# Properties

 Label 3648.1.l.a Level $3648$ Weight $1$ Character orbit 3648.l Analytic conductor $1.821$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -3, -152, 456 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,1,Mod(1823,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3648.1823");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3648.l (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: no Analytic conductor: $$1.82058916609$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{-38})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.1916338176.4

## $q$-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.

 $$f(q)$$ $$=$$ $$q + i q^{3} - i q^{7} - q^{9} +O(q^{10})$$ q + z * q^3 - z * q^7 - q^9 $$q + i q^{3} - i q^{7} - q^{9} - q^{13} + i q^{19} + 2 q^{21} - q^{25} - i q^{27} - q^{37} - 2 i q^{39} - 3 q^{49} - q^{57} + 2 i q^{63} - i q^{67} - q^{73} - i q^{75} + q^{81} + 4 i q^{91} +O(q^{100})$$ q + z * q^3 - z * q^7 - q^9 - q^13 + z * q^19 + 2 * q^21 - q^25 - z * q^27 - q^37 - 2*z * q^39 - 3 * q^49 - q^57 + 2*z * q^63 - z * q^67 - q^73 - z * q^75 + q^81 + 4*z * q^91 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 4 q^{13} + 4 q^{21} - 2 q^{25} - 4 q^{37} - 6 q^{49} - 2 q^{57} - 4 q^{73} + 2 q^{81}+O(q^{100})$$ 2 * q - 2 * q^9 - 4 * q^13 + 4 * q^21 - 2 * q^25 - 4 * q^37 - 6 * q^49 - 2 * q^57 - 4 * q^73 + 2 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times$$.

 $$n$$ $$1217$$ $$1921$$ $$2053$$ $$2623$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1823.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 2.00000i 0 −1.00000 0
1823.2 0 1.00000i 0 0 0 2.00000i 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 $$\Q(\sqrt{-3})$$
152.g odd 2 1 CM by $$\Q(\sqrt{-38})$$
456.p even 2 1 RM by $$\Q(\sqrt{114})$$
4.b odd 2 1 inner
12.b even 2 1 inner
152.b even 2 1 inner
456.l odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3648.1.l.a 2
3.b odd 2 1 CM 3648.1.l.a 2
4.b odd 2 1 inner 3648.1.l.a 2
8.b even 2 1 3648.1.l.b yes 2
8.d odd 2 1 3648.1.l.b yes 2
12.b even 2 1 inner 3648.1.l.a 2
19.b odd 2 1 3648.1.l.b yes 2
24.f even 2 1 3648.1.l.b yes 2
24.h odd 2 1 3648.1.l.b yes 2
57.d even 2 1 3648.1.l.b yes 2
76.d even 2 1 3648.1.l.b yes 2
152.b even 2 1 inner 3648.1.l.a 2
152.g odd 2 1 CM 3648.1.l.a 2
228.b odd 2 1 3648.1.l.b yes 2
456.l odd 2 1 inner 3648.1.l.a 2
456.p even 2 1 RM 3648.1.l.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3648.1.l.a 2 1.a even 1 1 trivial
3648.1.l.a 2 3.b odd 2 1 CM
3648.1.l.a 2 4.b odd 2 1 inner
3648.1.l.a 2 12.b even 2 1 inner
3648.1.l.a 2 152.b even 2 1 inner
3648.1.l.a 2 152.g odd 2 1 CM
3648.1.l.a 2 456.l odd 2 1 inner
3648.1.l.a 2 456.p even 2 1 RM
3648.1.l.b yes 2 8.b even 2 1
3648.1.l.b yes 2 8.d odd 2 1
3648.1.l.b yes 2 19.b odd 2 1
3648.1.l.b yes 2 24.f even 2 1
3648.1.l.b yes 2 24.h odd 2 1
3648.1.l.b yes 2 57.d even 2 1
3648.1.l.b yes 2 76.d even 2 1
3648.1.l.b yes 2 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3648, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 1$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$