# Properties

 Label 121.4.a.g Level $121$ Weight $4$ Character orbit 121.a Self dual yes Analytic conductor $7.139$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,4,Mod(1,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 121.a (trivial)

## 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: yes Analytic conductor: $$7.13923111069$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{5}, \sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 21x^{2} + 64$$ x^4 - 21*x^2 + 64 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (\beta_{2} + 2) q^{3} + (2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{4} + ( - 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 3) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 11) q^{6} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 11) q^{7} + (6 \beta_{3} - 3 \beta_{2} + 13 \beta_1 + 7) q^{8} + (3 \beta_{2} - 14) q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^2 + (b2 + 2) * q^3 + (2*b3 + b2 + b1 + 3) * q^4 + (-3*b3 + 2*b2 + 4*b1 - 3) * q^5 + (b3 + 2*b2 + b1 + 11) * q^6 + (-3*b3 + 5*b2 - 3*b1 + 11) * q^7 + (6*b3 - 3*b2 + 13*b1 + 7) * q^8 + (3*b2 - 14) * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (\beta_{2} + 2) q^{3} + (2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{4} + ( - 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 3) q^{5} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 11) q^{6} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 11) q^{7} + (6 \beta_{3} - 3 \beta_{2} + 13 \beta_1 + 7) q^{8} + (3 \beta_{2} - 14) q^{9} + ( - 6 \beta_{2} - 28 \beta_1 + 16) q^{10} + (5 \beta_{3} + 4 \beta_{2} + 19 \beta_1 + 15) q^{12} + ( - 7 \beta_{3} - 4 \beta_{2} + 14 \beta_1 - 1) q^{13} + ( - 4 \beta_{3} + 8 \beta_{2} - 24 \beta_1 + 50) q^{14} + ( - 2 \beta_{3} - \beta_{2} - 23 \beta_1 + 12) q^{15} + (6 \beta_{3} + 5 \beta_{2} + 69 \beta_1 - 25) q^{16} + ( - 2 \beta_{3} - 7 \beta_{2} - 16 \beta_1 + 63) q^{17} + (3 \beta_{3} - 14 \beta_{2} - 17 \beta_1 + 13) q^{18} + ( - 12 \beta_{3} + 6 \beta_{2} + 51 \beta_1 + 19) q^{19} + ( - 10 \beta_{3} - 38 \beta_1 - 42) q^{20} + ( - 9 \beta_{3} + 16 \beta_{2} - 30 \beta_1 + 67) q^{21} + ( - 10 \beta_{3} - 26 \beta_{2} - 16 \beta_1 - 4) q^{23} + (25 \beta_{3} + 4 \beta_{2} + 67 \beta_1 - 13) q^{24} + (19 \beta_{3} - 31 \beta_{2} - 51 \beta_1 + 2) q^{25} + ( - 4 \beta_{3} - 8 \beta_{2} - 46 \beta_1 - 30) q^{26} + ( - 38 \beta_{2} - 55) q^{27} + (6 \beta_{2} + 6 \beta_1 + 6) q^{28} + (31 \beta_{3} + \beta_{2} - 19 \beta_1 + 93) q^{29} + ( - 28 \beta_{3} + 10 \beta_{2} - 28 \beta_1 - 22) q^{30} + (41 \beta_{3} - 40 \beta_{2} + 48 \beta_1 - 45) q^{31} + (38 \beta_{3} + 5 \beta_{2} - 11 \beta_1 + 39) q^{32} + ( - 27 \beta_{3} + 61 \beta_{2} + 36 \beta_1 - 18) q^{34} + ( - 4 \beta_{3} + 3 \beta_{2} - 81 \beta_1 + 132) q^{35} + ( - 25 \beta_{3} - 8 \beta_{2} + 37 \beta_1 - 15) q^{36} + ( - 35 \beta_{3} + 17 \beta_{2} + 43 \beta_1 + 19) q^{37} + (33 \beta_{3} + 7 \beta_{2} - 44 \beta_1 + 112) q^{38} + ( - 5 \beta_{2} - 49 \beta_1 - 38) q^{39} + ( - 58 \beta_{3} - 4 \beta_{2} + 54 \beta_1 - 218) q^{40} + (62 \beta_{3} - 15 \beta_{2} - 22 \beta_1 + 26) q^{41} + ( - 32 \beta_{3} + 58 \beta_{2} - 60 \beta_1 + 172) q^{42} + ( - 31 \beta_{3} - 8 \beta_{2} + 199 \beta_1 - 177) q^{43} + (54 \beta_{3} - 43 \beta_{2} - 149 \beta_1 + 96) q^{45} + ( - 62 \beta_{3} - 14 \beta_{2} - 84 \beta_1 - 264) q^{46} + ( - 9 \beta_{3} + 37 \beta_{2} + 89 \beta_1 - 241) q^{47} + (81 \beta_{3} - 20 \beta_{2} + 123 \beta_1 - 5) q^{48} + ( - 69 \beta_{3} + 112 \beta_{2} - 216 \beta_1 + 120) q^{49} + ( - 44 \beta_{3} + 21 \beta_{2} + 153 \beta_1 - 309) q^{50} + ( - 20 \beta_{3} + 56 \beta_{2} - 34 \beta_1 + 63) q^{51} + ( - 6 \beta_{3} - 2 \beta_{2} - 216 \beta_1 - 144) q^{52} + ( - 41 \beta_{3} + 2 \beta_{2} + 238 \beta_1 + 95) q^{53} + ( - 38 \beta_{3} - 55 \beta_{2} - 17 \beta_1 - 397) q^{54} + (44 \beta_{3} - 58 \beta_{2} + 198 \beta_1 - 334) q^{56} + (27 \beta_{3} + 25 \beta_{2} - 57 \beta_1 + 92) q^{57} + (44 \beta_{3} + 124 \beta_{2} + 352 \beta_1 + 114) q^{58} + (32 \beta_{3} + 2 \beta_{2} + 137 \beta_1 - 31) q^{59} + ( - 58 \beta_{3} - 42 \beta_{2} - 128 \beta_1 - 84) q^{60} + (147 \beta_{3} - 42 \beta_{2} + 42 \beta_1 - 105) q^{61} + (90 \beta_{3} - 4 \beta_{2} + 412 \beta_1 - 316) q^{62} + (33 \beta_{3} - 52 \beta_{2} - 30 \beta_1 - 19) q^{63} + (22 \beta_{3} + 37 \beta_{2} - 187 \beta_1 + 311) q^{64} + ( - 13 \beta_{3} - 31 \beta_{2} + 169 \beta_1 + 127) q^{65} + ( - 63 \beta_{3} + 109 \beta_{2} - 246 \beta_1 + 204) q^{67} + (59 \beta_{3} + 11 \beta_{2} - 158 \beta_1 + 36) q^{68} + ( - 36 \beta_{3} - 30 \beta_{2} - 106 \beta_1 - 242) q^{69} + ( - 86 \beta_{3} + 128 \beta_{2} + 12 \beta_1 + 74) q^{70} + (77 \beta_{3} - 120 \beta_{2} - 330 \beta_1 + 243) q^{71} + ( - 45 \beta_{3} + 72 \beta_{2} - 59 \beta_1 - 179) q^{72} + ( - 44 \beta_{3} - 51 \beta_{2} - 436 \beta_1 + 136) q^{73} + ( - 10 \beta_{3} - 16 \beta_{2} - 270 \beta_1 + 180) q^{74} + ( - 13 \beta_{3} - 29 \beta_{2} + 120 \beta_1 - 275) q^{75} + (125 \beta_{3} + 97 \beta_{2} - 50 \beta_1 + 12) q^{76} + ( - 54 \beta_{3} - 38 \beta_{2} - 82 \beta_1 - 132) q^{78} + (13 \beta_{3} - 128 \beta_{2} + 68 \beta_1 + 143) q^{79} + (14 \beta_{3} - 276 \beta_{2} - 378 \beta_1 + 78) q^{80} + ( - 174 \beta_{2} - 74) q^{81} + (87 \beta_{3} + 88 \beta_{2} + 577 \beta_1 - 69) q^{82} + ( - 208 \beta_{3} + 14 \beta_{2} - 75 \beta_1 + 518) q^{83} + (6 \beta_{3} + 12 \beta_{2} + 6 \beta_1 + 66) q^{84} + ( - 197 \beta_{3} + 207 \beta_{2} + 503 \beta_1 - 279) q^{85} + (129 \beta_{3} - 208 \beta_{2} - 249 \beta_1 - 81) q^{86} + (43 \beta_{3} + 94 \beta_{2} + 260 \beta_1 + 195) q^{87} + (25 \beta_{3} + 32 \beta_{2} + 403 \beta_1 + 281) q^{89} + ( - 84 \beta_{3} + 150 \beta_{2} + 476 \beta_1 - 386) q^{90} + (8 \beta_{3} - 29 \beta_{2} + 15 \beta_1 - 44) q^{91} + ( - 142 \beta_{3} - 118 \beta_{2} - 764 \beta_1 - 504) q^{92} + (130 \beta_{3} - 85 \beta_{2} + 417 \beta_1 - 450) q^{93} + (108 \beta_{3} - 250 \beta_{2} - 270 \beta_1 + 172) q^{94} + ( - 60 \beta_{3} - 157 \beta_{2} - 53 \beta_1 + 414) q^{95} + (65 \beta_{3} + 44 \beta_{2} + 331 \beta_1 + 123) q^{96} + (50 \beta_{3} + 144 \beta_{2} + 421 \beta_1 - 594) q^{97} + ( - 242 \beta_{3} + 51 \beta_{2} - 829 \beta_1 + 843) q^{98}+O(q^{100})$$ q + (b2 + b1 + 1) * q^2 + (b2 + 2) * q^3 + (2*b3 + b2 + b1 + 3) * q^4 + (-3*b3 + 2*b2 + 4*b1 - 3) * q^5 + (b3 + 2*b2 + b1 + 11) * q^6 + (-3*b3 + 5*b2 - 3*b1 + 11) * q^7 + (6*b3 - 3*b2 + 13*b1 + 7) * q^8 + (3*b2 - 14) * q^9 + (-6*b2 - 28*b1 + 16) * q^10 + (5*b3 + 4*b2 + 19*b1 + 15) * q^12 + (-7*b3 - 4*b2 + 14*b1 - 1) * q^13 + (-4*b3 + 8*b2 - 24*b1 + 50) * q^14 + (-2*b3 - b2 - 23*b1 + 12) * q^15 + (6*b3 + 5*b2 + 69*b1 - 25) * q^16 + (-2*b3 - 7*b2 - 16*b1 + 63) * q^17 + (3*b3 - 14*b2 - 17*b1 + 13) * q^18 + (-12*b3 + 6*b2 + 51*b1 + 19) * q^19 + (-10*b3 - 38*b1 - 42) * q^20 + (-9*b3 + 16*b2 - 30*b1 + 67) * q^21 + (-10*b3 - 26*b2 - 16*b1 - 4) * q^23 + (25*b3 + 4*b2 + 67*b1 - 13) * q^24 + (19*b3 - 31*b2 - 51*b1 + 2) * q^25 + (-4*b3 - 8*b2 - 46*b1 - 30) * q^26 + (-38*b2 - 55) * q^27 + (6*b2 + 6*b1 + 6) * q^28 + (31*b3 + b2 - 19*b1 + 93) * q^29 + (-28*b3 + 10*b2 - 28*b1 - 22) * q^30 + (41*b3 - 40*b2 + 48*b1 - 45) * q^31 + (38*b3 + 5*b2 - 11*b1 + 39) * q^32 + (-27*b3 + 61*b2 + 36*b1 - 18) * q^34 + (-4*b3 + 3*b2 - 81*b1 + 132) * q^35 + (-25*b3 - 8*b2 + 37*b1 - 15) * q^36 + (-35*b3 + 17*b2 + 43*b1 + 19) * q^37 + (33*b3 + 7*b2 - 44*b1 + 112) * q^38 + (-5*b2 - 49*b1 - 38) * q^39 + (-58*b3 - 4*b2 + 54*b1 - 218) * q^40 + (62*b3 - 15*b2 - 22*b1 + 26) * q^41 + (-32*b3 + 58*b2 - 60*b1 + 172) * q^42 + (-31*b3 - 8*b2 + 199*b1 - 177) * q^43 + (54*b3 - 43*b2 - 149*b1 + 96) * q^45 + (-62*b3 - 14*b2 - 84*b1 - 264) * q^46 + (-9*b3 + 37*b2 + 89*b1 - 241) * q^47 + (81*b3 - 20*b2 + 123*b1 - 5) * q^48 + (-69*b3 + 112*b2 - 216*b1 + 120) * q^49 + (-44*b3 + 21*b2 + 153*b1 - 309) * q^50 + (-20*b3 + 56*b2 - 34*b1 + 63) * q^51 + (-6*b3 - 2*b2 - 216*b1 - 144) * q^52 + (-41*b3 + 2*b2 + 238*b1 + 95) * q^53 + (-38*b3 - 55*b2 - 17*b1 - 397) * q^54 + (44*b3 - 58*b2 + 198*b1 - 334) * q^56 + (27*b3 + 25*b2 - 57*b1 + 92) * q^57 + (44*b3 + 124*b2 + 352*b1 + 114) * q^58 + (32*b3 + 2*b2 + 137*b1 - 31) * q^59 + (-58*b3 - 42*b2 - 128*b1 - 84) * q^60 + (147*b3 - 42*b2 + 42*b1 - 105) * q^61 + (90*b3 - 4*b2 + 412*b1 - 316) * q^62 + (33*b3 - 52*b2 - 30*b1 - 19) * q^63 + (22*b3 + 37*b2 - 187*b1 + 311) * q^64 + (-13*b3 - 31*b2 + 169*b1 + 127) * q^65 + (-63*b3 + 109*b2 - 246*b1 + 204) * q^67 + (59*b3 + 11*b2 - 158*b1 + 36) * q^68 + (-36*b3 - 30*b2 - 106*b1 - 242) * q^69 + (-86*b3 + 128*b2 + 12*b1 + 74) * q^70 + (77*b3 - 120*b2 - 330*b1 + 243) * q^71 + (-45*b3 + 72*b2 - 59*b1 - 179) * q^72 + (-44*b3 - 51*b2 - 436*b1 + 136) * q^73 + (-10*b3 - 16*b2 - 270*b1 + 180) * q^74 + (-13*b3 - 29*b2 + 120*b1 - 275) * q^75 + (125*b3 + 97*b2 - 50*b1 + 12) * q^76 + (-54*b3 - 38*b2 - 82*b1 - 132) * q^78 + (13*b3 - 128*b2 + 68*b1 + 143) * q^79 + (14*b3 - 276*b2 - 378*b1 + 78) * q^80 + (-174*b2 - 74) * q^81 + (87*b3 + 88*b2 + 577*b1 - 69) * q^82 + (-208*b3 + 14*b2 - 75*b1 + 518) * q^83 + (6*b3 + 12*b2 + 6*b1 + 66) * q^84 + (-197*b3 + 207*b2 + 503*b1 - 279) * q^85 + (129*b3 - 208*b2 - 249*b1 - 81) * q^86 + (43*b3 + 94*b2 + 260*b1 + 195) * q^87 + (25*b3 + 32*b2 + 403*b1 + 281) * q^89 + (-84*b3 + 150*b2 + 476*b1 - 386) * q^90 + (8*b3 - 29*b2 + 15*b1 - 44) * q^91 + (-142*b3 - 118*b2 - 764*b1 - 504) * q^92 + (130*b3 - 85*b2 + 417*b1 - 450) * q^93 + (108*b3 - 250*b2 - 270*b1 + 172) * q^94 + (-60*b3 - 157*b2 - 53*b1 + 414) * q^95 + (65*b3 + 44*b2 + 331*b1 + 123) * q^96 + (50*b3 + 144*b2 + 421*b1 - 594) * q^97 + (-242*b3 + 51*b2 - 829*b1 + 843) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 6 q^{3} + 14 q^{4} - 11 q^{5} + 43 q^{6} + 25 q^{7} + 66 q^{8} - 62 q^{9}+O(q^{10})$$ 4 * q + 4 * q^2 + 6 * q^3 + 14 * q^4 - 11 * q^5 + 43 * q^6 + 25 * q^7 + 66 * q^8 - 62 * q^9 $$4 q + 4 q^{2} + 6 q^{3} + 14 q^{4} - 11 q^{5} + 43 q^{6} + 25 q^{7} + 66 q^{8} - 62 q^{9} + 20 q^{10} + 95 q^{12} + 25 q^{13} + 132 q^{14} + 2 q^{15} + 34 q^{16} + 232 q^{17} + 49 q^{18} + 154 q^{19} - 254 q^{20} + 167 q^{21} - 6 q^{23} + 99 q^{24} - 13 q^{25} - 200 q^{26} - 144 q^{27} + 24 q^{28} + 363 q^{29} - 192 q^{30} + 37 q^{31} + 162 q^{32} - 149 q^{34} + 356 q^{35} + 5 q^{36} + 93 q^{37} + 379 q^{38} - 240 q^{39} - 814 q^{40} + 152 q^{41} + 420 q^{42} - 325 q^{43} + 226 q^{45} - 1258 q^{46} - 869 q^{47} + 347 q^{48} - 245 q^{49} - 1016 q^{50} + 52 q^{51} - 1010 q^{52} + 811 q^{53} - 1550 q^{54} - 780 q^{56} + 231 q^{57} + 956 q^{58} + 178 q^{59} - 566 q^{60} - 105 q^{61} - 342 q^{62} + q^{63} + 818 q^{64} + 895 q^{65} + 43 q^{67} - 135 q^{68} - 1156 q^{69} - 22 q^{70} + 629 q^{71} - 1023 q^{72} - 270 q^{73} + 202 q^{74} - 815 q^{75} - 121 q^{76} - 670 q^{78} + 977 q^{79} + 122 q^{80} + 52 q^{81} + 789 q^{82} + 1686 q^{83} + 258 q^{84} - 721 q^{85} - 277 q^{86} + 1155 q^{87} + 1891 q^{89} - 976 q^{90} - 80 q^{91} - 3450 q^{92} - 666 q^{93} + 756 q^{94} + 1804 q^{95} + 1131 q^{96} - 1772 q^{97} + 1370 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 + 6 * q^3 + 14 * q^4 - 11 * q^5 + 43 * q^6 + 25 * q^7 + 66 * q^8 - 62 * q^9 + 20 * q^10 + 95 * q^12 + 25 * q^13 + 132 * q^14 + 2 * q^15 + 34 * q^16 + 232 * q^17 + 49 * q^18 + 154 * q^19 - 254 * q^20 + 167 * q^21 - 6 * q^23 + 99 * q^24 - 13 * q^25 - 200 * q^26 - 144 * q^27 + 24 * q^28 + 363 * q^29 - 192 * q^30 + 37 * q^31 + 162 * q^32 - 149 * q^34 + 356 * q^35 + 5 * q^36 + 93 * q^37 + 379 * q^38 - 240 * q^39 - 814 * q^40 + 152 * q^41 + 420 * q^42 - 325 * q^43 + 226 * q^45 - 1258 * q^46 - 869 * q^47 + 347 * q^48 - 245 * q^49 - 1016 * q^50 + 52 * q^51 - 1010 * q^52 + 811 * q^53 - 1550 * q^54 - 780 * q^56 + 231 * q^57 + 956 * q^58 + 178 * q^59 - 566 * q^60 - 105 * q^61 - 342 * q^62 + q^63 + 818 * q^64 + 895 * q^65 + 43 * q^67 - 135 * q^68 - 1156 * q^69 - 22 * q^70 + 629 * q^71 - 1023 * q^72 - 270 * q^73 + 202 * q^74 - 815 * q^75 - 121 * q^76 - 670 * q^78 + 977 * q^79 + 122 * q^80 + 52 * q^81 + 789 * q^82 + 1686 * q^83 + 258 * q^84 - 721 * q^85 - 277 * q^86 + 1155 * q^87 + 1891 * q^89 - 976 * q^90 - 80 * q^91 - 3450 * q^92 - 666 * q^93 + 756 * q^94 + 1804 * q^95 + 1131 * q^96 - 1772 * q^97 + 1370 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 21x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 13\nu + 8 ) / 16$$ (v^3 - 13*v + 8) / 16 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 29\nu - 8 ) / 16$$ (-v^3 + 29*v - 8) / 16 $$\beta_{3}$$ $$=$$ $$( \nu^{2} + \nu - 10 ) / 2$$ (v^2 + v - 10) / 2
 $$\nu$$ $$=$$ $$\beta_{2} + \beta_1$$ b2 + b1 $$\nu^{2}$$ $$=$$ $$2\beta_{3} - \beta_{2} - \beta _1 + 10$$ 2*b3 - b2 - b1 + 10 $$\nu^{3}$$ $$=$$ $$13\beta_{2} + 29\beta _1 - 8$$ 13*b2 + 29*b1 - 8

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.15942 −1.92335 1.92335 4.15942
−3.15942 −1.54138 1.98190 −17.2669 4.86986 −9.56478 19.0137 −24.6241 54.5532
1.2 −0.923347 −1.54138 −7.14743 8.72550 1.42323 0.775116 13.9863 −24.6241 −8.05666
1.3 2.92335 4.54138 0.545959 6.17671 13.2760 32.1271 −21.7907 −6.37586 18.0567
1.4 5.15942 4.54138 18.6196 −8.63533 23.4309 1.66258 54.7907 −6.37586 −44.5532
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.c.a 8 11.c even 5 2
99.4.f.c 8 33.h odd 10 2
121.4.a.f 4 11.b odd 2 1
121.4.a.g 4 1.a even 1 1 trivial
121.4.c.b 8 11.d odd 10 2
121.4.c.h 8 11.c even 5 2
121.4.c.i 8 11.d odd 10 2
176.4.m.c 8 44.h odd 10 2
1089.4.a.y 4 3.b odd 2 1
1089.4.a.bh 4 33.d even 2 1
1936.4.a.bk 4 4.b odd 2 1
1936.4.a.bl 4 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 4T_{2}^{3} - 15T_{2}^{2} + 38T_{2} + 44$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(121))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4 T^{3} - 15 T^{2} + 38 T + 44$$
$3$ $$(T^{2} - 3 T - 7)^{2}$$
$5$ $$T^{4} + 11 T^{3} - 183 T^{2} + \cdots + 8036$$
$7$ $$T^{4} - 25 T^{3} - 251 T^{2} + \cdots - 396$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 25 T^{3} - 2215 T^{2} + \cdots - 53900$$
$17$ $$T^{4} - 232 T^{3} + 18185 T^{2} + \cdots + 3099789$$
$19$ $$T^{4} - 154 T^{3} + 501 T^{2} + \cdots - 9492329$$
$23$ $$T^{4} + 6 T^{3} - 21180 T^{2} + \cdots + 49883584$$
$29$ $$T^{4} - 363 T^{3} + \cdots - 98528364$$
$31$ $$T^{4} - 37 T^{3} - 57125 T^{2} + \cdots - 33222196$$
$37$ $$T^{4} - 93 T^{3} + \cdots + 163244164$$
$41$ $$T^{4} - 152 T^{3} + \cdots + 1659084581$$
$43$ $$T^{4} + 325 T^{3} + \cdots - 1288748736$$
$47$ $$T^{4} + 869 T^{3} + \cdots + 837687536$$
$53$ $$T^{4} - 811 T^{3} + \cdots - 847714576$$
$59$ $$T^{4} - 178 T^{3} + \cdots + 258148219$$
$61$ $$T^{4} + 105 T^{3} + \cdots + 47885889744$$
$67$ $$T^{4} - 43 T^{3} + \cdots - 8869996224$$
$71$ $$T^{4} - 629 T^{3} + \cdots - 744521796$$
$73$ $$T^{4} + 270 T^{3} + \cdots + 38050128809$$
$79$ $$T^{4} - 977 T^{3} + \cdots - 210591964$$
$83$ $$T^{4} - 1686 T^{3} + \cdots - 181259356509$$
$89$ $$T^{4} - 1891 T^{3} + \cdots - 2046678844$$
$97$ $$T^{4} + 1772 T^{3} + \cdots - 357121332099$$