LMFDB - Newform orbit 150.12.a.g

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,-72464] 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 6)
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} - 72464 q^{7} + 32768 q^{8} + 59049 q^{9} - 408948 q^{11} + 248832 q^{12} - 1367558 q^{13} - 2318848 q^{14} + 1048576 q^{16} - 5422914 q^{17} + 1889568 q^{18}+ \cdots - 24147970452 q^{99}+O(q^{100}) $

q + 32 * q^2 + 243 * q^3 + 1024 * q^4 + 7776 * q^6 - 72464 * q^7 + 32768 * q^8 + 59049 * q^9 - 408948 * q^11 + 248832 * q^12 - 1367558 * q^13 - 2318848 * q^14 + 1048576 * q^16 - 5422914 * q^17 + 1889568 * q^18 + 15166100 * q^19 - 17608752 * q^21 - 13086336 * q^22 + 52194072 * q^23 + 7962624 * q^24 - 43761856 * q^26 + 14348907 * q^27 - 74203136 * q^28 + 118581150 * q^29 - 57652408 * q^31 + 33554432 * q^32 - 99374364 * q^33 - 173533248 * q^34 + 60466176 * q^36 + 375985186 * q^37 + 485315200 * q^38 - 332316594 * q^39 + 856316202 * q^41 - 563480064 * q^42 + 1245189172 * q^43 - 418762752 * q^44 + 1670210304 * q^46 + 1306762656 * q^47 + 254803968 * q^48 + 3273704553 * q^49 - 1317768102 * q^51 - 1400379392 * q^52 - 409556358 * q^53 + 459165024 * q^54 - 2374500352 * q^56 + 3685362300 * q^57 + 3794596800 * q^58 - 2882866260 * q^59 + 5731767302 * q^61 - 1844877056 * q^62 - 4278926736 * q^63 + 1073741824 * q^64 - 3179979648 * q^66 - 3893272244 * q^67 - 5553063936 * q^68 + 12683159496 * q^69 - 9075890088 * q^71 + 1934917632 * q^72 + 15571822822 * q^73 + 12031525952 * q^74 + 15530086400 * q^76 + 29634007872 * q^77 - 10634131008 * q^78 - 30196762600 * q^79 + 3486784401 * q^81 + 27402118464 * q^82 - 23135252628 * q^83 - 18031362048 * q^84 + 39846053504 * q^86 + 28815219450 * q^87 - 13400408064 * q^88 - 25614819990 * q^89 + 99098722912 * q^91 + 53446729728 * q^92 - 14009535144 * q^93 + 41816404992 * q^94 + 8153726976 * q^96 + 61937553406 * q^97 + 104758545696 * q^98 - 24147970452 * 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

−72464.0

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.g		1
5.b	even	2	1		6.12.a.a	✓	1
5.c	odd	4	2		150.12.c.f		2
15.d	odd	2	1		18.12.a.c		1
20.d	odd	2	1		48.12.a.h		1
40.e	odd	2	1		192.12.a.b		1
40.f	even	2	1		192.12.a.l		1
45.h	odd	6	2		162.12.c.d		2
45.j	even	6	2		162.12.c.g		2
60.h	even	2	1		144.12.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.12.a.a	✓	1	5.b	even	2	1
18.12.a.c		1	15.d	odd	2	1
48.12.a.h		1	20.d	odd	2	1
144.12.a.b		1	60.h	even	2	1
150.12.a.g		1	1.a	even	1	1	trivial
150.12.c.f		2	5.c	odd	4	2
162.12.c.d		2	45.h	odd	6	2
162.12.c.g		2	45.j	even	6	2
192.12.a.b		1	40.e	odd	2	1
192.12.a.l		1	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} + 72464 $ T7 + 72464 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 + 72464 $ T + 72464
$11$	$ T + 408948 $ T + 408948
$13$	$ T + 1367558 $ T + 1367558
$17$	$ T + 5422914 $ T + 5422914
$19$	$ T - 15166100 $ T - 15166100
$23$	$ T - 52194072 $ T - 52194072
$29$	$ T - 118581150 $ T - 118581150
$31$	$ T + 57652408 $ T + 57652408
$37$	$ T - 375985186 $ T - 375985186
$41$	$ T - 856316202 $ T - 856316202
$43$	$ T - 1245189172 $ T - 1245189172
$47$	$ T - 1306762656 $ T - 1306762656
$53$	$ T + 409556358 $ T + 409556358
$59$	$ T + 2882866260 $ T + 2882866260
$61$	$ T - 5731767302 $ T - 5731767302
$67$	$ T + 3893272244 $ T + 3893272244
$71$	$ T + 9075890088 $ T + 9075890088
$73$	$ T - 15571822822 $ T - 15571822822
$79$	$ T + 30196762600 $ T + 30196762600
$83$	$ T + 23135252628 $ T + 23135252628
$89$	$ T + 25614819990 $ T + 25614819990
$97$	$ T - 61937553406 $ T - 61937553406
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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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