# Properties

 Label 400.4.a.n 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 10) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + 26 q^{7} - 23 q^{9}+O(q^{10})$$ q + 2 * q^3 + 26 * q^7 - 23 * q^9 $$q + 2 q^{3} + 26 q^{7} - 23 q^{9} + 28 q^{11} - 12 q^{13} + 64 q^{17} + 60 q^{19} + 52 q^{21} - 58 q^{23} - 100 q^{27} + 90 q^{29} + 128 q^{31} + 56 q^{33} - 236 q^{37} - 24 q^{39} + 242 q^{41} + 362 q^{43} + 226 q^{47} + 333 q^{49} + 128 q^{51} + 108 q^{53} + 120 q^{57} + 20 q^{59} + 542 q^{61} - 598 q^{63} - 434 q^{67} - 116 q^{69} + 1128 q^{71} - 632 q^{73} + 728 q^{77} + 720 q^{79} + 421 q^{81} - 478 q^{83} + 180 q^{87} - 490 q^{89} - 312 q^{91} + 256 q^{93} - 1456 q^{97} - 644 q^{99}+O(q^{100})$$ q + 2 * q^3 + 26 * q^7 - 23 * q^9 + 28 * q^11 - 12 * q^13 + 64 * q^17 + 60 * q^19 + 52 * q^21 - 58 * q^23 - 100 * q^27 + 90 * q^29 + 128 * q^31 + 56 * q^33 - 236 * q^37 - 24 * q^39 + 242 * q^41 + 362 * q^43 + 226 * q^47 + 333 * q^49 + 128 * q^51 + 108 * q^53 + 120 * q^57 + 20 * q^59 + 542 * q^61 - 598 * q^63 - 434 * q^67 - 116 * q^69 + 1128 * q^71 - 632 * q^73 + 728 * q^77 + 720 * q^79 + 421 * q^81 - 478 * q^83 + 180 * q^87 - 490 * q^89 - 312 * q^91 + 256 * q^93 - 1456 * q^97 - 644 * 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 2.00000 0 0 0 26.0000 0 −23.0000 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.n 1
4.b odd 2 1 50.4.a.b 1
5.b even 2 1 400.4.a.h 1
5.c odd 4 2 80.4.c.a 2
8.b even 2 1 1600.4.a.t 1
8.d odd 2 1 1600.4.a.bh 1
12.b even 2 1 450.4.a.k 1
15.e even 4 2 720.4.f.f 2
20.d odd 2 1 50.4.a.d 1
20.e even 4 2 10.4.b.a 2
28.d even 2 1 2450.4.a.o 1
40.e odd 2 1 1600.4.a.u 1
40.f even 2 1 1600.4.a.bg 1
40.i odd 4 2 320.4.c.c 2
40.k even 4 2 320.4.c.d 2
60.h even 2 1 450.4.a.j 1
60.l odd 4 2 90.4.c.b 2
140.c even 2 1 2450.4.a.bb 1
140.j odd 4 2 490.4.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 20.e even 4 2
50.4.a.b 1 4.b odd 2 1
50.4.a.d 1 20.d odd 2 1
80.4.c.a 2 5.c odd 4 2
90.4.c.b 2 60.l odd 4 2
320.4.c.c 2 40.i odd 4 2
320.4.c.d 2 40.k even 4 2
400.4.a.h 1 5.b even 2 1
400.4.a.n 1 1.a even 1 1 trivial
450.4.a.j 1 60.h even 2 1
450.4.a.k 1 12.b even 2 1
490.4.c.b 2 140.j odd 4 2
720.4.f.f 2 15.e even 4 2
1600.4.a.t 1 8.b even 2 1
1600.4.a.u 1 40.e odd 2 1
1600.4.a.bg 1 40.f even 2 1
1600.4.a.bh 1 8.d odd 2 1
2450.4.a.o 1 28.d even 2 1
2450.4.a.bb 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(400))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{7} - 26$$ T7 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T - 26$$
$11$ $$T - 28$$
$13$ $$T + 12$$
$17$ $$T - 64$$
$19$ $$T - 60$$
$23$ $$T + 58$$
$29$ $$T - 90$$
$31$ $$T - 128$$
$37$ $$T + 236$$
$41$ $$T - 242$$
$43$ $$T - 362$$
$47$ $$T - 226$$
$53$ $$T - 108$$
$59$ $$T - 20$$
$61$ $$T - 542$$
$67$ $$T + 434$$
$71$ $$T - 1128$$
$73$ $$T + 632$$
$79$ $$T - 720$$
$83$ $$T + 478$$
$89$ $$T + 490$$
$97$ $$T + 1456$$