Properties

Label 403.2.be
Level 403
Weight 2
Character orbit be
Rep. character \(\chi_{403}(57,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 144
Newform subspaces 3
Sturm bound 74
Trace bound 6

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Defining parameters

Level: \( N \) = \( 403 = 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 403.be (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 403 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 3 \)
Sturm bound: \(74\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(403, [\chi])\).

Total New Old
Modular forms 160 160 0
Cusp forms 144 144 0
Eisenstein series 16 16 0

Trace form

\( 144q - 8q^{2} - 12q^{3} - 2q^{5} - 36q^{6} - 4q^{7} - 4q^{8} + 64q^{9} + O(q^{10}) \) \( 144q - 8q^{2} - 12q^{3} - 2q^{5} - 36q^{6} - 4q^{7} - 4q^{8} + 64q^{9} - 12q^{11} - 6q^{13} + 24q^{14} - 160q^{16} - 40q^{18} - 4q^{19} + 22q^{20} - 42q^{21} + 60q^{22} + 48q^{24} - 18q^{26} + 2q^{28} + 26q^{31} - 16q^{32} - 12q^{33} + 24q^{34} - 8q^{35} + 6q^{37} - 40q^{39} + 12q^{40} - 2q^{41} - 48q^{42} - 72q^{44} + 10q^{50} + 120q^{52} + 24q^{53} - 12q^{55} - 84q^{57} + 14q^{59} + 48q^{63} + 66q^{65} - 176q^{66} + 10q^{67} - 168q^{70} - 38q^{71} + 74q^{72} + 12q^{73} + 72q^{74} + 64q^{76} + 84q^{78} + 96q^{79} + 50q^{80} - 32q^{81} - 18q^{83} - 114q^{84} - 102q^{86} + 64q^{87} - 140q^{93} - 32q^{94} + 192q^{96} + 12q^{97} + 20q^{98} - 24q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(403, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
403.2.be.a \(4\) \(3.218\) \(\Q(\zeta_{12})\) None \(-2\) \(6\) \(-2\) \(-10\) \(q+(-1+\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+\cdots\)
403.2.be.b \(4\) \(3.218\) \(\Q(\zeta_{12})\) None \(-2\) \(6\) \(-2\) \(8\) \(q+(-\zeta_{12}-\zeta_{12}^{2})q^{2}+(1-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
403.2.be.c \(136\) \(3.218\) None \(-4\) \(-24\) \(2\) \(-2\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))
$3$ (\( 1 - 6 T + 20 T^{2} - 48 T^{3} + 91 T^{4} - 144 T^{5} + 180 T^{6} - 162 T^{7} + 81 T^{8} \))(\( 1 - 6 T + 20 T^{2} - 48 T^{3} + 91 T^{4} - 144 T^{5} + 180 T^{6} - 162 T^{7} + 81 T^{8} \))
$5$ (\( ( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} ) \))(\( ( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} ) \))
$7$ (\( ( 1 + 5 T + 7 T^{2} )^{2}( 1 + 2 T^{2} + 49 T^{4} ) \))(\( ( 1 - 4 T + 7 T^{2} )^{2}( 1 + 11 T^{2} + 49 T^{4} ) \))
$11$ (\( 1 - 6 T + 9 T^{2} + 66 T^{3} - 376 T^{4} + 726 T^{5} + 1089 T^{6} - 7986 T^{7} + 14641 T^{8} \))(\( 1 + 9 T^{2} - 48 T^{3} + 20 T^{4} - 528 T^{5} + 1089 T^{6} + 14641 T^{8} \))
$13$ (\( 1 - 4 T + 3 T^{2} - 52 T^{3} + 169 T^{4} \))(\( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \))
$17$ (\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))
$19$ (\( 1 - 6 T + 45 T^{2} - 126 T^{3} + 704 T^{4} - 2394 T^{5} + 16245 T^{6} - 41154 T^{7} + 130321 T^{8} \))(\( 1 - 12 T + 45 T^{2} + 108 T^{3} - 1348 T^{4} + 2052 T^{5} + 16245 T^{6} - 82308 T^{7} + 130321 T^{8} \))
$23$ (\( ( 1 + 2 T + 20 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 20 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( ( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} )( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} ) \))(\( ( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} )( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} ) \))
$31$ (\( 1 - 13 T^{2} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )^{2} \))
$37$ (\( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 + 20 T + 164 T^{2} + 668 T^{3} + 2335 T^{4} + 27388 T^{5} + 275684 T^{6} + 1378420 T^{7} + 2825761 T^{8} \))(\( 1 - 16 T + 164 T^{2} - 1204 T^{3} + 8239 T^{4} - 49364 T^{5} + 275684 T^{6} - 1102736 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 - 2 T - 56 T^{2} + 52 T^{3} + 1579 T^{4} + 2236 T^{5} - 103544 T^{6} - 159014 T^{7} + 3418801 T^{8} \))(\( 1 + 2 T - 56 T^{2} - 52 T^{3} + 1579 T^{4} - 2236 T^{5} - 103544 T^{6} + 159014 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 20 T + 200 T^{2} - 1460 T^{3} + 9982 T^{4} - 68620 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8} \))(\( 1 - 20 T + 200 T^{2} - 1460 T^{3} + 9982 T^{4} - 68620 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 + 30 T + 477 T^{2} + 5310 T^{3} + 44420 T^{4} + 281430 T^{5} + 1339893 T^{6} + 4466310 T^{7} + 7890481 T^{8} \))(\( 1 + 30 T + 477 T^{2} + 5310 T^{3} + 44420 T^{4} + 281430 T^{5} + 1339893 T^{6} + 4466310 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 22 T + 122 T^{2} - 1232 T^{3} - 20033 T^{4} - 72688 T^{5} + 424682 T^{6} + 4518338 T^{7} + 12117361 T^{8} \))(\( 1 - 2 T + 122 T^{2} - 608 T^{3} + 8287 T^{4} - 35872 T^{5} + 424682 T^{6} - 410758 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 182 T^{2} + 15291 T^{4} - 677222 T^{6} + 13845841 T^{8} \))(\( 1 - 182 T^{2} + 15291 T^{4} - 677222 T^{6} + 13845841 T^{8} \))
$67$ (\( 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 82008 T^{5} + 614993 T^{6} - 2406104 T^{7} + 20151121 T^{8} \))(\( 1 + 22 T + 137 T^{2} - 834 T^{3} - 16648 T^{4} - 55878 T^{5} + 614993 T^{6} + 6616786 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 - 16 T + 65 T^{2} + 1016 T^{3} - 15428 T^{4} + 72136 T^{5} + 327665 T^{6} - 5726576 T^{7} + 25411681 T^{8} \))(\( 1 + 2 T + 65 T^{2} + 818 T^{3} + 2464 T^{4} + 58078 T^{5} + 327665 T^{6} + 715822 T^{7} + 25411681 T^{8} \))
$73$ (\( 1 + 6 T + 90 T^{2} - 564 T^{3} - 97 T^{4} - 41172 T^{5} + 479610 T^{6} + 2334102 T^{7} + 28398241 T^{8} \))(\( 1 - 18 T + 90 T^{2} + 1356 T^{3} - 21121 T^{4} + 98988 T^{5} + 479610 T^{6} - 7002306 T^{7} + 28398241 T^{8} \))
$79$ (\( 1 + 36 T + 694 T^{2} + 9432 T^{3} + 96531 T^{4} + 745128 T^{5} + 4331254 T^{6} + 17749404 T^{7} + 38950081 T^{8} \))(\( 1 + 36 T + 694 T^{2} + 9432 T^{3} + 96531 T^{4} + 745128 T^{5} + 4331254 T^{6} + 17749404 T^{7} + 38950081 T^{8} \))
$83$ (\( 1 - 38 T + 650 T^{2} - 6944 T^{3} + 62479 T^{4} - 576352 T^{5} + 4477850 T^{6} - 21727906 T^{7} + 47458321 T^{8} \))(\( 1 - 34 T + 650 T^{2} - 8464 T^{3} + 86383 T^{4} - 702512 T^{5} + 4477850 T^{6} - 19440758 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 + 16 T + 128 T^{2} + 1840 T^{3} + 25774 T^{4} + 163760 T^{5} + 1013888 T^{6} + 11279504 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 128 T^{2} - 1840 T^{3} + 25774 T^{4} - 163760 T^{5} + 1013888 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))
$97$ (\( 1 + 20 T + 200 T^{2} + 2460 T^{3} + 29582 T^{4} + 238620 T^{5} + 1881800 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))(\( 1 + 20 T + 200 T^{2} + 2460 T^{3} + 29582 T^{4} + 238620 T^{5} + 1881800 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))
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