# Properties

 Label 3332.1.g.f Level $3332$ Weight $1$ Character orbit 3332.g Self dual yes Analytic conductor $1.663$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -68 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(883,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3332.883");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.g (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: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 476) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.188737808.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta q^{3} + q^{4} + \beta q^{6} - q^{8} + 2 q^{9} +O(q^{10})$$ q - q^2 - b * q^3 + q^4 + b * q^6 - q^8 + 2 * q^9 $$q - q^{2} - \beta q^{3} + q^{4} + \beta q^{6} - q^{8} + 2 q^{9} - \beta q^{11} - \beta q^{12} - q^{13} + q^{16} - q^{17} - 2 q^{18} + \beta q^{22} + \beta q^{24} + q^{25} + q^{26} - \beta q^{27} - q^{32} + 3 q^{33} + q^{34} + 2 q^{36} + \beta q^{39} - \beta q^{44} - \beta q^{48} - q^{50} + \beta q^{51} - q^{52} - q^{53} + \beta q^{54} + q^{64} - 3 q^{66} - q^{68} + \beta q^{71} - 2 q^{72} - \beta q^{75} - \beta q^{78} - \beta q^{79} + q^{81} + \beta q^{88} - q^{89} + \beta q^{96} - 2 \beta q^{99} +O(q^{100})$$ q - q^2 - b * q^3 + q^4 + b * q^6 - q^8 + 2 * q^9 - b * q^11 - b * q^12 - q^13 + q^16 - q^17 - 2 * q^18 + b * q^22 + b * q^24 + q^25 + q^26 - b * q^27 - q^32 + 3 * q^33 + q^34 + 2 * q^36 + b * q^39 - b * q^44 - b * q^48 - q^50 + b * q^51 - q^52 - q^53 + b * q^54 + q^64 - 3 * q^66 - q^68 + b * q^71 - 2 * q^72 - b * q^75 - b * q^78 - b * q^79 + q^81 + b * q^88 - q^89 + b * q^96 - 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{9} - 2 q^{13} + 2 q^{16} - 2 q^{17} - 4 q^{18} + 2 q^{25} + 2 q^{26} - 2 q^{32} + 6 q^{33} + 2 q^{34} + 4 q^{36} - 2 q^{50} - 2 q^{52} - 2 q^{53} + 2 q^{64} - 6 q^{66} - 2 q^{68} - 4 q^{72} + 2 q^{81} - 2 q^{89}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 4 * q^9 - 2 * q^13 + 2 * q^16 - 2 * q^17 - 4 * q^18 + 2 * q^25 + 2 * q^26 - 2 * q^32 + 6 * q^33 + 2 * q^34 + 4 * q^36 - 2 * q^50 - 2 * q^52 - 2 * q^53 + 2 * q^64 - 6 * q^66 - 2 * q^68 - 4 * q^72 + 2 * q^81 - 2 * q^89

## Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 1.73205 −1.73205
−1.00000 −1.73205 1.00000 0 1.73205 0 −1.00000 2.00000 0
883.2 −1.00000 1.73205 1.00000 0 −1.73205 0 −1.00000 2.00000 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
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
4.b odd 2 1 inner
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.g.f 2
4.b odd 2 1 inner 3332.1.g.f 2
7.b odd 2 1 3332.1.g.g 2
7.c even 3 2 3332.1.o.f 4
7.d odd 6 2 476.1.o.c 4
17.b even 2 1 inner 3332.1.g.f 2
28.d even 2 1 3332.1.g.g 2
28.f even 6 2 476.1.o.c 4
28.g odd 6 2 3332.1.o.f 4
68.d odd 2 1 CM 3332.1.g.f 2
119.d odd 2 1 3332.1.g.g 2
119.h odd 6 2 476.1.o.c 4
119.j even 6 2 3332.1.o.f 4
476.e even 2 1 3332.1.g.g 2
476.o odd 6 2 3332.1.o.f 4
476.q even 6 2 476.1.o.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.1.o.c 4 7.d odd 6 2
476.1.o.c 4 28.f even 6 2
476.1.o.c 4 119.h odd 6 2
476.1.o.c 4 476.q even 6 2
3332.1.g.f 2 1.a even 1 1 trivial
3332.1.g.f 2 4.b odd 2 1 inner
3332.1.g.f 2 17.b even 2 1 inner
3332.1.g.f 2 68.d odd 2 1 CM
3332.1.g.g 2 7.b odd 2 1
3332.1.g.g 2 28.d even 2 1
3332.1.g.g 2 119.d odd 2 1
3332.1.g.g 2 476.e even 2 1
3332.1.o.f 4 7.c even 3 2
3332.1.o.f 4 28.g odd 6 2
3332.1.o.f 4 119.j even 6 2
3332.1.o.f 4 476.o odd 6 2

## Hecke kernels

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

 $$T_{3}^{2} - 3$$ T3^2 - 3 $$T_{5}$$ T5 $$T_{11}^{2} - 3$$ T11^2 - 3 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

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