LMFDB - Embedded newform 38.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,8,Mod(1,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	38.a (trivial)

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:	yes
Analytic conductor:	$11.8706309684$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17953}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4488 $ x^2 - x - 4488
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$67.4944$ of defining polynomial
Character	$\chi$	$=$	38.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} -72.4944 q^{3} +64.0000 q^{4} +166.483 q^{5} +579.955 q^{6} +763.922 q^{7} -512.000 q^{8} +3068.44 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -72.4944 q^{3} +64.0000 q^{4} +166.483 q^{5} +579.955 q^{6} +763.922 q^{7} -512.000 q^{8} +3068.44 q^{9} -1331.87 q^{10} +1294.51 q^{11} -4639.64 q^{12} -8597.37 q^{13} -6111.37 q^{14} -12069.1 q^{15} +4096.00 q^{16} -11106.7 q^{17} -24547.5 q^{18} +6859.00 q^{19} +10654.9 q^{20} -55380.0 q^{21} -10356.0 q^{22} -11691.6 q^{23} +37117.1 q^{24} -50408.3 q^{25} +68779.0 q^{26} -63899.4 q^{27} +48891.0 q^{28} -49325.7 q^{29} +96552.8 q^{30} -129466. q^{31} -32768.0 q^{32} -93844.4 q^{33} +88853.5 q^{34} +127180. q^{35} +196380. q^{36} -492669. q^{37} -54872.0 q^{38} +623261. q^{39} -85239.4 q^{40} +441860. q^{41} +443040. q^{42} +638398. q^{43} +82848.4 q^{44} +510843. q^{45} +93532.9 q^{46} -649782. q^{47} -296937. q^{48} -239967. q^{49} +403267. q^{50} +805172. q^{51} -550232. q^{52} -2.07086e6 q^{53} +511195. q^{54} +215513. q^{55} -391128. q^{56} -497239. q^{57} +394606. q^{58} -142494. q^{59} -772422. q^{60} -3.41033e6 q^{61} +1.03573e6 q^{62} +2.34405e6 q^{63} +262144. q^{64} -1.43132e6 q^{65} +750755. q^{66} -3.63426e6 q^{67} -710828. q^{68} +847576. q^{69} -1.01744e6 q^{70} +3.90647e6 q^{71} -1.57104e6 q^{72} +2.50725e6 q^{73} +3.94135e6 q^{74} +3.65432e6 q^{75} +438976. q^{76} +988901. q^{77} -4.98609e6 q^{78} -4.34070e6 q^{79} +681915. q^{80} -2.07833e6 q^{81} -3.53488e6 q^{82} -2.81735e6 q^{83} -3.54432e6 q^{84} -1.84908e6 q^{85} -5.10718e6 q^{86} +3.57584e6 q^{87} -662787. q^{88} +1.21050e7 q^{89} -4.08675e6 q^{90} -6.56772e6 q^{91} -748263. q^{92} +9.38556e6 q^{93} +5.19825e6 q^{94} +1.14191e6 q^{95} +2.37550e6 q^{96} -7.17748e6 q^{97} +1.91973e6 q^{98} +3.97211e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{2} - 11 q^{3} + 128 q^{4} - 69 q^{5} + 88 q^{6} - 348 q^{7} - 1024 q^{8} + 4663 q^{9}+O(q^{10}) $ 2 * q - 16 * q^2 - 11 * q^3 + 128 * q^4 - 69 * q^5 + 88 * q^6 - 348 * q^7 - 1024 * q^8 + 4663 * q^9	\( 2 q - 16 q^{2} - 11 q^{3} + 128 q^{4} - 69 q^{5} + 88 q^{6} - 348 q^{7} - 1024 q^{8} + 4663 q^{9} + 552 q^{10} + 2723 q^{11} - 704 q^{12} - 14113 q^{13} + 2784 q^{14} - 26550 q^{15} + 8192 q^{16} - 38560 q^{17} - 37304 q^{18} + 13718 q^{19} - 4416 q^{20} - 123757 q^{21} - 21784 q^{22} - 73897 q^{23} + 5632 q^{24} - 73081 q^{25} + 112904 q^{26} - 100331 q^{27} - 22272 q^{28} + 159813 q^{29} + 212400 q^{30} - 259468 q^{31} - 65536 q^{32} - 6000 q^{33} + 308480 q^{34} + 389019 q^{35} + 298432 q^{36} - 528168 q^{37} - 109744 q^{38} + 284081 q^{39} + 35328 q^{40} + 1005650 q^{41} + 990056 q^{42} + 286217 q^{43} + 174272 q^{44} + 135351 q^{45} + 591176 q^{46} - 1397509 q^{47} - 45056 q^{48} + 172860 q^{49} + 584648 q^{50} - 883053 q^{51} - 903232 q^{52} - 1385969 q^{53} + 802648 q^{54} - 120873 q^{55} + 178176 q^{56} - 75449 q^{57} - 1278504 q^{58} + 2700953 q^{59} - 1699200 q^{60} - 3975947 q^{61} + 2075744 q^{62} + 571019 q^{63} + 524288 q^{64} - 132480 q^{65} + 48000 q^{66} - 134557 q^{67} - 2467840 q^{68} - 2977707 q^{69} - 3112152 q^{70} + 4202740 q^{71} - 2387456 q^{72} + 900498 q^{73} + 4225344 q^{74} + 2260081 q^{75} + 877952 q^{76} - 599473 q^{77} - 2272648 q^{78} - 6893730 q^{79} - 282624 q^{80} - 7805978 q^{81} - 8045200 q^{82} - 2465330 q^{83} - 7920448 q^{84} + 4615719 q^{85} - 2289736 q^{86} + 16436697 q^{87} - 1394176 q^{88} + 17431724 q^{89} - 1082808 q^{90} - 434771 q^{91} - 4729408 q^{92} + 1391168 q^{93} + 11180072 q^{94} - 473271 q^{95} + 360448 q^{96} + 6351934 q^{97} - 1382880 q^{98} + 6249933 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 11 * q^3 + 128 * q^4 - 69 * q^5 + 88 * q^6 - 348 * q^7 - 1024 * q^8 + 4663 * q^9 + 552 * q^10 + 2723 * q^11 - 704 * q^12 - 14113 * q^13 + 2784 * q^14 - 26550 * q^15 + 8192 * q^16 - 38560 * q^17 - 37304 * q^18 + 13718 * q^19 - 4416 * q^20 - 123757 * q^21 - 21784 * q^22 - 73897 * q^23 + 5632 * q^24 - 73081 * q^25 + 112904 * q^26 - 100331 * q^27 - 22272 * q^28 + 159813 * q^29 + 212400 * q^30 - 259468 * q^31 - 65536 * q^32 - 6000 * q^33 + 308480 * q^34 + 389019 * q^35 + 298432 * q^36 - 528168 * q^37 - 109744 * q^38 + 284081 * q^39 + 35328 * q^40 + 1005650 * q^41 + 990056 * q^42 + 286217 * q^43 + 174272 * q^44 + 135351 * q^45 + 591176 * q^46 - 1397509 * q^47 - 45056 * q^48 + 172860 * q^49 + 584648 * q^50 - 883053 * q^51 - 903232 * q^52 - 1385969 * q^53 + 802648 * q^54 - 120873 * q^55 + 178176 * q^56 - 75449 * q^57 - 1278504 * q^58 + 2700953 * q^59 - 1699200 * q^60 - 3975947 * q^61 + 2075744 * q^62 + 571019 * q^63 + 524288 * q^64 - 132480 * q^65 + 48000 * q^66 - 134557 * q^67 - 2467840 * q^68 - 2977707 * q^69 - 3112152 * q^70 + 4202740 * q^71 - 2387456 * q^72 + 900498 * q^73 + 4225344 * q^74 + 2260081 * q^75 + 877952 * q^76 - 599473 * q^77 - 2272648 * q^78 - 6893730 * q^79 - 282624 * q^80 - 7805978 * q^81 - 8045200 * q^82 - 2465330 * q^83 - 7920448 * q^84 + 4615719 * q^85 - 2289736 * q^86 + 16436697 * q^87 - 1394176 * q^88 + 17431724 * q^89 - 1082808 * q^90 - 434771 * q^91 - 4729408 * q^92 + 1391168 * q^93 + 11180072 * q^94 - 473271 * q^95 + 360448 * q^96 + 6351934 * q^97 - 1382880 * q^98 + 6249933 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−72.4944	−1.55017	−0.775086	−	0.631855i	$-0.782294\pi$
$3$	−72.4944	−1.55017	−0.775086	+	0.631855i	$0.782294\pi$
$4$	64.0000	0.500000
$5$	166.483	0.595628	0.297814	−	0.954624i	$-0.403742\pi$
$5$	166.483	0.595628	0.297814	+	0.954624i	$0.403742\pi$
$6$	579.955	1.09614
$7$	763.922	0.841794	0.420897	−	0.907109i	$-0.361715\pi$
$7$	763.922	0.841794	0.420897	+	0.907109i	$0.361715\pi$
$8$	−512.000	−0.353553
$9$	3068.44	1.40304
$10$	−1331.87	−0.421173
$11$	1294.51	0.293244	0.146622	−	0.989193i	$-0.453160\pi$
$11$	1294.51	0.293244	0.146622	+	0.989193i	$0.453160\pi$
$12$	−4639.64	−0.775086
$13$	−8597.37	−1.08534	−0.542668	−	0.839947i	$-0.682586\pi$
$13$	−8597.37	−1.08534	−0.542668	+	0.839947i	$0.682586\pi$
$14$	−6111.37	−0.595238
$15$	−12069.1	−0.923327
$16$	4096.00	0.250000
$17$	−11106.7	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$17$	−11106.7	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$18$	−24547.5	−0.992096
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	10654.9	0.297814
$21$	−55380.0	−1.30493
$22$	−10356.0	−0.207355
$23$	−11691.6	−0.200367	−0.100184	−	0.994969i	$-0.531943\pi$
$23$	−11691.6	−0.200367	−0.100184	+	0.994969i	$0.531943\pi$
$24$	37117.1	0.548069
$25$	−50408.3	−0.645227
$26$	68779.0	0.767448
$27$	−63899.4	−0.624774
$28$	48891.0	0.420897
$29$	−49325.7	−0.375561	−0.187780	−	0.982211i	$-0.560129\pi$
$29$	−49325.7	−0.375561	−0.187780	+	0.982211i	$0.560129\pi$
$30$	96552.8	0.652891
$31$	−129466.	−0.780530	−0.390265	−	0.920702i	$-0.627617\pi$
$31$	−129466.	−0.780530	−0.390265	+	0.920702i	$0.627617\pi$
$32$	−32768.0	−0.176777
$33$	−93844.4	−0.454579
$34$	88853.5	0.387702
$35$	127180.	0.501396
$36$	196380.	0.701518
$37$	−492669.	−1.59900	−0.799501	−	0.600664i	$-0.794903\pi$
$37$	−492669.	−1.59900	−0.799501	+	0.600664i	$0.794903\pi$
$38$	−54872.0	−0.162221
$39$	623261.	1.68246
$40$	−85239.4	−0.210586
$41$	441860.	1.00125	0.500624	−	0.865665i	$-0.333104\pi$
$41$	441860.	1.00125	0.500624	+	0.865665i	$0.333104\pi$
$42$	443040.	0.922722
$43$	638398.	1.22448	0.612240	−	0.790672i	$-0.290268\pi$
$43$	638398.	1.22448	0.612240	+	0.790672i	$0.290268\pi$
$44$	82848.4	0.146622
$45$	510843.	0.835688
$46$	93532.9	0.141681
$47$	−649782.	−0.912904	−0.456452	−	0.889748i	$-0.650880\pi$
$47$	−649782.	−0.912904	−0.456452	+	0.889748i	$0.650880\pi$
$48$	−296937.	−0.387543
$49$	−239967.	−0.291383
$50$	403267.	0.456244
$51$	805172.	0.849950
$52$	−550232.	−0.542668
$53$	−2.07086e6	−1.91067	−0.955333	−	0.295531i	$-0.904503\pi$
$53$	−2.07086e6	−1.91067	−0.955333	+	0.295531i	$0.904503\pi$
$54$	511195.	0.441782
$55$	215513.	0.174665
$56$	−391128.	−0.297619
$57$	−497239.	−0.355634
$58$	394606.	0.265562
$59$	−142494.	−0.0903262	−0.0451631	−	0.998980i	$-0.514381\pi$
$59$	−142494.	−0.0903262	−0.0451631	+	0.998980i	$0.514381\pi$
$60$	−772422.	−0.461663
$61$	−3.41033e6	−1.92372	−0.961861	−	0.273540i	$-0.911806\pi$
$61$	−3.41033e6	−1.92372	−0.961861	+	0.273540i	$0.911806\pi$
$62$	1.03573e6	0.551918
$63$	2.34405e6	1.18107
$64$	262144.	0.125000
$65$	−1.43132e6	−0.646457
$66$	750755.	0.321436
$67$	−3.63426e6	−1.47623	−0.738116	−	0.674674i	$-0.764284\pi$
$67$	−3.63426e6	−1.47623	−0.738116	+	0.674674i	$0.764284\pi$
$68$	−710828.	−0.274147
$69$	847576.	0.310604
$70$	−1.01744e6	−0.354541
$71$	3.90647e6	1.29533	0.647664	−	0.761926i	$-0.275746\pi$
$71$	3.90647e6	1.29533	0.647664	+	0.761926i	$0.275746\pi$
$72$	−1.57104e6	−0.496048
$73$	2.50725e6	0.754339	0.377170	−	0.926144i	$-0.376897\pi$
$73$	2.50725e6	0.754339	0.377170	+	0.926144i	$0.376897\pi$
$74$	3.94135e6	1.13067
$75$	3.65432e6	1.00021
$76$	438976.	0.114708
$77$	988901.	0.246851
$78$	−4.98609e6	−1.18968
$79$	−4.34070e6	−0.990525	−0.495262	−	0.868743i	$-0.664928\pi$
$79$	−4.34070e6	−0.990525	−0.495262	+	0.868743i	$0.664928\pi$
$80$	681915.	0.148907
$81$	−2.07833e6	−0.434527
$82$	−3.53488e6	−0.707989
$83$	−2.81735e6	−0.540838	−0.270419	−	0.962743i	$-0.587162\pi$
$83$	−2.81735e6	−0.540838	−0.270419	+	0.962743i	$0.587162\pi$
$84$	−3.54432e6	−0.652463
$85$	−1.84908e6	−0.326579
$86$	−5.10718e6	−0.865839
$87$	3.57584e6	0.582184
$88$	−662787.	−0.103678
$89$	1.21050e7	1.82012	0.910058	−	0.414482i	$-0.136037\pi$
$89$	1.21050e7	1.82012	0.910058	+	0.414482i	$0.136037\pi$
$90$	−4.08675e6	−0.590920
$91$	−6.56772e6	−0.913628
$92$	−748263.	−0.100184
$93$	9.38556e6	1.20996
$94$	5.19825e6	0.645520
$95$	1.14191e6	0.136647
$96$	2.37550e6	0.274034
$97$	−7.17748e6	−0.798493	−0.399246	−	0.916844i	$-0.630728\pi$
$97$	−7.17748e6	−0.798493	−0.399246	+	0.916844i	$0.630728\pi$
$98$	1.91973e6	0.206039
$99$	3.97211e6	0.411432
$100$	−3.22613e6	−0.322613
$101$	1.13270e7	1.09393	0.546966	−	0.837155i	$-0.315783\pi$
$101$	1.13270e7	1.09393	0.546966	+	0.837155i	$0.315783\pi$
$102$	−6.44138e6	−0.601005
$103$	1.72478e7	1.55526	0.777630	−	0.628722i	$-0.216421\pi$
$103$	1.72478e7	1.55526	0.777630	+	0.628722i	$0.216421\pi$
$104$	4.40185e6	0.383724
$105$	−9.21985e6	−0.777251
$106$	1.65669e7	1.35105
$107$	−761079.	−0.0600601	−0.0300301	−	0.999549i	$-0.509560\pi$
$107$	−761079.	−0.0600601	−0.0300301	+	0.999549i	$0.509560\pi$
$108$	−4.08956e6	−0.312387
$109$	−1.49002e6	−0.110204	−0.0551021	−	0.998481i	$-0.517548\pi$
$109$	−1.49002e6	−0.110204	−0.0551021	+	0.998481i	$0.517548\pi$
$110$	−1.72411e6	−0.123507
$111$	3.57157e7	2.47873
$112$	3.12902e6	0.210448
$113$	1.45254e7	0.947005	0.473503	−	0.880792i	$-0.342989\pi$
$113$	1.45254e7	0.947005	0.473503	+	0.880792i	$0.342989\pi$
$114$	3.97791e6	0.251471
$115$	−1.94646e6	−0.119344
$116$	−3.15684e6	−0.187780
$117$	−2.63805e7	−1.52276
$118$	1.13995e6	0.0638703
$119$	−8.48464e6	−0.461550
$120$	6.17938e6	0.326445
$121$	−1.78114e7	−0.914008
$122$	2.72827e7	1.36028
$123$	−3.20324e7	−1.55211
$124$	−8.28583e6	−0.390265
$125$	−2.13986e7	−0.979944
$126$	−1.87524e7	−0.835140
$127$	−2.48877e7	−1.07813	−0.539065	−	0.842264i	$-0.681222\pi$
$127$	−2.48877e7	−1.07813	−0.539065	+	0.842264i	$0.681222\pi$
$128$	−2.09715e6	−0.0883883
$129$	−4.62803e7	−1.89816
$130$	1.14505e7	0.457114
$131$	1.21840e6	0.0473522	0.0236761	−	0.999720i	$-0.492463\pi$
$131$	1.21840e6	0.0473522	0.0236761	+	0.999720i	$0.492463\pi$
$132$	−6.00604e6	−0.227290
$133$	5.23974e6	0.193121
$134$	2.90741e7	1.04385
$135$	−1.06382e7	−0.372133
$136$	5.68662e6	0.193851
$137$	−2.35343e7	−0.781949	−0.390975	−	0.920401i	$-0.627862\pi$
$137$	−2.35343e7	−0.781949	−0.390975	+	0.920401i	$0.627862\pi$
$138$	−6.78061e6	−0.219630
$139$	7.03322e6	0.222127	0.111064	−	0.993813i	$-0.464574\pi$
$139$	7.03322e6	0.222127	0.111064	+	0.993813i	$0.464574\pi$
$140$	8.13953e6	0.250698
$141$	4.71055e7	1.41516
$142$	−3.12517e7	−0.915936
$143$	−1.11293e7	−0.318268
$144$	1.25683e7	0.350759
$145$	−8.21190e6	−0.223695
$146$	−2.00580e7	−0.533399
$147$	1.73962e7	0.451695
$148$	−3.15308e7	−0.799501
$149$	7.07776e7	1.75285	0.876424	−	0.481541i	$-0.159923\pi$
$149$	7.07776e7	1.75285	0.876424	+	0.481541i	$0.159923\pi$
$150$	−2.92346e7	−0.707257
$151$	−3.64919e6	−0.0862536	−0.0431268	−	0.999070i	$-0.513732\pi$
$151$	−3.64919e6	−0.0862536	−0.0431268	+	0.999070i	$0.513732\pi$
$152$	−3.51181e6	−0.0811107
$153$	−3.40802e7	−0.769275
$154$	−7.91121e6	−0.174550
$155$	−2.15539e7	−0.464906
$156$	3.98887e7	0.841229
$157$	−3.41108e7	−0.703466	−0.351733	−	0.936100i	$-0.614407\pi$
$157$	−3.41108e7	−0.703466	−0.351733	+	0.936100i	$0.614407\pi$
$158$	3.47256e7	0.700407
$159$	1.50126e8	2.96186
$160$	−5.45532e6	−0.105293
$161$	−8.93147e6	−0.168668
$162$	1.66266e7	0.307257
$163$	−3.26683e7	−0.590841	−0.295420	−	0.955367i	$-0.595460\pi$
$163$	−3.26683e7	−0.590841	−0.295420	+	0.955367i	$0.595460\pi$
$164$	2.82790e7	0.500624
$165$	−1.56235e7	−0.270760
$166$	2.25388e7	0.382431
$167$	3.02934e6	0.0503316	0.0251658	−	0.999683i	$-0.491989\pi$
$167$	3.02934e6	0.0503316	0.0251658	+	0.999683i	$0.491989\pi$
$168$	2.83546e7	0.461361
$169$	1.11663e7	0.177953
$170$	1.47926e7	0.230926
$171$	2.10464e7	0.321878
$172$	4.08575e7	0.612240
$173$	−2.28024e7	−0.334826	−0.167413	−	0.985887i	$-0.553541\pi$
$173$	−2.28024e7	−0.334826	−0.167413	+	0.985887i	$0.553541\pi$
$174$	−2.86067e7	−0.411666
$175$	−3.85080e7	−0.543148
$176$	5.30229e6	0.0733111
$177$	1.03300e7	0.140021
$178$	−9.68398e7	−1.28702
$179$	6.66371e7	0.868421	0.434211	−	0.900811i	$-0.357027\pi$
$179$	6.66371e7	0.868421	0.434211	+	0.900811i	$0.357027\pi$
$180$	3.26940e7	0.417844
$181$	−1.32369e8	−1.65925	−0.829624	−	0.558322i	$-0.811445\pi$
$181$	−1.32369e8	−1.65925	−0.829624	+	0.558322i	$0.811445\pi$
$182$	5.25417e7	0.646033
$183$	2.47230e8	2.98210
$184$	5.98610e6	0.0708405
$185$	−8.20211e7	−0.952411
$186$	−7.50845e7	−0.855569
$187$	−1.43777e7	−0.160784
$188$	−4.15860e7	−0.456452
$189$	−4.88141e7	−0.525931
$190$	−9.13527e6	−0.0966237
$191$	6.39365e6	0.0663945	0.0331972	−	0.999449i	$-0.489431\pi$
$191$	6.39365e6	0.0663945	0.0331972	+	0.999449i	$0.489431\pi$
$192$	−1.90040e7	−0.193772
$193$	−1.05683e8	−1.05817	−0.529085	−	0.848569i	$-0.677465\pi$
$193$	−1.05683e8	−1.05817	−0.529085	+	0.848569i	$0.677465\pi$
$194$	5.74199e7	0.564620
$195$	1.03763e8	1.00212
$196$	−1.53579e7	−0.145692
$197$	−7.36537e7	−0.686377	−0.343189	−	0.939266i	$-0.611507\pi$
$197$	−7.36537e7	−0.686377	−0.343189	+	0.939266i	$0.611507\pi$
$198$	−3.17769e7	−0.290926
$199$	1.29375e8	1.16376	0.581880	−	0.813275i	$-0.302317\pi$
$199$	1.29375e8	1.16376	0.581880	+	0.813275i	$0.302317\pi$
$200$	2.58091e7	0.228122
$201$	2.63464e8	2.28841
$202$	−9.06161e7	−0.773527
$203$	−3.76810e7	−0.316145
$204$	5.15310e7	0.424975
$205$	7.35623e7	0.596371
$206$	−1.37982e8	−1.09974
$207$	−3.58750e7	−0.281122
$208$	−3.52148e7	−0.271334
$209$	8.87901e6	0.0672748
$210$	7.37588e7	0.549599
$211$	1.56501e8	1.14691	0.573453	−	0.819238i	$-0.305603\pi$
$211$	1.56501e8	1.14691	0.573453	+	0.819238i	$0.305603\pi$
$212$	−1.32535e8	−0.955333
$213$	−2.83197e8	−2.00798
$214$	6.08863e6	0.0424689
$215$	1.06283e8	0.729336
$216$	3.27165e7	0.220891
$217$	−9.89019e7	−0.657046
$218$	1.19201e7	0.0779261
$219$	−1.81761e8	−1.16936
$220$	1.37929e7	0.0873323
$221$	9.54883e7	0.595082
$222$	−2.85726e8	−1.75273
$223$	8.28997e7	0.500594	0.250297	−	0.968169i	$-0.419472\pi$
$223$	8.28997e7	0.500594	0.250297	+	0.968169i	$0.419472\pi$
$224$	−2.50322e7	−0.148810
$225$	−1.54675e8	−0.905276
$226$	−1.16203e8	−0.669634
$227$	−1.88411e8	−1.06909	−0.534547	−	0.845139i	$-0.679518\pi$
$227$	−1.88411e8	−1.06909	−0.534547	+	0.845139i	$0.679518\pi$
$228$	−3.18233e7	−0.177817
$229$	−1.12561e8	−0.619392	−0.309696	−	0.950836i	$-0.600227\pi$
$229$	−1.12561e8	−0.619392	−0.309696	+	0.950836i	$0.600227\pi$
$230$	1.55717e7	0.0843893
$231$	−7.16898e7	−0.382662
$232$	2.52548e7	0.132781
$233$	1.52695e7	0.0790820	0.0395410	−	0.999218i	$-0.487410\pi$
$233$	1.52695e7	0.0790820	0.0395410	+	0.999218i	$0.487410\pi$
$234$	2.11044e8	1.07676
$235$	−1.08178e8	−0.543751
$236$	−9.11960e6	−0.0451631
$237$	3.14677e8	1.53548
$238$	6.78771e7	0.326365
$239$	−1.87377e8	−0.887818	−0.443909	−	0.896072i	$-0.646409\pi$
$239$	−1.87377e8	−0.887818	−0.443909	+	0.896072i	$0.646409\pi$
$240$	−4.94350e7	−0.230832
$241$	1.52334e8	0.701031	0.350516	−	0.936557i	$-0.386006\pi$
$241$	1.52334e8	0.701031	0.350516	+	0.936557i	$0.386006\pi$
$242$	1.42491e8	0.646301
$243$	2.90415e8	1.29837
$244$	−2.18261e8	−0.961861
$245$	−3.99504e7	−0.173556
$246$	2.56259e8	1.09750
$247$	−5.89694e7	−0.248993
$248$	6.62866e7	0.275959
$249$	2.04242e8	0.838393
$250$	1.71189e8	0.692925
$251$	−2.48606e8	−0.992324	−0.496162	−	0.868230i	$-0.665258\pi$
$251$	−2.48606e8	−0.992324	−0.496162	+	0.868230i	$0.665258\pi$
$252$	1.50019e8	0.590533
$253$	−1.51349e7	−0.0587566
$254$	1.99101e8	0.762353
$255$	1.34048e8	0.506254
$256$	1.67772e7	0.0625000
$257$	3.73047e8	1.37087	0.685436	−	0.728133i	$-0.259611\pi$
$257$	3.73047e8	1.37087	0.685436	+	0.728133i	$0.259611\pi$
$258$	3.70242e8	1.34220
$259$	−3.76360e8	−1.34603
$260$	−9.16043e7	−0.323228
$261$	−1.51353e8	−0.526925
$262$	−9.74720e6	−0.0334831
$263$	−3.69713e7	−0.125320	−0.0626599	−	0.998035i	$-0.519958\pi$
$263$	−3.69713e7	−0.125320	−0.0626599	+	0.998035i	$0.519958\pi$
$264$	4.80483e7	0.160718
$265$	−3.44763e8	−1.13805
$266$	−4.19179e7	−0.136557
$267$	−8.77543e8	−2.82149
$268$	−2.32593e8	−0.738116
$269$	−3.07534e8	−0.963295	−0.481648	−	0.876365i	$-0.659962\pi$
$269$	−3.07534e8	−0.963295	−0.481648	+	0.876365i	$0.659962\pi$
$270$	8.51054e7	0.263138
$271$	5.58577e8	1.70487	0.852433	−	0.522836i	$-0.175126\pi$
$271$	5.58577e8	1.70487	0.852433	+	0.522836i	$0.175126\pi$
$272$	−4.54930e7	−0.137073
$273$	4.76123e8	1.41628
$274$	1.88274e8	0.552922
$275$	−6.52539e7	−0.189209
$276$	5.42449e7	0.155302
$277$	5.18452e8	1.46565	0.732824	−	0.680418i	$-0.238202\pi$
$277$	5.18452e8	1.46565	0.732824	+	0.680418i	$0.238202\pi$
$278$	−5.62658e7	−0.157068
$279$	−3.97259e8	−1.09511
$280$	−6.51162e7	−0.177270
$281$	6.83424e8	1.83746	0.918730	−	0.394886i	$-0.129216\pi$
$281$	6.83424e8	1.83746	0.918730	+	0.394886i	$0.129216\pi$
$282$	−3.76844e8	−1.00067
$283$	−4.14251e8	−1.08645	−0.543226	−	0.839586i	$-0.682797\pi$
$283$	−4.14251e8	−1.08645	−0.543226	+	0.839586i	$0.682797\pi$
$284$	2.50014e8	0.647664
$285$	−8.27820e7	−0.211826
$286$	8.90348e7	0.225050
$287$	3.37546e8	0.842844
$288$	−1.00547e8	−0.248024
$289$	−2.86980e8	−0.699374
$290$	6.56952e7	0.158176
$291$	5.20327e8	1.23780
$292$	1.60464e8	0.377170
$293$	4.17557e8	0.969794	0.484897	−	0.874571i	$-0.338857\pi$
$293$	4.17557e8	0.969794	0.484897	+	0.874571i	$0.338857\pi$
$294$	−1.39170e8	−0.319396
$295$	−2.37228e7	−0.0538009
$296$	2.52246e8	0.565333
$297$	−8.27181e7	−0.183212
$298$	−5.66221e8	−1.23945
$299$	1.00517e8	0.217466
$300$	2.33877e8	0.500106
$301$	4.87686e8	1.03076
$302$	2.91935e7	0.0609905
$303$	−8.21145e8	−1.69578
$304$	2.80945e7	0.0573539
$305$	−5.67763e8	−1.14582
$306$	2.72641e8	0.543960
$307$	1.45263e8	0.286531	0.143265	−	0.989684i	$-0.454240\pi$
$307$	1.45263e8	0.286531	0.143265	+	0.989684i	$0.454240\pi$
$308$	6.32897e7	0.123426
$309$	−1.25037e9	−2.41092
$310$	1.72431e8	0.328738
$311$	5.07825e8	0.957311	0.478655	−	0.878003i	$-0.341124\pi$
$311$	5.07825e8	0.957311	0.478655	+	0.878003i	$0.341124\pi$
$312$	−3.19110e8	−0.594838
$313$	8.98846e8	1.65684	0.828419	−	0.560108i	$-0.189240\pi$
$313$	8.98846e8	1.65684	0.828419	+	0.560108i	$0.189240\pi$
$314$	2.72886e8	0.497425
$315$	3.90244e8	0.703477
$316$	−2.77805e8	−0.495262
$317$	7.53754e7	0.132899	0.0664495	−	0.997790i	$-0.478833\pi$
$317$	7.53754e7	0.132899	0.0664495	+	0.997790i	$0.478833\pi$
$318$	−1.20101e9	−2.09435
$319$	−6.38524e7	−0.110131
$320$	4.36426e7	0.0744536
$321$	5.51739e7	0.0931036
$322$	7.14518e7	0.119266
$323$	−7.61807e7	−0.125787
$324$	−1.33013e8	−0.217264
$325$	4.33379e8	0.700287
$326$	2.61347e8	0.417788
$327$	1.08018e8	0.170836
$328$	−2.26232e8	−0.353994
$329$	−4.96382e8	−0.768476
$330$	1.24988e8	0.191456
$331$	2.03341e7	0.0308197	0.0154098	−	0.999881i	$-0.495095\pi$
$331$	2.03341e7	0.0308197	0.0154098	+	0.999881i	$0.495095\pi$
$332$	−1.80310e8	−0.270419
$333$	−1.51172e9	−2.24346
$334$	−2.42347e7	−0.0355898
$335$	−6.05044e8	−0.879285
$336$	−2.26837e8	−0.326231
$337$	4.59629e8	0.654189	0.327094	−	0.944992i	$-0.393931\pi$
$337$	4.59629e8	0.654189	0.327094	+	0.944992i	$0.393931\pi$
$338$	−8.93302e7	−0.125832
$339$	−1.05301e9	−1.46802
$340$	−1.18341e8	−0.163290
$341$	−1.67594e8	−0.228886
$342$	−1.68371e8	−0.227602
$343$	−8.12438e8	−1.08708
$344$	−3.26860e8	−0.432919
$345$	1.41107e8	0.185005
$346$	1.82419e8	0.236758
$347$	−1.32179e9	−1.69828	−0.849142	−	0.528165i	$-0.822880\pi$
$347$	−1.32179e9	−1.69828	−0.849142	+	0.528165i	$0.822880\pi$
$348$	2.28854e8	0.291092
$349$	7.87970e7	0.0992249	0.0496124	−	0.998769i	$-0.484201\pi$
$349$	7.87970e7	0.0992249	0.0496124	+	0.998769i	$0.484201\pi$
$350$	3.08064e8	0.384064
$351$	5.49366e8	0.678090
$352$	−4.24184e7	−0.0518388
$353$	7.85680e8	0.950680	0.475340	−	0.879802i	$-0.342325\pi$
$353$	7.85680e8	0.950680	0.475340	+	0.879802i	$0.342325\pi$
$354$	−8.26400e7	−0.0990100
$355$	6.50361e8	0.771535
$356$	7.74718e8	0.910058
$357$	6.15089e8	0.715482
$358$	−5.33097e8	−0.614066
$359$	9.83875e7	0.112230	0.0561151	−	0.998424i	$-0.482129\pi$
$359$	9.83875e7	0.112230	0.0561151	+	0.998424i	$0.482129\pi$
$360$	−2.61552e8	−0.295460
$361$	4.70459e7	0.0526316
$362$	1.05895e9	1.17327
$363$	1.29123e9	1.41687
$364$	−4.20334e8	−0.456814
$365$	4.17414e8	0.449306
$366$	−1.97784e9	−2.10866
$367$	1.15771e9	1.22255	0.611277	−	0.791416i	$-0.290656\pi$
$367$	1.15771e9	1.22255	0.611277	+	0.791416i	$0.290656\pi$
$368$	−4.78888e7	−0.0500918
$369$	1.35582e9	1.40479
$370$	6.56169e8	0.673456
$371$	−1.58197e9	−1.60839
$372$	6.00676e8	0.604979
$373$	2.33860e7	0.0233332	0.0116666	−	0.999932i	$-0.496286\pi$
$373$	2.33860e7	0.0233332	0.0116666	+	0.999932i	$0.496286\pi$
$374$	1.15021e8	0.113691
$375$	1.55128e9	1.51908
$376$	3.32688e8	0.322760
$377$	4.24071e8	0.407609
$378$	3.90513e8	0.371890
$379$	1.72018e9	1.62306	0.811532	−	0.584308i	$-0.198634\pi$
$379$	1.72018e9	1.62306	0.811532	+	0.584308i	$0.198634\pi$
$380$	7.30821e7	0.0683233
$381$	1.80422e9	1.67129
$382$	−5.11492e7	−0.0469480
$383$	−1.23762e9	−1.12562	−0.562809	−	0.826587i	$-0.690279\pi$
$383$	−1.23762e9	−1.12562	−0.562809	+	0.826587i	$0.690279\pi$
$384$	1.52032e8	0.137017
$385$	1.64635e8	0.147032
$386$	8.45466e8	0.748239
$387$	1.95889e9	1.71799
$388$	−4.59359e8	−0.399246
$389$	−1.85001e9	−1.59349	−0.796746	−	0.604314i	$-0.793447\pi$
$389$	−1.85001e9	−1.59349	−0.796746	+	0.604314i	$0.793447\pi$
$390$	−8.30100e8	−0.708605
$391$	1.29855e8	0.109860
$392$	1.22863e8	0.103020
$393$	−8.83271e7	−0.0734041
$394$	5.89230e8	0.485342
$395$	−7.22654e8	−0.589985
$396$	2.54215e8	0.205716
$397$	9.18688e8	0.736888	0.368444	−	0.929650i	$-0.379891\pi$
$397$	9.18688e8	0.736888	0.368444	+	0.929650i	$0.379891\pi$
$398$	−1.03500e9	−0.822902
$399$	−3.79852e8	−0.299370
$400$	−2.06473e8	−0.161307
$401$	−2.00949e9	−1.55626	−0.778128	−	0.628105i	$-0.783831\pi$
$401$	−2.00949e9	−1.55626	−0.778128	+	0.628105i	$0.783831\pi$
$402$	−2.10771e9	−1.61815
$403$	1.11307e9	0.847137
$404$	7.24929e8	0.546966
$405$	−3.46007e8	−0.258817
$406$	3.01448e8	0.223548
$407$	−6.37763e8	−0.468898
$408$	−4.12248e8	−0.300503
$409$	−2.38933e9	−1.72681	−0.863406	−	0.504509i	$-0.831673\pi$
$409$	−2.38933e9	−1.72681	−0.863406	+	0.504509i	$0.831673\pi$
$410$	−5.88498e8	−0.421698
$411$	1.70610e9	1.21216
$412$	1.10386e9	0.777630
$413$	−1.08854e8	−0.0760360
$414$	2.87000e8	0.198784
$415$	−4.69042e8	−0.322139
$416$	2.81719e8	0.191862
$417$	−5.09869e8	−0.344336
$418$	−7.10321e7	−0.0475705
$419$	−9.28481e8	−0.616629	−0.308315	−	0.951284i	$-0.599765\pi$
$419$	−9.28481e8	−0.616629	−0.308315	+	0.951284i	$0.599765\pi$
$420$	−5.90070e8	−0.388625
$421$	8.18254e7	0.0534442	0.0267221	−	0.999643i	$-0.491493\pi$
$421$	8.18254e7	0.0534442	0.0267221	+	0.999643i	$0.491493\pi$
$422$	−1.25201e9	−0.810986
$423$	−1.99381e9	−1.28084
$424$	1.06028e9	0.675523
$425$	5.59869e8	0.353774
$426$	2.26558e9	1.41986
$427$	−2.60523e9	−1.61938
$428$	−4.87090e7	−0.0300301
$429$	8.06815e8	0.493371
$430$	−8.50261e8	−0.515718
$431$	−2.15977e9	−1.29938	−0.649690	−	0.760199i	$-0.725101\pi$
$431$	−2.15977e9	−1.29938	−0.649690	+	0.760199i	$0.725101\pi$
$432$	−2.61732e8	−0.156194
$433$	1.90998e9	1.13063	0.565316	−	0.824874i	$-0.308754\pi$
$433$	1.90998e9	1.13063	0.565316	+	0.824874i	$0.308754\pi$
$434$	7.91215e8	0.464601
$435$	5.95317e8	0.346765
$436$	−9.53610e7	−0.0551021
$437$	−8.01928e7	−0.0459674
$438$	1.45409e9	0.826860
$439$	−1.86776e9	−1.05365	−0.526825	−	0.849974i	$-0.676618\pi$
$439$	−1.86776e9	−1.05365	−0.526825	+	0.849974i	$0.676618\pi$
$440$	−1.10343e8	−0.0617533
$441$	−7.36323e8	−0.408821
$442$	−7.63906e8	−0.420787
$443$	1.06690e9	0.583059	0.291529	−	0.956562i	$-0.405836\pi$
$443$	1.06690e9	0.583059	0.291529	+	0.956562i	$0.405836\pi$
$444$	2.28581e9	1.23936
$445$	2.01528e9	1.08411
$446$	−6.63198e8	−0.353974
$447$	−5.13098e9	−2.71722
$448$	2.00257e8	0.105224
$449$	−1.27020e9	−0.662231	−0.331115	−	0.943590i	$-0.607425\pi$
$449$	−1.27020e9	−0.662231	−0.331115	+	0.943590i	$0.607425\pi$
$450$	1.23740e9	0.640127
$451$	5.71990e8	0.293610
$452$	9.29623e8	0.473503
$453$	2.64546e8	0.133708
$454$	1.50729e9	0.755964
$455$	−1.09341e9	−0.544183
$456$	2.54586e8	0.125736
$457$	1.71850e9	0.842253	0.421126	−	0.907002i	$-0.361635\pi$
$457$	1.71850e9	0.842253	0.421126	+	0.907002i	$0.361635\pi$
$458$	9.00491e8	0.437976
$459$	7.09710e8	0.342560
$460$	−1.24573e8	−0.0596722
$461$	−1.62094e9	−0.770574	−0.385287	−	0.922797i	$-0.625898\pi$
$461$	−1.62094e9	−0.770574	−0.385287	+	0.922797i	$0.625898\pi$
$462$	5.73518e8	0.270583
$463$	−1.51310e9	−0.708490	−0.354245	−	0.935153i	$-0.615262\pi$
$463$	−1.51310e9	−0.708490	−0.354245	+	0.935153i	$0.615262\pi$
$464$	−2.02038e8	−0.0938902
$465$	1.56254e9	0.720685
$466$	−1.22156e8	−0.0559194
$467$	4.66037e8	0.211744	0.105872	−	0.994380i	$-0.466237\pi$
$467$	4.66037e8	0.211744	0.105872	+	0.994380i	$0.466237\pi$
$468$	−1.68835e9	−0.761382
$469$	−2.77629e9	−1.24268
$470$	8.65422e8	0.384490
$471$	2.47284e9	1.09049
$472$	7.29568e7	0.0319351
$473$	8.26410e8	0.359072
$474$	−2.51741e9	−1.08575
$475$	−3.45751e8	−0.148025
$476$	−5.43017e8	−0.230775
$477$	−6.35430e9	−2.68073
$478$	1.49902e9	0.627782
$479$	1.75752e9	0.730677	0.365339	−	0.930875i	$-0.380953\pi$
$479$	1.75752e9	0.730677	0.365339	+	0.930875i	$0.380953\pi$
$480$	3.95480e8	0.163223
$481$	4.23566e9	1.73545
$482$	−1.21867e9	−0.495704
$483$	6.47482e8	0.261464
$484$	−1.13993e9	−0.457004
$485$	−1.19493e9	−0.475605
$486$	−2.32332e9	−0.918084
$487$	−1.60783e9	−0.630794	−0.315397	−	0.948960i	$-0.602138\pi$
$487$	−1.60783e9	−0.630794	−0.315397	+	0.948960i	$0.602138\pi$
$488$	1.74609e9	0.680138
$489$	2.36827e9	0.915905
$490$	3.19603e8	0.122723
$491$	3.61019e9	1.37640	0.688199	−	0.725522i	$-0.258401\pi$
$491$	3.61019e9	1.37640	0.688199	+	0.725522i	$0.258401\pi$
$492$	−2.05007e9	−0.776053
$493$	5.47845e8	0.205918
$494$	4.71755e8	0.176065
$495$	6.61290e8	0.245061
$496$	−5.30293e8	−0.195133
$497$	2.98423e9	1.09040
$498$	−1.63394e9	−0.592833
$499$	6.25197e7	0.0225250	0.0112625	−	0.999937i	$-0.496415\pi$
$499$	6.25197e7	0.0225250	0.0112625	+	0.999937i	$0.496415\pi$
$500$	−1.36951e9	−0.489972
$501$	−2.19610e8	−0.0780227
$502$	1.98885e9	0.701679
$503$	−7.67606e8	−0.268937	−0.134468	−	0.990918i	$-0.542933\pi$
$503$	−7.67606e8	−0.268937	−0.134468	+	0.990918i	$0.542933\pi$
$504$	−1.20015e9	−0.417570
$505$	1.88576e9	0.651577
$506$	1.21079e8	0.0415472
$507$	−8.09492e8	−0.275858
$508$	−1.59281e9	−0.539065
$509$	−3.08040e9	−1.03537	−0.517684	−	0.855572i	$-0.673206\pi$
$509$	−3.08040e9	−1.03537	−0.517684	+	0.855572i	$0.673206\pi$
$510$	−1.07238e9	−0.357976
$511$	1.91534e9	0.634998
$512$	−1.34218e8	−0.0441942
$513$	−4.38286e8	−0.143333
$514$	−2.98437e9	−0.969353
$515$	2.87147e9	0.926358
$516$	−2.96194e9	−0.949078
$517$	−8.41146e8	−0.267704
$518$	3.01088e9	0.951787
$519$	1.65304e9	0.519038
$520$	7.32835e8	0.228557
$521$	5.34104e9	1.65460	0.827302	−	0.561758i	$-0.189875\pi$
$521$	5.34104e9	1.65460	0.827302	+	0.561758i	$0.189875\pi$
$522$	1.21082e9	0.372592
$523$	6.29313e9	1.92358	0.961792	−	0.273782i	$-0.0882747\pi$
$523$	6.29313e9	1.92358	0.961792	+	0.273782i	$0.0882747\pi$
$524$	7.79776e7	0.0236761
$525$	2.79162e9	0.841973
$526$	2.95770e8	0.0886144
$527$	1.43794e9	0.427960
$528$	−3.84387e8	−0.113645
$529$	−3.26813e9	−0.959853
$530$	2.75811e9	0.804721
$531$	−4.37233e8	−0.126731
$532$	3.35343e8	0.0965604
$533$	−3.79884e9	−1.08669
$534$	7.02034e9	1.99510
$535$	−1.26707e8	−0.0357735
$536$	1.86074e9	0.521927
$537$	−4.83082e9	−1.34620
$538$	2.46027e9	0.681153
$539$	−3.10638e8	−0.0854465
$540$	−6.80843e8	−0.186067
$541$	3.74563e9	1.01703	0.508516	−	0.861053i	$-0.330194\pi$
$541$	3.74563e9	1.01703	0.508516	+	0.861053i	$0.330194\pi$
$542$	−4.46861e9	−1.20552
$543$	9.59602e9	2.57212
$544$	3.63944e8	0.0969255
$545$	−2.48063e8	−0.0656408
$546$	−3.80898e9	−1.00146
$547$	−3.50690e9	−0.916152	−0.458076	−	0.888913i	$-0.651461\pi$
$547$	−3.50690e9	−0.916152	−0.458076	+	0.888913i	$0.651461\pi$
$548$	−1.50619e9	−0.390975
$549$	−1.04644e10	−2.69905
$550$	5.22031e8	0.133791
$551$	−3.38325e8	−0.0861596
$552$	−4.33959e8	−0.109815
$553$	−3.31596e9	−0.833818
$554$	−4.14762e9	−1.03637
$555$	5.94607e9	1.47640
$556$	4.50126e8	0.111064
$557$	−2.16099e9	−0.529859	−0.264929	−	0.964268i	$-0.585349\pi$
$557$	−2.16099e9	−0.529859	−0.264929	+	0.964268i	$0.585349\pi$
$558$	3.17807e9	0.774361
$559$	−5.48855e9	−1.32897
$560$	5.20930e8	0.125349
$561$	1.04230e9	0.249243
$562$	−5.46739e9	−1.29928
$563$	1.77561e9	0.419343	0.209671	−	0.977772i	$-0.432761\pi$
$563$	1.77561e9	0.419343	0.209671	+	0.977772i	$0.432761\pi$
$564$	3.01475e9	0.707579
$565$	2.41823e9	0.564063
$566$	3.31401e9	0.768238
$567$	−1.58768e9	−0.365782
$568$	−2.00011e9	−0.457968
$569$	7.34301e9	1.67102	0.835509	−	0.549477i	$-0.185173\pi$
$569$	7.34301e9	1.67102	0.835509	+	0.549477i	$0.185173\pi$
$570$	6.62256e8	0.149783
$571$	−5.26089e9	−1.18259	−0.591293	−	0.806457i	$-0.701382\pi$
$571$	−5.26089e9	−1.18259	−0.591293	+	0.806457i	$0.701382\pi$
$572$	−7.12278e8	−0.159134
$573$	−4.63504e8	−0.102923
$574$	−2.70037e9	−0.595980
$575$	5.89355e8	0.129282
$576$	8.04373e8	0.175379
$577$	−1.86698e9	−0.404598	−0.202299	−	0.979324i	$-0.564841\pi$
$577$	−1.86698e9	−0.404598	−0.202299	+	0.979324i	$0.564841\pi$
$578$	2.29584e9	0.494532
$579$	7.66144e9	1.64035
$580$	−5.25562e8	−0.111847
$581$	−2.15224e9	−0.455274
$582$	−4.16262e9	−0.875258
$583$	−2.68074e9	−0.560292
$584$	−1.28371e9	−0.266699
$585$	−4.39191e9	−0.907001
$586$	−3.34046e9	−0.685748
$587$	−4.57515e9	−0.933624	−0.466812	−	0.884357i	$-0.654597\pi$
$587$	−4.57515e9	−0.933624	−0.466812	+	0.884357i	$0.654597\pi$
$588$	1.11336e9	0.225847
$589$	−8.88007e8	−0.179066
$590$	1.89783e8	0.0380430
$591$	5.33948e9	1.06400
$592$	−2.01797e9	−0.399751
$593$	−2.49031e9	−0.490413	−0.245206	−	0.969471i	$-0.578856\pi$
$593$	−2.49031e9	−0.490413	−0.245206	+	0.969471i	$0.578856\pi$
$594$	6.61745e8	0.129550
$595$	−1.41255e9	−0.274912
$596$	4.52977e9	0.876424
$597$	−9.37894e9	−1.80403
$598$	−8.04137e8	−0.153771
$599$	7.43527e9	1.41352	0.706762	−	0.707452i	$-0.250155\pi$
$599$	7.43527e9	1.41352	0.706762	+	0.707452i	$0.250155\pi$
$600$	−1.87101e9	−0.353629
$601$	−1.14556e9	−0.215258	−0.107629	−	0.994191i	$-0.534326\pi$
$601$	−1.14556e9	−0.215258	−0.107629	+	0.994191i	$0.534326\pi$
$602$	−3.90149e9	−0.728858
$603$	−1.11515e10	−2.07120
$604$	−2.33548e8	−0.0431268
$605$	−2.96530e9	−0.544409
$606$	6.56916e9	1.19910
$607$	4.01272e9	0.728247	0.364124	−	0.931351i	$-0.381369\pi$
$607$	4.01272e9	0.728247	0.364124	+	0.931351i	$0.381369\pi$
$608$	−2.24756e8	−0.0405554
$609$	2.73166e9	0.490079
$610$	4.54210e9	0.810219
$611$	5.58641e9	0.990807
$612$	−2.18113e9	−0.384638
$613$	1.66749e9	0.292383	0.146192	−	0.989256i	$-0.453298\pi$
$613$	1.66749e9	0.292383	0.146192	+	0.989256i	$0.453298\pi$
$614$	−1.16211e9	−0.202608
$615$	−5.33285e9	−0.924478
$616$	−5.06317e8	−0.0872751
$617$	−3.52420e9	−0.604035	−0.302018	−	0.953302i	$-0.597660\pi$
$617$	−3.52420e9	−0.604035	−0.302018	+	0.953302i	$0.597660\pi$
$618$	1.00029e10	1.70478
$619$	2.13210e9	0.361318	0.180659	−	0.983546i	$-0.442177\pi$
$619$	2.13210e9	0.361318	0.180659	+	0.983546i	$0.442177\pi$
$620$	−1.37945e9	−0.232453
$621$	7.47086e8	0.125184
$622$	−4.06260e9	−0.676921
$623$	9.24725e9	1.53216
$624$	2.55288e9	0.420614
$625$	3.75637e8	0.0615444
$626$	−7.19077e9	−1.17156
$627$	−6.43679e8	−0.104288
$628$	−2.18309e9	−0.351733
$629$	5.47192e9	0.876723
$630$	−3.12196e9	−0.497433
$631$	−7.86195e9	−1.24574	−0.622870	−	0.782325i	$-0.714034\pi$
$631$	−7.86195e9	−1.24574	−0.622870	+	0.782325i	$0.714034\pi$
$632$	2.22244e9	0.350203
$633$	−1.13454e10	−1.77790
$634$	−6.03003e8	−0.0939738
$635$	−4.14338e9	−0.642165
$636$	9.60804e9	1.48093
$637$	2.06308e9	0.316249
$638$	5.10819e8	0.0778744
$639$	1.19868e10	1.81739
$640$	−3.49141e8	−0.0526466
$641$	9.58185e9	1.43697	0.718483	−	0.695545i	$-0.244837\pi$
$641$	9.58185e9	1.43697	0.718483	+	0.695545i	$0.244837\pi$
$642$	−4.41391e8	−0.0658342
$643$	−5.89382e8	−0.0874295	−0.0437148	−	0.999044i	$-0.513919\pi$
$643$	−5.89382e8	−0.0874295	−0.0437148	+	0.999044i	$0.513919\pi$
$644$	−5.71614e8	−0.0843340
$645$	−7.70489e9	−1.13060
$646$	6.09446e8	0.0889450
$647$	2.40894e9	0.349672	0.174836	−	0.984598i	$-0.444060\pi$
$647$	2.40894e9	0.349672	0.174836	+	0.984598i	$0.444060\pi$
$648$	1.06410e9	0.153629
$649$	−1.84459e8	−0.0264876
$650$	−3.46703e9	−0.495178
$651$	7.16983e9	1.01853
$652$	−2.09077e9	−0.295420
$653$	5.50718e9	0.773986	0.386993	−	0.922083i	$-0.373514\pi$
$653$	5.50718e9	0.773986	0.386993	+	0.922083i	$0.373514\pi$
$654$	−8.64142e8	−0.120799
$655$	2.02843e8	0.0282043
$656$	1.80986e9	0.250312
$657$	7.69333e9	1.05836
$658$	3.97106e9	0.543395
$659$	−1.21064e10	−1.64784	−0.823919	−	0.566708i	$-0.808217\pi$
$659$	−1.21064e10	−1.64784	−0.823919	+	0.566708i	$0.808217\pi$
$660$	−9.99905e8	−0.135380
$661$	−7.66197e9	−1.03190	−0.515948	−	0.856620i	$-0.672560\pi$
$661$	−7.66197e9	−1.03190	−0.515948	+	0.856620i	$0.672560\pi$
$662$	−1.62673e8	−0.0217928
$663$	−6.92237e9	−0.922480
$664$	1.44248e9	0.191215
$665$	8.72328e8	0.115028
$666$	1.20938e10	1.58636
$667$	5.76697e8	0.0752501
$668$	1.93878e8	0.0251658
$669$	−6.00976e9	−0.776008
$670$	4.84035e9	0.621749
$671$	−4.41469e9	−0.564120
$672$	1.81469e9	0.230680
$673$	3.64962e9	0.461525	0.230763	−	0.973010i	$-0.425878\pi$
$673$	3.64962e9	0.461525	0.230763	+	0.973010i	$0.425878\pi$
$674$	−3.67703e9	−0.462581
$675$	3.22106e9	0.403121
$676$	7.14642e8	0.0889764
$677$	−2.64750e9	−0.327926	−0.163963	−	0.986466i	$-0.552428\pi$
$677$	−2.64750e9	−0.327926	−0.163963	+	0.986466i	$0.552428\pi$
$678$	8.42406e9	1.03805
$679$	−5.48303e9	−0.672166
$680$	9.46727e8	0.115463
$681$	1.36587e10	1.65728
$682$	1.34076e9	0.161847
$683$	−7.02187e9	−0.843296	−0.421648	−	0.906760i	$-0.638548\pi$
$683$	−7.02187e9	−0.843296	−0.421648	+	0.906760i	$0.638548\pi$
$684$	1.34697e9	0.160939
$685$	−3.91806e9	−0.465751
$686$	6.49950e9	0.768680
$687$	8.16007e9	0.960164
$688$	2.61488e9	0.306120
$689$	1.78039e10	2.07371
$690$	−1.12886e9	−0.130818
$691$	−1.10917e10	−1.27886	−0.639432	−	0.768848i	$-0.720830\pi$
$691$	−1.10917e10	−1.27886	−0.639432	+	0.768848i	$0.720830\pi$
$692$	−1.45935e9	−0.167413
$693$	3.03438e9	0.346341
$694$	1.05743e10	1.20087
$695$	1.17091e9	0.132305
$696$	−1.83083e9	−0.205833
$697$	−4.90760e9	−0.548977
$698$	−6.30376e8	−0.0701626
$699$	−1.10695e9	−0.122591
$700$	−2.46451e9	−0.271574
$701$	2.64241e9	0.289725	0.144863	−	0.989452i	$-0.453726\pi$
$701$	2.64241e9	0.289725	0.144863	+	0.989452i	$0.453726\pi$
$702$	−4.39493e9	−0.479482
$703$	−3.37922e9	−0.366836
$704$	3.39347e8	0.0366555
$705$	7.84228e9	0.842908
$706$	−6.28544e9	−0.672232
$707$	8.65295e9	0.920865
$708$	6.61120e8	0.0700106
$709$	−1.32706e10	−1.39839	−0.699195	−	0.714931i	$-0.746458\pi$
$709$	−1.32706e10	−1.39839	−0.699195	+	0.714931i	$0.746458\pi$
$710$	−5.20289e9	−0.545557
$711$	−1.33192e10	−1.38974
$712$	−6.19775e9	−0.643508
$713$	1.51367e9	0.156393
$714$	−4.92071e9	−0.505922
$715$	−1.85285e9	−0.189570
$716$	4.26477e9	0.434211
$717$	1.35838e10	1.37627
$718$	−7.87100e8	−0.0793587
$719$	1.31391e10	1.31831	0.659153	−	0.752009i	$-0.270915\pi$
$719$	1.31391e10	1.31831	0.659153	+	0.752009i	$0.270915\pi$
$720$	2.09241e9	0.208922
$721$	1.31760e10	1.30921
$722$	−3.76367e8	−0.0372161
$723$	−1.10434e10	−1.08672
$724$	−8.47162e9	−0.829624
$725$	2.48643e9	0.242322
$726$	−1.03298e10	−1.00188
$727$	1.30042e10	1.25520	0.627599	−	0.778537i	$-0.284038\pi$
$727$	1.30042e10	1.25520	0.627599	+	0.778537i	$0.284038\pi$
$728$	3.36267e9	0.323016
$729$	−1.65082e10	−1.57817
$730$	−3.33931e9	−0.317707
$731$	−7.09049e9	−0.671375
$732$	1.58227e10	1.49105
$733$	−4.31002e9	−0.404218	−0.202109	−	0.979363i	$-0.564780\pi$
$733$	−4.31002e9	−0.404218	−0.202109	+	0.979363i	$0.564780\pi$
$734$	−9.26167e9	−0.864477
$735$	2.89618e9	0.269042
$736$	3.83111e8	0.0354203
$737$	−4.70457e9	−0.432896
$738$	−1.08466e10	−0.993333
$739$	−2.75286e9	−0.250916	−0.125458	−	0.992099i	$-0.540040\pi$
$739$	−2.75286e9	−0.250916	−0.125458	+	0.992099i	$0.540040\pi$
$740$	−5.24935e9	−0.476206
$741$	4.27495e9	0.385982
$742$	1.26558e10	1.13730
$743$	−1.09216e10	−0.976848	−0.488424	−	0.872606i	$-0.662428\pi$
$743$	−1.09216e10	−0.976848	−0.488424	+	0.872606i	$0.662428\pi$
$744$	−4.80541e9	−0.427784
$745$	1.17833e10	1.04405
$746$	−1.87088e8	−0.0164991
$747$	−8.64487e9	−0.758816
$748$	−9.20170e8	−0.0803920
$749$	−5.81404e8	−0.0505582
$750$	−1.24103e10	−1.07415
$751$	−1.71909e10	−1.48101	−0.740506	−	0.672050i	$-0.765414\pi$
$751$	−1.71909e10	−1.48101	−0.740506	+	0.672050i	$0.765414\pi$
$752$	−2.66151e9	−0.228226
$753$	1.80225e10	1.53827
$754$	−3.39257e9	−0.288223
$755$	−6.07529e8	−0.0513751
$756$	−3.12410e9	−0.262966
$757$	−1.50982e10	−1.26500	−0.632498	−	0.774562i	$-0.717971\pi$
$757$	−1.50982e10	−1.26500	−0.632498	+	0.774562i	$0.717971\pi$
$758$	−1.37614e10	−1.14768
$759$	1.09719e9	0.0910828
$760$	−5.84657e8	−0.0483118
$761$	−1.76556e10	−1.45223	−0.726116	−	0.687573i	$-0.758676\pi$
$761$	−1.76556e10	−1.45223	−0.726116	+	0.687573i	$0.758676\pi$
$762$	−1.44337e10	−1.18178
$763$	−1.13826e9	−0.0927692
$764$	4.09194e8	0.0331972
$765$	−5.67378e9	−0.458202
$766$	9.90095e9	0.795933
$767$	1.22507e9	0.0980342
$768$	−1.21625e9	−0.0968858
$769$	−7.19361e9	−0.570433	−0.285216	−	0.958463i	$-0.592065\pi$
$769$	−7.19361e9	−0.570433	−0.285216	+	0.958463i	$0.592065\pi$
$770$	−1.31708e9	−0.103967
$771$	−2.70438e10	−2.12509
$772$	−6.76373e9	−0.529085
$773$	9.46343e9	0.736920	0.368460	−	0.929644i	$-0.379885\pi$
$773$	9.46343e9	0.736920	0.368460	+	0.929644i	$0.379885\pi$
$774$	−1.56711e10	−1.21480
$775$	6.52617e9	0.503619
$776$	3.67487e9	0.282310
$777$	2.72840e10	2.08658
$778$	1.48001e10	1.12677
$779$	3.03072e9	0.229702
$780$	6.64080e9	0.501060
$781$	5.05694e9	0.379848
$782$	−1.03884e9	−0.0776828
$783$	3.15188e9	0.234641
$784$	−9.82904e8	−0.0728458
$785$	−5.67887e9	−0.419004
$786$	7.06617e8	0.0519045
$787$	−3.70164e9	−0.270696	−0.135348	−	0.990798i	$-0.543215\pi$
$787$	−3.70164e9	−0.270696	−0.135348	+	0.990798i	$0.543215\pi$
$788$	−4.71384e9	−0.343189
$789$	2.68021e9	0.194267
$790$	5.78124e9	0.417182
$791$	1.10962e10	0.797183
$792$	−2.03372e9	−0.145463
$793$	2.93199e10	2.08788
$794$	−7.34950e9	−0.521058
$795$	2.49934e10	1.76417
$796$	8.27998e9	0.581880
$797$	2.38156e10	1.66631	0.833157	−	0.553036i	$-0.186531\pi$
$797$	2.38156e10	1.66631	0.833157	+	0.553036i	$0.186531\pi$
$798$	3.03881e9	0.211687
$799$	7.21692e9	0.500539
$800$	1.65178e9	0.114061
$801$	3.71434e10	2.55369
$802$	1.60759e10	1.10044
$803$	3.24564e9	0.221206
$804$	1.68617e10	1.14421
$805$	−1.48694e9	−0.100463
$806$	−8.90454e9	−0.599017
$807$	2.22945e10	1.49327
$808$	−5.79943e9	−0.386763
$809$	−2.33184e10	−1.54838	−0.774191	−	0.632952i	$-0.781843\pi$
$809$	−2.33184e10	−1.54838	−0.774191	+	0.632952i	$0.781843\pi$
$810$	2.76806e9	0.183011
$811$	1.67858e10	1.10502	0.552509	−	0.833507i	$-0.313670\pi$
$811$	1.67858e10	1.10502	0.552509	+	0.833507i	$0.313670\pi$
$812$	−2.41158e9	−0.158072
$813$	−4.04937e10	−2.64284
$814$	5.10210e9	0.331561
$815$	−5.43873e9	−0.351922
$816$	3.29799e9	0.212487
$817$	4.37877e9	0.280915
$818$	1.91147e10	1.22104
$819$	−2.01526e10	−1.28185
$820$	4.70799e9	0.298186
$821$	−2.25645e10	−1.42306	−0.711532	−	0.702653i	$-0.751998\pi$
$821$	−2.25645e10	−1.42306	−0.711532	+	0.702653i	$0.751998\pi$
$822$	−1.36488e10	−0.857124
$823$	4.51535e9	0.282353	0.141176	−	0.989984i	$-0.454912\pi$
$823$	4.51535e9	0.282353	0.141176	+	0.989984i	$0.454912\pi$
$824$	−8.83087e9	−0.549868
$825$	4.73054e9	0.293307
$826$	8.70833e8	0.0537656
$827$	8.93847e9	0.549533	0.274767	−	0.961511i	$-0.411399\pi$
$827$	8.93847e9	0.549533	0.274767	+	0.961511i	$0.411399\pi$
$828$	−2.29600e9	−0.140561
$829$	9.36466e9	0.570888	0.285444	−	0.958395i	$-0.407859\pi$
$829$	9.36466e9	0.570888	0.285444	+	0.958395i	$0.407859\pi$
$830$	3.75233e9	0.227787
$831$	−3.75849e10	−2.27201
$832$	−2.25375e9	−0.135667
$833$	2.66523e9	0.159764
$834$	4.07895e9	0.243482
$835$	5.04335e8	0.0299789
$836$	5.68257e8	0.0336374
$837$	8.27280e9	0.487656
$838$	7.42785e9	0.436023
$839$	−1.43659e10	−0.839780	−0.419890	−	0.907575i	$-0.637931\pi$
$839$	−1.43659e10	−0.839780	−0.419890	+	0.907575i	$0.637931\pi$
$840$	4.72056e9	0.274800
$841$	−1.48169e10	−0.858954
$842$	−6.54603e8	−0.0377908
$843$	−4.95444e10	−2.84838
$844$	1.00161e10	0.573453
$845$	1.85900e9	0.105994
$846$	1.59505e10	0.905688
$847$	−1.36065e10	−0.769406
$848$	−8.48224e9	−0.477667
$849$	3.00309e10	1.68419
$850$	−4.47896e9	−0.250156
$851$	5.76009e9	0.320388
$852$	−1.81246e10	−1.00399
$853$	−5.03324e9	−0.277668	−0.138834	−	0.990316i	$-0.544335\pi$
$853$	−5.03324e9	−0.277668	−0.138834	+	0.990316i	$0.544335\pi$
$854$	2.08418e10	1.14507
$855$	3.50388e9	0.191720
$856$	3.89672e8	0.0212345
$857$	−2.80085e10	−1.52004	−0.760022	−	0.649897i	$-0.774812\pi$
$857$	−2.80085e10	−1.52004	−0.760022	+	0.649897i	$0.774812\pi$
$858$	−6.45452e9	−0.348866
$859$	3.20398e10	1.72470	0.862350	−	0.506313i	$-0.168992\pi$
$859$	3.20398e10	1.72470	0.862350	+	0.506313i	$0.168992\pi$
$860$	6.80208e9	0.364668
$861$	−2.44702e10	−1.30655
$862$	1.72781e10	0.918801
$863$	2.15424e10	1.14092	0.570462	−	0.821324i	$-0.306764\pi$
$863$	2.15424e10	1.14092	0.570462	+	0.821324i	$0.306764\pi$
$864$	2.09385e9	0.110446
$865$	−3.79621e9	−0.199432
$866$	−1.52798e10	−0.799478
$867$	2.08045e10	1.08415
$868$	−6.32972e9	−0.328523
$869$	−5.61907e9	−0.290466
$870$	−4.76254e9	−0.245200
$871$	3.12451e10	1.60221
$872$	7.62888e8	0.0389631
$873$	−2.20237e10	−1.12031
$874$	6.41542e8	0.0325039
$875$	−1.63469e10	−0.824911
$876$	−1.16327e10	−0.584678
$877$	2.11263e10	1.05761	0.528805	−	0.848743i	$-0.322640\pi$
$877$	2.11263e10	1.05761	0.528805	+	0.848743i	$0.322640\pi$
$878$	1.49421e10	0.745042
$879$	−3.02706e10	−1.50335
$880$	8.82743e8	0.0436662
$881$	3.56910e10	1.75850	0.879250	−	0.476360i	$-0.158044\pi$
$881$	3.56910e10	1.75850	0.879250	+	0.476360i	$0.158044\pi$
$882$	5.89059e9	0.289080
$883$	−2.49535e10	−1.21974	−0.609872	−	0.792500i	$-0.708779\pi$
$883$	−2.49535e10	−1.21974	−0.609872	+	0.792500i	$0.708779\pi$
$884$	6.11125e9	0.297541
$885$	1.71977e9	0.0834006
$886$	−8.53523e9	−0.412285
$887$	2.79507e10	1.34481	0.672404	−	0.740185i	$-0.265262\pi$
$887$	2.79507e10	1.34481	0.672404	+	0.740185i	$0.265262\pi$
$888$	−1.82865e10	−0.876363
$889$	−1.90122e10	−0.907563
$890$	−1.61222e10	−0.766583
$891$	−2.69041e9	−0.127423
$892$	5.30558e9	0.250297
$893$	−4.45685e9	−0.209434
$894$	4.10479e10	1.92136
$895$	1.10940e10	0.517256
$896$	−1.60206e9	−0.0744048
$897$	−7.28693e9	−0.337109
$898$	1.01616e10	0.468268
$899$	6.38600e9	0.293137
$900$	−9.89919e9	−0.452638
$901$	2.30004e10	1.04761
$902$	−4.57592e9	−0.207614
$903$	−3.53545e10	−1.59786
$904$	−7.43698e9	−0.334817
$905$	−2.20372e10	−0.988296
$906$	−2.11637e9	−0.0945459
$907$	2.19699e10	0.977693	0.488847	−	0.872370i	$-0.337418\pi$
$907$	2.19699e10	0.977693	0.488847	+	0.872370i	$0.337418\pi$
$908$	−1.20583e10	−0.534547
$909$	3.47562e10	1.53483
$910$	8.74732e9	0.384796
$911$	−5.96748e9	−0.261503	−0.130752	−	0.991415i	$-0.541739\pi$
$911$	−5.96748e9	−0.261503	−0.130752	+	0.991415i	$0.541739\pi$
$912$	−2.03669e9	−0.0889085
$913$	−3.64708e9	−0.158598
$914$	−1.37480e10	−0.595562
$915$	4.11596e10	1.77622
$916$	−7.20393e9	−0.309696
$917$	9.30762e8	0.0398608
$918$	−5.67768e9	−0.242226
$919$	−2.72390e10	−1.15768	−0.578839	−	0.815442i	$-0.696494\pi$
$919$	−2.72390e10	−1.15768	−0.578839	+	0.815442i	$0.696494\pi$
$920$	9.96586e8	0.0421946
$921$	−1.05308e10	−0.444172
$922$	1.29675e10	0.544878
$923$	−3.35853e10	−1.40587
$924$	−4.58815e9	−0.191331
$925$	2.48346e10	1.03172
$926$	1.21048e10	0.500978
$927$	5.29238e10	2.18209
$928$	1.61630e9	0.0663904
$929$	−3.27908e9	−0.134183	−0.0670914	−	0.997747i	$-0.521372\pi$
$929$	−3.27908e9	−0.134183	−0.0670914	+	0.997747i	$0.521372\pi$
$930$	−1.25003e10	−0.509601
$931$	−1.64593e9	−0.0668479
$932$	9.77245e8	0.0395410
$933$	−3.68145e10	−1.48400
$934$	−3.72830e9	−0.149726
$935$	−2.39364e9	−0.0957675
$936$	1.35068e10	0.538378
$937$	−1.83287e10	−0.727852	−0.363926	−	0.931428i	$-0.618564\pi$
$937$	−1.83287e10	−0.727852	−0.363926	+	0.931428i	$0.618564\pi$
$938$	2.22103e10	0.878709
$939$	−6.51613e10	−2.56839
$940$	−6.92337e9	−0.271876
$941$	−1.88117e10	−0.735975	−0.367988	−	0.929831i	$-0.619953\pi$
$941$	−1.88117e10	−0.735975	−0.367988	+	0.929831i	$0.619953\pi$
$942$	−1.97827e10	−0.771095
$943$	−5.16606e9	−0.200617
$944$	−5.83654e8	−0.0225816
$945$	−8.12673e9	−0.313260
$946$	−6.61128e9	−0.253902
$947$	2.11622e10	0.809721	0.404860	−	0.914378i	$-0.367320\pi$
$947$	2.11622e10	0.809721	0.404860	+	0.914378i	$0.367320\pi$
$948$	2.01393e10	0.767742
$949$	−2.15557e10	−0.818711
$950$	2.76601e9	0.104670
$951$	−5.46429e9	−0.206017
$952$	4.34413e9	0.163183
$953$	7.66744e9	0.286963	0.143481	−	0.989653i	$-0.454170\pi$
$953$	7.66744e9	0.286963	0.143481	+	0.989653i	$0.454170\pi$
$954$	5.08344e10	1.89556
$955$	1.06444e9	0.0395465
$956$	−1.19921e10	−0.443909
$957$	4.62894e9	0.170722
$958$	−1.40602e10	−0.516667
$959$	−1.79783e10	−0.658240
$960$	−3.16384e9	−0.115416
$961$	−1.07512e10	−0.390772
$962$	−3.38853e10	−1.22715
$963$	−2.33532e9	−0.0842665
$964$	9.74938e9	0.350516
$965$	−1.75945e10	−0.630276
$966$	−5.17985e9	−0.184883
$967$	−3.49256e10	−1.24208	−0.621042	−	0.783777i	$-0.713290\pi$
$967$	−3.49256e10	−1.24208	−0.621042	+	0.783777i	$0.713290\pi$
$968$	9.11945e9	0.323151
$969$	5.52268e9	0.194992
$970$	9.55944e9	0.336304
$971$	4.44861e10	1.55940	0.779700	−	0.626154i	$-0.215372\pi$
$971$	4.44861e10	1.55940	0.779700	+	0.626154i	$0.215372\pi$
$972$	1.85866e10	0.649183
$973$	5.37283e9	0.186986
$974$	1.28626e10	0.446039
$975$	−3.14176e10	−1.08557
$976$	−1.39687e10	−0.480930
$977$	−3.00033e10	−1.02929	−0.514645	−	0.857403i	$-0.672076\pi$
$977$	−3.00033e10	−1.02929	−0.514645	+	0.857403i	$0.672076\pi$
$978$	−1.89462e10	−0.647643
$979$	1.56700e10	0.533738
$980$	−2.55683e9	−0.0867781
$981$	−4.57202e9	−0.154620
$982$	−2.88815e10	−0.973261
$983$	2.91371e10	0.978383	0.489192	−	0.872176i	$-0.337292\pi$
$983$	2.91371e10	0.978383	0.489192	+	0.872176i	$0.337292\pi$
$984$	1.64006e10	0.548752
$985$	−1.22621e10	−0.408826
$986$	−4.38276e9	−0.145606
$987$	3.59849e10	1.19127
$988$	−3.77404e9	−0.124497
$989$	−7.46390e9	−0.245346
$990$	−5.29032e9	−0.173284
$991$	−3.89254e10	−1.27050	−0.635250	−	0.772306i	$-0.719103\pi$
$991$	−3.89254e10	−1.27050	−0.635250	+	0.772306i	$0.719103\pi$
$992$	4.24234e9	0.137980
$993$	−1.47411e9	−0.0477758
$994$	−2.38739e10	−0.771029
$995$	2.15387e10	0.693168
$996$	1.30715e10	0.419197
$997$	7.94798e9	0.253994	0.126997	−	0.991903i	$-0.459466\pi$
$997$	7.94798e9	0.253994	0.126997	+	0.991903i	$0.459466\pi$
$998$	−5.00158e8	−0.0159276
$999$	3.14812e10	0.999016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.a.c.1.1	✓	2
3.2	odd	2		342.8.a.i.1.1		2
4.3	odd	2		304.8.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.c.1.1	✓	2	1.1	even	1	trivial
304.8.a.b.1.2		2	4.3	odd	2
342.8.a.i.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -72.4944 q^{3} +64.0000 q^{4} +166.483 q^{5} +579.955 q^{6} +763.922 q^{7} -512.000 q^{8} +3068.44 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -72.4944 q^{3} +64.0000 q^{4} +166.483 q^{5} +579.955 q^{6} +763.922 q^{7} -512.000 q^{8} +3068.44 q^{9} -1331.87 q^{10} +1294.51 q^{11} -4639.64 q^{12} -8597.37 q^{13} -6111.37 q^{14} -12069.1 q^{15} +4096.00 q^{16} -11106.7 q^{17} -24547.5 q^{18} +6859.00 q^{19} +10654.9 q^{20} -55380.0 q^{21} -10356.0 q^{22} -11691.6 q^{23} +37117.1 q^{24} -50408.3 q^{25} +68779.0 q^{26} -63899.4 q^{27} +48891.0 q^{28} -49325.7 q^{29} +96552.8 q^{30} -129466. q^{31} -32768.0 q^{32} -93844.4 q^{33} +88853.5 q^{34} +127180. q^{35} +196380. q^{36} -492669. q^{37} -54872.0 q^{38} +623261. q^{39} -85239.4 q^{40} +441860. q^{41} +443040. q^{42} +638398. q^{43} +82848.4 q^{44} +510843. q^{45} +93532.9 q^{46} -649782. q^{47} -296937. q^{48} -239967. q^{49} +403267. q^{50} +805172. q^{51} -550232. q^{52} -2.07086e6 q^{53} +511195. q^{54} +215513. q^{55} -391128. q^{56} -497239. q^{57} +394606. q^{58} -142494. q^{59} -772422. q^{60} -3.41033e6 q^{61} +1.03573e6 q^{62} +2.34405e6 q^{63} +262144. q^{64} -1.43132e6 q^{65} +750755. q^{66} -3.63426e6 q^{67} -710828. q^{68} +847576. q^{69} -1.01744e6 q^{70} +3.90647e6 q^{71} -1.57104e6 q^{72} +2.50725e6 q^{73} +3.94135e6 q^{74} +3.65432e6 q^{75} +438976. q^{76} +988901. q^{77} -4.98609e6 q^{78} -4.34070e6 q^{79} +681915. q^{80} -2.07833e6 q^{81} -3.53488e6 q^{82} -2.81735e6 q^{83} -3.54432e6 q^{84} -1.84908e6 q^{85} -5.10718e6 q^{86} +3.57584e6 q^{87} -662787. q^{88} +1.21050e7 q^{89} -4.08675e6 q^{90} -6.56772e6 q^{91} -748263. q^{92} +9.38556e6 q^{93} +5.19825e6 q^{94} +1.14191e6 q^{95} +2.37550e6 q^{96} -7.17748e6 q^{97} +1.91973e6 q^{98} +3.97211e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} - 11 q^{3} + 128 q^{4} - 69 q^{5} + 88 q^{6} - 348 q^{7} - 1024 q^{8} + 4663 q^{9}+O(q^{10}) \) 2 * q - 16 * q^2 - 11 * q^3 + 128 * q^4 - 69 * q^5 + 88 * q^6 - 348 * q^7 - 1024 * q^8 + 4663 * q^9	\( 2 q - 16 q^{2} - 11 q^{3} + 128 q^{4} - 69 q^{5} + 88 q^{6} - 348 q^{7} - 1024 q^{8} + 4663 q^{9} + 552 q^{10} + 2723 q^{11} - 704 q^{12} - 14113 q^{13} + 2784 q^{14} - 26550 q^{15} + 8192 q^{16} - 38560 q^{17} - 37304 q^{18} + 13718 q^{19} - 4416 q^{20} - 123757 q^{21} - 21784 q^{22} - 73897 q^{23} + 5632 q^{24} - 73081 q^{25} + 112904 q^{26} - 100331 q^{27} - 22272 q^{28} + 159813 q^{29} + 212400 q^{30} - 259468 q^{31} - 65536 q^{32} - 6000 q^{33} + 308480 q^{34} + 389019 q^{35} + 298432 q^{36} - 528168 q^{37} - 109744 q^{38} + 284081 q^{39} + 35328 q^{40} + 1005650 q^{41} + 990056 q^{42} + 286217 q^{43} + 174272 q^{44} + 135351 q^{45} + 591176 q^{46} - 1397509 q^{47} - 45056 q^{48} + 172860 q^{49} + 584648 q^{50} - 883053 q^{51} - 903232 q^{52} - 1385969 q^{53} + 802648 q^{54} - 120873 q^{55} + 178176 q^{56} - 75449 q^{57} - 1278504 q^{58} + 2700953 q^{59} - 1699200 q^{60} - 3975947 q^{61} + 2075744 q^{62} + 571019 q^{63} + 524288 q^{64} - 132480 q^{65} + 48000 q^{66} - 134557 q^{67} - 2467840 q^{68} - 2977707 q^{69} - 3112152 q^{70} + 4202740 q^{71} - 2387456 q^{72} + 900498 q^{73} + 4225344 q^{74} + 2260081 q^{75} + 877952 q^{76} - 599473 q^{77} - 2272648 q^{78} - 6893730 q^{79} - 282624 q^{80} - 7805978 q^{81} - 8045200 q^{82} - 2465330 q^{83} - 7920448 q^{84} + 4615719 q^{85} - 2289736 q^{86} + 16436697 q^{87} - 1394176 q^{88} + 17431724 q^{89} - 1082808 q^{90} - 434771 q^{91} - 4729408 q^{92} + 1391168 q^{93} + 11180072 q^{94} - 473271 q^{95} + 360448 q^{96} + 6351934 q^{97} - 1382880 q^{98} + 6249933 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 11 * q^3 + 128 * q^4 - 69 * q^5 + 88 * q^6 - 348 * q^7 - 1024 * q^8 + 4663 * q^9 + 552 * q^10 + 2723 * q^11 - 704 * q^12 - 14113 * q^13 + 2784 * q^14 - 26550 * q^15 + 8192 * q^16 - 38560 * q^17 - 37304 * q^18 + 13718 * q^19 - 4416 * q^20 - 123757 * q^21 - 21784 * q^22 - 73897 * q^23 + 5632 * q^24 - 73081 * q^25 + 112904 * q^26 - 100331 * q^27 - 22272 * q^28 + 159813 * q^29 + 212400 * q^30 - 259468 * q^31 - 65536 * q^32 - 6000 * q^33 + 308480 * q^34 + 389019 * q^35 + 298432 * q^36 - 528168 * q^37 - 109744 * q^38 + 284081 * q^39 + 35328 * q^40 + 1005650 * q^41 + 990056 * q^42 + 286217 * q^43 + 174272 * q^44 + 135351 * q^45 + 591176 * q^46 - 1397509 * q^47 - 45056 * q^48 + 172860 * q^49 + 584648 * q^50 - 883053 * q^51 - 903232 * q^52 - 1385969 * q^53 + 802648 * q^54 - 120873 * q^55 + 178176 * q^56 - 75449 * q^57 - 1278504 * q^58 + 2700953 * q^59 - 1699200 * q^60 - 3975947 * q^61 + 2075744 * q^62 + 571019 * q^63 + 524288 * q^64 - 132480 * q^65 + 48000 * q^66 - 134557 * q^67 - 2467840 * q^68 - 2977707 * q^69 - 3112152 * q^70 + 4202740 * q^71 - 2387456 * q^72 + 900498 * q^73 + 4225344 * q^74 + 2260081 * q^75 + 877952 * q^76 - 599473 * q^77 - 2272648 * q^78 - 6893730 * q^79 - 282624 * q^80 - 7805978 * q^81 - 8045200 * q^82 - 2465330 * q^83 - 7920448 * q^84 + 4615719 * q^85 - 2289736 * q^86 + 16436697 * q^87 - 1394176 * q^88 + 17431724 * q^89 - 1082808 * q^90 - 434771 * q^91 - 4729408 * q^92 + 1391168 * q^93 + 11180072 * q^94 - 473271 * q^95 + 360448 * q^96 + 6351934 * q^97 - 1382880 * q^98 + 6249933 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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