# Properties

 Label 400.4.a.m 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 5) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + 6 q^{7} - 23 q^{9}+O(q^{10})$$ q + 2 * q^3 + 6 * q^7 - 23 * q^9 $$q + 2 q^{3} + 6 q^{7} - 23 q^{9} - 32 q^{11} + 38 q^{13} - 26 q^{17} - 100 q^{19} + 12 q^{21} - 78 q^{23} - 100 q^{27} - 50 q^{29} + 108 q^{31} - 64 q^{33} - 266 q^{37} + 76 q^{39} + 22 q^{41} + 442 q^{43} - 514 q^{47} - 307 q^{49} - 52 q^{51} - 2 q^{53} - 200 q^{57} - 500 q^{59} - 518 q^{61} - 138 q^{63} + 126 q^{67} - 156 q^{69} - 412 q^{71} + 878 q^{73} - 192 q^{77} - 600 q^{79} + 421 q^{81} + 282 q^{83} - 100 q^{87} - 150 q^{89} + 228 q^{91} + 216 q^{93} - 386 q^{97} + 736 q^{99}+O(q^{100})$$ q + 2 * q^3 + 6 * q^7 - 23 * q^9 - 32 * q^11 + 38 * q^13 - 26 * q^17 - 100 * q^19 + 12 * q^21 - 78 * q^23 - 100 * q^27 - 50 * q^29 + 108 * q^31 - 64 * q^33 - 266 * q^37 + 76 * q^39 + 22 * q^41 + 442 * q^43 - 514 * q^47 - 307 * q^49 - 52 * q^51 - 2 * q^53 - 200 * q^57 - 500 * q^59 - 518 * q^61 - 138 * q^63 + 126 * q^67 - 156 * q^69 - 412 * q^71 + 878 * q^73 - 192 * q^77 - 600 * q^79 + 421 * q^81 + 282 * q^83 - 100 * q^87 - 150 * q^89 + 228 * q^91 + 216 * q^93 - 386 * q^97 + 736 * 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 6.00000 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.m 1
4.b odd 2 1 25.4.a.c 1
5.b even 2 1 80.4.a.d 1
5.c odd 4 2 400.4.c.k 2
8.b even 2 1 1600.4.a.s 1
8.d odd 2 1 1600.4.a.bi 1
12.b even 2 1 225.4.a.b 1
15.d odd 2 1 720.4.a.u 1
20.d odd 2 1 5.4.a.a 1
20.e even 4 2 25.4.b.a 2
28.d even 2 1 1225.4.a.k 1
40.e odd 2 1 320.4.a.g 1
40.f even 2 1 320.4.a.h 1
60.h even 2 1 45.4.a.d 1
60.l odd 4 2 225.4.b.c 2
80.k odd 4 2 1280.4.d.e 2
80.q even 4 2 1280.4.d.l 2
140.c even 2 1 245.4.a.a 1
140.p odd 6 2 245.4.e.f 2
140.s even 6 2 245.4.e.g 2
180.n even 6 2 405.4.e.c 2
180.p odd 6 2 405.4.e.l 2
220.g even 2 1 605.4.a.d 1
260.g odd 2 1 845.4.a.b 1
340.d odd 2 1 1445.4.a.a 1
380.d even 2 1 1805.4.a.h 1
420.o odd 2 1 2205.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 20.d odd 2 1
25.4.a.c 1 4.b odd 2 1
25.4.b.a 2 20.e even 4 2
45.4.a.d 1 60.h even 2 1
80.4.a.d 1 5.b even 2 1
225.4.a.b 1 12.b even 2 1
225.4.b.c 2 60.l odd 4 2
245.4.a.a 1 140.c even 2 1
245.4.e.f 2 140.p odd 6 2
245.4.e.g 2 140.s even 6 2
320.4.a.g 1 40.e odd 2 1
320.4.a.h 1 40.f even 2 1
400.4.a.m 1 1.a even 1 1 trivial
400.4.c.k 2 5.c odd 4 2
405.4.e.c 2 180.n even 6 2
405.4.e.l 2 180.p odd 6 2
605.4.a.d 1 220.g even 2 1
720.4.a.u 1 15.d odd 2 1
845.4.a.b 1 260.g odd 2 1
1225.4.a.k 1 28.d even 2 1
1280.4.d.e 2 80.k odd 4 2
1280.4.d.l 2 80.q even 4 2
1445.4.a.a 1 340.d odd 2 1
1600.4.a.s 1 8.b even 2 1
1600.4.a.bi 1 8.d odd 2 1
1805.4.a.h 1 380.d even 2 1
2205.4.a.q 1 420.o odd 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 $$T_{7} - 6$$ T7 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T - 6$$
$11$ $$T + 32$$
$13$ $$T - 38$$
$17$ $$T + 26$$
$19$ $$T + 100$$
$23$ $$T + 78$$
$29$ $$T + 50$$
$31$ $$T - 108$$
$37$ $$T + 266$$
$41$ $$T - 22$$
$43$ $$T - 442$$
$47$ $$T + 514$$
$53$ $$T + 2$$
$59$ $$T + 500$$
$61$ $$T + 518$$
$67$ $$T - 126$$
$71$ $$T + 412$$
$73$ $$T - 878$$
$79$ $$T + 600$$
$83$ $$T - 282$$
$89$ $$T + 150$$
$97$ $$T + 386$$