LMFDB - Newform orbit 245.4.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, 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("245.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.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:	$14.4554679514$
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} - 2 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} - 23 q^{9}+O(q^{10}) $

$ q - 4 q^{2} - 2 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} - 23 q^{9} - 20 q^{10} + 32 q^{11} - 16 q^{12} + 38 q^{13} - 10 q^{15} - 64 q^{16} - 26 q^{17} + 92 q^{18} - 100 q^{19} + 40 q^{20} - 128 q^{22} - 78 q^{23} + 25 q^{25} - 152 q^{26} + 100 q^{27} - 50 q^{29} + 40 q^{30} + 108 q^{31} + 256 q^{32} - 64 q^{33} + 104 q^{34} - 184 q^{36} + 266 q^{37} + 400 q^{38} - 76 q^{39} - 22 q^{41} + 442 q^{43} + 256 q^{44} - 115 q^{45} + 312 q^{46} + 514 q^{47} + 128 q^{48} - 100 q^{50} + 52 q^{51} + 304 q^{52} + 2 q^{53} - 400 q^{54} + 160 q^{55} + 200 q^{57} + 200 q^{58} - 500 q^{59} - 80 q^{60} + 518 q^{61} - 432 q^{62} - 512 q^{64} + 190 q^{65} + 256 q^{66} + 126 q^{67} - 208 q^{68} + 156 q^{69} + 412 q^{71} + 878 q^{73} - 1064 q^{74} - 50 q^{75} - 800 q^{76} + 304 q^{78} + 600 q^{79} - 320 q^{80} + 421 q^{81} + 88 q^{82} - 282 q^{83} - 130 q^{85} - 1768 q^{86} + 100 q^{87} + 150 q^{89} + 460 q^{90} - 624 q^{92} - 216 q^{93} - 2056 q^{94} - 500 q^{95} - 512 q^{96} - 386 q^{97} - 736 q^{99}+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

−2.00000

8.00000

5.00000

8.00000

−23.0000

−20.0000

Atkin-Lehner signs

$ p $	Sign
$5$	$-1$
$7$	$-1$

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	245.4.a.a		1
3.b	odd	2	1		2205.4.a.q		1
5.b	even	2	1		1225.4.a.k		1
7.b	odd	2	1		5.4.a.a	✓	1
7.c	even	3	2		245.4.e.g		2
7.d	odd	6	2		245.4.e.f		2
21.c	even	2	1		45.4.a.d		1
28.d	even	2	1		80.4.a.d		1
35.c	odd	2	1		25.4.a.c		1
35.f	even	4	2		25.4.b.a		2
56.e	even	2	1		320.4.a.h		1
56.h	odd	2	1		320.4.a.g		1
63.l	odd	6	2		405.4.e.l		2
63.o	even	6	2		405.4.e.c		2
77.b	even	2	1		605.4.a.d		1
84.h	odd	2	1		720.4.a.u		1
91.b	odd	2	1		845.4.a.b		1
105.g	even	2	1		225.4.a.b		1
105.k	odd	4	2		225.4.b.c		2
112.j	even	4	2		1280.4.d.l		2
112.l	odd	4	2		1280.4.d.e		2
119.d	odd	2	1		1445.4.a.a		1
133.c	even	2	1		1805.4.a.h		1
140.c	even	2	1		400.4.a.m		1
140.j	odd	4	2		400.4.c.k		2
280.c	odd	2	1		1600.4.a.bi		1
280.n	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	7.b	odd	2	1
25.4.a.c		1	35.c	odd	2	1
25.4.b.a		2	35.f	even	4	2
45.4.a.d		1	21.c	even	2	1
80.4.a.d		1	28.d	even	2	1
225.4.a.b		1	105.g	even	2	1
225.4.b.c		2	105.k	odd	4	2
245.4.a.a		1	1.a	even	1	1	trivial
245.4.e.f		2	7.d	odd	6	2
245.4.e.g		2	7.c	even	3	2
320.4.a.g		1	56.h	odd	2	1
320.4.a.h		1	56.e	even	2	1
400.4.a.m		1	140.c	even	2	1
400.4.c.k		2	140.j	odd	4	2
405.4.e.c		2	63.o	even	6	2
405.4.e.l		2	63.l	odd	6	2
605.4.a.d		1	77.b	even	2	1
720.4.a.u		1	84.h	odd	2	1
845.4.a.b		1	91.b	odd	2	1
1225.4.a.k		1	5.b	even	2	1
1280.4.d.e		2	112.l	odd	4	2
1280.4.d.l		2	112.j	even	4	2
1445.4.a.a		1	119.d	odd	2	1
1600.4.a.s		1	280.n	even	2	1
1600.4.a.bi		1	280.c	odd	2	1
1805.4.a.h		1	133.c	even	2	1
2205.4.a.q		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(245))$:

$ T_{2} + 4 $

$ T_{3} + 2 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 4 $ T + 4
$3$	$ T + 2 $ T + 2
$5$	$ T - 5 $ T - 5
$7$	$ T $ T
$11$	$ 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
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

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