LMFDB - Newform orbit 70.6.a.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [70,6,Mod(1,70)] 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("70.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(70, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 70 = 2 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	70.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-4,11] 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:	$11.2268673869$
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 - 4 q^{2} + 11 q^{3} + 16 q^{4} - 25 q^{5} - 44 q^{6} - 49 q^{7} - 64 q^{8} - 122 q^{9} + 100 q^{10} + 83 q^{11} + 176 q^{12} - 83 q^{13} + 196 q^{14} - 275 q^{15} + 256 q^{16} - 177 q^{17} + 488 q^{18}+ \cdots - 10126 q^{99}+O(q^{100}) $

q - 4 * q^2 + 11 * q^3 + 16 * q^4 - 25 * q^5 - 44 * q^6 - 49 * q^7 - 64 * q^8 - 122 * q^9 + 100 * q^10 + 83 * q^11 + 176 * q^12 - 83 * q^13 + 196 * q^14 - 275 * q^15 + 256 * q^16 - 177 * q^17 + 488 * q^18 - 2082 * q^19 - 400 * q^20 - 539 * q^21 - 332 * q^22 - 3170 * q^23 - 704 * q^24 + 625 * q^25 + 332 * q^26 - 4015 * q^27 - 784 * q^28 - 8681 * q^29 + 1100 * q^30 + 1636 * q^31 - 1024 * q^32 + 913 * q^33 + 708 * q^34 + 1225 * q^35 - 1952 * q^36 + 4298 * q^37 + 8328 * q^38 - 913 * q^39 + 1600 * q^40 + 2356 * q^41 + 2156 * q^42 + 8738 * q^43 + 1328 * q^44 + 3050 * q^45 + 12680 * q^46 - 3641 * q^47 + 2816 * q^48 + 2401 * q^49 - 2500 * q^50 - 1947 * q^51 - 1328 * q^52 + 33268 * q^53 + 16060 * q^54 - 2075 * q^55 + 3136 * q^56 - 22902 * q^57 + 34724 * q^58 - 30968 * q^59 - 4400 * q^60 + 4560 * q^61 - 6544 * q^62 + 5978 * q^63 + 4096 * q^64 + 2075 * q^65 - 3652 * q^66 + 37788 * q^67 - 2832 * q^68 - 34870 * q^69 - 4900 * q^70 - 59304 * q^71 + 7808 * q^72 - 8910 * q^73 - 17192 * q^74 + 6875 * q^75 - 33312 * q^76 - 4067 * q^77 + 3652 * q^78 + 27589 * q^79 - 6400 * q^80 - 14519 * q^81 - 9424 * q^82 + 67676 * q^83 - 8624 * q^84 + 4425 * q^85 - 34952 * q^86 - 95491 * q^87 - 5312 * q^88 + 10700 * q^89 - 12200 * q^90 + 4067 * q^91 - 50720 * q^92 + 17996 * q^93 + 14564 * q^94 + 52050 * q^95 - 11264 * q^96 + 65075 * q^97 - 9604 * q^98 - 10126 * 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

−4.00000

11.0000

16.0000

−25.0000

−44.0000

−49.0000

−64.0000

−122.000

100.000

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	70.6.a.d	✓	1
3.b	odd	2	1		630.6.a.l		1
4.b	odd	2	1		560.6.a.a		1
5.b	even	2	1		350.6.a.h		1
5.c	odd	4	2		350.6.c.c		2
7.b	odd	2	1		490.6.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.6.a.d	✓	1	1.a	even	1	1	trivial
350.6.a.h		1	5.b	even	2	1
350.6.c.c		2	5.c	odd	4	2
490.6.a.c		1	7.b	odd	2	1
560.6.a.a		1	4.b	odd	2	1
630.6.a.l		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 11 $ T3 - 11 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(70))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 4 $ T + 4
$3$	$ T - 11 $ T - 11
$5$	$ T + 25 $ T + 25
$7$	$ T + 49 $ T + 49
$11$	$ T - 83 $ T - 83
$13$	$ T + 83 $ T + 83
$17$	$ T + 177 $ T + 177
$19$	$ T + 2082 $ T + 2082
$23$	$ T + 3170 $ T + 3170
$29$	$ T + 8681 $ T + 8681
$31$	$ T - 1636 $ T - 1636
$37$	$ T - 4298 $ T - 4298
$41$	$ T - 2356 $ T - 2356
$43$	$ T - 8738 $ T - 8738
$47$	$ T + 3641 $ T + 3641
$53$	$ T - 33268 $ T - 33268
$59$	$ T + 30968 $ T + 30968
$61$	$ T - 4560 $ T - 4560
$67$	$ T - 37788 $ T - 37788
$71$	$ T + 59304 $ T + 59304
$73$	$ T + 8910 $ T + 8910
$79$	$ T - 27589 $ T - 27589
$83$	$ T - 67676 $ T - 67676
$89$	$ T - 10700 $ T - 10700
$97$	$ T - 65075 $ T - 65075
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	70.a (trivial)

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