LMFDB - Embedded newform 138.7.b.a.91.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,7,Mod(91,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("138.91");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	138.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:	$31.7474635395$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.19
Character	$\chi$	$=$	138.91
Dual form			138.7.b.a.91.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685 q^{2} +15.5885 q^{3} +32.0000 q^{4} -227.787i q^{5} +88.1816 q^{6} +132.014i q^{7} +181.019 q^{8} +243.000 q^{9} +O(q^{10})$	\(q+5.65685 q^{2} +15.5885 q^{3} +32.0000 q^{4} -227.787i q^{5} +88.1816 q^{6} +132.014i q^{7} +181.019 q^{8} +243.000 q^{9} -1288.56i q^{10} -1009.98i q^{11} +498.831 q^{12} -597.885 q^{13} +746.783i q^{14} -3550.84i q^{15} +1024.00 q^{16} +500.961i q^{17} +1374.62 q^{18} -2114.23i q^{19} -7289.18i q^{20} +2057.89i q^{21} -5713.33i q^{22} +(2714.84 - 11860.3i) q^{23} +2821.81 q^{24} -36261.8 q^{25} -3382.15 q^{26} +3788.00 q^{27} +4224.44i q^{28} -12348.4 q^{29} -20086.6i q^{30} -39354.2 q^{31} +5792.62 q^{32} -15744.1i q^{33} +2833.86i q^{34} +30071.0 q^{35} +7776.00 q^{36} -26463.1i q^{37} -11959.9i q^{38} -9320.10 q^{39} -41233.8i q^{40} +101434. q^{41} +11641.2i q^{42} +18530.2i q^{43} -32319.5i q^{44} -55352.2i q^{45} +(15357.4 - 67091.7i) q^{46} +141083. q^{47} +15962.6 q^{48} +100221. q^{49} -205128. q^{50} +7809.21i q^{51} -19132.3 q^{52} -151919. i q^{53} +21428.1 q^{54} -230061. q^{55} +23897.1i q^{56} -32957.6i q^{57} -69852.8 q^{58} -226850. q^{59} -113627. i q^{60} +169657. i q^{61} -222621. q^{62} +32079.4i q^{63} +32768.0 q^{64} +136190. i q^{65} -89062.1i q^{66} -417920. i q^{67} +16030.7i q^{68} +(42320.2 - 184883. i) q^{69} +170107. q^{70} -102647. q^{71} +43987.7 q^{72} +548156. q^{73} -149698. i q^{74} -565266. q^{75} -67655.3i q^{76} +133332. q^{77} -52722.5 q^{78} +412836. i q^{79} -233254. i q^{80} +59049.0 q^{81} +573799. q^{82} +23165.2i q^{83} +65852.5i q^{84} +114112. q^{85} +104823. i q^{86} -192492. q^{87} -182827. i q^{88} -77950.4i q^{89} -313119. i q^{90} -78929.1i q^{91} +(86874.9 - 379528. i) q^{92} -613471. q^{93} +798084. q^{94} -481593. q^{95} +90298.0 q^{96} +461383. i q^{97} +566938. q^{98} -245426. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 768 q^{4} + 5832 q^{9}+O(q^{10}) $ 24 * q + 768 * q^4 + 5832 * q^9	$ 24 q + 768 q^{4} + 5832 q^{9} - 768 q^{13} + 24576 q^{16} - 44104 q^{23} - 119448 q^{25} - 53888 q^{26} + 3456 q^{29} + 50976 q^{31} + 149008 q^{35} + 186624 q^{36} + 11664 q^{39} - 3920 q^{41} - 150720 q^{46} + 441088 q^{47} - 32472 q^{49} + 8320 q^{50} - 24576 q^{52} + 826176 q^{55} - 307200 q^{58} - 1210160 q^{59} + 783744 q^{62} + 786432 q^{64} + 361584 q^{69} - 2480064 q^{70} + 1531264 q^{71} + 593472 q^{73} + 23328 q^{75} + 1068784 q^{77} + 171072 q^{78} + 1417176 q^{81} + 1454592 q^{82} - 1318272 q^{85} + 697248 q^{87} - 1411328 q^{92} - 983664 q^{93} + 1115712 q^{94} + 4047632 q^{95} - 2409344 q^{98}+O(q^{100}) $ 24 * q + 768 * q^4 + 5832 * q^9 - 768 * q^13 + 24576 * q^16 - 44104 * q^23 - 119448 * q^25 - 53888 * q^26 + 3456 * q^29 + 50976 * q^31 + 149008 * q^35 + 186624 * q^36 + 11664 * q^39 - 3920 * q^41 - 150720 * q^46 + 441088 * q^47 - 32472 * q^49 + 8320 * q^50 - 24576 * q^52 + 826176 * q^55 - 307200 * q^58 - 1210160 * q^59 + 783744 * q^62 + 786432 * q^64 + 361584 * q^69 - 2480064 * q^70 + 1531264 * q^71 + 593472 * q^73 + 23328 * q^75 + 1068784 * q^77 + 171072 * q^78 + 1417176 * q^81 + 1454592 * q^82 - 1318272 * q^85 + 697248 * q^87 - 1411328 * q^92 - 983664 * q^93 + 1115712 * q^94 + 4047632 * q^95 - 2409344 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	5.65685			0.707107
$2$	5.65685			0.707107
$3$	15.5885			0.577350
$3$	15.5885			0.577350
$4$	32.0000			0.500000
$5$		−	227.787i		−	1.82229i	−0.412081	−	0.911147i	$-0.635198\pi$
$5$		−	227.787i		−	1.82229i	0.412081	−	0.911147i	$-0.364802\pi$
$6$	88.1816			0.408248
$7$			132.014i			0.384880i	0.981309	+	0.192440i	$0.0616401\pi$
$7$			132.014i			0.384880i	−0.981309	+	0.192440i	$0.938360\pi$
$8$	181.019			0.353553
$9$	243.000			0.333333
$10$		−	1288.56i		−	1.28856i
$11$		−	1009.98i		−	0.758816i	−0.925229	−	0.379408i	$-0.876128\pi$
$11$		−	1009.98i		−	0.758816i	0.925229	−	0.379408i	$-0.123872\pi$
$12$	498.831			0.288675
$13$	−597.885			−0.272137			−0.136068	−	0.990699i	$-0.543447\pi$
$13$	−597.885			−0.272137			−0.136068	+	0.990699i	$0.543447\pi$
$14$			746.783i			0.272151i
$15$		−	3550.84i		−	1.05210i
$16$	1024.00			0.250000
$17$			500.961i			0.101966i	0.998700	+	0.0509832i	$0.0162355\pi$
$17$			500.961i			0.101966i	−0.998700	+	0.0509832i	$0.983765\pi$
$18$	1374.62			0.235702
$19$		−	2114.23i		−	0.308241i	−0.988052	−	0.154121i	$-0.950746\pi$
$19$		−	2114.23i		−	0.308241i	0.988052	−	0.154121i	$-0.0492545\pi$
$20$		−	7289.18i		−	0.911147i
$21$			2057.89i			0.222211i
$22$		−	5713.33i		−	0.536564i
$23$	2714.84	−	11860.3i	0.223131	−	0.974788i
$23$	2714.84	−	11860.3i	0.223131	−	0.974788i
$24$	2821.81			0.204124
$25$	−36261.8			−2.32076
$26$	−3382.15			−0.192430
$27$	3788.00			0.192450
$28$			4224.44i			0.192440i
$29$	−12348.4			−0.506308			−0.253154	−	0.967426i	$-0.581468\pi$
$29$	−12348.4			−0.506308			−0.253154	+	0.967426i	$0.581468\pi$
$30$		−	20086.6i		−	0.743949i
$31$	−39354.2			−1.32101			−0.660505	−	0.750822i	$-0.729658\pi$
$31$	−39354.2			−1.32101			−0.660505	+	0.750822i	$0.729658\pi$
$32$	5792.62			0.176777
$33$		−	15744.1i		−	0.438103i
$34$			2833.86i			0.0721011i
$35$	30071.0			0.701365
$36$	7776.00			0.166667
$37$		−	26463.1i		−	0.522439i	−0.965279	−	0.261219i	$-0.915875\pi$
$37$		−	26463.1i		−	0.522439i	0.965279	−	0.261219i	$-0.0841246\pi$
$38$		−	11959.9i		−	0.217960i
$39$	−9320.10			−0.157118
$40$		−	41233.8i		−	0.644278i
$41$	101434.			1.47175			0.735874	−	0.677118i	$-0.236772\pi$
$41$	101434.			1.47175			0.735874	+	0.677118i	$0.236772\pi$
$42$			11641.2i			0.157127i
$43$			18530.2i			0.233064i	0.993187	+	0.116532i	$0.0371777\pi$
$43$			18530.2i			0.233064i	−0.993187	+	0.116532i	$0.962822\pi$
$44$		−	32319.5i		−	0.379408i
$45$		−	55352.2i		−	0.607431i
$46$	15357.4	−	67091.7i	0.157778	−	0.689279i
$47$	141083.			1.35888			0.679438	−	0.733733i	$-0.262224\pi$
$47$	141083.			1.35888			0.679438	+	0.733733i	$0.262224\pi$
$48$	15962.6			0.144338
$49$	100221.			0.851867
$50$	−205128.			−1.64102
$51$			7809.21i			0.0588703i
$52$	−19132.3			−0.136068
$53$		−	151919.i		−	1.02043i	−0.860047	−	0.510215i	$-0.829566\pi$
$53$		−	151919.i		−	1.02043i	0.860047	−	0.510215i	$-0.170434\pi$
$54$	21428.1			0.136083
$55$	−230061.			−1.38279
$56$			23897.1i			0.136076i
$57$		−	32957.6i		−	0.177963i
$58$	−69852.8			−0.358014
$59$	−226850.			−1.10455			−0.552273	−	0.833663i	$-0.686239\pi$
$59$	−226850.			−1.10455			−0.552273	+	0.833663i	$0.686239\pi$
$60$		−	113627.i		−	0.526051i
$61$			169657.i			0.747450i	0.927540	+	0.373725i	$0.121920\pi$
$61$			169657.i			0.747450i	−0.927540	+	0.373725i	$0.878080\pi$
$62$	−222621.			−0.934095
$63$			32079.4i			0.128293i
$64$	32768.0			0.125000
$65$			136190.i			0.495914i
$66$		−	89062.1i		−	0.309785i
$67$		−	417920.i		−	1.38953i	−0.719236	−	0.694766i	$-0.755508\pi$
$67$		−	417920.i		−	1.38953i	0.719236	−	0.694766i	$-0.244492\pi$
$68$			16030.7i			0.0509832i
$69$	42320.2	−	184883.i	0.128825	−	0.562794i
$70$	170107.			0.495940
$71$	−102647.			−0.286795			−0.143398	−	0.989665i	$-0.545803\pi$
$71$	−102647.			−0.286795			−0.143398	+	0.989665i	$0.545803\pi$
$72$	43987.7			0.117851
$73$	548156.			1.40908			0.704540	−	0.709665i	$-0.251154\pi$
$73$	548156.			1.40908			0.704540	+	0.709665i	$0.251154\pi$
$74$		−	149698.i		−	0.369420i
$75$	−565266.			−1.33989
$76$		−	67655.3i		−	0.154121i
$77$	133332.			0.292053
$78$	−52722.5			−0.111099
$79$			412836.i			0.837330i	0.908141	+	0.418665i	$0.137502\pi$
$79$			412836.i			0.837330i	−0.908141	+	0.418665i	$0.862498\pi$
$80$		−	233254.i		−	0.455574i
$81$	59049.0			0.111111
$82$	573799.			1.04068
$83$			23165.2i			0.0405136i	0.999795	+	0.0202568i	$0.00644838\pi$
$83$			23165.2i			0.0405136i	−0.999795	+	0.0202568i	$0.993552\pi$
$84$			65852.5i			0.111105i
$85$	114112.			0.185813
$86$			104823.i			0.164801i
$87$	−192492.			−0.292317
$88$		−	182827.i		−	0.268282i
$89$		−	77950.4i		−	0.110573i	−0.998471	−	0.0552864i	$-0.982393\pi$
$89$		−	77950.4i		−	0.110573i	0.998471	−	0.0552864i	$-0.0176072\pi$
$90$		−	313119.i		−	0.429519i
$91$		−	78929.1i		−	0.104740i
$92$	86874.9	−	379528.i	0.111566	−	0.487394i
$93$	−613471.			−0.762686
$94$	798084.			0.960871
$95$	−481593.			−0.561707
$96$	90298.0			0.102062
$97$			461383.i			0.505529i	0.967528	+	0.252765i	$0.0813398\pi$
$97$			461383.i			0.505529i	−0.967528	+	0.252765i	$0.918660\pi$
$98$	566938.			0.602361
$99$		−	245426.i		−	0.252939i
$100$	−1.16038e6			−1.16038
$101$	1.79018e6			1.73753			0.868767	−	0.495222i	$-0.164913\pi$
$101$	1.79018e6			1.73753			0.868767	+	0.495222i	$0.164913\pi$
$102$			44175.5i			0.0416276i
$103$			1.61158e6i			1.47482i	0.675443	+	0.737412i	$0.263952\pi$
$103$			1.61158e6i			1.47482i	−0.675443	+	0.737412i	$0.736048\pi$
$104$	−108229.			−0.0962149
$105$	468761.			0.404933
$106$		−	859381.i		−	0.721553i
$107$			1.82754e6i			1.49182i	0.666049	+	0.745908i	$0.267984\pi$
$107$			1.82754e6i			1.49182i	−0.666049	+	0.745908i	$0.732016\pi$
$108$	121216.			0.0962250
$109$			368331.i			0.284419i	0.989837	+	0.142210i	$0.0454207\pi$
$109$			368331.i			0.284419i	−0.989837	+	0.142210i	$0.954579\pi$
$110$	−1.30142e6			−0.977778
$111$		−	412519.i		−	0.301630i
$112$			135182.i			0.0962200i
$113$		−	673269.i		−	0.466609i	−0.972404	−	0.233305i	$-0.925046\pi$
$113$		−	673269.i		−	0.466609i	0.972404	−	0.233305i	$-0.0749539\pi$
$114$		−	186436.i		−	0.125839i
$115$	−2.70161e6	−	618405.i	−1.77635	−	0.406611i
$116$	−395147.			−0.253154
$117$	−145286.			−0.0907123
$118$	−1.28326e6			−0.781032
$119$	−66133.8			−0.0392448
$120$		−	642772.i		−	0.371974i
$121$	751493.			0.424198
$122$			959725.i			0.528527i
$123$	1.58121e6			0.849714
$124$	−1.25933e6			−0.660505
$125$			4.70080e6i			2.40681i
$126$			181468.i			0.0907171i
$127$	273746.			0.133640			0.0668199	−	0.997765i	$-0.478715\pi$
$127$	273746.			0.133640			0.0668199	+	0.997765i	$0.478715\pi$
$128$	185364.			0.0883883
$129$			288857.i			0.134559i
$130$			770409.i			0.350664i
$131$	1.46927e6			0.653563			0.326781	−	0.945100i	$-0.394036\pi$
$131$	1.46927e6			0.653563			0.326781	+	0.945100i	$0.394036\pi$
$132$		−	503811.i		−	0.219051i
$133$	279107.			0.118636
$134$		−	2.36411e6i		−	0.982547i
$135$		−	862855.i		−	0.350701i
$136$			90683.6i			0.0360506i
$137$			1.93995e6i			0.754446i	0.926122	+	0.377223i	$0.123121\pi$
$137$			1.93995e6i			0.754446i	−0.926122	+	0.377223i	$0.876879\pi$
$138$	239399.	−	1.04586e6i	0.0910930	−	0.397956i
$139$	3.31464e6			1.23422			0.617110	−	0.786877i	$-0.288303\pi$
$139$	3.31464e6			1.23422			0.617110	+	0.786877i	$0.288303\pi$
$140$	962272.			0.350682
$141$	2.19926e6			0.784548
$142$	−580660.			−0.202795
$143$			603854.i			0.206502i
$144$	248832.			0.0833333
$145$			2.81279e6i			0.922643i
$146$	3.10084e6			0.996369
$147$	1.56230e6			0.491826
$148$		−	846819.i		−	0.261219i
$149$			5.75998e6i			1.74126i	0.491943	+	0.870628i	$0.336287\pi$
$149$			5.75998e6i			1.74126i	−0.491943	+	0.870628i	$0.663713\pi$
$150$	−3.19763e6			−0.947445
$151$	−1.39357e6			−0.404761			−0.202380	−	0.979307i	$-0.564868\pi$
$151$	−1.39357e6			−0.404761			−0.202380	+	0.979307i	$0.564868\pi$
$152$		−	382716.i		−	0.108980i
$153$			121733.i			0.0339888i
$154$	754239.			0.206513
$155$			8.96437e6i			2.40727i
$156$	−298243.			−0.0785592
$157$		−	5.11687e6i		−	1.32222i	−0.750287	−	0.661112i	$-0.770085\pi$
$157$		−	5.11687e6i		−	1.32222i	0.750287	−	0.661112i	$-0.229915\pi$
$158$			2.33536e6i			0.592082i
$159$		−	2.36818e6i		−	0.589146i
$160$		−	1.31948e6i		−	0.322139i
$161$	1.56572e6	+	358396.i	0.375177	+	0.0858788i
$162$	334032.			0.0785674
$163$	3.02765e6			0.699106			0.349553	−	0.936917i	$-0.386333\pi$
$163$	3.02765e6			0.699106			0.349553	+	0.936917i	$0.386333\pi$
$164$	3.24590e6			0.735874
$165$	−3.58630e6			−0.798352
$166$			131042.i			0.0286475i
$167$	−3.93229e6			−0.844298			−0.422149	−	0.906527i	$-0.638724\pi$
$167$	−3.93229e6			−0.844298			−0.422149	+	0.906527i	$0.638724\pi$
$168$			372518.i			0.0785633i
$169$	−4.46934e6			−0.925941
$170$	645517.			0.131389
$171$		−	513758.i		−	0.102747i
$172$			592967.i			0.116532i
$173$	−2.18275e6			−0.421567			−0.210783	−	0.977533i	$-0.567601\pi$
$173$	−2.18275e6			−0.421567			−0.210783	+	0.977533i	$0.567601\pi$
$174$	−1.08890e6			−0.206699
$175$		−	4.78706e6i		−	0.893213i
$176$		−	1.03422e6i		−	0.189704i
$177$	−3.53625e6			−0.637710
$178$		−	440954.i		−	0.0781868i
$179$	−151553.			−0.0264245			−0.0132122	−	0.999913i	$-0.504206\pi$
$179$	−151553.			−0.0264245			−0.0132122	+	0.999913i	$0.504206\pi$
$180$		−	1.77127e6i		−	0.303716i
$181$		−	523617.i		−	0.0883035i	−0.999025	−	0.0441518i	$-0.985941\pi$
$181$		−	523617.i		−	0.0883035i	0.999025	−	0.0441518i	$-0.0140585\pi$
$182$		−	446490.i		−	0.0740624i
$183$			2.64469e6i			0.431540i
$184$	491438.	−	2.14693e6i	0.0788888	−	0.344640i
$185$	−6.02794e6			−0.952038
$186$	−3.47032e6			−0.539300
$187$	505963.			0.0773737
$188$	4.51464e6			0.679438
$189$			500068.i			0.0740702i
$190$	−2.72430e6			−0.397187
$191$			4.36683e6i			0.626710i	0.949636	+	0.313355i	$0.101453\pi$
$191$			4.36683e6i			0.626710i	−0.949636	+	0.313355i	$0.898547\pi$
$192$	510803.			0.0721688
$193$	−5.06532e6			−0.704587			−0.352294	−	0.935890i	$-0.614598\pi$
$193$	−5.06532e6			−0.704587			−0.352294	+	0.935890i	$0.614598\pi$
$194$			2.60998e6i			0.357463i
$195$			2.12300e6i			0.286316i
$196$	3.20708e6			0.425934
$197$	7.19241e6			0.940753			0.470377	−	0.882466i	$-0.344118\pi$
$197$	7.19241e6			0.940753			0.470377	+	0.882466i	$0.344118\pi$
$198$		−	1.38834e6i		−	0.178855i
$199$		−	1.43873e7i		−	1.82567i	−0.408332	−	0.912833i	$-0.633889\pi$
$199$		−	1.43873e7i		−	1.82567i	0.408332	−	0.912833i	$-0.366111\pi$
$200$	−6.56409e6			−0.820512
$201$		−	6.51472e6i		−	0.802246i
$202$	1.01268e7			1.22862
$203$		−	1.63015e6i		−	0.194868i
$204$			249895.i			0.0294352i
$205$		−	2.31054e7i		−	2.68196i
$206$			9.11647e6i			1.04286i
$207$	659706.	−	2.88204e6i	0.0743771	−	0.324929i
$208$	−612234.			−0.0680342
$209$	−2.13534e6			−0.233899
$210$	2.65171e6			0.286331
$211$	8.89130e6			0.946494			0.473247	−	0.880930i	$-0.343082\pi$
$211$	8.89130e6			0.946494			0.473247	+	0.880930i	$0.343082\pi$
$212$		−	4.86139e6i		−	0.510215i
$213$	−1.60011e6			−0.165581
$214$			1.03381e7i			1.05487i
$215$	4.22094e6			0.424711
$216$	685700.			0.0680414
$217$		−	5.19530e6i		−	0.508430i
$218$			2.08360e6i			0.201115i
$219$	8.54490e6			0.813532
$220$	−7.36195e6			−0.691393
$221$		−	299517.i		−	0.0277488i
$222$		−	2.33356e6i		−	0.213285i
$223$	−4.58537e6			−0.413485			−0.206742	−	0.978395i	$-0.566286\pi$
$223$	−4.58537e6			−0.413485			−0.206742	+	0.978395i	$0.566286\pi$
$224$			764706.i			0.0680378i
$225$	−8.81162e6			−0.773586
$226$		−	3.80858e6i		−	0.329943i
$227$		−	2.12582e6i		−	0.181740i	−0.995863	−	0.0908698i	$-0.971035\pi$
$227$		−	2.12582e6i		−	0.181740i	0.995863	−	0.0908698i	$-0.0289647\pi$
$228$		−	1.05464e6i		−	0.0889817i
$229$		−	8.42951e6i		−	0.701934i	−0.936388	−	0.350967i	$-0.885853\pi$
$229$		−	8.42951e6i		−	0.701934i	0.936388	−	0.350967i	$-0.114147\pi$
$230$	−1.52826e7	−	3.49822e6i	−1.25607	−	0.287517i
$231$	2.07844e6			0.168617
$232$	−2.23529e6			−0.179007
$233$	1.70067e7			1.34447			0.672237	−	0.740336i	$-0.265334\pi$
$233$	1.70067e7			1.34447			0.672237	+	0.740336i	$0.265334\pi$
$234$	−821862.			−0.0641433
$235$		−	3.21368e7i		−	2.47627i
$236$	−7.25921e6			−0.552273
$237$			6.43548e6i			0.483433i
$238$	−374109.			−0.0277503
$239$	−5.59675e6			−0.409961			−0.204980	−	0.978766i	$-0.565713\pi$
$239$	−5.59675e6			−0.409961			−0.204980	+	0.978766i	$0.565713\pi$
$240$		−	3.63607e6i		−	0.263026i
$241$			1.52869e7i			1.09212i	0.837747	+	0.546059i	$0.183873\pi$
$241$			1.52869e7i			1.09212i	−0.837747	+	0.546059i	$0.816127\pi$
$242$	4.25108e6			0.299953
$243$	920483.			0.0641500
$244$			5.42902e6i			0.373725i
$245$		−	2.28291e7i		−	1.55235i
$246$	8.94465e6			0.600839
$247$			1.26407e6i			0.0838839i
$248$	−7.12387e6			−0.467048
$249$			361109.i			0.0233905i
$250$			2.65917e7i			1.70187i
$251$		−	2.67909e7i		−	1.69421i	−0.531429	−	0.847103i	$-0.678345\pi$
$251$		−	2.67909e7i		−	1.69421i	0.531429	−	0.847103i	$-0.321655\pi$
$252$			1.02654e6i			0.0641467i
$253$	−1.19787e7	−	2.74194e6i	−0.739685	−	0.169316i
$254$	1.54854e6			0.0944976
$255$	1.77883e6			0.107279
$256$	1.04858e6			0.0625000
$257$	−9.22223e6			−0.543296			−0.271648	−	0.962397i	$-0.587569\pi$
$257$	−9.22223e6			−0.543296			−0.271648	+	0.962397i	$0.587569\pi$
$258$			1.63402e6i			0.0951479i
$259$	3.49350e6			0.201076
$260$			4.35809e6i			0.247957i
$261$	−3.00065e6			−0.168769
$262$	8.31143e6			0.462138
$263$			2.33654e7i			1.28442i	0.766530	+	0.642209i	$0.221982\pi$
$263$			2.33654e7i			1.28442i	−0.766530	+	0.642209i	$0.778018\pi$
$264$		−	2.84999e6i		−	0.154893i
$265$	−3.46050e7			−1.85952
$266$	1.57887e6			0.0838883
$267$		−	1.21513e6i		−	0.0638392i
$268$		−	1.33734e7i		−	0.694766i
$269$	3.11306e7			1.59930			0.799652	−	0.600464i	$-0.205017\pi$
$269$	3.11306e7			1.59930			0.799652	+	0.600464i	$0.205017\pi$
$270$		−	4.88105e6i		−	0.247983i
$271$	−2.68015e7			−1.34664			−0.673321	−	0.739351i	$-0.735133\pi$
$271$	−2.68015e7			−1.34664			−0.673321	+	0.739351i	$0.735133\pi$
$272$			512984.i			0.0254916i
$273$		−	1.23038e6i		−	0.0604717i
$274$			1.09740e7i			0.533474i
$275$			3.66239e7i			1.76103i
$276$	1.35424e6	−	5.91626e6i	0.0644125	−	0.281397i
$277$	−1.51417e7			−0.712421			−0.356210	−	0.934406i	$-0.615931\pi$
$277$	−1.51417e7			−0.712421			−0.356210	+	0.934406i	$0.615931\pi$
$278$	1.87505e7			0.872725
$279$	−9.56307e6			−0.440337
$280$	5.44343e6			0.247970
$281$		−	3.03153e7i		−	1.36629i	−0.730283	−	0.683145i	$-0.760612\pi$
$281$		−	3.03153e7i		−	1.36629i	0.730283	−	0.683145i	$-0.239388\pi$
$282$	1.24409e7			0.554759
$283$			2.97687e7i			1.31341i	0.754147	+	0.656706i	$0.228051\pi$
$283$			2.97687e7i			1.31341i	−0.754147	+	0.656706i	$0.771949\pi$
$284$	−3.28471e6			−0.143398
$285$	−7.50730e6			−0.324302
$286$			3.41592e6i			0.146019i
$287$			1.33907e7i			0.566446i
$288$	1.40761e6			0.0589256
$289$	2.38866e7			0.989603
$290$			1.59116e7i			0.652407i
$291$			7.19225e6i			0.291867i
$292$	1.75410e7			0.704540
$293$			8.80463e6i			0.350032i	0.984566	+	0.175016i	$0.0559978\pi$
$293$			8.80463e6i			0.350032i	−0.984566	+	0.175016i	$0.944002\pi$
$294$	8.83768e6			0.347773
$295$			5.16735e7i			2.01281i
$296$		−	4.79033e6i		−	0.184710i
$297$		−	3.82582e6i		−	0.146034i
$298$			3.25834e7i			1.23125i
$299$	−1.62316e6	+	7.09106e6i	−0.0607223	+	0.265276i
$300$	−1.80885e7			−0.669945
$301$	−2.44624e6			−0.0897016
$302$	−7.88323e6			−0.286209
$303$	2.79062e7			1.00317
$304$		−	2.16497e6i		−	0.0770604i
$305$	3.86456e7			1.36207
$306$			688629.i			0.0240337i
$307$	3.78379e7			1.30771			0.653855	−	0.756620i	$-0.273151\pi$
$307$	3.78379e7			1.30771			0.653855	+	0.756620i	$0.273151\pi$
$308$	4.26662e6			0.146027
$309$			2.51220e7i			0.851490i
$310$			5.07101e7i			1.70220i
$311$	−3.38614e7			−1.12570			−0.562852	−	0.826558i	$-0.690296\pi$
$311$	−3.38614e7			−1.12570			−0.562852	+	0.826558i	$0.690296\pi$
$312$	−1.68712e6			−0.0555497
$313$		−	5.34585e7i		−	1.74335i	−0.490087	−	0.871674i	$-0.663035\pi$
$313$		−	5.34585e7i		−	1.74335i	0.490087	−	0.871674i	$-0.336965\pi$
$314$		−	2.89454e7i		−	0.934954i
$315$	7.30726e6			0.233788
$316$			1.32108e7i			0.418665i
$317$	2.82219e7			0.885949			0.442974	−	0.896534i	$-0.353923\pi$
$317$	2.82219e7			0.885949			0.442974	+	0.896534i	$0.353923\pi$
$318$		−	1.33964e7i		−	0.416589i
$319$			1.24716e7i			0.384195i
$320$		−	7.46412e6i		−	0.227787i
$321$			2.84885e7i			0.861301i
$322$	8.85703e6	+	2.02740e6i	0.265290	+	0.0607255i
$323$	1.05915e6			0.0314303
$324$	1.88957e6			0.0555556
$325$	2.16804e7			0.631564
$326$	1.71270e7			0.494343
$327$			5.74172e6i			0.164210i
$328$	1.83616e7			0.520342
$329$			1.86249e7i			0.523004i
$330$	−2.02872e7			−0.564520
$331$	5.10518e7			1.40775			0.703877	−	0.710322i	$-0.251450\pi$
$331$	5.10518e7			1.40775			0.703877	+	0.710322i	$0.251450\pi$
$332$			741285.i			0.0202568i
$333$		−	6.43053e6i		−	0.174146i
$334$	−2.22444e7			−0.597009
$335$	−9.51966e7			−2.53214
$336$			2.10728e6i			0.0555526i
$337$		−	1.43773e7i		−	0.375654i	−0.982202	−	0.187827i	$-0.939856\pi$
$337$		−	1.43773e7i		−	0.375654i	0.982202	−	0.187827i	$-0.0601445\pi$
$338$	−2.52824e7			−0.654740
$339$		−	1.04952e7i		−	0.269397i
$340$	3.65159e6			0.0929064
$341$			3.97471e7i			1.00240i
$342$		−	2.90625e6i		−	0.0726532i
$343$			2.87619e7i			0.712747i
$344$			3.35433e6i			0.0824005i
$345$	−4.21139e7	−	9.63997e6i	−1.02558	−	0.234757i
$346$	−1.23475e7			−0.298093
$347$	−3.90632e7			−0.934931			−0.467466	−	0.884011i	$-0.654833\pi$
$347$	−3.90632e7			−0.934931			−0.467466	+	0.884011i	$0.654833\pi$
$348$	−6.15974e6			−0.146159
$349$	−5.89226e6			−0.138614			−0.0693068	−	0.997595i	$-0.522079\pi$
$349$	−5.89226e6			−0.138614			−0.0693068	+	0.997595i	$0.522079\pi$
$350$		−	2.70797e7i		−	0.631597i
$351$	−2.26478e6			−0.0523728
$352$		−	5.85045e6i		−	0.134141i
$353$	−3.59468e7			−0.817214			−0.408607	−	0.912711i	$-0.633985\pi$
$353$	−3.59468e7			−0.817214			−0.408607	+	0.912711i	$0.633985\pi$
$354$	−2.00040e7			−0.450929
$355$			2.33817e7i			0.522626i
$356$		−	2.49441e6i		−	0.0552864i
$357$	−1.03092e6			−0.0226580
$358$	−857315.			−0.0186849
$359$		−	2.02194e7i		−	0.437003i	−0.975837	−	0.218501i	$-0.929883\pi$
$359$		−	2.02194e7i		−	0.437003i	0.975837	−	0.218501i	$-0.0701168\pi$
$360$		−	1.00198e7i		−	0.214759i
$361$	4.25759e7			0.904987
$362$		−	2.96203e6i		−	0.0624400i
$363$	1.17146e7			0.244911
$364$		−	2.52573e6i		−	0.0523700i
$365$		−	1.24863e8i		−	2.56776i
$366$			1.49606e7i			0.305145i
$367$		−	3.61729e7i		−	0.731789i	−0.930656	−	0.365894i	$-0.880763\pi$
$367$		−	3.61729e7i		−	0.731789i	0.930656	−	0.365894i	$-0.119237\pi$
$368$	2.78000e6	−	1.21449e7i	0.0557828	−	0.243697i
$369$	2.46485e7			0.490583
$370$	−3.40992e7			−0.673192
$371$	2.00554e7			0.392743
$372$	−1.96311e7			−0.381343
$373$		−	3.69240e7i		−	0.711512i	−0.934579	−	0.355756i	$-0.884223\pi$
$373$		−	3.69240e7i		−	0.711512i	0.934579	−	0.355756i	$-0.115777\pi$
$374$	2.86216e6			0.0547115
$375$			7.32782e7i			1.38957i
$376$	2.55387e7			0.480435
$377$	7.38289e6			0.137785
$378$			2.82881e6i			0.0523755i
$379$			3.42426e7i			0.628998i	0.949258	+	0.314499i	$0.101836\pi$
$379$			3.42426e7i			0.628998i	−0.949258	+	0.314499i	$0.898164\pi$
$380$	−1.54110e7			−0.280853
$381$	4.26727e6			0.0771570
$382$			2.47025e7i			0.443151i
$383$			1.53651e7i			0.273489i	0.990606	+	0.136745i	$0.0436640\pi$
$383$			1.53651e7i			0.273489i	−0.990606	+	0.136745i	$0.956336\pi$
$384$	2.88954e6			0.0510310
$385$		−	3.03712e7i		−	0.532207i
$386$	−2.86538e7			−0.498219
$387$			4.50284e6i			0.0776880i
$388$			1.47643e7i			0.252765i
$389$		−	1.06266e8i		−	1.80528i	−0.430395	−	0.902641i	$-0.641626\pi$
$389$		−	1.06266e8i		−	1.80528i	0.430395	−	0.902641i	$-0.358374\pi$
$390$			1.20095e7i			0.202456i
$391$	5.94152e6	+	1.36003e6i	0.0993957	+	0.0227519i
$392$	1.81420e7			0.301181
$393$	2.29036e7			0.377334
$394$	4.06864e7			0.665213
$395$	9.40387e7			1.52586
$396$		−	7.85364e6i		−	0.126469i
$397$	−8.84623e7			−1.41380			−0.706898	−	0.707316i	$-0.749906\pi$
$397$	−8.84623e7			−1.41380			−0.706898	+	0.707316i	$0.749906\pi$
$398$		−	8.13871e7i		−	1.29094i
$399$	4.35085e6			0.0684945
$400$	−3.71321e7			−0.580189
$401$		−	1.57114e7i		−	0.243659i	−0.992551	−	0.121830i	$-0.961124\pi$
$401$		−	1.57114e7i		−	0.243659i	0.992551	−	0.121830i	$-0.0388761\pi$
$402$		−	3.68528e7i		−	0.567274i
$403$	2.35293e7			0.359496
$404$	5.72858e7			0.868767
$405$		−	1.34506e7i		−	0.202477i
$406$		−	9.22154e6i		−	0.137792i
$407$	−2.67273e7			−0.396435
$408$			1.41362e6i			0.0208138i
$409$	−9.59153e6			−0.140190			−0.0700952	−	0.997540i	$-0.522330\pi$
$409$	−9.59153e6			−0.140190			−0.0700952	+	0.997540i	$0.522330\pi$
$410$		−	1.30704e8i		−	1.89643i
$411$			3.02408e7i			0.435580i
$412$			5.15706e7i			0.737412i
$413$		−	2.99474e7i		−	0.425117i
$414$	3.73186e6	−	1.63033e7i	0.0525926	−	0.229760i
$415$	5.27672e6			0.0738277
$416$	−3.46332e6			−0.0481075
$417$	5.16702e7			0.712577
$418$	−1.20793e7			−0.165391
$419$			1.22382e8i			1.66370i	0.555004	+	0.831848i	$0.312717\pi$
$419$			1.22382e8i			1.66370i	−0.555004	+	0.831848i	$0.687283\pi$
$420$	1.50003e7			0.202467
$421$			2.75232e7i			0.368853i	0.982846	+	0.184426i	$0.0590427\pi$
$421$			2.75232e7i			0.368853i	−0.982846	+	0.184426i	$0.940957\pi$
$422$	5.02968e7			0.669273
$423$	3.42831e7			0.452959
$424$		−	2.75002e7i		−	0.360777i
$425$		−	1.81658e7i		−	0.236639i
$426$	−9.05160e6			−0.117084
$427$	−2.23971e7			−0.287679
$428$			5.84813e7i			0.745908i
$429$			9.41316e6i			0.119224i
$430$	2.38772e7			0.300316
$431$			3.26440e7i			0.407728i	0.978999	+	0.203864i	$0.0653501\pi$
$431$			3.26440e7i			0.407728i	−0.978999	+	0.203864i	$0.934650\pi$
$432$	3.87891e6			0.0481125
$433$		−	2.70852e7i		−	0.333633i	−0.985988	−	0.166816i	$-0.946651\pi$
$433$		−	2.70852e7i		−	0.333633i	0.985988	−	0.166816i	$-0.0533487\pi$
$434$		−	2.93891e7i		−	0.359515i
$435$			4.38471e7i			0.532688i
$436$			1.17866e7i			0.142210i
$437$	−2.50753e7	−	5.73979e6i	−0.300470	−	0.0687783i
$438$	4.83373e7			0.575254
$439$	−1.40121e8			−1.65619			−0.828095	−	0.560587i	$-0.810575\pi$
$439$	−1.40121e8			−1.65619			−0.828095	+	0.560587i	$0.810575\pi$
$440$	−4.16455e7			−0.488889
$441$	2.43538e7			0.283956
$442$		−	1.69432e6i		−	0.0196214i
$443$	3.21354e7			0.369634			0.184817	−	0.982773i	$-0.440831\pi$
$443$	3.21354e7			0.369634			0.184817	+	0.982773i	$0.440831\pi$
$444$		−	1.32006e7i		−	0.150815i
$445$	−1.77561e7			−0.201496
$446$	−2.59388e7			−0.292378
$447$			8.97892e7i			1.00531i
$448$			4.32583e6i			0.0481100i
$449$	−8.93944e7			−0.987578			−0.493789	−	0.869582i	$-0.664388\pi$
$449$	−8.93944e7			−0.987578			−0.493789	+	0.869582i	$0.664388\pi$
$450$	−4.98461e7			−0.547008
$451$		−	1.02447e8i		−	1.11679i
$452$		−	2.15446e7i		−	0.233305i
$453$	−2.17236e7			−0.233689
$454$		−	1.20255e7i		−	0.128509i
$455$	−1.79790e7			−0.190867
$456$		−	5.96596e6i		−	0.0629195i
$457$			1.17493e8i			1.23101i	0.788132	+	0.615506i	$0.211048\pi$
$457$			1.17493e8i			1.23101i	−0.788132	+	0.615506i	$0.788952\pi$
$458$		−	4.76845e7i		−	0.496342i
$459$			1.89764e6i			0.0196234i
$460$	−8.64515e7	−	1.97889e7i	−0.888176	−	0.203306i
$461$	−1.15284e8			−1.17670			−0.588351	−	0.808605i	$-0.700223\pi$
$461$	−1.15284e8			−1.17670			−0.588351	+	0.808605i	$0.700223\pi$
$462$	1.17574e7			0.119230
$463$	−7.96990e7			−0.802990			−0.401495	−	0.915861i	$-0.631509\pi$
$463$	−7.96990e7			−0.802990			−0.401495	+	0.915861i	$0.631509\pi$
$464$	−1.26447e7			−0.126577
$465$			1.39741e8i			1.38984i
$466$	9.62045e7			0.950687
$467$		−	7.73301e7i		−	0.759273i	−0.925136	−	0.379637i	$-0.876049\pi$
$467$		−	7.73301e7i		−	0.759273i	0.925136	−	0.379637i	$-0.123951\pi$
$468$	−4.64915e6			−0.0453562
$469$	5.51712e7			0.534803
$470$		−	1.81793e8i		−	1.75099i
$471$		−	7.97641e7i		−	0.763387i
$472$	−4.10643e7			−0.390516
$473$	1.87152e7			0.176853
$474$			3.64046e7i			0.341839i
$475$			7.66658e7i			0.715354i
$476$	−2.11628e6			−0.0196224
$477$		−	3.69162e7i		−	0.340143i
$478$	−3.16600e7			−0.289886
$479$			1.12814e8i			1.02649i	0.858241	+	0.513246i	$0.171557\pi$
$479$			1.12814e8i			1.02649i	−0.858241	+	0.513246i	$0.828443\pi$
$480$		−	2.05687e7i		−	0.185987i
$481$			1.58219e7i			0.142175i
$482$			8.64760e7i			0.772244i
$483$	2.44071e7	+	5.58685e6i	0.216608	+	0.0495821i
$484$	2.40478e7			0.212099
$485$	1.05097e8			0.921223
$486$	5.20704e6			0.0453609
$487$	9.02156e7			0.781078			0.390539	−	0.920586i	$-0.372289\pi$
$487$	9.02156e7			0.781078			0.390539	+	0.920586i	$0.372289\pi$
$488$			3.07112e7i			0.264263i
$489$	4.71964e7			0.403629
$490$		−	1.29141e8i		−	1.09768i
$491$	1.19497e8			1.00952			0.504758	−	0.863261i	$-0.331582\pi$
$491$	1.19497e8			1.00952			0.504758	+	0.863261i	$0.331582\pi$
$492$	5.05986e7			0.424857
$493$		−	6.18604e6i		−	0.0516264i
$494$			7.15063e6i			0.0593149i
$495$	−5.59048e7			−0.460929
$496$	−4.02987e7			−0.330253
$497$		−	1.35509e7i		−	0.110382i
$498$			2.04274e6i			0.0165396i
$499$	8.83027e7			0.710677			0.355339	−	0.934738i	$-0.384366\pi$
$499$	8.83027e7			0.710677			0.355339	+	0.934738i	$0.384366\pi$
$500$			1.50426e8i			1.20340i
$501$	−6.12983e7			−0.487456
$502$		−	1.51552e8i		−	1.19798i
$503$		−	1.15847e8i		−	0.910292i	−0.890417	−	0.455146i	$-0.849587\pi$
$503$		−	1.15847e8i		−	0.910292i	0.890417	−	0.455146i	$-0.150413\pi$
$504$			5.80699e6i			0.0453585i
$505$		−	4.07780e8i		−	3.16630i
$506$	−6.77616e7	−	1.55108e7i	−0.523036	−	0.119724i
$507$	−6.96702e7			−0.534593
$508$	8.75986e6			0.0668199
$509$	−1.03994e8			−0.788597			−0.394299	−	0.918982i	$-0.629012\pi$
$509$	−1.03994e8			−0.788597			−0.394299	+	0.918982i	$0.629012\pi$
$510$	1.00626e7			0.0758578
$511$			7.23641e7i			0.542326i
$512$	5.93164e6			0.0441942
$513$		−	8.00869e6i		−	0.0593211i
$514$	−5.21688e7			−0.384168
$515$	3.67097e8			2.68756
$516$			9.24344e6i			0.0672797i
$517$		−	1.42491e8i		−	1.03114i
$518$	1.97622e7			0.142182
$519$	−3.40258e7			−0.243392
$520$			2.46531e7i			0.175332i
$521$			2.46287e8i			1.74152i	0.491710	+	0.870759i	$0.336372\pi$
$521$			2.46287e8i			1.74152i	−0.491710	+	0.870759i	$0.663628\pi$
$522$	−1.69742e7			−0.119338
$523$			1.38027e8i			0.964847i	0.875938	+	0.482423i	$0.160243\pi$
$523$			1.38027e8i			0.964847i	−0.875938	+	0.482423i	$0.839757\pi$
$524$	4.70166e7			0.326781
$525$		−	7.46229e7i		−	0.515697i
$526$			1.32175e8i			0.908220i
$527$		−	1.97149e7i		−	0.134699i
$528$		−	1.61220e7i		−	0.109526i
$529$	−1.33295e8	−	6.43973e7i	−0.900425	−	0.435012i
$530$	−1.95756e8			−1.31488
$531$	−5.51247e7			−0.368182
$532$	8.93144e6			0.0593180
$533$	−6.06461e7			−0.400517
$534$		−	6.87379e6i		−	0.0451411i
$535$	4.16289e8			2.71853
$536$		−	7.56515e7i		−	0.491273i
$537$	−2.36248e6			−0.0152562
$538$	1.76101e8			1.13088
$539$		−	1.01222e8i		−	0.646411i
$540$		−	2.76114e7i		−	0.175350i
$541$	−9.38636e6			−0.0592796			−0.0296398	−	0.999561i	$-0.509436\pi$
$541$	−9.38636e6			−0.0592796			−0.0296398	+	0.999561i	$0.509436\pi$
$542$	−1.51612e8			−0.952219
$543$		−	8.16238e6i		−	0.0509821i
$544$			2.90188e6i			0.0180253i
$545$	8.39010e7			0.518296
$546$		−	6.96009e6i		−	0.0427600i
$547$	−4.42899e7			−0.270609			−0.135305	−	0.990804i	$-0.543201\pi$
$547$	−4.42899e7			−0.270609			−0.135305	+	0.990804i	$0.543201\pi$
$548$			6.20783e7i			0.377223i
$549$			4.12266e7i			0.249150i
$550$			2.07176e8i			1.24523i
$551$			2.61072e7i			0.156065i
$552$	7.66077e6	−	3.34674e7i	0.0455465	−	0.198978i
$553$	−5.45001e7			−0.322272
$554$	−8.56547e7			−0.503758
$555$	−9.39664e7			−0.549659
$556$	1.06069e8			0.617110
$557$			2.70491e8i			1.56526i	0.622487	+	0.782630i	$0.286122\pi$
$557$			2.70491e8i			1.56526i	−0.622487	+	0.782630i	$0.713878\pi$
$558$	−5.40969e7			−0.311365
$559$		−	1.10789e7i		−	0.0634253i
$560$	3.07927e7			0.175341
$561$	7.88718e6			0.0446718
$562$		−	1.71489e8i		−	0.966113i
$563$		−	1.27514e8i		−	0.714550i	−0.933999	−	0.357275i	$-0.883706\pi$
$563$		−	1.27514e8i		−	0.714550i	0.933999	−	0.357275i	$-0.116294\pi$
$564$	7.03763e7			0.392274
$565$	−1.53362e8			−0.850299
$566$			1.68397e8i			0.928723i
$567$			7.79529e6i			0.0427644i
$568$	−1.85811e7			−0.101397
$569$			4.77781e6i			0.0259354i	0.999916	+	0.0129677i	$0.00412786\pi$
$569$			4.77781e6i			0.0259354i	−0.999916	+	0.0129677i	$0.995872\pi$
$570$	−4.24677e7			−0.229316
$571$			1.32081e8i			0.709466i	0.934968	+	0.354733i	$0.115428\pi$
$571$			1.32081e8i			0.709466i	−0.934968	+	0.354733i	$0.884572\pi$
$572$			1.93233e7i			0.103251i
$573$			6.80722e7i			0.361831i
$574$			7.57495e7i			0.400538i
$575$	−9.84450e7	+	4.30074e8i	−0.517834	+	2.26225i
$576$	7.96262e6			0.0416667
$577$	1.63153e8			0.849314			0.424657	−	0.905354i	$-0.360395\pi$
$577$	1.63153e8			0.849314			0.424657	+	0.905354i	$0.360395\pi$
$578$	1.35123e8			0.699755
$579$	−7.89605e7			−0.406794
$580$			9.00093e7i			0.461321i
$581$	−3.05812e6			−0.0155929
$582$			4.06855e7i			0.206381i
$583$	−1.53435e8			−0.774319
$584$	9.92268e7			0.498185
$585$			3.30942e7i			0.165305i
$586$			4.98065e7i			0.247510i
$587$	−4.43319e7			−0.219181			−0.109590	−	0.993977i	$-0.534954\pi$
$587$	−4.43319e7			−0.219181			−0.109590	+	0.993977i	$0.534954\pi$
$588$	4.99935e7			0.245913
$589$			8.32038e7i			0.407190i
$590$			2.92310e8i			1.42327i
$591$	1.12119e8			0.543144
$592$		−	2.70982e7i		−	0.130610i
$593$	−4.12294e8			−1.97717			−0.988583	−	0.150680i	$-0.951854\pi$
$593$	−4.12294e8			−1.97717			−0.988583	+	0.150680i	$0.951854\pi$
$594$		−	2.16421e7i		−	0.103262i
$595$			1.50644e7i			0.0715156i
$596$			1.84319e8i			0.870628i
$597$		−	2.24277e8i		−	1.05405i
$598$	−9.18199e6	+	4.01131e7i	−0.0429371	+	0.187578i
$599$	2.74253e8			1.27606			0.638030	−	0.770012i	$-0.279750\pi$
$599$	2.74253e8			1.27606			0.638030	+	0.770012i	$0.279750\pi$
$600$	−1.02324e8			−0.473723
$601$	−1.40431e8			−0.646903			−0.323452	−	0.946245i	$-0.604843\pi$
$601$	−1.40431e8			−0.646903			−0.323452	+	0.946245i	$0.604843\pi$
$602$	−1.38380e7			−0.0634286
$603$		−	1.01554e8i		−	0.463177i
$604$	−4.45943e7			−0.202380
$605$		−	1.71180e8i		−	0.773014i
$606$	1.57861e8			0.709345
$607$	−3.62972e8			−1.62296			−0.811478	−	0.584383i	$-0.801337\pi$
$607$	−3.62972e8			−1.62296			−0.811478	+	0.584383i	$0.801337\pi$
$608$		−	1.22469e7i		−	0.0544899i
$609$		−	2.54116e7i		−	0.112507i
$610$	2.18613e8			0.963132
$611$	−8.43512e7			−0.369801
$612$			3.89547e6i			0.0169944i
$613$		−	2.09438e7i		−	0.0909231i	−0.998966	−	0.0454615i	$-0.985524\pi$
$613$		−	2.09438e7i		−	0.0909231i	0.998966	−	0.0454615i	$-0.0144758\pi$
$614$	2.14043e8			0.924690
$615$		−	3.60178e8i		−	1.54843i
$616$	2.41357e7			0.103256
$617$			2.48534e8i			1.05811i	0.848587	+	0.529055i	$0.177454\pi$
$617$			2.48534e8i			1.05811i	−0.848587	+	0.529055i	$0.822546\pi$
$618$			1.42112e8i			0.602094i
$619$			3.60180e8i			1.51861i	0.650733	+	0.759307i	$0.274462\pi$
$619$			3.60180e8i			1.51861i	−0.650733	+	0.759307i	$0.725538\pi$
$620$			2.86860e8i			1.20363i
$621$	1.02838e7	−	4.49266e7i	0.0429416	−	0.187598i
$622$	−1.91549e8			−0.795993
$623$	1.02905e7			0.0425573
$624$	−9.54379e6			−0.0392796
$625$	5.04188e8			2.06516
$626$		−	3.02407e8i		−	1.23273i
$627$	−3.32866e7			−0.135041
$628$		−	1.63740e8i		−	0.661112i
$629$	1.32570e7			0.0532712
$630$	4.13361e7			0.165313
$631$			3.48966e8i			1.38898i	0.719504	+	0.694488i	$0.244369\pi$
$631$			3.48966e8i			1.38898i	−0.719504	+	0.694488i	$0.755631\pi$
$632$			7.47314e7i			0.296041i
$633$	1.38602e8			0.546459
$634$	1.59647e8			0.626461
$635$		−	6.23556e7i		−	0.243531i
$636$		−	7.57816e7i		−	0.294573i
$637$	−5.99208e7			−0.231825
$638$			7.05503e7i			0.271667i
$639$	−2.49433e7			−0.0955985
$640$		−	4.22234e7i		−	0.161070i
$641$		−	3.06750e8i		−	1.16469i	−0.812942	−	0.582345i	$-0.802135\pi$
$641$		−	3.06750e8i		−	1.16469i	0.812942	−	0.582345i	$-0.197865\pi$
$642$			1.61155e8i			0.609032i
$643$		−	4.80197e8i		−	1.80629i	−0.429339	−	0.903143i	$-0.641254\pi$
$643$		−	4.80197e8i		−	1.80629i	0.429339	−	0.903143i	$-0.358746\pi$
$644$	5.01030e7	+	1.14687e7i	0.187588	+	0.0429394i
$645$	6.57979e7			0.245207
$646$	5.99143e6			0.0222246
$647$	1.19745e8			0.442124			0.221062	−	0.975260i	$-0.429048\pi$
$647$	1.19745e8			0.442124			0.221062	+	0.975260i	$0.429048\pi$
$648$	1.06890e7			0.0392837
$649$			2.29115e8i			0.838147i
$650$	1.22643e8			0.446583
$651$		−	8.09867e7i		−	0.293542i
$652$	9.68849e7			0.349553
$653$	3.26136e8			1.17127			0.585637	−	0.810573i	$-0.300844\pi$
$653$	3.26136e8			1.17127			0.585637	+	0.810573i	$0.300844\pi$
$654$			3.24801e7i			0.116114i
$655$		−	3.34680e8i		−	1.19098i
$656$	1.03869e8			0.367937
$657$	1.33202e8			0.469693
$658$			1.05358e8i			0.369820i
$659$			3.62777e8i			1.26760i	0.773495	+	0.633802i	$0.218507\pi$
$659$			3.62777e8i			1.26760i	−0.773495	+	0.633802i	$0.781493\pi$
$660$	−1.14762e8			−0.399176
$661$		−	3.48794e8i		−	1.20772i	−0.797092	−	0.603858i	$-0.793629\pi$
$661$		−	3.48794e8i		−	1.20772i	0.797092	−	0.603858i	$-0.206371\pi$
$662$	2.88792e8			0.995433
$663$		−	4.66901e6i		−	0.0160208i
$664$			4.19334e6i			0.0143237i
$665$		−	6.35770e7i		−	0.216190i
$666$		−	3.63766e7i		−	0.123140i
$667$	−3.35238e7	+	1.46455e8i	−0.112973	+	0.493543i
$668$	−1.25833e8			−0.422149
$669$	−7.14788e7			−0.238726
$670$	−5.38513e8			−1.79049
$671$	1.71351e8			0.567177
$672$			1.19206e7i			0.0392817i
$673$	−3.07059e8			−1.00734			−0.503671	−	0.863895i	$-0.668018\pi$
$673$	−3.07059e8			−1.00734			−0.503671	+	0.863895i	$0.668018\pi$
$674$		−	8.13304e7i		−	0.265628i
$675$	−1.37360e8			−0.446630
$676$	−1.43019e8			−0.462971
$677$		−	5.13238e8i		−	1.65407i	−0.562153	−	0.827033i	$-0.690027\pi$
$677$		−	5.13238e8i		−	1.65407i	0.562153	−	0.827033i	$-0.309973\pi$
$678$		−	5.93700e7i		−	0.190492i
$679$	−6.09089e7			−0.194568
$680$	2.06565e7			0.0656947
$681$		−	3.31383e7i		−	0.104927i
$682$			2.24844e8i			0.708807i
$683$	−3.96100e8			−1.24320			−0.621602	−	0.783333i	$-0.713518\pi$
$683$	−3.96100e8			−1.24320			−0.621602	+	0.783333i	$0.713518\pi$
$684$		−	1.64402e7i		−	0.0513736i
$685$	4.41894e8			1.37482
$686$			1.62702e8i			0.503988i
$687$		−	1.31403e8i		−	0.405262i
$688$			1.89749e7i			0.0582660i
$689$			9.08298e7i			0.277697i
$690$	−2.38232e8	−	5.45319e7i	−0.725192	−	0.165998i
$691$	−5.49305e8			−1.66487			−0.832433	−	0.554126i	$-0.813052\pi$
$691$	−5.49305e8			−1.66487			−0.832433	+	0.554126i	$0.813052\pi$
$692$	−6.98481e7			−0.210783
$693$	3.23997e7			0.0973511
$694$	−2.20975e8			−0.661096
$695$		−	7.55032e8i		−	2.24911i
$696$	−3.48447e7			−0.103350
$697$			5.08146e7i			0.150069i
$698$	−3.33317e7			−0.0980146
$699$	2.65108e8			0.776233
$700$		−	1.53186e8i		−	0.446606i
$701$			2.10997e7i			0.0612524i	0.999531	+	0.0306262i	$0.00975015\pi$
$701$			2.10997e7i			0.0612524i	−0.999531	+	0.0306262i	$0.990250\pi$
$702$	−1.28116e7			−0.0370331
$703$	−5.59490e7			−0.161037
$704$		−	3.30952e7i		−	0.0948520i
$705$		−	5.00963e8i		−	1.42968i
$706$	−2.03346e8			−0.577857
$707$			2.36329e8i			0.668742i
$708$	−1.13160e8			−0.318855
$709$			2.12403e7i			0.0595966i	0.999556	+	0.0297983i	$0.00948649\pi$
$709$			2.12403e7i			0.0595966i	−0.999556	+	0.0297983i	$0.990514\pi$
$710$			1.32267e8i			0.369552i
$711$			1.00319e8i			0.279110i
$712$		−	1.41105e7i		−	0.0390934i
$713$	−1.06840e8	+	4.66751e8i	−0.294759	+	1.28771i
$714$	−5.83178e6			−0.0160216
$715$	1.37550e8			0.376307
$716$	−4.84970e6			−0.0132122
$717$	−8.72447e7			−0.236691
$718$		−	1.14378e8i		−	0.309008i
$719$	−4.19590e8			−1.12886			−0.564428	−	0.825482i	$-0.690903\pi$
$719$	−4.19590e8			−1.12886			−0.564428	+	0.825482i	$0.690903\pi$
$720$		−	5.66806e7i		−	0.151858i
$721$	−2.12751e8			−0.567630
$722$	2.40846e8			0.639923
$723$			2.38300e8i			0.630535i
$724$		−	1.67557e7i		−	0.0441518i
$725$	4.47774e8			1.17502
$726$	6.62679e7			0.173178
$727$			4.18453e8i			1.08904i	0.838748	+	0.544520i	$0.183288\pi$
$727$			4.18453e8i			1.08904i	−0.838748	+	0.544520i	$0.816712\pi$
$728$		−	1.42877e7i		−	0.0370312i
$729$	1.43489e7			0.0370370
$730$		−	7.06330e8i		−	1.81568i
$731$	−9.28291e6			−0.0237647
$732$			8.46301e7i			0.215770i
$733$		−	9.32685e7i		−	0.236823i	−0.992965	−	0.118411i	$-0.962220\pi$
$733$		−	9.32685e7i		−	0.236823i	0.992965	−	0.118411i	$-0.0377801\pi$
$734$		−	2.04625e8i		−	0.517453i
$735$		−	3.55870e8i		−	0.896252i
$736$	1.57260e7	−	6.87019e7i	0.0394444	−	0.172320i
$737$	−4.22092e8			−1.05440
$738$	1.39433e8			0.346894
$739$	−1.32637e8			−0.328648			−0.164324	−	0.986406i	$-0.552544\pi$
$739$	−1.32637e8			−0.328648			−0.164324	+	0.986406i	$0.552544\pi$
$740$	−1.92894e8			−0.476019
$741$			1.97048e7i			0.0484304i
$742$	1.13450e8			0.277711
$743$		−	2.98499e8i		−	0.727741i	−0.931449	−	0.363871i	$-0.881455\pi$
$743$		−	2.98499e8i		−	0.727741i	0.931449	−	0.363871i	$-0.118545\pi$
$744$	−1.11050e8			−0.269650
$745$	1.31205e9			3.17308
$746$		−	2.08874e8i		−	0.503115i
$747$			5.62913e6i			0.0135045i
$748$	1.61908e7			0.0386869
$749$	−2.41260e8			−0.574170
$750$			4.14524e8i			0.982575i
$751$		−	2.95690e8i		−	0.698098i	−0.937104	−	0.349049i	$-0.886505\pi$
$751$		−	2.95690e8i		−	0.698098i	0.937104	−	0.349049i	$-0.113495\pi$
$752$	1.44469e8			0.339719
$753$		−	4.17629e8i		−	0.978150i
$754$	4.17639e7			0.0974288
$755$			3.17437e8i			0.737593i
$756$			1.60022e7i			0.0370351i
$757$		−	5.07419e8i		−	1.16971i	−0.811137	−	0.584856i	$-0.801151\pi$
$757$		−	5.07419e8i		−	1.16971i	0.811137	−	0.584856i	$-0.198849\pi$
$758$			1.93705e8i			0.444768i
$759$	−1.86729e8	−	4.27427e7i	−0.427057	−	0.0977545i
$760$	−8.71777e7			−0.198593
$761$	−2.99126e8			−0.678736			−0.339368	−	0.940654i	$-0.610213\pi$
$761$	−2.99126e8			−0.678736			−0.339368	+	0.940654i	$0.610213\pi$
$762$	2.41393e7			0.0545582
$763$	−4.86248e7			−0.109467
$764$			1.39739e8i			0.313355i
$765$	2.77293e7			0.0619376
$766$			8.69184e7i			0.193386i
$767$	1.35630e8			0.300588
$768$	1.63457e7			0.0360844
$769$			2.47479e8i			0.544201i	0.962269	+	0.272100i	$0.0877183\pi$
$769$			2.47479e8i			0.544201i	−0.962269	+	0.272100i	$0.912282\pi$
$770$		−	1.71806e8i		−	0.376327i
$771$	−1.43760e8			−0.313672
$772$	−1.62090e8			−0.352294
$773$		−	3.34912e8i		−	0.725090i	−0.931966	−	0.362545i	$-0.881908\pi$
$773$		−	3.34912e8i		−	0.725090i	0.931966	−	0.362545i	$-0.118092\pi$
$774$			2.54719e7i			0.0549337i
$775$	1.42706e9			3.06574
$776$			8.35192e7i			0.178732i
$777$	5.44582e7			0.116091
$778$		−	6.01130e8i		−	1.27653i
$779$		−	2.14455e8i		−	0.453654i
$780$			6.79359e7i			0.143158i
$781$			1.03672e8i			0.217625i
$782$	3.36103e7	+	7.69348e6i	0.0702833	+	0.0160880i
$783$	−4.67755e7			−0.0974391
$784$	1.02627e8			0.212967
$785$	−1.16555e9			−2.40948
$786$	1.29562e8			0.266816
$787$			2.65076e8i			0.543809i	0.962324	+	0.271904i	$0.0876534\pi$
$787$			2.65076e8i			0.543809i	−0.962324	+	0.271904i	$0.912347\pi$
$788$	2.30157e8			0.470377
$789$			3.64231e8i			0.741559i
$790$	5.31963e8			1.07895
$791$	8.88808e7			0.179589
$792$		−	4.44269e7i		−	0.0894273i
$793$		−	1.01435e8i		−	0.203409i
$794$	−5.00418e8			−0.999704
$795$	−5.39439e8			−1.07360
$796$		−	4.60395e8i		−	0.912833i
$797$			4.66248e8i			0.920962i	0.887670	+	0.460481i	$0.152323\pi$
$797$			4.66248e8i			0.920962i	−0.887670	+	0.460481i	$0.847677\pi$
$798$	2.46121e7			0.0484329
$799$			7.06769e7i			0.138560i
$800$	−2.10051e8			−0.410256
$801$		−	1.89419e7i		−	0.0368576i
$802$		−	8.88773e7i		−	0.172293i
$803$		−	5.53629e8i		−	1.06923i
$804$		−	2.08471e8i		−	0.401123i
$805$	8.16380e7	−	3.56650e8i	0.156496	−	0.683682i
$806$	1.33102e8			0.254202
$807$	4.85279e8			0.923359
$808$	3.24058e8			0.614311
$809$	6.15981e8			1.16338			0.581690	−	0.813410i	$-0.302392\pi$
$809$	6.15981e8			1.16338			0.581690	+	0.813410i	$0.302392\pi$
$810$		−	7.60880e7i		−	0.143173i
$811$	−4.38821e8			−0.822668			−0.411334	−	0.911485i	$-0.634937\pi$
$811$	−4.38821e8			−0.822668			−0.411334	+	0.911485i	$0.634937\pi$
$812$		−	5.21649e7i		−	0.0974340i
$813$	−4.17795e8			−0.777484
$814$	−1.51193e8			−0.280322
$815$		−	6.89659e8i		−	1.27398i
$816$			7.99663e6i			0.0147176i
$817$	3.91771e7			0.0718400
$818$	−5.42579e7			−0.0991295
$819$		−	1.91798e7i		−	0.0349134i
$820$		−	7.39373e8i		−	1.34098i
$821$	9.59467e8			1.73381			0.866903	−	0.498477i	$-0.166107\pi$
$821$	9.59467e8			1.73381			0.866903	+	0.498477i	$0.166107\pi$
$822$			1.71068e8i			0.308001i
$823$	−5.41808e8			−0.971954			−0.485977	−	0.873972i	$-0.661536\pi$
$823$	−5.41808e8			−0.971954			−0.485977	+	0.873972i	$0.661536\pi$
$824$			2.91727e8i			0.521429i
$825$			5.70910e8i			1.01673i
$826$		−	1.69408e8i		−	0.300603i
$827$			1.02981e9i			1.82070i	0.413838	+	0.910350i	$0.364188\pi$
$827$			1.02981e9i			1.82070i	−0.413838	+	0.910350i	$0.635812\pi$
$828$	2.11106e7	−	9.22253e7i	0.0371886	−	0.162465i
$829$	−5.86875e8			−1.03011			−0.515053	−	0.857158i	$-0.672228\pi$
$829$	−5.86875e8			−1.03011			−0.515053	+	0.857158i	$0.672228\pi$
$830$	2.98496e7			0.0522041
$831$	−2.36036e8			−0.411316
$832$	−1.95915e7			−0.0340171
$833$			5.02070e7i			0.0868618i
$834$	2.92291e8			0.503868
$835$			8.95723e8i			1.53856i
$836$	−6.83308e7			−0.116949
$837$	−1.49074e8			−0.254229
$838$			6.92294e8i			1.17641i
$839$		−	1.48596e8i		−	0.251606i	−0.992055	−	0.125803i	$-0.959849\pi$
$839$		−	1.48596e8i		−	0.251606i	0.992055	−	0.125803i	$-0.0401507\pi$
$840$	8.48547e7			0.143165
$841$	−4.42342e8			−0.743652
$842$			1.55695e8i			0.260818i
$843$		−	4.72569e8i		−	0.788828i
$844$	2.84522e8			0.473247
$845$			1.01806e9i			1.68734i
$846$	1.93934e8			0.320290
$847$			9.92074e7i			0.163265i
$848$		−	1.55565e8i		−	0.255108i
$849$			4.64049e8i			0.758299i
$850$		−	1.02761e8i		−	0.167329i
$851$	−3.13859e8	−	7.18431e7i	−0.509267	−	0.116573i
$852$	−5.12036e7			−0.0827907
$853$	−9.14966e8			−1.47420			−0.737102	−	0.675781i	$-0.763806\pi$
$853$	−9.14966e8			−1.47420			−0.737102	+	0.675781i	$0.763806\pi$
$854$	−1.26697e8			−0.203419
$855$	−1.17027e8			−0.187236
$856$			3.30820e8i			0.527437i
$857$	3.61208e8			0.573871			0.286936	−	0.957950i	$-0.407363\pi$
$857$	3.61208e8			0.573871			0.286936	+	0.957950i	$0.407363\pi$
$858$			5.32489e7i			0.0843041i
$859$	−1.03446e8			−0.163205			−0.0816023	−	0.996665i	$-0.526004\pi$
$859$	−1.03446e8			−0.163205			−0.0816023	+	0.996665i	$0.526004\pi$
$860$	1.35070e8			0.212355
$861$			2.08741e8i			0.327038i
$862$			1.84662e8i			0.288307i
$863$	1.01800e9			1.58386			0.791928	−	0.610614i	$-0.209077\pi$
$863$	1.01800e9			1.58386			0.791928	+	0.610614i	$0.209077\pi$
$864$	2.19424e7			0.0340207
$865$			4.97203e8i			0.768219i
$866$		−	1.53217e8i		−	0.235914i
$867$	3.72355e8			0.571347
$868$		−	1.66250e8i		−	0.254215i
$869$	4.16958e8			0.635380
$870$			2.48037e8i			0.376667i
$871$			2.49868e8i			0.378143i
$872$			6.66751e7i			0.100557i
$873$			1.12116e8i			0.168510i
$874$	−1.41847e8	−	3.24692e7i	−0.212465	−	0.0486336i
$875$	−6.20570e8			−0.926332
$876$	2.73437e8			0.406766
$877$	−5.71311e8			−0.846981			−0.423491	−	0.905900i	$-0.639195\pi$
$877$	−5.71311e8			−0.846981			−0.423491	+	0.905900i	$0.639195\pi$
$878$	−7.92645e8			−1.17110
$879$			1.37251e8i			0.202091i
$880$	−2.35583e8			−0.345697
$881$			4.06783e8i			0.594888i	0.954739	+	0.297444i	$0.0961341\pi$
$881$			4.06783e8i			0.594888i	−0.954739	+	0.297444i	$0.903866\pi$
$882$	1.37766e8			0.200787
$883$	9.25628e8			1.34448			0.672240	−	0.740333i	$-0.265332\pi$
$883$	9.25628e8			1.34448			0.672240	+	0.740333i	$0.265332\pi$
$884$		−	9.58454e6i		−	0.0138744i
$885$			8.05511e8i			1.16209i
$886$	1.81785e8			0.261371
$887$	6.87701e8			0.985436			0.492718	−	0.870189i	$-0.336003\pi$
$887$	6.87701e8			0.985436			0.492718	+	0.870189i	$0.336003\pi$
$888$		−	7.46739e7i		−	0.106642i
$889$			3.61382e7i			0.0514353i
$890$	−1.00443e8			−0.142479
$891$		−	5.96386e7i		−	0.0843129i
$892$	−1.46732e8			−0.206742
$893$		−	2.98281e8i		−	0.418862i
$894$			5.07925e8i			0.710864i
$895$			3.45218e7i			0.0481531i
$896$			2.44706e7i			0.0340189i
$897$	−2.53026e7	+	1.10539e8i	−0.0350580	+	0.153157i
$898$	−5.05691e8			−0.698323
$899$	4.85960e8			0.668838
$900$	−2.81972e8			−0.386793
$901$	7.61053e7			0.104050
$902$		−	5.79528e8i		−	0.789687i
$903$	−3.81332e7			−0.0517893
$904$		−	1.21875e8i		−	0.164971i
$905$	−1.19273e8			−0.160915
$906$	−1.22887e8			−0.165243
$907$			1.40086e9i			1.87747i	0.344637	+	0.938736i	$0.388002\pi$
$907$			1.40086e9i			1.87747i	−0.344637	+	0.938736i	$0.611998\pi$
$908$		−	6.80263e7i		−	0.0908698i
$909$	4.35014e8			0.579178
$910$	−1.01705e8			−0.134964
$911$		−	5.01093e8i		−	0.662771i	−0.943495	−	0.331385i	$-0.892484\pi$
$911$		−	5.01093e8i		−	0.662771i	0.943495	−	0.331385i	$-0.107516\pi$
$912$		−	3.37485e7i		−	0.0444908i
$913$	2.33964e7			0.0307424
$914$			6.64639e8i			0.870457i
$915$	6.02425e8			0.786394
$916$		−	2.69744e8i		−	0.350967i
$917$			1.93964e8i			0.251543i
$918$			1.07347e7i			0.0138759i
$919$			1.43842e9i			1.85327i	0.375963	+	0.926635i	$0.377312\pi$
$919$			1.43842e9i			1.85327i	−0.375963	+	0.926635i	$0.622688\pi$
$920$	−4.89043e8	−	1.11943e8i	−0.628035	−	0.143759i
$921$	5.89834e8			0.755007
$922$	−6.52145e8			−0.832054
$923$	6.13712e7			0.0780476
$924$	6.65100e7			0.0843085
$925$			9.59600e8i			1.21245i
$926$	−4.50846e8			−0.567800
$927$			3.91614e8i			0.491608i
$928$	−7.15293e7			−0.0895035
$929$	1.08444e9			1.35256			0.676280	−	0.736645i	$-0.263591\pi$
$929$	1.08444e9			1.35256			0.676280	+	0.736645i	$0.263591\pi$
$930$			7.90493e8i			0.982764i
$931$		−	2.11891e8i		−	0.262581i
$932$	5.44215e8			0.672237
$933$	−5.27847e8			−0.649925
$934$		−	4.37445e8i		−	0.536887i
$935$		−	1.15252e8i		−	0.140998i
$936$	−2.62996e7			−0.0320716
$937$			1.36821e9i			1.66316i	0.555402	+	0.831582i	$0.312564\pi$
$937$			1.36821e9i			1.66316i	−0.555402	+	0.831582i	$0.687436\pi$
$938$	3.12095e8			0.378163
$939$		−	8.33336e8i		−	1.00652i
$940$		−	1.02838e9i		−	1.23814i
$941$		−	1.61102e9i		−	1.93345i	−0.255818	−	0.966725i	$-0.582345\pi$
$941$		−	1.61102e9i		−	1.93345i	0.255818	−	0.966725i	$-0.417655\pi$
$942$		−	4.51214e8i		−	0.539796i
$943$	2.75378e8	−	1.20304e9i	0.328393	−	1.43464i
$944$	−2.32295e8			−0.276136
$945$	1.13909e8			0.134978
$946$	1.05869e8			0.125054
$947$	−3.53025e8			−0.415677			−0.207838	−	0.978163i	$-0.566643\pi$
$947$	−3.53025e8			−0.415677			−0.207838	+	0.978163i	$0.566643\pi$
$948$			2.05935e8i			0.241716i
$949$	−3.27734e8			−0.383462
$950$			4.33687e8i			0.505831i
$951$	4.39936e8			0.511503
$952$	−1.19715e7			−0.0138751
$953$		−	1.49094e9i		−	1.72259i	−0.508104	−	0.861296i	$-0.669653\pi$
$953$		−	1.49094e9i		−	1.72259i	0.508104	−	0.861296i	$-0.330347\pi$
$954$		−	2.08830e8i		−	0.240518i
$955$	9.94707e8			1.14205
$956$	−1.79096e8			−0.204980
$957$			1.94414e8i			0.221815i
$958$			6.38172e8i			0.725840i
$959$	−2.56100e8			−0.290371
$960$		−	1.16354e8i		−	0.131513i
$961$	6.61250e8			0.745068
$962$			8.95021e7i			0.100533i
$963$			4.44092e8i			0.497272i
$964$			4.89182e8i			0.546059i
$965$			1.15381e9i			1.28397i
$966$	1.38067e8	+	3.16040e7i	0.153165	+	0.0350599i
$967$	2.52642e8			0.279400			0.139700	−	0.990194i	$-0.455386\pi$
$967$	2.52642e8			0.279400			0.139700	+	0.990194i	$0.455386\pi$
$968$	1.36035e8			0.149977
$969$	1.65104e7			0.0181463
$970$	5.94518e8			0.651403
$971$		−	1.28694e9i		−	1.40573i	−0.711324	−	0.702865i	$-0.751904\pi$
$971$		−	1.28694e9i		−	1.40573i	0.711324	−	0.702865i	$-0.248096\pi$
$972$	2.94555e7			0.0320750
$973$			4.37579e8i			0.475026i
$974$	5.10336e8			0.552306
$975$	3.37964e8			0.364633
$976$			1.73729e8i			0.186862i
$977$			1.84947e9i			1.98318i	0.129405	+	0.991592i	$0.458693\pi$
$977$			1.84947e9i			1.98318i	−0.129405	+	0.991592i	$0.541307\pi$
$978$	2.66983e8			0.285409
$979$	−7.87287e7			−0.0839044
$980$		−	7.30531e8i		−	0.776177i
$981$			8.95045e7i			0.0948065i
$982$	6.75978e8			0.713835
$983$		−	7.43938e7i		−	0.0783207i	−0.999233	−	0.0391603i	$-0.987532\pi$
$983$		−	7.43938e7i		−	0.0783207i	0.999233	−	0.0391603i	$-0.0124683\pi$
$984$	2.86229e8			0.300419
$985$		−	1.63834e9i		−	1.71433i
$986$		−	3.49935e7i		−	0.0365054i
$987$			2.90333e8i			0.301957i
$988$			4.04501e7i			0.0419420i
$989$	2.19773e8	+	5.03065e7i	0.227188	+	0.0520039i
$990$	−3.16246e8			−0.325926
$991$	−3.40394e8			−0.349753			−0.174876	−	0.984590i	$-0.555953\pi$
$991$	−3.40394e8			−0.349753			−0.174876	+	0.984590i	$0.555953\pi$
$992$	−2.27964e8			−0.233524
$993$	7.95819e8			0.812767
$994$		−	7.66552e7i		−	0.0780517i
$995$	−3.27725e9			−3.32690
$996$			1.15555e7i			0.0116953i
$997$	5.78347e8			0.583583			0.291792	−	0.956482i	$-0.405749\pi$
$997$	5.78347e8			0.583583			0.291792	+	0.956482i	$0.405749\pi$
$998$	4.99516e8			0.502525
$999$		−	1.00242e8i		−	0.100543i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.7.b.a.91.19	✓	24
3.2	odd	2		414.7.b.c.91.12		24
23.22	odd	2	inner	138.7.b.a.91.24	yes	24
69.68	even	2		414.7.b.c.91.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.7.b.a.91.19	✓	24	1.1	even	1	trivial
138.7.b.a.91.24	yes	24	23.22	odd	2	inner
414.7.b.c.91.1		24	69.68	even	2
414.7.b.c.91.12		24	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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.65685 q^{2} +15.5885 q^{3} +32.0000 q^{4} -227.787i q^{5} +88.1816 q^{6} +132.014i q^{7} +181.019 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q+5.65685 q^{2} +15.5885 q^{3} +32.0000 q^{4} -227.787i q^{5} +88.1816 q^{6} +132.014i q^{7} +181.019 q^{8} +243.000 q^{9} -1288.56i q^{10} -1009.98i q^{11} +498.831 q^{12} -597.885 q^{13} +746.783i q^{14} -3550.84i q^{15} +1024.00 q^{16} +500.961i q^{17} +1374.62 q^{18} -2114.23i q^{19} -7289.18i q^{20} +2057.89i q^{21} -5713.33i q^{22} +(2714.84 - 11860.3i) q^{23} +2821.81 q^{24} -36261.8 q^{25} -3382.15 q^{26} +3788.00 q^{27} +4224.44i q^{28} -12348.4 q^{29} -20086.6i q^{30} -39354.2 q^{31} +5792.62 q^{32} -15744.1i q^{33} +2833.86i q^{34} +30071.0 q^{35} +7776.00 q^{36} -26463.1i q^{37} -11959.9i q^{38} -9320.10 q^{39} -41233.8i q^{40} +101434. q^{41} +11641.2i q^{42} +18530.2i q^{43} -32319.5i q^{44} -55352.2i q^{45} +(15357.4 - 67091.7i) q^{46} +141083. q^{47} +15962.6 q^{48} +100221. q^{49} -205128. q^{50} +7809.21i q^{51} -19132.3 q^{52} -151919. i q^{53} +21428.1 q^{54} -230061. q^{55} +23897.1i q^{56} -32957.6i q^{57} -69852.8 q^{58} -226850. q^{59} -113627. i q^{60} +169657. i q^{61} -222621. q^{62} +32079.4i q^{63} +32768.0 q^{64} +136190. i q^{65} -89062.1i q^{66} -417920. i q^{67} +16030.7i q^{68} +(42320.2 - 184883. i) q^{69} +170107. q^{70} -102647. q^{71} +43987.7 q^{72} +548156. q^{73} -149698. i q^{74} -565266. q^{75} -67655.3i q^{76} +133332. q^{77} -52722.5 q^{78} +412836. i q^{79} -233254. i q^{80} +59049.0 q^{81} +573799. q^{82} +23165.2i q^{83} +65852.5i q^{84} +114112. q^{85} +104823. i q^{86} -192492. q^{87} -182827. i q^{88} -77950.4i q^{89} -313119. i q^{90} -78929.1i q^{91} +(86874.9 - 379528. i) q^{92} -613471. q^{93} +798084. q^{94} -481593. q^{95} +90298.0 q^{96} +461383. i q^{97} +566938. q^{98} -245426. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 768 q^{4} + 5832 q^{9}+O(q^{10}) \) 24 * q + 768 * q^4 + 5832 * q^9	\( 24 q + 768 q^{4} + 5832 q^{9} - 768 q^{13} + 24576 q^{16} - 44104 q^{23} - 119448 q^{25} - 53888 q^{26} + 3456 q^{29} + 50976 q^{31} + 149008 q^{35} + 186624 q^{36} + 11664 q^{39} - 3920 q^{41} - 150720 q^{46} + 441088 q^{47} - 32472 q^{49} + 8320 q^{50} - 24576 q^{52} + 826176 q^{55} - 307200 q^{58} - 1210160 q^{59} + 783744 q^{62} + 786432 q^{64} + 361584 q^{69} - 2480064 q^{70} + 1531264 q^{71} + 593472 q^{73} + 23328 q^{75} + 1068784 q^{77} + 171072 q^{78} + 1417176 q^{81} + 1454592 q^{82} - 1318272 q^{85} + 697248 q^{87} - 1411328 q^{92} - 983664 q^{93} + 1115712 q^{94} + 4047632 q^{95} - 2409344 q^{98}+O(q^{100}) \) 24 * q + 768 * q^4 + 5832 * q^9 - 768 * q^13 + 24576 * q^16 - 44104 * q^23 - 119448 * q^25 - 53888 * q^26 + 3456 * q^29 + 50976 * q^31 + 149008 * q^35 + 186624 * q^36 + 11664 * q^39 - 3920 * q^41 - 150720 * q^46 + 441088 * q^47 - 32472 * q^49 + 8320 * q^50 - 24576 * q^52 + 826176 * q^55 - 307200 * q^58 - 1210160 * q^59 + 783744 * q^62 + 786432 * q^64 + 361584 * q^69 - 2480064 * q^70 + 1531264 * q^71 + 593472 * q^73 + 23328 * q^75 + 1068784 * q^77 + 171072 * q^78 + 1417176 * q^81 + 1454592 * q^82 - 1318272 * q^85 + 697248 * q^87 - 1411328 * q^92 - 983664 * q^93 + 1115712 * q^94 + 4047632 * q^95 - 2409344 * q^98

Introduction

L-functions

Modular forms

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Database

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