LMFDB - Embedded newform 69.3.b.a.47.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,3,Mod(47,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("69.47");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	69.b (of order $2$, degree $1$, 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:	$1.88011382409$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 40x^{12} + 598x^{10} + 4207x^{8} + 14465x^{6} + 23786x^{4} + 17144x^{2} + 3887 $ x^14 + 40x^12 + 598x^10 + 4207x^8 + 14465x^6 + 23786x^4 + 17144x^2 + 3887
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			47.11
Root			$1.90763i$ of defining polynomial
Character	$\chi$	$=$	69.47
Dual form			69.3.b.a.47.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90763i q^{2} +(0.570904 - 2.94518i) q^{3} +0.360941 q^{4} -9.03892i q^{5} +(5.61831 + 1.08907i) q^{6} +1.87935 q^{7} +8.31907i q^{8} +(-8.34814 - 3.36283i) q^{9} +O(q^{10})$	\(q+1.90763i q^{2} +(0.570904 - 2.94518i) q^{3} +0.360941 q^{4} -9.03892i q^{5} +(5.61831 + 1.08907i) q^{6} +1.87935 q^{7} +8.31907i q^{8} +(-8.34814 - 3.36283i) q^{9} +17.2429 q^{10} +15.0455i q^{11} +(0.206063 - 1.06304i) q^{12} +4.90839 q^{13} +3.58511i q^{14} +(-26.6212 - 5.16035i) q^{15} -14.4260 q^{16} +10.8012i q^{17} +(6.41503 - 15.9252i) q^{18} +20.4660 q^{19} -3.26252i q^{20} +(1.07293 - 5.53503i) q^{21} -28.7012 q^{22} -4.79583i q^{23} +(24.5011 + 4.74939i) q^{24} -56.7020 q^{25} +9.36339i q^{26} +(-14.6701 + 22.6669i) q^{27} +0.678336 q^{28} -18.6576i q^{29} +(9.84405 - 50.7835i) q^{30} -20.8640 q^{31} +5.75686i q^{32} +(44.3116 + 8.58951i) q^{33} -20.6047 q^{34} -16.9873i q^{35} +(-3.01319 - 1.21378i) q^{36} +18.5135 q^{37} +39.0417i q^{38} +(2.80222 - 14.4561i) q^{39} +75.1954 q^{40} +71.5589i q^{41} +(10.5588 + 2.04676i) q^{42} +37.4329 q^{43} +5.43053i q^{44} +(-30.3963 + 75.4581i) q^{45} +9.14868 q^{46} -47.3869i q^{47} +(-8.23584 + 42.4870i) q^{48} -45.4680 q^{49} -108.167i q^{50} +(31.8115 + 6.16645i) q^{51} +1.77164 q^{52} +25.0321i q^{53} +(-43.2401 - 27.9852i) q^{54} +135.995 q^{55} +15.6345i q^{56} +(11.6841 - 60.2761i) q^{57} +35.5918 q^{58} -88.7116i q^{59} +(-9.60869 - 1.86258i) q^{60} -86.1718 q^{61} -39.8008i q^{62} +(-15.6891 - 6.31994i) q^{63} -68.6858 q^{64} -44.3665i q^{65} +(-16.3856 + 84.5301i) q^{66} -18.0430 q^{67} +3.89860i q^{68} +(-14.1246 - 2.73796i) q^{69} +32.4056 q^{70} -28.8589i q^{71} +(27.9756 - 69.4487i) q^{72} -27.2950 q^{73} +35.3169i q^{74} +(-32.3714 + 166.997i) q^{75} +7.38703 q^{76} +28.2757i q^{77} +(27.5768 + 5.34560i) q^{78} -9.77378 q^{79} +130.395i q^{80} +(58.3828 + 56.1467i) q^{81} -136.508 q^{82} +42.3276i q^{83} +(0.387265 - 1.99782i) q^{84} +97.6312 q^{85} +71.4081i q^{86} +(-54.9499 - 10.6517i) q^{87} -125.164 q^{88} -106.038i q^{89} +(-143.946 - 57.9850i) q^{90} +9.22459 q^{91} -1.73101i q^{92} +(-11.9113 + 61.4482i) q^{93} +90.3967 q^{94} -184.991i q^{95} +(16.9550 + 3.28662i) q^{96} +76.9222 q^{97} -86.7363i q^{98} +(50.5953 - 125.602i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) $ 14 * q - 4 * q^3 - 24 * q^4 + 11 * q^6 - 4 * q^7 - 8 * q^9	$ 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9} - 8 q^{10} + 19 q^{12} - 14 q^{15} + 72 q^{16} - 31 q^{18} + 8 q^{19} - 2 q^{21} - 84 q^{22} - 44 q^{24} + 38 q^{25} + 62 q^{27} + 76 q^{28} + 62 q^{30} - 144 q^{31} + 90 q^{33} - 68 q^{34} + 3 q^{36} + 48 q^{37} - 78 q^{39} + 120 q^{40} - 76 q^{42} - 48 q^{43} - 18 q^{45} - 317 q^{48} - 30 q^{49} + 18 q^{51} - 6 q^{52} + 312 q^{54} + 232 q^{55} + 76 q^{57} + 66 q^{58} - 36 q^{60} - 140 q^{61} - 206 q^{63} - 346 q^{64} + 398 q^{66} + 204 q^{67} + 80 q^{70} + 384 q^{72} - 224 q^{73} - 80 q^{75} + 100 q^{76} - 341 q^{78} - 344 q^{79} - 232 q^{81} - 62 q^{82} - 330 q^{84} + 480 q^{85} + 86 q^{87} + 436 q^{88} - 514 q^{90} - 172 q^{91} + 62 q^{93} + 514 q^{94} + 609 q^{96} - 24 q^{97} + 234 q^{99}+O(q^{100}) $ 14 * q - 4 * q^3 - 24 * q^4 + 11 * q^6 - 4 * q^7 - 8 * q^9 - 8 * q^10 + 19 * q^12 - 14 * q^15 + 72 * q^16 - 31 * q^18 + 8 * q^19 - 2 * q^21 - 84 * q^22 - 44 * q^24 + 38 * q^25 + 62 * q^27 + 76 * q^28 + 62 * q^30 - 144 * q^31 + 90 * q^33 - 68 * q^34 + 3 * q^36 + 48 * q^37 - 78 * q^39 + 120 * q^40 - 76 * q^42 - 48 * q^43 - 18 * q^45 - 317 * q^48 - 30 * q^49 + 18 * q^51 - 6 * q^52 + 312 * q^54 + 232 * q^55 + 76 * q^57 + 66 * q^58 - 36 * q^60 - 140 * q^61 - 206 * q^63 - 346 * q^64 + 398 * q^66 + 204 * q^67 + 80 * q^70 + 384 * q^72 - 224 * q^73 - 80 * q^75 + 100 * q^76 - 341 * q^78 - 344 * q^79 - 232 * q^81 - 62 * q^82 - 330 * q^84 + 480 * q^85 + 86 * q^87 + 436 * q^88 - 514 * q^90 - 172 * q^91 + 62 * q^93 + 514 * q^94 + 609 * q^96 - 24 * q^97 + 234 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.90763i			0.953816i	0.878953	+	0.476908i	$0.158242\pi$
$2$			1.90763i			0.953816i	−0.878953	+	0.476908i	$0.841758\pi$
$3$	0.570904	−	2.94518i	0.190301	−	0.981726i
$3$	0.570904	−	2.94518i	0.190301	−	0.981726i
$4$	0.360941			0.0902353
$5$		−	9.03892i		−	1.80778i	−0.427762	−	0.903892i	$-0.640698\pi$
$5$		−	9.03892i		−	1.80778i	0.427762	−	0.903892i	$-0.359302\pi$
$6$	5.61831	+	1.08907i	0.936386	+	0.181512i
$7$	1.87935			0.268479			0.134240	−	0.990949i	$-0.457141\pi$
$7$	1.87935			0.268479			0.134240	+	0.990949i	$0.457141\pi$
$8$			8.31907i			1.03988i
$9$	−8.34814	−	3.36283i	−0.927571	−	0.373647i
$10$	17.2429			1.72429
$11$			15.0455i			1.36777i	0.729590	+	0.683885i	$0.239711\pi$
$11$			15.0455i			1.36777i	−0.729590	+	0.683885i	$0.760289\pi$
$12$	0.206063	−	1.06304i	0.0171719	−	0.0885863i
$13$	4.90839			0.377568			0.188784	−	0.982019i	$-0.439545\pi$
$13$	4.90839			0.377568			0.188784	+	0.982019i	$0.439545\pi$
$14$			3.58511i			0.256080i
$15$	−26.6212	−	5.16035i	−1.77475	−	0.344024i
$16$	−14.4260			−0.901622
$17$			10.8012i			0.635365i	0.948197	+	0.317683i	$0.102905\pi$
$17$			10.8012i			0.635365i	−0.948197	+	0.317683i	$0.897095\pi$
$18$	6.41503	−	15.9252i	0.356391	−	0.884732i
$19$	20.4660			1.07716			0.538580	−	0.842574i	$-0.318961\pi$
$19$	20.4660			1.07716			0.538580	+	0.842574i	$0.318961\pi$
$20$		−	3.26252i		−	0.163126i
$21$	1.07293	−	5.53503i	0.0510919	−	0.263573i
$22$	−28.7012			−1.30460
$23$		−	4.79583i		−	0.208514i
$23$		−	4.79583i		−	0.208514i
$24$	24.5011	+	4.74939i	1.02088	+	0.197891i
$25$	−56.7020			−2.26808
$26$			9.36339i			0.360130i
$27$	−14.6701	+	22.6669i	−0.543337	+	0.839515i
$28$	0.678336			0.0242263
$29$		−	18.6576i		−	0.643365i	−0.946848	−	0.321683i	$-0.895752\pi$
$29$		−	18.6576i		−	0.643365i	0.946848	−	0.321683i	$-0.104248\pi$
$30$	9.84405	−	50.7835i	0.328135	−	1.69278i
$31$	−20.8640			−0.673032			−0.336516	−	0.941678i	$-0.609249\pi$
$31$	−20.8640			−0.673032			−0.336516	+	0.941678i	$0.609249\pi$
$32$			5.75686i			0.179902i
$33$	44.3116	+	8.58951i	1.34277	+	0.260288i
$34$	−20.6047			−0.606021
$35$		−	16.9873i		−	0.485352i
$36$	−3.01319	−	1.21378i	−0.0836996	−	0.0337162i
$37$	18.5135			0.500364			0.250182	−	0.968199i	$-0.419510\pi$
$37$	18.5135			0.500364			0.250182	+	0.968199i	$0.419510\pi$
$38$			39.0417i			1.02741i
$39$	2.80222	−	14.4561i	0.0718517	−	0.370668i
$40$	75.1954			1.87988
$41$			71.5589i			1.74534i	0.488311	+	0.872670i	$0.337613\pi$
$41$			71.5589i			1.74534i	−0.488311	+	0.872670i	$0.662387\pi$
$42$	10.5588	+	2.04676i	0.251400	+	0.0487323i
$43$	37.4329			0.870532			0.435266	−	0.900302i	$-0.356654\pi$
$43$	37.4329			0.870532			0.435266	+	0.900302i	$0.356654\pi$
$44$			5.43053i			0.123421i
$45$	−30.3963	+	75.4581i	−0.675473	+	1.67685i
$46$	9.14868			0.198884
$47$		−	47.3869i		−	1.00823i	−0.863636	−	0.504116i	$-0.831818\pi$
$47$		−	47.3869i		−	1.00823i	0.863636	−	0.504116i	$-0.168182\pi$
$48$	−8.23584	+	42.4870i	−0.171580	+	0.885146i
$49$	−45.4680			−0.927919
$50$		−	108.167i		−	2.16333i
$51$	31.8115	+	6.16645i	0.623754	+	0.120911i
$52$	1.77164			0.0340700
$53$			25.0321i			0.472303i	0.971716	+	0.236152i	$0.0758862\pi$
$53$			25.0321i			0.472303i	−0.971716	+	0.236152i	$0.924114\pi$
$54$	−43.2401	−	27.9852i	−0.800742	−	0.518244i
$55$	135.995			2.47263
$56$			15.6345i			0.279187i
$57$	11.6841	−	60.2761i	0.204985	−	1.05748i
$58$	35.5918			0.613652
$59$		−	88.7116i		−	1.50359i	−0.659398	−	0.751794i	$-0.729189\pi$
$59$		−	88.7116i		−	1.50359i	0.659398	−	0.751794i	$-0.270811\pi$
$60$	−9.60869	−	1.86258i	−0.160145	−	0.0310431i
$61$	−86.1718			−1.41265			−0.706326	−	0.707887i	$-0.749649\pi$
$61$	−86.1718			−1.41265			−0.706326	+	0.707887i	$0.749649\pi$
$62$		−	39.8008i		−	0.641949i
$63$	−15.6891	−	6.31994i	−0.249033	−	0.100317i
$64$	−68.6858			−1.07322
$65$		−	44.3665i		−	0.682561i
$66$	−16.3856	+	84.5301i	−0.248267	+	1.28076i
$67$	−18.0430			−0.269298			−0.134649	−	0.990893i	$-0.542991\pi$
$67$	−18.0430			−0.269298			−0.134649	+	0.990893i	$0.542991\pi$
$68$			3.89860i			0.0573323i
$69$	−14.1246	−	2.73796i	−0.204704	−	0.0396806i
$70$	32.4056			0.462936
$71$		−	28.8589i		−	0.406463i	−0.979131	−	0.203232i	$-0.934856\pi$
$71$		−	28.8589i		−	0.406463i	0.979131	−	0.203232i	$-0.0651444\pi$
$72$	27.9756	−	69.4487i	0.388550	−	0.964566i
$73$	−27.2950			−0.373904			−0.186952	−	0.982369i	$-0.559861\pi$
$73$	−27.2950			−0.373904			−0.186952	+	0.982369i	$0.559861\pi$
$74$			35.3169i			0.477255i
$75$	−32.3714	+	166.997i	−0.431619	+	2.22663i
$76$	7.38703			0.0971978
$77$			28.2757i			0.367218i
$78$	27.5768	+	5.34560i	0.353549	+	0.0685333i
$79$	−9.77378			−0.123719			−0.0618594	−	0.998085i	$-0.519703\pi$
$79$	−9.77378			−0.123719			−0.0618594	+	0.998085i	$0.519703\pi$
$80$			130.395i			1.62994i
$81$	58.3828	+	56.1467i	0.720775	+	0.693169i
$82$	−136.508			−1.66473
$83$			42.3276i			0.509971i	0.966945	+	0.254985i	$0.0820707\pi$
$83$			42.3276i			0.509971i	−0.966945	+	0.254985i	$0.917929\pi$
$84$	0.387265	−	1.99782i	0.00461029	−	0.0237836i
$85$	97.6312			1.14860
$86$			71.4081i			0.830327i
$87$	−54.9499	−	10.6517i	−0.631608	−	0.122433i
$88$	−125.164			−1.42232
$89$		−	106.038i		−	1.19144i	−0.803191	−	0.595721i	$-0.796866\pi$
$89$		−	106.038i		−	1.19144i	0.803191	−	0.595721i	$-0.203134\pi$
$90$	−143.946	−	57.9850i	−1.59940	−	0.644277i
$91$	9.22459			0.101369
$92$		−	1.73101i		−	0.0188154i
$93$	−11.9113	+	61.4482i	−0.128079	+	0.660733i
$94$	90.3967			0.961667
$95$		−	184.991i		−	1.94727i
$96$	16.9550	+	3.28662i	0.176614	+	0.0342356i
$97$	76.9222			0.793012			0.396506	−	0.918032i	$-0.370223\pi$
$97$	76.9222			0.793012			0.396506	+	0.918032i	$0.370223\pi$
$98$		−	86.7363i		−	0.885064i
$99$	50.5953	−	125.602i	0.511063	−	1.26870i
$100$	−20.4661			−0.204661
$101$		−	56.4146i		−	0.558560i	−0.960210	−	0.279280i	$-0.909904\pi$
$101$		−	56.4146i		−	0.558560i	0.960210	−	0.279280i	$-0.0900958\pi$
$102$	−11.7633	+	60.6846i	−0.115327	+	0.594947i
$103$	172.885			1.67850			0.839248	−	0.543750i	$-0.182996\pi$
$103$	172.885			1.67850			0.839248	+	0.543750i	$0.182996\pi$
$104$			40.8332i			0.392627i
$105$	−50.0307	−	9.69813i	−0.476483	−	0.0923631i
$106$	−47.7520			−0.450490
$107$			174.974i			1.63527i	0.575735	+	0.817637i	$0.304716\pi$
$107$			174.974i			1.63527i	−0.575735	+	0.817637i	$0.695284\pi$
$108$	−5.29504	+	8.18141i	−0.0490282	+	0.0757538i
$109$	−122.382			−1.12277			−0.561384	−	0.827555i	$-0.689731\pi$
$109$	−122.382			−1.12277			−0.561384	+	0.827555i	$0.689731\pi$
$110$			259.428i			2.35843i
$111$	10.5694	−	54.5254i	0.0952199	−	0.491220i
$112$	−27.1115			−0.242067
$113$		−	154.571i		−	1.36789i	−0.729534	−	0.683944i	$-0.760263\pi$
$113$		−	154.571i		−	1.36789i	0.729534	−	0.683944i	$-0.239737\pi$
$114$	114.985	+	22.2890i	1.00864	+	0.195518i
$115$	−43.3491			−0.376949
$116$		−	6.73429i		−	0.0580542i
$117$	−40.9759	−	16.5061i	−0.350221	−	0.141077i
$118$	169.229			1.43415
$119$			20.2993i			0.170582i
$120$	42.9293	−	221.464i	0.357744	−	1.84553i
$121$	−105.366			−0.870793
$122$		−	164.384i		−	1.34741i
$123$	210.754	+	40.8533i	1.71344	+	0.332140i
$124$	−7.53067			−0.0607312
$125$			286.552i			2.29241i
$126$	12.0561	−	29.9290i	0.0956835	−	0.237532i
$127$	48.0109			0.378039			0.189019	−	0.981973i	$-0.439469\pi$
$127$	48.0109			0.378039			0.189019	+	0.981973i	$0.439469\pi$
$128$		−	108.000i		−	0.843748i
$129$	21.3706	−	110.246i	0.165663	−	0.854623i
$130$	84.6349			0.651038
$131$		−	63.8008i		−	0.487029i	−0.969897	−	0.243515i	$-0.921700\pi$
$131$		−	63.8008i		−	0.487029i	0.969897	−	0.243515i	$-0.0783003\pi$
$132$	15.9939	+	3.10031i	0.121166	+	0.0234872i
$133$	38.4629			0.289195
$134$		−	34.4193i		−	0.256861i
$135$	204.884	+	132.602i	1.51766	+	0.982236i
$136$	−89.8560			−0.660706
$137$			16.2319i			0.118481i	0.998244	+	0.0592404i	$0.0188678\pi$
$137$			16.2319i			0.118481i	−0.998244	+	0.0592404i	$0.981132\pi$
$138$	5.22302	−	26.9445i	0.0378480	−	0.195250i
$139$	−76.5178			−0.550488			−0.275244	−	0.961374i	$-0.588759\pi$
$139$	−76.5178			−0.550488			−0.275244	+	0.961374i	$0.588759\pi$
$140$		−	6.13142i		−	0.0437959i
$141$	−139.563	−	27.0534i	−0.989807	−	0.191868i
$142$	55.0521			0.387691
$143$			73.8489i			0.516426i
$144$	120.430	+	48.5120i	0.836319	+	0.336889i
$145$	−168.644			−1.16306
$146$		−	52.0687i		−	0.356635i
$147$	−25.9579	+	133.911i	−0.176584	+	0.910962i
$148$	6.68227			0.0451505
$149$			119.010i			0.798727i	0.916793	+	0.399364i	$0.130769\pi$
$149$			119.010i			0.798727i	−0.916793	+	0.399364i	$0.869231\pi$
$150$	−318.570	−	61.7527i	−2.12380	−	0.411685i
$151$	20.7906			0.137686			0.0688430	−	0.997628i	$-0.478069\pi$
$151$	20.7906			0.137686			0.0688430	+	0.997628i	$0.478069\pi$
$152$			170.258i			1.12012i
$153$	36.3226	−	90.1700i	0.237403	−	0.589346i
$154$	−53.9397			−0.350258
$155$			188.588i			1.21670i
$156$	1.01144	−	5.21779i	0.00648356	−	0.0334474i
$157$	107.461			0.684467			0.342234	−	0.939615i	$-0.388817\pi$
$157$	107.461			0.684467			0.342234	+	0.939615i	$0.388817\pi$
$158$		−	18.6448i		−	0.118005i
$159$	73.7239	+	14.2909i	0.463672	+	0.0898799i
$160$	52.0358			0.325224
$161$		−	9.01306i		−	0.0559818i
$162$	−107.107	+	111.373i	−0.661155	+	0.687487i
$163$	50.1800			0.307853			0.153926	−	0.988082i	$-0.450808\pi$
$163$	50.1800			0.307853			0.153926	+	0.988082i	$0.450808\pi$
$164$			25.8286i			0.157491i
$165$	77.6399	−	400.528i	0.470545	−	2.42744i
$166$	−80.7454			−0.486418
$167$		−	104.691i		−	0.626891i	−0.949606	−	0.313446i	$-0.898517\pi$
$167$		−	104.691i		−	0.626891i	0.949606	−	0.313446i	$-0.101483\pi$
$168$	46.0463	+	8.92578i	0.274085	+	0.0531297i
$169$	−144.908			−0.857442
$170$			186.244i			1.09556i
$171$	−170.853	−	68.8237i	−0.999142	−	0.402478i
$172$	13.5111			0.0785527
$173$			139.326i			0.805354i	0.915342	+	0.402677i	$0.131920\pi$
$173$			139.326i			0.805354i	−0.915342	+	0.402677i	$0.868080\pi$
$174$	20.3195	−	104.824i	0.116779	−	0.602438i
$175$	−106.563			−0.608932
$176$		−	217.045i		−	1.23321i
$177$	−261.272	−	50.6458i	−1.47611	−	0.286135i
$178$	202.282			1.13642
$179$			5.73713i			0.0320510i	0.999872	+	0.0160255i	$0.00510129\pi$
$179$			5.73713i			0.0320510i	−0.999872	+	0.0160255i	$0.994899\pi$
$180$	−10.9713	+	27.2359i	−0.0609515	+	0.151311i
$181$	−99.8142			−0.551460			−0.275730	−	0.961235i	$-0.588920\pi$
$181$	−99.8142			−0.551460			−0.275730	+	0.961235i	$0.588920\pi$
$182$			17.5971i			0.0966875i
$183$	−49.1958	+	253.791i	−0.268830	+	1.38684i
$184$	39.8969			0.216831
$185$		−	167.342i		−	0.904549i
$186$	−117.220	−	22.7224i	−0.630218	−	0.122164i
$187$	−162.509			−0.869033
$188$		−	17.1039i		−	0.0909781i
$189$	−27.5703	+	42.5991i	−0.145875	+	0.225392i
$190$	352.894			1.85734
$191$		−	97.7010i		−	0.511523i	−0.966740	−	0.255762i	$-0.917674\pi$
$191$		−	97.7010i		−	0.511523i	0.966740	−	0.255762i	$-0.0823263\pi$
$192$	−39.2130	+	202.292i	−0.204234	+	1.05360i
$193$	−280.608			−1.45392			−0.726962	−	0.686677i	$-0.759069\pi$
$193$	−280.608			−1.45392			−0.726962	+	0.686677i	$0.759069\pi$
$194$			146.739i			0.756388i
$195$	−130.667	−	25.3290i	−0.670088	−	0.129892i
$196$	−16.4113			−0.0837310
$197$		−	85.5407i		−	0.434217i	−0.976148	−	0.217108i	$-0.930338\pi$
$197$		−	85.5407i		−	0.434217i	0.976148	−	0.217108i	$-0.0696625\pi$
$198$	239.602	+	96.5172i	1.21011	+	0.487460i
$199$	166.936			0.838875			0.419437	−	0.907784i	$-0.362227\pi$
$199$	166.936			0.838875			0.419437	+	0.907784i	$0.362227\pi$
$200$		−	471.708i		−	2.35854i
$201$	−10.3008	+	53.1397i	−0.0512477	+	0.264377i
$202$	107.618			0.532763
$203$		−	35.0642i		−	0.172730i
$204$	11.4821	+	2.22573i	0.0562846	+	0.0109104i
$205$	646.815			3.15520
$206$			329.801i			1.60098i
$207$	−16.1276	+	40.0363i	−0.0779109	+	0.193412i
$208$	−70.8082			−0.340424
$209$			307.921i			1.47331i
$210$	18.5005	−	95.4401i	0.0880974	−	0.454477i
$211$	−362.985			−1.72031			−0.860155	−	0.510033i	$-0.829633\pi$
$211$	−362.985			−1.72031			−0.860155	+	0.510033i	$0.829633\pi$
$212$			9.03510i			0.0426184i
$213$	−84.9945	−	16.4756i	−0.399035	−	0.0773505i
$214$	−333.786			−1.55975
$215$		−	338.352i		−	1.57373i
$216$	−188.567	−	122.042i	−0.872997	−	0.565008i
$217$	−39.2108			−0.180695
$218$		−	233.459i		−	1.07091i
$219$	−15.5828	+	80.3885i	−0.0711544	+	0.367071i
$220$	49.0861			0.223118
$221$			53.0165i			0.239894i
$222$	104.014	+	20.1625i	0.468533	+	0.0908222i
$223$	234.545			1.05177			0.525887	−	0.850555i	$-0.323734\pi$
$223$	234.545			1.05177			0.525887	+	0.850555i	$0.323734\pi$
$224$			10.8192i			0.0482999i
$225$	473.356	+	190.679i	2.10380	+	0.847462i
$226$	294.865			1.30471
$227$			35.5302i			0.156521i	0.996933	+	0.0782604i	$0.0249365\pi$
$227$			35.5302i			0.156521i	−0.996933	+	0.0782604i	$0.975063\pi$
$228$	4.21729	−	21.7561i	0.0184969	−	0.0954216i
$229$	−5.16193			−0.0225412			−0.0112706	−	0.999936i	$-0.503588\pi$
$229$	−5.16193			−0.0225412			−0.0112706	+	0.999936i	$0.503588\pi$
$230$		−	82.6941i		−	0.359540i
$231$	83.2771	+	16.1427i	0.360507	+	0.0698820i
$232$	155.214			0.669025
$233$		−	155.419i		−	0.667032i	−0.942744	−	0.333516i	$-0.891765\pi$
$233$		−	155.419i		−	0.667032i	0.942744	−	0.333516i	$-0.108235\pi$
$234$	31.4875	−	78.1669i	0.134562	−	0.334047i
$235$	−428.326			−1.82266
$236$		−	32.0197i		−	0.135677i
$237$	−5.57989	+	28.7855i	−0.0235438	+	0.121458i
$238$	−38.7236			−0.162704
$239$			148.044i			0.619429i	0.950830	+	0.309715i	$0.100234\pi$
$239$			148.044i			0.619429i	−0.950830	+	0.309715i	$0.899766\pi$
$240$	384.036	+	74.4430i	1.60015	+	0.310179i
$241$	−2.36786			−0.00982515			−0.00491257	−	0.999988i	$-0.501564\pi$
$241$	−2.36786			−0.00982515			−0.00491257	+	0.999988i	$0.501564\pi$
$242$		−	200.999i		−	0.830576i
$243$	198.693	−	139.893i	0.817666	−	0.575693i
$244$	−31.1029			−0.127471
$245$			410.982i			1.67748i
$246$	−77.9330	+	402.040i	−0.316801	+	1.63431i
$247$	100.455			0.406701
$248$		−	173.569i		−	0.699875i
$249$	124.662	+	24.1650i	0.500651	+	0.0970481i
$250$	−546.635			−2.18654
$251$		−	406.639i		−	1.62007i	−0.586378	−	0.810037i	$-0.699447\pi$
$251$		−	406.639i		−	1.62007i	0.586378	−	0.810037i	$-0.300553\pi$
$252$	−5.66284	−	2.28113i	−0.0224716	−	0.00905209i
$253$	72.1555			0.285200
$254$			91.5872i			0.360579i
$255$	55.7380	−	287.541i	0.218581	−	1.12761i
$256$	−68.7194			−0.268435
$257$			330.251i			1.28502i	0.766275	+	0.642512i	$0.222108\pi$
$257$			330.251i			1.28502i	−0.766275	+	0.642512i	$0.777892\pi$
$258$	210.310	+	40.7672i	0.815153	+	0.158012i
$259$	34.7933			0.134337
$260$		−	16.0137i		−	0.0615911i
$261$	−62.7422	+	155.756i	−0.240392	+	0.596767i
$262$	121.708			0.464536
$263$			178.633i			0.679213i	0.940568	+	0.339607i	$0.110294\pi$
$263$			178.633i			0.679213i	−0.940568	+	0.339607i	$0.889706\pi$
$264$	−71.4568	+	368.631i	−0.270670	+	1.39633i
$265$	226.263			0.853822
$266$			73.3731i			0.275839i
$267$	−312.302	−	60.5377i	−1.16967	−	0.226733i
$268$	−6.51244			−0.0243002
$269$			257.667i			0.957869i	0.877851	+	0.478934i	$0.158977\pi$
$269$			257.667i			0.957869i	−0.877851	+	0.478934i	$0.841023\pi$
$270$	−252.955	+	390.843i	−0.936872	+	1.44757i
$271$	133.934			0.494223			0.247111	−	0.968987i	$-0.420519\pi$
$271$	133.934			0.494223			0.247111	+	0.968987i	$0.420519\pi$
$272$		−	155.818i		−	0.572859i
$273$	5.26636	−	27.1681i	0.0192907	−	0.0995167i
$274$	−30.9644			−0.113009
$275$		−	853.108i		−	3.10221i
$276$	−5.09814	−	0.988242i	−0.0184715	−	0.00358059i
$277$	209.881			0.757692			0.378846	−	0.925460i	$-0.376321\pi$
$277$	209.881			0.757692			0.378846	+	0.925460i	$0.376321\pi$
$278$		−	145.968i		−	0.525064i
$279$	174.176	+	70.1620i	0.624285	+	0.251477i
$280$	141.319			0.504710
$281$		−	128.106i		−	0.455893i	−0.973674	−	0.227947i	$-0.926799\pi$
$281$		−	128.106i		−	0.455893i	0.973674	−	0.227947i	$-0.0732011\pi$
$282$	51.6079	−	266.234i	0.183007	−	0.944094i
$283$	25.6342			0.0905803			0.0452902	−	0.998974i	$-0.485579\pi$
$283$	25.6342			0.0905803			0.0452902	+	0.998974i	$0.485579\pi$
$284$		−	10.4164i		−	0.0366773i
$285$	−544.831	−	105.612i	−1.91169	−	0.370568i
$286$	−140.877			−0.492575
$287$			134.485i			0.468587i
$288$	19.3593	−	48.0591i	0.0672199	−	0.166872i
$289$	172.334			0.596311
$290$		−	321.711i		−	1.10935i
$291$	43.9152	−	226.550i	0.150911	−	0.778521i
$292$	−9.85188			−0.0337393
$293$		−	548.270i		−	1.87123i	−0.353025	−	0.935614i	$-0.614847\pi$
$293$		−	548.270i		−	1.87123i	0.353025	−	0.935614i	$-0.385153\pi$
$294$	−255.454	−	49.5181i	−0.868890	−	0.168429i
$295$	−801.857			−2.71816
$296$			154.015i			0.520320i
$297$	−341.034	−	220.719i	−1.14826	−	0.743160i
$298$	−227.028			−0.761839
$299$		−	23.5398i		−	0.0787284i
$300$	−11.6842	+	60.2762i	−0.0389472	+	0.200921i
$301$	70.3496			0.233720
$302$			39.6608i			0.131327i
$303$	−166.151	−	32.2073i	−0.548353	−	0.106295i
$304$	−295.242			−0.971191
$305$			778.899i			2.55377i
$306$	172.011	+	69.2901i	0.562128	+	0.226438i
$307$	−110.072			−0.358542			−0.179271	−	0.983800i	$-0.557374\pi$
$307$	−110.072			−0.358542			−0.179271	+	0.983800i	$0.557374\pi$
$308$			10.2059i			0.0331360i
$309$	98.7007	−	509.177i	0.319420	−	1.64782i
$310$	−359.756			−1.16050
$311$			248.464i			0.798919i	0.916751	+	0.399460i	$0.130802\pi$
$311$			248.464i			0.798919i	−0.916751	+	0.399460i	$0.869198\pi$
$312$	120.261	+	23.3118i	0.385452	+	0.0747174i
$313$	340.271			1.08713			0.543564	−	0.839368i	$-0.317075\pi$
$313$	340.271			1.08713			0.543564	+	0.839368i	$0.317075\pi$
$314$			204.997i			0.652856i
$315$	−57.1254	+	141.812i	−0.181351	+	0.450198i
$316$	−3.52776			−0.0111638
$317$		−	52.3363i		−	0.165099i	−0.996587	−	0.0825493i	$-0.973694\pi$
$317$		−	52.3363i		−	0.165099i	0.996587	−	0.0825493i	$-0.0263062\pi$
$318$	−27.2618	+	140.638i	−0.0857289	+	0.442258i
$319$	280.712			0.879975
$320$			620.845i			1.94014i
$321$	515.330	+	99.8935i	1.60539	+	0.311195i
$322$	17.1936			0.0533963
$323$			221.058i			0.684390i
$324$	21.0727	+	20.2656i	0.0650393	+	0.0625483i
$325$	−278.315			−0.856355
$326$			95.7250i			0.293635i
$327$	−69.8682	+	360.436i	−0.213664	+	1.10225i
$328$	−595.304			−1.81495
$329$		−	89.0567i		−	0.270689i
$330$	764.061	+	148.108i	2.31534	+	0.448813i
$331$	581.624			1.75717			0.878586	−	0.477584i	$-0.158487\pi$
$331$	581.624			1.75717			0.878586	+	0.477584i	$0.158487\pi$
$332$			15.2778i			0.0460173i
$333$	−154.553	−	62.2576i	−0.464123	−	0.186960i
$334$	199.712			0.597939
$335$			163.089i			0.486832i
$336$	−15.4780	+	79.8481i	−0.0460656	+	0.237643i
$337$	407.151			1.20816			0.604081	−	0.796923i	$-0.293540\pi$
$337$	407.151			1.20816			0.604081	+	0.796923i	$0.293540\pi$
$338$		−	276.431i		−	0.817842i
$339$	−455.240	−	88.2454i	−1.34289	−	0.260311i
$340$	35.2391			0.103644
$341$		−	313.909i		−	0.920553i
$342$	131.290	−	325.925i	0.383890	−	0.952998i
$343$	−177.539			−0.517606
$344$			311.407i			0.905252i
$345$	−24.7482	+	127.671i	−0.0717339	+	0.370060i
$346$	−265.783			−0.768159
$347$			130.165i			0.375116i	0.982254	+	0.187558i	$0.0600572\pi$
$347$			130.165i			0.375116i	−0.982254	+	0.187558i	$0.939943\pi$
$348$	−19.8337	−	3.84463i	−0.0569933	−	0.0110478i
$349$	572.203			1.63955			0.819774	−	0.572687i	$-0.194099\pi$
$349$	572.203			1.63955			0.819774	+	0.572687i	$0.194099\pi$
$350$		−	203.283i		−	0.580809i
$351$	−72.0065	+	111.258i	−0.205147	+	0.316974i
$352$	−86.6147			−0.246064
$353$			20.5209i			0.0581329i	0.999577	+	0.0290664i	$0.00925344\pi$
$353$			20.5209i			0.0581329i	−0.999577	+	0.0290664i	$0.990747\pi$
$354$	96.6136	−	498.410i	0.272920	−	1.40794i
$355$	−260.853			−0.734797
$356$		−	38.2736i		−	0.107510i
$357$	59.7850	+	11.5889i	0.167465	+	0.0324620i
$358$	−10.9443			−0.0305707
$359$			287.248i			0.800134i	0.916486	+	0.400067i	$0.131013\pi$
$359$			287.248i			0.800134i	−0.916486	+	0.400067i	$0.868987\pi$
$360$	−627.741	−	252.869i	−1.74373	−	0.702414i
$361$	57.8587			0.160273
$362$		−	190.409i		−	0.525991i
$363$	−60.1539	+	310.321i	−0.165713	+	0.854880i
$364$	3.32953			0.00914707
$365$			246.717i			0.675937i
$366$	−484.140	−	93.8475i	−1.32279	−	0.256414i
$367$	173.845			0.473691			0.236846	−	0.971547i	$-0.423886\pi$
$367$	173.845			0.473691			0.236846	+	0.971547i	$0.423886\pi$
$368$			69.1845i			0.188001i
$369$	240.640	−	597.384i	0.652142	−	1.61893i
$370$	319.226			0.862773
$371$			47.0441i			0.126804i
$372$	−4.29929	+	22.1792i	−0.0115572	+	0.0596214i
$373$	260.823			0.699257			0.349629	−	0.936888i	$-0.386308\pi$
$373$	260.823			0.699257			0.349629	+	0.936888i	$0.386308\pi$
$374$		−	310.008i		−	0.828897i
$375$	843.945	+	163.593i	2.25052	+	0.436249i
$376$	394.215			1.04844
$377$		−	91.5787i		−	0.242914i
$378$	−81.2634	−	52.5940i	−0.214983	−	0.139138i
$379$	167.051			0.440767			0.220384	−	0.975413i	$-0.429269\pi$
$379$	167.051			0.440767			0.220384	+	0.975413i	$0.429269\pi$
$380$		−	66.7708i		−	0.175713i
$381$	27.4096	−	141.401i	0.0719413	−	0.371130i
$382$	186.378			0.487899
$383$		−	235.175i		−	0.614034i	−0.951704	−	0.307017i	$-0.900669\pi$
$383$		−	235.175i		−	0.614034i	0.951704	−	0.307017i	$-0.0993309\pi$
$384$	−318.078	−	61.6575i	−0.828329	−	0.160566i
$385$	255.582			0.663850
$386$		−	535.296i		−	1.38678i
$387$	−312.495	−	125.880i	−0.807480	−	0.325272i
$388$	27.7644			0.0715577
$389$			626.052i			1.60939i	0.593689	+	0.804695i	$0.297671\pi$
$389$			626.052i			1.60939i	−0.593689	+	0.804695i	$0.702329\pi$
$390$	48.3184	−	249.265i	0.123893	−	0.639141i
$391$	51.8008			0.132483
$392$		−	378.252i		−	0.964928i
$393$	−187.905	−	36.4241i	−0.478129	−	0.0926823i
$394$	163.180			0.414163
$395$			88.3444i			0.223657i
$396$	18.2619	−	45.3348i	0.0461160	−	0.114482i
$397$	−733.268			−1.84702			−0.923512	−	0.383570i	$-0.874694\pi$
$397$	−733.268			−1.84702			−0.923512	+	0.383570i	$0.874694\pi$
$398$			318.453i			0.800132i
$399$	21.9586	−	113.280i	0.0550342	−	0.283910i
$400$	817.981			2.04495
$401$		−	309.654i		−	0.772205i	−0.922456	−	0.386102i	$-0.873821\pi$
$401$		−	309.654i		−	0.772205i	0.922456	−	0.386102i	$-0.126179\pi$
$402$	−101.371	−	19.6501i	−0.252167	−	0.0488809i
$403$	−102.409			−0.254116
$404$		−	20.3623i		−	0.0504018i
$405$	507.505	−	527.717i	1.25310	−	1.30301i
$406$	66.8896			0.164753
$407$			278.544i			0.684382i
$408$	−51.2991	+	264.642i	−0.125733	+	0.648632i
$409$	−692.773			−1.69382			−0.846911	−	0.531735i	$-0.821540\pi$
$409$	−692.773			−1.69382			−0.846911	+	0.531735i	$0.821540\pi$
$410$			1233.88i			3.00948i
$411$	47.8057	+	9.26683i	0.116316	+	0.0225470i
$412$	62.4013			0.151459
$413$		−	166.721i		−	0.403682i
$414$	−76.3744	−	30.7654i	−0.184479	−	0.0743126i
$415$	382.595			0.921916
$416$			28.2569i			0.0679253i
$417$	−43.6843	+	225.358i	−0.104759	+	0.540428i
$418$	−587.400			−1.40526
$419$			642.983i			1.53456i	0.641309	+	0.767282i	$0.278392\pi$
$419$			642.983i			1.53456i	−0.641309	+	0.767282i	$0.721608\pi$
$420$	−18.0581	−	3.50045i	−0.0429955	−	0.00833441i
$421$	189.894			0.451054			0.225527	−	0.974237i	$-0.427590\pi$
$421$	189.894			0.451054			0.225527	+	0.974237i	$0.427590\pi$
$422$		−	692.443i		−	1.64086i
$423$	−159.354	+	395.592i	−0.376723	+	0.935206i
$424$	−208.244			−0.491140
$425$		−	612.450i		−	1.44106i
$426$	31.4295	−	162.138i	0.0737781	−	0.380606i
$427$	−161.947			−0.379268
$428$			63.1554i			0.147559i
$429$	217.498	+	42.1607i	0.506989	+	0.0982766i
$430$	645.452			1.50105
$431$		−	420.623i		−	0.975925i	−0.872865	−	0.487962i	$-0.837740\pi$
$431$		−	420.623i		−	0.975925i	0.872865	−	0.487962i	$-0.162260\pi$
$432$	211.630	−	326.992i	0.489885	−	0.756925i
$433$	−193.157			−0.446090			−0.223045	−	0.974808i	$-0.571600\pi$
$433$	−193.157			−0.446090			−0.223045	+	0.974808i	$0.571600\pi$
$434$		−	74.7998i		−	0.172350i
$435$	−96.2798	+	496.688i	−0.221333	+	1.14181i
$436$	−44.1726			−0.101313
$437$		−	98.1517i		−	0.224603i
$438$	−153.352	−	29.7263i	−0.350118	−	0.0678682i
$439$	−46.0867			−0.104981			−0.0524905	−	0.998621i	$-0.516716\pi$
$439$	−46.0867			−0.104981			−0.0524905	+	0.998621i	$0.516716\pi$
$440$			1131.35i			2.57125i
$441$	379.573	+	152.901i	0.860711	+	0.346715i
$442$	−101.136			−0.228814
$443$		−	348.550i		−	0.786795i	−0.919368	−	0.393397i	$-0.871300\pi$
$443$		−	348.550i		−	0.786795i	0.919368	−	0.393397i	$-0.128700\pi$
$444$	3.81493	−	19.6805i	0.00859219	−	0.0443254i
$445$	−958.472			−2.15387
$446$			447.426i			1.00320i
$447$	350.507	+	67.9435i	0.784131	+	0.151999i
$448$	−129.085			−0.288136
$449$		−	486.012i		−	1.08243i	−0.840884	−	0.541216i	$-0.817964\pi$
$449$		−	486.012i		−	1.08243i	0.840884	−	0.541216i	$-0.182036\pi$
$450$	−363.745	+	902.989i	−0.808323	+	2.00664i
$451$	−1076.64			−2.38722
$452$		−	55.7912i		−	0.123432i
$453$	11.8694	−	61.2320i	0.0262018	−	0.135170i
$454$	−67.7785			−0.149292
$455$		−	83.3803i		−	0.183253i
$456$	501.441	+	97.2012i	1.09965	+	0.213161i
$457$	−418.489			−0.915731			−0.457866	−	0.889021i	$-0.651386\pi$
$457$	−418.489			−0.915731			−0.457866	+	0.889021i	$0.651386\pi$
$458$		−	9.84707i		−	0.0215002i
$459$	−244.830	−	158.455i	−0.533398	−	0.345217i
$460$	−15.6465			−0.0340141
$461$			456.247i			0.989690i	0.868981	+	0.494845i	$0.164775\pi$
$461$			456.247i			0.989690i	−0.868981	+	0.494845i	$0.835225\pi$
$462$	−30.7944	+	158.862i	−0.0666545	+	0.343857i
$463$	536.949			1.15972			0.579859	−	0.814717i	$-0.303108\pi$
$463$	536.949			1.15972			0.579859	+	0.814717i	$0.303108\pi$
$464$			269.154i			0.580072i
$465$	555.425	+	107.666i	1.19446	+	0.231539i
$466$	296.481			0.636226
$467$			450.722i			0.965143i	0.875856	+	0.482572i	$0.160297\pi$
$467$			450.722i			0.965143i	−0.875856	+	0.482572i	$0.839703\pi$
$468$	−14.7899	−	5.95771i	−0.0316023	−	0.0127302i
$469$	−33.9091			−0.0723008
$470$		−	817.088i		−	1.73849i
$471$	61.3501	−	316.493i	0.130255	−	0.671959i
$472$	737.998			1.56356
$473$			563.195i			1.19069i
$474$	−54.9122	−	10.6444i	−0.115848	−	0.0224565i
$475$	−1160.47			−2.44308
$476$			7.32685i			0.0153925i
$477$	84.1785	−	208.971i	0.176475	−	0.438095i
$478$	−282.413			−0.590821
$479$		−	268.531i		−	0.560607i	−0.959911	−	0.280303i	$-0.909565\pi$
$479$		−	268.531i		−	0.560607i	0.959911	−	0.280303i	$-0.0904351\pi$
$480$	29.7075	−	153.255i	0.0618905	−	0.319281i
$481$	90.8712			0.188921
$482$		−	4.51701i		−	0.00937138i
$483$	−26.5451	−	5.14559i	−0.0549587	−	0.0106534i
$484$	−38.0309			−0.0785763
$485$		−	695.293i		−	1.43359i
$486$	266.865	+	379.033i	0.549105	+	0.779903i
$487$	−627.920			−1.28936			−0.644681	−	0.764451i	$-0.723010\pi$
$487$	−627.920			−1.28936			−0.644681	+	0.764451i	$0.723010\pi$
$488$		−	716.869i		−	1.46899i
$489$	28.6480	−	147.789i	0.0585848	−	0.302227i
$490$	−784.002			−1.60000
$491$			874.666i			1.78140i	0.454594	+	0.890699i	$0.349784\pi$
$491$			874.666i			1.78140i	−0.454594	+	0.890699i	$0.650216\pi$
$492$	76.0697	+	14.7456i	0.154613	+	0.0299708i
$493$	201.525			0.408772
$494$			191.632i			0.387918i
$495$	−1135.30	−	457.327i	−2.29354	−	0.923892i
$496$	300.983			0.606821
$497$		−	54.2360i		−	0.109127i
$498$	−46.0979	+	237.810i	−0.0925660	+	0.477529i
$499$	276.685			0.554479			0.277239	−	0.960801i	$-0.410581\pi$
$499$	276.685			0.554479			0.277239	+	0.960801i	$0.410581\pi$
$500$			103.428i			0.206857i
$501$	−308.333	−	59.7684i	−0.615436	−	0.119298i
$502$	775.717			1.54525
$503$		−	697.904i		−	1.38748i	−0.720224	−	0.693742i	$-0.755961\pi$
$503$		−	697.904i		−	1.38748i	0.720224	−	0.693742i	$-0.244039\pi$
$504$	52.5760	−	130.519i	0.104318	−	0.258966i
$505$	−509.926			−1.00976
$506$			137.646i			0.272028i
$507$	−82.7284	+	426.779i	−0.163172	+	0.841773i
$508$	17.3291			0.0341124
$509$		−	689.468i		−	1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$509$		−	689.468i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$510$	548.523	+	106.328i	1.07553	+	0.208486i
$511$	−51.2969			−0.100385
$512$		−	563.090i		−	1.09979i
$513$	−300.239	+	463.901i	−0.585261	+	0.904291i
$514$	−629.998			−1.22568
$515$		−	1562.69i		−	3.03436i
$516$	7.71352	−	39.7925i	0.0149487	−	0.0771172i
$517$	712.958			1.37903
$518$			66.3729i			0.128133i
$519$	410.340	+	79.5419i	0.790637	+	0.153260i
$520$	369.088			0.709784
$521$		−	38.9691i		−	0.0747968i	−0.999300	−	0.0373984i	$-0.988093\pi$
$521$		−	38.9691i		−	0.0747968i	0.999300	−	0.0373984i	$-0.0119071\pi$
$522$	−297.125	−	119.689i	−0.569206	−	0.229289i
$523$	762.538			1.45801			0.729004	−	0.684510i	$-0.239984\pi$
$523$	762.538			1.45801			0.729004	+	0.684510i	$0.239984\pi$
$524$		−	23.0283i		−	0.0439472i
$525$	−60.8373	+	313.847i	−0.115881	+	0.597804i
$526$	−340.766			−0.647844
$527$		−	225.356i		−	0.427621i
$528$	−639.237	−	123.912i	−1.21068	−	0.234682i
$529$	−23.0000			−0.0434783
$530$			431.626i			0.814389i
$531$	−298.322	+	740.577i	−0.561811	+	1.39468i
$532$	13.8828			0.0260956
$533$			351.239i			0.658985i
$534$	115.484	−	595.757i	0.216262	−	1.11565i
$535$	1581.58			2.95622
$536$		−	150.101i		−	0.280038i
$537$	16.8969	+	3.27535i	0.0314653	+	0.00609935i
$538$	−491.533			−0.913631
$539$		−	684.088i		−	1.26918i
$540$	73.9511	+	47.8615i	0.136946	+	0.0886323i
$541$	706.247			1.30545			0.652723	−	0.757596i	$-0.273626\pi$
$541$	706.247			1.30545			0.652723	+	0.757596i	$0.273626\pi$
$542$			255.497i			0.471397i
$543$	−56.9843	+	293.970i	−0.104943	+	0.541382i
$544$	−62.1811			−0.114303
$545$			1106.20i			2.02972i
$546$	51.8267	+	10.0463i	0.0949206	+	0.0183998i
$547$	110.826			0.202606			0.101303	−	0.994856i	$-0.467699\pi$
$547$	110.826			0.202606			0.101303	+	0.994856i	$0.467699\pi$
$548$			5.85875i			0.0106911i
$549$	719.374	+	289.781i	1.31033	+	0.527834i
$550$	1627.42			2.95894
$551$		−	381.847i		−	0.693007i
$552$	22.7773	−	117.503i	0.0412632	−	0.212868i
$553$	−18.3684			−0.0332159
$554$			400.375i			0.722698i
$555$	−492.851	−	95.5360i	−0.888019	−	0.172137i
$556$	−27.6184			−0.0496734
$557$			5.93624i			0.0106575i	0.999986	+	0.00532876i	$0.00169620\pi$
$557$			5.93624i			0.0106575i	−0.999986	+	0.00532876i	$0.998304\pi$
$558$	−133.843	+	332.263i	−0.239863	+	0.595453i
$559$	183.735			0.328685
$560$			245.058i			0.437604i
$561$	−92.7771	+	478.618i	−0.165378	+	0.853152i
$562$	244.379			0.434838
$563$			248.803i			0.441924i	0.975282	+	0.220962i	$0.0709197\pi$
$563$			248.803i			0.441924i	−0.975282	+	0.220962i	$0.929080\pi$
$564$	−50.3739	−	9.76467i	−0.0893155	−	0.0173132i
$565$	−1397.16			−2.47285
$566$			48.9007i			0.0863970i
$567$	109.722	+	105.519i	0.193513	+	0.186101i
$568$	240.079			0.422674
$569$			231.447i			0.406761i	0.979100	+	0.203380i	$0.0651928\pi$
$569$			231.447i			0.406761i	−0.979100	+	0.203380i	$0.934807\pi$
$570$	201.469	−	1039.34i	0.353454	−	1.82340i
$571$	826.932			1.44822			0.724108	−	0.689686i	$-0.242252\pi$
$571$	826.932			1.44822			0.724108	+	0.689686i	$0.242252\pi$
$572$			26.6551i			0.0465999i
$573$	−287.747	−	55.7779i	−0.502176	−	0.0973436i
$574$	−256.547			−0.446946
$575$			271.933i			0.472927i
$576$	573.399	+	230.978i	0.995484	+	0.401004i
$577$	−414.273			−0.717978			−0.358989	−	0.933342i	$-0.616878\pi$
$577$	−414.273			−0.717978			−0.358989	+	0.933342i	$0.616878\pi$
$578$			328.750i			0.568771i
$579$	−160.200	+	826.439i	−0.276684	+	1.42736i
$580$	−60.8707			−0.104949
$581$			79.5485i			0.136916i
$582$	432.173	+	83.7740i	0.742565	+	0.143942i
$583$	−376.619			−0.646002
$584$		−	227.069i		−	0.388816i
$585$	−149.197	+	370.377i	−0.255037	+	0.633124i
$586$	1045.90			1.78481
$587$			701.075i			1.19434i	0.802116	+	0.597168i	$0.203707\pi$
$587$			701.075i			1.19434i	−0.802116	+	0.597168i	$0.796293\pi$
$588$	−9.36926	+	48.3341i	−0.0159341	+	0.0822009i
$589$	−427.003			−0.724963
$590$		−	1529.65i		−	2.59262i
$591$	−251.932	−	48.8355i	−0.426282	−	0.0826320i
$592$	−267.074			−0.451139
$593$		−	769.569i		−	1.29776i	−0.760893	−	0.648878i	$-0.775239\pi$
$593$		−	769.569i		−	1.29776i	0.760893	−	0.648878i	$-0.224761\pi$
$594$	421.050	−	650.567i	0.708838	−	1.09523i
$595$	183.484			0.308376
$596$			42.9557i			0.0720734i
$597$	95.3045	−	491.656i	0.159639	−	0.823545i
$598$	44.9053			0.0750924
$599$		−	777.924i		−	1.29870i	−0.760488	−	0.649352i	$-0.775040\pi$
$599$		−	777.924i		−	1.29870i	0.760488	−	0.649352i	$-0.224960\pi$
$600$	−1389.26	−	269.300i	−2.31544	−	0.448833i
$601$	247.428			0.411694			0.205847	−	0.978584i	$-0.434005\pi$
$601$	247.428			0.411694			0.205847	+	0.978584i	$0.434005\pi$
$602$			134.201i			0.222925i
$603$	150.625	+	60.6753i	0.249793	+	0.100622i
$604$	7.50418			0.0124241
$605$			952.394i			1.57421i
$606$	61.4397	−	316.955i	0.101386	−	0.523027i
$607$	24.7130			0.0407133			0.0203567	−	0.999793i	$-0.493520\pi$
$607$	24.7130			0.0407133			0.0203567	+	0.999793i	$0.493520\pi$
$608$			117.820i			0.193783i
$609$	−103.270	−	20.0183i	−0.169574	−	0.0328708i
$610$	−1485.85			−2.43583
$611$		−	232.593i		−	0.380676i
$612$	13.1103	−	32.5460i	0.0214221	−	0.0531798i
$613$	−36.4556			−0.0594708			−0.0297354	−	0.999558i	$-0.509466\pi$
$613$	−36.4556			−0.0594708			−0.0297354	+	0.999558i	$0.509466\pi$
$614$		−	209.977i		−	0.341983i
$615$	369.269	−	1904.98i	0.600438	−	3.09754i
$616$	−235.228			−0.381864
$617$		−	78.7615i		−	0.127652i	−0.997961	−	0.0638261i	$-0.979670\pi$
$617$		−	78.7615i		−	0.127652i	0.997961	−	0.0638261i	$-0.0203303\pi$
$618$	971.322	+	188.285i	1.57172	+	0.304668i
$619$	242.181			0.391245			0.195623	−	0.980679i	$-0.437327\pi$
$619$	242.181			0.391245			0.195623	+	0.980679i	$0.437327\pi$
$620$			68.0691i			0.109789i
$621$	108.707	+	70.3554i	0.175051	+	0.113294i
$622$	−473.978			−0.762022
$623$		−	199.284i		−	0.319877i
$624$	−40.4247	+	208.543i	−0.0647831	+	0.334203i
$625$	1172.57			1.87611
$626$			649.111i			1.03692i
$627$	906.882	+	175.793i	1.44638	+	0.280372i
$628$	38.7872			0.0617631
$629$			199.968i			0.317914i
$630$	−270.526	−	108.974i	−0.429406	−	0.172975i
$631$	−89.5586			−0.141931			−0.0709656	−	0.997479i	$-0.522608\pi$
$631$	−89.5586			−0.141931			−0.0709656	+	0.997479i	$0.522608\pi$
$632$		−	81.3088i		−	0.128653i
$633$	−207.230	+	1069.06i	−0.327377	+	1.68887i
$634$	99.8383			0.157474
$635$		−	433.967i		−	0.683412i
$636$	26.6100	+	5.15818i	0.0418396	+	0.00811034i
$637$	−223.175			−0.350353
$638$			535.495i			0.839334i
$639$	−97.0474	+	240.918i	−0.151874	+	0.377023i
$640$	−976.201			−1.52531
$641$		−	326.284i		−	0.509024i	−0.967070	−	0.254512i	$-0.918085\pi$
$641$		−	326.284i		−	0.509024i	0.967070	−	0.254512i	$-0.0819148\pi$
$642$	−190.560	+	983.060i	−0.296822	+	1.53125i
$643$	−1098.20			−1.70793			−0.853966	−	0.520328i	$-0.825810\pi$
$643$	−1098.20			−1.70793			−0.853966	+	0.520328i	$0.825810\pi$
$644$		−	3.25319i		−	0.00505153i
$645$	−996.508	−	193.167i	−1.54497	−	0.299483i
$646$	−421.697			−0.652782
$647$			882.777i			1.36442i	0.731158	+	0.682208i	$0.238980\pi$
$647$			882.777i			1.36442i	−0.731158	+	0.682208i	$0.761020\pi$
$648$	−467.088	+	485.691i	−0.720815	+	0.749522i
$649$	1334.71			2.05656
$650$		−	530.923i		−	0.816805i
$651$	−22.3856	+	115.483i	−0.0343865	+	0.177393i
$652$	18.1120			0.0277792
$653$			916.750i			1.40390i	0.712224	+	0.701952i	$0.247688\pi$
$653$			916.750i			1.40390i	−0.712224	+	0.701952i	$0.752312\pi$
$654$	−687.579	−	133.283i	−1.05134	−	0.203796i
$655$	−576.690			−0.880443
$656$		−	1032.31i		−	1.57364i
$657$	227.862	+	91.7882i	0.346822	+	0.139708i
$658$	169.887			0.258188
$659$			795.140i			1.20659i	0.797520	+	0.603293i	$0.206145\pi$
$659$			795.140i			1.20659i	−0.797520	+	0.603293i	$0.793855\pi$
$660$	28.0234	−	144.567i	0.0424597	−	0.219041i
$661$	−110.347			−0.166940			−0.0834699	−	0.996510i	$-0.526600\pi$
$661$	−110.347			−0.166940			−0.0834699	+	0.996510i	$0.526600\pi$
$662$			1109.52i			1.67602i
$663$	156.143	+	30.2673i	0.235510	+	0.0456521i
$664$	−352.126			−0.530310
$665$		−	347.663i		−	0.522802i
$666$	118.764	−	294.830i	0.178325	−	0.442688i
$667$	−89.4787			−0.134151
$668$		−	37.7872i		−	0.0565677i
$669$	133.903	−	690.778i	0.200154	−	1.03255i
$670$	−311.113			−0.464348
$671$		−	1296.49i		−	1.93218i
$672$	31.8644	+	6.17672i	0.0474173	+	0.00919154i
$673$	−423.939			−0.629924			−0.314962	−	0.949104i	$-0.601992\pi$
$673$	−423.939			−0.629924			−0.314962	+	0.949104i	$0.601992\pi$
$674$			776.694i			1.15236i
$675$	831.824	−	1285.26i	1.23233	−	1.90409i
$676$	−52.3032			−0.0773715
$677$		−	529.594i		−	0.782266i	−0.920334	−	0.391133i	$-0.872083\pi$
$677$		−	529.594i		−	0.782266i	0.920334	−	0.391133i	$-0.127917\pi$
$678$	168.340	−	868.430i	0.248289	−	1.28087i
$679$	144.564			0.212907
$680$			812.201i			1.19441i
$681$	104.643	+	20.2843i	0.153660	+	0.0297861i
$682$	598.822			0.878038
$683$			455.664i			0.667151i	0.942723	+	0.333576i	$0.108255\pi$
$683$			455.664i			0.667151i	−0.942723	+	0.333576i	$0.891745\pi$
$684$	−61.6680	−	24.8413i	−0.0901579	−	0.0363177i
$685$	146.718			0.214187
$686$		−	338.679i		−	0.493701i
$687$	−2.94697	+	15.2028i	−0.00428962	+	0.0221293i
$688$	−540.005			−0.784891
$689$			122.867i			0.178327i
$690$	−243.549	−	47.2104i	−0.352969	−	0.0684209i
$691$	−701.751			−1.01556			−0.507779	−	0.861487i	$-0.669534\pi$
$691$	−701.751			−1.01556			−0.507779	+	0.861487i	$0.669534\pi$
$692$			50.2886i			0.0726713i
$693$	95.0864	−	236.050i	0.137210	−	0.340620i
$694$	−248.307			−0.357791
$695$			691.638i			0.995162i
$696$	88.6122	−	457.132i	0.127316	−	0.656799i
$697$	−772.923			−1.10893
$698$			1091.55i			1.56383i
$699$	−457.735	−	88.7291i	−0.654843	−	0.126937i
$700$	−38.4630			−0.0549471
$701$		−	881.282i		−	1.25718i	−0.777737	−	0.628589i	$-0.783633\pi$
$701$		−	881.282i		−	1.25718i	0.777737	−	0.628589i	$-0.216367\pi$
$702$	−212.239	−	137.362i	−0.302335	−	0.195672i
$703$	378.897			0.538972
$704$		−	1033.41i		−	1.46791i
$705$	−244.533	+	1261.50i	−0.346855	+	1.78936i
$706$	−39.1463			−0.0554481
$707$		−	106.023i		−	0.149962i
$708$	−94.3036	−	18.2802i	−0.133197	−	0.0258194i
$709$	560.134			0.790034			0.395017	−	0.918674i	$-0.370739\pi$
$709$	560.134			0.790034			0.395017	+	0.918674i	$0.370739\pi$
$710$		−	497.611i		−	0.700861i
$711$	81.5929	+	32.8675i	0.114758	+	0.0462272i
$712$	882.141			1.23896
$713$			100.060i			0.140337i
$714$	−22.1074	+	114.048i	−0.0309628	+	0.159731i
$715$	667.514			0.933586
$716$			2.07077i			0.00289213i
$717$	436.015	+	84.5187i	0.608110	+	0.117878i
$718$	−547.964			−0.763181
$719$			852.701i			1.18595i	0.805219	+	0.592977i	$0.202048\pi$
$719$			852.701i			1.18595i	−0.805219	+	0.592977i	$0.797952\pi$
$720$	438.496	−	1088.56i	0.609022	−	1.51188i
$721$	324.912			0.450641
$722$			110.373i			0.152871i
$723$	−1.35182	+	6.97377i	−0.00186974	+	0.00964560i
$724$	−36.0270			−0.0497611
$725$			1057.92i			1.45920i
$726$	−591.979	−	114.751i	−0.815398	−	0.158060i
$727$	−265.998			−0.365885			−0.182943	−	0.983124i	$-0.558562\pi$
$727$	−265.998			−0.365885			−0.182943	+	0.983124i	$0.558562\pi$
$728$			76.7400i			0.105412i
$729$	−298.576	−	665.051i	−0.409569	−	0.912279i
$730$	−470.645			−0.644719
$731$			404.320i			0.553105i
$732$	−17.7568	+	91.6037i	−0.0242579	+	0.125142i
$733$	90.2590			0.123136			0.0615682	−	0.998103i	$-0.480390\pi$
$733$	90.2590			0.123136			0.0615682	+	0.998103i	$0.480390\pi$
$734$			331.631i			0.451814i
$735$	1210.41	+	234.631i	1.64682	+	0.319226i
$736$	27.6090			0.0375122
$737$		−	271.465i		−	0.368337i
$738$	1139.59	+	459.053i	1.54416	+	0.622023i
$739$	488.130			0.660527			0.330264	−	0.943889i	$-0.392862\pi$
$739$	488.130			0.660527			0.330264	+	0.943889i	$0.392862\pi$
$740$		−	60.4005i		−	0.0816222i
$741$	57.3503	−	295.858i	0.0773958	−	0.399269i
$742$	−89.7428			−0.120947
$743$		−	558.271i		−	0.751374i	−0.926747	−	0.375687i	$-0.877407\pi$
$743$		−	558.271i		−	0.751374i	0.926747	−	0.375687i	$-0.122593\pi$
$744$	−511.192	−	99.0913i	−0.687086	−	0.133187i
$745$	1075.72			1.44393
$746$			497.554i			0.666963i
$747$	142.340	−	353.356i	0.190549	−	0.473034i
$748$	−58.6562			−0.0784174
$749$			328.838i			0.439037i
$750$	−312.076	+	1609.94i	−0.416101	+	2.14658i
$751$	416.854			0.555065			0.277532	−	0.960716i	$-0.410483\pi$
$751$	416.854			0.555065			0.277532	+	0.960716i	$0.410483\pi$
$752$			683.601i			0.909044i
$753$	−1197.62	−	232.152i	−1.59047	−	0.308302i
$754$	174.698			0.231695
$755$		−	187.924i		−	0.248906i
$756$	−9.95126	+	15.3758i	−0.0131630	+	0.0203383i
$757$	−1191.21			−1.57359			−0.786795	−	0.617214i	$-0.788261\pi$
$757$	−1191.21			−1.57359			−0.786795	+	0.617214i	$0.788261\pi$
$758$			318.671i			0.420411i
$759$	41.1939	−	212.511i	0.0542739	−	0.279988i
$760$	1538.95			2.02494
$761$		−	659.277i		−	0.866330i	−0.901315	−	0.433165i	$-0.857397\pi$
$761$		−	659.277i		−	0.866330i	0.901315	−	0.433165i	$-0.142603\pi$
$762$	269.740	+	52.2875i	0.353990	+	0.0686187i
$763$	−229.999			−0.301440
$764$		−	35.2643i		−	0.0461575i
$765$	−815.039	−	328.317i	−1.06541	−	0.429172i
$766$	448.628			0.585676
$767$		−	435.431i		−	0.567707i
$768$	−39.2322	+	202.391i	−0.0510836	+	0.263530i
$769$	−941.346			−1.22412			−0.612059	−	0.790812i	$-0.709658\pi$
$769$	−941.346			−1.22412			−0.612059	+	0.790812i	$0.709658\pi$
$770$			487.557i			0.633190i
$771$	972.649	+	188.542i	1.26154	+	0.244542i
$772$	−101.283			−0.131195
$773$		−	348.331i		−	0.450622i	−0.974287	−	0.225311i	$-0.927660\pi$
$773$		−	348.331i		−	0.450622i	0.974287	−	0.225311i	$-0.0723399\pi$
$774$	240.133	−	596.125i	0.310249	−	0.770187i
$775$	1183.03			1.52649
$776$			639.921i			0.824641i
$777$	19.8637	−	102.473i	0.0255645	−	0.131882i
$778$	−1194.28			−1.53506
$779$			1464.53i			1.88001i
$780$	−47.1631	−	9.14228i	−0.0604656	−	0.0117209i
$781$	434.195			0.555948
$782$			98.8168i			0.126364i
$783$	422.910	+	273.709i	0.540114	+	0.349564i
$784$	655.920			0.836632
$785$		−	971.334i		−	1.23737i
$786$	69.4838	−	358.453i	0.0884018	−	0.456047i
$787$	−413.089			−0.524891			−0.262446	−	0.964947i	$-0.584529\pi$
$787$	−413.089			−0.524891			−0.262446	+	0.964947i	$0.584529\pi$
$788$		−	30.8751i		−	0.0391816i
$789$	526.106	+	101.982i	0.666801	+	0.129255i
$790$	−168.529			−0.213327
$791$		−	290.494i		−	0.367249i
$792$	1044.89	+	420.906i	1.31930	+	0.531447i
$793$	−422.964			−0.533372
$794$		−	1398.81i		−	1.76172i
$795$	129.174	−	666.384i	0.162483	−	0.838219i
$796$	60.2541			0.0756961
$797$		−	1453.09i		−	1.82319i	−0.411085	−	0.911597i	$-0.634850\pi$
$797$		−	1453.09i		−	1.82319i	0.411085	−	0.911597i	$-0.365150\pi$
$798$	216.097	+	41.8890i	0.270798	+	0.0524925i
$799$	511.836			0.640595
$800$		−	326.426i		−	0.408032i
$801$	−356.589	+	885.223i	−0.445180	+	1.10515i
$802$	590.706			0.736541
$803$		−	410.665i		−	0.511414i
$804$	−3.71798	+	19.1803i	−0.00462435	+	0.0238561i
$805$	−81.4683			−0.101203
$806$		−	195.358i		−	0.242379i
$807$	758.874	+	147.103i	0.940365	+	0.182284i
$808$	469.317			0.580837
$809$		−	890.041i		−	1.10017i	−0.835107	−	0.550087i	$-0.814594\pi$
$809$		−	890.041i		−	1.10017i	0.835107	−	0.550087i	$-0.185406\pi$
$810$	1006.69	+	968.133i	1.24283	+	1.19523i
$811$	−284.797			−0.351168			−0.175584	−	0.984464i	$-0.556181\pi$
$811$	−284.797			−0.351168			−0.175584	+	0.984464i	$0.556181\pi$
$812$		−	12.6561i		−	0.0155863i
$813$	76.4636	−	394.460i	0.0940512	−	0.485191i
$814$	−531.359			−0.652775
$815$		−	453.573i		−	0.556531i
$816$	−458.911	−	88.9570i	−0.562391	−	0.109016i
$817$	766.102			0.937702
$818$		−	1321.56i		−	1.61559i
$819$	−77.0082	−	31.0207i	−0.0940271	−	0.0378763i
$820$	233.462			0.284710
$821$			564.290i			0.687320i	0.939094	+	0.343660i	$0.111667\pi$
$821$			564.290i			0.687320i	−0.939094	+	0.343660i	$0.888333\pi$
$822$	−17.6777	+	91.1957i	−0.0215057	+	0.110944i
$823$	−391.675			−0.475912			−0.237956	−	0.971276i	$-0.576477\pi$
$823$	−391.675			−0.475912			−0.237956	+	0.971276i	$0.576477\pi$
$824$			1438.24i			1.74544i
$825$	−2512.55	−	487.043i	−3.04552	−	0.590355i
$826$	318.041			0.385038
$827$		−	254.849i		−	0.308161i	−0.988058	−	0.154081i	$-0.950758\pi$
$827$		−	254.849i		−	0.308161i	0.988058	−	0.154081i	$-0.0492416\pi$
$828$	−5.82110	+	14.4507i	−0.00703031	+	0.0174526i
$829$	−982.450			−1.18510			−0.592551	−	0.805533i	$-0.701879\pi$
$829$	−982.450			−1.18510			−0.592551	+	0.805533i	$0.701879\pi$
$830$			729.851i			0.879339i
$831$	119.822	−	618.136i	0.144190	−	0.743845i
$832$	−337.136			−0.405212
$833$		−	491.110i		−	0.589567i
$834$	−429.901	−	83.3336i	−0.515469	−	0.0999204i
$835$	−946.292			−1.13328
$836$			111.141i			0.132944i
$837$	306.077	−	472.922i	0.365683	−	0.565020i
$838$	−1226.57			−1.46369
$839$			1069.85i			1.27515i	0.770390	+	0.637573i	$0.220062\pi$
$839$			1069.85i			1.27515i	−0.770390	+	0.637573i	$0.779938\pi$
$840$	80.6794	−	416.209i	0.0960469	−	0.495486i
$841$	492.894			0.586081
$842$			362.248i			0.430223i
$843$	−377.295	−	73.1362i	−0.447562	−	0.0867571i
$844$	−131.016			−0.155233
$845$			1309.81i			1.55007i
$846$	−754.644	−	303.989i	−0.892015	−	0.359325i
$847$	−198.020			−0.233790
$848$		−	361.112i		−	0.425839i
$849$	14.6347	−	75.4974i	0.0172376	−	0.0889251i
$850$	1168.33			1.37450
$851$		−	88.7874i		−	0.104333i
$852$	−30.6780	−	5.94674i	−0.0360071	−	0.00697974i
$853$	1317.71			1.54480			0.772399	−	0.635138i	$-0.219057\pi$
$853$	1317.71			1.54480			0.772399	+	0.635138i	$0.219057\pi$
$854$		−	308.936i		−	0.361751i
$855$	−622.092	+	1544.33i	−0.727593	+	1.80623i
$856$	−1455.62			−1.70049
$857$			1041.96i			1.21582i	0.794005	+	0.607912i	$0.207993\pi$
$857$			1041.96i			1.21582i	−0.794005	+	0.607912i	$0.792007\pi$
$858$	−80.4270	+	414.906i	−0.0937378	+	0.483574i
$859$	1716.55			1.99832			0.999158	−	0.0410261i	$-0.0130627\pi$
$859$	1716.55			1.99832			0.999158	+	0.0410261i	$0.0130627\pi$
$860$		−	122.125i		−	0.142006i
$861$	396.081	+	76.7777i	0.460024	+	0.0891728i
$862$	802.395			0.930852
$863$		−	195.904i		−	0.227003i	−0.993538	−	0.113502i	$-0.963793\pi$
$863$		−	195.904i		−	0.227003i	0.993538	−	0.113502i	$-0.0362067\pi$
$864$	−130.490	−	84.4538i	−0.151030	−	0.0977475i
$865$	1259.36			1.45591
$866$		−	368.473i		−	0.425488i
$867$	98.3861	−	507.554i	0.113479	−	0.585414i
$868$	−14.1528			−0.0163051
$869$		−	147.051i		−	0.169219i
$870$	−947.497	−	183.666i	−1.08908	−	0.211111i
$871$	−88.5618			−0.101678
$872$		−	1018.10i		−	1.16755i
$873$	−642.157	−	258.676i	−0.735575	−	0.296307i
$874$	187.237			0.214230
$875$			538.532i			0.615465i
$876$	−5.62447	+	29.0155i	−0.00642063	+	0.0331227i
$877$	−1608.14			−1.83368			−0.916840	−	0.399254i	$-0.869269\pi$
$877$	−1608.14			−1.83368			−0.916840	+	0.399254i	$0.869269\pi$
$878$		−	87.9164i		−	0.100133i
$879$	−1614.75	−	313.009i	−1.83703	−	0.356097i
$880$	−1961.85			−2.22938
$881$		−	96.3410i		−	0.109354i	−0.998504	−	0.0546771i	$-0.982587\pi$
$881$		−	96.3410i		−	0.109354i	0.998504	−	0.0546771i	$-0.0174129\pi$
$882$	−291.679	+	724.086i	−0.330702	+	0.820959i
$883$	660.065			0.747526			0.373763	−	0.927524i	$-0.378067\pi$
$883$	660.065			0.747526			0.373763	+	0.927524i	$0.378067\pi$
$884$			19.1358i			0.0216469i
$885$	−457.783	+	2361.61i	−0.517269	+	2.66849i
$886$	664.905			0.750458
$887$			718.896i			0.810480i	0.914210	+	0.405240i	$0.132812\pi$
$887$			718.896i			0.810480i	−0.914210	+	0.405240i	$0.867188\pi$
$888$	453.601	+	87.9276i	0.510812	+	0.0990176i
$889$	90.2295			0.101496
$890$		−	1828.41i		−	2.05440i
$891$	−844.753	+	878.396i	−0.948095	+	0.985854i
$892$	84.6571			0.0949071
$893$		−	969.822i		−	1.08603i
$894$	−129.611	+	668.637i	−0.144979	+	0.747917i
$895$	51.8574			0.0579412
$896$		−	202.970i		−	0.226529i
$897$	−69.3289	−	13.4390i	−0.0772897	−	0.0149821i
$898$	927.131			1.03244
$899$			389.272i			0.433006i
$900$	170.854	+	68.8239i	0.189837	+	0.0764710i
$901$	−270.377			−0.300085
$902$		−	2053.83i		−	2.27697i
$903$	40.1629	−	207.192i	0.0444771	−	0.229448i
$904$	1285.89			1.42244
$905$			902.212i			0.996919i
$906$	116.808	+	22.6425i	0.128927	+	0.0249917i
$907$	348.937			0.384715			0.192358	−	0.981325i	$-0.438387\pi$
$907$	348.937			0.384715			0.192358	+	0.981325i	$0.438387\pi$
$908$			12.8243i			0.0141237i
$909$	−189.712	+	470.956i	−0.208704	+	0.518104i
$910$	159.059			0.174790
$911$			536.347i			0.588745i	0.955691	+	0.294373i	$0.0951107\pi$
$911$			536.347i			0.588745i	−0.955691	+	0.294373i	$0.904889\pi$
$912$	−168.555	+	869.541i	−0.184819	+	0.953444i
$913$	−636.838			−0.697522
$914$		−	798.323i		−	0.873439i
$915$	2294.00	+	444.677i	2.50710	+	0.485986i
$916$	−1.86315			−0.00203401
$917$		−	119.904i		−	0.130757i
$918$	302.273	−	467.045i	0.329274	−	0.508764i
$919$	−668.915			−0.727873			−0.363936	−	0.931424i	$-0.618567\pi$
$919$	−668.915			−0.727873			−0.363936	+	0.931424i	$0.618567\pi$
$920$		−	360.624i		−	0.391983i
$921$	−62.8407	+	324.182i	−0.0682310	+	0.351990i
$922$	−870.351			−0.943982
$923$		−	141.651i		−	0.153468i
$924$	30.0581	+	5.82658i	0.0325304	+	0.00630582i
$925$	−1049.75			−1.13486
$926$			1024.30i			1.10616i
$927$	−1443.27	−	581.382i	−1.55692	−	0.627165i
$928$	107.409			0.115743
$929$		−	529.279i		−	0.569730i	−0.958568	−	0.284865i	$-0.908051\pi$
$929$		−	529.279i		−	0.569730i	0.958568	−	0.284865i	$-0.0919488\pi$
$930$	−205.386	+	1059.55i	−0.220845	+	1.13930i
$931$	−930.550			−0.999517
$932$		−	56.0969i		−	0.0601898i
$933$	731.770	+	141.849i	0.784320	+	0.152035i
$934$	−859.811			−0.920569
$935$			1468.91i			1.57102i
$936$	137.315	−	340.881i	0.146704	−	0.364189i
$937$	248.737			0.265461			0.132730	−	0.991152i	$-0.457626\pi$
$937$	248.737			0.265461			0.132730	+	0.991152i	$0.457626\pi$
$938$		−	64.6861i		−	0.0689617i
$939$	194.262	−	1002.16i	0.206882	−	1.06726i
$940$	−154.600			−0.164469
$941$			908.767i			0.965746i	0.875690	+	0.482873i	$0.160407\pi$
$941$			908.767i			0.965746i	−0.875690	+	0.482873i	$0.839593\pi$
$942$	603.752	+	117.033i	0.640925	+	0.124239i
$943$	343.185			0.363928
$944$			1279.75i			1.35567i
$945$	385.050	+	249.206i	0.407460	+	0.263710i
$946$	−1074.37			−1.13570
$947$		−	671.532i		−	0.709116i	−0.935034	−	0.354558i	$-0.884631\pi$
$947$		−	671.532i		−	0.709116i	0.935034	−	0.354558i	$-0.115369\pi$
$948$	−2.01401	+	10.3899i	−0.00212449	+	0.0109598i
$949$	−133.974			−0.141174
$950$		−	2213.74i		−	2.33025i
$951$	−154.140	−	29.8790i	−0.162082	−	0.0314185i
$952$	−168.871			−0.177386
$953$			1197.49i			1.25655i	0.777991	+	0.628275i	$0.216239\pi$
$953$			1197.49i			1.25655i	−0.777991	+	0.628275i	$0.783761\pi$
$954$	398.640	+	160.582i	0.417862	+	0.168325i
$955$	−883.111			−0.924724
$956$			53.4350i			0.0558944i
$957$	160.260	−	826.747i	0.167460	−	0.863894i
$958$	512.258			0.534716
$959$			30.5054i			0.0318096i
$960$	1828.50	+	354.443i	1.90469	+	0.369211i
$961$	−525.694			−0.547028
$962$			173.349i			0.180196i
$963$	588.408	−	1460.71i	0.611016	−	1.51683i
$964$	−0.854658			−0.000886575
$965$			2536.39i			2.62838i
$966$	9.81590	−	50.6382i	0.0101614	−	0.0524205i
$967$	−426.374			−0.440925			−0.220462	−	0.975395i	$-0.570757\pi$
$967$	−426.374			−0.440925			−0.220462	+	0.975395i	$0.570757\pi$
$968$		−	876.547i		−	0.905524i
$969$	651.055	+	126.203i	0.671883	+	0.130240i
$970$	1326.36			1.36739
$971$		−	201.580i		−	0.207601i	−0.994598	−	0.103800i	$-0.966900\pi$
$971$		−	201.580i		−	0.207601i	0.994598	−	0.103800i	$-0.0331003\pi$
$972$	71.7164	−	50.4932i	0.0737823	−	0.0519478i
$973$	−143.804			−0.147794
$974$		−	1197.84i		−	1.22981i
$975$	−158.891	+	819.688i	−0.162965	+	0.840705i
$976$	1243.11			1.27368
$977$		−	747.081i		−	0.764668i	−0.924024	−	0.382334i	$-0.875120\pi$
$977$		−	747.081i		−	0.764668i	0.924024	−	0.382334i	$-0.124880\pi$
$978$	281.927	+	54.6498i	0.288269	+	0.0558791i
$979$	1595.40			1.62962
$980$			148.340i			0.151368i
$981$	1021.66	+	411.549i	1.04145	+	0.419519i
$982$	−1668.54			−1.69913
$983$		−	82.8134i		−	0.0842456i	−0.999112	−	0.0421228i	$-0.986588\pi$
$983$		−	82.8134i		−	0.0842456i	0.999112	−	0.0421228i	$-0.0134121\pi$
$984$	−339.861	+	1753.27i	−0.345387	+	1.78178i
$985$	−773.195			−0.784969
$986$			384.435i			0.389893i
$987$	−262.288	−	50.8428i	−0.265742	−	0.0515125i
$988$	36.2584			0.0366988
$989$		−	179.522i		−	0.181518i
$990$	872.411	−	2165.74i	0.881223	−	2.18761i
$991$	−1387.62			−1.40022			−0.700109	−	0.714036i	$-0.746865\pi$
$991$	−1387.62			−1.40022			−0.700109	+	0.714036i	$0.746865\pi$
$992$		−	120.111i		−	0.121080i
$993$	332.051	−	1712.99i	0.334392	−	1.72506i
$994$	103.462			0.104087
$995$		−	1508.92i		−	1.51650i
$996$	44.9957	+	8.72213i	0.0451764	+	0.00875716i
$997$	−141.507			−0.141933			−0.0709664	−	0.997479i	$-0.522608\pi$
$997$	−141.507			−0.141933			−0.0709664	+	0.997479i	$0.522608\pi$
$998$			527.813i			0.528871i
$999$	−271.594	+	419.643i	−0.271866	+	0.420063i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.3.b.a.47.11	yes	14
3.2	odd	2	inner	69.3.b.a.47.4	✓	14
4.3	odd	2		1104.3.g.b.737.6		14
12.11	even	2		1104.3.g.b.737.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.3.b.a.47.4	✓	14	3.2	odd	2	inner
69.3.b.a.47.11	yes	14	1.1	even	1	trivial
1104.3.g.b.737.5		14	12.11	even	2
1104.3.g.b.737.6		14	4.3	odd	2

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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	69.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.90763i q^{2} +(0.570904 - 2.94518i) q^{3} +0.360941 q^{4} -9.03892i q^{5} +(5.61831 + 1.08907i) q^{6} +1.87935 q^{7} +8.31907i q^{8} +(-8.34814 - 3.36283i) q^{9} +O(q^{10})\)	\(q+1.90763i q^{2} +(0.570904 - 2.94518i) q^{3} +0.360941 q^{4} -9.03892i q^{5} +(5.61831 + 1.08907i) q^{6} +1.87935 q^{7} +8.31907i q^{8} +(-8.34814 - 3.36283i) q^{9} +17.2429 q^{10} +15.0455i q^{11} +(0.206063 - 1.06304i) q^{12} +4.90839 q^{13} +3.58511i q^{14} +(-26.6212 - 5.16035i) q^{15} -14.4260 q^{16} +10.8012i q^{17} +(6.41503 - 15.9252i) q^{18} +20.4660 q^{19} -3.26252i q^{20} +(1.07293 - 5.53503i) q^{21} -28.7012 q^{22} -4.79583i q^{23} +(24.5011 + 4.74939i) q^{24} -56.7020 q^{25} +9.36339i q^{26} +(-14.6701 + 22.6669i) q^{27} +0.678336 q^{28} -18.6576i q^{29} +(9.84405 - 50.7835i) q^{30} -20.8640 q^{31} +5.75686i q^{32} +(44.3116 + 8.58951i) q^{33} -20.6047 q^{34} -16.9873i q^{35} +(-3.01319 - 1.21378i) q^{36} +18.5135 q^{37} +39.0417i q^{38} +(2.80222 - 14.4561i) q^{39} +75.1954 q^{40} +71.5589i q^{41} +(10.5588 + 2.04676i) q^{42} +37.4329 q^{43} +5.43053i q^{44} +(-30.3963 + 75.4581i) q^{45} +9.14868 q^{46} -47.3869i q^{47} +(-8.23584 + 42.4870i) q^{48} -45.4680 q^{49} -108.167i q^{50} +(31.8115 + 6.16645i) q^{51} +1.77164 q^{52} +25.0321i q^{53} +(-43.2401 - 27.9852i) q^{54} +135.995 q^{55} +15.6345i q^{56} +(11.6841 - 60.2761i) q^{57} +35.5918 q^{58} -88.7116i q^{59} +(-9.60869 - 1.86258i) q^{60} -86.1718 q^{61} -39.8008i q^{62} +(-15.6891 - 6.31994i) q^{63} -68.6858 q^{64} -44.3665i q^{65} +(-16.3856 + 84.5301i) q^{66} -18.0430 q^{67} +3.89860i q^{68} +(-14.1246 - 2.73796i) q^{69} +32.4056 q^{70} -28.8589i q^{71} +(27.9756 - 69.4487i) q^{72} -27.2950 q^{73} +35.3169i q^{74} +(-32.3714 + 166.997i) q^{75} +7.38703 q^{76} +28.2757i q^{77} +(27.5768 + 5.34560i) q^{78} -9.77378 q^{79} +130.395i q^{80} +(58.3828 + 56.1467i) q^{81} -136.508 q^{82} +42.3276i q^{83} +(0.387265 - 1.99782i) q^{84} +97.6312 q^{85} +71.4081i q^{86} +(-54.9499 - 10.6517i) q^{87} -125.164 q^{88} -106.038i q^{89} +(-143.946 - 57.9850i) q^{90} +9.22459 q^{91} -1.73101i q^{92} +(-11.9113 + 61.4482i) q^{93} +90.3967 q^{94} -184.991i q^{95} +(16.9550 + 3.28662i) q^{96} +76.9222 q^{97} -86.7363i q^{98} +(50.5953 - 125.602i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) 14 * q - 4 * q^3 - 24 * q^4 + 11 * q^6 - 4 * q^7 - 8 * q^9	\( 14 q - 4 q^{3} - 24 q^{4} + 11 q^{6} - 4 q^{7} - 8 q^{9} - 8 q^{10} + 19 q^{12} - 14 q^{15} + 72 q^{16} - 31 q^{18} + 8 q^{19} - 2 q^{21} - 84 q^{22} - 44 q^{24} + 38 q^{25} + 62 q^{27} + 76 q^{28} + 62 q^{30} - 144 q^{31} + 90 q^{33} - 68 q^{34} + 3 q^{36} + 48 q^{37} - 78 q^{39} + 120 q^{40} - 76 q^{42} - 48 q^{43} - 18 q^{45} - 317 q^{48} - 30 q^{49} + 18 q^{51} - 6 q^{52} + 312 q^{54} + 232 q^{55} + 76 q^{57} + 66 q^{58} - 36 q^{60} - 140 q^{61} - 206 q^{63} - 346 q^{64} + 398 q^{66} + 204 q^{67} + 80 q^{70} + 384 q^{72} - 224 q^{73} - 80 q^{75} + 100 q^{76} - 341 q^{78} - 344 q^{79} - 232 q^{81} - 62 q^{82} - 330 q^{84} + 480 q^{85} + 86 q^{87} + 436 q^{88} - 514 q^{90} - 172 q^{91} + 62 q^{93} + 514 q^{94} + 609 q^{96} - 24 q^{97} + 234 q^{99}+O(q^{100}) \) 14 * q - 4 * q^3 - 24 * q^4 + 11 * q^6 - 4 * q^7 - 8 * q^9 - 8 * q^10 + 19 * q^12 - 14 * q^15 + 72 * q^16 - 31 * q^18 + 8 * q^19 - 2 * q^21 - 84 * q^22 - 44 * q^24 + 38 * q^25 + 62 * q^27 + 76 * q^28 + 62 * q^30 - 144 * q^31 + 90 * q^33 - 68 * q^34 + 3 * q^36 + 48 * q^37 - 78 * q^39 + 120 * q^40 - 76 * q^42 - 48 * q^43 - 18 * q^45 - 317 * q^48 - 30 * q^49 + 18 * q^51 - 6 * q^52 + 312 * q^54 + 232 * q^55 + 76 * q^57 + 66 * q^58 - 36 * q^60 - 140 * q^61 - 206 * q^63 - 346 * q^64 + 398 * q^66 + 204 * q^67 + 80 * q^70 + 384 * q^72 - 224 * q^73 - 80 * q^75 + 100 * q^76 - 341 * q^78 - 344 * q^79 - 232 * q^81 - 62 * q^82 - 330 * q^84 + 480 * q^85 + 86 * q^87 + 436 * q^88 - 514 * q^90 - 172 * q^91 + 62 * q^93 + 514 * q^94 + 609 * q^96 - 24 * q^97 + 234 * q^99

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