LMFDB - Newform orbit 36.9.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,9,Mod(19,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("36.19");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 36 = 2^{2} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	36.d (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:	$14.6656299622$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4)
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} + 1054 q^{5} - 4096 q^{8}+O(q^{10}) $

$ q - 16 q^{2} + 256 q^{4} + 1054 q^{5} - 4096 q^{8} - 16864 q^{10} - 478 q^{13} + 65536 q^{16} + 63358 q^{17} + 269824 q^{20} + 720291 q^{25} + 7648 q^{26} + 1407838 q^{29} - 1048576 q^{32} - 1013728 q^{34} + 925922 q^{37} - 4317184 q^{40} - 3577922 q^{41} + 5764801 q^{49} - 11524656 q^{50} - 122368 q^{52} + 9620638 q^{53} - 22525408 q^{58} + 20722082 q^{61} + 16777216 q^{64} - 503812 q^{65} + 16219648 q^{68} - 54717118 q^{73} - 14814752 q^{74} + 69074944 q^{80} + 57246752 q^{82} + 66779332 q^{85} + 30265918 q^{89} - 173379838 q^{97} - 92236816 q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$.

$n$	$19$	$29$
$\chi(n)$	$-1$	$1$

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

1054.00

−4096.00

−16864.0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	36.9.d.a		1
3.b	odd	2	1		4.9.b.a	✓	1
4.b	odd	2	1	CM	36.9.d.a		1
12.b	even	2	1		4.9.b.a	✓	1
15.d	odd	2	1		100.9.b.a		1
15.e	even	4	2		100.9.d.a		2
24.f	even	2	1		64.9.c.a		1
24.h	odd	2	1		64.9.c.a		1
48.i	odd	4	2		256.9.d.a		2
48.k	even	4	2		256.9.d.a		2
60.h	even	2	1		100.9.b.a		1
60.l	odd	4	2		100.9.d.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.9.b.a	✓	1	3.b	odd	2	1
4.9.b.a	✓	1	12.b	even	2	1
36.9.d.a		1	1.a	even	1	1	trivial
36.9.d.a		1	4.b	odd	2	1	CM
64.9.c.a		1	24.f	even	2	1
64.9.c.a		1	24.h	odd	2	1
100.9.b.a		1	15.d	odd	2	1
100.9.b.a		1	60.h	even	2	1
100.9.d.a		2	15.e	even	4	2
100.9.d.a		2	60.l	odd	4	2
256.9.d.a		2	48.i	odd	4	2
256.9.d.a		2	48.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 1054 $ T5 - 1054 acting on $S_{9}^{\mathrm{new}}(36, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 16 $ T + 16
$3$	$ T $ T
$5$	$ T - 1054 $ T - 1054
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T + 478 $ T + 478
$17$	$ T - 63358 $ T - 63358
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T - 1407838 $ T - 1407838
$31$	$ T $ T
$37$	$ T - 925922 $ T - 925922
$41$	$ T + 3577922 $ T + 3577922
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T - 9620638 $ T - 9620638
$59$	$ T $ T
$61$	$ T - 20722082 $ T - 20722082
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T + 54717118 $ T + 54717118
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T - 30265918 $ T - 30265918
$97$	$ T + 173379838 $ T + 173379838
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Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	36.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(19\)	\(29\)
\(\chi(n)\)	\(-1\)	\(1\)

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