LMFDB - Newform orbit 336.4.bc.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(17,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.17"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.bc (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$ -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 $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - 3 \zeta_{6} - 3) q^{3} + ( - \zeta_{6} - 18) q^{7} + 27 \zeta_{6} q^{9} + (34 \zeta_{6} - 17) q^{13} + (17 \zeta_{6} - 34) q^{19} + (60 \zeta_{6} + 51) q^{21} + ( - 125 \zeta_{6} + 125) q^{25}+ \cdots + (1584 \zeta_{6} - 792) q^{97}+O(q^{100})$ q + (-3z - 3) q^3 + (-z - 18) * q^7 + 27z q^9 + (34z - 17) q^13 + (17z - 34) q^19 + (60z + 51) q^21 + (-125z + 125) q^25 + (-162z + 81) q^27 + (199z + 199) q^31 - 433z q^37 + (-153z + 153) q^39 + 449 * q^43 + (37z + 323) q^49 + 153 * q^57 + (-540z + 1080) q^61 + (-513z + 27) q^63 + (-1007z + 1007) q^67 + (487z + 487) q^73 + (375z - 750) q^75 + 503z q^79 + (729z - 729) q^81 + (-629z + 340) q^91 - 1791z q^93 + (1584z - 792) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 9 q^{3} - 37 q^{7} + 27 q^{9} - 51 q^{19} + 162 q^{21} + 125 q^{25} + 597 q^{31} - 433 q^{37} + 153 q^{39} + 898 q^{43} + 683 q^{49} + 306 q^{57} + 1620 q^{61} - 459 q^{63} + 1007 q^{67} + 1461 q^{73}+ \cdots - 1791 q^{93}+O(q^{100})$ 2 * q - 9 * q^3 - 37 * q^7 + 27 * q^9 - 51 * q^19 + 162 * q^21 + 125 * q^25 + 597 * q^31 - 433 * q^37 + 153 * q^39 + 898 * q^43 + 683 * q^49 + 306 * q^57 + 1620 * q^61 - 459 * q^63 + 1007 * q^67 + 1461 * q^73 - 1125 * q^75 + 503 * q^79 - 729 * q^81 + 51 * q^91 - 1791 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$ .

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$1$	$\zeta_{6}$

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}

17.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−4.50000

−

2.59808i

−18.5000

−

0.866025i

13.5000

23.3827i

257.1

−4.50000

2.59808i

−18.5000

0.866025i

13.5000

−

23.3827i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.4.bc.a		2
3.b	odd	2	1	CM	336.4.bc.a		2
4.b	odd	2	1		84.4.k.b	✓	2
7.d	odd	6	1	inner	336.4.bc.a		2
12.b	even	2	1		84.4.k.b	✓	2
21.g	even	6	1	inner	336.4.bc.a		2
28.d	even	2	1		588.4.k.a		2
28.f	even	6	1		84.4.k.b	✓	2
28.f	even	6	1		588.4.f.b		2
28.g	odd	6	1		588.4.f.b		2
28.g	odd	6	1		588.4.k.a		2
84.h	odd	2	1		588.4.k.a		2
84.j	odd	6	1		84.4.k.b	✓	2
84.j	odd	6	1		588.4.f.b		2
84.n	even	6	1		588.4.f.b		2
84.n	even	6	1		588.4.k.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.4.k.b	✓	2	4.b	odd	2	1
84.4.k.b	✓	2	12.b	even	2	1
84.4.k.b	✓	2	28.f	even	6	1
84.4.k.b	✓	2	84.j	odd	6	1
336.4.bc.a		2	1.a	even	1	1	trivial
336.4.bc.a		2	3.b	odd	2	1	CM
336.4.bc.a		2	7.d	odd	6	1	inner
336.4.bc.a		2	21.g	even	6	1	inner
588.4.f.b		2	28.f	even	6	1
588.4.f.b		2	28.g	odd	6	1
588.4.f.b		2	84.j	odd	6	1
588.4.f.b		2	84.n	even	6	1
588.4.k.a		2	28.d	even	2	1
588.4.k.a		2	28.g	odd	6	1
588.4.k.a		2	84.h	odd	2	1
588.4.k.a		2	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(336, [\chi])$ :

T_{5}

T5

T_{13}^{2} + 867

T13^2 + 867

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 9T + 27$ T^2 + 9*T + 27
$5$	$T^{2}$ T^2
$7$	$T^{2} + 37T + 343$ T^2 + 37*T + 343
$11$	$T^{2}$ T^2
$13$	$T^{2} + 867$ T^2 + 867
$17$	$T^{2}$ T^2
$19$	$T^{2} + 51T + 867$ T^2 + 51*T + 867
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} - 597T + 118803$ T^2 - 597*T + 118803
$37$	$T^{2} + 433T + 187489$ T^2 + 433*T + 187489
$41$	$T^{2}$ T^2
$43$	$(T - 449)^{2}$ (T - 449)^2
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$T^{2} - 1620 T + 874800$ T^2 - 1620*T + 874800
$67$	$T^{2} - 1007 T + 1014049$ T^2 - 1007*T + 1014049
$71$	$T^{2}$ T^2
$73$	$T^{2} - 1461 T + 711507$ T^2 - 1461*T + 711507
$79$	$T^{2} - 503T + 253009$ T^2 - 503*T + 253009
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2} + 1881792$ T^2 + 1881792
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 $\F_{q}$
Belyi maps

$n$ :		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