LMFDB - Newform orbit 1386.2.bk.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(703,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1386.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1386.bk (of order $6$ , degree $2$ , minimal)

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:	no
Analytic conductor:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + \cdots + 13417$ x^16 - 8x^15 + 74x^14 - 378x^13 + 1878x^12 - 6718x^11 + 22086x^10 - 56904x^9 + 130215x^8 - 239606x^7 + 378750x^6 - 477124x^5 + 493030x^4 - 386266x^3 + 223844x^2 - 82874*x + 13417
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 462)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{12} q^{2} + (\beta_{13} + 1) q^{4} + ( - \beta_{13} - \beta_{9} - 1) q^{5} + ( - \beta_{15} + \beta_{13} + \beta_{11} + \cdots + 1) q^{7} + (\beta_{12} + \beta_{11}) q^{8} + ( - \beta_{12} - \beta_{11} + \beta_{7}) q^{10}+ \cdots + (2 \beta_{15} - \beta_{14} + \beta_{12} + \cdots - 2) q^{98}+O(q^{100})$ q + b12 * q^2 + (b13 + 1) * q^4 + (-b13 - b9 - 1) * q^5 + (-b15 + b13 + b11 + b8 + b1 + 1) * q^7 + (b12 + b11) * q^8 + (-b12 - b11 + b7) * q^10 + (-b12 - b11 - b10 + b5 + b2 + b1 + 1) * q^11 + (b15 + b14 - b13 + b10 - b9 - b8 + b7 - b6 - b5 - 1) * q^13 + (b14 + b12 + b10 - b9 + b8 + b7 - b5 + b4 + b3 - 1) * q^14 + b13 * q^16 + (b15 - b14 + b13 + b9 - b8 + b7 - b6 - b3 + 1) * q^17 + (-2b13 - b12 - 2b11 + b9 - 2b8 - b4 - b1 - 1) q^19 + (-b13 - b9 + b8) * q^20 + (-b13 - b10 - b8 - b7 + b6 + b5 - b4 - b3 - b1) * q^22 + (-b15 - 2b13 - b12 - 2b10 + b9 - b8 - 2b7 + b6 + 2b5 - 3b4 - b3 - 2b1) * q^23 + (-b15 + b13 - b12 + 2b9 - b7 + 2b6 + b5 + b3 + b2 + b1) * q^25 + (-b14 - b12 - b10 + b9 + b6 + b5 - b3 + b2 + b1) * q^26 + (-b15 + b13 - b12 + b8 + b2 + b1 + 1) * q^28 + (b15 - b14 + b13 - b12 + 3b9 - 3b8 - b7 + 2b6 + 2b5 - 2b4 - b3 - b2 - b1 - 1) q^29 + (-b15 - b14 - b12 + b11 - 2b10 + b9 - 2b7 + b6 + b5 - b4 - b3 + 1) * q^31 + b11 * q^32 + (b15 - b14 + b12 + 2b11 - b7 - b3 - b2) q^34 + (b15 + b14 - 2b13 + b12 - b9 - 3b8 - 2b4 - b2 - 4b1 - 2) * q^35 + (2b14 - 2b13 - b12 - 4b11 - 3b9 + 2b8 + 2b7 - 2b6 - 2b5 + 3b4 + 2b3 + 2b2 + b1 + 1) q^37 + (-b13 - b12 - 2b11 - 2b10 - b7 + b5 + 1) * q^38 + (-b11 + b10 + b7) * q^40 + (b15 + b14 - b13 - b11 - b9 - b8 + b7 - 2b6 - b3 + b2 - b1 + 3) q^41 + (3b13 - b12 - b11 + b9 - b8 + 2b4 + b1 + 1) * q^43 + (b15 - b11 - b8 + b7 - b6 - b4 - b1) * q^44 + (b15 + b14 - b13 - 3b11 - b9 - b8 + b7 - b6 + b5 + b3 - 2b1 - 1) * q^46 + (3b15 + b14 - 2b13 + b10 - 2b9 - 3b8 - b5 - b4 - 2b3 - b2 - 3b1) * q^47 + (b15 + 2b11 - b10 + b7 - 2b6 - b5 - 2b4 - 2b3 - b1 + 1) * q^49 + (-b15 + b14 - b13 - b12 - 2b10 + b9 - b8 - 3b7 + b6 + b5 - 2b4 - b1 - 1) q^50 + (b15 - b13 - b12 - b8 - b7 - 2b4 - b3 - b2 - b1 - 1) q^52 + (b15 - b14 + 2b13 - 2b12 - 2b11 + b9 - b8 + 3b7 - 3b6 - b5 - b4 - b3 + 2) q^53 + (2b14 + b13 + 4b12 + b11 + 2b10 - 2b9 + 3b8 + b7 - 2b5 + b4 + b3 - 2b2) q^55 + (b14 - b13 - 1) * q^56 + (b15 - b14 - b13 + 2b11 - b10 + b9 - b8 - 2b7 + b6 + b5 - 2b4 - b2 - 2b1 - 1) * q^58 + (-2b15 + b14 - 2b12 + 2b11 - 4b10 - b9 + 2b8 - b7 - b6 + 2b5 + b4 - 2b3 + 3b2 + 3b1 + 4) q^59 + (-b15 + 3b13 + 3b11 - 3b10 + b9 - b8 - 3b7 - b4 - b3) * q^61 + (b15 + b14 - b13 + b12 - b11 + b10 - b9 - b8 + b7 - b6 - b5 + b3 - b2 - b1 - 2) * q^62 - q^64 + (-2b15 - b14 + 5b13 - 2b12 - b10 + b9 + 2b8 + b5 + b4 + b3 + b2 + 2b1 + 8) q^65 + (-b15 + b14 - 4b13 - 5b12 - 2b11 - b9 + b8 - 2b7 + 3b6 + b5 + b3 - 4) q^67 + (b15 - b14 + b13 + b12 + b11 + 2b10 + b9 - b8 + b7 - b5 - b2 - 1) q^68 + (-b14 + b13 - b12 - b11 - 2b10 + b7 - b6 + 2b5 + b2 + 1) * q^70 + (-b15 - b14 + b13 - 2b12 + 3b11 + b9 + b8 - b7 + 2b6 + 2b3 - 2b2 + 2) q^71 + (-2b14 + 6b13 + 2b12 + 2b11 + 2b9 + 2b6 + 2b5 + 2b2 + 2b1 + 6) q^73 + (-2b15 - b13 - b12 - 2b11 - 2b10 - b7 + b5 - 2b3 + 2b2 + 2b1 + 3) * q^74 + (-b13 + b12 - b11 - b9 - b8 - b1 + 1) * q^76 + (3b13 - 4b12 + b11 + b10 + b9 + b8 - b7 + 2b6 + b5 - b4 + 2b3 - b2 - b1 + 3) * q^77 + (-2b14 + 2b13 - b11 - 3b10 - 2b9 + b7 - 2b6 + 2b5 + 2b4 - 2b3 + 2b2 + 8) q^79 + (b8 + 1) * q^80 + (b15 - b14 + 2b12 - b10 + b9 - b8 - b6 + b5 - 2b3 + b2 + 1) * q^82 + (-b12 + 2b11 + b10 + b6 - b5 + 2b1 - 1) * q^83 + (-b15 + b14 - 6b13 + b12 + 3b11 - 3b10 + 2b9 - 2b8 - 5b7 + b6 + b5 - 2b4 - b3 - b2 - b1 - 4) q^85 + (-b13 + b12 + 3b11 - b10 - b7 - b6 - 2b5) * q^86 + (-b14 + b11 - b10 - b7 + b6 + 2b5 - b3 + 1) q^88 + (-2b15 + 2b13 + 2b8 + 4b6 + 2b4 + 2b3 + 2) * q^89 + (2b15 + b14 - 3b13 - 4b12 - 4b11 + 3b10 + b9 - 4b8 - b5 - 5b4 + b3 - 3b2 - 4b1 - 6) q^91 + (-b15 - b14 - b12 - 2b10 - b7 + 2b5 - b3 + b2 - b1 + 3) * q^92 + (2b15 - 3b14 + b12 + b11 + 3b9 - 2b8 + 2b7 - b6 + b5 - 2b4 - 2b3 + b2 + b1) q^94 + (-2b15 - 2b14 + 6b11 + 2b9 + 2b8 - 2b7 + 2b6 - 2b3 + 4b1 + 4) q^95 + (b15 - b14 - 7b13 - b12 - b11 + b10 + 3b9 - 3b8 + b7 + 2b6 + 2b5 - 2b4 - 2b3 - 2b2 - b1 - 5) * q^97 + (2b15 - b14 + b12 + b11 + 2b10 - b9 + b7 + b5 + b4 - b3 + b1 - 2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 8 q^{4} - 12 q^{5} + 6 q^{7} - 2 q^{10} + 4 q^{11} - 8 q^{14} - 8 q^{16} + 10 q^{19} + 2 q^{22} + 4 q^{23} + 10 q^{25} - 12 q^{26} + 6 q^{31} - 8 q^{35} + 14 q^{37} + 12 q^{38} + 2 q^{40} + 32 q^{41}+ \cdots - 40 q^{98}+O(q^{100})$ 16 * q + 8 * q^4 - 12 * q^5 + 6 * q^7 - 2 * q^10 + 4 * q^11 - 8 * q^14 - 8 * q^16 + 10 * q^19 + 2 * q^22 + 4 * q^23 + 10 * q^25 - 12 * q^26 + 6 * q^31 - 8 * q^35 + 14 * q^37 + 12 * q^38 + 2 * q^40 + 32 * q^41 - 4 * q^44 - 18 * q^46 + 24 * q^47 - 6 * q^49 + 14 * q^55 - 4 * q^56 - 28 * q^61 - 8 * q^62 - 16 * q^64 + 72 * q^65 - 16 * q^67 - 30 * q^70 + 56 * q^71 + 44 * q^73 + 24 * q^74 + 20 * q^76 + 52 * q^77 + 30 * q^79 + 12 * q^80 - 12 * q^82 + 8 * q^83 + 12 * q^86 - 2 * q^88 + 36 * q^89 - 8 * q^91 + 8 * q^92 - 14 * q^94 + 72 * q^95 - 40 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + \cdots + 13417$ x^16 - 8*x^15 + 74*x^14 - 378*x^13 + 1878*x^12 - 6718*x^11 + 22086*x^10 - 56904*x^9 + 130215*x^8 - 239606*x^7 + 378750*x^6 - 477124*x^5 + 493030*x^4 - 386266*x^3 + 223844*x^2 - 82874*x + 13417 :

$\beta_{1}$	$=$	$( - 3325577 \nu^{14} + 23279039 \nu^{13} - 218220364 \nu^{12} + 1006694677 \nu^{11} + \cdots - 13304183911 ) / 3261539424$ (-3325577v^14 + 23279039v^13 - 218220364v^12 + 1006694677v^11 - 4890679917v^10 + 15780182142v^9 - 48699356733v^8 + 110238860505v^7 - 222125567883v^6 + 336742486204v^5 - 415275809861v^4 + 371398223899v^3 - 230411582664v^2 + 86434816533v - 13304183911) / 3261539424
$\beta_{2}$	$=$	$( 14496748 \nu^{15} + 165765669 \nu^{14} - 1387066295 \nu^{13} + 16433893584 \nu^{12} + \cdots + 8734585660679 ) / 316369324128$ (14496748v^15 + 165765669v^14 - 1387066295v^13 + 16433893584v^12 - 89388557825v^11 + 460987458741v^10 - 1704964101954v^9 + 5511712419345v^8 - 14215412394141v^7 + 30601359218347v^6 - 53771673709840v^5 + 74664957471809v^4 - 81879734308403v^3 + 64273622949996v^2 - 34535277031465*v + 8734585660679) / 316369324128
$\beta_{3}$	$=$	$( 14496748 \nu^{15} - 383216889 \nu^{14} + 2455811611 \nu^{13} - 20082727968 \nu^{12} + \cdots + 1933997835005 ) / 316369324128$ (14496748v^15 - 383216889v^14 + 2455811611v^13 - 20082727968v^12 + 79753758709v^11 - 375114507753v^10 + 1016663085714v^9 - 3103062983781v^8 + 5617482562641v^7 - 11089350694487v^6 + 11620451848424v^5 - 12919822583029v^4 + 2845093331383v^3 + 825486570276v^2 - 5168168247283*v + 1933997835005) / 316369324128
$\beta_{4}$	$=$	$( - 28661011 \nu^{15} + 376248067 \nu^{14} - 3019246022 \nu^{13} + 19609879751 \nu^{12} + \cdots + 1766457326490 ) / 316369324128$ (-28661011v^15 + 376248067v^14 - 3019246022v^13 + 19609879751v^12 - 91540961349v^11 + 380021763240v^10 - 1215316946595v^9 + 3449204606697v^8 - 7754522847207v^7 + 14949580551542v^6 - 22774230237823v^5 + 27760712142815v^4 - 25755258117138v^3 + 16706236742943v^2 - 7914233731523*v + 1766457326490) / 316369324128
$\beta_{5}$	$=$	$( - 30920939 \nu^{15} - 47258327 \nu^{14} - 211956648 \nu^{13} - 7396232221 \nu^{12} + \cdots - 2047012171496 ) / 158184662064$ (-30920939v^15 - 47258327v^14 - 211956648v^13 - 7396232221v^12 + 30361391909v^11 - 217943855310v^10 + 703406190993v^9 - 2539649789049v^8 + 5922057053211v^7 - 13228223786684v^6 + 21168894176673v^5 - 29136194430747v^4 + 28877960180876v^3 - 21487115641149v^2 + 10220884783775*v - 2047012171496) / 158184662064
$\beta_{6}$	$=$	$( - 30920939 \nu^{15} + 511072412 \nu^{14} - 4120271821 \nu^{13} + 28521203647 \nu^{12} + \cdots + 1740262265133 ) / 158184662064$ (-30920939v^15 + 511072412v^14 - 4120271821v^13 + 28521203647v^12 - 134335126050v^11 + 572900637369v^10 - 1834246369401v^9 + 5281654047894v^8 - 11789222038392v^7 + 23033657253157v^6 - 34641417899267v^5 + 43240535558476v^4 - 39267317510445v^3 + 26479017902163v^2 - 10659357632440*v + 1740262265133) / 158184662064
$\beta_{7}$	$=$	$( 7915220 \nu^{15} - 40953259 \nu^{14} + 414028833 \nu^{13} - 1460644352 \nu^{12} + \cdots - 132462757465 ) / 28760847648$ (7915220v^15 - 40953259v^14 + 414028833v^13 - 1460644352v^12 + 7486016479v^11 - 18277094811v^10 + 59268665334v^9 - 96106377663v^8 + 208572400371v^7 - 200115377413v^6 + 311565411864v^5 - 169504852815v^4 + 313514371381v^3 - 241017203316v^2 + 288697703863*v - 132462757465) / 28760847648
$\beta_{8}$	$=$	$( - 7979 \nu^{15} + 68912 \nu^{14} - 628546 \nu^{13} + 3357412 \nu^{12} - 16625487 \nu^{11} + \cdots + 285645792 ) / 18627492$ (-7979v^15 + 68912v^14 - 628546v^13 + 3357412v^12 - 16625487v^11 + 61197354v^10 - 201214665v^9 + 526067616v^8 - 1200045765v^7 + 2196214414v^6 - 3408471857v^5 + 4104840184v^4 - 3987532800v^3 + 2721817314v^2 - 1298790385*v + 285645792) / 18627492
$\beta_{9}$	$=$	$( - 7979 \nu^{15} + 68912 \nu^{14} - 628546 \nu^{13} + 3357412 \nu^{12} - 16625487 \nu^{11} + \cdots + 304273284 ) / 18627492$ (-7979v^15 + 68912v^14 - 628546v^13 + 3357412v^12 - 16625487v^11 + 61197354v^10 - 201214665v^9 + 526067616v^8 - 1200045765v^7 + 2196214414v^6 - 3408471857v^5 + 4104840184v^4 - 3987532800v^3 + 2721817314v^2 - 1317417877*v + 304273284) / 18627492
$\beta_{10}$	$=$	$( - 6671 \nu^{15} + 41254 \nu^{14} - 402799 \nu^{13} + 1707247 \nu^{12} - 8612564 \nu^{11} + \cdots + 82787651 ) / 7787936$ (-6671v^15 + 41254v^14 - 402799v^13 + 1707247v^12 - 8612564v^11 + 26205779v^10 - 85020497v^9 + 186738624v^8 - 410011674v^7 + 634496219v^6 - 934157663v^5 + 967933718v^4 - 916480415v^3 + 580147099v^2 - 304374922*v + 82787651) / 7787936
$\beta_{11}$	$=$	$( 24636003 \nu^{15} - 180367241 \nu^{14} + 1683650240 \nu^{13} - 8042787727 \nu^{12} + \cdots - 463004009716 ) / 28760847648$ (24636003v^15 - 180367241v^14 + 1683650240v^13 - 8042787727v^12 + 39684270259v^11 - 133306414374v^10 + 429081442695v^9 - 1032074337351v^8 + 2272009170957v^7 - 3839497364232v^6 + 5691504149431v^5 - 6374489557805v^4 + 5941303854100v^3 - 3888260852079v^2 + 1867377806769*v - 463004009716) / 28760847648
$\beta_{12}$	$=$	$( 24636003 \nu^{15} - 189172804 \nu^{14} + 1745289181 \nu^{13} - 8640627827 \nu^{12} + \cdots - 503813289929 ) / 28760847648$ (24636003v^15 - 189172804v^14 + 1745289181v^13 - 8640627827v^12 + 42470004626v^11 - 147354845901v^10 + 475256763393v^9 - 1181342381406v^8 + 2620154110008v^7 - 4588380736521v^6 + 6885584559083v^5 - 8007122573884v^4 + 7544240352965v^3 - 5042857783047v^2 + 2373229705776*v - 503813289929) / 28760847648
$\beta_{13}$	$=$	$( 15958 \nu^{15} - 119685 \nu^{14} + 1130119 \nu^{13} - 5530551 \nu^{12} + 27795985 \nu^{11} + \cdots - 518381770 ) / 18627492$ (15958v^15 - 119685v^14 + 1130119v^13 - 5530551v^12 + 27795985v^11 - 96035346v^10 + 317610396v^9 - 791678301v^8 + 1812360285v^7 - 3211318028v^6 + 5028988886v^5 - 5993427763v^4 + 5979185161v^3 - 4172380467v^2 + 2121539399*v - 518381770) / 18627492
$\beta_{14}$	$=$	$( - 731076296 \nu^{15} + 5564679969 \nu^{14} - 51759660791 \nu^{13} + 255254450652 \nu^{12} + \cdots + 18336335470607 ) / 316369324128$ (-731076296v^15 + 5564679969v^14 - 51759660791v^13 + 255254450652v^12 - 1262781933077v^11 + 4374603032517v^10 - 14191982654406v^9 + 35285931287397v^8 - 78739325671977v^7 + 138259928421331v^6 - 209114172126772v^5 + 245045998417421v^4 - 234039065550755v^3 + 159802254505512v^2 - 77971936362277*v + 18336335470607) / 316369324128
$\beta_{15}$	$=$	$( 791321153 \nu^{15} - 5699211694 \nu^{14} + 53761281981 \nu^{13} - 254911441289 \nu^{12} + \cdots - 16525376998129 ) / 316369324128$ (791321153v^15 - 5699211694v^14 + 53761281981v^13 - 254911441289v^12 + 1268932240096v^11 - 4251240695721v^10 + 13809836508279v^9 - 33245971930596v^8 + 74020750115370v^7 - 125650206707929v^6 + 189347503067457v^5 - 214060374517662v^4 + 205152860375869v^3 - 136001913845709v^2 + 68927049496618*v - 16525376998129) / 316369324128

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$-\beta_{9} + \beta_{8} + 1$ -b9 + b8 + 1
$\nu^{2}$	$=$	$\beta_{15} + \beta_{14} - \beta_{13} - 2\beta_{11} - \beta_{10} - 2\beta_{9} + \beta_{7} - 2\beta_{6} - 2\beta _1 - 4$ b15 + b14 - b13 - 2b11 - b10 - 2b9 + b7 - 2b6 - 2b1 - 4
$\nu^{3}$	$=$	$2 \beta_{15} + \beta_{14} - 5 \beta_{13} - \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + 9 \beta_{9} + \cdots - 14$ 2b15 + b14 - 5b13 - b12 - 2b11 - 3b10 + 9b9 - 12b8 - 2b7 - 3b6 - 4b4 - b3 - b2 - 5b1 - 14
$\nu^{4}$	$=$	$- 8 \beta_{15} - 10 \beta_{14} + 2 \beta_{13} + 5 \beta_{12} + 17 \beta_{11} + 10 \beta_{10} + \cdots + 19$ -8b15 - 10b14 + 2b13 + 5b12 + 17b11 + 10b10 + 31b9 - 13b8 - 16b7 + 20b6 - 2b5 - 8b4 - 2b3 - 2b2 + 15*b1 + 19
$\nu^{5}$	$=$	$- 28 \beta_{15} - 22 \beta_{14} + 64 \beta_{13} + 24 \beta_{12} + 23 \beta_{11} + 58 \beta_{10} + \cdots + 162$ -28b15 - 22b14 + 64b13 + 24b12 + 23b11 + 58b10 - 69b9 + 119b8 + 24b7 + 52b6 - 8b5 + 43b4 + 12b3 + 12b2 + 74*b1 + 162
$\nu^{6}$	$=$	$52 \beta_{15} + 75 \beta_{14} + 61 \beta_{13} - 74 \beta_{12} - 170 \beta_{11} - 65 \beta_{10} + \cdots - 5$ 52b15 + 75b14 + 61b13 - 74b12 - 170b11 - 65b10 - 411b9 + 264b8 + 228b7 - 176b6 + 31b5 + 149b4 + 29b3 + 53b2 - 76*b1 - 5
$\nu^{7}$	$=$	$325 \beta_{15} + 298 \beta_{14} - 657 \beta_{13} - 450 \beta_{12} - 323 \beta_{11} - 852 \beta_{10} + \cdots - 1701$ 325b15 + 298b14 - 657b13 - 450b12 - 323b11 - 852b10 + 373b9 - 1066b8 - 167b7 - 719b6 + 219b5 - 344b4 - 139b3 - 55b2 - 886*b1 - 1701
$\nu^{8}$	$=$	$- 228 \beta_{15} - 448 \beta_{14} - 1368 \beta_{13} + 585 \beta_{12} + 1711 \beta_{11} - 150 \beta_{10} + \cdots - 1786$ -228b15 - 448b14 - 1368b13 + 585b12 + 1711b11 - 150b10 + 5023b9 - 3987b8 - 3036b7 + 1380b6 - 128b5 - 2090b4 - 422b3 - 746b2 - 67*b1 - 1786
$\nu^{9}$	$=$	$- 3506 \beta_{15} - 3490 \beta_{14} + 5934 \beta_{13} + 6780 \beta_{12} + 4769 \beta_{11} + \cdots + 16429$ -3506b15 - 3490b14 + 5934b13 + 6780b12 + 4769b11 + 10592b10 + 824b9 + 8212b8 - 336b7 + 9182b6 - 3490b5 + 1801b4 + 1418b3 - 544b2 + 9812*b1 + 16429
$\nu^{10}$	$=$	$- 775 \beta_{15} + 1126 \beta_{14} + 20300 \beta_{13} - 160 \beta_{12} - 15824 \beta_{11} + \cdots + 36293$ -775b15 + 1126b14 + 20300b13 - 160b12 - 15824b11 + 13530b10 - 56633b9 + 52830b8 + 37407b7 - 8116b6 - 3201b5 + 25763b4 + 5773b3 + 7951b2 + 10306*b1 + 36293
$\nu^{11}$	$=$	$35787 \beta_{15} + 37693 \beta_{14} - 47572 \beta_{13} - 87239 \beta_{12} - 66603 \beta_{11} + \cdots - 143099$ 35787b15 + 37693b14 - 47572b13 - 87239b12 - 66603b11 - 115639b10 - 67910b9 - 44026b8 + 36821b7 - 110918b6 + 43269b5 + 5326b4 - 12756b3 + 18132b2 - 101639*b1 - 143099
$\nu^{12}$	$=$	$39984 \beta_{15} + 26340 \beta_{14} - 258878 \beta_{13} - 94424 \beta_{12} + 124230 \beta_{11} + \cdots - 542199$ 39984b15 + 26340b14 - 258878b13 - 94424b12 + 124230b11 - 288350b10 + 581676b9 - 643364b8 - 424150b7 + 706b6 + 98482b5 - 289332b4 - 73746b3 - 66104b2 - 215666*b1 - 542199
$\nu^{13}$	$=$	$- 342678 \beta_{15} - 375858 \beta_{14} + 321008 \beta_{13} + 984284 \beta_{12} + 862898 \beta_{11} + \cdots + 1048389$ -342678b15 - 375858b14 + 321008b13 + 984284b12 + 862898b11 + 1100808b10 + 1390953b9 - 90875b8 - 832560b7 + 1270202b6 - 441664b5 - 348692b4 + 98730b3 - 303334b2 + 963960*b1 + 1048389
$\nu^{14}$	$=$	$- 729089 \beta_{15} - 672801 \beta_{14} + 3058731 \beta_{13} + 2237884 \beta_{12} - 659178 \beta_{11} + \cdots + 7051750$ -729089b15 - 672801b14 + 3058731b13 + 2237884b12 - 659178b11 + 4637781b10 - 5273480b9 + 7301078b8 + 4374105b7 + 1058408b6 - 1805654b5 + 2971940b4 + 895656b3 + 343122b2 + 3421446*b1 + 7051750
$\nu^{15}$	$=$	$3012048 \beta_{15} + 3370541 \beta_{14} - 1368219 \beta_{13} - 9668935 \beta_{12} - 10414656 \beta_{11} + \cdots - 4830300$ 3012048b15 + 3370541b14 - 1368219b13 - 9668935b12 - 10414656b11 - 8491987b10 - 21648505b9 + 7773878b8 + 14106296b7 - 13734847b6 + 3483128b5 + 6940034b4 - 565701b3 + 3983641b2 - 7907405*b1 - 4830300

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$ .

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$-\beta_{13}$	$-1$

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}

703.1

0.500000	+	2.40229i
0.500000	+	1.35798i
0.500000	−	1.56688i
0.500000	−	3.19339i
0.500000	+	3.43554i
0.500000	+	0.921602i
0.500000	−	0.0286340i
0.500000	−	3.32851i
0.500000	−	2.40229i
0.500000	−	1.35798i
0.500000	+	1.56688i
0.500000	+	3.19339i
0.500000	−	3.43554i
0.500000	−	0.921602i
0.500000	+	0.0286340i
0.500000	+	3.32851i

−0.866025

0.500000i

0.500000

−

0.866025i

−2.83045

1.63416i

2.54243

0.732142i

1.00000i

1.63416

−

2.83045i

703.2

−0.866025

0.500000i

0.500000

−

0.866025i

−1.92604

1.11200i

−2.45660

−

0.982398i

1.00000i

1.11200

−

1.92604i

703.3

−0.866025

0.500000i

0.500000

−

0.866025i

0.606961

−

0.350429i

1.82993

−

1.91085i

1.00000i

−0.350429

0.606961i

703.4

−0.866025

0.500000i

0.500000

−

0.866025i

2.01555

−

1.16368i

1.31629

2.29508i

1.00000i

−1.16368

2.01555i

703.5

0.866025

−

0.500000i

0.500000

−

0.866025i

−3.72526

2.15078i

1.43173

2.22489i

−

1.00000i

−2.15078

3.72526i

703.6

0.866025

−

0.500000i

0.500000

−

0.866025i

−1.54813

0.893814i

−0.165362

−

2.64058i

−

1.00000i

−0.893814

1.54813i

703.7

0.866025

−

0.500000i

0.500000

−

0.866025i

−0.725202

0.418696i

−2.44037

1.02205i

−

1.00000i

−0.418696

0.725202i

703.8

0.866025

−

0.500000i

0.500000

−

0.866025i

2.13257

−

1.23124i

0.941950

−

2.47239i

−

1.00000i

1.23124

−

2.13257i

901.1

−0.866025

−

0.500000i

0.500000

0.866025i

−2.83045

−

1.63416i

2.54243

−

0.732142i

−

1.00000i

1.63416

2.83045i

901.2

−0.866025

−

0.500000i

0.500000

0.866025i

−1.92604

−

1.11200i

−2.45660

0.982398i

−

1.00000i

1.11200

1.92604i

901.3

−0.866025

−

0.500000i

0.500000

0.866025i

0.606961

0.350429i

1.82993

1.91085i

−

1.00000i

−0.350429

−

0.606961i

901.4

−0.866025

−

0.500000i

0.500000

0.866025i

2.01555

1.16368i

1.31629

−

2.29508i

−

1.00000i

−1.16368

−

2.01555i

901.5

0.866025

0.500000i

0.500000

0.866025i

−3.72526

−

2.15078i

1.43173

−

2.22489i

1.00000i

−2.15078

−

3.72526i

901.6

0.866025

0.500000i

0.500000

0.866025i

−1.54813

−

0.893814i

−0.165362

2.64058i

1.00000i

−0.893814

−

1.54813i

901.7

0.866025

0.500000i

0.500000

0.866025i

−0.725202

−

0.418696i

−2.44037

−

1.02205i

1.00000i

−0.418696

−

0.725202i

901.8

0.866025

0.500000i

0.500000

0.866025i

2.13257

1.23124i

0.941950

2.47239i

1.00000i

1.23124

2.13257i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
77.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1386.2.bk.b		16
3.b	odd	2	1		462.2.p.b	yes	16
7.d	odd	6	1		1386.2.bk.a		16
11.b	odd	2	1		1386.2.bk.a		16
21.g	even	6	1		462.2.p.a	✓	16
21.g	even	6	1		3234.2.e.a		16
21.h	odd	6	1		3234.2.e.b		16
33.d	even	2	1		462.2.p.a	✓	16
77.i	even	6	1	inner	1386.2.bk.b		16
231.k	odd	6	1		462.2.p.b	yes	16
231.k	odd	6	1		3234.2.e.b		16
231.l	even	6	1		3234.2.e.a		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
462.2.p.a	✓	16	21.g	even	6	1
462.2.p.a	✓	16	33.d	even	2	1
462.2.p.b	yes	16	3.b	odd	2	1
462.2.p.b	yes	16	231.k	odd	6	1
1386.2.bk.a		16	7.d	odd	6	1
1386.2.bk.a		16	11.b	odd	2	1
1386.2.bk.b		16	1.a	even	1	1	trivial
1386.2.bk.b		16	77.i	even	6	1	inner
3234.2.e.a		16	21.g	even	6	1
3234.2.e.a		16	231.l	even	6	1
3234.2.e.b		16	21.h	odd	6	1
3234.2.e.b		16	231.k	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1386, [\chi])$ :

T_{5}^{16} + 12 T_{5}^{15} + 47 T_{5}^{14} - 12 T_{5}^{13} - 484 T_{5}^{12} - 312 T_{5}^{11} + \cdots + 35344

T_{13}^{8} - 64T_{13}^{6} + 8T_{13}^{5} + 836T_{13}^{4} - 1168T_{13}^{3} - 592T_{13}^{2} + 1216T_{13} - 128

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} - T^{2} + 1)^{4}$ (T^4 - T^2 + 1)^4
$3$	$T^{16}$ T^16
$5$	$T^{16} + 12 T^{15} + \cdots + 35344$ T^16 + 12T^15 + 47T^14 - 12T^13 - 484T^12 - 312T^11 + 4381T^10 + 8472T^9 - 11772T^8 - 39396T^7 + 25139T^6 + 154872T^5 + 118381T^4 - 62244T^3 - 66900T^2 + 29328*T + 35344
$7$	$T^{16} - 6 T^{15} + \cdots + 5764801$ T^16 - 6T^15 + 21T^14 - 10T^13 - 137T^12 + 548T^11 - 284T^10 - 4016T^9 + 16958T^8 - 28112T^7 - 13916T^6 + 187964T^5 - 328937T^4 - 168070T^3 + 2470629T^2 - 4941258*T + 5764801
$11$	$T^{16} + \cdots + 214358881$ T^16 - 4T^15 - 9T^14 + 20T^13 + 421T^12 - 960T^11 - 2794T^10 - 2664T^9 + 71702T^8 - 29304T^7 - 338074T^6 - 1277760T^5 + 6163861T^4 + 3221020T^3 - 15944049T^2 - 77948684*T + 214358881
$13$	$(T^{8} - 64 T^{6} + \cdots - 128)^{2}$ (T^8 - 64T^6 + 8T^5 + 836T^4 - 1168T^3 - 592T^2 + 1216T - 128)^2
$17$	$T^{16} + 53 T^{14} + \cdots + 576$ T^16 + 53T^14 + 136T^13 + 2082T^12 + 5340T^11 + 43709T^10 + 134304T^9 + 661234T^8 + 1314372T^7 + 2791005T^6 + 880544T^5 + 538417T^4 - 159780T^3 + 69528T^2 - 6624T + 576
$19$	$T^{16} - 10 T^{15} + \cdots + 2768896$ T^16 - 10T^15 + 107T^14 - 530T^13 + 3489T^12 - 13172T^11 + 72936T^10 - 205120T^9 + 860080T^8 - 1621056T^7 + 6396800T^6 - 8339456T^5 + 23727360T^4 + 2782208T^3 + 12875776T^2 - 3940352*T + 2768896
$23$	$T^{16} + \cdots + 24039882304$ T^16 - 4T^15 + 143T^14 - 532T^13 + 13392T^12 - 47696T^11 + 697889T^10 - 2171968T^9 + 25326352T^8 - 73933044T^7 + 569006503T^6 - 1395574872T^5 + 8371183769T^4 - 18938176388T^3 + 51979552888T^2 - 37979937888*T + 24039882304
$29$	$T^{16} + \cdots + 12810617856$ T^16 + 244T^14 + 23910T^12 + 1203204T^10 + 33087905T^8 + 491528720T^6 + 3667420672T^4 + 11931674112*T^2 + 12810617856
$31$	$T^{16} - 6 T^{15} + \cdots + 262144$ T^16 - 6T^15 - 37T^14 + 294T^13 + 1441T^12 - 11520T^11 + 3296T^10 + 108960T^9 - 81488T^8 - 866688T^7 + 1998848T^6 - 308736T^5 - 2094848T^4 + 473088T^3 + 2318336T^2 - 1376256*T + 262144
$37$	$T^{16} + \cdots + 125622042624$ T^16 - 14T^15 + 263T^14 - 2062T^13 + 27961T^12 - 199348T^11 + 1843896T^10 - 8801536T^9 + 54018160T^8 - 189887424T^7 + 994476160T^6 - 2631917056T^5 + 9620770048T^4 - 12484740096T^3 + 40451561472T^2 - 34637930496*T + 125622042624
$41$	$(T^{8} - 16 T^{7} + \cdots + 256)^{2}$ (T^8 - 16T^7 + 2T^6 + 952T^5 - 4815T^4 + 6824T^3 + 2848T^2 - 8064*T + 256)^2
$43$	$T^{16} + \cdots + 23084548096$ T^16 + 318T^14 + 38761T^12 + 2303384T^10 + 71086992T^8 + 1124077312T^6 + 8477803520T^4 + 25298960384*T^2 + 23084548096
$47$	$T^{16} + \cdots + 1344737098384$ T^16 - 24T^15 - 57T^14 + 5976T^13 - 2484T^12 - 1115364T^11 + 5374933T^10 + 90988980T^9 - 527953996T^8 - 5594306520T^7 + 44065903899T^6 + 50476519572T^5 - 363709353443T^4 - 326590505388T^3 + 3090667648540T^2 - 3457746300816*T + 1344737098384
$53$	$T^{16} + \cdots + 26488213504$ T^16 + 235T^14 + 232T^13 + 40865T^12 + 36036T^11 + 3016384T^10 + 2083056T^9 + 163718908T^8 + 171246848T^7 + 2778511712T^6 + 9183945376T^5 + 40171071120T^4 + 72713118848T^3 + 111058636544T^2 + 61179127808T + 26488213504
$59$	$T^{16} + \cdots + 109280491776$ T^16 - 266T^14 + 52915T^12 + 70032T^11 - 4201594T^10 - 8528784T^9 + 245570017T^8 + 1019215632T^7 - 2352978552T^6 - 15710742144T^5 + 24074195440T^4 + 201062028288T^3 + 190208538240T^2 - 294630488064*T + 109280491776
$61$	$T^{16} + \cdots + 3489501184$ T^16 + 28T^15 + 620T^14 + 6928T^13 + 69975T^12 + 399224T^11 + 3059300T^10 + 12716608T^9 + 100720417T^8 + 219443376T^7 + 1536277900T^6 + 2680338048T^5 + 16635270224T^4 + 9559013504T^3 + 11848985344T^2 - 3714447360*T + 3489501184
$67$	$T^{16} + \cdots + 2403352576$ T^16 + 16T^15 + 350T^14 + 2344T^13 + 38595T^12 + 195332T^11 + 3037910T^10 + 6732460T^9 + 86888329T^8 + 56306316T^7 + 1933517692T^6 + 129355824T^5 + 7183184528T^4 - 4344754048T^3 + 23480118784T^2 - 7497142272*T + 2403352576
$71$	$(T^{8} - 28 T^{7} + \cdots + 4193232)^{2}$ (T^8 - 28T^7 + 97T^6 + 4360T^5 - 58124T^4 + 251544T^3 - 45068T^2 - 2297712*T + 4193232)^2
$73$	$T^{16} + \cdots + 119793516544$ T^16 - 44T^15 + 1316T^14 - 24272T^13 + 349904T^12 - 3545344T^11 + 30397952T^10 - 189543936T^9 + 1260078592T^8 - 6019252224T^7 + 38692855808T^6 - 102344933376T^5 + 623445557248T^4 - 961323204608T^3 + 7545799049216T^2 + 942044872704*T + 119793516544
$79$	$T^{16} + \cdots + 1130708969104$ T^16 - 30T^15 + 87T^14 + 6390T^13 - 32868T^12 - 1615398T^11 + 25951621T^10 - 102519378T^9 - 620713636T^8 + 4732344210T^7 + 15304968387T^6 - 138506030502T^5 - 246923168675T^4 + 1712977494492T^3 + 6164867690476T^2 + 4406943704592*T + 1130708969104
$83$	$(T^{8} - 4 T^{7} + \cdots + 5604)^{2}$ (T^8 - 4T^7 - 141T^6 + 780T^5 + 1031T^4 - 8396T^3 - 263T^2 + 12516*T + 5604)^2
$89$	$T^{16} + \cdots + 96546188427264$ T^16 - 36T^15 + 256T^14 + 6336T^13 - 71984T^12 - 1587840T^11 + 37971200T^10 - 253334784T^9 - 350878976T^8 + 10521747456T^7 + 26915948544T^6 - 1178899292160T^5 + 9579425505280T^4 - 41364533084160T^3 + 105379288645632T^2 - 151241105866752*T + 96546188427264
$97$	$T^{16} + \cdots + 68597371921$ T^16 + 944T^14 + 340460T^12 + 58793584T^10 + 4910736438T^8 + 163854186512T^6 + 446119112140T^4 + 325271379408*T^2 + 68597371921
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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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 703.8
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1386.2.bk.b

Level	$1386$
Weight	$2$
Character orbit	1386.bk
Analytic conductor	$11.067$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$