LMFDB - Newform orbit 45.4.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	45.a (trivial)

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:	$2.65508595026$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q + 4 q^{2} + 8 q^{4} + 5 q^{5} + 6 q^{7}+O(q^{10}) $

$ q + 4 q^{2} + 8 q^{4} + 5 q^{5} + 6 q^{7} + 20 q^{10} - 32 q^{11} - 38 q^{13} + 24 q^{14} - 64 q^{16} - 26 q^{17} + 100 q^{19} + 40 q^{20} - 128 q^{22} + 78 q^{23} + 25 q^{25} - 152 q^{26} + 48 q^{28} + 50 q^{29} - 108 q^{31} - 256 q^{32} - 104 q^{34} + 30 q^{35} + 266 q^{37} + 400 q^{38} - 22 q^{41} + 442 q^{43} - 256 q^{44} + 312 q^{46} + 514 q^{47} - 307 q^{49} + 100 q^{50} - 304 q^{52} - 2 q^{53} - 160 q^{55} + 200 q^{58} - 500 q^{59} - 518 q^{61} - 432 q^{62} - 512 q^{64} - 190 q^{65} + 126 q^{67} - 208 q^{68} + 120 q^{70} - 412 q^{71} - 878 q^{73} + 1064 q^{74} + 800 q^{76} - 192 q^{77} + 600 q^{79} - 320 q^{80} - 88 q^{82} - 282 q^{83} - 130 q^{85} + 1768 q^{86} + 150 q^{89} - 228 q^{91} + 624 q^{92} + 2056 q^{94} + 500 q^{95} + 386 q^{97} - 1228 q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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} $

1.1

4.00000

8.00000

5.00000

6.00000

20.0000

Atkin-Lehner signs

$ p $	Sign
$3$	$-1$
$5$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	45.4.a.d		1
3.b	odd	2	1		5.4.a.a	✓	1
4.b	odd	2	1		720.4.a.u		1
5.b	even	2	1		225.4.a.b		1
5.c	odd	4	2		225.4.b.c		2
7.b	odd	2	1		2205.4.a.q		1
9.c	even	3	2		405.4.e.c		2
9.d	odd	6	2		405.4.e.l		2
12.b	even	2	1		80.4.a.d		1
15.d	odd	2	1		25.4.a.c		1
15.e	even	4	2		25.4.b.a		2
21.c	even	2	1		245.4.a.a		1
21.g	even	6	2		245.4.e.g		2
21.h	odd	6	2		245.4.e.f		2
24.f	even	2	1		320.4.a.h		1
24.h	odd	2	1		320.4.a.g		1
33.d	even	2	1		605.4.a.d		1
39.d	odd	2	1		845.4.a.b		1
48.i	odd	4	2		1280.4.d.e		2
48.k	even	4	2		1280.4.d.l		2
51.c	odd	2	1		1445.4.a.a		1
57.d	even	2	1		1805.4.a.h		1
60.h	even	2	1		400.4.a.m		1
60.l	odd	4	2		400.4.c.k		2
105.g	even	2	1		1225.4.a.k		1
120.i	odd	2	1		1600.4.a.bi		1
120.m	even	2	1		1600.4.a.s		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
5.4.a.a	✓	1	3.b	odd	2	1
25.4.a.c		1	15.d	odd	2	1
25.4.b.a		2	15.e	even	4	2
45.4.a.d		1	1.a	even	1	1	trivial
80.4.a.d		1	12.b	even	2	1
225.4.a.b		1	5.b	even	2	1
225.4.b.c		2	5.c	odd	4	2
245.4.a.a		1	21.c	even	2	1
245.4.e.f		2	21.h	odd	6	2
245.4.e.g		2	21.g	even	6	2
320.4.a.g		1	24.h	odd	2	1
320.4.a.h		1	24.f	even	2	1
400.4.a.m		1	60.h	even	2	1
400.4.c.k		2	60.l	odd	4	2
405.4.e.c		2	9.c	even	3	2
405.4.e.l		2	9.d	odd	6	2
605.4.a.d		1	33.d	even	2	1
720.4.a.u		1	4.b	odd	2	1
845.4.a.b		1	39.d	odd	2	1
1225.4.a.k		1	105.g	even	2	1
1280.4.d.e		2	48.i	odd	4	2
1280.4.d.l		2	48.k	even	4	2
1445.4.a.a		1	51.c	odd	2	1
1600.4.a.s		1	120.m	even	2	1
1600.4.a.bi		1	120.i	odd	2	1
1805.4.a.h		1	57.d	even	2	1
2205.4.a.q		1	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} - 4 $ T2 - 4 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(45))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 4 $ T - 4
$3$	$ T $ T
$5$	$ T - 5 $ T - 5
$7$	$ T - 6 $ T - 6
$11$	$ T + 32 $ T + 32
$13$	$ T + 38 $ T + 38
$17$	$ T + 26 $ T + 26
$19$	$ T - 100 $ T - 100
$23$	$ T - 78 $ T - 78
$29$	$ T - 50 $ T - 50
$31$	$ T + 108 $ T + 108
$37$	$ T - 266 $ T - 266
$41$	$ T + 22 $ T + 22
$43$	$ T - 442 $ T - 442
$47$	$ T - 514 $ T - 514
$53$	$ T + 2 $ T + 2
$59$	$ T + 500 $ T + 500
$61$	$ T + 518 $ T + 518
$67$	$ T - 126 $ T - 126
$71$	$ T + 412 $ T + 412
$73$	$ T + 878 $ T + 878
$79$	$ T - 600 $ T - 600
$83$	$ T + 282 $ T + 282
$89$	$ T - 150 $ T - 150
$97$	$ T - 386 $ T - 386
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Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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