# Properties

 Label 80.4.a.a Level $80$ Weight $4$ Character orbit 80.a Self dual yes Analytic conductor $4.720$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,4,Mod(1,80)]

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, 0]))

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

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 80.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: $$4.72015280046$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) 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)

 $$f(q)$$ $$=$$ $$q - 10 q^{3} - 5 q^{5} + 18 q^{7} + 73 q^{9}+O(q^{10})$$ q - 10 * q^3 - 5 * q^5 + 18 * q^7 + 73 * q^9 $$q - 10 q^{3} - 5 q^{5} + 18 q^{7} + 73 q^{9} + 16 q^{11} - 6 q^{13} + 50 q^{15} - 6 q^{17} + 124 q^{19} - 180 q^{21} - 42 q^{23} + 25 q^{25} - 460 q^{27} + 142 q^{29} + 188 q^{31} - 160 q^{33} - 90 q^{35} + 202 q^{37} + 60 q^{39} + 54 q^{41} - 66 q^{43} - 365 q^{45} - 38 q^{47} - 19 q^{49} + 60 q^{51} + 738 q^{53} - 80 q^{55} - 1240 q^{57} - 564 q^{59} - 262 q^{61} + 1314 q^{63} + 30 q^{65} + 554 q^{67} + 420 q^{69} - 140 q^{71} + 882 q^{73} - 250 q^{75} + 288 q^{77} + 1160 q^{79} + 2629 q^{81} - 642 q^{83} + 30 q^{85} - 1420 q^{87} - 854 q^{89} - 108 q^{91} - 1880 q^{93} - 620 q^{95} - 478 q^{97} + 1168 q^{99}+O(q^{100})$$ q - 10 * q^3 - 5 * q^5 + 18 * q^7 + 73 * q^9 + 16 * q^11 - 6 * q^13 + 50 * q^15 - 6 * q^17 + 124 * q^19 - 180 * q^21 - 42 * q^23 + 25 * q^25 - 460 * q^27 + 142 * q^29 + 188 * q^31 - 160 * q^33 - 90 * q^35 + 202 * q^37 + 60 * q^39 + 54 * q^41 - 66 * q^43 - 365 * q^45 - 38 * q^47 - 19 * q^49 + 60 * q^51 + 738 * q^53 - 80 * q^55 - 1240 * q^57 - 564 * q^59 - 262 * q^61 + 1314 * q^63 + 30 * q^65 + 554 * q^67 + 420 * q^69 - 140 * q^71 + 882 * q^73 - 250 * q^75 + 288 * q^77 + 1160 * q^79 + 2629 * q^81 - 642 * q^83 + 30 * q^85 - 1420 * q^87 - 854 * q^89 - 108 * q^91 - 1880 * q^93 - 620 * q^95 - 478 * q^97 + 1168 * 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
0 −10.0000 0 −5.00000 0 18.0000 0 73.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$5$$ $$+1$$

## 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 80.4.a.a 1
3.b odd 2 1 720.4.a.ba 1
4.b odd 2 1 40.4.a.c 1
5.b even 2 1 400.4.a.u 1
5.c odd 4 2 400.4.c.a 2
8.b even 2 1 320.4.a.n 1
8.d odd 2 1 320.4.a.a 1
12.b even 2 1 360.4.a.i 1
16.e even 4 2 1280.4.d.b 2
16.f odd 4 2 1280.4.d.o 2
20.d odd 2 1 200.4.a.a 1
20.e even 4 2 200.4.c.a 2
28.d even 2 1 1960.4.a.a 1
40.e odd 2 1 1600.4.a.ca 1
40.f even 2 1 1600.4.a.a 1
60.h even 2 1 1800.4.a.bd 1
60.l odd 4 2 1800.4.f.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.c 1 4.b odd 2 1
80.4.a.a 1 1.a even 1 1 trivial
200.4.a.a 1 20.d odd 2 1
200.4.c.a 2 20.e even 4 2
320.4.a.a 1 8.d odd 2 1
320.4.a.n 1 8.b even 2 1
360.4.a.i 1 12.b even 2 1
400.4.a.u 1 5.b even 2 1
400.4.c.a 2 5.c odd 4 2
720.4.a.ba 1 3.b odd 2 1
1280.4.d.b 2 16.e even 4 2
1280.4.d.o 2 16.f odd 4 2
1600.4.a.a 1 40.f even 2 1
1600.4.a.ca 1 40.e odd 2 1
1800.4.a.bd 1 60.h even 2 1
1800.4.f.n 2 60.l odd 4 2
1960.4.a.a 1 28.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(80))$$:

 $$T_{3} + 10$$ T3 + 10 $$T_{7} - 18$$ T7 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 10$$
$5$ $$T + 5$$
$7$ $$T - 18$$
$11$ $$T - 16$$
$13$ $$T + 6$$
$17$ $$T + 6$$
$19$ $$T - 124$$
$23$ $$T + 42$$
$29$ $$T - 142$$
$31$ $$T - 188$$
$37$ $$T - 202$$
$41$ $$T - 54$$
$43$ $$T + 66$$
$47$ $$T + 38$$
$53$ $$T - 738$$
$59$ $$T + 564$$
$61$ $$T + 262$$
$67$ $$T - 554$$
$71$ $$T + 140$$
$73$ $$T - 882$$
$79$ $$T - 1160$$
$83$ $$T + 642$$
$89$ $$T + 854$$
$97$ $$T + 478$$
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