# Properties

 Label 3332.1.w.a Level $3332$ Weight $1$ Character orbit 3332.w Analytic conductor $1.663$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(491,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 0, 3]))

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

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

chi := DirichletCharacter("3332.491");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.w (of order $$8$$, degree $$4$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.2.3089659810545728.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 1) q^{5} - \zeta_{8}^{3} q^{8} - \zeta_{8}^{3} q^{9} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 + (-z^3 - 1) * q^5 - z^3 * q^8 - z^3 * q^9 $$q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 1) q^{5} - \zeta_{8}^{3} q^{8} - \zeta_{8}^{3} q^{9} + (\zeta_{8} - 1) q^{10} - \zeta_{8}^{2} q^{13} - q^{16} - \zeta_{8}^{3} q^{17} - q^{18} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{20} + (\zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{25} + 2 \zeta_{8}^{3} q^{26} + (\zeta_{8}^{2} + \zeta_{8}) q^{29} + \zeta_{8} q^{32} - q^{34} + \zeta_{8} q^{36} + (\zeta_{8}^{3} - 1) q^{37} + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{40} + (\zeta_{8}^{3} + \zeta_{8}^{2}) q^{41} + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{45} + (\zeta_{8}^{3} - \zeta_{8} + 1) q^{50} + 2 q^{52} + ( - \zeta_{8}^{2} - 1) q^{53} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{58} + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{61} - \zeta_{8}^{2} q^{64} + (2 \zeta_{8}^{2} - 2 \zeta_{8}) q^{65} + \zeta_{8} q^{68} - \zeta_{8}^{2} q^{72} + ( - \zeta_{8}^{3} - 1) q^{73} + (\zeta_{8} + 1) q^{74} + (\zeta_{8}^{3} + 1) q^{80} - \zeta_{8}^{2} q^{81} + ( - \zeta_{8}^{3} + 1) q^{82} + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{85} + (\zeta_{8}^{3} + 1) q^{90} + ( - \zeta_{8}^{3} - 1) q^{97} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 + (-z^3 - 1) * q^5 - z^3 * q^8 - z^3 * q^9 + (z - 1) * q^10 - z^2 * q^13 - q^16 - z^3 * q^17 - q^18 + (-z^2 + z) * q^20 + (z^3 - z^2 + 1) * q^25 + 2*z^3 * q^26 + (z^2 + z) * q^29 + z * q^32 - q^34 + z * q^36 + (z^3 - 1) * q^37 + (z^3 - z^2) * q^40 + (z^3 + z^2) * q^41 + (z^3 - z^2) * q^45 + (z^3 - z + 1) * q^50 + 2 * q^52 + (-z^2 - 1) * q^53 + (-z^3 - z^2) * q^58 + (-z^3 - z^2) * q^61 - z^2 * q^64 + (2*z^2 - 2*z) * q^65 + z * q^68 - z^2 * q^72 + (-z^3 - 1) * q^73 + (z + 1) * q^74 + (z^3 + 1) * q^80 - z^2 * q^81 + (-z^3 + 1) * q^82 + (z^3 - z^2) * q^85 + (z^3 + 1) * q^90 + (-z^3 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5}+O(q^{10})$$ 4 * q - 4 * q^5 $$4 q - 4 q^{5} - 4 q^{10} - 4 q^{16} - 4 q^{18} + 4 q^{25} - 4 q^{34} - 4 q^{37} + 4 q^{50} + 8 q^{52} - 4 q^{53} - 4 q^{73} + 4 q^{74} + 4 q^{80} + 4 q^{82} + 4 q^{90} - 4 q^{97}+O(q^{100})$$ 4 * q - 4 * q^5 - 4 * q^10 - 4 * q^16 - 4 * q^18 + 4 * q^25 - 4 * q^34 - 4 * q^37 + 4 * q^50 + 8 * q^52 - 4 * q^53 - 4 * q^73 + 4 * q^74 + 4 * q^80 + 4 * q^82 + 4 * q^90 - 4 * q^97

## Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-\zeta_{8}^{3}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
491.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0.707107 + 0.707107i 0 1.00000i −1.70711 + 0.707107i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i −1.70711 0.707107i
1079.1 0.707107 0.707107i 0 1.00000i −1.70711 0.707107i 0 0 −0.707107 0.707107i −0.707107 0.707107i −1.70711 + 0.707107i
1471.1 −0.707107 + 0.707107i 0 1.00000i −0.292893 + 0.707107i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i −0.292893 0.707107i
2059.1 −0.707107 0.707107i 0 1.00000i −0.292893 0.707107i 0 0 0.707107 0.707107i 0.707107 0.707107i −0.292893 + 0.707107i
 $$n$$: 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.d even 8 1 inner
68.g odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.w.a 4
4.b odd 2 1 CM 3332.1.w.a 4
7.b odd 2 1 3332.1.w.d yes 4
7.c even 3 2 3332.1.bp.d 8
7.d odd 6 2 3332.1.bp.a 8
17.d even 8 1 inner 3332.1.w.a 4
28.d even 2 1 3332.1.w.d yes 4
28.f even 6 2 3332.1.bp.a 8
28.g odd 6 2 3332.1.bp.d 8
68.g odd 8 1 inner 3332.1.w.a 4
119.l odd 8 1 3332.1.w.d yes 4
119.q even 24 2 3332.1.bp.d 8
119.r odd 24 2 3332.1.bp.a 8
476.w even 8 1 3332.1.w.d yes 4
476.bg odd 24 2 3332.1.bp.d 8
476.bj even 24 2 3332.1.bp.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.w.a 4 1.a even 1 1 trivial
3332.1.w.a 4 4.b odd 2 1 CM
3332.1.w.a 4 17.d even 8 1 inner
3332.1.w.a 4 68.g odd 8 1 inner
3332.1.w.d yes 4 7.b odd 2 1
3332.1.w.d yes 4 28.d even 2 1
3332.1.w.d yes 4 119.l odd 8 1
3332.1.w.d yes 4 476.w even 8 1
3332.1.bp.a 8 7.d odd 6 2
3332.1.bp.a 8 28.f even 6 2
3332.1.bp.a 8 119.r odd 24 2
3332.1.bp.a 8 476.bj even 24 2
3332.1.bp.d 8 7.c even 3 2
3332.1.bp.d 8 28.g odd 6 2
3332.1.bp.d 8 119.q even 24 2
3332.1.bp.d 8 476.bg odd 24 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 4T_{5}^{3} + 6T_{5}^{2} + 4T_{5} + 2$$ acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$T^{4} + 1$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 2 T^{2} + 4 T + 2$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2$$
$41$ $$T^{4} + 2 T^{2} - 4 T + 2$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 2 T + 2)^{2}$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 2 T^{2} + 4 T + 2$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2$$