LMFDB - Embedded newform 354.8.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,8,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("354.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	354.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:	$110.584299021$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6162x^{3} - 12837x^{2} + 3760259x - 17264060 $ x^5 - x^4 - 6162x^3 - 12837x^2 + 3760259*x - 17264060
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$21.6805$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +32.4075 q^{5} +216.000 q^{6} -798.662 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +32.4075 q^{5} +216.000 q^{6} -798.662 q^{7} +512.000 q^{8} +729.000 q^{9} +259.260 q^{10} -937.000 q^{11} +1728.00 q^{12} +6726.51 q^{13} -6389.29 q^{14} +875.003 q^{15} +4096.00 q^{16} -29224.5 q^{17} +5832.00 q^{18} -9144.51 q^{19} +2074.08 q^{20} -21563.9 q^{21} -7496.00 q^{22} -9147.46 q^{23} +13824.0 q^{24} -77074.8 q^{25} +53812.1 q^{26} +19683.0 q^{27} -51114.3 q^{28} +30888.3 q^{29} +7000.02 q^{30} +116008. q^{31} +32768.0 q^{32} -25299.0 q^{33} -233796. q^{34} -25882.6 q^{35} +46656.0 q^{36} -200678. q^{37} -73156.1 q^{38} +181616. q^{39} +16592.6 q^{40} +713020. q^{41} -172511. q^{42} -944293. q^{43} -59968.0 q^{44} +23625.1 q^{45} -73179.7 q^{46} -490809. q^{47} +110592. q^{48} -185683. q^{49} -616598. q^{50} -789061. q^{51} +430496. q^{52} -1.69951e6 q^{53} +157464. q^{54} -30365.8 q^{55} -408915. q^{56} -246902. q^{57} +247107. q^{58} +205379. q^{59} +56000.2 q^{60} -200868. q^{61} +928064. q^{62} -582224. q^{63} +262144. q^{64} +217989. q^{65} -202392. q^{66} -2.86489e6 q^{67} -1.87037e6 q^{68} -246982. q^{69} -207061. q^{70} -3.28117e6 q^{71} +373248. q^{72} +643137. q^{73} -1.60542e6 q^{74} -2.08102e6 q^{75} -585249. q^{76} +748346. q^{77} +1.45293e6 q^{78} +4.50197e6 q^{79} +132741. q^{80} +531441. q^{81} +5.70416e6 q^{82} -735909. q^{83} -1.38009e6 q^{84} -947092. q^{85} -7.55434e6 q^{86} +833985. q^{87} -479744. q^{88} -5.25981e6 q^{89} +189001. q^{90} -5.37220e6 q^{91} -585438. q^{92} +3.13222e6 q^{93} -3.92647e6 q^{94} -296351. q^{95} +884736. q^{96} +1.59421e7 q^{97} -1.48546e6 q^{98} -683073. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 40 q^{2} + 135 q^{3} + 320 q^{4} + 164 q^{5} + 1080 q^{6} - 76 q^{7} + 2560 q^{8} + 3645 q^{9}+O(q^{10}) $ 5 * q + 40 * q^2 + 135 * q^3 + 320 * q^4 + 164 * q^5 + 1080 * q^6 - 76 * q^7 + 2560 * q^8 + 3645 * q^9	\( 5 q + 40 q^{2} + 135 q^{3} + 320 q^{4} + 164 q^{5} + 1080 q^{6} - 76 q^{7} + 2560 q^{8} + 3645 q^{9} + 1312 q^{10} - 15730 q^{11} + 8640 q^{12} - 21854 q^{13} - 608 q^{14} + 4428 q^{15} + 20480 q^{16} - 34548 q^{17} + 29160 q^{18} - 43828 q^{19} + 10496 q^{20} - 2052 q^{21} - 125840 q^{22} - 110582 q^{23} + 69120 q^{24} - 174577 q^{25} - 174832 q^{26} + 98415 q^{27} - 4864 q^{28} - 307558 q^{29} + 35424 q^{30} - 277994 q^{31} + 163840 q^{32} - 424710 q^{33} - 276384 q^{34} - 764338 q^{35} + 233280 q^{36} - 853778 q^{37} - 350624 q^{38} - 590058 q^{39} + 83968 q^{40} + 131342 q^{41} - 16416 q^{42} - 721996 q^{43} - 1006720 q^{44} + 119556 q^{45} - 884656 q^{46} - 358832 q^{47} + 552960 q^{48} - 207141 q^{49} - 1396616 q^{50} - 932796 q^{51} - 1398656 q^{52} + 1006180 q^{53} + 787320 q^{54} + 81944 q^{55} - 38912 q^{56} - 1183356 q^{57} - 2460464 q^{58} + 1026895 q^{59} + 283392 q^{60} + 101158 q^{61} - 2223952 q^{62} - 55404 q^{63} + 1310720 q^{64} - 3138378 q^{65} - 3397680 q^{66} - 6362512 q^{67} - 2211072 q^{68} - 2985714 q^{69} - 6114704 q^{70} - 8877414 q^{71} + 1866240 q^{72} - 4881862 q^{73} - 6830224 q^{74} - 4713579 q^{75} - 2804992 q^{76} - 2205694 q^{77} - 4720464 q^{78} - 2769432 q^{79} + 671744 q^{80} + 2657205 q^{81} + 1050736 q^{82} - 1430156 q^{83} - 131328 q^{84} - 7564814 q^{85} - 5775968 q^{86} - 8304066 q^{87} - 8053760 q^{88} - 17172176 q^{89} + 956448 q^{90} - 932474 q^{91} - 7077248 q^{92} - 7505838 q^{93} - 2870656 q^{94} + 15962708 q^{95} + 4423680 q^{96} + 20863830 q^{97} - 1657128 q^{98} - 11467170 q^{99}+O(q^{100}) \) 5 * q + 40 * q^2 + 135 * q^3 + 320 * q^4 + 164 * q^5 + 1080 * q^6 - 76 * q^7 + 2560 * q^8 + 3645 * q^9 + 1312 * q^10 - 15730 * q^11 + 8640 * q^12 - 21854 * q^13 - 608 * q^14 + 4428 * q^15 + 20480 * q^16 - 34548 * q^17 + 29160 * q^18 - 43828 * q^19 + 10496 * q^20 - 2052 * q^21 - 125840 * q^22 - 110582 * q^23 + 69120 * q^24 - 174577 * q^25 - 174832 * q^26 + 98415 * q^27 - 4864 * q^28 - 307558 * q^29 + 35424 * q^30 - 277994 * q^31 + 163840 * q^32 - 424710 * q^33 - 276384 * q^34 - 764338 * q^35 + 233280 * q^36 - 853778 * q^37 - 350624 * q^38 - 590058 * q^39 + 83968 * q^40 + 131342 * q^41 - 16416 * q^42 - 721996 * q^43 - 1006720 * q^44 + 119556 * q^45 - 884656 * q^46 - 358832 * q^47 + 552960 * q^48 - 207141 * q^49 - 1396616 * q^50 - 932796 * q^51 - 1398656 * q^52 + 1006180 * q^53 + 787320 * q^54 + 81944 * q^55 - 38912 * q^56 - 1183356 * q^57 - 2460464 * q^58 + 1026895 * q^59 + 283392 * q^60 + 101158 * q^61 - 2223952 * q^62 - 55404 * q^63 + 1310720 * q^64 - 3138378 * q^65 - 3397680 * q^66 - 6362512 * q^67 - 2211072 * q^68 - 2985714 * q^69 - 6114704 * q^70 - 8877414 * q^71 + 1866240 * q^72 - 4881862 * q^73 - 6830224 * q^74 - 4713579 * q^75 - 2804992 * q^76 - 2205694 * q^77 - 4720464 * q^78 - 2769432 * q^79 + 671744 * q^80 + 2657205 * q^81 + 1050736 * q^82 - 1430156 * q^83 - 131328 * q^84 - 7564814 * q^85 - 5775968 * q^86 - 8304066 * q^87 - 8053760 * q^88 - 17172176 * q^89 + 956448 * q^90 - 932474 * q^91 - 7077248 * q^92 - 7505838 * q^93 - 2870656 * q^94 + 15962708 * q^95 + 4423680 * q^96 + 20863830 * q^97 - 1657128 * q^98 - 11467170 * 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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	64.0000	0.500000
$5$	32.4075	0.115945	0.0579723	−	0.998318i	$-0.481536\pi$
$5$	32.4075	0.115945	0.0579723	+	0.998318i	$0.481536\pi$
$6$	216.000	0.408248
$7$	−798.662	−0.880075	−0.440037	−	0.897979i	$-0.645035\pi$
$7$	−798.662	−0.880075	−0.440037	+	0.897979i	$0.645035\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	259.260	0.0819852
$11$	−937.000	−0.212259	−0.106129	−	0.994352i	$-0.533846\pi$
$11$	−937.000	−0.212259	−0.106129	+	0.994352i	$0.533846\pi$
$12$	1728.00	0.288675
$13$	6726.51	0.849157	0.424578	−	0.905391i	$-0.360422\pi$
$13$	6726.51	0.849157	0.424578	+	0.905391i	$0.360422\pi$
$14$	−6389.29	−0.622307
$15$	875.003	0.0669407
$16$	4096.00	0.250000
$17$	−29224.5	−1.44270	−0.721349	−	0.692572i	$-0.756478\pi$
$17$	−29224.5	−1.44270	−0.721349	+	0.692572i	$0.756478\pi$
$18$	5832.00	0.235702
$19$	−9144.51	−0.305860	−0.152930	−	0.988237i	$-0.548871\pi$
$19$	−9144.51	−0.305860	−0.152930	+	0.988237i	$0.548871\pi$
$20$	2074.08	0.0579723
$21$	−21563.9	−0.508112
$22$	−7496.00	−0.150090
$23$	−9147.46	−0.156767	−0.0783833	−	0.996923i	$-0.524976\pi$
$23$	−9147.46	−0.156767	−0.0783833	+	0.996923i	$0.524976\pi$
$24$	13824.0	0.204124
$25$	−77074.8	−0.986557
$26$	53812.1	0.600445
$27$	19683.0	0.192450
$28$	−51114.3	−0.440037
$29$	30888.3	0.235181	0.117590	−	0.993062i	$-0.462483\pi$
$29$	30888.3	0.235181	0.117590	+	0.993062i	$0.462483\pi$
$30$	7000.02	0.0473342
$31$	116008.	0.699394	0.349697	−	0.936863i	$-0.386284\pi$
$31$	116008.	0.699394	0.349697	+	0.936863i	$0.386284\pi$
$32$	32768.0	0.176777
$33$	−25299.0	−0.122548
$34$	−233796.	−1.02014
$35$	−25882.6	−0.102040
$36$	46656.0	0.166667
$37$	−200678.	−0.651318	−0.325659	−	0.945487i	$-0.605586\pi$
$37$	−200678.	−0.651318	−0.325659	+	0.945487i	$0.605586\pi$
$38$	−73156.1	−0.216276
$39$	181616.	0.490261
$40$	16592.6	0.0409926
$41$	713020.	1.61569	0.807846	−	0.589394i	$-0.200633\pi$
$41$	713020.	1.61569	0.807846	+	0.589394i	$0.200633\pi$
$42$	−172511.	−0.359289
$43$	−944293.	−1.81120	−0.905601	−	0.424130i	$-0.860580\pi$
$43$	−944293.	−1.81120	−0.905601	+	0.424130i	$0.860580\pi$
$44$	−59968.0	−0.106129
$45$	23625.1	0.0386482
$46$	−73179.7	−0.110851
$47$	−490809.	−0.689556	−0.344778	−	0.938684i	$-0.612046\pi$
$47$	−490809.	−0.689556	−0.344778	+	0.938684i	$0.612046\pi$
$48$	110592.	0.144338
$49$	−185683.	−0.225468
$50$	−616598.	−0.697601
$51$	−789061.	−0.832942
$52$	430496.	0.424578
$53$	−1.69951e6	−1.56804	−0.784021	−	0.620735i	$-0.786834\pi$
$53$	−1.69951e6	−1.56804	−0.784021	+	0.620735i	$0.786834\pi$
$54$	157464.	0.136083
$55$	−30365.8	−0.0246102
$56$	−408915.	−0.311153
$57$	−246902.	−0.176588
$58$	247107.	0.166298
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	56000.2	0.0334703
$61$	−200868.	−0.113307	−0.0566536	−	0.998394i	$-0.518043\pi$
$61$	−200868.	−0.113307	−0.0566536	+	0.998394i	$0.518043\pi$
$62$	928064.	0.494547
$63$	−582224.	−0.293358
$64$	262144.	0.125000
$65$	217989.	0.0984552
$66$	−202392.	−0.0866542
$67$	−2.86489e6	−1.16371	−0.581856	−	0.813292i	$-0.697673\pi$
$67$	−2.86489e6	−1.16371	−0.581856	+	0.813292i	$0.697673\pi$
$68$	−1.87037e6	−0.721349
$69$	−246982.	−0.0905092
$70$	−207061.	−0.0721531
$71$	−3.28117e6	−1.08799	−0.543995	−	0.839089i	$-0.683089\pi$
$71$	−3.28117e6	−1.08799	−0.543995	+	0.839089i	$0.683089\pi$
$72$	373248.	0.117851
$73$	643137.	0.193497	0.0967483	−	0.995309i	$-0.469156\pi$
$73$	643137.	0.193497	0.0967483	+	0.995309i	$0.469156\pi$
$74$	−1.60542e6	−0.460552
$75$	−2.08102e6	−0.569589
$76$	−585249.	−0.152930
$77$	748346.	0.186803
$78$	1.45293e6	0.346667
$79$	4.50197e6	1.02733	0.513663	−	0.857992i	$-0.328288\pi$
$79$	4.50197e6	1.02733	0.513663	+	0.857992i	$0.328288\pi$
$80$	132741.	0.0289862
$81$	531441.	0.111111
$82$	5.70416e6	1.14247
$83$	−735909.	−0.141270	−0.0706352	−	0.997502i	$-0.522503\pi$
$83$	−735909.	−0.141270	−0.0706352	+	0.997502i	$0.522503\pi$
$84$	−1.38009e6	−0.254056
$85$	−947092.	−0.167273
$86$	−7.55434e6	−1.28071
$87$	833985.	0.135782
$88$	−479744.	−0.0750448
$89$	−5.25981e6	−0.790870	−0.395435	−	0.918494i	$-0.629406\pi$
$89$	−5.25981e6	−0.790870	−0.395435	+	0.918494i	$0.629406\pi$
$90$	189001.	0.0273284
$91$	−5.37220e6	−0.747322
$92$	−585438.	−0.0783833
$93$	3.13222e6	0.403796
$94$	−3.92647e6	−0.487590
$95$	−296351.	−0.0354628
$96$	884736.	0.102062
$97$	1.59421e7	1.77356	0.886778	−	0.462195i	$-0.152938\pi$
$97$	1.59421e7	1.77356	0.886778	+	0.462195i	$0.152938\pi$
$98$	−1.48546e6	−0.159430
$99$	−683073.	−0.0707529
$100$	−4.93278e6	−0.493278
$101$	−1.62080e7	−1.56532	−0.782661	−	0.622448i	$-0.786138\pi$
$101$	−1.62080e7	−1.56532	−0.782661	+	0.622448i	$0.786138\pi$
$102$	−6.31248e6	−0.588979
$103$	1.57063e7	1.41626	0.708129	−	0.706083i	$-0.249539\pi$
$103$	1.57063e7	1.41626	0.708129	+	0.706083i	$0.249539\pi$
$104$	3.44397e6	0.300222
$105$	−698831.	−0.0589128
$106$	−1.35961e7	−1.10877
$107$	−1.31830e6	−0.104033	−0.0520164	−	0.998646i	$-0.516565\pi$
$107$	−1.31830e6	−0.104033	−0.0520164	+	0.998646i	$0.516565\pi$
$108$	1.25971e6	0.0962250
$109$	−2.65953e7	−1.96704	−0.983518	−	0.180811i	$-0.942128\pi$
$109$	−2.65953e7	−1.96704	−0.983518	+	0.180811i	$0.942128\pi$
$110$	−242927.	−0.0174021
$111$	−5.41830e6	−0.376039
$112$	−3.27132e6	−0.220019
$113$	2.07521e7	1.35297	0.676483	−	0.736458i	$-0.263503\pi$
$113$	2.07521e7	1.35297	0.676483	+	0.736458i	$0.263503\pi$
$114$	−1.97521e6	−0.124867
$115$	−296447.	−0.0181762
$116$	1.97685e6	0.117590
$117$	4.90362e6	0.283052
$118$	1.64303e6	0.0920575
$119$	2.33405e7	1.26968
$120$	448001.	0.0236671
$121$	−1.86092e7	−0.954946
$122$	−1.60695e6	−0.0801202
$123$	1.92516e7	0.932820
$124$	7.42451e6	0.349697
$125$	−5.02964e6	−0.230331
$126$	−4.65779e6	−0.207436
$127$	−3.41430e7	−1.47907	−0.739535	−	0.673118i	$-0.764954\pi$
$127$	−3.41430e7	−1.47907	−0.739535	+	0.673118i	$0.764954\pi$
$128$	2.09715e6	0.0883883
$129$	−2.54959e7	−1.04570
$130$	1.74391e6	0.0696183
$131$	1.80511e7	0.701544	0.350772	−	0.936461i	$-0.385919\pi$
$131$	1.80511e7	0.701544	0.350772	+	0.936461i	$0.385919\pi$
$132$	−1.61914e6	−0.0612738
$133$	7.30337e6	0.269180
$134$	−2.29191e7	−0.822869
$135$	637877.	0.0223136
$136$	−1.49629e7	−0.510071
$137$	−2.93922e7	−0.976584	−0.488292	−	0.872680i	$-0.662380\pi$
$137$	−2.93922e7	−0.976584	−0.488292	+	0.872680i	$0.662380\pi$
$138$	−1.97585e6	−0.0639997
$139$	1.91531e7	0.604905	0.302453	−	0.953164i	$-0.402195\pi$
$139$	1.91531e7	0.604905	0.302453	+	0.953164i	$0.402195\pi$
$140$	−1.65649e6	−0.0510200
$141$	−1.32518e7	−0.398116
$142$	−2.62494e7	−0.769324
$143$	−6.30274e6	−0.180241
$144$	2.98598e6	0.0833333
$145$	1.00101e6	0.0272679
$146$	5.14509e6	0.136823
$147$	−5.01343e6	−0.130174
$148$	−1.28434e7	−0.325659
$149$	−5.32867e7	−1.31967	−0.659837	−	0.751409i	$-0.729375\pi$
$149$	−5.32867e7	−1.31967	−0.659837	+	0.751409i	$0.729375\pi$
$150$	−1.66481e7	−0.402760
$151$	3.85819e7	0.911937	0.455968	−	0.889996i	$-0.349293\pi$
$151$	3.85819e7	0.911937	0.455968	+	0.889996i	$0.349293\pi$
$152$	−4.68199e6	−0.108138
$153$	−2.13046e7	−0.480899
$154$	5.98677e6	0.132090
$155$	3.75953e6	0.0810910
$156$	1.16234e7	0.245130
$157$	2.66282e7	0.549153	0.274577	−	0.961565i	$-0.411462\pi$
$157$	2.66282e7	0.549153	0.274577	+	0.961565i	$0.411462\pi$
$158$	3.60158e7	0.726429
$159$	−4.58867e7	−0.905309
$160$	1.06193e6	0.0204963
$161$	7.30573e6	0.137966
$162$	4.25153e6	0.0785674
$163$	−7.57091e7	−1.36928	−0.684639	−	0.728883i	$-0.740040\pi$
$163$	−7.57091e7	−1.36928	−0.684639	+	0.728883i	$0.740040\pi$
$164$	4.56333e7	0.807846
$165$	−819878.	−0.0142087
$166$	−5.88727e6	−0.0998932
$167$	7.02008e6	0.116637	0.0583183	−	0.998298i	$-0.481426\pi$
$167$	7.02008e6	0.116637	0.0583183	+	0.998298i	$0.481426\pi$
$168$	−1.10407e7	−0.179645
$169$	−1.75026e7	−0.278933
$170$	−7.57674e6	−0.118280
$171$	−6.66635e6	−0.101953
$172$	−6.04347e7	−0.905601
$173$	3.26096e7	0.478833	0.239416	−	0.970917i	$-0.423044\pi$
$173$	3.26096e7	0.478833	0.239416	+	0.970917i	$0.423044\pi$
$174$	6.67188e6	0.0960121
$175$	6.15566e7	0.868244
$176$	−3.83795e6	−0.0530647
$177$	5.54523e6	0.0751646
$178$	−4.20785e7	−0.559230
$179$	−6.96853e6	−0.0908146	−0.0454073	−	0.998969i	$-0.514459\pi$
$179$	−6.96853e6	−0.0908146	−0.0454073	+	0.998969i	$0.514459\pi$
$180$	1.51200e6	0.0193241
$181$	−6.92296e7	−0.867794	−0.433897	−	0.900963i	$-0.642862\pi$
$181$	−6.92296e7	−0.867794	−0.433897	+	0.900963i	$0.642862\pi$
$182$	−4.29776e7	−0.528436
$183$	−5.42345e6	−0.0654179
$184$	−4.68350e6	−0.0554253
$185$	−6.50347e6	−0.0755168
$186$	2.50577e7	0.285527
$187$	2.73833e7	0.306225
$188$	−3.14118e7	−0.344778
$189$	−1.57201e7	−0.169371
$190$	−2.37081e6	−0.0250760
$191$	−2.14626e7	−0.222878	−0.111439	−	0.993771i	$-0.535546\pi$
$191$	−2.14626e7	−0.222878	−0.111439	+	0.993771i	$0.535546\pi$
$192$	7.07789e6	0.0721688
$193$	1.80130e8	1.80358	0.901792	−	0.432170i	$-0.142252\pi$
$193$	1.80130e8	1.80358	0.901792	+	0.432170i	$0.142252\pi$
$194$	1.27537e8	1.25409
$195$	5.88571e6	0.0568431
$196$	−1.18837e7	−0.112734
$197$	−2.55988e6	−0.0238554	−0.0119277	−	0.999929i	$-0.503797\pi$
$197$	−2.55988e6	−0.0238554	−0.0119277	+	0.999929i	$0.503797\pi$
$198$	−5.46459e6	−0.0500298
$199$	3.83164e7	0.344666	0.172333	−	0.985039i	$-0.444869\pi$
$199$	3.83164e7	0.344666	0.172333	+	0.985039i	$0.444869\pi$
$200$	−3.94623e7	−0.348801
$201$	−7.73519e7	−0.671870
$202$	−1.29664e8	−1.10685
$203$	−2.46693e7	−0.206977
$204$	−5.04999e7	−0.416471
$205$	2.31072e7	0.187331
$206$	1.25650e8	1.00145
$207$	−6.66850e6	−0.0522555
$208$	2.75518e7	0.212289
$209$	8.56841e6	0.0649214
$210$	−5.59065e6	−0.0416576
$211$	−1.99720e7	−0.146364	−0.0731818	−	0.997319i	$-0.523315\pi$
$211$	−1.99720e7	−0.146364	−0.0731818	+	0.997319i	$0.523315\pi$
$212$	−1.08768e8	−0.784021
$213$	−8.85916e7	−0.628151
$214$	−1.05464e7	−0.0735623
$215$	−3.06022e7	−0.209999
$216$	1.00777e7	0.0680414
$217$	−9.26512e7	−0.615519
$218$	−2.12762e8	−1.39090
$219$	1.73647e7	0.111715
$220$	−1.94341e6	−0.0123051
$221$	−1.96579e8	−1.22508
$222$	−4.33464e7	−0.265900
$223$	1.00374e8	0.606114	0.303057	−	0.952972i	$-0.401993\pi$
$223$	1.00374e8	0.606114	0.303057	+	0.952972i	$0.401993\pi$
$224$	−2.61705e7	−0.155577
$225$	−5.61875e7	−0.328852
$226$	1.66017e8	0.956691
$227$	7.81708e7	0.443562	0.221781	−	0.975097i	$-0.428813\pi$
$227$	7.81708e7	0.443562	0.221781	+	0.975097i	$0.428813\pi$
$228$	−1.58017e7	−0.0882942
$229$	−3.10614e8	−1.70922	−0.854608	−	0.519274i	$-0.826202\pi$
$229$	−3.10614e8	−1.70922	−0.854608	+	0.519274i	$0.826202\pi$
$230$	−2.37157e6	−0.0128525
$231$	2.02053e7	0.107851
$232$	1.58148e7	0.0831489
$233$	9.72386e7	0.503608	0.251804	−	0.967778i	$-0.418976\pi$
$233$	9.72386e7	0.503608	0.251804	+	0.967778i	$0.418976\pi$
$234$	3.92290e7	0.200148
$235$	−1.59059e7	−0.0799504
$236$	1.31443e7	0.0650945
$237$	1.21553e8	0.593126
$238$	1.86724e8	0.897801
$239$	−4.03151e8	−1.91018	−0.955092	−	0.296311i	$-0.904244\pi$
$239$	−4.03151e8	−1.91018	−0.955092	+	0.296311i	$0.904244\pi$
$240$	3.58401e6	0.0167352
$241$	−1.95842e8	−0.901251	−0.450625	−	0.892713i	$-0.648799\pi$
$241$	−1.95842e8	−0.901251	−0.450625	+	0.892713i	$0.648799\pi$
$242$	−1.48874e8	−0.675249
$243$	1.43489e7	0.0641500
$244$	−1.28556e7	−0.0566536
$245$	−6.01751e6	−0.0261418
$246$	1.54012e8	0.659603
$247$	−6.15106e7	−0.259723
$248$	5.93961e7	0.247273
$249$	−1.98696e7	−0.0815625
$250$	−4.02371e7	−0.162868
$251$	4.92241e8	1.96480	0.982402	−	0.186777i	$-0.0598042\pi$
$251$	4.92241e8	1.96480	0.982402	+	0.186777i	$0.0598042\pi$
$252$	−3.72624e7	−0.146679
$253$	8.57118e6	0.0332750
$254$	−2.73144e8	−1.04586
$255$	−2.55715e7	−0.0965751
$256$	1.67772e7	0.0625000
$257$	1.93576e8	0.711355	0.355677	−	0.934609i	$-0.384250\pi$
$257$	1.93576e8	0.711355	0.355677	+	0.934609i	$0.384250\pi$
$258$	−2.03967e8	−0.739420
$259$	1.60274e8	0.573209
$260$	1.39513e7	0.0492276
$261$	2.25176e7	0.0783936
$262$	1.44409e8	0.496067
$263$	−4.02613e8	−1.36472	−0.682358	−	0.731018i	$-0.739045\pi$
$263$	−4.02613e8	−1.36472	−0.682358	+	0.731018i	$0.739045\pi$
$264$	−1.29531e7	−0.0433271
$265$	−5.50768e7	−0.181806
$266$	5.84270e7	0.190339
$267$	−1.42015e8	−0.456609
$268$	−1.83353e8	−0.581856
$269$	−1.03196e8	−0.323243	−0.161621	−	0.986853i	$-0.551672\pi$
$269$	−1.03196e8	−0.323243	−0.161621	+	0.986853i	$0.551672\pi$
$270$	5.10302e6	0.0157781
$271$	2.62323e8	0.800652	0.400326	−	0.916373i	$-0.368897\pi$
$271$	2.62323e8	0.800652	0.400326	+	0.916373i	$0.368897\pi$
$272$	−1.19703e8	−0.360674
$273$	−1.45049e8	−0.431466
$274$	−2.35137e8	−0.690549
$275$	7.22191e7	0.209405
$276$	−1.58068e7	−0.0452546
$277$	4.06543e8	1.14928	0.574642	−	0.818405i	$-0.305141\pi$
$277$	4.06543e8	1.14928	0.574642	+	0.818405i	$0.305141\pi$
$278$	1.53225e8	0.427733
$279$	8.45699e7	0.233131
$280$	−1.32519e7	−0.0360766
$281$	2.19358e8	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$281$	2.19358e8	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$282$	−1.06015e8	−0.281510
$283$	4.49228e8	1.17819	0.589093	−	0.808065i	$-0.299485\pi$
$283$	4.49228e8	1.17819	0.589093	+	0.808065i	$0.299485\pi$
$284$	−2.09995e8	−0.543995
$285$	−8.00147e6	−0.0204745
$286$	−5.04219e7	−0.127450
$287$	−5.69462e8	−1.42193
$288$	2.38879e7	0.0589256
$289$	4.43731e8	1.08138
$290$	8.00812e6	0.0192813
$291$	4.30437e8	1.02396
$292$	4.11608e7	0.0967483
$293$	7.31207e8	1.69826	0.849129	−	0.528185i	$-0.177127\pi$
$293$	7.31207e8	1.69826	0.849129	+	0.528185i	$0.177127\pi$
$294$	−4.01075e7	−0.0920469
$295$	6.65582e6	0.0150947
$296$	−1.02747e8	−0.230276
$297$	−1.84430e7	−0.0408492
$298$	−4.26293e8	−0.933150
$299$	−6.15305e7	−0.133119
$300$	−1.33185e8	−0.284794
$301$	7.54170e8	1.59399
$302$	3.08655e8	0.644837
$303$	−4.37615e8	−0.903739
$304$	−3.74559e7	−0.0764650
$305$	−6.50964e6	−0.0131374
$306$	−1.70437e8	−0.340047
$307$	−4.93322e8	−0.973075	−0.486537	−	0.873660i	$-0.661740\pi$
$307$	−4.93322e8	−0.973075	−0.486537	+	0.873660i	$0.661740\pi$
$308$	4.78942e7	0.0934017
$309$	4.24069e8	0.817677
$310$	3.00762e7	0.0573400
$311$	−6.21606e8	−1.17180	−0.585900	−	0.810383i	$-0.699259\pi$
$311$	−6.21606e8	−1.17180	−0.585900	+	0.810383i	$0.699259\pi$
$312$	9.29872e7	0.173333
$313$	−4.74186e8	−0.874065	−0.437032	−	0.899446i	$-0.643970\pi$
$313$	−4.74186e8	−0.874065	−0.437032	+	0.899446i	$0.643970\pi$
$314$	2.13026e8	0.388310
$315$	−1.88684e7	−0.0340133
$316$	2.88126e8	0.513663
$317$	6.92992e8	1.22186	0.610929	−	0.791685i	$-0.290796\pi$
$317$	6.92992e8	1.22186	0.610929	+	0.791685i	$0.290796\pi$
$318$	−3.67094e8	−0.640150
$319$	−2.89424e7	−0.0499191
$320$	8.49543e6	0.0144931
$321$	−3.55940e7	−0.0600633
$322$	5.84458e7	0.0975569
$323$	2.67243e8	0.441264
$324$	3.40122e7	0.0555556
$325$	−5.18444e8	−0.837741
$326$	−6.05673e8	−0.968225
$327$	−7.18073e8	−1.13567
$328$	3.65066e8	0.571233
$329$	3.91990e8	0.606861
$330$	−6.55902e6	−0.0100471
$331$	8.52927e8	1.29275	0.646374	−	0.763021i	$-0.276285\pi$
$331$	8.52927e8	1.29275	0.646374	+	0.763021i	$0.276285\pi$
$332$	−4.70982e7	−0.0706352
$333$	−1.46294e8	−0.217106
$334$	5.61607e7	0.0824745
$335$	−9.28438e7	−0.134926
$336$	−8.83256e7	−0.127028
$337$	−4.35591e8	−0.619975	−0.309987	−	0.950741i	$-0.600325\pi$
$337$	−4.35591e8	−0.619975	−0.309987	+	0.950741i	$0.600325\pi$
$338$	−1.40021e8	−0.197235
$339$	5.60306e8	0.781135
$340$	−6.06139e7	−0.0836365
$341$	−1.08700e8	−0.148452
$342$	−5.33308e7	−0.0720919
$343$	8.06030e8	1.07850
$344$	−4.83478e8	−0.640357
$345$	−8.00406e6	−0.0104941
$346$	2.60877e8	0.338586
$347$	−2.61070e8	−0.335432	−0.167716	−	0.985835i	$-0.553639\pi$
$347$	−2.61070e8	−0.335432	−0.167716	+	0.985835i	$0.553639\pi$
$348$	5.33751e7	0.0678908
$349$	−1.37684e9	−1.73379	−0.866893	−	0.498494i	$-0.833887\pi$
$349$	−1.37684e9	−1.73379	−0.866893	+	0.498494i	$0.833887\pi$
$350$	4.92453e8	0.613941
$351$	1.32398e8	0.163420
$352$	−3.07036e7	−0.0375224
$353$	1.33426e7	0.0161446	0.00807232	−	0.999967i	$-0.497430\pi$
$353$	1.33426e7	0.0161446	0.00807232	+	0.999967i	$0.497430\pi$
$354$	4.43619e7	0.0531494
$355$	−1.06335e8	−0.126146
$356$	−3.36628e8	−0.395435
$357$	6.30192e8	0.733051
$358$	−5.57482e7	−0.0642156
$359$	−1.42709e8	−0.162787	−0.0813935	−	0.996682i	$-0.525937\pi$
$359$	−1.42709e8	−0.162787	−0.0813935	+	0.996682i	$0.525937\pi$
$360$	1.20960e7	0.0136642
$361$	−8.10250e8	−0.906450
$362$	−5.53836e8	−0.613623
$363$	−5.02448e8	−0.551338
$364$	−3.43821e8	−0.373661
$365$	2.08425e7	0.0224349
$366$	−4.33876e7	−0.0462574
$367$	8.41387e8	0.888515	0.444257	−	0.895899i	$-0.353468\pi$
$367$	8.41387e8	0.888515	0.444257	+	0.895899i	$0.353468\pi$
$368$	−3.74680e7	−0.0391916
$369$	5.19792e8	0.538564
$370$	−5.20277e7	−0.0533985
$371$	1.35733e9	1.37999
$372$	2.00462e8	0.201898
$373$	7.84247e8	0.782477	0.391239	−	0.920289i	$-0.372047\pi$
$373$	7.84247e8	0.782477	0.391239	+	0.920289i	$0.372047\pi$
$374$	2.19067e8	0.216534
$375$	−1.35800e8	−0.132981
$376$	−2.51294e8	−0.243795
$377$	2.07771e8	0.199705
$378$	−1.25760e8	−0.119763
$379$	−9.78582e8	−0.923336	−0.461668	−	0.887053i	$-0.652749\pi$
$379$	−9.78582e8	−0.923336	−0.461668	+	0.887053i	$0.652749\pi$
$380$	−1.89665e7	−0.0177314
$381$	−9.21861e8	−0.853941
$382$	−1.71701e8	−0.157598
$383$	1.68669e9	1.53405	0.767025	−	0.641617i	$-0.221736\pi$
$383$	1.68669e9	1.53405	0.767025	+	0.641617i	$0.221736\pi$
$384$	5.66231e7	0.0510310
$385$	2.42520e7	0.0216589
$386$	1.44104e9	1.27533
$387$	−6.88389e8	−0.603734
$388$	1.02030e9	0.886778
$389$	7.65596e8	0.659442	0.329721	−	0.944078i	$-0.393045\pi$
$389$	7.65596e8	0.659442	0.329721	+	0.944078i	$0.393045\pi$
$390$	4.70857e7	0.0401942
$391$	2.67330e8	0.226167
$392$	−9.50695e7	−0.0797150
$393$	4.87381e8	0.405037
$394$	−2.04790e7	−0.0168683
$395$	1.45898e8	0.119113
$396$	−4.37167e7	−0.0353764
$397$	1.18416e9	0.949822	0.474911	−	0.880034i	$-0.342480\pi$
$397$	1.18416e9	0.949822	0.474911	+	0.880034i	$0.342480\pi$
$398$	3.06531e8	0.243716
$399$	1.97191e8	0.155411
$400$	−3.15698e8	−0.246639
$401$	1.55411e9	1.20359	0.601793	−	0.798652i	$-0.294453\pi$
$401$	1.55411e9	1.20359	0.601793	+	0.798652i	$0.294453\pi$
$402$	−6.18816e8	−0.475084
$403$	7.80329e8	0.593896
$404$	−1.03731e9	−0.782661
$405$	1.72227e7	0.0128827
$406$	−1.97355e8	−0.146355
$407$	1.88035e8	0.138248
$408$	−4.03999e8	−0.294489
$409$	−2.85076e8	−0.206029	−0.103015	−	0.994680i	$-0.532849\pi$
$409$	−2.85076e8	−0.206029	−0.103015	+	0.994680i	$0.532849\pi$
$410$	1.84858e8	0.132463
$411$	−7.93588e8	−0.563831
$412$	1.00520e9	0.708129
$413$	−1.64028e8	−0.114576
$414$	−5.33480e7	−0.0369502
$415$	−2.38490e7	−0.0163795
$416$	2.20414e8	0.150111
$417$	5.17134e8	0.349242
$418$	6.85473e7	0.0459064
$419$	2.20855e9	1.46675	0.733377	−	0.679822i	$-0.237943\pi$
$419$	2.20855e9	1.46675	0.733377	+	0.679822i	$0.237943\pi$
$420$	−4.47252e7	−0.0294564
$421$	2.09804e9	1.37034	0.685168	−	0.728385i	$-0.259729\pi$
$421$	2.09804e9	1.37034	0.685168	+	0.728385i	$0.259729\pi$
$422$	−1.59776e8	−0.103495
$423$	−3.57800e8	−0.229852
$424$	−8.70148e8	−0.554386
$425$	2.25247e9	1.42330
$426$	−7.08733e8	−0.444170
$427$	1.60426e8	0.0997188
$428$	−8.43710e7	−0.0520164
$429$	−1.70174e8	−0.104062
$430$	−2.44817e8	−0.148492
$431$	2.82316e9	1.69850	0.849248	−	0.527993i	$-0.177055\pi$
$431$	2.82316e9	1.69850	0.849248	+	0.527993i	$0.177055\pi$
$432$	8.06216e7	0.0481125
$433$	6.31956e8	0.374093	0.187046	−	0.982351i	$-0.440109\pi$
$433$	6.31956e8	0.374093	0.187046	+	0.982351i	$0.440109\pi$
$434$	−7.41209e8	−0.435238
$435$	2.70274e7	0.0157432
$436$	−1.70210e9	−0.983518
$437$	8.36491e7	0.0479486
$438$	1.38918e8	0.0789947
$439$	−1.51968e9	−0.857285	−0.428643	−	0.903474i	$-0.641008\pi$
$439$	−1.51968e9	−0.857285	−0.428643	+	0.903474i	$0.641008\pi$
$440$	−1.55473e7	−0.00870104
$441$	−1.35363e8	−0.0751560
$442$	−1.57263e9	−0.866260
$443$	2.78707e9	1.52312	0.761560	−	0.648094i	$-0.224434\pi$
$443$	2.78707e9	1.52312	0.761560	+	0.648094i	$0.224434\pi$
$444$	−3.46771e8	−0.188019
$445$	−1.70457e8	−0.0916971
$446$	8.02992e8	0.428587
$447$	−1.43874e9	−0.761914
$448$	−2.09364e8	−0.110009
$449$	2.18517e9	1.13926	0.569630	−	0.821901i	$-0.307087\pi$
$449$	2.18517e9	1.13926	0.569630	+	0.821901i	$0.307087\pi$
$450$	−4.49500e8	−0.232534
$451$	−6.68100e8	−0.342944
$452$	1.32813e9	0.676483
$453$	1.04171e9	0.526507
$454$	6.25367e8	0.313646
$455$	−1.74100e8	−0.0866479
$456$	−1.26414e8	−0.0624334
$457$	1.44969e9	0.710506	0.355253	−	0.934770i	$-0.384395\pi$
$457$	1.44969e9	0.710506	0.355253	+	0.934770i	$0.384395\pi$
$458$	−2.48491e9	−1.20860
$459$	−5.75225e8	−0.277647
$460$	−1.89726e7	−0.00908812
$461$	−8.33014e8	−0.396003	−0.198002	−	0.980202i	$-0.563445\pi$
$461$	−8.33014e8	−0.396003	−0.198002	+	0.980202i	$0.563445\pi$
$462$	1.61643e8	0.0762622
$463$	3.07632e9	1.44045	0.720224	−	0.693742i	$-0.244039\pi$
$463$	3.07632e9	1.44045	0.720224	+	0.693742i	$0.244039\pi$
$464$	1.26519e8	0.0587952
$465$	1.01507e8	0.0468179
$466$	7.77909e8	0.356105
$467$	−1.93837e9	−0.880698	−0.440349	−	0.897827i	$-0.645145\pi$
$467$	−1.93837e9	−0.880698	−0.440349	+	0.897827i	$0.645145\pi$
$468$	3.13832e8	0.141526
$469$	2.28808e9	1.02415
$470$	−1.27247e8	−0.0565334
$471$	7.18962e8	0.317054
$472$	1.05154e8	0.0460287
$473$	8.84803e8	0.384443
$474$	9.72426e8	0.419404
$475$	7.04811e8	0.301748
$476$	1.49379e9	0.634841
$477$	−1.23894e9	−0.522680
$478$	−3.22521e9	−1.35070
$479$	−2.89923e9	−1.20534	−0.602669	−	0.797992i	$-0.705896\pi$
$479$	−2.89923e9	−1.20534	−0.602669	+	0.797992i	$0.705896\pi$
$480$	2.86721e7	0.0118335
$481$	−1.34986e9	−0.553071
$482$	−1.56673e9	−0.637281
$483$	1.97255e8	0.0796549
$484$	−1.19099e9	−0.477473
$485$	5.16644e8	0.205634
$486$	1.14791e8	0.0453609
$487$	−3.19468e9	−1.25336	−0.626680	−	0.779276i	$-0.715587\pi$
$487$	−3.19468e9	−1.25336	−0.626680	+	0.779276i	$0.715587\pi$
$488$	−1.02845e8	−0.0400601
$489$	−2.04415e9	−0.790553
$490$	−4.81401e7	−0.0184851
$491$	2.55725e9	0.974962	0.487481	−	0.873134i	$-0.337916\pi$
$491$	2.55725e9	0.974962	0.487481	+	0.873134i	$0.337916\pi$
$492$	1.23210e9	0.466410
$493$	−9.02695e8	−0.339295
$494$	−4.92085e8	−0.183652
$495$	−2.21367e7	−0.00820341
$496$	4.75169e8	0.174849
$497$	2.62054e9	0.957512
$498$	−1.58956e8	−0.0576734
$499$	3.98762e9	1.43669	0.718343	−	0.695689i	$-0.244901\pi$
$499$	3.98762e9	1.43669	0.718343	+	0.695689i	$0.244901\pi$
$500$	−3.21897e8	−0.115165
$501$	1.89542e8	0.0673401
$502$	3.93792e9	1.38933
$503$	5.33337e9	1.86859	0.934294	−	0.356502i	$-0.116031\pi$
$503$	5.33337e9	1.86859	0.934294	+	0.356502i	$0.116031\pi$
$504$	−2.98099e8	−0.103718
$505$	−5.25260e8	−0.181491
$506$	6.85694e7	0.0235290
$507$	−4.72571e8	−0.161042
$508$	−2.18515e9	−0.739535
$509$	1.04324e9	0.350648	0.175324	−	0.984511i	$-0.443903\pi$
$509$	1.04324e9	0.350648	0.175324	+	0.984511i	$0.443903\pi$
$510$	−2.04572e8	−0.0682889
$511$	−5.13649e8	−0.170292
$512$	1.34218e8	0.0441942
$513$	−1.79991e8	−0.0588628
$514$	1.54861e9	0.503004
$515$	5.09001e8	0.164208
$516$	−1.63174e9	−0.522849
$517$	4.59888e8	0.146364
$518$	1.28219e9	0.405320
$519$	8.80459e8	0.276454
$520$	1.11611e8	0.0348092
$521$	1.53601e9	0.475840	0.237920	−	0.971285i	$-0.423534\pi$
$521$	1.53601e9	0.475840	0.237920	+	0.971285i	$0.423534\pi$
$522$	1.80141e8	0.0554326
$523$	−4.20255e9	−1.28457	−0.642283	−	0.766467i	$-0.722013\pi$
$523$	−4.20255e9	−1.28457	−0.642283	+	0.766467i	$0.722013\pi$
$524$	1.15527e9	0.350772
$525$	1.66203e9	0.501281
$526$	−3.22090e9	−0.965000
$527$	−3.39027e9	−1.00901
$528$	−1.03625e8	−0.0306369
$529$	−3.32115e9	−0.975424
$530$	−4.40614e8	−0.128556
$531$	1.49721e8	0.0433963
$532$	4.67416e8	0.134590
$533$	4.79614e9	1.37198
$534$	−1.13612e9	−0.322871
$535$	−4.27227e7	−0.0120620
$536$	−1.46682e9	−0.411434
$537$	−1.88150e8	−0.0524318
$538$	−8.25566e8	−0.228567
$539$	1.73985e8	0.0478575
$540$	4.08241e7	0.0111568
$541$	−4.01144e9	−1.08920	−0.544602	−	0.838694i	$-0.683319\pi$
$541$	−4.01144e9	−1.08920	−0.544602	+	0.838694i	$0.683319\pi$
$542$	2.09858e9	0.566147
$543$	−1.86920e9	−0.501021
$544$	−9.57627e8	−0.255035
$545$	−8.61888e8	−0.228067
$546$	−1.16040e9	−0.305093
$547$	−2.69197e9	−0.703258	−0.351629	−	0.936140i	$-0.614372\pi$
$547$	−2.69197e9	−0.703258	−0.351629	+	0.936140i	$0.614372\pi$
$548$	−1.88110e9	−0.488292
$549$	−1.46433e8	−0.0377690
$550$	5.77753e8	0.148072
$551$	−2.82459e8	−0.0719324
$552$	−1.26455e8	−0.0319998
$553$	−3.59555e9	−0.904123
$554$	3.25235e9	0.812667
$555$	−1.75594e8	−0.0435997
$556$	1.22580e9	0.302453
$557$	−4.45329e9	−1.09191	−0.545956	−	0.837814i	$-0.683833\pi$
$557$	−4.45329e9	−1.09191	−0.545956	+	0.837814i	$0.683833\pi$
$558$	6.76559e8	0.164849
$559$	−6.35179e9	−1.53800
$560$	−1.06015e8	−0.0255100
$561$	7.39350e8	0.176799
$562$	1.75486e9	0.417029
$563$	−6.99977e9	−1.65312	−0.826560	−	0.562849i	$-0.809705\pi$
$563$	−6.99977e9	−1.65312	−0.826560	+	0.562849i	$0.809705\pi$
$564$	−8.48117e8	−0.199058
$565$	6.72523e8	0.156869
$566$	3.59382e9	0.833104
$567$	−4.24442e8	−0.0977861
$568$	−1.67996e9	−0.384662
$569$	3.05065e9	0.694223	0.347111	−	0.937824i	$-0.387163\pi$
$569$	3.05065e9	0.694223	0.347111	+	0.937824i	$0.387163\pi$
$570$	−6.40118e7	−0.0144776
$571$	−2.56852e9	−0.577373	−0.288687	−	0.957424i	$-0.593219\pi$
$571$	−2.56852e9	−0.577373	−0.288687	+	0.957424i	$0.593219\pi$
$572$	−4.03375e8	−0.0901204
$573$	−5.79491e8	−0.128678
$574$	−4.55570e9	−1.00546
$575$	7.05039e8	0.154659
$576$	1.91103e8	0.0416667
$577$	−3.36783e9	−0.729852	−0.364926	−	0.931037i	$-0.618906\pi$
$577$	−3.36783e9	−0.729852	−0.364926	+	0.931037i	$0.618906\pi$
$578$	3.54985e9	0.764649
$579$	4.86352e9	1.04130
$580$	6.40649e7	0.0136340
$581$	5.87743e8	0.124328
$582$	3.44350e9	0.724051
$583$	1.59244e9	0.332830
$584$	3.29286e8	0.0684114
$585$	1.58914e8	0.0328184
$586$	5.84966e9	1.20085
$587$	6.14997e9	1.25499	0.627494	−	0.778621i	$-0.284081\pi$
$587$	6.14997e9	1.25499	0.627494	+	0.778621i	$0.284081\pi$
$588$	−3.20860e8	−0.0650870
$589$	−1.06084e9	−0.213917
$590$	5.32466e7	0.0106736
$591$	−6.91167e7	−0.0137729
$592$	−8.21976e8	−0.162830
$593$	−1.64687e9	−0.324316	−0.162158	−	0.986765i	$-0.551845\pi$
$593$	−1.64687e9	−0.324316	−0.162158	+	0.986765i	$0.551845\pi$
$594$	−1.47544e8	−0.0288847
$595$	7.56406e8	0.147213
$596$	−3.41035e9	−0.659837
$597$	1.03454e9	0.198993
$598$	−4.92244e8	−0.0941296
$599$	3.26176e9	0.620095	0.310047	−	0.950721i	$-0.399655\pi$
$599$	3.26176e9	0.620095	0.310047	+	0.950721i	$0.399655\pi$
$600$	−1.06548e9	−0.201380
$601$	9.16054e9	1.72131	0.860657	−	0.509184i	$-0.170053\pi$
$601$	9.16054e9	1.72131	0.860657	+	0.509184i	$0.170053\pi$
$602$	6.03336e9	1.12712
$603$	−2.08850e9	−0.387904
$604$	2.46924e9	0.455968
$605$	−6.03078e8	−0.110721
$606$	−3.50092e9	−0.639040
$607$	3.31286e9	0.601234	0.300617	−	0.953745i	$-0.402807\pi$
$607$	3.31286e9	0.601234	0.300617	+	0.953745i	$0.402807\pi$
$608$	−2.99647e8	−0.0540689
$609$	−6.66072e8	−0.119498
$610$	−5.20772e7	−0.00928951
$611$	−3.30143e9	−0.585541
$612$	−1.36350e9	−0.240450
$613$	−8.93104e9	−1.56600	−0.782998	−	0.622025i	$-0.786310\pi$
$613$	−8.93104e9	−1.56600	−0.782998	+	0.622025i	$0.786310\pi$
$614$	−3.94658e9	−0.688068
$615$	6.23895e8	0.108155
$616$	3.83153e8	0.0660450
$617$	−6.56291e9	−1.12486	−0.562430	−	0.826845i	$-0.690133\pi$
$617$	−6.56291e9	−1.12486	−0.562430	+	0.826845i	$0.690133\pi$
$618$	3.39255e9	0.578185
$619$	−1.00445e10	−1.70220	−0.851099	−	0.525005i	$-0.824063\pi$
$619$	−1.00445e10	−1.70220	−0.851099	+	0.525005i	$0.824063\pi$
$620$	2.40610e8	0.0405455
$621$	−1.80050e8	−0.0301697
$622$	−4.97285e9	−0.828588
$623$	4.20081e9	0.696025
$624$	7.43898e8	0.122565
$625$	5.85847e9	0.959851
$626$	−3.79349e9	−0.618057
$627$	2.31347e8	0.0374824
$628$	1.70421e9	0.274577
$629$	5.86470e9	0.939655
$630$	−1.50948e8	−0.0240510
$631$	−7.31870e9	−1.15966	−0.579831	−	0.814737i	$-0.696881\pi$
$631$	−7.31870e9	−1.15966	−0.579831	+	0.814737i	$0.696881\pi$
$632$	2.30501e9	0.363214
$633$	−5.39244e8	−0.0845031
$634$	5.54394e9	0.863985
$635$	−1.10649e9	−0.171490
$636$	−2.93675e9	−0.452655
$637$	−1.24900e9	−0.191458
$638$	−2.31539e8	−0.0352982
$639$	−2.39197e9	−0.362663
$640$	6.79635e7	0.0102482
$641$	−8.57932e9	−1.28662	−0.643309	−	0.765606i	$-0.722439\pi$
$641$	−8.57932e9	−1.28662	−0.643309	+	0.765606i	$0.722439\pi$
$642$	−2.84752e8	−0.0424712
$643$	−8.32143e9	−1.23441	−0.617205	−	0.786802i	$-0.711735\pi$
$643$	−8.32143e9	−1.23441	−0.617205	+	0.786802i	$0.711735\pi$
$644$	4.67567e8	0.0689831
$645$	−8.26259e8	−0.121243
$646$	2.13795e9	0.312021
$647$	4.36540e9	0.633664	0.316832	−	0.948482i	$-0.397381\pi$
$647$	4.36540e9	0.633664	0.316832	+	0.948482i	$0.397381\pi$
$648$	2.72098e8	0.0392837
$649$	−1.92440e8	−0.0276337
$650$	−4.14755e9	−0.592373
$651$	−2.50158e9	−0.355370
$652$	−4.84538e9	−0.684639
$653$	−2.52065e9	−0.354255	−0.177128	−	0.984188i	$-0.556680\pi$
$653$	−2.52065e9	−0.354255	−0.177128	+	0.984188i	$0.556680\pi$
$654$	−5.74459e9	−0.803039
$655$	5.84992e8	0.0813403
$656$	2.92053e9	0.403923
$657$	4.68847e8	0.0644989
$658$	3.13592e9	0.429116
$659$	−1.42123e10	−1.93449	−0.967244	−	0.253847i	$-0.918304\pi$
$659$	−1.42123e10	−1.93449	−0.967244	+	0.253847i	$0.918304\pi$
$660$	−5.24722e7	−0.00710437
$661$	−8.76569e9	−1.18054	−0.590271	−	0.807205i	$-0.700979\pi$
$661$	−8.76569e9	−1.18054	−0.590271	+	0.807205i	$0.700979\pi$
$662$	6.82341e9	0.914111
$663$	−5.30762e9	−0.707298
$664$	−3.76786e8	−0.0499466
$665$	2.36684e8	0.0312100
$666$	−1.17035e9	−0.153517
$667$	−2.82550e8	−0.0368685
$668$	4.49285e8	0.0583183
$669$	2.71010e9	0.349940
$670$	−7.42751e8	−0.0954072
$671$	1.88214e8	0.0240504
$672$	−7.06605e8	−0.0898223
$673$	−6.69460e9	−0.846587	−0.423294	−	0.905993i	$-0.639126\pi$
$673$	−6.69460e9	−0.846587	−0.423294	+	0.905993i	$0.639126\pi$
$674$	−3.48473e9	−0.438389
$675$	−1.51706e9	−0.189863
$676$	−1.12017e9	−0.139466
$677$	1.51765e10	1.87980	0.939898	−	0.341455i	$-0.110920\pi$
$677$	1.51765e10	1.87980	0.939898	+	0.341455i	$0.110920\pi$
$678$	4.48245e9	0.552346
$679$	−1.27324e10	−1.56086
$680$	−4.84911e8	−0.0591400
$681$	2.11061e9	0.256091
$682$	−8.69596e8	−0.104972
$683$	4.71887e9	0.566716	0.283358	−	0.959014i	$-0.408552\pi$
$683$	4.71887e9	0.566716	0.283358	+	0.959014i	$0.408552\pi$
$684$	−4.26646e8	−0.0509767
$685$	−9.52527e8	−0.113230
$686$	6.44824e9	0.762617
$687$	−8.38658e9	−0.986816
$688$	−3.86782e9	−0.452801
$689$	−1.14318e10	−1.33151
$690$	−6.40324e7	−0.00742042
$691$	−1.49850e10	−1.72777	−0.863883	−	0.503693i	$-0.831974\pi$
$691$	−1.49850e10	−1.72777	−0.863883	+	0.503693i	$0.831974\pi$
$692$	2.08701e9	0.239416
$693$	5.45544e8	0.0622678
$694$	−2.08856e9	−0.237186
$695$	6.20704e8	0.0701355
$696$	4.27001e8	0.0480061
$697$	−2.08376e10	−2.33095
$698$	−1.10147e10	−1.22597
$699$	2.62544e9	0.290758
$700$	3.93963e9	0.434122
$701$	−1.25013e9	−0.137070	−0.0685348	−	0.997649i	$-0.521832\pi$
$701$	−1.25013e9	−0.137070	−0.0685348	+	0.997649i	$0.521832\pi$
$702$	1.05918e9	0.115556
$703$	1.83510e9	0.199212
$704$	−2.45629e8	−0.0265323
$705$	−4.29459e8	−0.0461594
$706$	1.06741e8	0.0114160
$707$	1.29447e10	1.37760
$708$	3.54895e8	0.0375823
$709$	5.62836e9	0.593089	0.296544	−	0.955019i	$-0.404166\pi$
$709$	5.62836e9	0.593089	0.296544	+	0.955019i	$0.404166\pi$
$710$	−8.50676e8	−0.0891990
$711$	3.28194e9	0.342442
$712$	−2.69302e9	−0.279615
$713$	−1.06118e9	−0.109642
$714$	5.04154e9	0.518346
$715$	−2.04256e8	−0.0208980
$716$	−4.45986e8	−0.0454073
$717$	−1.08851e10	−1.10284
$718$	−1.14167e9	−0.115108
$719$	−1.47435e10	−1.47927	−0.739637	−	0.673006i	$-0.765003\pi$
$719$	−1.47435e10	−1.47927	−0.739637	+	0.673006i	$0.765003\pi$
$720$	9.67683e7	0.00966205
$721$	−1.25440e10	−1.24641
$722$	−6.48200e9	−0.640957
$723$	−5.28773e9	−0.520337
$724$	−4.43069e9	−0.433897
$725$	−2.38071e9	−0.232019
$726$	−4.01959e9	−0.389855
$727$	1.22428e10	1.18171	0.590856	−	0.806777i	$-0.298790\pi$
$727$	1.22428e10	1.18171	0.590856	+	0.806777i	$0.298790\pi$
$728$	−2.75057e9	−0.264218
$729$	3.87420e8	0.0370370
$730$	1.66740e8	0.0158639
$731$	2.75965e10	2.61302
$732$	−3.47101e8	−0.0327090
$733$	7.62686e9	0.715289	0.357645	−	0.933858i	$-0.383580\pi$
$733$	7.62686e9	0.715289	0.357645	+	0.933858i	$0.383580\pi$
$734$	6.73110e9	0.628275
$735$	−1.62473e8	−0.0150930
$736$	−2.99744e8	−0.0277127
$737$	2.68440e9	0.247008
$738$	4.15834e9	0.380822
$739$	3.30374e9	0.301128	0.150564	−	0.988600i	$-0.451891\pi$
$739$	3.30374e9	0.301128	0.150564	+	0.988600i	$0.451891\pi$
$740$	−4.16222e8	−0.0377584
$741$	−1.66079e9	−0.149951
$742$	1.08587e10	0.975803
$743$	1.37871e10	1.23313	0.616567	−	0.787302i	$-0.288523\pi$
$743$	1.37871e10	1.23313	0.616567	+	0.787302i	$0.288523\pi$
$744$	1.60370e9	0.142763
$745$	−1.72689e9	−0.153009
$746$	6.27397e9	0.553295
$747$	−5.36478e8	−0.0470901
$748$	1.75253e9	0.153113
$749$	1.05287e9	0.0915566
$750$	−1.08640e9	−0.0940321
$751$	7.69783e9	0.663176	0.331588	−	0.943424i	$-0.392416\pi$
$751$	7.69783e9	0.663176	0.331588	+	0.943424i	$0.392416\pi$
$752$	−2.01035e9	−0.172389
$753$	1.32905e10	1.13438
$754$	1.66217e9	0.141213
$755$	1.25034e9	0.105734
$756$	−1.00608e9	−0.0846853
$757$	−1.70127e10	−1.42540	−0.712702	−	0.701467i	$-0.752529\pi$
$757$	−1.70127e10	−1.42540	−0.712702	+	0.701467i	$0.752529\pi$
$758$	−7.82866e9	−0.652897
$759$	2.31422e8	0.0192114
$760$	−1.51732e8	−0.0125380
$761$	−1.77546e10	−1.46038	−0.730188	−	0.683246i	$-0.760568\pi$
$761$	−1.77546e10	−1.46038	−0.730188	+	0.683246i	$0.760568\pi$
$762$	−7.37489e9	−0.603828
$763$	2.12407e10	1.73114
$764$	−1.37361e9	−0.111439
$765$	−6.90430e8	−0.0557577
$766$	1.34935e10	1.08474
$767$	1.38148e9	0.110551
$768$	4.52985e8	0.0360844
$769$	−1.89022e10	−1.49889	−0.749444	−	0.662067i	$-0.769679\pi$
$769$	−1.89022e10	−1.49889	−0.749444	+	0.662067i	$0.769679\pi$
$770$	1.94016e8	0.0153151
$771$	5.22656e9	0.410701
$772$	1.15283e10	0.901792
$773$	5.84390e9	0.455066	0.227533	−	0.973770i	$-0.426934\pi$
$773$	5.84390e9	0.455066	0.227533	+	0.973770i	$0.426934\pi$
$774$	−5.50712e9	−0.426905
$775$	−8.94129e9	−0.689992
$776$	8.16236e9	0.627047
$777$	4.32739e9	0.330942
$778$	6.12477e9	0.466296
$779$	−6.52022e9	−0.494176
$780$	3.76686e8	0.0284216
$781$	3.07446e9	0.230935
$782$	2.13864e9	0.159924
$783$	6.07975e8	0.0452605
$784$	−7.60556e8	−0.0563670
$785$	8.62955e8	0.0636714
$786$	3.89905e9	0.286404
$787$	1.32836e10	0.971410	0.485705	−	0.874123i	$-0.338563\pi$
$787$	1.32836e10	0.971410	0.485705	+	0.874123i	$0.338563\pi$
$788$	−1.63832e8	−0.0119277
$789$	−1.08705e10	−0.787919
$790$	1.16718e9	0.0842255
$791$	−1.65739e10	−1.19071
$792$	−3.49733e8	−0.0250149
$793$	−1.35114e9	−0.0962155
$794$	9.47325e9	0.671625
$795$	−1.48707e9	−0.104966
$796$	2.45225e9	0.172333
$797$	2.03924e10	1.42680	0.713402	−	0.700755i	$-0.247153\pi$
$797$	2.03924e10	1.42680	0.713402	+	0.700755i	$0.247153\pi$
$798$	1.57753e9	0.109892
$799$	1.43436e10	0.994821
$800$	−2.52559e9	−0.174400
$801$	−3.83440e9	−0.263623
$802$	1.24329e10	0.851064
$803$	−6.02619e8	−0.0410713
$804$	−4.95052e9	−0.335935
$805$	2.36760e8	0.0159964
$806$	6.24263e9	0.419948
$807$	−2.78628e9	−0.186624
$808$	−8.29848e9	−0.553425
$809$	−4.05663e9	−0.269368	−0.134684	−	0.990889i	$-0.543002\pi$
$809$	−4.05663e9	−0.269368	−0.134684	+	0.990889i	$0.543002\pi$
$810$	1.37781e8	0.00910947
$811$	−3.47538e7	−0.00228786	−0.00114393	−	0.999999i	$-0.500364\pi$
$811$	−3.47538e7	−0.00228786	−0.00114393	+	0.999999i	$0.500364\pi$
$812$	−1.57884e9	−0.103488
$813$	7.08272e9	0.462257
$814$	1.50428e9	0.0977560
$815$	−2.45354e9	−0.158760
$816$	−3.23199e9	−0.208235
$817$	8.63510e9	0.553975
$818$	−2.28061e9	−0.145685
$819$	−3.91634e9	−0.249107
$820$	1.47886e9	0.0936654
$821$	8.50364e9	0.536295	0.268148	−	0.963378i	$-0.413589\pi$
$821$	8.50364e9	0.536295	0.268148	+	0.963378i	$0.413589\pi$
$822$	−6.34871e9	−0.398689
$823$	1.96983e10	1.23177	0.615883	−	0.787838i	$-0.288799\pi$
$823$	1.96983e10	1.23177	0.615883	+	0.787838i	$0.288799\pi$
$824$	8.04160e9	0.500723
$825$	1.94991e9	0.120900
$826$	−1.31223e9	−0.0810175
$827$	−8.65486e9	−0.532097	−0.266048	−	0.963960i	$-0.585718\pi$
$827$	−8.65486e9	−0.532097	−0.266048	+	0.963960i	$0.585718\pi$
$828$	−4.26784e8	−0.0261278
$829$	−2.00409e10	−1.22173	−0.610867	−	0.791733i	$-0.709179\pi$
$829$	−2.00409e10	−1.22173	−0.610867	+	0.791733i	$0.709179\pi$
$830$	−1.90792e8	−0.0115821
$831$	1.09767e10	0.663540
$832$	1.76331e9	0.106145
$833$	5.42648e9	0.325282
$834$	4.13707e9	0.246951
$835$	2.27503e8	0.0135234
$836$	5.48378e8	0.0324607
$837$	2.28339e9	0.134599
$838$	1.76684e10	1.03715
$839$	1.31251e10	0.767249	0.383624	−	0.923489i	$-0.374676\pi$
$839$	1.31251e10	0.767249	0.383624	+	0.923489i	$0.374676\pi$
$840$	−3.57802e8	−0.0208288
$841$	−1.62958e10	−0.944690
$842$	1.67844e10	0.968974
$843$	5.92267e9	0.340503
$844$	−1.27821e9	−0.0731818
$845$	−5.67216e8	−0.0323407
$846$	−2.86240e9	−0.162530
$847$	1.48625e10	0.840424
$848$	−6.96118e9	−0.392010
$849$	1.21291e10	0.680226
$850$	1.80197e10	1.00643
$851$	1.83569e9	0.102105
$852$	−5.66986e9	−0.314075
$853$	1.37761e7	0.000759982 0	0.000379991	−	1.00000i	$-0.499879\pi$
$853$	1.37761e7	0.000759982 0	0.000379991		1.00000i	$0.499879\pi$
$854$	1.28341e9	0.0705118
$855$	−2.16040e8	−0.0118209
$856$	−6.74968e8	−0.0367811
$857$	−2.59507e10	−1.40837	−0.704184	−	0.710017i	$-0.748687\pi$
$857$	−2.59507e10	−1.40837	−0.704184	+	0.710017i	$0.748687\pi$
$858$	−1.36139e9	−0.0735830
$859$	−1.29464e10	−0.696904	−0.348452	−	0.937327i	$-0.613293\pi$
$859$	−1.29464e10	−0.696904	−0.348452	+	0.937327i	$0.613293\pi$
$860$	−1.95854e9	−0.105000
$861$	−1.53755e10	−0.820952
$862$	2.25853e10	1.20102
$863$	1.17347e10	0.621491	0.310746	−	0.950493i	$-0.399421\pi$
$863$	1.17347e10	0.621491	0.310746	+	0.950493i	$0.399421\pi$
$864$	6.44973e8	0.0340207
$865$	1.05680e9	0.0555181
$866$	5.05565e9	0.264524
$867$	1.19807e10	0.624333
$868$	−5.92967e9	−0.307760
$869$	−4.21835e9	−0.218059
$870$	2.16219e8	0.0111321
$871$	−1.92707e10	−0.988174
$872$	−1.36168e10	−0.695452
$873$	1.16218e10	0.591185
$874$	6.69193e8	0.0339048
$875$	4.01698e9	0.202708
$876$	1.11134e9	0.0558577
$877$	3.65758e9	0.183103	0.0915515	−	0.995800i	$-0.470817\pi$
$877$	3.65758e9	0.183103	0.0915515	+	0.995800i	$0.470817\pi$
$878$	−1.21574e10	−0.606192
$879$	1.97426e10	0.980490
$880$	−1.24378e8	−0.00615256
$881$	1.29938e10	0.640206	0.320103	−	0.947383i	$-0.396282\pi$
$881$	1.29938e10	0.640206	0.320103	+	0.947383i	$0.396282\pi$
$882$	−1.08290e9	−0.0531433
$883$	1.58519e10	0.774853	0.387426	−	0.921901i	$-0.373364\pi$
$883$	1.58519e10	0.774853	0.387426	+	0.921901i	$0.373364\pi$
$884$	−1.25810e10	−0.612538
$885$	1.79707e8	0.00871493
$886$	2.22965e10	1.07701
$887$	4.34813e9	0.209204	0.104602	−	0.994514i	$-0.466643\pi$
$887$	4.34813e9	0.209204	0.104602	+	0.994514i	$0.466643\pi$
$888$	−2.77417e9	−0.132950
$889$	2.72687e10	1.30169
$890$	−1.36366e9	−0.0648397
$891$	−4.97960e8	−0.0235843
$892$	6.42393e9	0.303057
$893$	4.48821e9	0.210908
$894$	−1.15099e10	−0.538755
$895$	−2.25833e8	−0.0105295
$896$	−1.67491e9	−0.0777884
$897$	−1.66132e9	−0.0768565
$898$	1.74813e10	0.805578
$899$	3.58330e9	0.164484
$900$	−3.59600e9	−0.164426
$901$	4.96672e10	2.26221
$902$	−5.34480e9	−0.242498
$903$	2.03626e10	0.920293
$904$	1.06251e10	0.478346
$905$	−2.24356e9	−0.100616
$906$	8.33370e9	0.372297
$907$	−2.09736e10	−0.933356	−0.466678	−	0.884427i	$-0.654549\pi$
$907$	−2.09736e10	−0.933356	−0.466678	+	0.884427i	$0.654549\pi$
$908$	5.00293e9	0.221781
$909$	−1.18156e10	−0.521774
$910$	−1.39280e9	−0.0612693
$911$	1.35913e10	0.595589	0.297795	−	0.954630i	$-0.403749\pi$
$911$	1.35913e10	0.595589	0.297795	+	0.954630i	$0.403749\pi$
$912$	−1.01131e9	−0.0441471
$913$	6.89547e8	0.0299858
$914$	1.15975e10	0.502404
$915$	−1.75760e8	−0.00758485
$916$	−1.98793e10	−0.854608
$917$	−1.44167e10	−0.617411
$918$	−4.60180e9	−0.196326
$919$	−1.55283e10	−0.659962	−0.329981	−	0.943987i	$-0.607042\pi$
$919$	−1.55283e10	−0.659962	−0.329981	+	0.943987i	$0.607042\pi$
$920$	−1.51781e8	−0.00642627
$921$	−1.33197e10	−0.561805
$922$	−6.66411e9	−0.280017
$923$	−2.20708e10	−0.923873
$924$	1.29314e9	0.0539255
$925$	1.54672e10	0.642562
$926$	2.46105e10	1.01855
$927$	1.14499e10	0.472086
$928$	1.01215e9	0.0415745
$929$	−1.58172e9	−0.0647253	−0.0323627	−	0.999476i	$-0.510303\pi$
$929$	−1.58172e9	−0.0647253	−0.0323627	+	0.999476i	$0.510303\pi$
$930$	8.12059e8	0.0331053
$931$	1.69798e9	0.0689617
$932$	6.22327e9	0.251804
$933$	−1.67834e10	−0.676539
$934$	−1.55069e10	−0.622748
$935$	8.87426e8	0.0355051
$936$	2.51066e9	0.100074
$937$	−4.28127e10	−1.70014	−0.850070	−	0.526670i	$-0.823440\pi$
$937$	−4.28127e10	−1.70014	−0.850070	+	0.526670i	$0.823440\pi$
$938$	1.83046e10	0.724186
$939$	−1.28030e10	−0.504641
$940$	−1.01798e9	−0.0399752
$941$	3.94354e10	1.54285	0.771424	−	0.636322i	$-0.219545\pi$
$941$	3.94354e10	1.54285	0.771424	+	0.636322i	$0.219545\pi$
$942$	5.75170e9	0.224191
$943$	−6.52233e9	−0.253286
$944$	8.41232e8	0.0325472
$945$	−5.09448e8	−0.0196376
$946$	7.07842e9	0.271843
$947$	−4.38792e10	−1.67893	−0.839467	−	0.543410i	$-0.817133\pi$
$947$	−4.38792e10	−1.67893	−0.839467	+	0.543410i	$0.817133\pi$
$948$	7.77941e9	0.296563
$949$	4.32607e9	0.164309
$950$	5.63849e9	0.213368
$951$	1.87108e10	0.705440
$952$	1.19503e10	0.448900
$953$	1.43561e9	0.0537294	0.0268647	−	0.999639i	$-0.491448\pi$
$953$	1.43561e9	0.0537294	0.0268647	+	0.999639i	$0.491448\pi$
$954$	−9.91153e9	−0.369591
$955$	−6.95550e8	−0.0258414
$956$	−2.58017e10	−0.955092
$957$	−7.81445e8	−0.0288208
$958$	−2.31938e10	−0.852302
$959$	2.34744e10	0.859467
$960$	2.29377e8	0.00836758
$961$	−1.40547e10	−0.510847
$962$	−1.07989e10	−0.391080
$963$	−9.61039e8	−0.0346776
$964$	−1.25339e10	−0.450625
$965$	5.83758e9	0.209116
$966$	1.57804e9	0.0563245
$967$	1.06351e10	0.378225	0.189112	−	0.981955i	$-0.439439\pi$
$967$	1.06351e10	0.378225	0.189112	+	0.981955i	$0.439439\pi$
$968$	−9.52791e9	−0.337624
$969$	7.21557e9	0.254764
$970$	4.13315e9	0.145405
$971$	7.34940e9	0.257623	0.128811	−	0.991669i	$-0.458884\pi$
$971$	7.34940e9	0.257623	0.128811	+	0.991669i	$0.458884\pi$
$972$	9.18330e8	0.0320750
$973$	−1.52968e10	−0.532362
$974$	−2.55574e10	−0.886260
$975$	−1.39980e10	−0.483670
$976$	−8.22757e8	−0.0283268
$977$	8.98474e9	0.308230	0.154115	−	0.988053i	$-0.450747\pi$
$977$	8.98474e9	0.308230	0.154115	+	0.988053i	$0.450747\pi$
$978$	−1.63532e10	−0.559005
$979$	4.92845e9	0.167869
$980$	−3.85121e8	−0.0130709
$981$	−1.93880e10	−0.655679
$982$	2.04580e10	0.689402
$983$	−7.14823e8	−0.0240027	−0.0120014	−	0.999928i	$-0.503820\pi$
$983$	−7.14823e8	−0.0240027	−0.0120014	+	0.999928i	$0.503820\pi$
$984$	9.85679e9	0.329802
$985$	−8.29592e7	−0.00276591
$986$	−7.22156e9	−0.239918
$987$	1.05837e10	0.350372
$988$	−3.93668e9	−0.129862
$989$	8.63788e9	0.283936
$990$	−1.77094e8	−0.00580069
$991$	−2.92600e9	−0.0955028	−0.0477514	−	0.998859i	$-0.515206\pi$
$991$	−2.92600e9	−0.0955028	−0.0477514	+	0.998859i	$0.515206\pi$
$992$	3.80135e9	0.123637
$993$	2.30290e10	0.746368
$994$	2.09644e10	0.677063
$995$	1.24174e9	0.0399622
$996$	−1.27165e9	−0.0407812
$997$	8.93864e9	0.285653	0.142826	−	0.989748i	$-0.454381\pi$
$997$	8.93864e9	0.285653	0.142826	+	0.989748i	$0.454381\pi$
$998$	3.19010e10	1.01589
$999$	−3.94994e9	−0.125346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.8.a.b.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.8.a.b.1.3	✓	5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +32.4075 q^{5} +216.000 q^{6} -798.662 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +32.4075 q^{5} +216.000 q^{6} -798.662 q^{7} +512.000 q^{8} +729.000 q^{9} +259.260 q^{10} -937.000 q^{11} +1728.00 q^{12} +6726.51 q^{13} -6389.29 q^{14} +875.003 q^{15} +4096.00 q^{16} -29224.5 q^{17} +5832.00 q^{18} -9144.51 q^{19} +2074.08 q^{20} -21563.9 q^{21} -7496.00 q^{22} -9147.46 q^{23} +13824.0 q^{24} -77074.8 q^{25} +53812.1 q^{26} +19683.0 q^{27} -51114.3 q^{28} +30888.3 q^{29} +7000.02 q^{30} +116008. q^{31} +32768.0 q^{32} -25299.0 q^{33} -233796. q^{34} -25882.6 q^{35} +46656.0 q^{36} -200678. q^{37} -73156.1 q^{38} +181616. q^{39} +16592.6 q^{40} +713020. q^{41} -172511. q^{42} -944293. q^{43} -59968.0 q^{44} +23625.1 q^{45} -73179.7 q^{46} -490809. q^{47} +110592. q^{48} -185683. q^{49} -616598. q^{50} -789061. q^{51} +430496. q^{52} -1.69951e6 q^{53} +157464. q^{54} -30365.8 q^{55} -408915. q^{56} -246902. q^{57} +247107. q^{58} +205379. q^{59} +56000.2 q^{60} -200868. q^{61} +928064. q^{62} -582224. q^{63} +262144. q^{64} +217989. q^{65} -202392. q^{66} -2.86489e6 q^{67} -1.87037e6 q^{68} -246982. q^{69} -207061. q^{70} -3.28117e6 q^{71} +373248. q^{72} +643137. q^{73} -1.60542e6 q^{74} -2.08102e6 q^{75} -585249. q^{76} +748346. q^{77} +1.45293e6 q^{78} +4.50197e6 q^{79} +132741. q^{80} +531441. q^{81} +5.70416e6 q^{82} -735909. q^{83} -1.38009e6 q^{84} -947092. q^{85} -7.55434e6 q^{86} +833985. q^{87} -479744. q^{88} -5.25981e6 q^{89} +189001. q^{90} -5.37220e6 q^{91} -585438. q^{92} +3.13222e6 q^{93} -3.92647e6 q^{94} -296351. q^{95} +884736. q^{96} +1.59421e7 q^{97} -1.48546e6 q^{98} -683073. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 40 q^{2} + 135 q^{3} + 320 q^{4} + 164 q^{5} + 1080 q^{6} - 76 q^{7} + 2560 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q + 40 * q^2 + 135 * q^3 + 320 * q^4 + 164 * q^5 + 1080 * q^6 - 76 * q^7 + 2560 * q^8 + 3645 * q^9	\( 5 q + 40 q^{2} + 135 q^{3} + 320 q^{4} + 164 q^{5} + 1080 q^{6} - 76 q^{7} + 2560 q^{8} + 3645 q^{9} + 1312 q^{10} - 15730 q^{11} + 8640 q^{12} - 21854 q^{13} - 608 q^{14} + 4428 q^{15} + 20480 q^{16} - 34548 q^{17} + 29160 q^{18} - 43828 q^{19} + 10496 q^{20} - 2052 q^{21} - 125840 q^{22} - 110582 q^{23} + 69120 q^{24} - 174577 q^{25} - 174832 q^{26} + 98415 q^{27} - 4864 q^{28} - 307558 q^{29} + 35424 q^{30} - 277994 q^{31} + 163840 q^{32} - 424710 q^{33} - 276384 q^{34} - 764338 q^{35} + 233280 q^{36} - 853778 q^{37} - 350624 q^{38} - 590058 q^{39} + 83968 q^{40} + 131342 q^{41} - 16416 q^{42} - 721996 q^{43} - 1006720 q^{44} + 119556 q^{45} - 884656 q^{46} - 358832 q^{47} + 552960 q^{48} - 207141 q^{49} - 1396616 q^{50} - 932796 q^{51} - 1398656 q^{52} + 1006180 q^{53} + 787320 q^{54} + 81944 q^{55} - 38912 q^{56} - 1183356 q^{57} - 2460464 q^{58} + 1026895 q^{59} + 283392 q^{60} + 101158 q^{61} - 2223952 q^{62} - 55404 q^{63} + 1310720 q^{64} - 3138378 q^{65} - 3397680 q^{66} - 6362512 q^{67} - 2211072 q^{68} - 2985714 q^{69} - 6114704 q^{70} - 8877414 q^{71} + 1866240 q^{72} - 4881862 q^{73} - 6830224 q^{74} - 4713579 q^{75} - 2804992 q^{76} - 2205694 q^{77} - 4720464 q^{78} - 2769432 q^{79} + 671744 q^{80} + 2657205 q^{81} + 1050736 q^{82} - 1430156 q^{83} - 131328 q^{84} - 7564814 q^{85} - 5775968 q^{86} - 8304066 q^{87} - 8053760 q^{88} - 17172176 q^{89} + 956448 q^{90} - 932474 q^{91} - 7077248 q^{92} - 7505838 q^{93} - 2870656 q^{94} + 15962708 q^{95} + 4423680 q^{96} + 20863830 q^{97} - 1657128 q^{98} - 11467170 q^{99}+O(q^{100}) \) 5 * q + 40 * q^2 + 135 * q^3 + 320 * q^4 + 164 * q^5 + 1080 * q^6 - 76 * q^7 + 2560 * q^8 + 3645 * q^9 + 1312 * q^10 - 15730 * q^11 + 8640 * q^12 - 21854 * q^13 - 608 * q^14 + 4428 * q^15 + 20480 * q^16 - 34548 * q^17 + 29160 * q^18 - 43828 * q^19 + 10496 * q^20 - 2052 * q^21 - 125840 * q^22 - 110582 * q^23 + 69120 * q^24 - 174577 * q^25 - 174832 * q^26 + 98415 * q^27 - 4864 * q^28 - 307558 * q^29 + 35424 * q^30 - 277994 * q^31 + 163840 * q^32 - 424710 * q^33 - 276384 * q^34 - 764338 * q^35 + 233280 * q^36 - 853778 * q^37 - 350624 * q^38 - 590058 * q^39 + 83968 * q^40 + 131342 * q^41 - 16416 * q^42 - 721996 * q^43 - 1006720 * q^44 + 119556 * q^45 - 884656 * q^46 - 358832 * q^47 + 552960 * q^48 - 207141 * q^49 - 1396616 * q^50 - 932796 * q^51 - 1398656 * q^52 + 1006180 * q^53 + 787320 * q^54 + 81944 * q^55 - 38912 * q^56 - 1183356 * q^57 - 2460464 * q^58 + 1026895 * q^59 + 283392 * q^60 + 101158 * q^61 - 2223952 * q^62 - 55404 * q^63 + 1310720 * q^64 - 3138378 * q^65 - 3397680 * q^66 - 6362512 * q^67 - 2211072 * q^68 - 2985714 * q^69 - 6114704 * q^70 - 8877414 * q^71 + 1866240 * q^72 - 4881862 * q^73 - 6830224 * q^74 - 4713579 * q^75 - 2804992 * q^76 - 2205694 * q^77 - 4720464 * q^78 - 2769432 * q^79 + 671744 * q^80 + 2657205 * q^81 + 1050736 * q^82 - 1430156 * q^83 - 131328 * q^84 - 7564814 * q^85 - 5775968 * q^86 - 8304066 * q^87 - 8053760 * q^88 - 17172176 * q^89 + 956448 * q^90 - 932474 * q^91 - 7077248 * q^92 - 7505838 * q^93 - 2870656 * q^94 + 15962708 * q^95 + 4423680 * q^96 + 20863830 * q^97 - 1657128 * q^98 - 11467170 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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