# Properties

 Label 400.2.c.d Level $400$ Weight $2$ Character orbit 400.c Analytic conductor $3.194$ 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] = [400,2,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, 2, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$3.19401608085$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 40) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{7} + 3 q^{9}+O(q^{10})$$ q + 2*i * q^7 + 3 * q^9 $$q + 2 i q^{7} + 3 q^{9} - 4 q^{11} + i q^{13} + i q^{17} + 4 q^{19} + 2 i q^{23} + 2 q^{29} + 8 q^{31} + 3 i q^{37} - 6 q^{41} - 4 i q^{43} - 2 i q^{47} - 9 q^{49} - 3 i q^{53} - 4 q^{59} - 2 q^{61} + 6 i q^{63} - 4 i q^{67} + 3 i q^{73} - 8 i q^{77} + 9 q^{81} - 8 i q^{83} + 6 q^{89} - 8 q^{91} - 7 i q^{97} - 12 q^{99} +O(q^{100})$$ q + 2*i * q^7 + 3 * q^9 - 4 * q^11 + i * q^13 + i * q^17 + 4 * q^19 + 2*i * q^23 + 2 * q^29 + 8 * q^31 + 3*i * q^37 - 6 * q^41 - 4*i * q^43 - 2*i * q^47 - 9 * q^49 - 3*i * q^53 - 4 * q^59 - 2 * q^61 + 6*i * q^63 - 4*i * q^67 + 3*i * q^73 - 8*i * q^77 + 9 * q^81 - 8*i * q^83 + 6 * q^89 - 8 * q^91 - 7*i * q^97 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{9}+O(q^{10})$$ 2 * q + 6 * q^9 $$2 q + 6 q^{9} - 8 q^{11} + 8 q^{19} + 4 q^{29} + 16 q^{31} - 12 q^{41} - 18 q^{49} - 8 q^{59} - 4 q^{61} + 18 q^{81} + 12 q^{89} - 16 q^{91} - 24 q^{99}+O(q^{100})$$ 2 * q + 6 * q^9 - 8 * q^11 + 8 * q^19 + 4 * q^29 + 16 * q^31 - 12 * q^41 - 18 * q^49 - 8 * q^59 - 4 * q^61 + 18 * q^81 + 12 * q^89 - 16 * q^91 - 24 * 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 0 0 0 0 4.00000i 0 3.00000 0
49.2 0 0 0 0 0 4.00000i 0 3.00000 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 400.2.c.d 2
3.b odd 2 1 3600.2.f.t 2
4.b odd 2 1 200.2.c.b 2
5.b even 2 1 inner 400.2.c.d 2
5.c odd 4 1 80.2.a.a 1
5.c odd 4 1 400.2.a.e 1
8.b even 2 1 1600.2.c.m 2
8.d odd 2 1 1600.2.c.k 2
12.b even 2 1 1800.2.f.a 2
15.d odd 2 1 3600.2.f.t 2
15.e even 4 1 720.2.a.e 1
15.e even 4 1 3600.2.a.h 1
20.d odd 2 1 200.2.c.b 2
20.e even 4 1 40.2.a.a 1
20.e even 4 1 200.2.a.c 1
35.f even 4 1 3920.2.a.s 1
40.e odd 2 1 1600.2.c.k 2
40.f even 2 1 1600.2.c.m 2
40.i odd 4 1 320.2.a.d 1
40.i odd 4 1 1600.2.a.k 1
40.k even 4 1 320.2.a.c 1
40.k even 4 1 1600.2.a.o 1
55.e even 4 1 9680.2.a.q 1
60.h even 2 1 1800.2.f.a 2
60.l odd 4 1 360.2.a.a 1
60.l odd 4 1 1800.2.a.v 1
80.i odd 4 1 1280.2.d.a 2
80.j even 4 1 1280.2.d.j 2
80.s even 4 1 1280.2.d.j 2
80.t odd 4 1 1280.2.d.a 2
120.q odd 4 1 2880.2.a.t 1
120.w even 4 1 2880.2.a.bg 1
140.j odd 4 1 1960.2.a.g 1
140.j odd 4 1 9800.2.a.x 1
140.w even 12 2 1960.2.q.h 2
140.x odd 12 2 1960.2.q.i 2
180.v odd 12 2 3240.2.q.x 2
180.x even 12 2 3240.2.q.k 2
220.i odd 4 1 4840.2.a.f 1
260.p even 4 1 6760.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 20.e even 4 1
80.2.a.a 1 5.c odd 4 1
200.2.a.c 1 20.e even 4 1
200.2.c.b 2 4.b odd 2 1
200.2.c.b 2 20.d odd 2 1
320.2.a.c 1 40.k even 4 1
320.2.a.d 1 40.i odd 4 1
360.2.a.a 1 60.l odd 4 1
400.2.a.e 1 5.c odd 4 1
400.2.c.d 2 1.a even 1 1 trivial
400.2.c.d 2 5.b even 2 1 inner
720.2.a.e 1 15.e even 4 1
1280.2.d.a 2 80.i odd 4 1
1280.2.d.a 2 80.t odd 4 1
1280.2.d.j 2 80.j even 4 1
1280.2.d.j 2 80.s even 4 1
1600.2.a.k 1 40.i odd 4 1
1600.2.a.o 1 40.k even 4 1
1600.2.c.k 2 8.d odd 2 1
1600.2.c.k 2 40.e odd 2 1
1600.2.c.m 2 8.b even 2 1
1600.2.c.m 2 40.f even 2 1
1800.2.a.v 1 60.l odd 4 1
1800.2.f.a 2 12.b even 2 1
1800.2.f.a 2 60.h even 2 1
1960.2.a.g 1 140.j odd 4 1
1960.2.q.h 2 140.w even 12 2
1960.2.q.i 2 140.x odd 12 2
2880.2.a.t 1 120.q odd 4 1
2880.2.a.bg 1 120.w even 4 1
3240.2.q.k 2 180.x even 12 2
3240.2.q.x 2 180.v odd 12 2
3600.2.a.h 1 15.e even 4 1
3600.2.f.t 2 3.b odd 2 1
3600.2.f.t 2 15.d odd 2 1
3920.2.a.s 1 35.f even 4 1
4840.2.a.f 1 220.i odd 4 1
6760.2.a.i 1 260.p even 4 1
9680.2.a.q 1 55.e even 4 1
9800.2.a.x 1 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 2)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 256$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 196$$