# Properties

 Label 130.2.g.a Level $130$ Weight $2$ Character orbit 130.g Analytic conductor $1.038$ Analytic rank $1$ 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: $$1$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( - 2 i - 2) q^{3} + q^{4} + ( - i - 2) q^{5} + (2 i + 2) q^{6} + 4 i q^{7} - q^{8} + 5 i q^{9}+O(q^{10})$$ q - q^2 + (-2*i - 2) * q^3 + q^4 + (-i - 2) * q^5 + (2*i + 2) * q^6 + 4*i * q^7 - q^8 + 5*i * q^9 $$q - q^{2} + ( - 2 i - 2) q^{3} + q^{4} + ( - i - 2) q^{5} + (2 i + 2) q^{6} + 4 i q^{7} - q^{8} + 5 i q^{9} + (i + 2) q^{10} + (2 i - 2) q^{11} + ( - 2 i - 2) q^{12} + ( - 3 i - 2) q^{13} - 4 i q^{14} + (6 i + 2) q^{15} + q^{16} + ( - 3 i - 3) q^{17} - 5 i q^{18} + (2 i - 2) q^{19} + ( - i - 2) q^{20} + ( - 8 i + 8) q^{21} + ( - 2 i + 2) q^{22} + (2 i - 2) q^{23} + (2 i + 2) q^{24} + (4 i + 3) q^{25} + (3 i + 2) q^{26} + ( - 4 i + 4) q^{27} + 4 i q^{28} - 6 i q^{29} + ( - 6 i - 2) q^{30} + ( - 6 i - 6) q^{31} - q^{32} + 8 q^{33} + (3 i + 3) q^{34} + ( - 8 i + 4) q^{35} + 5 i q^{36} + 4 i q^{37} + ( - 2 i + 2) q^{38} + (10 i - 2) q^{39} + (i + 2) q^{40} + ( - i - 1) q^{41} + (8 i - 8) q^{42} + ( - 2 i + 2) q^{43} + (2 i - 2) q^{44} + ( - 10 i + 5) q^{45} + ( - 2 i + 2) q^{46} + 4 i q^{47} + ( - 2 i - 2) q^{48} - 9 q^{49} + ( - 4 i - 3) q^{50} + 12 i q^{51} + ( - 3 i - 2) q^{52} + ( - i - 1) q^{53} + (4 i - 4) q^{54} + ( - 2 i + 6) q^{55} - 4 i q^{56} + 8 q^{57} + 6 i q^{58} + ( - 2 i - 2) q^{59} + (6 i + 2) q^{60} - 12 q^{61} + (6 i + 6) q^{62} - 20 q^{63} + q^{64} + (8 i + 1) q^{65} - 8 q^{66} + 4 q^{67} + ( - 3 i - 3) q^{68} + 8 q^{69} + (8 i - 4) q^{70} + ( - 2 i - 2) q^{71} - 5 i q^{72} + 10 q^{73} - 4 i q^{74} + ( - 14 i + 2) q^{75} + (2 i - 2) q^{76} + ( - 8 i - 8) q^{77} + ( - 10 i + 2) q^{78} - 4 i q^{79} + ( - i - 2) q^{80} - q^{81} + (i + 1) q^{82} + 8 i q^{83} + ( - 8 i + 8) q^{84} + (9 i + 3) q^{85} + (2 i - 2) q^{86} + (12 i - 12) q^{87} + ( - 2 i + 2) q^{88} + (5 i + 5) q^{89} + (10 i - 5) q^{90} + ( - 8 i + 12) q^{91} + (2 i - 2) q^{92} + 24 i q^{93} - 4 i q^{94} + ( - 2 i + 6) q^{95} + (2 i + 2) q^{96} + 9 q^{98} + ( - 10 i - 10) q^{99} +O(q^{100})$$ q - q^2 + (-2*i - 2) * q^3 + q^4 + (-i - 2) * q^5 + (2*i + 2) * q^6 + 4*i * q^7 - q^8 + 5*i * q^9 + (i + 2) * q^10 + (2*i - 2) * q^11 + (-2*i - 2) * q^12 + (-3*i - 2) * q^13 - 4*i * q^14 + (6*i + 2) * q^15 + q^16 + (-3*i - 3) * q^17 - 5*i * q^18 + (2*i - 2) * q^19 + (-i - 2) * q^20 + (-8*i + 8) * q^21 + (-2*i + 2) * q^22 + (2*i - 2) * q^23 + (2*i + 2) * q^24 + (4*i + 3) * q^25 + (3*i + 2) * q^26 + (-4*i + 4) * q^27 + 4*i * q^28 - 6*i * q^29 + (-6*i - 2) * q^30 + (-6*i - 6) * q^31 - q^32 + 8 * q^33 + (3*i + 3) * q^34 + (-8*i + 4) * q^35 + 5*i * q^36 + 4*i * q^37 + (-2*i + 2) * q^38 + (10*i - 2) * q^39 + (i + 2) * q^40 + (-i - 1) * q^41 + (8*i - 8) * q^42 + (-2*i + 2) * q^43 + (2*i - 2) * q^44 + (-10*i + 5) * q^45 + (-2*i + 2) * q^46 + 4*i * q^47 + (-2*i - 2) * q^48 - 9 * q^49 + (-4*i - 3) * q^50 + 12*i * q^51 + (-3*i - 2) * q^52 + (-i - 1) * q^53 + (4*i - 4) * q^54 + (-2*i + 6) * q^55 - 4*i * q^56 + 8 * q^57 + 6*i * q^58 + (-2*i - 2) * q^59 + (6*i + 2) * q^60 - 12 * q^61 + (6*i + 6) * q^62 - 20 * q^63 + q^64 + (8*i + 1) * q^65 - 8 * q^66 + 4 * q^67 + (-3*i - 3) * q^68 + 8 * q^69 + (8*i - 4) * q^70 + (-2*i - 2) * q^71 - 5*i * q^72 + 10 * q^73 - 4*i * q^74 + (-14*i + 2) * q^75 + (2*i - 2) * q^76 + (-8*i - 8) * q^77 + (-10*i + 2) * q^78 - 4*i * q^79 + (-i - 2) * q^80 - q^81 + (i + 1) * q^82 + 8*i * q^83 + (-8*i + 8) * q^84 + (9*i + 3) * q^85 + (2*i - 2) * q^86 + (12*i - 12) * q^87 + (-2*i + 2) * q^88 + (5*i + 5) * q^89 + (10*i - 5) * q^90 + (-8*i + 12) * q^91 + (2*i - 2) * q^92 + 24*i * q^93 - 4*i * q^94 + (-2*i + 6) * q^95 + (2*i + 2) * q^96 + 9 * q^98 + (-10*i - 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{5} + 4 q^{6} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^5 + 4 * q^6 - 2 * q^8 $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 4 q^{5} + 4 q^{6} - 2 q^{8} + 4 q^{10} - 4 q^{11} - 4 q^{12} - 4 q^{13} + 4 q^{15} + 2 q^{16} - 6 q^{17} - 4 q^{19} - 4 q^{20} + 16 q^{21} + 4 q^{22} - 4 q^{23} + 4 q^{24} + 6 q^{25} + 4 q^{26} + 8 q^{27} - 4 q^{30} - 12 q^{31} - 2 q^{32} + 16 q^{33} + 6 q^{34} + 8 q^{35} + 4 q^{38} - 4 q^{39} + 4 q^{40} - 2 q^{41} - 16 q^{42} + 4 q^{43} - 4 q^{44} + 10 q^{45} + 4 q^{46} - 4 q^{48} - 18 q^{49} - 6 q^{50} - 4 q^{52} - 2 q^{53} - 8 q^{54} + 12 q^{55} + 16 q^{57} - 4 q^{59} + 4 q^{60} - 24 q^{61} + 12 q^{62} - 40 q^{63} + 2 q^{64} + 2 q^{65} - 16 q^{66} + 8 q^{67} - 6 q^{68} + 16 q^{69} - 8 q^{70} - 4 q^{71} + 20 q^{73} + 4 q^{75} - 4 q^{76} - 16 q^{77} + 4 q^{78} - 4 q^{80} - 2 q^{81} + 2 q^{82} + 16 q^{84} + 6 q^{85} - 4 q^{86} - 24 q^{87} + 4 q^{88} + 10 q^{89} - 10 q^{90} + 24 q^{91} - 4 q^{92} + 12 q^{95} + 4 q^{96} + 18 q^{98} - 20 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 - 4 * q^5 + 4 * q^6 - 2 * q^8 + 4 * q^10 - 4 * q^11 - 4 * q^12 - 4 * q^13 + 4 * q^15 + 2 * q^16 - 6 * q^17 - 4 * q^19 - 4 * q^20 + 16 * q^21 + 4 * q^22 - 4 * q^23 + 4 * q^24 + 6 * q^25 + 4 * q^26 + 8 * q^27 - 4 * q^30 - 12 * q^31 - 2 * q^32 + 16 * q^33 + 6 * q^34 + 8 * q^35 + 4 * q^38 - 4 * q^39 + 4 * q^40 - 2 * q^41 - 16 * q^42 + 4 * q^43 - 4 * q^44 + 10 * q^45 + 4 * q^46 - 4 * q^48 - 18 * q^49 - 6 * q^50 - 4 * q^52 - 2 * q^53 - 8 * q^54 + 12 * q^55 + 16 * q^57 - 4 * q^59 + 4 * q^60 - 24 * q^61 + 12 * q^62 - 40 * q^63 + 2 * q^64 + 2 * q^65 - 16 * q^66 + 8 * q^67 - 6 * q^68 + 16 * q^69 - 8 * q^70 - 4 * q^71 + 20 * q^73 + 4 * q^75 - 4 * q^76 - 16 * q^77 + 4 * q^78 - 4 * q^80 - 2 * q^81 + 2 * q^82 + 16 * q^84 + 6 * q^85 - 4 * q^86 - 24 * q^87 + 4 * q^88 + 10 * q^89 - 10 * q^90 + 24 * q^91 - 4 * q^92 + 12 * q^95 + 4 * q^96 + 18 * q^98 - 20 * 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 −2.00000 + 2.00000i 1.00000 −2.00000 + 1.00000i 2.00000 2.00000i 4.00000i −1.00000 5.00000i 2.00000 1.00000i
73.1 −1.00000 −2.00000 2.00000i 1.00000 −2.00000 1.00000i 2.00000 + 2.00000i 4.00000i −1.00000 5.00000i 2.00000 + 1.00000i
 $$n$$: 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.a 2
3.b odd 2 1 1170.2.m.c 2
4.b odd 2 1 1040.2.bg.g 2
5.b even 2 1 650.2.g.e 2
5.c odd 4 1 130.2.j.a yes 2
5.c odd 4 1 650.2.j.e 2
13.d odd 4 1 130.2.j.a yes 2
15.e even 4 1 1170.2.w.b 2
20.e even 4 1 1040.2.cd.g 2
39.f even 4 1 1170.2.w.b 2
52.f even 4 1 1040.2.cd.g 2
65.f even 4 1 650.2.g.e 2
65.g odd 4 1 650.2.j.e 2
65.k even 4 1 inner 130.2.g.a 2
195.j odd 4 1 1170.2.m.c 2
260.s odd 4 1 1040.2.bg.g 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4T_{3} + 8$$ acting on $$S_{2}^{\mathrm{new}}(130, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 4T + 8$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 4T + 8$$
$13$ $$T^{2} + 4T + 13$$
$17$ $$T^{2} + 6T + 18$$
$19$ $$T^{2} + 4T + 8$$
$23$ $$T^{2} + 4T + 8$$
$29$ $$T^{2} + 36$$
$31$ $$T^{2} + 12T + 72$$
$37$ $$T^{2} + 16$$
$41$ $$T^{2} + 2T + 2$$
$43$ $$T^{2} - 4T + 8$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 2T + 2$$
$59$ $$T^{2} + 4T + 8$$
$61$ $$(T + 12)^{2}$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} + 4T + 8$$
$73$ $$(T - 10)^{2}$$
$79$ $$T^{2} + 16$$
$83$ $$T^{2} + 64$$
$89$ $$T^{2} - 10T + 50$$
$97$ $$T^{2}$$