# Properties

 Label 800.2.n.f Level $800$ Weight $2$ Character orbit 800.n Analytic conductor $6.388$ 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] = [800,2,Mod(543,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("800.543");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.n (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: $$6.38803216170$$ 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: no (minimal twist has level 160) 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.

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{3} + (3 i - 3) q^{7} - i q^{9} +O(q^{10})$$ q + (i + 1) * q^3 + (3*i - 3) * q^7 - i * q^9 $$q + (i + 1) q^{3} + (3 i - 3) q^{7} - i q^{9} + 2 i q^{11} + (3 i - 3) q^{13} + ( - i - 1) q^{17} - 4 q^{19} - 6 q^{21} + ( - i - 1) q^{23} + ( - 4 i + 4) q^{27} + 10 i q^{31} + (2 i - 2) q^{33} + (i + 1) q^{37} - 6 q^{39} - 10 q^{41} + (5 i + 5) q^{43} + (3 i - 3) q^{47} - 11 i q^{49} - 2 i q^{51} + ( - 5 i + 5) q^{53} + ( - 4 i - 4) q^{57} + 12 q^{59} + 2 q^{61} + (3 i + 3) q^{63} + (i - 1) q^{67} - 2 i q^{69} + 2 i q^{71} + (i - 1) q^{73} + ( - 6 i - 6) q^{77} - 8 q^{79} + 5 q^{81} + (5 i + 5) q^{83} + 16 i q^{89} - 18 i q^{91} + (10 i - 10) q^{93} + (3 i + 3) q^{97} + 2 q^{99} +O(q^{100})$$ q + (i + 1) * q^3 + (3*i - 3) * q^7 - i * q^9 + 2*i * q^11 + (3*i - 3) * q^13 + (-i - 1) * q^17 - 4 * q^19 - 6 * q^21 + (-i - 1) * q^23 + (-4*i + 4) * q^27 + 10*i * q^31 + (2*i - 2) * q^33 + (i + 1) * q^37 - 6 * q^39 - 10 * q^41 + (5*i + 5) * q^43 + (3*i - 3) * q^47 - 11*i * q^49 - 2*i * q^51 + (-5*i + 5) * q^53 + (-4*i - 4) * q^57 + 12 * q^59 + 2 * q^61 + (3*i + 3) * q^63 + (i - 1) * q^67 - 2*i * q^69 + 2*i * q^71 + (i - 1) * q^73 + (-6*i - 6) * q^77 - 8 * q^79 + 5 * q^81 + (5*i + 5) * q^83 + 16*i * q^89 - 18*i * q^91 + (10*i - 10) * q^93 + (3*i + 3) * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 6 q^{7}+O(q^{10})$$ 2 * q + 2 * q^3 - 6 * q^7 $$2 q + 2 q^{3} - 6 q^{7} - 6 q^{13} - 2 q^{17} - 8 q^{19} - 12 q^{21} - 2 q^{23} + 8 q^{27} - 4 q^{33} + 2 q^{37} - 12 q^{39} - 20 q^{41} + 10 q^{43} - 6 q^{47} + 10 q^{53} - 8 q^{57} + 24 q^{59} + 4 q^{61} + 6 q^{63} - 2 q^{67} - 2 q^{73} - 12 q^{77} - 16 q^{79} + 10 q^{81} + 10 q^{83} - 20 q^{93} + 6 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 6 * q^7 - 6 * q^13 - 2 * q^17 - 8 * q^19 - 12 * q^21 - 2 * q^23 + 8 * q^27 - 4 * q^33 + 2 * q^37 - 12 * q^39 - 20 * q^41 + 10 * q^43 - 6 * q^47 + 10 * q^53 - 8 * q^57 + 24 * q^59 + 4 * q^61 + 6 * q^63 - 2 * q^67 - 2 * q^73 - 12 * q^77 - 16 * q^79 + 10 * q^81 + 10 * q^83 - 20 * q^93 + 6 * q^97 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
543.1
 − 1.00000i 1.00000i
0 1.00000 1.00000i 0 0 0 −3.00000 3.00000i 0 1.00000i 0
607.1 0 1.00000 + 1.00000i 0 0 0 −3.00000 + 3.00000i 0 1.00000i 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
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.n.f 2
4.b odd 2 1 800.2.n.e 2
5.b even 2 1 160.2.n.c 2
5.c odd 4 1 160.2.n.d yes 2
5.c odd 4 1 800.2.n.e 2
8.b even 2 1 1600.2.n.c 2
8.d odd 2 1 1600.2.n.m 2
15.d odd 2 1 1440.2.x.f 2
15.e even 4 1 1440.2.x.a 2
20.d odd 2 1 160.2.n.d yes 2
20.e even 4 1 160.2.n.c 2
20.e even 4 1 inner 800.2.n.f 2
40.e odd 2 1 320.2.n.b 2
40.f even 2 1 320.2.n.g 2
40.i odd 4 1 320.2.n.b 2
40.i odd 4 1 1600.2.n.m 2
40.k even 4 1 320.2.n.g 2
40.k even 4 1 1600.2.n.c 2
60.h even 2 1 1440.2.x.a 2
60.l odd 4 1 1440.2.x.f 2
80.i odd 4 1 1280.2.o.f 2
80.j even 4 1 1280.2.o.c 2
80.k odd 4 1 1280.2.o.f 2
80.k odd 4 1 1280.2.o.l 2
80.q even 4 1 1280.2.o.c 2
80.q even 4 1 1280.2.o.m 2
80.s even 4 1 1280.2.o.m 2
80.t odd 4 1 1280.2.o.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.c 2 5.b even 2 1
160.2.n.c 2 20.e even 4 1
160.2.n.d yes 2 5.c odd 4 1
160.2.n.d yes 2 20.d odd 2 1
320.2.n.b 2 40.e odd 2 1
320.2.n.b 2 40.i odd 4 1
320.2.n.g 2 40.f even 2 1
320.2.n.g 2 40.k even 4 1
800.2.n.e 2 4.b odd 2 1
800.2.n.e 2 5.c odd 4 1
800.2.n.f 2 1.a even 1 1 trivial
800.2.n.f 2 20.e even 4 1 inner
1280.2.o.c 2 80.j even 4 1
1280.2.o.c 2 80.q even 4 1
1280.2.o.f 2 80.i odd 4 1
1280.2.o.f 2 80.k odd 4 1
1280.2.o.l 2 80.k odd 4 1
1280.2.o.l 2 80.t odd 4 1
1280.2.o.m 2 80.q even 4 1
1280.2.o.m 2 80.s even 4 1
1440.2.x.a 2 15.e even 4 1
1440.2.x.a 2 60.h even 2 1
1440.2.x.f 2 15.d odd 2 1
1440.2.x.f 2 60.l odd 4 1
1600.2.n.c 2 8.b even 2 1
1600.2.n.c 2 40.k even 4 1
1600.2.n.m 2 8.d odd 2 1
1600.2.n.m 2 40.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} - 2T_{3} + 2$$ T3^2 - 2*T3 + 2 $$T_{7}^{2} + 6T_{7} + 18$$ T7^2 + 6*T7 + 18 $$T_{11}^{2} + 4$$ T11^2 + 4 $$T_{13}^{2} + 6T_{13} + 18$$ T13^2 + 6*T13 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 18$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2} + 6T + 18$$
$17$ $$T^{2} + 2T + 2$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 2T + 2$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$T^{2} - 2T + 2$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} - 10T + 50$$
$47$ $$T^{2} + 6T + 18$$
$53$ $$T^{2} - 10T + 50$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 2T + 2$$
$71$ $$T^{2} + 4$$
$73$ $$T^{2} + 2T + 2$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} - 10T + 50$$
$89$ $$T^{2} + 256$$
$97$ $$T^{2} - 6T + 18$$