# Properties

 Label 3520.2.a.r Level $3520$ Weight $2$ Character orbit 3520.a Self dual yes Analytic conductor $28.107$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,2,Mod(1,3520)]

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, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3520 = 2^{6} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3520.a (trivial)

## 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: yes Analytic conductor: $$28.1073415115$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{5} + 4 q^{7} - 3 q^{9}+O(q^{10})$$ q - q^5 + 4 * q^7 - 3 * q^9 $$q - q^{5} + 4 q^{7} - 3 q^{9} + q^{11} - 6 q^{13} - 6 q^{17} - 4 q^{19} + 4 q^{23} + q^{25} + 2 q^{29} + 8 q^{31} - 4 q^{35} + 10 q^{37} + 10 q^{41} + 3 q^{45} + 4 q^{47} + 9 q^{49} + 10 q^{53} - q^{55} + 4 q^{59} + 2 q^{61} - 12 q^{63} + 6 q^{65} + 8 q^{67} - 14 q^{73} + 4 q^{77} - 16 q^{79} + 9 q^{81} + 8 q^{83} + 6 q^{85} - 6 q^{89} - 24 q^{91} + 4 q^{95} + 2 q^{97} - 3 q^{99}+O(q^{100})$$ q - q^5 + 4 * q^7 - 3 * q^9 + q^11 - 6 * q^13 - 6 * q^17 - 4 * q^19 + 4 * q^23 + q^25 + 2 * q^29 + 8 * q^31 - 4 * q^35 + 10 * q^37 + 10 * q^41 + 3 * q^45 + 4 * q^47 + 9 * q^49 + 10 * q^53 - q^55 + 4 * q^59 + 2 * q^61 - 12 * q^63 + 6 * q^65 + 8 * q^67 - 14 * q^73 + 4 * q^77 - 16 * q^79 + 9 * q^81 + 8 * q^83 + 6 * q^85 - 6 * q^89 - 24 * q^91 + 4 * q^95 + 2 * q^97 - 3 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 −1.00000 0 4.00000 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3520.2.a.r 1
4.b odd 2 1 3520.2.a.m 1
8.b even 2 1 440.2.a.c 1
8.d odd 2 1 880.2.a.g 1
24.f even 2 1 7920.2.a.b 1
24.h odd 2 1 3960.2.a.i 1
40.e odd 2 1 4400.2.a.r 1
40.f even 2 1 2200.2.a.e 1
40.i odd 4 2 2200.2.b.c 2
40.k even 4 2 4400.2.b.o 2
88.b odd 2 1 4840.2.a.e 1
88.g even 2 1 9680.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.c 1 8.b even 2 1
880.2.a.g 1 8.d odd 2 1
2200.2.a.e 1 40.f even 2 1
2200.2.b.c 2 40.i odd 4 2
3520.2.a.m 1 4.b odd 2 1
3520.2.a.r 1 1.a even 1 1 trivial
3960.2.a.i 1 24.h odd 2 1
4400.2.a.r 1 40.e odd 2 1
4400.2.b.o 2 40.k even 4 2
4840.2.a.e 1 88.b odd 2 1
7920.2.a.b 1 24.f even 2 1
9680.2.a.s 1 88.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3520))$$:

 $$T_{3}$$ T3 $$T_{7} - 4$$ T7 - 4 $$T_{13} + 6$$ T13 + 6 $$T_{17} + 6$$ T17 + 6 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 4$$
$11$ $$T - 1$$
$13$ $$T + 6$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T - 8$$
$37$ $$T - 10$$
$41$ $$T - 10$$
$43$ $$T$$
$47$ $$T - 4$$
$53$ $$T - 10$$
$59$ $$T - 4$$
$61$ $$T - 2$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T + 16$$
$83$ $$T - 8$$
$89$ $$T + 6$$
$97$ $$T - 2$$