LMFDB - Newform orbit 9.16.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9,16,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	9.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-216] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.8424154590$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1)
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 - 216 q^{2} + 13888 q^{4} - 52110 q^{5} + 2822456 q^{7} + 4078080 q^{8} + 11255760 q^{10} - 20586852 q^{11} - 190073338 q^{13} - 609650496 q^{14} - 1335947264 q^{16} - 1646527986 q^{17} + 1563257180 q^{19}+ \cdots - 695238414190488 q^{98}+O(q^{100}) $

q - 216 * q^2 + 13888 * q^4 - 52110 * q^5 + 2822456 * q^7 + 4078080 * q^8 + 11255760 * q^10 - 20586852 * q^11 - 190073338 * q^13 - 609650496 * q^14 - 1335947264 * q^16 - 1646527986 * q^17 + 1563257180 * q^19 - 723703680 * q^20 + 4446760032 * q^22 - 9451116072 * q^23 - 27802126025 * q^25 + 41055841008 * q^26 + 39198268928 * q^28 + 36902568330 * q^29 + 71588483552 * q^31 + 154934083584 * q^32 + 355650044976 * q^34 - 147078182160 * q^35 - 1033652081554 * q^37 - 337663550880 * q^38 - 212508748800 * q^40 - 1641974018202 * q^41 - 492403109308 * q^43 - 285910200576 * q^44 + 2041441071552 * q^46 + 3410684952624 * q^47 + 3218696361993 * q^49 + 6005259221400 * q^50 - 2639738518144 * q^52 - 6797151655902 * q^53 + 1072780857720 * q^55 + 11510201364480 * q^56 - 7970954759280 * q^58 - 9858856815540 * q^59 + 4931842626902 * q^61 - 15463112447232 * q^62 + 10310557892608 * q^64 + 9904721643180 * q^65 - 28837826625364 * q^67 - 22866980669568 * q^68 + 31768887346560 * q^70 - 125050114914552 * q^71 - 82171455513478 * q^73 + 223268849615664 * q^74 + 21710515715840 * q^76 - 58105483948512 * q^77 - 25413078694480 * q^79 + 69616211927040 * q^80 + 354666387931632 * q^82 + 281736730890468 * q^83 + 85800573350460 * q^85 + 106359071610528 * q^86 - 83954829404160 * q^88 - 715618564776810 * q^89 - 536473633278128 * q^91 - 131257100007936 * q^92 - 736707949766784 * q^94 - 81461331649800 * q^95 + 612786136081826 * q^97 - 695238414190488 * q^98

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

−216.000

13888.0

−52110.0

2.82246e6

4.07808e6

1.12558e7

Atkin-Lehner signs

$ p $	Sign
$3$	$ -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	9.16.a.a		1
3.b	odd	2	1		1.16.a.a	✓	1
4.b	odd	2	1		144.16.a.f		1
12.b	even	2	1		16.16.a.d		1
15.d	odd	2	1		25.16.a.a		1
15.e	even	4	2		25.16.b.a		2
21.c	even	2	1		49.16.a.a		1
21.g	even	6	2		49.16.c.b		2
21.h	odd	6	2		49.16.c.c		2
24.f	even	2	1		64.16.a.c		1
24.h	odd	2	1		64.16.a.i		1
33.d	even	2	1		121.16.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1.16.a.a	✓	1	3.b	odd	2	1
9.16.a.a		1	1.a	even	1	1	trivial
16.16.a.d		1	12.b	even	2	1
25.16.a.a		1	15.d	odd	2	1
25.16.b.a		2	15.e	even	4	2
49.16.a.a		1	21.c	even	2	1
49.16.c.b		2	21.g	even	6	2
49.16.c.c		2	21.h	odd	6	2
64.16.a.c		1	24.f	even	2	1
64.16.a.i		1	24.h	odd	2	1
121.16.a.a		1	33.d	even	2	1
144.16.a.f		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} + 216 $ T2 + 216 acting on $S_{16}^{\mathrm{new}}(\Gamma_0(9))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 216 $ T + 216
$3$	$ T $ T
$5$	$ T + 52110 $ T + 52110
$7$	$ T - 2822456 $ T - 2822456
$11$	$ T + 20586852 $ T + 20586852
$13$	$ T + 190073338 $ T + 190073338
$17$	$ T + 1646527986 $ T + 1646527986
$19$	$ T - 1563257180 $ T - 1563257180
$23$	$ T + 9451116072 $ T + 9451116072
$29$	$ T - 36902568330 $ T - 36902568330
$31$	$ T - 71588483552 $ T - 71588483552
$37$	$ T + 1033652081554 $ T + 1033652081554
$41$	$ T + 1641974018202 $ T + 1641974018202
$43$	$ T + 492403109308 $ T + 492403109308
$47$	$ T - 3410684952624 $ T - 3410684952624
$53$	$ T + 6797151655902 $ T + 6797151655902
$59$	$ T + 9858856815540 $ T + 9858856815540
$61$	$ T - 4931842626902 $ T - 4931842626902
$67$	$ T + 28837826625364 $ T + 28837826625364
$71$	$ T + 125050114914552 $ T + 125050114914552
$73$	$ T + 82171455513478 $ T + 82171455513478
$79$	$ T + 25413078694480 $ T + 25413078694480
$83$	$ T - 281736730890468 $ T - 281736730890468
$89$	$ T + 715618564776810 $ T + 715618564776810
$97$	$ T - 612786136081826 $ T - 612786136081826
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Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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