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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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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