Properties

Label 25.34.b.c
Level $25$
Weight $34$
Character orbit 25.b
Analytic conductor $172.457$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 20265063301 x^{10} + 151877130254993985540 x^{8} + 529361026408998369955851761920 x^{6} + 862054500541443268723145801151954288640 x^{4} + 544606882049987247786598582768552479276511789056 x^{2} + 39700341273686227371481766034965422500156905479886340096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{47}\cdot 3^{10}\cdot 5^{36}\cdot 7^{2}\cdot 11^{4} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{6} + \beta_{7} ) q^{2} + ( 158 \beta_{6} - 77 \beta_{7} + \beta_{8} ) q^{3} + ( -4920107609 + \beta_{1} - \beta_{2} ) q^{4} + ( 1102626541558 - 469 \beta_{1} - \beta_{2} + 24 \beta_{3} - \beta_{4} ) q^{6} + ( -81027079 \beta_{6} - 97559503 \beta_{7} + 216966 \beta_{8} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{7} + ( 17043855246 \beta_{6} - 3133923466 \beta_{7} + 3873134 \beta_{8} + 19 \beta_{9} + 65 \beta_{10} + 11 \beta_{11} ) q^{8} + ( -2267017097238785 + 192960 \beta_{1} - 43272 \beta_{2} + 4986 \beta_{3} - 336 \beta_{4} - 28 \beta_{5} ) q^{9} +O(q^{10})\) \( q +(-\beta_{6} + \beta_{7}) q^{2} +(158 \beta_{6} - 77 \beta_{7} + \beta_{8}) q^{3} +(-4920107609 + \beta_{1} - \beta_{2}) q^{4} +(1102626541558 - 469 \beta_{1} - \beta_{2} + 24 \beta_{3} - \beta_{4}) q^{6} +(-81027079 \beta_{6} - 97559503 \beta_{7} + 216966 \beta_{8} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11}) q^{7} +(17043855246 \beta_{6} - 3133923466 \beta_{7} + 3873134 \beta_{8} + 19 \beta_{9} + 65 \beta_{10} + 11 \beta_{11}) q^{8} +(-2267017097238785 + 192960 \beta_{1} - 43272 \beta_{2} + 4986 \beta_{3} - 336 \beta_{4} - 28 \beta_{5}) q^{9} +(25126816090199447 - 14413046 \beta_{1} - 7346728 \beta_{2} + 577206 \beta_{3} - 1350 \beta_{4} + 1323 \beta_{5}) q^{11} +(-3199256566908 \beta_{6} + 3321991441364 \beta_{7} - 6704302652 \beta_{8} - 21186 \beta_{9} - 22834 \beta_{10} - 1714 \beta_{11}) q^{12} +(-8387074752769 \beta_{6} + 4227611318652 \beta_{7} + 13181999140 \beta_{8} - 33220 \beta_{9} - 113336 \beta_{10} + 2572 \beta_{11}) q^{13} +(1181018334891663176 - 793421321 \beta_{1} + 180435773 \beta_{2} - 28744584 \beta_{3} - 333349 \beta_{4} - 58624 \beta_{5}) q^{14} +(10751920007426917244 - 4298613432 \beta_{1} + 19358092744 \beta_{2} + 345489912 \beta_{3} - 5261028 \beta_{4} - 1414392 \beta_{5}) q^{16} +(-731985235456705 \beta_{6} + 71979343991268 \beta_{7} + 1206195272012 \beta_{8} + 7165684 \beta_{9} - 6713080 \beta_{10} + 2320148 \beta_{11}) q^{17} +(4334947353973253 \beta_{6} - 3473895864045165 \beta_{7} - 3656013440552 \beta_{8} - 1626574 \beta_{9} - 790088 \beta_{10} + 2326922 \beta_{11}) q^{18} +(-\)\(28\!\cdots\!15\)\( - 34918200554 \beta_{1} - 342670207344 \beta_{2} + 5328047433 \beta_{3} - 9006490 \beta_{4} + 12206789 \beta_{5}) q^{19} +(-\)\(17\!\cdots\!60\)\( + 591204537160 \beta_{1} + 1005012220880 \beta_{2} - 8725120688 \beta_{3} - 150615128 \beta_{4} + 57788532 \beta_{5}) q^{21} +(-36431397220371032 \beta_{6} + 108546291928335100 \beta_{7} - 94929783983932 \beta_{8} - 325069925 \beta_{9} - 101378380 \beta_{10} + 970847 \beta_{11}) q^{22} +(45885434488382249 \beta_{6} - 42933048238174583 \beta_{7} + 639665020750 \beta_{8} - 1006032495 \beta_{9} - 788124893 \beta_{10} + 31884690 \beta_{11}) q^{23} +(-\)\(35\!\cdots\!64\)\( + 7376866278288 \beta_{1} - 6203339970864 \beta_{2} - 390398728368 \beta_{3} - 677882136 \beta_{4} + 881900208 \beta_{5}) q^{24} +(-\)\(60\!\cdots\!29\)\( + 10430594660524 \beta_{1} - 33963961882237 \beta_{2} - 824821320096 \beta_{3} - 11422638852 \beta_{4} + 2611370496 \beta_{5}) q^{26} +(2299088537948646532 \beta_{6} - 2250451666600822110 \beta_{7} + 918941113875110 \beta_{8} - 5712470624 \beta_{9} + 1397254448 \beta_{10} - 1264420976 \beta_{11}) q^{27} +(-10470715434620607320 \beta_{6} + 5300115833922306696 \beta_{7} - 4239827850235192 \beta_{8} - 6205640394 \beta_{9} - 22131550924 \beta_{10} - 727658106 \beta_{11}) q^{28} +(-\)\(21\!\cdots\!74\)\( - 9766658160152 \beta_{1} + 192624564103984 \beta_{2} + 2475437599560 \beta_{3} - 136047966712 \beta_{4} + 9668691956 \beta_{5}) q^{29} +(\)\(18\!\cdots\!38\)\( + 123262136993740 \beta_{1} - 46308791692232 \beta_{2} + 3329891184987 \beta_{3} + 33609440748 \beta_{4} + 38296391322 \beta_{5}) q^{31} +(-\)\(14\!\cdots\!08\)\( \beta_{6} + 23674690450925055848 \beta_{7} - 62354567493790968 \beta_{8} - 126374431156 \beta_{9} - 619422027204 \beta_{10} + 120225467308 \beta_{11}) q^{32} +(\)\(19\!\cdots\!36\)\( \beta_{6} - 54300684157922489532 \beta_{7} + 43575985890498164 \beta_{8} + 86508026172 \beta_{9} - 106773533776 \beta_{10} + 174075452276 \beta_{11}) q^{33} +(-\)\(14\!\cdots\!97\)\( - 1826974952460076 \beta_{1} - 2855190305417653 \beta_{2} + 66543944639904 \beta_{3} - 1663003909500 \beta_{4} - 64534718976 \beta_{5}) q^{34} +(\)\(28\!\cdots\!41\)\( - 348457560452445 \beta_{1} + 4051206253430301 \beta_{2} - 153790690023840 \beta_{3} + 786968922864 \beta_{4} - 171492358240 \beta_{5}) q^{36} +(-\)\(15\!\cdots\!99\)\( \beta_{6} + \)\(17\!\cdots\!56\)\( \beta_{7} + 65271889433181280 \beta_{8} - 2041651136528 \beta_{9} + 2224952922280 \beta_{10} + 1717470790328 \beta_{11}) q^{37} +(\)\(45\!\cdots\!28\)\( \beta_{6} - \)\(30\!\cdots\!40\)\( \beta_{7} - 445872585963821240 \beta_{8} - 1112876284394 \beta_{9} + 9854068980040 \beta_{10} + 1099217434910 \beta_{11}) q^{38} +(-\)\(96\!\cdots\!64\)\( - 5500102713832928 \beta_{1} + 26877784087632408 \beta_{2} + 120626671881017 \beta_{3} - 6554882764640 \beta_{4} - 1836074549232 \beta_{5}) q^{39} +(\)\(74\!\cdots\!18\)\( + 17945598000906768 \beta_{1} - 59056610166152808 \beta_{2} + 2609528006825850 \beta_{3} + 672376427456 \beta_{4} - 4908195482260 \beta_{5}) q^{41} +(-\)\(68\!\cdots\!36\)\( \beta_{6} - \)\(46\!\cdots\!48\)\( \beta_{7} + 625914534644549520 \beta_{8} + 16001358558156 \beta_{9} + 19118057601360 \beta_{10} + 3848883086172 \beta_{11}) q^{42} +(\)\(11\!\cdots\!84\)\( \beta_{6} + \)\(23\!\cdots\!75\)\( \beta_{7} + 1224143130911256203 \beta_{8} - 5317073654818 \beta_{9} - 20864986451446 \beta_{10} - 10524453770708 \beta_{11}) q^{43} +(-\)\(11\!\cdots\!36\)\( + 90506771014268372 \beta_{1} - 98705104469026996 \beta_{2} - 3952964213602416 \beta_{3} + 100945101032808 \beta_{4} + 21240874323312 \beta_{5}) q^{44} +(\)\(58\!\cdots\!80\)\( + 186668445437585743 \beta_{1} - 102534241352167627 \beta_{2} - 23208647589995208 \beta_{3} + 53144035859411 \beta_{4} + 38060732568320 \beta_{5}) q^{46} +(-\)\(24\!\cdots\!77\)\( \beta_{6} + \)\(17\!\cdots\!69\)\( \beta_{7} + 6187862865866354348 \beta_{8} - 5602146991521 \beta_{9} + 71852550727037 \beta_{10} - 121583657907426 \beta_{11}) q^{47} +(\)\(14\!\cdots\!08\)\( \beta_{6} - \)\(56\!\cdots\!44\)\( \beta_{7} - 23294897049850602256 \beta_{8} + 99967112743464 \beta_{9} + 624741905794312 \beta_{10} - 39715065577112 \beta_{11}) q^{48} +(\)\(11\!\cdots\!75\)\( - 118008759315157952 \beta_{1} - 148904432930537240 \beta_{2} - 34799533307815698 \beta_{3} + 187668726936336 \beta_{4} - 86937898310100 \beta_{5}) q^{49} +(-\)\(91\!\cdots\!28\)\( + 1322170820810425184 \beta_{1} - 223915002289509736 \beta_{2} + 92045816019627561 \beta_{3} - 860331326202400 \beta_{4} - 71449121381904 \beta_{5}) q^{51} +(\)\(50\!\cdots\!30\)\( \beta_{6} - \)\(11\!\cdots\!06\)\( \beta_{7} + 40540855938923921158 \beta_{8} + 193437343593464 \beta_{9} + 1327940357423113 \beta_{10} - 116082502607816 \beta_{11}) q^{52} +(\)\(25\!\cdots\!11\)\( \beta_{6} - \)\(41\!\cdots\!64\)\( \beta_{7} + 67930793307109529004 \beta_{8} - 177214407399756 \beta_{9} + 2946295216711704 \beta_{10} + 309915317090916 \beta_{11}) q^{53} +(\)\(30\!\cdots\!64\)\( - 1373982223565016366 \beta_{1} + 2770417577616340362 \beta_{2} - 41466438931903728 \beta_{3} - 571326939885942 \beta_{4} + 116188313679872 \beta_{5}) q^{54} +(-\)\(65\!\cdots\!20\)\( + 2481698750198519696 \beta_{1} - 12646055594711508880 \beta_{2} - 526597851876242496 \beta_{3} + 1742330990072096 \beta_{4} - 8913531261376 \beta_{5}) q^{56} +(\)\(13\!\cdots\!20\)\( \beta_{6} - \)\(11\!\cdots\!84\)\( \beta_{7} - \)\(30\!\cdots\!44\)\( \beta_{8} - 2255938113700188 \beta_{9} - 4575522205907384 \beta_{10} + 2666758033246372 \beta_{11}) q^{57} +(-\)\(23\!\cdots\!26\)\( \beta_{6} - \)\(23\!\cdots\!50\)\( \beta_{7} - \)\(18\!\cdots\!00\)\( \beta_{8} - 1035259462617260 \beta_{9} - 2505510186306768 \beta_{10} + 180928150185284 \beta_{11}) q^{58} +(\)\(29\!\cdots\!03\)\( - 5620821608272942550 \beta_{1} - 2867801225814653632 \beta_{2} - 1134982169053807539 \beta_{3} - 11623024768669862 \beta_{4} + 59309055537403 \beta_{5}) q^{59} +(-\)\(47\!\cdots\!10\)\( + 9352186873198468576 \beta_{1} + 25248306766384150432 \beta_{2} - 38414614471709496 \beta_{3} + 14347001668460576 \beta_{4} - 485676071454784 \beta_{5}) q^{61} +(-\)\(23\!\cdots\!40\)\( \beta_{6} + \)\(10\!\cdots\!84\)\( \beta_{7} + \)\(74\!\cdots\!92\)\( \beta_{8} + 4369113047835571 \beta_{9} + 17475879167177652 \beta_{10} + 2523588915690647 \beta_{11}) q^{62} +(-\)\(29\!\cdots\!21\)\( \beta_{6} - \)\(34\!\cdots\!25\)\( \beta_{7} - \)\(14\!\cdots\!02\)\( \beta_{8} + 2352224116026483 \beta_{9} - 4462357949270199 \beta_{10} - 4085618162340138 \beta_{11}) q^{63} +(-\)\(32\!\cdots\!80\)\( + 47776029829395225888 \beta_{1} - \)\(11\!\cdots\!56\)\( \beta_{2} - 7121551013745550176 \beta_{3} + 20540803111893712 \beta_{4} + 1409335781119840 \beta_{5}) q^{64} +(\)\(84\!\cdots\!64\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} + \)\(13\!\cdots\!92\)\( \beta_{2} - 3290845884109055712 \beta_{3} - 53551190176789580 \beta_{4} + 576494841742848 \beta_{5}) q^{66} +(-\)\(17\!\cdots\!20\)\( \beta_{6} - \)\(41\!\cdots\!37\)\( \beta_{7} + \)\(69\!\cdots\!27\)\( \beta_{8} + 2060362376056926 \beta_{9} - 46044127358382198 \beta_{10} - 33999454406991060 \beta_{11}) q^{67} +(\)\(18\!\cdots\!34\)\( \beta_{6} + \)\(83\!\cdots\!30\)\( \beta_{7} - \)\(19\!\cdots\!38\)\( \beta_{8} - 3854477678893304 \beta_{9} - 92799824619872423 \beta_{10} + 12640060714182344 \beta_{11}) q^{68} +(-\)\(52\!\cdots\!76\)\( - \)\(11\!\cdots\!56\)\( \beta_{1} + \)\(51\!\cdots\!00\)\( \beta_{2} - 10724692889675836128 \beta_{3} + 97860272292374952 \beta_{4} - 12028788184840812 \beta_{5}) q^{69} +(\)\(14\!\cdots\!40\)\( + 40756229004286498688 \beta_{1} - \)\(38\!\cdots\!12\)\( \beta_{2} - 20871986387770004487 \beta_{3} - 64648932742573312 \beta_{4} + 12974099236071008 \beta_{5}) q^{71} +(-\)\(47\!\cdots\!02\)\( \beta_{6} + \)\(31\!\cdots\!14\)\( \beta_{7} - \)\(17\!\cdots\!26\)\( \beta_{8} + 8022562076704617 \beta_{9} - 142622259972036813 \beta_{10} - 24308035566463791 \beta_{11}) q^{72} +(-\)\(14\!\cdots\!95\)\( \beta_{6} + \)\(25\!\cdots\!44\)\( \beta_{7} - \)\(24\!\cdots\!08\)\( \beta_{8} + 83403997468707356 \beta_{9} + 39246665920146360 \beta_{10} - 24174987547283300 \beta_{11}) q^{73} +(-\)\(34\!\cdots\!27\)\( + \)\(21\!\cdots\!36\)\( \beta_{1} - \)\(12\!\cdots\!31\)\( \beta_{2} - 84428321204415444672 \beta_{3} - 8000326596500792 \beta_{4} + 29200676197417984 \beta_{5}) q^{74} +(\)\(49\!\cdots\!52\)\( - \)\(81\!\cdots\!08\)\( \beta_{1} + \)\(35\!\cdots\!16\)\( \beta_{2} + 15165361889480071200 \beta_{3} + 391781296704836432 \beta_{4} - 11232365262671392 \beta_{5}) q^{76} +(\)\(14\!\cdots\!72\)\( \beta_{6} - \)\(40\!\cdots\!36\)\( \beta_{7} + \)\(63\!\cdots\!52\)\( \beta_{8} + 56920385204995084 \beta_{9} + 323370282703214552 \beta_{10} + 138535169128492748 \beta_{11}) q^{77} +(-\)\(29\!\cdots\!52\)\( \beta_{6} - \)\(44\!\cdots\!80\)\( \beta_{7} - \)\(10\!\cdots\!24\)\( \beta_{8} - 298738794508560873 \beta_{9} - 1495032169328928028 \beta_{10} - 55006730314444885 \beta_{11}) q^{78} +(\)\(36\!\cdots\!08\)\( + \)\(72\!\cdots\!96\)\( \beta_{1} + \)\(15\!\cdots\!24\)\( \beta_{2} + 34831780714467703974 \beta_{3} - 672035536007085472 \beta_{4} + 12106380026426672 \beta_{5}) q^{79} +(-\)\(22\!\cdots\!79\)\( + \)\(13\!\cdots\!32\)\( \beta_{1} + \)\(81\!\cdots\!84\)\( \beta_{2} - 69267784167453771606 \beta_{3} + 275380708775458608 \beta_{4} - 193653178709426492 \beta_{5}) q^{81} +(\)\(84\!\cdots\!14\)\( \beta_{6} - \)\(74\!\cdots\!74\)\( \beta_{7} - \)\(16\!\cdots\!76\)\( \beta_{8} - 567768247917274038 \beta_{9} + 35642888984067672 \beta_{10} + 898437994532814018 \beta_{11}) q^{82} +(-\)\(28\!\cdots\!74\)\( \beta_{6} + \)\(15\!\cdots\!83\)\( \beta_{7} + \)\(63\!\cdots\!01\)\( \beta_{8} - 94592695921901700 \beta_{9} + 3274640421804628836 \beta_{10} + 338664555292237800 \beta_{11}) q^{83} +(\)\(39\!\cdots\!88\)\( - \)\(44\!\cdots\!88\)\( \beta_{1} + \)\(47\!\cdots\!24\)\( \beta_{2} + \)\(19\!\cdots\!08\)\( \beta_{3} - 2910672867802930528 \beta_{4} - 181733637649734720 \beta_{5}) q^{84} +(-\)\(21\!\cdots\!06\)\( + \)\(53\!\cdots\!71\)\( \beta_{1} + \)\(86\!\cdots\!11\)\( \beta_{2} + \)\(18\!\cdots\!84\)\( \beta_{3} - 607001933156601661 \beta_{4} + 388100581603406336 \beta_{5}) q^{86} +(\)\(11\!\cdots\!04\)\( \beta_{6} - \)\(10\!\cdots\!74\)\( \beta_{7} - \)\(81\!\cdots\!74\)\( \beta_{8} + 114217035302273328 \beta_{9} - 2301131466859279360 \beta_{10} - 1860302711697518896 \beta_{11}) q^{87} +(\)\(28\!\cdots\!92\)\( \beta_{6} - \)\(88\!\cdots\!52\)\( \beta_{7} + \)\(94\!\cdots\!08\)\( \beta_{8} + 1922446003506137524 \beta_{9} + 12742540028338808716 \beta_{10} + 316279600449778388 \beta_{11}) q^{88} +(\)\(56\!\cdots\!62\)\( - \)\(15\!\cdots\!00\)\( \beta_{1} + \)\(28\!\cdots\!92\)\( \beta_{2} + \)\(26\!\cdots\!64\)\( \beta_{3} + 11503355727335767200 \beta_{4} + 407407482369246744 \beta_{5}) q^{89} +(\)\(32\!\cdots\!94\)\( + \)\(73\!\cdots\!56\)\( \beta_{1} + \)\(46\!\cdots\!56\)\( \beta_{2} + \)\(45\!\cdots\!63\)\( \beta_{3} - 8521128191865963564 \beta_{4} + 1019089708526794374 \beta_{5}) q^{91} +(\)\(24\!\cdots\!68\)\( \beta_{6} - \)\(10\!\cdots\!68\)\( \beta_{7} + \)\(22\!\cdots\!36\)\( \beta_{8} + 3021888556808186662 \beta_{9} + 20596761858891689380 \beta_{10} - 3742418900894733322 \beta_{11}) q^{92} +(\)\(10\!\cdots\!80\)\( \beta_{6} + \)\(43\!\cdots\!76\)\( \beta_{7} + \)\(20\!\cdots\!92\)\( \beta_{8} - 2593331968589303832 \beta_{9} - 11859366252315324968 \beta_{10} - 1604621984725072400 \beta_{11}) q^{93} +(-\)\(24\!\cdots\!40\)\( + \)\(30\!\cdots\!19\)\( \beta_{1} + \)\(89\!\cdots\!77\)\( \beta_{2} + \)\(42\!\cdots\!20\)\( \beta_{3} - 2188912169719735757 \beta_{4} - 1761899869229067008 \beta_{5}) q^{94} +(\)\(52\!\cdots\!72\)\( - \)\(26\!\cdots\!80\)\( \beta_{1} + \)\(22\!\cdots\!28\)\( \beta_{2} + \)\(16\!\cdots\!68\)\( \beta_{3} + 13058915242682354272 \beta_{4} - 4861568198274621120 \beta_{5}) q^{96} +(-\)\(13\!\cdots\!85\)\( \beta_{6} - \)\(26\!\cdots\!40\)\( \beta_{7} - \)\(12\!\cdots\!96\)\( \beta_{8} + 1951017452115648516 \beta_{9} - 31743670890463421248 \beta_{10} - 279181356653221956 \beta_{11}) q^{97} +(-\)\(34\!\cdots\!19\)\( \beta_{6} + \)\(17\!\cdots\!39\)\( \beta_{7} + \)\(33\!\cdots\!76\)\( \beta_{8} + 1099797529573664662 \beta_{9} - 21000924808603995864 \beta_{10} - 10607193509091705442 \beta_{11}) q^{98} +(-\)\(27\!\cdots\!71\)\( - \)\(95\!\cdots\!66\)\( \beta_{1} + \)\(69\!\cdots\!20\)\( \beta_{2} - \)\(18\!\cdots\!82\)\( \beta_{3} + 20694053187140059854 \beta_{4} + 7105173022048534321 \beta_{5}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + O(q^{10}) \) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + 301521793111780384q^{11} + 14172220017979782912q^{14} + \)\(12\!\cdots\!32\)\(q^{16} - \)\(34\!\cdots\!00\)\(q^{19} - \)\(21\!\cdots\!56\)\(q^{21} - \)\(42\!\cdots\!00\)\(q^{24} - \)\(72\!\cdots\!76\)\(q^{26} - \)\(26\!\cdots\!00\)\(q^{29} + \)\(21\!\cdots\!04\)\(q^{31} - \)\(17\!\cdots\!48\)\(q^{34} + \)\(33\!\cdots\!92\)\(q^{36} - \)\(11\!\cdots\!12\)\(q^{39} + \)\(89\!\cdots\!64\)\(q^{41} - \)\(14\!\cdots\!28\)\(q^{44} + \)\(69\!\cdots\!44\)\(q^{46} + \)\(14\!\cdots\!16\)\(q^{49} - \)\(10\!\cdots\!76\)\(q^{51} + \)\(36\!\cdots\!00\)\(q^{54} - \)\(78\!\cdots\!00\)\(q^{56} + \)\(35\!\cdots\!00\)\(q^{59} - \)\(56\!\cdots\!16\)\(q^{61} - \)\(38\!\cdots\!44\)\(q^{64} + \)\(10\!\cdots\!28\)\(q^{66} - \)\(63\!\cdots\!72\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} - \)\(40\!\cdots\!68\)\(q^{74} + \)\(58\!\cdots\!00\)\(q^{76} + \)\(44\!\cdots\!00\)\(q^{79} - \)\(26\!\cdots\!48\)\(q^{81} + \)\(47\!\cdots\!52\)\(q^{84} - \)\(25\!\cdots\!16\)\(q^{86} + \)\(67\!\cdots\!00\)\(q^{89} + \)\(38\!\cdots\!64\)\(q^{91} - \)\(29\!\cdots\!28\)\(q^{94} + \)\(63\!\cdots\!44\)\(q^{96} - \)\(33\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 20265063301 x^{10} + 151877130254993985540 x^{8} + 529361026408998369955851761920 x^{6} + 862054500541443268723145801151954288640 x^{4} + 544606882049987247786598582768552479276511789056 x^{2} + 39700341273686227371481766034965422500156905479886340096\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(7848608243207638774181 \nu^{10} + 109213289539878176731567001176505 \nu^{8} + 421129850774205585968397622485070495258260 \nu^{6} + 381336324335695060763529782512521527645565887970560 \nu^{4} + 81459543577587949471936977571247424351350986792107360911360 \nu^{2} + 785111750092814602956289868633733557315025731867482052503163514126336\)\()/ \)\(65\!\cdots\!20\)\( \)
\(\beta_{2}\)\(=\)\((\)\(7848608243207638774181 \nu^{10} + 109213289539878176731567001176505 \nu^{8} + 421129850774205585968397622485070495258260 \nu^{6} + 381336324335695060763529782512521527645565887970560 \nu^{4} - 182331702486824967493652637302002413650216267250347718737920 \nu^{2} - 105845966553333815167366271812502087039371044695831940423983755689984\)\()/ \)\(65\!\cdots\!20\)\( \)
\(\beta_{3}\)\(=\)\((\)\(779752462997331626941642252141933671097 \nu^{10} + 14672818909053308825667435805646958110478587041053 \nu^{8} + 91421103788768827293549552555160022965071709572377777478372 \nu^{6} + 206127504938446746433228654810638600563046039495876637084303269261568 \nu^{4} + 109537866708333871775635719606437171305047505021772092327545119908522069327872 \nu^{2} - 15603268507081392682905408953736073373661942030099639487442616777552989000249408225280\)\()/ \)\(49\!\cdots\!64\)\( \)
\(\beta_{4}\)\(=\)\((\)\(350121954757571366830303302940430164875383 \nu^{10} + 5930321400364670713635104696727227370104247784262355 \nu^{8} + 33820802615823330154452308433429840116220207365086042601427420 \nu^{6} + 77945238850822526911332130188006417390399749054184804469668844032638720 \nu^{4} + 61901518503950409976851528633913451342380819934807932614755613333109919113543680 \nu^{2} + 2688093683288572499035759923067300517762486156350574577888951867806370937897356915376128\)\()/ \)\(31\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-1260157163036734973356299516482954226011675 \nu^{10} - 22900200286228827686473732776956860686313477667476743 \nu^{8} - 144980767243686270440910960213045026975993319693153437596133484 \nu^{6} - 388426630249828903567866623629991497994373241922559938227693430213231360 \nu^{4} - 398838530929058664273405850353152822901361248471364156585016941696421113960071168 \nu^{2} - 71127929854873306295743175898375108398234865175452447802103135771770073552607610637647872\)\()/ \)\(49\!\cdots\!64\)\( \)
\(\beta_{6}\)\(=\)\((\)\(305741908930145219113367453699319307 \nu^{11} + 7920651499707618803034482542423463816736535735 \nu^{9} + 70435390443738777895157996530061258130201029617557170220 \nu^{7} + 254393319261217910034937430305239497400556441348296313134824600320 \nu^{5} + 347366823341471024896056257591738384815291360423060076094233818300426485760 \nu^{3} + 126440810607153327159583966138456550631963166294708474985847342207541548652777439232 \nu\)\()/ \)\(28\!\cdots\!20\)\( \)
\(\beta_{7}\)\(=\)\((\)\(305741908930145219113367453699319307 \nu^{11} + 7920651499707618803034482542423463816736535735 \nu^{9} + 70435390443738777895157996530061258130201029617557170220 \nu^{7} + 254393319261217910034937430305239497400556441348296313134824600320 \nu^{5} + 347366823341471024896056257591738384815291360423060076094233818300426485760 \nu^{3} + 184410304855167094315329602634795845312568168489133897461041620761090338089045327872 \nu\)\()/ \)\(28\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(19\!\cdots\!65\)\( \nu^{11} + \)\(32\!\cdots\!73\)\( \nu^{9} + \)\(18\!\cdots\!96\)\( \nu^{7} + \)\(42\!\cdots\!28\)\( \nu^{5} + \)\(31\!\cdots\!76\)\( \nu^{3} - \)\(23\!\cdots\!56\)\( \nu\)\()/ \)\(11\!\cdots\!96\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(63\!\cdots\!57\)\( \nu^{11} - \)\(12\!\cdots\!65\)\( \nu^{9} - \)\(63\!\cdots\!20\)\( \nu^{7} + \)\(12\!\cdots\!20\)\( \nu^{5} + \)\(69\!\cdots\!40\)\( \nu^{3} + \)\(95\!\cdots\!08\)\( \nu\)\()/ \)\(19\!\cdots\!20\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(15\!\cdots\!49\)\( \nu^{11} - \)\(28\!\cdots\!05\)\( \nu^{9} - \)\(18\!\cdots\!40\)\( \nu^{7} - \)\(50\!\cdots\!00\)\( \nu^{5} - \)\(59\!\cdots\!20\)\( \nu^{3} - \)\(20\!\cdots\!04\)\( \nu\)\()/ \)\(10\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(38\!\cdots\!33\)\( \nu^{11} + \)\(64\!\cdots\!45\)\( \nu^{9} + \)\(35\!\cdots\!00\)\( \nu^{7} + \)\(78\!\cdots\!80\)\( \nu^{5} + \)\(65\!\cdots\!20\)\( \nu^{3} + \)\(35\!\cdots\!28\)\( \nu\)\()/ \)\(19\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + \beta_{1} - 13510042201\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(11 \beta_{11} + 65 \beta_{10} + 19 \beta_{9} + 3873134 \beta_{8} - 20313792650 \beta_{7} + 34223724430 \beta_{6}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-353598 \beta_{5} - 1315257 \beta_{4} + 86372478 \beta_{3} + 11281974130 \beta_{2} - 7517104302 \beta_{1} + 71279020059459465439\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-64432913685 \beta_{11} - 713201255281 \beta_{10} - 194802365037 \beta_{9} - 48858609599499070 \beta_{8} + 125072590570552957914 \beta_{7} - 275666010830428356414 \beta_{6}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(3884813101147510 \beta_{5} + 15406264696581037 \beta_{4} - 1372514359070295606 \beta_{3} - 100846523046783227142 \beta_{2} + 56030178598050142674 \beta_{1} - 451329467252485042487243261559\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(302302303680406610109 \beta_{11} + 6291020796553159138737 \beta_{10} + 1737503527084330808661 \beta_{9} + 462270626762518886745869694 \beta_{8} - 864295663361844988811576106010 \beta_{7} + 2189494801640704527081309719806 \beta_{6}\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-34826697222155308897682262 \beta_{5} - 141945025749529519452582141 \beta_{4} + 14524064926120239218357213718 \beta_{3} + 838057657032840804403677768022 \beta_{2} - 429307619347051326839912691298 \beta_{1} + 3173445014358579648702364414818162511447\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-1308827057487898326272769587165 \beta_{11} - 52328675304277523873962965969713 \beta_{10} - 14715318205475152775591872619893 \beta_{9} - 3994706997395561252461180656310832510 \beta_{8} + 6385192872591047993133982381460753576474 \beta_{7} - 17322704023452545461401022100432491715326 \beta_{6}\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(293347189226059564099049597362386390 \beta_{5} + 1212418841447683022624856629610556605 \beta_{4} - 132654232683476356039849811941727093910 \beta_{3} - 6788250565998821936013799070911285070230 \beta_{2} + 3355880736056647789350722556570709917090 \beta_{1} - 23664738624126581050595468040571460103765650605399\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(5377901691575080707062150520272218883805 \beta_{11} + 425919645763689600736635713111689759798705 \beta_{10} + 121441050111342905695890958979800947852405 \beta_{9} + 33244953350490331060031239366836261782628619390 \beta_{8} - 48946322505401492959635457307899960822626594565914 \beta_{7} + 137260301687161478568731738226990526129473840912894 \beta_{6}\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
89431.3i
66460.8i
63200.4i
48092.0i
38204.6i
9129.22i
9129.22i
38204.6i
48092.0i
63200.4i
66460.8i
89431.3i
178863.i 8.55112e7i −2.34019e10 0 1.52948e13 8.09549e13i 2.64931e15i −1.75311e15 0
24.2 132922.i 1.17526e8i −9.07819e9 0 −1.56217e13 1.34853e13i 6.49000e13i −8.25322e15 0
24.3 126401.i 8.63211e7i −7.38725e9 0 1.09111e13 7.78650e13i 1.52021e14i −1.89228e15 0
24.4 96184.0i 1.07272e8i −6.61431e8 0 −1.03179e13 8.61302e13i 7.62595e14i −5.94832e15 0
24.5 76409.2i 7.43191e7i 2.75157e9 0 5.67866e12 1.36987e14i 8.66595e14i 3.57381e13 0
24.6 18258.4i 3.67420e7i 8.25656e9 0 6.70852e11 1.51681e13i 3.07591e14i 4.20908e15 0
24.7 18258.4i 3.67420e7i 8.25656e9 0 6.70852e11 1.51681e13i 3.07591e14i 4.20908e15 0
24.8 76409.2i 7.43191e7i 2.75157e9 0 5.67866e12 1.36987e14i 8.66595e14i 3.57381e13 0
24.9 96184.0i 1.07272e8i −6.61431e8 0 −1.03179e13 8.61302e13i 7.62595e14i −5.94832e15 0
24.10 126401.i 8.63211e7i −7.38725e9 0 1.09111e13 7.78650e13i 1.52021e14i −1.89228e15 0
24.11 132922.i 1.17526e8i −9.07819e9 0 −1.56217e13 1.34853e13i 6.49000e13i −8.25322e15 0
24.12 178863.i 8.55112e7i −2.34019e10 0 1.52948e13 8.09549e13i 2.64931e15i −1.75311e15 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 81060253204 T_{2}^{10} + \)\(24\!\cdots\!40\)\( T_{2}^{8} + \)\(33\!\cdots\!80\)\( T_{2}^{6} + \)\(22\!\cdots\!40\)\( T_{2}^{4} + \)\(55\!\cdots\!44\)\( T_{2}^{2} + \)\(16\!\cdots\!16\)\( \) acting on \(S_{34}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(16\!\cdots\!16\)\( + \)\(55\!\cdots\!44\)\( T^{2} + \)\(22\!\cdots\!40\)\( T^{4} + \)\(33\!\cdots\!80\)\( T^{6} + \)\(24\!\cdots\!40\)\( T^{8} + 81060253204 T^{10} + T^{12} \)
$3$ \( \)\(64\!\cdots\!96\)\( + \)\(87\!\cdots\!56\)\( T^{2} + \)\(38\!\cdots\!40\)\( T^{4} + \)\(80\!\cdots\!20\)\( T^{6} + \)\(87\!\cdots\!40\)\( T^{8} + 46956465982678576 T^{10} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( \)\(23\!\cdots\!36\)\( + \)\(23\!\cdots\!44\)\( T^{2} + \)\(67\!\cdots\!40\)\( T^{4} + \)\(30\!\cdots\!80\)\( T^{6} + \)\(52\!\cdots\!40\)\( T^{8} + \)\(39\!\cdots\!84\)\( T^{10} + T^{12} \)
$11$ \( ( \)\(24\!\cdots\!24\)\( - \)\(76\!\cdots\!92\)\( T - \)\(46\!\cdots\!60\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} - \)\(63\!\cdots\!40\)\( T^{4} - 150760896555890192 T^{5} + T^{6} )^{2} \)
$13$ \( \)\(26\!\cdots\!56\)\( + \)\(22\!\cdots\!56\)\( T^{2} + \)\(28\!\cdots\!40\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!40\)\( T^{8} + \)\(26\!\cdots\!36\)\( T^{10} + T^{12} \)
$17$ \( \)\(16\!\cdots\!76\)\( + \)\(63\!\cdots\!44\)\( T^{2} + \)\(62\!\cdots\!40\)\( T^{4} + \)\(25\!\cdots\!80\)\( T^{6} + \)\(45\!\cdots\!40\)\( T^{8} + \)\(35\!\cdots\!44\)\( T^{10} + T^{12} \)
$19$ \( ( -\)\(28\!\cdots\!00\)\( + \)\(85\!\cdots\!00\)\( T + \)\(81\!\cdots\!00\)\( T^{2} - \)\(76\!\cdots\!00\)\( T^{3} - \)\(53\!\cdots\!00\)\( T^{4} + \)\(17\!\cdots\!00\)\( T^{5} + T^{6} )^{2} \)
$23$ \( \)\(43\!\cdots\!16\)\( + \)\(35\!\cdots\!56\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!40\)\( T^{8} + \)\(59\!\cdots\!96\)\( T^{10} + T^{12} \)
$29$ \( ( -\)\(16\!\cdots\!00\)\( + \)\(55\!\cdots\!00\)\( T + \)\(12\!\cdots\!00\)\( T^{2} - \)\(56\!\cdots\!00\)\( T^{3} - \)\(71\!\cdots\!00\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + T^{6} )^{2} \)
$31$ \( ( \)\(11\!\cdots\!44\)\( + \)\(68\!\cdots\!08\)\( T - \)\(71\!\cdots\!60\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(33\!\cdots\!60\)\( T^{4} - \)\(10\!\cdots\!52\)\( T^{5} + T^{6} )^{2} \)
$37$ \( \)\(51\!\cdots\!56\)\( + \)\(45\!\cdots\!44\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{4} + \)\(21\!\cdots\!80\)\( T^{6} + \)\(16\!\cdots\!40\)\( T^{8} + \)\(64\!\cdots\!64\)\( T^{10} + T^{12} \)
$41$ \( ( \)\(96\!\cdots\!04\)\( + \)\(55\!\cdots\!08\)\( T + \)\(76\!\cdots\!40\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} - \)\(59\!\cdots\!40\)\( T^{4} - \)\(44\!\cdots\!32\)\( T^{5} + T^{6} )^{2} \)
$43$ \( \)\(52\!\cdots\!36\)\( + \)\(27\!\cdots\!56\)\( T^{2} + \)\(53\!\cdots\!40\)\( T^{4} + \)\(46\!\cdots\!20\)\( T^{6} + \)\(17\!\cdots\!40\)\( T^{8} + \)\(24\!\cdots\!16\)\( T^{10} + T^{12} \)
$47$ \( \)\(44\!\cdots\!96\)\( + \)\(55\!\cdots\!44\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{4} + \)\(15\!\cdots\!80\)\( T^{6} + \)\(73\!\cdots\!40\)\( T^{8} + \)\(14\!\cdots\!24\)\( T^{10} + T^{12} \)
$53$ \( \)\(47\!\cdots\!96\)\( + \)\(45\!\cdots\!56\)\( T^{2} + \)\(15\!\cdots\!40\)\( T^{4} + \)\(24\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!40\)\( T^{8} + \)\(71\!\cdots\!76\)\( T^{10} + T^{12} \)
$59$ \( ( \)\(95\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( T - \)\(63\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} - \)\(79\!\cdots\!00\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{5} + T^{6} )^{2} \)
$61$ \( ( \)\(77\!\cdots\!24\)\( + \)\(48\!\cdots\!08\)\( T + \)\(17\!\cdots\!40\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} - \)\(64\!\cdots\!40\)\( T^{4} + \)\(28\!\cdots\!08\)\( T^{5} + T^{6} )^{2} \)
$67$ \( \)\(32\!\cdots\!76\)\( + \)\(51\!\cdots\!44\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{4} + \)\(18\!\cdots\!80\)\( T^{6} + \)\(81\!\cdots\!40\)\( T^{8} + \)\(15\!\cdots\!44\)\( T^{10} + T^{12} \)
$71$ \( ( \)\(14\!\cdots\!84\)\( - \)\(10\!\cdots\!92\)\( T - \)\(12\!\cdots\!60\)\( T^{2} + \)\(81\!\cdots\!40\)\( T^{3} + \)\(51\!\cdots\!60\)\( T^{4} - \)\(84\!\cdots\!72\)\( T^{5} + T^{6} )^{2} \)
$73$ \( \)\(16\!\cdots\!16\)\( + \)\(77\!\cdots\!56\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{6} + \)\(56\!\cdots\!40\)\( T^{8} + \)\(12\!\cdots\!96\)\( T^{10} + T^{12} \)
$79$ \( ( -\)\(48\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( T + \)\(13\!\cdots\!00\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} - \)\(14\!\cdots\!00\)\( T^{4} - \)\(22\!\cdots\!00\)\( T^{5} + T^{6} )^{2} \)
$83$ \( \)\(15\!\cdots\!76\)\( + \)\(15\!\cdots\!56\)\( T^{2} + \)\(49\!\cdots\!40\)\( T^{4} + \)\(63\!\cdots\!20\)\( T^{6} + \)\(37\!\cdots\!40\)\( T^{8} + \)\(10\!\cdots\!56\)\( T^{10} + T^{12} \)
$89$ \( ( \)\(81\!\cdots\!00\)\( - \)\(15\!\cdots\!00\)\( T - \)\(41\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} - \)\(27\!\cdots\!00\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{5} + T^{6} )^{2} \)
$97$ \( \)\(89\!\cdots\!96\)\( + \)\(32\!\cdots\!44\)\( T^{2} + \)\(25\!\cdots\!40\)\( T^{4} + \)\(28\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!40\)\( T^{8} + \)\(24\!\cdots\!24\)\( T^{10} + T^{12} \)
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