# Properties

 Label 400.4.a.q Level $400$ Weight $4$ Character orbit 400.a Self dual yes Analytic conductor $23.601$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{3} + 2 q^{7} - 2 q^{9}+O(q^{10})$$ q + 5 * q^3 + 2 * q^7 - 2 * q^9 $$q + 5 q^{3} + 2 q^{7} - 2 q^{9} - 39 q^{11} - 84 q^{13} + 61 q^{17} - 151 q^{19} + 10 q^{21} - 58 q^{23} - 145 q^{27} + 192 q^{29} + 18 q^{31} - 195 q^{33} + 138 q^{37} - 420 q^{39} + 229 q^{41} - 164 q^{43} - 212 q^{47} - 339 q^{49} + 305 q^{51} - 578 q^{53} - 755 q^{57} + 336 q^{59} + 858 q^{61} - 4 q^{63} - 209 q^{67} - 290 q^{69} + 780 q^{71} + 403 q^{73} - 78 q^{77} + 230 q^{79} - 671 q^{81} - 1293 q^{83} + 960 q^{87} - 1369 q^{89} - 168 q^{91} + 90 q^{93} - 382 q^{97} + 78 q^{99}+O(q^{100})$$ q + 5 * q^3 + 2 * q^7 - 2 * q^9 - 39 * q^11 - 84 * q^13 + 61 * q^17 - 151 * q^19 + 10 * q^21 - 58 * q^23 - 145 * q^27 + 192 * q^29 + 18 * q^31 - 195 * q^33 + 138 * q^37 - 420 * q^39 + 229 * q^41 - 164 * q^43 - 212 * q^47 - 339 * q^49 + 305 * q^51 - 578 * q^53 - 755 * q^57 + 336 * q^59 + 858 * q^61 - 4 * q^63 - 209 * q^67 - 290 * q^69 + 780 * q^71 + 403 * q^73 - 78 * q^77 + 230 * q^79 - 671 * q^81 - 1293 * q^83 + 960 * q^87 - 1369 * q^89 - 168 * q^91 + 90 * q^93 - 382 * q^97 + 78 * 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 5.00000 0 0 0 2.00000 0 −2.00000 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 400.4.a.q 1
4.b odd 2 1 200.4.a.c 1
5.b even 2 1 400.4.a.f 1
5.c odd 4 2 400.4.c.g 2
8.b even 2 1 1600.4.a.n 1
8.d odd 2 1 1600.4.a.bn 1
12.b even 2 1 1800.4.a.p 1
20.d odd 2 1 200.4.a.h yes 1
20.e even 4 2 200.4.c.d 2
40.e odd 2 1 1600.4.a.m 1
40.f even 2 1 1600.4.a.bo 1
60.h even 2 1 1800.4.a.t 1
60.l odd 4 2 1800.4.f.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.c 1 4.b odd 2 1
200.4.a.h yes 1 20.d odd 2 1
200.4.c.d 2 20.e even 4 2
400.4.a.f 1 5.b even 2 1
400.4.a.q 1 1.a even 1 1 trivial
400.4.c.g 2 5.c odd 4 2
1600.4.a.m 1 40.e odd 2 1
1600.4.a.n 1 8.b even 2 1
1600.4.a.bn 1 8.d odd 2 1
1600.4.a.bo 1 40.f even 2 1
1800.4.a.p 1 12.b even 2 1
1800.4.a.t 1 60.h even 2 1
1800.4.f.c 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(400))$$:

 $$T_{3} - 5$$ T3 - 5 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 5$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T + 39$$
$13$ $$T + 84$$
$17$ $$T - 61$$
$19$ $$T + 151$$
$23$ $$T + 58$$
$29$ $$T - 192$$
$31$ $$T - 18$$
$37$ $$T - 138$$
$41$ $$T - 229$$
$43$ $$T + 164$$
$47$ $$T + 212$$
$53$ $$T + 578$$
$59$ $$T - 336$$
$61$ $$T - 858$$
$67$ $$T + 209$$
$71$ $$T - 780$$
$73$ $$T - 403$$
$79$ $$T - 230$$
$83$ $$T + 1293$$
$89$ $$T + 1369$$
$97$ $$T + 382$$