LMFDB - Newform orbit 7.59.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7,59,Mod(6,7)] 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("7.6"); S:= CuspForms(chi, 59); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 59, names="a")

Level:	$ N $	$=$	$ 7 $
Weight:	$ k $	$=$	$ 59 $
Character orbit:	$[\chi]$	$=$	7.b (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:	$149.134225691$
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 + 551073657 q^{2} + 15\!\cdots\!05 q^{4} - 32\!\cdots\!07 q^{7} - 15\!\cdots\!23 q^{8} + 47\!\cdots\!89 q^{9} - 31\!\cdots\!86 q^{11} - 17\!\cdots\!99 q^{14} - 87\!\cdots\!31 q^{16} + 25\!\cdots\!73 q^{18}+ \cdots - 14\!\cdots\!54 q^{99}+O(q^{100}) $

q + 551073657 * q^2 + 15451799287641905 * q^4 - 3219905755813179726837607 * q^7 - 150321087903738558081131223 * q^8 + 4710128697246244834921603689 * q^9 - 3133800701587005059955839894886 * q^11 - 1774405240051317960866661134618799 * q^14 - 87291669556229449023048276810524831 * q^16 + 2595627846132133970697609453201920673 * q^18 - 1726955012932716582067368949381323618102 * q^22 - 4869259682404532172343946296973829840942 * q^23 + 34694469519536141888238489627838134765625 * q^25 - 49753337463948160212537449739941903121335 * q^28 + 830148796882102345285852907621236899934506 * q^29 - 4777035857957838796950519512360655374294055 * q^32 + 72779963248811219965074919373347121758987545 * q^36 - 414398131882360431493476366095196068932059398 * q^37 + 235721000042039644901224797221907080083486749658 * q^43 - 48422859448393786893243245622296603154008797830 * q^44 - 2683320740065324097587732747684976443743636264894 * q^46 + 10367793076318844190248738727596255138212949486449 * q^49 + 19119208195805814654022469767369329929351806640625 * q^50 - 201333630806359706628309655297043423999447994526566 * q^53 + 484019736161346730380330212359724481107846978303361 * q^56 + 457473133351970337264951672166918189310251281908442 * q^58 - 15166170502884017563499771789145130511745788915132223 * q^63 + 22527612131237979881581310030968960007557006984106129 * q^64 + 32267003438865726285221854227417291975227228302224362 * q^67 + 920933808063292868914286470167401707486011794952133554 * q^71 - 708031669936674347479429406035672679558963278409881647 * q^72 - 228363893990380656775207992707150507838414056997078486 * q^74 + 10090542916611378424245950867471716474007697468071777802 * q^77 + 7793492627176906772525611464202595793266533921836792866 * q^79 + 22185312344622607535965183080365494317672538611578408721 * q^81 + 129899633524863940854699352784159775135798908445077559306 * q^86 + 471076330736057753052939372978435312786550831052292625578 * q^88 - 75238823291921798802626631734434955041685905252403874510 * q^92 + 5713417645586305565933576190253995138520050518253722373993 * q^98 - 14760604615995368211621704926359370003505748743144309834454 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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} $

6.1

5.51074e8

1.54518e16

−3.21991e24

−1.50321e26

4.71013e27

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7.59.b.a	✓	1
7.b	odd	2	1	CM	7.59.b.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.59.b.a	✓	1	1.a	even	1	1	trivial
7.59.b.a	✓	1	7.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} - 551073657 $ T2 - 551073657 acting on $S_{59}^{\mathrm{new}}(7, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 551073657 $ T - 551073657
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 32\!\cdots\!07 $ T + 3219905755813179726837607
$11$	$ T + 31\!\cdots\!86 $ T + 3133800701587005059955839894886
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T + 48\!\cdots\!42 $ T + 4869259682404532172343946296973829840942
$29$	$ T - 83\!\cdots\!06 $ T - 830148796882102345285852907621236899934506
$31$	$ T $ T
$37$	$ T + 41\!\cdots\!98 $ T + 414398131882360431493476366095196068932059398
$41$	$ T $ T
$43$	$ T - 23\!\cdots\!58 $ T - 235721000042039644901224797221907080083486749658
$47$	$ T $ T
$53$	$ T + 20\!\cdots\!66 $ T + 201333630806359706628309655297043423999447994526566
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T - 32\!\cdots\!62 $ T - 32267003438865726285221854227417291975227228302224362
$71$	$ T - 92\!\cdots\!54 $ T - 920933808063292868914286470167401707486011794952133554
$73$	$ T $ T
$79$	$ T - 77\!\cdots\!66 $ T - 7793492627176906772525611464202595793266533921836792866
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T $ T
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 59 \)
Character orbit:	\([\chi]\)	\(=\)	7.b (of order \(2\), degree \(1\), minimal)

\(n\)	\(3\)
\(\chi(n)\)	\(-1\)

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