Properties

Label 3.25.b.b
Level 3
Weight 25
Character orbit 3.b
Analytic conductor 10.949
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 25 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(10.9490145677\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{13}\cdot 3^{24}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -102807 + 16 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -10557800 + 8 \beta_{1} - 17 \beta_{2} - 8 \beta_{3} + \beta_{5} ) q^{4} \) \( + ( -16866 \beta_{1} - 112 \beta_{2} + 18 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{5} \) \( + ( -445648392 + 50290 \beta_{1} - 597 \beta_{2} - 757 \beta_{3} - 3 \beta_{4} - 515 \beta_{5} ) q^{6} \) \( + ( 331344146 + 1631 \beta_{1} + 238 \beta_{2} - 1631 \beta_{3} - 3500 \beta_{5} ) q^{7} \) \( + ( -3827162 \beta_{1} - 42512 \beta_{2} + 6786 \beta_{3} + 206 \beta_{4} - 1472 \beta_{5} ) q^{8} \) \( + ( -131864491647 + 7552794 \beta_{1} - 174132 \beta_{2} - 58602 \beta_{3} - 603 \beta_{4} + 31836 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -102807 + 16 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -10557800 + 8 \beta_{1} - 17 \beta_{2} - 8 \beta_{3} + \beta_{5} ) q^{4} \) \( + ( -16866 \beta_{1} - 112 \beta_{2} + 18 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{5} \) \( + ( -445648392 + 50290 \beta_{1} - 597 \beta_{2} - 757 \beta_{3} - 3 \beta_{4} - 515 \beta_{5} ) q^{6} \) \( + ( 331344146 + 1631 \beta_{1} + 238 \beta_{2} - 1631 \beta_{3} - 3500 \beta_{5} ) q^{7} \) \( + ( -3827162 \beta_{1} - 42512 \beta_{2} + 6786 \beta_{3} + 206 \beta_{4} - 1472 \beta_{5} ) q^{8} \) \( + ( -131864491647 + 7552794 \beta_{1} - 174132 \beta_{2} - 58602 \beta_{3} - 603 \beta_{4} + 31836 \beta_{5} ) q^{9} \) \( + ( 461803885200 - 738080 \beta_{1} + 1409738 \beta_{2} + 738080 \beta_{3} + 66422 \beta_{5} ) q^{10} \) \( + ( 26304201 \beta_{1} - 4409240 \beta_{2} + 696663 \beta_{3} - 5500 \beta_{4} - 145508 \beta_{5} ) q^{11} \) \( + ( -3082143640392 - 753496634 \beta_{1} - 11212685 \beta_{2} + 1600578 \beta_{3} + 19494 \beta_{4} - 969939 \beta_{5} ) q^{12} \) \( + ( -12257073010654 - 2561884 \beta_{1} + 9982216 \beta_{2} + 2561884 \beta_{3} - 4858448 \beta_{5} ) q^{13} \) \( + ( 166214776 \beta_{1} - 129502352 \beta_{2} + 20517714 \beta_{3} + 49406 \beta_{4} - 4329920 \beta_{5} ) q^{14} \) \( + ( 37858507509600 - 10277399665 \beta_{1} + 44153436 \beta_{2} + 72393745 \beta_{3} - 249900 \beta_{4} - 6262876 \beta_{5} ) q^{15} \) \( + ( -72222649817408 - 35880768 \beta_{1} + 27435576 \beta_{2} + 35880768 \beta_{3} + 44325960 \beta_{5} ) q^{16} \) \( + ( 94108902628 \beta_{1} + 140367008 \beta_{2} - 22237668 \beta_{3} - 48344 \beta_{4} + 4691792 \beta_{5} ) q^{17} \) \( + ( -204648146911440 - 221187774627 \beta_{1} - 255341970 \beta_{2} - 430975404 \beta_{3} + 1532844 \beta_{4} + 36009522 \beta_{5} ) q^{18} \) \( + ( 1015869915579026 + 463700073 \beta_{1} - 552650910 \beta_{2} - 463700073 \beta_{3} - 374749236 \beta_{5} ) q^{19} \) \( + ( 1025059861348 \beta_{1} + 7219950496 \beta_{2} - 1143837684 \beta_{3} - 2545708 \beta_{4} + 241343872 \beta_{5} ) q^{20} \) \( + ( 451018349822754 - 1276566714850 \beta_{1} + 2465034320 \beta_{2} - 1828515150 \beta_{3} - 1825929 \beta_{4} + 429316020 \beta_{5} ) q^{21} \) \( + ( -688791167483760 - 409673440 \beta_{1} - 2424375734 \beta_{2} + 409673440 \beta_{3} + 3243722614 \beta_{5} ) q^{22} \) \( + ( 2117603796670 \beta_{1} - 10861035344 \beta_{2} + 1725359490 \beta_{3} + 21358352 \beta_{4} - 367730072 \beta_{5} ) q^{23} \) \( + ( 13203927136580544 - 2212720931218 \beta_{1} + 11338648968 \beta_{2} + 15628933402 \beta_{3} - 43777482 \beta_{4} - 1744065688 \beta_{5} ) q^{24} \) \( + ( -34379117427161375 + 6148011500 \beta_{1} + 3462040600 \beta_{2} - 6148011500 \beta_{3} - 15758063600 \beta_{5} ) q^{25} \) \( + ( -6732493158710 \beta_{1} - 134440793024 \beta_{2} + 21271287672 \beta_{3} - 56892088 \beta_{4} - 4466188544 \beta_{5} ) q^{26} \) \( + ( -25614566682235767 + 6847468148043 \beta_{1} - 167952235023 \beta_{2} - 13009349979 \beta_{3} + 376561548 \beta_{4} - 9289798956 \beta_{5} ) q^{27} \) \( + ( 1904208712572656 - 98239719984 \beta_{1} + 160246682358 \beta_{2} + 98239719984 \beta_{3} + 36232757610 \beta_{5} ) q^{28} \) \( + ( -45148749137150 \beta_{1} + 230294278896 \beta_{2} - 36522361458 \beta_{3} - 221627373 \beta_{4} + 7735576596 \beta_{5} ) q^{29} \) \( + ( 279839042031013200 + 111133550564388 \beta_{1} + 425091625506 \beta_{2} - 55351629684 \beta_{3} - 1550020428 \beta_{4} + 62621868222 \beta_{5} ) q^{30} \) \( + ( -189640651829203438 + 242924626235 \beta_{1} - 450929879834 \beta_{2} - 242924626235 \beta_{3} - 34919372636 \beta_{5} ) q^{31} \) \( + ( -116188326026320 \beta_{1} + 1069849285504 \beta_{2} - 168743588976 \beta_{3} + 2434554608 \beta_{4} + 35012428288 \beta_{5} ) q^{32} \) \( + ( 1177968565249549920 + 33062879650042 \beta_{1} + 322217706228 \beta_{2} - 231847155562 \beta_{3} + 3291202167 \beta_{4} - 95627326652 \beta_{5} ) q^{33} \) \( + ( -2573431557520332096 + 887769767808 \beta_{1} - 1766916594696 \beta_{2} - 887769767808 \beta_{3} - 8622940920 \beta_{5} ) q^{34} \) \( + ( 339116566882870 \beta_{1} + 332328069360 \beta_{2} - 55012386870 \beta_{3} - 8976659580 \beta_{4} + 13471378200 \beta_{5} ) q^{35} \) \( + ( 3840312641784096024 - 514850100502380 \beta_{1} - 648680312097 \beta_{2} + 1631169759804 \beta_{3} - 905043492 \beta_{4} - 125700649359 \beta_{5} ) q^{36} \) \( + ( -2338770582619803934 - 4499220946884 \beta_{1} + 8465974516920 \beta_{2} + 4499220946884 \beta_{3} + 532467376848 \beta_{5} ) q^{37} \) \( + ( 653543486611964 \beta_{1} - 16580890160880 \beta_{2} + 2628722462238 \beta_{3} + 12805700370 \beta_{4} - 556111192128 \beta_{5} ) q^{38} \) \( + ( 4230122766030065394 - 1433509217531032 \beta_{1} - 9269438564374 \beta_{2} - 3152794087944 \beta_{3} - 13349511612 \beta_{4} + 1078923738672 \beta_{5} ) q^{39} \) \( + ( -20321787663204451200 + 2786671583360 \beta_{1} - 2428559011376 \beta_{2} - 2786671583360 \beta_{3} - 3144784155344 \beta_{5} ) q^{40} \) \( + ( 1655869609209524 \beta_{1} + 16439879169120 \beta_{2} - 2597694903828 \beta_{3} + 19822367310 \beta_{4} + 542710007688 \beta_{5} ) q^{41} \) \( + ( 34890648505566099120 - 953860695108286 \beta_{1} + 14545614248382 \beta_{2} + 5681924364256 \beta_{3} + 27021316224 \beta_{4} - 4877307186430 \beta_{5} ) q^{42} \) \( + ( -19100549691423725614 + 10246609828721 \beta_{1} - 25523634893582 \beta_{2} - 10246609828721 \beta_{3} + 5030415236140 \beta_{5} ) q^{43} \) \( + ( -1408205517255196 \beta_{1} + 35696087631776 \beta_{2} - 5681057513076 \beta_{3} - 109413642668 \beta_{4} + 1219046559104 \beta_{5} ) q^{44} \) \( + ( 51884160904988798400 + 7589101326006726 \beta_{1} + 49236839028792 \beta_{2} - 14460585118038 \beta_{3} + 2417447349 \beta_{4} + 12107578657644 \beta_{5} ) q^{45} \) \( + ( -57810150971792378784 - 2577167736128 \beta_{1} - 4936386721444 \beta_{2} + 2577167736128 \beta_{3} + 10090722193700 \beta_{5} ) q^{46} \) \( + ( -12098016788479372 \beta_{1} + 118596965862176 \beta_{2} - 18748105958772 \beta_{3} + 111551135272 \beta_{4} + 3923485226000 \beta_{5} ) q^{47} \) \( + ( 8527139873688788928 + 16788646005100720 \beta_{1} - 99163504889000 \beta_{2} + 21218452552080 \beta_{3} - 10994347728 \beta_{4} - 3913393440600 \beta_{5} ) q^{48} \) \( + ( 17991297016200975747 - 1313387671372 \beta_{1} + 42065644733224 \beta_{2} + 1313387671372 \beta_{3} - 39438869390480 \beta_{5} ) q^{49} \) \( + ( -33697523918881575 \beta_{1} - 571833190894400 \beta_{2} + 90591285738600 \beta_{3} + 191364426200 \beta_{4} - 19112136876800 \beta_{5} ) q^{50} \) \( + ( -79837023282789647232 + 5728919868821616 \beta_{1} - 71939990184912 \beta_{2} - 77238424842288 \beta_{3} - 313919442612 \beta_{4} - 43331146909392 \beta_{5} ) q^{51} \) \( + ( -20684538315553745104 - 172858099440624 \beta_{1} + 335706764534814 \beta_{2} + 172858099440624 \beta_{3} + 10009434346434 \beta_{5} ) q^{52} \) \( + ( 48677706490416630 \beta_{1} + 292397684695248 \beta_{2} - 46340112334086 \beta_{3} - 164295965019 \beta_{4} + 9790401747180 \beta_{5} ) q^{53} \) \( + ( -185565273089115224088 - 59798704300404246 \beta_{1} - 102414914693199 \beta_{2} + 301391881491987 \beta_{3} + 811998945621 \beta_{4} + 73296801686007 \beta_{5} ) q^{54} \) \( + ( 279878872011086779200 + 340899167709100 \beta_{1} - 796316322388840 \beta_{2} - 340899167709100 \beta_{3} + 114517986970640 \beta_{5} ) q^{55} \) \( + ( 103399714698678972 \beta_{1} - 67963997653664 \beta_{2} + 10281552134868 \beta_{3} - 1797803101108 \beta_{4} - 1786052428160 \beta_{5} ) q^{56} \) \( + ( -190066907744975995614 - 153462487379774926 \beta_{1} + 1242781353081380 \beta_{2} - 158968656725058 \beta_{3} + 452294560017 \beta_{4} + 17037567599724 \beta_{5} ) q^{57} \) \( + ( 1232561062310922938160 - 66520828470880 \beta_{1} + 346933249384078 \beta_{2} + 66520828470880 \beta_{3} - 213891592442318 \beta_{5} ) q^{58} \) \( + ( 213729180666636225 \beta_{1} + 3026239637921192 \beta_{2} - 478266668505441 \beta_{3} + 3329778120304 \beta_{4} + 99986713765292 \beta_{5} ) q^{59} \) \( + ( -2406129542103853065600 + 103320707600883860 \beta_{1} - 1423762021224528 \beta_{2} - 1087853767024100 \beta_{3} - 4680177984060 \beta_{4} - 263117172758032 \beta_{5} ) q^{60} \) \( + ( 1668846402432472695842 + 491097180743180 \beta_{1} - 1214195192957096 \beta_{2} - 491097180743180 \beta_{3} + 232000831470736 \beta_{5} ) q^{61} \) \( + ( -461518758353557040 \beta_{1} - 3420757702475984 \beta_{2} + 543322870632666 \beta_{3} + 6385879027382 \beta_{4} - 115728157823168 \beta_{5} ) q^{62} \) \( + ( -1387568338945367645934 - 56394260840967231 \beta_{1} + 496490598993942 \beta_{2} + 1041369638611071 \beta_{3} + 5586713612928 \beta_{4} + 685927818382380 \beta_{5} ) q^{63} \) \( + ( 1956982082496289007104 - 1947983782316544 \beta_{1} + 4047650379132864 \beta_{2} + 1947983782316544 \beta_{3} - 151682814499776 \beta_{5} ) q^{64} \) \( + ( 146265771973951060 \beta_{1} - 1616125527497760 \beta_{2} + 248358813502860 \beta_{3} - 28228981316070 \beta_{4} - 46343122565640 \beta_{5} ) q^{65} \) \( + ( -903145666255115920560 + 1053231484078592676 \beta_{1} - 2672138808587358 \beta_{2} + 1834954542343692 \beta_{3} + 4161591434484 \beta_{4} - 508158876952386 \beta_{5} ) q^{66} \) \( + ( -907514515405180474414 - 742910517139699 \beta_{1} + 1574124385937002 \beta_{2} + 742910517139699 \beta_{3} - 88303351657604 \beta_{5} ) q^{67} \) \( + ( -2048623373295188048 \beta_{1} - 6264280871337088 \beta_{2} + 997787986525584 \beta_{3} + 22288182114544 \beta_{4} - 214752877608448 \beta_{5} ) q^{68} \) \( + ( 1990548345509439976512 - 128806767538867084 \beta_{1} + 349196066054328 \beta_{2} - 134002044399764 \beta_{3} - 9876706901646 \beta_{4} - 1686738439289128 \beta_{5} ) q^{69} \) \( + ( -9272064513642363669600 + 7599859489678400 \beta_{1} - 15283467609508700 \beta_{2} - 7599859489678400 \beta_{3} + 83748630151900 \beta_{5} ) q^{70} \) \( + ( 756674531285537362 \beta_{1} - 1499530548451120 \beta_{2} + 250851726975726 \beta_{3} + 50347713016120 \beta_{4} - 63410408419336 \beta_{5} ) q^{71} \) \( + ( 10628875023037874098560 + 1517268121378814790 \beta_{1} + 16034769353841120 \beta_{2} - 2838433212858654 \beta_{3} - 4335616304658 \beta_{4} + 2618590389972048 \beta_{5} ) q^{72} \) \( + ( 1204831473200573840066 - 5883100163857584 \beta_{1} + 13178605767253920 \beta_{2} + 5883100163857584 \beta_{3} - 1412405439538752 \beta_{5} ) q^{73} \) \( + ( 2754799576919276282 \beta_{1} + 59627898835423680 \beta_{2} - 9472562563441464 \beta_{3} - 118044679372680 \beta_{4} + 2019075209013504 \beta_{5} ) q^{74} \) \( + ( 6287221229099664119625 - 6187187573070181400 \beta_{1} - 24766206079999175 \beta_{2} - 8384745460555800 \beta_{3} - 10899433826100 \beta_{4} + 2052531418496400 \beta_{5} ) q^{75} \) \( + ( -707352735833636040592 - 6583746123305648 \beta_{1} + 6656249176093670 \beta_{2} + 6583746123305648 \beta_{3} + 6511243070517626 \beta_{5} ) q^{76} \) \( + ( 6809499521535807404 \beta_{1} - 40296049918252128 \beta_{2} + 6397241902552116 \beta_{3} + 63877495608234 \beta_{4} - 1360235662770600 \beta_{5} ) q^{77} \) \( + ( 39286751689330640866800 - 365931035799395476 \beta_{1} + 16802044293553878 \beta_{2} + 4954941679501006 \beta_{3} + 85416180712194 \beta_{4} - 286142979388486 \beta_{5} ) q^{78} \) \( + ( -70580807351013804652654 + 12499954825131803 \beta_{1} - 19933107431287130 \beta_{2} - 12499954825131803 \beta_{3} - 5066802218976476 \beta_{5} ) q^{79} \) \( + ( -4847737014439945440 \beta_{1} - 7637188814160640 \beta_{2} + 1218830398639200 \beta_{3} + 36033136489120 \beta_{4} - 264181796869120 \beta_{5} ) q^{80} \) \( + ( 17121015115205290095681 - 9071660364253994376 \beta_{1} - 6391347743149200 \beta_{2} + 19645414636626792 \beta_{3} - 49322166531390 \beta_{4} - 13574586842175672 \beta_{5} ) q^{81} \) \( + ( -45375954220886731172640 + 15911776090584640 \beta_{1} - 21482319096270676 \beta_{2} - 15911776090584640 \beta_{3} - 10341233084898604 \beta_{5} ) q^{82} \) \( + ( 3078798376008577387 \beta_{1} - 157661954685329736 \beta_{2} + 24944579388311157 \beta_{3} - 69612888247692 \beta_{4} - 5236835052644940 \beta_{5} ) q^{83} \) \( + ( 33499468055634922346544 + 22437686350604876540 \beta_{1} - 19749288634353250 \beta_{2} + 8568592916868180 \beta_{3} - 163414714066884 \beta_{4} + 3697661545383330 \beta_{5} ) q^{84} \) \( + ( 53168137468156133164800 - 77900922818711280 \beta_{1} + 149709714481275168 \beta_{2} + 77900922818711280 \beta_{3} + 6092131156147392 \beta_{5} ) q^{85} \) \( + ( -33874828845440738804 \beta_{1} + 67876107152483728 \beta_{2} - 10678690025790066 \beta_{3} + 256351025074466 \beta_{4} + 2194176631729600 \beta_{5} ) q^{86} \) \( + ( -42071755904935635967200 + 641931438676705549 \beta_{1} - 2082210884421132 \beta_{2} + 16219427444302931 \beta_{3} + 131824769259444 \beta_{4} + 33282558642430924 \beta_{5} ) q^{87} \) \( + ( 26692076447258648722560 + 66221633552187520 \beta_{1} - 159197354697578032 \beta_{2} - 66221633552187520 \beta_{3} + 26754087593202992 \beta_{5} ) q^{88} \) \( + ( 14222732564527013960 \beta_{1} + 260927043593934208 \beta_{2} - 41461087139147304 \beta_{3} - 553644637903954 \beta_{4} + 8845206689905528 \beta_{5} ) q^{89} \) \( + ( -207801050276526937599600 + 42353915539310610120 \beta_{1} + 103581555150596202 \beta_{2} - 157075453037601000 \beta_{3} + 80972961775560 \beta_{4} - 31807948039769322 \beta_{5} ) q^{90} \) \( + ( 249505552703931899478628 + 23921562227721158 \beta_{1} - 44251088106086036 \beta_{2} - 23921562227721158 \beta_{3} - 3592036349356280 \beta_{5} ) q^{91} \) \( + ( -24340279692162944104 \beta_{1} + 171186544275060672 \beta_{2} - 27032230609049592 \beta_{3} + 271145879381304 \beta_{4} + 5633912574667008 \beta_{5} ) q^{92} \) \( + ( -83527603978194482875614 - 41278803469278023722 \beta_{1} - 169564585261225336 \beta_{2} + 10621128377600442 \beta_{3} + 489710349844659 \beta_{4} - 18400081725696636 \beta_{5} ) q^{93} \) \( + ( 329886139477680137709504 - 61352496984241792 \beta_{1} + 221392965094315288 \beta_{2} + 61352496984241792 \beta_{3} - 98687971125831704 \beta_{5} ) q^{94} \) \( + ( 52057188814845957042 \beta_{1} + 135529227579319504 \beta_{2} - 21404923856476146 \beta_{3} + 202014413435408 \beta_{4} + 4463770409061208 \beta_{5} ) q^{95} \) \( + ( -236766779203787433570816 - 23423707973197297680 \beta_{1} + 11985282130830912 \beta_{2} + 202125309427930704 \beta_{3} - 841819818012624 \beta_{4} + 67336616568620352 \beta_{5} ) q^{96} \) \( + ( -235267911709305408403774 + 361678218445291556 \beta_{1} - 833583073473213176 \beta_{2} - 361678218445291556 \beta_{3} + 110226636582630064 \beta_{5} ) q^{97} \) \( + ( 39608640832916730827 \beta_{1} - 1274318337985493696 \beta_{2} + 201778998090189912 \beta_{3} + 44729659325288 \beta_{4} - 42489205842003200 \beta_{5} ) q^{98} \) \( + ( -305743340580380408185920 - 59123724373906093935 \beta_{1} + 1245959431794803088 \beta_{2} + 105180306662778831 \beta_{3} - 1224875174028264 \beta_{4} - 33536178784867572 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 616842q^{3} \) \(\mathstrut -\mathstrut 63346800q^{4} \) \(\mathstrut -\mathstrut 2673890352q^{6} \) \(\mathstrut +\mathstrut 1988064876q^{7} \) \(\mathstrut -\mathstrut 791186949882q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 616842q^{3} \) \(\mathstrut -\mathstrut 63346800q^{4} \) \(\mathstrut -\mathstrut 2673890352q^{6} \) \(\mathstrut +\mathstrut 1988064876q^{7} \) \(\mathstrut -\mathstrut 791186949882q^{9} \) \(\mathstrut +\mathstrut 2770823311200q^{10} \) \(\mathstrut -\mathstrut 18492861842352q^{12} \) \(\mathstrut -\mathstrut 73542438063924q^{13} \) \(\mathstrut +\mathstrut 227151045057600q^{15} \) \(\mathstrut -\mathstrut 433335898904448q^{16} \) \(\mathstrut -\mathstrut 1227888881468640q^{18} \) \(\mathstrut +\mathstrut 6095219493474156q^{19} \) \(\mathstrut +\mathstrut 2706110098936524q^{21} \) \(\mathstrut -\mathstrut 4132747004902560q^{22} \) \(\mathstrut +\mathstrut 79223562819483264q^{24} \) \(\mathstrut -\mathstrut 206274704562968250q^{25} \) \(\mathstrut -\mathstrut 153687400093414602q^{27} \) \(\mathstrut +\mathstrut 11425252275435936q^{28} \) \(\mathstrut +\mathstrut 1679034252186079200q^{30} \) \(\mathstrut -\mathstrut 1137843910975220628q^{31} \) \(\mathstrut +\mathstrut 7067811391497299520q^{33} \) \(\mathstrut -\mathstrut 15440589345121992576q^{34} \) \(\mathstrut +\mathstrut 23041875850704576144q^{36} \) \(\mathstrut -\mathstrut 14032623495718823604q^{37} \) \(\mathstrut +\mathstrut 25380736596180392364q^{39} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!20\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!84\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!04\)\(q^{46} \) \(\mathstrut +\mathstrut 51162839242132733568q^{48} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!82\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!92\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!24\)\(q^{52} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!28\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!84\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!60\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!52\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!04\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!24\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!60\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!84\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!72\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(63\!\cdots\!60\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(72\!\cdots\!96\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!50\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!52\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!24\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!86\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!40\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!64\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!00\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!60\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!68\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!84\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!24\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!96\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!44\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!20\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut +\mathstrut \) \(253102\) \(x^{4}\mathstrut +\mathstrut \) \(17425276096\) \(x^{2}\mathstrut +\mathstrut \) \(250659115499520\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 18 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 927 \nu^{5} - 22464 \nu^{4} + 158192658 \nu^{3} - 8746814592 \nu^{2} + 4583765471040 \nu - 519220416540672 \)\()/\)\(1041302528\)
\(\beta_{3}\)\(=\)\((\)\( -927 \nu^{5} + 162432 \nu^{4} - 158192658 \nu^{3} + 11341271808 \nu^{2} - 4574393748288 \nu - 624717182509056 \)\()/\)\(520651264\)
\(\beta_{4}\)\(=\)\((\)\( 129501 \nu^{5} + 252288 \nu^{4} + 36839351862 \nu^{3} + 46328530176 \nu^{2} + 2340669617528256 \nu + 1452164483653632 \)\()/\)\(520651264\)
\(\beta_{5}\)\(=\)\((\)\( 927 \nu^{5} + 2217024 \nu^{4} + 158192658 \nu^{3} + 370146519936 \nu^{2} + 4583765471040 \nu + 9641799262384128 \)\()/\)\(1041302528\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/18\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut -\mathstrut \) \(27335016\)\()/324\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(736\) \(\beta_{5}\mathstrut +\mathstrut \) \(103\) \(\beta_{4}\mathstrut +\mathstrut \) \(3393\) \(\beta_{3}\mathstrut -\mathstrut \) \(21256\) \(\beta_{2}\mathstrut -\mathstrut \) \(18690797\) \(\beta_{1}\)\()/2916\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(250237\) \(\beta_{5}\mathstrut +\mathstrut \) \(18272248\) \(\beta_{3}\mathstrut +\mathstrut \) \(36794733\) \(\beta_{2}\mathstrut -\mathstrut \) \(18272248\) \(\beta_{1}\mathstrut +\mathstrut \) \(42588282369096\)\()/4374\)
\(\nu^{5}\)\(=\)\((\)\(103238176\) \(\beta_{5}\mathstrut -\mathstrut \) \(8788481\) \(\beta_{4}\mathstrut -\mathstrut \) \(481592855\) \(\beta_{3}\mathstrut +\mathstrut \) \(3026837432\) \(\beta_{2}\mathstrut +\mathstrut \) \(1194462231243\) \(\beta_{1}\)\()/1458\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
380.197i
298.476i
139.516i
139.516i
298.476i
380.197i
6843.54i 162008. 506145.i −3.00569e7 3.79015e8i −3.46383e9 1.10871e9i −1.14171e10 9.08798e10i −2.29937e11 1.63999e11i 2.59380e12
2.2 5372.56i −24323.9 + 530884.i −1.20872e7 5.28531e7i 2.85221e9 + 1.30681e8i 2.07202e10 2.51975e10i −2.81246e11 2.58263e10i −2.83957e11
2.3 2511.29i −446105. 288825.i 1.04706e7 3.68111e8i −7.25325e8 + 1.12030e9i −8.30906e9 6.84273e10i 1.15589e11 + 2.57693e11i −9.24434e11
2.4 2511.29i −446105. + 288825.i 1.04706e7 3.68111e8i −7.25325e8 1.12030e9i −8.30906e9 6.84273e10i 1.15589e11 2.57693e11i −9.24434e11
2.5 5372.56i −24323.9 530884.i −1.20872e7 5.28531e7i 2.85221e9 1.30681e8i 2.07202e10 2.51975e10i −2.81246e11 + 2.58263e10i −2.83957e11
2.6 6843.54i 162008. + 506145.i −3.00569e7 3.79015e8i −3.46383e9 + 1.10871e9i −1.14171e10 9.08798e10i −2.29937e11 + 1.63999e11i 2.59380e12
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.6
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{6} \) \(\mathstrut +\mathstrut 82005048 T_{2}^{4} \) \(\mathstrut +\mathstrut \)\(18\!\cdots\!96\)\( T_{2}^{2} \) \(\mathstrut +\mathstrut \)\(85\!\cdots\!80\)\( \) acting on \(S_{25}^{\mathrm{new}}(3, [\chi])\).