LMFDB - Newform orbit 1104.3.k.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,3,Mod(415,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1104.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1104.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.0818211854$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 138 x^{14} - 826 x^{13} + 6668 x^{12} - 29634 x^{11} + 142919 x^{10} + \cdots + 1513188 $ x^16 - 8x^15 + 138x^14 - 826x^13 + 6668x^12 - 29634x^11 + 142919x^10 - 461683x^9 + 1440096x^8 - 3284228x^7 + 6812887x^6 - 10590573x^5 + 14250513x^4 - 13904316x^3 + 10818534x^2 - 5200488*x + 1513188
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

$f(q)$	$=$	$ q + \beta_{8} q^{3} + \beta_{3} q^{5} + (\beta_{13} - \beta_{8}) q^{7} - 3 q^{9}+O(q^{10}) $ q + b8 * q^3 + b3 * q^5 + (b13 - b8) * q^7 - 3 * q^9	$ q + \beta_{8} q^{3} + \beta_{3} q^{5} + (\beta_{13} - \beta_{8}) q^{7} - 3 q^{9} + (\beta_{10} + \beta_{8}) q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{13} - \beta_{9} q^{15} + ( - \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{17} + (2 \beta_{12} - \beta_{9} + 2 \beta_{8}) q^{19} + (\beta_{5} + 2) q^{21} + \beta_{11} q^{23} + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 1) q^{25}+ \cdots + ( - 3 \beta_{10} - 3 \beta_{8}) q^{99}+O(q^{100}) $ q + b8 * q^3 + b3 * q^5 + (b13 - b8) * q^7 - 3 * q^9 + (b10 + b8) * q^11 + (-b2 + b1 - 1) * q^13 - b9 * q^15 + (-b6 + b4 + b2 - 1) * q^17 + (2b12 - b9 + 2b8) * q^19 + (b5 + 2) * q^21 + b11 * q^23 + (b6 + b5 - b4 - 2b3 + b1 + 1) q^25 - 3b8 q^27 + (b7 + b6 - b3 + b1) * q^29 + (b15 - b14 + b13 + 2b12 + 4b11 - b10 - b9 + 3b8) q^31 + (-b7 - 2) * q^33 + (-3b13 + b12 - b10 + 2b9 - 4b8) q^35 + (b7 - b6 - 3b5 + b4 + 2b3 + b2 + 4b1 + 3) q^37 + (b14 + b12 + b11 - b9 + b8) * q^39 + (-b6 - b5 + b4 + 2b3 + b2 + 2b1 + 3) * q^41 + (-b15 + 3b13 - 4b12 + b11 + b10 - 2b8) q^43 - 3b3 q^45 + (-b15 + 3b13 - 5b12 + 7b11 + b10 + 2b8) * q^47 + (-b7 - b5 - b4 + 3b3 + 4b1 + 5) * q^49 + (-b15 - b13 + 2b12 + b11 - b8) q^51 + (b7 - b6 - 2b5 + b4 + 4b3 - b2 + 2b1 - 1) q^53 + (b15 - 2b13 + 5b11 - 2b10 + 2b8) * q^55 + (-2b4 - b3 - 2) q^57 + (b14 - 8b13 - b12 + b11 + 3b9 - b8) * q^59 + (-b7 + 3b6 + b5 - 3b4 + 2b3 - b2 + 2b1 + 9) * q^61 + (-3b13 + 3b8) * q^63 + (b7 + 5b6 + 2b5 - b4 - 7b3 - 3b2 + 2b1 + 1) q^65 + (2b14 + 2b13 - 2b12 + 2b11 + 4b10 - b9 + 2b8) * q^67 - b1 * q^69 + (-b15 + b14 + b13 + 8b11 - 5b10 + 3b9 - 11b8) * q^71 + (-2b7 - 3b6 + b5 + b4 - 6b3 + b2 + 4b1 - 13) * q^73 + (b15 - b14 - 2b13 - 3b12 + 4b11 + 3b9) * q^75 + (-3b7 + 5b5 + 3b4 + b3 - 3b2 - b1 + 7) * q^77 + (b15 - 2b14 + 4b13 - 2b12 + 9b11 + 5b10 + 7b9 - 7b8) q^79 + 9 * q^81 + (2b15 - 2b14 - 2b13 - 2b12 + 8b11 - 3b10 - 4b9 + 9b8) * q^83 + (-b7 - 2b6 - 3b5 + 3b4 - 5b3 + 6b2 - 10) q^85 + (b15 - b14 + b13 + 4b11 + 3b10 + b9 + b8) * q^87 + (b7 - 2b5 - 6b4 + 5b3 - 4b2 + 4b1 - 10) q^89 + (-b15 - 3b13 - 2b12 + b11 - 5b10 + b9 - 4b8) * q^91 + (b7 - 3b6 - 2b4 - b3 - 3b1 - 6) q^93 + (b15 - b14 + 2b13 + 3b12 + 16b11 + 3b9 + 9b8) q^95 + (2b6 + 6b5 - 18b3 - 4b2 + 6b1 - 38) q^97 + (-3b10 - 3b8) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{5} - 48 q^{9}+O(q^{10}) $ 16 * q - 8 * q^5 - 48 * q^9	$ 16 q - 8 q^{5} - 48 q^{9} - 8 q^{13} - 28 q^{17} + 36 q^{21} + 40 q^{25} + 16 q^{29} - 36 q^{33} + 12 q^{37} + 16 q^{41} + 24 q^{45} + 48 q^{49} - 48 q^{53} - 24 q^{57} + 148 q^{61} + 128 q^{65} - 184 q^{73} + 136 q^{77} + 144 q^{81} - 192 q^{85} - 172 q^{89} - 96 q^{93} - 400 q^{97}+O(q^{100}) $ 16 * q - 8 * q^5 - 48 * q^9 - 8 * q^13 - 28 * q^17 + 36 * q^21 + 40 * q^25 + 16 * q^29 - 36 * q^33 + 12 * q^37 + 16 * q^41 + 24 * q^45 + 48 * q^49 - 48 * q^53 - 24 * q^57 + 148 * q^61 + 128 * q^65 - 184 * q^73 + 136 * q^77 + 144 * q^81 - 192 * q^85 - 172 * q^89 - 96 * q^93 - 400 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} - 8 x^{15} + 138 x^{14} - 826 x^{13} + 6668 x^{12} - 29634 x^{11} + 142919 x^{10} + \cdots + 1513188 $ x^16 - 8*x^15 + 138*x^14 - 826*x^13 + 6668*x^12 - 29634*x^11 + 142919*x^10 - 461683*x^9 + 1440096*x^8 - 3284228*x^7 + 6812887*x^6 - 10590573*x^5 + 14250513*x^4 - 13904316*x^3 + 10818534*x^2 - 5200488*x + 1513188 :

$\beta_{1}$	$=$	$ ( - 143240 \nu^{14} + 1002680 \nu^{13} - 21508999 \nu^{12} + 116019154 \nu^{11} + \cdots - 104230692531 ) / 36162037683 $ (-143240v^14 + 1002680v^13 - 21508999v^12 + 116019154v^11 - 1174847217v^10 + 4834624380v^9 - 29210011687v^8 + 89067544522v^7 - 333420058790v^6 + 707595997217v^5 - 1531283458128v^4 + 1978099521357v^3 - 2071686272763v^2 + 1187081591514v - 104230692531) / 36162037683
$\beta_{2}$	$=$	$ ( - 140512893561128 \nu^{14} + 983590254927896 \nu^{13} + \cdots - 26\!\cdots\!98 ) / 29\!\cdots\!65 $ (-140512893561128v^14 + 983590254927896v^13 - 18510868217912704v^12 + 98278535993413576v^11 - 850461389379413526v^10 + 3374862601366558038v^9 - 17116042836711069946v^8 + 49255872947013998284v^7 - 157244179596932520446v^6 + 312472538527996968506v^5 - 631487712247385162802v^4 + 793422541009332263604v^3 - 938201206719996961554v^2 + 586293176959558472202v - 265144162684272693498) / 2982588997745619765
$\beta_{3}$	$=$	$ ( - 61651462937429 \nu^{14} + 431560240562003 \nu^{13} + \cdots - 14\!\cdots\!20 ) / 11\!\cdots\!06 $ (-61651462937429v^14 + 431560240562003v^13 - 7815467491532791v^12 + 41282521821890707v^11 - 336809494333808661v^10 + 1315909874035106229v^9 - 6078235838289150178v^8 + 16853959530586656421v^7 - 46516441353098159039v^6 + 85651693367467937957v^5 - 141843815307189506922v^4 + 158183644771021559439v^3 - 139103351112463170294v^2 + 71839608599154552558v - 14552280079029593820) / 1193035599098247906
$\beta_{4}$	$=$	$ ( 736677511463 \nu^{14} - 5156742580241 \nu^{13} + 92448944196484 \nu^{12} + \cdots + 19\!\cdots\!08 ) / 12\!\cdots\!35 $ (736677511463v^14 - 5156742580241v^13 + 92448944196484v^12 - 487656011635771v^11 + 3912163603364916v^10 - 15213540274992423v^9 + 68137763136299071v^8 - 186423337254956809v^7 + 483982033848002771v^6 - 858218298435326516v^5 + 1261144211172973422v^4 - 1281561783175958934v^3 + 931583140113439659v^2 - 406942725600337092v + 190949628595272408) / 12479451873412635
$\beta_{5}$	$=$	$ ( - 173475766615451 \nu^{14} + \cdots - 62\!\cdots\!56 ) / 19\!\cdots\!10 $ (-173475766615451v^14 + 1214330366308157v^13 - 22146369274426213v^12 + 117091920884551237v^11 - 965746218404284227v^10 + 3784330024310045871v^9 - 17786552900171991322v^8 + 49678628515305734683v^7 - 140787776948808025457v^6 + 263146400787940753967v^5 - 448193545831150968714v^4 + 508817191975232301273v^3 - 457626329274894969918v^2 + 239837413464431586114v - 62942295370014168156) / 1988392665163746510
$\beta_{6}$	$=$	$ ( 596945386449457 \nu^{14} + \cdots + 44\!\cdots\!22 ) / 59\!\cdots\!30 $ (596945386449457v^14 - 4178617705146199v^13 + 75871078545894161v^12 - 400904441108464379v^11 + 3285141764533346979v^10 - 12850341834478462497v^9 + 59838076746697497104v^8 - 166489478451587776901v^7 + 467168946123552876859v^6 - 868534413342268617679v^5 + 1491881053620786804498v^4 - 1706856230571506447361v^3 + 1678799910654557699316v^2 - 945914049675405653358v + 446152584287023331022) / 5965177995491239530
$\beta_{7}$	$=$	$ ( 133672420258644 \nu^{14} - 935706941810508 \nu^{13} + \cdots + 10\!\cdots\!54 ) / 99\!\cdots\!55 $ (133672420258644v^14 - 935706941810508v^13 + 17171941233676402v^12 - 90867457158521808v^11 + 756877443119919358v^10 - 2973736540426297324v^9 + 14209279901573393938v^8 - 39956012080285559512v^7 + 116507751527240357178v^6 - 221207505569571992908v^5 + 395575251579884735316v^4 - 463618698920596016172v^3 + 463336523497315177842v^2 - 262555233287807320446v + 106927256919080268954) / 994196332581873255
$\beta_{8}$	$=$	$ ( - 364783933592 \nu^{15} + 2735879501940 \nu^{14} - 47785245441398 \nu^{13} + \cdots - 54\!\cdots\!79 ) / 57\!\cdots\!05 $ (-364783933592v^15 + 2735879501940v^14 - 47785245441398v^13 + 269109922922997v^12 - 2145874208179796v^11 + 8933386172217309v^10 - 41011100283214000v^9 + 121794253326673923v^8 - 340834264218128242v^7 + 681433889011663296v^6 - 1152050074258213375v^5 + 1420292966216860140v^4 - 1244645509413599559v^3 + 718313619593160561v^2 - 60592049111558046*v - 54856469300366079) / 57834524396273805
$\beta_{9}$	$=$	$ ( 10\!\cdots\!34 \nu^{15} + \cdots + 21\!\cdots\!68 ) / 12\!\cdots\!30 $ (1044153256528602334v^15 - 7831149423964517505v^14 + 128797421366924688011v^13 - 718410805954881956579v^12 + 5175295542860237437007v^11 - 20822905972673887115253v^10 + 78355300349435966843615v^9 - 207720907891849786713816v^8 + 342384095157667430604289v^7 - 359719298358252324081487v^6 - 724556509920181101411245v^5 + 2318862016011531920603100v^4 - 5625753970578771605979867v^3 + 6050567213534430780525318v^2 - 6149842594930221656423658*v + 2146834333718389709447868) / 122928399738418300487730
$\beta_{10}$	$=$	$ ( - 17\!\cdots\!44 \nu^{15} + \cdots - 32\!\cdots\!43 ) / 18\!\cdots\!95 $ (-1722065734930545844v^15 + 12915493011979093830v^14 - 216873600605906305376v^13 + 1213793426590041395189v^12 - 9052915177619062762922v^11 + 36870252734580758946453v^10 - 148516527595149345831410v^9 + 410901612457195541954721v^8 - 827178565521007296033004v^7 + 1210265275797529849550212v^6 + 181581588596255553108935v^5 - 2686149828938583768413940v^4 + 9388699372268429140149927v^3 - 11016167101045461861767373v^2 + 9964300193829100052180088*v - 3253280735329265372359743) / 184392599607627450731595
$\beta_{11}$	$=$	$ ( 59797395988 \nu^{15} - 448480469910 \nu^{14} + 7805099807922 \nu^{13} + \cdots - 34\!\cdots\!99 ) / 34\!\cdots\!45 $ (59797395988v^15 - 448480469910v^14 + 7805099807922v^13 - 43931194957858v^12 + 348635123444094v^11 - 1449214332752076v^10 + 6617237783394080v^9 - 19601260892838882v^8 + 54845019755651658v^7 - 109707925246762334v^6 + 192179916838007010v^5 - 245361037678811850v^4 + 261712142208796776v^3 - 190958126520367644v^2 + 119620660925447424*v - 34104766592492199) / 3405820681546245
$\beta_{12}$	$=$	$ ( 74\!\cdots\!66 \nu^{15} + \cdots + 33\!\cdots\!32 ) / 36\!\cdots\!90 $ (7424501752020233666v^15 - 55683763140151752495v^14 + 951986296990989942129v^13 - 5343373856149133044331v^12 + 41196252998759408739033v^11 - 169660260638979348051957v^10 + 735883721118657791887885v^9 - 2123209069803554798033874v^8 + 5379284958492603307423611v^7 - 9994829906635426218304393v^6 + 14324398855990884679846695v^5 - 15013718549657043129675600v^4 + 9805837739755222525549017v^3 - 3506036404446518754666288v^2 - 141504167804176681029162*v + 333398238725058745468032) / 368785199215254901463190
$\beta_{13}$	$=$	$ ( - 83\!\cdots\!26 \nu^{15} + \cdots + 41\!\cdots\!78 ) / 36\!\cdots\!90 $ (-8387773236519027026v^15 + 62908299273892702695v^14 - 1107602679058388553929v^13 + 6245308208225486276331v^12 - 50440066038059899297073v^11 + 210821013171287983607517v^10 - 989318096023402546457485v^9 + 2969323475358288777319074v^8 - 8650295041975881051454051v^7 + 17763837194285607899313273v^6 - 32457002201890941826150975v^5 + 42705505935067019039739600v^4 - 45575768800825827235796097v^3 + 32846970846284746451346828v^2 - 17000061464560696206868038*v + 4110617490546327071649678) / 368785199215254901463190
$\beta_{14}$	$=$	$ ( 31\!\cdots\!12 \nu^{15} + \cdots - 20\!\cdots\!96 ) / 18\!\cdots\!95 $ (31971685675174392712v^15 - 239787642563807945340v^14 + 4247950514778713596278v^13 - 23974899100510551204817v^12 + 195497640034726024232556v^11 - 819514044425589461802249v^10 + 3906900237260637444663500v^9 - 11813136753196059364256553v^8 + 35343710282363157618035142v^7 - 73808167747144639244697386v^6 + 140861966289547476711747825v^5 - 191478313830267286867571340v^4 + 218928892359780571067554749v^3 - 167058603760681400926292061v^2 + 86981418053892673560365976*v - 20610356981310823045409496) / 184392599607627450731595
$\beta_{15}$	$=$	$ ( 40\!\cdots\!18 \nu^{15} + \cdots - 33\!\cdots\!29 ) / 18\!\cdots\!95 $ (40203551511379922018v^15 - 301526636335349415135v^14 + 5341822814289059551937v^13 - 30148694308459420958043v^12 + 245914028199773320687949v^11 - 1030952456471422458730251v^10 + 4920817694055790147466395v^9 - 14887430501306990185747452v^8 + 44726238992160677994814303v^7 - 93650739536607437208185109v^6 + 181030841504415058317907495v^5 - 248507418159712400695892070v^4 + 293750937415473395373402651v^3 - 230422034440951844975973264v^2 + 131508109306323032726226234*v - 33829607825499319012538829) / 184392599607627450731595

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

$\nu$	$=$	$ ( \beta_{15} - \beta_{14} + \beta_{13} - 2\beta_{11} - 3\beta_{10} - \beta_{9} + \beta_{8} + 6 ) / 12 $ (b15 - b14 + b13 - 2b11 - 3b10 - b9 + b8 + 6) / 12
$\nu^{2}$	$=$	$ ( \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} + 7 \beta_{7} + \cdots - 156 ) / 12 $ (b15 - b14 + b13 - 2b11 - 3b10 - b9 + b8 + 7b7 + 3b6 + 4b5 - 4b4 + 7b3 + 6b2 + 3*b1 - 156) / 12
$\nu^{3}$	$=$	$ ( - 19 \beta_{15} + 13 \beta_{14} - 39 \beta_{13} - 38 \beta_{12} + 96 \beta_{11} + 67 \beta_{10} + \cdots - 158 ) / 8 $ (-19b15 + 13b14 - 39b13 - 38b12 + 96b11 + 67b10 + 29b9 - 73b8 + 7b7 + 3b6 + 4b5 - 4b4 + 7b3 + 6b2 + 3*b1 - 158) / 8
$\nu^{4}$	$=$	$ ( - 58 \beta_{15} + 40 \beta_{14} - 118 \beta_{13} - 114 \beta_{12} + 290 \beta_{11} + 204 \beta_{10} + \cdots + 4941 ) / 12 $ (-58b15 + 40b14 - 118b13 - 114b12 + 290b11 + 204b10 + 88b9 - 220b8 - 346b7 - 45b6 - 247b5 + 193b4 - 100b3 - 261b2 - 156*b1 + 4941) / 12
$\nu^{5}$	$=$	$ ( 1868 \beta_{15} - 773 \beta_{14} + 4721 \beta_{13} + 6198 \beta_{12} - 12919 \beta_{11} - 8619 \beta_{10} + \cdots + 25497 ) / 24 $ (1868b15 - 773b14 + 4721b13 + 6198b12 - 12919b11 - 8619b10 - 3809b9 + 17753b8 - 1765b7 - 240b6 - 1255b5 + 985b4 - 535b3 - 1335b2 - 795*b1 + 25497) / 24
$\nu^{6}$	$=$	$ ( 1965 \beta_{15} - 840 \beta_{14} + 4918 \beta_{13} + 6388 \beta_{12} - 13403 \beta_{11} - 8960 \beta_{10} + \cdots - 125049 ) / 8 $ (1965b15 - 840b14 + 4918b13 + 6388b12 - 13403b11 - 8960b10 - 3956b9 + 18120b8 + 10584b7 - 535b6 + 7305b5 - 5719b4 + 706b3 + 7055b2 + 6498*b1 - 125049) / 8
$\nu^{7}$	$=$	$ ( - 65267 \beta_{15} + 9608 \beta_{14} - 207098 \beta_{13} - 282936 \beta_{12} + 543925 \beta_{11} + \cdots - 1402809 ) / 24 $ (-65267b15 + 9608b14 - 207098b13 - 282936b12 + 543925b11 + 375966b10 + 161504b9 - 973418b8 + 117334b7 - 4767b6 + 81109b5 - 63511b4 + 9310b3 + 78771b2 + 71022*b1 - 1402809) / 24
$\nu^{8}$	$=$	$ ( - 144425 \beta_{15} + 25190 \beta_{14} - 448898 \beta_{13} - 610854 \beta_{12} + 1182349 \beta_{11} + \cdots + 7438467 ) / 12 $ (-144425b15 + 25190b14 - 448898b13 - 610854b12 + 1182349b11 + 815130b10 + 350906b9 - 2074190b8 - 700706b7 + 111243b6 - 447239b5 + 380903b4 + 15568b3 - 430689b2 - 527550*b1 + 7438467) / 12
$\nu^{9}$	$=$	$ ( 786941 \beta_{15} + 76505 \beta_{14} + 3066037 \beta_{13} + 4071420 \beta_{12} - 7438130 \beta_{11} + \cdots + 25157032 ) / 8 $ (786941b15 + 76505b14 + 3066037b13 + 4071420b12 - 7438130b11 - 5367093b10 - 2259193b9 + 15444777b8 - 2339271b7 + 342921b6 - 1505700b5 + 1271118b4 + 27321b3 - 1451490b2 - 1725813*b1 + 25157032) / 8
$\nu^{10}$	$=$	$ ( 7006169 \beta_{15} + 375841 \beta_{14} + 26414243 \beta_{13} + 35184699 \beta_{12} - 64795777 \beta_{11} + \cdots - 296847378 ) / 12 $ (7006169b15 + 375841b14 + 26414243b13 + 35184699b12 - 64795777b11 - 46461777b10 - 19617722b9 + 131583614b8 + 29841506b7 - 6708183b6 + 17733641b5 - 16486502b4 - 1942108b3 + 17409627b2 + 25440567*b1 - 296847378) / 12
$\nu^{11}$	$=$	$ ( - 86706556 \beta_{15} - 26239643 \beta_{14} - 395314939 \beta_{13} - 506910600 \beta_{12} + \cdots - 3972853521 ) / 24 $ (-86706556b15 - 26239643b14 - 395314939b13 - 506910600b12 + 895087745b11 + 668817015b10 + 278101585b9 - 2035263349b8 + 393896723b7 - 83270088b6 + 237382871b5 - 217016789b4 - 22006105b3 + 232303137b2 + 328096131*b1 - 3972853521) / 24
$\nu^{12}$	$=$	$ ( - 226424231 \beta_{15} - 54939440 \beta_{14} - 989381702 \beta_{13} - 1278703510 \beta_{12} + \cdots + 7771608171 ) / 8 $ (-226424231b15 - 54939440b14 - 989381702b13 - 1278703510b12 + 2278648423b11 + 1687519532b10 + 704014434b9 - 5058690354b8 - 813686592b7 + 216343567b6 - 457984457b5 + 456044425b4 + 71859130b3 - 459011083b2 - 748302528*b1 + 7771608171) / 8
$\nu^{13}$	$=$	$ ( 3164267743 \beta_{15} + 1465261079 \beta_{14} + 16128850513 \beta_{13} + 20125941786 \beta_{12} + \cdots + 205050601794 ) / 24 $ (3164267743b15 + 1465261079b14 + 16128850513b13 + 20125941786b12 - 34742323364b11 - 26618970363b10 - 10986061111b9 + 83650072795b8 - 21159673925b7 + 5325913827b6 - 12127699766b5 + 11807606876b4 + 1688882065b3 - 12077714454b2 - 18983480685*b1 + 205050601794) / 24
$\nu^{14}$	$=$	$ ( 32940973430 \beta_{15} + 12777792661 \beta_{14} + 159747218747 \beta_{13} + 201499882182 \beta_{12} + \cdots - 881172741837 ) / 24 $ (32940973430b15 + 12777792661b14 + 159747218747b13 + 201499882182b12 - 351371949883b11 - 266334765855b10 - 110295621443b9 + 824804017865b8 + 94690455053b7 - 27644337114b6 + 51288583355b5 - 53627158181b4 - 9656332447b3 + 52261989015b2 + 91270154877*b1 - 881172741837) / 24
$\nu^{15}$	$=$	$ ( - 37122678030 \beta_{15} - 21662030073 \beta_{14} - 204462583523 \beta_{13} - 250305255248 \beta_{12} + \cdots - 3462415442405 ) / 8 $ (-37122678030b15 - 21662030073b14 - 204462583523b13 - 250305255248b12 + 425589844477b11 + 331779133909b10 + 136280976853b9 - 1063434188961b8 + 366418956321b7 - 101482790158b6 + 202752004741b5 - 206393114617b4 - 34328951133b3 + 204810274377b2 + 344099280597*b1 - 3462415442405) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times$.

$n$	$97$	$277$	$415$	$737$
$\chi(n)$	$1$	$1$	$-1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

415.1

0.500000	−	2.34984i
0.500000	+	1.67743i
0.500000	+	5.51015i
0.500000	+	4.02698i
0.500000	−	6.86821i
0.500000	−	1.11200i
0.500000	−	2.10015i
0.500000	−	0.516427i
0.500000	+	2.34984i
0.500000	−	1.67743i
0.500000	−	5.51015i
0.500000	−	4.02698i
0.500000	+	6.86821i
0.500000	+	1.11200i
0.500000	+	2.10015i
0.500000	+	0.516427i

−

1.73205i

−9.29838

4.29519i

−3.00000

415.2

−

1.73205i

−4.22123

−

10.0250i

−3.00000

415.3

−

1.73205i

−3.55421

−

1.74513i

−3.00000

415.4

−

1.73205i

−2.80679

11.7691i

−3.00000

415.5

−

1.73205i

−0.789244

−

6.87634i

−3.00000

415.6

−

1.73205i

4.46245

2.74404i

−3.00000

415.7

−

1.73205i

5.81726

5.53259i

−3.00000

415.8

−

1.73205i

6.39015

4.69788i

−3.00000

415.9

1.73205i

−9.29838

−

4.29519i

−3.00000

415.10

1.73205i

−4.22123

10.0250i

−3.00000

415.11

1.73205i

−3.55421

1.74513i

−3.00000

415.12

1.73205i

−2.80679

−

11.7691i

−3.00000

415.13

1.73205i

−0.789244

6.87634i

−3.00000

415.14

1.73205i

4.46245

−

2.74404i

−3.00000

415.15

1.73205i

5.81726

−

5.53259i

−3.00000

415.16

1.73205i

6.39015

−

4.69788i

−3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1104.3.k.d	✓	16
4.b	odd	2	1	inner	1104.3.k.d	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1104.3.k.d	✓	16	1.a	even	1	1	trivial
1104.3.k.d	✓	16	4.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{8} + 4T_{5}^{7} - 102T_{5}^{6} - 296T_{5}^{5} + 2908T_{5}^{4} + 8712T_{5}^{3} - 22944T_{5}^{2} - 86976T_{5} - 51264 $ T5^8 + 4*T5^7 - 102*T5^6 - 296*T5^5 + 2908*T5^4 + 8712*T5^3 - 22944*T5^2 - 86976*T5 - 51264 acting on $S_{3}^{\mathrm{new}}(1104, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $ T^16
$3$	$ (T^{2} + 3)^{8} $ (T^2 + 3)^8
$5$	$ (T^{8} + 4 T^{7} + \cdots - 51264)^{2} $ (T^8 + 4T^7 - 102T^6 - 296T^5 + 2908T^4 + 8712T^3 - 22944T^2 - 86976*T - 51264)^2
$7$	$ T^{16} + \cdots + 188119977984 $ T^16 + 368T^14 + 51036T^12 + 3444032T^10 + 124073104T^8 + 2438308608T^6 + 25103467008T^4 + 118828560384*T^2 + 188119977984
$11$	$ T^{16} + \cdots + 5098202726400 $ T^16 + 924T^14 + 318240T^12 + 51035392T^10 + 3915118848T^8 + 129309327360T^6 + 1385704701952T^4 + 4861442654208*T^2 + 5098202726400
$13$	$ (T^{8} + 4 T^{7} + \cdots + 212320000)^{2} $ (T^8 + 4T^7 - 908T^6 - 7312T^5 + 225920T^4 + 2573696T^3 - 5861696T^2 - 83334656*T + 212320000)^2
$17$	$ (T^{8} + 14 T^{7} + \cdots - 2000526336)^{2} $ (T^8 + 14T^7 - 1370T^6 - 23628T^5 + 358692T^4 + 8616960T^3 + 20555136T^2 - 402312960*T - 2000526336)^2
$19$	$ T^{16} + \cdots + 45\!\cdots\!64 $ T^16 + 2340T^14 + 2168268T^12 + 1052008432T^10 + 291495525456T^8 + 46365932440128T^6 + 3931778533576960T^4 + 137615514906227712*T^2 + 45460980781092864
$23$	$ (T^{2} + 23)^{8} $ (T^2 + 23)^8
$29$	$ (T^{8} - 8 T^{7} + \cdots + 10109608704)^{2} $ (T^8 - 8T^7 - 2904T^6 + 8128T^5 + 2684992T^4 + 2830464T^3 - 735315072T^2 - 1237441536*T + 10109608704)^2
$31$	$ T^{16} + \cdots + 69\!\cdots\!00 $ T^16 + 14128T^14 + 83835328T^12 + 270925009920T^10 + 515867313475584T^8 + 581958371525197824T^6 + 366392984831583584256T^4 + 107117733548976714547200*T^2 + 6994364190929914429440000
$37$	$ (T^{8} + \cdots + 5328995500800)^{2} $ (T^8 - 6T^7 - 8396T^6 + 45768T^5 + 22210720T^4 - 88843680T^3 - 20743011264T^2 + 16508472192*T + 5328995500800)^2
$41$	$ (T^{8} - 8 T^{7} + \cdots + 1569554688)^{2} $ (T^8 - 8T^7 - 2704T^6 - 44192T^5 + 877664T^4 + 31338112T^3 + 326039296T^2 + 1350701568*T + 1569554688)^2
$43$	$ T^{16} + \cdots + 91\!\cdots\!00 $ T^16 + 18052T^14 + 132429292T^12 + 518385850800T^10 + 1185212766431952T^8 + 1617430422881837376T^6 + 1281722375880784071936T^4 + 537114782662228217852928*T^2 + 91233678821103140630630400
$47$	$ T^{16} + \cdots + 34\!\cdots\!00 $ T^16 + 29720T^14 + 349323696T^12 + 2045103195776T^10 + 6114128108691712T^8 + 8344544266556325888T^6 + 3534382691557125783552T^4 + 591310427257706986340352*T^2 + 34550840722885871822438400
$53$	$ (T^{8} + 24 T^{7} + \cdots + 8545699776)^{2} $ (T^8 + 24T^7 - 7494T^6 + 53800T^5 + 10160556T^4 - 156360120T^3 - 514496384T^2 + 4923416256*T + 8545699776)^2
$59$	$ T^{16} + \cdots + 99\!\cdots\!24 $ T^16 + 30744T^14 + 380953776T^12 + 2436064260736T^10 + 8498876869972224T^8 + 15533253939068583936T^6 + 12586066821239834804224T^4 + 2687529976277777856331776*T^2 + 99662151452515189678669824
$61$	$ (T^{8} + \cdots + 3752608538880)^{2} $ (T^8 - 74T^7 - 14308T^6 + 995032T^5 + 49157888T^4 - 2223440864T^3 - 80908188992T^2 + 60552208512*T + 3752608538880)^2
$67$	$ T^{16} + \cdots + 10\!\cdots\!44 $ T^16 + 39252T^14 + 610783308T^12 + 4778114314032T^10 + 19661710732940880T^8 + 40313814999716361024T^6 + 32586587502817240675584T^4 + 1374365467791338894668800*T^2 + 10331010286734582806482944
$71$	$ T^{16} + \cdots + 43\!\cdots\!24 $ T^16 + 57792T^14 + 1356962304T^12 + 16649543102464T^10 + 114115662011498496T^8 + 430272857724294266880T^6 + 807839403708778931027968T^4 + 574471360596698131479920640*T^2 + 43218217488803304608863617024
$73$	$ (T^{8} + \cdots + 35836917047040)^{2} $ (T^8 + 92T^7 - 22600T^6 - 1712848T^5 + 132600320T^4 + 6020346176T^3 - 200945597312T^2 - 2584599703296*T + 35836917047040)^2
$79$	$ T^{16} + \cdots + 19\!\cdots\!44 $ T^16 + 69088T^14 + 1817470012T^12 + 22408640555328T^10 + 125962843325259024T^8 + 251537730803067246336T^6 + 125537371826335127142912T^4 + 209944548091123635695616*T^2 + 19956162624485928505344
$83$	$ T^{16} + \cdots + 11\!\cdots\!64 $ T^16 + 70876T^14 + 2072165920T^12 + 32626353652992T^10 + 300268140091400448T^8 + 1623492213853842653184T^6 + 4823944668887997062430720T^4 + 6353272115611314916351279104*T^2 + 1178329960751632012920581259264
$89$	$ (T^{8} + \cdots - 11\!\cdots\!00)^{2} $ (T^8 + 86T^7 - 37138T^6 - 3056932T^5 + 412956788T^4 + 33783547280T^3 - 1124594951552T^2 - 110049407800320*T - 1153312016870400)^2
$97$	$ (T^{8} + \cdots + 55\!\cdots\!20)^{2} $ (T^8 + 200T^7 - 36536T^6 - 8072192T^5 + 272042816T^4 + 77748040576T^3 - 1130313371264T^2 - 214092689514496*T + 5591480314213120)^2
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1104.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{8} q^{3} + \beta_{3} q^{5} + (\beta_{13} - \beta_{8}) q^{7} - 3 q^{9}+O(q^{10}) \) q + b8 * q^3 + b3 * q^5 + (b13 - b8) * q^7 - 3 * q^9	\( q + \beta_{8} q^{3} + \beta_{3} q^{5} + (\beta_{13} - \beta_{8}) q^{7} - 3 q^{9} + (\beta_{10} + \beta_{8}) q^{11} + ( - \beta_{2} + \beta_1 - 1) q^{13} - \beta_{9} q^{15} + ( - \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{17} + (2 \beta_{12} - \beta_{9} + 2 \beta_{8}) q^{19} + (\beta_{5} + 2) q^{21} + \beta_{11} q^{23} + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 1) q^{25}+ \cdots + ( - 3 \beta_{10} - 3 \beta_{8}) q^{99}+O(q^{100}) \) q + b8 * q^3 + b3 * q^5 + (b13 - b8) * q^7 - 3 * q^9 + (b10 + b8) * q^11 + (-b2 + b1 - 1) * q^13 - b9 * q^15 + (-b6 + b4 + b2 - 1) * q^17 + (2b12 - b9 + 2b8) * q^19 + (b5 + 2) * q^21 + b11 * q^23 + (b6 + b5 - b4 - 2b3 + b1 + 1) q^25 - 3b8 q^27 + (b7 + b6 - b3 + b1) * q^29 + (b15 - b14 + b13 + 2b12 + 4b11 - b10 - b9 + 3b8) q^31 + (-b7 - 2) * q^33 + (-3b13 + b12 - b10 + 2b9 - 4b8) q^35 + (b7 - b6 - 3b5 + b4 + 2b3 + b2 + 4b1 + 3) q^37 + (b14 + b12 + b11 - b9 + b8) * q^39 + (-b6 - b5 + b4 + 2b3 + b2 + 2b1 + 3) * q^41 + (-b15 + 3b13 - 4b12 + b11 + b10 - 2b8) q^43 - 3b3 q^45 + (-b15 + 3b13 - 5b12 + 7b11 + b10 + 2b8) * q^47 + (-b7 - b5 - b4 + 3b3 + 4b1 + 5) * q^49 + (-b15 - b13 + 2b12 + b11 - b8) q^51 + (b7 - b6 - 2b5 + b4 + 4b3 - b2 + 2b1 - 1) q^53 + (b15 - 2b13 + 5b11 - 2b10 + 2b8) * q^55 + (-2b4 - b3 - 2) q^57 + (b14 - 8b13 - b12 + b11 + 3b9 - b8) * q^59 + (-b7 + 3b6 + b5 - 3b4 + 2b3 - b2 + 2b1 + 9) * q^61 + (-3b13 + 3b8) * q^63 + (b7 + 5b6 + 2b5 - b4 - 7b3 - 3b2 + 2b1 + 1) q^65 + (2b14 + 2b13 - 2b12 + 2b11 + 4b10 - b9 + 2b8) * q^67 - b1 * q^69 + (-b15 + b14 + b13 + 8b11 - 5b10 + 3b9 - 11b8) * q^71 + (-2b7 - 3b6 + b5 + b4 - 6b3 + b2 + 4b1 - 13) * q^73 + (b15 - b14 - 2b13 - 3b12 + 4b11 + 3b9) * q^75 + (-3b7 + 5b5 + 3b4 + b3 - 3b2 - b1 + 7) * q^77 + (b15 - 2b14 + 4b13 - 2b12 + 9b11 + 5b10 + 7b9 - 7b8) q^79 + 9 * q^81 + (2b15 - 2b14 - 2b13 - 2b12 + 8b11 - 3b10 - 4b9 + 9b8) * q^83 + (-b7 - 2b6 - 3b5 + 3b4 - 5b3 + 6b2 - 10) q^85 + (b15 - b14 + b13 + 4b11 + 3b10 + b9 + b8) * q^87 + (b7 - 2b5 - 6b4 + 5b3 - 4b2 + 4b1 - 10) q^89 + (-b15 - 3b13 - 2b12 + b11 - 5b10 + b9 - 4b8) * q^91 + (b7 - 3b6 - 2b4 - b3 - 3b1 - 6) q^93 + (b15 - b14 + 2b13 + 3b12 + 16b11 + 3b9 + 9b8) q^95 + (2b6 + 6b5 - 18b3 - 4b2 + 6b1 - 38) q^97 + (-3b10 - 3b8) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{5} - 48 q^{9}+O(q^{10}) \) 16 * q - 8 * q^5 - 48 * q^9	\( 16 q - 8 q^{5} - 48 q^{9} - 8 q^{13} - 28 q^{17} + 36 q^{21} + 40 q^{25} + 16 q^{29} - 36 q^{33} + 12 q^{37} + 16 q^{41} + 24 q^{45} + 48 q^{49} - 48 q^{53} - 24 q^{57} + 148 q^{61} + 128 q^{65} - 184 q^{73} + 136 q^{77} + 144 q^{81} - 192 q^{85} - 172 q^{89} - 96 q^{93} - 400 q^{97}+O(q^{100}) \) 16 * q - 8 * q^5 - 48 * q^9 - 8 * q^13 - 28 * q^17 + 36 * q^21 + 40 * q^25 + 16 * q^29 - 36 * q^33 + 12 * q^37 + 16 * q^41 + 24 * q^45 + 48 * q^49 - 48 * q^53 - 24 * q^57 + 148 * q^61 + 128 * q^65 - 184 * q^73 + 136 * q^77 + 144 * q^81 - 192 * q^85 - 172 * q^89 - 96 * q^93 - 400 * 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	1104.3.k.d

Level	$1104$
Weight	$3$
Character orbit	1104.k
Analytic conductor	$30.082$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(30.0818211854\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - 8 x^{15} + 138 x^{14} - 826 x^{13} + 6668 x^{12} - 29634 x^{11} + 142919 x^{10} + \cdots + 1513188 \) x^16 - 8x^15 + 138x^14 - 826x^13 + 6668x^12 - 29634x^11 + 142919x^10 - 461683x^9 + 1440096x^8 - 3284228x^7 + 6812887x^6 - 10590573x^5 + 14250513x^4 - 13904316x^3 + 10818534x^2 - 5200488*x + 1513188
Coefficient ring:	\(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index:	\( 2^{24}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 143240 \nu^{14} + 1002680 \nu^{13} - 21508999 \nu^{12} + 116019154 \nu^{11} + \cdots - 104230692531 ) / 36162037683 \) (-143240v^14 + 1002680v^13 - 21508999v^12 + 116019154v^11 - 1174847217v^10 + 4834624380v^9 - 29210011687v^8 + 89067544522v^7 - 333420058790v^6 + 707595997217v^5 - 1531283458128v^4 + 1978099521357v^3 - 2071686272763v^2 + 1187081591514v - 104230692531) / 36162037683
\(\beta_{2}\)	\(=\)	\( ( - 140512893561128 \nu^{14} + 983590254927896 \nu^{13} + \cdots - 26\!\cdots\!98 ) / 29\!\cdots\!65 \) (-140512893561128v^14 + 983590254927896v^13 - 18510868217912704v^12 + 98278535993413576v^11 - 850461389379413526v^10 + 3374862601366558038v^9 - 17116042836711069946v^8 + 49255872947013998284v^7 - 157244179596932520446v^6 + 312472538527996968506v^5 - 631487712247385162802v^4 + 793422541009332263604v^3 - 938201206719996961554v^2 + 586293176959558472202v - 265144162684272693498) / 2982588997745619765
\(\beta_{3}\)	\(=\)	\( ( - 61651462937429 \nu^{14} + 431560240562003 \nu^{13} + \cdots - 14\!\cdots\!20 ) / 11\!\cdots\!06 \) (-61651462937429v^14 + 431560240562003v^13 - 7815467491532791v^12 + 41282521821890707v^11 - 336809494333808661v^10 + 1315909874035106229v^9 - 6078235838289150178v^8 + 16853959530586656421v^7 - 46516441353098159039v^6 + 85651693367467937957v^5 - 141843815307189506922v^4 + 158183644771021559439v^3 - 139103351112463170294v^2 + 71839608599154552558v - 14552280079029593820) / 1193035599098247906
\(\beta_{4}\)	\(=\)	\( ( 736677511463 \nu^{14} - 5156742580241 \nu^{13} + 92448944196484 \nu^{12} + \cdots + 19\!\cdots\!08 ) / 12\!\cdots\!35 \) (736677511463v^14 - 5156742580241v^13 + 92448944196484v^12 - 487656011635771v^11 + 3912163603364916v^10 - 15213540274992423v^9 + 68137763136299071v^8 - 186423337254956809v^7 + 483982033848002771v^6 - 858218298435326516v^5 + 1261144211172973422v^4 - 1281561783175958934v^3 + 931583140113439659v^2 - 406942725600337092v + 190949628595272408) / 12479451873412635
\(\beta_{5}\)	\(=\)	\( ( - 173475766615451 \nu^{14} + \cdots - 62\!\cdots\!56 ) / 19\!\cdots\!10 \) (-173475766615451v^14 + 1214330366308157v^13 - 22146369274426213v^12 + 117091920884551237v^11 - 965746218404284227v^10 + 3784330024310045871v^9 - 17786552900171991322v^8 + 49678628515305734683v^7 - 140787776948808025457v^6 + 263146400787940753967v^5 - 448193545831150968714v^4 + 508817191975232301273v^3 - 457626329274894969918v^2 + 239837413464431586114v - 62942295370014168156) / 1988392665163746510
\(\beta_{6}\)	\(=\)	\( ( 596945386449457 \nu^{14} + \cdots + 44\!\cdots\!22 ) / 59\!\cdots\!30 \) (596945386449457v^14 - 4178617705146199v^13 + 75871078545894161v^12 - 400904441108464379v^11 + 3285141764533346979v^10 - 12850341834478462497v^9 + 59838076746697497104v^8 - 166489478451587776901v^7 + 467168946123552876859v^6 - 868534413342268617679v^5 + 1491881053620786804498v^4 - 1706856230571506447361v^3 + 1678799910654557699316v^2 - 945914049675405653358v + 446152584287023331022) / 5965177995491239530
\(\beta_{7}\)	\(=\)	\( ( 133672420258644 \nu^{14} - 935706941810508 \nu^{13} + \cdots + 10\!\cdots\!54 ) / 99\!\cdots\!55 \) (133672420258644v^14 - 935706941810508v^13 + 17171941233676402v^12 - 90867457158521808v^11 + 756877443119919358v^10 - 2973736540426297324v^9 + 14209279901573393938v^8 - 39956012080285559512v^7 + 116507751527240357178v^6 - 221207505569571992908v^5 + 395575251579884735316v^4 - 463618698920596016172v^3 + 463336523497315177842v^2 - 262555233287807320446v + 106927256919080268954) / 994196332581873255
\(\beta_{8}\)	\(=\)	\( ( - 364783933592 \nu^{15} + 2735879501940 \nu^{14} - 47785245441398 \nu^{13} + \cdots - 54\!\cdots\!79 ) / 57\!\cdots\!05 \) (-364783933592v^15 + 2735879501940v^14 - 47785245441398v^13 + 269109922922997v^12 - 2145874208179796v^11 + 8933386172217309v^10 - 41011100283214000v^9 + 121794253326673923v^8 - 340834264218128242v^7 + 681433889011663296v^6 - 1152050074258213375v^5 + 1420292966216860140v^4 - 1244645509413599559v^3 + 718313619593160561v^2 - 60592049111558046*v - 54856469300366079) / 57834524396273805
\(\beta_{9}\)	\(=\)	\( ( 10\!\cdots\!34 \nu^{15} + \cdots + 21\!\cdots\!68 ) / 12\!\cdots\!30 \) (1044153256528602334v^15 - 7831149423964517505v^14 + 128797421366924688011v^13 - 718410805954881956579v^12 + 5175295542860237437007v^11 - 20822905972673887115253v^10 + 78355300349435966843615v^9 - 207720907891849786713816v^8 + 342384095157667430604289v^7 - 359719298358252324081487v^6 - 724556509920181101411245v^5 + 2318862016011531920603100v^4 - 5625753970578771605979867v^3 + 6050567213534430780525318v^2 - 6149842594930221656423658*v + 2146834333718389709447868) / 122928399738418300487730
\(\beta_{10}\)	\(=\)	\( ( - 17\!\cdots\!44 \nu^{15} + \cdots - 32\!\cdots\!43 ) / 18\!\cdots\!95 \) (-1722065734930545844v^15 + 12915493011979093830v^14 - 216873600605906305376v^13 + 1213793426590041395189v^12 - 9052915177619062762922v^11 + 36870252734580758946453v^10 - 148516527595149345831410v^9 + 410901612457195541954721v^8 - 827178565521007296033004v^7 + 1210265275797529849550212v^6 + 181581588596255553108935v^5 - 2686149828938583768413940v^4 + 9388699372268429140149927v^3 - 11016167101045461861767373v^2 + 9964300193829100052180088*v - 3253280735329265372359743) / 184392599607627450731595
\(\beta_{11}\)	\(=\)	\( ( 59797395988 \nu^{15} - 448480469910 \nu^{14} + 7805099807922 \nu^{13} + \cdots - 34\!\cdots\!99 ) / 34\!\cdots\!45 \) (59797395988v^15 - 448480469910v^14 + 7805099807922v^13 - 43931194957858v^12 + 348635123444094v^11 - 1449214332752076v^10 + 6617237783394080v^9 - 19601260892838882v^8 + 54845019755651658v^7 - 109707925246762334v^6 + 192179916838007010v^5 - 245361037678811850v^4 + 261712142208796776v^3 - 190958126520367644v^2 + 119620660925447424*v - 34104766592492199) / 3405820681546245
\(\beta_{12}\)	\(=\)	\( ( 74\!\cdots\!66 \nu^{15} + \cdots + 33\!\cdots\!32 ) / 36\!\cdots\!90 \) (7424501752020233666v^15 - 55683763140151752495v^14 + 951986296990989942129v^13 - 5343373856149133044331v^12 + 41196252998759408739033v^11 - 169660260638979348051957v^10 + 735883721118657791887885v^9 - 2123209069803554798033874v^8 + 5379284958492603307423611v^7 - 9994829906635426218304393v^6 + 14324398855990884679846695v^5 - 15013718549657043129675600v^4 + 9805837739755222525549017v^3 - 3506036404446518754666288v^2 - 141504167804176681029162*v + 333398238725058745468032) / 368785199215254901463190
\(\beta_{13}\)	\(=\)	\( ( - 83\!\cdots\!26 \nu^{15} + \cdots + 41\!\cdots\!78 ) / 36\!\cdots\!90 \) (-8387773236519027026v^15 + 62908299273892702695v^14 - 1107602679058388553929v^13 + 6245308208225486276331v^12 - 50440066038059899297073v^11 + 210821013171287983607517v^10 - 989318096023402546457485v^9 + 2969323475358288777319074v^8 - 8650295041975881051454051v^7 + 17763837194285607899313273v^6 - 32457002201890941826150975v^5 + 42705505935067019039739600v^4 - 45575768800825827235796097v^3 + 32846970846284746451346828v^2 - 17000061464560696206868038*v + 4110617490546327071649678) / 368785199215254901463190
\(\beta_{14}\)	\(=\)	\( ( 31\!\cdots\!12 \nu^{15} + \cdots - 20\!\cdots\!96 ) / 18\!\cdots\!95 \) (31971685675174392712v^15 - 239787642563807945340v^14 + 4247950514778713596278v^13 - 23974899100510551204817v^12 + 195497640034726024232556v^11 - 819514044425589461802249v^10 + 3906900237260637444663500v^9 - 11813136753196059364256553v^8 + 35343710282363157618035142v^7 - 73808167747144639244697386v^6 + 140861966289547476711747825v^5 - 191478313830267286867571340v^4 + 218928892359780571067554749v^3 - 167058603760681400926292061v^2 + 86981418053892673560365976*v - 20610356981310823045409496) / 184392599607627450731595
\(\beta_{15}\)	\(=\)	\( ( 40\!\cdots\!18 \nu^{15} + \cdots - 33\!\cdots\!29 ) / 18\!\cdots\!95 \) (40203551511379922018v^15 - 301526636335349415135v^14 + 5341822814289059551937v^13 - 30148694308459420958043v^12 + 245914028199773320687949v^11 - 1030952456471422458730251v^10 + 4920817694055790147466395v^9 - 14887430501306990185747452v^8 + 44726238992160677994814303v^7 - 93650739536607437208185109v^6 + 181030841504415058317907495v^5 - 248507418159712400695892070v^4 + 293750937415473395373402651v^3 - 230422034440951844975973264v^2 + 131508109306323032726226234*v - 33829607825499319012538829) / 184392599607627450731595

\(\nu\)	\(=\)	\( ( \beta_{15} - \beta_{14} + \beta_{13} - 2\beta_{11} - 3\beta_{10} - \beta_{9} + \beta_{8} + 6 ) / 12 \) (b15 - b14 + b13 - 2b11 - 3b10 - b9 + b8 + 6) / 12
\(\nu^{2}\)	\(=\)	\( ( \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} + 7 \beta_{7} + \cdots - 156 ) / 12 \) (b15 - b14 + b13 - 2b11 - 3b10 - b9 + b8 + 7b7 + 3b6 + 4b5 - 4b4 + 7b3 + 6b2 + 3*b1 - 156) / 12
\(\nu^{3}\)	\(=\)	\( ( - 19 \beta_{15} + 13 \beta_{14} - 39 \beta_{13} - 38 \beta_{12} + 96 \beta_{11} + 67 \beta_{10} + \cdots - 158 ) / 8 \) (-19b15 + 13b14 - 39b13 - 38b12 + 96b11 + 67b10 + 29b9 - 73b8 + 7b7 + 3b6 + 4b5 - 4b4 + 7b3 + 6b2 + 3*b1 - 158) / 8
\(\nu^{4}\)	\(=\)	\( ( - 58 \beta_{15} + 40 \beta_{14} - 118 \beta_{13} - 114 \beta_{12} + 290 \beta_{11} + 204 \beta_{10} + \cdots + 4941 ) / 12 \) (-58b15 + 40b14 - 118b13 - 114b12 + 290b11 + 204b10 + 88b9 - 220b8 - 346b7 - 45b6 - 247b5 + 193b4 - 100b3 - 261b2 - 156*b1 + 4941) / 12
\(\nu^{5}\)	\(=\)	\( ( 1868 \beta_{15} - 773 \beta_{14} + 4721 \beta_{13} + 6198 \beta_{12} - 12919 \beta_{11} - 8619 \beta_{10} + \cdots + 25497 ) / 24 \) (1868b15 - 773b14 + 4721b13 + 6198b12 - 12919b11 - 8619b10 - 3809b9 + 17753b8 - 1765b7 - 240b6 - 1255b5 + 985b4 - 535b3 - 1335b2 - 795*b1 + 25497) / 24
\(\nu^{6}\)	\(=\)	\( ( 1965 \beta_{15} - 840 \beta_{14} + 4918 \beta_{13} + 6388 \beta_{12} - 13403 \beta_{11} - 8960 \beta_{10} + \cdots - 125049 ) / 8 \) (1965b15 - 840b14 + 4918b13 + 6388b12 - 13403b11 - 8960b10 - 3956b9 + 18120b8 + 10584b7 - 535b6 + 7305b5 - 5719b4 + 706b3 + 7055b2 + 6498*b1 - 125049) / 8
\(\nu^{7}\)	\(=\)	\( ( - 65267 \beta_{15} + 9608 \beta_{14} - 207098 \beta_{13} - 282936 \beta_{12} + 543925 \beta_{11} + \cdots - 1402809 ) / 24 \) (-65267b15 + 9608b14 - 207098b13 - 282936b12 + 543925b11 + 375966b10 + 161504b9 - 973418b8 + 117334b7 - 4767b6 + 81109b5 - 63511b4 + 9310b3 + 78771b2 + 71022*b1 - 1402809) / 24
\(\nu^{8}\)	\(=\)	\( ( - 144425 \beta_{15} + 25190 \beta_{14} - 448898 \beta_{13} - 610854 \beta_{12} + 1182349 \beta_{11} + \cdots + 7438467 ) / 12 \) (-144425b15 + 25190b14 - 448898b13 - 610854b12 + 1182349b11 + 815130b10 + 350906b9 - 2074190b8 - 700706b7 + 111243b6 - 447239b5 + 380903b4 + 15568b3 - 430689b2 - 527550*b1 + 7438467) / 12
\(\nu^{9}\)	\(=\)	\( ( 786941 \beta_{15} + 76505 \beta_{14} + 3066037 \beta_{13} + 4071420 \beta_{12} - 7438130 \beta_{11} + \cdots + 25157032 ) / 8 \) (786941b15 + 76505b14 + 3066037b13 + 4071420b12 - 7438130b11 - 5367093b10 - 2259193b9 + 15444777b8 - 2339271b7 + 342921b6 - 1505700b5 + 1271118b4 + 27321b3 - 1451490b2 - 1725813*b1 + 25157032) / 8
\(\nu^{10}\)	\(=\)	\( ( 7006169 \beta_{15} + 375841 \beta_{14} + 26414243 \beta_{13} + 35184699 \beta_{12} - 64795777 \beta_{11} + \cdots - 296847378 ) / 12 \) (7006169b15 + 375841b14 + 26414243b13 + 35184699b12 - 64795777b11 - 46461777b10 - 19617722b9 + 131583614b8 + 29841506b7 - 6708183b6 + 17733641b5 - 16486502b4 - 1942108b3 + 17409627b2 + 25440567*b1 - 296847378) / 12
\(\nu^{11}\)	\(=\)	\( ( - 86706556 \beta_{15} - 26239643 \beta_{14} - 395314939 \beta_{13} - 506910600 \beta_{12} + \cdots - 3972853521 ) / 24 \) (-86706556b15 - 26239643b14 - 395314939b13 - 506910600b12 + 895087745b11 + 668817015b10 + 278101585b9 - 2035263349b8 + 393896723b7 - 83270088b6 + 237382871b5 - 217016789b4 - 22006105b3 + 232303137b2 + 328096131*b1 - 3972853521) / 24
\(\nu^{12}\)	\(=\)	\( ( - 226424231 \beta_{15} - 54939440 \beta_{14} - 989381702 \beta_{13} - 1278703510 \beta_{12} + \cdots + 7771608171 ) / 8 \) (-226424231b15 - 54939440b14 - 989381702b13 - 1278703510b12 + 2278648423b11 + 1687519532b10 + 704014434b9 - 5058690354b8 - 813686592b7 + 216343567b6 - 457984457b5 + 456044425b4 + 71859130b3 - 459011083b2 - 748302528*b1 + 7771608171) / 8
\(\nu^{13}\)	\(=\)	\( ( 3164267743 \beta_{15} + 1465261079 \beta_{14} + 16128850513 \beta_{13} + 20125941786 \beta_{12} + \cdots + 205050601794 ) / 24 \) (3164267743b15 + 1465261079b14 + 16128850513b13 + 20125941786b12 - 34742323364b11 - 26618970363b10 - 10986061111b9 + 83650072795b8 - 21159673925b7 + 5325913827b6 - 12127699766b5 + 11807606876b4 + 1688882065b3 - 12077714454b2 - 18983480685*b1 + 205050601794) / 24
\(\nu^{14}\)	\(=\)	\( ( 32940973430 \beta_{15} + 12777792661 \beta_{14} + 159747218747 \beta_{13} + 201499882182 \beta_{12} + \cdots - 881172741837 ) / 24 \) (32940973430b15 + 12777792661b14 + 159747218747b13 + 201499882182b12 - 351371949883b11 - 266334765855b10 - 110295621443b9 + 824804017865b8 + 94690455053b7 - 27644337114b6 + 51288583355b5 - 53627158181b4 - 9656332447b3 + 52261989015b2 + 91270154877*b1 - 881172741837) / 24
\(\nu^{15}\)	\(=\)	\( ( - 37122678030 \beta_{15} - 21662030073 \beta_{14} - 204462583523 \beta_{13} - 250305255248 \beta_{12} + \cdots - 3462415442405 ) / 8 \) (-37122678030b15 - 21662030073b14 - 204462583523b13 - 250305255248b12 + 425589844477b11 + 331779133909b10 + 136280976853b9 - 1063434188961b8 + 366418956321b7 - 101482790158b6 + 202752004741b5 - 206393114617b4 - 34328951133b3 + 204810274377b2 + 344099280597*b1 - 3462415442405) / 8

\(n\)	\(97\)	\(277\)	\(415\)	\(737\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 415.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \) T^16
$3$	\( (T^{2} + 3)^{8} \) (T^2 + 3)^8
$5$	\( (T^{8} + 4 T^{7} + \cdots - 51264)^{2} \) (T^8 + 4T^7 - 102T^6 - 296T^5 + 2908T^4 + 8712T^3 - 22944T^2 - 86976*T - 51264)^2
$7$	\( T^{16} + \cdots + 188119977984 \) T^16 + 368T^14 + 51036T^12 + 3444032T^10 + 124073104T^8 + 2438308608T^6 + 25103467008T^4 + 118828560384*T^2 + 188119977984
$11$	\( T^{16} + \cdots + 5098202726400 \) T^16 + 924T^14 + 318240T^12 + 51035392T^10 + 3915118848T^8 + 129309327360T^6 + 1385704701952T^4 + 4861442654208*T^2 + 5098202726400
$13$	\( (T^{8} + 4 T^{7} + \cdots + 212320000)^{2} \) (T^8 + 4T^7 - 908T^6 - 7312T^5 + 225920T^4 + 2573696T^3 - 5861696T^2 - 83334656*T + 212320000)^2
$17$	\( (T^{8} + 14 T^{7} + \cdots - 2000526336)^{2} \) (T^8 + 14T^7 - 1370T^6 - 23628T^5 + 358692T^4 + 8616960T^3 + 20555136T^2 - 402312960*T - 2000526336)^2
$19$	\( T^{16} + \cdots + 45\!\cdots\!64 \) T^16 + 2340T^14 + 2168268T^12 + 1052008432T^10 + 291495525456T^8 + 46365932440128T^6 + 3931778533576960T^4 + 137615514906227712*T^2 + 45460980781092864
$23$	\( (T^{2} + 23)^{8} \) (T^2 + 23)^8
$29$	\( (T^{8} - 8 T^{7} + \cdots + 10109608704)^{2} \) (T^8 - 8T^7 - 2904T^6 + 8128T^5 + 2684992T^4 + 2830464T^3 - 735315072T^2 - 1237441536*T + 10109608704)^2
$31$	\( T^{16} + \cdots + 69\!\cdots\!00 \) T^16 + 14128T^14 + 83835328T^12 + 270925009920T^10 + 515867313475584T^8 + 581958371525197824T^6 + 366392984831583584256T^4 + 107117733548976714547200*T^2 + 6994364190929914429440000
$37$	\( (T^{8} + \cdots + 5328995500800)^{2} \) (T^8 - 6T^7 - 8396T^6 + 45768T^5 + 22210720T^4 - 88843680T^3 - 20743011264T^2 + 16508472192*T + 5328995500800)^2
$41$	\( (T^{8} - 8 T^{7} + \cdots + 1569554688)^{2} \) (T^8 - 8T^7 - 2704T^6 - 44192T^5 + 877664T^4 + 31338112T^3 + 326039296T^2 + 1350701568*T + 1569554688)^2
$43$	\( T^{16} + \cdots + 91\!\cdots\!00 \) T^16 + 18052T^14 + 132429292T^12 + 518385850800T^10 + 1185212766431952T^8 + 1617430422881837376T^6 + 1281722375880784071936T^4 + 537114782662228217852928*T^2 + 91233678821103140630630400
$47$	\( T^{16} + \cdots + 34\!\cdots\!00 \) T^16 + 29720T^14 + 349323696T^12 + 2045103195776T^10 + 6114128108691712T^8 + 8344544266556325888T^6 + 3534382691557125783552T^4 + 591310427257706986340352*T^2 + 34550840722885871822438400
$53$	\( (T^{8} + 24 T^{7} + \cdots + 8545699776)^{2} \) (T^8 + 24T^7 - 7494T^6 + 53800T^5 + 10160556T^4 - 156360120T^3 - 514496384T^2 + 4923416256*T + 8545699776)^2
$59$	\( T^{16} + \cdots + 99\!\cdots\!24 \) T^16 + 30744T^14 + 380953776T^12 + 2436064260736T^10 + 8498876869972224T^8 + 15533253939068583936T^6 + 12586066821239834804224T^4 + 2687529976277777856331776*T^2 + 99662151452515189678669824
$61$	\( (T^{8} + \cdots + 3752608538880)^{2} \) (T^8 - 74T^7 - 14308T^6 + 995032T^5 + 49157888T^4 - 2223440864T^3 - 80908188992T^2 + 60552208512*T + 3752608538880)^2
$67$	\( T^{16} + \cdots + 10\!\cdots\!44 \) T^16 + 39252T^14 + 610783308T^12 + 4778114314032T^10 + 19661710732940880T^8 + 40313814999716361024T^6 + 32586587502817240675584T^4 + 1374365467791338894668800*T^2 + 10331010286734582806482944
$71$	\( T^{16} + \cdots + 43\!\cdots\!24 \) T^16 + 57792T^14 + 1356962304T^12 + 16649543102464T^10 + 114115662011498496T^8 + 430272857724294266880T^6 + 807839403708778931027968T^4 + 574471360596698131479920640*T^2 + 43218217488803304608863617024
$73$	\( (T^{8} + \cdots + 35836917047040)^{2} \) (T^8 + 92T^7 - 22600T^6 - 1712848T^5 + 132600320T^4 + 6020346176T^3 - 200945597312T^2 - 2584599703296*T + 35836917047040)^2
$79$	\( T^{16} + \cdots + 19\!\cdots\!44 \) T^16 + 69088T^14 + 1817470012T^12 + 22408640555328T^10 + 125962843325259024T^8 + 251537730803067246336T^6 + 125537371826335127142912T^4 + 209944548091123635695616*T^2 + 19956162624485928505344
$83$	\( T^{16} + \cdots + 11\!\cdots\!64 \) T^16 + 70876T^14 + 2072165920T^12 + 32626353652992T^10 + 300268140091400448T^8 + 1623492213853842653184T^6 + 4823944668887997062430720T^4 + 6353272115611314916351279104*T^2 + 1178329960751632012920581259264
$89$	\( (T^{8} + \cdots - 11\!\cdots\!00)^{2} \) (T^8 + 86T^7 - 37138T^6 - 3056932T^5 + 412956788T^4 + 33783547280T^3 - 1124594951552T^2 - 110049407800320*T - 1153312016870400)^2
$97$	\( (T^{8} + \cdots + 55\!\cdots\!20)^{2} \) (T^8 + 200T^7 - 36536T^6 - 8072192T^5 + 272042816T^4 + 77748040576T^3 - 1130313371264T^2 - 214092689514496*T + 5591480314213120)^2
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