LMFDB - Newform orbit 400.2.u.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,2,Mod(81,400)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("400.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	400.u (of order $5$ , degree $4$ , 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:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + x^{2} - x + 1$ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$ -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 a primitive root of unity $\zeta_{10}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \zeta_{10}^{3} q^{3} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{7} + 2 \zeta_{10} q^{9} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + \cdots - 2) q^{11}+ \cdots + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4) q^{99}+O(q^{100})$ q + z^3 * q^3 + (-2z^2 + z - 2) q^5 + (-z^3 + z^2 + 1) * q^7 + 2z q^9 + (2z^3 - 4z^2 + 4z - 2) q^11 + (3z^2 + 3) q^13 + (-z^3 - z^2 + z + 1) * q^15 + (2z^3 - 4z^2 + 2z) q^17 + (3z^3 + z^2 + 3z) * q^19 + (z^3 + z - 1) * q^21 + (-7z^3 + 5z^2 - 5z + 7) q^23 + 5z^2 q^25 + (5z^3 - 5z^2 + 5z - 5) q^27 + (2z^3 - z + 1) q^29 + 3z^2 q^31 + (2z^3 - 4z^2 + 2z) q^33 + (-z^2 - 2z - 1) q^35 + (-2z^2 - z - 2) q^37 + (3z^3 - 3) q^39 + (2z^2 + 2z + 2) * q^41 + (3z^3 - 3z^2) * q^43 + (-4z^3 + 2z^2 - 4z) q^45 + (-z^3 - z + 1) * q^47 + (-z^3 + z^2 - 5) * q^49 + (2z^3 - 2z^2 + 2) * q^51 + (-3z^3 - 4z + 4) * q^53 + (-6z^3 + 6z^2 - 2) * q^55 + (3z^3 - 3z^2 - 4) * q^57 + (3z^2 - 9z + 3) * q^59 + (-z^3 - 5z^2 + 5z + 1) * q^61 + (2z^2 + 2) q^63 + (-3z^3 - 6z^2 - 3z) q^65 + (-2z^3 - 6z^2 - 2z) q^67 + (2z^3 + 5z^2 + 2z) q^69 + (-5z^3 + z - 1) q^71 + (-9z^3 + 9z^2 - 9z + 9) q^73 - 5 * q^75 + (2z^2 - 2z) * q^77 + (5z - 5) q^79 + z^2 * q^81 + (-2z^3 + 5z^2 - 2z) q^83 + (-2z^3 + 6z - 6) * q^85 + (z^2 - 3z + 1) q^87 + (4z^3 + 4z^2 - 4z - 4) q^89 + (3z^2 + 3z + 3) * q^91 - 3 * q^93 + (-10z^3 - 5z + 5) * q^95 + (-z^3 - 3z + 3) q^97 + (-4z^3 + 4z^2 - 4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q + q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{11} + 9 q^{13} + 5 q^{15} + 8 q^{17} + 5 q^{19} - 2 q^{21} + 11 q^{23} - 5 q^{25} - 5 q^{27} + 5 q^{29} - 3 q^{31} + 8 q^{33} - 5 q^{35} - 7 q^{37} - 9 q^{39}+ \cdots - 24 q^{99}+O(q^{100})$ 4 * q + q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^11 + 9 * q^13 + 5 * q^15 + 8 * q^17 + 5 * q^19 - 2 * q^21 + 11 * q^23 - 5 * q^25 - 5 * q^27 + 5 * q^29 - 3 * q^31 + 8 * q^33 - 5 * q^35 - 7 * q^37 - 9 * q^39 + 8 * q^41 + 6 * q^43 - 10 * q^45 + 2 * q^47 - 22 * q^49 + 12 * q^51 + 9 * q^53 - 20 * q^55 - 10 * q^57 + 13 * q^61 + 6 * q^63 + 2 * q^67 - q^69 - 8 * q^71 + 9 * q^73 - 20 * q^75 - 4 * q^77 - 15 * q^79 - q^81 - 9 * q^83 - 20 * q^85 - 20 * q^89 + 12 * q^91 - 12 * q^93 + 5 * q^95 + 8 * q^97 - 24 * q^99

Character values

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

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-\zeta_{10}^{3}$	$1$

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}

81.1

−0.309017	+	0.951057i
0.809017	+	0.587785i
0.809017	−	0.587785i
−0.309017	−	0.951057i

0.809017

−

0.587785i

−0.690983

2.12663i

−0.618034

1.90211i

161.1

−0.309017

0.951057i

−1.80902

−

1.31433i

1.61803

1.17557i

241.1

−0.309017

−

0.951057i

−1.80902

1.31433i

1.61803

−

1.17557i

321.1

0.809017

0.587785i

−0.690983

−

2.12663i

−0.618034

−

1.90211i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.d	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.2.u.b		4
4.b	odd	2	1		25.2.d.a	✓	4
12.b	even	2	1		225.2.h.b		4
20.d	odd	2	1		125.2.d.a		4
20.e	even	4	2		125.2.e.a		8
25.d	even	5	1	inner	400.2.u.b		4
25.d	even	5	1		10000.2.a.c		2
25.e	even	10	1		10000.2.a.l		2
100.h	odd	10	1		125.2.d.a		4
100.h	odd	10	1		625.2.a.c		2
100.h	odd	10	2		625.2.d.b		4
100.j	odd	10	1		25.2.d.a	✓	4
100.j	odd	10	1		625.2.a.b		2
100.j	odd	10	2		625.2.d.h		4
100.l	even	20	2		125.2.e.a		8
100.l	even	20	2		625.2.b.a		4
100.l	even	20	4		625.2.e.c		8
300.n	even	10	1		225.2.h.b		4
300.n	even	10	1		5625.2.a.f		2
300.r	even	10	1		5625.2.a.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
25.2.d.a	✓	4	4.b	odd	2	1
25.2.d.a	✓	4	100.j	odd	10	1
125.2.d.a		4	20.d	odd	2	1
125.2.d.a		4	100.h	odd	10	1
125.2.e.a		8	20.e	even	4	2
125.2.e.a		8	100.l	even	20	2
225.2.h.b		4	12.b	even	2	1
225.2.h.b		4	300.n	even	10	1
400.2.u.b		4	1.a	even	1	1	trivial
400.2.u.b		4	25.d	even	5	1	inner
625.2.a.b		2	100.j	odd	10	1
625.2.a.c		2	100.h	odd	10	1
625.2.b.a		4	100.l	even	20	2
625.2.d.b		4	100.h	odd	10	2
625.2.d.h		4	100.j	odd	10	2
625.2.e.c		8	100.l	even	20	4
5625.2.a.d		2	300.r	even	10	1
5625.2.a.f		2	300.n	even	10	1
10000.2.a.c		2	25.d	even	5	1
10000.2.a.l		2	25.e	even	10	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{4} - T_{3}^{3} + T_{3}^{2} - T_{3} + 1$ T3^4 - T3^3 + T3^2 - T3 + 1 acting on $S_{2}^{\mathrm{new}}(400, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} - T^{3} + T^{2} + \cdots + 1$ T^4 - T^3 + T^2 - T + 1
$5$	$T^{4} + 5 T^{3} + \cdots + 25$ T^4 + 5T^3 + 15T^2 + 25*T + 25
$7$	$(T^{2} - T - 1)^{2}$ (T^2 - T - 1)^2
$11$	$T^{4} - 2 T^{3} + \cdots + 16$ T^4 - 2T^3 + 24T^2 + 32*T + 16
$13$	$T^{4} - 9 T^{3} + \cdots + 81$ T^4 - 9T^3 + 36T^2 - 54*T + 81
$17$	$T^{4} - 8 T^{3} + \cdots + 16$ T^4 - 8T^3 + 24T^2 + 8*T + 16
$19$	$T^{4} - 5 T^{3} + \cdots + 25$ T^4 - 5T^3 + 40T^2 - 50*T + 25
$23$	$T^{4} - 11 T^{3} + \cdots + 961$ T^4 - 11T^3 + 51T^2 - 31*T + 961
$29$	$T^{4} - 5 T^{3} + \cdots + 25$ T^4 - 5T^3 + 10T^2 + 25
$31$	$T^{4} + 3 T^{3} + \cdots + 81$ T^4 + 3T^3 + 9T^2 + 27*T + 81
$37$	$T^{4} + 7 T^{3} + \cdots + 1$ T^4 + 7T^3 + 19T^2 + 3*T + 1
$41$	$T^{4} - 8 T^{3} + \cdots + 16$ T^4 - 8T^3 + 24T^2 + 8*T + 16
$43$	$(T^{2} - 3 T - 9)^{2}$ (T^2 - 3*T - 9)^2
$47$	$T^{4} - 2 T^{3} + \cdots + 1$ T^4 - 2T^3 + 4T^2 - 3*T + 1
$53$	$T^{4} - 9 T^{3} + \cdots + 361$ T^4 - 9T^3 + 61T^2 - 209*T + 361
$59$	$T^{4} + 90 T^{2} + \cdots + 2025$ T^4 + 90T^2 + 675T + 2025
$61$	$T^{4} - 13 T^{3} + \cdots + 1681$ T^4 - 13T^3 + 139T^2 - 697*T + 1681
$67$	$T^{4} - 2 T^{3} + \cdots + 1936$ T^4 - 2T^3 + 64T^2 - 528*T + 1936
$71$	$T^{4} + 8 T^{3} + \cdots + 841$ T^4 + 8T^3 + 34T^2 + 87*T + 841
$73$	$T^{4} - 9 T^{3} + \cdots + 6561$ T^4 - 9T^3 + 81T^2 - 729*T + 6561
$79$	$T^{4} + 15 T^{3} + \cdots + 625$ T^4 + 15T^3 + 100T^2 + 250*T + 625
$83$	$T^{4} + 9 T^{3} + \cdots + 121$ T^4 + 9T^3 + 31T^2 - 11*T + 121
$89$	$T^{4} + 20 T^{3} + \cdots + 6400$ T^4 + 20T^3 + 240T^2 + 1600*T + 6400
$97$	$T^{4} - 8 T^{3} + \cdots + 121$ T^4 - 8T^3 + 34T^2 - 77*T + 121
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more