LMFDB - Embedded newform 38.7.b.a.37.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,7,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	38.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:	$8.74205517755$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 5050x^{8} + 7354489x^{6} + 2475755792x^{4} + 232626987584x^{2} + 2900002611200 $ x^10 + 5050x^8 + 7354489x^6 + 2475755792x^4 + 232626987584x^2 + 2900002611200
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.6
Root			$-48.7374i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.7.b.a.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685i q^{2} -51.5658i q^{3} -32.0000 q^{4} +39.9185 q^{5} +291.700 q^{6} -25.1565 q^{7} -181.019i q^{8} -1930.03 q^{9} +O(q^{10})$	\(q+5.65685i q^{2} -51.5658i q^{3} -32.0000 q^{4} +39.9185 q^{5} +291.700 q^{6} -25.1565 q^{7} -181.019i q^{8} -1930.03 q^{9} +225.813i q^{10} -1368.34 q^{11} +1650.11i q^{12} -157.072i q^{13} -142.306i q^{14} -2058.43i q^{15} +1024.00 q^{16} -2735.86 q^{17} -10917.9i q^{18} +(-3468.77 - 5917.22i) q^{19} -1277.39 q^{20} +1297.21i q^{21} -7740.48i q^{22} +17315.8 q^{23} -9334.41 q^{24} -14031.5 q^{25} +888.533 q^{26} +61932.1i q^{27} +805.006 q^{28} -26064.3i q^{29} +11644.2 q^{30} +26833.9i q^{31} +5792.62i q^{32} +70559.4i q^{33} -15476.4i q^{34} -1004.21 q^{35} +61761.0 q^{36} -85898.9i q^{37} +(33472.9 - 19622.3i) q^{38} -8099.54 q^{39} -7226.03i q^{40} -53613.3i q^{41} -7338.14 q^{42} +111934. q^{43} +43786.8 q^{44} -77044.0 q^{45} +97952.8i q^{46} +60698.5 q^{47} -52803.4i q^{48} -117016. q^{49} -79374.2i q^{50} +141077. i q^{51} +5026.30i q^{52} +1359.96i q^{53} -350341. q^{54} -54622.0 q^{55} +4553.80i q^{56} +(-305126. + 178870. i) q^{57} +147442. q^{58} -238887. i q^{59} +65869.8i q^{60} +429630. q^{61} -151795. q^{62} +48552.7 q^{63} -32768.0 q^{64} -6270.08i q^{65} -399144. q^{66} -156985. i q^{67} +87547.6 q^{68} -892901. i q^{69} -5680.66i q^{70} -365008. i q^{71} +349373. i q^{72} -285370. q^{73} +485917. q^{74} +723546. i q^{75} +(111001. + 189351. i) q^{76} +34422.5 q^{77} -45817.9i q^{78} +606939. i q^{79} +40876.6 q^{80} +1.78659e6 q^{81} +303283. q^{82} -875034. q^{83} -41510.8i q^{84} -109212. q^{85} +633195. i q^{86} -1.34403e6 q^{87} +247695. i q^{88} -590538. i q^{89} -435827. i q^{90} +3951.37i q^{91} -554105. q^{92} +1.38371e6 q^{93} +343363. i q^{94} +(-138468. - 236207. i) q^{95} +298701. q^{96} +1.00906e6i q^{97} -661943. i q^{98} +2.64093e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 320 q^{4} - 112 q^{5} + 160 q^{6} - 224 q^{7} - 2890 q^{9}+O(q^{10}) $ 10 * q - 320 * q^4 - 112 * q^5 + 160 * q^6 - 224 * q^7 - 2890 * q^9	$ 10 q - 320 q^{4} - 112 q^{5} + 160 q^{6} - 224 q^{7} - 2890 q^{9} + 3644 q^{11} + 10240 q^{16} - 10420 q^{17} - 17230 q^{19} + 3584 q^{20} + 37712 q^{23} - 5120 q^{24} - 52078 q^{25} - 7104 q^{26} + 7168 q^{28} + 94688 q^{30} - 161720 q^{35} + 92480 q^{36} + 25152 q^{38} - 78876 q^{39} + 53792 q^{42} + 6308 q^{43} - 116608 q^{44} + 309808 q^{45} + 322220 q^{47} - 235770 q^{49} - 321728 q^{54} - 377880 q^{55} + 24228 q^{57} + 445920 q^{58} + 426304 q^{61} + 59424 q^{62} - 517916 q^{63} - 327680 q^{64} - 1417312 q^{66} + 333440 q^{68} - 786076 q^{73} - 293280 q^{74} + 551360 q^{76} + 2303716 q^{77} - 114688 q^{80} + 5261090 q^{81} - 455136 q^{82} - 101500 q^{83} - 1261380 q^{85} - 2460732 q^{87} - 1206784 q^{92} - 2827032 q^{93} + 3106292 q^{95} + 163840 q^{96} + 1061428 q^{99}+O(q^{100}) $ 10 * q - 320 * q^4 - 112 * q^5 + 160 * q^6 - 224 * q^7 - 2890 * q^9 + 3644 * q^11 + 10240 * q^16 - 10420 * q^17 - 17230 * q^19 + 3584 * q^20 + 37712 * q^23 - 5120 * q^24 - 52078 * q^25 - 7104 * q^26 + 7168 * q^28 + 94688 * q^30 - 161720 * q^35 + 92480 * q^36 + 25152 * q^38 - 78876 * q^39 + 53792 * q^42 + 6308 * q^43 - 116608 * q^44 + 309808 * q^45 + 322220 * q^47 - 235770 * q^49 - 321728 * q^54 - 377880 * q^55 + 24228 * q^57 + 445920 * q^58 + 426304 * q^61 + 59424 * q^62 - 517916 * q^63 - 327680 * q^64 - 1417312 * q^66 + 333440 * q^68 - 786076 * q^73 - 293280 * q^74 + 551360 * q^76 + 2303716 * q^77 - 114688 * q^80 + 5261090 * q^81 - 455136 * q^82 - 101500 * q^83 - 1261380 * q^85 - 2460732 * q^87 - 1206784 * q^92 - 2827032 * q^93 + 3106292 * q^95 + 163840 * q^96 + 1061428 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$.

$n$	$21$
$\chi(n)$	$-1$

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 $

$2$			5.65685i			0.707107i
$2$			5.65685i			0.707107i
$3$		−	51.5658i		−	1.90984i	−0.296857	−	0.954922i	$-0.595938\pi$
$3$		−	51.5658i		−	1.90984i	0.296857	−	0.954922i	$-0.404062\pi$
$4$	−32.0000			−0.500000
$5$	39.9185			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$5$	39.9185			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$6$	291.700			1.35046
$7$	−25.1565			−0.0733424			−0.0366712	−	0.999327i	$-0.511675\pi$
$7$	−25.1565			−0.0733424			−0.0366712	+	0.999327i	$0.511675\pi$
$8$		−	181.019i		−	0.353553i
$9$	−1930.03			−2.64750
$10$			225.813i			0.225813i
$11$	−1368.34			−1.02805			−0.514026	−	0.857775i	$-0.671846\pi$
$11$	−1368.34			−1.02805			−0.514026	+	0.857775i	$0.671846\pi$
$12$			1650.11i			0.954922i
$13$		−	157.072i		−	0.0714938i	−0.999361	−	0.0357469i	$-0.988619\pi$
$13$		−	157.072i		−	0.0714938i	0.999361	−	0.0357469i	$-0.0113810\pi$
$14$		−	142.306i		−	0.0518609i
$15$		−	2058.43i		−	0.609905i
$16$	1024.00			0.250000
$17$	−2735.86			−0.556862			−0.278431	−	0.960456i	$-0.589814\pi$
$17$	−2735.86			−0.556862			−0.278431	+	0.960456i	$0.589814\pi$
$18$		−	10917.9i		−	1.87207i
$19$	−3468.77	−	5917.22i	−0.505726	−	0.862694i
$19$	−3468.77	−	5917.22i	−0.505726	−	0.862694i
$20$	−1277.39			−0.159674
$21$			1297.21i			0.140073i
$22$		−	7740.48i		−	0.726942i
$23$	17315.8			1.42317			0.711587	−	0.702598i	$-0.247977\pi$
$23$	17315.8			1.42317			0.711587	+	0.702598i	$0.247977\pi$
$24$	−9334.41			−0.675232
$25$	−14031.5			−0.898017
$26$	888.533			0.0505538
$27$			61932.1i			3.14648i
$28$	805.006			0.0366712
$29$		−	26064.3i		−	1.06869i	−0.845266	−	0.534345i	$-0.820558\pi$
$29$		−	26064.3i		−	1.06869i	0.845266	−	0.534345i	$-0.179442\pi$
$30$	11644.2			0.431268
$31$			26833.9i			0.900739i	0.892842	+	0.450369i	$0.148708\pi$
$31$			26833.9i			0.900739i	−0.892842	+	0.450369i	$0.851292\pi$
$32$			5792.62i			0.176777i
$33$			70559.4i			1.96342i
$34$		−	15476.4i		−	0.393761i
$35$	−1004.21			−0.0234218
$36$	61761.0			1.32375
$37$		−	85898.9i		−	1.69583i	−0.530132	−	0.847915i	$-0.677858\pi$
$37$		−	85898.9i		−	1.69583i	0.530132	−	0.847915i	$-0.322142\pi$
$38$	33472.9	−	19622.3i	0.610017	−	0.357602i
$39$	−8099.54			−0.136542
$40$		−	7226.03i		−	0.112907i
$41$		−	53613.3i		−	0.777895i	−0.921260	−	0.388948i	$-0.872839\pi$
$41$		−	53613.3i		−	0.777895i	0.921260	−	0.388948i	$-0.127161\pi$
$42$	−7338.14			−0.0990463
$43$	111934.			1.40785			0.703926	−	0.710273i	$-0.251429\pi$
$43$	111934.			1.40785			0.703926	+	0.710273i	$0.251429\pi$
$44$	43786.8			0.514026
$45$	−77044.0			−0.845476
$46$			97952.8i			1.00634i
$47$	60698.5			0.584634			0.292317	−	0.956321i	$-0.405574\pi$
$47$	60698.5			0.584634			0.292317	+	0.956321i	$0.405574\pi$
$48$		−	52803.4i		−	0.477461i
$49$	−117016.			−0.994621
$50$		−	79374.2i		−	0.634994i
$51$			141077.i			1.06352i
$52$			5026.30i			0.0357469i
$53$			1359.96i			0.00913479i	0.999990	+	0.00456739i	$0.00145385\pi$
$53$			1359.96i			0.00913479i	−0.999990	+	0.00456739i	$0.998546\pi$
$54$	−350341.			−2.22489
$55$	−54622.0			−0.328306
$56$			4553.80i			0.0259305i
$57$	−305126.	+	178870.i	−1.64761	+	0.965858i
$58$	147442.			0.755678
$59$		−	238887.i		−	1.16315i	−0.813492	−	0.581575i	$-0.802436\pi$
$59$		−	238887.i		−	1.16315i	0.813492	−	0.581575i	$-0.197564\pi$
$60$			65869.8i			0.304953i
$61$	429630.			1.89280			0.946401	−	0.322995i	$-0.104690\pi$
$61$	429630.			1.89280			0.946401	+	0.322995i	$0.104690\pi$
$62$	−151795.			−0.636918
$63$	48552.7			0.194174
$64$	−32768.0			−0.125000
$65$		−	6270.08i		−	0.0228314i
$66$	−399144.			−1.38835
$67$		−	156985.i		−	0.521957i	−0.965345	−	0.260979i	$-0.915955\pi$
$67$		−	156985.i		−	0.521957i	0.965345	−	0.260979i	$-0.0840452\pi$
$68$	87547.6			0.278431
$69$		−	892901.i		−	2.71804i
$70$		−	5680.66i		−	0.0165617i
$71$		−	365008.i		−	1.01983i	−0.860225	−	0.509915i	$-0.829677\pi$
$71$		−	365008.i		−	1.01983i	0.860225	−	0.509915i	$-0.170323\pi$
$72$			349373.i			0.936034i
$73$	−285370.			−0.733566			−0.366783	−	0.930306i	$-0.619541\pi$
$73$	−285370.			−0.733566			−0.366783	+	0.930306i	$0.619541\pi$
$74$	485917.			1.19913
$75$			723546.i			1.71507i
$76$	111001.	+	189351.i	0.252863	+	0.431347i
$77$	34422.5			0.0753998
$78$		−	45817.9i		−	0.0965498i
$79$			606939.i			1.23102i	0.788131	+	0.615508i	$0.211049\pi$
$79$			606939.i			1.23102i	−0.788131	+	0.615508i	$0.788951\pi$
$80$	40876.6			0.0798371
$81$	1.78659e6			3.36177
$82$	303283.			0.550055
$83$	−875034.			−1.53035			−0.765175	−	0.643822i	$-0.777348\pi$
$83$	−875034.			−1.53035			−0.765175	+	0.643822i	$0.777348\pi$
$84$		−	41510.8i		−	0.0700363i
$85$	−109212.			−0.177833
$86$			633195.i			0.995502i
$87$	−1.34403e6			−2.04103
$88$			247695.i			0.363471i
$89$		−	590538.i		−	0.837680i	−0.908060	−	0.418840i	$-0.862437\pi$
$89$		−	590538.i		−	0.837680i	0.908060	−	0.418840i	$-0.137563\pi$
$90$		−	435827.i		−	0.597842i
$91$			3951.37i			0.00524353i
$92$	−554105.			−0.711587
$93$	1.38371e6			1.72027
$94$			343363.i			0.413399i
$95$	−138468.	−	236207.i	−0.161503	−	0.275500i
$96$	298701.			0.337616
$97$			1.00906e6i			1.10561i	0.833312	+	0.552803i	$0.186442\pi$
$97$			1.00906e6i			1.10561i	−0.833312	+	0.552803i	$0.813558\pi$
$98$		−	661943.i		−	0.703303i
$99$	2.64093e6			2.72177
$100$	449008.			0.449008
$101$	−534106.			−0.518398			−0.259199	−	0.965824i	$-0.583459\pi$
$101$	−534106.			−0.518398			−0.259199	+	0.965824i	$0.583459\pi$
$102$	−798052.			−0.752022
$103$		−	768218.i		−	0.703028i	−0.936183	−	0.351514i	$-0.885667\pi$
$103$		−	768218.i		−	0.703028i	0.936183	−	0.351514i	$-0.114333\pi$
$104$	−28433.1			−0.0252769
$105$			51782.8i			0.0447319i
$106$	−7693.09			−0.00645927
$107$			1.46546e6i			1.19625i	0.801402	+	0.598126i	$0.204088\pi$
$107$			1.46546e6i			1.19625i	−0.801402	+	0.598126i	$0.795912\pi$
$108$		−	1.98183e6i		−	1.57324i
$109$			1.89935e6i			1.46665i	0.679880	+	0.733323i	$0.262032\pi$
$109$			1.89935e6i			1.46665i	−0.679880	+	0.733323i	$0.737968\pi$
$110$		−	308989.i		−	0.232148i
$111$	−4.42944e6			−3.23877
$112$	−25760.2			−0.0183356
$113$		−	1.16465e6i		−	0.807161i	−0.914944	−	0.403581i	$-0.867765\pi$
$113$		−	1.16465e6i		−	0.807161i	0.914944	−	0.403581i	$-0.132235\pi$
$114$	−1.01184e6	−	1.72605e6i	−0.682964	−	1.16504i
$115$	691220.			0.454488
$116$			834057.i			0.534345i
$117$			303154.i			0.189280i
$118$	1.35135e6			0.822472
$119$	68824.6			0.0408416
$120$	−372616.			−0.215634
$121$	100784.			0.0568900
$122$			2.43035e6i			1.33841i
$123$	−2.76461e6			−1.48566
$124$		−	858685.i		−	0.450369i
$125$	−1.18384e6			−0.606128
$126$			274656.i			0.137302i
$127$			1.04931e6i			0.512264i	0.966642	+	0.256132i	$0.0824482\pi$
$127$			1.04931e6i			0.512264i	−0.966642	+	0.256132i	$0.917552\pi$
$128$		−	185364.i		−	0.0883883i
$129$		−	5.77197e6i		−	2.68878i
$130$	35468.9			0.0161443
$131$	265738.			0.118206			0.0591030	−	0.998252i	$-0.481176\pi$
$131$	265738.			0.118206			0.0591030	+	0.998252i	$0.481176\pi$
$132$		−	2.25790e6i		−	0.981709i
$133$	87262.0	+	148856.i	0.0370912	+	0.0632721i
$134$	888043.			0.369079
$135$			2.47224e6i			1.00482i
$136$			495244.i			0.196880i
$137$	−213121.			−0.0828827			−0.0414414	−	0.999141i	$-0.513195\pi$
$137$	−213121.			−0.0828827			−0.0414414	+	0.999141i	$0.513195\pi$
$138$	5.05101e6			1.92195
$139$	1.82687e6			0.680243			0.340122	−	0.940381i	$-0.389532\pi$
$139$	1.82687e6			0.680243			0.340122	+	0.940381i	$0.389532\pi$
$140$	32134.7			0.0117109
$141$		−	3.12997e6i		−	1.11656i
$142$	2.06480e6			0.721128
$143$			214927.i			0.0734993i
$144$	−1.97635e6			−0.661876
$145$		−	1.04045e6i		−	0.341284i
$146$		−	1.61430e6i		−	0.518710i
$147$			6.03403e6i			1.89957i
$148$			2.74876e6i			0.847915i
$149$	−2.41401e6			−0.729759			−0.364880	−	0.931055i	$-0.618890\pi$
$149$	−2.41401e6			−0.729759			−0.364880	+	0.931055i	$0.618890\pi$
$150$	−4.09299e6			−1.21274
$151$			366324.i			0.106398i	0.998584	+	0.0531991i	$0.0169418\pi$
$151$			366324.i			0.106398i	−0.998584	+	0.0531991i	$0.983058\pi$
$152$	−1.07113e6	+	627915.i	−0.305008	+	0.178801i
$153$	5.28030e6			1.47429
$154$			194723.i			0.0533157i
$155$			1.07117e6i			0.287649i
$156$	259185.			0.0682710
$157$	160372.			0.0414409			0.0207204	−	0.999785i	$-0.493404\pi$
$157$	160372.			0.0414409			0.0207204	+	0.999785i	$0.493404\pi$
$158$	−3.43336e6			−0.870460
$159$	70127.4			0.0174460
$160$			231233.i			0.0564533i
$161$	−435603.			−0.104379
$162$			1.01065e7i			2.37713i
$163$	7.65828e6			1.76835			0.884176	−	0.467154i	$-0.154721\pi$
$163$	7.65828e6			1.76835			0.884176	+	0.467154i	$0.154721\pi$
$164$			1.71563e6i			0.388948i
$165$			2.81663e6i			0.627014i
$166$		−	4.94994e6i		−	1.08212i
$167$		−	1.10021e6i		−	0.236226i	−0.993000	−	0.118113i	$-0.962316\pi$
$167$		−	1.10021e6i		−	0.236226i	0.993000	−	0.118113i	$-0.0376845\pi$
$168$	234820.			0.0495231
$169$	4.80214e6			0.994889
$170$		−	617794.i		−	0.125747i
$171$	6.69484e6	+	1.14204e7i	1.33891	+	2.28399i
$172$	−3.58189e6			−0.703926
$173$		−	5.74022e6i		−	1.10864i	−0.832304	−	0.554320i	$-0.812978\pi$
$173$		−	5.74022e6i		−	1.10864i	0.832304	−	0.554320i	$-0.187022\pi$
$174$		−	7.60296e6i		−	1.44323i
$175$	352983.			0.0658627
$176$	−1.40118e6			−0.257013
$177$	−1.23184e7			−2.22144
$178$	3.34059e6			0.592329
$179$		−	5.04968e6i		−	0.880449i	−0.897888	−	0.440225i	$-0.854899\pi$
$179$		−	5.04968e6i		−	0.880449i	0.897888	−	0.440225i	$-0.145101\pi$
$180$	2.46541e6			0.422738
$181$			1.02257e6i			0.172447i	0.996276	+	0.0862234i	$0.0274799\pi$
$181$			1.02257e6i			0.172447i	−0.996276	+	0.0862234i	$0.972520\pi$
$182$	−22352.3			−0.00370774
$183$		−	2.21542e7i		−	3.61496i
$184$		−	3.13449e6i		−	0.503168i
$185$		−	3.42896e6i		−	0.541560i
$186$			7.82745e6i			1.21641i
$187$	3.74358e6			0.572483
$188$	−1.94235e6			−0.292317
$189$		−	1.55799e6i		−	0.230770i
$190$	1.33619e6	−	783295.i	0.194808	−	0.114200i
$191$	−3.23276e6			−0.463952			−0.231976	−	0.972722i	$-0.574519\pi$
$191$	−3.23276e6			−0.463952			−0.231976	+	0.972722i	$0.574519\pi$
$192$			1.68971e6i			0.238731i
$193$			338219.i			0.0470464i	0.999723	+	0.0235232i	$0.00748836\pi$
$193$			338219.i			0.0470464i	−0.999723	+	0.0235232i	$0.992512\pi$
$194$	−5.70809e6			−0.781782
$195$	−323322.			−0.0436045
$196$	3.74452e6			0.497310
$197$	1.05750e7			1.38319			0.691593	−	0.722287i	$-0.256909\pi$
$197$	1.05750e7			1.38319			0.691593	+	0.722287i	$0.256909\pi$
$198$			1.49394e7i			1.92458i
$199$	−6.64038e6			−0.842624			−0.421312	−	0.906916i	$-0.638430\pi$
$199$	−6.64038e6			−0.842624			−0.421312	+	0.906916i	$0.638430\pi$
$200$			2.53997e6i			0.317497i
$201$	−8.09507e6			−0.996857
$202$		−	3.02136e6i		−	0.366563i
$203$			655685.i			0.0783803i
$204$		−	4.51446e6i		−	0.531760i
$205$		−	2.14017e6i		−	0.248420i
$206$	4.34570e6			0.497116
$207$	−3.34200e7			−3.76786
$208$		−	160842.i		−	0.0178735i
$209$	4.74645e6	+	8.09675e6i	0.519912	+	0.886894i
$210$	−292928.			−0.0316303
$211$			5.89794e6i			0.627846i	0.949448	+	0.313923i	$0.101643\pi$
$211$			5.89794e6i			0.627846i	−0.949448	+	0.313923i	$0.898357\pi$
$212$		−	43518.7i		−	0.00456739i
$213$	−1.88219e7			−1.94771
$214$	−8.28990e6			−0.845878
$215$	4.46825e6			0.449595
$216$	1.12109e7			1.11245
$217$		−	675046.i		−	0.0660624i
$218$	−1.07443e7			−1.03708
$219$			1.47153e7i			1.40100i
$220$	1.74790e6			0.164153
$221$			429727.i			0.0398122i
$222$		−	2.50567e7i		−	2.29016i
$223$			1.43507e7i			1.29407i	0.762460	+	0.647036i	$0.223992\pi$
$223$			1.43507e7i			1.29407i	−0.762460	+	0.647036i	$0.776008\pi$
$224$		−	145722.i		−	0.0129652i
$225$	2.70812e7			2.37750
$226$	6.58826e6			0.570749
$227$		−	9.39539e6i		−	0.803225i	−0.915810	−	0.401613i	$-0.868450\pi$
$227$		−	9.39539e6i		−	0.803225i	0.915810	−	0.401613i	$-0.131550\pi$
$228$	9.76404e6	−	5.72384e6i	0.823806	−	0.482929i
$229$	1.31738e7			1.09699			0.548497	−	0.836152i	$-0.315200\pi$
$229$	1.31738e7			1.09699			0.548497	+	0.836152i	$0.315200\pi$
$230$			3.91013e6i			0.321372i
$231$		−	1.77502e6i		−	0.144002i
$232$	−4.71814e6			−0.377839
$233$	−1.34530e7			−1.06353			−0.531767	−	0.846890i	$-0.678472\pi$
$233$	−1.34530e7			−1.06353			−0.531767	+	0.846890i	$0.678472\pi$
$234$	−1.71490e6			−0.133841
$235$	2.42299e6			0.186702
$236$			7.64438e6i			0.581575i
$237$	3.12973e7			2.35105
$238$			389331.i			0.0288794i
$239$	2.23712e7			1.63869			0.819344	−	0.573302i	$-0.194338\pi$
$239$	2.23712e7			1.63869			0.819344	+	0.573302i	$0.194338\pi$
$240$		−	2.10783e6i		−	0.152476i
$241$			5.17298e6i			0.369564i	0.982780	+	0.184782i	$0.0591579\pi$
$241$			5.17298e6i			0.369564i	−0.982780	+	0.184782i	$0.940842\pi$
$242$			570121.i			0.0402273i
$243$		−	4.69782e7i		−	3.27399i
$244$	−1.37482e7			−0.946401
$245$	−4.67111e6			−0.317630
$246$		−	1.56390e7i		−	1.05052i
$247$	−929429.	+	544847.i	−0.0616773	+	0.0361563i
$248$	4.85746e6			0.318459
$249$			4.51218e7i			2.92273i
$250$		−	6.69683e6i		−	0.428597i
$251$	1.08729e6			0.0687582			0.0343791	−	0.999409i	$-0.489055\pi$
$251$	1.08729e6			0.0687582			0.0343791	+	0.999409i	$0.489055\pi$
$252$	−1.55369e6			−0.0970872
$253$	−2.36938e7			−1.46310
$254$	−5.93581e6			−0.362226
$255$			5.63158e6i			0.339633i
$256$	1.04858e6			0.0625000
$257$		−	1.37915e7i		−	0.812480i	−0.913766	−	0.406240i	$-0.866840\pi$
$257$		−	1.37915e7i		−	0.812480i	0.913766	−	0.406240i	$-0.133160\pi$
$258$	3.26512e7			1.90125
$259$			2.16091e6i			0.124376i
$260$			200643.i			0.0114157i
$261$			5.03049e7i			2.82936i
$262$			1.50324e6i			0.0835843i
$263$	1.18420e7			0.650967			0.325484	−	0.945548i	$-0.394473\pi$
$263$	1.18420e7			0.650967			0.325484	+	0.945548i	$0.394473\pi$
$264$	1.27726e7			0.694173
$265$			54287.6i			0.00291718i
$266$	−842058.	+	493629.i	−0.0447401	+	0.0262274i
$267$	−3.04516e7			−1.59984
$268$			5.02353e6i			0.260979i
$269$			2.84915e7i			1.46372i	0.681453	+	0.731862i	$0.261348\pi$
$269$			2.84915e7i			1.46372i	−0.681453	+	0.731862i	$0.738652\pi$
$270$	−1.39851e7			−0.710516
$271$	242754.			0.0121971			0.00609857	−	0.999981i	$-0.498059\pi$
$271$	242754.			0.0121971			0.00609857	+	0.999981i	$0.498059\pi$
$272$	−2.80152e6			−0.139216
$273$	203756.			0.0100143
$274$		−	1.20559e6i		−	0.0586069i
$275$	1.91998e7			0.923207
$276$			2.85728e7i			1.35902i
$277$	−3.67776e7			−1.73039			−0.865195	−	0.501436i	$-0.832805\pi$
$277$	−3.67776e7			−1.73039			−0.865195	+	0.501436i	$0.832805\pi$
$278$			1.03344e7i			0.481005i
$279$		−	5.17903e7i		−	2.38471i
$280$			181781.i			0.00828085i
$281$		−	2.78086e7i		−	1.25331i	−0.779295	−	0.626657i	$-0.784423\pi$
$281$		−	2.78086e7i		−	1.25331i	0.779295	−	0.626657i	$-0.215577\pi$
$282$	1.77058e7			0.789527
$283$	−2.09382e7			−0.923803			−0.461901	−	0.886931i	$-0.652833\pi$
$283$	−2.09382e7			−0.923803			−0.461901	+	0.886931i	$0.652833\pi$
$284$			1.16803e7i			0.509915i
$285$	−1.21802e7	+	7.14023e6i	−0.526162	+	0.308445i
$286$	−1.21581e6			−0.0519719
$287$			1.34872e6i			0.0570527i
$288$		−	1.11799e7i		−	0.468017i
$289$	−1.66526e7			−0.689905
$290$	5.88566e6			0.241324
$291$	5.20328e7			2.11154
$292$	9.13183e6			0.366783
$293$		−	1.12853e7i		−	0.448651i	−0.974514	−	0.224325i	$-0.927982\pi$
$293$		−	1.12853e7i		−	0.448651i	0.974514	−	0.224325i	$-0.0720179\pi$
$294$	−3.41336e7			−1.34320
$295$		−	9.53601e6i		−	0.371450i
$296$	−1.55494e7			−0.599567
$297$		−	8.47440e7i		−	3.23474i
$298$		−	1.36557e7i		−	0.516018i
$299$		−	2.71982e6i		−	0.101748i
$300$		−	2.31535e7i		−	0.857536i
$301$	−2.81587e6			−0.103255
$302$	−2.07224e6			−0.0752350
$303$			2.75416e7i			0.990059i
$304$	−3.55202e6	−	6.05923e6i	−0.126431	−	0.215674i
$305$	1.71502e7			0.604463
$306$			2.98699e7i			1.04248i
$307$		−	762535.i		−	0.0263539i	−0.999913	−	0.0131769i	$-0.995806\pi$
$307$		−	762535.i		−	0.0263539i	0.999913	−	0.0131769i	$-0.00419447\pi$
$308$	−1.10152e6			−0.0376999
$309$	−3.96138e7			−1.34267
$310$	−6.05945e6			−0.203399
$311$	−1.78594e7			−0.593727			−0.296864	−	0.954920i	$-0.595941\pi$
$311$	−1.78594e7			−0.593727			−0.296864	+	0.954920i	$0.595941\pi$
$312$			1.46617e6i			0.0482749i
$313$	−2.41177e7			−0.786506			−0.393253	−	0.919430i	$-0.628650\pi$
$313$	−2.41177e7			−0.786506			−0.393253	+	0.919430i	$0.628650\pi$
$314$			907200.i			0.0293031i
$315$	1.93815e6			0.0620092
$316$		−	1.94220e7i		−	0.615508i
$317$		−	3.51314e7i		−	1.10285i	−0.834223	−	0.551427i	$-0.814084\pi$
$317$		−	3.51314e7i		−	1.10285i	0.834223	−	0.551427i	$-0.185916\pi$
$318$			396700.i			0.0123362i
$319$			3.56647e7i			1.09867i
$320$	−1.30805e6			−0.0399185
$321$	7.55676e7			2.28466
$322$		−	2.46414e6i		−	0.0738072i
$323$	9.49009e6	+	1.61887e7i	0.281620	+	0.480402i
$324$	−5.71707e7			−1.68089
$325$			2.20396e6i			0.0642027i
$326$			4.33218e7i			1.25041i
$327$	9.79415e7			2.80107
$328$	−9.70505e6			−0.275028
$329$	−1.52696e6			−0.0428785
$330$	−1.59332e7			−0.443366
$331$		−	2.71772e7i		−	0.749412i	−0.927144	−	0.374706i	$-0.877744\pi$
$331$		−	2.71772e7i		−	0.749412i	0.927144	−	0.374706i	$-0.122256\pi$
$332$	2.80011e7			0.765175
$333$			1.65787e8i			4.48972i
$334$	6.22375e6			0.167037
$335$		−	6.26663e6i		−	0.166686i
$336$			1.32835e6i			0.0350181i
$337$		−	6.30310e7i		−	1.64689i	−0.567397	−	0.823444i	$-0.692049\pi$
$337$		−	6.30310e7i		−	1.64689i	0.567397	−	0.823444i	$-0.307951\pi$
$338$			2.71650e7i			0.703492i
$339$	−6.00561e7			−1.54155
$340$	3.49477e6			0.0889165
$341$		−	3.67178e7i		−	0.926006i
$342$	−6.46036e7	+	3.78717e7i	−1.61502	+	0.946753i
$343$	5.90334e6			0.146290
$344$		−	2.02622e7i		−	0.497751i
$345$		−	3.56433e7i		−	0.868002i
$346$	3.24716e7			0.783927
$347$	3.67501e6			0.0879571			0.0439785	−	0.999032i	$-0.485997\pi$
$347$	3.67501e6			0.0879571			0.0439785	+	0.999032i	$0.485997\pi$
$348$	4.30088e7			1.02052
$349$	−4.34206e7			−1.02146			−0.510728	−	0.859742i	$-0.670624\pi$
$349$	−4.34206e7			−1.02146			−0.510728	+	0.859742i	$0.670624\pi$
$350$			1.99677e6i			0.0465720i
$351$	9.72779e6			0.224954
$352$		−	7.92625e6i		−	0.181736i
$353$	−1.09059e7			−0.247934			−0.123967	−	0.992286i	$-0.539562\pi$
$353$	−1.09059e7			−0.247934			−0.123967	+	0.992286i	$0.539562\pi$
$354$		−	6.96833e7i		−	1.57079i
$355$		−	1.45706e7i		−	0.325681i
$356$			1.88972e7i			0.418840i
$357$		−	3.54900e6i		−	0.0780011i
$358$	2.85653e7			0.622572
$359$	8.16663e7			1.76506			0.882530	−	0.470256i	$-0.155839\pi$
$359$	8.16663e7			1.76506			0.882530	+	0.470256i	$0.155839\pi$
$360$			1.39465e7i			0.298921i
$361$	−2.29811e7	+	4.10510e7i	−0.488483	+	0.872574i
$362$	−5.78450e6			−0.121938
$363$		−	5.19701e6i		−	0.108651i
$364$		−	126444.i		−	0.00262177i
$365$	−1.13915e7			−0.234263
$366$	1.25323e8			2.55616
$367$	5.31200e7			1.07463			0.537316	−	0.843381i	$-0.319438\pi$
$367$	5.31200e7			1.07463			0.537316	+	0.843381i	$0.319438\pi$
$368$	1.77313e7			0.355794
$369$			1.03475e8i			2.05948i
$370$	1.93971e7			0.382941
$371$		−	34211.8i		−	0.000669967i
$372$	−4.42788e7			−0.860135
$373$		−	2.35968e7i		−	0.454703i	−0.973813	−	0.227351i	$-0.926993\pi$
$373$		−	2.35968e7i		−	0.454703i	0.973813	−	0.227351i	$-0.0730066\pi$
$374$			2.11769e7i			0.404807i
$375$			6.10459e7i			1.15761i
$376$		−	1.09876e7i		−	0.206699i
$377$	−4.09397e6			−0.0764048
$378$	8.81333e6			0.163179
$379$		−	6.20101e7i		−	1.13906i	−0.821972	−	0.569528i	$-0.807126\pi$
$379$		−	6.20101e7i		−	1.13906i	0.821972	−	0.569528i	$-0.192874\pi$
$380$	4.43099e6	+	7.55861e6i	0.0807513	+	0.137750i
$381$	5.41087e7			0.978345
$382$		−	1.82872e7i		−	0.328064i
$383$		−	6.21448e7i		−	1.10614i	−0.833136	−	0.553068i	$-0.813457\pi$
$383$		−	6.21448e7i		−	1.10614i	0.833136	−	0.553068i	$-0.186543\pi$
$384$	−9.55843e6			−0.168808
$385$	1.37410e6			0.0240788
$386$	−1.91326e6			−0.0332668
$387$	−2.16036e8			−3.72730
$388$		−	3.22898e7i		−	0.552803i
$389$	−4.71506e7			−0.801011			−0.400506	−	0.916294i	$-0.631165\pi$
$389$	−4.71506e7			−0.801011			−0.400506	+	0.916294i	$0.631165\pi$
$390$		−	1.82898e6i		−	0.0308330i
$391$	−4.73736e7			−0.792512
$392$			2.11822e7i			0.351652i
$393$		−	1.37030e7i		−	0.225755i
$394$			5.98211e7i			0.978061i
$395$			2.42281e7i			0.393123i
$396$	−8.45098e7			−1.36089
$397$	5.12175e7			0.818553			0.409277	−	0.912410i	$-0.365781\pi$
$397$	5.12175e7			0.818553			0.409277	+	0.912410i	$0.365781\pi$
$398$		−	3.75637e7i		−	0.595825i
$399$	7.67589e6	−	4.49974e6i	0.120840	−	0.0708383i
$400$	−1.43683e7			−0.224504
$401$			6.30653e7i			0.978041i	0.872272	+	0.489021i	$0.162646\pi$
$401$			6.30653e7i			0.978041i	−0.872272	+	0.489021i	$0.837354\pi$
$402$		−	4.57927e7i		−	0.704884i
$403$	4.21485e6			0.0643973
$404$	1.70914e7			0.259199
$405$	7.13178e7			1.07358
$406$	−3.70911e6			−0.0554233
$407$			1.17539e8i			1.74340i
$408$	2.55377e7			0.376011
$409$		−	6.35675e7i		−	0.929107i	−0.885545	−	0.464553i	$-0.846215\pi$
$409$		−	6.35675e7i		−	0.929107i	0.885545	−	0.464553i	$-0.153785\pi$
$410$	1.21066e7			0.175659
$411$			1.09897e7i			0.158293i
$412$			2.45830e7i			0.351514i
$413$			6.00954e6i			0.0853083i
$414$		−	1.89052e8i		−	2.66428i
$415$	−3.49301e7			−0.488715
$416$	909858.			0.0126384
$417$		−	9.42042e7i		−	1.29916i
$418$	−4.58021e7	+	2.68500e7i	−0.627129	+	0.367633i
$419$	3.77509e7			0.513198			0.256599	−	0.966518i	$-0.417398\pi$
$419$	3.77509e7			0.513198			0.256599	+	0.966518i	$0.417398\pi$
$420$		−	1.65705e6i		−	0.0223660i
$421$			6.28689e7i			0.842537i	0.906936	+	0.421269i	$0.138415\pi$
$421$			6.28689e7i			0.842537i	−0.906936	+	0.421269i	$0.861585\pi$
$422$	−3.33638e7			−0.443954
$423$	−1.17150e8			−1.54782
$424$	246179.			0.00322964
$425$	3.83883e7			0.500071
$426$		−	1.06473e8i		−	1.37724i
$427$	−1.08080e7			−0.138823
$428$		−	4.68947e7i		−	0.598126i
$429$	1.10829e7			0.140372
$430$			2.52762e7i			0.317912i
$431$		−	7.64637e7i		−	0.955045i	−0.878620	−	0.477522i	$-0.841535\pi$
$431$		−	7.64637e7i		−	0.955045i	0.878620	−	0.477522i	$-0.158465\pi$
$432$			6.34185e7i			0.786619i
$433$		−	3.83030e7i		−	0.471812i	−0.971776	−	0.235906i	$-0.924194\pi$
$433$		−	3.83030e7i		−	0.471812i	0.971776	−	0.235906i	$-0.0758058\pi$
$434$	3.81864e6			0.0467131
$435$	−5.36515e7			−0.651800
$436$		−	6.07792e7i		−	0.733323i
$437$	−6.00645e7	−	1.02461e8i	−0.719736	−	1.22776i
$438$	−8.32424e7			−0.990655
$439$			5.64059e7i			0.666701i	0.942803	+	0.333350i	$0.108179\pi$
$439$			5.64059e7i			0.666701i	−0.942803	+	0.333350i	$0.891821\pi$
$440$			9.88764e6i			0.116074i
$441$	2.25845e8			2.63326
$442$	−2.43091e6			−0.0281515
$443$	−1.05839e8			−1.21740			−0.608701	−	0.793400i	$-0.708309\pi$
$443$	−1.05839e8			−1.21740			−0.608701	+	0.793400i	$0.708309\pi$
$444$	1.41742e8			1.61939
$445$		−	2.35734e7i		−	0.267512i
$446$	−8.11798e7			−0.915047
$447$			1.24480e8i			1.39373i
$448$	824327.			0.00916780
$449$			8.94076e7i			0.987724i	0.869540	+	0.493862i	$0.164415\pi$
$449$			8.94076e7i			0.987724i	−0.869540	+	0.493862i	$0.835585\pi$
$450$			1.53195e8i			1.68115i
$451$			7.33611e7i			0.799717i
$452$			3.72688e7i			0.403581i
$453$	1.88898e7			0.203204
$454$	5.31484e7			0.567966
$455$			157733.i			0.00167451i
$456$	3.23789e7	+	5.52337e7i	0.341482	+	0.582519i
$457$	−3.60536e7			−0.377747			−0.188873	−	0.982001i	$-0.560484\pi$
$457$	−3.60536e7			−0.377747			−0.188873	+	0.982001i	$0.560484\pi$
$458$			7.45223e7i			0.775693i
$459$		−	1.69438e8i		−	1.75215i
$460$	−2.21190e7			−0.227244
$461$	4.22650e7			0.431398			0.215699	−	0.976460i	$-0.430797\pi$
$461$	4.22650e7			0.431398			0.215699	+	0.976460i	$0.430797\pi$
$462$	1.00410e7			0.101825
$463$	1.05402e8			1.06195			0.530977	−	0.847386i	$-0.321825\pi$
$463$	1.05402e8			1.06195			0.530977	+	0.847386i	$0.321825\pi$
$464$		−	2.66898e7i		−	0.267173i
$465$	5.52357e7			0.549365
$466$		−	7.61017e7i		−	0.752033i
$467$	−1.16729e8			−1.14611			−0.573055	−	0.819517i	$-0.694242\pi$
$467$	−1.16729e8			−1.14611			−0.573055	+	0.819517i	$0.694242\pi$
$468$		−	9.70092e6i		−	0.0946401i
$469$			3.94919e6i			0.0382816i
$470$			1.37065e7i			0.132018i
$471$		−	8.26970e6i		−	0.0791456i
$472$	−4.32431e7			−0.411236
$473$	−1.53164e8			−1.44734
$474$			1.77044e8i			1.66244i
$475$	4.86721e7	+	8.30275e7i	0.454150	+	0.774714i
$476$	−2.20239e6			−0.0204208
$477$		−	2.62476e6i		−	0.0241844i
$478$			1.26551e8i			1.15873i
$479$	−2.50394e7			−0.227833			−0.113916	−	0.993490i	$-0.536340\pi$
$479$	−2.50394e7			−0.227833			−0.113916	+	0.993490i	$0.536340\pi$
$480$	1.19237e7			0.107817
$481$	−1.34923e7			−0.121241
$482$	−2.92628e7			−0.261321
$483$			2.24622e7i			0.199348i
$484$	−3.22509e6			−0.0284450
$485$			4.02801e7i			0.353074i
$486$	2.65749e8			2.31506
$487$			7.51465e7i			0.650612i	0.945609	+	0.325306i	$0.105467\pi$
$487$			7.51465e7i			0.650612i	−0.945609	+	0.325306i	$0.894533\pi$
$488$		−	7.77713e7i		−	0.669206i
$489$		−	3.94905e8i		−	3.37728i
$490$		−	2.64238e7i		−	0.224599i
$491$	1.74282e8			1.47234			0.736170	−	0.676797i	$-0.236633\pi$
$491$	1.74282e8			1.47234			0.736170	+	0.676797i	$0.236633\pi$
$492$	8.84676e7			0.742829
$493$			7.13083e7i			0.595113i
$494$	−3.08212e6	−	5.25765e6i	−0.0255663	−	0.0436124i
$495$	1.05422e8			0.869193
$496$			2.74779e7i			0.225185i
$497$			9.18231e6i			0.0747967i
$498$	−2.55248e8			−2.06668
$499$	1.36562e8			1.09908			0.549538	−	0.835469i	$-0.314804\pi$
$499$	1.36562e8			1.09908			0.549538	+	0.835469i	$0.314804\pi$
$500$	3.78830e7			0.303064
$501$	−5.67334e7			−0.451155
$502$			6.15064e6i			0.0486194i
$503$	1.51696e8			1.19198			0.595991	−	0.802991i	$-0.296759\pi$
$503$	1.51696e8			1.19198			0.595991	+	0.802991i	$0.296759\pi$
$504$		−	8.78898e6i		−	0.0686510i
$505$	−2.13207e7			−0.165549
$506$		−	1.34032e8i		−	1.03457i
$507$		−	2.47626e8i		−	1.90008i
$508$		−	3.35780e7i		−	0.256132i
$509$			2.19152e8i			1.66185i	0.556386	+	0.830924i	$0.312188\pi$
$509$			2.19152e8i			1.66185i	−0.556386	+	0.830924i	$0.687812\pi$
$510$	−3.18571e7			−0.240157
$511$	7.17889e6			0.0538015
$512$			5.93164e6i			0.0441942i
$513$	3.66466e8	−	2.14828e8i	2.71445	−	1.59125i
$514$	7.80166e7			0.574510
$515$		−	3.06661e7i		−	0.224511i
$516$			1.84703e8i			1.34439i
$517$	−8.30560e7			−0.601034
$518$	−1.22240e7			−0.0879473
$519$	−2.95999e8			−2.11733
$520$	−1.13501e6			−0.00807213
$521$		−	1.55803e8i		−	1.10170i	−0.834605	−	0.550849i	$-0.814304\pi$
$521$		−	1.55803e8i		−	1.10170i	0.834605	−	0.550849i	$-0.185696\pi$
$522$	−2.84567e8			−2.00066
$523$			3.83749e7i			0.268251i	0.990964	+	0.134126i	$0.0428226\pi$
$523$			3.83749e7i			0.268251i	−0.990964	+	0.134126i	$0.957177\pi$
$524$	−8.50362e6			−0.0591030
$525$		−	1.82018e7i		−	0.125788i
$526$			6.69886e7i			0.460303i
$527$		−	7.34139e7i		−	0.501587i
$528$			7.22528e7i			0.490855i
$529$	1.51800e8			1.02543
$530$	−307097.			−0.00206276
$531$			4.61059e8i			3.07945i
$532$	−2.79238e6	−	4.76340e6i	−0.0185456	−	0.0316360i
$533$	−8.42115e6			−0.0556147
$534$		−	1.72260e8i		−	1.13126i
$535$			5.84990e7i			0.382021i
$536$	−2.84174e7			−0.184540
$537$	−2.60391e8			−1.68152
$538$	−1.61172e8			−1.03501
$539$	1.60117e8			1.02252
$540$		−	7.91116e7i		−	0.502411i
$541$	2.36964e8			1.49655			0.748274	−	0.663390i	$-0.230883\pi$
$541$	2.36964e8			1.49655			0.748274	+	0.663390i	$0.230883\pi$
$542$			1.37322e6i			0.00862468i
$543$	5.27294e7			0.329347
$544$		−	1.58478e7i		−	0.0984402i
$545$			7.58193e7i			0.468371i
$546$			1.15262e6i			0.00708120i
$547$			1.85456e8i			1.13313i	0.824018	+	0.566563i	$0.191727\pi$
$547$			1.85456e8i			1.13313i	−0.824018	+	0.566563i	$0.808273\pi$
$548$	6.81986e6			0.0414414
$549$	−8.29199e8			−5.01120
$550$			1.08611e8i			0.652806i
$551$	−1.54228e8	+	9.04111e7i	−0.921953	+	0.540464i
$552$	−1.61632e8			−0.960973
$553$		−	1.52684e7i		−	0.0902857i
$554$		−	2.08045e8i		−	1.22357i
$555$	−1.76817e8			−1.03430
$556$	−5.84600e7			−0.340122
$557$	−4.79023e7			−0.277198			−0.138599	−	0.990349i	$-0.544260\pi$
$557$	−4.79023e7			−0.277198			−0.138599	+	0.990349i	$0.544260\pi$
$558$	2.92970e8			1.68624
$559$		−	1.75817e7i		−	0.100653i
$560$	−1.02831e6			−0.00585544
$561$		−	1.93041e8i		−	1.09335i
$562$	1.57309e8			0.886227
$563$			6.64627e7i			0.372437i	0.982508	+	0.186219i	$0.0596233\pi$
$563$			6.64627e7i			0.372437i	−0.982508	+	0.186219i	$0.940377\pi$
$564$			1.00159e8i			0.558280i
$565$		−	4.64911e7i		−	0.257766i
$566$		−	1.18444e8i		−	0.653227i
$567$	−4.49441e7			−0.246561
$568$	−6.60735e7			−0.360564
$569$		−	2.63899e8i		−	1.43252i	−0.697834	−	0.716259i	$-0.745853\pi$
$569$		−	2.63899e8i		−	1.43252i	0.697834	−	0.716259i	$-0.254147\pi$
$570$	−4.03912e7	−	6.89015e7i	−0.218103	−	0.372053i
$571$	2.72456e8			1.46348			0.731742	−	0.681581i	$-0.238707\pi$
$571$	2.72456e8			1.46348			0.731742	+	0.681581i	$0.238707\pi$
$572$		−	6.87767e6i		−	0.0367497i
$573$			1.66700e8i			0.886076i
$574$	−7.62952e6			−0.0403424
$575$	−2.42966e8			−1.27803
$576$	6.32432e7			0.330938
$577$	1.43860e8			0.748879			0.374439	−	0.927251i	$-0.377835\pi$
$577$	1.43860e8			0.748879			0.374439	+	0.927251i	$0.377835\pi$
$578$		−	9.42014e7i		−	0.487836i
$579$	1.74405e7			0.0898513
$580$			3.32943e7i			0.170642i
$581$	2.20128e7			0.112240
$582$			2.94342e8i			1.49308i
$583$		−	1.86088e6i		−	0.00939103i
$584$			5.16574e7i			0.259355i
$585$			1.21014e7i			0.0604463i
$586$	6.38391e7			0.317244
$587$	2.03952e8			1.00836			0.504179	−	0.863599i	$-0.331795\pi$
$587$	2.03952e8			1.00836			0.504179	+	0.863599i	$0.331795\pi$
$588$		−	1.93089e8i		−	0.949785i
$589$	1.58782e8	−	9.30807e7i	0.777062	−	0.455527i
$590$	5.39438e7			0.262655
$591$		−	5.45307e8i		−	2.64167i
$592$		−	8.79605e7i		−	0.423958i
$593$	−1.29678e7			−0.0621872			−0.0310936	−	0.999516i	$-0.509899\pi$
$593$	−1.29678e7			−0.0621872			−0.0310936	+	0.999516i	$0.509899\pi$
$594$	4.79384e8			2.28731
$595$	2.74738e6			0.0130427
$596$	7.72482e7			0.364880
$597$			3.42417e8i			1.60928i
$598$	1.53856e7			0.0719469
$599$		−	3.81320e8i		−	1.77423i	−0.461549	−	0.887114i	$-0.652706\pi$
$599$		−	3.81320e8i		−	1.77423i	0.461549	−	0.887114i	$-0.347294\pi$
$600$	1.30976e8			0.606369
$601$		−	1.47262e8i		−	0.678370i	−0.940720	−	0.339185i	$-0.889849\pi$
$601$		−	1.47262e8i		−	0.678370i	0.940720	−	0.339185i	$-0.110151\pi$
$602$		−	1.59289e7i		−	0.0730125i
$603$			3.02987e8i			1.38188i
$604$		−	1.17224e7i		−	0.0531991i
$605$	4.02315e6			0.0181677
$606$	−1.55799e8			−0.700078
$607$		−	9.07363e7i		−	0.405709i	−0.979209	−	0.202855i	$-0.934978\pi$
$607$		−	9.07363e7i		−	0.405709i	0.979209	−	0.202855i	$-0.0650219\pi$
$608$	3.42762e7	−	2.00933e7i	0.152504	−	0.0894005i
$609$	3.38109e7			0.149694
$610$			9.70161e7i			0.427420i
$611$		−	9.53403e6i		−	0.0417977i
$612$	−1.68970e8			−0.737147
$613$	−1.02413e8			−0.444603			−0.222302	−	0.974978i	$-0.571357\pi$
$613$	−1.02413e8			−0.444603			−0.222302	+	0.974978i	$0.571357\pi$
$614$	4.31355e6			0.0186350
$615$	−1.10359e8			−0.474443
$616$		−	6.23114e6i		−	0.0266579i
$617$	−2.66718e8			−1.13552			−0.567762	−	0.823193i	$-0.692191\pi$
$617$	−2.66718e8			−1.13552			−0.567762	+	0.823193i	$0.692191\pi$
$618$		−	2.24089e8i		−	0.949414i
$619$	−4.23626e8			−1.78612			−0.893061	−	0.449936i	$-0.851447\pi$
$619$	−4.23626e8			−1.78612			−0.893061	+	0.449936i	$0.851447\pi$
$620$		−	3.42774e7i		−	0.143825i
$621$			1.07240e9i			4.47799i
$622$		−	1.01028e8i		−	0.419828i
$623$			1.48558e7i			0.0614375i
$624$	−8.29393e6			−0.0341355
$625$	1.71985e8			0.704451
$626$		−	1.36430e8i		−	0.556144i
$627$	4.17515e8	−	2.44754e8i	1.69383	−	0.992951i
$628$	−5.13190e6			−0.0207204
$629$			2.35008e8i			0.944344i
$630$			1.09638e7i			0.0438472i
$631$	−1.14563e8			−0.455991			−0.227996	−	0.973662i	$-0.573217\pi$
$631$	−1.14563e8			−0.455991			−0.227996	+	0.973662i	$0.573217\pi$
$632$	1.09868e8			0.435230
$633$	3.04132e8			1.19909
$634$	1.98733e8			0.779835
$635$			4.18871e7i			0.163591i
$636$	−2.24408e6			−0.00872301
$637$			1.83800e7i			0.0711093i
$638$	−2.01750e8			−0.776876
$639$			7.04477e8i			2.70000i
$640$		−	7.39945e6i		−	0.0282267i
$641$			3.79080e8i			1.43932i	0.694328	+	0.719658i	$0.255702\pi$
$641$			3.79080e8i			1.43932i	−0.694328	+	0.719658i	$0.744298\pi$
$642$			4.27475e8i			1.61550i
$643$	1.05304e8			0.396107			0.198054	−	0.980191i	$-0.436538\pi$
$643$	1.05304e8			0.396107			0.198054	+	0.980191i	$0.436538\pi$
$644$	1.39393e7			0.0521895
$645$		−	2.30409e8i		−	0.858657i
$646$	−9.15771e7	+	5.36841e7i	−0.339695	+	0.199135i
$647$	4.50756e8			1.66429			0.832144	−	0.554560i	$-0.187113\pi$
$647$	4.50756e8			1.66429			0.832144	+	0.554560i	$0.187113\pi$
$648$		−	3.23406e8i		−	1.18857i
$649$			3.26878e8i			1.19578i
$650$	−1.24675e7			−0.0453981
$651$	−3.48093e7			−0.126169
$652$	−2.45065e8			−0.884176
$653$	−3.55740e8			−1.27759			−0.638797	−	0.769376i	$-0.720568\pi$
$653$	−3.55740e8			−1.27759			−0.638797	+	0.769376i	$0.720568\pi$
$654$			5.54041e8i			1.98065i
$655$	1.06079e7			0.0377489
$656$		−	5.49000e7i		−	0.194474i
$657$	5.50772e8			1.94212
$658$		−	8.63778e6i		−	0.0303197i
$659$		−	1.16578e6i		−	0.00407344i	−0.999998	−	0.00203672i	$-0.999352\pi$
$659$		−	1.16578e6i		−	0.00407344i	0.999998	−	0.00203672i	$-0.000648309\pi$
$660$		−	9.01320e7i		−	0.313507i
$661$		−	5.43258e8i		−	1.88106i	−0.339715	−	0.940528i	$-0.610331\pi$
$661$		−	5.43258e8i		−	1.88106i	0.339715	−	0.940528i	$-0.389669\pi$
$662$	1.53737e8			0.529914
$663$	2.21592e7			0.0760351
$664$			1.58398e8i			0.541060i
$665$	3.48337e6	+	5.94212e6i	0.0118450	+	0.0202058i
$666$	−9.37836e8			−3.17471
$667$		−	4.51323e8i		−	1.52093i
$668$			3.52068e7i			0.118113i
$669$	7.40005e8			2.47148
$670$	3.54494e7			0.117865
$671$	−5.87878e8			−1.94590
$672$	−7.51426e6			−0.0247616
$673$		−	1.90275e8i		−	0.624217i	−0.950046	−	0.312109i	$-0.898965\pi$
$673$		−	1.90275e8i		−	0.624217i	0.950046	−	0.312109i	$-0.101035\pi$
$674$	3.56557e8			1.16453
$675$		−	8.69001e8i		−	2.82559i
$676$	−1.53668e8			−0.497444
$677$			4.81730e7i			0.155252i	0.996983	+	0.0776261i	$0.0247340\pi$
$677$			4.81730e7i			0.155252i	−0.996983	+	0.0776261i	$0.975266\pi$
$678$		−	3.39729e8i		−	1.09004i
$679$		−	2.53843e7i		−	0.0810879i
$680$			1.97694e7i			0.0628734i
$681$	−4.84481e8			−1.53403
$682$	2.07707e8			0.654785
$683$			1.89512e8i			0.594806i	0.954752	+	0.297403i	$0.0961205\pi$
$683$			1.89512e8i			0.594806i	−0.954752	+	0.297403i	$0.903879\pi$
$684$	−2.14235e8	−	3.65453e8i	−0.669456	−	1.14199i
$685$	−8.50747e6			−0.0264684
$686$			3.33943e7i			0.103443i
$687$		−	6.79317e8i		−	2.09509i
$688$	1.14621e8			0.351963
$689$	213612.			0.000653081
$690$	2.01629e8			0.613770
$691$	6.88902e7			0.208796			0.104398	−	0.994536i	$-0.466708\pi$
$691$	6.88902e7			0.208796			0.104398	+	0.994536i	$0.466708\pi$
$692$			1.83687e8i			0.554320i
$693$	−6.64365e7			−0.199621
$694$			2.07890e7i			0.0621950i
$695$	7.29261e7			0.217234
$696$			2.43295e8i			0.721614i
$697$			1.46679e8i			0.433180i
$698$		−	2.45624e8i		−	0.722279i
$699$			6.93715e8i			2.03119i
$700$	−1.12955e7			−0.0329314
$701$	3.42895e8			0.995420			0.497710	−	0.867343i	$-0.334174\pi$
$701$	3.42895e8			0.995420			0.497710	+	0.867343i	$0.334174\pi$
$702$			5.50287e7i			0.159066i
$703$	−5.08283e8	+	2.97964e8i	−1.46298	+	0.857625i
$704$	4.48377e7			0.128506
$705$		−	1.24944e8i		−	0.356572i
$706$		−	6.16930e7i		−	0.175316i
$707$	1.34362e7			0.0380206
$708$	3.94188e8			1.11072
$709$	−1.95474e8			−0.548468			−0.274234	−	0.961663i	$-0.588424\pi$
$709$	−1.95474e8			−0.548468			−0.274234	+	0.961663i	$0.588424\pi$
$710$	8.24237e7			0.230291
$711$		−	1.17141e9i		−	3.25912i
$712$	−1.06899e8			−0.296164
$713$			4.64650e8i			1.28191i
$714$	2.00761e7			0.0551551
$715$			8.57958e6i			0.0234719i
$716$			1.61590e8i			0.440225i
$717$		−	1.15359e9i		−	3.12964i
$718$			4.61974e8i			1.24809i
$719$	−2.31962e8			−0.624066			−0.312033	−	0.950071i	$-0.601010\pi$
$719$	−2.31962e8			−0.624066			−0.312033	+	0.950071i	$0.601010\pi$
$720$	−7.88930e7			−0.211369
$721$			1.93256e7i			0.0515618i
$722$	−2.32219e8	−	1.30001e8i	−0.617003	−	0.345409i
$723$	2.66749e8			0.705810
$724$		−	3.27221e7i		−	0.0862234i
$725$			3.65721e8i			0.959702i
$726$	2.93987e7			0.0768278
$727$	−3.54964e8			−0.923806			−0.461903	−	0.886930i	$-0.652833\pi$
$727$	−3.54964e8			−0.923806			−0.461903	+	0.886930i	$0.652833\pi$
$728$	715275.			0.00185387
$729$	−1.12005e9			−2.89103
$730$		−	6.44403e7i		−	0.165649i
$731$	−3.06236e8			−0.783980
$732$			7.08935e8i			1.80748i
$733$	−4.30151e8			−1.09222			−0.546108	−	0.837715i	$-0.683891\pi$
$733$	−4.30151e8			−1.09222			−0.546108	+	0.837715i	$0.683891\pi$
$734$			3.00492e8i			0.759879i
$735$			2.40870e8i			0.606625i
$736$			1.00304e8i			0.251584i
$737$			2.14809e8i			0.536599i
$738$	−5.85345e8			−1.45627
$739$	2.90982e8			0.720995			0.360497	−	0.932760i	$-0.382607\pi$
$739$	2.90982e8			0.720995			0.360497	+	0.932760i	$0.382607\pi$
$740$			1.09727e8i			0.270780i
$741$	2.80955e7	+	4.79267e7i	0.0690529	+	0.117794i
$742$	193531.			0.000473739
$743$		−	3.06553e8i		−	0.747375i	−0.927555	−	0.373688i	$-0.878093\pi$
$743$		−	3.06553e8i		−	0.747375i	0.927555	−	0.373688i	$-0.121907\pi$
$744$		−	2.50479e8i		−	0.608207i
$745$	−9.63636e7			−0.233047
$746$	1.33484e8			0.321523
$747$	1.68884e9			4.05161
$748$	−1.19795e8			−0.286241
$749$		−	3.68658e7i		−	0.0877361i
$750$	−3.45328e8			−0.818554
$751$		−	4.56545e8i		−	1.07786i	−0.842349	−	0.538932i	$-0.818828\pi$
$751$		−	4.56545e8i		−	1.07786i	0.842349	−	0.538932i	$-0.181172\pi$
$752$	6.21553e7			0.146159
$753$		−	5.60670e7i		−	0.131317i
$754$		−	2.31590e7i		−	0.0540263i
$755$			1.46231e7i			0.0339781i
$756$			4.98557e7i			0.115385i
$757$	−2.12640e7			−0.0490182			−0.0245091	−	0.999700i	$-0.507802\pi$
$757$	−2.12640e7			−0.0490182			−0.0245091	+	0.999700i	$0.507802\pi$
$758$	3.50782e8			0.805434
$759$			1.22179e9i			2.79429i
$760$	−4.27580e7	+	2.50654e7i	−0.0974039	+	0.0570998i
$761$	2.83195e8			0.642586			0.321293	−	0.946980i	$-0.395883\pi$
$761$	2.83195e8			0.642586			0.321293	+	0.946980i	$0.395883\pi$
$762$			3.06085e8i			0.691794i
$763$		−	4.77809e7i		−	0.107567i
$764$	1.03448e8			0.231976
$765$	2.10782e8			0.470813
$766$	3.51544e8			0.782157
$767$	−3.75224e7			−0.0831581
$768$		−	5.40706e7i		−	0.119365i
$769$	1.63966e8			0.360557			0.180278	−	0.983616i	$-0.442300\pi$
$769$	1.63966e8			0.360557			0.180278	+	0.983616i	$0.442300\pi$
$770$			7.77306e6i			0.0170263i
$771$	−7.11170e8			−1.55171
$772$		−	1.08230e7i		−	0.0235232i
$773$			2.80416e8i			0.607106i	0.952815	+	0.303553i	$0.0981730\pi$
$773$			2.80416e8i			0.607106i	−0.952815	+	0.303553i	$0.901827\pi$
$774$		−	1.22209e9i		−	2.63560i
$775$		−	3.76520e8i		−	0.808878i
$776$	1.82659e8			0.390891
$777$	1.11429e8			0.237539
$778$		−	2.66724e8i		−	0.566400i
$779$	−3.17242e8	+	1.85972e8i	−0.671086	+	0.393402i
$780$	1.03463e7			0.0218022
$781$			4.99454e8i			1.04844i
$782$		−	2.67985e8i		−	0.560391i
$783$	1.61422e9			3.36261
$784$	−1.19825e8			−0.248655
$785$	6.40181e6			0.0132341
$786$	7.75158e7			0.159633
$787$		−	1.78603e8i		−	0.366407i	−0.983075	−	0.183203i	$-0.941353\pi$
$787$		−	1.78603e8i		−	0.366407i	0.983075	−	0.183203i	$-0.0586467\pi$
$788$	−3.38399e8			−0.691593
$789$		−	6.10644e8i		−	1.24325i
$790$	−1.37055e8			−0.277980
$791$			2.92985e7i			0.0591992i
$792$		−	4.78060e8i		−	0.962291i
$793$		−	6.74828e7i		−	0.135324i
$794$			2.89730e8i			0.578805i
$795$	2.79938e6			0.00557136
$796$	2.12492e8			0.421312
$797$			5.39911e8i			1.06647i	0.845968	+	0.533233i	$0.179023\pi$
$797$			5.39911e8i			1.06647i	−0.845968	+	0.533233i	$0.820977\pi$
$798$	2.54543e7	+	4.34214e7i	0.0500903	+	0.0854467i
$799$	−1.66063e8			−0.325561
$800$		−	8.12792e7i		−	0.158748i
$801$			1.13976e9i			2.21776i
$802$	−3.56751e8			−0.691580
$803$	3.90482e8			0.754144
$804$	2.59042e8			0.498428
$805$	−1.73886e7			−0.0333333
$806$			2.38428e7i			0.0455357i
$807$	1.46919e9			2.79548
$808$			9.66835e7i			0.183281i
$809$	−4.24059e8			−0.800904			−0.400452	−	0.916318i	$-0.631147\pi$
$809$	−4.24059e8			−0.800904			−0.400452	+	0.916318i	$0.631147\pi$
$810$			4.03435e8i			0.759133i
$811$			1.50159e8i			0.281507i	0.990045	+	0.140754i	$0.0449525\pi$
$811$			1.50159e8i			0.281507i	−0.990045	+	0.140754i	$0.955047\pi$
$812$		−	2.09819e7i		−	0.0391902i
$813$		−	1.25178e7i		−	0.0232946i
$814$	−6.64899e8			−1.23277
$815$	3.05707e8			0.564720
$816$			1.44463e8i			0.265880i
$817$	−3.88274e8	−	6.62339e8i	−0.711987	−	1.21455i
$818$	3.59592e8			0.656978
$819$		−	7.62627e6i		−	0.0138823i
$820$			6.84853e7i			0.124210i
$821$	−1.77116e8			−0.320057			−0.160029	−	0.987112i	$-0.551159\pi$
$821$	−1.77116e8			−0.320057			−0.160029	+	0.987112i	$0.551159\pi$
$822$	−6.21673e7			−0.111930
$823$	−5.42120e8			−0.972514			−0.486257	−	0.873816i	$-0.661638\pi$
$823$	−5.42120e8			−0.972514			−0.486257	+	0.873816i	$0.661638\pi$
$824$	−1.39062e8			−0.248558
$825$		−	9.90054e8i		−	1.76318i
$826$	−3.39951e7			−0.0603221
$827$			2.17728e8i			0.384945i	0.981302	+	0.192472i	$0.0616506\pi$
$827$			2.17728e8i			0.384945i	−0.981302	+	0.192472i	$0.938349\pi$
$828$	1.06944e9			1.88393
$829$			2.27590e8i			0.399474i	0.979850	+	0.199737i	$0.0640088\pi$
$829$			2.27590e8i			0.399474i	−0.979850	+	0.199737i	$0.935991\pi$
$830$		−	1.97594e8i		−	0.345573i
$831$			1.89647e9i			3.30477i
$832$			5.14693e6i			0.00893673i
$833$	3.20140e8			0.553867
$834$	5.32900e8			0.918644
$835$		−	4.39189e7i		−	0.0754383i
$836$	−1.51886e8	−	2.59096e8i	−0.259956	−	0.443447i
$837$	−1.66188e9			−2.83415
$838$			2.13551e8i			0.362886i
$839$		−	3.59559e8i		−	0.608814i	−0.952542	−	0.304407i	$-0.901542\pi$
$839$		−	3.59559e8i		−	0.608814i	0.952542	−	0.304407i	$-0.0984582\pi$
$840$	9.37369e6			0.0158151
$841$	−8.45237e7			−0.142099
$842$	−3.55640e8			−0.595764
$843$	−1.43397e9			−2.39364
$844$		−	1.88734e8i		−	0.313923i
$845$	1.91694e8			0.317716
$846$		−	6.62700e8i		−	1.09448i
$847$	−2.53537e6			−0.00417245
$848$			1.39260e6i			0.00228370i
$849$			1.07969e9i			1.76432i
$850$			2.17157e8i			0.353604i
$851$		−	1.48741e9i		−	2.41346i
$852$	6.02302e8			0.973857
$853$	8.96407e8			1.44430			0.722151	−	0.691736i	$-0.243154\pi$
$853$	8.96407e8			1.44430			0.722151	+	0.691736i	$0.243154\pi$
$854$		−	6.11391e7i		−	0.0981624i
$855$	2.67248e8	+	4.55886e8i	0.427579	+	0.729387i
$856$	2.65277e8			0.422939
$857$		−	2.39428e7i		−	0.0380394i	−0.999819	−	0.0190197i	$-0.993945\pi$
$857$		−	2.39428e7i		−	0.0380394i	0.999819	−	0.0190197i	$-0.00605452\pi$
$858$			6.26943e7i			0.0992582i
$859$	−5.73924e8			−0.905471			−0.452736	−	0.891645i	$-0.649552\pi$
$859$	−5.73924e8			−0.905471			−0.452736	+	0.891645i	$0.649552\pi$
$860$	−1.42984e8			−0.224798
$861$	6.95479e7			0.108962
$862$	4.32544e8			0.675319
$863$			9.59089e8i			1.49220i	0.665835	+	0.746099i	$0.268076\pi$
$863$			9.59089e8i			1.49220i	−0.665835	+	0.746099i	$0.731924\pi$
$864$	−3.58749e8			−0.556224
$865$		−	2.29141e8i		−	0.354042i
$866$	2.16675e8			0.333622
$867$			8.58706e8i			1.31761i
$868$			2.16015e7i			0.0330312i
$869$		−	8.30497e8i		−	1.26555i
$870$		−	3.03499e8i		−	0.460892i
$871$	−2.46580e7			−0.0373167
$872$	3.43819e8			0.518538
$873$		−	1.94751e9i		−	2.92710i
$874$	5.79608e8	−	3.39776e8i	0.868161	−	0.508930i
$875$	2.97813e7			0.0444549
$876$		−	4.70890e8i		−	0.700499i
$877$		−	2.35518e8i		−	0.349160i	−0.984643	−	0.174580i	$-0.944143\pi$
$877$		−	2.35518e8i		−	0.349160i	0.984643	−	0.174580i	$-0.0558568\pi$
$878$	−3.19080e8			−0.471429
$879$	−5.81933e8			−0.856853
$880$	−5.59329e7			−0.0820766
$881$	6.93176e8			1.01372			0.506858	−	0.862030i	$-0.330807\pi$
$881$	6.93176e8			1.01372			0.506858	+	0.862030i	$0.330807\pi$
$882$			1.27757e9i			1.86200i
$883$	−9.08954e8			−1.32026			−0.660130	−	0.751151i	$-0.729499\pi$
$883$	−9.08954e8			−1.32026			−0.660130	+	0.751151i	$0.729499\pi$
$884$		−	1.37513e7i		−	0.0199061i
$885$	−4.91732e8			−0.709412
$886$		−	5.98715e8i		−	0.860833i
$887$		−	1.21284e9i		−	1.73794i	−0.494868	−	0.868968i	$-0.664783\pi$
$887$		−	1.21284e9i		−	1.73794i	0.494868	−	0.868968i	$-0.335217\pi$
$888$			8.01815e8i			1.14508i
$889$		−	2.63970e7i		−	0.0375707i
$890$	1.33351e8			0.189159
$891$	−2.44465e9			−3.45608
$892$		−	4.59222e8i		−	0.647036i
$893$	−2.10549e8	−	3.59166e8i	−0.295665	−	0.504361i
$894$	−7.04166e8			−0.985514
$895$		−	2.01576e8i		−	0.281170i
$896$			4.66310e6i			0.00648262i
$897$	−1.40250e8			−0.194323
$898$	−5.05766e8			−0.698426
$899$	6.99407e8			0.962611
$900$	−8.66600e8			−1.18875
$901$		−	3.72066e6i		−	0.00508682i
$902$	−4.14993e8			−0.565485
$903$			1.45202e8i			0.197202i
$904$	−2.10824e8			−0.285375
$905$			4.08193e7i			0.0550706i
$906$			1.06857e8i			0.143687i
$907$			6.70720e8i			0.898916i	0.893301	+	0.449458i	$0.148383\pi$
$907$			6.70720e8i			0.898916i	−0.893301	+	0.449458i	$0.851617\pi$
$908$			3.00653e8i			0.401613i
$909$	1.03084e9			1.37246
$910$	−892272.			−0.00118406
$911$		−	4.11499e8i		−	0.544269i	−0.962259	−	0.272134i	$-0.912270\pi$
$911$		−	4.11499e8i		−	0.544269i	0.962259	−	0.272134i	$-0.0877296\pi$
$912$	−3.12449e8	+	1.83163e8i	−0.411903	+	0.241464i
$913$	1.19734e9			1.57328
$914$		−	2.03950e8i		−	0.267107i
$915$		−	8.84363e8i		−	1.15443i
$916$	−4.21562e8			−0.548497
$917$	−6.68502e6			−0.00866952
$918$	9.58485e8			1.23896
$919$	9.76999e8			1.25877			0.629387	−	0.777092i	$-0.283306\pi$
$919$	9.76999e8			1.25877			0.629387	+	0.777092i	$0.283306\pi$
$920$		−	1.25124e8i		−	0.160686i
$921$	−3.93207e7			−0.0503318
$922$			2.39087e8i			0.305044i
$923$	−5.73325e7			−0.0729115
$924$			5.68007e7i			0.0720009i
$925$			1.20529e9i			1.52288i
$926$			5.96243e8i			0.750914i
$927$			1.48268e9i			1.86127i
$928$	1.50980e8			0.188920
$929$	7.27336e8			0.907168			0.453584	−	0.891213i	$-0.350145\pi$
$929$	7.27336e8			0.907168			0.453584	+	0.891213i	$0.350145\pi$
$930$			3.12460e8i			0.388460i
$931$	4.05903e8	+	6.92410e8i	0.503005	+	0.858054i
$932$	4.30496e8			0.531767
$933$			9.20936e8i			1.13393i
$934$		−	6.60317e8i		−	0.810423i
$935$	1.49438e8			0.182821
$936$	5.48767e7			0.0669207
$937$	−7.48977e8			−0.910437			−0.455218	−	0.890380i	$-0.650439\pi$
$937$	−7.48977e8			−0.910437			−0.455218	+	0.890380i	$0.650439\pi$
$938$	−2.23400e7			−0.0270692
$939$			1.24365e9i			1.50210i
$940$	−7.75358e7			−0.0933510
$941$		−	7.23039e7i		−	0.0867746i	−0.999058	−	0.0433873i	$-0.986185\pi$
$941$		−	7.23039e7i		−	0.0867746i	0.999058	−	0.0433873i	$-0.0138149\pi$
$942$	4.67805e7			0.0559644
$943$		−	9.28356e8i		−	1.10708i
$944$		−	2.44620e8i		−	0.290788i
$945$		−	6.21927e7i		−	0.0736961i
$946$		−	8.66424e8i		−	1.02343i
$947$	1.18089e9			1.39046			0.695231	−	0.718786i	$-0.255302\pi$
$947$	1.18089e9			1.39046			0.695231	+	0.718786i	$0.255302\pi$
$948$	−1.00151e9			−1.17552
$949$			4.48236e7i			0.0524455i
$950$	−4.69675e8	+	2.75331e8i	−0.547805	+	0.321133i
$951$	−1.81158e9			−2.10628
$952$		−	1.24586e7i		−	0.0144397i
$953$			5.02689e8i			0.580792i	0.956907	+	0.290396i	$0.0937870\pi$
$953$			5.02689e8i			0.580792i	−0.956907	+	0.290396i	$0.906213\pi$
$954$	1.48479e7			0.0171009
$955$	−1.29047e8			−0.148162
$956$	−7.15880e8			−0.819344
$957$	1.83908e9			2.09829
$958$		−	1.41644e8i		−	0.161102i
$959$	5.36136e6			0.00607882
$960$			6.74506e7i			0.0762382i
$961$	1.67445e8			0.188670
$962$		−	7.63240e7i		−	0.0857306i
$963$		−	2.82838e9i		−	3.16708i
$964$		−	1.65535e8i		−	0.184782i
$965$			1.35012e7i			0.0150242i
$966$	−1.27066e8			−0.140960
$967$	−7.99455e8			−0.884127			−0.442063	−	0.896984i	$-0.645753\pi$
$967$	−7.99455e8			−0.884127			−0.442063	+	0.896984i	$0.645753\pi$
$968$		−	1.82439e7i		−	0.0201136i
$969$	8.34783e8	−	4.89364e8i	0.917492	−	0.537849i
$970$	−2.27859e8			−0.249661
$971$		−	1.39572e9i		−	1.52455i	−0.647255	−	0.762274i	$-0.724083\pi$
$971$		−	1.39572e9i		−	1.52455i	0.647255	−	0.762274i	$-0.275917\pi$
$972$			1.50330e9i			1.63699i
$973$	−4.59577e7			−0.0498907
$974$	−4.25093e8			−0.460052
$975$	1.13649e8			0.122617
$976$	4.39941e8			0.473200
$977$			1.42354e9i			1.52646i	0.646129	+	0.763228i	$0.276387\pi$
$977$			1.42354e9i			1.52646i	−0.646129	+	0.763228i	$0.723613\pi$
$978$	2.23392e9			2.38809
$979$			8.08055e8i			0.861178i
$980$	1.49476e8			0.158815
$981$		−	3.66580e9i		−	3.88295i
$982$			9.85887e8i			1.04110i
$983$		−	3.32507e8i		−	0.350058i	−0.984563	−	0.175029i	$-0.943998\pi$
$983$		−	3.32507e8i		−	0.350058i	0.984563	−	0.175029i	$-0.0560019\pi$
$984$			5.00449e8i			0.525260i
$985$	4.22138e8			0.441718
$986$	−4.03381e8			−0.420808
$987$			7.87388e7i			0.0818912i
$988$	2.97417e7	−	1.74351e7i	0.0308387	−	0.0180781i
$989$	1.93823e9			2.00362
$990$			5.96357e8i			0.614612i
$991$			8.50991e8i			0.874388i	0.899367	+	0.437194i	$0.144028\pi$
$991$			8.50991e8i			0.874388i	−0.899367	+	0.437194i	$0.855972\pi$
$992$	−1.55439e8			−0.159230
$993$	−1.40141e9			−1.43126
$994$	−5.19430e7			−0.0528893
$995$	−2.65074e8			−0.269091
$996$		−	1.44390e9i		−	1.46136i
$997$	1.54040e9			1.55435			0.777174	−	0.629286i	$-0.216653\pi$
$997$	1.54040e9			1.55435			0.777174	+	0.629286i	$0.216653\pi$
$998$			7.72510e8i			0.777164i
$999$	5.31990e9			5.33589

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.7.b.a.37.6	yes	10
3.2	odd	2		342.7.d.a.37.2		10
4.3	odd	2		304.7.e.e.113.10		10
19.18	odd	2	inner	38.7.b.a.37.5	✓	10
57.56	even	2		342.7.d.a.37.7		10
76.75	even	2		304.7.e.e.113.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.7.b.a.37.5	✓	10	19.18	odd	2	inner
38.7.b.a.37.6	yes	10	1.1	even	1	trivial
304.7.e.e.113.1		10	76.75	even	2
304.7.e.e.113.10		10	4.3	odd	2
342.7.d.a.37.2		10	3.2	odd	2
342.7.d.a.37.7		10	57.56	even	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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.65685i q^{2} -51.5658i q^{3} -32.0000 q^{4} +39.9185 q^{5} +291.700 q^{6} -25.1565 q^{7} -181.019i q^{8} -1930.03 q^{9} +O(q^{10})\)	\(q+5.65685i q^{2} -51.5658i q^{3} -32.0000 q^{4} +39.9185 q^{5} +291.700 q^{6} -25.1565 q^{7} -181.019i q^{8} -1930.03 q^{9} +225.813i q^{10} -1368.34 q^{11} +1650.11i q^{12} -157.072i q^{13} -142.306i q^{14} -2058.43i q^{15} +1024.00 q^{16} -2735.86 q^{17} -10917.9i q^{18} +(-3468.77 - 5917.22i) q^{19} -1277.39 q^{20} +1297.21i q^{21} -7740.48i q^{22} +17315.8 q^{23} -9334.41 q^{24} -14031.5 q^{25} +888.533 q^{26} +61932.1i q^{27} +805.006 q^{28} -26064.3i q^{29} +11644.2 q^{30} +26833.9i q^{31} +5792.62i q^{32} +70559.4i q^{33} -15476.4i q^{34} -1004.21 q^{35} +61761.0 q^{36} -85898.9i q^{37} +(33472.9 - 19622.3i) q^{38} -8099.54 q^{39} -7226.03i q^{40} -53613.3i q^{41} -7338.14 q^{42} +111934. q^{43} +43786.8 q^{44} -77044.0 q^{45} +97952.8i q^{46} +60698.5 q^{47} -52803.4i q^{48} -117016. q^{49} -79374.2i q^{50} +141077. i q^{51} +5026.30i q^{52} +1359.96i q^{53} -350341. q^{54} -54622.0 q^{55} +4553.80i q^{56} +(-305126. + 178870. i) q^{57} +147442. q^{58} -238887. i q^{59} +65869.8i q^{60} +429630. q^{61} -151795. q^{62} +48552.7 q^{63} -32768.0 q^{64} -6270.08i q^{65} -399144. q^{66} -156985. i q^{67} +87547.6 q^{68} -892901. i q^{69} -5680.66i q^{70} -365008. i q^{71} +349373. i q^{72} -285370. q^{73} +485917. q^{74} +723546. i q^{75} +(111001. + 189351. i) q^{76} +34422.5 q^{77} -45817.9i q^{78} +606939. i q^{79} +40876.6 q^{80} +1.78659e6 q^{81} +303283. q^{82} -875034. q^{83} -41510.8i q^{84} -109212. q^{85} +633195. i q^{86} -1.34403e6 q^{87} +247695. i q^{88} -590538. i q^{89} -435827. i q^{90} +3951.37i q^{91} -554105. q^{92} +1.38371e6 q^{93} +343363. i q^{94} +(-138468. - 236207. i) q^{95} +298701. q^{96} +1.00906e6i q^{97} -661943. i q^{98} +2.64093e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 320 q^{4} - 112 q^{5} + 160 q^{6} - 224 q^{7} - 2890 q^{9}+O(q^{10}) \) 10 * q - 320 * q^4 - 112 * q^5 + 160 * q^6 - 224 * q^7 - 2890 * q^9	\( 10 q - 320 q^{4} - 112 q^{5} + 160 q^{6} - 224 q^{7} - 2890 q^{9} + 3644 q^{11} + 10240 q^{16} - 10420 q^{17} - 17230 q^{19} + 3584 q^{20} + 37712 q^{23} - 5120 q^{24} - 52078 q^{25} - 7104 q^{26} + 7168 q^{28} + 94688 q^{30} - 161720 q^{35} + 92480 q^{36} + 25152 q^{38} - 78876 q^{39} + 53792 q^{42} + 6308 q^{43} - 116608 q^{44} + 309808 q^{45} + 322220 q^{47} - 235770 q^{49} - 321728 q^{54} - 377880 q^{55} + 24228 q^{57} + 445920 q^{58} + 426304 q^{61} + 59424 q^{62} - 517916 q^{63} - 327680 q^{64} - 1417312 q^{66} + 333440 q^{68} - 786076 q^{73} - 293280 q^{74} + 551360 q^{76} + 2303716 q^{77} - 114688 q^{80} + 5261090 q^{81} - 455136 q^{82} - 101500 q^{83} - 1261380 q^{85} - 2460732 q^{87} - 1206784 q^{92} - 2827032 q^{93} + 3106292 q^{95} + 163840 q^{96} + 1061428 q^{99}+O(q^{100}) \) 10 * q - 320 * q^4 - 112 * q^5 + 160 * q^6 - 224 * q^7 - 2890 * q^9 + 3644 * q^11 + 10240 * q^16 - 10420 * q^17 - 17230 * q^19 + 3584 * q^20 + 37712 * q^23 - 5120 * q^24 - 52078 * q^25 - 7104 * q^26 + 7168 * q^28 + 94688 * q^30 - 161720 * q^35 + 92480 * q^36 + 25152 * q^38 - 78876 * q^39 + 53792 * q^42 + 6308 * q^43 - 116608 * q^44 + 309808 * q^45 + 322220 * q^47 - 235770 * q^49 - 321728 * q^54 - 377880 * q^55 + 24228 * q^57 + 445920 * q^58 + 426304 * q^61 + 59424 * q^62 - 517916 * q^63 - 327680 * q^64 - 1417312 * q^66 + 333440 * q^68 - 786076 * q^73 - 293280 * q^74 + 551360 * q^76 + 2303716 * q^77 - 114688 * q^80 + 5261090 * q^81 - 455136 * q^82 - 101500 * q^83 - 1261380 * q^85 - 2460732 * q^87 - 1206784 * q^92 - 2827032 * q^93 + 3106292 * q^95 + 163840 * q^96 + 1061428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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