LMFDB - Newform orbit 30.8.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [30,8,Mod(1,30)] 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("30.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(30, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 30 = 2 \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	30.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,8,-27,64,125] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.37155076452$
Analytic rank:	$0$
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} - 27 q^{3} + 64 q^{4} + 125 q^{5} - 216 q^{6} + 512 q^{7} + 512 q^{8} + 729 q^{9} + 1000 q^{10} + 5460 q^{11} - 1728 q^{12} + 10166 q^{13} + 4096 q^{14} - 3375 q^{15} + 4096 q^{16} - 9918 q^{17}+ \cdots + 3980340 q^{99}+O(q^{100}) $

q + 8 * q^2 - 27 * q^3 + 64 * q^4 + 125 * q^5 - 216 * q^6 + 512 * q^7 + 512 * q^8 + 729 * q^9 + 1000 * q^10 + 5460 * q^11 - 1728 * q^12 + 10166 * q^13 + 4096 * q^14 - 3375 * q^15 + 4096 * q^16 - 9918 * q^17 + 5832 * q^18 - 12436 * q^19 + 8000 * q^20 - 13824 * q^21 + 43680 * q^22 + 33600 * q^23 - 13824 * q^24 + 15625 * q^25 + 81328 * q^26 - 19683 * q^27 + 32768 * q^28 - 187914 * q^29 - 27000 * q^30 - 42592 * q^31 + 32768 * q^32 - 147420 * q^33 - 79344 * q^34 + 64000 * q^35 + 46656 * q^36 - 544066 * q^37 - 99488 * q^38 - 274482 * q^39 + 64000 * q^40 + 374394 * q^41 - 110592 * q^42 - 540532 * q^43 + 349440 * q^44 + 91125 * q^45 + 268800 * q^46 + 1338360 * q^47 - 110592 * q^48 - 561399 * q^49 + 125000 * q^50 + 267786 * q^51 + 650624 * q^52 + 1308222 * q^53 - 157464 * q^54 + 682500 * q^55 + 262144 * q^56 + 335772 * q^57 - 1503312 * q^58 + 262740 * q^59 - 216000 * q^60 - 976330 * q^61 - 340736 * q^62 + 373248 * q^63 + 262144 * q^64 + 1270750 * q^65 - 1179360 * q^66 + 3559172 * q^67 - 634752 * q^68 - 907200 * q^69 + 512000 * q^70 - 2673720 * q^71 + 373248 * q^72 - 3032134 * q^73 - 4352528 * q^74 - 421875 * q^75 - 795904 * q^76 + 2795520 * q^77 - 2195856 * q^78 - 5475808 * q^79 + 512000 * q^80 + 531441 * q^81 + 2995152 * q^82 + 2231556 * q^83 - 884736 * q^84 - 1239750 * q^85 - 4324256 * q^86 + 5073678 * q^87 + 2795520 * q^88 - 10050678 * q^89 + 729000 * q^90 + 5204992 * q^91 + 2150400 * q^92 + 1149984 * q^93 + 10706880 * q^94 - 1554500 * q^95 - 884736 * q^96 + 5727554 * q^97 - 4491192 * q^98 + 3980340 * 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

−27.0000

64.0000

125.000

−216.000

512.000

729.000

1000.00

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$5$	$ -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	30.8.a.e	✓	1
3.b	odd	2	1		90.8.a.b		1
4.b	odd	2	1		240.8.a.l		1
5.b	even	2	1		150.8.a.g		1
5.c	odd	4	2		150.8.c.e		2
15.d	odd	2	1		450.8.a.r		1
15.e	even	4	2		450.8.c.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.8.a.e	✓	1	1.a	even	1	1	trivial
90.8.a.b		1	3.b	odd	2	1
150.8.a.g		1	5.b	even	2	1
150.8.c.e		2	5.c	odd	4	2
240.8.a.l		1	4.b	odd	2	1
450.8.a.r		1	15.d	odd	2	1
450.8.c.c		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} - 512 $ T7 - 512 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(30))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 8 $ T - 8
$3$	$ T + 27 $ T + 27
$5$	$ T - 125 $ T - 125
$7$	$ T - 512 $ T - 512
$11$	$ T - 5460 $ T - 5460
$13$	$ T - 10166 $ T - 10166
$17$	$ T + 9918 $ T + 9918
$19$	$ T + 12436 $ T + 12436
$23$	$ T - 33600 $ T - 33600
$29$	$ T + 187914 $ T + 187914
$31$	$ T + 42592 $ T + 42592
$37$	$ T + 544066 $ T + 544066
$41$	$ T - 374394 $ T - 374394
$43$	$ T + 540532 $ T + 540532
$47$	$ T - 1338360 $ T - 1338360
$53$	$ T - 1308222 $ T - 1308222
$59$	$ T - 262740 $ T - 262740
$61$	$ T + 976330 $ T + 976330
$67$	$ T - 3559172 $ T - 3559172
$71$	$ T + 2673720 $ T + 2673720
$73$	$ T + 3032134 $ T + 3032134
$79$	$ T + 5475808 $ T + 5475808
$83$	$ T - 2231556 $ T - 2231556
$89$	$ T + 10050678 $ T + 10050678
$97$	$ T - 5727554 $ T - 5727554
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	30.a (trivial)

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