LMFDB - Newform orbit 228.3.b.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [228,3,Mod(227,228)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(228, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("228.227");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 228 = 2^{2} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	228.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:	$6.21255002741$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 + 2 q^{2} - 3 q^{3} + 4 q^{4} - 6 q^{6} + 8 q^{8} + 9 q^{9}+O(q^{10}) $

$ q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 6 q^{6} + 8 q^{8} + 9 q^{9} + 16 q^{11} - 12 q^{12} + 16 q^{16} + 18 q^{18} + 19 q^{19} + 32 q^{22} - 8 q^{23} - 24 q^{24} + 25 q^{25} - 27 q^{27} - 56 q^{29} + 14 q^{31} + 32 q^{32} - 48 q^{33} + 36 q^{36} + 38 q^{38} - 32 q^{41} + 64 q^{44} - 16 q^{46} - 56 q^{47} - 48 q^{48} + 49 q^{49} + 50 q^{50} - 8 q^{53} - 54 q^{54} - 57 q^{57} - 112 q^{58} - 106 q^{61} + 28 q^{62} + 64 q^{64} - 96 q^{66} - 58 q^{67} + 24 q^{69} + 72 q^{72} - 82 q^{73} - 75 q^{75} + 76 q^{76} + 146 q^{79} + 81 q^{81} - 64 q^{82} - 128 q^{83} + 168 q^{87} + 128 q^{88} + 64 q^{89} - 32 q^{92} - 42 q^{93} - 112 q^{94} - 96 q^{96} + 98 q^{98} + 144 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/228\mathbb{Z}\right)^\times$.

$n$	$77$	$97$	$115$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

227.1

2.00000

−3.00000

4.00000

−6.00000

8.00000

9.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
228.b	odd	2	1	CM by $\Q(\sqrt{-57}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	228.3.b.c	yes	1
3.b	odd	2	1		228.3.b.a	✓	1
4.b	odd	2	1		228.3.b.d	yes	1
12.b	even	2	1		228.3.b.b	yes	1
19.b	odd	2	1		228.3.b.b	yes	1
57.d	even	2	1		228.3.b.d	yes	1
76.d	even	2	1		228.3.b.a	✓	1
228.b	odd	2	1	CM	228.3.b.c	yes	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
228.3.b.a	✓	1	3.b	odd	2	1
228.3.b.a	✓	1	76.d	even	2	1
228.3.b.b	yes	1	12.b	even	2	1
228.3.b.b	yes	1	19.b	odd	2	1
228.3.b.c	yes	1	1.a	even	1	1	trivial
228.3.b.c	yes	1	228.b	odd	2	1	CM
228.3.b.d	yes	1	4.b	odd	2	1
228.3.b.d	yes	1	57.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(228, [\chi])$:

$ T_{5} $

$ T_{11} - 16 $

$ T_{29} + 56 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T + 3 $ T + 3
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 16 $ T - 16
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T - 19 $ T - 19
$23$	$ T + 8 $ T + 8
$29$	$ T + 56 $ T + 56
$31$	$ T - 14 $ T - 14
$37$	$ T $ T
$41$	$ T + 32 $ T + 32
$43$	$ T $ T
$47$	$ T + 56 $ T + 56
$53$	$ T + 8 $ T + 8
$59$	$ T $ T
$61$	$ T + 106 $ T + 106
$67$	$ T + 58 $ T + 58
$71$	$ T $ T
$73$	$ T + 82 $ T + 82
$79$	$ T - 146 $ T - 146
$83$	$ T + 128 $ T + 128
$89$	$ T - 64 $ T - 64
$97$	$ T $ T
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	228.b (of order \(2\), degree \(1\), minimal)

\(n\)	\(77\)	\(97\)	\(115\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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