# Properties

 Label 3200.2.c.e Level $3200$ Weight $2$ Character orbit 3200.c Analytic conductor $25.552$ 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] = [3200,2,Mod(2049,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3200.2049");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.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: $$25.5521286468$$ 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 128) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 2 \beta q^{7} - q^{9}+O(q^{10})$$ q + b * q^3 + 2*b * q^7 - q^9 $$q + \beta q^{3} + 2 \beta q^{7} - q^{9} - 2 q^{11} + \beta q^{13} + \beta q^{17} - 2 q^{19} - 8 q^{21} + 2 \beta q^{23} + 2 \beta q^{27} + 6 q^{29} - 2 \beta q^{33} - 5 \beta q^{37} - 4 q^{39} - 6 q^{41} + 3 \beta q^{43} + 4 \beta q^{47} - 9 q^{49} - 4 q^{51} - 3 \beta q^{53} - 2 \beta q^{57} - 14 q^{59} + 2 q^{61} - 2 \beta q^{63} - 5 \beta q^{67} - 8 q^{69} + 12 q^{71} + 7 \beta q^{73} - 4 \beta q^{77} + 8 q^{79} - 11 q^{81} - 3 \beta q^{83} + 6 \beta q^{87} + 2 q^{89} - 8 q^{91} + \beta q^{97} + 2 q^{99} +O(q^{100})$$ q + b * q^3 + 2*b * q^7 - q^9 - 2 * q^11 + b * q^13 + b * q^17 - 2 * q^19 - 8 * q^21 + 2*b * q^23 + 2*b * q^27 + 6 * q^29 - 2*b * q^33 - 5*b * q^37 - 4 * q^39 - 6 * q^41 + 3*b * q^43 + 4*b * q^47 - 9 * q^49 - 4 * q^51 - 3*b * q^53 - 2*b * q^57 - 14 * q^59 + 2 * q^61 - 2*b * q^63 - 5*b * q^67 - 8 * q^69 + 12 * q^71 + 7*b * q^73 - 4*b * q^77 + 8 * q^79 - 11 * q^81 - 3*b * q^83 + 6*b * q^87 + 2 * q^89 - 8 * q^91 + b * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} - 4 q^{11} - 4 q^{19} - 16 q^{21} + 12 q^{29} - 8 q^{39} - 12 q^{41} - 18 q^{49} - 8 q^{51} - 28 q^{59} + 4 q^{61} - 16 q^{69} + 24 q^{71} + 16 q^{79} - 22 q^{81} + 4 q^{89} - 16 q^{91} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 - 4 * q^11 - 4 * q^19 - 16 * q^21 + 12 * q^29 - 8 * q^39 - 12 * q^41 - 18 * q^49 - 8 * q^51 - 28 * q^59 + 4 * q^61 - 16 * q^69 + 24 * q^71 + 16 * q^79 - 22 * q^81 + 4 * q^89 - 16 * q^91 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2049.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 4.00000i 0 −1.00000 0
2049.2 0 2.00000i 0 0 0 4.00000i 0 −1.00000 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 3200.2.c.e 2
4.b odd 2 1 3200.2.c.k 2
5.b even 2 1 inner 3200.2.c.e 2
5.c odd 4 1 128.2.a.d yes 1
5.c odd 4 1 3200.2.a.h 1
8.b even 2 1 3200.2.c.l 2
8.d odd 2 1 3200.2.c.f 2
15.e even 4 1 1152.2.a.c 1
20.d odd 2 1 3200.2.c.k 2
20.e even 4 1 128.2.a.b yes 1
20.e even 4 1 3200.2.a.u 1
35.f even 4 1 6272.2.a.a 1
40.e odd 2 1 3200.2.c.f 2
40.f even 2 1 3200.2.c.l 2
40.i odd 4 1 128.2.a.a 1
40.i odd 4 1 3200.2.a.x 1
40.k even 4 1 128.2.a.c yes 1
40.k even 4 1 3200.2.a.e 1
60.l odd 4 1 1152.2.a.h 1
80.i odd 4 1 256.2.b.c 2
80.j even 4 1 256.2.b.a 2
80.s even 4 1 256.2.b.a 2
80.t odd 4 1 256.2.b.c 2
120.q odd 4 1 1152.2.a.r 1
120.w even 4 1 1152.2.a.m 1
140.j odd 4 1 6272.2.a.g 1
160.u even 8 2 1024.2.e.m 4
160.v odd 8 2 1024.2.e.i 4
160.ba even 8 2 1024.2.e.m 4
160.bb odd 8 2 1024.2.e.i 4
240.z odd 4 1 2304.2.d.b 2
240.bb even 4 1 2304.2.d.r 2
240.bd odd 4 1 2304.2.d.b 2
240.bf even 4 1 2304.2.d.r 2
280.s even 4 1 6272.2.a.h 1
280.y odd 4 1 6272.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 40.i odd 4 1
128.2.a.b yes 1 20.e even 4 1
128.2.a.c yes 1 40.k even 4 1
128.2.a.d yes 1 5.c odd 4 1
256.2.b.a 2 80.j even 4 1
256.2.b.a 2 80.s even 4 1
256.2.b.c 2 80.i odd 4 1
256.2.b.c 2 80.t odd 4 1
1024.2.e.i 4 160.v odd 8 2
1024.2.e.i 4 160.bb odd 8 2
1024.2.e.m 4 160.u even 8 2
1024.2.e.m 4 160.ba even 8 2
1152.2.a.c 1 15.e even 4 1
1152.2.a.h 1 60.l odd 4 1
1152.2.a.m 1 120.w even 4 1
1152.2.a.r 1 120.q odd 4 1
2304.2.d.b 2 240.z odd 4 1
2304.2.d.b 2 240.bd odd 4 1
2304.2.d.r 2 240.bb even 4 1
2304.2.d.r 2 240.bf even 4 1
3200.2.a.e 1 40.k even 4 1
3200.2.a.h 1 5.c odd 4 1
3200.2.a.u 1 20.e even 4 1
3200.2.a.x 1 40.i odd 4 1
3200.2.c.e 2 1.a even 1 1 trivial
3200.2.c.e 2 5.b even 2 1 inner
3200.2.c.f 2 8.d odd 2 1
3200.2.c.f 2 40.e odd 2 1
3200.2.c.k 2 4.b odd 2 1
3200.2.c.k 2 20.d odd 2 1
3200.2.c.l 2 8.b even 2 1
3200.2.c.l 2 40.f even 2 1
6272.2.a.a 1 35.f even 4 1
6272.2.a.b 1 280.y odd 4 1
6272.2.a.g 1 140.j odd 4 1
6272.2.a.h 1 280.s even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} + 2$$ T11 + 2 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 14)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} + 4$$