LMFDB - Newform orbit 147.4.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [147,4,Mod(1,147)] 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("147.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	147.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-3,3,1,-3] 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:	$8.67328077084$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
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 - 3 q^{2} + 3 q^{3} + q^{4} - 3 q^{5} - 9 q^{6} + 21 q^{8} + 9 q^{9} + 9 q^{10} - 15 q^{11} + 3 q^{12} - 64 q^{13} - 9 q^{15} - 71 q^{16} + 84 q^{17} - 27 q^{18} - 16 q^{19} - 3 q^{20} + 45 q^{22} - 84 q^{23}+ \cdots - 135 q^{99}+O(q^{100}) $

q - 3 * q^2 + 3 * q^3 + q^4 - 3 * q^5 - 9 * q^6 + 21 * q^8 + 9 * q^9 + 9 * q^10 - 15 * q^11 + 3 * q^12 - 64 * q^13 - 9 * q^15 - 71 * q^16 + 84 * q^17 - 27 * q^18 - 16 * q^19 - 3 * q^20 + 45 * q^22 - 84 * q^23 + 63 * q^24 - 116 * q^25 + 192 * q^26 + 27 * q^27 - 297 * q^29 + 27 * q^30 - 253 * q^31 + 45 * q^32 - 45 * q^33 - 252 * q^34 + 9 * q^36 - 316 * q^37 + 48 * q^38 - 192 * q^39 - 63 * q^40 + 360 * q^41 + 26 * q^43 - 15 * q^44 - 27 * q^45 + 252 * q^46 - 30 * q^47 - 213 * q^48 + 348 * q^50 + 252 * q^51 - 64 * q^52 + 363 * q^53 - 81 * q^54 + 45 * q^55 - 48 * q^57 + 891 * q^58 - 15 * q^59 - 9 * q^60 - 118 * q^61 + 759 * q^62 + 433 * q^64 + 192 * q^65 + 135 * q^66 - 370 * q^67 + 84 * q^68 - 252 * q^69 - 342 * q^71 + 189 * q^72 + 362 * q^73 + 948 * q^74 - 348 * q^75 - 16 * q^76 + 576 * q^78 + 467 * q^79 + 213 * q^80 + 81 * q^81 - 1080 * q^82 + 477 * q^83 - 252 * q^85 - 78 * q^86 - 891 * q^87 - 315 * q^88 + 906 * q^89 + 81 * q^90 - 84 * q^92 - 759 * q^93 + 90 * q^94 + 48 * q^95 + 135 * q^96 + 503 * q^97 - 135 * 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

−3.00000

3.00000

1.00000

−3.00000

−9.00000

21.0000

9.00000

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$7$	$ +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	147.4.a.b		1
3.b	odd	2	1		441.4.a.l		1
4.b	odd	2	1		2352.4.a.i		1
7.b	odd	2	1		147.4.a.a		1
7.c	even	3	2		21.4.e.a	✓	2
7.d	odd	6	2		147.4.e.h		2
21.c	even	2	1		441.4.a.k		1
21.g	even	6	2		441.4.e.c		2
21.h	odd	6	2		63.4.e.a		2
28.d	even	2	1		2352.4.a.bd		1
28.g	odd	6	2		336.4.q.e		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.e.a	✓	2	7.c	even	3	2
63.4.e.a		2	21.h	odd	6	2
147.4.a.a		1	7.b	odd	2	1
147.4.a.b		1	1.a	even	1	1	trivial
147.4.e.h		2	7.d	odd	6	2
336.4.q.e		2	28.g	odd	6	2
441.4.a.k		1	21.c	even	2	1
441.4.a.l		1	3.b	odd	2	1
441.4.e.c		2	21.g	even	6	2
2352.4.a.i		1	4.b	odd	2	1
2352.4.a.bd		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(147))$:

$ T_{2} + 3 $

T2 + 3

$ T_{5} + 3 $

T5 + 3

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 3 $ T + 3
$3$	$ T - 3 $ T - 3
$5$	$ T + 3 $ T + 3
$7$	$ T $ T
$11$	$ T + 15 $ T + 15
$13$	$ T + 64 $ T + 64
$17$	$ T - 84 $ T - 84
$19$	$ T + 16 $ T + 16
$23$	$ T + 84 $ T + 84
$29$	$ T + 297 $ T + 297
$31$	$ T + 253 $ T + 253
$37$	$ T + 316 $ T + 316
$41$	$ T - 360 $ T - 360
$43$	$ T - 26 $ T - 26
$47$	$ T + 30 $ T + 30
$53$	$ T - 363 $ T - 363
$59$	$ T + 15 $ T + 15
$61$	$ T + 118 $ T + 118
$67$	$ T + 370 $ T + 370
$71$	$ T + 342 $ T + 342
$73$	$ T - 362 $ T - 362
$79$	$ T - 467 $ T - 467
$83$	$ T - 477 $ T - 477
$89$	$ T - 906 $ T - 906
$97$	$ T - 503 $ T - 503
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Elliptic curves over $\Q$
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Belyi maps

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
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