LMFDB - Embedded newform 38.7.b.a.37.1

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			37.1
Root			$-47.4187i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.7.b.a.37.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.65685i q^{2} -44.5903i q^{3} -32.0000 q^{4} -145.013 q^{5} -252.241 q^{6} +126.579 q^{7} +181.019i q^{8} -1259.29 q^{9} +O(q^{10})$	\(q-5.65685i q^{2} -44.5903i q^{3} -32.0000 q^{4} -145.013 q^{5} -252.241 q^{6} +126.579 q^{7} +181.019i q^{8} -1259.29 q^{9} +820.316i q^{10} +2263.01 q^{11} +1426.89i q^{12} -1379.25i q^{13} -716.042i q^{14} +6466.16i q^{15} +1024.00 q^{16} -7180.60 q^{17} +7123.64i q^{18} +(-6015.82 + 3294.81i) q^{19} +4640.41 q^{20} -5644.21i q^{21} -12801.5i q^{22} -7107.16 q^{23} +8071.70 q^{24} +5403.69 q^{25} -7802.20 q^{26} +23645.9i q^{27} -4050.54 q^{28} +22968.8i q^{29} +36578.1 q^{30} -49862.8i q^{31} -5792.62i q^{32} -100908. i q^{33} +40619.6i q^{34} -18355.6 q^{35} +40297.4 q^{36} -5829.43i q^{37} +(18638.2 + 34030.6i) q^{38} -61501.0 q^{39} -26250.1i q^{40} -54659.6i q^{41} -31928.5 q^{42} -32045.4 q^{43} -72416.2 q^{44} +182614. q^{45} +40204.2i q^{46} +155291. q^{47} -45660.4i q^{48} -101627. q^{49} -30567.9i q^{50} +320185. i q^{51} +44135.9i q^{52} -122867. i q^{53} +133762. q^{54} -328165. q^{55} +22913.3i q^{56} +(146916. + 268247. i) q^{57} +129931. q^{58} +54281.8i q^{59} -206917. i q^{60} -154441. q^{61} -282066. q^{62} -159401. q^{63} -32768.0 q^{64} +200008. i q^{65} -570822. q^{66} -554574. i q^{67} +229779. q^{68} +316910. i q^{69} +103835. i q^{70} +341576. i q^{71} -227956. i q^{72} +583990. q^{73} -32976.3 q^{74} -240952. i q^{75} +(192506. - 105434. i) q^{76} +286450. q^{77} +347902. i q^{78} -508526. i q^{79} -148493. q^{80} +136354. q^{81} -309202. q^{82} +301931. q^{83} +180615. i q^{84} +1.04128e6 q^{85} +181276. i q^{86} +1.02419e6 q^{87} +409648. i q^{88} -363648. i q^{89} -1.03302e6i q^{90} -174584. i q^{91} +227429. q^{92} -2.22340e6 q^{93} -878457. i q^{94} +(872371. - 477789. i) q^{95} -258295. q^{96} +495326. i q^{97} +574887. i q^{98} -2.84979e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	44.5903i		−	1.65149i	−0.564042	−	0.825746i	$-0.690754\pi$
$3$		−	44.5903i		−	1.65149i	0.564042	−	0.825746i	$-0.309246\pi$
$4$	−32.0000			−0.500000
$5$	−145.013			−1.16010			−0.580051	−	0.814580i	$-0.696967\pi$
$5$	−145.013			−1.16010			−0.580051	+	0.814580i	$0.696967\pi$
$6$	−252.241			−1.16778
$7$	126.579			0.369036			0.184518	−	0.982829i	$-0.440928\pi$
$7$	126.579			0.369036			0.184518	+	0.982829i	$0.440928\pi$
$8$			181.019i			0.353553i
$9$	−1259.29			−1.72743
$10$			820.316i			0.820316i
$11$	2263.01			1.70023			0.850115	−	0.526598i	$-0.176533\pi$
$11$	2263.01			1.70023			0.850115	+	0.526598i	$0.176533\pi$
$12$			1426.89i			0.825746i
$13$		−	1379.25i		−	0.627787i	−0.949458	−	0.313893i	$-0.898367\pi$
$13$		−	1379.25i		−	0.627787i	0.949458	−	0.313893i	$-0.101633\pi$
$14$		−	716.042i		−	0.260948i
$15$			6466.16i			1.91590i
$16$	1024.00			0.250000
$17$	−7180.60			−1.46155			−0.730775	−	0.682618i	$-0.760841\pi$
$17$	−7180.60			−1.46155			−0.730775	+	0.682618i	$0.760841\pi$
$18$			7123.64i			1.22147i
$19$	−6015.82	+	3294.81i	−0.877070	+	0.480362i
$19$	−6015.82	+	3294.81i	−0.877070	+	0.480362i
$20$	4640.41			0.580051
$21$		−	5644.21i		−	0.609461i
$22$		−	12801.5i		−	1.20224i
$23$	−7107.16			−0.584134			−0.292067	−	0.956398i	$-0.594343\pi$
$23$	−7107.16			−0.584134			−0.292067	+	0.956398i	$0.594343\pi$
$24$	8071.70			0.583891
$25$	5403.69			0.345836
$26$	−7802.20			−0.443912
$27$			23645.9i			1.20134i
$28$	−4050.54			−0.184518
$29$			22968.8i			0.941771i	0.882194	+	0.470885i	$0.156065\pi$
$29$			22968.8i			0.941771i	−0.882194	+	0.470885i	$0.843935\pi$
$30$	36578.1			1.35474
$31$		−	49862.8i		−	1.67375i	−0.547392	−	0.836876i	$-0.684379\pi$
$31$		−	49862.8i		−	1.67375i	0.547392	−	0.836876i	$-0.315621\pi$
$32$		−	5792.62i		−	0.176777i
$33$		−	100908.i		−	2.80792i
$34$			40619.6i			1.03347i
$35$	−18355.6			−0.428120
$36$	40297.4			0.863713
$37$		−	5829.43i		−	0.115086i	−0.998343	−	0.0575428i	$-0.981673\pi$
$37$		−	5829.43i		−	0.115086i	0.998343	−	0.0575428i	$-0.0183266\pi$
$38$	18638.2	+	34030.6i	0.339668	+	0.620182i
$39$	−61501.0			−1.03678
$40$		−	26250.1i		−	0.410158i
$41$		−	54659.6i		−	0.793077i	−0.918018	−	0.396538i	$-0.870211\pi$
$41$		−	54659.6i		−	0.793077i	0.918018	−	0.396538i	$-0.129789\pi$
$42$	−31928.5			−0.430954
$43$	−32045.4			−0.403052			−0.201526	−	0.979483i	$-0.564590\pi$
$43$	−32045.4			−0.403052			−0.201526	+	0.979483i	$0.564590\pi$
$44$	−72416.2			−0.850115
$45$	182614.			2.00399
$46$			40204.2i			0.413045i
$47$	155291.			1.49573			0.747863	−	0.663853i	$-0.231080\pi$
$47$	155291.			1.49573			0.747863	+	0.663853i	$0.231080\pi$
$48$		−	45660.4i		−	0.412873i
$49$	−101627.			−0.863812
$50$		−	30567.9i		−	0.244543i
$51$			320185.i			2.41374i
$52$			44135.9i			0.313893i
$53$		−	122867.i		−	0.825294i	−0.910891	−	0.412647i	$-0.864604\pi$
$53$		−	122867.i		−	0.825294i	0.910891	−	0.412647i	$-0.135396\pi$
$54$	133762.			0.849474
$55$	−328165.			−1.97244
$56$			22913.3i			0.130474i
$57$	146916.	+	268247.i	0.793315	+	1.44847i
$58$	129931.			0.665933
$59$			54281.8i			0.264301i	0.991230	+	0.132150i	$0.0421882\pi$
$59$			54281.8i			0.264301i	−0.991230	+	0.132150i	$0.957812\pi$
$60$		−	206917.i		−	0.957949i
$61$	−154441.			−0.680416			−0.340208	−	0.940350i	$-0.610497\pi$
$61$	−154441.			−0.680416			−0.340208	+	0.940350i	$0.610497\pi$
$62$	−282066.			−1.18352
$63$	−159401.			−0.637483
$64$	−32768.0			−0.125000
$65$			200008.i			0.728296i
$66$	−570822.			−1.98550
$67$		−	554574.i		−	1.84389i	−0.387319	−	0.921946i	$-0.626599\pi$
$67$		−	554574.i		−	1.84389i	0.387319	−	0.921946i	$-0.373401\pi$
$68$	229779.			0.730775
$69$			316910.i			0.964693i
$70$			103835.i			0.302726i
$71$			341576.i			0.954359i	0.878806	+	0.477180i	$0.158341\pi$
$71$			341576.i			0.954359i	−0.878806	+	0.477180i	$0.841659\pi$
$72$		−	227956.i		−	0.610737i
$73$	583990.			1.50119			0.750597	−	0.660760i	$-0.229766\pi$
$73$	583990.			1.50119			0.750597	+	0.660760i	$0.229766\pi$
$74$	−32976.3			−0.0813779
$75$		−	240952.i		−	0.571146i
$76$	192506.	−	105434.i	0.438535	−	0.240181i
$77$	286450.			0.627446
$78$			347902.i			0.733117i
$79$		−	508526.i		−	1.03141i	−0.856766	−	0.515706i	$-0.827530\pi$
$79$		−	508526.i		−	1.03141i	0.856766	−	0.515706i	$-0.172470\pi$
$80$	−148493.			−0.290025
$81$	136354.			0.256574
$82$	−309202.			−0.560790
$83$	301931.			0.528048			0.264024	−	0.964516i	$-0.414950\pi$
$83$	301931.			0.528048			0.264024	+	0.964516i	$0.414950\pi$
$84$			180615.i			0.304730i
$85$	1.04128e6			1.69555
$86$			181276.i			0.285000i
$87$	1.02419e6			1.55533
$88$			409648.i			0.601122i
$89$		−	363648.i		−	0.515836i	−0.966167	−	0.257918i	$-0.916964\pi$
$89$		−	363648.i		−	0.515836i	0.966167	−	0.257918i	$-0.0830364\pi$
$90$		−	1.03302e6i		−	1.41703i
$91$		−	174584.i		−	0.231676i
$92$	227429.			0.292067
$93$	−2.22340e6			−2.76419
$94$		−	878457.i		−	1.05764i
$95$	872371.	−	477789.i	1.01749	−	0.557269i
$96$	−258295.			−0.291945
$97$			495326.i			0.542720i	0.962478	+	0.271360i	$0.0874734\pi$
$97$			495326.i			0.542720i	−0.962478	+	0.271360i	$0.912527\pi$
$98$			574887.i			0.610807i
$99$	−2.84979e6			−2.93702
$100$	−172918.			−0.172918
$101$	−708197.			−0.687369			−0.343684	−	0.939085i	$-0.611675\pi$
$101$	−708197.			−0.687369			−0.343684	+	0.939085i	$0.611675\pi$
$102$	1.81124e6			1.70677
$103$		−	1.80274e6i		−	1.64976i	−0.565304	−	0.824882i	$-0.691241\pi$
$103$		−	1.80274e6i		−	1.64976i	0.565304	−	0.824882i	$-0.308759\pi$
$104$	249670.			0.221956
$105$			818483.i			0.707036i
$106$	−695042.			−0.583571
$107$			885646.i			0.722951i	0.932382	+	0.361475i	$0.117727\pi$
$107$			885646.i			0.722951i	−0.932382	+	0.361475i	$0.882273\pi$
$108$		−	756670.i		−	0.600669i
$109$		−	847426.i		−	0.654368i	−0.944961	−	0.327184i	$-0.893900\pi$
$109$		−	847426.i		−	0.654368i	0.944961	−	0.327184i	$-0.106100\pi$
$110$			1.85638e6i			1.39473i
$111$	−259936.			−0.190063
$112$	129617.			0.0922591
$113$			2.14982e6i			1.48993i	0.667102	+	0.744966i	$0.267535\pi$
$113$			2.14982e6i			1.48993i	−0.667102	+	0.744966i	$0.732465\pi$
$114$	1.51744e6	−	831084.i	1.02423	−	0.560958i
$115$	1.03063e6			0.677655
$116$		−	735003.i		−	0.470885i
$117$			1.73688e6i			1.08445i
$118$	307064.			0.186889
$119$	−908916.			−0.539365
$120$	−1.17050e6			−0.677372
$121$	3.34963e6			1.89078
$122$			873653.i			0.481127i
$123$	−2.43729e6			−1.30976
$124$			1.59561e6i			0.836876i
$125$	1.48222e6			0.758897
$126$			901706.i			0.450768i
$127$			839572.i			0.409871i	0.978776	+	0.204935i	$0.0656984\pi$
$127$			839572.i			0.409871i	−0.978776	+	0.204935i	$0.934302\pi$
$128$			185364.i			0.0883883i
$129$			1.42891e6i			0.665636i
$130$	1.13142e6			0.514983
$131$	−2.53503e6			−1.12764			−0.563818	−	0.825899i	$-0.690668\pi$
$131$	−2.53503e6			−1.12764			−0.563818	+	0.825899i	$0.690668\pi$
$132$			3.22906e6i			1.40396i
$133$	−761480.	+	417055.i	−0.323671	+	0.177271i
$134$	−3.13715e6			−1.30383
$135$		−	3.42896e6i		−	1.39367i
$136$		−	1.29983e6i		−	0.516736i
$137$	682706.			0.265505			0.132752	−	0.991149i	$-0.457618\pi$
$137$	682706.			0.265505			0.132752	+	0.991149i	$0.457618\pi$
$138$	1.79272e6			0.682141
$139$	−3.38986e6			−1.26222			−0.631112	−	0.775691i	$-0.717401\pi$
$139$	−3.38986e6			−1.26222			−0.631112	+	0.775691i	$0.717401\pi$
$140$	587380.			0.214060
$141$		−	6.92446e6i		−	2.47018i
$142$	1.93224e6			0.674834
$143$		−	3.12124e6i		−	1.06738i
$144$	−1.28952e6			−0.431856
$145$		−	3.33078e6i		−	1.09255i
$146$		−	3.30355e6i		−	1.06150i
$147$			4.53156e6i			1.42658i
$148$			186542.i			0.0575428i
$149$	−478299.			−0.144591			−0.0722954	−	0.997383i	$-0.523032\pi$
$149$	−478299.			−0.144591			−0.0722954	+	0.997383i	$0.523032\pi$
$150$	−1.36303e6			−0.403861
$151$		−	1.63991e6i		−	0.476309i	−0.971227	−	0.238155i	$-0.923458\pi$
$151$		−	1.63991e6i		−	0.476309i	0.971227	−	0.238155i	$-0.0765425\pi$
$152$	−596424.	−	1.08898e6i	−0.169834	−	0.310091i
$153$	9.04248e6			2.52472
$154$		−	1.62041e6i		−	0.443672i
$155$			7.23074e6i			1.94172i
$156$	1.96803e6			0.518392
$157$	598586.			0.154678			0.0773388	−	0.997005i	$-0.475358\pi$
$157$	598586.			0.154678			0.0773388	+	0.997005i	$0.475358\pi$
$158$	−2.87666e6			−0.729318
$159$	−5.47869e6			−1.36297
$160$			840003.i			0.205079i
$161$	−899621.			−0.215567
$162$		−	771333.i		−	0.181425i
$163$	4.58190e6			1.05799			0.528996	−	0.848624i	$-0.322569\pi$
$163$	4.58190e6			1.05799			0.528996	+	0.848624i	$0.322569\pi$
$164$			1.74911e6i			0.396538i
$165$			1.46330e7i			3.25747i
$166$		−	1.70798e6i		−	0.373387i
$167$		−	7.57589e6i		−	1.62661i	−0.581836	−	0.813306i	$-0.697665\pi$
$167$		−	7.57589e6i		−	1.62661i	0.581836	−	0.813306i	$-0.302335\pi$
$168$	1.02171e6			0.215477
$169$	2.92449e6			0.605884
$170$		−	5.89036e6i		−	1.19893i
$171$	7.57569e6	−	4.14913e6i	1.51507	−	0.829790i
$172$	1.02545e6			0.201526
$173$		−	6.77123e6i		−	1.30776i	−0.756597	−	0.653881i	$-0.773140\pi$
$173$		−	6.77123e6i		−	1.30776i	0.756597	−	0.653881i	$-0.226860\pi$
$174$		−	5.79368e6i		−	1.09978i
$175$	683996.			0.127626
$176$	2.31732e6			0.425057
$177$	2.42044e6			0.436491
$178$	−2.05710e6			−0.364751
$179$			1.06235e7i			1.85228i	0.377175	+	0.926142i	$0.376895\pi$
$179$			1.06235e7i			1.85228i	−0.377175	+	0.926142i	$0.623105\pi$
$180$	−5.84363e6			−1.00199
$181$		−	9.63814e6i		−	1.62539i	−0.582690	−	0.812695i	$-0.698000\pi$
$181$		−	9.63814e6i		−	1.62539i	0.582690	−	0.812695i	$-0.302000\pi$
$182$	−987598.			−0.163820
$183$			6.88659e6i			1.12370i
$184$		−	1.28653e6i		−	0.206523i
$185$			845342.i			0.133511i
$186$			1.25774e7i			1.95458i
$187$	−1.62497e7			−2.48497
$188$	−4.96930e6			−0.747863
$189$			2.99309e6i			0.443337i
$190$	−2.70278e6	−	4.93488e6i	−0.394049	−	0.719474i
$191$	−5.30824e6			−0.761816			−0.380908	−	0.924613i	$-0.624389\pi$
$191$	−5.30824e6			−0.761816			−0.380908	+	0.924613i	$0.624389\pi$
$192$			1.46113e6i			0.206436i
$193$		−	3.69587e6i		−	0.514097i	−0.966399	−	0.257048i	$-0.917250\pi$
$193$		−	3.69587e6i		−	0.514097i	0.966399	−	0.257048i	$-0.0827500\pi$
$194$	2.80199e6			0.383761
$195$	8.91843e6			1.20278
$196$	3.25205e6			0.431906
$197$	7.10926e6			0.929878			0.464939	−	0.885343i	$-0.346076\pi$
$197$	7.10926e6			0.929878			0.464939	+	0.885343i	$0.346076\pi$
$198$			1.61208e7i			2.07679i
$199$	9.74778e6			1.23693			0.618467	−	0.785811i	$-0.287754\pi$
$199$	9.74778e6			1.23693			0.618467	+	0.785811i	$0.287754\pi$
$200$			978172.i			0.122272i
$201$	−2.47286e7			−3.04517
$202$			4.00617e6i			0.486043i
$203$			2.90738e6i			0.347548i
$204$		−	1.02459e7i		−	1.20687i
$205$			7.92634e6i			0.920050i
$206$	−1.01979e7			−1.16656
$207$	8.95000e6			1.00905
$208$		−	1.41235e6i		−	0.156947i
$209$	−1.36138e7	+	7.45616e6i	−1.49122	+	0.816726i
$210$	4.63004e6			0.499950
$211$			168569.i			0.0179445i	0.999960	+	0.00897223i	$0.00285599\pi$
$211$			168569.i			0.0179445i	−0.999960	+	0.00897223i	$0.997144\pi$
$212$			3.93175e6i			0.412647i
$213$	1.52310e7			1.57612
$214$	5.00997e6			0.511204
$215$	4.64699e6			0.467581
$216$	−4.28037e6			−0.424737
$217$		−	6.31160e6i		−	0.617676i
$218$	−4.79376e6			−0.462708
$219$		−	2.60403e7i		−	2.47921i
$220$	1.05013e7			0.986220
$221$			9.90382e6i			0.917542i
$222$			1.47042e6i			0.134395i
$223$		−	3.22541e6i		−	0.290851i	−0.989369	−	0.145425i	$-0.953545\pi$
$223$		−	3.22541e6i		−	0.290851i	0.989369	−	0.145425i	$-0.0464551\pi$
$224$		−	733227.i		−	0.0652370i
$225$	−6.80483e6			−0.597406
$226$	1.21612e7			1.05354
$227$			1.24828e7i			1.06718i	0.845745	+	0.533588i	$0.179157\pi$
$227$			1.24828e7i			1.06718i	−0.845745	+	0.533588i	$0.820843\pi$
$228$	−4.70132e6	−	8.58391e6i	−0.396657	−	0.724237i
$229$	−1.56885e6			−0.130640			−0.0653198	−	0.997864i	$-0.520807\pi$
$229$	−1.56885e6			−0.130640			−0.0653198	+	0.997864i	$0.520807\pi$
$230$		−	5.83012e6i		−	0.479175i
$231$		−	1.27729e7i		−	1.03622i
$232$	−4.15781e6			−0.332966
$233$	7.65799e6			0.605407			0.302703	−	0.953085i	$-0.402111\pi$
$233$	7.65799e6			0.605407			0.302703	+	0.953085i	$0.402111\pi$
$234$	9.82526e6			0.766825
$235$	−2.25191e7			−1.73519
$236$		−	1.73702e6i		−	0.132150i
$237$	−2.26753e7			−1.70337
$238$			5.14161e6i			0.381389i
$239$	−3.30709e6			−0.242244			−0.121122	−	0.992638i	$-0.538649\pi$
$239$	−3.30709e6			−0.242244			−0.121122	+	0.992638i	$0.538649\pi$
$240$			6.62135e6i			0.478975i
$241$			1.16735e7i			0.833972i	0.908913	+	0.416986i	$0.136914\pi$
$241$			1.16735e7i			0.833972i	−0.908913	+	0.416986i	$0.863086\pi$
$242$		−	1.89484e7i		−	1.33698i
$243$			1.11578e7i			0.777608i
$244$	4.94213e6			0.340208
$245$	1.47372e7			1.00211
$246$			1.37874e7i			0.926140i
$247$	4.54435e6	+	8.29731e6i	0.301565	+	0.550613i
$248$	9.02613e6			0.591761
$249$		−	1.34632e7i		−	0.872068i
$250$		−	8.38470e6i		−	0.536621i
$251$	2.30216e7			1.45584			0.727922	−	0.685660i	$-0.240486\pi$
$251$	2.30216e7			1.45584			0.727922	+	0.685660i	$0.240486\pi$
$252$	5.10082e6			0.318741
$253$	−1.60835e7			−0.993162
$254$	4.74934e6			0.289822
$255$		−	4.64309e7i		−	2.80018i
$256$	1.04858e6			0.0625000
$257$		−	1.76929e7i		−	1.04232i	−0.853460	−	0.521159i	$-0.825500\pi$
$257$		−	1.76929e7i		−	1.04232i	0.853460	−	0.521159i	$-0.174500\pi$
$258$	8.08316e6			0.470676
$259$		−	737887.i		−	0.0424708i
$260$		−	6.40027e6i		−	0.364148i
$261$		−	2.89245e7i		−	1.62684i
$262$			1.43403e7i			0.797360i
$263$	3.07545e7			1.69060			0.845300	−	0.534292i	$-0.179422\pi$
$263$	3.07545e7			1.69060			0.845300	+	0.534292i	$0.179422\pi$
$264$	1.82663e7			0.992748
$265$			1.78173e7i			0.957425i
$266$	2.35922e6	+	4.30758e6i	0.125350	+	0.228870i
$267$	−1.62152e7			−0.851899
$268$			1.77464e7i			0.921946i
$269$			3.37828e7i			1.73556i	0.496951	+	0.867779i	$0.334453\pi$
$269$			3.37828e7i			1.73556i	−0.496951	+	0.867779i	$0.665547\pi$
$270$	−1.93971e7			−0.985476
$271$	9.42639e6			0.473628			0.236814	−	0.971555i	$-0.423897\pi$
$271$	9.42639e6			0.473628			0.236814	+	0.971555i	$0.423897\pi$
$272$	−7.35293e6			−0.365388
$273$	−7.78477e6			−0.382611
$274$		−	3.86197e6i		−	0.187740i
$275$	1.22286e7			0.588001
$276$		−	1.01411e7i		−	0.482347i
$277$	−9.62685e6			−0.452944			−0.226472	−	0.974018i	$-0.572719\pi$
$277$	−9.62685e6			−0.452944			−0.226472	+	0.974018i	$0.572719\pi$
$278$			1.91759e7i			0.892528i
$279$			6.27919e7i			2.89128i
$280$		−	3.32272e6i		−	0.151363i
$281$		−	2.35519e7i		−	1.06147i	−0.847538	−	0.530735i	$-0.821916\pi$
$281$		−	2.35519e7i		−	1.06147i	0.847538	−	0.530735i	$-0.178084\pi$
$282$	−3.91706e7			−1.74668
$283$	−3.18809e7			−1.40660			−0.703302	−	0.710891i	$-0.748292\pi$
$283$	−3.18809e7			−1.40660			−0.703302	+	0.710891i	$0.748292\pi$
$284$		−	1.09304e7i		−	0.477180i
$285$	−2.13047e7	−	3.88993e7i	−0.920326	−	1.68038i
$286$	−1.76564e7			−0.754753
$287$		−	6.91879e6i		−	0.292674i
$288$			7.29461e6i			0.305369i
$289$	2.74234e7			1.13613
$290$	−1.88417e7			−0.772550
$291$	2.20867e7			0.896298
$292$	−1.86877e7			−0.750597
$293$		−	1.90249e7i		−	0.756343i	−0.925736	−	0.378172i	$-0.876553\pi$
$293$		−	1.90249e7i		−	0.756343i	0.925736	−	0.378172i	$-0.123447\pi$
$294$	2.56344e7			1.00874
$295$		−	7.87155e6i		−	0.306616i
$296$	1.05524e6			0.0406889
$297$			5.35109e7i			2.04255i
$298$			2.70567e6i			0.102241i
$299$			9.80253e6i			0.366712i
$300$			7.71047e6i			0.285573i
$301$	−4.05629e6			−0.148741
$302$	−9.27673e6			−0.336801
$303$			3.15787e7i			1.13518i
$304$	−6.16020e6	+	3.37388e6i	−0.219268	+	0.120091i
$305$	2.23960e7			0.789352
$306$		−	5.11520e7i		−	1.78525i
$307$			1.22191e7i			0.422302i	0.977453	+	0.211151i	$0.0677212\pi$
$307$			1.22191e7i			0.422302i	−0.977453	+	0.211151i	$0.932279\pi$
$308$	−9.16640e6			−0.313723
$309$	−8.03848e7			−2.72457
$310$	4.09032e7			1.37301
$311$	−9.33928e6			−0.310479			−0.155239	−	0.987877i	$-0.549615\pi$
$311$	−9.33928e6			−0.310479			−0.155239	+	0.987877i	$0.549615\pi$
$312$		−	1.11329e7i		−	0.366559i
$313$	9.23202e6			0.301067			0.150534	−	0.988605i	$-0.451901\pi$
$313$	9.23202e6			0.301067			0.150534	+	0.988605i	$0.451901\pi$
$314$		−	3.38611e6i		−	0.109374i
$315$	2.31151e7			0.739545
$316$			1.62728e7i			0.515706i
$317$			2.10167e6i			0.0659762i	0.999456	+	0.0329881i	$0.0105024\pi$
$317$			2.10167e6i			0.0659762i	−0.999456	+	0.0329881i	$0.989498\pi$
$318$			3.09921e7i			0.963762i
$319$			5.19786e7i			1.60123i
$320$	4.75178e6			0.145013
$321$	3.94912e7			1.19395
$322$			5.08902e6i			0.152429i
$323$	4.31972e7	−	2.36587e7i	1.28188	−	0.702074i
$324$	−4.36332e6			−0.128287
$325$		−	7.45302e6i		−	0.217111i
$326$		−	2.59191e7i		−	0.748114i
$327$	−3.77870e7			−1.08068
$328$	9.89445e6			0.280395
$329$	1.96566e7			0.551977
$330$	8.27765e7			2.30338
$331$			2.80410e7i			0.773230i	0.922241	+	0.386615i	$0.126356\pi$
$331$			2.80410e7i			0.773230i	−0.922241	+	0.386615i	$0.873644\pi$
$332$	−9.66180e6			−0.264024
$333$			7.34097e6i			0.198802i
$334$	−4.28557e7			−1.15019
$335$			8.04203e7i			2.13910i
$336$		−	5.77967e6i		−	0.152365i
$337$			7.52666e6i			0.196658i	0.995154	+	0.0983292i	$0.0313498\pi$
$337$			7.52666e6i			0.196658i	−0.995154	+	0.0983292i	$0.968650\pi$
$338$		−	1.65434e7i		−	0.428425i
$339$	9.58611e7			2.46061
$340$	−3.33209e7			−0.847774
$341$		−	1.12840e8i		−	2.84576i
$342$	−2.34710e7	−	4.28546e7i	−0.586750	−	1.07132i
$343$	−2.77558e7			−0.687814
$344$		−	5.80084e6i		−	0.142500i
$345$		−	4.59560e7i		−	1.11914i
$346$	−3.83038e7			−0.924728
$347$	−3.34358e7			−0.800244			−0.400122	−	0.916462i	$-0.631032\pi$
$347$	−3.34358e7			−0.800244			−0.400122	+	0.916462i	$0.631032\pi$
$348$	−3.27740e7			−0.777663
$349$	−1.48335e7			−0.348953			−0.174476	−	0.984661i	$-0.555823\pi$
$349$	−1.48335e7			−0.348953			−0.174476	+	0.984661i	$0.555823\pi$
$350$		−	3.86927e6i		−	0.0902453i
$351$	3.26136e7			0.754184
$352$		−	1.31087e7i		−	0.300561i
$353$	2.94758e7			0.670102			0.335051	−	0.942200i	$-0.391246\pi$
$353$	2.94758e7			0.670102			0.335051	+	0.942200i	$0.391246\pi$
$354$		−	1.36921e7i		−	0.308645i
$355$		−	4.95328e7i		−	1.10715i
$356$			1.16367e7i			0.257918i
$357$			4.05288e7i			0.890757i
$358$	6.00955e7			1.30976
$359$	−6.23290e7			−1.34712			−0.673561	−	0.739132i	$-0.735236\pi$
$359$	−6.23290e7			−1.34712			−0.673561	+	0.739132i	$0.735236\pi$
$360$			3.30566e7i			0.708517i
$361$	2.53344e7	−	3.96419e7i	0.538504	−	0.842623i
$362$	−5.45215e7			−1.14932
$363$		−	1.49361e8i		−	3.12261i
$364$			5.58670e6i			0.115838i
$365$	−8.46860e7			−1.74154
$366$	3.89564e7			0.794577
$367$	−3.26138e7			−0.659787			−0.329893	−	0.944018i	$-0.607013\pi$
$367$	−3.26138e7			−0.659787			−0.329893	+	0.944018i	$0.607013\pi$
$368$	−7.27773e6			−0.146034
$369$			6.88325e7i			1.36998i
$370$	4.78198e6			0.0944066
$371$		−	1.55525e7i		−	0.304563i
$372$	7.11486e7			1.38209
$373$		−	3.13236e7i		−	0.603594i	−0.953372	−	0.301797i	$-0.902413\pi$
$373$		−	3.13236e7i		−	0.603594i	0.953372	−	0.301797i	$-0.0975865\pi$
$374$			9.19224e7i			1.75714i
$375$		−	6.60926e7i		−	1.25331i
$376$			2.81106e7i			0.528819i
$377$	3.16797e7			0.591231
$378$	1.69315e7			0.313487
$379$			3.69113e7i			0.678019i	0.940783	+	0.339009i	$0.110092\pi$
$379$			3.69113e7i			0.678019i	−0.940783	+	0.339009i	$0.889908\pi$
$380$	−2.79159e7	+	1.52892e7i	−0.508745	+	0.278635i
$381$	3.74368e7			0.676898
$382$			3.00279e7i			0.538685i
$383$		−	5.24493e6i		−	0.0933563i	−0.998910	−	0.0466781i	$-0.985136\pi$
$383$		−	5.24493e6i		−	0.0933563i	0.998910	−	0.0466781i	$-0.0148635\pi$
$384$	8.26542e6			0.145973
$385$	−4.15389e7			−0.727902
$386$	−2.09070e7			−0.363521
$387$	4.03546e7			0.696241
$388$		−	1.58504e7i		−	0.271360i
$389$	−6.19885e6			−0.105308			−0.0526541	−	0.998613i	$-0.516768\pi$
$389$	−6.19885e6			−0.105308			−0.0526541	+	0.998613i	$0.516768\pi$
$390$		−	5.04503e7i		−	0.850491i
$391$	5.10337e7			0.853742
$392$		−	1.83964e7i		−	0.305404i
$393$			1.13038e8i			1.86228i
$394$		−	4.02161e7i		−	0.657523i
$395$			7.37427e7i			1.19654i
$396$	9.11932e7			1.46851
$397$	4.07803e7			0.651746			0.325873	−	0.945414i	$-0.394342\pi$
$397$	4.07803e7			0.651746			0.325873	+	0.945414i	$0.394342\pi$
$398$		−	5.51418e7i		−	0.874644i
$399$	1.85966e7	+	3.39546e7i	0.292762	+	0.534540i
$400$	5.53338e6			0.0864590
$401$			2.01648e7i			0.312723i	0.987700	+	0.156362i	$0.0499765\pi$
$401$			2.01648e7i			0.312723i	−0.987700	+	0.156362i	$0.950024\pi$
$402$			1.39886e8i			2.15326i
$403$	−6.87731e7			−1.05076
$404$	2.26623e7			0.343684
$405$	−1.97730e7			−0.297651
$406$	1.64467e7			0.245753
$407$		−	1.31920e7i		−	0.195672i
$408$	−5.79597e7			−0.853386
$409$		−	7.90757e7i		−	1.15577i	−0.816117	−	0.577887i	$-0.803877\pi$
$409$		−	7.90757e7i		−	1.15577i	0.816117	−	0.577887i	$-0.196123\pi$
$410$	4.48382e7			0.650573
$411$		−	3.04421e7i		−	0.438479i
$412$			5.76878e7i			0.824882i
$413$			6.87096e6i			0.0975366i
$414$		−	5.06288e7i		−	0.713505i
$415$	−4.37839e7			−0.612590
$416$	−7.98945e6			−0.110978
$417$			1.51155e8i			2.08455i
$418$	4.21784e7	+	7.70115e7i	0.577513	+	1.05445i
$419$	1.05939e7			0.144017			0.0720087	−	0.997404i	$-0.477059\pi$
$419$	1.05939e7			0.144017			0.0720087	+	0.997404i	$0.477059\pi$
$420$		−	2.61914e7i		−	0.353518i
$421$			6.99258e7i			0.937112i	0.883434	+	0.468556i	$0.155226\pi$
$421$			6.99258e7i			0.937112i	−0.883434	+	0.468556i	$0.844774\pi$
$422$	953571.			0.0126887
$423$	−1.95557e8			−2.58375
$424$	2.22413e7			0.291785
$425$	−3.88017e7			−0.505457
$426$		−	8.61593e7i		−	1.11448i
$427$	−1.95491e7			−0.251098
$428$		−	2.83407e7i		−	0.361475i
$429$	−1.39177e8			−1.76277
$430$		−	2.62874e7i		−	0.330630i
$431$		−	1.17056e8i		−	1.46205i	−0.682350	−	0.731025i	$-0.739042\pi$
$431$		−	1.17056e8i		−	1.46205i	0.682350	−	0.731025i	$-0.260958\pi$
$432$			2.42134e7i			0.300334i
$433$			6.21462e7i			0.765510i	0.923850	+	0.382755i	$0.125025\pi$
$433$			6.21462e7i			0.765510i	−0.923850	+	0.382755i	$0.874975\pi$
$434$	−3.57038e7			−0.436763
$435$	−1.48520e8			−1.80434
$436$			2.71176e7i			0.327184i
$437$	4.27554e7	−	2.34167e7i	0.512327	−	0.280596i
$438$	−1.47306e8			−1.75307
$439$			1.75788e7i			0.207776i	0.994589	+	0.103888i	$0.0331283\pi$
$439$			1.75788e7i			0.207776i	−0.994589	+	0.103888i	$0.966872\pi$
$440$		−	5.94041e7i		−	0.697363i
$441$	1.27978e8			1.49217
$442$	5.60245e7			0.648800
$443$	1.48852e8			1.71215			0.856076	−	0.516849i	$-0.172895\pi$
$443$	1.48852e8			1.71215			0.856076	+	0.516849i	$0.172895\pi$
$444$	8.31796e6			0.0950315
$445$			5.27336e7i			0.598422i
$446$	−1.82457e7			−0.205663
$447$			2.13275e7i			0.238791i
$448$	−4.14776e6			−0.0461295
$449$			7.98608e6i			0.0882256i	0.999027	+	0.0441128i	$0.0140461\pi$
$449$			7.98608e6i			0.0882256i	−0.999027	+	0.0441128i	$0.985954\pi$
$450$			3.84939e7i			0.422430i
$451$		−	1.23695e8i		−	1.34841i
$452$		−	6.87942e7i		−	0.744966i
$453$	−7.31240e7			−0.786621
$454$	7.06136e7			0.754607
$455$			2.53170e7i			0.268768i
$456$	−4.85579e7	+	2.65947e7i	−0.512113	+	0.280479i
$457$	2.88637e7			0.302415			0.151207	−	0.988502i	$-0.451684\pi$
$457$	2.88637e7			0.302415			0.151207	+	0.988502i	$0.451684\pi$
$458$			8.87475e6i			0.0923761i
$459$		−	1.69792e8i		−	1.75582i
$460$	−3.29801e7			−0.338828
$461$	1.01419e8			1.03518			0.517591	−	0.855628i	$-0.326829\pi$
$461$	1.01419e8			1.03518			0.517591	+	0.855628i	$0.326829\pi$
$462$	−7.22544e7			−0.732720
$463$	1.34040e8			1.35049			0.675243	−	0.737595i	$-0.264039\pi$
$463$	1.34040e8			1.35049			0.675243	+	0.737595i	$0.264039\pi$
$464$			2.35201e7i			0.235443i
$465$	3.22421e8			3.20674
$466$		−	4.33202e7i		−	0.428087i
$467$	−6.65277e7			−0.653208			−0.326604	−	0.945161i	$-0.605904\pi$
$467$	−6.65277e7			−0.653208			−0.326604	+	0.945161i	$0.605904\pi$
$468$		−	5.55801e7i		−	0.542227i
$469$		−	7.01977e7i		−	0.680463i
$470$			1.27387e8i			1.22697i
$471$		−	2.66911e7i		−	0.255449i
$472$	−9.82606e6			−0.0934444
$473$	−7.25189e7			−0.685280
$474$			1.28271e8i			1.20446i
$475$	−3.25076e7	+	1.78041e7i	−0.303323	+	0.166127i
$476$	2.90853e7			0.269683
$477$			1.54726e8i			1.42563i
$478$			1.87077e7i			0.171292i
$479$	2.52760e6			0.0229986			0.0114993	−	0.999934i	$-0.496340\pi$
$479$	2.52760e6			0.0229986			0.0114993	+	0.999934i	$0.496340\pi$
$480$	3.74560e7			0.338686
$481$	−8.04023e6			−0.0722493
$482$	6.60355e7			0.589707
$483$			4.01143e7i			0.356007i
$484$	−1.07188e8			−0.945390
$485$		−	7.18286e7i		−	0.629611i
$486$	6.31182e7			0.549852
$487$			1.35097e8i			1.16966i	0.811156	+	0.584829i	$0.198838\pi$
$487$			1.35097e8i			1.16966i	−0.811156	+	0.584829i	$0.801162\pi$
$488$		−	2.79569e7i		−	0.240563i
$489$		−	2.04308e8i		−	1.74727i
$490$		−	8.33659e7i		−	0.708599i
$491$	2.73472e7			0.231030			0.115515	−	0.993306i	$-0.463148\pi$
$491$	2.73472e7			0.231030			0.115515	+	0.993306i	$0.463148\pi$
$492$	7.79932e7			0.654880
$493$		−	1.64930e8i		−	1.37645i
$494$	4.69367e7	−	2.57067e7i	0.389342	−	0.213239i
$495$	4.13255e8			3.40724
$496$		−	5.10595e7i		−	0.418438i
$497$			4.32365e7i			0.352193i
$498$	−7.61593e7			−0.616645
$499$	−2.93681e7			−0.236360			−0.118180	−	0.992992i	$-0.537706\pi$
$499$	−2.93681e7			−0.236360			−0.118180	+	0.992992i	$0.537706\pi$
$500$	−4.74310e7			−0.379448
$501$	−3.37811e8			−2.68634
$502$		−	1.30230e8i		−	1.02944i
$503$	−4.59648e7			−0.361178			−0.180589	−	0.983559i	$-0.557800\pi$
$503$	−4.59648e7			−0.361178			−0.180589	+	0.983559i	$0.557800\pi$
$504$		−	2.88546e7i		−	0.225384i
$505$	1.02698e8			0.797418
$506$			9.09823e7i			0.702272i
$507$		−	1.30404e8i		−	1.00061i
$508$		−	2.68663e7i		−	0.204935i
$509$		−	2.38227e7i		−	0.180650i	−0.995912	−	0.0903250i	$-0.971209\pi$
$509$		−	2.38227e7i		−	0.180650i	0.995912	−	0.0903250i	$-0.0287906\pi$
$510$	−2.62653e8			−1.98003
$511$	7.39211e7			0.553995
$512$		−	5.93164e6i		−	0.0441942i
$513$	−7.79087e7	−	1.42250e8i	−0.577077	−	1.05366i
$514$	−1.00086e8			−0.737030
$515$			2.61421e8i			1.91390i
$516$		−	4.57253e7i		−	0.332818i
$517$	3.51424e8			2.54308
$518$	−4.17412e6			−0.0300314
$519$	−3.01931e8			−2.15976
$520$	−3.62054e7			−0.257492
$521$			1.74523e8i			1.23407i	0.786936	+	0.617034i	$0.211666\pi$
$521$			1.74523e8i			1.23407i	−0.786936	+	0.617034i	$0.788334\pi$
$522$	−1.63622e8			−1.15035
$523$			1.72413e8i			1.20522i	0.798037	+	0.602608i	$0.205872\pi$
$523$			1.72413e8i			1.20522i	−0.798037	+	0.602608i	$0.794128\pi$
$524$	8.11210e7			0.563818
$525$		−	3.04996e7i		−	0.210773i
$526$		−	1.73974e8i		−	1.19543i
$527$			3.58045e8i			2.44627i
$528$		−	1.03330e8i		−	0.701979i
$529$	−9.75241e7			−0.658787
$530$	1.00790e8			0.677002
$531$		−	6.83567e7i		−	0.456560i
$532$	2.43674e7	−	1.33458e7i	0.161835	−	0.0886356i
$533$	−7.53892e7			−0.497883
$534$			9.17269e7i			0.602383i
$535$		−	1.28430e8i		−	0.838697i
$536$	1.00389e8			0.651914
$537$	4.73704e8			3.05903
$538$	1.91104e8			1.22722
$539$	−2.29982e8			−1.46868
$540$			1.09727e8i			0.696837i
$541$	−1.71366e8			−1.08227			−0.541133	−	0.840937i	$-0.682004\pi$
$541$	−1.71366e8			−1.08227			−0.541133	+	0.840937i	$0.682004\pi$
$542$		−	5.33237e7i		−	0.334906i
$543$	−4.29767e8			−2.68432
$544$			4.15945e7i			0.258368i
$545$			1.22888e8i			0.759134i
$546$			4.40373e7i			0.270547i
$547$		−	6.84113e7i		−	0.417990i	−0.977917	−	0.208995i	$-0.932981\pi$
$547$		−	6.84113e7i		−	0.417990i	0.977917	−	0.208995i	$-0.0670193\pi$
$548$	−2.18466e7			−0.132752
$549$	1.94487e8			1.17537
$550$		−	6.91753e7i		−	0.415779i
$551$	−7.56779e7	−	1.38177e8i	−0.452391	−	0.825999i
$552$	−5.73669e7			−0.341070
$553$		−	6.43689e7i		−	0.380628i
$554$			5.44577e7i			0.320280i
$555$	3.76940e7			0.220492
$556$	1.08475e8			0.631112
$557$	−1.07351e7			−0.0621216			−0.0310608	−	0.999517i	$-0.509889\pi$
$557$	−1.07351e7			−0.0621216			−0.0310608	+	0.999517i	$0.509889\pi$
$558$	3.55204e8			2.04445
$559$			4.41986e7i			0.253030i
$560$	−1.87962e7			−0.107030
$561$			7.24580e8i			4.10391i
$562$	−1.33230e8			−0.750573
$563$			3.13052e7i			0.175425i	0.996146	+	0.0877125i	$0.0279557\pi$
$563$			3.13052e7i			0.175425i	−0.996146	+	0.0877125i	$0.972044\pi$
$564$			2.21583e8i			1.23509i
$565$		−	3.11751e8i		−	1.72847i
$566$			1.80346e8i			0.994619i
$567$	1.72596e7			0.0946850
$568$	−6.18318e7			−0.337417
$569$		−	1.46303e8i		−	0.794173i	−0.917781	−	0.397087i	$-0.870021\pi$
$569$		−	1.46303e8i		−	0.794173i	0.917781	−	0.397087i	$-0.129979\pi$
$570$	−2.20047e8	+	1.20518e8i	−1.18821	+	0.650769i
$571$	1.81705e7			0.0976021			0.0488011	−	0.998809i	$-0.484460\pi$
$571$	1.81705e7			0.0976021			0.0488011	+	0.998809i	$0.484460\pi$
$572$			9.98798e7i			0.533691i
$573$			2.36696e8i			1.25813i
$574$	−3.91386e7			−0.206952
$575$	−3.84049e7			−0.202015
$576$	4.12645e7			0.215928
$577$	−1.99621e8			−1.03915			−0.519576	−	0.854424i	$-0.673910\pi$
$577$	−1.99621e8			−1.03915			−0.519576	+	0.854424i	$0.673910\pi$
$578$		−	1.55130e8i		−	0.803365i
$579$	−1.64800e8			−0.849027
$580$			1.06585e8i			0.546275i
$581$	3.82183e7			0.194869
$582$		−	1.24941e8i		−	0.633778i
$583$		−	2.78049e8i		−	1.40319i
$584$			1.05713e8i			0.530752i
$585$		−	2.51869e8i		−	1.25808i
$586$	−1.07621e8			−0.534815
$587$	−1.00352e8			−0.496147			−0.248073	−	0.968741i	$-0.579797\pi$
$587$	−1.00352e8			−0.496147			−0.248073	+	0.968741i	$0.579797\pi$
$588$		−	1.45010e8i		−	0.713289i
$589$	1.64288e8	+	2.99966e8i	0.804008	+	1.46800i
$590$	−4.45282e7			−0.216810
$591$		−	3.17004e8i		−	1.53569i
$592$		−	5.96934e6i		−	0.0287714i
$593$	2.86815e8			1.37543			0.687715	−	0.725981i	$-0.258614\pi$
$593$	2.86815e8			1.37543			0.687715	+	0.725981i	$0.258614\pi$
$594$	3.02703e8			1.44430
$595$	1.31804e8			0.625719
$596$	1.53056e7			0.0722954
$597$		−	4.34656e8i		−	2.04279i
$598$	5.54515e7			0.259304
$599$		−	4.18781e8i		−	1.94853i	−0.225411	−	0.974264i	$-0.572373\pi$
$599$		−	4.18781e8i		−	1.94853i	0.225411	−	0.974264i	$-0.427627\pi$
$600$	4.36170e7			0.201930
$601$		−	3.74442e8i		−	1.72489i	−0.506152	−	0.862444i	$-0.668933\pi$
$601$		−	3.74442e8i		−	1.72489i	0.506152	−	0.862444i	$-0.331067\pi$
$602$			2.29458e7i			0.105176i
$603$			6.98372e8i			3.18519i
$604$			5.24771e7i			0.238155i
$605$	−4.85739e8			−2.19350
$606$	1.78636e8			0.802696
$607$			1.51221e8i			0.676155i	0.941118	+	0.338077i	$0.109777\pi$
$607$			1.51221e8i			0.676155i	−0.941118	+	0.338077i	$0.890223\pi$
$608$	1.90856e7	+	3.48474e7i	0.0849169	+	0.155046i
$609$	1.29641e8			0.573972
$610$		−	1.26691e8i		−	0.558156i
$611$		−	2.14184e8i		−	0.938996i
$612$	−2.89359e8			−1.26236
$613$	−2.90931e8			−1.26301			−0.631507	−	0.775370i	$-0.717563\pi$
$613$	−2.90931e8			−1.26301			−0.631507	+	0.775370i	$0.717563\pi$
$614$	6.91216e7			0.298613
$615$	3.53438e8			1.51945
$616$			5.18530e7i			0.221836i
$617$	−2.65708e7			−0.113123			−0.0565614	−	0.998399i	$-0.518014\pi$
$617$	−2.65708e7			−0.113123			−0.0565614	+	0.998399i	$0.518014\pi$
$618$			4.54725e8i			1.92656i
$619$	1.47842e8			0.623340			0.311670	−	0.950190i	$-0.399112\pi$
$619$	1.47842e8			0.623340			0.311670	+	0.950190i	$0.399112\pi$
$620$		−	2.31384e8i		−	0.970862i
$621$		−	1.68055e8i		−	0.701742i
$622$			5.28309e7i			0.219542i
$623$		−	4.60304e7i		−	0.190362i
$624$	−6.29771e7			−0.259196
$625$	−2.99373e8			−1.22623
$626$		−	5.22242e7i		−	0.212887i
$627$	3.32472e8	+	6.07045e8i	1.34882	+	2.46274i
$628$	−1.91548e7			−0.0773388
$629$			4.18588e7i			0.168204i
$630$		−	1.30759e8i		−	0.522937i
$631$	2.71652e8			1.08125			0.540624	−	0.841265i	$-0.318188\pi$
$631$	2.71652e8			1.08125			0.540624	+	0.841265i	$0.318188\pi$
$632$	9.20530e7			0.364659
$633$	7.51654e6			0.0296351
$634$	1.18889e7			0.0466522
$635$		−	1.21749e8i		−	0.475492i
$636$	1.75318e8			0.681483
$637$			1.40168e8i			0.542290i
$638$	2.94036e8			1.13224
$639$		−	4.30144e8i		−	1.64858i
$640$		−	2.68801e7i		−	0.102539i
$641$			5.20556e7i			0.197648i	0.995105	+	0.0988242i	$0.0315081\pi$
$641$			5.20556e7i			0.197648i	−0.995105	+	0.0988242i	$0.968492\pi$
$642$		−	2.23396e8i		−	0.844249i
$643$	−2.93715e7			−0.110483			−0.0552413	−	0.998473i	$-0.517593\pi$
$643$	−2.93715e7			−0.110483			−0.0552413	+	0.998473i	$0.517593\pi$
$644$	2.87879e7			0.107783
$645$		−	2.07211e8i		−	0.772206i
$646$	−1.33834e8	−	2.44360e8i	−0.496441	−	0.906428i
$647$	1.80951e8			0.668109			0.334054	−	0.942554i	$-0.391583\pi$
$647$	1.80951e8			0.668109			0.334054	+	0.942554i	$0.391583\pi$
$648$			2.46827e7i			0.0907125i
$649$			1.22840e8i			0.449372i
$650$	−4.21607e7			−0.153521
$651$	−2.81436e8			−1.02009
$652$	−1.46621e8			−0.528996
$653$	−3.56579e8			−1.28061			−0.640304	−	0.768122i	$-0.721192\pi$
$653$	−3.56579e8			−1.28061			−0.640304	+	0.768122i	$0.721192\pi$
$654$			2.13755e8i			0.764159i
$655$	3.67612e8			1.30817
$656$		−	5.59715e7i		−	0.198269i
$657$	−7.35415e8			−2.59320
$658$		−	1.11195e8i		−	0.390307i
$659$			3.48175e7i			0.121658i	0.998148	+	0.0608290i	$0.0193744\pi$
$659$			3.48175e7i			0.121658i	−0.998148	+	0.0608290i	$0.980626\pi$
$660$		−	4.68254e8i		−	1.62873i
$661$			3.81847e8i			1.32216i	0.750315	+	0.661081i	$0.229902\pi$
$661$			3.81847e8i			1.32216i	−0.750315	+	0.661081i	$0.770098\pi$
$662$	1.58624e8			0.546756
$663$	4.41614e8			1.51531
$664$			5.46554e7i			0.186693i
$665$	1.10424e8	−	6.04783e7i	0.375491	−	0.205653i
$666$	4.15268e7			0.140574
$667$		−	1.63243e8i		−	0.550121i
$668$			2.42428e8i			0.813306i
$669$	−1.43822e8			−0.480338
$670$	4.54926e8			1.51257
$671$	−3.49502e8			−1.15686
$672$	−3.26948e7			−0.107738
$673$			1.09248e7i			0.0358400i	0.999839	+	0.0179200i	$0.00570442\pi$
$673$			1.09248e7i			0.0358400i	−0.999839	+	0.0179200i	$0.994296\pi$
$674$	4.25772e7			0.139058
$675$			1.27775e8i			0.415466i
$676$	−9.35835e7			−0.302942
$677$		−	4.18009e8i		−	1.34716i	−0.739113	−	0.673581i	$-0.764755\pi$
$677$		−	4.18009e8i		−	1.34716i	0.739113	−	0.673581i	$-0.235245\pi$
$678$		−	5.42272e8i		−	1.73992i
$679$			6.26981e7i			0.200283i
$680$			1.88491e8i			0.599467i
$681$	5.56614e8			1.76243
$682$	−6.38318e8			−2.01226
$683$		−	1.45637e8i		−	0.457098i	−0.973532	−	0.228549i	$-0.926602\pi$
$683$		−	1.45637e8i		−	0.457098i	0.973532	−	0.228549i	$-0.0733982\pi$
$684$	−2.42422e8	+	1.32772e8i	−0.757537	+	0.414895i
$685$	−9.90011e7			−0.308013
$686$			1.57010e8i			0.486358i
$687$			6.99554e7i			0.215750i
$688$	−3.28145e7			−0.100763
$689$	−1.69464e8			−0.518108
$690$	−2.59967e8			−0.791353
$691$	−5.23063e7			−0.158533			−0.0792666	−	0.996853i	$-0.525258\pi$
$691$	−5.23063e7			−0.158533			−0.0792666	+	0.996853i	$0.525258\pi$
$692$			2.16679e8i			0.653881i
$693$	−3.60725e8			−1.08387
$694$			1.89141e8i			0.565858i
$695$	4.91572e8			1.46431
$696$			1.85398e8i			0.549891i
$697$			3.92489e8i			1.15912i
$698$			8.39108e7i			0.246747i
$699$		−	3.41472e8i		−	0.999824i
$700$	−2.18879e7			−0.0638130
$701$	−2.76866e8			−0.803741			−0.401870	−	0.915697i	$-0.631640\pi$
$701$	−2.76866e8			−0.803741			−0.401870	+	0.915697i	$0.631640\pi$
$702$		−	1.84490e8i		−	0.533288i
$703$	1.92069e7	+	3.50688e7i	0.0552828	+	0.100938i
$704$	−7.41542e7			−0.212529
$705$			1.00413e9i			2.86566i
$706$		−	1.66740e8i		−	0.473834i
$707$	−8.96431e7			−0.253664
$708$	−7.74541e7			−0.218245
$709$	4.63177e8			1.29960			0.649798	−	0.760107i	$-0.274853\pi$
$709$	4.63177e8			1.29960			0.649798	+	0.760107i	$0.274853\pi$
$710$	−2.80200e8			−0.782876
$711$			6.40383e8i			1.78169i
$712$	6.58274e7			0.182375
$713$			3.54383e8i			0.977696i
$714$	2.29266e8			0.629861
$715$			4.52620e8i			1.23827i
$716$		−	3.39951e8i		−	0.926142i
$717$			1.47464e8i			0.400064i
$718$			3.52586e8i			0.952558i
$719$	4.49617e8			1.20964			0.604820	−	0.796362i	$-0.293245\pi$
$719$	4.49617e8			1.20964			0.604820	+	0.796362i	$0.293245\pi$
$720$	1.86996e8			0.500997
$721$		−	2.28190e8i		−	0.608823i
$722$	−2.24249e8	−	1.43313e8i	−0.595825	−	0.380780i
$723$	5.20526e8			1.37730
$724$			3.08420e8i			0.812695i
$725$			1.24117e8i			0.325698i
$726$	−8.44914e8			−2.20802
$727$	−6.25998e8			−1.62918			−0.814591	−	0.580036i	$-0.803038\pi$
$727$	−6.25998e8			−1.62918			−0.814591	+	0.580036i	$0.803038\pi$
$728$	3.16032e7			0.0819099
$729$	5.96933e8			1.54079
$730$			4.79056e8i			1.23145i
$731$	2.30105e8			0.589080
$732$		−	2.20371e8i		−	0.561851i
$733$	6.84776e8			1.73875			0.869374	−	0.494154i	$-0.164522\pi$
$733$	6.84776e8			1.73875			0.869374	+	0.494154i	$0.164522\pi$
$734$			1.84492e8i			0.466540i
$735$		−	6.57134e8i		−	1.65498i
$736$			4.11691e7i			0.103261i
$737$		−	1.25500e9i		−	3.13504i
$738$	3.89376e8			0.968723
$739$	−2.94973e8			−0.730885			−0.365443	−	0.930834i	$-0.619082\pi$
$739$	−2.94973e8			−0.730885			−0.365443	+	0.930834i	$0.619082\pi$
$740$		−	2.70509e7i		−	0.0667555i
$741$	3.69979e8	−	2.02634e8i	0.909333	−	0.498032i
$742$	−8.79781e7			−0.215359
$743$			2.91574e8i			0.710857i	0.934703	+	0.355429i	$0.115665\pi$
$743$			2.91574e8i			0.710857i	−0.934703	+	0.355429i	$0.884335\pi$
$744$		−	4.02478e8i		−	0.977289i
$745$	6.93595e7			0.167740
$746$	−1.77193e8			−0.426806
$747$	−3.80220e8			−0.912164
$748$	5.19991e8			1.24249
$749$			1.12105e8i			0.266795i
$750$	−3.73876e8			−0.886225
$751$			3.92028e7i			0.0925546i	0.998929	+	0.0462773i	$0.0147358\pi$
$751$			3.92028e7i			0.0925546i	−0.998929	+	0.0462773i	$0.985264\pi$
$752$	1.59018e8			0.373931
$753$		−	1.02654e9i		−	2.40431i
$754$		−	1.79208e8i		−	0.418064i
$755$			2.37808e8i			0.552567i
$756$		−	9.57788e7i		−	0.221669i
$757$	6.67071e7			0.153774			0.0768872	−	0.997040i	$-0.475502\pi$
$757$	6.67071e7			0.153774			0.0768872	+	0.997040i	$0.475502\pi$
$758$	2.08802e8			0.479432
$759$			7.17170e8i			1.64020i
$760$	8.64890e7	+	1.57916e8i	0.197024	+	0.359737i
$761$	−5.95378e8			−1.35095			−0.675475	−	0.737383i	$-0.736061\pi$
$761$	−5.95378e8			−1.35095			−0.675475	+	0.737383i	$0.736061\pi$
$762$		−	2.11774e8i		−	0.478639i
$763$		−	1.07267e8i		−	0.241486i
$764$	1.69864e8			0.380908
$765$	−1.31127e9			−2.92893
$766$	−2.96698e7			−0.0660128
$767$	7.48681e7			0.165924
$768$		−	4.67563e7i		−	0.103218i
$769$	2.23860e7			0.0492263			0.0246132	−	0.999697i	$-0.492165\pi$
$769$	2.23860e7			0.0492263			0.0246132	+	0.999697i	$0.492165\pi$
$770$			2.34979e8i			0.514704i
$771$	−7.88932e8			−1.72138
$772$			1.18268e8i			0.257048i
$773$			3.05640e8i			0.661717i	0.943680	+	0.330858i	$0.107338\pi$
$773$			3.05640e8i			0.661717i	−0.943680	+	0.330858i	$0.892662\pi$
$774$		−	2.28280e8i		−	0.492317i
$775$		−	2.69443e8i		−	0.578844i
$776$	−8.96636e7			−0.191881
$777$	−3.29026e7			−0.0701402
$778$			3.50660e7i			0.0744642i
$779$	1.80093e8	+	3.28823e8i	0.380964	+	0.695584i
$780$	−2.85390e8			−0.601388
$781$			7.72988e8i			1.62263i
$782$		−	2.88690e8i		−	0.603687i
$783$	−5.43120e8			−1.13138
$784$	−1.04066e8			−0.215953
$785$	−8.68026e7			−0.179442
$786$	6.39438e8			1.31683
$787$		−	4.92670e8i		−	1.01072i	−0.862908	−	0.505361i	$-0.831359\pi$
$787$		−	4.92670e8i		−	1.01072i	0.862908	−	0.505361i	$-0.168641\pi$
$788$	−2.27496e8			−0.464939
$789$		−	1.37135e9i		−	2.79201i
$790$	4.17152e8			0.846083
$791$			2.72123e8i			0.549839i
$792$		−	5.15867e8i		−	1.03839i
$793$			2.13013e8i			0.427156i
$794$		−	2.30688e8i		−	0.460854i
$795$	7.94479e8			1.58118
$796$	−3.11929e8			−0.618467
$797$		−	4.16098e8i		−	0.821904i	−0.911657	−	0.410952i	$-0.865196\pi$
$797$		−	4.16098e8i		−	0.821904i	0.911657	−	0.410952i	$-0.134804\pi$
$798$	1.92076e8	−	1.05198e8i	0.377977	−	0.207014i
$799$	−1.11508e9			−2.18608
$800$		−	3.13015e7i		−	0.0611358i
$801$			4.57940e8i			0.891068i
$802$	1.14069e8			0.221129
$803$	1.32157e9			2.55237
$804$	7.91316e8			1.52259
$805$	1.30456e8			0.250079
$806$			3.89039e8i			0.742999i
$807$	1.50639e9			2.86626
$808$		−	1.28197e8i		−	0.243022i
$809$	2.33716e7			0.0441410			0.0220705	−	0.999756i	$-0.492974\pi$
$809$	2.33716e7			0.0441410			0.0220705	+	0.999756i	$0.492974\pi$
$810$			1.11853e8i			0.210471i
$811$		−	8.62824e8i		−	1.61756i	−0.588113	−	0.808779i	$-0.700129\pi$
$811$		−	8.62824e8i		−	1.61756i	0.588113	−	0.808779i	$-0.299871\pi$
$812$		−	9.30363e7i		−	0.173774i
$813$		−	4.20325e8i		−	0.782193i
$814$	−7.46255e7			−0.138361
$815$	−6.64434e8			−1.22738
$816$			3.27869e8i			0.603435i
$817$	1.92780e8	−	1.05583e8i	0.353504	−	0.193611i
$818$	−4.47320e8			−0.817256
$819$			2.19853e8i			0.400203i
$820$		−	2.53643e8i		−	0.460025i
$821$	−2.90453e8			−0.524863			−0.262432	−	0.964951i	$-0.584524\pi$
$821$	−2.90453e8			−0.524863			−0.262432	+	0.964951i	$0.584524\pi$
$822$	−1.72206e8			−0.310051
$823$	1.96402e8			0.352328			0.176164	−	0.984361i	$-0.443631\pi$
$823$	1.96402e8			0.352328			0.176164	+	0.984361i	$0.443631\pi$
$824$	3.26331e8			0.583280
$825$		−	5.45276e8i		−	0.971078i
$826$	3.88680e7			0.0689688
$827$		−	5.04256e7i		−	0.0891528i	−0.999006	−	0.0445764i	$-0.985806\pi$
$827$		−	5.04256e7i		−	0.0891528i	0.999006	−	0.0445764i	$-0.0141938\pi$
$828$	−2.86400e8			−0.504524
$829$			4.36377e7i			0.0765946i	0.999266	+	0.0382973i	$0.0121934\pi$
$829$			4.36377e7i			0.0765946i	−0.999266	+	0.0382973i	$0.987807\pi$
$830$			2.47679e8i			0.433166i
$831$			4.29264e8i			0.748034i
$832$			4.51952e7i			0.0784733i
$833$	7.29740e8			1.26251
$834$	8.55060e8			1.47400
$835$			1.09860e9i			1.88704i
$836$	4.35643e8	−	2.38597e8i	0.745610	−	0.408363i
$837$	1.17905e9			2.01074
$838$		−	5.99283e7i		−	0.101836i
$839$		−	6.65465e7i		−	0.112678i	−0.998412	−	0.0563390i	$-0.982057\pi$
$839$		−	6.65465e7i		−	0.112678i	0.998412	−	0.0563390i	$-0.0179428\pi$
$840$	−1.48161e8			−0.249975
$841$	6.72553e7			0.113068
$842$	3.95560e8			0.662638
$843$	−1.05019e9			−1.75301
$844$		−	5.39421e6i		−	0.00897223i
$845$	−4.24088e8			−0.702887
$846$			1.10623e9i			1.82699i
$847$	4.23995e8			0.697767
$848$		−	1.25816e8i		−	0.206323i
$849$			1.42158e9i			2.32300i
$850$			2.19496e8i			0.357412i
$851$			4.14307e7i			0.0672255i
$852$	−4.87391e8			−0.788058
$853$	1.82272e8			0.293679			0.146840	−	0.989160i	$-0.453090\pi$
$853$	1.82272e8			0.293679			0.146840	+	0.989160i	$0.453090\pi$
$854$			1.10587e8i			0.177553i
$855$	−1.09857e9	+	6.01676e8i	−1.75764	+	0.962641i
$856$	−1.60319e8			−0.255602
$857$		−	1.05800e9i		−	1.68091i	−0.541884	−	0.840453i	$-0.682289\pi$
$857$		−	1.05800e9i		−	1.68091i	0.541884	−	0.840453i	$-0.317711\pi$
$858$			7.87305e8i			1.24647i
$859$	−9.80261e8			−1.54654			−0.773272	−	0.634074i	$-0.781381\pi$
$859$	−9.80261e8			−1.54654			−0.773272	+	0.634074i	$0.781381\pi$
$860$	−1.48704e8			−0.233790
$861$	−3.08511e8			−0.483349
$862$	−6.62170e8			−1.03383
$863$		−	9.42971e8i		−	1.46712i	−0.679624	−	0.733560i	$-0.737857\pi$
$863$		−	9.42971e8i		−	1.46712i	0.679624	−	0.733560i	$-0.262143\pi$
$864$	1.36972e8			0.212368
$865$			9.81914e8i			1.51714i
$866$	3.51552e8			0.541297
$867$		−	1.22282e9i		−	1.87631i
$868$			2.01971e8i			0.308838i
$869$		−	1.15080e9i		−	1.75364i
$870$			8.40157e8i			1.27586i
$871$	−7.64895e8			−1.15757
$872$	1.53400e8			0.231354
$873$		−	6.23761e8i		−	0.937509i
$874$	−1.32465e8	−	2.41861e8i	−0.198411	−	0.362270i
$875$	1.87619e8			0.280060
$876$			8.33289e8i			1.23960i
$877$			4.75095e8i			0.704339i	0.935936	+	0.352169i	$0.114556\pi$
$877$			4.75095e8i			0.704339i	−0.935936	+	0.352169i	$0.885444\pi$
$878$	9.94406e7			0.146920
$879$	−8.48325e8			−1.24909
$880$	−3.36041e8			−0.493110
$881$	−5.96553e8			−0.872412			−0.436206	−	0.899847i	$-0.643678\pi$
$881$	−5.96553e8			−0.872412			−0.436206	+	0.899847i	$0.643678\pi$
$882$		−	7.23951e8i		−	1.05512i
$883$	−4.57670e8			−0.664769			−0.332384	−	0.943144i	$-0.607853\pi$
$883$	−4.57670e8			−0.664769			−0.332384	+	0.943144i	$0.607853\pi$
$884$		−	3.16922e8i		−	0.458771i
$885$	−3.50995e8			−0.506373
$886$		−	8.42032e8i		−	1.21067i
$887$			2.17106e8i			0.311101i	0.987828	+	0.155551i	$0.0497152\pi$
$887$			2.17106e8i			0.311101i	−0.987828	+	0.155551i	$0.950285\pi$
$888$		−	4.70535e7i		−	0.0671974i
$889$			1.06273e8i			0.151257i
$890$	2.98306e8			0.423148
$891$	3.08569e8			0.436234
$892$			1.03213e8i			0.145425i
$893$	−9.34201e8	+	5.11653e8i	−1.31186	+	0.718490i
$894$	1.20647e8			0.168851
$895$		−	1.54054e9i		−	2.14884i
$896$			2.34632e7i			0.0326185i
$897$	4.37098e8			0.605621
$898$	4.51761e7			0.0623849
$899$	1.14529e9			1.57629
$900$	2.17755e8			0.298703
$901$			8.82260e8i			1.20621i
$902$	−6.99725e8			−0.953472
$903$			1.80871e8i			0.245644i
$904$	−3.89159e8			−0.526771
$905$			1.39765e9i			1.88562i
$906$			4.13652e8i			0.556225i
$907$			9.24193e8i			1.23863i	0.785144	+	0.619314i	$0.212589\pi$
$907$			9.24193e8i			1.23863i	−0.785144	+	0.619314i	$0.787411\pi$
$908$		−	3.99451e8i		−	0.533588i
$909$	8.91827e8			1.18738
$910$	1.43214e8			0.190048
$911$			1.84559e8i			0.244107i	0.992524	+	0.122053i	$0.0389479\pi$
$911$			1.84559e8i			0.244107i	−0.992524	+	0.122053i	$0.961052\pi$
$912$	1.50442e8	+	2.74685e8i	0.198329	+	0.362119i
$913$	6.83272e8			0.897803
$914$		−	1.63278e8i		−	0.213840i
$915$		−	9.98643e8i		−	1.30361i
$916$	5.02032e7			0.0653198
$917$	−3.20883e8			−0.416139
$918$	−9.60488e8			−1.24155
$919$	−1.15589e9			−1.48926			−0.744628	−	0.667479i	$-0.767373\pi$
$919$	−1.15589e9			−1.48926			−0.744628	+	0.667479i	$0.767373\pi$
$920$			1.86564e8i			0.239587i
$921$	5.44852e8			0.697429
$922$		−	5.73713e8i		−	0.731984i
$923$	4.71117e8			0.599134
$924$			4.08732e8i			0.518111i
$925$		−	3.15005e7i		−	0.0398008i
$926$		−	7.58242e8i		−	0.954938i
$927$			2.27018e9i			2.84985i
$928$	1.33050e8			0.166483
$929$	−1.02759e9			−1.28166			−0.640829	−	0.767684i	$-0.721409\pi$
$929$	−1.02759e9			−1.28166			−0.640829	+	0.767684i	$0.721409\pi$
$930$		−	1.82389e9i		−	2.26751i
$931$	6.11368e8	−	3.34840e8i	0.757624	−	0.414943i
$932$	−2.45056e8			−0.302703
$933$			4.16441e8i			0.512753i
$934$			3.76337e8i			0.461888i
$935$	2.35642e9			2.88282
$936$	−3.14408e8			−0.383413
$937$	6.03495e8			0.733593			0.366796	−	0.930301i	$-0.380455\pi$
$937$	6.03495e8			0.733593			0.366796	+	0.930301i	$0.380455\pi$
$938$	−3.97098e8			−0.481160
$939$		−	4.11658e8i		−	0.497210i
$940$	7.20612e8			0.867597
$941$			1.56791e9i			1.88171i	0.338806	+	0.940856i	$0.389977\pi$
$941$			1.56791e9i			1.88171i	−0.338806	+	0.940856i	$0.610023\pi$
$942$	−1.50988e8			−0.180630
$943$			3.88475e8i			0.463263i
$944$			5.55846e7i			0.0660752i
$945$		−	4.34036e8i		−	0.514316i
$946$			4.10229e8i			0.484566i
$947$	−7.52683e8			−0.886262			−0.443131	−	0.896457i	$-0.646132\pi$
$947$	−7.52683e8			−0.886262			−0.443131	+	0.896457i	$0.646132\pi$
$948$	7.25610e8			0.851684
$949$		−	8.05467e8i		−	0.942430i
$950$	1.00715e8	+	1.83891e8i	0.117469	+	0.214481i
$951$	9.37142e7			0.108959
$952$		−	1.64531e8i		−	0.190694i
$953$			5.63081e8i			0.650567i	0.945617	+	0.325283i	$0.105460\pi$
$953$			5.63081e8i			0.650567i	−0.945617	+	0.325283i	$0.894540\pi$
$954$	8.75262e8			1.00808
$955$	7.69762e8			0.883784
$956$	1.05827e8			0.121122
$957$	2.31774e9			2.64441
$958$		−	1.42983e7i		−	0.0162625i
$959$	8.64166e7			0.0979809
$960$		−	2.11883e8i		−	0.239487i
$961$	−1.59879e9			−1.80145
$962$			4.54824e7i			0.0510879i
$963$		−	1.11529e9i		−	1.24884i
$964$		−	3.73553e8i		−	0.416986i
$965$			5.35949e8i			0.596405i
$966$	2.26921e8			0.251735
$967$	1.10886e9			1.22630			0.613151	−	0.789966i	$-0.289902\pi$
$967$	1.10886e9			1.22630			0.613151	+	0.789966i	$0.289902\pi$
$968$			6.06348e8i			0.668492i
$969$	−1.05495e9	−	1.92618e9i	−1.15947	−	2.11702i
$970$	−4.06324e8			−0.445202
$971$		−	1.33702e9i		−	1.46043i	−0.683216	−	0.730216i	$-0.739419\pi$
$971$		−	1.33702e9i		−	1.46043i	0.683216	−	0.730216i	$-0.260581\pi$
$972$		−	3.57051e8i		−	0.388804i
$973$	−4.29086e8			−0.465807
$974$	7.64224e8			0.827073
$975$	−3.32332e8			−0.358558
$976$	−1.58148e8			−0.170104
$977$			1.39307e9i			1.49379i	0.664943	+	0.746895i	$0.268456\pi$
$977$			1.39307e9i			1.49379i	−0.664943	+	0.746895i	$0.731544\pi$
$978$	−1.15574e9			−1.23550
$979$		−	8.22938e8i		−	0.877039i
$980$	−4.71589e8			−0.501055
$981$			1.06716e9i			1.13037i
$982$		−	1.54699e8i		−	0.163363i
$983$		−	1.21056e7i		−	0.0127446i	−0.999980	−	0.00637228i	$-0.997972\pi$
$983$		−	1.21056e7i		−	0.0127446i	0.999980	−	0.00637228i	$-0.00202837\pi$
$984$		−	4.41196e8i		−	0.463070i
$985$	−1.03093e9			−1.07875
$986$	−9.32985e8			−0.973294
$987$		−	8.76494e8i		−	0.911586i
$988$	−1.45419e8	−	2.65514e8i	−0.150783	−	0.275306i
$989$	2.27752e8			0.235436
$990$		−	2.33773e9i		−	2.40928i
$991$		−	1.84529e9i		−	1.89602i	−0.318236	−	0.948011i	$-0.603091\pi$
$991$		−	1.84529e9i		−	1.89602i	0.318236	−	0.948011i	$-0.396909\pi$
$992$	−2.88836e8			−0.295881
$993$	1.25035e9			1.27698
$994$	2.44582e8			0.249038
$995$	−1.41355e9			−1.43497
$996$			4.30822e8i			0.436034i
$997$	−8.78938e7			−0.0886897			−0.0443448	−	0.999016i	$-0.514120\pi$
$997$	−8.78938e7			−0.0886897			−0.0443448	+	0.999016i	$0.514120\pi$
$998$			1.66131e8i			0.167132i
$999$	1.37842e8			0.138257

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.7.b.a.37.1	✓	10
3.2	odd	2		342.7.d.a.37.10		10
4.3	odd	2		304.7.e.e.113.9		10
19.18	odd	2	inner	38.7.b.a.37.10	yes	10
57.56	even	2		342.7.d.a.37.5		10
76.75	even	2		304.7.e.e.113.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.7.b.a.37.1	✓	10	1.1	even	1	trivial
38.7.b.a.37.10	yes	10	19.18	odd	2	inner
304.7.e.e.113.2		10	76.75	even	2
304.7.e.e.113.9		10	4.3	odd	2
342.7.d.a.37.5		10	57.56	even	2
342.7.d.a.37.10		10	3.2	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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.65685i q^{2} -44.5903i q^{3} -32.0000 q^{4} -145.013 q^{5} -252.241 q^{6} +126.579 q^{7} +181.019i q^{8} -1259.29 q^{9} +O(q^{10})\)	\(q-5.65685i q^{2} -44.5903i q^{3} -32.0000 q^{4} -145.013 q^{5} -252.241 q^{6} +126.579 q^{7} +181.019i q^{8} -1259.29 q^{9} +820.316i q^{10} +2263.01 q^{11} +1426.89i q^{12} -1379.25i q^{13} -716.042i q^{14} +6466.16i q^{15} +1024.00 q^{16} -7180.60 q^{17} +7123.64i q^{18} +(-6015.82 + 3294.81i) q^{19} +4640.41 q^{20} -5644.21i q^{21} -12801.5i q^{22} -7107.16 q^{23} +8071.70 q^{24} +5403.69 q^{25} -7802.20 q^{26} +23645.9i q^{27} -4050.54 q^{28} +22968.8i q^{29} +36578.1 q^{30} -49862.8i q^{31} -5792.62i q^{32} -100908. i q^{33} +40619.6i q^{34} -18355.6 q^{35} +40297.4 q^{36} -5829.43i q^{37} +(18638.2 + 34030.6i) q^{38} -61501.0 q^{39} -26250.1i q^{40} -54659.6i q^{41} -31928.5 q^{42} -32045.4 q^{43} -72416.2 q^{44} +182614. q^{45} +40204.2i q^{46} +155291. q^{47} -45660.4i q^{48} -101627. q^{49} -30567.9i q^{50} +320185. i q^{51} +44135.9i q^{52} -122867. i q^{53} +133762. q^{54} -328165. q^{55} +22913.3i q^{56} +(146916. + 268247. i) q^{57} +129931. q^{58} +54281.8i q^{59} -206917. i q^{60} -154441. q^{61} -282066. q^{62} -159401. q^{63} -32768.0 q^{64} +200008. i q^{65} -570822. q^{66} -554574. i q^{67} +229779. q^{68} +316910. i q^{69} +103835. i q^{70} +341576. i q^{71} -227956. i q^{72} +583990. q^{73} -32976.3 q^{74} -240952. i q^{75} +(192506. - 105434. i) q^{76} +286450. q^{77} +347902. i q^{78} -508526. i q^{79} -148493. q^{80} +136354. q^{81} -309202. q^{82} +301931. q^{83} +180615. i q^{84} +1.04128e6 q^{85} +181276. i q^{86} +1.02419e6 q^{87} +409648. i q^{88} -363648. i q^{89} -1.03302e6i q^{90} -174584. i q^{91} +227429. q^{92} -2.22340e6 q^{93} -878457. i q^{94} +(872371. - 477789. i) q^{95} -258295. q^{96} +495326. i q^{97} +574887. i q^{98} -2.84979e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 320 q^{4} - 112 q^{5} + 160 q^{6} - 224 q^{7} - 2890 q^{9}+O(q^{10}) \) 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

Properties

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