LMFDB - Newform orbit 22.6.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [22,6,Mod(1,22)] 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("22.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(22, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 22 = 2 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	22.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-4,1] 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:	yes
Analytic conductor:	$3.52844403589$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 4 q^{2} + q^{3} + 16 q^{4} - 51 q^{5} - 4 q^{6} - 166 q^{7} - 64 q^{8} - 242 q^{9} + 204 q^{10} - 121 q^{11} + 16 q^{12} + 692 q^{13} + 664 q^{14} - 51 q^{15} + 256 q^{16} - 738 q^{17} + 968 q^{18}+ \cdots + 29282 q^{99}+O(q^{100}) $

q - 4 * q^2 + q^3 + 16 * q^4 - 51 * q^5 - 4 * q^6 - 166 * q^7 - 64 * q^8 - 242 * q^9 + 204 * q^10 - 121 * q^11 + 16 * q^12 + 692 * q^13 + 664 * q^14 - 51 * q^15 + 256 * q^16 - 738 * q^17 + 968 * q^18 + 1424 * q^19 - 816 * q^20 - 166 * q^21 + 484 * q^22 - 1779 * q^23 - 64 * q^24 - 524 * q^25 - 2768 * q^26 - 485 * q^27 - 2656 * q^28 - 2064 * q^29 + 204 * q^30 + 6245 * q^31 - 1024 * q^32 - 121 * q^33 + 2952 * q^34 + 8466 * q^35 - 3872 * q^36 - 14785 * q^37 - 5696 * q^38 + 692 * q^39 + 3264 * q^40 + 5304 * q^41 + 664 * q^42 + 17798 * q^43 - 1936 * q^44 + 12342 * q^45 + 7116 * q^46 - 17184 * q^47 + 256 * q^48 + 10749 * q^49 + 2096 * q^50 - 738 * q^51 + 11072 * q^52 - 30726 * q^53 + 1940 * q^54 + 6171 * q^55 + 10624 * q^56 + 1424 * q^57 + 8256 * q^58 - 34989 * q^59 - 816 * q^60 - 45940 * q^61 - 24980 * q^62 + 40172 * q^63 + 4096 * q^64 - 35292 * q^65 + 484 * q^66 + 25343 * q^67 - 11808 * q^68 - 1779 * q^69 - 33864 * q^70 + 13311 * q^71 + 15488 * q^72 - 53260 * q^73 + 59140 * q^74 - 524 * q^75 + 22784 * q^76 + 20086 * q^77 - 2768 * q^78 + 77234 * q^79 - 13056 * q^80 + 58321 * q^81 - 21216 * q^82 + 55014 * q^83 - 2656 * q^84 + 37638 * q^85 - 71192 * q^86 - 2064 * q^87 + 7744 * q^88 + 125415 * q^89 - 49368 * q^90 - 114872 * q^91 - 28464 * q^92 + 6245 * q^93 + 68736 * q^94 - 72624 * q^95 - 1024 * q^96 - 88807 * q^97 - 42996 * q^98 + 29282 * q^99

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} $

1.1

−4.00000

1.00000

16.0000

−51.0000

−4.00000

−166.000

−64.0000

−242.000

204.000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$11$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	22.6.a.b	✓	1
3.b	odd	2	1		198.6.a.i		1
4.b	odd	2	1		176.6.a.b		1
5.b	even	2	1		550.6.a.f		1
5.c	odd	4	2		550.6.b.f		2
7.b	odd	2	1		1078.6.a.a		1
8.b	even	2	1		704.6.a.e		1
8.d	odd	2	1		704.6.a.f		1
11.b	odd	2	1		242.6.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
22.6.a.b	✓	1	1.a	even	1	1	trivial
176.6.a.b		1	4.b	odd	2	1
198.6.a.i		1	3.b	odd	2	1
242.6.a.d		1	11.b	odd	2	1
550.6.a.f		1	5.b	even	2	1
550.6.b.f		2	5.c	odd	4	2
704.6.a.e		1	8.b	even	2	1
704.6.a.f		1	8.d	odd	2	1
1078.6.a.a		1	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 1 $ T3 - 1 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(22))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 4 $ T + 4
$3$	$ T - 1 $ T - 1
$5$	$ T + 51 $ T + 51
$7$	$ T + 166 $ T + 166
$11$	$ T + 121 $ T + 121
$13$	$ T - 692 $ T - 692
$17$	$ T + 738 $ T + 738
$19$	$ T - 1424 $ T - 1424
$23$	$ T + 1779 $ T + 1779
$29$	$ T + 2064 $ T + 2064
$31$	$ T - 6245 $ T - 6245
$37$	$ T + 14785 $ T + 14785
$41$	$ T - 5304 $ T - 5304
$43$	$ T - 17798 $ T - 17798
$47$	$ T + 17184 $ T + 17184
$53$	$ T + 30726 $ T + 30726
$59$	$ T + 34989 $ T + 34989
$61$	$ T + 45940 $ T + 45940
$67$	$ T - 25343 $ T - 25343
$71$	$ T - 13311 $ T - 13311
$73$	$ T + 53260 $ T + 53260
$79$	$ T - 77234 $ T - 77234
$83$	$ T - 55014 $ T - 55014
$89$	$ T - 125415 $ T - 125415
$97$	$ T + 88807 $ T + 88807
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	22.a (trivial)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
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