LMFDB - Newform orbit 70.8.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [70,8,Mod(1,70)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(70, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("70.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 70 = 2 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	70.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,8,-93] 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:	$21.8669517839$
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 + 8 q^{2} - 93 q^{3} + 64 q^{4} + 125 q^{5} - 744 q^{6} + 343 q^{7} + 512 q^{8} + 6462 q^{9} + 1000 q^{10} - 2167 q^{11} - 5952 q^{12} - 1661 q^{13} + 2744 q^{14} - 11625 q^{15} + 4096 q^{16} - 35771 q^{17}+ \cdots - 14003154 q^{99}+O(q^{100}) $

q + 8 * q^2 - 93 * q^3 + 64 * q^4 + 125 * q^5 - 744 * q^6 + 343 * q^7 + 512 * q^8 + 6462 * q^9 + 1000 * q^10 - 2167 * q^11 - 5952 * q^12 - 1661 * q^13 + 2744 * q^14 - 11625 * q^15 + 4096 * q^16 - 35771 * q^17 + 51696 * q^18 + 20222 * q^19 + 8000 * q^20 - 31899 * q^21 - 17336 * q^22 - 42130 * q^23 - 47616 * q^24 + 15625 * q^25 - 13288 * q^26 - 397575 * q^27 + 21952 * q^28 - 111789 * q^29 - 93000 * q^30 - 269504 * q^31 + 32768 * q^32 + 201531 * q^33 - 286168 * q^34 + 42875 * q^35 + 413568 * q^36 + 532774 * q^37 + 161776 * q^38 + 154473 * q^39 + 64000 * q^40 + 158056 * q^41 - 255192 * q^42 - 521874 * q^43 - 138688 * q^44 + 807750 * q^45 - 337040 * q^46 - 939733 * q^47 - 380928 * q^48 + 117649 * q^49 + 125000 * q^50 + 3326703 * q^51 - 106304 * q^52 - 408384 * q^53 - 3180600 * q^54 - 270875 * q^55 + 175616 * q^56 - 1880646 * q^57 - 894312 * q^58 - 522172 * q^59 - 744000 * q^60 + 350080 * q^61 - 2156032 * q^62 + 2216466 * q^63 + 262144 * q^64 - 207625 * q^65 + 1612248 * q^66 - 3931176 * q^67 - 2289344 * q^68 + 3918090 * q^69 + 343000 * q^70 + 1194016 * q^71 + 3308544 * q^72 + 998350 * q^73 + 4262192 * q^74 - 1453125 * q^75 + 1294208 * q^76 - 743281 * q^77 + 1235784 * q^78 - 2120709 * q^79 + 512000 * q^80 + 22842081 * q^81 + 1264448 * q^82 - 1746708 * q^83 - 2041536 * q^84 - 4471375 * q^85 - 4174992 * q^86 + 10396377 * q^87 - 1109504 * q^88 - 10077740 * q^89 + 6462000 * q^90 - 569723 * q^91 - 2696320 * q^92 + 25063872 * q^93 - 7517864 * q^94 + 2527750 * q^95 - 3047424 * q^96 - 6238295 * q^97 + 941192 * q^98 - 14003154 * 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

8.00000

−93.0000

64.0000

125.000

−744.000

343.000

512.000

6462.00

1000.00

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	70.8.a.a	✓	1
4.b	odd	2	1		560.8.a.b		1
5.b	even	2	1		350.8.a.e		1
5.c	odd	4	2		350.8.c.a		2
7.b	odd	2	1		490.8.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.8.a.a	✓	1	1.a	even	1	1	trivial
350.8.a.e		1	5.b	even	2	1
350.8.c.a		2	5.c	odd	4	2
490.8.a.e		1	7.b	odd	2	1
560.8.a.b		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 93 $ T3 + 93 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(70))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 8 $ T - 8
$3$	$ T + 93 $ T + 93
$5$	$ T - 125 $ T - 125
$7$	$ T - 343 $ T - 343
$11$	$ T + 2167 $ T + 2167
$13$	$ T + 1661 $ T + 1661
$17$	$ T + 35771 $ T + 35771
$19$	$ T - 20222 $ T - 20222
$23$	$ T + 42130 $ T + 42130
$29$	$ T + 111789 $ T + 111789
$31$	$ T + 269504 $ T + 269504
$37$	$ T - 532774 $ T - 532774
$41$	$ T - 158056 $ T - 158056
$43$	$ T + 521874 $ T + 521874
$47$	$ T + 939733 $ T + 939733
$53$	$ T + 408384 $ T + 408384
$59$	$ T + 522172 $ T + 522172
$61$	$ T - 350080 $ T - 350080
$67$	$ T + 3931176 $ T + 3931176
$71$	$ T - 1194016 $ T - 1194016
$73$	$ T - 998350 $ T - 998350
$79$	$ T + 2120709 $ T + 2120709
$83$	$ T + 1746708 $ T + 1746708
$89$	$ T + 10077740 $ T + 10077740
$97$	$ T + 6238295 $ T + 6238295
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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Belyi maps

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	70.a (trivial)

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