LMFDB - Newform orbit 1080.4.q.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(361,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1080.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, 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:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
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} - 3 x^{15} + 2 x^{14} + 3 x^{13} - 32 x^{12} - 255 x^{11} + 558 x^{10} + 2079 x^{9} + \cdots + 43046721 $ x^16 - 3x^15 + 2x^14 + 3x^13 - 32x^12 - 255x^11 + 558x^10 + 2079x^9 - 6210x^8 + 18711x^7 + 45198x^6 - 185895x^5 - 209952x^4 + 177147x^3 + 1062882x^2 - 14348907*x + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{16} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 + ( - 5 \beta_1 + 5) q^{5} + ( - \beta_{4} + \beta_{2} - 4 \beta_1) q^{7}+O(q^{10}) $ q + (-5b1 + 5) q^5 + (-b4 + b2 - 4b1) q^7	$ q + ( - 5 \beta_1 + 5) q^{5} + ( - \beta_{4} + \beta_{2} - 4 \beta_1) q^{7} + (\beta_{7} - \beta_{4} + \cdots - 5 \beta_1) q^{11}+ \cdots + ( - 14 \beta_{9} - 3 \beta_{8} + \cdots + 21 \beta_1) q^{97}+O(q^{100}) $ q + (-5b1 + 5) q^5 + (-b4 + b2 - 4b1) q^7 + (b7 - b4 + b2 - 5b1) q^11 + (b15 + b14 - b12 + b11 - b9 + b3 + b2 - 3b1 + 3) q^13 + (-3b14 + b13 - b11 - b10 + b6 + b4 + 9) q^17 + (-2b15 - b14 - b12 - b11 + b10 + b8 + 2b5 + b4 - 8) * q^19 + (3b15 + b13 + 3b11 - 3b9 - b2 + 14b1 - 14) * q^23 - 25b1 q^25 + (b9 + 2b7 - 2b5 + b4 + 2b3 - b2 - 11b1) * q^29 + (-3b15 + b14 - 3b13 + 3b11 - 3b9 + b3 + b2 + 8b1 - 8) q^31 + (-5b4 - 20) q^35 + (b15 - b14 + 2b13 - 6b12 - b11 + 3b10 + 6b8 + 2b6 - b5 + 10b4 - 33) * q^37 + (5b15 + 2b14 - 5b12 + b10 + b7 + 2b3 - 4b2 + 29b1 - 29) * q^41 + (-5b9 + 6b8 + 2b7 + 6b6 - 2b5 - 3b3 + 14b1) q^43 + (-4b9 + 5b8 - 9b7 - 10b6 + 2b5 - 5b4 - 13b3 + 5b2 + 8b1) q^47 + (-5b15 + b14 - 2b12 - b10 - b7 + b3 - 7b2 - 31b1 + 31) * q^49 + (-3b15 + 15b14 - 15b13 - 4b12 - 6b11 + 13b10 + 4b8 - 15b6 + 3b5 + 9b4 + 77) * q^53 + (-5b10 - 5b4 - 25) * q^55 + (2b15 + 5b14 + 8b13 - 4b12 + b10 + b7 + 5b3 - 18b2 + 57b1 - 57) q^59 + (-11b9 - 2b8 + 8b7 + 6b6 - 5b5 - 22b4 + 22b2 + 105b1) * q^61 + (-5b9 - 5b8 + 5b5 - 5b4 + 5b3 + 5b2 - 15b1) q^65 + (-10b15 - 4b14 - 8b13 + 4b11 - 5b10 - 4b9 - 5b7 - 4b3 + 8b2 - 75b1 + 75) * q^67 + (12b15 - 16b14 + 10b13 - 5b12 - 7b11 - 18b10 + 5b8 + 10b6 - 12b5 + b4 + 15) q^71 + (3b15 + 4b14 + 11b13 - 9b12 - 4b11 - b10 + 9b8 + 11b6 - 3b5 - 6b4 - 126) q^73 + (-2b15 - 6b14 - b13 - 19b12 - 21b11 - 11b10 + 21b9 - 11b7 - 6b3 - 16b2 + 184b1 - 184) * q^77 + (-13b9 + 23b8 - 4b7 + 2b5 + 36b4 + b3 - 36b2 + 49b1) q^79 + (-12b9 - 3b8 + 9b7 - 2b5 + 8b4 + 26b3 - 8b2 + 12b1) * q^83 + (-15b14 + 5b13 - 5b11 - 5b10 + 5b9 - 5b7 - 15b3 + 5b2 - 45b1 + 45) q^85 + (b15 - 8b14 + 13b13 - 9b12 - 32b11 - 9b10 + 9b8 + 13b6 - b5 + 20b4 + 325) * q^89 + (-28b15 + b14 - 10b13 - 14b12 - 6b11 + 6b10 + 14b8 - 10b6 + 28b5 + b4 - 398) * q^91 + (-10b15 - 5b14 - 5b12 - 5b11 + 5b10 + 5b9 + 5b7 - 5b3 + 5b2 + 40b1 - 40) * q^95 + (-14b9 - 3b8 - 14b7 - 8b6 + 12b5 - 17b4 + 19b3 + 17b2 + 21b1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 40 q^{5} - 31 q^{7}+O(q^{10}) $ 16 * q + 40 * q^5 - 31 * q^7	$ 16 q + 40 q^{5} - 31 q^{7} - 41 q^{11} + 16 q^{13} + 158 q^{17} - 122 q^{19} - 113 q^{23} - 200 q^{25} - 89 q^{29} - 68 q^{31} - 310 q^{35} - 564 q^{37} - 252 q^{41} + 85 q^{43} + 25 q^{47} + 249 q^{49} + 1128 q^{53} - 410 q^{55} - 457 q^{59} + 853 q^{61} - 80 q^{65} + 596 q^{67} + 228 q^{71} - 2070 q^{73} - 1516 q^{77} + 300 q^{79} + 155 q^{83} + 395 q^{85} + 5142 q^{89} - 6400 q^{91} - 305 q^{95} + 283 q^{97}+O(q^{100}) $ 16 * q + 40 * q^5 - 31 * q^7 - 41 * q^11 + 16 * q^13 + 158 * q^17 - 122 * q^19 - 113 * q^23 - 200 * q^25 - 89 * q^29 - 68 * q^31 - 310 * q^35 - 564 * q^37 - 252 * q^41 + 85 * q^43 + 25 * q^47 + 249 * q^49 + 1128 * q^53 - 410 * q^55 - 457 * q^59 + 853 * q^61 - 80 * q^65 + 596 * q^67 + 228 * q^71 - 2070 * q^73 - 1516 * q^77 + 300 * q^79 + 155 * q^83 + 395 * q^85 + 5142 * q^89 - 6400 * q^91 - 305 * q^95 + 283 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 3 x^{15} + 2 x^{14} + 3 x^{13} - 32 x^{12} - 255 x^{11} + 558 x^{10} + 2079 x^{9} + \cdots + 43046721 $ x^16 - 3*x^15 + 2*x^14 + 3*x^13 - 32*x^12 - 255*x^11 + 558*x^10 + 2079*x^9 - 6210*x^8 + 18711*x^7 + 45198*x^6 - 185895*x^5 - 209952*x^4 + 177147*x^3 + 1062882*x^2 - 14348907*x + 43046721 :

$\beta_{1}$	$=$	$ ( 375 \nu^{15} + 9134 \nu^{14} + 3588 \nu^{13} - 13511 \nu^{12} - 88467 \nu^{11} + \cdots + 78655925205 ) / 44760086784 $ (375v^15 + 9134v^14 + 3588v^13 - 13511v^12 - 88467v^11 + 250088v^10 - 576438v^9 - 3200949v^8 + 30550473v^7 - 6645186v^6 - 109270296v^5 + 55869831v^4 + 485253747v^3 - 2974206276v^2 - 5926984326*v + 78655925205) / 44760086784
$\beta_{2}$	$=$	$ ( 19405 \nu^{15} - 1414470 \nu^{14} - 2995396 \nu^{13} - 2748093 \nu^{12} + \cdots + 4739496594759 ) / 402840781056 $ (19405v^15 - 1414470v^14 - 2995396v^13 - 2748093v^12 + 22617967v^11 + 126723336v^10 + 207115182v^9 + 856662777v^8 + 151802451v^7 - 6861725622v^6 - 48593244600v^5 - 64008225891v^4 - 46510764975v^3 - 68341659228v^2 + 2321639335134*v + 4739496594759) / 402840781056
$\beta_{3}$	$=$	$ ( 104485 \nu^{15} - 1292502 \nu^{14} - 751204 \nu^{13} + 17332203 \nu^{12} + \cdots - 4899563435313 ) / 402840781056 $ (104485v^15 - 1292502v^14 - 751204v^13 + 17332203v^12 + 50708023v^11 - 16095288v^10 + 359603838v^9 + 676175985v^8 - 5695808517v^7 - 2185468614v^6 + 2642601672v^5 + 36253669365v^4 + 231136845705v^3 + 642685773060v^2 + 2499973928622*v - 4899563435313) / 402840781056
$\beta_{4}$	$=$	$ ( - 7141 \nu^{15} - 20535 \nu^{14} - 158543 \nu^{13} - 1044534 \nu^{12} + 1182935 \nu^{11} + \cdots + 536916968064 ) / 25177548816 $ (-7141v^15 - 20535v^14 - 158543v^13 - 1044534v^12 + 1182935v^11 + 13084815v^10 + 9644301v^9 + 31933548v^8 + 260866629v^7 - 358985601v^6 - 3328888833v^5 - 2178046422v^4 - 5931399879v^3 - 22363923015v^2 + 94606595379*v + 536916968064) / 25177548816
$\beta_{5}$	$=$	$ ( - 129121 \nu^{15} - 1742658 \nu^{14} + 5078740 \nu^{13} + 29312817 \nu^{12} + \cdots - 6146512151427 ) / 402840781056 $ (-129121v^15 - 1742658v^14 + 5078740v^13 + 29312817v^12 + 12716117v^11 + 19268184v^10 + 130615290v^9 - 722055357v^8 - 6343337151v^7 - 2710054098v^6 + 23975742744v^5 - 69286860945v^4 + 199944080235v^3 + 1853098156620v^2 + 1114024693194*v - 6146512151427) / 402840781056
$\beta_{6}$	$=$	$ ( - 7373 \nu^{15} + 75566 \nu^{14} - 124948 \nu^{13} - 665771 \nu^{12} - 3935279 \nu^{11} + \cdots + 389083367889 ) / 22380043392 $ (-7373v^15 + 75566v^14 - 124948v^13 - 665771v^12 - 3935279v^11 + 401552v^10 - 27484422v^9 + 74974095v^8 + 554776749v^7 + 493956846v^6 + 287451504v^5 - 648492885v^4 - 18899220753v^3 - 69058855260v^2 - 144271705446*v + 389083367889) / 22380043392
$\beta_{7}$	$=$	$ ( - 30307 \nu^{15} + 31626 \nu^{14} + 697636 \nu^{13} - 2721645 \nu^{12} + 9494903 \nu^{11} + \cdots - 586396782369 ) / 67140130176 $ (-30307v^15 + 31626v^14 + 697636v^13 - 2721645v^12 + 9494903v^11 - 16983504v^10 - 14461506v^9 - 144127215v^8 + 259276275v^7 - 1316805174v^6 - 747020880v^5 + 9492190173v^4 - 25545325671v^3 + 16298862444v^2 + 195518206782*v - 586396782369) / 67140130176
$\beta_{8}$	$=$	$ ( - 68513 \nu^{15} + 25710 \nu^{14} + 944756 \nu^{13} + 918561 \nu^{12} - 5656187 \nu^{11} + \cdots + 79765574013 ) / 134280260352 $ (-68513v^15 + 25710v^14 + 944756v^13 + 918561v^12 - 5656187v^11 - 3056808v^10 + 62301546v^9 - 328912029v^8 + 11682657v^7 + 3933266526v^6 - 1727934120v^5 - 17224188705v^4 + 31966418907v^3 + 73089198828v^2 - 223174396422*v + 79765574013) / 134280260352
$\beta_{9}$	$=$	$ ( - 119455 \nu^{15} + 721650 \nu^{14} + 924412 \nu^{13} - 742305 \nu^{12} + \cdots + 1933443473715 ) / 134280260352 $ (-119455v^15 + 721650v^14 + 924412v^13 - 742305v^12 + 7699787v^11 + 32602584v^10 - 113487498v^9 - 237773043v^8 + 673681023v^7 - 149527134v^6 - 9518599656v^5 + 26744042241v^4 + 26150079285v^3 - 15836233212v^2 + 452246725062*v + 1933443473715) / 134280260352
$\beta_{10}$	$=$	$ ( - 8885 \nu^{15} + 1329 \nu^{14} - 163975 \nu^{13} + 111018 \nu^{12} + 19855 \nu^{11} + \cdots + 203155013952 ) / 8392516272 $ (-8885v^15 + 1329v^14 - 163975v^13 + 111018v^12 + 19855v^11 + 4608951v^10 + 8616717v^9 + 27086940v^8 + 18471429v^7 - 314431065v^6 - 416650473v^5 - 3471630678v^4 + 1878237153v^3 - 2672262495v^2 + 33784235811*v + 203155013952) / 8392516272
$\beta_{11}$	$=$	$ ( - 20285 \nu^{15} + 43845 \nu^{14} + 25445 \nu^{13} + 17634 \nu^{12} + 201595 \nu^{11} + \cdots + 187339329792 ) / 8392516272 $ (-20285v^15 + 43845v^14 + 25445v^13 + 17634v^12 + 201595v^11 + 3695235v^10 - 4593843v^9 - 51588036v^8 + 81613845v^7 - 45917037v^6 - 760298157v^5 + 1827715266v^4 + 5040507933v^3 + 3443560533v^2 - 27399503637*v + 187339329792) / 8392516272
$\beta_{12}$	$=$	$ ( 327935 \nu^{15} - 619314 \nu^{14} - 374828 \nu^{13} - 403743 \nu^{12} - 4021723 \nu^{11} + \cdots - 2692366730883 ) / 134280260352 $ (327935v^15 - 619314v^14 - 374828v^13 - 403743v^12 - 4021723v^11 - 56872968v^10 + 68313546v^9 + 796600035v^8 - 1030867263v^7 + 674865918v^6 + 11181337848v^5 - 28740615777v^4 - 76280843205v^3 - 81864825012v^2 + 1593571542426*v - 2692366730883) / 134280260352
$\beta_{13}$	$=$	$ ( - 58211 \nu^{15} - 73022 \nu^{14} + 588068 \nu^{13} + 527723 \nu^{12} + 1823839 \nu^{11} + \cdots + 139738690863 ) / 22380043392 $ (-58211v^15 - 73022v^14 + 588068v^13 + 527723v^12 + 1823839v^11 + 19442176v^10 + 15345654v^9 - 204563727v^8 - 252298557v^7 - 65280222v^6 - 2757571776v^5 - 334502379v^4 + 38805294753v^3 - 16451340084v^2 - 26427143754*v + 139738690863) / 22380043392
$\beta_{14}$	$=$	$ ( 89857 \nu^{15} - 29541 \nu^{14} + 465563 \nu^{13} - 378618 \nu^{12} - 3102467 \nu^{11} + \cdots - 1075364486208 ) / 25177548816 $ (89857v^15 - 29541v^14 + 465563v^13 - 378618v^12 - 3102467v^11 - 19998723v^10 - 62815833v^9 + 20341908v^8 - 66402801v^7 + 1451747853v^6 + 5370733269v^5 + 3669650406v^4 + 596598291v^3 - 86138614485v^2 - 29985495543*v - 1075364486208) / 25177548816
$\beta_{15}$	$=$	$ ( 1695151 \nu^{15} - 2117010 \nu^{14} - 11100220 \nu^{13} + 21907281 \nu^{12} + \cdots - 14728612560003 ) / 402840781056 $ (1695151v^15 - 2117010v^14 - 11100220v^13 + 21907281v^12 + 52214629v^11 - 394625496v^10 + 346512618v^9 + 4966328835v^8 - 7593612111v^7 - 16108603650v^6 + 91986205608v^5 - 124570651761v^4 - 835148778021v^3 + 1482775896060v^2 + 6908831848026*v - 14728612560003) / 402840781056

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

$\nu$	$=$	$ ( \beta_{12} + \beta_{11} - 3\beta _1 + 3 ) / 9 $ (b12 + b11 - 3*b1 + 3) / 9
$\nu^{2}$	$=$	$ ( -\beta_{15} - 2\beta_{13} - \beta_{6} + 2\beta_{5} + 2\beta_{4} - \beta_{2} - 8\beta _1 + 7 ) / 9 $ (-b15 - 2b13 - b6 + 2b5 + 2b4 - b2 - 8b1 + 7) / 9
$\nu^{3}$	$=$	$ ( - 6 \beta_{15} + 3 \beta_{14} - 6 \beta_{13} + \beta_{12} + 6 \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots - 21 ) / 9 $ (-6b15 + 3b14 - 6b13 + b12 + 6b10 - 2b9 - b8 + 3b7 - 3b6 + 3b5 + 6b3 + 39b1 - 21) / 9
$\nu^{4}$	$=$	$ ( - 2 \beta_{15} - 9 \beta_{14} + 2 \beta_{13} + 15 \beta_{12} + 12 \beta_{11} - 18 \beta_{10} + \cdots + 56 ) / 9 $ (-2b15 - 9b14 + 2b13 + 15b12 + 12b11 - 18b10 - 6b9 - 21b8 - 9b7 + 4b6 - 2b5 + 7b4 + 9b3 - 14b2 - 7*b1 + 56) / 9
$\nu^{5}$	$=$	$ ( 15 \beta_{15} + 15 \beta_{14} + 15 \beta_{13} + 9 \beta_{12} + 32 \beta_{11} + 21 \beta_{10} + \cdots + 1140 ) / 9 $ (15b15 + 15b14 + 15b13 + 9b12 + 32b11 + 21b10 - 32b9 - 7b8 + 6b7 + 12b6 + 6b5 + 72b4 + 21b3 - 108b2 - 396*b1 + 1140) / 9
$\nu^{6}$	$=$	$ ( 44 \beta_{15} + 63 \beta_{14} - 5 \beta_{13} - 84 \beta_{12} + 30 \beta_{11} - 9 \beta_{10} + \cdots + 2524 ) / 9 $ (44b15 + 63b14 - 5b13 - 84b12 + 30b11 - 9b10 + 30b9 + 240b8 - 45b7 + 32b6 - 49b5 + 131b4 + 99b3 - 31b2 - 2177*b1 + 2524) / 9
$\nu^{7}$	$=$	$ ( 12 \beta_{15} + 93 \beta_{14} - 78 \beta_{13} + 86 \beta_{12} + 377 \beta_{11} + 15 \beta_{10} + \cdots - 10449 ) / 9 $ (12b15 + 93b14 - 78b13 + 86b12 + 377b11 + 15b10 - 432b9 + 99b8 + 246b7 + 276b6 + 210b5 + 360b4 + 420b3 - 144b2 + 8043*b1 - 10449) / 9
$\nu^{8}$	$=$	$ ( - 836 \beta_{15} - 306 \beta_{14} - 592 \beta_{13} - 378 \beta_{12} - 2340 \beta_{11} + 468 \beta_{10} + \cdots + 1343 ) / 9 $ (-836b15 - 306b14 - 592b13 - 378b12 - 2340b11 + 468b10 + 1278b9 - 522b8 + 315b7 + 622b6 + 592b5 - 74b4 + 1035b3 - 368b2 - 3745*b1 + 1343) / 9
$\nu^{9}$	$=$	$ ( 546 \beta_{15} - 3081 \beta_{14} + 1788 \beta_{13} + 2717 \beta_{12} - 1215 \beta_{11} - 1977 \beta_{10} + \cdots - 65292 ) / 9 $ (546b15 - 3081b14 + 1788b13 + 2717b12 - 1215b11 - 1977b10 - 1690b9 + 838b8 - 759b7 + 300b6 - 1488b5 + 2484b4 + 1587b3 - 6156b2 + 56904*b1 - 65292) / 9
$\nu^{10}$	$=$	$ ( 5681 \beta_{15} + 819 \beta_{14} + 7198 \beta_{13} - 3408 \beta_{12} - 1074 \beta_{11} - 2817 \beta_{10} + \cdots - 65000 ) / 9 $ (5681b15 + 819b14 + 7198b13 - 3408b12 - 1074b11 - 2817b10 + 2724b9 - 4020b8 - 10602b7 + 707b6 + 1712b5 + 24464b4 - 1710b3 - 20173b2 - 76358*b1 - 65000) / 9
$\nu^{11}$	$=$	$ ( 24798 \beta_{15} + 26931 \beta_{14} + 10218 \beta_{13} - 69021 \beta_{12} - 22694 \beta_{11} + \cdots + 305067 ) / 9 $ (24798b15 + 26931b14 + 10218b13 - 69021b12 - 22694b11 + 20256b10 + 6332b9 + 45439b8 + 27588b7 + 8277b6 - 13047b5 + 30168b4 + 38481b3 - 1836b2 - 30438*b1 + 305067) / 9
$\nu^{12}$	$=$	$ ( 81082 \beta_{15} - 16992 \beta_{14} + 86576 \beta_{13} - 28167 \beta_{12} + 37698 \beta_{11} + \cdots - 2241409 ) / 9 $ (81082b15 - 16992b14 + 86576b13 - 28167b12 + 37698b11 - 53451b10 - 696b9 + 17733b8 - 38430b7 + 91192b6 - 17834b5 - 113351b4 + 39996b3 + 99922b2 + 1912976*b1 - 2241409) / 9
$\nu^{13}$	$=$	$ ( - 78555 \beta_{15} - 163686 \beta_{14} + 132999 \beta_{13} + 61897 \beta_{12} - 562472 \beta_{11} + \cdots - 2225583 ) / 9 $ (-78555b15 - 163686b14 + 132999b13 + 61897b12 - 562472b11 - 107124b10 + 313416b9 - 114309b8 + 120876b7 + 151950b6 + 189222b5 - 21312b4 - 32199b3 - 199548b2 + 4090692*b1 - 2225583) / 9
$\nu^{14}$	$=$	$ ( 110870 \beta_{15} - 244881 \beta_{14} - 31979 \beta_{13} - 294264 \beta_{12} - 1054638 \beta_{11} + \cdots - 21052172 ) / 9 $ (110870b15 - 244881b14 - 31979b13 - 294264b12 - 1054638b11 + 318267b10 + 1448316b9 + 446112b8 - 290538b7 - 484480b6 - 329119b5 - 180223b4 - 462330b3 - 783919b2 + 6812758*b1 - 21052172) / 9
$\nu^{15}$	$=$	$ ( 1040952 \beta_{15} + 747282 \beta_{14} + 171444 \beta_{13} - 1095524 \beta_{12} - 890919 \beta_{11} + \cdots - 58081098 ) / 9 $ (1040952b15 + 747282b14 + 171444b13 - 1095524b12 - 890919b11 - 1384554b10 - 429482b9 - 329905b8 - 1994865b7 - 2182602b6 + 159816b5 + 3290652b4 - 945129b3 - 1113372b2 + 67829142*b1 - 58081098) / 9

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$-1 + \beta_{1}$

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} $

361.1

1.06976	−	2.80279i
−2.93833	+	0.605173i
2.96167	+	0.478005i
−2.23411	−	2.00219i
1.37357	+	2.66708i
2.86381	−	0.893639i
0.285042	−	2.98643i
−1.88142	+	2.33672i
1.06976	+	2.80279i
−2.93833	−	0.605173i
2.96167	−	0.478005i
−2.23411	+	2.00219i
1.37357	−	2.66708i
2.86381	+	0.893639i
0.285042	+	2.98643i
−1.88142	−	2.33672i

2.50000

−

4.33013i

−15.1847

−

26.3006i

361.2

2.50000

−

4.33013i

−8.98058

−

15.5548i

361.3

2.50000

−

4.33013i

−8.94810

−

15.4986i

361.4

2.50000

−

4.33013i

−6.73761

−

11.6699i

361.5

2.50000

−

4.33013i

1.27941

2.21601i

361.6

2.50000

−

4.33013i

5.38667

9.32999i

361.7

2.50000

−

4.33013i

8.72105

15.1053i

361.8

2.50000

−

4.33013i

8.96385

15.5258i

721.1

2.50000

4.33013i

−15.1847

26.3006i

721.2

2.50000

4.33013i

−8.98058

15.5548i

721.3

2.50000

4.33013i

−8.94810

15.4986i

721.4

2.50000

4.33013i

−6.73761

11.6699i

721.5

2.50000

4.33013i

1.27941

−

2.21601i

721.6

2.50000

4.33013i

5.38667

−

9.32999i

721.7

2.50000

4.33013i

8.72105

−

15.1053i

721.8

2.50000

4.33013i

8.96385

−

15.5258i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1080.4.q.c		16
3.b	odd	2	1		360.4.q.c	✓	16
9.c	even	3	1	inner	1080.4.q.c		16
9.d	odd	6	1		360.4.q.c	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
360.4.q.c	✓	16	3.b	odd	2	1
360.4.q.c	✓	16	9.d	odd	6	1
1080.4.q.c		16	1.a	even	1	1	trivial
1080.4.q.c		16	9.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{16} + 31 T_{7}^{15} + 1728 T_{7}^{14} + 23873 T_{7}^{13} + 1128233 T_{7}^{12} + \cdots + 12\!\cdots\!00 $ T7^16 + 31*T7^15 + 1728*T7^14 + 23873*T7^13 + 1128233*T7^12 + 11747538*T7^11 + 497848285*T7^10 + 3057600289*T7^9 + 133296163569*T7^8 + 497019824830*T7^7 + 25797072685081*T7^6 + 19754485979625*T7^5 + 2711642195751296*T7^4 - 3981029622217945*T7^3 + 207957531729169485*T7^2 - 486082110166043300*T7 + 1285774702470403600 acting on $S_{4}^{\mathrm{new}}(1080, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ T^{16} $ T^16
$5$	$ (T^{2} - 5 T + 25)^{8} $ (T^2 - 5*T + 25)^8
$7$	$ T^{16} + \cdots + 12\!\cdots\!00 $ T^16 + 31T^15 + 1728T^14 + 23873T^13 + 1128233T^12 + 11747538T^11 + 497848285T^10 + 3057600289T^9 + 133296163569T^8 + 497019824830T^7 + 25797072685081T^6 + 19754485979625T^5 + 2711642195751296T^4 - 3981029622217945T^3 + 207957531729169485T^2 - 486082110166043300*T + 1285774702470403600
$11$	$ T^{16} + \cdots + 16\!\cdots\!00 $ T^16 + 41T^15 + 6417T^14 + 251032T^13 + 27234596T^12 + 938149056T^11 + 57768444640T^10 + 1180835285504T^9 + 60245532050976T^8 + 793708718888960T^7 + 44184446167685440T^6 + 160166858271897600T^5 + 14852868225533772800T^4 - 116398904677536224000T^3 + 4512690492648016032000T^2 - 25887649354568934400000*T + 167336853775102935040000
$13$	$ T^{16} + \cdots + 13\!\cdots\!00 $ T^16 - 16T^15 + 8712T^14 + 91696T^13 + 56427344T^12 + 381160224T^11 + 119728195552T^10 + 1637199545600T^9 + 184285293450624T^8 + 1486207323848960T^7 + 94279486002714880T^6 - 17892283473216000T^5 + 33247068003074105600T^4 + 29739622569916672000T^3 + 2485696859279009280000T^2 - 5050730538756812800000*T + 139414249354494976000000
$17$	$ (T^{8} + \cdots + 60216728526304)^{2} $ (T^8 - 79T^7 - 17858T^6 + 1136180T^5 + 82406704T^4 - 3741096928T^3 - 121756294160T^2 + 2410711039568*T + 60216728526304)^2
$19$	$ (T^{8} + \cdots + 103678903400800)^{2} $ (T^8 + 61T^7 - 25178T^6 - 1014104T^5 + 150993832T^4 + 5261220640T^3 - 230315322608T^2 - 4871698451120*T + 103678903400800)^2
$23$	$ T^{16} + \cdots + 69\!\cdots\!44 $ T^16 + 113T^15 + 48582T^14 + 1737481T^13 + 1393586615T^12 + 58876490052T^11 + 15075053142661T^10 - 365054867429557T^9 + 48059874753948315T^8 - 694632921592786624T^7 + 68220140447598993049T^6 - 991971795394513555461T^5 + 63489268169565135121130T^4 - 431012623720969066541177T^3 + 24126731273610135225795303T^2 - 74684963876642911226931058*T + 6986863233190250108879091844
$29$	$ T^{16} + \cdots + 21\!\cdots\!00 $ T^16 + 89T^15 + 55278T^14 - 5106191T^13 + 1891593131T^12 - 80861665104T^11 + 14839117118029T^10 - 290925308148229T^9 + 87507824498949483T^8 - 1192864565558255896T^7 + 94821097566514505401T^6 + 187553127607041819735T^5 + 67087471018028825902874T^4 - 358311127213884001555085T^3 + 1617264855169577058288015T^2 - 2037470587492083566381650*T + 2102835017487563118568900
$31$	$ T^{16} + \cdots + 32\!\cdots\!76 $ T^16 + 68T^15 + 80364T^14 + 6230464T^13 + 4758130112T^12 + 337702980768T^11 + 122831666901664T^10 + 5807357037179264T^9 + 2109045109189283712T^8 + 102917188731964790528T^7 + 11725515640421220114688T^6 + 208843516016555838743040T^5 + 26548688136617508666692864T^4 + 312962264107579647764700160T^3 + 43333779704634477386900438016T^2 - 347315501585014761829565059072*T + 3231268923526625155509219659776
$37$	$ (T^{8} + \cdots - 16\!\cdots\!96)^{2} $ (T^8 + 282T^7 - 143076T^6 - 18363312T^5 + 7656496848T^4 - 128554327584T^3 - 92313816314544T^2 + 7523678612137824*T - 168746749001421696)^2
$41$	$ T^{16} + \cdots + 42\!\cdots\!49 $ T^16 + 252T^15 + 163728T^14 + 29246040T^13 + 15927464718T^12 + 2699052039828T^11 + 769845638273904T^10 + 71752305648807252T^9 + 15829281527344823583T^8 + 1708668942286167671796T^7 + 192411477802835346361680T^6 + 11462054613264717337612116T^5 + 618419555132953652781243750T^4 + 15271546035556606610318626872T^3 + 501602723872687879050218821128T^2 + 3611387367934394383461377265084*T + 429403920803851027770320240755449
$43$	$ T^{16} + \cdots + 57\!\cdots\!44 $ T^16 - 85T^15 + 318741T^14 + 4524580T^13 + 70885305116T^12 + 1163632264320T^11 + 7523602537380880T^10 + 323347962893526368T^9 + 573695994373884308448T^8 + 12050738227577547446912T^7 + 14004334878570171167336896T^6 + 239409940661444142936215040T^5 + 265124536595134335252640413440T^4 + 2170799662051537391620826686720T^3 + 56867896301996797889806849553664T^2 - 318627819475717583389845546241024*T + 5776085414869720776783099962527744
$47$	$ T^{16} + \cdots + 92\!\cdots\!84 $ T^16 - 25T^15 + 566172T^14 + 16566997T^13 + 218115922697T^12 + 10775636171370T^11 + 46127630212821397T^10 + 4973837336172414317T^9 + 7137654816186038198637T^8 + 894853681709736381134894T^7 + 620524794594875939107027981T^6 + 114948792887732479520149188393T^5 + 37656396016491667365734326228796T^4 + 4164878002977388038587002742929195T^3 + 443993555242582930911226348273459257T^2 + 6809913726648722590817879414895895424*T + 92849304848562082284656460077619625984
$53$	$ (T^{8} + \cdots - 54\!\cdots\!44)^{2} $ (T^8 - 564T^7 - 917832T^6 + 570377808T^5 + 214435116864T^4 - 162076054051008T^3 - 2989652824388736T^2 + 10333542290342406912*T - 545272449044221350144)^2
$59$	$ T^{16} + \cdots + 62\!\cdots\!84 $ T^16 + 457T^15 + 915579T^14 + 282357662T^13 + 511221027032T^12 + 136695671418168T^11 + 145639779638371408T^10 + 11222663389492627456T^9 + 20574200999186214049728T^8 + 55683726893252856030592T^7 + 2262194930487445117843547392T^6 - 170866728702231453192302622720T^5 + 99288955563955151639122767322112T^4 + 816286574432082754748550641223680T^3 + 1306311896059568309092680938514456576T^2 - 61347800803993763210921684523737636864*T + 6268981326602431956397792748812369936384
$61$	$ T^{16} + \cdots + 80\!\cdots\!56 $ T^16 - 853T^15 + 1317654T^14 - 838267853T^13 + 1082381679179T^12 - 669426942667536T^11 + 363110975362713061T^10 - 108426743929381131655T^9 + 26817657092667770736123T^8 - 4532443371972391223178568T^7 + 741506661975428587821293089T^6 - 95413723957049229800587515411T^5 + 12541995573183997422349241433722T^4 - 1134730060029856024844201977584311T^3 + 87469965571911219051367522875157935T^2 - 3061386876461580231299383713980931694*T + 80749343297938314853058742911808776356
$67$	$ T^{16} + \cdots + 21\!\cdots\!01 $ T^16 - 596T^15 + 1148688T^14 - 146795152T^13 + 570373131158T^12 + 19699311452436T^11 + 217134825168502792T^10 + 48557647338821366416T^9 + 49642976395029584831139T^8 + 15053318977201041864519316T^7 + 9162829236631678158715183672T^6 + 2716381143933959426398008274296T^5 + 754402203114938412234293005174022T^4 + 112591883448977530753480825797863480T^3 + 13254684651846356868564139225342940868T^2 + 616599545044194427608901470730566209392*T + 21742162779120696232333956004991653377601
$71$	$ (T^{8} + \cdots + 16\!\cdots\!88)^{2} $ (T^8 - 114T^7 - 1678716T^6 + 105429456T^5 + 522683674992T^4 - 82525405914720T^3 - 12481051303983600T^2 - 13074727768513632*T + 16529781230746812288)^2
$73$	$ (T^{8} + \cdots + 29\!\cdots\!20)^{2} $ (T^8 + 1035T^7 - 313668T^6 - 659110464T^5 - 173206340352T^4 + 38407719877632T^3 + 21586031648243712T^2 + 2483157340396339200*T + 29958324604131409920)^2
$79$	$ T^{16} + \cdots + 25\!\cdots\!00 $ T^16 - 300T^15 + 1828584T^14 - 1703570688T^13 + 2814787953216T^12 - 2314196976259008T^11 + 2448718056017111808T^10 - 1953694832984144914176T^9 + 1513781826472160881396992T^8 - 859309344818227776696265728T^7 + 398070254550505441219200036864T^6 - 131642011405557463383699940749312T^5 + 33478193130428512896600564846084096T^4 - 5720840695470246841172902571954012160T^3 + 717576948256793649560354185333399388160T^2 - 53242752601278168837626397948938300620800*T + 2570368142150575403072645133272353421721600
$83$	$ T^{16} + \cdots + 75\!\cdots\!04 $ T^16 - 155T^15 + 2240616T^14 - 819201445T^13 + 3437716398401T^12 - 1355782101727650T^11 + 3055891054045826005T^10 - 1345282673460036395453T^9 + 1992119576506358875725753T^8 - 791203221870781886527716062T^7 + 757037150673454136763399629761T^6 - 261574025088776374820395346721045T^5 + 193882403590542453345382124031572840T^4 - 50346490594689178692518692291812746875T^3 + 19205235187681917372772666457069348913309T^2 - 390447345504824481845715146655779915660756*T + 7575827807579016064991171008516153821861904
$89$	$ (T^{8} + \cdots - 54\!\cdots\!42)^{2} $ (T^8 - 2571T^7 + 81759T^6 + 3927363273T^5 - 2671759314027T^4 + 134221938788511T^3 + 233098701379376805T^2 - 21266006155729491357*T - 5493570290889835450842)^2
$97$	$ T^{16} + \cdots + 39\!\cdots\!04 $ T^16 - 283T^15 + 4291215T^14 + 837658378T^13 + 12417782590808T^12 + 2666034882794280T^11 + 18468510372693634576T^10 + 4462334686938290652800T^9 + 19884369026023925475054528T^8 + 3797528506041831776828067968T^7 + 11625618685258619207752150186240T^6 + 1833755375501118923141125997211648T^5 + 4639010046357077442783942277735291904T^4 + 522965904019866633558507188210136266752T^3 + 60591047061828743744800107961454256365568T^2 - 48857051182913954089217199320501843304448*T + 39673588788228524972963565378680751407104
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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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - 5 \beta_1 + 5) q^{5} + ( - \beta_{4} + \beta_{2} - 4 \beta_1) q^{7}+O(q^{10}) \) q + (-5b1 + 5) q^5 + (-b4 + b2 - 4b1) q^7	\( q + ( - 5 \beta_1 + 5) q^{5} + ( - \beta_{4} + \beta_{2} - 4 \beta_1) q^{7} + (\beta_{7} - \beta_{4} + \cdots - 5 \beta_1) q^{11}+ \cdots + ( - 14 \beta_{9} - 3 \beta_{8} + \cdots + 21 \beta_1) q^{97}+O(q^{100}) \) q + (-5b1 + 5) q^5 + (-b4 + b2 - 4b1) q^7 + (b7 - b4 + b2 - 5b1) q^11 + (b15 + b14 - b12 + b11 - b9 + b3 + b2 - 3b1 + 3) q^13 + (-3b14 + b13 - b11 - b10 + b6 + b4 + 9) q^17 + (-2b15 - b14 - b12 - b11 + b10 + b8 + 2b5 + b4 - 8) * q^19 + (3b15 + b13 + 3b11 - 3b9 - b2 + 14b1 - 14) * q^23 - 25b1 q^25 + (b9 + 2b7 - 2b5 + b4 + 2b3 - b2 - 11b1) * q^29 + (-3b15 + b14 - 3b13 + 3b11 - 3b9 + b3 + b2 + 8b1 - 8) q^31 + (-5b4 - 20) q^35 + (b15 - b14 + 2b13 - 6b12 - b11 + 3b10 + 6b8 + 2b6 - b5 + 10b4 - 33) * q^37 + (5b15 + 2b14 - 5b12 + b10 + b7 + 2b3 - 4b2 + 29b1 - 29) * q^41 + (-5b9 + 6b8 + 2b7 + 6b6 - 2b5 - 3b3 + 14b1) q^43 + (-4b9 + 5b8 - 9b7 - 10b6 + 2b5 - 5b4 - 13b3 + 5b2 + 8b1) q^47 + (-5b15 + b14 - 2b12 - b10 - b7 + b3 - 7b2 - 31b1 + 31) * q^49 + (-3b15 + 15b14 - 15b13 - 4b12 - 6b11 + 13b10 + 4b8 - 15b6 + 3b5 + 9b4 + 77) * q^53 + (-5b10 - 5b4 - 25) * q^55 + (2b15 + 5b14 + 8b13 - 4b12 + b10 + b7 + 5b3 - 18b2 + 57b1 - 57) q^59 + (-11b9 - 2b8 + 8b7 + 6b6 - 5b5 - 22b4 + 22b2 + 105b1) * q^61 + (-5b9 - 5b8 + 5b5 - 5b4 + 5b3 + 5b2 - 15b1) q^65 + (-10b15 - 4b14 - 8b13 + 4b11 - 5b10 - 4b9 - 5b7 - 4b3 + 8b2 - 75b1 + 75) * q^67 + (12b15 - 16b14 + 10b13 - 5b12 - 7b11 - 18b10 + 5b8 + 10b6 - 12b5 + b4 + 15) q^71 + (3b15 + 4b14 + 11b13 - 9b12 - 4b11 - b10 + 9b8 + 11b6 - 3b5 - 6b4 - 126) q^73 + (-2b15 - 6b14 - b13 - 19b12 - 21b11 - 11b10 + 21b9 - 11b7 - 6b3 - 16b2 + 184b1 - 184) * q^77 + (-13b9 + 23b8 - 4b7 + 2b5 + 36b4 + b3 - 36b2 + 49b1) q^79 + (-12b9 - 3b8 + 9b7 - 2b5 + 8b4 + 26b3 - 8b2 + 12b1) * q^83 + (-15b14 + 5b13 - 5b11 - 5b10 + 5b9 - 5b7 - 15b3 + 5b2 - 45b1 + 45) q^85 + (b15 - 8b14 + 13b13 - 9b12 - 32b11 - 9b10 + 9b8 + 13b6 - b5 + 20b4 + 325) * q^89 + (-28b15 + b14 - 10b13 - 14b12 - 6b11 + 6b10 + 14b8 - 10b6 + 28b5 + b4 - 398) * q^91 + (-10b15 - 5b14 - 5b12 - 5b11 + 5b10 + 5b9 + 5b7 - 5b3 + 5b2 + 40b1 - 40) * q^95 + (-14b9 - 3b8 - 14b7 - 8b6 + 12b5 - 17b4 + 19b3 + 17b2 + 21b1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 40 q^{5} - 31 q^{7}+O(q^{10}) \) 16 * q + 40 * q^5 - 31 * q^7	\( 16 q + 40 q^{5} - 31 q^{7} - 41 q^{11} + 16 q^{13} + 158 q^{17} - 122 q^{19} - 113 q^{23} - 200 q^{25} - 89 q^{29} - 68 q^{31} - 310 q^{35} - 564 q^{37} - 252 q^{41} + 85 q^{43} + 25 q^{47} + 249 q^{49} + 1128 q^{53} - 410 q^{55} - 457 q^{59} + 853 q^{61} - 80 q^{65} + 596 q^{67} + 228 q^{71} - 2070 q^{73} - 1516 q^{77} + 300 q^{79} + 155 q^{83} + 395 q^{85} + 5142 q^{89} - 6400 q^{91} - 305 q^{95} + 283 q^{97}+O(q^{100}) \) 16 * q + 40 * q^5 - 31 * q^7 - 41 * q^11 + 16 * q^13 + 158 * q^17 - 122 * q^19 - 113 * q^23 - 200 * q^25 - 89 * q^29 - 68 * q^31 - 310 * q^35 - 564 * q^37 - 252 * q^41 + 85 * q^43 + 25 * q^47 + 249 * q^49 + 1128 * q^53 - 410 * q^55 - 457 * q^59 + 853 * q^61 - 80 * q^65 + 596 * q^67 + 228 * q^71 - 2070 * q^73 - 1516 * q^77 + 300 * q^79 + 155 * q^83 + 395 * q^85 + 5142 * q^89 - 6400 * q^91 - 305 * q^95 + 283 * q^97

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	1080.4.q.c

Level	$1080$
Weight	$4$
Character orbit	1080.q
Analytic conductor	$63.722$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(63.7220628062\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 3 x^{15} + 2 x^{14} + 3 x^{13} - 32 x^{12} - 255 x^{11} + 558 x^{10} + 2079 x^{9} + \cdots + 43046721 \) x^16 - 3x^15 + 2x^14 + 3x^13 - 32x^12 - 255x^11 + 558x^10 + 2079x^9 - 6210x^8 + 18711x^7 + 45198x^6 - 185895x^5 - 209952x^4 + 177147x^3 + 1062882x^2 - 14348907*x + 43046721
Coefficient ring:	\(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index:	\( 2^{12}\cdot 3^{16} \)
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( 375 \nu^{15} + 9134 \nu^{14} + 3588 \nu^{13} - 13511 \nu^{12} - 88467 \nu^{11} + \cdots + 78655925205 ) / 44760086784 \) (375v^15 + 9134v^14 + 3588v^13 - 13511v^12 - 88467v^11 + 250088v^10 - 576438v^9 - 3200949v^8 + 30550473v^7 - 6645186v^6 - 109270296v^5 + 55869831v^4 + 485253747v^3 - 2974206276v^2 - 5926984326*v + 78655925205) / 44760086784
\(\beta_{2}\)	\(=\)	\( ( 19405 \nu^{15} - 1414470 \nu^{14} - 2995396 \nu^{13} - 2748093 \nu^{12} + \cdots + 4739496594759 ) / 402840781056 \) (19405v^15 - 1414470v^14 - 2995396v^13 - 2748093v^12 + 22617967v^11 + 126723336v^10 + 207115182v^9 + 856662777v^8 + 151802451v^7 - 6861725622v^6 - 48593244600v^5 - 64008225891v^4 - 46510764975v^3 - 68341659228v^2 + 2321639335134*v + 4739496594759) / 402840781056
\(\beta_{3}\)	\(=\)	\( ( 104485 \nu^{15} - 1292502 \nu^{14} - 751204 \nu^{13} + 17332203 \nu^{12} + \cdots - 4899563435313 ) / 402840781056 \) (104485v^15 - 1292502v^14 - 751204v^13 + 17332203v^12 + 50708023v^11 - 16095288v^10 + 359603838v^9 + 676175985v^8 - 5695808517v^7 - 2185468614v^6 + 2642601672v^5 + 36253669365v^4 + 231136845705v^3 + 642685773060v^2 + 2499973928622*v - 4899563435313) / 402840781056
\(\beta_{4}\)	\(=\)	\( ( - 7141 \nu^{15} - 20535 \nu^{14} - 158543 \nu^{13} - 1044534 \nu^{12} + 1182935 \nu^{11} + \cdots + 536916968064 ) / 25177548816 \) (-7141v^15 - 20535v^14 - 158543v^13 - 1044534v^12 + 1182935v^11 + 13084815v^10 + 9644301v^9 + 31933548v^8 + 260866629v^7 - 358985601v^6 - 3328888833v^5 - 2178046422v^4 - 5931399879v^3 - 22363923015v^2 + 94606595379*v + 536916968064) / 25177548816
\(\beta_{5}\)	\(=\)	\( ( - 129121 \nu^{15} - 1742658 \nu^{14} + 5078740 \nu^{13} + 29312817 \nu^{12} + \cdots - 6146512151427 ) / 402840781056 \) (-129121v^15 - 1742658v^14 + 5078740v^13 + 29312817v^12 + 12716117v^11 + 19268184v^10 + 130615290v^9 - 722055357v^8 - 6343337151v^7 - 2710054098v^6 + 23975742744v^5 - 69286860945v^4 + 199944080235v^3 + 1853098156620v^2 + 1114024693194*v - 6146512151427) / 402840781056
\(\beta_{6}\)	\(=\)	\( ( - 7373 \nu^{15} + 75566 \nu^{14} - 124948 \nu^{13} - 665771 \nu^{12} - 3935279 \nu^{11} + \cdots + 389083367889 ) / 22380043392 \) (-7373v^15 + 75566v^14 - 124948v^13 - 665771v^12 - 3935279v^11 + 401552v^10 - 27484422v^9 + 74974095v^8 + 554776749v^7 + 493956846v^6 + 287451504v^5 - 648492885v^4 - 18899220753v^3 - 69058855260v^2 - 144271705446*v + 389083367889) / 22380043392
\(\beta_{7}\)	\(=\)	\( ( - 30307 \nu^{15} + 31626 \nu^{14} + 697636 \nu^{13} - 2721645 \nu^{12} + 9494903 \nu^{11} + \cdots - 586396782369 ) / 67140130176 \) (-30307v^15 + 31626v^14 + 697636v^13 - 2721645v^12 + 9494903v^11 - 16983504v^10 - 14461506v^9 - 144127215v^8 + 259276275v^7 - 1316805174v^6 - 747020880v^5 + 9492190173v^4 - 25545325671v^3 + 16298862444v^2 + 195518206782*v - 586396782369) / 67140130176
\(\beta_{8}\)	\(=\)	\( ( - 68513 \nu^{15} + 25710 \nu^{14} + 944756 \nu^{13} + 918561 \nu^{12} - 5656187 \nu^{11} + \cdots + 79765574013 ) / 134280260352 \) (-68513v^15 + 25710v^14 + 944756v^13 + 918561v^12 - 5656187v^11 - 3056808v^10 + 62301546v^9 - 328912029v^8 + 11682657v^7 + 3933266526v^6 - 1727934120v^5 - 17224188705v^4 + 31966418907v^3 + 73089198828v^2 - 223174396422*v + 79765574013) / 134280260352
\(\beta_{9}\)	\(=\)	\( ( - 119455 \nu^{15} + 721650 \nu^{14} + 924412 \nu^{13} - 742305 \nu^{12} + \cdots + 1933443473715 ) / 134280260352 \) (-119455v^15 + 721650v^14 + 924412v^13 - 742305v^12 + 7699787v^11 + 32602584v^10 - 113487498v^9 - 237773043v^8 + 673681023v^7 - 149527134v^6 - 9518599656v^5 + 26744042241v^4 + 26150079285v^3 - 15836233212v^2 + 452246725062*v + 1933443473715) / 134280260352
\(\beta_{10}\)	\(=\)	\( ( - 8885 \nu^{15} + 1329 \nu^{14} - 163975 \nu^{13} + 111018 \nu^{12} + 19855 \nu^{11} + \cdots + 203155013952 ) / 8392516272 \) (-8885v^15 + 1329v^14 - 163975v^13 + 111018v^12 + 19855v^11 + 4608951v^10 + 8616717v^9 + 27086940v^8 + 18471429v^7 - 314431065v^6 - 416650473v^5 - 3471630678v^4 + 1878237153v^3 - 2672262495v^2 + 33784235811*v + 203155013952) / 8392516272
\(\beta_{11}\)	\(=\)	\( ( - 20285 \nu^{15} + 43845 \nu^{14} + 25445 \nu^{13} + 17634 \nu^{12} + 201595 \nu^{11} + \cdots + 187339329792 ) / 8392516272 \) (-20285v^15 + 43845v^14 + 25445v^13 + 17634v^12 + 201595v^11 + 3695235v^10 - 4593843v^9 - 51588036v^8 + 81613845v^7 - 45917037v^6 - 760298157v^5 + 1827715266v^4 + 5040507933v^3 + 3443560533v^2 - 27399503637*v + 187339329792) / 8392516272
\(\beta_{12}\)	\(=\)	\( ( 327935 \nu^{15} - 619314 \nu^{14} - 374828 \nu^{13} - 403743 \nu^{12} - 4021723 \nu^{11} + \cdots - 2692366730883 ) / 134280260352 \) (327935v^15 - 619314v^14 - 374828v^13 - 403743v^12 - 4021723v^11 - 56872968v^10 + 68313546v^9 + 796600035v^8 - 1030867263v^7 + 674865918v^6 + 11181337848v^5 - 28740615777v^4 - 76280843205v^3 - 81864825012v^2 + 1593571542426*v - 2692366730883) / 134280260352
\(\beta_{13}\)	\(=\)	\( ( - 58211 \nu^{15} - 73022 \nu^{14} + 588068 \nu^{13} + 527723 \nu^{12} + 1823839 \nu^{11} + \cdots + 139738690863 ) / 22380043392 \) (-58211v^15 - 73022v^14 + 588068v^13 + 527723v^12 + 1823839v^11 + 19442176v^10 + 15345654v^9 - 204563727v^8 - 252298557v^7 - 65280222v^6 - 2757571776v^5 - 334502379v^4 + 38805294753v^3 - 16451340084v^2 - 26427143754*v + 139738690863) / 22380043392
\(\beta_{14}\)	\(=\)	\( ( 89857 \nu^{15} - 29541 \nu^{14} + 465563 \nu^{13} - 378618 \nu^{12} - 3102467 \nu^{11} + \cdots - 1075364486208 ) / 25177548816 \) (89857v^15 - 29541v^14 + 465563v^13 - 378618v^12 - 3102467v^11 - 19998723v^10 - 62815833v^9 + 20341908v^8 - 66402801v^7 + 1451747853v^6 + 5370733269v^5 + 3669650406v^4 + 596598291v^3 - 86138614485v^2 - 29985495543*v - 1075364486208) / 25177548816
\(\beta_{15}\)	\(=\)	\( ( 1695151 \nu^{15} - 2117010 \nu^{14} - 11100220 \nu^{13} + 21907281 \nu^{12} + \cdots - 14728612560003 ) / 402840781056 \) (1695151v^15 - 2117010v^14 - 11100220v^13 + 21907281v^12 + 52214629v^11 - 394625496v^10 + 346512618v^9 + 4966328835v^8 - 7593612111v^7 - 16108603650v^6 + 91986205608v^5 - 124570651761v^4 - 835148778021v^3 + 1482775896060v^2 + 6908831848026*v - 14728612560003) / 402840781056

\(\nu\)	\(=\)	\( ( \beta_{12} + \beta_{11} - 3\beta _1 + 3 ) / 9 \) (b12 + b11 - 3*b1 + 3) / 9
\(\nu^{2}\)	\(=\)	\( ( -\beta_{15} - 2\beta_{13} - \beta_{6} + 2\beta_{5} + 2\beta_{4} - \beta_{2} - 8\beta _1 + 7 ) / 9 \) (-b15 - 2b13 - b6 + 2b5 + 2b4 - b2 - 8b1 + 7) / 9
\(\nu^{3}\)	\(=\)	\( ( - 6 \beta_{15} + 3 \beta_{14} - 6 \beta_{13} + \beta_{12} + 6 \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots - 21 ) / 9 \) (-6b15 + 3b14 - 6b13 + b12 + 6b10 - 2b9 - b8 + 3b7 - 3b6 + 3b5 + 6b3 + 39b1 - 21) / 9
\(\nu^{4}\)	\(=\)	\( ( - 2 \beta_{15} - 9 \beta_{14} + 2 \beta_{13} + 15 \beta_{12} + 12 \beta_{11} - 18 \beta_{10} + \cdots + 56 ) / 9 \) (-2b15 - 9b14 + 2b13 + 15b12 + 12b11 - 18b10 - 6b9 - 21b8 - 9b7 + 4b6 - 2b5 + 7b4 + 9b3 - 14b2 - 7*b1 + 56) / 9
\(\nu^{5}\)	\(=\)	\( ( 15 \beta_{15} + 15 \beta_{14} + 15 \beta_{13} + 9 \beta_{12} + 32 \beta_{11} + 21 \beta_{10} + \cdots + 1140 ) / 9 \) (15b15 + 15b14 + 15b13 + 9b12 + 32b11 + 21b10 - 32b9 - 7b8 + 6b7 + 12b6 + 6b5 + 72b4 + 21b3 - 108b2 - 396*b1 + 1140) / 9
\(\nu^{6}\)	\(=\)	\( ( 44 \beta_{15} + 63 \beta_{14} - 5 \beta_{13} - 84 \beta_{12} + 30 \beta_{11} - 9 \beta_{10} + \cdots + 2524 ) / 9 \) (44b15 + 63b14 - 5b13 - 84b12 + 30b11 - 9b10 + 30b9 + 240b8 - 45b7 + 32b6 - 49b5 + 131b4 + 99b3 - 31b2 - 2177*b1 + 2524) / 9
\(\nu^{7}\)	\(=\)	\( ( 12 \beta_{15} + 93 \beta_{14} - 78 \beta_{13} + 86 \beta_{12} + 377 \beta_{11} + 15 \beta_{10} + \cdots - 10449 ) / 9 \) (12b15 + 93b14 - 78b13 + 86b12 + 377b11 + 15b10 - 432b9 + 99b8 + 246b7 + 276b6 + 210b5 + 360b4 + 420b3 - 144b2 + 8043*b1 - 10449) / 9
\(\nu^{8}\)	\(=\)	\( ( - 836 \beta_{15} - 306 \beta_{14} - 592 \beta_{13} - 378 \beta_{12} - 2340 \beta_{11} + 468 \beta_{10} + \cdots + 1343 ) / 9 \) (-836b15 - 306b14 - 592b13 - 378b12 - 2340b11 + 468b10 + 1278b9 - 522b8 + 315b7 + 622b6 + 592b5 - 74b4 + 1035b3 - 368b2 - 3745*b1 + 1343) / 9
\(\nu^{9}\)	\(=\)	\( ( 546 \beta_{15} - 3081 \beta_{14} + 1788 \beta_{13} + 2717 \beta_{12} - 1215 \beta_{11} - 1977 \beta_{10} + \cdots - 65292 ) / 9 \) (546b15 - 3081b14 + 1788b13 + 2717b12 - 1215b11 - 1977b10 - 1690b9 + 838b8 - 759b7 + 300b6 - 1488b5 + 2484b4 + 1587b3 - 6156b2 + 56904*b1 - 65292) / 9
\(\nu^{10}\)	\(=\)	\( ( 5681 \beta_{15} + 819 \beta_{14} + 7198 \beta_{13} - 3408 \beta_{12} - 1074 \beta_{11} - 2817 \beta_{10} + \cdots - 65000 ) / 9 \) (5681b15 + 819b14 + 7198b13 - 3408b12 - 1074b11 - 2817b10 + 2724b9 - 4020b8 - 10602b7 + 707b6 + 1712b5 + 24464b4 - 1710b3 - 20173b2 - 76358*b1 - 65000) / 9
\(\nu^{11}\)	\(=\)	\( ( 24798 \beta_{15} + 26931 \beta_{14} + 10218 \beta_{13} - 69021 \beta_{12} - 22694 \beta_{11} + \cdots + 305067 ) / 9 \) (24798b15 + 26931b14 + 10218b13 - 69021b12 - 22694b11 + 20256b10 + 6332b9 + 45439b8 + 27588b7 + 8277b6 - 13047b5 + 30168b4 + 38481b3 - 1836b2 - 30438*b1 + 305067) / 9
\(\nu^{12}\)	\(=\)	\( ( 81082 \beta_{15} - 16992 \beta_{14} + 86576 \beta_{13} - 28167 \beta_{12} + 37698 \beta_{11} + \cdots - 2241409 ) / 9 \) (81082b15 - 16992b14 + 86576b13 - 28167b12 + 37698b11 - 53451b10 - 696b9 + 17733b8 - 38430b7 + 91192b6 - 17834b5 - 113351b4 + 39996b3 + 99922b2 + 1912976*b1 - 2241409) / 9
\(\nu^{13}\)	\(=\)	\( ( - 78555 \beta_{15} - 163686 \beta_{14} + 132999 \beta_{13} + 61897 \beta_{12} - 562472 \beta_{11} + \cdots - 2225583 ) / 9 \) (-78555b15 - 163686b14 + 132999b13 + 61897b12 - 562472b11 - 107124b10 + 313416b9 - 114309b8 + 120876b7 + 151950b6 + 189222b5 - 21312b4 - 32199b3 - 199548b2 + 4090692*b1 - 2225583) / 9
\(\nu^{14}\)	\(=\)	\( ( 110870 \beta_{15} - 244881 \beta_{14} - 31979 \beta_{13} - 294264 \beta_{12} - 1054638 \beta_{11} + \cdots - 21052172 ) / 9 \) (110870b15 - 244881b14 - 31979b13 - 294264b12 - 1054638b11 + 318267b10 + 1448316b9 + 446112b8 - 290538b7 - 484480b6 - 329119b5 - 180223b4 - 462330b3 - 783919b2 + 6812758*b1 - 21052172) / 9
\(\nu^{15}\)	\(=\)	\( ( 1040952 \beta_{15} + 747282 \beta_{14} + 171444 \beta_{13} - 1095524 \beta_{12} - 890919 \beta_{11} + \cdots - 58081098 ) / 9 \) (1040952b15 + 747282b14 + 171444b13 - 1095524b12 - 890919b11 - 1384554b10 - 429482b9 - 329905b8 - 1994865b7 - 2182602b6 + 159816b5 + 3290652b4 - 945129b3 - 1113372b2 + 67829142*b1 - 58081098) / 9

\(n\)	\(217\)	\(271\)	\(541\)	\(1001\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(-1 + \beta_{1}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 361.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( T^{16} \) T^16
$5$	\( (T^{2} - 5 T + 25)^{8} \) (T^2 - 5*T + 25)^8
$7$	\( T^{16} + \cdots + 12\!\cdots\!00 \) T^16 + 31T^15 + 1728T^14 + 23873T^13 + 1128233T^12 + 11747538T^11 + 497848285T^10 + 3057600289T^9 + 133296163569T^8 + 497019824830T^7 + 25797072685081T^6 + 19754485979625T^5 + 2711642195751296T^4 - 3981029622217945T^3 + 207957531729169485T^2 - 486082110166043300*T + 1285774702470403600
$11$	\( T^{16} + \cdots + 16\!\cdots\!00 \) T^16 + 41T^15 + 6417T^14 + 251032T^13 + 27234596T^12 + 938149056T^11 + 57768444640T^10 + 1180835285504T^9 + 60245532050976T^8 + 793708718888960T^7 + 44184446167685440T^6 + 160166858271897600T^5 + 14852868225533772800T^4 - 116398904677536224000T^3 + 4512690492648016032000T^2 - 25887649354568934400000*T + 167336853775102935040000
$13$	\( T^{16} + \cdots + 13\!\cdots\!00 \) T^16 - 16T^15 + 8712T^14 + 91696T^13 + 56427344T^12 + 381160224T^11 + 119728195552T^10 + 1637199545600T^9 + 184285293450624T^8 + 1486207323848960T^7 + 94279486002714880T^6 - 17892283473216000T^5 + 33247068003074105600T^4 + 29739622569916672000T^3 + 2485696859279009280000T^2 - 5050730538756812800000*T + 139414249354494976000000
$17$	\( (T^{8} + \cdots + 60216728526304)^{2} \) (T^8 - 79T^7 - 17858T^6 + 1136180T^5 + 82406704T^4 - 3741096928T^3 - 121756294160T^2 + 2410711039568*T + 60216728526304)^2
$19$	\( (T^{8} + \cdots + 103678903400800)^{2} \) (T^8 + 61T^7 - 25178T^6 - 1014104T^5 + 150993832T^4 + 5261220640T^3 - 230315322608T^2 - 4871698451120*T + 103678903400800)^2
$23$	\( T^{16} + \cdots + 69\!\cdots\!44 \) T^16 + 113T^15 + 48582T^14 + 1737481T^13 + 1393586615T^12 + 58876490052T^11 + 15075053142661T^10 - 365054867429557T^9 + 48059874753948315T^8 - 694632921592786624T^7 + 68220140447598993049T^6 - 991971795394513555461T^5 + 63489268169565135121130T^4 - 431012623720969066541177T^3 + 24126731273610135225795303T^2 - 74684963876642911226931058*T + 6986863233190250108879091844
$29$	\( T^{16} + \cdots + 21\!\cdots\!00 \) T^16 + 89T^15 + 55278T^14 - 5106191T^13 + 1891593131T^12 - 80861665104T^11 + 14839117118029T^10 - 290925308148229T^9 + 87507824498949483T^8 - 1192864565558255896T^7 + 94821097566514505401T^6 + 187553127607041819735T^5 + 67087471018028825902874T^4 - 358311127213884001555085T^3 + 1617264855169577058288015T^2 - 2037470587492083566381650*T + 2102835017487563118568900
$31$	\( T^{16} + \cdots + 32\!\cdots\!76 \) T^16 + 68T^15 + 80364T^14 + 6230464T^13 + 4758130112T^12 + 337702980768T^11 + 122831666901664T^10 + 5807357037179264T^9 + 2109045109189283712T^8 + 102917188731964790528T^7 + 11725515640421220114688T^6 + 208843516016555838743040T^5 + 26548688136617508666692864T^4 + 312962264107579647764700160T^3 + 43333779704634477386900438016T^2 - 347315501585014761829565059072*T + 3231268923526625155509219659776
$37$	\( (T^{8} + \cdots - 16\!\cdots\!96)^{2} \) (T^8 + 282T^7 - 143076T^6 - 18363312T^5 + 7656496848T^4 - 128554327584T^3 - 92313816314544T^2 + 7523678612137824*T - 168746749001421696)^2
$41$	\( T^{16} + \cdots + 42\!\cdots\!49 \) T^16 + 252T^15 + 163728T^14 + 29246040T^13 + 15927464718T^12 + 2699052039828T^11 + 769845638273904T^10 + 71752305648807252T^9 + 15829281527344823583T^8 + 1708668942286167671796T^7 + 192411477802835346361680T^6 + 11462054613264717337612116T^5 + 618419555132953652781243750T^4 + 15271546035556606610318626872T^3 + 501602723872687879050218821128T^2 + 3611387367934394383461377265084*T + 429403920803851027770320240755449
$43$	\( T^{16} + \cdots + 57\!\cdots\!44 \) T^16 - 85T^15 + 318741T^14 + 4524580T^13 + 70885305116T^12 + 1163632264320T^11 + 7523602537380880T^10 + 323347962893526368T^9 + 573695994373884308448T^8 + 12050738227577547446912T^7 + 14004334878570171167336896T^6 + 239409940661444142936215040T^5 + 265124536595134335252640413440T^4 + 2170799662051537391620826686720T^3 + 56867896301996797889806849553664T^2 - 318627819475717583389845546241024*T + 5776085414869720776783099962527744
$47$	\( T^{16} + \cdots + 92\!\cdots\!84 \) T^16 - 25T^15 + 566172T^14 + 16566997T^13 + 218115922697T^12 + 10775636171370T^11 + 46127630212821397T^10 + 4973837336172414317T^9 + 7137654816186038198637T^8 + 894853681709736381134894T^7 + 620524794594875939107027981T^6 + 114948792887732479520149188393T^5 + 37656396016491667365734326228796T^4 + 4164878002977388038587002742929195T^3 + 443993555242582930911226348273459257T^2 + 6809913726648722590817879414895895424*T + 92849304848562082284656460077619625984
$53$	\( (T^{8} + \cdots - 54\!\cdots\!44)^{2} \) (T^8 - 564T^7 - 917832T^6 + 570377808T^5 + 214435116864T^4 - 162076054051008T^3 - 2989652824388736T^2 + 10333542290342406912*T - 545272449044221350144)^2
$59$	\( T^{16} + \cdots + 62\!\cdots\!84 \) T^16 + 457T^15 + 915579T^14 + 282357662T^13 + 511221027032T^12 + 136695671418168T^11 + 145639779638371408T^10 + 11222663389492627456T^9 + 20574200999186214049728T^8 + 55683726893252856030592T^7 + 2262194930487445117843547392T^6 - 170866728702231453192302622720T^5 + 99288955563955151639122767322112T^4 + 816286574432082754748550641223680T^3 + 1306311896059568309092680938514456576T^2 - 61347800803993763210921684523737636864*T + 6268981326602431956397792748812369936384
$61$	\( T^{16} + \cdots + 80\!\cdots\!56 \) T^16 - 853T^15 + 1317654T^14 - 838267853T^13 + 1082381679179T^12 - 669426942667536T^11 + 363110975362713061T^10 - 108426743929381131655T^9 + 26817657092667770736123T^8 - 4532443371972391223178568T^7 + 741506661975428587821293089T^6 - 95413723957049229800587515411T^5 + 12541995573183997422349241433722T^4 - 1134730060029856024844201977584311T^3 + 87469965571911219051367522875157935T^2 - 3061386876461580231299383713980931694*T + 80749343297938314853058742911808776356
$67$	\( T^{16} + \cdots + 21\!\cdots\!01 \) T^16 - 596T^15 + 1148688T^14 - 146795152T^13 + 570373131158T^12 + 19699311452436T^11 + 217134825168502792T^10 + 48557647338821366416T^9 + 49642976395029584831139T^8 + 15053318977201041864519316T^7 + 9162829236631678158715183672T^6 + 2716381143933959426398008274296T^5 + 754402203114938412234293005174022T^4 + 112591883448977530753480825797863480T^3 + 13254684651846356868564139225342940868T^2 + 616599545044194427608901470730566209392*T + 21742162779120696232333956004991653377601
$71$	\( (T^{8} + \cdots + 16\!\cdots\!88)^{2} \) (T^8 - 114T^7 - 1678716T^6 + 105429456T^5 + 522683674992T^4 - 82525405914720T^3 - 12481051303983600T^2 - 13074727768513632*T + 16529781230746812288)^2
$73$	\( (T^{8} + \cdots + 29\!\cdots\!20)^{2} \) (T^8 + 1035T^7 - 313668T^6 - 659110464T^5 - 173206340352T^4 + 38407719877632T^3 + 21586031648243712T^2 + 2483157340396339200*T + 29958324604131409920)^2
$79$	\( T^{16} + \cdots + 25\!\cdots\!00 \) T^16 - 300T^15 + 1828584T^14 - 1703570688T^13 + 2814787953216T^12 - 2314196976259008T^11 + 2448718056017111808T^10 - 1953694832984144914176T^9 + 1513781826472160881396992T^8 - 859309344818227776696265728T^7 + 398070254550505441219200036864T^6 - 131642011405557463383699940749312T^5 + 33478193130428512896600564846084096T^4 - 5720840695470246841172902571954012160T^3 + 717576948256793649560354185333399388160T^2 - 53242752601278168837626397948938300620800*T + 2570368142150575403072645133272353421721600
$83$	\( T^{16} + \cdots + 75\!\cdots\!04 \) T^16 - 155T^15 + 2240616T^14 - 819201445T^13 + 3437716398401T^12 - 1355782101727650T^11 + 3055891054045826005T^10 - 1345282673460036395453T^9 + 1992119576506358875725753T^8 - 791203221870781886527716062T^7 + 757037150673454136763399629761T^6 - 261574025088776374820395346721045T^5 + 193882403590542453345382124031572840T^4 - 50346490594689178692518692291812746875T^3 + 19205235187681917372772666457069348913309T^2 - 390447345504824481845715146655779915660756*T + 7575827807579016064991171008516153821861904
$89$	\( (T^{8} + \cdots - 54\!\cdots\!42)^{2} \) (T^8 - 2571T^7 + 81759T^6 + 3927363273T^5 - 2671759314027T^4 + 134221938788511T^3 + 233098701379376805T^2 - 21266006155729491357*T - 5493570290889835450842)^2
$97$	\( T^{16} + \cdots + 39\!\cdots\!04 \) T^16 - 283T^15 + 4291215T^14 + 837658378T^13 + 12417782590808T^12 + 2666034882794280T^11 + 18468510372693634576T^10 + 4462334686938290652800T^9 + 19884369026023925475054528T^8 + 3797528506041831776828067968T^7 + 11625618685258619207752150186240T^6 + 1833755375501118923141125997211648T^5 + 4639010046357077442783942277735291904T^4 + 522965904019866633558507188210136266752T^3 + 60591047061828743744800107961454256365568T^2 - 48857051182913954089217199320501843304448*T + 39673588788228524972963565378680751407104
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