# Properties

 Label 3332.1.o.c Level $3332$ Weight $1$ Character orbit 3332.o Analytic conductor $1.663$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -4, -68, 17 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(67,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4, 3]))

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

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

chi := DirichletCharacter("3332.67");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.o (of order $$6$$, degree $$2$$, 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: no Analytic conductor: $$1.66288462209$$ 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: $$\Z[a_1, a_2]$$ 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: $C_3\times D_4$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} + \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - q^{8} - \zeta_{6}^{2} q^{9} +O(q^{10})$$ q - z^2 * q^2 - z * q^4 - q^8 - z^2 * q^9 $$q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - q^{8} - \zeta_{6}^{2} q^{9} - 2 q^{13} + \zeta_{6}^{2} q^{16} - \zeta_{6} q^{17} - \zeta_{6} q^{18} - \zeta_{6} q^{25} + 2 \zeta_{6}^{2} q^{26} + \zeta_{6} q^{32} - q^{34} - q^{36} - q^{50} + 2 \zeta_{6} q^{52} - 2 \zeta_{6} q^{53} + q^{64} + \zeta_{6}^{2} q^{68} + \zeta_{6}^{2} q^{72} - \zeta_{6} q^{81} - 2 \zeta_{6}^{2} q^{89} +O(q^{100})$$ q - z^2 * q^2 - z * q^4 - q^8 - z^2 * q^9 - 2 * q^13 + z^2 * q^16 - z * q^17 - z * q^18 - z * q^25 + 2*z^2 * q^26 + z * q^32 - q^34 - q^36 - q^50 + 2*z * q^52 - 2*z * q^53 + q^64 + z^2 * q^68 + z^2 * q^72 - z * q^81 - 2*z^2 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{8} + q^{9}+O(q^{10})$$ 2 * q + q^2 - q^4 - 2 * q^8 + q^9 $$2 q + q^{2} - q^{4} - 2 q^{8} + q^{9} - 4 q^{13} - q^{16} - q^{17} - q^{18} - q^{25} - 2 q^{26} + q^{32} - 2 q^{34} - 2 q^{36} - 2 q^{50} + 2 q^{52} - 2 q^{53} + 2 q^{64} - q^{68} - q^{72} - q^{81} + 2 q^{89}+O(q^{100})$$ 2 * q + q^2 - q^4 - 2 * q^8 + q^9 - 4 * q^13 - q^16 - q^17 - q^18 - q^25 - 2 * q^26 + q^32 - 2 * q^34 - 2 * q^36 - 2 * q^50 + 2 * q^52 - 2 * q^53 + 2 * q^64 - q^68 - q^72 - q^81 + 2 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −1.00000 0.500000 + 0.866025i 0
2039.1 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −1.00000 0.500000 0.866025i 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 $$\Q(\sqrt{-17})$$
7.c even 3 1 inner
28.g odd 6 1 inner
119.j even 6 1 inner
476.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.o.c 2
4.b odd 2 1 CM 3332.1.o.c 2
7.b odd 2 1 3332.1.o.d 2
7.c even 3 1 68.1.d.a 1
7.c even 3 1 inner 3332.1.o.c 2
7.d odd 6 1 3332.1.g.a 1
7.d odd 6 1 3332.1.o.d 2
17.b even 2 1 RM 3332.1.o.c 2
21.h odd 6 1 612.1.e.a 1
28.d even 2 1 3332.1.o.d 2
28.f even 6 1 3332.1.g.a 1
28.f even 6 1 3332.1.o.d 2
28.g odd 6 1 68.1.d.a 1
28.g odd 6 1 inner 3332.1.o.c 2
35.j even 6 1 1700.1.h.d 1
35.l odd 12 2 1700.1.d.b 2
56.k odd 6 1 1088.1.g.a 1
56.p even 6 1 1088.1.g.a 1
68.d odd 2 1 CM 3332.1.o.c 2
84.n even 6 1 612.1.e.a 1
119.d odd 2 1 3332.1.o.d 2
119.h odd 6 1 3332.1.g.a 1
119.h odd 6 1 3332.1.o.d 2
119.j even 6 1 68.1.d.a 1
119.j even 6 1 inner 3332.1.o.c 2
119.n even 12 2 1156.1.c.a 1
119.q even 24 4 1156.1.f.a 2
119.t odd 48 8 1156.1.g.a 4
140.p odd 6 1 1700.1.h.d 1
140.w even 12 2 1700.1.d.b 2
357.q odd 6 1 612.1.e.a 1
476.e even 2 1 3332.1.o.d 2
476.o odd 6 1 68.1.d.a 1
476.o odd 6 1 inner 3332.1.o.c 2
476.q even 6 1 3332.1.g.a 1
476.q even 6 1 3332.1.o.d 2
476.bb odd 12 2 1156.1.c.a 1
476.bg odd 24 4 1156.1.f.a 2
476.bm even 48 8 1156.1.g.a 4
595.z even 6 1 1700.1.h.d 1
595.bp odd 12 2 1700.1.d.b 2
952.z even 6 1 1088.1.g.a 1
952.bi odd 6 1 1088.1.g.a 1
1428.be even 6 1 612.1.e.a 1
2380.bz odd 6 1 1700.1.h.d 1
2380.cz even 12 2 1700.1.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 7.c even 3 1
68.1.d.a 1 28.g odd 6 1
68.1.d.a 1 119.j even 6 1
68.1.d.a 1 476.o odd 6 1
612.1.e.a 1 21.h odd 6 1
612.1.e.a 1 84.n even 6 1
612.1.e.a 1 357.q odd 6 1
612.1.e.a 1 1428.be even 6 1
1088.1.g.a 1 56.k odd 6 1
1088.1.g.a 1 56.p even 6 1
1088.1.g.a 1 952.z even 6 1
1088.1.g.a 1 952.bi odd 6 1
1156.1.c.a 1 119.n even 12 2
1156.1.c.a 1 476.bb odd 12 2
1156.1.f.a 2 119.q even 24 4
1156.1.f.a 2 476.bg odd 24 4
1156.1.g.a 4 119.t odd 48 8
1156.1.g.a 4 476.bm even 48 8
1700.1.d.b 2 35.l odd 12 2
1700.1.d.b 2 140.w even 12 2
1700.1.d.b 2 595.bp odd 12 2
1700.1.d.b 2 2380.cz even 12 2
1700.1.h.d 1 35.j even 6 1
1700.1.h.d 1 140.p odd 6 1
1700.1.h.d 1 595.z even 6 1
1700.1.h.d 1 2380.bz odd 6 1
3332.1.g.a 1 7.d odd 6 1
3332.1.g.a 1 28.f even 6 1
3332.1.g.a 1 119.h odd 6 1
3332.1.g.a 1 476.q even 6 1
3332.1.o.c 2 1.a even 1 1 trivial
3332.1.o.c 2 4.b odd 2 1 CM
3332.1.o.c 2 7.c even 3 1 inner
3332.1.o.c 2 17.b even 2 1 RM
3332.1.o.c 2 28.g odd 6 1 inner
3332.1.o.c 2 68.d odd 2 1 CM
3332.1.o.c 2 119.j even 6 1 inner
3332.1.o.c 2 476.o odd 6 1 inner
3332.1.o.d 2 7.b odd 2 1
3332.1.o.d 2 7.d odd 6 1
3332.1.o.d 2 28.d even 2 1
3332.1.o.d 2 28.f even 6 1
3332.1.o.d 2 119.d odd 2 1
3332.1.o.d 2 119.h odd 6 1
3332.1.o.d 2 476.e even 2 1
3332.1.o.d 2 476.q even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}$$ T5 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

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