# Properties

 Label 171.1.p.a Level $171$ Weight $1$ Character orbit 171.p Analytic conductor $0.085$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,1,Mod(46,171)]

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, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("171.46");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 171.p (of order $$6$$, 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: $$0.0853401171602$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.22284891.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{4} + q^{7} +O(q^{10})$$ q - z * q^4 + q^7 $$q - \zeta_{6} q^{4} + q^{7} + (\zeta_{6}^{2} - 1) q^{13} + \zeta_{6}^{2} q^{16} - q^{19} + \zeta_{6} q^{25} - \zeta_{6} q^{28} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{31} + (\zeta_{6}^{2} + \zeta_{6}) q^{37} - \zeta_{6}^{2} q^{43} + (\zeta_{6} + 1) q^{52} - \zeta_{6} q^{61} + q^{64} + ( - \zeta_{6}^{2} + 1) q^{67} - \zeta_{6}^{2} q^{73} + \zeta_{6} q^{76} + ( - \zeta_{6} - 1) q^{79} + (\zeta_{6}^{2} - 1) q^{91} +O(q^{100})$$ q - z * q^4 + q^7 + (z^2 - 1) * q^13 + z^2 * q^16 - q^19 + z * q^25 - z * q^28 + (-z^2 - z) * q^31 + (z^2 + z) * q^37 - z^2 * q^43 + (z + 1) * q^52 - z * q^61 + q^64 + (-z^2 + 1) * q^67 - z^2 * q^73 + z * q^76 + (-z - 1) * q^79 + (z^2 - 1) * q^91 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{4} + 2 q^{7}+O(q^{10})$$ 2 * q - q^4 + 2 * q^7 $$2 q - q^{4} + 2 q^{7} - 3 q^{13} - q^{16} - 2 q^{19} + q^{25} - q^{28} + q^{43} + 3 q^{52} - q^{61} + 2 q^{64} + 3 q^{67} + q^{73} + q^{76} - 3 q^{79} - 3 q^{91}+O(q^{100})$$ 2 * q - q^4 + 2 * q^7 - 3 * q^13 - q^16 - 2 * q^19 + q^25 - q^28 + q^43 + 3 * q^52 - q^61 + 2 * q^64 + 3 * q^67 + q^73 + q^76 - 3 * q^79 - 3 * q^91

## Character values

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

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 −0.500000 0.866025i 0 0 1.00000 0 0 0
145.1 0 0 −0.500000 + 0.866025i 0 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-3})$$
19.d odd 6 1 inner
57.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.1.p.a 2
3.b odd 2 1 CM 171.1.p.a 2
4.b odd 2 1 2736.1.cd.a 2
9.c even 3 1 1539.1.i.a 2
9.c even 3 1 1539.1.s.a 2
9.d odd 6 1 1539.1.i.a 2
9.d odd 6 1 1539.1.s.a 2
12.b even 2 1 2736.1.cd.a 2
19.b odd 2 1 3249.1.p.b 2
19.c even 3 1 3249.1.c.a 2
19.c even 3 1 3249.1.p.b 2
19.d odd 6 1 inner 171.1.p.a 2
19.d odd 6 1 3249.1.c.a 2
19.e even 9 3 3249.1.ba.a 6
19.e even 9 3 3249.1.ba.b 6
19.f odd 18 3 3249.1.ba.a 6
19.f odd 18 3 3249.1.ba.b 6
57.d even 2 1 3249.1.p.b 2
57.f even 6 1 inner 171.1.p.a 2
57.f even 6 1 3249.1.c.a 2
57.h odd 6 1 3249.1.c.a 2
57.h odd 6 1 3249.1.p.b 2
57.j even 18 3 3249.1.ba.a 6
57.j even 18 3 3249.1.ba.b 6
57.l odd 18 3 3249.1.ba.a 6
57.l odd 18 3 3249.1.ba.b 6
76.f even 6 1 2736.1.cd.a 2
171.i odd 6 1 1539.1.s.a 2
171.k even 6 1 1539.1.i.a 2
171.s odd 6 1 1539.1.i.a 2
171.t even 6 1 1539.1.s.a 2
228.n odd 6 1 2736.1.cd.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.p.a 2 1.a even 1 1 trivial
171.1.p.a 2 3.b odd 2 1 CM
171.1.p.a 2 19.d odd 6 1 inner
171.1.p.a 2 57.f even 6 1 inner
1539.1.i.a 2 9.c even 3 1
1539.1.i.a 2 9.d odd 6 1
1539.1.i.a 2 171.k even 6 1
1539.1.i.a 2 171.s odd 6 1
1539.1.s.a 2 9.c even 3 1
1539.1.s.a 2 9.d odd 6 1
1539.1.s.a 2 171.i odd 6 1
1539.1.s.a 2 171.t even 6 1
2736.1.cd.a 2 4.b odd 2 1
2736.1.cd.a 2 12.b even 2 1
2736.1.cd.a 2 76.f even 6 1
2736.1.cd.a 2 228.n odd 6 1
3249.1.c.a 2 19.c even 3 1
3249.1.c.a 2 19.d odd 6 1
3249.1.c.a 2 57.f even 6 1
3249.1.c.a 2 57.h odd 6 1
3249.1.p.b 2 19.b odd 2 1
3249.1.p.b 2 19.c even 3 1
3249.1.p.b 2 57.d even 2 1
3249.1.p.b 2 57.h odd 6 1
3249.1.ba.a 6 19.e even 9 3
3249.1.ba.a 6 19.f odd 18 3
3249.1.ba.a 6 57.j even 18 3
3249.1.ba.a 6 57.l odd 18 3
3249.1.ba.b 6 19.e even 9 3
3249.1.ba.b 6 19.f odd 18 3
3249.1.ba.b 6 57.j even 18 3
3249.1.ba.b 6 57.l odd 18 3

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(171, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 3T + 3$$
$17$ $$T^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 3$$
$37$ $$T^{2} + 3$$
$41$ $$T^{2}$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} - 3T + 3$$
$71$ $$T^{2}$$
$73$ $$T^{2} - T + 1$$
$79$ $$T^{2} + 3T + 3$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$