LMFDB - Newform orbit 30.10.a.c

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:	$0$
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 - 16 q^{2} + 81 q^{3} + 256 q^{4} - 625 q^{5} - 1296 q^{6} + 7196 q^{7} - 4096 q^{8} + 6561 q^{9} + 10000 q^{10} - 45600 q^{11} + 20736 q^{12} - 34078 q^{13} - 115136 q^{14} - 50625 q^{15} + 65536 q^{16}+ \cdots - 299181600 q^{99}+O(q^{100}) $

q - 16 * q^2 + 81 * q^3 + 256 * q^4 - 625 * q^5 - 1296 * q^6 + 7196 * q^7 - 4096 * q^8 + 6561 * q^9 + 10000 * q^10 - 45600 * q^11 + 20736 * q^12 - 34078 * q^13 - 115136 * q^14 - 50625 * q^15 + 65536 * q^16 + 424326 * q^17 - 104976 * q^18 + 930716 * q^19 - 160000 * q^20 + 582876 * q^21 + 729600 * q^22 + 1572000 * q^23 - 331776 * q^24 + 390625 * q^25 + 545248 * q^26 + 531441 * q^27 + 1842176 * q^28 - 864486 * q^29 + 810000 * q^30 + 8703368 * q^31 - 1048576 * q^32 - 3693600 * q^33 - 6789216 * q^34 - 4497500 * q^35 + 1679616 * q^36 - 7526878 * q^37 - 14891456 * q^38 - 2760318 * q^39 + 2560000 * q^40 + 8562234 * q^41 - 9326016 * q^42 + 32831996 * q^43 - 11673600 * q^44 - 4100625 * q^45 - 25152000 * q^46 - 38536800 * q^47 + 5308416 * q^48 + 11428809 * q^49 - 6250000 * q^50 + 34370406 * q^51 - 8723968 * q^52 - 35746086 * q^53 - 8503056 * q^54 + 28500000 * q^55 - 29474816 * q^56 + 75387996 * q^57 + 13831776 * q^58 - 77109600 * q^59 - 12960000 * q^60 + 18400790 * q^61 - 139253888 * q^62 + 47212956 * q^63 + 16777216 * q^64 + 21298750 * q^65 + 59097600 * q^66 - 142510084 * q^67 + 108627456 * q^68 + 127332000 * q^69 + 71960000 * q^70 + 318643200 * q^71 - 26873856 * q^72 - 30899518 * q^73 + 120430048 * q^74 + 31640625 * q^75 + 238263296 * q^76 - 328137600 * q^77 + 44165088 * q^78 + 603013448 * q^79 - 40960000 * q^80 + 43046721 * q^81 - 136995744 * q^82 - 493844148 * q^83 + 149216256 * q^84 - 265203750 * q^85 - 525311936 * q^86 - 70023366 * q^87 + 186777600 * q^88 - 92882862 * q^89 + 65610000 * q^90 - 245225288 * q^91 + 402432000 * q^92 + 704972808 * q^93 + 616588800 * q^94 - 581697500 * q^95 - 84934656 * q^96 - 755725438 * q^97 - 182860944 * q^98 - 299181600 * 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

7196.00

−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.c	✓	1
3.b	odd	2	1		90.10.a.k		1
4.b	odd	2	1		240.10.a.a		1
5.b	even	2	1		150.10.a.f		1
5.c	odd	4	2		150.10.c.b		2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} - 7196 $ T7 - 7196 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 - 7196 $ T - 7196
$11$	$ T + 45600 $ T + 45600
$13$	$ T + 34078 $ T + 34078
$17$	$ T - 424326 $ T - 424326
$19$	$ T - 930716 $ T - 930716
$23$	$ T - 1572000 $ T - 1572000
$29$	$ T + 864486 $ T + 864486
$31$	$ T - 8703368 $ T - 8703368
$37$	$ T + 7526878 $ T + 7526878
$41$	$ T - 8562234 $ T - 8562234
$43$	$ T - 32831996 $ T - 32831996
$47$	$ T + 38536800 $ T + 38536800
$53$	$ T + 35746086 $ T + 35746086
$59$	$ T + 77109600 $ T + 77109600
$61$	$ T - 18400790 $ T - 18400790
$67$	$ T + 142510084 $ T + 142510084
$71$	$ T - 318643200 $ T - 318643200
$73$	$ T + 30899518 $ T + 30899518
$79$	$ T - 603013448 $ T - 603013448
$83$	$ T + 493844148 $ T + 493844148
$89$	$ T + 92882862 $ T + 92882862
$97$	$ T + 755725438 $ T + 755725438
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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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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