} + 3750000 T^{18} + 31250000 T^{20} + 136718750 T^{22} + 244140625 T^{24} \))
$7$ (\( ( 1 - 7 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 + T + 6 T^{2} + 11 T^{3} + 42 T^{4} + 49 T^{5} + 343 T^{6} )^{2} \))(\( ( 1 - T + 6 T^{2} - 11 T^{3} + 42 T^{4} - 49 T^{5} + 343 T^{6} )^{2} \))
$11$ (\( ( 1 - 8 T^{2} - 57 T^{4} - 968 T^{6} + 14641 T^{8} )^{2} \))(\( 1 - 38 T^{2} + 776 T^{4} - 9168 T^{6} + 57292 T^{8} + 170426 T^{10} - 5851922 T^{12} + 20621546 T^{14} + 838812172 T^{16} - 16241671248 T^{18} + 166342491656 T^{20} - 985622134838 T^{22} + 3138428376721 T^{24} \))(\( 1 - 38 T^{2} + 776 T^{4} - 9168 T^{6} + 57292 T^{8} + 170426 T^{10} - 5851922 T^{12} + 20621546 T^{14} + 838812172 T^{16} - 16241671248 T^{18} + 166342491656 T^{20} - 985622134838 T^{22} + 3138428376721 T^{24} \))
$13$ (\( ( 1 + T + 13 T^{2} )^{8} \))(\( ( 1 + T + 17 T^{2} + 40 T^{3} + 221 T^{4} + 169 T^{5} + 2197 T^{6} )^{4} \))(\( ( 1 + T + 17 T^{2} + 40 T^{3} + 221 T^{4} + 169 T^{5} + 2197 T^{6} )^{4} \))
$17$ (\( ( 1 + 26 T^{2} + 387 T^{4} + 7514 T^{6} + 83521 T^{8} )^{2} \))(\( 1 + 2 T^{2} + 161 T^{4} - 4674 T^{6} + 65662 T^{8} + 6274 T^{10} + 45569893 T^{12} + 1813186 T^{14} + 5484155902 T^{16} - 112818997506 T^{18} + 1123096948001 T^{20} + 4031987800898 T^{22} + 582622237229761 T^{24} \))(\( 1 + 2 T^{2} + 161 T^{4} - 4674 T^{6} + 65662 T^{8} + 6274 T^{10} + 45569893 T^{12} + 1813186 T^{14} + 5484155902 T^{16} - 112818997506 T^{18} + 1123096948001 T^{20} + 4031987800898 T^{22} + 582622237229761 T^{24} \))
$19$ (\( ( 1 + 31 T^{2} + 600 T^{4} + 11191 T^{6} + 130321 T^{8} )^{2} \))(\( ( 1 - 9 T + 74 T^{2} - 423 T^{3} + 2036 T^{4} - 8031 T^{5} + 37706 T^{6} - 152589 T^{7} + 734996 T^{8} - 2901357 T^{9} + 9643754 T^{10} - 22284891 T^{11} + 47045881 T^{12} )^{2} \))(\( ( 1 + 9 T + 74 T^{2} + 423 T^{3} + 2036 T^{4} + 8031 T^{5} + 37706 T^{6} + 152589 T^{7} + 734996 T^{8} + 2901357 T^{9} + 9643754 T^{10} + 22284891 T^{11} + 47045881 T^{12} )^{2} \))
$23$ (\( ( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} )^{2} \))(\( 1 - 62 T^{2} + 1073 T^{4} - 27042 T^{6} + 1725022 T^{8} - 36148606 T^{10} + 397332085 T^{12} - 19122612574 T^{14} + 482731881502 T^{16} - 4003186510338 T^{18} + 84027687206513 T^{20} - 2568443695246238 T^{22} + 21914624432020321 T^{24} \))(\( 1 - 62 T^{2} + 1073 T^{4} - 27042 T^{6} + 1725022 T^{8} - 36148606 T^{10} + 397332085 T^{12} - 19122612574 T^{14} + 482731881502 T^{16} - 4003186510338 T^{18} + 84027687206513 T^{20} - 2568443695246238 T^{22} + 21914624432020321 T^{24} \))
$29$ (\( ( 1 - 26 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 32 T^{2} - 448 T^{4} + 32182 T^{6} - 376768 T^{8} - 22632992 T^{10} + 594823321 T^{12} )^{2} \))(\( ( 1 - 32 T^{2} - 448 T^{4} + 32182 T^{6} - 376768 T^{8} - 22632992 T^{10} + 594823321 T^{12} )^{2} \))
$31$ (\( ( 1 - T^{2} - 960 T^{4} - 961 T^{6} + 923521 T^{8} )^{2} \))(\( ( 1 + 15 T + 147 T^{2} + 1080 T^{3} + 6567 T^{4} + 37641 T^{5} + 203290 T^{6} + 1166871 T^{7} + 6310887 T^{8} + 32174280 T^{9} + 135757587 T^{10} + 429437265 T^{11} + 887503681 T^{12} )^{2} \))(\( ( 1 - 15 T + 147 T^{2} - 1080 T^{3} + 6567 T^{4} - 37641 T^{5} + 203290 T^{6} - 1166871 T^{7} + 6310887 T^{8} - 32174280 T^{9} + 135757587 T^{10} - 429437265 T^{11} + 887503681 T^{12} )^{2} \))
$37$ (\( ( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{4} \))(\( ( 1 - 3 T - 48 T^{2} - 163 T^{3} + 1128 T^{4} + 8319 T^{5} - 31212 T^{6} + 307803 T^{7} + 1544232 T^{8} - 8256439 T^{9} - 89959728 T^{10} - 208031871 T^{11} + 2565726409 T^{12} )^{2} \))(\( ( 1 - 3 T - 48 T^{2} - 163 T^{3} + 1128 T^{4} + 8319 T^{5} - 31212 T^{6} + 307803 T^{7} + 1544232 T^{8} - 8256439 T^{9} - 89959728 T^{10} - 208031871 T^{11} + 2565726409 T^{12} )^{2} \))
$41$ (\( ( 1 - 80 T^{2} + 1681 T^{4} )^{4} \))(\( ( 1 - 140 T^{2} + 10775 T^{4} - 527384 T^{6} + 18112775 T^{8} - 395606540 T^{10} + 4750104241 T^{12} )^{2} \))(\( ( 1 - 140 T^{2} + 10775 T^{4} - 527384 T^{6} + 18112775 T^{8} - 395606540 T^{10} + 4750104241 T^{12} )^{2} \))
$43$ (\( ( 1 - 79 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 + 5 T^{2} + 2411 T^{4} + 80858 T^{6} + 4457939 T^{8} + 17094005 T^{10} + 6321363049 T^{12} )^{2} \))(\( ( 1 + 5 T^{2} + 2411 T^{4} + 80858 T^{6} + 4457939 T^{8} + 17094005 T^{10} + 6321363049 T^{12} )^{2} \))
$47$ (\( ( 1 + 32 T^{2} - 1185 T^{4} + 70688 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8} )^{3} \))(\( ( 1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8} )^{3} \))
$53$ (\( ( 1 + 98 T^{2} + 6795 T^{4} + 275282 T^{6} + 7890481 T^{8} )^{2} \))(\( 1 + 128 T^{2} + 12416 T^{4} + 514572 T^{6} + 5025664 T^{8} - 1856652416 T^{10} - 129165575306 T^{12} - 5215336636544 T^{14} + 39654906304384 T^{16} + 11405159634871788 T^{18} + 773016316147458176 T^{20} + 22385596206785670272 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 + 128 T^{2} + 12416 T^{4} + 514572 T^{6} + 5025664 T^{8} - 1856652416 T^{10} - 129165575306 T^{12} - 5215336636544 T^{14} + 39654906304384 T^{16} + 11405159634871788 T^{18} + 773016316147458176 T^{20} + 22385596206785670272 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( ( 1 - 62 T^{2} + 363 T^{4} - 215822 T^{6} + 12117361 T^{8} )^{2} \))(\( 1 - 140 T^{2} + 10976 T^{4} - 410532 T^{6} - 11424728 T^{8} + 3273287444 T^{10} - 253299214778 T^{12} + 11394313592564 T^{14} - 138437553502808 T^{16} - 17316458836707012 T^{18} + 1611610883145027296 T^{20} - 71556345462089796140 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 140 T^{2} + 10976 T^{4} - 410532 T^{6} - 11424728 T^{8} + 3273287444 T^{10} - 253299214778 T^{12} + 11394313592564 T^{14} - 138437553502808 T^{16} - 17316458836707012 T^{18} + 1611610883145027296 T^{20} - 71556345462089796140 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \))
$61$ (\( ( 1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4} )^{4} \))(\( ( 1 + 8 T - 125 T^{2} - 392 T^{3} + 16576 T^{4} + 25232 T^{5} - 1056859 T^{6} + 1539152 T^{7} + 61679296 T^{8} - 88976552 T^{9} - 1730730125 T^{10} + 6756770408 T^{11} + 51520374361 T^{12} )^{2} \))(\( ( 1 + 8 T - 125 T^{2} - 392 T^{3} + 16576 T^{4} + 25232 T^{5} - 1056859 T^{6} + 1539152 T^{7} + 61679296 T^{8} - 88976552 T^{9} - 1730730125 T^{10} + 6756770408 T^{11} + 51520374361 T^{12} )^{2} \))
$67$ (\( ( 1 + 71 T^{2} + 552 T^{4} + 318719 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 15 T + 228 T^{2} - 2295 T^{3} + 21768 T^{4} - 196131 T^{5} + 1618630 T^{6} - 13140777 T^{7} + 97716552 T^{8} - 690251085 T^{9} + 4594455588 T^{10} - 20251876605 T^{11} + 90458382169 T^{12} )^{2} \))(\( ( 1 + 15 T + 228 T^{2} + 2295 T^{3} + 21768 T^{4} + 196131 T^{5} + 1618630 T^{6} + 13140777 T^{7} + 97716552 T^{8} + 690251085 T^{9} + 4594455588 T^{10} + 20251876605 T^{11} + 90458382169 T^{12} )^{2} \))
$71$ (\( ( 1 + 128 T^{2} + 5041 T^{4} )^{4} \))(\( ( 1 - 40 T^{2} + 6227 T^{4} - 29008 T^{6} + 31390307 T^{8} - 1016467240 T^{10} + 128100283921 T^{12} )^{2} \))(\( ( 1 - 40 T^{2} + 6227 T^{4} - 29008 T^{6} + 31390307 T^{8} - 1016467240 T^{10} + 128100283921 T^{12} )^{2} \))
$73$ (\( ( 1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4} )^{4} \))(\( ( 1 + 9 T - 114 T^{2} - 377 T^{3} + 14196 T^{4} - 399 T^{5} - 1293084 T^{6} - 29127 T^{7} + 75650484 T^{8} - 146659409 T^{9} - 3237399474 T^{10} + 18657644337 T^{11} + 151334226289 T^{12} )^{2} \))(\( ( 1 + 9 T - 114 T^{2} - 377 T^{3} + 14196 T^{4} - 399 T^{5} - 1293084 T^{6} - 29127 T^{7} + 75650484 T^{8} - 146659409 T^{9} - 3237399474 T^{10} + 18657644337 T^{11} + 151334226289 T^{12} )^{2} \))
$79$ (\( ( 1 + 95 T^{2} + 2784 T^{4} + 592895 T^{6} + 38950081 T^{8} )^{2} \))(\( ( 1 + 3 T + 195 T^{2} + 576 T^{3} + 22143 T^{4} + 67101 T^{5} + 1974058 T^{6} + 5300979 T^{7} + 138194463 T^{8} + 283990464 T^{9} + 7595265795 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} )^{2} \))(\( ( 1 - 3 T + 195 T^{2} - 576 T^{3} + 22143 T^{4} - 67101 T^{5} + 1974058 T^{6} - 5300979 T^{7} + 138194463 T^{8} - 283990464 T^{9} + 7595265795 T^{10} - 9231169197 T^{11} + 243087455521 T^{12} )^{2} \))
$83$ (\( ( 1 + 152 T^{2} + 6889 T^{4} )^{4} \))(\( ( 1 + 278 T^{2} + 45932 T^{4} + 4583708 T^{6} + 316425548 T^{8} + 13193413238 T^{10} + 326940373369 T^{12} )^{2} \))(\( ( 1 + 278 T^{2} + 45932 T^{4} + 4583708 T^{6} + 316425548 T^{8} + 13193413238 T^{10} + 326940373369 T^{12} )^{2} \))
$89$ (\( ( 1 + 50 T^{2} - 5421 T^{4} + 396050 T^{6} + 62742241 T^{8} )^{2} \))(\( 1 + 206 T^{2} + 15413 T^{4} + 243402 T^{6} - 32110262 T^{8} - 3337107386 T^{10} - 368341687139 T^{12} - 26433227604506 T^{14} - 2014669796977142 T^{16} + 120966240182489322 T^{18} + 60674643262286174453 T^{20} + \)\(64\!\cdots\!06\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 + 206 T^{2} + 15413 T^{4} + 243402 T^{6} - 32110262 T^{8} - 3337107386 T^{10} - 368341687139 T^{12} - 26433227604506 T^{14} - 2014669796977142 T^{16} + 120966240182489322 T^{18} + 60674643262286174453 T^{20} + \)\(64\!\cdots\!06\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( ( 1 + 4 T + 97 T^{2} )^{8} \))(\( ( 1 - 14 T + 332 T^{2} - 2708 T^{3} + 32204 T^{4} - 131726 T^{5} + 912673 T^{6} )^{4} \))(\( ( 1 - 14 T + 332 T^{2} - 2708 T^{3} + 32204 T^{4} - 131726 T^{5} + 912673 T^{6} )^{4} \))
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