# Properties

 Label 130.3.h.b Level $130$ Weight $3$ Character orbit 130.h Analytic conductor $3.542$ 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] = [130,3,Mod(77,130)]

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, 2]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("130.77");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 130.h (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: $$3.54224343668$$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - i + 1) q^{2} + (2 i - 2) q^{3} - 2 i q^{4} + 5 i q^{5} + 4 i q^{6} + (2 i - 2) q^{7} + ( - 2 i - 2) q^{8} + i q^{9} +O(q^{10})$$ q + (-i + 1) * q^2 + (2*i - 2) * q^3 - 2*i * q^4 + 5*i * q^5 + 4*i * q^6 + (2*i - 2) * q^7 + (-2*i - 2) * q^8 + i * q^9 $$q + ( - i + 1) q^{2} + (2 i - 2) q^{3} - 2 i q^{4} + 5 i q^{5} + 4 i q^{6} + (2 i - 2) q^{7} + ( - 2 i - 2) q^{8} + i q^{9} + (5 i + 5) q^{10} + 20 i q^{11} + (4 i + 4) q^{12} - 13 i q^{13} + 4 i q^{14} + ( - 10 i - 10) q^{15} - 4 q^{16} + (17 i + 17) q^{17} + (i + 1) q^{18} - 8 q^{19} + 10 q^{20} - 8 i q^{21} + (20 i + 20) q^{22} + ( - 18 i + 18) q^{23} + 8 q^{24} - 25 q^{25} + ( - 13 i - 13) q^{26} + ( - 20 i - 20) q^{27} + (4 i + 4) q^{28} - 40 i q^{29} - 20 q^{30} + 40 i q^{31} + (4 i - 4) q^{32} + ( - 40 i - 40) q^{33} + 34 q^{34} + ( - 10 i - 10) q^{35} + 2 q^{36} + ( - 43 i + 43) q^{37} + (8 i - 8) q^{38} + (26 i + 26) q^{39} + ( - 10 i + 10) q^{40} + 10 i q^{41} + ( - 8 i - 8) q^{42} + (2 i - 2) q^{43} + 40 q^{44} - 5 q^{45} - 36 i q^{46} + (22 i - 22) q^{47} + ( - 8 i + 8) q^{48} + 41 i q^{49} + (25 i - 25) q^{50} - 68 q^{51} - 26 q^{52} + ( - 13 i + 13) q^{53} - 40 q^{54} - 100 q^{55} + 8 q^{56} + ( - 16 i + 16) q^{57} + ( - 40 i - 40) q^{58} + 32 q^{59} + (20 i - 20) q^{60} + 112 q^{61} + (40 i + 40) q^{62} + ( - 2 i - 2) q^{63} + 8 i q^{64} + 65 q^{65} - 80 q^{66} + ( - 18 i + 18) q^{67} + ( - 34 i + 34) q^{68} + 72 i q^{69} - 20 q^{70} - 40 i q^{71} + ( - 2 i + 2) q^{72} + (7 i + 7) q^{73} - 86 i q^{74} + ( - 50 i + 50) q^{75} + 16 i q^{76} + ( - 40 i - 40) q^{77} + 52 q^{78} + 120 i q^{79} - 20 i q^{80} + 71 q^{81} + (10 i + 10) q^{82} + (22 i + 22) q^{83} - 16 q^{84} + (85 i - 85) q^{85} + 4 i q^{86} + (80 i + 80) q^{87} + ( - 40 i + 40) q^{88} + 82 q^{89} + (5 i - 5) q^{90} + (26 i + 26) q^{91} + ( - 36 i - 36) q^{92} + ( - 80 i - 80) q^{93} + 44 i q^{94} - 40 i q^{95} - 16 i q^{96} + ( - 33 i + 33) q^{97} + (41 i + 41) q^{98} - 20 q^{99} +O(q^{100})$$ q + (-i + 1) * q^2 + (2*i - 2) * q^3 - 2*i * q^4 + 5*i * q^5 + 4*i * q^6 + (2*i - 2) * q^7 + (-2*i - 2) * q^8 + i * q^9 + (5*i + 5) * q^10 + 20*i * q^11 + (4*i + 4) * q^12 - 13*i * q^13 + 4*i * q^14 + (-10*i - 10) * q^15 - 4 * q^16 + (17*i + 17) * q^17 + (i + 1) * q^18 - 8 * q^19 + 10 * q^20 - 8*i * q^21 + (20*i + 20) * q^22 + (-18*i + 18) * q^23 + 8 * q^24 - 25 * q^25 + (-13*i - 13) * q^26 + (-20*i - 20) * q^27 + (4*i + 4) * q^28 - 40*i * q^29 - 20 * q^30 + 40*i * q^31 + (4*i - 4) * q^32 + (-40*i - 40) * q^33 + 34 * q^34 + (-10*i - 10) * q^35 + 2 * q^36 + (-43*i + 43) * q^37 + (8*i - 8) * q^38 + (26*i + 26) * q^39 + (-10*i + 10) * q^40 + 10*i * q^41 + (-8*i - 8) * q^42 + (2*i - 2) * q^43 + 40 * q^44 - 5 * q^45 - 36*i * q^46 + (22*i - 22) * q^47 + (-8*i + 8) * q^48 + 41*i * q^49 + (25*i - 25) * q^50 - 68 * q^51 - 26 * q^52 + (-13*i + 13) * q^53 - 40 * q^54 - 100 * q^55 + 8 * q^56 + (-16*i + 16) * q^57 + (-40*i - 40) * q^58 + 32 * q^59 + (20*i - 20) * q^60 + 112 * q^61 + (40*i + 40) * q^62 + (-2*i - 2) * q^63 + 8*i * q^64 + 65 * q^65 - 80 * q^66 + (-18*i + 18) * q^67 + (-34*i + 34) * q^68 + 72*i * q^69 - 20 * q^70 - 40*i * q^71 + (-2*i + 2) * q^72 + (7*i + 7) * q^73 - 86*i * q^74 + (-50*i + 50) * q^75 + 16*i * q^76 + (-40*i - 40) * q^77 + 52 * q^78 + 120*i * q^79 - 20*i * q^80 + 71 * q^81 + (10*i + 10) * q^82 + (22*i + 22) * q^83 - 16 * q^84 + (85*i - 85) * q^85 + 4*i * q^86 + (80*i + 80) * q^87 + (-40*i + 40) * q^88 + 82 * q^89 + (5*i - 5) * q^90 + (26*i + 26) * q^91 + (-36*i - 36) * q^92 + (-80*i - 80) * q^93 + 44*i * q^94 - 40*i * q^95 - 16*i * q^96 + (-33*i + 33) * q^97 + (41*i + 41) * q^98 - 20 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{3} - 4 q^{7} - 4 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^3 - 4 * q^7 - 4 * q^8 $$2 q + 2 q^{2} - 4 q^{3} - 4 q^{7} - 4 q^{8} + 10 q^{10} + 8 q^{12} - 20 q^{15} - 8 q^{16} + 34 q^{17} + 2 q^{18} - 16 q^{19} + 20 q^{20} + 40 q^{22} + 36 q^{23} + 16 q^{24} - 50 q^{25} - 26 q^{26} - 40 q^{27} + 8 q^{28} - 40 q^{30} - 8 q^{32} - 80 q^{33} + 68 q^{34} - 20 q^{35} + 4 q^{36} + 86 q^{37} - 16 q^{38} + 52 q^{39} + 20 q^{40} - 16 q^{42} - 4 q^{43} + 80 q^{44} - 10 q^{45} - 44 q^{47} + 16 q^{48} - 50 q^{50} - 136 q^{51} - 52 q^{52} + 26 q^{53} - 80 q^{54} - 200 q^{55} + 16 q^{56} + 32 q^{57} - 80 q^{58} + 64 q^{59} - 40 q^{60} + 224 q^{61} + 80 q^{62} - 4 q^{63} + 130 q^{65} - 160 q^{66} + 36 q^{67} + 68 q^{68} - 40 q^{70} + 4 q^{72} + 14 q^{73} + 100 q^{75} - 80 q^{77} + 104 q^{78} + 142 q^{81} + 20 q^{82} + 44 q^{83} - 32 q^{84} - 170 q^{85} + 160 q^{87} + 80 q^{88} + 164 q^{89} - 10 q^{90} + 52 q^{91} - 72 q^{92} - 160 q^{93} + 66 q^{97} + 82 q^{98} - 40 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^3 - 4 * q^7 - 4 * q^8 + 10 * q^10 + 8 * q^12 - 20 * q^15 - 8 * q^16 + 34 * q^17 + 2 * q^18 - 16 * q^19 + 20 * q^20 + 40 * q^22 + 36 * q^23 + 16 * q^24 - 50 * q^25 - 26 * q^26 - 40 * q^27 + 8 * q^28 - 40 * q^30 - 8 * q^32 - 80 * q^33 + 68 * q^34 - 20 * q^35 + 4 * q^36 + 86 * q^37 - 16 * q^38 + 52 * q^39 + 20 * q^40 - 16 * q^42 - 4 * q^43 + 80 * q^44 - 10 * q^45 - 44 * q^47 + 16 * q^48 - 50 * q^50 - 136 * q^51 - 52 * q^52 + 26 * q^53 - 80 * q^54 - 200 * q^55 + 16 * q^56 + 32 * q^57 - 80 * q^58 + 64 * q^59 - 40 * q^60 + 224 * q^61 + 80 * q^62 - 4 * q^63 + 130 * q^65 - 160 * q^66 + 36 * q^67 + 68 * q^68 - 40 * q^70 + 4 * q^72 + 14 * q^73 + 100 * q^75 - 80 * q^77 + 104 * q^78 + 142 * q^81 + 20 * q^82 + 44 * q^83 - 32 * q^84 - 170 * q^85 + 160 * q^87 + 80 * q^88 + 164 * q^89 - 10 * q^90 + 52 * q^91 - 72 * q^92 - 160 * q^93 + 66 * q^97 + 82 * q^98 - 40 * q^99

## Character values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$i$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
77.1
 1.00000i − 1.00000i
1.00000 1.00000i −2.00000 + 2.00000i 2.00000i 5.00000i 4.00000i −2.00000 + 2.00000i −2.00000 2.00000i 1.00000i 5.00000 + 5.00000i
103.1 1.00000 + 1.00000i −2.00000 2.00000i 2.00000i 5.00000i 4.00000i −2.00000 2.00000i −2.00000 + 2.00000i 1.00000i 5.00000 5.00000i
 $$n$$: 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.h odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.3.h.b yes 2
5.b even 2 1 650.3.h.a 2
5.c odd 4 1 130.3.h.a 2
5.c odd 4 1 650.3.h.b 2
13.b even 2 1 130.3.h.a 2
65.d even 2 1 650.3.h.b 2
65.h odd 4 1 inner 130.3.h.b yes 2
65.h odd 4 1 650.3.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.h.a 2 5.c odd 4 1
130.3.h.a 2 13.b even 2 1
130.3.h.b yes 2 1.a even 1 1 trivial
130.3.h.b yes 2 65.h odd 4 1 inner
650.3.h.a 2 5.b even 2 1
650.3.h.a 2 65.h odd 4 1
650.3.h.b 2 5.c odd 4 1
650.3.h.b 2 65.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(130, [\chi])$$:

 $$T_{3}^{2} + 4T_{3} + 8$$ T3^2 + 4*T3 + 8 $$T_{7}^{2} + 4T_{7} + 8$$ T7^2 + 4*T7 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2} + 4T + 8$$
$5$ $$T^{2} + 25$$
$7$ $$T^{2} + 4T + 8$$
$11$ $$T^{2} + 400$$
$13$ $$T^{2} + 169$$
$17$ $$T^{2} - 34T + 578$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} - 36T + 648$$
$29$ $$T^{2} + 1600$$
$31$ $$T^{2} + 1600$$
$37$ $$T^{2} - 86T + 3698$$
$41$ $$T^{2} + 100$$
$43$ $$T^{2} + 4T + 8$$
$47$ $$T^{2} + 44T + 968$$
$53$ $$T^{2} - 26T + 338$$
$59$ $$(T - 32)^{2}$$
$61$ $$(T - 112)^{2}$$
$67$ $$T^{2} - 36T + 648$$
$71$ $$T^{2} + 1600$$
$73$ $$T^{2} - 14T + 98$$
$79$ $$T^{2} + 14400$$
$83$ $$T^{2} - 44T + 968$$
$89$ $$(T - 82)^{2}$$
$97$ $$T^{2} - 66T + 2178$$