# Properties

 Label 3120.1.bx.a Level $3120$ Weight $1$ Character orbit 3120.bx Analytic conductor $1.557$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -39 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,1,Mod(389,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 2, 2, 2]))

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

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

chi := DirichletCharacter("3120.389");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3120.bx (of order $$4$$, degree $$2$$, 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: $$1.55708283941$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ 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: $$D_{8}$$ Projective field: Galois closure of 8.2.2021515591680000.22

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{3} q^{2} + \zeta_{16}^{2} q^{3} + \zeta_{16}^{6} q^{4} + \zeta_{16}^{3} q^{5} + \zeta_{16}^{5} q^{6} - \zeta_{16} q^{8} + \zeta_{16}^{4} q^{9} +O(q^{10})$$ q + z^3 * q^2 + z^2 * q^3 + z^6 * q^4 + z^3 * q^5 + z^5 * q^6 - z * q^8 + z^4 * q^9 $$q + \zeta_{16}^{3} q^{2} + \zeta_{16}^{2} q^{3} + \zeta_{16}^{6} q^{4} + \zeta_{16}^{3} q^{5} + \zeta_{16}^{5} q^{6} - \zeta_{16} q^{8} + \zeta_{16}^{4} q^{9} + \zeta_{16}^{6} q^{10} + (\zeta_{16}^{7} - \zeta_{16}^{5}) q^{11} - q^{12} + \zeta_{16}^{2} q^{13} + \zeta_{16}^{5} q^{15} - \zeta_{16}^{4} q^{16} + \zeta_{16}^{7} q^{18} - \zeta_{16} q^{20} + ( - \zeta_{16}^{2} + 1) q^{22} - \zeta_{16}^{3} q^{24} + \zeta_{16}^{6} q^{25} + \zeta_{16}^{5} q^{26} + \zeta_{16}^{6} q^{27} - q^{30} - \zeta_{16}^{7} q^{32} + ( - \zeta_{16}^{7} - \zeta_{16}) q^{33} - \zeta_{16}^{2} q^{36} + \zeta_{16}^{4} q^{39} - \zeta_{16}^{4} q^{40} + ( - \zeta_{16}^{7} - \zeta_{16}) q^{41} + (\zeta_{16}^{4} - 1) q^{43} + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{44} + \zeta_{16}^{7} q^{45} + (\zeta_{16}^{7} + \zeta_{16}) q^{47} - \zeta_{16}^{6} q^{48} + q^{49} - \zeta_{16} q^{50} - q^{52} - \zeta_{16} q^{54} + ( - \zeta_{16}^{2} + 1) q^{55} + ( - \zeta_{16}^{3} - \zeta_{16}) q^{59} - \zeta_{16}^{3} q^{60} - \zeta_{16}^{6} q^{61} + \zeta_{16}^{2} q^{64} + \zeta_{16}^{5} q^{65} + ( - \zeta_{16}^{4} + \zeta_{16}^{2}) q^{66} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{71} - \zeta_{16}^{5} q^{72} - q^{75} + \zeta_{16}^{7} q^{78} + ( - \zeta_{16}^{6} + \zeta_{16}^{2}) q^{79} - \zeta_{16}^{7} q^{80} - q^{81} + ( - \zeta_{16}^{4} + \zeta_{16}^{2}) q^{82} + ( - \zeta_{16}^{7} + \zeta_{16}^{5}) q^{83} + (\zeta_{16}^{7} - \zeta_{16}^{3}) q^{86} + (\zeta_{16}^{6} + 1) q^{88} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{89} - \zeta_{16}^{2} q^{90} + (\zeta_{16}^{4} - \zeta_{16}^{2}) q^{94} + \zeta_{16} q^{96} + \zeta_{16}^{3} q^{98} + ( - \zeta_{16}^{3} + \zeta_{16}) q^{99} +O(q^{100})$$ q + z^3 * q^2 + z^2 * q^3 + z^6 * q^4 + z^3 * q^5 + z^5 * q^6 - z * q^8 + z^4 * q^9 + z^6 * q^10 + (z^7 - z^5) * q^11 - q^12 + z^2 * q^13 + z^5 * q^15 - z^4 * q^16 + z^7 * q^18 - z * q^20 + (-z^2 + 1) * q^22 - z^3 * q^24 + z^6 * q^25 + z^5 * q^26 + z^6 * q^27 - q^30 - z^7 * q^32 + (-z^7 - z) * q^33 - z^2 * q^36 + z^4 * q^39 - z^4 * q^40 + (-z^7 - z) * q^41 + (z^4 - 1) * q^43 + (-z^5 + z^3) * q^44 + z^7 * q^45 + (z^7 + z) * q^47 - z^6 * q^48 + q^49 - z * q^50 - q^52 - z * q^54 + (-z^2 + 1) * q^55 + (-z^3 - z) * q^59 - z^3 * q^60 - z^6 * q^61 + z^2 * q^64 + z^5 * q^65 + (-z^4 + z^2) * q^66 + (-z^5 - z^3) * q^71 - z^5 * q^72 - q^75 + z^7 * q^78 + (-z^6 + z^2) * q^79 - z^7 * q^80 - q^81 + (-z^4 + z^2) * q^82 + (-z^7 + z^5) * q^83 + (z^7 - z^3) * q^86 + (z^6 + 1) * q^88 + (z^5 + z^3) * q^89 - z^2 * q^90 + (z^4 - z^2) * q^94 + z * q^96 + z^3 * q^98 + (-z^3 + z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 8 q^{12} + 8 q^{22} - 8 q^{30} - 8 q^{43} + 8 q^{49} - 8 q^{52} + 8 q^{55} - 8 q^{75} - 8 q^{81} + 8 q^{88}+O(q^{100})$$ 8 * q - 8 * q^12 + 8 * q^22 - 8 * q^30 - 8 * q^43 + 8 * q^49 - 8 * q^52 + 8 * q^55 - 8 * q^75 - 8 * q^81 + 8 * q^88

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$$.

 $$n$$ $$1951$$ $$2081$$ $$2341$$ $$2497$$ $$2641$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{16}^{4}$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
389.1
 0.382683 − 0.923880i −0.923880 − 0.382683i 0.923880 + 0.382683i −0.382683 + 0.923880i 0.382683 + 0.923880i −0.923880 + 0.382683i 0.923880 − 0.382683i −0.382683 − 0.923880i
−0.923880 + 0.382683i −0.707107 0.707107i 0.707107 0.707107i −0.923880 + 0.382683i 0.923880 + 0.382683i 0 −0.382683 + 0.923880i 1.00000i 0.707107 0.707107i
389.2 −0.382683 0.923880i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.382683 0.923880i 0.382683 0.923880i 0 0.923880 + 0.382683i 1.00000i −0.707107 + 0.707107i
389.3 0.382683 + 0.923880i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.382683 + 0.923880i −0.382683 + 0.923880i 0 −0.923880 0.382683i 1.00000i −0.707107 + 0.707107i
389.4 0.923880 0.382683i −0.707107 0.707107i 0.707107 0.707107i 0.923880 0.382683i −0.923880 0.382683i 0 0.382683 0.923880i 1.00000i 0.707107 0.707107i
1949.1 −0.923880 0.382683i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.923880 0.382683i 0.923880 0.382683i 0 −0.382683 0.923880i 1.00000i 0.707107 + 0.707107i
1949.2 −0.382683 + 0.923880i 0.707107 0.707107i −0.707107 0.707107i −0.382683 + 0.923880i 0.382683 + 0.923880i 0 0.923880 0.382683i 1.00000i −0.707107 0.707107i
1949.3 0.382683 0.923880i 0.707107 0.707107i −0.707107 0.707107i 0.382683 0.923880i −0.382683 0.923880i 0 −0.923880 + 0.382683i 1.00000i −0.707107 0.707107i
1949.4 0.923880 + 0.382683i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.923880 + 0.382683i −0.923880 + 0.382683i 0 0.382683 + 0.923880i 1.00000i 0.707107 + 0.707107i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 389.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
13.b even 2 1 inner
80.q even 4 1 inner
240.bm odd 4 1 inner
1040.be even 4 1 inner
3120.bx odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3120.1.bx.a 8
3.b odd 2 1 inner 3120.1.bx.a 8
5.b even 2 1 3120.1.bx.b yes 8
13.b even 2 1 inner 3120.1.bx.a 8
15.d odd 2 1 3120.1.bx.b yes 8
16.e even 4 1 3120.1.bx.b yes 8
39.d odd 2 1 CM 3120.1.bx.a 8
48.i odd 4 1 3120.1.bx.b yes 8
65.d even 2 1 3120.1.bx.b yes 8
80.q even 4 1 inner 3120.1.bx.a 8
195.e odd 2 1 3120.1.bx.b yes 8
208.p even 4 1 3120.1.bx.b yes 8
240.bm odd 4 1 inner 3120.1.bx.a 8
624.bi odd 4 1 3120.1.bx.b yes 8
1040.be even 4 1 inner 3120.1.bx.a 8
3120.bx odd 4 1 inner 3120.1.bx.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3120.1.bx.a 8 1.a even 1 1 trivial
3120.1.bx.a 8 3.b odd 2 1 inner
3120.1.bx.a 8 13.b even 2 1 inner
3120.1.bx.a 8 39.d odd 2 1 CM
3120.1.bx.a 8 80.q even 4 1 inner
3120.1.bx.a 8 240.bm odd 4 1 inner
3120.1.bx.a 8 1040.be even 4 1 inner
3120.1.bx.a 8 3120.bx odd 4 1 inner
3120.1.bx.b yes 8 5.b even 2 1
3120.1.bx.b yes 8 15.d odd 2 1
3120.1.bx.b yes 8 16.e even 4 1
3120.1.bx.b yes 8 48.i odd 4 1
3120.1.bx.b yes 8 65.d even 2 1
3120.1.bx.b yes 8 195.e odd 2 1
3120.1.bx.b yes 8 208.p even 4 1
3120.1.bx.b yes 8 624.bi odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{43}^{2} + 2T_{43} + 2$$ acting on $$S_{1}^{\mathrm{new}}(3120, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 1$$
$3$ $$(T^{4} + 1)^{2}$$
$5$ $$T^{8} + 1$$
$7$ $$T^{8}$$
$11$ $$T^{8} + 12T^{4} + 4$$
$13$ $$(T^{4} + 1)^{2}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$43$ $$(T^{2} + 2 T + 2)^{4}$$
$47$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$53$ $$T^{8}$$
$59$ $$T^{8} + 12T^{4} + 4$$
$61$ $$(T^{4} + 16)^{2}$$
$67$ $$T^{8}$$
$71$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$73$ $$T^{8}$$
$79$ $$(T^{2} - 2)^{4}$$
$83$ $$T^{8} + 12T^{4} + 4$$
$89$ $$(T^{4} + 4 T^{2} + 2)^{2}$$
$97$ $$T^{8}$$