LMFDB - Newform orbit 1080.4.q.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,4,Mod(361,1080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [16,0,0,0,40,0,-34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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} - 6 x^{15} + 16 x^{14} + 14 x^{13} - 284 x^{12} + 764 x^{11} + 19770 x^{10} + 55106 x^{9} + \cdots + 193472540143 $ x^16 - 6x^15 + 16x^14 + 14x^13 - 284x^12 + 764x^11 + 19770x^10 + 55106x^9 + 266991x^8 + 335444x^7 - 4573302x^6 + 2211506x^5 + 251845642x^4 + 1438820210x^3 + 12022873630x^2 + 75245140836*x + 193472540143
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{14} $
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_{5} - 4 \beta_1) q^{7} + ( - \beta_{13} + \beta_{2} + 8 \beta_1) q^{11} + (\beta_{12} - \beta_{8} + 6 \beta_1 - 6) q^{13} + ( - \beta_{14} - \beta_{9} + \beta_{5} - 13) q^{17}+ \cdots + (\beta_{14} - 7 \beta_{13} + \cdots + 30 \beta_1) q^{97}+O(q^{100}) $ q + (-5b1 + 5) q^5 + (b5 - 4b1) q^7 + (-b13 + b2 + 8b1) q^11 + (b12 - b8 + 6b1 - 6) q^13 + (-b14 - b9 + b5 - 13) * q^17 + (-b15 - b8 - b7 + b4 + 29) * q^19 + (-b15 - b13 - b12 - b10 + b9 + b8 + b6 + b3 - 21b1 + 21) q^23 - 25b1 q^25 + (-b14 + 2b13 + 2b12 - b11 - 2b10 + 2b7 - b6 + b4 + b3 - 2b2 + 14b1) * q^29 + (-2b13 + 2b12 - 3b10 + 2b9 - 2b8 + b6 + 21b1 - 21) * q^31 + (-5b9 + 5b5 - 20) * q^35 + (2b15 + 4b14 + 3b9 - b8 - b7 - 3b5 + 2b4 + 3b2 + 5) * q^37 + (-5b15 + 4b13 + 4b11 - 6b10 + 5b9 + b6 + 5b3 - 35b1 + 35) q^41 + (-4b14 + b13 - 3b12 - 4b11 - 2b10 + 2b7 - b6 - b5 + b4 - 5b3 - b2 + 20b1) q^43 + (6b13 - b12 - b10 + b7 + 4b6 - 4b4 + b3 - 6b2 + 73b1) q^47 + (7b15 - 5b13 - 7b12 - 3b11 - 3b10 - 3b9 + 7b8 + b6 - 7b3 + 140b1 - 140) q^49 + (-2b14 - 9b9 + 4b8 + 3b7 + 9b5 + 4b4 - 5b2 - 43) q^53 + (5b2 + 40) q^55 + (5b15 - 6b13 - 5b12 + 10b10 + 8b9 + 5b8 + 4b6 - 5b3 - 2b1 + 2) q^59 + (7b14 - 3b13 + 4b12 + 7b11 - 5b10 + 5b7 - 4b6 - 6b5 + 4b4 + 2b3 + 3b2 - 119b1) * q^61 + (5b12 + 30b1) * q^65 + (-2b15 + 3b13 + 8b12 - 4b11 + 2b10 + 15b9 - 8b8 + 6b6 + 2b3 + 54b1 - 54) * q^67 + (-4b14 - 20b9 - 2b8 + 9b7 + 20b5 - 3b4 - 6b2 + 15) q^71 + (2b15 + b14 + 19b9 - 9b8 + 2b7 - 19b5 + 4b4 + 8b2 + 15) q^73 + (2b15 + 9b13 - 7b12 - 2b11 + 5b10 + 19b9 + 7b8 + 10b6 - 2b3 + 89b1 - 89) * q^77 + (4b14 + 3b13 + 7b12 + 4b11 - b10 + b7 - b6 - 2b5 + b4 + 3b3 - 3b2 - 97b1) * q^79 + (12b14 + 3b13 - 6b12 + 12b11 - 13b10 + 13b7 + 16b6 - 2b5 - 16b4 + 8b3 - 3b2 + 165b1) * q^83 + (5b11 - 5b9 + 65b1 - 65) q^85 + (-10b15 - b14 - b9 + 5b8 + b5 + 8b4 + 2b2 + 18) * q^89 + (3b15 - 16b9 + 15b8 + 10b7 + 16b5 + 6b4 + 3b2 - 70) q^91 + (-5b15 + 5b12 - 5b10 - 5b8 + 5b6 + 5b3 - 145b1 + 145) q^95 + (b14 - 7b13 - 15b12 + b11 - 5b10 + 5b7 + 4b6 + 16b5 - 4b4 - 6b3 + 7b2 + 30b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 40 q^{5} - 34 q^{7} + 64 q^{11} - 48 q^{13} - 212 q^{17} + 456 q^{19} + 166 q^{23} - 200 q^{25} + 110 q^{29} - 160 q^{31} - 340 q^{35} + 104 q^{37} + 280 q^{41} + 136 q^{43} + 594 q^{47} - 1094 q^{49}+ \cdots + 182 q^{97}+O(q^{100}) $ 16 * q + 40 * q^5 - 34 * q^7 + 64 * q^11 - 48 * q^13 - 212 * q^17 + 456 * q^19 + 166 * q^23 - 200 * q^25 + 110 * q^29 - 160 * q^31 - 340 * q^35 + 104 * q^37 + 280 * q^41 + 136 * q^43 + 594 * q^47 - 1094 * q^49 - 752 * q^53 + 640 * q^55 + 24 * q^59 - 950 * q^61 + 240 * q^65 - 426 * q^67 + 136 * q^71 + 308 * q^73 - 696 * q^77 - 764 * q^79 + 1362 * q^83 - 530 * q^85 + 172 * q^89 - 1224 * q^91 + 1140 * q^95 + 182 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 6 x^{15} + 16 x^{14} + 14 x^{13} - 284 x^{12} + 764 x^{11} + 19770 x^{10} + 55106 x^{9} + \cdots + 193472540143 $ x^16 - 6*x^15 + 16*x^14 + 14*x^13 - 284*x^12 + 764*x^11 + 19770*x^10 + 55106*x^9 + 266991*x^8 + 335444*x^7 - 4573302*x^6 + 2211506*x^5 + 251845642*x^4 + 1438820210*x^3 + 12022873630*x^2 + 75245140836*x + 193472540143 :

$\beta_{1}$	$=$	$ ( 76 \nu^{15} - 77258 \nu^{14} + 527696 \nu^{13} + 274645 \nu^{12} - 16218838 \nu^{11} + \cdots + 900219652474679 ) / 11\!\cdots\!80 $ (76v^15 - 77258v^14 + 527696v^13 + 274645v^12 - 16218838v^11 + 99465007v^10 - 689327126v^9 + 2767453548v^8 - 7459039014v^7 + 3034174127v^6 + 80154392282v^5 + 371926108804v^4 - 4917806845786v^3 - 16882312431899v^2 + 25271278170362*v + 900219652474679) / 1157542685443980
$\beta_{2}$	$=$	$ ( - 157 \nu^{15} - 2095 \nu^{14} + 70300 \nu^{13} - 694036 \nu^{12} + 3516229 \nu^{11} + \cdots - 20917892904731 ) / 2301277704660 $ (-157v^15 - 2095v^14 + 70300v^13 - 694036v^12 + 3516229v^11 - 10138909v^10 - 397135v^9 + 122360205v^8 - 483221991v^7 + 1671657439v^6 - 10951325459v^5 + 35205089v^4 + 9763868746v^3 + 127664113826v^2 + 2442618374617*v - 20917892904731) / 2301277704660
$\beta_{3}$	$=$	$ ( - 13268 \nu^{15} + 321410 \nu^{14} - 2099393 \nu^{13} - 5510131 \nu^{12} + \cdots - 986312262228999 ) / 128615853938220 $ (-13268v^15 + 321410v^14 - 2099393v^13 - 5510131v^12 + 98024905v^11 - 454166437v^10 + 2309563812v^9 - 4601027166v^8 + 40170974577v^7 - 398750405437v^6 + 176004089992v^5 - 2285896513702v^4 + 10290367302673v^3 - 5711144014591v^2 + 666244296174565*v - 986312262228999) / 128615853938220
$\beta_{4}$	$=$	$ ( - 91 \nu^{15} + 887 \nu^{14} + 4318 \nu^{13} - 55384 \nu^{12} + 174931 \nu^{11} + \cdots + 5389336662397 ) / 383546284110 $ (-91v^15 + 887v^14 + 4318v^13 - 55384v^12 + 174931v^11 - 708265v^10 + 3610379v^9 + 16784391v^8 - 74074431v^7 + 480255805v^6 + 460296745v^5 - 14839616899v^4 - 32679237062v^3 + 426265088180v^2 + 2571419101291*v + 5389336662397) / 383546284110
$\beta_{5}$	$=$	$ ( 299945 \nu^{15} - 868339 \nu^{14} - 6297611 \nu^{13} + 81959675 \nu^{12} + \cdots + 24\!\cdots\!58 ) / 11\!\cdots\!80 $ (299945v^15 - 868339v^14 - 6297611v^13 + 81959675v^12 - 591062666v^11 + 2349441182v^10 + 2504269283v^9 + 42737622861v^8 - 97308355458v^7 + 941554338478v^6 - 9102751657241v^5 + 36963072251849v^4 + 28208037813313v^3 + 628873065589517v^2 + 4155480908460490*v + 24680020679939758) / 1157542685443980
$\beta_{6}$	$=$	$ ( 360499 \nu^{15} - 996851 \nu^{14} + 10938140 \nu^{13} - 71674496 \nu^{12} + \cdots + 46\!\cdots\!45 ) / 11\!\cdots\!80 $ (360499v^15 - 996851v^14 + 10938140v^13 - 71674496v^12 + 167434325v^11 - 1016227049v^10 + 16923057913v^9 + 57330215493v^8 + 73630746489v^7 + 1056169695011v^6 - 1405556159779v^5 - 23975405638799v^4 + 85095647791154v^3 + 1518060945727258v^2 + 15866763636336245*v + 46578585785987645) / 1157542685443980
$\beta_{7}$	$=$	$ ( - 434 \nu^{15} + 6094 \nu^{14} - 34564 \nu^{13} + 101911 \nu^{12} + 161186 \nu^{11} + \cdots + 4387912744523 ) / 1150638852330 $ (-434v^15 + 6094v^14 - 34564v^13 + 101911v^12 + 161186v^11 - 2836919v^10 + 4123678v^9 + 48886692v^8 - 308968368v^7 + 1361421659v^6 + 6481959746v^5 - 50745319544v^4 + 63119618042v^3 + 219068279461v^2 - 6101849944786*v + 4387912744523) / 1150638852330
$\beta_{8}$	$=$	$ ( - 449 \nu^{15} + 7087 \nu^{14} - 44152 \nu^{13} + 131047 \nu^{12} - 509443 \nu^{11} + \cdots + 2515559042558 ) / 1150638852330 $ (-449v^15 + 7087v^14 - 44152v^13 + 131047v^12 - 509443v^11 + 137224v^10 - 2183141v^9 + 34938861v^8 + 12537705v^7 - 588061594v^6 - 2385787741v^5 - 56727985013v^4 + 12787225922v^3 + 644206273519v^2 + 56591298119*v + 2515559042558) / 1150638852330
$\beta_{9}$	$=$	$ ( - 165770 \nu^{15} + 2137864 \nu^{14} - 17687509 \nu^{13} + 85202449 \nu^{12} + \cdots - 71\!\cdots\!89 ) / 385847561814660 $ (-165770v^15 + 2137864v^14 - 17687509v^13 + 85202449v^12 - 187968733v^11 + 104265901v^10 - 2963904770v^9 + 9684170808v^8 - 155614400709v^7 + 416302161389v^6 - 445550925850v^5 + 3048278070592v^4 + 5482113154661v^3 - 211680188712059v^2 - 2494134053225125*v - 7199949853631689) / 385847561814660
$\beta_{10}$	$=$	$ ( - 553543 \nu^{15} + 7632611 \nu^{14} - 53749274 \nu^{13} + 204757187 \nu^{12} + \cdots + 11\!\cdots\!82 ) / 11\!\cdots\!80 $ (-553543v^15 + 7632611v^14 - 53749274v^13 + 204757187v^12 + 2686531v^11 - 4982282404v^10 + 23383263461v^9 - 37121444187v^8 - 127659621405v^7 + 1574113480540v^6 - 945870989171v^5 - 15220983874651v^4 + 57078573489094v^3 - 150465173702023v^2 - 1794098900431625*v + 11411114397347182) / 1157542685443980
$\beta_{11}$	$=$	$ ( 572522 \nu^{15} - 10044223 \nu^{14} + 65299159 \nu^{13} - 146021446 \nu^{12} + \cdots + 12\!\cdots\!71 ) / 11\!\cdots\!80 $ (572522v^15 - 10044223v^14 + 65299159v^13 - 146021446v^12 + 451497547v^11 - 2010588637v^10 + 8581629500v^9 - 55519483767v^8 + 136789491291v^7 - 526025875325v^6 + 7383051627580v^5 + 60710337838121v^4 + 225001154268001v^3 - 860652338287174v^2 - 178186114762091*v + 12782313923398471) / 1157542685443980
$\beta_{12}$	$=$	$ ( 324121 \nu^{15} - 1493588 \nu^{14} + 1065959 \nu^{13} - 24751322 \nu^{12} + \cdots + 87\!\cdots\!50 ) / 578771342721990 $ (324121v^15 - 1493588v^14 + 1065959v^13 - 24751322v^12 - 1675108v^11 + 1409458090v^10 + 2280623059v^9 + 30017318730v^8 + 38230415472v^7 - 799604405842v^6 - 397721416645v^5 - 3959209481504v^4 + 219215004606611v^3 + 1102762896910468v^2 + 4128070554045620*v + 8790081047230850) / 578771342721990
$\beta_{13}$	$=$	$ ( 100305 \nu^{15} - 1188719 \nu^{14} + 7769075 \nu^{13} - 25953113 \nu^{12} + \cdots + 51\!\cdots\!26 ) / 128615853938220 $ (100305v^15 - 1188719v^14 + 7769075v^13 - 25953113v^12 + 21350948v^11 + 120125050v^10 + 804979295v^9 + 2719640481v^8 + 8221187316v^7 + 147441628338v^6 - 942069603289v^5 + 75815277937v^4 + 29912974802783v^3 - 61574862417947v^2 + 1178272029488396*v + 5165462484944026) / 128615853938220
$\beta_{14}$	$=$	$ ( - 1925 \nu^{15} + 20389 \nu^{14} - 84724 \nu^{13} + 219193 \nu^{12} - 1376377 \nu^{11} + \cdots - 28168704583270 ) / 2301277704660 $ (-1925v^15 + 20389v^14 - 84724v^13 + 219193v^12 - 1376377v^11 + 1113724v^10 - 23194757v^9 - 9829893v^8 + 134931999v^7 + 1305606536v^6 + 13491262919v^5 - 123016201937v^4 - 808254162520v^3 - 2936589815021v^2 - 5876193558805*v - 28168704583270) / 2301277704660
$\beta_{15}$	$=$	$ ( - 2747 \nu^{15} + 27019 \nu^{14} - 111784 \nu^{13} + 236452 \nu^{12} + \cdots - 82295219611021 ) / 2301277704660 $ (-2747v^15 + 27019v^14 - 111784v^13 + 236452v^12 - 136273v^11 + 3173065v^10 - 50629037v^9 + 8230767v^8 + 338818683v^7 - 2244680935v^6 + 4680969755v^5 - 9098655653v^4 - 64811663914v^3 - 1001518790930v^2 - 754247631973*v - 82295219611021) / 2301277704660

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

$\nu$	$=$	$ ( \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + 2\beta _1 + 2 ) / 9 $ (b9 - b7 + b6 - b5 + 2*b1 + 2) / 9
$\nu^{2}$	$=$	$ ( - 3 \beta_{14} - 3 \beta_{13} + \beta_{10} - 2 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + \beta_{6} + \cdots + 3 ) / 9 $ (-3b14 - 3b13 + b10 - 2b9 + 3b8 - 2b7 + b6 - 2b5 + b4 - 3*b3 + 3) / 9
$\nu^{3}$	$=$	$ ( 6 \beta_{15} + 3 \beta_{14} + 9 \beta_{13} + 15 \beta_{12} + 12 \beta_{11} + 8 \beta_{10} + 6 \beta_{9} + \cdots + 2 ) / 9 $ (6b15 + 3b14 + 9b13 + 15b12 + 12b11 + 8b10 + 6b9 + 9b8 - 2b7 - 6b6 - 5b5 - b4 - 6b3 + 3b2 - 139b1 + 2) / 9
$\nu^{4}$	$=$	$ ( 6 \beta_{15} + 81 \beta_{14} - 75 \beta_{13} + 78 \beta_{12} + 84 \beta_{11} + 59 \beta_{10} - 16 \beta_{9} + \cdots - 18 ) / 9 $ (6b15 + 81b14 - 75b13 + 78b12 + 84b11 + 59b10 - 16b9 - 51b8 - 52b7 - 16b6 + 125b5 - 22b4 - 18b3 + 66b2 + 390*b1 - 18) / 9
$\nu^{5}$	$=$	$ ( - 90 \beta_{15} + 525 \beta_{14} - 294 \beta_{13} + 594 \beta_{12} + 216 \beta_{11} + 253 \beta_{10} + \cdots - 2098 ) / 9 $ (-90b15 + 525b14 - 294b13 + 594b12 + 216b11 + 253b10 - 4b9 - 579b8 - 12b7 + 173b6 - 300b5 - 98b4 - 132b3 + 45b2 + 5990*b1 - 2098) / 9
$\nu^{6}$	$=$	$ 53 \beta_{15} + 104 \beta_{14} + 31 \beta_{13} + 200 \beta_{12} - 194 \beta_{11} + 449 \beta_{10} + \cdots - 8976 $ 53b15 + 104b14 + 31b13 + 200b12 - 194b11 + 449b10 - 332b9 - 328b8 + 8b7 + 45b6 - 140b5 - 222b4 - 319b3 + 94b2 + 3563*b1 - 8976
$\nu^{7}$	$=$	$ ( 7794 \beta_{15} + 1854 \beta_{14} - 234 \beta_{13} + 5310 \beta_{12} - 7686 \beta_{11} + 24687 \beta_{10} + \cdots - 944836 ) / 9 $ (7794b15 + 1854b14 - 234b13 + 5310b12 - 7686b11 + 24687b10 - 46637b9 - 6165b8 + 15326b7 - 5687b6 + 11510b5 - 22014b4 - 8109b3 + 4302b2 + 477011*b1 - 944836) / 9
$\nu^{8}$	$=$	$ ( 20754 \beta_{15} + 6171 \beta_{14} + 14856 \beta_{13} + 31230 \beta_{12} - 26802 \beta_{11} + \cdots - 6702432 ) / 9 $ (20754b15 + 6171b14 + 14856b13 + 31230b12 - 26802b11 + 98020b10 - 186599b9 - 41505b8 + 129364b7 - 85652b6 + 133081b5 - 38348b4 + 26016b3 + 15966b2 + 4355091*b1 - 6702432) / 9
$\nu^{9}$	$=$	$ ( 68280 \beta_{15} - 5433 \beta_{14} + 243234 \beta_{13} + 166245 \beta_{12} - 258018 \beta_{11} + \cdots - 22179121 ) / 9 $ (68280b15 - 5433b14 + 243234b13 + 166245b12 - 258018b11 + 486539b10 - 738444b9 - 578475b8 + 518992b7 - 826377b6 + 662323b5 + 396899b4 + 316695b3 - 278682b2 + 11919254*b1 - 22179121) / 9
$\nu^{10}$	$=$	$ ( 1268295 \beta_{15} - 1086111 \beta_{14} + 1811661 \beta_{13} + 173922 \beta_{12} - 2973552 \beta_{11} + \cdots - 23530851 ) / 9 $ (1268295b15 - 1086111b14 + 1811661b13 + 173922b12 - 2973552b11 + 266333b10 - 2260546b9 - 4557849b8 + 2961227b7 - 3058702b6 + 1619117b5 + 1776533b4 + 1374084b3 - 2271543b2 + 26200398*b1 - 23530851) / 9
$\nu^{11}$	$=$	$ ( 7432299 \beta_{15} - 15548703 \beta_{14} + 5758248 \beta_{13} - 13334823 \beta_{12} - 14396688 \beta_{11} + \cdots + 125828675 ) / 9 $ (7432299b15 - 15548703b14 + 5758248b13 - 13334823b12 - 14396688b11 - 10084760b10 - 3116497b9 - 16270653b8 + 18591360b7 + 2611451b6 - 342228b5 + 273124b4 + 7881429b3 - 9105453b2 - 49047583*b1 + 125828675) / 9
$\nu^{12}$	$=$	$ 4341634 \beta_{15} - 12547142 \beta_{14} + 2628071 \beta_{13} - 16316312 \beta_{12} - 1659250 \beta_{11} + \cdots + 56401067 $ 4341634b15 - 12547142b14 + 2628071b13 - 16316312b12 - 1659250b11 - 14759435b10 + 3706904b9 + 1805896b8 + 12717529b7 + 7086690b6 + 2672675b5 + 1155120b4 + 5358088b3 - 8344117b2 - 160802273*b1 + 56401067
$\nu^{13}$	$=$	$ ( 184488633 \beta_{15} - 396018414 \beta_{14} + 109709091 \beta_{13} - 911935980 \beta_{12} + \cdots + 13070072981 ) / 9 $ (184488633b15 - 396018414b14 + 109709091b13 - 911935980b12 + 360331434b11 - 1273269249b10 + 475622317b9 + 429904809b8 + 676338857b7 + 330792949b6 + 521842598b5 + 311611329b4 + 316491120b3 - 351916470b2 - 22546685257*b1 + 13070072981) / 9
$\nu^{14}$	$=$	$ ( 418349322 \beta_{15} + 295880235 \beta_{14} - 237851085 \beta_{13} - 4519304190 \beta_{12} + \cdots + 184026481212 ) / 9 $ (418349322b15 + 295880235b14 - 237851085b13 - 4519304190b12 + 3502722060b11 - 10010410253b10 + 5749013527b9 + 2636358447b8 + 3449286601b7 + 3499086529b6 + 4236748495b5 + 1157528161b4 + 2047026525b3 - 831578787b2 - 229072310730*b1 + 184026481212) / 9
$\nu^{15}$	$=$	$ ( - 7412079243 \beta_{15} + 10053860277 \beta_{14} - 7027379802 \beta_{13} - 22505590668 \beta_{12} + \cdots + 1351798631138 ) / 9 $ (-7412079243b15 + 10053860277b14 - 7027379802b13 - 22505590668b12 + 20743751136b11 - 66521645911b10 + 47706319689b9 + 15443262648b8 + 12611040631b7 + 36574557645b6 + 15231632098b5 - 7309537525b4 + 9259139055b3 + 2315812836b2 - 1589846574910*b1 + 1351798631138) / 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

2.04823	+	5.08932i
−2.94096	−	3.38657i
5.28147	−	2.94432i
−3.70856	+	2.19760i
5.78315	+	2.03015i
−0.887205	+	4.84133i
1.53665	−	5.16837i
−4.11278	−	0.927101i
2.04823	−	5.08932i
−2.94096	+	3.38657i
5.28147	+	2.94432i
−3.70856	−	2.19760i
5.78315	−	2.03015i
−0.887205	−	4.84133i
1.53665	+	5.16837i
−4.11278	+	0.927101i

2.50000

−

4.33013i

−17.3532

−

30.0566i

361.2

2.50000

−

4.33013i

−15.9276

−

27.5874i

361.3

2.50000

−

4.33013i

−5.91307

−

10.2417i

361.4

2.50000

−

4.33013i

−3.79996

−

6.58173i

361.5

2.50000

−

4.33013i

−2.04790

−

3.54707i

361.6

2.50000

−

4.33013i

4.29040

7.43119i

361.7

2.50000

−

4.33013i

6.84527

11.8564i

361.8

2.50000

−

4.33013i

16.9061

29.2822i

721.1

2.50000

4.33013i

−17.3532

30.0566i

721.2

2.50000

4.33013i

−15.9276

27.5874i

721.3

2.50000

4.33013i

−5.91307

10.2417i

721.4

2.50000

4.33013i

−3.79996

6.58173i

721.5

2.50000

4.33013i

−2.04790

3.54707i

721.6

2.50000

4.33013i

4.29040

−

7.43119i

721.7

2.50000

4.33013i

6.84527

−

11.8564i

721.8

2.50000

4.33013i

16.9061

−

29.2822i

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.b		16
3.b	odd	2	1		360.4.q.b	✓	16
9.c	even	3	1	inner	1080.4.q.b		16
9.d	odd	6	1		360.4.q.b	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
360.4.q.b	✓	16	3.b	odd	2	1
360.4.q.b	✓	16	9.d	odd	6	1
1080.4.q.b		16	1.a	even	1	1	trivial
1080.4.q.b		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} + 34 T_{7}^{15} + 2497 T_{7}^{14} + 48838 T_{7}^{13} + 3188398 T_{7}^{12} + \cdots + 26\!\cdots\!64 $ T7^16 + 34*T7^15 + 2497*T7^14 + 48838*T7^13 + 3188398*T7^12 + 58700818*T7^11 + 2201352109*T7^10 + 16632083158*T7^9 + 504023304658*T7^8 + 3890250850074*T7^7 + 79139175552405*T7^6 + 406499392754838*T7^5 + 5373493850910393*T7^4 + 26537771865843360*T7^3 + 256746621319921296*T7^2 + 787513253234213376*T7 + 2613380713166686464 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 + 26\!\cdots\!64 $ T^16 + 34T^15 + 2497T^14 + 48838T^13 + 3188398T^12 + 58700818T^11 + 2201352109T^10 + 16632083158T^9 + 504023304658T^8 + 3890250850074T^7 + 79139175552405T^6 + 406499392754838T^5 + 5373493850910393T^4 + 26537771865843360T^3 + 256746621319921296T^2 + 787513253234213376*T + 2613380713166686464
$11$	$ T^{16} + \cdots + 57\!\cdots\!64 $ T^16 - 64T^15 + 7273T^14 - 339064T^13 + 29214925T^12 - 1246959124T^11 + 63417918196T^10 - 1679483680192T^9 + 56209073475520T^8 - 1236179215633248T^7 + 32508897368710752T^6 - 494249259328399296T^5 + 7003814038424442432T^4 - 40793971557057479424T^3 + 249362606277104375808T^2 + 443858513869566443520*T + 5722542772348736196864
$13$	$ T^{16} + \cdots + 89\!\cdots\!44 $ T^16 + 48T^15 + 8664T^14 + 70128T^13 + 43131024T^12 + 676716768T^11 + 66688632288T^10 - 145085230080T^9 + 42695537228928T^8 - 253387123382016T^7 + 20903880536112384T^6 - 208667120684244480T^5 + 4428006853486678272T^4 - 15319026425336408064T^3 + 211729930018906189824T^2 - 78122622703191883776*T + 8991985545268306182144
$17$	$ (T^{8} + \cdots - 1974672949104)^{2} $ (T^8 + 106T^7 - 8259T^6 - 1309460T^5 - 20518556T^4 + 2864571144T^3 + 133003111680T^2 + 1610149733856*T - 1974672949104)^2
$19$	$ (T^{8} + \cdots + 5438892407536)^{2} $ (T^8 - 228T^7 + 4091T^6 + 1865688T^5 - 68406528T^4 - 4106026200T^3 + 82517584352T^2 + 1586325374496*T + 5438892407536)^2
$23$	$ T^{16} + \cdots + 79\!\cdots\!64 $ T^16 - 166T^15 + 45565T^14 - 3545818T^13 + 907960138T^12 - 81443234326T^11 + 9247040901973T^10 - 527364471329506T^9 + 39885962735049550T^8 - 2117849401909171350T^7 + 114024181539233325669T^6 - 3921929266645369188162T^5 + 105318095232765542265981T^4 - 1827955024752865691702016T^3 + 23513819471125118949549348T^2 - 165728040540867731001466944*T + 792528416815727764030562064
$29$	$ T^{16} + \cdots + 76\!\cdots\!56 $ T^16 - 110T^15 + 80929T^14 - 1174970T^13 + 3758369038T^12 + 21161453974T^11 + 98781131154565T^10 + 7100726682113074T^9 + 1540080118885305934T^8 + 132785915590401853194T^7 + 18963375271216255389261T^6 + 1601409399678929088183114T^5 + 122939151457634961537447045T^4 + 5930848240068093064043190300T^3 + 227504714824490338958623400700T^2 + 4936952733335042071181985864432*T + 76398544327160238101844788944656
$31$	$ T^{16} + \cdots + 29\!\cdots\!76 $ T^16 + 160T^15 + 106372T^14 + 8917360T^13 + 7112385664T^12 + 694828156768T^11 + 183872925161632T^10 + 16434956972547712T^9 + 3187661758930108288T^8 + 263208329545043070720T^7 + 29653284205785415104768T^6 + 1625442980063210285188608T^5 + 131037490169846531614008576T^4 + 6165382028633512517936480256T^3 + 369801823163826500791628276736T^2 + 10354531415454459814343730462720*T + 292533131580785400972855951691776
$37$	$ (T^{8} + \cdots + 55\!\cdots\!24)^{2} $ (T^8 - 52T^7 - 183012T^6 - 605368T^5 + 7702035136T^4 + 60570129120T^3 - 80327279213424T^2 - 2671603756636032*T + 55340712289108224)^2
$41$	$ T^{16} + \cdots + 52\!\cdots\!49 $ T^16 - 280T^15 + 448132T^14 - 96411712T^13 + 129944576290T^12 - 27558477184024T^11 + 19514263381784368T^10 - 3725858907539071072T^9 + 2033129392668611589595T^8 - 357256765089602681399016T^7 + 98894092863467786746001136T^6 - 9616325324318857550354566608T^5 + 1976169601735815352408310849490T^4 - 188291550491252782235322952063248T^3 + 25337720172612299228167638408097860T^2 - 1198233555300589629325231022058935136*T + 52794770246961604752809990012704952049
$43$	$ T^{16} + \cdots + 94\!\cdots\!44 $ T^16 - 136T^15 + 455881T^14 - 30426400T^13 + 145435124725T^12 - 7544085784012T^11 + 20907031003799116T^10 + 784064298350385440T^9 + 2018477708781867480784T^8 + 67984041268541227656192T^7 + 98037338726573822212989984T^6 + 6555005701967811633858741312T^5 + 3124595709547625249592538682304T^4 + 51589247430482910764079752371968T^3 + 1343042337980806330724172430573824T^2 - 8665329073736864056896671840460288*T + 94724919651659558892826993032409344
$47$	$ T^{16} + \cdots + 42\!\cdots\!36 $ T^16 - 594T^15 + 520573T^14 - 65617806T^13 + 63299475610T^12 + 2181729857598T^11 + 8646618044591485T^10 + 1051642049080704930T^9 + 214836545453673488230T^8 + 11378006086228424010006T^7 + 1685632336809777922028233T^6 + 58902244047528408829072674T^5 + 9791091243964080931235760361T^4 + 150421570433860382178720336000T^3 + 23099858514807864860909333263360T^2 + 94804194354927651419216818864128*T + 42167652362175646538512980310491136
$53$	$ (T^{8} + \cdots + 14\!\cdots\!96)^{2} $ (T^8 + 376T^7 - 373776T^6 - 49904000T^5 + 38686750624T^4 - 1038464493696T^3 - 483710269816320T^2 + 6911131952461824*T + 1442422409646295296)^2
$59$	$ T^{16} + \cdots + 78\!\cdots\!76 $ T^16 - 24T^15 + 1282545T^14 + 295206048T^13 + 1119294468333T^12 + 298168467089220T^11 + 561030782528170116T^10 + 184016251086515245296T^9 + 203642986430317623316128T^8 + 55677878016302788182379584T^7 + 39607512627691102231584449856T^6 + 8315553502815415869113269463808T^5 + 5052562674564902998411344528354816T^4 + 729774419993245579725589363918046208T^3 + 263475702719507370327130474311072678912T^2 - 12737379323148777783285158212115936956416*T + 782284191142452965828886499926701993496576
$61$	$ T^{16} + \cdots + 31\!\cdots\!36 $ T^16 + 950T^15 + 1510585T^14 + 930682034T^13 + 1094690311750T^12 + 602557878178274T^11 + 487137500421297349T^10 + 182994258156244132910T^9 + 109654526275122721523974T^8 + 33023725583183812073484870T^7 + 16328982267095636705291048421T^6 + 2872341860730181711333282137222T^5 + 543094580663455126830843920389605T^4 - 6387146551676352977448072173237052T^3 + 97028673278712046233723672018049980T^2 + 164061415716043625589094874735775120*T + 319214228022522669542479316939889936
$67$	$ T^{16} + \cdots + 22\!\cdots\!21 $ T^16 + 426T^15 + 1378714T^14 + 682376664T^13 + 1269467332516T^12 + 616122720480546T^11 + 695309228264589316T^10 + 325234267782030569694T^9 + 274639991225218096579057T^8 + 116089391155501193735250714T^7 + 68580742154391702723041049292T^6 + 24347081687034489151482983895606T^5 + 11276374384823509248976032039523180T^4 + 3283866047350319978302337655472379808T^3 + 1006988691505133222207459108561200287478T^2 + 158332788214857075069574504167152724507030*T + 22266216504137602592782177591977530290233121
$71$	$ (T^{8} + \cdots - 47\!\cdots\!36)^{2} $ (T^8 - 68T^7 - 936048T^6 - 25389992T^5 + 160133417392T^4 + 29299026862464T^3 - 288976713623472T^2 - 166180071125774592*T - 4707438249771164736)^2
$73$	$ (T^{8} + \cdots - 10\!\cdots\!92)^{2} $ (T^8 - 154T^7 - 1719639T^6 + 216618992T^5 + 907520755072T^4 - 101797465761024T^3 - 147298700273326080T^2 + 13277903701634629632*T - 108586850696560852992)^2
$79$	$ T^{16} + \cdots + 50\!\cdots\!04 $ T^16 + 764T^15 + 897048T^14 + 266699200T^13 + 254458469120T^12 + 45115361629632T^11 + 54899439527855872T^10 + 3036662638785463040T^9 + 6060185085207126401280T^8 + 11107752572020015877120T^7 + 498923596971851384365871104T^6 - 19651265383850625377258016768T^5 + 9532461103897235732459080318976T^4 + 474729362485777992074557754638336T^3 + 80696107675281068148846148547346432T^2 - 200125743359795701295850386472632320*T + 504288104004072809562341960023343104
$83$	$ T^{16} + \cdots + 41\!\cdots\!16 $ T^16 - 1362T^15 + 4072761T^14 - 3094070454T^13 + 7495957222398T^12 - 4665192586195626T^11 + 9276287481398131605T^10 - 4041772518993161734854T^9 + 7326989596565927669783970T^8 - 2726813236242623971309983954T^7 + 4098532291059658231741099476429T^6 - 1125248290352520505912932344848134T^5 + 1281149088298231707384557914669988025T^4 - 427647378507511766051700602069768768232T^3 + 227262799809265123355506499910525936639408T^2 - 31789053913211598524052885350199254105293440*T + 4108233369986950683603819679301578596370626816
$89$	$ (T^{8} + \cdots + 25\!\cdots\!48)^{2} $ (T^8 - 86T^7 - 1928957T^6 + 392124112T^5 + 1111003730443T^4 - 297575955644606T^3 - 162928649209008563T^2 + 32827201378290455908*T + 2591490799217186564548)^2
$97$	$ T^{16} + \cdots + 25\!\cdots\!44 $ T^16 - 182T^15 + 2367375T^14 + 334861226T^13 + 4003845110501T^12 + 458168561919924T^11 + 2930970616102953076T^10 + 285608869922118537040T^9 + 1555438265658436484735136T^8 + 29695163242475567810836288T^7 + 338861646216758327430644694592T^6 - 59182248030635627696777344663296T^5 + 53683050185973732063900776104240640T^4 - 4164434764466308657420733167465679872T^3 + 428149981214361274578066188896893121536T^2 + 8868014321894563682265460795847482765312*T + 253673147680271264996195338829810182918144
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\( q + ( - 5 \beta_1 + 5) q^{5} + (\beta_{5} - 4 \beta_1) q^{7} + ( - \beta_{13} + \beta_{2} + 8 \beta_1) q^{11} + (\beta_{12} - \beta_{8} + 6 \beta_1 - 6) q^{13} + ( - \beta_{14} - \beta_{9} + \beta_{5} - 13) q^{17}+ \cdots + (\beta_{14} - 7 \beta_{13} + \cdots + 30 \beta_1) q^{97}+O(q^{100}) \) q + (-5b1 + 5) q^5 + (b5 - 4b1) q^7 + (-b13 + b2 + 8b1) q^11 + (b12 - b8 + 6b1 - 6) q^13 + (-b14 - b9 + b5 - 13) * q^17 + (-b15 - b8 - b7 + b4 + 29) * q^19 + (-b15 - b13 - b12 - b10 + b9 + b8 + b6 + b3 - 21b1 + 21) q^23 - 25b1 q^25 + (-b14 + 2b13 + 2b12 - b11 - 2b10 + 2b7 - b6 + b4 + b3 - 2b2 + 14b1) * q^29 + (-2b13 + 2b12 - 3b10 + 2b9 - 2b8 + b6 + 21b1 - 21) * q^31 + (-5b9 + 5b5 - 20) * q^35 + (2b15 + 4b14 + 3b9 - b8 - b7 - 3b5 + 2b4 + 3b2 + 5) * q^37 + (-5b15 + 4b13 + 4b11 - 6b10 + 5b9 + b6 + 5b3 - 35b1 + 35) q^41 + (-4b14 + b13 - 3b12 - 4b11 - 2b10 + 2b7 - b6 - b5 + b4 - 5b3 - b2 + 20b1) q^43 + (6b13 - b12 - b10 + b7 + 4b6 - 4b4 + b3 - 6b2 + 73b1) q^47 + (7b15 - 5b13 - 7b12 - 3b11 - 3b10 - 3b9 + 7b8 + b6 - 7b3 + 140b1 - 140) q^49 + (-2b14 - 9b9 + 4b8 + 3b7 + 9b5 + 4b4 - 5b2 - 43) q^53 + (5b2 + 40) q^55 + (5b15 - 6b13 - 5b12 + 10b10 + 8b9 + 5b8 + 4b6 - 5b3 - 2b1 + 2) q^59 + (7b14 - 3b13 + 4b12 + 7b11 - 5b10 + 5b7 - 4b6 - 6b5 + 4b4 + 2b3 + 3b2 - 119b1) * q^61 + (5b12 + 30b1) * q^65 + (-2b15 + 3b13 + 8b12 - 4b11 + 2b10 + 15b9 - 8b8 + 6b6 + 2b3 + 54b1 - 54) * q^67 + (-4b14 - 20b9 - 2b8 + 9b7 + 20b5 - 3b4 - 6b2 + 15) q^71 + (2b15 + b14 + 19b9 - 9b8 + 2b7 - 19b5 + 4b4 + 8b2 + 15) q^73 + (2b15 + 9b13 - 7b12 - 2b11 + 5b10 + 19b9 + 7b8 + 10b6 - 2b3 + 89b1 - 89) * q^77 + (4b14 + 3b13 + 7b12 + 4b11 - b10 + b7 - b6 - 2b5 + b4 + 3b3 - 3b2 - 97b1) * q^79 + (12b14 + 3b13 - 6b12 + 12b11 - 13b10 + 13b7 + 16b6 - 2b5 - 16b4 + 8b3 - 3b2 + 165b1) * q^83 + (5b11 - 5b9 + 65b1 - 65) q^85 + (-10b15 - b14 - b9 + 5b8 + b5 + 8b4 + 2b2 + 18) * q^89 + (3b15 - 16b9 + 15b8 + 10b7 + 16b5 + 6b4 + 3b2 - 70) q^91 + (-5b15 + 5b12 - 5b10 - 5b8 + 5b6 + 5b3 - 145b1 + 145) q^95 + (b14 - 7b13 - 15b12 + b11 - 5b10 + 5b7 + 4b6 + 16b5 - 4b4 - 6b3 + 7b2 + 30b1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 40 q^{5} - 34 q^{7} + 64 q^{11} - 48 q^{13} - 212 q^{17} + 456 q^{19} + 166 q^{23} - 200 q^{25} + 110 q^{29} - 160 q^{31} - 340 q^{35} + 104 q^{37} + 280 q^{41} + 136 q^{43} + 594 q^{47} - 1094 q^{49}+ \cdots + 182 q^{97}+O(q^{100}) \) 16 * q + 40 * q^5 - 34 * q^7 + 64 * q^11 - 48 * q^13 - 212 * q^17 + 456 * q^19 + 166 * q^23 - 200 * q^25 + 110 * q^29 - 160 * q^31 - 340 * q^35 + 104 * q^37 + 280 * q^41 + 136 * q^43 + 594 * q^47 - 1094 * q^49 - 752 * q^53 + 640 * q^55 + 24 * q^59 - 950 * q^61 + 240 * q^65 - 426 * q^67 + 136 * q^71 + 308 * q^73 - 696 * q^77 - 764 * q^79 + 1362 * q^83 - 530 * q^85 + 172 * q^89 - 1224 * q^91 + 1140 * q^95 + 182 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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.b

Level	$1080$
Weight	$4$
Character orbit	1080.q
Analytic conductor	$63.722$
Analytic rank	$0$
Dimension	$16$
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:	\( x^{16} - 6 x^{15} + 16 x^{14} + 14 x^{13} - 284 x^{12} + 764 x^{11} + 19770 x^{10} + 55106 x^{9} + \cdots + 193472540143 \) x^16 - 6x^15 + 16x^14 + 14x^13 - 284x^12 + 764x^11 + 19770x^10 + 55106x^9 + 266991x^8 + 335444x^7 - 4573302x^6 + 2211506x^5 + 251845642x^4 + 1438820210x^3 + 12022873630x^2 + 75245140836*x + 193472540143
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{10}\cdot 3^{14} \)
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( 76 \nu^{15} - 77258 \nu^{14} + 527696 \nu^{13} + 274645 \nu^{12} - 16218838 \nu^{11} + \cdots + 900219652474679 ) / 11\!\cdots\!80 \) (76v^15 - 77258v^14 + 527696v^13 + 274645v^12 - 16218838v^11 + 99465007v^10 - 689327126v^9 + 2767453548v^8 - 7459039014v^7 + 3034174127v^6 + 80154392282v^5 + 371926108804v^4 - 4917806845786v^3 - 16882312431899v^2 + 25271278170362*v + 900219652474679) / 1157542685443980
\(\beta_{2}\)	\(=\)	\( ( - 157 \nu^{15} - 2095 \nu^{14} + 70300 \nu^{13} - 694036 \nu^{12} + 3516229 \nu^{11} + \cdots - 20917892904731 ) / 2301277704660 \) (-157v^15 - 2095v^14 + 70300v^13 - 694036v^12 + 3516229v^11 - 10138909v^10 - 397135v^9 + 122360205v^8 - 483221991v^7 + 1671657439v^6 - 10951325459v^5 + 35205089v^4 + 9763868746v^3 + 127664113826v^2 + 2442618374617*v - 20917892904731) / 2301277704660
\(\beta_{3}\)	\(=\)	\( ( - 13268 \nu^{15} + 321410 \nu^{14} - 2099393 \nu^{13} - 5510131 \nu^{12} + \cdots - 986312262228999 ) / 128615853938220 \) (-13268v^15 + 321410v^14 - 2099393v^13 - 5510131v^12 + 98024905v^11 - 454166437v^10 + 2309563812v^9 - 4601027166v^8 + 40170974577v^7 - 398750405437v^6 + 176004089992v^5 - 2285896513702v^4 + 10290367302673v^3 - 5711144014591v^2 + 666244296174565*v - 986312262228999) / 128615853938220
\(\beta_{4}\)	\(=\)	\( ( - 91 \nu^{15} + 887 \nu^{14} + 4318 \nu^{13} - 55384 \nu^{12} + 174931 \nu^{11} + \cdots + 5389336662397 ) / 383546284110 \) (-91v^15 + 887v^14 + 4318v^13 - 55384v^12 + 174931v^11 - 708265v^10 + 3610379v^9 + 16784391v^8 - 74074431v^7 + 480255805v^6 + 460296745v^5 - 14839616899v^4 - 32679237062v^3 + 426265088180v^2 + 2571419101291*v + 5389336662397) / 383546284110
\(\beta_{5}\)	\(=\)	\( ( 299945 \nu^{15} - 868339 \nu^{14} - 6297611 \nu^{13} + 81959675 \nu^{12} + \cdots + 24\!\cdots\!58 ) / 11\!\cdots\!80 \) (299945v^15 - 868339v^14 - 6297611v^13 + 81959675v^12 - 591062666v^11 + 2349441182v^10 + 2504269283v^9 + 42737622861v^8 - 97308355458v^7 + 941554338478v^6 - 9102751657241v^5 + 36963072251849v^4 + 28208037813313v^3 + 628873065589517v^2 + 4155480908460490*v + 24680020679939758) / 1157542685443980
\(\beta_{6}\)	\(=\)	\( ( 360499 \nu^{15} - 996851 \nu^{14} + 10938140 \nu^{13} - 71674496 \nu^{12} + \cdots + 46\!\cdots\!45 ) / 11\!\cdots\!80 \) (360499v^15 - 996851v^14 + 10938140v^13 - 71674496v^12 + 167434325v^11 - 1016227049v^10 + 16923057913v^9 + 57330215493v^8 + 73630746489v^7 + 1056169695011v^6 - 1405556159779v^5 - 23975405638799v^4 + 85095647791154v^3 + 1518060945727258v^2 + 15866763636336245*v + 46578585785987645) / 1157542685443980
\(\beta_{7}\)	\(=\)	\( ( - 434 \nu^{15} + 6094 \nu^{14} - 34564 \nu^{13} + 101911 \nu^{12} + 161186 \nu^{11} + \cdots + 4387912744523 ) / 1150638852330 \) (-434v^15 + 6094v^14 - 34564v^13 + 101911v^12 + 161186v^11 - 2836919v^10 + 4123678v^9 + 48886692v^8 - 308968368v^7 + 1361421659v^6 + 6481959746v^5 - 50745319544v^4 + 63119618042v^3 + 219068279461v^2 - 6101849944786*v + 4387912744523) / 1150638852330
\(\beta_{8}\)	\(=\)	\( ( - 449 \nu^{15} + 7087 \nu^{14} - 44152 \nu^{13} + 131047 \nu^{12} - 509443 \nu^{11} + \cdots + 2515559042558 ) / 1150638852330 \) (-449v^15 + 7087v^14 - 44152v^13 + 131047v^12 - 509443v^11 + 137224v^10 - 2183141v^9 + 34938861v^8 + 12537705v^7 - 588061594v^6 - 2385787741v^5 - 56727985013v^4 + 12787225922v^3 + 644206273519v^2 + 56591298119*v + 2515559042558) / 1150638852330
\(\beta_{9}\)	\(=\)	\( ( - 165770 \nu^{15} + 2137864 \nu^{14} - 17687509 \nu^{13} + 85202449 \nu^{12} + \cdots - 71\!\cdots\!89 ) / 385847561814660 \) (-165770v^15 + 2137864v^14 - 17687509v^13 + 85202449v^12 - 187968733v^11 + 104265901v^10 - 2963904770v^9 + 9684170808v^8 - 155614400709v^7 + 416302161389v^6 - 445550925850v^5 + 3048278070592v^4 + 5482113154661v^3 - 211680188712059v^2 - 2494134053225125*v - 7199949853631689) / 385847561814660
\(\beta_{10}\)	\(=\)	\( ( - 553543 \nu^{15} + 7632611 \nu^{14} - 53749274 \nu^{13} + 204757187 \nu^{12} + \cdots + 11\!\cdots\!82 ) / 11\!\cdots\!80 \) (-553543v^15 + 7632611v^14 - 53749274v^13 + 204757187v^12 + 2686531v^11 - 4982282404v^10 + 23383263461v^9 - 37121444187v^8 - 127659621405v^7 + 1574113480540v^6 - 945870989171v^5 - 15220983874651v^4 + 57078573489094v^3 - 150465173702023v^2 - 1794098900431625*v + 11411114397347182) / 1157542685443980
\(\beta_{11}\)	\(=\)	\( ( 572522 \nu^{15} - 10044223 \nu^{14} + 65299159 \nu^{13} - 146021446 \nu^{12} + \cdots + 12\!\cdots\!71 ) / 11\!\cdots\!80 \) (572522v^15 - 10044223v^14 + 65299159v^13 - 146021446v^12 + 451497547v^11 - 2010588637v^10 + 8581629500v^9 - 55519483767v^8 + 136789491291v^7 - 526025875325v^6 + 7383051627580v^5 + 60710337838121v^4 + 225001154268001v^3 - 860652338287174v^2 - 178186114762091*v + 12782313923398471) / 1157542685443980
\(\beta_{12}\)	\(=\)	\( ( 324121 \nu^{15} - 1493588 \nu^{14} + 1065959 \nu^{13} - 24751322 \nu^{12} + \cdots + 87\!\cdots\!50 ) / 578771342721990 \) (324121v^15 - 1493588v^14 + 1065959v^13 - 24751322v^12 - 1675108v^11 + 1409458090v^10 + 2280623059v^9 + 30017318730v^8 + 38230415472v^7 - 799604405842v^6 - 397721416645v^5 - 3959209481504v^4 + 219215004606611v^3 + 1102762896910468v^2 + 4128070554045620*v + 8790081047230850) / 578771342721990
\(\beta_{13}\)	\(=\)	\( ( 100305 \nu^{15} - 1188719 \nu^{14} + 7769075 \nu^{13} - 25953113 \nu^{12} + \cdots + 51\!\cdots\!26 ) / 128615853938220 \) (100305v^15 - 1188719v^14 + 7769075v^13 - 25953113v^12 + 21350948v^11 + 120125050v^10 + 804979295v^9 + 2719640481v^8 + 8221187316v^7 + 147441628338v^6 - 942069603289v^5 + 75815277937v^4 + 29912974802783v^3 - 61574862417947v^2 + 1178272029488396*v + 5165462484944026) / 128615853938220
\(\beta_{14}\)	\(=\)	\( ( - 1925 \nu^{15} + 20389 \nu^{14} - 84724 \nu^{13} + 219193 \nu^{12} - 1376377 \nu^{11} + \cdots - 28168704583270 ) / 2301277704660 \) (-1925v^15 + 20389v^14 - 84724v^13 + 219193v^12 - 1376377v^11 + 1113724v^10 - 23194757v^9 - 9829893v^8 + 134931999v^7 + 1305606536v^6 + 13491262919v^5 - 123016201937v^4 - 808254162520v^3 - 2936589815021v^2 - 5876193558805*v - 28168704583270) / 2301277704660
\(\beta_{15}\)	\(=\)	\( ( - 2747 \nu^{15} + 27019 \nu^{14} - 111784 \nu^{13} + 236452 \nu^{12} + \cdots - 82295219611021 ) / 2301277704660 \) (-2747v^15 + 27019v^14 - 111784v^13 + 236452v^12 - 136273v^11 + 3173065v^10 - 50629037v^9 + 8230767v^8 + 338818683v^7 - 2244680935v^6 + 4680969755v^5 - 9098655653v^4 - 64811663914v^3 - 1001518790930v^2 - 754247631973*v - 82295219611021) / 2301277704660

\(\nu\)	\(=\)	\( ( \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + 2\beta _1 + 2 ) / 9 \) (b9 - b7 + b6 - b5 + 2*b1 + 2) / 9
\(\nu^{2}\)	\(=\)	\( ( - 3 \beta_{14} - 3 \beta_{13} + \beta_{10} - 2 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + \beta_{6} + \cdots + 3 ) / 9 \) (-3b14 - 3b13 + b10 - 2b9 + 3b8 - 2b7 + b6 - 2b5 + b4 - 3*b3 + 3) / 9
\(\nu^{3}\)	\(=\)	\( ( 6 \beta_{15} + 3 \beta_{14} + 9 \beta_{13} + 15 \beta_{12} + 12 \beta_{11} + 8 \beta_{10} + 6 \beta_{9} + \cdots + 2 ) / 9 \) (6b15 + 3b14 + 9b13 + 15b12 + 12b11 + 8b10 + 6b9 + 9b8 - 2b7 - 6b6 - 5b5 - b4 - 6b3 + 3b2 - 139b1 + 2) / 9
\(\nu^{4}\)	\(=\)	\( ( 6 \beta_{15} + 81 \beta_{14} - 75 \beta_{13} + 78 \beta_{12} + 84 \beta_{11} + 59 \beta_{10} - 16 \beta_{9} + \cdots - 18 ) / 9 \) (6b15 + 81b14 - 75b13 + 78b12 + 84b11 + 59b10 - 16b9 - 51b8 - 52b7 - 16b6 + 125b5 - 22b4 - 18b3 + 66b2 + 390*b1 - 18) / 9
\(\nu^{5}\)	\(=\)	\( ( - 90 \beta_{15} + 525 \beta_{14} - 294 \beta_{13} + 594 \beta_{12} + 216 \beta_{11} + 253 \beta_{10} + \cdots - 2098 ) / 9 \) (-90b15 + 525b14 - 294b13 + 594b12 + 216b11 + 253b10 - 4b9 - 579b8 - 12b7 + 173b6 - 300b5 - 98b4 - 132b3 + 45b2 + 5990*b1 - 2098) / 9
\(\nu^{6}\)	\(=\)	\( 53 \beta_{15} + 104 \beta_{14} + 31 \beta_{13} + 200 \beta_{12} - 194 \beta_{11} + 449 \beta_{10} + \cdots - 8976 \) 53b15 + 104b14 + 31b13 + 200b12 - 194b11 + 449b10 - 332b9 - 328b8 + 8b7 + 45b6 - 140b5 - 222b4 - 319b3 + 94b2 + 3563*b1 - 8976
\(\nu^{7}\)	\(=\)	\( ( 7794 \beta_{15} + 1854 \beta_{14} - 234 \beta_{13} + 5310 \beta_{12} - 7686 \beta_{11} + 24687 \beta_{10} + \cdots - 944836 ) / 9 \) (7794b15 + 1854b14 - 234b13 + 5310b12 - 7686b11 + 24687b10 - 46637b9 - 6165b8 + 15326b7 - 5687b6 + 11510b5 - 22014b4 - 8109b3 + 4302b2 + 477011*b1 - 944836) / 9
\(\nu^{8}\)	\(=\)	\( ( 20754 \beta_{15} + 6171 \beta_{14} + 14856 \beta_{13} + 31230 \beta_{12} - 26802 \beta_{11} + \cdots - 6702432 ) / 9 \) (20754b15 + 6171b14 + 14856b13 + 31230b12 - 26802b11 + 98020b10 - 186599b9 - 41505b8 + 129364b7 - 85652b6 + 133081b5 - 38348b4 + 26016b3 + 15966b2 + 4355091*b1 - 6702432) / 9
\(\nu^{9}\)	\(=\)	\( ( 68280 \beta_{15} - 5433 \beta_{14} + 243234 \beta_{13} + 166245 \beta_{12} - 258018 \beta_{11} + \cdots - 22179121 ) / 9 \) (68280b15 - 5433b14 + 243234b13 + 166245b12 - 258018b11 + 486539b10 - 738444b9 - 578475b8 + 518992b7 - 826377b6 + 662323b5 + 396899b4 + 316695b3 - 278682b2 + 11919254*b1 - 22179121) / 9
\(\nu^{10}\)	\(=\)	\( ( 1268295 \beta_{15} - 1086111 \beta_{14} + 1811661 \beta_{13} + 173922 \beta_{12} - 2973552 \beta_{11} + \cdots - 23530851 ) / 9 \) (1268295b15 - 1086111b14 + 1811661b13 + 173922b12 - 2973552b11 + 266333b10 - 2260546b9 - 4557849b8 + 2961227b7 - 3058702b6 + 1619117b5 + 1776533b4 + 1374084b3 - 2271543b2 + 26200398*b1 - 23530851) / 9
\(\nu^{11}\)	\(=\)	\( ( 7432299 \beta_{15} - 15548703 \beta_{14} + 5758248 \beta_{13} - 13334823 \beta_{12} - 14396688 \beta_{11} + \cdots + 125828675 ) / 9 \) (7432299b15 - 15548703b14 + 5758248b13 - 13334823b12 - 14396688b11 - 10084760b10 - 3116497b9 - 16270653b8 + 18591360b7 + 2611451b6 - 342228b5 + 273124b4 + 7881429b3 - 9105453b2 - 49047583*b1 + 125828675) / 9
\(\nu^{12}\)	\(=\)	\( 4341634 \beta_{15} - 12547142 \beta_{14} + 2628071 \beta_{13} - 16316312 \beta_{12} - 1659250 \beta_{11} + \cdots + 56401067 \) 4341634b15 - 12547142b14 + 2628071b13 - 16316312b12 - 1659250b11 - 14759435b10 + 3706904b9 + 1805896b8 + 12717529b7 + 7086690b6 + 2672675b5 + 1155120b4 + 5358088b3 - 8344117b2 - 160802273*b1 + 56401067
\(\nu^{13}\)	\(=\)	\( ( 184488633 \beta_{15} - 396018414 \beta_{14} + 109709091 \beta_{13} - 911935980 \beta_{12} + \cdots + 13070072981 ) / 9 \) (184488633b15 - 396018414b14 + 109709091b13 - 911935980b12 + 360331434b11 - 1273269249b10 + 475622317b9 + 429904809b8 + 676338857b7 + 330792949b6 + 521842598b5 + 311611329b4 + 316491120b3 - 351916470b2 - 22546685257*b1 + 13070072981) / 9
\(\nu^{14}\)	\(=\)	\( ( 418349322 \beta_{15} + 295880235 \beta_{14} - 237851085 \beta_{13} - 4519304190 \beta_{12} + \cdots + 184026481212 ) / 9 \) (418349322b15 + 295880235b14 - 237851085b13 - 4519304190b12 + 3502722060b11 - 10010410253b10 + 5749013527b9 + 2636358447b8 + 3449286601b7 + 3499086529b6 + 4236748495b5 + 1157528161b4 + 2047026525b3 - 831578787b2 - 229072310730*b1 + 184026481212) / 9
\(\nu^{15}\)	\(=\)	\( ( - 7412079243 \beta_{15} + 10053860277 \beta_{14} - 7027379802 \beta_{13} - 22505590668 \beta_{12} + \cdots + 1351798631138 ) / 9 \) (-7412079243b15 + 10053860277b14 - 7027379802b13 - 22505590668b12 + 20743751136b11 - 66521645911b10 + 47706319689b9 + 15443262648b8 + 12611040631b7 + 36574557645b6 + 15231632098b5 - 7309537525b4 + 9259139055b3 + 2315812836b2 - 1589846574910*b1 + 1351798631138) / 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
$7$	\( T^{16} + \cdots + 26\!\cdots\!64 \) T^16 + 34T^15 + 2497T^14 + 48838T^13 + 3188398T^12 + 58700818T^11 + 2201352109T^10 + 16632083158T^9 + 504023304658T^8 + 3890250850074T^7 + 79139175552405T^6 + 406499392754838T^5 + 5373493850910393T^4 + 26537771865843360T^3 + 256746621319921296T^2 + 787513253234213376*T + 2613380713166686464
$11$	\( T^{16} + \cdots + 57\!\cdots\!64 \) T^16 - 64T^15 + 7273T^14 - 339064T^13 + 29214925T^12 - 1246959124T^11 + 63417918196T^10 - 1679483680192T^9 + 56209073475520T^8 - 1236179215633248T^7 + 32508897368710752T^6 - 494249259328399296T^5 + 7003814038424442432T^4 - 40793971557057479424T^3 + 249362606277104375808T^2 + 443858513869566443520*T + 5722542772348736196864
$13$	\( T^{16} + \cdots + 89\!\cdots\!44 \) T^16 + 48T^15 + 8664T^14 + 70128T^13 + 43131024T^12 + 676716768T^11 + 66688632288T^10 - 145085230080T^9 + 42695537228928T^8 - 253387123382016T^7 + 20903880536112384T^6 - 208667120684244480T^5 + 4428006853486678272T^4 - 15319026425336408064T^3 + 211729930018906189824T^2 - 78122622703191883776*T + 8991985545268306182144
$17$	\( (T^{8} + \cdots - 1974672949104)^{2} \) (T^8 + 106T^7 - 8259T^6 - 1309460T^5 - 20518556T^4 + 2864571144T^3 + 133003111680T^2 + 1610149733856*T - 1974672949104)^2
$19$	\( (T^{8} + \cdots + 5438892407536)^{2} \) (T^8 - 228T^7 + 4091T^6 + 1865688T^5 - 68406528T^4 - 4106026200T^3 + 82517584352T^2 + 1586325374496*T + 5438892407536)^2
$23$	\( T^{16} + \cdots + 79\!\cdots\!64 \) T^16 - 166T^15 + 45565T^14 - 3545818T^13 + 907960138T^12 - 81443234326T^11 + 9247040901973T^10 - 527364471329506T^9 + 39885962735049550T^8 - 2117849401909171350T^7 + 114024181539233325669T^6 - 3921929266645369188162T^5 + 105318095232765542265981T^4 - 1827955024752865691702016T^3 + 23513819471125118949549348T^2 - 165728040540867731001466944*T + 792528416815727764030562064
$29$	\( T^{16} + \cdots + 76\!\cdots\!56 \) T^16 - 110T^15 + 80929T^14 - 1174970T^13 + 3758369038T^12 + 21161453974T^11 + 98781131154565T^10 + 7100726682113074T^9 + 1540080118885305934T^8 + 132785915590401853194T^7 + 18963375271216255389261T^6 + 1601409399678929088183114T^5 + 122939151457634961537447045T^4 + 5930848240068093064043190300T^3 + 227504714824490338958623400700T^2 + 4936952733335042071181985864432*T + 76398544327160238101844788944656
$31$	\( T^{16} + \cdots + 29\!\cdots\!76 \) T^16 + 160T^15 + 106372T^14 + 8917360T^13 + 7112385664T^12 + 694828156768T^11 + 183872925161632T^10 + 16434956972547712T^9 + 3187661758930108288T^8 + 263208329545043070720T^7 + 29653284205785415104768T^6 + 1625442980063210285188608T^5 + 131037490169846531614008576T^4 + 6165382028633512517936480256T^3 + 369801823163826500791628276736T^2 + 10354531415454459814343730462720*T + 292533131580785400972855951691776
$37$	\( (T^{8} + \cdots + 55\!\cdots\!24)^{2} \) (T^8 - 52T^7 - 183012T^6 - 605368T^5 + 7702035136T^4 + 60570129120T^3 - 80327279213424T^2 - 2671603756636032*T + 55340712289108224)^2
$41$	\( T^{16} + \cdots + 52\!\cdots\!49 \) T^16 - 280T^15 + 448132T^14 - 96411712T^13 + 129944576290T^12 - 27558477184024T^11 + 19514263381784368T^10 - 3725858907539071072T^9 + 2033129392668611589595T^8 - 357256765089602681399016T^7 + 98894092863467786746001136T^6 - 9616325324318857550354566608T^5 + 1976169601735815352408310849490T^4 - 188291550491252782235322952063248T^3 + 25337720172612299228167638408097860T^2 - 1198233555300589629325231022058935136*T + 52794770246961604752809990012704952049
$43$	\( T^{16} + \cdots + 94\!\cdots\!44 \) T^16 - 136T^15 + 455881T^14 - 30426400T^13 + 145435124725T^12 - 7544085784012T^11 + 20907031003799116T^10 + 784064298350385440T^9 + 2018477708781867480784T^8 + 67984041268541227656192T^7 + 98037338726573822212989984T^6 + 6555005701967811633858741312T^5 + 3124595709547625249592538682304T^4 + 51589247430482910764079752371968T^3 + 1343042337980806330724172430573824T^2 - 8665329073736864056896671840460288*T + 94724919651659558892826993032409344
$47$	\( T^{16} + \cdots + 42\!\cdots\!36 \) T^16 - 594T^15 + 520573T^14 - 65617806T^13 + 63299475610T^12 + 2181729857598T^11 + 8646618044591485T^10 + 1051642049080704930T^9 + 214836545453673488230T^8 + 11378006086228424010006T^7 + 1685632336809777922028233T^6 + 58902244047528408829072674T^5 + 9791091243964080931235760361T^4 + 150421570433860382178720336000T^3 + 23099858514807864860909333263360T^2 + 94804194354927651419216818864128*T + 42167652362175646538512980310491136
$53$	\( (T^{8} + \cdots + 14\!\cdots\!96)^{2} \) (T^8 + 376T^7 - 373776T^6 - 49904000T^5 + 38686750624T^4 - 1038464493696T^3 - 483710269816320T^2 + 6911131952461824*T + 1442422409646295296)^2
$59$	\( T^{16} + \cdots + 78\!\cdots\!76 \) T^16 - 24T^15 + 1282545T^14 + 295206048T^13 + 1119294468333T^12 + 298168467089220T^11 + 561030782528170116T^10 + 184016251086515245296T^9 + 203642986430317623316128T^8 + 55677878016302788182379584T^7 + 39607512627691102231584449856T^6 + 8315553502815415869113269463808T^5 + 5052562674564902998411344528354816T^4 + 729774419993245579725589363918046208T^3 + 263475702719507370327130474311072678912T^2 - 12737379323148777783285158212115936956416*T + 782284191142452965828886499926701993496576
$61$	\( T^{16} + \cdots + 31\!\cdots\!36 \) T^16 + 950T^15 + 1510585T^14 + 930682034T^13 + 1094690311750T^12 + 602557878178274T^11 + 487137500421297349T^10 + 182994258156244132910T^9 + 109654526275122721523974T^8 + 33023725583183812073484870T^7 + 16328982267095636705291048421T^6 + 2872341860730181711333282137222T^5 + 543094580663455126830843920389605T^4 - 6387146551676352977448072173237052T^3 + 97028673278712046233723672018049980T^2 + 164061415716043625589094874735775120*T + 319214228022522669542479316939889936
$67$	\( T^{16} + \cdots + 22\!\cdots\!21 \) T^16 + 426T^15 + 1378714T^14 + 682376664T^13 + 1269467332516T^12 + 616122720480546T^11 + 695309228264589316T^10 + 325234267782030569694T^9 + 274639991225218096579057T^8 + 116089391155501193735250714T^7 + 68580742154391702723041049292T^6 + 24347081687034489151482983895606T^5 + 11276374384823509248976032039523180T^4 + 3283866047350319978302337655472379808T^3 + 1006988691505133222207459108561200287478T^2 + 158332788214857075069574504167152724507030*T + 22266216504137602592782177591977530290233121
$71$	\( (T^{8} + \cdots - 47\!\cdots\!36)^{2} \) (T^8 - 68T^7 - 936048T^6 - 25389992T^5 + 160133417392T^4 + 29299026862464T^3 - 288976713623472T^2 - 166180071125774592*T - 4707438249771164736)^2
$73$	\( (T^{8} + \cdots - 10\!\cdots\!92)^{2} \) (T^8 - 154T^7 - 1719639T^6 + 216618992T^5 + 907520755072T^4 - 101797465761024T^3 - 147298700273326080T^2 + 13277903701634629632*T - 108586850696560852992)^2
$79$	\( T^{16} + \cdots + 50\!\cdots\!04 \) T^16 + 764T^15 + 897048T^14 + 266699200T^13 + 254458469120T^12 + 45115361629632T^11 + 54899439527855872T^10 + 3036662638785463040T^9 + 6060185085207126401280T^8 + 11107752572020015877120T^7 + 498923596971851384365871104T^6 - 19651265383850625377258016768T^5 + 9532461103897235732459080318976T^4 + 474729362485777992074557754638336T^3 + 80696107675281068148846148547346432T^2 - 200125743359795701295850386472632320*T + 504288104004072809562341960023343104
$83$	\( T^{16} + \cdots + 41\!\cdots\!16 \) T^16 - 1362T^15 + 4072761T^14 - 3094070454T^13 + 7495957222398T^12 - 4665192586195626T^11 + 9276287481398131605T^10 - 4041772518993161734854T^9 + 7326989596565927669783970T^8 - 2726813236242623971309983954T^7 + 4098532291059658231741099476429T^6 - 1125248290352520505912932344848134T^5 + 1281149088298231707384557914669988025T^4 - 427647378507511766051700602069768768232T^3 + 227262799809265123355506499910525936639408T^2 - 31789053913211598524052885350199254105293440*T + 4108233369986950683603819679301578596370626816
$89$	\( (T^{8} + \cdots + 25\!\cdots\!48)^{2} \) (T^8 - 86T^7 - 1928957T^6 + 392124112T^5 + 1111003730443T^4 - 297575955644606T^3 - 162928649209008563T^2 + 32827201378290455908*T + 2591490799217186564548)^2
$97$	\( T^{16} + \cdots + 25\!\cdots\!44 \) T^16 - 182T^15 + 2367375T^14 + 334861226T^13 + 4003845110501T^12 + 458168561919924T^11 + 2930970616102953076T^10 + 285608869922118537040T^9 + 1555438265658436484735136T^8 + 29695163242475567810836288T^7 + 338861646216758327430644694592T^6 - 59182248030635627696777344663296T^5 + 53683050185973732063900776104240640T^4 - 4164434764466308657420733167465679872T^3 + 428149981214361274578066188896893121536T^2 + 8868014321894563682265460795847482765312*T + 253673147680271264996195338829810182918144
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