# Properties

 Label 171.2.f.c Level $171$ Weight $2$ Character orbit 171.f Analytic conductor $1.365$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,2,Mod(64,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("171.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 171.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: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.764411904.5 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81$$ x^8 - 6*x^6 + 21*x^4 - 54*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + ( - \beta_{6} + \beta_{4} - \beta_{3} - 1) q^{4} - \beta_1 q^{5} + ( - \beta_{6} - 1) q^{7} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{8}+O(q^{10})$$ q - b5 * q^2 + (-b6 + b4 - b3 - 1) * q^4 - b1 * q^5 + (-b6 - 1) * q^7 + (-b7 + b5 + b2 + b1) * q^8 $$q - \beta_{5} q^{2} + ( - \beta_{6} + \beta_{4} - \beta_{3} - 1) q^{4} - \beta_1 q^{5} + ( - \beta_{6} - 1) q^{7} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{8}+ \cdots + ( - 4 \beta_{5} - 2 \beta_1) q^{98}+O(q^{100})$$ q - b5 * q^2 + (-b6 + b4 - b3 - 1) * q^4 - b1 * q^5 + (-b6 - 1) * q^7 + (-b7 + b5 + b2 + b1) * q^8 + (-b6 + b4) * q^10 + (2*b7 - 2*b5 - b2 - b1) * q^11 + (b3 + 1) * q^13 + (3*b5 + b1) * q^14 + b3 * q^16 + 2*b1 * q^17 + (b6 - 2*b4 + 1) * q^19 + (-2*b7 + 2*b5 - b2 - b1) * q^20 + (3*b4 - 6*b3) * q^22 + (-2*b7 - b2) * q^23 + (2*b6 - 2*b4 - b3 - 1) * q^25 + (b7 - b5) * q^26 + (2*b6 - 2*b4 + 7*b3 + 7) * q^28 + (4*b7 + 2*b2) * q^29 + (-b6 - 7) * q^31 + (-b7 + 2*b2) * q^32 + (2*b6 - 2*b4) * q^34 + (2*b5 - b1) * q^35 + (2*b6 - 1) * q^37 + (4*b7 - 3*b5 - 2*b2 - b1) * q^38 + (b4 + 6*b3) * q^40 - 4*b5 * q^41 + (-3*b4 - b3) * q^43 + (-8*b7 + b2) * q^44 + (-b6 - 6) * q^46 - 4*b7 * q^47 + 2*b6 * q^49 + (3*b7 - 3*b5 - 2*b2 - 2*b1) * q^50 + (b4 - b3) * q^52 + b2 * q^53 - 6*b3 * q^55 + (5*b7 - 5*b5) * q^56 + (2*b6 + 12) * q^58 + (2*b5 + 3*b1) * q^59 + (-5*b3 - 5) * q^61 + (9*b5 + b1) * q^62 + (-3*b6 - 5) * q^64 + (-b2 - b1) * q^65 + (-3*b6 + 3*b4 + 7*b3 + 7) * q^67 + (4*b7 - 4*b5 + 2*b2 + 2*b1) * q^68 + (b6 - b4 + 6*b3 + 6) * q^70 + (4*b5 - 2*b1) * q^71 + 5*b3 * q^73 + (-3*b5 - 2*b1) * q^74 + (2*b4 - 7*b3 + 5) * q^76 + (-8*b7 + 8*b5 + b2 + b1) * q^77 + (-3*b4 - 7*b3) * q^79 - b2 * q^80 + (-4*b6 + 4*b4 - 12*b3 - 12) * q^82 + (-4*b7 + 4*b5 + 2*b2 + 2*b1) * q^83 + (-4*b6 + 4*b4 + 12*b3 + 12) * q^85 + (5*b7 - 3*b2) * q^86 + (-3*b6 - 12) * q^88 - 5*b2 * q^89 + (-b6 + b4 - b3 - 1) * q^91 + (4*b5 + 3*b1) * q^92 + (-4*b6 - 12) * q^94 + (4*b7 - 2*b5 + 4*b2 + b1) * q^95 + (2*b4 + 8*b3) * q^97 + (-4*b5 - 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{4} - 8 q^{7}+O(q^{10})$$ 8 * q - 4 * q^4 - 8 * q^7 $$8 q - 4 q^{4} - 8 q^{7} + 4 q^{13} - 4 q^{16} + 8 q^{19} + 24 q^{22} - 4 q^{25} + 28 q^{28} - 56 q^{31} - 8 q^{37} - 24 q^{40} + 4 q^{43} - 48 q^{46} + 4 q^{52} + 24 q^{55} + 96 q^{58} - 20 q^{61} - 40 q^{64} + 28 q^{67} + 24 q^{70} - 20 q^{73} + 68 q^{76} + 28 q^{79} - 48 q^{82} + 48 q^{85} - 96 q^{88} - 4 q^{91} - 96 q^{94} - 32 q^{97}+O(q^{100})$$ 8 * q - 4 * q^4 - 8 * q^7 + 4 * q^13 - 4 * q^16 + 8 * q^19 + 24 * q^22 - 4 * q^25 + 28 * q^28 - 56 * q^31 - 8 * q^37 - 24 * q^40 + 4 * q^43 - 48 * q^46 + 4 * q^52 + 24 * q^55 + 96 * q^58 - 20 * q^61 - 40 * q^64 + 28 * q^67 + 24 * q^70 - 20 * q^73 + 68 * q^76 + 28 * q^79 - 48 * q^82 + 48 * q^85 - 96 * q^88 - 4 * q^91 - 96 * q^94 - 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} - 21\nu^{5} + 87\nu^{3} - 162\nu ) / 135$$ (-v^7 - 21*v^5 + 87*v^3 - 162*v) / 135 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 24\nu^{5} - 48\nu^{3} - 27\nu ) / 135$$ (-v^7 + 24*v^5 - 48*v^3 - 27*v) / 135 $$\beta_{3}$$ $$=$$ $$( -2\nu^{6} + 3\nu^{4} - 6\nu^{2} - 9 ) / 45$$ (-2*v^6 + 3*v^4 - 6*v^2 - 9) / 45 $$\beta_{4}$$ $$=$$ $$( 2\nu^{6} - 3\nu^{4} + 51\nu^{2} - 81 ) / 45$$ (2*v^6 - 3*v^4 + 51*v^2 - 81) / 45 $$\beta_{5}$$ $$=$$ $$( -4\nu^{7} + 6\nu^{5} - 57\nu^{3} + 27\nu ) / 135$$ (-4*v^7 + 6*v^5 - 57*v^3 + 27*v) / 135 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} + 6\nu^{4} - 12\nu^{2} + 27 ) / 9$$ (-v^6 + 6*v^4 - 12*v^2 + 27) / 9 $$\beta_{7}$$ $$=$$ $$( -11\nu^{7} + 39\nu^{5} - 123\nu^{3} + 243\nu ) / 135$$ (-11*v^7 + 39*v^5 - 123*v^3 + 243*v) / 135
 $$\nu$$ $$=$$ $$( \beta_{7} - 2\beta_{5} - 2\beta_{2} - \beta_1 ) / 3$$ (b7 - 2*b5 - 2*b2 - b1) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 2$$ b4 + b3 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{7} - 3\beta_{5} + \beta_1$$ b7 - 3*b5 + b1 $$\nu^{4}$$ $$=$$ $$2\beta_{6} + 2\beta_{4} - 3\beta_{3} - 3$$ 2*b6 + 2*b4 - 3*b3 - 3 $$\nu^{5}$$ $$=$$ $$2\beta_{7} - 7\beta_{5} + 5\beta_{2} + \beta_1$$ 2*b7 - 7*b5 + 5*b2 + b1 $$\nu^{6}$$ $$=$$ $$3\beta_{6} - 30\beta_{3} - 15$$ 3*b6 - 30*b3 - 15 $$\nu^{7}$$ $$=$$ $$-9\beta_{7} - 6\beta_{5} + 3\beta_{2} - 15\beta_1$$ -9*b7 - 6*b5 + 3*b2 - 15*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −1.69185 − 0.370982i 1.27970 − 1.16721i −1.27970 + 1.16721i 1.69185 + 0.370982i −1.69185 + 0.370982i 1.27970 + 1.16721i −1.27970 − 1.16721i 1.69185 − 0.370982i
−1.16721 2.02166i 0 −1.72474 + 2.98735i −0.524648 0.908716i 0 −3.44949 3.38371 0 −1.22474 + 2.12132i
64.2 −0.370982 0.642559i 0 0.724745 1.25529i 1.65068 + 2.85906i 0 1.44949 −2.55940 0 1.22474 2.12132i
64.3 0.370982 + 0.642559i 0 0.724745 1.25529i −1.65068 2.85906i 0 1.44949 2.55940 0 1.22474 2.12132i
64.4 1.16721 + 2.02166i 0 −1.72474 + 2.98735i 0.524648 + 0.908716i 0 −3.44949 −3.38371 0 −1.22474 + 2.12132i
163.1 −1.16721 + 2.02166i 0 −1.72474 2.98735i −0.524648 + 0.908716i 0 −3.44949 3.38371 0 −1.22474 2.12132i
163.2 −0.370982 + 0.642559i 0 0.724745 + 1.25529i 1.65068 2.85906i 0 1.44949 −2.55940 0 1.22474 + 2.12132i
163.3 0.370982 0.642559i 0 0.724745 + 1.25529i −1.65068 + 2.85906i 0 1.44949 2.55940 0 1.22474 + 2.12132i
163.4 1.16721 2.02166i 0 −1.72474 2.98735i 0.524648 0.908716i 0 −3.44949 −3.38371 0 −1.22474 2.12132i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.f.c 8
3.b odd 2 1 inner 171.2.f.c 8
4.b odd 2 1 2736.2.s.bb 8
12.b even 2 1 2736.2.s.bb 8
19.c even 3 1 inner 171.2.f.c 8
19.c even 3 1 3249.2.a.bd 4
19.d odd 6 1 3249.2.a.be 4
57.f even 6 1 3249.2.a.be 4
57.h odd 6 1 inner 171.2.f.c 8
57.h odd 6 1 3249.2.a.bd 4
76.g odd 6 1 2736.2.s.bb 8
228.m even 6 1 2736.2.s.bb 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.f.c 8 1.a even 1 1 trivial
171.2.f.c 8 3.b odd 2 1 inner
171.2.f.c 8 19.c even 3 1 inner
171.2.f.c 8 57.h odd 6 1 inner
2736.2.s.bb 8 4.b odd 2 1
2736.2.s.bb 8 12.b even 2 1
2736.2.s.bb 8 76.g odd 6 1
2736.2.s.bb 8 228.m even 6 1
3249.2.a.bd 4 19.c even 3 1
3249.2.a.bd 4 57.h odd 6 1
3249.2.a.be 4 19.d odd 6 1
3249.2.a.be 4 57.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 6T_{2}^{6} + 33T_{2}^{4} + 18T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(171, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 6 T^{6} + \cdots + 9$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 12 T^{6} + \cdots + 144$$
$7$ $$(T^{2} + 2 T - 5)^{4}$$
$11$ $$(T^{4} - 36 T^{2} + 108)^{2}$$
$13$ $$(T^{2} - T + 1)^{4}$$
$17$ $$T^{8} + 48 T^{6} + \cdots + 36864$$
$19$ $$(T^{2} - 2 T + 19)^{4}$$
$23$ $$T^{8} + 36 T^{6} + \cdots + 90000$$
$29$ $$T^{8} + 144 T^{6} + \cdots + 23040000$$
$31$ $$(T^{2} + 14 T + 43)^{4}$$
$37$ $$(T^{2} + 2 T - 23)^{4}$$
$41$ $$T^{8} + 96 T^{6} + \cdots + 589824$$
$43$ $$(T^{4} - 2 T^{3} + \cdots + 2809)^{2}$$
$47$ $$T^{8} + 96 T^{6} + \cdots + 589824$$
$53$ $$T^{8} + 12 T^{6} + \cdots + 144$$
$59$ $$T^{8} + 132 T^{6} + \cdots + 18766224$$
$61$ $$(T^{2} + 5 T + 25)^{4}$$
$67$ $$(T^{4} - 14 T^{3} + \cdots + 25)^{2}$$
$71$ $$T^{8} + 144 T^{6} + \cdots + 23040000$$
$73$ $$(T^{2} + 5 T + 25)^{4}$$
$79$ $$(T^{4} - 14 T^{3} + \cdots + 25)^{2}$$
$83$ $$(T^{4} - 144 T^{2} + 1728)^{2}$$
$89$ $$T^{8} + 300 T^{6} + \cdots + 56250000$$
$97$ $$(T^{4} + 16 T^{3} + \cdots + 1600)^{2}$$