LMFDB - Embedded newform 688.2.bg.c.289.3

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			289.3
Character	$\chi$	$=$	688.289
Dual form			688.2.bg.c.369.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.41097 - 1.64377i) q^{3} +(-2.00127 - 1.85691i) q^{5} +(-1.01083 - 1.75082i) q^{7} +(2.01476 - 5.13352i) q^{9} +O(q^{10})$	\(q+(2.41097 - 1.64377i) q^{3} +(-2.00127 - 1.85691i) q^{5} +(-1.01083 - 1.75082i) q^{7} +(2.01476 - 5.13352i) q^{9} +(0.515533 + 0.646459i) q^{11} +(3.02110 - 0.931885i) q^{13} +(-7.87733 - 1.18732i) q^{15} +(-2.50278 + 2.32224i) q^{17} +(1.08222 + 2.75745i) q^{19} +(-5.31503 - 2.55958i) q^{21} +(-1.21045 + 0.182446i) q^{23} +(0.183328 + 2.44634i) q^{25} +(-1.63286 - 7.15403i) q^{27} +(-7.00539 - 4.77619i) q^{29} +(0.399005 - 5.32435i) q^{31} +(2.30556 + 0.711172i) q^{33} +(-1.22815 + 5.38089i) q^{35} +(-1.73217 + 3.00021i) q^{37} +(5.75196 - 7.21273i) q^{39} +(6.20561 - 2.98847i) q^{41} +(3.02108 - 5.82006i) q^{43} +(-13.5646 + 6.53235i) q^{45} +(-2.33951 + 2.93365i) q^{47} +(1.45643 - 2.52260i) q^{49} +(-2.21689 + 9.71283i) q^{51} +(8.47611 + 2.61454i) q^{53} +(0.168692 - 2.25104i) q^{55} +(7.14181 + 4.86920i) q^{57} +(0.633519 + 2.77563i) q^{59} +(0.573465 + 7.65235i) q^{61} +(-11.0245 + 1.66167i) q^{63} +(-7.77646 - 3.74495i) q^{65} +(3.76698 + 9.59812i) q^{67} +(-2.61846 + 2.42957i) q^{69} +(11.9877 + 1.80685i) q^{71} +(11.9045 - 3.67204i) q^{73} +(4.46321 + 5.59669i) q^{75} +(0.610712 - 1.55607i) q^{77} +(2.34287 + 4.05797i) q^{79} +(-3.56858 - 3.31116i) q^{81} +(3.94401 - 2.68898i) q^{83} +9.32092 q^{85} -24.7407 q^{87} +(-1.70700 + 1.16382i) q^{89} +(-4.68539 - 4.34741i) q^{91} +(-7.79001 - 13.4927i) q^{93} +(2.95452 - 7.52799i) q^{95} +(-7.51296 - 9.42095i) q^{97} +(4.35729 - 1.34404i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{17}{21}\right)$	$1$

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
$3$	2.41097	−	1.64377i	1.39197	−	0.949031i	0.392400	−	0.919795i	$-0.371645\pi$
$3$	2.41097	−	1.64377i	1.39197	−	0.949031i	0.999573	−	0.0292358i	$-0.00930737\pi$
$4$		0			0
$5$	−2.00127	−	1.85691i	−0.894996	−	0.830435i	0.0912316	−	0.995830i	$-0.470920\pi$
$5$	−2.00127	−	1.85691i	−0.894996	−	0.830435i	−0.986227	+	0.165395i	$0.947110\pi$
$6$		0			0
$7$	−1.01083	−	1.75082i	−0.382060	−	0.661747i	0.609297	−	0.792942i	$-0.291452\pi$
$7$	−1.01083	−	1.75082i	−0.382060	−	0.661747i	−0.991357	+	0.131195i	$0.958118\pi$
$8$		0			0
$9$	2.01476	−	5.13352i	0.671587	−	1.71117i
$10$		0			0
$11$	0.515533	+	0.646459i	0.155439	+	0.194915i	0.853453	−	0.521170i	$-0.174504\pi$
$11$	0.515533	+	0.646459i	0.155439	+	0.194915i	−0.698014	+	0.716084i	$0.745933\pi$
$12$		0			0
$13$	3.02110	−	0.931885i	0.837901	−	0.258458i	0.154035	−	0.988065i	$-0.450773\pi$
$13$	3.02110	−	0.931885i	0.837901	−	0.258458i	0.683867	+	0.729607i	$0.260297\pi$
$14$		0			0
$15$	−7.87733	−	1.18732i	−2.03392	−	0.306564i
$16$		0			0
$17$	−2.50278	+	2.32224i	−0.607013	+	0.563226i	−0.922642	−	0.385657i	$-0.873975\pi$
$17$	−2.50278	+	2.32224i	−0.607013	+	0.563226i	0.315629	+	0.948883i	$0.397784\pi$
$18$		0			0
$19$	1.08222	+	2.75745i	0.248278	+	0.632603i	0.999627	−	0.0273025i	$-0.00869174\pi$
$19$	1.08222	+	2.75745i	0.248278	+	0.632603i	−0.751349	+	0.659905i	$0.770597\pi$
$20$		0			0
$21$	−5.31503	−	2.55958i	−1.15983	−	0.558547i
$22$		0			0
$23$	−1.21045	+	0.182446i	−0.252396	+	0.0380426i	−0.274022	−	0.961724i	$-0.588354\pi$
$23$	−1.21045	+	0.182446i	−0.252396	+	0.0380426i	0.0216253	+	0.999766i	$0.493116\pi$
$24$		0			0
$25$	0.183328	+	2.44634i	0.0366655	+	0.489268i
$26$		0			0
$27$	−1.63286	−	7.15403i	−0.314244	−	1.37679i
$28$		0			0
$29$	−7.00539	−	4.77619i	−1.30087	−	0.886916i	−0.303052	−	0.952974i	$-0.598005\pi$
$29$	−7.00539	−	4.77619i	−1.30087	−	0.886916i	−0.997816	+	0.0660577i	$0.978958\pi$
$30$		0			0
$31$	0.399005	−	5.32435i	0.0716634	−	0.956281i	−0.839267	−	0.543720i	$-0.817015\pi$
$31$	0.399005	−	5.32435i	0.0716634	−	0.956281i	0.910930	−	0.412561i	$-0.135366\pi$
$32$		0			0
$33$	2.30556	+	0.711172i	0.401347	+	0.123799i
$34$		0			0
$35$	−1.22815	+	5.38089i	−0.207596	+	0.909536i
$36$		0			0
$37$	−1.73217	+	3.00021i	−0.284768	+	0.493232i	−0.972553	−	0.232682i	$-0.925250\pi$
$37$	−1.73217	+	3.00021i	−0.284768	+	0.493232i	0.687785	+	0.725914i	$0.258583\pi$
$38$		0			0
$39$	5.75196	−	7.21273i	0.921051	−	1.15496i
$40$		0			0
$41$	6.20561	−	2.98847i	0.969154	−	0.466720i	0.118793	−	0.992919i	$-0.462098\pi$
$41$	6.20561	−	2.98847i	0.969154	−	0.466720i	0.850361	+	0.526199i	$0.176383\pi$
$42$		0			0
$43$	3.02108	−	5.82006i	0.460710	−	0.887551i
$43$	3.02108	−	5.82006i	0.460710	−	0.887551i
$44$		0			0
$45$	−13.5646	+	6.53235i	−2.02209	+	0.973785i
$46$		0			0
$47$	−2.33951	+	2.93365i	−0.341252	+	0.427917i	−0.922612	−	0.385730i	$-0.873950\pi$
$47$	−2.33951	+	2.93365i	−0.341252	+	0.427917i	0.581359	+	0.813647i	$0.302521\pi$
$48$		0			0
$49$	1.45643	−	2.52260i	0.208061	−	0.360372i
$50$		0			0
$51$	−2.21689	+	9.71283i	−0.310427	+	1.36007i
$52$		0			0
$53$	8.47611	+	2.61454i	1.16428	+	0.359134i	0.815886	−	0.578213i	$-0.196250\pi$
$53$	8.47611	+	2.61454i	1.16428	+	0.359134i	0.348398	+	0.937347i	$0.386726\pi$
$54$		0			0
$55$	0.168692	−	2.25104i	0.0227464	−	0.303530i
$56$		0			0
$57$	7.14181	+	4.86920i	0.945956	+	0.644942i
$58$		0			0
$59$	0.633519	+	2.77563i	0.0824772	+	0.361356i	0.999278	−	0.0379881i	$-0.0120949\pi$
$59$	0.633519	+	2.77563i	0.0824772	+	0.361356i	−0.916801	+	0.399344i	$0.869238\pi$
$60$		0			0
$61$	0.573465	+	7.65235i	0.0734246	+	0.979783i	0.905287	+	0.424801i	$0.139656\pi$
$61$	0.573465	+	7.65235i	0.0734246	+	0.979783i	−0.831862	+	0.554982i	$0.812725\pi$
$62$		0			0
$63$	−11.0245	+	1.66167i	−1.38895	+	0.209351i
$64$		0			0
$65$	−7.77646	−	3.74495i	−0.964551	−	0.464503i
$66$		0			0
$67$	3.76698	+	9.59812i	0.460210	+	1.17260i	0.953141	+	0.302528i	$0.0978305\pi$
$67$	3.76698	+	9.59812i	0.460210	+	1.17260i	−0.492930	+	0.870069i	$0.664074\pi$
$68$		0			0
$69$	−2.61846	+	2.42957i	−0.315225	+	0.292486i
$70$		0			0
$71$	11.9877	+	1.80685i	1.42268	+	0.214434i	0.814867	−	0.579648i	$-0.196810\pi$
$71$	11.9877	+	1.80685i	1.42268	+	0.214434i	0.607811	+	0.794082i	$0.292048\pi$
$72$		0			0
$73$	11.9045	−	3.67204i	1.39331	−	0.429779i	0.494906	−	0.868947i	$-0.335203\pi$
$73$	11.9045	−	3.67204i	1.39331	−	0.429779i	0.898405	+	0.439167i	$0.144726\pi$
$74$		0			0
$75$	4.46321	+	5.59669i	0.515367	+	0.646250i
$76$		0			0
$77$	0.610712	−	1.55607i	0.0695971	−	0.177330i
$78$		0			0
$79$	2.34287	+	4.05797i	0.263593	+	0.456557i	0.967194	−	0.254038i	$-0.0817589\pi$
$79$	2.34287	+	4.05797i	0.263593	+	0.456557i	−0.703601	+	0.710596i	$0.748426\pi$
$80$		0			0
$81$	−3.56858	−	3.31116i	−0.396509	−	0.367907i
$82$		0			0
$83$	3.94401	−	2.68898i	0.432912	−	0.295154i	−0.327190	−	0.944959i	$-0.606102\pi$
$83$	3.94401	−	2.68898i	0.432912	−	0.295154i	0.760101	+	0.649805i	$0.225149\pi$
$84$		0			0
$85$	9.32092			1.01100
$86$		0			0
$87$	−24.7407			−2.65248
$88$		0			0
$89$	−1.70700	+	1.16382i	−0.180942	+	0.123364i	−0.650404	−	0.759589i	$-0.725400\pi$
$89$	−1.70700	+	1.16382i	−0.180942	+	0.123364i	0.469462	+	0.882953i	$0.344448\pi$
$90$		0			0
$91$	−4.68539	−	4.34741i	−0.491162	−	0.455732i
$92$		0			0
$93$	−7.79001	−	13.4927i	−0.807787	−	1.39913i
$94$		0			0
$95$	2.95452	−	7.52799i	0.303127	−	0.772355i
$96$		0			0
$97$	−7.51296	−	9.42095i	−0.762825	−	0.956552i	0.237063	−	0.971494i	$-0.423815\pi$
$97$	−7.51296	−	9.42095i	−0.762825	−	0.956552i	−0.999888	+	0.0149418i	$0.995244\pi$
$98$		0			0
$99$	4.35729	−	1.34404i	0.437924	−	0.135082i
$100$		0			0
$101$	−2.52020	−	0.379859i	−0.250769	−	0.0377974i	0.0224547	−	0.999748i	$-0.492852\pi$
$101$	−2.52020	−	0.379859i	−0.250769	−	0.0377974i	−0.273224	+	0.961951i	$0.588090\pi$
$102$		0			0
$103$	5.76522	−	5.34934i	0.568064	−	0.527086i	−0.343009	−	0.939332i	$-0.611446\pi$
$103$	5.76522	−	5.34934i	0.568064	−	0.527086i	0.911073	+	0.412246i	$0.135256\pi$
$104$		0			0
$105$	5.88391	+	14.9919i	0.574210	+	1.46306i
$106$		0			0
$107$	−7.55616	−	3.63886i	−0.730482	−	0.351782i	0.0313918	−	0.999507i	$-0.490006\pi$
$107$	−7.55616	−	3.63886i	−0.730482	−	0.351782i	−0.761874	+	0.647726i	$0.775720\pi$
$108$		0			0
$109$	5.60910	−	0.845435i	0.537254	−	0.0809780i	0.125190	−	0.992133i	$-0.460046\pi$
$109$	5.60910	−	0.845435i	0.537254	−	0.0809780i	0.412064	+	0.911155i	$0.364808\pi$
$110$		0			0
$111$	0.755445	+	10.0807i	0.0717037	+	0.956819i
$112$		0			0
$113$	3.99341	+	17.4963i	0.375669	+	1.64591i	0.710546	+	0.703651i	$0.248448\pi$
$113$	3.99341	+	17.4963i	0.375669	+	1.64591i	−0.334877	+	0.942262i	$0.608695\pi$
$114$		0			0
$115$	2.76123	+	1.88257i	0.257486	+	0.175551i
$116$		0			0
$117$	1.30293	−	17.3864i	0.120456	−	1.60737i
$118$		0			0
$119$	6.59571	+	2.03451i	0.604628	+	0.186503i
$120$		0			0
$121$	2.29560	−	10.0577i	0.208691	−	0.914333i
$122$		0			0
$123$	10.0492	−	17.4057i	0.906104	−	1.56942i
$124$		0			0
$125$	−4.33507	+	5.43600i	−0.387740	+	0.486211i
$126$		0			0
$127$	17.6508	−	8.50017i	1.56625	−	0.754268i	0.568593	−	0.822619i	$-0.307488\pi$
$127$	17.6508	−	8.50017i	1.56625	−	0.754268i	0.997661	+	0.0683509i	$0.0217737\pi$
$128$		0			0
$129$	−2.28311	−	18.9979i	−0.201017	−	1.67267i
$130$		0			0
$131$	−10.9256	+	5.26150i	−0.954576	+	0.459700i	−0.845288	−	0.534310i	$-0.820571\pi$
$131$	−10.9256	+	5.26150i	−0.954576	+	0.459700i	−0.109288	+	0.994010i	$0.534857\pi$
$132$		0			0
$133$	3.73385	−	4.68210i	0.323766	−	0.405989i
$134$		0			0
$135$	−10.0166	+	17.3492i	−0.862090	+	1.49318i
$136$		0			0
$137$	−1.32721	+	5.81487i	−0.113391	+	0.496798i	0.886057	+	0.463576i	$0.153434\pi$
$137$	−1.32721	+	5.81487i	−0.113391	+	0.496798i	−0.999448	+	0.0332220i	$0.989423\pi$
$138$		0			0
$139$	−2.97091	−	0.916404i	−0.251989	−	0.0777284i	0.166187	−	0.986094i	$-0.446854\pi$
$139$	−2.97091	−	0.916404i	−0.251989	−	0.0777284i	−0.418176	+	0.908366i	$0.637331\pi$
$140$		0			0
$141$	−0.818232	+	10.9185i	−0.0689075	+	0.919507i
$142$		0			0
$143$	2.15990	+	1.47260i	0.180620	+	0.123145i
$144$		0			0
$145$	5.15073	+	22.5668i	0.427745	+	1.87407i
$146$		0			0
$147$	−0.635184	−	8.47594i	−0.0523891	−	0.699084i
$148$		0			0
$149$	1.59287	−	0.240086i	0.130493	−	0.0196686i	−0.0834711	−	0.996510i	$-0.526601\pi$
$149$	1.59287	−	0.240086i	0.130493	−	0.0196686i	0.213964	+	0.976842i	$0.431363\pi$
$150$		0			0
$151$	−3.99279	−	1.92283i	−0.324929	−	0.156477i	0.264307	−	0.964439i	$-0.414857\pi$
$151$	−3.99279	−	1.92283i	−0.324929	−	0.156477i	−0.589236	+	0.807961i	$0.700571\pi$
$152$		0			0
$153$	6.87877	+	17.5268i	0.556116	+	1.41696i
$154$		0			0
$155$	−10.6853	+	9.91455i	−0.858267	+	0.796356i
$156$		0			0
$157$	−3.60721	−	0.543700i	−0.287887	−	0.0433920i	0.00351117	−	0.999994i	$-0.498882\pi$
$157$	−3.60721	−	0.543700i	−0.287887	−	0.0433920i	−0.291398	+	0.956602i	$0.594120\pi$
$158$		0			0
$159$	24.7333	−	7.62922i	1.96148	−	0.605036i
$160$		0			0
$161$	1.54300	+	1.93485i	0.121605	+	0.152488i
$162$		0			0
$163$	−2.81502	+	7.17257i	−0.220490	+	0.561799i	−0.997725	−	0.0674110i	$-0.978526\pi$
$163$	−2.81502	+	7.17257i	−0.220490	+	0.561799i	0.777236	+	0.629210i	$0.216621\pi$
$164$		0			0
$165$	−3.29348	−	5.70447i	−0.256397	−	0.444092i
$166$		0			0
$167$	−3.60763	−	3.34739i	−0.279167	−	0.259029i	0.528145	−	0.849154i	$-0.322888\pi$
$167$	−3.60763	−	3.34739i	−0.279167	−	0.259029i	−0.807312	+	0.590125i	$0.799078\pi$
$168$		0			0
$169$	−2.48249	+	1.69253i	−0.190961	+	0.130195i
$170$		0			0
$171$	16.3359			1.24923
$172$		0			0
$173$	−21.5624			−1.63936			−0.819680	−	0.572821i	$-0.805849\pi$
$173$	−21.5624			−1.63936			−0.819680	+	0.572821i	$0.805849\pi$
$174$		0			0
$175$	4.09778	−	2.79382i	0.309763	−	0.211193i
$176$		0			0
$177$	6.08989	+	5.65059i	0.457744	+	0.424725i
$178$		0			0
$179$	−8.40248	−	14.5535i	−0.628031	−	1.08778i	−0.987946	−	0.154797i	$-0.950528\pi$
$179$	−8.40248	−	14.5535i	−0.628031	−	1.08778i	0.359915	−	0.932985i	$-0.382806\pi$
$180$		0			0
$181$	0.294349	−	0.749988i	0.0218788	−	0.0557462i	−0.919517	−	0.393050i	$-0.871420\pi$
$181$	0.294349	−	0.749988i	0.0218788	−	0.0557462i	0.941396	+	0.337304i	$0.109515\pi$
$182$		0			0
$183$	13.9613	+	17.5069i	1.03205	+	1.29415i
$184$		0			0
$185$	9.03768	−	2.78775i	0.664463	−	0.204960i
$186$		0			0
$187$	−2.79150	−	0.420750i	−0.204134	−	0.0307683i
$188$		0			0
$189$	−10.8748	+	10.0904i	−0.791029	+	0.733968i
$190$		0			0
$191$	4.22806	+	10.7729i	0.305931	+	0.779500i	0.998318	+	0.0579738i	$0.0184640\pi$
$191$	4.22806	+	10.7729i	0.305931	+	0.779500i	−0.692387	+	0.721527i	$0.743441\pi$
$192$		0			0
$193$	5.28929	+	2.54719i	0.380732	+	0.183351i	0.614454	−	0.788953i	$-0.289376\pi$
$193$	5.28929	+	2.54719i	0.380732	+	0.183351i	−0.233722	+	0.972303i	$0.575091\pi$
$194$		0			0
$195$	−24.9046	+	3.75377i	−1.78346	+	0.268813i
$196$		0			0
$197$	0.412103	+	5.49913i	0.0293611	+	0.391797i	0.992453	+	0.122624i	$0.0391310\pi$
$197$	0.412103	+	5.49913i	0.0293611	+	0.391797i	−0.963092	+	0.269172i	$0.913250\pi$
$198$		0			0
$199$	4.37361	+	19.1620i	0.310037	+	1.35836i	0.854446	+	0.519540i	$0.173897\pi$
$199$	4.37361	+	19.1620i	0.310037	+	1.35836i	−0.544409	+	0.838820i	$0.683246\pi$
$200$		0			0
$201$	24.8592	+	16.9487i	1.75343	+	1.19547i
$202$		0			0
$203$	−1.28095	+	17.0931i	−0.0899051	+	1.19970i
$204$		0			0
$205$	−17.9684	−	5.54252i	−1.25497	−	0.387107i
$206$		0			0
$207$	−1.50218	+	6.58146i	−0.104408	+	0.457443i
$208$		0			0
$209$	−1.22466	+	2.12117i	−0.0847113	+	0.146724i
$210$		0			0
$211$	−10.8591	+	13.6169i	−0.747574	+	0.937428i	−0.999541	−	0.0303076i	$-0.990351\pi$
$211$	−10.8591	+	13.6169i	−0.747574	+	0.937428i	0.251966	+	0.967736i	$0.418923\pi$
$212$		0			0
$213$	31.8720	−	15.3487i	2.18383	−	1.05168i
$214$		0			0
$215$	−16.8533	+	6.03765i	−1.14939	+	0.411764i
$216$		0			0
$217$	−9.72529	+	4.68345i	−0.660196	+	0.317933i
$218$		0			0
$219$	22.6653	−	28.4213i	1.53158	−	1.92054i
$220$		0			0
$221$	−5.39708	+	9.34801i	−0.363047	+	0.628815i
$222$		0			0
$223$	−3.36182	+	14.7291i	−0.225124	+	0.986333i	0.728432	+	0.685118i	$0.240249\pi$
$223$	−3.36182	+	14.7291i	−0.225124	+	0.986333i	−0.953556	+	0.301215i	$0.902608\pi$
$224$		0			0
$225$	12.9277	+	3.98767i	0.861847	+	0.265844i
$226$		0			0
$227$	0.937088	−	12.5046i	0.0621967	−	0.829957i	−0.875659	−	0.482931i	$-0.839572\pi$
$227$	0.937088	−	12.5046i	0.0621967	−	0.829957i	0.937855	−	0.347027i	$-0.112809\pi$
$228$		0			0
$229$	16.8194	+	11.4673i	1.11146	+	0.757778i	0.972589	−	0.232533i	$-0.0747012\pi$
$229$	16.8194	+	11.4673i	1.11146	+	0.757778i	0.138868	+	0.990311i	$0.455654\pi$
$230$		0			0
$231$	−1.08541	−	4.75550i	−0.0714148	−	0.312889i
$232$		0			0
$233$	−2.20664	−	29.4455i	−0.144562	−	1.92904i	−0.325491	−	0.945545i	$-0.605530\pi$
$233$	−2.20664	−	29.4455i	−0.144562	−	1.92904i	0.180930	−	0.983496i	$-0.442089\pi$
$234$		0			0
$235$	10.1295	−	1.52678i	0.660776	−	0.0995960i
$236$		0			0
$237$	12.3189	+	5.93249i	0.800202	+	0.385357i
$238$		0			0
$239$	8.80723	+	22.4405i	0.569692	+	1.45155i	0.867748	+	0.497004i	$0.165567\pi$
$239$	8.80723	+	22.4405i	0.569692	+	1.45155i	−0.298056	+	0.954548i	$0.596338\pi$
$240$		0			0
$241$	4.34963	−	4.03587i	0.280184	−	0.259973i	−0.527544	−	0.849528i	$-0.676887\pi$
$241$	4.34963	−	4.03587i	0.280184	−	0.259973i	0.807729	+	0.589554i	$0.200697\pi$
$242$		0			0
$243$	7.72164	+	1.16385i	0.495343	+	0.0746610i
$244$		0			0
$245$	−7.59894	+	2.34396i	−0.485479	+	0.149750i
$246$		0			0
$247$	5.83912	+	7.32202i	0.371534	+	0.465889i
$248$		0			0
$249$	5.08882	−	12.9661i	0.322491	−	0.821693i
$250$		0			0
$251$	−7.19242	−	12.4576i	−0.453982	−	0.786319i	0.544647	−	0.838665i	$-0.316663\pi$
$251$	−7.19242	−	12.4576i	−0.453982	−	0.786319i	−0.998629	+	0.0523460i	$0.983330\pi$
$252$		0			0
$253$	−0.741972	−	0.688449i	−0.0466473	−	0.0432824i
$254$		0			0
$255$	22.4724	−	15.3214i	1.40728	−	0.959466i
$256$		0			0
$257$	−18.1706			−1.13345			−0.566724	−	0.823908i	$-0.691789\pi$
$257$	−18.1706			−1.13345			−0.566724	+	0.823908i	$0.691789\pi$
$258$		0			0
$259$	7.00377			0.435193
$260$		0			0
$261$	−38.6329	+	26.3394i	−2.39131	+	1.63037i
$262$		0			0
$263$	−11.6213	−	10.7830i	−0.716598	−	0.664906i	0.235089	−	0.971974i	$-0.424462\pi$
$263$	−11.6213	−	10.7830i	−0.716598	−	0.664906i	−0.951688	+	0.307068i	$0.900652\pi$
$264$		0			0
$265$	−12.1081	−	20.9718i	−0.743792	−	1.28829i
$266$		0			0
$267$	−2.20249	+	5.61184i	−0.134790	+	0.343439i
$268$		0			0
$269$	6.59147	+	8.26544i	0.401889	+	0.503953i	0.941058	−	0.338245i	$-0.109833\pi$
$269$	6.59147	+	8.26544i	0.401889	+	0.503953i	−0.539169	+	0.842197i	$0.681262\pi$
$270$		0			0
$271$	1.69682	−	0.523401i	0.103075	−	0.0317943i	−0.242789	−	0.970079i	$-0.578062\pi$
$271$	1.69682	−	0.523401i	0.103075	−	0.0317943i	0.345864	+	0.938285i	$0.387586\pi$
$272$		0			0
$273$	−18.4425	−	2.77975i	−1.11619	−	0.168238i
$274$		0			0
$275$	−1.48694	+	1.37968i	−0.0896661	+	0.0831980i
$276$		0			0
$277$	0.682551	+	1.73911i	0.0410105	+	0.104493i	0.949912	−	0.312519i	$-0.101173\pi$
$277$	0.682551	+	1.73911i	0.0410105	+	0.104493i	−0.908901	+	0.417012i	$0.863077\pi$
$278$		0			0
$279$	−26.5288	−	12.7756i	−1.58824	−	0.764854i
$280$		0			0
$281$	5.38800	−	0.812110i	0.321421	−	0.0484464i	0.0136498	−	0.999907i	$-0.495655\pi$
$281$	5.38800	−	0.812110i	0.321421	−	0.0484464i	0.307771	+	0.951460i	$0.400417\pi$
$282$		0			0
$283$	−0.707816	−	9.44514i	−0.0420753	−	0.561456i	−0.977911	−	0.209020i	$-0.932973\pi$
$283$	−0.707816	−	9.44514i	−0.0420753	−	0.561456i	0.935836	−	0.352436i	$-0.114646\pi$
$284$		0			0
$285$	−5.25103	−	23.0063i	−0.311044	−	1.36277i
$286$		0			0
$287$	−11.5051	−	7.84405i	−0.679125	−	0.463020i
$288$		0			0
$289$	−0.399306	+	5.32837i	−0.0234886	+	0.313433i
$290$		0			0
$291$	−33.5994	−	10.3640i	−1.96963	−	0.607550i
$292$		0			0
$293$	−2.12778	+	9.32241i	−0.124306	+	0.544621i	0.873973	+	0.485975i	$0.161535\pi$
$293$	−2.12778	+	9.32241i	−0.124306	+	0.544621i	−0.998279	+	0.0586458i	$0.981322\pi$
$294$		0			0
$295$	3.88625	−	6.73118i	0.226266	−	0.391904i
$296$		0			0
$297$	3.78299	−	4.74372i	0.219511	−	0.275258i
$298$		0			0
$299$	−3.48687	+	1.67919i	−0.201651	+	0.0971099i
$300$		0			0
$301$	−13.2437	+	0.593759i	−0.763353	+	0.0342237i
$302$		0			0
$303$	−6.70051	+	3.22680i	−0.384934	+	0.185375i
$304$		0			0
$305$	13.0621	−	16.3793i	0.747931	−	0.937876i
$306$		0			0
$307$	−8.96235	+	15.5232i	−0.511508	+	0.885959i	0.488403	+	0.872618i	$0.337580\pi$
$307$	−8.96235	+	15.5232i	−0.511508	+	0.885959i	−0.999911	+	0.0133402i	$0.995754\pi$
$308$		0			0
$309$	5.10667	−	22.3738i	0.290508	−	1.27280i
$310$		0			0
$311$	−11.9150	−	3.67529i	−0.675638	−	0.208407i	−0.0620976	−	0.998070i	$-0.519779\pi$
$311$	−11.9150	−	3.67529i	−0.675638	−	0.208407i	−0.613540	+	0.789663i	$0.710255\pi$
$312$		0			0
$313$	1.95694	−	26.1136i	0.110613	−	1.47603i	−0.615328	−	0.788271i	$-0.710976\pi$
$313$	1.95694	−	26.1136i	0.110613	−	1.47603i	0.725941	−	0.687757i	$-0.241405\pi$
$314$		0			0
$315$	25.1485	+	17.1460i	1.41696	+	0.966065i
$316$		0			0
$317$	−2.01390	−	8.82346i	−0.113112	−	0.495575i	−0.999469	−	0.0325735i	$-0.989630\pi$
$317$	−2.01390	−	8.82346i	−0.113112	−	0.495575i	0.886358	−	0.463001i	$-0.153227\pi$
$318$		0			0
$319$	−0.523901	−	6.99098i	−0.0293328	−	0.391420i
$320$		0			0
$321$	−24.1991	+	3.64743i	−1.35066	+	0.203580i
$322$		0			0
$323$	−9.11201	−	4.38811i	−0.507006	−	0.244161i
$324$		0			0
$325$	2.83356	+	7.21978i	0.157177	+	0.400482i
$326$		0			0
$327$	12.1336	−	11.2584i	0.670992	−	0.622590i
$328$		0			0
$329$	7.50114	+	1.13062i	0.413551	+	0.0623328i
$330$		0			0
$331$	3.33977	−	1.03018i	0.183571	−	0.0566240i	−0.201607	−	0.979466i	$-0.564616\pi$
$331$	3.33977	−	1.03018i	0.183571	−	0.0566240i	0.385178	+	0.922842i	$0.374140\pi$
$332$		0			0
$333$	11.9118	+	14.9369i	0.652760	+	0.818536i
$334$		0			0
$335$	10.2841	−	26.2034i	0.561879	−	1.43164i
$336$		0			0
$337$	−2.60035	−	4.50393i	−0.141650	−	0.245345i	0.786468	−	0.617631i	$-0.211907\pi$
$337$	−2.60035	−	4.50393i	−0.141650	−	0.245345i	−0.928118	+	0.372286i	$0.878574\pi$
$338$		0			0
$339$	38.3879	+	35.6187i	2.08494	+	1.93454i
$340$		0			0
$341$	3.64767	−	2.48694i	0.197532	−	0.134675i
$342$		0			0
$343$	−20.0405			−1.08209
$344$		0			0
$345$	9.75174			0.525016
$346$		0			0
$347$	−6.33995	+	4.32251i	−0.340346	+	0.232044i	−0.721416	−	0.692502i	$-0.756508\pi$
$347$	−6.33995	+	4.32251i	−0.340346	+	0.232044i	0.381069	+	0.924546i	$0.375556\pi$
$348$		0			0
$349$	19.3740	+	17.9765i	1.03707	+	0.962257i	0.999314	−	0.0370298i	$-0.0117896\pi$
$349$	19.3740	+	17.9765i	1.03707	+	0.962257i	0.0377526	+	0.999287i	$0.487980\pi$
$350$		0			0
$351$	−11.5998	−	20.0914i	−0.619150	−	1.07240i
$352$		0			0
$353$	−3.64148	+	9.27834i	−0.193817	+	0.493836i	−0.994437	−	0.105336i	$-0.966408\pi$
$353$	−3.64148	+	9.27834i	−0.193817	+	0.493836i	0.800620	+	0.599172i	$0.204504\pi$
$354$		0			0
$355$	−20.6355	−	25.8761i	−1.09522	−	1.37336i
$356$		0			0
$357$	19.2463	−	5.93670i	1.01862	−	0.314203i
$358$		0			0
$359$	−1.39816	−	0.210739i	−0.0737921	−	0.0111224i	0.112042	−	0.993703i	$-0.464261\pi$
$359$	−1.39816	−	0.210739i	−0.0737921	−	0.0111224i	−0.185835	+	0.982581i	$0.559499\pi$
$360$		0			0
$361$	7.49565	−	6.95495i	0.394508	−	0.366050i
$362$		0			0
$363$	−10.9979	−	28.0221i	−0.577239	−	1.47078i
$364$		0			0
$365$	−30.6427	−	14.7567i	−1.60391	−	0.772403i
$366$		0			0
$367$	28.0262	−	4.22427i	1.46295	−	0.220505i	0.631188	−	0.775630i	$-0.282568\pi$
$367$	28.0262	−	4.22427i	1.46295	−	0.220505i	0.831767	+	0.555125i	$0.187330\pi$
$368$		0			0
$369$	−2.83854	−	37.8777i	−0.147769	−	1.97183i
$370$		0			0
$371$	−3.99038	−	17.4830i	−0.207170	−	0.907672i
$372$		0			0
$373$	−19.4695	−	13.2741i	−1.00809	−	0.687304i	−0.0576903	−	0.998335i	$-0.518374\pi$
$373$	−19.4695	−	13.2741i	−1.00809	−	0.687304i	−0.950400	+	0.311030i	$0.899326\pi$
$374$		0			0
$375$	−1.51617	+	20.2319i	−0.0782946	+	1.04477i
$376$		0			0
$377$	−25.6148	−	7.90112i	−1.31923	−	0.406928i
$378$		0			0
$379$	−0.506526	+	2.21924i	−0.0260185	+	0.113995i	−0.986270	−	0.165143i	$-0.947191\pi$
$379$	−0.506526	+	2.21924i	−0.0260185	+	0.113995i	0.960251	+	0.279138i	$0.0900486\pi$
$380$		0			0
$381$	28.5831	−	49.5075i	1.46436	−	2.53634i
$382$		0			0
$383$	12.2178	−	15.3206i	0.624299	−	0.782846i	−0.364643	−	0.931147i	$-0.618809\pi$
$383$	12.2178	−	15.3206i	0.624299	−	0.782846i	0.988942	+	0.148301i	$0.0473805\pi$
$384$		0			0
$385$	−4.11168	+	1.98008i	−0.209550	+	0.100914i
$386$		0			0
$387$	−23.7907	−	27.2348i	−1.20935	−	1.38442i
$388$		0			0
$389$	−0.649494	+	0.312780i	−0.0329306	+	0.0158586i	−0.450277	−	0.892889i	$-0.648675\pi$
$389$	−0.649494	+	0.312780i	−0.0329306	+	0.0158586i	0.417346	+	0.908748i	$0.362960\pi$
$390$		0			0
$391$	2.60581	−	3.26758i	0.131781	−	0.165248i
$392$		0			0
$393$	−17.6926	+	30.6445i	−0.892475	+	1.54581i
$394$		0			0
$395$	2.84656	−	12.4716i	0.143226	−	0.627514i
$396$		0			0
$397$	−8.02091	−	2.47412i	−0.402558	−	0.124173i	0.0868631	−	0.996220i	$-0.472316\pi$
$397$	−8.02091	−	2.47412i	−0.402558	−	0.124173i	−0.489421	+	0.872048i	$0.662792\pi$
$398$		0			0
$399$	1.30590	−	17.4260i	0.0653765	−	0.872389i
$400$		0			0
$401$	23.6148	+	16.1003i	1.17927	+	0.804011i	0.984249	−	0.176789i	$-0.0565709\pi$
$401$	23.6148	+	16.1003i	1.17927	+	0.804011i	0.195019	+	0.980800i	$0.437523\pi$
$402$		0			0
$403$	−3.75625	−	16.4572i	−0.187112	−	0.819791i
$404$		0			0
$405$	0.993180	+	13.2531i	0.0493515	+	0.658550i
$406$		0			0
$407$	−2.83251	+	0.426932i	−0.140402	+	0.0211622i
$408$		0			0
$409$	9.56120	+	4.60443i	0.472771	+	0.227674i	0.655069	−	0.755569i	$-0.272639\pi$
$409$	9.56120	+	4.60443i	0.472771	+	0.227674i	−0.182299	+	0.983243i	$0.558354\pi$
$410$		0			0
$411$	6.35846	+	16.2011i	0.313640	+	0.799141i
$412$		0			0
$413$	4.21924	−	3.91488i	0.207615	−	0.192639i
$414$		0			0
$415$	−12.8862	−	1.94229i	−0.632560	−	0.0953431i
$416$		0			0
$417$	−8.66912	+	2.67407i	−0.424529	+	0.130950i
$418$		0			0
$419$	11.9692	+	15.0089i	0.584735	+	0.733235i	0.982912	−	0.184074i	$-0.0589285\pi$
$419$	11.9692	+	15.0089i	0.584735	+	0.733235i	−0.398177	+	0.917309i	$0.630357\pi$
$420$		0			0
$421$	−10.9213	+	27.8270i	−0.532272	+	1.35621i	0.371596	+	0.928395i	$0.378811\pi$
$421$	−10.9213	+	27.8270i	−0.532272	+	1.35621i	−0.903868	+	0.427812i	$0.859285\pi$
$422$		0			0
$423$	10.3464	+	17.9205i	0.503060	+	0.871326i
$424$		0			0
$425$	−6.13981	−	5.69691i	−0.297825	−	0.276341i
$426$		0			0
$427$	12.8182	−	8.73930i	0.620316	−	0.422924i
$428$		0			0
$429$	7.62806			0.368286
$430$		0			0
$431$	−20.3418			−0.979832			−0.489916	−	0.871770i	$-0.662972\pi$
$431$	−20.3418			−0.979832			−0.489916	+	0.871770i	$0.662972\pi$
$432$		0			0
$433$	−22.7969	+	15.5427i	−1.09555	+	0.746932i	−0.969500	−	0.245092i	$-0.921182\pi$
$433$	−22.7969	+	15.5427i	−1.09555	+	0.746932i	−0.126049	+	0.992024i	$0.540230\pi$
$434$		0			0
$435$	49.5129	+	45.9412i	2.37396	+	2.20271i
$436$		0			0
$437$	−1.81306	−	3.14031i	−0.0867304	−	0.150221i
$438$		0			0
$439$	9.38354	−	23.9089i	0.447852	−	1.14111i	−0.511597	−	0.859226i	$-0.670946\pi$
$439$	9.38354	−	23.9089i	0.447852	−	1.14111i	0.959449	−	0.281883i	$-0.0909590\pi$
$440$		0			0
$441$	−10.0155	−	12.5590i	−0.476928	−	0.598049i
$442$		0			0
$443$	−0.266159	+	0.0820992i	−0.0126456	+	0.00390065i	−0.301072	−	0.953602i	$-0.597344\pi$
$443$	−0.266159	+	0.0820992i	−0.0126456	+	0.00390065i	0.288426	+	0.957502i	$0.406868\pi$
$444$		0			0
$445$	5.57728	+	0.840639i	0.264388	+	0.0398501i
$446$		0			0
$447$	3.44570	−	3.19715i	0.162976	−	0.151220i
$448$		0			0
$449$	4.94037	+	12.5878i	0.233150	+	0.594057i	0.998788	−	0.0492145i	$-0.0156718\pi$
$449$	4.94037	+	12.5878i	0.233150	+	0.594057i	−0.765638	+	0.643272i	$0.777577\pi$
$450$		0			0
$451$	5.13112	+	2.47102i	0.241615	+	0.116356i
$452$		0			0
$453$	−12.7872	+	1.92735i	−0.600793	+	0.0905550i
$454$		0			0
$455$	1.30400	+	17.4007i	0.0611325	+	0.815757i
$456$		0			0
$457$	3.82680	+	16.7663i	0.179010	+	0.784295i	0.982088	+	0.188421i	$0.0603370\pi$
$457$	3.82680	+	16.7663i	0.179010	+	0.784295i	−0.803078	+	0.595874i	$0.796806\pi$
$458$		0			0
$459$	20.7001	+	14.1131i	0.966196	+	0.658741i
$460$		0			0
$461$	−0.242881	+	3.24102i	−0.0113121	+	0.150949i	0.988687	+	0.149990i	$0.0479243\pi$
$461$	−0.242881	+	3.24102i	−0.0113121	+	0.150949i	−1.00000		0.000958913i	$0.999695\pi$
$462$		0			0
$463$	−32.6129	−	10.0597i	−1.51565	−	0.467516i	−0.578275	−	0.815842i	$-0.696274\pi$
$463$	−32.6129	−	10.0597i	−1.51565	−	0.467516i	−0.937373	+	0.348326i	$0.886750\pi$
$464$		0			0
$465$	−9.46477	+	41.4679i	−0.438918	+	1.92303i
$466$		0			0
$467$	−17.3380	+	30.0303i	−0.802308	+	1.38964i	0.115786	+	0.993274i	$0.463061\pi$
$467$	−17.3380	+	30.0303i	−0.802308	+	1.38964i	−0.918094	+	0.396364i	$0.870272\pi$
$468$		0			0
$469$	12.9968	−	16.2974i	0.600134	−	0.752545i
$470$		0			0
$471$	−9.59059	+	4.61858i	−0.441911	+	0.212813i
$472$		0			0
$473$	5.31989	−	1.04743i	0.244609	−	0.0481610i
$474$		0			0
$475$	−6.54726	+	3.15299i	−0.300409	+	0.144669i
$476$		0			0
$477$	30.4991	−	38.2447i	1.39646	−	1.75110i
$478$		0			0
$479$	3.86193	−	6.68907i	0.176456	−	0.305631i	−0.764208	−	0.644970i	$-0.776870\pi$
$479$	3.86193	−	6.68907i	0.176456	−	0.305631i	0.940664	+	0.339339i	$0.110203\pi$
$480$		0			0
$481$	−2.43721	+	10.6781i	−0.111127	+	0.486881i
$482$		0			0
$483$	6.90057	+	2.12854i	0.313987	+	0.0968520i
$484$		0			0
$485$	−2.45838	+	32.8048i	−0.111629	+	1.48959i
$486$		0			0
$487$	19.0696	+	13.0014i	0.864125	+	0.589151i	0.912224	−	0.409692i	$-0.134364\pi$
$487$	19.0696	+	13.0014i	0.864125	+	0.589151i	−0.0480986	+	0.998843i	$0.515316\pi$
$488$		0			0
$489$	5.00311	+	21.9201i	0.226249	+	0.991260i
$490$		0			0
$491$	0.737039	+	9.83511i	0.0332621	+	0.443852i	0.988827	+	0.149070i	$0.0476280\pi$
$491$	0.737039	+	9.83511i	0.0332621	+	0.443852i	−0.955565	+	0.294782i	$0.904753\pi$
$492$		0			0
$493$	28.6244	−	4.31443i	1.28918	−	0.194312i
$494$		0			0
$495$	−11.2159	−	5.40128i	−0.504116	−	0.242770i
$496$		0			0
$497$	−8.95411	−	22.8147i	−0.401647	−	1.02338i
$498$		0			0
$499$	−30.3095	+	28.1231i	−1.35684	+	1.25896i	−0.420759	+	0.907172i	$0.638236\pi$
$499$	−30.3095	+	28.1231i	−1.35684	+	1.25896i	−0.936079	+	0.351790i	$0.885573\pi$
$500$		0			0
$501$	−14.2002	−	2.14034i	−0.634419	−	0.0956233i
$502$		0			0
$503$	−13.2126	+	4.07554i	−0.589120	+	0.181720i	−0.574962	−	0.818180i	$-0.694983\pi$
$503$	−13.2126	+	4.07554i	−0.589120	+	0.181720i	−0.0141584	+	0.999900i	$0.504507\pi$
$504$		0			0
$505$	4.33824	+	5.43998i	0.193049	+	0.242076i
$506$		0			0
$507$	−3.20307	+	8.16128i	−0.142253	+	0.362455i
$508$		0			0
$509$	12.1089	+	20.9732i	0.536716	+	0.929619i	0.999078	+	0.0429280i	$0.0136686\pi$
$509$	12.1089	+	20.9732i	0.536716	+	0.929619i	−0.462362	+	0.886691i	$0.652998\pi$
$510$		0			0
$511$	−18.4625	−	17.1307i	−0.816733	−	0.757818i
$512$		0			0
$513$	17.9598	−	12.2448i	0.792943	−	0.540620i
$514$		0			0
$515$	−21.4710			−0.946125
$516$		0			0
$517$	−3.10258			−0.136451
$518$		0			0
$519$	−51.9863	+	35.4436i	−2.28194	+	1.55580i
$520$		0			0
$521$	−4.41210	−	4.09383i	−0.193297	−	0.179354i	0.577586	−	0.816330i	$-0.303995\pi$
$521$	−4.41210	−	4.09383i	−0.193297	−	0.179354i	−0.770884	+	0.636976i	$0.780185\pi$
$522$		0			0
$523$	18.5046	+	32.0509i	0.809149	+	1.40149i	0.913454	+	0.406942i	$0.133405\pi$
$523$	18.5046	+	32.0509i	0.809149	+	1.40149i	−0.104305	+	0.994545i	$0.533262\pi$
$524$		0			0
$525$	5.28721	−	13.4716i	0.230753	−	0.587949i
$526$		0			0
$527$	11.3658	+	14.2522i	0.495101	+	0.620838i
$528$		0			0
$529$	−20.5463	+	6.33769i	−0.893316	+	0.275552i
$530$		0			0
$531$	15.5252	+	2.34004i	0.673734	+	0.101549i
$532$		0			0
$533$	15.9628	−	14.8114i	0.691428	−	0.641551i
$534$		0			0
$535$	8.36491	+	21.3134i	0.361647	+	0.921461i
$536$		0			0
$537$	−44.1808	−	21.2763i	−1.90654	−	0.918141i
$538$		0			0
$539$	2.38159	−	0.358968i	0.102583	−	0.0154618i
$540$		0			0
$541$	−0.253692	−	3.38529i	−0.0109071	−	0.145545i	0.989091	−	0.147303i	$-0.0470593\pi$
$541$	−0.253692	−	3.38529i	−0.0109071	−	0.145545i	−0.999998	+	0.00175845i	$0.999440\pi$
$542$		0			0
$543$	−0.523143	−	2.29204i	−0.0224502	−	0.0983608i
$544$		0			0
$545$	−12.7952	−	8.72363i	−0.548087	−	0.373679i
$546$		0			0
$547$	1.91085	−	25.4985i	0.0817021	−	1.09024i	−0.794170	−	0.607696i	$-0.792094\pi$
$547$	1.91085	−	25.4985i	0.0817021	−	1.09024i	0.875872	−	0.482543i	$-0.160287\pi$
$548$		0			0
$549$	40.4389	+	12.4738i	1.72589	+	0.532367i
$550$		0			0
$551$	5.58874	−	24.4859i	0.238089	−	1.04313i
$552$		0			0
$553$	4.73651	−	8.20387i	0.201417	−	0.348864i
$554$		0			0
$555$	17.2071	−	21.5770i	0.730401	−	0.915894i
$556$		0			0
$557$	−18.9148	+	9.10888i	−0.801445	+	0.385955i	−0.789328	−	0.613971i	$-0.789571\pi$
$557$	−18.9148	+	9.10888i	−0.801445	+	0.385955i	−0.0121162	+	0.999927i	$0.503857\pi$
$558$		0			0
$559$	3.70335	−	20.3983i	0.156635	−	0.862754i
$560$		0			0
$561$	−7.42182	+	3.57416i	−0.313350	+	0.150901i
$562$		0			0
$563$	16.8730	−	21.1580i	0.711111	−	0.891705i	−0.286688	−	0.958024i	$-0.592554\pi$
$563$	16.8730	−	21.1580i	0.711111	−	0.891705i	0.997798	+	0.0663195i	$0.0211257\pi$
$564$		0			0
$565$	24.4971	−	42.4302i	1.03060	−	1.78505i
$566$		0			0
$567$	−2.18999	+	9.59497i	−0.0919709	+	0.402951i
$568$		0			0
$569$	−26.2872	−	8.10851i	−1.10201	−	0.339927i	−0.310190	−	0.950675i	$-0.600393\pi$
$569$	−26.2872	−	8.10851i	−1.10201	−	0.339927i	−0.791825	+	0.610748i	$0.790869\pi$
$570$		0			0
$571$	−2.56665	+	34.2495i	−0.107411	+	1.43330i	0.639713	+	0.768614i	$0.279053\pi$
$571$	−2.56665	+	34.2495i	−0.107411	+	1.43330i	−0.747124	+	0.664685i	$0.768566\pi$
$572$		0			0
$573$	27.9019	+	19.0232i	1.16562	+	0.794705i
$574$		0			0
$575$	−0.668234	−	2.92772i	−0.0278673	−	0.122095i
$576$		0			0
$577$	−0.0405171	−	0.540663i	−0.00168675	−	0.0225081i	0.996297	−	0.0859794i	$-0.0274019\pi$
$577$	−0.0405171	−	0.540663i	−0.00168675	−	0.0225081i	−0.997984	+	0.0634713i	$0.979783\pi$
$578$		0			0
$579$	16.9393	−	2.55319i	0.703973	−	0.106107i
$580$		0			0
$581$	−8.69466	−	4.18713i	−0.360715	−	0.173711i
$582$		0			0
$583$	2.67953	+	6.82734i	0.110975	+	0.282759i
$584$		0			0
$585$	−34.8925	+	32.3755i	−1.44263	+	1.33856i
$586$		0			0
$587$	−9.72187	−	1.46534i	−0.401265	−	0.0604809i	−0.0546901	−	0.998503i	$-0.517417\pi$
$587$	−9.72187	−	1.46534i	−0.401265	−	0.0604809i	−0.346574	+	0.938022i	$0.612655\pi$
$588$		0			0
$589$	15.1134	−	4.66188i	0.622738	−	0.192089i
$590$		0			0
$591$	10.0329	+	12.5808i	0.412697	+	0.517505i
$592$		0			0
$593$	10.5629	−	26.9139i	0.433767	−	1.10522i	−0.532214	−	0.846610i	$-0.678640\pi$
$593$	10.5629	−	26.9139i	0.433767	−	1.10522i	0.965981	−	0.258611i	$-0.0832649\pi$
$594$		0			0
$595$	−9.42192	−	16.3192i	−0.386261	−	0.669024i
$596$		0			0
$597$	42.0426	+	39.0098i	1.72069	+	1.59657i
$598$		0			0
$599$	1.58885	−	1.08326i	0.0649188	−	0.0442609i	−0.530426	−	0.847731i	$-0.677968\pi$
$599$	1.58885	−	1.08326i	0.0649188	−	0.0442609i	0.595345	+	0.803470i	$0.297016\pi$
$600$		0			0
$601$	−45.9528			−1.87446			−0.937228	−	0.348718i	$-0.886617\pi$
$601$	−45.9528			−1.87446			−0.937228	+	0.348718i	$0.886617\pi$
$602$		0			0
$603$	56.8617			2.31559
$604$		0			0
$605$	−23.2703	+	15.8654i	−0.946071	+	0.645020i
$606$		0			0
$607$	−21.6805	−	20.1166i	−0.879986	−	0.816508i	0.104032	−	0.994574i	$-0.466825\pi$
$607$	−21.6805	−	20.1166i	−0.879986	−	0.816508i	−0.984019	+	0.178066i	$0.943016\pi$
$608$		0			0
$609$	25.0088	+	43.3165i	1.01341	+	1.75527i
$610$		0			0
$611$	−4.33406	+	11.0430i	−0.175337	+	0.446752i
$612$		0			0
$613$	−1.83476	−	2.30071i	−0.0741052	−	0.0929250i	0.743397	−	0.668850i	$-0.233213\pi$
$613$	−1.83476	−	2.30071i	−0.0741052	−	0.0929250i	−0.817502	+	0.575925i	$0.804642\pi$
$614$		0			0
$615$	−52.4319	+	16.1731i	−2.11426	+	0.652162i
$616$		0			0
$617$	9.18497	+	1.38441i	0.369773	+	0.0557343i	0.331300	−	0.943525i	$-0.392513\pi$
$617$	9.18497	+	1.38441i	0.369773	+	0.0557343i	0.0384726	+	0.999260i	$0.487751\pi$
$618$		0			0
$619$	28.9452	−	26.8572i	1.16341	−	1.07948i	0.167803	−	0.985821i	$-0.446333\pi$
$619$	28.9452	−	26.8572i	1.16341	−	1.07948i	0.995604	−	0.0936630i	$-0.0298576\pi$
$620$		0			0
$621$	3.28172	+	8.36169i	0.131691	+	0.335543i
$622$		0			0
$623$	3.76313	+	1.81223i	0.150767	+	0.0726053i
$624$		0			0
$625$	30.8988	−	4.65724i	1.23595	−	0.186290i
$626$		0			0
$627$	0.534104	+	7.12712i	0.0213300	+	0.284630i
$628$		0			0
$629$	−2.63197	−	11.5314i	−0.104943	−	0.459787i
$630$		0			0
$631$	18.1097	+	12.3470i	0.720936	+	0.491526i	0.867351	−	0.497698i	$-0.165821\pi$
$631$	18.1097	+	12.3470i	0.720936	+	0.491526i	−0.146415	+	0.989223i	$0.546773\pi$
$632$		0			0
$633$	−3.79793	+	50.6799i	−0.150954	+	2.01435i
$634$		0			0
$635$	−51.1081	−	15.7647i	−2.02816	−	0.625605i
$636$		0			0
$637$	2.04923	−	8.97825i	0.0811933	−	0.355731i
$638$		0			0
$639$	33.4279	−	57.8988i	1.32239	−	2.29044i
$640$		0			0
$641$	14.4602	−	18.1325i	0.571143	−	0.716191i	−0.409431	−	0.912341i	$-0.634273\pi$
$641$	14.4602	−	18.1325i	0.571143	−	0.716191i	0.980574	+	0.196151i	$0.0628441\pi$
$642$		0			0
$643$	18.3605	−	8.84194i	0.724067	−	0.348692i	−0.0352809	−	0.999377i	$-0.511233\pi$
$643$	18.3605	−	8.84194i	0.724067	−	0.348692i	0.759347	+	0.650685i	$0.225518\pi$
$644$		0			0
$645$	−30.7083	+	42.2595i	−1.20914	+	1.66397i
$646$		0			0
$647$	0.758290	−	0.365173i	0.0298114	−	0.0143564i	−0.418919	−	0.908024i	$-0.637591\pi$
$647$	0.758290	−	0.365173i	0.0298114	−	0.0143564i	0.448730	+	0.893667i	$0.351876\pi$
$648$		0			0
$649$	−1.46773	+	1.84047i	−0.0576134	+	0.0722449i
$650$		0			0
$651$	−15.7488	+	27.2778i	−0.617245	+	1.06910i
$652$		0			0
$653$	−8.94329	+	39.1831i	−0.349978	+	1.53335i	0.427250	+	0.904134i	$0.359483\pi$
$653$	−8.94329	+	39.1831i	−0.349978	+	1.53335i	−0.777227	+	0.629220i	$0.783375\pi$
$654$		0			0
$655$	31.6353	+	9.75819i	1.23609	+	0.381284i
$656$		0			0
$657$	5.13412	−	68.5101i	0.200301	−	2.67283i
$658$		0			0
$659$	16.9744	+	11.5729i	0.661229	+	0.450818i	0.846870	−	0.531800i	$-0.178484\pi$
$659$	16.9744	+	11.5729i	0.661229	+	0.450818i	−0.185642	+	0.982617i	$0.559436\pi$
$660$		0			0
$661$	−4.27999	−	18.7519i	−0.166472	−	0.729364i	−0.987389	−	0.158315i	$-0.949394\pi$
$661$	−4.27999	−	18.7519i	−0.166472	−	0.729364i	0.820916	−	0.571049i	$-0.193463\pi$
$662$		0			0
$663$	2.35380	+	31.4093i	0.0914140	+	1.21984i
$664$		0			0
$665$	−16.1667	+	2.43673i	−0.626916	+	0.0944925i
$666$		0			0
$667$	9.35107	+	4.50324i	0.362075	+	0.174366i
$668$		0			0
$669$	16.1060	+	41.0374i	0.622694	+	1.58660i
$670$		0			0
$671$	−4.65129	+	4.31576i	−0.179561	+	0.166608i
$672$		0			0
$673$	11.6700	+	1.75897i	0.449846	+	0.0678033i	0.370058	−	0.929009i	$-0.379338\pi$
$673$	11.6700	+	1.75897i	0.449846	+	0.0678033i	0.0797875	+	0.996812i	$0.474576\pi$
$674$		0			0
$675$	17.2018	−	5.30606i	0.662099	−	0.204230i
$676$		0			0
$677$	−5.08538	−	6.37687i	−0.195447	−	0.245083i	0.674445	−	0.738325i	$-0.264383\pi$
$677$	−5.08538	−	6.37687i	−0.195447	−	0.245083i	−0.869892	+	0.493242i	$0.835812\pi$
$678$		0			0
$679$	−8.90000	+	22.6768i	−0.341551	+	0.870257i
$680$		0			0
$681$	−18.2953	−	31.6885i	−0.701079	−	1.21430i
$682$		0			0
$683$	−4.30555	−	3.99496i	−0.164747	−	0.152863i	0.593491	−	0.804841i	$-0.297749\pi$
$683$	−4.30555	−	3.99496i	−0.164747	−	0.152863i	−0.758238	+	0.651977i	$0.773940\pi$
$684$		0			0
$685$	13.4538	−	9.17264i	0.514043	−	0.350469i
$686$		0			0
$687$	59.4005			2.26627
$688$		0			0
$689$	28.0436			1.06838
$690$		0			0
$691$	−30.4990	+	20.7939i	−1.16024	+	0.791037i	−0.981236	−	0.192812i	$-0.938239\pi$
$691$	−30.4990	+	20.7939i	−1.16024	+	0.791037i	−0.179002	+	0.983849i	$0.557287\pi$
$692$		0			0
$693$	−6.75768	−	6.27021i	−0.256703	−	0.238185i
$694$		0			0
$695$	4.24392	+	7.35068i	0.160981	+	0.278827i
$696$		0			0
$697$	−8.59134	+	21.8904i	−0.325420	+	0.829157i
$698$		0			0
$699$	−53.7218	−	67.3650i	−2.03194	−	2.54798i
$700$		0			0
$701$	−13.3574	+	4.12021i	−0.504502	+	0.155618i	−0.536551	−	0.843868i	$-0.680273\pi$
$701$	−13.3574	+	4.12021i	−0.504502	+	0.155618i	0.0320488	+	0.999486i	$0.489797\pi$
$702$		0			0
$703$	−10.1475	−	1.52950i	−0.382722	−	0.0576860i
$704$		0			0
$705$	21.9122	−	20.3316i	0.825263	−	0.765732i
$706$		0			0
$707$	1.88244	+	4.79638i	0.0707965	+	0.180387i
$708$		0			0
$709$	−26.8924	−	12.9507i	−1.00997	−	0.486374i	−0.145656	−	0.989335i	$-0.546529\pi$
$709$	−26.8924	−	12.9507i	−1.00997	−	0.486374i	−0.864309	+	0.502962i	$0.832244\pi$
$710$		0			0
$711$	25.5520	−	3.85134i	0.958275	−	0.144437i
$712$		0			0
$713$	0.488431	+	6.51765i	0.0182919	+	0.244088i
$714$		0			0
$715$	−1.58807	−	6.95780i	−0.0593906	−	0.260207i
$716$		0			0
$717$	58.1209	+	39.6261i	2.17056	+	1.47987i
$718$		0			0
$719$	−0.700233	+	9.34396i	−0.0261143	+	0.348471i	0.968807	+	0.247816i	$0.0797129\pi$
$719$	−0.700233	+	9.34396i	−0.0261143	+	0.348471i	−0.994921	+	0.100655i	$0.967906\pi$
$720$		0			0
$721$	−15.1934	−	4.68654i	−0.565832	−	0.174536i
$722$		0			0
$723$	3.85278	−	16.8801i	0.143286	−	0.627779i
$724$		0			0
$725$	10.3999	−	18.0132i	0.386243	−	0.668992i
$726$		0			0
$727$	32.3793	−	40.6024i	1.20088	−	1.50586i	0.389828	−	0.920888i	$-0.372534\pi$
$727$	32.3793	−	40.6024i	1.20088	−	1.50586i	0.811054	−	0.584971i	$-0.198894\pi$
$728$		0			0
$729$	33.6878	−	16.2232i	1.24770	−	0.600858i
$730$		0			0
$731$	5.95447	+	21.5820i	0.220234	+	0.798238i
$732$		0			0
$733$	14.5273	−	6.99597i	0.536577	−	0.258402i	−0.145912	−	0.989298i	$-0.546612\pi$
$733$	14.5273	−	6.99597i	0.536577	−	0.258402i	0.682489	+	0.730896i	$0.260897\pi$
$734$		0			0
$735$	−14.4679	+	18.1421i	−0.533655	+	0.669183i
$736$		0			0
$737$	−4.26278	+	7.38335i	−0.157021	+	0.271969i
$738$		0			0
$739$	2.96821	−	13.0046i	0.109187	−	0.478381i	−0.890537	−	0.454911i	$-0.849671\pi$
$739$	2.96821	−	13.0046i	0.109187	−	0.478381i	0.999725	−	0.0234707i	$-0.00747165\pi$
$740$		0			0
$741$	26.1136	+	8.05499i	0.959308	+	0.295907i
$742$		0			0
$743$	−0.266334	+	3.55398i	−0.00977085	+	0.130383i	−0.999956	−	0.00937874i	$-0.997015\pi$
$743$	−0.266334	+	3.55398i	−0.00977085	+	0.130383i	0.990185	+	0.139762i	$0.0446337\pi$
$744$		0			0
$745$	−3.63358	−	2.47733i	−0.133124	−	0.0907624i
$746$		0			0
$747$	−5.85772	−	25.6643i	−0.214323	−	0.939009i
$748$		0			0
$749$	1.26706	+	16.9077i	0.0462974	+	0.617796i
$750$		0			0
$751$	−0.730912	+	0.110167i	−0.0266714	+	0.00402006i	−0.162364	−	0.986731i	$-0.551912\pi$
$751$	−0.730912	+	0.110167i	−0.0266714	+	0.00402006i	0.135693	+	0.990751i	$0.456674\pi$
$752$		0			0
$753$	−37.8182	−	18.2123i	−1.37817	−	0.663692i
$754$		0			0
$755$	4.42014	+	11.2623i	0.160865	+	0.409879i
$756$		0			0
$757$	−4.16657	+	3.86601i	−0.151436	+	0.140513i	−0.752253	−	0.658875i	$-0.771033\pi$
$757$	−4.16657	+	3.86601i	−0.151436	+	0.140513i	0.600816	+	0.799387i	$0.294842\pi$
$758$		0			0
$759$	−2.92052	−	0.440198i	−0.106008	−	0.0159782i
$760$		0			0
$761$	−23.9356	+	7.38315i	−0.867665	+	0.267639i	−0.696459	−	0.717596i	$-0.745242\pi$
$761$	−23.9356	+	7.38315i	−0.867665	+	0.267639i	−0.171205	+	0.985235i	$0.554766\pi$
$762$		0			0
$763$	−7.15007	−	8.96591i	−0.258850	−	0.324588i
$764$		0			0
$765$	18.7794	−	47.8492i	0.678972	−	1.72999i
$766$		0			0
$767$	4.50049	+	7.79508i	0.162503	+	0.281464i
$768$		0			0
$769$	−19.8037	−	18.3752i	−0.714140	−	0.662625i	0.236956	−	0.971520i	$-0.423850\pi$
$769$	−19.8037	−	18.3752i	−0.714140	−	0.662625i	−0.951096	+	0.308895i	$0.900041\pi$
$770$		0			0
$771$	−43.8086	+	29.8682i	−1.57773	+	1.07568i
$772$		0			0
$773$	31.0614			1.11720			0.558600	−	0.829437i	$-0.311339\pi$
$773$	31.0614			1.11720			0.558600	+	0.829437i	$0.311339\pi$
$774$		0			0
$775$	13.0983			0.470505
$776$		0			0
$777$	16.8859	−	11.5126i	0.605777	−	0.413012i
$778$		0			0
$779$	14.9564	+	13.8775i	0.535868	+	0.497213i
$780$		0			0
$781$	5.01200	+	8.68104i	0.179344	+	0.310632i
$782$		0			0
$783$	−22.7302	+	57.9156i	−0.812311	+	2.06973i
$784$		0			0
$785$	6.20941	+	7.78636i	0.221623	+	0.277907i
$786$		0			0
$787$	10.8315	−	3.34108i	0.386102	−	0.119097i	−0.0956261	−	0.995417i	$-0.530485\pi$
$787$	10.8315	−	3.34108i	0.386102	−	0.119097i	0.481729	+	0.876321i	$0.340009\pi$
$788$		0			0
$789$	−45.7432	−	6.89468i	−1.62850	−	0.245457i
$790$		0			0
$791$	26.5961	−	24.6776i	0.945650	−	0.877435i
$792$		0			0
$793$	8.86360	+	22.5841i	0.314756	+	0.801985i
$794$		0			0
$795$	−63.6649	−	30.6594i	−2.25796	−	1.08738i
$796$		0			0
$797$	10.6905	−	1.61133i	0.378675	−	0.0570761i	0.0430532	−	0.999073i	$-0.486292\pi$
$797$	10.6905	−	1.61133i	0.378675	−	0.0570761i	0.335622	+	0.941997i	$0.391053\pi$
$798$		0			0
$799$	−0.957367	−	12.7752i	−0.0338692	−	0.451953i
$800$		0			0
$801$	2.53527	+	11.1078i	0.0895794	+	0.392473i
$802$		0			0
$803$	8.51096	+	5.80268i	0.300345	+	0.204772i
$804$		0			0
$805$	0.504896	−	6.73737i	0.0177953	−	0.237461i
$806$		0			0
$807$	29.4783	+	9.09284i	1.03768	+	0.320083i
$808$		0			0
$809$	8.65541	−	37.9218i	0.304308	−	1.33326i	−0.559245	−	0.829002i	$-0.688909\pi$
$809$	8.65541	−	37.9218i	0.304308	−	1.33326i	0.863553	−	0.504258i	$-0.168234\pi$
$810$		0			0
$811$	2.39529	−	4.14876i	0.0841100	−	0.145683i	−0.820902	−	0.571070i	$-0.806529\pi$
$811$	2.39529	−	4.14876i	0.0841100	−	0.145683i	0.905012	+	0.425387i	$0.139862\pi$
$812$		0			0
$813$	3.23063	−	4.05109i	0.113303	−	0.142078i
$814$		0			0
$815$	18.9524	−	9.12701i	0.663875	−	0.319705i
$816$		0			0
$817$	19.3180	+	2.03190i	0.675851	+	0.0710871i
$818$		0			0
$819$	−31.7575	+	15.2936i	−1.10970	+	0.534401i
$820$		0			0
$821$	−32.6933	+	40.9962i	−1.14101	+	1.43078i	−0.255097	+	0.966915i	$0.582107\pi$
$821$	−32.6933	+	40.9962i	−1.14101	+	1.43078i	−0.885908	+	0.463860i	$0.846464\pi$
$822$		0			0
$823$	−4.68633	+	8.11696i	−0.163355	+	0.282940i	−0.936070	−	0.351814i	$-0.885565\pi$
$823$	−4.68633	+	8.11696i	−0.163355	+	0.282940i	0.772715	+	0.634753i	$0.218898\pi$
$824$		0			0
$825$	−1.31709	+	5.77056i	−0.0458553	+	0.200905i
$826$		0			0
$827$	45.0728	+	13.9031i	1.56733	+	0.483459i	0.952331	−	0.305066i	$-0.0986784\pi$
$827$	45.0728	+	13.9031i	1.56733	+	0.483459i	0.615003	+	0.788524i	$0.289155\pi$
$828$		0			0
$829$	0.496542	−	6.62590i	0.0172456	−	0.230127i	−0.981920	−	0.189294i	$-0.939380\pi$
$829$	0.496542	−	6.62590i	0.0172456	−	0.230127i	0.999166	−	0.0408325i	$-0.0130010\pi$
$830$		0			0
$831$	4.50431	+	3.07099i	0.156253	+	0.106531i
$832$		0			0
$833$	2.21298	+	9.69568i	0.0766751	+	0.335935i
$834$		0			0
$835$	1.00405	+	13.3981i	0.0347465	+	0.463660i
$836$		0			0
$837$	−38.7421	+	5.83943i	−1.33912	+	0.201840i
$838$		0			0
$839$	−20.9640	−	10.0957i	−0.723757	−	0.348543i	0.0354681	−	0.999371i	$-0.488708\pi$
$839$	−20.9640	−	10.0957i	−0.723757	−	0.348543i	−0.759226	+	0.650828i	$0.774422\pi$
$840$		0			0
$841$	15.6685	+	39.9228i	0.540294	+	1.37665i
$842$		0			0
$843$	11.6554	−	10.8146i	0.401432	−	0.372474i
$844$		0			0
$845$	8.11101	+	1.22254i	0.279027	+	0.0420566i
$846$		0			0
$847$	−19.9296	+	6.14747i	−0.684789	+	0.211230i
$848$		0			0
$849$	−17.2322	−	21.6084i	−0.591406	−	0.741600i
$850$		0			0
$851$	1.54933	−	3.94764i	0.0531105	−	0.135323i
$852$		0			0
$853$	−22.6689	−	39.2637i	−0.776168	−	1.34436i	−0.934136	−	0.356918i	$-0.883828\pi$
$853$	−22.6689	−	39.2637i	−0.776168	−	1.34436i	0.157968	−	0.987444i	$-0.449506\pi$
$854$		0			0
$855$	−32.6925	−	30.3342i	−1.11806	−	1.03741i
$856$		0			0
$857$	24.0494	−	16.3966i	0.821511	−	0.560097i	−0.0780208	−	0.996952i	$-0.524860\pi$
$857$	24.0494	−	16.3966i	0.821511	−	0.560097i	0.899532	+	0.436855i	$0.143908\pi$
$858$		0			0
$859$	−34.6694			−1.18290			−0.591452	−	0.806340i	$-0.701445\pi$
$859$	−34.6694			−1.18290			−0.591452	+	0.806340i	$0.701445\pi$
$860$		0			0
$861$	−40.6322			−1.38474
$862$		0			0
$863$	8.65784	−	5.90282i	0.294716	−	0.200934i	−0.406930	−	0.913459i	$-0.633401\pi$
$863$	8.65784	−	5.90282i	0.294716	−	0.200934i	0.701647	+	0.712525i	$0.252448\pi$
$864$		0			0
$865$	43.1523	+	40.0394i	1.46722	+	1.36138i
$866$		0			0
$867$	7.79589	+	13.5029i	0.264762	+	0.458582i
$868$		0			0
$869$	−1.41548	+	3.60659i	−0.0480169	+	0.122345i
$870$		0			0
$871$	20.3248	+	25.4864i	0.688678	+	0.863575i
$872$		0			0
$873$	−63.4995	+	19.5870i	−2.14913	+	0.662919i
$874$		0			0
$875$	13.8995	+	2.09501i	0.469888	+	0.0708243i
$876$		0			0
$877$	8.34914	−	7.74687i	0.281930	−	0.261593i	−0.526513	−	0.850167i	$-0.676501\pi$
$877$	8.34914	−	7.74687i	0.281930	−	0.261593i	0.808443	+	0.588574i	$0.200310\pi$
$878$		0			0
$879$	10.1939	+	25.9736i	0.343831	+	0.876067i
$880$		0			0
$881$	−5.10037	−	2.45621i	−0.171836	−	0.0827518i	0.345988	−	0.938239i	$-0.387544\pi$
$881$	−5.10037	−	2.45621i	−0.171836	−	0.0827518i	−0.517824	+	0.855487i	$0.673258\pi$
$882$		0			0
$883$	−15.4143	+	2.32333i	−0.518732	+	0.0781863i	−0.403190	−	0.915116i	$-0.632099\pi$
$883$	−15.4143	+	2.32333i	−0.518732	+	0.0781863i	−0.115542	+	0.993303i	$0.536861\pi$
$884$		0			0
$885$	−1.69489	−	22.6167i	−0.0569731	−	0.760253i
$886$		0			0
$887$	−3.43965	−	15.0701i	−0.115492	−	0.506004i	−0.999274	−	0.0381046i	$-0.987868\pi$
$887$	−3.43965	−	15.0701i	−0.115492	−	0.506004i	0.883782	−	0.467900i	$-0.154989\pi$
$888$		0			0
$889$	−32.7243	−	22.3110i	−1.09754	−	0.748288i
$890$		0			0
$891$	0.300804	−	4.01395i	0.0100773	−	0.134473i
$892$		0			0
$893$	−10.6213	−	3.27622i	−0.355427	−	0.109635i
$894$		0			0
$895$	−10.2089	+	44.7282i	−0.341247	+	1.49510i
$896$		0			0
$897$	−5.64653	+	9.78007i	−0.188532	+	0.326547i
$898$		0			0
$899$	−28.2253	+	35.3934i	−0.941366	+	1.18044i
$900$		0			0
$901$	−27.2854	+	13.1400i	−0.909009	+	0.437756i
$902$		0			0
$903$	−30.9541	+	23.2011i	−1.03009	+	0.772083i
$904$		0			0
$905$	−1.98173	+	0.954352i	−0.0658750	+	0.0317237i
$906$		0			0
$907$	−30.3903	+	38.1082i	−1.00909	+	1.26536i	−0.0452260	+	0.998977i	$0.514401\pi$
$907$	−30.3903	+	38.1082i	−1.00909	+	1.26536i	−0.963867	+	0.266385i	$0.914171\pi$
$908$		0			0
$909$	−7.02761	+	12.1722i	−0.233091	+	0.403726i
$910$		0			0
$911$	−2.90180	+	12.7136i	−0.0961408	+	0.421220i	−0.999978	−	0.00665648i	$-0.997881\pi$
$911$	−2.90180	+	12.7136i	−0.0961408	+	0.421220i	0.903837	+	0.427877i	$0.140738\pi$
$912$		0			0
$913$	3.77159	+	1.16338i	0.124821	+	0.0385023i
$914$		0			0
$915$	4.56839	−	60.9610i	0.151026	−	2.01531i
$916$		0			0
$917$	20.2559	+	13.8103i	0.668910	+	0.456055i
$918$		0			0
$919$	4.83091	+	21.1656i	0.159357	+	0.698189i	0.989963	+	0.141328i	$0.0451373\pi$
$919$	4.83091	+	21.1656i	0.159357	+	0.698189i	−0.830606	+	0.556861i	$0.812006\pi$
$920$		0			0
$921$	3.90871	+	52.1581i	0.128796	+	1.71867i
$922$		0			0
$923$	37.8998	−	5.71247i	1.24749	−	0.188028i
$924$		0			0
$925$	−7.65710	−	3.68746i	−0.251764	−	0.121243i
$926$		0			0
$927$	−15.8454	−	40.3735i	−0.520433	−	1.32604i
$928$		0			0
$929$	23.6472	−	21.9414i	0.775839	−	0.719873i	−0.189212	−	0.981936i	$-0.560593\pi$
$929$	23.6472	−	21.9414i	0.775839	−	0.719873i	0.965051	+	0.262063i	$0.0844029\pi$
$930$		0			0
$931$	8.53212	+	1.28601i	0.279629	+	0.0421473i
$932$		0			0
$933$	−34.7680	+	10.7245i	−1.13825	+	0.351105i
$934$		0			0
$935$	4.80525	+	6.02559i	0.157148	+	0.197058i
$936$		0			0
$937$	−4.03376	+	10.2779i	−0.131777	+	0.335763i	−0.981499	−	0.191469i	$-0.938675\pi$
$937$	−4.03376	+	10.2779i	−0.131777	+	0.335763i	0.849721	+	0.527232i	$0.176770\pi$
$938$		0			0
$939$	−38.2066	−	66.1758i	−1.24683	−	2.15957i
$940$		0			0
$941$	24.4771	+	22.7114i	0.797931	+	0.740372i	0.969577	−	0.244787i	$-0.0787180\pi$
$941$	24.4771	+	22.7114i	0.797931	+	0.740372i	−0.171646	+	0.985159i	$0.554908\pi$
$942$		0			0
$943$	−6.96635	+	4.74958i	−0.226856	+	0.154668i
$944$		0			0
$945$	40.5005			1.31748
$946$		0			0
$947$	−14.9637			−0.486256			−0.243128	−	0.969994i	$-0.578173\pi$
$947$	−14.9637			−0.486256			−0.243128	+	0.969994i	$0.578173\pi$
$948$		0			0
$949$	32.5426	−	22.1872i	1.05638	−	0.720226i
$950$		0			0
$951$	−19.3592	−	17.9627i	−0.627764	−	0.582480i
$952$		0			0
$953$	−10.3374	−	17.9049i	−0.334862	−	0.579997i	0.648597	−	0.761132i	$-0.275356\pi$
$953$	−10.3374	−	17.9049i	−0.334862	−	0.579997i	−0.983458	+	0.181135i	$0.942023\pi$
$954$		0			0
$955$	11.5428	−	29.4106i	0.373517	−	0.951706i
$956$		0			0
$957$	−12.7547	−	15.9938i	−0.412300	−	0.517007i
$958$		0			0
$959$	11.5224	−	3.55418i	0.372077	−	0.114770i
$960$		0			0
$961$	2.46429	+	0.371432i	0.0794932	+	0.0119817i
$962$		0			0
$963$	−33.9040	+	31.4583i	−1.09254	+	1.01373i
$964$		0			0
$965$	−5.85542	−	14.9194i	−0.188492	−	0.480271i
$966$		0			0
$967$	−9.85181	−	4.74438i	−0.316813	−	0.152569i	0.268719	−	0.963219i	$-0.413400\pi$
$967$	−9.85181	−	4.74438i	−0.316813	−	0.152569i	−0.585531	+	0.810650i	$0.699114\pi$
$968$		0			0
$969$	−29.1818	+	4.39845i	−0.937455	+	0.141299i
$970$		0			0
$971$	1.61481	+	21.5482i	0.0518218	+	0.691513i	0.961255	+	0.275662i	$0.0888972\pi$
$971$	1.61481	+	21.5482i	0.0518218	+	0.691513i	−0.909433	+	0.415851i	$0.863484\pi$
$972$		0			0
$973$	1.39864	+	6.12785i	0.0448384	+	0.196450i
$974$		0			0
$975$	18.6993	+	12.7489i	0.598856	+	0.408293i
$976$		0			0
$977$	0.119768	−	1.59820i	0.00383172	−	0.0511308i	−0.994960	−	0.100271i	$-0.968029\pi$
$977$	0.119768	−	1.59820i	0.00383172	−	0.0511308i	0.998792	+	0.0491401i	$0.0156481\pi$
$978$		0			0
$979$	−1.63238	−	0.503521i	−0.0521710	−	0.0160926i
$980$		0			0
$981$	6.96092	−	30.4978i	0.222245	−	0.973719i
$982$		0			0
$983$	−0.512227	+	0.887204i	−0.0163375	+	0.0282974i	−0.874079	−	0.485785i	$-0.838534\pi$
$983$	−0.512227	+	0.887204i	−0.0163375	+	0.0282974i	0.857741	+	0.514082i	$0.171867\pi$
$984$		0			0
$985$	9.38665	−	11.7705i	0.299083	−	0.375039i
$986$		0			0
$987$	19.9435	−	9.60427i	0.634808	−	0.305707i
$988$		0			0
$989$	−2.59502	+	7.59608i	−0.0825169	+	0.241541i
$990$		0			0
$991$	13.1752	−	6.34486i	0.418525	−	0.201551i	−0.212756	−	0.977105i	$-0.568244\pi$
$991$	13.1752	−	6.34486i	0.418525	−	0.201551i	0.631281	+	0.775554i	$0.282530\pi$
$992$		0			0
$993$	6.35870	−	7.97355i	0.201787	−	0.253033i
$994$		0			0
$995$	26.8294	−	46.4698i	0.850548	−	1.47319i
$996$		0			0
$997$	−0.782739	+	3.42941i	−0.0247896	+	0.108610i	−0.985809	−	0.167870i	$-0.946311\pi$
$997$	−0.782739	+	3.42941i	−0.0247896	+	0.108610i	0.961019	+	0.276481i	$0.0891682\pi$
$998$		0			0
$999$	24.2920	+	7.49310i	0.768566	+	0.237071i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bg.c.289.3		36
4.3	odd	2		43.2.g.a.31.3	yes	36
12.11	even	2		387.2.y.c.289.1		36
43.25	even	21	inner	688.2.bg.c.369.3		36
172.91	even	42		1849.2.a.o.1.15		18
172.111	odd	42		43.2.g.a.25.3	✓	36
172.167	odd	42		1849.2.a.n.1.4		18
516.455	even	42		387.2.y.c.154.1		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.25.3	✓	36	172.111	odd	42
43.2.g.a.31.3	yes	36	4.3	odd	2
387.2.y.c.154.1		36	516.455	even	42
387.2.y.c.289.1		36	12.11	even	2
688.2.bg.c.289.3		36	1.1	even	1	trivial
688.2.bg.c.369.3		36	43.25	even	21	inner
1849.2.a.n.1.4		18	172.167	odd	42
1849.2.a.o.1.15		18	172.91	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)

\(f(q)\)	\(=\)	\(q+(2.41097 - 1.64377i) q^{3} +(-2.00127 - 1.85691i) q^{5} +(-1.01083 - 1.75082i) q^{7} +(2.01476 - 5.13352i) q^{9} +O(q^{10})\)	\(q+(2.41097 - 1.64377i) q^{3} +(-2.00127 - 1.85691i) q^{5} +(-1.01083 - 1.75082i) q^{7} +(2.01476 - 5.13352i) q^{9} +(0.515533 + 0.646459i) q^{11} +(3.02110 - 0.931885i) q^{13} +(-7.87733 - 1.18732i) q^{15} +(-2.50278 + 2.32224i) q^{17} +(1.08222 + 2.75745i) q^{19} +(-5.31503 - 2.55958i) q^{21} +(-1.21045 + 0.182446i) q^{23} +(0.183328 + 2.44634i) q^{25} +(-1.63286 - 7.15403i) q^{27} +(-7.00539 - 4.77619i) q^{29} +(0.399005 - 5.32435i) q^{31} +(2.30556 + 0.711172i) q^{33} +(-1.22815 + 5.38089i) q^{35} +(-1.73217 + 3.00021i) q^{37} +(5.75196 - 7.21273i) q^{39} +(6.20561 - 2.98847i) q^{41} +(3.02108 - 5.82006i) q^{43} +(-13.5646 + 6.53235i) q^{45} +(-2.33951 + 2.93365i) q^{47} +(1.45643 - 2.52260i) q^{49} +(-2.21689 + 9.71283i) q^{51} +(8.47611 + 2.61454i) q^{53} +(0.168692 - 2.25104i) q^{55} +(7.14181 + 4.86920i) q^{57} +(0.633519 + 2.77563i) q^{59} +(0.573465 + 7.65235i) q^{61} +(-11.0245 + 1.66167i) q^{63} +(-7.77646 - 3.74495i) q^{65} +(3.76698 + 9.59812i) q^{67} +(-2.61846 + 2.42957i) q^{69} +(11.9877 + 1.80685i) q^{71} +(11.9045 - 3.67204i) q^{73} +(4.46321 + 5.59669i) q^{75} +(0.610712 - 1.55607i) q^{77} +(2.34287 + 4.05797i) q^{79} +(-3.56858 - 3.31116i) q^{81} +(3.94401 - 2.68898i) q^{83} +9.32092 q^{85} -24.7407 q^{87} +(-1.70700 + 1.16382i) q^{89} +(-4.68539 - 4.34741i) q^{91} +(-7.79001 - 13.4927i) q^{93} +(2.95452 - 7.52799i) q^{95} +(-7.51296 - 9.42095i) q^{97} +(4.35729 - 1.34404i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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