# Properties

 Label 3520.2.a.a 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - q^{5} + q^{7} + 6 q^{9}+O(q^{10})$$ q - 3 * q^3 - q^5 + q^7 + 6 * q^9 $$q - 3 q^{3} - q^{5} + q^{7} + 6 q^{9} + q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 5 q^{19} - 3 q^{21} - 2 q^{23} + q^{25} - 9 q^{27} + 5 q^{29} + 5 q^{31} - 3 q^{33} - q^{35} + q^{37} - 18 q^{39} - 2 q^{41} - 12 q^{43} - 6 q^{45} - 2 q^{47} - 6 q^{49} - 9 q^{51} + 13 q^{53} - q^{55} - 15 q^{57} - 2 q^{59} - q^{61} + 6 q^{63} - 6 q^{65} - 16 q^{67} + 6 q^{69} + 15 q^{71} + 10 q^{73} - 3 q^{75} + q^{77} + 2 q^{79} + 9 q^{81} + 14 q^{83} - 3 q^{85} - 15 q^{87} + 9 q^{89} + 6 q^{91} - 15 q^{93} - 5 q^{95} - 16 q^{97} + 6 q^{99}+O(q^{100})$$ q - 3 * q^3 - q^5 + q^7 + 6 * q^9 + q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 5 * q^19 - 3 * q^21 - 2 * q^23 + q^25 - 9 * q^27 + 5 * q^29 + 5 * q^31 - 3 * q^33 - q^35 + q^37 - 18 * q^39 - 2 * q^41 - 12 * q^43 - 6 * q^45 - 2 * q^47 - 6 * q^49 - 9 * q^51 + 13 * q^53 - q^55 - 15 * q^57 - 2 * q^59 - q^61 + 6 * q^63 - 6 * q^65 - 16 * q^67 + 6 * q^69 + 15 * q^71 + 10 * q^73 - 3 * q^75 + q^77 + 2 * q^79 + 9 * q^81 + 14 * q^83 - 3 * q^85 - 15 * q^87 + 9 * q^89 + 6 * q^91 - 15 * q^93 - 5 * q^95 - 16 * q^97 + 6 * 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 −3.00000 0 −1.00000 0 1.00000 0 6.00000 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.a 1
4.b odd 2 1 3520.2.a.bh 1
8.b even 2 1 440.2.a.d 1
8.d odd 2 1 880.2.a.a 1
24.f even 2 1 7920.2.a.e 1
24.h odd 2 1 3960.2.a.f 1
40.e odd 2 1 4400.2.a.be 1
40.f even 2 1 2200.2.a.a 1
40.i odd 4 2 2200.2.b.b 2
40.k even 4 2 4400.2.b.a 2
88.b odd 2 1 4840.2.a.i 1
88.g even 2 1 9680.2.a.a 1

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

 $$T_{3} + 3$$ T3 + 3 $$T_{7} - 1$$ T7 - 1 $$T_{13} - 6$$ T13 - 6 $$T_{17} - 3$$ T17 - 3 $$T_{19} - 5$$ T19 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T - 1$$
$13$ $$T - 6$$
$17$ $$T - 3$$
$19$ $$T - 5$$
$23$ $$T + 2$$
$29$ $$T - 5$$
$31$ $$T - 5$$
$37$ $$T - 1$$
$41$ $$T + 2$$
$43$ $$T + 12$$
$47$ $$T + 2$$
$53$ $$T - 13$$
$59$ $$T + 2$$
$61$ $$T + 1$$
$67$ $$T + 16$$
$71$ $$T - 15$$
$73$ $$T - 10$$
$79$ $$T - 2$$
$83$ $$T - 14$$
$89$ $$T - 9$$
$97$ $$T + 16$$