Properties

Label 630.2.b
Level 630
Weight 2
Character orbit b
Rep. character \(\chi_{630}(251,\cdot)\)
Character field \(\Q\)
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.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(288\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(17\)

Dimensions

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

Total New Old
Modular forms 160 16 144
Cusp forms 128 16 112
Eisenstein series 32 0 32

Trace form

\( 16q - 16q^{4} - 8q^{7} + O(q^{10}) \) \( 16q - 16q^{4} - 8q^{7} + 16q^{16} + 16q^{25} + 8q^{28} - 16q^{37} - 32q^{43} - 16q^{46} + 8q^{49} - 16q^{58} - 16q^{64} + 64q^{67} + 16q^{79} - 8q^{91} + O(q^{100}) \)

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.b.a \(8\) \(5.031\) 8.0.7442857984.4 None \(0\) \(0\) \(-8\) \(-4\) \(q+\beta _{3}q^{2}-q^{4}-q^{5}+\beta _{6}q^{7}-\beta _{3}q^{8}+\cdots\)
630.2.b.b \(8\) \(5.031\) 8.0.7442857984.4 None \(0\) \(0\) \(8\) \(-4\) \(q-\beta _{3}q^{2}-q^{4}+q^{5}+\beta _{6}q^{7}+\beta _{3}q^{8}+\cdots\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 + T^{2} )^{4} \))
$3$ (\( \))(\( \))
$5$ (\( ( 1 + T )^{8} \))(\( ( 1 - T )^{8} \))
$7$ (\( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 28 T^{5} + 294 T^{6} + 1372 T^{7} + 2401 T^{8} \))(\( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 28 T^{5} + 294 T^{6} + 1372 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16} \))(\( 1 - 36 T^{2} + 776 T^{4} - 11916 T^{6} + 146094 T^{8} - 1441836 T^{10} + 11361416 T^{12} - 63776196 T^{14} + 214358881 T^{16} \))
$13$ (\( 1 - 44 T^{2} + 872 T^{4} - 9828 T^{6} + 102190 T^{8} - 1660932 T^{10} + 24905192 T^{12} - 212379596 T^{14} + 815730721 T^{16} \))(\( 1 - 44 T^{2} + 872 T^{4} - 9828 T^{6} + 102190 T^{8} - 1660932 T^{10} + 24905192 T^{12} - 212379596 T^{14} + 815730721 T^{16} \))
$17$ (\( ( 1 + 18 T^{2} + 120 T^{3} - 38 T^{4} + 2040 T^{5} + 5202 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 18 T^{2} - 120 T^{3} - 38 T^{4} - 2040 T^{5} + 5202 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16} \))(\( 1 - 44 T^{2} + 1256 T^{4} - 31140 T^{6} + 702382 T^{8} - 11241540 T^{10} + 163683176 T^{12} - 2070018764 T^{14} + 16983563041 T^{16} \))
$23$ (\( 1 - 80 T^{2} + 3740 T^{4} - 123312 T^{6} + 3185414 T^{8} - 65232048 T^{10} + 1046605340 T^{12} - 11842871120 T^{14} + 78310985281 T^{16} \))(\( 1 - 80 T^{2} + 3740 T^{4} - 123312 T^{6} + 3185414 T^{8} - 65232048 T^{10} + 1046605340 T^{12} - 11842871120 T^{14} + 78310985281 T^{16} \))
$29$ (\( 1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16} \))(\( 1 - 128 T^{2} + 8732 T^{4} - 399744 T^{6} + 13409894 T^{8} - 336184704 T^{10} + 6175977692 T^{12} - 76137385088 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16} \))(\( 1 - 32 T^{2} + 572 T^{4} - 21984 T^{6} + 1650310 T^{8} - 21126624 T^{10} + 528254012 T^{12} - 28400117792 T^{14} + 852891037441 T^{16} \))
$37$ (\( ( 1 + 4 T + 84 T^{2} + 532 T^{3} + 3602 T^{4} + 19684 T^{5} + 114996 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 4 T + 84 T^{2} + 532 T^{3} + 3602 T^{4} + 19684 T^{5} + 114996 T^{6} + 202612 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 + 4 T + 142 T^{2} + 468 T^{3} + 8354 T^{4} + 19188 T^{5} + 238702 T^{6} + 275684 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 4 T + 142 T^{2} - 468 T^{3} + 8354 T^{4} - 19188 T^{5} + 238702 T^{6} - 275684 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 + 8 T + 146 T^{2} + 984 T^{3} + 8842 T^{4} + 42312 T^{5} + 269954 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 146 T^{2} + 984 T^{3} + 8842 T^{4} + 42312 T^{5} + 269954 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 + 20 T + 166 T^{2} + 396 T^{3} - 838 T^{4} + 18612 T^{5} + 366694 T^{6} + 2076460 T^{7} + 4879681 T^{8} )^{2} \))(\( ( 1 - 20 T + 166 T^{2} - 396 T^{3} - 838 T^{4} - 18612 T^{5} + 366694 T^{6} - 2076460 T^{7} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 - 304 T^{2} + 43772 T^{4} - 3946320 T^{6} + 247218022 T^{8} - 11085212880 T^{10} + 345382134332 T^{12} - 6737965783216 T^{14} + 62259690411361 T^{16} \))(\( 1 - 304 T^{2} + 43772 T^{4} - 3946320 T^{6} + 247218022 T^{8} - 11085212880 T^{10} + 345382134332 T^{12} - 6737965783216 T^{14} + 62259690411361 T^{16} \))
$59$ (\( ( 1 + 42 T^{2} + 312 T^{3} + 3106 T^{4} + 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2} \))(\( ( 1 + 42 T^{2} - 312 T^{3} + 3106 T^{4} - 18408 T^{5} + 146202 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} \))(\( 1 - 236 T^{2} + 31208 T^{4} - 2804772 T^{6} + 194357422 T^{8} - 10436556612 T^{10} + 432101005928 T^{12} - 12158808349196 T^{14} + 191707312997281 T^{16} \))
$67$ (\( ( 1 - 16 T + 218 T^{2} - 1728 T^{3} + 15338 T^{4} - 115776 T^{5} + 978602 T^{6} - 4812208 T^{7} + 20151121 T^{8} )^{2} \))(\( ( 1 - 16 T + 218 T^{2} - 1728 T^{3} + 15338 T^{4} - 115776 T^{5} + 978602 T^{6} - 4812208 T^{7} + 20151121 T^{8} )^{2} \))
$71$ (\( 1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16} \))(\( 1 - 460 T^{2} + 98600 T^{4} - 12828132 T^{6} + 1106047246 T^{8} - 64666613412 T^{10} + 2505591746600 T^{12} - 58926130603660 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 - 232 T^{2} + 30332 T^{4} - 3074136 T^{6} + 255245510 T^{8} - 16382070744 T^{10} + 861375446012 T^{12} - 35109540499048 T^{14} + 806460091894081 T^{16} \))(\( 1 - 232 T^{2} + 30332 T^{4} - 3074136 T^{6} + 255245510 T^{8} - 16382070744 T^{10} + 861375446012 T^{12} - 35109540499048 T^{14} + 806460091894081 T^{16} \))
$79$ (\( ( 1 - 4 T + 48 T^{2} - 532 T^{3} + 9470 T^{4} - 42028 T^{5} + 299568 T^{6} - 1972156 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 4 T + 48 T^{2} - 532 T^{3} + 9470 T^{4} - 42028 T^{5} + 299568 T^{6} - 1972156 T^{7} + 38950081 T^{8} )^{2} \))
$83$ (\( ( 1 - 8 T + 244 T^{2} - 1800 T^{3} + 27878 T^{4} - 149400 T^{5} + 1680916 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 8 T + 244 T^{2} + 1800 T^{3} + 27878 T^{4} + 149400 T^{5} + 1680916 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 4 T + 118 T^{2} - 1308 T^{3} - 670 T^{4} - 116412 T^{5} + 934678 T^{6} + 2819876 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 4 T + 118 T^{2} + 1308 T^{3} - 670 T^{4} + 116412 T^{5} + 934678 T^{6} - 2819876 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 - 208 T^{2} + 38300 T^{4} - 4059696 T^{6} + 468613574 T^{8} - 38197679664 T^{10} + 3390671462300 T^{12} - 173258177025232 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 208 T^{2} + 38300 T^{4} - 4059696 T^{6} + 468613574 T^{8} - 38197679664 T^{10} + 3390671462300 T^{12} - 173258177025232 T^{14} + 7837433594376961 T^{16} \))
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