LMFDB - Newform orbit 7.39.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7,39,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, 39); 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, 39, names="a")

Level:	$ N $	$=$	$ 7 $
Weight:	$ k $	$=$	$ 39 $
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:	$64.0244283235$
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 - 413367 q^{2} - 104005630255 q^{4} - 11\!\cdots\!43 q^{7} + 15\!\cdots\!33 q^{8} + 13\!\cdots\!89 q^{9} - 10\!\cdots\!46 q^{11} + 47\!\cdots\!81 q^{14} - 36\!\cdots\!91 q^{16} - 55\!\cdots\!63 q^{18}+ \cdots - 14\!\cdots\!94 q^{99}+O(q^{100}) $

q - 413367 * q^2 - 104005630255 * q^4 - 11398895185373143 * q^7 + 156617951121339033 * q^8 + 1350851717672992089 * q^9 - 105169262610335033046 * q^11 + 4711927106092140002481 * q^14 - 36151842646288591063391 * q^16 - 558397521979331720853663 * q^18 + 43473502577446361605125882 * q^22 - 145793212693639450317344718 * q^23 + 363797880709171295166015625 * q^25 + 1185549277965418795065241465 * q^28 - 8377092508926125106925089894 * q^29 - 28106835854922995143548197655 * q^32 - 140496184277628864208074052695 * q^36 - 915056241452490562198383035702 * q^37 - 21664405013204310512940627763478 * q^43 + 10938195441241501588655482406730 * q^44 + 60266102951531658659329834045506 * q^46 + 129934811447123020117172145698449 * q^49 - 150382038555108010768890380859375 * q^50 + 1015774856582957233542544257805866 * q^53 - 1785271608980037746151525915790719 * q^56 + 3462813599137265557074303634213098 * q^58 - 15398217140735709790332844752065727 * q^63 + 21555781255622599774794749123141489 * q^64 + 99086060163779284647904187871379418 * q^67 + 271405679586296345090784519856199634 * q^71 + 211567628290685550167219509295909937 * q^72 + 378254053360491666224259000319028634 * q^74 + 1198813401218191713583073528745883578 * q^77 - 1007206758745985044694331420876567454 * q^79 + 1824800363140073127359051977856583921 * q^81 + 8955350107093226223802728476705610426 * q^86 - 16471394430972720925642917461824684518 * q^88 + 15163314973103237259987197153685243090 * q^92 - 53710763203462901456775098350930767783 * q^98 - 142068079043573063078159806861511573094 * 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

−413367.

−1.04006e11

−1.13989e16

1.56618e17

1.35085e18

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	7.39.b.a	✓	1
7.b	odd	2	1	CM	7.39.b.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.39.b.a	✓	1	1.a	even	1	1	trivial
7.39.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} + 413367 $ T2 + 413367 acting on $S_{39}^{\mathrm{new}}(7, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 413367 $ T + 413367
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 11\!\cdots\!43 $ T + 11398895185373143
$11$	$ T + 10\!\cdots\!46 $ T + 105169262610335033046
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T + 14\!\cdots\!18 $ T + 145793212693639450317344718
$29$	$ T + 83\!\cdots\!94 $ T + 8377092508926125106925089894
$31$	$ T $ T
$37$	$ T + 91\!\cdots\!02 $ T + 915056241452490562198383035702
$41$	$ T $ T
$43$	$ T + 21\!\cdots\!78 $ T + 21664405013204310512940627763478
$47$	$ T $ T
$53$	$ T - 10\!\cdots\!66 $ T - 1015774856582957233542544257805866
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T - 99\!\cdots\!18 $ T - 99086060163779284647904187871379418
$71$	$ T - 27\!\cdots\!34 $ T - 271405679586296345090784519856199634
$73$	$ T $ T
$79$	$ T + 10\!\cdots\!54 $ T + 1007206758745985044694331420876567454
$83$	$ 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:	\( k \)	\(=\)	\( 39 \)
Character orbit:	\([\chi]\)	\(=\)	7.b (of order \(2\), degree \(1\), minimal)

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

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