Properties

 Label 800.4.c.c Level $800$ Weight $4$ Character orbit 800.c Analytic conductor $47.202$ Analytic rank $1$ 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: $$1$$ 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.c 2
4.b odd 2 1 800.4.c.d 2
5.b even 2 1 inner 800.4.c.c 2
5.c odd 4 1 800.4.a.b 1
5.c odd 4 1 800.4.a.i yes 1
20.d odd 2 1 800.4.c.d 2
20.e even 4 1 800.4.a.c yes 1
20.e even 4 1 800.4.a.j yes 1
40.i odd 4 1 1600.4.a.l 1
40.i odd 4 1 1600.4.a.bq 1
40.k even 4 1 1600.4.a.k 1
40.k even 4 1 1600.4.a.bp 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.4.a.b 1 5.c odd 4 1
800.4.a.c yes 1 20.e even 4 1
800.4.a.i yes 1 5.c odd 4 1
800.4.a.j yes 1 20.e even 4 1
800.4.c.c 2 1.a even 1 1 trivial
800.4.c.c 2 5.b even 2 1 inner
800.4.c.d 2 4.b odd 2 1
800.4.c.d 2 20.d odd 2 1
1600.4.a.k 1 40.k even 4 1
1600.4.a.l 1 40.i odd 4 1
1600.4.a.bp 1 40.k even 4 1
1600.4.a.bq 1 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 $$S_{4}^{\mathrm{new}}(800, [\chi])$$:

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

Hecke characteristic polynomials

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