Properties

Label 210.2.m
Level 210
Weight 2
Character orbit m
Rep. character \(\chi_{210}(13,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 16
Newform subspaces 2
Sturm bound 96
Trace bound 7

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Defining parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 210.m (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(210, [\chi])\).

Total New Old
Modular forms 112 16 96
Cusp forms 80 16 64
Eisenstein series 32 0 32

Trace form

\( 16q + 12q^{7} + O(q^{10}) \) \( 16q + 12q^{7} + 16q^{11} - 8q^{15} - 16q^{16} - 16q^{21} + 16q^{22} + 32q^{23} - 8q^{25} - 12q^{28} + 8q^{30} - 32q^{35} - 16q^{36} - 56q^{37} - 4q^{42} - 16q^{46} + 32q^{51} - 16q^{53} + 8q^{56} - 8q^{57} - 24q^{58} - 12q^{63} - 32q^{65} + 36q^{70} + 16q^{71} + 32q^{78} - 16q^{81} + 56q^{85} + 16q^{86} - 16q^{88} - 16q^{91} + 32q^{92} - 8q^{93} - 16q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(210, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
210.2.m.a \(8\) \(1.677\) 8.0.1698758656.6 None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{3}q^{2}+\beta _{3}q^{3}-\beta _{5}q^{4}+(\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
210.2.m.b \(8\) \(1.677\) 8.0.1698758656.6 None \(0\) \(0\) \(0\) \(8\) \(q+\beta _{3}q^{2}-\beta _{3}q^{3}-\beta _{5}q^{4}+(-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(210, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(210, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))
$3$ (\( ( 1 + T^{4} )^{2} \))(\( ( 1 + T^{4} )^{2} \))
$5$ (\( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8} \))(\( 1 + 2 T^{2} + 16 T^{3} + 2 T^{4} + 80 T^{5} + 50 T^{6} + 625 T^{8} \))
$7$ (\( 1 - 4 T + 6 T^{2} + 28 T^{3} - 126 T^{4} + 196 T^{5} + 294 T^{6} - 1372 T^{7} + 2401 T^{8} \))(\( 1 - 8 T + 34 T^{2} - 112 T^{3} + 322 T^{4} - 784 T^{5} + 1666 T^{6} - 2744 T^{7} + 2401 T^{8} \))
$11$ (\( ( 1 - 4 T + 12 T^{2} + 4 T^{3} + 18 T^{4} + 44 T^{5} + 1452 T^{6} - 5324 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 - 4 T + 12 T^{2} + 4 T^{3} + 18 T^{4} + 44 T^{5} + 1452 T^{6} - 5324 T^{7} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 1768 T^{5} + 5408 T^{6} + 17576 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 136 T^{3} + 562 T^{4} - 1768 T^{5} + 5408 T^{6} - 17576 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 - 12 T + 72 T^{2} - 388 T^{3} + 1696 T^{4} - 4732 T^{5} + 9944 T^{6} - 2420 T^{7} - 98754 T^{8} - 41140 T^{9} + 2873816 T^{10} - 23248316 T^{11} + 141651616 T^{12} - 550904516 T^{13} + 1737904968 T^{14} - 4924064076 T^{15} + 6975757441 T^{16} \))(\( 1 + 12 T + 72 T^{2} + 388 T^{3} + 1696 T^{4} + 4732 T^{5} + 9944 T^{6} + 2420 T^{7} - 98754 T^{8} + 41140 T^{9} + 2873816 T^{10} + 23248316 T^{11} + 141651616 T^{12} + 550904516 T^{13} + 1737904968 T^{14} + 4924064076 T^{15} + 6975757441 T^{16} \))
$19$ (\( ( 1 + 4 T + 62 T^{2} + 212 T^{3} + 1666 T^{4} + 4028 T^{5} + 22382 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 4 T + 62 T^{2} - 212 T^{3} + 1666 T^{4} - 4028 T^{5} + 22382 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 16 T + 128 T^{2} - 656 T^{3} + 3300 T^{4} - 21520 T^{5} + 137088 T^{6} - 691344 T^{7} + 3238918 T^{8} - 15900912 T^{9} + 72519552 T^{10} - 261833840 T^{11} + 923475300 T^{12} - 4222241008 T^{13} + 18948593792 T^{14} - 54477207152 T^{15} + 78310985281 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 656 T^{3} + 3300 T^{4} - 21520 T^{5} + 137088 T^{6} - 691344 T^{7} + 3238918 T^{8} - 15900912 T^{9} + 72519552 T^{10} - 261833840 T^{11} + 923475300 T^{12} - 4222241008 T^{13} + 18948593792 T^{14} - 54477207152 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 4665868 T^{10} + 141456200 T^{12} - 2379293284 T^{14} + 500246412961 T^{16} \))(\( 1 - 4 T^{2} + 200 T^{4} - 5548 T^{6} + 1325934 T^{8} - 4665868 T^{10} + 141456200 T^{12} - 2379293284 T^{14} + 500246412961 T^{16} \))
$31$ (\( 1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 543168732 T^{10} + 10336047032 T^{12} - 131350544788 T^{14} + 852891037441 T^{16} \))(\( 1 - 148 T^{2} + 11192 T^{4} - 565212 T^{6} + 20507310 T^{8} - 543168732 T^{10} + 10336047032 T^{12} - 131350544788 T^{14} + 852891037441 T^{16} \))
$37$ (\( 1 + 28 T + 392 T^{2} + 4180 T^{3} + 38912 T^{4} + 311708 T^{5} + 2210520 T^{6} + 14742164 T^{7} + 92944158 T^{8} + 545460068 T^{9} + 3026201880 T^{10} + 15788945324 T^{11} + 72927352832 T^{12} + 289857740260 T^{13} + 1005764752328 T^{14} + 2658092559724 T^{15} + 3512479453921 T^{16} \))(\( 1 + 28 T + 392 T^{2} + 4180 T^{3} + 38912 T^{4} + 311708 T^{5} + 2210520 T^{6} + 14742164 T^{7} + 92944158 T^{8} + 545460068 T^{9} + 3026201880 T^{10} + 15788945324 T^{11} + 72927352832 T^{12} + 289857740260 T^{13} + 1005764752328 T^{14} + 2658092559724 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 1082429520 T^{10} + 38057349148 T^{12} - 836018346416 T^{14} + 7984925229121 T^{16} \))(\( 1 - 176 T^{2} + 13468 T^{4} - 643920 T^{6} + 26131718 T^{8} - 1082429520 T^{10} + 38057349148 T^{12} - 836018346416 T^{14} + 7984925229121 T^{16} \))
$43$ (\( 1 - 32 T^{3} - 6588 T^{4} - 1632 T^{5} + 512 T^{6} + 117440 T^{7} + 17663206 T^{8} + 5049920 T^{9} + 946688 T^{10} - 129755424 T^{11} - 22523060988 T^{12} - 4704270176 T^{13} + 11688200277601 T^{16} \))(\( 1 - 32 T^{3} - 6588 T^{4} - 1632 T^{5} + 512 T^{6} + 117440 T^{7} + 17663206 T^{8} + 5049920 T^{9} + 946688 T^{10} - 129755424 T^{11} - 22523060988 T^{12} - 4704270176 T^{13} + 11688200277601 T^{16} \))
$47$ (\( 1 - 24 T + 288 T^{2} - 2728 T^{3} + 26308 T^{4} - 243208 T^{5} + 1981280 T^{6} - 14819768 T^{7} + 104930310 T^{8} - 696529096 T^{9} + 4376647520 T^{10} - 25250584184 T^{11} + 128374647748 T^{12} - 625653179096 T^{13} + 3104414014752 T^{14} - 12158954891112 T^{15} + 23811286661761 T^{16} \))(\( 1 + 24 T + 288 T^{2} + 2728 T^{3} + 26308 T^{4} + 243208 T^{5} + 1981280 T^{6} + 14819768 T^{7} + 104930310 T^{8} + 696529096 T^{9} + 4376647520 T^{10} + 25250584184 T^{11} + 128374647748 T^{12} + 625653179096 T^{13} + 3104414014752 T^{14} + 12158954891112 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 8 T + 32 T^{2} + 264 T^{3} + 5828 T^{4} + 46136 T^{5} + 217440 T^{6} + 2204664 T^{7} + 21970150 T^{8} + 116847192 T^{9} + 610788960 T^{10} + 6868589272 T^{11} + 45985723268 T^{12} + 110403610152 T^{13} + 709259556128 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))(\( 1 + 8 T + 32 T^{2} + 264 T^{3} + 5828 T^{4} + 46136 T^{5} + 217440 T^{6} + 2204664 T^{7} + 21970150 T^{8} + 116847192 T^{9} + 610788960 T^{10} + 6868589272 T^{11} + 45985723268 T^{12} + 110403610152 T^{13} + 709259556128 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16} \))
$59$ (\( ( 1 - 4 T + 70 T^{2} - 116 T^{3} + 802 T^{4} - 6844 T^{5} + 243670 T^{6} - 821516 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 4 T + 70 T^{2} + 116 T^{3} + 802 T^{4} + 6844 T^{5} + 243670 T^{6} + 821516 T^{7} + 12117361 T^{8} )^{2} \))
$61$ (\( 1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 13451772216 T^{10} + 519662104412 T^{12} - 13601378831304 T^{14} + 191707312997281 T^{16} \))(\( 1 - 264 T^{2} + 37532 T^{4} - 3615096 T^{6} + 255247078 T^{8} - 13451772216 T^{10} + 519662104412 T^{12} - 13601378831304 T^{14} + 191707312997281 T^{16} \))
$67$ (\( 1 - 64 T^{3} + 9348 T^{4} + 10560 T^{5} + 2048 T^{6} + 428416 T^{7} + 51599014 T^{8} + 28703872 T^{9} + 9193472 T^{10} + 3176057280 T^{11} + 188372679108 T^{12} - 86408006848 T^{13} + 406067677556641 T^{16} \))(\( 1 - 64 T^{3} + 9348 T^{4} + 10560 T^{5} + 2048 T^{6} + 428416 T^{7} + 51599014 T^{8} + 28703872 T^{9} + 9193472 T^{10} + 3176057280 T^{11} + 188372679108 T^{12} - 86408006848 T^{13} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 4 T + 158 T^{2} + 220 T^{3} + 10018 T^{4} + 15620 T^{5} + 796478 T^{6} - 1431644 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 4 T + 158 T^{2} + 220 T^{3} + 10018 T^{4} + 15620 T^{5} + 796478 T^{6} - 1431644 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 + 28 T + 392 T^{2} + 4756 T^{3} + 46736 T^{4} + 302508 T^{5} + 1459480 T^{6} + 3207460 T^{7} - 21483554 T^{8} + 234144580 T^{9} + 7777568920 T^{10} + 117680754636 T^{11} + 1327220191376 T^{12} + 9859528496308 T^{13} + 59323016705288 T^{14} + 309327158534716 T^{15} + 806460091894081 T^{16} \))(\( 1 - 28 T + 392 T^{2} - 4756 T^{3} + 46736 T^{4} - 302508 T^{5} + 1459480 T^{6} - 3207460 T^{7} - 21483554 T^{8} - 234144580 T^{9} + 7777568920 T^{10} - 117680754636 T^{11} + 1327220191376 T^{12} - 9859528496308 T^{13} + 59323016705288 T^{14} - 309327158534716 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 25396875760 T^{10} + 1544137011164 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16} \))(\( 1 - 272 T^{2} + 39644 T^{4} - 4069360 T^{6} + 348070342 T^{8} - 25396875760 T^{10} + 1544137011164 T^{12} - 66119787901712 T^{14} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 16 T + 128 T^{2} + 976 T^{3} + 16964 T^{4} + 235664 T^{5} + 2075520 T^{6} + 16972368 T^{7} + 135669670 T^{8} + 1408706544 T^{9} + 14298257280 T^{10} + 134749611568 T^{11} + 805082957444 T^{12} + 3844503667568 T^{13} + 41848367791232 T^{14} + 434176815834032 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 16 T + 128 T^{2} - 976 T^{3} + 16964 T^{4} - 235664 T^{5} + 2075520 T^{6} - 16972368 T^{7} + 135669670 T^{8} - 1408706544 T^{9} + 14298257280 T^{10} - 134749611568 T^{11} + 805082957444 T^{12} - 3844503667568 T^{13} + 41848367791232 T^{14} - 434176815834032 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 + 32 T + 628 T^{2} + 8320 T^{3} + 88630 T^{4} + 740480 T^{5} + 4974388 T^{6} + 22559008 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 32 T + 628 T^{2} - 8320 T^{3} + 88630 T^{4} - 740480 T^{5} + 4974388 T^{6} - 22559008 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 + 28 T + 392 T^{2} + 4900 T^{3} + 61488 T^{4} + 704956 T^{5} + 7640472 T^{6} + 89011524 T^{7} + 981394654 T^{8} + 8634117828 T^{9} + 71889201048 T^{10} + 643394307388 T^{11} + 5443488430128 T^{12} + 42077967259300 T^{13} + 326525025932168 T^{14} + 2262351965387164 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 28 T + 392 T^{2} - 4900 T^{3} + 61488 T^{4} - 704956 T^{5} + 7640472 T^{6} - 89011524 T^{7} + 981394654 T^{8} - 8634117828 T^{9} + 71889201048 T^{10} - 643394307388 T^{11} + 5443488430128 T^{12} - 42077967259300 T^{13} + 326525025932168 T^{14} - 2262351965387164 T^{15} + 7837433594376961 T^{16} \))
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