# Properties

 Label 3520.1.o.a Level $3520$ Weight $1$ Character orbit 3520.o Analytic conductor $1.757$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -11, -40, 440 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,1,Mod(2529,3520)]

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, 1, 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("3520.2529");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3520 = 2^{6} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3520.o (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: no Analytic conductor: $$1.75670884447$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-10}, \sqrt{-11})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.23987814400.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{5} - q^{9}+O(q^{10})$$ q - q^5 - q^9 $$q - q^{5} - q^{9} - i q^{11} + 2 i q^{23} + q^{25} - 2 q^{37} + q^{45} + 2 i q^{47} - q^{49} - 2 q^{53} + i q^{55} + 2 i q^{59} + q^{81} - 2 q^{89} + i q^{99} +O(q^{100})$$ q - q^5 - q^9 - z * q^11 + 2*z * q^23 + q^25 - 2 * q^37 + q^45 + 2*z * q^47 - q^49 - 2 * q^53 + z * q^55 + 2*z * q^59 + q^81 - 2 * q^89 + z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^9 $$2 q - 2 q^{5} - 2 q^{9} + 2 q^{25} - 4 q^{37} + 2 q^{45} - 2 q^{49} - 4 q^{53} + 2 q^{81} - 4 q^{89}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^9 + 2 * q^25 - 4 * q^37 + 2 * q^45 - 2 * q^49 - 4 * q^53 + 2 * q^81 - 4 * q^89

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2529.1
 1.00000i − 1.00000i
0 0 0 −1.00000 0 0 0 −1.00000 0
2529.2 0 0 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
440.c even 2 1 RM by $$\Q(\sqrt{110})$$
4.b odd 2 1 inner
40.f even 2 1 inner
44.c even 2 1 inner
440.o 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.o.a 2
4.b odd 2 1 inner 3520.1.o.a 2
5.b even 2 1 3520.1.o.b yes 2
8.b even 2 1 3520.1.o.b yes 2
8.d odd 2 1 3520.1.o.b yes 2
11.b odd 2 1 CM 3520.1.o.a 2
20.d odd 2 1 3520.1.o.b yes 2
40.e odd 2 1 CM 3520.1.o.a 2
40.f even 2 1 inner 3520.1.o.a 2
44.c even 2 1 inner 3520.1.o.a 2
55.d odd 2 1 3520.1.o.b yes 2
88.b odd 2 1 3520.1.o.b yes 2
88.g even 2 1 3520.1.o.b yes 2
220.g even 2 1 3520.1.o.b yes 2
440.c even 2 1 RM 3520.1.o.a 2
440.o odd 2 1 inner 3520.1.o.a 2

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

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{37} + 2$$ T37 + 2

## Hecke characteristic polynomials

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