LMFDB - Embedded newform 23.8.a.b.1.8

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			1.8
Root			$19.9556$ 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)

$f(q)$	$=$	$q+19.9556 q^{2} -76.4323 q^{3} +270.224 q^{4} +522.229 q^{5} -1525.25 q^{6} +653.513 q^{7} +2838.16 q^{8} +3654.90 q^{9} +O(q^{10})$	\(q+19.9556 q^{2} -76.4323 q^{3} +270.224 q^{4} +522.229 q^{5} -1525.25 q^{6} +653.513 q^{7} +2838.16 q^{8} +3654.90 q^{9} +10421.4 q^{10} -4337.65 q^{11} -20653.9 q^{12} -1039.41 q^{13} +13041.2 q^{14} -39915.2 q^{15} +22048.4 q^{16} -4916.44 q^{17} +72935.5 q^{18} -16566.8 q^{19} +141119. q^{20} -49949.5 q^{21} -86560.2 q^{22} -12167.0 q^{23} -216927. q^{24} +194598. q^{25} -20741.9 q^{26} -112195. q^{27} +176595. q^{28} -105525. q^{29} -796530. q^{30} +6798.84 q^{31} +76703.4 q^{32} +331537. q^{33} -98110.2 q^{34} +341284. q^{35} +987642. q^{36} -407252. q^{37} -330599. q^{38} +79444.2 q^{39} +1.48217e6 q^{40} +614719. q^{41} -996770. q^{42} -607028. q^{43} -1.17214e6 q^{44} +1.90869e6 q^{45} -242799. q^{46} -907295. q^{47} -1.68521e6 q^{48} -396464. q^{49} +3.88332e6 q^{50} +375775. q^{51} -280872. q^{52} +881135. q^{53} -2.23891e6 q^{54} -2.26525e6 q^{55} +1.85478e6 q^{56} +1.26624e6 q^{57} -2.10581e6 q^{58} +1.60033e6 q^{59} -1.07860e7 q^{60} +1.21442e6 q^{61} +135675. q^{62} +2.38852e6 q^{63} -1.29154e6 q^{64} -542808. q^{65} +6.61600e6 q^{66} +783456. q^{67} -1.32854e6 q^{68} +929952. q^{69} +6.81050e6 q^{70} +354699. q^{71} +1.03732e7 q^{72} +2.41783e6 q^{73} -8.12693e6 q^{74} -1.48736e7 q^{75} -4.47674e6 q^{76} -2.83471e6 q^{77} +1.58535e6 q^{78} -4.43829e6 q^{79} +1.15143e7 q^{80} +582055. q^{81} +1.22671e7 q^{82} +8.13014e6 q^{83} -1.34976e7 q^{84} -2.56751e6 q^{85} -1.21136e7 q^{86} +8.06552e6 q^{87} -1.23110e7 q^{88} +4.11109e6 q^{89} +3.80891e7 q^{90} -679265. q^{91} -3.28782e6 q^{92} -519651. q^{93} -1.81056e7 q^{94} -8.65164e6 q^{95} -5.86262e6 q^{96} +3.08615e6 q^{97} -7.91165e6 q^{98} -1.58537e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	19.9556	1.76384	0.881919	−	0.471401i	$-0.156251\pi$
$2$	19.9556	1.76384	0.881919	+	0.471401i	$0.156251\pi$
$3$	−76.4323	−1.63438	−0.817189	−	0.576369i	$-0.804469\pi$
$3$	−76.4323	−1.63438	−0.817189	+	0.576369i	$0.804469\pi$
$4$	270.224	2.11113
$5$	522.229	1.86838	0.934192	−	0.356771i	$-0.116122\pi$
$5$	522.229	1.86838	0.934192	+	0.356771i	$0.116122\pi$
$6$	−1525.25	−2.88278
$7$	653.513	0.720130	0.360065	−	0.932927i	$-0.382754\pi$
$7$	653.513	0.720130	0.360065	+	0.932927i	$0.382754\pi$
$8$	2838.16	1.95985
$9$	3654.90	1.67119
$10$	10421.4	3.29553
$11$	−4337.65	−0.982608	−0.491304	−	0.870988i	$-0.663480\pi$
$11$	−4337.65	−0.982608	−0.491304	+	0.870988i	$0.663480\pi$
$12$	−20653.9	−3.45038
$13$	−1039.41	−0.131215	−0.0656075	−	0.997846i	$-0.520899\pi$
$13$	−1039.41	−0.131215	−0.0656075	+	0.997846i	$0.520899\pi$
$14$	13041.2	1.27019
$15$	−39915.2	−3.05365
$16$	22048.4	1.34573
$17$	−4916.44	−0.242705	−0.121353	−	0.992609i	$-0.538723\pi$
$17$	−4916.44	−0.242705	−0.121353	+	0.992609i	$0.538723\pi$
$18$	72935.5	2.94771
$19$	−16566.8	−0.554115	−0.277057	−	0.960853i	$-0.589359\pi$
$19$	−16566.8	−0.554115	−0.277057	+	0.960853i	$0.589359\pi$
$20$	141119.	3.94439
$21$	−49949.5	−1.17697
$22$	−86560.2	−1.73316
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	−216927.	−3.20313
$25$	194598.	2.49086
$26$	−20741.9	−0.231442
$27$	−112195.	−1.09698
$28$	176595.	1.52029
$29$	−105525.	−0.803457	−0.401728	−	0.915759i	$-0.631590\pi$
$29$	−105525.	−0.803457	−0.401728	+	0.915759i	$0.631590\pi$
$30$	−796530.	−5.38614
$31$	6798.84	0.0409891	0.0204946	−	0.999790i	$-0.493476\pi$
$31$	6798.84	0.0409891	0.0204946	+	0.999790i	$0.493476\pi$
$32$	76703.4	0.413799
$33$	331537.	1.60595
$34$	−98110.2	−0.428093
$35$	341284.	1.34548
$36$	987642.	3.52810
$37$	−407252.	−1.32177	−0.660886	−	0.750486i	$-0.729820\pi$
$37$	−407252.	−1.32177	−0.660886	+	0.750486i	$0.729820\pi$
$38$	−330599.	−0.977369
$39$	79444.2	0.214455
$40$	1.48217e6	3.66175
$41$	614719.	1.39294	0.696471	−	0.717585i	$-0.254752\pi$
$41$	614719.	1.39294	0.696471	+	0.717585i	$0.254752\pi$
$42$	−996770.	−2.07598
$43$	−607028.	−1.16431	−0.582156	−	0.813077i	$-0.697791\pi$
$43$	−607028.	−1.16431	−0.582156	+	0.813077i	$0.697791\pi$
$44$	−1.17214e6	−2.07441
$45$	1.90869e6	3.12243
$46$	−242799.	−0.367786
$47$	−907295.	−1.27469	−0.637347	−	0.770577i	$-0.719968\pi$
$47$	−907295.	−1.27469	−0.637347	+	0.770577i	$0.719968\pi$
$48$	−1.68521e6	−2.19943
$49$	−396464.	−0.481412
$50$	3.88332e6	4.39347
$51$	375775.	0.396672
$52$	−280872.	−0.277011
$53$	881135.	0.812974	0.406487	−	0.913657i	$-0.366754\pi$
$53$	881135.	0.812974	0.406487	+	0.913657i	$0.366754\pi$
$54$	−2.23891e6	−1.93490
$55$	−2.26525e6	−1.83589
$56$	1.85478e6	1.41135
$57$	1.26624e6	0.905633
$58$	−2.10581e6	−1.41717
$59$	1.60033e6	1.01444	0.507221	−	0.861816i	$-0.330673\pi$
$59$	1.60033e6	1.01444	0.507221	+	0.861816i	$0.330673\pi$
$60$	−1.07860e7	−6.44663
$61$	1.21442e6	0.685040	0.342520	−	0.939511i	$-0.388720\pi$
$61$	1.21442e6	0.685040	0.342520	+	0.939511i	$0.388720\pi$
$62$	135675.	0.0722982
$63$	2.38852e6	1.20348
$64$	−1.29154e6	−0.615853
$65$	−542808.	−0.245160
$66$	6.61600e6	2.83264
$67$	783456.	0.318239	0.159119	−	0.987259i	$-0.449135\pi$
$67$	783456.	0.318239	0.159119	+	0.987259i	$0.449135\pi$
$68$	−1.32854e6	−0.512381
$69$	929952.	0.340791
$70$	6.81050e6	2.37321
$71$	354699.	0.117613	0.0588066	−	0.998269i	$-0.481270\pi$
$71$	354699.	0.117613	0.0588066	+	0.998269i	$0.481270\pi$
$72$	1.03732e7	3.27528
$73$	2.41783e6	0.727439	0.363719	−	0.931509i	$-0.381507\pi$
$73$	2.41783e6	0.727439	0.363719	+	0.931509i	$0.381507\pi$
$74$	−8.12693e6	−2.33139
$75$	−1.48736e7	−4.07100
$76$	−4.47674e6	−1.16981
$77$	−2.83471e6	−0.707606
$78$	1.58535e6	0.378264
$79$	−4.43829e6	−1.01279	−0.506397	−	0.862301i	$-0.669023\pi$
$79$	−4.43829e6	−1.01279	−0.506397	+	0.862301i	$0.669023\pi$
$80$	1.15143e7	2.51434
$81$	582055.	0.121693
$82$	1.22671e7	2.45692
$83$	8.13014e6	1.56072	0.780360	−	0.625331i	$-0.215036\pi$
$83$	8.13014e6	1.56072	0.780360	+	0.625331i	$0.215036\pi$
$84$	−1.34976e7	−2.48472
$85$	−2.56751e6	−0.453467
$86$	−1.21136e7	−2.05366
$87$	8.06552e6	1.31315
$88$	−1.23110e7	−1.92576
$89$	4.11109e6	0.618147	0.309074	−	0.951038i	$-0.399981\pi$
$89$	4.11109e6	0.618147	0.309074	+	0.951038i	$0.399981\pi$
$90$	3.80891e7	5.50746
$91$	−679265.	−0.0944918
$92$	−3.28782e6	−0.440200
$93$	−519651.	−0.0669917
$94$	−1.81056e7	−2.24836
$95$	−8.65164e6	−1.03530
$96$	−5.86262e6	−0.676304
$97$	3.08615e6	0.343333	0.171667	−	0.985155i	$-0.445085\pi$
$97$	3.08615e6	0.343333	0.171667	+	0.985155i	$0.445085\pi$
$98$	−7.91165e6	−0.849134
$99$	−1.58537e7	−1.64213
$100$	5.25851e7	5.25851
$101$	−1.64582e7	−1.58949	−0.794746	−	0.606942i	$-0.792396\pi$
$101$	−1.64582e7	−1.58949	−0.794746	+	0.606942i	$0.792396\pi$
$102$	7.49879e6	0.699666
$103$	1.69239e7	1.52605	0.763026	−	0.646368i	$-0.223713\pi$
$103$	1.69239e7	1.52605	0.763026	+	0.646368i	$0.223713\pi$
$104$	−2.95000e6	−0.257161
$105$	−2.60851e7	−2.19902
$106$	1.75835e7	1.43396
$107$	−2.12029e6	−0.167321	−0.0836607	−	0.996494i	$-0.526661\pi$
$107$	−2.12029e6	−0.167321	−0.0836607	+	0.996494i	$0.526661\pi$
$108$	−3.03178e7	−2.31587
$109$	6.43760e6	0.476136	0.238068	−	0.971248i	$-0.423486\pi$
$109$	6.43760e6	0.476136	0.238068	+	0.971248i	$0.423486\pi$
$110$	−4.52043e7	−3.23821
$111$	3.11272e7	2.16028
$112$	1.44089e7	0.969099
$113$	−8.54511e6	−0.557113	−0.278557	−	0.960420i	$-0.589856\pi$
$113$	−8.54511e6	−0.557113	−0.278557	+	0.960420i	$0.589856\pi$
$114$	2.52684e7	1.59739
$115$	−6.35396e6	−0.389585
$116$	−2.85154e7	−1.69620
$117$	−3.79892e6	−0.219285
$118$	3.19354e7	1.78931
$119$	−3.21295e6	−0.174779
$120$	−1.13286e8	−5.98468
$121$	−671952.	−0.0344818
$122$	2.42345e7	1.20830
$123$	−4.69844e7	−2.27659
$124$	1.83721e6	0.0865332
$125$	6.08257e7	2.78549
$126$	4.76643e7	2.12274
$127$	3.53558e7	1.53161	0.765804	−	0.643074i	$-0.222341\pi$
$127$	3.53558e7	1.53161	0.765804	+	0.643074i	$0.222341\pi$
$128$	−3.55914e7	−1.50006
$129$	4.63966e7	1.90293
$130$	−1.08320e7	−0.432422
$131$	−2.98530e7	−1.16022	−0.580108	−	0.814539i	$-0.696990\pi$
$131$	−2.98530e7	−1.16022	−0.580108	+	0.814539i	$0.696990\pi$
$132$	8.95892e7	3.39037
$133$	−1.08266e7	−0.399035
$134$	1.56343e7	0.561322
$135$	−5.85915e7	−2.04959
$136$	−1.39536e7	−0.475665
$137$	1.62343e7	0.539400	0.269700	−	0.962944i	$-0.413075\pi$
$137$	1.62343e7	0.539400	0.269700	+	0.962944i	$0.413075\pi$
$138$	1.85577e7	0.601101
$139$	2.38753e7	0.754046	0.377023	−	0.926204i	$-0.376948\pi$
$139$	2.38753e7	0.754046	0.377023	+	0.926204i	$0.376948\pi$
$140$	9.22230e7	2.84048
$141$	6.93467e7	2.08333
$142$	7.07822e6	0.207451
$143$	4.50858e6	0.128933
$144$	8.05847e7	2.24897
$145$	−5.51082e7	−1.50117
$146$	4.82492e7	1.28308
$147$	3.03026e7	0.786810
$148$	−1.10049e8	−2.79043
$149$	−1.49109e7	−0.369277	−0.184638	−	0.982807i	$-0.559111\pi$
$149$	−1.49109e7	−0.369277	−0.184638	+	0.982807i	$0.559111\pi$
$150$	−2.96811e8	−7.18059
$151$	−7.01442e6	−0.165795	−0.0828977	−	0.996558i	$-0.526417\pi$
$151$	−7.01442e6	−0.165795	−0.0828977	+	0.996558i	$0.526417\pi$
$152$	−4.70191e7	−1.08598
$153$	−1.79691e7	−0.405607
$154$	−5.65682e7	−1.24810
$155$	3.55055e6	0.0765834
$156$	2.14677e7	0.452741
$157$	−2.71207e7	−0.559310	−0.279655	−	0.960101i	$-0.590220\pi$
$157$	−2.71207e7	−0.559310	−0.279655	+	0.960101i	$0.590220\pi$
$158$	−8.85686e7	−1.78640
$159$	−6.73472e7	−1.32871
$160$	4.00567e7	0.773135
$161$	−7.95129e6	−0.150158
$162$	1.16152e7	0.214647
$163$	5.06776e7	0.916557	0.458278	−	0.888809i	$-0.348466\pi$
$163$	5.06776e7	0.916557	0.458278	+	0.888809i	$0.348466\pi$
$164$	1.66112e8	2.94068
$165$	1.73138e8	3.00054
$166$	1.62242e8	2.75286
$167$	−2.07036e7	−0.343983	−0.171992	−	0.985098i	$-0.555020\pi$
$167$	−2.07036e7	−0.343983	−0.171992	+	0.985098i	$0.555020\pi$
$168$	−1.41765e8	−2.30667
$169$	−6.16682e7	−0.982783
$170$	−5.12360e7	−0.799842
$171$	−6.05498e7	−0.926033
$172$	−1.64034e8	−2.45801
$173$	5.36427e7	0.787679	0.393840	−	0.919179i	$-0.371147\pi$
$173$	5.36427e7	0.787679	0.393840	+	0.919179i	$0.371147\pi$
$174$	1.60952e8	2.31619
$175$	1.27172e8	1.79374
$176$	−9.56383e7	−1.32232
$177$	−1.22317e8	−1.65798
$178$	8.20391e7	1.09031
$179$	−3.38671e7	−0.441360	−0.220680	−	0.975346i	$-0.570828\pi$
$179$	−3.38671e7	−0.441360	−0.220680	+	0.975346i	$0.570828\pi$
$180$	5.15775e8	6.59184
$181$	1.04923e7	0.131521	0.0657607	−	0.997835i	$-0.479053\pi$
$181$	1.04923e7	0.131521	0.0657607	+	0.997835i	$0.479053\pi$
$182$	−1.35551e7	−0.166668
$183$	−9.28212e7	−1.11961
$184$	−3.45319e7	−0.408656
$185$	−2.12679e8	−2.46958
$186$	−1.03699e7	−0.118163
$187$	2.13258e7	0.238484
$188$	−2.45173e8	−2.69104
$189$	−7.33209e7	−0.789971
$190$	−1.72648e8	−1.82610
$191$	3.16922e7	0.329106	0.164553	−	0.986368i	$-0.447382\pi$
$191$	3.16922e7	0.329106	0.164553	+	0.986368i	$0.447382\pi$
$192$	9.87152e7	1.00654
$193$	1.66918e8	1.67130	0.835649	−	0.549264i	$-0.185092\pi$
$193$	1.66918e8	1.67130	0.835649	+	0.549264i	$0.185092\pi$
$194$	6.15858e7	0.605584
$195$	4.14881e7	0.400684
$196$	−1.07134e8	−1.01632
$197$	1.19450e8	1.11315	0.556577	−	0.830796i	$-0.312115\pi$
$197$	1.19450e8	1.11315	0.556577	+	0.830796i	$0.312115\pi$
$198$	−3.16369e8	−2.89645
$199$	2.02185e8	1.81871	0.909356	−	0.416019i	$-0.136575\pi$
$199$	2.02185e8	1.81871	0.909356	+	0.416019i	$0.136575\pi$
$200$	5.52301e8	4.88170
$201$	−5.98814e7	−0.520123
$202$	−3.28433e8	−2.80361
$203$	−6.89620e7	−0.578594
$204$	1.01543e8	0.837425
$205$	3.21024e8	2.60255
$206$	3.37725e8	2.69171
$207$	−4.44692e7	−0.348468
$208$	−2.29172e7	−0.176580
$209$	7.18608e7	0.544478
$210$	−5.20542e8	−3.87872
$211$	−1.01897e8	−0.746747	−0.373374	−	0.927681i	$-0.621799\pi$
$211$	−1.01897e8	−0.746747	−0.373374	+	0.927681i	$0.621799\pi$
$212$	2.38104e8	1.71629
$213$	−2.71105e7	−0.192224
$214$	−4.23115e7	−0.295128
$215$	−3.17008e8	−2.17538
$216$	−3.18427e8	−2.14992
$217$	4.44313e6	0.0295175
$218$	1.28466e8	0.839827
$219$	−1.84801e8	−1.18891
$220$	−6.12125e8	−3.87579
$221$	5.11017e6	0.0318466
$222$	6.21160e8	3.81038
$223$	1.66667e8	1.00643	0.503214	−	0.864162i	$-0.332151\pi$
$223$	1.66667e8	1.00643	0.503214	+	0.864162i	$0.332151\pi$
$224$	5.01266e7	0.297989
$225$	7.11237e8	4.16270
$226$	−1.70523e8	−0.982657
$227$	−1.88738e8	−1.07095	−0.535474	−	0.844552i	$-0.679867\pi$
$227$	−1.88738e8	−1.07095	−0.535474	+	0.844552i	$0.679867\pi$
$228$	3.42167e8	1.91191
$229$	−2.72204e8	−1.49785	−0.748927	−	0.662652i	$-0.769431\pi$
$229$	−2.72204e8	−1.49785	−0.748927	+	0.662652i	$0.769431\pi$
$230$	−1.26797e8	−0.687165
$231$	2.16664e8	1.15650
$232$	−2.99497e8	−1.57465
$233$	2.31254e8	1.19769	0.598844	−	0.800866i	$-0.295627\pi$
$233$	2.31254e8	1.19769	0.598844	+	0.800866i	$0.295627\pi$
$234$	−7.58096e7	−0.386784
$235$	−4.73816e8	−2.38162
$236$	4.32447e8	2.14161
$237$	3.39229e8	1.65529
$238$	−6.41163e7	−0.308283
$239$	−2.94824e8	−1.39692	−0.698458	−	0.715651i	$-0.746130\pi$
$239$	−2.94824e8	−1.39692	−0.698458	+	0.715651i	$0.746130\pi$
$240$	−8.80066e8	−4.10938
$241$	1.76524e8	0.812354	0.406177	−	0.913795i	$-0.366862\pi$
$241$	1.76524e8	0.812354	0.406177	+	0.913795i	$0.366862\pi$
$242$	−1.34092e7	−0.0608202
$243$	2.00883e8	0.898091
$244$	3.28167e8	1.44621
$245$	−2.07045e8	−0.899463
$246$	−9.37599e8	−4.01554
$247$	1.72196e7	0.0727082
$248$	1.92962e7	0.0803324
$249$	−6.21406e8	−2.55081
$250$	1.21381e9	4.91316
$251$	−3.25446e8	−1.29904	−0.649518	−	0.760346i	$-0.725029\pi$
$251$	−3.25446e8	−1.29904	−0.649518	+	0.760346i	$0.725029\pi$
$252$	6.45437e8	2.54069
$253$	5.27762e7	0.204888
$254$	7.05545e8	2.70151
$255$	1.96240e8	0.741136
$256$	−5.44929e8	−2.03002
$257$	−4.52380e8	−1.66241	−0.831203	−	0.555969i	$-0.812347\pi$
$257$	−4.52380e8	−1.66241	−0.831203	+	0.555969i	$0.812347\pi$
$258$	9.25869e8	3.35645
$259$	−2.66144e8	−0.951849
$260$	−1.46680e8	−0.517563
$261$	−3.85683e8	−1.34273
$262$	−5.95734e8	−2.04644
$263$	−8.90957e7	−0.302003	−0.151002	−	0.988534i	$-0.548250\pi$
$263$	−8.90957e7	−0.302003	−0.151002	+	0.988534i	$0.548250\pi$
$264$	9.40955e8	3.14742
$265$	4.60154e8	1.51895
$266$	−2.16051e8	−0.703833
$267$	−3.14220e8	−1.01029
$268$	2.11709e8	0.671842
$269$	2.72958e8	0.854993	0.427496	−	0.904017i	$-0.359396\pi$
$269$	2.72958e8	0.854993	0.427496	+	0.904017i	$0.359396\pi$
$270$	−1.16923e9	−3.61514
$271$	−1.53890e8	−0.469698	−0.234849	−	0.972032i	$-0.575460\pi$
$271$	−1.53890e8	−0.469698	−0.234849	+	0.972032i	$0.575460\pi$
$272$	−1.08400e8	−0.326615
$273$	5.19178e7	0.154435
$274$	3.23964e8	0.951414
$275$	−8.44099e8	−2.44754
$276$	2.51295e8	0.719454
$277$	−5.75372e8	−1.62656	−0.813279	−	0.581873i	$-0.802320\pi$
$277$	−5.75372e8	−1.62656	−0.813279	+	0.581873i	$0.802320\pi$
$278$	4.76446e8	1.33001
$279$	2.48491e7	0.0685007
$280$	9.68618e8	2.63693
$281$	5.26444e8	1.41540	0.707701	−	0.706512i	$-0.249732\pi$
$281$	5.26444e8	1.41540	0.707701	+	0.706512i	$0.249732\pi$
$282$	1.38385e9	3.67466
$283$	8.78058e7	0.230288	0.115144	−	0.993349i	$-0.463267\pi$
$283$	8.78058e7	0.230288	0.115144	+	0.993349i	$0.463267\pi$
$284$	9.58482e7	0.248296
$285$	6.61265e8	1.69207
$286$	8.99712e7	0.227417
$287$	4.01727e8	1.00310
$288$	2.80343e8	0.691538
$289$	−3.86167e8	−0.941094
$290$	−1.09972e9	−2.64781
$291$	−2.35882e8	−0.561137
$292$	6.53357e8	1.53572
$293$	2.26978e8	0.527167	0.263583	−	0.964637i	$-0.415096\pi$
$293$	2.26978e8	0.527167	0.263583	+	0.964637i	$0.415096\pi$
$294$	6.04706e8	1.38781
$295$	8.35738e8	1.89537
$296$	−1.15585e9	−2.59047
$297$	4.86663e8	1.07790
$298$	−2.97555e8	−0.651345
$299$	1.26464e7	0.0273602
$300$	−4.01920e9	−8.59440
$301$	−3.96701e8	−0.838456
$302$	−1.39977e8	−0.292436
$303$	1.25794e9	2.59783
$304$	−3.65270e8	−0.745688
$305$	6.34207e8	1.27992
$306$	−3.58583e8	−0.715426
$307$	3.04912e8	0.601437	0.300719	−	0.953713i	$-0.402773\pi$
$307$	3.04912e8	0.601437	0.300719	+	0.953713i	$0.402773\pi$
$308$	−7.66007e8	−1.49384
$309$	−1.29353e9	−2.49415
$310$	7.08532e7	0.135081
$311$	1.53838e8	0.290002	0.145001	−	0.989432i	$-0.453681\pi$
$311$	1.53838e8	0.290002	0.145001	+	0.989432i	$0.453681\pi$
$312$	2.25475e8	0.420299
$313$	2.04764e8	0.377440	0.188720	−	0.982031i	$-0.439566\pi$
$313$	2.04764e8	0.377440	0.188720	+	0.982031i	$0.439566\pi$
$314$	−5.41209e8	−0.986533
$315$	1.24736e9	2.24856
$316$	−1.19933e9	−2.13814
$317$	−7.27132e8	−1.28205	−0.641026	−	0.767519i	$-0.721491\pi$
$317$	−7.27132e8	−1.28205	−0.641026	+	0.767519i	$0.721491\pi$
$318$	−1.34395e9	−2.34363
$319$	4.57731e8	0.789483
$320$	−6.74478e8	−1.15065
$321$	1.62058e8	0.273466
$322$	−1.58672e8	−0.264854
$323$	8.14494e7	0.134487
$324$	1.57285e8	0.256910
$325$	−2.02266e8	−0.326838
$326$	1.01130e9	1.61666
$327$	−4.92040e8	−0.778186
$328$	1.74467e9	2.72995
$329$	−5.92929e8	−0.917946
$330$	3.45507e9	5.29246
$331$	8.28738e8	1.25609	0.628043	−	0.778179i	$-0.283856\pi$
$331$	8.28738e8	1.25609	0.628043	+	0.778179i	$0.283856\pi$
$332$	2.19696e9	3.29488
$333$	−1.48846e9	−2.20894
$334$	−4.13151e8	−0.606731
$335$	4.09144e8	0.594592
$336$	−1.10131e9	−1.58387
$337$	5.44518e8	0.775011	0.387505	−	0.921867i	$-0.373337\pi$
$337$	5.44518e8	0.775011	0.387505	+	0.921867i	$0.373337\pi$
$338$	−1.23062e9	−1.73347
$339$	6.53123e8	0.910534
$340$	−6.93802e8	−0.957325
$341$	−2.94910e7	−0.0402762
$342$	−1.20831e9	−1.63337
$343$	−7.97290e8	−1.06681
$344$	−1.72284e9	−2.28187
$345$	4.85648e8	0.636729
$346$	1.07047e9	1.38934
$347$	5.55611e8	0.713868	0.356934	−	0.934130i	$-0.383822\pi$
$347$	5.55611e8	0.713868	0.356934	+	0.934130i	$0.383822\pi$
$348$	2.17950e9	2.77223
$349$	−1.23854e9	−1.55963	−0.779816	−	0.626009i	$-0.784687\pi$
$349$	−1.23854e9	−1.55963	−0.779816	+	0.626009i	$0.784687\pi$
$350$	2.53780e9	3.16387
$351$	1.16616e8	0.143941
$352$	−3.32712e8	−0.406602
$353$	−9.18376e8	−1.11124	−0.555622	−	0.831435i	$-0.687520\pi$
$353$	−9.18376e8	−1.11124	−0.555622	+	0.831435i	$0.687520\pi$
$354$	−2.44090e9	−2.92441
$355$	1.85234e8	0.219746
$356$	1.11092e9	1.30499
$357$	2.45574e8	0.285656
$358$	−6.75837e8	−0.778487
$359$	1.34767e9	1.53728	0.768640	−	0.639682i	$-0.220934\pi$
$359$	1.34767e9	1.53728	0.768640	+	0.639682i	$0.220934\pi$
$360$	5.41718e9	6.11949
$361$	−6.19414e8	−0.692957
$362$	2.09380e8	0.231983
$363$	5.13588e7	0.0563562
$364$	−1.83554e8	−0.199484
$365$	1.26266e9	1.35913
$366$	−1.85230e9	−1.97482
$367$	7.77642e8	0.821199	0.410600	−	0.911816i	$-0.365319\pi$
$367$	7.77642e8	0.821199	0.410600	+	0.911816i	$0.365319\pi$
$368$	−2.68263e8	−0.280604
$369$	2.24674e9	2.32788
$370$	−4.24412e9	−4.35594
$371$	5.75833e8	0.585447
$372$	−1.40422e8	−0.141428
$373$	6.44024e8	0.642571	0.321285	−	0.946982i	$-0.395885\pi$
$373$	6.44024e8	0.642571	0.321285	+	0.946982i	$0.395885\pi$
$374$	4.25568e8	0.420647
$375$	−4.64905e9	−4.55255
$376$	−2.57505e9	−2.49821
$377$	1.09683e8	0.105426
$378$	−1.46316e9	−1.39338
$379$	1.46375e9	1.38112	0.690558	−	0.723277i	$-0.257365\pi$
$379$	1.46375e9	1.38112	0.690558	+	0.723277i	$0.257365\pi$
$380$	−2.33788e9	−2.18565
$381$	−2.70233e9	−2.50323
$382$	6.32435e8	0.580489
$383$	−1.05720e9	−0.961530	−0.480765	−	0.876850i	$-0.659641\pi$
$383$	−1.05720e9	−0.961530	−0.480765	+	0.876850i	$0.659641\pi$
$384$	2.72033e9	2.45167
$385$	−1.48037e9	−1.32208
$386$	3.33095e9	2.94790
$387$	−2.21863e9	−1.94579
$388$	8.33952e8	0.724820
$389$	2.67436e8	0.230354	0.115177	−	0.993345i	$-0.463256\pi$
$389$	2.67436e8	0.230354	0.115177	+	0.993345i	$0.463256\pi$
$390$	8.27917e8	0.706742
$391$	5.98183e7	0.0506075
$392$	−1.12523e9	−0.943495
$393$	2.28174e9	1.89623
$394$	2.38369e9	1.96342
$395$	−2.31780e9	−1.89229
$396$	−4.28405e9	−3.46674
$397$	3.69295e8	0.296215	0.148107	−	0.988971i	$-0.452682\pi$
$397$	3.69295e8	0.296215	0.148107	+	0.988971i	$0.452682\pi$
$398$	4.03472e9	3.20791
$399$	8.27501e8	0.652174
$400$	4.29058e9	3.35202
$401$	−1.98777e9	−1.53943	−0.769715	−	0.638387i	$-0.779602\pi$
$401$	−1.98777e9	−1.53943	−0.769715	+	0.638387i	$0.779602\pi$
$402$	−1.19497e9	−0.917412
$403$	−7.06675e6	−0.00537838
$404$	−4.44741e9	−3.35562
$405$	3.03966e8	0.227370
$406$	−1.37617e9	−1.02055
$407$	1.76652e9	1.29878
$408$	1.06651e9	0.777417
$409$	5.46117e8	0.394688	0.197344	−	0.980334i	$-0.436768\pi$
$409$	5.46117e8	0.394688	0.197344	+	0.980334i	$0.436768\pi$
$410$	6.40621e9	4.59048
$411$	−1.24082e9	−0.881583
$412$	4.57324e9	3.22169
$413$	1.04583e9	0.730530
$414$	−8.87407e8	−0.614641
$415$	4.24580e9	2.91602
$416$	−7.97259e7	−0.0542966
$417$	−1.82485e9	−1.23240
$418$	1.43402e9	0.960371
$419$	−1.16624e9	−0.774528	−0.387264	−	0.921969i	$-0.626580\pi$
$419$	−1.16624e9	−0.774528	−0.387264	+	0.921969i	$0.626580\pi$
$420$	−7.04882e9	−4.64242
$421$	−2.24993e9	−1.46954	−0.734769	−	0.678317i	$-0.762709\pi$
$421$	−2.24993e9	−1.46954	−0.734769	+	0.678317i	$0.762709\pi$
$422$	−2.03341e9	−1.31714
$423$	−3.31607e9	−2.13026
$424$	2.50080e9	1.59331
$425$	−9.56730e8	−0.604544
$426$	−5.41004e8	−0.339053
$427$	7.93641e8	0.493318
$428$	−5.72953e8	−0.353236
$429$	−3.44601e8	−0.210725
$430$	−6.32607e9	−3.83702
$431$	1.53220e9	0.921819	0.460909	−	0.887447i	$-0.347523\pi$
$431$	1.53220e9	0.921819	0.460909	+	0.887447i	$0.347523\pi$
$432$	−2.47372e9	−1.47624
$433$	9.81704e8	0.581129	0.290565	−	0.956855i	$-0.406157\pi$
$433$	9.81704e8	0.581129	0.290565	+	0.956855i	$0.406157\pi$
$434$	8.86651e7	0.0520641
$435$	4.21205e9	2.45347
$436$	1.73959e9	1.00518
$437$	2.01568e8	0.115541
$438$	−3.68780e9	−2.09705
$439$	−1.24756e9	−0.703777	−0.351889	−	0.936042i	$-0.614460\pi$
$439$	−1.24756e9	−0.703777	−0.351889	+	0.936042i	$0.614460\pi$
$440$	−6.42914e9	−3.59806
$441$	−1.44904e9	−0.804533
$442$	1.01976e8	0.0561722
$443$	−2.07638e9	−1.13473	−0.567367	−	0.823465i	$-0.692038\pi$
$443$	−2.07638e9	−1.13473	−0.567367	+	0.823465i	$0.692038\pi$
$444$	8.41132e9	4.56062
$445$	2.14693e9	1.15494
$446$	3.32594e9	1.77518
$447$	1.13967e9	0.603538
$448$	−8.44036e8	−0.443494
$449$	4.45747e8	0.232395	0.116197	−	0.993226i	$-0.462930\pi$
$449$	4.45747e8	0.232395	0.116197	+	0.993226i	$0.462930\pi$
$450$	1.41931e10	7.34234
$451$	−2.66644e9	−1.36872
$452$	−2.30910e9	−1.17614
$453$	5.36128e8	0.270972
$454$	−3.76637e9	−1.88898
$455$	−3.54732e8	−0.176547
$456$	3.59378e9	1.77490
$457$	2.31240e9	1.13333	0.566665	−	0.823948i	$-0.308233\pi$
$457$	2.31240e9	1.13333	0.566665	+	0.823948i	$0.308233\pi$
$458$	−5.43197e9	−2.64197
$459$	5.51599e8	0.266244
$460$	−1.71699e9	−0.822463
$461$	−1.38565e9	−0.658718	−0.329359	−	0.944205i	$-0.606833\pi$
$461$	−1.38565e9	−0.658718	−0.329359	+	0.944205i	$0.606833\pi$
$462$	4.32364e9	2.03987
$463$	1.70685e9	0.799212	0.399606	−	0.916687i	$-0.369147\pi$
$463$	1.70685e9	0.799212	0.399606	+	0.916687i	$0.369147\pi$
$464$	−2.32666e9	−1.08123
$465$	−2.71377e8	−0.125166
$466$	4.61480e9	2.11253
$467$	−3.29906e7	−0.0149893	−0.00749465	−	0.999972i	$-0.502386\pi$
$467$	−3.29906e7	−0.0149893	−0.00749465	+	0.999972i	$0.502386\pi$
$468$	−1.02656e9	−0.462939
$469$	5.11999e8	0.229173
$470$	−9.45526e9	−4.20079
$471$	2.07290e9	0.914125
$472$	4.54199e9	1.98815
$473$	2.63308e9	1.14406
$474$	6.76950e9	2.91966
$475$	−3.22386e9	−1.38022
$476$	−8.68218e8	−0.368981
$477$	3.22046e9	1.35864
$478$	−5.88338e9	−2.46393
$479$	1.56585e9	0.650993	0.325497	−	0.945543i	$-0.394469\pi$
$479$	1.56585e9	0.650993	0.325497	+	0.945543i	$0.394469\pi$
$480$	−3.06163e9	−1.26360
$481$	4.23300e8	0.173436
$482$	3.52264e9	1.43286
$483$	6.07736e8	0.245414
$484$	−1.81578e8	−0.0727953
$485$	1.61168e9	0.641478
$486$	4.00872e9	1.58409
$487$	−5.67843e7	−0.0222780	−0.0111390	−	0.999938i	$-0.503546\pi$
$487$	−5.67843e7	−0.0222780	−0.0111390	+	0.999938i	$0.503546\pi$
$488$	3.44673e9	1.34257
$489$	−3.87341e9	−1.49800
$490$	−4.13170e9	−1.58651
$491$	−2.55398e9	−0.973716	−0.486858	−	0.873481i	$-0.661857\pi$
$491$	−2.55398e9	−0.973716	−0.486858	+	0.873481i	$0.661857\pi$
$492$	−1.26963e10	−4.80618
$493$	5.18807e8	0.195003
$494$	3.43626e8	0.128245
$495$	−8.27925e9	−3.06812
$496$	1.49903e8	0.0551602
$497$	2.31800e8	0.0846968
$498$	−1.24005e10	−4.49921
$499$	−1.43002e8	−0.0515216	−0.0257608	−	0.999668i	$-0.508201\pi$
$499$	−1.43002e8	−0.0515216	−0.0257608	+	0.999668i	$0.508201\pi$
$500$	1.64366e10	5.88053
$501$	1.58242e9	0.562199
$502$	−6.49446e9	−2.29129
$503$	−5.23549e8	−0.183430	−0.0917148	−	0.995785i	$-0.529235\pi$
$503$	−5.23549e8	−0.183430	−0.0917148	+	0.995785i	$0.529235\pi$
$504$	6.77902e9	2.35863
$505$	−8.59497e9	−2.96978
$506$	1.05318e9	0.361389
$507$	4.71344e9	1.60624
$508$	9.55400e9	3.23342
$509$	−5.66266e9	−1.90330	−0.951652	−	0.307177i	$-0.900616\pi$
$509$	−5.66266e9	−1.90330	−0.951652	+	0.307177i	$0.900616\pi$
$510$	3.91609e9	1.30724
$511$	1.58009e9	0.523851
$512$	−6.31866e9	−2.08056
$513$	1.85871e9	0.607855
$514$	−9.02749e9	−2.93222
$515$	8.83813e9	2.85125
$516$	1.25375e10	4.01732
$517$	3.93553e9	1.25253
$518$	−5.31106e9	−1.67891
$519$	−4.10004e9	−1.28737
$520$	−1.54058e9	−0.480476
$521$	2.81823e9	0.873062	0.436531	−	0.899689i	$-0.356207\pi$
$521$	2.81823e9	0.873062	0.436531	+	0.899689i	$0.356207\pi$
$522$	−7.69653e9	−2.36836
$523$	2.41696e9	0.738776	0.369388	−	0.929275i	$-0.379567\pi$
$523$	2.41696e9	0.738776	0.369388	+	0.929275i	$0.379567\pi$
$524$	−8.06702e9	−2.44936
$525$	−9.72009e9	−2.93165
$526$	−1.77795e9	−0.532685
$527$	−3.34260e7	−0.00994827
$528$	7.30986e9	2.16118
$529$	1.48036e8	0.0434783
$530$	9.18263e9	2.67918
$531$	5.84904e9	1.69533
$532$	−2.92561e9	−0.842413
$533$	−6.38942e8	−0.182775
$534$	−6.27044e9	−1.78198
$535$	−1.10728e9	−0.312620
$536$	2.22358e9	0.623699
$537$	2.58854e9	0.721349
$538$	5.44702e9	1.50807
$539$	1.71972e9	0.473040
$540$	−1.58328e10	−4.32694
$541$	−3.24690e9	−0.881613	−0.440807	−	0.897602i	$-0.645308\pi$
$541$	−3.24690e9	−0.881613	−0.440807	+	0.897602i	$0.645308\pi$
$542$	−3.07097e9	−0.828472
$543$	−8.01952e8	−0.214956
$544$	−3.77107e8	−0.100431
$545$	3.36190e9	0.889605
$546$	1.03605e9	0.272399
$547$	2.67521e9	0.698878	0.349439	−	0.936959i	$-0.386372\pi$
$547$	2.67521e9	0.698878	0.349439	+	0.936959i	$0.386372\pi$
$548$	4.38689e9	1.13874
$549$	4.43860e9	1.14483
$550$	−1.68445e10	−4.31706
$551$	1.74821e9	0.445207
$552$	2.63935e9	0.667899
$553$	−2.90048e9	−0.729343
$554$	−1.14819e10	−2.86899
$555$	1.62555e10	4.03623
$556$	6.45169e9	1.59189
$557$	1.29427e9	0.317344	0.158672	−	0.987331i	$-0.449279\pi$
$557$	1.29427e9	0.317344	0.158672	+	0.987331i	$0.449279\pi$
$558$	4.95877e8	0.120824
$559$	6.30948e8	0.152775
$560$	7.52476e9	1.81065
$561$	−1.62998e9	−0.389773
$562$	1.05055e10	2.49654
$563$	−1.28811e8	−0.0304210	−0.0152105	−	0.999884i	$-0.504842\pi$
$563$	−1.28811e8	−0.0304210	−0.0152105	+	0.999884i	$0.504842\pi$
$564$	1.87392e10	4.39818
$565$	−4.46251e9	−1.04090
$566$	1.75221e9	0.406190
$567$	3.80381e8	0.0876350
$568$	1.00669e9	0.230504
$569$	−5.70575e9	−1.29843	−0.649217	−	0.760603i	$-0.724903\pi$
$569$	−5.70575e9	−1.29843	−0.649217	+	0.760603i	$0.724903\pi$
$570$	1.31959e10	2.98454
$571$	3.67874e9	0.826937	0.413469	−	0.910518i	$-0.364317\pi$
$571$	3.67874e9	0.826937	0.413469	+	0.910518i	$0.364317\pi$
$572$	1.21833e9	0.272193
$573$	−2.42231e9	−0.537883
$574$	8.01668e9	1.76931
$575$	−2.36768e9	−0.519380
$576$	−4.72044e9	−1.02921
$577$	6.01407e9	1.30333	0.651664	−	0.758508i	$-0.274071\pi$
$577$	6.01407e9	1.30333	0.651664	+	0.758508i	$0.274071\pi$
$578$	−7.70618e9	−1.65994
$579$	−1.27580e10	−2.73153
$580$	−1.48916e10	−3.16915
$581$	5.31315e9	1.12392
$582$	−4.70715e9	−0.989754
$583$	−3.82206e9	−0.798835
$584$	6.86220e9	1.42567
$585$	−1.98391e9	−0.409709
$586$	4.52948e9	0.929837
$587$	−5.32427e9	−1.08649	−0.543246	−	0.839574i	$-0.682805\pi$
$587$	−5.32427e9	−1.08649	−0.543246	+	0.839574i	$0.682805\pi$
$588$	8.18851e9	1.66106
$589$	−1.12635e8	−0.0227127
$590$	1.66776e10	3.34312
$591$	−9.12985e9	−1.81931
$592$	−8.97925e9	−1.77875
$593$	5.72975e8	0.112835	0.0564176	−	0.998407i	$-0.482032\pi$
$593$	5.72975e8	0.112835	0.0564176	+	0.998407i	$0.482032\pi$
$594$	9.71162e9	1.90125
$595$	−1.67790e9	−0.326555
$596$	−4.02929e9	−0.779590
$597$	−1.54535e10	−2.97246
$598$	2.52367e8	0.0482590
$599$	−1.42970e9	−0.271800	−0.135900	−	0.990723i	$-0.543393\pi$
$599$	−1.42970e9	−0.271800	−0.135900	+	0.990723i	$0.543393\pi$
$600$	−4.22137e10	−7.97855
$601$	4.44175e9	0.834629	0.417314	−	0.908762i	$-0.362971\pi$
$601$	4.44175e9	0.834629	0.417314	+	0.908762i	$0.362971\pi$
$602$	−7.91638e9	−1.47890
$603$	2.86345e9	0.531838
$604$	−1.89547e9	−0.350015
$605$	−3.50913e8	−0.0644251
$606$	2.51029e10	4.58216
$607$	−1.25069e9	−0.226980	−0.113490	−	0.993539i	$-0.536203\pi$
$607$	−1.25069e9	−0.226980	−0.113490	+	0.993539i	$0.536203\pi$
$608$	−1.27073e9	−0.229292
$609$	5.27092e9	0.945641
$610$	1.26560e10	2.25757
$611$	9.43048e8	0.167259
$612$	−4.85568e9	−0.856288
$613$	2.75470e9	0.483018	0.241509	−	0.970399i	$-0.422358\pi$
$613$	2.75470e9	0.483018	0.241509	+	0.970399i	$0.422358\pi$
$614$	6.08469e9	1.06084
$615$	−2.45366e10	−4.25355
$616$	−8.04537e9	−1.38680
$617$	9.76601e8	0.167386	0.0836930	−	0.996492i	$-0.473328\pi$
$617$	9.76601e8	0.167386	0.0836930	+	0.996492i	$0.473328\pi$
$618$	−2.58131e10	−4.39927
$619$	9.92338e9	1.68168	0.840838	−	0.541286i	$-0.182062\pi$
$619$	9.92338e9	1.68168	0.840838	+	0.541286i	$0.182062\pi$
$620$	9.59444e8	0.161677
$621$	1.36508e9	0.228737
$622$	3.06991e9	0.511517
$623$	2.68665e9	0.445147
$624$	1.75162e9	0.288598
$625$	1.65620e10	2.71351
$626$	4.08618e9	0.665744
$627$	−5.49249e9	−0.889883
$628$	−7.32868e9	−1.18077
$629$	2.00223e9	0.320801
$630$	2.48917e10	3.96609
$631$	1.02281e10	1.62067	0.810333	−	0.585969i	$-0.199286\pi$
$631$	1.02281e10	1.62067	0.810333	+	0.585969i	$0.199286\pi$
$632$	−1.25966e10	−1.98492
$633$	7.78824e9	1.22047
$634$	−1.45103e10	−2.26133
$635$	1.84638e10	2.86163
$636$	−1.81988e10	−2.80507
$637$	4.12087e8	0.0631685
$638$	9.13427e9	1.39252
$639$	1.29639e9	0.196554
$640$	−1.85869e10	−2.80270
$641$	−1.79873e9	−0.269750	−0.134875	−	0.990863i	$-0.543063\pi$
$641$	−1.79873e9	−0.269750	−0.134875	+	0.990863i	$0.543063\pi$
$642$	3.23397e9	0.482351
$643$	−1.13752e10	−1.68741	−0.843706	−	0.536806i	$-0.819631\pi$
$643$	−1.13752e10	−1.68741	−0.843706	+	0.536806i	$0.819631\pi$
$644$	−2.14863e9	−0.317002
$645$	2.42296e10	3.55540
$646$	1.62537e9	0.237213
$647$	1.94696e9	0.282613	0.141306	−	0.989966i	$-0.454870\pi$
$647$	1.94696e9	0.282613	0.141306	+	0.989966i	$0.454870\pi$
$648$	1.65197e9	0.238500
$649$	−6.94166e9	−0.996798
$650$	−4.03634e9	−0.576489
$651$	−3.39598e8	−0.0482428
$652$	1.36943e10	1.93497
$653$	−1.21092e8	−0.0170184	−0.00850919	−	0.999964i	$-0.502709\pi$
$653$	−1.21092e8	−0.0170184	−0.00850919	+	0.999964i	$0.502709\pi$
$654$	−9.81894e9	−1.37260
$655$	−1.55901e10	−2.16773
$656$	1.35536e10	1.87452
$657$	8.83694e9	1.21569
$658$	−1.18322e10	−1.61911
$659$	−5.44558e9	−0.741216	−0.370608	−	0.928789i	$-0.620851\pi$
$659$	−5.44558e9	−0.741216	−0.370608	+	0.928789i	$0.620851\pi$
$660$	4.67861e10	6.33451
$661$	−1.04355e9	−0.140542	−0.0702712	−	0.997528i	$-0.522386\pi$
$661$	−1.04355e9	−0.140542	−0.0702712	+	0.997528i	$0.522386\pi$
$662$	1.65379e10	2.21553
$663$	−3.90582e8	−0.0520493
$664$	2.30747e10	3.05877
$665$	−5.65396e9	−0.745550
$666$	−2.97031e10	−3.89621
$667$	1.28392e9	0.167532
$668$	−5.59460e9	−0.726192
$669$	−1.27388e10	−1.64489
$670$	8.16469e9	1.04876
$671$	−5.26775e9	−0.673125
$672$	−3.83130e9	−0.487027
$673$	−1.14044e9	−0.144218	−0.0721088	−	0.997397i	$-0.522973\pi$
$673$	−1.14044e9	−0.144218	−0.0721088	+	0.997397i	$0.522973\pi$
$674$	1.08662e10	1.36699
$675$	−2.18329e10	−2.73243
$676$	−1.66642e10	−2.07478
$677$	−1.61236e10	−1.99711	−0.998555	−	0.0537374i	$-0.982887\pi$
$677$	−1.61236e10	−1.99711	−0.998555	+	0.0537374i	$0.982887\pi$
$678$	1.30334e10	1.60603
$679$	2.01684e9	0.247245
$680$	−7.28700e9	−0.888725
$681$	1.44257e10	1.75033
$682$	−5.88509e8	−0.0710408
$683$	−1.72222e9	−0.206832	−0.103416	−	0.994638i	$-0.532977\pi$
$683$	−1.72222e9	−0.206832	−0.103416	+	0.994638i	$0.532977\pi$
$684$	−1.63620e10	−1.95497
$685$	8.47801e9	1.00781
$686$	−1.59104e10	−1.88168
$687$	2.08052e10	2.44806
$688$	−1.33840e10	−1.56685
$689$	−9.15856e8	−0.106674
$690$	9.69138e9	1.12309
$691$	−1.54378e10	−1.77997	−0.889985	−	0.455990i	$-0.849285\pi$
$691$	−1.54378e10	−1.77997	−0.889985	+	0.455990i	$0.849285\pi$
$692$	1.44956e10	1.66289
$693$	−1.03606e10	−1.18255
$694$	1.10875e10	1.25915
$695$	1.24684e10	1.40885
$696$	2.28913e10	2.57358
$697$	−3.02223e9	−0.338074
$698$	−2.47158e10	−2.75094
$699$	−1.76753e10	−1.95748
$700$	3.43651e10	3.78682
$701$	−1.02580e10	−1.12473	−0.562367	−	0.826888i	$-0.690109\pi$
$701$	−1.02580e10	−1.12473	−0.562367	+	0.826888i	$0.690109\pi$
$702$	2.32714e9	0.253888
$703$	6.74684e9	0.732414
$704$	5.60224e9	0.605142
$705$	3.62149e10	3.89247
$706$	−1.83267e10	−1.96005
$707$	−1.07557e10	−1.14464
$708$	−3.30529e10	−3.50021
$709$	−8.51027e9	−0.896771	−0.448386	−	0.893840i	$-0.648001\pi$
$709$	−8.51027e9	−0.896771	−0.448386	+	0.893840i	$0.648001\pi$
$710$	3.69645e9	0.387597
$711$	−1.62215e10	−1.69257
$712$	1.16679e10	1.21147
$713$	−8.27214e7	−0.00854682
$714$	4.90056e9	0.503851
$715$	2.35451e9	0.240896
$716$	−9.15171e9	−0.931766
$717$	2.25341e10	2.28309
$718$	2.68935e10	2.71151
$719$	−4.11020e9	−0.412394	−0.206197	−	0.978511i	$-0.566109\pi$
$719$	−4.11020e9	−0.412394	−0.206197	+	0.978511i	$0.566109\pi$
$720$	4.20837e10	4.20194
$721$	1.10600e10	1.09896
$722$	−1.23608e10	−1.22226
$723$	−1.34922e10	−1.32769
$724$	2.83528e9	0.277658
$725$	−2.05350e10	−2.00130
$726$	1.02489e9	0.0994033
$727$	−1.01166e9	−0.0976485	−0.0488242	−	0.998807i	$-0.515547\pi$
$727$	−1.01166e9	−0.0976485	−0.0488242	+	0.998807i	$0.515547\pi$
$728$	−1.92786e9	−0.185190
$729$	−1.66269e10	−1.58951
$730$	2.51971e10	2.39729
$731$	2.98442e9	0.282585
$732$	−2.50825e10	−2.36365
$733$	−1.31580e9	−0.123403	−0.0617017	−	0.998095i	$-0.519653\pi$
$733$	−1.31580e9	−0.123403	−0.0617017	+	0.998095i	$0.519653\pi$
$734$	1.55183e10	1.44846
$735$	1.58249e10	1.47006
$736$	−9.33250e8	−0.0862831
$737$	−3.39836e9	−0.312704
$738$	4.48349e10	4.10600
$739$	−7.35134e9	−0.670056	−0.335028	−	0.942208i	$-0.608746\pi$
$739$	−7.35134e9	−0.670056	−0.335028	+	0.942208i	$0.608746\pi$
$740$	−5.74709e10	−5.21359
$741$	−1.31613e9	−0.118833
$742$	1.14911e10	1.03263
$743$	1.06737e10	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$743$	1.06737e10	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$744$	−1.47485e9	−0.131294
$745$	−7.78691e9	−0.689951
$746$	1.28519e10	1.13339
$747$	2.97149e10	2.60826
$748$	5.76274e9	0.503470
$749$	−1.38563e9	−0.120493
$750$	−9.27744e10	−8.02997
$751$	−1.31746e10	−1.13501	−0.567504	−	0.823371i	$-0.692091\pi$
$751$	−1.31746e10	−1.13501	−0.567504	+	0.823371i	$0.692091\pi$
$752$	−2.00044e10	−1.71539
$753$	2.48746e10	2.12312
$754$	2.18879e9	0.185954
$755$	−3.66313e9	−0.309769
$756$	−1.98131e10	−1.66773
$757$	3.55107e9	0.297525	0.148762	−	0.988873i	$-0.452471\pi$
$757$	3.55107e9	0.297525	0.148762	+	0.988873i	$0.452471\pi$
$758$	2.92100e10	2.43607
$759$	−4.03381e9	−0.334864
$760$	−2.45548e10	−2.02903
$761$	−9.84137e9	−0.809486	−0.404743	−	0.914430i	$-0.632639\pi$
$761$	−9.84137e9	−0.809486	−0.404743	+	0.914430i	$0.632639\pi$
$762$	−5.39264e10	−4.41529
$763$	4.20705e9	0.342880
$764$	8.56399e9	0.694783
$765$	−9.38398e9	−0.757830
$766$	−2.10971e10	−1.69598
$767$	−1.66339e9	−0.133110
$768$	4.16502e10	3.31782
$769$	−2.42604e9	−0.192378	−0.0961892	−	0.995363i	$-0.530665\pi$
$769$	−2.42604e9	−0.192378	−0.0961892	+	0.995363i	$0.530665\pi$
$770$	−2.95416e10	−2.33193
$771$	3.45764e10	2.71700
$772$	4.51054e10	3.52832
$773$	1.53603e10	1.19611	0.598055	−	0.801455i	$-0.295940\pi$
$773$	1.53603e10	1.19611	0.598055	+	0.801455i	$0.295940\pi$
$774$	−4.42739e10	−3.43206
$775$	1.32304e9	0.102098
$776$	8.75899e9	0.672881
$777$	2.03420e10	1.55568
$778$	5.33683e9	0.406308
$779$	−1.01839e10	−0.771850
$780$	1.12111e10	0.845894
$781$	−1.53856e9	−0.115568
$782$	1.19371e9	0.0892635
$783$	1.18394e10	0.881379
$784$	−8.74139e9	−0.647850
$785$	−1.41632e10	−1.04501
$786$	4.55333e10	3.34465
$787$	1.68198e10	1.23001	0.615005	−	0.788523i	$-0.289154\pi$
$787$	1.68198e10	1.23001	0.615005	+	0.788523i	$0.289154\pi$
$788$	3.22783e10	2.35001
$789$	6.80979e9	0.493587
$790$	−4.62531e10	−3.33769
$791$	−5.58434e9	−0.401194
$792$	−4.49953e10	−3.21832
$793$	−1.26228e9	−0.0898874
$794$	7.36949e9	0.522475
$795$	−3.51707e10	−2.48254
$796$	5.46354e10	3.83953
$797$	−2.52755e10	−1.76846	−0.884232	−	0.467049i	$-0.845317\pi$
$797$	−2.52755e10	−1.76846	−0.884232	+	0.467049i	$0.845317\pi$
$798$	1.65132e10	1.15033
$799$	4.46066e9	0.309375
$800$	1.49263e10	1.03071
$801$	1.50256e10	1.03304
$802$	−3.96670e10	−2.71531
$803$	−1.04877e10	−0.714787
$804$	−1.61814e10	−1.09804
$805$	−4.15240e9	−0.280552
$806$	−1.41021e8	−0.00948660
$807$	−2.08628e10	−1.39738
$808$	−4.67111e10	−3.11516
$809$	−2.60561e10	−1.73017	−0.865085	−	0.501625i	$-0.832736\pi$
$809$	−2.60561e10	−1.73017	−0.865085	+	0.501625i	$0.832736\pi$
$810$	6.06581e9	0.401043
$811$	4.45254e8	0.0293113	0.0146557	−	0.999893i	$-0.495335\pi$
$811$	4.45254e8	0.0293113	0.0146557	+	0.999893i	$0.495335\pi$
$812$	−1.86352e10	−1.22148
$813$	1.17622e10	0.767665
$814$	3.52518e10	2.29085
$815$	2.64653e10	1.71248
$816$	8.28523e9	0.533813
$817$	1.00565e10	0.645162
$818$	1.08981e10	0.696166
$819$	−2.48265e9	−0.157914
$820$	8.67485e10	5.49431
$821$	1.81822e10	1.14669	0.573344	−	0.819314i	$-0.305646\pi$
$821$	1.81822e10	1.14669	0.573344	+	0.819314i	$0.305646\pi$
$822$	−2.47613e10	−1.55497
$823$	1.52481e10	0.953493	0.476746	−	0.879041i	$-0.341816\pi$
$823$	1.52481e10	0.953493	0.476746	+	0.879041i	$0.341816\pi$
$824$	4.80326e10	2.99083
$825$	6.45165e10	4.00020
$826$	2.08702e10	1.28854
$827$	1.68693e10	1.03712	0.518558	−	0.855042i	$-0.326469\pi$
$827$	1.68693e10	1.03712	0.518558	+	0.855042i	$0.326469\pi$
$828$	−1.20166e10	−0.735660
$829$	−2.98979e9	−0.182263	−0.0911317	−	0.995839i	$-0.529048\pi$
$829$	−2.98979e9	−0.182263	−0.0911317	+	0.995839i	$0.529048\pi$
$830$	8.47272e10	5.14339
$831$	4.39770e10	2.65841
$832$	1.34243e9	0.0808091
$833$	1.94919e9	0.116841
$834$	−3.64158e10	−2.17375
$835$	−1.08120e10	−0.642693
$836$	1.94185e10	1.14946
$837$	−7.62795e8	−0.0449644
$838$	−2.32729e10	−1.36614
$839$	3.09957e10	1.81190	0.905950	−	0.423384i	$-0.139158\pi$
$839$	3.09957e10	1.81190	0.905950	+	0.423384i	$0.139158\pi$
$840$	−7.40337e10	−4.30975
$841$	−6.11434e9	−0.354457
$842$	−4.48985e10	−2.59203
$843$	−4.02373e10	−2.31330
$844$	−2.75351e10	−1.57648
$845$	−3.22049e10	−1.83622
$846$	−6.61741e10	−3.75744
$847$	−4.39129e8	−0.0248314
$848$	1.94276e10	1.09404
$849$	−6.71120e9	−0.376377
$850$	−1.90921e10	−1.06632
$851$	4.95503e9	0.275609
$852$	−7.32590e9	−0.405810
$853$	−9.53699e9	−0.526126	−0.263063	−	0.964779i	$-0.584733\pi$
$853$	−9.53699e9	−0.526126	−0.263063	+	0.964779i	$0.584733\pi$
$854$	1.58376e10	0.870133
$855$	−3.16209e10	−1.73018
$856$	−6.01772e9	−0.327924
$857$	−2.04118e10	−1.10776	−0.553882	−	0.832595i	$-0.686854\pi$
$857$	−2.04118e10	−1.10776	−0.553882	+	0.832595i	$0.686854\pi$
$858$	−6.87671e9	−0.371685
$859$	1.51869e10	0.817510	0.408755	−	0.912644i	$-0.365963\pi$
$859$	1.51869e10	0.817510	0.408755	+	0.912644i	$0.365963\pi$
$860$	−8.56632e10	−4.59250
$861$	−3.07049e10	−1.63944
$862$	3.05759e10	1.62594
$863$	−2.59491e10	−1.37431	−0.687154	−	0.726512i	$-0.741140\pi$
$863$	−2.59491e10	−1.37431	−0.687154	+	0.726512i	$0.741140\pi$
$864$	−8.60573e9	−0.453931
$865$	2.80138e10	1.47169
$866$	1.95904e10	1.02502
$867$	2.95157e10	1.53810
$868$	1.20064e9	0.0623152
$869$	1.92518e10	0.995179
$870$	8.40538e10	4.32753
$871$	−8.14329e8	−0.0417577
$872$	1.82709e10	0.933154
$873$	1.12796e10	0.573776
$874$	4.02240e9	0.203796
$875$	3.97504e10	2.00592
$876$	−4.99376e10	−2.50994
$877$	−1.17297e10	−0.587202	−0.293601	−	0.955928i	$-0.594854\pi$
$877$	−1.17297e10	−0.587202	−0.293601	+	0.955928i	$0.594854\pi$
$878$	−2.48957e10	−1.24135
$879$	−1.73485e10	−0.861590
$880$	−4.99451e10	−2.47061
$881$	−2.57090e8	−0.0126669	−0.00633344	−	0.999980i	$-0.502016\pi$
$881$	−2.57090e8	−0.0126669	−0.00633344	+	0.999980i	$0.502016\pi$
$882$	−2.89163e10	−1.41907
$883$	−1.93060e10	−0.943692	−0.471846	−	0.881681i	$-0.656412\pi$
$883$	−1.93060e10	−0.943692	−0.471846	+	0.881681i	$0.656412\pi$
$884$	1.38089e9	0.0672321
$885$	−6.38774e10	−3.09774
$886$	−4.14353e10	−2.00149
$887$	−1.15896e9	−0.0557615	−0.0278807	−	0.999611i	$-0.508876\pi$
$887$	−1.15896e9	−0.0557615	−0.0278807	+	0.999611i	$0.508876\pi$
$888$	8.83440e10	4.23381
$889$	2.31055e10	1.10296
$890$	4.28432e10	2.03712
$891$	−2.52475e9	−0.119577
$892$	4.50375e10	2.12470
$893$	1.50309e10	0.706327
$894$	2.27428e10	1.06454
$895$	−1.76864e10	−0.824629
$896$	−2.32594e10	−1.08024
$897$	−9.66597e8	−0.0447169
$898$	8.89512e9	0.409907
$899$	−7.17447e8	−0.0329330
$900$	1.92193e11	8.78799
$901$	−4.33204e9	−0.197313
$902$	−5.32102e10	−2.41419
$903$	3.03208e10	1.37035
$904$	−2.42524e10	−1.09186
$905$	5.47940e9	0.245733
$906$	1.06987e10	0.477952
$907$	−3.72007e9	−0.165549	−0.0827743	−	0.996568i	$-0.526378\pi$
$907$	−3.72007e9	−0.165549	−0.0827743	+	0.996568i	$0.526378\pi$
$908$	−5.10015e10	−2.26091
$909$	−6.01532e10	−2.65635
$910$	−7.07887e9	−0.311400
$911$	−2.71597e9	−0.119017	−0.0595087	−	0.998228i	$-0.518953\pi$
$911$	−2.71597e9	−0.119017	−0.0595087	+	0.998228i	$0.518953\pi$
$912$	2.79185e10	1.21874
$913$	−3.52657e10	−1.53358
$914$	4.61452e10	1.99901
$915$	−4.84739e10	−2.09187
$916$	−7.35560e10	−3.16216
$917$	−1.95094e10	−0.835507
$918$	1.10075e10	0.469611
$919$	−2.47533e10	−1.05203	−0.526016	−	0.850475i	$-0.676315\pi$
$919$	−2.47533e10	−1.05203	−0.526016	+	0.850475i	$0.676315\pi$
$920$	−1.80336e10	−0.763527
$921$	−2.33051e10	−0.982977
$922$	−2.76514e10	−1.16187
$923$	−3.68676e8	−0.0154326
$924$	5.85477e10	2.44151
$925$	−7.92505e10	−3.29235
$926$	3.40612e10	1.40968
$927$	6.18550e10	2.55033
$928$	−8.09413e9	−0.332470
$929$	4.41128e10	1.80513	0.902567	−	0.430550i	$-0.141680\pi$
$929$	4.41128e10	1.80513	0.902567	+	0.430550i	$0.141680\pi$
$930$	−5.41547e9	−0.220773
$931$	6.56812e9	0.266758
$932$	6.24904e10	2.52847
$933$	−1.17582e10	−0.473973
$934$	−6.58346e8	−0.0264387
$935$	1.11369e10	0.445580
$936$	−1.07820e10	−0.429766
$937$	1.57657e10	0.626073	0.313036	−	0.949741i	$-0.398654\pi$
$937$	1.57657e10	0.626073	0.313036	+	0.949741i	$0.398654\pi$
$938$	1.02172e10	0.404225
$939$	−1.56506e10	−0.616880
$940$	−1.28037e11	−5.02790
$941$	4.32289e10	1.69126	0.845630	−	0.533769i	$-0.179225\pi$
$941$	4.32289e10	1.69126	0.845630	+	0.533769i	$0.179225\pi$
$942$	4.13659e10	1.61237
$943$	−7.47928e9	−0.290448
$944$	3.52847e10	1.36516
$945$	−3.82903e10	−1.47597
$946$	5.25445e10	2.01794
$947$	−2.21647e9	−0.0848081	−0.0424040	−	0.999101i	$-0.513502\pi$
$947$	−2.21647e9	−0.0848081	−0.0424040	+	0.999101i	$0.513502\pi$
$948$	9.16678e10	3.49452
$949$	−2.51311e9	−0.0954508
$950$	−6.43339e10	−2.43449
$951$	5.55764e10	2.09536
$952$	−9.11889e9	−0.342541
$953$	−2.41997e10	−0.905702	−0.452851	−	0.891586i	$-0.649593\pi$
$953$	−2.41997e10	−0.905702	−0.452851	+	0.891586i	$0.649593\pi$
$954$	6.42660e10	2.39642
$955$	1.65506e10	0.614895
$956$	−7.96686e10	−2.94907
$957$	−3.49854e10	−1.29031
$958$	3.12474e10	1.14825
$959$	1.06093e10	0.388438
$960$	5.15519e10	1.88060
$961$	−2.74664e10	−0.998320
$962$	8.44718e9	0.305914
$963$	−7.74943e9	−0.279626
$964$	4.77012e10	1.71498
$965$	8.71697e10	3.12262
$966$	1.21277e10	0.432871
$967$	9.31181e9	0.331163	0.165581	−	0.986196i	$-0.447050\pi$
$967$	9.31181e9	0.331163	0.165581	+	0.986196i	$0.447050\pi$
$968$	−1.90711e9	−0.0675790
$969$	−6.22537e9	−0.219802
$970$	3.21619e10	1.13146
$971$	−3.52264e10	−1.23481	−0.617407	−	0.786644i	$-0.711817\pi$
$971$	−3.52264e10	−1.23481	−0.617407	+	0.786644i	$0.711817\pi$
$972$	5.42833e10	1.89598
$973$	1.56028e10	0.543011
$974$	−1.13316e9	−0.0392948
$975$	1.54597e10	0.534177
$976$	2.67761e10	0.921877
$977$	−2.58727e10	−0.887588	−0.443794	−	0.896129i	$-0.646368\pi$
$977$	−2.58727e10	−0.887588	−0.443794	+	0.896129i	$0.646368\pi$
$978$	−7.72960e10	−2.64223
$979$	−1.78325e10	−0.607397
$980$	−5.59485e10	−1.89888
$981$	2.35288e10	0.795715
$982$	−5.09661e10	−1.71748
$983$	−3.46133e10	−1.16227	−0.581133	−	0.813808i	$-0.697391\pi$
$983$	−3.46133e10	−1.16227	−0.581133	+	0.813808i	$0.697391\pi$
$984$	−1.33349e11	−4.46178
$985$	6.23803e10	2.07980
$986$	1.03531e10	0.343954
$987$	4.53190e10	1.50027
$988$	4.65315e9	0.153496
$989$	7.38571e9	0.242776
$990$	−1.65217e11	−5.41168
$991$	−1.28569e10	−0.419641	−0.209820	−	0.977740i	$-0.567288\pi$
$991$	−1.28569e10	−0.419641	−0.209820	+	0.977740i	$0.567288\pi$
$992$	5.21494e8	0.0169613
$993$	−6.33424e10	−2.05292
$994$	4.62571e9	0.149391
$995$	1.05587e11	3.39805
$996$	−1.67919e11	−5.38507
$997$	3.61960e10	1.15672	0.578360	−	0.815782i	$-0.303693\pi$
$997$	3.61960e10	1.15672	0.578360	+	0.815782i	$0.303693\pi$
$998$	−2.85368e9	−0.0908758
$999$	4.56916e10	1.44996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.8.a.b.1.8	✓	8
3.2	odd	2		207.8.a.f.1.1		8
4.3	odd	2		368.8.a.h.1.7		8
5.4	even	2		575.8.a.b.1.1		8
23.22	odd	2		529.8.a.c.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.a.b.1.8	✓	8	1.1	even	1	trivial
207.8.a.f.1.1		8	3.2	odd	2
368.8.a.h.1.7		8	4.3	odd	2
529.8.a.c.1.8		8	23.22	odd	2
575.8.a.b.1.1		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)

\(f(q)\)	\(=\)	\(q+19.9556 q^{2} -76.4323 q^{3} +270.224 q^{4} +522.229 q^{5} -1525.25 q^{6} +653.513 q^{7} +2838.16 q^{8} +3654.90 q^{9} +O(q^{10})\)	\(q+19.9556 q^{2} -76.4323 q^{3} +270.224 q^{4} +522.229 q^{5} -1525.25 q^{6} +653.513 q^{7} +2838.16 q^{8} +3654.90 q^{9} +10421.4 q^{10} -4337.65 q^{11} -20653.9 q^{12} -1039.41 q^{13} +13041.2 q^{14} -39915.2 q^{15} +22048.4 q^{16} -4916.44 q^{17} +72935.5 q^{18} -16566.8 q^{19} +141119. q^{20} -49949.5 q^{21} -86560.2 q^{22} -12167.0 q^{23} -216927. q^{24} +194598. q^{25} -20741.9 q^{26} -112195. q^{27} +176595. q^{28} -105525. q^{29} -796530. q^{30} +6798.84 q^{31} +76703.4 q^{32} +331537. q^{33} -98110.2 q^{34} +341284. q^{35} +987642. q^{36} -407252. q^{37} -330599. q^{38} +79444.2 q^{39} +1.48217e6 q^{40} +614719. q^{41} -996770. q^{42} -607028. q^{43} -1.17214e6 q^{44} +1.90869e6 q^{45} -242799. q^{46} -907295. q^{47} -1.68521e6 q^{48} -396464. q^{49} +3.88332e6 q^{50} +375775. q^{51} -280872. q^{52} +881135. q^{53} -2.23891e6 q^{54} -2.26525e6 q^{55} +1.85478e6 q^{56} +1.26624e6 q^{57} -2.10581e6 q^{58} +1.60033e6 q^{59} -1.07860e7 q^{60} +1.21442e6 q^{61} +135675. q^{62} +2.38852e6 q^{63} -1.29154e6 q^{64} -542808. q^{65} +6.61600e6 q^{66} +783456. q^{67} -1.32854e6 q^{68} +929952. q^{69} +6.81050e6 q^{70} +354699. q^{71} +1.03732e7 q^{72} +2.41783e6 q^{73} -8.12693e6 q^{74} -1.48736e7 q^{75} -4.47674e6 q^{76} -2.83471e6 q^{77} +1.58535e6 q^{78} -4.43829e6 q^{79} +1.15143e7 q^{80} +582055. q^{81} +1.22671e7 q^{82} +8.13014e6 q^{83} -1.34976e7 q^{84} -2.56751e6 q^{85} -1.21136e7 q^{86} +8.06552e6 q^{87} -1.23110e7 q^{88} +4.11109e6 q^{89} +3.80891e7 q^{90} -679265. q^{91} -3.28782e6 q^{92} -519651. q^{93} -1.81056e7 q^{94} -8.65164e6 q^{95} -5.86262e6 q^{96} +3.08615e6 q^{97} -7.91165e6 q^{98} -1.58537e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9}+O(q^{10}) \) 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

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