LMFDB - Embedded newform 2016.3.l.h.433.23

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,3,Mod(433,2016)] 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, 1, 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("2016.433"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.23
Character	$\chi$	$=$	2016.433
Dual form			2016.3.l.h.433.24

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.24467 q^{5} +(6.31936 + 3.01094i) q^{7} -3.67957i q^{11} -12.5108 q^{13} -18.2226i q^{17} -4.94253 q^{19} +1.93648 q^{23} +2.50660 q^{25} -41.0446i q^{29} -22.7944i q^{31} +(33.1430 + 15.7914i) q^{35} -46.7736i q^{37} -66.3075i q^{41} +41.9792i q^{43} -53.3092i q^{47} +(30.8685 + 38.0544i) q^{49} -61.3279i q^{53} -19.2981i q^{55} +70.9445 q^{59} -82.0007 q^{61} -65.6152 q^{65} +1.63710i q^{67} +10.0937 q^{71} +64.4337i q^{73} +(11.0789 - 23.2525i) q^{77} +149.702 q^{79} +12.9895 q^{83} -95.5714i q^{85} +130.681i q^{89} +(-79.0604 - 37.6693i) q^{91} -25.9219 q^{95} -103.630i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{23} + 160 q^{25} - 16 q^{49} - 640 q^{71} + 64 q^{79} - 768 q^{95}+O(q^{100}) $ 32 * q + 64 * q^23 + 160 * q^25 - 16 * q^49 - 640 * q^71 + 64 * q^79 - 768 * q^95

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
$5$	5.24467			1.04893			0.524467	−	0.851431i	$-0.324264\pi$
$5$	5.24467			1.04893			0.524467	+	0.851431i	$0.324264\pi$
$6$		0			0
$7$	6.31936	+	3.01094i	0.902765	+	0.430134i
$7$	6.31936	+	3.01094i	0.902765	+	0.430134i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.67957i		−	0.334506i	−0.985914	−	0.167253i	$-0.946510\pi$
$11$		−	3.67957i		−	0.334506i	0.985914	−	0.167253i	$-0.0534897\pi$
$12$		0			0
$13$	−12.5108			−0.962372			−0.481186	−	0.876619i	$-0.659794\pi$
$13$	−12.5108			−0.962372			−0.481186	+	0.876619i	$0.659794\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	18.2226i		−	1.07192i	−0.844245	−	0.535958i	$-0.819951\pi$
$17$		−	18.2226i		−	1.07192i	0.844245	−	0.535958i	$-0.180049\pi$
$18$		0			0
$19$	−4.94253			−0.260133			−0.130067	−	0.991505i	$-0.541519\pi$
$19$	−4.94253			−0.260133			−0.130067	+	0.991505i	$0.541519\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.93648			0.0841948			0.0420974	−	0.999114i	$-0.486596\pi$
$23$	1.93648			0.0841948			0.0420974	+	0.999114i	$0.486596\pi$
$24$		0			0
$25$	2.50660			0.100264
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	41.0446i		−	1.41533i	−0.706548	−	0.707665i	$-0.749748\pi$
$29$		−	41.0446i		−	1.41533i	0.706548	−	0.707665i	$-0.250252\pi$
$30$		0			0
$31$		−	22.7944i		−	0.735303i	−0.929964	−	0.367652i	$-0.880162\pi$
$31$		−	22.7944i		−	0.735303i	0.929964	−	0.367652i	$-0.119838\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	33.1430	+	15.7914i	0.946942	+	0.451182i
$36$		0			0
$37$		−	46.7736i		−	1.26415i	−0.774907	−	0.632075i	$-0.782203\pi$
$37$		−	46.7736i		−	1.26415i	0.774907	−	0.632075i	$-0.217797\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	66.3075i		−	1.61725i	−0.588321	−	0.808627i	$-0.700211\pi$
$41$		−	66.3075i		−	1.61725i	0.588321	−	0.808627i	$-0.299789\pi$
$42$		0			0
$43$			41.9792i			0.976261i	0.872771	+	0.488131i	$0.162321\pi$
$43$			41.9792i			0.976261i	−0.872771	+	0.488131i	$0.837679\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	53.3092i		−	1.13424i	−0.823636	−	0.567119i	$-0.808058\pi$
$47$		−	53.3092i		−	1.13424i	0.823636	−	0.567119i	$-0.191942\pi$
$48$		0			0
$49$	30.8685	+	38.0544i	0.629970	+	0.776620i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	61.3279i		−	1.15713i	−0.815636	−	0.578565i	$-0.803613\pi$
$53$		−	61.3279i		−	1.15713i	0.815636	−	0.578565i	$-0.196387\pi$
$54$		0			0
$55$		−	19.2981i		−	0.350875i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	70.9445			1.20245			0.601225	−	0.799080i	$-0.294680\pi$
$59$	70.9445			1.20245			0.601225	+	0.799080i	$0.294680\pi$
$60$		0			0
$61$	−82.0007			−1.34427			−0.672137	−	0.740427i	$-0.734623\pi$
$61$	−82.0007			−1.34427			−0.672137	+	0.740427i	$0.734623\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−65.6152			−1.00947
$66$		0			0
$67$			1.63710i			0.0244344i	0.999925	+	0.0122172i	$0.00388895\pi$
$67$			1.63710i			0.0244344i	−0.999925	+	0.0122172i	$0.996111\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.0937			0.142165			0.0710823	−	0.997470i	$-0.477355\pi$
$71$	10.0937			0.142165			0.0710823	+	0.997470i	$0.477355\pi$
$72$		0			0
$73$			64.4337i			0.882653i	0.897347	+	0.441326i	$0.145492\pi$
$73$			64.4337i			0.882653i	−0.897347	+	0.441326i	$0.854508\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	11.0789	−	23.2525i	0.143882	−	0.301980i
$78$		0			0
$79$	149.702			1.89496			0.947479	−	0.319818i	$-0.103622\pi$
$79$	149.702			1.89496			0.947479	+	0.319818i	$0.103622\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.9895			0.156500			0.0782498	−	0.996934i	$-0.475067\pi$
$83$	12.9895			0.156500			0.0782498	+	0.996934i	$0.475067\pi$
$84$		0			0
$85$		−	95.5714i		−	1.12437i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			130.681i			1.46832i	0.678976	+	0.734160i	$0.262424\pi$
$89$			130.681i			1.46832i	−0.678976	+	0.734160i	$0.737576\pi$
$90$		0			0
$91$	−79.0604	−	37.6693i	−0.868796	−	0.413949i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−25.9219			−0.272863
$96$		0			0
$97$		−	103.630i		−	1.06835i	−0.845373	−	0.534176i	$-0.820622\pi$
$97$		−	103.630i		−	1.06835i	0.845373	−	0.534176i	$-0.179378\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	22.8199			0.225939			0.112970	−	0.993598i	$-0.463964\pi$
$101$	22.8199			0.225939			0.112970	+	0.993598i	$0.463964\pi$
$102$		0			0
$103$			158.219i			1.53611i	0.640386	+	0.768053i	$0.278774\pi$
$103$			158.219i			1.53611i	−0.640386	+	0.768053i	$0.721226\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			123.746i			1.15650i	0.815859	+	0.578251i	$0.196265\pi$
$107$			123.746i			1.15650i	−0.815859	+	0.578251i	$0.803735\pi$
$108$		0			0
$109$			39.5511i			0.362854i	0.983404	+	0.181427i	$0.0580716\pi$
$109$			39.5511i			0.362854i	−0.983404	+	0.181427i	$0.941928\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	53.1831			0.470647			0.235324	−	0.971917i	$-0.424385\pi$
$113$	53.1831			0.470647			0.235324	+	0.971917i	$0.424385\pi$
$114$		0			0
$115$	10.1562			0.0883148
$116$		0			0
$117$		0			0
$118$		0			0
$119$	54.8670	−	115.155i	0.461067	−	0.967688i
$120$		0			0
$121$	107.461			0.888106
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−117.971			−0.943764
$126$		0			0
$127$	44.1953			0.347994			0.173997	−	0.984746i	$-0.444332\pi$
$127$	44.1953			0.347994			0.173997	+	0.984746i	$0.444332\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−192.217			−1.46731			−0.733654	−	0.679523i	$-0.762187\pi$
$131$	−192.217			−1.46731			−0.733654	+	0.679523i	$0.762187\pi$
$132$		0			0
$133$	−31.2336	−	14.8816i	−0.234839	−	0.111892i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	265.473			1.93776			0.968879	−	0.247535i	$-0.0796206\pi$
$137$	265.473			1.93776			0.968879	+	0.247535i	$0.0796206\pi$
$138$		0			0
$139$	15.5158			0.111624			0.0558122	−	0.998441i	$-0.482225\pi$
$139$	15.5158			0.111624			0.0558122	+	0.998441i	$0.482225\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			46.0345i			0.321919i
$144$		0			0
$145$		−	215.265i		−	1.48459i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	156.124i		−	1.04782i	−0.851775	−	0.523908i	$-0.824474\pi$
$149$		−	156.124i		−	1.04782i	0.851775	−	0.523908i	$-0.175526\pi$
$150$		0			0
$151$	−83.2527			−0.551342			−0.275671	−	0.961252i	$-0.588900\pi$
$151$	−83.2527			−0.551342			−0.275671	+	0.961252i	$0.588900\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	119.549i		−	0.771285i
$156$		0			0
$157$	−165.066			−1.05138			−0.525689	−	0.850677i	$-0.676192\pi$
$157$	−165.066			−1.05138			−0.525689	+	0.850677i	$0.676192\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.2373	+	5.83062i	0.0760081	+	0.0362150i
$162$		0			0
$163$		−	207.397i		−	1.27238i	−0.771534	−	0.636188i	$-0.780510\pi$
$163$		−	207.397i		−	1.27238i	0.771534	−	0.636188i	$-0.219490\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			170.091i			1.01851i	0.860617	+	0.509253i	$0.170078\pi$
$167$			170.091i			1.01851i	−0.860617	+	0.509253i	$0.829922\pi$
$168$		0			0
$169$	−12.4791			−0.0738406
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−244.457			−1.41304			−0.706522	−	0.707691i	$-0.749737\pi$
$173$	−244.457			−1.41304			−0.706522	+	0.707691i	$0.749737\pi$
$174$		0			0
$175$	15.8401	+	7.54720i	0.0905147	+	0.0431269i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			51.4534i			0.287449i	0.989618	+	0.143725i	$0.0459079\pi$
$179$			51.4534i			0.287449i	−0.989618	+	0.143725i	$0.954092\pi$
$180$		0			0
$181$	281.237			1.55380			0.776898	−	0.629626i	$-0.216792\pi$
$181$	281.237			1.55380			0.776898	+	0.629626i	$0.216792\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	245.312i		−	1.32601i
$186$		0			0
$187$	−67.0512			−0.358562
$188$		0			0
$189$		0			0
$190$		0			0
$191$	126.618			0.662921			0.331460	−	0.943469i	$-0.392459\pi$
$191$	126.618			0.662921			0.331460	+	0.943469i	$0.392459\pi$
$192$		0			0
$193$	328.592			1.70255			0.851275	−	0.524719i	$-0.175830\pi$
$193$	328.592			1.70255			0.851275	+	0.524719i	$0.175830\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	26.1931i		−	0.132960i	−0.997788	−	0.0664799i	$-0.978823\pi$
$197$		−	26.1931i		−	0.132960i	0.997788	−	0.0664799i	$-0.0211768\pi$
$198$		0			0
$199$		−	341.606i		−	1.71661i	−0.513137	−	0.858307i	$-0.671517\pi$
$199$		−	341.606i		−	1.71661i	0.513137	−	0.858307i	$-0.328483\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	123.583	−	259.375i	0.608782	−	1.27771i
$204$		0			0
$205$		−	347.761i		−	1.69639i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			18.1864i			0.0870161i
$210$		0			0
$211$		−	89.5480i		−	0.424398i	−0.977226	−	0.212199i	$-0.931938\pi$
$211$		−	89.5480i		−	0.424398i	0.977226	−	0.212199i	$-0.0680625\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			220.167i			1.02403i
$216$		0			0
$217$	68.6325	−	144.046i	0.316279	−	0.663806i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			227.980i			1.03158i
$222$		0			0
$223$		−	391.913i		−	1.75746i	−0.477323	−	0.878728i	$-0.658393\pi$
$223$		−	391.913i		−	1.75746i	0.477323	−	0.878728i	$-0.341607\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−306.142			−1.34864			−0.674322	−	0.738437i	$-0.735564\pi$
$227$	−306.142			−1.34864			−0.674322	+	0.738437i	$0.735564\pi$
$228$		0			0
$229$	53.4932			0.233595			0.116797	−	0.993156i	$-0.462737\pi$
$229$	53.4932			0.233595			0.116797	+	0.993156i	$0.462737\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	28.7104			0.123221			0.0616103	−	0.998100i	$-0.480376\pi$
$233$	28.7104			0.123221			0.0616103	+	0.998100i	$0.480376\pi$
$234$		0			0
$235$		−	279.589i		−	1.18974i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	214.671			0.898207			0.449103	−	0.893480i	$-0.351744\pi$
$239$	214.671			0.898207			0.449103	+	0.893480i	$0.351744\pi$
$240$		0			0
$241$			355.937i			1.47692i	0.674298	+	0.738459i	$0.264446\pi$
$241$			355.937i			1.47692i	−0.674298	+	0.738459i	$0.735554\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	161.895	+	199.583i	0.660797	+	0.814623i
$246$		0			0
$247$	61.8351			0.250345
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−86.5294			−0.344739			−0.172369	−	0.985032i	$-0.555142\pi$
$251$	−86.5294			−0.344739			−0.172369	+	0.985032i	$0.555142\pi$
$252$		0			0
$253$		−	7.12541i		−	0.0281637i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			289.039i			1.12466i	0.826911	+	0.562332i	$0.190096\pi$
$257$			289.039i			1.12466i	−0.826911	+	0.562332i	$0.809904\pi$
$258$		0			0
$259$	140.832	−	295.579i	0.543754	−	1.14123i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−273.840			−1.04122			−0.520609	−	0.853795i	$-0.674295\pi$
$263$	−273.840			−1.04122			−0.520609	+	0.853795i	$0.674295\pi$
$264$		0			0
$265$		−	321.645i		−	1.21375i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	115.065			0.427752			0.213876	−	0.976861i	$-0.431391\pi$
$269$	115.065			0.427752			0.213876	+	0.976861i	$0.431391\pi$
$270$		0			0
$271$		−	85.2047i		−	0.314409i	−0.987566	−	0.157204i	$-0.949752\pi$
$271$		−	85.2047i		−	0.314409i	0.987566	−	0.157204i	$-0.0502481\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	9.22319i		−	0.0335389i
$276$		0			0
$277$		−	96.6558i		−	0.348938i	−0.984663	−	0.174469i	$-0.944179\pi$
$277$		−	96.6558i		−	0.348938i	0.984663	−	0.174469i	$-0.0558209\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−49.2889			−0.175405			−0.0877026	−	0.996147i	$-0.527953\pi$
$281$	−49.2889			−0.175405			−0.0877026	+	0.996147i	$0.527953\pi$
$282$		0			0
$283$	−435.567			−1.53911			−0.769553	−	0.638583i	$-0.779521\pi$
$283$	−435.567			−1.53911			−0.769553	+	0.638583i	$0.779521\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	199.648	−	419.020i	0.695636	−	1.46000i
$288$		0			0
$289$	−43.0622			−0.149004
$290$		0			0
$291$		0			0
$292$		0			0
$293$	92.1414			0.314476			0.157238	−	0.987561i	$-0.449741\pi$
$293$	92.1414			0.314476			0.157238	+	0.987561i	$0.449741\pi$
$294$		0			0
$295$	372.081			1.26129
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−24.2270			−0.0810267
$300$		0			0
$301$	−126.397	+	265.282i	−0.419923	+	0.881335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−430.067			−1.41005
$306$		0			0
$307$	554.865			1.80738			0.903689	−	0.428189i	$-0.140848\pi$
$307$	554.865			1.80738			0.903689	+	0.428189i	$0.140848\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	12.8088i		−	0.0411860i	−0.999788	−	0.0205930i	$-0.993445\pi$
$311$		−	12.8088i		−	0.0411860i	0.999788	−	0.0205930i	$-0.00655542\pi$
$312$		0			0
$313$			9.52099i			0.0304185i	0.999884	+	0.0152093i	$0.00484144\pi$
$313$			9.52099i			0.0304185i	−0.999884	+	0.0152093i	$0.995159\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	322.997i		−	1.01892i	−0.860495	−	0.509459i	$-0.829846\pi$
$317$		−	322.997i		−	1.01892i	0.860495	−	0.509459i	$-0.170154\pi$
$318$		0			0
$319$	−151.026			−0.473437
$320$		0			0
$321$		0			0
$322$		0			0
$323$			90.0656i			0.278841i
$324$		0			0
$325$	−31.3596			−0.0964911
$326$		0			0
$327$		0			0
$328$		0			0
$329$	160.511	−	336.880i	0.487874	−	1.02395i
$330$		0			0
$331$			605.748i			1.83005i	0.403392	+	0.915027i	$0.367831\pi$
$331$			605.748i			1.83005i	−0.403392	+	0.915027i	$0.632169\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			8.58608i			0.0256301i
$336$		0			0
$337$	−378.648			−1.12359			−0.561793	−	0.827278i	$-0.689888\pi$
$337$	−378.648			−1.12359			−0.561793	+	0.827278i	$0.689888\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−83.8735			−0.245963
$342$		0			0
$343$	80.4899	+	333.422i	0.234665	+	0.972076i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	297.236i		−	0.856586i	−0.903640	−	0.428293i	$-0.859115\pi$
$347$		−	297.236i		−	0.856586i	0.903640	−	0.428293i	$-0.140885\pi$
$348$		0			0
$349$	107.718			0.308649			0.154324	−	0.988020i	$-0.450680\pi$
$349$	107.718			0.308649			0.154324	+	0.988020i	$0.450680\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	158.792i		−	0.449836i	−0.974378	−	0.224918i	$-0.927789\pi$
$353$		−	158.792i		−	0.449836i	0.974378	−	0.224918i	$-0.0722115\pi$
$354$		0			0
$355$	52.9381			0.149121
$356$		0			0
$357$		0			0
$358$		0			0
$359$	202.772			0.564825			0.282412	−	0.959293i	$-0.408865\pi$
$359$	202.772			0.564825			0.282412	+	0.959293i	$0.408865\pi$
$360$		0			0
$361$	−336.571			−0.932331
$362$		0			0
$363$		0			0
$364$		0			0
$365$			337.934i			0.925845i
$366$		0			0
$367$			669.013i			1.82292i	0.411383	+	0.911462i	$0.365046\pi$
$367$			669.013i			1.82292i	−0.411383	+	0.911462i	$0.634954\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	184.654	−	387.553i	0.497721	−	1.04462i
$372$		0			0
$373$		−	148.124i		−	0.397116i	−0.980089	−	0.198558i	$-0.936374\pi$
$373$		−	148.124i		−	0.397116i	0.980089	−	0.198558i	$-0.0636258\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			513.502i			1.36207i
$378$		0			0
$379$		−	3.50008i		−	0.00923505i	−0.999989	−	0.00461753i	$-0.998530\pi$
$379$		−	3.50008i		−	0.00923505i	0.999989	−	0.00461753i	$-0.00146981\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	113.064i		−	0.295206i	−0.989047	−	0.147603i	$-0.952844\pi$
$383$		−	113.064i		−	0.295206i	0.989047	−	0.147603i	$-0.0471558\pi$
$384$		0			0
$385$	58.1054	−	121.952i	0.150923	−	0.316758i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			74.8109i			0.192316i	0.995366	+	0.0961580i	$0.0306554\pi$
$389$			74.8109i			0.192316i	−0.995366	+	0.0961580i	$0.969345\pi$
$390$		0			0
$391$		−	35.2877i		−	0.0902497i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	785.136			1.98769
$396$		0			0
$397$	119.449			0.300880			0.150440	−	0.988619i	$-0.451931\pi$
$397$	119.449			0.300880			0.150440	+	0.988619i	$0.451931\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	497.793			1.24138			0.620689	−	0.784057i	$-0.286853\pi$
$401$	497.793			1.24138			0.620689	+	0.784057i	$0.286853\pi$
$402$		0			0
$403$			285.177i			0.707635i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−172.107			−0.422866
$408$		0			0
$409$			241.893i			0.591425i	0.955277	+	0.295713i	$0.0955571\pi$
$409$			241.893i			0.591425i	−0.955277	+	0.295713i	$0.904443\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	448.324	+	213.609i	1.08553	+	0.517214i
$414$		0			0
$415$	68.1255			0.164158
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−719.154			−1.71636			−0.858179	−	0.513351i	$-0.828404\pi$
$419$	−719.154			−1.71636			−0.858179	+	0.513351i	$0.828404\pi$
$420$		0			0
$421$			20.4985i			0.0486901i	0.999704	+	0.0243450i	$0.00775003\pi$
$421$			20.4985i			0.0486901i	−0.999704	+	0.0243450i	$0.992250\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	45.6766i		−	0.107474i
$426$		0			0
$427$	−518.191	−	246.899i	−1.21356	−	0.578217i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−145.153			−0.336782			−0.168391	−	0.985720i	$-0.553857\pi$
$431$	−145.153			−0.336782			−0.168391	+	0.985720i	$0.553857\pi$
$432$		0			0
$433$			52.0462i			0.120199i	0.998192	+	0.0600996i	$0.0191418\pi$
$433$			52.0462i			0.120199i	−0.998192	+	0.0600996i	$0.980858\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−9.57111			−0.0219018
$438$		0			0
$439$		−	265.081i		−	0.603829i	−0.953335	−	0.301914i	$-0.902374\pi$
$439$		−	265.081i		−	0.603829i	0.953335	−	0.301914i	$-0.0976257\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			330.902i			0.746957i	0.927639	+	0.373478i	$0.121835\pi$
$443$			330.902i			0.746957i	−0.927639	+	0.373478i	$0.878165\pi$
$444$		0			0
$445$			685.377i			1.54017i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−625.698			−1.39354			−0.696768	−	0.717296i	$-0.745379\pi$
$449$	−625.698			−1.39354			−0.696768	+	0.717296i	$0.745379\pi$
$450$		0			0
$451$	−243.983			−0.540982
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−414.646	−	197.563i	−0.911310	−	0.434205i
$456$		0			0
$457$	397.689			0.870217			0.435108	−	0.900378i	$-0.356710\pi$
$457$	397.689			0.870217			0.435108	+	0.900378i	$0.356710\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−71.4288			−0.154943			−0.0774715	−	0.996995i	$-0.524685\pi$
$461$	−71.4288			−0.154943			−0.0774715	+	0.996995i	$0.524685\pi$
$462$		0			0
$463$	88.0481			0.190169			0.0950844	−	0.995469i	$-0.469688\pi$
$463$	88.0481			0.190169			0.0950844	+	0.995469i	$0.469688\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−118.536			−0.253824			−0.126912	−	0.991914i	$-0.540507\pi$
$467$	−118.536			−0.253824			−0.126912	+	0.991914i	$0.540507\pi$
$468$		0			0
$469$	−4.92922	+	10.3454i	−0.0105101	+	0.0220585i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	154.465			0.326565
$474$		0			0
$475$	−12.3889			−0.0260819
$476$		0			0
$477$		0			0
$478$		0			0
$479$			138.101i			0.288311i	0.989555	+	0.144156i	$0.0460466\pi$
$479$			138.101i			0.288311i	−0.989555	+	0.144156i	$0.953953\pi$
$480$		0			0
$481$			585.176i			1.21658i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	543.506i		−	1.12063i
$486$		0			0
$487$	−311.781			−0.640206			−0.320103	−	0.947383i	$-0.603718\pi$
$487$	−311.781			−0.640206			−0.320103	+	0.947383i	$0.603718\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	602.082i		−	1.22624i	−0.789991	−	0.613118i	$-0.789915\pi$
$491$		−	602.082i		−	1.22624i	0.789991	−	0.613118i	$-0.210085\pi$
$492$		0			0
$493$	−747.938			−1.51712
$494$		0			0
$495$		0			0
$496$		0			0
$497$	63.7856	+	30.3914i	0.128341	+	0.0611498i
$498$		0			0
$499$		−	209.915i		−	0.420671i	−0.977629	−	0.210335i	$-0.932544\pi$
$499$		−	209.915i		−	0.420671i	0.977629	−	0.210335i	$-0.0674556\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.12961i		−	0.0161622i	−0.999967	−	0.00808112i	$-0.997428\pi$
$503$		−	8.12961i		−	0.0161622i	0.999967	−	0.00808112i	$-0.00257233\pi$
$504$		0			0
$505$	119.683			0.236995
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−116.510			−0.228899			−0.114449	−	0.993429i	$-0.536510\pi$
$509$	−116.510			−0.228899			−0.114449	+	0.993429i	$0.536510\pi$
$510$		0			0
$511$	−194.006	+	407.179i	−0.379659	+	0.796828i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			829.807i			1.61128i
$516$		0			0
$517$	−196.155			−0.379410
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	459.362i		−	0.881693i	−0.897583	−	0.440846i	$-0.854678\pi$
$521$		−	459.362i		−	0.881693i	0.897583	−	0.440846i	$-0.145322\pi$
$522$		0			0
$523$	−392.738			−0.750933			−0.375467	−	0.926836i	$-0.622518\pi$
$523$	−392.738			−0.750933			−0.375467	+	0.926836i	$0.622518\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−415.373			−0.788183
$528$		0			0
$529$	−525.250			−0.992911
$530$		0			0
$531$		0			0
$532$		0			0
$533$			829.561i			1.55640i
$534$		0			0
$535$			649.006i			1.21309i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	140.024	−	113.583i	0.259784	−	0.210729i
$540$		0			0
$541$			224.719i			0.415378i	0.978195	+	0.207689i	$0.0665942\pi$
$541$			224.719i			0.415378i	−0.978195	+	0.207689i	$0.933406\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			207.432i			0.380610i
$546$		0			0
$547$			650.761i			1.18969i	0.803840	+	0.594845i	$0.202787\pi$
$547$			650.761i			1.18969i	−0.803840	+	0.594845i	$0.797213\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			202.864i			0.368174i
$552$		0			0
$553$	946.018	+	450.742i	1.71070	+	0.815086i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	269.480i		−	0.483806i	−0.970300	−	0.241903i	$-0.922228\pi$
$557$		−	269.480i		−	0.483806i	0.970300	−	0.241903i	$-0.0777715\pi$
$558$		0			0
$559$		−	525.195i		−	0.939526i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−114.814			−0.203932			−0.101966	−	0.994788i	$-0.532513\pi$
$563$	−114.814			−0.203932			−0.101966	+	0.994788i	$0.532513\pi$
$564$		0			0
$565$	278.928			0.493678
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−226.624			−0.398285			−0.199143	−	0.979971i	$-0.563816\pi$
$569$	−226.624			−0.398285			−0.199143	+	0.979971i	$0.563816\pi$
$570$		0			0
$571$			905.094i			1.58510i	0.609805	+	0.792551i	$0.291248\pi$
$571$			905.094i			1.58510i	−0.609805	+	0.792551i	$0.708752\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.85397			0.00844169
$576$		0			0
$577$		−	288.394i		−	0.499816i	−0.968270	−	0.249908i	$-0.919600\pi$
$577$		−	288.394i		−	0.499816i	0.968270	−	0.249908i	$-0.0804004\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	82.0851	+	39.1105i	0.141282	+	0.0673158i
$582$		0			0
$583$	−225.660			−0.387067
$584$		0			0
$585$		0			0
$586$		0			0
$587$	478.167			0.814594			0.407297	−	0.913296i	$-0.366471\pi$
$587$	478.167			0.814594			0.407297	+	0.913296i	$0.366471\pi$
$588$		0			0
$589$			112.662i			0.191277i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	350.708i		−	0.591414i	−0.955279	−	0.295707i	$-0.904445\pi$
$593$		−	350.708i		−	0.591414i	0.955279	−	0.295707i	$-0.0955552\pi$
$594$		0			0
$595$	287.760	−	603.950i	0.483629	−	1.01504i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	781.919			1.30537			0.652687	−	0.757627i	$-0.273642\pi$
$599$	781.919			1.30537			0.652687	+	0.757627i	$0.273642\pi$
$600$		0			0
$601$			125.321i			0.208521i	0.994550	+	0.104260i	$0.0332475\pi$
$601$			125.321i			0.208521i	−0.994550	+	0.104260i	$0.966752\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	563.597			0.931565
$606$		0			0
$607$		−	25.4854i		−	0.0419859i	−0.999780	−	0.0209929i	$-0.993317\pi$
$607$		−	25.4854i		−	0.0419859i	0.999780	−	0.0209929i	$-0.00668275\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			666.943i			1.09156i
$612$		0			0
$613$			1044.03i			1.70315i	0.524230	+	0.851577i	$0.324353\pi$
$613$			1044.03i			1.70315i	−0.524230	+	0.851577i	$0.675647\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	540.524			0.876052			0.438026	−	0.898962i	$-0.355678\pi$
$617$	540.524			0.876052			0.438026	+	0.898962i	$0.355678\pi$
$618$		0			0
$619$	295.065			0.476680			0.238340	−	0.971182i	$-0.423397\pi$
$619$	295.065			0.476680			0.238340	+	0.971182i	$0.423397\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−393.471	+	825.817i	−0.631574	+	1.32555i
$624$		0			0
$625$	−681.382			−1.09021
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−852.335			−1.35506
$630$		0			0
$631$	−787.813			−1.24852			−0.624258	−	0.781219i	$-0.714598\pi$
$631$	−787.813			−1.24852			−0.624258	+	0.781219i	$0.714598\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	231.790			0.365023
$636$		0			0
$637$	−386.191	−	476.092i	−0.606265	−	0.747397i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1170.55			−1.82613			−0.913066	−	0.407811i	$-0.866292\pi$
$641$	−1170.55			−1.82613			−0.913066	+	0.407811i	$0.866292\pi$
$642$		0			0
$643$	570.395			0.887085			0.443542	−	0.896253i	$-0.353722\pi$
$643$	570.395			0.887085			0.443542	+	0.896253i	$0.353722\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	869.330i		−	1.34363i	−0.740718	−	0.671816i	$-0.765515\pi$
$647$		−	869.330i		−	1.34363i	0.740718	−	0.671816i	$-0.234485\pi$
$648$		0			0
$649$		−	261.045i		−	0.402227i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1004.79i		−	1.53872i	−0.638813	−	0.769362i	$-0.720574\pi$
$653$		−	1004.79i		−	1.53872i	0.638813	−	0.769362i	$-0.279426\pi$
$654$		0			0
$655$	−1008.12			−1.53911
$656$		0			0
$657$		0			0
$658$		0			0
$659$			323.660i			0.491139i	0.969379	+	0.245569i	$0.0789749\pi$
$659$			323.660i			0.491139i	−0.969379	+	0.245569i	$0.921025\pi$
$660$		0			0
$661$	803.454			1.21551			0.607756	−	0.794124i	$-0.292070\pi$
$661$	803.454			1.21551			0.607756	+	0.794124i	$0.292070\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−163.810	−	78.0493i	−0.246331	−	0.117367i
$666$		0			0
$667$		−	79.4820i		−	0.119163i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			301.727i			0.449668i
$672$		0			0
$673$	−336.584			−0.500125			−0.250063	−	0.968230i	$-0.580451\pi$
$673$	−336.584			−0.500125			−0.250063	+	0.968230i	$0.580451\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	516.746			0.763289			0.381644	−	0.924309i	$-0.375358\pi$
$677$	516.746			0.763289			0.381644	+	0.924309i	$0.375358\pi$
$678$		0			0
$679$	312.024	−	654.876i	0.459534	−	0.964471i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			587.298i			0.859879i	0.902858	+	0.429940i	$0.141465\pi$
$683$			587.298i			0.859879i	−0.902858	+	0.429940i	$0.858535\pi$
$684$		0			0
$685$	1392.32			2.03258
$686$		0			0
$687$		0			0
$688$		0			0
$689$			767.263i			1.11359i
$690$		0			0
$691$	1102.12			1.59497			0.797483	−	0.603342i	$-0.206165\pi$
$691$	1102.12			1.59497			0.797483	+	0.603342i	$0.206165\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	81.3753			0.117087
$696$		0			0
$697$	−1208.29			−1.73356
$698$		0			0
$699$		0			0
$700$		0			0
$701$			12.2107i			0.0174189i	0.999962	+	0.00870946i	$0.00277234\pi$
$701$			12.2107i			0.0174189i	−0.999962	+	0.00870946i	$0.997228\pi$
$702$		0			0
$703$			231.180i			0.328847i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	144.207	+	68.7092i	0.203970	+	0.0971841i
$708$		0			0
$709$		−	1129.01i		−	1.59239i	−0.605037	−	0.796197i	$-0.706842\pi$
$709$		−	1129.01i		−	1.59239i	0.605037	−	0.796197i	$-0.293158\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	44.1409i		−	0.0619087i
$714$		0			0
$715$			241.436i			0.337672i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			878.237i			1.22147i	0.791835	+	0.610735i	$0.209126\pi$
$719$			878.237i			1.22147i	−0.791835	+	0.610735i	$0.790874\pi$
$720$		0			0
$721$	−476.387	+	999.842i	−0.660731	+	1.38674i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	102.882i		−	0.141907i
$726$		0			0
$727$			226.673i			0.311793i	0.987773	+	0.155896i	$0.0498266\pi$
$727$			226.673i			0.311793i	−0.987773	+	0.155896i	$0.950173\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	764.970			1.04647
$732$		0			0
$733$	103.641			0.141393			0.0706964	−	0.997498i	$-0.477478\pi$
$733$	103.641			0.141393			0.0706964	+	0.997498i	$0.477478\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.02383			0.00817345
$738$		0			0
$739$			708.603i			0.958867i	0.877578	+	0.479434i	$0.159158\pi$
$739$			708.603i			0.958867i	−0.877578	+	0.479434i	$0.840842\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−64.3198			−0.0865677			−0.0432838	−	0.999063i	$-0.513782\pi$
$743$	−64.3198			−0.0865677			−0.0432838	+	0.999063i	$0.513782\pi$
$744$		0			0
$745$		−	818.822i		−	1.09909i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−372.590	+	781.993i	−0.497451	+	1.04405i
$750$		0			0
$751$	500.993			0.667101			0.333551	−	0.942732i	$-0.391753\pi$
$751$	500.993			0.667101			0.333551	+	0.942732i	$0.391753\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−436.633			−0.578322
$756$		0			0
$757$			1207.70i			1.59537i	0.603074	+	0.797685i	$0.293942\pi$
$757$			1207.70i			1.59537i	−0.603074	+	0.797685i	$0.706058\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	389.162i		−	0.511382i	−0.966759	−	0.255691i	$-0.917697\pi$
$761$		−	389.162i		−	0.511382i	0.966759	−	0.255691i	$-0.0823030\pi$
$762$		0			0
$763$	−119.086	+	249.937i	−0.156076	+	0.327572i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−887.575			−1.15720
$768$		0			0
$769$		−	656.498i		−	0.853704i	−0.904322	−	0.426852i	$-0.859623\pi$
$769$		−	656.498i		−	0.853704i	0.904322	−	0.426852i	$-0.140377\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	369.776			0.478364			0.239182	−	0.970975i	$-0.423121\pi$
$773$	369.776			0.478364			0.239182	+	0.970975i	$0.423121\pi$
$774$		0			0
$775$		−	57.1364i		−	0.0737243i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			327.726i			0.420701i
$780$		0			0
$781$		−	37.1404i		−	0.0475549i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−865.718			−1.10283
$786$		0			0
$787$	590.922			0.750854			0.375427	−	0.926852i	$-0.377496\pi$
$787$	590.922			0.750854			0.375427	+	0.926852i	$0.377496\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	336.083	+	160.131i	0.424884	+	0.202441i
$792$		0			0
$793$	1025.90			1.29369
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1185.20			−1.48708			−0.743539	−	0.668692i	$-0.766854\pi$
$797$	−1185.20			−1.48708			−0.743539	+	0.668692i	$0.766854\pi$
$798$		0			0
$799$	−971.431			−1.21581
$800$		0			0
$801$		0			0
$802$		0			0
$803$	237.088			0.295253
$804$		0			0
$805$	64.1807	+	30.5797i	0.0797276	+	0.0379872i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	697.293			0.861920			0.430960	−	0.902371i	$-0.358175\pi$
$809$	697.293			0.861920			0.430960	+	0.902371i	$0.358175\pi$
$810$		0			0
$811$	−744.052			−0.917450			−0.458725	−	0.888578i	$-0.651694\pi$
$811$	−744.052			−0.917450			−0.458725	+	0.888578i	$0.651694\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	1087.73i		−	1.33464i
$816$		0			0
$817$		−	207.484i		−	0.253958i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	71.5020i		−	0.0870913i	−0.999051	−	0.0435457i	$-0.986135\pi$
$821$		−	71.5020i		−	0.0870913i	0.999051	−	0.0435457i	$-0.0138654\pi$
$822$		0			0
$823$	−1001.05			−1.21635			−0.608174	−	0.793804i	$-0.708098\pi$
$823$	−1001.05			−1.21635			−0.608174	+	0.793804i	$0.708098\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			504.214i			0.609691i	0.952402	+	0.304845i	$0.0986048\pi$
$827$			504.214i			0.609691i	−0.952402	+	0.304845i	$0.901395\pi$
$828$		0			0
$829$	709.973			0.856421			0.428210	−	0.903679i	$-0.359144\pi$
$829$	709.973			0.856421			0.428210	+	0.903679i	$0.359144\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	693.448	−	562.504i	0.832471	−	0.675275i
$834$		0			0
$835$			892.069i			1.06835i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			868.501i			1.03516i	0.855634	+	0.517581i	$0.173167\pi$
$839$			868.501i			1.03516i	−0.855634	+	0.517581i	$0.826833\pi$
$840$		0			0
$841$	−843.658			−1.00316
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−65.4486			−0.0774539
$846$		0			0
$847$	679.083	+	323.558i	0.801751	+	0.382004i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	90.5761i		−	0.106435i
$852$		0			0
$853$	−404.626			−0.474356			−0.237178	−	0.971466i	$-0.576222\pi$
$853$	−404.626			−0.474356			−0.237178	+	0.971466i	$0.576222\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1427.04i			1.66515i	0.553910	+	0.832577i	$0.313135\pi$
$857$			1427.04i			1.66515i	−0.553910	+	0.832577i	$0.686865\pi$
$858$		0			0
$859$	1004.69			1.16960			0.584801	−	0.811177i	$-0.301173\pi$
$859$	1004.69			1.16960			0.584801	+	0.811177i	$0.301173\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−328.400			−0.380533			−0.190267	−	0.981732i	$-0.560935\pi$
$863$	−328.400			−0.380533			−0.190267	+	0.981732i	$0.560935\pi$
$864$		0			0
$865$	−1282.10			−1.48219
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	550.837i		−	0.633875i
$870$		0			0
$871$		−	20.4815i		−	0.0235150i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−745.498	−	355.202i	−0.851998	−	0.405945i
$876$		0			0
$877$		−	924.919i		−	1.05464i	−0.849667	−	0.527320i	$-0.823197\pi$
$877$		−	924.919i		−	1.05464i	0.849667	−	0.527320i	$-0.176803\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1330.68i			1.51042i	0.655483	+	0.755210i	$0.272465\pi$
$881$			1330.68i			1.51042i	−0.655483	+	0.755210i	$0.727535\pi$
$882$		0			0
$883$		−	1431.52i		−	1.62120i	−0.585601	−	0.810600i	$-0.699141\pi$
$883$		−	1431.52i		−	1.62120i	0.585601	−	0.810600i	$-0.300859\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1257.50i		−	1.41770i	−0.705359	−	0.708850i	$-0.749214\pi$
$887$		−	1257.50i		−	1.41770i	0.705359	−	0.708850i	$-0.250786\pi$
$888$		0			0
$889$	279.286	+	133.069i	0.314157	+	0.149684i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			263.482i			0.295053i
$894$		0			0
$895$			269.856i			0.301515i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−935.587			−1.04070
$900$		0			0
$901$	−1117.55			−1.24035
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1475.00			1.62983
$906$		0			0
$907$			1630.83i			1.79805i	0.437899	+	0.899024i	$0.355723\pi$
$907$			1630.83i			1.79805i	−0.437899	+	0.899024i	$0.644277\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−778.173			−0.854197			−0.427098	−	0.904205i	$-0.640464\pi$
$911$	−778.173			−0.854197			−0.427098	+	0.904205i	$0.640464\pi$
$912$		0			0
$913$		−	47.7956i		−	0.0523501i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1214.69	−	578.754i	−1.32463	−	0.631139i
$918$		0			0
$919$	116.976			0.127286			0.0636429	−	0.997973i	$-0.479728\pi$
$919$	116.976			0.127286			0.0636429	+	0.997973i	$0.479728\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−126.280			−0.136815
$924$		0			0
$925$		−	117.242i		−	0.126749i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			631.449i			0.679709i	0.940478	+	0.339854i	$0.110378\pi$
$929$			631.449i			0.679709i	−0.940478	+	0.339854i	$0.889622\pi$
$930$		0			0
$931$	−152.569	−	188.085i	−0.163876	−	0.202024i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−351.662			−0.376109
$936$		0			0
$937$		−	50.0533i		−	0.0534187i	−0.999643	−	0.0267093i	$-0.991497\pi$
$937$		−	50.0533i		−	0.0534187i	0.999643	−	0.0267093i	$-0.00850286\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−272.536			−0.289623			−0.144812	−	0.989459i	$-0.546258\pi$
$941$	−272.536			−0.289623			−0.144812	+	0.989459i	$0.546258\pi$
$942$		0			0
$943$		−	128.403i		−	0.136164i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1505.37i			1.58962i	0.606858	+	0.794810i	$0.292430\pi$
$947$			1505.37i			1.58962i	−0.606858	+	0.794810i	$0.707570\pi$
$948$		0			0
$949$		−	806.119i		−	0.849440i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	714.817			0.750070			0.375035	−	0.927011i	$-0.377631\pi$
$953$	714.817			0.750070			0.375035	+	0.927011i	$0.377631\pi$
$954$		0			0
$955$	664.069			0.695360
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1677.62	+	799.322i	1.74934	+	0.833495i
$960$		0			0
$961$	441.415			0.459329
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1723.36			1.78586
$966$		0			0
$967$	982.093			1.01561			0.507804	−	0.861473i	$-0.330457\pi$
$967$	982.093			1.01561			0.507804	+	0.861473i	$0.330457\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	119.012			0.122566			0.0612832	−	0.998120i	$-0.480481\pi$
$971$	119.012			0.122566			0.0612832	+	0.998120i	$0.480481\pi$
$972$		0			0
$973$	98.0498	+	46.7171i	0.100771	+	0.0480134i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1313.25			−1.34417			−0.672083	−	0.740476i	$-0.734600\pi$
$977$	−1313.25			−1.34417			−0.672083	+	0.740476i	$0.734600\pi$
$978$		0			0
$979$	480.848			0.491162
$980$		0			0
$981$		0			0
$982$		0			0
$983$			59.3222i			0.0603482i	0.999545	+	0.0301741i	$0.00960617\pi$
$983$			59.3222i			0.0603482i	−0.999545	+	0.0301741i	$0.990394\pi$
$984$		0			0
$985$		−	137.374i		−	0.139466i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			81.2920i			0.0821961i
$990$		0			0
$991$	790.745			0.797927			0.398963	−	0.916967i	$-0.369370\pi$
$991$	790.745			0.797927			0.398963	+	0.916967i	$0.369370\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1791.61i		−	1.80062i
$996$		0			0
$997$	−1034.13			−1.03724			−0.518622	−	0.855004i	$-0.673555\pi$
$997$	−1034.13			−1.03724			−0.518622	+	0.855004i	$0.673555\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.l.h.433.23		32
3.2	odd	2		672.3.l.a.433.23		32
4.3	odd	2		504.3.l.h.181.6		32
7.6	odd	2	inner	2016.3.l.h.433.9		32
8.3	odd	2		504.3.l.h.181.7		32
8.5	even	2	inner	2016.3.l.h.433.10		32
12.11	even	2		168.3.l.a.13.27	yes	32
21.20	even	2		672.3.l.a.433.7		32
24.5	odd	2		672.3.l.a.433.10		32
24.11	even	2		168.3.l.a.13.26	yes	32
28.27	even	2		504.3.l.h.181.5		32
56.13	odd	2	inner	2016.3.l.h.433.24		32
56.27	even	2		504.3.l.h.181.8		32
84.83	odd	2		168.3.l.a.13.28	yes	32
168.83	odd	2		168.3.l.a.13.25	✓	32
168.125	even	2		672.3.l.a.433.26		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.l.a.13.25	✓	32	168.83	odd	2
168.3.l.a.13.26	yes	32	24.11	even	2
168.3.l.a.13.27	yes	32	12.11	even	2
168.3.l.a.13.28	yes	32	84.83	odd	2
504.3.l.h.181.5		32	28.27	even	2
504.3.l.h.181.6		32	4.3	odd	2
504.3.l.h.181.7		32	8.3	odd	2
504.3.l.h.181.8		32	56.27	even	2
672.3.l.a.433.7		32	21.20	even	2
672.3.l.a.433.10		32	24.5	odd	2
672.3.l.a.433.23		32	3.2	odd	2
672.3.l.a.433.26		32	168.125	even	2
2016.3.l.h.433.9		32	7.6	odd	2	inner
2016.3.l.h.433.10		32	8.5	even	2	inner
2016.3.l.h.433.23		32	1.1	even	1	trivial
2016.3.l.h.433.24		32	56.13	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.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.24467 q^{5} +(6.31936 + 3.01094i) q^{7} -3.67957i q^{11} -12.5108 q^{13} -18.2226i q^{17} -4.94253 q^{19} +1.93648 q^{23} +2.50660 q^{25} -41.0446i q^{29} -22.7944i q^{31} +(33.1430 + 15.7914i) q^{35} -46.7736i q^{37} -66.3075i q^{41} +41.9792i q^{43} -53.3092i q^{47} +(30.8685 + 38.0544i) q^{49} -61.3279i q^{53} -19.2981i q^{55} +70.9445 q^{59} -82.0007 q^{61} -65.6152 q^{65} +1.63710i q^{67} +10.0937 q^{71} +64.4337i q^{73} +(11.0789 - 23.2525i) q^{77} +149.702 q^{79} +12.9895 q^{83} -95.5714i q^{85} +130.681i q^{89} +(-79.0604 - 37.6693i) q^{91} -25.9219 q^{95} -103.630i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{23} + 160 q^{25} - 16 q^{49} - 640 q^{71} + 64 q^{79} - 768 q^{95}+O(q^{100}) \) 32 * q + 64 * q^23 + 160 * q^25 - 16 * q^49 - 640 * q^71 + 64 * q^79 - 768 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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