LMFDB - Newform orbit 700.4.a.d

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,-5,0,0,0,-7,0,-2,0,-65] 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 - 5 q^{3} - 7 q^{7} - 2 q^{9} - 65 q^{11} + 13 q^{13} - 113 q^{17} + 16 q^{19} + 35 q^{21} + 186 q^{23} + 145 q^{27} - 57 q^{29} - 258 q^{31} + 325 q^{33} - 134 q^{37} - 65 q^{39} - 414 q^{41} - 284 q^{43}+ \cdots + 130 q^{99}+O(q^{100}) $

q - 5 * q^3 - 7 * q^7 - 2 * q^9 - 65 * q^11 + 13 * q^13 - 113 * q^17 + 16 * q^19 + 35 * q^21 + 186 * q^23 + 145 * q^27 - 57 * q^29 - 258 * q^31 + 325 * q^33 - 134 * q^37 - 65 * q^39 - 414 * q^41 - 284 * q^43 + 419 * q^47 + 49 * q^49 + 565 * q^51 - 130 * q^53 - 80 * q^57 + 334 * q^59 - 56 * q^61 + 14 * q^63 + 534 * q^67 - 930 * q^69 + 952 * q^71 + 502 * q^73 + 455 * q^77 - 371 * q^79 - 671 * q^81 + 1220 * q^83 + 285 * q^87 + 880 * q^89 - 91 * q^91 + 1290 * q^93 - 241 * q^97 + 130 * 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

−5.00000

−7.00000

−2.00000

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

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

T3 + 5

$ T_{11} + 65 $

T11 + 65

$ T_{13} - 13 $

T13 - 13

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 5 $ T + 5
$5$	$ T $ T
$7$	$ T + 7 $ T + 7
$11$	$ T + 65 $ T + 65
$13$	$ T - 13 $ T - 13
$17$	$ T + 113 $ T + 113
$19$	$ T - 16 $ T - 16
$23$	$ T - 186 $ T - 186
$29$	$ T + 57 $ T + 57
$31$	$ T + 258 $ T + 258
$37$	$ T + 134 $ T + 134
$41$	$ T + 414 $ T + 414
$43$	$ T + 284 $ T + 284
$47$	$ T - 419 $ T - 419
$53$	$ T + 130 $ T + 130
$59$	$ T - 334 $ T - 334
$61$	$ T + 56 $ T + 56
$67$	$ T - 534 $ T - 534
$71$	$ T - 952 $ T - 952
$73$	$ T - 502 $ T - 502
$79$	$ T + 371 $ T + 371
$83$	$ T - 1220 $ T - 1220
$89$	$ T - 880 $ T - 880
$97$	$ T + 241 $ T + 241
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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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