LMFDB - Newform orbit 80.22.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,22,Mod(1,80)] 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("80.1"); S:= CuspForms(chi, 22); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 22, names="a")

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	80.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,21924] 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:	$223.581875430$
Analytic rank:	$1$
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 + 21924 q^{3} + 9765625 q^{5} + 722753248 q^{7} - 9979691427 q^{9} - 49976398572 q^{11} - 24351400354 q^{13} + 214101562500 q^{15} - 5768874283278 q^{17} + 30250225982620 q^{19} + 15845642209152 q^{21}+ \cdots + 49\!\cdots\!44 q^{99}+O(q^{100}) $

q + 21924 * q^3 + 9765625 * q^5 + 722753248 * q^7 - 9979691427 * q^9 - 49976398572 * q^11 - 24351400354 * q^13 + 214101562500 * q^15 - 5768874283278 * q^17 + 30250225982620 * q^19 + 15845642209152 * q^21 + 145464074718144 * q^23 + 95367431640625 * q^25 - 448127538468120 * q^27 - 1167107530943250 * q^29 + 7431907384909648 * q^31 - 1095682562292528 * q^33 + 7058137187500000 * q^35 - 54621676523378698 * q^37 - 533880101361096 * q^39 - 20889593910177078 * q^41 - 76193896658948996 * q^43 - 97457924091796875 * q^45 - 508989722403999912 * q^47 - 36173606588734503 * q^49 - 126476799786586872 * q^51 + 1341490659556639206 * q^53 - 488050767304687500 * q^55 + 663205954442960880 * q^57 - 2306403035584927500 * q^59 + 5268341017122878702 * q^61 - 7212854392902004896 * q^63 - 237806644082031250 * q^65 + 10904975405120900788 * q^67 + 3189154374120589056 * q^69 + 13367810994741254088 * q^71 - 17952853611107729014 * q^73 + 2090835571289062500 * q^75 - 36120604391255561856 * q^77 - 118985112237438248240 * q^79 + 94566349029996027801 * q^81 - 134353218394716671916 * q^83 - 56336662922636718750 * q^85 - 25587665508399813000 * q^87 + 347081364860908384890 * q^89 - 17600053699201849792 * q^91 + 162937137506759122752 * q^93 + 295412363111523437500 * q^95 - 376521927584699625598 * q^97 + 498749036381323442244 * 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

21924.0

9.76562e6

7.22753e8

−9.97969e9

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	80.22.a.a		1
4.b	odd	2	1		10.22.a.a	✓	1
20.d	odd	2	1		50.22.a.b		1
20.e	even	4	2		50.22.b.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.22.a.a	✓	1	4.b	odd	2	1
50.22.a.b		1	20.d	odd	2	1
50.22.b.b		2	20.e	even	4	2
80.22.a.a		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 21924 $ T3 - 21924 acting on $S_{22}^{\mathrm{new}}(\Gamma_0(80))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 21924 $ T - 21924
$5$	$ T - 9765625 $ T - 9765625
$7$	$ T - 722753248 $ T - 722753248
$11$	$ T + 49976398572 $ T + 49976398572
$13$	$ T + 24351400354 $ T + 24351400354
$17$	$ T + 5768874283278 $ T + 5768874283278
$19$	$ T - 30250225982620 $ T - 30250225982620
$23$	$ T - 145464074718144 $ T - 145464074718144
$29$	$ T + 1167107530943250 $ T + 1167107530943250
$31$	$ T - 7431907384909648 $ T - 7431907384909648
$37$	$ T + 54\!\cdots\!98 $ T + 54621676523378698
$41$	$ T + 20\!\cdots\!78 $ T + 20889593910177078
$43$	$ T + 76\!\cdots\!96 $ T + 76193896658948996
$47$	$ T + 50\!\cdots\!12 $ T + 508989722403999912
$53$	$ T - 13\!\cdots\!06 $ T - 1341490659556639206
$59$	$ T + 23\!\cdots\!00 $ T + 2306403035584927500
$61$	$ T - 52\!\cdots\!02 $ T - 5268341017122878702
$67$	$ T - 10\!\cdots\!88 $ T - 10904975405120900788
$71$	$ T - 13\!\cdots\!88 $ T - 13367810994741254088
$73$	$ T + 17\!\cdots\!14 $ T + 17952853611107729014
$79$	$ T + 11\!\cdots\!40 $ T + 118985112237438248240
$83$	$ T + 13\!\cdots\!16 $ T + 134353218394716671916
$89$	$ T - 34\!\cdots\!90 $ T - 347081364860908384890
$97$	$ T + 37\!\cdots\!98 $ T + 376521927584699625598
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(n\):		e.g. 2-40 or 80-90
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