LMFDB - Newform orbit 144.10.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,10,Mod(1,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	144.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,0,0,-830] 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:	$74.1651604076$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
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 - 830 q^{5} - 672 q^{7} - 73468 q^{11} - 78242 q^{13} + 161726 q^{17} + 653572 q^{19} - 1066696 q^{23} - 1264225 q^{25} - 3824838 q^{29} + 1579480 q^{31} + 557760 q^{35} + 16015590 q^{37} - 26268282 q^{41}+ \cdots + 1282496642 q^{97}+O(q^{100}) $

q - 830 * q^5 - 672 * q^7 - 73468 * q^11 - 78242 * q^13 + 161726 * q^17 + 653572 * q^19 - 1066696 * q^23 - 1264225 * q^25 - 3824838 * q^29 + 1579480 * q^31 + 557760 * q^35 + 16015590 * q^37 - 26268282 * q^41 + 44495228 * q^43 + 14324160 * q^47 - 39902023 * q^49 + 24386050 * q^53 + 60978440 * q^55 + 11942084 * q^59 - 189740258 * q^61 + 64940860 * q^65 + 106709572 * q^67 + 302754376 * q^71 + 81769546 * q^73 + 49370496 * q^77 - 315315352 * q^79 + 752833276 * q^83 - 134232580 * q^85 + 433284294 * q^89 + 52578624 * q^91 - 542464760 * q^95 + 1282496642 * q^97

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

−830.000

−672.000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ -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	144.10.a.e		1
3.b	odd	2	1		48.10.a.f		1
4.b	odd	2	1		72.10.a.b		1
12.b	even	2	1		24.10.a.a	✓	1
24.f	even	2	1		192.10.a.j		1
24.h	odd	2	1		192.10.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.10.a.a	✓	1	12.b	even	2	1
48.10.a.f		1	3.b	odd	2	1
72.10.a.b		1	4.b	odd	2	1
144.10.a.e		1	1.a	even	1	1	trivial
192.10.a.c		1	24.h	odd	2	1
192.10.a.j		1	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 830 $ T5 + 830 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(144))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T + 830 $ T + 830
$7$	$ T + 672 $ T + 672
$11$	$ T + 73468 $ T + 73468
$13$	$ T + 78242 $ T + 78242
$17$	$ T - 161726 $ T - 161726
$19$	$ T - 653572 $ T - 653572
$23$	$ T + 1066696 $ T + 1066696
$29$	$ T + 3824838 $ T + 3824838
$31$	$ T - 1579480 $ T - 1579480
$37$	$ T - 16015590 $ T - 16015590
$41$	$ T + 26268282 $ T + 26268282
$43$	$ T - 44495228 $ T - 44495228
$47$	$ T - 14324160 $ T - 14324160
$53$	$ T - 24386050 $ T - 24386050
$59$	$ T - 11942084 $ T - 11942084
$61$	$ T + 189740258 $ T + 189740258
$67$	$ T - 106709572 $ T - 106709572
$71$	$ T - 302754376 $ T - 302754376
$73$	$ T - 81769546 $ T - 81769546
$79$	$ T + 315315352 $ T + 315315352
$83$	$ T - 752833276 $ T - 752833276
$89$	$ T - 433284294 $ T - 433284294
$97$	$ T - 1282496642 $ T - 1282496642
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Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

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