# Properties

 Label 75.22.b.i Level $75$ Weight $22$ Character orbit 75.b Analytic conductor $209.608$ Analytic rank $0$ Dimension $12$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$209.608008215$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 1060747 x^{10} + 401129033703 x^{8} + \cdots + 46\!\cdots\!00$$ x^12 + 1060747*x^10 + 401129033703*x^8 + 65514516974855149*x^6 + 4457090535952523008900*x^4 + 90441605055305069382960000*x^2 + 469020395662241185850256000000 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{39}\cdot 3^{12}\cdot 5^{14}\cdot 7^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} - 15 \beta_{5}) q^{2} - 59049 \beta_{5} q^{3} + (\beta_{2} + 243 \beta_1 - 731822) q^{4} + ( - 59049 \beta_1 - 885735) q^{6} + ( - \beta_{11} + \cdots - 221197120 \beta_{5}) q^{7}+ \cdots - 3486784401 q^{9}+O(q^{10})$$ q + (b6 - 15*b5) * q^2 - 59049*b5 * q^3 + (b2 + 243*b1 - 731822) * q^4 + (-59049*b1 - 885735) * q^6 + (-b11 + 49*b8 - 19683*b6 - 221197120*b5) * q^7 + (-b11 + b9 + 406*b8 - 663939*b6 - 708933081*b5) * q^8 - 3486784401 * q^9 $$q + (\beta_{6} - 15 \beta_{5}) q^{2} - 59049 \beta_{5} q^{3} + (\beta_{2} + 243 \beta_1 - 731822) q^{4} + ( - 59049 \beta_1 - 885735) q^{6} + ( - \beta_{11} + \cdots - 221197120 \beta_{5}) q^{7}+ \cdots + ( - 76709256822 \beta_{7} + \cdots + 45\!\cdots\!69) q^{99}+O(q^{100})$$ q + (b6 - 15*b5) * q^2 - 59049*b5 * q^3 + (b2 + 243*b1 - 731822) * q^4 + (-59049*b1 - 885735) * q^6 + (-b11 + 49*b8 - 19683*b6 - 221197120*b5) * q^7 + (-b11 + b9 + 406*b8 - 663939*b6 - 708933081*b5) * q^8 - 3486784401 * q^9 + (22*b7 - 5*b4 + 7*b3 - 856*b2 - 1089973*b1 - 13051322369) * q^11 + (-59049*b8 + 14348907*b6 + 43213357278*b5) * q^12 + (192*b11 + 41*b10 - 24*b9 + 10225*b8 + 16269174*b6 + 19343717437*b5) * q^13 + (14*b7 - 68*b4 + 926*b3 - 125046*b2 - 321815001*b1 + 52411518679) * q^14 + (-420*b7 - 352*b4 + 988*b3 - 193608*b2 - 1167224796*b1 + 333172612836) * q^16 + (2477*b11 + 561*b10 - 38*b9 + 1581598*b8 - 1817817035*b6 - 897755498349*b5) * q^17 + (-3486784401*b6 + 52301766015*b5) * q^18 + (798*b7 - 2147*b4 - 15105*b3 + 3120522*b2 + 495339975*b1 - 1161770576808) * q^19 + (59049*b3 + 2893401*b2 + 1162261467*b1 - 13061468738880) * q^21 + (68404*b11 + 4724*b10 - 13668*b9 + 28444130*b8 - 11058029928*b6 + 3279959654446*b5) * q^22 + (19741*b11 - 6181*b10 - 38260*b9 - 18751382*b8 + 9502341945*b6 + 49295643731423*b5) * q^23 + (59049*b7 + 59049*b3 + 23973894*b2 + 39204934011*b1 - 41861789499969) * q^24 + (-135806*b7 + 3472*b4 - 629822*b3 + 42178124*b2 + 3243913169*b1 - 45720121878653) * q^26 + 205891132094649*b5 * q^27 + (-362922*b11 + 39744*b10 - 373590*b9 - 172299437*b8 + 340861210317*b6 + 445798159110084*b5) * q^28 + (-252640*b7 - 15905*b4 - 1093903*b3 - 285261600*b2 - 672270729223*b1 + 649494712225261) * q^29 + (-571692*b7 + 147244*b4 - 3620509*b3 - 427740317*b2 - 109726643871*b1 + 82134241435154) * q^31 + (2509728*b11 - 192896*b10 + 818784*b9 - 1090975040*b8 - 376868367136*b6 + 1810255507335328*b5) * q^32 + (413343*b11 + 295245*b10 - 1299078*b9 + 50545944*b8 - 64361815677*b6 + 770667534567081*b5) * q^33 + (-3312816*b7 - 45172*b4 - 9791456*b3 - 2137838986*b2 - 4479254852976*b1 + 5130372297566754) * q^34 + (-3486784401*b2 - 847288609443*b1 + 2551705533908622) * q^36 + (35939638*b11 + 54740*b10 + 3513864*b9 - 1353709002*b8 - 6053235920406*b6 + 4348206583309412*b5) * q^37 + (4602712*b11 - 1659308*b10 - 1905928*b9 - 673218678*b8 - 7437564645295*b6 - 1387111674973553*b5) * q^38 + (-1417176*b7 + 2421009*b4 - 11337408*b3 + 603776025*b2 - 960678455526*b1 + 1142227170937413) * q^39 + (-7767938*b7 - 6182151*b4 - 66201257*b3 + 9494465230*b2 + 918470060407*b1 + 18776843987443269) * q^41 + (54679374*b11 + 4015332*b10 - 826686*b9 + 7383841254*b8 - 19002853994049*b6 - 3094847766476271*b5) * q^42 + (-120771947*b11 - 3760761*b10 + 3067722*b9 - 24098195076*b8 - 1710617680719*b6 - 2230212264864090*b5) * q^43 + (1147586*b7 - 3842176*b4 - 134846782*b3 + 6148014130*b2 - 57867153740488*b1 + 3989416604598842) * q^44 + (37840872*b7 + 14680628*b4 + 47143448*b3 + 74421397722*b2 + 88569671492232*b1 - 26160280668027210) * q^46 + (-31728304*b11 - 14866322*b10 - 59951504*b9 - 48020087986*b8 + 16645600760068*b6 + 40539911597465698*b5) * q^47 + (58340412*b11 + 20785248*b10 + 24800580*b9 + 11432358792*b8 - 68923456979004*b6 - 19673509615352964*b5) * q^48 + (150983700*b7 - 23305646*b4 + 506176118*b3 - 67061604596*b2 - 156562255434186*b1 - 54464095812791112) * q^49 + (-2243862*b7 + 33126489*b4 - 146264373*b3 + 93391780302*b2 + 107340278099715*b1 - 53011564422010101) * q^51 + (-346269176*b11 + 5975232*b10 + 30834936*b9 - 226265665493*b8 - 96039481090185*b6 + 32030560581573486*b5) * q^52 + (548552517*b11 + 31693719*b10 + 240014172*b9 + 14170250238*b8 - 457869494067319*b6 + 92188076557950117*b5) * q^53 + (205891132094649*b1 + 3088366981419735) * q^54 + (141490209*b7 - 77083072*b4 + 1902173985*b3 + 592592409014*b2 + 197601932516931*b1 - 847793026998916537) * q^56 + (-891935145*b11 + 126778203*b10 - 47121102*b9 - 184263703578*b8 + 29249330183775*b6 + 68601390789935592*b5) * q^57 + (-723312580*b11 - 152497164*b10 - 280539756*b9 - 1205146845806*b8 + 1382778331083936*b6 + 1892245867303946310*b5) * q^58 + (-736420800*b7 + 382429144*b4 + 450027504*b3 - 480592048664*b2 + 187342105277152*b1 + 1167261537163076950) * q^59 + (-36522072*b7 - 255238947*b4 + 1714908382*b3 + 1987595684115*b2 - 768622219250436*b1 - 1705585442274439029) * q^61 + (-4858555350*b11 - 349657540*b10 + 170528838*b9 - 1274304558790*b8 + 948050315613231*b6 + 309609432102651485*b5) * q^62 + (3486784401*b11 - 170852435649*b8 + 68630377364883*b6 + 771266667562125120*b5) * q^63 + (531628416*b7 - 722912256*b4 + 3046068608*b3 - 1435328556032*b2 + 1409450318229888*b1 + 1790771638941245568) * q^64 + (-807081732*b7 + 278947476*b4 - 4039187796*b3 + 1679597432370*b2 + 652965609218472*b1 + 193678337635381854) * q^66 + (4807105613*b11 + 60620699*b10 + 2330873922*b9 - 2330566481472*b8 + 1753033449013857*b6 - 1495388408654534528*b5) * q^67 + (-3136938242*b11 - 448225280*b10 - 1982824702*b9 - 7340767298826*b8 + 6666631714409980*b6 + 10713271889799876802*b5) * q^68 + (-2259214740*b7 - 364981869*b4 - 1165686309*b3 - 1107250355718*b2 - 561103789510305*b1 + 2910858466696796727) * q^69 + (-4282719048*b7 + 1873775339*b4 - 10905027519*b3 + 12342344929028*b2 + 4352407507149649*b1 + 3000121344133000703) * q^71 + (3486784401*b11 - 3486784401*b9 - 1415634466806*b8 + 2315012148415539*b6 + 2471896808183669481*b5) * q^72 + (3988710098*b11 + 2615196796*b10 - 6099236292*b9 - 2325143951242*b8 + 1164975244939998*b6 + 13529113804116431132*b5) * q^73 + (-1335039340*b7 + 1424062008*b4 - 32779005772*b3 - 7509676628380*b2 + 5422598339589722*b1 + 17186901090541100754) * q^74 + (6871164978*b7 - 2984594176*b4 - 19197958606*b3 + 3515369820351*b2 - 948679677103617*b1 + 18581074491764773940) * q^76 + (69356598551*b11 + 4220719803*b10 - 3125009636*b9 - 14160894285392*b8 - 10715639362962537*b6 + 6799259639309524089*b5) * q^77 + (-37190359278*b11 - 205018128*b10 + 8019208494*b9 - 2490576044076*b8 + 191549828716281*b6 + 2699727476812580997*b5) * q^78 + (27162572412*b7 + 337321682*b4 - 4751197684*b3 + 3293544994810*b2 + 13268060385847560*b1 + 13187168464942142112) * q^79 + 12157665459056928801 * q^81 + (-13613119376*b11 - 9180640268*b10 - 4323377568*b9 - 19303335128406*b8 - 192888101315610*b6 - 2889979462196098508*b5) * q^82 + (2877328860*b11 - 6876039408*b10 + 41057878548*b9 + 22288259005224*b8 - 23832973295969916*b6 + 39777973966115832594*b5) * q^83 + (-22060115910*b7 + 2346843456*b4 + 21430181178*b3 - 10174109455413*b2 - 20127513608008533*b1 + 26323935497291350116) * q^84 + (42049633836*b7 - 7579421044*b4 + 181875817980*b3 - 11625004341202*b2 + 44822414324219267*b1 + 4779607108730411591) * q^86 + (-64593878247*b11 + 939174345*b10 + 14918139360*b9 + 16844412218400*b8 - 39696914289888927*b6 - 38352013262189436789*b5) * q^87 + (45889052338*b11 - 473591104*b10 - 24740189106*b9 - 11123875425708*b8 - 16470861706434186*b6 + 170503680461031476898*b5) * q^88 + (-47283516420*b7 - 17280743208*b4 + 84112477464*b3 + 20582166427116*b2 - 54924537499891992*b1 + 118740197357584871088) * q^89 + (-63060096330*b7 + 23540561889*b4 - 224804736765*b3 - 22290748535078*b2 + 113995919633889771*b1 + 111757711945456193460) * q^91 + (-122426481950*b11 + 6862316800*b10 + 35424678686*b9 + 130733035602426*b8 - 175364421351187692*b6 - 146847860924162950274*b5) * q^92 + (-213787435941*b11 - 8694610956*b10 + 33757840908*b9 + 25257637978533*b8 - 6479248593938679*b6 - 4849944822504408546*b5) * q^93 + (96454200732*b7 + 20815231264*b4 + 206973487132*b3 + 118622431597000*b2 + 139238056979899146*b1 - 46529384605727111890) * q^94 + (48348376416*b7 - 11390315904*b4 - 148196928672*b3 - 64420985136960*b2 + 22253700211013664*b1 + 106893777452643783072) * q^96 + (-430757643256*b11 + 9935314628*b10 - 149295927768*b9 + 105090571152148*b8 - 221943586458790080*b6 - 249832862838690670887*b5) * q^97 + (863916949392*b11 + 67977088424*b10 - 152883737328*b9 + 87810543632084*b8 + 118382482725271282*b6 + 443764453734233829806*b5) * q^98 + (-76709256822*b7 + 17433922005*b4 - 24407490807*b3 + 2984687447256*b2 + 3800500853911173*b1 + 45507147248651565969) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 8780896 q^{4} - 10865016 q^{6} - 41841412812 q^{9}+O(q^{10})$$ 12 * q - 8780896 * q^4 - 10865016 * q^6 - 41841412812 * q^9 $$12 q - 8780896 q^{4} - 10865016 q^{6} - 41841412812 q^{9} - 156620225032 q^{11} + 627651460296 q^{14} + 3993403225600 q^{16} - 13939277995244 q^{19} - 156732987630492 q^{21} - 502184750631552 q^{24} - 548628652527256 q^{26} + 77\!\cdots\!92 q^{29}+ \cdots + 54\!\cdots\!32 q^{99}+O(q^{100})$$ 12 * q - 8780896 * q^4 - 10865016 * q^6 - 41841412812 * q^9 - 156620225032 * q^11 + 627651460296 * q^14 + 3993403225600 * q^16 - 13939277995244 * q^19 - 156732987630492 * q^21 - 502184750631552 * q^24 - 548628652527256 * q^26 + 7791248610155192 * q^29 + 985173718965412 * q^31 + 61546559154981488 * q^34 + 30617091199603296 * q^36 + 13702880983025124 * q^39 + 225325764022848112 * q^41 + 47641506567595712 * q^44 - 313569387297163248 * q^46 - 654195133250673560 * q^49 - 635709784792304664 * q^51 + 37883968305415416 * q^54 - 10172728295109555840 * q^56 + 14007889739421516424 * q^59 - 20470107754287507580 * q^61 + 21494903192679653376 * q^64 + 2326745216172594672 * q^66 + 34928061626444619096 * q^69 + 36018816458490978544 * q^71 + 206264533660710509936 * q^74 + 222969085158358389216 * q^76 + 158299080558372904160 * q^79 + 145891985508683145612 * q^81 + 315806756621409101664 * q^84 + 57534620536059689528 * q^86 + 1424662587595914359616 * q^89 + 1341548617432625600852 * q^91 - 557796138660981960496 * q^94 + 1282814602270342944768 * q^96 + 546100957522687325832 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 1060747 x^{10} + 401129033703 x^{8} + \cdots + 46\!\cdots\!00$$ :

 $$\beta_{1}$$ $$=$$ $$( - 97404947564203 \nu^{10} + \cdots - 70\!\cdots\!00 ) / 34\!\cdots\!00$$ (-97404947564203*v^10 - 91586472500368328841*v^8 - 28039796744766220159196109*v^6 - 3061858100818538822851370598847*v^4 - 95859628496877981825934341037987500*v^2 - 705447514753029207649267722034490205000) / 344333923255995797638472346621675000 $$\beta_{2}$$ $$=$$ $$( 10\!\cdots\!33 \nu^{10} + \cdots + 46\!\cdots\!00 ) / 13\!\cdots\!00$$ (1071454423206233*v^10 + 1007451197504051617251*v^8 + 308437764192428421751157199*v^6 + 33680439109003927051365076587317*v^4 + 1274829624349495110573900053255734500*v^2 + 46721306078699245786935090962621918505000) / 13773356930239831905538893864867000 $$\beta_{3}$$ $$=$$ $$( 18\!\cdots\!23 \nu^{10} + \cdots - 46\!\cdots\!00 ) / 27\!\cdots\!00$$ (184026507977879810454623*v^10 + 144391830666910144952088460581*v^8 + 28147009212415490109128075095707369*v^6 - 934721196545358189719231966562902764573*v^4 - 390402318868392831838809227784123187710712500*v^2 - 4640472626372490984034307535250962284330583120000) / 2757426057434014347488886551746373400000 $$\beta_{4}$$ $$=$$ $$( 10\!\cdots\!37 \nu^{10} + \cdots - 15\!\cdots\!00 ) / 68\!\cdots\!00$$ (102278398555825935213637*v^10 + 67423137338380722020634393639*v^8 + 2028581496140101806866664992967411*v^6 - 5354194264521061890321959785655740172687*v^4 - 819797677217926167787866248900110776081337500*v^2 - 15581001548736958818293290932013115023397576130000) / 689356514358503586872221637936593350000 $$\beta_{5}$$ $$=$$ $$( 11\!\cdots\!23 \nu^{11} + \cdots + 30\!\cdots\!00 \nu ) / 10\!\cdots\!00$$ (11450864207076011242023*v^11 + 11405271628025035279515668381*v^9 + 3896351271373246513122444269389569*v^7 + 536830363298187763204274752717980618027*v^5 + 27738477683267916305457394660416943134327500*v^3 + 306195341657561002557634504932873628327423080000*v) / 10480769193299353552797015482122646112082920000000 $$\beta_{6}$$ $$=$$ $$( - 11\!\cdots\!23 \nu^{11} + \cdots + 41\!\cdots\!00 \nu ) / 10\!\cdots\!00$$ (-11450864207076011242023*v^11 - 11405271628025035279515668381*v^9 - 3896351271373246513122444269389569*v^7 - 536830363298187763204274752717980618027*v^5 - 27738477683267916305457394660416943134327500*v^3 + 41616881431539853208630427423557710820004256920000*v) / 10480769193299353552797015482122646112082920000000 $$\beta_{7}$$ $$=$$ $$( 20\!\cdots\!77 \nu^{10} + \cdots + 24\!\cdots\!00 ) / 27\!\cdots\!00$$ (2014155883688234880782777*v^10 + 1922784311253562795810414437219*v^8 + 612162168689971444304808080408304831*v^6 + 73955781774398061669933309586430030217173*v^4 + 2975169244432699530210928101806137700218612500*v^2 + 24211713395973636895486990078250033961473748720000) / 2757426057434014347488886551746373400000 $$\beta_{8}$$ $$=$$ $$( 12\!\cdots\!37 \nu^{11} + \cdots + 48\!\cdots\!00 \nu ) / 64\!\cdots\!00$$ (12633422464921699827050437*v^11 + 12990006460843902103901030599239*v^9 + 4681140046479117717218240988168149811*v^7 + 705000793223847749313274305560362912858513*v^5 + 41099423705335035791938477921711876681717966500*v^3 + 487177657434006745814821279599451432182805588920000*v) / 6449704118953448340182778758229320684358720000 $$\beta_{9}$$ $$=$$ $$( 29\!\cdots\!87 \nu^{11} + \cdots + 11\!\cdots\!00 \nu ) / 80\!\cdots\!00$$ (29409708363122711406286757116687*v^11 - 8391822829780376971474403515216868011*v^9 - 24111938046765067488139308522985961987918439*v^7 - 8172070976625722023927362758345903868446293388237*v^5 - 679495291417748017656608729479060365439091663392652500*v^3 + 11205434172613180883393897031154673069808203530172288520000*v) / 807019227884050223565370192123443750630384840000000 $$\beta_{10}$$ $$=$$ $$( 56\!\cdots\!31 \nu^{11} + \cdots - 22\!\cdots\!00 \nu ) / 80\!\cdots\!00$$ (56840875325027573525020701565931*v^11 - 515449994983929495952782446102147081143*v^9 - 499421264794051363429999730226671252472292707*v^7 - 146622004288423797082885614804657386922169752946081*v^5 - 13927359343253516300343470676555326433939189622960982500*v^3 - 225094671440949300128224406907742803641759941909042029240000*v) / 807019227884050223565370192123443750630384840000000 $$\beta_{11}$$ $$=$$ $$( 62\!\cdots\!83 \nu^{11} + \cdots + 48\!\cdots\!00 \nu ) / 80\!\cdots\!00$$ (62040310350016354900920349741283*v^11 + 47222673856984547864179934706437479601*v^9 + 8714263184563059460812721270221504085757349*v^7 - 238445984657188772907818880430766648839085900233*v^5 - 55838212849715150114513521614182770621265628891122500*v^3 + 4839761670370680172136519563266108112297232010975112680000*v) / 807019227884050223565370192123443750630384840000000
 $$\nu$$ $$=$$ $$( \beta_{6} + \beta_{5} ) / 4$$ (b6 + b5) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 275\beta _1 - 2828750 ) / 16$$ (b2 + 275*b1 - 2828750) / 16 $$\nu^{3}$$ $$=$$ $$( -\beta_{11} + \beta_{9} + 454\beta_{8} - 4870675\beta_{6} - 781801849\beta_{5} ) / 64$$ (-b11 + b9 + 454*b8 - 4870675*b6 - 781801849*b5) / 64 $$\nu^{4}$$ $$=$$ $$( -121\beta_{7} - 88\beta_{4} + 231\beta_{3} - 1628146\beta_{2} - 751841447\beta _1 + 3444795464265 ) / 64$$ (-121*b7 - 88*b4 + 231*b3 - 1628146*b2 - 751841447*b1 + 3444795464265) / 64 $$\nu^{5}$$ $$=$$ $$( 338183 \beta_{11} - 6908 \beta_{10} - 237687 \beta_{9} - 156769854 \beta_{8} + \cdots + 266494147996751 \beta_{5} ) / 32$$ (338183*b11 - 6908*b10 - 237687*b9 - 156769854*b8 + 856594358333*b6 + 266494147996751*b5) / 32 $$\nu^{6}$$ $$=$$ $$( 44250355 \beta_{7} + 23343320 \beta_{4} - 48994797 \beta_{3} + 317451890184 \beta_{2} + \cdots - 60\!\cdots\!47 ) / 32$$ (44250355*b7 + 23343320*b4 - 48994797*b3 + 317451890184*b2 + 202246292505811*b1 - 605880941995966447) / 32 $$\nu^{7}$$ $$=$$ $$( - 293178270541 \beta_{11} + 9473137872 \beta_{10} + 195159302285 \beta_{9} + \cdots - 28\!\cdots\!25 \beta_{5} ) / 64$$ (-293178270541*b11 + 9473137872*b10 + 195159302285*b9 + 166994244461430*b8 - 642794631044580031*b6 - 286524338981635559925*b5) / 64 $$\nu^{8}$$ $$=$$ $$( - 48330441340947 \beta_{7} - 19825547400120 \beta_{4} + 24588167856333 \beta_{3} + \cdots + 45\!\cdots\!43 ) / 64$$ (-48330441340947*b7 - 19825547400120*b4 + 24588167856333*b3 - 248216966565418678*b2 - 197655350209706380205*b1 + 454692252227294407826243) / 64 $$\nu^{9}$$ $$=$$ $$( 56\!\cdots\!69 \beta_{11} + \cdots + 69\!\cdots\!85 \beta_{5} ) / 32$$ (56971191048302669*b11 - 2537659705631028*b10 - 38753752127701661*b9 - 40387510549846666526*b8 + 124620276638149232047547*b6 + 69979619416730635725689885*b5) / 32 $$\nu^{10}$$ $$=$$ $$( 59\!\cdots\!27 \beta_{7} + \cdots - 44\!\cdots\!81 ) / 16$$ (5942607651702491827*b7 + 1991971629136795108*b4 - 543156397409671197*b3 + 24466049138856018697879*b2 + 22933165907061889895676068*b1 - 44079060654648971079778528881) / 16 $$\nu^{11}$$ $$=$$ $$( - 43\!\cdots\!97 \beta_{11} + \cdots - 64\!\cdots\!13 \beta_{5} ) / 64$$ (-43016283269646675707797*b11 + 2479426135780556802400*b10 + 30656346696262964362005*b9 + 37230132019603523657578446*b8 - 98471892547544965943264200159*b6 - 64942356066452429369534504481613*b5) / 64

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 646.238i − 571.187i − 388.716i − 370.009i − 144.571i − 89.2274i 89.2274i 144.571i 370.009i 388.716i 571.187i 646.238i
2568.95i 59049.0i −4.50235e6 0 1.51694e8 5.58323e8i 6.17885e9i −3.48678e9 0
49.2 2300.75i 59049.0i −3.19629e6 0 −1.35857e8 2.10374e8i 2.52885e9i −3.48678e9 0
49.3 1570.86i 59049.0i −370456. 0 −9.27578e7 1.39580e9i 2.71240e9i −3.48678e9 0
49.4 1464.04i 59049.0i −46252.4 0 8.64499e7 1.40408e8i 3.00259e9i −3.48678e9 0
49.5 594.286i 59049.0i 1.74398e6 0 −3.50920e7 9.49356e8i 2.28273e9i −3.48678e9 0
49.6 340.910i 59049.0i 1.98093e6 0 2.01304e7 6.73159e8i 1.39026e9i −3.48678e9 0
49.7 340.910i 59049.0i 1.98093e6 0 2.01304e7 6.73159e8i 1.39026e9i −3.48678e9 0
49.8 594.286i 59049.0i 1.74398e6 0 −3.50920e7 9.49356e8i 2.28273e9i −3.48678e9 0
49.9 1464.04i 59049.0i −46252.4 0 8.64499e7 1.40408e8i 3.00259e9i −3.48678e9 0
49.10 1570.86i 59049.0i −370456. 0 −9.27578e7 1.39580e9i 2.71240e9i −3.48678e9 0
49.11 2300.75i 59049.0i −3.19629e6 0 −1.35857e8 2.10374e8i 2.52885e9i −3.48678e9 0
49.12 2568.95i 59049.0i −4.50235e6 0 1.51694e8 5.58323e8i 6.17885e9i −3.48678e9 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.12 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 75.22.b.i 12
5.b even 2 1 inner 75.22.b.i 12
5.c odd 4 1 75.22.a.i 6
5.c odd 4 1 75.22.a.j yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.22.a.i 6 5.c odd 4 1
75.22.a.j yes 6 5.c odd 4 1
75.22.b.i 12 1.a even 1 1 trivial
75.22.b.i 12 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 16973360 T_{2}^{10} + 102849689765632 T_{2}^{8} + \cdots + 75\!\cdots\!84$$ acting on $$S_{22}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + \cdots + 75\!\cdots\!84$$
$3$ $$(T^{2} + 3486784401)^{6}$$
$5$ $$T^{12}$$
$7$ $$T^{12} + \cdots + 21\!\cdots\!25$$
$11$ $$(T^{6} + \cdots - 99\!\cdots\!52)^{2}$$
$13$ $$T^{12} + \cdots + 20\!\cdots\!09$$
$17$ $$T^{12} + \cdots + 72\!\cdots\!16$$
$19$ $$(T^{6} + \cdots - 69\!\cdots\!75)^{2}$$
$23$ $$T^{12} + \cdots + 53\!\cdots\!00$$
$29$ $$(T^{6} + \cdots - 38\!\cdots\!00)^{2}$$
$31$ $$(T^{6} + \cdots + 43\!\cdots\!25)^{2}$$
$37$ $$T^{12} + \cdots + 69\!\cdots\!00$$
$41$ $$(T^{6} + \cdots + 16\!\cdots\!00)^{2}$$
$43$ $$T^{12} + \cdots + 85\!\cdots\!61$$
$47$ $$T^{12} + \cdots + 20\!\cdots\!16$$
$53$ $$T^{12} + \cdots + 38\!\cdots\!00$$
$59$ $$(T^{6} + \cdots - 36\!\cdots\!00)^{2}$$
$61$ $$(T^{6} + \cdots + 14\!\cdots\!41)^{2}$$
$67$ $$T^{12} + \cdots + 92\!\cdots\!01$$
$71$ $$(T^{6} + \cdots + 54\!\cdots\!84)^{2}$$
$73$ $$T^{12} + \cdots + 74\!\cdots\!00$$
$79$ $$(T^{6} + \cdots - 37\!\cdots\!00)^{2}$$
$83$ $$T^{12} + \cdots + 26\!\cdots\!24$$
$89$ $$(T^{6} + \cdots - 85\!\cdots\!00)^{2}$$
$97$ $$T^{12} + \cdots + 64\!\cdots\!81$$