LMFDB - Embedded newform 688.2.bg.b.513.2

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:	$24$
Relative dimension:	$2$ over $\Q(\zeta_{21})$
Twist minimal:	no (minimal twist has level 86)
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			513.2
Character	$\chi$	$=$	688.513
Dual form			688.2.bg.b.401.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.00157347 + 0.0209965i) q^{3} +(2.31675 + 0.714623i) q^{5} +(0.898666 + 1.55654i) q^{7} +(2.96605 + 0.447061i) q^{9} +O(q^{10})$	\(q+(-0.00157347 + 0.0209965i) q^{3} +(2.31675 + 0.714623i) q^{5} +(0.898666 + 1.55654i) q^{7} +(2.96605 + 0.447061i) q^{9} +(-2.44477 + 3.06565i) q^{11} +(-4.21022 + 3.90652i) q^{13} +(-0.0186499 + 0.0475192i) q^{15} +(-2.78599 + 0.859363i) q^{17} +(5.68920 - 0.857509i) q^{19} +(-0.0340958 + 0.0164197i) q^{21} +(-0.955397 - 2.43431i) q^{23} +(0.725456 + 0.494608i) q^{25} +(-0.0281095 + 0.123156i) q^{27} +(-0.453309 - 6.04899i) q^{29} +(2.33820 - 1.59415i) q^{31} +(-0.0605210 - 0.0561553i) q^{33} +(0.969650 + 4.24831i) q^{35} +(-2.51504 + 4.35618i) q^{37} +(-0.0753985 - 0.0945467i) q^{39} +(2.36521 + 1.13903i) q^{41} +(-4.30267 + 4.94843i) q^{43} +(6.55213 + 3.15534i) q^{45} +(5.44891 + 6.83272i) q^{47} +(1.88480 - 3.26457i) q^{49} +(-0.0136599 - 0.0598481i) q^{51} +(7.59685 + 7.04884i) q^{53} +(-7.85471 + 5.35525i) q^{55} +(0.00905290 + 0.120803i) q^{57} +(1.94928 - 8.54036i) q^{59} +(-2.15646 - 1.47025i) q^{61} +(1.96963 + 5.01853i) q^{63} +(-12.5457 + 6.04170i) q^{65} +(12.5705 - 1.89470i) q^{67} +(0.0526153 - 0.0162297i) q^{69} +(-0.107431 + 0.273729i) q^{71} +(4.84493 - 4.49544i) q^{73} +(-0.0115265 + 0.0144538i) q^{75} +(-6.96882 - 1.05038i) q^{77} +(-6.93489 - 12.0116i) q^{79} +(8.59634 + 2.65162i) q^{81} +(0.441444 - 5.89067i) q^{83} -7.06856 q^{85} +0.127721 q^{87} +(0.787713 - 10.5113i) q^{89} +(-9.86422 - 3.04271i) q^{91} +(0.0297926 + 0.0516023i) q^{93} +(13.7933 + 2.07900i) q^{95} +(9.05823 - 11.3587i) q^{97} +(-8.62185 + 7.99991i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 6 q^{3} + 3 q^{5} - 3 q^{7} - 4 q^{9}+O(q^{10}) $ 24 * q + 6 * q^3 + 3 * q^5 - 3 * q^7 - 4 * q^9	$ 24 q + 6 q^{3} + 3 q^{5} - 3 q^{7} - 4 q^{9} - 2 q^{11} - 12 q^{13} + 42 q^{15} + 24 q^{17} + 18 q^{19} + 14 q^{21} + 4 q^{23} - 11 q^{25} - 23 q^{29} + 22 q^{31} + 50 q^{33} - 2 q^{35} + 2 q^{37} + 5 q^{39} - 14 q^{41} - 24 q^{43} + 71 q^{45} - 26 q^{47} + 13 q^{49} + 58 q^{51} + 31 q^{53} - 60 q^{55} - 61 q^{57} + 67 q^{59} + 15 q^{61} + 67 q^{63} - 17 q^{65} + 39 q^{67} - 15 q^{69} - 56 q^{71} - 81 q^{73} - 6 q^{75} + 53 q^{77} + 4 q^{79} + 70 q^{81} - 19 q^{83} - 38 q^{85} - 130 q^{87} - 21 q^{89} - 40 q^{91} + 50 q^{93} + 17 q^{95} + 39 q^{97} + 91 q^{99}+O(q^{100}) $ 24 * q + 6 * q^3 + 3 * q^5 - 3 * q^7 - 4 * q^9 - 2 * q^11 - 12 * q^13 + 42 * q^15 + 24 * q^17 + 18 * q^19 + 14 * q^21 + 4 * q^23 - 11 * q^25 - 23 * q^29 + 22 * q^31 + 50 * q^33 - 2 * q^35 + 2 * q^37 + 5 * q^39 - 14 * q^41 - 24 * q^43 + 71 * q^45 - 26 * q^47 + 13 * q^49 + 58 * q^51 + 31 * q^53 - 60 * q^55 - 61 * q^57 + 67 * q^59 + 15 * q^61 + 67 * q^63 - 17 * q^65 + 39 * q^67 - 15 * q^69 - 56 * q^71 - 81 * q^73 - 6 * q^75 + 53 * q^77 + 4 * q^79 + 70 * q^81 - 19 * q^83 - 38 * q^85 - 130 * q^87 - 21 * q^89 - 40 * q^91 + 50 * q^93 + 17 * q^95 + 39 * q^97 + 91 * 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
$3$	−0.00157347	+	0.0209965i	−0.000908443	+	0.0121223i	−0.997640	−	0.0686675i	$-0.978125\pi$
$3$	−0.00157347	+	0.0209965i	−0.000908443	+	0.0121223i	0.996731	+	0.0807899i	$0.0257443\pi$
$4$		0			0
$5$	2.31675	+	0.714623i	1.03608	+	0.319589i	0.765723	−	0.643170i	$-0.222381\pi$
$5$	2.31675	+	0.714623i	1.03608	+	0.319589i	0.270359	+	0.962759i	$0.412857\pi$
$6$		0			0
$7$	0.898666	+	1.55654i	0.339664	+	0.588315i	0.984369	−	0.176116i	$-0.0563534\pi$
$7$	0.898666	+	1.55654i	0.339664	+	0.588315i	−0.644706	+	0.764431i	$0.723020\pi$
$8$		0			0
$9$	2.96605	+	0.447061i	0.988685	+	0.149020i
$10$		0			0
$11$	−2.44477	+	3.06565i	−0.737126	+	0.924327i	−0.999170	−	0.0407291i	$-0.987032\pi$
$11$	−2.44477	+	3.06565i	−0.737126	+	0.924327i	0.262044	+	0.965056i	$0.415603\pi$
$12$		0			0
$13$	−4.21022	+	3.90652i	−1.16771	+	1.08347i	−0.172566	+	0.984998i	$0.555206\pi$
$13$	−4.21022	+	3.90652i	−1.16771	+	1.08347i	−0.995140	+	0.0984750i	$0.968604\pi$
$14$		0			0
$15$	−0.0186499	+	0.0475192i	−0.00481539	+	0.0122694i
$16$		0			0
$17$	−2.78599	+	0.859363i	−0.675701	+	0.208426i	−0.613568	−	0.789642i	$-0.710266\pi$
$17$	−2.78599	+	0.859363i	−0.675701	+	0.208426i	−0.0621329	+	0.998068i	$0.519790\pi$
$18$		0			0
$19$	5.68920	−	0.857509i	1.30519	−	0.196726i	0.540622	−	0.841266i	$-0.318189\pi$
$19$	5.68920	−	0.857509i	1.30519	−	0.196726i	0.764571	+	0.644540i	$0.222951\pi$
$20$		0			0
$21$	−0.0340958	+	0.0164197i	−0.00744031	+	0.00358307i
$22$		0			0
$23$	−0.955397	−	2.43431i	−0.199214	−	0.507589i	0.796001	−	0.605295i	$-0.206945\pi$
$23$	−0.955397	−	2.43431i	−0.199214	−	0.507589i	−0.995215	+	0.0977057i	$0.968850\pi$
$24$		0			0
$25$	0.725456	+	0.494608i	0.145091	+	0.0989215i
$26$		0			0
$27$	−0.0281095	+	0.123156i	−0.00540967	+	0.0237013i
$28$		0			0
$29$	−0.453309	−	6.04899i	−0.0841774	−	1.12327i	−0.866092	−	0.499884i	$-0.833376\pi$
$29$	−0.453309	−	6.04899i	−0.0841774	−	1.12327i	0.781915	−	0.623385i	$-0.214243\pi$
$30$		0			0
$31$	2.33820	−	1.59415i	0.419952	−	0.286319i	−0.334849	−	0.942272i	$-0.608685\pi$
$31$	2.33820	−	1.59415i	0.419952	−	0.286319i	0.754802	+	0.655953i	$0.227733\pi$
$32$		0			0
$33$	−0.0605210	−	0.0561553i	−0.0105354	−	0.00977538i
$34$		0			0
$35$	0.969650	+	4.24831i	0.163901	+	0.718096i
$36$		0			0
$37$	−2.51504	+	4.35618i	−0.413470	+	0.716152i	−0.995267	−	0.0971829i	$-0.969017\pi$
$37$	−2.51504	+	4.35618i	−0.413470	+	0.716152i	0.581796	+	0.813335i	$0.302350\pi$
$38$		0			0
$39$	−0.0753985	−	0.0945467i	−0.0120734	−	0.0151396i
$40$		0			0
$41$	2.36521	+	1.13903i	0.369384	+	0.177886i	0.609362	−	0.792893i	$-0.291426\pi$
$41$	2.36521	+	1.13903i	0.369384	+	0.177886i	−0.239978	+	0.970778i	$0.577140\pi$
$42$		0			0
$43$	−4.30267	+	4.94843i	−0.656152	+	0.754629i
$43$	−4.30267	+	4.94843i	−0.656152	+	0.754629i
$44$		0			0
$45$	6.55213	+	3.15534i	0.976734	+	0.470370i
$46$		0			0
$47$	5.44891	+	6.83272i	0.794806	+	0.996655i	0.999840	+	0.0179002i	$0.00569813\pi$
$47$	5.44891	+	6.83272i	0.794806	+	0.996655i	−0.205034	+	0.978755i	$0.565730\pi$
$48$		0			0
$49$	1.88480	−	3.26457i	0.269257	−	0.466367i
$50$		0			0
$51$	−0.0136599	−	0.0598481i	−0.00191277	−	0.00838041i
$52$		0			0
$53$	7.59685	+	7.04884i	1.04351	+	0.968233i	0.999526	−	0.0307991i	$-0.00980521\pi$
$53$	7.59685	+	7.04884i	1.04351	+	0.968233i	0.0439817	+	0.999032i	$0.485996\pi$
$54$		0			0
$55$	−7.85471	+	5.35525i	−1.05913	+	0.722101i
$56$		0			0
$57$	0.00905290	+	0.120803i	0.00119909	+	0.0160007i
$58$		0			0
$59$	1.94928	−	8.54036i	0.253775	−	1.11186i	−0.674004	−	0.738728i	$-0.735427\pi$
$59$	1.94928	−	8.54036i	0.253775	−	1.11186i	0.927778	−	0.373132i	$-0.121716\pi$
$60$		0			0
$61$	−2.15646	−	1.47025i	−0.276106	−	0.188246i	0.417348	−	0.908747i	$-0.362960\pi$
$61$	−2.15646	−	1.47025i	−0.276106	−	0.188246i	−0.693454	+	0.720501i	$0.743912\pi$
$62$		0			0
$63$	1.96963	+	5.01853i	0.248150	+	0.632275i
$64$		0			0
$65$	−12.5457	+	6.04170i	−1.55611	+	0.749381i
$66$		0			0
$67$	12.5705	−	1.89470i	1.53573	−	0.231474i	0.673959	−	0.738769i	$-0.264592\pi$
$67$	12.5705	−	1.89470i	1.53573	−	0.231474i	0.861773	+	0.507294i	$0.169354\pi$
$68$		0			0
$69$	0.0526153	−	0.0162297i	0.00633414	−	0.00195382i
$70$		0			0
$71$	−0.107431	+	0.273729i	−0.0127497	+	0.0324857i	−0.937107	−	0.349043i	$-0.886507\pi$
$71$	−0.107431	+	0.273729i	−0.0127497	+	0.0324857i	0.924357	+	0.381528i	$0.124602\pi$
$72$		0			0
$73$	4.84493	−	4.49544i	0.567056	−	0.526151i	−0.343709	−	0.939076i	$-0.611683\pi$
$73$	4.84493	−	4.49544i	0.567056	−	0.526151i	0.910765	+	0.412925i	$0.135493\pi$
$74$		0			0
$75$	−0.0115265	+	0.0144538i	−0.00133097	+	0.00166898i
$76$		0			0
$77$	−6.96882	−	1.05038i	−0.794170	−	0.119702i
$78$		0			0
$79$	−6.93489	−	12.0116i	−0.780236	−	1.35141i	−0.931804	−	0.362962i	$-0.881765\pi$
$79$	−6.93489	−	12.0116i	−0.780236	−	1.35141i	0.151568	−	0.988447i	$-0.451568\pi$
$80$		0			0
$81$	8.59634	+	2.65162i	0.955149	+	0.294625i
$82$		0			0
$83$	0.441444	−	5.89067i	0.0484548	−	0.646585i	−0.919116	−	0.393986i	$-0.871096\pi$
$83$	0.441444	−	5.89067i	0.0484548	−	0.646585i	0.967571	−	0.252598i	$-0.0812852\pi$
$84$		0			0
$85$	−7.06856			−0.766693
$86$		0			0
$87$	0.127721			0.0136931
$88$		0			0
$89$	0.787713	−	10.5113i	0.0834974	−	1.11420i	−0.785329	−	0.619079i	$-0.787506\pi$
$89$	0.787713	−	10.5113i	0.0834974	−	1.11420i	0.868826	−	0.495117i	$-0.164875\pi$
$90$		0			0
$91$	−9.86422	−	3.04271i	−1.03405	−	0.318962i
$92$		0			0
$93$	0.0297926	+	0.0516023i	0.00308935	+	0.00535090i
$94$		0			0
$95$	13.7933	+	2.07900i	1.41516	+	0.213301i
$96$		0			0
$97$	9.05823	−	11.3587i	0.919724	−	1.15330i	−0.0680929	−	0.997679i	$-0.521691\pi$
$97$	9.05823	−	11.3587i	0.919724	−	1.15330i	0.987817	−	0.155619i	$-0.0497371\pi$
$98$		0			0
$99$	−8.62185	+	7.99991i	−0.866529	+	0.804021i
$100$		0			0
$101$	−2.17270	+	5.53594i	−0.216191	+	0.550847i	−0.997291	−	0.0735599i	$-0.976564\pi$
$101$	−2.17270	+	5.53594i	−0.216191	+	0.550847i	0.781099	+	0.624407i	$0.214659\pi$
$102$		0			0
$103$	−6.78880	+	2.09407i	−0.668920	+	0.206334i	−0.610571	−	0.791962i	$-0.709060\pi$
$103$	−6.78880	+	2.09407i	−0.668920	+	0.206334i	−0.0583493	+	0.998296i	$0.518584\pi$
$104$		0			0
$105$	−0.0907254	+	0.0136746i	−0.00885389	+	0.00133451i
$106$		0			0
$107$	−3.96497	+	1.90943i	−0.383308	+	0.184591i	−0.615608	−	0.788053i	$-0.711089\pi$
$107$	−3.96497	+	1.90943i	−0.383308	+	0.184591i	0.232299	+	0.972644i	$0.425375\pi$
$108$		0			0
$109$	6.34700	+	16.1719i	0.607932	+	1.54899i	0.820383	+	0.571815i	$0.193760\pi$
$109$	6.34700	+	16.1719i	0.607932	+	1.54899i	−0.212450	+	0.977172i	$0.568144\pi$
$110$		0			0
$111$	−0.0875072	−	0.0596614i	−0.00830581	−	0.00566281i
$112$		0			0
$113$	2.91641	−	12.7776i	0.274353	−	1.20202i	−0.630464	−	0.776218i	$-0.717136\pi$
$113$	2.91641	−	12.7776i	0.274353	−	1.20202i	0.904817	−	0.425800i	$-0.140007\pi$
$114$		0			0
$115$	−0.473802	−	6.32245i	−0.0441822	−	0.589571i
$116$		0			0
$117$	−14.2342	+	9.70471i	−1.31595	+	0.897201i
$118$		0			0
$119$	−3.84130	−	3.56421i	−0.352132	−	0.326730i
$120$		0			0
$121$	−0.973547	−	4.26539i	−0.0885043	−	0.387763i
$122$		0			0
$123$	−0.0276371	+	0.0478689i	−0.00249196	+	0.00431619i
$124$		0			0
$125$	−6.23090	−	7.81330i	−0.557308	−	0.698843i
$126$		0			0
$127$	2.88229	+	1.38804i	0.255762	+	0.123168i	0.557372	−	0.830263i	$-0.311810\pi$
$127$	2.88229	+	1.38804i	0.255762	+	0.123168i	−0.301610	+	0.953431i	$0.597524\pi$
$128$		0			0
$129$	−0.0971296	−	0.0981272i	−0.00855179	−	0.00863962i
$130$		0			0
$131$	−19.8136	−	9.54173i	−1.73112	−	0.833665i	−0.986002	−	0.166735i	$-0.946677\pi$
$131$	−19.8136	−	9.54173i	−1.73112	−	0.833665i	−0.745121	−	0.666929i	$-0.767608\pi$
$132$		0			0
$133$	6.44744	+	8.08483i	0.559064	+	0.701044i
$134$		0			0
$135$	−0.153133	+	0.265233i	−0.0131796	+	0.0228277i
$136$		0			0
$137$	−3.65183	−	15.9997i	−0.311997	−	1.36695i	−0.851231	−	0.524791i	$-0.824144\pi$
$137$	−3.65183	−	15.9997i	−0.311997	−	1.36695i	0.539235	−	0.842156i	$-0.318714\pi$
$138$		0			0
$139$	13.4187	+	12.4507i	1.13816	+	1.05605i	0.997823	+	0.0659500i	$0.0210078\pi$
$139$	13.4187	+	12.4507i	1.13816	+	1.05605i	0.140333	+	0.990104i	$0.455183\pi$
$140$		0			0
$141$	−0.152037	+	0.103657i	−0.0128038	+	0.00872949i
$142$		0			0
$143$	−1.68296	−	22.4576i	−0.140736	−	1.87800i
$144$		0			0
$145$	3.27254	−	14.3380i	0.271770	−	1.19070i
$146$		0			0
$147$	0.0655788	+	0.0447109i	0.00540885	+	0.00368769i
$148$		0			0
$149$	−4.44495	−	11.3255i	−0.364144	−	0.927825i	−0.989111	−	0.147173i	$-0.952983\pi$
$149$	−4.44495	−	11.3255i	−0.364144	−	0.927825i	0.624966	−	0.780652i	$-0.285113\pi$
$150$		0			0
$151$	3.91238	−	1.88410i	0.318385	−	0.153326i	−0.267865	−	0.963457i	$-0.586318\pi$
$151$	3.91238	−	1.88410i	0.318385	−	0.153326i	0.586250	+	0.810130i	$0.300604\pi$
$152$		0			0
$153$	−8.64758	+	1.30341i	−0.699115	+	0.105375i
$154$		0			0
$155$	6.55624	−	2.02233i	0.526610	−	0.162438i
$156$		0			0
$157$	−6.36258	+	16.2116i	−0.507789	+	1.29383i	0.415456	+	0.909613i	$0.363622\pi$
$157$	−6.36258	+	16.2116i	−0.507789	+	1.29383i	−0.923245	+	0.384212i	$0.874473\pi$
$158$		0			0
$159$	−0.159954	+	0.148416i	−0.0126852	+	0.0117702i
$160$		0			0
$161$	2.93051	−	3.67474i	0.230957	−	0.289610i
$162$		0			0
$163$	−19.6342	−	2.95939i	−1.53787	−	0.231797i	−0.675229	−	0.737608i	$-0.735955\pi$
$163$	−19.6342	−	2.95939i	−1.53787	−	0.231797i	−0.862644	+	0.505811i	$0.831193\pi$
$164$		0			0
$165$	−0.100082	−	0.173348i	−0.00779139	−	0.0134951i
$166$		0			0
$167$	−4.70345	−	1.45082i	−0.363964	−	0.112268i	0.107379	−	0.994218i	$-0.465754\pi$
$167$	−4.70345	−	1.45082i	−0.363964	−	0.112268i	−0.471343	+	0.881950i	$0.656231\pi$
$168$		0			0
$169$	1.49362	−	19.9310i	0.114894	−	1.53315i
$170$		0			0
$171$	17.2578			1.31974
$172$		0			0
$173$	3.09098			0.235003			0.117502	−	0.993073i	$-0.462511\pi$
$173$	3.09098			0.235003			0.117502	+	0.993073i	$0.462511\pi$
$174$		0			0
$175$	−0.117931	+	1.57369i	−0.00891478	+	0.118959i
$176$		0			0
$177$	0.176250	+	0.0543660i	0.0132478	+	0.00408640i
$178$		0			0
$179$	2.06272	+	3.57274i	0.154175	+	0.267039i	0.932758	−	0.360502i	$-0.117395\pi$
$179$	2.06272	+	3.57274i	0.154175	+	0.267039i	−0.778583	+	0.627541i	$0.784061\pi$
$180$		0			0
$181$	−10.8798	−	1.63986i	−0.808686	−	0.121890i	−0.268331	−	0.963327i	$-0.586472\pi$
$181$	−10.8798	−	1.63986i	−0.808686	−	0.121890i	−0.540355	+	0.841437i	$0.681710\pi$
$182$		0			0
$183$	0.0342631	−	0.0429646i	0.00253280	−	0.00317604i
$184$		0			0
$185$	−8.93975	+	8.29488i	−0.657264	+	0.609852i
$186$		0			0
$187$	4.17660	−	10.6418i	0.305423	−	0.778205i
$188$		0			0
$189$	−0.216957	+	0.0669224i	−0.0157813	+	0.00486789i
$190$		0			0
$191$	17.2106	−	2.59408i	1.24531	−	0.187701i	0.506874	−	0.862020i	$-0.330801\pi$
$191$	17.2106	−	2.59408i	1.24531	−	0.187701i	0.738440	+	0.674319i	$0.235563\pi$
$192$		0			0
$193$	−2.67881	+	1.29005i	−0.192825	+	0.0928597i	−0.527805	−	0.849366i	$-0.676985\pi$
$193$	−2.67881	+	1.29005i	−0.192825	+	0.0928597i	0.334980	+	0.942225i	$0.391271\pi$
$194$		0			0
$195$	−0.107114	−	0.272923i	−0.00767061	−	0.0195444i
$196$		0			0
$197$	−0.955396	−	0.651378i	−0.0680691	−	0.0464087i	0.528807	−	0.848742i	$-0.322639\pi$
$197$	−0.955396	−	0.651378i	−0.0680691	−	0.0464087i	−0.596876	+	0.802333i	$0.703592\pi$
$198$		0			0
$199$	−2.19161	+	9.60206i	−0.155359	+	0.680672i	0.835915	+	0.548858i	$0.184937\pi$
$199$	−2.19161	+	9.60206i	−0.155359	+	0.680672i	−0.991274	+	0.131814i	$0.957920\pi$
$200$		0			0
$201$	0.0200027	+	0.266918i	0.00141088	+	0.0188269i
$202$		0			0
$203$	9.00810	−	6.14162i	0.632244	−	0.431057i
$204$		0			0
$205$	4.66563	+	4.32907i	0.325862	+	0.302356i
$206$		0			0
$207$	−1.74547	−	7.64742i	−0.121319	−	0.531533i
$208$		0			0
$209$	−11.2800	+	19.5375i	−0.780252	+	1.35144i
$210$		0			0
$211$	3.85057	+	4.82847i	0.265084	+	0.332405i	0.896504	−	0.443036i	$-0.146099\pi$
$211$	3.85057	+	4.82847i	0.265084	+	0.332405i	−0.631419	+	0.775442i	$0.717527\pi$
$212$		0			0
$213$	−0.00557832	−	0.00268637i	−0.000382220	−	0.000184067i
$214$		0			0
$215$	−13.5045	+	8.38950i	−0.920999	+	0.572159i
$216$		0			0
$217$	4.58262	+	2.20687i	0.311088	+	0.149812i
$218$		0			0
$219$	0.0867650	+	0.108800i	0.00586304	+	0.00735202i
$220$		0			0
$221$	8.37251	−	14.5016i	0.563196	−	0.975484i
$222$		0			0
$223$	−0.948442	−	4.15540i	−0.0635124	−	0.278266i	0.933193	−	0.359376i	$-0.117011\pi$
$223$	−0.948442	−	4.15540i	−0.0635124	−	0.278266i	−0.996705	+	0.0811103i	$0.974153\pi$
$224$		0			0
$225$	1.93062	+	1.79136i	0.128708	+	0.119424i
$226$		0			0
$227$	1.07952	−	0.736004i	0.0716502	−	0.0488503i	−0.526965	−	0.849887i	$-0.676670\pi$
$227$	1.07952	−	0.736004i	0.0716502	−	0.0488503i	0.598615	+	0.801037i	$0.295718\pi$
$228$		0			0
$229$	0.901045	+	12.0236i	0.0595427	+	0.794543i	0.944425	+	0.328727i	$0.106620\pi$
$229$	0.901045	+	12.0236i	0.0595427	+	0.794543i	−0.884882	+	0.465815i	$0.845761\pi$
$230$		0			0
$231$	0.0330195	−	0.144668i	0.00217252	−	0.00951845i
$232$		0			0
$233$	2.61973	+	1.78610i	0.171624	+	0.117011i	0.646086	−	0.763265i	$-0.276405\pi$
$233$	2.61973	+	1.78610i	0.171624	+	0.117011i	−0.474462	+	0.880276i	$0.657357\pi$
$234$		0			0
$235$	7.74096	+	19.7236i	0.504964	+	1.28663i
$236$		0			0
$237$	0.263113	−	0.126709i	0.0170910	−	0.00823060i
$238$		0			0
$239$	−20.6422	+	3.11130i	−1.33523	+	0.201254i	−0.777560	−	0.628809i	$-0.783543\pi$
$239$	−20.6422	+	3.11130i	−1.33523	+	0.201254i	−0.557670	+	0.830063i	$0.688305\pi$
$240$		0			0
$241$	0.116684	−	0.0359922i	0.00751627	−	0.00231846i	−0.290995	−	0.956725i	$-0.593986\pi$
$241$	0.116684	−	0.0359922i	0.00751627	−	0.00231846i	0.298511	+	0.954406i	$0.403510\pi$
$242$		0			0
$243$	−0.207654	+	0.529093i	−0.0133210	+	0.0339413i
$244$		0			0
$245$	6.69954	−	6.21627i	0.428018	−	0.397143i
$246$		0			0
$247$	−20.6029	+	25.8353i	−1.31093	+	1.64386i
$248$		0			0
$249$	0.122989	+	0.0185376i	0.00779409	+	0.00117477i
$250$		0			0
$251$	−8.61396	−	14.9198i	−0.543708	−	0.941730i	−0.998687	−	0.0512278i	$-0.983687\pi$
$251$	−8.61396	−	14.9198i	−0.543708	−	0.941730i	0.454979	−	0.890502i	$-0.349647\pi$
$252$		0			0
$253$	9.79847	+	3.02243i	0.616024	+	0.190018i
$254$		0			0
$255$	0.0111222	−	0.148415i	0.000696497	−	0.00929410i
$256$		0			0
$257$	21.6317			1.34935			0.674675	−	0.738115i	$-0.264284\pi$
$257$	21.6317			1.34935			0.674675	+	0.738115i	$0.264284\pi$
$258$		0			0
$259$	−9.04073			−0.561764
$260$		0			0
$261$	1.35973	−	18.1443i	0.0841650	−	1.12310i
$262$		0			0
$263$	−2.09734	−	0.646945i	−0.129328	−	0.0398923i	0.229416	−	0.973329i	$-0.426318\pi$
$263$	−2.09734	−	0.646945i	−0.129328	−	0.0398923i	−0.358743	+	0.933436i	$0.616795\pi$
$264$		0			0
$265$	12.5627	+	21.7593i	0.771723	+	1.33666i
$266$		0			0
$267$	0.219461	+	0.0330784i	0.0134308	+	0.00202437i
$268$		0			0
$269$	−3.17399	+	3.98006i	−0.193522	+	0.242669i	−0.869120	−	0.494601i	$-0.835314\pi$
$269$	−3.17399	+	3.98006i	−0.193522	+	0.242669i	0.675598	+	0.737270i	$0.263885\pi$
$270$		0			0
$271$	13.1299	−	12.1828i	0.797585	−	0.740051i	−0.171923	−	0.985110i	$-0.554998\pi$
$271$	13.1299	−	12.1828i	0.797585	−	0.740051i	0.969508	+	0.245059i	$0.0788074\pi$
$272$		0			0
$273$	0.0794072	−	0.202326i	0.00480594	−	0.0122453i
$274$		0			0
$275$	−3.28987	+	1.01479i	−0.198386	+	0.0611941i
$276$		0			0
$277$	20.1627	−	3.03904i	1.21146	−	0.182598i	0.487925	−	0.872886i	$-0.337754\pi$
$277$	20.1627	−	3.03904i	1.21146	−	0.182598i	0.723535	+	0.690288i	$0.242516\pi$
$278$		0			0
$279$	7.64790	−	3.68303i	0.457868	−	0.220497i
$280$		0			0
$281$	−1.27653	−	3.25254i	−0.0761513	−	0.194030i	0.887729	−	0.460367i	$-0.152282\pi$
$281$	−1.27653	−	3.25254i	−0.0761513	−	0.194030i	−0.963880	+	0.266336i	$0.914187\pi$
$282$		0			0
$283$	−2.94355	−	2.00688i	−0.174976	−	0.119296i	0.472665	−	0.881242i	$-0.343292\pi$
$283$	−2.94355	−	2.00688i	−0.174976	−	0.119296i	−0.647641	+	0.761946i	$0.724244\pi$
$284$		0			0
$285$	−0.0653550	+	0.286339i	−0.00387130	+	0.0169613i
$286$		0			0
$287$	0.352601	+	4.70514i	0.0208134	+	0.277736i
$288$		0			0
$289$	−7.02284	+	4.78809i	−0.413108	+	0.281652i
$290$		0			0
$291$	0.224239	+	0.208064i	0.0131451	+	0.0121969i
$292$		0			0
$293$	−6.82498	−	29.9022i	−0.398719	−	1.74690i	−0.632449	−	0.774602i	$-0.717950\pi$
$293$	−6.82498	−	29.9022i	−0.398719	−	1.74690i	0.233730	−	0.972302i	$-0.424907\pi$
$294$		0			0
$295$	10.6191	−	18.3929i	0.618270	−	1.07088i
$296$		0			0
$297$	−0.308830	−	0.387261i	−0.0179202	−	0.0224712i
$298$		0			0
$299$	13.5321	+	6.51673i	0.782583	+	0.376872i
$300$		0			0
$301$	−11.5691	−	2.25027i	−0.666831	−	0.129704i
$302$		0			0
$303$	−0.112817	−	0.0543296i	−0.00648115	−	0.00312116i
$304$		0			0
$305$	−3.94530	−	4.94725i	−0.225907	−	0.283279i
$306$		0			0
$307$	−9.07663	+	15.7212i	−0.518031	+	0.897255i	0.481750	+	0.876309i	$0.340001\pi$
$307$	−9.07663	+	15.7212i	−0.518031	+	0.897255i	−0.999781	+	0.0209466i	$0.993332\pi$
$308$		0			0
$309$	−0.0332861	−	0.145836i	−0.00189358	−	0.00829631i
$310$		0			0
$311$	−22.0670	−	20.4752i	−1.25131	−	1.16104i	−0.980096	−	0.198524i	$-0.936385\pi$
$311$	−22.0670	−	20.4752i	−1.25131	−	1.16104i	−0.271211	−	0.962520i	$-0.587424\pi$
$312$		0			0
$313$	−16.8134	+	11.4632i	−0.950351	+	0.647938i	−0.936015	−	0.351959i	$-0.885516\pi$
$313$	−16.8134	+	11.4632i	−0.950351	+	0.647938i	−0.0143357	+	0.999897i	$0.504563\pi$
$314$		0			0
$315$	0.976780	+	13.0342i	0.0550353	+	0.734395i
$316$		0			0
$317$	−3.46377	+	15.1758i	−0.194545	+	0.852356i	0.779572	+	0.626312i	$0.215436\pi$
$317$	−3.46377	+	15.1758i	−0.194545	+	0.852356i	−0.974117	+	0.226044i	$0.927421\pi$
$318$		0			0
$319$	19.6523	+	13.3987i	1.10032	+	0.750184i
$320$		0			0
$321$	−0.0338525	−	0.0862549i	−0.00188947	−	0.00481428i
$322$		0			0
$323$	−15.1131	+	7.27810i	−0.840917	+	0.404964i
$324$		0			0
$325$	−4.98653	+	0.751598i	−0.276603	+	0.0416912i
$326$		0			0
$327$	−0.349540	+	0.107819i	−0.0193296	+	0.00596239i
$328$		0			0
$329$	−5.73862	+	14.6218i	−0.316380	+	0.806124i
$330$		0			0
$331$	23.6094	−	21.9063i	1.29769	−	1.20408i	0.332326	−	0.943165i	$-0.392167\pi$
$331$	23.6094	−	21.9063i	1.29769	−	1.20408i	0.965362	−	0.260914i	$-0.0840239\pi$
$332$		0			0
$333$	−9.40723	+	11.7963i	−0.515513	+	0.646433i
$334$		0			0
$335$	30.4767	+	4.59363i	1.66512	+	0.250977i
$336$		0			0
$337$	4.09285	+	7.08903i	0.222952	+	0.386164i	0.955703	−	0.294333i	$-0.0950974\pi$
$337$	4.09285	+	7.08903i	0.222952	+	0.386164i	−0.732751	+	0.680497i	$0.761764\pi$
$338$		0			0
$339$	0.263697	+	0.0813396i	0.0143220	+	0.00441776i
$340$		0			0
$341$	−0.829239	+	11.0654i	−0.0449058	+	0.599226i
$342$		0			0
$343$	19.3565			1.04516
$344$		0			0
$345$	0.133495			0.00718711
$346$		0			0
$347$	1.88864	−	25.2022i	0.101388	−	1.35293i	−0.681982	−	0.731369i	$-0.738882\pi$
$347$	1.88864	−	25.2022i	0.101388	−	1.35293i	0.783369	−	0.621556i	$-0.213499\pi$
$348$		0			0
$349$	−15.0634	−	4.64644i	−0.806325	−	0.248718i	−0.135926	−	0.990719i	$-0.543401\pi$
$349$	−15.0634	−	4.64644i	−0.806325	−	0.248718i	−0.670399	+	0.742001i	$0.733877\pi$
$350$		0			0
$351$	−0.362763	−	0.628323i	−0.0193628	−	0.0335374i
$352$		0			0
$353$	2.71495	+	0.409212i	0.144502	+	0.0217802i	0.220895	−	0.975298i	$-0.429102\pi$
$353$	2.71495	+	0.409212i	0.144502	+	0.0217802i	−0.0763929	+	0.997078i	$0.524340\pi$
$354$		0			0
$355$	−0.444504	+	0.557390i	−0.0235918	+	0.0295832i
$356$		0			0
$357$	0.0808800	−	0.0750457i	0.00428062	−	0.00397184i
$358$		0			0
$359$	−11.6344	+	29.6440i	−0.614041	+	1.56455i	0.197529	+	0.980297i	$0.436708\pi$
$359$	−11.6344	+	29.6440i	−0.614041	+	1.56455i	−0.811571	+	0.584254i	$0.801387\pi$
$360$		0			0
$361$	13.4758	−	4.15674i	0.709254	−	0.218776i
$362$		0			0
$363$	0.0910900	−	0.0137296i	0.00478099	−	0.000720618i
$364$		0			0
$365$	14.4370	−	6.95251i	0.755669	−	0.363911i
$366$		0			0
$367$	9.26136	+	23.5976i	0.483439	+	1.23178i	0.939719	+	0.341947i	$0.111086\pi$
$367$	9.26136	+	23.5976i	0.483439	+	1.23178i	−0.456280	+	0.889836i	$0.650818\pi$
$368$		0			0
$369$	6.50613	+	4.43580i	0.338696	+	0.230919i
$370$		0			0
$371$	−4.14475	+	18.1593i	−0.215184	+	0.942785i
$372$		0			0
$373$	1.28659	+	17.1683i	0.0666171	+	0.888943i	0.925979	+	0.377575i	$0.123242\pi$
$373$	1.28659	+	17.1683i	0.0666171	+	0.888943i	−0.859362	+	0.511368i	$0.829139\pi$
$374$		0			0
$375$	0.173856	−	0.118533i	0.00897788	−	0.00612102i
$376$		0			0
$377$	25.5390	+	23.6968i	1.31533	+	1.22044i
$378$		0			0
$379$	7.78769	+	34.1201i	0.400027	+	1.75263i	0.627277	+	0.778796i	$0.284169\pi$
$379$	7.78769	+	34.1201i	0.400027	+	1.75263i	−0.227250	+	0.973836i	$0.572973\pi$
$380$		0			0
$381$	−0.0336791	+	0.0583339i	−0.00172543	+	0.00298854i
$382$		0			0
$383$	12.1296	+	15.2101i	0.619794	+	0.777198i	0.988316	−	0.152422i	$-0.0487072\pi$
$383$	12.1296	+	15.2101i	0.619794	+	0.777198i	−0.368521	+	0.929619i	$0.620136\pi$
$384$		0			0
$385$	−15.3944	−	7.41355i	−0.784571	−	0.377829i
$386$		0			0
$387$	−14.9742	+	12.7538i	−0.761182	+	0.648310i
$388$		0			0
$389$	−9.45418	−	4.55290i	−0.479346	−	0.230841i	0.178577	−	0.983926i	$-0.442851\pi$
$389$	−9.45418	−	4.55290i	−0.479346	−	0.230841i	−0.657923	+	0.753085i	$0.728565\pi$
$390$		0			0
$391$	4.75368	+	5.96093i	0.240404	+	0.301457i
$392$		0			0
$393$	0.231519	−	0.401003i	0.0116786	−	0.0202279i
$394$		0			0
$395$	−7.48266	−	32.7837i	−0.376494	−	1.64953i
$396$		0			0
$397$	−9.00299	−	8.35356i	−0.451847	−	0.419253i	0.421140	−	0.906996i	$-0.361630\pi$
$397$	−9.00299	−	8.35356i	−0.451847	−	0.419253i	−0.872988	+	0.487743i	$0.837820\pi$
$398$		0			0
$399$	−0.179898	+	0.122652i	−0.00900616	+	0.00614030i
$400$		0			0
$401$	−0.124266	−	1.65821i	−0.00620554	−	0.0828072i	0.993241	−	0.116069i	$-0.0370294\pi$
$401$	−0.124266	−	1.65821i	−0.00620554	−	0.0828072i	−0.999447	+	0.0332621i	$0.989410\pi$
$402$		0			0
$403$	−3.61673	+	15.8459i	−0.180162	+	0.789343i
$404$		0			0
$405$	18.0207	+	12.2863i	0.895455	+	0.610511i
$406$		0			0
$407$	−7.20580	−	18.3601i	−0.357178	−	0.910076i
$408$		0			0
$409$	−5.56701	+	2.68093i	−0.275271	+	0.132564i	−0.566427	−	0.824112i	$-0.691675\pi$
$409$	−5.56701	+	2.68093i	−0.275271	+	0.132564i	0.291156	+	0.956676i	$0.405960\pi$
$410$		0			0
$411$	0.341684	−	0.0515005i	0.0168540	−	0.00254033i
$412$		0			0
$413$	15.0451	−	4.64081i	0.740322	−	0.228359i
$414$		0			0
$415$	5.23232	−	13.3317i	0.256845	−	0.654429i
$416$		0			0
$417$	−0.282535	+	0.262154i	−0.0138358	+	0.0128377i
$418$		0			0
$419$	13.7407	−	17.2303i	0.671276	−	0.841753i	−0.323243	−	0.946316i	$-0.604773\pi$
$419$	13.7407	−	17.2303i	0.671276	−	0.841753i	0.994518	+	0.104563i	$0.0333444\pi$
$420$		0			0
$421$	18.4876	+	2.78656i	0.901032	+	0.135809i	0.583191	−	0.812335i	$-0.301804\pi$
$421$	18.4876	+	2.78656i	0.901032	+	0.135809i	0.317841	+	0.948144i	$0.397042\pi$
$422$		0			0
$423$	13.1071	+	22.7022i	0.637291	+	1.10382i
$424$		0			0
$425$	−2.44616	−	0.754540i	−0.118656	−	0.0366006i
$426$		0			0
$427$	0.350557	−	4.67786i	0.0169647	−	0.226378i
$428$		0			0
$429$	0.474179			0.0228936
$430$		0			0
$431$	−22.1719			−1.06798			−0.533990	−	0.845491i	$-0.679308\pi$
$431$	−22.1719			−1.06798			−0.533990	+	0.845491i	$0.679308\pi$
$432$		0			0
$433$	−1.53634	+	20.5011i	−0.0738319	+	0.985218i	0.830120	+	0.557584i	$0.188272\pi$
$433$	−1.53634	+	20.5011i	−0.0738319	+	0.985218i	−0.903952	+	0.427634i	$0.859347\pi$
$434$		0			0
$435$	0.295897	+	0.0912723i	0.0141872	+	0.00437617i
$436$		0			0
$437$	−7.52290	−	13.0300i	−0.359869	−	0.623311i
$438$		0			0
$439$	1.21091	+	0.182516i	0.0577937	+	0.00871100i	0.177875	−	0.984053i	$-0.443078\pi$
$439$	1.21091	+	0.182516i	0.0577937	+	0.00871100i	−0.120082	+	0.992764i	$0.538316\pi$
$440$		0			0
$441$	7.04987	−	8.84026i	0.335708	−	0.420965i
$442$		0			0
$443$	8.94107	−	8.29610i	0.424803	−	0.394160i	−0.438594	−	0.898686i	$-0.644523\pi$
$443$	8.94107	−	8.29610i	0.424803	−	0.394160i	0.863396	+	0.504526i	$0.168333\pi$
$444$		0			0
$445$	9.33656	−	23.7892i	0.442595	−	1.12771i
$446$		0			0
$447$	0.244791	−	0.0755079i	0.0115782	−	0.00357140i
$448$		0			0
$449$	−28.4688	+	4.29098i	−1.34353	+	0.202504i	−0.781132	−	0.624366i	$-0.785357\pi$
$449$	−28.4688	+	4.29098i	−1.34353	+	0.202504i	−0.562393	+	0.826870i	$0.690119\pi$
$450$		0			0
$451$	−9.27425	+	4.46624i	−0.436707	+	0.210307i
$452$		0			0
$453$	0.0334036	+	0.0851109i	0.00156944	+	0.00399886i
$454$		0			0
$455$	−20.6785	−	14.0984i	−0.969425	−	0.660943i
$456$		0			0
$457$	−6.13055	+	26.8597i	−0.286775	+	1.25644i	0.602147	+	0.798385i	$0.294312\pi$
$457$	−6.13055	+	26.8597i	−0.286775	+	1.25644i	−0.888922	+	0.458058i	$0.848545\pi$
$458$		0			0
$459$	−0.0275228	−	0.367266i	−0.00128465	−	0.0171425i
$460$		0			0
$461$	8.98990	−	6.12921i	0.418701	−	0.285466i	−0.335587	−	0.942009i	$-0.608935\pi$
$461$	8.98990	−	6.12921i	0.418701	−	0.285466i	0.754288	+	0.656544i	$0.227982\pi$
$462$		0			0
$463$	2.87488	+	2.66749i	0.133607	+	0.123969i	0.744146	−	0.668017i	$-0.232857\pi$
$463$	2.87488	+	2.66749i	0.133607	+	0.123969i	−0.610539	+	0.791986i	$0.709047\pi$
$464$		0			0
$465$	0.0321458	+	0.140840i	0.00149073	+	0.00653130i
$466$		0			0
$467$	−8.88428	+	15.3880i	−0.411116	+	0.712073i	−0.995012	−	0.0997552i	$-0.968194\pi$
$467$	−8.88428	+	15.3880i	−0.411116	+	0.712073i	0.583897	+	0.811828i	$0.301527\pi$
$468$		0			0
$469$	14.2459	+	17.8637i	0.657812	+	0.824871i
$470$		0			0
$471$	−0.330375	−	0.159100i	−0.0152229	−	0.00733095i
$472$		0			0
$473$	−4.65110	−	25.2883i	−0.213858	−	1.16276i
$474$		0			0
$475$	4.55140	+	2.19184i	0.208833	+	0.100568i
$476$		0			0
$477$	19.3814	+	24.3035i	0.887413	+	1.11278i
$478$		0			0
$479$	6.13237	−	10.6216i	0.280195	−	0.485312i	−0.691238	−	0.722628i	$-0.742934\pi$
$479$	6.13237	−	10.6216i	0.280195	−	0.485312i	0.971433	+	0.237315i	$0.0762675\pi$
$480$		0			0
$481$	−6.42860	−	28.1655i	−0.293119	−	1.28424i
$482$		0			0
$483$	0.0725456	+	0.0673125i	0.00330094	+	0.00306283i
$484$		0			0
$485$	29.1028	−	19.8420i	1.32149	−	0.900978i
$486$		0			0
$487$	−0.900979	−	12.0227i	−0.0408273	−	0.544802i	−0.979691	−	0.200513i	$-0.935739\pi$
$487$	−0.900979	−	12.0227i	−0.0408273	−	0.544802i	0.938864	−	0.344289i	$-0.111880\pi$
$488$		0			0
$489$	0.0930306	−	0.407594i	0.00420699	−	0.0184320i
$490$		0			0
$491$	11.6882	+	7.96886i	0.527480	+	0.359630i	0.797590	−	0.603200i	$-0.206108\pi$
$491$	11.6882	+	7.96886i	0.527480	+	0.359630i	−0.270110	+	0.962829i	$0.587060\pi$
$492$		0			0
$493$	6.46120	+	16.4629i	0.290998	+	0.741450i
$494$		0			0
$495$	−25.6916	+	12.3724i	−1.15475	+	0.556099i
$496$		0			0
$497$	−0.522614	+	0.0787713i	−0.0234424	+	0.00353338i
$498$		0			0
$499$	16.7921	−	5.17968i	0.751718	−	0.231874i	0.104863	−	0.994487i	$-0.466560\pi$
$499$	16.7921	−	5.17968i	0.751718	−	0.231874i	0.646856	+	0.762612i	$0.276084\pi$
$500$		0			0
$501$	0.0378629	−	0.0964732i	0.00169159	−	0.00431011i
$502$		0			0
$503$	1.23830	−	1.14898i	0.0552133	−	0.0512304i	−0.652084	−	0.758147i	$-0.726105\pi$
$503$	1.23830	−	1.14898i	0.0552133	−	0.0512304i	0.707297	+	0.706917i	$0.249914\pi$
$504$		0			0
$505$	−8.98971	+	11.2727i	−0.400037	+	0.501630i
$506$		0			0
$507$	0.416130	+	0.0627215i	0.0184810	+	0.00278556i
$508$		0			0
$509$	−3.91989	−	6.78945i	−0.173746	−	0.300937i	0.765981	−	0.642864i	$-0.222254\pi$
$509$	−3.91989	−	6.78945i	−0.173746	−	0.300937i	−0.939727	+	0.341927i	$0.888921\pi$
$510$		0			0
$511$	11.3513	+	3.50141i	0.502151	+	0.154893i
$512$		0			0
$513$	−0.0543134	+	0.724762i	−0.00239799	+	0.0319990i
$514$		0			0
$515$	−17.2244			−0.758999
$516$		0			0
$517$	−34.2680			−1.50711
$518$		0			0
$519$	−0.00486357	+	0.0648998i	−0.000213487	+	0.00284879i
$520$		0			0
$521$	−10.5911	−	3.26692i	−0.464004	−	0.143126i	0.0539338	−	0.998545i	$-0.482824\pi$
$521$	−10.5911	−	3.26692i	−0.464004	−	0.143126i	−0.517938	+	0.855418i	$0.673300\pi$
$522$		0			0
$523$	−15.2947	−	26.4913i	−0.668793	−	1.15838i	−0.978242	−	0.207467i	$-0.933478\pi$
$523$	−15.2947	−	26.4913i	−0.668793	−	1.15838i	0.309449	−	0.950916i	$-0.399855\pi$
$524$		0			0
$525$	−0.0328563	−	0.00495229i	−0.00143397	−	0.000216136i
$526$		0			0
$527$	−5.14422	+	6.45065i	−0.224086	+	0.280995i
$528$		0			0
$529$	11.8471	−	10.9925i	0.515091	−	0.477935i
$530$		0			0
$531$	9.59973	−	24.4597i	0.416593	−	1.06146i
$532$		0			0
$533$	−14.4077	+	4.44418i	−0.624066	+	0.192499i
$534$		0			0
$535$	−10.5504	+	1.59021i	−0.456132	+	0.0687509i
$536$		0			0
$537$	−0.0782606	+	0.0376883i	−0.00337720	+	0.00162637i
$538$		0			0
$539$	5.40010	+	13.7592i	0.232599	+	0.592652i
$540$		0			0
$541$	−30.2697	−	20.6375i	−1.30140	−	0.887276i	−0.303546	−	0.952817i	$-0.598170\pi$
$541$	−30.2697	−	20.6375i	−1.30140	−	0.887276i	−0.997850	+	0.0655405i	$0.979123\pi$
$542$		0			0
$543$	0.0515502	−	0.225856i	0.00221223	−	0.00969243i
$544$		0			0
$545$	3.14761	+	42.0020i	0.134829	+	1.79917i
$546$		0			0
$547$	−3.19386	+	2.17754i	−0.136560	+	0.0931047i	−0.629673	−	0.776860i	$-0.716811\pi$
$547$	−3.19386	+	2.17754i	−0.136560	+	0.0931047i	0.493113	+	0.869965i	$0.335859\pi$
$548$		0			0
$549$	−5.73888	−	5.32490i	−0.244929	−	0.227261i
$550$		0			0
$551$	−7.76604	−	34.0252i	−0.330844	−	1.44952i
$552$		0			0
$553$	12.4643	−	21.5888i	0.530036	−	0.918049i
$554$		0			0
$555$	−0.160097	−	0.200755i	−0.00679574	−	0.00852158i
$556$		0			0
$557$	−17.2943	−	8.32849i	−0.732783	−	0.352890i	0.0299952	−	0.999550i	$-0.490451\pi$
$557$	−17.2943	−	8.32849i	−0.732783	−	0.352890i	−0.762778	+	0.646660i	$0.776165\pi$
$558$		0			0
$559$	−1.21592	−	37.6425i	−0.0514281	−	1.59211i
$560$		0			0
$561$	0.216869	+	0.104438i	0.00915620	+	0.00440939i
$562$		0			0
$563$	7.52753	+	9.43922i	0.317248	+	0.397816i	0.914730	−	0.404066i	$-0.132404\pi$
$563$	7.52753	+	9.43922i	0.317248	+	0.397816i	−0.597482	+	0.801882i	$0.703832\pi$
$564$		0			0
$565$	15.8878	−	27.5185i	0.668404	−	1.15771i
$566$		0			0
$567$	3.59790	+	15.7634i	0.151098	+	0.662002i
$568$		0			0
$569$	−8.55751	−	7.94021i	−0.358749	−	0.332871i	0.480121	−	0.877202i	$-0.340593\pi$
$569$	−8.55751	−	7.94021i	−0.358749	−	0.332871i	−0.838870	+	0.544332i	$0.816783\pi$
$570$		0			0
$571$	14.8926	−	10.1536i	0.623238	−	0.424916i	−0.210100	−	0.977680i	$-0.567379\pi$
$571$	14.8926	−	10.1536i	0.623238	−	0.424916i	0.833338	+	0.552763i	$0.186427\pi$
$572$		0			0
$573$	0.0273862	+	0.365443i	0.00114407	+	0.0152666i
$574$		0			0
$575$	0.510931	−	2.23853i	0.0213073	−	0.0933533i
$576$		0			0
$577$	0.157348	+	0.107278i	0.00655047	+	0.00446604i	0.566591	−	0.823999i	$-0.308262\pi$
$577$	0.157348	+	0.107278i	0.00655047	+	0.00446604i	−0.560040	+	0.828465i	$0.689214\pi$
$578$		0			0
$579$	−0.0228715	−	0.0582755i	−0.000950505	−	0.00242185i
$580$		0			0
$581$	9.56574	−	4.60662i	0.396854	−	0.191115i
$582$		0			0
$583$	−40.1818	+	6.05643i	−1.66416	+	0.250832i
$584$		0			0
$585$	−39.9123	+	12.3113i	−1.65017	+	0.509010i
$586$		0			0
$587$	−4.43064	+	11.2891i	−0.182872	+	0.465951i	−0.992684	−	0.120739i	$-0.961474\pi$
$587$	−4.43064	+	11.2891i	−0.182872	+	0.465951i	0.809812	+	0.586689i	$0.199569\pi$
$588$		0			0
$589$	11.9355	−	11.0745i	0.491792	−	0.456317i
$590$		0			0
$591$	0.0151799	−	0.0190350i	0.000624419	−	0.000782997i
$592$		0			0
$593$	31.1516	+	4.69535i	1.27924	+	0.192815i	0.753284	−	0.657695i	$-0.228468\pi$
$593$	31.1516	+	4.69535i	1.27924	+	0.192815i	0.525959	+	0.850510i	$0.323707\pi$
$594$		0			0
$595$	−6.35227	−	11.0025i	−0.260418	−	0.451057i
$596$		0			0
$597$	−0.198161	−	0.0611246i	−0.00811020	−	0.00250166i
$598$		0			0
$599$	0.242931	−	3.24169i	0.00992589	−	0.132452i	−0.990039	−	0.140791i	$-0.955035\pi$
$599$	0.242931	−	3.24169i	0.00992589	−	0.132452i	0.999965	+	0.00833921i	$0.00265449\pi$
$600$		0			0
$601$	−2.89769			−0.118199			−0.0590996	−	0.998252i	$-0.518823\pi$
$601$	−2.89769			−0.118199			−0.0590996	+	0.998252i	$0.518823\pi$
$602$		0			0
$603$	38.1318			1.55285
$604$		0			0
$605$	0.792679	−	10.5776i	0.0322270	−	0.430039i
$606$		0			0
$607$	19.1849	+	5.91776i	0.778691	+	0.240194i	0.658524	−	0.752559i	$-0.271181\pi$
$607$	19.1849	+	5.91776i	0.778691	+	0.240194i	0.120167	+	0.992754i	$0.461657\pi$
$608$		0			0
$609$	0.114778	+	0.198802i	0.00465106	+	0.00805586i
$610$		0			0
$611$	−49.6333	−	7.48102i	−2.00795	−	0.302650i
$612$		0			0
$613$	−24.2823	+	30.4491i	−0.980753	+	1.22983i	−0.00752823	+	0.999972i	$0.502396\pi$
$613$	−24.2823	+	30.4491i	−0.980753	+	1.22983i	−0.973225	+	0.229854i	$0.926175\pi$
$614$		0			0
$615$	−0.0982366	+	0.0911502i	−0.00396128	+	0.00367553i
$616$		0			0
$617$	1.15594	−	2.94528i	0.0465363	−	0.118573i	−0.905716	−	0.423886i	$-0.860666\pi$
$617$	1.15594	−	2.94528i	0.0465363	−	0.118573i	0.952252	+	0.305313i	$0.0987611\pi$
$618$		0			0
$619$	−25.6669	+	7.91721i	−1.03164	+	0.318219i	−0.763943	−	0.645283i	$-0.776739\pi$
$619$	−25.6669	+	7.91721i	−1.03164	+	0.318219i	−0.267699	+	0.963503i	$0.586263\pi$
$620$		0			0
$621$	0.326655	−	0.0492353i	0.0131082	−	0.00197575i
$622$		0			0
$623$	17.0691	−	8.22005i	0.683859	−	0.329329i
$624$		0			0
$625$	−10.4558	−	26.6409i	−0.418231	−	1.06563i
$626$		0			0
$627$	−0.392470	−	0.267582i	−0.0156737	−	0.0106862i
$628$		0			0
$629$	3.26333	−	14.2976i	0.130118	−	0.570083i
$630$		0			0
$631$	0.799664	+	10.6708i	0.0318341	+	0.424797i	0.990239	+	0.139383i	$0.0445119\pi$
$631$	0.799664	+	10.6708i	0.0318341	+	0.424797i	−0.958404	+	0.285414i	$0.907869\pi$
$632$		0			0
$633$	−0.107440	+	0.0732511i	−0.00427034	+	0.00291147i
$634$		0			0
$635$	5.68563	+	5.27549i	0.225627	+	0.209351i
$636$		0			0
$637$	4.81766	+	21.1076i	0.190883	+	0.836312i
$638$		0			0
$639$	−0.441019	+	0.763868i	−0.0174465	+	0.0302181i
$640$		0			0
$641$	4.45675	+	5.58859i	0.176031	+	0.220736i	0.862018	−	0.506878i	$-0.169201\pi$
$641$	4.45675	+	5.58859i	0.176031	+	0.220736i	−0.685987	+	0.727614i	$0.740629\pi$
$642$		0			0
$643$	−7.28194	−	3.50680i	−0.287172	−	0.138295i	0.284752	−	0.958601i	$-0.408089\pi$
$643$	−7.28194	−	3.50680i	−0.287172	−	0.138295i	−0.571924	+	0.820306i	$0.693803\pi$
$644$		0			0
$645$	−0.154901	−	0.296747i	−0.00609923	−	0.0116844i
$646$		0			0
$647$	−7.01480	−	3.37815i	−0.275780	−	0.132809i	0.290882	−	0.956759i	$-0.406051\pi$
$647$	−7.01480	−	3.37815i	−0.275780	−	0.132809i	−0.566662	+	0.823950i	$0.691766\pi$
$648$		0			0
$649$	21.4162	+	26.8550i	0.840658	+	1.05415i
$650$		0			0
$651$	−0.0535472	+	0.0927464i	−0.00209868	+	0.00363502i
$652$		0			0
$653$	−8.39573	−	36.7841i	−0.328550	−	1.43947i	−0.821895	−	0.569639i	$-0.807083\pi$
$653$	−8.39573	−	36.7841i	−0.328550	−	1.43947i	0.493345	−	0.869834i	$-0.335774\pi$
$654$		0			0
$655$	−39.0845	−	36.2651i	−1.52716	−	1.41699i
$656$		0			0
$657$	16.3801	−	11.1677i	0.639047	−	0.435695i
$658$		0			0
$659$	1.89024	+	25.2235i	0.0736333	+	0.982569i	0.904604	+	0.426253i	$0.140167\pi$
$659$	1.89024	+	25.2235i	0.0736333	+	0.982569i	−0.830971	+	0.556316i	$0.812214\pi$
$660$		0			0
$661$	4.85084	−	21.2529i	0.188676	−	0.826643i	−0.788640	−	0.614855i	$-0.789214\pi$
$661$	4.85084	−	21.2529i	0.188676	−	0.826643i	0.977316	−	0.211788i	$-0.0679285\pi$
$662$		0			0
$663$	0.291309	+	0.198611i	0.0113135	+	0.00771342i
$664$		0			0
$665$	9.15950	+	23.3380i	0.355190	+	0.905010i
$666$		0			0
$667$	−14.2920	+	6.88269i	−0.553390	+	0.266499i
$668$		0			0
$669$	0.0887411	−	0.0133756i	0.00343093	−	0.000517129i
$670$		0			0
$671$	9.77929	−	3.01651i	0.377525	−	0.116451i
$672$		0			0
$673$	−0.0587432	+	0.149675i	−0.00226438	+	0.00576956i	−0.932001	−	0.362454i	$-0.881939\pi$
$673$	−0.0587432	+	0.149675i	−0.00226438	+	0.00576956i	0.929737	+	0.368224i	$0.120034\pi$
$674$		0			0
$675$	−0.0813059	+	0.0754409i	−0.00312947	+	0.00290372i
$676$		0			0
$677$	−8.41371	+	10.5505i	−0.323365	+	0.405487i	−0.916769	−	0.399418i	$-0.869212\pi$
$677$	−8.41371	+	10.5505i	−0.323365	+	0.405487i	0.593404	+	0.804905i	$0.297784\pi$
$678$		0			0
$679$	25.8205	+	3.89181i	0.990899	+	0.149354i
$680$		0			0
$681$	0.0137549	+	0.0238242i	0.000527089	+	0.000912945i
$682$		0			0
$683$	39.8767	+	12.3003i	1.52584	+	0.470660i	0.940404	−	0.340059i	$-0.110447\pi$
$683$	39.8767	+	12.3003i	1.52584	+	0.470660i	0.585437	+	0.810718i	$0.300923\pi$
$684$		0			0
$685$	2.97338	−	39.6770i	0.113607	−	1.51598i
$686$		0			0
$687$	−0.253871			−0.00968580
$688$		0			0
$689$	−59.5209			−2.26756
$690$		0			0
$691$	1.39044	−	18.5542i	0.0528950	−	0.705835i	−0.906218	−	0.422811i	$-0.861043\pi$
$691$	1.39044	−	18.5542i	0.0528950	−	0.705835i	0.959113	−	0.283024i	$-0.0913376\pi$
$692$		0			0
$693$	−20.2003	−	6.23097i	−0.767346	−	0.236695i
$694$		0			0
$695$	22.1901	+	38.4345i	0.841720	+	1.45790i
$696$		0			0
$697$	−7.56828	−	1.14074i	−0.286669	−	0.0432084i
$698$		0			0
$699$	−0.0416239	+	0.0521948i	−0.00157436	+	0.00197419i
$700$		0			0
$701$	21.4398	−	19.8932i	0.809770	−	0.751357i	−0.162125	−	0.986770i	$-0.551835\pi$
$701$	21.4398	−	19.8932i	0.809770	−	0.751357i	0.971895	+	0.235413i	$0.0756443\pi$
$702$		0			0
$703$	−10.5731	+	26.9399i	−0.398773	+	1.01606i
$704$		0			0
$705$	−0.426307	+	0.131498i	−0.0160557	+	0.00495252i
$706$		0			0
$707$	−10.5694	+	1.59308i	−0.397504	+	0.0599141i
$708$		0			0
$709$	20.4964	−	9.87055i	0.769758	−	0.370696i	−0.00742324	−	0.999972i	$-0.502363\pi$
$709$	20.4964	−	9.87055i	0.769758	−	0.370696i	0.777182	+	0.629276i	$0.216649\pi$
$710$		0			0
$711$	−15.1994	−	38.7273i	−0.570020	−	1.45239i
$712$		0			0
$713$	−6.11458	−	4.16885i	−0.228993	−	0.156125i
$714$		0			0
$715$	12.1497	−	53.2313i	0.454373	−	1.99074i
$716$		0			0
$717$	−0.0328467	−	0.438308i	−0.00122668	−	0.0163689i
$718$		0			0
$719$	13.1317	−	8.95305i	0.489730	−	0.333893i	−0.293148	−	0.956067i	$-0.594703\pi$
$719$	13.1317	−	8.95305i	0.489730	−	0.333893i	0.782879	+	0.622174i	$0.213751\pi$
$720$		0			0
$721$	−9.36035	−	8.68513i	−0.348598	−	0.323451i
$722$		0			0
$723$	0.000572111		0.00250658i	2.12770e−5		9.32208e-5i
$724$		0			0
$725$	2.66302	−	4.61249i	0.0989022	−	0.171304i
$726$		0			0
$727$	−16.8481	−	21.1268i	−0.624860	−	0.783550i	0.364159	−	0.931337i	$-0.381356\pi$
$727$	−16.8481	−	21.1268i	−0.624860	−	0.783550i	−0.989019	+	0.147787i	$0.952785\pi$
$728$		0			0
$729$	24.3046	+	11.7045i	0.900170	+	0.433499i
$730$		0			0
$731$	7.73469	−	17.4838i	0.286078	−	0.646663i
$732$		0			0
$733$	0.888398	+	0.427830i	0.0328137	+	0.0158023i	0.450219	−	0.892918i	$-0.351346\pi$
$733$	0.888398	+	0.427830i	0.0328137	+	0.0158023i	−0.417405	+	0.908721i	$0.637060\pi$
$734$		0			0
$735$	0.119978	+	0.150448i	0.00442547	+	0.00554936i
$736$		0			0
$737$	−24.9235	+	43.1688i	−0.918070	+	1.59014i
$738$		0			0
$739$	−4.33932	−	19.0118i	−0.159624	−	0.699360i	−0.989872	−	0.141965i	$-0.954658\pi$
$739$	−4.33932	−	19.0118i	−0.159624	−	0.699360i	0.830247	−	0.557395i	$-0.188199\pi$
$740$		0			0
$741$	−0.510032	−	0.473241i	−0.0187365	−	0.0173849i
$742$		0			0
$743$	23.0894	−	15.7421i	0.847068	−	0.577521i	−0.0601463	−	0.998190i	$-0.519157\pi$
$743$	23.0894	−	15.7421i	0.847068	−	0.577521i	0.907215	+	0.420668i	$0.138204\pi$
$744$		0			0
$745$	−2.20434	−	29.4149i	−0.0807609	−	1.07768i
$746$		0			0
$747$	3.94283	−	17.2747i	0.144261	−	0.632047i
$748$		0			0
$749$	−6.53528	−	4.45568i	−0.238794	−	0.162807i
$750$		0			0
$751$	11.3269	+	28.8604i	0.413323	+	1.05313i	0.974291	+	0.225292i	$0.0723337\pi$
$751$	11.3269	+	28.8604i	0.413323	+	1.05313i	−0.560968	+	0.827837i	$0.689571\pi$
$752$		0			0
$753$	0.326817	−	0.157387i	0.0119099	−	0.00573550i
$754$		0			0
$755$	10.4104	−	1.56912i	0.378875	−	0.0571062i
$756$		0			0
$757$	38.4184	−	11.8505i	1.39634	−	0.430713i	0.496918	−	0.867797i	$-0.334465\pi$
$757$	38.4184	−	11.8505i	1.39634	−	0.430713i	0.899421	+	0.437084i	$0.143989\pi$
$758$		0			0
$759$	−0.0788779	+	0.200978i	−0.00286309	+	0.00729503i
$760$		0			0
$761$	−32.3368	+	30.0041i	−1.17221	+	1.08765i	−0.177581	+	0.984106i	$0.556827\pi$
$761$	−32.3368	+	30.0041i	−1.17221	+	1.08765i	−0.994625	+	0.103542i	$0.966982\pi$
$762$		0			0
$763$	−19.4683	+	24.4125i	−0.704799	+	0.883790i
$764$		0			0
$765$	−20.9657	−	3.16008i	−0.758018	−	0.114253i
$766$		0			0
$767$	25.1561	+	43.5717i	0.908336	+	1.57328i
$768$		0			0
$769$	22.5974	+	6.97037i	0.814882	+	0.251358i	0.674060	−	0.738677i	$-0.264549\pi$
$769$	22.5974	+	6.97037i	0.814882	+	0.251358i	0.140823	+	0.990035i	$0.455025\pi$
$770$		0			0
$771$	−0.0340369	+	0.454191i	−0.00122581	+	0.0163573i
$772$		0			0
$773$	−21.2604			−0.764683			−0.382341	−	0.924021i	$-0.624882\pi$
$773$	−21.2604			−0.764683			−0.382341	+	0.924021i	$0.624882\pi$
$774$		0			0
$775$	2.48474			0.0892545
$776$		0			0
$777$	0.0142253	−	0.189824i	0.000510330	−	0.00680988i
$778$		0			0
$779$	14.4329	+	4.45196i	0.517112	+	0.159508i
$780$		0			0
$781$	−0.576513	−	0.998550i	−0.0206293	−	0.0357309i
$782$		0			0
$783$	0.757710	+	0.114206i	0.0270783	+	0.00408140i
$784$		0			0
$785$	−26.3257	+	33.0114i	−0.939604	+	1.17823i
$786$		0			0
$787$	−16.5242	+	15.3322i	−0.589024	+	0.546534i	−0.917376	−	0.398021i	$-0.869697\pi$
$787$	−16.5242	+	15.3322i	−0.589024	+	0.546534i	0.328352	+	0.944555i	$0.393507\pi$
$788$		0			0
$789$	0.0168837	−	0.0430189i	0.000601075	−	0.00153151i
$790$		0			0
$791$	22.5097	−	6.94333i	0.800353	−	0.246876i
$792$		0			0
$793$	14.8227	−	2.23416i	0.526370	−	0.0793375i
$794$		0			0
$795$	−0.476636	+	0.229536i	−0.0169045	+	0.00814080i
$796$		0			0
$797$	10.6397	+	27.1094i	0.376876	+	0.960264i	0.985934	+	0.167138i	$0.0534525\pi$
$797$	10.6397	+	27.1094i	0.376876	+	0.960264i	−0.609058	+	0.793126i	$0.708452\pi$
$798$		0			0
$799$	−21.0524	−	14.3533i	−0.744780	−	0.507783i
$800$		0			0
$801$	7.03559	−	30.8249i	0.248590	−	1.08915i
$802$		0			0
$803$	1.93668	+	25.8431i	0.0683438	+	0.911985i
$804$		0			0
$805$	9.41532	−	6.41926i	0.331846	−	0.226249i
$806$		0			0
$807$	−0.0785732	−	0.0729052i	−0.00276591	−	0.00256639i
$808$		0			0
$809$	9.32397	+	40.8510i	0.327813	+	1.43624i	0.823289	+	0.567622i	$0.192137\pi$
$809$	9.32397	+	40.8510i	0.327813	+	1.43624i	−0.495476	+	0.868622i	$0.665006\pi$
$810$		0			0
$811$	−9.91502	+	17.1733i	−0.348163	+	0.603037i	−0.985923	−	0.167199i	$-0.946528\pi$
$811$	−9.91502	+	17.1733i	−0.348163	+	0.603037i	0.637760	+	0.770235i	$0.279861\pi$
$812$		0			0
$813$	0.235136	+	0.294851i	0.00824658	+	0.0103409i
$814$		0			0
$815$	−43.3728	−	20.8872i	−1.51928	−	0.731648i
$816$		0			0
$817$	−20.2355	+	31.8422i	−0.707949	+	1.11402i
$818$		0			0
$819$	−27.8975	−	13.4347i	−0.974818	−	0.469448i
$820$		0			0
$821$	−5.10671	−	6.40361i	−0.178225	−	0.223488i	0.684692	−	0.728833i	$-0.259937\pi$
$821$	−5.10671	−	6.40361i	−0.178225	−	0.223488i	−0.862917	+	0.505345i	$0.831365\pi$
$822$		0			0
$823$	11.7343	−	20.3243i	0.409030	−	0.708462i	−0.585751	−	0.810491i	$-0.699200\pi$
$823$	11.7343	−	20.3243i	0.409030	−	0.708462i	0.994781	+	0.102030i	$0.0325336\pi$
$824$		0			0
$825$	−0.0161305	−	0.0706724i	−0.000561592	−	0.00246050i
$826$		0			0
$827$	11.5106	+	10.6803i	0.400263	+	0.371390i	0.854465	−	0.519509i	$-0.173885\pi$
$827$	11.5106	+	10.6803i	0.400263	+	0.371390i	−0.454202	+	0.890899i	$0.650075\pi$
$828$		0			0
$829$	−3.01056	+	2.05256i	−0.104561	+	0.0712885i	−0.614472	−	0.788939i	$-0.710631\pi$
$829$	−3.01056	+	2.05256i	−0.104561	+	0.0712885i	0.509911	+	0.860227i	$0.329678\pi$
$830$		0			0
$831$	0.0320837	+	0.428128i	0.00111297	+	0.0148516i
$832$		0			0
$833$	−2.44558	+	10.7148i	−0.0847342	+	0.371245i
$834$		0			0
$835$	−9.85994	−	6.72240i	−0.341217	−	0.232638i
$836$		0			0
$837$	0.130604	+	0.332773i	0.00451433	+	0.0115023i
$838$		0			0
$839$	−21.1982	+	10.2085i	−0.731844	+	0.352437i	−0.762409	−	0.647095i	$-0.775984\pi$
$839$	−21.1982	+	10.2085i	−0.731844	+	0.352437i	0.0305653	+	0.999533i	$0.490269\pi$
$840$		0			0
$841$	−7.70872	+	1.16190i	−0.265818	+	0.0400656i
$842$		0			0
$843$	0.0703005	−	0.0216848i	0.00242128	−	0.000746865i
$844$		0			0
$845$	17.7035	−	45.1077i	0.609018	−	1.55175i
$846$		0			0
$847$	5.76433	−	5.34852i	0.198065	−	0.183777i
$848$		0			0
$849$	0.0467689	−	0.0586464i	0.00160511	−	0.00201274i
$850$		0			0
$851$	13.0072	+	1.96052i	0.445880	+	0.0672056i
$852$		0			0
$853$	−14.0582	−	24.3495i	−0.481343	−	0.833710i	0.518428	−	0.855121i	$-0.326517\pi$
$853$	−14.0582	−	24.3495i	−0.481343	−	0.833710i	−0.999771	+	0.0214115i	$0.993184\pi$
$854$		0			0
$855$	39.9821	+	12.3329i	1.36736	+	0.421775i
$856$		0			0
$857$	1.84649	−	24.6397i	0.0630750	−	0.841678i	−0.872515	−	0.488588i	$-0.837512\pi$
$857$	1.84649	−	24.6397i	0.0630750	−	0.841678i	0.935590	−	0.353090i	$-0.114869\pi$
$858$		0			0
$859$	−41.7349			−1.42398			−0.711988	−	0.702191i	$-0.752205\pi$
$859$	−41.7349			−1.42398			−0.711988	+	0.702191i	$0.752205\pi$
$860$		0			0
$861$	−0.0993462			−0.00338571
$862$		0			0
$863$	2.57823	−	34.4041i	0.0877640	−	1.17113i	−0.763292	−	0.646054i	$-0.776418\pi$
$863$	2.57823	−	34.4041i	0.0877640	−	1.17113i	0.851056	−	0.525076i	$-0.175963\pi$
$864$		0			0
$865$	7.16104	+	2.20889i	0.243483	+	0.0751045i
$866$		0			0
$867$	−0.0894829	−	0.154989i	−0.00303900	−	0.00526370i
$868$		0			0
$869$	53.7775	+	8.10565i	1.82428	+	0.274965i
$870$		0			0
$871$	−45.5230	+	57.0840i	−1.54249	+	1.93422i
$872$		0			0
$873$	31.9452	−	29.6408i	1.08118	−	1.00319i
$874$		0			0
$875$	6.56218	−	16.7202i	0.221842	−	0.565244i
$876$		0			0
$877$	5.64788	−	1.74214i	0.190715	−	0.0588279i	−0.197926	−	0.980217i	$-0.563421\pi$
$877$	5.64788	−	1.74214i	0.190715	−	0.0588279i	0.388642	+	0.921389i	$0.372944\pi$
$878$		0			0
$879$	0.638580	−	0.0962504i	0.0215388	−	0.00324644i
$880$		0			0
$881$	2.59652	−	1.25042i	0.0874791	−	0.0421277i	−0.389633	−	0.920970i	$-0.627398\pi$
$881$	2.59652	−	1.25042i	0.0874791	−	0.0421277i	0.477112	+	0.878843i	$0.341684\pi$
$882$		0			0
$883$	−3.25164	−	8.28503i	−0.109426	−	0.278814i	0.865663	−	0.500628i	$-0.166897\pi$
$883$	−3.25164	−	8.28503i	−0.109426	−	0.278814i	−0.975089	+	0.221814i	$0.928802\pi$
$884$		0			0
$885$	0.369477	+	0.251905i	0.0124198	+	0.00846770i
$886$		0			0
$887$	−7.61588	+	33.3674i	−0.255716	+	1.12037i	0.670064	+	0.742304i	$0.266267\pi$
$887$	−7.61588	+	33.3674i	−0.255716	+	1.12037i	−0.925780	+	0.378063i	$0.876590\pi$
$888$		0			0
$889$	0.429687	+	5.73377i	0.0144112	+	0.192304i
$890$		0			0
$891$	−29.1450	+	19.8707i	−0.976395	+	0.665695i
$892$		0			0
$893$	36.8591	+	34.2003i	1.23344	+	1.14447i
$894$		0			0
$895$	2.22565	+	9.75122i	0.0743954	+	0.325947i
$896$		0			0
$897$	−0.158121	+	0.273873i	−0.00527950	+	0.00914436i
$898$		0			0
$899$	−10.7030	−	13.4211i	−0.356964	−	0.447618i
$900$		0			0
$901$	−27.2222	−	13.1095i	−0.906904	−	0.436742i
$902$		0			0
$903$	0.0654514	−	0.239369i	0.00217809	−	0.00796571i
$904$		0			0
$905$	−24.0338	−	11.5741i	−0.798911	−	0.384735i
$906$		0			0
$907$	−15.7915	−	19.8020i	−0.524349	−	0.657513i	0.447177	−	0.894445i	$-0.352429\pi$
$907$	−15.7915	−	19.8020i	−0.524349	−	0.657513i	−0.971526	+	0.236933i	$0.923858\pi$
$908$		0			0
$909$	−8.91924	+	15.4486i	−0.295833	+	0.512397i
$910$		0			0
$911$	−5.45282	−	23.8904i	−0.180660	−	0.791523i	−0.981317	−	0.192400i	$-0.938373\pi$
$911$	−5.45282	−	23.8904i	−0.180660	−	0.791523i	0.800657	−	0.599124i	$-0.204484\pi$
$912$		0			0
$913$	16.9795	+	15.7546i	0.561938	+	0.521402i
$914$		0			0
$915$	0.110083	−	0.0750531i	0.00363922	−	0.00248118i
$916$		0			0
$917$	−2.95378	−	39.4154i	−0.0975423	−	1.30161i
$918$		0			0
$919$	−0.662725	+	2.90359i	−0.0218613	+	0.0957806i	−0.984682	−	0.174362i	$-0.944214\pi$
$919$	−0.662725	+	2.90359i	−0.0218613	+	0.0957806i	0.962820	+	0.270143i	$0.0870708\pi$
$920$		0			0
$921$	−0.315808	−	0.215314i	−0.0104062	−	0.00709484i
$922$		0			0
$923$	−0.617020	−	1.57214i	−0.0203095	−	0.0517477i
$924$		0			0
$925$	−3.97915	+	1.91626i	−0.130834	+	0.0630062i
$926$		0			0
$927$	−21.0721	+	3.17611i	−0.692099	+	0.104317i
$928$		0			0
$929$	49.1278	−	15.1539i	1.61183	−	0.497184i	0.647540	−	0.762031i	$-0.275798\pi$
$929$	49.1278	−	15.1539i	1.61183	−	0.497184i	0.964290	+	0.264847i	$0.0853216\pi$
$930$		0			0
$931$	7.92361	−	20.1890i	0.259686	−	0.661668i
$932$		0			0
$933$	0.464630	−	0.431113i	0.0152113	−	0.0141140i
$934$		0			0
$935$	17.2810	−	21.6697i	0.565149	−	0.708675i
$936$		0			0
$937$	−27.2413	−	4.10597i	−0.889936	−	0.134136i	−0.311869	−	0.950125i	$-0.600955\pi$
$937$	−27.2413	−	4.10597i	−0.889936	−	0.134136i	−0.578067	+	0.815989i	$0.696193\pi$
$938$		0			0
$939$	−0.214232	−	0.371060i	−0.00699118	−	0.0121091i
$940$		0			0
$941$	−5.00466	−	1.54373i	−0.163147	−	0.0503242i	0.212106	−	0.977247i	$-0.431968\pi$
$941$	−5.00466	−	1.54373i	−0.163147	−	0.0503242i	−0.375253	+	0.926923i	$0.622444\pi$
$942$		0			0
$943$	0.513028	−	6.84588i	0.0167065	−	0.222933i
$944$		0			0
$945$	−0.550460			−0.0179065
$946$		0			0
$947$	5.92254			0.192457			0.0962283	−	0.995359i	$-0.469322\pi$
$947$	5.92254			0.192457			0.0962283	+	0.995359i	$0.469322\pi$
$948$		0			0
$949$	−2.83673	+	37.8536i	−0.0920843	+	1.22878i
$950$		0			0
$951$	−0.313188	−	0.0966056i	−0.0101558	−	0.00313265i
$952$		0			0
$953$	−6.42056	−	11.1207i	−0.207982	−	0.360236i	0.743096	−	0.669184i	$-0.233356\pi$
$953$	−6.42056	−	11.1207i	−0.207982	−	0.360236i	−0.951079	+	0.308948i	$0.900023\pi$
$954$		0			0
$955$	41.7264	+	6.28925i	1.35024	+	0.203515i
$956$		0			0
$957$	−0.312248	+	0.391547i	−0.0100936	+	0.0126569i
$958$		0			0
$959$	21.6223	−	20.0626i	0.698221	−	0.647855i
$960$		0			0
$961$	−8.39974	+	21.4022i	−0.270959	+	0.690393i
$962$		0			0
$963$	−12.6139	+	3.89089i	−0.406479	+	0.125382i
$964$		0			0
$965$	−7.12804	+	1.07438i	−0.229460	+	0.0345855i
$966$		0			0
$967$	−9.47397	+	4.56243i	−0.304662	+	0.146718i	−0.579968	−	0.814639i	$-0.696935\pi$
$967$	−9.47397	+	4.56243i	−0.304662	+	0.146718i	0.275306	+	0.961357i	$0.411221\pi$
$968$		0			0
$969$	−0.129035	−	0.328775i	−0.00414519	−	0.0105618i
$970$		0			0
$971$	37.3172	+	25.4424i	1.19757	+	0.816486i	0.986941	−	0.161081i	$-0.0514981\pi$
$971$	37.3172	+	25.4424i	1.19757	+	0.816486i	0.210624	+	0.977567i	$0.432450\pi$
$972$		0			0
$973$	−7.32105	+	32.0756i	−0.234702	+	1.02830i
$974$		0			0
$975$	−0.00793477	−	0.105882i	−0.000254116	−	0.00339094i
$976$		0			0
$977$	−7.77954	+	5.30400i	−0.248889	+	0.169690i	−0.681339	−	0.731968i	$-0.738602\pi$
$977$	−7.77954	+	5.30400i	−0.248889	+	0.169690i	0.432449	+	0.901658i	$0.357649\pi$
$978$		0			0
$979$	30.2981	+	28.1126i	0.968333	+	0.898482i
$980$		0			0
$981$	11.5957	+	50.8042i	0.370223	+	1.62205i
$982$		0			0
$983$	−23.1241	+	40.0522i	−0.737545	+	1.27747i	0.216052	+	0.976382i	$0.430682\pi$
$983$	−23.1241	+	40.0522i	−0.737545	+	1.27747i	−0.953598	+	0.301084i	$0.902651\pi$
$984$		0			0
$985$	−1.74792	−	2.19183i	−0.0556935	−	0.0698375i
$986$		0			0
$987$	−0.297976	−	0.143498i	−0.00948469	−	0.00456758i
$988$		0			0
$989$	16.1568	+	5.74633i	0.513756	+	0.182723i
$990$		0			0
$991$	44.5823	+	21.4697i	1.41620	+	0.682007i	0.976377	−	0.216076i	$-0.0693259\pi$
$991$	44.5823	+	21.4697i	1.41620	+	0.682007i	0.439826	+	0.898083i	$0.355040\pi$
$992$		0			0
$993$	0.422807	+	0.530183i	0.0134174	+	0.0168248i
$994$		0			0
$995$	−11.9393	+	20.6794i	−0.378500	+	0.655582i
$996$		0			0
$997$	7.02329	+	30.7711i	0.222430	+	0.974529i	0.955642	+	0.294530i	$0.0951630\pi$
$997$	7.02329	+	30.7711i	0.222430	+	0.974529i	−0.733212	+	0.680000i	$0.761980\pi$
$998$		0			0
$999$	−0.465792	−	0.432192i	−0.0147370	−	0.0136739i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.bg.b.513.2		24
4.3	odd	2		86.2.g.b.83.1	yes	24
12.11	even	2		774.2.z.f.685.1		24
43.14	even	21	inner	688.2.bg.b.401.2		24
172.119	even	42		3698.2.a.s.1.6		12
172.139	odd	42		3698.2.a.t.1.7		12
172.143	odd	42		86.2.g.b.57.1	✓	24
516.143	even	42		774.2.z.f.487.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
86.2.g.b.57.1	✓	24	172.143	odd	42
86.2.g.b.83.1	yes	24	4.3	odd	2
688.2.bg.b.401.2		24	43.14	even	21	inner
688.2.bg.b.513.2		24	1.1	even	1	trivial
774.2.z.f.487.1		24	516.143	even	42
774.2.z.f.685.1		24	12.11	even	2
3698.2.a.s.1.6		12	172.119	even	42
3698.2.a.t.1.7		12	172.139	odd	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)

\(f(q)\)	\(=\)	\(q+(-0.00157347 + 0.0209965i) q^{3} +(2.31675 + 0.714623i) q^{5} +(0.898666 + 1.55654i) q^{7} +(2.96605 + 0.447061i) q^{9} +O(q^{10})\)	\(q+(-0.00157347 + 0.0209965i) q^{3} +(2.31675 + 0.714623i) q^{5} +(0.898666 + 1.55654i) q^{7} +(2.96605 + 0.447061i) q^{9} +(-2.44477 + 3.06565i) q^{11} +(-4.21022 + 3.90652i) q^{13} +(-0.0186499 + 0.0475192i) q^{15} +(-2.78599 + 0.859363i) q^{17} +(5.68920 - 0.857509i) q^{19} +(-0.0340958 + 0.0164197i) q^{21} +(-0.955397 - 2.43431i) q^{23} +(0.725456 + 0.494608i) q^{25} +(-0.0281095 + 0.123156i) q^{27} +(-0.453309 - 6.04899i) q^{29} +(2.33820 - 1.59415i) q^{31} +(-0.0605210 - 0.0561553i) q^{33} +(0.969650 + 4.24831i) q^{35} +(-2.51504 + 4.35618i) q^{37} +(-0.0753985 - 0.0945467i) q^{39} +(2.36521 + 1.13903i) q^{41} +(-4.30267 + 4.94843i) q^{43} +(6.55213 + 3.15534i) q^{45} +(5.44891 + 6.83272i) q^{47} +(1.88480 - 3.26457i) q^{49} +(-0.0136599 - 0.0598481i) q^{51} +(7.59685 + 7.04884i) q^{53} +(-7.85471 + 5.35525i) q^{55} +(0.00905290 + 0.120803i) q^{57} +(1.94928 - 8.54036i) q^{59} +(-2.15646 - 1.47025i) q^{61} +(1.96963 + 5.01853i) q^{63} +(-12.5457 + 6.04170i) q^{65} +(12.5705 - 1.89470i) q^{67} +(0.0526153 - 0.0162297i) q^{69} +(-0.107431 + 0.273729i) q^{71} +(4.84493 - 4.49544i) q^{73} +(-0.0115265 + 0.0144538i) q^{75} +(-6.96882 - 1.05038i) q^{77} +(-6.93489 - 12.0116i) q^{79} +(8.59634 + 2.65162i) q^{81} +(0.441444 - 5.89067i) q^{83} -7.06856 q^{85} +0.127721 q^{87} +(0.787713 - 10.5113i) q^{89} +(-9.86422 - 3.04271i) q^{91} +(0.0297926 + 0.0516023i) q^{93} +(13.7933 + 2.07900i) q^{95} +(9.05823 - 11.3587i) q^{97} +(-8.62185 + 7.99991i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{3} + 3 q^{5} - 3 q^{7} - 4 q^{9}+O(q^{10}) \) 24 * q + 6 * q^3 + 3 * q^5 - 3 * q^7 - 4 * q^9	\( 24 q + 6 q^{3} + 3 q^{5} - 3 q^{7} - 4 q^{9} - 2 q^{11} - 12 q^{13} + 42 q^{15} + 24 q^{17} + 18 q^{19} + 14 q^{21} + 4 q^{23} - 11 q^{25} - 23 q^{29} + 22 q^{31} + 50 q^{33} - 2 q^{35} + 2 q^{37} + 5 q^{39} - 14 q^{41} - 24 q^{43} + 71 q^{45} - 26 q^{47} + 13 q^{49} + 58 q^{51} + 31 q^{53} - 60 q^{55} - 61 q^{57} + 67 q^{59} + 15 q^{61} + 67 q^{63} - 17 q^{65} + 39 q^{67} - 15 q^{69} - 56 q^{71} - 81 q^{73} - 6 q^{75} + 53 q^{77} + 4 q^{79} + 70 q^{81} - 19 q^{83} - 38 q^{85} - 130 q^{87} - 21 q^{89} - 40 q^{91} + 50 q^{93} + 17 q^{95} + 39 q^{97} + 91 q^{99}+O(q^{100}) \) 24 * q + 6 * q^3 + 3 * q^5 - 3 * q^7 - 4 * q^9 - 2 * q^11 - 12 * q^13 + 42 * q^15 + 24 * q^17 + 18 * q^19 + 14 * q^21 + 4 * q^23 - 11 * q^25 - 23 * q^29 + 22 * q^31 + 50 * q^33 - 2 * q^35 + 2 * q^37 + 5 * q^39 - 14 * q^41 - 24 * q^43 + 71 * q^45 - 26 * q^47 + 13 * q^49 + 58 * q^51 + 31 * q^53 - 60 * q^55 - 61 * q^57 + 67 * q^59 + 15 * q^61 + 67 * q^63 - 17 * q^65 + 39 * q^67 - 15 * q^69 - 56 * q^71 - 81 * q^73 - 6 * q^75 + 53 * q^77 + 4 * q^79 + 70 * q^81 - 19 * q^83 - 38 * q^85 - 130 * q^87 - 21 * q^89 - 40 * q^91 + 50 * q^93 + 17 * q^95 + 39 * q^97 + 91 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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