# Properties

 Label 400.4.a.t 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)

 $$f(q)$$ $$=$$ $$q + 9 q^{3} + 26 q^{7} + 54 q^{9}+O(q^{10})$$ q + 9 * q^3 + 26 * q^7 + 54 * q^9 $$q + 9 q^{3} + 26 q^{7} + 54 q^{9} + 59 q^{11} - 28 q^{13} - 5 q^{17} - 109 q^{19} + 234 q^{21} - 194 q^{23} + 243 q^{27} - 32 q^{29} - 10 q^{31} + 531 q^{33} + 198 q^{37} - 252 q^{39} + 117 q^{41} + 388 q^{43} - 68 q^{47} + 333 q^{49} - 45 q^{51} + 18 q^{53} - 981 q^{57} - 392 q^{59} - 710 q^{61} + 1404 q^{63} - 253 q^{67} - 1746 q^{69} + 612 q^{71} + 549 q^{73} + 1534 q^{77} - 414 q^{79} + 729 q^{81} - 121 q^{83} - 288 q^{87} - 81 q^{89} - 728 q^{91} - 90 q^{93} + 1502 q^{97} + 3186 q^{99}+O(q^{100})$$ q + 9 * q^3 + 26 * q^7 + 54 * q^9 + 59 * q^11 - 28 * q^13 - 5 * q^17 - 109 * q^19 + 234 * q^21 - 194 * q^23 + 243 * q^27 - 32 * q^29 - 10 * q^31 + 531 * q^33 + 198 * q^37 - 252 * q^39 + 117 * q^41 + 388 * q^43 - 68 * q^47 + 333 * q^49 - 45 * q^51 + 18 * q^53 - 981 * q^57 - 392 * q^59 - 710 * q^61 + 1404 * q^63 - 253 * q^67 - 1746 * q^69 + 612 * q^71 + 549 * q^73 + 1534 * q^77 - 414 * q^79 + 729 * q^81 - 121 * q^83 - 288 * q^87 - 81 * q^89 - 728 * q^91 - 90 * q^93 + 1502 * q^97 + 3186 * 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 9.00000 0 0 0 26.0000 0 54.0000 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.t 1
4.b odd 2 1 200.4.a.b 1
5.b even 2 1 400.4.a.a 1
5.c odd 4 2 400.4.c.b 2
8.b even 2 1 1600.4.a.b 1
8.d odd 2 1 1600.4.a.bz 1
12.b even 2 1 1800.4.a.c 1
20.d odd 2 1 200.4.a.j yes 1
20.e even 4 2 200.4.c.b 2
40.e odd 2 1 1600.4.a.c 1
40.f even 2 1 1600.4.a.by 1
60.h even 2 1 1800.4.a.bh 1
60.l odd 4 2 1800.4.f.w 2

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

 $$T_{3} - 9$$ T3 - 9 $$T_{7} - 26$$ T7 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 9$$
$5$ $$T$$
$7$ $$T - 26$$
$11$ $$T - 59$$
$13$ $$T + 28$$
$17$ $$T + 5$$
$19$ $$T + 109$$
$23$ $$T + 194$$
$29$ $$T + 32$$
$31$ $$T + 10$$
$37$ $$T - 198$$
$41$ $$T - 117$$
$43$ $$T - 388$$
$47$ $$T + 68$$
$53$ $$T - 18$$
$59$ $$T + 392$$
$61$ $$T + 710$$
$67$ $$T + 253$$
$71$ $$T - 612$$
$73$ $$T - 549$$
$79$ $$T + 414$$
$83$ $$T + 121$$
$89$ $$T + 81$$
$97$ $$T - 1502$$