# Properties

 Label 91.2.f.c Level $91$ Weight $2$ Character orbit 91.f Analytic conductor $0.727$ 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] = [91,2,Mod(22,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("91.22");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.f (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1$$ x^8 - x^7 + 7*x^6 + 38*x^4 - 16*x^3 + 15*x^2 + 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{5} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} + (\beta_{4} + 1) q^{7} + (\beta_{6} + 3 \beta_{3} + \beta_{2} - 1) q^{8} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} - 2) q^{9}+O(q^{10})$$ q + b1 * q^2 - b5 * q^3 + (-b7 - b4 + b3 + b2 + b1 - 1) * q^4 + (-b3 - 2) * q^5 + (b4 - b3 - b1 + 1) * q^6 + (b4 + 1) * q^7 + (b6 + 3*b3 + b2 - 1) * q^8 + (b7 + b6 - b5 - 2*b4 - b2 - 2) * q^9 $$q + \beta_1 q^{2} - \beta_{5} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{6} + (\beta_{4} + 1) q^{7} + (\beta_{6} + 3 \beta_{3} + \beta_{2} - 1) q^{8} + (\beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} - 2) q^{9} + ( - \beta_{7} - 3 \beta_{4} - \beta_1) q^{10} + \beta_{5} q^{11} + ( - 2 \beta_{6} - 2 \beta_{3} - \beta_{2} + 3) q^{12} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{13} - \beta_{3} q^{14} + (2 \beta_{5} + \beta_{4} - \beta_1) q^{15} + (\beta_{7} + \beta_{5} + 4 \beta_{4} - 4 \beta_1) q^{16} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{17} + ( - \beta_{6} - 2 \beta_{3} - 1) q^{18} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{19} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{20} - \beta_{6} q^{21} + ( - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{22} + (\beta_{7} + \beta_{5} + \beta_1) q^{23} + ( - 2 \beta_{7} - \beta_{5} + 4 \beta_1) q^{24} + (3 \beta_{3} - \beta_{2} + 2) q^{25} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - \beta_{2} - 3 \beta_1) q^{26} + (3 \beta_{6} + \beta_{3} - 7) q^{27} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{28} + ( - \beta_{7} - \beta_{5}) q^{29} + (\beta_{7} + \beta_{4} - \beta_{2} + 1) q^{30} + (\beta_{2} - 1) q^{31} + (2 \beta_{7} + \beta_{6} - \beta_{5} + 7 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 7) q^{32} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 5 \beta_{4} + \beta_{2} + 5) q^{33} + ( - \beta_{6} - 4 \beta_{3} - \beta_{2}) q^{34} + ( - 2 \beta_{4} - \beta_{3} - \beta_1 - 2) q^{35} + ( - 2 \beta_{5} - 9 \beta_{4}) q^{36} + (\beta_{5} - 2 \beta_{4} + \beta_1) q^{37} + (\beta_{6} + 4 \beta_{3} + 3 \beta_{2} - 5) q^{38} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{39} + ( - \beta_{6} + \beta_{2} - 4) q^{40} + (\beta_{7} - 6 \beta_{4} - 2 \beta_1) q^{41} + (\beta_{4} - \beta_1) q^{42} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{43} + (2 \beta_{6} + 2 \beta_{3} + \beta_{2} - 3) q^{44} + ( - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{45}+ \cdots + ( - 6 \beta_{6} - \beta_{3} + 7) q^{99}+O(q^{100})$$ q + b1 * q^2 - b5 * q^3 + (-b7 - b4 + b3 + b2 + b1 - 1) * q^4 + (-b3 - 2) * q^5 + (b4 - b3 - b1 + 1) * q^6 + (b4 + 1) * q^7 + (b6 + 3*b3 + b2 - 1) * q^8 + (b7 + b6 - b5 - 2*b4 - b2 - 2) * q^9 + (-b7 - 3*b4 - b1) * q^10 + b5 * q^11 + (-2*b6 - 2*b3 - b2 + 3) * q^12 + (b7 - b6 + b5 + b4 - b3 - b1 + 1) * q^13 - b3 * q^14 + (2*b5 + b4 - b1) * q^15 + (b7 + b5 + 4*b4 - 4*b1) * q^16 + (b7 - b6 + b5 + b4 - b3 - b2 - b1 + 1) * q^17 + (-b6 - 2*b3 - 1) * q^18 + (-b7 + 2*b6 - 2*b5 + 3*b3 + b2 + 3*b1) * q^19 + (b7 + b6 - b5 + b4 + 3*b3 - b2 + 3*b1 + 1) * q^20 - b6 * q^21 + (-b4 + b3 + b1 - 1) * q^22 + (b7 + b5 + b1) * q^23 + (-2*b7 - b5 + 4*b1) * q^24 + (3*b3 - b2 + 2) * q^25 + (-b6 + b5 - 2*b4 - 4*b3 - b2 - 3*b1) * q^26 + (3*b6 + b3 - 7) * q^27 + (-b7 - b4 + b1) * q^28 + (-b7 - b5) * q^29 + (b7 + b4 - b2 + 1) * q^30 + (b2 - 1) * q^31 + (2*b7 + b6 - b5 + 7*b4 - 4*b3 - 2*b2 - 4*b1 + 7) * q^32 + (-b7 - b6 + b5 + 5*b4 + b2 + 5) * q^33 + (-b6 - 4*b3 - b2) * q^34 + (-2*b4 - b3 - b1 - 2) * q^35 + (-2*b5 - 9*b4) * q^36 + (b5 - 2*b4 + b1) * q^37 + (b6 + 4*b3 + 3*b2 - 5) * q^38 + (-b7 + b6 - 3*b5 - 2*b4 + 2*b3 + 2*b2 + b1 + 2) * q^39 + (-b6 + b2 - 4) * q^40 + (b7 - 6*b4 - 2*b1) * q^41 + (b4 - b1) * q^42 + (-b4 - b3 - b1 - 1) * q^43 + (2*b6 + 2*b3 + b2 - 3) * q^44 + (-2*b7 - 3*b6 + 3*b5 + 3*b4 - 2*b3 + 2*b2 - 2*b1 + 3) * q^45 + (-b7 - b6 + b5 - 6*b4 - b3 + b2 - b1 - 6) * q^46 + (b6 + 3*b3 + 1) * q^47 + (-2*b7 - 2*b6 + 2*b5 - b4 + 5*b3 + 2*b2 + 5*b1 - 1) * q^48 + b4 * q^49 + (3*b7 - b5 + 11*b4 + 2*b1) * q^50 + (b6 + 2*b3 + 2*b2 + 2) * q^51 + (-b7 + b5 + b4 - b2 + 4*b1 + 8) * q^52 + (-4*b6 - 2*b3 - 2*b2 + 1) * q^53 + (b7 - 5*b1) * q^54 + (-2*b5 - b4 + b1) * q^55 + (-b7 + b6 - b5 - b4 + 3*b3 + b2 + 3*b1 - 1) * q^56 + (-b6 - 4*b3 - 3*b2 - 5) * q^57 + (b6 - b5 + 3*b4 + 2*b3 + 2*b1 + 3) * q^58 + (b7 + 2*b6 - 2*b5 + b4 - 2*b3 - b2 - 2*b1 + 1) * q^59 + (3*b6 - 2*b3 - 4) * q^60 + (-b7 + 2*b6 - 2*b5 - 2*b4 + 2*b3 + b2 + 2*b1 - 2) * q^61 + (b5 - 2*b4 - 4*b1) * q^62 + (b7 - b5 - 2*b4) * q^63 + (-10*b3 - 2*b2 + 1) * q^64 + (-b7 + 2*b6 - b5 - 2*b4 + b3 - b2 - 2*b1) * q^65 + (b6 - b3 + 1) * q^66 + (-b7 - 4*b5 - b4 + 6*b1) * q^67 + (-2*b7 + b5 - 7*b4 + 4*b1) * q^68 + (-2*b7 + 2*b6 - 2*b5 + 4*b4 + 2*b2 + 4) * q^69 + (b3 - b2 + 3) * q^70 + (2*b7 - 2*b6 + 2*b5 + 4*b4 - 2*b2 + 4) * q^71 + (-2*b6 + 2*b5 + 3*b3 + 3*b1) * q^72 + (-2*b6 - 2*b3 - 3*b2 - 2) * q^73 + (-b7 - 4*b4 + 4*b3 + b2 + 4*b1 - 4) * q^74 + (b7 + b5 - b4 + 2*b1) * q^75 + (2*b7 - b5 + 5*b4 - 11*b1) * q^76 + b6 * q^77 + (b7 + b6 + b5 + 3*b4 + 3*b3 + b2 - 2*b1 + 2) * q^78 + (-3*b6 - b3 + b2 - 6) * q^79 + (2*b7 - b5 + b4 - 2*b1) * q^80 + (7*b5 + 8*b4 + b1) * q^81 + (2*b7 - b6 + b5 + 4*b4 + b3 - 2*b2 + b1 + 4) * q^82 + (-4*b3 - b2 - 1) * q^83 + (b7 - 2*b6 + 2*b5 + 3*b4 - 2*b3 - b2 - 2*b1 + 3) * q^84 + (-b7 + b6 - b5 - 2*b4 - 2*b3 + b2 - 2*b1 - 2) * q^85 + (-b2 + 3) * q^86 + (2*b7 - 2*b6 + 2*b5 - 3*b4 - b3 - 2*b2 - b1 - 3) * q^87 + (2*b7 + b5 - 4*b1) * q^88 + (-b7 + 2*b5 - b1) * q^89 + (2*b6 + 4*b3 - 2*b2 + 7) * q^90 + (b5 + b4 + b2 - b1) * q^91 + (3*b6 + 7*b3 + b2 + 4) * q^92 + (-b7 - 2*b5 - 2*b4 + b1) * q^93 + (3*b7 + 8*b4 - b1) * q^94 + (-b7 - 3*b6 + 3*b5 - 5*b4 - 2*b3 + b2 - 2*b1 - 5) * q^95 + (6*b3 + b2 - 13) * q^96 + (b7 - 5*b6 + 5*b5 + b4 + 2*b3 - b2 + 2*b1 + 1) * q^97 + (-b3 - b1) * q^98 + (-6*b6 - b3 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} - q^{3} - 5 q^{4} - 14 q^{5} + 5 q^{6} + 4 q^{7} - 12 q^{8} - 7 q^{9}+O(q^{10})$$ 8 * q + q^2 - q^3 - 5 * q^4 - 14 * q^5 + 5 * q^6 + 4 * q^7 - 12 * q^8 - 7 * q^9 $$8 q + q^{2} - q^{3} - 5 q^{4} - 14 q^{5} + 5 q^{6} + 4 q^{7} - 12 q^{8} - 7 q^{9} + 11 q^{10} + q^{11} + 24 q^{12} + 4 q^{13} + 2 q^{14} - 3 q^{15} - 19 q^{16} + 4 q^{17} - 6 q^{18} - q^{19} + 2 q^{20} - 2 q^{21} - 5 q^{22} + 2 q^{23} + 3 q^{24} + 10 q^{25} + 12 q^{26} - 52 q^{27} + 5 q^{28} - q^{29} + 4 q^{30} - 8 q^{31} + 33 q^{32} + 19 q^{33} + 6 q^{34} - 7 q^{35} + 34 q^{36} + 10 q^{37} - 46 q^{38} + 20 q^{39} - 34 q^{40} + 22 q^{41} - 5 q^{42} - 3 q^{43} - 24 q^{44} + 11 q^{45} - 24 q^{46} + 4 q^{47} - 11 q^{48} - 4 q^{49} - 43 q^{50} + 14 q^{51} + 65 q^{52} + 4 q^{53} - 5 q^{54} + 3 q^{55} - 6 q^{56} - 34 q^{57} + 11 q^{58} + 8 q^{59} - 22 q^{60} - 8 q^{61} + 5 q^{62} + 7 q^{63} + 28 q^{64} + 7 q^{65} + 12 q^{66} + 6 q^{67} + 33 q^{68} + 18 q^{69} + 22 q^{70} + 14 q^{71} - 5 q^{72} - 16 q^{73} - 20 q^{74} + 7 q^{75} - 32 q^{76} + 2 q^{77} - q^{78} - 52 q^{79} - 7 q^{80} - 24 q^{81} + 14 q^{82} + 12 q^{84} - 5 q^{85} + 24 q^{86} - 13 q^{87} - 3 q^{88} + q^{89} + 52 q^{90} - 4 q^{91} + 24 q^{92} + 7 q^{93} - 33 q^{94} - 21 q^{95} - 116 q^{96} - 3 q^{97} + q^{98} + 46 q^{99}+O(q^{100})$$ 8 * q + q^2 - q^3 - 5 * q^4 - 14 * q^5 + 5 * q^6 + 4 * q^7 - 12 * q^8 - 7 * q^9 + 11 * q^10 + q^11 + 24 * q^12 + 4 * q^13 + 2 * q^14 - 3 * q^15 - 19 * q^16 + 4 * q^17 - 6 * q^18 - q^19 + 2 * q^20 - 2 * q^21 - 5 * q^22 + 2 * q^23 + 3 * q^24 + 10 * q^25 + 12 * q^26 - 52 * q^27 + 5 * q^28 - q^29 + 4 * q^30 - 8 * q^31 + 33 * q^32 + 19 * q^33 + 6 * q^34 - 7 * q^35 + 34 * q^36 + 10 * q^37 - 46 * q^38 + 20 * q^39 - 34 * q^40 + 22 * q^41 - 5 * q^42 - 3 * q^43 - 24 * q^44 + 11 * q^45 - 24 * q^46 + 4 * q^47 - 11 * q^48 - 4 * q^49 - 43 * q^50 + 14 * q^51 + 65 * q^52 + 4 * q^53 - 5 * q^54 + 3 * q^55 - 6 * q^56 - 34 * q^57 + 11 * q^58 + 8 * q^59 - 22 * q^60 - 8 * q^61 + 5 * q^62 + 7 * q^63 + 28 * q^64 + 7 * q^65 + 12 * q^66 + 6 * q^67 + 33 * q^68 + 18 * q^69 + 22 * q^70 + 14 * q^71 - 5 * q^72 - 16 * q^73 - 20 * q^74 + 7 * q^75 - 32 * q^76 + 2 * q^77 - q^78 - 52 * q^79 - 7 * q^80 - 24 * q^81 + 14 * q^82 + 12 * q^84 - 5 * q^85 + 24 * q^86 - 13 * q^87 - 3 * q^88 + q^89 + 52 * q^90 - 4 * q^91 + 24 * q^92 + 7 * q^93 - 33 * q^94 - 21 * q^95 - 116 * q^96 - 3 * q^97 + q^98 + 46 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -12\nu^{7} - 7\nu^{6} - 76\nu^{5} - 44\nu^{4} - 602\nu^{3} - 36\nu^{2} - 8\nu + 1249 ) / 458$$ (-12*v^7 - 7*v^6 - 76*v^5 - 44*v^4 - 602*v^3 - 36*v^2 - 8*v + 1249) / 458 $$\beta_{3}$$ $$=$$ $$( 57\nu^{7} - 24\nu^{6} + 361\nu^{5} + 209\nu^{4} + 2287\nu^{3} + 171\nu^{2} + 38\nu + 193 ) / 916$$ (57*v^7 - 24*v^6 + 361*v^5 + 209*v^4 + 2287*v^3 + 171*v^2 + 38*v + 193) / 916 $$\beta_{4}$$ $$=$$ $$( 193\nu^{7} - 250\nu^{6} + 1375\nu^{5} - 361\nu^{4} + 7125\nu^{3} - 5375\nu^{2} + 2724\nu - 375 ) / 916$$ (193*v^7 - 250*v^6 + 1375*v^5 - 361*v^4 + 7125*v^3 - 5375*v^2 + 2724*v - 375) / 916 $$\beta_{5}$$ $$=$$ $$( 174\nu^{7} - 242\nu^{6} + 1331\nu^{5} - 507\nu^{4} + 6897\nu^{3} - 5203\nu^{2} + 5383\nu - 363 ) / 458$$ (174*v^7 - 242*v^6 + 1331*v^5 - 507*v^4 + 6897*v^3 - 5203*v^2 + 5383*v - 363) / 458 $$\beta_{6}$$ $$=$$ $$( -375\nu^{7} + 182\nu^{6} - 2375\nu^{5} - 1375\nu^{4} - 13889\nu^{3} - 1125\nu^{2} - 250\nu - 2933 ) / 916$$ (-375*v^7 + 182*v^6 - 2375*v^5 - 1375*v^4 - 13889*v^3 - 1125*v^2 - 250*v - 2933) / 916 $$\beta_{7}$$ $$=$$ $$( -273\nu^{7} + 356\nu^{6} - 1958\nu^{5} + 602\nu^{4} - 10146\nu^{3} + 7654\nu^{2} - 3617\nu + 534 ) / 458$$ (-273*v^7 + 356*v^6 - 1958*v^5 + 602*v^4 - 10146*v^3 + 7654*v^2 - 3617*v + 534) / 458
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - 3\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3$$ -b7 - 3*b4 + b3 + b2 + b1 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{6} + 7\beta_{3} + \beta_{2} - 1$$ b6 + 7*b3 + b2 - 1 $$\nu^{4}$$ $$=$$ $$7\beta_{7} + \beta_{5} + 18\beta_{4} - 10\beta_1$$ 7*b7 + b5 + 18*b4 - 10*b1 $$\nu^{5}$$ $$=$$ $$10\beta_{7} - 7\beta_{6} + 7\beta_{5} + 15\beta_{4} - 48\beta_{3} - 10\beta_{2} - 48\beta _1 + 15$$ 10*b7 - 7*b6 + 7*b5 + 15*b4 - 48*b3 - 10*b2 - 48*b1 + 15 $$\nu^{6}$$ $$=$$ $$-10\beta_{6} - 86\beta_{3} - 48\beta_{2} + 117$$ -10*b6 - 86*b3 - 48*b2 + 117 $$\nu^{7}$$ $$=$$ $$-86\beta_{7} - 48\beta_{5} - 152\beta_{4} + 337\beta_1$$ -86*b7 - 48*b5 - 152*b4 + 337*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$-1 - \beta_{4}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 −1.11000 + 1.92258i −0.115680 + 0.200364i 0.355143 − 0.615126i 1.37054 − 2.37385i −1.11000 − 1.92258i −0.115680 − 0.200364i 0.355143 + 0.615126i 1.37054 + 2.37385i
−1.11000 + 1.92258i −0.274776 + 0.475925i −1.46422 2.53610i −4.22001 −0.610004 1.05656i 0.500000 + 0.866025i 2.06113 1.34900 + 2.33653i 4.68423 8.11332i
22.2 −0.115680 + 0.200364i 1.66113 2.87716i 0.973236 + 1.68569i −2.23136 0.384320 + 0.665661i 0.500000 + 0.866025i −0.913059 −4.01868 6.96056i 0.258125 0.447085i
22.3 0.355143 0.615126i −1.20394 + 2.08529i 0.747746 + 1.29513i −1.28971 0.855143 + 1.48115i 0.500000 + 0.866025i 2.48280 −1.39895 2.42305i −0.458033 + 0.793337i
22.4 1.37054 2.37385i −0.682410 + 1.18197i −2.75677 4.77486i 0.741082 1.87054 + 3.23987i 0.500000 + 0.866025i −9.63087 0.568634 + 0.984903i 1.01568 1.75921i
29.1 −1.11000 1.92258i −0.274776 0.475925i −1.46422 + 2.53610i −4.22001 −0.610004 + 1.05656i 0.500000 0.866025i 2.06113 1.34900 2.33653i 4.68423 + 8.11332i
29.2 −0.115680 0.200364i 1.66113 + 2.87716i 0.973236 1.68569i −2.23136 0.384320 0.665661i 0.500000 0.866025i −0.913059 −4.01868 + 6.96056i 0.258125 + 0.447085i
29.3 0.355143 + 0.615126i −1.20394 2.08529i 0.747746 1.29513i −1.28971 0.855143 1.48115i 0.500000 0.866025i 2.48280 −1.39895 + 2.42305i −0.458033 0.793337i
29.4 1.37054 + 2.37385i −0.682410 1.18197i −2.75677 + 4.77486i 0.741082 1.87054 3.23987i 0.500000 0.866025i −9.63087 0.568634 0.984903i 1.01568 + 1.75921i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 22.4 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.f.c 8
3.b odd 2 1 819.2.o.h 8
4.b odd 2 1 1456.2.s.q 8
7.b odd 2 1 637.2.f.i 8
7.c even 3 1 637.2.g.k 8
7.c even 3 1 637.2.h.h 8
7.d odd 6 1 637.2.g.j 8
7.d odd 6 1 637.2.h.i 8
13.c even 3 1 inner 91.2.f.c 8
13.c even 3 1 1183.2.a.k 4
13.e even 6 1 1183.2.a.l 4
13.f odd 12 2 1183.2.c.g 8
39.i odd 6 1 819.2.o.h 8
52.j odd 6 1 1456.2.s.q 8
91.g even 3 1 637.2.h.h 8
91.h even 3 1 637.2.g.k 8
91.m odd 6 1 637.2.h.i 8
91.n odd 6 1 637.2.f.i 8
91.n odd 6 1 8281.2.a.bp 4
91.t odd 6 1 8281.2.a.bt 4
91.v odd 6 1 637.2.g.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 1.a even 1 1 trivial
91.2.f.c 8 13.c even 3 1 inner
637.2.f.i 8 7.b odd 2 1
637.2.f.i 8 91.n odd 6 1
637.2.g.j 8 7.d odd 6 1
637.2.g.j 8 91.v odd 6 1
637.2.g.k 8 7.c even 3 1
637.2.g.k 8 91.h even 3 1
637.2.h.h 8 7.c even 3 1
637.2.h.h 8 91.g even 3 1
637.2.h.i 8 7.d odd 6 1
637.2.h.i 8 91.m odd 6 1
819.2.o.h 8 3.b odd 2 1
819.2.o.h 8 39.i odd 6 1
1183.2.a.k 4 13.c even 3 1
1183.2.a.l 4 13.e even 6 1
1183.2.c.g 8 13.f odd 12 2
1456.2.s.q 8 4.b odd 2 1
1456.2.s.q 8 52.j odd 6 1
8281.2.a.bp 4 91.n odd 6 1
8281.2.a.bt 4 91.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} + 38T_{2}^{4} - 16T_{2}^{3} + 15T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} + 7 T^{6} + 38 T^{4} + \cdots + 1$$
$3$ $$T^{8} + T^{7} + 10 T^{6} + 23 T^{5} + \cdots + 36$$
$5$ $$(T^{4} + 7 T^{3} + 12 T^{2} - T - 9)^{2}$$
$7$ $$(T^{2} - T + 1)^{4}$$
$11$ $$T^{8} - T^{7} + 10 T^{6} - 23 T^{5} + \cdots + 36$$
$13$ $$T^{8} - 4 T^{7} + 16 T^{6} + \cdots + 28561$$
$17$ $$T^{8} - 4 T^{7} + 28 T^{6} + \cdots + 2809$$
$19$ $$T^{8} + T^{7} + 56 T^{6} + \cdots + 250000$$
$23$ $$T^{8} - 2 T^{7} + 42 T^{6} + \cdots + 5184$$
$29$ $$T^{8} + T^{7} + 23 T^{6} - 64 T^{5} + \cdots + 25$$
$31$ $$(T^{4} + 4 T^{3} - 13 T^{2} - 26 T + 54)^{2}$$
$37$ $$T^{8} - 10 T^{7} + 83 T^{6} + \cdots + 256$$
$41$ $$T^{8} - 22 T^{7} + 335 T^{6} + \cdots + 318096$$
$43$ $$T^{8} + 3 T^{7} + 12 T^{6} + 7 T^{5} + \cdots + 4$$
$47$ $$(T^{4} - 2 T^{3} - 51 T^{2} - 12 T + 100)^{2}$$
$53$ $$(T^{4} - 2 T^{3} - 140 T^{2} + 890 T - 1389)^{2}$$
$59$ $$T^{8} - 8 T^{7} + 141 T^{6} + \cdots + 498436$$
$61$ $$T^{8} + 8 T^{7} + 79 T^{6} + \cdots + 10000$$
$67$ $$T^{8} - 6 T^{7} + 247 T^{6} + \cdots + 121220100$$
$71$ $$T^{8} - 14 T^{7} + 212 T^{6} + \cdots + 4129024$$
$73$ $$(T^{4} + 8 T^{3} - 119 T^{2} - 88 T + 1772)^{2}$$
$79$ $$(T^{4} + 26 T^{3} + 134 T^{2} - 1024 T - 7680)^{2}$$
$83$ $$(T^{4} - 97 T^{2} - 442 T - 426)^{2}$$
$89$ $$T^{8} - T^{7} + 72 T^{6} - 297 T^{5} + \cdots + 11664$$
$97$ $$T^{8} + 3 T^{7} + 314 T^{6} + \cdots + 246238864$$