Properties

Label 51.2.h
Level 51
Weight 2
Character orbit h
Rep. character \(\chi_{51}(19,\cdot)\)
Character field \(\Q(\zeta_{8})\)
Dimension 8
Newform subspaces 1
Sturm bound 12
Trace bound 0

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Defining parameters

Level: \( N \) = \( 51 = 3 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 51.h (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 17 \)
Character field: \(\Q(\zeta_{8})\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(51, [\chi])\).

Total New Old
Modular forms 32 8 24
Cusp forms 16 8 8
Eisenstein series 16 0 16

Trace form

\( 8q - 8q^{5} - 8q^{6} + O(q^{10}) \) \( 8q - 8q^{5} - 8q^{6} + 8q^{11} - 16q^{14} + 16q^{16} - 8q^{17} - 8q^{19} + 16q^{20} - 8q^{22} + 8q^{23} + 8q^{24} - 16q^{25} + 16q^{26} - 8q^{28} + 8q^{31} + 8q^{33} - 8q^{34} + 32q^{35} + 8q^{36} - 8q^{37} + 16q^{39} - 8q^{40} - 24q^{41} + 8q^{42} - 8q^{43} - 8q^{45} - 8q^{49} - 32q^{50} - 16q^{52} - 32q^{53} - 8q^{54} - 16q^{56} - 16q^{57} + 24q^{58} + 16q^{59} - 8q^{60} + 16q^{61} + 16q^{62} + 24q^{65} - 16q^{66} - 16q^{67} - 24q^{69} + 40q^{70} - 16q^{71} + 48q^{73} + 64q^{74} + 16q^{75} + 24q^{76} + 8q^{78} - 16q^{80} - 8q^{82} + 32q^{83} + 40q^{85} - 32q^{86} + 24q^{87} - 8q^{88} - 16q^{91} - 32q^{92} - 32q^{93} - 40q^{95} - 16q^{96} + 8q^{97} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(51, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
51.2.h.a \(8\) \(0.407\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(-8\) \(0\) \(q+(\zeta_{16}+\zeta_{16}^{3})q^{2}+\zeta_{16}^{7}q^{3}+(\zeta_{16}^{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(51, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(51, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(17, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{4} + 4 T^{8} - 64 T^{12} + 256 T^{16} \)
$3$ \( 1 + T^{8} \)
$5$ \( 1 + 8 T + 40 T^{2} + 160 T^{3} + 544 T^{4} + 1624 T^{5} + 4384 T^{6} + 10904 T^{7} + 25151 T^{8} + 54520 T^{9} + 109600 T^{10} + 203000 T^{11} + 340000 T^{12} + 500000 T^{13} + 625000 T^{14} + 625000 T^{15} + 390625 T^{16} \)
$7$ \( ( 1 + 4 T + 10 T^{2} + 28 T^{3} + 49 T^{4} )^{2}( 1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 616 T^{5} + 1568 T^{6} - 2744 T^{7} + 2401 T^{8} ) \)
$11$ \( 1 - 8 T + 16 T^{2} + 96 T^{3} - 640 T^{4} + 1240 T^{5} + 1952 T^{6} - 15496 T^{7} + 53087 T^{8} - 170456 T^{9} + 236192 T^{10} + 1650440 T^{11} - 9370240 T^{12} + 15460896 T^{13} + 28344976 T^{14} - 155897368 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 60 T^{2} + 1946 T^{4} - 41776 T^{6} + 637155 T^{8} - 7060144 T^{10} + 55579706 T^{12} - 289608540 T^{14} + 815730721 T^{16} \)
$17$ \( 1 + 8 T + 40 T^{2} + 88 T^{3} + 240 T^{4} + 1496 T^{5} + 11560 T^{6} + 39304 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 8 T + 32 T^{2} + 224 T^{3} + 734 T^{4} - 1320 T^{5} - 8960 T^{6} - 84232 T^{7} - 655901 T^{8} - 1600408 T^{9} - 3234560 T^{10} - 9053880 T^{11} + 95655614 T^{12} + 554646176 T^{13} + 1505468192 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 - 8 T + 88 T^{2} - 560 T^{3} + 3744 T^{4} - 22424 T^{5} + 123872 T^{6} - 722072 T^{7} + 3312623 T^{8} - 16607656 T^{9} + 65528288 T^{10} - 272832808 T^{11} + 1047724704 T^{12} - 3604352080 T^{13} + 13027158232 T^{14} - 27238603576 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 - 48 T^{2} + 1152 T^{4} + 8016 T^{6} - 960286 T^{8} + 6741456 T^{10} + 814787712 T^{12} - 28551519408 T^{14} + 500246412961 T^{16} \)
$31$ \( 1 - 8 T + 104 T^{2} - 344 T^{3} + 2528 T^{4} + 10072 T^{5} - 28728 T^{6} + 893640 T^{7} - 2291934 T^{8} + 27702840 T^{9} - 27607608 T^{10} + 300054952 T^{11} + 2334661088 T^{12} - 9848427944 T^{13} + 92300382824 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 8 T - 68 T^{2} - 520 T^{3} + 3528 T^{4} + 21736 T^{5} - 121324 T^{6} - 363944 T^{7} + 4087214 T^{8} - 13465928 T^{9} - 166092556 T^{10} + 1100993608 T^{11} + 6612040008 T^{12} - 36058857640 T^{13} - 174469395812 T^{14} + 759455017064 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 24 T + 184 T^{2} - 64 T^{3} - 7648 T^{4} - 17592 T^{5} + 256864 T^{6} + 1952264 T^{7} + 9209503 T^{8} + 80042824 T^{9} + 431788384 T^{10} - 1212458232 T^{11} - 21611420128 T^{12} - 7414796864 T^{13} + 874019180344 T^{14} + 4674102573144 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 8 T + 32 T^{2} + 128 T^{3} - 2338 T^{4} - 10376 T^{5} + 492952 T^{7} + 8953251 T^{8} + 21196936 T^{9} - 824964632 T^{11} - 7993156738 T^{12} + 18817080704 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 - 168 T^{2} + 16604 T^{4} - 1181336 T^{6} + 63379782 T^{8} - 2609571224 T^{10} + 81022223324 T^{12} - 1810908175272 T^{14} + 23811286661761 T^{16} \)
$53$ \( 1 + 32 T + 512 T^{2} + 6368 T^{3} + 74084 T^{4} + 768928 T^{5} + 6950400 T^{6} + 57590880 T^{7} + 440626726 T^{8} + 3052316640 T^{9} + 19523673600 T^{10} + 114475693856 T^{11} + 584558394404 T^{12} + 2663068899424 T^{13} + 11348152898048 T^{14} + 37590756474784 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 16 T + 128 T^{2} - 1024 T^{3} + 7072 T^{4} - 53760 T^{5} + 479232 T^{6} - 5119792 T^{7} + 50495970 T^{8} - 302067728 T^{9} + 1668206592 T^{10} - 11041175040 T^{11} + 85693976992 T^{12} - 732082482176 T^{13} + 5399108306048 T^{14} - 39818423757104 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 16 T + 324 T^{2} - 2576 T^{3} + 27144 T^{4} - 29808 T^{5} - 166612 T^{6} + 17565840 T^{7} - 104746898 T^{8} + 1071516240 T^{9} - 619963252 T^{10} - 6765849648 T^{11} + 375831508104 T^{12} - 2175680071376 T^{13} + 16692601292964 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \)
$67$ \( ( 1 + 8 T + 272 T^{2} + 1592 T^{3} + 27474 T^{4} + 106664 T^{5} + 1221008 T^{6} + 2406104 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( 1 + 16 T - 32 T^{2} - 1280 T^{3} + 4608 T^{4} + 133888 T^{5} + 581984 T^{6} - 1077392 T^{7} - 26627902 T^{8} - 76494832 T^{9} + 2933781344 T^{10} + 47919987968 T^{11} + 117097026048 T^{12} - 2309413569280 T^{13} - 4099209085472 T^{14} + 145521922534256 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 48 T + 1200 T^{2} - 21344 T^{3} + 306816 T^{4} - 3800544 T^{5} + 41743472 T^{6} - 412310704 T^{7} + 3694140226 T^{8} - 30098681392 T^{9} + 222450962288 T^{10} - 1478476225248 T^{11} + 8713034710656 T^{12} - 44247640080992 T^{13} + 181601071546800 T^{14} - 530275128916656 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 36 T^{2} - 1488 T^{3} + 648 T^{4} - 122448 T^{5} + 841644 T^{6} - 2436480 T^{7} + 130708238 T^{8} - 192481920 T^{9} + 5252700204 T^{10} - 60371639472 T^{11} + 25239652488 T^{12} - 4578659921712 T^{13} + 8751148398756 T^{14} + 1517108809906561 T^{16} \)
$83$ \( 1 - 32 T + 512 T^{2} - 6304 T^{3} + 73444 T^{4} - 833312 T^{5} + 8932864 T^{6} - 91493280 T^{7} + 877696230 T^{8} - 7593942240 T^{9} + 61538500096 T^{10} - 476476968544 T^{11} + 3485528927524 T^{12} - 24831712213472 T^{13} + 167393471164928 T^{14} - 868353631668064 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 416 T^{2} + 87680 T^{4} - 12136864 T^{6} + 1238928322 T^{8} - 96136099744 T^{10} + 5501239690880 T^{12} - 206744217039776 T^{14} + 3936588805702081 T^{16} \)
$97$ \( 1 - 8 T - 24 T^{2} + 2200 T^{3} - 14752 T^{4} + 58200 T^{5} + 1664072 T^{6} - 6187784 T^{7} + 82745634 T^{8} - 600215048 T^{9} + 15657253448 T^{10} + 53117568600 T^{11} - 1305983953312 T^{12} + 18892148565400 T^{13} - 19991328118296 T^{14} - 646386275824904 T^{15} + 7837433594376961 T^{16} \)
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