# Properties

 Label 121.4.c.i Level $121$ Weight $4$ Character orbit 121.c Analytic conductor $7.139$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,4,Mod(3,121)]

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([8]))

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

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

chi := DirichletCharacter("121.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 121.c (of order $$5$$, 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: $$7.13923111069$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.29283765625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561$$ x^8 - x^7 + 10*x^6 - 19*x^5 + 109*x^4 + 171*x^3 + 810*x^2 + 729*x + 6561 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{2} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 2) q^{3} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} - 2 \beta_1 + 3) q^{4} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + (2 \beta_{7} - 13 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{6} + ( - 5 \beta_{7} + 6 \beta_{6} - 3 \beta_{5} - 3 \beta_1 + 6) q^{7} + (26 \beta_{6} - 3 \beta_{5} + 26 \beta_{4} - 6 \beta_{3} + 7 \beta_{2} - 6 \beta_1 + 7) q^{8} + ( - 14 \beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q + (b4 - b2 - b1 + 1) * q^2 + (-2*b6 + b5 - 2*b4 - 2*b2 - 2) * q^3 + (b7 + 3*b6 - 2*b5 + 5*b4 - 2*b1 + 3) * q^4 + (-2*b7 - 2*b6 + b5 + 3*b4 + b3 + b2 - 2*b1) * q^5 + (2*b7 - 13*b6 + 3*b5 - 4*b4 + 3*b3 - 2*b2 + 2*b1) * q^6 + (-5*b7 + 6*b6 - 3*b5 - 3*b1 + 6) * q^7 + (26*b6 - 3*b5 + 26*b4 - 6*b3 + 7*b2 - 6*b1 + 7) * q^8 + (-14*b2 + 3*b1) * q^9 $$q + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{2} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 2) q^{3} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 5 \beta_{4} - 2 \beta_1 + 3) q^{4} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + (2 \beta_{7} - 13 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{6} + ( - 5 \beta_{7} + 6 \beta_{6} - 3 \beta_{5} - 3 \beta_1 + 6) q^{7} + (26 \beta_{6} - 3 \beta_{5} + 26 \beta_{4} - 6 \beta_{3} + 7 \beta_{2} - 6 \beta_1 + 7) q^{8} + ( - 14 \beta_{2} + 3 \beta_1) q^{9} + ( - 28 \beta_{6} + 6 \beta_{3} - 28 \beta_{2} - 16) q^{10} + (5 \beta_{7} - 24 \beta_{6} + 5 \beta_{5} - 5 \beta_{4} + 9 \beta_{3} - 24 \beta_{2} + \cdots + 15) q^{12}+ \cdots + (242 \beta_{7} - 1071 \beta_{6} + 242 \beta_{5} - 242 \beta_{4} + \cdots - 843) q^{98}+O(q^{100})$$ q + (b4 - b2 - b1 + 1) * q^2 + (-2*b6 + b5 - 2*b4 - 2*b2 - 2) * q^3 + (b7 + 3*b6 - 2*b5 + 5*b4 - 2*b1 + 3) * q^4 + (-2*b7 - 2*b6 + b5 + 3*b4 + b3 + b2 - 2*b1) * q^5 + (2*b7 - 13*b6 + 3*b5 - 4*b4 + 3*b3 - 2*b2 + 2*b1) * q^6 + (-5*b7 + 6*b6 - 3*b5 - 3*b1 + 6) * q^7 + (26*b6 - 3*b5 + 26*b4 - 6*b3 + 7*b2 - 6*b1 + 7) * q^8 + (-14*b2 + 3*b1) * q^9 + (-28*b6 + 6*b3 - 28*b2 - 16) * q^10 + (5*b7 - 24*b6 + 5*b5 - 5*b4 + 9*b3 - 24*b2 + 15) * q^12 + (-7*b7 + 14*b4 - 7*b3 + b2 + 4*b1 + 7) * q^13 + (-22*b6 + 4*b5 - 22*b4 - 4*b3 - 50*b2 - 4*b1 - 50) * q^14 + (-b7 - 25*b6 + 2*b5 - 12*b4 + 2*b1 - 25) * q^15 + (-5*b7 + 50*b6 - 11*b5 + 80*b4 - 11*b3 + 75*b2 - 5*b1) * q^16 + (-7*b7 - 45*b6 - 9*b5 + 25*b4 - 9*b3 + 18*b2 - 7*b1) * q^17 + (14*b7 + 14*b6 + 3*b5 - 13*b4 + 3*b1 + 14) * q^18 + (58*b6 + 6*b5 + 58*b4 + 12*b3 + 19*b2 + 12*b1 + 19) * q^19 + (10*b7 + 38*b4 + 10*b3 - 42*b2 + 48) * q^20 + (9*b7 - 39*b6 + 9*b5 - 9*b4 - 7*b3 - 39*b2 - 67) * q^21 + (-10*b7 + 26*b6 - 10*b5 + 10*b4 - 36*b3 + 26*b2 - 4) * q^23 + (25*b7 + 67*b4 + 25*b3 + 13*b2 - 4*b1 + 92) * q^24 + (30*b6 - 12*b5 + 30*b4 + 19*b3 - 2*b2 + 19*b1 - 2) * q^25 + (-8*b7 - 50*b6 + 4*b5 - 72*b4 + 4*b1 - 50) * q^26 + (38*b7 - 55*b6 + 38*b5 - 38*b4 + 38*b3 + 38*b1) * q^27 + (6*b7 - 12*b6 + 6*b5 - 12*b4 + 6*b3 - 6*b2 + 6*b1) * q^28 + (-b7 - 12*b6 + 31*b5 - 104*b4 + 31*b1 - 12) * q^29 + (-78*b6 + 18*b5 - 78*b4 + 28*b3 - 22*b2 + 28*b1 - 22) * q^30 + (-41*b7 - 48*b4 - 41*b3 - 45*b2 - 40*b1 - 89) * q^31 + (-38*b7 + 27*b6 - 38*b5 + 38*b4 - 43*b3 + 27*b2 - 39) * q^32 + (-27*b7 - 9*b6 - 27*b5 + 27*b4 + 34*b3 - 9*b2 - 18) * q^34 + (-4*b7 - 81*b4 - 4*b3 - 132*b2 - 3*b1 - 85) * q^35 + (3*b6 - 33*b5 + 3*b4 - 25*b3 + 15*b2 - 25*b1 + 15) * q^36 + (17*b7 + 8*b6 + 35*b5 + 10*b4 + 35*b1 + 8) * q^37 + (-7*b7 + 101*b6 - 40*b5 - 4*b4 - 40*b3 - 11*b2 - 7*b1) * q^38 + (-5*b7 + 87*b6 - 5*b5 + 54*b4 - 5*b3 + 49*b2 - 5*b1) * q^39 + (4*b7 + 4*b6 - 58*b5 + 218*b4 - 58*b1 + 4) * q^40 + (66*b6 - 47*b5 + 66*b4 - 62*b3 + 26*b2 - 62*b1 + 26) * q^41 + (32*b7 + 60*b4 + 32*b3 + 172*b2 + 58*b1 + 92) * q^42 + (31*b7 + 168*b6 + 31*b5 - 31*b4 + 39*b3 + 168*b2 + 177) * q^43 + (54*b7 + 95*b6 + 54*b5 - 54*b4 + 11*b3 + 95*b2 + 96) * q^45 + (-62*b7 - 84*b4 - 62*b3 + 264*b2 + 14*b1 - 146) * q^46 + (161*b6 + 28*b5 + 161*b4 - 9*b3 + 241*b2 - 9*b1 + 241) * q^47 + (-20*b7 + 204*b6 - 81*b5 + 219*b4 - 81*b1 + 204) * q^48 + (-112*b7 - 165*b6 - 43*b5 - 173*b4 - 43*b3 - 285*b2 - 112*b1) * q^49 + (21*b7 + 200*b6 - 23*b5 - 130*b4 - 23*b3 - 109*b2 + 21*b1) * q^50 + (-56*b7 + 54*b6 - 20*b5 + 47*b4 - 20*b1 + 54) * q^51 + (-366*b6 + 8*b5 - 366*b4 + 6*b3 - 144*b2 + 6*b1 - 144) * q^52 + (41*b7 - 238*b4 + 41*b3 + 95*b2 + 2*b1 - 197) * q^53 + (38*b7 - 55*b6 + 38*b5 - 38*b4 + 93*b3 - 55*b2 + 397) * q^54 + (44*b7 - 242*b6 + 44*b5 - 44*b4 - 14*b3 - 242*b2 - 334) * q^56 + (27*b7 - 57*b4 + 27*b3 - 92*b2 - 25*b1 - 30) * q^57 + (-510*b6 + 168*b5 - 510*b4 + 44*b3 - 114*b2 + 44*b1 - 114) * q^58 + (2*b7 + 169*b6 - 32*b5 + 136*b4 - 32*b1 + 169) * q^59 + (42*b7 - 270*b6 + 100*b5 - 228*b4 + 100*b3 - 186*b2 + 42*b1) * q^60 + (-42*b7 - 84*b6 + 105*b5 - 147*b4 + 105*b3 - 189*b2 - 42*b1) * q^61 + (4*b7 - 502*b6 + 90*b5 - 190*b4 + 90*b1 - 502) * q^62 + (-16*b6 + 19*b5 - 16*b4 - 33*b3 - 19*b2 - 33*b1 - 19) * q^63 + (-22*b7 + 187*b4 - 22*b3 + 311*b2 + 37*b1 + 165) * q^64 + (13*b7 + 156*b6 + 13*b5 - 13*b4 + 44*b3 + 156*b2 - 127) * q^65 + (-63*b7 + 309*b6 - 63*b5 + 63*b4 + 46*b3 + 309*b2 + 204) * q^67 + (59*b7 - 158*b4 + 59*b3 - 36*b2 - 11*b1 - 99) * q^68 + (384*b6 - 66*b5 + 384*b4 - 36*b3 + 242*b2 - 36*b1 + 242) * q^69 + (128*b7 - 74*b6 + 86*b5 - 128*b4 + 86*b1 - 74) * q^70 + (120*b7 - 10*b6 + 43*b5 - 373*b4 + 43*b3 - 253*b2 + 120*b1) * q^71 + (72*b7 + 283*b6 + 27*b5 + 32*b4 + 27*b3 + 104*b2 + 72*b1) * q^72 + (51*b7 + 480*b6 - 44*b5 + 293*b4 - 44*b1 + 480) * q^73 + (-100*b6 + 26*b5 - 100*b4 + 10*b3 + 180*b2 + 10*b1 + 180) * q^74 + (13*b7 - 120*b4 + 13*b3 - 275*b2 - 29*b1 - 107) * q^75 + (-125*b7 + 75*b6 - 125*b5 + 125*b4 - 222*b3 + 75*b2 - 12) * q^76 + (-54*b7 + 136*b6 - 54*b5 + 54*b4 - 92*b3 + 136*b2 - 132) * q^78 + (13*b7 + 68*b4 + 13*b3 - 143*b2 + 128*b1 + 81) * q^79 + (286*b6 - 262*b5 + 286*b4 + 14*b3 - 78*b2 + 14*b1 - 78) * q^80 + (-174*b7 + 100*b4) * q^81 + (-88*b7 + 595*b6 - 175*b5 + 752*b4 - 175*b3 + 664*b2 - 88*b1) * q^82 + (14*b7 - 235*b6 - 194*b5 + 269*b4 - 194*b3 + 283*b2 + 14*b1) * q^83 + (-12*b7 - 12*b6 + 6*b5 - 66*b4 + 6*b1 - 12) * q^84 + (27*b6 - 10*b5 + 27*b4 + 197*b3 - 279*b2 + 197*b1 - 279) * q^85 + (-129*b7 + 249*b4 - 129*b3 - 81*b2 - 208*b1 + 120) * q^86 + (-43*b7 + 303*b6 - 43*b5 + 43*b4 - 137*b3 + 303*b2 - 195) * q^87 + (25*b7 - 428*b6 + 25*b5 - 25*b4 + 57*b3 - 428*b2 + 281) * q^89 + (-84*b7 + 476*b4 - 84*b3 + 386*b2 - 150*b1 + 392) * q^90 + (21*b6 - 21*b5 + 21*b4 + 8*b3 + 44*b2 + 8*b1 + 44) * q^91 + (-118*b7 - 906*b6 + 142*b5 - 1292*b4 + 142*b1 - 906) * q^92 + (85*b7 + 97*b6 - 45*b5 + 462*b4 - 45*b3 + 547*b2 + 85*b1) * q^93 + (-250*b7 - 10*b6 - 142*b5 + 412*b4 - 142*b3 + 162*b2 - 250*b1) * q^94 + (157*b7 + 113*b6 - 60*b5 - 458*b4 - 60*b1 + 113) * q^95 + (519*b6 - 109*b5 + 519*b4 - 65*b3 + 123*b2 - 65*b1 + 123) * q^96 + (-50*b7 - 421*b4 - 50*b3 - 594*b2 + 144*b1 - 471) * q^97 + (242*b7 - 1071*b6 + 242*b5 - 242*b4 + 191*b3 - 1071*b2 - 843) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 7 q^{2} - 3 q^{3} + 3 q^{4} - 7 q^{5} + 29 q^{6} + 35 q^{7} - 47 q^{8} + 31 q^{9}+O(q^{10})$$ 8 * q + 7 * q^2 - 3 * q^3 + 3 * q^4 - 7 * q^5 + 29 * q^6 + 35 * q^7 - 47 * q^8 + 31 * q^9 $$8 q + 7 q^{2} - 3 q^{3} + 3 q^{4} - 7 q^{5} + 29 q^{6} + 35 q^{7} - 47 q^{8} + 31 q^{9} - 40 q^{10} + 190 q^{12} + 65 q^{13} - 196 q^{14} - 121 q^{15} - 377 q^{16} + 31 q^{17} + 102 q^{18} - 148 q^{19} + 342 q^{20} - 334 q^{21} - 12 q^{23} + 447 q^{24} - 201 q^{25} - 140 q^{26} + 72 q^{27} + 42 q^{28} + 199 q^{29} + 114 q^{30} - 361 q^{31} - 324 q^{32} - 298 q^{34} - 237 q^{35} + 120 q^{36} + 81 q^{37} - 52 q^{38} - 365 q^{39} - 532 q^{40} + 31 q^{41} + 170 q^{42} + 650 q^{43} + 452 q^{45} - 1204 q^{46} + 857 q^{47} + 644 q^{48} + 1375 q^{49} + 147 q^{50} + 246 q^{51} + 590 q^{52} - 1493 q^{53} + 3100 q^{54} - 1560 q^{56} - 102 q^{57} + 1392 q^{58} + 676 q^{59} + 1068 q^{60} + 525 q^{61} - 2456 q^{62} + 68 q^{63} + 471 q^{64} - 1790 q^{65} + 86 q^{67} - 710 q^{68} - 42 q^{69} - 144 q^{70} + 1143 q^{71} - 919 q^{72} + 2155 q^{73} + 1476 q^{74} - 160 q^{75} + 242 q^{76} - 1340 q^{78} + 861 q^{79} - 1916 q^{80} - 26 q^{81} - 3497 q^{82} - 52 q^{83} + 84 q^{84} - 2383 q^{85} + 1061 q^{86} - 2310 q^{87} + 3782 q^{89} + 1682 q^{90} + 135 q^{91} - 2450 q^{92} - 2077 q^{93} - 702 q^{94} + 1317 q^{95} - 1252 q^{96} - 1344 q^{97} - 2740 q^{98}+O(q^{100})$$ 8 * q + 7 * q^2 - 3 * q^3 + 3 * q^4 - 7 * q^5 + 29 * q^6 + 35 * q^7 - 47 * q^8 + 31 * q^9 - 40 * q^10 + 190 * q^12 + 65 * q^13 - 196 * q^14 - 121 * q^15 - 377 * q^16 + 31 * q^17 + 102 * q^18 - 148 * q^19 + 342 * q^20 - 334 * q^21 - 12 * q^23 + 447 * q^24 - 201 * q^25 - 140 * q^26 + 72 * q^27 + 42 * q^28 + 199 * q^29 + 114 * q^30 - 361 * q^31 - 324 * q^32 - 298 * q^34 - 237 * q^35 + 120 * q^36 + 81 * q^37 - 52 * q^38 - 365 * q^39 - 532 * q^40 + 31 * q^41 + 170 * q^42 + 650 * q^43 + 452 * q^45 - 1204 * q^46 + 857 * q^47 + 644 * q^48 + 1375 * q^49 + 147 * q^50 + 246 * q^51 + 590 * q^52 - 1493 * q^53 + 3100 * q^54 - 1560 * q^56 - 102 * q^57 + 1392 * q^58 + 676 * q^59 + 1068 * q^60 + 525 * q^61 - 2456 * q^62 + 68 * q^63 + 471 * q^64 - 1790 * q^65 + 86 * q^67 - 710 * q^68 - 42 * q^69 - 144 * q^70 + 1143 * q^71 - 919 * q^72 + 2155 * q^73 + 1476 * q^74 - 160 * q^75 + 242 * q^76 - 1340 * q^78 + 861 * q^79 - 1916 * q^80 - 26 * q^81 - 3497 * q^82 - 52 * q^83 + 84 * q^84 - 2383 * q^85 + 1061 * q^86 - 2310 * q^87 + 3782 * q^89 + 1682 * q^90 + 135 * q^91 - 2450 * q^92 - 2077 * q^93 - 702 * q^94 + 1317 * q^95 - 1252 * q^96 - 1344 * q^97 - 2740 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} + 280\nu ) / 981$$ (v^6 + 280*v) / 981 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + 171 ) / 109$$ (v^5 + 171) / 109 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 1261\nu^{2} ) / 8829$$ (v^7 + 1261*v^2) / 8829 $$\beta_{5}$$ $$=$$ $$( \nu^{7} - 10\nu^{6} + 19\nu^{5} - 109\nu^{4} + 1090\nu^{3} - 810\nu^{2} - 729\nu - 6561 ) / 8829$$ (v^7 - 10*v^6 + 19*v^5 - 109*v^4 + 1090*v^3 - 810*v^2 - 729*v - 6561) / 8829 $$\beta_{6}$$ $$=$$ $$( -19\nu^{7} + 19\nu^{6} - 190\nu^{5} + 1090\nu^{4} - 2071\nu^{3} - 3249\nu^{2} - 15390\nu - 13851 ) / 79461$$ (-19*v^7 + 19*v^6 - 190*v^5 + 1090*v^4 - 2071*v^3 - 3249*v^2 - 15390*v - 13851) / 79461 $$\beta_{7}$$ $$=$$ $$( 10\nu^{7} + 3781\nu^{2} ) / 8829$$ (10*v^7 + 3781*v^2) / 8829
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} + 10\beta_{4}$$ -b7 + 10*b4 $$\nu^{3}$$ $$=$$ $$9\beta_{6} + 10\beta_{5} + 9\beta_{4} + 9\beta_{2} + 9$$ 9*b6 + 10*b5 + 9*b4 + 9*b2 + 9 $$\nu^{4}$$ $$=$$ $$19\beta_{7} + 90\beta_{6} + 19\beta_{5} - 19\beta_{4} + 19\beta_{3} + 19\beta_1$$ 19*b7 + 90*b6 + 19*b5 - 19*b4 + 19*b3 + 19*b1 $$\nu^{5}$$ $$=$$ $$109\beta_{3} - 171$$ 109*b3 - 171 $$\nu^{6}$$ $$=$$ $$981\beta_{2} - 280\beta_1$$ 981*b2 - 280*b1 $$\nu^{7}$$ $$=$$ $$1261\beta_{7} - 3781\beta_{4}$$ 1261*b7 - 3781*b4

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 2.86504 − 2.08157i −2.05602 + 1.49379i 0.785330 − 2.41700i −1.09435 + 3.36805i 0.785330 + 2.41700i −1.09435 − 3.36805i 2.86504 + 2.08157i −2.05602 − 1.49379i
−0.747004 + 0.542730i −0.476313 1.46594i −2.20868 + 6.79761i −7.05908 5.12872i 1.15142 + 0.836554i −0.239524 + 0.737179i −4.32202 13.3018i 19.9214 14.4737i 8.05666
3.2 4.17405 3.03263i 1.40336 + 4.31911i 5.75376 17.7083i 6.98613 + 5.07572i 18.9560 + 13.7723i −0.513765 + 1.58121i −16.9313 52.1091i 5.15818 3.74763i 44.5532
9.1 −0.903364 + 2.78027i −3.67405 + 2.66936i −0.441690 0.320907i 1.90871 + 5.87440i −4.10252 12.6263i 25.9914 + 18.8838i −17.6291 + 12.8083i −1.97025 + 6.06380i −18.0567
9.2 0.976313 3.00478i 1.24700 0.906001i −1.60339 1.16493i −5.33576 16.4218i −1.50487 4.63152i −7.73807 5.62204i 15.3824 11.1760i −7.60928 + 23.4190i −54.5532
27.1 −0.903364 2.78027i −3.67405 2.66936i −0.441690 + 0.320907i 1.90871 5.87440i −4.10252 + 12.6263i 25.9914 18.8838i −17.6291 12.8083i −1.97025 6.06380i −18.0567
27.2 0.976313 + 3.00478i 1.24700 + 0.906001i −1.60339 + 1.16493i −5.33576 + 16.4218i −1.50487 + 4.63152i −7.73807 + 5.62204i 15.3824 + 11.1760i −7.60928 23.4190i −54.5532
81.1 −0.747004 0.542730i −0.476313 + 1.46594i −2.20868 6.79761i −7.05908 + 5.12872i 1.15142 0.836554i −0.239524 0.737179i −4.32202 + 13.3018i 19.9214 + 14.4737i 8.05666
81.2 4.17405 + 3.03263i 1.40336 4.31911i 5.75376 + 17.7083i 6.98613 5.07572i 18.9560 13.7723i −0.513765 1.58121i −16.9313 + 52.1091i 5.15818 + 3.74763i 44.5532
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.2 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.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.c.i 8
11.b odd 2 1 11.4.c.a 8
11.c even 5 1 121.4.a.f 4
11.c even 5 2 121.4.c.b 8
11.c even 5 1 inner 121.4.c.i 8
11.d odd 10 1 11.4.c.a 8
11.d odd 10 1 121.4.a.g 4
11.d odd 10 2 121.4.c.h 8
33.d even 2 1 99.4.f.c 8
33.f even 10 1 99.4.f.c 8
33.f even 10 1 1089.4.a.y 4
33.h odd 10 1 1089.4.a.bh 4
44.c even 2 1 176.4.m.c 8
44.g even 10 1 176.4.m.c 8
44.g even 10 1 1936.4.a.bk 4
44.h odd 10 1 1936.4.a.bl 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.c.a 8 11.b odd 2 1
11.4.c.a 8 11.d odd 10 1
99.4.f.c 8 33.d even 2 1
99.4.f.c 8 33.f even 10 1
121.4.a.f 4 11.c even 5 1
121.4.a.g 4 11.d odd 10 1
121.4.c.b 8 11.c even 5 2
121.4.c.h 8 11.d odd 10 2
121.4.c.i 8 1.a even 1 1 trivial
121.4.c.i 8 11.c even 5 1 inner
176.4.m.c 8 44.c even 2 1
176.4.m.c 8 44.g even 10 1
1089.4.a.y 4 33.f even 10 1
1089.4.a.bh 4 33.h odd 10 1
1936.4.a.bk 4 44.g even 10 1
1936.4.a.bl 4 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 7T_{2}^{7} + 31T_{2}^{6} - 71T_{2}^{5} + 319T_{2}^{4} - 78T_{2}^{3} + 1664T_{2}^{2} + 2816T_{2} + 1936$$ acting on $$S_{4}^{\mathrm{new}}(121, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 7 T^{7} + 31 T^{6} + \cdots + 1936$$
$3$ $$T^{8} + 3 T^{7} + 16 T^{6} + \cdots + 2401$$
$5$ $$T^{8} + 7 T^{7} + 250 T^{6} + \cdots + 64577296$$
$7$ $$T^{8} - 35 T^{7} + 268 T^{6} + \cdots + 156816$$
$11$ $$T^{8}$$
$13$ $$T^{8} - 65 T^{7} + \cdots + 2905210000$$
$17$ $$T^{8} - 31 T^{7} + \cdots + 9608691844521$$
$19$ $$T^{8} + 148 T^{7} + \cdots + 90104309844241$$
$23$ $$(T^{4} + 6 T^{3} - 21180 T^{2} + \cdots + 49883584)^{2}$$
$29$ $$T^{8} - 199 T^{7} + \cdots + 97\!\cdots\!96$$
$31$ $$T^{8} + 361 T^{7} + \cdots + 11\!\cdots\!16$$
$37$ $$T^{8} - 81 T^{7} + \cdots + 26\!\cdots\!96$$
$41$ $$T^{8} - 31 T^{7} + \cdots + 27\!\cdots\!61$$
$43$ $$(T^{4} - 325 T^{3} - 77011 T^{2} + \cdots - 1288748736)^{2}$$
$47$ $$T^{8} - 857 T^{7} + \cdots + 70\!\cdots\!96$$
$53$ $$T^{8} + 1493 T^{7} + \cdots + 71\!\cdots\!76$$
$59$ $$T^{8} - 676 T^{7} + \cdots + 66\!\cdots\!61$$
$61$ $$T^{8} - 525 T^{7} + \cdots + 22\!\cdots\!36$$
$67$ $$(T^{4} - 43 T^{3} - 394721 T^{2} + \cdots - 8869996224)^{2}$$
$71$ $$T^{8} - 1143 T^{7} + \cdots + 55\!\cdots\!16$$
$73$ $$T^{8} - 2155 T^{7} + \cdots + 14\!\cdots\!81$$
$79$ $$T^{8} - 861 T^{7} + \cdots + 44\!\cdots\!96$$
$83$ $$T^{8} + 52 T^{7} + \cdots + 32\!\cdots\!81$$
$89$ $$(T^{4} - 1891 T^{3} + \cdots - 2046678844)^{2}$$
$97$ $$T^{8} + 1344 T^{7} + \cdots + 12\!\cdots\!01$$