LMFDB - Embedded newform 688.2.bg.c.513.1

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			513.1
Character	$\chi$	$=$	688.513
Dual form			688.2.bg.c.401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.109070 + 1.45544i) q^{3} +(1.29085 + 0.398175i) q^{5} +(0.108163 + 0.187343i) q^{7} +(0.860089 + 0.129637i) q^{9} +O(q^{10})$	\(q+(-0.109070 + 1.45544i) q^{3} +(1.29085 + 0.398175i) q^{5} +(0.108163 + 0.187343i) q^{7} +(0.860089 + 0.129637i) q^{9} +(3.76031 - 4.71528i) q^{11} +(2.10767 - 1.95563i) q^{13} +(-0.720312 + 1.83532i) q^{15} +(0.270054 - 0.0833006i) q^{17} +(-1.12457 + 0.169502i) q^{19} +(-0.284464 + 0.136990i) q^{21} +(1.44040 + 3.67008i) q^{23} +(-2.62344 - 1.78863i) q^{25} +(-1.25681 + 5.50644i) q^{27} +(0.515946 + 6.88482i) q^{29} +(8.17225 - 5.57174i) q^{31} +(6.45267 + 5.98720i) q^{33} +(0.0650265 + 0.284900i) q^{35} +(-3.77129 + 6.53207i) q^{37} +(2.61642 + 3.28088i) q^{39} +(-4.62195 - 2.22581i) q^{41} +(-5.90092 + 2.85993i) q^{43} +(1.05863 + 0.509808i) q^{45} +(-0.288239 - 0.361440i) q^{47} +(3.47660 - 6.02165i) q^{49} +(0.0917841 + 0.402132i) q^{51} +(-6.12346 - 5.68174i) q^{53} +(6.73151 - 4.58946i) q^{55} +(-0.124042 - 1.65523i) q^{57} +(-1.85926 + 8.14595i) q^{59} +(11.0987 + 7.56700i) q^{61} +(0.0687427 + 0.175154i) q^{63} +(3.49937 - 1.68521i) q^{65} +(-6.22328 + 0.938008i) q^{67} +(-5.49868 + 1.69612i) q^{69} +(0.540324 - 1.37672i) q^{71} +(0.601198 - 0.557830i) q^{73} +(2.88938 - 3.62317i) q^{75} +(1.29010 + 0.194451i) q^{77} +(3.07798 + 5.33121i) q^{79} +(-5.38373 - 1.66066i) q^{81} +(0.538458 - 7.18523i) q^{83} +0.381767 q^{85} -10.0767 q^{87} +(0.331353 - 4.42159i) q^{89} +(0.594345 + 0.183331i) q^{91} +(7.21798 + 12.5019i) q^{93} +(-1.51914 - 0.228974i) q^{95} +(-2.78649 + 3.49415i) q^{97} +(3.84548 - 3.56808i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$e\left(\frac{11}{21}\right)$	$1$

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
$3$	−0.109070	+	1.45544i	−0.0629716	+	0.840298i	0.872887	+	0.487922i	$0.162245\pi$
$3$	−0.109070	+	1.45544i	−0.0629716	+	0.840298i	−0.935859	+	0.352375i	$0.885374\pi$
$4$		0			0
$5$	1.29085	+	0.398175i	0.577286	+	0.178069i	0.569629	−	0.821902i	$-0.307087\pi$
$5$	1.29085	+	0.398175i	0.577286	+	0.178069i	0.00765630	+	0.999971i	$0.497563\pi$
$6$		0			0
$7$	0.108163	+	0.187343i	0.0408816	+	0.0708091i	0.885742	−	0.464177i	$-0.153650\pi$
$7$	0.108163	+	0.187343i	0.0408816	+	0.0708091i	−0.844861	+	0.534987i	$0.820317\pi$
$8$		0			0
$9$	0.860089	+	0.129637i	0.286696	+	0.0432125i
$10$		0			0
$11$	3.76031	−	4.71528i	1.13378	−	1.42171i	0.241397	−	0.970426i	$-0.422394\pi$
$11$	3.76031	−	4.71528i	1.13378	−	1.42171i	0.892380	−	0.451285i	$-0.149034\pi$
$12$		0			0
$13$	2.10767	−	1.95563i	0.584562	−	0.542395i	−0.331487	−	0.943460i	$-0.607550\pi$
$13$	2.10767	−	1.95563i	0.584562	−	0.542395i	0.916050	+	0.401065i	$0.131360\pi$
$14$		0			0
$15$	−0.720312	+	1.83532i	−0.185984	+	0.473879i
$16$		0			0
$17$	0.270054	−	0.0833006i	0.0654977	−	0.0202034i	−0.261833	−	0.965113i	$-0.584327\pi$
$17$	0.270054	−	0.0833006i	0.0654977	−	0.0202034i	0.327331	+	0.944910i	$0.393851\pi$
$18$		0			0
$19$	−1.12457	+	0.169502i	−0.257994	+	0.0388863i	−0.276766	−	0.960937i	$-0.589263\pi$
$19$	−1.12457	+	0.169502i	−0.257994	+	0.0388863i	0.0187716	+	0.999824i	$0.494024\pi$
$20$		0			0
$21$	−0.284464	+	0.136990i	−0.0620751	+	0.0298938i
$22$		0			0
$23$	1.44040	+	3.67008i	0.300344	+	0.765265i	0.998770	+	0.0495821i	$0.0157890\pi$
$23$	1.44040	+	3.67008i	0.300344	+	0.765265i	−0.698426	+	0.715682i	$0.746116\pi$
$24$		0			0
$25$	−2.62344	−	1.78863i	−0.524688	−	0.357727i
$26$		0			0
$27$	−1.25681	+	5.50644i	−0.241873	+	1.05972i
$28$		0			0
$29$	0.515946	+	6.88482i	0.0958088	+	1.27848i	0.813272	+	0.581883i	$0.197684\pi$
$29$	0.515946	+	6.88482i	0.0958088	+	1.27848i	−0.717463	+	0.696596i	$0.754697\pi$
$30$		0			0
$31$	8.17225	−	5.57174i	1.46778	−	1.00071i	0.475100	−	0.879932i	$-0.342412\pi$
$31$	8.17225	−	5.57174i	1.46778	−	1.00071i	0.992679	−	0.120783i	$-0.0385405\pi$
$32$		0			0
$33$	6.45267	+	5.98720i	1.12326	+	1.04224i
$34$		0			0
$35$	0.0650265	+	0.284900i	0.0109915	+	0.0481568i
$36$		0			0
$37$	−3.77129	+	6.53207i	−0.619996	+	1.07387i	0.369489	+	0.929235i	$0.379533\pi$
$37$	−3.77129	+	6.53207i	−0.619996	+	1.07387i	−0.989486	+	0.144630i	$0.953801\pi$
$38$		0			0
$39$	2.61642	+	3.28088i	0.418962	+	0.525362i
$40$		0			0
$41$	−4.62195	−	2.22581i	−0.721827	−	0.347613i	0.0366371	−	0.999329i	$-0.488335\pi$
$41$	−4.62195	−	2.22581i	−0.721827	−	0.347613i	−0.758464	+	0.651715i	$0.774050\pi$
$42$		0			0
$43$	−5.90092	+	2.85993i	−0.899881	+	0.436135i
$43$	−5.90092	+	2.85993i	−0.899881	+	0.436135i
$44$		0			0
$45$	1.05863	+	0.509808i	0.157811	+	0.0759977i
$46$		0			0
$47$	−0.288239	−	0.361440i	−0.0420439	−	0.0527214i	0.760365	−	0.649496i	$-0.225020\pi$
$47$	−0.288239	−	0.361440i	−0.0420439	−	0.0527214i	−0.802409	+	0.596775i	$0.796449\pi$
$48$		0			0
$49$	3.47660	−	6.02165i	0.496657	−	0.860236i
$50$		0			0
$51$	0.0917841	+	0.402132i	0.0128523	+	0.0563098i
$52$		0			0
$53$	−6.12346	−	5.68174i	−0.841123	−	0.780448i	0.136539	−	0.990635i	$-0.456402\pi$
$53$	−6.12346	−	5.68174i	−0.841123	−	0.780448i	−0.977661	+	0.210187i	$0.932593\pi$
$54$		0			0
$55$	6.73151	−	4.58946i	0.907676	−	0.618843i
$56$		0			0
$57$	−0.124042	−	1.65523i	−0.0164298	−	0.219240i
$58$		0			0
$59$	−1.85926	+	8.14595i	−0.242055	+	1.06051i	0.697088	+	0.716986i	$0.254479\pi$
$59$	−1.85926	+	8.14595i	−0.242055	+	1.06051i	−0.939143	+	0.343527i	$0.888378\pi$
$60$		0			0
$61$	11.0987	+	7.56700i	1.42105	+	0.968855i	0.998030	+	0.0627337i	$0.0199819\pi$
$61$	11.0987	+	7.56700i	1.42105	+	0.968855i	0.423019	+	0.906121i	$0.360971\pi$
$62$		0			0
$63$	0.0687427	+	0.175154i	0.00866077	+	0.0220673i
$64$		0			0
$65$	3.49937	−	1.68521i	0.434043	−	0.209024i
$66$		0			0
$67$	−6.22328	+	0.938008i	−0.760294	+	0.114596i	−0.517736	−	0.855541i	$-0.673225\pi$
$67$	−6.22328	+	0.938008i	−0.760294	+	0.114596i	−0.242559	+	0.970137i	$0.577987\pi$
$68$		0			0
$69$	−5.49868	+	1.69612i	−0.661963	+	0.204189i
$70$		0			0
$71$	0.540324	−	1.37672i	0.0641247	−	0.163387i	−0.895220	−	0.445625i	$-0.852982\pi$
$71$	0.540324	−	1.37672i	0.0641247	−	0.163387i	0.959344	+	0.282238i	$0.0910768\pi$
$72$		0			0
$73$	0.601198	−	0.557830i	0.0703649	−	0.0652891i	−0.644206	−	0.764852i	$-0.722812\pi$
$73$	0.601198	−	0.557830i	0.0703649	−	0.0652891i	0.714571	+	0.699563i	$0.246622\pi$
$74$		0			0
$75$	2.88938	−	3.62317i	0.333637	−	0.418368i
$76$		0			0
$77$	1.29010	+	0.194451i	0.147021	+	0.0221598i
$78$		0			0
$79$	3.07798	+	5.33121i	0.346299	+	0.599808i	0.985589	−	0.169158i	$-0.0541048\pi$
$79$	3.07798	+	5.33121i	0.346299	+	0.599808i	−0.639290	+	0.768966i	$0.720772\pi$
$80$		0			0
$81$	−5.38373	−	1.66066i	−0.598192	−	0.184518i
$82$		0			0
$83$	0.538458	−	7.18523i	0.0591035	−	0.788681i	−0.886369	−	0.462979i	$-0.846780\pi$
$83$	0.538458	−	7.18523i	0.0591035	−	0.788681i	0.945473	−	0.325702i	$-0.105601\pi$
$84$		0			0
$85$	0.381767			0.0414085
$86$		0			0
$87$	−10.0767			−1.08034
$88$		0			0
$89$	0.331353	−	4.42159i	0.0351233	−	0.468688i	−0.951716	−	0.306981i	$-0.900681\pi$
$89$	0.331353	−	4.42159i	0.0351233	−	0.468688i	0.986839	−	0.161707i	$-0.0516999\pi$
$90$		0			0
$91$	0.594345	+	0.183331i	0.0623043	+	0.0192183i
$92$		0			0
$93$	7.21798	+	12.5019i	0.748470	+	1.29639i
$94$		0			0
$95$	−1.51914	−	0.228974i	−0.155861	−	0.0234922i
$96$		0			0
$97$	−2.78649	+	3.49415i	−0.282925	+	0.354777i	−0.902905	−	0.429841i	$-0.858570\pi$
$97$	−2.78649	+	3.49415i	−0.282925	+	0.354777i	0.619979	+	0.784618i	$0.287141\pi$
$98$		0			0
$99$	3.84548	−	3.56808i	0.386485	−	0.358606i
$100$		0			0
$101$	−1.09407	+	2.78765i	−0.108864	+	0.277382i	−0.974916	−	0.222572i	$-0.928555\pi$
$101$	−1.09407	+	2.78765i	−0.108864	+	0.277382i	0.866052	+	0.499954i	$0.166650\pi$
$102$		0			0
$103$	−8.07350	+	2.49034i	−0.795505	+	0.245381i	−0.665760	−	0.746166i	$-0.731892\pi$
$103$	−8.07350	+	2.49034i	−0.795505	+	0.245381i	−0.129746	+	0.991547i	$0.541416\pi$
$104$		0			0
$105$	−0.421746	+	0.0635680i	−0.0411582	+	0.00620360i
$106$		0			0
$107$	5.64595	−	2.71894i	0.545814	−	0.262850i	−0.140593	−	0.990067i	$-0.544901\pi$
$107$	5.64595	−	2.71894i	0.545814	−	0.262850i	0.686407	+	0.727217i	$0.259187\pi$
$108$		0			0
$109$	0.0776988	+	0.197973i	0.00744219	+	0.0189624i	0.934547	−	0.355841i	$-0.115805\pi$
$109$	0.0776988	+	0.197973i	0.00744219	+	0.0189624i	−0.927104	+	0.374803i	$0.877710\pi$
$110$		0			0
$111$	−9.09569	−	6.20133i	−0.863324	−	0.588605i
$112$		0			0
$113$	2.46387	−	10.7949i	0.231781	−	1.01550i	−0.716380	−	0.697710i	$-0.754202\pi$
$113$	2.46387	−	10.7949i	0.231781	−	1.01550i	0.948161	−	0.317789i	$-0.102940\pi$
$114$		0			0
$115$	0.398009	+	5.31105i	0.0371145	+	0.495258i
$116$		0			0
$117$	2.06631	−	1.40878i	0.191030	−	0.130242i
$118$		0			0
$119$	0.0448155	+	0.0415827i	0.00410823	+	0.00381188i
$120$		0			0
$121$	−5.64621	−	24.7377i	−0.513292	−	2.24888i
$122$		0			0
$123$	3.74365	−	6.48419i	0.337553	−	0.584659i
$124$		0			0
$125$	−6.88554	−	8.63419i	−0.615861	−	0.772265i
$126$		0			0
$127$	7.04038	+	3.39047i	0.624733	+	0.300855i	0.719339	−	0.694659i	$-0.244445\pi$
$127$	7.04038	+	3.39047i	0.624733	+	0.300855i	−0.0946062	+	0.995515i	$0.530159\pi$
$128$		0			0
$129$	−3.51883	−	8.90035i	−0.309816	−	0.783632i
$130$		0			0
$131$	−16.1797	−	7.79174i	−1.41363	−	0.680768i	−0.437753	−	0.899095i	$-0.644226\pi$
$131$	−16.1797	−	7.79174i	−1.41363	−	0.680768i	−0.975876	+	0.218327i	$0.929940\pi$
$132$		0			0
$133$	−0.153391	−	0.192347i	−0.0133007	−	0.0166786i
$134$		0			0
$135$	−3.81488	+	6.60757i	−0.328333	+	0.568689i
$136$		0			0
$137$	−2.23563	−	9.79494i	−0.191003	−	0.836839i	−0.976075	−	0.217434i	$-0.930231\pi$
$137$	−2.23563	−	9.79494i	−0.191003	−	0.836839i	0.785072	−	0.619404i	$-0.212626\pi$
$138$		0			0
$139$	−9.10473	−	8.44795i	−0.772253	−	0.716546i	0.192039	−	0.981387i	$-0.438490\pi$
$139$	−9.10473	−	8.44795i	−0.772253	−	0.716546i	−0.964292	+	0.264841i	$0.914680\pi$
$140$		0			0
$141$	0.557491	−	0.380091i	0.0469492	−	0.0320095i
$142$		0			0
$143$	−1.29586	−	17.2920i	−0.108365	−	1.44603i
$144$		0			0
$145$	−2.07535	+	9.09271i	−0.172349	+	0.755109i
$146$		0			0
$147$	8.38495	+	5.71676i	0.691579	+	0.471510i
$148$		0			0
$149$	−1.35074	−	3.44162i	−0.110657	−	0.281948i	0.864809	−	0.502101i	$-0.167440\pi$
$149$	−1.35074	−	3.44162i	−0.110657	−	0.281948i	−0.975465	+	0.220153i	$0.929344\pi$
$150$		0			0
$151$	−10.3773	+	4.99744i	−0.844492	+	0.406686i	−0.805530	−	0.592555i	$-0.798119\pi$
$151$	−10.3773	+	4.99744i	−0.844492	+	0.406686i	−0.0389621	+	0.999241i	$0.512405\pi$
$152$		0			0
$153$	0.243069	−	0.0366368i	0.0196510	−	0.00296191i
$154$		0			0
$155$	12.7677	−	3.93831i	1.02552	−	0.316332i
$156$		0			0
$157$	−7.52520	+	19.1739i	−0.600577	+	1.53024i	0.229914	+	0.973211i	$0.426156\pi$
$157$	−7.52520	+	19.1739i	−0.600577	+	1.53024i	−0.830490	+	0.557033i	$0.811940\pi$
$158$		0			0
$159$	8.93731	−	8.29262i	0.708775	−	0.657647i
$160$		0			0
$161$	−0.531767	+	0.666815i	−0.0419091	+	0.0525524i
$162$		0			0
$163$	6.83812	+	1.03068i	0.535603	+	0.0807292i	0.411274	−	0.911512i	$-0.365084\pi$
$163$	6.83812	+	1.03068i	0.535603	+	0.0807292i	0.124329	+	0.992241i	$0.460322\pi$
$164$		0			0
$165$	5.94547	+	10.2979i	0.462855	+	0.801688i
$166$		0			0
$167$	−3.39723	−	1.04791i	−0.262886	−	0.0810895i	0.160510	−	0.987034i	$-0.448686\pi$
$167$	−3.39723	−	1.04791i	−0.262886	−	0.0810895i	−0.423396	+	0.905945i	$0.639162\pi$
$168$		0			0
$169$	−0.353715	+	4.72000i	−0.0272089	+	0.363077i
$170$		0			0
$171$	−0.989203			−0.0756463
$172$		0			0
$173$	−24.0563			−1.82897			−0.914483	−	0.404623i	$-0.867403\pi$
$173$	−24.0563			−1.82897			−0.914483	+	0.404623i	$0.867403\pi$
$174$		0			0
$175$	0.0513297	−	0.684947i	0.00388016	−	0.0517771i
$176$		0			0
$177$	−11.6531	−	3.59452i	−0.875904	−	0.270181i
$178$		0			0
$179$	−2.60094	−	4.50496i	−0.194403	−	0.336717i	0.752301	−	0.658819i	$-0.228944\pi$
$179$	−2.60094	−	4.50496i	−0.194403	−	0.336717i	−0.946705	+	0.322103i	$0.895610\pi$
$180$		0			0
$181$	2.74306	+	0.413450i	0.203890	+	0.0307315i	0.250193	−	0.968196i	$-0.419506\pi$
$181$	2.74306	+	0.413450i	0.203890	+	0.0307315i	−0.0463029	+	0.998927i	$0.514744\pi$
$182$		0			0
$183$	−12.2238	+	15.3282i	−0.903612	+	1.13309i
$184$		0			0
$185$	−7.46908	+	6.93029i	−0.549137	+	0.509525i
$186$		0			0
$187$	0.622701	−	1.58662i	0.0455364	−	0.116025i
$188$		0			0
$189$	−1.16753	+	0.360137i	−0.0849256	+	0.0261961i
$190$		0			0
$191$	−3.41700	+	0.515030i	−0.247245	+	0.0372662i	−0.271496	−	0.962440i	$-0.587518\pi$
$191$	−3.41700	+	0.515030i	−0.247245	+	0.0372662i	0.0242502	+	0.999706i	$0.492280\pi$
$192$		0			0
$193$	16.4497	−	7.92174i	1.18407	−	0.570219i	0.264977	−	0.964255i	$-0.414636\pi$
$193$	16.4497	−	7.92174i	1.18407	−	0.570219i	0.919095	+	0.394035i	$0.128921\pi$
$194$		0			0
$195$	2.07104	+	5.27692i	0.148310	+	0.377888i
$196$		0			0
$197$	8.41116	+	5.73463i	0.599270	+	0.408575i	0.824607	−	0.565705i	$-0.191396\pi$
$197$	8.41116	+	5.73463i	0.599270	+	0.408575i	−0.225337	+	0.974281i	$0.572348\pi$
$198$		0			0
$199$	2.04642	−	8.96595i	0.145067	−	0.635579i	−0.849147	−	0.528157i	$-0.822883\pi$
$199$	2.04642	−	8.96595i	0.145067	−	0.635579i	0.994214	−	0.107422i	$-0.0342596\pi$
$200$		0			0
$201$	−0.686440	−	9.15990i	−0.0484177	−	0.646090i
$202$		0			0
$203$	−1.23402	+	0.841340i	−0.0866111	+	0.0590505i
$204$		0			0
$205$	−5.07998	−	4.71353i	−0.354801	−	0.329207i
$206$		0			0
$207$	0.763092	+	3.34332i	0.0530385	+	0.232377i
$208$		0			0
$209$	−3.42949	+	5.94004i	−0.237222	+	0.410881i
$210$		0			0
$211$	11.8825	+	14.9002i	0.818025	+	1.02577i	0.999105	+	0.0423025i	$0.0134693\pi$
$211$	11.8825	+	14.9002i	0.818025	+	1.02577i	−0.181080	+	0.983468i	$0.557959\pi$
$212$		0			0
$213$	1.94480	+	0.936567i	0.133256	+	0.0641725i
$214$		0			0
$215$	−8.75595	+	1.34214i	−0.597151	+	0.0915334i
$216$		0			0
$217$	1.92776	+	0.928360i	0.130865	+	0.0630212i
$218$		0			0
$219$	0.746315	+	0.935849i	0.0504313	+	0.0632388i
$220$		0			0
$221$	0.406279	−	0.703696i	0.0273293	−	0.0473357i
$222$		0			0
$223$	−1.62391	−	7.11481i	−0.108745	−	0.476443i	−0.999748	−	0.0224455i	$-0.992855\pi$
$223$	−1.62391	−	7.11481i	−0.108745	−	0.476443i	0.891003	−	0.453997i	$-0.150002\pi$
$224$		0			0
$225$	−2.02452	−	1.87848i	−0.134968	−	0.125232i
$226$		0			0
$227$	−12.9206	+	8.80914i	−0.857573	+	0.584683i	−0.910311	−	0.413925i	$-0.864158\pi$
$227$	−12.9206	+	8.80914i	−0.857573	+	0.584683i	0.0527382	+	0.998608i	$0.483205\pi$
$228$		0			0
$229$	−0.0803517	−	1.07222i	−0.00530979	−	0.0708542i	0.993919	−	0.110110i	$-0.0351203\pi$
$229$	−0.0803517	−	1.07222i	−0.00530979	−	0.0708542i	−0.999229	+	0.0392558i	$0.987501\pi$
$230$		0			0
$231$	−0.423723	+	1.85645i	−0.0278790	+	0.122146i
$232$		0			0
$233$	−1.10140	−	0.750923i	−0.0721552	−	0.0491946i	0.526705	−	0.850048i	$-0.323427\pi$
$233$	−1.10140	−	0.750923i	−0.0721552	−	0.0491946i	−0.598860	+	0.800853i	$0.704380\pi$
$234$		0			0
$235$	−0.228157	−	0.581334i	−0.0148833	−	0.0379220i
$236$		0			0
$237$	−8.09496	+	3.89833i	−0.525824	+	0.253224i
$238$		0			0
$239$	14.7110	−	2.21733i	0.951579	−	0.143427i	0.345134	−	0.938553i	$-0.387833\pi$
$239$	14.7110	−	2.21733i	0.951579	−	0.143427i	0.606444	+	0.795126i	$0.292595\pi$
$240$		0			0
$241$	22.8421	−	7.04584i	1.47139	−	0.453862i	0.547590	−	0.836747i	$-0.315545\pi$
$241$	22.8421	−	7.04584i	1.47139	−	0.453862i	0.923796	+	0.382885i	$0.125069\pi$
$242$		0			0
$243$	−3.18621	+	8.11832i	−0.204395	+	0.520790i
$244$		0			0
$245$	6.88544	−	6.38876i	0.439895	−	0.408163i
$246$		0			0
$247$	−2.03874	+	2.55650i	−0.129722	+	0.162666i
$248$		0			0
$249$	10.3989	+	1.56739i	0.659005	+	0.0993290i
$250$		0			0
$251$	−10.2680	−	17.7846i	−0.648108	−	1.12256i	−0.983574	−	0.180504i	$-0.942227\pi$
$251$	−10.2680	−	17.7846i	−0.648108	−	1.12256i	0.335466	−	0.942052i	$-0.391106\pi$
$252$		0			0
$253$	22.7218	+	7.00875i	1.42851	+	0.440637i
$254$		0			0
$255$	−0.0416394	+	0.555639i	−0.00260756	+	0.0347954i
$256$		0			0
$257$	−22.8388			−1.42464			−0.712322	−	0.701853i	$-0.752356\pi$
$257$	−22.8388			−1.42464			−0.712322	+	0.701853i	$0.752356\pi$
$258$		0			0
$259$	−1.63165			−0.101386
$260$		0			0
$261$	−0.448772	+	5.98844i	−0.0277783	+	0.370675i
$262$		0			0
$263$	−5.88459	−	1.81516i	−0.362860	−	0.111927i	0.107964	−	0.994155i	$-0.465567\pi$
$263$	−5.88459	−	1.81516i	−0.362860	−	0.111927i	−0.470823	+	0.882227i	$0.656043\pi$
$264$		0			0
$265$	−5.64215	−	9.77249i	−0.346594	−	0.600319i
$266$		0			0
$267$	6.39922	+	0.964527i	0.391626	+	0.0590281i
$268$		0			0
$269$	9.36364	−	11.7416i	0.570911	−	0.715900i	−0.409621	−	0.912256i	$-0.634339\pi$
$269$	9.36364	−	11.7416i	0.570911	−	0.715900i	0.980533	+	0.196355i	$0.0629106\pi$
$270$		0			0
$271$	2.26842	−	2.10479i	0.137797	−	0.127857i	−0.608263	−	0.793735i	$-0.708134\pi$
$271$	2.26842	−	2.10479i	0.137797	−	0.127857i	0.746060	+	0.665878i	$0.231943\pi$
$272$		0			0
$273$	−0.331652	+	0.845037i	−0.0200725	+	0.0511440i
$274$		0			0
$275$	−18.2989	+	5.64446i	−1.10346	+	0.340373i
$276$		0			0
$277$	−14.9446	+	2.25253i	−0.897932	+	0.135341i	−0.581760	−	0.813360i	$-0.697636\pi$
$277$	−14.9446	+	2.25253i	−0.897932	+	0.135341i	−0.316172	+	0.948702i	$0.602398\pi$
$278$		0			0
$279$	7.75116	−	3.73276i	0.464050	−	0.223475i
$280$		0			0
$281$	4.07782	+	10.3901i	0.243262	+	0.619822i	0.999402	−	0.0345883i	$-0.0110120\pi$
$281$	4.07782	+	10.3901i	0.243262	+	0.619822i	−0.756139	+	0.654411i	$0.772917\pi$
$282$		0			0
$283$	13.2949	+	9.06431i	0.790300	+	0.538818i	0.889871	−	0.456213i	$-0.150794\pi$
$283$	13.2949	+	9.06431i	0.790300	+	0.538818i	−0.0995704	+	0.995031i	$0.531747\pi$
$284$		0			0
$285$	0.498950	−	2.18604i	0.0295553	−	0.129490i
$286$		0			0
$287$	−0.0829312	−	1.10664i	−0.00489527	−	0.0653229i
$288$		0			0
$289$	−13.9801	+	9.53145i	−0.822357	+	0.560674i
$290$		0			0
$291$	−4.78160	−	4.43667i	−0.280302	−	0.260082i
$292$		0			0
$293$	3.26781	+	14.3172i	0.190908	+	0.836421i	0.976127	+	0.217203i	$0.0696933\pi$
$293$	3.26781	+	14.3172i	0.190908	+	0.836421i	−0.785219	+	0.619218i	$0.787450\pi$
$294$		0			0
$295$	−5.64354	+	9.77490i	−0.328580	+	0.569116i
$296$		0			0
$297$	21.2384	+	26.6322i	1.23238	+	1.54536i
$298$		0			0
$299$	10.2132	+	4.91842i	0.590645	+	0.284440i
$300$		0			0
$301$	−1.17405	−	0.796159i	−0.0676709	−	0.0458898i
$302$		0			0
$303$	−3.93792	−	1.89640i	−0.226228	−	0.108946i
$304$		0			0
$305$	11.3138	+	14.1871i	0.647828	+	0.812351i
$306$		0			0
$307$	0.0141820	−	0.0245640i	0.000809410	−	0.00140194i	−0.865620	−	0.500701i	$-0.833076\pi$
$307$	0.0141820	−	0.0245640i	0.000809410	−	0.00140194i	0.866430	+	0.499299i	$0.166409\pi$
$308$		0			0
$309$	−2.74397	−	12.0221i	−0.156099	−	0.683913i
$310$		0			0
$311$	−7.53405	−	6.99057i	−0.427217	−	0.396399i	0.437047	−	0.899439i	$-0.356024\pi$
$311$	−7.53405	−	6.99057i	−0.427217	−	0.396399i	−0.864263	+	0.503040i	$0.832215\pi$
$312$		0			0
$313$	14.8130	−	10.0994i	0.837283	−	0.570850i	−0.0670152	−	0.997752i	$-0.521348\pi$
$313$	14.8130	−	10.0994i	0.837283	−	0.570850i	0.904298	+	0.426902i	$0.140395\pi$
$314$		0			0
$315$	0.0189949	+	0.253469i	0.00107024	+	0.0142813i
$316$		0			0
$317$	4.83536	−	21.1851i	0.271581	−	1.18987i	−0.636566	−	0.771222i	$-0.719646\pi$
$317$	4.83536	−	21.1851i	0.271581	−	1.18987i	0.908147	−	0.418652i	$-0.137497\pi$
$318$		0			0
$319$	34.4040	+	23.4563i	1.92625	+	1.31330i
$320$		0			0
$321$	3.34145	+	8.51388i	0.186502	+	0.475199i
$322$		0			0
$323$	−0.289575	+	0.139452i	−0.0161124	+	0.00775931i
$324$		0			0
$325$	−9.02726	+	1.36064i	−0.500742	+	0.0754747i
$326$		0			0
$327$	−0.296612	+	0.0914928i	−0.0164027	+	0.00505956i
$328$		0			0
$329$	0.0365366	−	0.0930938i	0.00201433	−	0.00513243i
$330$		0			0
$331$	9.69617	−	8.99673i	0.532950	−	0.494505i	−0.367166	−	0.930155i	$-0.619672\pi$
$331$	9.69617	−	8.99673i	0.532950	−	0.494505i	0.900116	+	0.435650i	$0.143482\pi$
$332$		0			0
$333$	−4.09044	+	5.12926i	−0.224155	+	0.281081i
$334$		0			0
$335$	−8.40681	−	1.26712i	−0.459313	−	0.0692303i
$336$		0			0
$337$	4.71536	+	8.16725i	0.256862	+	0.444898i	0.965400	−	0.260775i	$-0.0839780\pi$
$337$	4.71536	+	8.16725i	0.256862	+	0.444898i	−0.708537	+	0.705673i	$0.750645\pi$
$338$		0			0
$339$	15.4426	+	4.76341i	0.838726	+	0.258713i
$340$		0			0
$341$	4.45786	−	59.4860i	0.241406	−	3.22135i
$342$		0			0
$343$	3.01843			0.162980
$344$		0			0
$345$	−7.77332			−0.418502
$346$		0			0
$347$	1.30647	−	17.4336i	0.0701351	−	0.935887i	−0.845526	−	0.533935i	$-0.820713\pi$
$347$	1.30647	−	17.4336i	0.0701351	−	0.935887i	0.915661	−	0.401952i	$-0.131668\pi$
$348$		0			0
$349$	8.91802	+	2.75084i	0.477370	+	0.147249i	0.524097	−	0.851658i	$-0.324403\pi$
$349$	8.91802	+	2.75084i	0.477370	+	0.147249i	−0.0467268	+	0.998908i	$0.514879\pi$
$350$		0			0
$351$	8.11964	+	14.0636i	0.433394	+	0.750661i
$352$		0			0
$353$	−10.2719	−	1.54824i	−0.546718	−	0.0824045i	−0.130125	−	0.991498i	$-0.541538\pi$
$353$	−10.2719	−	1.54824i	−0.546718	−	0.0824045i	−0.416593	+	0.909093i	$0.636776\pi$
$354$		0			0
$355$	1.24565	−	1.56200i	0.0661124	−	0.0829024i
$356$		0			0
$357$	−0.0654091	+	0.0606908i	−0.00346182	+	0.00321210i
$358$		0			0
$359$	−3.94742	+	10.0579i	−0.208337	+	0.530834i	−0.996402	−	0.0847587i	$-0.972988\pi$
$359$	−3.94742	+	10.0579i	−0.208337	+	0.530834i	0.788065	+	0.615593i	$0.211083\pi$
$360$		0			0
$361$	−16.9200	+	5.21912i	−0.890524	+	0.274690i
$362$		0			0
$363$	36.6200	−	5.51957i	1.92205	−	0.289703i
$364$		0			0
$365$	0.998170	−	0.480694i	0.0522466	−	0.0251606i
$366$		0			0
$367$	−7.07784	−	18.0340i	−0.369460	−	0.941369i	−0.987838	−	0.155488i	$-0.950305\pi$
$367$	−7.07784	−	18.0340i	−0.369460	−	0.941369i	0.618377	−	0.785881i	$-0.287790\pi$
$368$		0			0
$369$	−3.68673	−	2.51357i	−0.191924	−	0.130851i
$370$		0			0
$371$	0.402106	−	1.76174i	0.0208763	−	0.0914651i
$372$		0			0
$373$	2.56665	+	34.2496i	0.132896	+	1.77338i	0.522737	+	0.852494i	$0.324911\pi$
$373$	2.56665	+	34.2496i	0.132896	+	1.77338i	−0.389841	+	0.920882i	$0.627470\pi$
$374$		0			0
$375$	13.3175	−	9.07974i	0.687715	−	0.468876i
$376$		0			0
$377$	14.5516	+	13.5019i	0.749447	+	0.695385i
$378$		0			0
$379$	0.574248	+	2.51594i	0.0294971	+	0.129235i	0.987533	−	0.157414i	$-0.0503157\pi$
$379$	0.574248	+	2.51594i	0.0294971	+	0.129235i	−0.958036	+	0.286649i	$0.907459\pi$
$380$		0			0
$381$	−5.70251	+	9.87704i	−0.292148	+	0.506016i
$382$		0			0
$383$	−8.92040	−	11.1858i	−0.455811	−	0.571569i	0.499822	−	0.866128i	$-0.333399\pi$
$383$	−8.92040	−	11.1858i	−0.455811	−	0.571569i	−0.955633	+	0.294559i	$0.904827\pi$
$384$		0			0
$385$	1.58790	+	0.764693i	0.0809270	+	0.0389724i
$386$		0			0
$387$	−5.44606	+	1.69481i	−0.276839	+	0.0861521i
$388$		0			0
$389$	14.2240	+	6.84991i	0.721185	+	0.347304i	0.758210	−	0.652010i	$-0.226074\pi$
$389$	14.2240	+	6.84991i	0.721185	+	0.347304i	−0.0370254	+	0.999314i	$0.511788\pi$
$390$		0			0
$391$	0.694706	+	0.871133i	0.0351328	+	0.0440551i
$392$		0			0
$393$	13.1051	−	22.6987i	0.661066	−	1.14500i
$394$		0			0
$395$	1.85045	+	8.10736i	0.0931064	+	0.407926i
$396$		0			0
$397$	21.3361	+	19.7970i	1.07083	+	0.993585i	0.999992	−	0.00390325i	$-0.00124245\pi$
$397$	21.3361	+	19.7970i	1.07083	+	0.993585i	0.0708372	+	0.997488i	$0.477433\pi$
$398$		0			0
$399$	0.296679	−	0.202272i	0.0148525	−	0.0101263i
$400$		0			0
$401$	1.06942	+	14.2704i	0.0534043	+	0.712631i	0.958075	+	0.286519i	$0.0924980\pi$
$401$	1.06942	+	14.2704i	0.0534043	+	0.712631i	−0.904670	+	0.426112i	$0.859883\pi$
$402$		0			0
$403$	6.32812	−	27.7253i	0.315226	−	1.38110i
$404$		0			0
$405$	−6.28835	−	4.28733i	−0.312471	−	0.213039i
$406$		0			0
$407$	16.6193	+	42.3453i	0.823789	+	2.09898i
$408$		0			0
$409$	−27.4363	+	13.2126i	−1.35664	+	0.653322i	−0.963884	−	0.266321i	$-0.914192\pi$
$409$	−27.4363	+	13.2126i	−1.35664	+	0.653322i	−0.392754	+	0.919644i	$0.628478\pi$
$410$		0			0
$411$	14.4998	−	2.18549i	0.715221	−	0.107802i
$412$		0			0
$413$	−1.72719	+	0.532768i	−0.0849895	+	0.0262158i
$414$		0			0
$415$	3.55604	−	9.06065i	0.174559	−	0.444770i
$416$		0			0
$417$	13.2885	−	12.3300i	0.650742	−	0.603800i
$418$		0			0
$419$	−22.1749	+	27.8065i	−1.08332	+	1.35844i	−0.154459	+	0.987999i	$0.549363\pi$
$419$	−22.1749	+	27.8065i	−1.08332	+	1.35844i	−0.928858	+	0.370437i	$0.879208\pi$
$420$		0			0
$421$	−16.3130	−	2.45879i	−0.795047	−	0.119834i	−0.261056	−	0.965324i	$-0.584071\pi$
$421$	−16.3130	−	2.45879i	−0.795047	−	0.119834i	−0.533992	+	0.845490i	$0.679309\pi$
$422$		0			0
$423$	−0.201055	−	0.348237i	−0.00977561	−	0.0169318i
$424$		0			0
$425$	−0.857465	−	0.264493i	−0.0415932	−	0.0128298i
$426$		0			0
$427$	−0.217156	+	2.89774i	−0.0105089	+	0.140231i
$428$		0			0
$429$	25.3088			1.22192
$430$		0			0
$431$	9.05750			0.436285			0.218142	−	0.975917i	$-0.430000\pi$
$431$	9.05750			0.436285			0.218142	+	0.975917i	$0.430000\pi$
$432$		0			0
$433$	1.29146	−	17.2334i	0.0620638	−	0.828184i	−0.876130	−	0.482074i	$-0.839884\pi$
$433$	1.29146	−	17.2334i	0.0620638	−	0.828184i	0.938194	−	0.346110i	$-0.112497\pi$
$434$		0			0
$435$	−13.0075	−	4.01229i	−0.623663	−	0.192375i
$436$		0			0
$437$	−2.24191	−	3.88311i	−0.107245	−	0.185754i
$438$		0			0
$439$	18.2692	+	2.75364i	0.871943	+	0.131424i	0.569745	−	0.821821i	$-0.307042\pi$
$439$	18.2692	+	2.75364i	0.871943	+	0.131424i	0.302197	+	0.953245i	$0.402280\pi$
$440$		0			0
$441$	3.77082	−	4.72845i	0.179563	−	0.225165i
$442$		0			0
$443$	−6.55737	+	6.08435i	−0.311550	+	0.289076i	−0.820413	−	0.571771i	$-0.806256\pi$
$443$	−6.55737	+	6.08435i	−0.311550	+	0.289076i	0.508863	+	0.860848i	$0.330066\pi$
$444$		0			0
$445$	2.18829	−	5.57568i	0.103735	−	0.264313i
$446$		0			0
$447$	5.15639	−	1.59054i	0.243889	−	0.0752297i
$448$		0			0
$449$	20.0461	−	3.02146i	0.946032	−	0.142591i	0.342132	−	0.939652i	$-0.388851\pi$
$449$	20.0461	−	3.02146i	0.946032	−	0.142591i	0.603900	+	0.797060i	$0.293613\pi$
$450$		0			0
$451$	−27.8753	+	13.4240i	−1.31260	+	0.632113i
$452$		0			0
$453$	−6.14162	−	15.6486i	−0.288558	−	0.735235i
$454$		0			0
$455$	0.694213	+	0.473306i	0.0325452	+	0.0221889i
$456$		0			0
$457$	2.39474	−	10.4921i	0.112021	−	0.490797i	−0.887527	−	0.460755i	$-0.847579\pi$
$457$	2.39474	−	10.4921i	0.112021	−	0.490797i	0.999549	−	0.0300423i	$-0.00956421\pi$
$458$		0			0
$459$	0.119284	+	1.59173i	0.00556768	+	0.0742956i
$460$		0			0
$461$	−7.03749	+	4.79808i	−0.327769	+	0.223469i	−0.716012	−	0.698088i	$-0.754034\pi$
$461$	−7.03749	+	4.79808i	−0.327769	+	0.223469i	0.388243	+	0.921557i	$0.373082\pi$
$462$		0			0
$463$	−1.66719	−	1.54693i	−0.0774811	−	0.0718919i	0.640482	−	0.767973i	$-0.278735\pi$
$463$	−1.66719	−	1.54693i	−0.0774811	−	0.0718919i	−0.717963	+	0.696082i	$0.754925\pi$
$464$		0			0
$465$	4.33939	+	19.0121i	0.201234	+	0.881666i
$466$		0			0
$467$	−15.1256	+	26.1983i	−0.699930	+	1.21231i	0.268560	+	0.963263i	$0.413452\pi$
$467$	−15.1256	+	26.1983i	−0.699930	+	1.21231i	−0.968490	+	0.249052i	$0.919881\pi$
$468$		0			0
$469$	−0.848855	−	1.06443i	−0.0391965	−	0.0491509i
$470$		0			0
$471$	−27.0857	−	13.0438i	−1.24804	−	0.601025i
$472$		0			0
$473$	−8.70393	+	38.5787i	−0.400207	+	1.77385i
$474$		0			0
$475$	3.25342	+	1.56676i	0.149277	+	0.0718881i
$476$		0			0
$477$	−4.53015	−	5.68063i	−0.207422	−	0.260098i
$478$		0			0
$479$	6.78977	−	11.7602i	0.310233	−	0.537338i	−0.668180	−	0.744000i	$-0.732926\pi$
$479$	6.78977	−	11.7602i	0.310233	−	0.537338i	0.978413	+	0.206661i	$0.0662598\pi$
$480$		0			0
$481$	4.82568	+	21.1427i	0.220032	+	0.964024i
$482$		0			0
$483$	−0.912508	−	0.846683i	−0.0415205	−	0.0385254i
$484$		0			0
$485$	−4.98822	+	3.40091i	−0.226504	+	0.154428i
$486$		0			0
$487$	2.78410	+	37.1512i	0.126159	+	1.68348i	0.597765	+	0.801672i	$0.296056\pi$
$487$	2.78410	+	37.1512i	0.126159	+	1.68348i	−0.471605	+	0.881810i	$0.656325\pi$
$488$		0			0
$489$	−2.24593	+	9.84005i	−0.101564	+	0.444982i
$490$		0			0
$491$	−16.9652	−	11.5667i	−0.765629	−	0.521997i	0.116394	−	0.993203i	$-0.462866\pi$
$491$	−16.9652	−	11.5667i	−0.765629	−	0.521997i	−0.882023	+	0.471206i	$0.843819\pi$
$492$		0			0
$493$	0.712843	+	1.81629i	0.0321048	+	0.0818018i
$494$		0			0
$495$	6.38466	−	3.07469i	0.286969	−	0.138197i
$496$		0			0
$497$	0.316363	−	0.0476840i	0.0141908	−	0.00213892i
$498$		0			0
$499$	−6.66084	+	2.05460i	−0.298180	+	0.0919763i	−0.440235	−	0.897882i	$-0.645105\pi$
$499$	−6.66084	+	2.05460i	−0.298180	+	0.0919763i	0.142056	+	0.989859i	$0.454629\pi$
$500$		0			0
$501$	1.89570	−	4.83017i	0.0846937	−	0.215796i
$502$		0			0
$503$	9.51112	−	8.82503i	0.424080	−	0.393488i	−0.439057	−	0.898459i	$-0.644687\pi$
$503$	9.51112	−	8.82503i	0.424080	−	0.393488i	0.863136	+	0.504971i	$0.168497\pi$
$504$		0			0
$505$	−2.52226	+	3.16281i	−0.112239	+	0.140743i
$506$		0			0
$507$	−6.83109	−	1.02962i	−0.303379	−	0.0457271i
$508$		0			0
$509$	−14.9031	−	25.8129i	−0.660568	−	1.14414i	−0.980467	−	0.196686i	$-0.936982\pi$
$509$	−14.9031	−	25.8129i	−0.660568	−	1.14414i	0.319898	−	0.947452i	$-0.396351\pi$
$510$		0			0
$511$	0.169533	+	0.0522939i	0.00749969	+	0.00231335i
$512$		0			0
$513$	0.480019	−	6.40541i	0.0211934	−	0.282806i
$514$		0			0
$515$	−11.4133			−0.502929
$516$		0			0
$517$	−2.78816			−0.122623
$518$		0			0
$519$	2.62382	−	35.0125i	0.115173	−	1.53688i
$520$		0			0
$521$	−29.7906	−	9.18917i	−1.30515	−	0.402585i	−0.437256	−	0.899337i	$-0.644050\pi$
$521$	−29.7906	−	9.18917i	−1.30515	−	0.402585i	−0.867892	+	0.496752i	$0.834526\pi$
$522$		0			0
$523$	6.95091	+	12.0393i	0.303942	+	0.526443i	0.977025	−	0.213124i	$-0.0683637\pi$
$523$	6.95091	+	12.0393i	0.303942	+	0.526443i	−0.673083	+	0.739567i	$0.735030\pi$
$524$		0			0
$525$	0.991300	+	0.149414i	0.0432639	+	0.00652098i
$526$		0			0
$527$	1.74282	−	2.18542i	0.0759183	−	0.0951986i
$528$		0			0
$529$	5.46546	−	5.07120i	0.237629	−	0.220487i
$530$		0			0
$531$	−2.65515	+	6.76521i	−0.115224	+	0.293585i
$532$		0			0
$533$	−14.0944	+	4.34755i	−0.610496	+	0.188313i
$534$		0			0
$535$	8.37069	−	1.26168i	0.361896	−	0.0545471i
$536$		0			0
$537$	6.84038	−	3.29415i	0.295184	−	0.142153i
$538$		0			0
$539$	−15.3207	−	39.0365i	−0.659908	−	1.68142i
$540$		0			0
$541$	−17.1344	−	11.6820i	−0.736664	−	0.502249i	0.135916	−	0.990720i	$-0.456602\pi$
$541$	−17.1344	−	11.6820i	−0.736664	−	0.502249i	−0.872580	+	0.488472i	$0.837555\pi$
$542$		0			0
$543$	−0.900937	+	3.94726i	−0.0386629	+	0.169393i
$544$		0			0
$545$	0.0214696	+	0.286491i	0.000919655	+	0.0122719i
$546$		0			0
$547$	−16.9135	+	11.5314i	−0.723169	+	0.493049i	−0.868098	−	0.496394i	$-0.834657\pi$
$547$	−16.9135	+	11.5314i	−0.723169	+	0.493049i	0.144928	+	0.989442i	$0.453705\pi$
$548$		0			0
$549$	8.56494	+	7.94710i	0.365543	+	0.339174i
$550$		0			0
$551$	−1.74721	−	7.65501i	−0.0744335	−	0.326114i
$552$		0			0
$553$	−0.665844	+	1.15328i	−0.0283146	+	0.0490423i
$554$		0			0
$555$	−9.27195	−	11.6267i	−0.393573	−	0.493524i
$556$		0			0
$557$	15.3932	+	7.41296i	0.652229	+	0.314097i	0.730578	−	0.682829i	$-0.239251\pi$
$557$	15.3932	+	7.41296i	0.652229	+	0.314097i	−0.0783489	+	0.996926i	$0.524965\pi$
$558$		0			0
$559$	−6.84422	+	17.5678i	−0.289480	+	0.743039i
$560$		0			0
$561$	2.24130	+	1.07936i	0.0946280	+	0.0455704i
$562$		0			0
$563$	7.02348	+	8.80717i	0.296004	+	0.371178i	0.907487	−	0.420080i	$-0.137998\pi$
$563$	7.02348	+	8.80717i	0.296004	+	0.371178i	−0.611483	+	0.791258i	$0.709427\pi$
$564$		0			0
$565$	7.47874	−	12.9536i	0.314633	−	0.544960i
$566$		0			0
$567$	−0.271205	−	1.18823i	−0.0113895	−	0.0499008i
$568$		0			0
$569$	5.89078	+	5.46585i	0.246955	+	0.229140i	0.793931	−	0.608007i	$-0.208031\pi$
$569$	5.89078	+	5.46585i	0.246955	+	0.229140i	−0.546977	+	0.837148i	$0.684221\pi$
$570$		0			0
$571$	24.0538	−	16.3996i	1.00662	−	0.686302i	0.0565700	−	0.998399i	$-0.481984\pi$
$571$	24.0538	−	16.3996i	1.00662	−	0.686302i	0.950050	+	0.312096i	$0.101031\pi$
$572$		0			0
$573$	−0.376902	−	5.02941i	−0.0157453	−	0.210106i
$574$		0			0
$575$	2.78562	−	12.2046i	0.116168	−	0.508967i
$576$		0			0
$577$	−5.93710	−	4.04784i	−0.247165	−	0.168514i	0.433401	−	0.901201i	$-0.357313\pi$
$577$	−5.93710	−	4.04784i	−0.247165	−	0.168514i	−0.680565	+	0.732687i	$0.738266\pi$
$578$		0			0
$579$	9.73544	+	24.8055i	0.404591	+	1.03088i
$580$		0			0
$581$	1.40434	−	0.676297i	0.0582620	−	0.0280575i
$582$		0			0
$583$	−49.8172	+	7.50873i	−2.06322	+	0.310980i
$584$		0			0
$585$	3.22823	−	0.995778i	0.133471	−	0.0411703i
$586$		0			0
$587$	10.8541	−	27.6558i	0.447997	−	1.14148i	−0.511381	−	0.859354i	$-0.670866\pi$
$587$	10.8541	−	27.6558i	0.447997	−	1.14148i	0.959378	−	0.282123i	$-0.0910387\pi$
$588$		0			0
$589$	−8.24584	+	7.65102i	−0.339764	+	0.315255i
$590$		0			0
$591$	−9.26380	+	11.6164i	−0.381062	+	0.477837i
$592$		0			0
$593$	−31.8484	−	4.80037i	−1.30786	−	0.197127i	−0.542130	−	0.840295i	$-0.682382\pi$
$593$	−31.8484	−	4.80037i	−1.30786	−	0.197127i	−0.765726	+	0.643167i	$0.777620\pi$
$594$		0			0
$595$	0.0412930	+	0.0715215i	0.00169285	+	0.00293210i
$596$		0			0
$597$	12.8262	+	3.95635i	0.524940	+	0.161923i
$598$		0			0
$599$	1.13778	−	15.1826i	0.0464885	−	0.620346i	−0.924494	−	0.381196i	$-0.875512\pi$
$599$	1.13778	−	15.1826i	0.0464885	−	0.620346i	0.970983	−	0.239150i	$-0.0768687\pi$
$600$		0			0
$601$	−14.8489			−0.605699			−0.302849	−	0.953038i	$-0.597938\pi$
$601$	−14.8489			−0.605699			−0.302849	+	0.953038i	$0.597938\pi$
$602$		0			0
$603$	−5.47417			−0.222925
$604$		0			0
$605$	2.56150	−	34.1808i	0.104140	−	1.38965i
$606$		0			0
$607$	8.93135	+	2.75496i	0.362512	+	0.111820i	0.470660	−	0.882315i	$-0.344016\pi$
$607$	8.93135	+	2.75496i	0.362512	+	0.111820i	−0.108148	+	0.994135i	$0.534492\pi$
$608$		0			0
$609$	−1.08992	−	1.88780i	−0.0441659	−	0.0764976i
$610$		0			0
$611$	−1.31435	−	0.198107i	−0.0531731	−	0.00801455i
$612$		0			0
$613$	19.2070	−	24.0849i	0.775765	−	0.972779i	−0.224233	−	0.974536i	$-0.571988\pi$
$613$	19.2070	−	24.0849i	0.775765	−	0.972779i	0.999998	+	0.00175653i	$0.000559122\pi$
$614$		0			0
$615$	7.41433	−	6.87949i	0.298974	−	0.277408i
$616$		0			0
$617$	−1.40309	+	3.57502i	−0.0564864	+	0.143925i	−0.956331	−	0.292285i	$-0.905584\pi$
$617$	−1.40309	+	3.57502i	−0.0564864	+	0.143925i	0.899845	+	0.436210i	$0.143680\pi$
$618$		0			0
$619$	−32.6404	+	10.0682i	−1.31193	+	0.404676i	−0.870300	−	0.492521i	$-0.836075\pi$
$619$	−32.6404	+	10.0682i	−1.31193	+	0.404676i	−0.441629	+	0.897198i	$0.645599\pi$
$620$		0			0
$621$	−22.0194	+	3.31889i	−0.883608	+	0.133182i
$622$		0			0
$623$	0.864195	−	0.416175i	0.0346233	−	0.0166737i
$624$		0			0
$625$	0.349803	+	0.891285i	0.0139921	+	0.0356514i
$626$		0			0
$627$	−8.27131	−	5.63928i	−0.330324	−	0.225211i
$628$		0			0
$629$	−0.474327	+	2.07816i	−0.0189126	+	0.0828617i
$630$		0			0
$631$	0.687936	+	9.17986i	0.0273863	+	0.365445i	0.994013	+	0.109257i	$0.0348473\pi$
$631$	0.687936	+	9.17986i	0.0273863	+	0.365445i	−0.966627	+	0.256187i	$0.917534\pi$
$632$		0			0
$633$	−22.9823	+	15.6691i	−0.913465	+	0.622790i
$634$		0			0
$635$	7.73808	+	7.17989i	0.307076	+	0.284925i
$636$		0			0
$637$	−4.44860	−	19.4906i	−0.176260	−	0.772246i
$638$		0			0
$639$	0.643201	−	1.11406i	0.0254447	−	0.0440714i
$640$		0			0
$641$	2.10948	+	2.64521i	0.0833196	+	0.104479i	0.821744	−	0.569857i	$-0.193001\pi$
$641$	2.10948	+	2.64521i	0.0833196	+	0.104479i	−0.738424	+	0.674336i	$0.764430\pi$
$642$		0			0
$643$	10.1874	+	4.90598i	0.401751	+	0.193473i	0.623838	−	0.781554i	$-0.285573\pi$
$643$	10.1874	+	4.90598i	0.401751	+	0.193473i	−0.222087	+	0.975027i	$0.571287\pi$
$644$		0			0
$645$	−0.998394	−	12.8901i	−0.0393117	−	0.507548i
$646$		0			0
$647$	−20.4890	−	9.86700i	−0.805507	−	0.387912i	−0.0146350	−	0.999893i	$-0.504659\pi$
$647$	−20.4890	−	9.86700i	−0.805507	−	0.387912i	−0.790873	+	0.611981i	$0.790373\pi$
$648$		0			0
$649$	31.4191	+	39.3983i	1.23331	+	1.54652i
$650$		0			0
$651$	−1.56143	+	2.70448i	−0.0611973	+	0.105997i
$652$		0			0
$653$	−5.74350	−	25.1639i	−0.224760	−	0.984739i	−0.953841	−	0.300312i	$-0.902909\pi$
$653$	−5.74350	−	25.1639i	−0.224760	−	0.984739i	0.729081	−	0.684428i	$-0.239948\pi$
$654$		0			0
$655$	−17.7831	−	16.5003i	−0.694844	−	0.644721i
$656$		0			0
$657$	0.589399	−	0.401846i	0.0229947	−	0.0156775i
$658$		0			0
$659$	−0.761704	−	10.1642i	−0.0296718	−	0.395942i	−0.992191	−	0.124729i	$-0.960194\pi$
$659$	−0.761704	−	10.1642i	−0.0296718	−	0.395942i	0.962519	−	0.271214i	$-0.0874251\pi$
$660$		0			0
$661$	2.96298	−	12.9817i	0.115247	−	0.504928i	−0.884049	−	0.467395i	$-0.845193\pi$
$661$	2.96298	−	12.9817i	0.115247	−	0.504928i	0.999295	−	0.0375336i	$-0.0119501\pi$
$662$		0			0
$663$	0.979873	+	0.668066i	0.0380551	+	0.0259455i
$664$		0			0
$665$	−0.121418	−	0.309367i	−0.00470838	−	0.0119967i
$666$		0			0
$667$	−24.5247	+	11.8105i	−0.949600	+	0.457303i
$668$		0			0
$669$	10.5323	−	1.58749i	0.407201	−	0.0613757i
$670$		0			0
$671$	77.4153	−	23.8795i	2.98858	−	0.921856i
$672$		0			0
$673$	−2.30694	+	5.87798i	−0.0889259	+	0.226579i	−0.968417	−	0.249336i	$-0.919788\pi$
$673$	−2.30694	+	5.87798i	−0.0889259	+	0.226579i	0.879491	+	0.475915i	$0.157883\pi$
$674$		0			0
$675$	13.1462	−	12.1979i	0.505997	−	0.469496i
$676$		0			0
$677$	−16.2431	+	20.3682i	−0.624273	+	0.782814i	−0.988939	−	0.148325i	$-0.952612\pi$
$677$	−16.2431	+	20.3682i	−0.624273	+	0.782814i	0.364666	+	0.931139i	$0.381183\pi$
$678$		0			0
$679$	−0.955999	−	0.144094i	−0.0366879	−	0.00552981i
$680$		0			0
$681$	−11.4119	−	19.7660i	−0.437305	−	0.757435i
$682$		0			0
$683$	47.0692	+	14.5189i	1.80105	+	0.555551i	0.999123	−	0.0418665i	$-0.0133304\pi$
$683$	47.0692	+	14.5189i	1.80105	+	0.555551i	0.801930	+	0.597418i	$0.203807\pi$
$684$		0			0
$685$	1.01423	−	13.5340i	0.0387518	−	0.517107i
$686$		0			0
$687$	1.56931			0.0598730
$688$		0			0
$689$	−24.0176			−0.914999
$690$		0			0
$691$	3.68891	−	49.2251i	0.140333	−	1.87261i	−0.272978	−	0.962020i	$-0.588009\pi$
$691$	3.68891	−	49.2251i	0.140333	−	1.87261i	0.413311	−	0.910590i	$-0.364372\pi$
$692$		0			0
$693$	1.08439	+	0.334491i	0.0411927	+	0.0127063i
$694$		0			0
$695$	−8.38908	−	14.5303i	−0.318216	−	0.551166i
$696$		0			0
$697$	−1.43359	−	0.216078i	−0.0543009	−	0.00818455i
$698$		0			0
$699$	1.21305	−	1.52112i	0.0458818	−	0.0575340i
$700$		0			0
$701$	12.8277	−	11.9024i	0.484495	−	0.449546i	−0.399712	−	0.916641i	$-0.630890\pi$
$701$	12.8277	−	11.9024i	0.484495	−	0.449546i	0.884207	+	0.467095i	$0.154699\pi$
$702$		0			0
$703$	3.13388	−	7.98500i	0.118197	−	0.301160i
$704$		0			0
$705$	0.870981	−	0.268662i	0.0328030	−	0.0101184i
$706$		0			0
$707$	−0.640585	+	0.0965527i	−0.0240917	+	0.00363124i
$708$		0			0
$709$	−28.7314	+	13.8363i	−1.07903	+	0.519634i	−0.887007	−	0.461756i	$-0.847220\pi$
$709$	−28.7314	+	13.8363i	−1.07903	+	0.519634i	−0.192025	+	0.981390i	$0.561505\pi$
$710$		0			0
$711$	1.95621	+	4.98433i	0.0733635	+	0.186927i
$712$		0			0
$713$	32.2201	+	21.9673i	1.20665	+	0.822680i
$714$		0			0
$715$	5.21249	−	22.8374i	0.194936	−	0.854071i
$716$		0			0
$717$	1.62266	+	21.6529i	0.0605993	+	0.808641i
$718$		0			0
$719$	−35.7184	+	24.3524i	−1.33207	+	0.908190i	−0.999393	−	0.0348509i	$-0.988904\pi$
$719$	−35.7184	+	24.3524i	−1.33207	+	0.908190i	−0.332677	+	0.943041i	$0.607952\pi$
$720$		0			0
$721$	−1.33980	−	1.24315i	−0.0498968	−	0.0462974i
$722$		0			0
$723$	7.76340	+	34.0137i	0.288724	+	1.26498i
$724$		0			0
$725$	10.9609	−	18.9848i	0.407076	−	0.705077i
$726$		0			0
$727$	2.61281	+	3.27636i	0.0969039	+	0.121514i	0.827919	−	0.560848i	$-0.189525\pi$
$727$	2.61281	+	3.27636i	0.0969039	+	0.121514i	−0.731015	+	0.682361i	$0.760953\pi$
$728$		0			0
$729$	−26.6965	−	12.8563i	−0.988757	−	0.476161i
$730$		0			0
$731$	−1.35533	+	1.26388i	−0.0501288	+	0.0467464i
$732$		0			0
$733$	−44.3364	−	21.3513i	−1.63760	−	0.788629i	−0.999831	−	0.0184045i	$-0.994141\pi$
$733$	−44.3364	−	21.3513i	−1.63760	−	0.788629i	−0.637773	−	0.770224i	$-0.720144\pi$
$734$		0			0
$735$	8.54744	+	10.7182i	0.315277	+	0.395345i
$736$		0			0
$737$	−18.9785	+	32.8717i	−0.699082	+	1.21085i
$738$		0			0
$739$	−0.558133	−	2.44534i	−0.0205313	−	0.0899533i	0.963624	−	0.267261i	$-0.0861185\pi$
$739$	−0.558133	−	2.44534i	−0.0205313	−	0.0899533i	−0.984156	+	0.177307i	$0.943261\pi$
$740$		0			0
$741$	−3.49846	−	3.24609i	−0.128519	−	0.119248i
$742$		0			0
$743$	−14.7210	+	10.0366i	−0.540059	+	0.368206i	−0.802416	−	0.596765i	$-0.796453\pi$
$743$	−14.7210	+	10.0366i	−0.540059	+	0.368206i	0.262357	+	0.964971i	$0.415500\pi$
$744$		0			0
$745$	−0.373233	−	4.98044i	−0.0136742	−	0.182469i
$746$		0			0
$747$	1.39460	−	6.11013i	0.0510256	−	0.223558i
$748$		0			0
$749$	1.12006	+	0.763641i	0.0409260	+	0.0279028i
$750$		0			0
$751$	−14.1468	−	36.0454i	−0.516223	−	1.31532i	−0.916908	−	0.399099i	$-0.869323\pi$
$751$	−14.1468	−	36.0454i	−0.516223	−	1.31532i	0.400684	−	0.916216i	$-0.368772\pi$
$752$		0			0
$753$	27.0044	−	13.0046i	0.984094	−	0.473915i
$754$		0			0
$755$	−15.3854	+	2.31897i	−0.559932	+	0.0843961i
$756$		0			0
$757$	29.5522	−	9.11563i	1.07409	−	0.331313i	0.293246	−	0.956037i	$-0.405265\pi$
$757$	29.5522	−	9.11563i	1.07409	−	0.331313i	0.780846	+	0.624724i	$0.214788\pi$
$758$		0			0
$759$	−12.6791	+	32.3058i	−0.460221	+	1.17262i
$760$		0			0
$761$	31.4994	−	29.2271i	1.14185	−	1.05948i	0.144300	−	0.989534i	$-0.453907\pi$
$761$	31.4994	−	29.2271i	1.14185	−	1.05948i	0.997551	−	0.0699487i	$-0.0222836\pi$
$762$		0			0
$763$	−0.0286848	+	0.0359696i	−0.00103846	+	0.00130219i
$764$		0			0
$765$	0.328354	+	0.0494914i	0.0118717	+	0.00178936i
$766$		0			0
$767$	12.0118	+	20.8050i	0.433720	+	0.751225i
$768$		0			0
$769$	11.0134	+	3.39717i	0.397152	+	0.122505i	0.486896	−	0.873460i	$-0.338129\pi$
$769$	11.0134	+	3.39717i	0.397152	+	0.122505i	−0.0897448	+	0.995965i	$0.528605\pi$
$770$		0			0
$771$	2.49103	−	33.2404i	0.0897121	−	1.19712i
$772$		0			0
$773$	−24.2393			−0.871828			−0.435914	−	0.899988i	$-0.643575\pi$
$773$	−24.2393			−0.871828			−0.435914	+	0.899988i	$0.643575\pi$
$774$		0			0
$775$	−31.4052			−1.12811
$776$		0			0
$777$	0.177964	−	2.37477i	0.00638443	−	0.0851943i
$778$		0			0
$779$	5.57498	+	1.71965i	0.199744	+	0.0616130i
$780$		0			0
$781$	−4.45985	−	7.72469i	−0.159586	−	0.276411i
$782$		0			0
$783$	−38.5593	−	5.81189i	−1.37800	−	0.207700i
$784$		0			0
$785$	−17.3485	+	21.7543i	−0.619194	+	0.776444i
$786$		0			0
$787$	7.37365	−	6.84175i	0.262842	−	0.243882i	−0.537732	−	0.843116i	$-0.680719\pi$
$787$	7.37365	−	6.84175i	0.262842	−	0.243882i	0.800574	+	0.599234i	$0.204528\pi$
$788$		0			0
$789$	3.28368	−	8.36668i	0.116902	−	0.297862i
$790$		0			0
$791$	2.28885	−	0.706017i	0.0813821	−	0.0251031i
$792$		0			0
$793$	38.1907	−	5.75633i	1.35619	−	0.204413i
$794$		0			0
$795$	14.8386	−	7.14591i	0.526272	−	0.253439i
$796$		0			0
$797$	−14.9088	−	37.9869i	−0.528096	−	1.34557i	−0.907390	−	0.420290i	$-0.861928\pi$
$797$	−14.9088	−	37.9869i	−0.528096	−	1.34557i	0.379294	−	0.925276i	$-0.376167\pi$
$798$		0			0
$799$	−0.107948	−	0.0735978i	−0.00381893	−	0.00260370i
$800$		0			0
$801$	0.858197	−	3.76001i	0.0303229	−	0.132853i
$802$		0			0
$803$	−0.369635	−	4.93243i	−0.0130441	−	0.174062i
$804$		0			0
$805$	−0.951940	+	0.649022i	−0.0335515	+	0.0228750i
$806$		0			0
$807$	16.0679	+	14.9089i	0.565618	+	0.524817i
$808$		0			0
$809$	10.4359	+	45.7228i	0.366908	+	1.60753i	0.735221	+	0.677827i	$0.237078\pi$
$809$	10.4359	+	45.7228i	0.366908	+	1.60753i	−0.368313	+	0.929702i	$0.620065\pi$
$810$		0			0
$811$	17.1357	−	29.6798i	0.601714	−	1.04220i	−0.390847	−	0.920456i	$-0.627818\pi$
$811$	17.1357	−	29.6798i	0.601714	−	1.04220i	0.992562	−	0.121744i	$-0.0388487\pi$
$812$		0			0
$813$	2.81598	+	3.53112i	0.0987606	+	0.123842i
$814$		0			0
$815$	8.41660	+	4.05322i	0.294821	+	0.141978i
$816$		0			0
$817$	6.15123	−	4.21640i	0.215204	−	0.147513i
$818$		0			0
$819$	0.487423	+	0.234730i	0.0170319	+	0.00820215i
$820$		0			0
$821$	−11.5042	−	14.4258i	−0.401500	−	0.503465i	0.539447	−	0.842020i	$-0.318633\pi$
$821$	−11.5042	−	14.4258i	−0.401500	−	0.503465i	−0.940947	+	0.338555i	$0.890062\pi$
$822$		0			0
$823$	4.59427	−	7.95750i	0.160146	−	0.277381i	−0.774775	−	0.632237i	$-0.782137\pi$
$823$	4.59427	−	7.95750i	0.160146	−	0.277381i	0.934921	+	0.354856i	$0.115470\pi$
$824$		0			0
$825$	−6.21930	−	27.2485i	−0.216528	−	0.948672i
$826$		0			0
$827$	26.9067	+	24.9658i	0.935637	+	0.868145i	0.991475	−	0.130295i	$-0.0415925\pi$
$827$	26.9067	+	24.9658i	0.935637	+	0.868145i	−0.0558380	+	0.998440i	$0.517783\pi$
$828$		0			0
$829$	3.75599	−	2.56079i	0.130451	−	0.0889400i	−0.496336	−	0.868130i	$-0.665322\pi$
$829$	3.75599	−	2.56079i	0.130451	−	0.0889400i	0.626788	+	0.779190i	$0.284369\pi$
$830$		0			0
$831$	−1.64842	−	21.9966i	−0.0571829	−	0.763053i
$832$		0			0
$833$	0.437263	−	1.91577i	0.0151503	−	0.0663776i
$834$		0			0
$835$	−3.96807	−	2.70538i	−0.137321	−	0.0936236i
$836$		0			0
$837$	20.4095	+	52.0026i	0.705457	+	1.79747i
$838$		0			0
$839$	−21.6424	+	10.4224i	−0.747177	+	0.359822i	−0.768414	−	0.639953i	$-0.778954\pi$
$839$	−21.6424	+	10.4224i	−0.747177	+	0.359822i	0.0212370	+	0.999774i	$0.493240\pi$
$840$		0			0
$841$	−18.4585	+	2.78217i	−0.636500	+	0.0959369i
$842$		0			0
$843$	−15.5669	+	4.80176i	−0.536154	+	0.165382i
$844$		0			0
$845$	−2.33598	+	5.95197i	−0.0803601	+	0.204754i
$846$		0			0
$847$	4.02372	−	3.73347i	0.138257	−	0.128284i
$848$		0			0
$849$	−14.6426	+	18.3613i	−0.502534	+	0.630157i
$850$		0			0
$851$	−29.4054	−	4.43215i	−1.00800	−	0.151932i
$852$		0			0
$853$	24.8630	+	43.0639i	0.851292	+	1.47448i	0.880043	+	0.474894i	$0.157514\pi$
$853$	24.8630	+	43.0639i	0.851292	+	1.47448i	−0.0287512	+	0.999587i	$0.509153\pi$
$854$		0			0
$855$	−1.27691	−	0.393876i	−0.0436695	−	0.0134703i
$856$		0			0
$857$	−2.63975	+	35.2251i	−0.0901723	+	1.20327i	0.750183	+	0.661230i	$0.229965\pi$
$857$	−2.63975	+	35.2251i	−0.0901723	+	1.20327i	−0.840356	+	0.542036i	$0.817654\pi$
$858$		0			0
$859$	−8.58814			−0.293024			−0.146512	−	0.989209i	$-0.546805\pi$
$859$	−8.58814			−0.293024			−0.146512	+	0.989209i	$0.546805\pi$
$860$		0			0
$861$	1.61969			0.0551989
$862$		0			0
$863$	−2.53059	+	33.7684i	−0.0861424	+	1.14949i	0.771842	+	0.635814i	$0.219336\pi$
$863$	−2.53059	+	33.7684i	−0.0861424	+	1.14949i	−0.857984	+	0.513676i	$0.828283\pi$
$864$		0			0
$865$	−31.0531	−	9.57861i	−1.05584	−	0.325682i
$866$		0			0
$867$	−12.3476	−	21.3867i	−0.419347	−	0.726331i
$868$		0			0
$869$	36.7123	+	5.53349i	1.24538	+	0.187711i
$870$		0			0
$871$	−11.2822	+	14.1474i	−0.382283	+	0.479368i
$872$		0			0
$873$	−2.84960	+	2.64404i	−0.0964444	+	0.0894873i
$874$		0			0
$875$	0.872799	−	2.22386i	0.0295060	−	0.0751800i
$876$		0			0
$877$	−17.2010	+	5.30581i	−0.580837	+	0.179164i	−0.571231	−	0.820789i	$-0.693534\pi$
$877$	−17.2010	+	5.30581i	−0.580837	+	0.179164i	−0.00960602	+	0.999954i	$0.503058\pi$
$878$		0			0
$879$	−21.1943	+	3.19452i	−0.714864	+	0.107748i
$880$		0			0
$881$	23.6192	−	11.3744i	0.795750	−	0.383213i	0.00859074	−	0.999963i	$-0.497265\pi$
$881$	23.6192	−	11.3744i	0.795750	−	0.383213i	0.787159	+	0.616750i	$0.211551\pi$
$882$		0			0
$883$	18.1802	+	46.3225i	0.611814	+	1.55888i	0.814829	+	0.579701i	$0.196831\pi$
$883$	18.1802	+	46.3225i	0.611814	+	1.55888i	−0.203016	+	0.979176i	$0.565074\pi$
$884$		0			0
$885$	−13.6112	−	9.27997i	−0.457536	−	0.311943i
$886$		0			0
$887$	−1.90844	+	8.36143i	−0.0640792	+	0.280749i	−0.996809	−	0.0798277i	$-0.974563\pi$
$887$	−1.90844	+	8.36143i	−0.0640792	+	0.280749i	0.932730	+	0.360577i	$0.117420\pi$
$888$		0			0
$889$	0.126325	+	1.68569i	0.00423680	+	0.0565362i
$890$		0			0
$891$	−28.0750	+	19.1412i	−0.940547	+	0.641254i
$892$		0			0
$893$	0.385409	+	0.357607i	0.0128972	+	0.0119669i
$894$		0			0
$895$	−1.56366	−	6.85086i	−0.0522675	−	0.228999i
$896$		0			0
$897$	−8.27242	+	14.3282i	−0.276208	+	0.478406i
$898$		0			0
$899$	42.5769	+	53.3898i	1.42002	+	1.78065i
$900$		0			0
$901$	−2.12696	−	1.02429i	−0.0708592	−	0.0341240i
$902$		0			0
$903$	1.28681	−	1.62191i	0.0428225	−	0.0539739i
$904$		0			0
$905$	3.37626	+	1.62592i	0.112231	+	0.0540474i
$906$		0			0
$907$	17.5664	+	22.0275i	0.583282	+	0.731412i	0.982669	−	0.185370i	$-0.0593485\pi$
$907$	17.5664	+	22.0275i	0.583282	+	0.731412i	−0.399387	+	0.916782i	$0.630777\pi$
$908$		0			0
$909$	−1.30238	+	2.25579i	−0.0431973	+	0.0748200i
$910$		0			0
$911$	−5.43767	−	23.8240i	−0.180158	−	0.789324i	−0.981553	−	0.191189i	$-0.938766\pi$
$911$	−5.43767	−	23.8240i	−0.180158	−	0.789324i	0.801395	−	0.598135i	$-0.204091\pi$
$912$		0			0
$913$	−31.8556	−	29.5577i	−1.05427	−	0.978217i
$914$		0			0
$915$	−21.8824	+	14.9192i	−0.723411	+	0.493213i
$916$		0			0
$917$	−0.290311	−	3.87394i	−0.00958692	−	0.127929i
$918$		0			0
$919$	−3.55017	+	15.5543i	−0.117109	+	0.513089i	0.882014	+	0.471223i	$0.156187\pi$
$919$	−3.55017	+	15.5543i	−0.117109	+	0.513089i	−0.999123	+	0.0418659i	$0.986670\pi$
$920$		0			0
$921$	0.0342045	+	0.0233202i	0.00112708	+	0.000768427i
$922$		0			0
$923$	−1.55354	−	3.95835i	−0.0511353	−	0.130291i
$924$		0			0
$925$	21.5772	−	10.3910i	0.709455	−	0.341656i
$926$		0			0
$927$	−7.26677	+	1.09529i	−0.238672	+	0.0359740i
$928$		0			0
$929$	−5.97357	+	1.84260i	−0.195987	+	0.0604539i	−0.391195	−	0.920308i	$-0.627938\pi$
$929$	−5.97357	+	1.84260i	−0.195987	+	0.0604539i	0.195208	+	0.980762i	$0.437462\pi$
$930$		0			0
$931$	−2.88900	+	7.36105i	−0.0946832	+	0.241249i
$932$		0			0
$933$	10.9961	−	10.2029i	0.359996	−	0.334027i
$934$		0			0
$935$	1.43556	−	1.80014i	0.0469480	−	0.0588709i
$936$		0			0
$937$	53.4604	+	8.05785i	1.74647	+	0.263239i	0.943165	−	0.332324i	$-0.107833\pi$
$937$	53.4604	+	8.05785i	1.74647	+	0.263239i	0.803309	+	0.595563i	$0.203071\pi$
$938$		0			0
$939$	13.0833	+	22.6610i	0.426959	+	0.739514i
$940$		0			0
$941$	29.8574	+	9.20978i	0.973322	+	0.300230i	0.740326	−	0.672248i	$-0.234671\pi$
$941$	29.8574	+	9.20978i	0.973322	+	0.300230i	0.232996	+	0.972478i	$0.425147\pi$
$942$		0			0
$943$	1.51146	−	20.1690i	0.0492198	−	0.656792i
$944$		0			0
$945$	−1.65051			−0.0536911
$946$		0			0
$947$	38.7946			1.26066			0.630328	−	0.776329i	$-0.282920\pi$
$947$	38.7946			1.26066			0.630328	+	0.776329i	$0.282920\pi$
$948$		0			0
$949$	0.176216	−	2.35144i	0.00572022	−	0.0763311i
$950$		0			0
$951$	30.3062	+	9.34823i	0.982747	+	0.303137i
$952$		0			0
$953$	14.6148	+	25.3136i	0.473420	+	0.819987i	0.999537	−	0.0304252i	$-0.00968613\pi$
$953$	14.6148	+	25.3136i	0.473420	+	0.819987i	−0.526118	+	0.850412i	$0.676353\pi$
$954$		0			0
$955$	−4.61591	−	0.695736i	−0.149367	−	0.0225135i
$956$		0			0
$957$	−37.8916	+	47.5145i	−1.22486	+	1.53593i
$958$		0			0
$959$	1.59320	−	1.47828i	0.0514472	−	0.0477361i
$960$		0			0
$961$	24.4157	−	62.2103i	0.787604	−	2.00678i
$962$		0			0
$963$	5.20849	−	1.60661i	0.167841	−	0.0517722i
$964$		0			0
$965$	24.3883	−	3.67594i	0.785087	−	0.118333i
$966$		0			0
$967$	22.9889	−	11.0709i	0.739274	−	0.356015i	−0.0260504	−	0.999661i	$-0.508293\pi$
$967$	22.9889	−	11.0709i	0.739274	−	0.356015i	0.765324	+	0.643645i	$0.222579\pi$
$968$		0			0
$969$	−0.171380	−	0.436668i	−0.00550551	−	0.0140278i
$970$		0			0
$971$	−17.1865	−	11.7176i	−0.551542	−	0.376035i	0.255246	−	0.966876i	$-0.417844\pi$
$971$	−17.1865	−	11.7176i	−0.551542	−	0.376035i	−0.806788	+	0.590841i	$0.798796\pi$
$972$		0			0
$973$	0.597875	−	2.61946i	0.0191670	−	0.0839761i
$974$		0			0
$975$	−0.995724	−	13.2870i	−0.0318887	−	0.425525i
$976$		0			0
$977$	−15.5522	+	10.6033i	−0.497559	+	0.339230i	−0.785958	−	0.618280i	$-0.787830\pi$
$977$	−15.5522	+	10.6033i	−0.497559	+	0.339230i	0.288398	+	0.957510i	$0.406877\pi$
$978$		0			0
$979$	−19.6031	−	18.1890i	−0.626517	−	0.581323i
$980$		0			0
$981$	0.0411631	+	0.180347i	0.00131424	+	0.00575804i
$982$		0			0
$983$	−11.1156	+	19.2529i	−0.354534	+	0.614071i	−0.987038	−	0.160486i	$-0.948694\pi$
$983$	−11.1156	+	19.2529i	−0.354534	+	0.614071i	0.632504	+	0.774557i	$0.282027\pi$
$984$		0			0
$985$	8.57416	+	10.7517i	0.273195	+	0.342576i
$986$		0			0
$987$	0.131507	+	0.0633305i	0.00418592	+	0.00201583i
$988$		0			0
$989$	−18.9958	−	17.5374i	−0.604033	−	0.557657i
$990$		0			0
$991$	13.8081	+	6.64964i	0.438630	+	0.211233i	0.640148	−	0.768252i	$-0.278873\pi$
$991$	13.8081	+	6.64964i	0.438630	+	0.211233i	−0.201518	+	0.979485i	$0.564587\pi$
$992$		0			0
$993$	12.0366	+	15.0934i	0.381971	+	0.478976i
$994$		0			0
$995$	6.21163	−	10.7589i	0.196922	−	0.341079i
$996$		0			0
$997$	−1.57862	−	6.91637i	−0.0499953	−	0.219044i	0.943759	−	0.330634i	$-0.107263\pi$
$997$	−1.57862	−	6.91637i	−0.0499953	−	0.219044i	−0.993754	+	0.111591i	$0.964405\pi$
$998$		0			0
$999$	−31.2287	−	28.9760i	−0.988031	−	0.916759i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bg.c.513.1		36
4.3	odd	2		43.2.g.a.40.2	yes	36
12.11	even	2		387.2.y.c.298.2		36
43.14	even	21	inner	688.2.bg.c.401.1		36
172.119	even	42		1849.2.a.o.1.6		18
172.139	odd	42		1849.2.a.n.1.13		18
172.143	odd	42		43.2.g.a.14.2	✓	36
516.143	even	42		387.2.y.c.100.2		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.14.2	✓	36	172.143	odd	42
43.2.g.a.40.2	yes	36	4.3	odd	2
387.2.y.c.100.2		36	516.143	even	42
387.2.y.c.298.2		36	12.11	even	2
688.2.bg.c.401.1		36	43.14	even	21	inner
688.2.bg.c.513.1		36	1.1	even	1	trivial
1849.2.a.n.1.13		18	172.139	odd	42
1849.2.a.o.1.6		18	172.119	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)

\(f(q)\)	\(=\)	\(q+(-0.109070 + 1.45544i) q^{3} +(1.29085 + 0.398175i) q^{5} +(0.108163 + 0.187343i) q^{7} +(0.860089 + 0.129637i) q^{9} +O(q^{10})\)	\(q+(-0.109070 + 1.45544i) q^{3} +(1.29085 + 0.398175i) q^{5} +(0.108163 + 0.187343i) q^{7} +(0.860089 + 0.129637i) q^{9} +(3.76031 - 4.71528i) q^{11} +(2.10767 - 1.95563i) q^{13} +(-0.720312 + 1.83532i) q^{15} +(0.270054 - 0.0833006i) q^{17} +(-1.12457 + 0.169502i) q^{19} +(-0.284464 + 0.136990i) q^{21} +(1.44040 + 3.67008i) q^{23} +(-2.62344 - 1.78863i) q^{25} +(-1.25681 + 5.50644i) q^{27} +(0.515946 + 6.88482i) q^{29} +(8.17225 - 5.57174i) q^{31} +(6.45267 + 5.98720i) q^{33} +(0.0650265 + 0.284900i) q^{35} +(-3.77129 + 6.53207i) q^{37} +(2.61642 + 3.28088i) q^{39} +(-4.62195 - 2.22581i) q^{41} +(-5.90092 + 2.85993i) q^{43} +(1.05863 + 0.509808i) q^{45} +(-0.288239 - 0.361440i) q^{47} +(3.47660 - 6.02165i) q^{49} +(0.0917841 + 0.402132i) q^{51} +(-6.12346 - 5.68174i) q^{53} +(6.73151 - 4.58946i) q^{55} +(-0.124042 - 1.65523i) q^{57} +(-1.85926 + 8.14595i) q^{59} +(11.0987 + 7.56700i) q^{61} +(0.0687427 + 0.175154i) q^{63} +(3.49937 - 1.68521i) q^{65} +(-6.22328 + 0.938008i) q^{67} +(-5.49868 + 1.69612i) q^{69} +(0.540324 - 1.37672i) q^{71} +(0.601198 - 0.557830i) q^{73} +(2.88938 - 3.62317i) q^{75} +(1.29010 + 0.194451i) q^{77} +(3.07798 + 5.33121i) q^{79} +(-5.38373 - 1.66066i) q^{81} +(0.538458 - 7.18523i) q^{83} +0.381767 q^{85} -10.0767 q^{87} +(0.331353 - 4.42159i) q^{89} +(0.594345 + 0.183331i) q^{91} +(7.21798 + 12.5019i) q^{93} +(-1.51914 - 0.228974i) q^{95} +(-2.78649 + 3.49415i) q^{97} +(3.84548 - 3.56808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 16 q^{3} - 17 q^{5} - 6 q^{7} - q^{9}+O(q^{10}) \) 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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