# Properties

 Label 130.2.a.b Level $130$ Weight $2$ Character orbit 130.a Self dual yes Analytic conductor $1.038$ 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] = [130,2,Mod(1,130)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(130, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("130.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 130.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: $$1.03805522628$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} + q^{8} - 3 q^{9}+O(q^{10})$$ q + q^2 + q^4 + q^5 + q^8 - 3 * q^9 $$q + q^{2} + q^{4} + q^{5} + q^{8} - 3 q^{9} + q^{10} + q^{13} + q^{16} + 2 q^{17} - 3 q^{18} - 8 q^{19} + q^{20} - 4 q^{23} + q^{25} + q^{26} - 2 q^{29} - 4 q^{31} + q^{32} + 2 q^{34} - 3 q^{36} + 6 q^{37} - 8 q^{38} + q^{40} + 10 q^{41} - 3 q^{45} - 4 q^{46} + 8 q^{47} - 7 q^{49} + q^{50} + q^{52} + 6 q^{53} - 2 q^{58} + 8 q^{59} - 2 q^{61} - 4 q^{62} + q^{64} + q^{65} + 4 q^{67} + 2 q^{68} - 12 q^{71} - 3 q^{72} + 10 q^{73} + 6 q^{74} - 8 q^{76} - 8 q^{79} + q^{80} + 9 q^{81} + 10 q^{82} + 12 q^{83} + 2 q^{85} + 10 q^{89} - 3 q^{90} - 4 q^{92} + 8 q^{94} - 8 q^{95} - 14 q^{97} - 7 q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^5 + q^8 - 3 * q^9 + q^10 + q^13 + q^16 + 2 * q^17 - 3 * q^18 - 8 * q^19 + q^20 - 4 * q^23 + q^25 + q^26 - 2 * q^29 - 4 * q^31 + q^32 + 2 * q^34 - 3 * q^36 + 6 * q^37 - 8 * q^38 + q^40 + 10 * q^41 - 3 * q^45 - 4 * q^46 + 8 * q^47 - 7 * q^49 + q^50 + q^52 + 6 * q^53 - 2 * q^58 + 8 * q^59 - 2 * q^61 - 4 * q^62 + q^64 + q^65 + 4 * q^67 + 2 * q^68 - 12 * q^71 - 3 * q^72 + 10 * q^73 + 6 * q^74 - 8 * q^76 - 8 * q^79 + q^80 + 9 * q^81 + 10 * q^82 + 12 * q^83 + 2 * q^85 + 10 * q^89 - 3 * q^90 - 4 * q^92 + 8 * q^94 - 8 * q^95 - 14 * q^97 - 7 * q^98

## 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
1.00000 0 1.00000 1.00000 0 0 1.00000 −3.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## 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 130.2.a.b 1
3.b odd 2 1 1170.2.a.b 1
4.b odd 2 1 1040.2.a.e 1
5.b even 2 1 650.2.a.d 1
5.c odd 4 2 650.2.b.e 2
7.b odd 2 1 6370.2.a.r 1
8.b even 2 1 4160.2.a.i 1
8.d odd 2 1 4160.2.a.h 1
12.b even 2 1 9360.2.a.l 1
13.b even 2 1 1690.2.a.b 1
13.c even 3 2 1690.2.e.c 2
13.d odd 4 2 1690.2.d.d 2
13.e even 6 2 1690.2.e.i 2
13.f odd 12 4 1690.2.l.f 4
15.d odd 2 1 5850.2.a.bq 1
15.e even 4 2 5850.2.e.q 2
20.d odd 2 1 5200.2.a.r 1
65.d even 2 1 8450.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.a.b 1 1.a even 1 1 trivial
650.2.a.d 1 5.b even 2 1
650.2.b.e 2 5.c odd 4 2
1040.2.a.e 1 4.b odd 2 1
1170.2.a.b 1 3.b odd 2 1
1690.2.a.b 1 13.b even 2 1
1690.2.d.d 2 13.d odd 4 2
1690.2.e.c 2 13.c even 3 2
1690.2.e.i 2 13.e even 6 2
1690.2.l.f 4 13.f odd 12 4
4160.2.a.h 1 8.d odd 2 1
4160.2.a.i 1 8.b even 2 1
5200.2.a.r 1 20.d odd 2 1
5850.2.a.bq 1 15.d odd 2 1
5850.2.e.q 2 15.e even 4 2
6370.2.a.r 1 7.b odd 2 1
8450.2.a.r 1 65.d even 2 1
9360.2.a.l 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(130))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 1$$
$17$ $$T - 2$$
$19$ $$T + 8$$
$23$ $$T + 4$$
$29$ $$T + 2$$
$31$ $$T + 4$$
$37$ $$T - 6$$
$41$ $$T - 10$$
$43$ $$T$$
$47$ $$T - 8$$
$53$ $$T - 6$$
$59$ $$T - 8$$
$61$ $$T + 2$$
$67$ $$T - 4$$
$71$ $$T + 12$$
$73$ $$T - 10$$
$79$ $$T + 8$$
$83$ $$T - 12$$
$89$ $$T - 10$$
$97$ $$T + 14$$