Properties

Label 430.2.b
Level 430
Weight 2
Character orbit b
Rep. character \(\chi_{430}(259,\cdot)\)
Character field \(\Q\)
Dimension 22
Newform subspaces 2
Sturm bound 132
Trace bound 1

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Defining parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(132\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(430, [\chi])\).

Total New Old
Modular forms 70 22 48
Cusp forms 62 22 40
Eisenstein series 8 0 8

Trace form

\( 22q - 22q^{4} + 4q^{5} + 4q^{6} - 26q^{9} + O(q^{10}) \) \( 22q - 22q^{4} + 4q^{5} + 4q^{6} - 26q^{9} - 8q^{11} - 4q^{14} - 8q^{15} + 22q^{16} + 4q^{19} - 4q^{20} + 24q^{21} - 4q^{24} - 12q^{25} - 12q^{26} + 16q^{29} + 32q^{31} - 24q^{35} + 26q^{36} + 28q^{41} + 8q^{44} - 32q^{45} + 12q^{46} - 18q^{49} - 32q^{51} - 16q^{54} - 28q^{55} + 4q^{56} - 16q^{59} + 8q^{60} + 16q^{61} - 22q^{64} + 48q^{66} - 8q^{69} - 12q^{70} + 8q^{71} - 24q^{74} - 4q^{75} - 4q^{76} + 8q^{79} + 4q^{80} + 70q^{81} - 24q^{84} - 68q^{85} + 10q^{86} + 68q^{89} - 2q^{90} + 24q^{91} + 20q^{94} - 50q^{95} + 4q^{96} + 48q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(430, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
430.2.b.a \(6\) \(3.434\) 6.0.350464.1 None \(0\) \(0\) \(2\) \(0\) \(q-\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
430.2.b.b \(16\) \(3.434\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(2\) \(0\) \(q-\beta _{7}q^{2}+(\beta _{1}+\beta _{7})q^{3}-q^{4}-\beta _{5}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(430, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(430, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(215, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T^{2} )^{3} \))(\( ( 1 + T^{2} )^{8} \))
$3$ (\( 1 - 10 T^{2} + 51 T^{4} - 176 T^{6} + 459 T^{8} - 810 T^{10} + 729 T^{12} \))(\( 1 - 10 T^{2} + 51 T^{4} - 176 T^{6} + 667 T^{8} - 3040 T^{10} + 11605 T^{12} - 33410 T^{14} + 91072 T^{16} - 300690 T^{18} + 940005 T^{20} - 2216160 T^{22} + 4376187 T^{24} - 10392624 T^{26} + 27103491 T^{28} - 47829690 T^{30} + 43046721 T^{32} \))
$5$ (\( 1 - 2 T + 3 T^{2} - 12 T^{3} + 15 T^{4} - 50 T^{5} + 125 T^{6} \))(\( 1 - 2 T + 7 T^{2} - 37 T^{4} + 98 T^{5} - 159 T^{6} - 176 T^{7} + 728 T^{8} - 880 T^{9} - 3975 T^{10} + 12250 T^{11} - 23125 T^{12} + 109375 T^{14} - 156250 T^{15} + 390625 T^{16} \))
$7$ (\( 1 - 31 T^{2} + 454 T^{4} - 4003 T^{6} + 22246 T^{8} - 74431 T^{10} + 117649 T^{12} \))(\( 1 - 37 T^{2} + 669 T^{4} - 8530 T^{6} + 93904 T^{8} - 954556 T^{10} + 8661035 T^{12} - 68886013 T^{14} + 498263182 T^{16} - 3375414637 T^{18} + 20795145035 T^{20} - 112302558844 T^{22} + 541337873104 T^{24} - 2409513873970 T^{26} + 9259821137469 T^{28} - 25094253695413 T^{30} + 33232930569601 T^{32} \))
$11$ (\( ( 1 + 6 T + 41 T^{2} + 130 T^{3} + 451 T^{4} + 726 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 - 2 T + 28 T^{2} - 88 T^{3} + 612 T^{4} - 1532 T^{5} + 9668 T^{6} - 22990 T^{7} + 112406 T^{8} - 252890 T^{9} + 1169828 T^{10} - 2039092 T^{11} + 8960292 T^{12} - 14172488 T^{13} + 49603708 T^{14} - 38974342 T^{15} + 214358881 T^{16} )^{2} \))
$13$ (\( 1 - 43 T^{2} + 878 T^{4} - 12687 T^{6} + 148382 T^{8} - 1228123 T^{10} + 4826809 T^{12} \))(\( 1 - 57 T^{2} + 2221 T^{4} - 63010 T^{6} + 1470096 T^{8} - 29164612 T^{10} + 504883051 T^{12} - 7776883625 T^{14} + 106613731662 T^{16} - 1314293332625 T^{18} + 14419964819611 T^{20} - 140772011683108 T^{22} + 1199202470019216 T^{24} - 8686463571405490 T^{26} + 51745047057030301 T^{28} - 224430453984859473 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( 1 - 42 T^{2} + 1155 T^{4} - 24816 T^{6} + 333795 T^{8} - 3507882 T^{10} + 24137569 T^{12} \))(\( 1 - 82 T^{2} + 3475 T^{4} - 100800 T^{6} + 2208355 T^{8} - 36519332 T^{10} + 437841909 T^{12} - 3747068150 T^{14} + 36785708160 T^{16} - 1082902695350 T^{18} + 36568994081589 T^{20} - 881487895983908 T^{22} + 15404948823619555 T^{24} - 203212185165259200 T^{26} + 2024612274373419475 T^{28} - 13806981777870876178 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 - 17 T + 148 T^{2} - 797 T^{3} + 2812 T^{4} - 6137 T^{5} + 6859 T^{6} )^{2} \))(\( ( 1 + 15 T + 200 T^{2} + 1733 T^{3} + 13726 T^{4} + 86371 T^{5} + 508275 T^{6} + 2529942 T^{7} + 11896286 T^{8} + 48068898 T^{9} + 183487275 T^{10} + 592418689 T^{11} + 1788786046 T^{12} + 4291079567 T^{13} + 9409176200 T^{14} + 13408076085 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 - 118 T^{2} + 6135 T^{4} - 181696 T^{6} + 3245415 T^{8} - 33021238 T^{10} + 148035889 T^{12} \))(\( 1 - 214 T^{2} + 22679 T^{4} - 1596744 T^{6} + 84184463 T^{8} - 3537581064 T^{10} + 122756878141 T^{12} - 3589213546070 T^{14} + 89334432012112 T^{16} - 1898693965871030 T^{18} + 34352407535855581 T^{20} - 523688957718805896 T^{22} + 6592568242881889103 T^{24} - 66147533221326758856 T^{26} + \)\(49\!\cdots\!59\)\( T^{28} - \)\(24\!\cdots\!26\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 - 5 T + 82 T^{2} - 291 T^{3} + 2378 T^{4} - 4205 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 - 3 T + 78 T^{2} - 569 T^{3} + 3594 T^{4} - 31051 T^{5} + 181717 T^{6} - 931820 T^{7} + 6832480 T^{8} - 27022780 T^{9} + 152823997 T^{10} - 757302839 T^{11} + 2541967914 T^{12} - 11670843781 T^{13} + 46396219038 T^{14} - 51749628927 T^{15} + 500246412961 T^{16} )^{2} \))
$31$ (\( ( 1 + 9 T + 84 T^{2} + 423 T^{3} + 2604 T^{4} + 8649 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 - 25 T + 452 T^{2} - 5865 T^{3} + 63046 T^{4} - 563375 T^{5} + 4370993 T^{6} - 29418450 T^{7} + 174910256 T^{8} - 911971950 T^{9} + 4200524273 T^{10} - 16783504625 T^{11} + 58224304966 T^{12} - 167909970615 T^{13} + 401151663812 T^{14} - 687815352775 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 - 82 T^{2} + 4015 T^{4} - 159016 T^{6} + 5496535 T^{8} - 153681202 T^{10} + 2565726409 T^{12} \))(\( 1 - 394 T^{2} + 77383 T^{4} - 10014624 T^{6} + 953289279 T^{8} - 70658881624 T^{10} + 4214634576301 T^{12} - 206262118894570 T^{14} + 8368262409986752 T^{16} - 282372840766666330 T^{18} + 7898903752154858461 T^{20} - \)\(18\!\cdots\!16\)\( T^{22} + \)\(33\!\cdots\!59\)\( T^{24} - \)\(48\!\cdots\!76\)\( T^{26} + \)\(50\!\cdots\!23\)\( T^{28} - \)\(35\!\cdots\!66\)\( T^{30} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( ( 1 + 5 T + 46 T^{2} + 225 T^{3} + 1886 T^{4} + 8405 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 - 19 T + 406 T^{2} - 5039 T^{3} + 63478 T^{4} - 587281 T^{5} + 5386299 T^{6} - 39029276 T^{7} + 278582162 T^{8} - 1600200316 T^{9} + 9054368619 T^{10} - 40475993801 T^{11} + 179373656758 T^{12} - 583799396839 T^{13} + 1928542321846 T^{14} - 3700331203739 T^{15} + 7984925229121 T^{16} )^{2} \))
$43$ (\( ( 1 + T^{2} )^{3} \))(\( ( 1 + T^{2} )^{8} \))
$47$ (\( 1 - 170 T^{2} + 15279 T^{4} - 888968 T^{6} + 33751311 T^{8} - 829545770 T^{10} + 10779215329 T^{12} \))(\( 1 - 482 T^{2} + 116711 T^{4} - 18746928 T^{6} + 2228078775 T^{8} - 207244774732 T^{10} + 15578128441725 T^{12} - 964122051100670 T^{14} + 49615909496598816 T^{16} - 2129745610881380030 T^{18} + 76016297372645089725 T^{20} - \)\(22\!\cdots\!28\)\( T^{22} + \)\(53\!\cdots\!75\)\( T^{24} - \)\(98\!\cdots\!72\)\( T^{26} + \)\(13\!\cdots\!51\)\( T^{28} - \)\(12\!\cdots\!58\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 138 T^{2} + 14295 T^{4} - 854556 T^{6} + 40154655 T^{8} - 1088886378 T^{10} + 22164361129 T^{12} \))(\( 1 - 576 T^{2} + 154696 T^{4} - 25836080 T^{6} + 3031271836 T^{8} - 269432012400 T^{10} + 19302108479096 T^{12} - 1183214040569248 T^{14} + 65349852889172742 T^{16} - 3323648239959017632 T^{18} + \)\(15\!\cdots\!76\)\( T^{20} - \)\(59\!\cdots\!00\)\( T^{22} + \)\(18\!\cdots\!96\)\( T^{24} - \)\(45\!\cdots\!20\)\( T^{26} + \)\(75\!\cdots\!36\)\( T^{28} - \)\(79\!\cdots\!44\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( ( 1 + 20 T + 249 T^{2} + 2088 T^{3} + 14691 T^{4} + 69620 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 - 12 T + 376 T^{2} - 3324 T^{3} + 61004 T^{4} - 432892 T^{5} + 6064456 T^{6} - 36188204 T^{7} + 421065286 T^{8} - 2135104036 T^{9} + 21110371336 T^{10} - 88906926068 T^{11} + 739207490444 T^{12} - 2376408369876 T^{13} + 15859880649016 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16} )^{2} \))
$61$ (\( ( 1 + 21 T + 272 T^{2} + 2329 T^{3} + 16592 T^{4} + 78141 T^{5} + 226981 T^{6} )^{2} \))(\( ( 1 - 29 T + 693 T^{2} - 10726 T^{3} + 148982 T^{4} - 1635184 T^{5} + 16908739 T^{6} - 147864185 T^{7} + 1244942418 T^{8} - 9019715285 T^{9} + 62917417819 T^{10} - 371155699504 T^{11} + 2062781083862 T^{12} - 9059139924526 T^{13} + 35703619432173 T^{14} - 91139542244609 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( 1 - 87 T^{2} + 978 T^{4} + 259189 T^{6} + 4390242 T^{8} - 1753147527 T^{10} + 90458382169 T^{12} \))(\( 1 - 461 T^{2} + 124889 T^{4} - 23767026 T^{6} + 3523375680 T^{8} - 423196090940 T^{10} + 42444100674783 T^{12} - 3599926920976653 T^{14} + 260857140941399214 T^{16} - 16160071948264195317 T^{18} + \)\(85\!\cdots\!43\)\( T^{20} - \)\(38\!\cdots\!60\)\( T^{22} + \)\(14\!\cdots\!80\)\( T^{24} - \)\(43\!\cdots\!74\)\( T^{26} + \)\(10\!\cdots\!29\)\( T^{28} - \)\(16\!\cdots\!69\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 + 8 T + 159 T^{2} + 930 T^{3} + 11289 T^{4} + 40328 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 12 T + 350 T^{2} - 3178 T^{3} + 57608 T^{4} - 469734 T^{5} + 6497602 T^{6} - 47379416 T^{7} + 534124718 T^{8} - 3363938536 T^{9} + 32754411682 T^{10} - 168122965674 T^{11} + 1463916119048 T^{12} - 5733840877478 T^{13} + 44835099372350 T^{14} - 109141441900692 T^{15} + 645753531245761 T^{16} )^{2} \))
$73$ (\( 1 - 51 T^{2} + 5858 T^{4} - 385823 T^{6} + 31217282 T^{8} - 1448310291 T^{10} + 151334226289 T^{12} \))(\( 1 - 905 T^{2} + 395545 T^{4} - 111083010 T^{6} + 22471297760 T^{8} - 3475557609012 T^{10} + 425393441711119 T^{12} - 42042952532589265 T^{14} + 3391469041132093230 T^{16} - \)\(22\!\cdots\!85\)\( T^{18} + \)\(12\!\cdots\!79\)\( T^{20} - \)\(52\!\cdots\!68\)\( T^{22} + \)\(18\!\cdots\!60\)\( T^{24} - \)\(47\!\cdots\!90\)\( T^{26} + \)\(90\!\cdots\!45\)\( T^{28} - \)\(11\!\cdots\!45\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( ( 1 - 9 T + 234 T^{2} - 1361 T^{3} + 18486 T^{4} - 56169 T^{5} + 493039 T^{6} )^{2} \))(\( ( 1 + 5 T + 234 T^{2} + 995 T^{3} + 29332 T^{4} + 88335 T^{5} + 2715965 T^{6} + 3567410 T^{7} + 217842536 T^{8} + 281825390 T^{9} + 16950337565 T^{10} + 43552600065 T^{11} + 1142483775892 T^{12} + 3061671117005 T^{13} + 56882464591914 T^{14} + 96019544930795 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( 1 + 34 T^{2} + 7719 T^{4} + 448204 T^{6} + 53176191 T^{8} + 1613582914 T^{10} + 326940373369 T^{12} \))(\( 1 - 580 T^{2} + 173632 T^{4} - 35649596 T^{6} + 5673283660 T^{8} - 749699370772 T^{10} + 85546383012736 T^{12} - 8573159301594284 T^{14} + 757692623249501222 T^{16} - 59060494428683022476 T^{18} + \)\(40\!\cdots\!56\)\( T^{20} - \)\(24\!\cdots\!68\)\( T^{22} + \)\(12\!\cdots\!60\)\( T^{24} - \)\(55\!\cdots\!04\)\( T^{26} + \)\(18\!\cdots\!52\)\( T^{28} - \)\(42\!\cdots\!20\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 - 14 T - T^{2} + 1286 T^{3} - 89 T^{4} - 110894 T^{5} + 704969 T^{6} )^{2} \))(\( ( 1 - 20 T + 460 T^{2} - 5698 T^{3} + 87296 T^{4} - 929226 T^{5} + 12020020 T^{6} - 113397072 T^{7} + 1243372926 T^{8} - 10092339408 T^{9} + 95210578420 T^{10} - 655075523994 T^{11} + 5477146670336 T^{12} - 31817970740402 T^{13} + 228611393842060 T^{14} - 884626697910580 T^{15} + 3936588805702081 T^{16} )^{2} \))
$97$ (\( 1 - 510 T^{2} + 114171 T^{4} - 14374424 T^{6} + 1074234939 T^{8} - 45149933310 T^{10} + 832972004929 T^{12} \))(\( 1 - 774 T^{2} + 301691 T^{4} - 79641640 T^{6} + 16078651115 T^{8} - 2644465230904 T^{10} + 367166336275261 T^{12} - 43893175395348070 T^{14} + 4560761585591764624 T^{16} - \)\(41\!\cdots\!30\)\( T^{18} + \)\(32\!\cdots\!41\)\( T^{20} - \)\(22\!\cdots\!16\)\( T^{22} + \)\(12\!\cdots\!15\)\( T^{24} - \)\(58\!\cdots\!60\)\( T^{26} + \)\(20\!\cdots\!31\)\( T^{28} - \)\(50\!\cdots\!06\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \))
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