# Properties

 Label 3344.1.p.c Level $3344$ Weight $1$ Character orbit 3344.p Analytic conductor $1.669$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,1,Mod(3343,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("3344.3343");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3344 = 2^{4} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3344.p (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: $$1.66887340224$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.123005696.3

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{5} - q^{7} - q^{9}+O(q^{10})$$ q + q^5 - q^7 - q^9 $$q + q^{5} - q^{7} - q^{9} + \zeta_{6}^{2} q^{11} + (\zeta_{6}^{2} + \zeta_{6}) q^{17} - q^{19} - q^{35} - q^{43} - q^{45} + (\zeta_{6}^{2} + \zeta_{6}) q^{47} + \zeta_{6}^{2} q^{55} + (\zeta_{6}^{2} + \zeta_{6}) q^{61} + q^{63} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{73} - \zeta_{6}^{2} q^{77} + q^{81} + 2 q^{83} + (\zeta_{6}^{2} + \zeta_{6}) q^{85} - q^{95} - \zeta_{6}^{2} q^{99} +O(q^{100})$$ q + q^5 - q^7 - q^9 + z^2 * q^11 + (z^2 + z) * q^17 - q^19 - q^35 - q^43 - q^45 + (z^2 + z) * q^47 + z^2 * q^55 + (z^2 + z) * q^61 + q^63 + (-z^2 - z) * q^73 - z^2 * q^77 + q^81 + 2 * q^83 + (z^2 + z) * q^85 - q^95 - z^2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9 $$2 q + 2 q^{5} - 2 q^{7} - 2 q^{9} - q^{11} - 2 q^{19} - 2 q^{35} - 2 q^{43} - 2 q^{45} - q^{55} + 2 q^{63} + q^{77} + 2 q^{81} + 4 q^{83} - 2 q^{95} + q^{99}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^7 - 2 * q^9 - q^11 - 2 * q^19 - 2 * q^35 - 2 * q^43 - 2 * q^45 - q^55 + 2 * q^63 + q^77 + 2 * q^81 + 4 * q^83 - 2 * q^95 + q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3344\mathbb{Z}\right)^\times$$.

 $$n$$ $$705$$ $$837$$ $$2433$$ $$2927$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3343.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 1.00000 0 −1.00000 0 −1.00000 0
3343.2 0 0 0 1.00000 0 −1.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
44.c even 2 1 inner
836.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3344.1.p.c 2
4.b odd 2 1 3344.1.p.d yes 2
11.b odd 2 1 3344.1.p.d yes 2
19.b odd 2 1 CM 3344.1.p.c 2
44.c even 2 1 inner 3344.1.p.c 2
76.d even 2 1 3344.1.p.d yes 2
209.d even 2 1 3344.1.p.d yes 2
836.h odd 2 1 inner 3344.1.p.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3344.1.p.c 2 1.a even 1 1 trivial
3344.1.p.c 2 19.b odd 2 1 CM
3344.1.p.c 2 44.c even 2 1 inner
3344.1.p.c 2 836.h odd 2 1 inner
3344.1.p.d yes 2 4.b odd 2 1
3344.1.p.d yes 2 11.b odd 2 1
3344.1.p.d yes 2 76.d even 2 1
3344.1.p.d yes 2 209.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3344, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5} - 1$$ T5 - 1 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

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