# Properties

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

 $$f(q)$$ $$=$$ $$q - q^{2} - 2 q^{3} + q^{4} + q^{5} + 2 q^{6} - 4 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - 2 * q^3 + q^4 + q^5 + 2 * q^6 - 4 * q^7 - q^8 + q^9 $$q - q^{2} - 2 q^{3} + q^{4} + q^{5} + 2 q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{10} - 6 q^{11} - 2 q^{12} + q^{13} + 4 q^{14} - 2 q^{15} + q^{16} - 6 q^{17} - q^{18} + 2 q^{19} + q^{20} + 8 q^{21} + 6 q^{22} + 6 q^{23} + 2 q^{24} + q^{25} - q^{26} + 4 q^{27} - 4 q^{28} - 6 q^{29} + 2 q^{30} + 2 q^{31} - q^{32} + 12 q^{33} + 6 q^{34} - 4 q^{35} + q^{36} + 2 q^{37} - 2 q^{38} - 2 q^{39} - q^{40} - 6 q^{41} - 8 q^{42} + 2 q^{43} - 6 q^{44} + q^{45} - 6 q^{46} - 12 q^{47} - 2 q^{48} + 9 q^{49} - q^{50} + 12 q^{51} + q^{52} + 6 q^{53} - 4 q^{54} - 6 q^{55} + 4 q^{56} - 4 q^{57} + 6 q^{58} + 6 q^{59} - 2 q^{60} + 2 q^{61} - 2 q^{62} - 4 q^{63} + q^{64} + q^{65} - 12 q^{66} - 4 q^{67} - 6 q^{68} - 12 q^{69} + 4 q^{70} - 6 q^{71} - q^{72} - 10 q^{73} - 2 q^{74} - 2 q^{75} + 2 q^{76} + 24 q^{77} + 2 q^{78} - 4 q^{79} + q^{80} - 11 q^{81} + 6 q^{82} + 8 q^{84} - 6 q^{85} - 2 q^{86} + 12 q^{87} + 6 q^{88} - 6 q^{89} - q^{90} - 4 q^{91} + 6 q^{92} - 4 q^{93} + 12 q^{94} + 2 q^{95} + 2 q^{96} + 2 q^{97} - 9 q^{98} - 6 q^{99}+O(q^{100})$$ q - q^2 - 2 * q^3 + q^4 + q^5 + 2 * q^6 - 4 * q^7 - q^8 + q^9 - q^10 - 6 * q^11 - 2 * q^12 + q^13 + 4 * q^14 - 2 * q^15 + q^16 - 6 * q^17 - q^18 + 2 * q^19 + q^20 + 8 * q^21 + 6 * q^22 + 6 * q^23 + 2 * q^24 + q^25 - q^26 + 4 * q^27 - 4 * q^28 - 6 * q^29 + 2 * q^30 + 2 * q^31 - q^32 + 12 * q^33 + 6 * q^34 - 4 * q^35 + q^36 + 2 * q^37 - 2 * q^38 - 2 * q^39 - q^40 - 6 * q^41 - 8 * q^42 + 2 * q^43 - 6 * q^44 + q^45 - 6 * q^46 - 12 * q^47 - 2 * q^48 + 9 * q^49 - q^50 + 12 * q^51 + q^52 + 6 * q^53 - 4 * q^54 - 6 * q^55 + 4 * q^56 - 4 * q^57 + 6 * q^58 + 6 * q^59 - 2 * q^60 + 2 * q^61 - 2 * q^62 - 4 * q^63 + q^64 + q^65 - 12 * q^66 - 4 * q^67 - 6 * q^68 - 12 * q^69 + 4 * q^70 - 6 * q^71 - q^72 - 10 * q^73 - 2 * q^74 - 2 * q^75 + 2 * q^76 + 24 * q^77 + 2 * q^78 - 4 * q^79 + q^80 - 11 * q^81 + 6 * q^82 + 8 * q^84 - 6 * q^85 - 2 * q^86 + 12 * q^87 + 6 * q^88 - 6 * q^89 - q^90 - 4 * q^91 + 6 * q^92 - 4 * q^93 + 12 * q^94 + 2 * q^95 + 2 * q^96 + 2 * q^97 - 9 * q^98 - 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
−1.00000 −2.00000 1.00000 1.00000 2.00000 −4.00000 −1.00000 1.00000 −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.a 1
3.b odd 2 1 1170.2.a.i 1
4.b odd 2 1 1040.2.a.g 1
5.b even 2 1 650.2.a.l 1
5.c odd 4 2 650.2.b.f 2
7.b odd 2 1 6370.2.a.h 1
8.b even 2 1 4160.2.a.o 1
8.d odd 2 1 4160.2.a.b 1
12.b even 2 1 9360.2.a.z 1
13.b even 2 1 1690.2.a.f 1
13.c even 3 2 1690.2.e.j 2
13.d odd 4 2 1690.2.d.b 2
13.e even 6 2 1690.2.e.d 2
13.f odd 12 4 1690.2.l.h 4
15.d odd 2 1 5850.2.a.ba 1
15.e even 4 2 5850.2.e.bg 2
20.d odd 2 1 5200.2.a.e 1
65.d even 2 1 8450.2.a.i 1

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

## Hecke kernels

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

## Hecke characteristic polynomials

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