# Properties

 Label 80.4.c.a Level $80$ Weight $4$ Character orbit 80.c Analytic conductor $4.720$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,4,Mod(49,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("80.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 80.c (of order $$2$$, degree $$1$$, not 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: $$4.72015280046$$ 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: $$2$$ Twist minimal: no (minimal twist has level 10) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - 5 \beta - 5) q^{5} - 13 \beta q^{7} + 23 q^{9} +O(q^{10})$$ q + b * q^3 + (-5*b - 5) * q^5 - 13*b * q^7 + 23 * q^9 $$q + \beta q^{3} + ( - 5 \beta - 5) q^{5} - 13 \beta q^{7} + 23 q^{9} + 28 q^{11} - 6 \beta q^{13} + ( - 5 \beta + 20) q^{15} - 32 \beta q^{17} - 60 q^{19} + 52 q^{21} - 29 \beta q^{23} + (50 \beta - 75) q^{25} + 50 \beta q^{27} - 90 q^{29} + 128 q^{31} + 28 \beta q^{33} + (65 \beta - 260) q^{35} + 118 \beta q^{37} + 24 q^{39} + 242 q^{41} + 181 \beta q^{43} + ( - 115 \beta - 115) q^{45} - 113 \beta q^{47} - 333 q^{49} + 128 q^{51} + 54 \beta q^{53} + ( - 140 \beta - 140) q^{55} - 60 \beta q^{57} - 20 q^{59} + 542 q^{61} - 299 \beta q^{63} + (30 \beta - 120) q^{65} + 217 \beta q^{67} + 116 q^{69} + 1128 q^{71} - 316 \beta q^{73} + ( - 75 \beta - 200) q^{75} - 364 \beta q^{77} - 720 q^{79} + 421 q^{81} - 239 \beta q^{83} + (160 \beta - 640) q^{85} - 90 \beta q^{87} + 490 q^{89} - 312 q^{91} + 128 \beta q^{93} + (300 \beta + 300) q^{95} + 728 \beta q^{97} + 644 q^{99} +O(q^{100})$$ q + b * q^3 + (-5*b - 5) * q^5 - 13*b * q^7 + 23 * q^9 + 28 * q^11 - 6*b * q^13 + (-5*b + 20) * q^15 - 32*b * q^17 - 60 * q^19 + 52 * q^21 - 29*b * q^23 + (50*b - 75) * q^25 + 50*b * q^27 - 90 * q^29 + 128 * q^31 + 28*b * q^33 + (65*b - 260) * q^35 + 118*b * q^37 + 24 * q^39 + 242 * q^41 + 181*b * q^43 + (-115*b - 115) * q^45 - 113*b * q^47 - 333 * q^49 + 128 * q^51 + 54*b * q^53 + (-140*b - 140) * q^55 - 60*b * q^57 - 20 * q^59 + 542 * q^61 - 299*b * q^63 + (30*b - 120) * q^65 + 217*b * q^67 + 116 * q^69 + 1128 * q^71 - 316*b * q^73 + (-75*b - 200) * q^75 - 364*b * q^77 - 720 * q^79 + 421 * q^81 - 239*b * q^83 + (160*b - 640) * q^85 - 90*b * q^87 + 490 * q^89 - 312 * q^91 + 128*b * q^93 + (300*b + 300) * q^95 + 728*b * q^97 + 644 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{5} + 46 q^{9}+O(q^{10})$$ 2 * q - 10 * q^5 + 46 * q^9 $$2 q - 10 q^{5} + 46 q^{9} + 56 q^{11} + 40 q^{15} - 120 q^{19} + 104 q^{21} - 150 q^{25} - 180 q^{29} + 256 q^{31} - 520 q^{35} + 48 q^{39} + 484 q^{41} - 230 q^{45} - 666 q^{49} + 256 q^{51} - 280 q^{55} - 40 q^{59} + 1084 q^{61} - 240 q^{65} + 232 q^{69} + 2256 q^{71} - 400 q^{75} - 1440 q^{79} + 842 q^{81} - 1280 q^{85} + 980 q^{89} - 624 q^{91} + 600 q^{95} + 1288 q^{99}+O(q^{100})$$ 2 * q - 10 * q^5 + 46 * q^9 + 56 * q^11 + 40 * q^15 - 120 * q^19 + 104 * q^21 - 150 * q^25 - 180 * q^29 + 256 * q^31 - 520 * q^35 + 48 * q^39 + 484 * q^41 - 230 * q^45 - 666 * q^49 + 256 * q^51 - 280 * q^55 - 40 * q^59 + 1084 * q^61 - 240 * q^65 + 232 * q^69 + 2256 * q^71 - 400 * q^75 - 1440 * q^79 + 842 * q^81 - 1280 * q^85 + 980 * q^89 - 624 * q^91 + 600 * q^95 + 1288 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$21$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 2.00000i 0 −5.00000 + 10.0000i 0 26.0000i 0 23.0000 0
49.2 0 2.00000i 0 −5.00000 10.0000i 0 26.0000i 0 23.0000 0
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.4.c.a 2
3.b odd 2 1 720.4.f.f 2
4.b odd 2 1 10.4.b.a 2
5.b even 2 1 inner 80.4.c.a 2
5.c odd 4 1 400.4.a.h 1
5.c odd 4 1 400.4.a.n 1
8.b even 2 1 320.4.c.c 2
8.d odd 2 1 320.4.c.d 2
12.b even 2 1 90.4.c.b 2
15.d odd 2 1 720.4.f.f 2
20.d odd 2 1 10.4.b.a 2
20.e even 4 1 50.4.a.b 1
20.e even 4 1 50.4.a.d 1
28.d even 2 1 490.4.c.b 2
40.e odd 2 1 320.4.c.d 2
40.f even 2 1 320.4.c.c 2
40.i odd 4 1 1600.4.a.t 1
40.i odd 4 1 1600.4.a.bg 1
40.k even 4 1 1600.4.a.u 1
40.k even 4 1 1600.4.a.bh 1
60.h even 2 1 90.4.c.b 2
60.l odd 4 1 450.4.a.j 1
60.l odd 4 1 450.4.a.k 1
140.c even 2 1 490.4.c.b 2
140.j odd 4 1 2450.4.a.o 1
140.j odd 4 1 2450.4.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 4.b odd 2 1
10.4.b.a 2 20.d odd 2 1
50.4.a.b 1 20.e even 4 1
50.4.a.d 1 20.e even 4 1
80.4.c.a 2 1.a even 1 1 trivial
80.4.c.a 2 5.b even 2 1 inner
90.4.c.b 2 12.b even 2 1
90.4.c.b 2 60.h even 2 1
320.4.c.c 2 8.b even 2 1
320.4.c.c 2 40.f even 2 1
320.4.c.d 2 8.d odd 2 1
320.4.c.d 2 40.e odd 2 1
400.4.a.h 1 5.c odd 4 1
400.4.a.n 1 5.c odd 4 1
450.4.a.j 1 60.l odd 4 1
450.4.a.k 1 60.l odd 4 1
490.4.c.b 2 28.d even 2 1
490.4.c.b 2 140.c even 2 1
720.4.f.f 2 3.b odd 2 1
720.4.f.f 2 15.d odd 2 1
1600.4.a.t 1 40.i odd 4 1
1600.4.a.u 1 40.k even 4 1
1600.4.a.bg 1 40.i odd 4 1
1600.4.a.bh 1 40.k even 4 1
2450.4.a.o 1 140.j odd 4 1
2450.4.a.bb 1 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(80, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 10T + 125$$
$7$ $$T^{2} + 676$$
$11$ $$(T - 28)^{2}$$
$13$ $$T^{2} + 144$$
$17$ $$T^{2} + 4096$$
$19$ $$(T + 60)^{2}$$
$23$ $$T^{2} + 3364$$
$29$ $$(T + 90)^{2}$$
$31$ $$(T - 128)^{2}$$
$37$ $$T^{2} + 55696$$
$41$ $$(T - 242)^{2}$$
$43$ $$T^{2} + 131044$$
$47$ $$T^{2} + 51076$$
$53$ $$T^{2} + 11664$$
$59$ $$(T + 20)^{2}$$
$61$ $$(T - 542)^{2}$$
$67$ $$T^{2} + 188356$$
$71$ $$(T - 1128)^{2}$$
$73$ $$T^{2} + 399424$$
$79$ $$(T + 720)^{2}$$
$83$ $$T^{2} + 228484$$
$89$ $$(T - 490)^{2}$$
$97$ $$T^{2} + 2119936$$