LMFDB - Newform orbit 8004.2.a.j

Newspace parameters

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + 46229 x^{8} - 47547 x^{7} - 122604 x^{6} + 65278 x^{5} + 151028 x^{4} - 17988 x^{3} - 67608 x^{2} - 8424 x + 3888$
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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 + q^{3} + \beta_{1} q^{5} -\beta_{7} q^{7} + q^{9} +O(q^{10})$	\( q + q^{3} + \beta_{1} q^{5} -\beta_{7} q^{7} + q^{9} -\beta_{6} q^{11} + \beta_{4} q^{13} + \beta_{1} q^{15} + \beta_{9} q^{17} + ( 1 + \beta_{8} ) q^{19} -\beta_{7} q^{21} + q^{23} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{25} + q^{27} - q^{29} + ( 1 - \beta_{7} + \beta_{10} ) q^{31} -\beta_{6} q^{33} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{14} ) q^{35} + ( \beta_{1} + \beta_{14} ) q^{37} + \beta_{4} q^{39} + ( -\beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{41} + ( 1 + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{14} ) q^{43} + \beta_{1} q^{45} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{15} ) q^{47} + ( 2 - \beta_{1} - \beta_{2} - \beta_{8} - \beta_{11} - \beta_{13} ) q^{49} + \beta_{9} q^{51} + ( 1 - \beta_{8} - \beta_{11} - \beta_{12} ) q^{53} + ( 2 + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{12} - \beta_{14} ) q^{55} + ( 1 + \beta_{8} ) q^{57} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{12} - 2 \beta_{14} ) q^{59} + ( 2 + \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{13} ) q^{61} -\beta_{7} q^{63} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{7} - \beta_{12} + 2 \beta_{14} ) q^{65} + ( 3 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{15} ) q^{67} + q^{69} + ( \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{71} + ( 2 - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{13} - \beta_{14} ) q^{73} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{75} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{77} + ( 1 - \beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{79} + q^{81} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{83} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{85} - q^{87} + ( \beta_{7} - \beta_{11} + \beta_{12} ) q^{89} + ( 1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{91} + ( 1 - \beta_{7} + \beta_{10} ) q^{93} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{95} + ( 2 + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{97} -\beta_{6} q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16q + 16q^{3} + 3q^{5} + 4q^{7} + 16q^{9} + O(q^{10}) $	$ 16q + 16q^{3} + 3q^{5} + 4q^{7} + 16q^{9} + 5q^{11} + 6q^{13} + 3q^{15} + 3q^{17} + 11q^{19} + 4q^{21} + 16q^{23} + 27q^{25} + 16q^{27} - 16q^{29} + 14q^{31} + 5q^{33} + 11q^{35} + 4q^{37} + 6q^{39} + 11q^{41} + 23q^{43} + 3q^{45} - 2q^{47} + 34q^{49} + 3q^{51} + 19q^{53} + 31q^{55} + 11q^{57} + 32q^{59} + 19q^{61} + 4q^{63} + 6q^{65} + 33q^{67} + 16q^{69} - 5q^{71} + 23q^{73} + 27q^{75} + 42q^{77} + 24q^{79} + 16q^{81} + 7q^{83} - 16q^{87} - 2q^{89} + 25q^{91} + 14q^{93} + 7q^{95} + 33q^{97} + 5q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + 46229 x^{8} - 47547 x^{7} - 122604 x^{6} + 65278 x^{5} + 151028 x^{4} - 17988 x^{3} - 67608 x^{2} - 8424 x + 3888$:

$\beta_{0}$	$=$	$ 1 $
$\beta_{1}$	$=$	$ \nu $
$\beta_{2}$	$=$	$($$128089526929863711 \nu^{15} - 1780989647760093687 \nu^{14} - 2343590043301536281 \nu^{13} + 82428009307722432792 \nu^{12} - 39868293740417462800 \nu^{11} - 1434850612960902643364 \nu^{10} + 1061787358987892121509 \nu^{9} + 12059212146105833185194 \nu^{8} - 6849714181952244815581 \nu^{7} - 49998222735113343315999 \nu^{6} + 12943701821822810175386 \nu^{5} + 87862981358437561842546 \nu^{4} - 985493942779777495944 \nu^{3} - 44010301598073058173068 \nu^{2} - 5734576008998611033216 \nu + 1436096932911927060552$$)/ $$60\!\cdots\!08$
$\beta_{3}$	$=$	$($$-958087900246403592 \nu^{15} + 2000567955267874333 \nu^{14} + 49357290400200766149 \nu^{13} - 88974714684163101079 \nu^{12} - 963378769466844341132 \nu^{11} + 1415048553601951450082 \nu^{10} + 8893034704374805484880 \nu^{9} - 9960991537155374353069 \nu^{8} - 38488851452268663931116 \nu^{7} + 31123533131595448591055 \nu^{6} + 63124294156284455139465 \nu^{5} - 39843524562808100614170 \nu^{4} - 17036408551707978644276 \nu^{3} + 2778605399574854232452 \nu^{2} + 4140098642180225184060 \nu + 10187796734063718568848$$)/ $$18\!\cdots\!24$
$\beta_{4}$	$=$	$($$4486141389040236929 \nu^{15} - 16191524997021342828 \nu^{14} - 211861740633534736190 \nu^{13} + 721401882595358991197 \nu^{12} + 3820388029937201339968 \nu^{11} - 11804652191031635223666 \nu^{10} - 34218304176590055106445 \nu^{9} + 91025874133171328629977 \nu^{8} + 160820231928040018855981 \nu^{7} - 348051881480837120172498 \nu^{6} - 374464283660322591966429 \nu^{5} + 631086737220708958112600 \nu^{4} + 373450758905607661475524 \nu^{3} - 436002491424203250589680 \nu^{2} - 119911680097050427614324 \nu + 62507843402239991683728$$)/ $$54\!\cdots\!72$
$\beta_{5}$	$=$	$($$-2170353582856352493 \nu^{15} + 6394679216719266718 \nu^{14} + 105192616124931196248 \nu^{13} - 273011890888993264339 \nu^{12} - 1963933514692243560836 \nu^{11} + 4159658356999503153722 \nu^{10} + 18290876230917575384385 \nu^{9} - 28530527789909789588287 \nu^{8} - 88888441699606672234641 \nu^{7} + 90582649286250348855860 \nu^{6} + 210029308309103838088587 \nu^{5} - 128266188808560877480848 \nu^{4} - 209073282062352857065304 \nu^{3} + 66867424884872498482256 \nu^{2} + 63961903110921104602836 \nu - 6638130556491638529144$$)/ $$18\!\cdots\!24$
$\beta_{6}$	$=$	$($$1181647930064972053 \nu^{15} - 4488671540266881279 \nu^{14} - 53334067423612189198 \nu^{13} + 193579150520849398669 \nu^{12} + 901332302367398689427 \nu^{11} - 3015546321065278419096 \nu^{10} - 7391050976096937823363 \nu^{9} + 21665705786600716578792 \nu^{8} + 30526909733759416311410 \nu^{7} - 74930972648825634056871 \nu^{6} - 56119572043326164111031 \nu^{5} + 117837296847755457978037 \nu^{4} + 33444989655224619931886 \nu^{3} - 61721460041205402182268 \nu^{2} - 7863407220333139897548 \nu + 3287618491616138537532$$)/ $$90\!\cdots\!12$
$\beta_{7}$	$=$	$($$-8599410738229651769 \nu^{15} + 31111659866916402345 \nu^{14} + 397851089569802580887 \nu^{13} - 1357434095246951691440 \nu^{12} - 6950409487206069267520 \nu^{11} + 21480555688367312916876 \nu^{10} + 59301366750646584632885 \nu^{9} - 157147318825198786079022 \nu^{8} - 257042371188445739715205 \nu^{7} + 552656578883574074980929 \nu^{6} + 511728863085525731674458 \nu^{5} - 875606750046801999312014 \nu^{4} - 366980592422283677211496 \nu^{3} + 458996610472572057501996 \nu^{2} + 63732795840155393435880 \nu - 41744720482603951993704$$)/ $$54\!\cdots\!72$
$\beta_{8}$	$=$	$($$10166089516631326123 \nu^{15} - 39636043673251564446 \nu^{14} - 463149040304201162734 \nu^{13} + 1741577168182876800871 \nu^{12} + 7937358719018169248978 \nu^{11} - 27920257611472853652366 \nu^{10} - 66526615038749754213403 \nu^{9} + 209053689477697038444789 \nu^{8} + 286804021630695488156393 \nu^{7} - 763778398499676259472184 \nu^{6} - 590218313888419955236599 \nu^{5} + 1275883347371748259536286 \nu^{4} + 501014971463436711617372 \nu^{3} - 734875152379202721919584 \nu^{2} - 188592356674960101516348 \nu + 75381322683826038993408$$)/ $$54\!\cdots\!72$
$\beta_{9}$	$=$	$($$3833924860594975845 \nu^{15} - 15405971411407983416 \nu^{14} - 174748567588173763710 \nu^{13} + 683105279786040092525 \nu^{12} + 3000960197217652805056 \nu^{11} - 11102524477032892698670 \nu^{10} - 25318495554936950579361 \nu^{9} + 84691399626259266906305 \nu^{8} + 111080659953374901792933 \nu^{7} - 315727483540994364481618 \nu^{6} - 237361474884041971348545 \nu^{5} + 530394989727383036599272 \nu^{4} + 206406504455500178785888 \nu^{3} - 290455416050807336719528 \nu^{2} - 58482275477158954969404 \nu + 24368330568281801524128$$)/ $$18\!\cdots\!24$
$\beta_{10}$	$=$	$($$6055690973521494323 \nu^{15} - 21065591888740470630 \nu^{14} - 282138336758875451579 \nu^{13} + 904938036516128835734 \nu^{12} + 5019613746164753586637 \nu^{11} - 13994381788245243532422 \nu^{10} - 44614067074995671927885 \nu^{9} + 99213383165647751861865 \nu^{8} + 210015545205224104585822 \nu^{7} - 335507138819282232469092 \nu^{6} - 494990400061097230473426 \nu^{5} + 513309222385173256168109 \nu^{4} + 516577122221256491326894 \nu^{3} - 270657506289607141947036 \nu^{2} - 186121404779846364836280 \nu + 18853135064703346138200$$)/ $$27\!\cdots\!36$
$\beta_{11}$	$=$	$($$2103742732645796968 \nu^{15} - 7587807286478686338 \nu^{14} - 98318269336507859293 \nu^{13} + 333589527564189827935 \nu^{12} + 1746467731155074504273 \nu^{11} - 5342141610410401144176 \nu^{10} - 15313513871554454343490 \nu^{9} + 39771223605111128968242 \nu^{8} + 69581428527495728527523 \nu^{7} - 143065840395346199349174 \nu^{6} - 151733582643722727860283 \nu^{5} + 230632231887950394137473 \nu^{4} + 131479917376481033077250 \nu^{3} - 118630530405786786024672 \nu^{2} - 36722337309588432352164 \nu + 7696294671096247442904$$)/ $$90\!\cdots\!12$
$\beta_{12}$	$=$	$($$15643727938311237652 \nu^{15} - 56547423258025579917 \nu^{14} - 736811796288429578563 \nu^{13} + 2511604996502047287625 \nu^{12} + 13220075965007561551802 \nu^{11} - 40821134148960918199182 \nu^{10} - 117349102552007926729696 \nu^{9} + 310045122176464699783689 \nu^{8} + 543132566223574961590886 \nu^{7} - 1144928931914839786999539 \nu^{6} - 1234507192266499635469335 \nu^{5} + 1908106630392058166642008 \nu^{4} + 1205285884230039076834004 \nu^{3} - 1029539354898820273092228 \nu^{2} - 431698814922534389321508 \nu + 60226142682101943655944$$)/ $$54\!\cdots\!72$
$\beta_{13}$	$=$	$($$16902750278691257515 \nu^{15} - 66100600868765845944 \nu^{14} - 765501985844103172942 \nu^{13} + 2878724443289246873407 \nu^{12} + 13050836157168499238396 \nu^{11} - 45539064879024801708138 \nu^{10} - 109453059483503010262423 \nu^{9} + 334561429000218091255911 \nu^{8} + 478818628546944868872983 \nu^{7} - 1190123853407848150127478 \nu^{6} - 1026586069984405062827043 \nu^{5} + 1916731753201095741369916 \nu^{4} + 932234121437022270337808 \nu^{3} - 1043599212580558264999584 \nu^{2} - 292239039888885554789364 \nu + 83307899572874489830104$$)/ $$54\!\cdots\!72$
$\beta_{14}$	$=$	$($$18310254666719483314 \nu^{15} - 64997796606289655715 \nu^{14} - 854944160891350069327 \nu^{13} + 2842406557507315492831 \nu^{12} + 15152969014719198018848 \nu^{11} - 45130633610269192266030 \nu^{10} - 132199513307530952648122 \nu^{9} + 331680872425345273285917 \nu^{8} + 594566658421572741407942 \nu^{7} - 1173217721169586183526145 \nu^{6} - 1274906221868961120611337 \nu^{5} + 1870018247056491778135546 \nu^{4} + 1105580567571758766871040 \nu^{3} - 982373741782699986430740 \nu^{2} - 330829154134196740304148 \nu + 73093928798098161025416$$)/ $$54\!\cdots\!72$
$\beta_{15}$	$=$	$($$-28101973564733554969 \nu^{15} + 108682923995574038268 \nu^{14} + 1292569355600345279962 \nu^{13} - 4807355654391981949669 \nu^{12} - 22457115817186195403960 \nu^{11} + 77765457791517503993622 \nu^{10} + 191723502462768280879765 \nu^{9} - 588180818937536644168137 \nu^{8} - 848072989308638681641817 \nu^{7} + 2164766731936135949379246 \nu^{6} + 1814874225627403113533529 \nu^{5} - 3591447951250514794518160 \nu^{4} - 1594204885835619991562096 \nu^{3} + 1969768145562148310520360 \nu^{2} + 517658891675791265035020 \nu - 193169117447690193359208$$)/ $$54\!\cdots\!72$

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

$1$	$=$	$\beta_0$
$\nu$	$=$	$\beta_{1}$
$\nu^{2}$	$=$	$\beta_{15} - \beta_{14} + \beta_{11} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 7$
$\nu^{3}$	$=$	$\beta_{13} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + 13 \beta_{1} + 3$
$\nu^{4}$	$=$	$17 \beta_{15} - 16 \beta_{14} - 5 \beta_{13} + 16 \beta_{11} + \beta_{10} - \beta_{9} + 21 \beta_{8} - 22 \beta_{7} - 18 \beta_{5} + 16 \beta_{4} - 14 \beta_{3} + 18 \beta_{2} + \beta_{1} + 92$
$\nu^{5}$	$=$	$6 \beta_{14} + 27 \beta_{13} + 3 \beta_{12} + 42 \beta_{11} - 27 \beta_{10} - 43 \beta_{9} + 18 \beta_{8} + 45 \beta_{7} - 4 \beta_{6} + 5 \beta_{5} - 6 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} + 198 \beta_{1} + 48$
$\nu^{6}$	$=$	$292 \beta_{15} - 263 \beta_{14} - 140 \beta_{13} + 20 \beta_{12} + 250 \beta_{11} + 27 \beta_{10} - 22 \beta_{9} + 385 \beta_{8} - 435 \beta_{7} + 18 \beta_{6} - 310 \beta_{5} + 250 \beta_{4} - 196 \beta_{3} + 312 \beta_{2} + 2 \beta_{1} + 1422$
$\nu^{7}$	$=$	$-18 \beta_{15} + 218 \beta_{14} + 566 \beta_{13} + 97 \beta_{12} + 742 \beta_{11} - 578 \beta_{10} - 802 \beta_{9} + 301 \beta_{8} + 923 \beta_{7} - 114 \beta_{6} + 160 \beta_{5} - 222 \beta_{4} + 266 \beta_{3} - 204 \beta_{2} + 3255 \beta_{1} + 626$
$\nu^{8}$	$=$	$5100 \beta_{15} - 4491 \beta_{14} - 3030 \beta_{13} + 664 \beta_{12} + 3993 \beta_{11} + 555 \beta_{10} - 404 \beta_{9} + 6878 \beta_{8} - 8259 \beta_{7} + 636 \beta_{6} - 5445 \beta_{5} + 4089 \beta_{4} - 2905 \beta_{3} + 5467 \beta_{2} - 413 \beta_{1} + 23542$
$\nu^{9}$	$=$	$-819 \beta_{15} + 5553 \beta_{14} + 10895 \beta_{13} + 2315 \beta_{12} + 12537 \beta_{11} - 11449 \beta_{10} - 14473 \beta_{9} + 4797 \beta_{8} + 18229 \beta_{7} - 2238 \beta_{6} + 3752 \beta_{5} - 5663 \beta_{4} + 6505 \beta_{3} - 5024 \beta_{2} + 55717 \beta_{1} + 6965$
$\nu^{10}$	$=$	$90066 \beta_{15} - 78119 \beta_{14} - 60472 \beta_{13} + 15735 \beta_{12} + 65104 \beta_{11} + 10695 \beta_{10} - 6786 \beta_{9} + 122233 \beta_{8} - 153768 \beta_{7} + 16003 \beta_{6} - 97105 \beta_{5} + 69364 \beta_{4} - 45408 \beta_{3} + 96976 \beta_{2} - 15564 \beta_{1} + 403524$
$\nu^{11}$	$=$	$-24915 \beta_{15} + 124569 \beta_{14} + 202728 \beta_{13} + 48849 \beta_{12} + 209058 \beta_{11} - 218564 \beta_{10} - 259251 \beta_{9} + 72776 \beta_{8} + 352935 \beta_{7} - 38377 \beta_{6} + 80018 \beta_{5} - 125013 \beta_{4} + 141337 \beta_{3} - 109660 \beta_{2} + 975756 \beta_{1} + 55139$
$\nu^{12}$	$=$	$1602574 \beta_{15} - 1369817 \beta_{14} - 1167210 \beta_{13} + 327646 \beta_{12} + 1078384 \beta_{11} + 203896 \beta_{10} - 105992 \beta_{9} + 2172396 \beta_{8} - 2833616 \beta_{7} + 353182 \beta_{6} - 1745053 \beta_{5} + 1206681 \beta_{4} - 739735 \beta_{3} + 1735350 \beta_{2} - 414872 \beta_{1} + 7050086$
$\nu^{13}$	$=$	$-638200 \beta_{15} + 2634590 \beta_{14} + 3719888 \beta_{13} + 964071 \beta_{12} + 3477248 \beta_{11} - 4089626 \beta_{10} - 4643295 \beta_{9} + 1043482 \beta_{8} + 6758749 \beta_{7} - 620343 \beta_{6} + 1646430 \beta_{5} - 2570854 \beta_{4} + 2894128 \beta_{3} - 2258798 \beta_{2} + 17322068 \beta_{1} - 143779$
$\nu^{14}$	$=$	$28665480 \beta_{15} - 24111775 \beta_{14} - 22171021 \beta_{13} + 6402010 \beta_{12} + 18076637 \beta_{11} + 3905179 \beta_{10} - 1539030 \beta_{9} + 38666159 \beta_{8} - 51935720 \beta_{7} + 7302225 \beta_{6} - 31465887 \beta_{5} + 21356178 \beta_{4} - 12425342 \beta_{3} + 31239529 \beta_{2} - 9671295 \beta_{1} + 124565501$
$\nu^{15}$	$=$	$-14893147 \beta_{15} + 53944933 \beta_{14} + 67887756 \beta_{13} + 18262187 \beta_{12} + 57915120 \beta_{11} - 75643357 \beta_{10} - 83307736 \beta_{9} + 13834904 \beta_{8} + 128598654 \beta_{7} - 9769310 \beta_{6} + 33307089 \beta_{5} - 50852857 \beta_{4} + 57298007 \beta_{3} - 45079355 \beta_{2} + 310116137 \beta_{1} - 21232931$

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

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.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−4.29726
−3.12066
−2.45101
−2.34509
−1.86734
−0.900032
−0.812118
−0.434018
0.187793
0.897331
1.63007
1.72426
2.96949
3.50323
4.08902
4.22634

1.00000

−4.29726

3.54512

1.00000

1.2

1.00000

−3.12066

−0.760812

1.00000

1.3

1.00000

−2.45101

3.23607

1.00000

1.4

1.00000

−2.34509

−1.25807

1.00000

1.5

1.00000

−1.86734

−4.84365

1.00000

1.6

1.00000

−0.900032

0.226627

1.00000

1.7

1.00000

−0.812118

−4.94593

1.00000

1.8

1.00000

−0.434018

−2.25969

1.00000

1.9

1.00000

0.187793

3.11113

1.00000

1.10

1.00000

0.897331

3.81467

1.00000

1.11

1.00000

1.63007

−0.577524

1.00000

1.12

1.00000

1.72426

4.78449

1.00000

1.13

1.00000

2.96949

−3.61534

1.00000

1.14

1.00000

3.50323

0.271751

1.00000

1.15

1.00000

4.08902

2.57555

1.00000

1.16

1.00000

4.22634

0.695618

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$-1$
$23$	$-1$
$29$	$1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8004.2.a.j	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8004.2.a.j	✓	16	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8004))$:

$T_{5}^{16} - \cdots$

$T_{7}^{16} - \cdots$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} $
$3$	$ ( -1 + T )^{16} $
$5$	$ 3888 - 8424 T - 67608 T^{2} - 17988 T^{3} + 151028 T^{4} + 65278 T^{5} - 122604 T^{6} - 47547 T^{7} + 46229 T^{8} + 14400 T^{9} - 8965 T^{10} - 2028 T^{11} + 932 T^{12} + 130 T^{13} - 49 T^{14} - 3 T^{15} + T^{16} $
$7$	$ -7776 + 49248 T + 4752 T^{2} - 349536 T^{3} + 19952 T^{4} + 664560 T^{5} + 2976 T^{6} - 365696 T^{7} + 34702 T^{8} + 78570 T^{9} - 12919 T^{10} - 7172 T^{11} + 1462 T^{12} + 283 T^{13} - 65 T^{14} - 4 T^{15} + T^{16} $
$11$	$ 96 - 9152 T + 201056 T^{2} - 940904 T^{3} - 180890 T^{4} + 2104516 T^{5} - 67953 T^{6} - 1065245 T^{7} + 122711 T^{8} + 187935 T^{9} - 28595 T^{10} - 14228 T^{11} + 2474 T^{12} + 457 T^{13} - 86 T^{14} - 5 T^{15} + T^{16} $
$13$	$ -9853812 + 63373548 T - 157634445 T^{2} + 186078576 T^{3} - 94013054 T^{4} - 4261845 T^{5} + 23486159 T^{6} - 6445106 T^{7} - 1393983 T^{8} + 821286 T^{9} - 15979 T^{10} - 39328 T^{11} + 3784 T^{12} + 811 T^{13} - 111 T^{14} - 6 T^{15} + T^{16} $
$17$	$ -1290240 + 6945792 T + 63316992 T^{2} - 77722752 T^{3} - 91372568 T^{4} + 97925524 T^{5} + 11783121 T^{6} - 28230218 T^{7} + 4092278 T^{8} + 1870406 T^{9} - 396502 T^{10} - 51397 T^{11} + 13163 T^{12} + 640 T^{13} - 189 T^{14} - 3 T^{15} + T^{16} $
$19$	$ -19600384 + 6070784 T + 90889344 T^{2} - 75924064 T^{3} - 49459000 T^{4} + 46279172 T^{5} + 10643546 T^{6} - 10111423 T^{7} - 933335 T^{8} + 1043492 T^{9} + 15844 T^{10} - 54182 T^{11} + 1966 T^{12} + 1320 T^{13} - 94 T^{14} - 11 T^{15} + T^{16} $
$23$	$ ( -1 + T )^{16} $
$29$	$ ( 1 + T )^{16} $
$31$	$ -18887639648 + 806189872 T + 31043324680 T^{2} - 15765296740 T^{3} - 4585801416 T^{4} + 3486594912 T^{5} + 215132416 T^{6} - 301578311 T^{7} + 662722 T^{8} + 12931439 T^{9} - 407460 T^{10} - 290663 T^{11} + 14403 T^{12} + 3245 T^{13} - 202 T^{14} - 14 T^{15} + T^{16} $
$37$	$ -12141693396 + 17921156162 T + 154064384 T^{2} - 8041521331 T^{3} + 1480934780 T^{4} + 1290698097 T^{5} - 319460858 T^{6} - 93903410 T^{7} + 26313935 T^{8} + 3577482 T^{9} - 1071410 T^{10} - 75099 T^{11} + 22912 T^{12} + 839 T^{13} - 244 T^{14} - 4 T^{15} + T^{16} $
$41$	$ -235015087392 - 506362898160 T - 304246847496 T^{2} + 2501787660 T^{3} + 50601668172 T^{4} + 7359954796 T^{5} - 3261680268 T^{6} - 598754675 T^{7} + 117229541 T^{8} + 20818084 T^{9} - 2657465 T^{10} - 369902 T^{11} + 38004 T^{12} + 3256 T^{13} - 305 T^{14} - 11 T^{15} + T^{16} $
$43$	$ 1945609344 - 13007329216 T - 27225937696 T^{2} - 11620431568 T^{3} + 3524260216 T^{4} + 2831435060 T^{5} - 56856962 T^{6} - 256465437 T^{7} - 9898349 T^{8} + 11983530 T^{9} + 482243 T^{10} - 310444 T^{11} - 4810 T^{12} + 4206 T^{13} - 67 T^{14} - 23 T^{15} + T^{16} $
$47$	$ 632393025024 + 1339808740096 T + 981308958400 T^{2} + 222388056832 T^{3} - 61456983136 T^{4} - 34763458144 T^{5} - 1301284060 T^{6} + 1582425396 T^{7} + 181873256 T^{8} - 32605570 T^{9} - 5355813 T^{10} + 322617 T^{11} + 70571 T^{12} - 1414 T^{13} - 434 T^{14} + 2 T^{15} + T^{16} $
$53$	$ -3665936047104 + 2088850697856 T + 3777659039616 T^{2} - 1570923865920 T^{3} - 449124357984 T^{4} + 182551493240 T^{5} + 15052639504 T^{6} - 7826052244 T^{7} - 130027872 T^{8} + 162170026 T^{9} - 2410214 T^{10} - 1744601 T^{11} + 57857 T^{12} + 9286 T^{13} - 417 T^{14} - 19 T^{15} + T^{16} $
$59$	$ 1098890150400 + 3978832973696 T - 907836175296 T^{2} - 1913466277408 T^{3} - 196227290080 T^{4} + 159120835496 T^{5} + 19671072584 T^{6} - 6236000794 T^{7} - 628018821 T^{8} + 139887378 T^{9} + 8359298 T^{10} - 1797102 T^{11} - 36332 T^{12} + 12030 T^{13} - 122 T^{14} - 32 T^{15} + T^{16} $
$61$	$ -32058228160 + 81255345248 T - 22793629184 T^{2} - 45280259712 T^{3} + 9121571476 T^{4} + 8146175510 T^{5} - 1074124466 T^{6} - 666876435 T^{7} + 56479169 T^{8} + 27310917 T^{9} - 1541716 T^{10} - 556184 T^{11} + 25790 T^{12} + 5339 T^{13} - 252 T^{14} - 19 T^{15} + T^{16} $
$67$	$ 19755520512 - 2615034890496 T - 3035960092416 T^{2} + 4017855301360 T^{3} - 695228364762 T^{4} - 272153029050 T^{5} + 82073597111 T^{6} + 1247102108 T^{7} - 2193991730 T^{8} + 119558392 T^{9} + 22899496 T^{10} - 2199751 T^{11} - 82099 T^{12} + 14416 T^{13} - 123 T^{14} - 33 T^{15} + T^{16} $
$71$	$ -1463530442112 - 17855107761408 T - 11703559543428 T^{2} + 3204613721619 T^{3} + 1401881254849 T^{4} - 188626418559 T^{5} - 64294462118 T^{6} + 5362197501 T^{7} + 1514157509 T^{8} - 82328828 T^{9} - 20137338 T^{10} + 692771 T^{11} + 152609 T^{12} - 2973 T^{13} - 612 T^{14} + 5 T^{15} + T^{16} $
$73$	$ 417667777536 + 465033246720 T + 14383941888 T^{2} - 125420496896 T^{3} - 21432367392 T^{4} + 13239398640 T^{5} + 2458937784 T^{6} - 778247076 T^{7} - 112746688 T^{8} + 28458692 T^{9} + 2186080 T^{10} - 605271 T^{11} - 9456 T^{12} + 6177 T^{13} - 135 T^{14} - 23 T^{15} + T^{16} $
$79$	$ 179478303232 - 470212751680 T - 343145710016 T^{2} + 691594683760 T^{3} - 226445332592 T^{4} - 31456789068 T^{5} + 23946802672 T^{6} - 1487820279 T^{7} - 704415666 T^{8} + 89870609 T^{9} + 7405266 T^{10} - 1543561 T^{11} - 6953 T^{12} + 10631 T^{13} - 288 T^{14} - 24 T^{15} + T^{16} $
$83$	$ -5177854236672 + 10551707308032 T - 4242855666560 T^{2} - 1114815011776 T^{3} + 707201445248 T^{4} + 49685872480 T^{5} - 43002463608 T^{6} - 1839444348 T^{7} + 1233309628 T^{8} + 46256124 T^{9} - 18334040 T^{10} - 586337 T^{11} + 147071 T^{12} + 3386 T^{13} - 607 T^{14} - 7 T^{15} + T^{16} $
$89$	$ -1436811936000 + 4773442981888 T - 1727828413328 T^{2} - 1744551091552 T^{3} + 747627007512 T^{4} + 101869345092 T^{5} - 58075659897 T^{6} - 1261125480 T^{7} + 1695802816 T^{8} - 8169485 T^{9} - 24348682 T^{10} + 228412 T^{11} + 183601 T^{12} - 1237 T^{13} - 687 T^{14} + 2 T^{15} + T^{16} $
$97$	$ 10196394240 + 5628694144 T - 87629739008 T^{2} + 79968008192 T^{3} - 12919812272 T^{4} - 8769049080 T^{5} + 2806736920 T^{6} + 265767804 T^{7} - 165921676 T^{8} + 2784038 T^{9} + 4342088 T^{10} - 297579 T^{11} - 49679 T^{12} + 5509 T^{13} + 132 T^{14} - 33 T^{15} + T^{16} $
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Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about

Newspace parameters

Newform invariants

$q$-expansion

Embeddings

Atkin-Lehner signs

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	8004.2.a.j

Level	$8004$
Weight	$2$
Character orbit	8004.a
Self dual	yes
Analytic conductor	$63.912$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$1$

Self dual:	yes
Analytic conductor:	\(63.9122617778\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\(x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + 46229 x^{8} - 47547 x^{7} - 122604 x^{6} + 65278 x^{5} + 151028 x^{4} - 17988 x^{3} - 67608 x^{2} - 8424 x + 3888\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

\(f(q)\)	\(=\)	\( q + q^{3} + \beta_{1} q^{5} -\beta_{7} q^{7} + q^{9} +O(q^{10})\)	\( q + q^{3} + \beta_{1} q^{5} -\beta_{7} q^{7} + q^{9} -\beta_{6} q^{11} + \beta_{4} q^{13} + \beta_{1} q^{15} + \beta_{9} q^{17} + ( 1 + \beta_{8} ) q^{19} -\beta_{7} q^{21} + q^{23} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{25} + q^{27} - q^{29} + ( 1 - \beta_{7} + \beta_{10} ) q^{31} -\beta_{6} q^{33} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{14} ) q^{35} + ( \beta_{1} + \beta_{14} ) q^{37} + \beta_{4} q^{39} + ( -\beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{41} + ( 1 + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{14} ) q^{43} + \beta_{1} q^{45} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{15} ) q^{47} + ( 2 - \beta_{1} - \beta_{2} - \beta_{8} - \beta_{11} - \beta_{13} ) q^{49} + \beta_{9} q^{51} + ( 1 - \beta_{8} - \beta_{11} - \beta_{12} ) q^{53} + ( 2 + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{12} - \beta_{14} ) q^{55} + ( 1 + \beta_{8} ) q^{57} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{12} - 2 \beta_{14} ) q^{59} + ( 2 + \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{13} ) q^{61} -\beta_{7} q^{63} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{7} - \beta_{12} + 2 \beta_{14} ) q^{65} + ( 3 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{15} ) q^{67} + q^{69} + ( \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{71} + ( 2 - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{13} - \beta_{14} ) q^{73} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{75} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{77} + ( 1 - \beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{79} + q^{81} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{83} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{85} - q^{87} + ( \beta_{7} - \beta_{11} + \beta_{12} ) q^{89} + ( 1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{91} + ( 1 - \beta_{7} + \beta_{10} ) q^{93} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{95} + ( 2 + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{97} -\beta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q + 16q^{3} + 3q^{5} + 4q^{7} + 16q^{9} + O(q^{10}) \)	\( 16q + 16q^{3} + 3q^{5} + 4q^{7} + 16q^{9} + 5q^{11} + 6q^{13} + 3q^{15} + 3q^{17} + 11q^{19} + 4q^{21} + 16q^{23} + 27q^{25} + 16q^{27} - 16q^{29} + 14q^{31} + 5q^{33} + 11q^{35} + 4q^{37} + 6q^{39} + 11q^{41} + 23q^{43} + 3q^{45} - 2q^{47} + 34q^{49} + 3q^{51} + 19q^{53} + 31q^{55} + 11q^{57} + 32q^{59} + 19q^{61} + 4q^{63} + 6q^{65} + 33q^{67} + 16q^{69} - 5q^{71} + 23q^{73} + 27q^{75} + 42q^{77} + 24q^{79} + 16q^{81} + 7q^{83} - 16q^{87} - 2q^{89} + 25q^{91} + 14q^{93} + 7q^{95} + 33q^{97} + 5q^{99} + O(q^{100}) \)

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\((\)\(128089526929863711 \nu^{15} - 1780989647760093687 \nu^{14} - 2343590043301536281 \nu^{13} + 82428009307722432792 \nu^{12} - 39868293740417462800 \nu^{11} - 1434850612960902643364 \nu^{10} + 1061787358987892121509 \nu^{9} + 12059212146105833185194 \nu^{8} - 6849714181952244815581 \nu^{7} - 49998222735113343315999 \nu^{6} + 12943701821822810175386 \nu^{5} + 87862981358437561842546 \nu^{4} - 985493942779777495944 \nu^{3} - 44010301598073058173068 \nu^{2} - 5734576008998611033216 \nu + 1436096932911927060552\)\()/ \)\(60\!\cdots\!08\)\( \)
\(\beta_{3}\)	\(=\)	\((\)\(-958087900246403592 \nu^{15} + 2000567955267874333 \nu^{14} + 49357290400200766149 \nu^{13} - 88974714684163101079 \nu^{12} - 963378769466844341132 \nu^{11} + 1415048553601951450082 \nu^{10} + 8893034704374805484880 \nu^{9} - 9960991537155374353069 \nu^{8} - 38488851452268663931116 \nu^{7} + 31123533131595448591055 \nu^{6} + 63124294156284455139465 \nu^{5} - 39843524562808100614170 \nu^{4} - 17036408551707978644276 \nu^{3} + 2778605399574854232452 \nu^{2} + 4140098642180225184060 \nu + 10187796734063718568848\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{4}\)	\(=\)	\((\)\(4486141389040236929 \nu^{15} - 16191524997021342828 \nu^{14} - 211861740633534736190 \nu^{13} + 721401882595358991197 \nu^{12} + 3820388029937201339968 \nu^{11} - 11804652191031635223666 \nu^{10} - 34218304176590055106445 \nu^{9} + 91025874133171328629977 \nu^{8} + 160820231928040018855981 \nu^{7} - 348051881480837120172498 \nu^{6} - 374464283660322591966429 \nu^{5} + 631086737220708958112600 \nu^{4} + 373450758905607661475524 \nu^{3} - 436002491424203250589680 \nu^{2} - 119911680097050427614324 \nu + 62507843402239991683728\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{5}\)	\(=\)	\((\)\(-2170353582856352493 \nu^{15} + 6394679216719266718 \nu^{14} + 105192616124931196248 \nu^{13} - 273011890888993264339 \nu^{12} - 1963933514692243560836 \nu^{11} + 4159658356999503153722 \nu^{10} + 18290876230917575384385 \nu^{9} - 28530527789909789588287 \nu^{8} - 88888441699606672234641 \nu^{7} + 90582649286250348855860 \nu^{6} + 210029308309103838088587 \nu^{5} - 128266188808560877480848 \nu^{4} - 209073282062352857065304 \nu^{3} + 66867424884872498482256 \nu^{2} + 63961903110921104602836 \nu - 6638130556491638529144\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{6}\)	\(=\)	\((\)\(1181647930064972053 \nu^{15} - 4488671540266881279 \nu^{14} - 53334067423612189198 \nu^{13} + 193579150520849398669 \nu^{12} + 901332302367398689427 \nu^{11} - 3015546321065278419096 \nu^{10} - 7391050976096937823363 \nu^{9} + 21665705786600716578792 \nu^{8} + 30526909733759416311410 \nu^{7} - 74930972648825634056871 \nu^{6} - 56119572043326164111031 \nu^{5} + 117837296847755457978037 \nu^{4} + 33444989655224619931886 \nu^{3} - 61721460041205402182268 \nu^{2} - 7863407220333139897548 \nu + 3287618491616138537532\)\()/ \)\(90\!\cdots\!12\)\( \)
\(\beta_{7}\)	\(=\)	\((\)\(-8599410738229651769 \nu^{15} + 31111659866916402345 \nu^{14} + 397851089569802580887 \nu^{13} - 1357434095246951691440 \nu^{12} - 6950409487206069267520 \nu^{11} + 21480555688367312916876 \nu^{10} + 59301366750646584632885 \nu^{9} - 157147318825198786079022 \nu^{8} - 257042371188445739715205 \nu^{7} + 552656578883574074980929 \nu^{6} + 511728863085525731674458 \nu^{5} - 875606750046801999312014 \nu^{4} - 366980592422283677211496 \nu^{3} + 458996610472572057501996 \nu^{2} + 63732795840155393435880 \nu - 41744720482603951993704\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{8}\)	\(=\)	\((\)\(10166089516631326123 \nu^{15} - 39636043673251564446 \nu^{14} - 463149040304201162734 \nu^{13} + 1741577168182876800871 \nu^{12} + 7937358719018169248978 \nu^{11} - 27920257611472853652366 \nu^{10} - 66526615038749754213403 \nu^{9} + 209053689477697038444789 \nu^{8} + 286804021630695488156393 \nu^{7} - 763778398499676259472184 \nu^{6} - 590218313888419955236599 \nu^{5} + 1275883347371748259536286 \nu^{4} + 501014971463436711617372 \nu^{3} - 734875152379202721919584 \nu^{2} - 188592356674960101516348 \nu + 75381322683826038993408\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{9}\)	\(=\)	\((\)\(3833924860594975845 \nu^{15} - 15405971411407983416 \nu^{14} - 174748567588173763710 \nu^{13} + 683105279786040092525 \nu^{12} + 3000960197217652805056 \nu^{11} - 11102524477032892698670 \nu^{10} - 25318495554936950579361 \nu^{9} + 84691399626259266906305 \nu^{8} + 111080659953374901792933 \nu^{7} - 315727483540994364481618 \nu^{6} - 237361474884041971348545 \nu^{5} + 530394989727383036599272 \nu^{4} + 206406504455500178785888 \nu^{3} - 290455416050807336719528 \nu^{2} - 58482275477158954969404 \nu + 24368330568281801524128\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{10}\)	\(=\)	\((\)\(6055690973521494323 \nu^{15} - 21065591888740470630 \nu^{14} - 282138336758875451579 \nu^{13} + 904938036516128835734 \nu^{12} + 5019613746164753586637 \nu^{11} - 13994381788245243532422 \nu^{10} - 44614067074995671927885 \nu^{9} + 99213383165647751861865 \nu^{8} + 210015545205224104585822 \nu^{7} - 335507138819282232469092 \nu^{6} - 494990400061097230473426 \nu^{5} + 513309222385173256168109 \nu^{4} + 516577122221256491326894 \nu^{3} - 270657506289607141947036 \nu^{2} - 186121404779846364836280 \nu + 18853135064703346138200\)\()/ \)\(27\!\cdots\!36\)\( \)
\(\beta_{11}\)	\(=\)	\((\)\(2103742732645796968 \nu^{15} - 7587807286478686338 \nu^{14} - 98318269336507859293 \nu^{13} + 333589527564189827935 \nu^{12} + 1746467731155074504273 \nu^{11} - 5342141610410401144176 \nu^{10} - 15313513871554454343490 \nu^{9} + 39771223605111128968242 \nu^{8} + 69581428527495728527523 \nu^{7} - 143065840395346199349174 \nu^{6} - 151733582643722727860283 \nu^{5} + 230632231887950394137473 \nu^{4} + 131479917376481033077250 \nu^{3} - 118630530405786786024672 \nu^{2} - 36722337309588432352164 \nu + 7696294671096247442904\)\()/ \)\(90\!\cdots\!12\)\( \)
\(\beta_{12}\)	\(=\)	\((\)\(15643727938311237652 \nu^{15} - 56547423258025579917 \nu^{14} - 736811796288429578563 \nu^{13} + 2511604996502047287625 \nu^{12} + 13220075965007561551802 \nu^{11} - 40821134148960918199182 \nu^{10} - 117349102552007926729696 \nu^{9} + 310045122176464699783689 \nu^{8} + 543132566223574961590886 \nu^{7} - 1144928931914839786999539 \nu^{6} - 1234507192266499635469335 \nu^{5} + 1908106630392058166642008 \nu^{4} + 1205285884230039076834004 \nu^{3} - 1029539354898820273092228 \nu^{2} - 431698814922534389321508 \nu + 60226142682101943655944\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{13}\)	\(=\)	\((\)\(16902750278691257515 \nu^{15} - 66100600868765845944 \nu^{14} - 765501985844103172942 \nu^{13} + 2878724443289246873407 \nu^{12} + 13050836157168499238396 \nu^{11} - 45539064879024801708138 \nu^{10} - 109453059483503010262423 \nu^{9} + 334561429000218091255911 \nu^{8} + 478818628546944868872983 \nu^{7} - 1190123853407848150127478 \nu^{6} - 1026586069984405062827043 \nu^{5} + 1916731753201095741369916 \nu^{4} + 932234121437022270337808 \nu^{3} - 1043599212580558264999584 \nu^{2} - 292239039888885554789364 \nu + 83307899572874489830104\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{14}\)	\(=\)	\((\)\(18310254666719483314 \nu^{15} - 64997796606289655715 \nu^{14} - 854944160891350069327 \nu^{13} + 2842406557507315492831 \nu^{12} + 15152969014719198018848 \nu^{11} - 45130633610269192266030 \nu^{10} - 132199513307530952648122 \nu^{9} + 331680872425345273285917 \nu^{8} + 594566658421572741407942 \nu^{7} - 1173217721169586183526145 \nu^{6} - 1274906221868961120611337 \nu^{5} + 1870018247056491778135546 \nu^{4} + 1105580567571758766871040 \nu^{3} - 982373741782699986430740 \nu^{2} - 330829154134196740304148 \nu + 73093928798098161025416\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{15}\)	\(=\)	\((\)\(-28101973564733554969 \nu^{15} + 108682923995574038268 \nu^{14} + 1292569355600345279962 \nu^{13} - 4807355654391981949669 \nu^{12} - 22457115817186195403960 \nu^{11} + 77765457791517503993622 \nu^{10} + 191723502462768280879765 \nu^{9} - 588180818937536644168137 \nu^{8} - 848072989308638681641817 \nu^{7} + 2164766731936135949379246 \nu^{6} + 1814874225627403113533529 \nu^{5} - 3591447951250514794518160 \nu^{4} - 1594204885835619991562096 \nu^{3} + 1969768145562148310520360 \nu^{2} + 517658891675791265035020 \nu - 193169117447690193359208\)\()/ \)\(54\!\cdots\!72\)\( \)

\(1\)	\(=\)	\(\beta_0\)
\(\nu\)	\(=\)	\(\beta_{1}\)
\(\nu^{2}\)	\(=\)	\(\beta_{15} - \beta_{14} + \beta_{11} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 7\)
\(\nu^{3}\)	\(=\)	\(\beta_{13} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + 13 \beta_{1} + 3\)
\(\nu^{4}\)	\(=\)	\(17 \beta_{15} - 16 \beta_{14} - 5 \beta_{13} + 16 \beta_{11} + \beta_{10} - \beta_{9} + 21 \beta_{8} - 22 \beta_{7} - 18 \beta_{5} + 16 \beta_{4} - 14 \beta_{3} + 18 \beta_{2} + \beta_{1} + 92\)
\(\nu^{5}\)	\(=\)	\(6 \beta_{14} + 27 \beta_{13} + 3 \beta_{12} + 42 \beta_{11} - 27 \beta_{10} - 43 \beta_{9} + 18 \beta_{8} + 45 \beta_{7} - 4 \beta_{6} + 5 \beta_{5} - 6 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} + 198 \beta_{1} + 48\)
\(\nu^{6}\)	\(=\)	\(292 \beta_{15} - 263 \beta_{14} - 140 \beta_{13} + 20 \beta_{12} + 250 \beta_{11} + 27 \beta_{10} - 22 \beta_{9} + 385 \beta_{8} - 435 \beta_{7} + 18 \beta_{6} - 310 \beta_{5} + 250 \beta_{4} - 196 \beta_{3} + 312 \beta_{2} + 2 \beta_{1} + 1422\)
\(\nu^{7}\)	\(=\)	\(-18 \beta_{15} + 218 \beta_{14} + 566 \beta_{13} + 97 \beta_{12} + 742 \beta_{11} - 578 \beta_{10} - 802 \beta_{9} + 301 \beta_{8} + 923 \beta_{7} - 114 \beta_{6} + 160 \beta_{5} - 222 \beta_{4} + 266 \beta_{3} - 204 \beta_{2} + 3255 \beta_{1} + 626\)
\(\nu^{8}\)	\(=\)	\(5100 \beta_{15} - 4491 \beta_{14} - 3030 \beta_{13} + 664 \beta_{12} + 3993 \beta_{11} + 555 \beta_{10} - 404 \beta_{9} + 6878 \beta_{8} - 8259 \beta_{7} + 636 \beta_{6} - 5445 \beta_{5} + 4089 \beta_{4} - 2905 \beta_{3} + 5467 \beta_{2} - 413 \beta_{1} + 23542\)
\(\nu^{9}\)	\(=\)	\(-819 \beta_{15} + 5553 \beta_{14} + 10895 \beta_{13} + 2315 \beta_{12} + 12537 \beta_{11} - 11449 \beta_{10} - 14473 \beta_{9} + 4797 \beta_{8} + 18229 \beta_{7} - 2238 \beta_{6} + 3752 \beta_{5} - 5663 \beta_{4} + 6505 \beta_{3} - 5024 \beta_{2} + 55717 \beta_{1} + 6965\)
\(\nu^{10}\)	\(=\)	\(90066 \beta_{15} - 78119 \beta_{14} - 60472 \beta_{13} + 15735 \beta_{12} + 65104 \beta_{11} + 10695 \beta_{10} - 6786 \beta_{9} + 122233 \beta_{8} - 153768 \beta_{7} + 16003 \beta_{6} - 97105 \beta_{5} + 69364 \beta_{4} - 45408 \beta_{3} + 96976 \beta_{2} - 15564 \beta_{1} + 403524\)
\(\nu^{11}\)	\(=\)	\(-24915 \beta_{15} + 124569 \beta_{14} + 202728 \beta_{13} + 48849 \beta_{12} + 209058 \beta_{11} - 218564 \beta_{10} - 259251 \beta_{9} + 72776 \beta_{8} + 352935 \beta_{7} - 38377 \beta_{6} + 80018 \beta_{5} - 125013 \beta_{4} + 141337 \beta_{3} - 109660 \beta_{2} + 975756 \beta_{1} + 55139\)
\(\nu^{12}\)	\(=\)	\(1602574 \beta_{15} - 1369817 \beta_{14} - 1167210 \beta_{13} + 327646 \beta_{12} + 1078384 \beta_{11} + 203896 \beta_{10} - 105992 \beta_{9} + 2172396 \beta_{8} - 2833616 \beta_{7} + 353182 \beta_{6} - 1745053 \beta_{5} + 1206681 \beta_{4} - 739735 \beta_{3} + 1735350 \beta_{2} - 414872 \beta_{1} + 7050086\)
\(\nu^{13}\)	\(=\)	\(-638200 \beta_{15} + 2634590 \beta_{14} + 3719888 \beta_{13} + 964071 \beta_{12} + 3477248 \beta_{11} - 4089626 \beta_{10} - 4643295 \beta_{9} + 1043482 \beta_{8} + 6758749 \beta_{7} - 620343 \beta_{6} + 1646430 \beta_{5} - 2570854 \beta_{4} + 2894128 \beta_{3} - 2258798 \beta_{2} + 17322068 \beta_{1} - 143779\)
\(\nu^{14}\)	\(=\)	\(28665480 \beta_{15} - 24111775 \beta_{14} - 22171021 \beta_{13} + 6402010 \beta_{12} + 18076637 \beta_{11} + 3905179 \beta_{10} - 1539030 \beta_{9} + 38666159 \beta_{8} - 51935720 \beta_{7} + 7302225 \beta_{6} - 31465887 \beta_{5} + 21356178 \beta_{4} - 12425342 \beta_{3} + 31239529 \beta_{2} - 9671295 \beta_{1} + 124565501\)
\(\nu^{15}\)	\(=\)	\(-14893147 \beta_{15} + 53944933 \beta_{14} + 67887756 \beta_{13} + 18262187 \beta_{12} + 57915120 \beta_{11} - 75643357 \beta_{10} - 83307736 \beta_{9} + 13834904 \beta_{8} + 128598654 \beta_{7} - 9769310 \beta_{6} + 33307089 \beta_{5} - 50852857 \beta_{4} + 57298007 \beta_{3} - 45079355 \beta_{2} + 310116137 \beta_{1} - 21232931\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.16
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} \)
$3$	\( ( -1 + T )^{16} \)
$5$	\( 3888 - 8424 T - 67608 T^{2} - 17988 T^{3} + 151028 T^{4} + 65278 T^{5} - 122604 T^{6} - 47547 T^{7} + 46229 T^{8} + 14400 T^{9} - 8965 T^{10} - 2028 T^{11} + 932 T^{12} + 130 T^{13} - 49 T^{14} - 3 T^{15} + T^{16} \)
$7$	\( -7776 + 49248 T + 4752 T^{2} - 349536 T^{3} + 19952 T^{4} + 664560 T^{5} + 2976 T^{6} - 365696 T^{7} + 34702 T^{8} + 78570 T^{9} - 12919 T^{10} - 7172 T^{11} + 1462 T^{12} + 283 T^{13} - 65 T^{14} - 4 T^{15} + T^{16} \)
$11$	\( 96 - 9152 T + 201056 T^{2} - 940904 T^{3} - 180890 T^{4} + 2104516 T^{5} - 67953 T^{6} - 1065245 T^{7} + 122711 T^{8} + 187935 T^{9} - 28595 T^{10} - 14228 T^{11} + 2474 T^{12} + 457 T^{13} - 86 T^{14} - 5 T^{15} + T^{16} \)
$13$	\( -9853812 + 63373548 T - 157634445 T^{2} + 186078576 T^{3} - 94013054 T^{4} - 4261845 T^{5} + 23486159 T^{6} - 6445106 T^{7} - 1393983 T^{8} + 821286 T^{9} - 15979 T^{10} - 39328 T^{11} + 3784 T^{12} + 811 T^{13} - 111 T^{14} - 6 T^{15} + T^{16} \)
$17$	\( -1290240 + 6945792 T + 63316992 T^{2} - 77722752 T^{3} - 91372568 T^{4} + 97925524 T^{5} + 11783121 T^{6} - 28230218 T^{7} + 4092278 T^{8} + 1870406 T^{9} - 396502 T^{10} - 51397 T^{11} + 13163 T^{12} + 640 T^{13} - 189 T^{14} - 3 T^{15} + T^{16} \)
$19$	\( -19600384 + 6070784 T + 90889344 T^{2} - 75924064 T^{3} - 49459000 T^{4} + 46279172 T^{5} + 10643546 T^{6} - 10111423 T^{7} - 933335 T^{8} + 1043492 T^{9} + 15844 T^{10} - 54182 T^{11} + 1966 T^{12} + 1320 T^{13} - 94 T^{14} - 11 T^{15} + T^{16} \)
$23$	\( ( -1 + T )^{16} \)
$29$	\( ( 1 + T )^{16} \)
$31$	\( -18887639648 + 806189872 T + 31043324680 T^{2} - 15765296740 T^{3} - 4585801416 T^{4} + 3486594912 T^{5} + 215132416 T^{6} - 301578311 T^{7} + 662722 T^{8} + 12931439 T^{9} - 407460 T^{10} - 290663 T^{11} + 14403 T^{12} + 3245 T^{13} - 202 T^{14} - 14 T^{15} + T^{16} \)
$37$	\( -12141693396 + 17921156162 T + 154064384 T^{2} - 8041521331 T^{3} + 1480934780 T^{4} + 1290698097 T^{5} - 319460858 T^{6} - 93903410 T^{7} + 26313935 T^{8} + 3577482 T^{9} - 1071410 T^{10} - 75099 T^{11} + 22912 T^{12} + 839 T^{13} - 244 T^{14} - 4 T^{15} + T^{16} \)
$41$	\( -235015087392 - 506362898160 T - 304246847496 T^{2} + 2501787660 T^{3} + 50601668172 T^{4} + 7359954796 T^{5} - 3261680268 T^{6} - 598754675 T^{7} + 117229541 T^{8} + 20818084 T^{9} - 2657465 T^{10} - 369902 T^{11} + 38004 T^{12} + 3256 T^{13} - 305 T^{14} - 11 T^{15} + T^{16} \)
$43$	\( 1945609344 - 13007329216 T - 27225937696 T^{2} - 11620431568 T^{3} + 3524260216 T^{4} + 2831435060 T^{5} - 56856962 T^{6} - 256465437 T^{7} - 9898349 T^{8} + 11983530 T^{9} + 482243 T^{10} - 310444 T^{11} - 4810 T^{12} + 4206 T^{13} - 67 T^{14} - 23 T^{15} + T^{16} \)
$47$	\( 632393025024 + 1339808740096 T + 981308958400 T^{2} + 222388056832 T^{3} - 61456983136 T^{4} - 34763458144 T^{5} - 1301284060 T^{6} + 1582425396 T^{7} + 181873256 T^{8} - 32605570 T^{9} - 5355813 T^{10} + 322617 T^{11} + 70571 T^{12} - 1414 T^{13} - 434 T^{14} + 2 T^{15} + T^{16} \)
$53$	\( -3665936047104 + 2088850697856 T + 3777659039616 T^{2} - 1570923865920 T^{3} - 449124357984 T^{4} + 182551493240 T^{5} + 15052639504 T^{6} - 7826052244 T^{7} - 130027872 T^{8} + 162170026 T^{9} - 2410214 T^{10} - 1744601 T^{11} + 57857 T^{12} + 9286 T^{13} - 417 T^{14} - 19 T^{15} + T^{16} \)
$59$	\( 1098890150400 + 3978832973696 T - 907836175296 T^{2} - 1913466277408 T^{3} - 196227290080 T^{4} + 159120835496 T^{5} + 19671072584 T^{6} - 6236000794 T^{7} - 628018821 T^{8} + 139887378 T^{9} + 8359298 T^{10} - 1797102 T^{11} - 36332 T^{12} + 12030 T^{13} - 122 T^{14} - 32 T^{15} + T^{16} \)
$61$	\( -32058228160 + 81255345248 T - 22793629184 T^{2} - 45280259712 T^{3} + 9121571476 T^{4} + 8146175510 T^{5} - 1074124466 T^{6} - 666876435 T^{7} + 56479169 T^{8} + 27310917 T^{9} - 1541716 T^{10} - 556184 T^{11} + 25790 T^{12} + 5339 T^{13} - 252 T^{14} - 19 T^{15} + T^{16} \)
$67$	\( 19755520512 - 2615034890496 T - 3035960092416 T^{2} + 4017855301360 T^{3} - 695228364762 T^{4} - 272153029050 T^{5} + 82073597111 T^{6} + 1247102108 T^{7} - 2193991730 T^{8} + 119558392 T^{9} + 22899496 T^{10} - 2199751 T^{11} - 82099 T^{12} + 14416 T^{13} - 123 T^{14} - 33 T^{15} + T^{16} \)
$71$	\( -1463530442112 - 17855107761408 T - 11703559543428 T^{2} + 3204613721619 T^{3} + 1401881254849 T^{4} - 188626418559 T^{5} - 64294462118 T^{6} + 5362197501 T^{7} + 1514157509 T^{8} - 82328828 T^{9} - 20137338 T^{10} + 692771 T^{11} + 152609 T^{12} - 2973 T^{13} - 612 T^{14} + 5 T^{15} + T^{16} \)
$73$	\( 417667777536 + 465033246720 T + 14383941888 T^{2} - 125420496896 T^{3} - 21432367392 T^{4} + 13239398640 T^{5} + 2458937784 T^{6} - 778247076 T^{7} - 112746688 T^{8} + 28458692 T^{9} + 2186080 T^{10} - 605271 T^{11} - 9456 T^{12} + 6177 T^{13} - 135 T^{14} - 23 T^{15} + T^{16} \)
$79$	\( 179478303232 - 470212751680 T - 343145710016 T^{2} + 691594683760 T^{3} - 226445332592 T^{4} - 31456789068 T^{5} + 23946802672 T^{6} - 1487820279 T^{7} - 704415666 T^{8} + 89870609 T^{9} + 7405266 T^{10} - 1543561 T^{11} - 6953 T^{12} + 10631 T^{13} - 288 T^{14} - 24 T^{15} + T^{16} \)
$83$	\( -5177854236672 + 10551707308032 T - 4242855666560 T^{2} - 1114815011776 T^{3} + 707201445248 T^{4} + 49685872480 T^{5} - 43002463608 T^{6} - 1839444348 T^{7} + 1233309628 T^{8} + 46256124 T^{9} - 18334040 T^{10} - 586337 T^{11} + 147071 T^{12} + 3386 T^{13} - 607 T^{14} - 7 T^{15} + T^{16} \)
$89$	\( -1436811936000 + 4773442981888 T - 1727828413328 T^{2} - 1744551091552 T^{3} + 747627007512 T^{4} + 101869345092 T^{5} - 58075659897 T^{6} - 1261125480 T^{7} + 1695802816 T^{8} - 8169485 T^{9} - 24348682 T^{10} + 228412 T^{11} + 183601 T^{12} - 1237 T^{13} - 687 T^{14} + 2 T^{15} + T^{16} \)
$97$	\( 10196394240 + 5628694144 T - 87629739008 T^{2} + 79968008192 T^{3} - 12919812272 T^{4} - 8769049080 T^{5} + 2806736920 T^{6} + 265767804 T^{7} - 165921676 T^{8} + 2784038 T^{9} + 4342088 T^{10} - 297579 T^{11} - 49679 T^{12} + 5509 T^{13} + 132 T^{14} - 33 T^{15} + T^{16} \)
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