# Properties

 Label 612.1.e.a Level $612$ Weight $1$ Character orbit 612.e Self dual yes Analytic conductor $0.305$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -68, 17 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [612,1,Mod(271,612)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(612, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("612.271");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$612 = 2^{2} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 612.e (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: $$0.305427787731$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{17})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.2448.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^8 $$q + q^{2} + q^{4} + q^{8} - 2 q^{13} + q^{16} - q^{17} + q^{25} - 2 q^{26} + q^{32} - q^{34} - q^{49} + q^{50} - 2 q^{52} - 2 q^{53} + q^{64} - q^{68} + 2 q^{89} - q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^8 - 2 * q^13 + q^16 - q^17 + q^25 - 2 * q^26 + q^32 - q^34 - q^49 + q^50 - 2 * q^52 - 2 * q^53 + q^64 - q^68 + 2 * q^89 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/612\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$137$$ $$307$$ $$\chi(n)$$ $$1$$ $$0$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0
1.00000 0 1.00000 0 0 0 1.00000 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
17.b even 2 1 RM by $$\Q(\sqrt{17})$$
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 612.1.e.a 1
3.b odd 2 1 68.1.d.a 1
4.b odd 2 1 CM 612.1.e.a 1
12.b even 2 1 68.1.d.a 1
15.d odd 2 1 1700.1.h.d 1
15.e even 4 2 1700.1.d.b 2
17.b even 2 1 RM 612.1.e.a 1
21.c even 2 1 3332.1.g.a 1
21.g even 6 2 3332.1.o.d 2
21.h odd 6 2 3332.1.o.c 2
24.f even 2 1 1088.1.g.a 1
24.h odd 2 1 1088.1.g.a 1
51.c odd 2 1 68.1.d.a 1
51.f odd 4 2 1156.1.c.a 1
51.g odd 8 4 1156.1.f.a 2
51.i even 16 8 1156.1.g.a 4
60.h even 2 1 1700.1.h.d 1
60.l odd 4 2 1700.1.d.b 2
68.d odd 2 1 CM 612.1.e.a 1
84.h odd 2 1 3332.1.g.a 1
84.j odd 6 2 3332.1.o.d 2
84.n even 6 2 3332.1.o.c 2
204.h even 2 1 68.1.d.a 1
204.l even 4 2 1156.1.c.a 1
204.p even 8 4 1156.1.f.a 2
204.t odd 16 8 1156.1.g.a 4
255.h odd 2 1 1700.1.h.d 1
255.o even 4 2 1700.1.d.b 2
357.c even 2 1 3332.1.g.a 1
357.q odd 6 2 3332.1.o.c 2
357.s even 6 2 3332.1.o.d 2
408.b odd 2 1 1088.1.g.a 1
408.h even 2 1 1088.1.g.a 1
1020.b even 2 1 1700.1.h.d 1
1020.x odd 4 2 1700.1.d.b 2
1428.b odd 2 1 3332.1.g.a 1
1428.be even 6 2 3332.1.o.c 2
1428.bl odd 6 2 3332.1.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 3.b odd 2 1
68.1.d.a 1 12.b even 2 1
68.1.d.a 1 51.c odd 2 1
68.1.d.a 1 204.h even 2 1
612.1.e.a 1 1.a even 1 1 trivial
612.1.e.a 1 4.b odd 2 1 CM
612.1.e.a 1 17.b even 2 1 RM
612.1.e.a 1 68.d odd 2 1 CM
1088.1.g.a 1 24.f even 2 1
1088.1.g.a 1 24.h odd 2 1
1088.1.g.a 1 408.b odd 2 1
1088.1.g.a 1 408.h even 2 1
1156.1.c.a 1 51.f odd 4 2
1156.1.c.a 1 204.l even 4 2
1156.1.f.a 2 51.g odd 8 4
1156.1.f.a 2 204.p even 8 4
1156.1.g.a 4 51.i even 16 8
1156.1.g.a 4 204.t odd 16 8
1700.1.d.b 2 15.e even 4 2
1700.1.d.b 2 60.l odd 4 2
1700.1.d.b 2 255.o even 4 2
1700.1.d.b 2 1020.x odd 4 2
1700.1.h.d 1 15.d odd 2 1
1700.1.h.d 1 60.h even 2 1
1700.1.h.d 1 255.h odd 2 1
1700.1.h.d 1 1020.b even 2 1
3332.1.g.a 1 21.c even 2 1
3332.1.g.a 1 84.h odd 2 1
3332.1.g.a 1 357.c even 2 1
3332.1.g.a 1 1428.b odd 2 1
3332.1.o.c 2 21.h odd 6 2
3332.1.o.c 2 84.n even 6 2
3332.1.o.c 2 357.q odd 6 2
3332.1.o.c 2 1428.be even 6 2
3332.1.o.d 2 21.g even 6 2
3332.1.o.d 2 84.j odd 6 2
3332.1.o.d 2 357.s even 6 2
3332.1.o.d 2 1428.bl odd 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(612, [\chi])$$.

## Hecke characteristic polynomials

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