LMFDB - Embedded newform 2016.2.c.e.1009.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,2,Mod(1009,2016)] 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, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2016.1009"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.386672896.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 $ x^8 - x^6 - 2x^5 + 2x^4 - 4x^3 - 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.6
Root			$1.40961 - 0.114062i$ of defining polynomial
Character	$\chi$	$=$	2016.1009
Dual form			2016.2.c.e.1009.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.12875i q^{5} -1.00000 q^{7} -4.76717i q^{11} +0.456247i q^{13} -0.415006 q^{17} +7.63843i q^{19} +1.58499 q^{23} +3.72593 q^{25} -6.72593i q^{29} +5.89592 q^{31} -1.12875i q^{35} +5.89592i q^{37} +0.415006 q^{41} -9.43967i q^{43} +11.2769 q^{47} +1.00000 q^{49} -7.63843i q^{53} +5.38093 q^{55} +4.00000i q^{59} -1.80125i q^{61} -0.514988 q^{65} +8.09467i q^{67} +10.2068 q^{71} -3.34500 q^{73} +4.76717i q^{77} +4.83001 q^{79} +5.53434i q^{83} -0.468436i q^{85} -4.92999 q^{89} -0.456247i q^{91} -8.62185 q^{95} +16.4468 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} - 4 q^{17} + 12 q^{23} - 24 q^{25} - 8 q^{31} + 4 q^{41} + 8 q^{49} + 8 q^{55} + 16 q^{65} - 28 q^{71} - 8 q^{73} + 40 q^{79} - 20 q^{89} + 40 q^{95} + 40 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 4 * q^17 + 12 * q^23 - 24 * q^25 - 8 * q^31 + 4 * q^41 + 8 * q^49 + 8 * q^55 + 16 * q^65 - 28 * q^71 - 8 * q^73 + 40 * q^79 - 20 * q^89 + 40 * q^95 + 40 * q^97

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
$5$	1.12875i	0.504791i	0.967624	+	0.252395i	$0.0812184\pi$
$5$	1.12875i	0.504791i	−0.967624	+	0.252395i	$0.918782\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.76717i	− 1.43736i	−0.695343	−	0.718678i	$-0.744747\pi$
$11$	− 4.76717i	− 1.43736i	0.695343	−	0.718678i	$-0.255253\pi$
$12$	0	0
$13$	0.456247i	0.126540i	0.997996	+	0.0632701i	$0.0201530\pi$
$13$	0.456247i	0.126540i	−0.997996	+	0.0632701i	$0.979847\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.415006	−0.100654	−0.0503268	−	0.998733i	$-0.516026\pi$
$17$	−0.415006	−0.100654	−0.0503268	+	0.998733i	$0.516026\pi$
$18$	0	0
$19$	7.63843i	1.75237i	0.481970	+	0.876187i	$0.339921\pi$
$19$	7.63843i	1.75237i	−0.481970	+	0.876187i	$0.660079\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.58499	0.330494	0.165247	−	0.986252i	$-0.447158\pi$
$23$	1.58499	0.330494	0.165247	+	0.986252i	$0.447158\pi$
$24$	0	0
$25$	3.72593	0.745186
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.72593i	− 1.24897i	−0.781035	−	0.624487i	$-0.785308\pi$
$29$	− 6.72593i	− 1.24897i	0.781035	−	0.624487i	$-0.214692\pi$
$30$	0	0
$31$	5.89592	1.05894	0.529469	−	0.848329i	$-0.322391\pi$
$31$	5.89592	1.05894	0.529469	+	0.848329i	$0.322391\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.12875i	− 0.190793i
$36$	0	0
$37$	5.89592i	0.969283i	0.874713	+	0.484642i	$0.161050\pi$
$37$	5.89592i	0.969283i	−0.874713	+	0.484642i	$0.838950\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.415006	0.0648130	0.0324065	−	0.999475i	$-0.489683\pi$
$41$	0.415006	0.0648130	0.0324065	+	0.999475i	$0.489683\pi$
$42$	0	0
$43$	− 9.43967i	− 1.43954i	−0.694214	−	0.719768i	$-0.744248\pi$
$43$	− 9.43967i	− 1.43954i	0.694214	−	0.719768i	$-0.255752\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.2769	1.64490	0.822449	−	0.568839i	$-0.192607\pi$
$47$	11.2769	1.64490	0.822449	+	0.568839i	$0.192607\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.63843i	− 1.04922i	−0.851343	−	0.524609i	$-0.824211\pi$
$53$	− 7.63843i	− 1.04922i	0.851343	−	0.524609i	$-0.175789\pi$
$54$	0	0
$55$	5.38093	0.725565
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	− 1.80125i	− 0.230626i	−0.993329	−	0.115313i	$-0.963213\pi$
$61$	− 1.80125i	− 0.230626i	0.993329	−	0.115313i	$-0.0367871\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.514988	−0.0638764
$66$	0	0
$67$	8.09467i	0.988922i	0.869200	+	0.494461i	$0.164634\pi$
$67$	8.09467i	0.988922i	−0.869200	+	0.494461i	$0.835366\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.2068	1.21133	0.605665	−	0.795720i	$-0.292907\pi$
$71$	10.2068	1.21133	0.605665	+	0.795720i	$0.292907\pi$
$72$	0	0
$73$	−3.34500	−0.391503	−0.195751	−	0.980654i	$-0.562715\pi$
$73$	−3.34500	−0.391503	−0.195751	+	0.980654i	$0.562715\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.76717i	0.543270i
$78$	0	0
$79$	4.83001	0.543419	0.271709	−	0.962379i	$-0.412411\pi$
$79$	4.83001	0.543419	0.271709	+	0.962379i	$0.412411\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.53434i	0.607473i	0.952756	+	0.303737i	$0.0982343\pi$
$83$	5.53434i	0.607473i	−0.952756	+	0.303737i	$0.901766\pi$
$84$	0	0
$85$	− 0.468436i	− 0.0508090i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.92999	−0.522578	−0.261289	−	0.965261i	$-0.584148\pi$
$89$	−4.92999	−0.522578	−0.261289	+	0.965261i	$0.584148\pi$
$90$	0	0
$91$	− 0.456247i	− 0.0478277i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.62185	−0.884583
$96$	0	0
$97$	16.4468	1.66992	0.834962	−	0.550308i	$-0.185490\pi$
$97$	16.4468	1.66992	0.834962	+	0.550308i	$0.185490\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.4056i	1.63242i	0.577757	+	0.816209i	$0.303928\pi$
$101$	16.4056i	1.63242i	−0.577757	+	0.816209i	$0.696072\pi$
$102$	0	0
$103$	17.1728	1.69208	0.846042	−	0.533117i	$-0.178979\pi$
$103$	17.1728	1.69208	0.846042	+	0.533117i	$0.178979\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.7672i	1.23425i	0.786865	+	0.617125i	$0.211703\pi$
$107$	12.7672i	1.23425i	−0.786865	+	0.617125i	$0.788297\pi$
$108$	0	0
$109$	7.27685i	0.696996i	0.937310	+	0.348498i	$0.113308\pi$
$109$	7.27685i	0.696996i	−0.937310	+	0.348498i	$0.886692\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.34500	0.314671	0.157336	−	0.987545i	$-0.449710\pi$
$113$	3.34500	0.314671	0.157336	+	0.987545i	$0.449710\pi$
$114$	0	0
$115$	1.78906i	0.166830i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.415006	0.0380435
$120$	0	0
$121$	−11.7259	−1.06599
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.84937i	0.880954i
$126$	0	0
$127$	10.4468	0.927007	0.463504	−	0.886095i	$-0.346592\pi$
$127$	10.4468	0.927007	0.463504	+	0.886095i	$0.346592\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 6.25749i	− 0.546720i	−0.961912	−	0.273360i	$-0.911865\pi$
$131$	− 6.25749i	− 0.546720i	0.961912	−	0.273360i	$-0.0881350\pi$
$132$	0	0
$133$	− 7.63843i	− 0.662335i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.9618	1.10740	0.553702	−	0.832715i	$-0.313215\pi$
$137$	12.9618	1.10740	0.553702	+	0.832715i	$0.313215\pi$
$138$	0	0
$139$	− 18.3644i	− 1.55764i	−0.627245	−	0.778822i	$-0.715817\pi$
$139$	− 18.3644i	− 1.55764i	0.627245	−	0.778822i	$-0.284183\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.17501	0.181883
$144$	0	0
$145$	7.59187	0.630471
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 15.6384i	− 1.28115i	−0.767896	−	0.640575i	$-0.778696\pi$
$149$	− 15.6384i	− 1.28115i	0.767896	−	0.640575i	$-0.221304\pi$
$150$	0	0
$151$	−15.2769	−1.24321	−0.621606	−	0.783330i	$-0.713520\pi$
$151$	−15.2769	−1.24321	−0.621606	+	0.783330i	$0.713520\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.65500i	0.534543i
$156$	0	0
$157$	− 21.7824i	− 1.73843i	−0.494437	−	0.869214i	$-0.664626\pi$
$157$	− 21.7824i	− 1.73843i	0.494437	−	0.869214i	$-0.335374\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.58499	−0.124915
$162$	0	0
$163$	− 4.60966i	− 0.361056i	−0.983570	−	0.180528i	$-0.942219\pi$
$163$	− 4.60966i	− 0.361056i	0.983570	−	0.180528i	$-0.0577807\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.9618	−1.77684	−0.888420	−	0.459032i	$-0.848196\pi$
$167$	−22.9618	−1.77684	−0.888420	+	0.459032i	$0.848196\pi$
$168$	0	0
$169$	12.7918	0.983988
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 0.216252i	− 0.0164413i	−0.999966	−	0.00822067i	$-0.997383\pi$
$173$	− 0.216252i	− 0.0164413i	0.999966	−	0.00822067i	$-0.00261675\pi$
$174$	0	0
$175$	−3.72593	−0.281654
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 22.6165i	− 1.69044i	−0.534419	−	0.845220i	$-0.679470\pi$
$179$	− 22.6165i	− 1.69044i	0.534419	−	0.845220i	$-0.320530\pi$
$180$	0	0
$181$	− 7.73310i	− 0.574797i	−0.957811	−	0.287398i	$-0.907210\pi$
$181$	− 7.73310i	− 0.574797i	0.957811	−	0.287398i	$-0.0927903\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.65500	−0.489285
$186$	0	0
$187$	1.97840i	0.144675i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.2068	0.738541	0.369271	−	0.929322i	$-0.379608\pi$
$191$	10.2068	0.738541	0.369271	+	0.929322i	$0.379608\pi$
$192$	0	0
$193$	5.96407	0.429303	0.214651	−	0.976691i	$-0.431138\pi$
$193$	5.96407	0.429303	0.214651	+	0.976691i	$0.431138\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.8084i	1.34004i	0.742341	+	0.670022i	$0.233715\pi$
$197$	18.8084i	1.34004i	−0.742341	+	0.670022i	$0.766285\pi$
$198$	0	0
$199$	−3.27685	−0.232290	−0.116145	−	0.993232i	$-0.537054\pi$
$199$	−3.27685	−0.232290	−0.116145	+	0.993232i	$0.537054\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.72593i	0.472068i
$204$	0	0
$205$	0.468436i	0.0327170i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	36.4137	2.51879
$210$	0	0
$211$	10.1628i	0.699637i	0.936817	+	0.349819i	$0.113757\pi$
$211$	10.1628i	0.699637i	−0.936817	+	0.349819i	$0.886243\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.6550	0.726665
$216$	0	0
$217$	−5.89592	−0.400241
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 0.189345i	− 0.0127367i
$222$	0	0
$223$	−21.9668	−1.47101	−0.735504	−	0.677520i	$-0.763055\pi$
$223$	−21.9668	−1.47101	−0.735504	+	0.677520i	$0.763055\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 7.91752i	− 0.525504i	−0.964863	−	0.262752i	$-0.915370\pi$
$227$	− 7.91752i	− 0.525504i	0.964863	−	0.262752i	$-0.0846301\pi$
$228$	0	0
$229$	13.1606i	0.869676i	0.900509	+	0.434838i	$0.143194\pi$
$229$	13.1606i	0.869676i	−0.900509	+	0.434838i	$0.856806\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.2769	−1.13184	−0.565922	−	0.824459i	$-0.691480\pi$
$233$	−17.2769	−1.13184	−0.565922	+	0.824459i	$0.691480\pi$
$234$	0	0
$235$	12.7287i	0.830330i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.5219	−0.680603	−0.340301	−	0.940316i	$-0.610529\pi$
$239$	−10.5219	−0.680603	−0.340301	+	0.940316i	$0.610529\pi$
$240$	0	0
$241$	−6.10686	−0.393378	−0.196689	−	0.980466i	$-0.563019\pi$
$241$	−6.10686	−0.393378	−0.196689	+	0.980466i	$0.563019\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.12875i	0.0721130i
$246$	0	0
$247$	−3.48501	−0.221746
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.8743i	1.25446i	0.778836	+	0.627228i	$0.215811\pi$
$251$	19.8743i	1.25446i	−0.778836	+	0.627228i	$0.784189\pi$
$252$	0	0
$253$	− 7.55594i	− 0.475038i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.7550	0.920391	0.460195	−	0.887818i	$-0.347779\pi$
$257$	14.7550	0.920391	0.460195	+	0.887818i	$0.347779\pi$
$258$	0	0
$259$	− 5.89592i	− 0.366355i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.7600	−0.971804	−0.485902	−	0.874013i	$-0.661509\pi$
$263$	−15.7600	−0.971804	−0.485902	+	0.874013i	$0.661509\pi$
$264$	0	0
$265$	8.62185	0.529636
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 21.8331i	− 1.33119i	−0.746315	−	0.665593i	$-0.768179\pi$
$269$	− 21.8331i	− 1.33119i	0.746315	−	0.665593i	$-0.231821\pi$
$270$	0	0
$271$	18.8328	1.14401	0.572005	−	0.820250i	$-0.306166\pi$
$271$	18.8328	1.14401	0.572005	+	0.820250i	$0.306166\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 17.7622i	− 1.07110i
$276$	0	0
$277$	24.4684i	1.47017i	0.677977	+	0.735083i	$0.262857\pi$
$277$	24.4684i	1.47017i	−0.677977	+	0.735083i	$0.737143\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.5150	−0.865892	−0.432946	−	0.901420i	$-0.642526\pi$
$281$	−14.5150	−0.865892	−0.432946	+	0.901420i	$0.642526\pi$
$282$	0	0
$283$	− 16.5509i	− 0.983850i	−0.870638	−	0.491925i	$-0.836293\pi$
$283$	− 16.5509i	− 0.983850i	0.870638	−	0.491925i	$-0.163707\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.415006	−0.0244970
$288$	0	0
$289$	−16.8278	−0.989869
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.44377i	0.551711i	0.961199	+	0.275855i	$0.0889611\pi$
$293$	9.44377i	0.551711i	−0.961199	+	0.275855i	$0.911039\pi$
$294$	0	0
$295$	−4.51499	−0.262873
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.723150i	0.0418208i
$300$	0	0
$301$	9.43967i	0.544094i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.03315	0.116418
$306$	0	0
$307$	0.361575i	0.0206362i	0.999947	+	0.0103181i	$0.00328441\pi$
$307$	0.361575i	0.0206362i	−0.999947	+	0.0103181i	$0.996716\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.2769	−0.639452	−0.319726	−	0.947510i	$-0.603591\pi$
$311$	−11.2769	−0.639452	−0.319726	+	0.947510i	$0.603591\pi$
$312$	0	0
$313$	25.5837	1.44607	0.723037	−	0.690809i	$-0.242745\pi$
$313$	25.5837	1.44607	0.723037	+	0.690809i	$0.242745\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.361575i	0.0203081i	0.999948	+	0.0101540i	$0.00323218\pi$
$317$	0.361575i	0.0203081i	−0.999948	+	0.0101540i	$0.996768\pi$
$318$	0	0
$319$	−32.0637	−1.79522
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3.16999i	− 0.176383i
$324$	0	0
$325$	1.69995i	0.0942960i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.2769	−0.621713
$330$	0	0
$331$	7.80403i	0.428948i	0.976730	+	0.214474i	$0.0688037\pi$
$331$	7.80403i	0.428948i	−0.976730	+	0.214474i	$0.931196\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.13684	−0.499199
$336$	0	0
$337$	8.76186	0.477289	0.238645	−	0.971107i	$-0.423297\pi$
$337$	8.76186	0.477289	0.238645	+	0.971107i	$0.423297\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 28.1069i	− 1.52207i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.07717i	− 0.111509i	−0.998445	−	0.0557543i	$-0.982244\pi$
$347$	− 2.07717i	− 0.111509i	0.998445	−	0.0557543i	$-0.0177563\pi$
$348$	0	0
$349$	18.7381i	1.00303i	0.865149	+	0.501514i	$0.167224\pi$
$349$	18.7381i	1.00303i	−0.865149	+	0.501514i	$0.832776\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.2450	0.704961	0.352481	−	0.935819i	$-0.385338\pi$
$353$	13.2450	0.704961	0.352481	+	0.935819i	$0.385338\pi$
$354$	0	0
$355$	11.5209i	0.611468i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.69186	−0.300405	−0.150202	−	0.988655i	$-0.547993\pi$
$359$	−5.69186	−0.300405	−0.150202	+	0.988655i	$0.547993\pi$
$360$	0	0
$361$	−39.3455	−2.07082
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.77566i	− 0.197627i
$366$	0	0
$367$	−11.2769	−0.588647	−0.294323	−	0.955706i	$-0.595094\pi$
$367$	−11.2769	−0.588647	−0.294323	+	0.955706i	$0.595094\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.63843i	0.396567i
$372$	0	0
$373$	7.08751i	0.366977i	0.983022	+	0.183489i	$0.0587390\pi$
$373$	7.08751i	0.366977i	−0.983022	+	0.183489i	$0.941261\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.06869	0.158046
$378$	0	0
$379$	− 6.67781i	− 0.343016i	−0.985183	−	0.171508i	$-0.945136\pi$
$379$	− 6.67781i	− 0.343016i	0.985183	−	0.171508i	$-0.0548639\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−18.6550	−0.953226	−0.476613	−	0.879113i	$-0.658136\pi$
$383$	−18.6550	−0.953226	−0.476613	+	0.879113i	$0.658136\pi$
$384$	0	0
$385$	−5.38093	−0.274238
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 20.3428i	− 1.03142i	−0.856764	−	0.515709i	$-0.827528\pi$
$389$	− 20.3428i	− 1.03142i	0.856764	−	0.515709i	$-0.172472\pi$
$390$	0	0
$391$	−0.657782	−0.0332654
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.45186i	0.274313i
$396$	0	0
$397$	10.9031i	0.547210i	0.961842	+	0.273605i	$0.0882161\pi$
$397$	10.9031i	0.547210i	−0.961842	+	0.273605i	$0.911784\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−29.1756	−1.45696	−0.728479	−	0.685068i	$-0.759772\pi$
$401$	−29.1756	−1.45696	−0.728479	+	0.685068i	$0.759772\pi$
$402$	0	0
$403$	2.69000i	0.133998i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	28.1069	1.39321
$408$	0	0
$409$	−31.7237	−1.56864	−0.784318	−	0.620359i	$-0.786987\pi$
$409$	−31.7237	−1.56864	−0.784318	+	0.620359i	$0.786987\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	−6.24687	−0.306647
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 19.2769i	− 0.941736i	−0.882204	−	0.470868i	$-0.843941\pi$
$419$	− 19.2769i	− 0.941736i	0.882204	−	0.470868i	$-0.156059\pi$
$420$	0	0
$421$	− 19.3234i	− 0.941765i	−0.882196	−	0.470882i	$-0.843936\pi$
$421$	− 19.3234i	− 0.941765i	0.882196	−	0.470882i	$-0.156064\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.54628	−0.0750057
$426$	0	0
$427$	1.80125i	0.0871684i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.4150	0.694346	0.347173	−	0.937801i	$-0.387142\pi$
$431$	14.4150	0.694346	0.347173	+	0.937801i	$0.387142\pi$
$432$	0	0
$433$	−9.27685	−0.445817	−0.222908	−	0.974839i	$-0.571555\pi$
$433$	−9.27685	−0.445817	−0.222908	+	0.974839i	$0.571555\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.1069i	0.579150i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 15.6465i	− 0.743388i	−0.928355	−	0.371694i	$-0.878777\pi$
$443$	− 15.6465i	− 0.743388i	0.928355	−	0.371694i	$-0.121223\pi$
$444$	0	0
$445$	− 5.56472i	− 0.263793i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.6550	0.597226	0.298613	−	0.954374i	$-0.403476\pi$
$449$	12.6550	0.597226	0.298613	+	0.954374i	$0.403476\pi$
$450$	0	0
$451$	− 1.97840i	− 0.0931594i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.514988	0.0241430
$456$	0	0
$457$	−23.9237	−1.11910	−0.559551	−	0.828796i	$-0.689026\pi$
$457$	−23.9237	−1.11910	−0.559551	+	0.828796i	$0.689026\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.74558i	0.314173i	0.987585	+	0.157086i	$0.0502101\pi$
$461$	6.74558i	0.314173i	−0.987585	+	0.157086i	$0.949790\pi$
$462$	0	0
$463$	11.1700	0.519113	0.259557	−	0.965728i	$-0.416424\pi$
$463$	11.1700	0.519113	0.259557	+	0.965728i	$0.416424\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.70935i	− 0.171648i	−0.996310	−	0.0858242i	$-0.972648\pi$
$467$	− 3.70935i	− 0.171648i	0.996310	−	0.0858242i	$-0.0273523\pi$
$468$	0	0
$469$	− 8.09467i	− 0.373777i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−45.0005	−2.06913
$474$	0	0
$475$	28.4602i	1.30585i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.17501	−0.0993788	−0.0496894	−	0.998765i	$-0.515823\pi$
$479$	−2.17501	−0.0993788	−0.0496894	+	0.998765i	$0.515823\pi$
$480$	0	0
$481$	−2.69000	−0.122653
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.5643i	0.842962i
$486$	0	0
$487$	4.51499	0.204594	0.102297	−	0.994754i	$-0.467381\pi$
$487$	4.51499	0.204594	0.102297	+	0.994754i	$0.467381\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.91221i	0.0862967i	0.999069	+	0.0431483i	$0.0137388\pi$
$491$	1.91221i	0.0862967i	−0.999069	+	0.0431483i	$0.986261\pi$
$492$	0	0
$493$	2.79130i	0.125714i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.2068	−0.457840
$498$	0	0
$499$	27.4065i	1.22688i	0.789740	+	0.613442i	$0.210216\pi$
$499$	27.4065i	1.22688i	−0.789740	+	0.613442i	$0.789784\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.1368	0.764094	0.382047	−	0.924143i	$-0.375219\pi$
$503$	17.1368	0.764094	0.382047	+	0.924143i	$0.375219\pi$
$504$	0	0
$505$	−18.5178	−0.824030
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.6437i	0.959342i	0.877449	+	0.479671i	$0.159244\pi$
$509$	21.6437i	0.959342i	−0.877449	+	0.479671i	$0.840756\pi$
$510$	0	0
$511$	3.34500	0.147974
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.3837i	0.854148i
$516$	0	0
$517$	− 53.7587i	− 2.36430i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.3137	−1.24045	−0.620223	−	0.784426i	$-0.712958\pi$
$521$	−28.3137	−1.24045	−0.620223	+	0.784426i	$0.712958\pi$
$522$	0	0
$523$	− 15.0687i	− 0.658908i	−0.944172	−	0.329454i	$-0.893135\pi$
$523$	− 15.0687i	− 0.658908i	0.944172	−	0.329454i	$-0.106865\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.44684	−0.106586
$528$	0	0
$529$	−20.4878	−0.890774
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.189345i	0.00820145i
$534$	0	0
$535$	−14.4109	−0.623038
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.76717i	− 0.205337i
$540$	0	0
$541$	− 46.2990i	− 1.99055i	−0.0971017	−	0.995274i	$-0.530957\pi$
$541$	− 46.2990i	− 1.99055i	0.0971017	−	0.995274i	$-0.469043\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.21372	−0.351837
$546$	0	0
$547$	2.03714i	0.0871020i	0.999051	+	0.0435510i	$0.0138671\pi$
$547$	2.03714i	0.0871020i	−0.999051	+	0.0435510i	$0.986133\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	51.3755	2.18867
$552$	0	0
$553$	−4.83001	−0.205393
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.1234i	0.471315i	0.971836	+	0.235658i	$0.0757244\pi$
$557$	11.1234i	0.471315i	−0.971836	+	0.235658i	$0.924276\pi$
$558$	0	0
$559$	4.30683	0.182159
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.2437i	1.56963i	0.619727	+	0.784817i	$0.287243\pi$
$563$	37.2437i	1.56963i	−0.619727	+	0.784817i	$0.712757\pi$
$564$	0	0
$565$	3.77566i	0.158843i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.9369	−0.961564	−0.480782	−	0.876840i	$-0.659647\pi$
$569$	−22.9369	−0.961564	−0.480782	+	0.876840i	$0.659647\pi$
$570$	0	0
$571$	− 35.1634i	− 1.47154i	−0.677231	−	0.735770i	$-0.736820\pi$
$571$	− 35.1634i	− 1.47154i	0.677231	−	0.735770i	$-0.263180\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.90558	0.246280
$576$	0	0
$577$	13.7918	0.574162	0.287081	−	0.957906i	$-0.407315\pi$
$577$	13.7918	0.574162	0.287081	+	0.957906i	$0.407315\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 5.53434i	− 0.229603i
$582$	0	0
$583$	−36.4137	−1.50810
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 26.9862i	− 1.11384i	−0.830566	−	0.556920i	$-0.811983\pi$
$587$	− 26.9862i	− 1.11384i	0.830566	−	0.556920i	$-0.188017\pi$
$588$	0	0
$589$	45.0355i	1.85566i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.8337	1.30725	0.653627	−	0.756817i	$-0.273247\pi$
$593$	31.8337	1.30725	0.653627	+	0.756817i	$0.273247\pi$
$594$	0	0
$595$	0.468436i	0.0192040i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−34.9006	−1.42600	−0.712999	−	0.701165i	$-0.752664\pi$
$599$	−34.9006	−1.42600	−0.712999	+	0.701165i	$0.752664\pi$
$600$	0	0
$601$	−0.175010	−0.00713881	−0.00356941	−	0.999994i	$-0.501136\pi$
$601$	−0.175010	−0.00713881	−0.00356941	+	0.999994i	$0.501136\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 13.2356i	− 0.538104i
$606$	0	0
$607$	32.1705	1.30576	0.652881	−	0.757461i	$-0.273560\pi$
$607$	32.1705	1.30576	0.652881	+	0.757461i	$0.273560\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.14503i	0.208146i
$612$	0	0
$613$	− 8.37869i	− 0.338412i	−0.985581	−	0.169206i	$-0.945880\pi$
$613$	− 8.37869i	− 0.338412i	0.985581	−	0.169206i	$-0.0541203\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.9618	−1.16596	−0.582980	−	0.812487i	$-0.698113\pi$
$617$	−28.9618	−1.16596	−0.582980	+	0.812487i	$0.698113\pi$
$618$	0	0
$619$	− 6.85497i	− 0.275524i	−0.990465	−	0.137762i	$-0.956009\pi$
$619$	− 6.85497i	− 0.275524i	0.990465	−	0.137762i	$-0.0439910\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.92999	0.197516
$624$	0	0
$625$	7.51221	0.300488
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 2.44684i	− 0.0975619i
$630$	0	0
$631$	40.2055	1.60056	0.800278	−	0.599629i	$-0.204685\pi$
$631$	40.2055	1.60056	0.800278	+	0.599629i	$0.204685\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.7918i	0.467945i
$636$	0	0
$637$	0.456247i	0.0180772i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0737	0.950854	0.475427	−	0.879755i	$-0.342294\pi$
$641$	24.0737	0.950854	0.475427	+	0.879755i	$0.342294\pi$
$642$	0	0
$643$	− 23.7559i	− 0.936841i	−0.883506	−	0.468421i	$-0.844823\pi$
$643$	− 23.7559i	− 0.936841i	0.883506	−	0.468421i	$-0.155177\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.6218	−1.28250	−0.641249	−	0.767333i	$-0.721583\pi$
$647$	−32.6218	−1.28250	−0.641249	+	0.767333i	$0.721583\pi$
$648$	0	0
$649$	19.0687	0.748512
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.4109i	0.563942i	0.959423	+	0.281971i	$0.0909883\pi$
$653$	14.4109i	0.563942i	−0.959423	+	0.281971i	$0.909012\pi$
$654$	0	0
$655$	7.06313	0.275979
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23.3315i	0.908866i	0.890781	+	0.454433i	$0.150158\pi$
$659$	23.3315i	0.908866i	−0.890781	+	0.454433i	$0.849842\pi$
$660$	0	0
$661$	44.1468i	1.71711i	0.512721	+	0.858555i	$0.328638\pi$
$661$	44.1468i	1.71711i	−0.512721	+	0.858555i	$0.671362\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.62185	0.334341
$666$	0	0
$667$	− 10.6606i	− 0.412779i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.58685	−0.331492
$672$	0	0
$673$	11.6960	0.450846	0.225423	−	0.974261i	$-0.427624\pi$
$673$	11.6960	0.450846	0.225423	+	0.974261i	$0.427624\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 0.648756i	− 0.0249337i	−0.999922	−	0.0124669i	$-0.996032\pi$
$677$	− 0.648756i	− 0.0249337i	0.999922	−	0.0124669i	$-0.00396843\pi$
$678$	0	0
$679$	−16.4468	−0.631172
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.4909i	0.860589i	0.902689	+	0.430294i	$0.141590\pi$
$683$	22.4909i	0.860589i	−0.902689	+	0.430294i	$0.858410\pi$
$684$	0	0
$685$	14.6306i	0.559007i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.48501	0.132768
$690$	0	0
$691$	− 45.2681i	− 1.72208i	−0.508538	−	0.861039i	$-0.669814\pi$
$691$	− 45.2681i	− 1.72208i	0.508538	−	0.861039i	$-0.330186\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.7287	0.786285
$696$	0	0
$697$	−0.172230	−0.00652366
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.03091i	0.0389370i	0.999810	+	0.0194685i	$0.00619740\pi$
$701$	1.03091i	0.0389370i	−0.999810	+	0.0194685i	$0.993803\pi$
$702$	0	0
$703$	−45.0355	−1.69855
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 16.4056i	− 0.616996i
$708$	0	0
$709$	− 10.5481i	− 0.396144i	−0.980188	−	0.198072i	$-0.936532\pi$
$709$	− 10.5481i	− 0.396144i	0.980188	−	0.198072i	$-0.0634679\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.34500	0.349973
$714$	0	0
$715$	2.45504i	0.0918131i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7.00502	−0.261243	−0.130622	−	0.991432i	$-0.541697\pi$
$719$	−7.00502	−0.261243	−0.130622	+	0.991432i	$0.541697\pi$
$720$	0	0
$721$	−17.1728	−0.639547
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 25.0603i	− 0.930718i
$726$	0	0
$727$	17.8028	0.660270	0.330135	−	0.943934i	$-0.392906\pi$
$727$	17.8028	0.660270	0.330135	+	0.943934i	$0.392906\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.91752i	0.144895i
$732$	0	0
$733$	− 34.3300i	− 1.26801i	−0.773330	−	0.634004i	$-0.781410\pi$
$733$	− 34.3300i	− 1.26801i	0.773330	−	0.634004i	$-0.218590\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.5887	1.42143
$738$	0	0
$739$	− 0.0946726i	− 0.00348259i	−0.999998	−	0.00174129i	$-0.999446\pi$
$739$	− 0.0946726i	− 0.00348259i	0.999998	−	0.00174129i	$-0.000554271\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.2755	−0.487032	−0.243516	−	0.969897i	$-0.578301\pi$
$743$	−13.2755	−0.487032	−0.243516	+	0.969897i	$0.578301\pi$
$744$	0	0
$745$	17.6518	0.646713
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 12.7672i	− 0.466502i
$750$	0	0
$751$	−15.4187	−0.562637	−0.281318	−	0.959615i	$-0.590772\pi$
$751$	−15.4187	−0.562637	−0.281318	+	0.959615i	$0.590772\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 17.2437i	− 0.627562i
$756$	0	0
$757$	− 9.96685i	− 0.362251i	−0.983460	−	0.181126i	$-0.942026\pi$
$757$	− 9.96685i	− 0.362251i	0.983460	−	0.181126i	$-0.0579741\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.4837	−0.851283	−0.425642	−	0.904892i	$-0.639952\pi$
$761$	−23.4837	−0.851283	−0.425642	+	0.904892i	$0.639952\pi$
$762$	0	0
$763$	− 7.27685i	− 0.263440i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.82499	−0.0658966
$768$	0	0
$769$	22.9369	0.827125	0.413562	−	0.910476i	$-0.364284\pi$
$769$	22.9369	0.827125	0.413562	+	0.910476i	$0.364284\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35.0956i	1.26230i	0.775660	+	0.631150i	$0.217417\pi$
$773$	35.0956i	1.26230i	−0.775660	+	0.631150i	$0.782583\pi$
$774$	0	0
$775$	21.9678	0.789106
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.16999i	0.113577i
$780$	0	0
$781$	− 48.6578i	− 1.74111i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.5869	0.877542
$786$	0	0
$787$	17.9480i	0.639778i	0.947455	+	0.319889i	$0.103646\pi$
$787$	17.9480i	0.639778i	−0.947455	+	0.319889i	$0.896354\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.34500	−0.118934
$792$	0	0
$793$	0.821814	0.0291835
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 34.4199i	− 1.21922i	−0.792703	−	0.609608i	$-0.791327\pi$
$797$	− 34.4199i	− 1.21922i	0.792703	−	0.609608i	$-0.208673\pi$
$798$	0	0
$799$	−4.67996	−0.165565
$800$	0	0
$801$	0	0
$802$	0	0
$803$	15.9462i	0.562729i
$804$	0	0
$805$	− 1.78906i	− 0.0630560i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.89314	−0.0665592	−0.0332796	−	0.999446i	$-0.510595\pi$
$809$	−1.89314	−0.0665592	−0.0332796	+	0.999446i	$0.510595\pi$
$810$	0	0
$811$	10.3400i	0.363086i	0.983383	+	0.181543i	$0.0581091\pi$
$811$	10.3400i	0.363086i	−0.983383	+	0.181543i	$0.941891\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.20314	0.182258
$816$	0	0
$817$	72.1042	2.52261
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.40029i	0.0837706i	0.999122	+	0.0418853i	$0.0133364\pi$
$821$	2.40029i	0.0837706i	−0.999122	+	0.0418853i	$0.986664\pi$
$822$	0	0
$823$	−25.8655	−0.901616	−0.450808	−	0.892621i	$-0.648864\pi$
$823$	−25.8655	−0.901616	−0.450808	+	0.892621i	$0.648864\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 17.0928i	− 0.594375i	−0.954819	−	0.297188i	$-0.903951\pi$
$827$	− 17.0928i	− 0.594375i	0.954819	−	0.297188i	$-0.0960487\pi$
$828$	0	0
$829$	41.8893i	1.45488i	0.686174	+	0.727438i	$0.259289\pi$
$829$	41.8893i	1.45488i	−0.686174	+	0.727438i	$0.740711\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.415006	−0.0143791
$834$	0	0
$835$	− 25.9181i	− 0.896933i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.5500	−0.847560	−0.423780	−	0.905765i	$-0.639297\pi$
$839$	−24.5500	−0.847560	−0.423780	+	0.905765i	$0.639297\pi$
$840$	0	0
$841$	−16.2381	−0.559936
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.4387i	0.496708i
$846$	0	0
$847$	11.7259	0.402908
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.34500i	0.320342i
$852$	0	0
$853$	38.7193i	1.32572i	0.748742	+	0.662862i	$0.230658\pi$
$853$	38.7193i	1.32572i	−0.748742	+	0.662862i	$0.769342\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.2074	1.54425	0.772127	−	0.635468i	$-0.219193\pi$
$857$	45.2074	1.54425	0.772127	+	0.635468i	$0.219193\pi$
$858$	0	0
$859$	57.1890i	1.95126i	0.219418	+	0.975631i	$0.429584\pi$
$859$	57.1890i	1.95126i	−0.219418	+	0.975631i	$0.570416\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.1106	−1.53558	−0.767791	−	0.640701i	$-0.778644\pi$
$863$	−45.1106	−1.53558	−0.767791	+	0.640701i	$0.778644\pi$
$864$	0	0
$865$	0.244094	0.00829944
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 23.0255i	− 0.781086i
$870$	0	0
$871$	−3.69317	−0.125138
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 9.84937i	− 0.332969i
$876$	0	0
$877$	17.6600i	0.596337i	0.954513	+	0.298168i	$0.0963757\pi$
$877$	17.6600i	0.596337i	−0.954513	+	0.298168i	$0.903624\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6.78998	−0.228760	−0.114380	−	0.993437i	$-0.536488\pi$
$881$	−6.78998	−0.228760	−0.114380	+	0.993437i	$0.536488\pi$
$882$	0	0
$883$	28.6015i	0.962516i	0.876579	+	0.481258i	$0.159820\pi$
$883$	28.6015i	0.962516i	−0.876579	+	0.481258i	$0.840180\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.5919	0.792138	0.396069	−	0.918221i	$-0.370374\pi$
$887$	23.5919	0.792138	0.396069	+	0.918221i	$0.370374\pi$
$888$	0	0
$889$	−10.4468	−0.350376
$890$	0	0
$891$	0	0
$892$	0	0
$893$	86.1374i	2.88248i
$894$	0	0
$895$	25.5284	0.853319
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 39.6555i	− 1.32259i
$900$	0	0
$901$	3.16999i	0.105608i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.72871	0.290152
$906$	0	0
$907$	2.02096i	0.0671050i	0.999437	+	0.0335525i	$0.0106821\pi$
$907$	2.02096i	0.0671050i	−0.999437	+	0.0335525i	$0.989318\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	52.5524	1.74114	0.870569	−	0.492046i	$-0.163751\pi$
$911$	52.5524	1.74114	0.870569	+	0.492046i	$0.163751\pi$
$912$	0	0
$913$	26.3832	0.873156
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.25749i	0.206641i
$918$	0	0
$919$	−21.5237	−0.710002	−0.355001	−	0.934866i	$-0.615520\pi$
$919$	−21.5237	−0.710002	−0.355001	+	0.934866i	$0.615520\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.65685i	0.153282i
$924$	0	0
$925$	21.9678i	0.722296i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15.7500	0.516739	0.258370	−	0.966046i	$-0.416815\pi$
$929$	15.7500	0.516739	0.258370	+	0.966046i	$0.416815\pi$
$930$	0	0
$931$	7.63843i	0.250339i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.23312	−0.0730307
$936$	0	0
$937$	17.0687	0.557610	0.278805	−	0.960348i	$-0.410062\pi$
$937$	17.0687	0.557610	0.278805	+	0.960348i	$0.410062\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 26.5456i	− 0.865362i	−0.901547	−	0.432681i	$-0.857568\pi$
$941$	− 26.5456i	− 0.865362i	0.901547	−	0.432681i	$-0.142432\pi$
$942$	0	0
$943$	0.657782	0.0214203
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.3265i	0.660522i	0.943890	+	0.330261i	$0.107137\pi$
$947$	20.3265i	0.660522i	−0.943890	+	0.330261i	$0.892863\pi$
$948$	0	0
$949$	− 1.52615i	− 0.0495408i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.6855	1.25315	0.626573	−	0.779362i	$-0.284457\pi$
$953$	38.6855	1.25315	0.626573	+	0.779362i	$0.284457\pi$
$954$	0	0
$955$	11.5209i	0.372809i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.9618	−0.418559
$960$	0	0
$961$	3.76186	0.121350
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.73192i	0.216708i
$966$	0	0
$967$	26.6606	0.857346	0.428673	−	0.903460i	$-0.358981\pi$
$967$	26.6606	0.857346	0.428673	+	0.903460i	$0.358981\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.27685i	0.105159i	0.998617	+	0.0525796i	$0.0167443\pi$
$971$	3.27685i	0.105159i	−0.998617	+	0.0525796i	$0.983256\pi$
$972$	0	0
$973$	18.3644i	0.588734i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.0387	−0.385153	−0.192576	−	0.981282i	$-0.561684\pi$
$977$	−12.0387	−0.385153	−0.192576	+	0.981282i	$0.561684\pi$
$978$	0	0
$979$	23.5021i	0.751131i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.42188	0.141036	0.0705181	−	0.997510i	$-0.477535\pi$
$983$	4.42188	0.141036	0.0705181	+	0.997510i	$0.477535\pi$
$984$	0	0
$985$	−21.2299	−0.676442
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 14.9618i	− 0.475758i
$990$	0	0
$991$	9.10184	0.289129	0.144565	−	0.989495i	$-0.453822\pi$
$991$	9.10184	0.289129	0.144565	+	0.989495i	$0.453822\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 3.69873i	− 0.117258i
$996$	0	0
$997$	− 42.8556i	− 1.35725i	−0.734485	−	0.678625i	$-0.762576\pi$
$997$	− 42.8556i	− 1.35725i	0.734485	−	0.678625i	$-0.237424\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.c.e.1009.6		8
3.2	odd	2		672.2.c.b.337.6		8
4.3	odd	2		504.2.c.f.253.7		8
8.3	odd	2		504.2.c.f.253.8		8
8.5	even	2	inner	2016.2.c.e.1009.3		8
12.11	even	2		168.2.c.b.85.2	yes	8
21.20	even	2		4704.2.c.c.2353.3		8
24.5	odd	2		672.2.c.b.337.3		8
24.11	even	2		168.2.c.b.85.1	✓	8
48.5	odd	4		5376.2.a.bq.1.3		4
48.11	even	4		5376.2.a.bm.1.3		4
48.29	odd	4		5376.2.a.bl.1.2		4
48.35	even	4		5376.2.a.bp.1.2		4
84.83	odd	2		1176.2.c.c.589.2		8
168.83	odd	2		1176.2.c.c.589.1		8
168.125	even	2		4704.2.c.c.2353.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.b.85.1	✓	8	24.11	even	2
168.2.c.b.85.2	yes	8	12.11	even	2
504.2.c.f.253.7		8	4.3	odd	2
504.2.c.f.253.8		8	8.3	odd	2
672.2.c.b.337.3		8	24.5	odd	2
672.2.c.b.337.6		8	3.2	odd	2
1176.2.c.c.589.1		8	168.83	odd	2
1176.2.c.c.589.2		8	84.83	odd	2
2016.2.c.e.1009.3		8	8.5	even	2	inner
2016.2.c.e.1009.6		8	1.1	even	1	trivial
4704.2.c.c.2353.3		8	21.20	even	2
4704.2.c.c.2353.6		8	168.125	even	2
5376.2.a.bl.1.2		4	48.29	odd	4
5376.2.a.bm.1.3		4	48.11	even	4
5376.2.a.bp.1.2		4	48.35	even	4
5376.2.a.bq.1.3		4	48.5	odd	4

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.12875i q^{5} -1.00000 q^{7} -4.76717i q^{11} +0.456247i q^{13} -0.415006 q^{17} +7.63843i q^{19} +1.58499 q^{23} +3.72593 q^{25} -6.72593i q^{29} +5.89592 q^{31} -1.12875i q^{35} +5.89592i q^{37} +0.415006 q^{41} -9.43967i q^{43} +11.2769 q^{47} +1.00000 q^{49} -7.63843i q^{53} +5.38093 q^{55} +4.00000i q^{59} -1.80125i q^{61} -0.514988 q^{65} +8.09467i q^{67} +10.2068 q^{71} -3.34500 q^{73} +4.76717i q^{77} +4.83001 q^{79} +5.53434i q^{83} -0.468436i q^{85} -4.92999 q^{89} -0.456247i q^{91} -8.62185 q^{95} +16.4468 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} - 4 q^{17} + 12 q^{23} - 24 q^{25} - 8 q^{31} + 4 q^{41} + 8 q^{49} + 8 q^{55} + 16 q^{65} - 28 q^{71} - 8 q^{73} + 40 q^{79} - 20 q^{89} + 40 q^{95} + 40 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 4 * q^17 + 12 * q^23 - 24 * q^25 - 8 * q^31 + 4 * q^41 + 8 * q^49 + 8 * q^55 + 16 * q^65 - 28 * q^71 - 8 * q^73 + 40 * q^79 - 20 * q^89 + 40 * q^95 + 40 * q^97

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