LMFDB - Embedded newform 23.8.a.b.1.4

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 $ x^8 - 832x^6 - 1059x^5 + 203052x^4 + 678328x^3 - 13424272x^2 - 73308944x - 37372224
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-6.60982$ 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)

$f(q)$	$=$	$q-6.60982 q^{2} -84.4445 q^{3} -84.3103 q^{4} -124.345 q^{5} +558.163 q^{6} -780.885 q^{7} +1403.33 q^{8} +4943.87 q^{9} +O(q^{10})$	\(q-6.60982 q^{2} -84.4445 q^{3} -84.3103 q^{4} -124.345 q^{5} +558.163 q^{6} -780.885 q^{7} +1403.33 q^{8} +4943.87 q^{9} +821.899 q^{10} -5633.52 q^{11} +7119.55 q^{12} -1020.75 q^{13} +5161.51 q^{14} +10500.3 q^{15} +1515.96 q^{16} +15830.1 q^{17} -32678.1 q^{18} -41241.3 q^{19} +10483.6 q^{20} +65941.4 q^{21} +37236.6 q^{22} -12167.0 q^{23} -118504. q^{24} -62663.3 q^{25} +6746.98 q^{26} -232803. q^{27} +65836.7 q^{28} -11106.5 q^{29} -69404.8 q^{30} +262126. q^{31} -189647. q^{32} +475720. q^{33} -104634. q^{34} +97099.3 q^{35} -416820. q^{36} +501470. q^{37} +272597. q^{38} +86196.8 q^{39} -174498. q^{40} -313935. q^{41} -435861. q^{42} +463488. q^{43} +474964. q^{44} -614747. q^{45} +80421.6 q^{46} +905432. q^{47} -128014. q^{48} -213762. q^{49} +414193. q^{50} -1.33677e6 q^{51} +86059.9 q^{52} -850343. q^{53} +1.53878e6 q^{54} +700502. q^{55} -1.09584e6 q^{56} +3.48260e6 q^{57} +73411.8 q^{58} -1.69488e6 q^{59} -885281. q^{60} -125776. q^{61} -1.73261e6 q^{62} -3.86060e6 q^{63} +1.05949e6 q^{64} +126925. q^{65} -3.14442e6 q^{66} -1.98979e6 q^{67} -1.33464e6 q^{68} +1.02744e6 q^{69} -641808. q^{70} -5.28420e6 q^{71} +6.93790e6 q^{72} +5.74248e6 q^{73} -3.31463e6 q^{74} +5.29157e6 q^{75} +3.47707e6 q^{76} +4.39913e6 q^{77} -569745. q^{78} +3.18023e6 q^{79} -188502. q^{80} +8.84668e6 q^{81} +2.07505e6 q^{82} -4.67105e6 q^{83} -5.55955e6 q^{84} -1.96840e6 q^{85} -3.06357e6 q^{86} +937881. q^{87} -7.90570e6 q^{88} +1.04406e7 q^{89} +4.06336e6 q^{90} +797089. q^{91} +1.02580e6 q^{92} -2.21351e7 q^{93} -5.98474e6 q^{94} +5.12816e6 q^{95} +1.60146e7 q^{96} -2.16055e6 q^{97} +1.41293e6 q^{98} -2.78514e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9}+O(q^{10}) $ 8 * q + 40 * q^3 + 640 * q^4 + 444 * q^5 - 1745 * q^6 + 1446 * q^7 + 3177 * q^8 + 13878 * q^9	\( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9} + 19502 q^{10} + 7588 q^{11} + 22733 q^{12} + 19862 q^{13} + 17544 q^{14} - 12770 q^{15} + 64336 q^{16} + 42070 q^{17} - 59129 q^{18} + 1050 q^{19} + 3364 q^{20} - 7698 q^{21} - 128220 q^{22} - 97336 q^{23} - 621188 q^{24} + 49496 q^{25} - 371761 q^{26} - 69500 q^{27} + 143050 q^{28} - 102578 q^{29} - 671470 q^{30} + 304172 q^{31} - 612824 q^{32} + 747242 q^{33} - 524530 q^{34} + 531048 q^{35} + 1868983 q^{36} + 286472 q^{37} - 762932 q^{38} + 1032828 q^{39} + 2105286 q^{40} + 1324414 q^{41} - 1886168 q^{42} + 2052578 q^{43} - 867298 q^{44} + 2087442 q^{45} + 675556 q^{47} + 1411151 q^{48} - 55404 q^{49} + 1458528 q^{50} + 2775482 q^{51} - 1695409 q^{52} + 203654 q^{53} - 9897559 q^{54} - 1024444 q^{55} - 5766846 q^{56} + 3908648 q^{57} - 5039991 q^{58} - 748892 q^{59} - 18153300 q^{60} + 61822 q^{61} - 4939277 q^{62} + 1411632 q^{63} + 2702267 q^{64} - 1571618 q^{65} + 3791866 q^{66} + 3235604 q^{67} + 4914980 q^{68} - 486680 q^{69} + 10871764 q^{70} - 4951664 q^{71} - 7940241 q^{72} + 11019370 q^{73} + 356954 q^{74} - 13607220 q^{75} + 21973240 q^{76} - 5284888 q^{77} - 1506779 q^{78} + 4202464 q^{79} + 8785886 q^{80} + 10294096 q^{81} + 32636759 q^{82} + 518568 q^{83} + 7629190 q^{84} + 9854220 q^{85} - 14681386 q^{86} + 4862532 q^{87} + 20589740 q^{88} + 4203864 q^{89} + 49021076 q^{90} + 2488406 q^{91} - 7786880 q^{92} - 23367842 q^{93} + 12314327 q^{94} - 44485300 q^{95} - 45317009 q^{96} + 18621134 q^{97} + 35756 q^{98} - 64729930 q^{99}+O(q^{100}) \) 8 * q + 40 * q^3 + 640 * q^4 + 444 * q^5 - 1745 * q^6 + 1446 * q^7 + 3177 * q^8 + 13878 * q^9 + 19502 * q^10 + 7588 * q^11 + 22733 * q^12 + 19862 * q^13 + 17544 * q^14 - 12770 * q^15 + 64336 * q^16 + 42070 * q^17 - 59129 * q^18 + 1050 * q^19 + 3364 * q^20 - 7698 * q^21 - 128220 * q^22 - 97336 * q^23 - 621188 * q^24 + 49496 * q^25 - 371761 * q^26 - 69500 * q^27 + 143050 * q^28 - 102578 * q^29 - 671470 * q^30 + 304172 * q^31 - 612824 * q^32 + 747242 * q^33 - 524530 * q^34 + 531048 * q^35 + 1868983 * q^36 + 286472 * q^37 - 762932 * q^38 + 1032828 * q^39 + 2105286 * q^40 + 1324414 * q^41 - 1886168 * q^42 + 2052578 * q^43 - 867298 * q^44 + 2087442 * q^45 + 675556 * q^47 + 1411151 * q^48 - 55404 * q^49 + 1458528 * q^50 + 2775482 * q^51 - 1695409 * q^52 + 203654 * q^53 - 9897559 * q^54 - 1024444 * q^55 - 5766846 * q^56 + 3908648 * q^57 - 5039991 * q^58 - 748892 * q^59 - 18153300 * q^60 + 61822 * q^61 - 4939277 * q^62 + 1411632 * q^63 + 2702267 * q^64 - 1571618 * q^65 + 3791866 * q^66 + 3235604 * q^67 + 4914980 * q^68 - 486680 * q^69 + 10871764 * q^70 - 4951664 * q^71 - 7940241 * q^72 + 11019370 * q^73 + 356954 * q^74 - 13607220 * q^75 + 21973240 * q^76 - 5284888 * q^77 - 1506779 * q^78 + 4202464 * q^79 + 8785886 * q^80 + 10294096 * q^81 + 32636759 * q^82 + 518568 * q^83 + 7629190 * q^84 + 9854220 * q^85 - 14681386 * q^86 + 4862532 * q^87 + 20589740 * q^88 + 4203864 * q^89 + 49021076 * q^90 + 2488406 * q^91 - 7786880 * q^92 - 23367842 * q^93 + 12314327 * q^94 - 44485300 * q^95 - 45317009 * q^96 + 18621134 * q^97 + 35756 * q^98 - 64729930 * 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 $

$2$	−6.60982	−0.584231	−0.292115	−	0.956383i	$-0.594359\pi$
$2$	−6.60982	−0.584231	−0.292115	+	0.956383i	$0.594359\pi$
$3$	−84.4445	−1.80571	−0.902853	−	0.429949i	$-0.858531\pi$
$3$	−84.4445	−1.80571	−0.902853	+	0.429949i	$0.858531\pi$
$4$	−84.3103	−0.658675
$5$	−124.345	−0.444871	−0.222435	−	0.974947i	$-0.571401\pi$
$5$	−124.345	−0.444871	−0.222435	+	0.974947i	$0.571401\pi$
$6$	558.163	1.05495
$7$	−780.885	−0.860486	−0.430243	−	0.902713i	$-0.641572\pi$
$7$	−780.885	−0.860486	−0.430243	+	0.902713i	$0.641572\pi$
$8$	1403.33	0.969049
$9$	4943.87	2.26057
$10$	821.899	0.259907
$11$	−5633.52	−1.27616	−0.638081	−	0.769969i	$-0.720271\pi$
$11$	−5633.52	−1.27616	−0.638081	+	0.769969i	$0.720271\pi$
$12$	7119.55	1.18937
$13$	−1020.75	−0.128860	−0.0644300	−	0.997922i	$-0.520523\pi$
$13$	−1020.75	−0.128860	−0.0644300	+	0.997922i	$0.520523\pi$
$14$	5161.51	0.502722
$15$	10500.3	0.803306
$16$	1515.96	0.0925266
$17$	15830.1	0.781472	0.390736	−	0.920503i	$-0.372220\pi$
$17$	15830.1	0.781472	0.390736	+	0.920503i	$0.372220\pi$
$18$	−32678.1	−1.32070
$19$	−41241.3	−1.37941	−0.689707	−	0.724088i	$-0.742261\pi$
$19$	−41241.3	−1.37941	−0.689707	+	0.724088i	$0.742261\pi$
$20$	10483.6	0.293025
$21$	65941.4	1.55378
$22$	37236.6	0.745573
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	−118504.	−1.74982
$25$	−62663.3	−0.802090
$26$	6746.98	0.0752840
$27$	−232803.	−2.27623
$28$	65836.7	0.566780
$29$	−11106.5	−0.0845636	−0.0422818	−	0.999106i	$-0.513463\pi$
$29$	−11106.5	−0.0845636	−0.0422818	+	0.999106i	$0.513463\pi$
$30$	−69404.8	−0.469316
$31$	262126.	1.58032	0.790159	−	0.612902i	$-0.209998\pi$
$31$	262126.	1.58032	0.790159	+	0.612902i	$0.209998\pi$
$32$	−189647.	−1.02311
$33$	475720.	2.30437
$34$	−104634.	−0.456560
$35$	97099.3	0.382805
$36$	−416820.	−1.48898
$37$	501470.	1.62757	0.813784	−	0.581168i	$-0.197404\pi$
$37$	501470.	1.62757	0.813784	+	0.581168i	$0.197404\pi$
$38$	272597.	0.805896
$39$	86196.8	0.232683
$40$	−174498.	−0.431101
$41$	−313935.	−0.711370	−0.355685	−	0.934606i	$-0.615752\pi$
$41$	−313935.	−0.711370	−0.355685	+	0.934606i	$0.615752\pi$
$42$	−435861.	−0.907769
$43$	463488.	0.888995	0.444498	−	0.895780i	$-0.353382\pi$
$43$	463488.	0.888995	0.444498	+	0.895780i	$0.353382\pi$
$44$	474964.	0.840575
$45$	−614747.	−1.00566
$46$	80421.6	0.121821
$47$	905432.	1.27208	0.636038	−	0.771657i	$-0.280572\pi$
$47$	905432.	1.27208	0.636038	+	0.771657i	$0.280572\pi$
$48$	−128014.	−0.167076
$49$	−213762.	−0.259564
$50$	414193.	0.468606
$51$	−1.33677e6	−1.41111
$52$	86059.9	0.0848768
$53$	−850343.	−0.784564	−0.392282	−	0.919845i	$-0.628314\pi$
$53$	−850343.	−0.784564	−0.392282	+	0.919845i	$0.628314\pi$
$54$	1.53878e6	1.32984
$55$	700502.	0.567727
$56$	−1.09584e6	−0.833853
$57$	3.48260e6	2.49082
$58$	73411.8	0.0494046
$59$	−1.69488e6	−1.07438	−0.537190	−	0.843462i	$-0.680514\pi$
$59$	−1.69488e6	−1.07438	−0.537190	+	0.843462i	$0.680514\pi$
$60$	−885281.	−0.529117
$61$	−125776.	−0.0709484	−0.0354742	−	0.999371i	$-0.511294\pi$
$61$	−125776.	−0.0709484	−0.0354742	+	0.999371i	$0.511294\pi$
$62$	−1.73261e6	−0.923270
$63$	−3.86060e6	−1.94519
$64$	1.05949e6	0.505203
$65$	126925.	0.0573261
$66$	−3.14442e6	−1.34629
$67$	−1.98979e6	−0.808249	−0.404125	−	0.914704i	$-0.632424\pi$
$67$	−1.98979e6	−0.808249	−0.404125	+	0.914704i	$0.632424\pi$
$68$	−1.33464e6	−0.514736
$69$	1.02744e6	0.376516
$70$	−641808.	−0.223647
$71$	−5.28420e6	−1.75216	−0.876082	−	0.482163i	$-0.839851\pi$
$71$	−5.28420e6	−1.75216	−0.876082	+	0.482163i	$0.839851\pi$
$72$	6.93790e6	2.19061
$73$	5.74248e6	1.72771	0.863853	−	0.503745i	$-0.168045\pi$
$73$	5.74248e6	1.72771	0.863853	+	0.503745i	$0.168045\pi$
$74$	−3.31463e6	−0.950875
$75$	5.29157e6	1.44834
$76$	3.47707e6	0.908585
$77$	4.39913e6	1.09812
$78$	−569745.	−0.135941
$79$	3.18023e6	0.725710	0.362855	−	0.931846i	$-0.381802\pi$
$79$	3.18023e6	0.725710	0.362855	+	0.931846i	$0.381802\pi$
$80$	−188502.	−0.0411624
$81$	8.84668e6	1.84962
$82$	2.07505e6	0.415604
$83$	−4.67105e6	−0.896688	−0.448344	−	0.893861i	$-0.647986\pi$
$83$	−4.67105e6	−0.896688	−0.448344	+	0.893861i	$0.647986\pi$
$84$	−5.55955e6	−1.02344
$85$	−1.96840e6	−0.347654
$86$	−3.06357e6	−0.519378
$87$	937881.	0.152697
$88$	−7.90570e6	−1.23666
$89$	1.04406e7	1.56986	0.784930	−	0.619585i	$-0.212699\pi$
$89$	1.04406e7	1.56986	0.784930	+	0.619585i	$0.212699\pi$
$90$	4.06336e6	0.587539
$91$	797089.	0.110882
$92$	1.02580e6	0.137343
$93$	−2.21351e7	−2.85359
$94$	−5.98474e6	−0.743186
$95$	5.12816e6	0.613661
$96$	1.60146e7	1.84743
$97$	−2.16055e6	−0.240360	−0.120180	−	0.992752i	$-0.538347\pi$
$97$	−2.16055e6	−0.240360	−0.120180	+	0.992752i	$0.538347\pi$
$98$	1.41293e6	0.151645
$99$	−2.78514e7	−2.88486
$100$	5.28316e6	0.528316
$101$	−7.79027e6	−0.752363	−0.376182	−	0.926546i	$-0.622763\pi$
$101$	−7.79027e6	−0.752363	−0.376182	+	0.926546i	$0.622763\pi$
$102$	8.83579e6	0.824413
$103$	9.18330e6	0.828073	0.414036	−	0.910260i	$-0.364119\pi$
$103$	9.18330e6	0.828073	0.414036	+	0.910260i	$0.364119\pi$
$104$	−1.43245e6	−0.124872
$105$	−8.19950e6	−0.691234
$106$	5.62061e6	0.458367
$107$	1.41288e7	1.11496	0.557482	−	0.830189i	$-0.311768\pi$
$107$	1.41288e7	1.11496	0.557482	+	0.830189i	$0.311768\pi$
$108$	1.96277e7	1.49929
$109$	1.25459e7	0.927916	0.463958	−	0.885857i	$-0.346429\pi$
$109$	1.25459e7	0.927916	0.463958	+	0.885857i	$0.346429\pi$
$110$	−4.63019e6	−0.331684
$111$	−4.23464e7	−2.93891
$112$	−1.18379e6	−0.0796179
$113$	−1.83323e7	−1.19520	−0.597602	−	0.801793i	$-0.703880\pi$
$113$	−1.83323e7	−1.19520	−0.597602	+	0.801793i	$0.703880\pi$
$114$	−2.30194e7	−1.45521
$115$	1.51291e6	0.0927620
$116$	936391.	0.0556999
$117$	−5.04647e6	−0.291298
$118$	1.12029e7	0.627685
$119$	−1.23615e7	−0.672446
$120$	1.47354e7	0.778442
$121$	1.22494e7	0.628589
$122$	831355.	0.0414502
$123$	2.65101e7	1.28453
$124$	−2.20999e7	−1.04092
$125$	1.75063e7	0.801697
$126$	2.55178e7	1.13644
$127$	1.71030e7	0.740898	0.370449	−	0.928853i	$-0.379204\pi$
$127$	1.71030e7	0.740898	0.370449	+	0.928853i	$0.379204\pi$
$128$	1.72718e7	0.727950
$129$	−3.91391e7	−1.60526
$130$	−838954.	−0.0334916
$131$	−5.91356e6	−0.229826	−0.114913	−	0.993376i	$-0.536659\pi$
$131$	−5.91356e6	−0.229826	−0.114913	+	0.993376i	$0.536659\pi$
$132$	−4.01081e7	−1.51783
$133$	3.22047e7	1.18697
$134$	1.31521e7	0.472204
$135$	2.89479e7	1.01263
$136$	2.22149e7	0.757285
$137$	2.03662e7	0.676688	0.338344	−	0.941022i	$-0.390133\pi$
$137$	2.03662e7	0.676688	0.338344	+	0.941022i	$0.390133\pi$
$138$	−6.79116e6	−0.219972
$139$	4.36196e7	1.37762	0.688811	−	0.724941i	$-0.258133\pi$
$139$	4.36196e7	1.37762	0.688811	+	0.724941i	$0.258133\pi$
$140$	−8.18647e6	−0.252144
$141$	−7.64587e7	−2.29700
$142$	3.49276e7	1.02367
$143$	5.75042e6	0.164446
$144$	7.49470e6	0.209163
$145$	1.38104e6	0.0376199
$146$	−3.79567e7	−1.00938
$147$	1.80510e7	0.468695
$148$	−4.22791e7	−1.07204
$149$	2.01433e7	0.498860	0.249430	−	0.968393i	$-0.419757\pi$
$149$	2.01433e7	0.498860	0.249430	+	0.968393i	$0.419757\pi$
$150$	−3.49763e7	−0.846164
$151$	−6.22486e7	−1.47133	−0.735666	−	0.677345i	$-0.763131\pi$
$151$	−6.22486e7	−1.47133	−0.735666	+	0.677345i	$0.763131\pi$
$152$	−5.78753e7	−1.33672
$153$	7.82622e7	1.76658
$154$	−2.90775e7	−0.641555
$155$	−3.25941e7	−0.703037
$156$	−7.26728e6	−0.153263
$157$	6.99650e7	1.44289	0.721443	−	0.692474i	$-0.243479\pi$
$157$	6.99650e7	1.44289	0.721443	+	0.692474i	$0.243479\pi$
$158$	−2.10207e7	−0.423982
$159$	7.18068e7	1.41669
$160$	2.35817e7	0.455150
$161$	9.50103e6	0.179424
$162$	−5.84749e7	−1.08060
$163$	−1.99809e7	−0.361375	−0.180688	−	0.983541i	$-0.557832\pi$
$163$	−1.99809e7	−0.361375	−0.180688	+	0.983541i	$0.557832\pi$
$164$	2.64679e7	0.468561
$165$	−5.91535e7	−1.02515
$166$	3.08748e7	0.523873
$167$	3.20379e7	0.532299	0.266150	−	0.963932i	$-0.414248\pi$
$167$	3.20379e7	0.532299	0.266150	+	0.963932i	$0.414248\pi$
$168$	9.25378e7	1.50569
$169$	−6.17066e7	−0.983395
$170$	1.30108e7	0.203110
$171$	−2.03892e8	−3.11827
$172$	−3.90769e7	−0.585558
$173$	−1.75131e7	−0.257159	−0.128579	−	0.991699i	$-0.541042\pi$
$173$	−1.75131e7	−0.257159	−0.128579	+	0.991699i	$0.541042\pi$
$174$	−6.19922e6	−0.0892103
$175$	4.89328e7	0.690187
$176$	−8.54018e6	−0.118079
$177$	1.43124e8	1.94001
$178$	−6.90105e7	−0.917160
$179$	1.33542e7	0.174033	0.0870164	−	0.996207i	$-0.472267\pi$
$179$	1.33542e7	0.174033	0.0870164	+	0.996207i	$0.472267\pi$
$180$	5.18295e7	0.662405
$181$	−1.51280e7	−0.189630	−0.0948149	−	0.995495i	$-0.530226\pi$
$181$	−1.51280e7	−0.189630	−0.0948149	+	0.995495i	$0.530226\pi$
$182$	−5.26861e6	−0.0647808
$183$	1.06211e7	0.128112
$184$	−1.70743e7	−0.202061
$185$	−6.23554e7	−0.724057
$186$	1.46309e8	1.66715
$187$	−8.91795e7	−0.997285
$188$	−7.63373e7	−0.837884
$189$	1.81792e8	1.95866
$190$	−3.38962e7	−0.358520
$191$	6.37608e6	0.0662120	0.0331060	−	0.999452i	$-0.489460\pi$
$191$	6.37608e6	0.0662120	0.0331060	+	0.999452i	$0.489460\pi$
$192$	−8.94679e7	−0.912248
$193$	−8.39474e7	−0.840536	−0.420268	−	0.907400i	$-0.638064\pi$
$193$	−8.39474e7	−0.840536	−0.420268	+	0.907400i	$0.638064\pi$
$194$	1.42808e7	0.140426
$195$	−1.07182e7	−0.103514
$196$	1.80223e7	0.170968
$197$	4.98601e7	0.464645	0.232322	−	0.972639i	$-0.425368\pi$
$197$	4.98601e7	0.464645	0.232322	+	0.972639i	$0.425368\pi$
$198$	1.84093e8	1.68542
$199$	7.72013e6	0.0694447	0.0347223	−	0.999397i	$-0.488945\pi$
$199$	7.72013e6	0.0694447	0.0347223	+	0.999397i	$0.488945\pi$
$200$	−8.79374e7	−0.777264
$201$	1.68027e8	1.45946
$202$	5.14922e7	0.439554
$203$	8.67288e6	0.0727658
$204$	1.12703e8	0.929462
$205$	3.90363e7	0.316468
$206$	−6.06999e7	−0.483785
$207$	−6.01521e7	−0.471362
$208$	−1.54741e6	−0.0119230
$209$	2.32334e8	1.76036
$210$	5.41972e7	0.403840
$211$	−7.69209e7	−0.563710	−0.281855	−	0.959457i	$-0.590950\pi$
$211$	−7.69209e7	−0.563710	−0.281855	+	0.959457i	$0.590950\pi$
$212$	7.16927e7	0.516773
$213$	4.46221e8	3.16389
$214$	−9.33885e7	−0.651396
$215$	−5.76326e7	−0.395488
$216$	−3.26700e8	−2.20577
$217$	−2.04690e8	−1.35984
$218$	−8.29260e7	−0.542117
$219$	−4.84921e8	−3.11973
$220$	−5.90595e7	−0.373947
$221$	−1.61586e7	−0.100701
$222$	2.79902e8	1.71700
$223$	2.19768e8	1.32708	0.663542	−	0.748139i	$-0.269053\pi$
$223$	2.19768e8	1.32708	0.663542	+	0.748139i	$0.269053\pi$
$224$	1.48092e8	0.880368
$225$	−3.09799e8	−1.81318
$226$	1.21173e8	0.698275
$227$	−1.93329e8	−1.09700	−0.548501	−	0.836150i	$-0.684801\pi$
$227$	−1.93329e8	−1.09700	−0.548501	+	0.836150i	$0.684801\pi$
$228$	−2.93619e8	−1.64064
$229$	3.05007e7	0.167836	0.0839180	−	0.996473i	$-0.473257\pi$
$229$	3.05007e7	0.167836	0.0839180	+	0.996473i	$0.473257\pi$
$230$	−1.00000e7	−0.0541944
$231$	−3.71483e8	−1.98288
$232$	−1.55861e7	−0.0819462
$233$	−3.99757e7	−0.207038	−0.103519	−	0.994627i	$-0.533010\pi$
$233$	−3.99757e7	−0.207038	−0.103519	+	0.994627i	$0.533010\pi$
$234$	3.33562e7	0.170185
$235$	−1.12586e8	−0.565910
$236$	1.42896e8	0.707666
$237$	−2.68553e8	−1.31042
$238$	8.17074e7	0.392864
$239$	2.93752e8	1.39184	0.695918	−	0.718122i	$-0.254998\pi$
$239$	2.93752e8	1.39184	0.695918	+	0.718122i	$0.254998\pi$
$240$	1.59180e7	0.0743272
$241$	1.10724e8	0.509544	0.254772	−	0.967001i	$-0.418000\pi$
$241$	1.10724e8	0.509544	0.254772	+	0.967001i	$0.418000\pi$
$242$	−8.09664e7	−0.367241
$243$	−2.37913e8	−1.06364
$244$	1.06042e7	0.0467319
$245$	2.65802e7	0.115472
$246$	−1.75227e8	−0.750459
$247$	4.20971e7	0.177751
$248$	3.67850e8	1.53140
$249$	3.94445e8	1.61915
$250$	−1.15714e8	−0.468376
$251$	4.17393e8	1.66605	0.833024	−	0.553236i	$-0.186607\pi$
$251$	4.17393e8	1.66605	0.833024	+	0.553236i	$0.186607\pi$
$252$	3.25488e8	1.28125
$253$	6.85431e7	0.266098
$254$	−1.13048e8	−0.432856
$255$	1.66221e8	0.627761
$256$	−2.49778e8	−0.930494
$257$	2.22419e8	0.817345	0.408673	−	0.912681i	$-0.365992\pi$
$257$	2.22419e8	0.817345	0.408673	+	0.912681i	$0.365992\pi$
$258$	2.58702e8	0.937844
$259$	−3.91590e8	−1.40050
$260$	−1.07011e7	−0.0377592
$261$	−5.49090e7	−0.191162
$262$	3.90876e7	0.134272
$263$	−1.45957e8	−0.494742	−0.247371	−	0.968921i	$-0.579567\pi$
$263$	−1.45957e8	−0.494742	−0.247371	+	0.968921i	$0.579567\pi$
$264$	6.67593e8	2.23305
$265$	1.05736e8	0.349030
$266$	−2.12867e8	−0.693463
$267$	−8.81652e8	−2.83470
$268$	1.67760e8	0.532373
$269$	−2.32483e8	−0.728211	−0.364106	−	0.931358i	$-0.618625\pi$
$269$	−2.32483e8	−0.728211	−0.364106	+	0.931358i	$0.618625\pi$
$270$	−1.91340e8	−0.591607
$271$	−1.23728e8	−0.377638	−0.188819	−	0.982012i	$-0.560466\pi$
$271$	−1.23728e8	−0.377638	−0.188819	+	0.982012i	$0.560466\pi$
$272$	2.39978e7	0.0723070
$273$	−6.73098e7	−0.200221
$274$	−1.34617e8	−0.395342
$275$	3.53015e8	1.02360
$276$	−8.66235e7	−0.248001
$277$	2.85339e8	0.806644	0.403322	−	0.915058i	$-0.367855\pi$
$277$	2.85339e8	0.806644	0.403322	+	0.915058i	$0.367855\pi$
$278$	−2.88318e8	−0.804849
$279$	1.29592e9	3.57243
$280$	1.36263e8	0.370957
$281$	−2.80970e8	−0.755419	−0.377710	−	0.925924i	$-0.623288\pi$
$281$	−2.80970e8	−0.755419	−0.377710	+	0.925924i	$0.623288\pi$
$282$	5.05378e8	1.34198
$283$	−4.83688e8	−1.26857	−0.634283	−	0.773101i	$-0.718704\pi$
$283$	−4.83688e8	−1.26857	−0.634283	+	0.773101i	$0.718704\pi$
$284$	4.45512e8	1.15411
$285$	−4.33045e8	−1.10809
$286$	−3.80092e7	−0.0960745
$287$	2.45147e8	0.612124
$288$	−9.37590e8	−2.31281
$289$	−1.59745e8	−0.389301
$290$	−9.12840e6	−0.0219787
$291$	1.82446e8	0.434020
$292$	−4.84151e8	−1.13800
$293$	−7.09381e8	−1.64757	−0.823783	−	0.566905i	$-0.808141\pi$
$293$	−7.09381e8	−1.64757	−0.823783	+	0.566905i	$0.808141\pi$
$294$	−1.19314e8	−0.273826
$295$	2.10751e8	0.477960
$296$	7.03729e8	1.57719
$297$	1.31150e9	2.90483
$298$	−1.33143e8	−0.291449
$299$	1.24195e7	0.0268692
$300$	−4.46134e8	−0.953984
$301$	−3.61931e8	−0.764968
$302$	4.11452e8	0.859597
$303$	6.57846e8	1.35855
$304$	−6.25200e7	−0.127633
$305$	1.56396e7	0.0315629
$306$	−5.17299e8	−1.03209
$307$	3.59197e8	0.708514	0.354257	−	0.935148i	$-0.384734\pi$
$307$	3.59197e8	0.708514	0.354257	+	0.935148i	$0.384734\pi$
$308$	−3.70892e8	−0.723303
$309$	−7.75479e8	−1.49526
$310$	2.15441e8	0.410736
$311$	1.19374e8	0.225034	0.112517	−	0.993650i	$-0.464109\pi$
$311$	1.19374e8	0.225034	0.112517	+	0.993650i	$0.464109\pi$
$312$	1.20963e8	0.225481
$313$	4.51083e8	0.831478	0.415739	−	0.909484i	$-0.363523\pi$
$313$	4.51083e8	0.831478	0.415739	+	0.909484i	$0.363523\pi$
$314$	−4.62456e8	−0.842978
$315$	4.80047e8	0.865360
$316$	−2.68126e8	−0.478007
$317$	−6.71377e7	−0.118375	−0.0591874	−	0.998247i	$-0.518851\pi$
$317$	−6.71377e7	−0.118375	−0.0591874	+	0.998247i	$0.518851\pi$
$318$	−4.74630e8	−0.827675
$319$	6.25686e7	0.107917
$320$	−1.31742e8	−0.224750
$321$	−1.19310e9	−2.01330
$322$	−6.28000e7	−0.104825
$323$	−6.52856e8	−1.07797
$324$	−7.45866e8	−1.21830
$325$	6.39636e7	0.103357
$326$	1.32070e8	0.211126
$327$	−1.05943e9	−1.67554
$328$	−4.40555e8	−0.689352
$329$	−7.07038e8	−1.09460
$330$	3.90994e8	0.598923
$331$	−9.24320e8	−1.40096	−0.700478	−	0.713674i	$-0.747030\pi$
$331$	−9.24320e8	−1.40096	−0.700478	+	0.713674i	$0.747030\pi$
$332$	3.93818e8	0.590626
$333$	2.47921e9	3.67924
$334$	−2.11764e8	−0.310986
$335$	2.47421e8	0.359567
$336$	9.99644e7	0.143767
$337$	9.93530e8	1.41409	0.707044	−	0.707169i	$-0.250028\pi$
$337$	9.93530e8	1.41409	0.707044	+	0.707169i	$0.250028\pi$
$338$	4.07869e8	0.574530
$339$	1.54806e9	2.15819
$340$	1.65957e8	0.228991
$341$	−1.47669e9	−2.01674
$342$	1.34769e9	1.82179
$343$	8.10016e8	1.08384
$344$	6.50428e8	0.861479
$345$	−1.27757e8	−0.167501
$346$	1.15758e8	0.150240
$347$	3.51304e8	0.451367	0.225683	−	0.974201i	$-0.427539\pi$
$347$	3.51304e8	0.451367	0.225683	+	0.974201i	$0.427539\pi$
$348$	−7.90731e7	−0.100578
$349$	−5.77536e8	−0.727261	−0.363630	−	0.931543i	$-0.618463\pi$
$349$	−5.77536e8	−0.727261	−0.363630	+	0.931543i	$0.618463\pi$
$350$	−3.23437e8	−0.403229
$351$	2.37634e8	0.293314
$352$	1.06838e9	1.30565
$353$	−5.71113e8	−0.691052	−0.345526	−	0.938409i	$-0.612299\pi$
$353$	−5.71113e8	−0.691052	−0.345526	+	0.938409i	$0.612299\pi$
$354$	−9.46020e8	−1.13342
$355$	6.57064e8	0.779487
$356$	−8.80251e8	−1.03403
$357$	1.04386e9	1.21424
$358$	−8.82685e7	−0.101675
$359$	−5.07791e8	−0.579234	−0.289617	−	0.957143i	$-0.593528\pi$
$359$	−5.07791e8	−0.579234	−0.289617	+	0.957143i	$0.593528\pi$
$360$	−8.62694e8	−0.974537
$361$	8.06974e8	0.902784
$362$	9.99933e7	0.110787
$363$	−1.03440e9	−1.13505
$364$	−6.72029e7	−0.0730353
$365$	−7.14050e8	−0.768606
$366$	−7.02034e7	−0.0748469
$367$	1.20975e9	1.27751	0.638755	−	0.769410i	$-0.279450\pi$
$367$	1.20975e9	1.27751	0.638755	+	0.769410i	$0.279450\pi$
$368$	−1.84446e7	−0.0192931
$369$	−1.55205e9	−1.60811
$370$	4.12158e8	0.423017
$371$	6.64020e8	0.675107
$372$	1.86622e9	1.87959
$373$	−4.28549e8	−0.427583	−0.213791	−	0.976879i	$-0.568581\pi$
$373$	−4.28549e8	−0.427583	−0.213791	+	0.976879i	$0.568581\pi$
$374$	5.89460e8	0.582645
$375$	−1.47831e9	−1.44763
$376$	1.27062e9	1.23270
$377$	1.13369e7	0.0108969
$378$	−1.20161e9	−1.14431
$379$	1.25140e9	1.18075	0.590377	−	0.807128i	$-0.298979\pi$
$379$	1.25140e9	1.18075	0.590377	+	0.807128i	$0.298979\pi$
$380$	−4.32357e8	−0.404203
$381$	−1.44425e9	−1.33784
$382$	−4.21447e7	−0.0386831
$383$	4.37262e8	0.397692	0.198846	−	0.980031i	$-0.436281\pi$
$383$	4.37262e8	0.397692	0.198846	+	0.980031i	$0.436281\pi$
$384$	−1.45851e9	−1.31446
$385$	−5.47011e8	−0.488521
$386$	5.54877e8	0.491067
$387$	2.29143e9	2.00964
$388$	1.82156e8	0.158319
$389$	−6.31898e8	−0.544281	−0.272140	−	0.962258i	$-0.587732\pi$
$389$	−6.31898e8	−0.544281	−0.272140	+	0.962258i	$0.587732\pi$
$390$	7.08451e7	0.0604761
$391$	−1.92605e8	−0.162948
$392$	−2.99979e8	−0.251530
$393$	4.99368e8	0.414999
$394$	−3.29566e8	−0.271460
$395$	−3.95446e8	−0.322847
$396$	2.34816e9	1.90018
$397$	1.38373e9	1.10990	0.554951	−	0.831883i	$-0.312737\pi$
$397$	1.38373e9	1.10990	0.554951	+	0.831883i	$0.312737\pi$
$398$	−5.10286e7	−0.0405717
$399$	−2.71951e9	−2.14331
$400$	−9.49948e7	−0.0742147
$401$	5.70479e8	0.441809	0.220905	−	0.975295i	$-0.429099\pi$
$401$	5.70479e8	0.441809	0.220905	+	0.975295i	$0.429099\pi$
$402$	−1.11063e9	−0.852662
$403$	−2.67566e8	−0.203640
$404$	6.56800e8	0.495563
$405$	−1.10004e9	−0.822842
$406$	−5.73262e7	−0.0425120
$407$	−2.82504e9	−2.07704
$408$	−1.87593e9	−1.36743
$409$	2.05847e9	1.48769	0.743847	−	0.668350i	$-0.232999\pi$
$409$	2.05847e9	1.48769	0.743847	+	0.668350i	$0.232999\pi$
$410$	−2.58022e8	−0.184890
$411$	−1.71982e9	−1.22190
$412$	−7.74247e8	−0.545430
$413$	1.32351e9	0.924488
$414$	3.97594e8	0.275384
$415$	5.80823e8	0.398910
$416$	1.93582e8	0.131837
$417$	−3.68344e9	−2.48758
$418$	−1.53568e9	−1.02845
$419$	1.64434e9	1.09205	0.546024	−	0.837770i	$-0.316141\pi$
$419$	1.64434e9	1.09205	0.546024	+	0.837770i	$0.316141\pi$
$420$	6.91303e8	0.455298
$421$	7.16725e8	0.468129	0.234064	−	0.972221i	$-0.424797\pi$
$421$	7.16725e8	0.468129	0.234064	+	0.972221i	$0.424797\pi$
$422$	5.08433e8	0.329337
$423$	4.47634e9	2.87562
$424$	−1.19331e9	−0.760281
$425$	−9.91969e8	−0.626811
$426$	−2.94944e9	−1.84844
$427$	9.82164e7	0.0610501
$428$	−1.19120e9	−0.734398
$429$	−4.85592e8	−0.296941
$430$	3.80941e8	0.231056
$431$	2.15943e9	1.29918	0.649590	−	0.760284i	$-0.274940\pi$
$431$	2.15943e9	1.29918	0.649590	+	0.760284i	$0.274940\pi$
$432$	−3.52919e8	−0.210612
$433$	−1.25795e9	−0.744658	−0.372329	−	0.928101i	$-0.621441\pi$
$433$	−1.25795e9	−0.744658	−0.372329	+	0.928101i	$0.621441\pi$
$434$	1.35297e9	0.794461
$435$	−1.16621e8	−0.0679304
$436$	−1.05775e9	−0.611195
$437$	5.01783e8	0.287628
$438$	3.20524e9	1.82264
$439$	1.45755e9	0.822237	0.411119	−	0.911582i	$-0.365138\pi$
$439$	1.45755e9	0.822237	0.411119	+	0.911582i	$0.365138\pi$
$440$	9.83036e8	0.550155
$441$	−1.05681e9	−0.586762
$442$	1.06806e8	0.0588323
$443$	−2.26373e9	−1.23712	−0.618561	−	0.785737i	$-0.712284\pi$
$443$	−2.26373e9	−1.23712	−0.618561	+	0.785737i	$0.712284\pi$
$444$	3.57024e9	1.93578
$445$	−1.29824e9	−0.698385
$446$	−1.45263e9	−0.775323
$447$	−1.70099e9	−0.900794
$448$	−8.27338e8	−0.434720
$449$	2.35664e9	1.22866	0.614330	−	0.789049i	$-0.289426\pi$
$449$	2.35664e9	1.22866	0.614330	+	0.789049i	$0.289426\pi$
$450$	2.04772e9	1.05932
$451$	1.76856e9	0.907824
$452$	1.54560e9	0.787250
$453$	5.25655e9	2.65679
$454$	1.27787e9	0.640902
$455$	−9.91142e7	−0.0493283
$456$	4.88725e9	2.41372
$457$	−2.68601e9	−1.31644	−0.658220	−	0.752826i	$-0.728690\pi$
$457$	−2.68601e9	−1.31644	−0.658220	+	0.752826i	$0.728690\pi$
$458$	−2.01604e8	−0.0980549
$459$	−3.68530e9	−1.77881
$460$	−1.27554e8	−0.0611000
$461$	−8.96300e8	−0.426089	−0.213044	−	0.977043i	$-0.568338\pi$
$461$	−8.96300e8	−0.426089	−0.213044	+	0.977043i	$0.568338\pi$
$462$	2.45543e9	1.15846
$463$	−6.97024e8	−0.326373	−0.163186	−	0.986595i	$-0.552177\pi$
$463$	−6.97024e8	−0.326373	−0.163186	+	0.986595i	$0.552177\pi$
$464$	−1.68369e7	−0.00782439
$465$	2.75240e9	1.26948
$466$	2.64232e8	0.120958
$467$	−2.07682e9	−0.943606	−0.471803	−	0.881704i	$-0.656397\pi$
$467$	−2.07682e9	−0.943606	−0.471803	+	0.881704i	$0.656397\pi$
$468$	4.25469e8	0.191870
$469$	1.55380e9	0.695488
$470$	7.44173e8	0.330622
$471$	−5.90816e9	−2.60543
$472$	−2.37848e9	−1.04113
$473$	−2.61107e9	−1.13450
$474$	1.77508e9	0.765587
$475$	2.58432e9	1.10641
$476$	1.04220e9	0.442923
$477$	−4.20399e9	−1.77357
$478$	−1.94164e9	−0.813153
$479$	1.12171e9	0.466342	0.233171	−	0.972436i	$-0.425090\pi$
$479$	1.12171e9	0.466342	0.233171	+	0.972436i	$0.425090\pi$
$480$	−1.99134e9	−0.821867
$481$	−5.11876e8	−0.209728
$482$	−7.31864e8	−0.297691
$483$	−8.02310e8	−0.323987
$484$	−1.03275e9	−0.414035
$485$	2.68654e8	0.106929
$486$	1.57256e9	0.621414
$487$	−3.51560e9	−1.37927	−0.689633	−	0.724159i	$-0.742228\pi$
$487$	−3.51560e9	−1.37927	−0.689633	+	0.724159i	$0.742228\pi$
$488$	−1.76505e8	−0.0687525
$489$	1.68728e9	0.652537
$490$	−1.75690e8	−0.0674624
$491$	7.86318e7	0.0299787	0.0149894	−	0.999888i	$-0.495229\pi$
$491$	7.86318e7	0.0299787	0.0149894	+	0.999888i	$0.495229\pi$
$492$	−2.23507e9	−0.846084
$493$	−1.75817e8	−0.0660841
$494$	−2.78254e8	−0.103848
$495$	3.46319e9	1.28339
$496$	3.97372e8	0.146222
$497$	4.12635e9	1.50771
$498$	−2.60721e9	−0.945960
$499$	3.11788e9	1.12333	0.561665	−	0.827365i	$-0.310161\pi$
$499$	3.11788e9	1.12333	0.561665	+	0.827365i	$0.310161\pi$
$500$	−1.47597e9	−0.528058
$501$	−2.70542e9	−0.961176
$502$	−2.75889e9	−0.973357
$503$	1.89608e9	0.664305	0.332153	−	0.943226i	$-0.392225\pi$
$503$	1.89608e9	0.664305	0.332153	+	0.943226i	$0.392225\pi$
$504$	−5.41770e9	−1.88499
$505$	9.68683e8	0.334705
$506$	−4.53057e8	−0.155463
$507$	5.21078e9	1.77572
$508$	−1.44196e9	−0.488011
$509$	−5.07334e9	−1.70523	−0.852613	−	0.522543i	$-0.824984\pi$
$509$	−5.07334e9	−1.70523	−0.852613	+	0.522543i	$0.824984\pi$
$510$	−1.09869e9	−0.366757
$511$	−4.48422e9	−1.48667
$512$	−5.59802e8	−0.184327
$513$	9.60110e9	3.13986
$514$	−1.47015e9	−0.477518
$515$	−1.14190e9	−0.368385
$516$	3.29983e9	1.05735
$517$	−5.10077e9	−1.62338
$518$	2.58834e9	0.818215
$519$	1.47888e9	0.464353
$520$	1.78119e8	0.0555517
$521$	3.65476e9	1.13221	0.566105	−	0.824333i	$-0.308450\pi$
$521$	3.65476e9	1.13221	0.566105	+	0.824333i	$0.308450\pi$
$522$	3.62939e8	0.111683
$523$	5.86139e9	1.79162	0.895808	−	0.444442i	$-0.146598\pi$
$523$	5.86139e9	1.79162	0.895808	+	0.444442i	$0.146598\pi$
$524$	4.98575e8	0.151381
$525$	−4.13211e9	−1.24628
$526$	9.64747e8	0.289044
$527$	4.14950e9	1.23497
$528$	7.21171e8	0.213216
$529$	1.48036e8	0.0434783
$530$	−6.98896e8	−0.203914
$531$	−8.37929e9	−2.42871
$532$	−2.71519e9	−0.781825
$533$	3.20449e8	0.0916672
$534$	5.82756e9	1.65612
$535$	−1.75684e9	−0.496015
$536$	−2.79234e9	−0.783233
$537$	−1.12769e9	−0.314252
$538$	1.53667e9	0.425443
$539$	1.20423e9	0.331245
$540$	−2.44061e9	−0.666991
$541$	3.23517e9	0.878429	0.439215	−	0.898382i	$-0.355257\pi$
$541$	3.23517e9	0.878429	0.439215	+	0.898382i	$0.355257\pi$
$542$	8.17821e8	0.220628
$543$	1.27748e9	0.342415
$544$	−3.00213e9	−0.799529
$545$	−1.56002e9	−0.412803
$546$	4.44905e8	0.116975
$547$	6.58388e9	1.71999	0.859996	−	0.510301i	$-0.170466\pi$
$547$	6.58388e9	1.71999	0.859996	+	0.510301i	$0.170466\pi$
$548$	−1.71708e9	−0.445717
$549$	−6.21820e8	−0.160384
$550$	−2.33336e9	−0.598016
$551$	4.58046e8	0.116648
$552$	1.44183e9	0.364862
$553$	−2.48339e9	−0.624464
$554$	−1.88604e9	−0.471266
$555$	5.26557e9	1.30743
$556$	−3.67759e9	−0.907405
$557$	−6.33978e9	−1.55446	−0.777232	−	0.629214i	$-0.783377\pi$
$557$	−6.33978e9	−1.55446	−0.777232	+	0.629214i	$0.783377\pi$
$558$	−8.56579e9	−2.08712
$559$	−4.73106e8	−0.114556
$560$	1.47198e8	0.0354197
$561$	7.53072e9	1.80080
$562$	1.85716e9	0.441339
$563$	5.51536e9	1.30255	0.651275	−	0.758842i	$-0.274234\pi$
$563$	5.51536e9	1.30255	0.651275	+	0.758842i	$0.274234\pi$
$564$	6.44626e9	1.51297
$565$	2.27953e9	0.531711
$566$	3.19709e9	0.741135
$567$	−6.90824e9	−1.59157
$568$	−7.41548e9	−1.69793
$569$	−3.83008e9	−0.871594	−0.435797	−	0.900045i	$-0.643533\pi$
$569$	−3.83008e9	−0.871594	−0.435797	+	0.900045i	$0.643533\pi$
$570$	2.86235e9	0.647381
$571$	4.33170e9	0.973716	0.486858	−	0.873481i	$-0.338143\pi$
$571$	4.33170e9	0.973716	0.486858	+	0.873481i	$0.338143\pi$
$572$	−4.84820e8	−0.108317
$573$	−5.38425e8	−0.119559
$574$	−1.62038e9	−0.357622
$575$	7.62424e8	0.167247
$576$	5.23797e9	1.14205
$577$	2.83280e9	0.613904	0.306952	−	0.951725i	$-0.400691\pi$
$577$	2.83280e9	0.613904	0.306952	+	0.951725i	$0.400691\pi$
$578$	1.05589e9	0.227442
$579$	7.08889e9	1.51776
$580$	−1.16436e8	−0.0247793
$581$	3.64755e9	0.771588
$582$	−1.20594e9	−0.253568
$583$	4.79043e9	1.00123
$584$	8.05861e9	1.67423
$585$	6.27504e8	0.129590
$586$	4.68888e9	0.962559
$587$	−4.39108e8	−0.0896061	−0.0448031	−	0.998996i	$-0.514266\pi$
$587$	−4.39108e8	−0.0896061	−0.0448031	+	0.998996i	$0.514266\pi$
$588$	−1.52189e9	−0.308718
$589$	−1.08104e10	−2.17991
$590$	−1.39302e9	−0.279239
$591$	−4.21041e9	−0.839012
$592$	7.60207e8	0.150593
$593$	1.02984e9	0.202805	0.101402	−	0.994845i	$-0.467667\pi$
$593$	1.02984e9	0.202805	0.101402	+	0.994845i	$0.467667\pi$
$594$	−8.66878e9	−1.69709
$595$	1.53710e9	0.299152
$596$	−1.69829e9	−0.328586
$597$	−6.51923e8	−0.125397
$598$	−8.20905e7	−0.0156978
$599$	5.33042e9	1.01337	0.506685	−	0.862132i	$-0.330871\pi$
$599$	5.33042e9	1.01337	0.506685	+	0.862132i	$0.330871\pi$
$600$	7.42583e9	1.40351
$601$	−1.13823e9	−0.213880	−0.106940	−	0.994265i	$-0.534105\pi$
$601$	−1.13823e9	−0.213880	−0.106940	+	0.994265i	$0.534105\pi$
$602$	2.39230e9	0.446918
$603$	−9.83727e9	−1.82711
$604$	5.24820e9	0.969128
$605$	−1.52316e9	−0.279641
$606$	−4.34824e9	−0.793705
$607$	−4.90775e9	−0.890682	−0.445341	−	0.895361i	$-0.646917\pi$
$607$	−4.90775e9	−0.890682	−0.445341	+	0.895361i	$0.646917\pi$
$608$	7.82128e9	1.41129
$609$	−7.32377e8	−0.131394
$610$	−1.03375e8	−0.0184400
$611$	−9.24221e8	−0.163920
$612$	−6.59832e9	−1.16360
$613$	−7.80276e9	−1.36816	−0.684079	−	0.729408i	$-0.739796\pi$
$613$	−7.80276e9	−1.36816	−0.684079	+	0.729408i	$0.739796\pi$
$614$	−2.37423e9	−0.413936
$615$	−3.29640e9	−0.571448
$616$	6.17345e9	1.06413
$617$	−8.13084e9	−1.39360	−0.696798	−	0.717267i	$-0.745393\pi$
$617$	−8.13084e9	−1.39360	−0.696798	+	0.717267i	$0.745393\pi$
$618$	5.12577e9	0.873574
$619$	−6.52679e9	−1.10607	−0.553034	−	0.833158i	$-0.686530\pi$
$619$	−6.52679e9	−1.10607	−0.553034	+	0.833158i	$0.686530\pi$
$620$	2.74802e9	0.463073
$621$	2.83251e9	0.474626
$622$	−7.89041e8	−0.131472
$623$	−8.15291e9	−1.35084
$624$	1.30671e8	0.0215294
$625$	2.71874e9	0.445438
$626$	−2.98157e9	−0.485775
$627$	−1.96193e10	−3.17869
$628$	−5.89877e9	−0.950392
$629$	7.93834e9	1.27190
$630$	−3.17302e9	−0.505570
$631$	8.07715e9	1.27984	0.639920	−	0.768442i	$-0.278968\pi$
$631$	8.07715e9	1.27984	0.639920	+	0.768442i	$0.278968\pi$
$632$	4.46291e9	0.703249
$633$	6.49554e9	1.01789
$634$	4.43768e8	0.0691582
$635$	−2.12667e9	−0.329604
$636$	−6.05406e9	−0.933139
$637$	2.18197e8	0.0334473
$638$	−4.13567e8	−0.0630483
$639$	−2.61244e10	−3.96090
$640$	−2.14766e9	−0.323844
$641$	3.47450e7	0.00521062	0.00260531	−	0.999997i	$-0.499171\pi$
$641$	3.47450e7	0.00521062	0.00260531	+	0.999997i	$0.499171\pi$
$642$	7.88615e9	1.17623
$643$	1.43181e9	0.212397	0.106198	−	0.994345i	$-0.466132\pi$
$643$	1.43181e9	0.212397	0.106198	+	0.994345i	$0.466132\pi$
$644$	−8.01035e8	−0.118182
$645$	4.86675e9	0.714135
$646$	4.31526e9	0.629786
$647$	−2.51433e9	−0.364970	−0.182485	−	0.983209i	$-0.558414\pi$
$647$	−2.51433e9	−0.364970	−0.182485	+	0.983209i	$0.558414\pi$
$648$	1.24148e10	1.79237
$649$	9.54816e9	1.37108
$650$	−4.22788e8	−0.0603845
$651$	1.72850e10	2.45547
$652$	1.68460e9	0.238029
$653$	−5.66712e9	−0.796464	−0.398232	−	0.917285i	$-0.630376\pi$
$653$	−5.66712e9	−0.796464	−0.398232	+	0.917285i	$0.630376\pi$
$654$	7.00265e9	0.978904
$655$	7.35323e8	0.102243
$656$	−4.75911e8	−0.0658207
$657$	2.83901e10	3.90560
$658$	4.67339e9	0.639501
$659$	−1.19427e10	−1.62557	−0.812784	−	0.582566i	$-0.802049\pi$
$659$	−1.19427e10	−1.62557	−0.812784	+	0.582566i	$0.802049\pi$
$660$	4.98725e9	0.675239
$661$	1.95394e9	0.263152	0.131576	−	0.991306i	$-0.457996\pi$
$661$	1.95394e9	0.263152	0.131576	+	0.991306i	$0.457996\pi$
$662$	6.10958e9	0.818481
$663$	1.36451e9	0.181836
$664$	−6.55504e9	−0.868934
$665$	−4.00450e9	−0.528047
$666$	−1.63871e10	−2.14952
$667$	1.35133e8	0.0176327
$668$	−2.70112e9	−0.350612
$669$	−1.85582e10	−2.39632
$670$	−1.63541e9	−0.210070
$671$	7.08561e8	0.0905416
$672$	−1.25056e10	−1.58969
$673$	−5.07206e9	−0.641404	−0.320702	−	0.947180i	$-0.603919\pi$
$673$	−5.07206e9	−0.641404	−0.320702	+	0.947180i	$0.603919\pi$
$674$	−6.56705e9	−0.826154
$675$	1.45882e10	1.82574
$676$	5.20250e9	0.647737
$677$	8.21660e9	1.01773	0.508864	−	0.860847i	$-0.330066\pi$
$677$	8.21660e9	1.01773	0.508864	+	0.860847i	$0.330066\pi$
$678$	−1.02324e10	−1.26088
$679$	1.68714e9	0.206827
$680$	−2.76232e9	−0.336894
$681$	1.63256e10	1.98086
$682$	9.76068e9	1.17824
$683$	−3.61944e9	−0.434680	−0.217340	−	0.976096i	$-0.569738\pi$
$683$	−3.61944e9	−0.434680	−0.217340	+	0.976096i	$0.569738\pi$
$684$	1.71902e10	2.05392
$685$	−2.53244e9	−0.301039
$686$	−5.35405e9	−0.633211
$687$	−2.57561e9	−0.303062
$688$	7.02628e8	0.0822557
$689$	8.67988e8	0.101099
$690$	8.44449e8	0.0978591
$691$	−1.76538e9	−0.203547	−0.101774	−	0.994808i	$-0.532452\pi$
$691$	−1.76538e9	−0.203547	−0.101774	+	0.994808i	$0.532452\pi$
$692$	1.47653e9	0.169384
$693$	2.17488e10	2.48238
$694$	−2.32205e9	−0.263702
$695$	−5.42389e9	−0.612864
$696$	1.31616e9	0.147971
$697$	−4.96963e9	−0.555916
$698$	3.81741e9	0.424888
$699$	3.37573e9	0.373850
$700$	−4.12554e9	−0.454609
$701$	5.32097e7	0.00583415	0.00291708	−	0.999996i	$-0.499071\pi$
$701$	5.32097e7	0.00583415	0.00291708	+	0.999996i	$0.499071\pi$
$702$	−1.57072e9	−0.171363
$703$	−2.06813e10	−2.24509
$704$	−5.96865e9	−0.644721
$705$	9.50728e9	1.02187
$706$	3.77495e9	0.403734
$707$	6.08330e9	0.647398
$708$	−1.20668e10	−1.27784
$709$	5.41233e9	0.570326	0.285163	−	0.958479i	$-0.407952\pi$
$709$	5.41233e9	0.570326	0.285163	+	0.958479i	$0.407952\pi$
$710$	−4.34307e9	−0.455400
$711$	1.57226e10	1.64052
$712$	1.46516e10	1.52127
$713$	−3.18929e9	−0.329519
$714$	−6.89974e9	−0.709396
$715$	−7.15038e8	−0.0731573
$716$	−1.12589e9	−0.114631
$717$	−2.48057e10	−2.51324
$718$	3.35640e9	0.338406
$719$	5.70950e9	0.572858	0.286429	−	0.958102i	$-0.407532\pi$
$719$	5.70950e9	0.572858	0.286429	+	0.958102i	$0.407532\pi$
$720$	−9.31930e8	−0.0930507
$721$	−7.17110e9	−0.712545
$722$	−5.33395e9	−0.527434
$723$	−9.35002e9	−0.920086
$724$	1.27545e9	0.124904
$725$	6.95968e8	0.0678276
$726$	6.83717e9	0.663129
$727$	3.85712e9	0.372300	0.186150	−	0.982521i	$-0.440399\pi$
$727$	3.85712e9	0.372300	0.186150	+	0.982521i	$0.440399\pi$
$728$	1.11858e9	0.107450
$729$	7.42776e8	0.0710087
$730$	4.71974e9	0.449043
$731$	7.33709e9	0.694725
$732$	−8.95467e8	−0.0843841
$733$	1.49428e10	1.40142	0.700709	−	0.713448i	$-0.252867\pi$
$733$	1.49428e10	1.40142	0.700709	+	0.713448i	$0.252867\pi$
$734$	−7.99622e9	−0.746361
$735$	−2.24456e9	−0.208509
$736$	2.30743e9	0.213332
$737$	1.12095e10	1.03146
$738$	1.02588e10	0.939504
$739$	−3.87613e9	−0.353300	−0.176650	−	0.984274i	$-0.556526\pi$
$739$	−3.87613e9	−0.353300	−0.176650	+	0.984274i	$0.556526\pi$
$740$	5.25720e9	0.476918
$741$	−3.55487e9	−0.320967
$742$	−4.38905e9	−0.394418
$743$	3.56370e9	0.318743	0.159372	−	0.987219i	$-0.449053\pi$
$743$	3.56370e9	0.318743	0.159372	+	0.987219i	$0.449053\pi$
$744$	−3.10629e10	−2.76527
$745$	−2.50472e9	−0.221928
$746$	2.83263e9	0.249807
$747$	−2.30931e10	−2.02703
$748$	7.51875e9	0.656886
$749$	−1.10329e10	−0.959411
$750$	9.77139e9	0.845750
$751$	−4.65231e9	−0.400801	−0.200401	−	0.979714i	$-0.564224\pi$
$751$	−4.65231e9	−0.400801	−0.200401	+	0.979714i	$0.564224\pi$
$752$	1.37260e9	0.117701
$753$	−3.52466e10	−3.00839
$754$	−7.49351e7	−0.00636628
$755$	7.74032e9	0.654552
$756$	−1.53270e10	−1.29012
$757$	−5.59791e9	−0.469019	−0.234509	−	0.972114i	$-0.575348\pi$
$757$	−5.59791e9	−0.469019	−0.234509	+	0.972114i	$0.575348\pi$
$758$	−8.27153e9	−0.689832
$759$	−5.78809e9	−0.480495
$760$	7.19651e9	0.594668
$761$	2.20906e9	0.181703	0.0908515	−	0.995864i	$-0.471041\pi$
$761$	2.20906e9	0.181703	0.0908515	+	0.995864i	$0.471041\pi$
$762$	9.54624e9	0.781610
$763$	−9.79690e9	−0.798459
$764$	−5.37569e8	−0.0436122
$765$	−9.73153e9	−0.785898
$766$	−2.89022e9	−0.232344
$767$	1.73005e9	0.138444
$768$	2.10923e10	1.68020
$769$	6.42257e9	0.509292	0.254646	−	0.967034i	$-0.418041\pi$
$769$	6.42257e9	0.509292	0.254646	+	0.967034i	$0.418041\pi$
$770$	3.61564e9	0.285409
$771$	−1.87820e10	−1.47588
$772$	7.07763e9	0.553640
$773$	3.59464e9	0.279915	0.139958	−	0.990157i	$-0.455303\pi$
$773$	3.59464e9	0.279915	0.139958	+	0.990157i	$0.455303\pi$
$774$	−1.51459e10	−1.17409
$775$	−1.64257e10	−1.26756
$776$	−3.03196e9	−0.232921
$777$	3.30677e10	2.52889
$778$	4.17673e9	0.317986
$779$	1.29471e10	0.981275
$780$	9.03652e8	0.0681820
$781$	2.97686e10	2.23604
$782$	1.27309e9	0.0951994
$783$	2.58562e9	0.192486
$784$	−3.24053e8	−0.0240165
$785$	−8.69981e9	−0.641898
$786$	−3.30073e9	−0.242455
$787$	7.95613e9	0.581822	0.290911	−	0.956750i	$-0.406042\pi$
$787$	7.95613e9	0.581822	0.290911	+	0.956750i	$0.406042\pi$
$788$	−4.20372e9	−0.306050
$789$	1.23252e10	0.893359
$790$	2.61382e9	0.188617
$791$	1.43154e10	1.02846
$792$	−3.90848e10	−2.79557
$793$	1.28386e8	0.00914241
$794$	−9.14620e9	−0.648439
$795$	−8.92883e9	−0.630245
$796$	−6.50887e8	−0.0457414
$797$	2.78296e10	1.94716	0.973582	−	0.228338i	$-0.0733292\pi$
$797$	2.78296e10	1.94716	0.973582	+	0.228338i	$0.0733292\pi$
$798$	1.79755e10	1.25219
$799$	1.43331e10	0.994093
$800$	1.18839e10	0.820623
$801$	5.16171e10	3.54878
$802$	−3.77076e9	−0.258118
$803$	−3.23504e10	−2.20483
$804$	−1.41664e10	−0.961310
$805$	−1.18141e9	−0.0798204
$806$	1.76856e9	0.118973
$807$	1.96319e10	1.31494
$808$	−1.09323e10	−0.729077
$809$	−9.63781e9	−0.639968	−0.319984	−	0.947423i	$-0.603678\pi$
$809$	−9.63781e9	−0.639968	−0.319984	+	0.947423i	$0.603678\pi$
$810$	7.27107e9	0.480730
$811$	2.34453e10	1.54341	0.771706	−	0.635979i	$-0.219404\pi$
$811$	2.34453e10	1.54341	0.771706	+	0.635979i	$0.219404\pi$
$812$	−7.31214e8	−0.0479290
$813$	1.04482e10	0.681904
$814$	1.86730e10	1.21347
$815$	2.48453e9	0.160765
$816$	−2.02648e9	−0.130565
$817$	−1.91149e10	−1.22629
$818$	−1.36061e10	−0.869156
$819$	3.94071e9	0.250657
$820$	−3.29116e9	−0.208449
$821$	−1.43292e10	−0.903692	−0.451846	−	0.892096i	$-0.649234\pi$
$821$	−1.43292e10	−0.903692	−0.451846	+	0.892096i	$0.649234\pi$
$822$	1.13677e10	0.713872
$823$	−5.89700e8	−0.0368750	−0.0184375	−	0.999830i	$-0.505869\pi$
$823$	−5.89700e8	−0.0368750	−0.0184375	+	0.999830i	$0.505869\pi$
$824$	1.28872e10	0.802443
$825$	−2.98102e10	−1.84831
$826$	−8.74815e9	−0.540115
$827$	3.24688e9	0.199617	0.0998085	−	0.995007i	$-0.468177\pi$
$827$	3.24688e9	0.199617	0.0998085	+	0.995007i	$0.468177\pi$
$828$	5.07145e9	0.310474
$829$	1.94955e10	1.18849	0.594243	−	0.804286i	$-0.297452\pi$
$829$	1.94955e10	1.18849	0.594243	+	0.804286i	$0.297452\pi$
$830$	−3.83913e9	−0.233056
$831$	−2.40953e10	−1.45656
$832$	−1.08147e9	−0.0651004
$833$	−3.38388e9	−0.202842
$834$	2.43469e10	1.45332
$835$	−3.98375e9	−0.236805
$836$	−1.95881e10	−1.15950
$837$	−6.10237e10	−3.59716
$838$	−1.08688e10	−0.638008
$839$	2.67406e10	1.56316	0.781582	−	0.623803i	$-0.214413\pi$
$839$	2.67406e10	1.56316	0.781582	+	0.623803i	$0.214413\pi$
$840$	−1.15066e10	−0.669839
$841$	−1.71265e10	−0.992849
$842$	−4.73742e9	−0.273495
$843$	2.37264e10	1.36407
$844$	6.48522e9	0.371301
$845$	7.67292e9	0.437484
$846$	−2.95878e10	−1.68003
$847$	−9.56539e9	−0.540892
$848$	−1.28908e9	−0.0725931
$849$	4.08448e10	2.29066
$850$	6.55673e9	0.366202
$851$	−6.10139e9	−0.339371
$852$	−3.76211e10	−2.08398
$853$	1.05839e9	0.0583880	0.0291940	−	0.999574i	$-0.490706\pi$
$853$	1.05839e9	0.0583880	0.0291940	+	0.999574i	$0.490706\pi$
$854$	−6.49193e8	−0.0356674
$855$	2.53530e10	1.38723
$856$	1.98273e10	1.08045
$857$	−3.99239e9	−0.216670	−0.108335	−	0.994114i	$-0.534552\pi$
$857$	−3.99239e9	−0.216670	−0.108335	+	0.994114i	$0.534552\pi$
$858$	3.20967e9	0.173482
$859$	−1.45047e10	−0.780787	−0.390394	−	0.920648i	$-0.627661\pi$
$859$	−1.45047e10	−0.780787	−0.390394	+	0.920648i	$0.627661\pi$
$860$	4.85902e9	0.260498
$861$	−2.07013e10	−1.10532
$862$	−1.42735e10	−0.759021
$863$	1.69775e10	0.899156	0.449578	−	0.893241i	$-0.351574\pi$
$863$	1.69775e10	0.899156	0.449578	+	0.893241i	$0.351574\pi$
$864$	4.41503e10	2.32882
$865$	2.17767e9	0.114402
$866$	8.31484e9	0.435052
$867$	1.34896e10	0.702963
$868$	1.72575e10	0.895693
$869$	−1.79159e10	−0.926124
$870$	7.70843e8	0.0396870
$871$	2.03108e9	0.104151
$872$	1.76061e10	0.899196
$873$	−1.06815e10	−0.543352
$874$	−3.31669e9	−0.168041
$875$	−1.36704e10	−0.689849
$876$	4.08839e10	2.05488
$877$	−3.01948e10	−1.51159	−0.755794	−	0.654810i	$-0.772749\pi$
$877$	−3.01948e10	−1.51159	−0.755794	+	0.654810i	$0.772749\pi$
$878$	−9.63413e9	−0.480376
$879$	5.99033e10	2.97502
$880$	1.06193e9	0.0525299
$881$	−2.31767e9	−0.114192	−0.0570959	−	0.998369i	$-0.518184\pi$
$881$	−2.31767e9	−0.114192	−0.0570959	+	0.998369i	$0.518184\pi$
$882$	6.98533e9	0.342805
$883$	5.30440e9	0.259283	0.129642	−	0.991561i	$-0.458617\pi$
$883$	5.30440e9	0.259283	0.129642	+	0.991561i	$0.458617\pi$
$884$	1.36234e9	0.0663289
$885$	−1.77967e10	−0.863055
$886$	1.49629e10	0.722764
$887$	7.49349e9	0.360538	0.180269	−	0.983617i	$-0.442303\pi$
$887$	7.49349e9	0.360538	0.180269	+	0.983617i	$0.442303\pi$
$888$	−5.94261e10	−2.84794
$889$	−1.33555e10	−0.637533
$890$	8.58112e9	0.408018
$891$	−4.98380e10	−2.36041
$892$	−1.85287e10	−0.874116
$893$	−3.73412e10	−1.75472
$894$	1.12432e10	0.526272
$895$	−1.66053e9	−0.0774221
$896$	−1.34873e10	−0.626391
$897$	−1.04876e9	−0.0485178
$898$	−1.55770e10	−0.717821
$899$	−2.91130e9	−0.133637
$900$	2.61193e10	1.19430
$901$	−1.34611e10	−0.613115
$902$	−1.16898e10	−0.530378
$903$	3.05631e10	1.38131
$904$	−2.57263e10	−1.15821
$905$	1.88109e9	0.0843607
$906$	−3.47449e10	−1.55218
$907$	1.03049e10	0.458583	0.229292	−	0.973358i	$-0.426359\pi$
$907$	1.03049e10	0.458583	0.229292	+	0.973358i	$0.426359\pi$
$908$	1.62997e10	0.722567
$909$	−3.85141e10	−1.70077
$910$	6.55127e8	0.0288191
$911$	1.46565e10	0.642267	0.321133	−	0.947034i	$-0.395936\pi$
$911$	1.46565e10	0.642267	0.321133	+	0.947034i	$0.395936\pi$
$912$	5.27947e9	0.230467
$913$	2.63145e10	1.14432
$914$	1.77540e10	0.769104
$915$	−1.32068e9	−0.0569933
$916$	−2.57152e9	−0.110549
$917$	4.61781e9	0.197762
$918$	2.43592e10	1.03923
$919$	−9.36116e9	−0.397856	−0.198928	−	0.980014i	$-0.563746\pi$
$919$	−9.36116e9	−0.397856	−0.198928	+	0.980014i	$0.563746\pi$
$920$	2.12311e9	0.0898909
$921$	−3.03322e10	−1.27937
$922$	5.92438e9	0.248934
$923$	5.39385e9	0.225784
$924$	3.13198e10	1.30607
$925$	−3.14238e10	−1.30546
$926$	4.60720e9	0.190677
$927$	4.54011e10	1.87192
$928$	2.10631e9	0.0865175
$929$	−1.13489e10	−0.464406	−0.232203	−	0.972667i	$-0.574593\pi$
$929$	−1.13489e10	−0.464406	−0.232203	+	0.972667i	$0.574593\pi$
$930$	−1.81928e10	−0.741669
$931$	8.81581e9	0.358046
$932$	3.37037e9	0.136371
$933$	−1.00805e10	−0.406346
$934$	1.37274e10	0.551283
$935$	1.10890e10	0.443663
$936$	−7.08187e9	−0.282281
$937$	−8.31100e9	−0.330039	−0.165019	−	0.986290i	$-0.552769\pi$
$937$	−8.31100e9	−0.330039	−0.165019	+	0.986290i	$0.552769\pi$
$938$	−1.02703e10	−0.406325
$939$	−3.80914e10	−1.50141
$940$	9.49217e9	0.372750
$941$	−1.33666e10	−0.522947	−0.261474	−	0.965211i	$-0.584208\pi$
$941$	−1.33666e10	−0.522947	−0.261474	+	0.965211i	$0.584208\pi$
$942$	3.90518e10	1.52217
$943$	3.81964e9	0.148331
$944$	−2.56937e9	−0.0994087
$945$	−2.26050e10	−0.871351
$946$	1.72587e10	0.662811
$947$	−1.89737e10	−0.725985	−0.362993	−	0.931792i	$-0.618245\pi$
$947$	−1.89737e10	−0.725985	−0.362993	+	0.931792i	$0.618245\pi$
$948$	2.26418e10	0.863140
$949$	−5.86164e9	−0.222632
$950$	−1.70818e10	−0.646401
$951$	5.66941e9	0.213750
$952$	−1.73473e10	−0.651633
$953$	3.86212e10	1.44544	0.722721	−	0.691140i	$-0.242891\pi$
$953$	3.86212e10	1.44544	0.722721	+	0.691140i	$0.242891\pi$
$954$	2.77876e10	1.03617
$955$	−7.92835e8	−0.0294558
$956$	−2.47663e10	−0.916766
$957$	−5.28358e9	−0.194866
$958$	−7.41427e9	−0.272451
$959$	−1.59037e10	−0.582281
$960$	1.11249e10	0.405833
$961$	4.11975e10	1.49741
$962$	3.38341e9	0.122530
$963$	6.98508e10	2.52046
$964$	−9.33517e9	−0.335624
$965$	1.04385e10	0.373930
$966$	5.30312e9	0.189283
$967$	−2.03017e10	−0.722005	−0.361002	−	0.932565i	$-0.617565\pi$
$967$	−2.03017e10	−0.722005	−0.361002	+	0.932565i	$0.617565\pi$
$968$	1.71900e10	0.609133
$969$	5.51301e10	1.94650
$970$	−1.77575e9	−0.0624714
$971$	5.33502e10	1.87012	0.935058	−	0.354494i	$-0.115347\pi$
$971$	5.33502e10	1.87012	0.935058	+	0.354494i	$0.115347\pi$
$972$	2.00585e10	0.700595
$973$	−3.40619e10	−1.18543
$974$	2.32375e10	0.805810
$975$	−5.40137e9	−0.186633
$976$	−1.90671e8	−0.00656462
$977$	3.51948e10	1.20739	0.603695	−	0.797215i	$-0.293694\pi$
$977$	3.51948e10	1.20739	0.603695	+	0.797215i	$0.293694\pi$
$978$	−1.11526e10	−0.381232
$979$	−5.88174e10	−2.00339
$980$	−2.24099e9	−0.0760586
$981$	6.20253e10	2.09762
$982$	−5.19742e8	−0.0175145
$983$	−2.80193e10	−0.940851	−0.470425	−	0.882440i	$-0.655899\pi$
$983$	−2.80193e10	−0.940851	−0.470425	+	0.882440i	$0.655899\pi$
$984$	3.72024e10	1.24477
$985$	−6.19986e9	−0.206707
$986$	1.16212e9	0.0386084
$987$	5.97055e10	1.97653
$988$	−3.54922e9	−0.117080
$989$	−5.63926e9	−0.185368
$990$	−2.28911e10	−0.749795
$991$	−6.10530e10	−1.99273	−0.996366	−	0.0851693i	$-0.972857\pi$
$991$	−6.10530e10	−1.99273	−0.996366	+	0.0851693i	$0.972857\pi$
$992$	−4.97114e10	−1.61683
$993$	7.80537e10	2.52971
$994$	−2.72744e10	−0.880852
$995$	−9.59961e8	−0.0308939
$996$	−3.32558e10	−1.06650
$997$	2.11795e10	0.676833	0.338417	−	0.940996i	$-0.390109\pi$
$997$	2.11795e10	0.676833	0.338417	+	0.940996i	$0.390109\pi$
$998$	−2.06086e10	−0.656284
$999$	−1.16744e11	−3.70471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.8.a.b.1.4	✓	8
3.2	odd	2		207.8.a.f.1.5		8
4.3	odd	2		368.8.a.h.1.8		8
5.4	even	2		575.8.a.b.1.5		8
23.22	odd	2		529.8.a.c.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.a.b.1.4	✓	8	1.1	even	1	trivial
207.8.a.f.1.5		8	3.2	odd	2
368.8.a.h.1.8		8	4.3	odd	2
529.8.a.c.1.4		8	23.22	odd	2
575.8.a.b.1.5		8	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)

\(f(q)\)	\(=\)	\(q-6.60982 q^{2} -84.4445 q^{3} -84.3103 q^{4} -124.345 q^{5} +558.163 q^{6} -780.885 q^{7} +1403.33 q^{8} +4943.87 q^{9} +O(q^{10})\)	\(q-6.60982 q^{2} -84.4445 q^{3} -84.3103 q^{4} -124.345 q^{5} +558.163 q^{6} -780.885 q^{7} +1403.33 q^{8} +4943.87 q^{9} +821.899 q^{10} -5633.52 q^{11} +7119.55 q^{12} -1020.75 q^{13} +5161.51 q^{14} +10500.3 q^{15} +1515.96 q^{16} +15830.1 q^{17} -32678.1 q^{18} -41241.3 q^{19} +10483.6 q^{20} +65941.4 q^{21} +37236.6 q^{22} -12167.0 q^{23} -118504. q^{24} -62663.3 q^{25} +6746.98 q^{26} -232803. q^{27} +65836.7 q^{28} -11106.5 q^{29} -69404.8 q^{30} +262126. q^{31} -189647. q^{32} +475720. q^{33} -104634. q^{34} +97099.3 q^{35} -416820. q^{36} +501470. q^{37} +272597. q^{38} +86196.8 q^{39} -174498. q^{40} -313935. q^{41} -435861. q^{42} +463488. q^{43} +474964. q^{44} -614747. q^{45} +80421.6 q^{46} +905432. q^{47} -128014. q^{48} -213762. q^{49} +414193. q^{50} -1.33677e6 q^{51} +86059.9 q^{52} -850343. q^{53} +1.53878e6 q^{54} +700502. q^{55} -1.09584e6 q^{56} +3.48260e6 q^{57} +73411.8 q^{58} -1.69488e6 q^{59} -885281. q^{60} -125776. q^{61} -1.73261e6 q^{62} -3.86060e6 q^{63} +1.05949e6 q^{64} +126925. q^{65} -3.14442e6 q^{66} -1.98979e6 q^{67} -1.33464e6 q^{68} +1.02744e6 q^{69} -641808. q^{70} -5.28420e6 q^{71} +6.93790e6 q^{72} +5.74248e6 q^{73} -3.31463e6 q^{74} +5.29157e6 q^{75} +3.47707e6 q^{76} +4.39913e6 q^{77} -569745. q^{78} +3.18023e6 q^{79} -188502. q^{80} +8.84668e6 q^{81} +2.07505e6 q^{82} -4.67105e6 q^{83} -5.55955e6 q^{84} -1.96840e6 q^{85} -3.06357e6 q^{86} +937881. q^{87} -7.90570e6 q^{88} +1.04406e7 q^{89} +4.06336e6 q^{90} +797089. q^{91} +1.02580e6 q^{92} -2.21351e7 q^{93} -5.98474e6 q^{94} +5.12816e6 q^{95} +1.60146e7 q^{96} -2.16055e6 q^{97} +1.41293e6 q^{98} -2.78514e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9}+O(q^{10}) \) 8 * q + 40 * q^3 + 640 * q^4 + 444 * q^5 - 1745 * q^6 + 1446 * q^7 + 3177 * q^8 + 13878 * q^9	\( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9} + 19502 q^{10} + 7588 q^{11} + 22733 q^{12} + 19862 q^{13} + 17544 q^{14} - 12770 q^{15} + 64336 q^{16} + 42070 q^{17} - 59129 q^{18} + 1050 q^{19} + 3364 q^{20} - 7698 q^{21} - 128220 q^{22} - 97336 q^{23} - 621188 q^{24} + 49496 q^{25} - 371761 q^{26} - 69500 q^{27} + 143050 q^{28} - 102578 q^{29} - 671470 q^{30} + 304172 q^{31} - 612824 q^{32} + 747242 q^{33} - 524530 q^{34} + 531048 q^{35} + 1868983 q^{36} + 286472 q^{37} - 762932 q^{38} + 1032828 q^{39} + 2105286 q^{40} + 1324414 q^{41} - 1886168 q^{42} + 2052578 q^{43} - 867298 q^{44} + 2087442 q^{45} + 675556 q^{47} + 1411151 q^{48} - 55404 q^{49} + 1458528 q^{50} + 2775482 q^{51} - 1695409 q^{52} + 203654 q^{53} - 9897559 q^{54} - 1024444 q^{55} - 5766846 q^{56} + 3908648 q^{57} - 5039991 q^{58} - 748892 q^{59} - 18153300 q^{60} + 61822 q^{61} - 4939277 q^{62} + 1411632 q^{63} + 2702267 q^{64} - 1571618 q^{65} + 3791866 q^{66} + 3235604 q^{67} + 4914980 q^{68} - 486680 q^{69} + 10871764 q^{70} - 4951664 q^{71} - 7940241 q^{72} + 11019370 q^{73} + 356954 q^{74} - 13607220 q^{75} + 21973240 q^{76} - 5284888 q^{77} - 1506779 q^{78} + 4202464 q^{79} + 8785886 q^{80} + 10294096 q^{81} + 32636759 q^{82} + 518568 q^{83} + 7629190 q^{84} + 9854220 q^{85} - 14681386 q^{86} + 4862532 q^{87} + 20589740 q^{88} + 4203864 q^{89} + 49021076 q^{90} + 2488406 q^{91} - 7786880 q^{92} - 23367842 q^{93} + 12314327 q^{94} - 44485300 q^{95} - 45317009 q^{96} + 18621134 q^{97} + 35756 q^{98} - 64729930 q^{99}+O(q^{100}) \) 8 * q + 40 * q^3 + 640 * q^4 + 444 * q^5 - 1745 * q^6 + 1446 * q^7 + 3177 * q^8 + 13878 * q^9 + 19502 * q^10 + 7588 * q^11 + 22733 * q^12 + 19862 * q^13 + 17544 * q^14 - 12770 * q^15 + 64336 * q^16 + 42070 * q^17 - 59129 * q^18 + 1050 * q^19 + 3364 * q^20 - 7698 * q^21 - 128220 * q^22 - 97336 * q^23 - 621188 * q^24 + 49496 * q^25 - 371761 * q^26 - 69500 * q^27 + 143050 * q^28 - 102578 * q^29 - 671470 * q^30 + 304172 * q^31 - 612824 * q^32 + 747242 * q^33 - 524530 * q^34 + 531048 * q^35 + 1868983 * q^36 + 286472 * q^37 - 762932 * q^38 + 1032828 * q^39 + 2105286 * q^40 + 1324414 * q^41 - 1886168 * q^42 + 2052578 * q^43 - 867298 * q^44 + 2087442 * q^45 + 675556 * q^47 + 1411151 * q^48 - 55404 * q^49 + 1458528 * q^50 + 2775482 * q^51 - 1695409 * q^52 + 203654 * q^53 - 9897559 * q^54 - 1024444 * q^55 - 5766846 * q^56 + 3908648 * q^57 - 5039991 * q^58 - 748892 * q^59 - 18153300 * q^60 + 61822 * q^61 - 4939277 * q^62 + 1411632 * q^63 + 2702267 * q^64 - 1571618 * q^65 + 3791866 * q^66 + 3235604 * q^67 + 4914980 * q^68 - 486680 * q^69 + 10871764 * q^70 - 4951664 * q^71 - 7940241 * q^72 + 11019370 * q^73 + 356954 * q^74 - 13607220 * q^75 + 21973240 * q^76 - 5284888 * q^77 - 1506779 * q^78 + 4202464 * q^79 + 8785886 * q^80 + 10294096 * q^81 + 32636759 * q^82 + 518568 * q^83 + 7629190 * q^84 + 9854220 * q^85 - 14681386 * q^86 + 4862532 * q^87 + 20589740 * q^88 + 4203864 * q^89 + 49021076 * q^90 + 2488406 * q^91 - 7786880 * q^92 - 23367842 * q^93 + 12314327 * q^94 - 44485300 * q^95 - 45317009 * q^96 + 18621134 * q^97 + 35756 * q^98 - 64729930 * q^99

Introduction

L-functions

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