LMFDB - Newform orbit 126.6.g.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,6,Mod(37,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.37"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	126.g (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-4,0,-16,86] 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:	$20.2083612964$
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 42)
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 + (4 \zeta_{6} - 4) q^{2} - 16 \zeta_{6} q^{4} + ( - 86 \zeta_{6} + 86) q^{5} + (147 \zeta_{6} - 49) q^{7} + 64 q^{8} + 344 \zeta_{6} q^{10} + 34 \zeta_{6} q^{11} - 3 q^{13} + ( - 196 \zeta_{6} - 392) q^{14} + \cdots + ( - 76832 \zeta_{6} + 48020) q^{98} +O(q^{100}) $ q + (4z - 4) q^2 - 16z q^4 + (-86z + 86) q^5 + (147z - 49) q^7 + 64 * q^8 + 344z q^10 + 34z q^11 - 3 * q^13 + (-196z - 392) q^14 + (256z - 256) q^16 - 1904z q^17 + (-1489z + 1489) q^19 - 1376 * q^20 - 136 * q^22 + (224z - 224) q^23 - 4271z q^25 + (-12z + 12) q^26 + (-1568z + 2352) q^28 + 6508 * q^29 - 1731z q^31 - 1024z q^32 + 7616 * q^34 + (4214z + 8428) q^35 + (-7633z + 7633) q^37 + 5956z q^38 + (-5504z + 5504) q^40 - 15414 * q^41 + 18491 * q^43 + (-544z + 544) q^44 - 896z q^46 + (-18462z + 18462) q^47 + (7203z - 19208) q^49 + 17084 * q^50 + 48z q^52 - 19956z q^53 + 2924 * q^55 + (9408z - 3136) q^56 + (26032z - 26032) q^58 - 31828z q^59 + (-57654z + 57654) q^61 + 6924 * q^62 + 4096 * q^64 + (258z - 258) q^65 + 60563z q^67 + (30464z - 30464) q^68 + (33712z - 50568) q^70 + 44834 * q^71 - 20821z q^73 + 30532z q^74 - 23824 * q^76 + (3332z - 4998) q^77 + (-30531z + 30531) q^79 + 22016z q^80 + (-61656z + 61656) q^82 - 110602 * q^83 - 163744 * q^85 + (73964z - 73964) q^86 + 2176z q^88 + (58992z - 58992) q^89 + (-441z + 147) q^91 + 3584 * q^92 + 73848z q^94 - 128054z q^95 - 119846 * q^97 + (-76832z + 48020) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 16 q^{4} + 86 q^{5} + 49 q^{7} + 128 q^{8} + 344 q^{10} + 34 q^{11} - 6 q^{13} - 980 q^{14} - 256 q^{16} - 1904 q^{17} + 1489 q^{19} - 2752 q^{20} - 272 q^{22} - 224 q^{23} - 4271 q^{25}+ \cdots + 19208 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 - 16 * q^4 + 86 * q^5 + 49 * q^7 + 128 * q^8 + 344 * q^10 + 34 * q^11 - 6 * q^13 - 980 * q^14 - 256 * q^16 - 1904 * q^17 + 1489 * q^19 - 2752 * q^20 - 272 * q^22 - 224 * q^23 - 4271 * q^25 + 12 * q^26 + 3136 * q^28 + 13016 * q^29 - 1731 * q^31 - 1024 * q^32 + 15232 * q^34 + 21070 * q^35 + 7633 * q^37 + 5956 * q^38 + 5504 * q^40 - 30828 * q^41 + 36982 * q^43 + 544 * q^44 - 896 * q^46 + 18462 * q^47 - 31213 * q^49 + 34168 * q^50 + 48 * q^52 - 19956 * q^53 + 5848 * q^55 + 3136 * q^56 - 26032 * q^58 - 31828 * q^59 + 57654 * q^61 + 13848 * q^62 + 8192 * q^64 - 258 * q^65 + 60563 * q^67 - 30464 * q^68 - 67424 * q^70 + 89668 * q^71 - 20821 * q^73 + 30532 * q^74 - 47648 * q^76 - 6664 * q^77 + 30531 * q^79 + 22016 * q^80 + 61656 * q^82 - 221204 * q^83 - 327488 * q^85 - 73964 * q^86 + 2176 * q^88 - 58992 * q^89 - 147 * q^91 + 7168 * q^92 + 73848 * q^94 - 128054 * q^95 - 239692 * q^97 + 19208 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$.

$n$	$29$	$73$
$\chi(n)$	$1$	$-1 + \zeta_{6}$

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} $

37.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−2.00000

3.46410i

−8.00000

−

13.8564i

43.0000

−

74.4782i

24.5000

127.306i

64.0000

172.000

297.913i

109.1

−2.00000

−

3.46410i

−8.00000

13.8564i

43.0000

74.4782i

24.5000

−

127.306i

64.0000

172.000

−

297.913i

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	126.6.g.b		2
3.b	odd	2	1		42.6.e.b	✓	2
7.c	even	3	1	inner	126.6.g.b		2
7.c	even	3	1		882.6.a.m		1
7.d	odd	6	1		882.6.a.y		1
12.b	even	2	1		336.6.q.a		2
21.c	even	2	1		294.6.e.m		2
21.g	even	6	1		294.6.a.e		1
21.g	even	6	1		294.6.e.m		2
21.h	odd	6	1		42.6.e.b	✓	2
21.h	odd	6	1		294.6.a.d		1
84.n	even	6	1		336.6.q.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.e.b	✓	2	3.b	odd	2	1
42.6.e.b	✓	2	21.h	odd	6	1
126.6.g.b		2	1.a	even	1	1	trivial
126.6.g.b		2	7.c	even	3	1	inner
294.6.a.d		1	21.h	odd	6	1
294.6.a.e		1	21.g	even	6	1
294.6.e.m		2	21.c	even	2	1
294.6.e.m		2	21.g	even	6	1
336.6.q.a		2	12.b	even	2	1
336.6.q.a		2	84.n	even	6	1
882.6.a.m		1	7.c	even	3	1
882.6.a.y		1	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{2} - 86T_{5} + 7396 $ T5^2 - 86*T5 + 7396 acting on $S_{6}^{\mathrm{new}}(126, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 4T + 16 $ T^2 + 4*T + 16
$3$	$ T^{2} $ T^2
$5$	$ T^{2} - 86T + 7396 $ T^2 - 86*T + 7396
$7$	$ T^{2} - 49T + 16807 $ T^2 - 49*T + 16807
$11$	$ T^{2} - 34T + 1156 $ T^2 - 34*T + 1156
$13$	$ (T + 3)^{2} $ (T + 3)^2
$17$	$ T^{2} + 1904 T + 3625216 $ T^2 + 1904*T + 3625216
$19$	$ T^{2} - 1489 T + 2217121 $ T^2 - 1489*T + 2217121
$23$	$ T^{2} + 224T + 50176 $ T^2 + 224*T + 50176
$29$	$ (T - 6508)^{2} $ (T - 6508)^2
$31$	$ T^{2} + 1731 T + 2996361 $ T^2 + 1731*T + 2996361
$37$	$ T^{2} - 7633 T + 58262689 $ T^2 - 7633*T + 58262689
$41$	$ (T + 15414)^{2} $ (T + 15414)^2
$43$	$ (T - 18491)^{2} $ (T - 18491)^2
$47$	$ T^{2} - 18462 T + 340845444 $ T^2 - 18462*T + 340845444
$53$	$ T^{2} + 19956 T + 398241936 $ T^2 + 19956*T + 398241936
$59$	$ T^{2} + \cdots + 1013021584 $ T^2 + 31828*T + 1013021584
$61$	$ T^{2} + \cdots + 3323983716 $ T^2 - 57654*T + 3323983716
$67$	$ T^{2} + \cdots + 3667876969 $ T^2 - 60563*T + 3667876969
$71$	$ (T - 44834)^{2} $ (T - 44834)^2
$73$	$ T^{2} + 20821 T + 433514041 $ T^2 + 20821*T + 433514041
$79$	$ T^{2} - 30531 T + 932141961 $ T^2 - 30531*T + 932141961
$83$	$ (T + 110602)^{2} $ (T + 110602)^2
$89$	$ T^{2} + \cdots + 3480056064 $ T^2 + 58992*T + 3480056064
$97$	$ (T + 119846)^{2} $ (T + 119846)^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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	126.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (4 \zeta_{6} - 4) q^{2} - 16 \zeta_{6} q^{4} + ( - 86 \zeta_{6} + 86) q^{5} + (147 \zeta_{6} - 49) q^{7} + 64 q^{8} + 344 \zeta_{6} q^{10} + 34 \zeta_{6} q^{11} - 3 q^{13} + ( - 196 \zeta_{6} - 392) q^{14} + \cdots + ( - 76832 \zeta_{6} + 48020) q^{98} +O(q^{100}) \) q + (4z - 4) q^2 - 16z q^4 + (-86z + 86) q^5 + (147z - 49) q^7 + 64 * q^8 + 344z q^10 + 34z q^11 - 3 * q^13 + (-196z - 392) q^14 + (256z - 256) q^16 - 1904z q^17 + (-1489z + 1489) q^19 - 1376 * q^20 - 136 * q^22 + (224z - 224) q^23 - 4271z q^25 + (-12z + 12) q^26 + (-1568z + 2352) q^28 + 6508 * q^29 - 1731z q^31 - 1024z q^32 + 7616 * q^34 + (4214z + 8428) q^35 + (-7633z + 7633) q^37 + 5956z q^38 + (-5504z + 5504) q^40 - 15414 * q^41 + 18491 * q^43 + (-544z + 544) q^44 - 896z q^46 + (-18462z + 18462) q^47 + (7203z - 19208) q^49 + 17084 * q^50 + 48z q^52 - 19956z q^53 + 2924 * q^55 + (9408z - 3136) q^56 + (26032z - 26032) q^58 - 31828z q^59 + (-57654z + 57654) q^61 + 6924 * q^62 + 4096 * q^64 + (258z - 258) q^65 + 60563z q^67 + (30464z - 30464) q^68 + (33712z - 50568) q^70 + 44834 * q^71 - 20821z q^73 + 30532z q^74 - 23824 * q^76 + (3332z - 4998) q^77 + (-30531z + 30531) q^79 + 22016z q^80 + (-61656z + 61656) q^82 - 110602 * q^83 - 163744 * q^85 + (73964z - 73964) q^86 + 2176z q^88 + (58992z - 58992) q^89 + (-441z + 147) q^91 + 3584 * q^92 + 73848z q^94 - 128054z q^95 - 119846 * q^97 + (-76832z + 48020) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 16 q^{4} + 86 q^{5} + 49 q^{7} + 128 q^{8} + 344 q^{10} + 34 q^{11} - 6 q^{13} - 980 q^{14} - 256 q^{16} - 1904 q^{17} + 1489 q^{19} - 2752 q^{20} - 272 q^{22} - 224 q^{23} - 4271 q^{25}+ \cdots + 19208 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 - 16 * q^4 + 86 * q^5 + 49 * q^7 + 128 * q^8 + 344 * q^10 + 34 * q^11 - 6 * q^13 - 980 * q^14 - 256 * q^16 - 1904 * q^17 + 1489 * q^19 - 2752 * q^20 - 272 * q^22 - 224 * q^23 - 4271 * q^25 + 12 * q^26 + 3136 * q^28 + 13016 * q^29 - 1731 * q^31 - 1024 * q^32 + 15232 * q^34 + 21070 * q^35 + 7633 * q^37 + 5956 * q^38 + 5504 * q^40 - 30828 * q^41 + 36982 * q^43 + 544 * q^44 - 896 * q^46 + 18462 * q^47 - 31213 * q^49 + 34168 * q^50 + 48 * q^52 - 19956 * q^53 + 5848 * q^55 + 3136 * q^56 - 26032 * q^58 - 31828 * q^59 + 57654 * q^61 + 13848 * q^62 + 8192 * q^64 - 258 * q^65 + 60563 * q^67 - 30464 * q^68 - 67424 * q^70 + 89668 * q^71 - 20821 * q^73 + 30532 * q^74 - 47648 * q^76 - 6664 * q^77 + 30531 * q^79 + 22016 * q^80 + 61656 * q^82 - 221204 * q^83 - 327488 * q^85 - 73964 * q^86 + 2176 * q^88 - 58992 * q^89 - 147 * q^91 + 7168 * q^92 + 73848 * q^94 - 128054 * q^95 - 239692 * q^97 + 19208 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more