# Properties

 Label 400.4.a.k Level $400$ Weight $4$ Character orbit 400.a Self dual yes Analytic conductor $23.601$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,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, 4, names="a")

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$23.6007640023$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 6 q^{7} - 26 q^{9}+O(q^{10})$$ q + q^3 - 6 * q^7 - 26 * q^9 $$q + q^{3} - 6 q^{7} - 26 q^{9} + 19 q^{11} - 12 q^{13} + 75 q^{17} + 91 q^{19} - 6 q^{21} + 174 q^{23} - 53 q^{27} - 272 q^{29} + 230 q^{31} + 19 q^{33} + 182 q^{37} - 12 q^{39} + 117 q^{41} + 372 q^{43} - 52 q^{47} - 307 q^{49} + 75 q^{51} + 402 q^{53} + 91 q^{57} - 312 q^{59} + 170 q^{61} + 156 q^{63} + 763 q^{67} + 174 q^{69} + 52 q^{71} + 981 q^{73} - 114 q^{77} - 1054 q^{79} + 649 q^{81} + 351 q^{83} - 272 q^{87} + 799 q^{89} + 72 q^{91} + 230 q^{93} - 962 q^{97} - 494 q^{99}+O(q^{100})$$ q + q^3 - 6 * q^7 - 26 * q^9 + 19 * q^11 - 12 * q^13 + 75 * q^17 + 91 * q^19 - 6 * q^21 + 174 * q^23 - 53 * q^27 - 272 * q^29 + 230 * q^31 + 19 * q^33 + 182 * q^37 - 12 * q^39 + 117 * q^41 + 372 * q^43 - 52 * q^47 - 307 * q^49 + 75 * q^51 + 402 * q^53 + 91 * q^57 - 312 * q^59 + 170 * q^61 + 156 * q^63 + 763 * q^67 + 174 * q^69 + 52 * q^71 + 981 * q^73 - 114 * q^77 - 1054 * q^79 + 649 * q^81 + 351 * q^83 - 272 * q^87 + 799 * q^89 + 72 * q^91 + 230 * q^93 - 962 * q^97 - 494 * 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 1.00000 0 0 0 −6.00000 0 −26.0000 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 400.4.a.k 1
4.b odd 2 1 200.4.a.e 1
5.b even 2 1 400.4.a.j 1
5.c odd 4 2 400.4.c.m 2
8.b even 2 1 1600.4.a.v 1
8.d odd 2 1 1600.4.a.bf 1
12.b even 2 1 1800.4.a.w 1
20.d odd 2 1 200.4.a.f yes 1
20.e even 4 2 200.4.c.g 2
40.e odd 2 1 1600.4.a.w 1
40.f even 2 1 1600.4.a.be 1
60.h even 2 1 1800.4.a.l 1
60.l odd 4 2 1800.4.f.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.e 1 4.b odd 2 1
200.4.a.f yes 1 20.d odd 2 1
200.4.c.g 2 20.e even 4 2
400.4.a.j 1 5.b even 2 1
400.4.a.k 1 1.a even 1 1 trivial
400.4.c.m 2 5.c odd 4 2
1600.4.a.v 1 8.b even 2 1
1600.4.a.w 1 40.e odd 2 1
1600.4.a.be 1 40.f even 2 1
1600.4.a.bf 1 8.d odd 2 1
1800.4.a.l 1 60.h even 2 1
1800.4.a.w 1 12.b even 2 1
1800.4.f.p 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(400))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 6$$ T7 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 6$$
$11$ $$T - 19$$
$13$ $$T + 12$$
$17$ $$T - 75$$
$19$ $$T - 91$$
$23$ $$T - 174$$
$29$ $$T + 272$$
$31$ $$T - 230$$
$37$ $$T - 182$$
$41$ $$T - 117$$
$43$ $$T - 372$$
$47$ $$T + 52$$
$53$ $$T - 402$$
$59$ $$T + 312$$
$61$ $$T - 170$$
$67$ $$T - 763$$
$71$ $$T - 52$$
$73$ $$T - 981$$
$79$ $$T + 1054$$
$83$ $$T - 351$$
$89$ $$T - 799$$
$97$ $$T + 962$$