# Properties

 Label 2352.1.d.b Level $2352$ Weight $1$ Character orbit 2352.d Self dual yes Analytic conductor $1.174$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,1,Mod(785,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("2352.785");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2352.d (of order $$2$$, degree $$1$$, not 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: $$1.17380090971$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.588.1 Artin image: $D_6$ Artin field: Galois closure of 6.2.38723328.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{9}+O(q^{10})$$ q + q^3 + q^9 $$q + q^{3} + q^{9} + q^{13} - q^{19} + q^{25} + q^{27} - q^{31} - q^{37} + q^{39} + q^{43} - q^{57} - 2 q^{61} + q^{67} + q^{73} + q^{75} + q^{79} + q^{81} - q^{93} - 2 q^{97}+O(q^{100})$$ q + q^3 + q^9 + q^13 - q^19 + q^25 + q^27 - q^31 - q^37 + q^39 + q^43 - q^57 - 2 * q^61 + q^67 + q^73 + q^75 + q^79 + q^81 - q^93 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
785.1
 0
0 1.00000 0 0 0 0 0 1.00000 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 $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.1.d.b 1
3.b odd 2 1 CM 2352.1.d.b 1
4.b odd 2 1 588.1.c.a 1
7.b odd 2 1 2352.1.d.a 1
7.c even 3 2 2352.1.bn.a 2
7.d odd 6 2 336.1.bn.a 2
12.b even 2 1 588.1.c.a 1
21.c even 2 1 2352.1.d.a 1
21.g even 6 2 336.1.bn.a 2
21.h odd 6 2 2352.1.bn.a 2
28.d even 2 1 588.1.c.b 1
28.f even 6 2 84.1.p.a 2
28.g odd 6 2 588.1.p.a 2
56.j odd 6 2 1344.1.bn.a 2
56.m even 6 2 1344.1.bn.b 2
84.h odd 2 1 588.1.c.b 1
84.j odd 6 2 84.1.p.a 2
84.n even 6 2 588.1.p.a 2
140.s even 6 2 2100.1.bn.c 2
140.x odd 12 4 2100.1.bh.a 4
168.ba even 6 2 1344.1.bn.a 2
168.be odd 6 2 1344.1.bn.b 2
252.n even 6 2 2268.1.m.a 2
252.r odd 6 2 2268.1.bh.b 2
252.bj even 6 2 2268.1.bh.b 2
252.bn odd 6 2 2268.1.m.a 2
420.be odd 6 2 2100.1.bn.c 2
420.br even 12 4 2100.1.bh.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 28.f even 6 2
84.1.p.a 2 84.j odd 6 2
336.1.bn.a 2 7.d odd 6 2
336.1.bn.a 2 21.g even 6 2
588.1.c.a 1 4.b odd 2 1
588.1.c.a 1 12.b even 2 1
588.1.c.b 1 28.d even 2 1
588.1.c.b 1 84.h odd 2 1
588.1.p.a 2 28.g odd 6 2
588.1.p.a 2 84.n even 6 2
1344.1.bn.a 2 56.j odd 6 2
1344.1.bn.a 2 168.ba even 6 2
1344.1.bn.b 2 56.m even 6 2
1344.1.bn.b 2 168.be odd 6 2
2100.1.bh.a 4 140.x odd 12 4
2100.1.bh.a 4 420.br even 12 4
2100.1.bn.c 2 140.s even 6 2
2100.1.bn.c 2 420.be odd 6 2
2268.1.m.a 2 252.n even 6 2
2268.1.m.a 2 252.bn odd 6 2
2268.1.bh.b 2 252.r odd 6 2
2268.1.bh.b 2 252.bj even 6 2
2352.1.d.a 1 7.b odd 2 1
2352.1.d.a 1 21.c even 2 1
2352.1.d.b 1 1.a even 1 1 trivial
2352.1.d.b 1 3.b odd 2 1 CM
2352.1.bn.a 2 7.c even 3 2
2352.1.bn.a 2 21.h odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13} - 1$$ acting on $$S_{1}^{\mathrm{new}}(2352, [\chi])$$.

## Hecke characteristic polynomials

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