LMFDB - Newform orbit 2450.4.a.bc

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 70)
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} - q^{3} + 4 q^{4} - 2 q^{6} + 8 q^{8} - 26 q^{9}+O(q^{10}) $

$ q + 2 q^{2} - q^{3} + 4 q^{4} - 2 q^{6} + 8 q^{8} - 26 q^{9} - 65 q^{11} - 4 q^{12} + 13 q^{13} + 16 q^{16} - 73 q^{17} - 52 q^{18} + 142 q^{19} - 130 q^{22} - 130 q^{23} - 8 q^{24} + 26 q^{26} + 53 q^{27} + 111 q^{29} - 256 q^{31} + 32 q^{32} + 65 q^{33} - 146 q^{34} - 104 q^{36} + 266 q^{37} + 284 q^{38} - 13 q^{39} + 424 q^{41} - 534 q^{43} - 260 q^{44} - 260 q^{46} - 269 q^{47} - 16 q^{48} + 73 q^{51} + 52 q^{52} + 132 q^{53} + 106 q^{54} - 142 q^{57} + 222 q^{58} + 224 q^{59} + 572 q^{61} - 512 q^{62} + 64 q^{64} + 130 q^{66} + 108 q^{67} - 292 q^{68} + 130 q^{69} + 560 q^{71} - 208 q^{72} + 586 q^{73} + 532 q^{74} + 568 q^{76} - 26 q^{78} + 57 q^{79} + 649 q^{81} + 848 q^{82} + 252 q^{83} - 1068 q^{86} - 111 q^{87} - 520 q^{88} + 184 q^{89} - 520 q^{92} + 256 q^{93} - 538 q^{94} - 32 q^{96} - 605 q^{97} + 1690 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

−1.00000

4.00000

−2.00000

8.00000

−26.0000

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.bc		1
5.b	even	2	1		490.4.a.d		1
7.b	odd	2	1		350.4.a.r		1
35.c	odd	2	1		70.4.a.c	✓	1
35.f	even	4	2		350.4.c.h		2
35.i	odd	6	2		490.4.e.o		2
35.j	even	6	2		490.4.e.n		2
105.g	even	2	1		630.4.a.x		1
140.c	even	2	1		560.4.a.i		1
280.c	odd	2	1		2240.4.a.v		1
280.n	even	2	1		2240.4.a.r		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.c	✓	1	35.c	odd	2	1
350.4.a.r		1	7.b	odd	2	1
350.4.c.h		2	35.f	even	4	2
490.4.a.d		1	5.b	even	2	1
490.4.e.n		2	35.j	even	6	2
490.4.e.o		2	35.i	odd	6	2
560.4.a.i		1	140.c	even	2	1
630.4.a.x		1	105.g	even	2	1
2240.4.a.r		1	280.n	even	2	1
2240.4.a.v		1	280.c	odd	2	1
2450.4.a.bc		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} + 1 $

$ T_{11} + 65 $

$ T_{19} - 142 $

$ T_{23} + 130 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T + 1 $ T + 1
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T + 65 $ T + 65
$13$	$ T - 13 $ T - 13
$17$	$ T + 73 $ T + 73
$19$	$ T - 142 $ T - 142
$23$	$ T + 130 $ T + 130
$29$	$ T - 111 $ T - 111
$31$	$ T + 256 $ T + 256
$37$	$ T - 266 $ T - 266
$41$	$ T - 424 $ T - 424
$43$	$ T + 534 $ T + 534
$47$	$ T + 269 $ T + 269
$53$	$ T - 132 $ T - 132
$59$	$ T - 224 $ T - 224
$61$	$ T - 572 $ T - 572
$67$	$ T - 108 $ T - 108
$71$	$ T - 560 $ T - 560
$73$	$ T - 586 $ T - 586
$79$	$ T - 57 $ T - 57
$83$	$ T - 252 $ T - 252
$89$	$ T - 184 $ T - 184
$97$	$ T + 605 $ T + 605
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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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