# Properties

 Label 162.9.d.g Level $162$ Weight $9$ Character orbit 162.d Analytic conductor $65.995$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,9,Mod(53,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("162.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 162.d (of order $$6$$, degree $$2$$, 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: $$65.9953348299$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 11104 x^{14} + 77885403 x^{12} - 342246555016 x^{10} + \cdots + 28\!\cdots\!96$$ x^16 - 11104*x^14 + 77885403*x^12 - 342246555016*x^10 + 1109581168400533*x^8 - 2496682466448100572*x^6 + 4145143614138242101692*x^4 - 4313058873298631588690352*x^2 + 2834471900829569814904148496 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{40}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 \beta_{8} q^{2} + ( - 128 \beta_1 + 128) q^{4} + (\beta_{13} - \beta_{12} + \beta_{11} + 151 \beta_{9} + \beta_{8}) q^{5} + (\beta_{7} - \beta_{5} - \beta_{4} + 8 \beta_{2} - 186 \beta_1) q^{7} + (1024 \beta_{9} + 1024 \beta_{8}) q^{8}+O(q^{10})$$ q + 8*b8 * q^2 + (-128*b1 + 128) * q^4 + (b13 - b12 + b11 + 151*b9 + b8) * q^5 + (b7 - b5 - b4 + 8*b2 - 186*b1) * q^7 + (1024*b9 + 1024*b8) * q^8 $$q + 8 \beta_{8} q^{2} + ( - 128 \beta_1 + 128) q^{4} + (\beta_{13} - \beta_{12} + \beta_{11} + 151 \beta_{9} + \beta_{8}) q^{5} + (\beta_{7} - \beta_{5} - \beta_{4} + 8 \beta_{2} - 186 \beta_1) q^{7} + (1024 \beta_{9} + 1024 \beta_{8}) q^{8} + ( - 8 \beta_{5} + 8 \beta_{3} - 2400) q^{10} + ( - \beta_{14} + 14 \beta_{11} - 1640 \beta_{8}) q^{11} + (11 \beta_{6} - 16 \beta_{4} - 9 \beta_{3} + 9 \beta_{2} + 5853 \beta_1 - 5853) q^{13} + (8 \beta_{15} + 128 \beta_{13} - 8 \beta_{12} + 128 \beta_{11} + 1560 \beta_{9} + 128 \beta_{8}) q^{14} - 16384 \beta_1 q^{16} + ( - 13 \beta_{15} + 13 \beta_{14} - 227 \beta_{13} - 155 \beta_{12} + \cdots + 24690 \beta_{8}) q^{17}+ \cdots + (11888 \beta_{15} - 11888 \beta_{14} + 525088 \beta_{13} + \cdots - 23685112 \beta_{8}) q^{98}+O(q^{100})$$ q + 8*b8 * q^2 + (-128*b1 + 128) * q^4 + (b13 - b12 + b11 + 151*b9 + b8) * q^5 + (b7 - b5 - b4 + 8*b2 - 186*b1) * q^7 + (1024*b9 + 1024*b8) * q^8 + (-8*b5 + 8*b3 - 2400) * q^10 + (-b14 + 14*b11 - 1640*b8) * q^11 + (11*b6 - 16*b4 - 9*b3 + 9*b2 + 5853*b1 - 5853) * q^13 + (8*b15 + 128*b13 - 8*b12 + 128*b11 + 1560*b9 + 128*b8) * q^14 - 16384*b1 * q^16 + (-13*b15 + 13*b14 - 227*b13 - 155*b12 - 155*b10 + 24690*b9 + 24690*b8) * q^17 + (-9*b7 + 9*b6 - 70*b5 + 318*b3 - 80059) * q^19 + (128*b11 + 128*b10 - 19200*b8) * q^20 + (16*b6 - 8*b4 + 112*b3 - 112*b2 + 26360*b1 - 26360) * q^22 + (-10*b15 + 590*b13 - 369*b12 + 590*b11 + 41857*b9 + 590*b8) * q^23 + (-14*b7 + 405*b5 + 405*b4 - 2661*b2 + 19570*b1) * q^25 + (88*b15 - 88*b14 + 144*b13 - 168*b12 - 168*b10 - 46624*b9 - 46624*b8) * q^26 + (128*b7 - 128*b6 - 128*b5 + 1024*b3 - 23808) * q^28 + (78*b14 - 2737*b11 - 727*b10 + 3891*b8) * q^29 + (-218*b6 - 417*b4 - 350*b3 + 350*b2 + 13363*b1 - 13363) * q^31 + 131072*b9 * q^32 + (-208*b7 - 1136*b5 - 1136*b4 - 1816*b2 - 395720*b1) * q^34 + (-176*b15 + 176*b14 - 12350*b13 + 1551*b12 + 1551*b10 - 319411*b9 - 319411*b8) * q^35 + (-263*b7 + 263*b6 + 1726*b5 - 200*b3 - 182522) * q^37 + (-72*b14 + 5088*b11 + 1192*b10 - 638488*b8) * q^38 + (1024*b4 + 1024*b3 - 1024*b2 + 307200*b1 - 307200) * q^40 + (-119*b15 - 25028*b13 + 2403*b12 - 25028*b11 + 751718*b9 - 25028*b8) * q^41 + (561*b7 + 525*b5 + 525*b4 - 10052*b2 + 1058464*b1) * q^43 + (128*b15 - 128*b14 - 1792*b13 - 211712*b9 - 211712*b8) * q^44 + (-160*b7 + 160*b6 - 2872*b5 + 4720*b3 - 662120) * q^46 + (-743*b14 + 25332*b11 - 331*b10 - 3165391*b8) * q^47 + (1486*b6 + 3805*b4 - 32818*b3 + 32818*b2 + 2989652*b1 - 2989652) * q^49 + (-112*b15 - 42576*b13 + 6368*b12 - 42576*b11 - 181088*b9 - 42576*b8) * q^50 + (1408*b7 - 2048*b5 - 2048*b4 + 1152*b2 + 749184*b1) * q^52 + (-9*b15 + 9*b14 - 5416*b13 + 1783*b12 + 1783*b10 - 1284586*b9 - 1284586*b8) * q^53 + (77*b7 - 77*b6 + 2319*b5 + 3264*b3 + 512350) * q^55 + (1024*b14 + 16384*b11 + 1024*b10 - 183296*b8) * q^56 + (-1248*b6 - 5192*b4 - 21896*b3 + 21896*b2 - 78960*b1 + 78960) * q^58 + (1441*b15 - 85816*b13 - 13393*b12 - 85816*b11 + 3478743*b9 - 85816*b8) * q^59 + (-3623*b7 + 1401*b5 + 1401*b4 - 111138*b2 + 3540363*b1) * q^61 + (-1744*b15 + 1744*b14 + 5600*b13 - 8416*b12 - 8416*b10 - 100768*b9 - 100768*b8) * q^62 - 2097152 * q^64 + (1250*b14 + 125593*b11 + 1058*b10 - 3372980*b8) * q^65 + (-2919*b6 - 15807*b4 - 26400*b3 + 26400*b2 + 11625140*b1 - 11625140) * q^67 + (-1664*b15 - 29056*b13 - 19840*b12 - 29056*b11 + 3160320*b9 - 29056*b8) * q^68 + (-2816*b7 + 13816*b5 + 13816*b4 - 98800*b2 + 4997960*b1) * q^70 + (5648*b15 - 5648*b14 + 120622*b13 - 12497*b12 - 12497*b10 - 720879*b9 - 720879*b8) * q^71 + (5002*b7 - 5002*b6 - 13407*b5 - 73503*b3 - 17411363) * q^73 + (-2104*b14 - 3200*b11 - 25512*b10 - 1447968*b8) * q^74 + (1152*b6 + 8960*b4 + 40704*b3 - 40704*b2 + 10247552*b1 - 10247552) * q^76 + (-1649*b15 - 112288*b13 + 2099*b12 - 112288*b11 + 5802711*b9 - 112288*b8) * q^77 + (6033*b7 - 1202*b5 - 1202*b4 - 252474*b2 + 5114311*b1) * q^79 + (-16384*b13 + 16384*b12 + 16384*b10 - 2473984*b9 - 2473984*b8) * q^80 + (-1904*b7 + 1904*b6 + 20176*b5 - 200224*b3 - 12247888) * q^82 + (3298*b14 + 369888*b11 + 33848*b10 + 6591632*b8) * q^83 + (3907*b6 + 34917*b4 + 257874*b3 - 257874*b2 + 59945675*b1 - 59945675) * q^85 + (4488*b15 - 160832*b13 + 12888*b12 - 160832*b11 - 8552328*b9 - 160832*b8) * q^86 + (2048*b7 - 1024*b5 - 1024*b4 - 14336*b2 + 3374080*b1) * q^88 + (-16426*b15 + 16426*b14 + 513937*b13 - 13754*b12 - 13754*b10 + 22558158*b9 + 22558158*b8) * q^89 + (-14157*b7 + 14157*b6 - 22947*b5 - 395776*b3 - 84534458) * q^91 + (-1280*b14 + 75520*b11 + 47232*b10 - 5282176*b8) * q^92 + (11888*b6 - 8592*b4 + 202656*b3 - 202656*b2 + 50857504*b1 - 50857504) * q^94 + (-2321*b15 - 469018*b13 + 146144*b12 - 469018*b11 - 40986894*b9 - 469018*b8) * q^95 + (5014*b7 - 91877*b5 - 91877*b4 - 374538*b2 + 6182871*b1) * q^97 + (11888*b15 - 11888*b14 + 525088*b13 + 72768*b12 + 72768*b10 - 23685112*b9 - 23685112*b8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 1024 q^{4} - 1492 q^{7}+O(q^{10})$$ 16 * q + 1024 * q^4 - 1492 * q^7 $$16 q + 1024 q^{4} - 1492 q^{7} - 38400 q^{10} - 46780 q^{13} - 131072 q^{16} - 1280872 q^{19} - 210816 q^{22} + 156616 q^{25} - 381952 q^{28} - 107776 q^{31} - 3164928 q^{34} - 2918248 q^{37} - 2457600 q^{40} + 8465468 q^{43} - 10592640 q^{46} - 23911272 q^{49} + 5987840 q^{52} + 8196984 q^{55} + 626688 q^{58} + 28337396 q^{61} - 33554432 q^{64} - 93012796 q^{67} + 39994944 q^{70} - 278621824 q^{73} - 81975808 q^{76} + 40890356 q^{79} - 195950976 q^{82} - 479549772 q^{85} + 26984448 q^{88} - 1352438072 q^{91} - 406812480 q^{94} + 49442912 q^{97}+O(q^{100})$$ 16 * q + 1024 * q^4 - 1492 * q^7 - 38400 * q^10 - 46780 * q^13 - 131072 * q^16 - 1280872 * q^19 - 210816 * q^22 + 156616 * q^25 - 381952 * q^28 - 107776 * q^31 - 3164928 * q^34 - 2918248 * q^37 - 2457600 * q^40 + 8465468 * q^43 - 10592640 * q^46 - 23911272 * q^49 + 5987840 * q^52 + 8196984 * q^55 + 626688 * q^58 + 28337396 * q^61 - 33554432 * q^64 - 93012796 * q^67 + 39994944 * q^70 - 278621824 * q^73 - 81975808 * q^76 + 40890356 * q^79 - 195950976 * q^82 - 479549772 * q^85 + 26984448 * q^88 - 1352438072 * q^91 - 406812480 * q^94 + 49442912 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 11104 x^{14} + 77885403 x^{12} - 342246555016 x^{10} + \cdots + 28\!\cdots\!96$$ :

 $$\beta_{1}$$ $$=$$ $$( 14\!\cdots\!45 \nu^{14} + \cdots - 20\!\cdots\!48 ) / 21\!\cdots\!64$$ (140544788294957995067478064566745*v^14 - 1379897340561236374733006422179835633*v^12 + 9323820709230823336160048504740567508823*v^10 - 37393864440308303424825518340570552985895431*v^8 + 116853129461936012167008964449151273279383015745*v^6 - 233463111015511514541423279908907897915728083559409*v^4 + 416584835882637487748887050121607287020464213390905836*v^2 - 209879397411406740881359555575375270788522156957798029548) / 217001901726322888285864729982024947161483240013467528564 $$\beta_{2}$$ $$=$$ $$( - 19\!\cdots\!27 \nu^{14} + \cdots + 49\!\cdots\!60 ) / 64\!\cdots\!20$$ (-1901812596334426125695496809482289506068227*v^14 + 32638376989512261781818138139998125696049865985*v^12 - 220534068981338466739930532194983450024951455534535*v^10 + 1008309633161652889263331704762496862921582151438459803*v^8 - 2763898718894370204445679409574505596557132773907922771025*v^6 + 5522046319307669953298817068632066618026466457358796107835905*v^4 - 5957146610557462423083152238824293797025999268441875057365578646*v^2 + 4964226463585365464948452043810970195310948451529051709995189279660) / 6408867671916073919250935661676801902316770109150131047077583920 $$\beta_{3}$$ $$=$$ $$( - 17\!\cdots\!29 \nu^{14} + \cdots + 12\!\cdots\!28 ) / 18\!\cdots\!40$$ (-17522562544334129*v^14 + 172795285644834913532*v^12 - 1038204059146425849665823*v^10 + 3790587735094648266457419380*v^8 - 9551507752051231929819065297825*v^6 + 16095398174222096481384104274749328*v^4 - 17385325324744223358450041539520859696*v^2 + 12637157642427641555120234918884055189928) / 18058982503443929219042736553638172640 $$\beta_{4}$$ $$=$$ $$( 20\!\cdots\!51 \nu^{14} + \cdots - 58\!\cdots\!76 ) / 17\!\cdots\!40$$ (209494508666585051090541923866552532579107951*v^14 - 7326376988361760046923352601111779289852015211030*v^12 + 58056367000636182410433837103583343902060428796008605*v^10 - 367955441594427720083149488746006388162430699658995124578*v^8 + 1314114136490293243507482864869206363898030408917631180659603*v^6 - 3676669097544083720011827331674679184062467825424146169285396458*v^4 + 5461286371007969168251590636911629147525061719718810881176955869924*v^2 - 5864550793663103543645403498675840099168377300425282289774097336760176) / 173039427141733995819775262865273651362552792947053538271094765840 $$\beta_{5}$$ $$=$$ $$( 26\!\cdots\!63 \nu^{14} + \cdots - 46\!\cdots\!52 ) / 11\!\cdots\!60$$ (2660325702636236755686800128876063*v^14 - 24183656110447312030298639193383847076*v^12 + 157623117973779508904796681798210693057681*v^10 - 575497901876838687697196317521764428984550860*v^8 + 1578862596195001313471485643665349258328122381743*v^6 - 2443649514659190299815810398950548346219784168322416*v^4 + 2639489954342593945814692557115973909661939245885333712*v^2 - 466244440166480489455106963989493733348934931184131249352) / 117006904902842831554342577154547233891926875867478160 $$\beta_{6}$$ $$=$$ $$( - 20\!\cdots\!85 \nu^{14} + \cdots - 74\!\cdots\!76 ) / 12\!\cdots\!80$$ (-20931160306990498028225946407816203247018783785*v^14 + 66161829061227433696199214502252757148142431364561*v^12 - 43494170427229519147957446747422126489048549583343419*v^10 - 2687081921826386682300840182883810423097467501145485325913*v^8 + 12741075866736343935961710315457720418031420210404970442700975*v^6 - 42615861841543714266260381984629366558243405129780462791130367091*v^4 + 65953872478531820160723547348164924283162790564102779432777755143508*v^2 - 74678312566844498955387387489583651434468968349747241208054535177046176) / 129779570356300496864831447148955238521914594710290153703321074380 $$\beta_{7}$$ $$=$$ $$( 56\!\cdots\!27 \nu^{14} + \cdots - 10\!\cdots\!00 ) / 12\!\cdots\!80$$ (56419677199909854414675426449484275867303183927*v^14 - 671833586503048504798131521599925426010918699785075*v^12 + 4539508645220088644664569585461597007880007456195892325*v^10 - 19420087451760284809044236548115299874040783162458611630553*v^8 + 56892534504477959062407480032211154391846616015753906716619875*v^6 - 113666687063775767540028864591692819557463481362817041234667215475*v^4 + 142698895975892223372205888732981939948533883335062816872710724845396*v^2 - 102184433690302210429760730920976167074274085056031709719285086689541700) / 129779570356300496864831447148955238521914594710290153703321074380 $$\beta_{8}$$ $$=$$ $$( - 43\!\cdots\!91 \nu^{15} + \cdots - 19\!\cdots\!12 \nu ) / 42\!\cdots\!40$$ (-439767048107565213470878862116854240781699091*v^15 + 1024154083016808581438108333224960234989032789858*v^13 + 2247419938286084019643548356055773680631316623448423*v^11 - 78136419959570117932170486196603029290679468934247926666*v^9 + 346849368404048145060372610077809385825987431765017395591817*v^7 - 1120752911873865440039054291532315124435455944128624426286805930*v^5 + 1721815180594033883215929177370014848036003301061435596806824010220*v^3 - 1932054099359013240359499126995652164814649861785927129532956417216912*v) / 425640706740005502102735504819097218689283789796662682694773350639840 $$\beta_{9}$$ $$=$$ $$( 12\!\cdots\!61 \nu^{15} + \cdots - 42\!\cdots\!96 \nu ) / 28\!\cdots\!60$$ (1273907884186757407564336074260761*v^15 - 12088013272100269720764300691857748156*v^13 + 75478477134554603923527374787825983495607*v^11 - 275579532914841518721485753257097154154966420*v^9 + 730463895011395091090299138136543070336005061577*v^7 - 1170151602049587041652970572470866496313330637339152*v^5 + 1263930600579002210092350225600226571456561755152863664*v^3 - 426801315841532479267969972034695336378121427177511187496*v) / 287812451294778206086242532670287287054279193682162504160 $$\beta_{10}$$ $$=$$ $$( 85\!\cdots\!67 \nu^{15} + \cdots + 18\!\cdots\!80 \nu ) / 18\!\cdots\!80$$ (85685058367939942065662278073407436070261647134867*v^15 - 397098667199957477079158120397756942535293274887605182*v^13 + 1292961959248138081602217628961086837364087658286799364713*v^11 + 3519470182124970033381330826051969254670980502255870130326310*v^9 - 24845465223739203807986961420260921507593419377249114312536446809*v^7 + 95352209151094487051602078911167875913181756896522855589772544641806*v^5 - 154021215388092473065476301579588492709449846559392624571505441763385388*v^3 + 180441848222117706888003848709508682039396107250156843175334045712338508880*v) / 1893888324639654481606121628692573074557968222700250606650394023671968080 $$\beta_{11}$$ $$=$$ $$( - 79\!\cdots\!73 \nu^{15} + \cdots - 24\!\cdots\!40 \nu ) / 75\!\cdots\!20$$ (-796134326438068607888065175429014648760881980383773*v^15 + 2884315569312828580504670125844512995901297216403402278*v^13 - 4832529736429433773941190842008440284407310542989917533367*v^11 - 80413757353482631552186598334801000181728012774660346567601150*v^9 + 405053980639904814499279856198209032008422890535157449815798660071*v^7 - 1394396462962066106928065358819977934695410189941506788787549867634414*v^5 + 2170772364900748132946044481930623842837949250860786969071184295647630932*v^3 - 2475538518558335332098091507343589152349691122011626082179848311788030176240*v) / 7575553298558617926424486514770292298231872890801002426601576094687872320 $$\beta_{12}$$ $$=$$ $$( - 15\!\cdots\!85 \nu^{15} + \cdots + 62\!\cdots\!88 \nu ) / 12\!\cdots\!20$$ (-152856711039711343922101699154759013485*v^15 + 1479484572074184616821390341755765264136636*v^13 - 9056692334108098343819538501689142876216492195*v^11 + 33066897186301629995073115416893448277839718081700*v^9 - 85747935372162402381537371736515838929932959538490429*v^7 + 140406953695348185889030509697020782046927255552100829520*v^5 - 151659532832147708951268845250542543282699828536974338102640*v^3 + 62173876255455585066876606800581230328157769455625807882584088*v) / 1280621502036115627980736149116443283748015272288782062259920 $$\beta_{13}$$ $$=$$ $$( - 13\!\cdots\!79 \nu^{15} + \cdots + 24\!\cdots\!84 \nu ) / 51\!\cdots\!80$$ (-130832945568721323727963367329390291456609279*v^15 + 1581195828336984496870855189133056587536501326189*v^13 - 10683973348047418882011414704834561119472283021270459*v^11 + 45804720724383825519776605308545793131900376492023630047*v^9 - 133899584702573223877813914820241611446136700024262332649085*v^7 + 267520551244891040035680093071761243737720235783060717355852397*v^5 - 335071384283304355418240772554827619623762887062811421847756730630*v^3 + 240496461501854246087254010979953580037849582601981397764217876301884*v) / 518485927612647499425736577107379613502062423391240440303842462880 $$\beta_{14}$$ $$=$$ $$( 13\!\cdots\!89 \nu^{15} + \cdots - 76\!\cdots\!00 \nu ) / 42\!\cdots\!40$$ (134341174258606769808602093050458248241380137509889*v^15 - 985335561141740631759831842357235969879864687763987534*v^13 + 5490786758126906765070750443492873866990974893928641766531*v^11 - 15974490039163967964929618101081265038496012837293135059241210*v^9 + 39517311321336355824155595839685277132590169312014100230399189997*v^7 - 57887980874661993138817399572576400192501985332832573149197320928138*v^5 + 91718596355774848662891689674125615433270037311555337102688956568285084*v^3 - 76998607968643499196618433894752182634867553346743056542991694842124242000*v) / 420864072142145440356915917487238461012881827266722357033420894149326240 $$\beta_{15}$$ $$=$$ $$( 95\!\cdots\!35 \nu^{15} + \cdots - 95\!\cdots\!76 \nu ) / 28\!\cdots\!60$$ (95258568805449357248444844537691625035*v^15 - 952087106096776123189417666213804138265812*v^13 + 5644027952650964010379815317849652411641067045*v^11 - 20606915322060706073779053441504766648282059072700*v^9 + 50924727726523627252124533517926774528477776123015643*v^7 - 87500021218414961319699823800734678746577952063663939120*v^5 + 94512500923431683220785066742897340669331813121117443949840*v^3 - 95537498509123513786318255376930503118676757547075558808913176*v) / 284582556008025695106830255359209618610670060508618236057760
 $$\nu$$ $$=$$ $$( -\beta_{15} + 4\beta_{13} - 13\beta_{12} + 4\beta_{11} + 49\beta_{9} + 4\beta_{8} ) / 162$$ (-b15 + 4*b13 - 13*b12 + 4*b11 + 49*b9 + 4*b8) / 162 $$\nu^{2}$$ $$=$$ $$( -2\beta_{7} + 15\beta_{5} + 15\beta_{4} - 738\beta_{2} + 449711\beta_1 ) / 162$$ (-2*b7 + 15*b5 + 15*b4 - 738*b2 + 449711*b1) / 162 $$\nu^{3}$$ $$=$$ $$( - 1673 \beta_{15} + 1673 \beta_{14} + 18864 \beta_{13} - 41621 \beta_{12} - 41621 \beta_{10} + 501379 \beta_{9} + 501379 \beta_{8} ) / 162$$ (-1673*b15 + 1673*b14 + 18864*b13 - 41621*b12 - 41621*b10 + 501379*b9 + 501379*b8) / 162 $$\nu^{4}$$ $$=$$ $$( -11104\beta_{6} + 103125\beta_{4} + 4108806\beta_{3} - 4108806\beta_{2} + 1315110043\beta _1 - 1315110043 ) / 162$$ (-11104*b6 + 103125*b4 + 4108806*b3 - 4108806*b2 + 1315110043*b1 - 1315110043) / 162 $$\nu^{5}$$ $$=$$ $$( 1982509\beta_{14} - 81274620\beta_{11} - 136331191\beta_{10} + 2909594105\beta_{8} ) / 162$$ (1982509*b14 - 81274620*b11 - 136331191*b10 + 2909594105*b8) / 162 $$\nu^{6}$$ $$=$$ $$( 48681374\beta_{7} - 48681374\beta_{6} - 530474235\beta_{5} + 17472818586\beta_{3} - 4023019402499 ) / 162$$ (48681374*b7 - 48681374*b6 - 530474235*b5 + 17472818586*b3 - 4023019402499) / 162 $$\nu^{7}$$ $$=$$ $$( - 1232327353 \beta_{15} - 334702500192 \beta_{13} + 453769993415 \beta_{12} - 334702500192 \beta_{11} - 14776449049837 \beta_{9} - 334702500192 \beta_{8} ) / 162$$ (-1232327353*b15 - 334702500192*b13 + 453769993415*b12 - 334702500192*b11 - 14776449049837*b9 - 334702500192*b8) / 162 $$\nu^{8}$$ $$=$$ $$( 198311421880 \beta_{7} - 2422307561835 \beta_{5} - 2422307561835 \beta_{4} + 67210123467450 \beta_{2} - 12\!\cdots\!17 \beta_1 ) / 162$$ (198311421880*b7 - 2422307561835*b5 - 2422307561835*b4 + 67210123467450*b2 - 12755114716334917*b1) / 162 $$\nu^{9}$$ $$=$$ $$( - 21422959842329 \beta_{15} + 21422959842329 \beta_{14} + \cdots - 66\!\cdots\!59 \beta_{8} ) / 162$$ (-21422959842329*b15 + 21422959842329*b14 - 1340829850116804*b13 + 1527080559689173*b12 + 1527080559689173*b10 - 66261096708695159*b9 - 66261096708695159*b8) / 162 $$\nu^{10}$$ $$=$$ $$( 784340321652002 \beta_{6} + \cdots + 41\!\cdots\!01 ) / 162$$ (784340321652002*b6 - 10362422887128825*b4 - 246372498996383358*b3 + 246372498996383358*b2 - 41530930659415000601*b1 + 41530930659415000601) / 162 $$\nu^{11}$$ $$=$$ $$( 11\!\cdots\!87 \beta_{14} + \cdots - 27\!\cdots\!15 \beta_{8} ) / 162$$ (110379469080042187*b14 + 5268495797226183984*b11 + 5178027704278242941*b10 - 275048673363170921215*b8) / 162 $$\nu^{12}$$ $$=$$ $$( - 30\!\cdots\!44 \beta_{7} + \cdots + 13\!\cdots\!63 ) / 162$$ (-3055917202288340944*b7 + 3055917202288340944*b6 + 42543514109346994425*b5 - 880235124489644637246*b3 + 137802833793651688633663) / 162 $$\nu^{13}$$ $$=$$ $$( 45\!\cdots\!51 \beta_{15} + \cdots + 20\!\cdots\!60 \beta_{8} ) / 162$$ (458145495345932666051*b15 + 20397592523233377926460*b13 - 17649450616892757592111*b12 + 20397592523233377926460*b11 + 1141335730979917968861845*b9 + 20397592523233377926460*b8) / 162 $$\nu^{14}$$ $$=$$ $$( - 11\!\cdots\!54 \beta_{7} + \cdots + 46\!\cdots\!59 \beta_1 ) / 162$$ (-11787152269441976261054*b7 + 169803829708559161922715*b5 + 169803829708559161922715*b4 - 3100092078068641165408026*b2 + 463234022684708838412858259*b1) / 162 $$\nu^{15}$$ $$=$$ $$( 17\!\cdots\!53 \beta_{15} + \cdots + 45\!\cdots\!37 \beta_{8} ) / 162$$ (1744446993386349062877553*b15 - 1744446993386349062877553*b14 + 78038307898597342397168832*b13 - 60380821331804862929700815*b12 - 60380821331804862929700815*b10 + 4524481402875774264011516437*b9 + 4524481402875774264011516437*b8) / 162

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$1 - \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −49.7470 − 28.7215i −40.2810 − 23.2563i 39.8327 + 22.9974i 51.4201 + 29.6874i −51.4201 − 29.6874i −39.8327 − 22.9974i 40.2810 + 23.2563i 49.7470 + 28.7215i −49.7470 + 28.7215i −40.2810 + 23.2563i 39.8327 − 22.9974i 51.4201 − 29.6874i −51.4201 + 29.6874i −39.8327 + 22.9974i 40.2810 − 23.2563i 49.7470 − 28.7215i
−9.79796 + 5.65685i 0 64.0000 110.851i −467.559 269.945i 0 108.294 + 187.571i 1448.15i 0 6108.16
53.2 −9.79796 + 5.65685i 0 64.0000 110.851i −35.4801 20.4844i 0 −1314.30 2276.44i 1448.15i 0 463.510
53.3 −9.79796 + 5.65685i 0 64.0000 110.851i 269.260 + 155.458i 0 2242.38 + 3883.91i 1448.15i 0 −3517.60
53.4 −9.79796 + 5.65685i 0 64.0000 110.851i 968.625 + 559.236i 0 −1409.37 2441.10i 1448.15i 0 −12654.1
53.5 9.79796 5.65685i 0 64.0000 110.851i −968.625 559.236i 0 −1409.37 2441.10i 1448.15i 0 −12654.1
53.6 9.79796 5.65685i 0 64.0000 110.851i −269.260 155.458i 0 2242.38 + 3883.91i 1448.15i 0 −3517.60
53.7 9.79796 5.65685i 0 64.0000 110.851i 35.4801 + 20.4844i 0 −1314.30 2276.44i 1448.15i 0 463.510
53.8 9.79796 5.65685i 0 64.0000 110.851i 467.559 + 269.945i 0 108.294 + 187.571i 1448.15i 0 6108.16
107.1 −9.79796 5.65685i 0 64.0000 + 110.851i −467.559 + 269.945i 0 108.294 187.571i 1448.15i 0 6108.16
107.2 −9.79796 5.65685i 0 64.0000 + 110.851i −35.4801 + 20.4844i 0 −1314.30 + 2276.44i 1448.15i 0 463.510
107.3 −9.79796 5.65685i 0 64.0000 + 110.851i 269.260 155.458i 0 2242.38 3883.91i 1448.15i 0 −3517.60
107.4 −9.79796 5.65685i 0 64.0000 + 110.851i 968.625 559.236i 0 −1409.37 + 2441.10i 1448.15i 0 −12654.1
107.5 9.79796 + 5.65685i 0 64.0000 + 110.851i −968.625 + 559.236i 0 −1409.37 + 2441.10i 1448.15i 0 −12654.1
107.6 9.79796 + 5.65685i 0 64.0000 + 110.851i −269.260 + 155.458i 0 2242.38 3883.91i 1448.15i 0 −3517.60
107.7 9.79796 + 5.65685i 0 64.0000 + 110.851i 35.4801 20.4844i 0 −1314.30 + 2276.44i 1448.15i 0 463.510
107.8 9.79796 + 5.65685i 0 64.0000 + 110.851i 467.559 269.945i 0 108.294 187.571i 1448.15i 0 6108.16
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.9.d.g 16
3.b odd 2 1 inner 162.9.d.g 16
9.c even 3 1 162.9.b.b 8
9.c even 3 1 inner 162.9.d.g 16
9.d odd 6 1 162.9.b.b 8
9.d odd 6 1 inner 162.9.d.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.9.b.b 8 9.c even 3 1
162.9.b.b 8 9.d odd 6 1
162.9.d.g 16 1.a even 1 1 trivial
162.9.d.g 16 3.b odd 2 1 inner
162.9.d.g 16 9.c even 3 1 inner
162.9.d.g 16 9.d odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} - 1640808 T_{5}^{14} + 2175755301198 T_{5}^{12} + \cdots + 35\!\cdots\!25$$ acting on $$S_{9}^{\mathrm{new}}(162, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 128 T^{2} + 16384)^{4}$$
$3$ $$T^{16}$$
$5$ $$T^{16} - 1640808 T^{14} + \cdots + 35\!\cdots\!25$$
$7$ $$(T^{8} + 746 T^{7} + \cdots + 51\!\cdots\!96)^{2}$$
$11$ $$T^{16} - 86966028 T^{14} + \cdots + 97\!\cdots\!36$$
$13$ $$(T^{8} + 23390 T^{7} + \cdots + 45\!\cdots\!61)^{2}$$
$17$ $$(T^{8} + 50461667568 T^{6} + \cdots + 42\!\cdots\!25)^{2}$$
$19$ $$(T^{4} + 320218 T^{3} + \cdots - 10\!\cdots\!04)^{4}$$
$23$ $$T^{16} - 222640275348 T^{14} + \cdots + 10\!\cdots\!00$$
$29$ $$T^{16} - 1329947673696 T^{14} + \cdots + 27\!\cdots\!21$$
$31$ $$(T^{8} + 53888 T^{7} + \cdots + 22\!\cdots\!96)^{2}$$
$37$ $$(T^{4} + 729562 T^{3} + \cdots + 13\!\cdots\!53)^{4}$$
$41$ $$T^{16} - 37646502934896 T^{14} + \cdots + 14\!\cdots\!96$$
$43$ $$(T^{8} - 4232734 T^{7} + \cdots + 84\!\cdots\!64)^{2}$$
$47$ $$T^{16} - 138220620047856 T^{14} + \cdots + 64\!\cdots\!96$$
$53$ $$(T^{8} + 18704070936720 T^{6} + \cdots + 26\!\cdots\!36)^{2}$$
$59$ $$T^{16} - 799197825064464 T^{14} + \cdots + 40\!\cdots\!16$$
$61$ $$(T^{8} - 14168698 T^{7} + \cdots + 15\!\cdots\!89)^{2}$$
$67$ $$(T^{8} + 46506398 T^{7} + \cdots + 36\!\cdots\!36)^{2}$$
$71$ $$(T^{8} + \cdots + 13\!\cdots\!04)^{2}$$
$73$ $$(T^{4} + 69655456 T^{3} + \cdots - 13\!\cdots\!11)^{4}$$
$79$ $$(T^{8} - 20445178 T^{7} + \cdots + 41\!\cdots\!56)^{2}$$
$83$ $$T^{16} + \cdots + 15\!\cdots\!16$$
$89$ $$(T^{8} + \cdots + 23\!\cdots\!21)^{2}$$
$97$ $$(T^{8} - 24721456 T^{7} + \cdots + 38\!\cdots\!64)^{2}$$