# Properties

 Label 7.20.a.b Level $7$ Weight $20$ Character orbit 7.a Self dual yes Analytic conductor $16.017$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7,20,Mod(1,7)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 20, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$20$$ Character orbit: $$[\chi]$$ $$=$$ 7.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$16.0171687589$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2204112x^{3} - 22116602x^{2} + 863582048991x - 99444286717974$$ x^5 - 2204112*x^3 - 22116602*x^2 + 863582048991*x - 99444286717974 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}\cdot 3^{3}\cdot 7^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 23) q^{2} + (\beta_{2} + 4 \beta_1 - 6483) q^{3} + (\beta_{3} + 5 \beta_{2} + 62 \beta_1 + 357885) q^{4} + (\beta_{4} + \beta_{3} + 53 \beta_{2} - 812 \beta_1 + 1897109) q^{5} + ( - 5 \beta_{4} + 17 \beta_{3} + 510 \beta_{2} + 640 \beta_1 - 3151712) q^{6} - 40353607 q^{7} + ( - 5 \beta_{4} - 464 \beta_{3} + 11129 \beta_{2} + \cdots - 50000691) q^{8}+ \cdots + (205 \beta_{4} - 115 \beta_{3} - 22882 \beta_{2} - 697032 \beta_1 + 2918701) q^{9}+O(q^{10})$$ q + (-b1 - 23) * q^2 + (b2 + 4*b1 - 6483) * q^3 + (b3 + 5*b2 + 62*b1 + 357885) * q^4 + (b4 + b3 + 53*b2 - 812*b1 + 1897109) * q^5 + (-5*b4 + 17*b3 + 510*b2 + 640*b1 - 3151712) * q^6 - 40353607 * q^7 + (-5*b4 - 464*b3 + 11129*b2 - 371544*b1 - 50000691) * q^8 + (205*b4 - 115*b3 - 22882*b2 - 697032*b1 + 2918701) * q^9 $$q + ( - \beta_1 - 23) q^{2} + (\beta_{2} + 4 \beta_1 - 6483) q^{3} + (\beta_{3} + 5 \beta_{2} + 62 \beta_1 + 357885) q^{4} + (\beta_{4} + \beta_{3} + 53 \beta_{2} - 812 \beta_1 + 1897109) q^{5} + ( - 5 \beta_{4} + 17 \beta_{3} + 510 \beta_{2} + 640 \beta_1 - 3151712) q^{6} - 40353607 q^{7} + ( - 5 \beta_{4} - 464 \beta_{3} + 11129 \beta_{2} + \cdots - 50000691) q^{8}+ \cdots + (1449136201525 \beta_{4} + \cdots + 37\!\cdots\!80) q^{99}+O(q^{100})$$ q + (-b1 - 23) * q^2 + (b2 + 4*b1 - 6483) * q^3 + (b3 + 5*b2 + 62*b1 + 357885) * q^4 + (b4 + b3 + 53*b2 - 812*b1 + 1897109) * q^5 + (-5*b4 + 17*b3 + 510*b2 + 640*b1 - 3151712) * q^6 - 40353607 * q^7 + (-5*b4 - 464*b3 + 11129*b2 - 371544*b1 - 50000691) * q^8 + (205*b4 - 115*b3 - 22882*b2 - 697032*b1 + 2918701) * q^9 + (-321*b4 + 2109*b3 - 22458*b2 - 2500938*b1 + 683529626) * q^10 + (-415*b4 + 4961*b3 - 91082*b2 - 4960808*b1 + 1788799532) * q^11 + (-1830*b4 - 2004*b3 + 209210*b2 - 11356408*b1 + 3019329666) * q^12 + (8865*b4 - 29535*b3 - 311241*b2 - 208164*b1 + 180505279) * q^13 + (40353607*b1 + 928132961) * q^14 + (-7537*b4 - 82097*b3 + 3754944*b2 + 43450304*b1 + 43242007862) * q^15 + (-64545*b4 + 312758*b3 + 1373539*b2 + 203121004*b1 + 143735971207) * q^16 + (83810*b4 + 38114*b3 - 8431694*b2 + 246948936*b1 + 312551538744) * q^17 + (96530*b4 + 416470*b3 - 22649420*b2 + 248177931*b1 + 609252716765) * q^18 + (1530*b4 - 1922182*b3 + 26659459*b2 - 937510004*b1 + 695406264675) * q^19 + (-345422*b4 + 213958*b3 + 11701124*b2 - 1157549356*b1 + 1189021606492) * q^20 + (-40353607*b2 - 161414428*b1 + 261612434181) * q^21 + (586170*b4 + 246094*b3 + 46442564*b2 - 3664836340*b1 + 4310666295924) * q^22 + (-1363115*b4 + 5710357*b3 + 110155400*b2 + 482919648*b1 + 2454481195770) * q^23 + (1674390*b4 + 6588420*b3 - 1320186*b2 - 3402139608*b1 + 11643825835758) * q^24 + (1557141*b4 - 20612779*b3 + 242822098*b2 + 1677171208*b1 + 1729052973859) * q^25 + (291765*b4 + 14772063*b3 - 985514430*b2 + 18108377894*b1 + 114254076394) * q^26 + (-9460340*b4 - 5995252*b3 - 49186*b2 + 28660036344*b1 - 20174042564834) * q^27 + (-40353607*b3 - 201768035*b2 - 2501923634*b1 - 14441950641195) * q^28 + (18394170*b4 - 32314566*b3 + 88990158*b2 + 26554795576*b1 - 20315472369688) * q^29 + (-19843848*b4 + 71818632*b3 + 1529141456*b2 - 25964001424*b1 - 38427335950992) * q^30 + (21993690*b4 + 214466138*b3 + 1414367734*b2 + 3334163800*b1 + 34546835556410) * q^31 + (6914325*b4 - 134175954*b3 + 718358925*b2 - 134437452596*b1 - 155927752172455) * q^32 + (-52619620*b4 + 123535900*b3 + 4589058696*b2 - 8231058016*b1 - 131046740136160) * q^33 + (36551190*b4 - 385425598*b3 - 10681861124*b2 - 279059083706*b1 - 226858158346294) * q^34 + (-40353607*b4 - 40353607*b3 - 2138741171*b2 + 32767128884*b1 - 76555191022163) * q^35 + (6761180*b4 - 807970691*b3 - 3707681819*b2 - 319167867258*b1 - 239622439627027) * q^36 + (168193500*b4 + 334861404*b3 - 10882807530*b2 + 605156535000*b1 + 297241752162156) * q^37 + (-171857215*b4 + 2472962147*b3 + 1826008282*b2 + 164535943496*b1 + 817140289689880) * q^38 + (-58669335*b4 - 782456535*b3 + 25769070988*b2 + 1196445871216*b1 - 336758826876906) * q^39 + (140322060*b4 - 49613740*b3 + 47542997800*b2 - 67467279080*b1 + 637549938528440) * q^40 + (-48762460*b4 - 654540508*b3 - 9432404450*b2 + 317871898616*b1 + 1163983108777120) * q^41 + (201768035*b4 - 686011319*b3 - 20580339570*b2 - 25826308480*b1 + 127182947425184) * q^42 + (-333155325*b4 - 2562785149*b3 - 81872860430*b2 + 1305298247752*b1 + 28335284714380) * q^43 + (-54260340*b4 + 2319411228*b3 + 55693440912*b2 - 1854345126904*b1 + 2204265965006944) * q^44 + (78287527*b4 - 1320057433*b3 - 108225716359*b2 - 1255758354204*b1 + 1861439715907213) * q^45 + (-332973120*b4 - 2006719712*b3 + 192573493088*b2 - 6231286394320*b1 - 458483622213776) * q^46 + (623852820*b4 + 6042774804*b3 - 41765284734*b2 + 1986519622152*b1 + 3018123208217706) * q^47 + (970562730*b4 + 2241763404*b3 - 141620972470*b2 - 8600871442120*b1 + 1145756205676770) * q^48 + 1628413597910449 * q^49 + (-1744708786*b4 + 14948756234*b3 - 159573926708*b2 + 7546622482057*b1 - 1458167031985489) * q^50 + (-1298224890*b4 - 11597861754*b3 + 486052326678*b2 + 13056622729176*b1 - 10601705074135590) * q^51 + (553026150*b4 - 30607161170*b3 - 338349979960*b2 - 2547069199420*b1 - 16289364323577936) * q^52 + (2165899870*b4 + 16403865310*b3 + 453435144128*b2 + 7046607057792*b1 - 4117450112316150) * q^53 + (599326730*b4 - 32158571426*b3 + 403738654852*b2 + 20460022683168*b1 - 24798386680146400) * q^54 + (-3313618098*b4 + 28800198862*b3 - 508074934204*b2 - 24334130149744*b1 - 2518256066856952) * q^55 + (201768035*b4 + 18724073648*b3 - 449095292303*b2 + 14993140559208*b1 + 2017708234342437) * q^56 + (6120305715*b4 + 15917212275*b3 + 903294715282*b2 + 3849090465928*b1 + 22784934038983848) * q^57 + (-2489199030*b4 + 4407854174*b3 - 1535757110204*b2 + 38342609408242*b1 - 22922398304448578) * q^58 + (3483693020*b4 - 26119759012*b3 - 1227540980549*b2 - 45816180868500*b1 - 3889602842968425) * q^59 + (-749643536*b4 + 50994298544*b3 + 860847287232*b2 - 31888710267488*b1 + 1435584960764416) * q^60 + (-14275271235*b4 - 23412483907*b3 + 597328430005*b2 - 45500096076332*b1 + 12360690854307957) * q^61 + (-4454036350*b4 - 67169133562*b3 + 1223553001972*b2 - 147669784711320*b1 - 3486889787401896) * q^62 + (-8272489435*b4 + 4640664805*b3 + 923371235374*b2 + 28127755394424*b1 - 117780113104507) * q^63 + (27039366555*b4 + 58352730970*b3 - 1284832443733*b2 + 119827655429348*b1 + 46952216434529951) * q^64 + (28289327683*b4 - 174302437757*b3 - 615435011806*b2 + 68721937439944*b1 + 92018634600721052) * q^65 + (-16475484360*b4 + 5608431912*b3 + 6895039473200*b2 + 33521845703072*b1 + 11291674879533024) * q^66 + (-17926448865*b4 + 15208865183*b3 - 2524179772952*b2 - 233094402024416*b1 + 149872458152982594) * q^67 + (-1017674060*b4 + 255424937746*b3 - 5550196366570*b2 + 370787499537180*b1 + 85068932210030550) * q^68 + (19218048990*b4 + 108884101854*b3 - 2661027244032*b2 - 226986802393344*b1 + 105839667452547276) * q^69 + (12953507847*b4 - 85105757163*b3 + 906261306006*b2 + 100921869183366*b1 - 27582885900460982) * q^70 + (-75790788730*b4 - 323936268538*b3 + 2024519843524*b2 + 333379344523920*b1 - 13180269998760648) * q^71 + (-48744375045*b4 + 437269441416*b3 + 3975694153521*b2 + 553925689678008*b1 - 33121687403244795) * q^72 + (31249446720*b4 + 519211864768*b3 - 2868798931972*b2 - 309982299384976*b1 + 233967099060326278) * q^73 + (48328559730*b4 - 887248516458*b3 - 16513537105644*b2 - 399659718356022*b1 - 542992370434714266) * q^74 + (100576802208*b4 - 169525597152*b3 + 1749192662099*b2 + 161936281414604*b1 + 272100877614686967) * q^75 + (52588189230*b4 - 490957957252*b3 + 18797966401126*b2 - 1626173395632344*b1 - 528697242660347138) * q^76 + (16746746905*b4 - 200194244327*b3 + 3675487232774*b2 + 200186496434456*b1 - 72184513316111924) * q^77 + (-140035616180*b4 - 299130427708*b3 + 4517827612440*b2 + 530177481040336*b1 - 1041022115130902576) * q^78 + (-67231614720*b4 + 1098411128192*b3 + 7768654945516*b2 + 256169324376112*b1 + 168299471593652236) * q^79 + (-68271130824*b4 + 1075811330616*b3 + 10068313531008*b2 - 248970837546352*b1 - 567880000756323296) * q^80 + (20059222735*b4 + 527654584079*b3 - 29642554924102*b2 + 7250743596840*b1 + 226238322179295841) * q^81 + (37777159050*b4 - 217795214370*b3 - 9249896699772*b2 - 799260760031922*b1 - 308938310373138878) * q^82 + (-261145728490*b4 + 400261541654*b3 + 22725654467047*b2 - 69365812140900*b1 + 135525103619240367) * q^83 + (73847100810*b4 + 80868628428*b3 - 8442378120470*b2 + 458272025363656*b1 - 121840842745205262) * q^84 + (439313201544*b4 - 1396739098616*b3 - 15180481814398*b2 + 500891436722312*b1 + 1041240380771428366) * q^85 + (383428403870*b4 - 1955506575814*b3 - 51322377802196*b2 + 1628303476768108*b1 - 1169053831054332460) * q^86 + (60348067010*b4 - 2137556735678*b3 + 6628471418802*b2 + 1616698766496584*b1 + 328024141968550046) * q^87 + (-535277091120*b4 + 1681447867448*b3 + 38265218381320*b2 - 1712792378652464*b1 - 663945366181755064) * q^88 + (-450422743850*b4 + 8168770006*b3 + 26416462381400*b2 + 2862657890490976*b1 + 1712830543929440990) * q^89 + (508777581083*b4 - 293615889647*b3 - 67646017027106*b2 - 514047918758706*b1 + 1040243898722273282) * q^90 + (-357734726055*b4 + 1191843782745*b3 + 12559696996287*b2 + 8400168247548*b1 - 7284039090191353) * q^91 + (-263031065440*b4 + 8068780566416*b3 + 78939666292912*b2 + 313137872082144*b1 + 4261666228120293552) * q^92 + (-603719393910*b4 - 2315204451318*b3 + 52037983869420*b2 - 1064335962075600*b1 + 1263729403886542752) * q^93 + (282269105430*b4 - 5496195128574*b3 - 18508648496964*b2 - 5814401616976392*b1 - 1832280904413550584) * q^94 + (1723082671685*b4 + 251371724165*b3 + 178980720851800*b2 + 66901261609120*b1 + 3074133078678801530) * q^95 + (-198685221370*b4 + 1706686275892*b3 - 73190507356986*b2 + 809907567005768*b1 + 1418813449895033198) * q^96 + (-819104034000*b4 + 3673466424880*b3 - 153628925204566*b2 - 2228455474540120*b1 + 2935528967047047868) * q^97 + (-1628413597910449*b1 - 37453512751940327) * q^98 + (1449136201525*b4 - 1052489402827*b3 - 208846609242850*b2 - 3194034739893384*b1 + 3795770348196261580) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 115 q^{2} - 32414 q^{3} + 1789429 q^{4} + 9485596 q^{5} - 15758062 q^{6} - 201768035 q^{7} - 249991857 q^{8} + 14570533 q^{9}+O(q^{10})$$ 5 * q - 115 * q^2 - 32414 * q^3 + 1789429 * q^4 + 9485596 * q^5 - 15758062 * q^6 - 201768035 * q^7 - 249991857 * q^8 + 14570533 * q^9 $$5 q - 115 q^{2} - 32414 q^{3} + 1789429 q^{4} + 9485596 q^{5} - 15758062 q^{6} - 201768035 q^{7} - 249991857 q^{8} + 14570533 q^{9} + 3417623884 q^{10} + 8943902032 q^{11} + 15096861374 q^{12} + 902235824 q^{13} + 4640664805 q^{14} + 216213883888 q^{15} + 718680981361 q^{16} + 1562749140102 q^{17} + 3046240421405 q^{18} + 3477059903486 q^{19} + 5945119865048 q^{20} + 1308021817298 q^{21} + 21553377089920 q^{22} + 12272511787008 q^{23} + 58219119595794 q^{24} + 8645526747031 q^{25} + 570269803712 q^{26} - 100870197417764 q^{27} - 72209914620403 q^{28} - 101577258937886 q^{29} - 192135202588288 q^{30} + 172735355689956 q^{31} - 779637915241721 q^{32} - 655229182538384 q^{33} - 11\!\cdots\!86 q^{34}+ \cdots + 18\!\cdots\!52 q^{99}+O(q^{100})$$ 5 * q - 115 * q^2 - 32414 * q^3 + 1789429 * q^4 + 9485596 * q^5 - 15758062 * q^6 - 201768035 * q^7 - 249991857 * q^8 + 14570533 * q^9 + 3417623884 * q^10 + 8943902032 * q^11 + 15096861374 * q^12 + 902235824 * q^13 + 4640664805 * q^14 + 216213883888 * q^15 + 718680981361 * q^16 + 1562749140102 * q^17 + 3046240421405 * q^18 + 3477059903486 * q^19 + 5945119865048 * q^20 + 1308021817298 * q^21 + 21553377089920 * q^22 + 12272511787008 * q^23 + 58219119595794 * q^24 + 8645526747031 * q^25 + 570269803712 * q^26 - 100870197417764 * q^27 - 72209914620403 * q^28 - 101577258937886 * q^29 - 192135202588288 * q^30 + 172735355689956 * q^31 - 779637915241721 * q^32 - 655229182538384 * q^33 - 1134301124718186 * q^34 - 382778013144772 * q^35 - 1198115104607443 * q^36 + 1486197374948346 * q^37 + 4085700973352750 * q^38 - 1683767524187672 * q^39 + 3187797144931680 * q^40 + 5819906814784118 * q^41 + 635894641029634 * q^42 + 141597446651944 * q^43 + 11021383253324744 * q^44 + 9307091595589612 * q^45 - 2292223197882960 * q^46 + 15090567609176172 * q^47 + 5728636195085246 * q^48 + 8142067989552245 * q^49 - 7291007937901601 * q^50 - 53008026422264628 * q^51 - 81447129913734620 * q^52 - 20586815696201802 * q^53 - 123991498102832452 * q^54 - 12591813895799728 * q^55 + 10088073150578199 * q^56 + 113925551452116532 * q^57 - 114613529198008238 * q^58 - 19449219119756682 * q^59 + 7178735406454304 * q^60 + 61804089287724932 * q^61 - 17433153760837596 * q^62 - 587973562462531 * q^63 + 234759711948108497 * q^64 + 460092703581703528 * q^65 + 56465280304190768 * q^66 + 749359769302723700 * q^67 + 425338856446522494 * q^68 + 529195548133341504 * q^69 - 137913451088749588 * q^70 - 65898925746902448 * q^71 - 165604849847136825 * q^72 + 1169832076041387930 * q^73 - 2714977526790720246 * q^74 + 1360506206214891878 * q^75 - 2643466976965566542 * q^76 - 360918707645829424 * q^77 - 5205105618660856552 * q^78 + 841504095443693224 * q^79 - 2839390943008285264 * q^80 + 1131161420627748289 * q^81 - 1544700621744338842 * q^82 + 677648104634855718 * q^83 - 609212810819876018 * q^84 + 5206187680801224504 * q^85 - 5845318905571292552 * q^86 + 1640129415522837700 * q^87 - 3319689711861170328 * q^88 + 8564179578363560194 * q^89 + 5201151632432647868 * q^90 - 36408469863017168 * q^91 + 21308402274518259696 * q^92 + 6318701976340428408 * q^93 - 9161417816790226740 * q^94 + 15370842399660463600 * q^95 + 7093992550966754482 * q^96 + 14677488351947643894 * q^97 - 187267563759701635 * q^98 + 18978642497725266352 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2204112x^{3} - 22116602x^{2} + 863582048991x - 99444286717974$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{4} - 13001\nu^{3} + 6981503\nu^{2} + 18486426033\nu - 4730270045670 ) / 173803392$$ (-v^4 - 13001*v^3 + 6981503*v^2 + 18486426033*v - 4730270045670) / 173803392 $$\beta_{3}$$ $$=$$ $$( 5\nu^{4} + 65005\nu^{3} + 138895877\nu^{2} - 95212984437\nu - 129581367508098 ) / 173803392$$ (5*v^4 + 65005*v^3 + 138895877*v^2 - 95212984437*v - 129581367508098) / 173803392 $$\beta_{4}$$ $$=$$ $$( -4483\nu^{4} - 349019\nu^{3} + 8413964669\nu^{2} + 1122804345363\nu - 1799085118294194 ) / 289672320$$ (-4483*v^4 - 349019*v^3 + 8413964669*v^2 + 1122804345363*v - 1799085118294194) / 289672320
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5\beta_{2} + 16\beta _1 + 881644$$ b3 + 5*b2 + 16*b1 + 881644 $$\nu^{3}$$ $$=$$ $$5\beta_{4} + 395\beta_{3} - 11474\beta_{2} + 1417429\beta _1 + 13272336$$ 5*b4 + 395*b3 - 11474*b2 + 1417429*b1 + 13272336 $$\nu^{4}$$ $$=$$ $$-65005\beta_{4} + 1846108\beta_{3} + 10277597\beta_{2} + 170135652\beta _1 + 1252376544926$$ -65005*b4 + 1846108*b3 + 10277597*b2 + 170135652*b1 + 1252376544926

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1332.90 613.497 119.891 −802.831 −1263.45
−1355.90 −10527.9 1.31417e6 97087.9 1.42747e7 −4.03536e7 −1.07100e9 −1.05143e9 −1.31641e8
1.2 −636.497 31039.9 −119160. 7.19630e6 −1.97568e7 −4.03536e7 4.09552e8 −1.98785e8 −4.58042e9
1.3 −142.891 −20020.3 −503870. −5.07605e6 2.86072e6 −4.03536e7 1.46915e8 −7.61450e8 7.25322e8
1.4 779.831 −60095.5 83849.0 3.49867e6 −4.68644e7 −4.03536e7 −3.43468e8 2.44921e9 2.72837e9
1.5 1240.45 27189.8 1.01444e6 3.76958e6 3.37277e7 −4.03536e7 6.08010e8 −4.22978e8 4.67599e9
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$

## 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 7.20.a.b 5
3.b odd 2 1 63.20.a.d 5
7.b odd 2 1 49.20.a.d 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.20.a.b 5 1.a even 1 1 trivial
49.20.a.d 5 7.b odd 2 1
63.20.a.d 5 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} + 115T_{2}^{4} - 2198822T_{2}^{3} - 129845456T_{2}^{2} + 861102886144T_{2} + 119291562532864$$ acting on $$S_{20}^{\mathrm{new}}(\Gamma_0(7))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} + \cdots + 119291562532864$$
$3$ $$T^{5} + 32414 T^{4} + \cdots + 10\!\cdots\!52$$
$5$ $$T^{5} - 9485596 T^{4} + \cdots + 46\!\cdots\!00$$
$7$ $$(T + 40353607)^{5}$$
$11$ $$T^{5} - 8943902032 T^{4} + \cdots + 54\!\cdots\!72$$
$13$ $$T^{5} - 902235824 T^{4} + \cdots + 67\!\cdots\!44$$
$17$ $$T^{5} - 1562749140102 T^{4} + \cdots - 22\!\cdots\!52$$
$19$ $$T^{5} - 3477059903486 T^{4} + \cdots + 11\!\cdots\!00$$
$23$ $$T^{5} - 12272511787008 T^{4} + \cdots - 11\!\cdots\!52$$
$29$ $$T^{5} + 101577258937886 T^{4} + \cdots + 67\!\cdots\!00$$
$31$ $$T^{5} - 172735355689956 T^{4} + \cdots + 16\!\cdots\!00$$
$37$ $$T^{5} + \cdots - 69\!\cdots\!96$$
$41$ $$T^{5} + \cdots - 45\!\cdots\!52$$
$43$ $$T^{5} - 141597446651944 T^{4} + \cdots - 53\!\cdots\!24$$
$47$ $$T^{5} + \cdots + 44\!\cdots\!12$$
$53$ $$T^{5} + \cdots - 16\!\cdots\!92$$
$59$ $$T^{5} + \cdots + 49\!\cdots\!00$$
$61$ $$T^{5} + \cdots + 85\!\cdots\!92$$
$67$ $$T^{5} + \cdots - 66\!\cdots\!16$$
$71$ $$T^{5} + \cdots - 47\!\cdots\!12$$
$73$ $$T^{5} + \cdots - 15\!\cdots\!44$$
$79$ $$T^{5} + \cdots + 16\!\cdots\!00$$
$83$ $$T^{5} + \cdots - 87\!\cdots\!28$$
$89$ $$T^{5} + \cdots - 17\!\cdots\!00$$
$97$ $$T^{5} + \cdots + 11\!\cdots\!16$$