Properties

 Label 13.5.f.a Level $13$ Weight $5$ Character orbit 13.f Analytic conductor $1.344$ Analytic rank $0$ Dimension $16$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,5,Mod(2,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(12))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("13.2");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 13.f (of order $$12$$, degree $$4$$, 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: $$1.34380952009$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 152 x^{14} + 9190 x^{12} + 285720 x^{10} + 4862025 x^{8} + 43573680 x^{6} + 169417008 x^{4} + \cdots + 3779136$$ x^16 + 152*x^14 + 9190*x^12 + 285720*x^10 + 4862025*x^8 + 43573680*x^6 + 169417008*x^4 + 100636992*x^2 + 3779136 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{4}\cdot 3^{2}\cdot 13^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{12} q^{2} + ( - \beta_{14} + \beta_{12} + \beta_{9} + \cdots - 1) q^{3}+ \cdots + (3 \beta_{14} - 2 \beta_{13} + \cdots - 3) q^{9}+O(q^{10})$$ q - b12 * q^2 + (-b14 + b12 + b9 - b8 - 1) * q^3 + (b12 - b11 - b5 + b4 + 8*b3) * q^4 + (b15 - b14 - b13 - 2*b12 - b10 - b9 + b8 + b5 - 2*b3 - 5*b2 - 8*b1 + 4) * q^5 + (-2*b15 + b14 + b13 + 2*b11 + b10 - b9 + b6 - b4 - 13*b3 + 9*b2 + 14*b1 - 9) * q^6 + (b15 + 2*b14 + 2*b13 + b11 + b8 - b7 - 2*b6 - 6*b3 + 7*b2 - 7*b1 + 8) * q^7 + (-b15 + 12*b14 - b13 + 5*b12 + 4*b10 + 4*b8 + 4*b7 + b5 - 8*b4 - 4*b3 - 14*b2 + 22*b1) * q^8 + (3*b14 - 2*b13 - 6*b12 - b11 - 3*b10 - 4*b9 - 6*b7 + 12*b5 + 3*b4 + 11*b3 - 18*b2 - 21*b1 - 3) * q^9 $$q - \beta_{12} q^{2} + ( - \beta_{14} + \beta_{12} + \beta_{9} + \cdots - 1) q^{3}+ \cdots + (14 \beta_{15} - 254 \beta_{14} + \cdots + 608) q^{99}+O(q^{100})$$ q - b12 * q^2 + (-b14 + b12 + b9 - b8 - 1) * q^3 + (b12 - b11 - b5 + b4 + 8*b3) * q^4 + (b15 - b14 - b13 - 2*b12 - b10 - b9 + b8 + b5 - 2*b3 - 5*b2 - 8*b1 + 4) * q^5 + (-2*b15 + b14 + b13 + 2*b11 + b10 - b9 + b6 - b4 - 13*b3 + 9*b2 + 14*b1 - 9) * q^6 + (b15 + 2*b14 + 2*b13 + b11 + b8 - b7 - 2*b6 - 6*b3 + 7*b2 - 7*b1 + 8) * q^7 + (-b15 + 12*b14 - b13 + 5*b12 + 4*b10 + 4*b8 + 4*b7 + b5 - 8*b4 - 4*b3 - 14*b2 + 22*b1) * q^8 + (3*b14 - 2*b13 - 6*b12 - b11 - 3*b10 - 4*b9 - 6*b7 + 12*b5 + 3*b4 + 11*b3 - 18*b2 - 21*b1 - 3) * q^9 + (2*b15 - 18*b14 + b13 + 14*b12 - 3*b9 - 3*b8 + 5*b7 + 2*b6 - 20*b5 + 9*b4 + 21*b2 - 20*b1 - 23) * q^10 + (-b15 - 4*b14 + b13 - 2*b12 - 7*b10 + 4*b9 + 3*b8 - 4*b7 + b6 - 6*b5 - 4*b4 - 8*b3 + 33*b2 + 29*b1 - 26) * q^11 + (2*b15 - 17*b14 - 3*b13 - 9*b12 - 3*b11 + 2*b10 + 8*b9 - 4*b8 + 2*b7 - b6 + 19*b5 + 19*b4 + 30*b3 - 34*b2 - 2) * q^12 + (-2*b15 + 12*b14 + 9*b12 - 3*b11 + b10 + 3*b9 - 11*b8 + 3*b7 + 3*b5 - 28*b4 + 31*b3 + 17*b2 - 43*b1 + 37) * q^13 + (7*b14 + b13 - 7*b12 - b11 + 7*b10 - b8 - 7*b7 - 4*b6 - 18*b5 + 18*b4 - 29*b3 - 28*b2 + 60) * q^14 + (b15 + 35*b14 + b13 - 25*b12 + 2*b11 + 6*b10 + 7*b9 - b8 + 7*b7 + b6 - 7*b5 - 21*b4 + 68*b3 - 76*b2 + 83*b1 - 2) * q^15 + (-33*b14 + b13 - 26*b12 + 2*b11 - 8*b10 + 4*b9 - 4*b8 - 4*b7 + 25*b5 + 3*b4 - 148*b3 + 74*b2 - 28*b1 + 16) * q^16 + (-3*b15 - 17*b14 + 16*b12 + 9*b11 - 3*b10 - 3*b9 + 6*b8 + 6*b6 - 2*b5 + 25*b4 + 81*b3 - 3*b2 - 33*b1 + 69) * q^17 + (-5*b15 - 27*b14 + 7*b13 - 14*b12 + 2*b11 - 11*b10 - 14*b9 + 11*b8 + 3*b7 - 2*b6 + 3*b5 - 16*b4 - 109*b3 + 152*b2 + 32*b1 + 134) * q^18 + (12*b15 + 5*b14 - 8*b13 - 12*b11 + 10*b10 - 10*b9 + b8 + b7 - 8*b6 - 13*b5 + 2*b4 + 94*b3 + 41*b2 - 83*b1 - 40) * q^19 + (-4*b15 + 61*b14 - 13*b13 + 72*b12 - 4*b11 - 2*b10 - 2*b9 + 6*b8 - 6*b7 + 13*b6 - 36*b5 + 2*b4 + 88*b3 - 300*b2 + 80*b1 - 294) * q^20 + (7*b15 - 12*b14 + 5*b13 + 23*b12 - 2*b11 - 7*b9 - 7*b7 - 2*b6 + 37*b5 + 12*b4 - 89*b3 + 56*b2 + 33*b1 - 96) * q^21 + (-b15 + 52*b14 + 12*b13 + 8*b12 + 6*b11 + 7*b10 + 7*b9 + 14*b7 + 37*b5 - b4 + 43*b3 - 93*b2 - 168*b1 + 7) * q^22 + (-14*b15 - 21*b14 - 4*b13 + 47*b12 + 11*b9 + 11*b8 + 2*b7 - 14*b6 - 34*b5 - 24*b4 - 52*b2 - 56*b1 - 45) * q^23 + (3*b15 + 2*b14 - 3*b13 + 31*b12 + 16*b10 - 2*b9 - 14*b8 + 2*b7 - 3*b6 - 32*b5 - 55*b4 - 60*b3 + 376*b2 + 378*b1 - 328) * q^24 + (-12*b15 - 71*b14 + 21*b13 - 87*b12 + 21*b11 - 13*b10 - 16*b9 + 8*b8 - 13*b7 + 6*b6 + 75*b5 + 75*b4 - 74*b3 + 82*b2 + 182*b1 - 96) * q^25 + (17*b15 + 65*b14 - 3*b13 - 67*b12 + 16*b11 + 9*b10 - 8*b9 + 27*b8 - 5*b7 - 2*b6 + 7*b5 - 68*b4 - 113*b3 + 114*b2 - 598*b1 + 362) * q^26 + (-6*b14 - 5*b13 + 6*b12 + 5*b11 - 22*b10 - 5*b8 + 22*b7 + 27*b6 - 73*b5 + 73*b4 - 310*b3 - 305*b2 + 213) * q^27 + (-10*b15 - 12*b14 - 10*b13 - 54*b12 - 14*b11 - 22*b10 - 20*b9 - 2*b8 - 20*b7 - 4*b6 + 20*b5 + 16*b4 + 636*b3 - 498*b2 + 478*b1 + 116) * q^28 + (2*b15 + 59*b14 - 8*b13 - 11*b12 - 16*b11 + 22*b10 - 25*b9 + 25*b8 + 11*b7 - 2*b6 - 23*b5 - 31*b4 - 494*b3 + 247*b2 - 99*b1 + 146) * q^29 + (20*b15 - 35*b14 - 2*b12 - 30*b11 + 8*b10 + 14*b9 - 28*b8 - 40*b6 + 45*b5 - 38*b4 + 762*b3 + 14*b2 - 392*b1 + 764) * q^30 + (4*b15 + 21*b14 - 19*b13 + 51*b12 - 15*b11 + 55*b10 + 72*b9 - 55*b8 - 17*b7 + 15*b6 + 4*b5 - 34*b4 - 308*b3 + 355*b2 + 102*b1 + 181) * q^31 + (-15*b15 - 90*b14 + 21*b13 + 15*b11 - 60*b10 + 60*b9 + 21*b6 - 11*b5 + 120*b4 + 42*b3 + 306*b2 - 102*b1 - 306) * q^32 + (-4*b15 - 63*b14 + 25*b13 + 26*b12 - 4*b11 + 16*b10 + 16*b9 - 49*b8 + 49*b7 - 25*b6 - 13*b5 - 16*b4 + 269*b3 - 457*b2 + 334*b1 - 506) * q^33 + (-16*b15 - 45*b14 + b13 - 4*b12 + 17*b11 - 48*b10 + 30*b9 - 48*b8 - 18*b7 + 17*b6 - 16*b5 - 3*b4 - 438*b3 - 78*b2 + 468*b1 - 456) * q^34 + (6*b15 - 101*b14 - 16*b13 + 6*b12 - 8*b11 + b10 + 21*b9 + 2*b7 - 108*b5 - 5*b4 + 222*b3 - 465*b2 - 651*b1 + 1) * q^35 + (28*b15 + 7*b14 - 3*b13 - 180*b12 - 14*b9 - 14*b8 - 92*b7 + 28*b6 + 113*b5 + 81*b4 + 262*b2 - 658*b1 - 672) * q^36 + (14*b15 + 59*b14 - 14*b13 - 11*b12 + 2*b11 + 34*b10 - 59*b9 + 25*b8 + 59*b7 - 16*b6 + 31*b5 - b4 - 245*b3 + 801*b2 + 860*b1 - 472) * q^37 + (10*b15 + 250*b14 - 42*b13 + 188*b12 - 42*b11 + 45*b10 - 62*b9 + 31*b8 + 45*b7 - 5*b6 - 185*b5 - 185*b4 + 447*b3 - 416*b2 + 192*b1 - 20) * q^38 + (-52*b15 - 100*b14 + 19*b13 - 32*b12 - 7*b11 - 74*b10 - 9*b9 + 55*b8 - 28*b7 + 17*b6 + 91*b5 + 209*b4 + 284*b3 + 94*b2 - 790*b1 + 805) * q^39 + (-24*b14 + 24*b12 - 10*b10 + 58*b8 + 10*b7 - 53*b6 + 183*b5 - 183*b4 - 454*b3 - 512*b2 + 1146) * q^40 + (38*b15 - 30*b14 + 38*b13 + 110*b12 + 30*b11 + 8*b10 - 30*b9 + 38*b8 - 30*b7 - 8*b6 + 30*b5 + 30*b4 + 499*b3 - 514*b2 + 484*b1 - 7) * q^41 + (-12*b15 + 61*b14 + 23*b13 + 125*b12 + 46*b11 + 42*b10 + 5*b9 - 5*b8 + 21*b7 + 12*b6 - 45*b5 + 33*b4 - 1526*b3 + 763*b2 + 198*b1 - 161) * q^42 + (-35*b15 + 189*b14 + 6*b12 + 35*b11 + 14*b10 - 21*b9 + 42*b8 + 70*b6 - 181*b5 + 34*b4 + 924*b3 - 21*b2 + 50*b1 - 44) * q^43 + (8*b15 + 48*b14 + 28*b13 + 58*b12 + 36*b11 - 14*b10 - 50*b9 + 14*b8 + 36*b7 - 36*b6 - 72*b5 + 62*b4 - 712*b3 + 596*b2 - 130*b1 + 776) * q^44 + (-38*b15 + 199*b14 - 5*b13 + 38*b11 + 69*b10 - 69*b9 - 23*b8 - 23*b7 - 5*b6 + 134*b5 - 375*b4 - 319*b3 + 583*b2 + 365*b1 - 606) * q^45 + (26*b15 - 199*b14 - b13 - 424*b12 + 26*b11 - 59*b10 - 59*b9 + 90*b8 - 90*b7 + b6 + 212*b5 + 59*b4 - 161*b3 - 909*b2 - 310*b1 - 819) * q^46 + (6*b15 - 129*b14 - 43*b13 - 73*b12 - 49*b11 + b10 - 10*b9 + b8 - 9*b7 - 49*b6 - 54*b5 + 130*b4 + 94*b3 - 149*b2 + 56*b1 + 85) * q^47 + (-3*b15 + 57*b14 - 34*b13 + 84*b12 - 17*b11 + 48*b10 - 4*b9 + 96*b7 - 75*b5 - 36*b4 + 454*b3 - 904*b2 - 342*b1 + 48) * q^48 + (16*b15 + 304*b14 + 19*b13 + 7*b12 + 42*b9 + 42*b8 + 133*b7 + 16*b6 + 129*b5 - 178*b4 - 62*b2 - 448*b1 - 406) * q^49 + (-64*b15 - 103*b14 + 64*b13 - 78*b12 - 15*b11 - 64*b10 + 103*b9 - 39*b8 - 103*b7 + 79*b6 + 131*b5 + 346*b4 + 34*b3 + 1377*b2 + 1274*b1 - 1553) * q^50 + (72*b15 - 115*b14 - 24*b13 + 31*b12 - 24*b11 - 114*b10 + 146*b9 - 73*b8 - 114*b7 - 36*b6 - 156*b5 - 156*b4 + 390*b3 - 463*b2 - 168*b1 - 103) * q^51 + (50*b15 - 233*b14 - 23*b13 + 383*b12 - 91*b11 + 146*b10 + 18*b9 - 120*b8 + 54*b7 - 50*b6 - 452*b5 + 252*b4 + 342*b3 + 714*b2 - 906*b1 + 576) * q^52 + (-36*b14 + 36*b13 + 36*b12 - 36*b11 + 35*b10 - 154*b8 - 35*b7 - 13*b6 + 144*b5 - 144*b4 - 215*b3 - 61*b2 + 388) * q^53 + (-56*b15 - 499*b14 - 56*b13 + 365*b12 - b11 + 4*b10 + 105*b9 - 101*b8 + 105*b7 + 55*b6 - 105*b5 + 197*b4 + 1112*b3 - 1061*b2 + 1166*b1 + 55) * q^54 + (12*b15 + 29*b14 - 24*b13 + 68*b12 - 48*b11 - 126*b10 + 55*b9 - 55*b8 - 63*b7 - 12*b6 - 147*b5 - 8*b4 + 92*b3 - 46*b2 - 1035*b1 + 854) * q^55 + (-34*b15 - 32*b14 - 286*b12 + 34*b11 - 26*b10 + 34*b9 - 68*b8 + 68*b6 + 292*b5 - 328*b4 - 174*b3 + 34*b2 - 352*b1 + 610) * q^56 + (11*b15 + 186*b14 - 19*b13 - 437*b12 - 8*b11 - 144*b10 - 97*b9 + 144*b8 - 47*b7 + 8*b6 + 293*b5 + 330*b4 - 695*b3 + 770*b2 - 69*b1 + 936) * q^57 + (92*b15 - 126*b14 - 70*b13 - 92*b11 + 4*b10 - 4*b9 + 42*b8 + 42*b7 - 70*b6 - 83*b5 + 340*b4 + 812*b3 - 600*b2 - 766*b1 + 642) * q^58 + (29*b15 - 94*b14 - 26*b13 - 252*b12 + 29*b11 + 134*b10 + 134*b9 - 55*b8 + 55*b7 + 26*b6 + 126*b5 - 134*b4 + 366*b3 - 331*b2 + 555*b1 - 386) * q^59 + (28*b15 + 663*b14 + 53*b13 - 377*b12 + 25*b11 + 222*b10 - 92*b9 + 222*b8 + 130*b7 + 25*b6 - 415*b5 - 441*b4 - 518*b3 - 228*b2 + 968*b1 - 388) * q^60 + (-55*b15 - 391*b14 + 106*b13 - 104*b12 + 53*b11 - 160*b10 - 110*b9 - 320*b7 - 127*b5 - 56*b4 - 360*b3 + 830*b2 - 225*b1 - 160) * q^61 + (-110*b15 + 106*b14 + 38*b13 - 67*b12 - 88*b9 - 88*b8 + 98*b7 - 110*b6 + 96*b5 + 59*b4 + 632*b2 + 964*b1 + 876) * q^62 + (32*b15 - 22*b14 - 32*b13 - 72*b12 + 32*b11 - 78*b10 + 22*b9 + 56*b8 - 22*b7 - 64*b6 + 10*b5 + 210*b4 - 388*b3 - 652*b2 - 674*b1 + 1074) * q^63 + (-148*b15 + 359*b14 + 169*b13 + 399*b12 + 169*b11 + 148*b10 + 40*b9 - 20*b8 + 148*b7 + 74*b6 - 134*b5 - 134*b4 - 1306*b3 + 1286*b2 + 40*b1 + 108) * q^64 + (67*b15 - 237*b14 - 84*b13 + 49*b12 + 114*b11 - 44*b10 + 19*b9 - 187*b8 + 38*b7 + 35*b6 + 478*b5 - 38*b4 - 55*b3 - 1645*b2 + 27*b1 - 621) * q^65 + (115*b14 - 46*b13 - 115*b12 + 46*b11 + 149*b10 + 149*b8 - 149*b7 + 129*b6 + 272*b5 - 272*b4 + 295*b3 + 146*b2 - 1532) * q^66 + (-28*b15 + 758*b14 - 28*b13 + 325*b12 - 62*b11 + 125*b10 + 56*b9 + 69*b8 + 56*b7 - 34*b6 - 56*b5 - 407*b4 - 1164*b3 + 545*b2 - 489*b1 - 494) * q^67 + (51*b15 - 409*b14 - 4*b13 - 221*b12 - 8*b11 - 164*b10 - 42*b9 + 42*b8 - 82*b7 - 51*b6 + 369*b5 + 354*b4 - 288*b3 + 144*b2 + 858*b1 - 980) * q^68 + (132*b15 - 117*b14 + 120*b12 - 123*b11 - 105*b10 - 114*b9 + 228*b8 - 264*b6 - 108*b5 + 453*b4 + 693*b3 - 114*b2 - 102*b1 + 327) * q^69 + (-14*b15 - 176*b14 - 67*b13 + 806*b12 - 81*b11 + 14*b10 - 62*b9 - 14*b8 + 76*b7 + 81*b6 - 792*b5 - 190*b4 + 2454*b3 - 2240*b2 + 228*b1 - 2406) * q^70 + (-2*b15 + 407*b14 + 98*b13 + 2*b11 + 126*b10 - 126*b9 + 91*b8 + 91*b7 + 98*b6 - 163*b5 - 506*b4 - 220*b3 + 595*b2 + 437*b1 - 504) * q^71 + (-141*b15 + 350*b14 - 9*b13 + 910*b12 - 141*b11 - 182*b10 - 182*b9 + 28*b8 - 28*b7 + 9*b6 - 455*b5 + 182*b4 - 12*b3 + 976*b2 - 222*b1 + 1004) * q^72 + (-45*b15 - 782*b14 + 94*b13 + 317*b12 + 139*b11 - 31*b10 + 9*b9 - 31*b8 - 22*b7 + 139*b6 + 330*b5 + 751*b4 - 334*b3 + 2129*b2 - 1826*b1 - 356) * q^73 + (144*b15 + 191*b14 - 96*b13 - 481*b12 - 48*b11 - 42*b10 - 36*b9 - 84*b7 + 714*b5 + 439*b4 - 270*b3 + 576*b2 - 198*b1 - 42) * q^74 + (53*b15 - 484*b14 - 99*b13 + 554*b12 - 46*b9 - 46*b8 - 65*b7 + 53*b6 - 373*b5 - 135*b4 - 1444*b2 + 159*b1 + 113) * q^75 + (131*b15 - 48*b14 - 131*b13 - 285*b12 + 24*b11 - 38*b10 + 48*b9 - 10*b8 - 48*b7 - 155*b6 + 542*b5 - 785*b4 + 944*b3 - 2992*b2 - 3040*b1 + 1990) * q^76 + (-98*b15 - 297*b14 - 114*b13 - 395*b12 - 114*b11 + 84*b10 - 98*b9 + 49*b8 + 84*b7 + 49*b6 + 208*b5 + 208*b4 + 483*b3 - 434*b2 + 2958*b1 - 1346) * q^77 + (-173*b15 + 813*b14 + 202*b13 - 353*b12 + 170*b11 - 67*b10 + 171*b9 + 196*b8 + 8*b7 + 100*b6 - 254*b5 - 668*b4 - 2809*b3 + 2215*b2 + 3512*b1 - 2239) * q^78 + (232*b14 - 34*b13 - 232*b12 + 34*b11 - 164*b10 - 4*b8 + 164*b7 - 68*b6 - 744*b5 + 744*b4 - 90*b3 - 86*b2 - 774) * q^79 + (187*b15 - 350*b14 + 187*b13 - 1420*b12 + 29*b11 - 76*b10 - 176*b9 + 100*b8 - 176*b7 - 158*b6 + 176*b5 + 263*b4 - 810*b3 + 2818*b2 - 2994*b1 + 1932) * q^80 + (-78*b15 + 318*b14 + 22*b13 + 528*b12 + 44*b11 + 496*b10 + 256*b9 - 256*b8 + 248*b7 + 78*b6 - 326*b5 - 550*b4 + 4820*b3 - 2410*b2 - 141*b1 + 381) * q^81 + (8*b15 + 59*b14 + 101*b12 + 64*b11 + 116*b10 + 128*b9 - 256*b8 - 16*b6 - 44*b5 - 271*b4 - 4428*b3 + 128*b2 + 184*b1 - 508) * q^82 + (-156*b15 - 146*b14 + 216*b13 - 656*b12 + 60*b11 + 123*b10 + 216*b9 - 123*b8 - 93*b7 - 60*b6 + 779*b5 - 269*b4 - 912*b3 + 1335*b2 + 546*b1 + 573) * q^83 + (-54*b15 - 860*b14 + 30*b13 + 54*b11 - 156*b10 + 156*b9 - 338*b8 - 338*b7 + 30*b6 + 248*b5 + 888*b4 - 90*b3 - 1666*b2 - 404*b1 + 1328) * q^84 + (-44*b15 + 553*b14 - 73*b13 - 448*b12 - 44*b11 + 53*b10 + 53*b9 + 77*b8 - 77*b7 + 73*b6 + 224*b5 - 53*b4 + 456*b3 + 2111*b2 + 432*b1 + 2188) * q^85 + (29*b15 + 593*b14 - 195*b13 + 749*b12 - 224*b11 - 315*b10 + 322*b9 - 315*b8 + 7*b7 - 224*b6 + 420*b5 - 908*b4 + 3745*b3 - 4134*b2 + 74*b1 + 3752) * q^86 + (-54*b15 + 230*b14 + 44*b13 + 544*b12 + 22*b11 + 356*b10 + 181*b9 + 712*b7 - 670*b5 - 188*b4 - 1220*b3 + 2259*b2 + 3972*b1 + 356) * q^87 + (134*b15 - 370*b14 - 160*b13 - 192*b12 + 130*b9 + 130*b8 - 166*b7 + 134*b6 - 334*b5 + 396*b4 + 406*b2 + 1528*b1 + 1658) * q^88 + (-50*b15 + 226*b14 + 50*b13 + 889*b12 - 171*b11 + 217*b10 - 226*b9 + 9*b8 + 226*b7 + 221*b6 - 1570*b5 - 181*b4 + 116*b3 - 2685*b2 - 2459*b1 + 2804) * q^89 + (398*b15 - 1309*b14 - 96*b13 - 1395*b12 - 96*b11 - 485*b10 - 86*b9 + 43*b8 - 485*b7 - 199*b6 + 1205*b5 + 1205*b4 - 687*b3 + 730*b2 - 8288*b1 + 3702) * q^90 + (91*b15 - 208*b14 + 78*b13 - 1118*b12 - 195*b11 - 78*b10 - 442*b9 + 377*b8 - 273*b7 - 156*b6 + 676*b5 - 286*b4 + 2834*b3 - 2587*b2 + 2431*b1 - 442) * q^91 + (-845*b14 + 19*b13 + 845*b12 - 19*b11 - 270*b10 - 152*b8 + 270*b7 + 73*b6 - 881*b5 + 881*b4 + 4298*b3 + 4450*b2 - 3362) * q^92 + (-159*b15 + 587*b14 - 159*b13 - 244*b12 + 24*b11 - 358*b10 - 91*b9 - 267*b8 - 91*b7 + 183*b6 + 91*b5 - 248*b4 - 6558*b3 + 3640*b2 - 3731*b1 - 3276) * q^93 + (-128*b15 + 226*b14 - 74*b13 - 841*b12 - 148*b11 + 28*b10 - 376*b9 + 376*b8 + 14*b7 + 128*b6 + 164*b5 - 329*b4 + 3952*b3 - 1976*b2 + 3876*b1 - 3472) * q^94 + (-137*b15 - 584*b14 + 1226*b12 + 67*b11 + 283*b10 + 129*b9 - 258*b8 + 274*b6 - 359*b5 + 685*b4 - 2670*b3 + 129*b2 + 4349*b1 - 8673) * q^95 + (136*b15 - 405*b14 - 83*b13 - 209*b12 + 53*b11 + 188*b10 + 216*b9 - 188*b8 - 28*b7 - 53*b6 + 397*b5 - 593*b4 + 5198*b3 - 6458*b2 - 1072*b1 - 5602) * q^96 + (-54*b15 - 113*b14 - 113*b13 + 54*b11 - 366*b10 + 366*b9 + 247*b8 + 247*b7 - 113*b6 - 125*b5 + 354*b4 - 2656*b3 - 1629*b2 + 2537*b1 + 1876) * q^97 + (311*b15 + 67*b14 + 255*b13 + 1702*b12 + 311*b11 + 315*b10 + 315*b9 - 212*b8 + 212*b7 - 255*b6 - 851*b5 - 315*b4 - 5461*b3 + 6355*b2 - 4934*b1 + 6143) * q^98 + (14*b15 - 254*b14 - 26*b13 + 172*b12 - 40*b11 - 112*b10 - 266*b9 - 112*b8 - 378*b7 - 40*b6 + 816*b5 + 142*b4 + 986*b3 + 3656*b2 - 4754*b1 + 608) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{2} - 2 q^{3} - 6 q^{4} + 8 q^{5} - 38 q^{6} + 56 q^{7} + 90 q^{8} - 164 q^{9}+O(q^{10})$$ 16 * q - 4 * q^2 - 2 * q^3 - 6 * q^4 + 8 * q^5 - 38 * q^6 + 56 * q^7 + 90 * q^8 - 164 * q^9 $$16 q - 4 q^{2} - 2 q^{3} - 6 q^{4} + 8 q^{5} - 38 q^{6} + 56 q^{7} + 90 q^{8} - 164 q^{9} - 486 q^{10} - 100 q^{11} + 294 q^{13} + 808 q^{14} + 346 q^{15} + 230 q^{16} + 984 q^{17} + 2434 q^{18} - 1498 q^{19} - 3962 q^{20} - 1076 q^{21} - 1524 q^{22} - 1014 q^{23} - 2142 q^{24} + 614 q^{26} + 3352 q^{27} + 5764 q^{28} + 814 q^{29} + 9162 q^{30} + 4060 q^{31} - 4996 q^{32} - 5636 q^{33} - 2502 q^{34} - 4892 q^{35} - 15750 q^{36} - 1790 q^{37} + 6982 q^{39} + 18816 q^{40} + 4280 q^{41} - 1204 q^{42} - 1368 q^{43} + 10736 q^{44} - 6806 q^{45} - 15246 q^{46} + 1484 q^{47} - 3002 q^{48} - 11820 q^{49} - 13574 q^{50} + 1432 q^{52} + 7204 q^{53} + 13240 q^{54} + 6936 q^{55} + 8124 q^{56} + 12736 q^{57} + 3030 q^{58} - 2380 q^{59} - 6472 q^{60} - 162 q^{61} + 19614 q^{62} + 12004 q^{63} - 5248 q^{65} - 23872 q^{66} - 14854 q^{67} - 6444 q^{68} + 2412 q^{69} - 34524 q^{70} - 8050 q^{71} + 15420 q^{72} - 15448 q^{73} - 2882 q^{74} + 8280 q^{75} + 10622 q^{76} - 11672 q^{78} - 17064 q^{79} + 2564 q^{80} + 2128 q^{81} - 5346 q^{82} + 12788 q^{83} + 25948 q^{84} + 35382 q^{85} + 67260 q^{86} + 29342 q^{87} + 40836 q^{88} + 20492 q^{89} + 8996 q^{91} - 49884 q^{92} - 78920 q^{93} - 30606 q^{94} - 98574 q^{95} - 94664 q^{96} + 50944 q^{97} + 61484 q^{98} - 21632 q^{99}+O(q^{100})$$ 16 * q - 4 * q^2 - 2 * q^3 - 6 * q^4 + 8 * q^5 - 38 * q^6 + 56 * q^7 + 90 * q^8 - 164 * q^9 - 486 * q^10 - 100 * q^11 + 294 * q^13 + 808 * q^14 + 346 * q^15 + 230 * q^16 + 984 * q^17 + 2434 * q^18 - 1498 * q^19 - 3962 * q^20 - 1076 * q^21 - 1524 * q^22 - 1014 * q^23 - 2142 * q^24 + 614 * q^26 + 3352 * q^27 + 5764 * q^28 + 814 * q^29 + 9162 * q^30 + 4060 * q^31 - 4996 * q^32 - 5636 * q^33 - 2502 * q^34 - 4892 * q^35 - 15750 * q^36 - 1790 * q^37 + 6982 * q^39 + 18816 * q^40 + 4280 * q^41 - 1204 * q^42 - 1368 * q^43 + 10736 * q^44 - 6806 * q^45 - 15246 * q^46 + 1484 * q^47 - 3002 * q^48 - 11820 * q^49 - 13574 * q^50 + 1432 * q^52 + 7204 * q^53 + 13240 * q^54 + 6936 * q^55 + 8124 * q^56 + 12736 * q^57 + 3030 * q^58 - 2380 * q^59 - 6472 * q^60 - 162 * q^61 + 19614 * q^62 + 12004 * q^63 - 5248 * q^65 - 23872 * q^66 - 14854 * q^67 - 6444 * q^68 + 2412 * q^69 - 34524 * q^70 - 8050 * q^71 + 15420 * q^72 - 15448 * q^73 - 2882 * q^74 + 8280 * q^75 + 10622 * q^76 - 11672 * q^78 - 17064 * q^79 + 2564 * q^80 + 2128 * q^81 - 5346 * q^82 + 12788 * q^83 + 25948 * q^84 + 35382 * q^85 + 67260 * q^86 + 29342 * q^87 + 40836 * q^88 + 20492 * q^89 + 8996 * q^91 - 49884 * q^92 - 78920 * q^93 - 30606 * q^94 - 98574 * q^95 - 94664 * q^96 + 50944 * q^97 + 61484 * q^98 - 21632 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 152 x^{14} + 9190 x^{12} + 285720 x^{10} + 4862025 x^{8} + 43573680 x^{6} + 169417008 x^{4} + \cdots + 3779136$$ :

 $$\beta_{1}$$ $$=$$ $$( - 403105 \nu^{15} - 60681659 \nu^{13} - 3635180779 \nu^{11} - 112263516645 \nu^{9} + \cdots + 5545371476736 ) / 11090742953472$$ (-403105*v^15 - 60681659*v^13 - 3635180779*v^11 - 112263516645*v^9 - 1906294858020*v^7 - 17198065117872*v^5 - 68934581954784*v^3 - 50621014857024*v + 5545371476736) / 11090742953472 $$\beta_{2}$$ $$=$$ $$( - 450212 \nu^{15} - 12268161 \nu^{14} - 40141192 \nu^{13} - 1670382423 \nu^{12} + \cdots - 35916509215296 ) / 33272228860416$$ (-450212*v^15 - 12268161*v^14 - 40141192*v^13 - 1670382423*v^12 - 336194096*v^11 - 86530231647*v^10 + 64569965208*v^9 - 2167037463465*v^8 + 2514078351540*v^7 - 26905461929136*v^6 + 35960915423664*v^5 - 144684110994768*v^4 + 189141832996704*v^3 - 182218604617728*v^2 + 90475406179200*v - 35916509215296) / 33272228860416 $$\beta_{3}$$ $$=$$ $$( 450212 \nu^{15} - 12268161 \nu^{14} + 40141192 \nu^{13} - 1670382423 \nu^{12} + \cdots - 35916509215296 ) / 33272228860416$$ (450212*v^15 - 12268161*v^14 + 40141192*v^13 - 1670382423*v^12 + 336194096*v^11 - 86530231647*v^10 - 64569965208*v^9 - 2167037463465*v^8 - 2514078351540*v^7 - 26905461929136*v^6 - 35960915423664*v^5 - 144684110994768*v^4 - 189141832996704*v^3 - 182218604617728*v^2 - 90475406179200*v - 35916509215296) / 33272228860416 $$\beta_{4}$$ $$=$$ $$( - 53905 \nu^{15} - 3031335 \nu^{14} - 1946495 \nu^{13} - 399803697 \nu^{12} + \cdots + 22901572977984 ) / 3696914317824$$ (-53905*v^15 - 3031335*v^14 - 1946495*v^13 - 399803697*v^12 + 403691033*v^11 - 19722145833*v^10 + 33967214799*v^9 - 456891538383*v^8 + 1036401623208*v^7 - 4928117078304*v^6 + 14113769934816*v^5 - 18429203735856*v^4 + 73741688677440*v^3 + 20576677484160*v^2 + 32386072836672*v + 22901572977984) / 3696914317824 $$\beta_{5}$$ $$=$$ $$( - 53905 \nu^{15} + 3031335 \nu^{14} - 1946495 \nu^{13} + 399803697 \nu^{12} + \cdots - 22901572977984 ) / 3696914317824$$ (-53905*v^15 + 3031335*v^14 - 1946495*v^13 + 399803697*v^12 + 403691033*v^11 + 19722145833*v^10 + 33967214799*v^9 + 456891538383*v^8 + 1036401623208*v^7 + 4928117078304*v^6 + 14113769934816*v^5 + 18429203735856*v^4 + 73741688677440*v^3 - 20576677484160*v^2 + 32386072836672*v - 22901572977984) / 3696914317824 $$\beta_{6}$$ $$=$$ $$( 958933 \nu^{14} + 127064675 \nu^{12} + 6269679835 \nu^{10} + 142169221341 \nu^{8} + \cdots - 17722730943936 ) / 616152386304$$ (958933*v^14 + 127064675*v^12 + 6269679835*v^10 + 142169221341*v^8 + 1357351279200*v^6 + 1111509749520*v^4 - 41761008777600*v^2 - 17722730943936) / 616152386304 $$\beta_{7}$$ $$=$$ $$( 1611011 \nu^{15} + 20386710 \nu^{14} + 235643173 \nu^{13} + 2770864650 \nu^{12} + \cdots - 127810286301312 ) / 11090742953472$$ (1611011*v^15 + 20386710*v^14 + 235643173*v^13 + 2770864650*v^12 + 13401654797*v^11 + 141965775450*v^10 + 377367280059*v^9 + 3436503751926*v^8 + 5435904315744*v^7 + 38802395601648*v^6 + 35965207159632*v^5 + 147897464997600*v^4 + 69173551698048*v^3 - 236124309230208*v^2 - 38119690075968*v - 127810286301312) / 11090742953472 $$\beta_{8}$$ $$=$$ $$( - 64316 \nu^{15} + 10690839 \nu^{14} - 5734456 \nu^{13} + 1454826609 \nu^{12} + \cdots - 46660774261824 ) / 4753175551488$$ (-64316*v^15 + 10690839*v^14 - 5734456*v^13 + 1454826609*v^12 - 48027728*v^11 + 75309720489*v^10 + 9224280744*v^9 + 1881613092303*v^8 + 359154050220*v^7 + 23121020302128*v^6 + 5137273631952*v^5 + 118652121636336*v^4 + 27020261856672*v^3 + 95067845558784*v^2 + 12925058025600*v - 46660774261824) / 4753175551488 $$\beta_{9}$$ $$=$$ $$( 8483506 \nu^{15} - 22455 \nu^{14} + 1257390830 \nu^{13} - 34398657 \nu^{12} + \cdots - 121083795836352 ) / 33272228860416$$ (8483506*v^15 - 22455*v^14 + 1257390830*v^13 - 34398657*v^12 + 73776312574*v^11 - 3085996905*v^10 + 2217207089202*v^9 - 83390031711*v^8 + 36294499625328*v^7 - 575259720336*v^6 + 310555374841728*v^5 + 2193532745904*v^4 + 1153914652058112*v^3 - 10688655293568*v^2 + 921694995453312*v - 121083795836352) / 33272228860416 $$\beta_{10}$$ $$=$$ $$( 1611011 \nu^{15} - 20386710 \nu^{14} + 235643173 \nu^{13} - 2770864650 \nu^{12} + \cdots + 116719543347840 ) / 11090742953472$$ (1611011*v^15 - 20386710*v^14 + 235643173*v^13 - 2770864650*v^12 + 13401654797*v^11 - 141965775450*v^10 + 377367280059*v^9 - 3436503751926*v^8 + 5435904315744*v^7 - 38802395601648*v^6 + 35965207159632*v^5 - 147897464997600*v^4 + 69173551698048*v^3 + 236124309230208*v^2 - 38119690075968*v + 116719543347840) / 11090742953472 $$\beta_{11}$$ $$=$$ $$( 3060949 \nu^{15} + 30779505 \nu^{14} + 538883915 \nu^{13} + 4066826247 \nu^{12} + \cdots + 262228572037440 ) / 11090742953472$$ (3060949*v^15 + 30779505*v^14 + 538883915*v^13 + 4066826247*v^12 + 37885091971*v^11 + 202355357535*v^10 + 1360934568357*v^9 + 4811620057209*v^8 + 26434213763832*v^7 + 55871559620352*v^6 + 266224281763680*v^5 + 274948280891280*v^4 + 1144577610929088*v^3 + 313787054673792*v^2 + 765185781986496*v + 262228572037440) / 11090742953472 $$\beta_{12}$$ $$=$$ $$( 1668943 \nu^{15} + 3478479 \nu^{14} + 244791977 \nu^{13} + 476788953 \nu^{12} + \cdots - 4839006028608 ) / 3696914317824$$ (1668943*v^15 + 3478479*v^14 + 244791977*v^13 + 476788953*v^12 + 14182744225*v^11 + 24822766977*v^10 + 420766469319*v^9 + 620613013671*v^8 + 6825957638976*v^7 + 7560677757024*v^6 + 58482708936192*v^5 + 37860759720432*v^4 + 218257879045248*v^3 + 28035201382272*v^2 + 118564929748800*v - 4839006028608) / 3696914317824 $$\beta_{13}$$ $$=$$ $$( 3060949 \nu^{15} - 30779505 \nu^{14} + 538883915 \nu^{13} - 4066826247 \nu^{12} + \cdots - 262228572037440 ) / 11090742953472$$ (3060949*v^15 - 30779505*v^14 + 538883915*v^13 - 4066826247*v^12 + 37885091971*v^11 - 202355357535*v^10 + 1360934568357*v^9 - 4811620057209*v^8 + 26434213763832*v^7 - 55871559620352*v^6 + 266224281763680*v^5 - 274948280891280*v^4 + 1144577610929088*v^3 - 313787054673792*v^2 + 765185781986496*v - 262228572037440) / 11090742953472 $$\beta_{14}$$ $$=$$ $$( 1668943 \nu^{15} - 3478479 \nu^{14} + 244791977 \nu^{13} - 476788953 \nu^{12} + \cdots + 4839006028608 ) / 3696914317824$$ (1668943*v^15 - 3478479*v^14 + 244791977*v^13 - 476788953*v^12 + 14182744225*v^11 - 24822766977*v^10 + 420766469319*v^9 - 620613013671*v^8 + 6825957638976*v^7 - 7560677757024*v^6 + 58482708936192*v^5 - 37860759720432*v^4 + 218257879045248*v^3 - 28035201382272*v^2 + 118564929748800*v + 4839006028608) / 3696914317824 $$\beta_{15}$$ $$=$$ $$( 7320853 \nu^{15} + 2876799 \nu^{14} + 1089833795 \nu^{13} + 381194025 \nu^{12} + \cdots - 53168192831808 ) / 3696914317824$$ (7320853*v^15 + 2876799*v^14 + 1089833795*v^13 + 381194025*v^12 + 64086192235*v^11 + 18809039505*v^10 + 1921681816845*v^9 + 426507664023*v^8 + 31200573373200*v^7 + 4072053837600*v^6 + 262878864690816*v^5 + 3334529248560*v^4 + 938832132172032*v^3 - 125283026332800*v^2 + 452794700306880*v - 53168192831808) / 3696914317824
 $$\nu$$ $$=$$ $$( - 6 \beta_{15} + 8 \beta_{14} + \beta_{13} + 12 \beta_{12} + \beta_{11} + 8 \beta_{10} + 4 \beta_{9} + \cdots - 4 ) / 26$$ (-6*b15 + 8*b14 + b13 + 12*b12 + b11 + 8*b10 + 4*b9 - 2*b8 + 8*b7 + 3*b6 - 4*b5 - 4*b4 - 18*b3 + 16*b2 + 20*b1 - 4) / 26 $$\nu^{2}$$ $$=$$ $$( 7 \beta_{14} - 4 \beta_{13} - 7 \beta_{12} + 4 \beta_{11} - 7 \beta_{10} + 2 \beta_{8} + 7 \beta_{7} + \cdots - 242 ) / 13$$ (7*b14 - 4*b13 - 7*b12 + 4*b11 - 7*b10 + 2*b8 + 7*b7 - 3*b6 - 13*b5 + 13*b4 + 19*b3 + 17*b2 - 242) / 13 $$\nu^{3}$$ $$=$$ $$( 92 \beta_{15} - 114 \beta_{14} + 15 \beta_{13} - 184 \beta_{12} + 15 \beta_{11} - 140 \beta_{10} + \cdots - 242 ) / 13$$ (92*b15 - 114*b14 + 15*b13 - 184*b12 + 15*b11 - 140*b10 - 70*b9 + 35*b8 - 140*b7 - 46*b6 + 83*b5 + 83*b4 + 588*b3 - 553*b2 + 274*b1 - 242) / 13 $$\nu^{4}$$ $$=$$ $$( - 407 \beta_{14} + 175 \beta_{13} + 407 \beta_{12} - 175 \beta_{11} + 381 \beta_{10} - 380 \beta_{8} + \cdots + 7318 ) / 13$$ (-407*b14 + 175*b13 + 407*b12 - 175*b11 + 381*b10 - 380*b8 - 381*b7 + 258*b6 + 507*b5 - 507*b4 - 1855*b3 - 1475*b2 + 7318) / 13 $$\nu^{5}$$ $$=$$ $$( - 3476 \beta_{15} + 3430 \beta_{14} - 1479 \beta_{13} + 7212 \beta_{12} - 1479 \beta_{11} + \cdots + 17616 ) / 13$$ (-3476*b15 + 3430*b14 - 1479*b13 + 7212*b12 - 1479*b11 + 5588*b10 + 3782*b9 - 1891*b8 + 5588*b7 + 1738*b6 - 3977*b5 - 3977*b4 - 32190*b3 + 30299*b2 - 27838*b1 + 17616) / 13 $$\nu^{6}$$ $$=$$ $$( 17913 \beta_{14} - 8533 \beta_{13} - 17913 \beta_{12} + 8533 \beta_{11} - 18797 \beta_{10} + \cdots - 271710 ) / 13$$ (17913*b14 - 8533*b13 - 17913*b12 + 8533*b11 - 18797*b10 + 24644*b8 + 18797*b7 - 14814*b6 - 22607*b5 + 22607*b4 + 118171*b3 + 93527*b2 - 271710) / 13 $$\nu^{7}$$ $$=$$ $$11468 \beta_{15} - 8888 \beta_{14} + 7017 \beta_{13} - 24998 \beta_{12} + 7017 \beta_{11} - 19004 \beta_{10} + \cdots - 73312$$ 11468*b15 - 8888*b14 + 7017*b13 - 24998*b12 + 7017*b11 - 19004*b10 - 16110*b9 + 8055*b8 - 19004*b7 - 5734*b6 + 15173*b5 + 15173*b4 + 130698*b3 - 122643*b2 + 124726*b1 - 73312 $$\nu^{8}$$ $$=$$ $$( - 772903 \beta_{14} + 430867 \beta_{13} + 772903 \beta_{12} - 430867 \beta_{11} + 918997 \beta_{10} + \cdots + 11600462 ) / 13$$ (-772903*b14 + 430867*b13 + 772903*b12 - 430867*b11 + 918997*b10 - 1362716*b8 - 918997*b7 + 772518*b6 + 1088347*b5 - 1088347*b4 - 6639151*b3 - 5276435*b2 + 11600462) / 13 $$\nu^{9}$$ $$=$$ $$( - 6901292 \beta_{15} + 4408926 \beta_{14} - 5008851 \beta_{13} + 15652900 \beta_{12} - 5008851 \beta_{11} + \cdots + 48370256 ) / 13$$ (-6901292*b15 + 4408926*b14 - 5008851*b13 + 15652900*b12 - 5008851*b11 + 11583236*b10 + 11243974*b9 - 5621987*b8 + 11583236*b7 + 3450646*b6 - 9941777*b5 - 9941777*b4 - 88036518*b3 + 82414531*b2 - 84818014*b1 + 48370256) / 13 $$\nu^{10}$$ $$=$$ $$( 34597169 \beta_{14} - 21896869 \beta_{13} - 34597169 \beta_{12} + 21896869 \beta_{11} + \cdots - 536617966 ) / 13$$ (34597169*b14 - 21896869*b13 - 34597169*b12 + 21896869*b11 - 45236733*b10 + 71658740*b8 + 45236733*b7 - 39176646*b6 - 54100527*b5 + 54100527*b4 + 353441755*b3 + 281783015*b2 - 536617966) / 13 $$\nu^{11}$$ $$=$$ $$( 332946236 \beta_{15} - 187007200 \beta_{14} + 263030493 \beta_{13} - 775259574 \beta_{12} + \cdots - 2420792496 ) / 13$$ (332946236*b15 - 187007200*b14 + 263030493*b13 - 775259574*b12 + 263030493*b11 - 561086828*b10 - 588252374*b9 + 294126187*b8 - 561086828*b7 - 166473118*b6 + 503828801*b5 + 503828801*b4 + 4512195906*b3 - 4218069719*b2 + 4307663710*b1 - 2420792496) / 13 $$\nu^{12}$$ $$=$$ $$( - 1614876399 \beta_{14} + 1111784059 \beta_{13} + 1614876399 \beta_{12} - 1111784059 \beta_{11} + \cdots + 25885701678 ) / 13$$ (-1614876399*b14 + 1111784059*b13 + 1614876399*b12 - 1111784059*b11 + 2244399605*b10 - 3691353404*b8 - 2244399605*b7 + 1971703926*b6 + 2719026011*b5 - 2719026011*b4 - 18332662303*b3 - 14641308899*b2 + 25885701678) / 13 $$\nu^{13}$$ $$=$$ $$( - 16408659212 \beta_{15} + 8537243750 \beta_{14} - 13536977019 \beta_{13} + 38821589996 \beta_{12} + \cdots + 121041181744 ) / 13$$ (-16408659212*b15 + 8537243750*b14 - 13536977019*b13 + 38821589996*b12 - 13536977019*b11 + 27670764260*b10 + 30284346246*b9 - 15142173123*b8 + 27670764260*b7 + 8204329606*b6 - 25547886401*b5 - 25547886401*b4 - 229745363670*b3 + 214603190547*b2 - 217025181214*b1 + 121041181744) / 13 $$\nu^{14}$$ $$=$$ $$5983739437 \beta_{14} - 4333135633 \beta_{13} - 5983739437 \beta_{12} + 4333135633 \beta_{11} + \cdots - 98119308806$$ 5983739437*b14 - 4333135633*b13 - 5983739437*b12 + 4333135633*b11 - 8616856273*b10 + 14483635364*b8 + 8616856273*b7 - 7623329934*b6 - 10544524843*b5 + 10544524843*b4 + 72180049879*b3 + 57696414515*b2 - 98119308806 $$\nu^{15}$$ $$=$$ $$( 817505384348 \beta_{15} - 407544050424 \beta_{14} + 689836505013 \beta_{13} - 1952234100238 \beta_{12} + \cdots - 6064117949264 ) / 13$$ (817505384348*b15 - 407544050424*b14 + 689836505013*b13 - 1952234100238*b12 + 689836505013*b11 - 1378130038796*b10 - 1544690049814*b9 + 772345024907*b8 - 1378130038796*b7 - 408752692174*b6 + 1294054470737*b5 + 1294054470737*b4 + 11650966452498*b3 - 10878621427591*b2 + 10916665870750*b1 - 6064117949264) / 13

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{3}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 − 4.89748i 5.33868i 0.816521i − 3.98977i 4.34562i − 7.09996i 3.68702i − 0.200628i 4.89748i − 5.33868i − 0.816521i 3.98977i − 4.34562i 7.09996i − 3.68702i 0.200628i
−6.34141 1.69918i 7.28678 + 12.6211i 23.4699 + 13.5504i −20.2793 + 20.2793i −24.7631 92.4170i 21.3318 5.71584i −51.5321 51.5321i −65.6944 + 113.786i 163.058 94.1415i
2.2 −3.41998 0.916381i −5.16500 8.94604i −2.99988 1.73198i 8.65321 8.65321i 9.46622 + 35.3284i −2.94871 + 0.790103i 48.7300 + 48.7300i −12.8545 + 22.2646i −37.5235 + 21.6642i
2.3 2.38644 + 0.639444i 4.25709 + 7.37349i −8.57021 4.94801i 16.9748 16.9748i 5.44434 + 20.3185i −50.2192 + 13.4562i −45.2402 45.2402i 4.25442 7.36887i 51.3637 29.6548i
2.4 5.50893 + 1.47611i −4.28079 7.41455i 14.3130 + 8.26362i −29.3294 + 29.3294i −12.6379 47.1652i 45.8361 12.2817i 2.12627 + 2.12627i 3.84960 6.66770i −204.867 + 118.280i
6.1 −1.88469 7.03375i −0.939567 + 1.62738i −32.0652 + 18.5128i 26.6717 26.6717i 13.2174 + 3.54158i 7.85376 29.3106i 108.263 + 108.263i 38.7344 + 67.0900i −237.870 137.334i
6.2 −0.387376 1.44571i 3.29308 5.70378i 11.9164 6.87993i −10.5663 + 10.5663i −9.52166 2.55132i −14.3010 + 53.3722i −31.4958 31.4958i 18.8113 + 32.5821i 19.3688 + 11.1826i
6.3 0.540374 + 2.01670i −8.21171 + 14.2231i 10.0813 5.82045i 17.2508 17.2508i −33.1212 8.87479i −2.95016 + 11.0101i 40.8071 + 40.8071i −94.3643 163.444i 44.1117 + 25.4679i
6.4 1.59772 + 5.96275i 2.76012 4.78067i −19.1453 + 11.0536i −5.37551 + 5.37551i 32.9158 + 8.81977i 23.3974 87.3205i −26.6579 26.6579i 25.2635 + 43.7576i −40.6414 23.4643i
7.1 −6.34141 + 1.69918i 7.28678 12.6211i 23.4699 13.5504i −20.2793 20.2793i −24.7631 + 92.4170i 21.3318 + 5.71584i −51.5321 + 51.5321i −65.6944 113.786i 163.058 + 94.1415i
7.2 −3.41998 + 0.916381i −5.16500 + 8.94604i −2.99988 + 1.73198i 8.65321 + 8.65321i 9.46622 35.3284i −2.94871 0.790103i 48.7300 48.7300i −12.8545 22.2646i −37.5235 21.6642i
7.3 2.38644 0.639444i 4.25709 7.37349i −8.57021 + 4.94801i 16.9748 + 16.9748i 5.44434 20.3185i −50.2192 13.4562i −45.2402 + 45.2402i 4.25442 + 7.36887i 51.3637 + 29.6548i
7.4 5.50893 1.47611i −4.28079 + 7.41455i 14.3130 8.26362i −29.3294 29.3294i −12.6379 + 47.1652i 45.8361 + 12.2817i 2.12627 2.12627i 3.84960 + 6.66770i −204.867 118.280i
11.1 −1.88469 + 7.03375i −0.939567 1.62738i −32.0652 18.5128i 26.6717 + 26.6717i 13.2174 3.54158i 7.85376 + 29.3106i 108.263 108.263i 38.7344 67.0900i −237.870 + 137.334i
11.2 −0.387376 + 1.44571i 3.29308 + 5.70378i 11.9164 + 6.87993i −10.5663 10.5663i −9.52166 + 2.55132i −14.3010 53.3722i −31.4958 + 31.4958i 18.8113 32.5821i 19.3688 11.1826i
11.3 0.540374 2.01670i −8.21171 14.2231i 10.0813 + 5.82045i 17.2508 + 17.2508i −33.1212 + 8.87479i −2.95016 11.0101i 40.8071 40.8071i −94.3643 + 163.444i 44.1117 25.4679i
11.4 1.59772 5.96275i 2.76012 + 4.78067i −19.1453 11.0536i −5.37551 5.37551i 32.9158 8.81977i 23.3974 + 87.3205i −26.6579 + 26.6579i 25.2635 43.7576i −40.6414 + 23.4643i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.5.f.a 16
3.b odd 2 1 117.5.bd.c 16
13.c even 3 1 169.5.d.d 16
13.e even 6 1 169.5.d.c 16
13.f odd 12 1 inner 13.5.f.a 16
13.f odd 12 1 169.5.d.c 16
13.f odd 12 1 169.5.d.d 16
39.k even 12 1 117.5.bd.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.5.f.a 16 1.a even 1 1 trivial
13.5.f.a 16 13.f odd 12 1 inner
117.5.bd.c 16 3.b odd 2 1
117.5.bd.c 16 39.k even 12 1
169.5.d.c 16 13.e even 6 1
169.5.d.c 16 13.f odd 12 1
169.5.d.d 16 13.c even 3 1
169.5.d.d 16 13.f odd 12 1

Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(13, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + \cdots + 2116736064$$
$3$ $$T^{16} + \cdots + 151617258383616$$
$5$ $$T^{16} + \cdots + 13\!\cdots\!36$$
$7$ $$T^{16} + \cdots + 82\!\cdots\!44$$
$11$ $$T^{16} + \cdots + 22\!\cdots\!16$$
$13$ $$T^{16} + \cdots + 44\!\cdots\!81$$
$17$ $$T^{16} + \cdots + 16\!\cdots\!01$$
$19$ $$T^{16} + \cdots + 91\!\cdots\!64$$
$23$ $$T^{16} + \cdots + 45\!\cdots\!64$$
$29$ $$T^{16} + \cdots + 38\!\cdots\!41$$
$31$ $$T^{16} + \cdots + 22\!\cdots\!64$$
$37$ $$T^{16} + \cdots + 94\!\cdots\!29$$
$41$ $$T^{16} + \cdots + 87\!\cdots\!21$$
$43$ $$T^{16} + \cdots + 47\!\cdots\!36$$
$47$ $$T^{16} + \cdots + 17\!\cdots\!56$$
$53$ $$(T^{8} + \cdots - 20\!\cdots\!72)^{2}$$
$59$ $$T^{16} + \cdots + 63\!\cdots\!56$$
$61$ $$T^{16} + \cdots + 57\!\cdots\!89$$
$67$ $$T^{16} + \cdots + 31\!\cdots\!36$$
$71$ $$T^{16} + \cdots + 13\!\cdots\!24$$
$73$ $$T^{16} + \cdots + 42\!\cdots\!36$$
$79$ $$(T^{8} + \cdots + 77\!\cdots\!28)^{2}$$
$83$ $$T^{16} + \cdots + 13\!\cdots\!64$$
$89$ $$T^{16} + \cdots + 14\!\cdots\!96$$
$97$ $$T^{16} + \cdots + 37\!\cdots\!24$$