LMFDB - Newform orbit 7.73.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7,73,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, 73); 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, 73, names="a")

Level:	$ N $	$=$	$ 7 $
Weight:	$ k $	$=$	$ 73 $
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:	$229.811218987$
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 + 86965052897 q^{2} + 28\!\cdots\!13 q^{4} + 26\!\cdots\!01 q^{7} - 16\!\cdots\!51 q^{8} + 22\!\cdots\!41 q^{9} - 53\!\cdots\!78 q^{11} + 23\!\cdots\!97 q^{14} - 27\!\cdots\!95 q^{16} + 19\!\cdots\!77 q^{18}+ \cdots - 12\!\cdots\!98 q^{99}+O(q^{100}) $

q + 86965052897 * q^2 + 2840553942508362878913 * q^4 + 2651730845859653471779023381601 * q^7 - 163651927114756865735155258016351 * q^8 + 22528399544939174411840147874772641 * q^9 - 53672247167957504787434959799676658078 * q^11 + 230607913278791317611352465075930781548097 * q^14 - 27646135229115541422371180134065057663111295 * q^16 + 1959183458110386031373187323037904856413390977 * q^18 - 4667609814062282847287412019925949093249452351966 * q^22 - 6951677221648295093249057943662547525437231570878 * q^23 + 211758236813575084767080625169910490512847900390625 * q^25 + 7532384508677674574919508704783030547176224955079713 * q^28 + 56322130692099937812055539120814764613600994253351842 * q^29 - 1631423237133894368866757868787473246467279392816028319 * q^32 + 63993134145780580075745661343024828833875880384088219233 * q^36 + 482084921383319935446866716460077341326941886636128666082 * q^37 + 7291119889570242755705001039197022705371300781274008117602 * q^43 - 152458913296225004400465235307367477701355091143058817309214 * q^44 - 604552977303513976314269871666031491467425296614643674733566 * q^46 + 7031676478883553279994550741476882515263791803223057265323201 * q^49 + 18415566265868009974450425492165095420205034315586090087890625 * q^50 - 106969856276092507892872715780737824854032383382020172727033758 * q^53 - 433960863114576582852963231162656922058184287870974419370557951 * q^56 + 4898057074910218311835820403943508878360414230894535959082386274 * q^58 + 59739251981165789319143185091465521335682061421948187270857578241 * q^63 - 11321625727887145962627063034767776671604526431706770034778693823 * q^64 + 519146510589714733440574049890332236879334449448453239673135488162 * q^67 - 4735123027435915944066960513336594500093507076447739709320199588158 * q^71 - 3686816000340507512093249467017694857777170475375095843596785452991 * q^72 + 41924540688946504599611389329859326577208625342555906837119179739554 * q^74 - 142324353381876344710192975790062146422262248782424492460310193222878 * q^77 - 240856118488733957112164323012393891590663819545356410811238302008958 * q^79 + 507528786056415600719754159741696356908742250191663887263627442114881 * q^81 + 634072626874844959847016463901209049977525218469191542998465326792994 * q^86 + 8783566681615797174532822976090310127882993791238476383594387160233378 * q^88 - 19746614138998647010378921630419326178308517835631310441524193491095614 * q^92 + 611510116940698914498043957104287659842570064076088190168892338766363297 * q^98 - 1209149828674476743430733097304894261056135277618296387004238598946043998 * 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

8.69651e10

2.84055e21

2.65173e30

−1.63652e32

2.25284e34

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	7.73.b.a	✓	1
7.b	odd	2	1	CM	7.73.b.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.73.b.a	✓	1	1.a	even	1	1	trivial
7.73.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} - 86965052897 $ T2 - 86965052897 acting on $S_{73}^{\mathrm{new}}(7, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 86965052897 $ T - 86965052897
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 26\!\cdots\!01 $ T - 2651730845859653471779023381601
$11$	$ T + 53\!\cdots\!78 $ T + 53672247167957504787434959799676658078
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T + 69\!\cdots\!78 $ T + 6951677221648295093249057943662547525437231570878
$29$	$ T - 56\!\cdots\!42 $ T - 56322130692099937812055539120814764613600994253351842
$31$	$ T $ T
$37$	$ T - 48\!\cdots\!82 $ T - 482084921383319935446866716460077341326941886636128666082
$41$	$ T $ T
$43$	$ T - 72\!\cdots\!02 $ T - 7291119889570242755705001039197022705371300781274008117602
$47$	$ T $ T
$53$	$ T + 10\!\cdots\!58 $ T + 106969856276092507892872715780737824854032383382020172727033758
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T - 51\!\cdots\!62 $ T - 519146510589714733440574049890332236879334449448453239673135488162
$71$	$ T + 47\!\cdots\!58 $ T + 4735123027435915944066960513336594500093507076447739709320199588158
$73$	$ T $ T
$79$	$ T + 24\!\cdots\!58 $ T + 240856118488733957112164323012393891590663819545356410811238302008958
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T $ T
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 73 \)
Character orbit:	\([\chi]\)	\(=\)	7.b (of order \(2\), degree \(1\), minimal)

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

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