LMFDB - Newform orbit 104.1.h.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [104,1,Mod(51,104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("104.51");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 104 = 2^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	104.h (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:	$0.0519027613138$
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 3.1.104.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.86528.1

$q$-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}+O(q^{10}) $

$ q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} - q^{10} - q^{12} - q^{13} - q^{14} - q^{15} + q^{16} - q^{17} + q^{20} - q^{21} + q^{24} + q^{26} + q^{27} + q^{28} + q^{30} - 2 q^{31} - q^{32} + q^{34} + q^{35} + q^{37} + q^{39} - q^{40} + q^{42} - q^{43} + q^{47} - q^{48} + q^{51} - q^{52} - q^{54} - q^{56} - q^{60} + 2 q^{62} + q^{64} - q^{65} - q^{68} - q^{70} + q^{71} - q^{74} - q^{78} + q^{80} - q^{81} - q^{84} - q^{85} + q^{86} - q^{91} + 2 q^{93} - q^{94} + q^{96}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/104\mathbb{Z}\right)^\times$.

$n$	$41$	$53$	$79$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

51.1

−1.00000

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
104.h	odd	2	1	CM by $\Q(\sqrt{-26}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	104.1.h.a	✓	1
3.b	odd	2	1		936.1.o.b		1
4.b	odd	2	1		416.1.h.b		1
5.b	even	2	1		2600.1.o.d		1
5.c	odd	4	2		2600.1.b.b		2
8.b	even	2	1		416.1.h.a		1
8.d	odd	2	1		104.1.h.b	yes	1
12.b	even	2	1		3744.1.o.a		1
13.b	even	2	1		104.1.h.b	yes	1
13.c	even	3	2		1352.1.p.b		2
13.d	odd	4	2		1352.1.g.a		2
13.e	even	6	2		1352.1.p.a		2
13.f	odd	12	4		1352.1.n.a		4
16.e	even	4	2		3328.1.c.a		2
16.f	odd	4	2		3328.1.c.e		2
24.f	even	2	1		936.1.o.a		1
24.h	odd	2	1		3744.1.o.b		1
39.d	odd	2	1		936.1.o.a		1
40.e	odd	2	1		2600.1.o.b		1
40.k	even	4	2		2600.1.b.a		2
52.b	odd	2	1		416.1.h.a		1
65.d	even	2	1		2600.1.o.b		1
65.h	odd	4	2		2600.1.b.a		2
104.e	even	2	1		416.1.h.b		1
104.h	odd	2	1	CM	104.1.h.a	✓	1
104.m	even	4	2		1352.1.g.a		2
104.n	odd	6	2		1352.1.p.a		2
104.p	odd	6	2		1352.1.p.b		2
104.u	even	12	4		1352.1.n.a		4
156.h	even	2	1		3744.1.o.b		1
208.o	odd	4	2		3328.1.c.a		2
208.p	even	4	2		3328.1.c.e		2
312.b	odd	2	1		3744.1.o.a		1
312.h	even	2	1		936.1.o.b		1
520.b	odd	2	1		2600.1.o.d		1
520.bc	even	4	2		2600.1.b.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.1.h.a	✓	1	1.a	even	1	1	trivial
104.1.h.a	✓	1	104.h	odd	2	1	CM
104.1.h.b	yes	1	8.d	odd	2	1
104.1.h.b	yes	1	13.b	even	2	1
416.1.h.a		1	8.b	even	2	1
416.1.h.a		1	52.b	odd	2	1
416.1.h.b		1	4.b	odd	2	1
416.1.h.b		1	104.e	even	2	1
936.1.o.a		1	24.f	even	2	1
936.1.o.a		1	39.d	odd	2	1
936.1.o.b		1	3.b	odd	2	1
936.1.o.b		1	312.h	even	2	1
1352.1.g.a		2	13.d	odd	4	2
1352.1.g.a		2	104.m	even	4	2
1352.1.n.a		4	13.f	odd	12	4
1352.1.n.a		4	104.u	even	12	4
1352.1.p.a		2	13.e	even	6	2
1352.1.p.a		2	104.n	odd	6	2
1352.1.p.b		2	13.c	even	3	2
1352.1.p.b		2	104.p	odd	6	2
2600.1.b.a		2	40.k	even	4	2
2600.1.b.a		2	65.h	odd	4	2
2600.1.b.b		2	5.c	odd	4	2
2600.1.b.b		2	520.bc	even	4	2
2600.1.o.b		1	40.e	odd	2	1
2600.1.o.b		1	65.d	even	2	1
2600.1.o.d		1	5.b	even	2	1
2600.1.o.d		1	520.b	odd	2	1
3328.1.c.a		2	16.e	even	4	2
3328.1.c.a		2	208.o	odd	4	2
3328.1.c.e		2	16.f	odd	4	2
3328.1.c.e		2	208.p	even	4	2
3744.1.o.a		1	12.b	even	2	1
3744.1.o.a		1	312.b	odd	2	1
3744.1.o.b		1	24.h	odd	2	1
3744.1.o.b		1	156.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 1 $ T5 - 1 acting on $S_{1}^{\mathrm{new}}(104, [\chi])$.

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 $ T
$13$	$ T + 1 $ T + 1
$17$	$ T + 1 $ T + 1
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T + 2 $ T + 2
$37$	$ T - 1 $ T - 1
$41$	$ T $ T
$43$	$ T + 1 $ T + 1
$47$	$ T - 1 $ T - 1
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T $ T
$71$	$ T - 1 $ T - 1
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T $ T
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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:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	104.h (of order \(2\), degree \(1\), minimal)

\(n\)	\(41\)	\(53\)	\(79\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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