# Properties

 Label 3520.1.i.e Level $3520$ Weight $1$ Character orbit 3520.i Analytic conductor $1.757$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ RM discriminant 220 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,1,Mod(769,3520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3520.769");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3520 = 2^{6} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3520.i (of order $$2$$, degree $$1$$, 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: $$1.75670884447$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1760) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.4400.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{8}^{3} - \zeta_{8}) q^{3} - q^{5} - q^{9}+O(q^{10})$$ q + (-z^3 - z) * q^3 - q^5 - q^9 $$q + ( - \zeta_{8}^{3} - \zeta_{8}) q^{3} - q^{5} - q^{9} + \zeta_{8}^{2} q^{11} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{13} + (\zeta_{8}^{3} + \zeta_{8}) q^{15} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{17} - \zeta_{8}^{2} q^{19} + (\zeta_{8}^{3} + \zeta_{8}) q^{23} + q^{25} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{33} - \zeta_{8}^{2} q^{39} + q^{45} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{47} - q^{49} - \zeta_{8}^{2} q^{51} - \zeta_{8}^{2} q^{55} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{57} + (\zeta_{8}^{3} - \zeta_{8}) q^{65} + (\zeta_{8}^{3} + \zeta_{8}) q^{67} + (\zeta_{8}^{2} + 2) q^{69} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{73} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{75} - \zeta_{8}^{2} q^{79} - q^{81} + (\zeta_{8}^{3} - \zeta_{8}) q^{85} + 2 \zeta_{8}^{2} q^{95} - \zeta_{8}^{2} q^{99} +O(q^{100})$$ q + (-z^3 - z) * q^3 - q^5 - q^9 + z^2 * q^11 + (-z^3 + z) * q^13 + (z^3 + z) * q^15 + (-z^3 + z) * q^17 - z^2 * q^19 + (z^3 + z) * q^23 + q^25 + (-z^3 + z) * q^33 - z^2 * q^39 + q^45 + (-z^3 - z) * q^47 - q^49 - z^2 * q^51 - z^2 * q^55 + (2*z^3 - 2*z) * q^57 + (z^3 - z) * q^65 + (z^3 + z) * q^67 + (z^2 + 2) * q^69 + (-z^3 + z) * q^73 + (-z^3 - z) * q^75 - z^2 * q^79 - q^81 + (z^3 - z) * q^85 + 2*z^2 * q^95 - z^2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^5 - 4 * q^9 $$4 q - 4 q^{5} - 4 q^{9} + 4 q^{25} + 4 q^{45} - 4 q^{49} + 8 q^{69} - 4 q^{81}+O(q^{100})$$ 4 * q - 4 * q^5 - 4 * q^9 + 4 * q^25 + 4 * q^45 - 4 * q^49 + 8 * q^69 - 4 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times$$.

 $$n$$ $$321$$ $$1541$$ $$2751$$ $$2817$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
0 1.41421i 0 −1.00000 0 0 0 −1.00000 0
769.2 0 1.41421i 0 −1.00000 0 0 0 −1.00000 0
769.3 0 1.41421i 0 −1.00000 0 0 0 −1.00000 0
769.4 0 1.41421i 0 −1.00000 0 0 0 −1.00000 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
220.g even 2 1 RM by $$\Q(\sqrt{55})$$
4.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
20.d odd 2 1 inner
44.c even 2 1 inner
55.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3520.1.i.e 4
4.b odd 2 1 inner 3520.1.i.e 4
5.b even 2 1 inner 3520.1.i.e 4
8.b even 2 1 1760.1.i.a 4
8.d odd 2 1 1760.1.i.a 4
11.b odd 2 1 inner 3520.1.i.e 4
20.d odd 2 1 inner 3520.1.i.e 4
40.e odd 2 1 1760.1.i.a 4
40.f even 2 1 1760.1.i.a 4
44.c even 2 1 inner 3520.1.i.e 4
55.d odd 2 1 inner 3520.1.i.e 4
88.b odd 2 1 1760.1.i.a 4
88.g even 2 1 1760.1.i.a 4
220.g even 2 1 RM 3520.1.i.e 4
440.c even 2 1 1760.1.i.a 4
440.o odd 2 1 1760.1.i.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1760.1.i.a 4 8.b even 2 1
1760.1.i.a 4 8.d odd 2 1
1760.1.i.a 4 40.e odd 2 1
1760.1.i.a 4 40.f even 2 1
1760.1.i.a 4 88.b odd 2 1
1760.1.i.a 4 88.g even 2 1
1760.1.i.a 4 440.c even 2 1
1760.1.i.a 4 440.o odd 2 1
3520.1.i.e 4 1.a even 1 1 trivial
3520.1.i.e 4 4.b odd 2 1 inner
3520.1.i.e 4 5.b even 2 1 inner
3520.1.i.e 4 11.b odd 2 1 inner
3520.1.i.e 4 20.d odd 2 1 inner
3520.1.i.e 4 44.c even 2 1 inner
3520.1.i.e 4 55.d odd 2 1 inner
3520.1.i.e 4 220.g even 2 1 RM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3520, [\chi])$$:

 $$T_{3}^{2} + 2$$ T3^2 + 2 $$T_{31}$$ T31

## Hecke characteristic polynomials

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