LMFDB - Embedded newform 840.2.dd.a.313.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [840,2,Mod(73,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.73"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 0, 0, 9, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.dd (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$6.70743376979$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$12$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			313.7
Character	$\chi$	$=$	840.313
Dual form			840.2.dd.a.577.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.965926 + 0.258819i) q^{3} +(-2.16114 + 0.574011i) q^{5} +(2.47337 - 0.939373i) q^{7} +(0.866025 + 0.500000i) q^{9} +(-2.18212 - 3.77954i) q^{11} +(2.79969 - 2.79969i) q^{13} +(-2.23606 - 0.00489121i) q^{15} +(-0.0821940 + 0.306752i) q^{17} +(1.35172 - 2.34125i) q^{19} +(2.63222 - 0.267208i) q^{21} +(2.92383 - 0.783438i) q^{23} +(4.34102 - 2.48103i) q^{25} +(0.707107 + 0.707107i) q^{27} +4.52748i q^{29} +(1.89712 - 1.09530i) q^{31} +(-1.12955 - 4.21553i) q^{33} +(-4.80609 + 3.44986i) q^{35} +(0.692812 + 2.58561i) q^{37} +(3.42891 - 1.97968i) q^{39} -2.63418i q^{41} +(6.21198 + 6.21198i) q^{43} +(-2.15860 - 0.583460i) q^{45} +(6.07142 - 1.62683i) q^{47} +(5.23516 - 4.64684i) q^{49} +(-0.158787 + 0.275026i) q^{51} +(3.22704 - 12.0435i) q^{53} +(6.88536 + 6.91554i) q^{55} +(1.91162 - 1.91162i) q^{57} +(0.863327 + 1.49533i) q^{59} +(-8.84369 - 5.10591i) q^{61} +(2.61169 + 0.423166i) q^{63} +(-4.44347 + 7.65758i) q^{65} +(-7.38122 - 1.97779i) q^{67} +3.02697 q^{69} +5.39235 q^{71} +(-15.0908 - 4.04357i) q^{73} +(4.83524 - 1.27295i) q^{75} +(-8.94759 - 7.29840i) q^{77} +(11.6020 + 6.69839i) q^{79} +(0.500000 + 0.866025i) q^{81} +(-11.4560 + 11.4560i) q^{83} +(0.00155332 - 0.710113i) q^{85} +(-1.17180 + 4.37321i) q^{87} +(-3.03332 + 5.25387i) q^{89} +(4.29473 - 9.55465i) q^{91} +(2.11597 - 0.566971i) q^{93} +(-1.57735 + 5.83566i) q^{95} +(-7.89037 - 7.89037i) q^{97} -4.36424i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{11} + 16 q^{13} - 4 q^{15} - 4 q^{17} + 8 q^{19} - 24 q^{23} + 28 q^{25} + 12 q^{33} - 4 q^{37} - 12 q^{39} + 16 q^{43} - 4 q^{45} + 12 q^{47} - 12 q^{49} + 20 q^{53} - 56 q^{55} + 8 q^{57}+ \cdots + 44 q^{97}+O(q^{100}) $ 48 * q - 4 * q^11 + 16 * q^13 - 4 * q^15 - 4 * q^17 + 8 * q^19 - 24 * q^23 + 28 * q^25 + 12 * q^33 - 4 * q^37 - 12 * q^39 + 16 * q^43 - 4 * q^45 + 12 * q^47 - 12 * q^49 + 20 * q^53 - 56 * q^55 + 8 * q^57 + 8 * q^59 + 8 * q^63 + 12 * q^65 - 52 * q^67 - 8 * q^69 + 16 * q^71 - 4 * q^73 + 24 * q^75 + 8 * q^77 + 108 * q^79 + 24 * q^81 + 64 * q^83 + 32 * q^85 + 20 * q^87 + 64 * q^89 + 48 * q^91 - 24 * q^93 + 36 * q^95 + 44 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	0.965926	+	0.258819i	0.557678	+	0.149429i
$3$	0.965926	+	0.258819i	0.557678	+	0.149429i
$4$		0			0
$5$	−2.16114	+	0.574011i	−0.966490	+	0.256706i
$5$	−2.16114	+	0.574011i	−0.966490	+	0.256706i
$6$		0			0
$7$	2.47337	−	0.939373i	0.934848	−	0.355049i
$7$	2.47337	−	0.939373i	0.934848	−	0.355049i
$8$		0			0
$9$	0.866025	+	0.500000i	0.288675	+	0.166667i
$10$		0			0
$11$	−2.18212	−	3.77954i	−0.657934	−	1.13957i	−0.981150	−	0.193249i	$-0.938097\pi$
$11$	−2.18212	−	3.77954i	−0.657934	−	1.13957i	0.323216	−	0.946325i	$-0.395236\pi$
$12$		0			0
$13$	2.79969	−	2.79969i	0.776495	−	0.776495i	−0.202738	−	0.979233i	$-0.564984\pi$
$13$	2.79969	−	2.79969i	0.776495	−	0.776495i	0.979233	+	0.202738i	$0.0649839\pi$
$14$		0			0
$15$	−2.23606	−	0.00489121i	−0.577349	−	0.00126291i
$16$		0			0
$17$	−0.0821940	+	0.306752i	−0.0199350	+	0.0743983i	−0.975177	−	0.221428i	$-0.928928\pi$
$17$	−0.0821940	+	0.306752i	−0.0199350	+	0.0743983i	0.955242	+	0.295826i	$0.0955949\pi$
$18$		0			0
$19$	1.35172	−	2.34125i	0.310106	−	0.537120i	−0.668279	−	0.743911i	$-0.732969\pi$
$19$	1.35172	−	2.34125i	0.310106	−	0.537120i	0.978385	+	0.206791i	$0.0663021\pi$
$20$		0			0
$21$	2.63222	−	0.267208i	0.574398	−	0.0583096i
$22$		0			0
$23$	2.92383	−	0.783438i	0.609661	−	0.163358i	0.0592377	−	0.998244i	$-0.481133\pi$
$23$	2.92383	−	0.783438i	0.609661	−	0.163358i	0.550423	+	0.834886i	$0.314466\pi$
$24$		0			0
$25$	4.34102	−	2.48103i	0.868205	−	0.496207i
$26$		0			0
$27$	0.707107	+	0.707107i	0.136083	+	0.136083i
$28$		0			0
$29$			4.52748i			0.840732i	0.907355	+	0.420366i	$0.138098\pi$
$29$			4.52748i			0.840732i	−0.907355	+	0.420366i	$0.861902\pi$
$30$		0			0
$31$	1.89712	−	1.09530i	0.340733	−	0.196722i	−0.319863	−	0.947464i	$-0.603637\pi$
$31$	1.89712	−	1.09530i	0.340733	−	0.196722i	0.660596	+	0.750741i	$0.270304\pi$
$32$		0			0
$33$	−1.12955	−	4.21553i	−0.196629	−	0.733830i
$34$		0			0
$35$	−4.80609	+	3.44986i	−0.812377	+	0.583132i
$36$		0			0
$37$	0.692812	+	2.58561i	0.113898	+	0.425072i	0.999202	−	0.0399401i	$-0.0127167\pi$
$37$	0.692812	+	2.58561i	0.113898	+	0.425072i	−0.885304	+	0.465012i	$0.846050\pi$
$38$		0			0
$39$	3.42891	−	1.97968i	0.549065	−	0.317003i
$40$		0			0
$41$		−	2.63418i		−	0.411390i	−0.978616	−	0.205695i	$-0.934055\pi$
$41$		−	2.63418i		−	0.411390i	0.978616	−	0.205695i	$-0.0659454\pi$
$42$		0			0
$43$	6.21198	+	6.21198i	0.947319	+	0.947319i	0.998680	−	0.0513615i	$-0.0163561\pi$
$43$	6.21198	+	6.21198i	0.947319	+	0.947319i	−0.0513615	+	0.998680i	$0.516356\pi$
$44$		0			0
$45$	−2.15860	−	0.583460i	−0.321786	−	0.0869771i
$46$		0			0
$47$	6.07142	−	1.62683i	0.885608	−	0.237298i	0.212783	−	0.977100i	$-0.431747\pi$
$47$	6.07142	−	1.62683i	0.885608	−	0.237298i	0.672825	+	0.739802i	$0.265081\pi$
$48$		0			0
$49$	5.23516	−	4.64684i	0.747880	−	0.663834i
$50$		0			0
$51$	−0.158787	+	0.275026i	−0.0222346	+	0.0385114i
$52$		0			0
$53$	3.22704	−	12.0435i	0.443268	−	1.65430i	−0.277202	−	0.960812i	$-0.589407\pi$
$53$	3.22704	−	12.0435i	0.443268	−	1.65430i	0.720470	−	0.693486i	$-0.243926\pi$
$54$		0			0
$55$	6.88536	+	6.91554i	0.928421	+	0.932492i
$56$		0			0
$57$	1.91162	−	1.91162i	0.253201	−	0.253201i
$58$		0			0
$59$	0.863327	+	1.49533i	0.112396	+	0.194675i	0.916736	−	0.399494i	$-0.130814\pi$
$59$	0.863327	+	1.49533i	0.112396	+	0.194675i	−0.804340	+	0.594169i	$0.797481\pi$
$60$		0			0
$61$	−8.84369	−	5.10591i	−1.13232	−	0.653745i	−0.187802	−	0.982207i	$-0.560136\pi$
$61$	−8.84369	−	5.10591i	−1.13232	−	0.653745i	−0.944517	+	0.328462i	$0.893470\pi$
$62$		0			0
$63$	2.61169	+	0.423166i	0.329042	+	0.0533140i
$64$		0			0
$65$	−4.44347	+	7.65758i	−0.551144	+	0.949805i
$66$		0			0
$67$	−7.38122	−	1.97779i	−0.901759	−	0.241626i	−0.221988	−	0.975049i	$-0.571254\pi$
$67$	−7.38122	−	1.97779i	−0.901759	−	0.241626i	−0.679772	+	0.733424i	$0.737921\pi$
$68$		0			0
$69$	3.02697			0.364405
$70$		0			0
$71$	5.39235			0.639955			0.319977	−	0.947425i	$-0.396325\pi$
$71$	5.39235			0.639955			0.319977	+	0.947425i	$0.396325\pi$
$72$		0			0
$73$	−15.0908	−	4.04357i	−1.76624	−	0.473264i	−0.778276	−	0.627922i	$-0.783906\pi$
$73$	−15.0908	−	4.04357i	−1.76624	−	0.473264i	−0.987968	+	0.154658i	$0.950572\pi$
$74$		0			0
$75$	4.83524	−	1.27295i	0.558326	−	0.146988i
$76$		0			0
$77$	−8.94759	−	7.29840i	−1.01967	−	0.831729i
$78$		0			0
$79$	11.6020	+	6.69839i	1.30532	+	0.753628i	0.981312	−	0.192425i	$-0.0616353\pi$
$79$	11.6020	+	6.69839i	1.30532	+	0.753628i	0.324011	+	0.946053i	$0.394969\pi$
$80$		0			0
$81$	0.500000	+	0.866025i	0.0555556	+	0.0962250i
$82$		0			0
$83$	−11.4560	+	11.4560i	−1.25746	+	1.25746i	−0.305165	+	0.952299i	$0.598712\pi$
$83$	−11.4560	+	11.4560i	−1.25746	+	1.25746i	−0.952299	+	0.305165i	$0.901288\pi$
$84$		0			0
$85$	0.00155332	−	0.710113i	0.000168481	−	0.0770226i
$86$		0			0
$87$	−1.17180	+	4.37321i	−0.125630	+	0.468858i
$88$		0			0
$89$	−3.03332	+	5.25387i	−0.321532	+	0.556909i	−0.980804	−	0.194995i	$-0.937531\pi$
$89$	−3.03332	+	5.25387i	−0.321532	+	0.556909i	0.659273	+	0.751904i	$0.270864\pi$
$90$		0			0
$91$	4.29473	−	9.55465i	0.450210	−	1.00160i
$92$		0			0
$93$	2.11597	−	0.566971i	0.219415	−	0.0587922i
$94$		0			0
$95$	−1.57735	+	5.83566i	−0.161833	+	0.598726i
$96$		0			0
$97$	−7.89037	−	7.89037i	−0.801146	−	0.801146i	0.182129	−	0.983275i	$-0.441701\pi$
$97$	−7.89037	−	7.89037i	−0.801146	−	0.801146i	−0.983275	+	0.182129i	$0.941701\pi$
$98$		0			0
$99$		−	4.36424i		−	0.438622i
$100$		0			0
$101$	−2.94262	+	1.69892i	−0.292801	+	0.169049i	−0.639205	−	0.769037i	$-0.720736\pi$
$101$	−2.94262	+	1.69892i	−0.292801	+	0.169049i	0.346403	+	0.938086i	$0.387403\pi$
$102$		0			0
$103$	4.00916	+	14.9624i	0.395034	+	1.47429i	0.821722	+	0.569889i	$0.193014\pi$
$103$	4.00916	+	14.9624i	0.395034	+	1.47429i	−0.426687	+	0.904399i	$0.640319\pi$
$104$		0			0
$105$	−5.53521	+	2.08840i	−0.540182	+	0.203807i
$106$		0			0
$107$	−3.53035	−	13.1754i	−0.341291	−	1.27372i	−0.896885	−	0.442263i	$-0.854176\pi$
$107$	−3.53035	−	13.1754i	−0.341291	−	1.27372i	0.555594	−	0.831454i	$-0.312491\pi$
$108$		0			0
$109$	0.454868	−	0.262618i	0.0435685	−	0.0251543i	−0.478058	−	0.878329i	$-0.658659\pi$
$109$	0.454868	−	0.262618i	0.0435685	−	0.0251543i	0.521626	+	0.853174i	$0.325326\pi$
$110$		0			0
$111$			2.67682i			0.254073i
$112$		0			0
$113$	12.2952	+	12.2952i	1.15664	+	1.15664i	0.985194	+	0.171445i	$0.0548436\pi$
$113$	12.2952	+	12.2952i	1.15664	+	1.15664i	0.171445	+	0.985194i	$0.445156\pi$
$114$		0			0
$115$	−5.86910	+	3.37143i	−0.547296	+	0.314387i
$116$		0			0
$117$	3.82445	−	1.02476i	0.353571	−	0.0947390i
$118$		0			0
$119$	0.0848581	+	0.835923i	0.00777893	+	0.0766290i
$120$		0			0
$121$	−4.02329	+	6.96854i	−0.365754	+	0.633504i
$122$		0			0
$123$	0.681776	−	2.54442i	0.0614736	−	0.229423i
$124$		0			0
$125$	−7.95740	+	7.85365i	−0.711732	+	0.702451i
$126$		0			0
$127$	6.07887	−	6.07887i	0.539413	−	0.539413i	−0.383944	−	0.923357i	$-0.625434\pi$
$127$	6.07887	−	6.07887i	0.539413	−	0.539413i	0.923357	+	0.383944i	$0.125434\pi$
$128$		0			0
$129$	4.39254	+	7.60810i	0.386741	+	0.669855i
$130$		0			0
$131$	−4.99501	−	2.88387i	−0.436416	−	0.251965i	0.265660	−	0.964067i	$-0.414410\pi$
$131$	−4.99501	−	2.88387i	−0.436416	−	0.251965i	−0.702076	+	0.712102i	$0.747743\pi$
$132$		0			0
$133$	1.14401	−	7.06056i	0.0991979	−	0.612228i
$134$		0			0
$135$	−1.93404	−	1.12227i	−0.166456	−	0.0965894i
$136$		0			0
$137$	−18.3559	−	4.91846i	−1.56825	−	0.420212i	−0.632989	−	0.774161i	$-0.718172\pi$
$137$	−18.3559	−	4.91846i	−1.56825	−	0.420212i	−0.935265	+	0.353948i	$0.884839\pi$
$138$		0			0
$139$	0.316854			0.0268752			0.0134376	−	0.999910i	$-0.495723\pi$
$139$	0.316854			0.0268752			0.0134376	+	0.999910i	$0.495723\pi$
$140$		0			0
$141$	6.28560			0.529343
$142$		0			0
$143$	−16.6908	−	4.47229i	−1.39576	−	0.373992i
$144$		0			0
$145$	−2.59883	−	9.78451i	−0.215821	−	0.812559i
$146$		0			0
$147$	6.25946	−	3.13354i	0.516272	−	0.258450i
$148$		0			0
$149$	7.11870	+	4.10998i	0.583187	+	0.336703i	0.762399	−	0.647107i	$-0.224022\pi$
$149$	7.11870	+	4.10998i	0.583187	+	0.336703i	−0.179212	+	0.983810i	$0.557355\pi$
$150$		0			0
$151$	4.68283	+	8.11090i	0.381083	+	0.660055i	0.991217	−	0.132243i	$-0.0422178\pi$
$151$	4.68283	+	8.11090i	0.381083	+	0.660055i	−0.610134	+	0.792298i	$0.708884\pi$
$152$		0			0
$153$	−0.224558	+	0.224558i	−0.0181544	+	0.0181544i
$154$		0			0
$155$	−3.47123	+	3.45607i	−0.278816	+	0.277598i
$156$		0			0
$157$	3.96366	−	14.7926i	0.316334	−	1.18058i	−0.606407	−	0.795155i	$-0.707390\pi$
$157$	3.96366	−	14.7926i	0.316334	−	1.18058i	0.922741	−	0.385421i	$-0.125944\pi$
$158$		0			0
$159$	6.23416	−	10.7979i	0.494401	−	0.856328i
$160$		0			0
$161$	6.49579	−	4.68430i	0.511940	−	0.369175i
$162$		0			0
$163$	−17.7755	+	4.76292i	−1.39228	+	0.373061i	−0.875567	−	0.483097i	$-0.839512\pi$
$163$	−17.7755	+	4.76292i	−1.39228	+	0.373061i	−0.516715	+	0.856158i	$0.672845\pi$
$164$		0			0
$165$	4.86087	+	8.46196i	0.378418	+	0.658763i
$166$		0			0
$167$	7.47910	+	7.47910i	0.578750	+	0.578750i	0.934559	−	0.355809i	$-0.115795\pi$
$167$	7.47910	+	7.47910i	0.578750	+	0.578750i	−0.355809	+	0.934559i	$0.615795\pi$
$168$		0			0
$169$		−	2.67657i		−	0.205890i
$170$		0			0
$171$	2.34125	−	1.35172i	0.179040	−	0.103369i
$172$		0			0
$173$	1.41915	+	5.29635i	0.107896	+	0.402674i	0.998658	−	0.0517971i	$-0.0164949\pi$
$173$	1.41915	+	5.29635i	0.107896	+	0.402674i	−0.890761	+	0.454471i	$0.849828\pi$
$174$		0			0
$175$	8.40636	−	10.2144i	0.635461	−	0.772133i
$176$		0			0
$177$	0.446891	+	1.66782i	0.0335904	+	0.125361i
$178$		0			0
$179$	−8.91592	+	5.14761i	−0.666407	+	0.384750i	−0.794714	−	0.606984i	$-0.792379\pi$
$179$	−8.91592	+	5.14761i	−0.666407	+	0.384750i	0.128307	+	0.991735i	$0.459046\pi$
$180$		0			0
$181$			13.9242i			1.03498i	0.855689	+	0.517490i	$0.173134\pi$
$181$			13.9242i			1.03498i	−0.855689	+	0.517490i	$0.826866\pi$
$182$		0			0
$183$	−7.22085	−	7.22085i	−0.533780	−	0.533780i
$184$		0			0
$185$	−2.98143	−	5.19018i	−0.219199	−	0.381589i
$186$		0			0
$187$	1.33874	−	0.358714i	0.0978983	−	0.0262318i
$188$		0			0
$189$	2.41318	+	1.08470i	0.175533	+	0.0789005i
$190$		0			0
$191$	5.34601	−	9.25957i	0.386824	−	0.669999i	−0.605196	−	0.796076i	$-0.706905\pi$
$191$	5.34601	−	9.25957i	0.386824	−	0.669999i	0.992020	+	0.126077i	$0.0402387\pi$
$192$		0			0
$193$	−1.11606	+	4.16518i	−0.0803356	+	0.299816i	−0.994390	−	0.105775i	$-0.966268\pi$
$193$	−1.11606	+	4.16518i	−0.0803356	+	0.299816i	0.914055	+	0.405591i	$0.132934\pi$
$194$		0			0
$195$	−6.27399	+	6.24660i	−0.449289	+	0.447328i
$196$		0			0
$197$	−1.11483	+	1.11483i	−0.0794281	+	0.0794281i	−0.745705	−	0.666277i	$-0.767887\pi$
$197$	−1.11483	+	1.11483i	−0.0794281	+	0.0794281i	0.666277	+	0.745705i	$0.267887\pi$
$198$		0			0
$199$	11.7514	+	20.3540i	0.833034	+	1.44286i	0.895621	+	0.444817i	$0.146731\pi$
$199$	11.7514	+	20.3540i	0.833034	+	1.44286i	−0.0625877	+	0.998039i	$0.519935\pi$
$200$		0			0
$201$	−6.61782	−	3.82080i	−0.466785	−	0.269498i
$202$		0			0
$203$	4.25299	+	11.1982i	0.298502	+	0.785957i
$204$		0			0
$205$	1.51205	+	5.69282i	0.105606	+	0.397604i
$206$		0			0
$207$	2.92383	+	0.783438i	0.203220	+	0.0544527i
$208$		0			0
$209$	−11.7985			−0.816117
$210$		0			0
$211$	13.9609			0.961108			0.480554	−	0.876965i	$-0.340436\pi$
$211$	13.9609			0.961108			0.480554	+	0.876965i	$0.340436\pi$
$212$		0			0
$213$	5.20861	+	1.39564i	0.356888	+	0.0956280i
$214$		0			0
$215$	−16.9907	−	9.85920i	−1.15876	−	0.672392i
$216$		0			0
$217$	3.66340	−	4.49120i	0.248688	−	0.304883i
$218$		0			0
$219$	−13.5300	−	7.81157i	−0.914275	−	0.527857i
$220$		0			0
$221$	0.628694	+	1.08893i	0.0422905	+	0.0732493i
$222$		0			0
$223$	−19.7736	+	19.7736i	−1.32414	+	1.32414i	−0.413745	+	0.910393i	$0.635780\pi$
$223$	−19.7736	+	19.7736i	−1.32414	+	1.32414i	−0.910393	+	0.413745i	$0.864220\pi$
$224$		0			0
$225$	4.99995	+	0.0218741i	0.333330	+	0.00145827i
$226$		0			0
$227$	−2.45140	+	9.14875i	−0.162705	+	0.607224i	0.835617	+	0.549313i	$0.185111\pi$
$227$	−2.45140	+	9.14875i	−0.162705	+	0.607224i	−0.998322	+	0.0579110i	$0.981556\pi$
$228$		0			0
$229$	−0.927717	+	1.60685i	−0.0613053	+	0.106184i	−0.895049	−	0.445968i	$-0.852860\pi$
$229$	−0.927717	+	1.60685i	−0.0613053	+	0.106184i	0.833744	+	0.552151i	$0.186193\pi$
$230$		0			0
$231$	−6.75375	−	9.36552i	−0.444364	−	0.616206i
$232$		0			0
$233$	−26.5983	+	7.12699i	−1.74251	+	0.466905i	−0.983003	−	0.183591i	$-0.941228\pi$
$233$	−26.5983	+	7.12699i	−1.74251	+	0.466905i	−0.759510	+	0.650496i	$0.774561\pi$
$234$		0			0
$235$	−12.1873	+	7.00087i	−0.795015	+	0.456686i
$236$		0			0
$237$	9.47296	+	9.47296i	0.615335	+	0.615335i
$238$		0			0
$239$			14.3815i			0.930260i	0.885242	+	0.465130i	$0.153992\pi$
$239$			14.3815i			0.930260i	−0.885242	+	0.465130i	$0.846008\pi$
$240$		0			0
$241$	5.73574	−	3.31153i	0.369471	−	0.213314i	−0.303756	−	0.952750i	$-0.598241\pi$
$241$	5.73574	−	3.31153i	0.369471	−	0.213314i	0.673227	+	0.739435i	$0.264908\pi$
$242$		0			0
$243$	0.258819	+	0.965926i	0.0166032	+	0.0619642i
$244$		0			0
$245$	−8.64655	+	13.0475i	−0.552408	+	0.833574i
$246$		0			0
$247$	−2.77038	−	10.3392i	−0.176275	−	0.657867i
$248$		0			0
$249$	−14.0307	+	8.10065i	−0.889162	+	0.513358i
$250$		0			0
$251$		−	3.20820i		−	0.202500i	−0.994861	−	0.101250i	$-0.967716\pi$
$251$		−	3.20820i		−	0.202500i	0.994861	−	0.101250i	$-0.0322841\pi$
$252$		0			0
$253$	−9.34119	−	9.34119i	−0.587275	−	0.587275i
$254$		0			0
$255$	0.185291	−	0.685515i	0.0116034	−	0.0429286i
$256$		0			0
$257$	−15.1796	+	4.06736i	−0.946877	+	0.253715i	−0.699037	−	0.715086i	$-0.746388\pi$
$257$	−15.1796	+	4.06736i	−0.946877	+	0.253715i	−0.247841	+	0.968801i	$0.579721\pi$
$258$		0			0
$259$	4.14244	+	5.74437i	0.257398	+	0.356938i
$260$		0			0
$261$	−2.26374	+	3.92092i	−0.140122	+	0.242699i
$262$		0			0
$263$	3.76752	−	14.0606i	0.232315	−	0.867011i	−0.747026	−	0.664795i	$-0.768519\pi$
$263$	3.76752	−	14.0606i	0.232315	−	0.867011i	0.979341	−	0.202216i	$-0.0648145\pi$
$264$		0			0
$265$	−0.0609852	+	27.8799i	−0.00374629	+	1.71265i
$266$		0			0
$267$	−4.28977	+	4.28977i	−0.262529	+	0.262529i
$268$		0			0
$269$	−6.10002	−	10.5655i	−0.371925	−	0.644193i	0.617937	−	0.786228i	$-0.287969\pi$
$269$	−6.10002	−	10.5655i	−0.371925	−	0.644193i	−0.989862	+	0.142035i	$0.954635\pi$
$270$		0			0
$271$	−11.0842	−	6.39946i	−0.673317	−	0.388740i	0.124015	−	0.992280i	$-0.460423\pi$
$271$	−11.0842	−	6.39946i	−0.673317	−	0.388740i	−0.797332	+	0.603541i	$0.793756\pi$
$272$		0			0
$273$	6.62132	−	8.11752i	0.400740	−	0.491295i
$274$		0			0
$275$	−18.8498	−	10.9932i	−1.13669	−	0.662913i
$276$		0			0
$277$	−20.2295	−	5.42047i	−1.21547	−	0.325685i	−0.406565	−	0.913622i	$-0.633273\pi$
$277$	−20.2295	−	5.42047i	−1.21547	−	0.325685i	−0.808906	+	0.587937i	$0.799940\pi$
$278$		0			0
$279$	2.19061			0.131148
$280$		0			0
$281$	23.8868			1.42497			0.712483	−	0.701690i	$-0.247571\pi$
$281$	23.8868			1.42497			0.712483	+	0.701690i	$0.247571\pi$
$282$		0			0
$283$	6.24382	+	1.67303i	0.371157	+	0.0994511i	0.439576	−	0.898206i	$-0.355129\pi$
$283$	6.24382	+	1.67303i	0.371157	+	0.0994511i	−0.0684190	+	0.997657i	$0.521795\pi$
$284$		0			0
$285$	−3.03398	+	5.22857i	−0.179718	+	0.309714i
$286$		0			0
$287$	−2.47448	−	6.51531i	−0.146064	−	0.384586i
$288$		0			0
$289$	14.6351	+	8.44957i	0.860888	+	0.497034i
$290$		0			0
$291$	−5.57934	−	9.66369i	−0.327066	−	0.566496i
$292$		0			0
$293$	6.91665	−	6.91665i	0.404075	−	0.404075i	−0.475591	−	0.879666i	$-0.657766\pi$
$293$	6.91665	−	6.91665i	0.404075	−	0.404075i	0.879666	+	0.475591i	$0.157766\pi$
$294$		0			0
$295$	−2.72410	−	2.73604i	−0.158603	−	0.159299i
$296$		0			0
$297$	1.12955	−	4.21553i	0.0655430	−	0.244610i
$298$		0			0
$299$	5.99245	−	10.3792i	0.346552	−	0.600246i
$300$		0			0
$301$	21.1999	+	9.52919i	1.22194	+	0.549253i
$302$		0			0
$303$	−3.28206	+	0.879426i	−0.188550	+	0.0505217i
$304$		0			0
$305$	22.0433	+	5.95819i	1.26219	+	0.341165i
$306$		0			0
$307$	−7.31476	−	7.31476i	−0.417475	−	0.417475i	0.466857	−	0.884333i	$-0.345386\pi$
$307$	−7.31476	−	7.31476i	−0.417475	−	0.417475i	−0.884333	+	0.466857i	$0.845386\pi$
$308$		0			0
$309$			15.4902i			0.881207i
$310$		0			0
$311$	−18.8693	+	10.8942i	−1.06998	+	0.617752i	−0.928175	−	0.372143i	$-0.878623\pi$
$311$	−18.8693	+	10.8942i	−1.06998	+	0.617752i	−0.141803	+	0.989895i	$0.545290\pi$
$312$		0			0
$313$	−6.09895	−	22.7616i	−0.344733	−	1.28656i	−0.892924	−	0.450208i	$-0.851350\pi$
$313$	−6.09895	−	22.7616i	−0.344733	−	1.28656i	0.548191	−	0.836353i	$-0.315317\pi$
$314$		0			0
$315$	−5.88712	+	0.584619i	−0.331702	+	0.0329396i
$316$		0			0
$317$	5.71583	+	21.3318i	0.321033	+	1.19811i	0.918240	+	0.396024i	$0.129610\pi$
$317$	5.71583	+	21.3318i	0.321033	+	1.19811i	−0.597207	+	0.802087i	$0.703723\pi$
$318$		0			0
$319$	17.1118	−	9.87951i	0.958077	−	0.553146i
$320$		0			0
$321$		−	13.6402i		−	0.761322i
$322$		0			0
$323$	0.607080	+	0.607080i	0.0337788	+	0.0337788i
$324$		0			0
$325$	5.20740	−	19.0997i	0.288855	−	1.05946i
$326$		0			0
$327$	0.507340	−	0.135941i	0.0280560	−	0.00751757i
$328$		0			0
$329$	13.4887	−	9.72709i	0.743656	−	0.536272i
$330$		0			0
$331$	0.440888	−	0.763640i	0.0242334	−	0.0419735i	−0.853654	−	0.520840i	$-0.825619\pi$
$331$	0.440888	−	0.763640i	0.0242334	−	0.0419735i	0.877888	+	0.478866i	$0.158952\pi$
$332$		0			0
$333$	−0.692812	+	2.58561i	−0.0379659	+	0.141691i
$334$		0			0
$335$	17.0871	+	0.0373767i	0.933568	+	0.00204211i
$336$		0			0
$337$	17.8218	−	17.8218i	0.970813	−	0.970813i	−0.0287728	−	0.999586i	$-0.509160\pi$
$337$	17.8218	−	17.8218i	0.970813	−	0.970813i	0.999586	+	0.0287728i	$0.00915993\pi$
$338$		0			0
$339$	8.69405	+	15.0585i	0.472196	+	0.817867i
$340$		0			0
$341$	−8.27950	−	4.78017i	−0.448360	−	0.258861i
$342$		0			0
$343$	8.58339	−	16.4111i	0.463459	−	0.886118i
$344$		0			0
$345$	−6.54170	+	1.73752i	−0.352193	+	0.0935447i
$346$		0			0
$347$	21.5123	+	5.76421i	1.15484	+	0.309439i	0.784904	−	0.619617i	$-0.212712\pi$
$347$	21.5123	+	5.76421i	1.15484	+	0.309439i	0.369938	+	0.929056i	$0.379379\pi$
$348$		0			0
$349$	−1.52167			−0.0814531			−0.0407266	−	0.999170i	$-0.512967\pi$
$349$	−1.52167			−0.0814531			−0.0407266	+	0.999170i	$0.512967\pi$
$350$		0			0
$351$	3.95937			0.211335
$352$		0			0
$353$	18.7437	+	5.02235i	0.997625	+	0.267313i	0.720450	−	0.693506i	$-0.243935\pi$
$353$	18.7437	+	5.02235i	0.997625	+	0.267313i	0.277175	+	0.960819i	$0.410602\pi$
$354$		0			0
$355$	−11.6536	+	3.09527i	−0.618510	+	0.164280i
$356$		0			0
$357$	−0.134386	+	0.829403i	−0.00711247	+	0.0438966i
$358$		0			0
$359$	26.7834	+	15.4634i	1.41357	+	0.816126i	0.995723	−	0.0923909i	$-0.0294509\pi$
$359$	26.7834	+	15.4634i	1.41357	+	0.816126i	0.417849	+	0.908517i	$0.362784\pi$
$360$		0			0
$361$	5.84570	+	10.1250i	0.307668	+	0.532897i
$362$		0			0
$363$	−5.68979	+	5.68979i	−0.298637	+	0.298637i
$364$		0			0
$365$	34.9343	+	0.0764161i	1.82855	+	0.00399980i
$366$		0			0
$367$	−3.21383	+	11.9942i	−0.167761	+	0.626091i	0.829911	+	0.557895i	$0.188391\pi$
$367$	−3.21383	+	11.9942i	−0.167761	+	0.626091i	−0.997672	+	0.0681957i	$0.978276\pi$
$368$		0			0
$369$	1.31709	−	2.28127i	0.0685649	−	0.118758i
$370$		0			0
$371$	−3.33164	−	32.8194i	−0.172970	−	1.70390i
$372$		0			0
$373$	−5.86906	+	1.57261i	−0.303888	+	0.0814266i	−0.407541	−	0.913187i	$-0.633614\pi$
$373$	−5.86906	+	1.57261i	−0.303888	+	0.0814266i	0.103653	+	0.994614i	$0.466947\pi$
$374$		0			0
$375$	−9.71893	+	5.52651i	−0.501884	+	0.285388i
$376$		0			0
$377$	12.6756	+	12.6756i	0.652825	+	0.652825i
$378$		0			0
$379$		−	12.2273i		−	0.628072i	−0.949411	−	0.314036i	$-0.898319\pi$
$379$		−	12.2273i		−	0.628072i	0.949411	−	0.314036i	$-0.101681\pi$
$380$		0			0
$381$	7.44507	−	4.29841i	0.381423	−	0.220214i
$382$		0			0
$383$	−3.19999	−	11.9425i	−0.163512	−	0.610234i	−0.998225	−	0.0595499i	$-0.981033\pi$
$383$	−3.19999	−	11.9425i	−0.163512	−	0.610234i	0.834714	−	0.550684i	$-0.185633\pi$
$384$		0			0
$385$	23.5263	+	10.6368i	1.19901	+	0.542102i
$386$		0			0
$387$	2.27374	+	8.48573i	0.115581	+	0.431354i
$388$		0			0
$389$	25.2993	−	14.6065i	1.28272	−	0.740581i	0.305378	−	0.952231i	$-0.401217\pi$
$389$	25.2993	−	14.6065i	1.28272	−	0.740581i	0.977345	+	0.211651i	$0.0678839\pi$
$390$		0			0
$391$			0.961285i			0.0486143i
$392$		0			0
$393$	−4.07841	−	4.07841i	−0.205729	−	0.205729i
$394$		0			0
$395$	−28.9184	−	7.81649i	−1.45504	−	0.393290i
$396$		0			0
$397$	−35.6296	+	9.54692i	−1.78820	+	0.479146i	−0.992038	−	0.125939i	$-0.959805\pi$
$397$	−35.6296	+	9.54692i	−1.78820	+	0.479146i	−0.796160	+	0.605086i	$0.793139\pi$
$398$		0			0
$399$	2.93243	−	6.52388i	0.146805	−	0.326603i
$400$		0			0
$401$	17.7131	−	30.6799i	0.884549	−	1.53208i	0.0383185	−	0.999266i	$-0.487800\pi$
$401$	17.7131	−	30.6799i	0.884549	−	1.53208i	0.846230	−	0.532818i	$-0.178867\pi$
$402$		0			0
$403$	2.24485	−	8.37788i	0.111824	−	0.417332i
$404$		0			0
$405$	−1.57768	−	1.58459i	−0.0783954	−	0.0787391i
$406$		0			0
$407$	8.26062	−	8.26062i	0.409464	−	0.409464i
$408$		0			0
$409$	13.9895	+	24.2306i	0.691737	+	1.19812i	0.971268	+	0.237988i	$0.0764877\pi$
$409$	13.9895	+	24.2306i	0.691737	+	1.19812i	−0.279531	+	0.960137i	$0.590179\pi$
$410$		0			0
$411$	−16.4575	−	9.50174i	−0.811788	−	0.468686i
$412$		0			0
$413$	3.54000	+	2.88752i	0.174192	+	0.142085i
$414$		0			0
$415$	18.1822	−	31.3340i	0.892529	−	1.53812i
$416$		0			0
$417$	0.306058	+	0.0820080i	0.0149877	+	0.00401595i
$418$		0			0
$419$	6.49494			0.317299			0.158649	−	0.987335i	$-0.449286\pi$
$419$	6.49494			0.317299			0.158649	+	0.987335i	$0.449286\pi$
$420$		0			0
$421$	−16.6014			−0.809103			−0.404551	−	0.914515i	$-0.632572\pi$
$421$	−16.6014			−0.809103			−0.404551	+	0.914515i	$0.632572\pi$
$422$		0			0
$423$	6.07142	+	1.62683i	0.295203	+	0.0790993i
$424$		0			0
$425$	0.404256	+	1.53554i	0.0196093	+	0.0744848i
$426$		0			0
$427$	−26.6701	−	4.32130i	−1.29066	−	0.209122i
$428$		0			0
$429$	−14.9646	−	8.63981i	−0.722497	−	0.417134i
$430$		0			0
$431$	2.13950	+	3.70572i	0.103056	+	0.178498i	0.912942	−	0.408089i	$-0.133805\pi$
$431$	2.13950	+	3.70572i	0.103056	+	0.178498i	−0.809886	+	0.586587i	$0.800471\pi$
$432$		0			0
$433$	−11.5948	+	11.5948i	−0.557208	+	0.557208i	−0.928512	−	0.371303i	$-0.878911\pi$
$433$	−11.5948	+	11.5948i	−0.557208	+	0.557208i	0.371303	+	0.928512i	$0.378911\pi$
$434$		0			0
$435$	0.0221449	−	10.1237i	0.00106177	−	0.485396i
$436$		0			0
$437$	2.11798	−	7.90441i	0.101317	−	0.378119i
$438$		0			0
$439$	13.9971	−	24.2437i	0.668046	−	1.15709i	−0.310404	−	0.950605i	$-0.600464\pi$
$439$	13.9971	−	24.2437i	0.668046	−	1.15709i	0.978450	−	0.206484i	$-0.0662022\pi$
$440$		0			0
$441$	6.85720	−	1.40670i	0.326533	−	0.0669858i
$442$		0			0
$443$	8.79765	−	2.35732i	0.417989	−	0.112000i	−0.0436936	−	0.999045i	$-0.513913\pi$
$443$	8.79765	−	2.35732i	0.417989	−	0.112000i	0.461683	+	0.887045i	$0.347246\pi$
$444$		0			0
$445$	3.53965	−	13.0955i	0.167795	−	0.620786i
$446$		0			0
$447$	5.81239	+	5.81239i	0.274917	+	0.274917i
$448$		0			0
$449$			26.7133i			1.26068i	0.776319	+	0.630340i	$0.217084\pi$
$449$			26.7133i			1.26068i	−0.776319	+	0.630340i	$0.782916\pi$
$450$		0			0
$451$	−9.95599	+	5.74809i	−0.468809	+	0.270667i
$452$		0			0
$453$	2.42401	+	9.04653i	0.113890	+	0.425043i
$454$		0			0
$455$	−3.79704	+	23.1141i	−0.178008	+	1.08361i
$456$		0			0
$457$	0.910151	+	3.39673i	0.0425751	+	0.158892i	0.983941	−	0.178496i	$-0.0571231\pi$
$457$	0.910151	+	3.39673i	0.0425751	+	0.158892i	−0.941366	+	0.337388i	$0.890456\pi$
$458$		0			0
$459$	−0.275026	+	0.158787i	−0.0128371	+	0.00741152i
$460$		0			0
$461$		−	2.72235i		−	0.126793i	−0.997988	−	0.0633963i	$-0.979807\pi$
$461$		−	2.72235i		−	0.126793i	0.997988	−	0.0633963i	$-0.0201932\pi$
$462$		0			0
$463$	26.6530	+	26.6530i	1.23867	+	1.23867i	0.960544	+	0.278128i	$0.0897140\pi$
$463$	26.6530	+	26.6530i	1.23867	+	1.23867i	0.278128	+	0.960544i	$0.410286\pi$
$464$		0			0
$465$	−4.24744	+	2.43989i	−0.196970	+	0.113147i
$466$		0			0
$467$	41.2480	−	11.0524i	1.90873	−	0.511442i	0.914438	−	0.404725i	$-0.132633\pi$
$467$	41.2480	−	11.0524i	1.90873	−	0.511442i	0.994289	−	0.106717i	$-0.0340338\pi$
$468$		0			0
$469$	−20.1144	+	2.04190i	−0.928797	+	0.0942860i
$470$		0			0
$471$	7.65720	−	13.2627i	0.352825	−	0.611111i
$472$		0			0
$473$	9.92316	−	37.0337i	0.456267	−	1.70281i
$474$		0			0
$475$	0.0591354	−	13.5171i	0.00271332	−	0.620206i
$476$		0			0
$477$	8.81643	−	8.81643i	0.403677	−	0.403677i
$478$		0			0
$479$	−12.8201	−	22.2051i	−0.585766	−	1.01458i	−0.994779	−	0.102048i	$-0.967460\pi$
$479$	−12.8201	−	22.2051i	−0.585766	−	1.01458i	0.409013	−	0.912528i	$-0.365873\pi$
$480$		0			0
$481$	9.17858	+	5.29926i	0.418507	+	0.241625i
$482$		0			0
$483$	7.48684	−	2.84346i	0.340663	−	0.129382i
$484$		0			0
$485$	21.5813	+	12.5230i	0.979958	+	0.568641i
$486$		0			0
$487$	−23.5505	−	6.31034i	−1.06717	−	0.285949i	−0.317843	−	0.948143i	$-0.602959\pi$
$487$	−23.5505	−	6.31034i	−1.06717	−	0.285949i	−0.749332	+	0.662195i	$0.769625\pi$
$488$		0			0
$489$	−18.4025			−0.832190
$490$		0			0
$491$	7.86333			0.354867			0.177434	−	0.984133i	$-0.443221\pi$
$491$	7.86333			0.354867			0.177434	+	0.984133i	$0.443221\pi$
$492$		0			0
$493$	−1.38881	−	0.372132i	−0.0625491	−	0.0167600i
$494$		0			0
$495$	2.50512	+	9.43172i	0.112597	+	0.423924i
$496$		0			0
$497$	13.3373	−	5.06543i	0.598260	−	0.227216i
$498$		0			0
$499$	−2.03266	−	1.17356i	−0.0909945	−	0.0525357i	0.453812	−	0.891097i	$-0.350064\pi$
$499$	−2.03266	−	1.17356i	−0.0909945	−	0.0525357i	−0.544807	+	0.838562i	$0.683397\pi$
$500$		0			0
$501$	5.28852	+	9.15998i	0.236274	+	0.409238i
$502$		0			0
$503$	14.5260	−	14.5260i	0.647682	−	0.647682i	−0.304750	−	0.952432i	$-0.598573\pi$
$503$	14.5260	−	14.5260i	0.647682	−	0.647682i	0.952432	+	0.304750i	$0.0985729\pi$
$504$		0			0
$505$	5.38420	−	5.36070i	0.239594	−	0.238548i
$506$		0			0
$507$	0.692748	−	2.58537i	0.0307660	−	0.114820i
$508$		0			0
$509$	−8.29178	+	14.3618i	−0.367527	+	0.636575i	−0.989178	−	0.146719i	$-0.953129\pi$
$509$	−8.29178	+	14.3618i	−0.367527	+	0.636575i	0.621652	+	0.783294i	$0.286462\pi$
$510$		0			0
$511$	−41.1236	+	4.17463i	−1.81920	+	0.184675i
$512$		0			0
$513$	2.61132	−	0.699702i	0.115293	−	0.0308926i
$514$		0			0
$515$	−17.2529	−	30.0345i	−0.760255	−	1.32348i
$516$		0			0
$517$	−19.3972	−	19.3972i	−0.853090	−	0.853090i
$518$		0			0
$519$			5.48319i			0.240685i
$520$		0			0
$521$	−28.8215	+	16.6401i	−1.26269	+	0.729015i	−0.973594	−	0.228285i	$-0.926688\pi$
$521$	−28.8215	+	16.6401i	−1.26269	+	0.729015i	−0.289097	+	0.957300i	$0.593355\pi$
$522$		0			0
$523$	6.09013	+	22.7287i	0.266303	+	0.993855i	0.961448	+	0.274986i	$0.0886732\pi$
$523$	6.09013	+	22.7287i	0.266303	+	0.993855i	−0.695146	+	0.718869i	$0.744660\pi$
$524$		0			0
$525$	10.7636	−	7.69059i	0.469762	−	0.335645i
$526$		0			0
$527$	0.180055	+	0.671974i	0.00784331	+	0.0292716i
$528$		0			0
$529$	−11.9836	+	6.91872i	−0.521025	+	0.300814i
$530$		0			0
$531$			1.72665i			0.0749304i
$532$		0			0
$533$	−7.37489	−	7.37489i	−0.319442	−	0.319442i
$534$		0			0
$535$	15.1924	+	26.4475i	0.656825	+	1.14342i
$536$		0			0
$537$	−9.94441	+	2.66460i	−0.429133	+	0.114986i
$538$		0			0
$539$	−28.9867	−	9.64654i	−1.24854	−	0.415506i
$540$		0			0
$541$	7.54153	−	13.0623i	0.324236	−	0.561592i	−0.657122	−	0.753784i	$-0.728226\pi$
$541$	7.54153	−	13.0623i	0.324236	−	0.561592i	0.981357	+	0.192192i	$0.0615596\pi$
$542$		0			0
$543$	−3.60386	+	13.4498i	−0.154656	+	0.577185i
$544$		0			0
$545$	−0.832287	+	0.828654i	−0.0356513	+	0.0354956i
$546$		0			0
$547$	−24.4516	+	24.4516i	−1.04547	+	1.04547i	−0.0465590	+	0.998916i	$0.514826\pi$
$547$	−24.4516	+	24.4516i	−1.04547	+	1.04547i	−0.998916	+	0.0465590i	$0.985174\pi$
$548$		0			0
$549$	−5.10591	−	8.84369i	−0.217915	−	0.377440i
$550$		0			0
$551$	10.6000	+	6.11989i	0.451574	+	0.260716i
$552$		0			0
$553$	34.9883	+	5.66907i	1.48785	+	0.241073i
$554$		0			0
$555$	−1.53652	−	5.78498i	−0.0652219	−	0.245559i
$556$		0			0
$557$	18.3832	+	4.92576i	0.778920	+	0.208711i	0.626309	−	0.779575i	$-0.284565\pi$
$557$	18.3832	+	4.92576i	0.778920	+	0.208711i	0.152612	+	0.988286i	$0.451232\pi$
$558$		0			0
$559$	34.7833			1.47118
$560$		0			0
$561$	1.38596			0.0585155
$562$		0			0
$563$	−19.9179	−	5.33699i	−0.839441	−	0.224927i	−0.186613	−	0.982434i	$-0.559751\pi$
$563$	−19.9179	−	5.33699i	−0.839441	−	0.224927i	−0.652828	+	0.757506i	$0.726418\pi$
$564$		0			0
$565$	−33.6293	−	19.5141i	−1.41479	−	0.820964i
$566$		0			0
$567$	2.05021	+	1.67232i	0.0861006	+	0.0702308i
$568$		0			0
$569$	5.59023	+	3.22752i	0.234354	+	0.135305i	0.612579	−	0.790409i	$-0.290132\pi$
$569$	5.59023	+	3.22752i	0.234354	+	0.135305i	−0.378225	+	0.925714i	$0.623465\pi$
$570$		0			0
$571$	23.4901	+	40.6860i	0.983028	+	1.70265i	0.650390	+	0.759600i	$0.274605\pi$
$571$	23.4901	+	40.6860i	0.983028	+	1.70265i	0.332638	+	0.943055i	$0.392061\pi$
$572$		0			0
$573$	7.56041	−	7.56041i	0.315841	−	0.315841i
$574$		0			0
$575$	10.7487	−	10.6550i	0.448251	−	0.444346i
$576$		0			0
$577$	3.59166	−	13.4042i	0.149523	−	0.558026i	−0.849990	−	0.526799i	$-0.823392\pi$
$577$	3.59166	−	13.4042i	0.149523	−	0.558026i	0.999512	−	0.0312266i	$-0.00994136\pi$
$578$		0			0
$579$	−2.15606	+	3.73440i	−0.0896027	+	0.155196i
$580$		0			0
$581$	−17.5736	+	39.0966i	−0.729076	+	1.62200i
$582$		0			0
$583$	−52.5606	+	14.0836i	−2.17684	+	0.583282i
$584$		0			0
$585$	−7.67694	+	4.40992i	−0.317403	+	0.182328i
$586$		0			0
$587$	25.6597	+	25.6597i	1.05909	+	1.05909i	0.998141	+	0.0609461i	$0.0194118\pi$
$587$	25.6597	+	25.6597i	1.05909	+	1.05909i	0.0609461	+	0.998141i	$0.480588\pi$
$588$		0			0
$589$		−	5.92219i		−	0.244019i
$590$		0			0
$591$	−1.36538	+	0.788301i	−0.0561641	+	0.0324264i
$592$		0			0
$593$	−0.830494	−	3.09944i	−0.0341043	−	0.127279i	0.946775	−	0.321896i	$-0.104320\pi$
$593$	−0.830494	−	3.09944i	−0.0341043	−	0.127279i	−0.980879	+	0.194617i	$0.937654\pi$
$594$		0			0
$595$	−0.663219	−	1.75783i	−0.0271893	−	0.0720642i
$596$		0			0
$597$	6.08297	+	22.7019i	0.248959	+	0.929128i
$598$		0			0
$599$	35.6333	−	20.5729i	1.45594	−	0.840586i	0.457130	−	0.889400i	$-0.348877\pi$
$599$	35.6333	−	20.5729i	1.45594	−	0.840586i	0.998808	+	0.0488140i	$0.0155442\pi$
$600$		0			0
$601$			35.6457i			1.45402i	0.686628	+	0.727009i	$0.259090\pi$
$601$			35.6457i			1.45402i	−0.686628	+	0.727009i	$0.740910\pi$
$602$		0			0
$603$	−5.40343	−	5.40343i	−0.220045	−	0.220045i
$604$		0			0
$605$	4.69486	−	17.3694i	0.190873	−	0.706166i
$606$		0			0
$607$	31.2289	−	8.36776i	1.26754	−	0.339637i	0.438453	−	0.898754i	$-0.355527\pi$
$607$	31.2289	−	8.36776i	1.26754	−	0.339637i	0.829088	+	0.559117i	$0.188860\pi$
$608$		0			0
$609$	1.20978	+	11.9173i	0.0490227	+	0.482915i
$610$		0			0
$611$	12.4435	−	21.5527i	0.503410	−	0.871931i
$612$		0			0
$613$	−2.52780	+	9.43387i	−0.102097	+	0.381030i	−0.998000	−	0.0632197i	$-0.979863\pi$
$613$	−2.52780	+	9.43387i	−0.102097	+	0.381030i	0.895903	+	0.444250i	$0.146530\pi$
$614$		0			0
$615$	−0.0128843	+	5.89019i	−0.000519546	+	0.237515i
$616$		0			0
$617$	−9.23419	+	9.23419i	−0.371754	+	0.371754i	−0.868116	−	0.496362i	$-0.834669\pi$
$617$	−9.23419	+	9.23419i	−0.371754	+	0.371754i	0.496362	+	0.868116i	$0.334669\pi$
$618$		0			0
$619$	−12.1118	−	20.9783i	−0.486815	−	0.843188i	0.513070	−	0.858347i	$-0.328508\pi$
$619$	−12.1118	−	20.9783i	−0.486815	−	0.843188i	−0.999885	+	0.0151586i	$0.995175\pi$
$620$		0			0
$621$	2.62144	+	1.51349i	0.105195	+	0.0607341i
$622$		0			0
$623$	−2.56720	+	15.8442i	−0.102853	+	0.634785i
$624$		0			0
$625$	12.6890	−	21.5404i	0.507558	−	0.861617i
$626$		0			0
$627$	−11.3964	−	3.05367i	−0.455130	−	0.121952i
$628$		0			0
$629$	−0.850086			−0.0338952
$630$		0			0
$631$	−14.1379			−0.562819			−0.281410	−	0.959588i	$-0.590802\pi$
$631$	−14.1379			−0.562819			−0.281410	+	0.959588i	$0.590802\pi$
$632$		0			0
$633$	13.4852	+	3.61335i	0.535988	+	0.143618i
$634$		0			0
$635$	−9.64794	+	16.6266i	−0.382867	+	0.659807i
$636$		0			0
$637$	1.64711	−	27.6666i	0.0652609	−	1.09619i
$638$		0			0
$639$	4.66992	+	2.69618i	0.184739	+	0.106659i
$640$		0			0
$641$	−16.8927	−	29.2590i	−0.667222	−	1.15566i	−0.978678	−	0.205401i	$-0.934150\pi$
$641$	−16.8927	−	29.2590i	−0.667222	−	1.15566i	0.311456	−	0.950260i	$-0.399183\pi$
$642$		0			0
$643$	5.54153	−	5.54153i	0.218537	−	0.218537i	−0.589345	−	0.807882i	$-0.700614\pi$
$643$	5.54153	−	5.54153i	0.218537	−	0.218537i	0.807882	+	0.589345i	$0.200614\pi$
$644$		0			0
$645$	−13.8600	−	13.9208i	−0.545737	−	0.548130i
$646$		0			0
$647$	0.654586	−	2.44295i	0.0257344	−	0.0960422i	−0.951864	−	0.306520i	$-0.900835\pi$
$647$	0.654586	−	2.44295i	0.0257344	−	0.0960422i	0.977599	+	0.210478i	$0.0675020\pi$
$648$		0			0
$649$	3.76776	−	6.52596i	0.147898	−	0.256166i
$650$		0			0
$651$	4.70098	−	3.39001i	0.184246	−	0.132865i
$652$		0			0
$653$	9.22736	−	2.47246i	0.361094	−	0.0967550i	−0.0737104	−	0.997280i	$-0.523484\pi$
$653$	9.22736	−	2.47246i	0.361094	−	0.0967550i	0.434805	+	0.900525i	$0.356817\pi$
$654$		0			0
$655$	12.4503	+	3.36525i	0.486472	+	0.131491i
$656$		0			0
$657$	−11.0472	−	11.0472i	−0.430994	−	0.430994i
$658$		0			0
$659$			25.7019i			1.00120i	0.865678	+	0.500602i	$0.166888\pi$
$659$			25.7019i			1.00120i	−0.865678	+	0.500602i	$0.833112\pi$
$660$		0			0
$661$	−4.00560	+	2.31264i	−0.155800	+	0.0899511i	−0.575873	−	0.817539i	$-0.695338\pi$
$661$	−4.00560	+	2.31264i	−0.155800	+	0.0899511i	0.420073	+	0.907490i	$0.362004\pi$
$662$		0			0
$663$	0.325436	+	1.21454i	0.0126389	+	0.0471690i
$664$		0			0
$665$	1.58048	+	15.9155i	0.0612885	+	0.617177i
$666$		0			0
$667$	3.54700	+	13.2376i	0.137341	+	0.512562i
$668$		0			0
$669$	−24.2176	+	13.9820i	−0.936307	+	0.540577i
$670$		0			0
$671$			44.5668i			1.72048i
$672$		0			0
$673$	−19.3062	−	19.3062i	−0.744198	−	0.744198i	0.229185	−	0.973383i	$-0.426394\pi$
$673$	−19.3062	−	19.3062i	−0.744198	−	0.744198i	−0.973383	+	0.229185i	$0.926394\pi$
$674$		0			0
$675$	4.82392	+	1.31521i	0.185673	+	0.0506225i
$676$		0			0
$677$	11.4592	−	3.07047i	0.440411	−	0.118008i	−0.0317981	−	0.999494i	$-0.510123\pi$
$677$	11.4592	−	3.07047i	0.440411	−	0.118008i	0.472209	+	0.881487i	$0.343457\pi$
$678$		0			0
$679$	−26.9278	−	12.1038i	−1.03340	−	0.464503i
$680$		0			0
$681$	−4.73574	+	8.20255i	−0.181474	+	0.314322i
$682$		0			0
$683$	11.3685	−	42.4276i	0.435002	−	1.62345i	−0.306060	−	0.952012i	$-0.599011\pi$
$683$	11.3685	−	42.4276i	0.435002	−	1.62345i	0.741062	−	0.671437i	$-0.234322\pi$
$684$		0			0
$685$	42.4930	+	0.0929501i	1.62357	+	0.00355144i
$686$		0			0
$687$	−1.31199	+	1.31199i	−0.0500555	+	0.0500555i
$688$		0			0
$689$	−24.6833	−	42.7528i	−0.940359	−	1.62875i
$690$		0			0
$691$	7.97994	+	4.60722i	0.303571	+	0.175267i	0.644046	−	0.764987i	$-0.277254\pi$
$691$	7.97994	+	4.60722i	0.303571	+	0.175267i	−0.340475	+	0.940254i	$0.610588\pi$
$692$		0			0
$693$	−4.09965	−	10.7944i	−0.155733	−	0.410045i
$694$		0			0
$695$	−0.684766	+	0.181878i	−0.0259746	+	0.00689902i
$696$		0			0
$697$	0.808040	+	0.216514i	0.0306067	+	0.00820104i
$698$		0			0
$699$	−27.5366			−1.04153
$700$		0			0
$701$	16.6053			0.627174			0.313587	−	0.949559i	$-0.398469\pi$
$701$	16.6053			0.627174			0.313587	+	0.949559i	$0.398469\pi$
$702$		0			0
$703$	6.99005	+	1.87298i	0.263635	+	0.0706407i
$704$		0			0
$705$	−13.5840	+	3.60800i	−0.511604	+	0.135885i
$706$		0			0
$707$	−5.68228	+	6.96628i	−0.213704	+	0.261994i
$708$		0			0
$709$	−27.6385	−	15.9571i	−1.03799	−	0.599282i	−0.118725	−	0.992927i	$-0.537881\pi$
$709$	−27.6385	−	15.9571i	−1.03799	−	0.599282i	−0.919262	+	0.393645i	$0.871214\pi$
$710$		0			0
$711$	6.69839	+	11.6020i	0.251209	+	0.435107i
$712$		0			0
$713$	4.68876	−	4.68876i	0.175596	−	0.175596i
$714$		0			0
$715$	38.6383	+	0.0845183i	1.44499	+	0.00316081i
$716$		0			0
$717$	−3.72220	+	13.8914i	−0.139008	+	0.518785i
$718$		0			0
$719$	−8.63741	+	14.9604i	−0.322121	+	0.557930i	−0.980925	−	0.194384i	$-0.937729\pi$
$719$	−8.63741	+	14.9604i	−0.322121	+	0.557930i	0.658804	+	0.752314i	$0.271062\pi$
$720$		0			0
$721$	23.9714	+	33.2415i	0.892742	+	1.23798i
$722$		0			0
$723$	6.39738	−	1.71417i	0.237921	−	0.0637508i
$724$		0			0
$725$	11.2328	+	19.6539i	0.417177	+	0.729928i
$726$		0			0
$727$	−30.3766	−	30.3766i	−1.12661	−	1.12661i	−0.990725	−	0.135880i	$-0.956614\pi$
$727$	−30.3766	−	30.3766i	−1.12661	−	1.12661i	−0.135880	−	0.990725i	$-0.543386\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$	−2.41613	+	1.39495i	−0.0893637	+	0.0515941i
$732$		0			0
$733$	9.12882	+	34.0692i	0.337181	+	1.25838i	0.901486	+	0.432809i	$0.142478\pi$
$733$	9.12882	+	34.0692i	0.337181	+	1.25838i	−0.564305	+	0.825566i	$0.690856\pi$
$734$		0			0
$735$	−11.7289	+	10.3650i	−0.432626	+	0.382319i
$736$		0			0
$737$	8.63155	+	32.2134i	0.317947	+	1.18660i
$738$		0			0
$739$	−25.1311	+	14.5095i	−0.924463	+	0.533739i	−0.885056	−	0.465484i	$-0.845880\pi$
$739$	−25.1311	+	14.5095i	−0.924463	+	0.533739i	−0.0394067	+	0.999223i	$0.512547\pi$
$740$		0			0
$741$		−	10.7039i		−	0.393218i
$742$		0			0
$743$	6.98196	+	6.98196i	0.256143	+	0.256143i	0.823483	−	0.567340i	$-0.192028\pi$
$743$	6.98196	+	6.98196i	0.256143	+	0.256143i	−0.567340	+	0.823483i	$0.692028\pi$
$744$		0			0
$745$	−17.7437	−	4.79602i	−0.650077	−	0.175713i
$746$		0			0
$747$	−15.6493	+	4.19321i	−0.572576	+	0.153421i
$748$		0			0
$749$	−21.1085	−	29.2715i	−0.771288	−	1.06956i
$750$		0			0
$751$	−2.61368	+	4.52703i	−0.0953745	+	0.165194i	−0.909765	−	0.415124i	$-0.863738\pi$
$751$	−2.61368	+	4.52703i	−0.0953745	+	0.165194i	0.814390	+	0.580317i	$0.197072\pi$
$752$		0			0
$753$	0.830342	−	3.09888i	0.0302594	−	0.112929i
$754$		0			0
$755$	−14.7760	−	14.8408i	−0.537753	−	0.540111i
$756$		0			0
$757$	−33.4817	+	33.4817i	−1.21691	+	1.21691i	−0.248207	+	0.968707i	$0.579841\pi$
$757$	−33.4817	+	33.4817i	−1.21691	+	1.21691i	−0.968707	+	0.248207i	$0.920159\pi$
$758$		0			0
$759$	−6.60522	−	11.4406i	−0.239754	−	0.415266i
$760$		0			0
$761$	−45.5767	−	26.3137i	−1.65215	−	0.953872i	−0.976184	−	0.216942i	$-0.930392\pi$
$761$	−45.5767	−	26.3137i	−1.65215	−	0.953872i	−0.675970	−	0.736929i	$-0.736275\pi$
$762$		0			0
$763$	0.878363	−	1.07684i	0.0317989	−	0.0389844i
$764$		0			0
$765$	0.356402	−	0.614199i	0.0128857	−	0.0222064i
$766$		0			0
$767$	6.60351	+	1.76940i	0.238439	+	0.0638895i
$768$		0			0
$769$	−34.1069			−1.22993			−0.614963	−	0.788556i	$-0.710829\pi$
$769$	−34.1069			−1.22993			−0.614963	+	0.788556i	$0.710829\pi$
$770$		0			0
$771$	−15.7151			−0.565965
$772$		0			0
$773$	−2.25082	−	0.603106i	−0.0809564	−	0.0216922i	0.218113	−	0.975923i	$-0.430010\pi$
$773$	−2.25082	−	0.603106i	−0.0809564	−	0.0216922i	−0.299070	+	0.954231i	$0.596676\pi$
$774$		0			0
$775$	5.51797	−	9.46157i	0.198211	−	0.339869i
$776$		0			0
$777$	2.51453	+	6.62078i	0.0902084	+	0.237519i
$778$		0			0
$779$	−6.16727	−	3.56067i	−0.220965	−	0.127574i
$780$		0			0
$781$	−11.7668	−	20.3806i	−0.421048	−	0.729276i
$782$		0			0
$783$	−3.20141	+	3.20141i	−0.114409	+	0.114409i
$784$		0			0
$785$	−0.0749060	+	34.2439i	−0.00267351	+	1.22222i
$786$		0			0
$787$	6.20186	−	23.1456i	0.221072	−	0.825053i	−0.762868	−	0.646554i	$-0.776209\pi$
$787$	6.20186	−	23.1456i	0.221072	−	0.825053i	0.983940	−	0.178499i	$-0.0571240\pi$
$788$		0			0
$789$	7.27828	−	12.6064i	0.259114	−	0.448798i
$790$		0			0
$791$	41.9605	+	18.8609i	1.49194	+	0.670617i
$792$		0			0
$793$	−39.0546	+	10.4647i	−1.38687	+	0.371611i
$794$		0			0
$795$	−7.27477	+	26.9142i	−0.258009	+	0.954547i
$796$		0			0
$797$	−4.27731	−	4.27731i	−0.151510	−	0.151510i	0.627282	−	0.778792i	$-0.284167\pi$
$797$	−4.27731	−	4.27731i	−0.151510	−	0.151510i	−0.778792	+	0.627282i	$0.784167\pi$
$798$		0			0
$799$			1.99614i			0.0706182i
$800$		0			0
$801$	−5.25387	+	3.03332i	−0.185636	+	0.107177i
$802$		0			0
$803$	17.6471	+	65.8598i	0.622752	+	2.32414i
$804$		0			0
$805$	−11.3494	+	13.8521i	−0.400015	+	0.488221i
$806$		0			0
$807$	−3.15760	−	11.7843i	−0.111153	−	0.414828i
$808$		0			0
$809$	−24.3656	+	14.0675i	−0.856649	+	0.494587i	−0.862889	−	0.505394i	$-0.831347\pi$
$809$	−24.3656	+	14.0675i	−0.856649	+	0.494587i	0.00623939	+	0.999981i	$0.498014\pi$
$810$		0			0
$811$		−	14.6408i		−	0.514108i	−0.966397	−	0.257054i	$-0.917248\pi$
$811$		−	14.6408i		−	0.514108i	0.966397	−	0.257054i	$-0.0827518\pi$
$812$		0			0
$813$	−9.05021	−	9.05021i	−0.317405	−	0.317405i
$814$		0			0
$815$	35.6812	−	20.4966i	1.24986	−	0.717966i
$816$		0			0
$817$	22.9407	−	6.14694i	0.802593	−	0.215054i
$818$		0			0
$819$	8.49667	−	6.12720i	0.296898	−	0.214102i
$820$		0			0
$821$	−19.0342	+	32.9682i	−0.664298	+	1.15060i	0.315178	+	0.949033i	$0.397936\pi$
$821$	−19.0342	+	32.9682i	−0.664298	+	1.15060i	−0.979475	+	0.201565i	$0.935397\pi$
$822$		0			0
$823$	−7.30051	+	27.2459i	−0.254480	+	0.949731i	0.713900	+	0.700248i	$0.246927\pi$
$823$	−7.30051	+	27.2459i	−0.254480	+	0.949731i	−0.968379	+	0.249483i	$0.919739\pi$
$824$		0			0
$825$	−15.3623	−	15.4973i	−0.534845	−	0.539546i
$826$		0			0
$827$	−13.2575	+	13.2575i	−0.461008	+	0.461008i	−0.898986	−	0.437977i	$-0.855695\pi$
$827$	−13.2575	+	13.2575i	−0.461008	+	0.461008i	0.437977	+	0.898986i	$0.355695\pi$
$828$		0			0
$829$	23.4677	+	40.6472i	0.815067	+	1.41174i	0.909280	+	0.416185i	$0.136633\pi$
$829$	23.4677	+	40.6472i	0.815067	+	1.41174i	−0.0942131	+	0.995552i	$0.530033\pi$
$830$		0			0
$831$	−18.1373	−	10.4715i	−0.629174	−	0.363254i
$832$		0			0
$833$	0.995129	+	1.98784i	0.0344792	+	0.0688745i
$834$		0			0
$835$	−20.4564	−	11.8703i	−0.707924	−	0.410787i
$836$		0			0
$837$	2.11597	+	0.566971i	0.0731385	+	0.0195974i
$838$		0			0
$839$	−37.4760			−1.29382			−0.646908	−	0.762568i	$-0.723938\pi$
$839$	−37.4760			−1.29382			−0.646908	+	0.762568i	$0.723938\pi$
$840$		0			0
$841$	8.50190			0.293169
$842$		0			0
$843$	23.0728	+	6.18235i	0.794671	+	0.212931i
$844$		0			0
$845$	1.53638	+	5.78444i	0.0528532	+	0.198991i
$846$		0			0
$847$	−3.40504	+	21.0152i	−0.116999	+	0.722090i
$848$		0			0
$849$	5.59806	+	3.23204i	0.192125	+	0.110923i
$850$		0			0
$851$	4.05133	+	7.01711i	0.138878	+	0.240544i
$852$		0			0
$853$	3.02250	−	3.02250i	0.103488	−	0.103488i	−0.653467	−	0.756955i	$-0.726686\pi$
$853$	3.02250	−	3.02250i	0.103488	−	0.103488i	0.756955	+	0.653467i	$0.226686\pi$
$854$		0			0
$855$	−4.28386	+	4.26516i	−0.146505	+	0.145865i
$856$		0			0
$857$	3.41348	−	12.7393i	0.116602	−	0.435166i	−0.882799	−	0.469750i	$-0.844344\pi$
$857$	3.41348	−	12.7393i	0.116602	−	0.435166i	0.999402	+	0.0345842i	$0.0110107\pi$
$858$		0			0
$859$	3.65300	−	6.32718i	0.124639	−	0.215881i	−0.796953	−	0.604041i	$-0.793556\pi$
$859$	3.65300	−	6.32718i	0.124639	−	0.215881i	0.921592	+	0.388161i	$0.126889\pi$
$860$		0			0
$861$	−0.703874	−	6.93375i	−0.0239879	−	0.236301i
$862$		0			0
$863$	15.8416	−	4.24474i	0.539254	−	0.144493i	0.0210958	−	0.999777i	$-0.493284\pi$
$863$	15.8416	−	4.24474i	0.539254	−	0.144493i	0.518158	+	0.855285i	$0.326618\pi$
$864$		0			0
$865$	−6.10715	−	10.6315i	−0.207649	−	0.361483i
$866$		0			0
$867$	11.9495	+	11.9495i	0.405826	+	0.405826i
$868$		0			0
$869$		−	58.4668i		−	1.98335i
$870$		0			0
$871$	−26.2024	+	15.1279i	−0.887833	+	0.512591i
$872$		0			0
$873$	−2.88808	−	10.7785i	−0.0977466	−	0.364795i
$874$		0			0
$875$	−12.3041	+	26.9000i	−0.415956	+	0.909385i
$876$		0			0
$877$	1.28232	+	4.78568i	0.0433008	+	0.161601i	0.984191	−	0.177111i	$-0.0566753\pi$
$877$	1.28232	+	4.78568i	0.0433008	+	0.161601i	−0.940890	+	0.338712i	$0.890009\pi$
$878$		0			0
$879$	8.47113	−	4.89081i	0.285724	−	0.164963i
$880$		0			0
$881$		−	31.5578i		−	1.06321i	−0.846993	−	0.531605i	$-0.821589\pi$
$881$		−	31.5578i		−	1.06321i	0.846993	−	0.531605i	$-0.178411\pi$
$882$		0			0
$883$	−4.81686	−	4.81686i	−0.162100	−	0.162100i	0.621396	−	0.783496i	$-0.286566\pi$
$883$	−4.81686	−	4.81686i	−0.162100	−	0.162100i	−0.783496	+	0.621396i	$0.786566\pi$
$884$		0			0
$885$	−1.92314	−	3.34787i	−0.0646456	−	0.112537i
$886$		0			0
$887$	16.1074	−	4.31598i	0.540835	−	0.144916i	0.0219480	−	0.999759i	$-0.493013\pi$
$887$	16.1074	−	4.31598i	0.540835	−	0.144916i	0.518887	+	0.854843i	$0.326346\pi$
$888$		0			0
$889$	9.32500	−	20.7457i	0.312751	−	0.695787i
$890$		0			0
$891$	2.18212	−	3.77954i	0.0731037	−	0.126619i
$892$		0			0
$893$	4.39805	−	16.4137i	0.147175	−	0.549265i
$894$		0			0
$895$	16.3137	−	16.2425i	0.545308	−	0.542927i
$896$		0			0
$897$	8.47460	−	8.47460i	0.282959	−	0.282959i
$898$		0			0
$899$	4.95897	+	8.58919i	0.165391	+	0.286466i
$900$		0			0
$901$	3.42912	+	1.97980i	0.114240	+	0.0659567i
$902$		0			0
$903$	18.0112	+	14.6914i	0.599376	+	0.488900i
$904$		0			0
$905$	−7.99267	−	30.0922i	−0.265685	−	1.00030i
$906$		0			0
$907$	11.2030	+	3.00183i	0.371989	+	0.0996741i	0.439970	−	0.898013i	$-0.354989\pi$
$907$	11.2030	+	3.00183i	0.371989	+	0.0996741i	−0.0679811	+	0.997687i	$0.521656\pi$
$908$		0			0
$909$	−3.39784			−0.112699
$910$		0			0
$911$	−18.0104			−0.596712			−0.298356	−	0.954455i	$-0.596438\pi$
$911$	−18.0104			−0.596712			−0.298356	+	0.954455i	$0.596438\pi$
$912$		0			0
$913$	68.2971	+	18.3001i	2.26030	+	0.605646i
$914$		0			0
$915$	19.7501	+	11.4604i	0.652918	+	0.378869i
$916$		0			0
$917$	−15.0636	−	2.44071i	−0.497443	−	0.0805995i
$918$		0			0
$919$	−28.7287	−	16.5865i	−0.947672	−	0.547139i	−0.0553148	−	0.998469i	$-0.517616\pi$
$919$	−28.7287	−	16.5865i	−0.947672	−	0.547139i	−0.892357	+	0.451330i	$0.850950\pi$
$920$		0			0
$921$	−5.17232	−	8.95872i	−0.170434	−	0.295200i
$922$		0			0
$923$	15.0969	−	15.0969i	0.496922	−	0.496922i
$924$		0			0
$925$	9.42250	+	9.50531i	0.309810	+	0.312533i
$926$		0			0
$927$	−4.00916	+	14.9624i	−0.131678	+	0.491429i
$928$		0			0
$929$	28.5132	−	49.3864i	0.935489	−	1.62032i	0.161730	−	0.986835i	$-0.448293\pi$
$929$	28.5132	−	49.3864i	0.935489	−	1.62032i	0.773759	−	0.633480i	$-0.218374\pi$
$930$		0			0
$931$	−3.80294	−	18.5380i	−0.124636	−	0.607560i
$932$		0			0
$933$	−21.0459	+	5.63924i	−0.689013	+	0.184620i
$934$		0			0
$935$	−2.68729	+	1.54368i	−0.0878838	+	0.0504838i
$936$		0			0
$937$	18.4132	+	18.4132i	0.601534	+	0.601534i	0.940719	−	0.339186i	$-0.110152\pi$
$937$	18.4132	+	18.4132i	0.601534	+	0.601534i	−0.339186	+	0.940719i	$0.610152\pi$
$938$		0			0
$939$		−	23.5645i		−	0.769000i
$940$		0			0
$941$	2.15419	−	1.24372i	0.0702245	−	0.0405441i	−0.464477	−	0.885585i	$-0.653758\pi$
$941$	2.15419	−	1.24372i	0.0702245	−	0.0405441i	0.534701	+	0.845041i	$0.320424\pi$
$942$		0			0
$943$	−2.06372	−	7.70189i	−0.0672038	−	0.250808i
$944$		0			0
$945$	−5.83783	−	0.959001i	−0.189905	−	0.0311963i
$946$		0			0
$947$	13.1596	+	49.1124i	0.427630	+	1.59594i	0.758111	+	0.652126i	$0.226123\pi$
$947$	13.1596	+	49.1124i	0.427630	+	1.59594i	−0.330481	+	0.943813i	$0.607211\pi$
$948$		0			0
$949$	−53.5704	+	30.9289i	−1.73897	+	1.00399i
$950$		0			0
$951$			22.0843i			0.716131i
$952$		0			0
$953$	−9.31188	−	9.31188i	−0.301641	−	0.301641i	0.540014	−	0.841656i	$-0.318419\pi$
$953$	−9.31188	−	9.31188i	−0.301641	−	0.301641i	−0.841656	+	0.540014i	$0.818419\pi$
$954$		0			0
$955$	−6.23837	+	23.0799i	−0.201869	+	0.746847i
$956$		0			0
$957$	19.0857	−	5.11401i	0.616954	−	0.165312i
$958$		0			0
$959$	−50.0214	+	5.07788i	−1.61527	+	0.163973i
$960$		0			0
$961$	−13.1006	+	22.6909i	−0.422601	+	0.731966i
$962$		0			0
$963$	3.53035	−	13.1754i	0.113764	−	0.424572i
$964$		0			0
$965$	0.0210915	−	9.64216i	0.000678959	−	0.310392i
$966$		0			0
$967$	18.4525	−	18.4525i	0.593394	−	0.593394i	−0.345153	−	0.938547i	$-0.612173\pi$
$967$	18.4525	−	18.4525i	0.593394	−	0.593394i	0.938547	+	0.345153i	$0.112173\pi$
$968$		0			0
$969$	0.429270	+	0.743518i	0.0137901	+	0.0238852i
$970$		0			0
$971$	29.7627	+	17.1835i	0.955131	+	0.551445i	0.894671	−	0.446725i	$-0.147410\pi$
$971$	29.7627	+	17.1835i	0.955131	+	0.551445i	0.0604600	+	0.998171i	$0.480743\pi$
$972$		0			0
$973$	0.783700	−	0.297644i	0.0251243	−	0.00954204i
$974$		0			0
$975$	9.97332	−	17.1011i	0.319402	−	0.547673i
$976$		0			0
$977$	−17.3322	−	4.64414i	−0.554505	−	0.148579i	−0.0293246	−	0.999570i	$-0.509336\pi$
$977$	−17.3322	−	4.64414i	−0.554505	−	0.148579i	−0.525181	+	0.850991i	$0.676002\pi$
$978$		0			0
$979$	26.4763			0.846186
$980$		0			0
$981$	0.525237			0.0167695
$982$		0			0
$983$	−28.1996	−	7.55607i	−0.899428	−	0.241001i	−0.220658	−	0.975351i	$-0.570820\pi$
$983$	−28.1996	−	7.55607i	−0.899428	−	0.241001i	−0.678771	+	0.734350i	$0.737487\pi$
$984$		0			0
$985$	1.76937	−	3.04921i	0.0563768	−	0.0971560i
$986$		0			0
$987$	15.5466	−	5.90452i	0.494855	−	0.187943i
$988$		0			0
$989$	23.0295	+	13.2961i	0.732295	+	0.422791i
$990$		0			0
$991$	17.3733	+	30.0914i	0.551880	+	0.955885i	0.998139	+	0.0609809i	$0.0194229\pi$
$991$	17.3733	+	30.0914i	0.551880	+	0.955885i	−0.446258	+	0.894904i	$0.647244\pi$
$992$		0			0
$993$	0.623510	−	0.623510i	0.0197865	−	0.0197865i
$994$		0			0
$995$	−37.0798	−	37.2423i	−1.17551	−	1.18066i
$996$		0			0
$997$	−8.16415	+	30.4690i	−0.258561	+	0.964964i	0.707513	+	0.706700i	$0.249817\pi$
$997$	−8.16415	+	30.4690i	−0.258561	+	0.964964i	−0.966074	+	0.258264i	$0.916850\pi$
$998$		0			0
$999$	−1.33841	+	2.31820i	−0.0423454	+	0.0733445i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.dd.a.313.7	✓	48
5.2	odd	4		840.2.dd.b.817.9	yes	48
7.3	odd	6		840.2.dd.b.73.9	yes	48
35.17	even	12	inner	840.2.dd.a.577.7	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.dd.a.313.7	✓	48	1.1	even	1	trivial
840.2.dd.a.577.7	yes	48	35.17	even	12	inner
840.2.dd.b.73.9	yes	48	7.3	odd	6
840.2.dd.b.817.9	yes	48	5.2	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:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.dd (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.965926 + 0.258819i) q^{3} +(-2.16114 + 0.574011i) q^{5} +(2.47337 - 0.939373i) q^{7} +(0.866025 + 0.500000i) q^{9} +(-2.18212 - 3.77954i) q^{11} +(2.79969 - 2.79969i) q^{13} +(-2.23606 - 0.00489121i) q^{15} +(-0.0821940 + 0.306752i) q^{17} +(1.35172 - 2.34125i) q^{19} +(2.63222 - 0.267208i) q^{21} +(2.92383 - 0.783438i) q^{23} +(4.34102 - 2.48103i) q^{25} +(0.707107 + 0.707107i) q^{27} +4.52748i q^{29} +(1.89712 - 1.09530i) q^{31} +(-1.12955 - 4.21553i) q^{33} +(-4.80609 + 3.44986i) q^{35} +(0.692812 + 2.58561i) q^{37} +(3.42891 - 1.97968i) q^{39} -2.63418i q^{41} +(6.21198 + 6.21198i) q^{43} +(-2.15860 - 0.583460i) q^{45} +(6.07142 - 1.62683i) q^{47} +(5.23516 - 4.64684i) q^{49} +(-0.158787 + 0.275026i) q^{51} +(3.22704 - 12.0435i) q^{53} +(6.88536 + 6.91554i) q^{55} +(1.91162 - 1.91162i) q^{57} +(0.863327 + 1.49533i) q^{59} +(-8.84369 - 5.10591i) q^{61} +(2.61169 + 0.423166i) q^{63} +(-4.44347 + 7.65758i) q^{65} +(-7.38122 - 1.97779i) q^{67} +3.02697 q^{69} +5.39235 q^{71} +(-15.0908 - 4.04357i) q^{73} +(4.83524 - 1.27295i) q^{75} +(-8.94759 - 7.29840i) q^{77} +(11.6020 + 6.69839i) q^{79} +(0.500000 + 0.866025i) q^{81} +(-11.4560 + 11.4560i) q^{83} +(0.00155332 - 0.710113i) q^{85} +(-1.17180 + 4.37321i) q^{87} +(-3.03332 + 5.25387i) q^{89} +(4.29473 - 9.55465i) q^{91} +(2.11597 - 0.566971i) q^{93} +(-1.57735 + 5.83566i) q^{95} +(-7.89037 - 7.89037i) q^{97} -4.36424i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{11} + 16 q^{13} - 4 q^{15} - 4 q^{17} + 8 q^{19} - 24 q^{23} + 28 q^{25} + 12 q^{33} - 4 q^{37} - 12 q^{39} + 16 q^{43} - 4 q^{45} + 12 q^{47} - 12 q^{49} + 20 q^{53} - 56 q^{55} + 8 q^{57}+ \cdots + 44 q^{97}+O(q^{100}) \) 48 * q - 4 * q^11 + 16 * q^13 - 4 * q^15 - 4 * q^17 + 8 * q^19 - 24 * q^23 + 28 * q^25 + 12 * q^33 - 4 * q^37 - 12 * q^39 + 16 * q^43 - 4 * q^45 + 12 * q^47 - 12 * q^49 + 20 * q^53 - 56 * q^55 + 8 * q^57 + 8 * q^59 + 8 * q^63 + 12 * q^65 - 52 * q^67 - 8 * q^69 + 16 * q^71 - 4 * q^73 + 24 * q^75 + 8 * q^77 + 108 * q^79 + 24 * q^81 + 64 * q^83 + 32 * q^85 + 20 * q^87 + 64 * q^89 + 48 * q^91 - 24 * q^93 + 36 * q^95 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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