# Properties

 Label 130.4.a.a Level $130$ Weight $4$ Character orbit 130.a Self dual yes Analytic conductor $7.670$ 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] = [130,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("130.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$7.67024830075$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{5} + 4 q^{6} + 8 q^{7} - 8 q^{8} - 23 q^{9}+O(q^{10})$$ q - 2 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^5 + 4 * q^6 + 8 * q^7 - 8 * q^8 - 23 * q^9 $$q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{5} + 4 q^{6} + 8 q^{7} - 8 q^{8} - 23 q^{9} - 10 q^{10} + 6 q^{11} - 8 q^{12} + 13 q^{13} - 16 q^{14} - 10 q^{15} + 16 q^{16} + 114 q^{17} + 46 q^{18} + 38 q^{19} + 20 q^{20} - 16 q^{21} - 12 q^{22} + 150 q^{23} + 16 q^{24} + 25 q^{25} - 26 q^{26} + 100 q^{27} + 32 q^{28} + 114 q^{29} + 20 q^{30} - 34 q^{31} - 32 q^{32} - 12 q^{33} - 228 q^{34} + 40 q^{35} - 92 q^{36} + 146 q^{37} - 76 q^{38} - 26 q^{39} - 40 q^{40} - 30 q^{41} + 32 q^{42} + 122 q^{43} + 24 q^{44} - 115 q^{45} - 300 q^{46} + 336 q^{47} - 32 q^{48} - 279 q^{49} - 50 q^{50} - 228 q^{51} + 52 q^{52} - 570 q^{53} - 200 q^{54} + 30 q^{55} - 64 q^{56} - 76 q^{57} - 228 q^{58} + 66 q^{59} - 40 q^{60} - 502 q^{61} + 68 q^{62} - 184 q^{63} + 64 q^{64} + 65 q^{65} + 24 q^{66} + 728 q^{67} + 456 q^{68} - 300 q^{69} - 80 q^{70} + 582 q^{71} + 184 q^{72} - 994 q^{73} - 292 q^{74} - 50 q^{75} + 152 q^{76} + 48 q^{77} + 52 q^{78} - 988 q^{79} + 80 q^{80} + 421 q^{81} + 60 q^{82} - 84 q^{83} - 64 q^{84} + 570 q^{85} - 244 q^{86} - 228 q^{87} - 48 q^{88} + 906 q^{89} + 230 q^{90} + 104 q^{91} + 600 q^{92} + 68 q^{93} - 672 q^{94} + 190 q^{95} + 64 q^{96} + 290 q^{97} + 558 q^{98} - 138 q^{99}+O(q^{100})$$ q - 2 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^5 + 4 * q^6 + 8 * q^7 - 8 * q^8 - 23 * q^9 - 10 * q^10 + 6 * q^11 - 8 * q^12 + 13 * q^13 - 16 * q^14 - 10 * q^15 + 16 * q^16 + 114 * q^17 + 46 * q^18 + 38 * q^19 + 20 * q^20 - 16 * q^21 - 12 * q^22 + 150 * q^23 + 16 * q^24 + 25 * q^25 - 26 * q^26 + 100 * q^27 + 32 * q^28 + 114 * q^29 + 20 * q^30 - 34 * q^31 - 32 * q^32 - 12 * q^33 - 228 * q^34 + 40 * q^35 - 92 * q^36 + 146 * q^37 - 76 * q^38 - 26 * q^39 - 40 * q^40 - 30 * q^41 + 32 * q^42 + 122 * q^43 + 24 * q^44 - 115 * q^45 - 300 * q^46 + 336 * q^47 - 32 * q^48 - 279 * q^49 - 50 * q^50 - 228 * q^51 + 52 * q^52 - 570 * q^53 - 200 * q^54 + 30 * q^55 - 64 * q^56 - 76 * q^57 - 228 * q^58 + 66 * q^59 - 40 * q^60 - 502 * q^61 + 68 * q^62 - 184 * q^63 + 64 * q^64 + 65 * q^65 + 24 * q^66 + 728 * q^67 + 456 * q^68 - 300 * q^69 - 80 * q^70 + 582 * q^71 + 184 * q^72 - 994 * q^73 - 292 * q^74 - 50 * q^75 + 152 * q^76 + 48 * q^77 + 52 * q^78 - 988 * q^79 + 80 * q^80 + 421 * q^81 + 60 * q^82 - 84 * q^83 - 64 * q^84 + 570 * q^85 - 244 * q^86 - 228 * q^87 - 48 * q^88 + 906 * q^89 + 230 * q^90 + 104 * q^91 + 600 * q^92 + 68 * q^93 - 672 * q^94 + 190 * q^95 + 64 * q^96 + 290 * q^97 + 558 * q^98 - 138 * 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
−2.00000 −2.00000 4.00000 5.00000 4.00000 8.00000 −8.00000 −23.0000 −10.0000
 $$n$$: 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 130.4.a.a 1
3.b odd 2 1 1170.4.a.k 1
4.b odd 2 1 1040.4.a.c 1
5.b even 2 1 650.4.a.h 1
5.c odd 4 2 650.4.b.e 2
13.b even 2 1 1690.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.4.a.a 1 1.a even 1 1 trivial
650.4.a.h 1 5.b even 2 1
650.4.b.e 2 5.c odd 4 2
1040.4.a.c 1 4.b odd 2 1
1170.4.a.k 1 3.b odd 2 1
1690.4.a.i 1 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(130))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 2$$
$5$ $$T - 5$$
$7$ $$T - 8$$
$11$ $$T - 6$$
$13$ $$T - 13$$
$17$ $$T - 114$$
$19$ $$T - 38$$
$23$ $$T - 150$$
$29$ $$T - 114$$
$31$ $$T + 34$$
$37$ $$T - 146$$
$41$ $$T + 30$$
$43$ $$T - 122$$
$47$ $$T - 336$$
$53$ $$T + 570$$
$59$ $$T - 66$$
$61$ $$T + 502$$
$67$ $$T - 728$$
$71$ $$T - 582$$
$73$ $$T + 994$$
$79$ $$T + 988$$
$83$ $$T + 84$$
$89$ $$T - 906$$
$97$ $$T - 290$$