Properties

Label 336.2.bq
Level 336
Weight 2
Character orbit bq
Rep. character \(\chi_{336}(37,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 128
Newform subspaces 2
Sturm bound 128
Trace bound 1

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Defining parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.bq (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 112 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 2 \)
Sturm bound: \(128\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(336, [\chi])\).

Total New Old
Modular forms 272 128 144
Cusp forms 240 128 112
Eisenstein series 32 0 32

Trace form

\( 128q + 4q^{4} + 24q^{8} + O(q^{10}) \) \( 128q + 4q^{4} + 24q^{8} - 12q^{10} + 8q^{11} + 32q^{14} - 4q^{16} - 4q^{18} - 32q^{20} + 32q^{22} - 8q^{28} - 32q^{29} - 48q^{31} + 32q^{34} - 24q^{35} + 16q^{37} - 40q^{38} - 52q^{40} + 20q^{42} + 16q^{43} - 20q^{44} - 20q^{46} + 32q^{48} - 80q^{50} - 24q^{52} - 16q^{53} - 56q^{56} - 8q^{58} + 32q^{59} - 24q^{60} - 144q^{62} - 128q^{64} + 24q^{66} - 32q^{67} - 20q^{68} - 92q^{70} + 4q^{72} + 16q^{74} + 24q^{76} - 72q^{78} + 100q^{80} + 64q^{81} + 20q^{82} + 24q^{84} + 44q^{86} + 48q^{88} + 8q^{91} - 184q^{92} + 60q^{94} - 48q^{95} + 20q^{96} + 124q^{98} - 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(336, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
336.2.bq.a \(8\) \(2.683\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(8\) \(0\) \(q+(\zeta_{24}-\zeta_{24}^{7})q^{2}+\zeta_{24}^{5}q^{3}+(2-2\zeta_{24}^{4}+\cdots)q^{4}+\cdots\)
336.2.bq.b \(120\) \(2.683\) None \(0\) \(0\) \(-8\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))
$3$ (\( 1 - T^{4} + T^{8} \))
$5$ (\( 1 - 8 T + 32 T^{2} - 64 T^{3} + 17 T^{4} + 272 T^{5} - 672 T^{6} + 376 T^{7} + 816 T^{8} + 1880 T^{9} - 16800 T^{10} + 34000 T^{11} + 10625 T^{12} - 200000 T^{13} + 500000 T^{14} - 625000 T^{15} + 390625 T^{16} \))
$7$ (\( 1 + 10 T^{2} + 51 T^{4} + 490 T^{6} + 2401 T^{8} \))
$11$ (\( 1 - 4 T + 8 T^{2} + 64 T^{3} - 415 T^{4} + 1384 T^{5} - 168 T^{6} - 14524 T^{7} + 77232 T^{8} - 159764 T^{9} - 20328 T^{10} + 1842104 T^{11} - 6076015 T^{12} + 10307264 T^{13} + 14172488 T^{14} - 77948684 T^{15} + 214358881 T^{16} \))
$13$ (\( ( 1 + 4 T + 8 T^{2} + 44 T^{3} + 238 T^{4} + 572 T^{5} + 1352 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 - 8 T + 22 T^{2} - 64 T^{3} + 387 T^{4} - 1088 T^{5} + 6358 T^{6} - 39304 T^{7} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 4 T + 8 T^{2} + 120 T^{3} - 601 T^{4} + 2280 T^{5} + 2888 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 + 24 T^{2} + 526 T^{4} - 24192 T^{6} - 582045 T^{8} - 12797568 T^{10} + 147196366 T^{12} + 3552861336 T^{14} + 78310985281 T^{16} \))
$29$ (\( ( 1 - 16 T + 128 T^{2} - 832 T^{3} + 4879 T^{4} - 24128 T^{5} + 107648 T^{6} - 390224 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 2 T - 57 T^{2} + 2 T^{3} + 2636 T^{4} + 62 T^{5} - 54777 T^{6} - 59582 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 + 4 T + 8 T^{2} - 264 T^{3} - 1897 T^{4} - 9768 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 - 152 T^{2} + 9106 T^{4} - 255512 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 24 T + 288 T^{2} - 2664 T^{3} + 20018 T^{4} - 114552 T^{5} + 532512 T^{6} - 1908168 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 - 4 T - 64 T^{2} + 56 T^{3} + 3439 T^{4} + 2632 T^{5} - 141376 T^{6} - 415292 T^{7} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 + 16 T + 128 T^{2} - 384 T^{3} - 14095 T^{4} - 132640 T^{5} - 244352 T^{6} + 5040144 T^{7} + 65449648 T^{8} + 267127632 T^{9} - 686384768 T^{10} - 19747045280 T^{11} - 111216329695 T^{12} - 160587069312 T^{13} + 2837038224512 T^{14} + 18795378237392 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 20 T + 200 T^{2} + 640 T^{3} - 7087 T^{4} - 104360 T^{5} - 465000 T^{6} + 1041740 T^{7} + 28323408 T^{8} + 61462660 T^{9} - 1618665000 T^{10} - 21433352440 T^{11} - 85875737407 T^{12} + 457551551360 T^{13} + 8436106728200 T^{14} + 49773029696380 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 4 T + 8 T^{2} - 344 T^{3} - 4910 T^{4} - 20804 T^{5} + 15232 T^{6} + 1637244 T^{7} + 8848707 T^{8} + 99871884 T^{9} + 56678272 T^{10} - 4722112724 T^{11} - 67983079310 T^{12} - 290541127544 T^{13} + 412162994888 T^{14} + 12570971344084 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 32 T + 512 T^{2} + 4160 T^{3} + 8974 T^{4} - 146656 T^{5} - 634880 T^{6} + 15805728 T^{7} + 232306035 T^{8} + 1058983776 T^{9} - 2849976320 T^{10} - 44108698528 T^{11} + 180836159854 T^{12} + 5616520445120 T^{13} + 46314691670528 T^{14} + 193942771370336 T^{15} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 16 T^{2} - 3966 T^{4} - 80656 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 + 14 T + 7 T^{2} + 434 T^{3} + 13996 T^{4} + 34286 T^{5} + 43687 T^{6} + 6902546 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 + 4 T + 8 T^{2} - 144 T^{3} - 11569 T^{4} - 11952 T^{5} + 55112 T^{6} + 2287148 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 - 64 T^{2} - 3825 T^{4} - 506944 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 6 T + 75 T^{2} + 582 T^{3} + 9409 T^{4} )^{4} \))
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