LMFDB - Newform orbit 1764.2.b.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(1567,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1764.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.b (of order $2$, degree $1$, 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:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$16$
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} + 4 x^{14} + 54 x^{12} - 112 x^{11} - 104 x^{10} + 1312 x^{9} - 3159 x^{8} + 2544 x^{7} + \cdots + 782 $ x^16 + 4x^14 + 54x^12 - 112x^11 - 104x^10 + 1312x^9 - 3159x^8 + 2544x^7 + 4132x^6 - 16824x^5 + 27780x^4 - 26200x^3 + 14608x^2 - 4784*x + 782
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{7} q^{2} + ( - \beta_{10} + \beta_1 - 1) q^{4} + \beta_{12} q^{5} + \beta_{4} q^{8}+O(q^{10}) $ q + b7 * q^2 + (-b10 + b1 - 1) * q^4 + b12 * q^5 + b4 * q^8	$ q + \beta_{7} q^{2} + ( - \beta_{10} + \beta_1 - 1) q^{4} + \beta_{12} q^{5} + \beta_{4} q^{8} + \beta_{6} q^{10} + (\beta_{11} + \beta_{7} + \beta_{3}) q^{11} + (\beta_{9} + \beta_{6}) q^{13} + ( - \beta_{13} + \beta_{10} - 2 \beta_1 - 2) q^{16} + \beta_{12} q^{17} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_{6}) q^{19}+ \cdots + (2 \beta_{15} + 2 \beta_{14} + \cdots - 3 \beta_{6}) q^{97}+O(q^{100}) $ q + b7 * q^2 + (-b10 + b1 - 1) * q^4 + b12 * q^5 + b4 * q^8 + b6 * q^10 + (b11 + b7 + b3) * q^11 + (b9 + b6) * q^13 + (-b13 + b10 - 2b1 - 2) q^16 + b12 * q^17 + (-b15 + b14 + b9 - b6) * q^19 + (-b12 + b5 + b2) * q^20 + (b13 + 3b1 - 1) q^22 + (2b7 + b4 - b3) q^23 + (b1 + 1) * q^25 + (b12 + b5 + b2) * q^26 + (-2b11 + 2b7 + b4 + 3b3) q^29 + (-b15 + b14 - b9 + b6) * q^31 + (b11 - 4b7 - b4 - 2b3) * q^32 + b6 * q^34 + (3b1 - 4) q^37 + (4b12 + b8 - 3b5) * q^38 + (-b15 - b9 - b6) * q^40 + 3b5 q^41 + (4b10 - 2b1 + 2) * q^43 + (-3b11 + 2b3) * q^44 + (-b13 - 2b10 - b1 - 5) q^46 + (b12 + 2b8 - b5) q^47 + (-b11 + b7) * q^50 + (-b15 - b9 + b6) * q^52 + (2b4 + 2b3) * q^53 + (b15 - b14 - b9 + b6) * q^55 + (-3b13 + 2b10 + b1 + 1) * q^58 + (-b12 - 2b8 + b5) q^59 + (-b9 - b6) * q^61 + (-2b12 + b8 - b5 + 2b2) * q^62 + (2b13 + b10 - 3b1 + 3) * q^64 + (b11 - b7 + 2b4 + b3) q^65 + (-4b13 - 2b1) * q^67 + (-b12 + b5 + b2) * q^68 + (3b11 + 3b7 + 3b3) q^71 + (2b15 + 2b14 - b9 - b6) * q^73 + (-3b11 - 4b7) * q^74 + (3b15 + 2b14 - 3b9) q^76 + (-4b13 + 4b10 - 4b1 + 2) q^79 + (-b12 + b8 - 2b5 - b2) q^80 + (-3b15 + 3b9 + 3b6) q^82 + (-2b5 - 4b2) * q^83 + (b1 - 4) * q^85 + (-2b11 - 2b7 - 4b4) q^86 + (-3b13 + 2b10 - b1 + 5) * q^88 + (5b12 - 2b5) * q^89 + (3b11 - 4b7 + 2b4 - 2b3) * q^92 + (b15 + 4b14 - b9 - 2b6) * q^94 + (5b11 + 7b7 + b4 + 4b3) q^95 + (2b15 + 2b14 - 3b9 - 3b6) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4}+O(q^{10}) $ 16 * q - 8 * q^4	$ 16 q - 8 q^{4} - 40 q^{16} - 16 q^{22} + 16 q^{25} - 64 q^{37} - 64 q^{46} + 40 q^{64} - 64 q^{85} + 64 q^{88}+O(q^{100}) $ 16 * q - 8 * q^4 - 40 * q^16 - 16 * q^22 + 16 * q^25 - 64 * q^37 - 64 * q^46 + 40 * q^64 - 64 * q^85 + 64 * q^88

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 4 x^{14} + 54 x^{12} - 112 x^{11} - 104 x^{10} + 1312 x^{9} - 3159 x^{8} + 2544 x^{7} + \cdots + 782 $ x^16 + 4*x^14 + 54*x^12 - 112*x^11 - 104*x^10 + 1312*x^9 - 3159*x^8 + 2544*x^7 + 4132*x^6 - 16824*x^5 + 27780*x^4 - 26200*x^3 + 14608*x^2 - 4784*x + 782 :

$\beta_{1}$	$=$	$ ( 68460609096708 \nu^{15} - 4470673662808 \nu^{14} + 221129789385746 \nu^{13} + \cdots - 11\!\cdots\!24 ) / 14\!\cdots\!11 $ (68460609096708v^15 - 4470673662808v^14 + 221129789385746v^13 - 100263474949006v^12 + 3386328542280566v^11 - 8346277362814708v^10 - 9975218138394550v^9 + 91208721367956313v^8 - 213323163203640222v^7 + 132548534496325226v^6 + 345859102636619816v^5 - 1157428177735265097v^4 + 1741603948057338728v^3 - 1414776392578940182v^2 + 581795728509992600*v - 115526876966468324) / 14997061542603511
$\beta_{2}$	$=$	$ ( - 327417918828952 \nu^{15} - 912489737968762 \nu^{14} + \cdots - 47\!\cdots\!28 ) / 66\!\cdots\!63 $ (-327417918828952v^15 - 912489737968762v^14 - 2464634453668515v^13 - 4941371389549504v^12 - 23754738088595688v^11 - 19398225654807318v^10 + 66290267686202447v^9 - 283670421349742603v^8 + 13792080161554432v^7 + 733013280422225699v^6 - 1490194499653969402v^5 + 1222036940832765734v^4 + 954240083858638532v^3 - 3193825681085770178v^2 + 2032044391115129084*v - 472453024841471228) / 66415558260101263
$\beta_{3}$	$=$	$ ( 15\!\cdots\!87 \nu^{15} + \cdots - 18\!\cdots\!88 ) / 13\!\cdots\!21 $ (1555384052304487487v^15 + 1339604670703050332v^14 + 7531743358669999721v^13 + 6652426924136514709v^12 + 90531377909982386585v^11 - 95362399007409087852v^10 - 234348639348792239443v^9 + 1831404745105328437098v^8 - 3359157263983789674166v^7 + 1242927061655737977845v^6 + 7200798187189625037522v^5 - 19889480124946128542187v^4 + 26793508912799306677644v^3 - 19451040555700317738042v^2 + 8336186753002420522326*v - 1867599546248460713688) / 130639403097619184321
$\beta_{4}$	$=$	$ ( - 31\!\cdots\!55 \nu^{15} + \cdots + 48\!\cdots\!48 ) / 18\!\cdots\!03 $ (-318349714014117155v^15 - 169063652800363986v^14 - 1345989061870001853v^13 - 672215985049491150v^12 - 17423386714752992097v^11 + 26627463087863738529v^10 + 48407068192275527973v^9 - 391414509363635876612v^8 + 794214729102405745630v^7 - 373255694762441913900v^6 - 1523787615264866790274v^5 + 4520871847151325019132v^4 - 6370293219085152249418v^3 + 4835892465797678085258v^2 - 2036176636502016272590*v + 487073062122041378648) / 18662771871088454903
$\beta_{5}$	$=$	$ ( 473856923359748 \nu^{15} + 383840295516265 \nu^{14} + \cdots - 35\!\cdots\!00 ) / 14\!\cdots\!11 $ (473856923359748v^15 + 383840295516265v^14 + 2174756443991264v^13 + 1749715980623946v^12 + 26886219324590582v^11 - 31321770680120666v^10 - 76289123157682859v^9 + 563049199499742674v^8 - 1035407239122975332v^7 + 327835676897206279v^6 + 2307827064302842291v^5 - 6132682691001785589v^4 + 8024738563480367422v^3 - 5446994131157254103v^2 + 1919144838354128326*v - 358146927872032900) / 14997061542603511
$\beta_{6}$	$=$	$ ( 55\!\cdots\!27 \nu^{15} + \cdots - 50\!\cdots\!76 ) / 13\!\cdots\!21 $ (5500492503984100227v^15 + 4995786690360147679v^14 + 26478219286468497838v^13 + 23828334411265339223v^12 + 318208926056455796405v^11 - 328132761419129144004v^10 - 874281158143623647920v^9 + 6416571148191675290724v^8 - 11527558565646261639380v^7 + 3479438641302538497722v^6 + 25844146842968921774278v^5 - 68798329378779488859130v^4 + 90036644526399417686818v^3 - 62326650020669503237070v^2 + 24339909928772330731036*v - 5087914532726904122076) / 130639403097619184321
$\beta_{7}$	$=$	$ ( 59\!\cdots\!31 \nu^{15} + \cdots - 54\!\cdots\!74 ) / 13\!\cdots\!21 $ (5952725304690144831v^15 + 4913373504941792009v^14 + 27689331148288188771v^13 + 22550928196542473072v^12 + 339010718516081477989v^11 - 388470965209901181124v^10 - 950764010116474908521v^9 + 7026897442921298831606v^8 - 12970938757912323132088v^7 + 4252322434555389872600v^6 + 28339859091945305749350v^5 - 76671489479188666427654v^4 + 101240151222016002112922v^3 - 70636213458116768776102v^2 + 26778488614860426571854*v - 5496866961938737811274) / 130639403097619184321
$\beta_{8}$	$=$	$ ( 25\!\cdots\!96 \nu^{15} + \cdots - 21\!\cdots\!50 ) / 46\!\cdots\!41 $ (25902327859007496v^15 + 21577079064058868v^14 + 121685367454370832v^13 + 101698244496858797v^12 + 1483369668724706806v^11 - 1665246494940919959v^10 - 4078856972771178701v^9 + 30589633647423284589v^8 - 56420217111824698700v^7 + 19005489443147365107v^6 + 123088666360044540971v^5 - 334207906277498618299v^4 + 442421329376818071598v^3 - 309216748734375465357v^2 + 117379544690231687414*v - 21432466723742801850) / 464908907820708841
$\beta_{9}$	$=$	$ ( 73\!\cdots\!91 \nu^{15} + \cdots - 82\!\cdots\!86 ) / 13\!\cdots\!21 $ (7311390845285256291v^15 + 5144247759478178881v^14 + 33052494395288935583v^13 + 23464216531976602108v^12 + 412304745350307442286v^11 - 527646370388946126359v^10 - 1119851439923064088482v^9 + 8796643036979951155484v^8 - 16933879259688981418781v^7 + 6901825506220311144657v^6 + 34736847577918930053593v^5 - 98539036507797617339480v^4 + 134668567438981170920014v^3 - 98761332546754697649693v^2 + 39340148768838995107316*v - 8239028835957057526486) / 130639403097619184321
$\beta_{10}$	$=$	$ ( 26\!\cdots\!94 \nu^{15} + \cdots - 24\!\cdots\!37 ) / 46\!\cdots\!41 $ (26598978070628094v^15 + 20754861575511394v^14 + 121945297597469092v^13 + 94685759422792651v^12 + 1507636575022232512v^11 - 1803486805708866118v^10 - 4207419896042615164v^9 + 31667827257227959771v^8 - 59195530313521122086v^7 + 20748972299418636178v^6 + 127442752230905249876v^5 - 348322567732844885217v^4 + 464281601916429123172v^3 - 327276480336990007534v^2 + 123913467597404059828*v - 24957690805039740937) / 464908907820708841
$\beta_{11}$	$=$	$ ( 84\!\cdots\!72 \nu^{15} + \cdots - 87\!\cdots\!60 ) / 13\!\cdots\!21 $ (8448544333599963272v^15 + 6459958212915796841v^14 + 38949945226253425308v^13 + 30128663474405533398v^12 + 480593764804618359742v^11 - 577012814267616188201v^10 - 1305900993760500402942v^9 + 10081844323059287945776v^8 - 19013659824524795362304v^7 + 7168427247637577844956v^6 + 40100295800739031719712v^5 - 111495933370675259317678v^4 + 150268019875044278229648v^3 - 108297352271035265759942v^2 + 42615858527687211502860*v - 8762536382973257608560) / 130639403097619184321
$\beta_{12}$	$=$	$ ( - 12\!\cdots\!24 \nu^{15} + \cdots + 12\!\cdots\!32 ) / 14\!\cdots\!11 $ (-1244394735841324v^15 - 1015254487117297v^14 - 5815471027979606v^13 - 4747356334044193v^12 - 71097289196119918v^11 + 81359376829844749v^10 + 195369841406174707v^9 - 1472274308570746587v^8 + 2732180783018590880v^7 - 950033829183946954v^6 - 5888232645967983481v^5 + 16126119659137989335v^4 - 21484874797011548774v^3 + 15258438472298117941v^2 - 5959302325712781184*v + 1231061099793606132) / 14997061542603511
$\beta_{13}$	$=$	$ ( - 38\!\cdots\!06 \nu^{15} + \cdots + 40\!\cdots\!23 ) / 46\!\cdots\!41 $ (-38837071408467006v^15 - 30709761853974942v^14 - 180300822619285700v^13 - 143011704542588177v^12 - 2212341935495582864v^11 + 2600106172853103294v^10 + 6064169002183889708v^9 - 46102238907939724050v^8 + 86397532556884843782v^7 - 31309069072667929038v^6 - 183745007679448037388v^5 + 508101002775460820933v^4 - 681018271981825579836v^3 + 488427055994476300206v^2 - 192955113465912973604*v + 40106176486555581423) / 464908907820708841
$\beta_{14}$	$=$	$ ( 15\!\cdots\!17 \nu^{15} + \cdots - 15\!\cdots\!40 ) / 13\!\cdots\!21 $ (15998972551103279917v^15 + 13146545739463821814v^14 + 74380372178245371088v^13 + 60446949971898835049v^12 + 911342693405039975939v^11 - 1046558893404974928844v^10 - 2549704930836326331342v^9 + 18900925418135661838222v^8 - 34925501838727448675442v^7 + 11538713375264870681853v^6 + 76153154961304809095912v^5 - 206238287624443883630230v^4 + 272728671644159240844356v^3 - 190857346846836674542978v^2 + 73063629911885826958694*v - 15209681969564397709940) / 130639403097619184321
$\beta_{15}$	$=$	$ ( 24\!\cdots\!03 \nu^{15} + \cdots - 22\!\cdots\!96 ) / 18\!\cdots\!03 $ (2407697249379471303v^15 + 1962337849022174960v^14 + 11269452242416570801v^13 + 9250267627009115113v^12 + 137786151121143971234v^11 - 157048626672931238306v^10 - 375855880039528846246v^9 + 2851912517811313200749v^8 - 5288343572174182331263v^7 + 1853689065595543927095v^6 + 11414646474950630716223v^5 - 31237695433043179805246v^4 + 41586859446763682767190v^3 - 29458323721466198065049v^2 + 11345194533084915125680*v - 2256954365421484403296) / 18662771871088454903

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

$\nu$	$=$	$ ( 3\beta_{15} + 3\beta_{14} - 7\beta_{11} - \beta_{9} - 7\beta_{7} - \beta_{6} + 7\beta_{3} + 7\beta_1 ) / 14 $ (3b15 + 3b14 - 7b11 - b9 - 7b7 - b6 + 7b3 + 7b1) / 14
$\nu^{2}$	$=$	$ ( - 2 \beta_{15} - 2 \beta_{14} + 14 \beta_{13} - 18 \beta_{12} + 7 \beta_{11} - 14 \beta_{10} + \cdots - 14 ) / 14 $ (-2b15 - 2b14 + 14b13 - 18b12 + 7b11 - 14b10 - 4b9 + 6b8 + 7b7 - 4b6 + 2b5 - 7b4 - 4b2 + 14b1 - 14) / 14
$\nu^{3}$	$=$	$ ( 14 \beta_{15} - 7 \beta_{14} - 21 \beta_{13} + 39 \beta_{12} + 7 \beta_{11} + 42 \beta_{10} - 28 \beta_{9} + \cdots + 21 ) / 14 $ (14b15 - 7b14 - 21b13 + 39b12 + 7b11 + 42b10 - 28b9 - 6b8 - 28b7 + 35b6 - 9b5 - 28b4 - 42b3 + 18b2 - 49*b1 + 21) / 14
$\nu^{4}$	$=$	$ ( - 26 \beta_{15} + 16 \beta_{14} - 35 \beta_{13} + 68 \beta_{12} - 7 \beta_{11} + 49 \beta_{10} + \cdots - 56 ) / 7 $ (-26b15 + 16b14 - 35b13 + 68b12 - 7b11 + 49b10 + 46b9 + 24b8 + 21b7 - 38b6 - 48b5 + 49b4 + 56b3 - 30b2 + 7*b1 - 56) / 7
$\nu^{5}$	$=$	$ ( - 161 \beta_{15} - 196 \beta_{14} + 315 \beta_{13} - 590 \beta_{12} + 350 \beta_{11} - 490 \beta_{10} + \cdots + 245 ) / 14 $ (-161b15 - 196b14 + 315b13 - 590b12 + 350b11 - 490b10 - 105b9 - 130b8 + 637b7 + 400b5 + 161b4 + 196b3 + 180b2 + 63b1 + 245) / 14
$\nu^{6}$	$=$	$ ( 422 \beta_{15} + 317 \beta_{14} - 420 \beta_{13} + 610 \beta_{12} - 672 \beta_{11} + 560 \beta_{10} + \cdots + 1064 ) / 7 $ (422b15 + 317b14 - 420b13 + 610b12 - 672b11 + 560b10 + 60b9 - 110b8 - 1442b7 + 270b6 - 130b5 - 546b4 - 763b3 + 141b2 - 1015*b1 + 1064) / 7
$\nu^{7}$	$=$	$ ( - 1152 \beta_{15} + 416 \beta_{14} + 1617 \beta_{13} - 1638 \beta_{12} - 595 \beta_{11} - 2254 \beta_{10} + \cdots - 11907 ) / 14 $ (-1152b15 + 416b14 + 1617b13 - 1638b12 - 595b11 - 2254b10 + 1854b9 + 1869b8 + 2814b7 - 2164b6 - 1239b5 + 3388b4 + 4914b3 - 2716b2 + 9709*b1 - 11907) / 14
$\nu^{8}$	$=$	$ ( - 680 \beta_{15} - 4208 \beta_{14} + 3115 \beta_{13} - 6244 \beta_{12} + 7336 \beta_{11} - 4263 \beta_{10} + \cdots + 9772 ) / 7 $ (-680b15 - 4208b14 + 3115b13 - 6244b12 + 7336b11 - 4263b10 - 5280b9 - 2072b8 + 6188b7 + 3148b6 + 4564b5 - 4256b4 - 5908b3 + 2884b2 - 4634*b1 + 9772) / 7
$\nu^{9}$	$=$	$ ( 20667 \beta_{15} + 24972 \beta_{14} - 52143 \beta_{13} + 87744 \beta_{12} - 47943 \beta_{11} + \cdots + 24885 ) / 14 $ (20667b15 + 24972b14 - 52143b13 + 87744b12 - 47943b11 + 74214b10 + 14209b9 + 1440b8 - 78106b7 + 4150b6 - 37992b5 - 9716b4 - 13580b3 - 1968b2 - 54719*b1 + 24885) / 14
$\nu^{10}$	$=$	$ ( - 49773 \beta_{15} - 15858 \beta_{14} + 61257 \beta_{13} - 91989 \beta_{12} + 42483 \beta_{11} + \cdots - 143297 ) / 7 $ (-49773b15 - 15858b14 + 61257b13 - 91989b12 + 42483b11 - 86562b10 + 26937b9 + 20058b8 + 139713b7 - 54333b6 + 18145b5 + 80458b4 + 113596b3 - 28093b2 + 143997*b1 - 143297) / 7
$\nu^{11}$	$=$	$ ( 168752 \beta_{15} - 125850 \beta_{14} + 45353 \beta_{13} - 193248 \beta_{12} + 158557 \beta_{11} + \cdots + 1215907 ) / 14 $ (168752b15 - 125850b14 + 45353b13 - 193248b12 + 158557b11 - 65296b10 - 346956b9 - 242275b8 - 273000b7 + 365910b6 + 323389b5 - 495488b4 - 702338b3 + 343596b2 - 829759*b1 + 1215907) / 14
$\nu^{12}$	$=$	$ ( 248970 \beta_{15} + 554352 \beta_{14} - 683242 \beta_{13} + 1268366 \beta_{12} - 1002050 \beta_{11} + \cdots - 989653 ) / 7 $ (248970b15 + 554352b14 - 683242b13 + 1268366b12 - 1002050b11 + 964677b10 + 536090b9 + 285378b8 - 1250830b7 - 201262b6 - 809932b5 + 165634b4 + 232960b3 - 403542b2 + 216783*b1 - 989653) / 7
$\nu^{13}$	$=$	$ ( - 3723771 \beta_{15} - 3352582 \beta_{14} + 6092905 \beta_{13} - 9815208 \beta_{12} + 6721715 \beta_{11} + \cdots - 7018557 ) / 14 $ (-3723771b15 - 3352582b14 + 6092905b13 - 9815208b12 + 6721715b11 - 8620248b10 - 1012029b9 + 494416b8 + 13076630b7 - 1914294b6 + 3571360b5 + 3561852b4 + 5037298b3 - 702468b2 + 9279081*b1 - 7018557) / 14
$\nu^{14}$	$=$	$ ( 5365394 \beta_{15} + 517824 \beta_{14} - 6184157 \beta_{13} + 8209161 \beta_{12} - 2486463 \beta_{11} + \cdots + 25481323 ) / 7 $ (5365394b15 + 517824b14 - 6184157b13 + 8209161b12 - 2486463b11 + 8747046b10 - 4630516b9 - 3998767b8 - 13988520b7 + 7065441b6 + 595888b5 - 10480421b4 - 14816676b3 + 5651431b2 - 22382745*b1 + 25481323) / 7
$\nu^{15}$	$=$	$ ( - 7914484 \beta_{15} + 31441000 \beta_{14} - 20980519 \beta_{13} + 49492304 \beta_{12} + \cdots - 139549431 ) / 14 $ (-7914484b15 + 31441000b14 - 20980519b13 + 49492304b12 - 50219043b11 + 29690458b10 + 52372088b9 + 29316639b8 - 15267476b7 - 42652016b6 - 49829581b5 + 55745130b4 + 78857142b3 - 41465038b2 + 83849479*b1 - 139549431) / 14

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$-1$	$-1$

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} $

1567.1

0.271975	+	0.405613i
0.271975	+	1.48800i
0.271975	−	1.48800i
0.271975	−	0.405613i
0.812979	+	2.57288i
0.812979	−	0.0402481i
0.812979	+	0.0402481i
0.812979	−	2.57288i
−2.22719	−	0.0402481i
−2.22719	+	2.57288i
−2.22719	−	2.57288i
−2.22719	+	0.0402481i
1.14224	+	1.48800i
1.14224	+	0.405613i
1.14224	−	0.405613i
1.14224	−	1.48800i

−1.05050

−

0.946809i

0.207107

1.98925i

−

1.60804i

1.66587

−

2.28580i

−1.52250

1.68925i

1567.2

−1.05050

−

0.946809i

0.207107

1.98925i

1.60804i

1.66587

−

2.28580i

1.52250

−

1.68925i

1567.3

−1.05050

0.946809i

0.207107

−

1.98925i

−

1.60804i

1.66587

2.28580i

1.52250

1.68925i

1567.4

−1.05050

0.946809i

0.207107

−

1.98925i

1.60804i

1.66587

2.28580i

−1.52250

−

1.68925i

1567.5

−0.629640

−

1.26631i

−1.20711

1.59465i

−

2.32685i

2.77937

0.524525i

−2.94652

1.46508i

1567.6

−0.629640

−

1.26631i

−1.20711

1.59465i

2.32685i

2.77937

0.524525i

2.94652

−

1.46508i

1567.7

−0.629640

1.26631i

−1.20711

−

1.59465i

−

2.32685i

2.77937

−

0.524525i

2.94652

1.46508i

1567.8

−0.629640

1.26631i

−1.20711

−

1.59465i

2.32685i

2.77937

−

0.524525i

−2.94652

−

1.46508i

1567.9

0.629640

−

1.26631i

−1.20711

−

1.59465i

−

2.32685i

−2.77937

0.524525i

−2.94652

−

1.46508i

1567.10

0.629640

−

1.26631i

−1.20711

−

1.59465i

2.32685i

−2.77937

0.524525i

2.94652

1.46508i

1567.11

0.629640

1.26631i

−1.20711

1.59465i

−

2.32685i

−2.77937

−

0.524525i

2.94652

−

1.46508i

1567.12

0.629640

1.26631i

−1.20711

1.59465i

2.32685i

−2.77937

−

0.524525i

−2.94652

1.46508i

1567.13

1.05050

−

0.946809i

0.207107

−

1.98925i

−

1.60804i

−1.66587

−

2.28580i

−1.52250

−

1.68925i

1567.14

1.05050

−

0.946809i

0.207107

−

1.98925i

1.60804i

−1.66587

−

2.28580i

1.52250

1.68925i

1567.15

1.05050

0.946809i

0.207107

1.98925i

−

1.60804i

−1.66587

2.28580i

1.52250

−

1.68925i

1567.16

1.05050

0.946809i

0.207107

1.98925i

1.60804i

−1.66587

2.28580i

−1.52250

1.68925i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
7.b	odd	2	1	inner
12.b	even	2	1	inner
21.c	even	2	1	inner
28.d	even	2	1	inner
84.h	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1764.2.b.n	✓	16
3.b	odd	2	1	inner	1764.2.b.n	✓	16
4.b	odd	2	1	inner	1764.2.b.n	✓	16
7.b	odd	2	1	inner	1764.2.b.n	✓	16
12.b	even	2	1	inner	1764.2.b.n	✓	16
21.c	even	2	1	inner	1764.2.b.n	✓	16
28.d	even	2	1	inner	1764.2.b.n	✓	16
84.h	odd	2	1	inner	1764.2.b.n	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1764.2.b.n	✓	16	1.a	even	1	1	trivial
1764.2.b.n	✓	16	3.b	odd	2	1	inner
1764.2.b.n	✓	16	4.b	odd	2	1	inner
1764.2.b.n	✓	16	7.b	odd	2	1	inner
1764.2.b.n	✓	16	12.b	even	2	1	inner
1764.2.b.n	✓	16	21.c	even	2	1	inner
1764.2.b.n	✓	16	28.d	even	2	1	inner
1764.2.b.n	✓	16	84.h	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1764, [\chi])$:

$ T_{5}^{4} + 8T_{5}^{2} + 14 $

$ T_{11}^{4} + 20T_{11}^{2} + 92 $

$ T_{19}^{4} - 88T_{19}^{2} + 1288 $

$ T_{29}^{4} - 84T_{29}^{2} + 1372 $

$ T_{53}^{4} - 80T_{53}^{2} + 448 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{8} + 2 T^{6} + 7 T^{4} + \cdots + 16)^{2} $ (T^8 + 2T^6 + 7T^4 + 8*T^2 + 16)^2
$3$	$ T^{16} $ T^16
$5$	$ (T^{4} + 8 T^{2} + 14)^{4} $ (T^4 + 8*T^2 + 14)^4
$7$	$ T^{16} $ T^16
$11$	$ (T^{4} + 20 T^{2} + 92)^{4} $ (T^4 + 20*T^2 + 92)^4
$13$	$ (T^{4} + 20 T^{2} + 98)^{4} $ (T^4 + 20*T^2 + 98)^4
$17$	$ (T^{4} + 8 T^{2} + 14)^{4} $ (T^4 + 8*T^2 + 14)^4
$19$	$ (T^{4} - 88 T^{2} + 1288)^{4} $ (T^4 - 88*T^2 + 1288)^4
$23$	$ (T^{4} + 44 T^{2} + 92)^{4} $ (T^4 + 44*T^2 + 92)^4
$29$	$ (T^{4} - 84 T^{2} + 1372)^{4} $ (T^4 - 84*T^2 + 1372)^4
$31$	$ (T^{4} - 120 T^{2} + 1288)^{4} $ (T^4 - 120*T^2 + 1288)^4
$37$	$ (T^{2} + 8 T - 2)^{8} $ (T^2 + 8*T - 2)^8
$41$	$ (T^{4} + 144 T^{2} + 1134)^{4} $ (T^4 + 144*T^2 + 1134)^4
$43$	$ (T^{4} + 104 T^{2} + 2576)^{4} $ (T^4 + 104*T^2 + 2576)^4
$47$	$ (T^{4} - 192 T^{2} + 9016)^{4} $ (T^4 - 192*T^2 + 9016)^4
$53$	$ (T^{4} - 80 T^{2} + 448)^{4} $ (T^4 - 80*T^2 + 448)^4
$59$	$ (T^{4} - 192 T^{2} + 9016)^{4} $ (T^4 - 192*T^2 + 9016)^4
$61$	$ (T^{4} + 20 T^{2} + 98)^{4} $ (T^4 + 20*T^2 + 98)^4
$67$	$ (T^{4} + 208 T^{2} + 10304)^{4} $ (T^4 + 208*T^2 + 10304)^4
$71$	$ (T^{4} + 180 T^{2} + 7452)^{4} $ (T^4 + 180*T^2 + 7452)^4
$73$	$ (T^{4} + 164 T^{2} + 4802)^{4} $ (T^4 + 164*T^2 + 4802)^4
$79$	$ (T^{4} + 248 T^{2} + 2576)^{4} $ (T^4 + 248*T^2 + 2576)^4
$83$	$ (T^{4} - 384 T^{2} + 36064)^{4} $ (T^4 - 384*T^2 + 36064)^4
$89$	$ (T^{4} + 184 T^{2} + 14)^{4} $ (T^4 + 184*T^2 + 14)^4
$97$	$ (T^{4} + 196 T^{2} + 4802)^{4} $ (T^4 + 196*T^2 + 4802)^4
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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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{7} q^{2} + ( - \beta_{10} + \beta_1 - 1) q^{4} + \beta_{12} q^{5} + \beta_{4} q^{8}+O(q^{10}) \) q + b7 * q^2 + (-b10 + b1 - 1) * q^4 + b12 * q^5 + b4 * q^8	\( q + \beta_{7} q^{2} + ( - \beta_{10} + \beta_1 - 1) q^{4} + \beta_{12} q^{5} + \beta_{4} q^{8} + \beta_{6} q^{10} + (\beta_{11} + \beta_{7} + \beta_{3}) q^{11} + (\beta_{9} + \beta_{6}) q^{13} + ( - \beta_{13} + \beta_{10} - 2 \beta_1 - 2) q^{16} + \beta_{12} q^{17} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_{6}) q^{19}+ \cdots + (2 \beta_{15} + 2 \beta_{14} + \cdots - 3 \beta_{6}) q^{97}+O(q^{100}) \) q + b7 * q^2 + (-b10 + b1 - 1) * q^4 + b12 * q^5 + b4 * q^8 + b6 * q^10 + (b11 + b7 + b3) * q^11 + (b9 + b6) * q^13 + (-b13 + b10 - 2b1 - 2) q^16 + b12 * q^17 + (-b15 + b14 + b9 - b6) * q^19 + (-b12 + b5 + b2) * q^20 + (b13 + 3b1 - 1) q^22 + (2b7 + b4 - b3) q^23 + (b1 + 1) * q^25 + (b12 + b5 + b2) * q^26 + (-2b11 + 2b7 + b4 + 3b3) q^29 + (-b15 + b14 - b9 + b6) * q^31 + (b11 - 4b7 - b4 - 2b3) * q^32 + b6 * q^34 + (3b1 - 4) q^37 + (4b12 + b8 - 3b5) * q^38 + (-b15 - b9 - b6) * q^40 + 3b5 q^41 + (4b10 - 2b1 + 2) * q^43 + (-3b11 + 2b3) * q^44 + (-b13 - 2b10 - b1 - 5) q^46 + (b12 + 2b8 - b5) q^47 + (-b11 + b7) * q^50 + (-b15 - b9 + b6) * q^52 + (2b4 + 2b3) * q^53 + (b15 - b14 - b9 + b6) * q^55 + (-3b13 + 2b10 + b1 + 1) * q^58 + (-b12 - 2b8 + b5) q^59 + (-b9 - b6) * q^61 + (-2b12 + b8 - b5 + 2b2) * q^62 + (2b13 + b10 - 3b1 + 3) * q^64 + (b11 - b7 + 2b4 + b3) q^65 + (-4b13 - 2b1) * q^67 + (-b12 + b5 + b2) * q^68 + (3b11 + 3b7 + 3b3) q^71 + (2b15 + 2b14 - b9 - b6) * q^73 + (-3b11 - 4b7) * q^74 + (3b15 + 2b14 - 3b9) q^76 + (-4b13 + 4b10 - 4b1 + 2) q^79 + (-b12 + b8 - 2b5 - b2) q^80 + (-3b15 + 3b9 + 3b6) q^82 + (-2b5 - 4b2) * q^83 + (b1 - 4) * q^85 + (-2b11 - 2b7 - 4b4) q^86 + (-3b13 + 2b10 - b1 + 5) * q^88 + (5b12 - 2b5) * q^89 + (3b11 - 4b7 + 2b4 - 2b3) * q^92 + (b15 + 4b14 - b9 - 2b6) * q^94 + (5b11 + 7b7 + b4 + 4b3) q^95 + (2b15 + 2b14 - 3b9 - 3b6) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4}+O(q^{10}) \) 16 * q - 8 * q^4	\( 16 q - 8 q^{4} - 40 q^{16} - 16 q^{22} + 16 q^{25} - 64 q^{37} - 64 q^{46} + 40 q^{64} - 64 q^{85} + 64 q^{88}+O(q^{100}) \) 16 * q - 8 * q^4 - 40 * q^16 - 16 * q^22 + 16 * q^25 - 64 * q^37 - 64 * q^46 + 40 * q^64 - 64 * q^85 + 64 * q^88

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	1764.2.b.n

Level	$1764$
Weight	$2$
Character orbit	1764.b
Analytic conductor	$14.086$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(14.0856109166\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 4 x^{14} + 54 x^{12} - 112 x^{11} - 104 x^{10} + 1312 x^{9} - 3159 x^{8} + 2544 x^{7} + \cdots + 782 \) x^16 + 4x^14 + 54x^12 - 112x^11 - 104x^10 + 1312x^9 - 3159x^8 + 2544x^7 + 4132x^6 - 16824x^5 + 27780x^4 - 26200x^3 + 14608x^2 - 4784*x + 782
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{5}\cdot 7^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 68460609096708 \nu^{15} - 4470673662808 \nu^{14} + 221129789385746 \nu^{13} + \cdots - 11\!\cdots\!24 ) / 14\!\cdots\!11 \) (68460609096708v^15 - 4470673662808v^14 + 221129789385746v^13 - 100263474949006v^12 + 3386328542280566v^11 - 8346277362814708v^10 - 9975218138394550v^9 + 91208721367956313v^8 - 213323163203640222v^7 + 132548534496325226v^6 + 345859102636619816v^5 - 1157428177735265097v^4 + 1741603948057338728v^3 - 1414776392578940182v^2 + 581795728509992600*v - 115526876966468324) / 14997061542603511
\(\beta_{2}\)	\(=\)	\( ( - 327417918828952 \nu^{15} - 912489737968762 \nu^{14} + \cdots - 47\!\cdots\!28 ) / 66\!\cdots\!63 \) (-327417918828952v^15 - 912489737968762v^14 - 2464634453668515v^13 - 4941371389549504v^12 - 23754738088595688v^11 - 19398225654807318v^10 + 66290267686202447v^9 - 283670421349742603v^8 + 13792080161554432v^7 + 733013280422225699v^6 - 1490194499653969402v^5 + 1222036940832765734v^4 + 954240083858638532v^3 - 3193825681085770178v^2 + 2032044391115129084*v - 472453024841471228) / 66415558260101263
\(\beta_{3}\)	\(=\)	\( ( 15\!\cdots\!87 \nu^{15} + \cdots - 18\!\cdots\!88 ) / 13\!\cdots\!21 \) (1555384052304487487v^15 + 1339604670703050332v^14 + 7531743358669999721v^13 + 6652426924136514709v^12 + 90531377909982386585v^11 - 95362399007409087852v^10 - 234348639348792239443v^9 + 1831404745105328437098v^8 - 3359157263983789674166v^7 + 1242927061655737977845v^6 + 7200798187189625037522v^5 - 19889480124946128542187v^4 + 26793508912799306677644v^3 - 19451040555700317738042v^2 + 8336186753002420522326*v - 1867599546248460713688) / 130639403097619184321
\(\beta_{4}\)	\(=\)	\( ( - 31\!\cdots\!55 \nu^{15} + \cdots + 48\!\cdots\!48 ) / 18\!\cdots\!03 \) (-318349714014117155v^15 - 169063652800363986v^14 - 1345989061870001853v^13 - 672215985049491150v^12 - 17423386714752992097v^11 + 26627463087863738529v^10 + 48407068192275527973v^9 - 391414509363635876612v^8 + 794214729102405745630v^7 - 373255694762441913900v^6 - 1523787615264866790274v^5 + 4520871847151325019132v^4 - 6370293219085152249418v^3 + 4835892465797678085258v^2 - 2036176636502016272590*v + 487073062122041378648) / 18662771871088454903
\(\beta_{5}\)	\(=\)	\( ( 473856923359748 \nu^{15} + 383840295516265 \nu^{14} + \cdots - 35\!\cdots\!00 ) / 14\!\cdots\!11 \) (473856923359748v^15 + 383840295516265v^14 + 2174756443991264v^13 + 1749715980623946v^12 + 26886219324590582v^11 - 31321770680120666v^10 - 76289123157682859v^9 + 563049199499742674v^8 - 1035407239122975332v^7 + 327835676897206279v^6 + 2307827064302842291v^5 - 6132682691001785589v^4 + 8024738563480367422v^3 - 5446994131157254103v^2 + 1919144838354128326*v - 358146927872032900) / 14997061542603511
\(\beta_{6}\)	\(=\)	\( ( 55\!\cdots\!27 \nu^{15} + \cdots - 50\!\cdots\!76 ) / 13\!\cdots\!21 \) (5500492503984100227v^15 + 4995786690360147679v^14 + 26478219286468497838v^13 + 23828334411265339223v^12 + 318208926056455796405v^11 - 328132761419129144004v^10 - 874281158143623647920v^9 + 6416571148191675290724v^8 - 11527558565646261639380v^7 + 3479438641302538497722v^6 + 25844146842968921774278v^5 - 68798329378779488859130v^4 + 90036644526399417686818v^3 - 62326650020669503237070v^2 + 24339909928772330731036*v - 5087914532726904122076) / 130639403097619184321
\(\beta_{7}\)	\(=\)	\( ( 59\!\cdots\!31 \nu^{15} + \cdots - 54\!\cdots\!74 ) / 13\!\cdots\!21 \) (5952725304690144831v^15 + 4913373504941792009v^14 + 27689331148288188771v^13 + 22550928196542473072v^12 + 339010718516081477989v^11 - 388470965209901181124v^10 - 950764010116474908521v^9 + 7026897442921298831606v^8 - 12970938757912323132088v^7 + 4252322434555389872600v^6 + 28339859091945305749350v^5 - 76671489479188666427654v^4 + 101240151222016002112922v^3 - 70636213458116768776102v^2 + 26778488614860426571854*v - 5496866961938737811274) / 130639403097619184321
\(\beta_{8}\)	\(=\)	\( ( 25\!\cdots\!96 \nu^{15} + \cdots - 21\!\cdots\!50 ) / 46\!\cdots\!41 \) (25902327859007496v^15 + 21577079064058868v^14 + 121685367454370832v^13 + 101698244496858797v^12 + 1483369668724706806v^11 - 1665246494940919959v^10 - 4078856972771178701v^9 + 30589633647423284589v^8 - 56420217111824698700v^7 + 19005489443147365107v^6 + 123088666360044540971v^5 - 334207906277498618299v^4 + 442421329376818071598v^3 - 309216748734375465357v^2 + 117379544690231687414*v - 21432466723742801850) / 464908907820708841
\(\beta_{9}\)	\(=\)	\( ( 73\!\cdots\!91 \nu^{15} + \cdots - 82\!\cdots\!86 ) / 13\!\cdots\!21 \) (7311390845285256291v^15 + 5144247759478178881v^14 + 33052494395288935583v^13 + 23464216531976602108v^12 + 412304745350307442286v^11 - 527646370388946126359v^10 - 1119851439923064088482v^9 + 8796643036979951155484v^8 - 16933879259688981418781v^7 + 6901825506220311144657v^6 + 34736847577918930053593v^5 - 98539036507797617339480v^4 + 134668567438981170920014v^3 - 98761332546754697649693v^2 + 39340148768838995107316*v - 8239028835957057526486) / 130639403097619184321
\(\beta_{10}\)	\(=\)	\( ( 26\!\cdots\!94 \nu^{15} + \cdots - 24\!\cdots\!37 ) / 46\!\cdots\!41 \) (26598978070628094v^15 + 20754861575511394v^14 + 121945297597469092v^13 + 94685759422792651v^12 + 1507636575022232512v^11 - 1803486805708866118v^10 - 4207419896042615164v^9 + 31667827257227959771v^8 - 59195530313521122086v^7 + 20748972299418636178v^6 + 127442752230905249876v^5 - 348322567732844885217v^4 + 464281601916429123172v^3 - 327276480336990007534v^2 + 123913467597404059828*v - 24957690805039740937) / 464908907820708841
\(\beta_{11}\)	\(=\)	\( ( 84\!\cdots\!72 \nu^{15} + \cdots - 87\!\cdots\!60 ) / 13\!\cdots\!21 \) (8448544333599963272v^15 + 6459958212915796841v^14 + 38949945226253425308v^13 + 30128663474405533398v^12 + 480593764804618359742v^11 - 577012814267616188201v^10 - 1305900993760500402942v^9 + 10081844323059287945776v^8 - 19013659824524795362304v^7 + 7168427247637577844956v^6 + 40100295800739031719712v^5 - 111495933370675259317678v^4 + 150268019875044278229648v^3 - 108297352271035265759942v^2 + 42615858527687211502860*v - 8762536382973257608560) / 130639403097619184321
\(\beta_{12}\)	\(=\)	\( ( - 12\!\cdots\!24 \nu^{15} + \cdots + 12\!\cdots\!32 ) / 14\!\cdots\!11 \) (-1244394735841324v^15 - 1015254487117297v^14 - 5815471027979606v^13 - 4747356334044193v^12 - 71097289196119918v^11 + 81359376829844749v^10 + 195369841406174707v^9 - 1472274308570746587v^8 + 2732180783018590880v^7 - 950033829183946954v^6 - 5888232645967983481v^5 + 16126119659137989335v^4 - 21484874797011548774v^3 + 15258438472298117941v^2 - 5959302325712781184*v + 1231061099793606132) / 14997061542603511
\(\beta_{13}\)	\(=\)	\( ( - 38\!\cdots\!06 \nu^{15} + \cdots + 40\!\cdots\!23 ) / 46\!\cdots\!41 \) (-38837071408467006v^15 - 30709761853974942v^14 - 180300822619285700v^13 - 143011704542588177v^12 - 2212341935495582864v^11 + 2600106172853103294v^10 + 6064169002183889708v^9 - 46102238907939724050v^8 + 86397532556884843782v^7 - 31309069072667929038v^6 - 183745007679448037388v^5 + 508101002775460820933v^4 - 681018271981825579836v^3 + 488427055994476300206v^2 - 192955113465912973604*v + 40106176486555581423) / 464908907820708841
\(\beta_{14}\)	\(=\)	\( ( 15\!\cdots\!17 \nu^{15} + \cdots - 15\!\cdots\!40 ) / 13\!\cdots\!21 \) (15998972551103279917v^15 + 13146545739463821814v^14 + 74380372178245371088v^13 + 60446949971898835049v^12 + 911342693405039975939v^11 - 1046558893404974928844v^10 - 2549704930836326331342v^9 + 18900925418135661838222v^8 - 34925501838727448675442v^7 + 11538713375264870681853v^6 + 76153154961304809095912v^5 - 206238287624443883630230v^4 + 272728671644159240844356v^3 - 190857346846836674542978v^2 + 73063629911885826958694*v - 15209681969564397709940) / 130639403097619184321
\(\beta_{15}\)	\(=\)	\( ( 24\!\cdots\!03 \nu^{15} + \cdots - 22\!\cdots\!96 ) / 18\!\cdots\!03 \) (2407697249379471303v^15 + 1962337849022174960v^14 + 11269452242416570801v^13 + 9250267627009115113v^12 + 137786151121143971234v^11 - 157048626672931238306v^10 - 375855880039528846246v^9 + 2851912517811313200749v^8 - 5288343572174182331263v^7 + 1853689065595543927095v^6 + 11414646474950630716223v^5 - 31237695433043179805246v^4 + 41586859446763682767190v^3 - 29458323721466198065049v^2 + 11345194533084915125680*v - 2256954365421484403296) / 18662771871088454903

\(\nu\)	\(=\)	\( ( 3\beta_{15} + 3\beta_{14} - 7\beta_{11} - \beta_{9} - 7\beta_{7} - \beta_{6} + 7\beta_{3} + 7\beta_1 ) / 14 \) (3b15 + 3b14 - 7b11 - b9 - 7b7 - b6 + 7b3 + 7b1) / 14
\(\nu^{2}\)	\(=\)	\( ( - 2 \beta_{15} - 2 \beta_{14} + 14 \beta_{13} - 18 \beta_{12} + 7 \beta_{11} - 14 \beta_{10} + \cdots - 14 ) / 14 \) (-2b15 - 2b14 + 14b13 - 18b12 + 7b11 - 14b10 - 4b9 + 6b8 + 7b7 - 4b6 + 2b5 - 7b4 - 4b2 + 14b1 - 14) / 14
\(\nu^{3}\)	\(=\)	\( ( 14 \beta_{15} - 7 \beta_{14} - 21 \beta_{13} + 39 \beta_{12} + 7 \beta_{11} + 42 \beta_{10} - 28 \beta_{9} + \cdots + 21 ) / 14 \) (14b15 - 7b14 - 21b13 + 39b12 + 7b11 + 42b10 - 28b9 - 6b8 - 28b7 + 35b6 - 9b5 - 28b4 - 42b3 + 18b2 - 49*b1 + 21) / 14
\(\nu^{4}\)	\(=\)	\( ( - 26 \beta_{15} + 16 \beta_{14} - 35 \beta_{13} + 68 \beta_{12} - 7 \beta_{11} + 49 \beta_{10} + \cdots - 56 ) / 7 \) (-26b15 + 16b14 - 35b13 + 68b12 - 7b11 + 49b10 + 46b9 + 24b8 + 21b7 - 38b6 - 48b5 + 49b4 + 56b3 - 30b2 + 7*b1 - 56) / 7
\(\nu^{5}\)	\(=\)	\( ( - 161 \beta_{15} - 196 \beta_{14} + 315 \beta_{13} - 590 \beta_{12} + 350 \beta_{11} - 490 \beta_{10} + \cdots + 245 ) / 14 \) (-161b15 - 196b14 + 315b13 - 590b12 + 350b11 - 490b10 - 105b9 - 130b8 + 637b7 + 400b5 + 161b4 + 196b3 + 180b2 + 63b1 + 245) / 14
\(\nu^{6}\)	\(=\)	\( ( 422 \beta_{15} + 317 \beta_{14} - 420 \beta_{13} + 610 \beta_{12} - 672 \beta_{11} + 560 \beta_{10} + \cdots + 1064 ) / 7 \) (422b15 + 317b14 - 420b13 + 610b12 - 672b11 + 560b10 + 60b9 - 110b8 - 1442b7 + 270b6 - 130b5 - 546b4 - 763b3 + 141b2 - 1015*b1 + 1064) / 7
\(\nu^{7}\)	\(=\)	\( ( - 1152 \beta_{15} + 416 \beta_{14} + 1617 \beta_{13} - 1638 \beta_{12} - 595 \beta_{11} - 2254 \beta_{10} + \cdots - 11907 ) / 14 \) (-1152b15 + 416b14 + 1617b13 - 1638b12 - 595b11 - 2254b10 + 1854b9 + 1869b8 + 2814b7 - 2164b6 - 1239b5 + 3388b4 + 4914b3 - 2716b2 + 9709*b1 - 11907) / 14
\(\nu^{8}\)	\(=\)	\( ( - 680 \beta_{15} - 4208 \beta_{14} + 3115 \beta_{13} - 6244 \beta_{12} + 7336 \beta_{11} - 4263 \beta_{10} + \cdots + 9772 ) / 7 \) (-680b15 - 4208b14 + 3115b13 - 6244b12 + 7336b11 - 4263b10 - 5280b9 - 2072b8 + 6188b7 + 3148b6 + 4564b5 - 4256b4 - 5908b3 + 2884b2 - 4634*b1 + 9772) / 7
\(\nu^{9}\)	\(=\)	\( ( 20667 \beta_{15} + 24972 \beta_{14} - 52143 \beta_{13} + 87744 \beta_{12} - 47943 \beta_{11} + \cdots + 24885 ) / 14 \) (20667b15 + 24972b14 - 52143b13 + 87744b12 - 47943b11 + 74214b10 + 14209b9 + 1440b8 - 78106b7 + 4150b6 - 37992b5 - 9716b4 - 13580b3 - 1968b2 - 54719*b1 + 24885) / 14
\(\nu^{10}\)	\(=\)	\( ( - 49773 \beta_{15} - 15858 \beta_{14} + 61257 \beta_{13} - 91989 \beta_{12} + 42483 \beta_{11} + \cdots - 143297 ) / 7 \) (-49773b15 - 15858b14 + 61257b13 - 91989b12 + 42483b11 - 86562b10 + 26937b9 + 20058b8 + 139713b7 - 54333b6 + 18145b5 + 80458b4 + 113596b3 - 28093b2 + 143997*b1 - 143297) / 7
\(\nu^{11}\)	\(=\)	\( ( 168752 \beta_{15} - 125850 \beta_{14} + 45353 \beta_{13} - 193248 \beta_{12} + 158557 \beta_{11} + \cdots + 1215907 ) / 14 \) (168752b15 - 125850b14 + 45353b13 - 193248b12 + 158557b11 - 65296b10 - 346956b9 - 242275b8 - 273000b7 + 365910b6 + 323389b5 - 495488b4 - 702338b3 + 343596b2 - 829759*b1 + 1215907) / 14
\(\nu^{12}\)	\(=\)	\( ( 248970 \beta_{15} + 554352 \beta_{14} - 683242 \beta_{13} + 1268366 \beta_{12} - 1002050 \beta_{11} + \cdots - 989653 ) / 7 \) (248970b15 + 554352b14 - 683242b13 + 1268366b12 - 1002050b11 + 964677b10 + 536090b9 + 285378b8 - 1250830b7 - 201262b6 - 809932b5 + 165634b4 + 232960b3 - 403542b2 + 216783*b1 - 989653) / 7
\(\nu^{13}\)	\(=\)	\( ( - 3723771 \beta_{15} - 3352582 \beta_{14} + 6092905 \beta_{13} - 9815208 \beta_{12} + 6721715 \beta_{11} + \cdots - 7018557 ) / 14 \) (-3723771b15 - 3352582b14 + 6092905b13 - 9815208b12 + 6721715b11 - 8620248b10 - 1012029b9 + 494416b8 + 13076630b7 - 1914294b6 + 3571360b5 + 3561852b4 + 5037298b3 - 702468b2 + 9279081*b1 - 7018557) / 14
\(\nu^{14}\)	\(=\)	\( ( 5365394 \beta_{15} + 517824 \beta_{14} - 6184157 \beta_{13} + 8209161 \beta_{12} - 2486463 \beta_{11} + \cdots + 25481323 ) / 7 \) (5365394b15 + 517824b14 - 6184157b13 + 8209161b12 - 2486463b11 + 8747046b10 - 4630516b9 - 3998767b8 - 13988520b7 + 7065441b6 + 595888b5 - 10480421b4 - 14816676b3 + 5651431b2 - 22382745*b1 + 25481323) / 7
\(\nu^{15}\)	\(=\)	\( ( - 7914484 \beta_{15} + 31441000 \beta_{14} - 20980519 \beta_{13} + 49492304 \beta_{12} + \cdots - 139549431 ) / 14 \) (-7914484b15 + 31441000b14 - 20980519b13 + 49492304b12 - 50219043b11 + 29690458b10 + 52372088b9 + 29316639b8 - 15267476b7 - 42652016b6 - 49829581b5 + 55745130b4 + 78857142b3 - 41465038b2 + 83849479*b1 - 139549431) / 14

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1567.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{8} + 2 T^{6} + 7 T^{4} + \cdots + 16)^{2} \) (T^8 + 2T^6 + 7T^4 + 8*T^2 + 16)^2
$3$	\( T^{16} \) T^16
$5$	\( (T^{4} + 8 T^{2} + 14)^{4} \) (T^4 + 8*T^2 + 14)^4
$7$	\( T^{16} \) T^16
$11$	\( (T^{4} + 20 T^{2} + 92)^{4} \) (T^4 + 20*T^2 + 92)^4
$13$	\( (T^{4} + 20 T^{2} + 98)^{4} \) (T^4 + 20*T^2 + 98)^4
$17$	\( (T^{4} + 8 T^{2} + 14)^{4} \) (T^4 + 8*T^2 + 14)^4
$19$	\( (T^{4} - 88 T^{2} + 1288)^{4} \) (T^4 - 88*T^2 + 1288)^4
$23$	\( (T^{4} + 44 T^{2} + 92)^{4} \) (T^4 + 44*T^2 + 92)^4
$29$	\( (T^{4} - 84 T^{2} + 1372)^{4} \) (T^4 - 84*T^2 + 1372)^4
$31$	\( (T^{4} - 120 T^{2} + 1288)^{4} \) (T^4 - 120*T^2 + 1288)^4
$37$	\( (T^{2} + 8 T - 2)^{8} \) (T^2 + 8*T - 2)^8
$41$	\( (T^{4} + 144 T^{2} + 1134)^{4} \) (T^4 + 144*T^2 + 1134)^4
$43$	\( (T^{4} + 104 T^{2} + 2576)^{4} \) (T^4 + 104*T^2 + 2576)^4
$47$	\( (T^{4} - 192 T^{2} + 9016)^{4} \) (T^4 - 192*T^2 + 9016)^4
$53$	\( (T^{4} - 80 T^{2} + 448)^{4} \) (T^4 - 80*T^2 + 448)^4
$59$	\( (T^{4} - 192 T^{2} + 9016)^{4} \) (T^4 - 192*T^2 + 9016)^4
$61$	\( (T^{4} + 20 T^{2} + 98)^{4} \) (T^4 + 20*T^2 + 98)^4
$67$	\( (T^{4} + 208 T^{2} + 10304)^{4} \) (T^4 + 208*T^2 + 10304)^4
$71$	\( (T^{4} + 180 T^{2} + 7452)^{4} \) (T^4 + 180*T^2 + 7452)^4
$73$	\( (T^{4} + 164 T^{2} + 4802)^{4} \) (T^4 + 164*T^2 + 4802)^4
$79$	\( (T^{4} + 248 T^{2} + 2576)^{4} \) (T^4 + 248*T^2 + 2576)^4
$83$	\( (T^{4} - 384 T^{2} + 36064)^{4} \) (T^4 - 384*T^2 + 36064)^4
$89$	\( (T^{4} + 184 T^{2} + 14)^{4} \) (T^4 + 184*T^2 + 14)^4
$97$	\( (T^{4} + 196 T^{2} + 4802)^{4} \) (T^4 + 196*T^2 + 4802)^4
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\(n\)	\(785\)	\(883\)	\(1081\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)