LMFDB - Newform orbit 242.6.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [242,6,Mod(1,242)] 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("242.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(242, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	242.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.8128843947$
Analytic rank:	$1$
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 + 4 q^{2} + 4 q^{3} + 16 q^{4} - 9 q^{5} + 16 q^{6} - 32 q^{7} + 64 q^{8} - 227 q^{9} - 36 q^{10} + 64 q^{12} + 145 q^{13} - 128 q^{14} - 36 q^{15} + 256 q^{16} - 603 q^{17} - 908 q^{18} - 1448 q^{19}+ \cdots - 63132 q^{98}+O(q^{100}) $

q + 4 * q^2 + 4 * q^3 + 16 * q^4 - 9 * q^5 + 16 * q^6 - 32 * q^7 + 64 * q^8 - 227 * q^9 - 36 * q^10 + 64 * q^12 + 145 * q^13 - 128 * q^14 - 36 * q^15 + 256 * q^16 - 603 * q^17 - 908 * q^18 - 1448 * q^19 - 144 * q^20 - 128 * q^21 - 60 * q^23 + 256 * q^24 - 3044 * q^25 + 580 * q^26 - 1880 * q^27 - 512 * q^28 - 4407 * q^29 - 144 * q^30 - 2104 * q^31 + 1024 * q^32 - 2412 * q^34 + 288 * q^35 - 3632 * q^36 + 13055 * q^37 - 5792 * q^38 + 580 * q^39 - 576 * q^40 - 1215 * q^41 - 512 * q^42 - 6476 * q^43 + 2043 * q^45 - 240 * q^46 - 24300 * q^47 + 1024 * q^48 - 15783 * q^49 - 12176 * q^50 - 2412 * q^51 + 2320 * q^52 + 12363 * q^53 - 7520 * q^54 - 2048 * q^56 - 5792 * q^57 - 17628 * q^58 + 32340 * q^59 - 576 * q^60 - 42782 * q^61 - 8416 * q^62 + 7264 * q^63 + 4096 * q^64 - 1305 * q^65 + 56288 * q^67 - 9648 * q^68 - 240 * q^69 + 1152 * q^70 - 54084 * q^71 - 14528 * q^72 - 16394 * q^73 + 52220 * q^74 - 12176 * q^75 - 23168 * q^76 + 2320 * q^78 - 76700 * q^79 - 2304 * q^80 + 47641 * q^81 - 4860 * q^82 - 71928 * q^83 - 2048 * q^84 + 5427 * q^85 - 25904 * q^86 - 17628 * q^87 + 97539 * q^89 + 8172 * q^90 - 4640 * q^91 - 960 * q^92 - 8416 * q^93 - 97200 * q^94 + 13032 * q^95 + 4096 * q^96 + 93467 * q^97 - 63132 * 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

4.00000

16.0000

−9.00000

16.0000

−32.0000

64.0000

−227.000

−36.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$11$	$ -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	242.6.a.e	yes	1
11.b	odd	2	1		242.6.a.b	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
242.6.a.b	✓	1	11.b	odd	2	1
242.6.a.e	yes	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 $S_{6}^{\mathrm{new}}(\Gamma_0(242))$:

$ T_{3} - 4 $

T3 - 4

$ T_{7} + 32 $

T7 + 32

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 4 $ T - 4
$3$	$ T - 4 $ T - 4
$5$	$ T + 9 $ T + 9
$7$	$ T + 32 $ T + 32
$11$	$ T $ T
$13$	$ T - 145 $ T - 145
$17$	$ T + 603 $ T + 603
$19$	$ T + 1448 $ T + 1448
$23$	$ T + 60 $ T + 60
$29$	$ T + 4407 $ T + 4407
$31$	$ T + 2104 $ T + 2104
$37$	$ T - 13055 $ T - 13055
$41$	$ T + 1215 $ T + 1215
$43$	$ T + 6476 $ T + 6476
$47$	$ T + 24300 $ T + 24300
$53$	$ T - 12363 $ T - 12363
$59$	$ T - 32340 $ T - 32340
$61$	$ T + 42782 $ T + 42782
$67$	$ T - 56288 $ T - 56288
$71$	$ T + 54084 $ T + 54084
$73$	$ T + 16394 $ T + 16394
$79$	$ T + 76700 $ T + 76700
$83$	$ T + 71928 $ T + 71928
$89$	$ T - 97539 $ T - 97539
$97$	$ T - 93467 $ T - 93467
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Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)

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