# Properties

 Label 95.2.k.b 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: $$x^{18} + 12 x^{16} - 8 x^{15} + 96 x^{14} - 75 x^{13} + 448 x^{12} - 405 x^{11} + 1521 x^{10} + \cdots + 361$$ x^18 + 12*x^16 - 8*x^15 + 96*x^14 - 75*x^13 + 448*x^12 - 405*x^11 + 1521*x^10 - 1294*x^9 + 3333*x^8 - 2616*x^7 + 5113*x^6 - 3126*x^5 + 4032*x^4 - 1359*x^3 + 1698*x^2 - 513*x + 361 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 12 x^{16} - 8 x^{15} + 96 x^{14} - 75 x^{13} + 448 x^{12} - 405 x^{11} + 1521 x^{10} + \cdots + 361$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 136164871502623 \nu^{17} + \cdots - 12\!\cdots\!58 ) / 23\!\cdots\!13$$ (-136164871502623*v^17 - 24539850127465139*v^16 - 15475762038850846*v^15 - 278387493146710423*v^14 + 57114523551811041*v^13 - 2034802237447217245*v^12 + 755780565816668612*v^11 - 8643822113053672298*v^10 + 5395265824707751653*v^9 - 26540269945879200040*v^8 + 16118541568209638063*v^7 - 51490846595808251984*v^6 + 32949301580318741461*v^5 - 68799337664864459525*v^4 + 28965642747827808589*v^3 - 38566636741667052372*v^2 + 8340705133450433900*v - 12713186363580574958) / 2363313751376612913 $$\beta_{3}$$ $$=$$ $$( - 305669388244104 \nu^{17} + 493873766181026 \nu^{16} + \cdots + 47\!\cdots\!16 ) / 78\!\cdots\!71$$ (-305669388244104*v^17 + 493873766181026*v^16 - 3880691063159639*v^15 + 7968889393851538*v^14 - 35512529776538063*v^13 + 67530840922095229*v^12 - 187990470308389149*v^11 + 323224172461819959*v^10 - 708615722180805182*v^9 + 1061204782303381103*v^8 - 1753639286959150144*v^7 + 2134926077072574208*v^6 - 2885622013327482014*v^5 + 2855869998225035874*v^4 - 2894326577431631545*v^3 + 1371327034244450974*v^2 - 1311641429597997762*v + 475150684820058216) / 787771250458870971 $$\beta_{4}$$ $$=$$ $$( - 13\!\cdots\!19 \nu^{17} + \cdots + 40\!\cdots\!81 ) / 23\!\cdots\!13$$ (-1365784748015119*v^17 - 1373794800955595*v^16 - 29794953034609280*v^15 - 26620230113148110*v^14 - 281280084655676041*v^13 - 137185723270135532*v^12 - 1561503944080813321*v^11 - 572552850128153362*v^10 - 5560668611293666895*v^9 - 919186295759062923*v^8 - 12852323826200522587*v^7 - 1330055455523271005*v^6 - 18735117324885579084*v^5 + 334980033043877088*v^4 - 14654463655353182629*v^3 - 1688955502281513037*v^2 - 2254140586374967780*v + 404864180772791181) / 2363313751376612913 $$\beta_{5}$$ $$=$$ $$( - 13\!\cdots\!95 \nu^{17} + \cdots + 49\!\cdots\!59 ) / 23\!\cdots\!13$$ (-1373794800955595*v^17 - 13405536058427852*v^16 - 37546508097269062*v^15 - 150164748846224617*v^14 - 239619579371269457*v^13 - 949632376970040009*v^12 - 1125695673074276557*v^11 - 3483310009562670896*v^10 - 2686511759690626909*v^9 - 8300163261066130960*v^8 - 4902948356330822309*v^7 - 11751859908284275637*v^6 - 3934463089251384906*v^5 - 9147619551356222821*v^4 - 3545056974834059758*v^3 + 64961915754704282*v^2 - 295783394958964866*v + 493048294033457959) / 2363313751376612913 $$\beta_{6}$$ $$=$$ $$( - 38\!\cdots\!82 \nu^{17} + \cdots + 73\!\cdots\!88 ) / 23\!\cdots\!13$$ (-3821497981088482*v^17 - 2486336156102819*v^16 - 60510905073974959*v^15 - 9894754734681145*v^14 - 509501528649318297*v^13 + 77946254382002165*v^12 - 2638706167903118716*v^11 + 887747573331774484*v^10 - 9149472805995407940*v^9 + 4616558327094384365*v^8 - 21323954668151248570*v^7 + 12469640235719854456*v^6 - 31837869359996258159*v^5 + 22149232909835934805*v^4 - 26501304396415488623*v^3 + 17863733309230704777*v^2 - 2744800892072834308*v + 7301259660632472688) / 2363313751376612913 $$\beta_{7}$$ $$=$$ $$( 13\!\cdots\!56 \nu^{17} + 305669388244104 \nu^{16} + \cdots - 15\!\cdots\!37 ) / 78\!\cdots\!71$$ (1316206883158056*v^17 + 305669388244104*v^16 + 15300608831715646*v^15 - 6648964002104809*v^14 + 118386971389321838*v^13 - 63202986460316137*v^12 + 522129842732713859*v^11 - 345073317370623531*v^10 + 1678726496821583217*v^9 - 994555984625719282*v^8 + 3325712759262419545*v^7 - 1689557919382324352*v^6 + 4594839716514566120*v^5 - 1228840703424601042*v^4 + 2451076154668245918*v^3 + 1105601423219833441*v^2 + 863592253357928114*v - 151343951920955937) / 787771250458870971 $$\beta_{8}$$ $$=$$ $$( 42\!\cdots\!70 \nu^{17} + \cdots - 79\!\cdots\!20 ) / 23\!\cdots\!13$$ (4263268715769470*v^17 + 8457548837445544*v^16 + 61247608976156191*v^15 + 69138387783314911*v^14 + 433874328606196583*v^13 + 380877728603667745*v^12 + 1872955328631131189*v^11 + 1157179500078806039*v^10 + 5114203147106148517*v^9 + 2367273831481700889*v^8 + 8844406095767234201*v^7 + 2915707886574672496*v^6 + 8365658335605219288*v^5 + 2218844617339251399*v^4 + 2036234158376448173*v^3 + 1088798398418223410*v^2 - 2524946775203921152*v - 791795258602032720) / 2363313751376612913 $$\beta_{9}$$ $$=$$ $$( - 116934812256397 \nu^{17} - 400329673137974 \nu^{16} + \cdots - 19\!\cdots\!38 ) / 63\!\cdots\!49$$ (-116934812256397*v^17 - 400329673137974*v^16 - 2128850997697097*v^15 - 4687402177384319*v^14 - 16482913440421588*v^13 - 31523850998316788*v^12 - 80012426739975022*v^11 - 131895294695077462*v^10 - 247925063622357167*v^9 - 370055452165070628*v^8 - 521090564283915070*v^7 - 705854963648750474*v^6 - 697344780135723768*v^5 - 871890492989000946*v^4 - 567458240342481703*v^3 - 640580280306084166*v^2 - 171974561020272844*v - 197333589053007138) / 63873344631800349 $$\beta_{10}$$ $$=$$ $$( 47\!\cdots\!03 \nu^{17} + \cdots + 52\!\cdots\!54 ) / 23\!\cdots\!13$$ (4785929688950203*v^17 + 17994909435137720*v^16 + 64753023529443092*v^15 + 154272805835837465*v^14 + 361776262948934794*v^13 + 1028457065358737192*v^12 + 1262057723734120030*v^11 + 3763965626999821126*v^10 + 1962960094436407709*v^9 + 11195319724831197969*v^8 + 814973947936715863*v^7 + 20443213806190380140*v^6 - 5834921042750809410*v^5 + 29102676427842391179*v^4 - 8268649772449030736*v^3 + 15717515903286395077*v^2 - 2679740142811021208*v + 5284901620360032954) / 2363313751376612913 $$\beta_{11}$$ $$=$$ $$( - 61\!\cdots\!77 \nu^{17} + 828266836359098 \nu^{16} + \cdots + 60\!\cdots\!85 ) / 23\!\cdots\!13$$ (-6194705453707877*v^17 + 828266836359098*v^16 - 83367742292924762*v^15 + 38324000046280303*v^14 - 728701165601510295*v^13 + 373992166879518313*v^12 - 3703626960967606784*v^11 + 1887987830399360111*v^10 - 13414255907811757605*v^9 + 6224701970246744260*v^8 - 31375345171161971747*v^7 + 11814910314894911588*v^6 - 50251521538706392489*v^5 + 14641160425862497394*v^4 - 42524480716536293605*v^3 + 2614293290552294073*v^2 - 15503997389545927664*v + 6094047996318719285) / 2363313751376612913 $$\beta_{12}$$ $$=$$ $$( 84\!\cdots\!25 \nu^{17} + \cdots - 58\!\cdots\!05 ) / 23\!\cdots\!13$$ (8415587854105225*v^17 - 9292642129044856*v^16 + 79443562791874058*v^15 - 184902616685114542*v^14 + 657167974186164952*v^13 - 1367584144063781320*v^12 + 2857108775199131491*v^11 - 6262533437436660725*v^10 + 10050176092564259657*v^9 - 17690009620923466737*v^8 + 20640804020965884865*v^7 - 33019415996230257943*v^6 + 32005685555634494898*v^5 - 35714722354143733911*v^4 + 18849342455135910337*v^3 - 17079528084795008093*v^2 + 9109842231458219395*v - 5866741202393917305) / 2363313751376612913 $$\beta_{13}$$ $$=$$ $$( 84\!\cdots\!44 \nu^{17} + \cdots - 15\!\cdots\!70 ) / 23\!\cdots\!13$$ (8457548837445544*v^17 + 10088384386922551*v^16 + 103244537509470671*v^15 + 24600531892327463*v^14 + 700622882286377995*v^13 - 36989056033591371*v^12 + 2883803329965441389*v^11 - 1370228569579215353*v^10 + 7883943549687395069*v^9 - 5365068533892409309*v^8 + 14068418847027606016*v^7 - 13432434608124080822*v^6 + 15545822622834614619*v^5 - 15153265303606054867*v^4 + 6882580583148933140*v^3 - 9763977054580481212*v^2 + 1395261592587705390*v - 1539040006392778670) / 2363313751376612913 $$\beta_{14}$$ $$=$$ $$( - 87\!\cdots\!64 \nu^{17} + \cdots + 47\!\cdots\!08 ) / 23\!\cdots\!13$$ (-8702267306092864*v^17 + 14221624195347986*v^16 - 74982329495166347*v^15 + 248401446998221342*v^14 - 650986737024112972*v^13 + 1795113360355580223*v^12 - 2848046794500608741*v^11 + 8066371995986638622*v^10 - 10588073804621455064*v^9 + 22544880002101140796*v^8 - 21958967702593121845*v^7 + 41328594684758717876*v^6 - 34655897913289925196*v^5 + 42647826050912606095*v^4 - 16578850159503713636*v^3 + 15986074699460918557*v^2 - 6190538917553550213*v + 4765747833043009508) / 2363313751376612913 $$\beta_{15}$$ $$=$$ $$( 92\!\cdots\!56 \nu^{17} + \cdots + 30\!\cdots\!25 ) / 23\!\cdots\!13$$ (9292642129044856*v^17 + 21543491457388642*v^16 + 117577913852272742*v^15 + 150728459807936648*v^14 + 736415055005889445*v^13 + 913074583440009309*v^12 + 2854220356524044600*v^11 + 2749933033529787568*v^10 + 6800238937711305587*v^9 + 7408350296766830060*v^8 + 11004238169890989343*v^7 + 11023215142405520527*v^6 + 9407594722210800561*v^5 + 15082307772616356863*v^4 + 5642744191066007318*v^3 + 5179825944812452655*v^2 + 1549544633237936880*v + 3038027215331986225) / 2363313751376612913 $$\beta_{16}$$ $$=$$ $$( 316478299376213 \nu^{17} - 588887179277519 \nu^{16} + \cdots - 38\!\cdots\!43 ) / 63\!\cdots\!49$$ (316478299376213*v^17 - 588887179277519*v^16 + 3303505876934864*v^15 - 9141301869341992*v^14 + 30340466650154259*v^13 - 70368354475407094*v^12 + 147431058119218739*v^11 - 317553566740250921*v^10 + 554156872867621638*v^9 - 959223616419207394*v^8 + 1244593416258686864*v^7 - 1802618050547274896*v^6 + 2008185379160806393*v^5 - 2158348756640158619*v^4 + 1379062922698433023*v^3 - 837284802272031468*v^2 + 435487443612750998*v - 380413025158626443) / 63873344631800349 $$\beta_{17}$$ $$=$$ $$( 17\!\cdots\!19 \nu^{17} + \cdots - 11\!\cdots\!95 ) / 23\!\cdots\!13$$ (17924896300275019*v^17 - 4499256512921017*v^16 + 192044400084805108*v^15 - 214598336312124923*v^14 + 1503924491838736770*v^13 - 1743300156274298741*v^12 + 6599826966823468927*v^11 - 8805655803798270706*v^10 + 22001645173958880027*v^9 - 26056201587940490639*v^8 + 45153108522930576424*v^7 - 51113768682332337226*v^6 + 67330488050959221206*v^5 - 56810286570525635647*v^4 + 42899194197704035433*v^3 - 28423311584312662812*v^2 + 19825858771860031108*v - 11104061301097034995) / 2363313751376612913
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{16} + 3\beta_{7} + \beta_{2}$$ -b16 + 3*b7 + b2 $$\nu^{3}$$ $$=$$ $$\beta_{17} - \beta_{12} - 2\beta_{10} + \beta_{9} - \beta_{4} - 3\beta_{3} - 2\beta_{2} - 3\beta _1 + 1$$ b17 - b12 - 2*b10 + b9 - b4 - 3*b3 - 2*b2 - 3*b1 + 1 $$\nu^{4}$$ $$=$$ $$- \beta_{17} + 6 \beta_{16} + \beta_{14} - \beta_{13} + 2 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} + \cdots - 12$$ -b17 + 6*b16 + b14 - b13 + 2*b12 + 6*b11 + 6*b10 - 5*b9 - 6*b8 - 11*b7 - b6 - b5 + b4 + b1 - 12 $$\nu^{5}$$ $$=$$ $$- 9 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - 7 \beta_{11} + 9 \beta_{10} + \cdots + 7$$ -9*b16 - b15 - b14 - b13 - b12 - 7*b11 + 9*b10 + 10*b8 + 9*b7 + 7*b6 + b5 + b4 + 11*b3 + 16*b2 + 7 $$\nu^{6}$$ $$=$$ $$3 \beta_{17} - 8 \beta_{16} + 10 \beta_{15} - 9 \beta_{14} + 10 \beta_{13} - 12 \beta_{12} - 33 \beta_{11} + \cdots + 57$$ 3*b17 - 8*b16 + 10*b15 - 9*b14 + 10*b13 - 12*b12 - 33*b11 - 58*b10 + 36*b9 + 24*b8 + 8*b6 + b5 - 22*b4 - 9*b3 - 39*b2 - 9*b1 + 57 $$\nu^{7}$$ $$=$$ $$- 46 \beta_{17} + 69 \beta_{16} - 10 \beta_{15} + 22 \beta_{14} - 12 \beta_{13} + 77 \beta_{12} + \cdots - 112$$ -46*b17 + 69*b16 - 10*b15 + 22*b14 - 12*b13 + 77*b12 + 69*b11 + 79*b10 - 65*b9 - 79*b8 - 66*b7 - 46*b6 - 22*b5 + 63*b4 + 49*b1 - 112 $$\nu^{8}$$ $$=$$ $$52 \beta_{17} - 176 \beta_{16} - 63 \beta_{15} - 14 \beta_{14} - 14 \beta_{13} - 76 \beta_{12} + \cdots + 39$$ 52*b17 - 176*b16 - 63*b15 - 14*b14 - 14*b13 - 76*b12 - 39*b11 + 157*b10 - 52*b9 + 129*b8 + 258*b7 + 39*b6 + 63*b5 + 76*b4 + 66*b3 + 267*b2 + 39 $$\nu^{9}$$ $$=$$ $$252 \beta_{17} - 53 \beta_{16} + 181 \beta_{15} - 105 \beta_{14} + 181 \beta_{13} - 417 \beta_{12} + \cdots + 518$$ 252*b17 - 53*b16 + 181*b15 - 105*b14 + 181*b13 - 417*b12 - 249*b11 - 1051*b10 + 501*b9 + 84*b8 + 53*b6 + 76*b5 - 550*b4 - 258*b3 - 753*b2 - 258*b1 + 518 $$\nu^{10}$$ $$=$$ $$- 682 \beta_{17} + 1497 \beta_{16} - 133 \beta_{15} + 550 \beta_{14} - 417 \beta_{13} + 1349 \beta_{12} + \cdots - 2209$$ -682*b17 + 1497*b16 - 133*b15 + 550*b14 - 417*b13 + 1349*b12 + 1497*b11 + 1702*b10 - 1169*b9 - 1702*b8 - 1527*b7 - 682*b6 - 550*b5 + 816*b4 + 465*b1 - 2209 $$\nu^{11}$$ $$=$$ $$487 \beta_{17} - 3061 \beta_{16} - 816 \beta_{15} - 533 \beta_{14} - 533 \beta_{13} - 1081 \beta_{12} + \cdots + 1560$$ 487*b17 - 3061*b16 - 816*b15 - 533*b14 - 533*b13 - 1081*b12 - 1560*b11 + 3235*b10 - 487*b9 + 3342*b8 + 3246*b7 + 1560*b6 + 816*b5 + 1081*b4 + 1527*b3 + 5108*b2 + 1560 $$\nu^{12}$$ $$=$$ $$2804 \beta_{17} - 2093 \beta_{16} + 3829 \beta_{15} - 2748 \beta_{14} + 3829 \beta_{13} - 5996 \beta_{12} + \cdots + 11782$$ 2804*b17 - 2093*b16 + 3829*b15 - 2748*b14 + 3829*b13 - 5996*b12 - 7227*b11 - 19673*b10 + 10031*b9 + 4035*b8 + 2093*b6 + 1081*b5 - 9642*b4 - 3246*b3 - 12835*b2 - 3246*b1 + 11782 $$\nu^{13}$$ $$=$$ $$- 13860 \beta_{17} + 24807 \beta_{16} - 3646 \beta_{15} + 9642 \beta_{14} - 5996 \beta_{13} + \cdots - 36433$$ -13860*b17 + 24807*b16 - 3646*b15 + 9642*b14 - 5996*b13 + 26415*b12 + 24807*b11 + 29062*b10 - 20922*b9 - 29062*b8 - 22573*b7 - 13860*b6 - 9642*b5 + 18275*b4 + 9689*b1 - 36433 $$\nu^{14}$$ $$=$$ $$13621 \beta_{17} - 54352 \beta_{16} - 18275 \beta_{15} - 8140 \beta_{14} - 8140 \beta_{13} + \cdots + 20828$$ 13621*b17 - 54352*b16 - 18275*b15 - 8140*b14 - 8140*b13 - 24798*b12 - 20828*b11 + 56416*b10 - 13621*b9 + 53972*b8 + 63849*b7 + 20828*b6 + 18275*b5 + 24798*b4 + 22573*b3 + 88801*b2 + 20828 $$\nu^{15}$$ $$=$$ $$65420 \beta_{17} - 28968 \beta_{16} + 67593 \beta_{15} - 42795 \beta_{14} + 67593 \beta_{13} + \cdots + 185488$$ 65420*b17 - 28968*b16 + 67593*b15 - 42795*b14 + 67593*b13 - 122798*b12 - 106818*b11 - 353905*b10 + 172238*b9 + 49440*b8 + 28968*b6 + 24798*b5 - 181667*b4 - 63849*b3 - 237658*b2 - 63849*b1 + 185488 $$\nu^{16}$$ $$=$$ $$- 239831 \beta_{17} + 463488 \beta_{16} - 58869 \beta_{15} + 181667 \beta_{14} - 122798 \beta_{13} + \cdots - 669036$$ -239831*b17 + 463488*b16 - 58869*b15 + 181667*b14 - 122798*b13 + 469405*b12 + 463488*b11 + 542052*b10 - 373270*b9 - 542052*b8 - 429205*b7 - 239831*b6 - 181667*b5 + 300623*b4 + 156520*b1 - 669036 $$\nu^{17}$$ $$=$$ $$208482 \beta_{17} - 982637 \beta_{16} - 300623 \beta_{15} - 168782 \beta_{14} - 168782 \beta_{13} + \cdots + 436673$$ 208482*b17 - 982637*b16 - 300623*b15 - 168782*b14 - 168782*b13 - 416438*b12 - 436673*b11 + 1040364*b10 - 208482*b9 + 1039838*b8 + 1082661*b7 + 436673*b6 + 300623*b5 + 416438*b4 + 429205*b3 + 1627792*b2 + 436673

## Character values

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

 $$n$$ $$21$$ $$77$$ $$\chi(n)$$ $$\beta_{4} + \beta_{8}$$ $$1$$

## 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
 0.785237 + 1.36007i 0.296923 + 0.514286i −0.908512 − 1.57359i 0.785237 − 1.36007i 0.296923 − 0.514286i −0.908512 + 1.57359i 0.731154 − 1.26640i −0.359728 + 0.623068i −1.31112 + 2.27092i −0.841804 − 1.45805i 0.597039 + 1.03410i 1.01081 + 1.75077i 0.731154 + 1.26640i −0.359728 − 0.623068i −1.31112 − 2.27092i −0.841804 + 1.45805i 0.597039 − 1.03410i 1.01081 − 1.75077i
−1.20305 + 1.00948i −2.14722 0.781523i 0.0809872 0.459301i −0.173648 0.984808i 3.37214 1.22736i 2.00668 3.47568i −1.20425 2.08582i 1.70162 + 1.42783i 1.20305 + 1.00948i
6.2 −0.454913 + 0.381717i 1.81364 + 0.660111i −0.286059 + 1.62232i −0.173648 0.984808i −1.07702 + 0.392004i −0.530259 + 0.918436i −1.08298 1.87578i 0.555408 + 0.466042i 0.454913 + 0.381717i
6.3 1.39192 1.16796i −0.166424 0.0605732i 0.226016 1.28180i −0.173648 0.984808i −0.302396 + 0.110063i −0.536732 + 0.929646i 0.634528 + 1.09903i −2.27411 1.90820i −1.39192 1.16796i
16.1 −1.20305 1.00948i −2.14722 + 0.781523i 0.0809872 + 0.459301i −0.173648 + 0.984808i 3.37214 + 1.22736i 2.00668 + 3.47568i −1.20425 + 2.08582i 1.70162 1.42783i 1.20305 1.00948i
16.2 −0.454913 0.381717i 1.81364 0.660111i −0.286059 1.62232i −0.173648 + 0.984808i −1.07702 0.392004i −0.530259 0.918436i −1.08298 + 1.87578i 0.555408 0.466042i 0.454913 0.381717i
16.3 1.39192 + 1.16796i −0.166424 + 0.0605732i 0.226016 + 1.28180i −0.173648 + 0.984808i −0.302396 0.110063i −0.536732 0.929646i 0.634528 1.09903i −2.27411 + 1.90820i −1.39192 + 1.16796i
36.1 −0.253927 + 1.44009i 1.15700 + 0.970838i −0.130002 0.0473169i 0.939693 0.342020i −1.69189 + 1.41966i −2.03586 3.52622i −1.36116 + 2.35759i −0.124823 0.707907i 0.253927 + 1.44009i
36.2 0.124932 0.708527i −0.945515 0.793382i 1.39298 + 0.507004i 0.939693 0.342020i −0.680257 + 0.570804i −0.645970 1.11885i 1.25271 2.16976i −0.256400 1.45411i −0.124932 0.708527i
36.3 0.455347 2.58240i −0.711484 0.597006i −4.58206 1.66773i 0.939693 0.342020i −1.86568 + 1.56549i 1.91579 + 3.31824i −3.77094 + 6.53146i −0.371151 2.10490i −0.455347 2.58240i
61.1 −1.58207 0.575828i −0.564707 3.20261i 0.639290 + 0.536428i −0.766044 + 0.642788i −0.950745 + 5.39194i 0.274194 0.474919i 0.981094 + 1.69930i −7.11876 + 2.59101i 1.58207 0.575828i
61.2 1.12207 + 0.408399i 0.394733 + 2.23864i −0.439845 0.369074i −0.766044 + 0.642788i −0.471342 + 2.67312i 1.09825 1.90222i −1.53688 2.66196i −2.03663 + 0.741274i −1.12207 + 0.408399i
61.3 1.89970 + 0.691434i −0.330026 1.87167i 1.59869 + 1.34146i −0.766044 + 0.642788i 0.667187 3.78381i −1.54609 + 2.67790i 0.0878797 + 0.152212i −0.575162 + 0.209342i −1.89970 + 0.691434i
66.1 −0.253927 1.44009i 1.15700 0.970838i −0.130002 + 0.0473169i 0.939693 + 0.342020i −1.69189 1.41966i −2.03586 + 3.52622i −1.36116 2.35759i −0.124823 + 0.707907i 0.253927 1.44009i
66.2 0.124932 + 0.708527i −0.945515 + 0.793382i 1.39298 0.507004i 0.939693 + 0.342020i −0.680257 0.570804i −0.645970 + 1.11885i 1.25271 + 2.16976i −0.256400 + 1.45411i −0.124932 + 0.708527i
66.3 0.455347 + 2.58240i −0.711484 + 0.597006i −4.58206 + 1.66773i 0.939693 + 0.342020i −1.86568 1.56549i 1.91579 3.31824i −3.77094 6.53146i −0.371151 + 2.10490i −0.455347 + 2.58240i
81.1 −1.58207 + 0.575828i −0.564707 + 3.20261i 0.639290 0.536428i −0.766044 0.642788i −0.950745 5.39194i 0.274194 + 0.474919i 0.981094 1.69930i −7.11876 2.59101i 1.58207 + 0.575828i
81.2 1.12207 0.408399i 0.394733 2.23864i −0.439845 + 0.369074i −0.766044 0.642788i −0.471342 2.67312i 1.09825 + 1.90222i −1.53688 + 2.66196i −2.03663 0.741274i −1.12207 0.408399i
81.3 1.89970 0.691434i −0.330026 + 1.87167i 1.59869 1.34146i −0.766044 0.642788i 0.667187 + 3.78381i −1.54609 2.67790i 0.0878797 0.152212i −0.575162 0.209342i −1.89970 0.691434i
 $$n$$: 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.b 18
3.b odd 2 1 855.2.bs.b 18
5.b even 2 1 475.2.l.b 18
5.c odd 4 2 475.2.u.c 36
19.e even 9 1 inner 95.2.k.b 18
19.e even 9 1 1805.2.a.t 9
19.f odd 18 1 1805.2.a.u 9
57.l odd 18 1 855.2.bs.b 18
95.o odd 18 1 9025.2.a.cd 9
95.p even 18 1 475.2.l.b 18
95.p even 18 1 9025.2.a.ce 9
95.q odd 36 2 475.2.u.c 36

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{18} - 3 T_{2}^{17} + 6 T_{2}^{16} - 5 T_{2}^{15} - 15 T_{2}^{14} + 24 T_{2}^{13} + 70 T_{2}^{12} + \cdots + 361$$ acting on $$S_{2}^{\mathrm{new}}(95, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} - 3 T^{17} + \cdots + 361$$
$3$ $$T^{18} + 3 T^{17} + \cdots + 361$$
$5$ $$(T^{6} - T^{3} + 1)^{3}$$
$7$ $$T^{18} + 33 T^{16} + \cdots + 117649$$
$11$ $$T^{18} + 36 T^{16} + \cdots + 361$$
$13$ $$T^{18} + 3 T^{17} + \cdots + 98743969$$
$17$ $$T^{18} + 24 T^{17} + \cdots + 1$$
$19$ $$T^{18} + \cdots + 322687697779$$
$23$ $$T^{18} + \cdots + 3993860809$$
$29$ $$T^{18} + \cdots + 492805404001$$
$31$ $$T^{18} + \cdots + 993257404129$$
$37$ $$(T^{9} + 30 T^{8} + \cdots + 27721)^{2}$$
$41$ $$T^{18} + \cdots + 132479256529$$
$43$ $$T^{18} + \cdots + 101989801$$
$47$ $$T^{18} + \cdots + 32737816559809$$
$53$ $$T^{18} + \cdots + 19250147475001$$
$59$ $$T^{18} + \cdots + 333749999521$$
$61$ $$T^{18} + \cdots + 3468418292161$$
$67$ $$T^{18} + \cdots + 36\!\cdots\!81$$
$71$ $$T^{18} + \cdots + 8590138489$$
$73$ $$T^{18} + \cdots + 1047660743809$$
$79$ $$T^{18} + \cdots + 316957384070209$$
$83$ $$T^{18} + 171 T^{16} + \cdots + 96609241$$
$89$ $$T^{18} + \cdots + 143304841$$
$97$ $$T^{18} + \cdots + 36\!\cdots\!09$$