LMFDB - Newform orbit 50.12.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [50,12,Mod(1,50)] 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("50.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	50.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-32,318] 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:	$38.4171590280$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
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 - 32 q^{2} + 318 q^{3} + 1024 q^{4} - 10176 q^{6} + 70714 q^{7} - 32768 q^{8} - 76023 q^{9} + 238272 q^{11} + 325632 q^{12} + 2097478 q^{13} - 2262848 q^{14} + 1048576 q^{16} - 5955546 q^{17} + 2432736 q^{18}+ \cdots - 18114152256 q^{99}+O(q^{100}) $

q - 32 * q^2 + 318 * q^3 + 1024 * q^4 - 10176 * q^6 + 70714 * q^7 - 32768 * q^8 - 76023 * q^9 + 238272 * q^11 + 325632 * q^12 + 2097478 * q^13 - 2262848 * q^14 + 1048576 * q^16 - 5955546 * q^17 + 2432736 * q^18 + 10210820 * q^19 + 22487052 * q^21 - 7624704 * q^22 + 3535758 * q^23 - 10420224 * q^24 - 67119296 * q^26 - 80508060 * q^27 + 72411136 * q^28 - 139304850 * q^29 - 101002348 * q^31 - 33554432 * q^32 + 75770496 * q^33 + 190577472 * q^34 - 77847552 * q^36 + 524913814 * q^37 - 326746240 * q^38 + 666998004 * q^39 + 284590422 * q^41 - 719585664 * q^42 + 1253635078 * q^43 + 243990528 * q^44 - 113144256 * q^46 + 216106434 * q^47 + 333447168 * q^48 + 3023143053 * q^49 - 1893863628 * q^51 + 2147817472 * q^52 + 4881275358 * q^53 + 2576257920 * q^54 - 2317156352 * q^56 + 3247040760 * q^57 + 4457755200 * q^58 + 8692473300 * q^59 + 3296491802 * q^61 + 3232075136 * q^62 - 5375890422 * q^63 + 1073741824 * q^64 - 2424655872 * q^66 - 18275027966 * q^67 - 6098479104 * q^68 + 1124371044 * q^69 - 13287447588 * q^71 + 2491121664 * q^72 + 32505250798 * q^73 - 16797242048 * q^74 + 10455879680 * q^76 + 16849166208 * q^77 - 21343936128 * q^78 + 9297455960 * q^79 - 12134316699 * q^81 - 9106893504 * q^82 + 22741484838 * q^83 + 23026741248 * q^84 - 40116322496 * q^86 - 44298942300 * q^87 - 7807696896 * q^88 - 93378882390 * q^89 + 148321059292 * q^91 + 3620616192 * q^92 - 32118746664 * q^93 - 6915405888 * q^94 - 10670309376 * q^96 + 5811134014 * q^97 - 96740577696 * q^98 - 18114152256 * 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

−32.0000

318.000

1024.00

−10176.0

70714.0

−32768.0

−76023.0

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$5$	$ +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	50.12.a.b		1
5.b	even	2	1		10.12.a.c	✓	1
5.c	odd	4	2		50.12.b.b		2
15.d	odd	2	1		90.12.a.b		1
20.d	odd	2	1		80.12.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.12.a.c	✓	1	5.b	even	2	1
50.12.a.b		1	1.a	even	1	1	trivial
50.12.b.b		2	5.c	odd	4	2
80.12.a.e		1	20.d	odd	2	1
90.12.a.b		1	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 318 $ T3 - 318 acting on $S_{12}^{\mathrm{new}}(\Gamma_0(50))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 32 $ T + 32
$3$	$ T - 318 $ T - 318
$5$	$ T $ T
$7$	$ T - 70714 $ T - 70714
$11$	$ T - 238272 $ T - 238272
$13$	$ T - 2097478 $ T - 2097478
$17$	$ T + 5955546 $ T + 5955546
$19$	$ T - 10210820 $ T - 10210820
$23$	$ T - 3535758 $ T - 3535758
$29$	$ T + 139304850 $ T + 139304850
$31$	$ T + 101002348 $ T + 101002348
$37$	$ T - 524913814 $ T - 524913814
$41$	$ T - 284590422 $ T - 284590422
$43$	$ T - 1253635078 $ T - 1253635078
$47$	$ T - 216106434 $ T - 216106434
$53$	$ T - 4881275358 $ T - 4881275358
$59$	$ T - 8692473300 $ T - 8692473300
$61$	$ T - 3296491802 $ T - 3296491802
$67$	$ T + 18275027966 $ T + 18275027966
$71$	$ T + 13287447588 $ T + 13287447588
$73$	$ T - 32505250798 $ T - 32505250798
$79$	$ T - 9297455960 $ T - 9297455960
$83$	$ T - 22741484838 $ T - 22741484838
$89$	$ T + 93378882390 $ T + 93378882390
$97$	$ T - 5811134014 $ T - 5811134014
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Classical	Maass
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

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