# Properties

 Label 3520.2.a.l 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 110) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + q^5 + 5 * q^7 - 2 * q^9 $$q - q^{3} + q^{5} + 5 q^{7} - 2 q^{9} - q^{11} - 2 q^{13} - q^{15} + 3 q^{17} + 7 q^{19} - 5 q^{21} - 6 q^{23} + q^{25} + 5 q^{27} + 3 q^{29} - 7 q^{31} + q^{33} + 5 q^{35} + 7 q^{37} + 2 q^{39} + 6 q^{41} - 8 q^{43} - 2 q^{45} + 6 q^{47} + 18 q^{49} - 3 q^{51} + 3 q^{53} - q^{55} - 7 q^{57} + 6 q^{59} + q^{61} - 10 q^{63} - 2 q^{65} - 8 q^{67} + 6 q^{69} + 3 q^{71} + 2 q^{73} - q^{75} - 5 q^{77} - 10 q^{79} + q^{81} + 6 q^{83} + 3 q^{85} - 3 q^{87} + 9 q^{89} - 10 q^{91} + 7 q^{93} + 7 q^{95} - 4 q^{97} + 2 q^{99}+O(q^{100})$$ q - q^3 + q^5 + 5 * q^7 - 2 * q^9 - q^11 - 2 * q^13 - q^15 + 3 * q^17 + 7 * q^19 - 5 * q^21 - 6 * q^23 + q^25 + 5 * q^27 + 3 * q^29 - 7 * q^31 + q^33 + 5 * q^35 + 7 * q^37 + 2 * q^39 + 6 * q^41 - 8 * q^43 - 2 * q^45 + 6 * q^47 + 18 * q^49 - 3 * q^51 + 3 * q^53 - q^55 - 7 * q^57 + 6 * q^59 + q^61 - 10 * q^63 - 2 * q^65 - 8 * q^67 + 6 * q^69 + 3 * q^71 + 2 * q^73 - q^75 - 5 * q^77 - 10 * q^79 + q^81 + 6 * q^83 + 3 * q^85 - 3 * q^87 + 9 * q^89 - 10 * q^91 + 7 * q^93 + 7 * q^95 - 4 * q^97 + 2 * 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 −1.00000 0 1.00000 0 5.00000 0 −2.00000 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.l 1
4.b odd 2 1 3520.2.a.z 1
8.b even 2 1 110.2.a.a 1
8.d odd 2 1 880.2.a.c 1
24.f even 2 1 7920.2.a.s 1
24.h odd 2 1 990.2.a.l 1
40.e odd 2 1 4400.2.a.w 1
40.f even 2 1 550.2.a.i 1
40.i odd 4 2 550.2.b.b 2
40.k even 4 2 4400.2.b.g 2
56.h odd 2 1 5390.2.a.h 1
88.b odd 2 1 1210.2.a.k 1
88.g even 2 1 9680.2.a.j 1
120.i odd 2 1 4950.2.a.a 1
120.w even 4 2 4950.2.c.a 2
440.o odd 2 1 6050.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.a 1 8.b even 2 1
550.2.a.i 1 40.f even 2 1
550.2.b.b 2 40.i odd 4 2
880.2.a.c 1 8.d odd 2 1
990.2.a.l 1 24.h odd 2 1
1210.2.a.k 1 88.b odd 2 1
3520.2.a.l 1 1.a even 1 1 trivial
3520.2.a.z 1 4.b odd 2 1
4400.2.a.w 1 40.e odd 2 1
4400.2.b.g 2 40.k even 4 2
4950.2.a.a 1 120.i odd 2 1
4950.2.c.a 2 120.w even 4 2
5390.2.a.h 1 56.h odd 2 1
6050.2.a.i 1 440.o odd 2 1
7920.2.a.s 1 24.f even 2 1
9680.2.a.j 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3520))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7} - 5$$ T7 - 5 $$T_{13} + 2$$ T13 + 2 $$T_{17} - 3$$ T17 - 3 $$T_{19} - 7$$ T19 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T - 5$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T - 3$$
$19$ $$T - 7$$
$23$ $$T + 6$$
$29$ $$T - 3$$
$31$ $$T + 7$$
$37$ $$T - 7$$
$41$ $$T - 6$$
$43$ $$T + 8$$
$47$ $$T - 6$$
$53$ $$T - 3$$
$59$ $$T - 6$$
$61$ $$T - 1$$
$67$ $$T + 8$$
$71$ $$T - 3$$
$73$ $$T - 2$$
$79$ $$T + 10$$
$83$ $$T - 6$$
$89$ $$T - 9$$
$97$ $$T + 4$$