# Properties

 Label 400.4.a.p 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 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{3} + 16 q^{7} - 11 q^{9}+O(q^{10})$$ q + 4 * q^3 + 16 * q^7 - 11 * q^9 $$q + 4 q^{3} + 16 q^{7} - 11 q^{9} - 36 q^{11} + 42 q^{13} + 110 q^{17} + 116 q^{19} + 64 q^{21} + 16 q^{23} - 152 q^{27} + 198 q^{29} - 240 q^{31} - 144 q^{33} + 258 q^{37} + 168 q^{39} + 442 q^{41} - 292 q^{43} + 392 q^{47} - 87 q^{49} + 440 q^{51} - 142 q^{53} + 464 q^{57} + 348 q^{59} - 570 q^{61} - 176 q^{63} + 692 q^{67} + 64 q^{69} - 168 q^{71} + 134 q^{73} - 576 q^{77} - 784 q^{79} - 311 q^{81} + 564 q^{83} + 792 q^{87} + 1034 q^{89} + 672 q^{91} - 960 q^{93} + 382 q^{97} + 396 q^{99}+O(q^{100})$$ q + 4 * q^3 + 16 * q^7 - 11 * q^9 - 36 * q^11 + 42 * q^13 + 110 * q^17 + 116 * q^19 + 64 * q^21 + 16 * q^23 - 152 * q^27 + 198 * q^29 - 240 * q^31 - 144 * q^33 + 258 * q^37 + 168 * q^39 + 442 * q^41 - 292 * q^43 + 392 * q^47 - 87 * q^49 + 440 * q^51 - 142 * q^53 + 464 * q^57 + 348 * q^59 - 570 * q^61 - 176 * q^63 + 692 * q^67 + 64 * q^69 - 168 * q^71 + 134 * q^73 - 576 * q^77 - 784 * q^79 - 311 * q^81 + 564 * q^83 + 792 * q^87 + 1034 * q^89 + 672 * q^91 - 960 * q^93 + 382 * q^97 + 396 * 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 4.00000 0 0 0 16.0000 0 −11.0000 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.p 1
4.b odd 2 1 200.4.a.d 1
5.b even 2 1 80.4.a.b 1
5.c odd 4 2 400.4.c.h 2
8.b even 2 1 1600.4.a.q 1
8.d odd 2 1 1600.4.a.bk 1
12.b even 2 1 1800.4.a.h 1
15.d odd 2 1 720.4.a.d 1
20.d odd 2 1 40.4.a.b 1
20.e even 4 2 200.4.c.f 2
40.e odd 2 1 320.4.a.e 1
40.f even 2 1 320.4.a.j 1
60.h even 2 1 360.4.a.f 1
60.l odd 4 2 1800.4.f.d 2
80.k odd 4 2 1280.4.d.d 2
80.q even 4 2 1280.4.d.m 2
140.c even 2 1 1960.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.b 1 20.d odd 2 1
80.4.a.b 1 5.b even 2 1
200.4.a.d 1 4.b odd 2 1
200.4.c.f 2 20.e even 4 2
320.4.a.e 1 40.e odd 2 1
320.4.a.j 1 40.f even 2 1
360.4.a.f 1 60.h even 2 1
400.4.a.p 1 1.a even 1 1 trivial
400.4.c.h 2 5.c odd 4 2
720.4.a.d 1 15.d odd 2 1
1280.4.d.d 2 80.k odd 4 2
1280.4.d.m 2 80.q even 4 2
1600.4.a.q 1 8.b even 2 1
1600.4.a.bk 1 8.d odd 2 1
1800.4.a.h 1 12.b even 2 1
1800.4.f.d 2 60.l odd 4 2
1960.4.a.e 1 140.c even 2 1

## Hecke kernels

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

 $$T_{3} - 4$$ T3 - 4 $$T_{7} - 16$$ T7 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 4$$
$5$ $$T$$
$7$ $$T - 16$$
$11$ $$T + 36$$
$13$ $$T - 42$$
$17$ $$T - 110$$
$19$ $$T - 116$$
$23$ $$T - 16$$
$29$ $$T - 198$$
$31$ $$T + 240$$
$37$ $$T - 258$$
$41$ $$T - 442$$
$43$ $$T + 292$$
$47$ $$T - 392$$
$53$ $$T + 142$$
$59$ $$T - 348$$
$61$ $$T + 570$$
$67$ $$T - 692$$
$71$ $$T + 168$$
$73$ $$T - 134$$
$79$ $$T + 784$$
$83$ $$T - 564$$
$89$ $$T - 1034$$
$97$ $$T - 382$$