# Properties

 Label 400.4.c.k Level $400$ Weight $4$ Character orbit 400.c Analytic conductor $23.601$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$23.6007640023$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 3 i q^{7} + 23 q^{9} +O(q^{10})$$ q + i * q^3 - 3*i * q^7 + 23 * q^9 $$q + i q^{3} - 3 i q^{7} + 23 q^{9} - 32 q^{11} + 19 i q^{13} + 13 i q^{17} + 100 q^{19} + 12 q^{21} - 39 i q^{23} + 50 i q^{27} + 50 q^{29} + 108 q^{31} - 32 i q^{33} + 133 i q^{37} - 76 q^{39} + 22 q^{41} + 221 i q^{43} + 257 i q^{47} + 307 q^{49} - 52 q^{51} - i q^{53} + 100 i q^{57} + 500 q^{59} - 518 q^{61} - 69 i q^{63} - 63 i q^{67} + 156 q^{69} - 412 q^{71} + 439 i q^{73} + 96 i q^{77} + 600 q^{79} + 421 q^{81} + 141 i q^{83} + 50 i q^{87} + 150 q^{89} + 228 q^{91} + 108 i q^{93} + 193 i q^{97} - 736 q^{99} +O(q^{100})$$ q + i * q^3 - 3*i * q^7 + 23 * q^9 - 32 * q^11 + 19*i * q^13 + 13*i * q^17 + 100 * q^19 + 12 * q^21 - 39*i * q^23 + 50*i * q^27 + 50 * q^29 + 108 * q^31 - 32*i * q^33 + 133*i * q^37 - 76 * q^39 + 22 * q^41 + 221*i * q^43 + 257*i * q^47 + 307 * q^49 - 52 * q^51 - i * q^53 + 100*i * q^57 + 500 * q^59 - 518 * q^61 - 69*i * q^63 - 63*i * q^67 + 156 * q^69 - 412 * q^71 + 439*i * q^73 + 96*i * q^77 + 600 * q^79 + 421 * q^81 + 141*i * q^83 + 50*i * q^87 + 150 * q^89 + 228 * q^91 + 108*i * q^93 + 193*i * q^97 - 736 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 46 q^{9}+O(q^{10})$$ 2 * q + 46 * q^9 $$2 q + 46 q^{9} - 64 q^{11} + 200 q^{19} + 24 q^{21} + 100 q^{29} + 216 q^{31} - 152 q^{39} + 44 q^{41} + 614 q^{49} - 104 q^{51} + 1000 q^{59} - 1036 q^{61} + 312 q^{69} - 824 q^{71} + 1200 q^{79} + 842 q^{81} + 300 q^{89} + 456 q^{91} - 1472 q^{99}+O(q^{100})$$ 2 * q + 46 * q^9 - 64 * q^11 + 200 * q^19 + 24 * q^21 + 100 * q^29 + 216 * q^31 - 152 * q^39 + 44 * q^41 + 614 * q^49 - 104 * q^51 + 1000 * q^59 - 1036 * q^61 + 312 * q^69 - 824 * q^71 + 1200 * q^79 + 842 * q^81 + 300 * q^89 + 456 * q^91 - 1472 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 6.00000i 0 23.0000 0
49.2 0 2.00000i 0 0 0 6.00000i 0 23.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.4.c.k 2
4.b odd 2 1 25.4.b.a 2
5.b even 2 1 inner 400.4.c.k 2
5.c odd 4 1 80.4.a.d 1
5.c odd 4 1 400.4.a.m 1
12.b even 2 1 225.4.b.c 2
15.e even 4 1 720.4.a.u 1
20.d odd 2 1 25.4.b.a 2
20.e even 4 1 5.4.a.a 1
20.e even 4 1 25.4.a.c 1
40.i odd 4 1 320.4.a.h 1
40.i odd 4 1 1600.4.a.s 1
40.k even 4 1 320.4.a.g 1
40.k even 4 1 1600.4.a.bi 1
60.h even 2 1 225.4.b.c 2
60.l odd 4 1 45.4.a.d 1
60.l odd 4 1 225.4.a.b 1
80.i odd 4 1 1280.4.d.l 2
80.j even 4 1 1280.4.d.e 2
80.s even 4 1 1280.4.d.e 2
80.t odd 4 1 1280.4.d.l 2
140.j odd 4 1 245.4.a.a 1
140.j odd 4 1 1225.4.a.k 1
140.w even 12 2 245.4.e.f 2
140.x odd 12 2 245.4.e.g 2
180.v odd 12 2 405.4.e.c 2
180.x even 12 2 405.4.e.l 2
220.i odd 4 1 605.4.a.d 1
260.p even 4 1 845.4.a.b 1
340.r even 4 1 1445.4.a.a 1
380.j odd 4 1 1805.4.a.h 1
420.w even 4 1 2205.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 20.e even 4 1
25.4.a.c 1 20.e even 4 1
25.4.b.a 2 4.b odd 2 1
25.4.b.a 2 20.d odd 2 1
45.4.a.d 1 60.l odd 4 1
80.4.a.d 1 5.c odd 4 1
225.4.a.b 1 60.l odd 4 1
225.4.b.c 2 12.b even 2 1
225.4.b.c 2 60.h even 2 1
245.4.a.a 1 140.j odd 4 1
245.4.e.f 2 140.w even 12 2
245.4.e.g 2 140.x odd 12 2
320.4.a.g 1 40.k even 4 1
320.4.a.h 1 40.i odd 4 1
400.4.a.m 1 5.c odd 4 1
400.4.c.k 2 1.a even 1 1 trivial
400.4.c.k 2 5.b even 2 1 inner
405.4.e.c 2 180.v odd 12 2
405.4.e.l 2 180.x even 12 2
605.4.a.d 1 220.i odd 4 1
720.4.a.u 1 15.e even 4 1
845.4.a.b 1 260.p even 4 1
1225.4.a.k 1 140.j odd 4 1
1280.4.d.e 2 80.j even 4 1
1280.4.d.e 2 80.s even 4 1
1280.4.d.l 2 80.i odd 4 1
1280.4.d.l 2 80.t odd 4 1
1445.4.a.a 1 340.r even 4 1
1600.4.a.s 1 40.i odd 4 1
1600.4.a.bi 1 40.k even 4 1
1805.4.a.h 1 380.j odd 4 1
2205.4.a.q 1 420.w even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}^{2} + 36$$ T7^2 + 36 $$T_{11} + 32$$ T11 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 36$$
$11$ $$(T + 32)^{2}$$
$13$ $$T^{2} + 1444$$
$17$ $$T^{2} + 676$$
$19$ $$(T - 100)^{2}$$
$23$ $$T^{2} + 6084$$
$29$ $$(T - 50)^{2}$$
$31$ $$(T - 108)^{2}$$
$37$ $$T^{2} + 70756$$
$41$ $$(T - 22)^{2}$$
$43$ $$T^{2} + 195364$$
$47$ $$T^{2} + 264196$$
$53$ $$T^{2} + 4$$
$59$ $$(T - 500)^{2}$$
$61$ $$(T + 518)^{2}$$
$67$ $$T^{2} + 15876$$
$71$ $$(T + 412)^{2}$$
$73$ $$T^{2} + 770884$$
$79$ $$(T - 600)^{2}$$
$83$ $$T^{2} + 79524$$
$89$ $$(T - 150)^{2}$$
$97$ $$T^{2} + 148996$$