LMFDB - Embedded newform 38.8.a.b.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2737}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 684 $ x^2 - x - 684
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			$26.6582$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} -56.6582 q^{3} +64.0000 q^{4} -304.873 q^{5} +453.265 q^{6} -1452.95 q^{7} -512.000 q^{8} +1023.15 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -56.6582 q^{3} +64.0000 q^{4} -304.873 q^{5} +453.265 q^{6} -1452.95 q^{7} -512.000 q^{8} +1023.15 q^{9} +2438.98 q^{10} -6618.68 q^{11} -3626.12 q^{12} +9599.26 q^{13} +11623.6 q^{14} +17273.5 q^{15} +4096.00 q^{16} +12453.8 q^{17} -8185.19 q^{18} -6859.00 q^{19} -19511.8 q^{20} +82321.4 q^{21} +52949.4 q^{22} -1184.25 q^{23} +29009.0 q^{24} +14822.3 q^{25} -76794.1 q^{26} +65941.7 q^{27} -92988.7 q^{28} -84164.1 q^{29} -138188. q^{30} +28058.5 q^{31} -32768.0 q^{32} +375002. q^{33} -99630.1 q^{34} +442964. q^{35} +65481.5 q^{36} -412795. q^{37} +54872.0 q^{38} -543877. q^{39} +156095. q^{40} -539470. q^{41} -658571. q^{42} +318505. q^{43} -423596. q^{44} -311930. q^{45} +9474.02 q^{46} -1.24384e6 q^{47} -232072. q^{48} +1.28752e6 q^{49} -118578. q^{50} -705608. q^{51} +614353. q^{52} +2.06709e6 q^{53} -527534. q^{54} +2.01785e6 q^{55} +743910. q^{56} +388618. q^{57} +673313. q^{58} -2.06604e6 q^{59} +1.10551e6 q^{60} +1.56677e6 q^{61} -224468. q^{62} -1.48658e6 q^{63} +262144. q^{64} -2.92655e6 q^{65} -3.00002e6 q^{66} +2.00776e6 q^{67} +797041. q^{68} +67097.6 q^{69} -3.54371e6 q^{70} +1.70325e6 q^{71} -523852. q^{72} +4.48077e6 q^{73} +3.30236e6 q^{74} -839804. q^{75} -438976. q^{76} +9.61661e6 q^{77} +4.35101e6 q^{78} -8.64298e6 q^{79} -1.24876e6 q^{80} -5.97376e6 q^{81} +4.31576e6 q^{82} -4.34087e6 q^{83} +5.26857e6 q^{84} -3.79681e6 q^{85} -2.54804e6 q^{86} +4.76859e6 q^{87} +3.38876e6 q^{88} +1.69658e6 q^{89} +2.49544e6 q^{90} -1.39472e7 q^{91} -75792.2 q^{92} -1.58974e6 q^{93} +9.95071e6 q^{94} +2.09112e6 q^{95} +1.85658e6 q^{96} +4.33665e6 q^{97} -1.03001e7 q^{98} -6.77189e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{2} - 61 q^{3} + 128 q^{4} + 175 q^{5} + 488 q^{6} - 2592 q^{7} - 1024 q^{8} - 1145 q^{9}+O(q^{10}) $ 2 * q - 16 * q^2 - 61 * q^3 + 128 * q^4 + 175 * q^5 + 488 * q^6 - 2592 * q^7 - 1024 * q^8 - 1145 * q^9	\( 2 q - 16 q^{2} - 61 q^{3} + 128 q^{4} + 175 q^{5} + 488 q^{6} - 2592 q^{7} - 1024 q^{8} - 1145 q^{9} - 1400 q^{10} + 1045 q^{11} - 3904 q^{12} + 14647 q^{13} + 20736 q^{14} + 15190 q^{15} + 8192 q^{16} + 29616 q^{17} + 9160 q^{18} - 13718 q^{19} + 11200 q^{20} + 87267 q^{21} - 8360 q^{22} + 69985 q^{23} + 31232 q^{24} + 166975 q^{25} - 117176 q^{26} + 84851 q^{27} - 165888 q^{28} + 138821 q^{29} - 121520 q^{30} - 199396 q^{31} - 65536 q^{32} + 341728 q^{33} - 236928 q^{34} - 103635 q^{35} - 73280 q^{36} - 67840 q^{37} + 109744 q^{38} - 565793 q^{39} - 89600 q^{40} - 539350 q^{41} - 698136 q^{42} + 602639 q^{43} + 66880 q^{44} - 1352365 q^{45} - 559880 q^{46} - 1031975 q^{47} - 249856 q^{48} + 1761412 q^{49} - 1335800 q^{50} - 780123 q^{51} + 937408 q^{52} + 2138263 q^{53} - 678808 q^{54} + 5695445 q^{55} + 1327104 q^{56} + 418399 q^{57} - 1110568 q^{58} - 3936369 q^{59} + 972160 q^{60} - 1027655 q^{61} + 1595168 q^{62} + 983049 q^{63} + 524288 q^{64} - 504280 q^{65} - 2733824 q^{66} + 764949 q^{67} + 1895424 q^{68} - 241907 q^{69} + 829080 q^{70} - 3572084 q^{71} + 586240 q^{72} + 9069522 q^{73} + 542720 q^{74} - 1500425 q^{75} - 877952 q^{76} + 887283 q^{77} + 4526344 q^{78} - 2753414 q^{79} + 716800 q^{80} - 1314122 q^{81} + 4314800 q^{82} - 7643046 q^{83} + 5585088 q^{84} + 4438875 q^{85} - 4821112 q^{86} + 3800423 q^{87} - 535040 q^{88} + 1393620 q^{89} + 10818920 q^{90} - 19696869 q^{91} + 4479040 q^{92} - 602176 q^{93} + 8255800 q^{94} - 1200325 q^{95} + 1998848 q^{96} - 6921466 q^{97} - 14091296 q^{98} - 23387893 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 61 * q^3 + 128 * q^4 + 175 * q^5 + 488 * q^6 - 2592 * q^7 - 1024 * q^8 - 1145 * q^9 - 1400 * q^10 + 1045 * q^11 - 3904 * q^12 + 14647 * q^13 + 20736 * q^14 + 15190 * q^15 + 8192 * q^16 + 29616 * q^17 + 9160 * q^18 - 13718 * q^19 + 11200 * q^20 + 87267 * q^21 - 8360 * q^22 + 69985 * q^23 + 31232 * q^24 + 166975 * q^25 - 117176 * q^26 + 84851 * q^27 - 165888 * q^28 + 138821 * q^29 - 121520 * q^30 - 199396 * q^31 - 65536 * q^32 + 341728 * q^33 - 236928 * q^34 - 103635 * q^35 - 73280 * q^36 - 67840 * q^37 + 109744 * q^38 - 565793 * q^39 - 89600 * q^40 - 539350 * q^41 - 698136 * q^42 + 602639 * q^43 + 66880 * q^44 - 1352365 * q^45 - 559880 * q^46 - 1031975 * q^47 - 249856 * q^48 + 1761412 * q^49 - 1335800 * q^50 - 780123 * q^51 + 937408 * q^52 + 2138263 * q^53 - 678808 * q^54 + 5695445 * q^55 + 1327104 * q^56 + 418399 * q^57 - 1110568 * q^58 - 3936369 * q^59 + 972160 * q^60 - 1027655 * q^61 + 1595168 * q^62 + 983049 * q^63 + 524288 * q^64 - 504280 * q^65 - 2733824 * q^66 + 764949 * q^67 + 1895424 * q^68 - 241907 * q^69 + 829080 * q^70 - 3572084 * q^71 + 586240 * q^72 + 9069522 * q^73 + 542720 * q^74 - 1500425 * q^75 - 877952 * q^76 + 887283 * q^77 + 4526344 * q^78 - 2753414 * q^79 + 716800 * q^80 - 1314122 * q^81 + 4314800 * q^82 - 7643046 * q^83 + 5585088 * q^84 + 4438875 * q^85 - 4821112 * q^86 + 3800423 * q^87 - 535040 * q^88 + 1393620 * q^89 + 10818920 * q^90 - 19696869 * q^91 + 4479040 * q^92 - 602176 * q^93 + 8255800 * q^94 - 1200325 * q^95 + 1998848 * q^96 - 6921466 * q^97 - 14091296 * q^98 - 23387893 * 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
$3$	−56.6582	−1.21154	−0.605771	−	0.795639i	$-0.707135\pi$
$3$	−56.6582	−1.21154	−0.605771	+	0.795639i	$0.707135\pi$
$4$	64.0000	0.500000
$5$	−304.873	−1.09075	−0.545373	−	0.838194i	$-0.683612\pi$
$5$	−304.873	−1.09075	−0.545373	+	0.838194i	$0.683612\pi$
$6$	453.265	0.856689
$7$	−1452.95	−1.60106	−0.800529	−	0.599294i	$-0.795448\pi$
$7$	−1452.95	−1.60106	−0.800529	+	0.599294i	$0.795448\pi$
$8$	−512.000	−0.353553
$9$	1023.15	0.467832
$10$	2438.98	0.771273
$11$	−6618.68	−1.49933	−0.749665	−	0.661818i	$-0.769785\pi$
$11$	−6618.68	−1.49933	−0.749665	+	0.661818i	$0.769785\pi$
$12$	−3626.12	−0.605771
$13$	9599.26	1.21181	0.605907	−	0.795535i	$-0.292810\pi$
$13$	9599.26	1.21181	0.605907	+	0.795535i	$0.292810\pi$
$14$	11623.6	1.13212
$15$	17273.5	1.32148
$16$	4096.00	0.250000
$17$	12453.8	0.614794	0.307397	−	0.951581i	$-0.400542\pi$
$17$	12453.8	0.614794	0.307397	+	0.951581i	$0.400542\pi$
$18$	−8185.19	−0.330807
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	−19511.8	−0.545373
$21$	82321.4	1.93975
$22$	52949.4	1.06019
$23$	−1184.25	−0.0202954	−0.0101477	−	0.999949i	$-0.503230\pi$
$23$	−1184.25	−0.0202954	−0.0101477	+	0.999949i	$0.503230\pi$
$24$	29009.0	0.428344
$25$	14822.3	0.189725
$26$	−76794.1	−0.856882
$27$	65941.7	0.644743
$28$	−92988.7	−0.800529
$29$	−84164.1	−0.640817	−0.320409	−	0.947279i	$-0.603820\pi$
$29$	−84164.1	−0.640817	−0.320409	+	0.947279i	$0.603820\pi$
$30$	−138188.	−0.934430
$31$	28058.5	0.169160	0.0845802	−	0.996417i	$-0.473045\pi$
$31$	28058.5	0.169160	0.0845802	+	0.996417i	$0.473045\pi$
$32$	−32768.0	−0.176777
$33$	375002.	1.81650
$34$	−99630.1	−0.434725
$35$	442964.	1.74635
$36$	65481.5	0.233916
$37$	−412795.	−1.33976	−0.669882	−	0.742467i	$-0.733655\pi$
$37$	−412795.	−1.33976	−0.669882	+	0.742467i	$0.733655\pi$
$38$	54872.0	0.162221
$39$	−543877.	−1.46816
$40$	156095.	0.385637
$41$	−539470.	−1.22243	−0.611215	−	0.791464i	$-0.709319\pi$
$41$	−539470.	−1.22243	−0.611215	+	0.791464i	$0.709319\pi$
$42$	−658571.	−1.37161
$43$	318505.	0.610910	0.305455	−	0.952207i	$-0.401191\pi$
$43$	318505.	0.610910	0.305455	+	0.952207i	$0.401191\pi$
$44$	−423596.	−0.749665
$45$	−311930.	−0.510286
$46$	9474.02	0.0143510
$47$	−1.24384e6	−1.74752	−0.873759	−	0.486360i	$-0.838325\pi$
$47$	−1.24384e6	−1.74752	−0.873759	+	0.486360i	$0.838325\pi$
$48$	−232072.	−0.302885
$49$	1.28752e6	1.56339
$50$	−118578.	−0.134156
$51$	−705608.	−0.744848
$52$	614353.	0.605907
$53$	2.06709e6	1.90719	0.953596	−	0.301090i	$-0.0973506\pi$
$53$	2.06709e6	1.90719	0.953596	+	0.301090i	$0.0973506\pi$
$54$	−527534.	−0.455902
$55$	2.01785e6	1.63539
$56$	743910.	0.566060
$57$	388618.	0.277947
$58$	673313.	0.453126
$59$	−2.06604e6	−1.30966	−0.654828	−	0.755778i	$-0.727259\pi$
$59$	−2.06604e6	−1.30966	−0.654828	+	0.755778i	$0.727259\pi$
$60$	1.10551e6	0.660741
$61$	1.56677e6	0.883792	0.441896	−	0.897066i	$-0.354306\pi$
$61$	1.56677e6	0.883792	0.441896	+	0.897066i	$0.354306\pi$
$62$	−224468.	−0.119614
$63$	−1.48658e6	−0.749026
$64$	262144.	0.125000
$65$	−2.92655e6	−1.32178
$66$	−3.00002e6	−1.28446
$67$	2.00776e6	0.815549	0.407774	−	0.913083i	$-0.366305\pi$
$67$	2.00776e6	0.815549	0.407774	+	0.913083i	$0.366305\pi$
$68$	797041.	0.307397
$69$	67097.6	0.0245887
$70$	−3.54371e6	−1.23485
$71$	1.70325e6	0.564773	0.282386	−	0.959301i	$-0.408874\pi$
$71$	1.70325e6	0.564773	0.282386	+	0.959301i	$0.408874\pi$
$72$	−523852.	−0.165404
$73$	4.48077e6	1.34810	0.674051	−	0.738685i	$-0.264553\pi$
$73$	4.48077e6	1.34810	0.674051	+	0.738685i	$0.264553\pi$
$74$	3.30236e6	0.947356
$75$	−839804.	−0.229860
$76$	−438976.	−0.114708
$77$	9.61661e6	2.40051
$78$	4.35101e6	1.03815
$79$	−8.64298e6	−1.97228	−0.986140	−	0.165916i	$-0.946942\pi$
$79$	−8.64298e6	−1.97228	−0.986140	+	0.165916i	$0.946942\pi$
$80$	−1.24876e6	−0.272686
$81$	−5.97376e6	−1.24897
$82$	4.31576e6	0.864389
$83$	−4.34087e6	−0.833303	−0.416652	−	0.909066i	$-0.636797\pi$
$83$	−4.34087e6	−0.833303	−0.416652	+	0.909066i	$0.636797\pi$
$84$	5.26857e6	0.969874
$85$	−3.79681e6	−0.670583
$86$	−2.54804e6	−0.431979
$87$	4.76859e6	0.776376
$88$	3.38876e6	0.530093
$89$	1.69658e6	0.255099	0.127549	−	0.991832i	$-0.459289\pi$
$89$	1.69658e6	0.255099	0.127549	+	0.991832i	$0.459289\pi$
$90$	2.49544e6	0.360826
$91$	−1.39472e7	−1.94019
$92$	−75792.2	−0.0101477
$93$	−1.58974e6	−0.204945
$94$	9.95071e6	1.23568
$95$	2.09112e6	0.250234
$96$	1.85658e6	0.214172
$97$	4.33665e6	0.482451	0.241226	−	0.970469i	$-0.422451\pi$
$97$	4.33665e6	0.482451	0.241226	+	0.970469i	$0.422451\pi$
$98$	−1.03001e7	−1.10548
$99$	−6.77189e6	−0.701434
$100$	948627.	0.0948627
$101$	−8.30296e6	−0.801878	−0.400939	−	0.916105i	$-0.631316\pi$
$101$	−8.30296e6	−0.801878	−0.400939	+	0.916105i	$0.631316\pi$
$102$	5.64486e6	0.526687
$103$	9.00718e6	0.812192	0.406096	−	0.913830i	$-0.366890\pi$
$103$	9.00718e6	0.812192	0.406096	+	0.913830i	$0.366890\pi$
$104$	−4.91482e6	−0.428441
$105$	−2.50975e7	−2.11577
$106$	−1.65367e7	−1.34859
$107$	−2.34480e6	−0.185038	−0.0925192	−	0.995711i	$-0.529492\pi$
$107$	−2.34480e6	−0.185038	−0.0925192	+	0.995711i	$0.529492\pi$
$108$	4.22027e6	0.322372
$109$	−3.58778e6	−0.265359	−0.132679	−	0.991159i	$-0.542358\pi$
$109$	−3.58778e6	−0.265359	−0.132679	+	0.991159i	$0.542358\pi$
$110$	−1.61428e7	−1.15639
$111$	2.33882e7	1.62318
$112$	−5.95128e6	−0.400265
$113$	−1.35645e7	−0.884363	−0.442182	−	0.896926i	$-0.645795\pi$
$113$	−1.35645e7	−0.884363	−0.442182	+	0.896926i	$0.645795\pi$
$114$	−3.10895e6	−0.196538
$115$	361046.	0.0221371
$116$	−5.38650e6	−0.320409
$117$	9.82147e6	0.566925
$118$	1.65283e7	0.926066
$119$	−1.80947e7	−0.984321
$120$	−8.84404e6	−0.467215
$121$	2.43198e7	1.24799
$122$	−1.25341e7	−0.624935
$123$	3.05654e7	1.48102
$124$	1.79574e6	0.0845802
$125$	1.92993e7	0.883803
$126$	1.18927e7	0.529642
$127$	3.70127e7	1.60339	0.801693	−	0.597736i	$-0.203933\pi$
$127$	3.70127e7	1.60339	0.801693	+	0.597736i	$0.203933\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.80459e7	−0.740143
$130$	2.34124e7	0.934640
$131$	3.46796e6	0.134780	0.0673898	−	0.997727i	$-0.478533\pi$
$131$	3.46796e6	0.134780	0.0673898	+	0.997727i	$0.478533\pi$
$132$	2.40002e7	0.908250
$133$	9.96578e6	0.367308
$134$	−1.60621e7	−0.576680
$135$	−2.01038e7	−0.703251
$136$	−6.37633e6	−0.217362
$137$	−2.27216e6	−0.0754947	−0.0377473	−	0.999287i	$-0.512018\pi$
$137$	−2.27216e6	−0.0754947	−0.0377473	+	0.999287i	$0.512018\pi$
$138$	−536781.	−0.0173868
$139$	3.46122e7	1.09314	0.546572	−	0.837412i	$-0.315932\pi$
$139$	3.46122e7	1.09314	0.546572	+	0.837412i	$0.315932\pi$
$140$	2.83497e7	0.873174
$141$	7.04736e7	2.11719
$142$	−1.36260e7	−0.399355
$143$	−6.35344e7	−1.81691
$144$	4.19082e6	0.116958
$145$	2.56593e7	0.698968
$146$	−3.58462e7	−0.953252
$147$	−7.29484e7	−1.89411
$148$	−2.64189e7	−0.669882
$149$	7.18613e7	1.77968	0.889842	−	0.456268i	$-0.150814\pi$
$149$	7.18613e7	1.77968	0.889842	+	0.456268i	$0.150814\pi$
$150$	6.71843e6	0.162536
$151$	7.71355e6	0.182320	0.0911602	−	0.995836i	$-0.470942\pi$
$151$	7.71355e6	0.182320	0.0911602	+	0.995836i	$0.470942\pi$
$152$	3.51181e6	0.0811107
$153$	1.27421e7	0.287620
$154$	−7.69329e7	−1.69742
$155$	−8.55427e6	−0.184511
$156$	−3.48081e7	−0.734081
$157$	4.62544e7	0.953904	0.476952	−	0.878929i	$-0.341742\pi$
$157$	4.62544e7	0.953904	0.476952	+	0.878929i	$0.341742\pi$
$158$	6.91438e7	1.39461
$159$	−1.17118e8	−2.31064
$160$	9.99007e6	0.192818
$161$	1.72066e6	0.0324941
$162$	4.77901e7	0.883152
$163$	2.14964e7	0.388785	0.194393	−	0.980924i	$-0.437726\pi$
$163$	2.14964e7	0.388785	0.194393	+	0.980924i	$0.437726\pi$
$164$	−3.45261e7	−0.611215
$165$	−1.14328e8	−1.98134
$166$	3.47269e7	0.589235
$167$	−7.98540e7	−1.32675	−0.663375	−	0.748287i	$-0.730876\pi$
$167$	−7.98540e7	−1.32675	−0.663375	+	0.748287i	$0.730876\pi$
$168$	−4.21486e7	−0.685805
$169$	2.93973e7	0.468494
$170$	3.03745e7	0.474174
$171$	−7.01778e6	−0.107328
$172$	2.03843e7	0.305455
$173$	−1.51721e7	−0.222784	−0.111392	−	0.993777i	$-0.535531\pi$
$173$	−1.51721e7	−0.222784	−0.111392	+	0.993777i	$0.535531\pi$
$174$	−3.81487e7	−0.548981
$175$	−2.15360e7	−0.303761
$176$	−2.71101e7	−0.374832
$177$	1.17058e8	1.58670
$178$	−1.35726e7	−0.180382
$179$	6.33798e7	0.825972	0.412986	−	0.910737i	$-0.364486\pi$
$179$	6.33798e7	0.825972	0.412986	+	0.910737i	$0.364486\pi$
$180$	−1.99635e7	−0.255143
$181$	3.81245e7	0.477892	0.238946	−	0.971033i	$-0.423198\pi$
$181$	3.81245e7	0.477892	0.238946	+	0.971033i	$0.423198\pi$
$182$	1.11578e8	1.37192
$183$	−8.87702e7	−1.07075
$184$	606337.	0.00717550
$185$	1.25850e8	1.46134
$186$	1.27180e7	0.144918
$187$	−8.24275e7	−0.921778
$188$	−7.96057e7	−0.873759
$189$	−9.58099e7	−1.03227
$190$	−1.67290e7	−0.176942
$191$	−1.62341e8	−1.68582	−0.842910	−	0.538054i	$-0.819160\pi$
$191$	−1.62341e8	−1.68582	−0.842910	+	0.538054i	$0.819160\pi$
$192$	−1.48526e7	−0.151443
$193$	−1.20741e8	−1.20894	−0.604468	−	0.796629i	$-0.706614\pi$
$193$	−1.20741e8	−1.20894	−0.604468	+	0.796629i	$0.706614\pi$
$194$	−3.46932e7	−0.341145
$195$	1.65813e8	1.60139
$196$	8.24011e7	0.781694
$197$	1.41739e7	0.132086	0.0660430	−	0.997817i	$-0.478963\pi$
$197$	1.41739e7	0.132086	0.0660430	+	0.997817i	$0.478963\pi$
$198$	5.41752e7	0.495989
$199$	−7.34170e7	−0.660405	−0.330203	−	0.943910i	$-0.607117\pi$
$199$	−7.34170e7	−0.660405	−0.330203	+	0.943910i	$0.607117\pi$
$200$	−7.58902e6	−0.0670781
$201$	−1.13756e8	−0.988071
$202$	6.64237e7	0.567013
$203$	1.22286e8	1.02599
$204$	−4.51589e7	−0.372424
$205$	1.64470e8	1.33336
$206$	−7.20575e7	−0.574307
$207$	−1.21167e6	−0.00949482
$208$	3.93186e7	0.302954
$209$	4.53975e7	0.343970
$210$	2.00780e8	1.49608
$211$	1.50771e8	1.10492	0.552460	−	0.833540i	$-0.313689\pi$
$211$	1.50771e8	1.10492	0.552460	+	0.833540i	$0.313689\pi$
$212$	1.32294e8	0.953596
$213$	−9.65030e7	−0.684246
$214$	1.87584e7	0.130842
$215$	−9.71036e7	−0.666347
$216$	−3.37621e7	−0.227951
$217$	−4.07676e7	−0.270836
$218$	2.87023e7	0.187637
$219$	−2.53872e8	−1.63328
$220$	1.29143e8	0.817693
$221$	1.19547e8	0.745016
$222$	−1.87106e8	−1.14776
$223$	2.07808e8	1.25486	0.627429	−	0.778674i	$-0.284107\pi$
$223$	2.07808e8	1.25486	0.627429	+	0.778674i	$0.284107\pi$
$224$	4.76102e7	0.283030
$225$	1.51654e7	0.0887596
$226$	1.08516e8	0.625339
$227$	4.11125e7	0.233283	0.116641	−	0.993174i	$-0.462787\pi$
$227$	4.11125e7	0.233283	0.116641	+	0.993174i	$0.462787\pi$
$228$	2.48716e7	0.138973
$229$	−2.39111e8	−1.31575	−0.657877	−	0.753125i	$-0.728545\pi$
$229$	−2.39111e8	−1.31575	−0.657877	+	0.753125i	$0.728545\pi$
$230$	−2.88837e6	−0.0156533
$231$	−5.44859e8	−2.90832
$232$	4.30920e7	0.226563
$233$	1.79003e8	0.927076	0.463538	−	0.886077i	$-0.346580\pi$
$233$	1.79003e8	0.927076	0.463538	+	0.886077i	$0.346580\pi$
$234$	−7.85718e7	−0.400877
$235$	3.79212e8	1.90610
$236$	−1.32227e8	−0.654828
$237$	4.89695e8	2.38950
$238$	1.44757e8	0.696020
$239$	3.32179e8	1.57391	0.786953	−	0.617013i	$-0.211657\pi$
$239$	3.32179e8	1.57391	0.786953	+	0.617013i	$0.211657\pi$
$240$	7.07524e7	0.330371
$241$	−2.85949e8	−1.31592	−0.657960	−	0.753053i	$-0.728580\pi$
$241$	−2.85949e8	−1.31592	−0.657960	+	0.753053i	$0.728580\pi$
$242$	−1.94558e8	−0.882461
$243$	1.94248e8	0.868429
$244$	1.00273e8	0.441896
$245$	−3.92529e8	−1.70526
$246$	−2.44523e8	−1.04724
$247$	−6.58413e7	−0.278009
$248$	−1.43660e7	−0.0598072
$249$	2.45946e8	1.00958
$250$	−1.54394e8	−0.624943
$251$	−3.75035e8	−1.49697	−0.748487	−	0.663150i	$-0.769219\pi$
$251$	−3.75035e8	−1.49697	−0.748487	+	0.663150i	$0.769219\pi$
$252$	−9.51413e7	−0.374513
$253$	7.83819e6	0.0304294
$254$	−2.96102e8	−1.13377
$255$	2.15120e8	0.812439
$256$	1.67772e7	0.0625000
$257$	−2.37807e8	−0.873892	−0.436946	−	0.899488i	$-0.643940\pi$
$257$	−2.37807e8	−0.873892	−0.436946	+	0.899488i	$0.643940\pi$
$258$	1.44367e8	0.523360
$259$	5.99770e8	2.14504
$260$	−1.87299e8	−0.660890
$261$	−8.61124e7	−0.299795
$262$	−2.77437e7	−0.0953036
$263$	−5.17576e7	−0.175440	−0.0877201	−	0.996145i	$-0.527958\pi$
$263$	−5.17576e7	−0.175440	−0.0877201	+	0.996145i	$0.527958\pi$
$264$	−1.92001e8	−0.642229
$265$	−6.30200e8	−2.08026
$266$	−7.97262e7	−0.259726
$267$	−9.61249e7	−0.309062
$268$	1.28497e8	0.407774
$269$	5.73561e8	1.79658	0.898290	−	0.439404i	$-0.144810\pi$
$269$	5.73561e8	1.79658	0.898290	+	0.439404i	$0.144810\pi$
$270$	1.60831e8	0.497273
$271$	−6.28079e8	−1.91700	−0.958498	−	0.285098i	$-0.907974\pi$
$271$	−6.28079e8	−1.91700	−0.958498	+	0.285098i	$0.907974\pi$
$272$	5.10106e7	0.153698
$273$	7.90225e8	2.35061
$274$	1.81773e7	0.0533828
$275$	−9.81041e7	−0.284461
$276$	4.29425e6	0.0122943
$277$	1.03535e8	0.292691	0.146346	−	0.989234i	$-0.453249\pi$
$277$	1.03535e8	0.292691	0.146346	+	0.989234i	$0.453249\pi$
$278$	−2.76898e8	−0.772969
$279$	2.87080e7	0.0791387
$280$	−2.26798e8	−0.617427
$281$	3.14594e7	0.0845821	0.0422910	−	0.999105i	$-0.486534\pi$
$281$	3.14594e7	0.0845821	0.0422910	+	0.999105i	$0.486534\pi$
$282$	−5.63789e8	−1.49708
$283$	−2.63638e8	−0.691442	−0.345721	−	0.938337i	$-0.612366\pi$
$283$	−2.63638e8	−0.691442	−0.345721	+	0.938337i	$0.612366\pi$
$284$	1.09008e8	0.282386
$285$	−1.18479e8	−0.303169
$286$	5.08276e8	1.28475
$287$	7.83823e8	1.95718
$288$	−3.35265e7	−0.0827018
$289$	−2.55242e8	−0.622029
$290$	−2.05275e8	−0.494245
$291$	−2.45707e8	−0.584510
$292$	2.86769e8	0.674051
$293$	−3.68983e8	−0.856977	−0.428488	−	0.903547i	$-0.640954\pi$
$293$	−3.68983e8	−0.856977	−0.428488	+	0.903547i	$0.640954\pi$
$294$	5.83587e8	1.33934
$295$	6.29880e8	1.42850
$296$	2.11351e8	0.473678
$297$	−4.36447e8	−0.966683
$298$	−5.74890e8	−1.25843
$299$	−1.13680e7	−0.0245942
$300$	−5.37475e7	−0.114930
$301$	−4.62772e8	−0.978103
$302$	−6.17084e7	−0.128920
$303$	4.70430e8	0.971508
$304$	−2.80945e7	−0.0573539
$305$	−4.77664e8	−0.963992
$306$	−1.01936e8	−0.203378
$307$	−4.79924e8	−0.946647	−0.473323	−	0.880889i	$-0.656946\pi$
$307$	−4.79924e8	−0.946647	−0.473323	+	0.880889i	$0.656946\pi$
$308$	6.15463e8	1.20026
$309$	−5.10331e8	−0.984004
$310$	6.84342e7	0.130469
$311$	−1.03167e7	−0.0194482	−0.00972412	−	0.999953i	$-0.503095\pi$
$311$	−1.03167e7	−0.0194482	−0.00972412	+	0.999953i	$0.503095\pi$
$312$	2.78465e8	0.519074
$313$	−3.34381e8	−0.616362	−0.308181	−	0.951328i	$-0.599720\pi$
$313$	−3.34381e8	−0.616362	−0.308181	+	0.951328i	$0.599720\pi$
$314$	−3.70036e8	−0.674512
$315$	4.53218e8	0.816997
$316$	−5.53150e8	−0.986140
$317$	2.88274e6	0.00508274	0.00254137	−	0.999997i	$-0.499191\pi$
$317$	2.88274e6	0.00508274	0.00254137	+	0.999997i	$0.499191\pi$
$318$	9.36941e8	1.63387
$319$	5.57056e8	0.960796
$320$	−7.99205e7	−0.136343
$321$	1.32852e8	0.224182
$322$	−1.37653e7	−0.0229768
$323$	−8.54204e7	−0.141043
$324$	−3.82321e8	−0.624483
$325$	1.42283e8	0.229912
$326$	−1.71972e8	−0.274913
$327$	2.03277e8	0.321493
$328$	2.76209e8	0.432194
$329$	1.80723e9	2.79788
$330$	9.14624e8	1.40102
$331$	1.23651e9	1.87414	0.937069	−	0.349146i	$-0.113528\pi$
$331$	1.23651e9	1.87414	0.937069	+	0.349146i	$0.113528\pi$
$332$	−2.77816e8	−0.416652
$333$	−4.22351e8	−0.626785
$334$	6.38832e8	0.938153
$335$	−6.12111e8	−0.889556
$336$	3.37189e8	0.484937
$337$	−2.27624e8	−0.323977	−0.161988	−	0.986793i	$-0.551791\pi$
$337$	−2.27624e8	−0.323977	−0.161988	+	0.986793i	$0.551791\pi$
$338$	−2.35178e8	−0.331275
$339$	7.68542e8	1.07144
$340$	−2.42996e8	−0.335292
$341$	−1.85710e8	−0.253627
$342$	5.61422e7	0.0758924
$343$	−6.74132e8	−0.902019
$344$	−1.63075e8	−0.215989
$345$	−2.04562e7	−0.0268200
$346$	1.21377e8	0.157532
$347$	5.73303e6	0.00736598	0.00368299	−	0.999993i	$-0.498828\pi$
$347$	5.73303e6	0.00736598	0.00368299	+	0.999993i	$0.498828\pi$
$348$	3.05189e8	0.388188
$349$	−2.61887e8	−0.329780	−0.164890	−	0.986312i	$-0.552727\pi$
$349$	−2.61887e8	−0.329780	−0.164890	+	0.986312i	$0.552727\pi$
$350$	1.72288e8	0.214792
$351$	6.32992e8	0.781309
$352$	2.16881e8	0.265046
$353$	8.01960e8	0.970378	0.485189	−	0.874409i	$-0.338751\pi$
$353$	8.01960e8	0.970378	0.485189	+	0.874409i	$0.338751\pi$
$354$	−9.36465e8	−1.12197
$355$	−5.19274e8	−0.616024
$356$	1.08581e8	0.127549
$357$	1.02521e9	1.19255
$358$	−5.07038e8	−0.584050
$359$	1.15339e9	1.31567	0.657833	−	0.753163i	$-0.271473\pi$
$359$	1.15339e9	1.31567	0.657833	+	0.753163i	$0.271473\pi$
$360$	1.59708e8	0.180413
$361$	4.70459e7	0.0526316
$362$	−3.04996e8	−0.337920
$363$	−1.37791e9	−1.51199
$364$	−8.92623e8	−0.970093
$365$	−1.36606e9	−1.47044
$366$	7.10161e8	0.757135
$367$	−3.13172e8	−0.330713	−0.165357	−	0.986234i	$-0.552878\pi$
$367$	−3.13172e8	−0.330713	−0.165357	+	0.986234i	$0.552878\pi$
$368$	−4.85070e6	−0.00507384
$369$	−5.51958e8	−0.571892
$370$	−1.00680e9	−1.03332
$371$	−3.00338e9	−3.05353
$372$	−1.01744e8	−0.102472
$373$	−1.08785e8	−0.108539	−0.0542697	−	0.998526i	$-0.517283\pi$
$373$	−1.08785e8	−0.108539	−0.0542697	+	0.998526i	$0.517283\pi$
$374$	6.59420e8	0.651796
$375$	−1.09346e9	−1.07076
$376$	6.36845e8	0.617841
$377$	−8.07913e8	−0.776551
$378$	7.66479e8	0.729927
$379$	−8.72499e8	−0.823243	−0.411621	−	0.911355i	$-0.635037\pi$
$379$	−8.72499e8	−0.823243	−0.411621	+	0.911355i	$0.635037\pi$
$380$	1.33832e8	0.125117
$381$	−2.09707e9	−1.94257
$382$	1.29873e9	1.19206
$383$	−8.68555e7	−0.0789954	−0.0394977	−	0.999220i	$-0.512576\pi$
$383$	−8.68555e7	−0.0789954	−0.0394977	+	0.999220i	$0.512576\pi$
$384$	1.18821e8	0.107086
$385$	−2.93184e9	−2.61835
$386$	9.65926e8	0.854847
$387$	3.25878e8	0.285803
$388$	2.77546e8	0.241226
$389$	1.34986e9	1.16269	0.581347	−	0.813656i	$-0.302526\pi$
$389$	1.34986e9	1.16269	0.581347	+	0.813656i	$0.302526\pi$
$390$	−1.32650e9	−1.13236
$391$	−1.47484e7	−0.0124775
$392$	−6.59209e8	−0.552741
$393$	−1.96488e8	−0.163291
$394$	−1.13391e8	−0.0933989
$395$	2.63501e9	2.15125
$396$	−4.33401e8	−0.350717
$397$	2.40300e9	1.92746	0.963732	−	0.266872i	$-0.0859902\pi$
$397$	2.40300e9	1.92746	0.963732	+	0.266872i	$0.0859902\pi$
$398$	5.87336e8	0.466977
$399$	−5.64643e8	−0.445009
$400$	6.07121e7	0.0474313
$401$	−1.68831e9	−1.30751	−0.653757	−	0.756704i	$-0.726808\pi$
$401$	−1.68831e9	−1.30751	−0.653757	+	0.756704i	$0.726808\pi$
$402$	9.10048e8	0.698672
$403$	2.69341e8	0.204991
$404$	−5.31389e8	−0.400939
$405$	1.82124e9	1.36230
$406$	−9.78290e8	−0.725481
$407$	2.73216e9	2.00875
$408$	3.61271e8	0.263343
$409$	−5.86269e8	−0.423706	−0.211853	−	0.977301i	$-0.567950\pi$
$409$	−5.86269e8	−0.423706	−0.211853	+	0.977301i	$0.567950\pi$
$410$	−1.31576e9	−0.942828
$411$	1.28736e8	0.0914649
$412$	5.76460e8	0.406096
$413$	3.00185e9	2.09684
$414$	9.69333e6	0.00671385
$415$	1.32341e9	0.908922
$416$	−3.14549e8	−0.214221
$417$	−1.96106e9	−1.32439
$418$	−3.63180e8	−0.243223
$419$	2.82375e9	1.87532	0.937662	−	0.347547i	$-0.112985\pi$
$419$	2.82375e9	1.87532	0.937662	+	0.347547i	$0.112985\pi$
$420$	−1.60624e9	−1.05789
$421$	5.94837e8	0.388518	0.194259	−	0.980950i	$-0.437770\pi$
$421$	5.94837e8	0.388518	0.194259	+	0.980950i	$0.437770\pi$
$422$	−1.20617e9	−0.781296
$423$	−1.27263e9	−0.817545
$424$	−1.05835e9	−0.674294
$425$	1.84593e8	0.116642
$426$	7.72024e8	0.483835
$427$	−2.27643e9	−1.41500
$428$	−1.50067e8	−0.0925192
$429$	3.59975e9	2.20126
$430$	7.76829e8	0.471179
$431$	1.39546e9	0.839550	0.419775	−	0.907628i	$-0.362109\pi$
$431$	1.39546e9	0.839550	0.419775	+	0.907628i	$0.362109\pi$
$432$	2.70097e8	0.161186
$433$	−1.70411e9	−1.00877	−0.504383	−	0.863480i	$-0.668280\pi$
$433$	−1.70411e9	−1.00877	−0.504383	+	0.863480i	$0.668280\pi$
$434$	3.26141e8	0.191510
$435$	−1.45381e9	−0.846829
$436$	−2.29618e8	−0.132679
$437$	8.12279e6	0.00465608
$438$	2.03098e9	1.15490
$439$	−2.93207e9	−1.65405	−0.827026	−	0.562164i	$-0.809969\pi$
$439$	−2.93207e9	−1.65405	−0.827026	+	0.562164i	$0.809969\pi$
$440$	−1.03314e9	−0.578196
$441$	1.31732e9	0.731403
$442$	−9.56375e8	−0.526806
$443$	1.76473e9	0.964420	0.482210	−	0.876056i	$-0.339834\pi$
$443$	1.76473e9	0.964420	0.482210	+	0.876056i	$0.339834\pi$
$444$	1.49685e9	0.811590
$445$	−5.17239e8	−0.278248
$446$	−1.66246e9	−0.887319
$447$	−4.07153e9	−2.15616
$448$	−3.80882e8	−0.200132
$449$	9.59818e8	0.500411	0.250205	−	0.968193i	$-0.419502\pi$
$449$	9.59818e8	0.500411	0.250205	+	0.968193i	$0.419502\pi$
$450$	−1.21323e8	−0.0627625
$451$	3.57058e9	1.83283
$452$	−8.68131e8	−0.442182
$453$	−4.37036e8	−0.220889
$454$	−3.28900e8	−0.164956
$455$	4.25213e9	2.11625
$456$	−1.98973e8	−0.0982690
$457$	1.66471e9	0.815889	0.407944	−	0.913007i	$-0.366246\pi$
$457$	1.66471e9	0.815889	0.407944	+	0.913007i	$0.366246\pi$
$458$	1.91288e9	0.930378
$459$	8.21222e8	0.396384
$460$	2.31070e7	0.0110685
$461$	4.19324e8	0.199341	0.0996705	−	0.995020i	$-0.468221\pi$
$461$	4.19324e8	0.199341	0.0996705	+	0.995020i	$0.468221\pi$
$462$	4.35887e9	2.05649
$463$	2.98154e9	1.39607	0.698035	−	0.716063i	$-0.254058\pi$
$463$	2.98154e9	1.39607	0.698035	+	0.716063i	$0.254058\pi$
$464$	−3.44736e8	−0.160204
$465$	4.84669e8	0.223543
$466$	−1.43203e9	−0.655542
$467$	−7.20080e8	−0.327168	−0.163584	−	0.986529i	$-0.552306\pi$
$467$	−7.20080e8	−0.327168	−0.163584	+	0.986529i	$0.552306\pi$
$468$	6.28574e8	0.283463
$469$	−2.91717e9	−1.30574
$470$	−3.03370e9	−1.34781
$471$	−2.62069e9	−1.15569
$472$	1.05781e9	0.463033
$473$	−2.10809e9	−0.915955
$474$	−3.91756e9	−1.68963
$475$	−1.01666e8	−0.0435260
$476$	−1.15806e9	−0.492160
$477$	2.11494e9	0.892245
$478$	−2.65743e9	−1.11292
$479$	3.06418e9	1.27391	0.636957	−	0.770899i	$-0.280193\pi$
$479$	3.06418e9	1.27391	0.636957	+	0.770899i	$0.280193\pi$
$480$	−5.66019e8	−0.233607
$481$	−3.96253e9	−1.62355
$482$	2.28759e9	0.930495
$483$	−9.74894e7	−0.0393679
$484$	1.55647e9	0.623994
$485$	−1.32213e9	−0.526232
$486$	−1.55398e9	−0.614072
$487$	−8.67010e8	−0.340152	−0.170076	−	0.985431i	$-0.554401\pi$
$487$	−8.67010e8	−0.340152	−0.170076	+	0.985431i	$0.554401\pi$
$488$	−8.02185e8	−0.312468
$489$	−1.21795e9	−0.471030
$490$	3.14023e9	1.20580
$491$	−1.47869e9	−0.563758	−0.281879	−	0.959450i	$-0.590958\pi$
$491$	−1.47869e9	−0.563758	−0.281879	+	0.959450i	$0.590958\pi$
$492$	1.95619e9	0.740512
$493$	−1.04816e9	−0.393970
$494$	5.26731e8	0.196582
$495$	2.06456e9	0.765086
$496$	1.14928e8	0.0422901
$497$	−2.47473e9	−0.904235
$498$	−1.96756e9	−0.713882
$499$	−3.01343e8	−0.108570	−0.0542850	−	0.998525i	$-0.517288\pi$
$499$	−3.01343e8	−0.108570	−0.0542850	+	0.998525i	$0.517288\pi$
$500$	1.23515e9	0.441902
$501$	4.52438e9	1.60741
$502$	3.00028e9	1.05852
$503$	6.35428e8	0.222627	0.111314	−	0.993785i	$-0.464494\pi$
$503$	6.35428e8	0.222627	0.111314	+	0.993785i	$0.464494\pi$
$504$	7.61130e8	0.264821
$505$	2.53134e9	0.874644
$506$	−6.27055e7	−0.0215169
$507$	−1.66560e9	−0.567600
$508$	2.36882e9	0.801693
$509$	−8.58196e8	−0.288452	−0.144226	−	0.989545i	$-0.546069\pi$
$509$	−8.58196e8	−0.288452	−0.144226	+	0.989545i	$0.546069\pi$
$510$	−1.72096e9	−0.574481
$511$	−6.51033e9	−2.15839
$512$	−1.34218e8	−0.0441942
$513$	−4.52294e8	−0.147914
$514$	1.90245e9	0.617935
$515$	−2.74604e9	−0.885895
$516$	−1.15494e9	−0.370071
$517$	8.23257e9	2.62010
$518$	−4.79816e9	−1.51677
$519$	8.59623e8	0.269912
$520$	1.49839e9	0.467320
$521$	−5.01411e8	−0.155332	−0.0776662	−	0.996979i	$-0.524747\pi$
$521$	−5.01411e8	−0.155332	−0.0776662	+	0.996979i	$0.524747\pi$
$522$	6.88899e8	0.211987
$523$	−4.66146e9	−1.42484	−0.712420	−	0.701753i	$-0.752401\pi$
$523$	−4.66146e9	−1.42484	−0.712420	+	0.701753i	$0.752401\pi$
$524$	2.21949e8	0.0673898
$525$	1.22019e9	0.368020
$526$	4.14061e8	0.124055
$527$	3.49434e8	0.103999
$528$	1.53601e9	0.454125
$529$	−3.40342e9	−0.999588
$530$	5.04160e9	1.47097
$531$	−2.11387e9	−0.612699
$532$	6.37810e8	0.183654
$533$	−5.17852e9	−1.48136
$534$	7.68999e8	0.218540
$535$	7.14864e8	0.201830
$536$	−1.02797e9	−0.288340
$537$	−3.59098e9	−1.00070
$538$	−4.58849e9	−1.27037
$539$	−8.52167e9	−2.34403
$540$	−1.28664e9	−0.351625
$541$	−6.32136e9	−1.71641	−0.858204	−	0.513309i	$-0.828419\pi$
$541$	−6.32136e9	−1.71641	−0.858204	+	0.513309i	$0.828419\pi$
$542$	5.02463e9	1.35552
$543$	−2.16007e9	−0.578986
$544$	−4.08085e8	−0.108681
$545$	1.09382e9	0.289439
$546$	−6.32180e9	−1.66214
$547$	4.31567e9	1.12744	0.563719	−	0.825967i	$-0.309370\pi$
$547$	4.31567e9	1.12744	0.563719	+	0.825967i	$0.309370\pi$
$548$	−1.45418e8	−0.0377473
$549$	1.60304e9	0.413466
$550$	7.84832e8	0.201144
$551$	5.77282e8	0.147014
$552$	−3.43540e7	−0.00869341
$553$	1.25578e10	3.15774
$554$	−8.28283e8	−0.206964
$555$	−7.13042e9	−1.77048
$556$	2.21518e9	0.546572
$557$	−5.30631e8	−0.130107	−0.0650533	−	0.997882i	$-0.520722\pi$
$557$	−5.30631e8	−0.130107	−0.0650533	+	0.997882i	$0.520722\pi$
$558$	−2.29664e8	−0.0559595
$559$	3.05742e9	0.740309
$560$	1.81438e9	0.436587
$561$	4.67019e9	1.11677
$562$	−2.51675e8	−0.0598086
$563$	−1.51472e9	−0.357729	−0.178864	−	0.983874i	$-0.557242\pi$
$563$	−1.51472e9	−0.357729	−0.178864	+	0.983874i	$0.557242\pi$
$564$	4.51031e9	1.05859
$565$	4.13546e9	0.964615
$566$	2.10910e9	0.488923
$567$	8.67957e9	1.99967
$568$	−8.72063e8	−0.199677
$569$	3.00545e8	0.0683939	0.0341969	−	0.999415i	$-0.489113\pi$
$569$	3.00545e8	0.0683939	0.0341969	+	0.999415i	$0.489113\pi$
$570$	9.47833e8	0.214373
$571$	−3.38671e9	−0.761293	−0.380646	−	0.924721i	$-0.624298\pi$
$571$	−3.38671e9	−0.761293	−0.380646	+	0.924721i	$0.624298\pi$
$572$	−4.06620e9	−0.908454
$573$	9.19794e9	2.04244
$574$	−6.27058e9	−1.38394
$575$	−1.75533e7	−0.00385055
$576$	2.68212e8	0.0584790
$577$	−2.29584e9	−0.497537	−0.248769	−	0.968563i	$-0.580026\pi$
$577$	−2.29584e9	−0.497537	−0.248769	+	0.968563i	$0.580026\pi$
$578$	2.04194e9	0.439841
$579$	6.84095e9	1.46468
$580$	1.64220e9	0.349484
$581$	6.30706e9	1.33417
$582$	1.96565e9	0.413311
$583$	−1.36814e10	−2.85951
$584$	−2.29415e9	−0.476626
$585$	−2.99430e9	−0.618371
$586$	2.95186e9	0.605974
$587$	3.22236e9	0.657568	0.328784	−	0.944405i	$-0.393361\pi$
$587$	3.22236e9	0.657568	0.328784	+	0.944405i	$0.393361\pi$
$588$	−4.66870e9	−0.947055
$589$	−1.92453e8	−0.0388081
$590$	−5.03904e9	−1.01010
$591$	−8.03066e8	−0.160028
$592$	−1.69081e9	−0.334941
$593$	−4.52343e9	−0.890791	−0.445396	−	0.895334i	$-0.646937\pi$
$593$	−4.52343e9	−0.890791	−0.445396	+	0.895334i	$0.646937\pi$
$594$	3.49158e9	0.683548
$595$	5.51657e9	1.07364
$596$	4.59912e9	0.889842
$597$	4.15967e9	0.800108
$598$	9.09436e7	0.0173907
$599$	3.02560e9	0.575197	0.287599	−	0.957751i	$-0.407143\pi$
$599$	3.02560e9	0.575197	0.287599	+	0.957751i	$0.407143\pi$
$600$	4.29980e8	0.0812678
$601$	4.24372e8	0.0797418	0.0398709	−	0.999205i	$-0.487305\pi$
$601$	4.24372e8	0.0797418	0.0398709	+	0.999205i	$0.487305\pi$
$602$	3.70218e9	0.691623
$603$	2.05424e9	0.381540
$604$	4.93667e8	0.0911602
$605$	−7.41443e9	−1.36124
$606$	−3.76344e9	−0.686960
$607$	7.92597e9	1.43844	0.719221	−	0.694782i	$-0.244499\pi$
$607$	7.92597e9	1.43844	0.719221	+	0.694782i	$0.244499\pi$
$608$	2.24756e8	0.0405554
$609$	−6.92851e9	−1.24302
$610$	3.82132e9	0.681645
$611$	−1.19399e10	−2.11767
$612$	8.15491e8	0.143810
$613$	1.00101e10	1.75519	0.877596	−	0.479401i	$-0.159146\pi$
$613$	1.00101e10	1.75519	0.877596	+	0.479401i	$0.159146\pi$
$614$	3.83939e9	0.669380
$615$	−9.31855e9	−1.61542
$616$	−4.92370e9	−0.848710
$617$	7.21813e9	1.23716	0.618581	−	0.785721i	$-0.287708\pi$
$617$	7.21813e9	1.23716	0.618581	+	0.785721i	$0.287708\pi$
$618$	4.08264e9	0.695796
$619$	6.42938e9	1.08956	0.544781	−	0.838579i	$-0.316613\pi$
$619$	6.42938e9	1.08956	0.544781	+	0.838579i	$0.316613\pi$
$620$	−5.47473e8	−0.0922555
$621$	−7.80916e7	−0.0130853
$622$	8.25338e7	0.0137520
$623$	−2.46504e9	−0.408428
$624$	−2.22772e9	−0.367041
$625$	−7.04181e9	−1.15373
$626$	2.67505e9	0.435834
$627$	−2.57214e9	−0.416734
$628$	2.96028e9	0.476952
$629$	−5.14085e9	−0.823679
$630$	−3.62575e9	−0.577704
$631$	6.68363e9	1.05903	0.529516	−	0.848300i	$-0.322373\pi$
$631$	6.68363e9	1.05903	0.529516	+	0.848300i	$0.322373\pi$
$632$	4.42520e9	0.697306
$633$	−8.54244e9	−1.33866
$634$	−2.30619e7	−0.00359404
$635$	−1.12842e10	−1.74889
$636$	−7.49553e9	−1.15532
$637$	1.23592e10	1.89454
$638$	−4.45644e9	−0.679385
$639$	1.74268e9	0.264219
$640$	6.39364e8	0.0964092
$641$	5.78610e9	0.867726	0.433863	−	0.900979i	$-0.357150\pi$
$641$	5.78610e9	0.867726	0.433863	+	0.900979i	$0.357150\pi$
$642$	−1.06281e9	−0.158520
$643$	6.94495e9	1.03022	0.515111	−	0.857124i	$-0.327751\pi$
$643$	6.94495e9	1.03022	0.515111	+	0.857124i	$0.327751\pi$
$644$	1.10122e8	0.0162470
$645$	5.50171e9	0.807307
$646$	6.83363e8	0.0997327
$647$	−3.92658e9	−0.569966	−0.284983	−	0.958533i	$-0.591988\pi$
$647$	−3.92658e9	−0.569966	−0.284983	+	0.958533i	$0.591988\pi$
$648$	3.05857e9	0.441576
$649$	1.36745e10	1.96361
$650$	−1.13826e9	−0.162572
$651$	2.30982e9	0.328129
$652$	1.37577e9	0.194393
$653$	−4.31183e9	−0.605990	−0.302995	−	0.952992i	$-0.597987\pi$
$653$	−4.31183e9	−0.605990	−0.302995	+	0.952992i	$0.597987\pi$
$654$	−1.62622e9	−0.227330
$655$	−1.05729e9	−0.147010
$656$	−2.20967e9	−0.305608
$657$	4.58449e9	0.630685
$658$	−1.44579e10	−1.97840
$659$	3.80602e9	0.518050	0.259025	−	0.965871i	$-0.416599\pi$
$659$	3.80602e9	0.518050	0.259025	+	0.965871i	$0.416599\pi$
$660$	−7.31699e9	−0.990669
$661$	−1.16549e9	−0.156965	−0.0784824	−	0.996916i	$-0.525007\pi$
$661$	−1.16549e9	−0.156965	−0.0784824	+	0.996916i	$0.525007\pi$
$662$	−9.89212e9	−1.32521
$663$	−6.77331e9	−0.902617
$664$	2.22252e9	0.294617
$665$	−3.03829e9	−0.400640
$666$	3.37880e9	0.443204
$667$	9.96716e7	0.0130056
$668$	−5.11065e9	−0.663375
$669$	−1.17740e10	−1.52031
$670$	4.89689e9	0.629011
$671$	−1.03699e10	−1.32510
$672$	−2.69751e9	−0.342902
$673$	−9.09368e9	−1.14997	−0.574986	−	0.818164i	$-0.694992\pi$
$673$	−9.09368e9	−1.14997	−0.574986	+	0.818164i	$0.694992\pi$
$674$	1.82099e9	0.229086
$675$	9.77407e8	0.122324
$676$	1.88143e9	0.234247
$677$	9.19265e7	0.0113862	0.00569312	−	0.999984i	$-0.498188\pi$
$677$	9.19265e7	0.0113862	0.00569312	+	0.999984i	$0.498188\pi$
$678$	−6.14834e9	−0.757624
$679$	−6.30093e9	−0.772433
$680$	1.94397e9	0.237087
$681$	−2.32936e9	−0.282632
$682$	1.48568e9	0.179342
$683$	4.34387e7	0.00521680	0.00260840	−	0.999997i	$-0.499170\pi$
$683$	4.34387e7	0.00521680	0.00260840	+	0.999997i	$0.499170\pi$
$684$	−4.49138e8	−0.0536640
$685$	6.92718e8	0.0823455
$686$	5.39306e9	0.637823
$687$	1.35476e10	1.59409
$688$	1.30460e9	0.152727
$689$	1.98426e10	2.31116
$690$	1.63650e8	0.0189646
$691$	5.26770e9	0.607362	0.303681	−	0.952774i	$-0.401784\pi$
$691$	5.26770e9	0.607362	0.303681	+	0.952774i	$0.401784\pi$
$692$	−9.71014e8	−0.111392
$693$	9.83922e9	1.12304
$694$	−4.58642e7	−0.00520854
$695$	−1.05523e10	−1.19234
$696$	−2.44152e9	−0.274490
$697$	−6.71844e9	−0.751542
$698$	2.09510e9	0.233190
$699$	−1.01420e10	−1.12319
$700$	−1.37831e9	−0.151881
$701$	9.91617e9	1.08725	0.543627	−	0.839327i	$-0.317051\pi$
$701$	9.91617e9	1.08725	0.543627	+	0.839327i	$0.317051\pi$
$702$	−5.06393e9	−0.552469
$703$	2.83136e9	0.307363
$704$	−1.73505e9	−0.187416
$705$	−2.14855e10	−2.30931
$706$	−6.41568e9	−0.686161
$707$	1.20638e10	1.28385
$708$	7.49172e9	0.793351
$709$	−7.42816e9	−0.782743	−0.391372	−	0.920233i	$-0.627999\pi$
$709$	−7.42816e9	−0.782743	−0.391372	+	0.920233i	$0.627999\pi$
$710$	4.15419e9	0.435594
$711$	−8.84305e9	−0.922696
$712$	−8.68647e8	−0.0901910
$713$	−3.32284e7	−0.00343317
$714$	−8.20169e9	−0.843257
$715$	1.93699e10	1.98178
$716$	4.05631e9	0.412986
$717$	−1.88206e10	−1.90685
$718$	−9.22713e9	−0.930317
$719$	6.31538e9	0.633649	0.316824	−	0.948484i	$-0.397383\pi$
$719$	6.31538e9	0.633649	0.316824	+	0.948484i	$0.397383\pi$
$720$	−1.27767e9	−0.127571
$721$	−1.30870e10	−1.30037
$722$	−3.76367e8	−0.0372161
$723$	1.62014e10	1.59429
$724$	2.43997e9	0.238946
$725$	−1.24751e9	−0.121579
$726$	1.10233e10	1.06914
$727$	−3.40136e9	−0.328309	−0.164154	−	0.986435i	$-0.552490\pi$
$727$	−3.40136e9	−0.328309	−0.164154	+	0.986435i	$0.552490\pi$
$728$	7.14099e9	0.685959
$729$	2.05888e9	0.196827
$730$	1.09285e10	1.03976
$731$	3.96659e9	0.375584
$732$	−5.68129e9	−0.535375
$733$	−6.67349e9	−0.625877	−0.312938	−	0.949773i	$-0.601313\pi$
$733$	−6.67349e9	−0.625877	−0.312938	+	0.949773i	$0.601313\pi$
$734$	2.50538e9	0.233850
$735$	2.22400e10	2.06599
$736$	3.88056e7	0.00358775
$737$	−1.32887e10	−1.22278
$738$	4.41567e9	0.404389
$739$	1.61449e10	1.47157	0.735784	−	0.677216i	$-0.236814\pi$
$739$	1.61449e10	1.47157	0.735784	+	0.677216i	$0.236814\pi$
$740$	8.05439e9	0.730671
$741$	3.73045e9	0.336820
$742$	2.40270e10	2.15917
$743$	1.58087e10	1.41395	0.706977	−	0.707237i	$-0.250058\pi$
$743$	1.58087e10	1.41395	0.706977	+	0.707237i	$0.250058\pi$
$744$	8.13949e8	0.0724589
$745$	−2.19085e10	−1.94118
$746$	8.70279e8	0.0767490
$747$	−4.44135e9	−0.389846
$748$	−5.27536e9	−0.460889
$749$	3.40687e9	0.296257
$750$	8.74769e9	0.757144
$751$	3.19115e9	0.274921	0.137460	−	0.990507i	$-0.456106\pi$
$751$	3.19115e9	0.274921	0.137460	+	0.990507i	$0.456106\pi$
$752$	−5.09476e9	−0.436879
$753$	2.12488e10	1.81365
$754$	6.46331e9	0.549105
$755$	−2.35165e9	−0.198865
$756$	−6.13184e9	−0.516136
$757$	4.35759e9	0.365099	0.182549	−	0.983197i	$-0.441565\pi$
$757$	4.35759e9	0.365099	0.182549	+	0.983197i	$0.441565\pi$
$758$	6.97999e9	0.582120
$759$	−4.44098e8	−0.0368665
$760$	−1.07065e9	−0.0884711
$761$	−1.35474e10	−1.11432	−0.557162	−	0.830404i	$-0.688110\pi$
$761$	−1.35474e10	−1.11432	−0.557162	+	0.830404i	$0.688110\pi$
$762$	1.67766e10	1.37360
$763$	5.21286e9	0.424855
$764$	−1.03898e10	−0.842910
$765$	−3.88470e9	−0.313720
$766$	6.94844e8	0.0558582
$767$	−1.98325e10	−1.58706
$768$	−9.50566e8	−0.0757213
$769$	1.00893e9	0.0800053	0.0400026	−	0.999200i	$-0.487263\pi$
$769$	1.00893e9	0.0800053	0.0400026	+	0.999200i	$0.487263\pi$
$770$	2.34547e10	1.85145
$771$	1.34737e10	1.05876
$772$	−7.72741e9	−0.604468
$773$	6.74942e9	0.525579	0.262790	−	0.964853i	$-0.415358\pi$
$773$	6.74942e9	0.525579	0.262790	+	0.964853i	$0.415358\pi$
$774$	−2.60703e9	−0.202093
$775$	4.15892e8	0.0320940
$776$	−2.22037e9	−0.170572
$777$	−3.39819e10	−2.59881
$778$	−1.07989e10	−0.822148
$779$	3.70023e9	0.280445
$780$	1.06120e10	0.800696
$781$	−1.12733e10	−0.846781
$782$	1.17987e8	0.00882290
$783$	−5.54993e9	−0.413163
$784$	5.27367e9	0.390847
$785$	−1.41017e10	−1.04047
$786$	1.57190e9	0.115464
$787$	−2.14479e10	−1.56846	−0.784229	−	0.620471i	$-0.786941\pi$
$787$	−2.14479e10	−1.56846	−0.784229	+	0.620471i	$0.786941\pi$
$788$	9.07128e8	0.0660430
$789$	2.93249e9	0.212553
$790$	−2.10801e10	−1.52117
$791$	1.97086e10	1.41592
$792$	3.46721e9	0.247994
$793$	1.50398e10	1.07099
$794$	−1.92240e10	−1.36292
$795$	3.57060e10	2.52032
$796$	−4.69869e9	−0.330203
$797$	1.01066e10	0.707132	0.353566	−	0.935410i	$-0.384969\pi$
$797$	1.01066e10	0.707132	0.353566	+	0.935410i	$0.384969\pi$
$798$	4.51714e9	0.314669
$799$	−1.54905e10	−1.07436
$800$	−4.85697e8	−0.0335390
$801$	1.73585e9	0.119343
$802$	1.35065e10	0.924553
$803$	−2.96568e10	−2.02125
$804$	−7.28039e9	−0.494036
$805$	−5.24582e8	−0.0354428
$806$	−2.15473e9	−0.144951
$807$	−3.24969e10	−2.17663
$808$	4.25112e9	0.283507
$809$	−1.88120e10	−1.24915	−0.624575	−	0.780965i	$-0.714728\pi$
$809$	−1.88120e10	−1.24915	−0.624575	+	0.780965i	$0.714728\pi$
$810$	−1.45699e10	−0.963294
$811$	−6.71783e9	−0.442238	−0.221119	−	0.975247i	$-0.570971\pi$
$811$	−6.71783e9	−0.442238	−0.221119	+	0.975247i	$0.570971\pi$
$812$	7.82632e9	0.512993
$813$	3.55858e10	2.32252
$814$	−2.18573e10	−1.42040
$815$	−6.55368e9	−0.424066
$816$	−2.89017e9	−0.186212
$817$	−2.18463e9	−0.140152
$818$	4.69015e9	0.299606
$819$	−1.42701e10	−0.907681
$820$	1.05261e10	0.666680
$821$	1.78198e10	1.12383	0.561916	−	0.827195i	$-0.310065\pi$
$821$	1.78198e10	1.12383	0.561916	+	0.827195i	$0.310065\pi$
$822$	−1.02989e9	−0.0646754
$823$	1.39878e10	0.874682	0.437341	−	0.899296i	$-0.355920\pi$
$823$	1.39878e10	0.874682	0.437341	+	0.899296i	$0.355920\pi$
$824$	−4.61168e9	−0.287153
$825$	5.55840e9	0.344636
$826$	−2.40148e10	−1.48269
$827$	−2.37193e10	−1.45825	−0.729125	−	0.684380i	$-0.760073\pi$
$827$	−2.37193e10	−1.45825	−0.729125	+	0.684380i	$0.760073\pi$
$828$	−7.75467e7	−0.00474741
$829$	−8.03327e9	−0.489724	−0.244862	−	0.969558i	$-0.578743\pi$
$829$	−8.03327e9	−0.489724	−0.244862	+	0.969558i	$0.578743\pi$
$830$	−1.05873e10	−0.642705
$831$	−5.86613e9	−0.354607
$832$	2.51639e9	0.151477
$833$	1.60344e10	0.961162
$834$	1.56885e10	0.936484
$835$	2.43453e10	1.44715
$836$	2.90544e9	0.171985
$837$	1.85023e9	0.109065
$838$	−2.25900e10	−1.32605
$839$	1.19133e10	0.696408	0.348204	−	0.937419i	$-0.386792\pi$
$839$	1.19133e10	0.696408	0.348204	+	0.937419i	$0.386792\pi$
$840$	1.28499e10	0.748038
$841$	−1.01663e10	−0.589354
$842$	−4.75870e9	−0.274724
$843$	−1.78243e9	−0.102475
$844$	9.64938e9	0.552460
$845$	−8.96243e9	−0.511008
$846$	1.01811e10	0.578091
$847$	−3.53354e10	−1.99810
$848$	8.46681e9	0.476798
$849$	1.49372e10	0.837710
$850$	−1.47675e9	−0.0824783
$851$	4.88854e8	0.0271910
$852$	−6.17619e9	−0.342123
$853$	1.32637e10	0.731719	0.365859	−	0.930670i	$-0.380775\pi$
$853$	1.32637e10	0.731719	0.365859	+	0.930670i	$0.380775\pi$
$854$	1.82115e10	1.00056
$855$	2.13953e9	0.117068
$856$	1.20054e9	0.0654209
$857$	1.47133e10	0.798506	0.399253	−	0.916841i	$-0.369270\pi$
$857$	1.47133e10	0.798506	0.399253	+	0.916841i	$0.369270\pi$
$858$	−2.87980e10	−1.55653
$859$	2.48645e10	1.33846	0.669228	−	0.743057i	$-0.266625\pi$
$859$	2.48645e10	1.33846	0.669228	+	0.743057i	$0.266625\pi$
$860$	−6.21463e9	−0.333174
$861$	−4.44100e10	−2.37121
$862$	−1.11637e10	−0.593652
$863$	−2.25587e9	−0.119475	−0.0597375	−	0.998214i	$-0.519026\pi$
$863$	−2.25587e9	−0.119475	−0.0597375	+	0.998214i	$0.519026\pi$
$864$	−2.16078e9	−0.113976
$865$	4.62555e9	0.243001
$866$	1.36329e10	0.713305
$867$	1.44616e10	0.753613
$868$	−2.60913e9	−0.135418
$869$	5.72051e10	2.95710
$870$	1.16305e10	0.598798
$871$	1.92730e10	0.988294
$872$	1.83694e9	0.0938185
$873$	4.43704e9	0.225706
$874$	−6.49823e7	−0.00329234
$875$	−2.80408e10	−1.41502
$876$	−1.62478e10	−0.816640
$877$	2.31658e10	1.15971	0.579854	−	0.814721i	$-0.303110\pi$
$877$	2.31658e10	1.15971	0.579854	+	0.814721i	$0.303110\pi$
$878$	2.34566e10	1.16959
$879$	2.09059e10	1.03826
$880$	8.26513e9	0.408847
$881$	−2.21776e10	−1.09270	−0.546348	−	0.837558i	$-0.683982\pi$
$881$	−2.21776e10	−1.09270	−0.546348	+	0.837558i	$0.683982\pi$
$882$	−1.05386e10	−0.517180
$883$	−3.37482e10	−1.64964	−0.824818	−	0.565399i	$-0.808722\pi$
$883$	−3.37482e10	−1.64964	−0.824818	+	0.565399i	$0.808722\pi$
$884$	7.65100e9	0.372508
$885$	−3.56878e10	−1.73069
$886$	−1.41179e10	−0.681948
$887$	3.80869e9	0.183250	0.0916248	−	0.995794i	$-0.470794\pi$
$887$	3.80869e9	0.183250	0.0916248	+	0.995794i	$0.470794\pi$
$888$	−1.19748e10	−0.573881
$889$	−5.37776e10	−2.56712
$890$	4.13791e9	0.196751
$891$	3.95384e10	1.87261
$892$	1.32997e10	0.627429
$893$	8.53149e9	0.400908
$894$	3.25722e10	1.52464
$895$	−1.93228e10	−0.900925
$896$	3.04705e9	0.141515
$897$	6.44087e8	0.0297969
$898$	−7.67854e9	−0.353844
$899$	−2.36152e9	−0.108401
$900$	9.70586e8	0.0443798
$901$	2.57431e10	1.17253
$902$	−2.85647e10	−1.29600
$903$	2.62198e10	1.18501
$904$	6.94505e9	0.312670
$905$	−1.16231e10	−0.521258
$906$	3.49629e9	0.156192
$907$	−2.47690e10	−1.10226	−0.551128	−	0.834421i	$-0.685802\pi$
$907$	−2.47690e10	−1.10226	−0.551128	+	0.834421i	$0.685802\pi$
$908$	2.63120e9	0.116641
$909$	−8.49516e9	−0.375144
$910$	−3.40170e10	−1.49641
$911$	−2.79428e9	−0.122449	−0.0612245	−	0.998124i	$-0.519501\pi$
$911$	−2.79428e9	−0.122449	−0.0612245	+	0.998124i	$0.519501\pi$
$912$	1.59178e9	0.0694867
$913$	2.87308e10	1.24940
$914$	−1.33177e10	−0.576920
$915$	2.70636e10	1.16792
$916$	−1.53031e10	−0.657877
$917$	−5.03876e9	−0.215790
$918$	−6.56978e9	−0.280286
$919$	−2.46502e10	−1.04765	−0.523825	−	0.851826i	$-0.675495\pi$
$919$	−2.46502e10	−1.04765	−0.523825	+	0.851826i	$0.675495\pi$
$920$	−1.84856e8	−0.00782664
$921$	2.71916e10	1.14690
$922$	−3.35459e9	−0.140955
$923$	1.63499e10	0.684400
$924$	−3.48710e10	−1.45416
$925$	−6.11857e9	−0.254187
$926$	−2.38523e10	−0.987171
$927$	9.21569e9	0.379969
$928$	2.75789e9	0.113282
$929$	−2.54613e10	−1.04190	−0.520950	−	0.853587i	$-0.674422\pi$
$929$	−2.54613e10	−1.04190	−0.520950	+	0.853587i	$0.674422\pi$
$930$	−3.87736e9	−0.158068
$931$	−8.83109e9	−0.358666
$932$	1.14562e10	0.463538
$933$	5.84526e8	0.0235623
$934$	5.76064e9	0.231343
$935$	2.51299e10	1.00543
$936$	−5.02859e9	−0.200438
$937$	−3.73474e9	−0.148311	−0.0741553	−	0.997247i	$-0.523626\pi$
$937$	−3.73474e9	−0.148311	−0.0741553	+	0.997247i	$0.523626\pi$
$938$	2.33374e10	0.923299
$939$	1.89454e10	0.746748
$940$	2.42696e10	0.953048
$941$	−1.81296e10	−0.709290	−0.354645	−	0.935001i	$-0.615398\pi$
$941$	−1.81296e10	−0.709290	−0.354645	+	0.935001i	$0.615398\pi$
$942$	2.09655e10	0.817199
$943$	6.38869e8	0.0248097
$944$	−8.46251e9	−0.327414
$945$	2.92098e10	1.12595
$946$	1.68647e10	0.647678
$947$	−1.65653e9	−0.0633831	−0.0316916	−	0.999498i	$-0.510089\pi$
$947$	−1.65653e9	−0.0633831	−0.0316916	+	0.999498i	$0.510089\pi$
$948$	3.13405e10	1.19475
$949$	4.30121e10	1.63365
$950$	8.13329e8	0.0307775
$951$	−1.63331e8	−0.00615794
$952$	9.26448e9	0.348010
$953$	1.48333e10	0.555153	0.277577	−	0.960704i	$-0.410469\pi$
$953$	1.48333e10	0.555153	0.277577	+	0.960704i	$0.410469\pi$
$954$	−1.69195e10	−0.630913
$955$	4.94933e10	1.83880
$956$	2.12594e10	0.786953
$957$	−3.15617e10	−1.16404
$958$	−2.45135e10	−0.900794
$959$	3.30133e9	0.120871
$960$	4.52815e9	0.165185
$961$	−2.67253e10	−0.971385
$962$	3.17002e10	1.14802
$963$	−2.39907e9	−0.0865669
$964$	−1.83007e10	−0.657960
$965$	3.68105e10	1.31864
$966$	7.79915e8	0.0278373
$967$	3.82632e10	1.36078	0.680391	−	0.732850i	$-0.261810\pi$
$967$	3.82632e10	1.36078	0.680391	+	0.732850i	$0.261810\pi$
$968$	−1.24517e10	−0.441231
$969$	4.83976e9	0.170880
$970$	1.05770e10	0.372102
$971$	4.22581e10	1.48130	0.740650	−	0.671891i	$-0.234518\pi$
$971$	4.22581e10	1.48130	0.740650	+	0.671891i	$0.234518\pi$
$972$	1.24319e10	0.434215
$973$	−5.02898e10	−1.75019
$974$	6.93608e9	0.240524
$975$	−8.06150e9	−0.278548
$976$	6.41748e9	0.220948
$977$	−1.68344e10	−0.577520	−0.288760	−	0.957402i	$-0.593243\pi$
$977$	−1.68344e10	−0.577520	−0.288760	+	0.957402i	$0.593243\pi$
$978$	9.74359e9	0.333068
$979$	−1.12291e10	−0.382477
$980$	−2.51219e10	−0.852630
$981$	−3.67083e9	−0.124143
$982$	1.18295e10	0.398637
$983$	4.65458e9	0.156294	0.0781472	−	0.996942i	$-0.475100\pi$
$983$	4.65458e9	0.156294	0.0781472	+	0.996942i	$0.475100\pi$
$984$	−1.56495e10	−0.523621
$985$	−4.32123e9	−0.144072
$986$	8.38528e9	0.278579
$987$	−1.02395e11	−3.38974
$988$	−4.21385e9	−0.139005
$989$	−3.77191e8	−0.0123986
$990$	−1.65165e10	−0.540998
$991$	−9.27680e9	−0.302789	−0.151395	−	0.988473i	$-0.548376\pi$
$991$	−9.27680e9	−0.302789	−0.151395	+	0.988473i	$0.548376\pi$
$992$	−9.19421e8	−0.0299036
$993$	−7.00587e10	−2.27059
$994$	1.97979e10	0.639390
$995$	2.23828e10	0.720334
$996$	1.57405e10	0.504791
$997$	2.47112e10	0.789698	0.394849	−	0.918746i	$-0.370797\pi$
$997$	2.47112e10	0.789698	0.394849	+	0.918746i	$0.370797\pi$
$998$	2.41075e9	0.0767705
$999$	−2.72204e10	−0.863804

Twists

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

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

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -56.6582 q^{3} +64.0000 q^{4} -304.873 q^{5} +453.265 q^{6} -1452.95 q^{7} -512.000 q^{8} +1023.15 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -56.6582 q^{3} +64.0000 q^{4} -304.873 q^{5} +453.265 q^{6} -1452.95 q^{7} -512.000 q^{8} +1023.15 q^{9} +2438.98 q^{10} -6618.68 q^{11} -3626.12 q^{12} +9599.26 q^{13} +11623.6 q^{14} +17273.5 q^{15} +4096.00 q^{16} +12453.8 q^{17} -8185.19 q^{18} -6859.00 q^{19} -19511.8 q^{20} +82321.4 q^{21} +52949.4 q^{22} -1184.25 q^{23} +29009.0 q^{24} +14822.3 q^{25} -76794.1 q^{26} +65941.7 q^{27} -92988.7 q^{28} -84164.1 q^{29} -138188. q^{30} +28058.5 q^{31} -32768.0 q^{32} +375002. q^{33} -99630.1 q^{34} +442964. q^{35} +65481.5 q^{36} -412795. q^{37} +54872.0 q^{38} -543877. q^{39} +156095. q^{40} -539470. q^{41} -658571. q^{42} +318505. q^{43} -423596. q^{44} -311930. q^{45} +9474.02 q^{46} -1.24384e6 q^{47} -232072. q^{48} +1.28752e6 q^{49} -118578. q^{50} -705608. q^{51} +614353. q^{52} +2.06709e6 q^{53} -527534. q^{54} +2.01785e6 q^{55} +743910. q^{56} +388618. q^{57} +673313. q^{58} -2.06604e6 q^{59} +1.10551e6 q^{60} +1.56677e6 q^{61} -224468. q^{62} -1.48658e6 q^{63} +262144. q^{64} -2.92655e6 q^{65} -3.00002e6 q^{66} +2.00776e6 q^{67} +797041. q^{68} +67097.6 q^{69} -3.54371e6 q^{70} +1.70325e6 q^{71} -523852. q^{72} +4.48077e6 q^{73} +3.30236e6 q^{74} -839804. q^{75} -438976. q^{76} +9.61661e6 q^{77} +4.35101e6 q^{78} -8.64298e6 q^{79} -1.24876e6 q^{80} -5.97376e6 q^{81} +4.31576e6 q^{82} -4.34087e6 q^{83} +5.26857e6 q^{84} -3.79681e6 q^{85} -2.54804e6 q^{86} +4.76859e6 q^{87} +3.38876e6 q^{88} +1.69658e6 q^{89} +2.49544e6 q^{90} -1.39472e7 q^{91} -75792.2 q^{92} -1.58974e6 q^{93} +9.95071e6 q^{94} +2.09112e6 q^{95} +1.85658e6 q^{96} +4.33665e6 q^{97} -1.03001e7 q^{98} -6.77189e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} - 61 q^{3} + 128 q^{4} + 175 q^{5} + 488 q^{6} - 2592 q^{7} - 1024 q^{8} - 1145 q^{9}+O(q^{10}) \) 2 * q - 16 * q^2 - 61 * q^3 + 128 * q^4 + 175 * q^5 + 488 * q^6 - 2592 * q^7 - 1024 * q^8 - 1145 * q^9	\( 2 q - 16 q^{2} - 61 q^{3} + 128 q^{4} + 175 q^{5} + 488 q^{6} - 2592 q^{7} - 1024 q^{8} - 1145 q^{9} - 1400 q^{10} + 1045 q^{11} - 3904 q^{12} + 14647 q^{13} + 20736 q^{14} + 15190 q^{15} + 8192 q^{16} + 29616 q^{17} + 9160 q^{18} - 13718 q^{19} + 11200 q^{20} + 87267 q^{21} - 8360 q^{22} + 69985 q^{23} + 31232 q^{24} + 166975 q^{25} - 117176 q^{26} + 84851 q^{27} - 165888 q^{28} + 138821 q^{29} - 121520 q^{30} - 199396 q^{31} - 65536 q^{32} + 341728 q^{33} - 236928 q^{34} - 103635 q^{35} - 73280 q^{36} - 67840 q^{37} + 109744 q^{38} - 565793 q^{39} - 89600 q^{40} - 539350 q^{41} - 698136 q^{42} + 602639 q^{43} + 66880 q^{44} - 1352365 q^{45} - 559880 q^{46} - 1031975 q^{47} - 249856 q^{48} + 1761412 q^{49} - 1335800 q^{50} - 780123 q^{51} + 937408 q^{52} + 2138263 q^{53} - 678808 q^{54} + 5695445 q^{55} + 1327104 q^{56} + 418399 q^{57} - 1110568 q^{58} - 3936369 q^{59} + 972160 q^{60} - 1027655 q^{61} + 1595168 q^{62} + 983049 q^{63} + 524288 q^{64} - 504280 q^{65} - 2733824 q^{66} + 764949 q^{67} + 1895424 q^{68} - 241907 q^{69} + 829080 q^{70} - 3572084 q^{71} + 586240 q^{72} + 9069522 q^{73} + 542720 q^{74} - 1500425 q^{75} - 877952 q^{76} + 887283 q^{77} + 4526344 q^{78} - 2753414 q^{79} + 716800 q^{80} - 1314122 q^{81} + 4314800 q^{82} - 7643046 q^{83} + 5585088 q^{84} + 4438875 q^{85} - 4821112 q^{86} + 3800423 q^{87} - 535040 q^{88} + 1393620 q^{89} + 10818920 q^{90} - 19696869 q^{91} + 4479040 q^{92} - 602176 q^{93} + 8255800 q^{94} - 1200325 q^{95} + 1998848 q^{96} - 6921466 q^{97} - 14091296 q^{98} - 23387893 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 61 * q^3 + 128 * q^4 + 175 * q^5 + 488 * q^6 - 2592 * q^7 - 1024 * q^8 - 1145 * q^9 - 1400 * q^10 + 1045 * q^11 - 3904 * q^12 + 14647 * q^13 + 20736 * q^14 + 15190 * q^15 + 8192 * q^16 + 29616 * q^17 + 9160 * q^18 - 13718 * q^19 + 11200 * q^20 + 87267 * q^21 - 8360 * q^22 + 69985 * q^23 + 31232 * q^24 + 166975 * q^25 - 117176 * q^26 + 84851 * q^27 - 165888 * q^28 + 138821 * q^29 - 121520 * q^30 - 199396 * q^31 - 65536 * q^32 + 341728 * q^33 - 236928 * q^34 - 103635 * q^35 - 73280 * q^36 - 67840 * q^37 + 109744 * q^38 - 565793 * q^39 - 89600 * q^40 - 539350 * q^41 - 698136 * q^42 + 602639 * q^43 + 66880 * q^44 - 1352365 * q^45 - 559880 * q^46 - 1031975 * q^47 - 249856 * q^48 + 1761412 * q^49 - 1335800 * q^50 - 780123 * q^51 + 937408 * q^52 + 2138263 * q^53 - 678808 * q^54 + 5695445 * q^55 + 1327104 * q^56 + 418399 * q^57 - 1110568 * q^58 - 3936369 * q^59 + 972160 * q^60 - 1027655 * q^61 + 1595168 * q^62 + 983049 * q^63 + 524288 * q^64 - 504280 * q^65 - 2733824 * q^66 + 764949 * q^67 + 1895424 * q^68 - 241907 * q^69 + 829080 * q^70 - 3572084 * q^71 + 586240 * q^72 + 9069522 * q^73 + 542720 * q^74 - 1500425 * q^75 - 877952 * q^76 + 887283 * q^77 + 4526344 * q^78 - 2753414 * q^79 + 716800 * q^80 - 1314122 * q^81 + 4314800 * q^82 - 7643046 * q^83 + 5585088 * q^84 + 4438875 * q^85 - 4821112 * q^86 + 3800423 * q^87 - 535040 * q^88 + 1393620 * q^89 + 10818920 * q^90 - 19696869 * q^91 + 4479040 * q^92 - 602176 * q^93 + 8255800 * q^94 - 1200325 * q^95 + 1998848 * q^96 - 6921466 * q^97 - 14091296 * q^98 - 23387893 * q^99

Introduction

L-functions

Modular forms

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