# Properties

 Label 400.2.n.a Level $400$ Weight $2$ Character orbit 400.n Analytic conductor $3.194$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,2,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, 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("400.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.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: $$3.19401608085$$ 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: no (minimal twist has level 80) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - 3 i q^{9} +O(q^{10})$$ q - 3*i * q^9 $$q - 3 i q^{9} + ( - 5 i + 5) q^{13} + (5 i + 5) q^{17} - 4 i q^{29} + ( - 5 i - 5) q^{37} + 8 q^{41} + 7 i q^{49} + (5 i - 5) q^{53} - 12 q^{61} + (5 i - 5) q^{73} - 9 q^{81} + 16 i q^{89} + ( - 5 i - 5) q^{97} +O(q^{100})$$ q - 3*i * q^9 + (-5*i + 5) * q^13 + (5*i + 5) * q^17 - 4*i * q^29 + (-5*i - 5) * q^37 + 8 * q^41 + 7*i * q^49 + (5*i - 5) * q^53 - 12 * q^61 + (5*i - 5) * q^73 - 9 * q^81 + 16*i * q^89 + (-5*i - 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 10 q^{13} + 10 q^{17} - 10 q^{37} + 16 q^{41} - 10 q^{53} - 24 q^{61} - 10 q^{73} - 18 q^{81} - 10 q^{97}+O(q^{100})$$ 2 * q + 10 * q^13 + 10 * q^17 - 10 * q^37 + 16 * q^41 - 10 * q^53 - 24 * q^61 - 10 * q^73 - 18 * q^81 - 10 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$i$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 − 1.00000i 1.00000i
0 0 0 0 0 0 0 3.00000i 0
207.1 0 0 0 0 0 0 0 3.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
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 400.2.n.a 2
3.b odd 2 1 3600.2.x.c 2
4.b odd 2 1 CM 400.2.n.a 2
5.b even 2 1 80.2.n.a 2
5.c odd 4 1 80.2.n.a 2
5.c odd 4 1 inner 400.2.n.a 2
8.b even 2 1 1600.2.n.g 2
8.d odd 2 1 1600.2.n.g 2
12.b even 2 1 3600.2.x.c 2
15.d odd 2 1 720.2.x.a 2
15.e even 4 1 720.2.x.a 2
15.e even 4 1 3600.2.x.c 2
20.d odd 2 1 80.2.n.a 2
20.e even 4 1 80.2.n.a 2
20.e even 4 1 inner 400.2.n.a 2
40.e odd 2 1 320.2.n.d 2
40.f even 2 1 320.2.n.d 2
40.i odd 4 1 320.2.n.d 2
40.i odd 4 1 1600.2.n.g 2
40.k even 4 1 320.2.n.d 2
40.k even 4 1 1600.2.n.g 2
60.h even 2 1 720.2.x.a 2
60.l odd 4 1 720.2.x.a 2
60.l odd 4 1 3600.2.x.c 2
80.i odd 4 1 1280.2.o.i 2
80.j even 4 1 1280.2.o.h 2
80.k odd 4 1 1280.2.o.h 2
80.k odd 4 1 1280.2.o.i 2
80.q even 4 1 1280.2.o.h 2
80.q even 4 1 1280.2.o.i 2
80.s even 4 1 1280.2.o.i 2
80.t odd 4 1 1280.2.o.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.a 2 5.b even 2 1
80.2.n.a 2 5.c odd 4 1
80.2.n.a 2 20.d odd 2 1
80.2.n.a 2 20.e even 4 1
320.2.n.d 2 40.e odd 2 1
320.2.n.d 2 40.f even 2 1
320.2.n.d 2 40.i odd 4 1
320.2.n.d 2 40.k even 4 1
400.2.n.a 2 1.a even 1 1 trivial
400.2.n.a 2 4.b odd 2 1 CM
400.2.n.a 2 5.c odd 4 1 inner
400.2.n.a 2 20.e even 4 1 inner
720.2.x.a 2 15.d odd 2 1
720.2.x.a 2 15.e even 4 1
720.2.x.a 2 60.h even 2 1
720.2.x.a 2 60.l odd 4 1
1280.2.o.h 2 80.j even 4 1
1280.2.o.h 2 80.k odd 4 1
1280.2.o.h 2 80.q even 4 1
1280.2.o.h 2 80.t odd 4 1
1280.2.o.i 2 80.i odd 4 1
1280.2.o.i 2 80.k odd 4 1
1280.2.o.i 2 80.q even 4 1
1280.2.o.i 2 80.s even 4 1
1600.2.n.g 2 8.b even 2 1
1600.2.n.g 2 8.d odd 2 1
1600.2.n.g 2 40.i odd 4 1
1600.2.n.g 2 40.k even 4 1
3600.2.x.c 2 3.b odd 2 1
3600.2.x.c 2 12.b even 2 1
3600.2.x.c 2 15.e even 4 1
3600.2.x.c 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 10T + 50$$
$17$ $$T^{2} - 10T + 50$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 10T + 50$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 10T + 50$$
$59$ $$T^{2}$$
$61$ $$(T + 12)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 10T + 50$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 256$$
$97$ $$T^{2} + 10T + 50$$