LMFDB - Newform orbit 72.11.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,11,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, 11, names="a")

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

chi := DirichletCharacter("72.19");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 11 $
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:	$45.7457221925$
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 + 32 q^{2} + 1024 q^{4} + 32768 q^{8}+O(q^{10}) $

$ q + 32 q^{2} + 1024 q^{4} + 32768 q^{8} + 97426 q^{11} + 1048576 q^{16} - 823682 q^{17} + 3353726 q^{19} + 3117632 q^{22} + 9765625 q^{25} + 33554432 q^{32} - 26357824 q^{34} + 107319232 q^{38} + 37778926 q^{41} + 214485614 q^{43} + 99764224 q^{44} + 282475249 q^{49} + 312500000 q^{50} - 921043598 q^{59} + 1073741824 q^{64} + 1813708382 q^{67} - 843450368 q^{68} - 1605781582 q^{73} + 3434215424 q^{76} + 1208925632 q^{82} - 96051518 q^{83} + 6863539648 q^{86} + 3192455168 q^{88} + 11116019374 q^{89} - 9872978014 q^{97} + 9039207968 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

32.0000

1024.00

32768.0

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	72.11.b.a		1
3.b	odd	2	1		8.11.d.a	✓	1
4.b	odd	2	1		288.11.b.a		1
8.b	even	2	1		288.11.b.a		1
8.d	odd	2	1	CM	72.11.b.a		1
12.b	even	2	1		32.11.d.a		1
24.f	even	2	1		8.11.d.a	✓	1
24.h	odd	2	1		32.11.d.a		1
48.i	odd	4	2		256.11.c.c		2
48.k	even	4	2		256.11.c.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.11.d.a	✓	1	3.b	odd	2	1
8.11.d.a	✓	1	24.f	even	2	1
32.11.d.a		1	12.b	even	2	1
32.11.d.a		1	24.h	odd	2	1
72.11.b.a		1	1.a	even	1	1	trivial
72.11.b.a		1	8.d	odd	2	1	CM
256.11.c.c		2	48.i	odd	4	2
256.11.c.c		2	48.k	even	4	2
288.11.b.a		1	4.b	odd	2	1
288.11.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 $S_{11}^{\mathrm{new}}(72, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 32 $ T - 32
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 97426 $ T - 97426
$13$	$ T $ T
$17$	$ T + 823682 $ T + 823682
$19$	$ T - 3353726 $ T - 3353726
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T $ T
$37$	$ T $ T
$41$	$ T - 37778926 $ T - 37778926
$43$	$ T - 214485614 $ T - 214485614
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T + 921043598 $ T + 921043598
$61$	$ T $ T
$67$	$ T - 1813708382 $ T - 1813708382
$71$	$ T $ T
$73$	$ T + 1605781582 $ T + 1605781582
$79$	$ T $ T
$83$	$ T + 96051518 $ T + 96051518
$89$	$ T - 11116019374 $ T - 11116019374
$97$	$ T + 9872978014 $ T + 9872978014
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	72.b (of order \(2\), degree \(1\), minimal)

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

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