LMFDB - Newform orbit 1638.4.a.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,4,Mod(1,1638)] 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("1638.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1638.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,0,4,14,0,-7,-8,0,-28,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$96.6451285894$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
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 - 2 q^{2} + 4 q^{4} + 14 q^{5} - 7 q^{7} - 8 q^{8} - 28 q^{10} - 8 q^{11} + 13 q^{13} + 14 q^{14} + 16 q^{16} + 98 q^{17} - 28 q^{19} + 56 q^{20} + 16 q^{22} + 52 q^{23} + 71 q^{25} - 26 q^{26} - 28 q^{28}+ \cdots - 98 q^{98}+O(q^{100}) $

q - 2 * q^2 + 4 * q^4 + 14 * q^5 - 7 * q^7 - 8 * q^8 - 28 * q^10 - 8 * q^11 + 13 * q^13 + 14 * q^14 + 16 * q^16 + 98 * q^17 - 28 * q^19 + 56 * q^20 + 16 * q^22 + 52 * q^23 + 71 * q^25 - 26 * q^26 - 28 * q^28 + 2 * q^29 - 168 * q^31 - 32 * q^32 - 196 * q^34 - 98 * q^35 - 146 * q^37 + 56 * q^38 - 112 * q^40 + 514 * q^41 - 236 * q^43 - 32 * q^44 - 104 * q^46 + 216 * q^47 + 49 * q^49 - 142 * q^50 + 52 * q^52 + 66 * q^53 - 112 * q^55 + 56 * q^56 - 4 * q^58 + 84 * q^59 + 446 * q^61 + 336 * q^62 + 64 * q^64 + 182 * q^65 + 292 * q^67 + 392 * q^68 + 196 * q^70 - 100 * q^71 + 450 * q^73 + 292 * q^74 - 112 * q^76 + 56 * q^77 + 392 * q^79 + 224 * q^80 - 1028 * q^82 + 292 * q^83 + 1372 * q^85 + 472 * q^86 + 64 * q^88 + 402 * q^89 - 91 * q^91 + 208 * q^92 - 432 * q^94 - 392 * q^95 + 314 * q^97 - 98 * 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

−2.00000

4.00000

14.0000

−7.00000

−8.00000

−28.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ -1 $
$7$	$ +1 $
$13$	$ -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	1638.4.a.h		1
3.b	odd	2	1		546.4.a.c	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
546.4.a.c	✓	1	3.b	odd	2	1
1638.4.a.h		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1638))$:

$ T_{5} - 14 $

T5 - 14

$ T_{11} + 8 $

T11 + 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T - 14 $ T - 14
$7$	$ T + 7 $ T + 7
$11$	$ T + 8 $ T + 8
$13$	$ T - 13 $ T - 13
$17$	$ T - 98 $ T - 98
$19$	$ T + 28 $ T + 28
$23$	$ T - 52 $ T - 52
$29$	$ T - 2 $ T - 2
$31$	$ T + 168 $ T + 168
$37$	$ T + 146 $ T + 146
$41$	$ T - 514 $ T - 514
$43$	$ T + 236 $ T + 236
$47$	$ T - 216 $ T - 216
$53$	$ T - 66 $ T - 66
$59$	$ T - 84 $ T - 84
$61$	$ T - 446 $ T - 446
$67$	$ T - 292 $ T - 292
$71$	$ T + 100 $ T + 100
$73$	$ T - 450 $ T - 450
$79$	$ T - 392 $ T - 392
$83$	$ T - 292 $ T - 292
$89$	$ T - 402 $ T - 402
$97$	$ T - 314 $ T - 314
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Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

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