LMFDB - Newform orbit 336.4.bl.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(31,336)] 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([3, 0, 0, 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("336.31"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.bl (of order $6$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -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.

$f(q)$	$=$	$q + ( - 3 \zeta_{6} + 3) q^{3} + (\zeta_{6} - 2) q^{5} + (21 \zeta_{6} - 7) q^{7} - 9 \zeta_{6} q^{9} + ( - 23 \zeta_{6} - 23) q^{11} + (40 \zeta_{6} - 20) q^{13} + (6 \zeta_{6} - 3) q^{15} + ( - 54 \zeta_{6} - 54) q^{17} + \cdots + (414 \zeta_{6} - 207) q^{99} +O(q^{100})$ q + (-3z + 3) q^3 + (z - 2) * q^5 + (21z - 7) q^7 - 9z q^9 + (-23z - 23) q^11 + (40z - 20) q^13 + (6z - 3) q^15 + (-54z - 54) q^17 - 94z q^19 + (21z + 42) q^21 + (-52z + 104) q^23 + (122z - 122) q^25 - 27 * q^27 - 225 * q^29 + (11z - 11) q^31 + (69z - 138) q^33 + (-28z - 7) q^35 - 358z q^37 + (60z + 60) q^39 + (164z - 82) q^41 + (460z - 230) q^43 + (9z + 9) q^45 + 192z q^47 + (147z - 392) q^49 + (162z - 324) q^51 + (417z - 417) q^53 + 69 * q^55 - 282 * q^57 + (-561z + 561) q^59 + (308z - 616) q^61 + (-126z + 189) q^63 - 60z q^65 + (56z + 56) q^67 + (-312z + 156) q^69 + (-968z + 484) q^71 + (-12z - 12) q^73 + 366z q^75 + (-805z + 644) q^77 + (11z - 22) q^79 + (81z - 81) q^81 - 693 * q^83 + 162 * q^85 + (675z - 675) q^87 + (590z - 1180) q^89 + (140z - 700) q^91 + 33z q^93 + (94z + 94) q^95 + (950z - 475) q^97 + (414z - 207) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 3 q^{3} - 3 q^{5} + 7 q^{7} - 9 q^{9} - 69 q^{11} - 162 q^{17} - 94 q^{19} + 105 q^{21} + 156 q^{23} - 122 q^{25} - 54 q^{27} - 450 q^{29} - 11 q^{31} - 207 q^{33} - 42 q^{35} - 358 q^{37} + 180 q^{39}+ \cdots + 282 q^{95}+O(q^{100})$ 2 * q + 3 * q^3 - 3 * q^5 + 7 * q^7 - 9 * q^9 - 69 * q^11 - 162 * q^17 - 94 * q^19 + 105 * q^21 + 156 * q^23 - 122 * q^25 - 54 * q^27 - 450 * q^29 - 11 * q^31 - 207 * q^33 - 42 * q^35 - 358 * q^37 + 180 * q^39 + 27 * q^45 + 192 * q^47 - 637 * q^49 - 486 * q^51 - 417 * q^53 + 138 * q^55 - 564 * q^57 + 561 * q^59 - 924 * q^61 + 252 * q^63 - 60 * q^65 + 168 * q^67 - 36 * q^73 + 366 * q^75 + 483 * q^77 - 33 * q^79 - 81 * q^81 - 1386 * q^83 + 324 * q^85 - 675 * q^87 - 1770 * q^89 - 1260 * q^91 + 33 * q^93 + 282 * q^95

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}

31.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.50000

−

2.59808i

−1.50000

0.866025i

3.50000

18.1865i

−4.50000

−

7.79423i

271.1

1.50000

2.59808i

−1.50000

−

0.866025i

3.50000

−

18.1865i

−4.50000

7.79423i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.f	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.bl.d	yes	2
4.b	odd	2	1		336.4.bl.b	✓	2
7.d	odd	6	1		336.4.bl.b	✓	2
28.f	even	6	1	inner	336.4.bl.d	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.4.bl.b	✓	2	4.b	odd	2	1
336.4.bl.b	✓	2	7.d	odd	6	1
336.4.bl.d	yes	2	1.a	even	1	1	trivial
336.4.bl.d	yes	2	28.f	even	6	1	inner

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}^{2} + 3T_{5} + 3

T5^2 + 3*T5 + 3

T_{11}^{2} + 69T_{11} + 1587

T11^2 + 69*T11 + 1587

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$5$	$T^{2} + 3T + 3$ T^2 + 3*T + 3
$7$	$T^{2} - 7T + 343$ T^2 - 7*T + 343
$11$	$T^{2} + 69T + 1587$ T^2 + 69*T + 1587
$13$	$T^{2} + 1200$ T^2 + 1200
$17$	$T^{2} + 162T + 8748$ T^2 + 162*T + 8748
$19$	$T^{2} + 94T + 8836$ T^2 + 94*T + 8836
$23$	$T^{2} - 156T + 8112$ T^2 - 156*T + 8112
$29$	$(T + 225)^{2}$ (T + 225)^2
$31$	$T^{2} + 11T + 121$ T^2 + 11*T + 121
$37$	$T^{2} + 358T + 128164$ T^2 + 358*T + 128164
$41$	$T^{2} + 20172$ T^2 + 20172
$43$	$T^{2} + 158700$ T^2 + 158700
$47$	$T^{2} - 192T + 36864$ T^2 - 192*T + 36864
$53$	$T^{2} + 417T + 173889$ T^2 + 417*T + 173889
$59$	$T^{2} - 561T + 314721$ T^2 - 561*T + 314721
$61$	$T^{2} + 924T + 284592$ T^2 + 924*T + 284592
$67$	$T^{2} - 168T + 9408$ T^2 - 168*T + 9408
$71$	$T^{2} + 702768$ T^2 + 702768
$73$	$T^{2} + 36T + 432$ T^2 + 36*T + 432
$79$	$T^{2} + 33T + 363$ T^2 + 33*T + 363
$83$	$(T + 693)^{2}$ (T + 693)^2
$89$	$T^{2} + 1770 T + 1044300$ T^2 + 1770*T + 1044300
$97$	$T^{2} + 676875$ T^2 + 676875
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