Properties

Label 2400.2.m
Level 2400
Weight 2
Character orbit m
Rep. character \(\chi_{2400}(1199,\cdot)\)
Character field \(\Q\)
Dimension 68
Newform subspaces 5
Sturm bound 960
Trace bound 21

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Defining parameters

Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.m (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 120 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(960\)
Trace bound: \(21\)
Distinguishing \(T_p\): \(7\), \(11\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2400, [\chi])\).

Total New Old
Modular forms 528 76 452
Cusp forms 432 68 364
Eisenstein series 96 8 88

Trace form

\( 68q + 4q^{9} + O(q^{10}) \) \( 68q + 4q^{9} - 24q^{19} + 52q^{49} - 8q^{51} + 12q^{81} - 48q^{91} + 96q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(2400, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
2400.2.m.a \(4\) \(19.164\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{2}q^{3}+(1+\zeta_{8}^{3})q^{9}+\zeta_{8}^{3}q^{11}+\cdots\)
2400.2.m.b \(8\) \(19.164\) \(\Q(\zeta_{24})\) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{3}q^{3}+(-\zeta_{24}^{5}-\zeta_{24}^{6})q^{9}+\cdots\)
2400.2.m.c \(16\) \(19.164\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{4}q^{3}+\beta _{9}q^{7}+\beta _{7}q^{9}+\beta _{10}q^{11}+\cdots\)
2400.2.m.d \(16\) \(19.164\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{4}q^{3}+\beta _{9}q^{7}+\beta _{7}q^{9}+\beta _{10}q^{11}+\cdots\)
2400.2.m.e \(24\) \(19.164\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(2400, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2400, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(480, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(600, [\chi])\)\(^{\oplus 3}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \))(\( 1 - 4 T^{4} + 16 T^{6} + 70 T^{8} + 144 T^{10} - 324 T^{12} + 6561 T^{16} \))(\( 1 - 4 T^{4} + 16 T^{6} + 70 T^{8} + 144 T^{10} - 324 T^{12} + 6561 T^{16} \))
$5$ 1
$7$ (\( ( 1 + 7 T^{2} )^{4} \))(\( ( 1 + 7 T^{2} )^{8} \))(\( ( 1 + 20 T^{2} + 244 T^{4} + 2364 T^{6} + 18678 T^{8} + 115836 T^{10} + 585844 T^{12} + 2352980 T^{14} + 5764801 T^{16} )^{2} \))(\( ( 1 + 20 T^{2} + 244 T^{4} + 2364 T^{6} + 18678 T^{8} + 115836 T^{10} + 585844 T^{12} + 2352980 T^{14} + 5764801 T^{16} )^{2} \))
$11$ (\( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 - 40 T^{2} + 892 T^{4} - 14424 T^{6} + 178918 T^{8} - 1745304 T^{10} + 13059772 T^{12} - 70862440 T^{14} + 214358881 T^{16} )^{2} \))(\( ( 1 - 40 T^{2} + 892 T^{4} - 14424 T^{6} + 178918 T^{8} - 1745304 T^{10} + 13059772 T^{12} - 70862440 T^{14} + 214358881 T^{16} )^{2} \))
$13$ (\( ( 1 + 13 T^{2} )^{4} \))(\( ( 1 + 13 T^{2} )^{8} \))(\( ( 1 + 52 T^{2} + 1556 T^{4} + 31660 T^{6} + 477814 T^{8} + 5350540 T^{10} + 44440916 T^{12} + 250994068 T^{14} + 815730721 T^{16} )^{2} \))(\( ( 1 + 52 T^{2} + 1556 T^{4} + 31660 T^{6} + 477814 T^{8} + 5350540 T^{10} + 44440916 T^{12} + 250994068 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( ( 1 + 2 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 84 T^{2} + 3604 T^{4} + 101164 T^{6} + 2018902 T^{8} + 29236396 T^{10} + 301009684 T^{12} + 2027555796 T^{14} + 6975757441 T^{16} )^{2} \))(\( ( 1 + 84 T^{2} + 3604 T^{4} + 101164 T^{6} + 2018902 T^{8} + 29236396 T^{10} + 301009684 T^{12} + 2027555796 T^{14} + 6975757441 T^{16} )^{2} \))
$19$ (\( ( 1 - 2 T + 19 T^{2} )^{4} \))(\( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{4} \))(\( ( 1 + 2 T + 40 T^{2} + 42 T^{3} + 766 T^{4} + 798 T^{5} + 14440 T^{6} + 13718 T^{7} + 130321 T^{8} )^{4} \))(\( ( 1 + 2 T + 40 T^{2} + 42 T^{3} + 766 T^{4} + 798 T^{5} + 14440 T^{6} + 13718 T^{7} + 130321 T^{8} )^{4} \))
$23$ (\( ( 1 - 23 T^{2} )^{4} \))(\( ( 1 - 23 T^{2} )^{8} \))(\( ( 1 - 92 T^{2} + 4420 T^{4} - 144852 T^{6} + 3702742 T^{8} - 76626708 T^{10} + 1236897220 T^{12} - 13619301788 T^{14} + 78310985281 T^{16} )^{2} \))(\( ( 1 - 92 T^{2} + 4420 T^{4} - 144852 T^{6} + 3702742 T^{8} - 76626708 T^{10} + 1236897220 T^{12} - 13619301788 T^{14} + 78310985281 T^{16} )^{2} \))
$29$ (\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{8} \))(\( ( 1 + 52 T^{2} + 112 T^{3} + 1286 T^{4} + 3248 T^{5} + 43732 T^{6} + 707281 T^{8} )^{4} \))(\( ( 1 + 52 T^{2} - 112 T^{3} + 1286 T^{4} - 3248 T^{5} + 43732 T^{6} + 707281 T^{8} )^{4} \))
$31$ (\( ( 1 - 31 T^{2} )^{4} \))(\( ( 1 - 31 T^{2} )^{8} \))(\( ( 1 - 108 T^{2} + 5412 T^{4} - 170900 T^{6} + 4790966 T^{8} - 164234900 T^{10} + 4998095652 T^{12} - 95850397548 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 - 108 T^{2} + 5412 T^{4} - 170900 T^{6} + 4790966 T^{8} - 164234900 T^{10} + 4998095652 T^{12} - 95850397548 T^{14} + 852891037441 T^{16} )^{2} \))
$37$ (\( ( 1 + 37 T^{2} )^{4} \))(\( ( 1 + 37 T^{2} )^{8} \))(\( ( 1 + 68 T^{2} + 4788 T^{4} + 250460 T^{6} + 9311478 T^{8} + 342879740 T^{10} + 8973482868 T^{12} + 174469395812 T^{14} + 3512479453921 T^{16} )^{2} \))(\( ( 1 + 68 T^{2} + 4788 T^{4} + 250460 T^{6} + 9311478 T^{8} + 342879740 T^{10} + 8973482868 T^{12} + 174469395812 T^{14} + 3512479453921 T^{16} )^{2} \))
$41$ (\( ( 1 - 6 T + 41 T^{2} )^{2}( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} )^{2}( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 - 264 T^{2} + 32668 T^{4} - 2456248 T^{6} + 122337670 T^{8} - 4128952888 T^{10} + 92311960348 T^{12} - 1254027519624 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 264 T^{2} + 32668 T^{4} - 2456248 T^{6} + 122337670 T^{8} - 4128952888 T^{10} + 92311960348 T^{12} - 1254027519624 T^{14} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 - 320 T^{2} + 45756 T^{4} - 3816624 T^{6} + 203071110 T^{8} - 7056937776 T^{10} + 156430658556 T^{12} - 2022836175680 T^{14} + 11688200277601 T^{16} )^{2} \))(\( ( 1 - 320 T^{2} + 45756 T^{4} - 3816624 T^{6} + 203071110 T^{8} - 7056937776 T^{10} + 156430658556 T^{12} - 2022836175680 T^{14} + 11688200277601 T^{16} )^{2} \))
$47$ (\( ( 1 - 47 T^{2} )^{4} \))(\( ( 1 - 47 T^{2} )^{8} \))(\( ( 1 - 140 T^{2} + 13348 T^{4} - 854628 T^{6} + 45549910 T^{8} - 1887873252 T^{10} + 65133981988 T^{12} - 1509090146060 T^{14} + 23811286661761 T^{16} )^{2} \))(\( ( 1 - 140 T^{2} + 13348 T^{4} - 854628 T^{6} + 45549910 T^{8} - 1887873252 T^{10} + 65133981988 T^{12} - 1509090146060 T^{14} + 23811286661761 T^{16} )^{2} \))
$53$ (\( ( 1 - 53 T^{2} )^{4} \))(\( ( 1 - 53 T^{2} )^{8} \))(\( ( 1 - 312 T^{2} + 45948 T^{4} - 4212040 T^{6} + 265479398 T^{8} - 11831620360 T^{10} + 362551820988 T^{12} - 6915280672248 T^{14} + 62259690411361 T^{16} )^{2} \))(\( ( 1 - 312 T^{2} + 45948 T^{4} - 4212040 T^{6} + 265479398 T^{8} - 11831620360 T^{10} + 362551820988 T^{12} - 6915280672248 T^{14} + 62259690411361 T^{16} )^{2} \))
$59$ (\( ( 1 - 6 T + 59 T^{2} )^{2}( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{4}( 1 + 6 T + 59 T^{2} )^{4} \))(\( ( 1 - 312 T^{2} + 41500 T^{4} - 3304648 T^{6} + 204947494 T^{8} - 11503479688 T^{10} + 502870481500 T^{12} - 13160326495992 T^{14} + 146830437604321 T^{16} )^{2} \))(\( ( 1 - 312 T^{2} + 41500 T^{4} - 3304648 T^{6} + 204947494 T^{8} - 11503479688 T^{10} + 502870481500 T^{12} - 13160326495992 T^{14} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 61 T^{2} )^{8} \))(\( ( 1 - 280 T^{2} + 39452 T^{4} - 3783016 T^{6} + 267380710 T^{8} - 14076602536 T^{10} + 546246119132 T^{12} - 14425704821080 T^{14} + 191707312997281 T^{16} )^{2} \))(\( ( 1 - 280 T^{2} + 39452 T^{4} - 3783016 T^{6} + 267380710 T^{8} - 14076602536 T^{10} + 546246119132 T^{12} - 14425704821080 T^{14} + 191707312997281 T^{16} )^{2} \))
$67$ (\( ( 1 + 62 T^{2} + 4489 T^{4} )^{2} \))(\( ( 1 - 62 T^{2} - 645 T^{4} - 278318 T^{6} + 20151121 T^{8} )^{2} \))(\( ( 1 - 176 T^{2} + 15964 T^{4} - 993216 T^{6} + 63719302 T^{8} - 4458546624 T^{10} + 321692495644 T^{12} - 15920675261744 T^{14} + 406067677556641 T^{16} )^{2} \))(\( ( 1 - 176 T^{2} + 15964 T^{4} - 993216 T^{6} + 63719302 T^{8} - 4458546624 T^{10} + 321692495644 T^{12} - 15920675261744 T^{14} + 406067677556641 T^{16} )^{2} \))
$71$ (\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 + 71 T^{2} )^{8} \))(\( ( 1 - 12 T + 220 T^{2} - 1564 T^{3} + 18854 T^{4} - 111044 T^{5} + 1109020 T^{6} - 4294932 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 + 12 T + 220 T^{2} + 1564 T^{3} + 18854 T^{4} + 111044 T^{5} + 1109020 T^{6} + 4294932 T^{7} + 25411681 T^{8} )^{4} \))
$73$ (\( ( 1 - 142 T^{2} + 5329 T^{4} )^{2} \))(\( ( 1 + 142 T^{2} + 14835 T^{4} + 756718 T^{6} + 28398241 T^{8} )^{2} \))(\( ( 1 - 344 T^{2} + 60316 T^{4} - 7095528 T^{6} + 604376710 T^{8} - 37812068712 T^{10} + 1712868304156 T^{12} - 52058973843416 T^{14} + 806460091894081 T^{16} )^{2} \))(\( ( 1 - 344 T^{2} + 60316 T^{4} - 7095528 T^{6} + 604376710 T^{8} - 37812068712 T^{10} + 1712868304156 T^{12} - 52058973843416 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( ( 1 - 79 T^{2} )^{4} \))(\( ( 1 - 79 T^{2} )^{8} \))(\( ( 1 - 524 T^{2} + 125604 T^{4} - 18135668 T^{6} + 1736460342 T^{8} - 113184703988 T^{10} + 4892285973924 T^{12} - 127377826693004 T^{14} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 - 524 T^{2} + 125604 T^{4} - 18135668 T^{6} + 1736460342 T^{8} - 113184703988 T^{10} + 4892285973924 T^{12} - 127377826693004 T^{14} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 158 T^{2} + 18075 T^{4} - 1088462 T^{6} + 47458321 T^{8} )^{2} \))(\( ( 1 + 512 T^{2} + 125148 T^{4} + 18790160 T^{6} + 1886502918 T^{8} + 129445412240 T^{10} + 5939313956508 T^{12} + 167393471164928 T^{14} + 2252292232139041 T^{16} )^{2} \))(\( ( 1 + 512 T^{2} + 125148 T^{4} + 18790160 T^{6} + 1886502918 T^{8} + 129445412240 T^{10} + 5939313956508 T^{12} + 167393471164928 T^{14} + 2252292232139041 T^{16} )^{2} \))
$89$ (\( ( 1 - 18 T + 89 T^{2} )^{2}( 1 + 18 T + 89 T^{2} )^{2} \))(\( ( 1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2}( 1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 328 T^{2} + 61788 T^{4} - 8336760 T^{6} + 843121542 T^{8} - 66035475960 T^{10} + 3876717586908 T^{12} - 163009863435208 T^{14} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 328 T^{2} + 61788 T^{4} - 8336760 T^{6} + 843121542 T^{8} - 66035475960 T^{10} + 3876717586908 T^{12} - 163009863435208 T^{14} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( ( 1 - 94 T^{2} + 9409 T^{4} )^{2} \))(\( ( 1 - 94 T^{2} + 9409 T^{4} )^{4} \))(\( ( 1 - 408 T^{2} + 72412 T^{4} - 7810088 T^{6} + 720805318 T^{8} - 73485117992 T^{10} + 6410582295772 T^{12} - 339852578011032 T^{14} + 7837433594376961 T^{16} )^{2} \))(\( ( 1 - 408 T^{2} + 72412 T^{4} - 7810088 T^{6} + 720805318 T^{8} - 73485117992 T^{10} + 6410582295772 T^{12} - 339852578011032 T^{14} + 7837433594376961 T^{16} )^{2} \))
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