# Properties

 Label 800.4.c.d Level $800$ Weight $4$ Character orbit 800.c Analytic conductor $47.202$ 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] = [800,4,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, 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("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.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: $$47.2015280046$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 i q^{3} + 10 i q^{7} + 2 q^{9}+O(q^{10})$$ q + 5*i * q^3 + 10*i * q^7 + 2 * q^9 $$q + 5 i q^{3} + 10 i q^{7} + 2 q^{9} + 15 q^{11} + 8 i q^{13} + 21 i q^{17} + 105 q^{19} - 50 q^{21} - 10 i q^{23} + 145 i q^{27} + 20 q^{29} + 230 q^{31} + 75 i q^{33} + 54 i q^{37} - 40 q^{39} - 195 q^{41} - 300 i q^{43} + 480 i q^{47} + 243 q^{49} - 105 q^{51} + 322 i q^{53} + 525 i q^{57} + 560 q^{59} - 730 q^{61} + 20 i q^{63} - 255 i q^{67} + 50 q^{69} + 40 q^{71} + 317 i q^{73} + 150 i q^{77} - 830 q^{79} - 671 q^{81} + 75 i q^{83} + 100 i q^{87} + 705 q^{89} - 80 q^{91} + 1150 i q^{93} + 1434 i q^{97} + 30 q^{99} +O(q^{100})$$ q + 5*i * q^3 + 10*i * q^7 + 2 * q^9 + 15 * q^11 + 8*i * q^13 + 21*i * q^17 + 105 * q^19 - 50 * q^21 - 10*i * q^23 + 145*i * q^27 + 20 * q^29 + 230 * q^31 + 75*i * q^33 + 54*i * q^37 - 40 * q^39 - 195 * q^41 - 300*i * q^43 + 480*i * q^47 + 243 * q^49 - 105 * q^51 + 322*i * q^53 + 525*i * q^57 + 560 * q^59 - 730 * q^61 + 20*i * q^63 - 255*i * q^67 + 50 * q^69 + 40 * q^71 + 317*i * q^73 + 150*i * q^77 - 830 * q^79 - 671 * q^81 + 75*i * q^83 + 100*i * q^87 + 705 * q^89 - 80 * q^91 + 1150*i * q^93 + 1434*i * q^97 + 30 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 30 q^{11} + 210 q^{19} - 100 q^{21} + 40 q^{29} + 460 q^{31} - 80 q^{39} - 390 q^{41} + 486 q^{49} - 210 q^{51} + 1120 q^{59} - 1460 q^{61} + 100 q^{69} + 80 q^{71} - 1660 q^{79} - 1342 q^{81} + 1410 q^{89} - 160 q^{91} + 60 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 30 * q^11 + 210 * q^19 - 100 * q^21 + 40 * q^29 + 460 * q^31 - 80 * q^39 - 390 * q^41 + 486 * q^49 - 210 * q^51 + 1120 * q^59 - 1460 * q^61 + 100 * q^69 + 80 * q^71 - 1660 * q^79 - 1342 * q^81 + 1410 * q^89 - 160 * q^91 + 60 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 5.00000i 0 0 0 10.0000i 0 2.00000 0
449.2 0 5.00000i 0 0 0 10.0000i 0 2.00000 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 800.4.c.d 2
4.b odd 2 1 800.4.c.c 2
5.b even 2 1 inner 800.4.c.d 2
5.c odd 4 1 800.4.a.c yes 1
5.c odd 4 1 800.4.a.j yes 1
20.d odd 2 1 800.4.c.c 2
20.e even 4 1 800.4.a.b 1
20.e even 4 1 800.4.a.i yes 1
40.i odd 4 1 1600.4.a.k 1
40.i odd 4 1 1600.4.a.bp 1
40.k even 4 1 1600.4.a.l 1
40.k even 4 1 1600.4.a.bq 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.4.a.b 1 20.e even 4 1
800.4.a.c yes 1 5.c odd 4 1
800.4.a.i yes 1 20.e even 4 1
800.4.a.j yes 1 5.c odd 4 1
800.4.c.c 2 4.b odd 2 1
800.4.c.c 2 20.d odd 2 1
800.4.c.d 2 1.a even 1 1 trivial
800.4.c.d 2 5.b even 2 1 inner
1600.4.a.k 1 40.i odd 4 1
1600.4.a.l 1 40.k even 4 1
1600.4.a.bp 1 40.i odd 4 1
1600.4.a.bq 1 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} + 25$$ T3^2 + 25 $$T_{11} - 15$$ T11 - 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 25$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 100$$
$11$ $$(T - 15)^{2}$$
$13$ $$T^{2} + 64$$
$17$ $$T^{2} + 441$$
$19$ $$(T - 105)^{2}$$
$23$ $$T^{2} + 100$$
$29$ $$(T - 20)^{2}$$
$31$ $$(T - 230)^{2}$$
$37$ $$T^{2} + 2916$$
$41$ $$(T + 195)^{2}$$
$43$ $$T^{2} + 90000$$
$47$ $$T^{2} + 230400$$
$53$ $$T^{2} + 103684$$
$59$ $$(T - 560)^{2}$$
$61$ $$(T + 730)^{2}$$
$67$ $$T^{2} + 65025$$
$71$ $$(T - 40)^{2}$$
$73$ $$T^{2} + 100489$$
$79$ $$(T + 830)^{2}$$
$83$ $$T^{2} + 5625$$
$89$ $$(T - 705)^{2}$$
$97$ $$T^{2} + 2056356$$