LMFDB - Newform orbit 8.51.d.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8,51,Mod(3,8)] 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("8.3"); S:= CuspForms(chi, 51); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 51, names="a")

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 51 $
Character orbit:	$[\chi]$	$=$	8.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$126.668362818$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 33554432 q^{2} - 1357246255682 q^{3} + 11\!\cdots\!24 q^{4} + 45\!\cdots\!24 q^{6} - 37\!\cdots\!68 q^{8} + 11\!\cdots\!75 q^{9} - 21\!\cdots\!26 q^{11} - 15\!\cdots\!68 q^{12} + 12\!\cdots\!76 q^{16}+ \cdots - 24\!\cdots\!50 q^{99}+O(q^{100}) $

q - 33554432 * q^2 - 1357246255682 * q^3 + 1125899906842624 * q^4 + 45541627193536282624 * q^6 - 37778931862957161709568 * q^8 + 1124219410870956328514875 * q^9 - 216566032867586063515304626 * q^11 - 1528123432834864034839789568 * q^12 + 1267650600228229401496703205376 * q^16 + 11484333194425479818498080615682 * q^17 - 37722543775149564900122034176000 * q^18 - 101750491668668099890091086635874 * q^19 + 7266750223365181572371970032402432 * q^22 + 51275313814664012486277349953765376 * q^24 + 88817841970012523233890533447265625 * q^25 - 551478230213219955633037045381464932 * q^27 - 42535295865117307932921825928971026432 * q^32 + 293933437217436130012033034140113384932 * q^33 - 385350277237692541637166188149419802624 * q^34 + 1265758529970279365270731587460268032000 * q^36 + 3414179953662890288331268840329542893568 * q^38 - 30490943576570305325097526558911137086126 * q^41 - 111409311987059022761513629469659116638386 * q^43 - 243831676230891796237808347158285201178624 * q^44 - 1720514030672804209817944272163823452946432 * q^48 + 1798465042647412146620280340569649349251249 * q^49 - 2980232238769531250000000000000000000000000 * q^50 - 15587068227138484598832776876538179750805124 * q^51 + 18504538775169834502331758492733279121178624 * q^54 + 138100473831102314731823683070462656237536068 * q^57 - 130885310902277473186937781461662575640682002 * q^59 + 1427247692705959881058285969449495136382746624 * q^64 - 9862769531638729838831921625808113046990618624 * q^66 - 4619012503980257606885070447298279180095842818 * q^67 + 12930209673753302225271461532958912606600429568 * q^68 - 42471808522307700982979924641687616381517824000 * q^72 - 69994247752061998294033733980392006291536999182 * q^73 - 120547683451555087685846956446766853332519531250 * q^75 - 114560869090944603103271953776556504613310693376 * q^76 - 58583089741350785482953115116573116007765210251 * q^81 + 1023106292855865103250222848289177743399093010432 * q^82 + 115552090148733990836824537145935185395928788318 * q^83 + 3738276183236556859237661297112872892422791626752 * q^86 + 8181633399535475076219396013755074019554458861568 * q^88 - 9615826823133397140723005163574167862048168091374 * q^89 + 57730871047256523107649943460110506911896252186624 * q^96 - 7412272971804212073669633541486217764746038123614 * q^97 - 60346472977889690849744226508581140353295285485568 * q^98 - 243467737885057769316782796887953523314815797311750 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$.

$n$	$5$	$7$
$\chi(n)$	$-1$	$-1$

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} $

3.1

−3.35544e7

−1.35725e12

1.12590e15

4.55416e19

−3.77789e22

1.12422e24

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8.51.d.a	✓	1
8.d	odd	2	1	CM	8.51.d.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.51.d.a	✓	1	1.a	even	1	1	trivial
8.51.d.a	✓	1	8.d	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 1357246255682 $ T3 + 1357246255682 acting on $S_{51}^{\mathrm{new}}(8, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 33554432 $ T + 33554432
$3$	$ T + 1357246255682 $ T + 1357246255682
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T + 21\!\cdots\!26 $ T + 216566032867586063515304626
$13$	$ T $ T
$17$	$ T - 11\!\cdots\!82 $ T - 11484333194425479818498080615682
$19$	$ T + 10\!\cdots\!74 $ T + 101750491668668099890091086635874
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T $ T
$37$	$ T $ T
$41$	$ T + 30\!\cdots\!26 $ T + 30490943576570305325097526558911137086126
$43$	$ T + 11\!\cdots\!86 $ T + 111409311987059022761513629469659116638386
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T + 13\!\cdots\!02 $ T + 130885310902277473186937781461662575640682002
$61$	$ T $ T
$67$	$ T + 46\!\cdots\!18 $ T + 4619012503980257606885070447298279180095842818
$71$	$ T $ T
$73$	$ T + 69\!\cdots\!82 $ T + 69994247752061998294033733980392006291536999182
$79$	$ T $ T
$83$	$ T - 11\!\cdots\!18 $ T - 115552090148733990836824537145935185395928788318
$89$	$ T + 96\!\cdots\!74 $ T + 9615826823133397140723005163574167862048168091374
$97$	$ T + 74\!\cdots\!14 $ T + 7412272971804212073669633541486217764746038123614
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 51 \)
Character orbit:	\([\chi]\)	\(=\)	8.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(-1\)

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