# Properties

 Label 3520.2.a.ba Level $3520$ Weight $2$ Character orbit 3520.a Self dual yes Analytic conductor $28.107$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + q^5 + q^7 - 2 * q^9 $$q + q^{3} + q^{5} + q^{7} - 2 q^{9} - q^{11} - 2 q^{13} + q^{15} - 3 q^{17} - q^{19} + q^{21} - 6 q^{23} + q^{25} - 5 q^{27} + 9 q^{29} - 5 q^{31} - q^{33} + q^{35} - 5 q^{37} - 2 q^{39} - 6 q^{41} + 8 q^{43} - 2 q^{45} - 6 q^{47} - 6 q^{49} - 3 q^{51} - 9 q^{53} - q^{55} - q^{57} + 6 q^{59} - 5 q^{61} - 2 q^{63} - 2 q^{65} + 8 q^{67} - 6 q^{69} + 9 q^{71} - 10 q^{73} + q^{75} - q^{77} - 14 q^{79} + q^{81} - 6 q^{83} - 3 q^{85} + 9 q^{87} - 15 q^{89} - 2 q^{91} - 5 q^{93} - q^{95} + 8 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^3 + q^5 + q^7 - 2 * q^9 - q^11 - 2 * q^13 + q^15 - 3 * q^17 - q^19 + q^21 - 6 * q^23 + q^25 - 5 * q^27 + 9 * q^29 - 5 * q^31 - q^33 + q^35 - 5 * q^37 - 2 * q^39 - 6 * q^41 + 8 * q^43 - 2 * q^45 - 6 * q^47 - 6 * q^49 - 3 * q^51 - 9 * q^53 - q^55 - q^57 + 6 * q^59 - 5 * q^61 - 2 * q^63 - 2 * q^65 + 8 * q^67 - 6 * q^69 + 9 * q^71 - 10 * q^73 + q^75 - q^77 - 14 * q^79 + q^81 - 6 * q^83 - 3 * q^85 + 9 * q^87 - 15 * q^89 - 2 * q^91 - 5 * q^93 - q^95 + 8 * 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 1.00000 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.ba 1
4.b odd 2 1 3520.2.a.k 1
8.b even 2 1 880.2.a.d 1
8.d odd 2 1 110.2.a.c 1
24.f even 2 1 990.2.a.f 1
24.h odd 2 1 7920.2.a.bc 1
40.e odd 2 1 550.2.a.d 1
40.f even 2 1 4400.2.a.t 1
40.i odd 4 2 4400.2.b.j 2
40.k even 4 2 550.2.b.c 2
56.e even 2 1 5390.2.a.x 1
88.b odd 2 1 9680.2.a.g 1
88.g even 2 1 1210.2.a.e 1
120.m even 2 1 4950.2.a.bm 1
120.q odd 4 2 4950.2.c.s 2
440.c even 2 1 6050.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.c 1 8.d odd 2 1
550.2.a.d 1 40.e odd 2 1
550.2.b.c 2 40.k even 4 2
880.2.a.d 1 8.b even 2 1
990.2.a.f 1 24.f even 2 1
1210.2.a.e 1 88.g even 2 1
3520.2.a.k 1 4.b odd 2 1
3520.2.a.ba 1 1.a even 1 1 trivial
4400.2.a.t 1 40.f even 2 1
4400.2.b.j 2 40.i odd 4 2
4950.2.a.bm 1 120.m even 2 1
4950.2.c.s 2 120.q odd 4 2
5390.2.a.x 1 56.e even 2 1
6050.2.a.bc 1 440.c even 2 1
7920.2.a.bc 1 24.h odd 2 1
9680.2.a.g 1 88.b odd 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 $$T_{7} - 1$$ T7 - 1 $$T_{13} + 2$$ T13 + 2 $$T_{17} + 3$$ T17 + 3 $$T_{19} + 1$$ T19 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T + 3$$
$19$ $$T + 1$$
$23$ $$T + 6$$
$29$ $$T - 9$$
$31$ $$T + 5$$
$37$ $$T + 5$$
$41$ $$T + 6$$
$43$ $$T - 8$$
$47$ $$T + 6$$
$53$ $$T + 9$$
$59$ $$T - 6$$
$61$ $$T + 5$$
$67$ $$T - 8$$
$71$ $$T - 9$$
$73$ $$T + 10$$
$79$ $$T + 14$$
$83$ $$T + 6$$
$89$ $$T + 15$$
$97$ $$T - 8$$