LMFDB - Embedded newform 768.2.s.a.431.23

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,2,Mod(47,768)] 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("768.47"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 11, 8])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.s (of order $16$, degree $8$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$240$
Relative dimension:	$30$ over $\Q(\zeta_{16})$
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			431.23
Character	$\chi$	$=$	768.431
Dual form			768.2.s.a.335.23

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.25794 - 1.19062i) q^{3} +(1.50066 - 2.24589i) q^{5} +(0.0860024 + 0.207628i) q^{7} +(0.164844 - 2.99547i) q^{9} +(-3.68379 + 0.732751i) q^{11} +(-1.53272 + 1.02413i) q^{13} +(-0.786264 - 4.61192i) q^{15} +(4.44652 - 4.44652i) q^{17} +(3.66999 - 2.45221i) q^{19} +(0.355392 + 0.158788i) q^{21} +(1.88615 - 4.55357i) q^{23} +(-0.878641 - 2.12123i) q^{25} +(-3.35910 - 3.96440i) q^{27} +(-1.40435 + 7.06016i) q^{29} -6.44241 q^{31} +(-3.76157 + 5.30775i) q^{33} +(0.595370 + 0.118426i) q^{35} +(-5.00124 + 7.48488i) q^{37} +(-0.708723 + 3.11318i) q^{39} +(4.55750 + 1.88778i) q^{41} +(1.54503 - 0.307325i) q^{43} +(-6.48012 - 4.86539i) q^{45} +(4.26705 + 4.26705i) q^{47} +(4.91403 - 4.91403i) q^{49} +(0.299352 - 10.8876i) q^{51} +(-1.02142 - 5.13504i) q^{53} +(-3.88242 + 9.37299i) q^{55} +(1.69699 - 7.45431i) q^{57} +(-7.47264 - 4.99306i) q^{59} +(0.815852 - 4.10157i) q^{61} +(0.636120 - 0.223391i) q^{63} +4.97918i q^{65} +(8.05661 + 1.60256i) q^{67} +(-3.04891 - 7.97383i) q^{69} +(-2.52914 + 1.04761i) q^{71} +(5.90887 + 2.44753i) q^{73} +(-3.63086 - 1.62225i) q^{75} +(-0.468954 - 0.701839i) q^{77} +(8.79908 + 8.79908i) q^{79} +(-8.94565 - 0.987568i) q^{81} +(4.93898 + 7.39171i) q^{83} +(-3.31370 - 16.6591i) q^{85} +(6.63938 + 10.5533i) q^{87} +(5.44855 - 2.25687i) q^{89} +(-0.344455 - 0.230158i) q^{91} +(-8.10419 + 7.67047i) q^{93} -11.9223i q^{95} +16.3503i q^{97} +(1.58768 + 11.1555i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 240 q + 8 q^{3} + 16 q^{7} - 8 q^{9} - 16 q^{13} + 8 q^{15} + 16 q^{19} - 8 q^{21} - 16 q^{25} + 8 q^{27} + 32 q^{31} - 16 q^{37} + 8 q^{39} + 16 q^{43} - 8 q^{45} - 16 q^{49} + 8 q^{51} + 80 q^{55} - 8 q^{57}+ \cdots + 8 q^{99}+O(q^{100}) $ 240 * q + 8 * q^3 + 16 * q^7 - 8 * q^9 - 16 * q^13 + 8 * q^15 + 16 * q^19 - 8 * q^21 - 16 * q^25 + 8 * q^27 + 32 * q^31 - 16 * q^37 + 8 * q^39 + 16 * q^43 - 8 * q^45 - 16 * q^49 + 8 * q^51 + 80 * q^55 - 8 * q^57 - 16 * q^61 + 144 * q^67 - 8 * q^69 - 16 * q^73 + 8 * q^75 + 48 * q^79 - 8 * q^81 - 16 * q^85 + 8 * q^87 + 16 * q^91 - 32 * q^93 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{16}\right)$

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
$3$	1.25794	−	1.19062i	0.726274	−	0.687405i
$3$	1.25794	−	1.19062i	0.726274	−	0.687405i
$4$		0			0
$5$	1.50066	−	2.24589i	0.671114	−	1.00439i	−0.327120	−	0.944983i	$-0.606078\pi$
$5$	1.50066	−	2.24589i	0.671114	−	1.00439i	0.998234	−	0.0594102i	$-0.0189220\pi$
$6$		0			0
$7$	0.0860024	+	0.207628i	0.0325058	+	0.0784760i	0.939299	−	0.343100i	$-0.111477\pi$
$7$	0.0860024	+	0.207628i	0.0325058	+	0.0784760i	−0.906793	+	0.421576i	$0.861477\pi$
$8$		0			0
$9$	0.164844	−	2.99547i	0.0549479	−	0.998489i
$10$		0			0
$11$	−3.68379	+	0.732751i	−1.11070	+	0.220933i	−0.716154	−	0.697942i	$-0.754099\pi$
$11$	−3.68379	+	0.732751i	−1.11070	+	0.220933i	−0.394549	+	0.918875i	$0.629099\pi$
$12$		0			0
$13$	−1.53272	+	1.02413i	−0.425099	+	0.284042i	−0.749661	−	0.661822i	$-0.769783\pi$
$13$	−1.53272	+	1.02413i	−0.425099	+	0.284042i	0.324561	+	0.945865i	$0.394783\pi$
$14$		0			0
$15$	−0.786264	−	4.61192i	−0.203012	−	1.19079i
$16$		0			0
$17$	4.44652	−	4.44652i	1.07844	−	1.07844i	0.0817905	−	0.996650i	$-0.473936\pi$
$17$	4.44652	−	4.44652i	1.07844	−	1.07844i	0.996650	−	0.0817905i	$-0.0260639\pi$
$18$		0			0
$19$	3.66999	−	2.45221i	0.841953	−	0.562575i	−0.0581221	−	0.998309i	$-0.518511\pi$
$19$	3.66999	−	2.45221i	0.841953	−	0.562575i	0.900075	+	0.435734i	$0.143511\pi$
$20$		0			0
$21$	0.355392	+	0.158788i	0.0775530	+	0.0346504i
$22$		0			0
$23$	1.88615	−	4.55357i	0.393290	−	0.949486i	−0.595928	−	0.803038i	$-0.703216\pi$
$23$	1.88615	−	4.55357i	0.393290	−	0.949486i	0.989218	−	0.146448i	$-0.0467842\pi$
$24$		0			0
$25$	−0.878641	−	2.12123i	−0.175728	−	0.424245i
$26$		0			0
$27$	−3.35910	−	3.96440i	−0.646460	−	0.762948i
$28$		0			0
$29$	−1.40435	+	7.06016i	−0.260782	+	1.31104i	0.599153	+	0.800635i	$0.295504\pi$
$29$	−1.40435	+	7.06016i	−0.260782	+	1.31104i	−0.859935	+	0.510404i	$0.829496\pi$
$30$		0			0
$31$	−6.44241			−1.15709			−0.578546	−	0.815650i	$-0.696380\pi$
$31$	−6.44241			−1.15709			−0.578546	+	0.815650i	$0.696380\pi$
$32$		0			0
$33$	−3.76157	+	5.30775i	−0.654805	+	0.923961i
$34$		0			0
$35$	0.595370	+	0.118426i	0.100636	+	0.0200177i
$36$		0			0
$37$	−5.00124	+	7.48488i	−0.822198	+	1.23051i	0.148191	+	0.988959i	$0.452655\pi$
$37$	−5.00124	+	7.48488i	−0.822198	+	1.23051i	−0.970389	+	0.241548i	$0.922345\pi$
$38$		0			0
$39$	−0.708723	+	3.11318i	−0.113486	+	0.498508i
$40$		0			0
$41$	4.55750	+	1.88778i	0.711762	+	0.294821i	0.709033	−	0.705175i	$-0.249132\pi$
$41$	4.55750	+	1.88778i	0.711762	+	0.294821i	0.00272832	+	0.999996i	$0.499132\pi$
$42$		0			0
$43$	1.54503	−	0.307325i	0.235615	−	0.0468667i	−0.0758705	−	0.997118i	$-0.524174\pi$
$43$	1.54503	−	0.307325i	0.235615	−	0.0468667i	0.311485	+	0.950251i	$0.399174\pi$
$44$		0			0
$45$	−6.48012	−	4.86539i	−0.965999	−	0.725289i
$46$		0			0
$47$	4.26705	+	4.26705i	0.622413	+	0.622413i	0.946148	−	0.323735i	$-0.104938\pi$
$47$	4.26705	+	4.26705i	0.622413	+	0.622413i	−0.323735	+	0.946148i	$0.604938\pi$
$48$		0			0
$49$	4.91403	−	4.91403i	0.702005	−	0.702005i
$50$		0			0
$51$	0.299352	−	10.8876i	0.0419176	−	1.52457i
$52$		0			0
$53$	−1.02142	−	5.13504i	−0.140303	−	0.705351i	−0.985334	−	0.170635i	$-0.945418\pi$
$53$	−1.02142	−	5.13504i	−0.140303	−	0.705351i	0.845031	−	0.534717i	$-0.179582\pi$
$54$		0			0
$55$	−3.88242	+	9.37299i	−0.523506	+	1.26385i
$56$		0			0
$57$	1.69699	−	7.45431i	0.224772	−	0.987347i
$58$		0			0
$59$	−7.47264	−	4.99306i	−0.972854	−	0.650040i	−0.0358495	−	0.999357i	$-0.511414\pi$
$59$	−7.47264	−	4.99306i	−0.972854	−	0.650040i	−0.937005	+	0.349317i	$0.886414\pi$
$60$		0			0
$61$	0.815852	−	4.10157i	0.104459	−	0.525152i	−0.892754	−	0.450545i	$-0.851230\pi$
$61$	0.815852	−	4.10157i	0.104459	−	0.525152i	0.997213	−	0.0746070i	$-0.0237702\pi$
$62$		0			0
$63$	0.636120	−	0.223391i	0.0801436	−	0.0281446i
$64$		0			0
$65$			4.97918i			0.617592i
$66$		0			0
$67$	8.05661	+	1.60256i	0.984271	+	0.195784i	0.660888	−	0.750485i	$-0.270180\pi$
$67$	8.05661	+	1.60256i	0.984271	+	0.195784i	0.323383	+	0.946268i	$0.395180\pi$
$68$		0			0
$69$	−3.04891	−	7.97383i	−0.367045	−	0.959936i
$70$		0			0
$71$	−2.52914	+	1.04761i	−0.300154	+	0.124328i	−0.527678	−	0.849445i	$-0.676937\pi$
$71$	−2.52914	+	1.04761i	−0.300154	+	0.124328i	0.227524	+	0.973773i	$0.426937\pi$
$72$		0			0
$73$	5.90887	+	2.44753i	0.691581	+	0.286462i	0.700659	−	0.713497i	$-0.252890\pi$
$73$	5.90887	+	2.44753i	0.691581	+	0.286462i	−0.00907776	+	0.999959i	$0.502890\pi$
$74$		0			0
$75$	−3.63086	−	1.62225i	−0.419255	−	0.187322i
$76$		0			0
$77$	−0.468954	−	0.701839i	−0.0534423	−	0.0799820i
$78$		0			0
$79$	8.79908	+	8.79908i	0.989974	+	0.989974i	0.999950	−	0.00997611i	$-0.00317555\pi$
$79$	8.79908	+	8.79908i	0.989974	+	0.989974i	−0.00997611	+	0.999950i	$0.503176\pi$
$80$		0			0
$81$	−8.94565	−	0.987568i	−0.993961	−	0.109730i
$82$		0			0
$83$	4.93898	+	7.39171i	0.542123	+	0.811345i	0.996852	−	0.0792850i	$-0.0252637\pi$
$83$	4.93898	+	7.39171i	0.542123	+	0.811345i	−0.454729	+	0.890630i	$0.650264\pi$
$84$		0			0
$85$	−3.31370	−	16.6591i	−0.359422	−	1.80693i
$86$		0			0
$87$	6.63938	+	10.5533i	0.711816	+	1.13144i
$88$		0			0
$89$	5.44855	−	2.25687i	0.577546	−	0.239227i	−0.0747368	−	0.997203i	$-0.523812\pi$
$89$	5.44855	−	2.25687i	0.577546	−	0.239227i	0.652282	+	0.757976i	$0.273812\pi$
$90$		0			0
$91$	−0.344455	−	0.230158i	−0.0361087	−	0.0241271i
$92$		0			0
$93$	−8.10419	+	7.67047i	−0.840365	+	0.795391i
$94$		0			0
$95$		−	11.9223i		−	1.22320i
$96$		0			0
$97$			16.3503i			1.66012i	0.557671	+	0.830062i	$0.311695\pi$
$97$			16.3503i			1.66012i	−0.557671	+	0.830062i	$0.688305\pi$
$98$		0			0
$99$	1.58768	+	11.1555i	0.159568	+	1.12117i
$100$		0			0
$101$	−6.96357	−	4.65291i	−0.692901	−	0.462982i	0.158594	−	0.987344i	$-0.449304\pi$
$101$	−6.96357	−	4.65291i	−0.692901	−	0.462982i	−0.851495	+	0.524362i	$0.824304\pi$
$102$		0			0
$103$	3.52118	−	1.45852i	0.346952	−	0.143712i	−0.202401	−	0.979303i	$-0.564874\pi$
$103$	3.52118	−	1.45852i	0.346952	−	0.143712i	0.549353	+	0.835591i	$0.314874\pi$
$104$		0			0
$105$	0.889943	−	0.559886i	0.0868495	−	0.0546393i
$106$		0			0
$107$	−2.68194	−	13.4830i	−0.259273	−	1.30345i	−0.862570	−	0.505937i	$-0.831147\pi$
$107$	−2.68194	−	13.4830i	−0.259273	−	1.30345i	0.603297	−	0.797516i	$-0.293853\pi$
$108$		0			0
$109$	8.00490	+	11.9802i	0.766730	+	1.14749i	0.985160	+	0.171640i	$0.0549066\pi$
$109$	8.00490	+	11.9802i	0.766730	+	1.14749i	−0.218429	+	0.975853i	$0.570093\pi$
$110$		0			0
$111$	2.62038	+	15.3701i	0.248715	+	1.45887i
$112$		0			0
$113$	−6.18271	−	6.18271i	−0.581620	−	0.581620i	0.353728	−	0.935348i	$-0.384914\pi$
$113$	−6.18271	−	6.18271i	−0.581620	−	0.581620i	−0.935348	+	0.353728i	$0.884914\pi$
$114$		0			0
$115$	−7.39637	−	11.0694i	−0.689715	−	1.03223i
$116$		0			0
$117$	2.81509	+	4.76003i	0.260255	+	0.440065i
$118$		0			0
$119$	1.30563	+	0.540811i	0.119687	+	0.0495761i
$120$		0			0
$121$	2.87069	−	1.18908i	0.260972	−	0.108098i
$122$		0			0
$123$	7.98070	−	3.05154i	0.719596	−	0.275148i
$124$		0			0
$125$	7.16347	+	1.42490i	0.640720	+	0.127447i
$126$		0			0
$127$		−	5.87144i		−	0.521006i	−0.965473	−	0.260503i	$-0.916112\pi$
$127$		−	5.87144i		−	0.521006i	0.965473	−	0.260503i	$-0.0838884\pi$
$128$		0			0
$129$	1.57765	−	2.22614i	0.138904	−	0.196001i
$130$		0			0
$131$	−1.84224	+	9.26156i	−0.160957	+	0.809187i	0.812966	+	0.582311i	$0.197851\pi$
$131$	−1.84224	+	9.26156i	−0.160957	+	0.809187i	−0.973924	+	0.226876i	$0.927149\pi$
$132$		0			0
$133$	0.824775	+	0.551097i	0.0715171	+	0.0477862i
$134$		0			0
$135$	−13.9445	+	1.59498i	−1.20015	+	0.137274i
$136$		0			0
$137$	2.68881	−	6.49136i	0.229721	−	0.554594i	−0.766423	−	0.642337i	$-0.777965\pi$
$137$	2.68881	−	6.49136i	0.229721	−	0.554594i	0.996143	+	0.0877421i	$0.0279651\pi$
$138$		0			0
$139$	2.66596	+	13.4027i	0.226124	+	1.13680i	0.912349	+	0.409413i	$0.134266\pi$
$139$	2.66596	+	13.4027i	0.226124	+	1.13680i	−0.686225	+	0.727389i	$0.740734\pi$
$140$		0			0
$141$	10.4481	+	0.287269i	0.879893	+	0.0241924i
$142$		0			0
$143$	4.89577	−	4.89577i	0.409405	−	0.409405i
$144$		0			0
$145$	13.7489	+	13.7489i	1.14178	+	1.14178i
$146$		0			0
$147$	0.330826	−	12.0323i	0.0272860	−	0.992410i
$148$		0			0
$149$	−12.8724	+	2.56049i	−1.05455	+	0.209763i	−0.691775	−	0.722113i	$-0.743171\pi$
$149$	−12.8724	+	2.56049i	−1.05455	+	0.209763i	−0.362776	+	0.931876i	$0.618171\pi$
$150$		0			0
$151$	12.1990	+	5.05300i	0.992742	+	0.411207i	0.819131	−	0.573607i	$-0.194456\pi$
$151$	12.1990	+	5.05300i	0.992742	+	0.411207i	0.173612	+	0.984814i	$0.444456\pi$
$152$		0			0
$153$	−12.5864	−	14.0524i	−1.01755	−	1.13607i
$154$		0			0
$155$	−9.66785	+	14.4690i	−0.776540	+	1.16217i
$156$		0			0
$157$	13.9852	+	2.78184i	1.11614	+	0.222015i	0.718501	−	0.695526i	$-0.244828\pi$
$157$	13.9852	+	2.78184i	1.11614	+	0.222015i	0.397642	+	0.917541i	$0.369828\pi$
$158$		0			0
$159$	−7.39877	−	5.24346i	−0.586761	−	0.415833i
$160$		0			0
$161$	1.10766			0.0872961
$162$		0			0
$163$	1.85551	−	9.32826i	0.145334	−	0.730646i	−0.837541	−	0.546375i	$-0.816008\pi$
$163$	1.85551	−	9.32826i	0.145334	−	0.730646i	0.982875	−	0.184271i	$-0.0589925\pi$
$164$		0			0
$165$	6.27582	+	16.4132i	0.488572	+	1.27776i
$166$		0			0
$167$	6.77662	+	16.3602i	0.524391	+	1.26599i	0.935152	+	0.354248i	$0.115263\pi$
$167$	6.77662	+	16.3602i	0.524391	+	1.26599i	−0.410761	+	0.911743i	$0.634737\pi$
$168$		0			0
$169$	−3.67450	+	8.87103i	−0.282654	+	0.682387i
$170$		0			0
$171$	−6.74054	−	11.3976i	−0.515462	−	0.871594i
$172$		0			0
$173$	−2.35470	+	1.57336i	−0.179024	+	0.119620i	−0.641854	−	0.766827i	$-0.721834\pi$
$173$	−2.35470	+	1.57336i	−0.179024	+	0.119620i	0.462829	+	0.886447i	$0.346834\pi$
$174$		0			0
$175$	0.364861	−	0.364861i	0.0275809	−	0.0275809i
$176$		0			0
$177$	−15.3450	+	2.61609i	−1.15340	+	0.196638i
$178$		0			0
$179$	−14.8704	+	9.93607i	−1.11146	+	0.742657i	−0.968980	−	0.247140i	$-0.920509\pi$
$179$	−14.8704	+	9.93607i	−1.11146	+	0.742657i	−0.142485	+	0.989797i	$0.545509\pi$
$180$		0			0
$181$	−3.21012	+	0.638533i	−0.238606	+	0.0474618i	−0.312945	−	0.949771i	$-0.601315\pi$
$181$	−3.21012	+	0.638533i	−0.238606	+	0.0474618i	0.0743385	+	0.997233i	$0.476315\pi$
$182$		0			0
$183$	−3.85712	−	6.13091i	−0.285126	−	0.453210i
$184$		0			0
$185$	9.30509	+	22.4645i	0.684124	+	1.65162i
$186$		0			0
$187$	−13.1218	+	19.6382i	−0.959565	+	1.43609i
$188$		0			0
$189$	0.534229	−	1.03839i	0.0388594	−	0.0755319i
$190$		0			0
$191$	9.71747			0.703131			0.351566	−	0.936163i	$-0.385649\pi$
$191$	9.71747			0.703131			0.351566	+	0.936163i	$0.385649\pi$
$192$		0			0
$193$	−13.1295			−0.945086			−0.472543	−	0.881308i	$-0.656664\pi$
$193$	−13.1295			−0.945086			−0.472543	+	0.881308i	$0.656664\pi$
$194$		0			0
$195$	5.92832	+	6.26353i	0.424536	+	0.448541i
$196$		0			0
$197$	−4.09830	+	6.13354i	−0.291992	+	0.436997i	−0.948243	−	0.317546i	$-0.897141\pi$
$197$	−4.09830	+	6.13354i	−0.291992	+	0.436997i	0.656251	+	0.754543i	$0.272141\pi$
$198$		0			0
$199$	−5.31197	−	12.8242i	−0.376556	−	0.909086i	−0.992606	−	0.121379i	$-0.961268\pi$
$199$	−5.31197	−	12.8242i	−0.376556	−	0.909086i	0.616050	−	0.787707i	$-0.288732\pi$
$200$		0			0
$201$	12.0428	−	7.57643i	0.849433	−	0.534400i
$202$		0			0
$203$	−1.58667	+	0.315607i	−0.111362	+	0.0221513i
$204$		0			0
$205$	11.0790	−	7.40274i	0.773790	−	0.517030i
$206$		0			0
$207$	−13.3292	−	6.40054i	−0.926441	−	0.444868i
$208$		0			0
$209$	−11.7226	+	11.7226i	−0.810869	+	0.810869i
$210$		0			0
$211$	3.26507	−	2.18165i	0.224777	−	0.150191i	−0.438083	−	0.898934i	$-0.644343\pi$
$211$	3.26507	−	2.18165i	0.224777	−	0.150191i	0.662860	+	0.748743i	$0.269343\pi$
$212$		0			0
$213$	−1.93422	+	4.32908i	−0.132530	+	0.296624i
$214$		0			0
$215$	1.62834	−	3.93116i	0.111052	−	0.268103i
$216$		0			0
$217$	−0.554063	−	1.33763i	−0.0376122	−	0.0908039i
$218$		0			0
$219$	10.3471	−	3.95636i	0.699193	−	0.267346i
$220$		0			0
$221$	−2.26145	+	11.3691i	−0.152122	+	0.764767i
$222$		0			0
$223$	21.8404			1.46254			0.731272	−	0.682086i	$-0.238927\pi$
$223$	21.8404			1.46254			0.731272	+	0.682086i	$0.238927\pi$
$224$		0			0
$225$	−6.49890	+	2.28227i	−0.433260	+	0.152151i
$226$		0			0
$227$	15.0713	+	2.99786i	1.00031	+	0.198975i	0.667974	−	0.744184i	$-0.267162\pi$
$227$	15.0713	+	2.99786i	1.00031	+	0.198975i	0.332341	+	0.943159i	$0.392162\pi$
$228$		0			0
$229$	−8.04172	+	12.0353i	−0.531412	+	0.795314i	−0.995919	−	0.0902557i	$-0.971232\pi$
$229$	−8.04172	+	12.0353i	−0.531412	+	0.795314i	0.464507	+	0.885569i	$0.346232\pi$
$230$		0			0
$231$	−1.42554	−	0.324528i	−0.0937938	−	0.0213524i
$232$		0			0
$233$	6.45313	+	2.67297i	0.422758	+	0.175112i	0.583912	−	0.811817i	$-0.301521\pi$
$233$	6.45313	+	2.67297i	0.422758	+	0.175112i	−0.161153	+	0.986929i	$0.551521\pi$
$234$		0			0
$235$	15.9867	−	3.17995i	1.04286	−	0.207437i
$236$		0			0
$237$	21.5451	+	0.592377i	1.39951	+	0.0384790i
$238$		0			0
$239$	−14.9125	−	14.9125i	−0.964607	−	0.964607i	0.0347874	−	0.999395i	$-0.488925\pi$
$239$	−14.9125	−	14.9125i	−0.964607	−	0.964607i	−0.999395	+	0.0347874i	$0.988925\pi$
$240$		0			0
$241$	21.5764	−	21.5764i	1.38986	−	1.38986i	0.564261	−	0.825597i	$-0.309161\pi$
$241$	21.5764	−	21.5764i	1.38986	−	1.38986i	0.825597	−	0.564261i	$-0.190839\pi$
$242$		0			0
$243$	−12.4289	+	9.40858i	−0.797317	+	0.603560i
$244$		0			0
$245$	−3.66211	−	18.4107i	−0.233964	−	1.17621i
$246$		0			0
$247$	−3.11368	+	7.51709i	−0.198119	+	0.478301i
$248$		0			0
$249$	15.0137	+	3.41790i	0.951453	+	0.216600i
$250$		0			0
$251$	14.4199	+	9.63510i	0.910179	+	0.608162i	0.920055	−	0.391789i	$-0.128144\pi$
$251$	14.4199	+	9.63510i	0.910179	+	0.608162i	−0.00987642	+	0.999951i	$0.503144\pi$
$252$		0			0
$253$	−3.61155	+	18.1565i	−0.227056	+	1.14149i
$254$		0			0
$255$	−24.0031	−	17.0109i	−1.50313	−	1.06526i
$256$		0			0
$257$			21.8375i			1.36219i	0.732196	+	0.681094i	$0.238496\pi$
$257$			21.8375i			1.36219i	−0.732196	+	0.681094i	$0.761504\pi$
$258$		0			0
$259$	−1.98419	−	0.394680i	−0.123292	−	0.0245242i
$260$		0			0
$261$	20.9170	+	5.37052i	1.29473	+	0.332427i
$262$		0			0
$263$	−13.6376	+	5.64887i	−0.840929	+	0.348324i	−0.761220	−	0.648494i	$-0.775399\pi$
$263$	−13.6376	+	5.64887i	−0.840929	+	0.348324i	−0.0797089	+	0.996818i	$0.525399\pi$
$264$		0			0
$265$	−13.0655	−	5.41192i	−0.802610	−	0.332452i
$266$		0			0
$267$	4.16690	−	9.32617i	0.255010	−	0.570752i
$268$		0			0
$269$	−12.5566	−	18.7923i	−0.765591	−	1.14579i	−0.985402	−	0.170246i	$-0.945544\pi$
$269$	−12.5566	−	18.7923i	−0.765591	−	1.14579i	0.219810	−	0.975543i	$-0.429456\pi$
$270$		0			0
$271$	−0.680826	−	0.680826i	−0.0413572	−	0.0413572i	0.686126	−	0.727483i	$-0.259310\pi$
$271$	−0.680826	−	0.680826i	−0.0413572	−	0.0413572i	−0.727483	+	0.686126i	$0.759310\pi$
$272$		0			0
$273$	−0.707336	+	0.120590i	−0.0428099	+	0.00729846i
$274$		0			0
$275$	4.79105	+	7.17032i	0.288911	+	0.432387i
$276$		0			0
$277$	−3.03373	−	15.2516i	−0.182279	−	0.916378i	−0.958320	−	0.285696i	$-0.907775\pi$
$277$	−3.03373	−	15.2516i	−0.182279	−	0.916378i	0.776041	−	0.630682i	$-0.217225\pi$
$278$		0			0
$279$	−1.06199	+	19.2980i	−0.0635798	+	1.15534i
$280$		0			0
$281$	2.89583	−	1.19949i	0.172751	−	0.0715557i	−0.294632	−	0.955611i	$-0.595197\pi$
$281$	2.89583	−	1.19949i	0.172751	−	0.0715557i	0.467383	+	0.884055i	$0.345197\pi$
$282$		0			0
$283$	−12.6363	−	8.44328i	−0.751147	−	0.501901i	0.120090	−	0.992763i	$-0.461682\pi$
$283$	−12.6363	−	8.44328i	−0.751147	−	0.501901i	−0.871237	+	0.490862i	$0.836682\pi$
$284$		0			0
$285$	−14.1950	−	14.9976i	−0.840837	−	0.888381i
$286$		0			0
$287$			1.10862i			0.0654397i
$288$		0			0
$289$		−	22.5431i		−	1.32607i
$290$		0			0
$291$	19.4670	+	20.5678i	1.14118	+	1.20570i
$292$		0			0
$293$	−0.788441	−	0.526819i	−0.0460612	−	0.0307771i	0.532326	−	0.846539i	$-0.321318\pi$
$293$	−0.788441	−	0.526819i	−0.0460612	−	0.0307771i	−0.578388	+	0.815762i	$0.696318\pi$
$294$		0			0
$295$	−22.4277	+	9.28987i	−1.30579	+	0.540877i
$296$		0			0
$297$	15.2791	+	12.1426i	0.886585	+	0.704585i
$298$		0			0
$299$	1.77251	+	8.91101i	0.102507	+	0.515337i
$300$		0			0
$301$	0.196686	+	0.294361i	0.0113368	+	0.0169667i
$302$		0			0
$303$	−14.2996	+	2.43788i	−0.821492	+	0.140052i
$304$		0			0
$305$	−7.98736	−	7.98736i	−0.457355	−	0.457355i
$306$		0			0
$307$	4.10368	+	6.14160i	0.234210	+	0.350519i	0.929894	−	0.367828i	$-0.119899\pi$
$307$	4.10368	+	6.14160i	0.234210	+	0.350519i	−0.695684	+	0.718348i	$0.744899\pi$
$308$		0			0
$309$	2.69290	−	6.02713i	0.153194	−	0.342871i
$310$		0			0
$311$	4.51135	+	1.86866i	0.255815	+	0.105962i	0.506906	−	0.862002i	$-0.330789\pi$
$311$	4.51135	+	1.86866i	0.255815	+	0.105962i	−0.251091	+	0.967964i	$0.580789\pi$
$312$		0			0
$313$	−29.8873	+	12.3797i	−1.68933	+	0.699743i	−0.999702	−	0.0244212i	$-0.992226\pi$
$313$	−29.8873	+	12.3797i	−1.68933	+	0.699743i	−0.689627	+	0.724164i	$0.742226\pi$
$314$		0			0
$315$	0.452886	−	1.76389i	0.0255172	−	0.0993839i
$316$		0			0
$317$	−20.5647	−	4.09058i	−1.15503	−	0.229750i	−0.419831	−	0.907602i	$-0.637911\pi$
$317$	−20.5647	−	4.09058i	−1.15503	−	0.229750i	−0.735198	+	0.677853i	$0.762911\pi$
$318$		0			0
$319$		−	27.0372i		−	1.51379i
$320$		0			0
$321$	−19.4269	−	13.7677i	−1.08430	−	0.768439i
$322$		0			0
$323$	5.41489	−	27.2225i	0.301293	−	1.51470i
$324$		0			0
$325$	3.51912	+	2.35140i	0.195205	+	0.130432i
$326$		0			0
$327$	24.3336	+	5.53959i	1.34565	+	0.306340i
$328$		0			0
$329$	−0.518983	+	1.25294i	−0.0286125	+	0.0690766i
$330$		0			0
$331$	1.37125	+	6.89372i	0.0753705	+	0.378913i	0.999998	−	0.00195950i	$-0.000623727\pi$
$331$	1.37125	+	6.89372i	0.0753705	+	0.378913i	−0.924628	+	0.380872i	$0.875624\pi$
$332$		0			0
$333$	21.5963	+	16.2149i	1.18347	+	0.888570i
$334$		0			0
$335$	15.6894	−	15.6894i	0.857202	−	0.857202i
$336$		0			0
$337$	−12.1364	−	12.1364i	−0.661114	−	0.661114i	0.294529	−	0.955643i	$-0.404837\pi$
$337$	−12.1364	−	12.1364i	−0.661114	−	0.661114i	−0.955643	+	0.294529i	$0.904837\pi$
$338$		0			0
$339$	−15.1388	−	0.416236i	−0.822224	−	0.0226068i
$340$		0			0
$341$	23.7325	−	4.72068i	1.28519	−	0.255639i
$342$		0			0
$343$	2.89631	+	1.19969i	0.156386	+	0.0647771i
$344$		0			0
$345$	−22.4837	−	5.11847i	−1.21048	−	0.275569i
$346$		0			0
$347$	−4.55745	+	6.82071i	−0.244657	+	0.366155i	−0.933392	−	0.358858i	$-0.883166\pi$
$347$	−4.55745	+	6.82071i	−0.244657	+	0.366155i	0.688735	+	0.725013i	$0.258166\pi$
$348$		0			0
$349$	−18.8023	−	3.74002i	−1.00647	−	0.200198i	−0.335780	−	0.941940i	$-0.609000\pi$
$349$	−18.8023	−	3.74002i	−1.00647	−	0.200198i	−0.670685	+	0.741742i	$0.734000\pi$
$350$		0			0
$351$	9.20861	+	2.63614i	0.491519	+	0.140707i
$352$		0			0
$353$	2.39915			0.127694			0.0638471	−	0.997960i	$-0.479663\pi$
$353$	2.39915			0.127694			0.0638471	+	0.997960i	$0.479663\pi$
$354$		0			0
$355$	−1.44257	+	7.25228i	−0.0765635	+	0.384911i
$356$		0			0
$357$	2.28632	−	0.874205i	0.121005	−	0.0462678i
$358$		0			0
$359$	−7.68944	−	18.5640i	−0.405833	−	0.979768i	−0.986222	−	0.165428i	$-0.947099\pi$
$359$	−7.68944	−	18.5640i	−0.405833	−	0.979768i	0.580389	−	0.814340i	$-0.302901\pi$
$360$		0			0
$361$	0.184509	−	0.445444i	0.00971100	−	0.0234444i
$362$		0			0
$363$	2.19542	−	4.91370i	0.115230	−	0.257902i
$364$		0			0
$365$	14.3641	−	9.59777i	0.751850	−	0.502370i
$366$		0			0
$367$	−13.4524	+	13.4524i	−0.702207	+	0.702207i	−0.964884	−	0.262677i	$-0.915395\pi$
$367$	−13.4524	+	13.4524i	−0.702207	+	0.702207i	0.262677	+	0.964884i	$0.415395\pi$
$368$		0			0
$369$	6.40605	−	13.3407i	0.333486	−	0.694487i
$370$		0			0
$371$	0.978333	−	0.653701i	0.0507925	−	0.0339385i
$372$		0			0
$373$	0.738592	−	0.146915i	0.0382428	−	0.00760697i	−0.175932	−	0.984402i	$-0.556294\pi$
$373$	0.738592	−	0.146915i	0.0382428	−	0.00760697i	0.214175	+	0.976795i	$0.431294\pi$
$374$		0			0
$375$	10.7078	−	6.73653i	0.552946	−	0.347873i
$376$		0			0
$377$	−5.07804	−	12.2595i	−0.261532	−	0.631395i
$378$		0			0
$379$	−20.6000	+	30.8301i	−1.05815	+	1.58364i	−0.275370	+	0.961338i	$0.588800\pi$
$379$	−20.6000	+	30.8301i	−1.05815	+	1.58364i	−0.782781	+	0.622297i	$0.786200\pi$
$380$		0			0
$381$	−6.99066	−	7.38594i	−0.358142	−	0.378393i
$382$		0			0
$383$	6.06226			0.309767			0.154883	−	0.987933i	$-0.450500\pi$
$383$	6.06226			0.309767			0.154883	+	0.987933i	$0.450500\pi$
$384$		0			0
$385$	−2.27999			−0.116199
$386$		0			0
$387$	−0.665895	−	4.67875i	−0.0338493	−	0.237834i
$388$		0			0
$389$	−2.01124	+	3.01003i	−0.101974	+	0.152614i	−0.878948	−	0.476918i	$-0.841754\pi$
$389$	−2.01124	+	3.01003i	−0.101974	+	0.152614i	0.776974	+	0.629533i	$0.216754\pi$
$390$		0			0
$391$	−11.8608	−	28.6344i	−0.599824	−	1.44810i
$392$		0			0
$393$	8.70958	+	13.8439i	0.439340	+	0.698334i
$394$		0			0
$395$	32.9662	−	6.55738i	1.65871	−	0.329938i
$396$		0			0
$397$	11.5530	−	7.71947i	0.579828	−	0.387429i	−0.230790	−	0.973004i	$-0.574131\pi$
$397$	11.5530	−	7.71947i	0.579828	−	0.387429i	0.810619	+	0.585575i	$0.199131\pi$
$398$		0			0
$399$	1.69367	−	0.288745i	0.0847895	−	0.0144554i
$400$		0			0
$401$	19.3426	−	19.3426i	0.965925	−	0.965925i	−0.0335131	−	0.999438i	$-0.510670\pi$
$401$	19.3426	−	19.3426i	0.965925	−	0.965925i	0.999438	+	0.0335131i	$0.0106696\pi$
$402$		0			0
$403$	9.87440	−	6.59786i	0.491879	−	0.328663i
$404$		0			0
$405$	−15.6423	+	18.6090i	−0.777273	+	0.924687i
$406$		0			0
$407$	12.9389	−	31.2374i	0.641359	−	1.54838i
$408$		0			0
$409$	−6.29782	−	15.2043i	−0.311407	−	0.751803i	−0.999653	−	0.0263261i	$-0.991619\pi$
$409$	−6.29782	−	15.2043i	−0.311407	−	0.751803i	0.688246	−	0.725477i	$-0.258381\pi$
$410$		0			0
$411$	−4.34638	−	11.3671i	−0.214391	−	0.560699i
$412$		0			0
$413$	0.394034	−	1.98094i	0.0193892	−	0.0974759i
$414$		0			0
$415$	24.0127			1.17874
$416$		0			0
$417$	19.3112	+	13.6857i	0.945671	+	0.670191i
$418$		0			0
$419$	−17.1733	−	3.41598i	−0.838971	−	0.166882i	−0.243139	−	0.969991i	$-0.578177\pi$
$419$	−17.1733	−	3.41598i	−0.838971	−	0.166882i	−0.595831	+	0.803110i	$0.703177\pi$
$420$		0			0
$421$	15.5605	−	23.2879i	0.758372	−	1.13498i	−0.228510	−	0.973542i	$-0.573385\pi$
$421$	15.5605	−	23.2879i	0.758372	−	1.13498i	0.986882	−	0.161442i	$-0.0516146\pi$
$422$		0			0
$423$	13.4852	−	12.0784i	0.655673	−	0.587273i
$424$		0			0
$425$	−13.3390	−	5.52518i	−0.647035	−	0.268011i
$426$		0			0
$427$	0.921766	−	0.183351i	0.0446074	−	0.00887296i
$428$		0			0
$429$	0.329596	−	11.9876i	0.0159131	−	0.578768i
$430$		0			0
$431$	8.82179	+	8.82179i	0.424931	+	0.424931i	0.886897	−	0.461966i	$-0.152856\pi$
$431$	8.82179	+	8.82179i	0.424931	+	0.424931i	−0.461966	+	0.886897i	$0.652856\pi$
$432$		0			0
$433$	−17.1259	+	17.1259i	−0.823018	+	0.823018i	−0.986540	−	0.163521i	$-0.947715\pi$
$433$	−17.1259	+	17.1259i	−0.823018	+	0.823018i	0.163521	+	0.986540i	$0.447715\pi$
$434$		0			0
$435$	33.6651	+	0.925612i	1.61412	+	0.0443797i
$436$		0			0
$437$	−4.24415	−	21.3368i	−0.203025	−	1.02068i
$438$		0			0
$439$	8.82799	−	21.3126i	0.421337	−	1.01720i	−0.560617	−	0.828075i	$-0.689436\pi$
$439$	8.82799	−	21.3126i	0.421337	−	1.01720i	0.981954	−	0.189122i	$-0.0605641\pi$
$440$		0			0
$441$	−13.9098	−	15.5299i	−0.662371	−	0.739518i
$442$		0			0
$443$	−0.625397	−	0.417877i	−0.0297135	−	0.0198539i	0.540625	−	0.841264i	$-0.318188\pi$
$443$	−0.625397	−	0.417877i	−0.0297135	−	0.0198539i	−0.570338	+	0.821410i	$0.693188\pi$
$444$		0			0
$445$	3.10774	−	15.6236i	0.147321	−	0.740632i
$446$		0			0
$447$	−13.1442	+	18.5471i	−0.621701	+	0.877249i
$448$		0			0
$449$			11.2059i			0.528838i	0.964408	+	0.264419i	$0.0851802\pi$
$449$			11.2059i			0.528838i	−0.964408	+	0.264419i	$0.914820\pi$
$450$		0			0
$451$	−18.1721	−	3.61466i	−0.855692	−	0.170208i
$452$		0			0
$453$	21.3619	−	8.16802i	1.00367	−	0.383767i
$454$		0			0
$455$	−1.03382	+	0.428222i	−0.0484661	+	0.0200753i
$456$		0			0
$457$	7.55593	+	3.12977i	0.353451	+	0.146404i	0.552343	−	0.833617i	$-0.313734\pi$
$457$	7.55593	+	3.12977i	0.353451	+	0.146404i	−0.198891	+	0.980022i	$0.563734\pi$
$458$		0			0
$459$	−32.5641	−	2.69145i	−1.51996	−	0.125626i
$460$		0			0
$461$	−15.4154	−	23.0707i	−0.717965	−	1.07451i	−0.993568	−	0.113239i	$-0.963877\pi$
$461$	−15.4154	−	23.0707i	−0.717965	−	1.07451i	0.275602	−	0.961272i	$-0.411123\pi$
$462$		0			0
$463$	−7.21675	−	7.21675i	−0.335391	−	0.335391i	0.519239	−	0.854629i	$-0.326216\pi$
$463$	−7.21675	−	7.21675i	−0.335391	−	0.335391i	−0.854629	+	0.519239i	$0.826216\pi$
$464$		0			0
$465$	5.06544	+	29.7119i	0.234904	+	1.37786i
$466$		0			0
$467$	−2.23916	−	3.35114i	−0.103616	−	0.155072i	0.776039	−	0.630685i	$-0.217226\pi$
$467$	−2.23916	−	3.35114i	−0.103616	−	0.155072i	−0.879654	+	0.475613i	$0.842226\pi$
$468$		0			0
$469$	0.360151	+	1.81060i	0.0166302	+	0.0836058i
$470$		0			0
$471$	20.9048	−	13.1517i	0.963240	−	0.605999i
$472$		0			0
$473$	−5.46637	+	2.26424i	−0.251344	+	0.104110i
$474$		0			0
$475$	−8.42629	−	5.63027i	−0.386625	−	0.258334i
$476$		0			0
$477$	−15.5502	+	2.21316i	−0.711995	+	0.101334i
$478$		0			0
$479$			7.63624i			0.348909i	0.984665	+	0.174454i	$0.0558161\pi$
$479$			7.63624i			0.348909i	−0.984665	+	0.174454i	$0.944184\pi$
$480$		0			0
$481$		−	16.5941i		−	0.756627i
$482$		0			0
$483$	1.39338	−	1.31881i	0.0634009	−	0.0600078i
$484$		0			0
$485$	36.7211	+	24.5362i	1.66742	+	1.11413i
$486$		0			0
$487$	−11.0892	+	4.59329i	−0.502499	+	0.208142i	−0.619510	−	0.784989i	$-0.712669\pi$
$487$	−11.0892	+	4.59329i	−0.502499	+	0.208142i	0.117011	+	0.993131i	$0.462669\pi$
$488$		0			0
$489$	−8.77230	−	13.9436i	−0.396697	−	0.630553i
$490$		0			0
$491$	−3.44475	−	17.3179i	−0.155460	−	0.781548i	−0.977305	−	0.211837i	$-0.932055\pi$
$491$	−3.44475	−	17.3179i	−0.155460	−	0.781548i	0.821845	−	0.569711i	$-0.192945\pi$
$492$		0			0
$493$	25.1487	+	37.6377i	1.13264	+	1.69511i
$494$		0			0
$495$	27.4365	+	13.1747i	1.23318	+	0.592161i
$496$		0			0
$497$	−0.435025	−	0.435025i	−0.0195135	−	0.0195135i
$498$		0			0
$499$	−15.1695	−	22.7027i	−0.679080	−	1.01631i	−0.997657	−	0.0684175i	$-0.978205\pi$
$499$	−15.1695	−	22.7027i	−0.679080	−	1.01631i	0.318577	−	0.947897i	$-0.396795\pi$
$500$		0			0
$501$	28.0034	+	12.5118i	1.25110	+	0.558988i
$502$		0			0
$503$	−20.3314	−	8.42152i	−0.906530	−	0.375497i	−0.119803	−	0.992798i	$-0.538226\pi$
$503$	−20.3314	−	8.42152i	−0.906530	−	0.375497i	−0.786727	+	0.617300i	$0.788226\pi$
$504$		0			0
$505$	−20.8999	+	8.65701i	−0.930032	+	0.385232i
$506$		0			0
$507$	5.93972	+	15.5342i	0.263792	+	0.689898i
$508$		0			0
$509$	32.8678	+	6.53780i	1.45684	+	0.289783i	0.859058	−	0.511878i	$-0.171050\pi$
$509$	32.8678	+	6.53780i	1.45684	+	0.289783i	0.597779	+	0.801661i	$0.296050\pi$
$510$		0			0
$511$			1.43734i			0.0635842i
$512$		0			0
$513$	−22.0494	−	6.31207i	−0.973504	−	0.278685i
$514$		0			0
$515$	2.00840	−	10.0969i	0.0885008	−	0.444924i
$516$		0			0
$517$	−18.8456	−	12.5922i	−0.828828	−	0.553805i
$518$		0			0
$519$	−1.08880	+	4.78275i	−0.0477931	+	0.209939i
$520$		0			0
$521$	0.588808	−	1.42151i	0.0257962	−	0.0622774i	−0.910457	−	0.413605i	$-0.864270\pi$
$521$	0.588808	−	1.42151i	0.0257962	−	0.0622774i	0.936253	+	0.351327i	$0.114270\pi$
$522$		0			0
$523$	−2.29528	−	11.5391i	−0.100365	−	0.504571i	−0.997965	−	0.0637598i	$-0.979691\pi$
$523$	−2.29528	−	11.5391i	−0.100365	−	0.504571i	0.897600	−	0.440811i	$-0.145309\pi$
$524$		0			0
$525$	0.0245634	−	0.893385i	0.00107203	−	0.0389905i
$526$		0			0
$527$	−28.6463	+	28.6463i	−1.24785	+	1.24785i
$528$		0			0
$529$	−0.914009	−	0.914009i	−0.0397395	−	0.0397395i
$530$		0			0
$531$	−16.1884	+	21.5610i	−0.702515	+	0.935666i
$532$		0			0
$533$	−8.91869	+	1.77404i	−0.386311	+	0.0768421i
$534$		0			0
$535$	−34.3061	−	14.2100i	−1.48318	−	0.614354i
$536$		0			0
$537$	−6.87601	+	30.2040i	−0.296722	+	1.30340i
$538$		0			0
$539$	−14.5015	+	21.7030i	−0.624624	+	0.934815i
$540$		0			0
$541$	−12.5139	−	2.48917i	−0.538015	−	0.107018i	−0.0813987	−	0.996682i	$-0.525939\pi$
$541$	−12.5139	−	2.48917i	−0.538015	−	0.107018i	−0.456616	+	0.889664i	$0.650939\pi$
$542$		0			0
$543$	−3.27790	+	4.62527i	−0.140668	+	0.198489i
$544$		0			0
$545$	38.9188			1.66710
$546$		0			0
$547$	−1.34057	+	6.73951i	−0.0573187	+	0.288161i	−0.998804	−	0.0488934i	$-0.984431\pi$
$547$	−1.34057	+	6.73951i	−0.0573187	+	0.288161i	0.941485	+	0.337054i	$0.109431\pi$
$548$		0			0
$549$	−12.1516	−	3.11998i	−0.518619	−	0.133157i
$550$		0			0
$551$	12.1590	+	29.3545i	0.517992	+	1.25054i
$552$		0			0
$553$	−1.07019	+	2.58368i	−0.0455093	+	0.109869i
$554$		0			0
$555$	38.4519	+	17.1802i	1.63219	+	0.729259i
$556$		0			0
$557$	−21.4271	+	14.3171i	−0.907894	+	0.606635i	−0.919411	−	0.393298i	$-0.871334\pi$
$557$	−21.4271	+	14.3171i	−0.907894	+	0.606635i	0.0115170	+	0.999934i	$0.496334\pi$
$558$		0			0
$559$	−2.05335	+	2.05335i	−0.0868476	+	0.0868476i
$560$		0			0
$561$	6.87515	+	40.3269i	0.290269	+	1.70260i
$562$		0			0
$563$	−17.0640	+	11.4018i	−0.719162	+	0.480528i	−0.860511	−	0.509431i	$-0.829856\pi$
$563$	−17.0640	+	11.4018i	−0.719162	+	0.480528i	0.141350	+	0.989960i	$0.454856\pi$
$564$		0			0
$565$	−23.1638	+	4.60757i	−0.974509	+	0.193842i
$566$		0			0
$567$	−0.564300	−	1.94230i	−0.0236984	−	0.0815690i
$568$		0			0
$569$	−4.14149	−	9.99843i	−0.173620	−	0.419156i	0.812985	−	0.582285i	$-0.197841\pi$
$569$	−4.14149	−	9.99843i	−0.173620	−	0.419156i	−0.986605	+	0.163129i	$0.947841\pi$
$570$		0			0
$571$	24.6168	−	36.8416i	1.03018	−	1.54177i	0.203518	−	0.979071i	$-0.434762\pi$
$571$	24.6168	−	36.8416i	1.03018	−	1.54177i	0.826661	−	0.562701i	$-0.190238\pi$
$572$		0			0
$573$	12.2240	−	11.5698i	0.510666	−	0.483336i
$574$		0			0
$575$	−11.3164			−0.471927
$576$		0			0
$577$	5.58949			0.232693			0.116347	−	0.993209i	$-0.462882\pi$
$577$	5.58949			0.232693			0.116347	+	0.993209i	$0.462882\pi$
$578$		0			0
$579$	−16.5162	+	15.6323i	−0.686391	+	0.649657i
$580$		0			0
$581$	−1.10996	+	1.66118i	−0.0460490	+	0.0689172i
$582$		0			0
$583$	7.52540	+	18.1679i	0.311670	+	0.752439i
$584$		0			0
$585$	14.9150	+	0.820787i	0.616659	+	0.0339354i
$586$		0			0
$587$	−1.57303	+	0.312895i	−0.0649258	+	0.0129145i	−0.227446	−	0.973791i	$-0.573038\pi$
$587$	−1.57303	+	0.312895i	−0.0649258	+	0.0129145i	0.162521	+	0.986705i	$0.448038\pi$
$588$		0			0
$589$	−23.6436	+	15.7981i	−0.974217	+	0.650951i
$590$		0			0
$591$	2.14729	+	12.5952i	0.0883278	+	0.518096i
$592$		0			0
$593$	4.33456	−	4.33456i	0.177999	−	0.177999i	−0.612484	−	0.790483i	$-0.709830\pi$
$593$	4.33456	−	4.33456i	0.177999	−	0.177999i	0.790483	+	0.612484i	$0.209830\pi$
$594$		0			0
$595$	3.17391	−	2.12074i	0.130118	−	0.0869419i
$596$		0			0
$597$	−21.9510	−	9.80762i	−0.898393	−	0.401399i
$598$		0			0
$599$	9.62407	−	23.2346i	0.393229	−	0.949338i	−0.596003	−	0.802982i	$-0.703245\pi$
$599$	9.62407	−	23.2346i	0.393229	−	0.949338i	0.989232	−	0.146356i	$-0.0467546\pi$
$600$		0			0
$601$	0.673957	+	1.62708i	0.0274913	+	0.0663699i	0.937029	−	0.349252i	$-0.113564\pi$
$601$	0.673957	+	1.62708i	0.0274913	+	0.0663699i	−0.909538	+	0.415621i	$0.863564\pi$
$602$		0			0
$603$	6.12849	−	23.8691i	0.249572	−	0.972026i
$604$		0			0
$605$	1.63738	−	8.23165i	0.0665689	−	0.334664i
$606$		0			0
$607$	−11.3664			−0.461349			−0.230674	−	0.973031i	$-0.574093\pi$
$607$	−11.3664			−0.461349			−0.230674	+	0.973031i	$0.574093\pi$
$608$		0			0
$609$	−1.62017	+	2.28613i	−0.0656525	+	0.0926388i
$610$		0			0
$611$	−10.9102	−	2.17017i	−0.441379	−	0.0877958i
$612$		0			0
$613$	1.91745	−	2.86967i	0.0774453	−	0.115905i	−0.790725	−	0.612172i	$-0.790296\pi$
$613$	1.91745	−	2.86967i	0.0774453	−	0.115905i	0.868170	+	0.496267i	$0.165296\pi$
$614$		0			0
$615$	5.12288	−	22.5031i	0.206574	−	0.907413i
$616$		0			0
$617$	−13.8035	−	5.71758i	−0.555706	−	0.230181i	0.0871137	−	0.996198i	$-0.472236\pi$
$617$	−13.8035	−	5.71758i	−0.555706	−	0.230181i	−0.642820	+	0.766017i	$0.722236\pi$
$618$		0			0
$619$	6.75340	−	1.34333i	0.271442	−	0.0539932i	−0.0574924	−	0.998346i	$-0.518310\pi$
$619$	6.75340	−	1.34333i	0.271442	−	0.0539932i	0.328934	+	0.944353i	$0.393310\pi$
$620$		0			0
$621$	−24.3879	+	7.81847i	−0.978655	+	0.313744i
$622$		0			0
$623$	0.937177	+	0.937177i	0.0375472	+	0.0375472i
$624$		0			0
$625$	22.0677	−	22.0677i	0.882706	−	0.882706i
$626$		0			0
$627$	−0.789196	+	28.7035i	−0.0315175	+	1.14631i
$628$		0			0
$629$	11.0436	+	55.5198i	0.440336	+	2.21372i
$630$		0			0
$631$	1.24136	−	2.99691i	0.0494179	−	0.119305i	−0.897243	−	0.441538i	$-0.854433\pi$
$631$	1.24136	−	2.99691i	0.0494179	−	0.119305i	0.946661	+	0.322232i	$0.104433\pi$
$632$		0			0
$633$	1.50975	−	6.63185i	0.0600074	−	0.263592i
$634$		0			0
$635$	−13.1866	−	8.81102i	−0.523295	−	0.349655i
$636$		0			0
$637$	−2.49922	+	12.5644i	−0.0990227	+	0.497821i
$638$		0			0
$639$	2.72115	+	7.74866i	0.107647	+	0.306532i
$640$		0			0
$641$			34.0351i			1.34431i	0.740412	+	0.672153i	$0.234630\pi$
$641$			34.0351i			1.34431i	−0.740412	+	0.672153i	$0.765370\pi$
$642$		0			0
$643$	−8.02330	−	1.59593i	−0.316408	−	0.0629374i	0.0343316	−	0.999410i	$-0.489070\pi$
$643$	−8.02330	−	1.59593i	−0.316408	−	0.0629374i	−0.350740	+	0.936473i	$0.614070\pi$
$644$		0			0
$645$	−2.63216	−	6.88391i	−0.103641	−	0.271054i
$646$		0			0
$647$	−42.9113	+	17.7744i	−1.68702	+	0.698785i	−0.999622	−	0.0275003i	$-0.991245\pi$
$647$	−42.9113	+	17.7744i	−1.68702	+	0.698785i	−0.687394	+	0.726285i	$0.741245\pi$
$648$		0			0
$649$	31.1863	+	12.9178i	1.22417	+	0.507067i
$650$		0			0
$651$	−2.28958	−	1.02298i	−0.0897359	−	0.0400937i
$652$		0			0
$653$	−4.61269	−	6.90338i	−0.180509	−	0.270150i	0.730170	−	0.683265i	$-0.239441\pi$
$653$	−4.61269	−	6.90338i	−0.180509	−	0.270150i	−0.910679	+	0.413115i	$0.864441\pi$
$654$		0			0
$655$	18.0359	+	18.0359i	0.704721	+	0.704721i
$656$		0			0
$657$	8.30555	−	17.2964i	0.324030	−	0.674796i
$658$		0			0
$659$	−8.44152	−	12.6336i	−0.328835	−	0.492136i	0.629807	−	0.776752i	$-0.283134\pi$
$659$	−8.44152	−	12.6336i	−0.328835	−	0.492136i	−0.958642	+	0.284615i	$0.908134\pi$
$660$		0			0
$661$	3.84431	+	19.3267i	0.149526	+	0.751720i	0.980671	+	0.195664i	$0.0626861\pi$
$661$	3.84431	+	19.3267i	0.149526	+	0.751720i	−0.831145	+	0.556056i	$0.812314\pi$
$662$		0			0
$663$	10.6915	+	16.9942i	0.415223	+	0.659999i
$664$		0			0
$665$	2.47541	−	1.02535i	0.0959922	−	0.0397613i
$666$		0			0
$667$	29.5001	+	19.7114i	1.14225	+	0.763227i
$668$		0			0
$669$	27.4740	−	26.0037i	1.06221	−	1.00536i
$670$		0			0
$671$			15.7071i			0.606367i
$672$		0			0
$673$			3.92280i			0.151213i	0.997138	+	0.0756064i	$0.0240893\pi$
$673$			3.92280i			0.151213i	−0.997138	+	0.0756064i	$0.975911\pi$
$674$		0			0
$675$	−5.45794	+	10.6087i	−0.210076	+	0.408329i
$676$		0			0
$677$	−11.1725	−	7.46522i	−0.429394	−	0.286912i	0.322033	−	0.946728i	$-0.395634\pi$
$677$	−11.1725	−	7.46522i	−0.429394	−	0.286912i	−0.751426	+	0.659817i	$0.770634\pi$
$678$		0			0
$679$	−3.39479	+	1.40617i	−0.130280	+	0.0539637i
$680$		0			0
$681$	22.5281	−	14.1730i	0.863279	−	0.543111i
$682$		0			0
$683$	7.51550	+	37.7830i	0.287573	+	1.44572i	0.806666	+	0.591007i	$0.201270\pi$
$683$	7.51550	+	37.7830i	0.287573	+	1.44572i	−0.519094	+	0.854717i	$0.673730\pi$
$684$		0			0
$685$	−10.5439	−	15.7801i	−0.402862	−	0.602926i
$686$		0			0
$687$	4.21343	+	24.7143i	0.160752	+	0.942911i
$688$		0			0
$689$	6.82449	+	6.82449i	0.259992	+	0.259992i
$690$		0			0
$691$	−18.6673	−	27.9376i	−0.710137	−	1.06280i	−0.994567	−	0.104099i	$-0.966804\pi$
$691$	−18.6673	−	27.9376i	−0.710137	−	1.06280i	0.284430	−	0.958697i	$-0.408196\pi$
$692$		0			0
$693$	−2.17964	+	1.28904i	−0.0827977	+	0.0489667i
$694$		0			0
$695$	34.1017	+	14.1254i	1.29355	+	0.535806i
$696$		0			0
$697$	28.6591	−	11.8710i	1.08554	−	0.449645i
$698$		0			0
$699$	11.3002	−	4.32078i	0.427412	−	0.163427i
$700$		0			0
$701$	12.3453	+	2.45562i	0.466274	+	0.0927476i	0.422635	−	0.906300i	$-0.361105\pi$
$701$	12.3453	+	2.45562i	0.466274	+	0.0927476i	0.0436384	+	0.999047i	$0.486105\pi$
$702$		0			0
$703$			39.7335i			1.49858i
$704$		0			0
$705$	16.3243	−	23.0343i	0.614807	−	0.867523i
$706$		0			0
$707$	0.367191	−	1.84599i	0.0138096	−	0.0694258i
$708$		0			0
$709$	−0.942734	−	0.629915i	−0.0354051	−	0.0236569i	0.537742	−	0.843110i	$-0.319278\pi$
$709$	−0.942734	−	0.629915i	−0.0354051	−	0.0236569i	−0.573147	+	0.819453i	$0.694278\pi$
$710$		0			0
$711$	27.8078	−	24.9069i	1.04288	−	0.934081i
$712$		0			0
$713$	−12.1514	+	29.3360i	−0.455072	+	1.09864i
$714$		0			0
$715$	−3.64850	−	18.3423i	−0.136446	−	0.685961i
$716$		0			0
$717$	−36.5141	−	1.00395i	−1.36365	−	0.0374931i
$718$		0			0
$719$	−22.8751	+	22.8751i	−0.853097	+	0.853097i	−0.990513	−	0.137416i	$-0.956120\pi$
$719$	−22.8751	+	22.8751i	−0.853097	+	0.853097i	0.137416	+	0.990513i	$0.456120\pi$
$720$		0			0
$721$	0.605660	+	0.605660i	0.0225559	+	0.0225559i
$722$		0			0
$723$	1.45258	−	52.8312i	0.0540220	−	1.96481i
$724$		0			0
$725$	16.2101	−	3.22439i	0.602029	−	0.119751i
$726$		0			0
$727$	−30.7003	−	12.7165i	−1.13861	−	0.471628i	−0.267911	−	0.963444i	$-0.586333\pi$
$727$	−30.7003	−	12.7165i	−1.13861	−	0.471628i	−0.870699	+	0.491815i	$0.836333\pi$
$728$		0			0
$729$	−4.43286	+	26.6336i	−0.164180	+	0.986430i
$730$		0			0
$731$	5.50348	−	8.23654i	0.203554	−	0.304639i
$732$		0			0
$733$	−23.5980	−	4.69393i	−0.871610	−	0.173374i	−0.261032	−	0.965330i	$-0.584063\pi$
$733$	−23.5980	−	4.69393i	−0.871610	−	0.173374i	−0.610578	+	0.791956i	$0.709063\pi$
$734$		0			0
$735$	−26.5269	−	18.7994i	−0.978458	−	0.693426i
$736$		0			0
$737$	−30.8531			−1.13649
$738$		0			0
$739$	−2.09191	+	10.5168i	−0.0769523	+	0.386865i	0.923045	+	0.384691i	$0.125692\pi$
$739$	−2.09191	+	10.5168i	−0.0769523	+	0.386865i	−0.999998	+	0.00217415i	$0.999308\pi$
$740$		0			0
$741$	5.03317	+	13.1633i	0.184898	+	0.483565i
$742$		0			0
$743$	−6.77485	−	16.3559i	−0.248545	−	0.600041i	0.749536	−	0.661964i	$-0.230277\pi$
$743$	−6.77485	−	16.3559i	−0.248545	−	0.600041i	−0.998081	+	0.0619227i	$0.980277\pi$
$744$		0			0
$745$	−13.5665	+	32.7525i	−0.497039	+	1.19996i
$746$		0			0
$747$	22.9558	−	13.5761i	0.839908	−	0.496723i
$748$		0			0
$749$	2.56880	−	1.71642i	0.0938620	−	0.0627166i
$750$		0			0
$751$	14.3761	−	14.3761i	0.524591	−	0.524591i	−0.394364	−	0.918954i	$-0.629035\pi$
$751$	14.3761	−	14.3761i	0.524591	−	0.524591i	0.918954	+	0.394364i	$0.129035\pi$
$752$		0			0
$753$	29.6112	−	5.04828i	1.07909	−	0.183969i
$754$		0			0
$755$	29.6550	−	19.8149i	1.07926	−	0.721137i
$756$		0			0
$757$	47.9560	−	9.53904i	1.74299	−	0.346702i	0.781999	−	0.623280i	$-0.214200\pi$
$757$	47.9560	−	9.53904i	1.74299	−	0.346702i	0.960992	+	0.276578i	$0.0892003\pi$
$758$		0			0
$759$	17.0744	+	27.1398i	0.619760	+	0.985113i
$760$		0			0
$761$	−1.03255	−	2.49280i	−0.0374300	−	0.0903641i	0.904059	−	0.427408i	$-0.140573\pi$
$761$	−1.03255	−	2.49280i	−0.0374300	−	0.0903641i	−0.941489	+	0.337043i	$0.890573\pi$
$762$		0			0
$763$	−1.79898	+	2.69237i	−0.0651275	+	0.0974702i
$764$		0			0
$765$	−50.4481	+	7.17994i	−1.82395	+	0.259591i
$766$		0			0
$767$	16.5670			0.598199
$768$		0			0
$769$	35.0025			1.26222			0.631111	−	0.775693i	$-0.282599\pi$
$769$	35.0025			1.26222			0.631111	+	0.775693i	$0.282599\pi$
$770$		0			0
$771$	26.0002	+	27.4704i	0.936376	+	0.989322i
$772$		0			0
$773$	−21.1773	+	31.6940i	−0.761694	+	1.13996i	0.224519	+	0.974470i	$0.427919\pi$
$773$	−21.1773	+	31.6940i	−0.761694	+	1.13996i	−0.986212	+	0.165485i	$0.947081\pi$
$774$		0			0
$775$	5.66056	+	13.6658i	0.203333	+	0.490890i
$776$		0			0
$777$	−2.96591	+	1.86593i	−0.106401	+	0.0669399i
$778$		0			0
$779$	21.3552	−	4.24781i	0.765129	−	0.152194i
$780$		0			0
$781$	8.54919	−	5.71239i	0.305914	−	0.204405i
$782$		0			0
$783$	32.7066	−	18.1484i	1.16884	−	0.648571i
$784$		0			0
$785$	27.2347	−	27.2347i	0.972050	−	0.972050i
$786$		0			0
$787$	−3.91417	+	2.61536i	−0.139525	+	0.0932276i	−0.623374	−	0.781924i	$-0.714238\pi$
$787$	−3.91417	+	2.61536i	−0.139525	+	0.0932276i	0.483849	+	0.875152i	$0.339238\pi$
$788$		0			0
$789$	−10.4296	+	23.3431i	−0.371305	+	0.831037i
$790$		0			0
$791$	0.751976	−	1.81543i	0.0267372	−	0.0645493i
$792$		0			0
$793$	2.95006	+	7.12208i	0.104760	+	0.252913i
$794$		0			0
$795$	−22.8793	+	8.74821i	−0.811444	+	0.310267i
$796$		0			0
$797$	−6.71269	+	33.7469i	−0.237776	+	1.19538i	0.658742	+	0.752369i	$0.271089\pi$
$797$	−6.71269	+	33.7469i	−0.237776	+	1.19538i	−0.896517	+	0.443009i	$0.853911\pi$
$798$		0			0
$799$	37.9471			1.34247
$800$		0			0
$801$	−5.86221	−	16.6930i	−0.207131	−	0.589818i
$802$		0			0
$803$	−23.5605	−	4.68647i	−0.831430	−	0.165382i
$804$		0			0
$805$	1.66222	−	2.48769i	0.0585856	−	0.0876796i
$806$		0			0
$807$	−38.1701	−	8.68950i	−1.34365	−	0.305885i
$808$		0			0
$809$	−51.7697	−	21.4437i	−1.82013	−	0.753921i	−0.975997	−	0.217784i	$-0.930117\pi$
$809$	−51.7697	−	21.4437i	−1.82013	−	0.753921i	−0.844130	−	0.536138i	$-0.819883\pi$
$810$		0			0
$811$	−7.49147	+	1.49015i	−0.263061	+	0.0523261i	−0.324859	−	0.945763i	$-0.605317\pi$
$811$	−7.49147	+	1.49015i	−0.263061	+	0.0523261i	0.0617975	+	0.998089i	$0.480317\pi$
$812$		0			0
$813$	−1.66705	−	0.0458350i	−0.0584659	−	0.00160750i
$814$		0			0
$815$	−18.1658	−	18.1658i	−0.636320	−	0.636320i
$816$		0			0
$817$	4.91662	−	4.91662i	0.172011	−	0.172011i
$818$		0			0
$819$	−0.746211	+	0.993865i	−0.0260747	+	0.0347284i
$820$		0			0
$821$	−0.780146	−	3.92206i	−0.0272273	−	0.136881i	0.964781	−	0.263055i	$-0.0847301\pi$
$821$	−0.780146	−	3.92206i	−0.0272273	−	0.136881i	−0.992008	+	0.126174i	$0.959730\pi$
$822$		0			0
$823$	−1.40574	+	3.39375i	−0.0490010	+	0.118299i	−0.946485	−	0.322749i	$-0.895393\pi$
$823$	−1.40574	+	3.39375i	−0.0490010	+	0.118299i	0.897484	+	0.441048i	$0.145393\pi$
$824$		0			0
$825$	14.5640	+	3.31553i	0.507054	+	0.115432i
$826$		0			0
$827$	−24.9397	−	16.6642i	−0.867240	−	0.579471i	0.0404176	−	0.999183i	$-0.487131\pi$
$827$	−24.9397	−	16.6642i	−0.867240	−	0.579471i	−0.907657	+	0.419712i	$0.862131\pi$
$828$		0			0
$829$	−3.24515	+	16.3145i	−0.112709	+	0.566625i	0.882620	+	0.470086i	$0.155777\pi$
$829$	−3.24515	+	16.3145i	−0.112709	+	0.566625i	−0.995329	+	0.0965390i	$0.969223\pi$
$830$		0			0
$831$	−21.9751	−	15.5736i	−0.762308	−	0.540242i
$832$		0			0
$833$		−	43.7007i		−	1.51414i
$834$		0			0
$835$	46.9126	+	9.33150i	1.62348	+	0.322930i
$836$		0			0
$837$	21.6407	+	25.5403i	0.748013	+	0.882801i
$838$		0			0
$839$	36.1705	−	14.9823i	1.24874	−	0.517246i	0.342307	−	0.939588i	$-0.388792\pi$
$839$	36.1705	−	14.9823i	1.24874	−	0.517246i	0.906437	+	0.422342i	$0.138792\pi$
$840$		0			0
$841$	−21.0812	−	8.73210i	−0.726937	−	0.301107i
$842$		0			0
$843$	2.21465	−	4.95673i	0.0762766	−	0.170719i
$844$		0			0
$845$	14.4092	+	21.5649i	0.495692	+	0.741855i
$846$		0			0
$847$	0.493772	+	0.493772i	0.0169662	+	0.0169662i
$848$		0			0
$849$	−25.9484	+	4.42383i	−0.890548	+	0.151825i
$850$		0			0
$851$	24.6499	+	36.8911i	0.844986	+	1.26461i
$852$		0			0
$853$	−6.90740	−	34.7258i	−0.236505	−	1.18899i	−0.898326	−	0.439329i	$-0.855216\pi$
$853$	−6.90740	−	34.7258i	−0.236505	−	1.18899i	0.661821	−	0.749662i	$-0.269784\pi$
$854$		0			0
$855$	−35.7129	−	1.96532i	−1.22136	−	0.0672125i
$856$		0			0
$857$	31.3906	−	13.0024i	1.07228	−	0.444154i	0.224487	−	0.974477i	$-0.427929\pi$
$857$	31.3906	−	13.0024i	1.07228	−	0.444154i	0.847796	+	0.530323i	$0.177929\pi$
$858$		0			0
$859$	9.06812	+	6.05912i	0.309400	+	0.206735i	0.700570	−	0.713584i	$-0.252929\pi$
$859$	9.06812	+	6.05912i	0.309400	+	0.206735i	−0.391169	+	0.920319i	$0.627929\pi$
$860$		0			0
$861$	1.31994	+	1.39458i	0.0449836	+	0.0475271i
$862$		0			0
$863$			38.2716i			1.30278i	0.758743	+	0.651390i	$0.225814\pi$
$863$			38.2716i			1.30278i	−0.758743	+	0.651390i	$0.774186\pi$
$864$		0			0
$865$			7.64946i			0.260089i
$866$		0			0
$867$	−26.8403	−	28.3580i	−0.911545	−	0.963087i
$868$		0			0
$869$	−38.8615	−	25.9664i	−1.31829	−	0.880850i
$870$		0			0
$871$	−13.9897	+	5.79473i	−0.474024	+	0.196347i
$872$		0			0
$873$	48.9769	+	2.69525i	1.65762	+	0.0912204i
$874$		0			0
$875$	0.320225	+	1.60988i	0.0108256	+	0.0544239i
$876$		0			0
$877$	−31.1713	−	46.6511i	−1.05258	−	1.57530i	−0.792658	−	0.609666i	$-0.791304\pi$
$877$	−31.1713	−	46.6511i	−1.05258	−	1.57530i	−0.259921	−	0.965630i	$-0.583696\pi$
$878$		0			0
$879$	−1.61906	+	0.276025i	−0.0546094	+	0.00931010i
$880$		0			0
$881$	−10.8757	−	10.8757i	−0.366411	−	0.366411i	0.499756	−	0.866166i	$-0.333423\pi$
$881$	−10.8757	−	10.8757i	−0.366411	−	0.366411i	−0.866166	+	0.499756i	$0.833423\pi$
$882$		0			0
$883$	27.8748	+	41.7176i	0.938062	+	1.40391i	0.914679	+	0.404182i	$0.132444\pi$
$883$	27.8748	+	41.7176i	0.938062	+	1.40391i	0.0233832	+	0.999727i	$0.492556\pi$
$884$		0			0
$885$	−17.1521	+	38.3890i	−0.576561	+	1.29043i
$886$		0			0
$887$	1.11945	+	0.463689i	0.0375873	+	0.0155692i	0.401398	−	0.915904i	$-0.368524\pi$
$887$	1.11945	+	0.463689i	0.0375873	+	0.0155692i	−0.363811	+	0.931473i	$0.618524\pi$
$888$		0			0
$889$	1.21908	−	0.504958i	0.0408865	−	0.0169357i
$890$		0			0
$891$	33.6775	−	2.91694i	1.12824	−	0.0977213i
$892$		0			0
$893$	26.1237	+	5.19633i	0.874197	+	0.173889i
$894$		0			0
$895$			48.3079i			1.61476i
$896$		0			0
$897$	12.8393	+	9.09916i	0.428693	+	0.303812i
$898$		0			0
$899$	9.04742	−	45.4845i	0.301748	−	1.51699i
$900$		0			0
$901$	−27.3748	−	18.2913i	−0.911988	−	0.609371i
$902$		0			0
$903$	0.597892	+	0.136111i	0.0198966	+	0.00452950i
$904$		0			0
$905$	−3.38321	+	8.16780i	−0.112462	+	0.271507i
$906$		0			0
$907$	5.34813	+	26.8869i	0.177582	+	0.892763i	0.962108	+	0.272668i	$0.0879061\pi$
$907$	5.34813	+	26.8869i	0.177582	+	0.892763i	−0.784527	+	0.620095i	$0.787094\pi$
$908$		0			0
$909$	−15.0855	+	20.0922i	−0.500356	+	0.666415i
$910$		0			0
$911$	−0.887056	+	0.887056i	−0.0293895	+	0.0293895i	−0.721649	−	0.692259i	$-0.756615\pi$
$911$	−0.887056	+	0.887056i	−0.0293895	+	0.0293895i	0.692259	+	0.721649i	$0.256615\pi$
$912$		0			0
$913$	−23.6104	−	23.6104i	−0.781391	−	0.781391i
$914$		0			0
$915$	−19.5576	−	0.537730i	−0.646553	−	0.0177768i
$916$		0			0
$917$	−2.08140	+	0.414016i	−0.0687338	+	0.0136720i
$918$		0			0
$919$	28.5724	+	11.8351i	0.942517	+	0.390403i	0.800413	−	0.599448i	$-0.204613\pi$
$919$	28.5724	+	11.8351i	0.942517	+	0.390403i	0.142104	+	0.989852i	$0.454613\pi$
$920$		0			0
$921$	12.4745	+	2.83985i	0.411049	+	0.0935762i
$922$		0			0
$923$	2.80358	−	4.19585i	0.0922809	−	0.138108i
$924$		0			0
$925$	20.2714	+	4.03223i	0.666520	+	0.132579i
$926$		0			0
$927$	−3.78851	−	10.7880i	−0.124431	−	0.354325i
$928$		0			0
$929$	−6.51999			−0.213914			−0.106957	−	0.994264i	$-0.534111\pi$
$929$	−6.51999			−0.213914			−0.106957	+	0.994264i	$0.534111\pi$
$930$		0			0
$931$	5.98422	−	30.0847i	0.196125	−	0.985986i
$932$		0			0
$933$	7.89988	−	3.02063i	0.258631	−	0.0988911i
$934$		0			0
$935$	24.4140	+	58.9405i	0.798422	+	1.92756i
$936$		0			0
$937$	−11.7888	+	28.4606i	−0.385122	+	0.929767i	0.605835	+	0.795590i	$0.292839\pi$
$937$	−11.7888	+	28.4606i	−0.385122	+	0.929767i	−0.990957	+	0.134177i	$0.957161\pi$
$938$		0			0
$939$	−22.8570	+	51.1574i	−0.745909	+	1.66946i
$940$		0			0
$941$	25.9536	−	17.3417i	0.846064	−	0.565322i	−0.0552578	−	0.998472i	$-0.517598\pi$
$941$	25.9536	−	17.3417i	0.846064	−	0.565322i	0.901322	+	0.433150i	$0.142598\pi$
$942$		0			0
$943$	17.1923	−	17.1923i	0.559857	−	0.559857i
$944$		0			0
$945$	−1.53042	−	2.75809i	−0.0497846	−	0.0897207i
$946$		0			0
$947$	−24.8159	+	16.5814i	−0.806408	+	0.538824i	−0.889080	−	0.457752i	$-0.848655\pi$
$947$	−24.8159	+	16.5814i	−0.806408	+	0.538824i	0.0826720	+	0.996577i	$0.473655\pi$
$948$		0			0
$949$	−11.5632	+	2.30007i	−0.375358	+	0.0746634i
$950$		0			0
$951$	−30.7396	+	19.3391i	−0.996799	+	0.627112i
$952$		0			0
$953$	1.96929	+	4.75430i	0.0637917	+	0.154007i	0.952561	−	0.304348i	$-0.0984387\pi$
$953$	1.96929	+	4.75430i	0.0637917	+	0.154007i	−0.888769	+	0.458355i	$0.848439\pi$
$954$		0			0
$955$	14.5826	−	21.8244i	0.471881	−	0.706220i
$956$		0			0
$957$	−32.1910	−	34.0112i	−1.04059	−	1.09943i
$958$		0			0
$959$	1.57903			0.0509896
$960$		0			0
$961$	10.5047			0.338860
$962$		0			0
$963$	−40.8301	+	5.81107i	−1.31573	+	0.187259i
$964$		0			0
$965$	−19.7029	+	29.4875i	−0.634260	+	0.949237i
$966$		0			0
$967$	8.73678	+	21.0925i	0.280956	+	0.678287i	0.999858	−	0.0168233i	$-0.00535529\pi$
$967$	8.73678	+	21.0925i	0.280956	+	0.678287i	−0.718903	+	0.695111i	$0.755355\pi$
$968$		0			0
$969$	−25.6000	−	40.6914i	−0.822392	−	1.30720i
$970$		0			0
$971$	4.59135	−	0.913277i	0.147344	−	0.0293085i	−0.120867	−	0.992669i	$-0.538568\pi$
$971$	4.59135	−	0.913277i	0.147344	−	0.0293085i	0.268211	+	0.963360i	$0.413568\pi$
$972$		0			0
$973$	−2.55350	+	1.70619i	−0.0818613	+	0.0546980i
$974$		0			0
$975$	7.22648	−	1.23201i	0.231432	−	0.0394558i
$976$		0			0
$977$	2.06409	−	2.06409i	0.0660360	−	0.0660360i	−0.673317	−	0.739353i	$-0.735131\pi$
$977$	2.06409	−	2.06409i	0.0660360	−	0.0660360i	0.739353	+	0.673317i	$0.235131\pi$
$978$		0			0
$979$	−18.4176	+	12.3062i	−0.588629	+	0.393309i
$980$		0			0
$981$	37.2058	−	22.0036i	1.18789	−	0.702520i
$982$		0			0
$983$	−12.7549	+	30.7931i	−0.406818	+	0.982146i	0.579151	+	0.815220i	$0.303384\pi$
$983$	−12.7549	+	30.7931i	−0.406818	+	0.982146i	−0.985969	+	0.166926i	$0.946616\pi$
$984$		0			0
$985$	7.62513	+	18.4087i	0.242957	+	0.586549i
$986$		0			0
$987$	0.838920	+	2.19404i	0.0267031	+	0.0698369i
$988$		0			0
$989$	1.51473	−	7.61507i	0.0481657	−	0.242145i
$990$		0			0
$991$	−49.3048			−1.56622			−0.783109	−	0.621885i	$-0.786367\pi$
$991$	−49.3048			−1.56622			−0.783109	+	0.621885i	$0.786367\pi$
$992$		0			0
$993$	9.93275	+	7.03927i	0.315206	+	0.223385i
$994$		0			0
$995$	−36.7733	−	7.31466i	−1.16579	−	0.231890i
$996$		0			0
$997$	0.00817348	−	0.0122325i	0.000258857	−	0.000387407i	−0.831340	−	0.555764i	$-0.812426\pi$
$997$	0.00817348	−	0.0122325i	0.000258857	−	0.000387407i	0.831599	+	0.555377i	$0.187426\pi$
$998$		0			0
$999$	46.4727	−	5.31559i	1.47033	−	0.168178i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.s.a.431.23		240
3.2	odd	2	inner	768.2.s.a.431.20		240
4.3	odd	2		192.2.s.a.131.22	yes	240
12.11	even	2		192.2.s.a.131.9	yes	240
64.21	even	16		192.2.s.a.107.9	✓	240
64.43	odd	16	inner	768.2.s.a.335.20		240
192.107	even	16	inner	768.2.s.a.335.23		240
192.149	odd	16		192.2.s.a.107.22	yes	240

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.2.s.a.107.9	✓	240	64.21	even	16
192.2.s.a.107.22	yes	240	192.149	odd	16
192.2.s.a.131.9	yes	240	12.11	even	2
192.2.s.a.131.22	yes	240	4.3	odd	2
768.2.s.a.335.20		240	64.43	odd	16	inner
768.2.s.a.335.23		240	192.107	even	16	inner
768.2.s.a.431.20		240	3.2	odd	2	inner
768.2.s.a.431.23		240	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.s (of order \(16\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.25794 - 1.19062i) q^{3} +(1.50066 - 2.24589i) q^{5} +(0.0860024 + 0.207628i) q^{7} +(0.164844 - 2.99547i) q^{9} +(-3.68379 + 0.732751i) q^{11} +(-1.53272 + 1.02413i) q^{13} +(-0.786264 - 4.61192i) q^{15} +(4.44652 - 4.44652i) q^{17} +(3.66999 - 2.45221i) q^{19} +(0.355392 + 0.158788i) q^{21} +(1.88615 - 4.55357i) q^{23} +(-0.878641 - 2.12123i) q^{25} +(-3.35910 - 3.96440i) q^{27} +(-1.40435 + 7.06016i) q^{29} -6.44241 q^{31} +(-3.76157 + 5.30775i) q^{33} +(0.595370 + 0.118426i) q^{35} +(-5.00124 + 7.48488i) q^{37} +(-0.708723 + 3.11318i) q^{39} +(4.55750 + 1.88778i) q^{41} +(1.54503 - 0.307325i) q^{43} +(-6.48012 - 4.86539i) q^{45} +(4.26705 + 4.26705i) q^{47} +(4.91403 - 4.91403i) q^{49} +(0.299352 - 10.8876i) q^{51} +(-1.02142 - 5.13504i) q^{53} +(-3.88242 + 9.37299i) q^{55} +(1.69699 - 7.45431i) q^{57} +(-7.47264 - 4.99306i) q^{59} +(0.815852 - 4.10157i) q^{61} +(0.636120 - 0.223391i) q^{63} +4.97918i q^{65} +(8.05661 + 1.60256i) q^{67} +(-3.04891 - 7.97383i) q^{69} +(-2.52914 + 1.04761i) q^{71} +(5.90887 + 2.44753i) q^{73} +(-3.63086 - 1.62225i) q^{75} +(-0.468954 - 0.701839i) q^{77} +(8.79908 + 8.79908i) q^{79} +(-8.94565 - 0.987568i) q^{81} +(4.93898 + 7.39171i) q^{83} +(-3.31370 - 16.6591i) q^{85} +(6.63938 + 10.5533i) q^{87} +(5.44855 - 2.25687i) q^{89} +(-0.344455 - 0.230158i) q^{91} +(-8.10419 + 7.67047i) q^{93} -11.9223i q^{95} +16.3503i q^{97} +(1.58768 + 11.1555i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 8 q^{3} + 16 q^{7} - 8 q^{9} - 16 q^{13} + 8 q^{15} + 16 q^{19} - 8 q^{21} - 16 q^{25} + 8 q^{27} + 32 q^{31} - 16 q^{37} + 8 q^{39} + 16 q^{43} - 8 q^{45} - 16 q^{49} + 8 q^{51} + 80 q^{55} - 8 q^{57}+ \cdots + 8 q^{99}+O(q^{100}) \) 240 * q + 8 * q^3 + 16 * q^7 - 8 * q^9 - 16 * q^13 + 8 * q^15 + 16 * q^19 - 8 * q^21 - 16 * q^25 + 8 * q^27 + 32 * q^31 - 16 * q^37 + 8 * q^39 + 16 * q^43 - 8 * q^45 - 16 * q^49 + 8 * q^51 + 80 * q^55 - 8 * q^57 - 16 * q^61 + 144 * q^67 - 8 * q^69 - 16 * q^73 + 8 * q^75 + 48 * q^79 - 8 * q^81 - 16 * q^85 + 8 * q^87 + 16 * q^91 - 32 * q^93 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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