# Properties

 Label 400.2.a.e Level $400$ Weight $2$ Character orbit 400.a Self dual yes Analytic conductor $3.194$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,2,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

## 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: yes Analytic conductor: $$3.19401608085$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 4 q^{7} - 3 q^{9}+O(q^{10})$$ q - 4 * q^7 - 3 * q^9 $$q - 4 q^{7} - 3 q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} + 4 q^{23} - 2 q^{29} + 8 q^{31} - 6 q^{37} - 6 q^{41} - 8 q^{43} + 4 q^{47} + 9 q^{49} - 6 q^{53} + 4 q^{59} - 2 q^{61} + 12 q^{63} + 8 q^{67} + 6 q^{73} + 16 q^{77} + 9 q^{81} - 16 q^{83} - 6 q^{89} - 8 q^{91} + 14 q^{97} + 12 q^{99}+O(q^{100})$$ q - 4 * q^7 - 3 * q^9 - 4 * q^11 + 2 * q^13 - 2 * q^17 - 4 * q^19 + 4 * q^23 - 2 * q^29 + 8 * q^31 - 6 * q^37 - 6 * q^41 - 8 * q^43 + 4 * q^47 + 9 * q^49 - 6 * q^53 + 4 * q^59 - 2 * q^61 + 12 * q^63 + 8 * q^67 + 6 * q^73 + 16 * q^77 + 9 * q^81 - 16 * q^83 - 6 * q^89 - 8 * q^91 + 14 * q^97 + 12 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 −4.00000 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$5$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.a.e 1
3.b odd 2 1 3600.2.a.h 1
4.b odd 2 1 200.2.a.c 1
5.b even 2 1 80.2.a.a 1
5.c odd 4 2 400.2.c.d 2
8.b even 2 1 1600.2.a.k 1
8.d odd 2 1 1600.2.a.o 1
12.b even 2 1 1800.2.a.v 1
15.d odd 2 1 720.2.a.e 1
15.e even 4 2 3600.2.f.t 2
20.d odd 2 1 40.2.a.a 1
20.e even 4 2 200.2.c.b 2
28.d even 2 1 9800.2.a.x 1
35.c odd 2 1 3920.2.a.s 1
40.e odd 2 1 320.2.a.c 1
40.f even 2 1 320.2.a.d 1
40.i odd 4 2 1600.2.c.m 2
40.k even 4 2 1600.2.c.k 2
55.d odd 2 1 9680.2.a.q 1
60.h even 2 1 360.2.a.a 1
60.l odd 4 2 1800.2.f.a 2
80.k odd 4 2 1280.2.d.j 2
80.q even 4 2 1280.2.d.a 2
120.i odd 2 1 2880.2.a.bg 1
120.m even 2 1 2880.2.a.t 1
140.c even 2 1 1960.2.a.g 1
140.p odd 6 2 1960.2.q.h 2
140.s even 6 2 1960.2.q.i 2
180.n even 6 2 3240.2.q.x 2
180.p odd 6 2 3240.2.q.k 2
220.g even 2 1 4840.2.a.f 1
260.g odd 2 1 6760.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 20.d odd 2 1
80.2.a.a 1 5.b even 2 1
200.2.a.c 1 4.b odd 2 1
200.2.c.b 2 20.e even 4 2
320.2.a.c 1 40.e odd 2 1
320.2.a.d 1 40.f even 2 1
360.2.a.a 1 60.h even 2 1
400.2.a.e 1 1.a even 1 1 trivial
400.2.c.d 2 5.c odd 4 2
720.2.a.e 1 15.d odd 2 1
1280.2.d.a 2 80.q even 4 2
1280.2.d.j 2 80.k odd 4 2
1600.2.a.k 1 8.b even 2 1
1600.2.a.o 1 8.d odd 2 1
1600.2.c.k 2 40.k even 4 2
1600.2.c.m 2 40.i odd 4 2
1800.2.a.v 1 12.b even 2 1
1800.2.f.a 2 60.l odd 4 2
1960.2.a.g 1 140.c even 2 1
1960.2.q.h 2 140.p odd 6 2
1960.2.q.i 2 140.s even 6 2
2880.2.a.t 1 120.m even 2 1
2880.2.a.bg 1 120.i odd 2 1
3240.2.q.k 2 180.p odd 6 2
3240.2.q.x 2 180.n even 6 2
3600.2.a.h 1 3.b odd 2 1
3600.2.f.t 2 15.e even 4 2
3920.2.a.s 1 35.c odd 2 1
4840.2.a.f 1 220.g even 2 1
6760.2.a.i 1 260.g odd 2 1
9680.2.a.q 1 55.d odd 2 1
9800.2.a.x 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(400))$$:

 $$T_{3}$$ T3 $$T_{7} + 4$$ T7 + 4

## Hecke characteristic polynomials

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