# Properties

 Label 95.2.k.a Level $95$ Weight $2$ Character orbit 95.k Analytic conductor $0.759$ Analytic rank $0$ Dimension $18$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,2,Mod(6,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 14]))

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

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

chi := DirichletCharacter("95.6");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 95.k (of order $$9$$, degree $$6$$, 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: $$0.758578819202$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} - 3 x^{17} + 15 x^{16} - 14 x^{15} + 72 x^{14} - 51 x^{13} + 231 x^{12} - 93 x^{11} + 438 x^{10} + \cdots + 1$$ x^18 - 3*x^17 + 15*x^16 - 14*x^15 + 72*x^14 - 51*x^13 + 231*x^12 - 93*x^11 + 438*x^10 - 156*x^9 + 582*x^8 - 138*x^7 + 437*x^6 - 132*x^5 + 198*x^4 - 16*x^3 + 15*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{15} - \beta_{4}) q^{2} + ( - \beta_{15} - \beta_{11} + \cdots + \beta_{5}) q^{3}+ \cdots + (\beta_{16} + \beta_{15} + \cdots + \beta_{2}) q^{9}+O(q^{10})$$ q + (b15 - b4) * q^2 + (-b15 - b11 + b8 + b6 + b5) * q^3 + (b14 - b12 + b10 + b9 + b7 + b5) * q^4 - b6 * q^5 + (b17 - 2*b13 - b10 + b9 - b8 - b7 + b3 + b1) * q^6 + (b13 - b11 + b10 + b6 - b4 - b1) * q^7 + (-b17 + b12 - b11 - b10 - b9 + 2*b8 + b7 + 2*b5 - b1) * q^8 + (b16 + b15 - b12 - b9 - 2*b5 + b4 + b2) * q^9 $$q + (\beta_{15} - \beta_{4}) q^{2} + ( - \beta_{15} - \beta_{11} + \cdots + \beta_{5}) q^{3}+ \cdots + (2 \beta_{17} + 4 \beta_{16} + \cdots - 6) q^{99}+O(q^{100})$$ q + (b15 - b4) * q^2 + (-b15 - b11 + b8 + b6 + b5) * q^3 + (b14 - b12 + b10 + b9 + b7 + b5) * q^4 - b6 * q^5 + (b17 - 2*b13 - b10 + b9 - b8 - b7 + b3 + b1) * q^6 + (b13 - b11 + b10 + b6 - b4 - b1) * q^7 + (-b17 + b12 - b11 - b10 - b9 + 2*b8 + b7 + 2*b5 - b1) * q^8 + (b16 + b15 - b12 - b9 - 2*b5 + b4 + b2) * q^9 + (b16 - b3) * q^10 + (-b17 - b16 - b15 - 2*b14 + b13 + 2*b12 + b11 - b10 + b7 + b6 - 2*b5 - b4 + b3 - 2*b1) * q^11 + (-b13 - 2*b10 - b9 - b8 - b6 + 2*b4 + b3 + 2*b2 + b1) * q^12 + (b12 + b10 - b9 - b8 - 2*b7 - 2*b6 - b5 + b4 - b3 - 2*b2 - b1 - 2) * q^13 + (-2*b16 + b13 + b11 + b9 + b7 + b5 - b4 - 2*b2 - 1) * q^14 + (-b17 - b16 + b13 + b8 + b7 + b5) * q^15 + (b17 - b15 + b14 - b12 + b11 + b10 + b9 - 2*b8 + b6 - b3 + b1) * q^16 + (-b16 - b15 + b13 + b11 - b10 + b8 + b7 + b4 + b3 + b1 + 1) * q^17 + (-3*b17 + b15 - 3*b14 + b13 + 3*b11 + b10 - b9 - b6 - 3*b5 - 2*b4 - 3*b3 + 3) * q^18 + (2*b17 - b16 + b15 + b14 + b13 - b9 - b8 - 2*b7 - 2*b6 + b5 + b4 + b2 + 2*b1) * q^19 + (-b12 + b10 - b9 - b7 - b5 + b4) * q^20 + (2*b17 + 2*b16 + 2*b15 + 2*b14 - 2*b13 - b11 - 2*b10 - b8 - b7 + 2*b4 + 2*b3 + 2*b1 - 2) * q^21 + (2*b17 + b16 - 2*b15 - b13 - 2*b12 + 2*b11 + 2*b10 + 2*b9 - 2*b8 - 2*b7 + 2*b6 - 2*b5 - 2*b3 - b2 + 2*b1) * q^22 + (b16 - b14 - b13 - b12 + b10 + b8 - 2*b7 - 2*b6 - b5 + b4 + 2*b2 + b1) * q^23 + (3*b16 - b11 - b9 - b5 + b4 + 3*b2) * q^24 - b13 * q^25 + (b16 - b15 + b13 + b10 + b8 + 2*b6 - b4 + b2 + 2*b1) * q^26 + (b17 - b16 + 2*b15 + b14 - 2*b13 + 2*b12 + 2*b10 + 3*b9 - b8 - b7 + b6 - b4 + b3 - 4*b2 - 2*b1 - 4) * q^27 + (b17 - 2*b15 + 2*b14 - 2*b11 + b8 + b6 + 4*b5 + b4 - b3 - 2*b2 - b1 - 1) * q^28 + (-2*b17 - 2*b15 - 2*b14 + 2*b12 + 2*b11 + 2*b9 - 2*b8 - 3*b5 - 2*b4 - b3 - 2*b2) * q^29 + (b17 + b14 + b12 + b10 + b9 + b5 - 2*b2 - b1 - 2) * q^30 + (b17 - 2*b16 + 2*b15 - b14 + b13 - b9 - 2*b8 - b7 - b6 - b5 + b3 + b2 - 3*b1) * q^31 + (-2*b17 + 2*b12 + 2*b11 - b10 - 2*b9 + b8 + 2*b7 - b6 - b5 + 2*b4 + b3 - b2 - 2*b1 - 1) * q^32 + (-b16 + 2*b14 + 3*b13 - 2*b11 - b9 + 2*b8 + 2*b7 + b6 + 2*b5 + b4 - b3 - b2 + b1 - 3) * q^33 + (3*b16 - b14 - 3*b13 + b12 - 2*b10 + b9 - b7 + b6 - b5 - 2*b4 - b2 - 2*b1) * q^34 + (b13 - b12 - b11 - b9 + b8 + b5 - b3 + b2 + 1) * q^35 + (-4*b16 + 4*b15 + b14 + 4*b13 - 4*b12 + b11 + 4*b10 + b8 + b7 - 2*b4 + 4) * q^36 + (2*b17 + 2*b16 - b15 + b14 - 3*b13 - b12 - 2*b11 - 2*b10 + 2*b9 + 3*b6 + 3*b5 - b4 + b3 + 4) * q^37 + (b16 + b13 - 2*b11 - 3*b10 - 2*b9 + b8 + 2*b7 - 2*b6 + b5 + b4 + 3*b3 + b2 - b1 - 1) * q^38 + (-2*b17 - 3*b16 - 2*b15 - 3*b14 + b13 + 2*b12 + 2*b11 - b6 - b5 + 2) * q^39 + (-2*b17 - b14 + b12 + b11 - b10 + b8 + b7 - b4) * q^40 + (-2*b17 - b16 - 2*b15 - b13 + b11 + 2*b9 - b8 + 2*b7 + 2*b6 - b5 - b2 - 2) * q^41 + (-2*b17 - b16 - 3*b15 - 4*b14 + b13 - 2*b10 - b9 - b8 + b7 - 2*b6 - 2*b5 + b4 + 5*b2 + b1 + 3) * q^42 + (3*b16 + 2*b14 + b12 + b10 + 2*b9 + 2*b8 + 3*b7 + b6 + 4*b5 - 2*b4 - b3 + 3*b2) * q^43 + (4*b15 - 4*b13 - b12 - 4*b11 - 2*b10 - 3*b9 + 2*b8 - b6 + 2*b5 - b4 + 2*b3 - b2 - 3*b1 + 3) * q^44 + (b16 - b15 - 2*b11 - b10 - b9 + b6 + b4 + b3 + b2) * q^45 + (-b17 + 2*b16 + 3*b15 - 3*b13 - b12 - b11 - 2*b10 - 2*b9 - b7 - 2*b6 + 2*b2 + b1 + 2) * q^46 + (b16 + 3*b15 + b14 - b12 - b11 - b9 - 3*b6 + b5 - 2*b4 - 2*b3 + 3*b2 + 3*b1 + 3) * q^47 + (-b17 - 3*b16 - b14 + b12 + b11 + b9 - b8 - 2*b6 - b5 - b4 + 2) * q^48 + (3*b17 - b16 + b15 + 3*b14 - b13 - 2*b12 - b9 - b8 - b7 + b6 + 2*b5 + b4 - b3 + b2 + 2*b1 + 1) * q^49 + (b2 + b1) * q^50 + (-2*b15 + 2*b13 - b12 - 2*b11 + 2*b10 - b9 + 2*b8 + 2*b7 + 2*b5 - b4 - 2*b3 - b1 - 2) * q^51 + (-2*b16 + b14 + 3*b13 - 2*b12 - 2*b11 - 2*b10 - b9 + b8 - b7 - 2*b6 + b4 + 3*b3 - 2*b2 - b1 - 3) * q^52 + (b17 - b14 - 4*b12 + b10 + 3*b9 - 2*b8 - b7 + b6 - 2*b5 + b4 - b2 + b1) * q^53 + (b17 - 5*b16 - 6*b15 + b14 + b13 + 4*b12 - 4*b11 + 2*b9 + 3*b8 + 6*b6 + 5*b5 + 4*b3 + b2 - 4) * q^54 + (-b16 - b14 + b13 + b12 - b11 - b8 - b7 - 2*b4 - b3 - b1 + 1) * q^55 + (2*b17 + 3*b16 + 2*b15 + 2*b14 - b13 + 2*b12 - 2*b11 - b10 + b9 - b7 + b6 + b5 + b4 + 2*b3 - 1) * q^56 + (-2*b17 - 3*b16 + b15 - 2*b14 + b13 - b12 + b11 + 2*b10 - 3*b6 - b5 - 4*b4 - 2*b3 + 2*b2 + 6) * q^57 + (b17 + 3*b14 - 4*b12 - b11 + b10 - b9 + b7 + 2*b4 + b3) * q^58 + (-b17 + b15 - 2*b14 + 2*b13 + b11 + b8 + b7 + 3*b4 - 2*b2) * q^59 + (b17 - 2*b15 - b13 + 2*b12 - b10 + b9 - b7 + 2*b6 + 2*b3 - b2 - b1 - 1) * q^60 + (-2*b17 + 3*b16 - b15 + b14 - 3*b13 - 2*b10 + 2*b9 + b8 + 4*b7 + 3*b5 - 2*b4 + b2 - 2*b1 + 1) * q^61 + (-b14 + 3*b13 + 3*b12 + b11 + 3*b10 - b9 - b8 - b7 + b6 - b5 + b4 - 4*b3 + b1 - 3) * q^62 + (5*b15 - 4*b13 - 2*b12 + 6*b11 + 3*b10 - 3*b8 - 4*b6 - 3*b5 - 2*b4 - 3*b3 - 4*b2 + 1) * q^63 + (-b17 + 3*b16 - 3*b15 + b14 + b13 - b11 + 2*b10 + b9 + 3*b8 + b7 + 4*b6 + b5 - 2*b4 - b3 - 3*b2) * q^64 + (b17 + 2*b15 - 2*b13 - b12 + b11 - b10 + b7 - b4 + b3 + b1) * q^65 + (-5*b15 - 3*b14 + 2*b12 + 3*b11 + 2*b9 - 2*b5 - 5*b2 - 2*b1) * q^66 + (2*b17 + 7*b16 + b15 + 2*b14 - 2*b11 + 2*b8 + 3*b5 + b4 + 3*b3 + b2 - b1) * q^67 + (b17 - b16 + 2*b15 - 2*b13 - 2*b12 + b11 + 4*b10 + 4*b9 - 3*b8 - 2*b7 + b6 - 3*b5 + 4*b2 + 2*b1 + 4) * q^68 + (-b17 + 2*b16 - 2*b15 + b14 + 2*b13 + b11 - b10 - b9 + 3*b8 + b7 + 4*b6 + b5 + b4 + b3 - b2 - 2*b1) * q^69 + (2*b15 + b11 + b10 - b8 - b7 - b6 - b5 - b3 - b2 + 1) * q^70 + (-3*b16 + 2*b14 + 4*b13 + 3*b12 - b11 + 3*b10 - b9 + 2*b8 - 2*b7 - 2*b6 + 3*b5 + b4 - 4*b3 - 3*b2 + b1 - 4) * q^71 + (b17 + 7*b15 + 2*b14 + 3*b12 + b9 + b5 - 4*b4 - 7*b2 - 4*b1 - 7) * q^72 + (-2*b17 - b16 - 4*b15 - 2*b14 + b13 + 6*b12 + 2*b11 - 4*b10 - b9 + 4*b6 - 4*b5 + 6*b3 + b2 - 4*b1) * q^73 + (b17 - 2*b16 + 3*b15 - 3*b13 + b12 - 3*b11 - 2*b10 - 3*b8 - 3*b7 + b4 + b3 + 5*b2 + b1 + 2) * q^74 + (-b17 - b14 + b11 - b5 + 1) * q^75 + (-b17 + 3*b16 - b15 - 2*b14 + b12 + 3*b11 + b10 + 4*b9 - b8 - b7 - 3*b6 - 4*b5 - 5*b3 + 2*b2 + 3*b1 - 1) * q^76 + (b17 + 3*b16 - 2*b15 + 2*b14 - 5*b13 - 3*b12 - b11 + b10 - b9 + 5*b6 - b4 - 2*b3 - 3) * q^77 + (4*b17 + 4*b16 + 2*b15 + 3*b14 - 3*b13 - 2*b12 - b11 - b8 - b7 + 2*b3 - b2 + 2*b1 - 4) * q^78 + (3*b17 - b16 + b14 - 3*b13 - 3*b12 + b11 + 2*b10 - b9 - 2*b8 - 2*b7 - 3*b3 - 3*b2 + 2*b1 - 4) * q^79 + (2*b17 - b16 + b14 + b13 - b12 + b10 - b8 - 2*b7 - b5 + b4 + b1) * q^80 + (3*b16 - 2*b14 - 3*b13 - 4*b12 + b11 - 4*b10 - 6*b9 - 2*b8 - 4*b7 - 5*b6 - 3*b5 + 6*b4 + 3*b2 + 4*b1 + 3) * q^81 + (-b17 - 2*b15 - b13 + 3*b12 - 2*b9 + b8 + b7 - b6 + 3*b4 - b2 - 2*b1 - 3) * q^82 + (2*b17 + b16 - b15 - 2*b14 + 3*b11 - 2*b10 - b9 - 7*b8 - 2*b7 + b6 - 2*b5 + 2*b4 + b3 - 2*b2 + b1) * q^83 + (-2*b17 + 5*b16 - 4*b14 + 2*b11 - 2*b9 - 2*b8 - 5*b6 - 6*b5 + 2*b4 - 2*b3 + 4*b2 + 4) * q^84 + (-b17 - b16 + b12 + b9 - b8 - b6 - b5 + b3 - b1 + 1) * q^85 + (3*b17 - 2*b16 - 4*b15 + 3*b14 - 2*b12 - 3*b11 - 2*b9 + 3*b8 - b6 + 5*b5 + 5*b4 + 4*b3 - 4*b2 - 3*b1 + 1) * q^86 + (-3*b17 - 4*b16 - 3*b15 - b14 + 3*b13 - b12 - 2*b11 + 4*b8 + 2*b7 + 4*b6 + 3*b5 + b1) * q^87 + (b17 - 2*b16 + 2*b15 - b14 + 6*b13 + b11 + 3*b10 + 5*b9 - 2*b8 - b7 + 4*b6 - b5 - 3*b4 - 5*b3 + 2*b2 + 6*b1) * q^88 + (-3*b17 - 4*b15 + 3*b13 - b12 + 2*b11 + b10 + 2*b9 + b8 + b7 + b6 - 2*b5 - b4 - b3 + b2 + 2*b1 - 3) * q^89 + (b16 - 3*b14 - b13 - b12 + 3*b11 - b10 - 3*b8 - 3*b6 - 3*b5 - 2*b3 + b2 + 3*b1 + 1) * q^90 + (-3*b17 - 5*b16 - 2*b15 - 3*b14 + 5*b13 + 2*b12 - 3*b9 + 4*b8 - b7 - b6 + b4 + 3*b2 + b1 + 2) * q^91 + (b17 - 7*b16 + 3*b15 - b14 + 6*b13 - 2*b12 + 4*b11 + 3*b10 - 2*b9 - 3*b8 - 2*b7 - 3*b6 - 5*b5 - 2*b3 + 6*b2 + 3*b1 - 1) * q^92 + (-2*b17 - 5*b16 + 3*b13 + 3*b12 - 3*b10 + 2*b2 + 5) * q^93 + (-b17 - 3*b16 + 4*b15 + 2*b14 + 7*b13 - 5*b12 + b11 + 2*b10 - 2*b9 + 5*b7 - 7*b6 + b5 + b4 - b3 + 2) * q^94 + (b17 + b16 - b15 + 2*b14 - 2*b13 + b11 - b10 + 2*b9 + b7 + b6 + b5 + b3 + 1) * q^95 + (b17 - 5*b16 - 4*b15 + b13 + 3*b12 - b11 + b10 - b9 + b7 - b6 + 3*b5 - b3 - 4) * q^96 + (-3*b17 + b15 - b14 + 3*b13 - 3*b12 + 3*b10 + 2*b4 - 3*b2) * q^97 + (-3*b17 - 2*b16 + b15 - b14 + 2*b13 + 4*b12 - 2*b11 - 3*b10 - 3*b9 + 3*b8 + 2*b7 - b6 + b5 + 4*b3 + 2*b2 - 3*b1) * q^98 + (2*b17 + 4*b16 + 6*b15 + 3*b14 - 4*b13 + 2*b12 - b10 - 2*b9 - b7 - 7*b6 + b5 + b2 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 6 q^{6} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 18 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 - 6 * q^6 - 6 * q^8 + 3 * q^9 $$18 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 6 q^{6} - 6 q^{8} + 3 q^{9} - 3 q^{10} - 18 q^{12} - 3 q^{13} - 12 q^{14} - 3 q^{15} - 3 q^{16} + 24 q^{17} + 48 q^{18} + 12 q^{20} - 21 q^{21} + 9 q^{22} - 9 q^{23} - 15 q^{24} + 3 q^{26} - 24 q^{27} - 12 q^{28} + 15 q^{29} - 12 q^{30} - 18 q^{31} + 15 q^{32} - 33 q^{33} - 12 q^{34} + 75 q^{36} + 36 q^{37} - 33 q^{38} + 36 q^{39} - 6 q^{40} - 30 q^{41} - 9 q^{42} - 36 q^{43} + 42 q^{44} - 6 q^{45} + 9 q^{46} + 21 q^{47} + 33 q^{48} + 9 q^{49} - 6 q^{50} - 45 q^{51} - 39 q^{52} - 12 q^{53} - 66 q^{54} + 3 q^{55} + 72 q^{57} + 12 q^{58} + 18 q^{59} - 3 q^{60} - 30 q^{61} - 24 q^{62} + 54 q^{63} + 36 q^{64} - 9 q^{65} + 39 q^{66} + 51 q^{68} + 15 q^{69} + 33 q^{70} - 12 q^{71} - 66 q^{72} + 24 q^{73} - 15 q^{74} + 18 q^{75} - 33 q^{76} - 60 q^{77} - 48 q^{78} - 51 q^{79} + 15 q^{80} + 27 q^{81} - 15 q^{82} + 48 q^{84} + 24 q^{85} + 63 q^{86} - 15 q^{87} - 27 q^{88} - 54 q^{89} - 9 q^{90} + 30 q^{91} - 42 q^{92} + 72 q^{93} + 30 q^{94} + 15 q^{95} - 66 q^{96} + 27 q^{97} - 3 q^{98} - 93 q^{99}+O(q^{100})$$ 18 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 - 6 * q^6 - 6 * q^8 + 3 * q^9 - 3 * q^10 - 18 * q^12 - 3 * q^13 - 12 * q^14 - 3 * q^15 - 3 * q^16 + 24 * q^17 + 48 * q^18 + 12 * q^20 - 21 * q^21 + 9 * q^22 - 9 * q^23 - 15 * q^24 + 3 * q^26 - 24 * q^27 - 12 * q^28 + 15 * q^29 - 12 * q^30 - 18 * q^31 + 15 * q^32 - 33 * q^33 - 12 * q^34 + 75 * q^36 + 36 * q^37 - 33 * q^38 + 36 * q^39 - 6 * q^40 - 30 * q^41 - 9 * q^42 - 36 * q^43 + 42 * q^44 - 6 * q^45 + 9 * q^46 + 21 * q^47 + 33 * q^48 + 9 * q^49 - 6 * q^50 - 45 * q^51 - 39 * q^52 - 12 * q^53 - 66 * q^54 + 3 * q^55 + 72 * q^57 + 12 * q^58 + 18 * q^59 - 3 * q^60 - 30 * q^61 - 24 * q^62 + 54 * q^63 + 36 * q^64 - 9 * q^65 + 39 * q^66 + 51 * q^68 + 15 * q^69 + 33 * q^70 - 12 * q^71 - 66 * q^72 + 24 * q^73 - 15 * q^74 + 18 * q^75 - 33 * q^76 - 60 * q^77 - 48 * q^78 - 51 * q^79 + 15 * q^80 + 27 * q^81 - 15 * q^82 + 48 * q^84 + 24 * q^85 + 63 * q^86 - 15 * q^87 - 27 * q^88 - 54 * q^89 - 9 * q^90 + 30 * q^91 - 42 * q^92 + 72 * q^93 + 30 * q^94 + 15 * q^95 - 66 * q^96 + 27 * q^97 - 3 * q^98 - 93 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 3 x^{17} + 15 x^{16} - 14 x^{15} + 72 x^{14} - 51 x^{13} + 231 x^{12} - 93 x^{11} + 438 x^{10} + \cdots + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 2362215094 \nu^{17} - 176902925324 \nu^{16} + 523927730528 \nu^{15} + \cdots - 2601214274036 ) / 2269131592089$$ (-2362215094*v^17 - 176902925324*v^16 + 523927730528*v^15 - 2732196133634*v^14 + 2458535955743*v^13 - 12964718137852*v^12 + 9051939676612*v^11 - 41495881520992*v^10 + 16563882463922*v^9 - 77292885590832*v^8 + 27960663596715*v^7 - 101945748763594*v^6 + 24300805314434*v^5 - 73622429378663*v^4 + 22913565282806*v^3 - 34340446173148*v^2 + 1348014480151*v - 2601214274036) / 2269131592089 $$\beta_{3}$$ $$=$$ $$( - 17485732004 \nu^{17} - 88297854546 \nu^{16} + 51093388053 \nu^{15} - 1446829381050 \nu^{14} + \cdots + 1249494973861 ) / 6807394776267$$ (-17485732004*v^17 - 88297854546*v^16 + 51093388053*v^15 - 1446829381050*v^14 - 1289873408104*v^13 - 6148599460869*v^12 - 6799863255329*v^11 - 19595360087749*v^10 - 25750669196294*v^9 - 32654560143716*v^8 - 45248146421483*v^7 - 41806244150830*v^6 - 53683022385418*v^5 - 21178028146029*v^4 - 24543770185394*v^3 - 6310322562086*v^2 - 4173984429332*v + 1249494973861) / 6807394776267 $$\beta_{4}$$ $$=$$ $$( 31170269891 \nu^{17} + 67213359096 \nu^{16} + 77458832787 \nu^{15} + 1646328025947 \nu^{14} + \cdots + 2036474776337 ) / 6807394776267$$ (31170269891*v^17 + 67213359096*v^16 + 77458832787*v^15 + 1646328025947*v^14 + 1576807416328*v^13 + 7658100229362*v^12 + 7321752947336*v^11 + 23929992121897*v^10 + 25358952088733*v^9 + 39943209775688*v^8 + 44417859093215*v^7 + 44701878155230*v^6 + 51736757215426*v^5 + 24440209046322*v^4 + 25432415690738*v^3 + 6559174797101*v^2 + 4174852913747*v + 2036474776337) / 6807394776267 $$\beta_{5}$$ $$=$$ $$( - 150661038729 \nu^{17} + 22583410048 \nu^{16} - 1207986331145 \nu^{15} + \cdots + 9924998536452 ) / 6807394776267$$ (-150661038729*v^17 + 22583410048*v^16 - 1207986331145*v^15 - 3440777225226*v^14 - 9048902407408*v^13 - 16935494273702*v^12 - 33578770190668*v^11 - 61173284253370*v^10 - 91399807598455*v^9 - 107347453764758*v^8 - 139987308514533*v^7 - 140150638887798*v^6 - 150604628667508*v^5 - 71368680699863*v^4 - 65671130584969*v^3 - 18574098462028*v^2 - 25239051620405*v + 9924998536452) / 6807394776267 $$\beta_{6}$$ $$=$$ $$( 271722740134 \nu^{17} - 484705722592 \nu^{16} + 3183507161538 \nu^{15} + \cdots - 2451247768033 ) / 6807394776267$$ (271722740134*v^17 - 484705722592*v^16 + 3183507161538*v^15 + 840321900394*v^14 + 16364383205847*v^13 + 8550245559592*v^12 + 52145226330891*v^11 + 44506175702300*v^10 + 108555610980184*v^9 + 86539068699317*v^8 + 143914939792586*v^7 + 117052142258774*v^6 + 128192309299746*v^5 + 63296274710146*v^4 + 52866415784337*v^3 + 15495186910162*v^2 + 23094455322128*v - 2451247768033) / 6807394776267 $$\beta_{7}$$ $$=$$ $$( 14393251623 \nu^{17} - 18779546977 \nu^{16} + 163974948404 \nu^{15} + 84439865154 \nu^{14} + \cdots - 139873473393 ) / 358283935593$$ (14393251623*v^17 - 18779546977*v^16 + 163974948404*v^15 + 84439865154*v^14 + 1054848843451*v^13 + 517821809207*v^12 + 3667695351382*v^11 + 2401081552594*v^10 + 8811394522903*v^9 + 4204775784857*v^8 + 12178339879671*v^7 + 5750184114669*v^6 + 11453985494683*v^5 + 2591457050228*v^4 + 3507803893486*v^3 + 907619400199*v^2 + 1344049384007*v - 139873473393) / 358283935593 $$\beta_{8}$$ $$=$$ $$( - 99295558576 \nu^{17} + 144550753822 \nu^{16} - 988488698886 \nu^{15} + \cdots - 2743148781339 ) / 2269131592089$$ (-99295558576*v^17 + 144550753822*v^16 - 988488698886*v^15 - 1021310336031*v^14 - 4366731414611*v^13 - 6588252818956*v^12 - 11452733474463*v^11 - 28551737645641*v^10 - 17195400090500*v^9 - 56855271575000*v^8 - 8530715161662*v^7 - 84690957416629*v^6 + 8235247858388*v^5 - 64199831043919*v^4 + 22920186337367*v^3 - 36148782515349*v^2 + 1289655835611*v - 2743148781339) / 2269131592089 $$\beta_{9}$$ $$=$$ $$( - 330462497810 \nu^{17} + 892333940472 \nu^{16} - 4644440262270 \nu^{15} + \cdots + 271722740134 ) / 6807394776267$$ (-330462497810*v^17 + 892333940472*v^16 - 4644440262270*v^15 + 3199654083801*v^14 - 22408105306426*v^13 + 10622726640063*v^12 - 69776390534762*v^11 + 10458949198508*v^10 - 128927816160221*v^9 + 14227694965402*v^8 - 154549880397266*v^7 - 9449471861188*v^6 - 99163676407834*v^5 + 934686762195*v^4 - 19842750752306*v^3 - 19018614220118*v^2 + 2451247768033*v + 271722740134) / 6807394776267 $$\beta_{10}$$ $$=$$ $$( 335323421585 \nu^{17} - 1203461029533 \nu^{16} + 5639724390405 \nu^{15} + \cdots - 759862203949 ) / 6807394776267$$ (335323421585*v^17 - 1203461029533*v^16 + 5639724390405*v^15 - 7682766129021*v^14 + 26906316005605*v^13 - 30567017722290*v^12 + 85618508403902*v^11 - 73041253362959*v^10 + 156613392824897*v^9 - 126748060068652*v^8 + 199076210749085*v^7 - 141640168905788*v^6 + 135207369075700*v^5 - 109625290547904*v^4 + 54155545522772*v^3 - 30581574139987*v^2 - 422383778863*v - 759862203949) / 6807394776267 $$\beta_{11}$$ $$=$$ $$( 169369527342 \nu^{17} - 512091105045 \nu^{16} + 2467349014493 \nu^{15} - 2179007028305 \nu^{14} + \cdots + 671726483761 ) / 2269131592089$$ (169369527342*v^17 - 512091105045*v^16 + 2467349014493*v^15 - 2179007028305*v^14 + 10956780516730*v^13 - 7678713505092*v^12 + 32807739705559*v^11 - 11980132322574*v^10 + 53548739509621*v^9 - 19086289356322*v^8 + 58786967689962*v^7 - 10254441293754*v^6 + 22431948273852*v^5 - 10070536517332*v^4 - 3857061703044*v^3 + 8766054588512*v^2 - 15266507142983*v + 671726483761) / 2269131592089 $$\beta_{12}$$ $$=$$ $$( - 183989570606 \nu^{17} + 559360956938 \nu^{16} - 2765267144950 \nu^{15} + \cdots + 2362215094 ) / 2269131592089$$ (-183989570606*v^17 + 559360956938*v^16 - 2765267144950*v^15 + 2628615442511*v^14 - 13085191107646*v^13 + 9597611363326*v^12 - 41715567524734*v^11 + 17598532675094*v^10 - 77661391145496*v^9 + 29335472781423*v^8 - 102271734446566*v^7 + 25333093310512*v^6 - 73934241771071*v^5 + 23381283871418*v^4 - 34378241614652*v^3 + 1383447706561*v^2 - 332082681947*v + 2362215094) / 2269131592089 $$\beta_{13}$$ $$=$$ $$( - 759862203949 \nu^{17} + 1944263190262 \nu^{16} - 10194472029702 \nu^{15} + \cdots + 422383778863 ) / 6807394776267$$ (-759862203949*v^17 + 1944263190262*v^16 - 10194472029702*v^15 + 4998346464881*v^14 - 47027312555307*v^13 + 11846656395794*v^12 - 144961151389929*v^11 - 14951323436645*v^10 - 259778391966703*v^9 - 38074889008853*v^8 - 315491742629666*v^7 - 94215226604123*v^6 - 190419614219925*v^5 - 34905558154432*v^4 - 40827425833998*v^3 - 41997750259588*v^2 + 19183641080752*v + 422383778863) / 6807394776267 $$\beta_{14}$$ $$=$$ $$( - 1201752794172 \nu^{17} + 3351021374386 \nu^{16} - 17453288335442 \nu^{15} + \cdots - 12113076116529 ) / 6807394776267$$ (-1201752794172*v^17 + 3351021374386*v^16 - 17453288335442*v^15 + 13589938568817*v^14 - 85919693699728*v^13 + 46214882705029*v^12 - 280006541552455*v^11 + 66417646782434*v^10 - 551278539461887*v^9 + 104668494106942*v^8 - 753304634763705*v^7 + 67175092224633*v^6 - 600411869161108*v^5 + 84122081916376*v^4 - 280065299810173*v^3 - 9094600892191*v^2 - 27211156260656*v - 12113076116529) / 6807394776267 $$\beta_{15}$$ $$=$$ $$( - 1764752036203 \nu^{17} + 5655888876310 \nu^{16} - 27296401124421 \nu^{15} + \cdots + 1723605145714 ) / 6807394776267$$ (-1764752036203*v^17 + 5655888876310*v^16 - 27296401124421*v^15 + 29428427601899*v^14 - 128615472664470*v^13 + 113987266569107*v^12 - 410622346773594*v^11 + 241220082848977*v^10 - 759490348933525*v^9 + 429588085896622*v^8 - 1001370170259860*v^7 + 442503520486495*v^6 - 717045289804293*v^5 + 383847702402056*v^4 - 325915380884067*v^3 + 73511199022292*v^2 - 893491525826*v + 1723605145714) / 6807394776267 $$\beta_{16}$$ $$=$$ $$( - 2009357177810 \nu^{17} + 5675262379841 \nu^{16} - 29025194492163 \nu^{15} + \cdots - 3751600650469 ) / 6807394776267$$ (-2009357177810*v^17 + 5675262379841*v^16 - 29025194492163*v^15 + 22542369486988*v^14 - 138437780054349*v^13 + 74281026654601*v^12 - 439743089812689*v^11 + 94451845877099*v^10 - 826652550605570*v^9 + 131095657717169*v^8 - 1075352377560484*v^7 + 32966933367212*v^6 - 778255161948012*v^5 + 76344756009802*v^4 - 309405458804505*v^3 - 46549600863206*v^2 - 5869106089249*v - 3751600650469) / 6807394776267 $$\beta_{17}$$ $$=$$ $$( - 2169011193168 \nu^{17} + 7176985755886 \nu^{16} - 34139016228998 \nu^{15} + \cdots + 2187209095662 ) / 6807394776267$$ (-2169011193168*v^17 + 7176985755886*v^16 - 34139016228998*v^15 + 39022236800535*v^14 - 158773504414729*v^13 + 150026353889968*v^12 - 501373370047309*v^11 + 321726275203346*v^10 - 904040446392028*v^9 + 552412980832141*v^8 - 1163466071525892*v^7 + 559762487588955*v^6 - 790741872567937*v^5 + 454020967394587*v^4 - 354767905666813*v^3 + 83122854265979*v^2 - 620019744989*v + 2187209095662) / 6807394776267
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{12} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 1$$ -b12 - b10 + b8 + b7 + b5 - b4 + b3 - b2 + b1 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{14} - 5\beta_{12} - 3\beta_{10} + 3\beta_{9} + 3\beta_{7} + 2\beta_{5} - 2\beta_{4} + \beta_{3}$$ b14 - 5*b12 - 3*b10 + 3*b9 + 3*b7 + 2*b5 - 2*b4 + b3 $$\nu^{4}$$ $$=$$ $$- 3 \beta_{17} + 3 \beta_{14} - 2 \beta_{11} - 3 \beta_{10} + 9 \beta_{9} - 5 \beta_{8} + \cdots - 11 \beta_1$$ -3*b17 + 3*b14 - 2*b11 - 3*b10 + 9*b9 - 5*b8 + 3*b7 + 3*b5 + 3*b4 - 9*b3 + 2*b2 - 11*b1 $$\nu^{5}$$ $$=$$ $$- 9 \beta_{17} + \beta_{15} + 3 \beta_{14} - \beta_{13} + 39 \beta_{12} - 12 \beta_{11} + 25 \beta_{10} + \cdots + 1$$ -9*b17 + b15 + 3*b14 - b13 + 39*b12 - 12*b11 + 25*b10 - 13*b9 - 11*b8 - 23*b7 - 8*b5 + 38*b4 - 38*b3 + b2 - 39*b1 + 1 $$\nu^{6}$$ $$=$$ $$13 \beta_{17} + 2 \beta_{16} - \beta_{15} - 25 \beta_{14} - 3 \beta_{13} + 114 \beta_{12} - 13 \beta_{11} + \cdots + 7$$ 13*b17 + 2*b16 - b15 - 25*b14 - 3*b13 + 114*b12 - 13*b11 + 128*b10 - 128*b9 - 115*b7 + 3*b6 - 64*b5 + 87*b4 - 41*b3 + 7 $$\nu^{7}$$ $$=$$ $$128 \beta_{17} + 13 \beta_{16} - 13 \beta_{15} - 128 \beta_{14} - 3 \beta_{13} + 87 \beta_{11} + \cdots + 375 \beta_1$$ 128*b17 + 13*b16 - 13*b15 - 128*b14 - 3*b13 + 87*b11 + 142*b10 - 269*b9 + 114*b8 - 128*b7 + 10*b6 - 128*b5 - 142*b4 + 269*b3 - 12*b2 + 375*b1 $$\nu^{8}$$ $$=$$ $$269 \beta_{17} + 14 \beta_{16} - 27 \beta_{15} - 142 \beta_{14} + 27 \beta_{13} - 1181 \beta_{12} + \cdots - 46$$ 269*b17 + 14*b16 - 27*b15 - 142*b14 + 27*b13 - 1181*b12 + 411*b11 - 883*b10 + 455*b9 + 375*b8 + 786*b7 - 14*b6 + 233*b5 - 1338*b4 + 1338*b3 - 46*b2 + 1181*b1 - 46 $$\nu^{9}$$ $$=$$ $$- 455 \beta_{17} - 97 \beta_{16} + 44 \beta_{15} + 883 \beta_{14} + 141 \beta_{13} - 3823 \beta_{12} + \cdots - 126$$ -455*b17 - 97*b16 + 44*b15 + 883*b14 + 141*b13 - 3823*b12 + 455*b11 - 4309*b10 + 4309*b9 + 3857*b7 - 141*b6 + 2064*b5 - 2815*b4 + 1494*b3 - 126 $$\nu^{10}$$ $$=$$ $$- 4309 \beta_{17} - 452 \beta_{16} + 452 \beta_{15} + 4309 \beta_{14} + 156 \beta_{13} + \cdots - 12254 \beta_1$$ -4309*b17 - 452*b16 + 452*b15 + 4309*b14 + 156*b13 - 2815*b11 - 4808*b10 + 9112*b9 - 3823*b8 + 4309*b7 - 296*b6 + 4309*b5 + 4808*b4 - 9112*b3 + 421*b2 - 12254*b1 $$\nu^{11}$$ $$=$$ $$- 9112 \beta_{17} - 499 \beta_{16} + 980 \beta_{15} + 4808 \beta_{14} - 980 \beta_{13} + 39530 \beta_{12} + \cdots + 1307$$ -9112*b17 - 499*b16 + 980*b15 + 4808*b14 - 980*b13 + 39530*b12 - 13920*b11 + 29285*b10 - 15570*b9 - 12254*b8 - 26174*b7 + 499*b6 - 7446*b5 + 44855*b4 - 44855*b3 + 1307*b2 - 39530*b1 + 1307 $$\nu^{12}$$ $$=$$ $$15570 \beta_{17} + 3111 \beta_{16} - 1650 \beta_{15} - 29285 \beta_{14} - 4761 \beta_{13} + 127231 \beta_{12} + \cdots + 4244$$ 15570*b17 + 3111*b16 - 1650*b15 - 29285*b14 - 4761*b13 + 127231*b12 - 15570*b11 + 144639*b10 - 144639*b9 - 129240*b7 + 4761*b6 - 68815*b5 + 94477*b4 - 50162*b3 + 4244 $$\nu^{13}$$ $$=$$ $$144639 \beta_{17} + 15399 \beta_{16} - 15399 \beta_{15} - 144639 \beta_{14} - 5307 \beta_{13} + \cdots + 410121 \beta_1$$ 144639*b17 + 15399*b16 - 15399*b15 - 144639*b14 - 5307*b13 + 94477*b11 + 161859*b10 - 304266*b9 + 127231*b8 - 144639*b7 + 10092*b6 - 144639*b5 - 161859*b4 + 304266*b3 - 13561*b2 + 410121*b1 $$\nu^{14}$$ $$=$$ $$304266 \beta_{17} + 17220 \beta_{16} - 32396 \beta_{15} - 161859 \beta_{14} + 32396 \beta_{13} + \cdots - 43774$$ 304266*b17 + 17220*b16 - 32396*b15 - 161859*b14 + 32396*b13 - 1321304*b12 + 466125*b11 - 980815*b10 + 521594*b9 + 410121*b8 + 876246*b7 - 17220*b6 + 248262*b5 - 1502409*b4 + 1502409*b3 - 43774*b2 + 1321304*b1 - 43774 $$\nu^{15}$$ $$=$$ $$- 521594 \beta_{17} - 104569 \beta_{16} + 55469 \beta_{15} + 980815 \beta_{14} + 160038 \beta_{13} + \cdots - 140792$$ -521594*b17 - 104569*b16 + 55469*b15 + 980815*b14 + 160038*b13 - 4258423*b12 + 521594*b11 - 4841863*b10 + 4841863*b9 + 4326122*b7 - 160038*b6 + 2302119*b5 - 3160375*b4 + 1681488*b3 - 140792 $$\nu^{16}$$ $$=$$ $$- 4841863 \beta_{17} - 515741 \beta_{16} + 515741 \beta_{15} + 4841863 \beta_{14} + \cdots - 13722757 \beta_1$$ -4841863*b17 - 515741*b16 + 515741*b15 + 4841863*b14 + 179079*b13 - 3160375*b11 - 5418926*b10 + 10185670*b9 - 4258423*b8 + 4841863*b7 - 336662*b6 + 4841863*b5 + 5418926*b4 - 10185670*b3 + 453895*b2 - 13722757*b1 $$\nu^{17}$$ $$=$$ $$- 10185670 \beta_{17} - 577063 \beta_{16} + 1085384 \beta_{15} + 5418926 \beta_{14} - 1085384 \beta_{13} + \cdots + 1462096$$ -10185670*b17 - 577063*b16 + 1085384*b15 + 5418926*b14 - 1085384*b13 + 44225341*b12 - 15604596*b11 + 32824390*b10 - 17465163*b9 - 13722757*b8 - 29327353*b7 + 577063*b6 - 8303831*b5 + 50289553*b4 - 50289553*b3 + 1462096*b2 - 44225341*b1 + 1462096

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$\beta_{13} - \beta_{16}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 0.816390 − 1.41403i 0.154946 − 0.268374i −0.644984 + 1.11715i 0.816390 + 1.41403i 0.154946 + 0.268374i −0.644984 − 1.11715i 1.61137 + 2.79097i 0.394508 + 0.683308i −0.566185 − 0.980662i −0.791558 + 1.37102i −0.128481 + 0.222535i 0.653994 − 1.13275i 1.61137 − 2.79097i 0.394508 − 0.683308i −0.566185 + 0.980662i −0.791558 − 1.37102i −0.128481 − 0.222535i 0.653994 + 1.13275i
−0.484738 + 0.406743i −1.80387 0.656554i −0.277766 + 1.57529i 0.173648 + 0.984808i 1.14145 0.415455i −2.04448 + 3.54114i −1.13887 1.97259i 0.524744 + 0.440313i −0.484738 0.406743i
6.2 0.528654 0.443593i 0.652945 + 0.237653i −0.264596 + 1.50060i 0.173648 + 0.984808i 0.450603 0.164006i 1.16732 2.02186i 1.21589 + 2.10598i −1.92827 1.61801i 0.528654 + 0.443593i
6.3 1.75422 1.47196i −3.10785 1.13116i 0.563307 3.19467i 0.173648 + 0.984808i −7.11687 + 2.59033i 1.46955 2.54534i −1.42431 2.46697i 6.08105 + 5.10261i 1.75422 + 1.47196i
16.1 −0.484738 0.406743i −1.80387 + 0.656554i −0.277766 1.57529i 0.173648 0.984808i 1.14145 + 0.415455i −2.04448 3.54114i −1.13887 + 1.97259i 0.524744 0.440313i −0.484738 + 0.406743i
16.2 0.528654 + 0.443593i 0.652945 0.237653i −0.264596 1.50060i 0.173648 0.984808i 0.450603 + 0.164006i 1.16732 + 2.02186i 1.21589 2.10598i −1.92827 + 1.61801i 0.528654 0.443593i
16.3 1.75422 + 1.47196i −3.10785 + 1.13116i 0.563307 + 3.19467i 0.173648 0.984808i −7.11687 2.59033i 1.46955 + 2.54534i −1.42431 + 2.46697i 6.08105 5.10261i 1.75422 1.47196i
36.1 −0.385975 + 2.18897i 0.794389 + 0.666572i −2.76323 1.00573i −0.939693 + 0.342020i −1.76572 + 1.48162i 1.01254 + 1.75377i 1.04532 1.81055i −0.334208 1.89539i −0.385975 2.18897i
36.2 0.0366369 0.207778i 0.0612035 + 0.0513559i 1.83756 + 0.668816i −0.939693 + 0.342020i 0.0129129 0.0108352i 0.843614 + 1.46118i 0.417271 0.722735i −0.519836 2.94814i 0.0366369 + 0.207778i
36.3 0.370282 2.09998i 1.70859 + 1.43367i −2.39340 0.871127i −0.939693 + 0.342020i 3.64334 3.05712i −0.742812 1.28659i −0.583208 + 1.01015i 0.342900 + 1.94468i 0.370282 + 2.09998i
61.1 −2.42733 0.883478i −0.0430161 0.243956i 3.57933 + 3.00342i 0.766044 0.642788i −0.111115 + 0.630167i 0.200820 0.347830i −3.45167 5.97847i 2.76141 1.00507i −2.42733 + 0.883478i
61.2 −1.18116 0.429906i 0.523072 + 2.96649i −0.321776 0.270002i 0.766044 0.642788i 0.657481 3.72876i −1.86196 + 3.22501i 1.52095 + 2.63437i −5.70738 + 2.07732i −1.18116 + 0.429906i
61.3 0.289414 + 0.105338i −0.285463 1.61894i −1.45942 1.22460i 0.766044 0.642788i 0.0879194 0.498616i −0.0445979 + 0.0772459i −0.601369 1.04160i 0.279590 0.101762i 0.289414 0.105338i
66.1 −0.385975 2.18897i 0.794389 0.666572i −2.76323 + 1.00573i −0.939693 0.342020i −1.76572 1.48162i 1.01254 1.75377i 1.04532 + 1.81055i −0.334208 + 1.89539i −0.385975 + 2.18897i
66.2 0.0366369 + 0.207778i 0.0612035 0.0513559i 1.83756 0.668816i −0.939693 0.342020i 0.0129129 + 0.0108352i 0.843614 1.46118i 0.417271 + 0.722735i −0.519836 + 2.94814i 0.0366369 0.207778i
66.3 0.370282 + 2.09998i 1.70859 1.43367i −2.39340 + 0.871127i −0.939693 0.342020i 3.64334 + 3.05712i −0.742812 + 1.28659i −0.583208 1.01015i 0.342900 1.94468i 0.370282 2.09998i
81.1 −2.42733 + 0.883478i −0.0430161 + 0.243956i 3.57933 3.00342i 0.766044 + 0.642788i −0.111115 0.630167i 0.200820 + 0.347830i −3.45167 + 5.97847i 2.76141 + 1.00507i −2.42733 0.883478i
81.2 −1.18116 + 0.429906i 0.523072 2.96649i −0.321776 + 0.270002i 0.766044 + 0.642788i 0.657481 + 3.72876i −1.86196 3.22501i 1.52095 2.63437i −5.70738 2.07732i −1.18116 0.429906i
81.3 0.289414 0.105338i −0.285463 + 1.61894i −1.45942 + 1.22460i 0.766044 + 0.642788i 0.0879194 + 0.498616i −0.0445979 0.0772459i −0.601369 + 1.04160i 0.279590 + 0.101762i 0.289414 + 0.105338i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 6.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.2.k.a 18
3.b odd 2 1 855.2.bs.c 18
5.b even 2 1 475.2.l.c 18
5.c odd 4 2 475.2.u.b 36
19.e even 9 1 inner 95.2.k.a 18
19.e even 9 1 1805.2.a.v 9
19.f odd 18 1 1805.2.a.s 9
57.l odd 18 1 855.2.bs.c 18
95.o odd 18 1 9025.2.a.cf 9
95.p even 18 1 475.2.l.c 18
95.p even 18 1 9025.2.a.cc 9
95.q odd 36 2 475.2.u.b 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.a 18 1.a even 1 1 trivial
95.2.k.a 18 19.e even 9 1 inner
475.2.l.c 18 5.b even 2 1
475.2.l.c 18 95.p even 18 1
475.2.u.b 36 5.c odd 4 2
475.2.u.b 36 95.q odd 36 2
855.2.bs.c 18 3.b odd 2 1
855.2.bs.c 18 57.l odd 18 1
1805.2.a.s 9 19.f odd 18 1
1805.2.a.v 9 19.e even 9 1
9025.2.a.cc 9 95.p even 18 1
9025.2.a.cf 9 95.o odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{18} + 3 T_{2}^{17} + 6 T_{2}^{16} + 19 T_{2}^{15} + 33 T_{2}^{14} + 66 T_{2}^{13} + 268 T_{2}^{12} + \cdots + 1$$ acting on $$S_{2}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} + 3 T^{17} + \cdots + 1$$
$3$ $$T^{18} + 3 T^{17} + \cdots + 1$$
$5$ $$(T^{6} + T^{3} + 1)^{3}$$
$7$ $$T^{18} + 27 T^{16} + \cdots + 361$$
$11$ $$T^{18} + 60 T^{16} + \cdots + 597529$$
$13$ $$T^{18} + 3 T^{17} + \cdots + 2809$$
$17$ $$T^{18} - 24 T^{17} + \cdots + 2859481$$
$19$ $$T^{18} + \cdots + 322687697779$$
$23$ $$T^{18} + \cdots + 2085474889$$
$29$ $$T^{18} + \cdots + 44269422409$$
$31$ $$T^{18} + \cdots + 47085094081$$
$37$ $$(T^{9} - 18 T^{8} + \cdots - 11125)^{2}$$
$41$ $$T^{18} + 30 T^{17} + \cdots + 130321$$
$43$ $$T^{18} + \cdots + 2531341458361$$
$47$ $$T^{18} + \cdots + 2033313439249$$
$53$ $$T^{18} + \cdots + 44884235881$$
$59$ $$T^{18} + \cdots + 538419880441$$
$61$ $$T^{18} + \cdots + 3789756721$$
$67$ $$T^{18} + \cdots + 42166698100921$$
$71$ $$T^{18} + \cdots + 28\!\cdots\!49$$
$73$ $$T^{18} + \cdots + 585504633023209$$
$79$ $$T^{18} + 51 T^{17} + \cdots + 14691889$$
$83$ $$T^{18} + \cdots + 36\!\cdots\!49$$
$89$ $$T^{18} + \cdots + 916651141561$$
$97$ $$T^{18} + \cdots + 3665788890625$$