LMFDB - Embedded newform 69.8.a.a.1.1

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:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016 $ x^5 - 455x^3 - 474x^2 + 42284*x + 127016
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			1.1
Root			$-17.8260$ 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)

$f(q)$	$=$	$q-17.8260 q^{2} -27.0000 q^{3} +189.767 q^{4} -31.0428 q^{5} +481.302 q^{6} -1247.35 q^{7} -1101.05 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-17.8260 q^{2} -27.0000 q^{3} +189.767 q^{4} -31.0428 q^{5} +481.302 q^{6} -1247.35 q^{7} -1101.05 q^{8} +729.000 q^{9} +553.369 q^{10} -939.114 q^{11} -5123.70 q^{12} +10151.8 q^{13} +22235.2 q^{14} +838.155 q^{15} -4662.78 q^{16} +33965.5 q^{17} -12995.2 q^{18} +18716.1 q^{19} -5890.88 q^{20} +33678.3 q^{21} +16740.7 q^{22} +12167.0 q^{23} +29728.4 q^{24} -77161.3 q^{25} -180965. q^{26} -19683.0 q^{27} -236704. q^{28} +91722.7 q^{29} -14941.0 q^{30} -187422. q^{31} +224053. q^{32} +25356.1 q^{33} -605469. q^{34} +38721.0 q^{35} +138340. q^{36} -112278. q^{37} -333633. q^{38} -274098. q^{39} +34179.6 q^{40} +218882. q^{41} -600350. q^{42} -910269. q^{43} -178212. q^{44} -22630.2 q^{45} -216889. q^{46} +54041.9 q^{47} +125895. q^{48} +732327. q^{49} +1.37548e6 q^{50} -917067. q^{51} +1.92647e6 q^{52} -135797. q^{53} +350869. q^{54} +29152.7 q^{55} +1.37339e6 q^{56} -505335. q^{57} -1.63505e6 q^{58} -2.26099e6 q^{59} +159054. q^{60} +124600. q^{61} +3.34098e6 q^{62} -909315. q^{63} -3.39714e6 q^{64} -315139. q^{65} -451998. q^{66} +3.30689e6 q^{67} +6.44551e6 q^{68} -328509. q^{69} -690242. q^{70} -740248. q^{71} -802666. q^{72} +2.68349e6 q^{73} +2.00147e6 q^{74} +2.08336e6 q^{75} +3.55169e6 q^{76} +1.17140e6 q^{77} +4.88607e6 q^{78} -6.55101e6 q^{79} +144746. q^{80} +531441. q^{81} -3.90179e6 q^{82} +3.21765e6 q^{83} +6.39102e6 q^{84} -1.05438e6 q^{85} +1.62265e7 q^{86} -2.47651e6 q^{87} +1.03401e6 q^{88} -1.14227e7 q^{89} +403406. q^{90} -1.26628e7 q^{91} +2.30889e6 q^{92} +5.06038e6 q^{93} -963351. q^{94} -580999. q^{95} -6.04944e6 q^{96} -1.41253e7 q^{97} -1.30545e7 q^{98} -684614. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9}+O(q^{10}) $ 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99}+O(q^{100}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9 + 1452 * q^10 - 1148 * q^11 - 7290 * q^12 - 642 * q^13 + 5756 * q^14 + 7182 * q^15 - 22606 * q^16 - 5798 * q^17 - 6036 * q^19 - 27376 * q^20 + 13392 * q^21 - 97896 * q^22 + 60835 * q^23 - 38394 * q^24 - 262477 * q^25 - 355992 * q^26 - 98415 * q^27 - 507124 * q^28 - 169162 * q^29 - 39204 * q^30 - 199640 * q^31 - 284794 * q^32 + 30996 * q^33 - 1027740 * q^34 - 137680 * q^35 + 196830 * q^36 - 202002 * q^37 - 554924 * q^38 + 17334 * q^39 - 340904 * q^40 + 541282 * q^41 - 155412 * q^42 - 909596 * q^43 - 1236032 * q^44 - 193914 * q^45 + 80208 * q^47 + 610362 * q^48 + 850589 * q^49 - 941416 * q^50 + 156546 * q^51 + 146940 * q^52 - 278138 * q^53 - 933560 * q^55 - 539932 * q^56 + 162972 * q^57 - 3522712 * q^58 - 3177380 * q^59 + 739152 * q^60 + 147782 * q^61 + 4606456 * q^62 - 361584 * q^63 - 4142622 * q^64 + 3877332 * q^65 + 2643192 * q^66 - 464916 * q^67 + 7513072 * q^68 - 1642545 * q^69 + 2093200 * q^70 + 1576792 * q^71 + 1036638 * q^72 - 38190 * q^73 + 12164864 * q^74 + 7086879 * q^75 + 6889436 * q^76 + 10332384 * q^77 + 9611784 * q^78 - 3913336 * q^79 + 6334776 * q^80 + 2657205 * q^81 + 6799360 * q^82 + 15774716 * q^83 + 13692348 * q^84 - 8520740 * q^85 + 24874084 * q^86 + 4567374 * q^87 + 53216 * q^88 + 1116482 * q^89 + 1058508 * q^90 - 27369552 * q^91 + 3285090 * q^92 + 5390280 * q^93 - 7153744 * q^94 - 6067832 * q^95 + 7689438 * q^96 - 15738566 * q^97 + 11730488 * q^98 - 836892 * 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 $

$2$	−17.8260	−1.57561	−0.787806	−	0.615924i	$-0.788783\pi$
$2$	−17.8260	−1.57561	−0.787806	+	0.615924i	$0.788783\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	189.767	1.48255
$5$	−31.0428	−0.111062	−0.0555310	−	0.998457i	$-0.517685\pi$
$5$	−31.0428	−0.111062	−0.0555310	+	0.998457i	$0.517685\pi$
$6$	481.302	0.909680
$7$	−1247.35	−1.37450	−0.687248	−	0.726423i	$-0.741181\pi$
$7$	−1247.35	−1.37450	−0.687248	+	0.726423i	$0.741181\pi$
$8$	−1101.05	−0.760313
$9$	729.000	0.333333
$10$	553.369	0.174991
$11$	−939.114	−0.212737	−0.106369	−	0.994327i	$-0.533922\pi$
$11$	−939.114	−0.212737	−0.106369	+	0.994327i	$0.533922\pi$
$12$	−5123.70	−0.855951
$13$	10151.8	1.28156	0.640781	−	0.767723i	$-0.278610\pi$
$13$	10151.8	1.28156	0.640781	+	0.767723i	$0.278610\pi$
$14$	22235.2	2.16567
$15$	838.155	0.0641217
$16$	−4662.78	−0.284594
$17$	33965.5	1.67674	0.838371	−	0.545100i	$-0.183508\pi$
$17$	33965.5	1.67674	0.838371	+	0.545100i	$0.183508\pi$
$18$	−12995.2	−0.525204
$19$	18716.1	0.626005	0.313002	−	0.949752i	$-0.398665\pi$
$19$	18716.1	0.626005	0.313002	+	0.949752i	$0.398665\pi$
$20$	−5890.88	−0.164655
$21$	33678.3	0.793566
$22$	16740.7	0.335192
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	29728.4	0.438967
$25$	−77161.3	−0.987665
$26$	−180965.	−2.01925
$27$	−19683.0	−0.192450
$28$	−236704.	−2.03776
$29$	91722.7	0.698367	0.349183	−	0.937054i	$-0.386459\pi$
$29$	91722.7	0.698367	0.349183	+	0.937054i	$0.386459\pi$
$30$	−14941.0	−0.101031
$31$	−187422.	−1.12994	−0.564968	−	0.825113i	$-0.691111\pi$
$31$	−187422.	−1.12994	−0.564968	+	0.825113i	$0.691111\pi$
$32$	224053.	1.20872
$33$	25356.1	0.122824
$34$	−605469.	−2.64189
$35$	38721.0	0.152654
$36$	138340.	0.494184
$37$	−112278.	−0.364408	−0.182204	−	0.983261i	$-0.558323\pi$
$37$	−112278.	−0.364408	−0.182204	+	0.983261i	$0.558323\pi$
$38$	−333633.	−0.986340
$39$	−274098.	−0.739911
$40$	34179.6	0.0844418
$41$	218882.	0.495983	0.247991	−	0.968762i	$-0.420230\pi$
$41$	218882.	0.495983	0.247991	+	0.968762i	$0.420230\pi$
$42$	−600350.	−1.25035
$43$	−910269.	−1.74594	−0.872971	−	0.487772i	$-0.837810\pi$
$43$	−910269.	−1.74594	−0.872971	+	0.487772i	$0.837810\pi$
$44$	−178212.	−0.315394
$45$	−22630.2	−0.0370207
$46$	−216889.	−0.328538
$47$	54041.9	0.0759256	0.0379628	−	0.999279i	$-0.487913\pi$
$47$	54041.9	0.0759256	0.0379628	+	0.999279i	$0.487913\pi$
$48$	125895.	0.164310
$49$	732327.	0.889239
$50$	1.37548e6	1.55618
$51$	−917067.	−0.968068
$52$	1.92647e6	1.89998
$53$	−135797.	−0.125292	−0.0626462	−	0.998036i	$-0.519954\pi$
$53$	−135797.	−0.125292	−0.0626462	+	0.998036i	$0.519954\pi$
$54$	350869.	0.303227
$55$	29152.7	0.0236270
$56$	1.37339e6	1.04505
$57$	−505335.	−0.361424
$58$	−1.63505e6	−1.10035
$59$	−2.26099e6	−1.43323	−0.716616	−	0.697468i	$-0.754310\pi$
$59$	−2.26099e6	−1.43323	−0.716616	+	0.697468i	$0.754310\pi$
$60$	159054.	0.0950636
$61$	124600.	0.0702851	0.0351426	−	0.999382i	$-0.488811\pi$
$61$	124600.	0.0702851	0.0351426	+	0.999382i	$0.488811\pi$
$62$	3.34098e6	1.78034
$63$	−909315.	−0.458165
$64$	−3.39714e6	−1.61988
$65$	−315139.	−0.142333
$66$	−451998.	−0.193523
$67$	3.30689e6	1.34325	0.671627	−	0.740890i	$-0.265596\pi$
$67$	3.30689e6	1.34325	0.671627	+	0.740890i	$0.265596\pi$
$68$	6.44551e6	2.48586
$69$	−328509.	−0.120386
$70$	−690242.	−0.240524
$71$	−740248.	−0.245456	−0.122728	−	0.992440i	$-0.539164\pi$
$71$	−740248.	−0.245456	−0.122728	+	0.992440i	$0.539164\pi$
$72$	−802666.	−0.253438
$73$	2.68349e6	0.807365	0.403683	−	0.914899i	$-0.367730\pi$
$73$	2.68349e6	0.807365	0.403683	+	0.914899i	$0.367730\pi$
$74$	2.00147e6	0.574166
$75$	2.08336e6	0.570229
$76$	3.55169e6	0.928084
$77$	1.17140e6	0.292407
$78$	4.88607e6	1.16581
$79$	−6.55101e6	−1.49490	−0.747452	−	0.664315i	$-0.768723\pi$
$79$	−6.55101e6	−1.49490	−0.747452	+	0.664315i	$0.768723\pi$
$80$	144746.	0.0316076
$81$	531441.	0.111111
$82$	−3.90179e6	−0.781476
$83$	3.21765e6	0.617682	0.308841	−	0.951114i	$-0.400059\pi$
$83$	3.21765e6	0.617682	0.308841	+	0.951114i	$0.400059\pi$
$84$	6.39102e6	1.17650
$85$	−1.05438e6	−0.186222
$86$	1.62265e7	2.75093
$87$	−2.47651e6	−0.403202
$88$	1.03401e6	0.161747
$89$	−1.14227e7	−1.71753	−0.858765	−	0.512370i	$-0.828768\pi$
$89$	−1.14227e7	−1.71753	−0.858765	+	0.512370i	$0.828768\pi$
$90$	403406.	0.0583302
$91$	−1.26628e7	−1.76150
$92$	2.30889e6	0.309133
$93$	5.06038e6	0.652369
$94$	−963351.	−0.119629
$95$	−580999.	−0.0695253
$96$	−6.04944e6	−0.697856
$97$	−1.41253e7	−1.57143	−0.785717	−	0.618586i	$-0.787706\pi$
$97$	−1.41253e7	−1.57143	−0.785717	+	0.618586i	$0.787706\pi$
$98$	−1.30545e7	−1.40110
$99$	−684614.	−0.0709125
$100$	−1.46426e7	−1.46426
$101$	−1.20112e7	−1.16001	−0.580006	−	0.814612i	$-0.696950\pi$
$101$	−1.20112e7	−1.16001	−0.580006	+	0.814612i	$0.696950\pi$
$102$	1.63476e7	1.52530
$103$	−7.74503e6	−0.698382	−0.349191	−	0.937052i	$-0.613544\pi$
$103$	−7.74503e6	−0.698382	−0.349191	+	0.937052i	$0.613544\pi$
$104$	−1.11776e7	−0.974388
$105$	−1.04547e6	−0.0881350
$106$	2.42072e6	0.197412
$107$	4.90393e6	0.386991	0.193495	−	0.981101i	$-0.438018\pi$
$107$	4.90393e6	0.386991	0.193495	+	0.981101i	$0.438018\pi$
$108$	−3.73517e6	−0.285317
$109$	2.12306e7	1.57025	0.785127	−	0.619335i	$-0.212598\pi$
$109$	2.12306e7	1.57025	0.785127	+	0.619335i	$0.212598\pi$
$110$	−519676.	−0.0372270
$111$	3.03150e6	0.210391
$112$	5.81610e6	0.391173
$113$	7.90376e6	0.515299	0.257649	−	0.966238i	$-0.417052\pi$
$113$	7.90376e6	0.515299	0.257649	+	0.966238i	$0.417052\pi$
$114$	9.00810e6	0.569464
$115$	−377697.	−0.0231580
$116$	1.74059e7	1.03536
$117$	7.40064e6	0.427188
$118$	4.03044e7	2.25822
$119$	−4.23667e7	−2.30468
$120$	−922850.	−0.0487525
$121$	−1.86052e7	−0.954743
$122$	−2.22112e6	−0.110742
$123$	−5.90982e6	−0.286356
$124$	−3.55663e7	−1.67519
$125$	4.82052e6	0.220754
$126$	1.62094e7	0.721890
$127$	3.73010e7	1.61587	0.807936	−	0.589270i	$-0.200585\pi$
$127$	3.73010e7	1.61587	0.807936	+	0.589270i	$0.200585\pi$
$128$	3.18786e7	1.34358
$129$	2.45773e7	1.00802
$130$	5.61767e6	0.224261
$131$	−1.67942e6	−0.0652694	−0.0326347	−	0.999467i	$-0.510390\pi$
$131$	−1.67942e6	−0.0652694	−0.0326347	+	0.999467i	$0.510390\pi$
$132$	4.81174e6	0.182093
$133$	−2.33454e7	−0.860441
$134$	−5.89486e7	−2.11644
$135$	611015.	0.0213739
$136$	−3.73977e7	−1.27485
$137$	8.56278e6	0.284507	0.142253	−	0.989830i	$-0.454565\pi$
$137$	8.56278e6	0.284507	0.142253	+	0.989830i	$0.454565\pi$
$138$	5.85600e6	0.189681
$139$	−1.00144e7	−0.316281	−0.158140	−	0.987417i	$-0.550550\pi$
$139$	−1.00144e7	−0.316281	−0.158140	+	0.987417i	$0.550550\pi$
$140$	7.34796e6	0.226318
$141$	−1.45913e6	−0.0438357
$142$	1.31957e7	0.386743
$143$	−9.53367e6	−0.272636
$144$	−3.39917e6	−0.0948646
$145$	−2.84733e6	−0.0775620
$146$	−4.78359e7	−1.27209
$147$	−1.97728e7	−0.513403
$148$	−2.13066e7	−0.540254
$149$	1.78249e7	0.441443	0.220722	−	0.975337i	$-0.429159\pi$
$149$	1.78249e7	0.441443	0.220722	+	0.975337i	$0.429159\pi$
$150$	−3.71379e7	−0.898459
$151$	−1.24711e7	−0.294771	−0.147386	−	0.989079i	$-0.547086\pi$
$151$	−1.24711e7	−0.294771	−0.147386	+	0.989079i	$0.547086\pi$
$152$	−2.06074e7	−0.475959
$153$	2.47608e7	0.558914
$154$	−2.08814e7	−0.460719
$155$	5.81808e6	0.125493
$156$	−5.20146e7	−1.09696
$157$	−2.73688e7	−0.564426	−0.282213	−	0.959352i	$-0.591069\pi$
$157$	−2.73688e7	−0.564426	−0.282213	+	0.959352i	$0.591069\pi$
$158$	1.16778e8	2.35539
$159$	3.66652e6	0.0723376
$160$	−6.95523e6	−0.134243
$161$	−1.51764e7	−0.286602
$162$	−9.47347e6	−0.175068
$163$	8.32721e7	1.50606	0.753031	−	0.657985i	$-0.228591\pi$
$163$	8.32721e7	1.50606	0.753031	+	0.657985i	$0.228591\pi$
$164$	4.15365e7	0.735320
$165$	−787123.	−0.0136411
$166$	−5.73578e7	−0.973227
$167$	9.35501e6	0.155431	0.0777153	−	0.996976i	$-0.475237\pi$
$167$	9.35501e6	0.155431	0.0777153	+	0.996976i	$0.475237\pi$
$168$	−3.70815e7	−0.603358
$169$	4.03099e7	0.642404
$170$	1.87954e7	0.293414
$171$	1.36440e7	0.208668
$172$	−1.72738e8	−2.58845
$173$	−8.62670e7	−1.26673	−0.633364	−	0.773854i	$-0.718326\pi$
$173$	−8.62670e7	−1.26673	−0.633364	+	0.773854i	$0.718326\pi$
$174$	4.41463e7	0.635290
$175$	9.62468e7	1.35754
$176$	4.37889e6	0.0605438
$177$	6.10467e7	0.827476
$178$	2.03621e8	2.70616
$179$	−9.64475e7	−1.25691	−0.628456	−	0.777845i	$-0.716313\pi$
$179$	−9.64475e7	−1.25691	−0.628456	+	0.777845i	$0.716313\pi$
$180$	−4.29445e6	−0.0548850
$181$	−2.62936e7	−0.329591	−0.164796	−	0.986328i	$-0.552696\pi$
$181$	−2.62936e7	−0.329591	−0.164796	+	0.986328i	$0.552696\pi$
$182$	2.25726e8	2.77544
$183$	−3.36420e6	−0.0405791
$184$	−1.33965e7	−0.158536
$185$	3.48542e6	0.0404719
$186$	−9.02064e7	−1.02788
$187$	−3.18974e7	−0.356706
$188$	1.02553e7	0.112564
$189$	2.45515e7	0.264522
$190$	1.03569e7	0.109545
$191$	5.41983e7	0.562819	0.281410	−	0.959588i	$-0.409198\pi$
$191$	5.41983e7	0.562819	0.281410	+	0.959588i	$0.409198\pi$
$192$	9.17227e7	0.935239
$193$	−1.54438e8	−1.54633	−0.773166	−	0.634203i	$-0.781328\pi$
$193$	−1.54438e8	−1.54633	−0.773166	+	0.634203i	$0.781328\pi$
$194$	2.51797e8	2.47597
$195$	8.50875e6	0.0821759
$196$	1.38971e8	1.31834
$197$	1.17189e8	1.09208	0.546041	−	0.837759i	$-0.316134\pi$
$197$	1.17189e8	1.09208	0.546041	+	0.837759i	$0.316134\pi$
$198$	1.22039e7	0.111731
$199$	−9.17056e7	−0.824917	−0.412458	−	0.910976i	$-0.635330\pi$
$199$	−9.17056e7	−0.824917	−0.412458	+	0.910976i	$0.635330\pi$
$200$	8.49585e7	0.750934
$201$	−8.92860e7	−0.775528
$202$	2.14112e8	1.82773
$203$	−1.14410e8	−0.959903
$204$	−1.74029e8	−1.43521
$205$	−6.79470e6	−0.0550848
$206$	1.38063e8	1.10038
$207$	8.86974e6	0.0695048
$208$	−4.73355e7	−0.364725
$209$	−1.75766e7	−0.133175
$210$	1.86365e7	0.138866
$211$	3.23106e7	0.236786	0.118393	−	0.992967i	$-0.462226\pi$
$211$	3.23106e7	0.236786	0.118393	+	0.992967i	$0.462226\pi$
$212$	−2.57697e7	−0.185752
$213$	1.99867e7	0.141714
$214$	−8.74174e7	−0.609747
$215$	2.82573e7	0.193908
$216$	2.16720e7	0.146322
$217$	2.33779e8	1.55309
$218$	−3.78457e8	−2.47411
$219$	−7.24543e7	−0.466133
$220$	5.53221e6	0.0350283
$221$	3.44809e8	2.14885
$222$	−5.40396e7	−0.331495
$223$	1.25273e6	0.00756468	0.00378234	−	0.999993i	$-0.498796\pi$
$223$	1.25273e6	0.00756468	0.00378234	+	0.999993i	$0.498796\pi$
$224$	−2.79472e8	−1.66138
$225$	−5.62506e7	−0.329222
$226$	−1.40892e8	−0.811911
$227$	−1.41797e8	−0.804594	−0.402297	−	0.915509i	$-0.631788\pi$
$227$	−1.41797e8	−0.804594	−0.402297	+	0.915509i	$0.631788\pi$
$228$	−9.58956e7	−0.535829
$229$	−7.55158e7	−0.415541	−0.207770	−	0.978178i	$-0.566621\pi$
$229$	−7.55158e7	−0.415541	−0.207770	+	0.978178i	$0.566621\pi$
$230$	6.73284e6	0.0364880
$231$	−3.16278e7	−0.168821
$232$	−1.00991e8	−0.530977
$233$	−2.66330e8	−1.37935	−0.689674	−	0.724120i	$-0.742246\pi$
$233$	−2.66330e8	−1.37935	−0.689674	+	0.724120i	$0.742246\pi$
$234$	−1.31924e8	−0.673082
$235$	−1.67761e6	−0.00843244
$236$	−4.29060e8	−2.12484
$237$	1.76877e8	0.863084
$238$	7.55228e8	3.63127
$239$	−3.47869e8	−1.64825	−0.824125	−	0.566408i	$-0.808333\pi$
$239$	−3.47869e8	−1.64825	−0.824125	+	0.566408i	$0.808333\pi$
$240$	−3.90814e6	−0.0182486
$241$	−1.34331e8	−0.618184	−0.309092	−	0.951032i	$-0.600025\pi$
$241$	−1.34331e8	−0.618184	−0.309092	+	0.951032i	$0.600025\pi$
$242$	3.31657e8	1.50430
$243$	−1.43489e7	−0.0641500
$244$	2.36449e7	0.104201
$245$	−2.27335e7	−0.0987607
$246$	1.05348e8	0.451185
$247$	1.90001e8	0.802265
$248$	2.06361e8	0.859104
$249$	−8.68765e7	−0.356619
$250$	−8.59306e7	−0.347823
$251$	−1.25191e8	−0.499707	−0.249854	−	0.968284i	$-0.580382\pi$
$251$	−1.25191e8	−0.499707	−0.249854	+	0.968284i	$0.580382\pi$
$252$	−1.72557e8	−0.679253
$253$	−1.14262e7	−0.0443588
$254$	−6.64927e8	−2.54599
$255$	2.84683e7	0.107515
$256$	−1.33434e8	−0.497082
$257$	−3.83946e8	−1.41093	−0.705463	−	0.708746i	$-0.749261\pi$
$257$	−3.83946e8	−1.41093	−0.705463	+	0.708746i	$0.749261\pi$
$258$	−4.38114e8	−1.58825
$259$	1.40049e8	0.500878
$260$	−5.98028e7	−0.211016
$261$	6.68658e7	0.232789
$262$	2.99373e7	0.102839
$263$	−5.17704e8	−1.75483	−0.877417	−	0.479729i	$-0.840735\pi$
$263$	−5.17704e8	−1.75483	−0.877417	+	0.479729i	$0.840735\pi$
$264$	−2.79183e7	−0.0933847
$265$	4.21552e6	0.0139152
$266$	4.16156e8	1.35572
$267$	3.08413e8	0.991616
$268$	6.27537e8	1.99144
$269$	−2.04806e8	−0.641518	−0.320759	−	0.947161i	$-0.603938\pi$
$269$	−2.04806e8	−0.641518	−0.320759	+	0.947161i	$0.603938\pi$
$270$	−1.08920e7	−0.0336769
$271$	−2.30872e8	−0.704659	−0.352329	−	0.935876i	$-0.614610\pi$
$271$	−2.30872e8	−0.704659	−0.352329	+	0.935876i	$0.614610\pi$
$272$	−1.58374e8	−0.477190
$273$	3.41894e8	1.01700
$274$	−1.52640e8	−0.448272
$275$	7.24633e7	0.210113
$276$	−6.23400e7	−0.178478
$277$	−3.59723e8	−1.01693	−0.508463	−	0.861084i	$-0.669786\pi$
$277$	−3.59723e8	−1.01693	−0.508463	+	0.861084i	$0.669786\pi$
$278$	1.78517e8	0.498335
$279$	−1.36630e8	−0.376645
$280$	−4.26338e7	−0.116065
$281$	5.59198e8	1.50346	0.751732	−	0.659469i	$-0.229219\pi$
$281$	5.59198e8	1.50346	0.751732	+	0.659469i	$0.229219\pi$
$282$	2.60105e7	0.0690679
$283$	−2.74952e8	−0.721114	−0.360557	−	0.932737i	$-0.617413\pi$
$283$	−2.74952e8	−0.721114	−0.360557	+	0.932737i	$0.617413\pi$
$284$	−1.40474e8	−0.363900
$285$	1.56870e7	0.0401405
$286$	1.69947e8	0.429569
$287$	−2.73021e8	−0.681726
$288$	1.63335e8	0.402907
$289$	7.43314e8	1.81146
$290$	5.07564e7	0.122208
$291$	3.81383e8	0.907268
$292$	5.09237e8	1.19696
$293$	2.01642e8	0.468321	0.234161	−	0.972198i	$-0.424766\pi$
$293$	2.01642e8	0.468321	0.234161	+	0.972198i	$0.424766\pi$
$294$	3.52471e8	0.808923
$295$	7.01873e7	0.159178
$296$	1.23624e8	0.277064
$297$	1.84846e7	0.0409413
$298$	−3.17747e8	−0.695543
$299$	1.23517e8	0.267224
$300$	3.95351e8	0.845393
$301$	1.13542e9	2.39979
$302$	2.22310e8	0.464445
$303$	3.24303e8	0.669734
$304$	−8.72691e7	−0.178157
$305$	−3.86793e6	−0.00780600
$306$	−4.41387e8	−0.880631
$307$	−4.38553e8	−0.865043	−0.432521	−	0.901624i	$-0.642376\pi$
$307$	−4.38553e8	−0.865043	−0.432521	+	0.901624i	$0.642376\pi$
$308$	2.22292e8	0.433508
$309$	2.09116e8	0.403211
$310$	−1.03713e8	−0.197728
$311$	6.05423e8	1.14129	0.570647	−	0.821196i	$-0.306693\pi$
$311$	6.05423e8	1.14129	0.570647	+	0.821196i	$0.306693\pi$
$312$	3.01795e8	0.562563
$313$	−1.07052e7	−0.0197329	−0.00986647	−	0.999951i	$-0.503141\pi$
$313$	−1.07052e7	−0.0197329	−0.00986647	+	0.999951i	$0.503141\pi$
$314$	4.87877e8	0.889316
$315$	2.82276e7	0.0508848
$316$	−1.24316e9	−2.21627
$317$	9.99712e8	1.76266	0.881328	−	0.472505i	$-0.156650\pi$
$317$	9.99712e8	1.76266	0.881328	+	0.472505i	$0.156650\pi$
$318$	−6.53594e7	−0.113976
$319$	−8.61381e7	−0.148569
$320$	1.05457e8	0.179907
$321$	−1.32406e8	−0.223429
$322$	2.70535e8	0.451574
$323$	6.35701e8	1.04965
$324$	1.00850e8	0.164728
$325$	−7.83324e8	−1.26576
$326$	−1.48441e9	−2.37297
$327$	−5.73227e8	−0.906587
$328$	−2.41000e8	−0.377102
$329$	−6.74089e7	−0.104359
$330$	1.40313e7	0.0214930
$331$	−8.39788e8	−1.27283	−0.636417	−	0.771345i	$-0.719584\pi$
$331$	−8.39788e8	−1.27283	−0.636417	+	0.771345i	$0.719584\pi$
$332$	6.10602e8	0.915745
$333$	−8.18506e7	−0.121469
$334$	−1.66762e8	−0.244898
$335$	−1.02655e8	−0.149184
$336$	−1.57035e8	−0.225844
$337$	−5.53660e8	−0.788022	−0.394011	−	0.919106i	$-0.628913\pi$
$337$	−5.53660e8	−0.788022	−0.394011	+	0.919106i	$0.628913\pi$
$338$	−7.18564e8	−1.01218
$339$	−2.13402e8	−0.297508
$340$	−2.00086e8	−0.276084
$341$	1.76010e8	0.240380
$342$	−2.43219e8	−0.328780
$343$	1.13778e8	0.152240
$344$	1.00225e9	1.32746
$345$	1.01978e7	0.0133703
$346$	1.53780e9	1.99587
$347$	8.05434e8	1.03485	0.517424	−	0.855729i	$-0.326891\pi$
$347$	8.05434e8	1.03485	0.517424	+	0.855729i	$0.326891\pi$
$348$	−4.69959e8	−0.597768
$349$	6.32931e8	0.797017	0.398509	−	0.917165i	$-0.369528\pi$
$349$	6.32931e8	0.797017	0.398509	+	0.917165i	$0.369528\pi$
$350$	−1.71570e9	−2.13896
$351$	−1.99817e8	−0.246637
$352$	−2.10412e8	−0.257140
$353$	3.46589e8	0.419376	0.209688	−	0.977768i	$-0.432755\pi$
$353$	3.46589e8	0.419376	0.209688	+	0.977768i	$0.432755\pi$
$354$	−1.08822e9	−1.30378
$355$	2.29793e7	0.0272608
$356$	−2.16765e9	−2.54633
$357$	1.14390e9	1.33060
$358$	1.71927e9	1.98041
$359$	−2.14794e8	−0.245014	−0.122507	−	0.992468i	$-0.539093\pi$
$359$	−2.14794e8	−0.245014	−0.122507	+	0.992468i	$0.539093\pi$
$360$	2.49170e7	0.0281473
$361$	−5.43580e8	−0.608118
$362$	4.68710e8	0.519307
$363$	5.02341e8	0.551221
$364$	−2.40297e9	−2.61152
$365$	−8.33030e7	−0.0896676
$366$	5.99702e7	0.0639369
$367$	1.71463e9	1.81067	0.905335	−	0.424699i	$-0.139620\pi$
$367$	1.71463e9	1.81067	0.905335	+	0.424699i	$0.139620\pi$
$368$	−5.67321e7	−0.0593419
$369$	1.59565e8	0.165328
$370$	−6.21311e7	−0.0637680
$371$	1.69386e8	0.172214
$372$	9.60291e8	0.967170
$373$	−8.64659e8	−0.862709	−0.431354	−	0.902183i	$-0.641964\pi$
$373$	−8.64659e8	−0.862709	−0.431354	+	0.902183i	$0.641964\pi$
$374$	5.68604e8	0.562030
$375$	−1.30154e8	−0.127452
$376$	−5.95028e7	−0.0577272
$377$	9.31147e8	0.895001
$378$	−4.37655e8	−0.416784
$379$	−5.13537e7	−0.0484546	−0.0242273	−	0.999706i	$-0.507713\pi$
$379$	−5.13537e7	−0.0484546	−0.0242273	+	0.999706i	$0.507713\pi$
$380$	−1.10254e8	−0.103075
$381$	−1.00713e9	−0.932924
$382$	−9.66139e8	−0.886784
$383$	2.01952e9	1.83676	0.918379	−	0.395703i	$-0.129499\pi$
$383$	2.01952e9	1.83676	0.918379	+	0.395703i	$0.129499\pi$
$384$	−8.60722e8	−0.775718
$385$	−3.63635e7	−0.0324753
$386$	2.75301e9	2.43642
$387$	−6.63586e8	−0.581981
$388$	−2.68051e9	−2.32973
$389$	1.39868e9	1.20474	0.602370	−	0.798217i	$-0.294223\pi$
$389$	1.39868e9	1.20474	0.602370	+	0.798217i	$0.294223\pi$
$390$	−1.51677e8	−0.129477
$391$	4.13258e8	0.349625
$392$	−8.06329e8	−0.676100
$393$	4.53443e7	0.0376833
$394$	−2.08901e9	−1.72070
$395$	2.03361e8	0.166027
$396$	−1.29917e8	−0.105131
$397$	−1.96246e9	−1.57411	−0.787055	−	0.616883i	$-0.788395\pi$
$397$	−1.96246e9	−1.57411	−0.787055	+	0.616883i	$0.788395\pi$
$398$	1.63474e9	1.29975
$399$	6.30327e8	0.496776
$400$	3.59787e8	0.281083
$401$	−2.26678e9	−1.75552	−0.877758	−	0.479104i	$-0.840962\pi$
$401$	−2.26678e9	−1.75552	−0.877758	+	0.479104i	$0.840962\pi$
$402$	1.59161e9	1.22193
$403$	−1.90266e9	−1.44808
$404$	−2.27933e9	−1.71978
$405$	−1.64974e7	−0.0123402
$406$	2.03947e9	1.51243
$407$	1.05442e8	0.0775233
$408$	1.00974e9	0.736034
$409$	2.28154e9	1.64891	0.824453	−	0.565931i	$-0.191483\pi$
$409$	2.28154e9	1.64891	0.824453	+	0.565931i	$0.191483\pi$
$410$	1.21122e8	0.0867923
$411$	−2.31195e8	−0.164260
$412$	−1.46975e9	−1.03539
$413$	2.82023e9	1.96997
$414$	−1.58112e8	−0.109513
$415$	−9.98847e7	−0.0686010
$416$	2.27454e9	1.54905
$417$	2.70389e8	0.182605
$418$	3.13320e8	0.209832
$419$	6.99241e7	0.0464384	0.0232192	−	0.999730i	$-0.492608\pi$
$419$	6.99241e7	0.0464384	0.0232192	+	0.999730i	$0.492608\pi$
$420$	−1.98395e8	−0.130665
$421$	−1.08544e9	−0.708952	−0.354476	−	0.935065i	$-0.615341\pi$
$421$	−1.08544e9	−0.708952	−0.354476	+	0.935065i	$0.615341\pi$
$422$	−5.75969e8	−0.373083
$423$	3.93965e7	0.0253085
$424$	1.49519e8	0.0952614
$425$	−2.62082e9	−1.65606
$426$	−3.56283e8	−0.223286
$427$	−1.55419e8	−0.0966066
$428$	9.30601e8	0.573734
$429$	2.57409e8	0.157407
$430$	−5.03714e8	−0.305523
$431$	−3.16872e9	−1.90640	−0.953198	−	0.302348i	$-0.902229\pi$
$431$	−3.16872e9	−1.90640	−0.953198	+	0.302348i	$0.902229\pi$
$432$	9.17776e7	0.0547701
$433$	3.26047e8	0.193007	0.0965033	−	0.995333i	$-0.469234\pi$
$433$	3.26047e8	0.193007	0.0965033	+	0.995333i	$0.469234\pi$
$434$	−4.16735e9	−2.44707
$435$	7.68778e7	0.0447805
$436$	4.02886e9	2.32798
$437$	2.27719e8	0.130531
$438$	1.29157e9	0.734444
$439$	−2.63160e9	−1.48455	−0.742273	−	0.670097i	$-0.766252\pi$
$439$	−2.63160e9	−1.48455	−0.742273	+	0.670097i	$0.766252\pi$
$440$	−3.20986e7	−0.0179639
$441$	5.33866e8	0.296413
$442$	−6.14658e9	−3.38575
$443$	−3.10154e8	−0.169498	−0.0847489	−	0.996402i	$-0.527009\pi$
$443$	−3.10154e8	−0.169498	−0.0847489	+	0.996402i	$0.527009\pi$
$444$	5.75278e8	0.311916
$445$	3.54593e8	0.190752
$446$	−2.23312e7	−0.0119190
$447$	−4.81272e8	−0.254868
$448$	4.23740e9	2.22652
$449$	−1.98075e9	−1.03268	−0.516341	−	0.856383i	$-0.672706\pi$
$449$	−1.98075e9	−1.03268	−0.516341	+	0.856383i	$0.672706\pi$
$450$	1.00272e9	0.518725
$451$	−2.05555e8	−0.105514
$452$	1.49987e9	0.763957
$453$	3.36719e8	0.170186
$454$	2.52767e9	1.26773
$455$	3.93087e8	0.195636
$456$	5.56399e8	0.274795
$457$	−1.57211e9	−0.770504	−0.385252	−	0.922811i	$-0.625885\pi$
$457$	−1.57211e9	−0.770504	−0.385252	+	0.922811i	$0.625885\pi$
$458$	1.34615e9	0.654731
$459$	−6.68542e8	−0.322689
$460$	−7.16743e7	−0.0343329
$461$	2.80297e9	1.33249	0.666247	−	0.745731i	$-0.267899\pi$
$461$	2.80297e9	1.33249	0.666247	+	0.745731i	$0.267899\pi$
$462$	5.63797e8	0.265997
$463$	3.29626e9	1.54343	0.771717	−	0.635966i	$-0.219398\pi$
$463$	3.29626e9	1.54343	0.771717	+	0.635966i	$0.219398\pi$
$464$	−4.27683e8	−0.198751
$465$	−1.57088e8	−0.0724533
$466$	4.74759e9	2.17332
$467$	−2.22099e9	−1.00911	−0.504554	−	0.863380i	$-0.668343\pi$
$467$	−2.22099e9	−1.00911	−0.504554	+	0.863380i	$0.668343\pi$
$468$	1.40439e9	0.633327
$469$	−4.12483e9	−1.84630
$470$	2.99051e7	0.0132863
$471$	7.38958e8	0.325872
$472$	2.48946e9	1.08970
$473$	8.54846e8	0.371427
$474$	−3.15302e9	−1.35988
$475$	−1.44416e9	−0.618283
$476$	−8.03977e9	−3.41680
$477$	−9.89961e7	−0.0417642
$478$	6.20111e9	2.59700
$479$	4.08881e9	1.69990	0.849948	−	0.526867i	$-0.176633\pi$
$479$	4.08881e9	1.69990	0.849948	+	0.526867i	$0.176633\pi$
$480$	1.87791e8	0.0775053
$481$	−1.13982e9	−0.467012
$482$	2.39459e9	0.974017
$483$	4.09764e8	0.165470
$484$	−3.53065e9	−1.41545
$485$	4.38488e8	0.174527
$486$	2.55784e8	0.101076
$487$	−2.20516e9	−0.865144	−0.432572	−	0.901599i	$-0.642394\pi$
$487$	−2.20516e9	−0.865144	−0.432572	+	0.901599i	$0.642394\pi$
$488$	−1.37191e8	−0.0534387
$489$	−2.24835e9	−0.869525
$490$	4.05247e8	0.155608
$491$	−2.44960e9	−0.933921	−0.466961	−	0.884278i	$-0.654651\pi$
$491$	−2.44960e9	−0.933921	−0.466961	+	0.884278i	$0.654651\pi$
$492$	−1.12148e9	−0.424537
$493$	3.11540e9	1.17098
$494$	−3.38697e9	−1.26406
$495$	2.12523e7	0.00787568
$496$	8.73906e8	0.321573
$497$	9.23345e8	0.337378
$498$	1.54866e9	0.561893
$499$	−4.09114e9	−1.47398	−0.736991	−	0.675903i	$-0.763754\pi$
$499$	−4.09114e9	−1.47398	−0.736991	+	0.675903i	$0.763754\pi$
$500$	9.14773e8	0.327279
$501$	−2.52585e8	−0.0897379
$502$	2.23166e9	0.787344
$503$	4.53054e8	0.158731	0.0793655	−	0.996846i	$-0.474711\pi$
$503$	4.53054e8	0.158731	0.0793655	+	0.996846i	$0.474711\pi$
$504$	1.00120e9	0.348349
$505$	3.72862e8	0.128833
$506$	2.03684e8	0.0698923
$507$	−1.08837e9	−0.370892
$508$	7.07847e9	2.39561
$509$	−2.23184e9	−0.750156	−0.375078	−	0.926993i	$-0.622384\pi$
$509$	−2.23184e9	−0.750156	−0.375078	+	0.926993i	$0.622384\pi$
$510$	−5.07476e8	−0.169403
$511$	−3.34724e9	−1.10972
$512$	−1.70186e9	−0.560375
$513$	−3.68389e8	−0.120475
$514$	6.84423e9	2.22307
$515$	2.40427e8	0.0775636
$516$	4.66394e9	1.49444
$517$	−5.07515e7	−0.0161522
$518$	−2.49652e9	−0.789189
$519$	2.32921e9	0.731346
$520$	3.46984e8	0.108218
$521$	3.73130e9	1.15592	0.577960	−	0.816065i	$-0.303849\pi$
$521$	3.73130e9	1.15592	0.577960	+	0.816065i	$0.303849\pi$
$522$	−1.19195e9	−0.366785
$523$	−3.65832e9	−1.11822	−0.559108	−	0.829095i	$-0.688856\pi$
$523$	−3.65832e9	−1.11822	−0.559108	+	0.829095i	$0.688856\pi$
$524$	−3.18698e8	−0.0967652
$525$	−2.59866e9	−0.783777
$526$	9.22859e9	2.76494
$527$	−6.36586e9	−1.89461
$528$	−1.18230e8	−0.0349550
$529$	1.48036e8	0.0434783
$530$	−7.51458e7	−0.0219250
$531$	−1.64826e9	−0.477744
$532$	−4.43018e9	−1.27565
$533$	2.22204e9	0.635633
$534$	−5.49778e9	−1.56240
$535$	−1.52231e8	−0.0429800
$536$	−3.64105e9	−1.02129
$537$	2.60408e9	0.725679
$538$	3.65087e9	1.01078
$539$	−6.87739e8	−0.189175
$540$	1.15950e8	0.0316879
$541$	2.44725e9	0.664490	0.332245	−	0.943193i	$-0.392194\pi$
$541$	2.44725e9	0.664490	0.332245	+	0.943193i	$0.392194\pi$
$542$	4.11553e9	1.11027
$543$	7.09928e8	0.190289
$544$	7.61007e9	2.02671
$545$	−6.59057e8	−0.174396
$546$	−6.09461e9	−1.60240
$547$	3.76239e9	0.982897	0.491449	−	0.870907i	$-0.336468\pi$
$547$	3.76239e9	0.982897	0.491449	+	0.870907i	$0.336468\pi$
$548$	1.62493e9	0.421796
$549$	9.08333e7	0.0234284
$550$	−1.29173e9	−0.331057
$551$	1.71669e9	0.437181
$552$	3.61705e8	0.0915309
$553$	8.17137e9	2.05474
$554$	6.41242e9	1.60228
$555$	−9.41063e7	−0.0233665
$556$	−1.90040e9	−0.468902
$557$	6.50679e9	1.59541	0.797707	−	0.603045i	$-0.206046\pi$
$557$	6.50679e9	1.59541	0.797707	+	0.603045i	$0.206046\pi$
$558$	2.43557e9	0.593446
$559$	−9.24084e9	−2.23754
$560$	−1.80548e8	−0.0434445
$561$	8.61231e8	0.205944
$562$	−9.96826e9	−2.36888
$563$	2.23762e9	0.528453	0.264226	−	0.964461i	$-0.414883\pi$
$563$	2.23762e9	0.528453	0.264226	+	0.964461i	$0.414883\pi$
$564$	−2.76894e8	−0.0649886
$565$	−2.45355e8	−0.0572301
$566$	4.90129e9	1.13620
$567$	−6.62890e8	−0.152722
$568$	8.15050e8	0.186623
$569$	3.55887e9	0.809877	0.404939	−	0.914344i	$-0.367293\pi$
$569$	3.55887e9	0.809877	0.404939	+	0.914344i	$0.367293\pi$
$570$	−2.79636e8	−0.0632458
$571$	−2.69760e9	−0.606389	−0.303194	−	0.952929i	$-0.598053\pi$
$571$	−2.69760e9	−0.606389	−0.303194	+	0.952929i	$0.598053\pi$
$572$	−1.80917e9	−0.404197
$573$	−1.46335e9	−0.324944
$574$	4.86688e9	1.07414
$575$	−9.38822e8	−0.205942
$576$	−2.47651e9	−0.539961
$577$	−2.73131e9	−0.591910	−0.295955	−	0.955202i	$-0.595638\pi$
$577$	−2.73131e9	−0.591910	−0.295955	+	0.955202i	$0.595638\pi$
$578$	−1.32503e10	−2.85416
$579$	4.16982e9	0.892776
$580$	−5.40327e8	−0.114990
$581$	−4.01352e9	−0.849002
$582$	−6.79853e9	−1.42950
$583$	1.27529e8	0.0266544
$584$	−2.95466e9	−0.613850
$585$	−2.29736e8	−0.0474443
$586$	−3.59447e9	−0.737892
$587$	−4.25360e9	−0.868008	−0.434004	−	0.900911i	$-0.642900\pi$
$587$	−4.25360e9	−0.868008	−0.434004	+	0.900911i	$0.642900\pi$
$588$	−3.75222e9	−0.761145
$589$	−3.50780e9	−0.707345
$590$	−1.25116e9	−0.250802
$591$	−3.16410e9	−0.630513
$592$	5.23528e8	0.103708
$593$	8.85467e9	1.74374	0.871869	−	0.489740i	$-0.162908\pi$
$593$	8.85467e9	1.74374	0.871869	+	0.489740i	$0.162908\pi$
$594$	−3.29506e8	−0.0645076
$595$	1.31518e9	0.255962
$596$	3.38257e9	0.654462
$597$	2.47605e9	0.476266
$598$	−2.20181e9	−0.421042
$599$	−8.77964e9	−1.66910	−0.834551	−	0.550930i	$-0.814273\pi$
$599$	−8.77964e9	−1.66910	−0.834551	+	0.550930i	$0.814273\pi$
$600$	−2.29388e9	−0.433552
$601$	4.36354e9	0.819933	0.409967	−	0.912101i	$-0.365540\pi$
$601$	4.36354e9	0.819933	0.409967	+	0.912101i	$0.365540\pi$
$602$	−2.02400e10	−3.78114
$603$	2.41072e9	0.447751
$604$	−2.36659e9	−0.437013
$605$	5.77558e8	0.106036
$606$	−5.78103e9	−1.05524
$607$	−1.00264e10	−1.81963	−0.909815	−	0.415015i	$-0.863776\pi$
$607$	−1.00264e10	−1.81963	−0.909815	+	0.415015i	$0.863776\pi$
$608$	4.19340e9	0.756666
$609$	3.08907e9	0.554200
$610$	6.89497e7	0.0122992
$611$	5.48621e8	0.0973034
$612$	4.69877e9	0.828618
$613$	−1.06422e10	−1.86603	−0.933013	−	0.359842i	$-0.882830\pi$
$613$	−1.06422e10	−1.86603	−0.933013	+	0.359842i	$0.882830\pi$
$614$	7.81765e9	1.36297
$615$	1.83457e8	0.0318032
$616$	−1.28977e9	−0.222321
$617$	−1.31952e9	−0.226161	−0.113080	−	0.993586i	$-0.536072\pi$
$617$	−1.31952e9	−0.226161	−0.113080	+	0.993586i	$0.536072\pi$
$618$	−3.72770e9	−0.635304
$619$	8.51725e9	1.44339	0.721693	−	0.692214i	$-0.243364\pi$
$619$	8.51725e9	1.44339	0.721693	+	0.692214i	$0.243364\pi$
$620$	1.10408e9	0.186050
$621$	−2.39483e8	−0.0401286
$622$	−1.07923e10	−1.79823
$623$	1.42481e10	2.36074
$624$	1.27806e9	0.210574
$625$	5.87859e9	0.963148
$626$	1.90832e8	0.0310914
$627$	4.74567e8	0.0768884
$628$	−5.19368e9	−0.836791
$629$	−3.81357e9	−0.611019
$630$	−5.03186e8	−0.0801746
$631$	1.32684e9	0.210240	0.105120	−	0.994460i	$-0.466477\pi$
$631$	1.32684e9	0.210240	0.105120	+	0.994460i	$0.466477\pi$
$632$	7.21299e9	1.13659
$633$	−8.72386e8	−0.136708
$634$	−1.78209e10	−2.77726
$635$	−1.15792e9	−0.179462
$636$	6.95783e8	0.107244
$637$	7.43441e9	1.13962
$638$	1.53550e9	0.234087
$639$	−5.39641e8	−0.0818185
$640$	−9.89600e8	−0.149221
$641$	3.00273e9	0.450312	0.225156	−	0.974323i	$-0.427711\pi$
$641$	3.00273e9	0.450312	0.225156	+	0.974323i	$0.427711\pi$
$642$	2.36027e9	0.352038
$643$	−1.08587e10	−1.61080	−0.805398	−	0.592734i	$-0.798049\pi$
$643$	−1.08587e10	−1.61080	−0.805398	+	0.592734i	$0.798049\pi$
$644$	−2.87998e9	−0.424902
$645$	−7.62946e8	−0.111953
$646$	−1.13320e10	−1.65384
$647$	−8.53415e9	−1.23878	−0.619392	−	0.785082i	$-0.712621\pi$
$647$	−8.53415e9	−1.23878	−0.619392	+	0.785082i	$0.712621\pi$
$648$	−5.85143e8	−0.0844792
$649$	2.12333e9	0.304902
$650$	1.39635e10	1.99434
$651$	−6.31204e9	−0.896678
$652$	1.58022e10	2.23281
$653$	−2.44692e9	−0.343893	−0.171947	−	0.985106i	$-0.555006\pi$
$653$	−2.44692e9	−0.343893	−0.171947	+	0.985106i	$0.555006\pi$
$654$	1.02183e10	1.42843
$655$	5.21338e7	0.00724895
$656$	−1.02060e9	−0.141154
$657$	1.95627e9	0.269122
$658$	1.20163e9	0.164430
$659$	7.66941e9	1.04391	0.521955	−	0.852973i	$-0.325203\pi$
$659$	7.66941e9	1.04391	0.521955	+	0.852973i	$0.325203\pi$
$660$	−1.49370e8	−0.0202236
$661$	1.06623e10	1.43597	0.717987	−	0.696057i	$-0.245064\pi$
$661$	1.06623e10	1.43597	0.717987	+	0.696057i	$0.245064\pi$
$662$	1.49701e10	2.00549
$663$	−9.30986e9	−1.24064
$664$	−3.54279e9	−0.469632
$665$	7.24707e8	0.0955623
$666$	1.45907e9	0.191389
$667$	1.11599e9	0.145620
$668$	1.77527e9	0.230434
$669$	−3.38237e7	−0.00436747
$670$	1.82993e9	0.235057
$671$	−1.17014e8	−0.0149523
$672$	7.54574e9	0.959200
$673$	1.13404e10	1.43408	0.717042	−	0.697030i	$-0.245495\pi$
$673$	1.13404e10	1.43408	0.717042	+	0.697030i	$0.245495\pi$
$674$	9.86954e9	1.24162
$675$	1.51877e9	0.190076
$676$	7.64946e9	0.952396
$677$	−2.59963e9	−0.321996	−0.160998	−	0.986955i	$-0.551471\pi$
$677$	−2.59963e9	−0.321996	−0.160998	+	0.986955i	$0.551471\pi$
$678$	3.80410e9	0.468757
$679$	1.76191e10	2.15993
$680$	1.16093e9	0.141587
$681$	3.82852e9	0.464532
$682$	−3.13756e9	−0.378745
$683$	1.25557e10	1.50789	0.753943	−	0.656940i	$-0.228149\pi$
$683$	1.25557e10	1.50789	0.753943	+	0.656940i	$0.228149\pi$
$684$	2.58918e9	0.309361
$685$	−2.65812e8	−0.0315979
$686$	−2.02821e9	−0.239871
$687$	2.03893e9	0.239913
$688$	4.24439e9	0.496884
$689$	−1.37858e9	−0.160570
$690$	−1.81787e8	−0.0210664
$691$	4.66084e9	0.537392	0.268696	−	0.963225i	$-0.413407\pi$
$691$	4.66084e9	0.537392	0.268696	+	0.963225i	$0.413407\pi$
$692$	−1.63706e10	−1.87799
$693$	8.53950e8	0.0974689
$694$	−1.43577e10	−1.63052
$695$	3.10874e8	0.0351268
$696$	2.72676e9	0.306560
$697$	7.43443e9	0.831635
$698$	−1.12826e10	−1.25579
$699$	7.19090e9	0.796367
$700$	1.82644e10	2.01263
$701$	−4.37132e9	−0.479292	−0.239646	−	0.970860i	$-0.577031\pi$
$701$	−4.37132e9	−0.479292	−0.239646	+	0.970860i	$0.577031\pi$
$702$	3.56194e9	0.388604
$703$	−2.10140e9	−0.228121
$704$	3.19030e9	0.344610
$705$	4.52955e7	0.00486847
$706$	−6.17830e9	−0.660774
$707$	1.49822e10	1.59443
$708$	1.15846e10	1.22678
$709$	−1.06398e10	−1.12117	−0.560585	−	0.828097i	$-0.689424\pi$
$709$	−1.06398e10	−1.12117	−0.560585	+	0.828097i	$0.689424\pi$
$710$	−4.09630e8	−0.0429524
$711$	−4.77569e9	−0.498302
$712$	1.25770e10	1.30586
$713$	−2.28036e9	−0.235608
$714$	−2.03912e10	−2.09652
$715$	2.95951e8	0.0302795
$716$	−1.83025e10	−1.86344
$717$	9.39246e9	0.951617
$718$	3.82892e9	0.386048
$719$	−1.04952e10	−1.05302	−0.526512	−	0.850168i	$-0.676501\pi$
$719$	−1.04952e10	−1.05302	−0.526512	+	0.850168i	$0.676501\pi$
$720$	1.05520e8	0.0105359
$721$	9.66073e9	0.959923
$722$	9.68985e9	0.958158
$723$	3.62694e9	0.356908
$724$	−4.98965e9	−0.488635
$725$	−7.07744e9	−0.689753
$726$	−8.95474e9	−0.868510
$727$	1.98822e9	0.191908	0.0959541	−	0.995386i	$-0.469410\pi$
$727$	1.98822e9	0.191908	0.0959541	+	0.995386i	$0.469410\pi$
$728$	1.39423e10	1.33929
$729$	3.87420e8	0.0370370
$730$	1.48496e9	0.141281
$731$	−3.09177e10	−2.92750
$732$	−6.38412e8	−0.0601606
$733$	2.85161e9	0.267440	0.133720	−	0.991019i	$-0.457308\pi$
$733$	2.85161e9	0.267440	0.133720	+	0.991019i	$0.457308\pi$
$734$	−3.05650e10	−2.85291
$735$	6.13803e8	0.0570195
$736$	2.72606e9	0.252036
$737$	−3.10555e9	−0.285760
$738$	−2.84441e9	−0.260492
$739$	−1.35553e10	−1.23553	−0.617766	−	0.786362i	$-0.711962\pi$
$739$	−1.35553e10	−1.23553	−0.617766	+	0.786362i	$0.711962\pi$
$740$	6.61416e8	0.0600017
$741$	−5.13004e9	−0.463188
$742$	−3.01947e9	−0.271342
$743$	−6.48538e9	−0.580062	−0.290031	−	0.957017i	$-0.593666\pi$
$743$	−6.48538e9	−0.580062	−0.290031	+	0.957017i	$0.593666\pi$
$744$	−5.57173e9	−0.496004
$745$	−5.53334e8	−0.0490276
$746$	1.54134e10	1.35929
$747$	2.34567e9	0.205894
$748$	−6.05307e9	−0.528835
$749$	−6.11689e9	−0.531917
$750$	2.32013e9	0.200815
$751$	8.10260e9	0.698047	0.349024	−	0.937114i	$-0.386513\pi$
$751$	8.10260e9	0.698047	0.349024	+	0.937114i	$0.386513\pi$
$752$	−2.51986e8	−0.0216079
$753$	3.38016e9	0.288506
$754$	−1.65986e10	−1.41017
$755$	3.87137e8	0.0327379
$756$	4.65905e9	0.392167
$757$	3.72306e9	0.311935	0.155968	−	0.987762i	$-0.450150\pi$
$757$	3.72306e9	0.311935	0.155968	+	0.987762i	$0.450150\pi$
$758$	9.15432e8	0.0763456
$759$	3.08507e8	0.0256106
$760$	6.39709e8	0.0528610
$761$	1.48338e10	1.22013	0.610065	−	0.792351i	$-0.291143\pi$
$761$	1.48338e10	1.22013	0.610065	+	0.792351i	$0.291143\pi$
$762$	1.79530e10	1.46993
$763$	−2.64819e10	−2.15831
$764$	1.02850e10	0.834408
$765$	−7.68644e8	−0.0620741
$766$	−3.59999e10	−2.89402
$767$	−2.29530e10	−1.83678
$768$	3.60273e9	0.286990
$769$	9.76344e9	0.774213	0.387107	−	0.922035i	$-0.373475\pi$
$769$	9.76344e9	0.774213	0.387107	+	0.922035i	$0.373475\pi$
$770$	6.48216e8	0.0511684
$771$	1.03665e10	0.814599
$772$	−2.93071e10	−2.29252
$773$	1.51981e10	1.18348	0.591740	−	0.806129i	$-0.298441\pi$
$773$	1.51981e10	1.18348	0.591740	+	0.806129i	$0.298441\pi$
$774$	1.18291e10	0.916976
$775$	1.44617e10	1.11600
$776$	1.55526e10	1.19478
$777$	−3.78133e9	−0.289182
$778$	−2.49328e10	−1.89820
$779$	4.09662e9	0.310488
$780$	1.61468e9	0.121830
$781$	6.95177e8	0.0522176
$782$	−7.36674e9	−0.550873
$783$	−1.80538e9	−0.134401
$784$	−3.41468e9	−0.253072
$785$	8.49604e8	0.0626863
$786$	−8.08308e8	−0.0593743
$787$	−5.97907e8	−0.0437242	−0.0218621	−	0.999761i	$-0.506959\pi$
$787$	−5.97907e8	−0.0437242	−0.0218621	+	0.999761i	$0.506959\pi$
$788$	2.22385e10	1.61907
$789$	1.39780e10	1.01315
$790$	−3.62512e9	−0.261594
$791$	−9.85872e9	−0.708276
$792$	7.53795e8	0.0539157
$793$	1.26491e9	0.0900748
$794$	3.49829e10	2.48018
$795$	−1.13819e8	−0.00803396
$796$	−1.74026e10	−1.22298
$797$	−6.05571e9	−0.423703	−0.211851	−	0.977302i	$-0.567949\pi$
$797$	−6.05571e9	−0.423703	−0.211851	+	0.977302i	$0.567949\pi$
$798$	−1.12362e10	−0.782726
$799$	1.83556e9	0.127308
$800$	−1.72883e10	−1.19381
$801$	−8.32716e9	−0.572510
$802$	4.04077e10	2.76601
$803$	−2.52010e9	−0.171757
$804$	−1.69435e10	−1.14976
$805$	4.71119e8	0.0318306
$806$	3.39168e10	2.28162
$807$	5.52975e9	0.370381
$808$	1.32250e10	0.881972
$809$	−7.82042e9	−0.519290	−0.259645	−	0.965704i	$-0.583606\pi$
$809$	−7.82042e9	−0.519290	−0.259645	+	0.965704i	$0.583606\pi$
$810$	2.94083e8	0.0194434
$811$	2.15298e10	1.41731	0.708657	−	0.705553i	$-0.249301\pi$
$811$	2.15298e10	1.41731	0.708657	+	0.705553i	$0.249301\pi$
$812$	−2.17112e10	−1.42310
$813$	6.23355e9	0.406835
$814$	−1.87961e9	−0.122147
$815$	−2.58500e9	−0.167266
$816$	4.27609e9	0.275506
$817$	−1.70367e10	−1.09297
$818$	−4.06707e10	−2.59803
$819$	−9.23115e9	−0.587168
$820$	−1.28941e9	−0.0816661
$821$	−2.41869e10	−1.52539	−0.762693	−	0.646761i	$-0.776123\pi$
$821$	−2.41869e10	−1.52539	−0.762693	+	0.646761i	$0.776123\pi$
$822$	4.12128e9	0.258810
$823$	−3.08772e8	−0.0193081	−0.00965403	−	0.999953i	$-0.503073\pi$
$823$	−3.08772e8	−0.0193081	−0.00965403	+	0.999953i	$0.503073\pi$
$824$	8.52767e9	0.530988
$825$	−1.95651e9	−0.121309
$826$	−5.02735e10	−3.10391
$827$	−1.13854e10	−0.699967	−0.349984	−	0.936756i	$-0.613813\pi$
$827$	−1.13854e10	−0.699967	−0.349984	+	0.936756i	$0.613813\pi$
$828$	1.68318e9	0.103044
$829$	−5.67599e9	−0.346019	−0.173010	−	0.984920i	$-0.555349\pi$
$829$	−5.67599e9	−0.346019	−0.173010	+	0.984920i	$0.555349\pi$
$830$	1.78054e9	0.108089
$831$	9.71252e9	0.587122
$832$	−3.44870e10	−2.07598
$833$	2.48738e10	1.49103
$834$	−4.81995e9	−0.287714
$835$	−2.90405e8	−0.0172624
$836$	−3.33544e9	−0.197438
$837$	3.68902e9	0.217456
$838$	−1.24647e9	−0.0731689
$839$	2.26083e10	1.32160	0.660801	−	0.750561i	$-0.270217\pi$
$839$	2.26083e10	1.32160	0.660801	+	0.750561i	$0.270217\pi$
$840$	1.15111e9	0.0670101
$841$	−8.83683e9	−0.512284
$842$	1.93490e10	1.11703
$843$	−1.50983e10	−0.868026
$844$	6.13147e9	0.351047
$845$	−1.25133e9	−0.0713466
$846$	−7.02283e8	−0.0398764
$847$	2.32072e10	1.31229
$848$	6.33193e8	0.0356575
$849$	7.42370e9	0.416336
$850$	4.67188e10	2.60931
$851$	−1.36609e9	−0.0759844
$852$	3.79280e9	0.210098
$853$	−3.13840e10	−1.73136	−0.865679	−	0.500599i	$-0.833113\pi$
$853$	−3.13840e10	−1.73136	−0.865679	+	0.500599i	$0.833113\pi$
$854$	2.77050e9	0.152214
$855$	−4.23549e8	−0.0231751
$856$	−5.39947e9	−0.294234
$857$	6.44481e9	0.349766	0.174883	−	0.984589i	$-0.444045\pi$
$857$	6.44481e9	0.349766	0.174883	+	0.984589i	$0.444045\pi$
$858$	−4.58858e9	−0.248012
$859$	−2.36238e10	−1.27167	−0.635834	−	0.771826i	$-0.719344\pi$
$859$	−2.36238e10	−1.27167	−0.635834	+	0.771826i	$0.719344\pi$
$860$	5.36228e9	0.287478
$861$	7.37158e9	0.393595
$862$	5.64856e10	3.00374
$863$	−1.31698e10	−0.697496	−0.348748	−	0.937217i	$-0.613393\pi$
$863$	−1.31698e10	−0.697496	−0.348748	+	0.937217i	$0.613393\pi$
$864$	−4.41004e9	−0.232619
$865$	2.67797e9	0.140685
$866$	−5.81211e9	−0.304103
$867$	−2.00695e10	−1.04585
$868$	4.43635e10	2.30254
$869$	6.15215e9	0.318022
$870$	−1.37042e9	−0.0705566
$871$	3.35708e10	1.72146
$872$	−2.33760e10	−1.19388
$873$	−1.02973e10	−0.523811
$874$	−4.05932e9	−0.205666
$875$	−6.01285e9	−0.303426
$876$	−1.37494e10	−0.691065
$877$	−1.72997e10	−0.866046	−0.433023	−	0.901383i	$-0.642553\pi$
$877$	−1.72997e10	−0.866046	−0.433023	+	0.901383i	$0.642553\pi$
$878$	4.69109e10	2.33907
$879$	−5.44433e9	−0.270385
$880$	−1.35933e8	−0.00672411
$881$	2.51040e10	1.23688	0.618439	−	0.785832i	$-0.287765\pi$
$881$	2.51040e10	1.23688	0.618439	+	0.785832i	$0.287765\pi$
$882$	−9.51670e9	−0.467032
$883$	2.96678e10	1.45018	0.725091	−	0.688653i	$-0.241797\pi$
$883$	2.96678e10	1.45018	0.725091	+	0.688653i	$0.241797\pi$
$884$	6.54333e10	3.18578
$885$	−1.89506e9	−0.0919012
$886$	5.52881e9	0.267063
$887$	1.46350e10	0.704140	0.352070	−	0.935974i	$-0.385478\pi$
$887$	1.46350e10	0.704140	0.352070	+	0.935974i	$0.385478\pi$
$888$	−3.33784e9	−0.159963
$889$	−4.65272e10	−2.22101
$890$	−6.32097e9	−0.300551
$891$	−4.99084e8	−0.0236375
$892$	2.37726e8	0.0112150
$893$	1.01145e9	0.0475298
$894$	8.57916e9	0.401572
$895$	2.99400e9	0.139595
$896$	−3.97636e10	−1.84675
$897$	−3.33495e9	−0.154282
$898$	3.53088e10	1.62711
$899$	−1.71908e10	−0.789110
$900$	−1.06745e10	−0.488088
$901$	−4.61241e9	−0.210083
$902$	3.66423e9	0.166249
$903$	−3.06563e10	−1.38552
$904$	−8.70244e9	−0.391788
$905$	8.16227e8	0.0366050
$906$	−6.00236e9	−0.268147
$907$	1.17379e10	0.522353	0.261177	−	0.965291i	$-0.415889\pi$
$907$	1.17379e10	0.522353	0.261177	+	0.965291i	$0.415889\pi$
$908$	−2.69083e10	−1.19285
$909$	−8.75619e9	−0.386671
$910$	−7.00717e9	−0.308246
$911$	−3.54807e9	−0.155481	−0.0777405	−	0.996974i	$-0.524771\pi$
$911$	−3.54807e9	−0.155481	−0.0777405	+	0.996974i	$0.524771\pi$
$912$	2.35627e9	0.102859
$913$	−3.02174e9	−0.131404
$914$	2.80244e10	1.21401
$915$	1.04434e8	0.00450680
$916$	−1.43304e10	−0.616060
$917$	2.09482e9	0.0897126
$918$	1.19174e10	0.508433
$919$	3.70231e10	1.57350	0.786752	−	0.617269i	$-0.211761\pi$
$919$	3.70231e10	1.57350	0.786752	+	0.617269i	$0.211761\pi$
$920$	4.15864e8	0.0176073
$921$	1.18409e10	0.499433
$922$	−4.99658e10	−2.09949
$923$	−7.51482e9	−0.314567
$924$	−6.00189e9	−0.250286
$925$	8.66352e9	0.359914
$926$	−5.87592e10	−2.43185
$927$	−5.64613e9	−0.232794
$928$	2.05508e10	0.844131
$929$	2.67577e10	1.09495	0.547475	−	0.836822i	$-0.315589\pi$
$929$	2.67577e10	1.09495	0.547475	+	0.836822i	$0.315589\pi$
$930$	2.80026e9	0.114158
$931$	1.37063e10	0.556668
$932$	−5.05405e10	−2.04495
$933$	−1.63464e10	−0.658926
$934$	3.95914e10	1.58996
$935$	9.90185e8	0.0396165
$936$	−8.14847e9	−0.324796
$937$	−3.93013e9	−0.156069	−0.0780347	−	0.996951i	$-0.524865\pi$
$937$	−3.93013e9	−0.156069	−0.0780347	+	0.996951i	$0.524865\pi$
$938$	7.35293e10	2.90904
$939$	2.89042e8	0.0113928
$940$	−3.18354e8	−0.0125015
$941$	−2.54150e10	−0.994320	−0.497160	−	0.867659i	$-0.665624\pi$
$941$	−2.54150e10	−0.994320	−0.497160	+	0.867659i	$0.665624\pi$
$942$	−1.31727e10	−0.513447
$943$	2.66314e9	0.103420
$944$	1.05425e10	0.407889
$945$	−7.62146e8	−0.0293783
$946$	−1.52385e10	−0.585225
$947$	−2.76072e10	−1.05632	−0.528161	−	0.849144i	$-0.677118\pi$
$947$	−2.76072e10	−1.05632	−0.528161	+	0.849144i	$0.677118\pi$
$948$	3.35654e10	1.27957
$949$	2.72422e10	1.03469
$950$	2.57436e10	0.974174
$951$	−2.69922e10	−1.01767
$952$	4.66478e10	1.75227
$953$	−2.15615e10	−0.806962	−0.403481	−	0.914988i	$-0.632200\pi$
$953$	−2.15615e10	−0.806962	−0.403481	+	0.914988i	$0.632200\pi$
$954$	1.76470e9	0.0658041
$955$	−1.68246e9	−0.0625078
$956$	−6.60139e10	−2.44361
$957$	2.32573e9	0.0857763
$958$	−7.28871e10	−2.67837
$959$	−1.06807e10	−0.391053
$960$	−2.84733e9	−0.103870
$961$	7.61423e9	0.276754
$962$	2.03184e10	0.735830
$963$	3.57496e9	0.128997
$964$	−2.54916e10	−0.916489
$965$	4.79418e9	0.171739
$966$	−7.30446e9	−0.260716
$967$	8.89528e9	0.316349	0.158175	−	0.987411i	$-0.449439\pi$
$967$	8.89528e9	0.316349	0.158175	+	0.987411i	$0.449439\pi$
$968$	2.04853e10	0.725903
$969$	−1.71639e10	−0.606015
$970$	−7.81649e9	−0.274986
$971$	−9.39030e9	−0.329164	−0.164582	−	0.986363i	$-0.552628\pi$
$971$	−9.39030e9	−0.329164	−0.164582	+	0.986363i	$0.552628\pi$
$972$	−2.72294e9	−0.0951057
$973$	1.24914e10	0.434727
$974$	3.93092e10	1.36313
$975$	2.11498e10	0.730784
$976$	−5.80983e8	−0.0200027
$977$	4.04777e10	1.38863	0.694313	−	0.719673i	$-0.255708\pi$
$977$	4.04777e10	1.38863	0.694313	+	0.719673i	$0.255708\pi$
$978$	4.00790e10	1.37003
$979$	1.07272e10	0.365383
$980$	−4.31405e9	−0.146418
$981$	1.54771e10	0.523418
$982$	4.36666e10	1.47150
$983$	2.17203e10	0.729336	0.364668	−	0.931138i	$-0.381182\pi$
$983$	2.17203e10	0.729336	0.364668	+	0.931138i	$0.381182\pi$
$984$	6.50700e9	0.217720
$985$	−3.63787e9	−0.121289
$986$	−5.55352e10	−1.84501
$987$	1.82004e9	0.0602519
$988$	3.60559e10	1.18940
$989$	−1.10752e10	−0.364054
$990$	−3.78844e8	−0.0124090
$991$	4.02205e10	1.31277	0.656387	−	0.754425i	$-0.272084\pi$
$991$	4.02205e10	1.31277	0.656387	+	0.754425i	$0.272084\pi$
$992$	−4.19924e10	−1.36578
$993$	2.26743e10	0.734871
$994$	−1.64595e10	−0.531576
$995$	2.84680e9	0.0916169
$996$	−1.64862e10	−0.528706
$997$	1.02432e10	0.327342	0.163671	−	0.986515i	$-0.447666\pi$
$997$	1.02432e10	0.327342	0.163671	+	0.986515i	$0.447666\pi$
$998$	7.29287e10	2.32242
$999$	2.20997e9	0.0701304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.a.1.1	✓	5
3.2	odd	2		207.8.a.a.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.a.1.1	✓	5	1.1	even	1	trivial
207.8.a.a.1.5		5	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)

\(f(q)\)	\(=\)	\(q-17.8260 q^{2} -27.0000 q^{3} +189.767 q^{4} -31.0428 q^{5} +481.302 q^{6} -1247.35 q^{7} -1101.05 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-17.8260 q^{2} -27.0000 q^{3} +189.767 q^{4} -31.0428 q^{5} +481.302 q^{6} -1247.35 q^{7} -1101.05 q^{8} +729.000 q^{9} +553.369 q^{10} -939.114 q^{11} -5123.70 q^{12} +10151.8 q^{13} +22235.2 q^{14} +838.155 q^{15} -4662.78 q^{16} +33965.5 q^{17} -12995.2 q^{18} +18716.1 q^{19} -5890.88 q^{20} +33678.3 q^{21} +16740.7 q^{22} +12167.0 q^{23} +29728.4 q^{24} -77161.3 q^{25} -180965. q^{26} -19683.0 q^{27} -236704. q^{28} +91722.7 q^{29} -14941.0 q^{30} -187422. q^{31} +224053. q^{32} +25356.1 q^{33} -605469. q^{34} +38721.0 q^{35} +138340. q^{36} -112278. q^{37} -333633. q^{38} -274098. q^{39} +34179.6 q^{40} +218882. q^{41} -600350. q^{42} -910269. q^{43} -178212. q^{44} -22630.2 q^{45} -216889. q^{46} +54041.9 q^{47} +125895. q^{48} +732327. q^{49} +1.37548e6 q^{50} -917067. q^{51} +1.92647e6 q^{52} -135797. q^{53} +350869. q^{54} +29152.7 q^{55} +1.37339e6 q^{56} -505335. q^{57} -1.63505e6 q^{58} -2.26099e6 q^{59} +159054. q^{60} +124600. q^{61} +3.34098e6 q^{62} -909315. q^{63} -3.39714e6 q^{64} -315139. q^{65} -451998. q^{66} +3.30689e6 q^{67} +6.44551e6 q^{68} -328509. q^{69} -690242. q^{70} -740248. q^{71} -802666. q^{72} +2.68349e6 q^{73} +2.00147e6 q^{74} +2.08336e6 q^{75} +3.55169e6 q^{76} +1.17140e6 q^{77} +4.88607e6 q^{78} -6.55101e6 q^{79} +144746. q^{80} +531441. q^{81} -3.90179e6 q^{82} +3.21765e6 q^{83} +6.39102e6 q^{84} -1.05438e6 q^{85} +1.62265e7 q^{86} -2.47651e6 q^{87} +1.03401e6 q^{88} -1.14227e7 q^{89} +403406. q^{90} -1.26628e7 q^{91} +2.30889e6 q^{92} +5.06038e6 q^{93} -963351. q^{94} -580999. q^{95} -6.04944e6 q^{96} -1.41253e7 q^{97} -1.30545e7 q^{98} -684614. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99}+O(q^{100}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9 + 1452 * q^10 - 1148 * q^11 - 7290 * q^12 - 642 * q^13 + 5756 * q^14 + 7182 * q^15 - 22606 * q^16 - 5798 * q^17 - 6036 * q^19 - 27376 * q^20 + 13392 * q^21 - 97896 * q^22 + 60835 * q^23 - 38394 * q^24 - 262477 * q^25 - 355992 * q^26 - 98415 * q^27 - 507124 * q^28 - 169162 * q^29 - 39204 * q^30 - 199640 * q^31 - 284794 * q^32 + 30996 * q^33 - 1027740 * q^34 - 137680 * q^35 + 196830 * q^36 - 202002 * q^37 - 554924 * q^38 + 17334 * q^39 - 340904 * q^40 + 541282 * q^41 - 155412 * q^42 - 909596 * q^43 - 1236032 * q^44 - 193914 * q^45 + 80208 * q^47 + 610362 * q^48 + 850589 * q^49 - 941416 * q^50 + 156546 * q^51 + 146940 * q^52 - 278138 * q^53 - 933560 * q^55 - 539932 * q^56 + 162972 * q^57 - 3522712 * q^58 - 3177380 * q^59 + 739152 * q^60 + 147782 * q^61 + 4606456 * q^62 - 361584 * q^63 - 4142622 * q^64 + 3877332 * q^65 + 2643192 * q^66 - 464916 * q^67 + 7513072 * q^68 - 1642545 * q^69 + 2093200 * q^70 + 1576792 * q^71 + 1036638 * q^72 - 38190 * q^73 + 12164864 * q^74 + 7086879 * q^75 + 6889436 * q^76 + 10332384 * q^77 + 9611784 * q^78 - 3913336 * q^79 + 6334776 * q^80 + 2657205 * q^81 + 6799360 * q^82 + 15774716 * q^83 + 13692348 * q^84 - 8520740 * q^85 + 24874084 * q^86 + 4567374 * q^87 + 53216 * q^88 + 1116482 * q^89 + 1058508 * q^90 - 27369552 * q^91 + 3285090 * q^92 + 5390280 * q^93 - 7153744 * q^94 - 6067832 * q^95 + 7689438 * q^96 - 15738566 * q^97 + 11730488 * q^98 - 836892 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more