Properties

Label 4.19.b.a
Level 4
Weight 19
Character orbit 4.b
Analytic conductor 8.215
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(8.21544550839\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{56}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( 11 - \beta_{1} ) q^{2} \) \( -\beta_{2} q^{3} \) \( + ( 44007 - 9 \beta_{1} - \beta_{2} + \beta_{4} ) q^{4} \) \( + ( 107404 + 412 \beta_{1} - \beta_{4} + \beta_{5} ) q^{5} \) \( + ( 38865 + 9 \beta_{1} + 43 \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{6} \) \( + ( 830 - 1625 \beta_{1} - 212 \beta_{2} + 37 \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{7} \) \( + ( -19247535 - 44966 \beta_{1} - 1024 \beta_{2} + \beta_{3} + 25 \beta_{4} - 52 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{8} \) \( + ( -91144375 - 97680 \beta_{1} + 224 \beta_{2} - 4 \beta_{3} - 1062 \beta_{4} + 2 \beta_{5} - 16 \beta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 11 - \beta_{1} ) q^{2} \) \( -\beta_{2} q^{3} \) \( + ( 44007 - 9 \beta_{1} - \beta_{2} + \beta_{4} ) q^{4} \) \( + ( 107404 + 412 \beta_{1} - \beta_{4} + \beta_{5} ) q^{5} \) \( + ( 38865 + 9 \beta_{1} + 43 \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{6} \) \( + ( 830 - 1625 \beta_{1} - 212 \beta_{2} + 37 \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{7} \) \( + ( -19247535 - 44966 \beta_{1} - 1024 \beta_{2} + \beta_{3} + 25 \beta_{4} - 52 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{8} \) \( + ( -91144375 - 97680 \beta_{1} + 224 \beta_{2} - 4 \beta_{3} - 1062 \beta_{4} + 2 \beta_{5} - 16 \beta_{6} ) q^{9} \) \( + ( -106644262 - 109346 \beta_{1} + 16640 \beta_{2} + 28 \beta_{3} - 308 \beta_{4} + 240 \beta_{5} - 48 \beta_{6} - 32 \beta_{7} ) q^{10} \) \( + ( 144228 - 281582 \beta_{1} - 7027 \beta_{2} - 96 \beta_{3} + 6454 \beta_{4} + 94 \beta_{5} - 94 \beta_{6} + 136 \beta_{7} ) q^{11} \) \( + ( -372929132 - 81696 \beta_{1} - 191448 \beta_{2} + 364 \beta_{3} - 1116 \beta_{4} + 1168 \beta_{5} - 176 \beta_{6} - 144 \beta_{7} ) q^{12} \) \( + ( -829399428 - 5906116 \beta_{1} + 2240 \beta_{2} - 1064 \beta_{3} - 11565 \beta_{4} - 59 \beta_{5} - 160 \beta_{6} ) q^{13} \) \( + ( -416333818 + 106422 \beta_{1} + 847682 \beta_{2} + 2912 \beta_{3} + 6880 \beta_{4} - 5364 \beta_{5} + 128 \beta_{6} + 454 \beta_{7} ) q^{14} \) \( + ( -15257750 + 30453165 \beta_{1} + 33420 \beta_{2} - 7232 \beta_{3} - 65961 \beta_{4} - 757 \beta_{5} + 757 \beta_{6} - 2092 \beta_{7} ) q^{15} \) \( + ( 4400783636 + 19188808 \beta_{1} - 1758144 \beta_{2} + 16020 \beta_{3} + 31028 \beta_{4} - 7184 \beta_{5} + 2480 \beta_{6} + 2384 \beta_{7} ) q^{16} \) \( + ( 26801771634 - 139732656 \beta_{1} - 57120 \beta_{2} - 33796 \beta_{3} + 231506 \beta_{4} + 3978 \beta_{5} + 4080 \beta_{6} ) q^{17} \) \( + ( 24631415323 + 92349543 \beta_{1} + 795136 \beta_{2} + 64120 \beta_{3} + 64728 \beta_{4} + 49888 \beta_{5} + 4768 \beta_{6} - 3648 \beta_{7} ) q^{18} \) \( + ( -198258484 + 396408006 \beta_{1} + 808391 \beta_{2} - 115360 \beta_{3} - 235566 \beta_{4} - 3830 \beta_{5} + 3830 \beta_{6} + 18904 \beta_{7} ) q^{19} \) \( + ( -52781106314 + 106001878 \beta_{1} + 9440550 \beta_{2} + 192192 \beta_{3} + 154522 \beta_{4} - 20224 \beta_{5} - 6912 \beta_{6} - 23808 \beta_{7} ) q^{20} \) \( + ( -95702039696 - 1552780416 \beta_{1} + 274624 \beta_{2} - 296744 \beta_{3} - 1502964 \beta_{4} - 88436 \beta_{5} - 19616 \beta_{6} ) q^{21} \) \( + ( -73339868993 - 8487609 \beta_{1} - 36799163 \beta_{2} + 433472 \beta_{3} + 69184 \beta_{4} - 244322 \beta_{5} - 37120 \beta_{6} + 16975 \beta_{7} ) q^{22} \) \( + ( -870391550 + 1743739865 \beta_{1} - 7363444 \beta_{2} - 572864 \beta_{3} + 2373659 \beta_{4} + 58879 \beta_{5} - 58879 \beta_{6} - 107516 \beta_{7} ) q^{23} \) \( + ( -492755993080 + 361894608 \beta_{1} + 56107776 \beta_{2} + 704648 \beta_{3} + 493128 \beta_{4} + 474208 \beta_{5} - 37408 \beta_{6} + 157216 \beta_{7} ) q^{24} \) \( + ( 67769441755 - 3308521600 \beta_{1} + 230720 \beta_{2} - 751640 \beta_{3} - 2731500 \beta_{4} + 892180 \beta_{5} - 16480 \beta_{6} ) q^{25} \) \( + ( 1538829419002 + 894384558 \beta_{1} + 2442496 \beta_{2} + 638988 \beta_{3} + 4865916 \beta_{4} + 479920 \beta_{5} + 51472 \beta_{6} - 33952 \beta_{7} ) q^{26} \) \( + ( -789149100 + 1569176010 \beta_{1} + 8261274 \beta_{2} - 273504 \beta_{3} - 9390114 \beta_{4} - 185370 \beta_{5} + 185370 \beta_{6} + 365160 \beta_{7} ) q^{27} \) \( + ( -2607645334920 + 277832000 \beta_{1} - 221767952 \beta_{2} - 533624 \beta_{3} + 807640 \beta_{4} - 2449568 \beta_{5} + 225248 \beta_{6} - 698208 \beta_{7} ) q^{28} \) \( + ( 1400584209868 + 8369709052 \beta_{1} - 5166336 \beta_{2} + 1812576 \beta_{3} + 31335047 \beta_{4} - 5166887 \beta_{5} + 369024 \beta_{6} ) q^{29} \) \( + ( 7976597103130 + 406323690 \beta_{1} + 556588190 \beta_{2} - 3801824 \beta_{3} - 32987232 \beta_{4} + 1457716 \beta_{5} + 394624 \beta_{6} - 88614 \beta_{7} ) q^{30} \) \( + ( 10135097264 - 20285879176 \beta_{1} + 173527704 \beta_{2} + 6368320 \beta_{3} - 8476504 \beta_{4} - 200120 \beta_{5} + 200120 \beta_{6} - 439584 \beta_{7} ) q^{31} \) \( + ( -16701494249392 - 5156399840 \beta_{1} - 526666496 \beta_{2} - 9557424 \beta_{3} - 9936688 \beta_{4} + 4583360 \beta_{5} + 42688 \beta_{6} + 1938752 \beta_{7} ) q^{32} \) \( + ( -2726301419488 + 67077490896 \beta_{1} + 8775200 \beta_{2} + 13052900 \beta_{3} - 47170458 \beta_{4} + 18854558 \beta_{5} - 626800 \beta_{6} ) q^{33} \) \( + ( 36899441453710 - 25323284082 \beta_{1} - 270685696 \beta_{2} - 16224936 \beta_{3} + 124718392 \beta_{4} - 11644320 \beta_{5} - 1431264 \beta_{6} + 786624 \beta_{7} ) q^{34} \) \( + ( 36855057160 - 73588962940 \beta_{1} - 1184934820 \beta_{2} + 18854080 \beta_{3} + 137846540 \beta_{4} + 2164060 \beta_{5} - 2164060 \beta_{6} - 2169200 \beta_{7} ) q^{35} \) \( + ( -74471350965985 - 27422977233 \beta_{1} + 1647200375 \beta_{2} - 19074688 \beta_{3} - 33424119 \beta_{4} + 6574592 \beta_{5} - 2524672 \beta_{6} - 2025984 \beta_{7} ) q^{36} \) \( + ( -19886537299364 + 51176592220 \beta_{1} + 33557440 \beta_{2} + 16942904 \beta_{3} - 97681077 \beta_{4} - 43575379 \beta_{5} - 2396960 \beta_{6} ) q^{37} \) \( + ( 103831259447549 + 3691668117 \beta_{1} - 3588192241 \beta_{2} - 10034752 \beta_{3} - 500155200 \beta_{4} + 22551130 \beta_{5} - 834304 \beta_{6} - 1824851 \beta_{7} ) q^{38} \) \( + ( -3705805422 + 7367518593 \beta_{1} + 4311217108 \beta_{2} - 2221056 \beta_{3} - 54406701 \beta_{4} - 2056041 \beta_{5} + 2056041 \beta_{6} + 12205476 \beta_{7} ) q^{39} \) \( + ( -201196283674118 + 45321426596 \beta_{1} + 4429793280 \beta_{2} + 20222426 \beta_{3} + 49040970 \beta_{4} - 44703816 \beta_{5} + 3192984 \beta_{6} - 8026264 \beta_{7} ) q^{40} \) \( + ( 56517846273858 - 157662503232 \beta_{1} - 114985920 \beta_{2} - 45941560 \beta_{3} + 444474836 \beta_{4} + 51660084 \beta_{5} + 8213280 \beta_{6} ) q^{41} \) \( + ( 405856574582480 + 112398389280 \beta_{1} - 1732916224 \beta_{2} + 76066256 \beta_{3} + 1302726672 \beta_{4} + 39349568 \beta_{5} + 10208192 \beta_{6} - 1564032 \beta_{7} ) q^{42} \) \( + ( -226124931400 + 451522490140 \beta_{1} - 12491192927 \beta_{2} - 112812096 \beta_{3} - 799969196 \beta_{4} - 9099260 \beta_{5} + 9099260 \beta_{6} - 22017040 \beta_{7} ) q^{43} \) \( + ( -428678154191700 + 55263239456 \beta_{1} - 6358050728 \beta_{2} + 148823124 \beta_{3} + 216470428 \beta_{4} + 46168176 \beta_{5} + 14424496 \beta_{6} + 41616272 \beta_{7} ) q^{44} \) \( + ( 49706994943676 - 843077909892 \beta_{1} - 67291840 \beta_{2} - 185284120 \beta_{3} + 85576491 \beta_{4} + 46372349 \beta_{5} + 4806560 \beta_{6} ) q^{45} \) \( + ( 457265991665346 + 21859481298 \beta_{1} + 22957070550 \beta_{2} + 211391648 \beta_{3} - 1987443424 \beta_{4} - 254640764 \beta_{5} - 4751488 \beta_{6} + 18412162 \beta_{7} ) q^{46} \) \( + ( -354956213780 + 710552737510 \beta_{1} + 35009178912 \beta_{2} - 221856320 \beta_{3} + 414851794 \beta_{4} + 12553002 \beta_{5} - 12553002 \beta_{6} - 21504168 \beta_{7} ) q^{47} \) \( + ( -839404352865376 + 454693081152 \beta_{1} - 30280449536 \beta_{2} + 213899168 \beta_{3} - 68365152 \beta_{4} + 109314944 \beta_{5} - 33635968 \beta_{6} - 77703552 \beta_{7} ) q^{48} \) \( + ( -280816969565967 - 1169178339776 \beta_{1} + 610417024 \beta_{2} - 165319504 \beta_{3} - 2649031704 \beta_{4} - 393968056 \beta_{5} - 43601216 \beta_{6} ) q^{49} \) \( + ( 867829689628905 - 29727308395 \beta_{1} + 12568744960 \beta_{2} + 90966960 \beta_{3} + 2854354800 \beta_{4} + 265013440 \beta_{5} - 37814720 \beta_{6} - 32241280 \beta_{7} ) q^{50} \) \( + ( 234087231652 - 464693088078 \beta_{1} - 88609444998 \beta_{2} + 35336608 \beta_{3} + 3264085782 \beta_{4} + 34959038 \beta_{5} - 34959038 \beta_{6} + 182707976 \beta_{7} ) q^{51} \) \( + ( -809179166712858 - 1570716626362 \beta_{1} + 11431232534 \beta_{2} - 201735744 \beta_{3} - 241590358 \beta_{4} - 150760192 \beta_{5} - 41477888 \beta_{6} - 1601792 \beta_{7} ) q^{52} \) \( + ( 150825467277404 + 2446020459740 \beta_{1} - 164767680 \beta_{2} + 412305736 \beta_{3} + 179388467 \beta_{4} + 1009679189 \beta_{5} + 11769120 \beta_{6} ) q^{53} \) \( + ( 410571284053938 + 6741212130 \beta_{1} - 69184929978 \beta_{2} - 657209280 \beta_{3} - 2260142784 \beta_{4} + 846412452 \beta_{5} + 12399360 \beta_{6} - 43712814 \beta_{7} ) q^{54} \) \( + ( 1384018062430 - 2771677554505 \beta_{1} + 180425733260 \beta_{2} + 927349248 \beta_{3} - 2374240171 \beta_{4} - 25766287 \beta_{5} + 25766287 \beta_{6} - 288307652 \beta_{7} ) q^{55} \) \( + ( 927693436609072 + 2664748326112 \beta_{1} + 128689869312 \beta_{2} - 1200107216 \beta_{3} + 214434736 \beta_{4} - 599420352 \beta_{5} + 163273536 \beta_{6} + 329555136 \beta_{7} ) q^{56} \) \( + ( 329320081268064 + 6757929973872 \beta_{1} - 1561099680 \beta_{2} + 1416359340 \beta_{3} + 10155167154 \beta_{4} - 1379337894 \beta_{5} + 111507120 \beta_{6} ) q^{57} \) \( + ( -2179265614511350 - 1507106144402 \beta_{1} - 96502015744 \beta_{2} - 1622244932 \beta_{3} - 6962123540 \beta_{4} - 2379598992 \beta_{5} + 135827280 \beta_{6} + 248001760 \beta_{7} ) q^{58} \) \( + ( 2171804976848 - 4358862449752 \beta_{1} - 309143961027 \beta_{2} + 1669289216 \beta_{3} - 12908193160 \beta_{4} - 214844904 \beta_{5} + 214844904 \beta_{6} - 245323872 \beta_{7} ) q^{59} \) \( + ( 4552905293286920 - 7788914591040 \beta_{1} + 57729268240 \beta_{2} - 1620147208 \beta_{3} - 2853595224 \beta_{4} + 832604832 \beta_{5} + 31535648 \beta_{6} - 677872288 \beta_{7} ) q^{60} \) \( + ( 19584100735068 + 6979854447548 \beta_{1} + 37231040 \beta_{2} + 1419556600 \beta_{3} + 747126531 \beta_{4} + 496912309 \beta_{5} - 2659360 \beta_{6} ) q^{61} \) \( + ( -5322427561980424 - 204162860232 \beta_{1} + 80366262056 \beta_{2} - 969649408 \beta_{3} + 25154236160 \beta_{4} + 119283440 \beta_{5} + 93430784 \beta_{6} - 190955144 \beta_{7} ) q^{62} \) \( + ( 1590200739750 - 3160331586405 \beta_{1} + 501689657364 \beta_{2} + 305643072 \beta_{3} + 18588144897 \beta_{4} + 154248045 \beta_{5} - 154248045 \beta_{6} + 1478895180 \beta_{7} ) q^{63} \) \( + ( 9253678242966848 + 17034913942656 \beta_{1} - 331780140032 \beta_{2} + 552056128 \beta_{3} - 3431485632 \beta_{4} + 3655421696 \beta_{5} - 378115328 \beta_{6} + 309757184 \beta_{7} ) q^{64} \) \( + ( -704885829632556 - 7810163082048 \beta_{1} + 1951266240 \beta_{2} - 1677555080 \beta_{3} - 13068979796 \beta_{4} + 2192598156 \beta_{5} - 139376160 \beta_{6} ) q^{65} \) \( + ( -17602254776324216 + 2056188067152 \beta_{1} + 397692075520 \beta_{2} + 3037634824 \beta_{3} - 57353874264 \beta_{4} + 6460652320 \beta_{5} - 714471584 \beta_{6} - 743749056 \beta_{7} ) q^{66} \) \( + ( -5658952202900 + 11360854443350 \beta_{1} - 668400677341 \beta_{2} - 4246117920 \beta_{3} + 38377256450 \beta_{4} + 829865210 \beta_{5} - 829865210 \beta_{6} - 1333067240 \beta_{7} ) q^{67} \) \( + ( 16419707541016622 - 36327235764786 \beta_{1} - 523695744034 \beta_{2} + 5869209472 \beta_{3} + 12781752226 \beta_{4} - 9182815744 \beta_{5} + 42344960 \beta_{6} + 980200960 \beta_{7} ) q^{68} \) \( + ( -3334332556957040 - 41194224764928 \beta_{1} + 9518415424 \beta_{2} - 7258442744 \beta_{3} - 44578950588 \beta_{4} - 7552022012 \beta_{5} - 679886816 \beta_{6} ) q^{69} \) \( + ( -19232452244350500 - 735849623940 \beta_{1} + 314661867060 \beta_{2} + 8325699200 \beta_{3} + 89779710080 \beta_{4} - 6535941320 \beta_{5} - 345756160 \beta_{6} + 1546461340 \beta_{7} ) q^{70} \) \( + ( -19575023727274 + 39075469379571 \beta_{1} + 362461952036 \beta_{2} - 9450370624 \beta_{3} - 79911935223 \beta_{4} - 933928619 \beta_{5} + 933928619 \beta_{6} - 2248653140 \beta_{7} ) q^{71} \) \( + ( 19471413403377241 + 75359072297514 \beta_{1} + 435914030080 \beta_{2} + 9831777673 \beta_{3} + 15492754017 \beta_{4} - 4214628308 \beta_{5} - 37623332 \beta_{6} - 1824090588 \beta_{7} ) q^{72} \) \( + ( 3953987020103122 - 34348076057936 \beta_{1} - 7396837280 \beta_{2} - 9214838164 \beta_{3} + 8432281890 \beta_{4} + 17223684266 \beta_{5} + 528345520 \beta_{6} ) q^{73} \) \( + ( -13628361803292566 + 19248114037310 \beta_{1} - 539108037376 \beta_{2} + 8377317228 \beta_{3} - 48642220452 \beta_{4} - 3056278480 \beta_{5} + 2820294032 \beta_{6} + 857493088 \beta_{7} ) q^{74} \) \( + ( -12218415973560 + 24402760245540 \beta_{1} + 697979620845 \beta_{2} - 6760918080 \beta_{3} - 43358053140 \beta_{4} - 1056117060 \beta_{5} + 1056117060 \beta_{6} + 4637667600 \beta_{7} ) q^{75} \) \( + ( 19063715874555140 - 102889718694048 \beta_{1} + 1413852178184 \beta_{2} + 3001075196 \beta_{3} - 13958212140 \beta_{4} + 35854112080 \beta_{5} + 952584464 \beta_{6} + 1532305072 \beta_{7} ) q^{76} \) \( + ( 22597166670141520 + 8700638993920 \beta_{1} - 51043634880 \beta_{2} - 178487960 \beta_{3} + 261168189300 \beta_{4} - 20712398540 \beta_{5} + 3645973920 \beta_{6} ) q^{77} \) \( + ( 1760812564544682 - 342213149574 \beta_{1} - 2804913296818 \beta_{2} - 4745022048 \beta_{3} - 24676279776 \beta_{4} + 5499503188 \beta_{5} - 645379200 \beta_{6} - 4908311830 \beta_{7} ) q^{78} \) \( + ( 23559674457084 - 47041367624306 \beta_{1} - 2211792569896 \beta_{2} + 11779375360 \beta_{3} + 86170108426 \beta_{4} + 1078518210 \beta_{5} - 1078518210 \beta_{6} + 1433520376 \beta_{7} ) q^{79} \) \( + ( -19136841504616696 + 200628173823312 \beta_{1} + 947571912320 \beta_{2} - 16719073784 \beta_{3} - 22842754488 \beta_{4} - 24379405728 \beta_{5} + 2659495904 \beta_{6} - 2663481824 \beta_{7} ) q^{80} \) \( + ( -32603108658804111 + 89292668453136 \beta_{1} + 51994317984 \beta_{2} + 23256341556 \beta_{3} - 213377692770 \beta_{4} - 8482036170 \beta_{5} - 3713879856 \beta_{6} ) q^{81} \) \( + ( 41920649389672054 - 54765012720306 \beta_{1} + 271953228800 \beta_{2} - 31439490768 \beta_{3} + 147132687088 \beta_{4} - 12964188480 \beta_{5} - 4976521152 \beta_{6} + 186652032 \beta_{7} ) q^{82} \) \( + ( 54003124111680 - 108130665729440 \beta_{1} + 4989695541347 \beta_{2} + 35206512768 \beta_{3} - 81115932512 \beta_{4} - 1382695520 \beta_{5} + 1382695520 \beta_{6} - 5329669760 \beta_{7} ) q^{83} \) \( + ( -106065703846391936 - 410367006685056 \beta_{1} - 62632727936 \beta_{2} - 39658137344 \beta_{3} - 25135749504 \beta_{4} - 52261016576 \beta_{5} - 7248536576 \beta_{6} + 4461511680 \beta_{7} ) q^{84} \) \( + ( -25688971261557944 + 212350449936888 \beta_{1} + 108036882240 \beta_{2} + 47814913320 \beta_{3} - 514859038974 \beta_{4} + 53357221734 \beta_{5} - 7716920160 \beta_{6} ) q^{85} \) \( + ( 118776406997135447 + 5820306821727 \beta_{1} + 6639789172877 \beta_{2} - 44722238080 \beta_{3} - 503652185216 \beta_{4} + 45053846094 \beta_{5} + 4443046400 \beta_{6} + 11749524007 \beta_{7} ) q^{86} \) \( + ( 91325252283610 - 182191812573915 \beta_{1} - 9897476037396 \beta_{2} + 43437763264 \beta_{3} + 496747660095 \beta_{4} + 8592190163 \beta_{5} - 8592190163 \beta_{6} - 11802283852 \beta_{7} ) q^{87} \) \( + ( -171337192479558536 + 420798938934832 \beta_{1} - 8072665903872 \beta_{2} - 44566595080 \beta_{3} + 9583542328 \beta_{4} + 60818153888 \beta_{5} - 6244685792 \beta_{6} + 3605709792 \beta_{7} ) q^{88} \) \( + ( 20163314821388082 + 174377798622384 \beta_{1} - 148870386336 \beta_{2} + 29305132556 \beta_{3} + 776056236146 \beta_{4} - 44933568006 \beta_{5} + 10633599024 \beta_{6} ) q^{89} \) \( + ( 221451914368033882 - 40548225829074 \beta_{1} - 221078946560 \beta_{2} - 17947040468 \beta_{3} + 729932998428 \beta_{4} - 3713293520 \beta_{5} - 3687066992 \beta_{6} - 407245728 \beta_{7} ) q^{90} \) \( + ( 33317683943640 - 66469471258900 \beta_{1} + 12224879526700 \beta_{2} + 13754737472 \beta_{3} + 159786893636 \beta_{4} + 547863732 \beta_{5} - 547863732 \beta_{6} + 18768744752 \beta_{7} ) q^{91} \) \( + ( -148965216562381016 - 464448124854848 \beta_{1} - 6770802314160 \beta_{2} + 15389260504 \beta_{3} + 28600746824 \beta_{4} + 39570630944 \beta_{5} + 20901043872 \beta_{6} - 23017734432 \beta_{7} ) q^{92} \) \( + ( 83596656587789696 - 153463795585152 \beta_{1} - 28063517440 \beta_{2} - 38795828320 \beta_{3} + 39830437296 \beta_{4} + 53673173744 \beta_{5} + 2004536960 \beta_{6} ) q^{93} \) \( + ( 184836780942624404 + 7946668961460 \beta_{1} + 3407801422556 \beta_{2} + 45511187904 \beta_{3} - 832693655360 \beta_{4} - 126656158680 \beta_{5} - 1149168384 \beta_{6} - 32689048428 \beta_{7} ) q^{94} \) \( + ( -229283340110630 + 456692718997125 \beta_{1} - 8753625926300 \beta_{2} - 92129717504 \beta_{3} - 1935833352417 \beta_{4} - 33921593869 \beta_{5} + 33921593869 \beta_{6} + 37585598516 \beta_{7} ) q^{95} \) \( + ( -253961104369492352 + 832396116181248 \beta_{1} + 20132139788288 \beta_{2} + 116008971904 \beta_{3} - 223112396160 \beta_{4} + 12532305408 \beta_{5} + 2384774656 \beta_{6} + 29201144320 \beta_{7} ) q^{96} \) \( + ( -151545495406221806 - 653457354805616 \beta_{1} + 55987358560 \beta_{2} - 113144315956 \beta_{3} - 152420362614 \beta_{4} - 224664357982 \beta_{5} - 3999097040 \beta_{6} ) q^{97} \) \( + ( 303323828938056059 + 292836042944271 \beta_{1} - 5112013993984 \beta_{2} + 163548163296 \beta_{3} + 931910570592 \beta_{4} + 40088221568 \beta_{5} + 32165236352 \beta_{6} + 2840305408 \beta_{7} ) q^{98} \) \( + ( -299966892248208 + 600524689345272 \beta_{1} + 8804651775177 \beta_{2} - 182956684800 \beta_{3} + 451799256168 \beta_{4} + 18323724360 \beta_{5} - 18323724360 \beta_{6} - 80498540832 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 84q^{2} \) \(\mathstrut +\mathstrut 352016q^{4} \) \(\mathstrut +\mathstrut 860880q^{5} \) \(\mathstrut +\mathstrut 310944q^{6} \) \(\mathstrut -\mathstrut 154160064q^{8} \) \(\mathstrut -\mathstrut 729541560q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 84q^{2} \) \(\mathstrut +\mathstrut 352016q^{4} \) \(\mathstrut +\mathstrut 860880q^{5} \) \(\mathstrut +\mathstrut 310944q^{6} \) \(\mathstrut -\mathstrut 154160064q^{8} \) \(\mathstrut -\mathstrut 729541560q^{9} \) \(\mathstrut -\mathstrut 853591160q^{10} \) \(\mathstrut -\mathstrut 2983758720q^{12} \) \(\mathstrut -\mathstrut 6658778288q^{13} \) \(\mathstrut -\mathstrut 3330240576q^{14} \) \(\mathstrut +\mathstrut 35282993408q^{16} \) \(\mathstrut +\mathstrut 213854181648q^{17} \) \(\mathstrut +\mathstrut 197420552436q^{18} \) \(\mathstrut -\mathstrut 421824543840q^{20} \) \(\mathstrut -\mathstrut 771822339072q^{21} \) \(\mathstrut -\mathstrut 586750684320q^{22} \) \(\mathstrut -\mathstrut 3940602195456q^{24} \) \(\mathstrut +\mathstrut 528925732440q^{25} \) \(\mathstrut +\mathstrut 12314194404552q^{26} \) \(\mathstrut -\mathstrut 20860043224320q^{28} \) \(\mathstrut +\mathstrut 11238056568912q^{29} \) \(\mathstrut +\mathstrut 63814514963520q^{30} \) \(\mathstrut -\mathstrut 133632603995136q^{32} \) \(\mathstrut -\mathstrut 21541938424320q^{33} \) \(\mathstrut +\mathstrut 295093712425768q^{34} \) \(\mathstrut -\mathstrut 595880470532208q^{36} \) \(\mathstrut -\mathstrut 158886968816432q^{37} \) \(\mathstrut +\mathstrut 830666716492320q^{38} \) \(\mathstrut -\mathstrut 1609388875268480q^{40} \) \(\mathstrut +\mathstrut 451509984725136q^{41} \) \(\mathstrut +\mathstrut 3247297173265920q^{42} \) \(\mathstrut -\mathstrut 3429204744604800q^{44} \) \(\mathstrut +\mathstrut 394282398204240q^{45} \) \(\mathstrut +\mathstrut 3658225092496704q^{46} \) \(\mathstrut -\mathstrut 6713415182530560q^{48} \) \(\mathstrut -\mathstrut 2251201133570680q^{49} \) \(\mathstrut +\mathstrut 6942506471898780q^{50} \) \(\mathstrut -\mathstrut 6479715597253472q^{52} \) \(\mathstrut +\mathstrut 1216384760086992q^{53} \) \(\mathstrut +\mathstrut 3284600487812928q^{54} \) \(\mathstrut +\mathstrut 7432202560564224q^{56} \) \(\mathstrut +\mathstrut 2661563378188800q^{57} \) \(\mathstrut -\mathstrut 17440122911455928q^{58} \) \(\mathstrut +\mathstrut 36392091128935680q^{60} \) \(\mathstrut +\mathstrut 184592915104336q^{61} \) \(\mathstrut -\mathstrut 42580340982416640q^{62} \) \(\mathstrut +\mathstrut 74097564160495616q^{64} \) \(\mathstrut -\mathstrut 5670291051587040q^{65} \) \(\mathstrut -\mathstrut 140809597617788160q^{66} \) \(\mathstrut +\mathstrut 131212356714741792q^{68} \) \(\mathstrut -\mathstrut 26839260584143872q^{69} \) \(\mathstrut -\mathstrut 153862868594448000q^{70} \) \(\mathstrut +\mathstrut 156072744876684864q^{72} \) \(\mathstrut +\mathstrut 31494366486758032q^{73} \) \(\mathstrut -\mathstrut 108949653815722872q^{74} \) \(\mathstrut +\mathstrut 152098090223483520q^{76} \) \(\mathstrut +\mathstrut 180811187963888640q^{77} \) \(\mathstrut +\mathstrut 14085206442507840q^{78} \) \(\mathstrut -\mathstrut 152292076037383680q^{80} \) \(\mathstrut -\mathstrut 260466732987857784q^{81} \) \(\mathstrut +\mathstrut 335145451981844968q^{82} \) \(\mathstrut -\mathstrut 850166994683553792q^{84} \) \(\mathstrut -\mathstrut 204660361893474400q^{85} \) \(\mathstrut +\mathstrut 950236163482864224q^{86} \) \(\mathstrut -\mathstrut 1369014843355476480q^{88} \) \(\mathstrut +\mathstrut 162001265029847952q^{89} \) \(\mathstrut +\mathstrut 1771450132254688200q^{90} \) \(\mathstrut -\mathstrut 1193579560451823360q^{92} \) \(\mathstrut +\mathstrut 668158876340367360q^{93} \) \(\mathstrut +\mathstrut 1478730179820369024q^{94} \) \(\mathstrut -\mathstrut 2028358051400441856q^{96} \) \(\mathstrut -\mathstrut 1214976752903766512q^{97} \) \(\mathstrut +\mathstrut 2427758859173433876q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut -\mathstrut \) \(10969\) \(x^{6}\mathstrut -\mathstrut \) \(835887\) \(x^{5}\mathstrut +\mathstrut \) \(20786973\) \(x^{4}\mathstrut -\mathstrut \) \(17082875145\) \(x^{3}\mathstrut -\mathstrut \) \(2740871824195\) \(x^{2}\mathstrut +\mathstrut \) \(109263554839035\) \(x\mathstrut +\mathstrut \) \(71754930984829630\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + 10969 \nu^{5} + 868794 \nu^{4} - 18180591 \nu^{3} + 17028333372 \nu^{2} + 2791956824311 \nu - 88793056460566 \)\()/\)\(1099511627776\)
\(\beta_{2}\)\(=\)\((\)\(349\) \(\nu^{7}\mathstrut +\mathstrut \) \(53248\) \(\nu^{6}\mathstrut +\mathstrut \) \(525867\) \(\nu^{5}\mathstrut +\mathstrut \) \(212100462\) \(\nu^{4}\mathstrut -\mathstrut \) \(9178232109\) \(\nu^{3}\mathstrut -\mathstrut \) \(5640029463756\) \(\nu^{2}\mathstrut -\mathstrut \) \(1862342586261691\) \(\nu\mathstrut -\mathstrut \) \(183524160977116930\)\()/\)\(5772436045824\)
\(\beta_{3}\)\(=\)\((\)\(11435\) \(\nu^{7}\mathstrut +\mathstrut \) \(1163264\) \(\nu^{6}\mathstrut -\mathstrut \) \(138717939\) \(\nu^{5}\mathstrut -\mathstrut \) \(27079663422\) \(\nu^{4}\mathstrut -\mathstrut \) \(970946160987\) \(\nu^{3}\mathstrut -\mathstrut \) \(174632082951444\) \(\nu^{2}\mathstrut +\mathstrut \) \(326556690092287747\) \(\nu\mathstrut -\mathstrut \) \(1178329490020766798\)\()/\)\(23089744183296\)
\(\beta_{4}\)\(=\)\((\)\(2971\) \(\nu^{7}\mathstrut -\mathstrut \) \(1163264\) \(\nu^{6}\mathstrut -\mathstrut \) \(19301475\) \(\nu^{5}\mathstrut +\mathstrut \) \(14563817058\) \(\nu^{4}\mathstrut +\mathstrut \) \(1232855754933\) \(\nu^{3}\mathstrut -\mathstrut \) \(70678087605588\) \(\nu^{2}\mathstrut +\mathstrut \) \(11524748595809651\) \(\nu\mathstrut +\mathstrut \) \(2323538655384380498\)\()/\)\(23089744183296\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(27353\) \(\nu^{7}\mathstrut +\mathstrut \) \(3555328\) \(\nu^{6}\mathstrut -\mathstrut \) \(477828111\) \(\nu^{5}\mathstrut -\mathstrut \) \(15638480214\) \(\nu^{4}\mathstrut +\mathstrut \) \(18432717602409\) \(\nu^{3}\mathstrut +\mathstrut \) \(162963069743580\) \(\nu^{2}\mathstrut +\mathstrut \) \(14732116446484319\) \(\nu\mathstrut -\mathstrut \) \(3111102958052283814\)\()/\)\(23089744183296\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(3817\) \(\nu^{7}\mathstrut -\mathstrut \) \(3604480\) \(\nu^{6}\mathstrut -\mathstrut \) \(1646666367\) \(\nu^{5}\mathstrut -\mathstrut \) \(190583341494\) \(\nu^{4}\mathstrut +\mathstrut \) \(18428744008569\) \(\nu^{3}\mathstrut +\mathstrut \) \(23405771210053020\) \(\nu^{2}\mathstrut -\mathstrut \) \(51067895636057393\) \(\nu\mathstrut -\mathstrut \) \(29061431548193611526\)\()/\)\(23089744183296\)
\(\beta_{7}\)\(=\)\((\)\(683035\) \(\nu^{7}\mathstrut +\mathstrut \) \(75218944\) \(\nu^{6}\mathstrut +\mathstrut \) \(7547727645\) \(\nu^{5}\mathstrut -\mathstrut \) \(422692546206\) \(\nu^{4}\mathstrut +\mathstrut \) \(33370477920309\) \(\nu^{3}\mathstrut -\mathstrut \) \(13315004464282452\) \(\nu^{2}\mathstrut -\mathstrut \) \(3116216988267892237\) \(\nu\mathstrut -\mathstrut \) \(386322136343476179118\)\()/\)\(23089744183296\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(686\) \(\beta_{1}\mathstrut +\mathstrut \) \(5801\)\()/16384\)
\(\nu^{2}\)\(=\)\((\)\(16\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(53\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(800\) \(\beta_{2}\mathstrut +\mathstrut \) \(81334\) \(\beta_{1}\mathstrut +\mathstrut \) \(44906797\)\()/16384\)
\(\nu^{3}\)\(=\)\((\)\(1024\) \(\beta_{7}\mathstrut +\mathstrut \) \(320\) \(\beta_{6}\mathstrut +\mathstrut \) \(14016\) \(\beta_{5}\mathstrut +\mathstrut \) \(58307\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(271744\) \(\beta_{2}\mathstrut +\mathstrut \) \(5177930\) \(\beta_{1}\mathstrut +\mathstrut \) \(5335373179\)\()/16384\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(140288\) \(\beta_{7}\mathstrut -\mathstrut \) \(8464\) \(\beta_{6}\mathstrut -\mathstrut \) \(120560\) \(\beta_{5}\mathstrut +\mathstrut \) \(10208305\) \(\beta_{4}\mathstrut +\mathstrut \) \(1889\) \(\beta_{3}\mathstrut +\mathstrut \) \(107155168\) \(\beta_{2}\mathstrut +\mathstrut \) \(4164164766\) \(\beta_{1}\mathstrut +\mathstrut \) \(341814123305\)\()/16384\)
\(\nu^{5}\)\(=\)\((\)\(28823552\) \(\beta_{7}\mathstrut +\mathstrut \) \(1727008\) \(\beta_{6}\mathstrut -\mathstrut \) \(78969376\) \(\beta_{5}\mathstrut +\mathstrut \) \(366137859\) \(\beta_{4}\mathstrut -\mathstrut \) \(3783773\) \(\beta_{3}\mathstrut -\mathstrut \) \(6811399616\) \(\beta_{2}\mathstrut +\mathstrut \) \(636988879210\) \(\beta_{1}\mathstrut +\mathstrut \) \(271636965004379\)\()/16384\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(700712960\) \(\beta_{7}\mathstrut -\mathstrut \) \(57019888\) \(\beta_{6}\mathstrut +\mathstrut \) \(11801042416\) \(\beta_{5}\mathstrut -\mathstrut \) \(95357690295\) \(\beta_{4}\mathstrut +\mathstrut \) \(16939288601\) \(\beta_{3}\mathstrut +\mathstrut \) \(1212327792416\) \(\beta_{2}\mathstrut +\mathstrut \) \(38172139074894\) \(\beta_{1}\mathstrut +\mathstrut \) \(41881243398273633\)\()/16384\)
\(\nu^{7}\)\(=\)\((\)\(175667244032\) \(\beta_{7}\mathstrut +\mathstrut \) \(278225623168\) \(\beta_{6}\mathstrut -\mathstrut \) \(1498229387392\) \(\beta_{5}\mathstrut +\mathstrut \) \(15519483083131\) \(\beta_{4}\mathstrut +\mathstrut \) \(2837180895227\) \(\beta_{3}\mathstrut +\mathstrut \) \(36944657857792\) \(\beta_{2}\mathstrut -\mathstrut \) \(4203339109316326\) \(\beta_{1}\mathstrut +\mathstrut \) \(2505650305448552883\)\()/16384\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
130.763 7.78570i
130.763 + 7.78570i
54.8957 117.008i
54.8957 + 117.008i
−67.8767 106.586i
−67.8767 + 106.586i
−116.282 46.4306i
−116.282 + 46.4306i
−511.052 31.1428i 30905.2i 260204. + 31831.2i 991387. 962474. 1.57942e7i 6.49702e7i −1.31987e8 2.43709e7i −5.67711e8 −5.06650e8 3.08746e7i
3.2 −511.052 + 31.1428i 30905.2i 260204. 31831.2i 991387. 962474. + 1.57942e7i 6.49702e7i −1.31987e8 + 2.43709e7i −5.67711e8 −5.06650e8 + 3.08746e7i
3.3 −207.583 468.031i 4128.11i −175963. + 194311.i −1.32216e6 −1.93208e6 + 856924.i 1.88890e7i 1.27470e8 + 4.20206e7i 3.70379e8 2.74457e8 + 6.18811e8i
3.4 −207.583 + 468.031i 4128.11i −175963. 194311.i −1.32216e6 −1.93208e6 856924.i 1.88890e7i 1.27470e8 4.20206e7i 3.70379e8 2.74457e8 6.18811e8i
3.5 283.507 426.343i 14439.2i −101392. 241742.i 2.88086e6 6.15603e6 + 4.09360e6i 4.40005e7i −1.31810e8 2.53077e7i 1.78931e8 8.16744e8 1.22823e9i
3.6 283.507 + 426.343i 14439.2i −101392. + 241742.i 2.88086e6 6.15603e6 4.09360e6i 4.40005e7i −1.31810e8 + 2.53077e7i 1.78931e8 8.16744e8 + 1.22823e9i
3.7 477.128 185.722i 27088.6i 193159. 177227.i −2.11965e6 −5.03095e6 1.29247e7i 3.35455e7i 5.92464e7 1.20434e8i −3.46370e8 −1.01135e9 + 3.93667e8i
3.8 477.128 + 185.722i 27088.6i 193159. + 177227.i −2.11965e6 −5.03095e6 + 1.29247e7i 3.35455e7i 5.92464e7 + 1.20434e8i −3.46370e8 −1.01135e9 3.93667e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{19}^{\mathrm{new}}(4, [\chi])\).