LMFDB - Embedded newform 69.6.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,6,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, 6, names="a")

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
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:	$11.0664835671$
Analytic rank:	$0$
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} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$11.0973$ 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)

$f(q)$	$=$	$q-3.54286 q^{2} +9.00000 q^{3} -19.4482 q^{4} -92.1306 q^{5} -31.8857 q^{6} -8.98894 q^{7} +182.273 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-3.54286 q^{2} +9.00000 q^{3} -19.4482 q^{4} -92.1306 q^{5} -31.8857 q^{6} -8.98894 q^{7} +182.273 q^{8} +81.0000 q^{9} +326.406 q^{10} +612.746 q^{11} -175.034 q^{12} -496.413 q^{13} +31.8465 q^{14} -829.175 q^{15} -23.4274 q^{16} +745.481 q^{17} -286.971 q^{18} +1299.52 q^{19} +1791.77 q^{20} -80.9005 q^{21} -2170.87 q^{22} -529.000 q^{23} +1640.46 q^{24} +5363.05 q^{25} +1758.72 q^{26} +729.000 q^{27} +174.818 q^{28} +7754.82 q^{29} +2937.65 q^{30} -1066.62 q^{31} -5749.75 q^{32} +5514.72 q^{33} -2641.13 q^{34} +828.157 q^{35} -1575.30 q^{36} -840.279 q^{37} -4604.02 q^{38} -4467.72 q^{39} -16793.0 q^{40} +9821.00 q^{41} +286.619 q^{42} -13529.8 q^{43} -11916.8 q^{44} -7462.58 q^{45} +1874.17 q^{46} +20485.1 q^{47} -210.846 q^{48} -16726.2 q^{49} -19000.5 q^{50} +6709.33 q^{51} +9654.33 q^{52} +4816.85 q^{53} -2582.74 q^{54} -56452.7 q^{55} -1638.45 q^{56} +11695.7 q^{57} -27474.2 q^{58} +30528.3 q^{59} +16125.9 q^{60} +9.53050 q^{61} +3778.88 q^{62} -728.104 q^{63} +21120.2 q^{64} +45734.8 q^{65} -19537.8 q^{66} +9996.72 q^{67} -14498.2 q^{68} -4761.00 q^{69} -2934.04 q^{70} -73725.9 q^{71} +14764.2 q^{72} +55169.6 q^{73} +2976.99 q^{74} +48267.4 q^{75} -25273.3 q^{76} -5507.94 q^{77} +15828.5 q^{78} +62397.4 q^{79} +2158.38 q^{80} +6561.00 q^{81} -34794.4 q^{82} -82435.2 q^{83} +1573.37 q^{84} -68681.6 q^{85} +47934.2 q^{86} +69793.4 q^{87} +111687. q^{88} +95173.0 q^{89} +26438.8 q^{90} +4462.23 q^{91} +10288.1 q^{92} -9599.57 q^{93} -72575.8 q^{94} -119726. q^{95} -51747.8 q^{96} +47134.4 q^{97} +59258.5 q^{98} +49632.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9}+O(q^{10}) $ 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9	\( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99}+O(q^{100}) \) 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9 - 172 * q^10 + 1100 * q^11 + 1062 * q^12 - 978 * q^13 - 344 * q^14 + 846 * q^15 + 1218 * q^16 + 2522 * q^17 + 648 * q^18 + 2060 * q^19 + 7720 * q^20 + 2448 * q^21 - 2572 * q^22 - 2645 * q^23 + 2322 * q^24 + 12035 * q^25 + 9280 * q^26 + 3645 * q^27 + 8072 * q^28 + 1526 * q^29 - 1548 * q^30 - 7392 * q^31 - 5086 * q^32 + 9900 * q^33 - 15608 * q^34 + 6056 * q^35 + 9558 * q^36 - 8210 * q^37 - 14276 * q^38 - 8802 * q^39 - 37472 * q^40 + 21250 * q^41 - 3096 * q^42 - 4548 * q^43 - 4260 * q^44 + 7614 * q^45 - 4232 * q^46 + 536 * q^47 + 10962 * q^48 - 27979 * q^49 - 81872 * q^50 + 22698 * q^51 - 76380 * q^52 - 11482 * q^53 + 5832 * q^54 - 77064 * q^55 - 28624 * q^56 + 18540 * q^57 - 79680 * q^58 + 74676 * q^59 + 69480 * q^60 - 44618 * q^61 + 64880 * q^62 + 22032 * q^63 - 137382 * q^64 - 24388 * q^65 - 23148 * q^66 - 1412 * q^67 + 80196 * q^68 - 23805 * q^69 - 222304 * q^70 + 37912 * q^71 + 20898 * q^72 + 46546 * q^73 + 111604 * q^74 + 108315 * q^75 - 79548 * q^76 + 157008 * q^77 + 83520 * q^78 + 50544 * q^79 + 69424 * q^80 + 32805 * q^81 - 233720 * q^82 + 89588 * q^83 + 72648 * q^84 + 147892 * q^85 + 77428 * q^86 + 13734 * q^87 + 54484 * q^88 + 280410 * q^89 - 13932 * q^90 - 27416 * q^91 - 62422 * q^92 - 66528 * q^93 + 113632 * q^94 + 203120 * q^95 - 45774 * q^96 + 90074 * q^97 + 32976 * q^98 + 89100 * 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 $

$2$	−3.54286	−0.626294	−0.313147	−	0.949705i	$-0.601383\pi$
$2$	−3.54286	−0.626294	−0.313147	+	0.949705i	$0.601383\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−19.4482	−0.607755
$5$	−92.1306	−1.64808	−0.824041	−	0.566530i	$-0.808286\pi$
$5$	−92.1306	−1.64808	−0.824041	+	0.566530i	$0.808286\pi$
$6$	−31.8857	−0.361591
$7$	−8.98894	−0.0693368	−0.0346684	−	0.999399i	$-0.511038\pi$
$7$	−8.98894	−0.0693368	−0.0346684	+	0.999399i	$0.511038\pi$
$8$	182.273	1.00693
$9$	81.0000	0.333333
$10$	326.406	1.03218
$11$	612.746	1.52686	0.763430	−	0.645891i	$-0.223514\pi$
$11$	612.746	1.52686	0.763430	+	0.645891i	$0.223514\pi$
$12$	−175.034	−0.350888
$13$	−496.413	−0.814676	−0.407338	−	0.913278i	$-0.633543\pi$
$13$	−496.413	−0.814676	−0.407338	+	0.913278i	$0.633543\pi$
$14$	31.8465	0.0434252
$15$	−829.175	−0.951521
$16$	−23.4274	−0.0228783
$17$	745.481	0.625625	0.312813	−	0.949815i	$-0.398729\pi$
$17$	745.481	0.625625	0.312813	+	0.949815i	$0.398729\pi$
$18$	−286.971	−0.208765
$19$	1299.52	0.825847	0.412924	−	0.910766i	$-0.364508\pi$
$19$	1299.52	0.825847	0.412924	+	0.910766i	$0.364508\pi$
$20$	1791.77	1.00163
$21$	−80.9005	−0.0400316
$22$	−2170.87	−0.956263
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	1640.46	0.581350
$25$	5363.05	1.71618
$26$	1758.72	0.510227
$27$	729.000	0.192450
$28$	174.818	0.0421398
$29$	7754.82	1.71229	0.856144	−	0.516738i	$-0.172854\pi$
$29$	7754.82	1.71229	0.856144	+	0.516738i	$0.172854\pi$
$30$	2937.65	0.595932
$31$	−1066.62	−0.199345	−0.0996724	−	0.995020i	$-0.531779\pi$
$31$	−1066.62	−0.199345	−0.0996724	+	0.995020i	$0.531779\pi$
$32$	−5749.75	−0.992600
$33$	5514.72	0.881533
$34$	−2641.13	−0.391826
$35$	828.157	0.114273
$36$	−1575.30	−0.202585
$37$	−840.279	−0.100907	−0.0504533	−	0.998726i	$-0.516067\pi$
$37$	−840.279	−0.100907	−0.0504533	+	0.998726i	$0.516067\pi$
$38$	−4604.02	−0.517224
$39$	−4467.72	−0.470353
$40$	−16793.0	−1.65950
$41$	9821.00	0.912422	0.456211	−	0.889872i	$-0.349206\pi$
$41$	9821.00	0.912422	0.456211	+	0.889872i	$0.349206\pi$
$42$	286.619	0.0250716
$43$	−13529.8	−1.11589	−0.557944	−	0.829879i	$-0.688410\pi$
$43$	−13529.8	−1.11589	−0.557944	+	0.829879i	$0.688410\pi$
$44$	−11916.8	−0.927957
$45$	−7462.58	−0.549361
$46$	1874.17	0.130591
$47$	20485.1	1.35268	0.676338	−	0.736591i	$-0.263566\pi$
$47$	20485.1	1.35268	0.676338	+	0.736591i	$0.263566\pi$
$48$	−210.846	−0.0132088
$49$	−16726.2	−0.995192
$50$	−19000.5	−1.07483
$51$	6709.33	0.361205
$52$	9654.33	0.495124
$53$	4816.85	0.235545	0.117772	−	0.993041i	$-0.462425\pi$
$53$	4816.85	0.235545	0.117772	+	0.993041i	$0.462425\pi$
$54$	−2582.74	−0.120530
$55$	−56452.7	−2.51639
$56$	−1638.45	−0.0698171
$57$	11695.7	0.476803
$58$	−27474.2	−1.07240
$59$	30528.3	1.14175	0.570877	−	0.821036i	$-0.306603\pi$
$59$	30528.3	1.14175	0.570877	+	0.821036i	$0.306603\pi$
$60$	16125.9	0.578292
$61$	9.53050	0.000327937 0	0.000163969	−	1.00000i	$-0.499948\pi$
$61$	9.53050	0.000327937 0	0.000163969		1.00000i	$0.499948\pi$
$62$	3778.88	0.124849
$63$	−728.104	−0.0231123
$64$	21120.2	0.644538
$65$	45734.8	1.34265
$66$	−19537.8	−0.552099
$67$	9996.72	0.272064	0.136032	−	0.990704i	$-0.456565\pi$
$67$	9996.72	0.272064	0.136032	+	0.990704i	$0.456565\pi$
$68$	−14498.2	−0.380227
$69$	−4761.00	−0.120386
$70$	−2934.04	−0.0715684
$71$	−73725.9	−1.73570	−0.867849	−	0.496828i	$-0.834498\pi$
$71$	−73725.9	−1.73570	−0.867849	+	0.496828i	$0.834498\pi$
$72$	14764.2	0.335643
$73$	55169.6	1.21169	0.605847	−	0.795581i	$-0.292834\pi$
$73$	55169.6	1.21169	0.605847	+	0.795581i	$0.292834\pi$
$74$	2976.99	0.0631972
$75$	48267.4	0.990834
$76$	−25273.3	−0.501913
$77$	−5507.94	−0.105867
$78$	15828.5	0.294580
$79$	62397.4	1.12486	0.562431	−	0.826844i	$-0.309866\pi$
$79$	62397.4	1.12486	0.562431	+	0.826844i	$0.309866\pi$
$80$	2158.38	0.0377053
$81$	6561.00	0.111111
$82$	−34794.4	−0.571445
$83$	−82435.2	−1.31346	−0.656731	−	0.754125i	$-0.728061\pi$
$83$	−82435.2	−1.31346	−0.656731	+	0.754125i	$0.728061\pi$
$84$	1573.37	0.0243294
$85$	−68681.6	−1.03108
$86$	47934.2	0.698875
$87$	69793.4	0.988589
$88$	111687.	1.53744
$89$	95173.0	1.27362	0.636808	−	0.771022i	$-0.280254\pi$
$89$	95173.0	1.27362	0.636808	+	0.771022i	$0.280254\pi$
$90$	26438.8	0.344062
$91$	4462.23	0.0564870
$92$	10288.1	0.126726
$93$	−9599.57	−0.115092
$94$	−72575.8	−0.847174
$95$	−119726.	−1.36106
$96$	−51747.8	−0.573078
$97$	47134.4	0.508638	0.254319	−	0.967120i	$-0.418149\pi$
$97$	47134.4	0.508638	0.254319	+	0.967120i	$0.418149\pi$
$98$	59258.5	0.623283
$99$	49632.4	0.508953
$100$	−104301.	−1.04301
$101$	−15830.2	−0.154413	−0.0772065	−	0.997015i	$-0.524600\pi$
$101$	−15830.2	−0.154413	−0.0772065	+	0.997015i	$0.524600\pi$
$102$	−23770.2	−0.226221
$103$	−90782.5	−0.843158	−0.421579	−	0.906792i	$-0.638524\pi$
$103$	−90782.5	−0.843158	−0.421579	+	0.906792i	$0.638524\pi$
$104$	−90482.9	−0.820320
$105$	7453.41	0.0659754
$106$	−17065.4	−0.147520
$107$	209377.	1.76795	0.883973	−	0.467538i	$-0.154859\pi$
$107$	209377.	1.76795	0.883973	+	0.467538i	$0.154859\pi$
$108$	−14177.7	−0.116963
$109$	40116.0	0.323408	0.161704	−	0.986839i	$-0.448301\pi$
$109$	40116.0	0.323408	0.161704	+	0.986839i	$0.448301\pi$
$110$	200004.	1.57600
$111$	−7562.51	−0.0582584
$112$	210.587	0.00158631
$113$	248398.	1.83000	0.915001	−	0.403453i	$-0.132190\pi$
$113$	248398.	1.83000	0.915001	+	0.403453i	$0.132190\pi$
$114$	−41436.2	−0.298619
$115$	48737.1	0.343649
$116$	−150817.	−1.04065
$117$	−40209.5	−0.271559
$118$	−108157.	−0.715074
$119$	−6701.09	−0.0433788
$120$	−151137.	−0.958113
$121$	214407.	1.33130
$122$	−33.7652	−0.000205385 0
$123$	88389.0	0.526787
$124$	20743.8	0.121153
$125$	−206193.	−1.18032
$126$	2579.57	0.0144751
$127$	128385.	0.706328	0.353164	−	0.935561i	$-0.385106\pi$
$127$	128385.	0.706328	0.353164	+	0.935561i	$0.385106\pi$
$128$	109166.	0.588929
$129$	−121768.	−0.644258
$130$	−162032.	−0.840896
$131$	−361743.	−1.84171	−0.920857	−	0.389901i	$-0.872509\pi$
$131$	−361743.	−1.84171	−0.920857	+	0.389901i	$0.872509\pi$
$132$	−107251.	−0.535756
$133$	−11681.3	−0.0572616
$134$	−35416.9	−0.170392
$135$	−67163.2	−0.317174
$136$	135881.	0.629960
$137$	−116630.	−0.530897	−0.265448	−	0.964125i	$-0.585520\pi$
$137$	−116630.	−0.530897	−0.265448	+	0.964125i	$0.585520\pi$
$138$	16867.5	0.0753970
$139$	94540.9	0.415033	0.207517	−	0.978231i	$-0.433462\pi$
$139$	94540.9	0.415033	0.207517	+	0.978231i	$0.433462\pi$
$140$	−16106.1	−0.0694498
$141$	184366.	0.780968
$142$	261200.	1.08706
$143$	−304175.	−1.24390
$144$	−1897.62	−0.00762610
$145$	−714456.	−2.82199
$146$	−195458.	−0.758877
$147$	−150536.	−0.574575
$148$	16341.9	0.0613265
$149$	275055.	1.01497	0.507486	−	0.861660i	$-0.330575\pi$
$149$	275055.	1.01497	0.507486	+	0.861660i	$0.330575\pi$
$150$	−171005.	−0.620554
$151$	−47405.4	−0.169194	−0.0845971	−	0.996415i	$-0.526960\pi$
$151$	−47405.4	−0.169194	−0.0845971	+	0.996415i	$0.526960\pi$
$152$	236868.	0.831569
$153$	60383.9	0.208542
$154$	19513.8	0.0663042
$155$	98268.2	0.328537
$156$	86888.9	0.285860
$157$	88519.4	0.286609	0.143304	−	0.989679i	$-0.454227\pi$
$157$	88519.4	0.286609	0.143304	+	0.989679i	$0.454227\pi$
$158$	−221065.	−0.704494
$159$	43351.7	0.135992
$160$	529728.	1.63589
$161$	4755.15	0.0144577
$162$	−23244.7	−0.0695883
$163$	522220.	1.53952	0.769759	−	0.638335i	$-0.220377\pi$
$163$	522220.	1.53952	0.769759	+	0.638335i	$0.220377\pi$
$164$	−191000.	−0.554530
$165$	−508074.	−1.45284
$166$	292056.	0.822614
$167$	−234704.	−0.651223	−0.325611	−	0.945504i	$-0.605570\pi$
$167$	−234704.	−0.651223	−0.325611	+	0.945504i	$0.605570\pi$
$168$	−14746.0	−0.0403089
$169$	−124867.	−0.336303
$170$	243329.	0.645761
$171$	105261.	0.275282
$172$	263130.	0.678187
$173$	−249399.	−0.633547	−0.316773	−	0.948501i	$-0.602599\pi$
$173$	−249399.	−0.633547	−0.316773	+	0.948501i	$0.602599\pi$
$174$	−247268.	−0.619148
$175$	−48208.1	−0.118994
$176$	−14355.0	−0.0349319
$177$	274755.	0.659192
$178$	−337184.	−0.797659
$179$	353019.	0.823503	0.411751	−	0.911296i	$-0.364917\pi$
$179$	353019.	0.823503	0.411751	+	0.911296i	$0.364917\pi$
$180$	145133.	0.333877
$181$	−188804.	−0.428367	−0.214183	−	0.976793i	$-0.568709\pi$
$181$	−188804.	−0.428367	−0.214183	+	0.976793i	$0.568709\pi$
$182$	−15809.0	−0.0353775
$183$	85.7745	0.000189335 0
$184$	−96422.7	−0.209959
$185$	77415.4	0.166302
$186$	34009.9	0.0720813
$187$	456791.	0.955241
$188$	−398398.	−0.822096
$189$	−6552.94	−0.0133439
$190$	424171.	0.852427
$191$	374966.	0.743719	0.371859	−	0.928289i	$-0.378720\pi$
$191$	374966.	0.743719	0.371859	+	0.928289i	$0.378720\pi$
$192$	190082.	0.372124
$193$	−761894.	−1.47232	−0.736158	−	0.676810i	$-0.763362\pi$
$193$	−761894.	−1.47232	−0.736158	+	0.676810i	$0.763362\pi$
$194$	−166991.	−0.318557
$195$	411614.	0.775181
$196$	325294.	0.604833
$197$	−772963.	−1.41904	−0.709518	−	0.704688i	$-0.751087\pi$
$197$	−772963.	−1.41904	−0.709518	+	0.704688i	$0.751087\pi$
$198$	−175841.	−0.318754
$199$	863431.	1.54559	0.772796	−	0.634654i	$-0.218857\pi$
$199$	863431.	1.54559	0.772796	+	0.634654i	$0.218857\pi$
$200$	977541.	1.72807
$201$	89970.5	0.157076
$202$	56084.2	0.0967080
$203$	−69707.6	−0.118724
$204$	−130484.	−0.219524
$205$	−904814.	−1.50375
$206$	321629.	0.528065
$207$	−42849.0	−0.0695048
$208$	11629.7	0.0186384
$209$	796278.	1.26095
$210$	−26406.4	−0.0413200
$211$	973725.	1.50567	0.752835	−	0.658209i	$-0.228686\pi$
$211$	973725.	1.50567	0.752835	+	0.658209i	$0.228686\pi$
$212$	−93679.0	−0.143154
$213$	−663533.	−1.00211
$214$	−741792.	−1.10725
$215$	1.24651e6	1.83908
$216$	132877.	0.193783
$217$	9587.78	0.0138219
$218$	−142125.	−0.202549
$219$	496527.	0.699572
$220$	1.09790e6	1.52935
$221$	−370066.	−0.509682
$222$	26792.9	0.0364869
$223$	−1.07660e6	−1.44974	−0.724870	−	0.688885i	$-0.758100\pi$
$223$	−1.07660e6	−1.44974	−0.724870	+	0.688885i	$0.758100\pi$
$224$	51684.2	0.0688236
$225$	434407.	0.572059
$226$	−880037.	−1.14612
$227$	−313794.	−0.404185	−0.202092	−	0.979366i	$-0.564774\pi$
$227$	−313794.	−0.404185	−0.202092	+	0.979366i	$0.564774\pi$
$228$	−227460.	−0.289780
$229$	1.11820e6	1.40907	0.704534	−	0.709671i	$-0.251156\pi$
$229$	1.11820e6	1.40907	0.704534	+	0.709671i	$0.251156\pi$
$230$	−172669.	−0.215225
$231$	−49571.5	−0.0611226
$232$	1.41350e6	1.72415
$233$	−562380.	−0.678641	−0.339321	−	0.940671i	$-0.610197\pi$
$233$	−562380.	−0.678641	−0.339321	+	0.940671i	$0.610197\pi$
$234$	142456.	0.170076
$235$	−1.88731e6	−2.22932
$236$	−593719.	−0.693907
$237$	561577.	0.649439
$238$	23741.0	0.0271679
$239$	−560026.	−0.634181	−0.317090	−	0.948395i	$-0.602706\pi$
$239$	−560026.	−0.634181	−0.317090	+	0.948395i	$0.602706\pi$
$240$	19425.4	0.0217692
$241$	−1.62461e6	−1.80180	−0.900902	−	0.434023i	$-0.857093\pi$
$241$	−1.62461e6	−1.80180	−0.900902	+	0.434023i	$0.857093\pi$
$242$	−759613.	−0.833785
$243$	59049.0	0.0641500
$244$	−185.351	−0.000199306 0
$245$	1.54099e6	1.64016
$246$	−313149.	−0.329924
$247$	−645100.	−0.672798
$248$	−194416.	−0.200726
$249$	−741917.	−0.758327
$250$	730511.	0.739226
$251$	−1.09921e6	−1.10127	−0.550637	−	0.834745i	$-0.685615\pi$
$251$	−1.09921e6	−1.10127	−0.550637	+	0.834745i	$0.685615\pi$
$252$	14160.3	0.0140466
$253$	−324143.	−0.318372
$254$	−454851.	−0.442369
$255$	−618134.	−0.595295
$256$	−1.06261e6	−1.01338
$257$	−1.36203e6	−1.28633	−0.643167	−	0.765726i	$-0.722380\pi$
$257$	−1.36203e6	−1.28633	−0.643167	+	0.765726i	$0.722380\pi$
$258$	431408.	0.403495
$259$	7553.22	0.00699653
$260$	−889459.	−0.816004
$261$	628140.	0.570762
$262$	1.28160e6	1.15346
$263$	−802156.	−0.715104	−0.357552	−	0.933893i	$-0.616389\pi$
$263$	−802156.	−0.715104	−0.357552	+	0.933893i	$0.616389\pi$
$264$	1.00519e6	0.887640
$265$	−443780.	−0.388197
$266$	41385.3	0.0358626
$267$	856557.	0.735323
$268$	−194418.	−0.165348
$269$	−352123.	−0.296698	−0.148349	−	0.988935i	$-0.547396\pi$
$269$	−352123.	−0.296698	−0.148349	+	0.988935i	$0.547396\pi$
$270$	237950.	0.198644
$271$	−1.51322e6	−1.25164	−0.625819	−	0.779968i	$-0.715235\pi$
$271$	−1.51322e6	−1.25164	−0.625819	+	0.779968i	$0.715235\pi$
$272$	−17464.7	−0.0143132
$273$	40160.1	0.0326128
$274$	413205.	0.332498
$275$	3.28619e6	2.62036
$276$	92592.7	0.0731651
$277$	−1.26262e6	−0.988720	−0.494360	−	0.869257i	$-0.664598\pi$
$277$	−1.26262e6	−0.988720	−0.494360	+	0.869257i	$0.664598\pi$
$278$	−334945.	−0.259933
$279$	−86396.1	−0.0664483
$280$	150951.	0.115064
$281$	1.24571e6	0.941133	0.470567	−	0.882365i	$-0.344050\pi$
$281$	1.24571e6	0.941133	0.470567	+	0.882365i	$0.344050\pi$
$282$	−653183.	−0.489116
$283$	372353.	0.276368	0.138184	−	0.990407i	$-0.455873\pi$
$283$	372353.	0.276368	0.138184	+	0.990407i	$0.455873\pi$
$284$	1.43383e6	1.05488
$285$	−1.07753e6	−0.785811
$286$	1.07765e6	0.779045
$287$	−88280.4	−0.0632644
$288$	−465730.	−0.330867
$289$	−864115.	−0.608593
$290$	2.53122e6	1.76740
$291$	424210.	0.293662
$292$	−1.07295e6	−0.736413
$293$	1.74374e6	1.18662	0.593311	−	0.804974i	$-0.297821\pi$
$293$	1.74374e6	1.18662	0.593311	+	0.804974i	$0.297821\pi$
$294$	533327.	0.359853
$295$	−2.81259e6	−1.88170
$296$	−153161.	−0.101606
$297$	446692.	0.293844
$298$	−974480.	−0.635671
$299$	262603.	0.169872
$300$	−938713.	−0.602185
$301$	121619.	0.0773721
$302$	167951.	0.105965
$303$	−142472.	−0.0891504
$304$	−30444.4	−0.0188940
$305$	−878.050	−0.000540468 0
$306$	−213932.	−0.130609
$307$	−1.60710e6	−0.973188	−0.486594	−	0.873628i	$-0.661761\pi$
$307$	−1.60710e6	−0.973188	−0.486594	+	0.873628i	$0.661761\pi$
$308$	107119.	0.0643415
$309$	−817042.	−0.486797
$310$	−348150.	−0.205761
$311$	2.21364e6	1.29780	0.648898	−	0.760876i	$-0.275230\pi$
$311$	2.21364e6	1.29780	0.648898	+	0.760876i	$0.275230\pi$
$312$	−814346.	−0.473612
$313$	480737.	0.277362	0.138681	−	0.990337i	$-0.455714\pi$
$313$	480737.	0.277362	0.138681	+	0.990337i	$0.455714\pi$
$314$	−313612.	−0.179501
$315$	67080.7	0.0380909
$316$	−1.21352e6	−0.683640
$317$	3.23334e6	1.80719	0.903593	−	0.428393i	$-0.140920\pi$
$317$	3.23334e6	1.80719	0.903593	+	0.428393i	$0.140920\pi$
$318$	−153589.	−0.0851710
$319$	4.75174e6	2.61442
$320$	−1.94582e6	−1.06225
$321$	1.88439e6	1.02072
$322$	−16846.8	−0.00905479
$323$	968769.	0.516671
$324$	−127599.	−0.0675284
$325$	−2.66229e6	−1.39813
$326$	−1.85015e6	−0.964191
$327$	361044.	0.186720
$328$	1.79011e6	0.918744
$329$	−184140.	−0.0937902
$330$	1.80003e6	0.909905
$331$	2.61134e6	1.31007	0.655034	−	0.755600i	$-0.272654\pi$
$331$	2.61134e6	1.31007	0.655034	+	0.755600i	$0.272654\pi$
$332$	1.60321e6	0.798263
$333$	−68062.6	−0.0336355
$334$	831523.	0.407857
$335$	−921004.	−0.448383
$336$	1895.29	0.000915855 0
$337$	2.76804e6	1.32769	0.663846	−	0.747870i	$-0.268923\pi$
$337$	2.76804e6	1.32769	0.663846	+	0.747870i	$0.268923\pi$
$338$	442386.	0.210625
$339$	2.23558e6	1.05655
$340$	1.33573e6	0.626645
$341$	−653567.	−0.304371
$342$	−372926.	−0.172408
$343$	301428.	0.138340
$344$	−2.46613e6	−1.12362
$345$	438634.	0.198406
$346$	883583.	0.396787
$347$	1.39621e6	0.622484	0.311242	−	0.950331i	$-0.399255\pi$
$347$	1.39621e6	0.622484	0.311242	+	0.950331i	$0.399255\pi$
$348$	−1.35735e6	−0.600820
$349$	−1.00509e6	−0.441714	−0.220857	−	0.975306i	$-0.570885\pi$
$349$	−1.00509e6	−0.441714	−0.220857	+	0.975306i	$0.570885\pi$
$350$	170795.	0.0745253
$351$	−361885.	−0.156784
$352$	−3.52314e6	−1.51556
$353$	−4.24269e6	−1.81219	−0.906096	−	0.423073i	$-0.860952\pi$
$353$	−4.24269e6	−1.81219	−0.906096	+	0.423073i	$0.860952\pi$
$354$	−973416.	−0.412848
$355$	6.79241e6	2.86057
$356$	−1.85094e6	−0.774047
$357$	−60309.8	−0.0250448
$358$	−1.25069e6	−0.515755
$359$	313836.	0.128519	0.0642595	−	0.997933i	$-0.479531\pi$
$359$	313836.	0.128519	0.0642595	+	0.997933i	$0.479531\pi$
$360$	−1.36023e6	−0.553167
$361$	−787340.	−0.317976
$362$	668907.	0.268284
$363$	1.92966e6	0.768626
$364$	−86782.2	−0.0343303
$365$	−5.08281e6	−1.99697
$366$	−303.887	−0.000118579 0
$367$	290172.	0.112458	0.0562290	−	0.998418i	$-0.482092\pi$
$367$	290172.	0.112458	0.0562290	+	0.998418i	$0.482092\pi$
$368$	12393.1	0.00477045
$369$	795501.	0.304141
$370$	−274272.	−0.104154
$371$	−43298.4	−0.0163319
$372$	186694.	0.0699476
$373$	−5.14109e6	−1.91330	−0.956649	−	0.291243i	$-0.905931\pi$
$373$	−5.14109e6	−1.91330	−0.956649	+	0.291243i	$0.905931\pi$
$374$	−1.61834e6	−0.598262
$375$	−1.85573e6	−0.681456
$376$	3.73389e6	1.36205
$377$	−3.84959e6	−1.39496
$378$	23216.1	0.00835719
$379$	−631911.	−0.225974	−0.112987	−	0.993596i	$-0.536042\pi$
$379$	−631911.	−0.225974	−0.112987	+	0.993596i	$0.536042\pi$
$380$	2.32845e6	0.827194
$381$	1.15547e6	0.407799
$382$	−1.32845e6	−0.465787
$383$	−118799.	−0.0413823	−0.0206911	−	0.999786i	$-0.506587\pi$
$383$	−118799.	−0.0413823	−0.0206911	+	0.999786i	$0.506587\pi$
$384$	982495.	0.340018
$385$	507450.	0.174478
$386$	2.69928e6	0.922103
$387$	−1.09591e6	−0.371963
$388$	−916678.	−0.309128
$389$	2.58745e6	0.866957	0.433478	−	0.901164i	$-0.357286\pi$
$389$	2.58745e6	0.866957	0.433478	+	0.901164i	$0.357286\pi$
$390$	−1.45829e6	−0.485492
$391$	−394359.	−0.130452
$392$	−3.04874e6	−1.00209
$393$	−3.25569e6	−1.06331
$394$	2.73850e6	0.888734
$395$	−5.74871e6	−1.85386
$396$	−965260.	−0.309319
$397$	3.56801e6	1.13619	0.568093	−	0.822964i	$-0.307681\pi$
$397$	3.56801e6	1.13619	0.568093	+	0.822964i	$0.307681\pi$
$398$	−3.05901e6	−0.967996
$399$	−105132.	−0.0330600
$400$	−125642.	−0.0392632
$401$	4.28994e6	1.33226	0.666132	−	0.745833i	$-0.267949\pi$
$401$	4.28994e6	1.33226	0.666132	+	0.745833i	$0.267949\pi$
$402$	−318753.	−0.0983759
$403$	529483.	0.162401
$404$	307869.	0.0938453
$405$	−604469.	−0.183120
$406$	246964.	0.0743565
$407$	−514878.	−0.154070
$408$	1.22293e6	0.363707
$409$	3.14276e6	0.928973	0.464486	−	0.885580i	$-0.346239\pi$
$409$	3.14276e6	0.928973	0.464486	+	0.885580i	$0.346239\pi$
$410$	3.20563e6	0.941789
$411$	−1.04967e6	−0.306513
$412$	1.76555e6	0.512434
$413$	−274417.	−0.0791655
$414$	151808.	0.0435305
$415$	7.59480e6	2.16469
$416$	2.85425e6	0.808647
$417$	850868.	0.239620
$418$	−2.82110e6	−0.789728
$419$	−257827.	−0.0717454	−0.0358727	−	0.999356i	$-0.511421\pi$
$419$	−257827.	−0.0717454	−0.0358727	+	0.999356i	$0.511421\pi$
$420$	−144955.	−0.0400969
$421$	−1.48632e6	−0.408701	−0.204351	−	0.978898i	$-0.565508\pi$
$421$	−1.48632e6	−0.408701	−0.204351	+	0.978898i	$0.565508\pi$
$422$	−3.44977e6	−0.942993
$423$	1.65929e6	0.450892
$424$	877985.	0.237177
$425$	3.99805e6	1.07368
$426$	2.35080e6	0.627613
$427$	−85.6691	−2.27381e−5 0
$428$	−4.07199e6	−1.07448
$429$	−2.73758e6	−0.718163
$430$	−4.41621e6	−1.15180
$431$	4.98067e6	1.29150	0.645750	−	0.763549i	$-0.276545\pi$
$431$	4.98067e6	1.29150	0.645750	+	0.763549i	$0.276545\pi$
$432$	−17078.6	−0.00440293
$433$	1.44742e6	0.371001	0.185500	−	0.982644i	$-0.440609\pi$
$433$	1.44742e6	0.371001	0.185500	+	0.982644i	$0.440609\pi$
$434$	−33968.1	−0.00865659
$435$	−6.43010e6	−1.62928
$436$	−780182.	−0.196553
$437$	−687447.	−0.172201
$438$	−1.75912e6	−0.438138
$439$	−4.31837e6	−1.06944	−0.534722	−	0.845028i	$-0.679584\pi$
$439$	−4.31837e6	−1.06944	−0.534722	+	0.845028i	$0.679584\pi$
$440$	−1.02898e7	−2.53382
$441$	−1.35482e6	−0.331731
$442$	1.31109e6	0.319211
$443$	5.65863e6	1.36994	0.684971	−	0.728570i	$-0.259815\pi$
$443$	5.65863e6	1.36994	0.684971	+	0.728570i	$0.259815\pi$
$444$	147077.	0.0354068
$445$	−8.76834e6	−2.09902
$446$	3.81422e6	0.907965
$447$	2.47550e6	0.585994
$448$	−189848.	−0.0446902
$449$	−1.72793e6	−0.404492	−0.202246	−	0.979335i	$-0.564824\pi$
$449$	−1.72793e6	−0.404492	−0.202246	+	0.979335i	$0.564824\pi$
$450$	−1.53904e6	−0.358277
$451$	6.01778e6	1.39314
$452$	−4.83088e6	−1.11219
$453$	−426649.	−0.0976843
$454$	1.11173e6	0.253139
$455$	−411108.	−0.0930952
$456$	2.13182e6	0.480107
$457$	−3.19765e6	−0.716210	−0.358105	−	0.933681i	$-0.616577\pi$
$457$	−3.19765e6	−0.716210	−0.358105	+	0.933681i	$0.616577\pi$
$458$	−3.96163e6	−0.882491
$459$	543456.	0.120402
$460$	−947847.	−0.208854
$461$	7.02590e6	1.53975	0.769874	−	0.638196i	$-0.220319\pi$
$461$	7.02590e6	1.53975	0.769874	+	0.638196i	$0.220319\pi$
$462$	175625.	0.0382808
$463$	−9.07013e6	−1.96635	−0.983176	−	0.182663i	$-0.941528\pi$
$463$	−9.07013e6	−1.96635	−0.983176	+	0.182663i	$0.941528\pi$
$464$	−181675.	−0.0391742
$465$	884414.	0.189681
$466$	1.99243e6	0.425029
$467$	−446092.	−0.0946525	−0.0473263	−	0.998879i	$-0.515070\pi$
$467$	−446092.	−0.0946525	−0.0473263	+	0.998879i	$0.515070\pi$
$468$	782000.	0.165041
$469$	−89860.0	−0.0188640
$470$	6.68646e6	1.39621
$471$	796675.	0.165474
$472$	5.56450e6	1.14966
$473$	−8.29034e6	−1.70380
$474$	−1.98959e6	−0.406740
$475$	6.96940e6	1.41730
$476$	130324.	0.0263637
$477$	390165.	0.0785150
$478$	1.98409e6	0.397184
$479$	−2.86711e6	−0.570960	−0.285480	−	0.958385i	$-0.592153\pi$
$479$	−2.86711e6	−0.570960	−0.285480	+	0.958385i	$0.592153\pi$
$480$	4.76755e6	0.944479
$481$	417125.	0.0822061
$482$	5.75577e6	1.12846
$483$	42796.4	0.00834717
$484$	−4.16982e6	−0.809104
$485$	−4.34252e6	−0.838278
$486$	−209202.	−0.0401768
$487$	3.38408e6	0.646574	0.323287	−	0.946301i	$-0.395212\pi$
$487$	3.38408e6	0.646574	0.323287	+	0.946301i	$0.395212\pi$
$488$	1737.16	0.000330209 0
$489$	4.69998e6	0.888841
$490$	−5.45952e6	−1.02722
$491$	2.36070e6	0.441913	0.220956	−	0.975284i	$-0.429082\pi$
$491$	2.36070e6	0.441913	0.220956	+	0.975284i	$0.429082\pi$
$492$	−1.71900e6	−0.320158
$493$	5.78107e6	1.07125
$494$	2.28550e6	0.421370
$495$	−4.57267e6	−0.838797
$496$	24988.1	0.00456067
$497$	662718.	0.120348
$498$	2.62850e6	0.474936
$499$	−3.42412e6	−0.615598	−0.307799	−	0.951451i	$-0.599592\pi$
$499$	−3.42412e6	−0.615598	−0.307799	+	0.951451i	$0.599592\pi$
$500$	4.01007e6	0.717343
$501$	−2.11234e6	−0.375984
$502$	3.89434e6	0.689722
$503$	7.49119e6	1.32017	0.660087	−	0.751190i	$-0.270520\pi$
$503$	7.49119e6	1.32017	0.660087	+	0.751190i	$0.270520\pi$
$504$	−132714.	−0.0232724
$505$	1.45845e6	0.254485
$506$	1.14839e6	0.199395
$507$	−1.12380e6	−0.194165
$508$	−2.49686e6	−0.429275
$509$	−5.34876e6	−0.915079	−0.457539	−	0.889189i	$-0.651269\pi$
$509$	−5.34876e6	−0.915079	−0.457539	+	0.889189i	$0.651269\pi$
$510$	2.18996e6	0.372830
$511$	−495917.	−0.0840149
$512$	271348.	0.0457458
$513$	947352.	0.158934
$514$	4.82548e6	0.805624
$515$	8.36384e6	1.38959
$516$	2.36817e6	0.391551
$517$	1.25522e7	2.06535
$518$	−26760.0	−0.00438189
$519$	−2.24459e6	−0.365778
$520$	8.33625e6	1.35196
$521$	−4.73330e6	−0.763958	−0.381979	−	0.924171i	$-0.624757\pi$
$521$	−4.73330e6	−0.763958	−0.381979	+	0.924171i	$0.624757\pi$
$522$	−2.22541e6	−0.357465
$523$	−2.28134e6	−0.364701	−0.182350	−	0.983234i	$-0.558370\pi$
$523$	−2.28134e6	−0.364701	−0.182350	+	0.983234i	$0.558370\pi$
$524$	7.03524e6	1.11931
$525$	−433873.	−0.0687013
$526$	2.84192e6	0.447866
$527$	−795144.	−0.124715
$528$	−129195.	−0.0201680
$529$	279841.	0.0434783
$530$	1.57225e6	0.243126
$531$	2.47279e6	0.380584
$532$	227181.	0.0348010
$533$	−4.87527e6	−0.743329
$534$	−3.03466e6	−0.460528
$535$	−1.92900e7	−2.91372
$536$	1.82214e6	0.273949
$537$	3.17717e6	0.475449
$538$	1.24752e6	0.185820
$539$	−1.02489e7	−1.51952
$540$	1.30620e6	0.192764
$541$	−2.57849e6	−0.378767	−0.189383	−	0.981903i	$-0.560649\pi$
$541$	−2.57849e6	−0.378767	−0.189383	+	0.981903i	$0.560649\pi$
$542$	5.36112e6	0.783894
$543$	−1.69924e6	−0.247318
$544$	−4.28633e6	−0.620995
$545$	−3.69591e6	−0.533004
$546$	−142281.	−0.0204252
$547$	6.20267e6	0.886360	0.443180	−	0.896433i	$-0.353850\pi$
$547$	6.20267e6	0.886360	0.443180	+	0.896433i	$0.353850\pi$
$548$	2.26825e6	0.322655
$549$	771.970	0.000109312 0
$550$	−1.16425e7	−1.64112
$551$	1.00776e7	1.41409
$552$	−867804.	−0.121220
$553$	−560887.	−0.0779942
$554$	4.47328e6	0.619230
$555$	696739.	0.0960146
$556$	−1.83865e6	−0.252239
$557$	−2.67729e6	−0.365643	−0.182821	−	0.983146i	$-0.558523\pi$
$557$	−2.67729e6	−0.365643	−0.182821	+	0.983146i	$0.558523\pi$
$558$	306089.	0.0416162
$559$	6.71638e6	0.909087
$560$	−19401.5	−0.00261436
$561$	4.11112e6	0.551509
$562$	−4.41337e6	−0.589426
$563$	−7.74164e6	−1.02935	−0.514673	−	0.857386i	$-0.672087\pi$
$563$	−7.74164e6	−1.02935	−0.514673	+	0.857386i	$0.672087\pi$
$564$	−3.58558e6	−0.474637
$565$	−2.28850e7	−3.01599
$566$	−1.31919e6	−0.173088
$567$	−58976.5	−0.00770408
$568$	−1.34383e7	−1.74772
$569$	1.38677e7	1.79565	0.897827	−	0.440348i	$-0.145145\pi$
$569$	1.38677e7	1.79565	0.897827	+	0.440348i	$0.145145\pi$
$570$	3.81754e6	0.492149
$571$	−3.83164e6	−0.491807	−0.245904	−	0.969294i	$-0.579085\pi$
$571$	−3.83164e6	−0.491807	−0.245904	+	0.969294i	$0.579085\pi$
$572$	5.91565e6	0.755984
$573$	3.37470e6	0.429386
$574$	312765.	0.0396222
$575$	−2.83705e6	−0.357847
$576$	1.71074e6	0.214846
$577$	−2.35210e6	−0.294114	−0.147057	−	0.989128i	$-0.546980\pi$
$577$	−2.35210e6	−0.294114	−0.147057	+	0.989128i	$0.546980\pi$
$578$	3.06144e6	0.381159
$579$	−6.85704e6	−0.850042
$580$	1.38949e7	1.71508
$581$	741005.	0.0910712
$582$	−1.50291e6	−0.183919
$583$	2.95151e6	0.359644
$584$	1.00560e7	1.22009
$585$	3.70452e6	0.447551
$586$	−6.17781e6	−0.743174
$587$	−5.80653e6	−0.695539	−0.347770	−	0.937580i	$-0.613061\pi$
$587$	−5.80653e6	−0.695539	−0.347770	+	0.937580i	$0.613061\pi$
$588$	2.92765e6	0.349201
$589$	−1.38610e6	−0.164628
$590$	9.96460e6	1.17850
$591$	−6.95667e6	−0.819281
$592$	19685.5	0.00230857
$593$	−5.28179e6	−0.616800	−0.308400	−	0.951257i	$-0.599794\pi$
$593$	−5.28179e6	−0.616800	−0.308400	+	0.951257i	$0.599794\pi$
$594$	−1.58257e6	−0.184033
$595$	617375.	0.0714919
$596$	−5.34932e6	−0.616854
$597$	7.77088e6	0.892348
$598$	−930363.	−0.106390
$599$	−9.88981e6	−1.12621	−0.563107	−	0.826384i	$-0.690394\pi$
$599$	−9.88981e6	−1.12621	−0.563107	+	0.826384i	$0.690394\pi$
$600$	8.79787e6	0.997699
$601$	4.37165e6	0.493695	0.246848	−	0.969054i	$-0.420605\pi$
$601$	4.37165e6	0.493695	0.246848	+	0.969054i	$0.420605\pi$
$602$	−430878.	−0.0484577
$603$	809734.	0.0906879
$604$	921948.	0.102829
$605$	−1.97534e7	−2.19409
$606$	504758.	0.0558344
$607$	−778883.	−0.0858026	−0.0429013	−	0.999079i	$-0.513660\pi$
$607$	−778883.	−0.0858026	−0.0429013	+	0.999079i	$0.513660\pi$
$608$	−7.47193e6	−0.819736
$609$	−627369.	−0.0685456
$610$	3110.81	0.000338492 0
$611$	−1.01691e7	−1.10199
$612$	−1.17436e6	−0.126742
$613$	−5.29772e6	−0.569427	−0.284713	−	0.958613i	$-0.591898\pi$
$613$	−5.29772e6	−0.569427	−0.284713	+	0.958613i	$0.591898\pi$
$614$	5.69372e6	0.609502
$615$	−8.14333e6	−0.868189
$616$	−1.00395e6	−0.106601
$617$	−1.13961e7	−1.20515	−0.602577	−	0.798060i	$-0.705860\pi$
$617$	−1.13961e7	−1.20515	−0.602577	+	0.798060i	$0.705860\pi$
$618$	2.89466e6	0.304878
$619$	−1.65105e7	−1.73194	−0.865971	−	0.500094i	$-0.833299\pi$
$619$	−1.65105e7	−1.73194	−0.865971	+	0.500094i	$0.833299\pi$
$620$	−1.91114e6	−0.199670
$621$	−385641.	−0.0401286
$622$	−7.84261e6	−0.812802
$623$	−855504.	−0.0883084
$624$	104667.	0.0107609
$625$	2.23714e6	0.229083
$626$	−1.70318e6	−0.173710
$627$	7.16650e6	0.728011
$628$	−1.72154e6	−0.174188
$629$	−626412.	−0.0631296
$630$	−237657.	−0.0238561
$631$	1.21667e7	1.21646	0.608232	−	0.793760i	$-0.291879\pi$
$631$	1.21667e7	1.21646	0.608232	+	0.793760i	$0.291879\pi$
$632$	1.13734e7	1.13265
$633$	8.76352e6	0.869299
$634$	−1.14552e7	−1.13183
$635$	−1.18282e7	−1.16409
$636$	−843111.	−0.0826498
$637$	8.30310e6	0.810759
$638$	−1.68347e7	−1.63740
$639$	−5.97180e6	−0.578566
$640$	−1.00575e7	−0.970604
$641$	1.41113e7	1.35651	0.678253	−	0.734829i	$-0.262737\pi$
$641$	1.41113e7	1.35651	0.678253	+	0.734829i	$0.262737\pi$
$642$	−6.67613e6	−0.639274
$643$	5.63004e6	0.537012	0.268506	−	0.963278i	$-0.413470\pi$
$643$	5.63004e6	0.537012	0.268506	+	0.963278i	$0.413470\pi$
$644$	−92479.0	−0.00878675
$645$	1.12186e7	1.06179
$646$	−3.43221e6	−0.323588
$647$	7.21927e6	0.678004	0.339002	−	0.940786i	$-0.389911\pi$
$647$	7.21927e6	0.678004	0.339002	+	0.940786i	$0.389911\pi$
$648$	1.19590e6	0.111881
$649$	1.87061e7	1.74330
$650$	9.43210e6	0.875639
$651$	86290.0	0.00798009
$652$	−1.01562e7	−0.935650
$653$	1.37204e7	1.25917	0.629586	−	0.776931i	$-0.283225\pi$
$653$	1.37204e7	1.25917	0.629586	+	0.776931i	$0.283225\pi$
$654$	−1.27913e6	−0.116942
$655$	3.33276e7	3.03530
$656$	−230080.	−0.0208747
$657$	4.46874e6	0.403898
$658$	652380.	0.0587403
$659$	1.35053e7	1.21141	0.605705	−	0.795689i	$-0.292891\pi$
$659$	1.35053e7	1.21141	0.605705	+	0.795689i	$0.292891\pi$
$660$	9.88111e6	0.882970
$661$	6.74829e6	0.600745	0.300373	−	0.953822i	$-0.402889\pi$
$661$	6.74829e6	0.600745	0.300373	+	0.953822i	$0.402889\pi$
$662$	−9.25161e6	−0.820488
$663$	−3.33060e6	−0.294265
$664$	−1.50257e7	−1.32256
$665$	1.07621e6	0.0943718
$666$	241136.	0.0210657
$667$	−4.10230e6	−0.357037
$668$	4.56456e6	0.395784
$669$	−9.68936e6	−0.837008
$670$	3.26298e6	0.280820
$671$	5839.78	0.000500714 0
$672$	465158.	0.0397354
$673$	−1.24941e7	−1.06333	−0.531665	−	0.846955i	$-0.678434\pi$
$673$	−1.24941e7	−1.06333	−0.531665	+	0.846955i	$0.678434\pi$
$674$	−9.80676e6	−0.831526
$675$	3.90966e6	0.330278
$676$	2.42844e6	0.204390
$677$	9.64155e6	0.808491	0.404246	−	0.914650i	$-0.367534\pi$
$677$	9.64155e6	0.808491	0.404246	+	0.914650i	$0.367534\pi$
$678$	−7.92033e6	−0.661712
$679$	−423689.	−0.0352673
$680$	−1.25188e7	−1.03823
$681$	−2.82415e6	−0.233356
$682$	2.31549e6	0.190626
$683$	4.95512e6	0.406445	0.203223	−	0.979133i	$-0.434858\pi$
$683$	4.95512e6	0.406445	0.203223	+	0.979133i	$0.434858\pi$
$684$	−2.04714e6	−0.167304
$685$	1.07452e7	0.874962
$686$	−1.06792e6	−0.0866417
$687$	1.00638e7	0.813525
$688$	316968.	0.0255296
$689$	−2.39115e6	−0.191893
$690$	−1.55402e6	−0.124260
$691$	−1.18245e7	−0.942079	−0.471039	−	0.882112i	$-0.656121\pi$
$691$	−1.18245e7	−0.942079	−0.471039	+	0.882112i	$0.656121\pi$
$692$	4.85035e6	0.385041
$693$	−446143.	−0.0352892
$694$	−4.94658e6	−0.389858
$695$	−8.71011e6	−0.684009
$696$	1.27215e7	0.995439
$697$	7.32137e6	0.570834
$698$	3.56089e6	0.276643
$699$	−5.06142e6	−0.391814
$700$	937560.	0.0723193
$701$	8.50355e6	0.653590	0.326795	−	0.945095i	$-0.394031\pi$
$701$	8.50355e6	0.653590	0.326795	+	0.945095i	$0.394031\pi$
$702$	1.28211e6	0.0981932
$703$	−1.09196e6	−0.0833334
$704$	1.29413e7	0.984119
$705$	−1.69858e7	−1.28710
$706$	1.50312e7	1.13497
$707$	142297.	0.0107065
$708$	−5.34347e6	−0.400627
$709$	1.18862e7	0.888029	0.444015	−	0.896020i	$-0.353554\pi$
$709$	1.18862e7	0.888029	0.444015	+	0.896020i	$0.353554\pi$
$710$	−2.40645e7	−1.79156
$711$	5.05419e6	0.374954
$712$	1.73475e7	1.28244
$713$	564241.	0.0415663
$714$	213669.	0.0156854
$715$	2.80239e7	2.05004
$716$	−6.86556e6	−0.500488
$717$	−5.04023e6	−0.366145
$718$	−1.11188e6	−0.0804907
$719$	−3.08137e6	−0.222291	−0.111145	−	0.993804i	$-0.535452\pi$
$719$	−3.08137e6	−0.222291	−0.111145	+	0.993804i	$0.535452\pi$
$720$	174829.	0.0125684
$721$	816038.	0.0584618
$722$	2.78943e6	0.199147
$723$	−1.46215e7	−1.04027
$724$	3.67190e6	0.260342
$725$	4.15895e7	2.93859
$726$	−6.83652e6	−0.481386
$727$	−4.11888e6	−0.289030	−0.144515	−	0.989503i	$-0.546162\pi$
$727$	−4.11888e6	−0.289030	−0.144515	+	0.989503i	$0.546162\pi$
$728$	813346.	0.0568783
$729$	531441.	0.0370370
$730$	1.80077e7	1.25069
$731$	−1.00862e7	−0.698128
$732$	−1668.16	−0.000115069 0
$733$	−2.07922e6	−0.142935	−0.0714677	−	0.997443i	$-0.522768\pi$
$733$	−2.07922e6	−0.142935	−0.0714677	+	0.997443i	$0.522768\pi$
$734$	−1.02804e6	−0.0704318
$735$	1.38690e7	0.946946
$736$	3.04162e6	0.206971
$737$	6.12545e6	0.415403
$738$	−2.81835e6	−0.190482
$739$	−9.96819e6	−0.671437	−0.335719	−	0.941962i	$-0.608979\pi$
$739$	−9.96819e6	−0.671437	−0.335719	+	0.941962i	$0.608979\pi$
$740$	−1.50559e6	−0.101071
$741$	−5.80590e6	−0.388440
$742$	153400.	0.0102286
$743$	−1.21883e7	−0.809977	−0.404988	−	0.914322i	$-0.632724\pi$
$743$	−1.21883e7	−0.809977	−0.404988	+	0.914322i	$0.632724\pi$
$744$	−1.74975e6	−0.115889
$745$	−2.53410e7	−1.67276
$746$	1.82141e7	1.19829
$747$	−6.67725e6	−0.437821
$748$	−8.88374e6	−0.580553
$749$	−1.88208e6	−0.122584
$750$	6.57460e6	0.426792
$751$	1.30656e7	0.845334	0.422667	−	0.906285i	$-0.361094\pi$
$751$	1.30656e7	0.845334	0.422667	+	0.906285i	$0.361094\pi$
$752$	−479913.	−0.0309469
$753$	−9.89287e6	−0.635821
$754$	1.36386e7	0.873655
$755$	4.36749e6	0.278846
$756$	127443.	0.00810980
$757$	1.30510e7	0.827761	0.413881	−	0.910331i	$-0.364173\pi$
$757$	1.30510e7	0.827761	0.413881	+	0.910331i	$0.364173\pi$
$758$	2.23877e6	0.141526
$759$	−2.91729e6	−0.183812
$760$	−2.18228e7	−1.37049
$761$	8.03065e6	0.502677	0.251338	−	0.967899i	$-0.419129\pi$
$761$	8.03065e6	0.502677	0.251338	+	0.967899i	$0.419129\pi$
$762$	−4.09366e6	−0.255402
$763$	−360600.	−0.0224241
$764$	−7.29241e6	−0.451999
$765$	−5.56321e6	−0.343694
$766$	420886.	0.0259175
$767$	−1.51546e7	−0.930159
$768$	−9.56346e6	−0.585076
$769$	6.12813e6	0.373691	0.186845	−	0.982389i	$-0.440174\pi$
$769$	6.12813e6	0.373691	0.186845	+	0.982389i	$0.440174\pi$
$770$	−1.79782e6	−0.109275
$771$	−1.22583e7	−0.742665
$772$	1.48174e7	0.894808
$773$	8.04760e6	0.484415	0.242208	−	0.970224i	$-0.422128\pi$
$773$	8.04760e6	0.484415	0.242208	+	0.970224i	$0.422128\pi$
$774$	3.88267e6	0.232958
$775$	−5.72033e6	−0.342111
$776$	8.59136e6	0.512162
$777$	67979.0	0.00403945
$778$	−9.16695e6	−0.542970
$779$	1.27626e7	0.753522
$780$	−8.00513e6	−0.471120
$781$	−4.51753e7	−2.65017
$782$	1.39716e6	0.0817013
$783$	5.65326e6	0.329530
$784$	391851.	0.0227683
$785$	−8.15535e6	−0.472355
$786$	1.15344e7	0.665948
$787$	−3.49427e6	−0.201103	−0.100552	−	0.994932i	$-0.532061\pi$
$787$	−3.49427e6	−0.201103	−0.100552	+	0.994932i	$0.532061\pi$
$788$	1.50327e7	0.862426
$789$	−7.21940e6	−0.412866
$790$	2.03669e7	1.16106
$791$	−2.23283e6	−0.126886
$792$	9.04668e6	0.512479
$793$	−4731.06	−0.000267163 0
$794$	−1.26409e7	−0.711587
$795$	−3.99402e6	−0.224126
$796$	−1.67922e7	−0.939342
$797$	−1.33411e7	−0.743951	−0.371976	−	0.928242i	$-0.621320\pi$
$797$	−1.33411e7	−0.743951	−0.371976	+	0.928242i	$0.621320\pi$
$798$	372468.	0.0207053
$799$	1.52713e7	0.846268
$800$	−3.08362e7	−1.70348
$801$	7.70901e6	0.424539
$802$	−1.51987e7	−0.834390
$803$	3.38050e7	1.85009
$804$	−1.74976e6	−0.0954638
$805$	−438095.	−0.0238275
$806$	−1.87588e6	−0.101711
$807$	−3.16911e6	−0.171298
$808$	−2.88543e6	−0.155483
$809$	−5.57231e6	−0.299339	−0.149670	−	0.988736i	$-0.547821\pi$
$809$	−5.57231e6	−0.299339	−0.149670	+	0.988736i	$0.547821\pi$
$810$	2.14155e6	0.114687
$811$	8.30913e6	0.443612	0.221806	−	0.975091i	$-0.428805\pi$
$811$	8.30913e6	0.443612	0.221806	+	0.975091i	$0.428805\pi$
$812$	1.35569e6	0.0721554
$813$	−1.36190e7	−0.722633
$814$	1.82414e6	0.0964932
$815$	−4.81125e7	−2.53725
$816$	−157182.	−0.00826375
$817$	−1.75823e7	−0.921553
$818$	−1.11344e7	−0.581811
$819$	361441.	0.0188290
$820$	1.75970e7	0.913910
$821$	1.35323e7	0.700669	0.350334	−	0.936625i	$-0.386068\pi$
$821$	1.35323e7	0.700669	0.350334	+	0.936625i	$0.386068\pi$
$822$	3.71884e6	0.191968
$823$	1.20061e7	0.617879	0.308939	−	0.951082i	$-0.400026\pi$
$823$	1.20061e7	0.617879	0.308939	+	0.951082i	$0.400026\pi$
$824$	−1.65472e7	−0.848999
$825$	2.95757e7	1.51286
$826$	972220.	0.0495809
$827$	−2.71374e7	−1.37976	−0.689881	−	0.723923i	$-0.742337\pi$
$827$	−2.71374e7	−1.37976	−0.689881	+	0.723923i	$0.742337\pi$
$828$	833335.	0.0422419
$829$	−6.90219e6	−0.348819	−0.174410	−	0.984673i	$-0.555802\pi$
$829$	−6.90219e6	−0.348819	−0.174410	+	0.984673i	$0.555802\pi$
$830$	−2.69073e7	−1.35574
$831$	−1.13636e7	−0.570838
$832$	−1.04844e7	−0.525090
$833$	−1.24691e7	−0.622617
$834$	−3.01450e6	−0.150072
$835$	2.16234e7	1.07327
$836$	−1.54861e7	−0.766351
$837$	−777565.	−0.0383639
$838$	913445.	0.0449337
$839$	−1.57032e6	−0.0770164	−0.0385082	−	0.999258i	$-0.512261\pi$
$839$	−1.57032e6	−0.0770164	−0.0385082	+	0.999258i	$0.512261\pi$
$840$	1.35856e6	0.0664325
$841$	3.96260e7	1.93193
$842$	5.26580e6	0.255967
$843$	1.12114e7	0.543363
$844$	−1.89372e7	−0.915079
$845$	1.15041e7	0.554255
$846$	−5.87864e6	−0.282391
$847$	−1.92729e6	−0.0923079
$848$	−112846.	−0.00538887
$849$	3.35118e6	0.159561
$850$	−1.41645e7	−0.672441
$851$	444508.	0.0210405
$852$	1.29045e7	0.609035
$853$	−2.20053e7	−1.03551	−0.517755	−	0.855529i	$-0.673232\pi$
$853$	−2.20053e7	−1.03551	−0.517755	+	0.855529i	$0.673232\pi$
$854$	303.513	1.42408e−5 0
$855$	−9.69779e6	−0.453688
$856$	3.81638e7	1.78019
$857$	9.95578e6	0.463045	0.231523	−	0.972830i	$-0.425629\pi$
$857$	9.95578e6	0.463045	0.231523	+	0.972830i	$0.425629\pi$
$858$	9.69884e6	0.449782
$859$	1.38922e7	0.642374	0.321187	−	0.947016i	$-0.395918\pi$
$859$	1.38922e7	0.642374	0.321187	+	0.947016i	$0.395918\pi$
$860$	−2.42423e7	−1.11771
$861$	−794524.	−0.0365257
$862$	−1.76458e7	−0.808860
$863$	7.55200e6	0.345171	0.172586	−	0.984995i	$-0.444788\pi$
$863$	7.55200e6	0.345171	0.172586	+	0.984995i	$0.444788\pi$
$864$	−4.19157e6	−0.191026
$865$	2.29772e7	1.04414
$866$	−5.12800e6	−0.232356
$867$	−7.77704e6	−0.351371
$868$	−186465.	−0.00840035
$869$	3.82338e7	1.71750
$870$	2.27809e7	1.02041
$871$	−4.96250e6	−0.221644
$872$	7.31208e6	0.325649
$873$	3.81789e6	0.169546
$874$	2.43553e6	0.107849
$875$	1.85346e6	0.0818393
$876$	−9.65653e6	−0.425168
$877$	−2.70277e6	−0.118662	−0.0593309	−	0.998238i	$-0.518897\pi$
$877$	−2.70277e6	−0.118662	−0.0593309	+	0.998238i	$0.518897\pi$
$878$	1.52994e7	0.669787
$879$	1.56936e7	0.685096
$880$	1.32254e6	0.0575707
$881$	−4.41216e7	−1.91519	−0.957595	−	0.288117i	$-0.906971\pi$
$881$	−4.41216e7	−1.91519	−0.957595	+	0.288117i	$0.906971\pi$
$882$	4.79994e6	0.207761
$883$	2.72497e7	1.17614	0.588071	−	0.808809i	$-0.299888\pi$
$883$	2.72497e7	1.17614	0.588071	+	0.808809i	$0.299888\pi$
$884$	7.19711e6	0.309762
$885$	−2.53133e7	−1.08640
$886$	−2.00477e7	−0.857987
$887$	1.11988e7	0.477930	0.238965	−	0.971028i	$-0.423192\pi$
$887$	1.11988e7	0.477930	0.238965	+	0.971028i	$0.423192\pi$
$888$	−1.37845e6	−0.0586620
$889$	−1.15405e6	−0.0489745
$890$	3.10650e7	1.31461
$891$	4.02023e6	0.169651
$892$	2.09378e7	0.881088
$893$	2.66209e7	1.11710
$894$	−8.77032e6	−0.367005
$895$	−3.25238e7	−1.35720
$896$	−981288.	−0.0408344
$897$	2.36342e6	0.0980754
$898$	6.12181e6	0.253331
$899$	−8.27143e6	−0.341336
$900$	−8.44842e6	−0.347672
$901$	3.59087e6	0.147363
$902$	−2.13201e7	−0.872516
$903$	1.09457e6	0.0446708
$904$	4.52763e7	1.84268
$905$	1.73947e7	0.705984
$906$	1.51156e6	0.0611792
$907$	2.65141e7	1.07018	0.535092	−	0.844794i	$-0.320277\pi$
$907$	2.65141e7	1.07018	0.535092	+	0.844794i	$0.320277\pi$
$908$	6.10272e6	0.245645
$909$	−1.28225e6	−0.0514710
$910$	1.45650e6	0.0583050
$911$	−1.62499e7	−0.648715	−0.324357	−	0.945935i	$-0.605148\pi$
$911$	−1.62499e7	−0.648715	−0.324357	+	0.945935i	$0.605148\pi$
$912$	−274000.	−0.0109084
$913$	−5.05119e7	−2.00547
$914$	1.13288e7	0.448558
$915$	−7902.45	−0.000312039 0
$916$	−2.17470e7	−0.856368
$917$	3.25169e6	0.127698
$918$	−1.92538e6	−0.0754069
$919$	−2.66141e7	−1.03950	−0.519748	−	0.854319i	$-0.673974\pi$
$919$	−2.66141e7	−1.03950	−0.519748	+	0.854319i	$0.673974\pi$
$920$	8.88348e6	0.346030
$921$	−1.44639e7	−0.561870
$922$	−2.48917e7	−0.964335
$923$	3.65985e7	1.41403
$924$	964074.	0.0371476
$925$	−4.50646e6	−0.173173
$926$	3.21342e7	1.23151
$927$	−7.35338e6	−0.281053
$928$	−4.45883e7	−1.69962
$929$	−1.98073e7	−0.752984	−0.376492	−	0.926420i	$-0.622870\pi$
$929$	−1.98073e7	−0.752984	−0.376492	+	0.926420i	$0.622870\pi$
$930$	−3.13335e6	−0.118796
$931$	−2.17361e7	−0.821877
$932$	1.09373e7	0.412448
$933$	1.99228e7	0.749282
$934$	1.58044e6	0.0592804
$935$	−4.20844e7	−1.57432
$936$	−7.32912e6	−0.273440
$937$	−9.15888e6	−0.340795	−0.170398	−	0.985375i	$-0.554505\pi$
$937$	−9.15888e6	−0.340795	−0.170398	+	0.985375i	$0.554505\pi$
$938$	318361.	0.0118144
$939$	4.32664e6	0.160135
$940$	3.67047e7	1.35488
$941$	5.38492e7	1.98247	0.991233	−	0.132128i	$-0.0421811\pi$
$941$	5.38492e7	1.98247	0.991233	+	0.132128i	$0.0421811\pi$
$942$	−2.82250e6	−0.103635
$943$	−5.19531e6	−0.190253
$944$	−715198.	−0.0261214
$945$	603726.	0.0219918
$946$	2.93715e7	1.06708
$947$	−1.89117e7	−0.685261	−0.342631	−	0.939470i	$-0.611318\pi$
$947$	−1.89117e7	−0.685261	−0.342631	+	0.939470i	$0.611318\pi$
$948$	−1.09216e7	−0.394700
$949$	−2.73869e7	−0.987138
$950$	−2.46916e7	−0.887647
$951$	2.91000e7	1.04338
$952$	−1.22143e6	−0.0436794
$953$	3.40143e7	1.21319	0.606595	−	0.795011i	$-0.292535\pi$
$953$	3.40143e7	1.21319	0.606595	+	0.795011i	$0.292535\pi$
$954$	−1.38230e6	−0.0491735
$955$	−3.45459e7	−1.22571
$956$	1.08915e7	0.385427
$957$	4.27656e7	1.50944
$958$	1.01578e7	0.357589
$959$	1.04838e6	0.0368107
$960$	−1.75124e7	−0.613291
$961$	−2.74915e7	−0.960262
$962$	−1.47782e6	−0.0514852
$963$	1.69595e7	0.589315
$964$	3.15957e7	1.09506
$965$	7.01937e7	2.42650
$966$	−151621.	−0.00522778
$967$	−6.27887e6	−0.215931	−0.107966	−	0.994155i	$-0.534434\pi$
$967$	−6.27887e6	−0.215931	−0.107966	+	0.994155i	$0.534434\pi$
$968$	3.90807e7	1.34052
$969$	8.71892e6	0.298300
$970$	1.53849e7	0.525009
$971$	−2.00595e7	−0.682765	−0.341383	−	0.939924i	$-0.610895\pi$
$971$	−2.00595e7	−0.682765	−0.341383	+	0.939924i	$0.610895\pi$
$972$	−1.14839e6	−0.0389875
$973$	−849823.	−0.0287771
$974$	−1.19893e7	−0.404945
$975$	−2.39606e7	−0.807209
$976$	−223.274	−7.50265e−6 0
$977$	−4.81426e7	−1.61359	−0.806795	−	0.590831i	$-0.798800\pi$
$977$	−4.81426e7	−1.61359	−0.806795	+	0.590831i	$0.798800\pi$
$978$	−1.66514e7	−0.556676
$979$	5.83169e7	1.94463
$980$	−2.99695e7	−0.996815
$981$	3.24939e6	0.107803
$982$	−8.36361e6	−0.276768
$983$	−5.45980e7	−1.80216	−0.901079	−	0.433656i	$-0.857223\pi$
$983$	−5.45980e7	−1.80216	−0.901079	+	0.433656i	$0.857223\pi$
$984$	1.61110e7	0.530437
$985$	7.12136e7	2.33869
$986$	−2.04815e7	−0.670918
$987$	−1.65726e6	−0.0541498
$988$	1.25460e7	0.408896
$989$	7.15727e6	0.232679
$990$	1.62003e7	0.525334
$991$	1.21280e7	0.392288	0.196144	−	0.980575i	$-0.437158\pi$
$991$	1.21280e7	0.392288	0.196144	+	0.980575i	$0.437158\pi$
$992$	6.13279e6	0.197870
$993$	2.35021e7	0.756368
$994$	−2.34791e6	−0.0753731
$995$	−7.95484e7	−2.54726
$996$	1.44289e7	0.460877
$997$	−4.49074e6	−0.143080	−0.0715401	−	0.997438i	$-0.522791\pi$
$997$	−4.49074e6	−0.143080	−0.0715401	+	0.997438i	$0.522791\pi$
$998$	1.21312e7	0.385546
$999$	−612563.	−0.0194195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.e.1.2	✓	5
3.2	odd	2		207.6.a.f.1.4		5
4.3	odd	2		1104.6.a.r.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.2	✓	5	1.1	even	1	trivial
207.6.a.f.1.4		5	3.2	odd	2
1104.6.a.r.1.1		5	4.3	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q-3.54286 q^{2} +9.00000 q^{3} -19.4482 q^{4} -92.1306 q^{5} -31.8857 q^{6} -8.98894 q^{7} +182.273 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-3.54286 q^{2} +9.00000 q^{3} -19.4482 q^{4} -92.1306 q^{5} -31.8857 q^{6} -8.98894 q^{7} +182.273 q^{8} +81.0000 q^{9} +326.406 q^{10} +612.746 q^{11} -175.034 q^{12} -496.413 q^{13} +31.8465 q^{14} -829.175 q^{15} -23.4274 q^{16} +745.481 q^{17} -286.971 q^{18} +1299.52 q^{19} +1791.77 q^{20} -80.9005 q^{21} -2170.87 q^{22} -529.000 q^{23} +1640.46 q^{24} +5363.05 q^{25} +1758.72 q^{26} +729.000 q^{27} +174.818 q^{28} +7754.82 q^{29} +2937.65 q^{30} -1066.62 q^{31} -5749.75 q^{32} +5514.72 q^{33} -2641.13 q^{34} +828.157 q^{35} -1575.30 q^{36} -840.279 q^{37} -4604.02 q^{38} -4467.72 q^{39} -16793.0 q^{40} +9821.00 q^{41} +286.619 q^{42} -13529.8 q^{43} -11916.8 q^{44} -7462.58 q^{45} +1874.17 q^{46} +20485.1 q^{47} -210.846 q^{48} -16726.2 q^{49} -19000.5 q^{50} +6709.33 q^{51} +9654.33 q^{52} +4816.85 q^{53} -2582.74 q^{54} -56452.7 q^{55} -1638.45 q^{56} +11695.7 q^{57} -27474.2 q^{58} +30528.3 q^{59} +16125.9 q^{60} +9.53050 q^{61} +3778.88 q^{62} -728.104 q^{63} +21120.2 q^{64} +45734.8 q^{65} -19537.8 q^{66} +9996.72 q^{67} -14498.2 q^{68} -4761.00 q^{69} -2934.04 q^{70} -73725.9 q^{71} +14764.2 q^{72} +55169.6 q^{73} +2976.99 q^{74} +48267.4 q^{75} -25273.3 q^{76} -5507.94 q^{77} +15828.5 q^{78} +62397.4 q^{79} +2158.38 q^{80} +6561.00 q^{81} -34794.4 q^{82} -82435.2 q^{83} +1573.37 q^{84} -68681.6 q^{85} +47934.2 q^{86} +69793.4 q^{87} +111687. q^{88} +95173.0 q^{89} +26438.8 q^{90} +4462.23 q^{91} +10288.1 q^{92} -9599.57 q^{93} -72575.8 q^{94} -119726. q^{95} -51747.8 q^{96} +47134.4 q^{97} +59258.5 q^{98} +49632.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9}+O(q^{10}) \) 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9	\( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99}+O(q^{100}) \) 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9 - 172 * q^10 + 1100 * q^11 + 1062 * q^12 - 978 * q^13 - 344 * q^14 + 846 * q^15 + 1218 * q^16 + 2522 * q^17 + 648 * q^18 + 2060 * q^19 + 7720 * q^20 + 2448 * q^21 - 2572 * q^22 - 2645 * q^23 + 2322 * q^24 + 12035 * q^25 + 9280 * q^26 + 3645 * q^27 + 8072 * q^28 + 1526 * q^29 - 1548 * q^30 - 7392 * q^31 - 5086 * q^32 + 9900 * q^33 - 15608 * q^34 + 6056 * q^35 + 9558 * q^36 - 8210 * q^37 - 14276 * q^38 - 8802 * q^39 - 37472 * q^40 + 21250 * q^41 - 3096 * q^42 - 4548 * q^43 - 4260 * q^44 + 7614 * q^45 - 4232 * q^46 + 536 * q^47 + 10962 * q^48 - 27979 * q^49 - 81872 * q^50 + 22698 * q^51 - 76380 * q^52 - 11482 * q^53 + 5832 * q^54 - 77064 * q^55 - 28624 * q^56 + 18540 * q^57 - 79680 * q^58 + 74676 * q^59 + 69480 * q^60 - 44618 * q^61 + 64880 * q^62 + 22032 * q^63 - 137382 * q^64 - 24388 * q^65 - 23148 * q^66 - 1412 * q^67 + 80196 * q^68 - 23805 * q^69 - 222304 * q^70 + 37912 * q^71 + 20898 * q^72 + 46546 * q^73 + 111604 * q^74 + 108315 * q^75 - 79548 * q^76 + 157008 * q^77 + 83520 * q^78 + 50544 * q^79 + 69424 * q^80 + 32805 * q^81 - 233720 * q^82 + 89588 * q^83 + 72648 * q^84 + 147892 * q^85 + 77428 * q^86 + 13734 * q^87 + 54484 * q^88 + 280410 * q^89 - 13932 * q^90 - 27416 * q^91 - 62422 * q^92 - 66528 * q^93 + 113632 * q^94 + 203120 * q^95 - 45774 * q^96 + 90074 * q^97 + 32976 * q^98 + 89100 * q^99

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