LMFDB - Newform orbit 150.12.a.b

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,-10556] 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:	$0$
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} - 10556 q^{7} - 32768 q^{8} + 59049 q^{9} + 516912 q^{11} - 248832 q^{12} - 1478402 q^{13} + 337792 q^{14} + 1048576 q^{16} + 1265874 q^{17} - 1889568 q^{18}+ \cdots + 30523136688 q^{99}+O(q^{100}) $

q - 32 * q^2 - 243 * q^3 + 1024 * q^4 + 7776 * q^6 - 10556 * q^7 - 32768 * q^8 + 59049 * q^9 + 516912 * q^11 - 248832 * q^12 - 1478402 * q^13 + 337792 * q^14 + 1048576 * q^16 + 1265874 * q^17 - 1889568 * q^18 + 7842380 * q^19 + 2565108 * q^21 - 16541184 * q^22 - 1561032 * q^23 + 7962624 * q^24 + 47308864 * q^26 - 14348907 * q^27 - 10809344 * q^28 + 33939570 * q^29 + 14610032 * q^31 - 33554432 * q^32 - 125609616 * q^33 - 40507968 * q^34 + 60466176 * q^36 - 290433266 * q^37 - 250956160 * q^38 + 359251686 * q^39 + 820995642 * q^41 - 82083456 * q^42 + 328352908 * q^43 + 529317888 * q^44 + 49953024 * q^46 + 19932264 * q^47 - 254803968 * q^48 - 1865897607 * q^49 - 307607382 * q^51 - 1513883648 * q^52 + 1077746478 * q^53 + 459165024 * q^54 + 345899008 * q^56 - 1905698340 * q^57 - 1086066240 * q^58 - 10780120560 * q^59 - 9856407538 * q^61 - 467521024 * q^62 - 623321244 * q^63 + 1073741824 * q^64 + 4019507712 * q^66 + 13111086124 * q^67 + 1296254976 * q^68 + 379330776 * q^69 - 28727516928 * q^71 - 1934917632 * q^72 - 13379419682 * q^73 + 9293864512 * q^74 + 8030597120 * q^76 - 5456523072 * q^77 - 11496053952 * q^78 + 51583351520 * q^79 + 3486784401 * q^81 - 26271860544 * q^82 - 4311034212 * q^83 + 2626670592 * q^84 - 10507293056 * q^86 - 8247315510 * q^87 - 16938172416 * q^88 - 8526268590 * q^89 + 15606011512 * q^91 - 1598496768 * q^92 - 3550237776 * q^93 - 637832448 * q^94 + 8153726976 * q^96 - 72302803586 * q^97 + 59708723424 * q^98 + 30523136688 * 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

−10556.0

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} + 10556 $ T7 + 10556 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 + 10556 $ T + 10556
$11$	$ T - 516912 $ T - 516912
$13$	$ T + 1478402 $ T + 1478402
$17$	$ T - 1265874 $ T - 1265874
$19$	$ T - 7842380 $ T - 7842380
$23$	$ T + 1561032 $ T + 1561032
$29$	$ T - 33939570 $ T - 33939570
$31$	$ T - 14610032 $ T - 14610032
$37$	$ T + 290433266 $ T + 290433266
$41$	$ T - 820995642 $ T - 820995642
$43$	$ T - 328352908 $ T - 328352908
$47$	$ T - 19932264 $ T - 19932264
$53$	$ T - 1077746478 $ T - 1077746478
$59$	$ T + 10780120560 $ T + 10780120560
$61$	$ T + 9856407538 $ T + 9856407538
$67$	$ T - 13111086124 $ T - 13111086124
$71$	$ T + 28727516928 $ T + 28727516928
$73$	$ T + 13379419682 $ T + 13379419682
$79$	$ T - 51583351520 $ T - 51583351520
$83$	$ T + 4311034212 $ T + 4311034212
$89$	$ T + 8526268590 $ T + 8526268590
$97$	$ T + 72302803586 $ T + 72302803586
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Classical	Maass
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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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