LMFDB - Newform orbit 18.14.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [18,14,Mod(1,18)] 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("18.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	18.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,64,0,4096,15936] 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:	$19.3015672113$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 + 64 q^{2} + 4096 q^{4} + 15936 q^{5} + 98252 q^{7} + 262144 q^{8} + 1019904 q^{10} - 1630464 q^{11} + 22316306 q^{13} + 6288128 q^{14} + 16777216 q^{16} + 122937984 q^{17} - 74272984 q^{19} + 65273856 q^{20}+ \cdots - 5583075513792 q^{98}+O(q^{100}) $

q + 64 * q^2 + 4096 * q^4 + 15936 * q^5 + 98252 * q^7 + 262144 * q^8 + 1019904 * q^10 - 1630464 * q^11 + 22316306 * q^13 + 6288128 * q^14 + 16777216 * q^16 + 122937984 * q^17 - 74272984 * q^19 + 65273856 * q^20 - 104349696 * q^22 + 1069509120 * q^23 - 966747029 * q^25 + 1428243584 * q^26 + 402440192 * q^28 + 5607090624 * q^29 + 2162031116 * q^31 + 1073741824 * q^32 + 7868030976 * q^34 + 1565743872 * q^35 - 5959452922 * q^37 - 4753470976 * q^38 + 4177526784 * q^40 - 21676851840 * q^41 - 61101030232 * q^43 - 6678380544 * q^44 + 68448583680 * q^46 - 136471948800 * q^47 - 87235554903 * q^49 - 61871809856 * q^50 + 91407589376 * q^52 - 557015616 * q^53 - 25983074304 * q^55 + 25756172288 * q^56 + 358853799936 * q^58 + 302211949056 * q^59 - 190535454658 * q^61 + 138369991424 * q^62 + 68719476736 * q^64 + 355632652416 * q^65 - 918343123024 * q^67 + 503553982464 * q^68 + 100207607808 * q^70 + 1086593292288 * q^71 - 77275903210 * q^73 - 381404987008 * q^74 - 304222142464 * q^76 - 160196348928 * q^77 + 2624363498636 * q^79 + 267361714176 * q^80 - 1387318517760 * q^82 - 3269608182528 * q^83 + 1959139713024 * q^85 - 3910465934848 * q^86 - 427416354816 * q^88 - 5922230600448 * q^89 + 2192621697112 * q^91 + 4380709355520 * q^92 - 8734204723200 * q^94 - 1183614273024 * q^95 + 5340133325582 * q^97 - 5583075513792 * q^98

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

64.0000

4096.00

15936.0

98252.0

262144.

1.01990e6

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	18.14.a.e	yes	1
3.b	odd	2	1		18.14.a.b	✓	1
4.b	odd	2	1		144.14.a.h		1
12.b	even	2	1		144.14.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.14.a.b	✓	1	3.b	odd	2	1
18.14.a.e	yes	1	1.a	even	1	1	trivial
144.14.a.c		1	12.b	even	2	1
144.14.a.h		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 15936 $ T5 - 15936 acting on $S_{14}^{\mathrm{new}}(\Gamma_0(18))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 64 $ T - 64
$3$	$ T $ T
$5$	$ T - 15936 $ T - 15936
$7$	$ T - 98252 $ T - 98252
$11$	$ T + 1630464 $ T + 1630464
$13$	$ T - 22316306 $ T - 22316306
$17$	$ T - 122937984 $ T - 122937984
$19$	$ T + 74272984 $ T + 74272984
$23$	$ T - 1069509120 $ T - 1069509120
$29$	$ T - 5607090624 $ T - 5607090624
$31$	$ T - 2162031116 $ T - 2162031116
$37$	$ T + 5959452922 $ T + 5959452922
$41$	$ T + 21676851840 $ T + 21676851840
$43$	$ T + 61101030232 $ T + 61101030232
$47$	$ T + 136471948800 $ T + 136471948800
$53$	$ T + 557015616 $ T + 557015616
$59$	$ T - 302211949056 $ T - 302211949056
$61$	$ T + 190535454658 $ T + 190535454658
$67$	$ T + 918343123024 $ T + 918343123024
$71$	$ T - 1086593292288 $ T - 1086593292288
$73$	$ T + 77275903210 $ T + 77275903210
$79$	$ T - 2624363498636 $ T - 2624363498636
$83$	$ T + 3269608182528 $ T + 3269608182528
$89$	$ T + 5922230600448 $ T + 5922230600448
$97$	$ T - 5340133325582 $ T - 5340133325582
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Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

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