LMFDB - Newform orbit 2450.4.a.b

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:	$1$
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} - 8 q^{3} + 4 q^{4} + 16 q^{6} - 8 q^{8} + 37 q^{9}+O(q^{10}) $

$ q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 16 q^{6} - 8 q^{8} + 37 q^{9} + 12 q^{11} - 32 q^{12} - 58 q^{13} + 16 q^{16} + 66 q^{17} - 74 q^{18} + 100 q^{19} - 24 q^{22} - 132 q^{23} + 64 q^{24} + 116 q^{26} - 80 q^{27} - 90 q^{29} - 152 q^{31} - 32 q^{32} - 96 q^{33} - 132 q^{34} + 148 q^{36} + 34 q^{37} - 200 q^{38} + 464 q^{39} + 438 q^{41} - 32 q^{43} + 48 q^{44} + 264 q^{46} - 204 q^{47} - 128 q^{48} - 528 q^{51} - 232 q^{52} - 222 q^{53} + 160 q^{54} - 800 q^{57} + 180 q^{58} - 420 q^{59} - 902 q^{61} + 304 q^{62} + 64 q^{64} + 192 q^{66} + 1024 q^{67} + 264 q^{68} + 1056 q^{69} + 432 q^{71} - 296 q^{72} + 362 q^{73} - 68 q^{74} + 400 q^{76} - 928 q^{78} - 160 q^{79} - 359 q^{81} - 876 q^{82} + 72 q^{83} + 64 q^{86} + 720 q^{87} - 96 q^{88} - 810 q^{89} - 528 q^{92} + 1216 q^{93} + 408 q^{94} + 256 q^{96} + 1106 q^{97} + 444 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

−8.00000

4.00000

16.0000

−8.00000

37.0000

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.b		1
5.b	even	2	1		490.4.a.o		1
7.b	odd	2	1		50.4.a.c		1
21.c	even	2	1		450.4.a.q		1
28.d	even	2	1		400.4.a.b		1
35.c	odd	2	1		10.4.a.a	✓	1
35.f	even	4	2		50.4.b.a		2
35.i	odd	6	2		490.4.e.i		2
35.j	even	6	2		490.4.e.a		2
56.e	even	2	1		1600.4.a.bx		1
56.h	odd	2	1		1600.4.a.d		1
105.g	even	2	1		90.4.a.a		1
105.k	odd	4	2		450.4.c.d		2
140.c	even	2	1		80.4.a.f		1
140.j	odd	4	2		400.4.c.c		2
280.c	odd	2	1		320.4.a.m		1
280.n	even	2	1		320.4.a.b		1
315.z	even	6	2		810.4.e.w		2
315.bg	odd	6	2		810.4.e.c		2
385.h	even	2	1		1210.4.a.b		1
420.o	odd	2	1		720.4.a.j		1
455.h	odd	2	1		1690.4.a.a		1
560.be	even	4	2		1280.4.d.g		2
560.bf	odd	4	2		1280.4.d.j		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.4.a.a	✓	1	35.c	odd	2	1
50.4.a.c		1	7.b	odd	2	1
50.4.b.a		2	35.f	even	4	2
80.4.a.f		1	140.c	even	2	1
90.4.a.a		1	105.g	even	2	1
320.4.a.b		1	280.n	even	2	1
320.4.a.m		1	280.c	odd	2	1
400.4.a.b		1	28.d	even	2	1
400.4.c.c		2	140.j	odd	4	2
450.4.a.q		1	21.c	even	2	1
450.4.c.d		2	105.k	odd	4	2
490.4.a.o		1	5.b	even	2	1
490.4.e.a		2	35.j	even	6	2
490.4.e.i		2	35.i	odd	6	2
720.4.a.j		1	420.o	odd	2	1
810.4.e.c		2	315.bg	odd	6	2
810.4.e.w		2	315.z	even	6	2
1210.4.a.b		1	385.h	even	2	1
1280.4.d.g		2	560.be	even	4	2
1280.4.d.j		2	560.bf	odd	4	2
1600.4.a.d		1	56.h	odd	2	1
1600.4.a.bx		1	56.e	even	2	1
1690.4.a.a		1	455.h	odd	2	1
2450.4.a.b		1	1.a	even	1	1	trivial

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} + 8 $

$ T_{11} - 12 $

$ T_{19} - 100 $

$ T_{23} + 132 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T + 8 $ T + 8
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 12 $ T - 12
$13$	$ T + 58 $ T + 58
$17$	$ T - 66 $ T - 66
$19$	$ T - 100 $ T - 100
$23$	$ T + 132 $ T + 132
$29$	$ T + 90 $ T + 90
$31$	$ T + 152 $ T + 152
$37$	$ T - 34 $ T - 34
$41$	$ T - 438 $ T - 438
$43$	$ T + 32 $ T + 32
$47$	$ T + 204 $ T + 204
$53$	$ T + 222 $ T + 222
$59$	$ T + 420 $ T + 420
$61$	$ T + 902 $ T + 902
$67$	$ T - 1024 $ T - 1024
$71$	$ T - 432 $ T - 432
$73$	$ T - 362 $ T - 362
$79$	$ T + 160 $ T + 160
$83$	$ T - 72 $ T - 72
$89$	$ T + 810 $ T + 810
$97$	$ T - 1106 $ T - 1106
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Overview	Random
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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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