Properties

Label 672.2.b
Level 672
Weight 2
Character orbit b
Rep. character \(\chi_{672}(223,\cdot)\)
Character field \(\Q\)
Dimension 16
Newform subspaces 2
Sturm bound 256
Trace bound 3

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Defining parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(256\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(672, [\chi])\).

Total New Old
Modular forms 144 16 128
Cusp forms 112 16 96
Eisenstein series 32 0 32

Trace form

\( 16q + 16q^{9} + O(q^{10}) \) \( 16q + 16q^{9} + 8q^{21} - 32q^{25} + 16q^{37} + 32q^{53} + 16q^{57} + 32q^{65} + 64q^{77} + 16q^{81} - 112q^{85} - 32q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(672, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
672.2.b.a \(8\) \(5.366\) 8.0.836829184.2 None \(0\) \(-8\) \(0\) \(-4\) \(q-q^{3}+\beta _{1}q^{5}+\beta _{4}q^{7}+q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
672.2.b.b \(8\) \(5.366\) 8.0.836829184.2 None \(0\) \(8\) \(0\) \(4\) \(q+q^{3}-\beta _{1}q^{5}+\beta _{2}q^{7}+q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(672, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(672, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + T )^{8} \))(\( ( 1 - T )^{8} \))
$5$ (\( 1 - 12 T^{2} + 56 T^{4} + 28 T^{6} - 1074 T^{8} + 700 T^{10} + 35000 T^{12} - 187500 T^{14} + 390625 T^{16} \))(\( 1 - 12 T^{2} + 56 T^{4} + 28 T^{6} - 1074 T^{8} + 700 T^{10} + 35000 T^{12} - 187500 T^{14} + 390625 T^{16} \))
$7$ (\( 1 + 4 T + 8 T^{2} + 20 T^{3} + 46 T^{4} + 140 T^{5} + 392 T^{6} + 1372 T^{7} + 2401 T^{8} \))(\( 1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 140 T^{5} + 392 T^{6} - 1372 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 20 T^{2} + 296 T^{4} - 2268 T^{6} + 20718 T^{8} - 274428 T^{10} + 4333736 T^{12} - 35431220 T^{14} + 214358881 T^{16} \))(\( 1 - 20 T^{2} + 296 T^{4} - 2268 T^{6} + 20718 T^{8} - 274428 T^{10} + 4333736 T^{12} - 35431220 T^{14} + 214358881 T^{16} \))
$13$ (\( ( 1 - 18 T^{2} + 169 T^{4} )^{4} \))(\( ( 1 - 18 T^{2} + 169 T^{4} )^{4} \))
$17$ (\( 1 - 108 T^{2} + 5432 T^{4} - 166628 T^{6} + 3420078 T^{8} - 48155492 T^{10} + 453686072 T^{12} - 2606857452 T^{14} + 6975757441 T^{16} \))(\( 1 - 108 T^{2} + 5432 T^{4} - 166628 T^{6} + 3420078 T^{8} - 48155492 T^{10} + 453686072 T^{12} - 2606857452 T^{14} + 6975757441 T^{16} \))
$19$ (\( ( 1 + 4 T + 32 T^{2} + 132 T^{3} + 558 T^{4} + 2508 T^{5} + 11552 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 4 T + 32 T^{2} - 132 T^{3} + 558 T^{4} - 2508 T^{5} + 11552 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 100 T^{2} + 5000 T^{4} - 174764 T^{6} + 4622926 T^{8} - 92450156 T^{10} + 1399205000 T^{12} - 14803588900 T^{14} + 78310985281 T^{16} \))(\( 1 - 100 T^{2} + 5000 T^{4} - 174764 T^{6} + 4622926 T^{8} - 92450156 T^{10} + 1399205000 T^{12} - 14803588900 T^{14} + 78310985281 T^{16} \))
$29$ (\( ( 1 + 36 T^{2} + 96 T^{3} + 390 T^{4} + 2784 T^{5} + 30276 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 + 36 T^{2} + 96 T^{3} + 390 T^{4} + 2784 T^{5} + 30276 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 8 T + 100 T^{2} - 616 T^{3} + 4534 T^{4} - 19096 T^{5} + 96100 T^{6} - 238328 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 8 T + 100 T^{2} + 616 T^{3} + 4534 T^{4} + 19096 T^{5} + 96100 T^{6} + 238328 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( ( 1 - 4 T + 40 T^{2} + 292 T^{3} - 866 T^{4} + 10804 T^{5} + 54760 T^{6} - 202612 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 - 4 T + 40 T^{2} + 292 T^{3} - 866 T^{4} + 10804 T^{5} + 54760 T^{6} - 202612 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( 1 - 204 T^{2} + 21176 T^{4} - 1439172 T^{6} + 69360622 T^{8} - 2419248132 T^{10} + 59838314936 T^{12} - 969021265164 T^{14} + 7984925229121 T^{16} \))(\( 1 - 204 T^{2} + 21176 T^{4} - 1439172 T^{6} + 69360622 T^{8} - 2419248132 T^{10} + 59838314936 T^{12} - 969021265164 T^{14} + 7984925229121 T^{16} \))
$43$ (\( 1 - 192 T^{2} + 18396 T^{4} - 1162560 T^{6} + 55940774 T^{8} - 2149573440 T^{10} + 62892263196 T^{12} - 1213701705408 T^{14} + 11688200277601 T^{16} \))(\( 1 - 192 T^{2} + 18396 T^{4} - 1162560 T^{6} + 55940774 T^{8} - 2149573440 T^{10} + 62892263196 T^{12} - 1213701705408 T^{14} + 11688200277601 T^{16} \))
$47$ (\( ( 1 + 8 T + 108 T^{2} + 872 T^{3} + 6758 T^{4} + 40984 T^{5} + 238572 T^{6} + 830584 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 8 T + 108 T^{2} - 872 T^{3} + 6758 T^{4} - 40984 T^{5} + 238572 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 - 8 T + 132 T^{2} - 632 T^{3} + 7718 T^{4} - 33496 T^{5} + 370788 T^{6} - 1191016 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 - 8 T + 132 T^{2} - 632 T^{3} + 7718 T^{4} - 33496 T^{5} + 370788 T^{6} - 1191016 T^{7} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 8 T + 102 T^{2} + 472 T^{3} + 3481 T^{4} )^{4} \))(\( ( 1 - 8 T + 102 T^{2} - 472 T^{3} + 3481 T^{4} )^{4} \))
$61$ (\( 1 - 184 T^{2} + 16284 T^{4} - 814280 T^{6} + 38728934 T^{8} - 3029935880 T^{10} + 225465674844 T^{12} - 9479748882424 T^{14} + 191707312997281 T^{16} \))(\( 1 - 184 T^{2} + 16284 T^{4} - 814280 T^{6} + 38728934 T^{8} - 3029935880 T^{10} + 225465674844 T^{12} - 9479748882424 T^{14} + 191707312997281 T^{16} \))
$67$ (\( 1 - 240 T^{2} + 37436 T^{4} - 3815056 T^{6} + 299829030 T^{8} - 17125786384 T^{10} + 754377365756 T^{12} - 21710011720560 T^{14} + 406067677556641 T^{16} \))(\( 1 - 240 T^{2} + 37436 T^{4} - 3815056 T^{6} + 299829030 T^{8} - 17125786384 T^{10} + 754377365756 T^{12} - 21710011720560 T^{14} + 406067677556641 T^{16} \))
$71$ (\( 1 - 484 T^{2} + 107912 T^{4} - 14382508 T^{6} + 1250123214 T^{8} - 72502222828 T^{10} + 2742225320072 T^{12} - 62000537417764 T^{14} + 645753531245761 T^{16} \))(\( 1 - 484 T^{2} + 107912 T^{4} - 14382508 T^{6} + 1250123214 T^{8} - 72502222828 T^{10} + 2742225320072 T^{12} - 62000537417764 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 - 408 T^{2} + 80380 T^{4} - 10037032 T^{6} + 869620166 T^{8} - 53487343528 T^{10} + 2282650611580 T^{12} - 61744364325912 T^{14} + 806460091894081 T^{16} \))(\( 1 - 408 T^{2} + 80380 T^{4} - 10037032 T^{6} + 869620166 T^{8} - 53487343528 T^{10} + 2282650611580 T^{12} - 61744364325912 T^{14} + 806460091894081 T^{16} \))
$79$ (\( 1 - 368 T^{2} + 73436 T^{4} - 9593744 T^{6} + 892827078 T^{8} - 59874556304 T^{10} + 2860338148316 T^{12} - 89456183631728 T^{14} + 1517108809906561 T^{16} \))(\( 1 - 368 T^{2} + 73436 T^{4} - 9593744 T^{6} + 892827078 T^{8} - 59874556304 T^{10} + 2860338148316 T^{12} - 89456183631728 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 + 8 T + 124 T^{2} + 1480 T^{3} + 7318 T^{4} + 122840 T^{5} + 854236 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 8 T + 124 T^{2} - 1480 T^{3} + 7318 T^{4} - 122840 T^{5} + 854236 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 + 52 T^{2} - 328 T^{4} + 280764 T^{6} + 120770670 T^{8} + 2223931644 T^{10} - 20579455048 T^{12} + 25843027129972 T^{14} + 3936588805702081 T^{16} \))(\( 1 + 52 T^{2} - 328 T^{4} + 280764 T^{6} + 120770670 T^{8} + 2223931644 T^{10} - 20579455048 T^{12} + 25843027129972 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 344 T^{2} + 58172 T^{4} - 6404840 T^{6} + 615495942 T^{8} - 60263139560 T^{10} + 5149925334332 T^{12} - 286542369695576 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 344 T^{2} + 58172 T^{4} - 6404840 T^{6} + 615495942 T^{8} - 60263139560 T^{10} + 5149925334332 T^{12} - 286542369695576 T^{14} + 7837433594376961 T^{16} \))
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