Properties

 Label 3332.1.bn.a Level $3332$ Weight $1$ Character orbit 3332.bn Analytic conductor $1.663$ Analytic rank $0$ Dimension $8$ Projective image $D_{16}$ 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(979,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 8, 3]))

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

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

chi := DirichletCharacter("3332.979");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.bn (of order $$16$$, degree $$8$$, 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: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 1$$ x^8 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{16}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{5} q^{2} - \zeta_{16}^{2} q^{4} + (\zeta_{16}^{5} - \zeta_{16}^{2}) q^{5} - \zeta_{16}^{7} q^{8} + \zeta_{16}^{3} q^{9} +O(q^{10})$$ q + z^5 * q^2 - z^2 * q^4 + (z^5 - z^2) * q^5 - z^7 * q^8 + z^3 * q^9 $$q + \zeta_{16}^{5} q^{2} - \zeta_{16}^{2} q^{4} + (\zeta_{16}^{5} - \zeta_{16}^{2}) q^{5} - \zeta_{16}^{7} q^{8} + \zeta_{16}^{3} q^{9} + ( - \zeta_{16}^{7} - \zeta_{16}^{2}) q^{10} + ( - \zeta_{16}^{4} - 1) q^{13} + \zeta_{16}^{4} q^{16} - \zeta_{16}^{5} q^{17} - q^{18} + ( - \zeta_{16}^{7} + \zeta_{16}^{4}) q^{20} + ( - \zeta_{16}^{7} + \cdots - \zeta_{16}^{2}) q^{25} + \cdots + ( - \zeta_{16}^{5} - \zeta_{16}^{2}) q^{97} +O(q^{100})$$ q + z^5 * q^2 - z^2 * q^4 + (z^5 - z^2) * q^5 - z^7 * q^8 + z^3 * q^9 + (-z^7 - z^2) * q^10 + (-z^4 - 1) * q^13 + z^4 * q^16 - z^5 * q^17 - q^18 + (-z^7 + z^4) * q^20 + (-z^7 + z^4 - z^2) * q^25 + (-z^5 + z) * q^26 + (z^6 - z) * q^29 - z * q^32 + z^2 * q^34 - z^5 * q^36 + (-z^3 + 1) * q^37 + (z^4 - z) * q^40 + (z^5 - z^4) * q^41 + (-z^5 - 1) * q^45 + (-z^7 + z^4 - z) * q^50 + (z^6 + z^2) * q^52 + (-z^6 - z^4) * q^53 + (-z^6 - z^3) * q^58 + (-z^5 - z^4) * q^61 - z^6 * q^64 + (z^6 - z^5 + z^2 + z) * q^65 + z^7 * q^68 + z^2 * q^72 + (-z^6 + z) * q^73 + (z^5 + 1) * q^74 + (-z^6 - z) * q^80 + z^6 * q^81 + (-z^2 + z) * q^82 + (z^7 + z^2) * q^85 + 2*z^6 * q^89 + (-z^5 + z^2) * q^90 + (-z^5 - z^2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 8 q^{13} - 8 q^{18} + 8 q^{37} - 8 q^{45} + 8 q^{74}+O(q^{100})$$ 8 * q - 8 * q^13 - 8 * q^18 + 8 * q^37 - 8 * q^45 + 8 * q^74

Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-\zeta_{16}^{3}$$ $$-1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
979.1
 −0.923880 − 0.382683i −0.382683 + 0.923880i 0.382683 + 0.923880i −0.923880 + 0.382683i 0.923880 − 0.382683i −0.382683 − 0.923880i 0.382683 − 0.923880i 0.923880 + 0.382683i
0.382683 0.923880i 0 −0.707107 0.707107i −0.324423 1.63099i 0 0 −0.923880 + 0.382683i −0.382683 0.923880i −1.63099 0.324423i
1371.1 −0.923880 0.382683i 0 0.707107 + 0.707107i −0.216773 + 0.324423i 0 0 −0.382683 0.923880i 0.923880 0.382683i 0.324423 0.216773i
1567.1 0.923880 0.382683i 0 0.707107 0.707107i 1.63099 1.08979i 0 0 0.382683 0.923880i −0.923880 0.382683i 1.08979 1.63099i
1763.1 0.382683 + 0.923880i 0 −0.707107 + 0.707107i −0.324423 + 1.63099i 0 0 −0.923880 0.382683i −0.382683 + 0.923880i −1.63099 + 0.324423i
2351.1 −0.382683 0.923880i 0 −0.707107 + 0.707107i −1.08979 0.216773i 0 0 0.923880 + 0.382683i 0.382683 0.923880i 0.216773 + 1.08979i
2547.1 −0.923880 + 0.382683i 0 0.707107 0.707107i −0.216773 0.324423i 0 0 −0.382683 + 0.923880i 0.923880 + 0.382683i 0.324423 + 0.216773i
2743.1 0.923880 + 0.382683i 0 0.707107 + 0.707107i 1.63099 + 1.08979i 0 0 0.382683 + 0.923880i −0.923880 + 0.382683i 1.08979 + 1.63099i
3135.1 −0.382683 + 0.923880i 0 −0.707107 0.707107i −1.08979 + 0.216773i 0 0 0.923880 0.382683i 0.382683 + 0.923880i 0.216773 1.08979i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 979.1 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 $$\Q(\sqrt{-1})$$
119.p even 16 1 inner
476.bf odd 16 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.bn.a 8
4.b odd 2 1 CM 3332.1.bn.a 8
7.b odd 2 1 3332.1.bn.c yes 8
7.c even 3 2 3332.1.ce.b 16
7.d odd 6 2 3332.1.ce.d 16
17.e odd 16 1 3332.1.bn.c yes 8
28.d even 2 1 3332.1.bn.c yes 8
28.f even 6 2 3332.1.ce.d 16
28.g odd 6 2 3332.1.ce.b 16
68.i even 16 1 3332.1.bn.c yes 8
119.p even 16 1 inner 3332.1.bn.a 8
119.s even 48 2 3332.1.ce.b 16
119.t odd 48 2 3332.1.ce.d 16
476.bf odd 16 1 inner 3332.1.bn.a 8
476.bk odd 48 2 3332.1.ce.b 16
476.bm even 48 2 3332.1.ce.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.bn.a 8 1.a even 1 1 trivial
3332.1.bn.a 8 4.b odd 2 1 CM
3332.1.bn.a 8 119.p even 16 1 inner
3332.1.bn.a 8 476.bf odd 16 1 inner
3332.1.bn.c yes 8 7.b odd 2 1
3332.1.bn.c yes 8 17.e odd 16 1
3332.1.bn.c yes 8 28.d even 2 1
3332.1.bn.c yes 8 68.i even 16 1
3332.1.ce.b 16 7.c even 3 2
3332.1.ce.b 16 28.g odd 6 2
3332.1.ce.b 16 119.s even 48 2
3332.1.ce.b 16 476.bk odd 48 2
3332.1.ce.d 16 7.d odd 6 2
3332.1.ce.d 16 28.f even 6 2
3332.1.ce.d 16 119.t odd 48 2
3332.1.ce.d 16 476.bm even 48 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 2T_{5}^{4} + 16T_{5}^{3} + 20T_{5}^{2} + 8T_{5} + 2$$ acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 1$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 2 T^{4} + \cdots + 2$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$(T^{2} + 2 T + 2)^{4}$$
$17$ $$T^{8} + 1$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8} + 8 T^{5} + \cdots + 2$$
$31$ $$T^{8}$$
$37$ $$T^{8} - 8 T^{7} + \cdots + 2$$
$41$ $$T^{8} + 4 T^{6} + \cdots + 2$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} + 2 T^{2} + 4 T + 2)^{2}$$
$59$ $$T^{8}$$
$61$ $$T^{8} + 4 T^{6} + \cdots + 2$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8} - 8 T^{5} + \cdots + 2$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$(T^{4} + 16)^{2}$$
$97$ $$T^{8} + 2 T^{4} + \cdots + 2$$