LMFDB - Newform orbit 1050.6.g.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1050,6,Mod(799,1050)] 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("1050.799"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1050.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-32,0,-72,0,0,-162,0,-828] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$168.403010804$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 4 i q^{2} + 9 i q^{3} - 16 q^{4} - 36 q^{6} - 49 i q^{7} - 64 i q^{8} - 81 q^{9} - 414 q^{11} - 144 i q^{12} - 1054 i q^{13} + 196 q^{14} + 256 q^{16} + 1848 i q^{17} - 324 i q^{18} - 236 q^{19} + 441 q^{21} + \cdots + 33534 q^{99} +O(q^{100}) $ q + 4i q^2 + 9i q^3 - 16 * q^4 - 36 * q^6 - 49i q^7 - 64i q^8 - 81 * q^9 - 414 * q^11 - 144i q^12 - 1054i q^13 + 196 * q^14 + 256 * q^16 + 1848i q^17 - 324i q^18 - 236 * q^19 + 441 * q^21 - 1656i q^22 + 2898i q^23 + 576 * q^24 + 4216 * q^26 - 729i q^27 + 784i q^28 + 6522 * q^29 + 6200 * q^31 + 1024i q^32 - 3726i q^33 - 7392 * q^34 + 1296 * q^36 - 9650i q^37 - 944i q^38 + 9486 * q^39 + 8484 * q^41 + 1764i q^42 - 10804i q^43 + 6624 * q^44 - 11592 * q^46 - 60i q^47 + 2304i q^48 - 2401 * q^49 - 16632 * q^51 + 16864i q^52 + 22506i q^53 + 2916 * q^54 - 3136 * q^56 - 2124i q^57 + 26088i q^58 + 28176 * q^59 - 35194 * q^61 + 24800i q^62 + 3969i q^63 - 4096 * q^64 + 14904 * q^66 + 28216i q^67 - 29568i q^68 - 26082 * q^69 - 6642 * q^71 + 5184i q^72 - 52090i q^73 + 38600 * q^74 + 3776 * q^76 + 20286i q^77 + 37944i q^78 - 43340 * q^79 + 6561 * q^81 + 33936i q^82 + 25716i q^83 - 7056 * q^84 + 43216 * q^86 + 58698i q^87 + 26496i q^88 - 98724 * q^89 - 51646 * q^91 - 46368i q^92 + 55800i q^93 + 240 * q^94 - 9216 * q^96 + 148954i q^97 - 9604i q^98 + 33534 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} - 72 q^{6} - 162 q^{9} - 828 q^{11} + 392 q^{14} + 512 q^{16} - 472 q^{19} + 882 q^{21} + 1152 q^{24} + 8432 q^{26} + 13044 q^{29} + 12400 q^{31} - 14784 q^{34} + 2592 q^{36} + 18972 q^{39}+ \cdots + 67068 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9 - 828 * q^11 + 392 * q^14 + 512 * q^16 - 472 * q^19 + 882 * q^21 + 1152 * q^24 + 8432 * q^26 + 13044 * q^29 + 12400 * q^31 - 14784 * q^34 + 2592 * q^36 + 18972 * q^39 + 16968 * q^41 + 13248 * q^44 - 23184 * q^46 - 4802 * q^49 - 33264 * q^51 + 5832 * q^54 - 6272 * q^56 + 56352 * q^59 - 70388 * q^61 - 8192 * q^64 + 29808 * q^66 - 52164 * q^69 - 13284 * q^71 + 77200 * q^74 + 7552 * q^76 - 86680 * q^79 + 13122 * q^81 - 14112 * q^84 + 86432 * q^86 - 197448 * q^89 - 103292 * q^91 + 480 * q^94 - 18432 * q^96 + 67068 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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} $

799.1

	−	1.00000i
		1.00000i

−

4.00000i

−

9.00000i

−16.0000

−36.0000

49.0000i

64.0000i

−81.0000

799.2

4.00000i

9.00000i

−16.0000

−36.0000

−

49.0000i

−

64.0000i

−81.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1050.6.g.b		2
5.b	even	2	1	inner	1050.6.g.b		2
5.c	odd	4	1		42.6.a.c	✓	1
5.c	odd	4	1		1050.6.a.g		1
15.e	even	4	1		126.6.a.l		1
20.e	even	4	1		336.6.a.b		1
35.f	even	4	1		294.6.a.c		1
35.k	even	12	2		294.6.e.n		2
35.l	odd	12	2		294.6.e.l		2
60.l	odd	4	1		1008.6.a.ba		1
105.k	odd	4	1		882.6.a.n		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.a.c	✓	1	5.c	odd	4	1
126.6.a.l		1	15.e	even	4	1
294.6.a.c		1	35.f	even	4	1
294.6.e.l		2	35.l	odd	12	2
294.6.e.n		2	35.k	even	12	2
336.6.a.b		1	20.e	even	4	1
882.6.a.n		1	105.k	odd	4	1
1008.6.a.ba		1	60.l	odd	4	1
1050.6.a.g		1	5.c	odd	4	1
1050.6.g.b		2	1.a	even	1	1	trivial
1050.6.g.b		2	5.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11} + 414 $ T11 + 414 acting on $S_{6}^{\mathrm{new}}(1050, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 16 $ T^2 + 16
$3$	$ T^{2} + 81 $ T^2 + 81
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 2401 $ T^2 + 2401
$11$	$ (T + 414)^{2} $ (T + 414)^2
$13$	$ T^{2} + 1110916 $ T^2 + 1110916
$17$	$ T^{2} + 3415104 $ T^2 + 3415104
$19$	$ (T + 236)^{2} $ (T + 236)^2
$23$	$ T^{2} + 8398404 $ T^2 + 8398404
$29$	$ (T - 6522)^{2} $ (T - 6522)^2
$31$	$ (T - 6200)^{2} $ (T - 6200)^2
$37$	$ T^{2} + 93122500 $ T^2 + 93122500
$41$	$ (T - 8484)^{2} $ (T - 8484)^2
$43$	$ T^{2} + 116726416 $ T^2 + 116726416
$47$	$ T^{2} + 3600 $ T^2 + 3600
$53$	$ T^{2} + 506520036 $ T^2 + 506520036
$59$	$ (T - 28176)^{2} $ (T - 28176)^2
$61$	$ (T + 35194)^{2} $ (T + 35194)^2
$67$	$ T^{2} + 796142656 $ T^2 + 796142656
$71$	$ (T + 6642)^{2} $ (T + 6642)^2
$73$	$ T^{2} + 2713368100 $ T^2 + 2713368100
$79$	$ (T + 43340)^{2} $ (T + 43340)^2
$83$	$ T^{2} + 661312656 $ T^2 + 661312656
$89$	$ (T + 98724)^{2} $ (T + 98724)^2
$97$	$ T^{2} + 22187294116 $ T^2 + 22187294116
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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 4 i q^{2} + 9 i q^{3} - 16 q^{4} - 36 q^{6} - 49 i q^{7} - 64 i q^{8} - 81 q^{9} - 414 q^{11} - 144 i q^{12} - 1054 i q^{13} + 196 q^{14} + 256 q^{16} + 1848 i q^{17} - 324 i q^{18} - 236 q^{19} + 441 q^{21} + \cdots + 33534 q^{99} +O(q^{100}) \) q + 4i q^2 + 9i q^3 - 16 * q^4 - 36 * q^6 - 49i q^7 - 64i q^8 - 81 * q^9 - 414 * q^11 - 144i q^12 - 1054i q^13 + 196 * q^14 + 256 * q^16 + 1848i q^17 - 324i q^18 - 236 * q^19 + 441 * q^21 - 1656i q^22 + 2898i q^23 + 576 * q^24 + 4216 * q^26 - 729i q^27 + 784i q^28 + 6522 * q^29 + 6200 * q^31 + 1024i q^32 - 3726i q^33 - 7392 * q^34 + 1296 * q^36 - 9650i q^37 - 944i q^38 + 9486 * q^39 + 8484 * q^41 + 1764i q^42 - 10804i q^43 + 6624 * q^44 - 11592 * q^46 - 60i q^47 + 2304i q^48 - 2401 * q^49 - 16632 * q^51 + 16864i q^52 + 22506i q^53 + 2916 * q^54 - 3136 * q^56 - 2124i q^57 + 26088i q^58 + 28176 * q^59 - 35194 * q^61 + 24800i q^62 + 3969i q^63 - 4096 * q^64 + 14904 * q^66 + 28216i q^67 - 29568i q^68 - 26082 * q^69 - 6642 * q^71 + 5184i q^72 - 52090i q^73 + 38600 * q^74 + 3776 * q^76 + 20286i q^77 + 37944i q^78 - 43340 * q^79 + 6561 * q^81 + 33936i q^82 + 25716i q^83 - 7056 * q^84 + 43216 * q^86 + 58698i q^87 + 26496i q^88 - 98724 * q^89 - 51646 * q^91 - 46368i q^92 + 55800i q^93 + 240 * q^94 - 9216 * q^96 + 148954i q^97 - 9604i q^98 + 33534 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} - 72 q^{6} - 162 q^{9} - 828 q^{11} + 392 q^{14} + 512 q^{16} - 472 q^{19} + 882 q^{21} + 1152 q^{24} + 8432 q^{26} + 13044 q^{29} + 12400 q^{31} - 14784 q^{34} + 2592 q^{36} + 18972 q^{39}+ \cdots + 67068 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 - 72 * q^6 - 162 * q^9 - 828 * q^11 + 392 * q^14 + 512 * q^16 - 472 * q^19 + 882 * q^21 + 1152 * q^24 + 8432 * q^26 + 13044 * q^29 + 12400 * q^31 - 14784 * q^34 + 2592 * q^36 + 18972 * q^39 + 16968 * q^41 + 13248 * q^44 - 23184 * q^46 - 4802 * q^49 - 33264 * q^51 + 5832 * q^54 - 6272 * q^56 + 56352 * q^59 - 70388 * q^61 - 8192 * q^64 + 29808 * q^66 - 52164 * q^69 - 13284 * q^71 + 77200 * q^74 + 7552 * q^76 - 86680 * q^79 + 13122 * q^81 - 14112 * q^84 + 86432 * q^86 - 197448 * q^89 - 103292 * q^91 + 480 * q^94 - 18432 * q^96 + 67068 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more