LMFDB - Newform orbit 150.12.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,12,Mod(1,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.1"); S:= CuspForms(chi, 12); N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	150.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-32,243,1024,0,-7776,-29348] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$115.251477084$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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 - 32 q^{2} + 243 q^{3} + 1024 q^{4} - 7776 q^{6} - 29348 q^{7} - 32768 q^{8} + 59049 q^{9} - 538680 q^{11} + 248832 q^{12} + 25606 q^{13} + 939136 q^{14} + 1048576 q^{16} + 9807162 q^{17} - 1889568 q^{18}+ \cdots - 31808515320 q^{99}+O(q^{100}) $

q - 32 * q^2 + 243 * q^3 + 1024 * q^4 - 7776 * q^6 - 29348 * q^7 - 32768 * q^8 + 59049 * q^9 - 538680 * q^11 + 248832 * q^12 + 25606 * q^13 + 939136 * q^14 + 1048576 * q^16 + 9807162 * q^17 - 1889568 * q^18 - 6599596 * q^19 - 7131564 * q^21 + 17237760 * q^22 + 35008200 * q^23 - 7962624 * q^24 - 819392 * q^26 + 14348907 * q^27 - 30052352 * q^28 + 1288146 * q^29 - 50347432 * q^31 - 33554432 * q^32 - 130899240 * q^33 - 313829184 * q^34 + 60466176 * q^36 + 650167894 * q^37 + 211187072 * q^38 + 6222258 * q^39 - 678700566 * q^41 + 228210048 * q^42 - 354979292 * q^43 - 551608320 * q^44 - 1120262400 * q^46 - 1215951480 * q^47 + 254803968 * q^48 - 1116021639 * q^49 + 2383140366 * q^51 + 26220544 * q^52 - 4566555138 * q^53 - 459165024 * q^54 + 961675264 * q^56 - 1603701828 * q^57 - 41220672 * q^58 - 2196450120 * q^59 + 9925999550 * q^61 + 1611117824 * q^62 - 1732970052 * q^63 + 1073741824 * q^64 + 4188775680 * q^66 + 674495812 * q^67 + 10042533888 * q^68 + 8506992600 * q^69 + 17538228960 * q^71 - 1934917632 * q^72 - 19619940914 * q^73 - 20805372608 * q^74 - 6757986304 * q^76 + 15809180640 * q^77 - 199112256 * q^78 - 1369906648 * q^79 + 3486784401 * q^81 + 21718418112 * q^82 + 62181646116 * q^83 - 7302721536 * q^84 + 11359337344 * q^86 + 313019478 * q^87 + 17651466240 * q^88 - 64990633758 * q^89 - 751484888 * q^91 + 35848396800 * q^92 - 12234425976 * q^93 + 38910447360 * q^94 - 8153726976 * q^96 - 101104524386 * q^97 + 35712692448 * q^98 - 31808515320 * 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

−32.0000

243.000

1024.00

−7776.00

−29348.0

−32768.0

59049.0

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	150.12.a.c		1
5.b	even	2	1		30.12.a.d	✓	1
5.c	odd	4	2		150.12.c.a		2
15.d	odd	2	1		90.12.a.e		1
20.d	odd	2	1		240.12.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.12.a.d	✓	1	5.b	even	2	1
90.12.a.e		1	15.d	odd	2	1
150.12.a.c		1	1.a	even	1	1	trivial
150.12.c.a		2	5.c	odd	4	2
240.12.a.c		1	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} + 29348 $ T7 + 29348 acting on $S_{12}^{\mathrm{new}}(\Gamma_0(150))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 32 $ T + 32
$3$	$ T - 243 $ T - 243
$5$	$ T $ T
$7$	$ T + 29348 $ T + 29348
$11$	$ T + 538680 $ T + 538680
$13$	$ T - 25606 $ T - 25606
$17$	$ T - 9807162 $ T - 9807162
$19$	$ T + 6599596 $ T + 6599596
$23$	$ T - 35008200 $ T - 35008200
$29$	$ T - 1288146 $ T - 1288146
$31$	$ T + 50347432 $ T + 50347432
$37$	$ T - 650167894 $ T - 650167894
$41$	$ T + 678700566 $ T + 678700566
$43$	$ T + 354979292 $ T + 354979292
$47$	$ T + 1215951480 $ T + 1215951480
$53$	$ T + 4566555138 $ T + 4566555138
$59$	$ T + 2196450120 $ T + 2196450120
$61$	$ T - 9925999550 $ T - 9925999550
$67$	$ T - 674495812 $ T - 674495812
$71$	$ T - 17538228960 $ T - 17538228960
$73$	$ T + 19619940914 $ T + 19619940914
$79$	$ T + 1369906648 $ T + 1369906648
$83$	$ T - 62181646116 $ T - 62181646116
$89$	$ T + 64990633758 $ T + 64990633758
$97$	$ T + 101104524386 $ T + 101104524386
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

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