# Properties

 Label 121.5.d.e Level $121$ Weight $5$ Character orbit 121.d Analytic conductor $12.508$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,5,Mod(40,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([7]))

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

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

chi := DirichletCharacter("121.40");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 121.d (of order $$10$$, degree $$4$$, 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: $$12.5077655331$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205$$ x^12 + 115*x^10 + 5030*x^8 + 102975*x^6 + 953170*x^4 + 2910655*x^2 + 73205 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 + 1) q^{2} + (\beta_{11} - \beta_{9} + 2 \beta_{4} - \beta_{2} + 2) q^{3} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - \beta_1) q^{4} + (\beta_{11} - \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{2} + 4) q^{5} + (3 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + 6 \beta_{7} + \beta_{5} + 6 \beta_{4} + \cdots + 7) q^{6}+ \cdots + ( - \beta_{11} - 4 \beta_{10} + 6 \beta_{8} + 6 \beta_{7} - 9 \beta_{6} + 31 \beta_{4} + \cdots - 7) q^{9}+O(q^{10})$$ q + (-b8 + b7 + b6 - b3 - b1 + 1) * q^2 + (b11 - b9 + 2*b4 - b2 + 2) * q^3 + (-b9 - b8 + 2*b7 + b6 + b5 - 2*b4 - 7*b3 - 2*b2 - b1) * q^4 + (b11 - b9 - b8 + 2*b7 + 2*b6 - b5 + 4*b2 + 4) * q^5 + (3*b11 + 2*b10 - 2*b9 - 6*b8 + 6*b7 + b5 + 6*b4 - 6*b3 - 7*b2 - 6*b1 + 7) * q^6 + (-2*b11 + 2*b10 + b9 - 5*b6 - b5 - 7*b4 - 12*b3 - 17*b2 - 24) * q^7 + (-2*b11 + b10 - 2*b9 - 5*b8 - 2*b7 + 5*b6 + 5*b5 - 26*b4 - 38*b3 - 19*b2 - 12) * q^8 + (-b11 - 4*b10 + 6*b8 + 6*b7 - 9*b6 + 31*b4 - 7*b3 - 9*b1 - 7) * q^9 $$q + ( - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 + 1) q^{2} + (\beta_{11} - \beta_{9} + 2 \beta_{4} - \beta_{2} + 2) q^{3} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - \beta_1) q^{4} + (\beta_{11} - \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{2} + 4) q^{5} + (3 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + 6 \beta_{7} + \beta_{5} + 6 \beta_{4} + \cdots + 7) q^{6}+ \cdots + (92 \beta_{11} + 313 \beta_{10} - 313 \beta_{9} - 39 \beta_{7} - 39 \beta_{6} + \cdots - 630) q^{98}+O(q^{100})$$ q + (-b8 + b7 + b6 - b3 - b1 + 1) * q^2 + (b11 - b9 + 2*b4 - b2 + 2) * q^3 + (-b9 - b8 + 2*b7 + b6 + b5 - 2*b4 - 7*b3 - 2*b2 - b1) * q^4 + (b11 - b9 - b8 + 2*b7 + 2*b6 - b5 + 4*b2 + 4) * q^5 + (3*b11 + 2*b10 - 2*b9 - 6*b8 + 6*b7 + b5 + 6*b4 - 6*b3 - 7*b2 - 6*b1 + 7) * q^6 + (-2*b11 + 2*b10 + b9 - 5*b6 - b5 - 7*b4 - 12*b3 - 17*b2 - 24) * q^7 + (-2*b11 + b10 - 2*b9 - 5*b8 - 2*b7 + 5*b6 + 5*b5 - 26*b4 - 38*b3 - 19*b2 - 12) * q^8 + (-b11 - 4*b10 + 6*b8 + 6*b7 - 9*b6 + 31*b4 - 7*b3 - 9*b1 - 7) * q^9 + (4*b11 + 5*b10 - 5*b9 + 5*b7 + 5*b6 - 2*b5 - 34*b4 - 41*b3 + 7*b2 - 9*b1 - 17) * q^10 + (-b10 - b9 - 18*b8 + 24*b7 - 6*b6 + 6*b5 - 127*b3 - 127*b2 - 9*b1 - 2) * q^12 + (-3*b11 + b10 + 8*b9 + 15*b8 - 15*b7 + 13*b6 - 2*b5 - 22*b4 - 28*b3 - 44*b2 - 13*b1 - 16) * q^13 + (-5*b11 + 5*b9 + 24*b8 - 19*b7 - 14*b6 - 42*b4 - 101*b2 + 38*b1 - 42) * q^14 + (-b11 - b10 - 3*b9 - 9*b8 + 36*b7 + 18*b6 + 4*b5 + 94*b4 - 51*b3 + 94*b2 - 27*b1) * q^15 + (3*b11 - 12*b8 + 33*b7 + 24*b6 - 3*b5 + 15*b4 + 15*b3 + 145*b2 + 9*b1 + 145) * q^16 + (-9*b11 - 2*b10 + 8*b9 - 24*b8 + 29*b7 - 7*b5 - 88*b4 + 88*b3 + 17*b2 - 29*b1 - 17) * q^17 + (10*b11 - 10*b10 - 7*b9 - 6*b8 - 30*b6 + 3*b5 + 113*b4 - 100*b3 - 313*b2 - 6*b1 - 200) * q^18 + (11*b11 - 5*b10 + 11*b9 - 15*b8 - 28*b7 + 15*b6 - 27*b5 - 20*b4 - 136*b3 - 68*b2 - 116) * q^19 + (-2*b11 + 14*b10 + 22*b8 + 22*b7 + 2*b6 - 24*b4 - 120*b3 + 2*b1 - 120) * q^20 + (-28*b11 - 27*b10 + 27*b9 - 12*b7 - 12*b6 + 14*b5 - 344*b4 - 238*b3 - 106*b2 + 21*b1 - 172) * q^21 + (8*b10 + 8*b9 + 16*b8 - 8*b7 - 8*b6 - 28*b5 - 362*b3 - 362*b2 + 8*b1 - 40) * q^23 + (19*b11 - 3*b10 - 44*b9 - 51*b8 + 51*b7 - 3*b6 + 6*b5 - 349*b4 - 585*b3 - 698*b2 + 3*b1 - 113) * q^24 + (9*b11 + 10*b10 - 9*b9 - 23*b8 + 33*b7 + 43*b6 - 10*b5 + 86*b4 - 297*b2 - 66*b1 + 86) * q^25 + (13*b11 + 13*b10 + 35*b9 - 9*b8 - 26*b7 - 13*b6 - 48*b5 + 525*b4 + 352*b3 + 525*b2 + 35*b1) * q^26 + (-46*b11 + 23*b9 + 21*b8 - 72*b7 - 42*b6 + 46*b5 + 220*b4 + 220*b3 + 402*b2 - 30*b1 + 402) * q^27 + (-18*b11 - 32*b10 + 2*b9 + 36*b8 - 62*b7 + 14*b5 - 236*b4 + 236*b3 + 160*b2 + 62*b1 - 160) * q^28 + (-11*b11 + 11*b10 + 18*b9 - b8 + 96*b6 + 7*b5 - 138*b4 - 320*b3 - 502*b2 - b1 - 640) * q^29 + (-14*b11 + 9*b10 - 14*b9 + 81*b8 + 75*b7 - 81*b6 + 37*b5 - 199*b4 - 860*b3 - 430*b2 - 661) * q^30 + (38*b11 + 25*b10 - 104*b8 - 104*b7 + 7*b6 + 293*b4 + 35*b3 + 7*b1 + 35) * q^31 + (66*b11 + 31*b10 - 31*b9 + 41*b7 + 41*b6 - 33*b5 - 178*b4 - 344*b3 + 166*b2 - 45*b1 - 89) * q^32 + (-29*b10 - 29*b9 + 74*b8 - 158*b7 + 84*b6 + 13*b5 - 290*b3 - 290*b2 + 37*b1 - 36) * q^34 + (-43*b11 - 11*b10 + 64*b9 + 99*b8 - 99*b7 - 96*b6 + 22*b5 - 223*b4 - 272*b3 - 446*b2 + 96*b1 - 174) * q^35 + (-23*b11 - 67*b10 + 23*b9 - 144*b8 + 33*b7 - 78*b6 + 67*b5 + 407*b4 - 558*b2 - 66*b1 + 407) * q^36 + (-61*b11 - 61*b10 - 32*b9 + 177*b8 - 332*b7 - 166*b6 + 93*b5 + 34*b4 + 6*b3 + 34*b2 + 155*b1) * q^37 + (108*b11 - 40*b9 + 76*b8 - 105*b7 - 152*b6 - 108*b5 + 757*b4 + 757*b3 + 790*b2 + 47*b1 + 790) * q^38 + (59*b11 + 56*b10 - 31*b9 + 267*b8 - 183*b7 + 3*b5 - 467*b4 + 467*b3 + 149*b2 + 183*b1 - 149) * q^39 + (-6*b11 + 6*b10 - 18*b9 + 102*b8 + 42*b6 - 24*b5 - 106*b4 - 220*b3 - 334*b2 + 102*b1 - 440) * q^40 + (-18*b11 - 17*b10 - 18*b9 - 79*b8 + 128*b7 + 79*b6 + 19*b5 - 711*b4 - 660*b3 - 330*b2 + 51) * q^41 + (-79*b11 - 147*b10 + 189*b6 - 710*b4 + 15*b3 + 189*b1 + 15) * q^42 + (-28*b11 + 61*b10 - 61*b9 - 188*b7 - 188*b6 + 14*b5 - 1032*b4 - 417*b3 - 615*b2 + 189*b1 - 516) * q^43 + (61*b10 + 61*b9 + 18*b8 + 39*b7 - 57*b6 + 87*b5 - 146*b3 - 146*b2 + 9*b1 + 350) * q^45 + (52*b11 + 56*b10 + 8*b9 - 206*b8 + 206*b7 - 52*b6 - 112*b5 - 242*b4 - 260*b3 - 484*b2 + 52*b1 - 224) * q^46 + (80*b11 + 129*b10 - 80*b9 + 96*b8 + 99*b7 + 294*b6 - 129*b5 - 439*b4 + 503*b2 - 198*b1 - 439) * q^47 + (121*b11 + 121*b10 - 145*b9 - 288*b8 + 378*b7 + 189*b6 + 24*b5 - 103*b4 - 443*b3 - 103*b2 - 90*b1) * q^48 + (-21*b11 - 67*b9 - 133*b8 + 112*b7 + 266*b6 + 21*b5 - 481*b4 - 481*b3 - 851*b2 - 154*b1 - 851) * q^49 + (70*b11 + 78*b10 - 31*b9 - 340*b8 + 92*b7 - 8*b5 + 699*b4 - 699*b3 - 172*b2 - 92*b1 + 172) * q^50 + (-25*b11 + 25*b10 - 2*b9 - 192*b8 + 45*b6 - 27*b5 - 614*b4 + 208*b3 + 1030*b2 - 192*b1 + 416) * q^51 + (36*b11 + 30*b10 + 36*b9 + 234*b8 - 396*b7 - 234*b6 - 42*b5 + 1682*b4 + 2036*b3 + 1018*b2 + 354) * q^52 + (-29*b11 + 99*b10 + 105*b8 + 105*b7 - 157*b6 + 520*b4 - 236*b3 - 157*b1 - 236) * q^53 + (-128*b11 - 136*b10 + 136*b9 + 228*b7 + 228*b6 + 64*b5 + 1970*b4 + 1161*b3 + 809*b2 - 195*b1 + 985) * q^54 + (-44*b10 - 44*b9 + 44*b8 + 22*b7 - 66*b6 - 28*b5 + 1134*b3 + 1134*b2 + 22*b1 - 1148) * q^56 + (-78*b11 - 73*b10 + 10*b9 + 261*b8 - 261*b7 + 258*b6 + 146*b5 + 1683*b4 + 2476*b3 + 3366*b2 - 258*b1 + 890) * q^57 + (-89*b11 + 60*b10 + 89*b9 + 194*b8 - 371*b7 - 548*b6 - 60*b5 - 638*b4 + 1127*b2 + 742*b1 - 638) * q^58 + (-43*b11 - 43*b10 + 171*b9 + 193*b8 + 230*b7 + 115*b6 - 128*b5 - 1636*b4 - 766*b3 - 1636*b2 - 423*b1) * q^59 + (-142*b11 + 142*b9 + 54*b8 + 222*b7 - 108*b6 + 142*b5 - 2162*b4 - 2162*b3 - 1894*b2 + 330*b1 - 1894) * q^60 + (-169*b11 - 40*b10 + 149*b9 + 43*b8 + 426*b7 - 129*b5 + 418*b4 - 418*b3 - 534*b2 - 426*b1 + 534) * q^61 + (85*b11 - 85*b10 + 9*b9 - 49*b8 - 435*b6 + 94*b5 + 2517*b4 + 2056*b3 + 1595*b2 - 49*b1 + 4112) * q^62 + (11*b11 + 44*b10 + 11*b9 - 429*b8 + 72*b7 + 429*b6 + 22*b5 - 809*b4 + 912*b3 + 456*b2 + 1721) * q^63 + (121*b11 + 196*b10 + 88*b8 + 88*b7 - 385*b6 - 295*b4 + 862*b3 - 385*b1 + 862) * q^64 + (186*b11 + 32*b10 - 32*b9 + 149*b7 + 149*b6 - 93*b5 + 756*b4 + 1124*b3 - 368*b2 + 22*b1 + 378) * q^65 + (-152*b10 - 152*b9 - 832*b8 + 911*b7 - 79*b6 - 139*b5 + 1521*b3 + 1521*b2 - 416*b1 + 2285) * q^67 + (54*b11 + 33*b10 - 42*b9 - 25*b8 + 25*b7 + 176*b6 - 66*b5 + 586*b4 + 753*b3 + 1172*b2 - 176*b1 + 419) * q^68 + (-100*b11 - 430*b10 + 100*b9 + 252*b8 + 252*b6 + 430*b5 + 418*b4 + 1508*b2 + 418) * q^69 + (-212*b11 - 212*b10 + 259*b9 - 253*b8 - 92*b7 - 46*b6 - 47*b5 - 343*b4 + 1821*b3 - 343*b2 + 345*b1) * q^70 + (64*b11 + 129*b9 - 56*b8 - 83*b7 + 112*b6 - 64*b5 + 659*b4 + 659*b3 - 2110*b2 - 195*b1 - 2110) * q^71 + (-215*b11 - 440*b10 - 5*b9 - 789*b8 + 420*b7 + 225*b5 + 2921*b4 - 2921*b3 - 254*b2 - 420*b1 + 254) * q^72 + (26*b11 - 26*b10 + 41*b9 + 393*b8 - 505*b6 + 67*b5 - 1885*b4 + 87*b3 + 2059*b2 + 393*b1 + 174) * q^73 + (23*b11 - 140*b10 + 23*b9 - 306*b8 + 401*b7 + 306*b6 - 186*b5 + 2874*b4 + 5674*b3 + 2837*b2 + 2800) * q^74 + (168*b11 - 192*b10 + 204*b8 + 204*b7 - 189*b6 + 1608*b4 - 261*b3 - 189*b1 - 261) * q^75 + (58*b11 + 72*b10 - 72*b9 - 371*b7 - 371*b6 - 29*b5 + 6410*b4 + 4149*b3 + 2261*b2 - 574*b1 + 3205) * q^76 + (407*b10 + 407*b9 + 336*b8 - 483*b7 + 147*b6 - 295*b5 + 4567*b3 + 4567*b2 + 168*b1 - 1710) * q^78 + (268*b11 + 17*b10 - 502*b9 + 40*b8 - 40*b7 + 29*b6 - 34*b5 + 2849*b4 + 3797*b3 + 5698*b2 - 29*b1 + 1901) * q^79 + (212*b11 + 232*b10 - 212*b9 - 164*b8 + 132*b7 + 100*b6 - 232*b5 - 656*b4 + 2204*b2 - 264*b1 - 656) * q^80 + (354*b11 + 354*b10 - 158*b9 + 12*b8 + 456*b7 + 228*b6 - 196*b5 - 3378*b4 - 2734*b3 - 3378*b2 - 468*b1) * q^81 + (-133*b11 - 164*b9 - 452*b8 + 840*b7 + 904*b6 + 133*b5 - 4310*b4 - 4310*b3 - 1369*b2 - 64*b1 - 1369) * q^82 + (219*b11 + 34*b10 - 202*b9 + 81*b8 - 361*b7 + 185*b5 - 1078*b4 + 1078*b3 - 2992*b2 + 361*b1 + 2992) * q^83 + (-182*b11 + 182*b10 - 88*b9 - 174*b8 + 852*b6 - 270*b5 - 628*b4 + 1154*b3 + 2936*b2 - 174*b1 + 2308) * q^84 + (-87*b11 - 96*b10 - 87*b9 + 255*b8 - 693*b7 - 255*b6 + 78*b5 - 1796*b4 - 388*b3 - 194*b2 + 1408) * q^85 + (-71*b11 + 117*b10 + 187*b8 + 187*b7 + 797*b6 - 2649*b4 + 2297*b3 + 797*b1 + 2297) * q^86 + (-338*b11 - 234*b10 + 234*b9 + 369*b7 + 369*b6 + 169*b5 - 3328*b4 - 1037*b3 - 2291*b2 + 447*b1 - 1664) * q^87 + (-185*b10 - 185*b9 + 994*b8 - 904*b7 - 90*b6 + 466*b5 - 437*b3 - 437*b2 + 497*b1 - 80) * q^89 + (-553*b11 - 274*b10 + 558*b9 - 513*b8 + 513*b7 - 42*b6 + 548*b5 - 1397*b4 - 1865*b3 - 2794*b2 + 42*b1 - 929) * q^90 + (-37*b11 + 301*b10 + 37*b9 - 973*b8 + 245*b7 - 483*b6 - 301*b5 - 158*b4 - 1279*b2 - 490*b1 - 158) * q^91 + (-318*b11 - 318*b10 - 314*b9 + 16*b8 - 76*b7 - 38*b6 + 632*b5 + 870*b4 + 1422*b3 + 870*b2 + 60*b1) * q^92 + (429*b11 - 368*b9 + 450*b8 - 921*b7 - 900*b6 - 429*b5 + 2802*b4 + 2802*b3 - 2288*b2 - 21*b1 - 2288) * q^93 + (468*b11 + 1066*b10 + 65*b9 + 1605*b8 - 506*b7 - 598*b5 + 1051*b4 - 1051*b3 + 5455*b2 + 506*b1 - 5455) * q^94 + (181*b11 - 181*b10 - 77*b9 - 537*b8 + 798*b6 + 104*b5 + 592*b4 + 1319*b3 + 2046*b2 - 537*b1 + 2638) * q^95 + (-143*b11 + 105*b10 - 143*b9 + 810*b8 + 279*b7 - 810*b6 + 391*b5 + 2093*b4 - 2444*b3 - 1222*b2 - 4537) * q^96 + (-728*b11 - 348*b10 - 1248*b8 - 1248*b7 + 212*b6 - 1429*b4 - 2929*b3 + 212*b1 - 2929) * q^97 + (92*b11 + 313*b10 - 313*b9 - 39*b7 - 39*b6 - 46*b5 - 1260*b4 - 3118*b3 + 1858*b2 + 868*b1 - 630) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 15 q^{2} + 19 q^{3} + 27 q^{4} + 32 q^{5} + 80 q^{6} - 200 q^{7} + 145 q^{8} - 244 q^{9}+O(q^{10})$$ 12 * q + 15 * q^2 + 19 * q^3 + 27 * q^4 + 32 * q^5 + 80 * q^6 - 200 * q^7 + 145 * q^8 - 244 * q^9 $$12 q + 15 q^{2} + 19 q^{3} + 27 q^{4} + 32 q^{5} + 80 q^{6} - 200 q^{7} + 145 q^{8} - 244 q^{9} + 594 q^{12} + 250 q^{13} - 40 q^{14} - 506 q^{15} + 1167 q^{16} - 405 q^{17} - 1685 q^{18} - 530 q^{19} - 1068 q^{20} + 1684 q^{23} + 3160 q^{24} + 1707 q^{25} - 4080 q^{26} + 2563 q^{27} - 2170 q^{28} - 4230 q^{29} - 4200 q^{30} + 104 q^{31} + 2370 q^{34} + 910 q^{35} + 4761 q^{36} + 454 q^{37} + 2105 q^{38} - 1180 q^{39} - 3180 q^{40} + 5385 q^{41} + 2690 q^{42} + 5136 q^{45} - 1000 q^{46} - 4516 q^{47} + 1099 q^{48} - 4183 q^{49} + 2315 q^{50} + 3335 q^{51} - 9000 q^{52} - 4726 q^{53} - 21340 q^{56} - 9275 q^{57} - 9770 q^{58} + 11624 q^{59} - 5156 q^{60} + 5780 q^{61} + 28710 q^{62} + 20980 q^{63} + 6987 q^{64} + 12154 q^{67} - 1855 q^{68} + 268 q^{69} - 3340 q^{70} - 21646 q^{71} + 610 q^{72} - 1015 q^{73} - 1730 q^{74} - 9546 q^{75} - 42920 q^{78} - 15130 q^{79} - 12868 q^{80} + 27880 q^{81} + 13500 q^{82} + 46770 q^{83} + 21130 q^{84} + 25740 q^{85} + 31950 q^{86} + 5554 q^{89} + 5640 q^{90} - 850 q^{91} - 10246 q^{92} - 38832 q^{93} - 76630 q^{94} + 23390 q^{95} - 54645 q^{96} - 16546 q^{97}+O(q^{100})$$ 12 * q + 15 * q^2 + 19 * q^3 + 27 * q^4 + 32 * q^5 + 80 * q^6 - 200 * q^7 + 145 * q^8 - 244 * q^9 + 594 * q^12 + 250 * q^13 - 40 * q^14 - 506 * q^15 + 1167 * q^16 - 405 * q^17 - 1685 * q^18 - 530 * q^19 - 1068 * q^20 + 1684 * q^23 + 3160 * q^24 + 1707 * q^25 - 4080 * q^26 + 2563 * q^27 - 2170 * q^28 - 4230 * q^29 - 4200 * q^30 + 104 * q^31 + 2370 * q^34 + 910 * q^35 + 4761 * q^36 + 454 * q^37 + 2105 * q^38 - 1180 * q^39 - 3180 * q^40 + 5385 * q^41 + 2690 * q^42 + 5136 * q^45 - 1000 * q^46 - 4516 * q^47 + 1099 * q^48 - 4183 * q^49 + 2315 * q^50 + 3335 * q^51 - 9000 * q^52 - 4726 * q^53 - 21340 * q^56 - 9275 * q^57 - 9770 * q^58 + 11624 * q^59 - 5156 * q^60 + 5780 * q^61 + 28710 * q^62 + 20980 * q^63 + 6987 * q^64 + 12154 * q^67 - 1855 * q^68 + 268 * q^69 - 3340 * q^70 - 21646 * q^71 + 610 * q^72 - 1015 * q^73 - 1730 * q^74 - 9546 * q^75 - 42920 * q^78 - 15130 * q^79 - 12868 * q^80 + 27880 * q^81 + 13500 * q^82 + 46770 * q^83 + 21130 * q^84 + 25740 * q^85 + 31950 * q^86 + 5554 * q^89 + 5640 * q^90 - 850 * q^91 - 10246 * q^92 - 38832 * q^93 - 76630 * q^94 + 23390 * q^95 - 54645 * q^96 - 16546 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 13052 \nu^{11} - 117304 \nu^{10} - 1219776 \nu^{9} - 10477907 \nu^{8} - 40135806 \nu^{7} - 321264966 \nu^{6} - 549385674 \nu^{5} + \cdots - 1427754383 ) / 1339827192$$ (-13052*v^11 - 117304*v^10 - 1219776*v^9 - 10477907*v^8 - 40135806*v^7 - 321264966*v^6 - 549385674*v^5 - 3857093625*v^4 - 2823880454*v^3 - 13654448170*v^2 - 5016493020*v - 1427754383) / 1339827192 $$\beta_{3}$$ $$=$$ $$( 13052 \nu^{11} - 117304 \nu^{10} + 1219776 \nu^{9} - 10477907 \nu^{8} + 40135806 \nu^{7} - 321264966 \nu^{6} + 549385674 \nu^{5} + \cdots - 1427754383 ) / 1339827192$$ (13052*v^11 - 117304*v^10 + 1219776*v^9 - 10477907*v^8 + 40135806*v^7 - 321264966*v^6 + 549385674*v^5 - 3857093625*v^4 + 2823880454*v^3 - 13654448170*v^2 + 5016493020*v - 1427754383) / 1339827192 $$\beta_{4}$$ $$=$$ $$( - 31452 \nu^{11} + 117304 \nu^{10} - 2737673 \nu^{9} + 10477907 \nu^{8} - 79586472 \nu^{7} + 321264966 \nu^{6} - 827426403 \nu^{5} + \cdots + 757840787 ) / 1339827192$$ (-31452*v^11 + 117304*v^10 - 2737673*v^9 + 10477907*v^8 - 79586472*v^7 + 321264966*v^6 - 827426403*v^5 + 3857093625*v^4 - 1134660378*v^3 + 13654448170*v^2 + 7973305285*v + 757840787) / 1339827192 $$\beta_{5}$$ $$=$$ $$( 45643 \nu^{10} + 4016197 \nu^{8} + 120327945 \nu^{6} + 1386775659 \nu^{4} + 4387794733 \nu^{2} - 676101899 ) / 60901236$$ (45643*v^10 + 4016197*v^8 + 120327945*v^6 + 1386775659*v^4 + 4387794733*v^2 - 676101899) / 60901236 $$\beta_{6}$$ $$=$$ $$( 10664 \nu^{11} + 25564 \nu^{10} + 952537 \nu^{9} + 2319614 \nu^{8} + 29205906 \nu^{7} + 72240366 \nu^{6} + 350644875 \nu^{5} + 874263126 \nu^{4} + \cdots + 86861060 ) / 121802472$$ (10664*v^11 + 25564*v^10 + 952537*v^9 + 2319614*v^8 + 29205906*v^7 + 72240366*v^6 + 350644875*v^5 + 874263126*v^4 + 1241313470*v^3 + 2997579640*v^2 + 129795853*v + 86861060) / 121802472 $$\beta_{7}$$ $$=$$ $$( 10664 \nu^{11} - 25564 \nu^{10} + 952537 \nu^{9} - 2319614 \nu^{8} + 29205906 \nu^{7} - 72240366 \nu^{6} + 350644875 \nu^{5} - 874263126 \nu^{4} + \cdots - 86861060 ) / 121802472$$ (10664*v^11 - 25564*v^10 + 952537*v^9 - 2319614*v^8 + 29205906*v^7 - 72240366*v^6 + 350644875*v^5 - 874263126*v^4 + 1241313470*v^3 - 2997579640*v^2 + 129795853*v - 86861060) / 121802472 $$\beta_{8}$$ $$=$$ $$( 10664 \nu^{11} - 79937 \nu^{10} + 952537 \nu^{9} - 7147008 \nu^{8} + 29205906 \nu^{7} - 219213027 \nu^{6} + 350644875 \nu^{5} - 2622222042 \nu^{4} + \cdots - 209313060 ) / 121802472$$ (10664*v^11 - 79937*v^10 + 952537*v^9 - 7147008*v^8 + 29205906*v^7 - 219213027*v^6 + 350644875*v^5 - 2622222042*v^4 + 1241313470*v^3 - 9047202395*v^2 + 68894617*v - 209313060) / 121802472 $$\beta_{9}$$ $$=$$ $$( - 169940 \nu^{11} - 259930 \nu^{10} - 15878131 \nu^{9} - 23961014 \nu^{8} - 530879748 \nu^{7} - 766470936 \nu^{6} - 7708194267 \nu^{5} + \cdots + 4416548158 ) / 1339827192$$ (-169940*v^11 - 259930*v^10 - 15878131*v^9 - 23961014*v^8 - 530879748*v^7 - 766470936*v^6 - 7708194267*v^5 - 9732816588*v^4 - 46854588548*v^3 - 36929122516*v^2 - 104516796175*v + 4416548158) / 1339827192 $$\beta_{10}$$ $$=$$ $$( 169940 \nu^{11} - 259930 \nu^{10} + 15878131 \nu^{9} - 23961014 \nu^{8} + 530879748 \nu^{7} - 766470936 \nu^{6} + 7708194267 \nu^{5} + \cdots + 4416548158 ) / 1339827192$$ (169940*v^11 - 259930*v^10 + 15878131*v^9 - 23961014*v^8 + 530879748*v^7 - 766470936*v^6 + 7708194267*v^5 - 9732816588*v^4 + 46854588548*v^3 - 36929122516*v^2 + 104516796175*v + 4416548158) / 1339827192 $$\beta_{11}$$ $$=$$ $$( - 380623 \nu^{11} + 502073 \nu^{10} - 35561311 \nu^{9} + 44178167 \nu^{8} - 1181014629 \nu^{7} + 1323607395 \nu^{6} - 16702393701 \nu^{5} + \cdots - 7437120889 ) / 1339827192$$ (-380623*v^11 + 502073*v^10 - 35561311*v^9 + 44178167*v^8 - 1181014629*v^7 + 1323607395*v^6 - 16702393701*v^5 + 15254532249*v^4 - 93304347409*v^3 + 48265742063*v^2 - 169552623955*v - 7437120889) / 1339827192
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta _1 - 19$$ b10 + b9 - 2*b8 + b7 + b6 - b5 + b3 + b2 - b1 - 19 $$\nu^{3}$$ $$=$$ $$\beta_{10} - \beta_{9} - 5\beta_{7} - 5\beta_{6} - 24\beta_{4} - 9\beta_{3} - 15\beta_{2} - 25\beta _1 - 12$$ b10 - b9 - 5*b7 - 5*b6 - 24*b4 - 9*b3 - 15*b2 - 25*b1 - 12 $$\nu^{4}$$ $$=$$ $$- 31 \beta_{10} - 31 \beta_{9} + 90 \beta_{8} - 51 \beta_{7} - 39 \beta_{6} + 33 \beta_{5} - 113 \beta_{3} - 113 \beta_{2} + 45 \beta _1 + 476$$ -31*b10 - 31*b9 + 90*b8 - 51*b7 - 39*b6 + 33*b5 - 113*b3 - 113*b2 + 45*b1 + 476 $$\nu^{5}$$ $$=$$ $$32 \beta_{11} - 33 \beta_{10} + 33 \beta_{9} + 271 \beta_{7} + 271 \beta_{6} - 16 \beta_{5} + 1300 \beta_{4} + 677 \beta_{3} + 623 \beta_{2} + 682 \beta _1 + 650$$ 32*b11 - 33*b10 + 33*b9 + 271*b7 + 271*b6 - 16*b5 + 1300*b4 + 677*b3 + 623*b2 + 682*b1 + 650 $$\nu^{6}$$ $$=$$ $$954 \beta_{10} + 954 \beta_{9} - 3628 \beta_{8} + 2292 \beta_{7} + 1336 \beta_{6} - 1142 \beta_{5} + 5450 \beta_{3} + 5450 \beta_{2} - 1814 \beta _1 - 12905$$ 954*b10 + 954*b9 - 3628*b8 + 2292*b7 + 1336*b6 - 1142*b5 + 5450*b3 + 5450*b2 - 1814*b1 - 12905 $$\nu^{7}$$ $$=$$ $$- 2096 \beta_{11} + 1056 \beta_{10} - 1056 \beta_{9} - 11834 \beta_{7} - 11834 \beta_{6} + 1048 \beta_{5} - 56748 \beta_{4} - 34702 \beta_{3} - 22046 \beta_{2} - 19683 \beta _1 - 28374$$ -2096*b11 + 1056*b10 - 1056*b9 - 11834*b7 - 11834*b6 + 1048*b5 - 56748*b4 - 34702*b3 - 22046*b2 - 19683*b1 - 28374 $$\nu^{8}$$ $$=$$ $$- 30477 \beta_{10} - 30477 \beta_{9} + 140790 \beta_{8} - 95925 \beta_{7} - 44865 \beta_{6} + 40191 \beta_{5} - 226923 \beta_{3} - 226923 \beta_{2} + 70395 \beta _1 + 369011$$ -30477*b10 - 30477*b9 + 140790*b8 - 95925*b7 - 44865*b6 + 40191*b5 - 226923*b3 - 226923*b2 + 70395*b1 + 369011 $$\nu^{9}$$ $$=$$ $$99264 \beta_{11} - 36579 \beta_{10} + 36579 \beta_{9} + 478665 \beta_{7} + 478665 \beta_{6} - 49632 \beta_{5} + 2301516 \beta_{4} + 1545459 \beta_{3} + 756057 \beta_{2} + 596465 \beta _1 + 1150758$$ 99264*b11 - 36579*b10 + 36579*b9 + 478665*b7 + 478665*b6 - 49632*b5 + 2301516*b4 + 1545459*b3 + 756057*b2 + 596465*b1 + 1150758 $$\nu^{10}$$ $$=$$ $$1012445 \beta_{10} + 1012445 \beta_{9} - 5366098 \beta_{8} + 3851627 \beta_{7} + 1514471 \beta_{6} - 1431005 \beta_{5} + 8936705 \beta_{3} + 8936705 \beta_{2} - 2683049 \beta _1 - 11069600$$ 1012445*b10 + 1012445*b9 - 5366098*b8 + 3851627*b7 + 1514471*b6 - 1431005*b5 + 8936705*b3 + 8936705*b2 - 2683049*b1 - 11069600 $$\nu^{11}$$ $$=$$ $$- 4178328 \beta_{11} + 1343903 \beta_{10} - 1343903 \beta_{9} - 18668485 \beta_{7} - 18668485 \beta_{6} + 2089164 \beta_{5} - 90111516 \beta_{4} - 64217769 \beta_{3} + \cdots - 45055758$$ -4178328*b11 + 1343903*b10 - 1343903*b9 - 18668485*b7 - 18668485*b6 + 2089164*b5 - 90111516*b4 - 64217769*b3 - 25893747*b2 - 18898340*b1 - 45055758

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/121\mathbb{Z}\right)^\times$$.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
40.1
 − 5.04186i 2.38108i 4.56289i 6.10049i 0.159251i − 5.08417i − 6.10049i − 0.159251i 5.08417i 5.04186i − 2.38108i − 4.56289i
−2.27255 + 3.12789i 0.628596 1.93462i 0.325034 + 1.00035i 4.62630 3.36120i 4.62277 + 6.36269i −4.09081 + 1.32918i −62.7006 20.3726i 62.1828 + 45.1784i 22.1090i
40.2 2.09055 2.87739i −4.88370 + 15.0305i 1.03528 + 3.18628i −14.3611 + 10.4339i 33.0390 + 45.4743i −31.7711 + 10.3231i 65.4537 + 21.2672i −136.535 99.1984i 63.1351i
40.3 3.37298 4.64251i 3.97396 12.2306i −5.23164 16.1013i 24.4430 17.7589i −43.3765 59.7027i −54.3873 + 17.6715i −5.07527 1.64906i −68.2643 49.5969i 173.377i
94.1 −3.99290 + 1.29737i 5.92412 + 4.30413i 1.31577 0.955966i −5.62746 + 17.3195i −29.2385 9.50015i −29.6650 40.8304i 35.4704 48.8208i −8.46066 26.0392i 76.4560i
94.2 1.65756 0.538574i −7.02074 5.10087i −10.4868 + 7.61913i 3.07887 9.47577i −14.3845 4.67381i 34.6308 + 47.6652i −29.6699 + 40.8372i −1.75841 5.41183i 17.3649i
94.3 6.64435 2.15888i 10.8778 + 7.90316i 26.5424 19.2842i 3.84039 11.8195i 89.3377 + 29.0276i −14.7166 20.2556i 69.0217 95.0002i 30.8355 + 94.9021i 86.8239i
112.1 −3.99290 1.29737i 5.92412 4.30413i 1.31577 + 0.955966i −5.62746 17.3195i −29.2385 + 9.50015i −29.6650 + 40.8304i 35.4704 + 48.8208i −8.46066 + 26.0392i 76.4560i
112.2 1.65756 + 0.538574i −7.02074 + 5.10087i −10.4868 7.61913i 3.07887 + 9.47577i −14.3845 + 4.67381i 34.6308 47.6652i −29.6699 40.8372i −1.75841 + 5.41183i 17.3649i
112.3 6.64435 + 2.15888i 10.8778 7.90316i 26.5424 + 19.2842i 3.84039 + 11.8195i 89.3377 29.0276i −14.7166 + 20.2556i 69.0217 + 95.0002i 30.8355 94.9021i 86.8239i
118.1 −2.27255 3.12789i 0.628596 + 1.93462i 0.325034 1.00035i 4.62630 + 3.36120i 4.62277 6.36269i −4.09081 1.32918i −62.7006 + 20.3726i 62.1828 45.1784i 22.1090i
118.2 2.09055 + 2.87739i −4.88370 15.0305i 1.03528 3.18628i −14.3611 10.4339i 33.0390 45.4743i −31.7711 10.3231i 65.4537 21.2672i −136.535 + 99.1984i 63.1351i
118.3 3.37298 + 4.64251i 3.97396 + 12.2306i −5.23164 + 16.1013i 24.4430 + 17.7589i −43.3765 + 59.7027i −54.3873 17.6715i −5.07527 + 1.64906i −68.2643 + 49.5969i 173.377i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 40.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.5.d.e 12
11.b odd 2 1 121.5.d.c 12
11.c even 5 1 11.5.d.a 12
11.c even 5 1 121.5.b.b 12
11.c even 5 1 121.5.d.c 12
11.c even 5 1 121.5.d.d 12
11.d odd 10 1 11.5.d.a 12
11.d odd 10 1 121.5.b.b 12
11.d odd 10 1 121.5.d.d 12
11.d odd 10 1 inner 121.5.d.e 12
33.f even 10 1 99.5.k.a 12
33.h odd 10 1 99.5.k.a 12
44.g even 10 1 176.5.n.a 12
44.h odd 10 1 176.5.n.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.5.d.a 12 11.c even 5 1
11.5.d.a 12 11.d odd 10 1
99.5.k.a 12 33.f even 10 1
99.5.k.a 12 33.h odd 10 1
121.5.b.b 12 11.c even 5 1
121.5.b.b 12 11.d odd 10 1
121.5.d.c 12 11.b odd 2 1
121.5.d.c 12 11.c even 5 1
121.5.d.d 12 11.c even 5 1
121.5.d.d 12 11.d odd 10 1
121.5.d.e 12 1.a even 1 1 trivial
121.5.d.e 12 11.d odd 10 1 inner
176.5.n.a 12 44.g even 10 1
176.5.n.a 12 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 15 T_{2}^{11} + 75 T_{2}^{10} + 5 T_{2}^{9} - 1305 T_{2}^{8} + 1950 T_{2}^{7} + 21730 T_{2}^{6} - 10180 T_{2}^{5} - 172260 T_{2}^{4} + 27560 T_{2}^{3} + 6145160 T_{2}^{2} - 18581200 T_{2} + 16272080$$ acting on $$S_{5}^{\mathrm{new}}(121, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 15 T^{11} + 75 T^{10} + \cdots + 16272080$$
$3$ $$T^{12} - 19 T^{11} + \cdots + 124779204081$$
$5$ $$T^{12} - 32 T^{11} + \cdots + 47826408560896$$
$7$ $$T^{12} + 200 T^{11} + \cdots + 37\!\cdots\!00$$
$11$ $$T^{12}$$
$13$ $$T^{12} - 250 T^{11} + \cdots + 10\!\cdots\!80$$
$17$ $$T^{12} + 405 T^{11} + \cdots + 16\!\cdots\!05$$
$19$ $$T^{12} + 530 T^{11} + \cdots + 31\!\cdots\!05$$
$23$ $$(T^{6} - 842 T^{5} + \cdots - 10\!\cdots\!96)^{2}$$
$29$ $$T^{12} + 4230 T^{11} + \cdots + 61\!\cdots\!80$$
$31$ $$T^{12} - 104 T^{11} + \cdots + 41\!\cdots\!36$$
$37$ $$T^{12} - 454 T^{11} + \cdots + 49\!\cdots\!16$$
$41$ $$T^{12} - 5385 T^{11} + \cdots + 72\!\cdots\!05$$
$43$ $$T^{12} + 17669305 T^{10} + \cdots + 16\!\cdots\!00$$
$47$ $$T^{12} + 4516 T^{11} + \cdots + 84\!\cdots\!96$$
$53$ $$T^{12} + 4726 T^{11} + \cdots + 11\!\cdots\!76$$
$59$ $$T^{12} - 11624 T^{11} + \cdots + 38\!\cdots\!41$$
$61$ $$T^{12} - 5780 T^{11} + \cdots + 15\!\cdots\!80$$
$67$ $$(T^{6} - 6077 T^{5} + \cdots + 15\!\cdots\!84)^{2}$$
$71$ $$T^{12} + 21646 T^{11} + \cdots + 56\!\cdots\!56$$
$73$ $$T^{12} + 1015 T^{11} + \cdots + 14\!\cdots\!05$$
$79$ $$T^{12} + 15130 T^{11} + \cdots + 36\!\cdots\!80$$
$83$ $$T^{12} - 46770 T^{11} + \cdots + 18\!\cdots\!05$$
$89$ $$(T^{6} - 2777 T^{5} + \cdots + 66\!\cdots\!44)^{2}$$
$97$ $$T^{12} + 16546 T^{11} + \cdots + 18\!\cdots\!01$$