LMFDB - Embedded newform 688.2.bg.c.17.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(17,688)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 0, 38]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("688.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.bg (of order $21$, degree $12$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$3$ over $\Q(\zeta_{21})$
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			17.3
Character	$\chi$	$=$	688.17
Dual form			688.2.bg.c.81.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.63701 - 0.504949i) q^{3} +(0.140805 - 0.358765i) q^{5} +(-1.74586 + 3.02391i) q^{7} +(-0.0539035 + 0.0367508i) q^{9} +O(q^{10})$	\(q+(1.63701 - 0.504949i) q^{3} +(0.140805 - 0.358765i) q^{5} +(-1.74586 + 3.02391i) q^{7} +(-0.0539035 + 0.0367508i) q^{9} +(3.90633 + 1.88119i) q^{11} +(1.26408 - 0.190529i) q^{13} +(0.0493402 - 0.658399i) q^{15} +(0.205594 + 0.523844i) q^{17} +(6.30490 + 4.29861i) q^{19} +(-1.33105 + 5.83172i) q^{21} +(-0.553943 - 7.39185i) q^{23} +(3.55637 + 3.29983i) q^{25} +(-3.27401 + 4.10548i) q^{27} +(3.26143 + 1.00602i) q^{29} +(0.717059 - 0.665333i) q^{31} +(7.34459 + 1.10702i) q^{33} +(0.839048 + 1.05213i) q^{35} +(-2.19799 - 3.80703i) q^{37} +(1.97309 - 0.950190i) q^{39} +(-1.07956 - 4.72986i) q^{41} +(-3.30784 - 5.66200i) q^{43} +(0.00559502 + 0.0245134i) q^{45} +(-3.93826 + 1.89657i) q^{47} +(-2.59602 - 4.49644i) q^{49} +(0.601072 + 0.753721i) q^{51} +(-9.56301 - 1.44139i) q^{53} +(1.22493 - 1.13657i) q^{55} +(12.4917 + 3.85319i) q^{57} +(2.92222 - 3.66435i) q^{59} +(3.96588 + 3.67980i) q^{61} +(-0.0170234 - 0.227161i) q^{63} +(0.109633 - 0.480333i) q^{65} +(-10.8402 - 7.39075i) q^{67} +(-4.63932 - 11.8208i) q^{69} +(-0.570335 + 7.61059i) q^{71} +(-2.05301 + 0.309442i) q^{73} +(7.48805 + 3.60605i) q^{75} +(-12.5084 + 8.52811i) q^{77} +(-4.14234 + 7.17474i) q^{79} +(-3.21501 + 8.19171i) q^{81} +(11.1319 - 3.43375i) q^{83} +0.216885 q^{85} +5.84697 q^{87} +(2.60667 - 0.804052i) q^{89} +(-1.63075 + 4.15509i) q^{91} +(0.837869 - 1.45123i) q^{93} +(2.42995 - 1.65671i) q^{95} +(0.441741 + 0.212731i) q^{97} +(-0.279700 + 0.0421580i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	$ 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) $ 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{19}{21}\right)$	$1$

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
$3$	1.63701	−	0.504949i	0.945125	−	0.291533i	0.216374	−	0.976311i	$-0.430577\pi$
$3$	1.63701	−	0.504949i	0.945125	−	0.291533i	0.728752	+	0.684778i	$0.240101\pi$
$4$		0			0
$5$	0.140805	−	0.358765i	0.0629698	−	0.160444i	−0.895926	−	0.444204i	$-0.853487\pi$
$5$	0.140805	−	0.358765i	0.0629698	−	0.160444i	0.958895	+	0.283760i	$0.0915818\pi$
$6$		0			0
$7$	−1.74586	+	3.02391i	−0.659871	+	1.14293i	0.320777	+	0.947155i	$0.396056\pi$
$7$	−1.74586	+	3.02391i	−0.659871	+	1.14293i	−0.980649	+	0.195776i	$0.937277\pi$
$8$		0			0
$9$	−0.0539035	+	0.0367508i	−0.0179678	+	0.0122503i
$10$		0			0
$11$	3.90633	+	1.88119i	1.17780	+	0.567200i	0.917270	−	0.398265i	$-0.130388\pi$
$11$	3.90633	+	1.88119i	1.17780	+	0.567200i	0.260533	+	0.965465i	$0.416102\pi$
$12$		0			0
$13$	1.26408	−	0.190529i	0.350591	−	0.0528432i	0.0286141	−	0.999591i	$-0.490891\pi$
$13$	1.26408	−	0.190529i	0.350591	−	0.0528432i	0.321977	+	0.946747i	$0.395653\pi$
$14$		0			0
$15$	0.0493402	−	0.658399i	0.0127396	−	0.169998i
$16$		0			0
$17$	0.205594	+	0.523844i	0.0498638	+	0.127051i	0.953636	−	0.300964i	$-0.0973084\pi$
$17$	0.205594	+	0.523844i	0.0498638	+	0.127051i	−0.903772	+	0.428015i	$0.859213\pi$
$18$		0			0
$19$	6.30490	+	4.29861i	1.44644	+	0.986169i	0.995614	+	0.0935586i	$0.0298242\pi$
$19$	6.30490	+	4.29861i	1.44644	+	0.986169i	0.450830	+	0.892610i	$0.351128\pi$
$20$		0			0
$21$	−1.33105	+	5.83172i	−0.290460	+	1.27259i
$22$		0			0
$23$	−0.553943	−	7.39185i	−0.115505	−	1.54131i	−0.690610	−	0.723228i	$-0.742658\pi$
$23$	−0.553943	−	7.39185i	−0.115505	−	1.54131i	0.575105	−	0.818080i	$-0.304961\pi$
$24$		0			0
$25$	3.55637	+	3.29983i	0.711275	+	0.659966i
$26$		0			0
$27$	−3.27401	+	4.10548i	−0.630084	+	0.790100i
$28$		0			0
$29$	3.26143	+	1.00602i	0.605633	+	0.186813i	0.582380	−	0.812917i	$-0.302122\pi$
$29$	3.26143	+	1.00602i	0.605633	+	0.186813i	0.0232528	+	0.999730i	$0.492598\pi$
$30$		0			0
$31$	0.717059	−	0.665333i	0.128788	−	0.119497i	−0.613150	−	0.789967i	$-0.710098\pi$
$31$	0.717059	−	0.665333i	0.128788	−	0.119497i	0.741937	+	0.670469i	$0.233907\pi$
$32$		0			0
$33$	7.34459	+	1.10702i	1.27853	+	0.192707i
$34$		0			0
$35$	0.839048	+	1.05213i	0.141825	+	0.177843i
$36$		0			0
$37$	−2.19799	−	3.80703i	−0.361348	−	0.625872i	0.626835	−	0.779152i	$-0.284350\pi$
$37$	−2.19799	−	3.80703i	−0.361348	−	0.625872i	−0.988183	+	0.153279i	$0.951017\pi$
$38$		0			0
$39$	1.97309	−	0.950190i	0.315947	−	0.152152i
$40$		0			0
$41$	−1.07956	−	4.72986i	−0.168599	−	0.738679i	−0.986559	−	0.163405i	$-0.947752\pi$
$41$	−1.07956	−	4.72986i	−0.168599	−	0.738679i	0.817960	−	0.575275i	$-0.195105\pi$
$42$		0			0
$43$	−3.30784	−	5.66200i	−0.504440	−	0.863447i
$43$	−3.30784	−	5.66200i	−0.504440	−	0.863447i
$44$		0			0
$45$	0.00559502	+	0.0245134i	0.000834056	+	0.00365424i
$46$		0			0
$47$	−3.93826	+	1.89657i	−0.574455	+	0.276643i	−0.698472	−	0.715637i	$-0.746136\pi$
$47$	−3.93826	+	1.89657i	−0.574455	+	0.276643i	0.124017	+	0.992280i	$0.460422\pi$
$48$		0			0
$49$	−2.59602	−	4.49644i	−0.370860	−	0.642348i
$50$		0			0
$51$	0.601072	+	0.753721i	0.0841670	+	0.105542i
$52$		0			0
$53$	−9.56301	−	1.44139i	−1.31358	−	0.197990i	−0.545374	−	0.838193i	$-0.683613\pi$
$53$	−9.56301	−	1.44139i	−1.31358	−	0.197990i	−0.768207	+	0.640202i	$0.778851\pi$
$54$		0			0
$55$	1.22493	−	1.13657i	0.165170	−	0.153256i
$56$		0			0
$57$	12.4917	+	3.85319i	1.65457	+	0.510368i
$58$		0			0
$59$	2.92222	−	3.66435i	0.380441	−	0.477058i	−0.554336	−	0.832293i	$-0.687028\pi$
$59$	2.92222	−	3.66435i	0.380441	−	0.477058i	0.934777	+	0.355235i	$0.115599\pi$
$60$		0			0
$61$	3.96588	+	3.67980i	0.507779	+	0.471150i	0.891964	−	0.452106i	$-0.149327\pi$
$61$	3.96588	+	3.67980i	0.507779	+	0.471150i	−0.384185	+	0.923256i	$0.625518\pi$
$62$		0			0
$63$	−0.0170234	−	0.227161i	−0.00214474	−	0.0286196i
$64$		0			0
$65$	0.109633	−	0.480333i	0.0135983	−	0.0595780i
$66$		0			0
$67$	−10.8402	−	7.39075i	−1.32435	−	0.902924i	−0.325259	−	0.945625i	$-0.605451\pi$
$67$	−10.8402	−	7.39075i	−1.32435	−	0.902924i	−0.999088	+	0.0427010i	$0.986404\pi$
$68$		0			0
$69$	−4.63932	−	11.8208i	−0.558508	−	1.42306i
$70$		0			0
$71$	−0.570335	+	7.61059i	−0.0676863	+	0.903211i	0.855239	+	0.518233i	$0.173410\pi$
$71$	−0.570335	+	7.61059i	−0.0676863	+	0.903211i	−0.922926	+	0.384978i	$0.874209\pi$
$72$		0			0
$73$	−2.05301	+	0.309442i	−0.240287	+	0.0362175i	−0.268082	−	0.963396i	$-0.586390\pi$
$73$	−2.05301	+	0.309442i	−0.240287	+	0.0362175i	0.0277945	+	0.999614i	$0.491152\pi$
$74$		0			0
$75$	7.48805	+	3.60605i	0.864646	+	0.416391i
$76$		0			0
$77$	−12.5084	+	8.52811i	−1.42547	+	0.971868i
$78$		0			0
$79$	−4.14234	+	7.17474i	−0.466049	+	0.807221i	−0.999248	−	0.0387687i	$-0.987656\pi$
$79$	−4.14234	+	7.17474i	−0.466049	+	0.807221i	0.533199	+	0.845990i	$0.320990\pi$
$80$		0			0
$81$	−3.21501	+	8.19171i	−0.357223	+	0.910190i
$82$		0			0
$83$	11.1319	−	3.43375i	1.22189	−	0.376903i	0.384302	−	0.923207i	$-0.374442\pi$
$83$	11.1319	−	3.43375i	1.22189	−	0.376903i	0.837587	+	0.546305i	$0.183966\pi$
$84$		0			0
$85$	0.216885			0.0235245
$86$		0			0
$87$	5.84697			0.626861
$88$		0			0
$89$	2.60667	−	0.804052i	0.276307	−	0.0852293i	−0.153505	−	0.988148i	$-0.549056\pi$
$89$	2.60667	−	0.804052i	0.276307	−	0.0852293i	0.429812	+	0.902919i	$0.358580\pi$
$90$		0			0
$91$	−1.63075	+	4.15509i	−0.170949	+	0.435571i
$92$		0			0
$93$	0.837869	−	1.45123i	0.0868830	−	0.150486i
$94$		0			0
$95$	2.42995	−	1.65671i	0.249308	−	0.169975i
$96$		0			0
$97$	0.441741	+	0.212731i	0.0448520	+	0.0215996i	0.456175	−	0.889890i	$-0.349219\pi$
$97$	0.441741	+	0.212731i	0.0448520	+	0.0215996i	−0.411323	+	0.911490i	$0.634933\pi$
$98$		0			0
$99$	−0.279700	+	0.0421580i	−0.0281109	+	0.00423704i
$100$		0			0
$101$	0.164575	−	2.19611i	0.0163759	−	0.218521i	−0.983012	−	0.183541i	$-0.941244\pi$
$101$	0.164575	−	2.19611i	0.0163759	−	0.218521i	0.999388	−	0.0349802i	$-0.0111368\pi$
$102$		0			0
$103$	1.86980	+	4.76417i	0.184237	+	0.469427i	0.992915	−	0.118823i	$-0.0379121\pi$
$103$	1.86980	+	4.76417i	0.184237	+	0.469427i	−0.808679	+	0.588250i	$0.799817\pi$
$104$		0			0
$105$	1.90480	+	1.29867i	0.185889	+	0.126737i
$106$		0			0
$107$	−2.28640	+	10.0174i	−0.221035	+	0.968418i	0.735666	+	0.677345i	$0.236869\pi$
$107$	−2.28640	+	10.0174i	−0.221035	+	0.968418i	−0.956701	+	0.291073i	$0.905988\pi$
$108$		0			0
$109$	−0.0839896	−	1.12076i	−0.00804474	−	0.107350i	0.991736	−	0.128297i	$-0.0409509\pi$
$109$	−0.0839896	−	1.12076i	−0.00804474	−	0.107350i	−0.999781	+	0.0209470i	$0.993332\pi$
$110$		0			0
$111$	−5.52048	−	5.12226i	−0.523981	−	0.486183i
$112$		0			0
$113$	9.95806	−	12.4870i	0.936776	−	1.17468i	−0.0476475	−	0.998864i	$-0.515172\pi$
$113$	9.95806	−	12.4870i	0.936776	−	1.17468i	0.984423	−	0.175815i	$-0.0562562\pi$
$114$		0			0
$115$	−2.72993	−	0.842073i	−0.254568	−	0.0785237i
$116$		0			0
$117$	−0.0611361	+	0.0567260i	−0.00565203	+	0.00524432i
$118$		0			0
$119$	−1.94299	−	0.292859i	−0.178114	−	0.0268463i
$120$		0			0
$121$	4.86216	+	6.09695i	0.442014	+	0.554268i
$122$		0			0
$123$	−4.15558	−	7.19768i	−0.374696	−	0.648993i
$124$		0			0
$125$	3.42081	−	1.64738i	0.305967	−	0.147346i
$126$		0			0
$127$	−0.749226	−	3.28258i	−0.0664831	−	0.291281i	0.930747	−	0.365665i	$-0.119158\pi$
$127$	−0.749226	−	3.28258i	−0.0664831	−	0.291281i	−0.997230	+	0.0743833i	$0.976301\pi$
$128$		0			0
$129$	−8.27397	−	7.59843i	−0.728482	−	0.669005i
$130$		0			0
$131$	−0.342281	−	1.49963i	−0.0299053	−	0.131024i	0.957772	−	0.287530i	$-0.0928340\pi$
$131$	−0.342281	−	1.49963i	−0.0299053	−	0.131024i	−0.987677	+	0.156506i	$0.949977\pi$
$132$		0			0
$133$	−24.0060	+	11.5607i	−2.08159	+	1.00244i
$134$		0			0
$135$	1.01191	+	1.75267i	0.0870909	+	0.150846i
$136$		0			0
$137$	−1.98722	−	2.49190i	−0.169780	−	0.212897i	0.689661	−	0.724132i	$-0.257760\pi$
$137$	−1.98722	−	2.49190i	−0.169780	−	0.212897i	−0.859441	+	0.511235i	$0.829188\pi$
$138$		0			0
$139$	−19.8520	−	2.99220i	−1.68382	−	0.253795i	−0.763741	−	0.645523i	$-0.776640\pi$
$139$	−19.8520	−	2.99220i	−1.68382	−	0.253795i	−0.920081	+	0.391727i	$0.871878\pi$
$140$		0			0
$141$	−5.48929	+	5.09332i	−0.462282	+	0.428935i
$142$		0			0
$143$	5.29632	+	1.63370i	0.442900	+	0.136617i
$144$		0			0
$145$	0.820149	−	1.02843i	0.0681097	−	0.0854069i
$146$		0			0
$147$	−6.52017	−	6.04984i	−0.537775	−	0.498982i
$148$		0			0
$149$	−0.944726	−	12.6065i	−0.0773949	−	1.03276i	−0.891789	−	0.452451i	$-0.850550\pi$
$149$	−0.944726	−	12.6065i	−0.0773949	−	1.03276i	0.814394	−	0.580312i	$-0.197069\pi$
$150$		0			0
$151$	0.223758	−	0.980350i	0.0182092	−	0.0797798i	−0.965007	−	0.262224i	$-0.915544\pi$
$151$	0.223758	−	0.980350i	0.0182092	−	0.0797798i	0.983216	+	0.182445i	$0.0584011\pi$
$152$		0			0
$153$	−0.0303339	−	0.0206813i	−0.00245235	−	0.00167198i
$154$		0			0
$155$	−0.137733	−	0.350937i	−0.0110630	−	0.0281880i
$156$		0			0
$157$	0.355966	−	4.75004i	0.0284092	−	0.379095i	−0.964819	−	0.262915i	$-0.915316\pi$
$157$	0.355966	−	4.75004i	0.0284092	−	0.379095i	0.993228	−	0.116180i	$-0.0370648\pi$
$158$		0			0
$159$	−16.3825	+	2.46927i	−1.29922	+	0.195826i
$160$		0			0
$161$	23.3194	+	11.2300i	1.83783	+	0.885050i
$162$		0			0
$163$	7.84688	−	5.34991i	0.614615	−	0.419037i	−0.215599	−	0.976482i	$-0.569171\pi$
$163$	7.84688	−	5.34991i	0.614615	−	0.419037i	0.830214	+	0.557445i	$0.188218\pi$
$164$		0			0
$165$	1.43131	−	2.47911i	0.111428	−	0.192998i
$166$		0			0
$167$	8.26817	−	21.0670i	0.639810	−	1.63021i	−0.129684	−	0.991555i	$-0.541396\pi$
$167$	8.26817	−	21.0670i	0.639810	−	1.63021i	0.769494	−	0.638654i	$-0.220509\pi$
$168$		0			0
$169$	−10.8609	+	3.35013i	−0.835451	+	0.257702i
$170$		0			0
$171$	−0.497834			−0.0380703
$172$		0			0
$173$	9.30235			0.707245			0.353622	−	0.935388i	$-0.384950\pi$
$173$	9.30235			0.707245			0.353622	+	0.935388i	$0.384950\pi$
$174$		0			0
$175$	−16.1873	+	4.99312i	−1.22365	+	0.377445i
$176$		0			0
$177$	2.93338	−	7.47413i	0.220486	−	0.561790i
$178$		0			0
$179$	3.61481	−	6.26104i	0.270184	−	0.467972i	−0.698725	−	0.715390i	$-0.746249\pi$
$179$	3.61481	−	6.26104i	0.270184	−	0.467972i	0.968909	+	0.247418i	$0.0795822\pi$
$180$		0			0
$181$	−15.5329	+	10.5902i	−1.15455	+	0.787162i	−0.980295	−	0.197538i	$-0.936705\pi$
$181$	−15.5329	+	10.5902i	−1.15455	+	0.787162i	−0.174259	+	0.984700i	$0.555753\pi$
$182$		0			0
$183$	8.35028	+	4.02129i	0.617271	+	0.297262i
$184$		0			0
$185$	−1.67532	+	0.252513i	−0.123172	+	0.0185652i
$186$		0			0
$187$	−0.182333	+	2.43307i	−0.0133335	+	0.177924i
$188$		0			0
$189$	−6.69865	−	17.0679i	−0.487255	−	1.24151i
$190$		0			0
$191$	5.81055	+	3.96157i	0.420437	+	0.286649i	0.755001	−	0.655724i	$-0.227637\pi$
$191$	5.81055	+	3.96157i	0.420437	+	0.286649i	−0.334564	+	0.942373i	$0.608589\pi$
$192$		0			0
$193$	0.916048	−	4.01347i	0.0659385	−	0.288896i	−0.931198	−	0.364513i	$-0.881236\pi$
$193$	0.916048	−	4.01347i	0.0659385	−	0.288896i	0.997137	+	0.0756172i	$0.0240927\pi$
$194$		0			0
$195$	−0.0630742	−	0.841667i	−0.00451684	−	0.0602730i
$196$		0			0
$197$	−1.32247	−	1.22707i	−0.0942220	−	0.0874252i	0.631656	−	0.775249i	$-0.282376\pi$
$197$	−1.32247	−	1.22707i	−0.0942220	−	0.0874252i	−0.725878	+	0.687824i	$0.758566\pi$
$198$		0			0
$199$	14.1954	−	17.8005i	1.00629	−	1.26185i	0.0414119	−	0.999142i	$-0.486814\pi$
$199$	14.1954	−	17.8005i	1.00629	−	1.26185i	0.964877	−	0.262704i	$-0.0846142\pi$
$200$		0			0
$201$	−21.4775	−	6.62493i	−1.51491	−	0.467286i
$202$		0			0
$203$	−8.73610	+	8.10592i	−0.613154	+	0.568924i
$204$		0			0
$205$	−1.84891	−	0.278679i	−0.129134	−	0.0194638i
$206$		0			0
$207$	0.301516	+	0.378089i	0.0209568	+	0.0262790i
$208$		0			0
$209$	16.5425	+	28.6525i	1.14427	+	1.98194i
$210$		0			0
$211$	−7.10545	+	3.42180i	−0.489159	+	0.235567i	−0.662170	−	0.749353i	$-0.730365\pi$
$211$	−7.10545	+	3.42180i	−0.489159	+	0.235567i	0.173011	+	0.984920i	$0.444650\pi$
$212$		0			0
$213$	2.90932	+	12.7466i	0.199343	+	0.873381i
$214$		0			0
$215$	−2.49708	+	0.389499i	−0.170300	+	0.0265636i
$216$		0			0
$217$	0.760027	+	3.32990i	0.0515940	+	0.226048i
$218$		0			0
$219$	−3.20454	+	1.54323i	−0.216543	+	0.104282i
$220$		0			0
$221$	0.359693	+	0.623007i	0.0241956	+	0.0419080i
$222$		0			0
$223$	−5.36853	−	6.73192i	−0.359503	−	0.450803i	0.568884	−	0.822418i	$-0.307375\pi$
$223$	−5.36853	−	6.73192i	−0.359503	−	0.450803i	−0.928387	+	0.371615i	$0.878804\pi$
$224$		0			0
$225$	−0.312973	−	0.0471730i	−0.0208648	−	0.00314487i
$226$		0			0
$227$	−5.97274	+	5.54190i	−0.396425	+	0.367829i	−0.853049	−	0.521831i	$-0.825249\pi$
$227$	−5.97274	+	5.54190i	−0.396425	+	0.367829i	0.456624	+	0.889660i	$0.349059\pi$
$228$		0			0
$229$	−23.6766	−	7.30327i	−1.56460	−	0.482614i	−0.613030	−	0.790060i	$-0.710049\pi$
$229$	−23.6766	−	7.30327i	−1.56460	−	0.482614i	−0.951565	+	0.307446i	$0.900526\pi$
$230$		0			0
$231$	−16.1701	+	20.2767i	−1.06392	+	1.33411i
$232$		0			0
$233$	−6.32022	−	5.86431i	−0.414051	−	0.384183i	0.445459	−	0.895303i	$-0.353041\pi$
$233$	−6.32022	−	5.86431i	−0.414051	−	0.384183i	−0.859510	+	0.511119i	$0.829231\pi$
$234$		0			0
$235$	0.125895	+	1.67996i	0.00821251	+	0.109588i
$236$		0			0
$237$	−3.15815	+	13.8368i	−0.205144	+	0.898794i
$238$		0			0
$239$	−2.78125	−	1.89623i	−0.179904	−	0.122657i	0.470020	−	0.882656i	$-0.344247\pi$
$239$	−2.78125	−	1.89623i	−0.179904	−	0.122657i	−0.649924	+	0.759999i	$0.725199\pi$
$240$		0			0
$241$	−9.87013	−	25.1487i	−0.635791	−	1.61997i	−0.776645	−	0.629939i	$-0.783080\pi$
$241$	−9.87013	−	25.1487i	−0.635791	−	1.61997i	0.140854	−	0.990030i	$-0.455015\pi$
$242$		0			0
$243$	0.0506572	−	0.675974i	0.00324966	−	0.0433637i
$244$		0			0
$245$	−1.97870	+	0.298240i	−0.126414	+	0.0190539i
$246$		0			0
$247$	8.78888	+	4.23250i	0.559223	+	0.269308i
$248$		0			0
$249$	16.4892	−	11.2421i	1.04496	−	0.712441i
$250$		0			0
$251$	−5.38285	+	9.32338i	−0.339763	+	0.588486i	−0.984388	−	0.176012i	$-0.943680\pi$
$251$	−5.38285	+	9.32338i	−0.339763	+	0.588486i	0.644625	+	0.764499i	$0.277013\pi$
$252$		0			0
$253$	11.7416	−	29.9171i	0.738188	−	1.88087i
$254$		0			0
$255$	0.355042	−	0.109516i	0.0222336	−	0.00685816i
$256$		0			0
$257$	12.7546			0.795610			0.397805	−	0.917470i	$-0.369772\pi$
$257$	12.7546			0.795610			0.397805	+	0.917470i	$0.369772\pi$
$258$		0			0
$259$	15.3495			0.953772
$260$		0			0
$261$	−0.212775	+	0.0656323i	−0.0131704	+	0.00406254i
$262$		0			0
$263$	5.41921	−	13.8079i	0.334163	−	0.851433i	−0.660742	−	0.750613i	$-0.729758\pi$
$263$	5.41921	−	13.8079i	0.334163	−	0.851433i	0.994905	−	0.100820i	$-0.0321465\pi$
$264$		0			0
$265$	−1.86364	+	3.22792i	−0.114482	+	0.198289i
$266$		0			0
$267$	3.86113	−	2.63247i	0.236297	−	0.161105i
$268$		0			0
$269$	−9.64831	−	4.64638i	−0.588268	−	0.283295i	0.115977	−	0.993252i	$-0.463000\pi$
$269$	−9.64831	−	4.64638i	−0.588268	−	0.283295i	−0.704245	+	0.709957i	$0.748714\pi$
$270$		0			0
$271$	14.3276	−	2.15955i	0.870343	−	0.131183i	0.301338	−	0.953517i	$-0.402567\pi$
$271$	14.3276	−	2.15955i	0.870343	−	0.131183i	0.569005	+	0.822334i	$0.307329\pi$
$272$		0			0
$273$	−0.571441	+	7.62534i	−0.0345852	+	0.461507i
$274$		0			0
$275$	7.68476	+	19.5804i	0.463408	+	1.18075i
$276$		0			0
$277$	16.8697	+	11.5016i	1.01360	+	0.691061i	0.951703	−	0.307021i	$-0.0993322\pi$
$277$	16.8697	+	11.5016i	1.01360	+	0.691061i	0.0618982	+	0.998082i	$0.480285\pi$
$278$		0			0
$279$	−0.0142005	+	0.0622163i	−0.000850159	+	0.00372479i
$280$		0			0
$281$	−0.541902	−	7.23118i	−0.0323272	−	0.431376i	−0.989762	−	0.142727i	$-0.954413\pi$
$281$	−0.541902	−	7.23118i	−0.0323272	−	0.431376i	0.957435	−	0.288649i	$-0.0932061\pi$
$282$		0			0
$283$	−22.3431	−	20.7313i	−1.32816	−	1.23235i	−0.952113	−	0.305748i	$-0.901094\pi$
$283$	−22.3431	−	20.7313i	−1.32816	−	1.23235i	−0.376044	−	0.926602i	$-0.622716\pi$
$284$		0			0
$285$	3.14129	−	3.93905i	0.186074	−	0.233329i
$286$		0			0
$287$	16.1874	+	4.99316i	0.955513	+	0.294737i
$288$		0			0
$289$	12.2297	−	11.3475i	0.719396	−	0.667502i
$290$		0			0
$291$	0.830552	+	0.125186i	0.0486878	+	0.00733850i
$292$		0			0
$293$	16.0151	+	20.0823i	0.935611	+	1.17322i	0.984671	+	0.174424i	$0.0558063\pi$
$293$	16.0151	+	20.0823i	0.935611	+	1.17322i	−0.0490592	+	0.998796i	$0.515622\pi$
$294$		0			0
$295$	−0.903177	−	1.56435i	−0.0525850	−	0.0910799i
$296$		0			0
$297$	−20.5126	+	9.87833i	−1.19026	+	0.573199i
$298$		0			0
$299$	−2.10859	−	9.23832i	−0.121943	−	0.534266i
$300$		0			0
$301$	22.8964	−	0.117571i	1.31973	−	0.00677671i
$302$		0			0
$303$	−0.839512	−	3.67814i	−0.0482287	−	0.211304i
$304$		0			0
$305$	1.87860	−	0.904685i	0.107568	−	0.0518021i
$306$		0			0
$307$	7.92491	+	13.7263i	0.452298	+	0.783404i	0.998528	−	0.0542314i	$-0.0172709\pi$
$307$	7.92491	+	13.7263i	0.452298	+	0.783404i	−0.546230	+	0.837635i	$0.683938\pi$
$308$		0			0
$309$	5.46653	+	6.85481i	0.310980	+	0.389957i
$310$		0			0
$311$	26.7139	+	4.02647i	1.51481	+	0.228320i	0.853221	−	0.521550i	$-0.174646\pi$
$311$	26.7139	+	4.02647i	1.51481	+	0.228320i	0.661585	+	0.749870i	$0.269884\pi$
$312$		0			0
$313$	−14.1284	+	13.1093i	−0.798586	+	0.740979i	−0.969707	−	0.244271i	$-0.921452\pi$
$313$	−14.1284	+	13.1093i	−0.798586	+	0.740979i	0.171121	+	0.985250i	$0.445261\pi$
$314$		0			0
$315$	−0.0838943	−	0.0258780i	−0.00472691	−	0.00145806i
$316$		0			0
$317$	−8.01912	+	10.0557i	−0.450399	+	0.564782i	−0.954251	−	0.299008i	$-0.903344\pi$
$317$	−8.01912	+	10.0557i	−0.450399	+	0.564782i	0.503852	+	0.863790i	$0.331916\pi$
$318$		0			0
$319$	10.8477	+	10.0652i	0.607356	+	0.563544i
$320$		0			0
$321$	1.31542	+	17.5530i	0.0734195	+	0.979715i
$322$		0			0
$323$	−0.955553	+	4.18655i	−0.0531684	+	0.232946i
$324$		0			0
$325$	5.12424	+	3.49365i	0.284242	+	0.193793i
$326$		0			0
$327$	−0.703420	−	1.79228i	−0.0388992	−	0.0991135i
$328$		0			0
$329$	1.14059	−	15.2201i	0.0628827	−	0.839111i
$330$		0			0
$331$	−2.51633	+	0.379275i	−0.138310	+	0.0208469i	−0.217832	−	0.975986i	$-0.569899\pi$
$331$	−2.51633	+	0.379275i	−0.138310	+	0.0208469i	0.0795224	+	0.996833i	$0.474660\pi$
$332$		0			0
$333$	0.258391	+	0.124435i	0.0141597	+	0.00681897i
$334$		0			0
$335$	−4.17790	+	2.84844i	−0.228263	+	0.155627i
$336$		0			0
$337$	−0.368739	+	0.638675i	−0.0200865	+	0.0347908i	−0.875894	−	0.482504i	$-0.839727\pi$
$337$	−0.368739	+	0.638675i	−0.0200865	+	0.0347908i	0.855807	+	0.517295i	$0.173061\pi$
$338$		0			0
$339$	9.99609	−	25.4696i	0.542913	−	1.38332i
$340$		0			0
$341$	4.05269	−	1.25009i	0.219465	−	0.0676961i
$342$		0			0
$343$	−6.31287			−0.340863
$344$		0			0
$345$	−4.89412			−0.263491
$346$		0			0
$347$	5.00949	−	1.54522i	0.268923	−	0.0829519i	−0.157361	−	0.987541i	$-0.550299\pi$
$347$	5.00949	−	1.54522i	0.268923	−	0.0829519i	0.426284	+	0.904589i	$0.359822\pi$
$348$		0			0
$349$	3.64385	−	9.28439i	0.195051	−	0.496982i	−0.799569	−	0.600575i	$-0.794939\pi$
$349$	3.64385	−	9.28439i	0.195051	−	0.496982i	0.994620	+	0.103593i	$0.0330338\pi$
$350$		0			0
$351$	−3.35638	+	5.81343i	−0.179151	+	0.310298i
$352$		0			0
$353$	20.6981	−	14.1117i	1.10165	−	0.751090i	0.130948	−	0.991389i	$-0.458198\pi$
$353$	20.6981	−	14.1117i	1.10165	−	0.751090i	0.970699	+	0.240299i	$0.0772454\pi$
$354$		0			0
$355$	2.65011	+	1.27622i	0.140653	+	0.0677349i
$356$		0			0
$357$	−3.32857	+	0.501701i	−0.176167	+	0.0265528i
$358$		0			0
$359$	−1.02841	+	13.7231i	−0.0542772	+	0.724279i	0.901985	+	0.431768i	$0.142110\pi$
$359$	−1.02841	+	13.7231i	−0.0542772	+	0.724279i	−0.956262	+	0.292511i	$0.905509\pi$
$360$		0			0
$361$	14.3323	+	36.5180i	0.754330	+	1.92200i
$362$		0			0
$363$	11.0380	+	7.52560i	0.579346	+	0.394992i
$364$		0			0
$365$	−0.178057	+	0.780120i	−0.00931995	+	0.0408334i
$366$		0			0
$367$	2.05378	+	27.4058i	0.107206	+	1.43057i	0.748429	+	0.663215i	$0.230809\pi$
$367$	2.05378	+	27.4058i	0.107206	+	1.43057i	−0.641222	+	0.767355i	$0.721572\pi$
$368$		0			0
$369$	0.232018	+	0.215281i	0.0120784	+	0.0112071i
$370$		0			0
$371$	21.0543	−	26.4012i	1.09308	−	1.37068i
$372$		0			0
$373$	15.0169	+	4.63210i	0.777545	+	0.239841i	0.658030	−	0.752992i	$-0.271390\pi$
$373$	15.0169	+	4.63210i	0.777545	+	0.239841i	0.119515	+	0.992832i	$0.461866\pi$
$374$		0			0
$375$	4.76805	−	4.42410i	0.246221	−	0.228460i
$376$		0			0
$377$	4.31437	+	0.650287i	0.222201	+	0.0334915i
$378$		0			0
$379$	16.0442	+	20.1188i	0.824136	+	1.03343i	0.998808	+	0.0488070i	$0.0155419\pi$
$379$	16.0442	+	20.1188i	0.824136	+	1.03343i	−0.174672	+	0.984627i	$0.555887\pi$
$380$		0			0
$381$	−2.88402	−	4.99527i	−0.147753	−	0.255916i
$382$		0			0
$383$	−16.2586	+	7.82975i	−0.830777	+	0.400081i	−0.800407	−	0.599457i	$-0.795383\pi$
$383$	−16.2586	+	7.82975i	−0.830777	+	0.400081i	−0.0303708	+	0.999539i	$0.509669\pi$
$384$		0			0
$385$	1.29834	+	5.68838i	0.0661694	+	0.289907i
$386$		0			0
$387$	0.386387	+	0.183636i	0.0196412	+	0.00933474i
$388$		0			0
$389$	5.62992	+	24.6663i	0.285448	+	1.25063i	0.890698	+	0.454596i	$0.150216\pi$
$389$	5.62992	+	24.6663i	0.285448	+	1.25063i	−0.605249	+	0.796036i	$0.706927\pi$
$390$		0			0
$391$	3.75829	−	1.80990i	0.190065	−	0.0915304i
$392$		0			0
$393$	−1.31756	−	2.28207i	−0.0664619	−	0.115115i
$394$		0			0
$395$	1.99078	+	2.49636i	0.100167	+	0.125606i
$396$		0			0
$397$	−2.61362	−	0.393940i	−0.131174	−	0.0197713i	0.0831270	−	0.996539i	$-0.473509\pi$
$397$	−2.61362	−	0.393940i	−0.131174	−	0.0197713i	−0.214301	+	0.976768i	$0.568747\pi$
$398$		0			0
$399$	−33.4605	+	31.0468i	−1.67512	+	1.55428i
$400$		0			0
$401$	28.1025	+	8.66848i	1.40337	+	0.432883i	0.901771	−	0.432215i	$-0.142268\pi$
$401$	28.1025	+	8.66848i	1.40337	+	0.432883i	0.501603	+	0.865098i	$0.332744\pi$
$402$		0			0
$403$	0.779651	−	0.977651i	0.0388372	−	0.0487003i
$404$		0			0
$405$	2.48621	+	2.30686i	0.123541	+	0.114629i
$406$		0			0
$407$	−1.42433	−	19.0064i	−0.0706014	−	0.942111i
$408$		0			0
$409$	0.0843680	−	0.369640i	0.00417173	−	0.0182775i	−0.972799	−	0.231650i	$-0.925588\pi$
$409$	0.0843680	−	0.369640i	0.00417173	−	0.0182775i	0.976971	+	0.213372i	$0.0684448\pi$
$410$		0			0
$411$	−4.51138	−	3.07581i	−0.222530	−	0.151718i
$412$		0			0
$413$	5.97889	+	15.2340i	0.294202	+	0.749614i
$414$		0			0
$415$	0.335522	−	4.47723i	0.0164701	−	0.219779i
$416$		0			0
$417$	−34.0087	+	5.12599i	−1.66541	+	0.251021i
$418$		0			0
$419$	14.9002	+	7.17556i	0.727922	+	0.350549i	0.760867	−	0.648908i	$-0.224774\pi$
$419$	14.9002	+	7.17556i	0.727922	+	0.350549i	−0.0329444	+	0.999457i	$0.510488\pi$
$420$		0			0
$421$	−7.57559	+	5.16495i	−0.369212	+	0.251724i	−0.733685	−	0.679489i	$-0.762201\pi$
$421$	−7.57559	+	5.16495i	−0.369212	+	0.251724i	0.364473	+	0.931214i	$0.381249\pi$
$422$		0			0
$423$	0.142586	−	0.246966i	0.00693277	−	0.0120079i
$424$		0			0
$425$	−0.997429	+	2.54141i	−0.0483824	+	0.123276i
$426$		0			0
$427$	−18.0512	+	5.56807i	−0.873561	+	0.269458i
$428$		0			0
$429$	9.49503			0.458425
$430$		0			0
$431$	−9.52715			−0.458907			−0.229453	−	0.973320i	$-0.573694\pi$
$431$	−9.52715			−0.458907			−0.229453	+	0.973320i	$0.573694\pi$
$432$		0			0
$433$	−27.5381	+	8.49437i	−1.32339	+	0.408213i	−0.874350	−	0.485296i	$-0.838712\pi$
$433$	−27.5381	+	8.49437i	−1.32339	+	0.408213i	−0.449045	+	0.893509i	$0.648236\pi$
$434$		0			0
$435$	0.823282	−	2.09769i	0.0394733	−	0.100576i
$436$		0			0
$437$	28.2821	−	48.9861i	1.35292	−	2.34332i
$438$		0			0
$439$	5.56118	−	3.79155i	0.265421	−	0.180961i	−0.423296	−	0.905992i	$-0.639127\pi$
$439$	5.56118	−	3.79155i	0.265421	−	0.180961i	0.688716	+	0.725031i	$0.258175\pi$
$440$		0			0
$441$	0.305182	+	0.146968i	0.0145325	+	0.00699848i
$442$		0			0
$443$	3.28651	−	0.495362i	0.156147	−	0.0235354i	−0.0705035	−	0.997512i	$-0.522461\pi$
$443$	3.28651	−	0.495362i	0.156147	−	0.0235354i	0.226650	+	0.973976i	$0.427222\pi$
$444$		0			0
$445$	0.0785665	−	1.04840i	0.00372441	−	0.0496988i
$446$		0			0
$447$	−7.91216	−	20.1598i	−0.374232	−	0.953528i
$448$		0			0
$449$	−3.54373	−	2.41607i	−0.167239	−	0.114022i	0.476808	−	0.879007i	$-0.341794\pi$
$449$	−3.54373	−	2.41607i	−0.167239	−	0.114022i	−0.644047	+	0.764986i	$0.722746\pi$
$450$		0			0
$451$	4.68064	−	20.5072i	0.220403	−	0.965648i
$452$		0			0
$453$	−0.128733	−	1.71782i	−0.00604841	−	0.0807105i
$454$		0			0
$455$	1.26108	+	1.17011i	0.0591204	+	0.0548557i
$456$		0			0
$457$	−1.37989	+	1.73032i	−0.0645484	+	0.0809412i	−0.813058	−	0.582183i	$-0.802199\pi$
$457$	−1.37989	+	1.73032i	−0.0645484	+	0.0809412i	0.748509	+	0.663125i	$0.230770\pi$
$458$		0			0
$459$	−2.82375	−	0.871010i	−0.131801	−	0.0406553i
$460$		0			0
$461$	−11.1729	+	10.3670i	−0.520375	+	0.482837i	−0.896074	−	0.443904i	$-0.853593\pi$
$461$	−11.1729	+	10.3670i	−0.520375	+	0.482837i	0.375699	+	0.926742i	$0.377403\pi$
$462$		0			0
$463$	−10.6302	−	1.60224i	−0.494027	−	0.0744626i	−0.102697	−	0.994713i	$-0.532747\pi$
$463$	−10.6302	−	1.60224i	−0.494027	−	0.0744626i	−0.391331	+	0.920250i	$0.627985\pi$
$464$		0			0
$465$	−0.402675	−	0.504938i	−0.0186736	−	0.0234160i
$466$		0			0
$467$	−7.41391	−	12.8413i	−0.343075	−	0.594223i	0.641927	−	0.766766i	$-0.278135\pi$
$467$	−7.41391	−	12.8413i	−0.343075	−	0.594223i	−0.985002	+	0.172543i	$0.944802\pi$
$468$		0			0
$469$	41.2745	−	19.8767i	1.90588	−	0.917822i
$470$		0			0
$471$	−1.81581	−	7.95559i	−0.0836682	−	0.366574i
$472$		0			0
$473$	−2.27021	−	28.3403i	−0.104384	−	1.30309i
$474$		0			0
$475$	8.23789	+	36.0926i	0.377981	+	1.65604i
$476$		0			0
$477$	0.568453	−	0.273752i	0.0260277	−	0.0125343i
$478$		0			0
$479$	−17.1592	−	29.7205i	−0.784022	−	1.35797i	−0.929581	−	0.368617i	$-0.879831\pi$
$479$	−17.1592	−	29.7205i	−0.784022	−	1.35797i	0.145559	−	0.989350i	$-0.453502\pi$
$480$		0			0
$481$	−3.50378	−	4.39360i	−0.159758	−	0.200331i
$482$		0			0
$483$	43.8446	+	6.60851i	1.99500	+	0.300697i
$484$		0			0
$485$	0.138520	−	0.128528i	0.00628986	−	0.00583614i
$486$		0			0
$487$	6.21312	+	1.91649i	0.281543	+	0.0868446i	0.432311	−	0.901725i	$-0.357698\pi$
$487$	6.21312	+	1.91649i	0.281543	+	0.0868446i	−0.150768	+	0.988569i	$0.548175\pi$
$488$		0			0
$489$	10.1439	−	12.7201i	0.458725	−	0.575223i
$490$		0			0
$491$	−5.97382	−	5.54289i	−0.269595	−	0.250147i	0.533777	−	0.845625i	$-0.320772\pi$
$491$	−5.97382	−	5.54289i	−0.269595	−	0.250147i	−0.803372	+	0.595478i	$0.796963\pi$
$492$		0			0
$493$	0.143533	+	1.91531i	0.00646439	+	0.0862613i
$494$		0			0
$495$	−0.0242583	+	0.106283i	−0.00109033	+	0.00477705i
$496$		0			0
$497$	−22.0180	−	15.0116i	−0.987643	−	0.673364i
$498$		0			0
$499$	−2.88373	−	7.34763i	−0.129094	−	0.328925i	0.851684	−	0.524056i	$-0.175582\pi$
$499$	−2.88373	−	7.34763i	−0.129094	−	0.328925i	−0.980777	+	0.195131i	$0.937487\pi$
$500$		0			0
$501$	2.89730	−	38.6617i	0.129442	−	1.72728i
$502$		0			0
$503$	24.2178	−	3.65024i	1.07982	−	0.162756i	0.415044	−	0.909801i	$-0.363766\pi$
$503$	24.2178	−	3.65024i	1.07982	−	0.162756i	0.664773	+	0.747045i	$0.268528\pi$
$504$		0			0
$505$	−0.764713	−	0.368266i	−0.0340293	−	0.0163876i
$506$		0			0
$507$	−16.0876	+	10.9684i	−0.714477	+	0.487122i
$508$		0			0
$509$	−15.1556	+	26.2503i	−0.671760	+	1.16352i	0.305645	+	0.952146i	$0.401128\pi$
$509$	−15.1556	+	26.2503i	−0.671760	+	1.16352i	−0.977405	+	0.211376i	$0.932205\pi$
$510$		0			0
$511$	2.64854	−	6.74837i	0.117165	−	0.298530i
$512$		0			0
$513$	−38.2902	+	11.8110i	−1.69055	+	0.521466i
$514$		0			0
$515$	1.97249			0.0869184
$516$		0			0
$517$	−18.9520			−0.833507
$518$		0			0
$519$	15.2280	−	4.69722i	0.668435	−	0.206185i
$520$		0			0
$521$	4.59482	−	11.7074i	0.201303	−	0.512911i	−0.794199	−	0.607658i	$-0.792109\pi$
$521$	4.59482	−	11.7074i	0.201303	−	0.512911i	0.995501	+	0.0947472i	$0.0302043\pi$
$522$		0			0
$523$	−18.0801	+	31.3157i	−0.790589	+	1.36934i	0.135014	+	0.990844i	$0.456892\pi$
$523$	−18.0801	+	31.3157i	−0.790589	+	1.36934i	−0.925603	+	0.378496i	$0.876441\pi$
$524$		0			0
$525$	−23.9774	+	16.3475i	−1.04646	+	0.713465i
$526$		0			0
$527$	0.495953	+	0.238838i	0.0216041	+	0.0104040i
$528$		0			0
$529$	−31.5895	+	4.76135i	−1.37346	+	0.207015i
$530$		0			0
$531$	−0.0228502	+	0.304915i	−0.000991616	+	0.0132322i
$532$		0			0
$533$	−2.26582	−	5.77321i	−0.0981434	−	0.250065i
$534$		0			0
$535$	3.27195	+	2.23078i	0.141459	+	0.0964450i
$536$		0			0
$537$	2.75596	−	12.0747i	0.118929	−	0.521060i
$538$		0			0
$539$	−1.68226	−	22.4482i	−0.0724600	−	0.966912i
$540$		0			0
$541$	3.25662	+	3.02170i	0.140013	+	0.129913i	0.747070	−	0.664745i	$-0.231460\pi$
$541$	3.25662	+	3.02170i	0.140013	+	0.129913i	−0.607057	+	0.794658i	$0.707650\pi$
$542$		0			0
$543$	−20.0800	+	25.1795i	−0.861715	+	1.08056i
$544$		0			0
$545$	−0.413916	−	0.127676i	−0.0177302	−	0.00546905i
$546$		0			0
$547$	0.724131	−	0.671896i	0.0309616	−	0.0287282i	−0.664540	−	0.747252i	$-0.731373\pi$
$547$	0.724131	−	0.671896i	0.0309616	−	0.0287282i	0.695502	+	0.718524i	$0.255182\pi$
$548$		0			0
$549$	−0.349011	−	0.0526049i	−0.0148954	−	0.00224512i
$550$		0			0
$551$	16.2385	+	20.3625i	0.691785	+	0.867470i
$552$		0			0
$553$	−14.4638	−	25.0521i	−0.615065	−	1.06532i
$554$		0			0
$555$	−2.61500	+	1.25932i	−0.111000	+	0.0534550i
$556$		0			0
$557$	−2.10042	−	9.20253i	−0.0889976	−	0.389924i	0.910736	−	0.412988i	$-0.135515\pi$
$557$	−2.10042	−	9.20253i	−0.0889976	−	0.389924i	−0.999734	+	0.0230643i	$0.992658\pi$
$558$		0			0
$559$	−5.26013	−	6.52695i	−0.222480	−	0.276061i
$560$		0			0
$561$	0.930095	+	4.07501i	0.0392687	+	0.172047i
$562$		0			0
$563$	−24.8446	+	11.9645i	−1.04708	+	0.504245i	−0.876652	−	0.481125i	$-0.840228\pi$
$563$	−24.8446	+	11.9645i	−1.04708	+	0.504245i	−0.170425	+	0.985371i	$0.554514\pi$
$564$		0			0
$565$	−3.07776	−	5.33083i	−0.129482	−	0.224270i
$566$		0			0
$567$	−19.1581	−	24.0234i	−0.804563	−	1.00889i
$568$		0			0
$569$	20.0473	+	3.02164i	0.840425	+	0.126674i	0.555127	−	0.831765i	$-0.312670\pi$
$569$	20.0473	+	3.02164i	0.840425	+	0.126674i	0.285297	+	0.958439i	$0.407908\pi$
$570$		0			0
$571$	−13.1590	+	12.2098i	−0.550688	+	0.510964i	−0.905712	−	0.423893i	$-0.860663\pi$
$571$	−13.1590	+	12.2098i	−0.550688	+	0.510964i	0.355024	+	0.934857i	$0.384473\pi$
$572$		0			0
$573$	11.5123	+	3.55107i	0.480933	+	0.148348i
$574$		0			0
$575$	22.4218	−	28.1161i	0.935055	−	1.17252i
$576$		0			0
$577$	−0.766879	−	0.711560i	−0.0319256	−	0.0296226i	0.664049	−	0.747689i	$-0.268837\pi$
$577$	−0.766879	−	0.711560i	−0.0319256	−	0.0296226i	−0.695974	+	0.718067i	$0.745027\pi$
$578$		0			0
$579$	−0.527022	−	7.03262i	−0.0219023	−	0.292266i
$580$		0			0
$581$	−9.05140	+	39.6568i	−0.375516	+	1.64524i
$582$		0			0
$583$	−34.6448	−	23.6204i	−1.43484	−	0.978257i
$584$		0			0
$585$	0.0117430	+	0.0299208i	0.000485514	+	0.00123707i
$586$		0			0
$587$	1.23653	−	16.5003i	0.0510370	−	0.681042i	−0.911744	−	0.410759i	$-0.865264\pi$
$587$	1.23653	−	16.5003i	0.0510370	−	0.681042i	0.962781	−	0.270283i	$-0.0871172\pi$
$588$		0			0
$589$	7.38099	−	1.11251i	0.304128	−	0.0458400i
$590$		0			0
$591$	−2.78450	−	1.34094i	−0.114539	−	0.0551590i
$592$		0			0
$593$	−8.32969	+	5.67909i	−0.342059	+	0.233212i	−0.722149	−	0.691737i	$-0.756846\pi$
$593$	−8.32969	+	5.67909i	−0.342059	+	0.233212i	0.380090	+	0.924950i	$0.375893\pi$
$594$		0			0
$595$	−0.378650	+	0.655841i	−0.0155231	+	0.0268869i
$596$		0			0
$597$	14.2497	−	36.3075i	0.583200	−	1.48597i
$598$		0			0
$599$	−32.4927	+	10.0227i	−1.32761	+	0.409515i	−0.875833	−	0.482614i	$-0.839688\pi$
$599$	−32.4927	+	10.0227i	−1.32761	+	0.409515i	−0.451782	+	0.892129i	$0.649211\pi$
$600$		0			0
$601$	−27.2454			−1.11136			−0.555681	−	0.831396i	$-0.687542\pi$
$601$	−27.2454			−1.11136			−0.555681	+	0.831396i	$0.687542\pi$
$602$		0			0
$603$	0.855944			0.0348567
$604$		0			0
$605$	2.87199	−	0.885890i	0.116763	−	0.0360166i
$606$		0			0
$607$	8.72876	−	22.2405i	0.354289	−	0.902715i	−0.636982	−	0.770879i	$-0.719817\pi$
$607$	8.72876	−	22.2405i	0.354289	−	0.902715i	0.991272	−	0.131836i	$-0.0420873\pi$
$608$		0			0
$609$	−10.2080	+	17.6807i	−0.413648	+	0.716459i
$610$		0			0
$611$	−4.61691	+	3.14776i	−0.186780	+	0.127345i
$612$		0			0
$613$	30.1578	+	14.5232i	1.21806	+	0.586588i	0.928771	−	0.370654i	$-0.120866\pi$
$613$	30.1578	+	14.5232i	1.21806	+	0.586588i	0.289291	+	0.957241i	$0.406581\pi$
$614$		0			0
$615$	−3.16740	+	0.477409i	−0.127722	+	0.0192510i
$616$		0			0
$617$	−1.17887	+	15.7309i	−0.0474594	+	0.633302i	0.921864	+	0.387513i	$0.126666\pi$
$617$	−1.17887	+	15.7309i	−0.0474594	+	0.633302i	−0.969323	+	0.245789i	$0.920953\pi$
$618$		0			0
$619$	−7.08652	−	18.0562i	−0.284831	−	0.725738i	−0.999651	−	0.0264292i	$-0.991586\pi$
$619$	−7.08652	−	18.0562i	−0.284831	−	0.725738i	0.714819	−	0.699309i	$-0.246509\pi$
$620$		0			0
$621$	32.1607	+	21.9268i	1.29056	+	0.879892i
$622$		0			0
$623$	−2.11949	+	9.28610i	−0.0849156	+	0.372040i
$624$		0			0
$625$	1.70340	+	22.7303i	0.0681359	+	0.909210i
$626$		0			0
$627$	41.5483	+	38.5512i	1.65928	+	1.53959i
$628$		0			0
$629$	1.54240	−	1.93411i	0.0614994	−	0.0771178i
$630$		0			0
$631$	1.52150	+	0.469320i	0.0605699	+	0.0186833i	0.324892	−	0.945751i	$-0.394672\pi$
$631$	1.52150	+	0.469320i	0.0605699	+	0.0186833i	−0.264322	+	0.964434i	$0.585148\pi$
$632$		0			0
$633$	−9.90382	+	9.18940i	−0.393641	+	0.365246i
$634$		0			0
$635$	−1.28317	−	0.193406i	−0.0509209	−	0.00767510i
$636$		0			0
$637$	−4.13827	−	5.18922i	−0.163964	−	0.205604i
$638$		0			0
$639$	−0.248952	−	0.431198i	−0.00984840	−	0.0170579i
$640$		0			0
$641$	6.14244	−	2.95804i	0.242612	−	0.116836i	−0.308627	−	0.951183i	$-0.599870\pi$
$641$	6.14244	−	2.95804i	0.242612	−	0.116836i	0.551239	+	0.834347i	$0.314155\pi$
$642$		0			0
$643$	3.83478	+	16.8013i	0.151229	+	0.662578i	0.992529	+	0.122009i	$0.0389337\pi$
$643$	3.83478	+	16.8013i	0.151229	+	0.662578i	−0.841300	+	0.540569i	$0.818209\pi$
$644$		0			0
$645$	−3.89106	+	1.89851i	−0.153210	+	0.0747539i
$646$		0			0
$647$	3.00033	+	13.1453i	0.117955	+	0.516795i	0.999039	+	0.0438349i	$0.0139575\pi$
$647$	3.00033	+	13.1453i	0.117955	+	0.516795i	−0.881084	+	0.472961i	$0.843185\pi$
$648$		0			0
$649$	18.3085	−	8.81691i	0.718671	−	0.346094i
$650$		0			0
$651$	2.92560	+	5.06728i	0.114663	+	0.198602i
$652$		0			0
$653$	−23.9528	−	30.0358i	−0.937344	−	1.17539i	−0.984301	−	0.176496i	$-0.943524\pi$
$653$	−23.9528	−	30.0358i	−0.937344	−	1.17539i	0.0469569	−	0.998897i	$-0.485048\pi$
$654$		0			0
$655$	−0.586210	−	0.0883570i	−0.0229051	−	0.00345239i
$656$		0			0
$657$	0.0992925	−	0.0921300i	0.00387377	−	0.00359433i
$658$		0			0
$659$	6.80245	+	2.09828i	0.264986	+	0.0817373i	0.424401	−	0.905474i	$-0.360485\pi$
$659$	6.80245	+	2.09828i	0.264986	+	0.0817373i	−0.159415	+	0.987212i	$0.550961\pi$
$660$		0			0
$661$	13.9581	−	17.5029i	0.542908	−	0.680785i	−0.432388	−	0.901688i	$-0.642329\pi$
$661$	13.9581	−	17.5029i	0.542908	−	0.680785i	0.975296	+	0.220903i	$0.0709003\pi$
$662$		0			0
$663$	0.903406	+	0.838238i	0.0350854	+	0.0325545i
$664$		0			0
$665$	0.767406	+	10.2403i	0.0297587	+	0.397103i
$666$		0			0
$667$	5.62969	−	24.6653i	0.217983	−	0.955044i
$668$		0			0
$669$	−12.1876	−	8.30936i	−0.471199	−	0.321258i
$670$		0			0
$671$	8.56964	+	21.8351i	0.330827	+	0.842935i
$672$		0			0
$673$	0.148095	−	1.97619i	0.00570864	−	0.0761765i	−0.993622	−	0.112764i	$-0.964030\pi$
$673$	0.148095	−	1.97619i	0.00570864	−	0.0761765i	0.999330	+	0.0365873i	$0.0116487\pi$
$674$		0			0
$675$	−25.1910	+	3.79693i	−0.969602	+	0.146144i
$676$		0			0
$677$	6.11443	+	2.94456i	0.234997	+	0.113168i	0.547677	−	0.836690i	$-0.315512\pi$
$677$	6.11443	+	2.94456i	0.234997	+	0.113168i	−0.312680	+	0.949859i	$0.601227\pi$
$678$		0			0
$679$	−1.41450	+	0.964388i	−0.0542834	+	0.0370098i
$680$		0			0
$681$	−6.97904	+	12.0880i	−0.267437	+	0.463215i
$682$		0			0
$683$	−3.77933	+	9.62958i	−0.144612	+	0.368466i	−0.984767	−	0.173881i	$-0.944369\pi$
$683$	−3.77933	+	9.62958i	−0.144612	+	0.368466i	0.840154	+	0.542347i	$0.182464\pi$
$684$		0			0
$685$	−1.17382	+	0.362074i	−0.0448492	+	0.0138342i
$686$		0			0
$687$	−42.4465			−1.61944
$688$		0			0
$689$	−12.3630			−0.470993
$690$		0			0
$691$	10.6685	−	3.29080i	0.405850	−	0.125188i	−0.0851075	−	0.996372i	$-0.527123\pi$
$691$	10.6685	−	3.29080i	0.405850	−	0.125188i	0.490957	+	0.871184i	$0.336647\pi$
$692$		0			0
$693$	0.360834	−	0.919390i	0.0137070	−	0.0349247i
$694$		0			0
$695$	−3.86875	+	6.70087i	−0.146750	+	0.254179i
$696$		0			0
$697$	2.25576	−	1.53795i	0.0854428	−	0.0582539i
$698$		0			0
$699$	−13.3074	−	6.40851i	−0.503332	−	0.242392i
$700$		0			0
$701$	45.9912	−	6.93206i	1.73706	−	0.261820i	0.797285	−	0.603603i	$-0.206269\pi$
$701$	45.9912	−	6.93206i	1.73706	−	0.261820i	0.939779	+	0.341783i	$0.111031\pi$
$702$		0			0
$703$	2.50683	−	33.4513i	0.0945467	−	1.26164i
$704$		0			0
$705$	1.05438	+	2.68653i	0.0397104	+	0.101180i
$706$		0			0
$707$	6.35350	+	4.33175i	0.238948	+	0.162912i
$708$		0			0
$709$	−1.61013	+	7.05446i	−0.0604699	+	0.264936i	−0.996121	−	0.0879894i	$-0.971956\pi$
$709$	−1.61013	+	7.05446i	−0.0604699	+	0.264936i	0.935652	+	0.352925i	$0.114813\pi$
$710$		0			0
$711$	−0.0403908	−	0.538978i	−0.00151477	−	0.0202133i
$712$		0			0
$713$	−5.31525	−	4.93183i	−0.199058	−	0.184699i
$714$		0			0
$715$	1.33186	−	1.67010i	0.0498087	−	0.0624582i
$716$		0			0
$717$	−5.51043	−	1.69974i	−0.205791	−	0.0634780i
$718$		0			0
$719$	−26.6874	+	24.7623i	−0.995274	+	0.923479i	−0.997068	−	0.0765194i	$-0.975619\pi$
$719$	−26.6874	+	24.7623i	−0.995274	+	0.923479i	0.00179446	+	0.999998i	$0.499429\pi$
$720$		0			0
$721$	−17.6708	−	2.66345i	−0.658095	−	0.0991919i
$722$		0			0
$723$	−28.8563	−	36.1846i	−1.07318	−	1.34572i
$724$		0			0
$725$	8.27918	+	14.3400i	0.307481	+	0.532573i
$726$		0			0
$727$	38.1354	−	18.3650i	1.41436	−	0.681121i	0.438343	−	0.898808i	$-0.355565\pi$
$727$	38.1354	−	18.3650i	1.41436	−	0.681121i	0.976019	+	0.217687i	$0.0698511\pi$
$728$		0			0
$729$	−6.13298	−	26.8703i	−0.227147	−	0.995197i
$730$		0			0
$731$	2.28593	−	2.89686i	0.0845483	−	0.107144i
$732$		0			0
$733$	4.08534	+	17.8990i	0.150895	+	0.661116i	0.992626	+	0.121218i	$0.0386800\pi$
$733$	4.08534	+	17.8990i	0.150895	+	0.661116i	−0.841731	+	0.539898i	$0.818463\pi$
$734$		0			0
$735$	−3.08854	+	1.48736i	−0.113923	+	0.0548622i
$736$		0			0
$737$	−28.4422	−	49.2633i	−1.04768	−	1.81464i
$738$		0			0
$739$	−19.5007	−	24.4531i	−0.717345	−	0.899522i	0.280839	−	0.959755i	$-0.409387\pi$
$739$	−19.5007	−	24.4531i	−0.717345	−	0.899522i	−0.998184	+	0.0602326i	$0.980816\pi$
$740$		0			0
$741$	16.5246	+	2.49069i	0.607048	+	0.0914977i
$742$		0			0
$743$	17.5715	−	16.3039i	0.644635	−	0.598134i	−0.288579	−	0.957456i	$-0.593183\pi$
$743$	17.5715	−	16.3039i	0.644635	−	0.598134i	0.933213	+	0.359323i	$0.116992\pi$
$744$		0			0
$745$	−4.65579	−	1.43612i	−0.170575	−	0.0526153i
$746$		0			0
$747$	−0.473858	+	0.594199i	−0.0173375	+	0.0217406i
$748$		0			0
$749$	−26.3000	−	24.4028i	−0.960980	−	0.891659i
$750$		0			0
$751$	3.81330	+	50.8850i	0.139149	+	1.85682i	0.433540	+	0.901134i	$0.357264\pi$
$751$	3.81330	+	50.8850i	0.139149	+	1.85682i	−0.294391	+	0.955685i	$0.595117\pi$
$752$		0			0
$753$	−4.10393	+	17.9805i	−0.149555	+	0.655245i
$754$		0			0
$755$	−0.320209	−	0.218315i	−0.0116536	−	0.00794528i
$756$		0			0
$757$	8.24045	+	20.9963i	0.299504	+	0.763125i	0.998832	+	0.0483232i	$0.0153878\pi$
$757$	8.24045	+	20.9963i	0.299504	+	0.763125i	−0.699327	+	0.714802i	$0.746517\pi$
$758$		0			0
$759$	4.11444	−	54.9033i	0.149345	−	1.99286i
$760$		0			0
$761$	−35.6862	+	5.37882i	−1.29362	+	0.194982i	−0.759546	−	0.650454i	$-0.774579\pi$
$761$	−35.6862	+	5.37882i	−1.29362	+	0.194982i	−0.534076	+	0.845436i	$0.679341\pi$
$762$		0			0
$763$	3.53572	+	1.70271i	0.128002	+	0.0616423i
$764$		0			0
$765$	−0.0116909	+	0.00797071i	−0.000422685	+	0.000288182i
$766$		0			0
$767$	2.99574	−	5.18878i	0.108170	−	0.187356i
$768$		0			0
$769$	−9.37295	+	23.8819i	−0.337997	+	0.861203i	0.656294	+	0.754505i	$0.272123\pi$
$769$	−9.37295	+	23.8819i	−0.337997	+	0.861203i	−0.994292	+	0.106698i	$0.965972\pi$
$770$		0			0
$771$	20.8793	−	6.44043i	0.751951	−	0.231946i
$772$		0			0
$773$	−30.1582			−1.08472			−0.542358	−	0.840148i	$-0.682468\pi$
$773$	−30.1582			−1.08472			−0.542358	+	0.840148i	$0.682468\pi$
$774$		0			0
$775$	4.74562			0.170468
$776$		0			0
$777$	25.1272	−	7.75072i	0.901434	−	0.278055i
$778$		0			0
$779$	13.5253	−	34.4619i	0.484594	−	1.23472i
$780$		0			0
$781$	−16.5449	+	28.6566i	−0.592023	+	1.02541i
$782$		0			0
$783$	−14.8082	+	10.0960i	−0.529200	+	0.360803i
$784$		0			0
$785$	−1.65403	−	0.796537i	−0.0590347	−	0.0284296i
$786$		0			0
$787$	20.5239	−	3.09348i	0.731598	−	0.110271i	0.227330	−	0.973818i	$-0.427000\pi$
$787$	20.5239	−	3.09348i	0.731598	−	0.110271i	0.504268	+	0.863547i	$0.331762\pi$
$788$		0			0
$789$	1.89898	−	25.3401i	0.0676053	−	0.902130i
$790$		0			0
$791$	20.3743	+	51.9128i	0.724426	+	1.84581i
$792$		0			0
$793$	5.71428	+	3.89593i	0.202920	+	0.138349i
$794$		0			0
$795$	−1.42085	+	6.22516i	−0.0503924	+	0.220784i
$796$		0			0
$797$	1.25760	+	16.7815i	0.0445463	+	0.594430i	0.974155	+	0.225881i	$0.0725261\pi$
$797$	1.25760	+	16.7815i	0.0445463	+	0.594430i	−0.929609	+	0.368549i	$0.879855\pi$
$798$		0			0
$799$	−1.80319	−	1.67311i	−0.0637922	−	0.0591905i
$800$		0			0
$801$	−0.110959	+	0.139139i	−0.00392055	+	0.00491622i
$802$		0			0
$803$	−8.60187	−	2.65333i	−0.303554	−	0.0936339i
$804$		0			0
$805$	7.31242	−	6.78494i	0.257729	−	0.239138i
$806$		0			0
$807$	−18.1405	−	2.73424i	−0.638577	−	0.0962499i
$808$		0			0
$809$	21.2357	+	26.6287i	0.746608	+	0.936217i	0.999511	−	0.0312750i	$-0.00995677\pi$
$809$	21.2357	+	26.6287i	0.746608	+	0.936217i	−0.252903	+	0.967492i	$0.581385\pi$
$810$		0			0
$811$	6.69823	+	11.6017i	0.235207	+	0.407390i	0.959333	−	0.282278i	$-0.0910900\pi$
$811$	6.69823	+	11.6017i	0.235207	+	0.407390i	−0.724126	+	0.689668i	$0.757757\pi$
$812$		0			0
$813$	22.3640	−	10.7699i	0.784339	−	0.377718i
$814$		0			0
$815$	−0.814481	−	3.56848i	−0.0285300	−	0.124998i
$816$		0			0
$817$	3.48313	−	49.9174i	0.121859	−	1.74639i
$818$		0			0
$819$	−0.0647995	−	0.283905i	−0.00226428	−	0.00992045i
$820$		0			0
$821$	−11.8336	+	5.69878i	−0.412997	+	0.198889i	−0.628832	−	0.777541i	$-0.716467\pi$
$821$	−11.8336	+	5.69878i	−0.412997	+	0.198889i	0.215836	+	0.976430i	$0.430752\pi$
$822$		0			0
$823$	9.61010	+	16.6452i	0.334987	+	0.580215i	0.983482	−	0.181004i	$-0.0579347\pi$
$823$	9.61010	+	16.6452i	0.334987	+	0.580215i	−0.648495	+	0.761219i	$0.724601\pi$
$824$		0			0
$825$	22.4671	+	28.1729i	0.782205	+	0.980854i
$826$		0			0
$827$	−32.5196	−	4.90154i	−1.13082	−	0.170443i	−0.443135	−	0.896455i	$-0.646134\pi$
$827$	−32.5196	−	4.90154i	−1.13082	−	0.170443i	−0.687682	+	0.726012i	$0.741372\pi$
$828$		0			0
$829$	8.73426	−	8.10421i	0.303353	−	0.281471i	−0.513777	−	0.857924i	$-0.671754\pi$
$829$	8.73426	−	8.10421i	0.303353	−	0.281471i	0.817130	+	0.576453i	$0.195564\pi$
$830$		0			0
$831$	33.4235	+	10.3098i	1.15945	+	0.357642i
$832$		0			0
$833$	1.82171	−	2.28435i	0.0631184	−	0.0791480i
$834$		0			0
$835$	−6.39388	−	5.93266i	−0.221269	−	0.205308i
$836$		0			0
$837$	0.383854	+	5.12218i	0.0132679	+	0.177048i
$838$		0			0
$839$	−11.8997	+	52.1359i	−0.410822	+	1.79993i	0.169486	+	0.985533i	$0.445789\pi$
$839$	−11.8997	+	52.1359i	−0.410822	+	1.79993i	−0.580308	+	0.814397i	$0.697068\pi$
$840$		0			0
$841$	−14.3361	−	9.77416i	−0.494347	−	0.337040i
$842$		0			0
$843$	−4.53848	−	11.5638i	−0.156313	−	0.398280i
$844$		0			0
$845$	−0.327352	+	4.36821i	−0.0112613	+	0.150271i
$846$		0			0
$847$	−26.9253	+	4.05833i	−0.925163	+	0.139446i
$848$		0			0
$849$	−47.0440	−	22.6552i	−1.61455	−	0.777524i
$850$		0			0
$851$	−26.9235	+	18.3561i	−0.922924	+	0.629239i
$852$		0			0
$853$	−14.4537	+	25.0346i	−0.494886	+	0.857168i	−0.999983	−	0.00589491i	$-0.998124\pi$
$853$	−14.4537	+	25.0346i	−0.494886	+	0.857168i	0.505096	+	0.863063i	$0.331457\pi$
$854$		0			0
$855$	−0.0700974	+	0.178605i	−0.00239728	+	0.00610817i
$856$		0			0
$857$	36.3470	−	11.2116i	1.24159	−	0.382980i	0.396714	−	0.917942i	$-0.370151\pi$
$857$	36.3470	−	11.2116i	1.24159	−	0.382980i	0.844876	+	0.534963i	$0.179674\pi$
$858$		0			0
$859$	−38.9400			−1.32861			−0.664307	−	0.747460i	$-0.731273\pi$
$859$	−38.9400			−1.32861			−0.664307	+	0.747460i	$0.731273\pi$
$860$		0			0
$861$	29.0202			0.989005
$862$		0			0
$863$	45.6356	−	14.0767i	1.55345	−	0.479177i	0.605035	−	0.796199i	$-0.293159\pi$
$863$	45.6356	−	14.0767i	1.55345	−	0.479177i	0.948418	+	0.317022i	$0.102683\pi$
$864$		0			0
$865$	1.30982	−	3.33736i	0.0445351	−	0.113474i
$866$		0			0
$867$	14.2902	−	24.7514i	0.485321	−	0.840601i
$868$		0			0
$869$	−29.6784	+	20.2344i	−1.00677	+	0.686404i
$870$		0			0
$871$	−15.1110	−	7.27709i	−0.512018	−	0.246575i
$872$		0			0
$873$	−0.0316295	+	0.00476738i	−0.00107050	+	0.000161351i
$874$		0			0
$875$	−0.990726	+	13.2203i	−0.0334926	+	0.446928i
$876$		0			0
$877$	−12.6609	−	32.2594i	−0.427527	−	1.08932i	−0.968661	−	0.248385i	$-0.920100\pi$
$877$	−12.6609	−	32.2594i	−0.427527	−	1.08932i	0.541134	−	0.840936i	$-0.317995\pi$
$878$		0			0
$879$	36.3573	+	24.7880i	1.22630	+	0.836079i
$880$		0			0
$881$	11.6579	−	51.0764i	0.392763	−	1.72081i	−0.262081	−	0.965046i	$-0.584409\pi$
$881$	11.6579	−	51.0764i	0.392763	−	1.72081i	0.654845	−	0.755763i	$-0.272734\pi$
$882$		0			0
$883$	2.16026	+	28.8267i	0.0726985	+	0.970094i	0.907638	+	0.419753i	$0.137883\pi$
$883$	2.16026	+	28.8267i	0.0726985	+	0.970094i	−0.834940	+	0.550341i	$0.814498\pi$
$884$		0			0
$885$	−2.26842	−	2.10479i	−0.0762522	−	0.0707517i
$886$		0			0
$887$	28.3829	−	35.5910i	0.953004	−	1.19503i	−0.0277167	−	0.999616i	$-0.508824\pi$
$887$	28.3829	−	35.5910i	0.953004	−	1.19503i	0.980721	−	0.195414i	$-0.0626049\pi$
$888$		0			0
$889$	11.2343	+	3.46531i	0.376785	+	0.116223i
$890$		0			0
$891$	−27.9691	+	25.9515i	−0.936999	+	0.869408i
$892$		0			0
$893$	−32.9830	−	4.97138i	−1.10373	−	0.166361i
$894$		0			0
$895$	−1.73726	−	2.17845i	−0.0580701	−	0.0728176i
$896$		0			0
$897$	−8.11665	−	14.0584i	−0.271007	−	0.469398i
$898$		0			0
$899$	3.00798	−	1.44856i	0.100322	−	0.0483123i
$900$		0			0
$901$	−1.21103	−	5.30587i	−0.0403452	−	0.176764i
$902$		0			0
$903$	37.4221	−	11.7540i	1.24533	−	0.391148i
$904$		0			0
$905$	1.61227	+	7.06382i	0.0535937	+	0.234809i
$906$		0			0
$907$	19.7496	−	9.51092i	0.655775	−	0.315805i	−0.0762434	−	0.997089i	$-0.524293\pi$
$907$	19.7496	−	9.51092i	0.655775	−	0.315805i	0.732019	+	0.681284i	$0.238578\pi$
$908$		0			0
$909$	0.0718375	+	0.124426i	0.00238270	+	0.00412696i
$910$		0			0
$911$	28.0122	+	35.1262i	0.928087	+	1.16378i	0.986214	+	0.165475i	$0.0529157\pi$
$911$	28.0122	+	35.1262i	0.928087	+	1.16378i	−0.0581272	+	0.998309i	$0.518513\pi$
$912$		0			0
$913$	49.9446	+	7.52793i	1.65292	+	0.249138i
$914$		0			0
$915$	2.61846	−	2.42957i	0.0865635	−	0.0803192i
$916$		0			0
$917$	5.13233	+	1.58311i	0.169484	+	0.0522790i
$918$		0			0
$919$	21.8856	−	27.4436i	0.721938	−	0.905282i	−0.276508	−	0.961012i	$-0.589177\pi$
$919$	21.8856	−	27.4436i	0.721938	−	0.905282i	0.998446	+	0.0557299i	$0.0177486\pi$
$920$		0			0
$921$	19.9042	+	18.4684i	0.655867	+	0.608555i
$922$		0			0
$923$	0.729090	+	9.72903i	0.0239983	+	0.320235i
$924$		0			0
$925$	4.74569	−	20.7922i	0.156037	−	0.683644i
$926$		0			0
$927$	−0.275876	−	0.188089i	−0.00906095	−	0.00617765i
$928$		0			0
$929$	3.19040	+	8.12900i	0.104674	+	0.266704i	0.973609	−	0.228221i	$-0.0732908\pi$
$929$	3.19040	+	8.12900i	0.104674	+	0.266704i	−0.868936	+	0.494925i	$0.835196\pi$
$930$		0			0
$931$	2.96078	−	39.5089i	0.0970357	−	1.29485i
$932$		0			0
$933$	45.7639	−	6.89780i	1.49824	−	0.225824i
$934$		0			0
$935$	0.847226	+	0.408002i	0.0277072	+	0.0133431i
$936$		0			0
$937$	−23.8766	+	16.2788i	−0.780015	+	0.531806i	−0.886621	−	0.462496i	$-0.846954\pi$
$937$	−23.8766	+	16.2788i	−0.780015	+	0.531806i	0.106606	+	0.994301i	$0.466002\pi$
$938$		0			0
$939$	−16.5088	+	28.5941i	−0.538744	+	0.933132i
$940$		0			0
$941$	−1.16024	+	2.95624i	−0.0378226	+	0.0963705i	−0.948537	−	0.316666i	$-0.897437\pi$
$941$	−1.16024	+	2.95624i	−0.0378226	+	0.0963705i	0.910714	+	0.413037i	$0.135532\pi$
$942$		0			0
$943$	−34.3644	+	10.6000i	−1.11906	+	0.345184i
$944$		0			0
$945$	−7.06656			−0.229875
$946$		0			0
$947$	−47.5288			−1.54448			−0.772239	−	0.635332i	$-0.780863\pi$
$947$	−47.5288			−1.54448			−0.772239	+	0.635332i	$0.780863\pi$
$948$		0			0
$949$	−2.53621	+	0.782316i	−0.0823288	+	0.0253951i
$950$		0			0
$951$	−8.04974	+	20.5104i	−0.261031	+	0.665096i
$952$		0			0
$953$	−20.3266	+	35.2067i	−0.658443	+	1.14046i	0.322576	+	0.946544i	$0.395451\pi$
$953$	−20.3266	+	35.2067i	−0.658443	+	1.14046i	−0.981019	+	0.193913i	$0.937882\pi$
$954$		0			0
$955$	2.23943	−	1.52681i	0.0724661	−	0.0494065i
$956$		0			0
$957$	22.8402	+	10.9993i	0.738319	+	0.355556i
$958$		0			0
$959$	11.0047	−	1.65869i	0.355360	−	0.0535619i
$960$		0			0
$961$	−2.24513	+	29.9592i	−0.0724235	+	0.966424i
$962$		0			0
$963$	−0.244902	−	0.624000i	−0.00789186	−	0.0201081i
$964$		0			0
$965$	−1.31091	−	0.893761i	−0.0421996	−	0.0287712i
$966$		0			0
$967$	−1.68074	+	7.36380i	−0.0540490	+	0.236804i	−0.994736	−	0.102472i	$-0.967325\pi$
$967$	−1.68074	+	7.36380i	−0.0540490	+	0.236804i	0.940687	+	0.339276i	$0.110182\pi$
$968$		0			0
$969$	0.549750	+	7.33591i	0.0176605	+	0.235663i
$970$		0			0
$971$	−15.3469	−	14.2398i	−0.492505	−	0.456978i	0.394394	−	0.918942i	$-0.370955\pi$
$971$	−15.3469	−	14.2398i	−0.492505	−	0.456978i	−0.886899	+	0.461964i	$0.847145\pi$
$972$		0			0
$973$	43.7068	−	54.8066i	1.40118	−	1.75702i
$974$		0			0
$975$	10.1525	+	3.13164i	0.325141	+	0.100293i
$976$		0			0
$977$	−17.9851	+	16.6877i	−0.575395	+	0.533888i	−0.913298	−	0.407293i	$-0.866473\pi$
$977$	−17.9851	+	16.6877i	−0.575395	+	0.533888i	0.337903	+	0.941181i	$0.390282\pi$
$978$		0			0
$979$	11.6951	+	1.76275i	0.373777	+	0.0563378i
$980$		0			0
$981$	0.0457163	+	0.0573264i	0.00145961	+	0.00183029i
$982$		0			0
$983$	0.981609	+	1.70020i	0.0313085	+	0.0542279i	0.881255	−	0.472641i	$-0.156699\pi$
$983$	0.981609	+	1.70020i	0.0313085	+	0.0542279i	−0.849947	+	0.526869i	$0.823366\pi$
$984$		0			0
$985$	−0.626440	+	0.301678i	−0.0199600	+	0.00961225i
$986$		0			0
$987$	−5.81823	−	25.4913i	−0.185196	−	0.811397i
$988$		0			0
$989$	−40.0203	+	27.5875i	−1.27257	+	0.877230i
$990$		0			0
$991$	3.67174	+	16.0869i	0.116637	+	0.511019i	0.999169	+	0.0407664i	$0.0129799\pi$
$991$	3.67174	+	16.0869i	0.116637	+	0.511019i	−0.882532	+	0.470252i	$0.844163\pi$
$992$		0			0
$993$	−3.92773	+	1.89149i	−0.124643	+	0.0600247i
$994$		0			0
$995$	−4.38741	−	7.59922i	−0.139090	−	0.240912i
$996$		0			0
$997$	−20.7198	−	25.9819i	−0.656204	−	0.822854i	0.336720	−	0.941605i	$-0.390682\pi$
$997$	−20.7198	−	25.9819i	−0.656204	−	0.822854i	−0.992924	+	0.118751i	$0.962111\pi$
$998$		0			0
$999$	22.8259	+	3.44046i	0.722181	+	0.108851i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bg.c.17.3		36
4.3	odd	2		43.2.g.a.17.3	✓	36
12.11	even	2		387.2.y.c.361.1		36
43.38	even	21	inner	688.2.bg.c.81.3		36
172.95	odd	42		1849.2.a.n.1.6		18
172.163	even	42		1849.2.a.o.1.13		18
172.167	odd	42		43.2.g.a.38.3	yes	36
516.167	even	42		387.2.y.c.253.1		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.17.3	✓	36	4.3	odd	2
43.2.g.a.38.3	yes	36	172.167	odd	42
387.2.y.c.253.1		36	516.167	even	42
387.2.y.c.361.1		36	12.11	even	2
688.2.bg.c.17.3		36	1.1	even	1	trivial
688.2.bg.c.81.3		36	43.38	even	21	inner
1849.2.a.n.1.6		18	172.95	odd	42
1849.2.a.o.1.13		18	172.163	even	42

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.bg (of order \(21\), degree \(12\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.63701 - 0.504949i) q^{3} +(0.140805 - 0.358765i) q^{5} +(-1.74586 + 3.02391i) q^{7} +(-0.0539035 + 0.0367508i) q^{9} +O(q^{10})\)	\(q+(1.63701 - 0.504949i) q^{3} +(0.140805 - 0.358765i) q^{5} +(-1.74586 + 3.02391i) q^{7} +(-0.0539035 + 0.0367508i) q^{9} +(3.90633 + 1.88119i) q^{11} +(1.26408 - 0.190529i) q^{13} +(0.0493402 - 0.658399i) q^{15} +(0.205594 + 0.523844i) q^{17} +(6.30490 + 4.29861i) q^{19} +(-1.33105 + 5.83172i) q^{21} +(-0.553943 - 7.39185i) q^{23} +(3.55637 + 3.29983i) q^{25} +(-3.27401 + 4.10548i) q^{27} +(3.26143 + 1.00602i) q^{29} +(0.717059 - 0.665333i) q^{31} +(7.34459 + 1.10702i) q^{33} +(0.839048 + 1.05213i) q^{35} +(-2.19799 - 3.80703i) q^{37} +(1.97309 - 0.950190i) q^{39} +(-1.07956 - 4.72986i) q^{41} +(-3.30784 - 5.66200i) q^{43} +(0.00559502 + 0.0245134i) q^{45} +(-3.93826 + 1.89657i) q^{47} +(-2.59602 - 4.49644i) q^{49} +(0.601072 + 0.753721i) q^{51} +(-9.56301 - 1.44139i) q^{53} +(1.22493 - 1.13657i) q^{55} +(12.4917 + 3.85319i) q^{57} +(2.92222 - 3.66435i) q^{59} +(3.96588 + 3.67980i) q^{61} +(-0.0170234 - 0.227161i) q^{63} +(0.109633 - 0.480333i) q^{65} +(-10.8402 - 7.39075i) q^{67} +(-4.63932 - 11.8208i) q^{69} +(-0.570335 + 7.61059i) q^{71} +(-2.05301 + 0.309442i) q^{73} +(7.48805 + 3.60605i) q^{75} +(-12.5084 + 8.52811i) q^{77} +(-4.14234 + 7.17474i) q^{79} +(-3.21501 + 8.19171i) q^{81} +(11.1319 - 3.43375i) q^{83} +0.216885 q^{85} +5.84697 q^{87} +(2.60667 - 0.804052i) q^{89} +(-1.63075 + 4.15509i) q^{91} +(0.837869 - 1.45123i) q^{93} +(2.42995 - 1.65671i) q^{95} +(0.441741 + 0.212731i) q^{97} +(-0.279700 + 0.0421580i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9} + 4 q^{11} + 3 q^{15} - 10 q^{17} - 10 q^{19} - 21 q^{21} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 9 q^{29} - 40 q^{31} - 11 q^{33} - 11 q^{35} - 19 q^{37} + q^{39} - 28 q^{41} + 8 q^{43} - 46 q^{45} + 30 q^{47} + 6 q^{49} - 57 q^{51} - 24 q^{53} - 14 q^{55} + 52 q^{57} + q^{59} - 14 q^{61} - 47 q^{63} + 38 q^{65} - 66 q^{67} - 7 q^{69} + 33 q^{71} + 29 q^{73} + 55 q^{75} - 27 q^{77} + 17 q^{79} + 38 q^{81} + 23 q^{83} - 56 q^{85} + 86 q^{87} - 19 q^{89} + 13 q^{91} - 30 q^{93} - q^{95} - 31 q^{97} + 31 q^{99}+O(q^{100}) \) 36 * q + 16 * q^3 - 17 * q^5 - 6 * q^7 - q^9 + 4 * q^11 + 3 * q^15 - 10 * q^17 - 10 * q^19 - 21 * q^21 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 9 * q^29 - 40 * q^31 - 11 * q^33 - 11 * q^35 - 19 * q^37 + q^39 - 28 * q^41 + 8 * q^43 - 46 * q^45 + 30 * q^47 + 6 * q^49 - 57 * q^51 - 24 * q^53 - 14 * q^55 + 52 * q^57 + q^59 - 14 * q^61 - 47 * q^63 + 38 * q^65 - 66 * q^67 - 7 * q^69 + 33 * q^71 + 29 * q^73 + 55 * q^75 - 27 * q^77 + 17 * q^79 + 38 * q^81 + 23 * q^83 - 56 * q^85 + 86 * q^87 - 19 * q^89 + 13 * q^91 - 30 * q^93 - q^95 - 31 * q^97 + 31 * q^99

Introduction

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