LMFDB - Newform orbit 700.4.a.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,4,Mod(1,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("700.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	700.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:	$41.3013370040$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
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^{3} + 7 q^{7} - 11 q^{9}+O(q^{10}) $

$ q + 4 q^{3} + 7 q^{7} - 11 q^{9} + 68 q^{11} - 22 q^{13} + 30 q^{17} + 108 q^{19} + 28 q^{21} - 184 q^{23} - 152 q^{27} + 166 q^{29} - 32 q^{31} + 272 q^{33} + 370 q^{37} - 88 q^{39} + 154 q^{41} - 212 q^{43} + 512 q^{47} + 49 q^{49} + 120 q^{51} + 98 q^{53} + 432 q^{57} - 860 q^{59} + 390 q^{61} - 77 q^{63} - 60 q^{67} - 736 q^{69} + 840 q^{71} + 630 q^{73} + 476 q^{77} + 1312 q^{79} - 311 q^{81} + 436 q^{83} + 664 q^{87} - 598 q^{89} - 154 q^{91} - 128 q^{93} - 914 q^{97} - 748 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

7.00000

−11.0000

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	700.4.a.i		1
5.b	even	2	1		140.4.a.c	✓	1
5.c	odd	4	2		700.4.e.g		2
15.d	odd	2	1		1260.4.a.a		1
20.d	odd	2	1		560.4.a.m		1
35.c	odd	2	1		980.4.a.k		1
35.i	odd	6	2		980.4.i.f		2
35.j	even	6	2		980.4.i.m		2
40.e	odd	2	1		2240.4.a.n		1
40.f	even	2	1		2240.4.a.x		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.4.a.c	✓	1	5.b	even	2	1
560.4.a.m		1	20.d	odd	2	1
700.4.a.i		1	1.a	even	1	1	trivial
700.4.e.g		2	5.c	odd	4	2
980.4.a.k		1	35.c	odd	2	1
980.4.i.f		2	35.i	odd	6	2
980.4.i.m		2	35.j	even	6	2
1260.4.a.a		1	15.d	odd	2	1
2240.4.a.n		1	40.e	odd	2	1
2240.4.a.x		1	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(700))$:

$ T_{3} - 4 $

$ T_{11} - 68 $

$ T_{13} + 22 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 4 $ T - 4
$5$	$ T $ T
$7$	$ T - 7 $ T - 7
$11$	$ T - 68 $ T - 68
$13$	$ T + 22 $ T + 22
$17$	$ T - 30 $ T - 30
$19$	$ T - 108 $ T - 108
$23$	$ T + 184 $ T + 184
$29$	$ T - 166 $ T - 166
$31$	$ T + 32 $ T + 32
$37$	$ T - 370 $ T - 370
$41$	$ T - 154 $ T - 154
$43$	$ T + 212 $ T + 212
$47$	$ T - 512 $ T - 512
$53$	$ T - 98 $ T - 98
$59$	$ T + 860 $ T + 860
$61$	$ T - 390 $ T - 390
$67$	$ T + 60 $ T + 60
$71$	$ T - 840 $ T - 840
$73$	$ T - 630 $ T - 630
$79$	$ T - 1312 $ T - 1312
$83$	$ T - 436 $ T - 436
$89$	$ T + 598 $ T + 598
$97$	$ T + 914 $ T + 914
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	700.a (trivial)

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