LMFDB - Embedded newform 37.8.b.a.36.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,8,Mod(36,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("37.36");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 37 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	37.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:	$11.5582459429$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 1702 x^{18} + 1194509 x^{16} + 450999516 x^{14} + 100204783492 x^{12} + 13461378480848 x^{10} + \cdots + 13\!\cdots\!96 $ x^20 + 1702x^18 + 1194509x^16 + 450999516x^14 + 100204783492x^12 + 13461378480848x^10 + 1081011973644416x^8 + 49304995250225664x^6 + 1131877572418003968x^4 + 9402469145336696832*x^2 + 130757963535876096
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			36.1
Root			$-21.7545i$ of defining polynomial
Character	$\chi$	$=$	37.36
Dual form			37.8.b.a.36.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.7545i q^{2} +44.4956 q^{3} -345.259 q^{4} -104.884i q^{5} -967.981i q^{6} -454.964 q^{7} +4726.36i q^{8} -207.137 q^{9} +O(q^{10})$	\(q-21.7545i q^{2} +44.4956 q^{3} -345.259 q^{4} -104.884i q^{5} -967.981i q^{6} -454.964 q^{7} +4726.36i q^{8} -207.137 q^{9} -2281.70 q^{10} -4886.81 q^{11} -15362.5 q^{12} +2167.39i q^{13} +9897.53i q^{14} -4666.88i q^{15} +58626.6 q^{16} -33286.0i q^{17} +4506.17i q^{18} +24585.0i q^{19} +36212.1i q^{20} -20243.9 q^{21} +106310. i q^{22} -62365.5i q^{23} +210303. i q^{24} +67124.3 q^{25} +47150.4 q^{26} -106529. q^{27} +157081. q^{28} -61128.3i q^{29} -101526. q^{30} -34740.4i q^{31} -670419. i q^{32} -217442. q^{33} -724120. q^{34} +47718.5i q^{35} +71516.0 q^{36} +(-29898.6 - 306656. i) q^{37} +534835. q^{38} +96439.3i q^{39} +495720. q^{40} -645608. q^{41} +440397. i q^{42} +524254. i q^{43} +1.68721e6 q^{44} +21725.4i q^{45} -1.35673e6 q^{46} -1.03728e6 q^{47} +2.60863e6 q^{48} -616550. q^{49} -1.46026e6i q^{50} -1.48108e6i q^{51} -748310. i q^{52} +2.01545e6 q^{53} +2.31748e6i q^{54} +512548. i q^{55} -2.15033e6i q^{56} +1.09393e6i q^{57} -1.32982e6 q^{58} -2.51705e6i q^{59} +1.61128e6i q^{60} +981203. i q^{61} -755761. q^{62} +94240.0 q^{63} -7.08043e6 q^{64} +227324. q^{65} +4.73034e6i q^{66} +941977. q^{67} +1.14923e7i q^{68} -2.77499e6i q^{69} +1.03809e6 q^{70} -2.67347e6 q^{71} -979006. i q^{72} +4.46976e6 q^{73} +(-6.67115e6 + 650429. i) q^{74} +2.98674e6 q^{75} -8.48819e6i q^{76} +2.22332e6 q^{77} +2.09799e6 q^{78} -4.48534e6i q^{79} -6.14899e6i q^{80} -4.28705e6 q^{81} +1.40449e7i q^{82} +6.80298e6 q^{83} +6.98940e6 q^{84} -3.49116e6 q^{85} +1.14049e7 q^{86} -2.71994e6i q^{87} -2.30968e7i q^{88} -746269. i q^{89} +472625. q^{90} -986084. i q^{91} +2.15322e7i q^{92} -1.54580e6i q^{93} +2.25655e7i q^{94} +2.57857e6 q^{95} -2.98307e7i q^{96} -3.11369e6i q^{97} +1.34128e7i q^{98} +1.01224e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9}+O(q^{10}) $ 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9	$ 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9} - 882 q^{10} + 3498 q^{11} - 30374 q^{12} + 36116 q^{16} + 113482 q^{21} - 108112 q^{25} + 49278 q^{26} - 304110 q^{27} - 41192 q^{28} + 429776 q^{30} + 305646 q^{33} - 960356 q^{34} + 484758 q^{36} + 108732 q^{37} + 1049916 q^{38} - 496346 q^{40} - 1577742 q^{41} + 685266 q^{44} - 2906298 q^{46} - 1512786 q^{47} + 1522958 q^{48} + 3269246 q^{49} + 2999358 q^{53} + 405946 q^{58} + 3728310 q^{62} - 11995292 q^{63} - 11109700 q^{64} + 4251792 q^{65} + 3562224 q^{67} + 21605644 q^{70} - 15259086 q^{71} + 11088018 q^{73} - 2036544 q^{74} + 14882062 q^{75} - 2419122 q^{77} - 12178734 q^{78} - 17764972 q^{81} - 12873822 q^{83} + 9944396 q^{84} - 2698920 q^{85} + 15345336 q^{86} - 13219100 q^{90} + 48981192 q^{95} + 43111380 q^{99}+O(q^{100}) $ 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9 - 882 * q^10 + 3498 * q^11 - 30374 * q^12 + 36116 * q^16 + 113482 * q^21 - 108112 * q^25 + 49278 * q^26 - 304110 * q^27 - 41192 * q^28 + 429776 * q^30 + 305646 * q^33 - 960356 * q^34 + 484758 * q^36 + 108732 * q^37 + 1049916 * q^38 - 496346 * q^40 - 1577742 * q^41 + 685266 * q^44 - 2906298 * q^46 - 1512786 * q^47 + 1522958 * q^48 + 3269246 * q^49 + 2999358 * q^53 + 405946 * q^58 + 3728310 * q^62 - 11995292 * q^63 - 11109700 * q^64 + 4251792 * q^65 + 3562224 * q^67 + 21605644 * q^70 - 15259086 * q^71 + 11088018 * q^73 - 2036544 * q^74 + 14882062 * q^75 - 2419122 * q^77 - 12178734 * q^78 - 17764972 * q^81 - 12873822 * q^83 + 9944396 * q^84 - 2698920 * q^85 + 15345336 * q^86 - 13219100 * q^90 + 48981192 * q^95 + 43111380 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$		−	21.7545i		−	1.92285i	−0.275075	−	0.961423i	$-0.588703\pi$
$2$		−	21.7545i		−	1.92285i	0.275075	−	0.961423i	$-0.411297\pi$
$3$	44.4956			0.951466			0.475733	−	0.879590i	$-0.342183\pi$
$3$	44.4956			0.951466			0.475733	+	0.879590i	$0.342183\pi$
$4$	−345.259			−2.69734
$5$		−	104.884i		−	0.375244i	−0.982241	−	0.187622i	$-0.939922\pi$
$5$		−	104.884i		−	0.375244i	0.982241	−	0.187622i	$-0.0600781\pi$
$6$		−	967.981i		−	1.82952i
$7$	−454.964			−0.501342			−0.250671	−	0.968072i	$-0.580651\pi$
$7$	−454.964			−0.501342			−0.250671	+	0.968072i	$0.580651\pi$
$8$			4726.36i			3.26371i
$9$	−207.137			−0.0947129
$10$	−2281.70			−0.721537
$11$	−4886.81			−1.10701			−0.553504	−	0.832846i	$-0.686710\pi$
$11$	−4886.81			−1.10701			−0.553504	+	0.832846i	$0.686710\pi$
$12$	−15362.5			−2.56642
$13$			2167.39i			0.273612i	0.990598	+	0.136806i	$0.0436836\pi$
$13$			2167.39i			0.273612i	−0.990598	+	0.136806i	$0.956316\pi$
$14$			9897.53i			0.964004i
$15$		−	4666.88i		−	0.357032i
$16$	58626.6			3.57828
$17$		−	33286.0i		−	1.64320i	−0.570066	−	0.821599i	$-0.693082\pi$
$17$		−	33286.0i		−	1.64320i	0.570066	−	0.821599i	$-0.306918\pi$
$18$			4506.17i			0.182118i
$19$			24585.0i			0.822304i	0.911567	+	0.411152i	$0.134873\pi$
$19$			24585.0i			0.822304i	−0.911567	+	0.411152i	$0.865127\pi$
$20$			36212.1i			1.01216i
$21$	−20243.9			−0.477010
$22$			106310.i			2.12861i
$23$		−	62365.5i		−	1.06880i	−0.845231	−	0.534401i	$-0.820537\pi$
$23$		−	62365.5i		−	1.06880i	0.845231	−	0.534401i	$-0.179463\pi$
$24$			210303.i			3.10531i
$25$	67124.3			0.859192
$26$	47150.4			0.526113
$27$	−106529.			−1.04158
$28$	157081.			1.35229
$29$		−	61128.3i		−	0.465424i	−0.972546	−	0.232712i	$-0.925240\pi$
$29$		−	61128.3i		−	0.465424i	0.972546	−	0.232712i	$-0.0747600\pi$
$30$	−101526.			−0.686518
$31$		−	34740.4i		−	0.209445i	−0.994502	−	0.104722i	$-0.966605\pi$
$31$		−	34740.4i		−	0.209445i	0.994502	−	0.104722i	$-0.0333954\pi$
$32$		−	670419.i		−	3.61677i
$33$	−217442.			−1.05328
$34$	−724120.			−3.15962
$35$			47718.5i			0.188126i
$36$	71516.0			0.255473
$37$	−29898.6	−	306656.i	−0.0970385	−	0.995281i
$37$	−29898.6	−	306656.i	−0.0970385	−	0.995281i
$38$	534835.			1.58116
$39$			96439.3i			0.260332i
$40$	495720.			1.22469
$41$	−645608.			−1.46294			−0.731468	−	0.681876i	$-0.761165\pi$
$41$	−645608.			−1.46294			−0.731468	+	0.681876i	$0.761165\pi$
$42$			440397.i			0.917216i
$43$			524254.i			1.00555i	0.864419	+	0.502773i	$0.167687\pi$
$43$			524254.i			1.00555i	−0.864419	+	0.502773i	$0.832313\pi$
$44$	1.68721e6			2.98597
$45$			21725.4i			0.0355405i
$46$	−1.35673e6			−2.05514
$47$	−1.03728e6			−1.45731			−0.728656	−	0.684880i	$-0.759855\pi$
$47$	−1.03728e6			−1.45731			−0.728656	+	0.684880i	$0.759855\pi$
$48$	2.60863e6			3.40461
$49$	−616550.			−0.748656
$50$		−	1.46026e6i		−	1.65209i
$51$		−	1.48108e6i		−	1.56345i
$52$		−	748310.i		−	0.738022i
$53$	2.01545e6			1.85955			0.929773	−	0.368133i	$-0.120003\pi$
$53$	2.01545e6			1.85955			0.929773	+	0.368133i	$0.120003\pi$
$54$			2.31748e6i			2.00280i
$55$			512548.i			0.415399i
$56$		−	2.15033e6i		−	1.63624i
$57$			1.09393e6i			0.782394i
$58$	−1.32982e6			−0.894939
$59$		−	2.51705e6i		−	1.59555i	−0.602957	−	0.797774i	$-0.706011\pi$
$59$		−	2.51705e6i		−	1.59555i	0.602957	−	0.797774i	$-0.293989\pi$
$60$			1.61128e6i			0.963035i
$61$			981203.i			0.553483i	0.960944	+	0.276742i	$0.0892546\pi$
$61$			981203.i			0.553483i	−0.960944	+	0.276742i	$0.910745\pi$
$62$	−755761.			−0.402730
$63$	94240.0			0.0474836
$64$	−7.08043e6			−3.37621
$65$	227324.			0.102671
$66$			4.73034e6i			2.02530i
$67$	941977.			0.382630			0.191315	−	0.981529i	$-0.438725\pi$
$67$	941977.			0.382630			0.191315	+	0.981529i	$0.438725\pi$
$68$			1.14923e7i			4.43226i
$69$		−	2.77499e6i		−	1.01693i
$70$	1.03809e6			0.361737
$71$	−2.67347e6			−0.886485			−0.443242	−	0.896402i	$-0.646172\pi$
$71$	−2.67347e6			−0.886485			−0.443242	+	0.896402i	$0.646172\pi$
$72$		−	979006.i		−	0.309116i
$73$	4.46976e6			1.34479			0.672394	−	0.740193i	$-0.265266\pi$
$73$	4.46976e6			1.34479			0.672394	+	0.740193i	$0.265266\pi$
$74$	−6.67115e6	+	650429.i	−1.91377	+	0.186590i
$75$	2.98674e6			0.817491
$76$		−	8.48819e6i		−	2.21803i
$77$	2.22332e6			0.554990
$78$	2.09799e6			0.500578
$79$		−	4.48534e6i		−	1.02353i	−0.859126	−	0.511765i	$-0.828992\pi$
$79$		−	4.48534e6i		−	1.02353i	0.859126	−	0.511765i	$-0.171008\pi$
$80$		−	6.14899e6i		−	1.34273i
$81$	−4.28705e6			−0.896317
$82$			1.40449e7i			2.81300i
$83$	6.80298e6			1.30595			0.652974	−	0.757380i	$-0.273521\pi$
$83$	6.80298e6			1.30595			0.652974	+	0.757380i	$0.273521\pi$
$84$	6.98940e6			1.28666
$85$	−3.49116e6			−0.616601
$86$	1.14049e7			1.93351
$87$		−	2.71994e6i		−	0.442835i
$88$		−	2.30968e7i		−	3.61296i
$89$		−	746269.i		−	0.112210i	−0.998425	−	0.0561049i	$-0.982132\pi$
$89$		−	746269.i		−	0.112210i	0.998425	−	0.0561049i	$-0.0178681\pi$
$90$	472625.			0.0683389
$91$		−	986084.i		−	0.137173i
$92$			2.15322e7i			2.88292i
$93$		−	1.54580e6i		−	0.199279i
$94$			2.25655e7i			2.80219i
$95$	2.57857e6			0.308565
$96$		−	2.98307e7i		−	3.44123i
$97$		−	3.11369e6i		−	0.346398i	−0.984887	−	0.173199i	$-0.944590\pi$
$97$		−	3.11369e6i		−	0.346398i	0.984887	−	0.173199i	$-0.0554103\pi$
$98$			1.34128e7i			1.43955i
$99$	1.01224e6			0.104848
$100$	−2.31753e7			−2.31753
$101$	6.76405e6			0.653254			0.326627	−	0.945153i	$-0.394088\pi$
$101$	6.76405e6			0.653254			0.326627	+	0.945153i	$0.394088\pi$
$102$	−3.22202e7			−3.00627
$103$		−	8.86414e6i		−	0.799293i	−0.916669	−	0.399647i	$-0.869133\pi$
$103$		−	8.86414e6i		−	0.799293i	0.916669	−	0.399647i	$-0.130867\pi$
$104$	−1.02439e7			−0.892990
$105$			2.12326e6i			0.178995i
$106$		−	4.38452e7i		−	3.57562i
$107$	3.59428e6			0.283641			0.141820	−	0.989892i	$-0.454704\pi$
$107$	3.59428e6			0.283641			0.141820	+	0.989892i	$0.454704\pi$
$108$	3.67800e7			2.80950
$109$			1.73029e7i			1.27975i	0.768477	+	0.639877i	$0.221015\pi$
$109$			1.73029e7i			1.27975i	−0.768477	+	0.639877i	$0.778985\pi$
$110$	1.11502e7			0.798748
$111$	−1.33036e6	−	1.36449e7i	−0.0923288	−	0.946975i
$112$	−2.66730e7			−1.79394
$113$			3.00271e6i			0.195767i	0.995198	+	0.0978834i	$0.0312072\pi$
$113$			3.00271e6i			0.195767i	−0.995198	+	0.0978834i	$0.968793\pi$
$114$	2.37978e7			1.50442
$115$	−6.54114e6			−0.401062
$116$			2.11051e7i			1.25541i
$117$		−	448946.i		−	0.0259146i
$118$	−5.47572e7			−3.06799
$119$			1.51439e7i			0.823804i
$120$	2.20574e7			1.16525
$121$	4.39374e6			0.225469
$122$	2.13456e7			1.06426
$123$	−2.87268e7			−1.39193
$124$			1.19944e7i			0.564942i
$125$		−	1.52343e7i		−	0.697651i
$126$		−	2.05015e6i		−	0.0913036i
$127$	−1.72118e6			−0.0745612			−0.0372806	−	0.999305i	$-0.511870\pi$
$127$	−1.72118e6			−0.0745612			−0.0372806	+	0.999305i	$0.511870\pi$
$128$			6.82177e7i			2.87516i
$129$			2.33270e7i			0.956742i
$130$		−	4.94533e6i		−	0.197421i
$131$			2.53833e6i			0.0986503i	0.998783	+	0.0493252i	$0.0157071\pi$
$131$			2.53833e6i			0.0986503i	−0.998783	+	0.0493252i	$0.984293\pi$
$132$	7.50737e7			2.84105
$133$		−	1.11853e7i		−	0.412256i
$134$		−	2.04923e7i		−	0.735738i
$135$			1.11732e7i			0.390848i
$136$	1.57322e8			5.36293
$137$	−8.62321e6			−0.286515			−0.143257	−	0.989685i	$-0.545758\pi$
$137$	−8.62321e6			−0.286515			−0.143257	+	0.989685i	$0.545758\pi$
$138$	−6.03686e7			−1.95539
$139$	6.63374e6			0.209511			0.104755	−	0.994498i	$-0.466594\pi$
$139$	6.63374e6			0.209511			0.104755	+	0.994498i	$0.466594\pi$
$140$		−	1.64752e7i		−	0.507438i
$141$	−4.61543e7			−1.38658
$142$			5.81601e7i			1.70457i
$143$		−	1.05916e7i		−	0.302891i
$144$	−1.21437e7			−0.338910
$145$	−6.41138e6			−0.174648
$146$		−	9.72374e7i		−	2.58582i
$147$	−2.74338e7			−0.712321
$148$	1.03227e7	+	1.05876e8i	0.261745	+	2.68461i
$149$	−6.28548e7			−1.55663			−0.778317	−	0.627872i	$-0.783926\pi$
$149$	−6.28548e7			−1.55663			−0.778317	+	0.627872i	$0.783926\pi$
$150$		−	6.49751e7i		−	1.57191i
$151$	2.77490e6			0.0655886			0.0327943	−	0.999462i	$-0.489559\pi$
$151$	2.77490e6			0.0655886			0.0327943	+	0.999462i	$0.489559\pi$
$152$	−1.16198e8			−2.68377
$153$			6.89476e6i			0.155632i
$154$		−	4.83673e7i		−	1.06716i
$155$	−3.64371e6			−0.0785929
$156$		−	3.32965e7i		−	0.702203i
$157$	−3.15080e6			−0.0649789			−0.0324894	−	0.999472i	$-0.510344\pi$
$157$	−3.15080e6			−0.0649789			−0.0324894	+	0.999472i	$0.510344\pi$
$158$	−9.75764e7			−1.96809
$159$	8.96788e7			1.76929
$160$	−7.03162e7			−1.35717
$161$			2.83741e7i			0.535835i
$162$			9.32628e7i			1.72348i
$163$			7.61252e7i			1.37680i	0.725330	+	0.688401i	$0.241687\pi$
$163$			7.61252e7i			1.37680i	−0.725330	+	0.688401i	$0.758313\pi$
$164$	2.22902e8			3.94603
$165$			2.28062e7i			0.395238i
$166$		−	1.47996e8i		−	2.51114i
$167$			5.09259e7i			0.846118i	0.906102	+	0.423059i	$0.139044\pi$
$167$			5.09259e7i			0.846118i	−0.906102	+	0.423059i	$0.860956\pi$
$168$		−	9.56802e7i		−	1.55682i
$169$	5.80510e7			0.925137
$170$			7.59486e7i			1.18563i
$171$		−	5.09247e6i		−	0.0778828i
$172$		−	1.81003e8i		−	2.71229i
$173$	9.72088e7			1.42739			0.713697	−	0.700454i	$-0.247019\pi$
$173$	9.72088e7			1.42739			0.713697	+	0.700454i	$0.247019\pi$
$174$	−5.91710e7			−0.851504
$175$	−3.05392e7			−0.430749
$176$	−2.86497e8			−3.96119
$177$		−	1.11998e8i		−	1.51811i
$178$	−1.62347e7			−0.215762
$179$		−	1.19521e8i		−	1.55761i	−0.627265	−	0.778806i	$-0.715826\pi$
$179$		−	1.19521e8i		−	1.55761i	0.627265	−	0.778806i	$-0.284174\pi$
$180$		−	7.50088e6i		−	0.0958646i
$181$	3.00909e7			0.377191			0.188595	−	0.982055i	$-0.439607\pi$
$181$	3.00909e7			0.377191			0.188595	+	0.982055i	$0.439607\pi$
$182$	−2.14518e7			−0.263763
$183$			4.36593e7i			0.526621i
$184$	2.94762e8			3.48826
$185$	−3.21633e7	+	3.13588e6i	−0.373473	+	0.0364132i
$186$	−3.36281e7			−0.383183
$187$			1.62662e8i			1.81903i
$188$	3.58129e8			3.93086
$189$	4.84668e7			0.522189
$190$		−	5.60956e7i		−	0.593323i
$191$			1.24539e8i			1.29327i	0.762800	+	0.646634i	$0.223824\pi$
$191$			1.24539e8i			1.29327i	−0.762800	+	0.646634i	$0.776176\pi$
$192$	−3.15048e8			−3.21235
$193$		−	1.77832e8i		−	1.78057i	−0.455401	−	0.890286i	$-0.650504\pi$
$193$		−	1.77832e8i		−	1.78057i	0.455401	−	0.890286i	$-0.349496\pi$
$194$	−6.77369e7			−0.666069
$195$	1.01149e7			0.0976882
$196$	2.12870e8			2.01938
$197$	1.29509e8			1.20689			0.603445	−	0.797405i	$-0.293794\pi$
$197$	1.29509e8			1.20689			0.603445	+	0.797405i	$0.293794\pi$
$198$		−	2.20208e7i		−	0.201607i
$199$			6.68192e7i			0.601056i	0.953773	+	0.300528i	$0.0971630\pi$
$199$			6.68192e7i			0.601056i	−0.953773	+	0.300528i	$0.902837\pi$
$200$			3.17254e8i			2.80416i
$201$	4.19139e7			0.364059
$202$		−	1.47149e8i		−	1.25611i
$203$			2.78112e7i			0.233337i
$204$			5.11356e8i			4.21714i
$205$			6.77139e7i			0.548959i
$206$	−1.92835e8			−1.53692
$207$			1.29182e7i			0.101229i
$208$			1.27066e8i			0.979060i
$209$		−	1.20142e8i		−	0.910298i
$210$	4.61906e7			0.344180
$211$	−2.32513e8			−1.70396			−0.851978	−	0.523578i	$-0.824597\pi$
$211$	−2.32513e8			−1.70396			−0.851978	+	0.523578i	$0.824597\pi$
$212$	−6.95853e8			−5.01582
$213$	−1.18958e8			−0.843460
$214$		−	7.81917e7i		−	0.545397i
$215$	5.49858e7			0.377325
$216$		−	5.03493e8i		−	3.39943i
$217$			1.58056e7i			0.105003i
$218$	3.76417e8			2.46077
$219$	1.98885e8			1.27952
$220$		−	1.76962e8i		−	1.12047i
$221$	7.21435e7			0.449598
$222$	−2.96837e8	+	2.89412e7i	−1.82089	+	0.177534i
$223$	−8.13089e7			−0.490988			−0.245494	−	0.969398i	$-0.578950\pi$
$223$	−8.13089e7			−0.490988			−0.245494	+	0.969398i	$0.578950\pi$
$224$			3.05017e8i			1.81324i
$225$	−1.39039e7			−0.0813766
$226$	6.53226e7			0.376429
$227$		−	8.95466e7i		−	0.508111i	−0.967190	−	0.254056i	$-0.918235\pi$
$227$		−	8.95466e7i		−	0.508111i	0.967190	−	0.254056i	$-0.0817646\pi$
$228$		−	3.77687e8i		−	2.11038i
$229$	−1.46451e8			−0.805877			−0.402938	−	0.915227i	$-0.632011\pi$
$229$	−1.46451e8			−0.805877			−0.402938	+	0.915227i	$0.632011\pi$
$230$			1.42299e8i			0.771180i
$231$	9.89283e7			0.528054
$232$	2.88914e8			1.51901
$233$	−1.67528e8			−0.867645			−0.433822	−	0.900998i	$-0.642835\pi$
$233$	−1.67528e8			−0.867645			−0.433822	+	0.900998i	$0.642835\pi$
$234$	−9.76661e6			−0.0498297
$235$			1.08794e8i			0.546848i
$236$			8.69034e8i			4.30373i
$237$		−	1.99578e8i		−	0.973854i
$238$	3.29449e8			1.58405
$239$		−	4.76626e7i		−	0.225832i	−0.993605	−	0.112916i	$-0.963981\pi$
$239$		−	4.76626e7i		−	0.225832i	0.993605	−	0.112916i	$-0.0360191\pi$
$240$		−	2.73603e8i		−	1.27756i
$241$		−	2.47131e8i		−	1.13728i	−0.822587	−	0.568639i	$-0.807470\pi$
$241$		−	2.47131e8i		−	1.13728i	0.822587	−	0.568639i	$-0.192530\pi$
$242$		−	9.55838e7i		−	0.433541i
$243$	4.22230e7			0.188767
$244$		−	3.38769e8i		−	1.49293i
$245$			6.46663e7i			0.280929i
$246$			6.24937e8i			2.67647i
$247$	−5.32852e7			−0.224992
$248$	1.64196e8			0.683567
$249$	3.02703e8			1.24257
$250$	−3.31415e8			−1.34148
$251$			3.41733e8i			1.36405i	0.731331	+	0.682023i	$0.238899\pi$
$251$			3.41733e8i			1.36405i	−0.731331	+	0.682023i	$0.761101\pi$
$252$	−3.25372e7			−0.128079
$253$			3.04768e8i			1.18317i
$254$			3.74434e7i			0.143370i
$255$	−1.55342e8			−0.586674
$256$	5.77748e8			2.15228
$257$		−	3.73033e8i		−	1.37082i	−0.728156	−	0.685411i	$-0.759623\pi$
$257$		−	3.73033e8i		−	1.37082i	0.728156	−	0.685411i	$-0.240377\pi$
$258$	5.07468e8			1.83967
$259$	1.36028e7	+	1.39518e8i	0.0486495	+	0.498976i
$260$	−7.84857e7			−0.276939
$261$			1.26619e7i			0.0440817i
$262$	5.52201e7			0.189689
$263$	1.08376e8			0.367358			0.183679	−	0.982986i	$-0.441199\pi$
$263$	1.08376e8			0.367358			0.183679	+	0.982986i	$0.441199\pi$
$264$		−	1.02771e9i		−	3.43761i
$265$		−	2.11389e8i		−	0.697784i
$266$	−2.43331e8			−0.792704
$267$		−	3.32057e7i		−	0.106764i
$268$	−3.25226e8			−1.03208
$269$	−1.95365e8			−0.611948			−0.305974	−	0.952040i	$-0.598982\pi$
$269$	−1.95365e8			−0.611948			−0.305974	+	0.952040i	$0.598982\pi$
$270$	2.43067e8			0.751540
$271$	−4.54585e8			−1.38747			−0.693733	−	0.720232i	$-0.744035\pi$
$271$	−4.54585e8			−1.38747			−0.693733	+	0.720232i	$0.744035\pi$
$272$		−	1.95144e9i		−	5.87983i
$273$		−	4.38764e7i		−	0.130515i
$274$			1.87594e8i			0.550924i
$275$	−3.28024e8			−0.951133
$276$			9.58091e8i			2.74300i
$277$		−	3.64988e8i		−	1.03181i	−0.856646	−	0.515905i	$-0.827456\pi$
$277$		−	3.64988e8i		−	1.03181i	0.856646	−	0.515905i	$-0.172544\pi$
$278$		−	1.44314e8i		−	0.402857i
$279$			7.19603e6i			0.0198371i
$280$	−2.25535e8			−0.613989
$281$		−	5.72410e8i		−	1.53899i	−0.638655	−	0.769493i	$-0.720509\pi$
$281$		−	5.72410e8i		−	1.53899i	0.638655	−	0.769493i	$-0.279491\pi$
$282$			1.00407e9i			2.66618i
$283$		−	1.27520e7i		−	0.0334446i	−0.999860	−	0.0167223i	$-0.994677\pi$
$283$		−	1.27520e7i		−	0.0334446i	0.999860	−	0.0167223i	$-0.00532312\pi$
$284$	9.23040e8			2.39115
$285$	1.14735e8			0.293589
$286$	−2.30415e8			−0.582412
$287$	2.93729e8			0.733432
$288$			1.38869e8i			0.342555i
$289$	−6.97616e8			−1.70010
$290$			1.39476e8i			0.335821i
$291$		−	1.38546e8i		−	0.329585i
$292$	−1.54322e9			−3.62734
$293$	−3.96104e8			−0.919968			−0.459984	−	0.887927i	$-0.652145\pi$
$293$	−3.96104e8			−0.919968			−0.459984	+	0.887927i	$0.652145\pi$
$294$			5.96809e8i			1.36968i
$295$	−2.63998e8			−0.598720
$296$	1.44937e9	−	1.41311e8i	3.24831	−	0.316706i
$297$	5.20585e8			1.15304
$298$			1.36738e9i			2.99317i
$299$	1.35170e8			0.292436
$300$	−1.03120e9			−2.20505
$301$		−	2.38517e8i		−	0.504123i
$302$		−	6.03666e7i		−	0.126117i
$303$	3.00971e8			0.621549
$304$			1.44133e9i			2.94244i
$305$	1.02913e8			0.207692
$306$	1.49992e8			0.299257
$307$	−2.49480e8			−0.492097			−0.246049	−	0.969257i	$-0.579132\pi$
$307$	−2.49480e8			−0.492097			−0.246049	+	0.969257i	$0.579132\pi$
$308$	−7.67623e8			−1.49699
$309$		−	3.94416e8i		−	0.760500i
$310$			7.92672e7i			0.151122i
$311$			4.75503e8i			0.896380i	0.893938	+	0.448190i	$0.147931\pi$
$311$			4.75503e8i			0.896380i	−0.893938	+	0.448190i	$0.852069\pi$
$312$	−4.55807e8			−0.849650
$313$		−	4.15781e8i		−	0.766406i	−0.923664	−	0.383203i	$-0.874821\pi$
$313$		−	4.15781e8i		−	0.766406i	0.923664	−	0.383203i	$-0.125179\pi$
$314$			6.85442e7i			0.124944i
$315$		−	9.88427e6i		−	0.0178179i
$316$			1.54860e9i			2.76080i
$317$	8.03197e8			1.41617			0.708084	−	0.706128i	$-0.249560\pi$
$317$	8.03197e8			1.41617			0.708084	+	0.706128i	$0.249560\pi$
$318$		−	1.95092e9i		−	3.40208i
$319$			2.98722e8i			0.515229i
$320$			7.42624e8i			1.26690i
$321$	1.59930e8			0.269874
$322$	6.17264e8			1.03033
$323$	8.18335e8			1.35121
$324$	1.48014e9			2.41767
$325$			1.45484e8i			0.235085i
$326$	1.65607e9			2.64738
$327$			7.69905e8i			1.21764i
$328$		−	3.05138e9i		−	4.77461i
$329$	4.71924e8			0.730612
$330$	4.96137e8			0.759981
$331$			2.23396e8i			0.338593i	0.985565	+	0.169296i	$0.0541495\pi$
$331$			2.23396e8i			0.338593i	−0.985565	+	0.169296i	$0.945850\pi$
$332$	−2.34879e9			−3.52258
$333$	6.19310e6	+	6.35199e7i	0.00919080	+	0.0942660i
$334$	1.10787e9			1.62696
$335$		−	9.87984e7i		−	0.143580i
$336$	−1.18683e9			−1.70688
$337$	−4.04797e8			−0.576146			−0.288073	−	0.957608i	$-0.593015\pi$
$337$	−4.04797e8			−0.576146			−0.288073	+	0.957608i	$0.593015\pi$
$338$		−	1.26287e9i		−	1.77890i
$339$			1.33608e8i			0.186265i
$340$	1.20536e9			1.66318
$341$			1.69770e8i			0.231857i
$342$	−1.10784e8			−0.149757
$343$	6.55191e8			0.876675
$344$	−2.47781e9			−3.28181
$345$	−2.91052e8			−0.381596
$346$		−	2.11473e9i		−	2.74466i
$347$		−	8.74771e7i		−	0.112394i	−0.998420	−	0.0561968i	$-0.982103\pi$
$347$		−	8.74771e7i		−	0.112394i	0.998420	−	0.0561968i	$-0.0178974\pi$
$348$			9.39084e8i			1.19448i
$349$	1.35954e8			0.171200			0.0856000	−	0.996330i	$-0.472719\pi$
$349$	1.35954e8			0.171200			0.0856000	+	0.996330i	$0.472719\pi$
$350$			6.64365e8i			0.828264i
$351$		−	2.30889e8i		−	0.284989i
$352$			3.27621e9i			4.00380i
$353$			1.11183e9i			1.34533i	0.739948	+	0.672664i	$0.234850\pi$
$353$			1.11183e9i			1.34533i	−0.739948	+	0.672664i	$0.765150\pi$
$354$	−2.43646e9			−2.91909
$355$			2.80404e8i			0.332648i
$356$			2.57656e8i			0.302667i
$357$			6.73839e8i			0.783822i
$358$	−2.60012e9			−2.99505
$359$	7.81492e8			0.891444			0.445722	−	0.895172i	$-0.352947\pi$
$359$	7.81492e8			0.891444			0.445722	+	0.895172i	$0.352947\pi$
$360$	−1.02682e8			−0.115994
$361$	2.89450e8			0.323816
$362$		−	6.54614e8i		−	0.725279i
$363$	1.95502e8			0.214526
$364$			3.40454e8i			0.370002i
$365$		−	4.68806e8i		−	0.504624i
$366$	9.49787e8			1.01261
$367$	8.22805e8			0.868892			0.434446	−	0.900698i	$-0.356944\pi$
$367$	8.22805e8			0.868892			0.434446	+	0.900698i	$0.356944\pi$
$368$		−	3.65628e9i		−	3.82447i
$369$	1.33729e8			0.138559
$370$	6.82195e7	+	6.99697e8i	0.0700169	+	0.718132i
$371$	−9.16959e8			−0.932269
$372$			5.33700e8i			0.537523i
$373$	−4.90669e8			−0.489562			−0.244781	−	0.969578i	$-0.578716\pi$
$373$	−4.90669e8			−0.489562			−0.244781	+	0.969578i	$0.578716\pi$
$374$	3.53864e9			3.49772
$375$		−	6.77861e8i		−	0.663791i
$376$		−	4.90255e9i		−	4.75625i
$377$	1.32489e8			0.127346
$378$		−	1.05437e9i		−	1.00409i
$379$	−1.17562e9			−1.10926			−0.554628	−	0.832099i	$-0.687139\pi$
$379$	−1.17562e9			−1.10926			−0.554628	+	0.832099i	$0.687139\pi$
$380$	−8.90275e8			−0.832303
$381$	−7.65849e7			−0.0709424
$382$	2.70929e9			2.48676
$383$			1.66864e9i			1.51764i	0.651303	+	0.758818i	$0.274223\pi$
$383$			1.66864e9i			1.51764i	−0.651303	+	0.758818i	$0.725777\pi$
$384$			3.03539e9i			2.73562i
$385$		−	2.33191e8i		−	0.208257i
$386$	−3.86865e9			−3.42377
$387$		−	1.08592e8i		−	0.0952382i
$388$			1.07503e9i			0.934351i
$389$			2.49220e8i			0.214664i	0.994223	+	0.107332i	$0.0342308\pi$
$389$			2.49220e8i			0.214664i	−0.994223	+	0.107332i	$0.965769\pi$
$390$		−	2.20045e8i		−	0.187839i
$391$	−2.07590e9			−1.75625
$392$		−	2.91404e9i		−	2.44340i
$393$			1.12945e8i			0.0938624i
$394$		−	2.81740e9i		−	2.32066i
$395$	−4.70440e8			−0.384074
$396$	−3.49485e8			−0.282810
$397$	−5.57512e8			−0.447185			−0.223592	−	0.974683i	$-0.571778\pi$
$397$	−5.57512e8			−0.447185			−0.223592	+	0.974683i	$0.571778\pi$
$398$	1.45362e9			1.15574
$399$		−	4.97697e8i		−	0.392247i
$400$	3.93527e9			3.07443
$401$			3.66785e8i			0.284057i	0.989863	+	0.142029i	$0.0453625\pi$
$401$			3.66785e8i			0.284057i	−0.989863	+	0.142029i	$0.954637\pi$
$402$		−	9.11817e8i		−	0.700029i
$403$	7.52959e7			0.0573065
$404$	−2.33535e9			−1.76205
$405$			4.49643e8i			0.336338i
$406$	6.05019e8			0.448671
$407$	1.46109e8	+	1.49857e9i	0.107422	+	1.10178i
$408$	7.00012e9			5.10264
$409$		−	6.12382e7i		−	0.0442579i	−0.999755	−	0.0221289i	$-0.992956\pi$
$409$		−	6.12382e7i		−	0.0442579i	0.999755	−	0.0221289i	$-0.00704444\pi$
$410$	1.47308e9			1.05556
$411$	−3.83695e8			−0.272609
$412$			3.06042e9i			2.15596i
$413$			1.14517e9i			0.799915i
$414$	2.81029e8			0.194648
$415$		−	7.13524e8i		−	0.490050i
$416$	1.45306e9			0.989591
$417$	2.95173e8			0.199342
$418$	−2.61364e9			−1.75036
$419$	1.50934e9			1.00239			0.501196	−	0.865334i	$-0.332894\pi$
$419$	1.50934e9			1.00239			0.501196	+	0.865334i	$0.332894\pi$
$420$		−	7.33076e8i		−	0.482810i
$421$			5.95797e8i			0.389144i	0.980888	+	0.194572i	$0.0623319\pi$
$421$			5.95797e8i			0.389144i	−0.980888	+	0.194572i	$0.937668\pi$
$422$			5.05820e9i			3.27644i
$423$	2.14859e8			0.138026
$424$			9.52576e9i			6.06903i
$425$		−	2.23430e9i		−	1.41182i
$426$			2.58787e9i			1.62184i
$427$		−	4.46413e8i		−	0.277485i
$428$	−1.24096e9			−0.765074
$429$		−	4.71280e8i		−	0.288190i
$430$		−	1.19619e9i		−	0.725539i
$431$		−	2.58548e9i		−	1.55550i	−0.628574	−	0.777750i	$-0.716361\pi$
$431$		−	2.58548e9i		−	1.55550i	0.628574	−	0.777750i	$-0.283639\pi$
$432$	−6.24541e9			−3.72707
$433$	−2.04513e9			−1.21063			−0.605316	−	0.795985i	$-0.706953\pi$
$433$	−2.04513e9			−1.21063			−0.605316	+	0.795985i	$0.706953\pi$
$434$	3.43844e8			0.201905
$435$	−2.85278e8			−0.166171
$436$		−	5.97399e9i		−	3.45193i
$437$	1.53325e9			0.878880
$438$		−	4.32664e9i		−	2.46032i
$439$			1.41260e9i			0.796879i	0.917195	+	0.398440i	$0.130448\pi$
$439$			1.41260e9i			0.796879i	−0.917195	+	0.398440i	$0.869552\pi$
$440$	−2.42249e9			−1.35574
$441$	1.27711e8			0.0709074
$442$		−	1.56945e9i		−	0.864508i
$443$	−2.39107e9			−1.30671			−0.653356	−	0.757050i	$-0.726640\pi$
$443$	−2.39107e9			−1.30671			−0.653356	+	0.757050i	$0.726640\pi$
$444$	4.59317e8	+	4.71101e9i	0.249042	+	2.55431i
$445$	−7.82717e7			−0.0421061
$446$			1.76884e9i			0.944094i
$447$	−2.79676e9			−1.48108
$448$	3.22134e9			1.69264
$449$		−	3.25453e9i		−	1.69678i	−0.529371	−	0.848390i	$-0.677572\pi$
$449$		−	3.25453e9i		−	1.69678i	0.529371	−	0.848390i	$-0.322428\pi$
$450$			3.02474e8i			0.156475i
$451$	3.15496e9			1.61948
$452$		−	1.03671e9i		−	0.528049i
$453$	1.23471e8			0.0624053
$454$	−1.94804e9			−0.977019
$455$	−1.03424e8			−0.0514734
$456$	−5.17029e9			−2.55351
$457$			1.33143e9i			0.652546i	0.945276	+	0.326273i	$0.105793\pi$
$457$			1.33143e9i			0.652546i	−0.945276	+	0.326273i	$0.894207\pi$
$458$			3.18597e9i			1.54958i
$459$			3.54591e9i			1.71153i
$460$	2.25839e9			1.08180
$461$		−	2.82943e9i		−	1.34507i	−0.740063	−	0.672537i	$-0.765205\pi$
$461$		−	2.82943e9i		−	1.34507i	0.740063	−	0.672537i	$-0.234795\pi$
$462$		−	2.15214e9i		−	1.01537i
$463$			1.56953e9i			0.734914i	0.930040	+	0.367457i	$0.119772\pi$
$463$			1.56953e9i			0.734914i	−0.930040	+	0.367457i	$0.880228\pi$
$464$		−	3.58374e9i		−	1.66542i
$465$	−1.62129e8			−0.0747784
$466$			3.64449e9i			1.66835i
$467$		−	1.74773e9i		−	0.794084i	−0.917800	−	0.397042i	$-0.870037\pi$
$467$		−	1.74773e9i		−	0.794084i	0.917800	−	0.397042i	$-0.129963\pi$
$468$			1.55003e8i			0.0699003i
$469$	−4.28566e8			−0.191828
$470$	2.36676e9			1.05150
$471$	−1.40197e8			−0.0618252
$472$	1.18965e10			5.20741
$473$		−	2.56193e9i		−	1.11315i
$474$	−4.34172e9			−1.87257
$475$			1.65025e9i			0.706517i
$476$		−	5.22858e9i		−	2.22208i
$477$	−4.17475e8			−0.176123
$478$	−1.03688e9			−0.434240
$479$			3.15497e9i			1.31166i	0.754910	+	0.655829i	$0.227681\pi$
$479$			3.15497e9i			1.31166i	−0.754910	+	0.655829i	$0.772319\pi$
$480$	−3.12876e9			−1.29130
$481$	6.64642e8	−	6.48017e7i	0.272320	−	0.0265509i
$482$	−5.37621e9			−2.18681
$483$			1.26252e9i			0.509829i
$484$	−1.51698e9			−0.608164
$485$	−3.26577e8			−0.129984
$486$		−	9.18541e8i		−	0.362971i
$487$		−	1.60592e9i		−	0.630045i	−0.949084	−	0.315022i	$-0.897988\pi$
$487$		−	1.60592e9i		−	0.630045i	0.949084	−	0.315022i	$-0.102012\pi$
$488$	−4.63752e9			−1.80641
$489$			3.38724e9i			1.30998i
$490$	1.40678e9			0.540183
$491$	1.43396e9			0.546703			0.273352	−	0.961914i	$-0.411868\pi$
$491$	1.43396e9			0.546703			0.273352	+	0.961914i	$0.411868\pi$
$492$	9.91817e9			3.75451
$493$	−2.03471e9			−0.764784
$494$			1.15919e9i			0.432625i
$495$		−	1.06168e8i		−	0.0393436i
$496$		−	2.03671e9i		−	0.749452i
$497$	1.21633e9			0.444432
$498$		−	6.58516e9i		−	2.38926i
$499$		−	3.31967e9i		−	1.19603i	−0.801484	−	0.598016i	$-0.795956\pi$
$499$		−	3.31967e9i		−	1.19603i	0.801484	−	0.598016i	$-0.204044\pi$
$500$			5.25979e9i			1.88180i
$501$			2.26598e9i			0.805053i
$502$	7.43423e9			2.62285
$503$		−	3.12094e8i		−	0.109345i	−0.998504	−	0.0546723i	$-0.982589\pi$
$503$		−	3.12094e8i		−	0.109345i	0.998504	−	0.0546723i	$-0.0174114\pi$
$504$			4.45413e8i			0.154973i
$505$		−	7.09441e8i		−	0.245130i
$506$	6.63009e9			2.27506
$507$	2.58301e9			0.880236
$508$	5.94252e8			0.201116
$509$	−9.19098e8			−0.308922			−0.154461	−	0.987999i	$-0.549364\pi$
$509$	−9.19098e8			−0.308922			−0.154461	+	0.987999i	$0.549364\pi$
$510$			3.37938e9i			1.12808i
$511$	−2.03358e9			−0.674199
$512$		−	3.83677e9i		−	1.26334i
$513$		−	2.61901e9i		−	0.856497i
$514$	−8.11515e9			−2.63588
$515$	−9.29706e8			−0.299930
$516$		−	8.05386e9i		−	2.58066i
$517$	5.06898e9			1.61326
$518$	3.03514e9	−	2.95922e8i	0.959454	−	0.0935455i
$519$	4.32537e9			1.35812
$520$			1.07442e9i			0.335090i
$521$	3.12185e9			0.967118			0.483559	−	0.875312i	$-0.339344\pi$
$521$	3.12185e9			0.967118			0.483559	+	0.875312i	$0.339344\pi$
$522$	2.75454e8			0.0847623
$523$		−	2.00814e9i		−	0.613816i	−0.951739	−	0.306908i	$-0.900706\pi$
$523$		−	2.00814e9i		−	0.613816i	0.951739	−	0.306908i	$-0.0992944\pi$
$524$		−	8.76381e8i		−	0.266093i
$525$	−1.35886e9			−0.409843
$526$		−	2.35768e9i		−	0.706374i
$527$	−1.15637e9			−0.344159
$528$	−1.27479e10			−3.76894
$529$	−4.84629e8			−0.142336
$530$	−4.59866e9			−1.34173
$531$			5.21375e8i			0.151119i
$532$			3.86182e9i			1.11199i
$533$		−	1.39928e9i		−	0.400276i
$534$	−7.22375e8			−0.205290
$535$		−	3.76982e8i		−	0.106435i
$536$			4.45213e9i			1.24879i
$537$		−	5.31817e9i		−	1.48201i
$538$			4.25008e9i			1.17668i
$539$	3.01296e9			0.828769
$540$		−	3.85763e9i		−	1.05425i
$541$			2.86739e9i			0.778569i	0.921118	+	0.389284i	$0.127278\pi$
$541$			2.86739e9i			0.778569i	−0.921118	+	0.389284i	$0.872722\pi$
$542$			9.88928e9i			2.66788i
$543$	1.33892e9			0.358884
$544$	−2.23155e10			−5.94307
$545$	1.81480e9			0.480221
$546$	−9.54510e8			−0.250961
$547$			5.00927e8i			0.130864i	0.997857	+	0.0654318i	$0.0208425\pi$
$547$			5.00927e8i			0.130864i	−0.997857	+	0.0654318i	$0.979158\pi$
$548$	2.97724e9			0.772827
$549$		−	2.03244e8i		−	0.0524220i
$550$			7.13600e9i			1.82888i
$551$	1.50284e9			0.382720
$552$	1.31156e10			3.31896
$553$			2.04067e9i			0.513139i
$554$	−7.94015e9			−1.98401
$555$	−1.43113e9	+	1.39533e8i	−0.355347	+	0.0346459i
$556$	−2.29036e9			−0.565121
$557$			2.30833e9i			0.565985i	0.959122	+	0.282993i	$0.0913272\pi$
$557$			2.30833e9i			0.565985i	−0.959122	+	0.282993i	$0.908673\pi$
$558$	1.56546e8			0.0381437
$559$	−1.13626e9			−0.275129
$560$			2.79757e9i			0.673167i
$561$			7.23776e9i			1.73075i
$562$	−1.24525e10			−2.95923
$563$		−	4.87705e9i		−	1.15180i	−0.817520	−	0.575900i	$-0.804652\pi$
$563$		−	4.87705e9i		−	1.15180i	0.817520	−	0.575900i	$-0.195348\pi$
$564$	1.59352e10			3.74008
$565$	3.14936e8			0.0734604
$566$	−2.77414e8			−0.0643088
$567$	1.95046e9			0.449361
$568$		−	1.26358e10i		−	2.89323i
$569$		−	6.83355e9i		−	1.55508i	−0.628831	−	0.777542i	$-0.716466\pi$
$569$		−	6.83355e9i		−	1.55508i	0.628831	−	0.777542i	$-0.283534\pi$
$570$		−	2.49601e9i		−	0.564526i
$571$	1.74276e9			0.391753			0.195876	−	0.980629i	$-0.437245\pi$
$571$	1.74276e9			0.391753			0.195876	+	0.980629i	$0.437245\pi$
$572$			3.65685e9i			0.816997i
$573$			5.54145e9i			1.23050i
$574$		−	6.38992e9i		−	1.41028i
$575$		−	4.18624e9i		−	0.918305i
$576$	1.46662e9			0.319771
$577$		−	7.52277e9i		−	1.63028i	−0.579263	−	0.815141i	$-0.696660\pi$
$577$		−	7.52277e9i		−	1.63028i	0.579263	−	0.815141i	$-0.303340\pi$
$578$			1.51763e10i			3.26903i
$579$		−	7.91276e9i		−	1.69415i
$580$	2.21359e9			0.471084
$581$	−3.09511e9			−0.654727
$582$	−3.01400e9			−0.633742
$583$	−9.84913e9			−2.05853
$584$			2.11257e10i			4.38900i
$585$	−4.70873e7			−0.00972429
$586$			8.61706e9i			1.76896i
$587$			2.16464e9i			0.441726i	0.975305	+	0.220863i	$0.0708873\pi$
$587$			2.16464e9i			0.441726i	−0.975305	+	0.220863i	$0.929113\pi$
$588$	9.47177e9			1.92137
$589$	8.54093e8			0.172227
$590$			5.74315e9i			1.15125i
$591$	5.76258e9			1.14831
$592$	−1.75285e9	−	1.79782e10i	−0.347231	−	3.56140i
$593$	−4.03004e9			−0.793630			−0.396815	−	0.917899i	$-0.629885\pi$
$593$	−4.03004e9			−0.793630			−0.396815	+	0.917899i	$0.629885\pi$
$594$		−	1.13251e10i		−	2.21712i
$595$	1.58836e9			0.309128
$596$	2.17012e10			4.19876
$597$			2.97316e9i			0.571885i
$598$		−	2.94056e9i		−	0.562310i
$599$	−3.02810e9			−0.575674			−0.287837	−	0.957679i	$-0.592936\pi$
$599$	−3.02810e9			−0.575674			−0.287837	+	0.957679i	$0.592936\pi$
$600$			1.41164e10i			2.66806i
$601$	2.35007e9			0.441590			0.220795	−	0.975320i	$-0.429135\pi$
$601$	2.35007e9			0.441590			0.220795	+	0.975320i	$0.429135\pi$
$602$	−5.18882e9			−0.969350
$603$	−1.95119e8			−0.0362400
$604$	−9.58059e8			−0.176914
$605$		−	4.60833e8i		−	0.0846058i
$606$		−	6.54748e9i		−	1.19514i
$607$		−	3.02142e9i		−	0.548341i	−0.961681	−	0.274171i	$-0.911597\pi$
$607$		−	3.02142e9i		−	0.548341i	0.961681	−	0.274171i	$-0.0884033\pi$
$608$	1.64822e10			2.97409
$609$			1.23748e9i			0.222012i
$610$		−	2.23881e9i		−	0.399359i
$611$		−	2.24818e9i		−	0.398737i
$612$		−	2.38048e9i		−	0.419792i
$613$	3.24180e9			0.568428			0.284214	−	0.958761i	$-0.408267\pi$
$613$	3.24180e9			0.568428			0.284214	+	0.958761i	$0.408267\pi$
$614$			5.42731e9i			0.946228i
$615$			3.01298e9i			0.522315i
$616$			1.05082e10i			1.81133i
$617$	−1.02520e10			−1.75716			−0.878582	−	0.477592i	$-0.841510\pi$
$617$	−1.02520e10			−1.75716			−0.878582	+	0.477592i	$0.841510\pi$
$618$	−8.58032e9			−1.46232
$619$	−5.38103e9			−0.911901			−0.455951	−	0.890005i	$-0.650701\pi$
$619$	−5.38103e9			−0.911901			−0.455951	+	0.890005i	$0.650701\pi$
$620$	1.25802e9			0.211991
$621$			6.64371e9i			1.11324i
$622$	1.03443e10			1.72360
$623$			3.39526e8i			0.0562555i
$624$			5.65391e9i			0.931542i
$625$	3.64625e9			0.597402
$626$	−9.04511e9			−1.47368
$627$		−	5.34580e9i		−	0.866117i
$628$	1.08784e9			0.175270
$629$	−1.02073e10	+	9.95202e8i	−1.63544	+	0.159453i
$630$	−2.15028e8			−0.0342612
$631$			8.47871e9i			1.34347i	0.740793	+	0.671733i	$0.234450\pi$
$631$			8.47871e9i			1.34347i	−0.740793	+	0.671733i	$0.765550\pi$
$632$	2.11993e10			3.34051
$633$	−1.03458e10			−1.62126
$634$		−	1.74732e10i		−	2.72307i
$635$			1.80524e8i			0.0279787i
$636$	−3.09624e10			−4.77238
$637$		−	1.33630e9i		−	0.204841i
$638$	6.49856e9			0.990706
$639$	5.53775e8			0.0839616
$640$	7.15495e9			1.07889
$641$	4.21164e9			0.631609			0.315804	−	0.948824i	$-0.397726\pi$
$641$	4.21164e9			0.631609			0.315804	+	0.948824i	$0.397726\pi$
$642$		−	3.47919e9i		−	0.518927i
$643$			9.74205e8i			0.144515i	0.997386	+	0.0722573i	$0.0230203\pi$
$643$			9.74205e8i			0.144515i	−0.997386	+	0.0722573i	$0.976980\pi$
$644$		−	9.79640e9i		−	1.44533i
$645$	2.44663e9			0.359012
$646$		−	1.78025e10i		−	2.59816i
$647$		−	2.45205e9i		−	0.355930i	−0.984037	−	0.177965i	$-0.943049\pi$
$647$		−	2.45205e9i		−	0.355930i	0.984037	−	0.177965i	$-0.0569514\pi$
$648$		−	2.02622e10i		−	2.92532i
$649$			1.23003e10i			1.76628i
$650$	3.16494e9			0.452032
$651$			7.03283e8i			0.0999071i
$652$		−	2.62829e10i		−	3.71370i
$653$			1.17683e10i			1.65393i	0.562256	+	0.826963i	$0.309934\pi$
$653$			1.17683e10i			1.65393i	−0.562256	+	0.826963i	$0.690066\pi$
$654$	1.67489e10			2.34134
$655$	2.66230e8			0.0370180
$656$	−3.78498e10			−5.23480
$657$	−9.25853e8			−0.127369
$658$		−	1.02665e10i		−	1.40485i
$659$	−3.61797e8			−0.0492454			−0.0246227	−	0.999697i	$-0.507838\pi$
$659$	−3.61797e8			−0.0492454			−0.0246227	+	0.999697i	$0.507838\pi$
$660$		−	7.87403e9i		−	1.06609i
$661$			3.26094e9i			0.439175i	0.975593	+	0.219588i	$0.0704712\pi$
$661$			3.26094e9i			0.439175i	−0.975593	+	0.219588i	$0.929529\pi$
$662$	4.85988e9			0.651062
$663$	3.21007e9			0.427777
$664$			3.21534e10i			4.26224i
$665$	−1.17316e9			−0.154697
$666$	1.38184e9	−	1.34728e8i	0.181259	−	0.0176725i
$667$	−3.81230e9			−0.497446
$668$		−	1.75826e10i		−	2.28227i
$669$	−3.61789e9			−0.467158
$670$	−2.14931e9			−0.276082
$671$		−	4.79496e9i		−	0.612711i
$672$			1.35719e10i			1.72524i
$673$	1.13675e10			1.43751			0.718756	−	0.695263i	$-0.244712\pi$
$673$	1.13675e10			1.43751			0.718756	+	0.695263i	$0.244712\pi$
$674$			8.80617e9i			1.10784i
$675$	−7.15067e9			−0.894918
$676$	−2.00426e10			−2.49540
$677$	1.33905e10			1.65858			0.829291	−	0.558817i	$-0.188745\pi$
$677$	1.33905e10			1.65858			0.829291	+	0.558817i	$0.188745\pi$
$678$	2.90657e9			0.358160
$679$			1.41662e9i			0.173664i
$680$		−	1.65005e10i		−	2.01241i
$681$		−	3.98443e9i		−	0.483450i
$682$	3.69326e9			0.445825
$683$		−	6.45078e9i		−	0.774712i	−0.921930	−	0.387356i	$-0.873389\pi$
$683$		−	6.45078e9i		−	0.774712i	0.921930	−	0.387356i	$-0.126611\pi$
$684$			1.75822e9i			0.210076i
$685$			9.04437e8i			0.107513i
$686$		−	1.42534e10i		−	1.68571i
$687$	−6.51644e9			−0.766764
$688$			3.07352e10i			3.59813i
$689$			4.36826e9i			0.508793i
$690$			6.33170e9i			0.733751i
$691$	7.44952e9			0.858924			0.429462	−	0.903085i	$-0.358703\pi$
$691$	7.44952e9			0.858924			0.429462	+	0.903085i	$0.358703\pi$
$692$	−3.35622e10			−3.85016
$693$	−4.60533e8			−0.0525648
$694$	−1.90302e9			−0.216115
$695$		−	6.95773e8i		−	0.0786178i
$696$	1.28554e10			1.44529
$697$			2.14897e10i			2.40389i
$698$		−	2.95762e9i		−	0.329191i
$699$	−7.45427e9			−0.825534
$700$	1.05439e10			1.16187
$701$		−	5.42774e9i		−	0.595121i	−0.954703	−	0.297561i	$-0.903827\pi$
$701$		−	5.42774e9i		−	0.595121i	0.954703	−	0.297561i	$-0.0961731\pi$
$702$	−5.02287e9			−0.547990
$703$	7.53914e9	−	7.35056e8i	0.818423	−	0.0797952i
$704$	3.46007e10			3.73750
$705$			4.84085e9i			0.520307i
$706$	2.41874e10			2.58686
$707$	−3.07740e9			−0.327504
$708$			3.86682e10i			4.09485i
$709$			1.51095e9i			0.159216i	0.996826	+	0.0796082i	$0.0253669\pi$
$709$			1.51095e9i			0.159216i	−0.996826	+	0.0796082i	$0.974633\pi$
$710$	6.10006e9			0.639632
$711$			9.29081e8i			0.0969415i
$712$	3.52714e9			0.366220
$713$	−2.16660e9			−0.223855
$714$	1.46590e10			1.50717
$715$	−1.11089e9			−0.113658
$716$			4.12657e10i			4.20140i
$717$		−	2.12078e9i		−	0.214871i
$718$		−	1.70010e10i		−	1.71411i
$719$	1.00886e10			1.01223			0.506114	−	0.862467i	$-0.331082\pi$
$719$	1.00886e10			1.01223			0.506114	+	0.862467i	$0.331082\pi$
$720$			1.27368e9i			0.127174i
$721$			4.03287e9i			0.400719i
$722$		−	6.29684e9i		−	0.622648i
$723$		−	1.09962e10i		−	1.08208i
$724$	−1.03892e10			−1.01741
$725$		−	4.10320e9i		−	0.399889i
$726$		−	4.25306e9i		−	0.412500i
$727$		−	1.71908e10i		−	1.65930i	−0.558285	−	0.829649i	$-0.688540\pi$
$727$		−	1.71908e10i		−	1.65930i	0.558285	−	0.829649i	$-0.311460\pi$
$728$	4.66059e9			0.447694
$729$	1.12545e10			1.07592
$730$	−1.01986e10			−0.970314
$731$	1.74503e10			1.65231
$732$		−	1.50738e10i		−	1.42047i
$733$	−8.12481e9			−0.761990			−0.380995	−	0.924577i	$-0.624419\pi$
$733$	−8.12481e9			−0.761990			−0.380995	+	0.924577i	$0.624419\pi$
$734$		−	1.78997e10i		−	1.67074i
$735$			2.87737e9i			0.267294i
$736$	−4.18110e10			−3.86561
$737$	−4.60327e9			−0.423574
$738$		−	2.90922e9i		−	0.266428i
$739$	5.94655e9			0.542013			0.271006	−	0.962578i	$-0.412644\pi$
$739$	5.94655e9			0.542013			0.271006	+	0.962578i	$0.412644\pi$
$740$	1.11047e10	−	1.08269e9i	1.00738	−	0.0982185i
$741$	−2.37096e9			−0.214072
$742$			1.99480e10i			1.79261i
$743$	−1.45771e10			−1.30380			−0.651899	−	0.758306i	$-0.726027\pi$
$743$	−1.45771e10			−1.30380			−0.651899	+	0.758306i	$0.726027\pi$
$744$	7.30600e9			0.650391
$745$			6.59246e9i			0.584118i
$746$			1.06743e10i			0.941352i
$747$	−1.40915e9			−0.123690
$748$		−	5.61606e10i		−	4.90655i
$749$	−1.63527e9			−0.142201
$750$	−1.47465e10			−1.27637
$751$	4.58555e9			0.395050			0.197525	−	0.980298i	$-0.436710\pi$
$751$	4.58555e9			0.395050			0.197525	+	0.980298i	$0.436710\pi$
$752$	−6.08120e10			−5.21467
$753$			1.52056e10i			1.29784i
$754$		−	2.88223e9i		−	0.244866i
$755$		−	2.91043e8i		−	0.0246117i
$756$	−1.67336e10			−1.40852
$757$		−	2.34942e9i		−	0.196845i	−0.995145	−	0.0984227i	$-0.968620\pi$
$757$		−	2.34942e9i		−	0.196845i	0.995145	−	0.0984227i	$-0.0313797\pi$
$758$			2.55751e10i			2.13293i
$759$			1.35609e10i			1.12575i
$760$			1.21873e10i			1.00707i
$761$	7.10749e9			0.584615			0.292307	−	0.956324i	$-0.405577\pi$
$761$	7.10749e9			0.584615			0.292307	+	0.956324i	$0.405577\pi$
$762$			1.66607e9i			0.136411i
$763$		−	7.87221e9i		−	0.641595i
$764$		−	4.29982e10i		−	3.48838i
$765$	7.23150e8			0.0584001
$766$	3.63005e10			2.91818
$767$	5.45542e9			0.436560
$768$	2.57073e10			2.04782
$769$		−	3.75931e9i		−	0.298103i	−0.988829	−	0.149051i	$-0.952378\pi$
$769$		−	3.75931e9i		−	0.298103i	0.988829	−	0.149051i	$-0.0476220\pi$
$770$	−5.07296e9			−0.400446
$771$		−	1.65983e10i		−	1.30429i
$772$			6.13981e10i			4.80280i
$773$	−4.84455e9			−0.377247			−0.188623	−	0.982049i	$-0.560403\pi$
$773$	−4.84455e9			−0.377247			−0.188623	+	0.982049i	$0.560403\pi$
$774$	−2.36238e9			−0.183128
$775$		−	2.33193e9i		−	0.179953i
$776$	1.47164e10			1.13054
$777$	6.05264e8	+	6.20793e9i	0.0462883	+	0.474759i
$778$	5.42166e9			0.412766
$779$		−	1.58723e10i		−	1.20298i
$780$	−3.49227e9			−0.263498
$781$	1.30647e10			0.981347
$782$			4.51601e10i			3.37700i
$783$			6.51192e9i			0.484778i
$784$	−3.61463e10			−2.67890
$785$			3.30469e8i			0.0243830i
$786$	2.45706e9			0.180483
$787$	2.27702e10			1.66516			0.832579	−	0.553906i	$-0.186863\pi$
$787$	2.27702e10			1.66516			0.832579	+	0.553906i	$0.186863\pi$
$788$	−4.47141e10			−3.25539
$789$	4.82228e9			0.349529
$790$			1.02342e10i			0.738515i
$791$		−	1.36613e9i		−	0.0981462i
$792$			4.78421e9i			0.342194i
$793$	−2.12665e9			−0.151440
$794$			1.21284e10i			0.859867i
$795$		−	9.40588e9i		−	0.663918i
$796$		−	2.30699e10i		−	1.62125i
$797$		−	2.27331e10i		−	1.59058i	−0.606230	−	0.795290i	$-0.707319\pi$
$797$		−	2.27331e10i		−	1.59058i	0.606230	−	0.795290i	$-0.292681\pi$
$798$	−1.08272e10			−0.754231
$799$			3.45268e10i			2.39465i
$800$		−	4.50014e10i		−	3.10750i
$801$			1.54580e8i			0.0106277i
$802$	7.97922e9			0.546198
$803$	−2.18428e10			−1.48869
$804$	−1.44711e10			−0.981989
$805$	2.97599e9			0.201069
$806$		−	1.63803e9i		−	0.110191i
$807$	−8.69291e9			−0.582248
$808$			3.19694e10i			2.13204i
$809$		−	1.13452e10i		−	0.753345i	−0.926347	−	0.376672i	$-0.877068\pi$
$809$		−	1.13452e10i		−	0.753345i	0.926347	−	0.376672i	$-0.122932\pi$
$810$	9.78177e9			0.646726
$811$	1.40064e10			0.922045			0.461023	−	0.887388i	$-0.347483\pi$
$811$	1.40064e10			0.922045			0.461023	+	0.887388i	$0.347483\pi$
$812$		−	9.60206e9i		−	0.629388i
$813$	−2.02271e10			−1.32013
$814$	3.26007e10	−	3.17852e9i	2.11856	−	0.206557i
$815$	7.98431e9			0.516637
$816$		−	8.68307e10i		−	5.59445i
$817$	−1.28888e10			−0.826864
$818$	−1.33221e9			−0.0851011
$819$			2.04255e8i			0.0129921i
$820$		−	2.33788e10i		−	1.48073i
$821$	1.18355e10			0.746421			0.373211	−	0.927747i	$-0.378257\pi$
$821$	1.18355e10			0.746421			0.373211	+	0.927747i	$0.378257\pi$
$822$			8.34711e9i			0.524185i
$823$	6.47971e9			0.405188			0.202594	−	0.979263i	$-0.435063\pi$
$823$	6.47971e9			0.405188			0.202594	+	0.979263i	$0.435063\pi$
$824$	4.18951e10			2.60866
$825$	−1.45956e10			−0.904970
$826$	2.49126e10			1.53811
$827$			1.43752e10i			0.883782i	0.897069	+	0.441891i	$0.145692\pi$
$827$			1.43752e10i			0.883782i	−0.897069	+	0.441891i	$0.854308\pi$
$828$		−	4.46013e9i		−	0.273049i
$829$		−	2.63001e10i		−	1.60331i	−0.597789	−	0.801654i	$-0.703954\pi$
$829$		−	2.63001e10i		−	1.60331i	0.597789	−	0.801654i	$-0.296046\pi$
$830$	−1.55224e10			−0.942290
$831$		−	1.62404e10i		−	0.981732i
$832$		−	1.53460e10i		−	0.923771i
$833$			2.05225e10i			1.23019i
$834$		−	6.42134e9i		−	0.383305i
$835$	5.34131e9			0.317501
$836$			4.14802e10i			2.45538i
$837$			3.70085e9i			0.218154i
$838$		−	3.28349e10i		−	1.92744i
$839$	−1.52010e10			−0.888601			−0.444300	−	0.895878i	$-0.646548\pi$
$839$	−1.52010e10			−0.888601			−0.444300	+	0.895878i	$0.646548\pi$
$840$	−1.00353e10			−0.584189
$841$	1.35132e10			0.783380
$842$	1.29613e10			0.748265
$843$		−	2.54697e10i		−	1.46429i
$844$	8.02771e10			4.59614
$845$		−	6.08862e9i		−	0.347152i
$846$		−	4.67415e9i		−	0.265403i
$847$	−1.99900e9			−0.113037
$848$	1.18159e11			6.65398
$849$		−	5.67409e8i		−	0.0318214i
$850$	−4.86061e10			−2.71472
$851$	−1.91248e10	+	1.86464e9i	−1.06376	+	0.103715i
$852$	4.10713e10			2.27509
$853$			8.22207e9i			0.453586i	0.973943	+	0.226793i	$0.0728241\pi$
$853$			8.22207e9i			0.453586i	−0.973943	+	0.226793i	$0.927176\pi$
$854$	−9.71149e9			−0.533560
$855$	−5.34118e8			−0.0292251
$856$			1.69879e10i			0.925722i
$857$		−	1.78926e8i		−	0.00971048i	−0.999988	−	0.00485524i	$-0.998455\pi$
$857$		−	1.78926e8i		−	0.00971048i	0.999988	−	0.00485524i	$-0.00154548\pi$
$858$	−1.02525e10			−0.554145
$859$			2.16162e10i			1.16360i	0.813333	+	0.581799i	$0.197651\pi$
$859$			2.16162e10i			1.16360i	−0.813333	+	0.581799i	$0.802349\pi$
$860$	−1.89843e10			−1.01777
$861$	1.30696e10			0.697835
$862$	−5.62458e10			−2.99099
$863$	−2.23399e10			−1.18316			−0.591580	−	0.806246i	$-0.701496\pi$
$863$	−2.23399e10			−1.18316			−0.591580	+	0.806246i	$0.701496\pi$
$864$			7.14188e10i			3.76716i
$865$		−	1.01956e10i		−	0.535622i
$866$			4.44907e10i			2.32786i
$867$	−3.10409e10			−1.61759
$868$		−	5.45704e9i		−	0.283229i
$869$			2.19190e10i			1.13306i
$870$			6.20609e9i			0.319522i
$871$			2.04163e9i			0.104692i
$872$	−8.17799e10			−4.17675
$873$			6.44962e8i			0.0328083i
$874$		−	3.33552e10i		−	1.68995i
$875$			6.93108e9i			0.349762i
$876$	−6.86667e10			−3.45129
$877$	2.04101e9			0.102175			0.0510876	−	0.998694i	$-0.483731\pi$
$877$	2.04101e9			0.102175			0.0510876	+	0.998694i	$0.483731\pi$
$878$	3.07304e10			1.53228
$879$	−1.76249e10			−0.875318
$880$			3.00490e10i			1.48641i
$881$	−2.83294e10			−1.39580			−0.697898	−	0.716197i	$-0.745881\pi$
$881$	−2.83294e10			−1.39580			−0.697898	+	0.716197i	$0.745881\pi$
$882$		−	2.77828e9i		−	0.136344i
$883$		−	1.99398e9i		−	0.0974672i	−0.998812	−	0.0487336i	$-0.984481\pi$
$883$		−	1.99398e9i		−	0.0974672i	0.998812	−	0.0487336i	$-0.0155185\pi$
$884$	−2.49082e10			−1.21272
$885$	−1.17468e10			−0.569662
$886$			5.20167e10i			2.51261i
$887$	3.53659e10			1.70158			0.850789	−	0.525508i	$-0.176125\pi$
$887$	3.53659e10			1.70158			0.850789	+	0.525508i	$0.176125\pi$
$888$	6.44906e10	−	6.28774e9i	3.09066	−	0.301335i
$889$	7.83074e8			0.0373807
$890$			1.70276e9i			0.0809635i
$891$	2.09500e10			0.992230
$892$	2.80726e10			1.32436
$893$		−	2.55015e10i		−	1.19835i
$894$			6.08423e10i			2.84790i
$895$	−1.25359e10			−0.584485
$896$		−	3.10366e10i		−	1.44144i
$897$	6.01448e9			0.278243
$898$	−7.08007e10			−3.26265
$899$	−2.12362e9			−0.0974806
$900$	4.80046e9			0.219500
$901$		−	6.70863e10i		−	3.05560i
$902$		−	6.86347e10i		−	3.11402i
$903$		−	1.06130e10i		−	0.479655i
$904$	−1.41919e10			−0.638927
$905$		−	3.15606e9i		−	0.141539i
$906$		−	2.68605e9i		−	0.119996i
$907$			2.43851e10i			1.08517i	0.840000	+	0.542586i	$0.182555\pi$
$907$			2.43851e10i			1.08517i	−0.840000	+	0.542586i	$0.817445\pi$
$908$			3.09168e10i			1.37055i
$909$	−1.40109e9			−0.0618716
$910$			2.24995e9i			0.0989754i
$911$			1.61283e10i			0.706766i	0.935479	+	0.353383i	$0.114969\pi$
$911$			1.61283e10i			0.706766i	−0.935479	+	0.353383i	$0.885031\pi$
$912$			6.41331e10i			2.79963i
$913$	−3.32449e10			−1.44570
$914$	2.89646e10			1.25474
$915$	4.57916e9			0.197611
$916$	5.05636e10			2.17372
$917$		−	1.15485e9i		−	0.0494576i
$918$	7.71395e10			3.29100
$919$			5.84343e9i			0.248350i	0.992260	+	0.124175i	$0.0396284\pi$
$919$			5.84343e9i			0.248350i	−0.992260	+	0.124175i	$0.960372\pi$
$920$		−	3.09158e10i		−	1.30895i
$921$	−1.11008e10			−0.468214
$922$	−6.15530e10			−2.58637
$923$		−	5.79445e9i		−	0.242553i
$924$	−3.41559e10			−1.42434
$925$	−2.00692e9	−	2.05841e10i	−0.0833747	−	0.855137i
$926$	3.41444e10			1.41313
$927$			1.83609e9i			0.0757034i
$928$	−4.09815e10			−1.68333
$929$	1.63783e10			0.670213			0.335106	−	0.942180i	$-0.391228\pi$
$929$	1.63783e10			0.670213			0.335106	+	0.942180i	$0.391228\pi$
$930$			3.52705e9i			0.143787i
$931$		−	1.51579e10i		−	0.615623i
$932$	5.78406e10			2.34033
$933$			2.11578e10i			0.852875i
$934$	−3.80211e10			−1.52690
$935$	1.70607e10			0.682582
$936$	2.12188e9			0.0845777
$937$	8.69763e9			0.345392			0.172696	−	0.984975i	$-0.444752\pi$
$937$	8.69763e9			0.345392			0.172696	+	0.984975i	$0.444752\pi$
$938$			9.32325e9i			0.368856i
$939$		−	1.85004e10i		−	0.729209i
$940$		−	3.75620e10i		−	1.47503i
$941$	−3.36939e9			−0.131822			−0.0659110	−	0.997826i	$-0.520995\pi$
$941$	−3.36939e9			−0.131822			−0.0659110	+	0.997826i	$0.520995\pi$
$942$			3.04992e9i			0.118880i
$943$			4.02637e10i			1.56359i
$944$		−	1.47566e11i		−	5.70932i
$945$		−	5.08339e9i		−	0.195948i
$946$	−5.57335e10			−2.14041
$947$		−	2.64380e10i		−	1.01159i	−0.862655	−	0.505794i	$-0.831200\pi$
$947$		−	2.64380e10i		−	1.01159i	0.862655	−	0.505794i	$-0.168800\pi$
$948$			6.89061e10i			2.62681i
$949$			9.68769e9i			0.367950i
$950$	3.59004e10			1.35852
$951$	3.57388e10			1.34744
$952$	−7.15757e10			−2.68866
$953$	2.70801e10			1.01350			0.506752	−	0.862092i	$-0.330846\pi$
$953$	2.70801e10			1.01350			0.506752	+	0.862092i	$0.330846\pi$
$954$			9.08197e9i			0.338657i
$955$	1.30622e10			0.485292
$956$			1.64559e10i			0.609144i
$957$			1.32918e10i			0.490223i
$958$	6.86348e10			2.52211
$959$	3.92325e9			0.143642
$960$			3.30435e10i			1.20542i
$961$	2.63057e10			0.956133
$962$	−1.40973e9	−	1.44590e10i	−0.0510532	−	0.523630i
$963$	−7.44508e8			−0.0268644
$964$			8.53241e10i			3.06762i
$965$	−1.86517e10			−0.668150
$966$	2.74656e10			0.980322
$967$			4.09717e10i			1.45711i	0.684990	+	0.728553i	$0.259807\pi$
$967$			4.09717e10i			1.45711i	−0.684990	+	0.728553i	$0.740193\pi$
$968$			2.07664e10i			0.735865i
$969$	3.64123e10			1.28563
$970$			7.10452e9i			0.249939i
$971$	−2.06611e10			−0.724247			−0.362123	−	0.932130i	$-0.617948\pi$
$971$	−2.06611e10			−0.724247			−0.362123	+	0.932130i	$0.617948\pi$
$972$	−1.45779e10			−0.509169
$973$	−3.01811e9			−0.105037
$974$	−3.49359e10			−1.21148
$975$			6.47342e9i			0.223675i
$976$			5.75246e10i			1.98052i
$977$		−	2.77000e10i		−	0.950272i	−0.879912	−	0.475136i	$-0.842399\pi$
$977$		−	2.77000e10i		−	0.950272i	0.879912	−	0.475136i	$-0.157601\pi$
$978$	7.36878e10			2.51889
$979$			3.64688e9i			0.124217i
$980$		−	2.23266e10i		−	0.757760i
$981$		−	3.58408e9i		−	0.121209i
$982$		−	3.11951e10i		−	1.05123i
$983$	−1.35597e8			−0.00455317			−0.00227658	−	0.999997i	$-0.500725\pi$
$983$	−1.35597e8			−0.00455317			−0.00227658	+	0.999997i	$0.500725\pi$
$984$		−	1.35773e11i		−	4.54287i
$985$		−	1.35834e10i		−	0.452879i
$986$			4.42642e10i			1.47056i
$987$	2.09986e10			0.695152
$988$	1.83972e10			0.606879
$989$	3.26953e10			1.07473
$990$	−2.30963e9			−0.0756518
$991$			6.02898e9i			0.196782i	0.995148	+	0.0983911i	$0.0313696\pi$
$991$			6.02898e9i			0.196782i	−0.995148	+	0.0983911i	$0.968630\pi$
$992$	−2.32906e10			−0.757513
$993$			9.94016e9i			0.322160i
$994$		−	2.64608e10i		−	0.854575i
$995$	7.00826e9			0.225543
$996$	−1.04511e11			−3.35161
$997$		−	2.38603e10i		−	0.762505i	−0.924471	−	0.381253i	$-0.875493\pi$
$997$		−	2.38603e10i		−	0.762505i	0.924471	−	0.381253i	$-0.124507\pi$
$998$	−7.22178e10			−2.29979
$999$	3.18505e9	+	3.26677e10i	0.101074	+	1.03667i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.b.a.36.1	✓	20
37.36	even	2	inner	37.8.b.a.36.20	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.b.a.36.1	✓	20	1.1	even	1	trivial
37.8.b.a.36.20	yes	20	37.36	even	2	inner

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:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-21.7545i q^{2} +44.4956 q^{3} -345.259 q^{4} -104.884i q^{5} -967.981i q^{6} -454.964 q^{7} +4726.36i q^{8} -207.137 q^{9} +O(q^{10})\)	\(q-21.7545i q^{2} +44.4956 q^{3} -345.259 q^{4} -104.884i q^{5} -967.981i q^{6} -454.964 q^{7} +4726.36i q^{8} -207.137 q^{9} -2281.70 q^{10} -4886.81 q^{11} -15362.5 q^{12} +2167.39i q^{13} +9897.53i q^{14} -4666.88i q^{15} +58626.6 q^{16} -33286.0i q^{17} +4506.17i q^{18} +24585.0i q^{19} +36212.1i q^{20} -20243.9 q^{21} +106310. i q^{22} -62365.5i q^{23} +210303. i q^{24} +67124.3 q^{25} +47150.4 q^{26} -106529. q^{27} +157081. q^{28} -61128.3i q^{29} -101526. q^{30} -34740.4i q^{31} -670419. i q^{32} -217442. q^{33} -724120. q^{34} +47718.5i q^{35} +71516.0 q^{36} +(-29898.6 - 306656. i) q^{37} +534835. q^{38} +96439.3i q^{39} +495720. q^{40} -645608. q^{41} +440397. i q^{42} +524254. i q^{43} +1.68721e6 q^{44} +21725.4i q^{45} -1.35673e6 q^{46} -1.03728e6 q^{47} +2.60863e6 q^{48} -616550. q^{49} -1.46026e6i q^{50} -1.48108e6i q^{51} -748310. i q^{52} +2.01545e6 q^{53} +2.31748e6i q^{54} +512548. i q^{55} -2.15033e6i q^{56} +1.09393e6i q^{57} -1.32982e6 q^{58} -2.51705e6i q^{59} +1.61128e6i q^{60} +981203. i q^{61} -755761. q^{62} +94240.0 q^{63} -7.08043e6 q^{64} +227324. q^{65} +4.73034e6i q^{66} +941977. q^{67} +1.14923e7i q^{68} -2.77499e6i q^{69} +1.03809e6 q^{70} -2.67347e6 q^{71} -979006. i q^{72} +4.46976e6 q^{73} +(-6.67115e6 + 650429. i) q^{74} +2.98674e6 q^{75} -8.48819e6i q^{76} +2.22332e6 q^{77} +2.09799e6 q^{78} -4.48534e6i q^{79} -6.14899e6i q^{80} -4.28705e6 q^{81} +1.40449e7i q^{82} +6.80298e6 q^{83} +6.98940e6 q^{84} -3.49116e6 q^{85} +1.14049e7 q^{86} -2.71994e6i q^{87} -2.30968e7i q^{88} -746269. i q^{89} +472625. q^{90} -986084. i q^{91} +2.15322e7i q^{92} -1.54580e6i q^{93} +2.25655e7i q^{94} +2.57857e6 q^{95} -2.98307e7i q^{96} -3.11369e6i q^{97} +1.34128e7i q^{98} +1.01224e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9}+O(q^{10}) \) 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9	\( 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9} - 882 q^{10} + 3498 q^{11} - 30374 q^{12} + 36116 q^{16} + 113482 q^{21} - 108112 q^{25} + 49278 q^{26} - 304110 q^{27} - 41192 q^{28} + 429776 q^{30} + 305646 q^{33} - 960356 q^{34} + 484758 q^{36} + 108732 q^{37} + 1049916 q^{38} - 496346 q^{40} - 1577742 q^{41} + 685266 q^{44} - 2906298 q^{46} - 1512786 q^{47} + 1522958 q^{48} + 3269246 q^{49} + 2999358 q^{53} + 405946 q^{58} + 3728310 q^{62} - 11995292 q^{63} - 11109700 q^{64} + 4251792 q^{65} + 3562224 q^{67} + 21605644 q^{70} - 15259086 q^{71} + 11088018 q^{73} - 2036544 q^{74} + 14882062 q^{75} - 2419122 q^{77} - 12178734 q^{78} - 17764972 q^{81} - 12873822 q^{83} + 9944396 q^{84} - 2698920 q^{85} + 15345336 q^{86} - 13219100 q^{90} + 48981192 q^{95} + 43111380 q^{99}+O(q^{100}) \) 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9 - 882 * q^10 + 3498 * q^11 - 30374 * q^12 + 36116 * q^16 + 113482 * q^21 - 108112 * q^25 + 49278 * q^26 - 304110 * q^27 - 41192 * q^28 + 429776 * q^30 + 305646 * q^33 - 960356 * q^34 + 484758 * q^36 + 108732 * q^37 + 1049916 * q^38 - 496346 * q^40 - 1577742 * q^41 + 685266 * q^44 - 2906298 * q^46 - 1512786 * q^47 + 1522958 * q^48 + 3269246 * q^49 + 2999358 * q^53 + 405946 * q^58 + 3728310 * q^62 - 11995292 * q^63 - 11109700 * q^64 + 4251792 * q^65 + 3562224 * q^67 + 21605644 * q^70 - 15259086 * q^71 + 11088018 * q^73 - 2036544 * q^74 + 14882062 * q^75 - 2419122 * q^77 - 12178734 * q^78 - 17764972 * q^81 - 12873822 * q^83 + 9944396 * q^84 - 2698920 * q^85 + 15345336 * q^86 - 13219100 * q^90 + 48981192 * q^95 + 43111380 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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