Properties

Label 672.2.c
Level 672
Weight 2
Character orbit c
Rep. character \(\chi_{672}(337,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 2
Sturm bound 256
Trace bound 1

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Defining parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(256\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 144 12 132
Cusp forms 112 12 100
Eisenstein series 32 0 32

Trace form

\( 12q - 4q^{7} - 12q^{9} + O(q^{10}) \) \( 12q - 4q^{7} - 12q^{9} + 8q^{15} + 8q^{17} + 8q^{23} - 20q^{25} - 16q^{31} - 16q^{39} - 8q^{41} + 12q^{49} + 32q^{55} + 4q^{63} - 32q^{65} - 8q^{71} + 40q^{73} + 48q^{79} + 12q^{81} + 40q^{89} - 80q^{95} + 8q^{97} + O(q^{100}) \)

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.c.a \(4\) \(5.366\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(4\) \(q-\zeta_{12}q^{3}+(\zeta_{12}-\zeta_{12}^{2})q^{5}+q^{7}+\cdots\)
672.2.c.b \(8\) \(5.366\) 8.0.386672896.3 None \(0\) \(0\) \(0\) \(-8\) \(q+\beta _{2}q^{3}+\beta _{1}q^{5}-q^{7}-q^{9}+(-2\beta _{2}+\cdots)q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(672, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{4} \))
$5$ (\( 1 - 12 T^{2} + 74 T^{4} - 300 T^{6} + 625 T^{8} \))(\( 1 - 8 T^{2} + 16 T^{4} - 168 T^{6} + 1694 T^{8} - 4200 T^{10} + 10000 T^{12} - 125000 T^{14} + 390625 T^{16} \))
$7$ (\( ( 1 - T )^{4} \))(\( ( 1 + T )^{8} \))
$11$ (\( 1 - 20 T^{2} + 234 T^{4} - 2420 T^{6} + 14641 T^{8} \))(\( 1 - 24 T^{2} + 592 T^{4} - 8312 T^{6} + 114206 T^{8} - 1005752 T^{10} + 8667472 T^{12} - 42517464 T^{14} + 214358881 T^{16} \))
$13$ (\( 1 - 20 T^{2} + 246 T^{4} - 3380 T^{6} + 28561 T^{8} \))(\( 1 - 48 T^{2} + 1340 T^{4} - 26704 T^{6} + 396198 T^{8} - 4512976 T^{10} + 38271740 T^{12} - 231686832 T^{14} + 815730721 T^{16} \))
$17$ (\( ( 1 - 2 T + 8 T^{2} - 34 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 2 T + 38 T^{2} - 70 T^{3} + 706 T^{4} - 1190 T^{5} + 10982 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - 20 T^{2} + 54 T^{4} - 7220 T^{6} + 130321 T^{8} \))(\( 1 - 64 T^{2} + 2012 T^{4} - 42432 T^{6} + 793446 T^{8} - 15317952 T^{10} + 262205852 T^{12} - 3010936384 T^{14} + 16983563041 T^{16} \))
$23$ (\( ( 1 - 10 T + 68 T^{2} - 230 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 6 T + 74 T^{2} + 334 T^{3} + 2282 T^{4} + 7682 T^{5} + 39146 T^{6} + 73002 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 46 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 16 T^{2} + 2876 T^{4} - 34928 T^{6} + 3439654 T^{8} - 29374448 T^{10} + 2034140156 T^{12} - 9517173136 T^{14} + 500246412961 T^{16} \))
$31$ (\( ( 1 + 2 T + 31 T^{2} )^{4} \))(\( ( 1 + 4 T + 80 T^{2} + 244 T^{3} + 3294 T^{4} + 7564 T^{5} + 76880 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2} \))
$37$ (\( 1 - 44 T^{2} + 2454 T^{4} - 60236 T^{6} + 1874161 T^{8} \))(\( 1 - 192 T^{2} + 18716 T^{4} - 1175872 T^{6} + 51538086 T^{8} - 1609768768 T^{10} + 35076797276 T^{12} - 492619470528 T^{14} + 3512479453921 T^{16} \))
$41$ (\( ( 1 + 2 T + 8 T^{2} + 82 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 2 T + 134 T^{2} + 214 T^{3} + 7618 T^{4} + 8774 T^{5} + 225254 T^{6} + 137842 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8} \))(\( 1 - 168 T^{2} + 15676 T^{4} - 1026520 T^{6} + 50753126 T^{8} - 1898035480 T^{10} + 53593124476 T^{12} - 1061988992232 T^{14} + 11688200277601 T^{16} \))
$47$ (\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 44 T^{2} + 128 T^{3} + 3302 T^{4} + 6016 T^{5} + 97196 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 + 2 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 336 T^{2} + 52604 T^{4} - 5027376 T^{6} + 321653350 T^{8} - 14121899184 T^{10} + 415070862524 T^{12} - 7447225339344 T^{14} + 62259690411361 T^{16} \))
$59$ (\( 1 - 12 T^{2} - 5290 T^{4} - 41772 T^{6} + 12117361 T^{8} \))(\( ( 1 - 102 T^{2} + 3481 T^{4} )^{4} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \))(\( 1 - 272 T^{2} + 35516 T^{4} - 3030896 T^{6} + 201654822 T^{8} - 11277964016 T^{10} + 491748888956 T^{12} - 14013541826192 T^{14} + 191707312997281 T^{16} \))
$67$ (\( ( 1 - 98 T^{2} + 4489 T^{4} )^{2} \))(\( 1 - 344 T^{2} + 59996 T^{4} - 6783272 T^{6} + 536949606 T^{8} - 30450108008 T^{10} + 1208986655516 T^{12} - 31117683466136 T^{14} + 406067677556641 T^{16} \))
$71$ (\( ( 1 + 18 T + 220 T^{2} + 1278 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 - 14 T + 194 T^{2} - 1686 T^{3} + 14330 T^{4} - 119706 T^{5} + 977954 T^{6} - 5010754 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 24 T + 278 T^{2} - 1752 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 + 4 T + 92 T^{2} - 580 T^{3} + 166 T^{4} - 42340 T^{5} + 490268 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 - 4 T + 54 T^{2} - 316 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 - 20 T + 340 T^{2} - 3972 T^{3} + 38678 T^{4} - 313788 T^{5} + 2121940 T^{6} - 9860780 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( 1 - 300 T^{2} + 36086 T^{4} - 2066700 T^{6} + 47458321 T^{8} \))(\( 1 - 248 T^{2} + 44156 T^{4} - 5204488 T^{6} + 499369126 T^{8} - 35853717832 T^{10} + 2095569622076 T^{12} - 81081212595512 T^{14} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 - 10 T + 200 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 10 T + 198 T^{2} - 494 T^{3} + 13122 T^{4} - 43966 T^{5} + 1568358 T^{6} - 7049690 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 16 T + 246 T^{2} + 1552 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 249484 T^{5} + 2822700 T^{6} - 18253460 T^{7} + 88529281 T^{8} )^{2} \))
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