# Properties

 Label 3520.1.y.a Level $3520$ Weight $1$ Character orbit 3520.y Analytic conductor $1.757$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -11 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,1,Mod(703,3520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3, 2]))

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

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

chi := DirichletCharacter("3520.703");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3520 = 2^{6} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3520.y (of order $$4$$, degree $$2$$, 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: $$2$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 880) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.242000.1 Artin image: $C_4^2:C_2^2$ Artin 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.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{3} + i q^{5} + i q^{9}+O(q^{10})$$ q + (-z - 1) * q^3 + z * q^5 + z * q^9 $$q + ( - i - 1) q^{3} + i q^{5} + i q^{9} - i q^{11} + ( - i + 1) q^{15} + (i + 1) q^{23} - q^{25} - q^{27} - i q^{31} + (i - 1) q^{33} + (i + 1) q^{37} - q^{45} + (i - 1) q^{47} - i q^{49} + ( - i + 1) q^{53} + q^{55} + q^{59} + ( - i + 1) q^{67} + ( - 2 i + 1) q^{69} + (i + 1) q^{75} + q^{81} - i q^{89} + (2 i - 2) q^{93} + (i + 1) q^{97} + q^{99} +O(q^{100})$$ q + (-z - 1) * q^3 + z * q^5 + z * q^9 - z * q^11 + (-z + 1) * q^15 + (z + 1) * q^23 - q^25 - q^27 - z * q^31 + (z - 1) * q^33 + (z + 1) * q^37 - q^45 + (z - 1) * q^47 - z * q^49 + (-z + 1) * q^53 + q^55 + q^59 + (-z + 1) * q^67 + (-2*z + 1) * q^69 + (z + 1) * q^75 + q^81 - z * q^89 + (2*z - 2) * q^93 + (z + 1) * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3}+O(q^{10})$$ 2 * q - 2 * q^3 $$2 q - 2 q^{3} + 2 q^{15} + 2 q^{23} - 2 q^{25} - 2 q^{33} + 2 q^{37} - 2 q^{45} - 2 q^{47} + 2 q^{53} + 2 q^{55} + 4 q^{59} + 2 q^{67} + 2 q^{75} + 2 q^{81} - 4 q^{93} + 2 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^15 + 2 * q^23 - 2 * q^25 - 2 * q^33 + 2 * q^37 - 2 * q^45 - 2 * q^47 + 2 * q^53 + 2 * q^55 + 4 * q^59 + 2 * q^67 + 2 * q^75 + 2 * q^81 - 4 * q^93 + 2 * q^97 + 2 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 − 1.00000i 1.00000i
0 −1.00000 + 1.00000i 0 1.00000i 0 0 0 1.00000i 0
1407.1 0 −1.00000 1.00000i 0 1.00000i 0 0 0 1.00000i 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 $$\Q(\sqrt{-11})$$
20.e even 4 1 inner
220.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3520.1.y.a 2
4.b odd 2 1 3520.1.y.b 2
5.c odd 4 1 3520.1.y.b 2
8.b even 2 1 880.1.y.b yes 2
8.d odd 2 1 880.1.y.a 2
11.b odd 2 1 CM 3520.1.y.a 2
20.e even 4 1 inner 3520.1.y.a 2
40.i odd 4 1 880.1.y.a 2
40.k even 4 1 880.1.y.b yes 2
44.c even 2 1 3520.1.y.b 2
55.e even 4 1 3520.1.y.b 2
88.b odd 2 1 880.1.y.b yes 2
88.g even 2 1 880.1.y.a 2
220.i odd 4 1 inner 3520.1.y.a 2
440.t even 4 1 880.1.y.a 2
440.w odd 4 1 880.1.y.b yes 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} + 2$$ acting on $$S_{1}^{\mathrm{new}}(3520, [\chi])$$.

## Hecke characteristic polynomials

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