LMFDB - Embedded newform 69.6.a.d.1.1

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:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 75x^{2} - 42x + 736 $ x^4 - 75x^2 - 42x + 736
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.33314$ 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)

$f(q)$	$=$	$q-7.33314 q^{2} -9.00000 q^{3} +21.7749 q^{4} -0.408582 q^{5} +65.9983 q^{6} -4.34307 q^{7} +74.9818 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-7.33314 q^{2} -9.00000 q^{3} +21.7749 q^{4} -0.408582 q^{5} +65.9983 q^{6} -4.34307 q^{7} +74.9818 q^{8} +81.0000 q^{9} +2.99619 q^{10} +428.488 q^{11} -195.975 q^{12} +76.4506 q^{13} +31.8484 q^{14} +3.67724 q^{15} -1246.65 q^{16} -71.8173 q^{17} -593.984 q^{18} -1731.89 q^{19} -8.89685 q^{20} +39.0877 q^{21} -3142.17 q^{22} -529.000 q^{23} -674.836 q^{24} -3124.83 q^{25} -560.623 q^{26} -729.000 q^{27} -94.5702 q^{28} -1666.22 q^{29} -26.9657 q^{30} -3620.99 q^{31} +6742.44 q^{32} -3856.40 q^{33} +526.646 q^{34} +1.77450 q^{35} +1763.77 q^{36} -1469.54 q^{37} +12700.2 q^{38} -688.055 q^{39} -30.6362 q^{40} +7590.26 q^{41} -286.635 q^{42} +5304.35 q^{43} +9330.31 q^{44} -33.0952 q^{45} +3879.23 q^{46} -4933.30 q^{47} +11219.9 q^{48} -16788.1 q^{49} +22914.8 q^{50} +646.356 q^{51} +1664.71 q^{52} +3095.71 q^{53} +5345.86 q^{54} -175.073 q^{55} -325.651 q^{56} +15587.1 q^{57} +12218.6 q^{58} -10750.0 q^{59} +80.0717 q^{60} -21156.5 q^{61} +26553.2 q^{62} -351.789 q^{63} -9550.48 q^{64} -31.2364 q^{65} +28279.5 q^{66} -26405.9 q^{67} -1563.82 q^{68} +4761.00 q^{69} -13.0127 q^{70} -58175.6 q^{71} +6073.52 q^{72} +411.889 q^{73} +10776.4 q^{74} +28123.5 q^{75} -37711.9 q^{76} -1860.96 q^{77} +5045.61 q^{78} -62132.2 q^{79} +509.359 q^{80} +6561.00 q^{81} -55660.5 q^{82} -91527.4 q^{83} +851.132 q^{84} +29.3433 q^{85} -38897.5 q^{86} +14996.0 q^{87} +32128.8 q^{88} -80408.4 q^{89} +242.691 q^{90} -332.031 q^{91} -11518.9 q^{92} +32588.9 q^{93} +36176.6 q^{94} +707.621 q^{95} -60682.0 q^{96} +118915. q^{97} +123110. q^{98} +34707.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 36 q^{3} + 26 q^{4} + 22 q^{5} - 36 q^{6} - 62 q^{7} + 72 q^{8} + 324 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 36 * q^3 + 26 * q^4 + 22 * q^5 - 36 * q^6 - 62 * q^7 + 72 * q^8 + 324 * q^9	\( 4 q + 4 q^{2} - 36 q^{3} + 26 q^{4} + 22 q^{5} - 36 q^{6} - 62 q^{7} + 72 q^{8} + 324 q^{9} - 496 q^{10} - 1076 q^{11} - 234 q^{12} - 396 q^{13} - 1806 q^{14} - 198 q^{15} - 1982 q^{16} + 70 q^{17} + 324 q^{18} - 6366 q^{19} - 5240 q^{20} + 558 q^{21} - 6974 q^{22} - 2116 q^{23} - 648 q^{24} + 1264 q^{25} + 2464 q^{26} - 2916 q^{27} - 6474 q^{28} + 3948 q^{29} + 4464 q^{30} + 3092 q^{31} - 3672 q^{32} + 9684 q^{33} + 11682 q^{34} + 1304 q^{35} + 2106 q^{36} - 17464 q^{37} - 12628 q^{38} + 3564 q^{39} - 14108 q^{40} + 18680 q^{41} + 16254 q^{42} - 25846 q^{43} + 20746 q^{44} + 1782 q^{45} - 2116 q^{46} + 18392 q^{47} + 17838 q^{48} + 7952 q^{49} + 69444 q^{50} - 630 q^{51} + 8844 q^{52} - 26518 q^{53} - 2916 q^{54} - 40848 q^{55} + 54890 q^{56} + 57294 q^{57} + 568 q^{58} - 14520 q^{59} + 47160 q^{60} - 13688 q^{61} + 120136 q^{62} - 5022 q^{63} - 30190 q^{64} + 38324 q^{65} + 62766 q^{66} - 11098 q^{67} + 112138 q^{68} + 19044 q^{69} - 29596 q^{70} - 57496 q^{71} + 5832 q^{72} - 112272 q^{73} - 21226 q^{74} - 11376 q^{75} - 76240 q^{76} - 4792 q^{77} - 22176 q^{78} - 240754 q^{79} + 41200 q^{80} + 26244 q^{81} + 49976 q^{82} - 93268 q^{83} + 58266 q^{84} - 323204 q^{85} - 88224 q^{86} - 35532 q^{87} + 42382 q^{88} - 107582 q^{89} - 40176 q^{90} - 301532 q^{91} - 13754 q^{92} - 27828 q^{93} + 79360 q^{94} - 18640 q^{95} + 33048 q^{96} - 53076 q^{97} + 59664 q^{98} - 87156 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 36 * q^3 + 26 * q^4 + 22 * q^5 - 36 * q^6 - 62 * q^7 + 72 * q^8 + 324 * q^9 - 496 * q^10 - 1076 * q^11 - 234 * q^12 - 396 * q^13 - 1806 * q^14 - 198 * q^15 - 1982 * q^16 + 70 * q^17 + 324 * q^18 - 6366 * q^19 - 5240 * q^20 + 558 * q^21 - 6974 * q^22 - 2116 * q^23 - 648 * q^24 + 1264 * q^25 + 2464 * q^26 - 2916 * q^27 - 6474 * q^28 + 3948 * q^29 + 4464 * q^30 + 3092 * q^31 - 3672 * q^32 + 9684 * q^33 + 11682 * q^34 + 1304 * q^35 + 2106 * q^36 - 17464 * q^37 - 12628 * q^38 + 3564 * q^39 - 14108 * q^40 + 18680 * q^41 + 16254 * q^42 - 25846 * q^43 + 20746 * q^44 + 1782 * q^45 - 2116 * q^46 + 18392 * q^47 + 17838 * q^48 + 7952 * q^49 + 69444 * q^50 - 630 * q^51 + 8844 * q^52 - 26518 * q^53 - 2916 * q^54 - 40848 * q^55 + 54890 * q^56 + 57294 * q^57 + 568 * q^58 - 14520 * q^59 + 47160 * q^60 - 13688 * q^61 + 120136 * q^62 - 5022 * q^63 - 30190 * q^64 + 38324 * q^65 + 62766 * q^66 - 11098 * q^67 + 112138 * q^68 + 19044 * q^69 - 29596 * q^70 - 57496 * q^71 + 5832 * q^72 - 112272 * q^73 - 21226 * q^74 - 11376 * q^75 - 76240 * q^76 - 4792 * q^77 - 22176 * q^78 - 240754 * q^79 + 41200 * q^80 + 26244 * q^81 + 49976 * q^82 - 93268 * q^83 + 58266 * q^84 - 323204 * q^85 - 88224 * q^86 - 35532 * q^87 + 42382 * q^88 - 107582 * q^89 - 40176 * q^90 - 301532 * q^91 - 13754 * q^92 - 27828 * q^93 + 79360 * q^94 - 18640 * q^95 + 33048 * q^96 - 53076 * q^97 + 59664 * q^98 - 87156 * 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 $

$2$	−7.33314	−1.29633	−0.648164	−	0.761501i	$-0.724463\pi$
$2$	−7.33314	−1.29633	−0.648164	+	0.761501i	$0.724463\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	21.7749	0.680467
$5$	−0.408582	−0.00730894	−0.00365447	−	0.999993i	$-0.501163\pi$
$5$	−0.408582	−0.00730894	−0.00365447	+	0.999993i	$0.501163\pi$
$6$	65.9983	0.748435
$7$	−4.34307	−0.0335006	−0.0167503	−	0.999860i	$-0.505332\pi$
$7$	−4.34307	−0.0335006	−0.0167503	+	0.999860i	$0.505332\pi$
$8$	74.9818	0.414220
$9$	81.0000	0.333333
$10$	2.99619	0.00947479
$11$	428.488	1.06772	0.533860	−	0.845573i	$-0.320741\pi$
$11$	428.488	1.06772	0.533860	+	0.845573i	$0.320741\pi$
$12$	−195.975	−0.392868
$13$	76.4506	0.125465	0.0627325	−	0.998030i	$-0.480019\pi$
$13$	76.4506	0.125465	0.0627325	+	0.998030i	$0.480019\pi$
$14$	31.8484	0.0434277
$15$	3.67724	0.00421982
$16$	−1246.65	−1.21743
$17$	−71.8173	−0.0602708	−0.0301354	−	0.999546i	$-0.509594\pi$
$17$	−71.8173	−0.0602708	−0.0301354	+	0.999546i	$0.509594\pi$
$18$	−593.984	−0.432109
$19$	−1731.89	−1.10062	−0.550310	−	0.834960i	$-0.685491\pi$
$19$	−1731.89	−1.10062	−0.550310	+	0.834960i	$0.685491\pi$
$20$	−8.89685	−0.00497349
$21$	39.0877	0.0193416
$22$	−3142.17	−1.38412
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	−674.836	−0.239150
$25$	−3124.83	−0.999947
$26$	−560.623	−0.162644
$27$	−729.000	−0.192450
$28$	−94.5702	−0.0227960
$29$	−1666.22	−0.367907	−0.183953	−	0.982935i	$-0.558890\pi$
$29$	−1666.22	−0.367907	−0.183953	+	0.982935i	$0.558890\pi$
$30$	−26.9657	−0.00547027
$31$	−3620.99	−0.676742	−0.338371	−	0.941013i	$-0.609876\pi$
$31$	−3620.99	−0.676742	−0.338371	+	0.941013i	$0.609876\pi$
$32$	6742.44	1.16397
$33$	−3856.40	−0.616448
$34$	526.646	0.0781307
$35$	1.77450	0.000244854 0
$36$	1763.77	0.226822
$37$	−1469.54	−0.176473	−0.0882365	−	0.996100i	$-0.528123\pi$
$37$	−1469.54	−0.176473	−0.0882365	+	0.996100i	$0.528123\pi$
$38$	12700.2	1.42677
$39$	−688.055	−0.0724372
$40$	−30.6362	−0.00302751
$41$	7590.26	0.705176	0.352588	−	0.935779i	$-0.385302\pi$
$41$	7590.26	0.705176	0.352588	+	0.935779i	$0.385302\pi$
$42$	−286.635	−0.0250730
$43$	5304.35	0.437483	0.218741	−	0.975783i	$-0.429805\pi$
$43$	5304.35	0.437483	0.218741	+	0.975783i	$0.429805\pi$
$44$	9330.31	0.726548
$45$	−33.0952	−0.00243631
$46$	3879.23	0.270303
$47$	−4933.30	−0.325756	−0.162878	−	0.986646i	$-0.552078\pi$
$47$	−4933.30	−0.325756	−0.162878	+	0.986646i	$0.552078\pi$
$48$	11219.9	0.702884
$49$	−16788.1	−0.998878
$50$	22914.8	1.29626
$51$	646.356	0.0347973
$52$	1664.71	0.0853748
$53$	3095.71	0.151381	0.0756904	−	0.997131i	$-0.475884\pi$
$53$	3095.71	0.151381	0.0756904	+	0.997131i	$0.475884\pi$
$54$	5345.86	0.249478
$55$	−175.073	−0.00780390
$56$	−325.651	−0.0138766
$57$	15587.1	0.635443
$58$	12218.6	0.476928
$59$	−10750.0	−0.402047	−0.201024	−	0.979586i	$-0.564427\pi$
$59$	−10750.0	−0.402047	−0.201024	+	0.979586i	$0.564427\pi$
$60$	80.0717	0.00287145
$61$	−21156.5	−0.727980	−0.363990	−	0.931403i	$-0.618586\pi$
$61$	−21156.5	−0.727980	−0.363990	+	0.931403i	$0.618586\pi$
$62$	26553.2	0.877280
$63$	−351.789	−0.0111669
$64$	−9550.48	−0.291457
$65$	−31.2364	−0.000917016 0
$66$	28279.5	0.799119
$67$	−26405.9	−0.718645	−0.359322	−	0.933214i	$-0.616992\pi$
$67$	−26405.9	−0.718645	−0.359322	+	0.933214i	$0.616992\pi$
$68$	−1563.82	−0.0410123
$69$	4761.00	0.120386
$70$	−13.0127	−0.000317411 0
$71$	−58175.6	−1.36960	−0.684802	−	0.728730i	$-0.740111\pi$
$71$	−58175.6	−1.36960	−0.684802	+	0.728730i	$0.740111\pi$
$72$	6073.52	0.138073
$73$	411.889	0.00904633	0.00452317	−	0.999990i	$-0.498560\pi$
$73$	411.889	0.00904633	0.00452317	+	0.999990i	$0.498560\pi$
$74$	10776.4	0.228767
$75$	28123.5	0.577319
$76$	−37711.9	−0.748936
$77$	−1860.96	−0.0357692
$78$	5045.61	0.0939025
$79$	−62132.2	−1.12008	−0.560040	−	0.828465i	$-0.689214\pi$
$79$	−62132.2	−1.12008	−0.560040	+	0.828465i	$0.689214\pi$
$80$	509.359	0.00889814
$81$	6561.00	0.111111
$82$	−55660.5	−0.914139
$83$	−91527.4	−1.45833	−0.729165	−	0.684338i	$-0.760091\pi$
$83$	−91527.4	−1.45833	−0.729165	+	0.684338i	$0.760091\pi$
$84$	851.132	0.0131613
$85$	29.3433	0.000440515 0
$86$	−38897.5	−0.567121
$87$	14996.0	0.212411
$88$	32128.8	0.442271
$89$	−80408.4	−1.07603	−0.538017	−	0.842934i	$-0.680827\pi$
$89$	−80408.4	−1.07603	−0.538017	+	0.842934i	$0.680827\pi$
$90$	242.691	0.00315826
$91$	−332.031	−0.00420315
$92$	−11518.9	−0.141887
$93$	32588.9	0.390717
$94$	36176.6	0.422287
$95$	707.621	0.00804437
$96$	−60682.0	−0.672019
$97$	118915.	1.28323	0.641617	−	0.767025i	$-0.278264\pi$
$97$	118915.	1.28323	0.641617	+	0.767025i	$0.278264\pi$
$98$	123110.	1.29487
$99$	34707.6	0.355907
$100$	−68043.1	−0.680431
$101$	102784.	1.00258	0.501292	−	0.865278i	$-0.332858\pi$
$101$	102784.	1.00258	0.501292	+	0.865278i	$0.332858\pi$
$102$	−4739.82	−0.0451088
$103$	120582.	1.11993	0.559965	−	0.828516i	$-0.310814\pi$
$103$	120582.	1.11993	0.559965	+	0.828516i	$0.310814\pi$
$104$	5732.40	0.0519701
$105$	−15.9705	−0.000141366 0
$106$	−22701.3	−0.196239
$107$	−31567.6	−0.266552	−0.133276	−	0.991079i	$-0.542550\pi$
$107$	−31567.6	−0.266552	−0.133276	+	0.991079i	$0.542550\pi$
$108$	−15873.9	−0.130956
$109$	−119904.	−0.966642	−0.483321	−	0.875443i	$-0.660570\pi$
$109$	−119904.	−0.966642	−0.483321	+	0.875443i	$0.660570\pi$
$110$	1283.83	0.0101164
$111$	13225.9	0.101887
$112$	5414.29	0.0407847
$113$	−13937.4	−0.102680	−0.0513399	−	0.998681i	$-0.516349\pi$
$113$	−13937.4	−0.102680	−0.0513399	+	0.998681i	$0.516349\pi$
$114$	−114302.	−0.823743
$115$	216.140	0.00152402
$116$	−36281.9	−0.250349
$117$	6192.50	0.0418217
$118$	78831.0	0.521185
$119$	311.908	0.00201911
$120$	275.726	0.00174793
$121$	22551.3	0.140026
$122$	155144.	0.943701
$123$	−68312.4	−0.407133
$124$	−78846.8	−0.460500
$125$	2553.57	0.0146175
$126$	2579.72	0.0144759
$127$	−51910.3	−0.285591	−0.142795	−	0.989752i	$-0.545609\pi$
$127$	−51910.3	−0.285591	−0.142795	+	0.989752i	$0.545609\pi$
$128$	−145723.	−0.786147
$129$	−47739.1	−0.252581
$130$	229.061	0.00118875
$131$	−37036.5	−0.188561	−0.0942805	−	0.995546i	$-0.530055\pi$
$131$	−37036.5	−0.188561	−0.0942805	+	0.995546i	$0.530055\pi$
$132$	−83972.8	−0.419473
$133$	7521.75	0.0368714
$134$	193638.	0.931599
$135$	297.856	0.00140661
$136$	−5384.99	−0.0249653
$137$	−20689.2	−0.0941762	−0.0470881	−	0.998891i	$-0.514994\pi$
$137$	−20689.2	−0.0941762	−0.0470881	+	0.998891i	$0.514994\pi$
$138$	−34913.1	−0.156060
$139$	45101.7	0.197996	0.0989978	−	0.995088i	$-0.468436\pi$
$139$	45101.7	0.197996	0.0989978	+	0.995088i	$0.468436\pi$
$140$	38.6397	0.000166615 0
$141$	44399.7	0.188075
$142$	426610.	1.77546
$143$	32758.2	0.133961
$144$	−100979.	−0.405811
$145$	680.789	0.00268901
$146$	−3020.44	−0.0117270
$147$	151093.	0.576702
$148$	−31999.3	−0.120084
$149$	429347.	1.58432	0.792159	−	0.610314i	$-0.208957\pi$
$149$	429347.	1.58432	0.792159	+	0.610314i	$0.208957\pi$
$150$	−206234.	−0.748396
$151$	−153762.	−0.548791	−0.274395	−	0.961617i	$-0.588478\pi$
$151$	−153762.	−0.548791	−0.274395	+	0.961617i	$0.588478\pi$
$152$	−129861.	−0.455899
$153$	−5817.20	−0.0200903
$154$	13646.7	0.0463687
$155$	1479.47	0.00494627
$156$	−14982.4	−0.0492912
$157$	469295.	1.51949	0.759743	−	0.650224i	$-0.225325\pi$
$157$	469295.	1.51949	0.759743	+	0.650224i	$0.225325\pi$
$158$	455624.	1.45199
$159$	−27861.4	−0.0873997
$160$	−2754.84	−0.00850740
$161$	2297.49	0.00698535
$162$	−48112.7	−0.144036
$163$	−300211.	−0.885030	−0.442515	−	0.896761i	$-0.645914\pi$
$163$	−300211.	−0.885030	−0.442515	+	0.896761i	$0.645914\pi$
$164$	165278.	0.479849
$165$	1575.65	0.00450558
$166$	671183.	1.89047
$167$	83003.5	0.230306	0.115153	−	0.993348i	$-0.463264\pi$
$167$	83003.5	0.230306	0.115153	+	0.993348i	$0.463264\pi$
$168$	2930.86	0.00801165
$169$	−365448.	−0.984259
$170$	−215.178	−0.000571053 0
$171$	−140283.	−0.366873
$172$	115502.	0.297693
$173$	−246304.	−0.625686	−0.312843	−	0.949805i	$-0.601281\pi$
$173$	−246304.	−0.625686	−0.312843	+	0.949805i	$0.601281\pi$
$174$	−109968.	−0.275355
$175$	13571.4	0.0334988
$176$	−534175.	−1.29988
$177$	96749.7	0.232122
$178$	589646.	1.39489
$179$	−778110.	−1.81513	−0.907567	−	0.419907i	$-0.862063\pi$
$179$	−778110.	−1.81513	−0.907567	+	0.419907i	$0.862063\pi$
$180$	−720.645	−0.00165783
$181$	81753.5	0.185485	0.0927427	−	0.995690i	$-0.470437\pi$
$181$	81753.5	0.185485	0.0927427	+	0.995690i	$0.470437\pi$
$182$	2434.83	0.00544866
$183$	190409.	0.420299
$184$	−39665.4	−0.0863708
$185$	600.430	0.00128983
$186$	−238979.	−0.506498
$187$	−30772.9	−0.0643523
$188$	−107422.	−0.221666
$189$	3166.10	0.00644719
$190$	−5189.09	−0.0104281
$191$	−194867.	−0.386505	−0.193252	−	0.981149i	$-0.561904\pi$
$191$	−194867.	−0.386505	−0.193252	+	0.981149i	$0.561904\pi$
$192$	85954.3	0.168273
$193$	622869.	1.20366	0.601830	−	0.798624i	$-0.294439\pi$
$193$	622869.	1.20366	0.601830	+	0.798624i	$0.294439\pi$
$194$	−872018.	−1.66349
$195$	281.127	0.000529440 0
$196$	−365561.	−0.679703
$197$	−279380.	−0.512896	−0.256448	−	0.966558i	$-0.582552\pi$
$197$	−279380.	−0.512896	−0.256448	+	0.966558i	$0.582552\pi$
$198$	−254515.	−0.461372
$199$	−715118.	−1.28010	−0.640052	−	0.768332i	$-0.721087\pi$
$199$	−715118.	−1.28010	−0.640052	+	0.768332i	$0.721087\pi$
$200$	−234305.	−0.414197
$201$	237653.	0.414910
$202$	−753727.	−1.29968
$203$	7236.53	0.0123251
$204$	14074.4	0.0236784
$205$	−3101.25	−0.00515409
$206$	−884248.	−1.45180
$207$	−42849.0	−0.0695048
$208$	−95307.2	−0.152745
$209$	−742097.	−1.17515
$210$	117.114	0.000183257 0
$211$	236093.	0.365071	0.182535	−	0.983199i	$-0.441570\pi$
$211$	236093.	0.365071	0.182535	+	0.983199i	$0.441570\pi$
$212$	67408.9	0.103010
$213$	523580.	0.790741
$214$	231490.	0.345539
$215$	−2167.26	−0.00319754
$216$	−54661.7	−0.0797166
$217$	15726.2	0.0226712
$218$	879270.	1.25309
$219$	−3707.00	−0.00522290
$220$	−3812.20	−0.00531030
$221$	−5490.48	−0.00756187
$222$	−96987.4	−0.132079
$223$	312172.	0.420370	0.210185	−	0.977662i	$-0.432593\pi$
$223$	312172.	0.420370	0.210185	+	0.977662i	$0.432593\pi$
$224$	−29282.9	−0.0389937
$225$	−253111.	−0.333316
$226$	102205.	0.133107
$227$	706271.	0.909718	0.454859	−	0.890564i	$-0.349690\pi$
$227$	706271.	0.909718	0.454859	+	0.890564i	$0.349690\pi$
$228$	339407.	0.432398
$229$	−524546.	−0.660990	−0.330495	−	0.943808i	$-0.607216\pi$
$229$	−524546.	−0.660990	−0.330495	+	0.943808i	$0.607216\pi$
$230$	−1584.98	−0.00197563
$231$	16748.6	0.0206514
$232$	−124936.	−0.152394
$233$	1.44203e6	1.74014	0.870072	−	0.492925i	$-0.164072\pi$
$233$	1.44203e6	1.74014	0.870072	+	0.492925i	$0.164072\pi$
$234$	−45410.5	−0.0542146
$235$	2015.66	0.00238093
$236$	−234080.	−0.273580
$237$	559190.	0.646679
$238$	−2287.26	−0.00261742
$239$	447167.	0.506379	0.253189	−	0.967417i	$-0.418520\pi$
$239$	447167.	0.506379	0.253189	+	0.967417i	$0.418520\pi$
$240$	−4584.23	−0.00513734
$241$	443110.	0.491438	0.245719	−	0.969341i	$-0.420976\pi$
$241$	443110.	0.491438	0.245719	+	0.969341i	$0.420976\pi$
$242$	−165372.	−0.181519
$243$	−59049.0	−0.0641500
$244$	−460682.	−0.495366
$245$	6859.33	0.00730074
$246$	500944.	0.527778
$247$	−132404.	−0.138089
$248$	−271508.	−0.280320
$249$	823746.	0.841967
$250$	−18725.7	−0.0189491
$251$	516201.	0.517172	0.258586	−	0.965988i	$-0.416744\pi$
$251$	516201.	0.517172	0.258586	+	0.965988i	$0.416744\pi$
$252$	−7660.19	−0.00759868
$253$	−226670.	−0.222635
$254$	380666.	0.370220
$255$	−264.089	−0.000254332 0
$256$	1.37422e6	1.31056
$257$	950802.	0.897961	0.448980	−	0.893542i	$-0.351787\pi$
$257$	950802.	0.897961	0.448980	+	0.893542i	$0.351787\pi$
$258$	350078.	0.327428
$259$	6382.34	0.00591195
$260$	−680.170	−0.000623999 0
$261$	−134964.	−0.122636
$262$	271594.	0.244437
$263$	1.96344e6	1.75036	0.875180	−	0.483797i	$-0.160743\pi$
$263$	1.96344e6	1.75036	0.875180	+	0.483797i	$0.160743\pi$
$264$	−289159.	−0.255345
$265$	−1264.85	−0.00110643
$266$	−55158.0	−0.0477974
$267$	723675.	0.621249
$268$	−574987.	−0.489014
$269$	2.10595e6	1.77447	0.887234	−	0.461320i	$-0.152624\pi$
$269$	2.10595e6	1.77447	0.887234	+	0.461320i	$0.152624\pi$
$270$	−2184.22	−0.00182342
$271$	−2.30306e6	−1.90494	−0.952471	−	0.304629i	$-0.901468\pi$
$271$	−2.30306e6	−1.90494	−0.952471	+	0.304629i	$0.901468\pi$
$272$	89531.0	0.0733755
$273$	2988.28	0.00242669
$274$	151716.	0.122083
$275$	−1.33895e6	−1.06766
$276$	103671.	0.0819186
$277$	−1.76069e6	−1.37875	−0.689374	−	0.724406i	$-0.742114\pi$
$277$	−1.76069e6	−1.37875	−0.689374	+	0.724406i	$0.742114\pi$
$278$	−330737.	−0.256667
$279$	−293300.	−0.225581
$280$	133.055	0.000101423 0
$281$	1.82573e6	1.37934	0.689669	−	0.724125i	$-0.257756\pi$
$281$	1.82573e6	1.37934	0.689669	+	0.724125i	$0.257756\pi$
$282$	−325589.	−0.243808
$283$	1.72696e6	1.28179	0.640893	−	0.767631i	$-0.278564\pi$
$283$	1.72696e6	1.28179	0.640893	+	0.767631i	$0.278564\pi$
$284$	−1.26677e6	−0.931970
$285$	−6368.59	−0.00464442
$286$	−240220.	−0.173658
$287$	−32965.1	−0.0236238
$288$	546138.	0.387990
$289$	−1.41470e6	−0.996367
$290$	−4992.32	−0.00348584
$291$	−1.07023e6	−0.740876
$292$	8968.85	0.00615573
$293$	405581.	0.276000	0.138000	−	0.990432i	$-0.455933\pi$
$293$	405581.	0.276000	0.138000	+	0.990432i	$0.455933\pi$
$294$	−1.10799e6	−0.747596
$295$	4392.25	0.00293854
$296$	−110189.	−0.0730986
$297$	−312368.	−0.205483
$298$	−3.14846e6	−2.05380
$299$	−40442.4	−0.0261613
$300$	612388.	0.392847
$301$	−23037.2	−0.0146559
$302$	1.12756e6	0.711413
$303$	−925053.	−0.578842
$304$	2.15907e6	1.33993
$305$	8644.17	0.00532076
$306$	42658.3	0.0260436
$307$	−2.61976e6	−1.58641	−0.793205	−	0.608954i	$-0.791589\pi$
$307$	−2.61976e6	−1.58641	−0.793205	+	0.608954i	$0.791589\pi$
$308$	−40522.2	−0.0243398
$309$	−1.08524e6	−0.646592
$310$	−10849.2	−0.00641198
$311$	1.54756e6	0.907293	0.453646	−	0.891182i	$-0.350123\pi$
$311$	1.54756e6	0.907293	0.453646	+	0.891182i	$0.350123\pi$
$312$	−51591.6	−0.0300049
$313$	369596.	0.213239	0.106619	−	0.994300i	$-0.465997\pi$
$313$	369596.	0.213239	0.106619	+	0.994300i	$0.465997\pi$
$314$	−3.44140e6	−1.96975
$315$	143.735	8.16179e−5 0
$316$	−1.35293e6	−0.762178
$317$	−2.00158e6	−1.11873	−0.559364	−	0.828922i	$-0.688955\pi$
$317$	−2.00158e6	−1.11873	−0.559364	+	0.828922i	$0.688955\pi$
$318$	204312.	0.113299
$319$	−713957.	−0.392822
$320$	3902.16	0.00213025
$321$	284109.	0.153894
$322$	−16847.8	−0.00905531
$323$	124380.	0.0663352
$324$	142865.	0.0756074
$325$	−238895.	−0.125458
$326$	2.20149e6	1.14729
$327$	1.07913e6	0.558091
$328$	569131.	0.292098
$329$	21425.7	0.0109130
$330$	−11554.5	−0.00584072
$331$	−201176.	−0.100927	−0.0504635	−	0.998726i	$-0.516070\pi$
$331$	−201176.	−0.100927	−0.0504635	+	0.998726i	$0.516070\pi$
$332$	−1.99300e6	−0.992346
$333$	−119033.	−0.0588244
$334$	−608676.	−0.298552
$335$	10789.0	0.00525253
$336$	−48728.6	−0.0235470
$337$	1.33440e6	0.640044	0.320022	−	0.947410i	$-0.396310\pi$
$337$	1.33440e6	0.640044	0.320022	+	0.947410i	$0.396310\pi$
$338$	2.67988e6	1.27592
$339$	125436.	0.0592822
$340$	638.948	0.000299756 0
$341$	−1.55155e6	−0.722571
$342$	1.02872e6	0.475588
$343$	145906.	0.0669635
$344$	397729.	0.181214
$345$	−1945.26	−0.000879893 0
$346$	1.80618e6	0.811095
$347$	−4.16940e6	−1.85887	−0.929436	−	0.368983i	$-0.879706\pi$
$347$	−4.16940e6	−1.85887	−0.929436	+	0.368983i	$0.879706\pi$
$348$	326537.	0.144539
$349$	−1.02682e6	−0.451263	−0.225631	−	0.974213i	$-0.572444\pi$
$349$	−1.02682e6	−0.451263	−0.225631	+	0.974213i	$0.572444\pi$
$350$	−99520.9	−0.0434254
$351$	−55732.5	−0.0241457
$352$	2.88906e6	1.24280
$353$	4.39424e6	1.87693	0.938463	−	0.345379i	$-0.112250\pi$
$353$	4.39424e6	1.87693	0.938463	+	0.345379i	$0.112250\pi$
$354$	−709479.	−0.300906
$355$	23769.5	0.0100103
$356$	−1.75089e6	−0.732206
$357$	−2807.17	−0.00116573
$358$	5.70599e6	2.35301
$359$	−2.65847e6	−1.08867	−0.544334	−	0.838869i	$-0.683217\pi$
$359$	−2.65847e6	−1.08867	−0.544334	+	0.838869i	$0.683217\pi$
$360$	−2481.53	−0.00100917
$361$	523361.	0.211365
$362$	−599510.	−0.240450
$363$	−202961.	−0.0808439
$364$	−7229.95	−0.00286010
$365$	−168.290	−6.61191e−5 0
$366$	−1.39629e6	−0.544846
$367$	−3.96638e6	−1.53720	−0.768598	−	0.639733i	$-0.779045\pi$
$367$	−3.96638e6	−1.53720	−0.768598	+	0.639733i	$0.779045\pi$
$368$	659478.	0.253852
$369$	614811.	0.235059
$370$	−4403.04	−0.00167204
$371$	−13444.9	−0.00507134
$372$	709622.	0.265870
$373$	996522.	0.370864	0.185432	−	0.982657i	$-0.440632\pi$
$373$	996522.	0.370864	0.185432	+	0.982657i	$0.440632\pi$
$374$	225662.	0.0834217
$375$	−22982.1	−0.00843941
$376$	−369907.	−0.134935
$377$	−127384.	−0.0461594
$378$	−23217.5	−0.00835767
$379$	−2.86660e6	−1.02511	−0.512554	−	0.858655i	$-0.671301\pi$
$379$	−2.86660e6	−1.02511	−0.512554	+	0.858655i	$0.671301\pi$
$380$	15408.4	0.00547393
$381$	467193.	0.164886
$382$	1.42899e6	0.501037
$383$	−1.89544e6	−0.660256	−0.330128	−	0.943936i	$-0.607092\pi$
$383$	−1.89544e6	−0.660256	−0.330128	+	0.943936i	$0.607092\pi$
$384$	1.31151e6	0.453882
$385$	760.354	0.000261435 0
$386$	−4.56759e6	−1.56034
$387$	429652.	0.145828
$388$	2.58936e6	0.873198
$389$	1.32321e6	0.443357	0.221678	−	0.975120i	$-0.428847\pi$
$389$	1.32321e6	0.443357	0.221678	+	0.975120i	$0.428847\pi$
$390$	−2061.55	−0.000686327 0
$391$	37991.3	0.0125673
$392$	−1.25880e6	−0.413755
$393$	333329.	0.108866
$394$	2.04873e6	0.664881
$395$	25386.1	0.00818660
$396$	755755.	0.242183
$397$	−2.73318e6	−0.870346	−0.435173	−	0.900347i	$-0.643313\pi$
$397$	−2.73318e6	−0.870346	−0.435173	+	0.900347i	$0.643313\pi$
$398$	5.24406e6	1.65943
$399$	−67695.7	−0.0212877
$400$	3.89557e6	1.21737
$401$	−3.51245e6	−1.09081	−0.545405	−	0.838173i	$-0.683624\pi$
$401$	−3.51245e6	−1.09081	−0.545405	+	0.838173i	$0.683624\pi$
$402$	−1.74274e6	−0.537859
$403$	−276827.	−0.0849074
$404$	2.23811e6	0.682225
$405$	−2680.71	−0.000812104 0
$406$	−53066.5	−0.0159774
$407$	−629683.	−0.188424
$408$	48464.9	0.0144137
$409$	1.01219e6	0.299195	0.149597	−	0.988747i	$-0.452202\pi$
$409$	1.01219e6	0.299195	0.149597	+	0.988747i	$0.452202\pi$
$410$	22741.9	0.00668139
$411$	186202.	0.0543727
$412$	2.62568e6	0.762076
$413$	46687.9	0.0134688
$414$	314218.	0.0901010
$415$	37396.5	0.0106588
$416$	515464.	0.146038
$417$	−405915.	−0.114313
$418$	5.44190e6	1.52339
$419$	−4.96595e6	−1.38187	−0.690935	−	0.722917i	$-0.742801\pi$
$419$	−4.96595e6	−1.38187	−0.690935	+	0.722917i	$0.742801\pi$
$420$	−347.757	−9.61951e−5 0
$421$	6.28423e6	1.72801	0.864007	−	0.503480i	$-0.167947\pi$
$421$	6.28423e6	1.72801	0.864007	+	0.503480i	$0.167947\pi$
$422$	−1.73130e6	−0.473252
$423$	−399597.	−0.108585
$424$	232122.	0.0627049
$425$	224417.	0.0602675
$426$	−3.83949e6	−1.02506
$427$	91884.3	0.0243877
$428$	−687383.	−0.181380
$429$	−294824.	−0.0773427
$430$	15892.8	0.00414506
$431$	−2.44198e6	−0.633212	−0.316606	−	0.948557i	$-0.602543\pi$
$431$	−2.44198e6	−0.633212	−0.316606	+	0.948557i	$0.602543\pi$
$432$	908808.	0.234295
$433$	3.22990e6	0.827885	0.413942	−	0.910303i	$-0.364151\pi$
$433$	3.22990e6	0.827885	0.413942	+	0.910303i	$0.364151\pi$
$434$	−115323.	−0.0293894
$435$	−6127.10	−0.00155250
$436$	−2.61089e6	−0.657768
$437$	916172.	0.229495
$438$	27183.9	0.00677060
$439$	1.41329e6	0.350002	0.175001	−	0.984568i	$-0.444007\pi$
$439$	1.41329e6	0.350002	0.175001	+	0.984568i	$0.444007\pi$
$440$	−13127.3	−0.00323253
$441$	−1.35984e6	−0.332959
$442$	40262.4	0.00980267
$443$	1.95693e6	0.473769	0.236885	−	0.971538i	$-0.423874\pi$
$443$	1.95693e6	0.473769	0.236885	+	0.971538i	$0.423874\pi$
$444$	287993.	0.0693306
$445$	32853.4	0.00786467
$446$	−2.28920e6	−0.544938
$447$	−3.86412e6	−0.914707
$448$	41478.4	0.00976399
$449$	3.44271e6	0.805907	0.402954	−	0.915220i	$-0.367984\pi$
$449$	3.44271e6	0.805907	0.402954	+	0.915220i	$0.367984\pi$
$450$	1.85610e6	0.432086
$451$	3.25234e6	0.752930
$452$	−303486.	−0.0698702
$453$	1.38386e6	0.316844
$454$	−5.17918e6	−1.17929
$455$	135.662	3.07206e−5 0
$456$	1.16874e6	0.263213
$457$	−5.91669e6	−1.32522	−0.662610	−	0.748964i	$-0.730551\pi$
$457$	−5.91669e6	−1.32522	−0.662610	+	0.748964i	$0.730551\pi$
$458$	3.84657e6	0.856861
$459$	52354.8	0.0115991
$460$	4706.44	0.00103704
$461$	1.76919e6	0.387724	0.193862	−	0.981029i	$-0.437899\pi$
$461$	1.76919e6	0.387724	0.193862	+	0.981029i	$0.437899\pi$
$462$	−122820.	−0.0267710
$463$	4.37655e6	0.948811	0.474405	−	0.880307i	$-0.342663\pi$
$463$	4.37655e6	0.948811	0.474405	+	0.880307i	$0.342663\pi$
$464$	2.07720e6	0.447902
$465$	−13315.2	−0.00285573
$466$	−1.05746e7	−2.25580
$467$	−2.11837e6	−0.449478	−0.224739	−	0.974419i	$-0.572153\pi$
$467$	−2.11837e6	−0.449478	−0.224739	+	0.974419i	$0.572153\pi$
$468$	134841.	0.0284583
$469$	114683.	0.0240750
$470$	−14781.1	−0.00308647
$471$	−4.22365e6	−0.877276
$472$	−806052.	−0.166536
$473$	2.27285e6	0.467109
$474$	−4.10062e6	−0.838308
$475$	5.41188e6	1.10056
$476$	6791.78	0.00137393
$477$	250753.	0.0504603
$478$	−3.27914e6	−0.656433
$479$	3.77749e6	0.752255	0.376127	−	0.926568i	$-0.377256\pi$
$479$	3.77749e6	0.752255	0.376127	+	0.926568i	$0.377256\pi$
$480$	24793.6	0.00491175
$481$	−112348.	−0.0221412
$482$	−3.24939e6	−0.637065
$483$	−20677.4	−0.00403299
$484$	491053.	0.0952829
$485$	−48586.4	−0.00937908
$486$	433015.	0.0831595
$487$	1.00545e7	1.92105	0.960524	−	0.278197i	$-0.0897367\pi$
$487$	1.00545e7	1.92105	0.960524	+	0.278197i	$0.0897367\pi$
$488$	−1.58635e6	−0.301543
$489$	2.70190e6	0.510972
$490$	−50300.5	−0.00946415
$491$	4.65796e6	0.871950	0.435975	−	0.899959i	$-0.356404\pi$
$491$	4.65796e6	0.871950	0.435975	+	0.899959i	$0.356404\pi$
$492$	−1.48750e6	−0.277041
$493$	119664.	0.0221740
$494$	970940.	0.179009
$495$	−14180.9	−0.00260130
$496$	4.51411e6	0.823887
$497$	252661.	0.0458825
$498$	−6.04065e6	−1.09147
$499$	−6.85708e6	−1.23279	−0.616393	−	0.787439i	$-0.711407\pi$
$499$	−6.85708e6	−1.23279	−0.616393	+	0.787439i	$0.711407\pi$
$500$	55603.9	0.00994672
$501$	−747032.	−0.132967
$502$	−3.78538e6	−0.670424
$503$	−5.51954e6	−0.972709	−0.486354	−	0.873762i	$-0.661674\pi$
$503$	−5.51954e6	−0.972709	−0.486354	+	0.873762i	$0.661674\pi$
$504$	−26377.8	−0.00462553
$505$	−41995.6	−0.00732782
$506$	1.66221e6	0.288608
$507$	3.28903e6	0.568262
$508$	−1.13034e6	−0.194335
$509$	9.87497e6	1.68943	0.844717	−	0.535213i	$-0.179769\pi$
$509$	9.87497e6	1.68943	0.844717	+	0.535213i	$0.179769\pi$
$510$	1936.60	0.000329697 0
$511$	−1788.86	−0.000303057 0
$512$	−5.41423e6	−0.912772
$513$	1.26255e6	0.211814
$514$	−6.97237e6	−1.16405
$515$	−49267.8	−0.00818551
$516$	−1.03952e6	−0.171873
$517$	−2.11386e6	−0.347816
$518$	−46802.6	−0.00766383
$519$	2.21674e6	0.361240
$520$	−2342.16	−0.000379846 0
$521$	−709124.	−0.114453	−0.0572266	−	0.998361i	$-0.518226\pi$
$521$	−709124.	−0.114453	−0.0572266	+	0.998361i	$0.518226\pi$
$522$	989710.	0.158976
$523$	−1.02536e7	−1.63916	−0.819581	−	0.572964i	$-0.805793\pi$
$523$	−1.02536e7	−1.63916	−0.819581	+	0.572964i	$0.805793\pi$
$524$	−806468.	−0.128310
$525$	−122142.	−0.0193405
$526$	−1.43982e7	−2.26904
$527$	260050.	0.0407877
$528$	4.80758e6	0.750484
$529$	279841.	0.0434783
$530$	9275.34	0.00143430
$531$	−870747.	−0.134016
$532$	163786.	0.0250898
$533$	580280.	0.0884748
$534$	−5.30681e6	−0.805342
$535$	12898.0	0.00194821
$536$	−1.97996e6	−0.297677
$537$	7.00299e6	1.04797
$538$	−1.54433e7	−2.30029
$539$	−7.19352e6	−1.06652
$540$	6485.81	0.000957149 0
$541$	−2.50498e6	−0.367969	−0.183984	−	0.982929i	$-0.558900\pi$
$541$	−2.50498e6	−0.367969	−0.183984	+	0.982929i	$0.558900\pi$
$542$	1.68887e7	2.46943
$543$	−735781.	−0.107090
$544$	−484224.	−0.0701535
$545$	48990.5	0.00706513
$546$	−21913.4	−0.00314579
$547$	5.52668e6	0.789762	0.394881	−	0.918732i	$-0.370786\pi$
$547$	5.52668e6	0.789762	0.394881	+	0.918732i	$0.370786\pi$
$548$	−450505.	−0.0640838
$549$	−1.71368e6	−0.242660
$550$	9.81874e6	1.38404
$551$	2.88572e6	0.404926
$552$	356988.	0.0498662
$553$	269845.	0.0375233
$554$	1.29114e7	1.78731
$555$	−5403.87	−0.000744684 0
$556$	982086.	0.134729
$557$	−1.08390e7	−1.48031	−0.740156	−	0.672436i	$-0.765248\pi$
$557$	−1.08390e7	−1.48031	−0.740156	+	0.672436i	$0.765248\pi$
$558$	2.15081e6	0.292427
$559$	405521.	0.0548888
$560$	−2212.18	−0.000298093 0
$561$	276956.	0.0371538
$562$	−1.33883e7	−1.78808
$563$	−5.90889e6	−0.785661	−0.392831	−	0.919611i	$-0.628504\pi$
$563$	−5.90889e6	−0.785661	−0.392831	+	0.919611i	$0.628504\pi$
$564$	966801.	0.127979
$565$	5694.56	0.000750480 0
$566$	−1.26640e7	−1.66161
$567$	−28494.9	−0.00372229
$568$	−4.36211e6	−0.567316
$569$	−6.51286e6	−0.843317	−0.421659	−	0.906755i	$-0.638552\pi$
$569$	−6.51286e6	−0.843317	−0.421659	+	0.906755i	$0.638552\pi$
$570$	46701.8	0.00602069
$571$	1.08281e7	1.38984	0.694918	−	0.719089i	$-0.255440\pi$
$571$	1.08281e7	1.38984	0.694918	+	0.719089i	$0.255440\pi$
$572$	713308.	0.0911564
$573$	1.75380e6	0.223148
$574$	241738.	0.0306242
$575$	1.65304e6	0.208503
$576$	−773589.	−0.0971525
$577$	9.82825e6	1.22896	0.614478	−	0.788934i	$-0.289367\pi$
$577$	9.82825e6	1.22896	0.614478	+	0.788934i	$0.289367\pi$
$578$	1.03742e7	1.29162
$579$	−5.60582e6	−0.694933
$580$	14824.1	0.00182978
$581$	397510.	0.0488549
$582$	7.84816e6	0.960418
$583$	1.32648e6	0.161632
$584$	30884.1	0.00374717
$585$	−2530.14	−0.000305672 0
$586$	−2.97418e6	−0.357786
$587$	−5.37142e6	−0.643419	−0.321710	−	0.946838i	$-0.604258\pi$
$587$	−5.37142e6	−0.643419	−0.321710	+	0.946838i	$0.604258\pi$
$588$	3.29005e6	0.392427
$589$	6.27117e6	0.744836
$590$	−32208.9	−0.00380931
$591$	2.51442e6	0.296120
$592$	1.83201e6	0.214844
$593$	5.42476e6	0.633496	0.316748	−	0.948510i	$-0.397409\pi$
$593$	5.42476e6	0.633496	0.316748	+	0.948510i	$0.397409\pi$
$594$	2.29064e6	0.266373
$595$	−127.440	−1.47575e−5 0
$596$	9.34900e6	1.07808
$597$	6.43606e6	0.739068
$598$	296570.	0.0339136
$599$	7.31392e6	0.832882	0.416441	−	0.909163i	$-0.363277\pi$
$599$	7.31392e6	0.832882	0.416441	+	0.909163i	$0.363277\pi$
$600$	2.10875e6	0.239137
$601$	−2.65146e6	−0.299433	−0.149716	−	0.988729i	$-0.547836\pi$
$601$	−2.65146e6	−0.299433	−0.149716	+	0.988729i	$0.547836\pi$
$602$	168935.	0.0189989
$603$	−2.13888e6	−0.239548
$604$	−3.34816e6	−0.373434
$605$	−9214.05	−0.00102344
$606$	6.78354e6	0.750369
$607$	−894452.	−0.0985338	−0.0492669	−	0.998786i	$-0.515688\pi$
$607$	−894452.	−0.0985338	−0.0492669	+	0.998786i	$0.515688\pi$
$608$	−1.16772e7	−1.28109
$609$	−65128.7	−0.00711589
$610$	−63388.9	−0.00689745
$611$	−377154.	−0.0408710
$612$	−126669.	−0.0136708
$613$	1.51504e7	1.62845	0.814223	−	0.580553i	$-0.197164\pi$
$613$	1.51504e7	1.62845	0.814223	+	0.580553i	$0.197164\pi$
$614$	1.92111e7	2.05651
$615$	27911.2	0.00297571
$616$	−139538.	−0.0148163
$617$	−3.64371e6	−0.385328	−0.192664	−	0.981265i	$-0.561713\pi$
$617$	−3.64371e6	−0.385328	−0.192664	+	0.981265i	$0.561713\pi$
$618$	7.95823e6	0.838196
$619$	1.06699e7	1.11927	0.559633	−	0.828741i	$-0.310942\pi$
$619$	1.06699e7	1.11927	0.559633	+	0.828741i	$0.310942\pi$
$620$	32215.4	0.00336577
$621$	385641.	0.0401286
$622$	−1.13485e7	−1.17615
$623$	349220.	0.0360478
$624$	857764.	0.0881874
$625$	9.76406e6	0.999840
$626$	−2.71030e6	−0.276427
$627$	6.67887e6	0.678476
$628$	1.02189e7	1.03396
$629$	105539.	0.0106362
$630$	−1054.03	−0.000105804 0
$631$	1.27764e7	1.27742	0.638712	−	0.769445i	$-0.279467\pi$
$631$	1.27764e7	1.27742	0.638712	+	0.769445i	$0.279467\pi$
$632$	−4.65878e6	−0.463959
$633$	−2.12484e6	−0.210774
$634$	1.46779e7	1.45024
$635$	21209.6	0.00208737
$636$	−606680.	−0.0594726
$637$	−1.28346e6	−0.125324
$638$	5.23555e6	0.509226
$639$	−4.71222e6	−0.456534
$640$	59539.9	0.00574590
$641$	−463451.	−0.0445511	−0.0222756	−	0.999752i	$-0.507091\pi$
$641$	−463451.	−0.0445511	−0.0222756	+	0.999752i	$0.507091\pi$
$642$	−2.08341e6	−0.199497
$643$	−1.12992e7	−1.07775	−0.538877	−	0.842385i	$-0.681151\pi$
$643$	−1.12992e7	−1.07775	−0.538877	+	0.842385i	$0.681151\pi$
$644$	50027.6	0.00475330
$645$	19505.4	0.00184610
$646$	−912096.	−0.0859922
$647$	−6.27215e6	−0.589055	−0.294528	−	0.955643i	$-0.595162\pi$
$647$	−6.27215e6	−0.589055	−0.294528	+	0.955643i	$0.595162\pi$
$648$	491955.	0.0460244
$649$	−4.60624e6	−0.429274
$650$	1.75185e6	0.162635
$651$	−141536.	−0.0130892
$652$	−6.53709e6	−0.602234
$653$	−6.55444e6	−0.601524	−0.300762	−	0.953699i	$-0.597241\pi$
$653$	−6.55444e6	−0.601524	−0.300762	+	0.953699i	$0.597241\pi$
$654$	−7.91343e6	−0.723469
$655$	15132.5	0.00137818
$656$	−9.46240e6	−0.858503
$657$	33363.0	0.00301544
$658$	−157118.	−0.0141469
$659$	5.49875e6	0.493231	0.246616	−	0.969113i	$-0.420682\pi$
$659$	5.49875e6	0.493231	0.246616	+	0.969113i	$0.420682\pi$
$660$	34309.8	0.00306590
$661$	−1.94455e7	−1.73107	−0.865537	−	0.500845i	$-0.833023\pi$
$661$	−1.94455e7	−1.73107	−0.865537	+	0.500845i	$0.833023\pi$
$662$	1.47525e6	0.130834
$663$	49414.3	0.00436585
$664$	−6.86288e6	−0.604069
$665$	−3073.25	−0.000269491 0
$666$	872886.	0.0762557
$667$	881432.	0.0767139
$668$	1.80740e6	0.156716
$669$	−2.80955e6	−0.242701
$670$	−79117.1	−0.00680900
$671$	−9.06531e6	−0.777278
$672$	263546.	0.0225130
$673$	−1.99517e7	−1.69802	−0.849008	−	0.528381i	$-0.822799\pi$
$673$	−1.99517e7	−1.69802	−0.849008	+	0.528381i	$0.822799\pi$
$674$	−9.78532e6	−0.829708
$675$	2.27800e6	0.192440
$676$	−7.95762e6	−0.669755
$677$	−2.03746e7	−1.70851	−0.854254	−	0.519856i	$-0.825986\pi$
$677$	−2.03746e7	−1.70851	−0.854254	+	0.519856i	$0.825986\pi$
$678$	−919842.	−0.0768492
$679$	−516455.	−0.0429891
$680$	2200.21	0.000182470 0
$681$	−6.35644e6	−0.525226
$682$	1.13777e7	0.936689
$683$	8.14142e6	0.667803	0.333901	−	0.942608i	$-0.391635\pi$
$683$	8.14142e6	0.667803	0.333901	+	0.942608i	$0.391635\pi$
$684$	−3.05467e6	−0.249645
$685$	8453.22	0.000688328 0
$686$	−1.06995e6	−0.0868067
$687$	4.72092e6	0.381623
$688$	−6.61267e6	−0.532605
$689$	236669.	0.0189930
$690$	14264.9	0.00114063
$691$	−1.69238e7	−1.34835	−0.674174	−	0.738573i	$-0.735500\pi$
$691$	−1.69238e7	−1.34835	−0.674174	+	0.738573i	$0.735500\pi$
$692$	−5.36326e6	−0.425759
$693$	−150738.	−0.0119231
$694$	3.05748e7	2.40971
$695$	−18427.7	−0.00144714
$696$	1.12443e6	0.0879849
$697$	−545112.	−0.0425015
$698$	7.52979e6	0.584984
$699$	−1.29783e7	−1.00467
$700$	295516.	0.0227948
$701$	−1.16539e7	−0.895728	−0.447864	−	0.894102i	$-0.647815\pi$
$701$	−1.16539e7	−0.895728	−0.447864	+	0.894102i	$0.647815\pi$
$702$	408694.	0.0313008
$703$	2.54510e6	0.194230
$704$	−4.09227e6	−0.311195
$705$	−18140.9	−0.00137463
$706$	−3.22236e7	−2.43311
$707$	−446397.	−0.0335871
$708$	2.10672e6	0.157951
$709$	−1.45958e7	−1.09046	−0.545232	−	0.838285i	$-0.683558\pi$
$709$	−1.45958e7	−1.09046	−0.545232	+	0.838285i	$0.683558\pi$
$710$	−174305.	−0.0129767
$711$	−5.03271e6	−0.373360
$712$	−6.02916e6	−0.445715
$713$	1.91550e6	0.141110
$714$	20585.4	0.00151117
$715$	−13384.4	−0.000979116 0
$716$	−1.69433e7	−1.23514
$717$	−4.02451e6	−0.292358
$718$	1.94949e7	1.41127
$719$	1.97942e7	1.42796	0.713979	−	0.700168i	$-0.246891\pi$
$719$	1.97942e7	1.42796	0.713979	+	0.700168i	$0.246891\pi$
$720$	41258.1	0.00296605
$721$	−523699.	−0.0375183
$722$	−3.83788e6	−0.273998
$723$	−3.98799e6	−0.283732
$724$	1.78018e6	0.126217
$725$	5.20667e6	0.367887
$726$	1.48834e6	0.104800
$727$	2.42059e7	1.69857	0.849287	−	0.527932i	$-0.177032\pi$
$727$	2.42059e7	1.69857	0.849287	+	0.527932i	$0.177032\pi$
$728$	−24896.2	−0.00174103
$729$	531441.	0.0370370
$730$	1234.10	8.57121e−5 0
$731$	−380944.	−0.0263674
$732$	4.14613e6	0.286000
$733$	−1.38705e7	−0.953522	−0.476761	−	0.879033i	$-0.658189\pi$
$733$	−1.38705e7	−0.953522	−0.476761	+	0.879033i	$0.658189\pi$
$734$	2.90860e7	1.99271
$735$	−61734.0	−0.00421508
$736$	−3.56675e6	−0.242705
$737$	−1.13146e7	−0.767311
$738$	−4.50850e6	−0.304713
$739$	1.59262e7	1.07276	0.536379	−	0.843977i	$-0.319792\pi$
$739$	1.59262e7	1.07276	0.536379	+	0.843977i	$0.319792\pi$
$740$	13074.3	0.000877688 0
$741$	1.19164e6	0.0797259
$742$	98593.4	0.00657412
$743$	3.17168e6	0.210774	0.105387	−	0.994431i	$-0.466392\pi$
$743$	3.17168e6	0.210774	0.105387	+	0.994431i	$0.466392\pi$
$744$	2.44357e6	0.161843
$745$	−175423.	−0.0115797
$746$	−7.30764e6	−0.480762
$747$	−7.41372e6	−0.486110
$748$	−670078.	−0.0437896
$749$	137101.	0.00892965
$750$	168531.	0.0109402
$751$	2.49776e7	1.61604	0.808018	−	0.589157i	$-0.200540\pi$
$751$	2.49776e7	1.61604	0.808018	+	0.589157i	$0.200540\pi$
$752$	6.15010e6	0.396586
$753$	−4.64581e6	−0.298589
$754$	934123.	0.0598378
$755$	62824.4	0.00401108
$756$	68941.7	0.00438710
$757$	−9.09279e6	−0.576710	−0.288355	−	0.957524i	$-0.593108\pi$
$757$	−9.09279e6	−0.576710	−0.288355	+	0.957524i	$0.593108\pi$
$758$	2.10212e7	1.32888
$759$	2.04003e6	0.128538
$760$	53058.7	0.00333214
$761$	1.21933e7	0.763238	0.381619	−	0.924320i	$-0.375367\pi$
$761$	1.21933e7	0.763238	0.381619	+	0.924320i	$0.375367\pi$
$762$	−3.42599e6	−0.213746
$763$	520750.	0.0323831
$764$	−4.24322e6	−0.263004
$765$	2376.80	0.000146838 0
$766$	1.38995e7	0.855909
$767$	−821842.	−0.0504429
$768$	−1.23680e7	−0.756653
$769$	7.54269e6	0.459950	0.229975	−	0.973197i	$-0.426136\pi$
$769$	7.54269e6	0.459950	0.229975	+	0.973197i	$0.426136\pi$
$770$	−5575.78	−0.000338906 0
$771$	−8.55722e6	−0.518438
$772$	1.35629e7	0.819050
$773$	2.69009e7	1.61927	0.809634	−	0.586935i	$-0.199666\pi$
$773$	2.69009e7	1.61927	0.809634	+	0.586935i	$0.199666\pi$
$774$	−3.15070e6	−0.189040
$775$	1.13150e7	0.676706
$776$	8.91643e6	0.531541
$777$	−57441.1	−0.00341327
$778$	−9.70325e6	−0.574736
$779$	−1.31455e7	−0.776131
$780$	6121.53	0.000360266 0
$781$	−2.49276e7	−1.46235
$782$	−278596.	−0.0162914
$783$	1.21468e6	0.0708037
$784$	2.09289e7	1.21607
$785$	−191746.	−0.0111058
$786$	−2.44435e6	−0.141126
$787$	1.24123e7	0.714356	0.357178	−	0.934036i	$-0.383739\pi$
$787$	1.24123e7	0.714356	0.357178	+	0.934036i	$0.383739\pi$
$788$	−6.08347e6	−0.349009
$789$	−1.76709e7	−1.01057
$790$	−186160.	−0.0106125
$791$	60531.1	0.00343983
$792$	2.60243e6	0.147424
$793$	−1.61743e6	−0.0913360
$794$	2.00428e7	1.12825
$795$	11383.7	0.000638799 0
$796$	−1.55717e7	−0.871068
$797$	−2.76236e6	−0.154040	−0.0770202	−	0.997030i	$-0.524541\pi$
$797$	−2.76236e6	−0.154040	−0.0770202	+	0.997030i	$0.524541\pi$
$798$	496422.	0.0275959
$799$	354296.	0.0196336
$800$	−2.10690e7	−1.16391
$801$	−6.51308e6	−0.358678
$802$	2.57573e7	1.41405
$803$	176489.	0.00965895
$804$	5.17489e6	0.282332
$805$	−938.712	−5.10555e−5 0
$806$	2.03001e6	0.110068
$807$	−1.89536e7	−1.02449
$808$	7.70690e6	0.415290
$809$	−2.05022e7	−1.10136	−0.550681	−	0.834716i	$-0.685632\pi$
$809$	−2.05022e7	−1.10136	−0.550681	+	0.834716i	$0.685632\pi$
$810$	19658.0	0.00105275
$811$	1.35473e7	0.723272	0.361636	−	0.932319i	$-0.382218\pi$
$811$	1.35473e7	0.723272	0.361636	+	0.932319i	$0.382218\pi$
$812$	157575.	0.00838682
$813$	2.07275e7	1.09982
$814$	4.61755e6	0.244259
$815$	122661.	0.00646863
$816$	−805779.	−0.0423634
$817$	−9.18657e6	−0.481502
$818$	−7.42254e6	−0.387855
$819$	−26894.5	−0.00140105
$820$	−67529.5	−0.00350719
$821$	3.14628e7	1.62907	0.814535	−	0.580115i	$-0.196992\pi$
$821$	3.14628e7	1.62907	0.814535	+	0.580115i	$0.196992\pi$
$822$	−1.36545e6	−0.0704848
$823$	1.73389e7	0.892322	0.446161	−	0.894953i	$-0.352791\pi$
$823$	1.73389e7	0.892322	0.446161	+	0.894953i	$0.352791\pi$
$824$	9.04149e6	0.463897
$825$	1.20506e7	0.616415
$826$	−342369.	−0.0174600
$827$	−6.54365e6	−0.332703	−0.166351	−	0.986067i	$-0.553199\pi$
$827$	−6.54365e6	−0.332703	−0.166351	+	0.986067i	$0.553199\pi$
$828$	−933035.	−0.0472957
$829$	−6.28583e6	−0.317670	−0.158835	−	0.987305i	$-0.550774\pi$
$829$	−6.28583e6	−0.317670	−0.158835	+	0.987305i	$0.550774\pi$
$830$	−274233.	−0.0138174
$831$	1.58463e7	0.796020
$832$	−730140.	−0.0365677
$833$	1.20568e6	0.0602031
$834$	2.97663e6	0.148187
$835$	−33913.8	−0.00168329
$836$	−1.61591e7	−0.799654
$837$	2.63970e6	0.130239
$838$	3.64160e7	1.79136
$839$	7.73114e6	0.379174	0.189587	−	0.981864i	$-0.439285\pi$
$839$	7.73114e6	0.379174	0.189587	+	0.981864i	$0.439285\pi$
$840$	−1197.50	−5.85567e−5 0
$841$	−1.77349e7	−0.864645
$842$	−4.60832e7	−2.24007
$843$	−1.64316e7	−0.796361
$844$	5.14092e6	0.248419
$845$	149316.	0.00719389
$846$	2.93030e6	0.140762
$847$	−97941.9	−0.00469094
$848$	−3.85927e6	−0.184296
$849$	−1.55426e7	−0.740039
$850$	−1.64568e6	−0.0781265
$851$	777389.	0.0367972
$852$	1.14009e7	0.538073
$853$	−1.25054e7	−0.588471	−0.294236	−	0.955733i	$-0.595065\pi$
$853$	−1.25054e7	−0.588471	−0.294236	+	0.955733i	$0.595065\pi$
$854$	−673800.	−0.0316145
$855$	57317.3	0.00268146
$856$	−2.36700e6	−0.110411
$857$	2.09830e7	0.975924	0.487962	−	0.872865i	$-0.337740\pi$
$857$	2.09830e7	0.975924	0.487962	+	0.872865i	$0.337740\pi$
$858$	2.16198e6	0.100262
$859$	−9.91769e6	−0.458593	−0.229297	−	0.973357i	$-0.573643\pi$
$859$	−9.91769e6	−0.458593	−0.229297	+	0.973357i	$0.573643\pi$
$860$	−47192.0	−0.00217582
$861$	296686.	0.0136392
$862$	1.79074e7	0.820851
$863$	6.48533e6	0.296418	0.148209	−	0.988956i	$-0.452649\pi$
$863$	6.48533e6	0.296418	0.148209	+	0.988956i	$0.452649\pi$
$864$	−4.91524e6	−0.224006
$865$	100636.	0.00457310
$866$	−2.36853e7	−1.07321
$867$	1.27323e7	0.575253
$868$	342438.	0.0154270
$869$	−2.66229e7	−1.19593
$870$	44930.9	0.00201255
$871$	−2.01875e6	−0.0901647
$872$	−8.99058e6	−0.400402
$873$	9.63208e6	0.427745
$874$	−6.71842e6	−0.297501
$875$	−11090.3	−0.000489694 0
$876$	−80719.7	−0.00355401
$877$	1.00228e7	0.440037	0.220018	−	0.975496i	$-0.429388\pi$
$877$	1.00228e7	0.440037	0.220018	+	0.975496i	$0.429388\pi$
$878$	−1.03639e7	−0.453717
$879$	−3.65023e6	−0.159349
$880$	218254.	0.00950072
$881$	−945657.	−0.0410482	−0.0205241	−	0.999789i	$-0.506533\pi$
$881$	−945657.	−0.0410482	−0.0205241	+	0.999789i	$0.506533\pi$
$882$	9.97189e6	0.431624
$883$	8.36917e6	0.361227	0.180614	−	0.983554i	$-0.442192\pi$
$883$	8.36917e6	0.361227	0.180614	+	0.983554i	$0.442192\pi$
$884$	−119555.	−0.00514560
$885$	−39530.2	−0.00169657
$886$	−1.43505e7	−0.614160
$887$	−2.98445e7	−1.27367	−0.636833	−	0.771002i	$-0.719756\pi$
$887$	−2.98445e7	−1.27367	−0.636833	+	0.771002i	$0.719756\pi$
$888$	991701.	0.0422035
$889$	225450.	0.00956746
$890$	−240919.	−0.0101952
$891$	2.81131e6	0.118636
$892$	6.79753e6	0.286048
$893$	8.54396e6	0.358534
$894$	2.83361e7	1.18576
$895$	317922.	0.0132667
$896$	632887.	0.0263364
$897$	363981.	0.0151042
$898$	−2.52459e7	−1.04472
$899$	6.03337e6	0.248978
$900$	−5.51149e6	−0.226810
$901$	−222326.	−0.00912384
$902$	−2.38499e7	−0.976044
$903$	207335.	0.00846160
$904$	−1.04505e6	−0.0425320
$905$	−33403.0	−0.00135570
$906$	−1.01480e7	−0.410734
$907$	1.98050e7	0.799387	0.399694	−	0.916649i	$-0.369117\pi$
$907$	1.98050e7	0.799387	0.399694	+	0.916649i	$0.369117\pi$
$908$	1.53790e7	0.619033
$909$	8.32548e6	0.334195
$910$	−994.827	−3.98239e−5 0
$911$	2.34551e7	0.936356	0.468178	−	0.883634i	$-0.344911\pi$
$911$	2.34551e7	0.936356	0.468178	+	0.883634i	$0.344911\pi$
$912$	−1.94316e7	−0.773609
$913$	−3.92184e7	−1.55709
$914$	4.33879e7	1.71792
$915$	−77797.5	−0.00307194
$916$	−1.14220e7	−0.449782
$917$	160852.	0.00631690
$918$	−383925.	−0.0150363
$919$	−2.57051e7	−1.00399	−0.501996	−	0.864870i	$-0.667401\pi$
$919$	−2.57051e7	−1.00399	−0.501996	+	0.864870i	$0.667401\pi$
$920$	16206.6	0.000631279 0
$921$	2.35778e7	0.915915
$922$	−1.29737e7	−0.502618
$923$	−4.44756e6	−0.171837
$924$	364700.	0.0140526
$925$	4.59208e6	0.176464
$926$	−3.20939e7	−1.22997
$927$	9.76718e6	0.373310
$928$	−1.12344e7	−0.428233
$929$	3.96038e7	1.50556	0.752780	−	0.658273i	$-0.228713\pi$
$929$	3.96038e7	1.50556	0.752780	+	0.658273i	$0.228713\pi$
$930$	97642.6	0.00370196
$931$	2.90753e7	1.09939
$932$	3.14002e7	1.18411
$933$	−1.39281e7	−0.523826
$934$	1.55343e7	0.582671
$935$	12573.2	0.000470347 0
$936$	464325.	0.0173234
$937$	−1.91246e7	−0.711611	−0.355805	−	0.934560i	$-0.615793\pi$
$937$	−1.91246e7	−0.711611	−0.355805	+	0.934560i	$0.615793\pi$
$938$	−840985.	−0.0312091
$939$	−3.32636e6	−0.123113
$940$	43890.8	0.00162015
$941$	−8.52219e6	−0.313745	−0.156873	−	0.987619i	$-0.550141\pi$
$941$	−8.52219e6	−0.313745	−0.156873	+	0.987619i	$0.550141\pi$
$942$	3.09726e7	1.13724
$943$	−4.01525e6	−0.147039
$944$	1.34014e7	0.489465
$945$	−1293.61	−4.71221e−5 0
$946$	−1.66671e7	−0.605527
$947$	−3.79789e7	−1.37616	−0.688078	−	0.725636i	$-0.741546\pi$
$947$	−3.79789e7	−1.37616	−0.688078	+	0.725636i	$0.741546\pi$
$948$	1.21763e7	0.440044
$949$	31489.1	0.00113500
$950$	−3.96861e7	−1.42669
$951$	1.80142e7	0.645898
$952$	23387.4	0.000836353 0
$953$	236188.	0.00842415	0.00421208	−	0.999991i	$-0.498659\pi$
$953$	236188.	0.00842415	0.00421208	+	0.999991i	$0.498659\pi$
$954$	−1.83880e6	−0.0654131
$955$	79619.1	0.00282494
$956$	9.73705e6	0.344574
$957$	6.42561e6	0.226796
$958$	−2.77009e7	−0.975169
$959$	89854.5	0.00315496
$960$	−35119.4	−0.00122990
$961$	−1.55176e7	−0.542021
$962$	823861.	0.0287023
$963$	−2.55698e6	−0.0888508
$964$	9.64870e6	0.334407
$965$	−254493.	−0.00879747
$966$	151630.	0.00522809
$967$	2.02260e7	0.695573	0.347787	−	0.937574i	$-0.386933\pi$
$967$	2.02260e7	0.695573	0.347787	+	0.937574i	$0.386933\pi$
$968$	1.69093e6	0.0580014
$969$	−1.11942e6	−0.0382987
$970$	356291.	0.0121584
$971$	4.63134e7	1.57637	0.788185	−	0.615438i	$-0.211021\pi$
$971$	4.63134e7	1.57637	0.788185	+	0.615438i	$0.211021\pi$
$972$	−1.28579e6	−0.0436520
$973$	−195880.	−0.00663297
$974$	−7.37311e7	−2.49031
$975$	2.15006e6	0.0724334
$976$	2.63748e7	0.886265
$977$	−2.97234e7	−0.996237	−0.498118	−	0.867109i	$-0.665976\pi$
$977$	−2.97234e7	−0.996237	−0.498118	+	0.867109i	$0.665976\pi$
$978$	−1.98134e7	−0.662388
$979$	−3.44541e7	−1.14890
$980$	149362.	0.00496791
$981$	−9.71219e6	−0.322214
$982$	−3.41574e7	−1.13033
$983$	3.58345e7	1.18282	0.591408	−	0.806372i	$-0.298572\pi$
$983$	3.58345e7	1.18282	0.591408	+	0.806372i	$0.298572\pi$
$984$	−5.12218e6	−0.168643
$985$	114150.	0.00374872
$986$	−877510.	−0.0287448
$987$	−192831.	−0.00630064
$988$	−2.88310e6	−0.0939652
$989$	−2.80600e6	−0.0912215
$990$	103990.	0.00337214
$991$	−2.93191e7	−0.948346	−0.474173	−	0.880432i	$-0.657253\pi$
$991$	−2.93191e7	−0.948346	−0.474173	+	0.880432i	$0.657253\pi$
$992$	−2.44143e7	−0.787708
$993$	1.81059e6	0.0582702
$994$	−1.85280e6	−0.0594788
$995$	292185.	0.00935620
$996$	1.79370e7	0.572931
$997$	−5.37510e6	−0.171257	−0.0856286	−	0.996327i	$-0.527290\pi$
$997$	−5.37510e6	−0.171257	−0.0856286	+	0.996327i	$0.527290\pi$
$998$	5.02839e7	1.59810
$999$	1.07130e6	0.0339623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.d.1.1	✓	4
3.2	odd	2		207.6.a.e.1.4		4
4.3	odd	2		1104.6.a.o.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.d.1.1	✓	4	1.1	even	1	trivial
207.6.a.e.1.4		4	3.2	odd	2
1104.6.a.o.1.2		4	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)

\(f(q)\)	\(=\)	\(q-7.33314 q^{2} -9.00000 q^{3} +21.7749 q^{4} -0.408582 q^{5} +65.9983 q^{6} -4.34307 q^{7} +74.9818 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-7.33314 q^{2} -9.00000 q^{3} +21.7749 q^{4} -0.408582 q^{5} +65.9983 q^{6} -4.34307 q^{7} +74.9818 q^{8} +81.0000 q^{9} +2.99619 q^{10} +428.488 q^{11} -195.975 q^{12} +76.4506 q^{13} +31.8484 q^{14} +3.67724 q^{15} -1246.65 q^{16} -71.8173 q^{17} -593.984 q^{18} -1731.89 q^{19} -8.89685 q^{20} +39.0877 q^{21} -3142.17 q^{22} -529.000 q^{23} -674.836 q^{24} -3124.83 q^{25} -560.623 q^{26} -729.000 q^{27} -94.5702 q^{28} -1666.22 q^{29} -26.9657 q^{30} -3620.99 q^{31} +6742.44 q^{32} -3856.40 q^{33} +526.646 q^{34} +1.77450 q^{35} +1763.77 q^{36} -1469.54 q^{37} +12700.2 q^{38} -688.055 q^{39} -30.6362 q^{40} +7590.26 q^{41} -286.635 q^{42} +5304.35 q^{43} +9330.31 q^{44} -33.0952 q^{45} +3879.23 q^{46} -4933.30 q^{47} +11219.9 q^{48} -16788.1 q^{49} +22914.8 q^{50} +646.356 q^{51} +1664.71 q^{52} +3095.71 q^{53} +5345.86 q^{54} -175.073 q^{55} -325.651 q^{56} +15587.1 q^{57} +12218.6 q^{58} -10750.0 q^{59} +80.0717 q^{60} -21156.5 q^{61} +26553.2 q^{62} -351.789 q^{63} -9550.48 q^{64} -31.2364 q^{65} +28279.5 q^{66} -26405.9 q^{67} -1563.82 q^{68} +4761.00 q^{69} -13.0127 q^{70} -58175.6 q^{71} +6073.52 q^{72} +411.889 q^{73} +10776.4 q^{74} +28123.5 q^{75} -37711.9 q^{76} -1860.96 q^{77} +5045.61 q^{78} -62132.2 q^{79} +509.359 q^{80} +6561.00 q^{81} -55660.5 q^{82} -91527.4 q^{83} +851.132 q^{84} +29.3433 q^{85} -38897.5 q^{86} +14996.0 q^{87} +32128.8 q^{88} -80408.4 q^{89} +242.691 q^{90} -332.031 q^{91} -11518.9 q^{92} +32588.9 q^{93} +36176.6 q^{94} +707.621 q^{95} -60682.0 q^{96} +118915. q^{97} +123110. q^{98} +34707.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 36 q^{3} + 26 q^{4} + 22 q^{5} - 36 q^{6} - 62 q^{7} + 72 q^{8} + 324 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 36 * q^3 + 26 * q^4 + 22 * q^5 - 36 * q^6 - 62 * q^7 + 72 * q^8 + 324 * q^9	\( 4 q + 4 q^{2} - 36 q^{3} + 26 q^{4} + 22 q^{5} - 36 q^{6} - 62 q^{7} + 72 q^{8} + 324 q^{9} - 496 q^{10} - 1076 q^{11} - 234 q^{12} - 396 q^{13} - 1806 q^{14} - 198 q^{15} - 1982 q^{16} + 70 q^{17} + 324 q^{18} - 6366 q^{19} - 5240 q^{20} + 558 q^{21} - 6974 q^{22} - 2116 q^{23} - 648 q^{24} + 1264 q^{25} + 2464 q^{26} - 2916 q^{27} - 6474 q^{28} + 3948 q^{29} + 4464 q^{30} + 3092 q^{31} - 3672 q^{32} + 9684 q^{33} + 11682 q^{34} + 1304 q^{35} + 2106 q^{36} - 17464 q^{37} - 12628 q^{38} + 3564 q^{39} - 14108 q^{40} + 18680 q^{41} + 16254 q^{42} - 25846 q^{43} + 20746 q^{44} + 1782 q^{45} - 2116 q^{46} + 18392 q^{47} + 17838 q^{48} + 7952 q^{49} + 69444 q^{50} - 630 q^{51} + 8844 q^{52} - 26518 q^{53} - 2916 q^{54} - 40848 q^{55} + 54890 q^{56} + 57294 q^{57} + 568 q^{58} - 14520 q^{59} + 47160 q^{60} - 13688 q^{61} + 120136 q^{62} - 5022 q^{63} - 30190 q^{64} + 38324 q^{65} + 62766 q^{66} - 11098 q^{67} + 112138 q^{68} + 19044 q^{69} - 29596 q^{70} - 57496 q^{71} + 5832 q^{72} - 112272 q^{73} - 21226 q^{74} - 11376 q^{75} - 76240 q^{76} - 4792 q^{77} - 22176 q^{78} - 240754 q^{79} + 41200 q^{80} + 26244 q^{81} + 49976 q^{82} - 93268 q^{83} + 58266 q^{84} - 323204 q^{85} - 88224 q^{86} - 35532 q^{87} + 42382 q^{88} - 107582 q^{89} - 40176 q^{90} - 301532 q^{91} - 13754 q^{92} - 27828 q^{93} + 79360 q^{94} - 18640 q^{95} + 33048 q^{96} - 53076 q^{97} + 59664 q^{98} - 87156 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 36 * q^3 + 26 * q^4 + 22 * q^5 - 36 * q^6 - 62 * q^7 + 72 * q^8 + 324 * q^9 - 496 * q^10 - 1076 * q^11 - 234 * q^12 - 396 * q^13 - 1806 * q^14 - 198 * q^15 - 1982 * q^16 + 70 * q^17 + 324 * q^18 - 6366 * q^19 - 5240 * q^20 + 558 * q^21 - 6974 * q^22 - 2116 * q^23 - 648 * q^24 + 1264 * q^25 + 2464 * q^26 - 2916 * q^27 - 6474 * q^28 + 3948 * q^29 + 4464 * q^30 + 3092 * q^31 - 3672 * q^32 + 9684 * q^33 + 11682 * q^34 + 1304 * q^35 + 2106 * q^36 - 17464 * q^37 - 12628 * q^38 + 3564 * q^39 - 14108 * q^40 + 18680 * q^41 + 16254 * q^42 - 25846 * q^43 + 20746 * q^44 + 1782 * q^45 - 2116 * q^46 + 18392 * q^47 + 17838 * q^48 + 7952 * q^49 + 69444 * q^50 - 630 * q^51 + 8844 * q^52 - 26518 * q^53 - 2916 * q^54 - 40848 * q^55 + 54890 * q^56 + 57294 * q^57 + 568 * q^58 - 14520 * q^59 + 47160 * q^60 - 13688 * q^61 + 120136 * q^62 - 5022 * q^63 - 30190 * q^64 + 38324 * q^65 + 62766 * q^66 - 11098 * q^67 + 112138 * q^68 + 19044 * q^69 - 29596 * q^70 - 57496 * q^71 + 5832 * q^72 - 112272 * q^73 - 21226 * q^74 - 11376 * q^75 - 76240 * q^76 - 4792 * q^77 - 22176 * q^78 - 240754 * q^79 + 41200 * q^80 + 26244 * q^81 + 49976 * q^82 - 93268 * q^83 + 58266 * q^84 - 323204 * q^85 - 88224 * q^86 - 35532 * q^87 + 42382 * q^88 - 107582 * q^89 - 40176 * q^90 - 301532 * q^91 - 13754 * q^92 - 27828 * q^93 + 79360 * q^94 - 18640 * q^95 + 33048 * q^96 - 53076 * q^97 + 59664 * q^98 - 87156 * q^99

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