LMFDB - Embedded newform 69.8.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.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:	$21.5545667584$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 775x^{5} - 474x^{4} + 167184x^{3} - 33920x^{2} - 9348928x + 28965760 $ x^7 - 775x^5 - 474x^4 + 167184x^3 - 33920x^2 - 9348928*x + 28965760
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.39427$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.39427 q^{2} -27.0000 q^{3} -108.690 q^{4} -380.251 q^{5} -118.645 q^{6} -1696.01 q^{7} -1040.08 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+4.39427 q^{2} -27.0000 q^{3} -108.690 q^{4} -380.251 q^{5} -118.645 q^{6} -1696.01 q^{7} -1040.08 q^{8} +729.000 q^{9} -1670.93 q^{10} +8255.22 q^{11} +2934.64 q^{12} -2531.03 q^{13} -7452.74 q^{14} +10266.8 q^{15} +9341.97 q^{16} -7207.96 q^{17} +3203.42 q^{18} -24136.0 q^{19} +41329.6 q^{20} +45792.4 q^{21} +36275.7 q^{22} -12167.0 q^{23} +28082.2 q^{24} +66465.7 q^{25} -11122.0 q^{26} -19683.0 q^{27} +184340. q^{28} -122247. q^{29} +45115.0 q^{30} -209857. q^{31} +174182. q^{32} -222891. q^{33} -31673.7 q^{34} +644911. q^{35} -79235.3 q^{36} +34537.4 q^{37} -106060. q^{38} +68337.7 q^{39} +395492. q^{40} +545394. q^{41} +201224. q^{42} -387267. q^{43} -897263. q^{44} -277203. q^{45} -53465.1 q^{46} +398639. q^{47} -252233. q^{48} +2.05292e6 q^{49} +292068. q^{50} +194615. q^{51} +275098. q^{52} -658129. q^{53} -86492.4 q^{54} -3.13905e6 q^{55} +1.76399e6 q^{56} +651673. q^{57} -537184. q^{58} -707752. q^{59} -1.11590e6 q^{60} +1.80508e6 q^{61} -922170. q^{62} -1.23639e6 q^{63} -430371. q^{64} +962425. q^{65} -979443. q^{66} +4.26695e6 q^{67} +783435. q^{68} +328509. q^{69} +2.83391e6 q^{70} +1.92112e6 q^{71} -758220. q^{72} -6.42145e6 q^{73} +151767. q^{74} -1.79457e6 q^{75} +2.62335e6 q^{76} -1.40010e7 q^{77} +300295. q^{78} +2.30552e6 q^{79} -3.55229e6 q^{80} +531441. q^{81} +2.39661e6 q^{82} -9.08815e6 q^{83} -4.97719e6 q^{84} +2.74083e6 q^{85} -1.70176e6 q^{86} +3.30066e6 q^{87} -8.58610e6 q^{88} -2.68687e6 q^{89} -1.21810e6 q^{90} +4.29266e6 q^{91} +1.32244e6 q^{92} +5.66615e6 q^{93} +1.75173e6 q^{94} +9.17775e6 q^{95} -4.70290e6 q^{96} +1.13566e6 q^{97} +9.02108e6 q^{98} +6.01805e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9}+O(q^{10}) $ 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9} - 15310 q^{10} + 9040 q^{11} - 17658 q^{12} + 3774 q^{13} + 4536 q^{14} + 13932 q^{15} + 52002 q^{16} - 40760 q^{17} + 81598 q^{19} - 88946 q^{20} - 27486 q^{21} + 245034 q^{22} - 85169 q^{23} - 38394 q^{24} + 321325 q^{25} + 412748 q^{26} - 137781 q^{27} + 965948 q^{28} + 154126 q^{29} + 413370 q^{30} + 243132 q^{31} + 1278286 q^{32} - 244080 q^{33} + 984836 q^{34} - 130296 q^{35} + 476766 q^{36} + 582114 q^{37} + 772558 q^{38} - 101898 q^{39} - 132618 q^{40} + 113062 q^{41} - 122472 q^{42} - 659778 q^{43} + 659390 q^{44} - 376164 q^{45} - 591032 q^{47} - 1404054 q^{48} + 3263235 q^{49} - 702684 q^{50} + 1100520 q^{51} + 1793280 q^{52} + 207128 q^{53} + 184664 q^{55} + 5390508 q^{56} - 2203146 q^{57} - 1142916 q^{58} + 447148 q^{59} + 2401542 q^{60} + 2248970 q^{61} - 5729060 q^{62} + 742122 q^{63} + 7212922 q^{64} - 827096 q^{65} - 6615918 q^{66} + 4467570 q^{67} - 5477620 q^{68} + 2299563 q^{69} - 12744284 q^{70} - 5154608 q^{71} + 1036638 q^{72} - 13239250 q^{73} - 2827426 q^{74} - 8675775 q^{75} - 527434 q^{76} - 18415912 q^{77} - 11144196 q^{78} + 9594446 q^{79} - 55932394 q^{80} + 3720087 q^{81} - 20889952 q^{82} - 573720 q^{83} - 26080596 q^{84} + 7477272 q^{85} - 28416910 q^{86} - 4161402 q^{87} + 26555702 q^{88} - 3810540 q^{89} - 11160990 q^{90} + 36092068 q^{91} - 7957218 q^{92} - 6564564 q^{93} + 33545768 q^{94} + 10497320 q^{95} - 34513722 q^{96} + 49497978 q^{97} - 1023376 q^{98} + 6590160 q^{99}+O(q^{100}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9 - 15310 * q^10 + 9040 * q^11 - 17658 * q^12 + 3774 * q^13 + 4536 * q^14 + 13932 * q^15 + 52002 * q^16 - 40760 * q^17 + 81598 * q^19 - 88946 * q^20 - 27486 * q^21 + 245034 * q^22 - 85169 * q^23 - 38394 * q^24 + 321325 * q^25 + 412748 * q^26 - 137781 * q^27 + 965948 * q^28 + 154126 * q^29 + 413370 * q^30 + 243132 * q^31 + 1278286 * q^32 - 244080 * q^33 + 984836 * q^34 - 130296 * q^35 + 476766 * q^36 + 582114 * q^37 + 772558 * q^38 - 101898 * q^39 - 132618 * q^40 + 113062 * q^41 - 122472 * q^42 - 659778 * q^43 + 659390 * q^44 - 376164 * q^45 - 591032 * q^47 - 1404054 * q^48 + 3263235 * q^49 - 702684 * q^50 + 1100520 * q^51 + 1793280 * q^52 + 207128 * q^53 + 184664 * q^55 + 5390508 * q^56 - 2203146 * q^57 - 1142916 * q^58 + 447148 * q^59 + 2401542 * q^60 + 2248970 * q^61 - 5729060 * q^62 + 742122 * q^63 + 7212922 * q^64 - 827096 * q^65 - 6615918 * q^66 + 4467570 * q^67 - 5477620 * q^68 + 2299563 * q^69 - 12744284 * q^70 - 5154608 * q^71 + 1036638 * q^72 - 13239250 * q^73 - 2827426 * q^74 - 8675775 * q^75 - 527434 * q^76 - 18415912 * q^77 - 11144196 * q^78 + 9594446 * q^79 - 55932394 * q^80 + 3720087 * q^81 - 20889952 * q^82 - 573720 * q^83 - 26080596 * q^84 + 7477272 * q^85 - 28416910 * q^86 - 4161402 * q^87 + 26555702 * q^88 - 3810540 * q^89 - 11160990 * q^90 + 36092068 * q^91 - 7957218 * q^92 - 6564564 * q^93 + 33545768 * q^94 + 10497320 * q^95 - 34513722 * q^96 + 49497978 * q^97 - 1023376 * q^98 + 6590160 * 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 $

$2$	4.39427	0.388402	0.194201	−	0.980962i	$-0.437789\pi$
$2$	4.39427	0.388402	0.194201	+	0.980962i	$0.437789\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	−108.690	−0.849144
$5$	−380.251	−1.36043	−0.680213	−	0.733014i	$-0.738113\pi$
$5$	−380.251	−1.36043	−0.680213	+	0.733014i	$0.738113\pi$
$6$	−118.645	−0.224244
$7$	−1696.01	−1.86890	−0.934450	−	0.356094i	$-0.884108\pi$
$7$	−1696.01	−1.86890	−0.934450	+	0.356094i	$0.884108\pi$
$8$	−1040.08	−0.718212
$9$	729.000	0.333333
$10$	−1670.93	−0.528393
$11$	8255.22	1.87005	0.935027	−	0.354576i	$-0.115375\pi$
$11$	8255.22	1.87005	0.935027	+	0.354576i	$0.115375\pi$
$12$	2934.64	0.490253
$13$	−2531.03	−0.319518	−0.159759	−	0.987156i	$-0.551072\pi$
$13$	−2531.03	−0.319518	−0.159759	+	0.987156i	$0.551072\pi$
$14$	−7452.74	−0.725885
$15$	10266.8	0.785443
$16$	9341.97	0.570188
$17$	−7207.96	−0.355829	−0.177914	−	0.984046i	$-0.556935\pi$
$17$	−7207.96	−0.355829	−0.177914	+	0.984046i	$0.556935\pi$
$18$	3203.42	0.129467
$19$	−24136.0	−0.807288	−0.403644	−	0.914916i	$-0.632256\pi$
$19$	−24136.0	−0.807288	−0.403644	+	0.914916i	$0.632256\pi$
$20$	41329.6	1.15520
$21$	45792.4	1.07901
$22$	36275.7	0.726333
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	28082.2	0.414660
$25$	66465.7	0.850761
$26$	−11122.0	−0.124101
$27$	−19683.0	−0.192450
$28$	184340.	1.58696
$29$	−122247.	−0.930772	−0.465386	−	0.885108i	$-0.654085\pi$
$29$	−122247.	−0.930772	−0.465386	+	0.885108i	$0.654085\pi$
$30$	45115.0	0.305068
$31$	−209857.	−1.26520	−0.632599	−	0.774480i	$-0.718012\pi$
$31$	−209857.	−1.26520	−0.632599	+	0.774480i	$0.718012\pi$
$32$	174182.	0.939674
$33$	−222891.	−1.07968
$34$	−31673.7	−0.138205
$35$	644911.	2.54250
$36$	−79235.3	−0.283048
$37$	34537.4	0.112094	0.0560472	−	0.998428i	$-0.482150\pi$
$37$	34537.4	0.112094	0.0560472	+	0.998428i	$0.482150\pi$
$38$	−106060.	−0.313552
$39$	68337.7	0.184474
$40$	395492.	0.977075
$41$	545394.	1.23585	0.617927	−	0.786236i	$-0.287973\pi$
$41$	545394.	1.23585	0.617927	+	0.786236i	$0.287973\pi$
$42$	201224.	0.419090
$43$	−387267.	−0.742798	−0.371399	−	0.928473i	$-0.621122\pi$
$43$	−387267.	−0.742798	−0.371399	+	0.928473i	$0.621122\pi$
$44$	−897263.	−1.58794
$45$	−277203.	−0.453476
$46$	−53465.1	−0.0809875
$47$	398639.	0.560064	0.280032	−	0.959991i	$-0.409655\pi$
$47$	398639.	0.560064	0.280032	+	0.959991i	$0.409655\pi$
$48$	−252233.	−0.329198
$49$	2.05292e6	2.49279
$50$	292068.	0.330438
$51$	194615.	0.205438
$52$	275098.	0.271317
$53$	−658129.	−0.607219	−0.303610	−	0.952796i	$-0.598192\pi$
$53$	−658129.	−0.607219	−0.303610	+	0.952796i	$0.598192\pi$
$54$	−86492.4	−0.0747481
$55$	−3.13905e6	−2.54407
$56$	1.76399e6	1.34227
$57$	651673.	0.466088
$58$	−537184.	−0.361514
$59$	−707752.	−0.448641	−0.224320	−	0.974515i	$-0.572016\pi$
$59$	−707752.	−0.448641	−0.224320	+	0.974515i	$0.572016\pi$
$60$	−1.11590e6	−0.666954
$61$	1.80508e6	1.01822	0.509112	−	0.860700i	$-0.329974\pi$
$61$	1.80508e6	1.01822	0.509112	+	0.860700i	$0.329974\pi$
$62$	−922170.	−0.491406
$63$	−1.23639e6	−0.622967
$64$	−430371.	−0.205217
$65$	962425.	0.434681
$66$	−979443.	−0.419349
$67$	4.26695e6	1.73323	0.866613	−	0.498980i	$-0.166292\pi$
$67$	4.26695e6	1.73323	0.866613	+	0.498980i	$0.166292\pi$
$68$	783435.	0.302150
$69$	328509.	0.120386
$70$	2.83391e6	0.987514
$71$	1.92112e6	0.637016	0.318508	−	0.947920i	$-0.396818\pi$
$71$	1.92112e6	0.637016	0.318508	+	0.947920i	$0.396818\pi$
$72$	−758220.	−0.239404
$73$	−6.42145e6	−1.93198	−0.965992	−	0.258573i	$-0.916748\pi$
$73$	−6.42145e6	−1.93198	−0.965992	+	0.258573i	$0.916748\pi$
$74$	151767.	0.0435377
$75$	−1.79457e6	−0.491187
$76$	2.62335e6	0.685503
$77$	−1.40010e7	−3.49494
$78$	300295.	0.0716500
$79$	2.30552e6	0.526108	0.263054	−	0.964781i	$-0.415270\pi$
$79$	2.30552e6	0.526108	0.263054	+	0.964781i	$0.415270\pi$
$80$	−3.55229e6	−0.775700
$81$	531441.	0.111111
$82$	2.39661e6	0.480009
$83$	−9.08815e6	−1.74462	−0.872312	−	0.488949i	$-0.837380\pi$
$83$	−9.08815e6	−1.74462	−0.872312	+	0.488949i	$0.837380\pi$
$84$	−4.97719e6	−0.916235
$85$	2.74083e6	0.484079
$86$	−1.70176e6	−0.288505
$87$	3.30066e6	0.537382
$88$	−8.58610e6	−1.34309
$89$	−2.68687e6	−0.404000	−0.202000	−	0.979386i	$-0.564744\pi$
$89$	−2.68687e6	−0.404000	−0.202000	+	0.979386i	$0.564744\pi$
$90$	−1.21810e6	−0.176131
$91$	4.29266e6	0.597147
$92$	1.32244e6	0.177059
$93$	5.66615e6	0.730462
$94$	1.75173e6	0.217530
$95$	9.17775e6	1.09826
$96$	−4.70290e6	−0.542521
$97$	1.13566e6	0.126342	0.0631710	−	0.998003i	$-0.479879\pi$
$97$	1.13566e6	0.126342	0.0631710	+	0.998003i	$0.479879\pi$
$98$	9.02108e6	0.968205
$99$	6.01805e6	0.623351
$100$	−7.22419e6	−0.722419
$101$	−6.46223e6	−0.624105	−0.312053	−	0.950065i	$-0.601017\pi$
$101$	−6.46223e6	−0.624105	−0.312053	+	0.950065i	$0.601017\pi$
$102$	855190.	0.0797925
$103$	8.33354e6	0.751448	0.375724	−	0.926732i	$-0.377394\pi$
$103$	8.33354e6	0.751448	0.375724	+	0.926732i	$0.377394\pi$
$104$	2.63248e6	0.229481
$105$	−1.74126e7	−1.46791
$106$	−2.89200e6	−0.235845
$107$	1.19338e7	0.941752	0.470876	−	0.882199i	$-0.343938\pi$
$107$	1.19338e7	0.941752	0.470876	+	0.882199i	$0.343938\pi$
$108$	2.13935e6	0.163418
$109$	1.56240e7	1.15558	0.577790	−	0.816186i	$-0.303915\pi$
$109$	1.56240e7	1.15558	0.577790	+	0.816186i	$0.303915\pi$
$110$	−1.37939e7	−0.988124
$111$	−932510.	−0.0647177
$112$	−1.58441e7	−1.06563
$113$	5.94665e6	0.387702	0.193851	−	0.981031i	$-0.437902\pi$
$113$	5.94665e6	0.387702	0.193851	+	0.981031i	$0.437902\pi$
$114$	2.86363e6	0.181030
$115$	4.62651e6	0.283669
$116$	1.32870e7	0.790359
$117$	−1.84512e6	−0.106506
$118$	−3.11005e6	−0.174253
$119$	1.22248e7	0.665008
$120$	−1.06783e7	−0.564114
$121$	4.86615e7	2.49710
$122$	7.93203e6	0.395480
$123$	−1.47256e7	−0.713521
$124$	2.28095e7	1.07433
$125$	4.43345e6	0.203028
$126$	−5.43305e6	−0.241962
$127$	−8.29111e6	−0.359170	−0.179585	−	0.983742i	$-0.557475\pi$
$127$	−8.29111e6	−0.359170	−0.179585	+	0.983742i	$0.557475\pi$
$128$	−2.41864e7	−1.01938
$129$	1.04562e7	0.428855
$130$	4.22916e6	0.168831
$131$	−2.84556e7	−1.10591	−0.552953	−	0.833213i	$-0.686499\pi$
$131$	−2.84556e7	−1.10591	−0.552953	+	0.833213i	$0.686499\pi$
$132$	2.42261e7	0.916800
$133$	4.09350e7	1.50874
$134$	1.87501e7	0.673189
$135$	7.48448e6	0.261814
$136$	7.49686e6	0.255560
$137$	−6.32359e6	−0.210108	−0.105054	−	0.994467i	$-0.533501\pi$
$137$	−6.32359e6	−0.210108	−0.105054	+	0.994467i	$0.533501\pi$
$138$	1.44356e6	0.0467581
$139$	5.91795e7	1.86905	0.934523	−	0.355904i	$-0.115827\pi$
$139$	5.91795e7	1.86905	0.934523	+	0.355904i	$0.115827\pi$
$140$	−7.00956e7	−2.15895
$141$	−1.07633e7	−0.323353
$142$	8.44192e6	0.247418
$143$	−2.08942e7	−0.597516
$144$	6.81029e6	0.190063
$145$	4.64843e7	1.26625
$146$	−2.82176e7	−0.750387
$147$	−5.54288e7	−1.43921
$148$	−3.75388e6	−0.0951842
$149$	−4.50223e7	−1.11500	−0.557501	−	0.830176i	$-0.688240\pi$
$149$	−4.50223e7	−1.11500	−0.557501	+	0.830176i	$0.688240\pi$
$150$	−7.88585e6	−0.190778
$151$	2.59950e7	0.614427	0.307214	−	0.951641i	$-0.400603\pi$
$151$	2.59950e7	0.614427	0.307214	+	0.951641i	$0.400603\pi$
$152$	2.51034e7	0.579803
$153$	−5.25460e6	−0.118610
$154$	−6.15240e7	−1.35744
$155$	7.97985e7	1.72121
$156$	−7.42765e6	−0.156645
$157$	1.11763e7	0.230489	0.115245	−	0.993337i	$-0.463235\pi$
$157$	1.11763e7	0.230489	0.115245	+	0.993337i	$0.463235\pi$
$158$	1.01311e7	0.204342
$159$	1.77695e7	0.350578
$160$	−6.62327e7	−1.27836
$161$	2.06354e7	0.389693
$162$	2.33530e6	0.0431558
$163$	−7.13634e7	−1.29068	−0.645340	−	0.763895i	$-0.723284\pi$
$163$	−7.13634e7	−1.29068	−0.645340	+	0.763895i	$0.723284\pi$
$164$	−5.92791e7	−1.04942
$165$	8.47545e7	1.46882
$166$	−3.99358e7	−0.677616
$167$	−3.85665e7	−0.640770	−0.320385	−	0.947287i	$-0.603812\pi$
$167$	−3.85665e7	−0.640770	−0.320385	+	0.947287i	$0.603812\pi$
$168$	−4.76278e7	−0.774958
$169$	−5.63424e7	−0.897908
$170$	1.20440e7	0.188017
$171$	−1.75952e7	−0.269096
$172$	4.20922e7	0.630742
$173$	4.13589e7	0.607305	0.303653	−	0.952783i	$-0.401794\pi$
$173$	4.13589e7	0.607305	0.303653	+	0.952783i	$0.401794\pi$
$174$	1.45040e7	0.208720
$175$	−1.12727e8	−1.58999
$176$	7.71200e7	1.06628
$177$	1.91093e7	0.259023
$178$	−1.18068e7	−0.156914
$179$	1.03462e8	1.34832	0.674161	−	0.738584i	$-0.264505\pi$
$179$	1.03462e8	1.34832	0.674161	+	0.738584i	$0.264505\pi$
$180$	3.01293e7	0.385066
$181$	−2.20652e7	−0.276587	−0.138294	−	0.990391i	$-0.544162\pi$
$181$	−2.20652e7	−0.276587	−0.138294	+	0.990391i	$0.544162\pi$
$182$	1.88631e7	0.231933
$183$	−4.87373e7	−0.587872
$184$	1.26547e7	0.149757
$185$	−1.31329e7	−0.152496
$186$	2.48986e7	0.283713
$187$	−5.95033e7	−0.665419
$188$	−4.33283e7	−0.475575
$189$	3.33826e7	0.359670
$190$	4.03295e7	0.426565
$191$	−1.97858e7	−0.205465	−0.102732	−	0.994709i	$-0.532759\pi$
$191$	−1.97858e7	−0.205465	−0.102732	+	0.994709i	$0.532759\pi$
$192$	1.16200e7	0.118482
$193$	1.37488e8	1.37662	0.688312	−	0.725415i	$-0.258352\pi$
$193$	1.37488e8	1.37662	0.688312	+	0.725415i	$0.258352\pi$
$194$	4.99041e6	0.0490716
$195$	−2.59855e7	−0.250963
$196$	−2.23132e8	−2.11674
$197$	−7.68103e7	−0.715793	−0.357897	−	0.933761i	$-0.616506\pi$
$197$	−7.68103e7	−0.715793	−0.357897	+	0.933761i	$0.616506\pi$
$198$	2.64450e7	0.242111
$199$	−7.05622e7	−0.634726	−0.317363	−	0.948304i	$-0.602797\pi$
$199$	−7.05622e7	−0.634726	−0.317363	+	0.948304i	$0.602797\pi$
$200$	−6.91298e7	−0.611027
$201$	−1.15208e8	−1.00068
$202$	−2.83968e7	−0.242404
$203$	2.07332e8	1.73952
$204$	−2.11528e7	−0.174446
$205$	−2.07387e8	−1.68129
$206$	3.66198e7	0.291864
$207$	−8.86974e6	−0.0695048
$208$	−2.36448e7	−0.182185
$209$	−1.99248e8	−1.50967
$210$	−7.65156e7	−0.570141
$211$	−1.03646e8	−0.759561	−0.379780	−	0.925077i	$-0.624000\pi$
$211$	−1.03646e8	−0.759561	−0.379780	+	0.925077i	$0.624000\pi$
$212$	7.15323e7	0.515616
$213$	−5.18702e7	−0.367781
$214$	5.24405e7	0.365779
$215$	1.47259e8	1.01052
$216$	2.04719e7	0.138220
$217$	3.55921e8	2.36453
$218$	6.86562e7	0.448830
$219$	1.73379e8	1.11543
$220$	3.41185e8	2.16028
$221$	1.82435e7	0.113694
$222$	−4.09770e6	−0.0251365
$223$	−2.54215e7	−0.153509	−0.0767547	−	0.997050i	$-0.524456\pi$
$223$	−2.54215e7	−0.153509	−0.0767547	+	0.997050i	$0.524456\pi$
$224$	−2.95414e8	−1.75616
$225$	4.84535e7	0.283587
$226$	2.61312e7	0.150584
$227$	1.50542e8	0.854213	0.427107	−	0.904201i	$-0.359533\pi$
$227$	1.50542e8	0.854213	0.427107	+	0.904201i	$0.359533\pi$
$228$	−7.08306e7	−0.395775
$229$	−2.82901e8	−1.55672	−0.778359	−	0.627820i	$-0.783947\pi$
$229$	−2.82901e8	−1.55672	−0.778359	+	0.627820i	$0.783947\pi$
$230$	2.03301e7	0.110178
$231$	3.78026e8	2.01781
$232$	1.27146e8	0.668492
$233$	1.06393e8	0.551019	0.275509	−	0.961298i	$-0.411154\pi$
$233$	1.06393e8	0.551019	0.275509	+	0.961298i	$0.411154\pi$
$234$	−8.10795e6	−0.0413672
$235$	−1.51583e8	−0.761926
$236$	7.69258e7	0.380961
$237$	−6.22492e7	−0.303749
$238$	5.37190e7	0.258291
$239$	1.46749e8	0.695315	0.347657	−	0.937622i	$-0.386977\pi$
$239$	1.46749e8	0.695315	0.347657	+	0.937622i	$0.386977\pi$
$240$	9.59119e7	0.447850
$241$	1.19903e8	0.551784	0.275892	−	0.961189i	$-0.411027\pi$
$241$	1.19903e8	0.551784	0.275892	+	0.961189i	$0.411027\pi$
$242$	2.13832e8	0.969880
$243$	−1.43489e7	−0.0641500
$244$	−1.96195e8	−0.864618
$245$	−7.80624e8	−3.39126
$246$	−6.47085e7	−0.277133
$247$	6.10890e7	0.257943
$248$	2.18269e8	0.908680
$249$	2.45380e8	1.00726
$250$	1.94818e7	0.0788566
$251$	2.88319e8	1.15084	0.575420	−	0.817858i	$-0.304839\pi$
$251$	2.88319e8	1.15084	0.575420	+	0.817858i	$0.304839\pi$
$252$	1.34384e8	0.528988
$253$	−1.00441e8	−0.389933
$254$	−3.64334e7	−0.139502
$255$	−7.40025e7	−0.279483
$256$	−5.11942e7	−0.190713
$257$	2.35811e8	0.866558	0.433279	−	0.901260i	$-0.357356\pi$
$257$	2.35811e8	0.866558	0.433279	+	0.901260i	$0.357356\pi$
$258$	4.59474e7	0.166568
$259$	−5.85759e7	−0.209493
$260$	−1.04606e8	−0.369106
$261$	−8.91177e7	−0.310257
$262$	−1.25042e8	−0.429536
$263$	−1.66295e8	−0.563683	−0.281842	−	0.959461i	$-0.590945\pi$
$263$	−1.66295e8	−0.563683	−0.281842	+	0.959461i	$0.590945\pi$
$264$	2.31825e8	0.775436
$265$	2.50254e8	0.826078
$266$	1.79880e8	0.585998
$267$	7.25454e7	0.233249
$268$	−4.63776e8	−1.47176
$269$	−5.60938e8	−1.75704	−0.878520	−	0.477706i	$-0.841468\pi$
$269$	−5.60938e8	−1.75704	−0.878520	+	0.477706i	$0.841468\pi$
$270$	3.28888e7	0.101689
$271$	5.78070e8	1.76436	0.882182	−	0.470909i	$-0.156074\pi$
$271$	5.78070e8	1.76436	0.882182	+	0.470909i	$0.156074\pi$
$272$	−6.73365e7	−0.202889
$273$	−1.15902e8	−0.344763
$274$	−2.77876e7	−0.0816063
$275$	5.48689e8	1.59097
$276$	−3.57058e7	−0.102225
$277$	−5.36416e8	−1.51643	−0.758215	−	0.652004i	$-0.773929\pi$
$277$	−5.36416e8	−1.51643	−0.758215	+	0.652004i	$0.773929\pi$
$278$	2.60051e8	0.725942
$279$	−1.52986e8	−0.421733
$280$	−6.70760e8	−1.82605
$281$	6.15298e8	1.65430	0.827148	−	0.561984i	$-0.189962\pi$
$281$	6.15298e8	1.65430	0.827148	+	0.561984i	$0.189962\pi$
$282$	−4.72967e7	−0.125591
$283$	1.51773e8	0.398055	0.199028	−	0.979994i	$-0.436222\pi$
$283$	1.51773e8	0.398055	0.199028	+	0.979994i	$0.436222\pi$
$284$	−2.08807e8	−0.540918
$285$	−2.47799e8	−0.634078
$286$	−9.18147e7	−0.232076
$287$	−9.24996e8	−2.30969
$288$	1.26978e8	0.313225
$289$	−3.58384e8	−0.873386
$290$	2.04265e8	0.491814
$291$	−3.06629e7	−0.0729436
$292$	6.97950e8	1.64053
$293$	−2.07040e8	−0.480858	−0.240429	−	0.970667i	$-0.577288\pi$
$293$	−2.07040e8	−0.480858	−0.240429	+	0.970667i	$0.577288\pi$
$294$	−2.43569e8	−0.558993
$295$	2.69123e8	0.610343
$296$	−3.59217e7	−0.0805075
$297$	−1.62487e8	−0.359892
$298$	−1.97840e8	−0.433070
$299$	3.07950e7	0.0666241
$300$	1.95053e8	0.417089
$301$	6.56810e8	1.38822
$302$	1.14229e8	0.238645
$303$	1.74480e8	0.360327
$304$	−2.25478e8	−0.460306
$305$	−6.86385e8	−1.38522
$306$	−2.30901e7	−0.0460682
$307$	6.60705e8	1.30324	0.651619	−	0.758547i	$-0.274090\pi$
$307$	6.60705e8	1.30324	0.651619	+	0.758547i	$0.274090\pi$
$308$	1.52177e9	2.96771
$309$	−2.25006e8	−0.433849
$310$	3.50656e8	0.668522
$311$	3.85192e8	0.726133	0.363066	−	0.931763i	$-0.381730\pi$
$311$	3.85192e8	0.726133	0.363066	+	0.931763i	$0.381730\pi$
$312$	−7.10768e7	−0.132491
$313$	−2.51118e8	−0.462885	−0.231443	−	0.972849i	$-0.574345\pi$
$313$	−2.51118e8	−0.462885	−0.231443	+	0.972849i	$0.574345\pi$
$314$	4.91118e7	0.0895225
$315$	4.70140e8	0.847501
$316$	−2.50588e8	−0.446741
$317$	−8.90008e8	−1.56923	−0.784615	−	0.619983i	$-0.787139\pi$
$317$	−8.90008e8	−1.56923	−0.784615	+	0.619983i	$0.787139\pi$
$318$	7.80839e7	0.136165
$319$	−1.00917e9	−1.74059
$320$	1.63649e8	0.279182
$321$	−3.22213e8	−0.543721
$322$	9.06775e7	0.151358
$323$	1.73971e8	0.287256
$324$	−5.77625e7	−0.0943493
$325$	−1.68227e8	−0.271833
$326$	−3.13590e8	−0.501303
$327$	−4.21848e8	−0.667174
$328$	−5.67255e8	−0.887605
$329$	−6.76098e8	−1.04670
$330$	3.72434e8	0.570493
$331$	7.44639e8	1.12862	0.564310	−	0.825563i	$-0.309142\pi$
$331$	7.44639e8	1.12862	0.564310	+	0.825563i	$0.309142\pi$
$332$	9.87794e8	1.48144
$333$	2.51778e7	0.0373648
$334$	−1.69472e8	−0.248877
$335$	−1.62251e9	−2.35793
$336$	4.27791e8	0.615239
$337$	3.86500e8	0.550104	0.275052	−	0.961429i	$-0.411305\pi$
$337$	3.86500e8	0.550104	0.275052	+	0.961429i	$0.411305\pi$
$338$	−2.47584e8	−0.348750
$339$	−1.60559e8	−0.223840
$340$	−2.97902e8	−0.411052
$341$	−1.73242e9	−2.36599
$342$	−7.73179e7	−0.104517
$343$	−2.08504e9	−2.78987
$344$	4.02789e8	0.533486
$345$	−1.24916e8	−0.163776
$346$	1.81742e8	0.235879
$347$	−9.30082e8	−1.19500	−0.597500	−	0.801869i	$-0.703839\pi$
$347$	−9.30082e8	−1.19500	−0.597500	+	0.801869i	$0.703839\pi$
$348$	−3.58750e8	−0.456314
$349$	1.37822e9	1.73552	0.867760	−	0.496983i	$-0.165559\pi$
$349$	1.37822e9	1.73552	0.867760	+	0.496983i	$0.165559\pi$
$350$	−4.95352e8	−0.617555
$351$	4.98182e7	0.0614912
$352$	1.43791e9	1.75724
$353$	−6.21887e8	−0.752488	−0.376244	−	0.926521i	$-0.622785\pi$
$353$	−6.21887e8	−0.752488	−0.376244	+	0.926521i	$0.622785\pi$
$354$	8.39714e7	0.100605
$355$	−7.30507e8	−0.866613
$356$	2.92037e8	0.343054
$357$	−3.30069e8	−0.383943
$358$	4.54638e8	0.523691
$359$	5.76331e8	0.657417	0.328709	−	0.944431i	$-0.393387\pi$
$359$	5.76331e8	0.657417	0.328709	+	0.944431i	$0.393387\pi$
$360$	2.88314e8	0.325692
$361$	−3.11324e8	−0.348287
$362$	−9.69603e7	−0.107427
$363$	−1.31386e9	−1.44170
$364$	−4.66570e8	−0.507064
$365$	2.44176e9	2.62832
$366$	−2.14165e8	−0.228331
$367$	7.04985e8	0.744473	0.372236	−	0.928138i	$-0.378591\pi$
$367$	7.04985e8	0.744473	0.372236	+	0.928138i	$0.378591\pi$
$368$	−1.13664e8	−0.118893
$369$	3.97592e8	0.411951
$370$	−5.77094e7	−0.0592299
$371$	1.11620e9	1.13483
$372$	−6.15856e8	−0.620267
$373$	1.32904e9	1.32604	0.663021	−	0.748601i	$-0.269274\pi$
$373$	1.32904e9	1.32604	0.663021	+	0.748601i	$0.269274\pi$
$374$	−2.61473e8	−0.258450
$375$	−1.19703e8	−0.117218
$376$	−4.14618e8	−0.402245
$377$	3.09409e8	0.297398
$378$	1.46692e8	0.139697
$379$	1.08570e9	1.02441	0.512206	−	0.858863i	$-0.328829\pi$
$379$	1.08570e9	1.02441	0.512206	+	0.858863i	$0.328829\pi$
$380$	−9.97533e8	−0.932577
$381$	2.23860e8	0.207367
$382$	−8.69442e7	−0.0798029
$383$	1.93558e9	1.76041	0.880206	−	0.474591i	$-0.157404\pi$
$383$	1.93558e9	1.76041	0.880206	+	0.474591i	$0.157404\pi$
$384$	6.53033e8	0.588540
$385$	5.32388e9	4.75462
$386$	6.04161e8	0.534684
$387$	−2.82318e8	−0.247599
$388$	−1.23436e8	−0.107283
$389$	8.59112e8	0.739990	0.369995	−	0.929034i	$-0.379359\pi$
$389$	8.59112e8	0.739990	0.369995	+	0.929034i	$0.379359\pi$
$390$	−1.14187e8	−0.0974746
$391$	8.76992e7	0.0741954
$392$	−2.13520e9	−1.79035
$393$	7.68301e8	0.638495
$394$	−3.37525e8	−0.278016
$395$	−8.76678e8	−0.715731
$396$	−6.54105e8	−0.529315
$397$	−8.15479e8	−0.654103	−0.327051	−	0.945007i	$-0.606055\pi$
$397$	−8.15479e8	−0.654103	−0.327051	+	0.945007i	$0.606055\pi$
$398$	−3.10070e8	−0.246529
$399$	−1.10525e9	−0.871071
$400$	6.20921e8	0.485094
$401$	−2.24092e8	−0.173548	−0.0867742	−	0.996228i	$-0.527656\pi$
$401$	−2.24092e8	−0.173548	−0.0867742	+	0.996228i	$0.527656\pi$
$402$	−5.06253e8	−0.388666
$403$	5.31155e8	0.404253
$404$	7.02383e8	0.529955
$405$	−2.02081e8	−0.151159
$406$	9.11072e8	0.675634
$407$	2.85114e8	0.209623
$408$	−2.02415e8	−0.147548
$409$	−9.61531e8	−0.694915	−0.347458	−	0.937696i	$-0.612955\pi$
$409$	−9.61531e8	−0.694915	−0.347458	+	0.937696i	$0.612955\pi$
$410$	−9.11313e8	−0.653017
$411$	1.70737e8	0.121306
$412$	−9.05776e8	−0.638088
$413$	1.20036e9	0.838465
$414$	−3.89761e7	−0.0269958
$415$	3.45578e9	2.37343
$416$	−4.40858e8	−0.300243
$417$	−1.59785e9	−1.07909
$418$	−8.75551e8	−0.586360
$419$	3.49956e8	0.232415	0.116208	−	0.993225i	$-0.462926\pi$
$419$	3.49956e8	0.232415	0.116208	+	0.993225i	$0.462926\pi$
$420$	1.89258e9	1.24647
$421$	−6.11982e8	−0.399716	−0.199858	−	0.979825i	$-0.564048\pi$
$421$	−6.11982e8	−0.399716	−0.199858	+	0.979825i	$0.564048\pi$
$422$	−4.55447e8	−0.295015
$423$	2.90608e8	0.186688
$424$	6.84508e8	0.436112
$425$	−4.79082e8	−0.302725
$426$	−2.27932e8	−0.142847
$427$	−3.06145e9	−1.90296
$428$	−1.29709e9	−0.799683
$429$	5.64143e8	0.344976
$430$	6.47094e8	0.392489
$431$	−4.65095e8	−0.279815	−0.139908	−	0.990165i	$-0.544681\pi$
$431$	−4.65095e8	−0.279815	−0.139908	+	0.990165i	$0.544681\pi$
$432$	−1.83878e8	−0.109733
$433$	−1.89221e9	−1.12012	−0.560058	−	0.828454i	$-0.689221\pi$
$433$	−1.89221e9	−1.12012	−0.560058	+	0.828454i	$0.689221\pi$
$434$	1.56401e9	0.918388
$435$	−1.25508e9	−0.731069
$436$	−1.69818e9	−0.981253
$437$	2.93663e8	0.168331
$438$	7.61875e8	0.433236
$439$	−1.34468e9	−0.758568	−0.379284	−	0.925280i	$-0.623830\pi$
$439$	−1.34468e9	−0.758568	−0.379284	+	0.925280i	$0.623830\pi$
$440$	3.26487e9	1.82718
$441$	1.49658e9	0.830929
$442$	8.01670e7	0.0441589
$443$	5.88728e8	0.321738	0.160869	−	0.986976i	$-0.448570\pi$
$443$	5.88728e8	0.321738	0.160869	+	0.986976i	$0.448570\pi$
$444$	1.01355e8	0.0549546
$445$	1.02168e9	0.549612
$446$	−1.11709e8	−0.0596234
$447$	1.21560e9	0.643747
$448$	7.29915e8	0.383530
$449$	3.31247e9	1.72699	0.863494	−	0.504358i	$-0.168271\pi$
$449$	3.31247e9	1.72699	0.863494	+	0.504358i	$0.168271\pi$
$450$	2.12918e8	0.110146
$451$	4.50235e9	2.31111
$452$	−6.46343e8	−0.329214
$453$	−7.01865e8	−0.354740
$454$	6.61521e8	0.331778
$455$	−1.63229e9	−0.812375
$456$	−6.77793e8	−0.334750
$457$	−1.96359e9	−0.962374	−0.481187	−	0.876618i	$-0.659794\pi$
$457$	−1.96359e9	−0.962374	−0.481187	+	0.876618i	$0.659794\pi$
$458$	−1.24314e9	−0.604633
$459$	1.41874e8	0.0684793
$460$	−5.02857e8	−0.240875
$461$	3.88159e8	0.184525	0.0922627	−	0.995735i	$-0.470590\pi$
$461$	3.88159e8	0.184525	0.0922627	+	0.995735i	$0.470590\pi$
$462$	1.66115e9	0.783721
$463$	−1.02703e9	−0.480894	−0.240447	−	0.970662i	$-0.577294\pi$
$463$	−1.02703e9	−0.480894	−0.240447	+	0.970662i	$0.577294\pi$
$464$	−1.14202e9	−0.530716
$465$	−2.15456e9	−0.993740
$466$	4.67519e8	0.214017
$467$	2.09348e9	0.951174	0.475587	−	0.879669i	$-0.342236\pi$
$467$	2.09348e9	0.951174	0.475587	+	0.879669i	$0.342236\pi$
$468$	2.00547e8	0.0904388
$469$	−7.23680e9	−3.23923
$470$	−6.66097e8	−0.295934
$471$	−3.01761e8	−0.133073
$472$	7.36120e8	0.322219
$473$	−3.19697e9	−1.38907
$474$	−2.73540e8	−0.117977
$475$	−1.60422e9	−0.686809
$476$	−1.32872e9	−0.564688
$477$	−4.79776e8	−0.202406
$478$	6.44853e8	0.270062
$479$	−2.50915e9	−1.04317	−0.521583	−	0.853201i	$-0.674658\pi$
$479$	−2.50915e9	−1.04317	−0.521583	+	0.853201i	$0.674658\pi$
$480$	1.78828e9	0.738060
$481$	−8.74151e7	−0.0358161
$482$	5.26885e8	0.214314
$483$	−5.57156e8	−0.224989
$484$	−5.28903e9	−2.12040
$485$	−4.31836e8	−0.171879
$486$	−6.30530e7	−0.0249160
$487$	−8.90103e8	−0.349212	−0.174606	−	0.984638i	$-0.555865\pi$
$487$	−8.90103e8	−0.349212	−0.174606	+	0.984638i	$0.555865\pi$
$488$	−1.87744e9	−0.731300
$489$	1.92681e9	0.745175
$490$	−3.43027e9	−1.31717
$491$	−9.02657e8	−0.344142	−0.172071	−	0.985085i	$-0.555046\pi$
$491$	−9.02657e8	−0.344142	−0.172071	+	0.985085i	$0.555046\pi$
$492$	1.60054e9	0.605881
$493$	8.81147e8	0.331195
$494$	2.68441e8	0.100186
$495$	−2.28837e9	−0.848024
$496$	−1.96048e9	−0.721401
$497$	−3.25824e9	−1.19052
$498$	1.07827e9	0.391222
$499$	3.35336e9	1.20817	0.604086	−	0.796919i	$-0.293538\pi$
$499$	3.35336e9	1.20817	0.604086	+	0.796919i	$0.293538\pi$
$500$	−4.81873e8	−0.172400
$501$	1.04130e9	0.369949
$502$	1.26695e9	0.446989
$503$	1.01550e9	0.355790	0.177895	−	0.984049i	$-0.443071\pi$
$503$	1.01550e9	0.355790	0.177895	+	0.984049i	$0.443071\pi$
$504$	1.28595e9	0.447422
$505$	2.45727e9	0.849050
$506$	−4.41366e8	−0.151451
$507$	1.52125e9	0.518408
$508$	9.01164e8	0.304987
$509$	−1.46602e9	−0.492751	−0.246376	−	0.969174i	$-0.579240\pi$
$509$	−1.46602e9	−0.492751	−0.246376	+	0.969174i	$0.579240\pi$
$510$	−3.25187e8	−0.108552
$511$	1.08909e10	3.61068
$512$	2.87090e9	0.945308
$513$	4.75070e8	0.155363
$514$	1.03622e9	0.336573
$515$	−3.16884e9	−1.02229
$516$	−1.13649e9	−0.364159
$517$	3.29086e9	1.04735
$518$	−2.57398e8	−0.0813676
$519$	−1.11669e9	−0.350628
$520$	−1.00100e9	−0.312193
$521$	−5.80894e9	−1.79955	−0.899777	−	0.436350i	$-0.856271\pi$
$521$	−5.80894e9	−1.79955	−0.899777	+	0.436350i	$0.856271\pi$
$522$	−3.91607e8	−0.120505
$523$	−3.21788e8	−0.0983590	−0.0491795	−	0.998790i	$-0.515661\pi$
$523$	−3.21788e8	−0.0983590	−0.0491795	+	0.998790i	$0.515661\pi$
$524$	3.09285e9	0.939073
$525$	3.04362e9	0.917980
$526$	−7.30747e8	−0.218936
$527$	1.51264e9	0.450194
$528$	−2.08224e9	−0.615619
$529$	1.48036e8	0.0434783
$530$	1.09968e9	0.320850
$531$	−5.15951e8	−0.149547
$532$	−4.44924e9	−1.28114
$533$	−1.38041e9	−0.394877
$534$	3.18784e8	0.0905946
$535$	−4.53785e9	−1.28118
$536$	−4.43797e9	−1.24482
$537$	−2.79346e9	−0.778454
$538$	−2.46491e9	−0.682438
$539$	1.69473e10	4.66165
$540$	−8.13491e8	−0.222318
$541$	−3.62728e9	−0.984897	−0.492448	−	0.870342i	$-0.663898\pi$
$541$	−3.62728e9	−0.984897	−0.492448	+	0.870342i	$0.663898\pi$
$542$	2.54020e9	0.685283
$543$	5.95759e8	0.159688
$544$	−1.25549e9	−0.334363
$545$	−5.94105e9	−1.57208
$546$	−5.09304e8	−0.133907
$547$	6.12918e9	1.60120	0.800602	−	0.599196i	$-0.204513\pi$
$547$	6.12918e9	1.60120	0.800602	+	0.599196i	$0.204513\pi$
$548$	6.87314e8	0.178412
$549$	1.31591e9	0.339408
$550$	2.41109e9	0.617936
$551$	2.95055e9	0.751401
$552$	−3.41676e8	−0.0864625
$553$	−3.91020e9	−0.983243
$554$	−2.35716e9	−0.588985
$555$	3.54588e8	0.0880437
$556$	−6.43225e9	−1.58709
$557$	8.91017e8	0.218470	0.109235	−	0.994016i	$-0.465160\pi$
$557$	8.91017e8	0.218470	0.109235	+	0.994016i	$0.465160\pi$
$558$	−6.72262e8	−0.163802
$559$	9.80183e8	0.237337
$560$	6.02473e9	1.44971
$561$	1.60659e9	0.384180
$562$	2.70379e9	0.642533
$563$	5.43184e9	1.28283	0.641413	−	0.767196i	$-0.278348\pi$
$563$	5.43184e9	1.28283	0.641413	+	0.767196i	$0.278348\pi$
$564$	1.16986e9	0.274573
$565$	−2.26122e9	−0.527440
$566$	6.66933e8	0.154606
$567$	−9.01331e8	−0.207656
$568$	−1.99812e9	−0.457512
$569$	9.75425e8	0.221973	0.110987	−	0.993822i	$-0.464599\pi$
$569$	9.75425e8	0.221973	0.110987	+	0.993822i	$0.464599\pi$
$570$	−1.08890e9	−0.246277
$571$	7.11712e9	1.59984	0.799922	−	0.600104i	$-0.204874\pi$
$571$	7.11712e9	1.59984	0.799922	+	0.600104i	$0.204874\pi$
$572$	2.27100e9	0.507377
$573$	5.34217e8	0.118625
$574$	−4.06468e9	−0.897088
$575$	−8.08689e8	−0.177396
$576$	−3.13740e8	−0.0684056
$577$	−8.44988e9	−1.83120	−0.915599	−	0.402092i	$-0.868283\pi$
$577$	−8.44988e9	−1.83120	−0.915599	+	0.402092i	$0.868283\pi$
$578$	−1.57484e9	−0.339225
$579$	−3.71219e9	−0.794794
$580$	−5.05240e9	−1.07523
$581$	1.54136e10	3.26053
$582$	−1.34741e8	−0.0283315
$583$	−5.43300e9	−1.13553
$584$	6.67884e9	1.38757
$585$	7.01608e8	0.144894
$586$	−9.09788e8	−0.186766
$587$	5.11213e8	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$587$	5.11213e8	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$588$	6.02458e9	1.22210
$589$	5.06513e9	1.02138
$590$	1.18260e9	0.237059
$591$	2.07388e9	0.413263
$592$	3.22647e8	0.0639149
$593$	7.96058e7	0.0156766	0.00783832	−	0.999969i	$-0.497505\pi$
$593$	7.96058e7	0.0156766	0.00783832	+	0.999969i	$0.497505\pi$
$594$	−7.14014e8	−0.139783
$595$	−4.64849e9	−0.904695
$596$	4.89349e9	0.946797
$597$	1.90518e9	0.366459
$598$	1.35322e8	0.0258769
$599$	−9.51346e9	−1.80861	−0.904304	−	0.426889i	$-0.859610\pi$
$599$	−9.51346e9	−1.80861	−0.904304	+	0.426889i	$0.859610\pi$
$600$	1.86650e9	0.352777
$601$	−2.90990e9	−0.546786	−0.273393	−	0.961902i	$-0.588146\pi$
$601$	−2.90990e9	−0.546786	−0.273393	+	0.961902i	$0.588146\pi$
$602$	2.88620e9	0.539186
$603$	3.11060e9	0.577742
$604$	−2.82541e9	−0.521737
$605$	−1.85036e10	−3.39713
$606$	7.66714e8	0.139952
$607$	8.21259e9	1.49046	0.745229	−	0.666809i	$-0.232340\pi$
$607$	8.21259e9	1.49046	0.745229	+	0.666809i	$0.232340\pi$
$608$	−4.20405e9	−0.758587
$609$	−5.59796e9	−1.00431
$610$	−3.01616e9	−0.538022
$611$	−1.00897e9	−0.178950
$612$	5.71124e8	0.100717
$613$	1.10993e10	1.94619	0.973096	−	0.230398i	$-0.0740029\pi$
$613$	1.10993e10	1.94619	0.973096	+	0.230398i	$0.0740029\pi$
$614$	2.90332e9	0.506180
$615$	5.59944e9	0.970693
$616$	1.45621e10	2.51011
$617$	7.42842e9	1.27321	0.636603	−	0.771192i	$-0.280339\pi$
$617$	7.42842e9	1.27321	0.636603	+	0.771192i	$0.280339\pi$
$618$	−9.88735e8	−0.168508
$619$	−1.80311e9	−0.305566	−0.152783	−	0.988260i	$-0.548823\pi$
$619$	−1.80311e9	−0.305566	−0.152783	+	0.988260i	$0.548823\pi$
$620$	−8.67333e9	−1.46155
$621$	2.39483e8	0.0401286
$622$	1.69264e9	0.282032
$623$	4.55696e9	0.755035
$624$	6.38409e8	0.105185
$625$	−6.87846e9	−1.12697
$626$	−1.10348e9	−0.179786
$627$	5.37970e9	0.871609
$628$	−1.21476e9	−0.195718
$629$	−2.48944e8	−0.0398864
$630$	2.06592e9	0.329171
$631$	−8.61808e9	−1.36555	−0.682776	−	0.730628i	$-0.739227\pi$
$631$	−8.61808e9	−1.36555	−0.682776	+	0.730628i	$0.739227\pi$
$632$	−2.39793e9	−0.377857
$633$	2.79843e9	0.438533
$634$	−3.91094e9	−0.609493
$635$	3.15270e9	0.488624
$636$	−1.93137e9	−0.297691
$637$	−5.19599e9	−0.796490
$638$	−4.43457e9	−0.676051
$639$	1.40050e9	0.212339
$640$	9.19690e9	1.38679
$641$	2.12141e9	0.318143	0.159071	−	0.987267i	$-0.449150\pi$
$641$	2.12141e9	0.318143	0.159071	+	0.987267i	$0.449150\pi$
$642$	−1.41589e9	−0.211182
$643$	8.49719e9	1.26048	0.630241	−	0.776399i	$-0.282956\pi$
$643$	8.49719e9	1.26048	0.630241	+	0.776399i	$0.282956\pi$
$644$	−2.24287e9	−0.330905
$645$	−3.97598e9	−0.583426
$646$	7.64478e8	0.111571
$647$	−8.51590e9	−1.23613	−0.618067	−	0.786125i	$-0.712084\pi$
$647$	−8.51590e9	−1.23613	−0.618067	+	0.786125i	$0.712084\pi$
$648$	−5.52742e8	−0.0798013
$649$	−5.84265e9	−0.838983
$650$	−7.39233e8	−0.105581
$651$	−9.60987e9	−1.36516
$652$	7.75651e9	1.09597
$653$	−6.60595e9	−0.928409	−0.464204	−	0.885728i	$-0.653660\pi$
$653$	−6.60595e9	−0.928409	−0.464204	+	0.885728i	$0.653660\pi$
$654$	−1.85372e9	−0.259132
$655$	1.08203e10	1.50450
$656$	5.09506e9	0.704670
$657$	−4.68124e9	−0.643994
$658$	−2.97096e9	−0.406542
$659$	7.39305e9	1.00629	0.503147	−	0.864201i	$-0.332175\pi$
$659$	7.39305e9	1.00629	0.503147	+	0.864201i	$0.332175\pi$
$660$	−9.21199e9	−1.24724
$661$	4.78815e9	0.644856	0.322428	−	0.946594i	$-0.395501\pi$
$661$	4.78815e9	0.644856	0.322428	+	0.946594i	$0.395501\pi$
$662$	3.27214e9	0.438358
$663$	−4.92575e8	−0.0656410
$664$	9.45241e9	1.25301
$665$	−1.55656e10	−2.05253
$666$	1.10638e8	0.0145126
$667$	1.48737e9	0.194079
$668$	4.19181e9	0.544106
$669$	6.86381e8	0.0886286
$670$	−7.12975e9	−0.915825
$671$	1.49014e10	1.90413
$672$	7.97619e9	1.01392
$673$	−2.29110e9	−0.289729	−0.144864	−	0.989452i	$-0.546275\pi$
$673$	−2.29110e9	−0.289729	−0.144864	+	0.989452i	$0.546275\pi$
$674$	1.69838e9	0.213662
$675$	−1.30825e9	−0.163729
$676$	6.12388e9	0.762453
$677$	−6.26212e9	−0.775641	−0.387821	−	0.921735i	$-0.626772\pi$
$677$	−6.26212e9	−0.775641	−0.387821	+	0.921735i	$0.626772\pi$
$678$	−7.05542e8	−0.0869399
$679$	−1.92610e9	−0.236121
$680$	−2.85069e9	−0.347671
$681$	−4.06463e9	−0.493180
$682$	−7.61272e9	−0.918955
$683$	−5.24593e9	−0.630014	−0.315007	−	0.949089i	$-0.602007\pi$
$683$	−5.24593e9	−0.630014	−0.315007	+	0.949089i	$0.602007\pi$
$684$	1.91243e9	0.228501
$685$	2.40455e9	0.285836
$686$	−9.16222e9	−1.08359
$687$	7.63832e9	0.898771
$688$	−3.61784e9	−0.423535
$689$	1.66574e9	0.194017
$690$	−5.48914e8	−0.0636110
$691$	−2.06738e9	−0.238367	−0.119184	−	0.992872i	$-0.538028\pi$
$691$	−2.06738e9	−0.238367	−0.119184	+	0.992872i	$0.538028\pi$
$692$	−4.49531e9	−0.515690
$693$	−1.02067e10	−1.16498
$694$	−4.08703e9	−0.464141
$695$	−2.25031e10	−2.54270
$696$	−3.43295e9	−0.385954
$697$	−3.93118e9	−0.439752
$698$	6.05627e9	0.674080
$699$	−2.87260e9	−0.318131
$700$	1.22523e10	1.35013
$701$	1.27196e10	1.39463	0.697315	−	0.716765i	$-0.254378\pi$
$701$	1.27196e10	1.39463	0.697315	+	0.716765i	$0.254378\pi$
$702$	2.18915e8	0.0238833
$703$	−8.33596e8	−0.0904924
$704$	−3.55281e9	−0.383767
$705$	4.09274e9	0.439898
$706$	−2.73274e9	−0.292268
$707$	1.09600e10	1.16639
$708$	−2.07700e9	−0.219948
$709$	−1.18222e10	−1.24576	−0.622882	−	0.782315i	$-0.714039\pi$
$709$	−1.18222e10	−1.24576	−0.622882	+	0.782315i	$0.714039\pi$
$710$	−3.21005e9	−0.336595
$711$	1.68073e9	0.175369
$712$	2.79456e9	0.290157
$713$	2.55334e9	0.263812
$714$	−1.45041e9	−0.149124
$715$	7.94503e9	0.812876
$716$	−1.12453e10	−1.14492
$717$	−3.96221e9	−0.401440
$718$	2.53255e9	0.255342
$719$	−1.46693e10	−1.47183	−0.735916	−	0.677073i	$-0.763248\pi$
$719$	−1.46693e10	−1.47183	−0.735916	+	0.677073i	$0.763248\pi$
$720$	−2.58962e9	−0.258567
$721$	−1.41338e10	−1.40438
$722$	−1.36804e9	−0.135275
$723$	−3.23737e9	−0.318573
$724$	2.39827e9	0.234862
$725$	−8.12520e9	−0.791865
$726$	−5.77345e9	−0.559961
$727$	−4.84256e9	−0.467417	−0.233708	−	0.972307i	$-0.575086\pi$
$727$	−4.84256e9	−0.467417	−0.233708	+	0.972307i	$0.575086\pi$
$728$	−4.46471e9	−0.428878
$729$	3.87420e8	0.0370370
$730$	1.07298e10	1.02085
$731$	2.79140e9	0.264309
$732$	5.29728e9	0.499188
$733$	−2.13950e8	−0.0200654	−0.0100327	−	0.999950i	$-0.503194\pi$
$733$	−2.13950e8	−0.0200654	−0.0100327	+	0.999950i	$0.503194\pi$
$734$	3.09790e9	0.289155
$735$	2.10768e10	1.95794
$736$	−2.11927e9	−0.195936
$737$	3.52246e10	3.24123
$738$	1.74713e9	0.160003
$739$	5.64170e9	0.514226	0.257113	−	0.966381i	$-0.417229\pi$
$739$	5.64170e9	0.514226	0.257113	+	0.966381i	$0.417229\pi$
$740$	1.42742e9	0.129491
$741$	−1.64940e9	−0.148923
$742$	4.90487e9	0.440772
$743$	−4.68943e9	−0.419430	−0.209715	−	0.977763i	$-0.567254\pi$
$743$	−4.68943e9	−0.419430	−0.209715	+	0.977763i	$0.567254\pi$
$744$	−5.89326e9	−0.524627
$745$	1.71198e10	1.51688
$746$	5.84016e9	0.515038
$747$	−6.62526e9	−0.581541
$748$	6.46743e9	0.565036
$749$	−2.02399e10	−1.76004
$750$	−5.26007e8	−0.0455279
$751$	1.10397e10	0.951082	0.475541	−	0.879694i	$-0.342252\pi$
$751$	1.10397e10	0.951082	0.475541	+	0.879694i	$0.342252\pi$
$752$	3.72408e9	0.319342
$753$	−7.78461e9	−0.664438
$754$	1.35963e9	0.115510
$755$	−9.88462e9	−0.835883
$756$	−3.62837e9	−0.305412
$757$	8.27621e9	0.693419	0.346710	−	0.937972i	$-0.387299\pi$
$757$	8.27621e9	0.693419	0.346710	+	0.937972i	$0.387299\pi$
$758$	4.77088e9	0.397884
$759$	2.71191e9	0.225128
$760$	−9.54561e9	−0.788780
$761$	−8.43273e9	−0.693621	−0.346810	−	0.937935i	$-0.612735\pi$
$761$	−8.43273e9	−0.693621	−0.346810	+	0.937935i	$0.612735\pi$
$762$	9.83702e8	0.0805418
$763$	−2.64985e10	−2.15966
$764$	2.15053e9	0.174469
$765$	1.99807e9	0.161360
$766$	8.50545e9	0.683749
$767$	1.79134e9	0.143349
$768$	1.38224e9	0.110108
$769$	8.41447e9	0.667244	0.333622	−	0.942707i	$-0.391729\pi$
$769$	8.41447e9	0.667244	0.333622	+	0.942707i	$0.391729\pi$
$770$	2.33946e10	1.84670
$771$	−6.36689e9	−0.500307
$772$	−1.49437e10	−1.16895
$773$	2.62024e9	0.204039	0.102019	−	0.994782i	$-0.467470\pi$
$773$	2.62024e9	0.204039	0.102019	+	0.994782i	$0.467470\pi$
$774$	−1.24058e9	−0.0961682
$775$	−1.39483e10	−1.07638
$776$	−1.18118e9	−0.0907404
$777$	1.58155e9	0.120951
$778$	3.77517e9	0.287414
$779$	−1.31637e10	−0.997689
$780$	2.82437e9	0.213104
$781$	1.58593e10	1.19125
$782$	3.85374e8	0.0288177
$783$	2.40618e9	0.179127
$784$	1.91783e10	1.42136
$785$	−4.24981e9	−0.313564
$786$	3.37612e9	0.247993
$787$	−2.25378e9	−0.164817	−0.0824083	−	0.996599i	$-0.526261\pi$
$787$	−2.25378e9	−0.164817	−0.0824083	+	0.996599i	$0.526261\pi$
$788$	8.34854e9	0.607811
$789$	4.48998e9	0.325443
$790$	−3.85236e9	−0.277992
$791$	−1.00856e10	−0.724576
$792$	−6.25927e9	−0.447698
$793$	−4.56872e9	−0.325341
$794$	−3.58344e9	−0.254055
$795$	−6.75686e9	−0.476936
$796$	7.66944e9	0.538974
$797$	2.06342e10	1.44372	0.721861	−	0.692038i	$-0.243287\pi$
$797$	2.06342e10	1.44372	0.721861	+	0.692038i	$0.243287\pi$
$798$	−4.85675e9	−0.338326
$799$	−2.87338e9	−0.199287
$800$	1.15771e10	0.799439
$801$	−1.95873e9	−0.134667
$802$	−9.84720e8	−0.0674066
$803$	−5.30105e10	−3.61291
$804$	1.25220e10	0.849720
$805$	−7.84663e9	−0.530148
$806$	2.33404e9	0.157013
$807$	1.51453e10	1.01443
$808$	6.72125e9	0.448240
$809$	−1.40485e8	−0.00932849	−0.00466424	−	0.999989i	$-0.501485\pi$
$809$	−1.40485e8	−0.00932849	−0.00466424	+	0.999989i	$0.501485\pi$
$810$	−8.87998e8	−0.0587103
$811$	2.82662e10	1.86078	0.930389	−	0.366573i	$-0.119469\pi$
$811$	2.82662e10	1.86078	0.930389	+	0.366573i	$0.119469\pi$
$812$	−2.25350e10	−1.47710
$813$	−1.56079e10	−1.01866
$814$	1.25287e9	0.0814179
$815$	2.71360e10	1.75588
$816$	1.81809e9	0.117138
$817$	9.34709e9	0.599652
$818$	−4.22523e9	−0.269907
$819$	3.12935e9	0.199049
$820$	2.25409e10	1.42766
$821$	−3.06226e9	−0.193126	−0.0965629	−	0.995327i	$-0.530785\pi$
$821$	−3.06226e9	−0.193126	−0.0965629	+	0.995327i	$0.530785\pi$
$822$	7.50265e8	0.0471154
$823$	1.34157e10	0.838910	0.419455	−	0.907776i	$-0.362221\pi$
$823$	1.34157e10	0.838910	0.419455	+	0.907776i	$0.362221\pi$
$824$	−8.66756e9	−0.539699
$825$	−1.48146e10	−0.918547
$826$	5.27469e9	0.325662
$827$	2.27540e10	1.39890	0.699452	−	0.714679i	$-0.253427\pi$
$827$	2.27540e10	1.39890	0.699452	+	0.714679i	$0.253427\pi$
$828$	9.64056e8	0.0590196
$829$	1.30559e10	0.795911	0.397956	−	0.917405i	$-0.369720\pi$
$829$	1.30559e10	0.795911	0.397956	+	0.917405i	$0.369720\pi$
$830$	1.51856e10	0.921847
$831$	1.44832e10	0.875512
$832$	1.08928e9	0.0655704
$833$	−1.47973e10	−0.887005
$834$	−7.02138e9	−0.419123
$835$	1.46649e10	0.871721
$836$	2.16564e10	1.28193
$837$	4.13062e9	0.243487
$838$	1.53780e9	0.0902707
$839$	2.18667e10	1.27825	0.639125	−	0.769103i	$-0.279296\pi$
$839$	2.18667e10	1.27825	0.639125	+	0.769103i	$0.279296\pi$
$840$	1.81105e10	1.05427
$841$	−2.30567e9	−0.133663
$842$	−2.68921e9	−0.155250
$843$	−1.66130e10	−0.955109
$844$	1.12653e10	0.644976
$845$	2.14243e10	1.22154
$846$	1.27701e9	0.0725101
$847$	−8.25305e10	−4.66684
$848$	−6.14822e9	−0.346229
$849$	−4.09788e9	−0.229817
$850$	−2.10522e9	−0.117579
$851$	−4.20217e8	−0.0233733
$852$	5.63779e9	0.312299
$853$	−3.58664e9	−0.197863	−0.0989317	−	0.995094i	$-0.531543\pi$
$853$	−3.58664e9	−0.197863	−0.0989317	+	0.995094i	$0.531543\pi$
$854$	−1.34528e10	−0.739114
$855$	6.69058e9	0.366085
$856$	−1.24122e10	−0.676377
$857$	−1.45611e10	−0.790242	−0.395121	−	0.918629i	$-0.629297\pi$
$857$	−1.45611e10	−0.790242	−0.395121	+	0.918629i	$0.629297\pi$
$858$	2.47900e9	0.133989
$859$	7.60133e9	0.409179	0.204590	−	0.978848i	$-0.434414\pi$
$859$	7.60133e9	0.409179	0.204590	+	0.978848i	$0.434414\pi$
$860$	−1.60056e10	−0.858079
$861$	2.49749e10	1.33350
$862$	−2.04376e9	−0.108681
$863$	2.02701e10	1.07354	0.536769	−	0.843729i	$-0.319645\pi$
$863$	2.02701e10	1.07354	0.536769	+	0.843729i	$0.319645\pi$
$864$	−3.42842e9	−0.180840
$865$	−1.57267e10	−0.826195
$866$	−8.31490e9	−0.435055
$867$	9.67637e9	0.504250
$868$	−3.86852e10	−2.00782
$869$	1.90326e10	0.983850
$870$	−5.51515e9	−0.283949
$871$	−1.07998e10	−0.553797
$872$	−1.62503e10	−0.829951
$873$	8.27898e8	0.0421140
$874$	1.29044e9	0.0653802
$875$	−7.51918e9	−0.379439
$876$	−1.88447e10	−0.947161
$877$	9.69405e9	0.485296	0.242648	−	0.970114i	$-0.421984\pi$
$877$	9.69405e9	0.485296	0.242648	+	0.970114i	$0.421984\pi$
$878$	−5.90891e9	−0.294630
$879$	5.59007e9	0.277624
$880$	−2.93249e10	−1.45060
$881$	1.45932e10	0.719009	0.359505	−	0.933143i	$-0.382946\pi$
$881$	1.45932e10	0.719009	0.359505	+	0.933143i	$0.382946\pi$
$882$	6.57637e9	0.322735
$883$	1.53827e10	0.751917	0.375959	−	0.926636i	$-0.377313\pi$
$883$	1.53827e10	0.751917	0.375959	+	0.926636i	$0.377313\pi$
$884$	−1.98290e9	−0.0965422
$885$	−7.26633e9	−0.352382
$886$	2.58703e9	0.124964
$887$	1.14814e10	0.552413	0.276207	−	0.961098i	$-0.410923\pi$
$887$	1.14814e10	0.552413	0.276207	+	0.961098i	$0.410923\pi$
$888$	9.69887e8	0.0464810
$889$	1.40618e10	0.671253
$890$	4.48955e9	0.213471
$891$	4.38716e9	0.207784
$892$	2.76308e9	0.130351
$893$	−9.62157e9	−0.452133
$894$	5.34169e9	0.250033
$895$	−3.93414e10	−1.83429
$896$	4.10205e10	1.90512
$897$	−8.31465e8	−0.0384654
$898$	1.45559e10	0.670767
$899$	2.56543e10	1.17761
$900$	−5.26643e9	−0.240806
$901$	4.74377e9	0.216066
$902$	1.97845e10	0.897642
$903$	−1.77339e10	−0.801487
$904$	−6.18500e9	−0.278452
$905$	8.39030e9	0.376277
$906$	−3.08418e9	−0.137782
$907$	1.34820e10	0.599967	0.299984	−	0.953944i	$-0.403019\pi$
$907$	1.34820e10	0.599967	0.299984	+	0.953944i	$0.403019\pi$
$908$	−1.63624e10	−0.725350
$909$	−4.71097e9	−0.208035
$910$	−7.17271e9	−0.315528
$911$	−1.94284e10	−0.851378	−0.425689	−	0.904869i	$-0.639968\pi$
$911$	−1.94284e10	−0.851378	−0.425689	+	0.904869i	$0.639968\pi$
$912$	6.08791e9	0.265758
$913$	−7.50246e10	−3.26254
$914$	−8.62854e9	−0.373788
$915$	1.85324e10	0.799757
$916$	3.07486e10	1.32188
$917$	4.82611e10	2.06683
$918$	6.23434e8	0.0265975
$919$	2.91697e10	1.23973	0.619865	−	0.784708i	$-0.287187\pi$
$919$	2.91697e10	1.23973	0.619865	+	0.784708i	$0.287187\pi$
$920$	−4.81195e9	−0.203734
$921$	−1.78390e10	−0.752424
$922$	1.70567e9	0.0716701
$923$	−4.86240e9	−0.203538
$924$	−4.10878e10	−1.71341
$925$	2.29555e9	0.0953656
$926$	−4.51304e9	−0.186780
$927$	6.07515e9	0.250483
$928$	−2.12931e10	−0.874623
$929$	5.38416e9	0.220324	0.110162	−	0.993914i	$-0.464863\pi$
$929$	5.38416e9	0.220324	0.110162	+	0.993914i	$0.464863\pi$
$930$	−9.46771e9	−0.385971
$931$	−4.95493e10	−2.01240
$932$	−1.15639e10	−0.467894
$933$	−1.04002e10	−0.419233
$934$	9.19932e9	0.369438
$935$	2.26262e10	0.905254
$936$	1.91907e9	0.0764938
$937$	3.75570e10	1.49143	0.745715	−	0.666265i	$-0.232108\pi$
$937$	3.75570e10	1.49143	0.745715	+	0.666265i	$0.232108\pi$
$938$	−3.18005e10	−1.25812
$939$	6.78020e9	0.267247
$940$	1.64756e10	0.646985
$941$	−6.70025e9	−0.262136	−0.131068	−	0.991373i	$-0.541841\pi$
$941$	−6.70025e9	−0.262136	−0.131068	+	0.991373i	$0.541841\pi$
$942$	−1.32602e9	−0.0516859
$943$	−6.63581e9	−0.257693
$944$	−6.61179e9	−0.255810
$945$	−1.26938e10	−0.489305
$946$	−1.40484e10	−0.539519
$947$	−2.69155e10	−1.02986	−0.514930	−	0.857232i	$-0.672182\pi$
$947$	−2.69155e10	−1.02986	−0.514930	+	0.857232i	$0.672182\pi$
$948$	6.76588e9	0.257926
$949$	1.62529e10	0.617303
$950$	−7.04937e9	−0.266758
$951$	2.40302e10	0.905995
$952$	−1.27148e10	−0.477617
$953$	−4.42943e9	−0.165777	−0.0828883	−	0.996559i	$-0.526414\pi$
$953$	−4.42943e9	−0.165777	−0.0828883	+	0.996559i	$0.526414\pi$
$954$	−2.10827e9	−0.0786151
$955$	7.52357e9	0.279520
$956$	−1.59502e10	−0.590422
$957$	2.72476e10	1.00493
$958$	−1.10259e10	−0.405168
$959$	1.07249e10	0.392670
$960$	−4.41852e9	−0.161186
$961$	1.65275e10	0.600725
$962$	−3.84126e8	−0.0139111
$963$	8.69976e9	0.313917
$964$	−1.30323e10	−0.468544
$965$	−5.22801e10	−1.87280
$966$	−2.44829e9	−0.0873863
$967$	−2.76528e10	−0.983438	−0.491719	−	0.870754i	$-0.663631\pi$
$967$	−2.76528e10	−0.983438	−0.491719	+	0.870754i	$0.663631\pi$
$968$	−5.06119e10	−1.79345
$969$	−4.69723e9	−0.165847
$970$	−1.89761e9	−0.0667583
$971$	−3.12834e10	−1.09660	−0.548298	−	0.836283i	$-0.684724\pi$
$971$	−3.12834e10	−1.09660	−0.548298	+	0.836283i	$0.684724\pi$
$972$	1.55959e9	0.0544726
$973$	−1.00369e11	−3.49306
$974$	−3.91136e9	−0.135635
$975$	4.54212e9	0.156943
$976$	1.68630e10	0.580579
$977$	−3.58056e10	−1.22834	−0.614172	−	0.789172i	$-0.710510\pi$
$977$	−3.58056e10	−1.22834	−0.614172	+	0.789172i	$0.710510\pi$
$978$	8.46693e9	0.289428
$979$	−2.21807e10	−0.755501
$980$	8.48463e10	2.87966
$981$	1.13899e10	0.385193
$982$	−3.96652e9	−0.133666
$983$	1.35443e10	0.454798	0.227399	−	0.973802i	$-0.426978\pi$
$983$	1.35443e10	0.454798	0.227399	+	0.973802i	$0.426978\pi$
$984$	1.53159e10	0.512459
$985$	2.92072e10	0.973784
$986$	3.87200e9	0.128637
$987$	1.82546e10	0.604315
$988$	−6.63978e9	−0.219030
$989$	4.71188e9	0.154884
$990$	−1.00557e10	−0.329375
$991$	5.35491e10	1.74781	0.873905	−	0.486097i	$-0.161580\pi$
$991$	5.35491e10	1.74781	0.873905	+	0.486097i	$0.161580\pi$
$992$	−3.65533e10	−1.18887
$993$	−2.01052e10	−0.651609
$994$	−1.43176e10	−0.462400
$995$	2.68313e10	0.863499
$996$	−2.66704e10	−0.855308
$997$	1.55752e10	0.497737	0.248868	−	0.968537i	$-0.419941\pi$
$997$	1.55752e10	0.497737	0.248868	+	0.968537i	$0.419941\pi$
$998$	1.47356e10	0.469257
$999$	−6.79800e8	−0.0215726

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.c.1.4	✓	7
3.2	odd	2		207.8.a.d.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.c.1.4	✓	7	1.1	even	1	trivial
207.8.a.d.1.4		7	3.2	odd	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+4.39427 q^{2} -27.0000 q^{3} -108.690 q^{4} -380.251 q^{5} -118.645 q^{6} -1696.01 q^{7} -1040.08 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+4.39427 q^{2} -27.0000 q^{3} -108.690 q^{4} -380.251 q^{5} -118.645 q^{6} -1696.01 q^{7} -1040.08 q^{8} +729.000 q^{9} -1670.93 q^{10} +8255.22 q^{11} +2934.64 q^{12} -2531.03 q^{13} -7452.74 q^{14} +10266.8 q^{15} +9341.97 q^{16} -7207.96 q^{17} +3203.42 q^{18} -24136.0 q^{19} +41329.6 q^{20} +45792.4 q^{21} +36275.7 q^{22} -12167.0 q^{23} +28082.2 q^{24} +66465.7 q^{25} -11122.0 q^{26} -19683.0 q^{27} +184340. q^{28} -122247. q^{29} +45115.0 q^{30} -209857. q^{31} +174182. q^{32} -222891. q^{33} -31673.7 q^{34} +644911. q^{35} -79235.3 q^{36} +34537.4 q^{37} -106060. q^{38} +68337.7 q^{39} +395492. q^{40} +545394. q^{41} +201224. q^{42} -387267. q^{43} -897263. q^{44} -277203. q^{45} -53465.1 q^{46} +398639. q^{47} -252233. q^{48} +2.05292e6 q^{49} +292068. q^{50} +194615. q^{51} +275098. q^{52} -658129. q^{53} -86492.4 q^{54} -3.13905e6 q^{55} +1.76399e6 q^{56} +651673. q^{57} -537184. q^{58} -707752. q^{59} -1.11590e6 q^{60} +1.80508e6 q^{61} -922170. q^{62} -1.23639e6 q^{63} -430371. q^{64} +962425. q^{65} -979443. q^{66} +4.26695e6 q^{67} +783435. q^{68} +328509. q^{69} +2.83391e6 q^{70} +1.92112e6 q^{71} -758220. q^{72} -6.42145e6 q^{73} +151767. q^{74} -1.79457e6 q^{75} +2.62335e6 q^{76} -1.40010e7 q^{77} +300295. q^{78} +2.30552e6 q^{79} -3.55229e6 q^{80} +531441. q^{81} +2.39661e6 q^{82} -9.08815e6 q^{83} -4.97719e6 q^{84} +2.74083e6 q^{85} -1.70176e6 q^{86} +3.30066e6 q^{87} -8.58610e6 q^{88} -2.68687e6 q^{89} -1.21810e6 q^{90} +4.29266e6 q^{91} +1.32244e6 q^{92} +5.66615e6 q^{93} +1.75173e6 q^{94} +9.17775e6 q^{95} -4.70290e6 q^{96} +1.13566e6 q^{97} +9.02108e6 q^{98} +6.01805e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9}+O(q^{10}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9} - 15310 q^{10} + 9040 q^{11} - 17658 q^{12} + 3774 q^{13} + 4536 q^{14} + 13932 q^{15} + 52002 q^{16} - 40760 q^{17} + 81598 q^{19} - 88946 q^{20} - 27486 q^{21} + 245034 q^{22} - 85169 q^{23} - 38394 q^{24} + 321325 q^{25} + 412748 q^{26} - 137781 q^{27} + 965948 q^{28} + 154126 q^{29} + 413370 q^{30} + 243132 q^{31} + 1278286 q^{32} - 244080 q^{33} + 984836 q^{34} - 130296 q^{35} + 476766 q^{36} + 582114 q^{37} + 772558 q^{38} - 101898 q^{39} - 132618 q^{40} + 113062 q^{41} - 122472 q^{42} - 659778 q^{43} + 659390 q^{44} - 376164 q^{45} - 591032 q^{47} - 1404054 q^{48} + 3263235 q^{49} - 702684 q^{50} + 1100520 q^{51} + 1793280 q^{52} + 207128 q^{53} + 184664 q^{55} + 5390508 q^{56} - 2203146 q^{57} - 1142916 q^{58} + 447148 q^{59} + 2401542 q^{60} + 2248970 q^{61} - 5729060 q^{62} + 742122 q^{63} + 7212922 q^{64} - 827096 q^{65} - 6615918 q^{66} + 4467570 q^{67} - 5477620 q^{68} + 2299563 q^{69} - 12744284 q^{70} - 5154608 q^{71} + 1036638 q^{72} - 13239250 q^{73} - 2827426 q^{74} - 8675775 q^{75} - 527434 q^{76} - 18415912 q^{77} - 11144196 q^{78} + 9594446 q^{79} - 55932394 q^{80} + 3720087 q^{81} - 20889952 q^{82} - 573720 q^{83} - 26080596 q^{84} + 7477272 q^{85} - 28416910 q^{86} - 4161402 q^{87} + 26555702 q^{88} - 3810540 q^{89} - 11160990 q^{90} + 36092068 q^{91} - 7957218 q^{92} - 6564564 q^{93} + 33545768 q^{94} + 10497320 q^{95} - 34513722 q^{96} + 49497978 q^{97} - 1023376 q^{98} + 6590160 q^{99}+O(q^{100}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9 - 15310 * q^10 + 9040 * q^11 - 17658 * q^12 + 3774 * q^13 + 4536 * q^14 + 13932 * q^15 + 52002 * q^16 - 40760 * q^17 + 81598 * q^19 - 88946 * q^20 - 27486 * q^21 + 245034 * q^22 - 85169 * q^23 - 38394 * q^24 + 321325 * q^25 + 412748 * q^26 - 137781 * q^27 + 965948 * q^28 + 154126 * q^29 + 413370 * q^30 + 243132 * q^31 + 1278286 * q^32 - 244080 * q^33 + 984836 * q^34 - 130296 * q^35 + 476766 * q^36 + 582114 * q^37 + 772558 * q^38 - 101898 * q^39 - 132618 * q^40 + 113062 * q^41 - 122472 * q^42 - 659778 * q^43 + 659390 * q^44 - 376164 * q^45 - 591032 * q^47 - 1404054 * q^48 + 3263235 * q^49 - 702684 * q^50 + 1100520 * q^51 + 1793280 * q^52 + 207128 * q^53 + 184664 * q^55 + 5390508 * q^56 - 2203146 * q^57 - 1142916 * q^58 + 447148 * q^59 + 2401542 * q^60 + 2248970 * q^61 - 5729060 * q^62 + 742122 * q^63 + 7212922 * q^64 - 827096 * q^65 - 6615918 * q^66 + 4467570 * q^67 - 5477620 * q^68 + 2299563 * q^69 - 12744284 * q^70 - 5154608 * q^71 + 1036638 * q^72 - 13239250 * q^73 - 2827426 * q^74 - 8675775 * q^75 - 527434 * q^76 - 18415912 * q^77 - 11144196 * q^78 + 9594446 * q^79 - 55932394 * q^80 + 3720087 * q^81 - 20889952 * q^82 - 573720 * q^83 - 26080596 * q^84 + 7477272 * q^85 - 28416910 * q^86 - 4161402 * q^87 + 26555702 * q^88 - 3810540 * q^89 - 11160990 * q^90 + 36092068 * q^91 - 7957218 * q^92 - 6564564 * q^93 + 33545768 * q^94 + 10497320 * q^95 - 34513722 * q^96 + 49497978 * q^97 - 1023376 * q^98 + 6590160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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