LMFDB - Newform orbit 30.8.a.d

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:	$1$
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} - 27 q^{3} + 64 q^{4} - 125 q^{5} - 216 q^{6} - 988 q^{7} + 512 q^{8} + 729 q^{9} - 1000 q^{10} - 8040 q^{11} - 1728 q^{12} - 3334 q^{13} - 7904 q^{14} + 3375 q^{15} + 4096 q^{16} + 6582 q^{17}+ \cdots - 5861160 q^{99}+O(q^{100}) $

q + 8 * q^2 - 27 * q^3 + 64 * q^4 - 125 * q^5 - 216 * q^6 - 988 * q^7 + 512 * q^8 + 729 * q^9 - 1000 * q^10 - 8040 * q^11 - 1728 * q^12 - 3334 * q^13 - 7904 * q^14 + 3375 * q^15 + 4096 * q^16 + 6582 * q^17 + 5832 * q^18 - 27436 * q^19 - 8000 * q^20 + 26676 * q^21 - 64320 * q^22 + 48600 * q^23 - 13824 * q^24 + 15625 * q^25 - 26672 * q^26 - 19683 * q^27 - 63232 * q^28 - 132414 * q^29 + 27000 * q^30 + 254408 * q^31 + 32768 * q^32 + 217080 * q^33 + 52656 * q^34 + 123500 * q^35 + 46656 * q^36 + 519434 * q^37 - 219488 * q^38 + 90018 * q^39 - 64000 * q^40 + 92394 * q^41 + 213408 * q^42 - 234532 * q^43 - 514560 * q^44 - 91125 * q^45 + 388800 * q^46 - 1277640 * q^47 - 110592 * q^48 + 152601 * q^49 + 125000 * q^50 - 177714 * q^51 - 213376 * q^52 - 835278 * q^53 - 157464 * q^54 + 1005000 * q^55 - 505856 * q^56 + 740772 * q^57 - 1059312 * q^58 - 3068760 * q^59 + 216000 * q^60 - 1009330 * q^61 + 2035264 * q^62 - 720252 * q^63 + 262144 * q^64 + 416750 * q^65 + 1736640 * q^66 + 3082172 * q^67 + 421248 * q^68 - 1312200 * q^69 + 988000 * q^70 - 3666720 * q^71 + 373248 * q^72 + 1122866 * q^73 + 4155472 * q^74 - 421875 * q^75 - 1755904 * q^76 + 7943520 * q^77 + 720144 * q^78 - 4128808 * q^79 - 512000 * q^80 + 531441 * q^81 + 739152 * q^82 + 4586556 * q^83 + 1707264 * q^84 - 822750 * q^85 - 1876256 * q^86 + 3575178 * q^87 - 4116480 * q^88 - 5763678 * q^89 - 729000 * q^90 + 3293992 * q^91 + 3110400 * q^92 - 6869016 * q^93 - 10221120 * q^94 + 3429500 * q^95 - 884736 * q^96 + 6747554 * q^97 + 1220808 * q^98 - 5861160 * 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

−988.000

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.d	✓	1
3.b	odd	2	1		90.8.a.c		1
4.b	odd	2	1		240.8.a.j		1
5.b	even	2	1		150.8.a.i		1
5.c	odd	4	2		150.8.c.a		2
15.d	odd	2	1		450.8.a.x		1
15.e	even	4	2		450.8.c.r		2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} + 988 $ T7 + 988 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 + 988 $ T + 988
$11$	$ T + 8040 $ T + 8040
$13$	$ T + 3334 $ T + 3334
$17$	$ T - 6582 $ T - 6582
$19$	$ T + 27436 $ T + 27436
$23$	$ T - 48600 $ T - 48600
$29$	$ T + 132414 $ T + 132414
$31$	$ T - 254408 $ T - 254408
$37$	$ T - 519434 $ T - 519434
$41$	$ T - 92394 $ T - 92394
$43$	$ T + 234532 $ T + 234532
$47$	$ T + 1277640 $ T + 1277640
$53$	$ T + 835278 $ T + 835278
$59$	$ T + 3068760 $ T + 3068760
$61$	$ T + 1009330 $ T + 1009330
$67$	$ T - 3082172 $ T - 3082172
$71$	$ T + 3666720 $ T + 3666720
$73$	$ T - 1122866 $ T - 1122866
$79$	$ T + 4128808 $ T + 4128808
$83$	$ T - 4586556 $ T - 4586556
$89$	$ T + 5763678 $ T + 5763678
$97$	$ T - 6747554 $ T - 6747554
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
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 80-90
Significant digits:
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