LMFDB - Newform orbit 6.26.a.a

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:	$1$
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} - 292754850 q^{5} + 2176782336 q^{6} + 3580644032 q^{7} - 68719476736 q^{8} + 282429536481 q^{9}+O(q^{10}) $

$ q - 4096 q^{2} - 531441 q^{3} + 16777216 q^{4} - 292754850 q^{5} + 2176782336 q^{6} + 3580644032 q^{7} - 68719476736 q^{8} + 282429536481 q^{9} + 1199123865600 q^{10} + 15111573238212 q^{11} - 8916100448256 q^{12} + 1221071681246 q^{13} - 14666317955072 q^{14} + 155581930238850 q^{15} + 281474976710656 q^{16} + 25\!\cdots\!82 q^{17}+ \cdots + 42\!\cdots\!72 q^{99}+O(q^{100}) $

q - 4096 * q^2 - 531441 * q^3 + 16777216 * q^4 - 292754850 * q^5 + 2176782336 * q^6 + 3580644032 * q^7 - 68719476736 * q^8 + 282429536481 * q^9 + 1199123865600 * q^10 + 15111573238212 * q^11 - 8916100448256 * q^12 + 1221071681246 * q^13 - 14666317955072 * q^14 + 155581930238850 * q^15 + 281474976710656 * q^16 + 2518250853863682 * q^17 - 1156831381426176 * q^18 - 7992693407413060 * q^19 - 4911611353497600 * q^20 - 1902901045010112 * q^21 - 61897003983716352 * q^22 - 99645642629247624 * q^23 + 36520347436056576 * q^24 - 212317821678430625 * q^25 - 5001509606383616 * q^26 - 150094635296999121 * q^27 + 60073238343974912 * q^28 - 2080672742244316890 * q^29 - 637263586258329600 * q^30 - 4937672075835729208 * q^31 - 1152921504606846976 * q^32 - 8030909593288623492 * q^33 - 10314755497425641472 * q^34 - 1048250906491555200 * q^35 + 4738381338321616896 * q^36 + 19829154107621718182 * q^37 + 32738072196763893760 * q^38 - 648927555353055486 * q^39 + 20117960103926169600 * q^40 + 224696060863159376442 * q^41 + 7794282680361418752 * q^42 - 72221008334482349884 * q^43 + 253530128317302177792 * q^44 - 82682616588064682850 * q^45 + 408148552209398267904 * q^46 + 189872435947262116992 * q^47 - 149587343098087735296 * q^48 - 1328247607980067683783 * q^49 + 869653797594851840000 * q^50 - 1338301752028169025762 * q^51 + 20486183347747291136 * q^52 - 2645676034335389555874 * q^53 + 614787626176508399616 * q^54 - 4423986356616768328200 * q^55 - 246059984256921239552 * q^56 + 4247644977129004019460 * q^57 + 8522435552232721981440 * q^58 - 16454608826354674865340 * q^59 + 2610231649314118041600 * q^60 - 35546954389065591688738 * q^61 + 20224704822623146835968 * q^62 + 1011279634261218931392 * q^63 + 4722366482869645213696 * q^64 - 357474656882420543100 * q^65 + 32894605694110201823232 * q^66 + 106703750402023286661692 * q^67 + 42249238517455427469312 * q^68 + 52955779964529986546184 * q^69 + 4293635712989410099200 * q^70 + 73672004836753334994312 * q^71 - 19408409961765342806016 * q^72 - 262402855870448192600374 * q^73 - 81220215224818557673472 * q^74 + 112834395470606849780625 * q^75 - 134095143717944908840960 * q^76 + 54109164529534712150784 * q^77 + 2658007266726115270656 * q^78 - 1002642123108883497568840 * q^79 - 82403164585681590681600 * q^80 + 79766443076872509863361 * q^81 - 920355065295500805906432 * q^82 + 1558588706101147601425596 * q^83 - 31925381858760371208192 * q^84 - 737230150985234144357700 * q^85 + 295817250138039705124864 * q^86 + 1105754802811062012338490 * q^87 - 1038459405587669720236032 * q^88 + 2181670205644666928498490 * q^89 + 338667997544712940953600 * q^90 + 4372223028097696223872 * q^91 - 1671776469849695305334784 * q^92 + 2624081385654215766028728 * q^93 - 777717497639985631199232 * q^94 + 2339899759583199268341000 * q^95 + 612709757329767363772416 * q^96 - 440165308375605500117758 * q^97 + 5440502202286357232775168 * q^98 + 4267954625166899357211972 * 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

−2.92755e8

2.17678e9

3.58064e9

−6.87195e10

2.82430e11

1.19912e12

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.a	✓	1
3.b	odd	2	1		18.26.a.d		1
4.b	odd	2	1		48.26.a.c		1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 292754850 $ T5 + 292754850 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 + 292754850 $ T + 292754850
$7$	$ T - 3580644032 $ T - 3580644032
$11$	$ T - 15111573238212 $ T - 15111573238212
$13$	$ T - 1221071681246 $ T - 1221071681246
$17$	$ T - 2518250853863682 $ T - 2518250853863682
$19$	$ T + 7992693407413060 $ T + 7992693407413060
$23$	$ T + 99\!\cdots\!24 $ T + 99645642629247624
$29$	$ T + 20\!\cdots\!90 $ T + 2080672742244316890
$31$	$ T + 49\!\cdots\!08 $ T + 4937672075835729208
$37$	$ T - 19\!\cdots\!82 $ T - 19829154107621718182
$41$	$ T - 22\!\cdots\!42 $ T - 224696060863159376442
$43$	$ T + 72\!\cdots\!84 $ T + 72221008334482349884
$47$	$ T - 18\!\cdots\!92 $ T - 189872435947262116992
$53$	$ T + 26\!\cdots\!74 $ T + 2645676034335389555874
$59$	$ T + 16\!\cdots\!40 $ T + 16454608826354674865340
$61$	$ T + 35\!\cdots\!38 $ T + 35546954389065591688738
$67$	$ T - 10\!\cdots\!92 $ T - 106703750402023286661692
$71$	$ T - 73\!\cdots\!12 $ T - 73672004836753334994312
$73$	$ T + 26\!\cdots\!74 $ T + 262402855870448192600374
$79$	$ T + 10\!\cdots\!40 $ T + 1002642123108883497568840
$83$	$ T - 15\!\cdots\!96 $ T - 1558588706101147601425596
$89$	$ T - 21\!\cdots\!90 $ T - 2181670205644666928498490
$97$	$ T + 44\!\cdots\!58 $ T + 440165308375605500117758
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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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