LMFDB - Newform orbit 50.12.a.a

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,-207] 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:	$1$
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 - 32 q^{2} - 207 q^{3} + 1024 q^{4} + 6624 q^{6} + 19514 q^{7} - 32768 q^{8} - 134298 q^{9} - 259203 q^{11} - 211968 q^{12} + 1140428 q^{13} - 624448 q^{14} + 1048576 q^{16} + 4543029 q^{17} + 4297536 q^{18}+ \cdots + 34810444494 q^{99}+O(q^{100}) $

q - 32 * q^2 - 207 * q^3 + 1024 * q^4 + 6624 * q^6 + 19514 * q^7 - 32768 * q^8 - 134298 * q^9 - 259203 * q^11 - 211968 * q^12 + 1140428 * q^13 - 624448 * q^14 + 1048576 * q^16 + 4543029 * q^17 + 4297536 * q^18 - 294955 * q^19 - 4039398 * q^21 + 8294496 * q^22 + 12917358 * q^23 + 6782976 * q^24 - 36493696 * q^26 + 64469115 * q^27 + 19982336 * q^28 + 135279600 * q^29 - 170173798 * q^31 - 33554432 * q^32 + 53655021 * q^33 - 145376928 * q^34 - 137521152 * q^36 + 18577514 * q^37 + 9438560 * q^38 - 236068596 * q^39 - 903441003 * q^41 + 129260736 * q^42 - 1583839372 * q^43 - 265423872 * q^44 - 413355456 * q^46 + 2293409484 * q^47 - 217055232 * q^48 - 1596530547 * q^49 - 940407003 * q^51 + 1167798272 * q^52 - 4139435442 * q^53 - 2063011680 * q^54 - 639434752 * q^56 + 61055685 * q^57 - 4328947200 * q^58 - 2537617800 * q^59 - 12389028598 * q^61 + 5445561536 * q^62 - 2620691172 * q^63 + 1073741824 * q^64 - 1716960672 * q^66 - 299957941 * q^67 + 4652061696 * q^68 - 2673893106 * q^69 - 21161111988 * q^71 + 4400676864 * q^72 + 1820066123 * q^73 - 594480448 * q^74 - 302033920 * q^76 - 5058087342 * q^77 + 7554195072 * q^78 + 25997905310 * q^79 + 10445381001 * q^81 + 28910112096 * q^82 + 4410812463 * q^83 - 4136343552 * q^84 + 50682859904 * q^86 - 28002877200 * q^87 + 8493563904 * q^88 + 27987108735 * q^89 + 22254311992 * q^91 + 13227374592 * q^92 + 35225976186 * q^93 - 73389103488 * q^94 + 6945767424 * q^96 - 121928647486 * q^97 + 51088977504 * q^98 + 34810444494 * 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

−207.000

1024.00

6624.00

19514.0

−32768.0

−134298.

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.a	✓	1
5.b	even	2	1		50.12.a.e	yes	1
5.c	odd	4	2		50.12.b.d		2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 207 $ T3 + 207 acting on $S_{12}^{\mathrm{new}}(\Gamma_0(50))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 32 $ T + 32
$3$	$ T + 207 $ T + 207
$5$	$ T $ T
$7$	$ T - 19514 $ T - 19514
$11$	$ T + 259203 $ T + 259203
$13$	$ T - 1140428 $ T - 1140428
$17$	$ T - 4543029 $ T - 4543029
$19$	$ T + 294955 $ T + 294955
$23$	$ T - 12917358 $ T - 12917358
$29$	$ T - 135279600 $ T - 135279600
$31$	$ T + 170173798 $ T + 170173798
$37$	$ T - 18577514 $ T - 18577514
$41$	$ T + 903441003 $ T + 903441003
$43$	$ T + 1583839372 $ T + 1583839372
$47$	$ T - 2293409484 $ T - 2293409484
$53$	$ T + 4139435442 $ T + 4139435442
$59$	$ T + 2537617800 $ T + 2537617800
$61$	$ T + 12389028598 $ T + 12389028598
$67$	$ T + 299957941 $ T + 299957941
$71$	$ T + 21161111988 $ T + 21161111988
$73$	$ T - 1820066123 $ T - 1820066123
$79$	$ T - 25997905310 $ T - 25997905310
$83$	$ T - 4410812463 $ T - 4410812463
$89$	$ T - 27987108735 $ T - 27987108735
$97$	$ T + 121928647486 $ T + 121928647486
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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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