LMFDB - Newform orbit 700.6.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [700,6,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, 6); 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, 6, names="a")

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	700.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$112.268673869$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 + 28 q^{3} - 49 q^{7} + 541 q^{9} - 577 q^{11} - 974 q^{13} + 1700 q^{17} - 878 q^{19} - 1372 q^{21} + 1297 q^{23} + 8344 q^{27} + 109 q^{29} - 2922 q^{31} - 16156 q^{33} - 14175 q^{37} - 27272 q^{39}+ \cdots - 312157 q^{99}+O(q^{100}) $

q + 28 * q^3 - 49 * q^7 + 541 * q^9 - 577 * q^11 - 974 * q^13 + 1700 * q^17 - 878 * q^19 - 1372 * q^21 + 1297 * q^23 + 8344 * q^27 + 109 * q^29 - 2922 * q^31 - 16156 * q^33 - 14175 * q^37 - 27272 * q^39 + 13464 * q^41 - 12229 * q^43 - 16974 * q^47 + 2401 * q^49 + 47600 * q^51 + 24506 * q^53 - 24584 * q^57 - 24570 * q^59 - 5080 * q^61 - 26509 * q^63 - 58575 * q^67 + 36316 * q^69 - 20705 * q^71 - 53540 * q^73 + 28273 * q^77 - 101027 * q^79 + 102169 * q^81 - 61638 * q^83 + 3052 * q^87 + 47038 * q^89 + 47726 * q^91 - 81816 * q^93 - 82 * q^97 - 312157 * 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

28.0000

−49.0000

541.000

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	700.6.a.f	yes	1
5.b	even	2	1		700.6.a.a	✓	1
5.c	odd	4	2		700.6.e.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.6.a.a	✓	1	5.b	even	2	1
700.6.a.f	yes	1	1.a	even	1	1	trivial
700.6.e.a		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 28 $ T3 - 28 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(700))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 28 $ T - 28
$5$	$ T $ T
$7$	$ T + 49 $ T + 49
$11$	$ T + 577 $ T + 577
$13$	$ T + 974 $ T + 974
$17$	$ T - 1700 $ T - 1700
$19$	$ T + 878 $ T + 878
$23$	$ T - 1297 $ T - 1297
$29$	$ T - 109 $ T - 109
$31$	$ T + 2922 $ T + 2922
$37$	$ T + 14175 $ T + 14175
$41$	$ T - 13464 $ T - 13464
$43$	$ T + 12229 $ T + 12229
$47$	$ T + 16974 $ T + 16974
$53$	$ T - 24506 $ T - 24506
$59$	$ T + 24570 $ T + 24570
$61$	$ T + 5080 $ T + 5080
$67$	$ T + 58575 $ T + 58575
$71$	$ T + 20705 $ T + 20705
$73$	$ T + 53540 $ T + 53540
$79$	$ T + 101027 $ T + 101027
$83$	$ T + 61638 $ T + 61638
$89$	$ T - 47038 $ T - 47038
$97$	$ T + 82 $ T + 82
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Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	700.a (trivial)

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