LMFDB - Newform orbit 30.8.a.b

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} + 416 q^{7} - 512 q^{8} + 729 q^{9} - 1000 q^{10} - 3948 q^{11} - 1728 q^{12} - 7978 q^{13} - 3328 q^{14} - 3375 q^{15} + 4096 q^{16} - 34014 q^{17}+ \cdots - 2878092 q^{99}+O(q^{100}) $

q - 8 * q^2 - 27 * q^3 + 64 * q^4 + 125 * q^5 + 216 * q^6 + 416 * q^7 - 512 * q^8 + 729 * q^9 - 1000 * q^10 - 3948 * q^11 - 1728 * q^12 - 7978 * q^13 - 3328 * q^14 - 3375 * q^15 + 4096 * q^16 - 34014 * q^17 - 5832 * q^18 + 3020 * q^19 + 8000 * q^20 - 11232 * q^21 + 31584 * q^22 - 66528 * q^23 + 13824 * q^24 + 15625 * q^25 + 63824 * q^26 - 19683 * q^27 + 26624 * q^28 - 18570 * q^29 + 27000 * q^30 + 208832 * q^31 - 32768 * q^32 + 106596 * q^33 + 272112 * q^34 + 52000 * q^35 + 46656 * q^36 - 301474 * q^37 - 24160 * q^38 + 215406 * q^39 - 64000 * q^40 - 460998 * q^41 + 89856 * q^42 + 343052 * q^43 - 252672 * q^44 + 91125 * q^45 + 532224 * q^46 - 1356264 * q^47 - 110592 * q^48 - 650487 * q^49 - 125000 * q^50 + 918378 * q^51 - 510592 * q^52 + 617982 * q^53 + 157464 * q^54 - 493500 * q^55 - 212992 * q^56 - 81540 * q^57 + 148560 * q^58 + 939540 * q^59 - 216000 * q^60 - 204298 * q^61 - 1670656 * q^62 + 303264 * q^63 + 262144 * q^64 - 997250 * q^65 - 852768 * q^66 - 758524 * q^67 - 2176896 * q^68 + 1796256 * q^69 - 416000 * q^70 + 912072 * q^71 - 373248 * q^72 + 3043322 * q^73 + 2411792 * q^74 - 421875 * q^75 + 193280 * q^76 - 1642368 * q^77 - 1723248 * q^78 + 6010880 * q^79 + 512000 * q^80 + 531441 * q^81 + 3687984 * q^82 + 9723012 * q^83 - 718848 * q^84 - 4251750 * q^85 - 2744416 * q^86 + 501390 * q^87 + 2021376 * q^88 + 7160010 * q^89 - 729000 * q^90 - 3318848 * q^91 - 4257792 * q^92 - 5638464 * q^93 + 10850112 * q^94 + 377500 * q^95 + 884736 * q^96 - 16785214 * q^97 + 5203896 * q^98 - 2878092 * 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

416.000

−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.b	✓	1
3.b	odd	2	1		90.8.a.h		1
4.b	odd	2	1		240.8.a.m		1
5.b	even	2	1		150.8.a.n		1
5.c	odd	4	2		150.8.c.g		2
15.d	odd	2	1		450.8.a.e		1
15.e	even	4	2		450.8.c.n		2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} - 416 $ T7 - 416 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 - 416 $ T - 416
$11$	$ T + 3948 $ T + 3948
$13$	$ T + 7978 $ T + 7978
$17$	$ T + 34014 $ T + 34014
$19$	$ T - 3020 $ T - 3020
$23$	$ T + 66528 $ T + 66528
$29$	$ T + 18570 $ T + 18570
$31$	$ T - 208832 $ T - 208832
$37$	$ T + 301474 $ T + 301474
$41$	$ T + 460998 $ T + 460998
$43$	$ T - 343052 $ T - 343052
$47$	$ T + 1356264 $ T + 1356264
$53$	$ T - 617982 $ T - 617982
$59$	$ T - 939540 $ T - 939540
$61$	$ T + 204298 $ T + 204298
$67$	$ T + 758524 $ T + 758524
$71$	$ T - 912072 $ T - 912072
$73$	$ T - 3043322 $ T - 3043322
$79$	$ T - 6010880 $ T - 6010880
$83$	$ T - 9723012 $ T - 9723012
$89$	$ T - 7160010 $ T - 7160010
$97$	$ T + 16785214 $ T + 16785214
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Classical	Maass
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Elliptic curves over $\Q$
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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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