LMFDB - Embedded newform 23.8.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,8,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, 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("23.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	23.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:	$7.18485558613$
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} - 2x^{4} - 104x^{3} + 200x^{2} + 2037x - 3704 $ x^5 - 2x^4 - 104x^3 + 200x^2 + 2037x - 3704
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.79887$ of defining polynomial
Character	$\chi$	$=$	23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.402251 q^{2} +50.9104 q^{3} -127.838 q^{4} -384.733 q^{5} -20.4788 q^{6} -88.4463 q^{7} +102.911 q^{8} +404.864 q^{9} +O(q^{10})$	\(q-0.402251 q^{2} +50.9104 q^{3} -127.838 q^{4} -384.733 q^{5} -20.4788 q^{6} -88.4463 q^{7} +102.911 q^{8} +404.864 q^{9} +154.759 q^{10} -1542.65 q^{11} -6508.29 q^{12} -10703.0 q^{13} +35.5776 q^{14} -19586.9 q^{15} +16321.9 q^{16} +1333.62 q^{17} -162.857 q^{18} +2046.37 q^{19} +49183.5 q^{20} -4502.83 q^{21} +620.531 q^{22} +12167.0 q^{23} +5239.25 q^{24} +69894.1 q^{25} +4305.31 q^{26} -90729.2 q^{27} +11306.8 q^{28} +83684.7 q^{29} +7878.84 q^{30} +64855.0 q^{31} -19738.1 q^{32} -78536.6 q^{33} -536.450 q^{34} +34028.2 q^{35} -51757.1 q^{36} +453317. q^{37} -823.156 q^{38} -544896. q^{39} -39593.3 q^{40} -345040. q^{41} +1811.27 q^{42} -931824. q^{43} +197209. q^{44} -155764. q^{45} -4894.19 q^{46} -526950. q^{47} +830953. q^{48} -815720. q^{49} -28115.0 q^{50} +67895.1 q^{51} +1.36826e6 q^{52} +1.69960e6 q^{53} +36495.9 q^{54} +593506. q^{55} -9102.12 q^{56} +104182. q^{57} -33662.3 q^{58} -2.43749e6 q^{59} +2.50395e6 q^{60} -685871. q^{61} -26088.0 q^{62} -35808.7 q^{63} -2.08126e6 q^{64} +4.11781e6 q^{65} +31591.5 q^{66} -1.97121e6 q^{67} -170488. q^{68} +619426. q^{69} -13687.9 q^{70} +4.72714e6 q^{71} +41665.1 q^{72} -5.25959e6 q^{73} -182347. q^{74} +3.55834e6 q^{75} -261605. q^{76} +136441. q^{77} +219185. q^{78} -6.35079e6 q^{79} -6.27956e6 q^{80} -5.50449e6 q^{81} +138793. q^{82} +4.47655e6 q^{83} +575634. q^{84} -513087. q^{85} +374827. q^{86} +4.26042e6 q^{87} -158756. q^{88} -7.76279e6 q^{89} +62656.4 q^{90} +946645. q^{91} -1.55541e6 q^{92} +3.30179e6 q^{93} +211966. q^{94} -787306. q^{95} -1.00488e6 q^{96} +1.41073e7 q^{97} +328124. q^{98} -624562. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9}+O(q^{10}) $ 5 * q - 16 * q^2 - 68 * q^3 + 256 * q^4 - 56 * q^5 + 538 * q^6 - 1156 * q^7 - 5952 * q^8 - 2195 * q^9	\( 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9} - 11260 q^{10} - 1318 q^{11} - 28012 q^{12} - 19662 q^{13} - 6848 q^{14} - 24600 q^{15} + 32448 q^{16} - 5002 q^{17} - 24650 q^{18} - 38314 q^{19} + 104440 q^{20} + 33280 q^{21} - 74872 q^{22} + 60835 q^{23} + 361992 q^{24} + 54959 q^{25} + 345430 q^{26} + 100900 q^{27} + 90800 q^{28} - 150634 q^{29} + 515724 q^{30} - 179940 q^{31} - 404032 q^{32} - 619688 q^{33} + 32116 q^{34} - 374032 q^{35} + 510524 q^{36} - 752672 q^{37} + 456808 q^{38} - 207996 q^{39} - 1082576 q^{40} - 1192910 q^{41} - 250520 q^{42} - 932646 q^{43} + 2467104 q^{44} - 389952 q^{45} - 194672 q^{46} - 1008460 q^{47} - 1916464 q^{48} - 2005219 q^{49} + 1571224 q^{50} - 211520 q^{51} - 1740516 q^{52} + 897104 q^{53} + 1844686 q^{54} + 1203168 q^{55} + 3050144 q^{56} + 3137192 q^{57} + 5685090 q^{58} + 1020972 q^{59} - 1479384 q^{60} - 2758364 q^{61} + 2661794 q^{62} + 1135132 q^{63} + 5173248 q^{64} - 1350472 q^{65} + 11693212 q^{66} - 1523138 q^{67} + 2501304 q^{68} - 827356 q^{69} - 2794240 q^{70} + 3044884 q^{71} - 6740904 q^{72} - 8872022 q^{73} + 1408492 q^{74} + 1960276 q^{75} - 17963952 q^{76} - 3501672 q^{77} - 15280362 q^{78} - 4437540 q^{79} + 12197536 q^{80} - 7995203 q^{81} - 7738154 q^{82} - 4637362 q^{83} + 2663744 q^{84} - 8625728 q^{85} - 3025868 q^{86} + 17151068 q^{87} - 41815040 q^{88} + 6381402 q^{89} - 4970376 q^{90} + 3240808 q^{91} + 3114752 q^{92} + 7185076 q^{93} - 13893974 q^{94} + 15762704 q^{95} + 40696544 q^{96} - 6432034 q^{97} + 22652640 q^{98} + 30201754 q^{99}+O(q^{100}) \) 5 * q - 16 * q^2 - 68 * q^3 + 256 * q^4 - 56 * q^5 + 538 * q^6 - 1156 * q^7 - 5952 * q^8 - 2195 * q^9 - 11260 * q^10 - 1318 * q^11 - 28012 * q^12 - 19662 * q^13 - 6848 * q^14 - 24600 * q^15 + 32448 * q^16 - 5002 * q^17 - 24650 * q^18 - 38314 * q^19 + 104440 * q^20 + 33280 * q^21 - 74872 * q^22 + 60835 * q^23 + 361992 * q^24 + 54959 * q^25 + 345430 * q^26 + 100900 * q^27 + 90800 * q^28 - 150634 * q^29 + 515724 * q^30 - 179940 * q^31 - 404032 * q^32 - 619688 * q^33 + 32116 * q^34 - 374032 * q^35 + 510524 * q^36 - 752672 * q^37 + 456808 * q^38 - 207996 * q^39 - 1082576 * q^40 - 1192910 * q^41 - 250520 * q^42 - 932646 * q^43 + 2467104 * q^44 - 389952 * q^45 - 194672 * q^46 - 1008460 * q^47 - 1916464 * q^48 - 2005219 * q^49 + 1571224 * q^50 - 211520 * q^51 - 1740516 * q^52 + 897104 * q^53 + 1844686 * q^54 + 1203168 * q^55 + 3050144 * q^56 + 3137192 * q^57 + 5685090 * q^58 + 1020972 * q^59 - 1479384 * q^60 - 2758364 * q^61 + 2661794 * q^62 + 1135132 * q^63 + 5173248 * q^64 - 1350472 * q^65 + 11693212 * q^66 - 1523138 * q^67 + 2501304 * q^68 - 827356 * q^69 - 2794240 * q^70 + 3044884 * q^71 - 6740904 * q^72 - 8872022 * q^73 + 1408492 * q^74 + 1960276 * q^75 - 17963952 * q^76 - 3501672 * q^77 - 15280362 * q^78 - 4437540 * q^79 + 12197536 * q^80 - 7995203 * q^81 - 7738154 * q^82 - 4637362 * q^83 + 2663744 * q^84 - 8625728 * q^85 - 3025868 * q^86 + 17151068 * q^87 - 41815040 * q^88 + 6381402 * q^89 - 4970376 * q^90 + 3240808 * q^91 + 3114752 * q^92 + 7185076 * q^93 - 13893974 * q^94 + 15762704 * q^95 + 40696544 * q^96 - 6432034 * q^97 + 22652640 * q^98 + 30201754 * 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 $

$2$	−0.402251	−0.0355543	−0.0177772	−	0.999842i	$-0.505659\pi$
$2$	−0.402251	−0.0355543	−0.0177772	+	0.999842i	$0.505659\pi$
$3$	50.9104	1.08863	0.544317	−	0.838880i	$-0.316789\pi$
$3$	50.9104	1.08863	0.544317	+	0.838880i	$0.316789\pi$
$4$	−127.838	−0.998736
$5$	−384.733	−1.37646	−0.688231	−	0.725492i	$-0.741612\pi$
$5$	−384.733	−1.37646	−0.688231	+	0.725492i	$0.741612\pi$
$6$	−20.4788	−0.0387056
$7$	−88.4463	−0.0974623	−0.0487312	−	0.998812i	$-0.515518\pi$
$7$	−88.4463	−0.0974623	−0.0487312	+	0.998812i	$0.515518\pi$
$8$	102.911	0.0710637
$9$	404.864	0.185123
$10$	154.759	0.0489391
$11$	−1542.65	−0.349455	−0.174728	−	0.984617i	$-0.555905\pi$
$11$	−1542.65	−0.349455	−0.174728	+	0.984617i	$0.555905\pi$
$12$	−6508.29	−1.08726
$13$	−10703.0	−1.35116	−0.675578	−	0.737288i	$-0.736106\pi$
$13$	−10703.0	−1.35116	−0.675578	+	0.737288i	$0.736106\pi$
$14$	35.5776	0.00346521
$15$	−19586.9	−1.49846
$16$	16321.9	0.996209
$17$	1333.62	0.0658356	0.0329178	−	0.999458i	$-0.489520\pi$
$17$	1333.62	0.0658356	0.0329178	+	0.999458i	$0.489520\pi$
$18$	−162.857	−0.00658192
$19$	2046.37	0.0684459	0.0342229	−	0.999414i	$-0.489104\pi$
$19$	2046.37	0.0684459	0.0342229	+	0.999414i	$0.489104\pi$
$20$	49183.5	1.37472
$21$	−4502.83	−0.106101
$22$	620.531	0.0124247
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	5239.25	0.0773623
$25$	69894.1	0.894645
$26$	4305.31	0.0480395
$27$	−90729.2	−0.887102
$28$	11306.8	0.0973391
$29$	83684.7	0.637166	0.318583	−	0.947895i	$-0.396793\pi$
$29$	83684.7	0.637166	0.318583	+	0.947895i	$0.396793\pi$
$30$	7878.84	0.0532768
$31$	64855.0	0.391001	0.195500	−	0.980704i	$-0.437367\pi$
$31$	64855.0	0.391001	0.195500	+	0.980704i	$0.437367\pi$
$32$	−19738.1	−0.106483
$33$	−78536.6	−0.380429
$34$	−536.450	−0.00234074
$35$	34028.2	0.134153
$36$	−51757.1	−0.184889
$37$	453317.	1.47128	0.735641	−	0.677371i	$-0.236881\pi$
$37$	453317.	1.47128	0.735641	+	0.677371i	$0.236881\pi$
$38$	−823.156	−0.00243355
$39$	−544896.	−1.47091
$40$	−39593.3	−0.0978164
$41$	−345040.	−0.781856	−0.390928	−	0.920421i	$-0.627846\pi$
$41$	−345040.	−0.781856	−0.390928	+	0.920421i	$0.627846\pi$
$42$	1811.27	0.00377234
$43$	−931824.	−1.78729	−0.893643	−	0.448778i	$-0.851859\pi$
$43$	−931824.	−1.78729	−0.893643	+	0.448778i	$0.851859\pi$
$44$	197209.	0.349014
$45$	−155764.	−0.254815
$46$	−4894.19	−0.00741359
$47$	−526950.	−0.740332	−0.370166	−	0.928966i	$-0.620699\pi$
$47$	−526950.	−0.740332	−0.370166	+	0.928966i	$0.620699\pi$
$48$	830953.	1.08451
$49$	−815720.	−0.990501
$50$	−28115.0	−0.0318085
$51$	67895.1	0.0716709
$52$	1.36826e6	1.34945
$53$	1.69960e6	1.56813	0.784064	−	0.620680i	$-0.213143\pi$
$53$	1.69960e6	1.56813	0.784064	+	0.620680i	$0.213143\pi$
$54$	36495.9	0.0315403
$55$	593506.	0.481012
$56$	−9102.12	−0.00692603
$57$	104182.	0.0745125
$58$	−33662.3	−0.0226540
$59$	−2.43749e6	−1.54512	−0.772559	−	0.634943i	$-0.781024\pi$
$59$	−2.43749e6	−1.54512	−0.772559	+	0.634943i	$0.781024\pi$
$60$	2.50395e6	1.49657
$61$	−685871.	−0.386891	−0.193445	−	0.981111i	$-0.561966\pi$
$61$	−685871.	−0.386891	−0.193445	+	0.981111i	$0.561966\pi$
$62$	−26088.0	−0.0139018
$63$	−35808.7	−0.0180425
$64$	−2.08126e6	−0.992423
$65$	4.11781e6	1.85981
$66$	31591.5	0.0135259
$67$	−1.97121e6	−0.800704	−0.400352	−	0.916361i	$-0.631112\pi$
$67$	−1.97121e6	−0.800704	−0.400352	+	0.916361i	$0.631112\pi$
$68$	−170488.	−0.0657524
$69$	619426.	0.226996
$70$	−13687.9	−0.00476972
$71$	4.72714e6	1.56745	0.783726	−	0.621107i	$-0.213317\pi$
$71$	4.72714e6	1.56745	0.783726	+	0.621107i	$0.213317\pi$
$72$	41665.1	0.0131555
$73$	−5.25959e6	−1.58242	−0.791210	−	0.611545i	$-0.790548\pi$
$73$	−5.25959e6	−1.58242	−0.791210	+	0.611545i	$0.790548\pi$
$74$	−182347.	−0.0523104
$75$	3.55834e6	0.973941
$76$	−261605.	−0.0683593
$77$	136441.	0.0340587
$78$	219185.	0.0522974
$79$	−6.35079e6	−1.44922	−0.724608	−	0.689161i	$-0.757979\pi$
$79$	−6.35079e6	−1.44922	−0.724608	+	0.689161i	$0.757979\pi$
$80$	−6.27956e6	−1.37124
$81$	−5.50449e6	−1.15085
$82$	138793.	0.0277983
$83$	4.47655e6	0.859351	0.429676	−	0.902983i	$-0.358628\pi$
$83$	4.47655e6	0.859351	0.429676	+	0.902983i	$0.358628\pi$
$84$	575634.	0.105967
$85$	−513087.	−0.0906202
$86$	374827.	0.0635458
$87$	4.26042e6	0.693641
$88$	−158756.	−0.0248336
$89$	−7.76279e6	−1.16722	−0.583610	−	0.812034i	$-0.698360\pi$
$89$	−7.76279e6	−1.16722	−0.583610	+	0.812034i	$0.698360\pi$
$90$	62656.4	0.00905976
$91$	946645.	0.131687
$92$	−1.55541e6	−0.208251
$93$	3.30179e6	0.425657
$94$	211966.	0.0263220
$95$	−787306.	−0.0942131
$96$	−1.00488e6	−0.115921
$97$	1.41073e7	1.56944	0.784719	−	0.619851i	$-0.212807\pi$
$97$	1.41073e7	1.56944	0.784719	+	0.619851i	$0.212807\pi$
$98$	328124.	0.0352166
$99$	−624562.	−0.0646922
$100$	−8.93514e6	−0.893514
$101$	6.21725e6	0.600445	0.300223	−	0.953869i	$-0.402939\pi$
$101$	6.21725e6	0.600445	0.300223	+	0.953869i	$0.402939\pi$
$102$	−27310.9	−0.00254821
$103$	−8.01103e6	−0.722368	−0.361184	−	0.932495i	$-0.617627\pi$
$103$	−8.01103e6	−0.722368	−0.361184	+	0.932495i	$0.617627\pi$
$104$	−1.10146e6	−0.0960182
$105$	1.73239e6	0.146044
$106$	−683667.	−0.0557537
$107$	2.33961e7	1.84629	0.923146	−	0.384450i	$-0.125609\pi$
$107$	2.33961e7	1.84629	0.923146	+	0.384450i	$0.125609\pi$
$108$	1.15987e7	0.885981
$109$	3.56295e6	0.263522	0.131761	−	0.991282i	$-0.457937\pi$
$109$	3.56295e6	0.263522	0.131761	+	0.991282i	$0.457937\pi$
$110$	−238739.	−0.0171020
$111$	2.30785e7	1.60169
$112$	−1.44361e6	−0.0970929
$113$	−4.54201e6	−0.296124	−0.148062	−	0.988978i	$-0.547304\pi$
$113$	−4.54201e6	−0.296124	−0.148062	+	0.988978i	$0.547304\pi$
$114$	−41907.2	−0.00264924
$115$	−4.68104e6	−0.287012
$116$	−1.06981e7	−0.636361
$117$	−4.33328e6	−0.250130
$118$	980485.	0.0549356
$119$	−117954.	−0.00641649
$120$	−2.01571e6	−0.106486
$121$	−1.71074e7	−0.877881
$122$	275893.	0.0137556
$123$	−1.75661e7	−0.851154
$124$	−8.29095e6	−0.390507
$125$	3.16668e6	0.145017
$126$	14404.1	0.000641489 0
$127$	−2.06956e7	−0.896528	−0.448264	−	0.893901i	$-0.647958\pi$
$127$	−2.06956e7	−0.896528	−0.448264	+	0.893901i	$0.647958\pi$
$128$	3.36367e6	0.141768
$129$	−4.74395e7	−1.94570
$130$	−1.65639e6	−0.0661245
$131$	2.24937e7	0.874201	0.437100	−	0.899413i	$-0.356005\pi$
$131$	2.24937e7	0.874201	0.437100	+	0.899413i	$0.356005\pi$
$132$	1.00400e7	0.379948
$133$	−180994.	−0.00667089
$134$	792923.	0.0284685
$135$	3.49065e7	1.22106
$136$	137245.	0.00467852
$137$	2.21841e7	0.737089	0.368544	−	0.929610i	$-0.379856\pi$
$137$	2.21841e7	0.737089	0.368544	+	0.929610i	$0.379856\pi$
$138$	−249165.	−0.00807068
$139$	1.39982e7	0.442101	0.221051	−	0.975262i	$-0.429051\pi$
$139$	1.39982e7	0.442101	0.221051	+	0.975262i	$0.429051\pi$
$140$	−4.35010e6	−0.133983
$141$	−2.68272e7	−0.805950
$142$	−1.90150e6	−0.0557297
$143$	1.65110e7	0.472169
$144$	6.60815e6	0.184421
$145$	−3.21962e7	−0.877035
$146$	2.11568e6	0.0562619
$147$	−4.15286e7	−1.07829
$148$	−5.79512e7	−1.46942
$149$	2.56936e7	0.636316	0.318158	−	0.948038i	$-0.396936\pi$
$149$	2.56936e7	0.636316	0.318158	+	0.948038i	$0.396936\pi$
$150$	−1.43135e6	−0.0346278
$151$	1.74107e7	0.411525	0.205762	−	0.978602i	$-0.434033\pi$
$151$	1.74107e7	0.411525	0.205762	+	0.978602i	$0.434033\pi$
$152$	210595.	0.00486402
$153$	539935.	0.0121877
$154$	−54883.7	−0.00121094
$155$	−2.49518e7	−0.538197
$156$	6.96585e7	1.46906
$157$	−7.30771e7	−1.50707	−0.753534	−	0.657409i	$-0.771652\pi$
$157$	−7.30771e7	−1.50707	−0.753534	+	0.657409i	$0.771652\pi$
$158$	2.55462e6	0.0515259
$159$	8.65273e7	1.70712
$160$	7.59391e6	0.146570
$161$	−1.07613e6	−0.0203223
$162$	2.21419e6	0.0409178
$163$	2.11254e7	0.382075	0.191038	−	0.981583i	$-0.438815\pi$
$163$	2.11254e7	0.382075	0.191038	+	0.981583i	$0.438815\pi$
$164$	4.41093e7	0.780867
$165$	3.02156e7	0.523646
$166$	−1.80070e6	−0.0305536
$167$	5.37163e7	0.892480	0.446240	−	0.894913i	$-0.352763\pi$
$167$	5.37163e7	0.892480	0.446240	+	0.894913i	$0.352763\pi$
$168$	−463392.	−0.00753991
$169$	5.18067e7	0.825625
$170$	206390.	0.00322194
$171$	828503.	0.0126709
$172$	1.19123e8	1.78503
$173$	−2.31665e7	−0.340172	−0.170086	−	0.985429i	$-0.554405\pi$
$173$	−2.31665e7	−0.340172	−0.170086	+	0.985429i	$0.554405\pi$
$174$	−1.71376e6	−0.0246619
$175$	−6.18188e6	−0.0871942
$176$	−2.51789e7	−0.348131
$177$	−1.24094e8	−1.68207
$178$	3.12259e6	0.0414997
$179$	−1.29109e8	−1.68257	−0.841283	−	0.540595i	$-0.818199\pi$
$179$	−1.29109e8	−1.68257	−0.841283	+	0.540595i	$0.818199\pi$
$180$	1.99126e7	0.254492
$181$	6.39727e7	0.801899	0.400950	−	0.916100i	$-0.368680\pi$
$181$	6.39727e7	0.801899	0.400950	+	0.916100i	$0.368680\pi$
$182$	−380789.	−0.00468204
$183$	−3.49180e7	−0.421182
$184$	1.25212e6	0.0148178
$185$	−1.74406e8	−2.02516
$186$	−1.32815e6	−0.0151339
$187$	−2.05730e6	−0.0230066
$188$	6.73643e7	0.739396
$189$	8.02466e6	0.0864591
$190$	316695.	0.00334968
$191$	−1.71991e7	−0.178604	−0.0893018	−	0.996005i	$-0.528464\pi$
$191$	−1.71991e7	−0.178604	−0.0893018	+	0.996005i	$0.528464\pi$
$192$	−1.05958e8	−1.08039
$193$	−1.42571e8	−1.42752	−0.713758	−	0.700393i	$-0.753008\pi$
$193$	−1.42571e8	−1.42752	−0.713758	+	0.700393i	$0.753008\pi$
$194$	−5.67470e6	−0.0558003
$195$	2.09639e8	2.02466
$196$	1.04280e8	0.989249
$197$	−1.89065e8	−1.76189	−0.880945	−	0.473218i	$-0.843092\pi$
$197$	−1.89065e8	−1.76189	−0.880945	+	0.473218i	$0.843092\pi$
$198$	251231.	0.00230009
$199$	6.82579e7	0.613998	0.306999	−	0.951710i	$-0.400675\pi$
$199$	6.82579e7	0.613998	0.306999	+	0.951710i	$0.400675\pi$
$200$	7.19289e6	0.0635768
$201$	−1.00355e8	−0.871673
$202$	−2.50090e6	−0.0213484
$203$	−7.40160e6	−0.0620997
$204$	−8.67958e6	−0.0715803
$205$	1.32748e8	1.07619
$206$	3.22245e6	0.0256833
$207$	4.92598e6	0.0386008
$208$	−1.74694e8	−1.34603
$209$	−3.15683e6	−0.0239188
$210$	−696855.	−0.00519248
$211$	9.78232e7	0.716892	0.358446	−	0.933551i	$-0.383307\pi$
$211$	9.78232e7	0.716892	0.358446	+	0.933551i	$0.383307\pi$
$212$	−2.17274e8	−1.56615
$213$	2.40660e8	1.70638
$214$	−9.41111e6	−0.0656437
$215$	3.58503e8	2.46013
$216$	−9.33705e6	−0.0630408
$217$	−5.73619e6	−0.0381078
$218$	−1.43320e6	−0.00936935
$219$	−2.67767e8	−1.72268
$220$	−7.58727e7	−0.480404
$221$	−1.42738e7	−0.0889543
$222$	−9.28337e6	−0.0569469
$223$	9.79866e7	0.591697	0.295849	−	0.955235i	$-0.404398\pi$
$223$	9.79866e7	0.591697	0.295849	+	0.955235i	$0.404398\pi$
$224$	1.74577e6	0.0103781
$225$	2.82976e7	0.165619
$226$	1.82703e6	0.0105285
$227$	−5.74109e7	−0.325765	−0.162882	−	0.986646i	$-0.552079\pi$
$227$	−5.74109e7	−0.325765	−0.162882	+	0.986646i	$0.552079\pi$
$228$	−1.33184e7	−0.0744183
$229$	8.75822e7	0.481938	0.240969	−	0.970533i	$-0.422535\pi$
$229$	8.75822e7	0.481938	0.240969	+	0.970533i	$0.422535\pi$
$230$	1.88295e6	0.0102045
$231$	6.94628e6	0.0370775
$232$	8.61209e6	0.0452794
$233$	2.74007e8	1.41911	0.709555	−	0.704650i	$-0.248896\pi$
$233$	2.74007e8	1.41911	0.709555	+	0.704650i	$0.248896\pi$
$234$	1.74307e6	0.00889321
$235$	2.02735e8	1.01904
$236$	3.11605e8	1.54316
$237$	−3.23321e8	−1.57767
$238$	47447.1	0.000228134 0
$239$	−1.69836e8	−0.804706	−0.402353	−	0.915485i	$-0.631807\pi$
$239$	−1.69836e8	−0.804706	−0.402353	+	0.915485i	$0.631807\pi$
$240$	−3.19695e8	−1.49278
$241$	1.66358e8	0.765569	0.382784	−	0.923838i	$-0.374965\pi$
$241$	1.66358e8	0.765569	0.382784	+	0.923838i	$0.374965\pi$
$242$	6.88148e6	0.0312125
$243$	−8.18109e7	−0.365754
$244$	8.76806e7	0.386402
$245$	3.13834e8	1.36339
$246$	7.06600e6	0.0302622
$247$	−2.19024e7	−0.0924811
$248$	6.67431e6	0.0277860
$249$	2.27903e8	0.935518
$250$	−1.27380e6	−0.00515598
$251$	−3.12623e7	−0.124785	−0.0623926	−	0.998052i	$-0.519873\pi$
$251$	−3.12623e7	−0.124785	−0.0623926	+	0.998052i	$0.519873\pi$
$252$	4.57772e6	0.0180197
$253$	−1.87694e7	−0.0728665
$254$	8.32481e6	0.0318755
$255$	−2.61214e7	−0.0986522
$256$	2.65049e8	0.987383
$257$	−6.86194e7	−0.252163	−0.126081	−	0.992020i	$-0.540240\pi$
$257$	−6.86194e7	−0.252163	−0.126081	+	0.992020i	$0.540240\pi$
$258$	1.90826e7	0.0691780
$259$	−4.00942e7	−0.143395
$260$	−5.26414e8	−1.85746
$261$	3.38809e7	0.117954
$262$	−9.04812e6	−0.0310816
$263$	−2.36522e8	−0.801727	−0.400863	−	0.916138i	$-0.631290\pi$
$263$	−2.36522e8	−0.801727	−0.400863	+	0.916138i	$0.631290\pi$
$264$	−8.08230e6	−0.0270347
$265$	−6.53892e8	−2.15847
$266$	72805.1	0.000237179 0
$267$	−3.95206e8	−1.27067
$268$	2.51996e8	0.799692
$269$	−9.84375e7	−0.308338	−0.154169	−	0.988044i	$-0.549270\pi$
$269$	−9.84375e7	−0.308338	−0.154169	+	0.988044i	$0.549270\pi$
$270$	−1.40412e7	−0.0434140
$271$	1.54836e8	0.472585	0.236293	−	0.971682i	$-0.424068\pi$
$271$	1.54836e8	0.472585	0.236293	+	0.971682i	$0.424068\pi$
$272$	2.17672e7	0.0655861
$273$	4.81941e7	0.143359
$274$	−8.92358e6	−0.0262067
$275$	−1.07822e8	−0.312639
$276$	−7.91863e7	−0.226709
$277$	−3.56376e8	−1.00746	−0.503731	−	0.863860i	$-0.668040\pi$
$277$	−3.56376e8	−1.00746	−0.503731	+	0.863860i	$0.668040\pi$
$278$	−5.63081e6	−0.0157186
$279$	2.62575e7	0.0723832
$280$	3.50188e6	0.00953341
$281$	5.82862e8	1.56709	0.783544	−	0.621336i	$-0.213410\pi$
$281$	5.82862e8	1.56709	0.783544	+	0.621336i	$0.213410\pi$
$282$	1.07913e7	0.0286550
$283$	2.08290e8	0.546282	0.273141	−	0.961974i	$-0.411938\pi$
$283$	2.08290e8	0.546282	0.273141	+	0.961974i	$0.411938\pi$
$284$	−6.04309e8	−1.56547
$285$	−4.00820e7	−0.102563
$286$	−6.64158e6	−0.0167877
$287$	3.05176e7	0.0762014
$288$	−7.99126e6	−0.0197125
$289$	−4.08560e8	−0.995666
$290$	1.29510e7	0.0311824
$291$	7.18210e8	1.70854
$292$	6.72376e8	1.58042
$293$	−3.22973e8	−0.750117	−0.375059	−	0.927001i	$-0.622377\pi$
$293$	−3.22973e8	−0.750117	−0.375059	+	0.927001i	$0.622377\pi$
$294$	1.67049e7	0.0383380
$295$	9.37783e8	2.12679
$296$	4.66514e7	0.104555
$297$	1.39963e8	0.310003
$298$	−1.03353e7	−0.0226238
$299$	−1.30224e8	−0.281736
$300$	−4.54891e8	−0.972710
$301$	8.24164e7	0.174193
$302$	−7.00346e6	−0.0146315
$303$	3.16522e8	0.653665
$304$	3.34007e7	0.0681864
$305$	2.63877e8	0.532540
$306$	−217189.	−0.000433325 0
$307$	−1.44882e8	−0.285779	−0.142890	−	0.989739i	$-0.545639\pi$
$307$	−1.44882e8	−0.285779	−0.142890	+	0.989739i	$0.545639\pi$
$308$	−1.74424e7	−0.0340157
$309$	−4.07845e8	−0.786394
$310$	1.00369e7	0.0191352
$311$	−7.99308e8	−1.50679	−0.753395	−	0.657568i	$-0.771585\pi$
$311$	−7.99308e8	−1.50679	−0.753395	+	0.657568i	$0.771585\pi$
$312$	−5.60759e7	−0.104529
$313$	−3.89384e8	−0.717750	−0.358875	−	0.933386i	$-0.616840\pi$
$313$	−3.89384e8	−0.717750	−0.358875	+	0.933386i	$0.616840\pi$
$314$	2.93954e7	0.0535828
$315$	1.37768e7	0.0248348
$316$	8.11874e8	1.44738
$317$	9.88291e7	0.174252	0.0871260	−	0.996197i	$-0.472232\pi$
$317$	9.88291e7	0.174252	0.0871260	+	0.996197i	$0.472232\pi$
$318$	−3.48057e7	−0.0606954
$319$	−1.29096e8	−0.222661
$320$	8.00729e8	1.36603
$321$	1.19110e9	2.00994
$322$	432873.	0.000722546 0
$323$	2.72908e6	0.00450618
$324$	7.03684e8	1.14940
$325$	−7.48080e8	−1.20881
$326$	−8.49773e6	−0.0135844
$327$	1.81391e8	0.286879
$328$	−3.55085e7	−0.0555616
$329$	4.66068e7	0.0721545
$330$	−1.21543e7	−0.0186179
$331$	−1.22800e9	−1.86124	−0.930618	−	0.365991i	$-0.880730\pi$
$331$	−1.22800e9	−1.86124	−0.930618	+	0.365991i	$0.880730\pi$
$332$	−5.72275e8	−0.858265
$333$	1.83532e8	0.272368
$334$	−2.16075e7	−0.0317315
$335$	7.58390e8	1.10214
$336$	−7.34948e7	−0.105699
$337$	−1.34977e8	−0.192113	−0.0960564	−	0.995376i	$-0.530623\pi$
$337$	−1.34977e8	−0.192113	−0.0960564	+	0.995376i	$0.530623\pi$
$338$	−2.08393e7	−0.0293545
$339$	−2.31235e8	−0.322371
$340$	6.55921e7	0.0905056
$341$	−1.00048e8	−0.136637
$342$	−333266.	−0.000450505 0
$343$	1.44987e8	0.193999
$344$	−9.58951e7	−0.127011
$345$	−2.38313e8	−0.312451
$346$	9.31874e6	0.0120946
$347$	3.93209e8	0.505208	0.252604	−	0.967570i	$-0.418713\pi$
$347$	3.93209e8	0.505208	0.252604	+	0.967570i	$0.418713\pi$
$348$	−5.44644e8	−0.692764
$349$	−1.34667e9	−1.69580	−0.847898	−	0.530160i	$-0.822132\pi$
$349$	−1.34667e9	−1.69580	−0.847898	+	0.530160i	$0.822132\pi$
$350$	2.48667e6	0.00310013
$351$	9.71079e8	1.19861
$352$	3.04490e7	0.0372111
$353$	−4.92850e8	−0.596353	−0.298176	−	0.954511i	$-0.596378\pi$
$353$	−4.92850e8	−0.596353	−0.298176	+	0.954511i	$0.596378\pi$
$354$	4.99168e7	0.0598048
$355$	−1.81868e9	−2.15754
$356$	9.92381e8	1.16574
$357$	−6.00507e6	−0.00698521
$358$	5.19344e7	0.0598225
$359$	1.19631e8	0.136463	0.0682315	−	0.997670i	$-0.478264\pi$
$359$	1.19631e8	0.136463	0.0682315	+	0.997670i	$0.478264\pi$
$360$	−1.60299e7	−0.0181081
$361$	−8.89684e8	−0.995315
$362$	−2.57331e7	−0.0285110
$363$	−8.70945e8	−0.955691
$364$	−1.21017e8	−0.131520
$365$	2.02353e9	2.17814
$366$	1.40458e7	0.0149748
$367$	−4.60690e8	−0.486495	−0.243247	−	0.969964i	$-0.578213\pi$
$367$	−4.60690e8	−0.486495	−0.243247	+	0.969964i	$0.578213\pi$
$368$	1.98588e8	0.207724
$369$	−1.39694e8	−0.144739
$370$	7.01550e7	0.0720033
$371$	−1.50323e8	−0.152833
$372$	−4.22095e8	−0.425118
$373$	9.79539e8	0.977329	0.488664	−	0.872472i	$-0.337484\pi$
$373$	9.79539e8	0.977329	0.488664	+	0.872472i	$0.337484\pi$
$374$	827553.	0.000817985 0
$375$	1.61217e8	0.157870
$376$	−5.42290e7	−0.0526107
$377$	−8.95681e8	−0.860912
$378$	−3.22793e6	−0.00307399
$379$	1.08268e9	1.02156	0.510779	−	0.859712i	$-0.329357\pi$
$379$	1.08268e9	1.02156	0.510779	+	0.859712i	$0.329357\pi$
$380$	1.00648e8	0.0940940
$381$	−1.05362e9	−0.975991
$382$	6.91838e6	0.00635013
$383$	−8.15469e8	−0.741672	−0.370836	−	0.928698i	$-0.620929\pi$
$383$	−8.15469e8	−0.741672	−0.370836	+	0.928698i	$0.620929\pi$
$384$	1.71246e8	0.154334
$385$	−5.24934e7	−0.0468805
$386$	5.73494e7	0.0507543
$387$	−3.77262e8	−0.330868
$388$	−1.80346e9	−1.56745
$389$	8.18454e8	0.704970	0.352485	−	0.935818i	$-0.385337\pi$
$389$	8.18454e8	0.704970	0.352485	+	0.935818i	$0.385337\pi$
$390$	−8.43276e7	−0.0719853
$391$	1.62262e7	0.0137277
$392$	−8.39468e7	−0.0703887
$393$	1.14516e9	0.951685
$394$	7.60516e7	0.0626428
$395$	2.44336e9	1.99479
$396$	7.98428e7	0.0646105
$397$	1.22294e8	0.0980932	0.0490466	−	0.998796i	$-0.484382\pi$
$397$	1.22294e8	0.0980932	0.0490466	+	0.998796i	$0.484382\pi$
$398$	−2.74568e7	−0.0218303
$399$	−9.21448e6	−0.00726216
$400$	1.14080e9	0.891254
$401$	1.84198e9	1.42653	0.713263	−	0.700896i	$-0.247216\pi$
$401$	1.84198e9	1.42653	0.713263	+	0.700896i	$0.247216\pi$
$402$	4.03680e7	0.0309918
$403$	−6.94146e8	−0.528303
$404$	−7.94802e8	−0.599686
$405$	2.11776e9	1.58410
$406$	2.97730e6	0.00220791
$407$	−6.99308e8	−0.514148
$408$	6.98717e6	0.00509320
$409$	1.51116e9	1.09214	0.546069	−	0.837740i	$-0.316123\pi$
$409$	1.51116e9	1.09214	0.546069	+	0.837740i	$0.316123\pi$
$410$	−5.33982e7	−0.0382633
$411$	1.12940e9	0.802420
$412$	1.02412e9	0.721455
$413$	2.15587e8	0.150591
$414$	−1.98148e6	−0.00137243
$415$	−1.72228e9	−1.18286
$416$	2.11258e8	0.143876
$417$	7.12656e8	0.481286
$418$	1.26984e6	0.000850416 0
$419$	−2.24117e9	−1.48842	−0.744211	−	0.667945i	$-0.767174\pi$
$419$	−2.24117e9	−1.48842	−0.744211	+	0.667945i	$0.767174\pi$
$420$	−2.21465e8	−0.145859
$421$	1.63358e9	1.06697	0.533486	−	0.845809i	$-0.320882\pi$
$421$	1.63358e9	1.06697	0.533486	+	0.845809i	$0.320882\pi$
$422$	−3.93495e7	−0.0254886
$423$	−2.13343e8	−0.137052
$424$	1.74908e8	0.111437
$425$	9.32123e7	0.0588995
$426$	−9.68059e7	−0.0606692
$427$	6.06628e7	0.0377073
$428$	−2.99092e9	−1.84396
$429$	8.40581e8	0.514019
$430$	−1.44208e8	−0.0874683
$431$	−1.50743e9	−0.906917	−0.453458	−	0.891277i	$-0.649810\pi$
$431$	−1.50743e9	−0.906917	−0.453458	+	0.891277i	$0.649810\pi$
$432$	−1.48087e9	−0.883740
$433$	1.26852e8	0.0750916	0.0375458	−	0.999295i	$-0.488046\pi$
$433$	1.26852e8	0.0750916	0.0375458	+	0.999295i	$0.488046\pi$
$434$	2.30739e6	0.00135490
$435$	−1.63912e9	−0.954769
$436$	−4.55481e8	−0.263189
$437$	2.48982e7	0.0142719
$438$	1.07710e8	0.0612486
$439$	−3.49859e9	−1.97364	−0.986819	−	0.161825i	$-0.948262\pi$
$439$	−3.49859e9	−1.97364	−0.986819	+	0.161825i	$0.948262\pi$
$440$	6.10784e7	0.0341825
$441$	−3.30256e8	−0.183365
$442$	5.74166e6	0.00316271
$443$	−1.74067e8	−0.0951269	−0.0475635	−	0.998868i	$-0.515146\pi$
$443$	−1.74067e8	−0.0951269	−0.0475635	+	0.998868i	$0.515146\pi$
$444$	−2.95032e9	−1.59966
$445$	2.98660e9	1.60663
$446$	−3.94152e7	−0.0210374
$447$	1.30807e9	0.692715
$448$	1.84080e8	0.0967239
$449$	3.10269e9	1.61762	0.808810	−	0.588070i	$-0.200112\pi$
$449$	3.10269e9	1.61762	0.808810	+	0.588070i	$0.200112\pi$
$450$	−1.13828e7	−0.00588849
$451$	5.32275e8	0.273224
$452$	5.80642e8	0.295750
$453$	8.86383e8	0.448000
$454$	2.30936e7	0.0115823
$455$	−3.64205e8	−0.181262
$456$	1.07215e7	0.00529513
$457$	−3.21350e9	−1.57497	−0.787485	−	0.616334i	$-0.788617\pi$
$457$	−3.21350e9	−1.57497	−0.787485	+	0.616334i	$0.788617\pi$
$458$	−3.52300e7	−0.0171350
$459$	−1.20998e8	−0.0584029
$460$	5.98416e8	0.286649
$461$	7.95845e8	0.378334	0.189167	−	0.981945i	$-0.439421\pi$
$461$	7.95845e8	0.378334	0.189167	+	0.981945i	$0.439421\pi$
$462$	−2.79415e6	−0.00131826
$463$	3.52592e9	1.65097	0.825484	−	0.564426i	$-0.190902\pi$
$463$	3.52592e9	1.65097	0.825484	+	0.564426i	$0.190902\pi$
$464$	1.36589e9	0.634751
$465$	−1.27031e9	−0.585900
$466$	−1.10220e8	−0.0504555
$467$	−1.91403e9	−0.869639	−0.434820	−	0.900518i	$-0.643188\pi$
$467$	−1.91403e9	−0.869639	−0.434820	+	0.900518i	$0.643188\pi$
$468$	5.53959e8	0.249814
$469$	1.74347e8	0.0780385
$470$	−8.15503e7	−0.0362312
$471$	−3.72038e9	−1.64064
$472$	−2.50846e8	−0.109802
$473$	1.43747e9	0.624577
$474$	1.30056e8	0.0560928
$475$	1.43029e8	0.0612347
$476$	1.50790e7	0.00640838
$477$	6.88107e8	0.290297
$478$	6.83167e7	0.0286108
$479$	−1.93092e9	−0.802769	−0.401385	−	0.915910i	$-0.631471\pi$
$479$	−1.93092e9	−0.802769	−0.401385	+	0.915910i	$0.631471\pi$
$480$	3.86608e8	0.159561
$481$	−4.85187e9	−1.98793
$482$	−6.69178e7	−0.0272193
$483$	−5.47860e7	−0.0221235
$484$	2.18698e9	0.876771
$485$	−5.42756e9	−2.16027
$486$	3.29085e7	0.0130041
$487$	−1.16957e9	−0.458854	−0.229427	−	0.973326i	$-0.573685\pi$
$487$	−1.16957e9	−0.458854	−0.229427	+	0.973326i	$0.573685\pi$
$488$	−7.05839e7	−0.0274939
$489$	1.07550e9	0.415940
$490$	−1.26240e8	−0.0484743
$491$	3.11569e9	1.18787	0.593935	−	0.804513i	$-0.297574\pi$
$491$	3.11569e9	1.18787	0.593935	+	0.804513i	$0.297574\pi$
$492$	2.24562e9	0.850078
$493$	1.11604e8	0.0419483
$494$	8.81028e6	0.00328810
$495$	2.40289e8	0.0890463
$496$	1.05856e9	0.389519
$497$	−4.18098e8	−0.152767
$498$	−9.16743e7	−0.0332617
$499$	2.66804e9	0.961258	0.480629	−	0.876924i	$-0.340408\pi$
$499$	2.66804e9	0.961258	0.480629	+	0.876924i	$0.340408\pi$
$500$	−4.04822e8	−0.144834
$501$	2.73472e9	0.971584
$502$	1.25753e7	0.00443665
$503$	1.59344e9	0.558274	0.279137	−	0.960251i	$-0.409952\pi$
$503$	1.59344e9	0.558274	0.279137	+	0.960251i	$0.409952\pi$
$504$	−3.68512e6	−0.00128217
$505$	−2.39198e9	−0.826490
$506$	7.55000e6	0.00259072
$507$	2.63750e9	0.898803
$508$	2.64568e9	0.895395
$509$	3.32887e9	1.11888	0.559441	−	0.828870i	$-0.311016\pi$
$509$	3.32887e9	1.11888	0.559441	+	0.828870i	$0.311016\pi$
$510$	1.05074e7	0.00350751
$511$	4.65191e8	0.154226
$512$	−5.37166e8	−0.176874
$513$	−1.85666e8	−0.0607185
$514$	2.76022e7	0.00896548
$515$	3.08211e9	0.994311
$516$	6.06458e9	1.94324
$517$	8.12896e8	0.258713
$518$	1.61280e7	0.00509830
$519$	−1.17941e9	−0.370322
$520$	4.23769e8	0.132165
$521$	−4.42347e9	−1.37035	−0.685174	−	0.728380i	$-0.740274\pi$
$521$	−4.42347e9	−1.37035	−0.685174	+	0.728380i	$0.740274\pi$
$522$	−1.36286e7	−0.00419378
$523$	2.17666e9	0.665327	0.332664	−	0.943046i	$-0.392053\pi$
$523$	2.17666e9	0.665327	0.332664	+	0.943046i	$0.392053\pi$
$524$	−2.87555e9	−0.873096
$525$	−3.14722e8	−0.0949225
$526$	9.51413e7	0.0285049
$527$	8.64920e7	0.0257418
$528$	−1.28187e9	−0.378987
$529$	1.48036e8	0.0434783
$530$	2.63029e8	0.0767428
$531$	−9.86854e8	−0.286037
$532$	2.31380e7	0.00666246
$533$	3.69298e9	1.05641
$534$	1.58972e8	0.0451780
$535$	−9.00124e9	−2.54135
$536$	−2.02860e8	−0.0569010
$537$	−6.57300e9	−1.83170
$538$	3.95966e7	0.0109628
$539$	1.25837e9	0.346136
$540$	−4.46238e9	−1.21952
$541$	5.15044e8	0.139847	0.0699236	−	0.997552i	$-0.477724\pi$
$541$	5.15044e8	0.139847	0.0699236	+	0.997552i	$0.477724\pi$
$542$	−6.22831e7	−0.0168025
$543$	3.25687e9	0.872975
$544$	−2.63232e7	−0.00701039
$545$	−1.37078e9	−0.362728
$546$	−1.93861e7	−0.00509702
$547$	4.45529e9	1.16391	0.581956	−	0.813221i	$-0.302288\pi$
$547$	4.45529e9	1.16391	0.581956	+	0.813221i	$0.302288\pi$
$548$	−2.83598e9	−0.736157
$549$	−2.77685e8	−0.0716224
$550$	4.33715e7	0.0111157
$551$	1.71250e8	0.0436114
$552$	6.37459e7	0.0161312
$553$	5.61705e8	0.141244
$554$	1.43353e8	0.0358197
$555$	−8.87906e9	−2.20466
$556$	−1.78951e9	−0.441542
$557$	−4.80050e9	−1.17705	−0.588523	−	0.808481i	$-0.700290\pi$
$557$	−4.80050e9	−1.17705	−0.588523	+	0.808481i	$0.700290\pi$
$558$	−1.05621e7	−0.00257354
$559$	9.97336e9	2.41490
$560$	5.55404e8	0.133645
$561$	−1.04738e8	−0.0250458
$562$	−2.34457e8	−0.0557168
$563$	−3.83067e9	−0.904681	−0.452340	−	0.891845i	$-0.649411\pi$
$563$	−3.83067e9	−0.904681	−0.452340	+	0.891845i	$0.649411\pi$
$564$	3.42954e9	0.804931
$565$	1.74746e9	0.407603
$566$	−8.37850e7	−0.0194227
$567$	4.86852e8	0.112165
$568$	4.86476e8	0.111389
$569$	−6.63078e9	−1.50894	−0.754469	−	0.656335i	$-0.772106\pi$
$569$	−6.63078e9	−1.50894	−0.754469	+	0.656335i	$0.772106\pi$
$570$	1.61231e7	0.00364658
$571$	1.69931e9	0.381985	0.190993	−	0.981591i	$-0.438829\pi$
$571$	1.69931e9	0.381985	0.190993	+	0.981591i	$0.438829\pi$
$572$	−2.11074e9	−0.471572
$573$	−8.75615e8	−0.194434
$574$	−1.22757e7	−0.00270929
$575$	8.50402e8	0.186546
$576$	−8.42628e8	−0.183720
$577$	5.62886e9	1.21985	0.609923	−	0.792461i	$-0.291200\pi$
$577$	5.62886e9	1.21985	0.609923	+	0.792461i	$0.291200\pi$
$578$	1.64344e8	0.0354002
$579$	−7.25834e9	−1.55404
$580$	4.11591e9	0.875926
$581$	−3.95935e8	−0.0837543
$582$	−2.88901e8	−0.0607461
$583$	−2.62188e9	−0.547991
$584$	−5.41271e8	−0.112453
$585$	1.66715e9	0.344295
$586$	1.29916e8	0.0266699
$587$	7.62160e9	1.55530	0.777648	−	0.628700i	$-0.216413\pi$
$587$	7.62160e9	1.55530	0.777648	+	0.628700i	$0.216413\pi$
$588$	5.30894e9	1.07693
$589$	1.32718e8	0.0267624
$590$	−3.77225e8	−0.0756167
$591$	−9.62536e9	−1.91805
$592$	7.39899e9	1.46571
$593$	−7.47338e9	−1.47172	−0.735860	−	0.677133i	$-0.763222\pi$
$593$	−7.47338e9	−1.47172	−0.735860	+	0.677133i	$0.763222\pi$
$594$	−5.63003e7	−0.0110219
$595$	4.53807e7	0.00883205
$596$	−3.28462e9	−0.635511
$597$	3.47503e9	0.668419
$598$	5.23828e7	0.0100169
$599$	5.84854e9	1.11187	0.555934	−	0.831226i	$-0.312361\pi$
$599$	5.84854e9	1.11187	0.555934	+	0.831226i	$0.312361\pi$
$600$	3.66193e8	0.0692118
$601$	−6.61342e9	−1.24270	−0.621349	−	0.783534i	$-0.713415\pi$
$601$	−6.61342e9	−1.24270	−0.621349	+	0.783534i	$0.713415\pi$
$602$	−3.31521e7	−0.00619332
$603$	−7.98074e8	−0.148229
$604$	−2.22575e9	−0.411005
$605$	6.58178e9	1.20837
$606$	−1.27322e8	−0.0232406
$607$	6.35793e9	1.15387	0.576933	−	0.816792i	$-0.304249\pi$
$607$	6.35793e9	1.15387	0.576933	+	0.816792i	$0.304249\pi$
$608$	−4.03916e7	−0.00728834
$609$	−3.76818e8	−0.0676038
$610$	−1.06145e8	−0.0189341
$611$	5.63997e9	1.00030
$612$	−6.90243e7	−0.0121723
$613$	−1.00817e10	−1.76775	−0.883875	−	0.467724i	$-0.845074\pi$
$613$	−1.00817e10	−1.76775	−0.883875	+	0.467724i	$0.845074\pi$
$614$	5.82791e7	0.0101607
$615$	6.75826e9	1.17158
$616$	1.40413e7	0.00242034
$617$	−1.28533e9	−0.220301	−0.110150	−	0.993915i	$-0.535133\pi$
$617$	−1.28533e9	−0.220301	−0.110150	+	0.993915i	$0.535133\pi$
$618$	1.64056e8	0.0279597
$619$	6.97489e9	1.18201	0.591004	−	0.806669i	$-0.298732\pi$
$619$	6.97489e9	1.18201	0.591004	+	0.806669i	$0.298732\pi$
$620$	3.18980e9	0.537517
$621$	−1.10390e9	−0.184974
$622$	3.21523e8	0.0535729
$623$	6.86590e8	0.113760
$624$	−8.89373e9	−1.46534
$625$	−6.67880e9	−1.09426
$626$	1.56630e8	0.0255191
$627$	−1.60715e8	−0.0260388
$628$	9.34205e9	1.50516
$629$	6.04553e8	0.0968628
$630$	−5.54173e6	−0.000882985 0
$631$	−5.67839e9	−0.899751	−0.449876	−	0.893091i	$-0.648532\pi$
$631$	−5.67839e9	−0.899751	−0.449876	+	0.893091i	$0.648532\pi$
$632$	−6.53568e8	−0.102987
$633$	4.98022e9	0.780432
$634$	−3.97541e7	−0.00619541
$635$	7.96225e9	1.23404
$636$	−1.10615e10	−1.70496
$637$	8.73069e9	1.33832
$638$	5.19289e7	0.00791657
$639$	1.91385e9	0.290171
$640$	−1.29411e9	−0.195138
$641$	−1.37747e9	−0.206576	−0.103288	−	0.994652i	$-0.532936\pi$
$641$	−1.37747e9	−0.206576	−0.103288	+	0.994652i	$0.532936\pi$
$642$	−4.79123e8	−0.0714619
$643$	3.86369e8	0.0573144	0.0286572	−	0.999589i	$-0.490877\pi$
$643$	3.86369e8	0.0573144	0.0286572	+	0.999589i	$0.490877\pi$
$644$	1.37570e8	0.0202966
$645$	1.82515e10	2.67818
$646$	−1.09778e6	−0.000160214 0
$647$	−2.22898e9	−0.323550	−0.161775	−	0.986828i	$-0.551722\pi$
$647$	−2.22898e9	−0.323550	−0.161775	+	0.986828i	$0.551722\pi$
$648$	−5.66474e8	−0.0817838
$649$	3.76019e9	0.539950
$650$	3.00916e8	0.0429783
$651$	−2.92031e8	−0.0414855
$652$	−2.70064e9	−0.381592
$653$	2.61965e9	0.368169	0.184085	−	0.982910i	$-0.441068\pi$
$653$	2.61965e9	0.368169	0.184085	+	0.982910i	$0.441068\pi$
$654$	−7.29648e7	−0.0101998
$655$	−8.65406e9	−1.20330
$656$	−5.63171e9	−0.778892
$657$	−2.12942e9	−0.292942
$658$	−1.87476e7	−0.00256540
$659$	2.75214e9	0.374603	0.187301	−	0.982303i	$-0.440026\pi$
$659$	2.75214e9	0.374603	0.187301	+	0.982303i	$0.440026\pi$
$660$	−3.86271e9	−0.522984
$661$	−3.04011e9	−0.409434	−0.204717	−	0.978821i	$-0.565627\pi$
$661$	−3.04011e9	−0.409434	−0.204717	+	0.978821i	$0.565627\pi$
$662$	4.93966e8	0.0661750
$663$	−7.26684e8	−0.0968386
$664$	4.60688e8	0.0610687
$665$	6.96344e7	0.00918222
$666$	−7.38259e7	−0.00968387
$667$	1.01819e9	0.132858
$668$	−6.86700e9	−0.891352
$669$	4.98853e9	0.644142
$670$	−3.05063e8	−0.0391858
$671$	1.05806e9	0.135201
$672$	8.88776e7	0.0112980
$673$	−9.40002e9	−1.18871	−0.594355	−	0.804203i	$-0.702593\pi$
$673$	−9.40002e9	−1.18871	−0.594355	+	0.804203i	$0.702593\pi$
$674$	5.42948e7	0.00683044
$675$	−6.34144e9	−0.793642
$676$	−6.62288e9	−0.824581
$677$	1.10150e10	1.36434	0.682170	−	0.731194i	$-0.261036\pi$
$677$	1.10150e10	1.36434	0.682170	+	0.731194i	$0.261036\pi$
$678$	9.30147e7	0.0114617
$679$	−1.24774e9	−0.152961
$680$	−5.28024e7	−0.00643981
$681$	−2.92281e9	−0.354638
$682$	4.02446e7	0.00485805
$683$	9.36716e8	0.112496	0.0562478	−	0.998417i	$-0.482086\pi$
$683$	9.36716e8	0.112496	0.0562478	+	0.998417i	$0.482086\pi$
$684$	−1.05914e8	−0.0126549
$685$	−8.53495e9	−1.01457
$686$	−5.83211e7	−0.00689750
$687$	4.45884e9	0.524654
$688$	−1.52091e10	−1.78051
$689$	−1.81909e10	−2.11879
$690$	9.58619e7	0.0111090
$691$	−3.94317e9	−0.454645	−0.227323	−	0.973820i	$-0.572997\pi$
$691$	−3.94317e9	−0.454645	−0.227323	+	0.973820i	$0.572997\pi$
$692$	2.96156e9	0.339742
$693$	5.52402e7	0.00630505
$694$	−1.58169e8	−0.0179623
$695$	−5.38558e9	−0.608535
$696$	4.38445e8	0.0492927
$697$	−4.60153e8	−0.0514740
$698$	5.41701e8	0.0602929
$699$	1.39498e10	1.54489
$700$	7.90281e8	0.0870840
$701$	−5.21559e9	−0.571861	−0.285931	−	0.958250i	$-0.592303\pi$
$701$	−5.21559e9	−0.571861	−0.285931	+	0.958250i	$0.592303\pi$
$702$	−3.90618e8	−0.0426159
$703$	9.27656e8	0.100703
$704$	3.21065e9	0.346808
$705$	1.03213e10	1.10936
$706$	1.98250e8	0.0212029
$707$	−5.49893e8	−0.0585208
$708$	1.58639e10	1.67994
$709$	5.25555e9	0.553805	0.276902	−	0.960898i	$-0.410692\pi$
$709$	5.25555e9	0.553805	0.276902	+	0.960898i	$0.410692\pi$
$710$	7.31568e8	0.0767097
$711$	−2.57121e9	−0.268283
$712$	−7.98878e8	−0.0829470
$713$	7.89091e8	0.0815293
$714$	2.41555e6	0.000248354 0
$715$	−6.35232e9	−0.649922
$716$	1.65051e10	1.68044
$717$	−8.64641e9	−0.876030
$718$	−4.81219e7	−0.00485185
$719$	−6.95125e9	−0.697448	−0.348724	−	0.937226i	$-0.613385\pi$
$719$	−6.95125e9	−0.697448	−0.348724	+	0.937226i	$0.613385\pi$
$720$	−2.54237e9	−0.253849
$721$	7.08547e8	0.0704036
$722$	3.57877e8	0.0353878
$723$	8.46935e9	0.833424
$724$	−8.17816e9	−0.800886
$725$	5.84907e9	0.570038
$726$	3.50339e8	0.0339789
$727$	2.90274e9	0.280180	0.140090	−	0.990139i	$-0.455261\pi$
$727$	2.90274e9	0.280180	0.140090	+	0.990139i	$0.455261\pi$
$728$	9.74205e7	0.00935816
$729$	7.87330e9	0.752680
$730$	−8.13969e8	−0.0774423
$731$	−1.24270e9	−0.117667
$732$	4.46385e9	0.420650
$733$	−1.52984e10	−1.43477	−0.717384	−	0.696678i	$-0.754661\pi$
$733$	−1.52984e10	−1.43477	−0.717384	+	0.696678i	$0.754661\pi$
$734$	1.85313e8	0.0172970
$735$	1.59774e10	1.48423
$736$	−2.40154e8	−0.0222033
$737$	3.04088e9	0.279810
$738$	5.61923e7	0.00514611
$739$	3.76980e9	0.343607	0.171804	−	0.985131i	$-0.445041\pi$
$739$	3.76980e9	0.343607	0.171804	+	0.985131i	$0.445041\pi$
$740$	2.22957e10	2.02260
$741$	−1.11506e9	−0.100678
$742$	6.04678e7	0.00543389
$743$	−1.48758e10	−1.33052	−0.665259	−	0.746613i	$-0.731679\pi$
$743$	−1.48758e10	−1.33052	−0.665259	+	0.746613i	$0.731679\pi$
$744$	3.39791e8	0.0302487
$745$	−9.88516e9	−0.875864
$746$	−3.94021e8	−0.0347483
$747$	1.81240e9	0.159086
$748$	2.63002e8	0.0229775
$749$	−2.06930e9	−0.179944
$750$	−6.48496e7	−0.00561297
$751$	1.61429e10	1.39073	0.695364	−	0.718658i	$-0.255243\pi$
$751$	1.61429e10	1.39073	0.695364	+	0.718658i	$0.255243\pi$
$752$	−8.60081e9	−0.737526
$753$	−1.59158e9	−0.135845
$754$	3.60289e8	0.0306091
$755$	−6.69845e9	−0.566448
$756$	−1.02586e9	−0.0863498
$757$	1.61028e10	1.34917	0.674583	−	0.738199i	$-0.264323\pi$
$757$	1.61028e10	1.34917	0.674583	+	0.738199i	$0.264323\pi$
$758$	−4.35510e8	−0.0363208
$759$	−9.55555e8	−0.0793249
$760$	−8.10227e7	−0.00669513
$761$	−2.13114e10	−1.75293	−0.876466	−	0.481464i	$-0.840105\pi$
$761$	−2.13114e10	−1.75293	−0.876466	+	0.481464i	$0.840105\pi$
$762$	4.23819e8	0.0347007
$763$	−3.15130e8	−0.0256835
$764$	2.19871e9	0.178378
$765$	−2.07731e8	−0.0167759
$766$	3.28024e8	0.0263697
$767$	2.60886e10	2.08770
$768$	1.34937e10	1.07490
$769$	−1.82130e9	−0.144424	−0.0722121	−	0.997389i	$-0.523006\pi$
$769$	−1.82130e9	−0.144424	−0.0722121	+	0.997389i	$0.523006\pi$
$770$	2.11155e7	0.00166681
$771$	−3.49344e9	−0.274513
$772$	1.82260e10	1.42571
$773$	−1.16843e10	−0.909861	−0.454930	−	0.890527i	$-0.650336\pi$
$773$	−1.16843e10	−0.909861	−0.454930	+	0.890527i	$0.650336\pi$
$774$	1.51754e8	0.0117638
$775$	4.53299e9	0.349807
$776$	1.45180e9	0.111530
$777$	−2.04121e9	−0.156104
$778$	−3.29224e8	−0.0250647
$779$	−7.06081e8	−0.0535148
$780$	−2.67999e10	−2.02210
$781$	−7.29230e9	−0.547754
$782$	−6.52699e6	−0.000488078 0
$783$	−7.59264e9	−0.565232
$784$	−1.33141e10	−0.986746
$785$	2.81151e10	2.07442
$786$	−4.60643e8	−0.0338365
$787$	−9.02441e9	−0.659944	−0.329972	−	0.943991i	$-0.607039\pi$
$787$	−9.02441e9	−0.659944	−0.329972	+	0.943991i	$0.607039\pi$
$788$	2.41697e10	1.75966
$789$	−1.20414e10	−0.872787
$790$	−9.82844e8	−0.0709234
$791$	4.01724e8	0.0288609
$792$	−6.42744e7	−0.00459727
$793$	7.34092e9	0.522750
$794$	−4.91930e7	−0.00348764
$795$	−3.32899e10	−2.34978
$796$	−8.72596e9	−0.613222
$797$	−1.38932e10	−0.972072	−0.486036	−	0.873939i	$-0.661558\pi$
$797$	−1.38932e10	−0.972072	−0.486036	+	0.873939i	$0.661558\pi$
$798$	3.70653e6	0.000258201 0
$799$	−7.02751e8	−0.0487402
$800$	−1.37958e9	−0.0952647
$801$	−3.14287e9	−0.216079
$802$	−7.40939e8	−0.0507192
$803$	8.11368e9	0.552985
$804$	1.28292e10	0.870571
$805$	4.14021e8	0.0279728
$806$	2.79221e8	0.0187835
$807$	−5.01149e9	−0.335668
$808$	6.39825e8	0.0426699
$809$	−6.66037e9	−0.442261	−0.221130	−	0.975244i	$-0.570975\pi$
$809$	−6.66037e9	−0.442261	−0.221130	+	0.975244i	$0.570975\pi$
$810$	−8.51871e8	−0.0563217
$811$	1.27667e10	0.840436	0.420218	−	0.907423i	$-0.361954\pi$
$811$	1.27667e10	0.840436	0.420218	+	0.907423i	$0.361954\pi$
$812$	9.46207e8	0.0620212
$813$	7.88277e9	0.514472
$814$	2.81297e8	0.0182802
$815$	−8.12764e9	−0.525912
$816$	1.10818e9	0.0713992
$817$	−1.90686e9	−0.122332
$818$	−6.07865e8	−0.0388303
$819$	3.83263e8	0.0243783
$820$	−1.69703e10	−1.07483
$821$	1.80980e10	1.14138	0.570690	−	0.821166i	$-0.306676\pi$
$821$	1.80980e10	1.14138	0.570690	+	0.821166i	$0.306676\pi$
$822$	−4.54303e8	−0.0285295
$823$	−6.53913e9	−0.408903	−0.204452	−	0.978877i	$-0.565541\pi$
$823$	−6.53913e9	−0.408903	−0.204452	+	0.978877i	$0.565541\pi$
$824$	−8.24425e8	−0.0513341
$825$	−5.48925e9	−0.340349
$826$	−8.67203e7	−0.00535415
$827$	−1.84244e10	−1.13273	−0.566363	−	0.824156i	$-0.691650\pi$
$827$	−1.84244e10	−1.13273	−0.566363	+	0.824156i	$0.691650\pi$
$828$	−6.29728e8	−0.0385520
$829$	1.57006e9	0.0957142	0.0478571	−	0.998854i	$-0.484761\pi$
$829$	1.57006e9	0.0957142	0.0478571	+	0.998854i	$0.484761\pi$
$830$	6.92788e8	0.0420559
$831$	−1.81432e10	−1.09676
$832$	2.22759e10	1.34092
$833$	−1.08786e9	−0.0652103
$834$	−2.86667e8	−0.0171118
$835$	−2.06664e10	−1.22846
$836$	4.03563e8	0.0238885
$837$	−5.88424e9	−0.346858
$838$	9.01514e8	0.0529198
$839$	3.01177e10	1.76058	0.880289	−	0.474438i	$-0.157349\pi$
$839$	3.01177e10	1.76058	0.880289	+	0.474438i	$0.157349\pi$
$840$	1.78282e8	0.0103784
$841$	−1.02468e10	−0.594019
$842$	−6.57110e8	−0.0379355
$843$	2.96737e10	1.70598
$844$	−1.25055e10	−0.715985
$845$	−1.99317e10	−1.13644
$846$	8.58174e7	0.00487281
$847$	1.51309e9	0.0855603
$848$	2.77407e10	1.56218
$849$	1.06041e10	0.594700
$850$	−3.74947e7	−0.00209413
$851$	5.51551e9	0.306784
$852$	−3.07656e10	−1.70422
$853$	1.76390e10	0.973087	0.486544	−	0.873656i	$-0.338257\pi$
$853$	1.76390e10	0.973087	0.486544	+	0.873656i	$0.338257\pi$
$854$	−2.44017e7	−0.00134066
$855$	−3.18752e8	−0.0174410
$856$	2.40772e9	0.131204
$857$	−1.54494e10	−0.838456	−0.419228	−	0.907881i	$-0.637699\pi$
$857$	−1.54494e10	−0.838456	−0.419228	+	0.907881i	$0.637699\pi$
$858$	−3.38125e8	−0.0182756
$859$	2.02861e8	0.0109200	0.00545999	−	0.999985i	$-0.498262\pi$
$859$	2.02861e8	0.0109200	0.00545999	+	0.999985i	$0.498262\pi$
$860$	−4.58304e10	−2.45702
$861$	1.55366e9	0.0829555
$862$	6.06367e8	0.0322448
$863$	−1.28373e10	−0.679884	−0.339942	−	0.940446i	$-0.610407\pi$
$863$	−1.28373e10	−0.679884	−0.339942	+	0.940446i	$0.610407\pi$
$864$	1.79083e9	0.0944616
$865$	8.91289e9	0.468233
$866$	−5.10266e7	−0.00266983
$867$	−2.07999e10	−1.08392
$868$	7.33304e8	0.0380597
$869$	9.79703e9	0.506437
$870$	6.59338e8	0.0339462
$871$	2.10980e10	1.08188
$872$	3.66668e8	0.0187269
$873$	5.71156e9	0.290539
$874$	−1.00153e7	−0.000507429 0
$875$	−2.80081e8	−0.0141337
$876$	3.42309e10	1.72050
$877$	2.06902e10	1.03578	0.517890	−	0.855448i	$-0.326718\pi$
$877$	2.06902e10	1.03578	0.517890	+	0.855448i	$0.326718\pi$
$878$	1.40731e9	0.0701714
$879$	−1.64427e10	−0.816603
$880$	9.68714e9	0.479188
$881$	2.52509e10	1.24412	0.622058	−	0.782971i	$-0.286297\pi$
$881$	2.52509e10	1.24412	0.622058	+	0.782971i	$0.286297\pi$
$882$	1.32846e8	0.00651940
$883$	2.50476e8	0.0122434	0.00612172	−	0.999981i	$-0.498051\pi$
$883$	2.50476e8	0.0122434	0.00612172	+	0.999981i	$0.498051\pi$
$884$	1.82474e9	0.0888418
$885$	4.77429e10	2.31530
$886$	7.00187e7	0.00338217
$887$	1.31505e10	0.632719	0.316360	−	0.948639i	$-0.397539\pi$
$887$	1.31505e10	0.632719	0.316360	+	0.948639i	$0.397539\pi$
$888$	2.37504e9	0.113822
$889$	1.83045e9	0.0873777
$890$	−1.20136e9	−0.0571227
$891$	8.49148e9	0.402172
$892$	−1.25264e10	−0.590949
$893$	−1.07834e9	−0.0506727
$894$	−5.26172e8	−0.0246290
$895$	4.96726e10	2.31599
$896$	−2.97504e8	−0.0138171
$897$	−6.62975e9	−0.306707
$898$	−1.24806e9	−0.0575134
$899$	5.42737e9	0.249133
$900$	−3.61752e9	−0.165410
$901$	2.26662e9	0.103239
$902$	−2.14108e8	−0.00971428
$903$	4.19585e9	0.189632
$904$	−4.67424e8	−0.0210437
$905$	−2.46124e10	−1.10378
$906$	−3.56549e8	−0.0159283
$907$	−3.75292e10	−1.67010	−0.835052	−	0.550170i	$-0.814563\pi$
$907$	−3.75292e10	−1.67010	−0.835052	+	0.550170i	$0.814563\pi$
$908$	7.33930e9	0.325353
$909$	2.51714e9	0.111156
$910$	1.46502e8	0.00644464
$911$	−4.85320e9	−0.212674	−0.106337	−	0.994330i	$-0.533912\pi$
$911$	−4.85320e9	−0.212674	−0.106337	+	0.994330i	$0.533912\pi$
$912$	1.70044e9	0.0742300
$913$	−6.90574e9	−0.300305
$914$	1.29264e9	0.0559970
$915$	1.34341e10	0.579741
$916$	−1.11963e10	−0.481329
$917$	−1.98949e9	−0.0852016
$918$	4.86717e7	0.00207648
$919$	−7.39351e8	−0.0314229	−0.0157114	−	0.999877i	$-0.505001\pi$
$919$	−7.39351e8	−0.0314229	−0.0157114	+	0.999877i	$0.505001\pi$
$920$	−4.81732e8	−0.0203961
$921$	−7.37601e9	−0.311109
$922$	−3.20130e8	−0.0134514
$923$	−5.05948e10	−2.11787
$924$	−8.88000e8	−0.0370306
$925$	3.16842e10	1.31628
$926$	−1.41830e9	−0.0586990
$927$	−3.24338e9	−0.133727
$928$	−1.65178e9	−0.0678476
$929$	−8.85297e9	−0.362271	−0.181136	−	0.983458i	$-0.557977\pi$
$929$	−8.85297e9	−0.362271	−0.181136	+	0.983458i	$0.557977\pi$
$930$	5.10982e8	0.0208313
$931$	−1.66927e9	−0.0677957
$932$	−3.50286e10	−1.41732
$933$	−4.06931e10	−1.64034
$934$	7.69920e8	0.0309194
$935$	7.91512e8	0.0316677
$936$	−4.45943e8	−0.0177752
$937$	−2.94224e10	−1.16839	−0.584197	−	0.811612i	$-0.698590\pi$
$937$	−2.94224e10	−1.16839	−0.584197	+	0.811612i	$0.698590\pi$
$938$	−7.01312e7	−0.00277460
$939$	−1.98237e10	−0.781367
$940$	−2.59172e10	−1.01775
$941$	3.75226e10	1.46801	0.734006	−	0.679143i	$-0.237648\pi$
$941$	3.75226e10	1.46801	0.734006	+	0.679143i	$0.237648\pi$
$942$	1.49653e9	0.0583320
$943$	−4.19811e9	−0.163028
$944$	−3.97845e10	−1.53926
$945$	−3.08735e9	−0.119008
$946$	−5.78226e8	−0.0222064
$947$	6.07906e9	0.232601	0.116300	−	0.993214i	$-0.462896\pi$
$947$	6.07906e9	0.232601	0.116300	+	0.993214i	$0.462896\pi$
$948$	4.13328e10	1.57567
$949$	5.62936e10	2.13810
$950$	−5.75338e7	−0.00217716
$951$	5.03143e9	0.189696
$952$	−1.21388e7	−0.000455980 0
$953$	−3.80772e10	−1.42508	−0.712540	−	0.701631i	$-0.752455\pi$
$953$	−3.80772e10	−1.42508	−0.712540	+	0.701631i	$0.752455\pi$
$954$	−2.76792e8	−0.0103213
$955$	6.61707e9	0.245841
$956$	2.17115e10	0.803688
$957$	−6.57231e9	−0.242397
$958$	7.76716e8	0.0285419
$959$	−1.96210e9	−0.0718384
$960$	4.07654e10	1.48711
$961$	−2.33064e10	−0.847118
$962$	1.95167e9	0.0706796
$963$	9.47224e9	0.341791
$964$	−2.12669e10	−0.764601
$965$	5.48517e10	1.96492
$966$	2.20377e7	0.000786587 0
$967$	9.52122e9	0.338610	0.169305	−	0.985564i	$-0.445848\pi$
$967$	9.52122e9	0.338610	0.169305	+	0.985564i	$0.445848\pi$
$968$	−1.76055e9	−0.0623855
$969$	1.38939e8	0.00490557
$970$	2.18324e9	0.0768070
$971$	−1.13764e10	−0.398783	−0.199391	−	0.979920i	$-0.563896\pi$
$971$	−1.13764e10	−0.398783	−0.199391	+	0.979920i	$0.563896\pi$
$972$	1.04586e10	0.365292
$973$	−1.23809e9	−0.0430882
$974$	4.70461e8	0.0163143
$975$	−3.80850e10	−1.31595
$976$	−1.11947e10	−0.385424
$977$	1.12524e10	0.386023	0.193011	−	0.981197i	$-0.438175\pi$
$977$	1.12524e10	0.386023	0.193011	+	0.981197i	$0.438175\pi$
$978$	−4.32622e8	−0.0147885
$979$	1.19752e10	0.407891
$980$	−4.01200e10	−1.36166
$981$	1.44251e9	0.0487840
$982$	−1.25329e9	−0.0422339
$983$	−2.57837e9	−0.0865780	−0.0432890	−	0.999063i	$-0.513784\pi$
$983$	−2.57837e9	−0.0865780	−0.0432890	+	0.999063i	$0.513784\pi$
$984$	−1.80775e9	−0.0604862
$985$	7.27394e10	2.42517
$986$	−4.48927e7	−0.00149144
$987$	2.37277e9	0.0785498
$988$	2.79997e9	0.0923642
$989$	−1.13375e10	−0.372675
$990$	−9.66566e7	−0.00316598
$991$	3.72886e9	0.121708	0.0608540	−	0.998147i	$-0.480618\pi$
$991$	3.72886e9	0.121708	0.0608540	+	0.998147i	$0.480618\pi$
$992$	−1.28012e9	−0.0416350
$993$	−6.25181e10	−2.02620
$994$	1.68180e8	0.00543154
$995$	−2.62610e10	−0.845144
$996$	−2.91347e10	−0.934336
$997$	−4.69533e10	−1.50049	−0.750246	−	0.661159i	$-0.770065\pi$
$997$	−4.69533e10	−1.50049	−0.750246	+	0.661159i	$0.770065\pi$
$998$	−1.07322e9	−0.0341769
$999$	−4.11291e10	−1.30518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.8.a.a.1.3	✓	5
3.2	odd	2		207.8.a.b.1.3		5
4.3	odd	2		368.8.a.e.1.1		5
5.4	even	2		575.8.a.a.1.3		5
23.22	odd	2		529.8.a.b.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.a.a.1.3	✓	5	1.1	even	1	trivial
207.8.a.b.1.3		5	3.2	odd	2
368.8.a.e.1.1		5	4.3	odd	2
529.8.a.b.1.3		5	23.22	odd	2
575.8.a.a.1.3		5	5.4	even	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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	23.a (trivial)

\(f(q)\)	\(=\)	\(q-0.402251 q^{2} +50.9104 q^{3} -127.838 q^{4} -384.733 q^{5} -20.4788 q^{6} -88.4463 q^{7} +102.911 q^{8} +404.864 q^{9} +O(q^{10})\)	\(q-0.402251 q^{2} +50.9104 q^{3} -127.838 q^{4} -384.733 q^{5} -20.4788 q^{6} -88.4463 q^{7} +102.911 q^{8} +404.864 q^{9} +154.759 q^{10} -1542.65 q^{11} -6508.29 q^{12} -10703.0 q^{13} +35.5776 q^{14} -19586.9 q^{15} +16321.9 q^{16} +1333.62 q^{17} -162.857 q^{18} +2046.37 q^{19} +49183.5 q^{20} -4502.83 q^{21} +620.531 q^{22} +12167.0 q^{23} +5239.25 q^{24} +69894.1 q^{25} +4305.31 q^{26} -90729.2 q^{27} +11306.8 q^{28} +83684.7 q^{29} +7878.84 q^{30} +64855.0 q^{31} -19738.1 q^{32} -78536.6 q^{33} -536.450 q^{34} +34028.2 q^{35} -51757.1 q^{36} +453317. q^{37} -823.156 q^{38} -544896. q^{39} -39593.3 q^{40} -345040. q^{41} +1811.27 q^{42} -931824. q^{43} +197209. q^{44} -155764. q^{45} -4894.19 q^{46} -526950. q^{47} +830953. q^{48} -815720. q^{49} -28115.0 q^{50} +67895.1 q^{51} +1.36826e6 q^{52} +1.69960e6 q^{53} +36495.9 q^{54} +593506. q^{55} -9102.12 q^{56} +104182. q^{57} -33662.3 q^{58} -2.43749e6 q^{59} +2.50395e6 q^{60} -685871. q^{61} -26088.0 q^{62} -35808.7 q^{63} -2.08126e6 q^{64} +4.11781e6 q^{65} +31591.5 q^{66} -1.97121e6 q^{67} -170488. q^{68} +619426. q^{69} -13687.9 q^{70} +4.72714e6 q^{71} +41665.1 q^{72} -5.25959e6 q^{73} -182347. q^{74} +3.55834e6 q^{75} -261605. q^{76} +136441. q^{77} +219185. q^{78} -6.35079e6 q^{79} -6.27956e6 q^{80} -5.50449e6 q^{81} +138793. q^{82} +4.47655e6 q^{83} +575634. q^{84} -513087. q^{85} +374827. q^{86} +4.26042e6 q^{87} -158756. q^{88} -7.76279e6 q^{89} +62656.4 q^{90} +946645. q^{91} -1.55541e6 q^{92} +3.30179e6 q^{93} +211966. q^{94} -787306. q^{95} -1.00488e6 q^{96} +1.41073e7 q^{97} +328124. q^{98} -624562. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9}+O(q^{10}) \) 5 * q - 16 * q^2 - 68 * q^3 + 256 * q^4 - 56 * q^5 + 538 * q^6 - 1156 * q^7 - 5952 * q^8 - 2195 * q^9	\( 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9} - 11260 q^{10} - 1318 q^{11} - 28012 q^{12} - 19662 q^{13} - 6848 q^{14} - 24600 q^{15} + 32448 q^{16} - 5002 q^{17} - 24650 q^{18} - 38314 q^{19} + 104440 q^{20} + 33280 q^{21} - 74872 q^{22} + 60835 q^{23} + 361992 q^{24} + 54959 q^{25} + 345430 q^{26} + 100900 q^{27} + 90800 q^{28} - 150634 q^{29} + 515724 q^{30} - 179940 q^{31} - 404032 q^{32} - 619688 q^{33} + 32116 q^{34} - 374032 q^{35} + 510524 q^{36} - 752672 q^{37} + 456808 q^{38} - 207996 q^{39} - 1082576 q^{40} - 1192910 q^{41} - 250520 q^{42} - 932646 q^{43} + 2467104 q^{44} - 389952 q^{45} - 194672 q^{46} - 1008460 q^{47} - 1916464 q^{48} - 2005219 q^{49} + 1571224 q^{50} - 211520 q^{51} - 1740516 q^{52} + 897104 q^{53} + 1844686 q^{54} + 1203168 q^{55} + 3050144 q^{56} + 3137192 q^{57} + 5685090 q^{58} + 1020972 q^{59} - 1479384 q^{60} - 2758364 q^{61} + 2661794 q^{62} + 1135132 q^{63} + 5173248 q^{64} - 1350472 q^{65} + 11693212 q^{66} - 1523138 q^{67} + 2501304 q^{68} - 827356 q^{69} - 2794240 q^{70} + 3044884 q^{71} - 6740904 q^{72} - 8872022 q^{73} + 1408492 q^{74} + 1960276 q^{75} - 17963952 q^{76} - 3501672 q^{77} - 15280362 q^{78} - 4437540 q^{79} + 12197536 q^{80} - 7995203 q^{81} - 7738154 q^{82} - 4637362 q^{83} + 2663744 q^{84} - 8625728 q^{85} - 3025868 q^{86} + 17151068 q^{87} - 41815040 q^{88} + 6381402 q^{89} - 4970376 q^{90} + 3240808 q^{91} + 3114752 q^{92} + 7185076 q^{93} - 13893974 q^{94} + 15762704 q^{95} + 40696544 q^{96} - 6432034 q^{97} + 22652640 q^{98} + 30201754 q^{99}+O(q^{100}) \) 5 * q - 16 * q^2 - 68 * q^3 + 256 * q^4 - 56 * q^5 + 538 * q^6 - 1156 * q^7 - 5952 * q^8 - 2195 * q^9 - 11260 * q^10 - 1318 * q^11 - 28012 * q^12 - 19662 * q^13 - 6848 * q^14 - 24600 * q^15 + 32448 * q^16 - 5002 * q^17 - 24650 * q^18 - 38314 * q^19 + 104440 * q^20 + 33280 * q^21 - 74872 * q^22 + 60835 * q^23 + 361992 * q^24 + 54959 * q^25 + 345430 * q^26 + 100900 * q^27 + 90800 * q^28 - 150634 * q^29 + 515724 * q^30 - 179940 * q^31 - 404032 * q^32 - 619688 * q^33 + 32116 * q^34 - 374032 * q^35 + 510524 * q^36 - 752672 * q^37 + 456808 * q^38 - 207996 * q^39 - 1082576 * q^40 - 1192910 * q^41 - 250520 * q^42 - 932646 * q^43 + 2467104 * q^44 - 389952 * q^45 - 194672 * q^46 - 1008460 * q^47 - 1916464 * q^48 - 2005219 * q^49 + 1571224 * q^50 - 211520 * q^51 - 1740516 * q^52 + 897104 * q^53 + 1844686 * q^54 + 1203168 * q^55 + 3050144 * q^56 + 3137192 * q^57 + 5685090 * q^58 + 1020972 * q^59 - 1479384 * q^60 - 2758364 * q^61 + 2661794 * q^62 + 1135132 * q^63 + 5173248 * q^64 - 1350472 * q^65 + 11693212 * q^66 - 1523138 * q^67 + 2501304 * q^68 - 827356 * q^69 - 2794240 * q^70 + 3044884 * q^71 - 6740904 * q^72 - 8872022 * q^73 + 1408492 * q^74 + 1960276 * q^75 - 17963952 * q^76 - 3501672 * q^77 - 15280362 * q^78 - 4437540 * q^79 + 12197536 * q^80 - 7995203 * q^81 - 7738154 * q^82 - 4637362 * q^83 + 2663744 * q^84 - 8625728 * q^85 - 3025868 * q^86 + 17151068 * q^87 - 41815040 * q^88 + 6381402 * q^89 - 4970376 * q^90 + 3240808 * q^91 + 3114752 * q^92 + 7185076 * q^93 - 13893974 * q^94 + 15762704 * q^95 + 40696544 * q^96 - 6432034 * q^97 + 22652640 * q^98 + 30201754 * q^99

Introduction

L-functions

Modular forms

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