LMFDB - Newform orbit 30.10.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [30,10,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, 10); 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, 10, names="a")

Level:	$ N $	$=$	$ 30 = 2 \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	30.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,16,-81,256,625] 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:	$15.4510750849$
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 + 16 q^{2} - 81 q^{3} + 256 q^{4} + 625 q^{5} - 1296 q^{6} - 10336 q^{7} + 4096 q^{8} + 6561 q^{9} + 10000 q^{10} + 27420 q^{11} - 20736 q^{12} - 169762 q^{13} - 165376 q^{14} - 50625 q^{15} + 65536 q^{16}+ \cdots + 179902620 q^{99}+O(q^{100}) $

q + 16 * q^2 - 81 * q^3 + 256 * q^4 + 625 * q^5 - 1296 * q^6 - 10336 * q^7 + 4096 * q^8 + 6561 * q^9 + 10000 * q^10 + 27420 * q^11 - 20736 * q^12 - 169762 * q^13 - 165376 * q^14 - 50625 * q^15 + 65536 * q^16 - 385086 * q^17 + 104976 * q^18 - 637084 * q^19 + 160000 * q^20 + 837216 * q^21 + 438720 * q^22 - 1298400 * q^23 - 331776 * q^24 + 390625 * q^25 - 2716192 * q^26 - 531441 * q^27 - 2646016 * q^28 + 7162974 * q^29 - 810000 * q^30 - 7031872 * q^31 + 1048576 * q^32 - 2221020 * q^33 - 6161376 * q^34 - 6460000 * q^35 + 1679616 * q^36 + 1926038 * q^37 - 10193344 * q^38 + 13750722 * q^39 + 2560000 * q^40 + 8896074 * q^41 + 13395456 * q^42 + 32429444 * q^43 + 7019520 * q^44 + 4100625 * q^45 - 20774400 * q^46 + 17206440 * q^47 - 5308416 * q^48 + 66479289 * q^49 + 6250000 * q^50 + 31191966 * q^51 - 43459072 * q^52 - 20642154 * q^53 - 8503056 * q^54 + 17137500 * q^55 - 42336256 * q^56 + 51603804 * q^57 + 114607584 * q^58 - 63193380 * q^59 - 12960000 * q^60 - 63758050 * q^61 - 112509952 * q^62 - 67814496 * q^63 + 16777216 * q^64 - 106101250 * q^65 - 35536320 * q^66 + 145261964 * q^67 - 98582016 * q^68 + 105170400 * q^69 - 103360000 * q^70 - 367656840 * q^71 + 26873856 * q^72 + 252486218 * q^73 + 30816608 * q^74 - 31640625 * q^75 - 163093504 * q^76 - 283413120 * q^77 + 220011552 * q^78 - 185523712 * q^79 + 40960000 * q^80 + 43046721 * q^81 + 142337184 * q^82 - 467897652 * q^83 + 214327296 * q^84 - 240678750 * q^85 + 518871104 * q^86 - 580200894 * q^87 + 112312320 * q^88 + 579096378 * q^89 + 65610000 * q^90 + 1754660032 * q^91 - 332390400 * q^92 + 569581632 * q^93 + 275303040 * q^94 - 398177500 * q^95 - 84934656 * q^96 - 1314516862 * q^97 + 1063668624 * q^98 + 179902620 * 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

16.0000

−81.0000

256.000

625.000

−1296.00

−10336.0

4096.00

6561.00

10000.0

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	30.10.a.e	✓	1
3.b	odd	2	1		90.10.a.a		1
4.b	odd	2	1		240.10.a.j		1
5.b	even	2	1		150.10.a.e		1
5.c	odd	4	2		150.10.c.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.10.a.e	✓	1	1.a	even	1	1	trivial
90.10.a.a		1	3.b	odd	2	1
150.10.a.e		1	5.b	even	2	1
150.10.c.c		2	5.c	odd	4	2
240.10.a.j		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} + 10336 $ T7 + 10336 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(30))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 16 $ T - 16
$3$	$ T + 81 $ T + 81
$5$	$ T - 625 $ T - 625
$7$	$ T + 10336 $ T + 10336
$11$	$ T - 27420 $ T - 27420
$13$	$ T + 169762 $ T + 169762
$17$	$ T + 385086 $ T + 385086
$19$	$ T + 637084 $ T + 637084
$23$	$ T + 1298400 $ T + 1298400
$29$	$ T - 7162974 $ T - 7162974
$31$	$ T + 7031872 $ T + 7031872
$37$	$ T - 1926038 $ T - 1926038
$41$	$ T - 8896074 $ T - 8896074
$43$	$ T - 32429444 $ T - 32429444
$47$	$ T - 17206440 $ T - 17206440
$53$	$ T + 20642154 $ T + 20642154
$59$	$ T + 63193380 $ T + 63193380
$61$	$ T + 63758050 $ T + 63758050
$67$	$ T - 145261964 $ T - 145261964
$71$	$ T + 367656840 $ T + 367656840
$73$	$ T - 252486218 $ T - 252486218
$79$	$ T + 185523712 $ T + 185523712
$83$	$ T + 467897652 $ T + 467897652
$89$	$ T - 579096378 $ T - 579096378
$97$	$ T + 1314516862 $ T + 1314516862
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	30.a (trivial)

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