Properties

Label 4.21.b.b
Level 4
Weight 21
Character orbit 4.b
Analytic conductor 10.141
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 21 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(10.1405506041\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{56}\cdot 3^{5}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( 49 + \beta_{1} ) q^{2} \) \( + ( 6 - 12 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -164539 + 57 \beta_{1} - \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 2322148 - 2279 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{5} \) \( + ( 12432864 - 571 \beta_{1} + 102 \beta_{2} - 10 \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{7} ) q^{6} \) \( + ( -12010 + 24072 \beta_{1} + 820 \beta_{2} - 44 \beta_{3} + 5 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{7} \) \( + ( 272248205 - 177853 \beta_{1} - 826 \beta_{2} + 163 \beta_{3} + 218 \beta_{4} - 16 \beta_{5} - 17 \beta_{6} + 3 \beta_{7} ) q^{8} \) \( + ( -1301762023 + 682854 \beta_{1} - 1420 \beta_{2} + 1156 \beta_{3} - 344 \beta_{4} - 506 \beta_{5} + 28 \beta_{6} + 28 \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 49 + \beta_{1} ) q^{2} \) \( + ( 6 - 12 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -164539 + 57 \beta_{1} - \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 2322148 - 2279 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{5} \) \( + ( 12432864 - 571 \beta_{1} + 102 \beta_{2} - 10 \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{7} ) q^{6} \) \( + ( -12010 + 24072 \beta_{1} + 820 \beta_{2} - 44 \beta_{3} + 5 \beta_{4} - 8 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{7} \) \( + ( 272248205 - 177853 \beta_{1} - 826 \beta_{2} + 163 \beta_{3} + 218 \beta_{4} - 16 \beta_{5} - 17 \beta_{6} + 3 \beta_{7} ) q^{8} \) \( + ( -1301762023 + 682854 \beta_{1} - 1420 \beta_{2} + 1156 \beta_{3} - 344 \beta_{4} - 506 \beta_{5} + 28 \beta_{6} + 28 \beta_{7} ) q^{9} \) \( + ( -2270968858 + 2429638 \beta_{1} - 35112 \beta_{2} - 3908 \beta_{3} - 3792 \beta_{4} + 136 \beta_{5} - 84 \beta_{6} + 20 \beta_{7} ) q^{10} \) \( + ( -1863370 + 3748556 \beta_{1} + 15573 \beta_{2} - 19080 \beta_{3} - 1458 \beta_{4} - 2736 \beta_{5} - 104 \beta_{6} - 24 \beta_{7} ) q^{11} \) \( + ( -5228846876 + 12754428 \beta_{1} + 430344 \beta_{2} + 1340 \beta_{3} + 2792 \beta_{4} - 11776 \beta_{5} + 284 \beta_{6} - 340 \beta_{7} ) q^{12} \) \( + ( -5405468860 + 16025725 \beta_{1} - 23298 \beta_{2} + 70380 \beta_{3} + 2228 \beta_{4} - 7117 \beta_{5} - 1352 \beta_{6} - 1352 \beta_{7} ) q^{13} \) \( + ( -25070012736 + 875094 \beta_{1} - 2496140 \beta_{2} + 3284 \beta_{3} - 8738 \beta_{4} - 47018 \beta_{5} + 1408 \beta_{6} - 1374 \beta_{7} ) q^{14} \) \( + ( -63460814 + 127220344 \beta_{1} + 242404 \beta_{2} - 261444 \beta_{3} + 44479 \beta_{4} - 37272 \beta_{5} - 1972 \beta_{6} + 180 \beta_{7} ) q^{15} \) \( + ( -128811702500 + 280254756 \beta_{1} + 12393096 \beta_{2} - 256348 \beta_{3} - 458824 \beta_{4} + 53824 \beta_{5} - 5900 \beta_{6} + 3044 \beta_{7} ) q^{16} \) \( + ( -304162445350 - 395317762 \beta_{1} + 869380 \beta_{2} - 187068 \beta_{3} + 426632 \beta_{4} - 62322 \beta_{5} + 14076 \beta_{6} + 14076 \beta_{7} ) q^{17} \) \( + ( 651456417289 - 1341133479 \beta_{1} - 48726416 \beta_{2} + 342104 \beta_{3} - 813088 \beta_{4} + 351056 \beta_{5} - 24136 \beta_{6} + 13256 \beta_{7} ) q^{18} \) \( + ( -1150485454 + 2296672196 \beta_{1} + 5907271 \beta_{2} + 3767016 \beta_{3} + 2820170 \beta_{4} + 531696 \beta_{5} + 41800 \beta_{6} - 15048 \beta_{7} ) q^{19} \) \( + ( 2561929904922 - 2389524606 \beta_{1} + 69251214 \beta_{2} + 2711186 \beta_{3} - 2171584 \beta_{4} + 480256 \beta_{5} + 54240 \beta_{6} - 24480 \beta_{7} ) q^{20} \) \( + ( 4165849127296 - 11264046108 \beta_{1} + 25277176 \beta_{2} - 5704792 \beta_{3} + 11032496 \beta_{4} + 1933172 \beta_{5} - 65416 \beta_{6} - 65416 \beta_{7} ) q^{21} \) \( + ( -3915710395872 + 179931155 \beta_{1} + 4471914 \beta_{2} - 2243750 \beta_{3} - 20261641 \beta_{4} + 731859 \beta_{5} + 264448 \beta_{6} - 35319 \beta_{7} ) q^{22} \) \( + ( -16225758022 + 32453619992 \beta_{1} - 2943580 \beta_{2} - 1950228 \beta_{3} + 27211915 \beta_{4} - 153720 \beta_{5} - 326468 \beta_{6} + 291396 \beta_{7} ) q^{23} \) \( + ( -5942868023296 - 5040979456 \beta_{1} - 600356608 \beta_{2} - 5208576 \beta_{3} - 53292544 \beta_{4} - 3499008 \beta_{5} - 153344 \beta_{6} + 334080 \beta_{7} ) q^{24} \) \( + ( -34723557620205 - 38850961460 \beta_{1} + 95350440 \beta_{2} + 29883960 \beta_{3} + 55663760 \beta_{4} - 8929540 \beta_{5} + 188200 \beta_{6} + 188200 \beta_{7} ) q^{25} \) \( + ( 16526902102774 - 6098766122 \beta_{1} + 1101066744 \beta_{2} + 4140044 \beta_{3} - 80375440 \beta_{4} - 12660248 \beta_{5} - 1484740 \beta_{6} - 9084 \beta_{7} ) q^{26} \) \( + ( -96061105464 + 192248131656 \beta_{1} + 816303546 \beta_{2} - 109337832 \beta_{3} + 132563766 \beta_{4} - 16582896 \beta_{5} + 1671864 \beta_{6} - 2247480 \beta_{7} ) q^{27} \) \( + ( 48249099844024 - 27712066424 \beta_{1} - 2186687248 \beta_{2} - 58690040 \beta_{3} - 201687760 \beta_{4} + 6511616 \beta_{5} - 531896 \beta_{6} - 2894552 \beta_{7} ) q^{28} \) \( + ( 12700820270980 - 211330499063 \beta_{1} + 504793638 \beta_{2} + 73656868 \beta_{3} + 269828164 \beta_{4} - 15644305 \beta_{5} - 357408 \beta_{6} - 357408 \beta_{7} ) q^{29} \) \( + ( -132985369544384 + 6230834682 \beta_{1} + 317592812 \beta_{2} + 35263628 \beta_{3} - 282534862 \beta_{4} + 16315898 \beta_{5} + 3520640 \beta_{6} - 318130 \beta_{7} ) q^{30} \) \( + ( -74379199072 + 148359315424 \beta_{1} - 1806966536 \beta_{2} + 346050848 \beta_{3} + 221587688 \beta_{4} + 53031872 \beta_{5} - 6664544 \beta_{6} + 8390752 \beta_{7} ) q^{31} \) \( + ( 197623168817552 - 137903207056 \beta_{1} + 4455841248 \beta_{2} + 370648944 \beta_{3} + 72471840 \beta_{4} - 35059968 \beta_{5} + 4153904 \beta_{6} + 13938288 \beta_{7} ) q^{32} \) \( + ( 141385017837464 - 9080243958 \beta_{1} - 134595028 \beta_{2} - 900285572 \beta_{3} - 296499176 \beta_{4} + 236711242 \beta_{5} - 1501916 \beta_{6} - 1501916 \beta_{7} ) q^{33} \) \( + ( -428760696733878 - 286361674902 \beta_{1} - 16351557712 \beta_{2} - 478080392 \beta_{3} + 319508064 \beta_{4} + 150361872 \beta_{5} + 3998808 \beta_{6} + 2822952 \beta_{7} ) q^{34} \) \( + ( 507889067264 - 1016494236304 \beta_{1} + 3039670076 \beta_{2} + 630914544 \beta_{3} - 745247204 \beta_{4} + 85187232 \beta_{5} + 16689712 \beta_{6} - 11534640 \beta_{7} ) q^{35} \) \( + ( 1582777931886677 + 582002623497 \beta_{1} + 33182664431 \beta_{2} - 612841391 \beta_{3} + 1324602496 \beta_{4} + 316033024 \beta_{5} - 4583744 \beta_{6} - 39457856 \beta_{7} ) q^{36} \) \( + ( 258802178796164 + 1633589484317 \beta_{1} - 3441513666 \beta_{2} + 2157271308 \beta_{3} - 1136518988 \beta_{4} - 665910605 \beta_{5} + 16334552 \beta_{6} + 16334552 \beta_{7} ) q^{37} \) \( + ( -2401824527302816 + 112180585745 \beta_{1} - 16012358642 \beta_{2} + 3036234846 \beta_{3} + 4190784989 \beta_{4} - 389141679 \beta_{5} - 48207104 \beta_{6} + 557859 \beta_{7} ) q^{38} \) \( + ( 2351557761786 - 4698684927432 \beta_{1} - 16941316724 \beta_{2} - 3866457204 \beta_{3} - 4759218093 \beta_{4} - 564138936 \beta_{5} + 3260892 \beta_{6} - 27505116 \beta_{7} ) q^{39} \) \( + ( 2678670247555338 + 2438056332182 \beta_{1} - 27113119828 \beta_{2} - 1156625322 \beta_{3} + 3824447892 \beta_{4} - 833141536 \beta_{5} - 28732626 \beta_{6} + 64689430 \beta_{7} ) q^{40} \) \( + ( -2734340676566782 + 9177609812876 \beta_{1} - 21399705944 \beta_{2} - 598222056 \beta_{3} - 10957853680 \beta_{4} + 455173788 \beta_{5} - 39434616 \beta_{6} - 39434616 \beta_{7} ) q^{41} \) \( + ( -11589550275786640 + 4740534567792 \beta_{1} + 153758790944 \beta_{2} - 11414748080 \beta_{3} + 4747441216 \beta_{4} - 922304672 \beta_{5} + 119473552 \beta_{6} - 45990032 \beta_{7} ) q^{42} \) \( + ( 3627959664578 - 7254545469684 \beta_{1} + 26367275449 \beta_{2} - 1262513648 \beta_{3} - 6934779868 \beta_{4} - 111345824 \beta_{5} - 181669424 \beta_{6} + 161030960 \beta_{7} ) q^{43} \) \( + ( 19744168355580924 - 4878386176476 \beta_{1} - 181600563528 \beta_{2} + 4861557988 \beta_{3} + 9633998296 \beta_{4} - 953213440 \beta_{5} + 94159364 \beta_{6} - 19519500 \beta_{7} ) q^{44} \) \( + ( 11203346392082276 - 7033116902043 \beta_{1} + 16657229102 \beta_{2} + 1156314268 \beta_{3} + 8395712788 \beta_{4} + 336502363 \beta_{5} - 80720120 \beta_{6} - 80720120 \beta_{7} ) q^{45} \) \( + ( -33941727550371776 + 1616279688338 \beta_{1} + 158728371260 \beta_{2} + 27775121756 \beta_{3} - 4898675734 \beta_{4} + 3713511570 \beta_{5} - 32427392 \beta_{6} + 90581910 \beta_{7} ) q^{46} \) \( + ( -2933704962036 + 5841332820144 \beta_{1} - 190666964464 \beta_{2} + 22944650760 \beta_{3} + 12714698242 \beta_{4} + 3132453168 \beta_{5} + 520624872 \beta_{6} - 339159528 \beta_{7} ) q^{47} \) \( + ( 60666500507134208 - 8836526004480 \beta_{1} - 40014459392 \beta_{2} + 31164160 \beta_{3} - 15142678016 \beta_{4} + 6946238464 \beta_{5} - 3792128 \beta_{6} - 347920640 \beta_{7} ) q^{48} \) \( + ( -14588652291503423 - 22321119506728 \beta_{1} + 46962999888 \beta_{2} - 24484228272 \beta_{3} + 18847618720 \beta_{4} + 1929925144 \beta_{5} + 540791216 \beta_{6} + 540791216 \beta_{7} ) q^{49} \) \( + ( -42364884846623245 - 32867002088605 \beta_{1} - 484905086880 \beta_{2} - 56028568720 \beta_{3} - 26429174080 \beta_{4} + 3111473440 \beta_{5} - 626622160 \beta_{6} + 199669200 \beta_{7} ) q^{50} \) \( + ( -37507099682984 + 75009562542664 \beta_{1} + 915079569898 \beta_{2} + 3976856376 \beta_{3} + 55779703102 \beta_{4} + 659966928 \beta_{5} - 203286952 \beta_{6} + 214304040 \beta_{7} ) q^{51} \) \( + ( 76611319654322874 + 12977808436386 \beta_{1} + 619406845934 \beta_{2} - 17122973518 \beta_{3} - 101089840064 \beta_{4} - 2411281408 \beta_{5} - 514648480 \beta_{6} + 1564780768 \beta_{7} ) q^{52} \) \( + ( -16330643588920668 - 56202357495387 \beta_{1} + 135503175470 \beta_{2} + 24581514812 \beta_{3} + 72777375636 \beta_{4} - 2990168261 \beta_{5} - 407123928 \beta_{6} - 407123928 \beta_{7} ) q^{53} \) \( + ( -200948132658083136 + 9261159651114 \beta_{1} - 1458312503412 \beta_{2} + 131734800300 \beta_{3} - 95825222622 \beta_{4} - 22032052566 \beta_{5} + 1942344960 \beta_{6} - 858993954 \beta_{7} ) q^{54} \) \( + ( -44499624172970 + 89122808747400 \beta_{1} - 2330111231020 \beta_{2} - 108516363820 \beta_{3} + 73916973445 \beta_{4} - 15044037640 \beta_{5} - 1887593980 \beta_{6} + 1069366780 \beta_{7} ) q^{55} \) \( + ( 218645126818271232 + 38162208316416 \beta_{1} + 1596635188736 \beta_{2} - 258812928 \beta_{3} - 48295912448 \beta_{4} - 20916910080 \beta_{5} + 1113632256 \beta_{6} - 3051532800 \beta_{7} ) q^{56} \) \( + ( 55090270966139016 - 70354452722706 \beta_{1} + 188567717604 \beta_{2} + 124361413716 \beta_{3} + 118207388424 \beta_{4} - 10892940882 \beta_{5} - 2606117748 \beta_{6} - 2606117748 \beta_{7} ) q^{57} \) \( + ( -220594507985211130 + 23116597209574 \beta_{1} + 27158051928 \beta_{2} - 272019056516 \beta_{3} - 73436531664 \beta_{4} - 1475047160 \beta_{5} - 1548581268 \beta_{6} + 272856404 \beta_{7} ) q^{58} \) \( + ( -29276308407438 + 58585622465916 \beta_{1} + 6449346076933 \beta_{2} - 27425543328 \beta_{3} - 19540548904 \beta_{4} - 5580107712 \beta_{5} + 3982055904 \beta_{6} - 3878403552 \beta_{7} ) q^{59} \) \( + ( 285011664146480264 - 147015200339144 \beta_{1} - 2315365543024 \beta_{2} + 86572791224 \beta_{3} + 175525970256 \beta_{4} - 15269444608 \beta_{5} + 246253432 \beta_{6} + 301141400 \beta_{7} ) q^{60} \) \( + ( -189492003326769628 + 258222785811709 \beta_{1} - 644631539586 \beta_{2} - 212142907764 \beta_{3} - 359648291788 \beta_{4} + 8781240275 \beta_{5} + 5710256344 \beta_{6} + 5710256344 \beta_{7} ) q^{61} \) \( + ( -155430548424253184 + 7973443185784 \beta_{1} + 4794195674512 \beta_{2} + 237061168912 \beta_{3} + 291238258200 \beta_{4} + 86790805112 \beta_{5} - 6405960704 \beta_{6} + 2880142824 \beta_{7} ) q^{62} \) \( + ( 338249710852350 - 676938898381368 \beta_{1} - 14166471208644 \beta_{2} + 382851907812 \beta_{3} - 348120077655 \beta_{4} + 56624768856 \beta_{5} - 2239993644 \beta_{6} + 4507158060 \beta_{7} ) q^{63} \) \( + ( 22475577279690176 + 196119990529600 \beta_{1} - 8651653263232 \beta_{2} - 105394043328 \beta_{3} + 283230004096 \beta_{4} + 119509165056 \beta_{5} - 6658806976 \beta_{6} + 10562857536 \beta_{7} ) q^{64} \) \( + ( 356889539384768740 + 186577489920340 \beta_{1} - 457443712360 \beta_{2} - 109162550040 \beta_{3} - 246037638800 \beta_{4} - 820317180 \beta_{5} + 3627219960 \beta_{6} + 3627219960 \beta_{7} ) q^{65} \) \( + ( -2820878252095880 + 143252474734584 \beta_{1} + 10507967823376 \beta_{2} + 264760646888 \beta_{3} + 838301370656 \beta_{4} - 47332042192 \beta_{5} + 18898487432 \beta_{6} - 4902439432 \beta_{7} ) q^{66} \) \( + ( 316687980044362 - 633620055139116 \beta_{1} + 21495212626643 \beta_{2} + 212618011688 \beta_{3} - 717468556838 \beta_{4} + 31477038704 \beta_{5} - 1319965816 \beta_{6} + 2573753080 \beta_{7} ) q^{67} \) \( + ( -303338467238110710 - 414785390765454 \beta_{1} + 3604203953726 \beta_{2} - 122773915646 \beta_{3} + 777622055552 \beta_{4} + 89050032128 \beta_{5} + 11088981952 \beta_{6} - 19336402752 \beta_{7} ) q^{68} \) \( + ( 430996625629375936 + 661557498584268 \beta_{1} - 1545826559576 \beta_{2} - 213751095688 \beta_{3} - 893954534512 \beta_{4} + 243956674364 \beta_{5} - 24583083928 \beta_{6} - 24583083928 \beta_{7} ) q^{69} \) \( + ( 1063365989335210624 - 50783208384052 \beta_{1} - 7421850749272 \beta_{2} - 711202225048 \beta_{3} + 785965129052 \beta_{4} - 190136903988 \beta_{5} - 6249950720 \beta_{6} - 128436060 \beta_{7} ) q^{70} \) \( + ( 386694690016478 - 772550772698872 \beta_{1} - 38082108215444 \beta_{2} - 729433470492 \beta_{3} - 435184341079 \beta_{4} - 109173863592 \beta_{5} + 7500009940 \beta_{6} - 11594524884 \beta_{7} ) q^{71} \) \( + ( -1967534643819343907 + 1682154014313459 \beta_{1} + 22160823162406 \beta_{2} + 522459072851 \beta_{3} + 461559554426 \beta_{4} - 613560105232 \beta_{5} + 16069248863 \beta_{6} + 12246548147 \beta_{7} ) q^{72} \) \( + ( -1294180520081416054 + 601851051581342 \beta_{1} - 1154347187196 \beta_{2} + 1517623427988 \beta_{3} - 152203166264 \beta_{4} - 507815100674 \beta_{5} + 16568934668 \beta_{6} + 16568934668 \beta_{7} ) q^{73} \) \( + ( 1723453116382949334 + 172698207719862 \beta_{1} - 43919425725960 \beta_{2} + 1227568721932 \beta_{3} - 1880114224784 \beta_{4} + 255216126440 \beta_{5} - 45221024708 \beta_{6} + 15147681924 \beta_{7} ) q^{74} \) \( + ( -1256305778576310 + 2514832233789660 \beta_{1} + 58446469838085 \beta_{2} - 1951356191760 \beta_{3} + 1076098053660 \beta_{4} - 269320445280 \beta_{5} - 36961302480 \beta_{6} + 22037691600 \beta_{7} ) q^{75} \) \( + ( -3320913667191981868 - 2247852273897844 \beta_{1} + 24850344106984 \beta_{2} - 894873906996 \beta_{3} - 2469431954872 \beta_{4} + 191884478976 \beta_{5} - 65718984404 \beta_{6} - 2765908356 \beta_{7} ) q^{76} \) \( + ( -658694146210939456 - 2689932415798852 \beta_{1} + 6304951661768 \beta_{2} + 610259432152 \beta_{3} + 3434246132432 \beta_{4} - 516689289748 \beta_{5} + 46983797640 \beta_{6} + 46983797640 \beta_{7} ) q^{77} \) \( + ( 4912709276734586688 - 233131953977318 \beta_{1} - 7271803799892 \beta_{2} - 5181918309812 \beta_{3} - 3872297433646 \beta_{4} - 53384883686 \beta_{5} + 58170248832 \beta_{6} - 30150337106 \beta_{7} ) q^{78} \) \( + ( -1920003896778036 + 3839275319468944 \beta_{1} - 52982516799576 \beta_{2} + 664611914344 \beta_{3} + 4053735606330 \beta_{4} + 67862172784 \beta_{5} + 72442075080 \beta_{6} - 63192234952 \beta_{7} ) q^{79} \) \( + ( -4032801608454867592 + 2873530647967496 \beta_{1} - 20370084156784 \beta_{2} + 749635385864 \beta_{3} - 3723731479056 \beta_{4} + 1075654946944 \beta_{5} - 550492760 \beta_{6} - 6317080440 \beta_{7} ) q^{80} \) \( + ( 1966219254080973849 - 4569096840006798 \beta_{1} + 9851526106716 \beta_{2} - 5032824458772 \beta_{3} + 3428329532280 \beta_{4} + 1932392441778 \beta_{5} - 86991784140 \beta_{6} - 86991784140 \beta_{7} ) q^{81} \) \( + ( 9473507016065901906 - 3182257058510862 \beta_{1} + 46813586615648 \beta_{2} + 11283543652336 \beta_{3} + 168824887488 \beta_{4} - 459404564448 \beta_{5} + 12365490096 \beta_{6} - 13520152752 \beta_{7} ) q^{82} \) \( + ( -2164276662213106 + 4316914176981860 \beta_{1} + 56622815002995 \beta_{2} + 10162995815808 \beta_{3} + 5495871426720 \beta_{4} + 1476151628544 \beta_{5} + 8205499264 \beta_{6} + 56688528000 \beta_{7} ) q^{83} \) \( + ( -10698945760357126784 - 11124204382280064 \beta_{1} - 122544292959488 \beta_{2} + 1391242907264 \beta_{3} - 4417763162368 \beta_{4} - 882214014976 \beta_{5} + 200331692672 \beta_{6} + 69918240896 \beta_{7} ) q^{84} \) \( + ( 2516432368626200680 - 5187645648111330 \beta_{1} + 12471218303220 \beta_{2} + 2211143294880 \beta_{3} + 6743508237720 \beta_{4} - 426286835910 \beta_{5} - 16322450040 \beta_{6} - 16322450040 \beta_{7} ) q^{85} \) \( + ( 7590300745980241440 - 345740733923541 \beta_{1} + 103068142629562 \beta_{2} - 6440094975958 \beta_{3} - 2977369045457 \beta_{4} + 1703087971051 \beta_{5} - 15405969920 \beta_{6} + 121596707793 \beta_{7} ) q^{86} \) \( + ( 320302805198290 - 636978406424840 \beta_{1} - 46791442639772 \beta_{2} - 3252968722308 \beta_{3} - 1009225847633 \beta_{4} - 374235249432 \beta_{5} - 249033511156 \beta_{6} + 211107325428 \beta_{7} ) q^{87} \) \( + ( -4520994440665327616 + 19993799740419584 \beta_{1} - 82748627816704 \beta_{2} - 6668276266496 \beta_{3} - 909949723136 \beta_{4} + 808608891904 \beta_{5} - 81886261504 \beta_{6} - 196484210944 \beta_{7} ) q^{88} \) \( + ( -5211572477018818486 + 7677668483617454 \beta_{1} - 19121268796700 \beta_{2} - 5997526320204 \beta_{3} - 10573177176952 \beta_{4} + 127381321422 \beta_{5} + 177032291436 \beta_{6} + 177032291436 \beta_{7} ) q^{89} \) \( + ( -6813705935904132266 + 11561596264426806 \beta_{1} + 104052991715896 \beta_{2} - 8356964687956 \beta_{3} - 1911459138064 \beta_{4} - 859423130968 \beta_{5} - 24686200228 \beta_{6} - 15770700700 \beta_{7} ) q^{90} \) \( + ( 4936308277971888 - 9845905172322832 \beta_{1} - 55768664276436 \beta_{2} - 23204492548912 \beta_{3} - 13039422800748 \beta_{4} - 3506891072032 \beta_{5} + 323578742160 \beta_{6} - 447914985104 \beta_{7} ) q^{91} \) \( + ( 8133811484909345896 - 34389531958969256 \beta_{1} + 90366076946768 \beta_{2} + 10377714130136 \beta_{3} + 23916784185360 \beta_{4} - 1229397773312 \beta_{5} - 421126334568 \beta_{6} + 295529991480 \beta_{7} ) q^{92} \) \( + ( -6667884188983845568 + 21753212560559376 \beta_{1} - 46828411331104 \beta_{2} + 21082537007584 \beta_{3} - 18303135576128 \beta_{4} - 4820953430384 \beta_{5} - 58023331808 \beta_{6} - 58023331808 \beta_{7} ) q^{93} \) \( + ( -6134688200716945280 + 269237464216788 \beta_{1} - 222569094401512 \beta_{2} + 11258363937112 \beta_{3} + 28175121222084 \beta_{4} - 4096877863212 \beta_{5} - 243546534144 \beta_{6} - 343571483844 \beta_{7} ) q^{94} \) \( + ( 12022140835017970 - 24058587206391080 \beta_{1} + 353738414007260 \beta_{2} + 12543364645020 \beta_{3} - 21276683359145 \beta_{4} + 1762171710120 \beta_{5} + 159908835500 \beta_{6} - 69387558060 \beta_{7} ) q^{95} \) \( + ( 18582017747630402560 + 59969141057229824 \beta_{1} + 340976770115584 \beta_{2} + 2064422474752 \beta_{3} + 25944907241472 \beta_{4} - 5037989740544 \beta_{5} + 121821209600 \beta_{6} - 179466447872 \beta_{7} ) q^{96} \) \( + ( 11651735149239067898 + 19370600445404774 \beta_{1} - 41184761775564 \beta_{2} + 20208259679796 \beta_{3} - 16274422737944 \beta_{4} - 2983218567818 \beta_{5} - 266947916404 \beta_{6} - 266947916404 \beta_{7} ) q^{97} \) \( + ( -24087137037789281743 - 13528496091307519 \beta_{1} - 456154024775232 \beta_{2} - 21696162468512 \beta_{3} + 28673039683456 \beta_{4} + 5188705637696 \beta_{5} + 516872749792 \beta_{6} + 21970113312 \beta_{7} ) q^{98} \) \( + ( 18085530154820538 - 36168393014135124 \beta_{1} - 940626140071959 \beta_{2} - 2531381967072 \beta_{3} - 22403435705016 \beta_{4} - 135913538880 \beta_{5} - 583292795232 \beta_{6} + 526662398304 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 396q^{2} \) \(\mathstrut -\mathstrut 1316080q^{4} \) \(\mathstrut +\mathstrut 18568080q^{5} \) \(\mathstrut +\mathstrut 99460608q^{6} \) \(\mathstrut +\mathstrut 2177274816q^{8} \) \(\mathstrut -\mathstrut 10411362168q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 396q^{2} \) \(\mathstrut -\mathstrut 1316080q^{4} \) \(\mathstrut +\mathstrut 18568080q^{5} \) \(\mathstrut +\mathstrut 99460608q^{6} \) \(\mathstrut +\mathstrut 2177274816q^{8} \) \(\mathstrut -\mathstrut 10411362168q^{9} \) \(\mathstrut -\mathstrut 18158047400q^{10} \) \(\mathstrut -\mathstrut 41779799040q^{12} \) \(\mathstrut -\mathstrut 43179394928q^{13} \) \(\mathstrut -\mathstrut 200556776448q^{14} \) \(\mathstrut -\mathstrut 1029373411072q^{16} \) \(\mathstrut -\mathstrut 2434881831408q^{17} \) \(\mathstrut +\mathstrut 5206289577036q^{18} \) \(\mathstrut +\mathstrut 20485893906720q^{20} \) \(\mathstrut +\mathstrut 33281721747456q^{21} \) \(\mathstrut -\mathstrut 31324969489920q^{22} \) \(\mathstrut -\mathstrut 47563142934528q^{24} \) \(\mathstrut -\mathstrut 277943780989800q^{25} \) \(\mathstrut +\mathstrut 132190787676888q^{26} \) \(\mathstrut +\mathstrut 385881741772800q^{28} \) \(\mathstrut +\mathstrut 100761472221840q^{29} \) \(\mathstrut -\mathstrut 1063857826698240q^{30} \) \(\mathstrut +\mathstrut 1580435080068096q^{32} \) \(\mathstrut +\mathstrut 1131041167426560q^{33} \) \(\mathstrut -\mathstrut 3431232331444712q^{34} \) \(\mathstrut +\mathstrut 12664550278353936q^{36} \) \(\mathstrut +\mathstrut 2076957753749392q^{37} \) \(\mathstrut -\mathstrut 19214136907706880q^{38} \) \(\mathstrut +\mathstrut 21439106246704000q^{40} \) \(\mathstrut -\mathstrut 21838015545475824q^{41} \) \(\mathstrut -\mathstrut 92697489416232960q^{42} \) \(\mathstrut +\mathstrut 157933848933319680q^{44} \) \(\mathstrut +\mathstrut 89598644640316560q^{45} \) \(\mathstrut -\mathstrut 271527229329687552q^{46} \) \(\mathstrut +\mathstrut 485296685862666240q^{48} \) \(\mathstrut -\mathstrut 116798593027266808q^{49} \) \(\mathstrut -\mathstrut 339050758449721500q^{50} \) \(\mathstrut +\mathstrut 612942390331308832q^{52} \) \(\mathstrut -\mathstrut 130869871775960688q^{53} \) \(\mathstrut -\mathstrut 1607547577815069696q^{54} \) \(\mathstrut +\mathstrut 1749313578676543488q^{56} \) \(\mathstrut +\mathstrut 440441203792112640q^{57} \) \(\mathstrut -\mathstrut 1764664691469265448q^{58} \) \(\mathstrut +\mathstrut 2279505537583872000q^{60} \) \(\mathstrut -\mathstrut 1514903948917580144q^{61} \) \(\mathstrut -\mathstrut 1243411198213386240q^{62} \) \(\mathstrut +\mathstrut 180589154660126720q^{64} \) \(\mathstrut +\mathstrut 2855862185106362400q^{65} \) \(\mathstrut -\mathstrut 21993146403409920q^{66} \) \(\mathstrut -\mathstrut 2428367014363481568q^{68} \) \(\mathstrut +\mathstrut 3450619355851659264q^{69} \) \(\mathstrut +\mathstrut 8506721176491632640q^{70} \) \(\mathstrut -\mathstrut 15733548898901626944q^{72} \) \(\mathstrut -\mathstrut 10351032717211693808q^{73} \) \(\mathstrut +\mathstrut 13788321655033867608q^{74} \) \(\mathstrut -\mathstrut 26576303558589158400q^{76} \) \(\mathstrut -\mathstrut 5280312525070141440q^{77} \) \(\mathstrut +\mathstrut 39300720744848010240q^{78} \) \(\mathstrut -\mathstrut 32250911443885739520q^{80} \) \(\mathstrut +\mathstrut 15711465243559695624q^{81} \) \(\mathstrut +\mathstrut 75775370396849523352q^{82} \) \(\mathstrut -\mathstrut 85636060864270565376q^{84} \) \(\mathstrut +\mathstrut 20110715505842996000q^{85} \) \(\mathstrut +\mathstrut 60721004056878217728q^{86} \) \(\mathstrut -\mathstrut 36088003765030440960q^{88} \) \(\mathstrut -\mathstrut 41661892622796073200q^{89} \) \(\mathstrut -\mathstrut 54463437967726626600q^{90} \) \(\mathstrut +\mathstrut 64932970344704317440q^{92} \) \(\mathstrut -\mathstrut 53255995615294218240q^{93} \) \(\mathstrut -\mathstrut 49076400009934399488q^{94} \) \(\mathstrut +\mathstrut 148896006651003076608q^{96} \) \(\mathstrut +\mathstrut 93291432495858610192q^{97} \) \(\mathstrut -\mathstrut 192751276316506807284q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut +\mathstrut \) \(41744\) \(x^{6}\mathstrut -\mathstrut \) \(12461904\) \(x^{5}\mathstrut +\mathstrut \) \(2339673984\) \(x^{4}\mathstrut -\mathstrut \) \(888904297728\) \(x^{3}\mathstrut +\mathstrut \) \(163438296621056\) \(x^{2}\mathstrut -\mathstrut \) \(23541970369228800\) \(x\mathstrut +\mathstrut \) \(18139117546354278400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\(18041\) \(\nu^{7}\mathstrut -\mathstrut \) \(3891479\) \(\nu^{6}\mathstrut +\mathstrut \) \(787005348\) \(\nu^{5}\mathstrut +\mathstrut \) \(302936330112\) \(\nu^{4}\mathstrut +\mathstrut \) \(51485437452672\) \(\nu^{3}\mathstrut -\mathstrut \) \(10758414598985472\) \(\nu^{2}\mathstrut +\mathstrut \) \(717876375920309248\) \(\nu\mathstrut -\mathstrut \) \(25585037522225176576\)\()/\)\(36785261018873856\)
\(\beta_{3}\)\(=\)\((\)\(18041\) \(\nu^{7}\mathstrut -\mathstrut \) \(3891479\) \(\nu^{6}\mathstrut +\mathstrut \) \(787005348\) \(\nu^{5}\mathstrut +\mathstrut \) \(302936330112\) \(\nu^{4}\mathstrut +\mathstrut \) \(51485437452672\) \(\nu^{3}\mathstrut +\mathstrut \) \(577805761702996224\) \(\nu^{2}\mathstrut +\mathstrut \) \(6456377094864630784\) \(\nu\mathstrut +\mathstrut \) \(6113875026527821389824\)\()/\)\(36785261018873856\)
\(\beta_{4}\)\(=\)\((\)\(249607\) \(\nu^{7}\mathstrut -\mathstrut \) \(123241\) \(\nu^{6}\mathstrut +\mathstrut \) \(10433869404\) \(\nu^{5}\mathstrut -\mathstrut \) \(3772990508928\) \(\nu^{4}\mathstrut +\mathstrut \) \(616364275162752\) \(\nu^{3}\mathstrut -\mathstrut \) \(235169239430703360\) \(\nu^{2}\mathstrut +\mathstrut \) \(33911623072277728256\) \(\nu\mathstrut -\mathstrut \) \(5898906786242128142336\)\()/\)\(18392630509436928\)
\(\beta_{5}\)\(=\)\((\)\(492821\) \(\nu^{7}\mathstrut +\mathstrut \) \(351728581\) \(\nu^{6}\mathstrut -\mathstrut \) \(22119906540\) \(\nu^{5}\mathstrut +\mathstrut \) \(10760216187264\) \(\nu^{4}\mathstrut -\mathstrut \) \(2897202560558208\) \(\nu^{3}\mathstrut +\mathstrut \) \(461344745751250176\) \(\nu^{2}\mathstrut -\mathstrut \) \(179093594736954825728\) \(\nu\mathstrut +\mathstrut \) \(43353880487245306191872\)\()/\)\(18392630509436928\)
\(\beta_{6}\)\(=\)\((\)\(1155067\) \(\nu^{7}\mathstrut +\mathstrut \) \(11657291\) \(\nu^{6}\mathstrut +\mathstrut \) \(211723652396\) \(\nu^{5}\mathstrut -\mathstrut \) \(32382180710784\) \(\nu^{4}\mathstrut -\mathstrut \) \(39709814857450368\) \(\nu^{3}\mathstrut -\mathstrut \) \(2241326693447300352\) \(\nu^{2}\mathstrut -\mathstrut \) \(126798519347604939776\) \(\nu\mathstrut +\mathstrut \) \(109626834523451544641536\)\()/\)\(12261753672957952\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(616355\) \(\nu^{7}\mathstrut +\mathstrut \) \(258986637\) \(\nu^{6}\mathstrut +\mathstrut \) \(168407965236\) \(\nu^{5}\mathstrut +\mathstrut \) \(14964065291648\) \(\nu^{4}\mathstrut +\mathstrut \) \(48491294451584\) \(\nu^{3}\mathstrut -\mathstrut \) \(425733037190296320\) \(\nu^{2}\mathstrut -\mathstrut \) \(7113770810203978752\) \(\nu\mathstrut -\mathstrut \) \(41240951986505981370368\)\()/\)\(3065438418239488\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(39\) \(\beta_{1}\mathstrut -\mathstrut \) \(166939\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(17\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(218\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut -\mathstrut \) \(682\) \(\beta_{2}\mathstrut -\mathstrut \) \(179149\) \(\beta_{1}\mathstrut +\mathstrut \) \(296169917\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(617\) \(\beta_{7}\mathstrut -\mathstrut \) \(659\) \(\beta_{6}\mathstrut +\mathstrut \) \(14224\) \(\beta_{5}\mathstrut -\mathstrut \) \(125170\) \(\beta_{4}\mathstrut -\mathstrut \) \(68455\) \(\beta_{3}\mathstrut +\mathstrut \) \(3134466\) \(\beta_{2}\mathstrut +\mathstrut \) \(78687033\) \(\beta_{1}\mathstrut -\mathstrut \) \(45843578153\)\()/64\)
\(\nu^{5}\)\(=\)\((\)\(829803\) \(\beta_{7}\mathstrut +\mathstrut \) \(323639\) \(\beta_{6}\mathstrut -\mathstrut \) \(3021648\) \(\beta_{5}\mathstrut +\mathstrut \) \(11725770\) \(\beta_{4}\mathstrut +\mathstrut \) \(27176379\) \(\beta_{3}\mathstrut +\mathstrut \) \(91473318\) \(\beta_{2}\mathstrut -\mathstrut \) \(13081161061\) \(\beta_{1}\mathstrut +\mathstrut \) \(14687099299349\)\()/64\)
\(\nu^{6}\)\(=\)\((\)\(103862433\) \(\beta_{7}\mathstrut -\mathstrut \) \(125334907\) \(\beta_{6}\mathstrut +\mathstrut \) \(2054718480\) \(\beta_{5}\mathstrut +\mathstrut \) \(3844046494\) \(\beta_{4}\mathstrut -\mathstrut \) \(3457519215\) \(\beta_{3}\mathstrut -\mathstrut \) \(148513793614\) \(\beta_{2}\mathstrut +\mathstrut \) \(3842470480945\) \(\beta_{1}\mathstrut -\mathstrut \) \(617296274243617\)\()/64\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(32717086077\) \(\beta_{7}\mathstrut +\mathstrut \) \(18427121327\) \(\beta_{6}\mathstrut +\mathstrut \) \(381838456368\) \(\beta_{5}\mathstrut +\mathstrut \) \(1797322630810\) \(\beta_{4}\mathstrut +\mathstrut \) \(1549255724883\) \(\beta_{3}\mathstrut +\mathstrut \) \(41398159122518\) \(\beta_{2}\mathstrut -\mathstrut \) \(140241348295885\) \(\beta_{1}\mathstrut -\mathstrut \) \(1157353423587302659\)\()/64\)

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
−205.349 167.786i
−205.349 + 167.786i
−98.5328 240.932i
−98.5328 + 240.932i
77.2239 239.948i
77.2239 + 239.948i
228.158 88.6584i
228.158 + 88.6584i
−773.395 671.146i 44571.4i 147703. + 1.03812e6i 1.43704e7 −2.99139e7 + 3.44713e7i 2.43808e8i 5.82498e8 9.02008e8i 1.50018e9 −1.11140e10 9.64462e9i
3.2 −773.395 + 671.146i 44571.4i 147703. 1.03812e6i 1.43704e7 −2.99139e7 3.44713e7i 2.43808e8i 5.82498e8 + 9.02008e8i 1.50018e9 −1.11140e10 + 9.64462e9i
3.3 −346.131 963.727i 109031.i −808962. + 667152.i −3.29975e6 1.05076e8 3.77390e7i 3.27338e8i 9.22959e8 + 5.48696e8i −8.40096e9 1.14215e9 + 3.18005e9i
3.4 −346.131 + 963.727i 109031.i −808962. 667152.i −3.29975e6 1.05076e8 + 3.77390e7i 3.27338e8i 9.22959e8 5.48696e8i −8.40096e9 1.14215e9 3.18005e9i
3.5 356.896 959.792i 46975.6i −793827. 685091.i −4.32154e6 −4.50868e7 1.67654e7i 1.10972e8i −9.40859e8 + 5.17403e8i 1.28008e9 −1.54234e9 + 4.14778e9i
3.6 356.896 + 959.792i 46975.6i −793827. + 685091.i −4.32154e6 −4.50868e7 + 1.67654e7i 1.10972e8i −9.40859e8 5.17403e8i 1.28008e9 −1.54234e9 4.14778e9i
3.7 960.631 354.634i 55423.5i 797046. 681344.i 2.53495e6 1.96550e7 + 5.32415e7i 4.45714e8i 5.24039e8 9.37179e8i 4.15025e8 2.43515e9 8.98977e8i
3.8 960.631 + 354.634i 55423.5i 797046. + 681344.i 2.53495e6 1.96550e7 5.32415e7i 4.45714e8i 5.24039e8 + 9.37179e8i 4.15025e8 2.43515e9 + 8.98977e8i
\(n\): 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

This newform can be constructed as the kernel of the linear operator \(T_{3}^{8} \) \(\mathstrut +\mathstrut 19152818688 T_{3}^{6} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!36\)\( T_{3}^{4} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!20\)\( T_{3}^{2} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!80\)\( \) acting on \(S_{21}^{\mathrm{new}}(4, [\chi])\).