Properties

Label 630.2.be
Level 630
Weight 2
Character orbit be
Rep. character \(\chi_{630}(341,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 16
Newform subspaces 2
Sturm bound 288
Trace bound 5

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Defining parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.be (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(288\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(630, [\chi])\).

Total New Old
Modular forms 320 16 304
Cusp forms 256 16 240
Eisenstein series 64 0 64

Trace form

\( 16q + 8q^{4} + O(q^{10}) \) \( 16q + 8q^{4} - 8q^{16} - 8q^{25} + 48q^{37} - 32q^{43} + 16q^{46} - 32q^{58} - 16q^{64} - 48q^{67} - 48q^{79} - 40q^{91} - 24q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(630, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
630.2.be.a \(8\) \(5.031\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(-4\) \(0\) \(q+(\zeta_{24}^{2}-\zeta_{24}^{6})q^{2}+(1-\zeta_{24}^{4})q^{4}+\cdots\)
630.2.be.b \(8\) \(5.031\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(4\) \(0\) \(q-\zeta_{24}^{2}q^{2}+\zeta_{24}^{4}q^{4}+(1-\zeta_{24}^{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(630, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(630, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(315, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - T^{2} + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} )^{2} \))
$3$ (\( \))(\( \))
$5$ (\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))
$7$ (\( 1 - 94 T^{4} + 2401 T^{8} \))(\( 1 - 94 T^{4} + 2401 T^{8} \))
$11$ (\( 1 + 24 T + 304 T^{2} + 2688 T^{3} + 18481 T^{4} + 104232 T^{5} + 496816 T^{6} + 2036016 T^{7} + 7239280 T^{8} + 22396176 T^{9} + 60114736 T^{10} + 138732792 T^{11} + 270580321 T^{12} + 432905088 T^{13} + 538554544 T^{14} + 467692104 T^{15} + 214358881 T^{16} \))(\( 1 - 24 T + 304 T^{2} - 2688 T^{3} + 18481 T^{4} - 104232 T^{5} + 496816 T^{6} - 2036016 T^{7} + 7239280 T^{8} - 22396176 T^{9} + 60114736 T^{10} - 138732792 T^{11} + 270580321 T^{12} - 432905088 T^{13} + 538554544 T^{14} - 467692104 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 11756992 T^{10} + 85854366 T^{12} - 386144720 T^{14} + 815730721 T^{16} \))(\( 1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 11756992 T^{10} + 85854366 T^{12} - 386144720 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 - 28 T^{2} - 192 T^{3} + 442 T^{4} + 4320 T^{5} + 15824 T^{6} - 58560 T^{7} - 349805 T^{8} - 995520 T^{9} + 4573136 T^{10} + 21224160 T^{11} + 36916282 T^{12} - 272612544 T^{13} - 675851932 T^{14} + 6975757441 T^{16} \))(\( 1 - 28 T^{2} + 192 T^{3} + 442 T^{4} - 4320 T^{5} + 15824 T^{6} + 58560 T^{7} - 349805 T^{8} + 995520 T^{9} + 4573136 T^{10} - 21224160 T^{11} + 36916282 T^{12} + 272612544 T^{13} - 675851932 T^{14} + 6975757441 T^{16} \))
$19$ (\( 1 + 58 T^{2} + 1881 T^{4} - 1872 T^{5} + 44906 T^{6} - 71184 T^{7} + 869156 T^{8} - 1352496 T^{9} + 16211066 T^{10} - 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16} \))(\( 1 + 58 T^{2} + 1881 T^{4} - 1872 T^{5} + 44906 T^{6} - 71184 T^{7} + 869156 T^{8} - 1352496 T^{9} + 16211066 T^{10} - 12840048 T^{11} + 245133801 T^{12} + 2728661098 T^{14} + 16983563041 T^{16} \))
$23$ (\( ( 1 + 6 T + 57 T^{2} + 270 T^{3} + 1772 T^{4} + 6210 T^{5} + 30153 T^{6} + 73002 T^{7} + 279841 T^{8} )^{2} \))(\( ( 1 - 6 T + 57 T^{2} - 270 T^{3} + 1772 T^{4} - 6210 T^{5} + 30153 T^{6} - 73002 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( 1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16} \))(\( 1 + 100 T^{2} + 5706 T^{4} + 237200 T^{6} + 7904915 T^{8} + 227949200 T^{10} + 5269610826 T^{12} + 88750368100 T^{14} + 852891037441 T^{16} \))
$37$ (\( 1 - 24 T + 216 T^{2} - 1680 T^{3} + 21025 T^{4} - 178920 T^{5} + 986040 T^{6} - 7528032 T^{7} + 58723920 T^{8} - 278537184 T^{9} + 1349888760 T^{10} - 9062834760 T^{11} + 39404235025 T^{12} - 116497847760 T^{13} + 554196904344 T^{14} - 2278365051192 T^{15} + 3512479453921 T^{16} \))(\( 1 - 24 T + 216 T^{2} - 1680 T^{3} + 21025 T^{4} - 178920 T^{5} + 986040 T^{6} - 7528032 T^{7} + 58723920 T^{8} - 278537184 T^{9} + 1349888760 T^{10} - 9062834760 T^{11} + 39404235025 T^{12} - 116497847760 T^{13} + 554196904344 T^{14} - 2278365051192 T^{15} + 3512479453921 T^{16} \))
$41$ (\( ( 1 + 16 T + 232 T^{2} + 2000 T^{3} + 15591 T^{4} + 82000 T^{5} + 389992 T^{6} + 1102736 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 16 T + 232 T^{2} - 2000 T^{3} + 15591 T^{4} - 82000 T^{5} + 389992 T^{6} - 1102736 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 + 8 T + 180 T^{2} + 1000 T^{3} + 11750 T^{4} + 43000 T^{5} + 332820 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 180 T^{2} + 1000 T^{3} + 11750 T^{4} + 43000 T^{5} + 332820 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 - 8 T - 78 T^{2} + 240 T^{3} + 6305 T^{4} + 3648 T^{5} - 344398 T^{6} + 95176 T^{7} + 11426244 T^{8} + 4473272 T^{9} - 760775182 T^{10} + 378746304 T^{11} + 30766388705 T^{12} + 55042801680 T^{13} - 840778795662 T^{14} - 4052984963704 T^{15} + 23811286661761 T^{16} \))(\( 1 + 8 T - 78 T^{2} - 240 T^{3} + 6305 T^{4} - 3648 T^{5} - 344398 T^{6} - 95176 T^{7} + 11426244 T^{8} - 4473272 T^{9} - 760775182 T^{10} - 378746304 T^{11} + 30766388705 T^{12} - 55042801680 T^{13} - 840778795662 T^{14} + 4052984963704 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 24 T + 362 T^{2} + 4080 T^{3} + 37881 T^{4} + 336960 T^{5} + 2978650 T^{6} + 25456152 T^{7} + 197950532 T^{8} + 1349176056 T^{9} + 8367027850 T^{10} + 50165593920 T^{11} + 298899310761 T^{12} + 1706237611440 T^{13} + 8023498728698 T^{14} + 28193067356088 T^{15} + 62259690411361 T^{16} \))(\( 1 - 24 T + 362 T^{2} - 4080 T^{3} + 37881 T^{4} - 336960 T^{5} + 2978650 T^{6} - 25456152 T^{7} + 197950532 T^{8} - 1349176056 T^{9} + 8367027850 T^{10} - 50165593920 T^{11} + 298899310761 T^{12} - 1706237611440 T^{13} + 8023498728698 T^{14} - 28193067356088 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 24 T + 224 T^{2} - 1392 T^{3} + 9358 T^{4} - 16056 T^{5} - 613888 T^{6} + 6806952 T^{7} - 48734957 T^{8} + 401610168 T^{9} - 2136944128 T^{10} - 3297565224 T^{11} + 113394264238 T^{12} - 995174624208 T^{13} + 9448439535584 T^{14} - 59727635635656 T^{15} + 146830437604321 T^{16} \))(\( 1 + 24 T + 224 T^{2} + 1392 T^{3} + 9358 T^{4} + 16056 T^{5} - 613888 T^{6} - 6806952 T^{7} - 48734957 T^{8} - 401610168 T^{9} - 2136944128 T^{10} + 3297565224 T^{11} + 113394264238 T^{12} + 995174624208 T^{13} + 9448439535584 T^{14} + 59727635635656 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16} \))(\( 1 + 124 T^{2} + 6186 T^{4} + 30240 T^{5} + 293552 T^{6} + 4411200 T^{7} + 18080963 T^{8} + 269083200 T^{9} + 1092306992 T^{10} + 6863905440 T^{11} + 85650372426 T^{12} + 6388526420764 T^{14} + 191707312997281 T^{16} \))
$67$ (\( 1 + 24 T + 332 T^{2} + 3696 T^{3} + 30922 T^{4} + 206568 T^{5} + 963056 T^{6} + 1103736 T^{7} - 11820413 T^{8} + 73950312 T^{9} + 4323158384 T^{10} + 62128011384 T^{11} + 623112963562 T^{12} + 4990062395472 T^{13} + 30032182880108 T^{14} + 145457078527752 T^{15} + 406067677556641 T^{16} \))(\( 1 + 24 T + 332 T^{2} + 3696 T^{3} + 30922 T^{4} + 206568 T^{5} + 963056 T^{6} + 1103736 T^{7} - 11820413 T^{8} + 73950312 T^{9} + 4323158384 T^{10} + 62128011384 T^{11} + 623112963562 T^{12} + 4990062395472 T^{13} + 30032182880108 T^{14} + 145457078527752 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16} \))(\( 1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 + 156 T^{2} + 12202 T^{4} + 230256 T^{6} - 10850829 T^{8} + 1227034224 T^{10} + 346515336682 T^{12} + 23608139301084 T^{14} + 806460091894081 T^{16} \))(\( 1 + 156 T^{2} + 12202 T^{4} + 230256 T^{6} - 10850829 T^{8} + 1227034224 T^{10} + 346515336682 T^{12} + 23608139301084 T^{14} + 806460091894081 T^{16} \))
$79$ (\( 1 + 24 T + 252 T^{2} + 1584 T^{3} + 5050 T^{4} + 17832 T^{5} - 542160 T^{6} - 18801768 T^{7} - 226869549 T^{8} - 1485339672 T^{9} - 3383620560 T^{10} + 8791871448 T^{11} + 196697909050 T^{12} + 4874057336016 T^{13} + 61258038791292 T^{14} + 460893815667816 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 24 T + 252 T^{2} + 1584 T^{3} + 5050 T^{4} + 17832 T^{5} - 542160 T^{6} - 18801768 T^{7} - 226869549 T^{8} - 1485339672 T^{9} - 3383620560 T^{10} + 8791871448 T^{11} + 196697909050 T^{12} + 4874057336016 T^{13} + 61258038791292 T^{14} + 460893815667816 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 - 8 T + 148 T^{2} - 1192 T^{3} + 18438 T^{4} - 98936 T^{5} + 1019572 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 8 T + 148 T^{2} + 1192 T^{3} + 18438 T^{4} + 98936 T^{5} + 1019572 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 - 16 T - 48 T^{2} + 1824 T^{3} + 2078 T^{4} - 135504 T^{5} + 38912 T^{6} + 7970384 T^{7} - 72309597 T^{8} + 709364176 T^{9} + 308221952 T^{10} - 95526119376 T^{11} + 130378376798 T^{12} + 10185324434976 T^{13} - 23855101966128 T^{14} - 707701358328464 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 16 T - 48 T^{2} - 1824 T^{3} + 2078 T^{4} + 135504 T^{5} + 38912 T^{6} - 7970384 T^{7} - 72309597 T^{8} - 709364176 T^{9} + 308221952 T^{10} + 95526119376 T^{11} + 130378376798 T^{12} - 10185324434976 T^{13} - 23855101966128 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 6147840600 T^{10} + 1685243393116 T^{12} + 19991328118296 T^{14} + 7837433594376961 T^{16} \))(\( 1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 6147840600 T^{10} + 1685243393116 T^{12} + 19991328118296 T^{14} + 7837433594376961 T^{16} \))
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