LMFDB - Newform orbit 2450.4.a.o

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
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 - 2 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{8} - 23 q^{9}+O(q^{10}) $

$ q - 2 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{8} - 23 q^{9} - 28 q^{11} + 8 q^{12} + 12 q^{13} + 16 q^{16} - 64 q^{17} + 46 q^{18} + 60 q^{19} + 56 q^{22} + 58 q^{23} - 16 q^{24} - 24 q^{26} - 100 q^{27} + 90 q^{29} + 128 q^{31} - 32 q^{32} - 56 q^{33} + 128 q^{34} - 92 q^{36} - 236 q^{37} - 120 q^{38} + 24 q^{39} - 242 q^{41} - 362 q^{43} - 112 q^{44} - 116 q^{46} + 226 q^{47} + 32 q^{48} - 128 q^{51} + 48 q^{52} + 108 q^{53} + 200 q^{54} + 120 q^{57} - 180 q^{58} + 20 q^{59} - 542 q^{61} - 256 q^{62} + 64 q^{64} + 112 q^{66} + 434 q^{67} - 256 q^{68} + 116 q^{69} - 1128 q^{71} + 184 q^{72} + 632 q^{73} + 472 q^{74} + 240 q^{76} - 48 q^{78} - 720 q^{79} + 421 q^{81} + 484 q^{82} - 478 q^{83} + 724 q^{86} + 180 q^{87} + 224 q^{88} + 490 q^{89} + 232 q^{92} + 256 q^{93} - 452 q^{94} - 64 q^{96} + 1456 q^{97} + 644 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

−2.00000

2.00000

4.00000

−4.00000

−8.00000

−23.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$1$
$5$	$-1$
$7$	$-1$

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	2450.4.a.o		1
5.b	even	2	1		2450.4.a.bb		1
5.c	odd	4	2		490.4.c.b		2
7.b	odd	2	1		50.4.a.b		1
21.c	even	2	1		450.4.a.k		1
28.d	even	2	1		400.4.a.n		1
35.c	odd	2	1		50.4.a.d		1
35.f	even	4	2		10.4.b.a	✓	2
56.e	even	2	1		1600.4.a.t		1
56.h	odd	2	1		1600.4.a.bh		1
105.g	even	2	1		450.4.a.j		1
105.k	odd	4	2		90.4.c.b		2
140.c	even	2	1		400.4.a.h		1
140.j	odd	4	2		80.4.c.a		2
280.c	odd	2	1		1600.4.a.u		1
280.n	even	2	1		1600.4.a.bg		1
280.s	even	4	2		320.4.c.d		2
280.y	odd	4	2		320.4.c.c		2
420.w	even	4	2		720.4.f.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.4.b.a	✓	2	35.f	even	4	2
50.4.a.b		1	7.b	odd	2	1
50.4.a.d		1	35.c	odd	2	1
80.4.c.a		2	140.j	odd	4	2
90.4.c.b		2	105.k	odd	4	2
320.4.c.c		2	280.y	odd	4	2
320.4.c.d		2	280.s	even	4	2
400.4.a.h		1	140.c	even	2	1
400.4.a.n		1	28.d	even	2	1
450.4.a.j		1	105.g	even	2	1
450.4.a.k		1	21.c	even	2	1
490.4.c.b		2	5.c	odd	4	2
720.4.f.f		2	420.w	even	4	2
1600.4.a.t		1	56.e	even	2	1
1600.4.a.u		1	280.c	odd	2	1
1600.4.a.bg		1	280.n	even	2	1
1600.4.a.bh		1	56.h	odd	2	1
2450.4.a.o		1	1.a	even	1	1	trivial
2450.4.a.bb		1	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2450))$:

$ T_{3} - 2 $

$ T_{11} + 28 $

$ T_{19} - 60 $

$ T_{23} - 58 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T - 2 $ T - 2
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T + 28 $ T + 28
$13$	$ T - 12 $ T - 12
$17$	$ T + 64 $ T + 64
$19$	$ T - 60 $ T - 60
$23$	$ T - 58 $ T - 58
$29$	$ T - 90 $ T - 90
$31$	$ T - 128 $ T - 128
$37$	$ T + 236 $ T + 236
$41$	$ T + 242 $ T + 242
$43$	$ T + 362 $ T + 362
$47$	$ T - 226 $ T - 226
$53$	$ T - 108 $ T - 108
$59$	$ T - 20 $ T - 20
$61$	$ T + 542 $ T + 542
$67$	$ T - 434 $ T - 434
$71$	$ T + 1128 $ T + 1128
$73$	$ T - 632 $ T - 632
$79$	$ T + 720 $ T + 720
$83$	$ T + 478 $ T + 478
$89$	$ T - 490 $ T - 490
$97$	$ T - 1456 $ T - 1456
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

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