LMFDB - Newform orbit 72.9.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,9,Mod(19,72)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("72.19");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	72.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$29.3312599244$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - 16 q^{2} + 256 q^{4} - 4096 q^{8}+O(q^{10}) $

$ q - 16 q^{2} + 256 q^{4} - 4096 q^{8} + 27166 q^{11} + 65536 q^{16} - 162434 q^{17} - 72286 q^{19} - 434656 q^{22} + 390625 q^{25} - 1048576 q^{32} + 2598944 q^{34} + 1156576 q^{38} + 4099006 q^{41} + 5426402 q^{43} + 6954496 q^{44} + 5764801 q^{49} - 6250000 q^{50} + 24178078 q^{59} + 16777216 q^{64} - 13944286 q^{67} - 41583104 q^{68} + 33567554 q^{73} - 18505216 q^{76} - 65584096 q^{82} - 30209954 q^{83} - 86822432 q^{86} - 111271936 q^{88} + 95519806 q^{89} - 77418238 q^{97} - 92236816 q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$1$

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} $

19.1

−16.0000

256.000

−4096.00

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	72.9.b.a		1
3.b	odd	2	1		8.9.d.a	✓	1
4.b	odd	2	1		288.9.b.a		1
8.b	even	2	1		288.9.b.a		1
8.d	odd	2	1	CM	72.9.b.a		1
12.b	even	2	1		32.9.d.a		1
24.f	even	2	1		8.9.d.a	✓	1
24.h	odd	2	1		32.9.d.a		1
48.i	odd	4	2		256.9.c.f		2
48.k	even	4	2		256.9.c.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.9.d.a	✓	1	3.b	odd	2	1
8.9.d.a	✓	1	24.f	even	2	1
32.9.d.a		1	12.b	even	2	1
32.9.d.a		1	24.h	odd	2	1
72.9.b.a		1	1.a	even	1	1	trivial
72.9.b.a		1	8.d	odd	2	1	CM
256.9.c.f		2	48.i	odd	4	2
256.9.c.f		2	48.k	even	4	2
288.9.b.a		1	4.b	odd	2	1
288.9.b.a		1	8.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{9}^{\mathrm{new}}(72, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 16 $ T + 16
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 27166 $ T - 27166
$13$	$ T $ T
$17$	$ T + 162434 $ T + 162434
$19$	$ T + 72286 $ T + 72286
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T $ T
$37$	$ T $ T
$41$	$ T - 4099006 $ T - 4099006
$43$	$ T - 5426402 $ T - 5426402
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T - 24178078 $ T - 24178078
$61$	$ T $ T
$67$	$ T + 13944286 $ T + 13944286
$71$	$ T $ T
$73$	$ T - 33567554 $ T - 33567554
$79$	$ T $ T
$83$	$ T + 30209954 $ T + 30209954
$89$	$ T - 95519806 $ T - 95519806
$97$	$ T + 77418238 $ T + 77418238
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	72.b (of order \(2\), degree \(1\), minimal)

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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