LMFDB - Embedded newform 23.8.a.b.1.6

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			1.6
Root			$11.0962$ 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)

$f(q)$	$=$	$q+11.0962 q^{2} +60.8046 q^{3} -4.87416 q^{4} +165.526 q^{5} +674.701 q^{6} +952.148 q^{7} -1474.40 q^{8} +1510.20 q^{9} +O(q^{10})$	\(q+11.0962 q^{2} +60.8046 q^{3} -4.87416 q^{4} +165.526 q^{5} +674.701 q^{6} +952.148 q^{7} -1474.40 q^{8} +1510.20 q^{9} +1836.71 q^{10} -4863.13 q^{11} -296.371 q^{12} +12899.5 q^{13} +10565.2 q^{14} +10064.8 q^{15} -15736.4 q^{16} -18595.8 q^{17} +16757.5 q^{18} -8378.65 q^{19} -806.801 q^{20} +57895.0 q^{21} -53962.3 q^{22} -12167.0 q^{23} -89650.3 q^{24} -50726.1 q^{25} +143136. q^{26} -41152.4 q^{27} -4640.92 q^{28} -133283. q^{29} +111681. q^{30} -107642. q^{31} +14109.3 q^{32} -295701. q^{33} -206343. q^{34} +157606. q^{35} -7360.97 q^{36} +422577. q^{37} -92971.3 q^{38} +784351. q^{39} -244052. q^{40} -85366.0 q^{41} +642415. q^{42} +410360. q^{43} +23703.7 q^{44} +249978. q^{45} -135008. q^{46} +1.11839e6 q^{47} -956843. q^{48} +83043.7 q^{49} -562867. q^{50} -1.13071e6 q^{51} -62874.3 q^{52} +275790. q^{53} -456635. q^{54} -804976. q^{55} -1.40385e6 q^{56} -509461. q^{57} -1.47893e6 q^{58} -182891. q^{59} -49057.2 q^{60} +2.49804e6 q^{61} -1.19442e6 q^{62} +1.43794e6 q^{63} +2.17081e6 q^{64} +2.13521e6 q^{65} -3.28116e6 q^{66} -705106. q^{67} +90638.9 q^{68} -739810. q^{69} +1.74882e6 q^{70} +4.54627e6 q^{71} -2.22664e6 q^{72} -4.46832e6 q^{73} +4.68901e6 q^{74} -3.08438e6 q^{75} +40838.9 q^{76} -4.63042e6 q^{77} +8.70332e6 q^{78} +4.92889e6 q^{79} -2.60478e6 q^{80} -5.80507e6 q^{81} -947239. q^{82} +369529. q^{83} -282190. q^{84} -3.07809e6 q^{85} +4.55344e6 q^{86} -8.10420e6 q^{87} +7.17020e6 q^{88} +2.70162e6 q^{89} +2.77381e6 q^{90} +1.22823e7 q^{91} +59303.9 q^{92} -6.54512e6 q^{93} +1.24099e7 q^{94} -1.38689e6 q^{95} +857910. q^{96} +8.20102e6 q^{97} +921470. q^{98} -7.34432e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	11.0962	0.980776	0.490388	−	0.871504i	$-0.336855\pi$
$2$	11.0962	0.980776	0.490388	+	0.871504i	$0.336855\pi$
$3$	60.8046	1.30021	0.650103	−	0.759846i	$-0.274726\pi$
$3$	60.8046	1.30021	0.650103	+	0.759846i	$0.274726\pi$
$4$	−4.87416	−0.0380794
$5$	165.526	0.592205	0.296102	−	0.955156i	$-0.404313\pi$
$5$	165.526	0.592205	0.296102	+	0.955156i	$0.404313\pi$
$6$	674.701	1.27521
$7$	952.148	1.04921	0.524604	−	0.851346i	$-0.324213\pi$
$7$	952.148	1.04921	0.524604	+	0.851346i	$0.324213\pi$
$8$	−1474.40	−1.01812
$9$	1510.20	0.690537
$10$	1836.71	0.580820
$11$	−4863.13	−1.10164	−0.550822	−	0.834622i	$-0.685686\pi$
$11$	−4863.13	−1.10164	−0.550822	+	0.834622i	$0.685686\pi$
$12$	−296.371	−0.0495110
$13$	12899.5	1.62844	0.814220	−	0.580556i	$-0.197165\pi$
$13$	12899.5	1.62844	0.814220	+	0.580556i	$0.197165\pi$
$14$	10565.2	1.02904
$15$	10064.8	0.769988
$16$	−15736.4	−0.960471
$17$	−18595.8	−0.918002	−0.459001	−	0.888436i	$-0.651793\pi$
$17$	−18595.8	−0.918002	−0.459001	+	0.888436i	$0.651793\pi$
$18$	16757.5	0.677261
$19$	−8378.65	−0.280244	−0.140122	−	0.990134i	$-0.544750\pi$
$19$	−8378.65	−0.280244	−0.140122	+	0.990134i	$0.544750\pi$
$20$	−806.801	−0.0225508
$21$	57895.0	1.36419
$22$	−53962.3	−1.08047
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	−89650.3	−1.32377
$25$	−50726.1	−0.649294
$26$	143136.	1.59714
$27$	−41152.4	−0.402366
$28$	−4640.92	−0.0399532
$29$	−133283.	−1.01480	−0.507400	−	0.861711i	$-0.669393\pi$
$29$	−133283.	−1.01480	−0.507400	+	0.861711i	$0.669393\pi$
$30$	111681.	0.755185
$31$	−107642.	−0.648956	−0.324478	−	0.945893i	$-0.605189\pi$
$31$	−107642.	−0.648956	−0.324478	+	0.945893i	$0.605189\pi$
$32$	14109.3	0.0761168
$33$	−295701.	−1.43237
$34$	−206343.	−0.900354
$35$	157606.	0.621346
$36$	−7360.97	−0.0262952
$37$	422577.	1.37151	0.685757	−	0.727831i	$-0.259471\pi$
$37$	422577.	1.37151	0.685757	+	0.727831i	$0.259471\pi$
$38$	−92971.3	−0.274857
$39$	784351.	2.11731
$40$	−244052.	−0.602937
$41$	−85366.0	−0.193438	−0.0967189	−	0.995312i	$-0.530835\pi$
$41$	−85366.0	−0.193438	−0.0967189	+	0.995312i	$0.530835\pi$
$42$	642415.	1.33796
$43$	410360.	0.787091	0.393546	−	0.919305i	$-0.371248\pi$
$43$	410360.	0.787091	0.393546	+	0.919305i	$0.371248\pi$
$44$	23703.7	0.0419499
$45$	249978.	0.408939
$46$	−135008.	−0.204506
$47$	1.11839e6	1.57127	0.785635	−	0.618691i	$-0.212337\pi$
$47$	1.11839e6	1.57127	0.785635	+	0.618691i	$0.212337\pi$
$48$	−956843.	−1.24881
$49$	83043.7	0.100837
$50$	−562867.	−0.636811
$51$	−1.13071e6	−1.19359
$52$	−62874.3	−0.0620100
$53$	275790.	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$53$	275790.	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$54$	−456635.	−0.394631
$55$	−804976.	−0.652399
$56$	−1.40385e6	−1.06822
$57$	−509461.	−0.364375
$58$	−1.47893e6	−0.995291
$59$	−182891.	−0.115934	−0.0579670	−	0.998318i	$-0.518462\pi$
$59$	−182891.	−0.115934	−0.0579670	+	0.998318i	$0.518462\pi$
$60$	−49057.2	−0.0293206
$61$	2.49804e6	1.40911	0.704556	−	0.709648i	$-0.251146\pi$
$61$	2.49804e6	1.40911	0.704556	+	0.709648i	$0.251146\pi$
$62$	−1.19442e6	−0.636480
$63$	1.43794e6	0.724516
$64$	2.17081e6	1.03512
$65$	2.13521e6	0.964370
$66$	−3.28116e6	−1.40483
$67$	−705106.	−0.286413	−0.143207	−	0.989693i	$-0.545741\pi$
$67$	−705106.	−0.286413	−0.143207	+	0.989693i	$0.545741\pi$
$68$	90638.9	0.0349569
$69$	−739810.	−0.271112
$70$	1.74882e6	0.609401
$71$	4.54627e6	1.50748	0.753740	−	0.657173i	$-0.228248\pi$
$71$	4.54627e6	1.50748	0.753740	+	0.657173i	$0.228248\pi$
$72$	−2.22664e6	−0.703051
$73$	−4.46832e6	−1.34435	−0.672177	−	0.740390i	$-0.734641\pi$
$73$	−4.46832e6	−1.34435	−0.672177	+	0.740390i	$0.734641\pi$
$74$	4.68901e6	1.34515
$75$	−3.08438e6	−0.844216
$76$	40838.9	0.0106715
$77$	−4.63042e6	−1.15585
$78$	8.70332e6	2.07661
$79$	4.92889e6	1.12475	0.562373	−	0.826884i	$-0.309889\pi$
$79$	4.92889e6	1.12475	0.562373	+	0.826884i	$0.309889\pi$
$80$	−2.60478e6	−0.568795
$81$	−5.80507e6	−1.21370
$82$	−947239.	−0.189719
$83$	369529.	0.0709373	0.0354687	−	0.999371i	$-0.488708\pi$
$83$	369529.	0.0709373	0.0354687	+	0.999371i	$0.488708\pi$
$84$	−282190.	−0.0519473
$85$	−3.07809e6	−0.543645
$86$	4.55344e6	0.771960
$87$	−8.10420e6	−1.31945
$88$	7.17020e6	1.12161
$89$	2.70162e6	0.406217	0.203109	−	0.979156i	$-0.434896\pi$
$89$	2.70162e6	0.406217	0.203109	+	0.979156i	$0.434896\pi$
$90$	2.77381e6	0.401077
$91$	1.22823e7	1.70857
$92$	59303.9	0.00794009
$93$	−6.54512e6	−0.843777
$94$	1.24099e7	1.54106
$95$	−1.38689e6	−0.165962
$96$	857910.	0.0989675
$97$	8.20102e6	0.912362	0.456181	−	0.889887i	$-0.349217\pi$
$97$	8.20102e6	0.912362	0.456181	+	0.889887i	$0.349217\pi$
$98$	921470.	0.0988985
$99$	−7.34432e6	−0.760726
$100$	247247.	0.0247247
$101$	−1.65975e7	−1.60294	−0.801471	−	0.598034i	$-0.795949\pi$
$101$	−1.65975e7	−1.60294	−0.801471	+	0.598034i	$0.795949\pi$
$102$	−1.25466e7	−1.17065
$103$	−1.26773e7	−1.14313	−0.571565	−	0.820556i	$-0.693664\pi$
$103$	−1.26773e7	−1.14313	−0.571565	+	0.820556i	$0.693664\pi$
$104$	−1.90191e7	−1.65795
$105$	9.58315e6	0.807878
$106$	3.06023e6	0.249565
$107$	−2.30763e7	−1.82105	−0.910526	−	0.413453i	$-0.864323\pi$
$107$	−2.30763e7	−1.82105	−0.910526	+	0.413453i	$0.864323\pi$
$108$	200583.	0.0153218
$109$	2.67537e7	1.97875	0.989376	−	0.145377i	$-0.0464396\pi$
$109$	2.67537e7	1.97875	0.989376	+	0.145377i	$0.0464396\pi$
$110$	−8.93218e6	−0.639857
$111$	2.56947e7	1.78325
$112$	−1.49833e7	−1.00773
$113$	−1.63549e6	−0.106629	−0.0533143	−	0.998578i	$-0.516979\pi$
$113$	−1.63549e6	−0.106629	−0.0533143	+	0.998578i	$0.516979\pi$
$114$	−5.65308e6	−0.357370
$115$	−2.01396e6	−0.123483
$116$	649640.	0.0386429
$117$	1.94809e7	1.12450
$118$	−2.02940e6	−0.113705
$119$	−1.77060e7	−0.963175
$120$	−1.48395e7	−0.783942
$121$	4.16288e6	0.213622
$122$	2.77188e7	1.38202
$123$	−5.19065e6	−0.251509
$124$	524663.	0.0247118
$125$	−2.13282e7	−0.976719
$126$	1.59557e7	0.710588
$127$	−9.72199e6	−0.421155	−0.210578	−	0.977577i	$-0.567534\pi$
$127$	−9.72199e6	−0.421155	−0.210578	+	0.977577i	$0.567534\pi$
$128$	2.22818e7	0.939108
$129$	2.49518e7	1.02338
$130$	2.36927e7	0.945831
$131$	−1.23717e7	−0.480815	−0.240408	−	0.970672i	$-0.577281\pi$
$131$	−1.23717e7	−0.480815	−0.240408	+	0.970672i	$0.577281\pi$
$132$	1.44129e6	0.0545436
$133$	−7.97772e6	−0.294034
$134$	−7.82401e6	−0.280907
$135$	−6.81179e6	−0.238283
$136$	2.74176e7	0.934639
$137$	2.11350e7	0.702231	0.351116	−	0.936332i	$-0.385802\pi$
$137$	2.11350e7	0.702231	0.351116	+	0.936332i	$0.385802\pi$
$138$	−8.20909e6	−0.265900
$139$	1.65004e7	0.521126	0.260563	−	0.965457i	$-0.416092\pi$
$139$	1.65004e7	0.521126	0.260563	+	0.965457i	$0.416092\pi$
$140$	−768194.	−0.0236604
$141$	6.80033e7	2.04297
$142$	5.04464e7	1.47850
$143$	−6.27321e7	−1.79396
$144$	−2.37651e7	−0.663240
$145$	−2.20618e7	−0.600969
$146$	−4.95814e7	−1.31851
$147$	5.04944e6	0.131109
$148$	−2.05971e6	−0.0522264
$149$	7.87547e7	1.95040	0.975202	−	0.221315i	$-0.0710348\pi$
$149$	7.87547e7	1.95040	0.975202	+	0.221315i	$0.0710348\pi$
$150$	−3.42249e7	−0.827986
$151$	−6.23367e7	−1.47341	−0.736707	−	0.676212i	$-0.763620\pi$
$151$	−6.23367e7	−1.47341	−0.736707	+	0.676212i	$0.763620\pi$
$152$	1.23535e7	0.285323
$153$	−2.80835e7	−0.633914
$154$	−5.13801e7	−1.13363
$155$	−1.78175e7	−0.384315
$156$	−3.82305e6	−0.0806258
$157$	6.46758e7	1.33381	0.666903	−	0.745144i	$-0.267619\pi$
$157$	6.46758e7	1.33381	0.666903	+	0.745144i	$0.267619\pi$
$158$	5.46920e7	1.10312
$159$	1.67693e7	0.330846
$160$	2.33546e6	0.0450767
$161$	−1.15848e7	−0.218775
$162$	−6.44143e7	−1.19036
$163$	−5.34198e7	−0.966153	−0.483077	−	0.875578i	$-0.660481\pi$
$163$	−5.34198e7	−0.966153	−0.483077	+	0.875578i	$0.660481\pi$
$164$	416087.	0.00736599
$165$	−4.89463e7	−0.848253
$166$	4.10037e6	0.0695736
$167$	−2.35239e7	−0.390843	−0.195421	−	0.980719i	$-0.562607\pi$
$167$	−2.35239e7	−0.390843	−0.195421	+	0.980719i	$0.562607\pi$
$168$	−8.53604e7	−1.38891
$169$	1.03649e8	1.65182
$170$	−3.41552e7	−0.533194
$171$	−1.26535e7	−0.193519
$172$	−2.00016e6	−0.0299719
$173$	−4.02988e7	−0.591739	−0.295870	−	0.955228i	$-0.595609\pi$
$173$	−4.02988e7	−0.591739	−0.295870	+	0.955228i	$0.595609\pi$
$174$	−8.99259e7	−1.29408
$175$	−4.82988e7	−0.681244
$176$	7.65279e7	1.05810
$177$	−1.11206e7	−0.150738
$178$	2.99777e7	0.398408
$179$	9.45318e7	1.23195	0.615974	−	0.787767i	$-0.288763\pi$
$179$	9.45318e7	1.23195	0.615974	+	0.787767i	$0.288763\pi$
$180$	−1.21843e6	−0.0155721
$181$	2.13500e7	0.267623	0.133811	−	0.991007i	$-0.457278\pi$
$181$	2.13500e7	0.267623	0.133811	+	0.991007i	$0.457278\pi$
$182$	1.36287e8	1.67573
$183$	1.51893e8	1.83214
$184$	1.79390e7	0.212293
$185$	6.99476e7	0.812217
$186$	−7.26260e7	−0.827555
$187$	9.04338e7	1.01131
$188$	−5.45121e6	−0.0598329
$189$	−3.91832e7	−0.422166
$190$	−1.53892e7	−0.162771
$191$	−1.38843e8	−1.44181	−0.720903	−	0.693036i	$-0.756273\pi$
$191$	−1.38843e8	−1.44181	−0.720903	+	0.693036i	$0.756273\pi$
$192$	1.31995e8	1.34587
$193$	−1.26393e8	−1.26553	−0.632767	−	0.774342i	$-0.718081\pi$
$193$	−1.26393e8	−1.26553	−0.632767	+	0.774342i	$0.718081\pi$
$194$	9.10003e7	0.894822
$195$	1.29831e8	1.25388
$196$	−404768.	−0.00383981
$197$	1.25589e8	1.17036	0.585181	−	0.810903i	$-0.301023\pi$
$197$	1.25589e8	1.17036	0.585181	+	0.810903i	$0.301023\pi$
$198$	−8.14941e7	−0.746102
$199$	−6.73412e7	−0.605752	−0.302876	−	0.953030i	$-0.597947\pi$
$199$	−6.73412e7	−0.605752	−0.302876	+	0.953030i	$0.597947\pi$
$200$	7.47905e7	0.661061
$201$	−4.28737e7	−0.372396
$202$	−1.84169e8	−1.57213
$203$	−1.26905e8	−1.06474
$204$	5.51126e6	0.0454512
$205$	−1.41303e7	−0.114555
$206$	−1.40670e8	−1.12115
$207$	−1.83746e7	−0.143987
$208$	−2.02991e8	−1.56407
$209$	4.07465e7	0.308730
$210$	1.06337e8	0.792347
$211$	2.55103e8	1.86950	0.934752	−	0.355300i	$-0.115621\pi$
$211$	2.55103e8	1.86950	0.934752	+	0.355300i	$0.115621\pi$
$212$	−1.34425e6	−0.00968954
$213$	2.76435e8	1.96003
$214$	−2.56059e8	−1.78604
$215$	6.79253e7	0.466119
$216$	6.06750e7	0.409658
$217$	−1.02491e8	−0.680890
$218$	2.96865e8	1.94071
$219$	−2.71694e8	−1.74794
$220$	3.92358e6	0.0248429
$221$	−2.39877e8	−1.49491
$222$	2.85113e8	1.74897
$223$	−5.74660e6	−0.0347012	−0.0173506	−	0.999849i	$-0.505523\pi$
$223$	−5.74660e6	−0.0347012	−0.0173506	+	0.999849i	$0.505523\pi$
$224$	1.34341e7	0.0798623
$225$	−7.66067e7	−0.448361
$226$	−1.81478e7	−0.104579
$227$	5.38756e7	0.305704	0.152852	−	0.988249i	$-0.451154\pi$
$227$	5.38756e7	0.305704	0.152852	+	0.988249i	$0.451154\pi$
$228$	2.48319e6	0.0138752
$229$	−1.29368e8	−0.711876	−0.355938	−	0.934510i	$-0.615838\pi$
$229$	−1.29368e8	−0.711876	−0.355938	+	0.934510i	$0.615838\pi$
$230$	−2.23473e7	−0.121109
$231$	−2.81551e8	−1.50285
$232$	1.96512e8	1.03319
$233$	4.55108e7	0.235705	0.117852	−	0.993031i	$-0.462399\pi$
$233$	4.55108e7	0.235705	0.117852	+	0.993031i	$0.462399\pi$
$234$	2.16164e8	1.10288
$235$	1.85123e8	0.930513
$236$	891441.	0.00441469
$237$	2.99699e8	1.46240
$238$	−1.96469e8	−0.944659
$239$	−3.62185e8	−1.71608	−0.858041	−	0.513582i	$-0.828318\pi$
$239$	−3.62185e8	−1.71608	−0.858041	+	0.513582i	$0.828318\pi$
$240$	−1.58383e8	−0.739551
$241$	−1.34139e8	−0.617297	−0.308648	−	0.951176i	$-0.599877\pi$
$241$	−1.34139e8	−0.617297	−0.308648	+	0.951176i	$0.599877\pi$
$242$	4.61922e7	0.209515
$243$	−2.62975e8	−1.17569
$244$	−1.21759e7	−0.0536581
$245$	1.37459e7	0.0597162
$246$	−5.75965e7	−0.246674
$247$	−1.08081e8	−0.456361
$248$	1.58707e8	0.660717
$249$	2.24691e7	0.0922332
$250$	−2.36662e8	−0.957942
$251$	−5.80152e7	−0.231571	−0.115785	−	0.993274i	$-0.536938\pi$
$251$	−5.80152e7	−0.231571	−0.115785	+	0.993274i	$0.536938\pi$
$252$	−7.00874e6	−0.0275891
$253$	5.91697e7	0.229709
$254$	−1.07877e8	−0.413059
$255$	−1.87162e8	−0.706851
$256$	−3.06205e7	−0.114070
$257$	1.24570e8	0.457770	0.228885	−	0.973453i	$-0.426492\pi$
$257$	1.24570e8	0.457770	0.228885	+	0.973453i	$0.426492\pi$
$258$	2.76870e8	1.00371
$259$	4.02356e8	1.43900
$260$	−1.04073e7	−0.0367226
$261$	−2.01284e8	−0.700756
$262$	−1.37278e8	−0.471572
$263$	−3.79795e8	−1.28737	−0.643686	−	0.765289i	$-0.722596\pi$
$263$	−3.79795e8	−1.28737	−0.643686	+	0.765289i	$0.722596\pi$
$264$	4.35981e8	1.45832
$265$	4.56505e7	0.150690
$266$	−8.85225e7	−0.288382
$267$	1.64271e8	0.528166
$268$	3.43680e6	0.0109064
$269$	2.77479e8	0.869154	0.434577	−	0.900635i	$-0.356898\pi$
$269$	2.77479e8	0.869154	0.434577	+	0.900635i	$0.356898\pi$
$270$	−7.55851e7	−0.233702
$271$	4.63332e8	1.41416	0.707082	−	0.707131i	$-0.250011\pi$
$271$	4.63332e8	1.41416	0.707082	+	0.707131i	$0.250011\pi$
$272$	2.92630e8	0.881714
$273$	7.46819e8	2.22150
$274$	2.34518e8	0.688731
$275$	2.46688e8	0.715291
$276$	3.60595e6	0.0103238
$277$	−4.78837e8	−1.35366	−0.676828	−	0.736141i	$-0.736646\pi$
$277$	−4.78837e8	−1.35366	−0.676828	+	0.736141i	$0.736646\pi$
$278$	1.83092e8	0.511107
$279$	−1.62561e8	−0.448128
$280$	−2.32373e8	−0.632606
$281$	7.01149e7	0.188512	0.0942559	−	0.995548i	$-0.469953\pi$
$281$	7.01149e7	0.188512	0.0942559	+	0.995548i	$0.469953\pi$
$282$	7.54578e8	2.00370
$283$	−2.76444e8	−0.725027	−0.362513	−	0.931979i	$-0.618081\pi$
$283$	−2.76444e8	−0.725027	−0.362513	+	0.931979i	$0.618081\pi$
$284$	−2.21593e7	−0.0574038
$285$	−8.43291e7	−0.215785
$286$	−6.96088e8	−1.75948
$287$	−8.12811e7	−0.202957
$288$	2.13079e7	0.0525614
$289$	−6.45348e7	−0.157272
$290$	−2.44802e8	−0.589416
$291$	4.98660e8	1.18626
$292$	2.17793e7	0.0511922
$293$	−6.73610e8	−1.56449	−0.782243	−	0.622973i	$-0.785925\pi$
$293$	−6.73610e8	−1.56449	−0.782243	+	0.622973i	$0.785925\pi$
$294$	5.60296e7	0.128589
$295$	−3.02733e7	−0.0686567
$296$	−6.23048e8	−1.39637
$297$	2.00129e8	0.443265
$298$	8.73879e8	1.91291
$299$	−1.56949e8	−0.339553
$300$	1.50338e7	0.0321472
$301$	3.90723e8	0.825822
$302$	−6.91701e8	−1.44509
$303$	−1.00920e9	−2.08416
$304$	1.31849e8	0.269166
$305$	4.13492e8	0.834483
$306$	−3.11620e8	−0.621727
$307$	−2.02723e7	−0.0399869	−0.0199934	−	0.999800i	$-0.506365\pi$
$307$	−2.02723e7	−0.0399869	−0.0199934	+	0.999800i	$0.506365\pi$
$308$	2.25694e7	0.0440142
$309$	−7.70838e8	−1.48631
$310$	−1.97707e8	−0.376926
$311$	−3.68701e8	−0.695046	−0.347523	−	0.937672i	$-0.612977\pi$
$311$	−3.68701e8	−0.695046	−0.347523	+	0.937672i	$0.612977\pi$
$312$	−1.15645e9	−2.15568
$313$	5.51670e8	1.01689	0.508445	−	0.861094i	$-0.330220\pi$
$313$	5.51670e8	1.01689	0.508445	+	0.861094i	$0.330220\pi$
$314$	7.17656e8	1.30816
$315$	2.38016e8	0.429062
$316$	−2.40242e7	−0.0428296
$317$	−1.59775e8	−0.281709	−0.140855	−	0.990030i	$-0.544985\pi$
$317$	−1.59775e8	−0.281709	−0.140855	+	0.990030i	$0.544985\pi$
$318$	1.86076e8	0.324486
$319$	6.48171e8	1.11795
$320$	3.59326e8	0.613005
$321$	−1.40314e9	−2.36774
$322$	−1.28547e8	−0.214569
$323$	1.55808e8	0.257265
$324$	2.82948e7	0.0462168
$325$	−6.54342e8	−1.05734
$326$	−5.92758e8	−0.947579
$327$	1.62675e9	2.57279
$328$	1.25864e8	0.196944
$329$	1.06487e9	1.64859
$330$	−5.43118e8	−0.831946
$331$	−2.39933e8	−0.363658	−0.181829	−	0.983330i	$-0.558202\pi$
$331$	−2.39933e8	−0.363658	−0.181829	+	0.983330i	$0.558202\pi$
$332$	−1.80114e6	−0.00270125
$333$	6.38178e8	0.947080
$334$	−2.61026e8	−0.383329
$335$	−1.16714e8	−0.169615
$336$	−9.11057e8	−1.31026
$337$	1.35378e9	1.92682	0.963412	−	0.268026i	$-0.0863713\pi$
$337$	1.35378e9	1.92682	0.963412	+	0.268026i	$0.0863713\pi$
$338$	1.15011e9	1.62006
$339$	−9.94455e7	−0.138639
$340$	1.50031e7	0.0207017
$341$	5.23476e8	0.714919
$342$	−1.40406e8	−0.189799
$343$	−7.05065e8	−0.943409
$344$	−6.05034e8	−0.801356
$345$	−1.22458e8	−0.160554
$346$	−4.47164e8	−0.580364
$347$	4.09155e7	0.0525695	0.0262848	−	0.999654i	$-0.491632\pi$
$347$	4.09155e7	0.0525695	0.0262848	+	0.999654i	$0.491632\pi$
$348$	3.95011e7	0.0502438
$349$	1.24915e9	1.57298	0.786491	−	0.617601i	$-0.211895\pi$
$349$	1.24915e9	1.57298	0.786491	+	0.617601i	$0.211895\pi$
$350$	−5.35933e8	−0.668148
$351$	−5.30846e8	−0.655230
$352$	−6.86153e7	−0.0838536
$353$	9.60797e8	1.16257	0.581287	−	0.813699i	$-0.302550\pi$
$353$	9.60797e8	1.16257	0.581287	+	0.813699i	$0.302550\pi$
$354$	−1.23397e8	−0.147840
$355$	7.52527e8	0.892736
$356$	−1.31681e7	−0.0154685
$357$	−1.07660e9	−1.25233
$358$	1.04894e9	1.20826
$359$	7.94256e8	0.906003	0.453002	−	0.891510i	$-0.350353\pi$
$359$	7.94256e8	0.906003	0.453002	+	0.891510i	$0.350353\pi$
$360$	−3.68568e8	−0.416350
$361$	−8.23670e8	−0.921463
$362$	2.36904e8	0.262478
$363$	2.53122e8	0.277752
$364$	−5.98657e7	−0.0650614
$365$	−7.39623e8	−0.796133
$366$	1.68543e9	1.79691
$367$	8.04891e8	0.849974	0.424987	−	0.905199i	$-0.360279\pi$
$367$	8.04891e8	0.849974	0.424987	+	0.905199i	$0.360279\pi$
$368$	1.91464e8	0.200272
$369$	−1.28920e8	−0.133576
$370$	7.76153e8	0.796602
$371$	2.62593e8	0.266978
$372$	3.19020e7	0.0321305
$373$	1.19314e9	1.19045	0.595223	−	0.803561i	$-0.297064\pi$
$373$	1.19314e9	1.19045	0.595223	+	0.803561i	$0.297064\pi$
$374$	1.00347e9	0.991871
$375$	−1.29686e9	−1.26994
$376$	−1.64895e9	−1.59974
$377$	−1.71928e9	−1.65254
$378$	−4.34784e8	−0.414050
$379$	−6.14932e8	−0.580216	−0.290108	−	0.956994i	$-0.593691\pi$
$379$	−6.14932e8	−0.580216	−0.290108	+	0.956994i	$0.593691\pi$
$380$	6.75990e6	0.00631972
$381$	−5.91142e8	−0.547588
$382$	−1.54063e9	−1.41409
$383$	1.05026e9	0.955214	0.477607	−	0.878574i	$-0.341504\pi$
$383$	1.05026e9	0.955214	0.477607	+	0.878574i	$0.341504\pi$
$384$	1.35484e9	1.22103
$385$	−7.66456e8	−0.684502
$386$	−1.40249e9	−1.24121
$387$	6.19727e8	0.543515
$388$	−3.99731e7	−0.0347421
$389$	−9.88396e8	−0.851349	−0.425674	−	0.904876i	$-0.639963\pi$
$389$	−9.88396e8	−0.851349	−0.425674	+	0.904876i	$0.639963\pi$
$390$	1.44063e9	1.22977
$391$	2.26255e8	0.191417
$392$	−1.22440e8	−0.102665
$393$	−7.52254e8	−0.625159
$394$	1.39356e9	1.14786
$395$	8.15861e8	0.666080
$396$	3.57974e7	0.0289680
$397$	1.79703e8	0.144141	0.0720707	−	0.997400i	$-0.477039\pi$
$397$	1.79703e8	0.144141	0.0720707	+	0.997400i	$0.477039\pi$
$398$	−7.47231e8	−0.594107
$399$	−4.85082e8	−0.382305
$400$	7.98243e8	0.623628
$401$	−2.18971e9	−1.69583	−0.847914	−	0.530133i	$-0.822142\pi$
$401$	−2.18971e9	−1.69583	−0.847914	+	0.530133i	$0.822142\pi$
$402$	−4.75736e8	−0.365237
$403$	−1.38853e9	−1.05679
$404$	8.08988e7	0.0610390
$405$	−9.60891e8	−0.718756
$406$	−1.40816e9	−1.04427
$407$	−2.05505e9	−1.51092
$408$	1.66712e9	1.21522
$409$	8.31329e8	0.600816	0.300408	−	0.953811i	$-0.402877\pi$
$409$	8.31329e8	0.600816	0.300408	+	0.953811i	$0.402877\pi$
$410$	−1.56793e8	−0.112353
$411$	1.28511e9	0.913045
$412$	6.17911e7	0.0435297
$413$	−1.74140e8	−0.121639
$414$	−2.03889e8	−0.141219
$415$	6.11667e7	0.0420094
$416$	1.82003e8	0.123952
$417$	1.00330e9	0.677571
$418$	4.52132e8	0.302794
$419$	−1.68006e9	−1.11578	−0.557888	−	0.829916i	$-0.688388\pi$
$419$	−1.68006e9	−1.11578	−0.557888	+	0.829916i	$0.688388\pi$
$420$	−4.67098e7	−0.0307635
$421$	1.78087e8	0.116317	0.0581586	−	0.998307i	$-0.481477\pi$
$421$	1.78087e8	0.116317	0.0581586	+	0.998307i	$0.481477\pi$
$422$	2.83067e9	1.83356
$423$	1.68900e9	1.08502
$424$	−4.06625e8	−0.259068
$425$	9.43292e8	0.596053
$426$	3.06737e9	1.92235
$427$	2.37851e9	1.47845
$428$	1.12477e8	0.0693445
$429$	−3.81440e9	−2.33252
$430$	7.53713e8	0.457158
$431$	5.82450e8	0.350419	0.175210	−	0.984531i	$-0.443940\pi$
$431$	5.82450e8	0.350419	0.175210	+	0.984531i	$0.443940\pi$
$432$	6.47588e8	0.386461
$433$	−2.04988e9	−1.21345	−0.606723	−	0.794913i	$-0.707516\pi$
$433$	−2.04988e9	−1.21345	−0.606723	+	0.794913i	$0.707516\pi$
$434$	−1.13726e9	−0.667800
$435$	−1.34146e9	−0.781384
$436$	−1.30402e8	−0.0753496
$437$	1.01943e8	0.0584349
$438$	−3.01478e9	−1.71434
$439$	3.26179e6	0.00184005	0.000920027	−	1.00000i	$-0.499707\pi$
$439$	3.26179e6	0.00184005	0.000920027		1.00000i	$0.499707\pi$
$440$	1.18686e9	0.664222
$441$	1.25413e8	0.0696317
$442$	−2.66173e9	−1.46617
$443$	−1.85183e9	−1.01202	−0.506010	−	0.862528i	$-0.668880\pi$
$443$	−1.85183e9	−1.01202	−0.506010	+	0.862528i	$0.668880\pi$
$444$	−1.25240e8	−0.0679050
$445$	4.47188e8	0.240564
$446$	−6.37655e7	−0.0340340
$447$	4.78865e9	2.53593
$448$	2.06694e9	1.08606
$449$	−9.26659e8	−0.483123	−0.241562	−	0.970385i	$-0.577660\pi$
$449$	−9.26659e8	−0.483123	−0.241562	+	0.970385i	$0.577660\pi$
$450$	−8.50044e8	−0.439742
$451$	4.15146e8	0.213100
$452$	7.97165e6	0.00406035
$453$	−3.79036e9	−1.91574
$454$	5.97815e8	0.299827
$455$	2.03304e9	1.01182
$456$	7.51149e8	0.370979
$457$	1.41700e9	0.694483	0.347242	−	0.937776i	$-0.387118\pi$
$457$	1.41700e9	0.694483	0.347242	+	0.937776i	$0.387118\pi$
$458$	−1.43550e9	−0.698190
$459$	7.65261e8	0.369373
$460$	9.81634e6	0.00470216
$461$	2.11900e9	1.00735	0.503673	−	0.863894i	$-0.331982\pi$
$461$	2.11900e9	1.00735	0.503673	+	0.863894i	$0.331982\pi$
$462$	−3.12415e9	−1.47396
$463$	2.03922e9	0.954841	0.477421	−	0.878675i	$-0.341572\pi$
$463$	2.03922e9	0.954841	0.477421	+	0.878675i	$0.341572\pi$
$464$	2.09738e9	0.974685
$465$	−1.08339e9	−0.499688
$466$	5.04997e8	0.231174
$467$	−1.27874e9	−0.580996	−0.290498	−	0.956876i	$-0.593821\pi$
$467$	−1.27874e9	−0.580996	−0.290498	+	0.956876i	$0.593821\pi$
$468$	−9.49530e7	−0.0428202
$469$	−6.71366e8	−0.300507
$470$	2.05416e9	0.912624
$471$	3.93259e9	1.73422
$472$	2.69655e8	0.118035
$473$	−1.99563e9	−0.867095
$474$	3.32553e9	1.43429
$475$	4.25016e8	0.181961
$476$	8.63017e7	0.0366771
$477$	4.16500e8	0.175712
$478$	−4.01888e9	−1.68309
$479$	4.76473e8	0.198091	0.0990453	−	0.995083i	$-0.468421\pi$
$479$	4.76473e8	0.198091	0.0990453	+	0.995083i	$0.468421\pi$
$480$	1.42007e8	0.0586090
$481$	5.45105e9	2.23343
$482$	−1.48843e9	−0.605430
$483$	−7.04409e8	−0.284453
$484$	−2.02905e7	−0.00813457
$485$	1.35748e9	0.540305
$486$	−2.91802e9	−1.15309
$487$	−9.49824e8	−0.372642	−0.186321	−	0.982489i	$-0.559656\pi$
$487$	−9.49824e8	−0.372642	−0.186321	+	0.982489i	$0.559656\pi$
$488$	−3.68311e9	−1.43465
$489$	−3.24817e9	−1.25620
$490$	1.52527e8	0.0585682
$491$	−1.10186e8	−0.0420089	−0.0210044	−	0.999779i	$-0.506686\pi$
$491$	−1.10186e8	−0.0420089	−0.0210044	+	0.999779i	$0.506686\pi$
$492$	2.53000e7	0.00957731
$493$	2.47850e9	0.931589
$494$	−1.19929e9	−0.447588
$495$	−1.21568e9	−0.450505
$496$	1.69389e9	0.623303
$497$	4.32873e9	1.58166
$498$	2.49321e8	0.0904600
$499$	−5.66434e8	−0.204079	−0.102039	−	0.994780i	$-0.532537\pi$
$499$	−5.66434e8	−0.204079	−0.102039	+	0.994780i	$0.532537\pi$
$500$	1.03957e8	0.0371928
$501$	−1.43036e9	−0.508176
$502$	−6.43748e8	−0.227119
$503$	−3.11228e8	−0.109041	−0.0545207	−	0.998513i	$-0.517363\pi$
$503$	−3.11228e8	−0.109041	−0.0545207	+	0.998513i	$0.517363\pi$
$504$	−2.12009e9	−0.737647
$505$	−2.74732e9	−0.949269
$506$	6.56560e8	0.225293
$507$	6.30236e9	2.14771
$508$	4.73865e7	0.0160373
$509$	5.06352e8	0.170193	0.0850963	−	0.996373i	$-0.472880\pi$
$509$	5.06352e8	0.170193	0.0850963	+	0.996373i	$0.472880\pi$
$510$	−2.07679e9	−0.693262
$511$	−4.25450e9	−1.41051
$512$	−3.19184e9	−1.05098
$513$	3.44801e8	0.112761
$514$	1.38225e9	0.448970
$515$	−2.09842e9	−0.676967
$516$	−1.21619e8	−0.0389697
$517$	−5.43888e9	−1.73098
$518$	4.46463e9	1.41134
$519$	−2.45035e9	−0.769383
$520$	−3.14815e9	−0.981847
$521$	−3.18059e9	−0.985317	−0.492658	−	0.870223i	$-0.663975\pi$
$521$	−3.18059e9	−0.985317	−0.492658	+	0.870223i	$0.663975\pi$
$522$	−2.23349e9	−0.687285
$523$	−4.24695e9	−1.29814	−0.649069	−	0.760730i	$-0.724841\pi$
$523$	−4.24695e9	−1.29814	−0.649069	+	0.760730i	$0.724841\pi$
$524$	6.03014e7	0.0183091
$525$	−2.93679e9	−0.885758
$526$	−4.21429e9	−1.26262
$527$	2.00169e9	0.595743
$528$	4.65325e9	1.37575
$529$	1.48036e8	0.0434783
$530$	5.06548e8	0.147793
$531$	−2.76203e8	−0.0800567
$532$	3.88847e7	0.0111966
$533$	−1.10118e9	−0.315002
$534$	1.82278e9	0.518013
$535$	−3.81973e9	−1.07843
$536$	1.03961e9	0.291604
$537$	5.74797e9	1.60179
$538$	3.07896e9	0.852445
$539$	−4.03852e8	−0.111087
$540$	3.32018e7	0.00907367
$541$	1.47556e9	0.400651	0.200325	−	0.979729i	$-0.435800\pi$
$541$	1.47556e9	0.400651	0.200325	+	0.979729i	$0.435800\pi$
$542$	5.14123e9	1.38698
$543$	1.29818e9	0.347965
$544$	−2.62374e8	−0.0698754
$545$	4.42844e9	1.17183
$546$	8.28685e9	2.17879
$547$	−3.24569e9	−0.847914	−0.423957	−	0.905682i	$-0.639359\pi$
$547$	−3.24569e9	−0.847914	−0.423957	+	0.905682i	$0.639359\pi$
$548$	−1.03015e8	−0.0267405
$549$	3.77255e9	0.973044
$550$	2.73730e9	0.701540
$551$	1.11673e9	0.284392
$552$	1.09078e9	0.276025
$553$	4.69304e9	1.18009
$554$	−5.31327e9	−1.32763
$555$	4.25314e9	1.05605
$556$	−8.04255e7	−0.0198441
$557$	5.12858e9	1.25749	0.628744	−	0.777612i	$-0.283569\pi$
$557$	5.12858e9	1.25749	0.628744	+	0.777612i	$0.283569\pi$
$558$	−1.80381e9	−0.439513
$559$	5.29345e9	1.28173
$560$	−2.48014e9	−0.596784
$561$	5.49880e9	1.31491
$562$	7.78010e8	0.184888
$563$	−1.95702e9	−0.462185	−0.231092	−	0.972932i	$-0.574230\pi$
$563$	−1.95702e9	−0.462185	−0.231092	+	0.972932i	$0.574230\pi$
$564$	−3.31459e8	−0.0777951
$565$	−2.70717e8	−0.0631460
$566$	−3.06748e9	−0.711088
$567$	−5.52729e9	−1.27342
$568$	−6.70302e9	−1.53480
$569$	8.49926e8	0.193414	0.0967070	−	0.995313i	$-0.469169\pi$
$569$	8.49926e8	0.193414	0.0967070	+	0.995313i	$0.469169\pi$
$570$	−9.35734e8	−0.211636
$571$	3.26079e9	0.732988	0.366494	−	0.930421i	$-0.380558\pi$
$571$	3.26079e9	0.732988	0.366494	+	0.930421i	$0.380558\pi$
$572$	3.05766e8	0.0683130
$573$	−8.44229e9	−1.87465
$574$	−9.01912e8	−0.199055
$575$	6.17184e8	0.135387
$576$	3.27837e9	0.714791
$577$	2.86087e9	0.619988	0.309994	−	0.950739i	$-0.399673\pi$
$577$	2.86087e9	0.619988	0.309994	+	0.950739i	$0.399673\pi$
$578$	−7.16091e8	−0.154248
$579$	−7.68531e9	−1.64546
$580$	1.07532e8	0.0228845
$581$	3.51846e8	0.0744280
$582$	5.53324e9	1.16345
$583$	−1.34121e9	−0.280321
$584$	6.58808e9	1.36872
$585$	3.22460e9	0.665933
$586$	−7.47452e9	−1.53441
$587$	−8.05972e9	−1.64470	−0.822349	−	0.568983i	$-0.807337\pi$
$587$	−8.05972e9	−1.64470	−0.822349	+	0.568983i	$0.807337\pi$
$588$	−2.46118e7	−0.00499255
$589$	9.01893e8	0.181866
$590$	−3.35919e8	−0.0673368
$591$	7.63640e9	1.52171
$592$	−6.64982e9	−1.31730
$593$	2.06892e9	0.407429	0.203714	−	0.979030i	$-0.434699\pi$
$593$	2.06892e9	0.407429	0.203714	+	0.979030i	$0.434699\pi$
$594$	2.22068e9	0.434743
$595$	−2.93080e9	−0.570397
$596$	−3.83863e8	−0.0742702
$597$	−4.09465e9	−0.787602
$598$	−1.74153e9	−0.333026
$599$	−2.27549e9	−0.432595	−0.216298	−	0.976328i	$-0.569398\pi$
$599$	−2.27549e9	−0.432595	−0.216298	+	0.976328i	$0.569398\pi$
$600$	4.54761e9	0.859516
$601$	−4.39655e9	−0.826136	−0.413068	−	0.910700i	$-0.635543\pi$
$601$	−4.39655e9	−0.826136	−0.413068	+	0.910700i	$0.635543\pi$
$602$	4.33555e9	0.809946
$603$	−1.06485e9	−0.197779
$604$	3.03839e8	0.0561066
$605$	6.89066e8	0.126508
$606$	−1.11983e10	−2.04409
$607$	−1.55818e9	−0.282785	−0.141393	−	0.989954i	$-0.545158\pi$
$607$	−1.55818e9	−0.282785	−0.141393	+	0.989954i	$0.545158\pi$
$608$	−1.18217e8	−0.0213313
$609$	−7.71640e9	−1.38438
$610$	4.58819e9	0.818440
$611$	1.44267e10	2.55872
$612$	1.36883e8	0.0241390
$613$	2.08002e9	0.364717	0.182359	−	0.983232i	$-0.441627\pi$
$613$	2.08002e9	0.364717	0.182359	+	0.983232i	$0.441627\pi$
$614$	−2.24945e8	−0.0392182
$615$	−8.59188e8	−0.148945
$616$	6.82709e9	1.17680
$617$	1.48880e9	0.255175	0.127588	−	0.991827i	$-0.459277\pi$
$617$	1.48880e9	0.255175	0.127588	+	0.991827i	$0.459277\pi$
$618$	−8.55338e9	−1.45773
$619$	−4.92878e9	−0.835261	−0.417630	−	0.908617i	$-0.637139\pi$
$619$	−4.92878e9	−0.835261	−0.417630	+	0.908617i	$0.637139\pi$
$620$	8.68455e7	0.0146345
$621$	5.00701e8	0.0838992
$622$	−4.09119e9	−0.681684
$623$	2.57234e9	0.426206
$624$	−1.23428e10	−2.03361
$625$	4.32594e8	0.0708763
$626$	6.12145e9	0.997342
$627$	2.47758e9	0.401412
$628$	−3.15240e8	−0.0507905
$629$	−7.85816e9	−1.25905
$630$	2.64108e9	0.420813
$631$	−3.18924e9	−0.505340	−0.252670	−	0.967552i	$-0.581309\pi$
$631$	−3.18924e9	−0.505340	−0.252670	+	0.967552i	$0.581309\pi$
$632$	−7.26715e9	−1.14513
$633$	1.55114e10	2.43074
$634$	−1.77290e9	−0.276294
$635$	−1.60924e9	−0.249410
$636$	−8.17364e7	−0.0125984
$637$	1.07122e9	0.164207
$638$	7.19224e9	1.09646
$639$	6.86580e9	1.04097
$640$	3.68822e9	0.556144
$641$	3.87554e9	0.581204	0.290602	−	0.956844i	$-0.406144\pi$
$641$	3.87554e9	0.581204	0.290602	+	0.956844i	$0.406144\pi$
$642$	−1.55696e10	−2.32222
$643$	−9.82602e9	−1.45760	−0.728801	−	0.684725i	$-0.759922\pi$
$643$	−9.82602e9	−1.45760	−0.728801	+	0.684725i	$0.759922\pi$
$644$	5.64661e7	0.00833081
$645$	4.13017e9	0.606051
$646$	1.72888e9	0.252319
$647$	−3.65079e9	−0.529934	−0.264967	−	0.964258i	$-0.585361\pi$
$647$	−3.65079e9	−0.529934	−0.264967	+	0.964258i	$0.585361\pi$
$648$	8.55899e9	1.23569
$649$	8.89425e8	0.127718
$650$	−7.26072e9	−1.03701
$651$	−6.23193e9	−0.885297
$652$	2.60377e8	0.0367905
$653$	1.36014e9	0.191156	0.0955779	−	0.995422i	$-0.469530\pi$
$653$	1.36014e9	0.191156	0.0955779	+	0.995422i	$0.469530\pi$
$654$	1.80508e10	2.52333
$655$	−2.04783e9	−0.284741
$656$	1.34335e9	0.185791
$657$	−6.74807e9	−0.928326
$658$	1.18161e10	1.61689
$659$	−4.55486e9	−0.619978	−0.309989	−	0.950740i	$-0.600325\pi$
$659$	−4.55486e9	−0.619978	−0.309989	+	0.950740i	$0.600325\pi$
$660$	2.38572e8	0.0323009
$661$	7.93234e9	1.06831	0.534154	−	0.845387i	$-0.320630\pi$
$661$	7.93234e9	1.06831	0.534154	+	0.845387i	$0.320630\pi$
$662$	−2.66235e9	−0.356667
$663$	−1.45856e10	−1.94369
$664$	−5.44833e8	−0.0722229
$665$	−1.32052e9	−0.174128
$666$	7.08135e9	0.928873
$667$	1.62165e9	0.211600
$668$	1.14659e8	0.0148830
$669$	−3.49420e8	−0.0451187
$670$	−1.29508e9	−0.166354
$671$	−1.21483e10	−1.55234
$672$	8.16858e8	0.103837
$673$	−1.94604e9	−0.246093	−0.123046	−	0.992401i	$-0.539266\pi$
$673$	−1.94604e9	−0.246093	−0.123046	+	0.992401i	$0.539266\pi$
$674$	1.50218e10	1.88978
$675$	2.08750e9	0.261254
$676$	−5.05203e8	−0.0629002
$677$	−1.37845e10	−1.70738	−0.853692	−	0.520778i	$-0.825642\pi$
$677$	−1.37845e10	−1.70738	−0.853692	+	0.520778i	$0.825642\pi$
$678$	−1.10347e9	−0.135974
$679$	7.80859e9	0.957257
$680$	4.53834e9	0.553497
$681$	3.27589e9	0.397479
$682$	5.80860e9	0.701175
$683$	7.27273e9	0.873424	0.436712	−	0.899601i	$-0.356143\pi$
$683$	7.27273e9	0.873424	0.436712	+	0.899601i	$0.356143\pi$
$684$	6.16750e7	0.00736907
$685$	3.49839e9	0.415864
$686$	−7.82355e9	−0.925272
$687$	−7.86620e9	−0.925585
$688$	−6.45756e9	−0.755978
$689$	3.55757e9	0.414368
$690$	−1.35882e9	−0.157467
$691$	1.50396e10	1.73405	0.867027	−	0.498261i	$-0.166028\pi$
$691$	1.50396e10	1.73405	0.867027	+	0.498261i	$0.166028\pi$
$692$	1.96423e8	0.0225331
$693$	−6.99288e9	−0.798160
$694$	4.54006e8	0.0515589
$695$	2.73125e9	0.308613
$696$	1.19488e10	1.34336
$697$	1.58745e9	0.177576
$698$	1.38608e10	1.54274
$699$	2.76727e9	0.306465
$700$	2.35416e8	0.0259413
$701$	8.30205e7	0.00910274	0.00455137	−	0.999990i	$-0.498551\pi$
$701$	8.30205e7	0.00910274	0.00455137	+	0.999990i	$0.498551\pi$
$702$	−5.89038e9	−0.642633
$703$	−3.54063e9	−0.384359
$704$	−1.05569e10	−1.14034
$705$	1.12563e10	1.20986
$706$	1.06612e10	1.14022
$707$	−1.58033e10	−1.68182
$708$	5.42037e7	0.00574001
$709$	5.57061e9	0.587004	0.293502	−	0.955959i	$-0.405179\pi$
$709$	5.57061e9	0.587004	0.293502	+	0.955959i	$0.405179\pi$
$710$	8.35020e9	0.875574
$711$	7.44363e9	0.776678
$712$	−3.98326e9	−0.413579
$713$	1.30968e9	0.135317
$714$	−1.19462e10	−1.22825
$715$	−1.03838e10	−1.06239
$716$	−4.60763e8	−0.0469118
$717$	−2.20225e10	−2.23126
$718$	8.81323e9	0.888586
$719$	6.00718e9	0.602726	0.301363	−	0.953510i	$-0.402558\pi$
$719$	6.00718e9	0.602726	0.301363	+	0.953510i	$0.402558\pi$
$720$	−3.93375e9	−0.392774
$721$	−1.20707e10	−1.19938
$722$	−9.13961e9	−0.903749
$723$	−8.15625e9	−0.802613
$724$	−1.04063e8	−0.0101909
$725$	6.76090e9	0.658903
$726$	2.80870e9	0.272413
$727$	1.41144e10	1.36236	0.681182	−	0.732114i	$-0.261466\pi$
$727$	1.41144e10	1.36236	0.681182	+	0.732114i	$0.261466\pi$
$728$	−1.81090e10	−1.73954
$729$	−3.29441e9	−0.314942
$730$	−8.20702e9	−0.780828
$731$	−7.63097e9	−0.722552
$732$	−7.40349e8	−0.0697666
$733$	1.56763e10	1.47021	0.735104	−	0.677954i	$-0.237133\pi$
$733$	1.56763e10	1.47021	0.735104	+	0.677954i	$0.237133\pi$
$734$	8.93124e9	0.833634
$735$	8.35815e8	0.0776433
$736$	−1.71668e8	−0.0158714
$737$	3.42903e9	0.315526
$738$	−1.43052e9	−0.131008
$739$	5.76779e9	0.525720	0.262860	−	0.964834i	$-0.415334\pi$
$739$	5.76779e9	0.525720	0.262860	+	0.964834i	$0.415334\pi$
$740$	−3.40936e8	−0.0309287
$741$	−6.57180e9	−0.593364
$742$	2.91379e9	0.261845
$743$	1.11257e10	0.995096	0.497548	−	0.867436i	$-0.334234\pi$
$743$	1.11257e10	0.995096	0.497548	+	0.867436i	$0.334234\pi$
$744$	9.65012e9	0.859068
$745$	1.30360e10	1.15504
$746$	1.32393e10	1.16756
$747$	5.58063e8	0.0489848
$748$	−4.40789e8	−0.0385101
$749$	−2.19720e10	−1.91066
$750$	−1.43902e10	−1.24552
$751$	2.96634e9	0.255553	0.127777	−	0.991803i	$-0.459216\pi$
$751$	2.96634e9	0.255553	0.127777	+	0.991803i	$0.459216\pi$
$752$	−1.75994e10	−1.50916
$753$	−3.52759e9	−0.301090
$754$	−1.90775e10	−1.62077
$755$	−1.03184e10	−0.872562
$756$	1.90985e8	0.0160758
$757$	1.45781e10	1.22142	0.610710	−	0.791854i	$-0.290884\pi$
$757$	1.45781e10	1.22142	0.610710	+	0.791854i	$0.290884\pi$
$758$	−6.82341e9	−0.569062
$759$	3.59779e9	0.298669
$760$	2.04482e9	0.168970
$761$	−1.30095e10	−1.07007	−0.535036	−	0.844829i	$-0.679702\pi$
$761$	−1.30095e10	−1.07007	−0.535036	+	0.844829i	$0.679702\pi$
$762$	−6.55943e9	−0.537061
$763$	2.54735e10	2.07612
$764$	6.76742e8	0.0549031
$765$	−4.64855e9	−0.375407
$766$	1.16539e10	0.936850
$767$	−2.35921e9	−0.188792
$768$	−1.86187e9	−0.148315
$769$	1.68664e9	0.133746	0.0668728	−	0.997762i	$-0.478698\pi$
$769$	1.68664e9	0.133746	0.0668728	+	0.997762i	$0.478698\pi$
$770$	−8.50476e9	−0.671343
$771$	7.57443e9	0.595196
$772$	6.16062e8	0.0481907
$773$	−1.63352e10	−1.27203	−0.636013	−	0.771678i	$-0.719418\pi$
$773$	−1.63352e10	−1.27203	−0.636013	+	0.771678i	$0.719418\pi$
$774$	6.87662e9	0.533067
$775$	5.46025e9	0.421363
$776$	−1.20916e10	−0.928896
$777$	2.44651e10	1.87100
$778$	−1.09674e10	−0.834982
$779$	7.15252e8	0.0542098
$780$	−6.32815e8	−0.0477469
$781$	−2.21091e10	−1.66071
$782$	2.51057e9	0.187737
$783$	5.48489e9	0.408321
$784$	−1.30680e9	−0.0968510
$785$	1.07055e10	0.789886
$786$	−8.34717e9	−0.613141
$787$	−1.13744e10	−0.831795	−0.415898	−	0.909411i	$-0.636533\pi$
$787$	−1.13744e10	−0.831795	−0.415898	+	0.909411i	$0.636533\pi$
$788$	−6.12141e8	−0.0445666
$789$	−2.30933e10	−1.67385
$790$	9.05296e9	0.653275
$791$	−1.55723e9	−0.111876
$792$	1.08285e10	0.774513
$793$	3.22236e10	2.29466
$794$	1.99402e9	0.141370
$795$	2.77576e9	0.195929
$796$	3.28231e8	0.0230666
$797$	1.05659e10	0.739267	0.369633	−	0.929178i	$-0.379483\pi$
$797$	1.05659e10	0.739267	0.369633	+	0.929178i	$0.379483\pi$
$798$	−5.38258e9	−0.374956
$799$	−2.07974e10	−1.44243
$800$	−7.15709e8	−0.0494221
$801$	4.07999e9	0.280508
$802$	−2.42975e10	−1.66323
$803$	2.17300e10	1.48100
$804$	2.08973e8	0.0141806
$805$	−1.91759e9	−0.129560
$806$	−1.54074e10	−1.03647
$807$	1.68720e10	1.13008
$808$	2.44713e10	1.63199
$809$	1.96297e10	1.30345	0.651724	−	0.758457i	$-0.274046\pi$
$809$	1.96297e10	1.30345	0.651724	+	0.758457i	$0.274046\pi$
$810$	−1.06622e10	−0.704938
$811$	−2.53973e10	−1.67191	−0.835957	−	0.548795i	$-0.815087\pi$
$811$	−2.53973e10	−1.67191	−0.835957	+	0.548795i	$0.815087\pi$
$812$	6.18554e8	0.0405445
$813$	2.81727e10	1.83871
$814$	−2.28033e10	−1.48187
$815$	−8.84238e9	−0.572160
$816$	1.77933e10	1.14641
$817$	−3.43826e9	−0.220578
$818$	9.22461e9	0.589266
$819$	1.85487e10	1.17983
$820$	6.88734e7	0.00436217
$821$	3.97732e9	0.250836	0.125418	−	0.992104i	$-0.459973\pi$
$821$	3.97732e9	0.250836	0.125418	+	0.992104i	$0.459973\pi$
$822$	1.42598e10	0.895492
$823$	−9.82551e9	−0.614407	−0.307203	−	0.951644i	$-0.599393\pi$
$823$	−9.82551e9	−0.614407	−0.307203	+	0.951644i	$0.599393\pi$
$824$	1.86914e10	1.16385
$825$	1.49998e10	0.930026
$826$	−1.93229e9	−0.119300
$827$	1.33644e10	0.821636	0.410818	−	0.911717i	$-0.365243\pi$
$827$	1.33644e10	0.821636	0.410818	+	0.911717i	$0.365243\pi$
$828$	8.95609e7	0.00548293
$829$	−1.05176e10	−0.641173	−0.320586	−	0.947219i	$-0.603880\pi$
$829$	−1.05176e10	−0.641173	−0.320586	+	0.947219i	$0.603880\pi$
$830$	6.78718e8	0.0412018
$831$	−2.91155e10	−1.76003
$832$	2.80025e10	1.68564
$833$	−1.54426e9	−0.0925687
$834$	1.11328e10	0.664545
$835$	−3.89383e9	−0.231459
$836$	−1.98605e8	−0.0117562
$837$	4.42971e9	0.261118
$838$	−1.86423e10	−1.09433
$839$	1.44529e10	0.844865	0.422433	−	0.906394i	$-0.361176\pi$
$839$	1.44529e10	0.844865	0.422433	+	0.906394i	$0.361176\pi$
$840$	−1.41294e10	−0.822519
$841$	5.14370e8	0.0298188
$842$	1.97609e9	0.114081
$843$	4.26331e9	0.245104
$844$	−1.24341e9	−0.0711895
$845$	1.71567e10	0.978215
$846$	1.87414e10	1.06416
$847$	3.96368e9	0.224133
$848$	−4.33994e9	−0.244398
$849$	−1.68091e10	−0.942684
$850$	1.04670e10	0.584594
$851$	−5.14150e9	−0.285980
$852$	−1.34739e9	−0.0746368
$853$	−2.49192e10	−1.37471	−0.687356	−	0.726321i	$-0.741229\pi$
$853$	−2.49192e10	−1.37471	−0.687356	+	0.726321i	$0.741229\pi$
$854$	2.63924e10	1.45003
$855$	−2.09448e9	−0.114603
$856$	3.40236e10	1.85405
$857$	1.98477e10	1.07715	0.538577	−	0.842576i	$-0.318962\pi$
$857$	1.98477e10	1.07715	0.538577	+	0.842576i	$0.318962\pi$
$858$	−4.23254e10	−2.28768
$859$	−2.59097e10	−1.39472	−0.697359	−	0.716722i	$-0.745642\pi$
$859$	−2.59097e10	−1.39472	−0.697359	+	0.716722i	$0.745642\pi$
$860$	−3.31078e8	−0.0177495
$861$	−4.94227e9	−0.263885
$862$	6.46298e9	0.343683
$863$	−2.02220e9	−0.107099	−0.0535496	−	0.998565i	$-0.517054\pi$
$863$	−2.02220e9	−0.107099	−0.0535496	+	0.998565i	$0.517054\pi$
$864$	−5.80631e8	−0.0306268
$865$	−6.67050e9	−0.350431
$866$	−2.27459e10	−1.19012
$867$	−3.92401e9	−0.204486
$868$	4.99557e8	0.0259278
$869$	−2.39698e10	−1.23907
$870$	−1.48851e10	−0.766362
$871$	−9.09554e9	−0.466407
$872$	−3.94457e10	−2.01461
$873$	1.23852e10	0.630019
$874$	1.13118e9	0.0573116
$875$	−2.03076e10	−1.02478
$876$	1.32428e9	0.0665604
$877$	9.24606e9	0.462869	0.231435	−	0.972850i	$-0.425658\pi$
$877$	9.24606e9	0.462869	0.231435	+	0.972850i	$0.425658\pi$
$878$	3.61935e7	0.00180468
$879$	−4.09586e10	−2.03416
$880$	1.26674e10	0.626610
$881$	−1.84268e10	−0.907893	−0.453946	−	0.891029i	$-0.649984\pi$
$881$	−1.84268e10	−0.907893	−0.453946	+	0.891029i	$0.649984\pi$
$882$	1.39161e9	0.0682931
$883$	−1.07940e10	−0.527618	−0.263809	−	0.964575i	$-0.584979\pi$
$883$	−1.07940e10	−0.527618	−0.263809	+	0.964575i	$0.584979\pi$
$884$	1.16920e9	0.0569253
$885$	−1.84076e9	−0.0892678
$886$	−2.05483e10	−0.992564
$887$	−1.27893e10	−0.615339	−0.307670	−	0.951493i	$-0.599549\pi$
$887$	−1.27893e10	−0.615339	−0.307670	+	0.951493i	$0.599549\pi$
$888$	−3.78842e10	−1.81557
$889$	−9.25677e9	−0.441879
$890$	4.96209e9	0.235939
$891$	2.82308e10	1.33706
$892$	2.80098e7	0.00132140
$893$	−9.37060e9	−0.440339
$894$	5.31359e10	2.48718
$895$	1.56475e10	0.729565
$896$	2.12156e10	0.985319
$897$	−9.54320e9	−0.441490
$898$	−1.02824e10	−0.473835
$899$	1.43468e10	0.658560
$900$	3.73393e8	0.0170733
$901$	−5.12855e9	−0.233592
$902$	4.60655e9	0.209003
$903$	2.37578e10	1.07374
$904$	2.41137e9	0.108561
$905$	3.53398e9	0.158487
$906$	−4.20587e10	−1.87891
$907$	1.90825e10	0.849199	0.424599	−	0.905381i	$-0.360415\pi$
$907$	1.90825e10	0.849199	0.424599	+	0.905381i	$0.360415\pi$
$908$	−2.62598e8	−0.0116410
$909$	−2.50656e10	−1.10689
$910$	2.25590e10	0.992373
$911$	5.20460e9	0.228072	0.114036	−	0.993477i	$-0.463622\pi$
$911$	5.20460e9	0.228072	0.114036	+	0.993477i	$0.463622\pi$
$912$	8.01706e9	0.349972
$913$	−1.79707e9	−0.0781477
$914$	1.57233e10	0.681132
$915$	2.51422e10	1.08500
$916$	6.30562e8	0.0271078
$917$	−1.17797e10	−0.504475
$918$	8.49150e9	0.362272
$919$	−1.65425e10	−0.703066	−0.351533	−	0.936176i	$-0.614339\pi$
$919$	−1.65425e10	−0.703066	−0.351533	+	0.936176i	$0.614339\pi$
$920$	2.96938e9	0.125721
$921$	−1.23265e9	−0.0519912
$922$	2.35129e10	0.987980
$923$	5.86448e10	2.45484
$924$	1.37232e9	0.0572275
$925$	−2.14357e10	−0.890515
$926$	2.26276e10	0.936485
$927$	−1.91453e10	−0.789374
$928$	−1.88052e9	−0.0772433
$929$	2.36746e10	0.968786	0.484393	−	0.874850i	$-0.339040\pi$
$929$	2.36746e10	0.968786	0.484393	+	0.874850i	$0.339040\pi$
$930$	−1.20215e10	−0.490082
$931$	−6.95794e8	−0.0282590
$932$	−2.21827e8	−0.00897549
$933$	−2.24187e10	−0.903703
$934$	−1.41892e10	−0.569826
$935$	1.49692e10	0.598904
$936$	−2.87226e10	−1.14488
$937$	3.19714e10	1.26962	0.634809	−	0.772669i	$-0.281079\pi$
$937$	3.19714e10	1.26962	0.634809	+	0.772669i	$0.281079\pi$
$938$	−7.44962e9	−0.294730
$939$	3.35441e10	1.32217
$940$	−9.02317e8	−0.0354333
$941$	3.81681e10	1.49326	0.746632	−	0.665237i	$-0.231669\pi$
$941$	3.81681e10	1.49326	0.746632	+	0.665237i	$0.231669\pi$
$942$	4.36368e10	1.70088
$943$	1.03865e9	0.0403346
$944$	2.87804e9	0.111351
$945$	−6.48584e9	−0.250009
$946$	−2.21440e10	−0.850426
$947$	−3.14336e10	−1.20273	−0.601366	−	0.798974i	$-0.705377\pi$
$947$	−3.14336e10	−1.20273	−0.601366	+	0.798974i	$0.705377\pi$
$948$	−1.46078e9	−0.0556873
$949$	−5.76392e10	−2.18920
$950$	4.71607e9	0.178463
$951$	−9.71505e9	−0.366280
$952$	2.61057e10	0.980631
$953$	−1.41899e10	−0.531072	−0.265536	−	0.964101i	$-0.585549\pi$
$953$	−1.41899e10	−0.531072	−0.265536	+	0.964101i	$0.585549\pi$
$954$	4.62157e9	0.172334
$955$	−2.29821e10	−0.853844
$956$	1.76535e9	0.0653473
$957$	3.94118e10	1.45356
$958$	5.28704e9	0.194282
$959$	2.01236e10	0.736786
$960$	2.18487e10	0.797033
$961$	−1.59259e10	−0.578856
$962$	6.04860e10	2.19049
$963$	−3.48498e10	−1.25750
$964$	6.53813e8	0.0235063
$965$	−2.09214e10	−0.749455
$966$	−7.81627e9	−0.278984
$967$	−4.79184e10	−1.70416	−0.852078	−	0.523415i	$-0.824658\pi$
$967$	−4.79184e10	−1.70416	−0.852078	+	0.523415i	$0.824658\pi$
$968$	−6.13775e9	−0.217493
$969$	9.47383e9	0.334497
$970$	1.50629e10	0.529918
$971$	3.34597e10	1.17288	0.586442	−	0.809991i	$-0.300528\pi$
$971$	3.34597e10	1.17288	0.586442	+	0.809991i	$0.300528\pi$
$972$	1.28178e9	0.0447695
$973$	1.57108e10	0.546769
$974$	−1.05394e10	−0.365478
$975$	−3.97871e10	−1.37476
$976$	−3.93101e10	−1.35341
$977$	−6.62901e9	−0.227414	−0.113707	−	0.993514i	$-0.536273\pi$
$977$	−6.62901e9	−0.227414	−0.113707	+	0.993514i	$0.536273\pi$
$978$	−3.60424e10	−1.23205
$979$	−1.31383e10	−0.447507
$980$	−6.69997e7	−0.00227395
$981$	4.04036e10	1.36640
$982$	−1.22265e9	−0.0412013
$983$	−5.89365e10	−1.97900	−0.989502	−	0.144519i	$-0.953836\pi$
$983$	−5.89365e10	−1.97900	−0.989502	+	0.144519i	$0.953836\pi$
$984$	7.65309e9	0.256067
$985$	2.07883e10	0.693094
$986$	2.75019e10	0.913679
$987$	6.47492e10	2.14350
$988$	5.26802e8	0.0173779
$989$	−4.99285e9	−0.164120
$990$	−1.34894e10	−0.441845
$991$	4.22516e10	1.37907	0.689533	−	0.724254i	$-0.257816\pi$
$991$	4.22516e10	1.37907	0.689533	+	0.724254i	$0.257816\pi$
$992$	−1.51875e9	−0.0493964
$993$	−1.45891e10	−0.472830
$994$	4.80325e10	1.55125
$995$	−1.11467e10	−0.358729
$996$	−1.09518e8	−0.00351218
$997$	−4.76627e10	−1.52316	−0.761581	−	0.648070i	$-0.775576\pi$
$997$	−4.76627e10	−1.52316	−0.761581	+	0.648070i	$0.775576\pi$
$998$	−6.28527e9	−0.200155
$999$	−1.73901e10	−0.551851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.8.a.b.1.6	✓	8
3.2	odd	2		207.8.a.f.1.3		8
4.3	odd	2		368.8.a.h.1.3		8
5.4	even	2		575.8.a.b.1.3		8
23.22	odd	2		529.8.a.c.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.a.b.1.6	✓	8	1.1	even	1	trivial
207.8.a.f.1.3		8	3.2	odd	2
368.8.a.h.1.3		8	4.3	odd	2
529.8.a.c.1.6		8	23.22	odd	2
575.8.a.b.1.3		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)

\(f(q)\)	\(=\)	\(q+11.0962 q^{2} +60.8046 q^{3} -4.87416 q^{4} +165.526 q^{5} +674.701 q^{6} +952.148 q^{7} -1474.40 q^{8} +1510.20 q^{9} +O(q^{10})\)	\(q+11.0962 q^{2} +60.8046 q^{3} -4.87416 q^{4} +165.526 q^{5} +674.701 q^{6} +952.148 q^{7} -1474.40 q^{8} +1510.20 q^{9} +1836.71 q^{10} -4863.13 q^{11} -296.371 q^{12} +12899.5 q^{13} +10565.2 q^{14} +10064.8 q^{15} -15736.4 q^{16} -18595.8 q^{17} +16757.5 q^{18} -8378.65 q^{19} -806.801 q^{20} +57895.0 q^{21} -53962.3 q^{22} -12167.0 q^{23} -89650.3 q^{24} -50726.1 q^{25} +143136. q^{26} -41152.4 q^{27} -4640.92 q^{28} -133283. q^{29} +111681. q^{30} -107642. q^{31} +14109.3 q^{32} -295701. q^{33} -206343. q^{34} +157606. q^{35} -7360.97 q^{36} +422577. q^{37} -92971.3 q^{38} +784351. q^{39} -244052. q^{40} -85366.0 q^{41} +642415. q^{42} +410360. q^{43} +23703.7 q^{44} +249978. q^{45} -135008. q^{46} +1.11839e6 q^{47} -956843. q^{48} +83043.7 q^{49} -562867. q^{50} -1.13071e6 q^{51} -62874.3 q^{52} +275790. q^{53} -456635. q^{54} -804976. q^{55} -1.40385e6 q^{56} -509461. q^{57} -1.47893e6 q^{58} -182891. q^{59} -49057.2 q^{60} +2.49804e6 q^{61} -1.19442e6 q^{62} +1.43794e6 q^{63} +2.17081e6 q^{64} +2.13521e6 q^{65} -3.28116e6 q^{66} -705106. q^{67} +90638.9 q^{68} -739810. q^{69} +1.74882e6 q^{70} +4.54627e6 q^{71} -2.22664e6 q^{72} -4.46832e6 q^{73} +4.68901e6 q^{74} -3.08438e6 q^{75} +40838.9 q^{76} -4.63042e6 q^{77} +8.70332e6 q^{78} +4.92889e6 q^{79} -2.60478e6 q^{80} -5.80507e6 q^{81} -947239. q^{82} +369529. q^{83} -282190. q^{84} -3.07809e6 q^{85} +4.55344e6 q^{86} -8.10420e6 q^{87} +7.17020e6 q^{88} +2.70162e6 q^{89} +2.77381e6 q^{90} +1.22823e7 q^{91} +59303.9 q^{92} -6.54512e6 q^{93} +1.24099e7 q^{94} -1.38689e6 q^{95} +857910. q^{96} +8.20102e6 q^{97} +921470. q^{98} -7.34432e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more