LMFDB - Newform orbit 336.6.q.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,6,Mod(193,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.193"); S:= CuspForms(chi, 6); N := Newforms(S);

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

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

Newform invariants

comment:select newform

sage:traces = [2,0,-9,0,-11] 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:	$53.8889634572$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 + (9 \zeta_{6} - 9) q^{3} - 11 \zeta_{6} q^{5} + (7 \zeta_{6} - 133) q^{7} - 81 \zeta_{6} q^{9} + ( - 269 \zeta_{6} + 269) q^{11} - 308 q^{13} + 99 q^{15} + (1896 \zeta_{6} - 1896) q^{17} - 164 \zeta_{6} q^{19} + \cdots - 21789 q^{99} +O(q^{100}) $ q + (9z - 9) q^3 - 11z q^5 + (7z - 133) q^7 - 81z q^9 + (-269z + 269) q^11 - 308 * q^13 + 99 * q^15 + (1896z - 1896) q^17 - 164z q^19 + (-1197z + 1134) q^21 - 3264z q^23 + (-3004z + 3004) q^25 + 729 * q^27 + 2417 * q^29 + (-2841z + 2841) q^31 + 2421z q^33 + (1386z + 77) q^35 + 11328z q^37 + (-2772z + 2772) q^39 - 16856 * q^41 + 7894 * q^43 + (891z - 891) q^45 + 21102z q^47 + (-1813z + 17640) q^49 - 17064z q^51 + (-29691z + 29691) q^53 - 2959 * q^55 + 1476 * q^57 + (8163z - 8163) q^59 - 15166z q^61 + (10206z + 567) q^63 + 3388z q^65 + (32078z - 32078) q^67 + 29376 * q^69 + 38274 * q^71 + (34866z - 34866) q^73 + 27036z q^75 + (35777z - 33894) q^77 + 13529z q^79 + (6561z - 6561) q^81 + 68103 * q^83 + 20856 * q^85 + (21753z - 21753) q^87 + 114922z q^89 + (-2156z + 40964) q^91 + 25569z q^93 + (1804z - 1804) q^95 + 154959 * q^97 - 21789 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{3} - 11 q^{5} - 259 q^{7} - 81 q^{9} + 269 q^{11} - 616 q^{13} + 198 q^{15} - 1896 q^{17} - 164 q^{19} + 1071 q^{21} - 3264 q^{23} + 3004 q^{25} + 1458 q^{27} + 4834 q^{29} + 2841 q^{31} + 2421 q^{33}+ \cdots - 43578 q^{99}+O(q^{100}) $ 2 * q - 9 * q^3 - 11 * q^5 - 259 * q^7 - 81 * q^9 + 269 * q^11 - 616 * q^13 + 198 * q^15 - 1896 * q^17 - 164 * q^19 + 1071 * q^21 - 3264 * q^23 + 3004 * q^25 + 1458 * q^27 + 4834 * q^29 + 2841 * q^31 + 2421 * q^33 + 1540 * q^35 + 11328 * q^37 + 2772 * q^39 - 33712 * q^41 + 15788 * q^43 - 891 * q^45 + 21102 * q^47 + 33467 * q^49 - 17064 * q^51 + 29691 * q^53 - 5918 * q^55 + 2952 * q^57 - 8163 * q^59 - 15166 * q^61 + 11340 * q^63 + 3388 * q^65 - 32078 * q^67 + 58752 * q^69 + 76548 * q^71 - 34866 * q^73 + 27036 * q^75 - 32011 * q^77 + 13529 * q^79 - 6561 * q^81 + 136206 * q^83 + 41712 * q^85 - 21753 * q^87 + 114922 * q^89 + 79772 * q^91 + 25569 * q^93 - 1804 * q^95 + 309918 * q^97 - 43578 * q^99

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} $

193.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−4.50000

7.79423i

−5.50000

−

9.52628i

−129.500

6.06218i

−40.5000

−

70.1481i

289.1

−4.50000

−

7.79423i

−5.50000

9.52628i

−129.500

−

6.06218i

−40.5000

70.1481i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.6.q.b		2
4.b	odd	2	1		21.6.e.a	✓	2
7.c	even	3	1	inner	336.6.q.b		2
12.b	even	2	1		63.6.e.a		2
28.d	even	2	1		147.6.e.g		2
28.f	even	6	1		147.6.a.d		1
28.f	even	6	1		147.6.e.g		2
28.g	odd	6	1		21.6.e.a	✓	2
28.g	odd	6	1		147.6.a.c		1
84.j	odd	6	1		441.6.a.h		1
84.n	even	6	1		63.6.e.a		2
84.n	even	6	1		441.6.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.6.e.a	✓	2	4.b	odd	2	1
21.6.e.a	✓	2	28.g	odd	6	1
63.6.e.a		2	12.b	even	2	1
63.6.e.a		2	84.n	even	6	1
147.6.a.c		1	28.g	odd	6	1
147.6.a.d		1	28.f	even	6	1
147.6.e.g		2	28.d	even	2	1
147.6.e.g		2	28.f	even	6	1
336.6.q.b		2	1.a	even	1	1	trivial
336.6.q.b		2	7.c	even	3	1	inner
441.6.a.g		1	84.n	even	6	1
441.6.a.h		1	84.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{2} + 11T_{5} + 121 $ T5^2 + 11*T5 + 121 acting on $S_{6}^{\mathrm{new}}(336, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 9T + 81 $ T^2 + 9*T + 81
$5$	$ T^{2} + 11T + 121 $ T^2 + 11*T + 121
$7$	$ T^{2} + 259T + 16807 $ T^2 + 259*T + 16807
$11$	$ T^{2} - 269T + 72361 $ T^2 - 269*T + 72361
$13$	$ (T + 308)^{2} $ (T + 308)^2
$17$	$ T^{2} + 1896 T + 3594816 $ T^2 + 1896*T + 3594816
$19$	$ T^{2} + 164T + 26896 $ T^2 + 164*T + 26896
$23$	$ T^{2} + 3264 T + 10653696 $ T^2 + 3264*T + 10653696
$29$	$ (T - 2417)^{2} $ (T - 2417)^2
$31$	$ T^{2} - 2841 T + 8071281 $ T^2 - 2841*T + 8071281
$37$	$ T^{2} - 11328 T + 128323584 $ T^2 - 11328*T + 128323584
$41$	$ (T + 16856)^{2} $ (T + 16856)^2
$43$	$ (T - 7894)^{2} $ (T - 7894)^2
$47$	$ T^{2} - 21102 T + 445294404 $ T^2 - 21102*T + 445294404
$53$	$ T^{2} - 29691 T + 881555481 $ T^2 - 29691*T + 881555481
$59$	$ T^{2} + 8163 T + 66634569 $ T^2 + 8163*T + 66634569
$61$	$ T^{2} + 15166 T + 230007556 $ T^2 + 15166*T + 230007556
$67$	$ T^{2} + \cdots + 1028998084 $ T^2 + 32078*T + 1028998084
$71$	$ (T - 38274)^{2} $ (T - 38274)^2
$73$	$ T^{2} + \cdots + 1215637956 $ T^2 + 34866*T + 1215637956
$79$	$ T^{2} - 13529 T + 183033841 $ T^2 - 13529*T + 183033841
$83$	$ (T - 68103)^{2} $ (T - 68103)^2
$89$	$ T^{2} + \cdots + 13207066084 $ T^2 - 114922*T + 13207066084
$97$	$ (T - 154959)^{2} $ (T - 154959)^2
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

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

\(f(q)\)	\(=\)	\( q + (9 \zeta_{6} - 9) q^{3} - 11 \zeta_{6} q^{5} + (7 \zeta_{6} - 133) q^{7} - 81 \zeta_{6} q^{9} + ( - 269 \zeta_{6} + 269) q^{11} - 308 q^{13} + 99 q^{15} + (1896 \zeta_{6} - 1896) q^{17} - 164 \zeta_{6} q^{19} + \cdots - 21789 q^{99} +O(q^{100}) \) q + (9z - 9) q^3 - 11z q^5 + (7z - 133) q^7 - 81z q^9 + (-269z + 269) q^11 - 308 * q^13 + 99 * q^15 + (1896z - 1896) q^17 - 164z q^19 + (-1197z + 1134) q^21 - 3264z q^23 + (-3004z + 3004) q^25 + 729 * q^27 + 2417 * q^29 + (-2841z + 2841) q^31 + 2421z q^33 + (1386z + 77) q^35 + 11328z q^37 + (-2772z + 2772) q^39 - 16856 * q^41 + 7894 * q^43 + (891z - 891) q^45 + 21102z q^47 + (-1813z + 17640) q^49 - 17064z q^51 + (-29691z + 29691) q^53 - 2959 * q^55 + 1476 * q^57 + (8163z - 8163) q^59 - 15166z q^61 + (10206z + 567) q^63 + 3388z q^65 + (32078z - 32078) q^67 + 29376 * q^69 + 38274 * q^71 + (34866z - 34866) q^73 + 27036z q^75 + (35777z - 33894) q^77 + 13529z q^79 + (6561z - 6561) q^81 + 68103 * q^83 + 20856 * q^85 + (21753z - 21753) q^87 + 114922z q^89 + (-2156z + 40964) q^91 + 25569z q^93 + (1804z - 1804) q^95 + 154959 * q^97 - 21789 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{3} - 11 q^{5} - 259 q^{7} - 81 q^{9} + 269 q^{11} - 616 q^{13} + 198 q^{15} - 1896 q^{17} - 164 q^{19} + 1071 q^{21} - 3264 q^{23} + 3004 q^{25} + 1458 q^{27} + 4834 q^{29} + 2841 q^{31} + 2421 q^{33}+ \cdots - 43578 q^{99}+O(q^{100}) \) 2 * q - 9 * q^3 - 11 * q^5 - 259 * q^7 - 81 * q^9 + 269 * q^11 - 616 * q^13 + 198 * q^15 - 1896 * q^17 - 164 * q^19 + 1071 * q^21 - 3264 * q^23 + 3004 * q^25 + 1458 * q^27 + 4834 * q^29 + 2841 * q^31 + 2421 * q^33 + 1540 * q^35 + 11328 * q^37 + 2772 * q^39 - 33712 * q^41 + 15788 * q^43 - 891 * q^45 + 21102 * q^47 + 33467 * q^49 - 17064 * q^51 + 29691 * q^53 - 5918 * q^55 + 2952 * q^57 - 8163 * q^59 - 15166 * q^61 + 11340 * q^63 + 3388 * q^65 - 32078 * q^67 + 58752 * q^69 + 76548 * q^71 - 34866 * q^73 + 27036 * q^75 - 32011 * q^77 + 13529 * q^79 - 6561 * q^81 + 136206 * q^83 + 41712 * q^85 - 21753 * q^87 + 114922 * q^89 + 79772 * q^91 + 25569 * q^93 - 1804 * q^95 + 309918 * q^97 - 43578 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more