# Properties

 Label 130.2.g.b Level $130$ Weight $2$ Character orbit 130.g Analytic conductor $1.038$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [130,2,Mod(57,130)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(130, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("130.57");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 130.g (of order $$4$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$1.03805522628$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + (i + 1) q^{3} + q^{4} + (2 i - 1) q^{5} + ( - i - 1) q^{6} + 2 i q^{7} - q^{8} - i q^{9} +O(q^{10})$$ q - q^2 + (i + 1) * q^3 + q^4 + (2*i - 1) * q^5 + (-i - 1) * q^6 + 2*i * q^7 - q^8 - i * q^9 $$q - q^{2} + (i + 1) q^{3} + q^{4} + (2 i - 1) q^{5} + ( - i - 1) q^{6} + 2 i q^{7} - q^{8} - i q^{9} + ( - 2 i + 1) q^{10} + (i - 1) q^{11} + (i + 1) q^{12} + (2 i + 3) q^{13} - 2 i q^{14} + (i - 3) q^{15} + q^{16} + (i + 1) q^{17} + i q^{18} + ( - 3 i + 3) q^{19} + (2 i - 1) q^{20} + (2 i - 2) q^{21} + ( - i + 1) q^{22} + (i - 1) q^{23} + ( - i - 1) q^{24} + ( - 4 i - 3) q^{25} + ( - 2 i - 3) q^{26} + ( - 4 i + 4) q^{27} + 2 i q^{28} - 8 i q^{29} + ( - i + 3) q^{30} + (i + 1) q^{31} - q^{32} - 2 q^{33} + ( - i - 1) q^{34} + ( - 2 i - 4) q^{35} - i q^{36} - 8 i q^{37} + (3 i - 3) q^{38} + (5 i + 1) q^{39} + ( - 2 i + 1) q^{40} + ( - 7 i - 7) q^{41} + ( - 2 i + 2) q^{42} + (i - 1) q^{43} + (i - 1) q^{44} + (i + 2) q^{45} + ( - i + 1) q^{46} + 10 i q^{47} + (i + 1) q^{48} + 3 q^{49} + (4 i + 3) q^{50} + 2 i q^{51} + (2 i + 3) q^{52} + (i + 1) q^{53} + (4 i - 4) q^{54} + ( - 3 i - 1) q^{55} - 2 i q^{56} + 6 q^{57} + 8 i q^{58} + (9 i + 9) q^{59} + (i - 3) q^{60} + 2 q^{61} + ( - i - 1) q^{62} + 2 q^{63} + q^{64} + (4 i - 7) q^{65} + 2 q^{66} - 12 q^{67} + (i + 1) q^{68} - 2 q^{69} + (2 i + 4) q^{70} + (5 i + 5) q^{71} + i q^{72} + 6 q^{73} + 8 i q^{74} + ( - 7 i + 1) q^{75} + ( - 3 i + 3) q^{76} + ( - 2 i - 2) q^{77} + ( - 5 i - 1) q^{78} + 10 i q^{79} + (2 i - 1) q^{80} + 5 q^{81} + (7 i + 7) q^{82} - 18 i q^{83} + (2 i - 2) q^{84} + (i - 3) q^{85} + ( - i + 1) q^{86} + ( - 8 i + 8) q^{87} + ( - i + 1) q^{88} + ( - 11 i - 11) q^{89} + ( - i - 2) q^{90} + (6 i - 4) q^{91} + (i - 1) q^{92} + 2 i q^{93} - 10 i q^{94} + (9 i + 3) q^{95} + ( - i - 1) q^{96} - 14 q^{97} - 3 q^{98} + (i + 1) q^{99} +O(q^{100})$$ q - q^2 + (i + 1) * q^3 + q^4 + (2*i - 1) * q^5 + (-i - 1) * q^6 + 2*i * q^7 - q^8 - i * q^9 + (-2*i + 1) * q^10 + (i - 1) * q^11 + (i + 1) * q^12 + (2*i + 3) * q^13 - 2*i * q^14 + (i - 3) * q^15 + q^16 + (i + 1) * q^17 + i * q^18 + (-3*i + 3) * q^19 + (2*i - 1) * q^20 + (2*i - 2) * q^21 + (-i + 1) * q^22 + (i - 1) * q^23 + (-i - 1) * q^24 + (-4*i - 3) * q^25 + (-2*i - 3) * q^26 + (-4*i + 4) * q^27 + 2*i * q^28 - 8*i * q^29 + (-i + 3) * q^30 + (i + 1) * q^31 - q^32 - 2 * q^33 + (-i - 1) * q^34 + (-2*i - 4) * q^35 - i * q^36 - 8*i * q^37 + (3*i - 3) * q^38 + (5*i + 1) * q^39 + (-2*i + 1) * q^40 + (-7*i - 7) * q^41 + (-2*i + 2) * q^42 + (i - 1) * q^43 + (i - 1) * q^44 + (i + 2) * q^45 + (-i + 1) * q^46 + 10*i * q^47 + (i + 1) * q^48 + 3 * q^49 + (4*i + 3) * q^50 + 2*i * q^51 + (2*i + 3) * q^52 + (i + 1) * q^53 + (4*i - 4) * q^54 + (-3*i - 1) * q^55 - 2*i * q^56 + 6 * q^57 + 8*i * q^58 + (9*i + 9) * q^59 + (i - 3) * q^60 + 2 * q^61 + (-i - 1) * q^62 + 2 * q^63 + q^64 + (4*i - 7) * q^65 + 2 * q^66 - 12 * q^67 + (i + 1) * q^68 - 2 * q^69 + (2*i + 4) * q^70 + (5*i + 5) * q^71 + i * q^72 + 6 * q^73 + 8*i * q^74 + (-7*i + 1) * q^75 + (-3*i + 3) * q^76 + (-2*i - 2) * q^77 + (-5*i - 1) * q^78 + 10*i * q^79 + (2*i - 1) * q^80 + 5 * q^81 + (7*i + 7) * q^82 - 18*i * q^83 + (2*i - 2) * q^84 + (i - 3) * q^85 + (-i + 1) * q^86 + (-8*i + 8) * q^87 + (-i + 1) * q^88 + (-11*i - 11) * q^89 + (-i - 2) * q^90 + (6*i - 4) * q^91 + (i - 1) * q^92 + 2*i * q^93 - 10*i * q^94 + (9*i + 3) * q^95 + (-i - 1) * q^96 - 14 * q^97 - 3 * q^98 + (i + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} - 6 q^{15} + 2 q^{16} + 2 q^{17} + 6 q^{19} - 2 q^{20} - 4 q^{21} + 2 q^{22} - 2 q^{23} - 2 q^{24} - 6 q^{25} - 6 q^{26} + 8 q^{27} + 6 q^{30} + 2 q^{31} - 2 q^{32} - 4 q^{33} - 2 q^{34} - 8 q^{35} - 6 q^{38} + 2 q^{39} + 2 q^{40} - 14 q^{41} + 4 q^{42} - 2 q^{43} - 2 q^{44} + 4 q^{45} + 2 q^{46} + 2 q^{48} + 6 q^{49} + 6 q^{50} + 6 q^{52} + 2 q^{53} - 8 q^{54} - 2 q^{55} + 12 q^{57} + 18 q^{59} - 6 q^{60} + 4 q^{61} - 2 q^{62} + 4 q^{63} + 2 q^{64} - 14 q^{65} + 4 q^{66} - 24 q^{67} + 2 q^{68} - 4 q^{69} + 8 q^{70} + 10 q^{71} + 12 q^{73} + 2 q^{75} + 6 q^{76} - 4 q^{77} - 2 q^{78} - 2 q^{80} + 10 q^{81} + 14 q^{82} - 4 q^{84} - 6 q^{85} + 2 q^{86} + 16 q^{87} + 2 q^{88} - 22 q^{89} - 4 q^{90} - 8 q^{91} - 2 q^{92} + 6 q^{95} - 2 q^{96} - 28 q^{97} - 6 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^8 + 2 * q^10 - 2 * q^11 + 2 * q^12 + 6 * q^13 - 6 * q^15 + 2 * q^16 + 2 * q^17 + 6 * q^19 - 2 * q^20 - 4 * q^21 + 2 * q^22 - 2 * q^23 - 2 * q^24 - 6 * q^25 - 6 * q^26 + 8 * q^27 + 6 * q^30 + 2 * q^31 - 2 * q^32 - 4 * q^33 - 2 * q^34 - 8 * q^35 - 6 * q^38 + 2 * q^39 + 2 * q^40 - 14 * q^41 + 4 * q^42 - 2 * q^43 - 2 * q^44 + 4 * q^45 + 2 * q^46 + 2 * q^48 + 6 * q^49 + 6 * q^50 + 6 * q^52 + 2 * q^53 - 8 * q^54 - 2 * q^55 + 12 * q^57 + 18 * q^59 - 6 * q^60 + 4 * q^61 - 2 * q^62 + 4 * q^63 + 2 * q^64 - 14 * q^65 + 4 * q^66 - 24 * q^67 + 2 * q^68 - 4 * q^69 + 8 * q^70 + 10 * q^71 + 12 * q^73 + 2 * q^75 + 6 * q^76 - 4 * q^77 - 2 * q^78 - 2 * q^80 + 10 * q^81 + 14 * q^82 - 4 * q^84 - 6 * q^85 + 2 * q^86 + 16 * q^87 + 2 * q^88 - 22 * q^89 - 4 * q^90 - 8 * q^91 - 2 * q^92 + 6 * q^95 - 2 * q^96 - 28 * q^97 - 6 * q^98 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/130\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-i$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 − 1.00000i 1.00000i
−1.00000 1.00000 1.00000i 1.00000 −1.00000 2.00000i −1.00000 + 1.00000i 2.00000i −1.00000 1.00000i 1.00000 + 2.00000i
73.1 −1.00000 1.00000 + 1.00000i 1.00000 −1.00000 + 2.00000i −1.00000 1.00000i 2.00000i −1.00000 1.00000i 1.00000 2.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.g.b 2
3.b odd 2 1 1170.2.m.b 2
4.b odd 2 1 1040.2.bg.b 2
5.b even 2 1 650.2.g.c 2
5.c odd 4 1 130.2.j.c yes 2
5.c odd 4 1 650.2.j.a 2
13.d odd 4 1 130.2.j.c yes 2
15.e even 4 1 1170.2.w.a 2
20.e even 4 1 1040.2.cd.c 2
39.f even 4 1 1170.2.w.a 2
52.f even 4 1 1040.2.cd.c 2
65.f even 4 1 650.2.g.c 2
65.g odd 4 1 650.2.j.a 2
65.k even 4 1 inner 130.2.g.b 2
195.j odd 4 1 1170.2.m.b 2
260.s odd 4 1 1040.2.bg.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.g.b 2 1.a even 1 1 trivial
130.2.g.b 2 65.k even 4 1 inner
130.2.j.c yes 2 5.c odd 4 1
130.2.j.c yes 2 13.d odd 4 1
650.2.g.c 2 5.b even 2 1
650.2.g.c 2 65.f even 4 1
650.2.j.a 2 5.c odd 4 1
650.2.j.a 2 65.g odd 4 1
1040.2.bg.b 2 4.b odd 2 1
1040.2.bg.b 2 260.s odd 4 1
1040.2.cd.c 2 20.e even 4 1
1040.2.cd.c 2 52.f even 4 1
1170.2.m.b 2 3.b odd 2 1
1170.2.m.b 2 195.j odd 4 1
1170.2.w.a 2 15.e even 4 1
1170.2.w.a 2 39.f even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 2T + 2$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 2T + 2$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$T^{2} - 2T + 2$$
$19$ $$T^{2} - 6T + 18$$
$23$ $$T^{2} + 2T + 2$$
$29$ $$T^{2} + 64$$
$31$ $$T^{2} - 2T + 2$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 14T + 98$$
$43$ $$T^{2} + 2T + 2$$
$47$ $$T^{2} + 100$$
$53$ $$T^{2} - 2T + 2$$
$59$ $$T^{2} - 18T + 162$$
$61$ $$(T - 2)^{2}$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2} - 10T + 50$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} + 100$$
$83$ $$T^{2} + 324$$
$89$ $$T^{2} + 22T + 242$$
$97$ $$(T + 14)^{2}$$