LMFDB - Newform orbit 7.91.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7,91,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, 91); 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, 91, names="a")

Level:	$ N $	$=$	$ 7 $
Weight:	$ k $	$=$	$ 91 $
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:	$359.071719566$
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 - 31472005452039 q^{2} - 24\!\cdots\!03 q^{4} - 10\!\cdots\!07 q^{7} + 46\!\cdots\!53 q^{8} + 87\!\cdots\!49 q^{9} + 19\!\cdots\!02 q^{11} + 33\!\cdots\!73 q^{14} - 11\!\cdots\!95 q^{16} - 27\!\cdots\!11 q^{18}+ \cdots + 17\!\cdots\!98 q^{99}+O(q^{100}) $

q - 31472005452039 * q^2 - 247452912112207734169866703 * q^4 - 107006904423598033356356300384937784807 * q^7 + 46748295064805191166173613831036224650153 * q^8 + 8727963568087712425891397479476727340041449 * q^9 + 19790026180117550061787577790564298756261221402 * q^11 + 3367721879425293492520243667870209557841719341371473 * q^14 - 1164930729411608658309728526486029320780626115877198495 * q^16 - 274686517000054249261017628218600385454601533200241564511 * q^18 - 622831811836654080566486169542923440147441986751417271338678 * q^22 - 26046435544030421392519870598898104072151495949941239288887086 * q^23 + 807793566946316088741610050849573099185363389551639556884765625 * q^25 + 26479170115732017157406921742245444075666973387840109703288581321 * q^28 + 580412802836801697056191804292414337576559040577984350399354230698 * q^29 - 21208879961759570431453662693972605420979302097288110670735082124967 * q^32 - 2159760001732559727097677821814068782386969362212608296849366824972647 * q^36 + 56453630842781392956606177001280321032168747892715304490237080395544186 * q^37 + 60856270988117847188006371765924165337154860427933045648536426829889825114 * q^43 - 4897099609046918261928448695266658325136066138979894528823817354310777606 * q^44 + 819733561447907819105459985941163911504572307965320510321538681936659468354 * q^46 + 11450477594321044359340126713545146077054004823284978858214566372120240027249 * q^49 - 25422883543056490885302173667056684601417426705438629142008721828460693359375 * q^50 - 437300056706579856794766108045574581035543205742280080269234463273076188109286 * q^53 - 5002390341965768722034523927328307757510329748820854069839258616102304373625471 * q^56 - 18266754895313060175305956081233772514819838751635698509521696115472296580493222 * q^58 - 933952363343007509784849246607720613968056605597111123983144280064188809622465343 * q^63 + 2109600378720691419769001469779598523763697204493685805065707610328495643633800593 * q^64 + 29065819855380227506271998224512668233453389125775671434403943720420468425859130986 * q^67 - 361016447746128253436575954668384053018863837740769604769687032555652650641087269198 * q^71 + 408017416195834313882059643340560421464290263083101474130352880105079490758544191697 * q^72 - 1776708977671413045577332476326478915407775033583804155168397371856248184434328295254 * q^74 - 2117669439996341558145495834047444982886465145719263818230565508032072387922058839414 * q^77 + 19218343878996286145339351596908940646293175331160698373466437245523698217602476337698 * q^79 + 76177348045866392339289727720615561750424801402395196724001565744957137343033038019601 * q^81 - 1915268892328807708487975997268236447808358228004356905298392689791184573193500224707446 * q^86 + 925149983208354794751812316255904535798902664379088184487428045043237746825567926174506 * q^88 + 6445266325513243505763269939846758087456544213422914127864251297186594663660437838097458 * q^92 - 360369493276922320941664835781077944527008312593864257363541682024654423025641845922610711 * q^98 + 172726627511568014101410504467724652577934007449611963835015236630082465601274802045891498 * 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

−3.14720e13

−2.47453e26

−1.07007e38

4.67483e40

8.72796e42

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	7.91.b.a	✓	1
7.b	odd	2	1	CM	7.91.b.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.91.b.a	✓	1	1.a	even	1	1	trivial
7.91.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} + 31472005452039 $ T2 + 31472005452039 acting on $S_{91}^{\mathrm{new}}(7, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 31472005452039 $ T + 31472005452039
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 10\!\cdots\!07 $ T + 107006904423598033356356300384937784807
$11$	$ T - 19\!\cdots\!02 $ T - 19790026180117550061787577790564298756261221402
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T + 26\!\cdots\!86 $ T + 26046435544030421392519870598898104072151495949941239288887086
$29$	$ T - 58\!\cdots\!98 $ T - 580412802836801697056191804292414337576559040577984350399354230698
$31$	$ T $ T
$37$	$ T - 56\!\cdots\!86 $ T - 56453630842781392956606177001280321032168747892715304490237080395544186
$41$	$ T $ T
$43$	$ T - 60\!\cdots\!14 $ T - 60856270988117847188006371765924165337154860427933045648536426829889825114
$47$	$ T $ T
$53$	$ T + 43\!\cdots\!86 $ T + 437300056706579856794766108045574581035543205742280080269234463273076188109286
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T - 29\!\cdots\!86 $ T - 29065819855380227506271998224512668233453389125775671434403943720420468425859130986
$71$	$ T + 36\!\cdots\!98 $ T + 361016447746128253436575954668384053018863837740769604769687032555652650641087269198
$73$	$ T $ T
$79$	$ T - 19\!\cdots\!98 $ T - 19218343878996286145339351596908940646293175331160698373466437245523698217602476337698
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T $ T
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Overview	Random
Universe	Knowledge

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:	\( k \)	\(=\)	\( 91 \)
Character orbit:	\([\chi]\)	\(=\)	7.b (of order \(2\), degree \(1\), minimal)

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

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