LMFDB - Newform orbit 700.4.a.m

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [700,4,Mod(1,700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [1,0,7,0,0,0,-7,0,22,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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 + 7 q^{3} - 7 q^{7} + 22 q^{9} - 7 q^{11} + 3 q^{13} + 61 q^{17} + 48 q^{19} - 49 q^{21} + 58 q^{23} - 35 q^{27} + 219 q^{29} + 298 q^{31} - 49 q^{33} - 170 q^{37} + 21 q^{39} + 50 q^{41} + 484 q^{43}+ \cdots - 154 q^{99}+O(q^{100}) $

q + 7 * q^3 - 7 * q^7 + 22 * q^9 - 7 * q^11 + 3 * q^13 + 61 * q^17 + 48 * q^19 - 49 * q^21 + 58 * q^23 - 35 * q^27 + 219 * q^29 + 298 * q^31 - 49 * q^33 - 170 * q^37 + 21 * q^39 + 50 * q^41 + 484 * q^43 + 131 * q^47 + 49 * q^49 + 427 * q^51 + 210 * q^53 + 336 * q^57 - 782 * q^59 + 488 * q^61 - 154 * q^63 + 494 * q^67 + 406 * q^69 - 240 * q^71 + 58 * q^73 + 49 * q^77 - 1065 * q^79 - 839 * q^81 + 1036 * q^83 + 1533 * q^87 + 608 * q^89 - 21 * q^91 + 2086 * q^93 - 1339 * q^97 - 154 * q^99

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

7.00000

−7.00000

22.0000

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.m		1
5.b	even	2	1		700.4.a.c		1
5.c	odd	4	2		140.4.e.b	✓	2
15.e	even	4	2		1260.4.k.b		2
20.e	even	4	2		560.4.g.b		2
35.f	even	4	2		980.4.e.b		2

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

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} - 7 $

T3 - 7

$ T_{11} + 7 $

T11 + 7

$ T_{13} - 3 $

T13 - 3

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 7 $ T - 7
$5$	$ T $ T
$7$	$ T + 7 $ T + 7
$11$	$ T + 7 $ T + 7
$13$	$ T - 3 $ T - 3
$17$	$ T - 61 $ T - 61
$19$	$ T - 48 $ T - 48
$23$	$ T - 58 $ T - 58
$29$	$ T - 219 $ T - 219
$31$	$ T - 298 $ T - 298
$37$	$ T + 170 $ T + 170
$41$	$ T - 50 $ T - 50
$43$	$ T - 484 $ T - 484
$47$	$ T - 131 $ T - 131
$53$	$ T - 210 $ T - 210
$59$	$ T + 782 $ T + 782
$61$	$ T - 488 $ T - 488
$67$	$ T - 494 $ T - 494
$71$	$ T + 240 $ T + 240
$73$	$ T - 58 $ T - 58
$79$	$ T + 1065 $ T + 1065
$83$	$ T - 1036 $ T - 1036
$89$	$ T - 608 $ T - 608
$97$	$ T + 1339 $ T + 1339
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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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