# Properties

 Label 400.1.x.a Level $400$ Weight $1$ Character orbit 400.x Analytic conductor $0.200$ Analytic rank $0$ Dimension $4$ Projective image $D_{10}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,1,Mod(79,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("400.79");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 400.x (of order $$10$$, degree $$4$$, 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: $$0.199626005053$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.2.195312500000000.4

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{3} q^{5} - \zeta_{10}^{4} q^{9} +O(q^{10})$$ q + z^3 * q^5 - z^4 * q^9 $$q + \zeta_{10}^{3} q^{5} - \zeta_{10}^{4} q^{9} + (\zeta_{10}^{2} + \zeta_{10}) q^{13} + (\zeta_{10}^{4} - \zeta_{10}^{2}) q^{17} - \zeta_{10} q^{25} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{29} + ( - \zeta_{10}^{3} - 1) q^{37} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{41} + \zeta_{10}^{2} q^{45} - q^{49} + (\zeta_{10}^{4} - 1) q^{53} + ( - \zeta_{10}^{4} + \zeta_{10}^{3}) q^{61} + (\zeta_{10}^{4} - 1) q^{65} + ( - \zeta_{10}^{4} - \zeta_{10}^{3}) q^{73} - \zeta_{10}^{3} q^{81} + ( - \zeta_{10}^{2} + 1) q^{85} + (\zeta_{10}^{2} + 1) q^{89} + (\zeta_{10}^{3} - \zeta_{10}) q^{97} +O(q^{100})$$ q + z^3 * q^5 - z^4 * q^9 + (z^2 + z) * q^13 + (z^4 - z^2) * q^17 - z * q^25 + (-z^3 - z) * q^29 + (-z^3 - 1) * q^37 + (-z^2 + z) * q^41 + z^2 * q^45 - q^49 + (z^4 - 1) * q^53 + (-z^4 + z^3) * q^61 + (z^4 - 1) * q^65 + (-z^4 - z^3) * q^73 - z^3 * q^81 + (-z^2 + 1) * q^85 + (z^2 + 1) * q^89 + (z^3 - z) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{5} + q^{9}+O(q^{10})$$ 4 * q + q^5 + q^9 $$4 q + q^{5} + q^{9} - q^{25} - 2 q^{29} - 5 q^{37} + 2 q^{41} - q^{45} - 4 q^{49} - 5 q^{53} + 2 q^{61} - 5 q^{65} - q^{81} + 5 q^{85} + 3 q^{89}+O(q^{100})$$ 4 * q + q^5 + q^9 - q^25 - 2 * q^29 - 5 * q^37 + 2 * q^41 - q^45 - 4 * q^49 - 5 * q^53 + 2 * q^61 - 5 * q^65 - q^81 + 5 * q^85 + 3 * q^89

## Character values

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

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
0 0 0 0.809017 0.587785i 0 0 0 −0.309017 0.951057i 0
159.1 0 0 0 −0.309017 + 0.951057i 0 0 0 0.809017 0.587785i 0
239.1 0 0 0 −0.309017 0.951057i 0 0 0 0.809017 + 0.587785i 0
319.1 0 0 0 0.809017 + 0.587785i 0 0 0 −0.309017 + 0.951057i 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 $$\Q(\sqrt{-1})$$
25.e even 10 1 inner
100.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.1.x.a 4
3.b odd 2 1 3600.1.ct.a 4
4.b odd 2 1 CM 400.1.x.a 4
5.b even 2 1 2000.1.x.a 4
5.c odd 4 2 2000.1.z.a 8
8.b even 2 1 1600.1.bf.a 4
8.d odd 2 1 1600.1.bf.a 4
12.b even 2 1 3600.1.ct.a 4
20.d odd 2 1 2000.1.x.a 4
20.e even 4 2 2000.1.z.a 8
25.d even 5 1 2000.1.x.a 4
25.e even 10 1 inner 400.1.x.a 4
25.f odd 20 2 2000.1.z.a 8
75.h odd 10 1 3600.1.ct.a 4
100.h odd 10 1 inner 400.1.x.a 4
100.j odd 10 1 2000.1.x.a 4
100.l even 20 2 2000.1.z.a 8
200.o even 10 1 1600.1.bf.a 4
200.s odd 10 1 1600.1.bf.a 4
300.r even 10 1 3600.1.ct.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.1.x.a 4 1.a even 1 1 trivial
400.1.x.a 4 4.b odd 2 1 CM
400.1.x.a 4 25.e even 10 1 inner
400.1.x.a 4 100.h odd 10 1 inner
1600.1.bf.a 4 8.b even 2 1
1600.1.bf.a 4 8.d odd 2 1
1600.1.bf.a 4 200.o even 10 1
1600.1.bf.a 4 200.s odd 10 1
2000.1.x.a 4 5.b even 2 1
2000.1.x.a 4 20.d odd 2 1
2000.1.x.a 4 25.d even 5 1
2000.1.x.a 4 100.j odd 10 1
2000.1.z.a 8 5.c odd 4 2
2000.1.z.a 8 20.e even 4 2
2000.1.z.a 8 25.f odd 20 2
2000.1.z.a 8 100.l even 20 2
3600.1.ct.a 4 3.b odd 2 1
3600.1.ct.a 4 12.b even 2 1
3600.1.ct.a 4 75.h odd 10 1
3600.1.ct.a 4 300.r even 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - T^{3} + T^{2} + \cdots + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 5T + 5$$
$17$ $$T^{4} + 5T + 5$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 2 T^{3} + \cdots + 1$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 5 T^{3} + \cdots + 5$$
$41$ $$T^{4} - 2 T^{3} + \cdots + 1$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 5 T^{3} + \cdots + 5$$
$59$ $$T^{4}$$
$61$ $$T^{4} - 2 T^{3} + \cdots + 1$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 5T + 5$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} - 3 T^{3} + \cdots + 1$$
$97$ $$T^{4} + 5T + 5$$