LMFDB - Newform orbit 6.26.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6,26,Mod(1,6)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 26, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6.1");

S:= CuspForms(chi, 26);

N := Newforms(S);

Level:	$ N $	$=$	$ 6 = 2 \cdot 3 $
Weight:	$ k $	$=$	$ 26 $
Character orbit:	$[\chi]$	$=$	6.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$23.7598067971$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 - 4096 q^{2} + 531441 q^{3} + 16777216 q^{4} + 590425734 q^{5} - 2176782336 q^{6} + 57857417576 q^{7} - 68719476736 q^{8} + 282429536481 q^{9}+O(q^{10}) $

$ q - 4096 q^{2} + 531441 q^{3} + 16777216 q^{4} + 590425734 q^{5} - 2176782336 q^{6} + 57857417576 q^{7} - 68719476736 q^{8} + 282429536481 q^{9} - 2418383806464 q^{10} + 9494266240140 q^{11} + 8916100448256 q^{12} - 134968021061458 q^{13} - 236983982391296 q^{14} + 313776442502694 q^{15} + 281474976710656 q^{16} - 25\!\cdots\!14 q^{17}+ \cdots + 26\!\cdots\!40 q^{99}+O(q^{100}) $

q - 4096 * q^2 + 531441 * q^3 + 16777216 * q^4 + 590425734 * q^5 - 2176782336 * q^6 + 57857417576 * q^7 - 68719476736 * q^8 + 282429536481 * q^9 - 2418383806464 * q^10 + 9494266240140 * q^11 + 8916100448256 * q^12 - 134968021061458 * q^13 - 236983982391296 * q^14 + 313776442502694 * q^15 + 281474976710656 * q^16 - 2526114016804014 * q^17 - 1156831381426176 * q^18 + 11468758872260756 * q^19 + 9905700071276544 * q^20 + 30747803854007016 * q^21 - 38888514519613440 * q^22 + 113342630802000600 * q^23 - 36520347436056576 * q^24 + 50579323492485631 * q^25 + 552829014267731968 * q^26 + 150094635296999121 * q^27 + 970686391874748416 * q^28 + 1081348899350530974 * q^29 - 1285228308491034624 * q^30 + 4649090467326833408 * q^31 - 1152921504606846976 * q^32 + 5045642344926241740 * q^33 + 10346963012829241344 * q^34 + 34160508239654300784 * q^35 + 4738381338321616896 * q^36 - 46093370056702003258 * q^37 - 46976036340780056576 * q^38 - 71727540080922300978 * q^39 - 40573747491948724224 * q^40 + 51449233931826001194 * q^41 - 125943004586012737536 * q^42 - 369342639690619984084 * q^43 + 159287355472336650240 * q^44 + 166753666380074202054 * q^45 - 464251415764994457600 * q^46 - 49106637730499080080 * q^47 + 149587343098087735296 * q^48 + 2006412148899668814969 * q^49 - 207172909025221144576 * q^50 - 1342480559204342004174 * q^51 - 2264387642440630140928 * q^52 + 4440077625909370178934 * q^53 - 614787626176508399616 * q^54 + 5605659113626079762760 * q^55 - 3975931461118969511936 * q^56 + 6094968683833128429396 * q^57 - 4429205091739774869504 * q^58 + 22549343358865698156540 * q^59 + 5264295151579277819904 * q^60 - 12310641025418994171490 * q^61 - 19042674554170709639168 * q^62 + 16340643627977342590056 * q^63 + 4722366482869645213696 * q^64 - 79688592901738798760172 * q^65 - 20666951044817886167040 * q^66 + 6385919851217871016196 * q^67 - 42381160500548572545024 * q^68 + 60234921056046000864600 * q^69 - 139921441749624016011264 * q^70 + 60276767662482517683720 * q^71 - 19408409961765342806016 * q^72 - 268812100727467655130358 * q^73 + 188798443752251405344768 * q^74 + 26879926256170056224271 * q^75 + 192413844851835111735296 * q^76 + 549313726433499472700640 * q^77 + 293796004171457744805888 * q^78 - 262861199604551219808592 * q^79 + 166190069727021974421504 * q^80 + 79766443076872509863361 * q^81 - 210736062184759300890624 * q^82 + 688505757189913419282132 * q^83 + 515862546784308172947456 * q^84 - 1491482722539198300096276 * q^85 + 1512827452172779454808064 * q^86 + 574673140419745531353534 * q^87 - 652441008014690919383040 * q^88 - 2558749855718408389837542 * q^89 - 683023017492783931613184 * q^90 - 7808901153959138265385808 * q^91 + 1901573798973417298329600 * q^92 + 2470717287046639673180928 * q^93 + 201140788144124232007680 * q^94 + 6771450375223569100694904 * q^95 - 612709757329767363772416 * q^96 + 2595207926228515475808482 * q^97 - 8218264161893043466113024 * q^98 + 2681461213429946836547340 * q^99

Display 100 coefficients 10 coefficients

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

−4096.00

531441.

1.67772e7

5.90426e8

−2.17678e9

5.78574e10

−6.87195e10

2.82430e11

−2.41838e12

Atkin-Lehner signs

$ p $	Sign
$2$	$1$
$3$	$-1$

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	6.26.a.b	✓	1
3.b	odd	2	1		18.26.a.b		1
4.b	odd	2	1		48.26.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.26.a.b	✓	1	1.a	even	1	1	trivial
18.26.a.b		1	3.b	odd	2	1
48.26.a.a		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 590425734 $ T5 - 590425734 acting on $S_{26}^{\mathrm{new}}(\Gamma_0(6))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 4096 $ T + 4096
$3$	$ T - 531441 $ T - 531441
$5$	$ T - 590425734 $ T - 590425734
$7$	$ T - 57857417576 $ T - 57857417576
$11$	$ T - 9494266240140 $ T - 9494266240140
$13$	$ T + 134968021061458 $ T + 134968021061458
$17$	$ T + 2526114016804014 $ T + 2526114016804014
$19$	$ T - 11\!\cdots\!56 $ T - 11468758872260756
$23$	$ T - 11\!\cdots\!00 $ T - 113342630802000600
$29$	$ T - 10\!\cdots\!74 $ T - 1081348899350530974
$31$	$ T - 46\!\cdots\!08 $ T - 4649090467326833408
$37$	$ T + 46\!\cdots\!58 $ T + 46093370056702003258
$41$	$ T - 51\!\cdots\!94 $ T - 51449233931826001194
$43$	$ T + 36\!\cdots\!84 $ T + 369342639690619984084
$47$	$ T + 49\!\cdots\!80 $ T + 49106637730499080080
$53$	$ T - 44\!\cdots\!34 $ T - 4440077625909370178934
$59$	$ T - 22\!\cdots\!40 $ T - 22549343358865698156540
$61$	$ T + 12\!\cdots\!90 $ T + 12310641025418994171490
$67$	$ T - 63\!\cdots\!96 $ T - 6385919851217871016196
$71$	$ T - 60\!\cdots\!20 $ T - 60276767662482517683720
$73$	$ T + 26\!\cdots\!58 $ T + 268812100727467655130358
$79$	$ T + 26\!\cdots\!92 $ T + 262861199604551219808592
$83$	$ T - 68\!\cdots\!32 $ T - 688505757189913419282132
$89$	$ T + 25\!\cdots\!42 $ T + 2558749855718408389837542
$97$	$ T - 25\!\cdots\!82 $ T - 2595207926228515475808482
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6 = 2 \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	6.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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