LMFDB - Newform orbit 212.1.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [212,1,Mod(211,212)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(212, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("212.211");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 212 = 2^{2} \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	212.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:	$0.105801782678$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.212.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.179776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8}+O(q^{10}) $

$ q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{12} - q^{13} + q^{16} - q^{17} + q^{19} + q^{23} - q^{24} + q^{25} + q^{26} - q^{27} - q^{29} - 2 q^{31} - q^{32} + q^{34} - q^{37} - q^{38} - q^{39} - q^{46} + q^{48} + q^{49} - q^{50} - q^{51} - q^{52} + q^{53} + q^{54} + q^{57} + q^{58} + 2 q^{62} + q^{64} - 2 q^{67} - q^{68} + q^{69} + q^{71} + q^{74} + q^{75} + q^{76} + q^{78} + q^{79} - q^{81} + q^{83} - q^{87} + 2 q^{89} + q^{92} - 2 q^{93} - q^{96} - q^{97} - q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/212\mathbb{Z}\right)^\times$.

$n$	$107$	$161$
$\chi(n)$	$-1$	$-1$

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} $

211.1

−1.00000

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
212.d	odd	2	1	CM by $\Q(\sqrt{-53}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	212.1.d.a	✓	1
3.b	odd	2	1		1908.1.b.b		1
4.b	odd	2	1		212.1.d.b	yes	1
8.b	even	2	1		3392.1.f.a		1
8.d	odd	2	1		3392.1.f.c		1
12.b	even	2	1		1908.1.b.a		1
53.b	even	2	1		212.1.d.b	yes	1
159.d	odd	2	1		1908.1.b.a		1
212.d	odd	2	1	CM	212.1.d.a	✓	1
424.e	odd	2	1		3392.1.f.a		1
424.h	even	2	1		3392.1.f.c		1
636.g	even	2	1		1908.1.b.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
212.1.d.a	✓	1	1.a	even	1	1	trivial
212.1.d.a	✓	1	212.d	odd	2	1	CM
212.1.d.b	yes	1	4.b	odd	2	1
212.1.d.b	yes	1	53.b	even	2	1
1908.1.b.a		1	12.b	even	2	1
1908.1.b.a		1	159.d	odd	2	1
1908.1.b.b		1	3.b	odd	2	1
1908.1.b.b		1	636.g	even	2	1
3392.1.f.a		1	8.b	even	2	1
3392.1.f.a		1	424.e	odd	2	1
3392.1.f.c		1	8.d	odd	2	1
3392.1.f.c		1	424.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 1 $ T3 - 1 acting on $S_{1}^{\mathrm{new}}(212, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T - 1 $ T - 1
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T + 1 $ T + 1
$17$	$ T + 1 $ T + 1
$19$	$ T - 1 $ T - 1
$23$	$ T - 1 $ T - 1
$29$	$ T + 1 $ T + 1
$31$	$ T + 2 $ T + 2
$37$	$ T + 1 $ T + 1
$41$	$ T $ T
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T - 1 $ T - 1
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T + 2 $ T + 2
$71$	$ T - 1 $ T - 1
$73$	$ T $ T
$79$	$ T - 1 $ T - 1
$83$	$ T - 1 $ T - 1
$89$	$ T - 2 $ T - 2
$97$	$ T + 1 $ T + 1
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Level:	\( N \)	\(=\)	\( 212 = 2^{2} \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	212.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(107\)	\(161\)
\(\chi(n)\)	\(-1\)	\(-1\)

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