# Properties

 Label 684.1.h.a Level $684$ Weight $1$ Character orbit 684.h Self dual yes Analytic conductor $0.341$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -19 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,1,Mod(37,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("684.37");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 684.h (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: $$0.341360468641$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.76.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.155952.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{5} - q^{7}+O(q^{10})$$ q + q^5 - q^7 $$q + q^{5} - q^{7} + q^{11} + q^{17} + q^{19} - 2 q^{23} - q^{35} - q^{43} + q^{47} + q^{55} - q^{61} - q^{73} - q^{77} - 2 q^{83} + q^{85} + q^{95}+O(q^{100})$$ q + q^5 - q^7 + q^11 + q^17 + q^19 - 2 * q^23 - q^35 - q^43 + q^47 + q^55 - q^61 - q^73 - q^77 - 2 * q^83 + q^85 + q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$1$$ $$0$$ $$0$$

## 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 0 1.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.1.h.a 1
3.b odd 2 1 76.1.c.a 1
4.b odd 2 1 2736.1.o.b 1
12.b even 2 1 304.1.e.a 1
15.d odd 2 1 1900.1.e.a 1
15.e even 4 2 1900.1.g.a 2
19.b odd 2 1 CM 684.1.h.a 1
21.c even 2 1 3724.1.e.c 1
21.g even 6 2 3724.1.bc.b 2
21.h odd 6 2 3724.1.bc.c 2
24.f even 2 1 1216.1.e.b 1
24.h odd 2 1 1216.1.e.a 1
57.d even 2 1 76.1.c.a 1
57.f even 6 2 1444.1.h.a 2
57.h odd 6 2 1444.1.h.a 2
57.j even 18 6 1444.1.j.a 6
57.l odd 18 6 1444.1.j.a 6
76.d even 2 1 2736.1.o.b 1
228.b odd 2 1 304.1.e.a 1
285.b even 2 1 1900.1.e.a 1
285.j odd 4 2 1900.1.g.a 2
399.h odd 2 1 3724.1.e.c 1
399.s odd 6 2 3724.1.bc.b 2
399.w even 6 2 3724.1.bc.c 2
456.l odd 2 1 1216.1.e.b 1
456.p even 2 1 1216.1.e.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 3.b odd 2 1
76.1.c.a 1 57.d even 2 1
304.1.e.a 1 12.b even 2 1
304.1.e.a 1 228.b odd 2 1
684.1.h.a 1 1.a even 1 1 trivial
684.1.h.a 1 19.b odd 2 1 CM
1216.1.e.a 1 24.h odd 2 1
1216.1.e.a 1 456.p even 2 1
1216.1.e.b 1 24.f even 2 1
1216.1.e.b 1 456.l odd 2 1
1444.1.h.a 2 57.f even 6 2
1444.1.h.a 2 57.h odd 6 2
1444.1.j.a 6 57.j even 18 6
1444.1.j.a 6 57.l odd 18 6
1900.1.e.a 1 15.d odd 2 1
1900.1.e.a 1 285.b even 2 1
1900.1.g.a 2 15.e even 4 2
1900.1.g.a 2 285.j odd 4 2
2736.1.o.b 1 4.b odd 2 1
2736.1.o.b 1 76.d even 2 1
3724.1.e.c 1 21.c even 2 1
3724.1.e.c 1 399.h odd 2 1
3724.1.bc.b 2 21.g even 6 2
3724.1.bc.b 2 399.s odd 6 2
3724.1.bc.c 2 21.h odd 6 2
3724.1.bc.c 2 399.w even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1$$ acting on $$S_{1}^{\mathrm{new}}(684, [\chi])$$.

## Hecke characteristic polynomials

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