# Properties

 Label 117.4.g.f Level $117$ Weight $4$ Character orbit 117.g Analytic conductor $6.903$ 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] = [117,4,Mod(55,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("117.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.g (of order $$3$$, degree $$2$$, 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: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 52 x^{14} + 1899 x^{12} + 33440 x^{10} + 424113 x^{8} + 2869882 x^{6} + 13705540 x^{4} + \cdots + 24920064$$ x^16 + 52*x^14 + 1899*x^12 + 33440*x^10 + 424113*x^8 + 2869882*x^6 + 13705540*x^4 + 21016320*x^2 + 24920064 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{6} + \beta_{3} - 5 \beta_{2} - 5) q^{4} + (\beta_{12} + \beta_{9}) q^{5} + ( - \beta_{7} - \beta_{4} + 3 \beta_{2} + 3) q^{7} + (\beta_{13} + \beta_{12} + \cdots - 5 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b6 + b3 - 5*b2 - 5) * q^4 + (b12 + b9) * q^5 + (-b7 - b4 + 3*b2 + 3) * q^7 + (b13 + b12 - b11 + b9 - 5*b5 - 5*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{6} + \beta_{3} - 5 \beta_{2} - 5) q^{4} + (\beta_{12} + \beta_{9}) q^{5} + ( - \beta_{7} - \beta_{4} + 3 \beta_{2} + 3) q^{7} + (\beta_{13} + \beta_{12} + \cdots - 5 \beta_1) q^{8}+ \cdots + (8 \beta_{15} + 25 \beta_{13} + \cdots - 33 \beta_{5}) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b6 + b3 - 5*b2 - 5) * q^4 + (b12 + b9) * q^5 + (-b7 - b4 + 3*b2 + 3) * q^7 + (b13 + b12 - b11 + b9 - 5*b5 - 5*b1) * q^8 + (b10 - 3*b6 + b4 + 4*b2) * q^10 + (b14 - b9 + b1) * q^11 + (b10 - b8 - b4 - 2*b3 - 4*b2) * q^13 + (-b15 - b14 - b13 + 2*b12 + b11 + 2*b9 + 7*b5 + 7*b1) * q^14 + (2*b10 - 9*b6 + 25*b2) * q^16 + (b15 + 2*b13 - 13*b5) * q^17 + (b10 - b8 + b7 + 6*b6 + b4 + 6*b3 - 31*b2 - 31) * q^19 + (b15 - 3*b13 - 6*b12 + 18*b5) * q^20 + (-b10 + b8 - 7*b7 + 8*b6 - 7*b4 + 8*b3 - 15*b2 - 15) * q^22 + (-b14 - 2*b11 - 5*b9 - 11*b1) * q^23 + (b8 - 6*b7 + 46) * q^25 + (-b15 - 2*b13 - 7*b12 + 3*b11 - 11*b9 - 6*b5 + 10*b1) * q^26 + (b10 + 6*b6 + b4 - 57*b2) * q^28 + (6*b11 + b9 + 10*b1) * q^29 + (-3*b8 + 6*b7 - 8*b3 + 28) * q^31 + (-3*b13 - 19*b12 + 45*b5) * q^32 + (2*b8 + 8*b7 + 27*b3 - 163) * q^34 + (-b15 + 17*b12 + 27*b5) * q^35 + (-2*b10 - 12*b6 + 7*b4 + 126*b2) * q^37 + (b15 + b14 + 6*b13 - 5*b12 - 6*b11 - 5*b9 - 89*b5 - 89*b1) * q^38 + (-b8 + b7 - 43*b3 + 212) * q^40 + (b14 - 4*b11 - 6*b9 + 49*b1) * q^41 + (7*b7 - 26*b6 + 7*b4 - 26*b3 - 11*b2 - 11) * q^43 + (b15 + b14 + 2*b13 + 23*b12 - 2*b11 + 23*b9 - 37*b5 - 37*b1) * q^44 + (-3*b10 + 3*b8 - b7 - 18*b6 - b4 - 18*b3 + 113*b2 + 113) * q^46 + (-b15 - b14 + 2*b13 + 11*b12 - 2*b11 + 11*b9 + 33*b5 + 33*b1) * q^47 + (-7*b10 + 10*b6 + 8*b4 + 37*b2) * q^49 + (-6*b14 + 5*b11 + 21*b9 + 76*b1) * q^50 + (-6*b10 + 5*b8 - b7 + 54*b6 - 8*b4 + 25*b3 - 162*b2 - 186) * q^52 + (6*b15 + 6*b14 + 2*b13 + 19*b12 - 2*b11 + 19*b9 + 60*b5 + 60*b1) * q^53 + (-11*b10 - 24*b6 - 7*b4 + 167*b2) * q^55 + (-7*b15 - 2*b13 + 11*b12 - 59*b5) * q^56 + (-5*b10 + 5*b8 + 7*b7 + 61*b6 + 7*b4 + 61*b3 - 102*b2 - 102) * q^58 + (-6*b15 + 4*b13 - 6*b12 + 2*b5) * q^59 + (3*b10 - 3*b8 + 15*b7 - 10*b6 + 15*b4 - 10*b3 - 148*b2 - 148) * q^61 + (6*b14 + 5*b11 - 47*b9 + 50*b1) * q^62 + (-6*b8 + 16*b7 - 57*b3 + 449) * q^64 + (6*b15 - b14 + 2*b13 - 12*b12 - 6*b11 - 8*b9 - 156*b5 - 63*b1) * q^65 + (-8*b10 + 30*b6 + b4 - 273*b2) * q^67 + (-21*b11 + 29*b9 - 295*b1) * q^68 + (17*b8 - 23*b7 + 28*b3 + 285) * q^70 + (b15 + 14*b13 - 11*b12 + 131*b5) * q^71 + (-12*b8 - 8*b7 - 6*b3 - 273) * q^73 + (7*b15 - 3*b13 - 8*b12 + 206*b5) * q^74 + (9*b10 - 76*b6 - 9*b4 + 867*b2) * q^76 + (-6*b15 - 6*b14 - 26*b13 - 28*b12 + 26*b11 - 28*b9 - 176*b5 - 176*b1) * q^77 + (5*b8 - 16*b3 + 234) * q^79 + (-7*b14 + 19*b11 - 6*b9 + 402*b1) * q^80 + (-2*b10 + 2*b8 - 16*b7 + 35*b6 - 16*b4 + 35*b3 - 675*b2 - 675) * q^82 + (-7*b15 - 7*b14 - 8*b13 - 13*b12 + 8*b11 - 13*b9 - 51*b5 - 51*b1) * q^83 + (18*b10 - 18*b8 + 19*b7 - 74*b6 + 19*b4 - 74*b3 + 102*b2 + 102) * q^85 + (7*b15 + 7*b14 - 19*b13 - 40*b12 + 19*b11 - 40*b9 + 169*b5 + 169*b1) * q^86 + (17*b10 - 56*b6 - 41*b4 + 447*b2) * q^88 + (8*b14 - 24*b11 - 32*b9 - 44*b1) * q^89 + (11*b10 - 5*b8 - 7*b7 - 12*b6 + 34*b4 - 22*b3 + 32*b2 - 275) * q^91 + (-9*b15 - 9*b14 - 29*b12 - 29*b9 + 191*b5 + 191*b1) * q^92 + (13*b10 - 22*b6 + 15*b4 - 395*b2) * q^94 + (15*b15 - 22*b13 - 65*b12 - 39*b5) * q^95 + (b10 - b8 - 48*b7 + 58*b6 - 48*b4 + 58*b3 + 558*b2 + 558) * q^97 + (8*b15 + 25*b13 + 57*b12 - 33*b5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 40 q^{4} + 22 q^{7}+O(q^{10})$$ 16 * q - 40 * q^4 + 22 * q^7 $$16 q - 40 q^{4} + 22 q^{7} - 36 q^{10} + 36 q^{13} - 204 q^{16} - 244 q^{19} - 136 q^{22} + 708 q^{25} + 452 q^{28} + 484 q^{31} - 2584 q^{34} - 1018 q^{37} + 3400 q^{40} - 74 q^{43} + 896 q^{46} - 298 q^{49} - 1676 q^{52} - 1300 q^{55} - 812 q^{58} - 1148 q^{61} + 7272 q^{64} + 2198 q^{67} + 4400 q^{70} - 4352 q^{73} - 6936 q^{76} + 3724 q^{79} - 5436 q^{82} + 890 q^{85} - 3528 q^{88} - 4754 q^{91} + 3104 q^{94} + 4370 q^{97}+O(q^{100})$$ 16 * q - 40 * q^4 + 22 * q^7 - 36 * q^10 + 36 * q^13 - 204 * q^16 - 244 * q^19 - 136 * q^22 + 708 * q^25 + 452 * q^28 + 484 * q^31 - 2584 * q^34 - 1018 * q^37 + 3400 * q^40 - 74 * q^43 + 896 * q^46 - 298 * q^49 - 1676 * q^52 - 1300 * q^55 - 812 * q^58 - 1148 * q^61 + 7272 * q^64 + 2198 * q^67 + 4400 * q^70 - 4352 * q^73 - 6936 * q^76 + 3724 * q^79 - 5436 * q^82 + 890 * q^85 - 3528 * q^88 - 4754 * q^91 + 3104 * q^94 + 4370 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 52 x^{14} + 1899 x^{12} + 33440 x^{10} + 424113 x^{8} + 2869882 x^{6} + 13705540 x^{4} + \cdots + 24920064$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 1400516976091 \nu^{14} - 70185147718972 \nu^{12} + \cdots - 25\!\cdots\!88 ) / 19\!\cdots\!08$$ (-1400516976091*v^14 - 70185147718972*v^12 - 2537753710344969*v^10 - 42518671468817696*v^8 - 531022140611344875*v^6 - 3269311083458423326*v^4 - 16925010973113963820*v^2 - 25929136118756652288) / 19347670437452567808 $$\beta_{3}$$ $$=$$ $$( - 1058387435 \nu^{14} - 48809305790 \nu^{12} - 1728612264289 \nu^{10} + \cdots + 86\!\cdots\!80 ) / 77\!\cdots\!48$$ (-1058387435*v^14 - 48809305790*v^12 - 1728612264289*v^10 - 25222482239398*v^8 - 300483724727391*v^6 - 909387204879920*v^4 - 1404077250081792*v^2 + 86786355372439580) / 7751470527825548 $$\beta_{4}$$ $$=$$ $$( - 3479246660543 \nu^{14} + 32435361605764 \nu^{12} + \cdots + 20\!\cdots\!40 ) / 19\!\cdots\!08$$ (-3479246660543*v^14 + 32435361605764*v^12 + 3580765159783899*v^10 + 232118184426369680*v^8 + 3608907912699869841*v^6 + 40174000892884504378*v^4 + 135634859424724819780*v^2 + 209921486006331191040) / 19347670437452567808 $$\beta_{5}$$ $$=$$ $$( - 1400516976091 \nu^{15} - 70185147718972 \nu^{13} + \cdots - 25\!\cdots\!88 \nu ) / 19\!\cdots\!08$$ (-1400516976091*v^15 - 70185147718972*v^13 - 2537753710344969*v^11 - 42518671468817696*v^9 - 531022140611344875*v^7 - 3269311083458423326*v^5 - 16925010973113963820*v^3 - 25929136118756652288*v) / 19347670437452567808 $$\beta_{6}$$ $$=$$ $$( - 1197306588571 \nu^{14} - 60813761007292 \nu^{12} + \cdots - 23\!\cdots\!40 ) / 14\!\cdots\!16$$ (-1197306588571*v^14 - 60813761007292*v^12 - 2205860155601481*v^10 - 37675954878853280*v^8 - 473329265463685803*v^6 - 3094708740121478686*v^4 - 15167145799755754540*v^2 - 23244445912812483840) / 1488282341342505216 $$\beta_{7}$$ $$=$$ $$( 45779722905 \nu^{14} + 2174587212793 \nu^{12} + 74769775086507 \nu^{10} + \cdots - 12\!\cdots\!00 ) / 31\!\cdots\!92$$ (45779722905*v^14 + 2174587212793*v^12 + 74769775086507*v^10 + 1090978794448674*v^8 + 11062740994006466*v^6 + 39334834178898960*v^4 + 60732266200759296*v^2 - 127620970309043600) / 31005882111302192 $$\beta_{8}$$ $$=$$ $$( 28699944525 \nu^{14} + 1329441616294 \nu^{12} + 46874211134535 \nu^{10} + \cdots - 77\!\cdots\!76 ) / 15\!\cdots\!96$$ (28699944525*v^14 + 1329441616294*v^12 + 46874211134535*v^10 + 683949768407370*v^8 + 7787903072151153*v^6 + 24659554213054800*v^4 + 38073901724052480*v^2 - 774173355811127376) / 15502941055651096 $$\beta_{9}$$ $$=$$ $$( 30816719395 \nu^{15} + 1427060227874 \nu^{13} + 50331435663113 \nu^{11} + \cdots - 11\!\cdots\!80 \nu ) / 12\!\cdots\!68$$ (30816719395*v^15 + 1427060227874*v^13 + 50331435663113*v^11 + 734394732886166*v^9 + 8388870521605935*v^7 + 26478328622814640*v^5 + 40882056224216064*v^3 - 1164787241335121880*v) / 124023528445208768 $$\beta_{10}$$ $$=$$ $$( - 841688410411 \nu^{14} - 44413834261852 \nu^{12} + \cdots - 18\!\cdots\!56 ) / 24\!\cdots\!36$$ (-841688410411*v^14 - 44413834261852*v^12 - 1625046434800377*v^10 - 29201200846415552*v^8 - 372366733955282427*v^6 - 2665131110836616798*v^4 - 12090881746378888300*v^2 - 18546238052410189056) / 248047056890417536 $$\beta_{11}$$ $$=$$ $$( 47750918355 \nu^{15} + 2208009120514 \nu^{13} + 77989231891737 \nu^{11} + \cdots - 35\!\cdots\!04 \nu ) / 12\!\cdots\!68$$ (47750918355*v^15 + 2208009120514*v^13 + 77989231891737*v^11 + 1137954448716534*v^9 + 13196610117244191*v^7 + 41028523900893360*v^5 + 63347292225524736*v^3 - 3545557154855825304*v) / 124023528445208768 $$\beta_{12}$$ $$=$$ $$( 33607973109585 \nu^{15} + \cdots + 70\!\cdots\!40 \nu ) / 51\!\cdots\!88$$ (33607973109585*v^15 + 1745816677861620*v^13 + 63652785294026075*v^11 + 1120914160002984480*v^9 + 14210731889366257665*v^7 + 97960598478417753242*v^5 + 459070608412657863300*v^3 + 703930756872383389440*v) / 51593787833206847488 $$\beta_{13}$$ $$=$$ $$( - 8076330538717 \nu^{15} - 411639246983524 \nu^{13} + \cdots - 15\!\cdots\!08 \nu ) / 39\!\cdots\!76$$ (-8076330538717*v^15 - 411639246983524*v^13 - 14943180757095135*v^11 - 256467609267026336*v^9 - 3226765545524312013*v^7 - 21153010608954516306*v^5 - 103533222838252967860*v^3 - 158684086080771134208*v) / 3968752910246680576 $$\beta_{14}$$ $$=$$ $$( 292503248765 \nu^{15} + 13760478427434 \nu^{13} + 477731203563991 \nu^{11} + \cdots - 58\!\cdots\!92 \nu ) / 12\!\cdots\!68$$ (292503248765*v^15 + 13760478427434*v^13 + 477731203563991*v^11 + 6970659092283562*v^9 + 74225315136481925*v^7 + 251324517862118480*v^5 + 388040469476994048*v^3 - 5889521404981408392*v) / 124023528445208768 $$\beta_{15}$$ $$=$$ $$( 147991404382739 \nu^{15} + \cdots + 37\!\cdots\!52 \nu ) / 51\!\cdots\!88$$ (147991404382739*v^15 + 8180796288455388*v^13 + 304046254612699633*v^11 + 5741356569946908384*v^9 + 74328934137893879075*v^7 + 543168352364046381326*v^5 + 2445232621808654880780*v^3 + 3753967810211603741952*v) / 51593787833206847488
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{3} - 13\beta_{2} - 13$$ b6 + b3 - 13*b2 - 13 $$\nu^{3}$$ $$=$$ $$\beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} - 21\beta_{5} - 21\beta_1$$ b13 + b12 - b11 + b9 - 21*b5 - 21*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{10} - 33\beta_{6} + 273\beta_{2}$$ 2*b10 - 33*b6 + 273*b2 $$\nu^{5}$$ $$=$$ $$-35\beta_{13} - 51\beta_{12} + 525\beta_{5}$$ -35*b13 - 51*b12 + 525*b5 $$\nu^{6}$$ $$=$$ $$-86\beta_{8} + 16\beta_{7} - 993\beta_{3} + 6889$$ -86*b8 + 16*b7 - 993*b3 + 6889 $$\nu^{7}$$ $$=$$ $$16\beta_{14} + 1063\beta_{11} - 1799\beta_{9} + 14253\beta_1$$ 16*b14 + 1063*b11 - 1799*b9 + 14253*b1 $$\nu^{8}$$ $$=$$ $$- 2862 \beta_{10} + 2862 \beta_{8} - 832 \beta_{7} + 29281 \beta_{6} - 832 \beta_{4} + 29281 \beta_{3} + \cdots - 188201$$ -2862*b10 + 2862*b8 - 832*b7 + 29281*b6 - 832*b4 + 29281*b3 - 188201*b2 - 188201 $$\nu^{9}$$ $$=$$ $$- 832 \beta_{15} - 832 \beta_{14} + 31311 \beta_{13} + 56703 \beta_{12} - 31311 \beta_{11} + \cdots - 401949 \beta_1$$ -832*b15 - 832*b14 + 31311*b13 + 56703*b12 - 31311*b11 + 56703*b9 - 401949*b5 - 401949*b1 $$\nu^{10}$$ $$=$$ $$88014\beta_{10} - 857185\beta_{6} + 30384\beta_{4} + 5325241\beta_{2}$$ 88014*b10 - 857185*b6 + 30384*b4 + 5325241*b2 $$\nu^{11}$$ $$=$$ $$30384\beta_{15} - 914815\beta_{13} - 1710079\beta_{12} + 11533101\beta_{5}$$ 30384*b15 - 914815*b13 - 1710079*b12 + 11533101*b5 $$\nu^{12}$$ $$=$$ $$-2624894\beta_{8} + 977568\beta_{7} - 25018209\beta_{3} + 153050601$$ -2624894*b8 + 977568*b7 - 25018209*b3 + 153050601 $$\nu^{13}$$ $$=$$ $$977568\beta_{14} + 26665535\beta_{11} - 50597391\beta_{9} + 333536637\beta_1$$ 977568*b14 + 26665535*b11 - 50597391*b9 + 333536637*b1 $$\nu^{14}$$ $$=$$ $$- 77262926 \beta_{10} + 77262926 \beta_{8} - 29797264 \beta_{7} + 729228897 \beta_{6} + \cdots - 4429748569$$ -77262926*b10 + 77262926*b8 - 29797264*b7 + 729228897*b6 - 29797264*b4 + 729228897*b3 - 4429748569*b2 - 4429748569 $$\nu^{15}$$ $$=$$ $$- 29797264 \beta_{15} - 29797264 \beta_{14} + 776694559 \beta_{13} + 1484189759 \beta_{12} + \cdots - 9680813133 \beta_1$$ -29797264*b15 - 29797264*b14 + 776694559*b13 + 1484189759*b12 - 776694559*b11 + 1484189759*b9 - 9680813133*b5 - 9680813133*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −2.69794 − 4.67298i −1.84606 − 3.19747i −1.37814 − 2.38701i −0.643348 − 1.11431i 0.643348 + 1.11431i 1.37814 + 2.38701i 1.84606 + 3.19747i 2.69794 + 4.67298i −2.69794 + 4.67298i −1.84606 + 3.19747i −1.37814 + 2.38701i −0.643348 + 1.11431i 0.643348 − 1.11431i 1.37814 − 2.38701i 1.84606 − 3.19747i 2.69794 − 4.67298i
−2.69794 4.67298i 0 −10.5578 + 18.2867i 13.0421 0 3.21247 5.56416i 70.7705 0 −35.1870 60.9457i
55.2 −1.84606 3.19747i 0 −2.81586 + 4.87721i −18.7574 0 12.0691 20.9044i −8.74396 0 34.6273 + 59.9763i
55.3 −1.37814 2.38701i 0 0.201468 0.348954i 0.313209 0 −14.2830 + 24.7388i −23.1608 0 −0.431646 0.747632i
55.4 −0.643348 1.11431i 0 3.17221 5.49442i 12.4484 0 4.50137 7.79659i −18.4569 0 −8.00866 13.8714i
55.5 0.643348 + 1.11431i 0 3.17221 5.49442i −12.4484 0 4.50137 7.79659i 18.4569 0 −8.00866 13.8714i
55.6 1.37814 + 2.38701i 0 0.201468 0.348954i −0.313209 0 −14.2830 + 24.7388i 23.1608 0 −0.431646 0.747632i
55.7 1.84606 + 3.19747i 0 −2.81586 + 4.87721i 18.7574 0 12.0691 20.9044i 8.74396 0 34.6273 + 59.9763i
55.8 2.69794 + 4.67298i 0 −10.5578 + 18.2867i −13.0421 0 3.21247 5.56416i −70.7705 0 −35.1870 60.9457i
100.1 −2.69794 + 4.67298i 0 −10.5578 18.2867i 13.0421 0 3.21247 + 5.56416i 70.7705 0 −35.1870 + 60.9457i
100.2 −1.84606 + 3.19747i 0 −2.81586 4.87721i −18.7574 0 12.0691 + 20.9044i −8.74396 0 34.6273 59.9763i
100.3 −1.37814 + 2.38701i 0 0.201468 + 0.348954i 0.313209 0 −14.2830 24.7388i −23.1608 0 −0.431646 + 0.747632i
100.4 −0.643348 + 1.11431i 0 3.17221 + 5.49442i 12.4484 0 4.50137 + 7.79659i −18.4569 0 −8.00866 + 13.8714i
100.5 0.643348 1.11431i 0 3.17221 + 5.49442i −12.4484 0 4.50137 + 7.79659i 18.4569 0 −8.00866 + 13.8714i
100.6 1.37814 2.38701i 0 0.201468 + 0.348954i −0.313209 0 −14.2830 24.7388i 23.1608 0 −0.431646 + 0.747632i
100.7 1.84606 3.19747i 0 −2.81586 4.87721i 18.7574 0 12.0691 + 20.9044i 8.74396 0 34.6273 59.9763i
100.8 2.69794 4.67298i 0 −10.5578 18.2867i −13.0421 0 3.21247 + 5.56416i −70.7705 0 −35.1870 + 60.9457i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.8 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
13.c even 3 1 inner
39.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.g.f 16
3.b odd 2 1 inner 117.4.g.f 16
13.c even 3 1 inner 117.4.g.f 16
13.c even 3 1 1521.4.a.bc 8
13.e even 6 1 1521.4.a.bd 8
39.h odd 6 1 1521.4.a.bd 8
39.i odd 6 1 inner 117.4.g.f 16
39.i odd 6 1 1521.4.a.bc 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.g.f 16 1.a even 1 1 trivial
117.4.g.f 16 3.b odd 2 1 inner
117.4.g.f 16 13.c even 3 1 inner
117.4.g.f 16 39.i odd 6 1 inner
1521.4.a.bc 8 13.c even 3 1
1521.4.a.bc 8 39.i odd 6 1
1521.4.a.bd 8 13.e even 6 1
1521.4.a.bd 8 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 52 T_{2}^{14} + 1899 T_{2}^{12} + 33440 T_{2}^{10} + 424113 T_{2}^{8} + 2869882 T_{2}^{6} + \cdots + 24920064$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 52 T^{14} + \cdots + 24920064$$
$3$ $$T^{16}$$
$5$ $$(T^{8} - 677 T^{6} + \cdots + 909792)^{2}$$
$7$ $$(T^{8} - 11 T^{7} + \cdots + 1590733456)^{2}$$
$11$ $$T^{16} + \cdots + 30\!\cdots\!04$$
$13$ $$(T^{8} + \cdots + 23298085122481)^{2}$$
$17$ $$T^{16} + \cdots + 90\!\cdots\!64$$
$19$ $$(T^{8} + \cdots + 210844339430400)^{2}$$
$23$ $$T^{16} + \cdots + 41\!\cdots\!24$$
$29$ $$T^{16} + \cdots + 59\!\cdots\!04$$
$31$ $$(T^{4} - 121 T^{3} + \cdots - 349762400)^{4}$$
$37$ $$(T^{8} + \cdots + 13\!\cdots\!84)^{2}$$
$41$ $$T^{16} + \cdots + 19\!\cdots\!64$$
$43$ $$(T^{8} + \cdots + 17\!\cdots\!64)^{2}$$
$47$ $$(T^{8} + \cdots + 36\!\cdots\!52)^{2}$$
$53$ $$(T^{8} + \cdots + 44\!\cdots\!28)^{2}$$
$59$ $$T^{16} + \cdots + 33\!\cdots\!44$$
$61$ $$(T^{8} + \cdots + 46\!\cdots\!21)^{2}$$
$67$ $$(T^{8} + \cdots + 72\!\cdots\!36)^{2}$$
$71$ $$T^{16} + \cdots + 15\!\cdots\!00$$
$73$ $$(T^{4} + 1088 T^{3} + \cdots + 20997802657)^{4}$$
$79$ $$(T^{4} - 931 T^{3} + \cdots - 9861650240)^{4}$$
$83$ $$(T^{8} + \cdots + 23\!\cdots\!32)^{2}$$
$89$ $$T^{16} + \cdots + 19\!\cdots\!24$$
$97$ $$(T^{8} + \cdots + 84\!\cdots\!36)^{2}$$