LMFDB - Newform orbit 440.1.o.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [440,1,Mod(109,440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("440.109"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	440.o (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$0.219588605559$
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 3.1.440.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.2129600.1
Stark unit:	Root of $x^{6} - 2x^{5} - 37x^{4} - 84x^{3} - 37x^{2} - 2x + 1$

$q$-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^{5} + q^{6} + q^{7} - q^{8} - q^{10} - q^{11} - q^{12} - q^{14} - q^{15} + q^{16} + q^{17} + q^{19} + q^{20} - q^{21} + q^{22} + q^{24} + q^{25} + q^{27}+ \cdots + q^{96}+O(q^{100}) $

q - q^2 - q^3 + q^4 + q^5 + q^6 + q^7 - q^8 - q^10 - q^11 - q^12 - q^14 - q^15 + q^16 + q^17 + q^19 + q^20 - q^21 + q^22 + q^24 + q^25 + q^27 + q^28 + q^29 + q^30 - q^31 - q^32 + q^33 - q^34 + q^35 - q^37 - q^38 - q^40 + q^42 - q^44 - q^48 - q^50 - q^51 - q^53 - q^54 - q^55 - q^56 - q^57 - q^58 - q^60 + q^61 + q^62 + q^64 - q^66 + 2 * q^67 + q^68 - q^70 - q^71 - 2 * q^73 + q^74 - q^75 + q^76 - q^77 + q^80 - q^81 - q^84 + q^85 - q^87 + q^88 - q^89 + q^93 + q^95 + q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$221$	$321$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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} $

109.1

−1.00000

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
440.o	odd	2	1	CM by $\Q(\sqrt{-110}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	440.1.o.a	✓	1
3.b	odd	2	1		3960.1.x.c		1
4.b	odd	2	1		1760.1.o.c		1
5.b	even	2	1		440.1.o.d	yes	1
5.c	odd	4	2		2200.1.d.f		2
8.b	even	2	1		440.1.o.b	yes	1
8.d	odd	2	1		1760.1.o.a		1
11.b	odd	2	1		440.1.o.c	yes	1
15.d	odd	2	1		3960.1.x.b		1
20.d	odd	2	1		1760.1.o.b		1
24.h	odd	2	1		3960.1.x.d		1
33.d	even	2	1		3960.1.x.a		1
40.e	odd	2	1		1760.1.o.d		1
40.f	even	2	1		440.1.o.c	yes	1
40.i	odd	4	2		2200.1.d.e		2
44.c	even	2	1		1760.1.o.d		1
55.d	odd	2	1		440.1.o.b	yes	1
55.e	even	4	2		2200.1.d.e		2
88.b	odd	2	1		440.1.o.d	yes	1
88.g	even	2	1		1760.1.o.b		1
120.i	odd	2	1		3960.1.x.a		1
165.d	even	2	1		3960.1.x.d		1
220.g	even	2	1		1760.1.o.a		1
264.m	even	2	1		3960.1.x.b		1
440.c	even	2	1		1760.1.o.c		1
440.o	odd	2	1	CM	440.1.o.a	✓	1
440.t	even	4	2		2200.1.d.f		2
1320.u	even	2	1		3960.1.x.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
440.1.o.a	✓	1	1.a	even	1	1	trivial
440.1.o.a	✓	1	440.o	odd	2	1	CM
440.1.o.b	yes	1	8.b	even	2	1
440.1.o.b	yes	1	55.d	odd	2	1
440.1.o.c	yes	1	11.b	odd	2	1
440.1.o.c	yes	1	40.f	even	2	1
440.1.o.d	yes	1	5.b	even	2	1
440.1.o.d	yes	1	88.b	odd	2	1
1760.1.o.a		1	8.d	odd	2	1
1760.1.o.a		1	220.g	even	2	1
1760.1.o.b		1	20.d	odd	2	1
1760.1.o.b		1	88.g	even	2	1
1760.1.o.c		1	4.b	odd	2	1
1760.1.o.c		1	440.c	even	2	1
1760.1.o.d		1	40.e	odd	2	1
1760.1.o.d		1	44.c	even	2	1
2200.1.d.e		2	40.i	odd	4	2
2200.1.d.e		2	55.e	even	4	2
2200.1.d.f		2	5.c	odd	4	2
2200.1.d.f		2	440.t	even	4	2
3960.1.x.a		1	33.d	even	2	1
3960.1.x.a		1	120.i	odd	2	1
3960.1.x.b		1	15.d	odd	2	1
3960.1.x.b		1	264.m	even	2	1
3960.1.x.c		1	3.b	odd	2	1
3960.1.x.c		1	1320.u	even	2	1
3960.1.x.d		1	24.h	odd	2	1
3960.1.x.d		1	165.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 $S_{1}^{\mathrm{new}}(440, [\chi])$:

$ T_{3} + 1 $

T3 + 1

$ T_{7} - 1 $

T7 - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T + 1 $ T + 1
$5$	$ T - 1 $ T - 1
$7$	$ T - 1 $ T - 1
$11$	$ T + 1 $ T + 1
$13$	$ T $ T
$17$	$ T - 1 $ T - 1
$19$	$ T - 1 $ T - 1
$23$	$ T $ T
$29$	$ T - 1 $ T - 1
$31$	$ T + 1 $ T + 1
$37$	$ T + 1 $ T + 1
$41$	$ T $ T
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T + 1 $ T + 1
$59$	$ T $ T
$61$	$ T - 1 $ T - 1
$67$	$ T - 2 $ T - 2
$71$	$ T + 1 $ T + 1
$73$	$ T + 2 $ T + 2
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T + 1 $ T + 1
$97$	$ T $ T
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Level:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	440.o (of order \(2\), degree \(1\), minimal)

\(n\)	\(111\)	\(177\)	\(221\)	\(321\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(-1\)

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