Properties

Label 25.34.a.c
Level $25$
Weight $34$
Character orbit 25.a
Self dual yes
Analytic conductor $172.457$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.457072203\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 9680197848 x^{4} + 8052423720422 x^{3} + 24239866893261762265 x^{2} - 69081627028404093368325 x - 10572274201725134136583265250\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{5}\cdot 5^{6}\cdot 7\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( -24558 - \beta_{1} ) q^{2} + ( -4418958 - 77 \beta_{1} - \beta_{2} ) q^{3} + ( 4920092068 + 46623 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( 1102624263657 + 6864255 \beta_{1} + 30552 \beta_{2} + 468 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{6} + ( -3185671255582 + 97559148 \beta_{1} + 216147 \beta_{2} + 3312 \beta_{3} - 17 \beta_{4} + 5 \beta_{5} ) q^{7} + ( -511635642610246 - 3133914612 \beta_{1} - 3852490 \beta_{2} - 83674 \beta_{3} + 134 \beta_{4} + 40 \beta_{5} ) q^{8} + ( 2267015634483853 + 4382146760 \beta_{1} - 6075156 \beta_{2} + 193296 \beta_{3} - 28 \beta_{4} + 756 \beta_{5} ) q^{9} +O(q^{10})\) \( q +(-24558 - \beta_{1}) q^{2} +(-4418958 - 77 \beta_{1} - \beta_{2}) q^{3} +(4920092068 + 46623 \beta_{1} + \beta_{2} + \beta_{3}) q^{4} +(1102624263657 + 6864255 \beta_{1} + 30552 \beta_{2} + 468 \beta_{3} + \beta_{4} - 2 \beta_{5}) q^{6} +(-3185671255582 + 97559148 \beta_{1} + 216147 \beta_{2} + 3312 \beta_{3} - 17 \beta_{4} + 5 \beta_{5}) q^{7} +(-511635642610246 - 3133914612 \beta_{1} - 3852490 \beta_{2} - 83674 \beta_{3} + 134 \beta_{4} + 40 \beta_{5}) q^{8} +(2267015634483853 + 4382146760 \beta_{1} - 6075156 \beta_{2} + 193296 \beta_{3} - 28 \beta_{4} + 756 \beta_{5}) q^{9} +(25126821644470344 - 15919797219 \beta_{1} + 735667448 \beta_{2} + 14411696 \beta_{3} + 15903 \beta_{4} + 1269 \beta_{5}) q^{11} +(-77724158528649828 - 3321987710240 \beta_{1} - 6695550484 \beta_{2} - 35444916 \beta_{3} + 75556 \beta_{4} + 97360 \beta_{5}) q^{12} +(233847859187027282 + 4227595028852 \beta_{1} - 13220112692 \beta_{2} + 154586416 \beta_{3} - 81656 \beta_{4} - 178960 \beta_{5}) q^{13} +(-1181012244499862693 - 18235901054591 \beta_{1} + 35093460488 \beta_{2} - 793087972 \beta_{3} + 311515 \beta_{4} + 842570 \beta_{5}) q^{14} +(10751698031801611492 + 666344497775648 \beta_{1} + 436988040948 \beta_{2} + 4293352404 \beta_{3} - 10297284 \beta_{4} - 14765232 \beta_{5}) q^{16} +(-22396782737615808818 - 71978632509244 \beta_{1} + 1207850209132 \beta_{2} - 6749454032 \beta_{3} - 37738088 \beta_{4} - 24227680 \beta_{5}) q^{17} +(-\)\(11\!\cdots\!50\)\( - 3473896119269977 \beta_{1} + 3655399616928 \beta_{2} + 2496038448 \beta_{3} + 11408732 \beta_{4} - 19767480 \beta_{5}) q^{18} +(\)\(28\!\cdots\!68\)\( + 10244681461820999 \beta_{1} - 6692355956710 \beta_{2} - 34909194064 \beta_{3} - 143281169 \beta_{4} - 18607387 \beta_{5}) q^{19} +(-\)\(17\!\cdots\!16\)\( + 22440764316024556 \beta_{1} - 11501053976368 \beta_{2} - 591355152288 \beta_{3} + 786288980 \beta_{4} - 127864660 \beta_{5}) q^{21} +(-\)\(41\!\cdots\!22\)\( - 108546266480938682 \beta_{1} - 94869534047408 \beta_{2} - 243484405256 \beta_{3} + 968413846 \beta_{4} + 1630203860 \beta_{5}) q^{22} +(-\)\(11\!\cdots\!90\)\( - 42933190737365684 \beta_{1} - 975037194779 \beta_{2} + 1358005856080 \beta_{3} - 2794904655 \beta_{4} - 5189585925 \beta_{5}) q^{23} +(\)\(35\!\cdots\!96\)\( + 301874229418111968 \beta_{1} + 495562416978408 \beta_{2} + 7377544160424 \beta_{3} - 10378784424 \beta_{4} - 1289936352 \beta_{5}) q^{24} +(-\)\(60\!\cdots\!48\)\( - 1210185937538331270 \beta_{1} - 1041229904012704 \beta_{2} - 10442017299376 \beta_{3} + 40147714308 \beta_{4} - 15011166216 \beta_{5}) q^{26} +(\)\(56\!\cdots\!44\)\( + 2250451644212229998 \beta_{1} + 918896520555990 \beta_{2} + 204743125056 \beta_{3} + 25988358704 \beta_{4} + 22240248240 \beta_{5}) q^{27} +(\)\(29\!\cdots\!64\)\( + 5300112711327556284 \beta_{1} + 4232529613391816 \beta_{2} + 29596449989864 \beta_{3} - 23710527924 \beta_{4} - 27389911440 \beta_{5}) q^{28} +(\)\(21\!\cdots\!94\)\( - 6137808882409723068 \beta_{1} - 3112687079867392 \beta_{2} - 9630610193440 \beta_{3} - 242403578228 \beta_{4} + 243089857556 \beta_{5}) q^{29} +(\)\(18\!\cdots\!68\)\( - 3388099694918505812 \beta_{1} + 4025742456653054 \beta_{2} - 123228527552992 \beta_{3} + 387650863794 \beta_{4} + 182108055462 \beta_{5}) q^{31} +(-\)\(44\!\cdots\!40\)\( - 23674599322570287136 \beta_{1} - 62140876314710376 \beta_{2} - 867369316347432 \beta_{3} - 462454977688 \beta_{4} + 1232999492320 \beta_{5}) q^{32} +(-\)\(58\!\cdots\!64\)\( - 54300701840174618364 \beta_{1} - 43618058243442636 \beta_{2} + 171779073856272 \beta_{3} + 1478052244448 \beta_{4} - 437837130520 \beta_{5}) q^{33} +(\)\(14\!\cdots\!20\)\( + 62028152501636042878 \beta_{1} - 85162746659265952 \beta_{2} - 1825311948550576 \beta_{3} - 953122000764 \beta_{4} + 3519611975928 \beta_{5}) q^{34} +(\)\(28\!\cdots\!12\)\( + \)\(13\!\cdots\!43\)\( \beta_{1} - 191907891499071027 \beta_{2} + 349244529375309 \beta_{3} - 2673384863504 \beta_{4} + 1059460771008 \beta_{5}) q^{36} +(-\)\(46\!\cdots\!78\)\( - \)\(17\!\cdots\!88\)\( \beta_{1} + 64934905550931848 \beta_{2} + 1366726671422432 \beta_{3} - 5897342122712 \beta_{4} + 18795609634280 \beta_{5}) q^{37} +(-\)\(13\!\cdots\!56\)\( - \)\(30\!\cdots\!28\)\( \beta_{1} + 448701969129176224 \beta_{2} - 11481687976142768 \beta_{3} + 4355893191188 \beta_{4} - 11060468596520 \beta_{5}) q^{38} +(\)\(96\!\cdots\!52\)\( - \)\(92\!\cdots\!82\)\( \beta_{1} - 157334254410157330 \beta_{2} - 5493547831068288 \beta_{3} + 13641937276912 \beta_{4} + 18617989176976 \beta_{5}) q^{39} +(\)\(74\!\cdots\!58\)\( - \)\(21\!\cdots\!32\)\( \beta_{1} + 3245263265241030644 \beta_{2} - 17944925624479312 \beta_{3} - 54662526732316 \beta_{4} - 13379833591868 \beta_{5}) q^{41} +(-\)\(24\!\cdots\!48\)\( + \)\(46\!\cdots\!64\)\( \beta_{1} + 618975393651013440 \beta_{2} + 28094202897223392 \beta_{3} - 74946257277672 \beta_{4} - 60762377359920 \beta_{5}) q^{42} +(-\)\(37\!\cdots\!82\)\( + \)\(23\!\cdots\!49\)\( \beta_{1} - 1230021894277821697 \beta_{2} + 23796012831990560 \beta_{3} - 89622397359410 \beta_{4} + 26036900579450 \beta_{5}) q^{43} +(\)\(11\!\cdots\!80\)\( + \)\(44\!\cdots\!88\)\( \beta_{1} + 5045952962716245548 \beta_{2} + 90405825913235564 \beta_{3} - 132704516523624 \beta_{4} - 265612825035552 \beta_{5}) q^{44} +(\)\(58\!\cdots\!53\)\( - \)\(57\!\cdots\!17\)\( \beta_{1} - 29212393650096783176 \beta_{2} - 186615301401726332 \beta_{3} + 365524022392109 \beta_{4} + 220470269423782 \beta_{5}) q^{46} +(-\)\(63\!\cdots\!62\)\( - \)\(17\!\cdots\!90\)\( \beta_{1} + 6154437256507651999 \beta_{2} + 136200415632987280 \beta_{3} + 867892046326545 \beta_{4} - 579907554579525 \beta_{5}) q^{47} +(-\)\(40\!\cdots\!28\)\( - \)\(56\!\cdots\!24\)\( \beta_{1} + 23500066991230398512 \beta_{2} - 832522204704740688 \beta_{3} + 21895879190608 \beta_{4} + 698410891602880 \beta_{5}) q^{48} +(-\)\(11\!\cdots\!39\)\( + \)\(23\!\cdots\!52\)\( \beta_{1} + 43372989519856592836 \beta_{2} - 118196428042094288 \beta_{3} + 1143985608347436 \beta_{4} - 114523758942372 \beta_{5}) q^{49} +(-\)\(91\!\cdots\!00\)\( - \)\(28\!\cdots\!86\)\( \beta_{1} + \)\(11\!\cdots\!70\)\( \beta_{2} - 1323031152136627584 \beta_{3} + 74390991001456 \beta_{4} - 1935010016550512 \beta_{5}) q^{51} +(\)\(15\!\cdots\!36\)\( + \)\(11\!\cdots\!98\)\( \beta_{1} + 40103374382504825906 \beta_{2} + 1775376851823262130 \beta_{3} + 232265487474320 \beta_{4} - 1547599231006400 \beta_{5}) q^{52} +(-\)\(78\!\cdots\!62\)\( - \)\(41\!\cdots\!32\)\( \beta_{1} - 67069168248378721308 \beta_{2} - 3496079966838332784 \beta_{3} + 1637763997437144 \beta_{4} - 2435648622453360 \beta_{5}) q^{53} +(-\)\(30\!\cdots\!22\)\( - \)\(10\!\cdots\!46\)\( \beta_{1} + 50443714279879755120 \beta_{2} - 1373410896625130424 \beta_{3} - 1849398390364534 \beta_{4} + 794088938732268 \beta_{5}) q^{54} +(-\)\(65\!\cdots\!04\)\( - \)\(42\!\cdots\!12\)\( \beta_{1} - \)\(66\!\cdots\!88\)\( \beta_{2} - 2479956419208447600 \beta_{3} - 1840379833947232 \beta_{4} + 3457921386360064 \beta_{5}) q^{56} +(\)\(40\!\cdots\!36\)\( + \)\(11\!\cdots\!36\)\( \beta_{1} - \)\(30\!\cdots\!56\)\( \beta_{2} - 7588398299320953360 \beta_{3} - 11899491891624040 \beta_{4} + 24613480734732800 \beta_{5}) q^{57} +(\)\(73\!\cdots\!40\)\( - \)\(23\!\cdots\!22\)\( \beta_{1} + \)\(18\!\cdots\!68\)\( \beta_{2} + 3574520024274182112 \beta_{3} - 1839281336554792 \beta_{4} - 6080938064012720 \beta_{5}) q^{58} +(-\)\(29\!\cdots\!28\)\( - \)\(12\!\cdots\!19\)\( \beta_{1} + \)\(14\!\cdots\!82\)\( \beta_{2} - 5609198583504272688 \beta_{3} - 12275424379581295 \beta_{4} + 23068122370727515 \beta_{5}) q^{59} +(-\)\(47\!\cdots\!18\)\( + \)\(64\!\cdots\!00\)\( \beta_{1} - 58427209505596046000 \beta_{2} - 9337839871530008000 \beta_{3} - 19689438454463200 \beta_{4} + 27236975122556800 \beta_{5}) q^{61} +(-\)\(57\!\cdots\!54\)\( - \)\(10\!\cdots\!42\)\( \beta_{1} + \)\(73\!\cdots\!16\)\( \beta_{2} + 22404603041652321912 \beta_{3} - 30772461553341242 \beta_{4} - 9227620660724620 \beta_{5}) q^{62} +(\)\(93\!\cdots\!26\)\( - \)\(34\!\cdots\!40\)\( \beta_{1} + \)\(14\!\cdots\!83\)\( \beta_{2} + 3109775174438892912 \beta_{3} - 21542654788301517 \beta_{4} + 32189211391833105 \beta_{5}) q^{63} +(\)\(32\!\cdots\!72\)\( + \)\(43\!\cdots\!00\)\( \beta_{1} + \)\(89\!\cdots\!24\)\( \beta_{2} + 47755489026283332176 \beta_{3} + 5038109519575472 \beta_{4} - 45309613567146944 \beta_{5}) q^{64} +(\)\(84\!\cdots\!28\)\( + \)\(60\!\cdots\!12\)\( \beta_{1} - \)\(40\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!08\)\( \beta_{3} + 59892633435960908 \beta_{4} - 105372895828350616 \beta_{5}) q^{66} +(-\)\(55\!\cdots\!58\)\( + \)\(41\!\cdots\!71\)\( \beta_{1} + \)\(70\!\cdots\!77\)\( \beta_{2} - 43285766613366926112 \beta_{3} + 231815093720766642 \beta_{4} - 180299083915239930 \beta_{5}) q^{67} +(-\)\(64\!\cdots\!80\)\( + \)\(83\!\cdots\!30\)\( \beta_{1} + \)\(19\!\cdots\!30\)\( \beta_{2} + \)\(12\!\cdots\!94\)\( \beta_{3} + 76916991962596496 \beta_{4} - 82472691965378240 \beta_{5}) q^{68} +(\)\(52\!\cdots\!96\)\( - \)\(17\!\cdots\!40\)\( \beta_{1} + \)\(13\!\cdots\!48\)\( \beta_{2} - \)\(11\!\cdots\!08\)\( \beta_{3} + 230176942325623884 \beta_{4} - 159634180030227468 \beta_{5}) q^{69} +(\)\(14\!\cdots\!12\)\( - \)\(12\!\cdots\!50\)\( \beta_{1} - \)\(26\!\cdots\!50\)\( \beta_{2} - 40820877937029072000 \beta_{3} + 207364024339354400 \beta_{4} - 90375567776933600 \beta_{5}) q^{71} +(-\)\(14\!\cdots\!42\)\( - \)\(31\!\cdots\!76\)\( \beta_{1} - \)\(16\!\cdots\!62\)\( \beta_{2} - \)\(16\!\cdots\!46\)\( \beta_{3} + 146088562735132686 \beta_{4} - 161652988215842040 \beta_{5}) q^{72} +(\)\(28\!\cdots\!22\)\( + \)\(25\!\cdots\!36\)\( \beta_{1} + \)\(24\!\cdots\!72\)\( \beta_{2} - 88538608537798936848 \beta_{3} + 80987079575138968 \beta_{4} + 537894925079953280 \beta_{5}) q^{73} +(\)\(34\!\cdots\!40\)\( + \)\(42\!\cdots\!38\)\( \beta_{1} + \)\(10\!\cdots\!84\)\( \beta_{2} + \)\(21\!\cdots\!28\)\( \beta_{3} - 329207764768098616 \beta_{4} - 71601375399252368 \beta_{5}) q^{74} +(\)\(49\!\cdots\!72\)\( + \)\(12\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!44\)\( \beta_{2} + \)\(81\!\cdots\!40\)\( \beta_{3} - 515337314594221744 \beta_{4} + 749865497621658688 \beta_{5}) q^{76} +(\)\(19\!\cdots\!00\)\( + \)\(40\!\cdots\!36\)\( \beta_{1} + \)\(63\!\cdots\!16\)\( \beta_{2} + \)\(36\!\cdots\!68\)\( \beta_{3} - 1140507339514434488 \beta_{4} + 408073919617488320 \beta_{5}) q^{77} +(\)\(95\!\cdots\!90\)\( - \)\(44\!\cdots\!62\)\( \beta_{1} + \)\(10\!\cdots\!48\)\( \beta_{2} + \)\(19\!\cdots\!04\)\( \beta_{3} - 1281263495726796814 \beta_{4} - 1218660320970579940 \beta_{5}) q^{78} +(-\)\(36\!\cdots\!40\)\( - \)\(37\!\cdots\!60\)\( \beta_{1} - \)\(42\!\cdots\!08\)\( \beta_{2} + \)\(72\!\cdots\!68\)\( \beta_{3} - 805205716297778864 \beta_{4} + 1307751931934890928 \beta_{5}) q^{79} +(-\)\(22\!\cdots\!43\)\( + \)\(51\!\cdots\!28\)\( \beta_{1} - \)\(87\!\cdots\!04\)\( \beta_{2} - \)\(13\!\cdots\!24\)\( \beta_{3} - 2405565674579150020 \beta_{4} - 30198118577362260 \beta_{5}) q^{81} +(\)\(26\!\cdots\!76\)\( + \)\(74\!\cdots\!78\)\( \beta_{1} - \)\(16\!\cdots\!24\)\( \beta_{2} - \)\(51\!\cdots\!68\)\( \beta_{3} - 4585761217977876012 \beta_{4} + 7331031212250440280 \beta_{5}) q^{82} +(\)\(78\!\cdots\!34\)\( + \)\(15\!\cdots\!35\)\( \beta_{1} - \)\(62\!\cdots\!13\)\( \beta_{2} - \)\(39\!\cdots\!00\)\( \beta_{3} + 2086873799279959500 \beta_{4} - 2166286256070697500 \beta_{5}) q^{83} +(-\)\(39\!\cdots\!32\)\( - \)\(21\!\cdots\!08\)\( \beta_{1} - \)\(24\!\cdots\!32\)\( \beta_{2} - \)\(44\!\cdots\!60\)\( \beta_{3} - 911602853655848608 \beta_{4} + 6366546648555065216 \beta_{5}) q^{84} +(-\)\(21\!\cdots\!71\)\( + \)\(18\!\cdots\!51\)\( \beta_{1} + \)\(22\!\cdots\!56\)\( \beta_{2} - \)\(53\!\cdots\!32\)\( \beta_{3} + 4876108330794071357 \beta_{4} - 49702121502984314 \beta_{5}) q^{86} +(\)\(29\!\cdots\!56\)\( + \)\(10\!\cdots\!14\)\( \beta_{1} - \)\(81\!\cdots\!98\)\( \beta_{2} - \)\(20\!\cdots\!68\)\( \beta_{3} + 12679467875975812288 \beta_{4} - 9872598734998961120 \beta_{5}) q^{87} +(-\)\(82\!\cdots\!72\)\( - \)\(88\!\cdots\!84\)\( \beta_{1} - \)\(93\!\cdots\!80\)\( \beta_{2} - \)\(16\!\cdots\!68\)\( \beta_{3} + 7981295213666861288 \beta_{4} + 8030832015281795680 \beta_{5}) q^{88} +(-\)\(56\!\cdots\!98\)\( - \)\(91\!\cdots\!44\)\( \beta_{1} - \)\(33\!\cdots\!56\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3} + 7021873421274053016 \beta_{4} - 24228933901779274632 \beta_{5}) q^{89} +(\)\(32\!\cdots\!68\)\( + \)\(13\!\cdots\!88\)\( \beta_{1} + \)\(56\!\cdots\!82\)\( \beta_{2} - \)\(73\!\cdots\!20\)\( \beta_{3} + 19731114985660701678 \beta_{4} - 13984987258151544006 \beta_{5}) q^{91} +(\)\(69\!\cdots\!56\)\( + \)\(10\!\cdots\!56\)\( \beta_{1} + \)\(22\!\cdots\!44\)\( \beta_{2} + \)\(28\!\cdots\!52\)\( \beta_{3} + 17131266635838573268 \beta_{4} - 33821537288514599920 \beta_{5}) q^{92} +(-\)\(35\!\cdots\!20\)\( + \)\(43\!\cdots\!16\)\( \beta_{1} - \)\(20\!\cdots\!84\)\( \beta_{2} + \)\(15\!\cdots\!56\)\( \beta_{3} - 19012349798843418296 \beta_{4} - 4943549919321157160 \beta_{5}) q^{93} +(\)\(24\!\cdots\!27\)\( + \)\(24\!\cdots\!69\)\( \beta_{1} - \)\(52\!\cdots\!20\)\( \beta_{2} + \)\(30\!\cdots\!76\)\( \beta_{3} + 17191986391800001331 \beta_{4} + 9663523947126672538 \beta_{5}) q^{94} +(\)\(52\!\cdots\!40\)\( + \)\(75\!\cdots\!24\)\( \beta_{1} + \)\(20\!\cdots\!24\)\( \beta_{2} + \)\(26\!\cdots\!52\)\( \beta_{3} - 66536165423703186592 \beta_{4} + 11533125890540845184 \beta_{5}) q^{96} +(-\)\(42\!\cdots\!26\)\( + \)\(26\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!68\)\( \beta_{2} - \)\(38\!\cdots\!84\)\( \beta_{3} - 3898782859774391856 \beta_{4} - 11150994043844352360 \beta_{5}) q^{97} +(-\)\(11\!\cdots\!58\)\( + \)\(17\!\cdots\!23\)\( \beta_{1} - \)\(33\!\cdots\!96\)\( \beta_{2} + \)\(21\!\cdots\!88\)\( \beta_{3} - 70950961974920944108 \beta_{4} + 58534955193326850520 \beta_{5}) q^{98} +(\)\(27\!\cdots\!72\)\( - \)\(23\!\cdots\!67\)\( \beta_{1} + \)\(23\!\cdots\!12\)\( \beta_{2} - \)\(95\!\cdots\!20\)\( \beta_{3} - 57462850055393817677 \beta_{4} - 62703625440425722671 \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 147350q^{2} - 26513900q^{3} + 29520645652q^{4} + 6615759249352q^{6} - 19113832847500q^{7} - 3069820115785800q^{8} + 13602102583345438q^{9} + O(q^{10}) \) \( 6q - 147350q^{2} - 26513900q^{3} + 29520645652q^{4} + 6615759249352q^{6} - 19113832847500q^{7} - 3069820115785800q^{8} + 13602102583345438q^{9} + 150760896555890192q^{11} - 466351581756413200q^{12} + 1403095636752804700q^{13} - 7086110008989891456q^{14} + 64511520005858668816q^{16} - \)\(13\!\cdots\!00\)\(q^{17} - \)\(67\!\cdots\!50\)\(q^{18} + \)\(17\!\cdots\!00\)\(q^{19} - \)\(10\!\cdots\!28\)\(q^{21} - \)\(24\!\cdots\!00\)\(q^{22} - \)\(68\!\cdots\!00\)\(q^{23} + \)\(21\!\cdots\!00\)\(q^{24} - \)\(36\!\cdots\!88\)\(q^{26} + \)\(34\!\cdots\!00\)\(q^{27} + \)\(17\!\cdots\!00\)\(q^{28} + \)\(13\!\cdots\!00\)\(q^{29} + \)\(10\!\cdots\!52\)\(q^{31} - \)\(26\!\cdots\!00\)\(q^{32} - \)\(34\!\cdots\!00\)\(q^{33} + \)\(88\!\cdots\!24\)\(q^{34} + \)\(16\!\cdots\!96\)\(q^{36} - \)\(27\!\cdots\!00\)\(q^{37} - \)\(83\!\cdots\!00\)\(q^{38} + \)\(57\!\cdots\!56\)\(q^{39} + \)\(44\!\cdots\!32\)\(q^{41} - \)\(14\!\cdots\!00\)\(q^{42} - \)\(22\!\cdots\!00\)\(q^{43} + \)\(71\!\cdots\!64\)\(q^{44} + \)\(34\!\cdots\!72\)\(q^{46} - \)\(38\!\cdots\!00\)\(q^{47} - \)\(24\!\cdots\!00\)\(q^{48} - \)\(71\!\cdots\!58\)\(q^{49} - \)\(54\!\cdots\!88\)\(q^{51} + \)\(90\!\cdots\!00\)\(q^{52} - \)\(47\!\cdots\!00\)\(q^{53} - \)\(18\!\cdots\!00\)\(q^{54} - \)\(39\!\cdots\!00\)\(q^{56} + \)\(24\!\cdots\!00\)\(q^{57} + \)\(44\!\cdots\!00\)\(q^{58} - \)\(17\!\cdots\!00\)\(q^{59} - \)\(28\!\cdots\!08\)\(q^{61} - \)\(34\!\cdots\!00\)\(q^{62} + \)\(55\!\cdots\!00\)\(q^{63} + \)\(19\!\cdots\!72\)\(q^{64} + \)\(50\!\cdots\!64\)\(q^{66} - \)\(33\!\cdots\!00\)\(q^{67} - \)\(38\!\cdots\!00\)\(q^{68} + \)\(31\!\cdots\!36\)\(q^{69} + \)\(84\!\cdots\!72\)\(q^{71} - \)\(87\!\cdots\!00\)\(q^{72} + \)\(17\!\cdots\!00\)\(q^{73} + \)\(20\!\cdots\!84\)\(q^{74} + \)\(29\!\cdots\!00\)\(q^{76} + \)\(11\!\cdots\!00\)\(q^{77} + \)\(57\!\cdots\!00\)\(q^{78} - \)\(22\!\cdots\!00\)\(q^{79} - \)\(13\!\cdots\!74\)\(q^{81} + \)\(15\!\cdots\!00\)\(q^{82} + \)\(47\!\cdots\!00\)\(q^{83} - \)\(23\!\cdots\!76\)\(q^{84} - \)\(12\!\cdots\!08\)\(q^{86} + \)\(17\!\cdots\!00\)\(q^{87} - \)\(49\!\cdots\!00\)\(q^{88} - \)\(33\!\cdots\!00\)\(q^{89} + \)\(19\!\cdots\!32\)\(q^{91} + \)\(41\!\cdots\!00\)\(q^{92} - \)\(21\!\cdots\!00\)\(q^{93} + \)\(14\!\cdots\!64\)\(q^{94} + \)\(31\!\cdots\!72\)\(q^{96} - \)\(25\!\cdots\!00\)\(q^{97} - \)\(69\!\cdots\!50\)\(q^{98} + \)\(16\!\cdots\!16\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 9680197848 x^{4} + 8052423720422 x^{3} + 24239866893261762265 x^{2} - 69081627028404093368325 x - 10572274201725134136583265250\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(64759 \nu^{5} + 178753080 \nu^{4} - 443366379644016 \nu^{3} - 2219345891051097862 \nu^{2} + 497177359501705207052073 \nu + 3158210254432945978141147422\)\()/ \)\(11\!\cdots\!72\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-64759 \nu^{5} - 178753080 \nu^{4} + 443366379644016 \nu^{3} + 467687168437307494150 \nu^{2} + 83028281302203390920919 \nu - 1505098511780245718200582504734\)\()/ \)\(11\!\cdots\!72\)\( \)
\(\beta_{4}\)\(=\)\((\)\(693353137 \nu^{5} + 22711573265160 \nu^{4} - 7564015938151063056 \nu^{3} - 142065885137099881758346 \nu^{2} + 18384255498505988786988504111 \nu + 90683077682865758752943810163378\)\()/ 50910543090996793344 \)
\(\beta_{5}\)\(=\)\((\)\(2773802021 \nu^{5} - 549722510060376 \nu^{4} - 24950793531285037392 \nu^{3} + 3482773264826083164386926 \nu^{2} + 52334398719922215930068580411 \nu - 3030648596667910248295146115485222\)\()/ \)\(40\!\cdots\!52\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2493 \beta_{1} + 12906931296\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-20 \beta_{5} - 67 \beta_{4} + 5000 \beta_{3} + 1889408 \beta_{2} + 9344083493 \beta_{1} - 16088601109849\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-2708988 \beta_{5} + 716451 \beta_{4} + 6365565999 \beta_{3} + 21984654807 \beta_{2} - 4547030871598 \beta_{1} + 60301975528888621041\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-64725270960 \beta_{5} - 230343390564 \beta_{4} + 25466060786969 \beta_{3} + 10048466840966201 \beta_{2} + 24272901966106437312 \beta_{1} - 14671356594135807142800\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
77152.3
54181.8
25925.6
−21408.2
−60371.0
−75479.4
−178863. −8.55112e7 2.34019e10 0 1.52948e13 8.09549e13 −2.64931e15 1.75311e15 0
1.2 −132922. 1.17526e8 9.07819e9 0 −1.56217e13 1.34853e13 −6.49000e13 8.25322e15 0
1.3 −76409.2 −7.43191e7 −2.75157e9 0 5.67866e12 −1.36987e14 8.66595e14 −3.57381e13 0
1.4 18258.4 3.67420e7 −8.25656e9 0 6.70852e11 1.51681e13 −3.07591e14 −4.20908e15 0
1.5 96184.0 −1.07272e8 6.61431e8 0 −1.03179e13 8.61302e13 −7.62595e14 5.94832e15 0
1.6 126401. 8.63211e7 7.38725e9 0 1.09111e13 −7.78650e13 −1.52021e14 1.89228e15 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.34.a.c 6
5.b even 2 1 5.34.a.b 6
5.c odd 4 2 25.34.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.34.a.b 6 5.b even 2 1
25.34.a.c 6 1.a even 1 1 trivial
25.34.b.c 12 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 147350 T_{2}^{5} - 29674115352 T_{2}^{4} - \)3571811788892800

'>\(35\!\cdots\!00\)\( T_{2}^{3} + \)248434013984696762368
'>\(24\!\cdots\!68\)\( T_{2}^{2} + \)18902755381238535133593600'>\(18\!\cdots\!00\)\( T_{2} - \)403252523683384760678328827904'>\(40\!\cdots\!04\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(25))\).