LMFDB - Embedded newform 2016.3.f.c.1441.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,3,Mod(1441,2016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2016, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2016.1441"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,112] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4694952902656.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1 $ x^8 + 20x^6 + 56x^4 + 20*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1441.7
Root			$-0.244539i$ of defining polynomial
Character	$\chi$	$=$	2016.1441
Dual form			2016.3.f.c.1441.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.12900i q^{5} +(-6.21005 + 3.23038i) q^{7} +15.2485 q^{11} -3.00789i q^{13} -21.6213i q^{17} -11.8981i q^{19} +1.72205 q^{23} -12.5647 q^{25} +41.1293 q^{29} +8.50759i q^{31} +(-19.7990 - 38.0614i) q^{35} +53.1293 q^{37} -43.0162i q^{41} -36.6446 q^{43} -30.2569i q^{47} +(28.1293 - 40.1216i) q^{49} +30.0000 q^{53} +93.4582i q^{55} -73.0525i q^{59} +5.67609i q^{61} +18.4353 q^{65} +28.7750 q^{67} +5.53191 q^{71} +94.9429i q^{73} +(-94.6940 + 49.2585i) q^{77} -81.5337 q^{79} +86.2943i q^{83} +132.517 q^{85} +27.6371i q^{89} +(9.71661 + 18.6791i) q^{91} +72.9234 q^{95} +100.959i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{25} + 112 q^{29} + 208 q^{37} + 8 q^{49} + 240 q^{53} + 256 q^{65} - 432 q^{77} + 192 q^{85}+O(q^{100}) $ 8 * q + 8 * q^25 + 112 * q^29 + 208 * q^37 + 8 * q^49 + 240 * q^53 + 256 * q^65 - 432 * q^77 + 192 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			6.12900i			1.22580i	0.790160	+	0.612900i	$0.209997\pi$
$5$			6.12900i			1.22580i	−0.790160	+	0.612900i	$0.790003\pi$
$6$		0			0
$7$	−6.21005	+	3.23038i	−0.887149	+	0.461483i
$7$	−6.21005	+	3.23038i	−0.887149	+	0.461483i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	15.2485			1.38623			0.693114	−	0.720828i	$-0.256238\pi$
$11$	15.2485			1.38623			0.693114	+	0.720828i	$0.256238\pi$
$12$		0			0
$13$		−	3.00789i		−	0.231376i	−0.993286	−	0.115688i	$-0.963093\pi$
$13$		−	3.00789i		−	0.231376i	0.993286	−	0.115688i	$-0.0369073\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	21.6213i		−	1.27184i	−0.771753	−	0.635922i	$-0.780620\pi$
$17$		−	21.6213i		−	1.27184i	0.771753	−	0.635922i	$-0.219380\pi$
$18$		0			0
$19$		−	11.8981i		−	0.626216i	−0.949718	−	0.313108i	$-0.898630\pi$
$19$		−	11.8981i		−	0.626216i	0.949718	−	0.313108i	$-0.101370\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.72205			0.0748715			0.0374358	−	0.999299i	$-0.488081\pi$
$23$	1.72205			0.0748715			0.0374358	+	0.999299i	$0.488081\pi$
$24$		0			0
$25$	−12.5647			−0.502586
$26$		0			0
$27$		0			0
$28$		0			0
$29$	41.1293			1.41825			0.709126	−	0.705082i	$-0.249090\pi$
$29$	41.1293			1.41825			0.709126	+	0.705082i	$0.249090\pi$
$30$		0			0
$31$			8.50759i			0.274438i	0.990541	+	0.137219i	$0.0438165\pi$
$31$			8.50759i			0.274438i	−0.990541	+	0.137219i	$0.956184\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−19.7990	−	38.0614i	−0.565685	−	1.08747i
$36$		0			0
$37$	53.1293			1.43593			0.717964	−	0.696080i	$-0.245074\pi$
$37$	53.1293			1.43593			0.717964	+	0.696080i	$0.245074\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	43.0162i		−	1.04918i	−0.851356	−	0.524588i	$-0.824219\pi$
$41$		−	43.0162i		−	1.04918i	0.851356	−	0.524588i	$-0.175781\pi$
$42$		0			0
$43$	−36.6446			−0.852200			−0.426100	−	0.904676i	$-0.640113\pi$
$43$	−36.6446			−0.852200			−0.426100	+	0.904676i	$0.640113\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	30.2569i		−	0.643765i	−0.946780	−	0.321882i	$-0.895684\pi$
$47$		−	30.2569i		−	0.643765i	0.946780	−	0.321882i	$-0.104316\pi$
$48$		0			0
$49$	28.1293	−	40.1216i	0.574068	−	0.818808i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	30.0000			0.566038			0.283019	−	0.959114i	$-0.408664\pi$
$53$	30.0000			0.566038			0.283019	+	0.959114i	$0.408664\pi$
$54$		0			0
$55$			93.4582i			1.69924i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	73.0525i		−	1.23818i	−0.785321	−	0.619089i	$-0.787502\pi$
$59$		−	73.0525i		−	1.23818i	0.785321	−	0.619089i	$-0.212498\pi$
$60$		0			0
$61$			5.67609i			0.0930506i	0.998917	+	0.0465253i	$0.0148148\pi$
$61$			5.67609i			0.0930506i	−0.998917	+	0.0465253i	$0.985185\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	18.4353			0.283621
$66$		0			0
$67$	28.7750			0.429477			0.214739	−	0.976672i	$-0.431110\pi$
$67$	28.7750			0.429477			0.214739	+	0.976672i	$0.431110\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.53191			0.0779142			0.0389571	−	0.999241i	$-0.487596\pi$
$71$	5.53191			0.0779142			0.0389571	+	0.999241i	$0.487596\pi$
$72$		0			0
$73$			94.9429i			1.30059i	0.759683	+	0.650294i	$0.225354\pi$
$73$			94.9429i			1.30059i	−0.759683	+	0.650294i	$0.774646\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−94.6940	+	49.2585i	−1.22979	+	0.639720i
$78$		0			0
$79$	−81.5337			−1.03207			−0.516036	−	0.856567i	$-0.672593\pi$
$79$	−81.5337			−1.03207			−0.516036	+	0.856567i	$0.672593\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			86.2943i			1.03969i	0.854261	+	0.519845i	$0.174010\pi$
$83$			86.2943i			1.03969i	−0.854261	+	0.519845i	$0.825990\pi$
$84$		0			0
$85$	132.517			1.55903
$86$		0			0
$87$		0			0
$88$		0			0
$89$			27.6371i			0.310529i	0.987873	+	0.155265i	$0.0496231\pi$
$89$			27.6371i			0.310529i	−0.987873	+	0.155265i	$0.950377\pi$
$90$		0			0
$91$	9.71661	+	18.6791i	0.106776	+	0.205265i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	72.9234			0.767615
$96$		0			0
$97$			100.959i			1.04081i	0.853919	+	0.520406i	$0.174219\pi$
$97$			100.959i			1.04081i	−0.853919	+	0.520406i	$0.825781\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			107.541i			1.06476i	0.846506	+	0.532379i	$0.178702\pi$
$101$			107.541i			1.06476i	−0.846506	+	0.532379i	$0.821298\pi$
$102$		0			0
$103$			59.1076i			0.573860i	0.957952	+	0.286930i	$0.0926347\pi$
$103$			59.1076i			0.573860i	−0.957952	+	0.286930i	$0.907365\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	45.9954			0.429864			0.214932	−	0.976629i	$-0.431047\pi$
$107$	45.9954			0.429864			0.214932	+	0.976629i	$0.431047\pi$
$108$		0			0
$109$	−23.7414			−0.217811			−0.108905	−	0.994052i	$-0.534735\pi$
$109$	−23.7414			−0.217811			−0.108905	+	0.994052i	$0.534735\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	187.211			1.65674			0.828368	−	0.560184i	$-0.189269\pi$
$113$	187.211			1.65674			0.828368	+	0.560184i	$0.189269\pi$
$114$		0			0
$115$			10.5544i			0.0917776i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	69.8451	+	134.270i	0.586934	+	1.12832i
$120$		0			0
$121$	111.517			0.921630
$122$		0			0
$123$		0			0
$124$		0			0
$125$			76.2162i			0.609730i
$126$		0			0
$127$	206.350			1.62480			0.812402	−	0.583097i	$-0.198159\pi$
$127$	206.350			1.62480			0.812402	+	0.583097i	$0.198159\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	186.979i		−	1.42732i	−0.700492	−	0.713660i	$-0.747036\pi$
$131$		−	186.979i		−	1.42732i	0.700492	−	0.713660i	$-0.252964\pi$
$132$		0			0
$133$	38.4353	+	73.8877i	0.288988	+	0.555547i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.51728			0.0183743			0.00918715	−	0.999958i	$-0.497076\pi$
$137$	2.51728			0.0183743			0.00918715	+	0.999958i	$0.497076\pi$
$138$		0			0
$139$		−	18.6791i		−	0.134382i	−0.997740	−	0.0671910i	$-0.978596\pi$
$139$		−	18.6791i		−	0.134382i	0.997740	−	0.0671910i	$-0.0214037\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	45.8658i		−	0.320740i
$144$		0			0
$145$			252.082i			1.73849i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.51728			0.0168945			0.00844725	−	0.999964i	$-0.497311\pi$
$149$	2.51728			0.0168945			0.00844725	+	0.999964i	$0.497311\pi$
$150$		0			0
$151$	−78.4553			−0.519572			−0.259786	−	0.965666i	$-0.583652\pi$
$151$	−78.4553			−0.519572			−0.259786	+	0.965666i	$0.583652\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−52.1430			−0.336407
$156$		0			0
$157$		−	298.559i		−	1.90165i	−0.309732	−	0.950824i	$-0.600239\pi$
$157$		−	298.559i		−	1.90165i	0.309732	−	0.950824i	$-0.399761\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.6940	+	5.56286i	−0.0664222	+	0.0345519i
$162$		0			0
$163$	−8.86012			−0.0543566			−0.0271783	−	0.999631i	$-0.508652\pi$
$163$	−8.86012			−0.0543566			−0.0271783	+	0.999631i	$0.508652\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			252.930i			1.51455i	0.653095	+	0.757276i	$0.273470\pi$
$167$			252.930i			1.51455i	−0.653095	+	0.757276i	$0.726530\pi$
$168$		0			0
$169$	159.953			0.946465
$170$		0			0
$171$		0			0
$172$		0			0
$173$			7.89137i			0.0456149i	0.999740	+	0.0228074i	$0.00726046\pi$
$173$			7.89137i			0.0456149i	−0.999740	+	0.0228074i	$0.992740\pi$
$174$		0			0
$175$	78.0271	−	40.5886i	0.445869	−	0.231935i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−298.573			−1.66800			−0.834002	−	0.551761i	$-0.813956\pi$
$179$	−298.573			−1.66800			−0.834002	+	0.551761i	$0.813956\pi$
$180$		0			0
$181$			281.821i			1.55702i	0.627631	+	0.778511i	$0.284025\pi$
$181$			281.821i			1.55702i	−0.627631	+	0.778511i	$0.715975\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			325.630i			1.76016i
$186$		0			0
$187$		−	329.694i		−	1.76307i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.5396			0.107537			0.0537686	−	0.998553i	$-0.482877\pi$
$191$	20.5396			0.107537			0.0537686	+	0.998553i	$0.482877\pi$
$192$		0			0
$193$	191.565			0.992563			0.496282	−	0.868162i	$-0.334698\pi$
$193$	191.565			0.992563			0.496282	+	0.868162i	$0.334698\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	113.129			0.574261			0.287130	−	0.957892i	$-0.407299\pi$
$197$	113.129			0.574261			0.287130	+	0.957892i	$0.407299\pi$
$198$		0			0
$199$		−	132.543i		−	0.666045i	−0.942919	−	0.333023i	$-0.891931\pi$
$199$		−	132.543i		−	0.666045i	0.942919	−	0.333023i	$-0.108069\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−255.415	+	132.863i	−1.25820	+	0.654499i
$204$		0			0
$205$	263.647			1.28608
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	181.428i		−	0.868078i
$210$		0			0
$211$	−235.366			−1.11548			−0.557739	−	0.830016i	$-0.688331\pi$
$211$	−235.366			−1.11548			−0.557739	+	0.830016i	$0.688331\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	224.595i		−	1.04463i
$216$		0			0
$217$	−27.4827	−	52.8325i	−0.126648	−	0.243468i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−65.0346			−0.294274
$222$		0			0
$223$		−	209.306i		−	0.938593i	−0.883041	−	0.469297i	$-0.844508\pi$
$223$		−	209.306i		−	0.938593i	0.883041	−	0.469297i	$-0.155492\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			217.751i			0.959256i	0.877472	+	0.479628i	$0.159228\pi$
$227$			217.751i			0.959256i	−0.877472	+	0.479628i	$0.840772\pi$
$228$		0			0
$229$			51.1341i			0.223293i	0.993748	+	0.111646i	$0.0356124\pi$
$229$			51.1341i			0.223293i	−0.993748	+	0.111646i	$0.964388\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	242.517			1.04085			0.520423	−	0.853908i	$-0.325774\pi$
$233$	242.517			1.04085			0.520423	+	0.853908i	$0.325774\pi$
$234$		0			0
$235$	185.445			0.789127
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.4613			0.0730598			0.0365299	−	0.999333i	$-0.488370\pi$
$239$	17.4613			0.0730598			0.0365299	+	0.999333i	$0.488370\pi$
$240$		0			0
$241$			68.8417i			0.285650i	0.989748	+	0.142825i	$0.0456186\pi$
$241$			68.8417i			0.285650i	−0.989748	+	0.142825i	$0.954381\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	245.905	+	172.405i	1.00369	+	0.703692i
$246$		0			0
$247$	−35.7881			−0.144891
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	122.246i		−	0.487037i	−0.969896	−	0.243518i	$-0.921698\pi$
$251$		−	122.246i		−	0.487037i	0.969896	−	0.243518i	$-0.0783016\pi$
$252$		0			0
$253$	26.2586			0.103789
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	312.466i		−	1.21582i	−0.794006	−	0.607910i	$-0.792008\pi$
$257$		−	312.466i		−	1.21582i	0.794006	−	0.607910i	$-0.207992\pi$
$258$		0			0
$259$	−329.935	+	171.628i	−1.27388	+	0.662655i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	373.959			1.42190			0.710949	−	0.703244i	$-0.248266\pi$
$263$	373.959			1.42190			0.710949	+	0.703244i	$0.248266\pi$
$264$		0			0
$265$			183.870i			0.693849i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	180.636i		−	0.671508i	−0.941950	−	0.335754i	$-0.891009\pi$
$269$		−	180.636i		−	0.671508i	0.941950	−	0.335754i	$-0.108991\pi$
$270$		0			0
$271$		−	275.195i		−	1.01548i	−0.861511	−	0.507740i	$-0.830481\pi$
$271$		−	275.195i		−	1.01548i	0.861511	−	0.507740i	$-0.169519\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−191.592			−0.696700
$276$		0			0
$277$	376.940			1.36079			0.680397	−	0.732844i	$-0.261807\pi$
$277$	376.940			1.36079			0.680397	+	0.732844i	$0.261807\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−62.5173			−0.222481			−0.111241	−	0.993793i	$-0.535482\pi$
$281$	−62.5173			−0.222481			−0.111241	+	0.993793i	$0.535482\pi$
$282$		0			0
$283$		−	103.435i		−	0.365494i	−0.983160	−	0.182747i	$-0.941501\pi$
$283$		−	103.435i		−	0.365494i	0.983160	−	0.182747i	$-0.0584989\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	138.959	+	267.133i	0.484177	+	0.930776i
$288$		0			0
$289$	−178.483			−0.617587
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	507.624i		−	1.73251i	−0.499606	−	0.866253i	$-0.666522\pi$
$293$		−	507.624i		−	1.73251i	0.499606	−	0.866253i	$-0.333478\pi$
$294$		0			0
$295$	447.739			1.51776
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	5.17972i		−	0.0173235i
$300$		0			0
$301$	227.565	−	118.376i	0.756029	−	0.393275i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−34.7887			−0.114061
$306$		0			0
$307$			419.316i			1.36585i	0.730489	+	0.682925i	$0.239292\pi$
$307$			419.316i			1.36585i	−0.730489	+	0.682925i	$0.760708\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			171.628i			0.551858i	0.961178	+	0.275929i	$0.0889854\pi$
$311$			171.628i			0.551858i	−0.961178	+	0.275929i	$0.911015\pi$
$312$		0			0
$313$		−	503.484i		−	1.60857i	−0.594240	−	0.804287i	$-0.702547\pi$
$313$		−	503.484i		−	1.60857i	0.594240	−	0.804287i	$-0.297453\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	205.647			0.648727			0.324364	−	0.945932i	$-0.394850\pi$
$317$	205.647			0.648727			0.324364	+	0.945932i	$0.394850\pi$
$318$		0			0
$319$	627.161			1.96602
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−257.253			−0.796448
$324$		0			0
$325$			37.7931i			0.116286i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	97.7414	+	187.897i	0.297086	+	0.571115i
$330$		0			0
$331$	−19.6740			−0.0594382			−0.0297191	−	0.999558i	$-0.509461\pi$
$331$	−19.6740			−0.0594382			−0.0297191	+	0.999558i	$0.509461\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			176.362i			0.526454i
$336$		0			0
$337$	317.823			0.943096			0.471548	−	0.881840i	$-0.343695\pi$
$337$	317.823			0.943096			0.471548	+	0.881840i	$0.343695\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			129.728i			0.380434i
$342$		0			0
$343$	−45.0765	+	340.025i	−0.131418	+	0.991327i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	416.385			1.19996			0.599979	−	0.800016i	$-0.295176\pi$
$347$	416.385			1.19996			0.599979	+	0.800016i	$0.295176\pi$
$348$		0			0
$349$			333.344i			0.955140i	0.878594	+	0.477570i	$0.158482\pi$
$349$			333.344i			0.955140i	−0.878594	+	0.477570i	$0.841518\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			230.864i			0.654005i	0.945023	+	0.327003i	$0.106039\pi$
$353$			230.864i			0.654005i	−0.945023	+	0.327003i	$0.893961\pi$
$354$		0			0
$355$			33.9051i			0.0955073i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.9796			−0.0472970			−0.0236485	−	0.999720i	$-0.507528\pi$
$359$	−16.9796			−0.0472970			−0.0236485	+	0.999720i	$0.507528\pi$
$360$		0			0
$361$	219.435			0.607854
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−581.905			−1.59426
$366$		0			0
$367$			457.377i			1.24626i	0.782119	+	0.623129i	$0.214139\pi$
$367$			457.377i			1.24626i	−0.782119	+	0.623129i	$0.785861\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−186.301	+	96.9113i	−0.502160	+	0.261217i
$372$		0			0
$373$	−541.974			−1.45301			−0.726507	−	0.687159i	$-0.758858\pi$
$373$	−541.974			−1.45301			−0.726507	+	0.687159i	$0.758858\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	123.712i		−	0.328149i
$378$		0			0
$379$	−254.799			−0.672294			−0.336147	−	0.941810i	$-0.609124\pi$
$379$	−254.799			−0.672294			−0.336147	+	0.941810i	$0.609124\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	295.983i		−	0.772803i	−0.922331	−	0.386401i	$-0.873718\pi$
$383$		−	295.983i		−	0.772803i	0.922331	−	0.386401i	$-0.126282\pi$
$384$		0			0
$385$	−301.905	−	580.380i	−0.784169	−	1.50748i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	538.164			1.38345			0.691727	−	0.722159i	$-0.256850\pi$
$389$	538.164			1.38345			0.691727	+	0.722159i	$0.256850\pi$
$390$		0			0
$391$		−	37.2329i		−	0.0952249i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	499.720i		−	1.26511i
$396$		0			0
$397$			547.293i			1.37857i	0.724490	+	0.689286i	$0.242075\pi$
$397$			547.293i			1.37857i	−0.724490	+	0.689286i	$0.757925\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−203.375			−0.507170			−0.253585	−	0.967313i	$-0.581610\pi$
$401$	−203.375			−0.507170			−0.253585	+	0.967313i	$0.581610\pi$
$402$		0			0
$403$	25.5899			0.0634984
$404$		0			0
$405$		0			0
$406$		0			0
$407$	810.143			1.99052
$408$		0			0
$409$		−	103.627i		−	0.253366i	−0.991943	−	0.126683i	$-0.959567\pi$
$409$		−	103.627i		−	0.253366i	0.991943	−	0.126683i	$-0.0404332\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	235.987	+	453.659i	0.571398	+	1.09845i
$414$		0			0
$415$	−528.898			−1.27445
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	438.503i		−	1.04655i	−0.852165	−	0.523273i	$-0.824711\pi$
$419$		−	438.503i		−	1.04655i	0.852165	−	0.523273i	$-0.175289\pi$
$420$		0			0
$421$	−741.293			−1.76079			−0.880396	−	0.474240i	$-0.842723\pi$
$421$	−741.293			−1.76079			−0.880396	+	0.474240i	$0.842723\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			271.665i			0.639212i
$426$		0			0
$427$	−18.3359	−	35.2488i	−0.0429412	−	0.0825498i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	82.8809			0.192299			0.0961495	−	0.995367i	$-0.469347\pi$
$431$	82.8809			0.192299			0.0961495	+	0.995367i	$0.469347\pi$
$432$		0			0
$433$			607.840i			1.40379i	0.712282	+	0.701893i	$0.247662\pi$
$433$			607.840i			1.40379i	−0.712282	+	0.701893i	$0.752338\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	20.4891i		−	0.0468857i
$438$		0			0
$439$			654.207i			1.49022i	0.666941	+	0.745111i	$0.267603\pi$
$439$			654.207i			1.49022i	−0.666941	+	0.745111i	$0.732397\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−566.176			−1.27805			−0.639025	−	0.769186i	$-0.720662\pi$
$443$	−566.176			−1.27805			−0.639025	+	0.769186i	$0.720662\pi$
$444$		0			0
$445$	−169.388			−0.380647
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−348.776			−0.776784			−0.388392	−	0.921494i	$-0.626969\pi$
$449$	−348.776			−0.776784			−0.388392	+	0.921494i	$0.626969\pi$
$450$		0			0
$451$		−	655.934i		−	1.45440i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−114.484	+	59.5531i	−0.251614	+	0.130886i
$456$		0			0
$457$	354.694			0.776136			0.388068	−	0.921631i	$-0.373143\pi$
$457$	354.694			0.776136			0.388068	+	0.921631i	$0.373143\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	220.531i		−	0.478375i	−0.970973	−	0.239187i	$-0.923119\pi$
$461$		−	220.531i		−	0.478375i	0.970973	−	0.239187i	$-0.0768810\pi$
$462$		0			0
$463$	−497.178			−1.07382			−0.536910	−	0.843640i	$-0.680408\pi$
$463$	−497.178			−1.07382			−0.536910	+	0.843640i	$0.680408\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	299.499i		−	0.641326i	−0.947193	−	0.320663i	$-0.896094\pi$
$467$		−	299.499i		−	0.641326i	0.947193	−	0.320663i	$-0.103906\pi$
$468$		0			0
$469$	−178.694	+	92.9541i	−0.381011	+	0.198196i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−558.776			−1.18134
$474$		0			0
$475$			149.496i			0.314727i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	524.867i		−	1.09576i	−0.836558	−	0.547878i	$-0.815436\pi$
$479$		−	524.867i		−	1.09576i	0.836558	−	0.547878i	$-0.184564\pi$
$480$		0			0
$481$		−	159.807i		−	0.332239i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−618.776			−1.27583
$486$		0			0
$487$	62.2345			0.127792			0.0638958	−	0.997957i	$-0.479647\pi$
$487$	62.2345			0.127792			0.0638958	+	0.997957i	$0.479647\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−127.136			−0.258933			−0.129467	−	0.991584i	$-0.541326\pi$
$491$	−127.136			−0.258933			−0.129467	+	0.991584i	$0.541326\pi$
$492$		0			0
$493$		−	889.271i		−	1.80380i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−34.3534	+	17.8702i	−0.0691215	+	0.0359560i
$498$		0			0
$499$	−699.460			−1.40172			−0.700862	−	0.713297i	$-0.747201\pi$
$499$	−699.460			−1.40172			−0.700862	+	0.713297i	$0.747201\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			856.217i			1.70222i	0.524986	+	0.851111i	$0.324070\pi$
$503$			856.217i			1.70222i	−0.524986	+	0.851111i	$0.675930\pi$
$504$		0			0
$505$	−659.116			−1.30518
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	83.6547i		−	0.164351i	−0.996618	−	0.0821755i	$-0.973813\pi$
$509$		−	83.6547i		−	0.164351i	0.996618	−	0.0821755i	$-0.0261868\pi$
$510$		0			0
$511$	−306.701	−	589.600i	−0.600199	−	1.15382i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−362.270			−0.703438
$516$		0			0
$517$		−	461.374i		−	0.892405i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			563.287i			1.08117i	0.841291	+	0.540583i	$0.181796\pi$
$521$			563.287i			1.08117i	−0.841291	+	0.540583i	$0.818204\pi$
$522$		0			0
$523$		−	414.011i		−	0.791607i	−0.918335	−	0.395804i	$-0.870466\pi$
$523$		−	414.011i		−	0.791607i	0.918335	−	0.395804i	$-0.129534\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	183.946			0.349043
$528$		0			0
$529$	−526.035			−0.994394
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−129.388			−0.242754
$534$		0			0
$535$			281.906i			0.526927i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	428.930	−	611.795i	0.795789	−	1.13505i
$540$		0			0
$541$	−86.7068			−0.160271			−0.0801357	−	0.996784i	$-0.525535\pi$
$541$	−86.7068			−0.160271			−0.0801357	+	0.996784i	$0.525535\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	145.511i		−	0.266992i
$546$		0			0
$547$	827.355			1.51253			0.756266	−	0.654265i	$-0.227022\pi$
$547$	827.355			1.51253			0.756266	+	0.654265i	$0.227022\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	489.361i		−	0.888132i
$552$		0			0
$553$	506.328	−	263.385i	0.915602	−	0.476283i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	343.741			0.617130			0.308565	−	0.951203i	$-0.400151\pi$
$557$	343.741			0.617130			0.308565	+	0.951203i	$0.400151\pi$
$558$		0			0
$559$			110.223i			0.197179i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	308.968i		−	0.548788i	−0.961617	−	0.274394i	$-0.911523\pi$
$563$		−	308.968i		−	0.548788i	0.961617	−	0.274394i	$-0.0884772\pi$
$564$		0			0
$565$			1147.42i			2.03083i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−182.177			−0.320170			−0.160085	−	0.987103i	$-0.551177\pi$
$569$	−182.177			−0.320170			−0.160085	+	0.987103i	$0.551177\pi$
$570$		0			0
$571$	378.037			0.662061			0.331031	−	0.943620i	$-0.392604\pi$
$571$	378.037			0.662061			0.331031	+	0.943620i	$0.392604\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−21.6369			−0.0376294
$576$		0			0
$577$		−	81.6019i		−	0.141424i	−0.997497	−	0.0707122i	$-0.977473\pi$
$577$		−	81.6019i		−	0.141424i	0.997497	−	0.0707122i	$-0.0225272\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−278.763	−	535.891i	−0.479799	−	0.922360i
$582$		0			0
$583$	457.456			0.784658
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	37.9917i		−	0.0647217i	−0.999476	−	0.0323609i	$-0.989697\pi$
$587$		−	37.9917i		−	0.0647217i	0.999476	−	0.0323609i	$-0.0103026\pi$
$588$		0			0
$589$	101.224			0.171858
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	406.099i		−	0.684822i	−0.939550	−	0.342411i	$-0.888757\pi$
$593$		−	406.099i		−	0.684822i	0.939550	−	0.342411i	$-0.111243\pi$
$594$		0			0
$595$	−822.938	+	428.081i	−1.38309	+	0.719464i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	292.800			0.488815			0.244407	−	0.969673i	$-0.421407\pi$
$599$	292.800			0.488815			0.244407	+	0.969673i	$0.421407\pi$
$600$		0			0
$601$			14.6997i			0.0244588i	0.999925	+	0.0122294i	$0.00389284\pi$
$601$			14.6997i			0.0244588i	−0.999925	+	0.0122294i	$0.996107\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			683.490i			1.12973i
$606$		0			0
$607$			959.534i			1.58078i	0.612604	+	0.790390i	$0.290122\pi$
$607$			959.534i			1.58078i	−0.612604	+	0.790390i	$0.709878\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−91.0095			−0.148952
$612$		0			0
$613$	−896.397			−1.46231			−0.731156	−	0.682211i	$-0.761019\pi$
$613$	−896.397			−1.46231			−0.731156	+	0.682211i	$0.761019\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−137.280			−0.222497			−0.111248	−	0.993793i	$-0.535485\pi$
$617$	−137.280			−0.222497			−0.111248	+	0.993793i	$0.535485\pi$
$618$		0			0
$619$			703.714i			1.13686i	0.822733	+	0.568428i	$0.192448\pi$
$619$			703.714i			1.13686i	−0.822733	+	0.568428i	$0.807552\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−89.2783	−	171.628i	−0.143304	−	0.275486i
$624$		0			0
$625$	−781.246			−1.24999
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1148.73i		−	1.82628i
$630$		0			0
$631$	−87.6903			−0.138970			−0.0694852	−	0.997583i	$-0.522136\pi$
$631$	−87.6903			−0.138970			−0.0694852	+	0.997583i	$0.522136\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1264.72i			1.99169i
$636$		0			0
$637$	−120.681	−	84.6098i	−0.189452	−	0.132825i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	380.953			0.594310			0.297155	−	0.954829i	$-0.403962\pi$
$641$	380.953			0.594310			0.297155	+	0.954829i	$0.403962\pi$
$642$		0			0
$643$		−	441.511i		−	0.686642i	−0.939218	−	0.343321i	$-0.888448\pi$
$643$		−	441.511i		−	0.686642i	0.939218	−	0.343321i	$-0.111552\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			844.897i			1.30587i	0.757414	+	0.652934i	$0.226462\pi$
$647$			844.897i			1.30587i	−0.757414	+	0.652934i	$0.773538\pi$
$648$		0			0
$649$		−	1113.94i		−	1.71640i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−478.845			−0.733300			−0.366650	−	0.930359i	$-0.619495\pi$
$653$	−478.845			−0.733300			−0.366650	+	0.930359i	$0.619495\pi$
$654$		0			0
$655$	1145.99			1.74961
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−257.012			−0.390003			−0.195002	−	0.980803i	$-0.562471\pi$
$659$	−257.012			−0.390003			−0.195002	+	0.980803i	$0.562471\pi$
$660$		0			0
$661$		−	788.652i		−	1.19312i	−0.802568	−	0.596560i	$-0.796534\pi$
$661$		−	788.652i		−	1.19312i	0.802568	−	0.596560i	$-0.203466\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−452.858	+	235.570i	−0.680989	+	0.354241i
$666$		0			0
$667$	70.8266			0.106187
$668$		0			0
$669$		0			0
$670$		0			0
$671$			86.5519i			0.128989i
$672$		0			0
$673$	−511.362			−0.759825			−0.379913	−	0.925022i	$-0.624046\pi$
$673$	−511.362			−0.759825			−0.379913	+	0.925022i	$0.624046\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	543.542i		−	0.802868i	−0.915888	−	0.401434i	$-0.868512\pi$
$677$		−	543.542i		−	0.802868i	0.915888	−	0.401434i	$-0.131488\pi$
$678$		0			0
$679$	−326.135	−	626.958i	−0.480316	−	0.923355i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	785.312			1.14980			0.574899	−	0.818224i	$-0.305041\pi$
$683$	785.312			1.14980			0.574899	+	0.818224i	$0.305041\pi$
$684$		0			0
$685$			15.4284i			0.0225232i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	90.2366i		−	0.130967i
$690$		0			0
$691$		−	1063.98i		−	1.53978i	−0.638179	−	0.769888i	$-0.720312\pi$
$691$		−	1063.98i		−	1.53978i	0.638179	−	0.769888i	$-0.279688\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	114.484			0.164726
$696$		0			0
$697$	−930.069			−1.33439
$698$		0			0
$699$		0			0
$700$		0			0
$701$	113.810			0.162354			0.0811772	−	0.996700i	$-0.474132\pi$
$701$	113.810			0.162354			0.0811772	+	0.996700i	$0.474132\pi$
$702$		0			0
$703$		−	632.138i		−	0.899200i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−347.397	−	667.832i	−0.491367	−	0.944600i
$708$		0			0
$709$	418.000			0.589563			0.294781	−	0.955565i	$-0.404753\pi$
$709$	418.000			0.589563			0.294781	+	0.955565i	$0.404753\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			14.6505i			0.0205476i
$714$		0			0
$715$	281.112			0.393163
$716$		0			0
$717$		0			0
$718$		0			0
$719$			752.720i			1.04690i	0.852057	+	0.523449i	$0.175355\pi$
$719$			752.720i			1.04690i	−0.852057	+	0.523449i	$0.824645\pi$
$720$		0			0
$721$	−190.940	−	367.061i	−0.264826	−	0.509099i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−516.776			−0.712794
$726$		0			0
$727$			269.945i			0.371314i	0.982615	+	0.185657i	$0.0594414\pi$
$727$			269.945i			0.371314i	−0.982615	+	0.185657i	$0.940559\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			792.306i			1.08387i
$732$		0			0
$733$		−	805.971i		−	1.09955i	−0.835312	−	0.549776i	$-0.814713\pi$
$733$		−	805.971i		−	1.09955i	0.835312	−	0.549776i	$-0.185287\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	438.776			0.595354
$738$		0			0
$739$	672.425			0.909912			0.454956	−	0.890514i	$-0.349655\pi$
$739$	672.425			0.909912			0.454956	+	0.890514i	$0.349655\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	367.686			0.494867			0.247434	−	0.968905i	$-0.420413\pi$
$743$	367.686			0.494867			0.247434	+	0.968905i	$0.420413\pi$
$744$		0			0
$745$			15.4284i			0.0207093i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−285.634	+	148.583i	−0.381354	+	0.198375i
$750$		0			0
$751$	−181.778			−0.242048			−0.121024	−	0.992650i	$-0.538618\pi$
$751$	−181.778			−0.242048			−0.121024	+	0.992650i	$0.538618\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	480.853i		−	0.636891i
$756$		0			0
$757$	−63.7414			−0.0842026			−0.0421013	−	0.999113i	$-0.513405\pi$
$757$	−63.7414			−0.0842026			−0.0421013	+	0.999113i	$0.513405\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1279.94i		−	1.68192i	−0.541096	−	0.840961i	$-0.681990\pi$
$761$		−	1279.94i		−	1.68192i	0.541096	−	0.840961i	$-0.318010\pi$
$762$		0			0
$763$	147.435	−	76.6936i	0.193231	−	0.100516i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−219.734			−0.286485
$768$		0			0
$769$			321.701i			0.418337i	0.977880	+	0.209169i	$0.0670758\pi$
$769$			321.701i			0.418337i	−0.977880	+	0.209169i	$0.932924\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	697.333i		−	0.902112i	−0.892496	−	0.451056i	$-0.851047\pi$
$773$		−	697.333i		−	0.902112i	0.892496	−	0.451056i	$-0.148953\pi$
$774$		0			0
$775$		−	106.895i		−	0.137929i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−511.811			−0.657011
$780$		0			0
$781$	84.3534			0.108007
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1829.87			2.33104
$786$		0			0
$787$		−	330.202i		−	0.419570i	−0.977748	−	0.209785i	$-0.932724\pi$
$787$		−	330.202i		−	0.419570i	0.977748	−	0.209785i	$-0.0672764\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1162.59	+	604.763i	−1.46977	+	0.764555i
$792$		0			0
$793$	17.0730			0.0215297
$794$		0			0
$795$		0			0
$796$		0			0
$797$			456.879i			0.573248i	0.958043	+	0.286624i	$0.0925331\pi$
$797$			456.879i			0.573248i	−0.958043	+	0.286624i	$0.907467\pi$
$798$		0			0
$799$	−654.196			−0.818768
$800$		0			0
$801$		0			0
$802$		0			0
$803$			1447.74i			1.80291i
$804$		0			0
$805$	−34.0948	−	65.5434i	−0.0423537	−	0.0814204i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−175.211			−0.216578			−0.108289	−	0.994119i	$-0.534537\pi$
$809$	−175.211			−0.216578			−0.108289	+	0.994119i	$0.534537\pi$
$810$		0			0
$811$			149.621i			0.184489i	0.995736	+	0.0922446i	$0.0294042\pi$
$811$			149.621i			0.184489i	−0.995736	+	0.0922446i	$0.970596\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	54.3037i		−	0.0666303i
$816$		0			0
$817$			436.001i			0.533661i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	880.750			1.07278			0.536389	−	0.843971i	$-0.319788\pi$
$821$	880.750			1.07278			0.536389	+	0.843971i	$0.319788\pi$
$822$		0			0
$823$	1149.93			1.39724			0.698621	−	0.715492i	$-0.253798\pi$
$823$	1149.93			1.39724			0.698621	+	0.715492i	$0.253798\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	633.041			0.765466			0.382733	−	0.923859i	$-0.374983\pi$
$827$	633.041			0.765466			0.382733	+	0.923859i	$0.374983\pi$
$828$		0			0
$829$		−	900.963i		−	1.08681i	−0.839472	−	0.543404i	$-0.817135\pi$
$829$		−	900.963i		−	1.08681i	0.839472	−	0.543404i	$-0.182865\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−867.483	−	608.194i	−1.04140	−	0.730125i
$834$		0			0
$835$	−1550.21			−1.85654
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1587.19i		−	1.89176i	−0.324514	−	0.945881i	$-0.605201\pi$
$839$		−	1587.19i		−	1.89176i	0.324514	−	0.945881i	$-0.394799\pi$
$840$		0			0
$841$	850.621			1.01144
$842$		0			0
$843$		0			0
$844$		0			0
$845$			980.350i			1.16018i
$846$		0			0
$847$	−692.527	+	360.243i	−0.817624	+	0.425316i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	91.4911			0.107510
$852$		0			0
$853$		−	614.776i		−	0.720722i	−0.932813	−	0.360361i	$-0.882654\pi$
$853$		−	614.776i		−	0.720722i	0.932813	−	0.360361i	$-0.117346\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	741.142i		−	0.864809i	−0.901680	−	0.432405i	$-0.857665\pi$
$857$		−	741.142i		−	0.864809i	0.901680	−	0.432405i	$-0.142335\pi$
$858$		0			0
$859$			686.448i			0.799125i	0.916706	+	0.399563i	$0.130838\pi$
$859$			686.448i			0.799125i	−0.916706	+	0.399563i	$0.869162\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1375.44			−1.59378			−0.796892	−	0.604122i	$-0.793524\pi$
$863$	−1375.44			−1.59378			−0.796892	+	0.604122i	$0.793524\pi$
$864$		0			0
$865$	−48.3662			−0.0559147
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1243.27			−1.43069
$870$		0			0
$871$		−	86.5519i		−	0.0993707i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−246.207	−	473.306i	−0.281380	−	0.540921i
$876$		0			0
$877$	−1319.39			−1.50443			−0.752217	−	0.658916i	$-0.771015\pi$
$877$	−1319.39			−1.50443			−0.752217	+	0.658916i	$0.771015\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	270.582i		−	0.307130i	−0.988139	−	0.153565i	$-0.950924\pi$
$881$		−	270.582i		−	0.307130i	0.988139	−	0.153565i	$-0.0490755\pi$
$882$		0			0
$883$	1491.67			1.68932			0.844660	−	0.535303i	$-0.179803\pi$
$883$	1491.67			1.68932			0.844660	+	0.535303i	$0.179803\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			79.8264i			0.0899960i	0.998987	+	0.0449980i	$0.0143281\pi$
$887$			79.8264i			0.0899960i	−0.998987	+	0.0449980i	$0.985672\pi$
$888$		0			0
$889$	−1281.44	+	666.589i	−1.44144	+	0.749819i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−360.000			−0.403135
$894$		0			0
$895$		−	1829.95i		−	2.04464i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			349.911i			0.389223i
$900$		0			0
$901$		−	648.640i		−	0.719912i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1727.28			−1.90860
$906$		0			0
$907$	845.557			0.932256			0.466128	−	0.884717i	$-0.345649\pi$
$907$	845.557			0.932256			0.466128	+	0.884717i	$0.345649\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1377.51			−1.51209			−0.756045	−	0.654519i	$-0.772871\pi$
$911$	−1377.51			−1.51209			−0.756045	+	0.654519i	$0.772871\pi$
$912$		0			0
$913$			1315.86i			1.44125i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	604.013	+	1161.15i	0.658684	+	1.26625i
$918$		0			0
$919$	−648.950			−0.706148			−0.353074	−	0.935595i	$-0.614864\pi$
$919$	−648.950			−0.706148			−0.353074	+	0.935595i	$0.614864\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	16.6394i		−	0.0180275i
$924$		0			0
$925$	−667.552			−0.721678
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1394.62i			1.50120i	0.660756	+	0.750601i	$0.270236\pi$
$929$			1394.62i			1.50120i	−0.660756	+	0.750601i	$0.729764\pi$
$930$		0			0
$931$	−477.370	−	334.685i	−0.512750	−	0.359490i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2020.69			2.16117
$936$		0			0
$937$		−	433.382i		−	0.462521i	−0.972892	−	0.231260i	$-0.925715\pi$
$937$		−	433.382i		−	0.462521i	0.972892	−	0.231260i	$-0.0742850\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	891.147i		−	0.947021i	−0.880788	−	0.473511i	$-0.842987\pi$
$941$		−	891.147i		−	0.947021i	0.880788	−	0.473511i	$-0.157013\pi$
$942$		0			0
$943$		−	74.0759i		−	0.0785535i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1021.88			−1.07907			−0.539537	−	0.841962i	$-0.681401\pi$
$947$	−1021.88			−1.07907			−0.539537	+	0.841962i	$0.681401\pi$
$948$		0			0
$949$	285.577			0.300925
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1322.52			1.38774			0.693871	−	0.720100i	$-0.255904\pi$
$953$	1322.52			1.38774			0.693871	+	0.720100i	$0.255904\pi$
$954$		0			0
$955$			125.887i			0.131819i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15.6324	+	8.13176i	−0.0163008	+	0.00847942i
$960$		0			0
$961$	888.621			0.924684
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1174.10i			1.21668i
$966$		0			0
$967$	1681.56			1.73894			0.869471	−	0.493983i	$-0.164460\pi$
$967$	1681.56			1.73894			0.869471	+	0.493983i	$0.164460\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1177.91i			1.21309i	0.795049	+	0.606546i	$0.207445\pi$
$971$			1177.91i			1.21309i	−0.795049	+	0.606546i	$0.792555\pi$
$972$		0			0
$973$	60.3406	+	115.998i	0.0620150	+	0.119217i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1440.21			−1.47411			−0.737056	−	0.675832i	$-0.763785\pi$
$977$	−1440.21			−1.47411			−0.737056	+	0.675832i	$0.763785\pi$
$978$		0			0
$979$			421.425i			0.430465i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	283.257i		−	0.288156i	−0.989566	−	0.144078i	$-0.953978\pi$
$983$		−	283.257i		−	0.288156i	0.989566	−	0.144078i	$-0.0460215\pi$
$984$		0			0
$985$			693.370i			0.703929i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−63.1037			−0.0638055
$990$		0			0
$991$	1079.85			1.08966			0.544830	−	0.838547i	$-0.316594\pi$
$991$	1079.85			1.08966			0.544830	+	0.838547i	$0.316594\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	812.356			0.816438
$996$		0			0
$997$			1572.31i			1.57704i	0.615010	+	0.788520i	$0.289152\pi$
$997$			1572.31i			1.57704i	−0.615010	+	0.788520i	$0.710848\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.f.c.1441.7		8
3.2	odd	2		224.3.c.a.97.7	yes	8
4.3	odd	2	inner	2016.3.f.c.1441.8		8
7.6	odd	2	inner	2016.3.f.c.1441.1		8
12.11	even	2		224.3.c.a.97.1	✓	8
21.20	even	2		224.3.c.a.97.2	yes	8
24.5	odd	2		448.3.c.g.321.2		8
24.11	even	2		448.3.c.g.321.8		8
28.27	even	2	inner	2016.3.f.c.1441.2		8
84.83	odd	2		224.3.c.a.97.8	yes	8
168.83	odd	2		448.3.c.g.321.1		8
168.125	even	2		448.3.c.g.321.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.c.a.97.1	✓	8	12.11	even	2
224.3.c.a.97.2	yes	8	21.20	even	2
224.3.c.a.97.7	yes	8	3.2	odd	2
224.3.c.a.97.8	yes	8	84.83	odd	2
448.3.c.g.321.1		8	168.83	odd	2
448.3.c.g.321.2		8	24.5	odd	2
448.3.c.g.321.7		8	168.125	even	2
448.3.c.g.321.8		8	24.11	even	2
2016.3.f.c.1441.1		8	7.6	odd	2	inner
2016.3.f.c.1441.2		8	28.27	even	2	inner
2016.3.f.c.1441.7		8	1.1	even	1	trivial
2016.3.f.c.1441.8		8	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.12900i q^{5} +(-6.21005 + 3.23038i) q^{7} +15.2485 q^{11} -3.00789i q^{13} -21.6213i q^{17} -11.8981i q^{19} +1.72205 q^{23} -12.5647 q^{25} +41.1293 q^{29} +8.50759i q^{31} +(-19.7990 - 38.0614i) q^{35} +53.1293 q^{37} -43.0162i q^{41} -36.6446 q^{43} -30.2569i q^{47} +(28.1293 - 40.1216i) q^{49} +30.0000 q^{53} +93.4582i q^{55} -73.0525i q^{59} +5.67609i q^{61} +18.4353 q^{65} +28.7750 q^{67} +5.53191 q^{71} +94.9429i q^{73} +(-94.6940 + 49.2585i) q^{77} -81.5337 q^{79} +86.2943i q^{83} +132.517 q^{85} +27.6371i q^{89} +(9.71661 + 18.6791i) q^{91} +72.9234 q^{95} +100.959i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{25} + 112 q^{29} + 208 q^{37} + 8 q^{49} + 240 q^{53} + 256 q^{65} - 432 q^{77} + 192 q^{85}+O(q^{100}) \) 8 * q + 8 * q^25 + 112 * q^29 + 208 * q^37 + 8 * q^49 + 240 * q^53 + 256 * q^65 - 432 * q^77 + 192 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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