LMFDB - Newform orbit 1680.2.di.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(289,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3, 2]))

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

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

chi := DirichletCharacter("1680.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.di (of order $6$, degree $2$, not 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:	$13.4148675396$
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} - 2 x^{15} + 2 x^{14} - 4 x^{13} - 14 x^{12} + 38 x^{11} - 40 x^{10} + 64 x^{9} + 291 x^{8} + \cdots + 1 $ x^16 - 2x^15 + 2x^14 - 4x^13 - 14x^12 + 38x^11 - 40x^10 + 64x^9 + 291x^8 - 630x^7 + 584x^6 - 800x^5 + 734x^4 - 188x^3 + 32x^2 - 8*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
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_{3} q^{3} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{7} + 1) q^{9}+O(q^{10}) $ q + b3 * q^3 + (-b14 - b13 + b12 - b11 - b8 + b4 - b2 + 1) * q^5 + (b13 + b5 - b4 + b2 - 1) * q^7 + (-b7 + 1) * q^9	$ q + \beta_{3} q^{3} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 1) q^{5}+ \cdots + (\beta_{13} - \beta_{12} - \beta_{2} + \cdots + 1) q^{99}+O(q^{100}) $ q + b3 * q^3 + (-b14 - b13 + b12 - b11 - b8 + b4 - b2 + 1) * q^5 + (b13 + b5 - b4 + b2 - 1) * q^7 + (-b7 + 1) * q^9 + (-b14 - b10 + 2b9 - b8 - b7 - 2b2 + 2) * q^11 + (b15 - 2b12 - 2b6 - 2b4 - 2b3) * q^13 + (-b12 + b1 + 1) * q^15 + (2b14 + 2b11 + b5 + b3) * q^17 + (-b10 + b9 - 4b7 - b1 + 3) q^19 + (b14 + b10 - b9 + b8 + b7 - 1) * q^21 + (3b15 + 2b14 - 2b13 - 2b12 - 2b8 - 5b6 + 3b5 - 2b2 + 2) * q^23 + (-2b14 - b11 - b10 + 3b9 - b8 - 2b7 - b5 - 2b3 - 3b2 + 3) q^25 + (b6 + b3) * q^27 + (2b13 - 2b12 + 2b2 + 3b1 + 2) * q^29 + (2b10 - b7) q^31 + (2b15 - b13 - b11 - b8 - b6 + 2b5 + b4 - b2 + 1) * q^33 + (-b15 + b14 + 2b13 + b11 - 2b9 + b8 - 2b7 - b6 - b5 + 2b3 + 2b2 + b1) q^35 + (b13 + b11 + b8 - 6b6 - b4 + b2 - 1) q^37 + (2b10 + b9 + b7 - b2 + 1) q^39 + (-b13 + b12 + b2 - 3b1 - 1) q^41 + (-2b15 + b13 - 2b12 + 2b6 - 3b4 + 2b3 + b2 - 1) q^43 + (-b14 - b11 - b8) * q^45 + (-b14 + b13 + b12 + b8 - 2b6 + b2 - 1) q^47 + (3b10 + b9 + 2b7 + b2 - 3) * q^49 + (-2b10 - b9 - 2b1 - 2) * q^51 + (2b14 + 2b11 + 6b3) q^53 + (2b15 - b13 - 2b6 + 3b4 - 2b3 - 2b2 - b1 + 1) q^55 + (b15 + b12 + 3b6 + b4 + 3b3) * q^57 + (-b10 - b7) * q^59 + (-2b14 - 2b13 + 2b12 - 3b10 + 5b9 - 2b8 - 5b7 - 2b2 - 3b1 + 4) q^61 + (-b15 + b13 + b11 + b8 - b4 + b2 - 1) * q^63 + (4b15 + b14 - b13 - b12 + b11 - b10 - 2b9 - b8 - 3b7 + 4b5 - b4 - b2 - b1 + 3) * q^65 + (-2b14 - 4b11 - 2b8 - b5) q^67 + (2b13 - 2b12 - b2 + 3) * q^69 + (-2b13 + 2b12 + 2b2 - 5b1) * q^71 + (-2b14 - 3b11 - b8 + 4b5 + 4b3) * q^73 + (3b15 - b13 - b11 + b10 + b9 - b8 + b7 - b6 + 3b5 + b4 - b2 + b1 + 1) * q^75 + (-b15 - 3b14 - 4b11 - b8 - 3b6 + b3) q^77 + (-4b14 - 4b13 + 4b12 - 3b10 + b9 - 4b8 - 2b7 - 4b2 - 3b1 + 3) * q^79 - b7 * q^81 + (-b15 - 2b13 - b6 + 2b4 - b3 - 2b2 + 2) q^83 + (-4b15 + b13 + 2b12 + b4 + 3b2 - 6) q^85 + (-b14 - 3b11 - 2b8 + b3) * q^87 + (b14 + b13 - b12 - 3b10 - 6b9 + b8 - b7 + b2 - 3b1 - 3) q^89 + (2b14 + b10 + b9 + 2b8 + 5b2 + 2b1 - 2) * q^91 + (2b14 - 2b12 + 2b11 + b6 - 2b4) * q^93 + (-2b14 - b11 + b10 - 2b9 - 2b8 + 5b7 - b5 - b3 + 2b2 - 2) q^95 + (-b15 + b13 - b4 + b2 - 1) * q^97 + (b13 - b12 - b2 + b1 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{5} + 8 q^{9}+O(q^{10}) $ 16 * q + 2 * q^5 + 8 * q^9	$ 16 q + 2 q^{5} + 8 q^{9} + 4 q^{15} + 24 q^{19} - 4 q^{21} - 4 q^{25} + 24 q^{29} - 16 q^{31} + 10 q^{35} + 4 q^{39} + 16 q^{41} - 2 q^{45} - 40 q^{49} - 4 q^{51} - 8 q^{55} - 4 q^{59} + 16 q^{61} + 30 q^{65} + 40 q^{69} + 56 q^{71} - 8 q^{75} + 16 q^{79} - 8 q^{81} - 64 q^{85} + 16 q^{89} - 8 q^{91} + 22 q^{95}+O(q^{100}) $ 16 * q + 2 * q^5 + 8 * q^9 + 4 * q^15 + 24 * q^19 - 4 * q^21 - 4 * q^25 + 24 * q^29 - 16 * q^31 + 10 * q^35 + 4 * q^39 + 16 * q^41 - 2 * q^45 - 40 * q^49 - 4 * q^51 - 8 * q^55 - 4 * q^59 + 16 * q^61 + 30 * q^65 + 40 * q^69 + 56 * q^71 - 8 * q^75 + 16 * q^79 - 8 * q^81 - 64 * q^85 + 16 * q^89 - 8 * q^91 + 22 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} - 14 x^{12} + 38 x^{11} - 40 x^{10} + 64 x^{9} + 291 x^{8} + \cdots + 1 $ x^16 - 2*x^15 + 2*x^14 - 4*x^13 - 14*x^12 + 38*x^11 - 40*x^10 + 64*x^9 + 291*x^8 - 630*x^7 + 584*x^6 - 800*x^5 + 734*x^4 - 188*x^3 + 32*x^2 - 8*x + 1 :

$\beta_{1}$	$=$	$ ( - 38414168495 \nu^{15} + 46999484762 \nu^{14} - 56685616158 \nu^{13} + \cdots - 33746305280705 ) / 11431248166457 $ (-38414168495v^15 + 46999484762v^14 - 56685616158v^13 + 131873366880v^12 + 642956778028v^11 - 939694854143v^10 + 1093318205564v^9 - 2230952782978v^8 - 12780049004340v^7 + 14111475820386v^6 - 17131346947484v^5 + 27207219807940v^4 - 7455199259279v^3 + 1191125364091v^2 - 321416591346*v - 33746305280705) / 11431248166457
$\beta_{2}$	$=$	$ ( 145306983361 \nu^{15} - 177522986678 \nu^{14} + 213949914242 \nu^{13} + \cdots + 31632456480199 ) / 11431248166457 $ (145306983361v^15 - 177522986678v^14 + 213949914242v^13 - 520543533819v^12 - 2426061321940v^11 + 3549177780569v^10 - 4097251614084v^9 + 8418948255726v^8 + 48225232348268v^7 - 53280290889126v^6 + 64651815055380v^5 - 102672086774396v^4 + 28133974246241v^3 - 4494960282621v^2 + 1212926356606*v + 31632456480199) / 11431248166457
$\beta_{3}$	$=$	$ ( 760172030748 \nu^{15} - 826140524550 \nu^{14} + 311805690170 \nu^{13} - 1979665829518 \nu^{12} + \cdots - 575089767434 ) / 11431248166457 $ (760172030748v^15 - 826140524550v^14 + 311805690170v^13 - 1979665829518v^12 - 13114010371696v^11 + 18515621292532v^10 - 6663971281019v^9 + 27229891583895v^8 + 259493064521626v^7 - 266716860192172v^6 + 60835212617387v^5 - 306079675769764v^4 + 90642215592396v^3 + 246128680839936v^2 + 3861979127591*v - 575089767434) / 11431248166457
$\beta_{4}$	$=$	$ ( - 1159675947687 \nu^{15} + 2123382923210 \nu^{14} - 1790860341485 \nu^{13} + \cdots + 29672440598565 ) / 11431248166457 $ (-1159675947687v^15 + 2123382923210v^14 - 1790860341485v^13 + 4132499467647v^12 + 17082017216494v^11 - 41638199149391v^10 + 36484081923634v^9 - 63733593693012v^8 - 351285551153804v^7 + 677606774159518v^6 - 505106942012337v^5 + 778126373623118v^4 - 678461323755244v^3 + 34074325119351v^2 - 9164323089196*v + 29672440598565) / 11431248166457
$\beta_{5}$	$=$	$ ( - 2241001108620 \nu^{15} + 2440547176294 \nu^{14} - 921260080696 \nu^{13} + \cdots + 1695440450074 ) / 11431248166457 $ (-2241001108620v^15 + 2440547176294v^14 - 921260080696v^13 + 5836557871079v^12 + 38651783264660v^11 - 54581263789178v^10 + 19684393272760v^9 - 80280420418569v^8 - 764989544114834v^7 + 786254645863424v^6 - 179383356485812v^5 + 902375388086909v^4 - 267226659358818v^3 - 682424450829525v^2 - 11385720733444*v + 1695440450074) / 11431248166457
$\beta_{6}$	$=$	$ ( 4058097157982 \nu^{15} - 7568755097298 \nu^{14} + 7059730047127 \nu^{13} + \cdots - 16137847630635 ) / 11431248166457 $ (4058097157982v^15 - 7568755097298v^14 + 7059730047127v^13 - 15184509679358v^12 - 58946431470254v^11 + 146411669575112v^10 - 142164152788000v^9 + 238760344218351v^8 + 1214849064761822v^7 - 2395065444113876v^6 + 2037822081506248v^5 - 2940884698505042v^4 + 2557823285973472v^3 - 388309099007712v^2 + 34395735205495*v - 16137847630635) / 11431248166457
$\beta_{7}$	$=$	$ ( - 2575730834 \nu^{15} + 4699915310 \nu^{14} - 4379029892 \nu^{13} + 9614611682 \nu^{12} + \cdots + 17015828908 ) / 7008735847 $ (-2575730834v^15 + 4699915310v^14 - 4379029892v^13 + 9614611682v^12 + 37669504970v^11 - 91091321784v^10 + 87843289680v^9 - 151013347873v^8 - 774495170890v^7 + 1484105110236v^6 - 1259913976952v^5 + 1863977222846v^4 - 1585945295448v^3 + 240744226192v^2 - 69083699784*v + 17015828908) / 7008735847
$\beta_{8}$	$=$	$ ( 1063280569755 \nu^{15} - 2138087245431 \nu^{14} + 2140603616666 \nu^{13} + \cdots - 8602022998880 ) / 1633035452351 $ (1063280569755v^15 - 2138087245431v^14 + 2140603616666v^13 - 4270022226214v^12 - 14845214238260v^11 + 40596196036702v^10 - 42811945511246v^9 + 68369173152460v^8 + 308749841689177v^7 - 673674444501418v^6 + 625167345362715v^5 - 855730190069746v^4 + 788555489457606v^3 - 202118395383557v^2 + 34379925210470*v - 8602022998880) / 1633035452351
$\beta_{9}$	$=$	$ ( 7502690467678 \nu^{15} - 13694199839130 \nu^{14} + 12758334474904 \nu^{13} + \cdots - 29172119880252 ) / 11431248166457 $ (7502690467678v^15 - 13694199839130v^14 + 12758334474904v^13 - 28102531513444v^12 - 109693802936905v^11 + 265404776190093v^10 - 255814696983230v^9 + 441714091122816v^8 + 2255188372760505v^7 - 4324366285850692v^6 + 3669645519916604v^5 - 5466340585268175v^4 + 4623119858303861v^3 - 701766532319694v^2 + 201383747371308*v - 29172119880252) / 11431248166457
$\beta_{10}$	$=$	$ ( - 8183751109447 \nu^{15} + 14956739476040 \nu^{14} - 13922497684986 \nu^{13} + \cdots + 54263991142284 ) / 11431248166457 $ (-8183751109447v^15 + 14956739476040v^14 - 13922497684986v^13 + 30578923581156v^12 + 119598143979065v^11 - 289853173711402v^10 + 279305469940010v^9 - 480196298050966v^8 - 2459380223502305v^7 + 4723682477021618v^6 - 4004669271289776v^5 + 5932582107617908v^4 - 5055851615402594v^3 + 767408313770491v^2 - 220236155839242*v + 54263991142284) / 11431248166457
$\beta_{11}$	$=$	$ ( - 9494932205673 \nu^{15} + 17203785936492 \nu^{14} - 15830728042254 \nu^{13} + \cdots + 61766681609962 ) / 11431248166457 $ (-9494932205673v^15 + 17203785936492v^14 - 15830728042254v^13 + 35235059970569v^12 + 139295605560736v^11 - 334146095435292v^10 + 317763568748586v^9 - 552101744717173v^8 - 2861708932288753v^7 + 5435608190228966v^6 - 4540481060164001v^5 + 6816437052589699v^4 - 5764590891006364v^3 + 806110661364879v^2 - 251085559700414*v + 61766681609962) / 11431248166457
$\beta_{12}$	$=$	$ ( 10583106332007 \nu^{15} - 18562866254784 \nu^{14} + 16212768119001 \nu^{13} + \cdots - 62361843284805 ) / 11431248166457 $ (10583106332007v^15 - 18562866254784v^14 + 16212768119001v^13 - 37702760224023v^12 - 157985234866350v^11 + 364596213484152v^10 - 327711794585876v^9 + 583962180212718v^8 + 3234466606384584v^7 - 5891738412332556v^6 + 4610113536311955v^5 - 7128621954756198v^4 + 5858692599566580v^3 - 312154621077708v^2 + 83988991866438*v - 62361843284805) / 11431248166457
$\beta_{13}$	$=$	$ ( 10663789073454 \nu^{15} - 18661163594876 \nu^{14} + 16331067749359 \nu^{13} + \cdots - 32056382963606 ) / 11431248166457 $ (10663789073454v^15 - 18661163594876v^14 + 16331067749359v^13 - 37987756392193v^12 - 159325975568434v^11 + 366561272531474v^10 - 329946311402856v^9 + 588615808979414v^8 + 3261120405274960v^7 - 5921218860713121v^6 + 4645853676172177v^5 - 7185374813558250v^4 + 5874244137453899v^3 - 314639249925878v^2 + 84659440952318*v - 32056382963606) / 11431248166457
$\beta_{14}$	$=$	$ ( 11015276267169 \nu^{15} - 18856066985592 \nu^{14} + 16454339422594 \nu^{13} + \cdots - 62916861144830 ) / 11431248166457 $ (11015276267169v^15 - 18856066985592v^14 + 16454339422594v^13 - 39194391629605v^12 - 165523626304128v^11 + 371177338020356v^10 - 331091511310624v^9 + 606561527884963v^8 + 3380695061332005v^7 - 5969041910613310v^6 + 4662151485398775v^5 - 7428596404129227v^4 + 5945875322191156v^3 - 325284547851464v^2 + 258809517955596*v - 62916861144830) / 11431248166457
$\beta_{15}$	$=$	$ ( 13399447651346 \nu^{15} - 23430444102746 \nu^{14} + 20519536606234 \nu^{13} + \cdots - 46488804269431 ) / 11431248166457 $ (13399447651346v^15 - 23430444102746v^14 + 20519536606234v^13 - 47737126077397v^12 - 200273669782708v^11 + 460259833552415v^10 - 414511080319944v^9 + 739757034059786v^8 + 4098887977021204v^7 - 7433685405000834v^6 + 5838337123809904v^5 - 9030367022903148v^4 + 7367017819324544v^3 - 395429090524619v^2 + 106398295849754*v - 46488804269431) / 11431248166457

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

$\nu$	$=$	$ ( -\beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_1 ) / 2 $ (-b15 - b14 + b12 + b11 - b10 + b9 - b7 - b5 - b4 - b1) / 2
$\nu^{2}$	$=$	$ -\beta_{14} - \beta_{11} + 2\beta_{3} $ -b14 - b11 + 2*b3
$\nu^{3}$	$=$	$ ( 5\beta_{15} - 8\beta_{13} + 3\beta_{12} - 2\beta_{6} + 3\beta_{4} - 2\beta_{3} - 3\beta_{2} - 3\beta _1 + 3 ) / 2 $ (5b15 - 8b13 + 3b12 - 2b6 + 3b4 - 2b3 - 3b2 - 3b1 + 3) / 2
$\nu^{4}$	$=$	$ 5\beta_{10} + \beta_{9} - 8\beta_{7} + 5\beta _1 + 13 $ 5b10 + b9 - 8b7 + 5*b1 + 13
$\nu^{5}$	$=$	$ ( - 23 \beta_{14} + 11 \beta_{11} - 11 \beta_{10} + 23 \beta_{9} - 13 \beta_{7} - 23 \beta_{5} + \cdots + 23 ) / 2 $ (-23b14 + 11b11 - 11b10 + 23b9 - 13b7 - 23b5 - 10b3 - 23b2 + 23) / 2
$\nu^{6}$	$=$	$ 7\beta_{15} - \beta_{13} - 23\beta_{12} + 28\beta_{6} - 22\beta_{4} + 28\beta_{3} - \beta_{2} + 1 $ 7b15 - b13 - 23b12 + 28b6 - 22b4 + 28*b3 - b2 + 1
$\nu^{7}$	$=$	$ ( 103 \beta_{15} - 43 \beta_{14} - 148 \beta_{13} + 43 \beta_{12} - 43 \beta_{11} - 43 \beta_{10} + \cdots + 166 ) / 2 $ (103b15 - 43b14 - 148b13 + 43b12 - 43b11 - 43b10 + 103b9 - 148b8 - 61b7 - 42b6 + 103b5 + 43b4 - 148b2 - 43b1 + 166) / 2
$\nu^{8}$	$=$	$ -10\beta_{14} + 94\beta_{10} + 40\beta_{9} - 10\beta_{8} - 157\beta_{7} - 40\beta_{2} + 40 $ -10b14 + 94b10 + 40b9 - 10b8 - 157b7 - 40b2 + 40
$\nu^{9}$	$=$	$ ( 459\beta_{15} - 479\beta_{12} - 168\beta_{6} + 171\beta_{4} - 168\beta_{3} - 459\beta_{2} + 171\beta _1 + 339 ) / 2 $ (459b15 - 479b12 - 168b6 + 171b4 - 168b3 - 459b2 + 171*b1 + 339) / 2
$\nu^{10}$	$=$	$ 214 \beta_{15} + 469 \beta_{14} - 70 \beta_{13} - 469 \beta_{12} + 399 \beta_{11} - 70 \beta_{8} + \cdots + 70 $ 214b15 + 469b14 - 70b13 - 469b12 + 399b11 - 70b8 + 496b6 + 214b5 - 399b4 - 70b2 + 70
$\nu^{11}$	$=$	$ ( - 681 \beta_{14} - 681 \beta_{11} - 681 \beta_{10} + 2047 \beta_{9} - 2868 \beta_{8} - 1389 \beta_{7} + \cdots + 2047 ) / 2 $ (-681b14 - 681b11 - 681b10 + 2047b9 - 2868b8 - 1389b7 + 2047b5 + 658b3 - 2047*b2 + 2047) / 2
$\nu^{12}$	$=$	$ 424\beta_{13} - 424\beta_{12} - 683\beta_{2} - 1693\beta _1 - 4232 $ 424b13 - 424b12 - 683b2 - 1693b1 - 4232
$\nu^{13}$	$=$	$ ( 9149 \beta_{15} + 9997 \beta_{14} - 9997 \beta_{12} - 2701 \beta_{11} + 2701 \beta_{10} - 9149 \beta_{9} + \cdots - 3910 ) / 2 $ (9149b15 + 9997b14 - 9997b12 - 2701b11 + 2701b10 - 9149b9 + 6611b7 - 2538b6 + 9149b5 + 2701b4 + 2701*b1 - 3910) / 2
$\nu^{14}$	$=$	$ 9573\beta_{14} + 7194\beta_{11} - 2379\beta_{8} + 5603\beta_{5} - 9050\beta_{3} $ 9573b14 + 7194b11 - 2379b8 + 5603b5 - 9050*b3
$\nu^{15}$	$=$	$ ( - 40993 \beta_{15} + 56392 \beta_{13} - 10641 \beta_{12} + 9630 \beta_{6} - 10641 \beta_{4} + \cdots - 36121 ) / 2 $ (-40993b15 + 56392b13 - 10641b12 + 9630b6 - 10641b4 + 9630b3 + 15399b2 + 10641b1 - 36121) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1 + \beta_{7}$	$-1$	$1$	$1$	$1$

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} $

289.1

0.526774	+	1.96595i
−0.0543258	−	0.202747i
−0.556918	−	2.07845i
−0.281555	−	1.05078i
1.05078	−	0.281555i
0.202747	−	0.0543258i
2.07845	−	0.556918i
−1.96595	+	0.526774i
0.526774	−	1.96595i
−0.0543258	+	0.202747i
−0.556918	+	2.07845i
−0.281555	+	1.05078i
1.05078	+	0.281555i
0.202747	+	0.0543258i
2.07845	+	0.556918i
−1.96595	−	0.526774i

−0.866025

0.500000i

−1.98074

−

1.03763i

−0.478401

2.60214i

0.500000

−

0.866025i

289.2

−0.866025

0.500000i

−1.33920

1.79069i

2.40898

1.09398i

0.500000

−

0.866025i

289.3

−0.866025

0.500000i

0.717839

−

2.11771i

1.11487

−

2.39939i

0.500000

−

0.866025i

289.4

−0.866025

0.500000i

2.23607

−

0.00136408i

−1.31340

−

2.29673i

0.500000

−

0.866025i

289.5

0.866025

−

0.500000i

−1.11685

−

1.93717i

1.31340

2.29673i

0.500000

−

0.866025i

289.6

0.866025

−

0.500000i

−0.881181

2.05512i

−2.40898

−

1.09398i

0.500000

−

0.866025i

289.7

0.866025

−

0.500000i

1.47507

−

1.68052i

−1.11487

2.39939i

0.500000

−

0.866025i

289.8

0.866025

−

0.500000i

1.88899

1.19655i

0.478401

−

2.60214i

0.500000

−

0.866025i

529.1

−0.866025

−

0.500000i

−1.98074

1.03763i

−0.478401

−

2.60214i

0.500000

0.866025i

529.2

−0.866025

−

0.500000i

−1.33920

−

1.79069i

2.40898

−

1.09398i

0.500000

0.866025i

529.3

−0.866025

−

0.500000i

0.717839

2.11771i

1.11487

2.39939i

0.500000

0.866025i

529.4

−0.866025

−

0.500000i

2.23607

0.00136408i

−1.31340

2.29673i

0.500000

0.866025i

529.5

0.866025

0.500000i

−1.11685

1.93717i

1.31340

−

2.29673i

0.500000

0.866025i

529.6

0.866025

0.500000i

−0.881181

−

2.05512i

−2.40898

1.09398i

0.500000

0.866025i

529.7

0.866025

0.500000i

1.47507

1.68052i

−1.11487

−

2.39939i

0.500000

0.866025i

529.8

0.866025

0.500000i

1.88899

−

1.19655i

0.478401

2.60214i

0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.c	even	3	1	inner
35.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1680.2.di.d		16
4.b	odd	2	1		105.2.q.a	✓	16
5.b	even	2	1	inner	1680.2.di.d		16
7.c	even	3	1	inner	1680.2.di.d		16
12.b	even	2	1		315.2.bf.b		16
20.d	odd	2	1		105.2.q.a	✓	16
20.e	even	4	1		525.2.i.h		8
20.e	even	4	1		525.2.i.k		8
28.d	even	2	1		735.2.q.g		16
28.f	even	6	1		735.2.d.e		8
28.f	even	6	1		735.2.q.g		16
28.g	odd	6	1		105.2.q.a	✓	16
28.g	odd	6	1		735.2.d.d		8
35.j	even	6	1	inner	1680.2.di.d		16
60.h	even	2	1		315.2.bf.b		16
84.j	odd	6	1		2205.2.d.o		8
84.n	even	6	1		315.2.bf.b		16
84.n	even	6	1		2205.2.d.s		8
140.c	even	2	1		735.2.q.g		16
140.p	odd	6	1		105.2.q.a	✓	16
140.p	odd	6	1		735.2.d.d		8
140.s	even	6	1		735.2.d.e		8
140.s	even	6	1		735.2.q.g		16
140.w	even	12	1		525.2.i.h		8
140.w	even	12	1		525.2.i.k		8
140.w	even	12	1		3675.2.a.bp		4
140.w	even	12	1		3675.2.a.bz		4
140.x	odd	12	1		3675.2.a.bn		4
140.x	odd	12	1		3675.2.a.cb		4
420.ba	even	6	1		315.2.bf.b		16
420.ba	even	6	1		2205.2.d.s		8
420.be	odd	6	1		2205.2.d.o		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.q.a	✓	16	4.b	odd	2	1
105.2.q.a	✓	16	20.d	odd	2	1
105.2.q.a	✓	16	28.g	odd	6	1
105.2.q.a	✓	16	140.p	odd	6	1
315.2.bf.b		16	12.b	even	2	1
315.2.bf.b		16	60.h	even	2	1
315.2.bf.b		16	84.n	even	6	1
315.2.bf.b		16	420.ba	even	6	1
525.2.i.h		8	20.e	even	4	1
525.2.i.h		8	140.w	even	12	1
525.2.i.k		8	20.e	even	4	1
525.2.i.k		8	140.w	even	12	1
735.2.d.d		8	28.g	odd	6	1
735.2.d.d		8	140.p	odd	6	1
735.2.d.e		8	28.f	even	6	1
735.2.d.e		8	140.s	even	6	1
735.2.q.g		16	28.d	even	2	1
735.2.q.g		16	28.f	even	6	1
735.2.q.g		16	140.c	even	2	1
735.2.q.g		16	140.s	even	6	1
1680.2.di.d		16	1.a	even	1	1	trivial
1680.2.di.d		16	5.b	even	2	1	inner
1680.2.di.d		16	7.c	even	3	1	inner
1680.2.di.d		16	35.j	even	6	1	inner
2205.2.d.o		8	84.j	odd	6	1
2205.2.d.o		8	420.be	odd	6	1
2205.2.d.s		8	84.n	even	6	1
2205.2.d.s		8	420.ba	even	6	1
3675.2.a.bn		4	140.x	odd	12	1
3675.2.a.bp		4	140.w	even	12	1
3675.2.a.bz		4	140.w	even	12	1
3675.2.a.cb		4	140.x	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{8} + 18T_{11}^{6} + 28T_{11}^{5} + 294T_{11}^{4} + 252T_{11}^{3} + 736T_{11}^{2} - 420T_{11} + 900 $ T11^8 + 18*T11^6 + 28*T11^5 + 294*T11^4 + 252*T11^3 + 736*T11^2 - 420*T11 + 900 acting on $S_{2}^{\mathrm{new}}(1680, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{4} - T^{2} + 1)^{4} $ (T^4 - T^2 + 1)^4
$5$	$ T^{16} - 2 T^{15} + \cdots + 390625 $ T^16 - 2T^15 + 4T^14 - 20T^13 + 62T^12 - 118T^11 + 200T^10 - 670T^9 + 1639T^8 - 3350T^7 + 5000T^6 - 14750T^5 + 38750T^4 - 62500T^3 + 62500T^2 - 156250*T + 390625
$7$	$ T^{16} + 20 T^{14} + \cdots + 5764801 $ T^16 + 20T^14 + 202T^12 + 1244T^10 + 7267T^8 + 60956T^6 + 485002T^4 + 2352980*T^2 + 5764801
$11$	$ (T^{8} + 18 T^{6} + \cdots + 900)^{2} $ (T^8 + 18T^6 + 28T^5 + 294T^4 + 252T^3 + 736T^2 - 420T + 900)^2
$13$	$ (T^{8} + 60 T^{6} + \cdots + 16129)^{2} $ (T^8 + 60T^6 + 1182T^4 + 8408*T^2 + 16129)^2
$17$	$ T^{16} + \cdots + 322417936 $ T^16 - 60T^14 + 2460T^12 - 52424T^10 + 802364T^8 - 6951600T^6 + 43338304T^4 - 143432528*T^2 + 322417936
$19$	$ (T^{8} - 12 T^{7} + \cdots + 81)^{2} $ (T^8 - 12T^7 + 106T^6 - 376T^5 + 955T^4 - 1304T^3 + 1258T^2 - 360*T + 81)^2
$23$	$ T^{16} + \cdots + 5143987297296 $ T^16 - 164T^14 + 17164T^12 - 1102024T^10 + 51933820T^8 - 1660004976T^6 + 38942401792T^4 - 560232108432*T^2 + 5143987297296
$29$	$ (T^{4} - 6 T^{3} - 38 T^{2} + \cdots - 22)^{4} $ (T^4 - 6T^3 - 38T^2 + 190*T - 22)^4
$31$	$ (T^{8} + 8 T^{7} + \cdots + 3721)^{2} $ (T^8 + 8T^7 + 70T^6 + 144T^5 + 743T^4 - 400T^3 + 9582T^2 - 5856*T + 3721)^2
$37$	$ T^{16} + \cdots + 676751377201 $ T^16 - 168T^14 + 19450T^12 - 1171576T^10 + 50754123T^8 - 1050464408T^6 + 15651985658T^4 - 124407562972*T^2 + 676751377201
$41$	$ (T^{4} - 4 T^{3} - 50 T^{2} + \cdots - 10)^{4} $ (T^4 - 4T^3 - 50T^2 + 146*T - 10)^4
$43$	$ (T^{8} + 128 T^{6} + \cdots + 2401)^{2} $ (T^8 + 128T^6 + 3390T^4 + 20580*T^2 + 2401)^2
$47$	$ T^{16} + \cdots + 207360000 $ T^16 - 68T^14 + 3328T^12 - 72256T^10 + 1125568T^8 - 8326656T^6 + 44317696T^4 - 114278400*T^2 + 207360000
$53$	$ T^{16} - 160 T^{14} + \cdots + 84934656 $ T^16 - 160T^14 + 19328T^12 - 963072T^10 + 36092928T^8 - 123895808T^6 + 351207424T^4 - 186384384*T^2 + 84934656
$59$	$ (T^{8} + 2 T^{7} + \cdots + 100)^{2} $ (T^8 + 2T^7 + 10T^6 + 38T^4 - 4T^3 + 96T^2 - 60T + 100)^2
$61$	$ (T^{8} - 8 T^{7} + \cdots + 250000)^{2} $ (T^8 - 8T^7 + 164T^6 - 600T^5 + 15100T^4 - 62000T^3 + 540000T^2 + 350000*T + 250000)^2
$67$	$ T^{16} + \cdots + 670801950625 $ T^16 - 180T^14 + 22498T^12 - 1451128T^10 + 67419699T^8 - 1345080632T^6 + 19318673906T^4 - 135643644400*T^2 + 670801950625
$71$	$ (T^{4} - 14 T^{3} + \cdots - 3202)^{4} $ (T^4 - 14T^3 - 90T^2 + 1334*T - 3202)^4
$73$	$ T^{16} + \cdots + 3722279521041 $ T^16 - 272T^14 + 53610T^12 - 4636440T^10 + 290051387T^8 - 8172618232T^6 + 165578604682T^4 - 873295574724*T^2 + 3722279521041
$79$	$ (T^{8} - 8 T^{7} + \cdots + 50140561)^{2} $ (T^8 - 8T^7 + 218T^6 - 136T^5 + 22107T^4 + 7960T^3 + 1558330T^2 + 4843404*T + 50140561)^2
$83$	$ (T^{8} + 148 T^{6} + \cdots + 131044)^{2} $ (T^8 + 148T^6 + 6716T^4 + 97188*T^2 + 131044)^2
$89$	$ (T^{8} - 8 T^{7} + \cdots + 285156)^{2} $ (T^8 - 8T^7 + 258T^6 - 420T^5 + 46058T^4 - 199828T^3 + 868600T^2 - 526524*T + 285156)^2
$97$	$ (T^{8} + 20 T^{6} + \cdots + 16)^{2} $ (T^8 + 20T^6 + 120T^4 + 192*T^2 + 16)^2
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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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.di (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{3} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{7} + 1) q^{9}+O(q^{10}) \) q + b3 * q^3 + (-b14 - b13 + b12 - b11 - b8 + b4 - b2 + 1) * q^5 + (b13 + b5 - b4 + b2 - 1) * q^7 + (-b7 + 1) * q^9	\( q + \beta_{3} q^{3} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 1) q^{5}+ \cdots + (\beta_{13} - \beta_{12} - \beta_{2} + \cdots + 1) q^{99}+O(q^{100}) \) q + b3 * q^3 + (-b14 - b13 + b12 - b11 - b8 + b4 - b2 + 1) * q^5 + (b13 + b5 - b4 + b2 - 1) * q^7 + (-b7 + 1) * q^9 + (-b14 - b10 + 2b9 - b8 - b7 - 2b2 + 2) * q^11 + (b15 - 2b12 - 2b6 - 2b4 - 2b3) * q^13 + (-b12 + b1 + 1) * q^15 + (2b14 + 2b11 + b5 + b3) * q^17 + (-b10 + b9 - 4b7 - b1 + 3) q^19 + (b14 + b10 - b9 + b8 + b7 - 1) * q^21 + (3b15 + 2b14 - 2b13 - 2b12 - 2b8 - 5b6 + 3b5 - 2b2 + 2) * q^23 + (-2b14 - b11 - b10 + 3b9 - b8 - 2b7 - b5 - 2b3 - 3b2 + 3) q^25 + (b6 + b3) * q^27 + (2b13 - 2b12 + 2b2 + 3b1 + 2) * q^29 + (2b10 - b7) q^31 + (2b15 - b13 - b11 - b8 - b6 + 2b5 + b4 - b2 + 1) * q^33 + (-b15 + b14 + 2b13 + b11 - 2b9 + b8 - 2b7 - b6 - b5 + 2b3 + 2b2 + b1) q^35 + (b13 + b11 + b8 - 6b6 - b4 + b2 - 1) q^37 + (2b10 + b9 + b7 - b2 + 1) q^39 + (-b13 + b12 + b2 - 3b1 - 1) q^41 + (-2b15 + b13 - 2b12 + 2b6 - 3b4 + 2b3 + b2 - 1) q^43 + (-b14 - b11 - b8) * q^45 + (-b14 + b13 + b12 + b8 - 2b6 + b2 - 1) q^47 + (3b10 + b9 + 2b7 + b2 - 3) * q^49 + (-2b10 - b9 - 2b1 - 2) * q^51 + (2b14 + 2b11 + 6b3) q^53 + (2b15 - b13 - 2b6 + 3b4 - 2b3 - 2b2 - b1 + 1) q^55 + (b15 + b12 + 3b6 + b4 + 3b3) * q^57 + (-b10 - b7) * q^59 + (-2b14 - 2b13 + 2b12 - 3b10 + 5b9 - 2b8 - 5b7 - 2b2 - 3b1 + 4) q^61 + (-b15 + b13 + b11 + b8 - b4 + b2 - 1) * q^63 + (4b15 + b14 - b13 - b12 + b11 - b10 - 2b9 - b8 - 3b7 + 4b5 - b4 - b2 - b1 + 3) * q^65 + (-2b14 - 4b11 - 2b8 - b5) q^67 + (2b13 - 2b12 - b2 + 3) * q^69 + (-2b13 + 2b12 + 2b2 - 5b1) * q^71 + (-2b14 - 3b11 - b8 + 4b5 + 4b3) * q^73 + (3b15 - b13 - b11 + b10 + b9 - b8 + b7 - b6 + 3b5 + b4 - b2 + b1 + 1) * q^75 + (-b15 - 3b14 - 4b11 - b8 - 3b6 + b3) q^77 + (-4b14 - 4b13 + 4b12 - 3b10 + b9 - 4b8 - 2b7 - 4b2 - 3b1 + 3) * q^79 - b7 * q^81 + (-b15 - 2b13 - b6 + 2b4 - b3 - 2b2 + 2) q^83 + (-4b15 + b13 + 2b12 + b4 + 3b2 - 6) q^85 + (-b14 - 3b11 - 2b8 + b3) * q^87 + (b14 + b13 - b12 - 3b10 - 6b9 + b8 - b7 + b2 - 3b1 - 3) q^89 + (2b14 + b10 + b9 + 2b8 + 5b2 + 2b1 - 2) * q^91 + (2b14 - 2b12 + 2b11 + b6 - 2b4) * q^93 + (-2b14 - b11 + b10 - 2b9 - 2b8 + 5b7 - b5 - b3 + 2b2 - 2) q^95 + (-b15 + b13 - b4 + b2 - 1) * q^97 + (b13 - b12 - b2 + b1 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{5} + 8 q^{9}+O(q^{10}) \) 16 * q + 2 * q^5 + 8 * q^9	\( 16 q + 2 q^{5} + 8 q^{9} + 4 q^{15} + 24 q^{19} - 4 q^{21} - 4 q^{25} + 24 q^{29} - 16 q^{31} + 10 q^{35} + 4 q^{39} + 16 q^{41} - 2 q^{45} - 40 q^{49} - 4 q^{51} - 8 q^{55} - 4 q^{59} + 16 q^{61} + 30 q^{65} + 40 q^{69} + 56 q^{71} - 8 q^{75} + 16 q^{79} - 8 q^{81} - 64 q^{85} + 16 q^{89} - 8 q^{91} + 22 q^{95}+O(q^{100}) \) 16 * q + 2 * q^5 + 8 * q^9 + 4 * q^15 + 24 * q^19 - 4 * q^21 - 4 * q^25 + 24 * q^29 - 16 * q^31 + 10 * q^35 + 4 * q^39 + 16 * q^41 - 2 * q^45 - 40 * q^49 - 4 * q^51 - 8 * q^55 - 4 * q^59 + 16 * q^61 + 30 * q^65 + 40 * q^69 + 56 * q^71 - 8 * q^75 + 16 * q^79 - 8 * q^81 - 64 * q^85 + 16 * q^89 - 8 * q^91 + 22 * q^95

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	1680.2.di.d

Level	$1680$
Weight	$2$
Character orbit	1680.di
Analytic conductor	$13.415$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(13.4148675396\)
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:	\( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} - 14 x^{12} + 38 x^{11} - 40 x^{10} + 64 x^{9} + 291 x^{8} + \cdots + 1 \) x^16 - 2x^15 + 2x^14 - 4x^13 - 14x^12 + 38x^11 - 40x^10 + 64x^9 + 291x^8 - 630x^7 + 584x^6 - 800x^5 + 734x^4 - 188x^3 + 32x^2 - 8*x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( - 38414168495 \nu^{15} + 46999484762 \nu^{14} - 56685616158 \nu^{13} + \cdots - 33746305280705 ) / 11431248166457 \) (-38414168495v^15 + 46999484762v^14 - 56685616158v^13 + 131873366880v^12 + 642956778028v^11 - 939694854143v^10 + 1093318205564v^9 - 2230952782978v^8 - 12780049004340v^7 + 14111475820386v^6 - 17131346947484v^5 + 27207219807940v^4 - 7455199259279v^3 + 1191125364091v^2 - 321416591346*v - 33746305280705) / 11431248166457
\(\beta_{2}\)	\(=\)	\( ( 145306983361 \nu^{15} - 177522986678 \nu^{14} + 213949914242 \nu^{13} + \cdots + 31632456480199 ) / 11431248166457 \) (145306983361v^15 - 177522986678v^14 + 213949914242v^13 - 520543533819v^12 - 2426061321940v^11 + 3549177780569v^10 - 4097251614084v^9 + 8418948255726v^8 + 48225232348268v^7 - 53280290889126v^6 + 64651815055380v^5 - 102672086774396v^4 + 28133974246241v^3 - 4494960282621v^2 + 1212926356606*v + 31632456480199) / 11431248166457
\(\beta_{3}\)	\(=\)	\( ( 760172030748 \nu^{15} - 826140524550 \nu^{14} + 311805690170 \nu^{13} - 1979665829518 \nu^{12} + \cdots - 575089767434 ) / 11431248166457 \) (760172030748v^15 - 826140524550v^14 + 311805690170v^13 - 1979665829518v^12 - 13114010371696v^11 + 18515621292532v^10 - 6663971281019v^9 + 27229891583895v^8 + 259493064521626v^7 - 266716860192172v^6 + 60835212617387v^5 - 306079675769764v^4 + 90642215592396v^3 + 246128680839936v^2 + 3861979127591*v - 575089767434) / 11431248166457
\(\beta_{4}\)	\(=\)	\( ( - 1159675947687 \nu^{15} + 2123382923210 \nu^{14} - 1790860341485 \nu^{13} + \cdots + 29672440598565 ) / 11431248166457 \) (-1159675947687v^15 + 2123382923210v^14 - 1790860341485v^13 + 4132499467647v^12 + 17082017216494v^11 - 41638199149391v^10 + 36484081923634v^9 - 63733593693012v^8 - 351285551153804v^7 + 677606774159518v^6 - 505106942012337v^5 + 778126373623118v^4 - 678461323755244v^3 + 34074325119351v^2 - 9164323089196*v + 29672440598565) / 11431248166457
\(\beta_{5}\)	\(=\)	\( ( - 2241001108620 \nu^{15} + 2440547176294 \nu^{14} - 921260080696 \nu^{13} + \cdots + 1695440450074 ) / 11431248166457 \) (-2241001108620v^15 + 2440547176294v^14 - 921260080696v^13 + 5836557871079v^12 + 38651783264660v^11 - 54581263789178v^10 + 19684393272760v^9 - 80280420418569v^8 - 764989544114834v^7 + 786254645863424v^6 - 179383356485812v^5 + 902375388086909v^4 - 267226659358818v^3 - 682424450829525v^2 - 11385720733444*v + 1695440450074) / 11431248166457
\(\beta_{6}\)	\(=\)	\( ( 4058097157982 \nu^{15} - 7568755097298 \nu^{14} + 7059730047127 \nu^{13} + \cdots - 16137847630635 ) / 11431248166457 \) (4058097157982v^15 - 7568755097298v^14 + 7059730047127v^13 - 15184509679358v^12 - 58946431470254v^11 + 146411669575112v^10 - 142164152788000v^9 + 238760344218351v^8 + 1214849064761822v^7 - 2395065444113876v^6 + 2037822081506248v^5 - 2940884698505042v^4 + 2557823285973472v^3 - 388309099007712v^2 + 34395735205495*v - 16137847630635) / 11431248166457
\(\beta_{7}\)	\(=\)	\( ( - 2575730834 \nu^{15} + 4699915310 \nu^{14} - 4379029892 \nu^{13} + 9614611682 \nu^{12} + \cdots + 17015828908 ) / 7008735847 \) (-2575730834v^15 + 4699915310v^14 - 4379029892v^13 + 9614611682v^12 + 37669504970v^11 - 91091321784v^10 + 87843289680v^9 - 151013347873v^8 - 774495170890v^7 + 1484105110236v^6 - 1259913976952v^5 + 1863977222846v^4 - 1585945295448v^3 + 240744226192v^2 - 69083699784*v + 17015828908) / 7008735847
\(\beta_{8}\)	\(=\)	\( ( 1063280569755 \nu^{15} - 2138087245431 \nu^{14} + 2140603616666 \nu^{13} + \cdots - 8602022998880 ) / 1633035452351 \) (1063280569755v^15 - 2138087245431v^14 + 2140603616666v^13 - 4270022226214v^12 - 14845214238260v^11 + 40596196036702v^10 - 42811945511246v^9 + 68369173152460v^8 + 308749841689177v^7 - 673674444501418v^6 + 625167345362715v^5 - 855730190069746v^4 + 788555489457606v^3 - 202118395383557v^2 + 34379925210470*v - 8602022998880) / 1633035452351
\(\beta_{9}\)	\(=\)	\( ( 7502690467678 \nu^{15} - 13694199839130 \nu^{14} + 12758334474904 \nu^{13} + \cdots - 29172119880252 ) / 11431248166457 \) (7502690467678v^15 - 13694199839130v^14 + 12758334474904v^13 - 28102531513444v^12 - 109693802936905v^11 + 265404776190093v^10 - 255814696983230v^9 + 441714091122816v^8 + 2255188372760505v^7 - 4324366285850692v^6 + 3669645519916604v^5 - 5466340585268175v^4 + 4623119858303861v^3 - 701766532319694v^2 + 201383747371308*v - 29172119880252) / 11431248166457
\(\beta_{10}\)	\(=\)	\( ( - 8183751109447 \nu^{15} + 14956739476040 \nu^{14} - 13922497684986 \nu^{13} + \cdots + 54263991142284 ) / 11431248166457 \) (-8183751109447v^15 + 14956739476040v^14 - 13922497684986v^13 + 30578923581156v^12 + 119598143979065v^11 - 289853173711402v^10 + 279305469940010v^9 - 480196298050966v^8 - 2459380223502305v^7 + 4723682477021618v^6 - 4004669271289776v^5 + 5932582107617908v^4 - 5055851615402594v^3 + 767408313770491v^2 - 220236155839242*v + 54263991142284) / 11431248166457
\(\beta_{11}\)	\(=\)	\( ( - 9494932205673 \nu^{15} + 17203785936492 \nu^{14} - 15830728042254 \nu^{13} + \cdots + 61766681609962 ) / 11431248166457 \) (-9494932205673v^15 + 17203785936492v^14 - 15830728042254v^13 + 35235059970569v^12 + 139295605560736v^11 - 334146095435292v^10 + 317763568748586v^9 - 552101744717173v^8 - 2861708932288753v^7 + 5435608190228966v^6 - 4540481060164001v^5 + 6816437052589699v^4 - 5764590891006364v^3 + 806110661364879v^2 - 251085559700414*v + 61766681609962) / 11431248166457
\(\beta_{12}\)	\(=\)	\( ( 10583106332007 \nu^{15} - 18562866254784 \nu^{14} + 16212768119001 \nu^{13} + \cdots - 62361843284805 ) / 11431248166457 \) (10583106332007v^15 - 18562866254784v^14 + 16212768119001v^13 - 37702760224023v^12 - 157985234866350v^11 + 364596213484152v^10 - 327711794585876v^9 + 583962180212718v^8 + 3234466606384584v^7 - 5891738412332556v^6 + 4610113536311955v^5 - 7128621954756198v^4 + 5858692599566580v^3 - 312154621077708v^2 + 83988991866438*v - 62361843284805) / 11431248166457
\(\beta_{13}\)	\(=\)	\( ( 10663789073454 \nu^{15} - 18661163594876 \nu^{14} + 16331067749359 \nu^{13} + \cdots - 32056382963606 ) / 11431248166457 \) (10663789073454v^15 - 18661163594876v^14 + 16331067749359v^13 - 37987756392193v^12 - 159325975568434v^11 + 366561272531474v^10 - 329946311402856v^9 + 588615808979414v^8 + 3261120405274960v^7 - 5921218860713121v^6 + 4645853676172177v^5 - 7185374813558250v^4 + 5874244137453899v^3 - 314639249925878v^2 + 84659440952318*v - 32056382963606) / 11431248166457
\(\beta_{14}\)	\(=\)	\( ( 11015276267169 \nu^{15} - 18856066985592 \nu^{14} + 16454339422594 \nu^{13} + \cdots - 62916861144830 ) / 11431248166457 \) (11015276267169v^15 - 18856066985592v^14 + 16454339422594v^13 - 39194391629605v^12 - 165523626304128v^11 + 371177338020356v^10 - 331091511310624v^9 + 606561527884963v^8 + 3380695061332005v^7 - 5969041910613310v^6 + 4662151485398775v^5 - 7428596404129227v^4 + 5945875322191156v^3 - 325284547851464v^2 + 258809517955596*v - 62916861144830) / 11431248166457
\(\beta_{15}\)	\(=\)	\( ( 13399447651346 \nu^{15} - 23430444102746 \nu^{14} + 20519536606234 \nu^{13} + \cdots - 46488804269431 ) / 11431248166457 \) (13399447651346v^15 - 23430444102746v^14 + 20519536606234v^13 - 47737126077397v^12 - 200273669782708v^11 + 460259833552415v^10 - 414511080319944v^9 + 739757034059786v^8 + 4098887977021204v^7 - 7433685405000834v^6 + 5838337123809904v^5 - 9030367022903148v^4 + 7367017819324544v^3 - 395429090524619v^2 + 106398295849754*v - 46488804269431) / 11431248166457

\(\nu\)	\(=\)	\( ( -\beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_1 ) / 2 \) (-b15 - b14 + b12 + b11 - b10 + b9 - b7 - b5 - b4 - b1) / 2
\(\nu^{2}\)	\(=\)	\( -\beta_{14} - \beta_{11} + 2\beta_{3} \) -b14 - b11 + 2*b3
\(\nu^{3}\)	\(=\)	\( ( 5\beta_{15} - 8\beta_{13} + 3\beta_{12} - 2\beta_{6} + 3\beta_{4} - 2\beta_{3} - 3\beta_{2} - 3\beta _1 + 3 ) / 2 \) (5b15 - 8b13 + 3b12 - 2b6 + 3b4 - 2b3 - 3b2 - 3b1 + 3) / 2
\(\nu^{4}\)	\(=\)	\( 5\beta_{10} + \beta_{9} - 8\beta_{7} + 5\beta _1 + 13 \) 5b10 + b9 - 8b7 + 5*b1 + 13
\(\nu^{5}\)	\(=\)	\( ( - 23 \beta_{14} + 11 \beta_{11} - 11 \beta_{10} + 23 \beta_{9} - 13 \beta_{7} - 23 \beta_{5} + \cdots + 23 ) / 2 \) (-23b14 + 11b11 - 11b10 + 23b9 - 13b7 - 23b5 - 10b3 - 23b2 + 23) / 2
\(\nu^{6}\)	\(=\)	\( 7\beta_{15} - \beta_{13} - 23\beta_{12} + 28\beta_{6} - 22\beta_{4} + 28\beta_{3} - \beta_{2} + 1 \) 7b15 - b13 - 23b12 + 28b6 - 22b4 + 28*b3 - b2 + 1
\(\nu^{7}\)	\(=\)	\( ( 103 \beta_{15} - 43 \beta_{14} - 148 \beta_{13} + 43 \beta_{12} - 43 \beta_{11} - 43 \beta_{10} + \cdots + 166 ) / 2 \) (103b15 - 43b14 - 148b13 + 43b12 - 43b11 - 43b10 + 103b9 - 148b8 - 61b7 - 42b6 + 103b5 + 43b4 - 148b2 - 43b1 + 166) / 2
\(\nu^{8}\)	\(=\)	\( -10\beta_{14} + 94\beta_{10} + 40\beta_{9} - 10\beta_{8} - 157\beta_{7} - 40\beta_{2} + 40 \) -10b14 + 94b10 + 40b9 - 10b8 - 157b7 - 40b2 + 40
\(\nu^{9}\)	\(=\)	\( ( 459\beta_{15} - 479\beta_{12} - 168\beta_{6} + 171\beta_{4} - 168\beta_{3} - 459\beta_{2} + 171\beta _1 + 339 ) / 2 \) (459b15 - 479b12 - 168b6 + 171b4 - 168b3 - 459b2 + 171*b1 + 339) / 2
\(\nu^{10}\)	\(=\)	\( 214 \beta_{15} + 469 \beta_{14} - 70 \beta_{13} - 469 \beta_{12} + 399 \beta_{11} - 70 \beta_{8} + \cdots + 70 \) 214b15 + 469b14 - 70b13 - 469b12 + 399b11 - 70b8 + 496b6 + 214b5 - 399b4 - 70b2 + 70
\(\nu^{11}\)	\(=\)	\( ( - 681 \beta_{14} - 681 \beta_{11} - 681 \beta_{10} + 2047 \beta_{9} - 2868 \beta_{8} - 1389 \beta_{7} + \cdots + 2047 ) / 2 \) (-681b14 - 681b11 - 681b10 + 2047b9 - 2868b8 - 1389b7 + 2047b5 + 658b3 - 2047*b2 + 2047) / 2
\(\nu^{12}\)	\(=\)	\( 424\beta_{13} - 424\beta_{12} - 683\beta_{2} - 1693\beta _1 - 4232 \) 424b13 - 424b12 - 683b2 - 1693b1 - 4232
\(\nu^{13}\)	\(=\)	\( ( 9149 \beta_{15} + 9997 \beta_{14} - 9997 \beta_{12} - 2701 \beta_{11} + 2701 \beta_{10} - 9149 \beta_{9} + \cdots - 3910 ) / 2 \) (9149b15 + 9997b14 - 9997b12 - 2701b11 + 2701b10 - 9149b9 + 6611b7 - 2538b6 + 9149b5 + 2701b4 + 2701*b1 - 3910) / 2
\(\nu^{14}\)	\(=\)	\( 9573\beta_{14} + 7194\beta_{11} - 2379\beta_{8} + 5603\beta_{5} - 9050\beta_{3} \) 9573b14 + 7194b11 - 2379b8 + 5603b5 - 9050*b3
\(\nu^{15}\)	\(=\)	\( ( - 40993 \beta_{15} + 56392 \beta_{13} - 10641 \beta_{12} + 9630 \beta_{6} - 10641 \beta_{4} + \cdots - 36121 ) / 2 \) (-40993b15 + 56392b13 - 10641b12 + 9630b6 - 10641b4 + 9630b3 + 15399b2 + 10641b1 - 36121) / 2

\(n\)	\(241\)	\(337\)	\(421\)	\(1121\)	\(1471\)
\(\chi(n)\)	\(-1 + \beta_{7}\)	\(-1\)	\(1\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 289.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{4} - T^{2} + 1)^{4} \) (T^4 - T^2 + 1)^4
$5$	\( T^{16} - 2 T^{15} + \cdots + 390625 \) T^16 - 2T^15 + 4T^14 - 20T^13 + 62T^12 - 118T^11 + 200T^10 - 670T^9 + 1639T^8 - 3350T^7 + 5000T^6 - 14750T^5 + 38750T^4 - 62500T^3 + 62500T^2 - 156250*T + 390625
$7$	\( T^{16} + 20 T^{14} + \cdots + 5764801 \) T^16 + 20T^14 + 202T^12 + 1244T^10 + 7267T^8 + 60956T^6 + 485002T^4 + 2352980*T^2 + 5764801
$11$	\( (T^{8} + 18 T^{6} + \cdots + 900)^{2} \) (T^8 + 18T^6 + 28T^5 + 294T^4 + 252T^3 + 736T^2 - 420T + 900)^2
$13$	\( (T^{8} + 60 T^{6} + \cdots + 16129)^{2} \) (T^8 + 60T^6 + 1182T^4 + 8408*T^2 + 16129)^2
$17$	\( T^{16} + \cdots + 322417936 \) T^16 - 60T^14 + 2460T^12 - 52424T^10 + 802364T^8 - 6951600T^6 + 43338304T^4 - 143432528*T^2 + 322417936
$19$	\( (T^{8} - 12 T^{7} + \cdots + 81)^{2} \) (T^8 - 12T^7 + 106T^6 - 376T^5 + 955T^4 - 1304T^3 + 1258T^2 - 360*T + 81)^2
$23$	\( T^{16} + \cdots + 5143987297296 \) T^16 - 164T^14 + 17164T^12 - 1102024T^10 + 51933820T^8 - 1660004976T^6 + 38942401792T^4 - 560232108432*T^2 + 5143987297296
$29$	\( (T^{4} - 6 T^{3} - 38 T^{2} + \cdots - 22)^{4} \) (T^4 - 6T^3 - 38T^2 + 190*T - 22)^4
$31$	\( (T^{8} + 8 T^{7} + \cdots + 3721)^{2} \) (T^8 + 8T^7 + 70T^6 + 144T^5 + 743T^4 - 400T^3 + 9582T^2 - 5856*T + 3721)^2
$37$	\( T^{16} + \cdots + 676751377201 \) T^16 - 168T^14 + 19450T^12 - 1171576T^10 + 50754123T^8 - 1050464408T^6 + 15651985658T^4 - 124407562972*T^2 + 676751377201
$41$	\( (T^{4} - 4 T^{3} - 50 T^{2} + \cdots - 10)^{4} \) (T^4 - 4T^3 - 50T^2 + 146*T - 10)^4
$43$	\( (T^{8} + 128 T^{6} + \cdots + 2401)^{2} \) (T^8 + 128T^6 + 3390T^4 + 20580*T^2 + 2401)^2
$47$	\( T^{16} + \cdots + 207360000 \) T^16 - 68T^14 + 3328T^12 - 72256T^10 + 1125568T^8 - 8326656T^6 + 44317696T^4 - 114278400*T^2 + 207360000
$53$	\( T^{16} - 160 T^{14} + \cdots + 84934656 \) T^16 - 160T^14 + 19328T^12 - 963072T^10 + 36092928T^8 - 123895808T^6 + 351207424T^4 - 186384384*T^2 + 84934656
$59$	\( (T^{8} + 2 T^{7} + \cdots + 100)^{2} \) (T^8 + 2T^7 + 10T^6 + 38T^4 - 4T^3 + 96T^2 - 60T + 100)^2
$61$	\( (T^{8} - 8 T^{7} + \cdots + 250000)^{2} \) (T^8 - 8T^7 + 164T^6 - 600T^5 + 15100T^4 - 62000T^3 + 540000T^2 + 350000*T + 250000)^2
$67$	\( T^{16} + \cdots + 670801950625 \) T^16 - 180T^14 + 22498T^12 - 1451128T^10 + 67419699T^8 - 1345080632T^6 + 19318673906T^4 - 135643644400*T^2 + 670801950625
$71$	\( (T^{4} - 14 T^{3} + \cdots - 3202)^{4} \) (T^4 - 14T^3 - 90T^2 + 1334*T - 3202)^4
$73$	\( T^{16} + \cdots + 3722279521041 \) T^16 - 272T^14 + 53610T^12 - 4636440T^10 + 290051387T^8 - 8172618232T^6 + 165578604682T^4 - 873295574724*T^2 + 3722279521041
$79$	\( (T^{8} - 8 T^{7} + \cdots + 50140561)^{2} \) (T^8 - 8T^7 + 218T^6 - 136T^5 + 22107T^4 + 7960T^3 + 1558330T^2 + 4843404*T + 50140561)^2
$83$	\( (T^{8} + 148 T^{6} + \cdots + 131044)^{2} \) (T^8 + 148T^6 + 6716T^4 + 97188*T^2 + 131044)^2
$89$	\( (T^{8} - 8 T^{7} + \cdots + 285156)^{2} \) (T^8 - 8T^7 + 258T^6 - 420T^5 + 46058T^4 - 199828T^3 + 868600T^2 - 526524*T + 285156)^2
$97$	\( (T^{8} + 20 T^{6} + \cdots + 16)^{2} \) (T^8 + 20T^6 + 120T^4 + 192*T^2 + 16)^2
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