# Properties

 Label 171.1.c.a Level $171$ Weight $1$ Character orbit 171.c Self dual yes Analytic conductor $0.085$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -19, 57 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,1,Mod(37,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(2))

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("171.37");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 171.c (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$0.0853401171602$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{-19})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.513.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{4} - 2 q^{7}+O(q^{10})$$ q + q^4 - 2 * q^7 $$q + q^{4} - 2 q^{7} + q^{16} - q^{19} - q^{25} - 2 q^{28} + 2 q^{43} + 3 q^{49} - 2 q^{61} + q^{64} + 2 q^{73} - q^{76}+O(q^{100})$$ q + q^4 - 2 * q^7 + q^16 - q^19 - q^25 - 2 * q^28 + 2 * q^43 + 3 * q^49 - 2 * q^61 + q^64 + 2 * q^73 - q^76

## Character values

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

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0
0 0 1.00000 0 0 −2.00000 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.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
57.d even 2 1 RM by $$\Q(\sqrt{57})$$

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

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