Properties

Label 5.20.b.a
Level 5
Weight 20
Character orbit 5.b
Analytic conductor 11.441
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 5.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(11.4408348278\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{8}\cdot 5^{13} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( -202593 + \beta_{3} ) q^{4} + ( 18375 + 94 \beta_{1} + 27 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{5} + ( 420717 - 3 \beta_{3} + \beta_{4} + \beta_{6} ) q^{6} + ( -30616 \beta_{1} + 322 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{7} + ( -92622 \beta_{1} + 656 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{8} + ( 43169823 + 27 \beta_{1} - 9 \beta_{2} - 387 \beta_{3} + 66 \beta_{4} + 12 \beta_{6} + 9 \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( -202593 + \beta_{3} ) q^{4} + ( 18375 + 94 \beta_{1} + 27 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{5} + ( 420717 - 3 \beta_{3} + \beta_{4} + \beta_{6} ) q^{6} + ( -30616 \beta_{1} + 322 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{7} + ( -92622 \beta_{1} + 656 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{8} + ( 43169823 + 27 \beta_{1} - 9 \beta_{2} - 387 \beta_{3} + 66 \beta_{4} + 12 \beta_{6} + 9 \beta_{7} ) q^{9} + ( -76336375 + 739033 \beta_{1} - 16336 \beta_{2} + 11 \beta_{3} + 37 \beta_{4} - 25 \beta_{5} + 75 \beta_{6} + 45 \beta_{7} ) q^{10} + ( -422446908 + 462 \beta_{1} - 154 \beta_{2} - 358 \beta_{3} + 552 \beta_{4} - 372 \beta_{6} + 154 \beta_{7} ) q^{11} + ( 1408182 \beta_{1} - 102192 \beta_{2} + 522 \beta_{3} - 1962 \beta_{4} - 63 \beta_{5} + 459 \beta_{7} ) q^{12} + ( -2594783 \beta_{1} - 337345 \beta_{2} + 1051 \beta_{3} - 4340 \beta_{4} + 68 \beta_{5} + 1119 \beta_{7} ) q^{13} + ( 22156323879 + 7296 \beta_{1} - 2432 \beta_{2} - 154145 \beta_{3} + 15447 \beta_{4} + 855 \beta_{6} + 2432 \beta_{7} ) q^{14} + ( -30290578500 - 25056336 \beta_{1} - 1131738 \beta_{2} + 109263 \beta_{3} + 7346 \beta_{4} + 675 \beta_{5} + 1100 \beta_{6} + 4410 \beta_{7} ) q^{15} + ( -39091284604 + 21312 \beta_{1} - 7104 \beta_{2} + 62492 \beta_{3} + 44400 \beta_{4} + 1776 \beta_{6} + 7104 \beta_{7} ) q^{16} + ( -67282226 \beta_{1} + 5212914 \beta_{2} + 7446 \beta_{3} - 32232 \beta_{4} + 1224 \beta_{5} + 8670 \beta_{7} ) q^{17} + ( 200741955 \beta_{1} - 9104352 \beta_{2} + 11196 \beta_{3} - 37692 \beta_{4} - 3546 \beta_{5} + 7650 \beta_{7} ) q^{18} + ( 568398547580 - 10026 \beta_{1} + 3342 \beta_{2} - 2724542 \beta_{3} - 39936 \beta_{4} - 19884 \beta_{6} - 3342 \beta_{7} ) q^{19} + ( -522553678875 - 3388162 \beta_{1} - 35536496 \beta_{2} + 2863421 \beta_{3} - 87698 \beta_{4} - 7875 \beta_{5} - 42000 \beta_{6} - 25825 \beta_{7} ) q^{20} + ( -372894291828 - 218457 \beta_{1} + 72819 \beta_{2} + 3069513 \beta_{3} - 376914 \beta_{4} + 60000 \beta_{6} - 72819 \beta_{7} ) q^{21} + ( -476682692 \beta_{1} + 208875200 \beta_{2} - 118856 \beta_{3} + 496840 \beta_{4} - 10708 \beta_{5} - 129564 \beta_{7} ) q^{22} + ( 2372966902 \beta_{1} - 32886096 \beta_{2} - 267675 \beta_{3} + 961470 \beta_{4} + 54615 \beta_{5} - 213060 \beta_{7} ) q^{23} + ( -772080347460 - 724032 \beta_{1} + 241344 \beta_{2} + 8145444 \beta_{3} - 1279408 \beta_{4} + 168656 \beta_{6} - 241344 \beta_{7} ) q^{24} + ( 2214463703125 - 8526270125 \beta_{1} - 324670375 \beta_{2} - 8873375 \beta_{3} + 284500 \beta_{4} + 50000 \beta_{5} + 412500 \beta_{6} - 249375 \beta_{7} ) q^{25} + ( 1988653516062 - 148416 \beta_{1} + 49472 \beta_{2} - 9388906 \beta_{3} - 1133934 \beta_{4} - 837102 \beta_{6} - 49472 \beta_{7} ) q^{26} + ( -15209647812 \beta_{1} + 969750936 \beta_{2} + 173718 \beta_{3} - 753948 \beta_{4} + 29538 \beta_{5} + 203256 \beta_{7} ) q^{27} + ( 68579683938 \beta_{1} - 832768272 \beta_{2} + 1368222 \beta_{3} - 4565502 \beta_{4} - 453693 \beta_{5} + 914529 \beta_{7} ) q^{28} + ( -23527787543130 + 4145916 \beta_{1} - 1381972 \beta_{2} + 94661300 \beta_{3} + 7593984 \beta_{4} - 697848 \beta_{6} + 1381972 \beta_{7} ) q^{29} + ( 18562685367375 - 76147903172 \beta_{1} - 957112576 \beta_{2} - 102630249 \beta_{3} + 259887 \beta_{4} - 171000 \beta_{5} - 1802625 \beta_{6} + 2467800 \beta_{7} ) q^{30} + ( 9014375835872 + 7538148 \beta_{1} - 2512716 \beta_{2} - 116458820 \beta_{3} + 19518360 \beta_{4} + 4442064 \beta_{6} + 2512716 \beta_{7} ) q^{31} + ( -117797392328 \beta_{1} - 1794806848 \beta_{2} + 3081736 \beta_{3} - 12913672 \beta_{4} + 293364 \beta_{5} + 3375100 \beta_{7} ) q^{32} + ( 321132842304 \beta_{1} + 1498957536 \beta_{2} - 1125612 \beta_{3} - 31824 \beta_{4} + 2267136 \beta_{5} + 1141524 \beta_{7} ) q^{33} + ( 47305027748224 + 3936384 \beta_{1} - 1312128 \beta_{2} - 221243984 \beta_{3} + 9585144 \beta_{4} + 1712376 \beta_{6} + 1312128 \beta_{7} ) q^{34} + ( -37389469489500 - 456511133517 \beta_{1} + 6635187389 \beta_{2} + 319182536 \beta_{3} - 1790988 \beta_{4} + 205350 \beta_{5} + 2552700 \beta_{6} - 4822130 \beta_{7} ) q^{35} + ( -120502531654839 - 16122240 \beta_{1} + 5374080 \beta_{2} + 457867863 \beta_{3} - 40611744 \beta_{4} - 8367264 \beta_{6} - 5374080 \beta_{7} ) q^{36} + ( -310999080351 \beta_{1} - 15941339241 \beta_{2} - 12944049 \beta_{3} + 59191044 \beta_{4} - 3707424 \beta_{5} - 16651473 \beta_{7} ) q^{37} + ( 1694103591924 \beta_{1} + 11134149824 \beta_{2} - 6381800 \beta_{3} + 38436968 \beta_{4} - 6454884 \beta_{5} - 12836684 \beta_{7} ) q^{38} + ( 376456723506216 - 92020212 \beta_{1} + 30673404 \beta_{2} - 977119164 \beta_{3} - 192461712 \beta_{4} - 8421288 \beta_{6} - 30673404 \beta_{7} ) q^{39} + ( -26701428492500 - 1325833212730 \beta_{1} + 21571026160 \beta_{2} + 997654590 \beta_{3} - 7077370 \beta_{4} + 336625 \beta_{5} + 5562000 \beta_{6} - 13385925 \beta_{7} ) q^{40} + ( -414283678607238 - 100322985 \beta_{1} + 33440995 \beta_{2} + 888631729 \beta_{3} - 218857566 \beta_{4} - 18211596 \beta_{6} - 33440995 \beta_{7} ) q^{41} + ( -1580018751888 \beta_{1} - 24483004896 \beta_{2} - 9403956 \beta_{3} - 2429388 \beta_{4} + 20022606 \beta_{5} + 10618650 \beta_{7} ) q^{42} + ( 2464856493413 \beta_{1} + 10380791911 \beta_{2} - 21680494 \beta_{3} + 75693596 \beta_{4} + 5514190 \beta_{5} - 16166304 \beta_{7} ) q^{43} + ( 61139913108444 + 238724352 \beta_{1} - 79574784 \beta_{2} + 130980132 \beta_{3} + 547642560 \beta_{4} + 70193856 \beta_{6} + 79574784 \beta_{7} ) q^{44} + ( 1274769651407625 - 1296847056003 \beta_{1} - 19698928824 \beta_{2} - 2146282101 \beta_{3} + 31707513 \beta_{4} + 1615500 \beta_{5} + 4491000 \beta_{6} + 23040225 \beta_{7} ) q^{45} + ( -1714608789353993 + 521193600 \beta_{1} - 173731200 \beta_{2} - 4441581761 \beta_{3} + 1140666639 \beta_{4} + 98279439 \beta_{6} + 173731200 \beta_{7} ) q^{46} + ( 1726140963416 \beta_{1} + 87173371058 \beta_{2} + 117420763 \beta_{3} - 351324142 \beta_{4} - 59179455 \beta_{5} + 58241308 \beta_{7} ) q^{47} + ( -3202508585144 \beta_{1} - 117190817728 \beta_{2} + 230982264 \beta_{3} - 981685368 \beta_{4} + 28878156 \beta_{5} + 259860420 \beta_{7} ) q^{48} + ( -1391157251477093 - 31086945 \beta_{1} + 10362315 \beta_{2} + 16421004921 \beta_{3} - 407713494 \beta_{4} - 345539604 \beta_{6} - 10362315 \beta_{7} ) q^{49} + ( 6296880458343750 + 6010531514125 \beta_{1} - 244993312000 \beta_{2} - 13948494250 \beta_{3} - 2844750 \beta_{4} - 24712500 \beta_{5} - 237393750 \beta_{6} + 293882500 \beta_{7} ) q^{50} + ( -5858383926425568 - 660007116 \beta_{1} + 220002372 \beta_{2} - 8318186292 \beta_{3} - 1270440056 \beta_{4} + 49574176 \beta_{6} - 220002372 \beta_{7} ) q^{51} + ( 4326157893924 \beta_{1} + 347210814816 \beta_{2} + 114608028 \beta_{3} - 582905052 \beta_{4} + 62236470 \beta_{5} + 176844498 \beta_{7} ) q^{52} + ( -20425881248321 \beta_{1} - 78489932271 \beta_{2} - 481245867 \beta_{3} + 2131758852 \beta_{4} - 103387692 \beta_{5} - 584633559 \beta_{7} ) q^{53} + ( 10758581354273130 + 98433792 \beta_{1} - 32811264 \beta_{2} - 20617744806 \beta_{3} + 1085002650 \beta_{4} + 888135066 \beta_{6} + 32811264 \beta_{7} ) q^{54} + ( 8727653290765500 + 21756865543598 \beta_{1} + 54448735834 \beta_{2} + 35503786466 \beta_{3} - 5023108 \beta_{4} + 90677500 \beta_{5} + 927217500 \beta_{6} - 1147538250 \beta_{7} ) q^{55} + ( -37978972354113420 - 11707584 \beta_{1} + 3902528 \beta_{2} + 45906132524 \beta_{3} - 1087794960 \beta_{4} - 1064379792 \beta_{6} - 3902528 \beta_{7} ) q^{56} + ( 15945938344168 \beta_{1} + 510245356232 \beta_{2} - 1758624876 \beta_{3} + 6579789984 \beta_{4} + 227354760 \beta_{5} - 1531270116 \beta_{7} ) q^{57} + ( -63855575922794 \beta_{1} + 289643554688 \beta_{2} + 100114672 \beta_{3} - 510352880 \beta_{4} + 54947096 \beta_{5} + 155061768 \beta_{7} ) q^{58} + ( 32456004097251540 - 4537909902 \beta_{1} + 1512636634 \beta_{2} - 26947931882 \beta_{3} - 9811139904 \beta_{4} - 735320100 \beta_{6} - 1512636634 \beta_{7} ) q^{59} + ( 39763827581980500 + 47228005100178 \beta_{1} + 279807187824 \beta_{2} + 4897187226 \beta_{3} - 229793358 \beta_{4} - 97809525 \beta_{5} - 1064380800 \beta_{6} + 1061297145 \beta_{7} ) q^{60} + ( -72861695344153258 - 3751274601 \beta_{1} + 1250424867 \beta_{2} - 32361154967 \beta_{3} - 5621672442 \beta_{4} + 1880876760 \beta_{6} - 1250424867 \beta_{7} ) q^{61} + ( 56785763851536 \beta_{1} - 3229195610752 \beta_{2} + 3705999184 \beta_{3} - 12675592528 \beta_{4} - 1074202104 \beta_{5} + 2631797080 \beta_{7} ) q^{62} + ( -72389943308574 \beta_{1} + 453052943784 \beta_{2} - 595215189 \beta_{3} + 1700024274 \beta_{4} + 340418241 \beta_{5} - 254796948 \beta_{7} ) q^{63} + ( 65676968409513712 + 11369952000 \beta_{1} - 3789984000 \beta_{2} - 114817553776 \beta_{3} + 20415922752 \beta_{4} - 2323981248 \beta_{6} + 3789984000 \beta_{7} ) q^{64} + ( 93443165586129000 + 18419130686859 \beta_{1} + 2323256757697 \beta_{2} + 65238367453 \beta_{3} - 2107136124 \beta_{4} - 315571200 \beta_{5} - 2159973900 \beta_{6} + 842006285 \beta_{7} ) q^{65} + ( -233883448499997036 + 15883506432 \beta_{1} - 5294502144 \beta_{2} + 40443148404 \beta_{3} + 33927563892 \beta_{4} + 2160551028 \beta_{6} + 5294502144 \beta_{7} ) q^{66} + ( -57036762632819 \beta_{1} - 2035417736185 \beta_{2} - 1527263570 \beta_{3} + 2874188932 \beta_{4} + 1617432674 \beta_{5} + 90169104 \beta_{7} ) q^{67} + ( 103281295677184 \beta_{1} + 1320789903104 \beta_{2} + 5846622592 \beta_{3} - 22651065472 \beta_{4} - 367712448 \beta_{5} + 5478910144 \beta_{7} ) q^{68} + ( 37768037815476876 - 448175997 \beta_{1} + 149391999 \beta_{2} + 427947102429 \beta_{3} + 7162882510 \beta_{4} + 8059234504 \beta_{6} - 149391999 \beta_{7} ) q^{69} + ( 329795133229954125 - 168428687008484 \beta_{1} - 1021714622272 \beta_{2} - 497377703003 \beta_{3} + 12924738289 \beta_{4} + 1291599500 \beta_{5} + 7900379625 \beta_{6} + 1798800900 \beta_{7} ) q^{70} + ( -327125950075365768 - 7083580608 \beta_{1} + 2361193536 \beta_{2} - 477858165936 \beta_{3} - 23998997736 \beta_{4} - 9831836520 \beta_{6} - 2361193536 \beta_{7} ) q^{71} + ( -203628270856578 \beta_{1} + 1562410944240 \beta_{2} - 1889133102 \beta_{3} + 5549516334 \beta_{4} + 1003508037 \beta_{5} - 885625065 \beta_{7} ) q^{72} + ( 534468205048404 \beta_{1} - 5032854950484 \beta_{2} - 6285488880 \beta_{3} + 28639789248 \beta_{4} - 1748916864 \beta_{5} - 8034405744 \beta_{7} ) q^{73} + ( 230948876702776914 - 17458117056 \beta_{1} + 5819372352 \beta_{2} + 163265778282 \beta_{3} - 44857574778 \beta_{4} - 9941340666 \beta_{6} - 5819372352 \beta_{7} ) q^{74} + ( 359741553526125000 - 379787592965125 \beta_{1} - 1286440167875 \beta_{2} + 455557671000 \beta_{3} - 19987008000 \beta_{4} - 1673325000 \beta_{5} - 7597275000 \beta_{6} - 8139690000 \beta_{7} ) q^{75} + ( -936811787482284540 - 44957415168 \beta_{1} + 14985805056 \beta_{2} + 1026419546108 \beta_{3} - 86497304640 \beta_{4} + 3417525696 \beta_{6} - 14985805056 \beta_{7} ) q^{76} + ( -15669136821204 \beta_{1} + 18112960868940 \beta_{2} + 1445291256 \beta_{3} + 9800845608 \beta_{4} - 7791005316 \beta_{5} - 6345714060 \beta_{7} ) q^{77} + ( 799459214978904 \beta_{1} + 9240809916288 \beta_{2} - 16807646160 \beta_{3} + 60343866576 \beta_{4} + 3443359032 \beta_{5} - 13364287128 \beta_{7} ) q^{78} + ( 491886511388732720 - 25177492548 \beta_{1} + 8392497516 \beta_{2} - 340189147964 \beta_{3} - 47964744360 \beta_{4} + 2390240736 \beta_{6} - 8392497516 \beta_{7} ) q^{79} + ( 683142438232411500 - 438784422643416 \beta_{1} - 20440370172928 \beta_{2} + 83354298828 \beta_{3} - 7276592264 \beta_{4} - 124714500 \beta_{5} + 2933106000 \beta_{6} - 9712193900 \beta_{7} ) q^{80} + ( -1041022161707588019 + 38600241147 \beta_{1} - 12866747049 \beta_{2} - 931831506675 \beta_{3} + 89768409354 \beta_{4} + 12567927060 \beta_{6} + 12866747049 \beta_{7} ) q^{81} + ( -765081886050418 \beta_{1} + 16995994365856 \beta_{2} - 26891714644 \beta_{3} + 87359991764 \beta_{4} + 10103433406 \beta_{5} - 16788281238 \beta_{7} ) q^{82} + ( 306559976658421 \beta_{1} - 36093084535137 \beta_{2} + 27030707790 \beta_{3} - 120710308860 \beta_{4} + 6293738850 \beta_{5} + 33324446640 \beta_{7} ) q^{83} + ( 960484467353016204 + 25434207360 \beta_{1} - 8478069120 \beta_{2} - 2373801259212 \beta_{3} + 63428405472 \beta_{4} + 12559990752 \beta_{6} + 8478069120 \beta_{7} ) q^{84} + ( 469048376152448000 - 379496427984842 \beta_{1} + 14320364181914 \beta_{2} + 1558416634886 \beta_{3} + 37596403312 \beta_{4} + 3285391100 \beta_{5} - 7960885800 \beta_{6} + 35867344770 \beta_{7} ) q^{85} + ( -1794820384893261903 + 49543321344 \beta_{1} - 16514440448 \beta_{2} + 1753663792369 \beta_{3} + 120150157485 \beta_{4} + 21063514797 \beta_{6} + 16514440448 \beta_{7} ) q^{86} + ( 491371999659330 \beta_{1} - 20813661384162 \beta_{2} + 74686193928 \beta_{3} - 307378557504 \beta_{4} + 4316890896 \beta_{5} + 79003084824 \beta_{7} ) q^{87} + ( -277176466649080 \beta_{1} + 53767540177984 \beta_{2} + 11043443384 \beta_{3} + 4565300552 \beta_{4} - 24369537044 \beta_{5} - 13326093660 \beta_{7} ) q^{88} + ( 265017637704615210 + 268064209044 \beta_{1} - 89354736348 \beta_{2} - 609354529668 \beta_{3} + 501606971904 \beta_{4} - 34521446184 \beta_{6} + 89354736348 \beta_{7} ) q^{89} + ( 948699590768283375 + 2161552120645779 \beta_{1} - 5971187951568 \beta_{2} - 1434879079407 \beta_{3} - 44336858169 \beta_{4} - 8952416325 \beta_{5} - 24299107275 \beta_{6} + 4871189385 \beta_{7} ) q^{90} + ( -419829777496727208 + 94329252576 \beta_{1} - 31443084192 \beta_{2} - 2730176310576 \beta_{3} + 133647650760 \beta_{4} - 55010854392 \beta_{6} + 31443084192 \beta_{7} ) q^{91} + ( 1280441541903298 \beta_{1} - 107714138908432 \beta_{2} + 910227646 \beta_{3} + 43415024354 \beta_{4} - 23527967469 \beta_{5} - 22617739823 \beta_{7} ) q^{92} + ( -4974358274605976 \beta_{1} - 67700351268472 \beta_{2} + 58042608840 \beta_{3} - 241568252496 \beta_{4} + 4698908568 \beta_{5} + 62741517408 \beta_{7} ) q^{93} + ( -1281471456872213741 - 465320297088 \beta_{1} + 155106765696 \beta_{2} + 8870246906219 \beta_{3} - 902907459165 \beta_{4} + 27733135011 \beta_{6} - 155106765696 \beta_{7} ) q^{94} + ( 763258223735602500 + 1666290159168790 \beta_{1} + 125665768006570 \beta_{2} - 5966078530070 \beta_{3} + 240271016460 \beta_{4} + 24990238500 \beta_{5} + 148553584500 \beta_{6} + 28638966950 \beta_{7} ) q^{95} + ( 1958770190435419152 - 318585864960 \beta_{1} + 106195288320 \beta_{2} - 2040191292816 \beta_{3} - 775019314752 \beta_{4} - 137847584832 \beta_{6} - 106195288320 \beta_{7} ) q^{96} + ( 4390030698345722 \beta_{1} - 2475407522666 \beta_{2} - 145344523462 \beta_{3} + 554882220632 \beta_{4} + 13247936608 \beta_{5} - 132096586854 \beta_{7} ) q^{97} + ( -8280638985291473 \beta_{1} + 228127450721184 \beta_{2} - 216010733556 \beta_{3} + 724198773876 \beta_{4} + 69922080174 \beta_{5} - 146088653382 \beta_{7} ) q^{98} + ( -2032118078154385284 - 6369181794 \beta_{1} + 2123060598 \beta_{2} + 3717187558074 \beta_{3} + 84002658480 \beta_{4} + 96741022068 \beta_{6} - 2123060598 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 1620744q^{4} + 147000q^{5} + 3365736q^{6} + 345358584q^{9} + O(q^{10}) \) \( 8q - 1620744q^{4} + 147000q^{5} + 3365736q^{6} + 345358584q^{9} - 610691000q^{10} - 3379575264q^{11} + 177250591032q^{14} - 242324628000q^{15} - 312730276832q^{16} + 4547188380640q^{19} - 4180429431000q^{20} - 2983154334624q^{21} - 6176642779680q^{24} + 17715709625000q^{25} + 15909228128496q^{26} - 188222300345040q^{29} + 148501482939000q^{30} + 72115006686976q^{31} + 378440221985792q^{34} - 299115755916000q^{35} - 964020253238712q^{36} + 3011653788049728q^{39} - 213611427940000q^{40} - 3314269428857904q^{41} + 489119304867552q^{44} + 10198157211261000q^{45} - 13716870314831944q^{46} - 11129258011816744q^{49} + 50375043666750000q^{50} - 46867071411404544q^{51} + 86068650834185040q^{54} + 69821226326124000q^{55} - 303831778832907360q^{56} + 259648032778012320q^{59} + 318110620655844000q^{60} - 582893562753226064q^{61} + 525415747276109696q^{64} + 747545324689032000q^{65} - 1871067587999976288q^{66} + 302144302523815008q^{69} + 2638361065839633000q^{70} - 2617007600602926144q^{71} + 1847591013622215312q^{74} + 2877932428209000000q^{75} - 7494494299858276320q^{76} + 3935092091109861760q^{79} + 5465139505859292000q^{80} - 8328177293660704152q^{81} + 7683875738824129632q^{84} + 3752387009219584000q^{85} - 14358563079146095224q^{86} + 2120141101636921680q^{89} + 7589596726146267000q^{90} - 3358638219973817664q^{91} - 10251771654977709928q^{94} + 6106065789884820000q^{95} + 15670161523483353216q^{96} - 16256944625235082272q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 726881 x^{6} + 160513523376 x^{4} + 10607307647230976 x^{2} + 32429098232548950016\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 173 \nu^{7} + 111026445 \nu^{5} + 20083149107568 \nu^{3} + 947168012636830720 \nu \)\()/ 1505285197258752 \)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 726881 \)
\(\beta_{4}\)\(=\)\((\)\(-24489 \nu^{7} + 1840496 \nu^{6} - 19390777353 \nu^{5} + 1808281798512 \nu^{4} - 4679503473675312 \nu^{3} + 431776354413212928 \nu^{2} - 324881951467425186816 \nu + 15034918824126539505664\)\()/ 3763212993146880 \)
\(\beta_{5}\)\(=\)\((\)\(-254739 \nu^{7} - 7361984 \nu^{6} - 173282696883 \nu^{5} - 7233127194048 \nu^{4} - 14399254905804432 \nu^{3} - 1696999713707676672 \nu^{2} + 3822609063572486667264 \nu - 54668859249162963460096\)\()/ 7526425986293760 \)
\(\beta_{6}\)\(=\)\((\)\(24489 \nu^{7} - 75460336 \nu^{6} + 19390777353 \nu^{5} - 40236733980912 \nu^{4} + 4679503473675312 \nu^{3} - 4841151702138678528 \nu^{2} + 324881951467425186816 \nu - 36463112405104365338624\)\()/ 3763212993146880 \)
\(\beta_{7}\)\(=\)\((\)\(294733 \nu^{7} + 14723968 \nu^{6} + 233244460461 \nu^{5} + 14466254388096 \nu^{4} + 56254457429641584 \nu^{3} + 3424105131360528384 \nu^{2} + 3903274099116368632832 \nu + 114808534545669121482752\)\()/ 7526425986293760 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 726881\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} + 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 656 \beta_{2} - 1141198 \beta_{1}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(1776 \beta_{7} + 444 \beta_{6} + 11100 \beta_{4} - 377593 \beta_{3} - 1776 \beta_{2} + 5328 \beta_{1} + 207328941409\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(319487 \beta_{7} - 1499523 \beta_{5} - 4276994 \beta_{4} + 1819010 \beta_{3} - 792634640 \beta_{2} + 362708638734 \beta_{1}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-927045360 \beta_{7} - 436228668 \beta_{6} - 5998500828 \beta_{4} + 137204036265 \beta_{3} + 927045360 \beta_{2} - 2781136080 \beta_{1} - 65851160816938001\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-321125173071 \beta_{7} + 614088211347 \beta_{5} + 2512677114978 \beta_{4} - 935213384418 \beta_{3} + 502145329633296 \beta_{2} - 122196954824937902 \beta_{1}\)\()/8\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
607.943i
490.322i
337.134i
56.6657i
56.6657i
337.134i
490.322i
607.943i
1215.89i 18754.1i −954092. 4.05746e6 + 1.61570e6i 2.28029e7 1.79878e8i 5.22593e8i 8.10544e8 1.96451e9 4.93341e9i
4.2 980.644i 42067.9i −437374. −2.12371e6 3.81619e6i −4.12537e7 4.16374e7i 8.52313e7i −6.07450e8 −3.74233e9 + 2.08261e9i
4.3 674.268i 35433.1i 69650.8 −4.08230e6 + 1.55186e6i 2.38914e7 1.27830e8i 4.00474e8i −9.32466e7 1.04637e9 + 2.75257e9i
4.4 113.331i 33157.6i 511444. 2.22206e6 + 3.75978e6i −3.75780e6 2.70208e7i 1.17381e8i 6.28322e7 4.26102e8 2.51829e8i
4.5 113.331i 33157.6i 511444. 2.22206e6 3.75978e6i −3.75780e6 2.70208e7i 1.17381e8i 6.28322e7 4.26102e8 + 2.51829e8i
4.6 674.268i 35433.1i 69650.8 −4.08230e6 1.55186e6i 2.38914e7 1.27830e8i 4.00474e8i −9.32466e7 1.04637e9 2.75257e9i
4.7 980.644i 42067.9i −437374. −2.12371e6 + 3.81619e6i −4.12537e7 4.16374e7i 8.52313e7i −6.07450e8 −3.74233e9 2.08261e9i
4.8 1215.89i 18754.1i −954092. 4.05746e6 1.61570e6i 2.28029e7 1.79878e8i 5.22593e8i 8.10544e8 1.96451e9 + 4.93341e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{20}^{\mathrm{new}}(5, [\chi])\).