LMFDB - Embedded newform 69.6.a.d.1.3

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			1.3
Root			$-3.86863$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.86863 q^{2} -9.00000 q^{3} -8.29644 q^{4} +66.7344 q^{5} -43.8177 q^{6} -185.299 q^{7} -196.188 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+4.86863 q^{2} -9.00000 q^{3} -8.29644 q^{4} +66.7344 q^{5} -43.8177 q^{6} -185.299 q^{7} -196.188 q^{8} +81.0000 q^{9} +324.905 q^{10} -563.729 q^{11} +74.6679 q^{12} +767.333 q^{13} -902.153 q^{14} -600.610 q^{15} -689.683 q^{16} -1737.11 q^{17} +394.359 q^{18} -2283.22 q^{19} -553.658 q^{20} +1667.69 q^{21} -2744.59 q^{22} -529.000 q^{23} +1765.70 q^{24} +1328.48 q^{25} +3735.86 q^{26} -729.000 q^{27} +1537.32 q^{28} +3574.17 q^{29} -2924.15 q^{30} +2830.44 q^{31} +2920.22 q^{32} +5073.56 q^{33} -8457.37 q^{34} -12365.8 q^{35} -672.011 q^{36} +4495.35 q^{37} -11116.2 q^{38} -6906.00 q^{39} -13092.5 q^{40} +16207.8 q^{41} +8119.38 q^{42} +7058.96 q^{43} +4676.94 q^{44} +5405.49 q^{45} -2575.51 q^{46} -9336.01 q^{47} +6207.15 q^{48} +17528.8 q^{49} +6467.88 q^{50} +15634.0 q^{51} -6366.13 q^{52} -20013.2 q^{53} -3549.23 q^{54} -37620.1 q^{55} +36353.6 q^{56} +20549.0 q^{57} +17401.3 q^{58} +11065.7 q^{59} +4982.92 q^{60} -43813.3 q^{61} +13780.4 q^{62} -15009.2 q^{63} +36287.3 q^{64} +51207.5 q^{65} +24701.3 q^{66} +34523.5 q^{67} +14411.9 q^{68} +4761.00 q^{69} -60204.7 q^{70} -58611.8 q^{71} -15891.3 q^{72} -62694.7 q^{73} +21886.2 q^{74} -11956.3 q^{75} +18942.6 q^{76} +104459. q^{77} -33622.8 q^{78} -57510.8 q^{79} -46025.6 q^{80} +6561.00 q^{81} +78909.8 q^{82} +55180.2 q^{83} -13835.9 q^{84} -115925. q^{85} +34367.5 q^{86} -32167.6 q^{87} +110597. q^{88} -81686.4 q^{89} +26317.3 q^{90} -142186. q^{91} +4388.81 q^{92} -25474.0 q^{93} -45453.6 q^{94} -152370. q^{95} -26282.0 q^{96} -171497. q^{97} +85341.1 q^{98} -45662.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.86863	0.860660	0.430330	−	0.902672i	$-0.358397\pi$
$2$	4.86863	0.860660	0.430330	+	0.902672i	$0.358397\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−8.29644	−0.259264
$5$	66.7344	1.19378	0.596891	−	0.802323i	$-0.296403\pi$
$5$	66.7344	1.19378	0.596891	+	0.802323i	$0.296403\pi$
$6$	−43.8177	−0.496903
$7$	−185.299	−1.42932	−0.714658	−	0.699474i	$-0.753418\pi$
$7$	−185.299	−1.42932	−0.714658	+	0.699474i	$0.753418\pi$
$8$	−196.188	−1.08380
$9$	81.0000	0.333333
$10$	324.905	1.02744
$11$	−563.729	−1.40472	−0.702359	−	0.711823i	$-0.747870\pi$
$11$	−563.729	−1.40472	−0.702359	+	0.711823i	$0.747870\pi$
$12$	74.6679	0.149686
$13$	767.333	1.25929	0.629645	−	0.776883i	$-0.283200\pi$
$13$	767.333	1.25929	0.629645	+	0.776883i	$0.283200\pi$
$14$	−902.153	−1.23016
$15$	−600.610	−0.689230
$16$	−689.683	−0.673519
$17$	−1737.11	−1.45783	−0.728913	−	0.684606i	$-0.759974\pi$
$17$	−1737.11	−1.45783	−0.728913	+	0.684606i	$0.759974\pi$
$18$	394.359	0.286887
$19$	−2283.22	−1.45099	−0.725495	−	0.688227i	$-0.758389\pi$
$19$	−2283.22	−1.45099	−0.725495	+	0.688227i	$0.758389\pi$
$20$	−553.658	−0.309504
$21$	1667.69	0.825216
$22$	−2744.59	−1.20898
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	1765.70	0.625731
$25$	1328.48	0.425114
$26$	3735.86	1.08382
$27$	−729.000	−0.192450
$28$	1537.32	0.370570
$29$	3574.17	0.789188	0.394594	−	0.918856i	$-0.370885\pi$
$29$	3574.17	0.789188	0.394594	+	0.918856i	$0.370885\pi$
$30$	−2924.15	−0.593193
$31$	2830.44	0.528993	0.264497	−	0.964387i	$-0.414794\pi$
$31$	2830.44	0.528993	0.264497	+	0.964387i	$0.414794\pi$
$32$	2920.22	0.504127
$33$	5073.56	0.811014
$34$	−8457.37	−1.25469
$35$	−12365.8	−1.70629
$36$	−672.011	−0.0864212
$37$	4495.35	0.539833	0.269916	−	0.962884i	$-0.413004\pi$
$37$	4495.35	0.539833	0.269916	+	0.962884i	$0.413004\pi$
$38$	−11116.2	−1.24881
$39$	−6906.00	−0.727051
$40$	−13092.5	−1.29382
$41$	16207.8	1.50579	0.752895	−	0.658140i	$-0.228657\pi$
$41$	16207.8	1.50579	0.752895	+	0.658140i	$0.228657\pi$
$42$	8119.38	0.710231
$43$	7058.96	0.582196	0.291098	−	0.956693i	$-0.405979\pi$
$43$	7058.96	0.582196	0.291098	+	0.956693i	$0.405979\pi$
$44$	4676.94	0.364192
$45$	5405.49	0.397927
$46$	−2575.51	−0.179460
$47$	−9336.01	−0.616476	−0.308238	−	0.951309i	$-0.599739\pi$
$47$	−9336.01	−0.616476	−0.308238	+	0.951309i	$0.599739\pi$
$48$	6207.15	0.388856
$49$	17528.8	1.04294
$50$	6467.88	0.365879
$51$	15634.0	0.841677
$52$	−6366.13	−0.326488
$53$	−20013.2	−0.978647	−0.489323	−	0.872102i	$-0.662756\pi$
$53$	−20013.2	−0.978647	−0.489323	+	0.872102i	$0.662756\pi$
$54$	−3549.23	−0.165634
$55$	−37620.1	−1.67692
$56$	36353.6	1.54909
$57$	20549.0	0.837730
$58$	17401.3	0.679223
$59$	11065.7	0.413857	0.206928	−	0.978356i	$-0.433653\pi$
$59$	11065.7	0.413857	0.206928	+	0.978356i	$0.433653\pi$
$60$	4982.92	0.178692
$61$	−43813.3	−1.50758	−0.753792	−	0.657113i	$-0.771778\pi$
$61$	−43813.3	−1.50758	−0.753792	+	0.657113i	$0.771778\pi$
$62$	13780.4	0.455284
$63$	−15009.2	−0.476439
$64$	36287.3	1.10740
$65$	51207.5	1.50332
$66$	24701.3	0.698007
$67$	34523.5	0.939567	0.469783	−	0.882782i	$-0.344332\pi$
$67$	34523.5	0.939567	0.469783	+	0.882782i	$0.344332\pi$
$68$	14411.9	0.377961
$69$	4761.00	0.120386
$70$	−60204.7	−1.46854
$71$	−58611.8	−1.37987	−0.689936	−	0.723870i	$-0.742361\pi$
$71$	−58611.8	−1.37987	−0.689936	+	0.723870i	$0.742361\pi$
$72$	−15891.3	−0.361266
$73$	−62694.7	−1.37697	−0.688484	−	0.725252i	$-0.741723\pi$
$73$	−62694.7	−1.37697	−0.688484	+	0.725252i	$0.741723\pi$
$74$	21886.2	0.464613
$75$	−11956.3	−0.245440
$76$	18942.6	0.376189
$77$	104459.	2.00778
$78$	−33622.8	−0.625744
$79$	−57510.8	−1.03677	−0.518384	−	0.855148i	$-0.673466\pi$
$79$	−57510.8	−1.03677	−0.518384	+	0.855148i	$0.673466\pi$
$80$	−46025.6	−0.804034
$81$	6561.00	0.111111
$82$	78909.8	1.29597
$83$	55180.2	0.879201	0.439601	−	0.898193i	$-0.355120\pi$
$83$	55180.2	0.879201	0.439601	+	0.898193i	$0.355120\pi$
$84$	−13835.9	−0.213949
$85$	−115925.	−1.74033
$86$	34367.5	0.501073
$87$	−32167.6	−0.455638
$88$	110597.	1.52243
$89$	−81686.4	−1.09314	−0.546568	−	0.837414i	$-0.684066\pi$
$89$	−81686.4	−1.09314	−0.546568	+	0.837414i	$0.684066\pi$
$90$	26317.3	0.342480
$91$	−142186.	−1.79992
$92$	4388.81	0.0540602
$93$	−25474.0	−0.305414
$94$	−45453.6	−0.530577
$95$	−152370.	−1.73216
$96$	−26282.0	−0.291058
$97$	−171497.	−1.85066	−0.925332	−	0.379158i	$-0.876214\pi$
$97$	−171497.	−1.85066	−0.925332	+	0.379158i	$0.876214\pi$
$98$	85341.1	0.897621
$99$	−45662.1	−0.468239
$100$	−11021.7	−0.110217
$101$	−43082.5	−0.420240	−0.210120	−	0.977676i	$-0.567386\pi$
$101$	−43082.5	−0.420240	−0.210120	+	0.977676i	$0.567386\pi$
$102$	76116.3	0.724398
$103$	41385.5	0.384375	0.192187	−	0.981358i	$-0.438442\pi$
$103$	41385.5	0.384375	0.192187	+	0.981358i	$0.438442\pi$
$104$	−150542.	−1.36482
$105$	111292.	0.985128
$106$	−97436.7	−0.842282
$107$	165235.	1.39522	0.697611	−	0.716477i	$-0.254246\pi$
$107$	165235.	1.39522	0.697611	+	0.716477i	$0.254246\pi$
$108$	6048.10	0.0498953
$109$	136414.	1.09974	0.549872	−	0.835249i	$-0.314676\pi$
$109$	136414.	1.09974	0.549872	+	0.835249i	$0.314676\pi$
$110$	−183159.	−1.44326
$111$	−40458.1	−0.311673
$112$	127798.	0.962671
$113$	74143.6	0.546232	0.273116	−	0.961981i	$-0.411946\pi$
$113$	74143.6	0.546232	0.273116	+	0.961981i	$0.411946\pi$
$114$	100046.	0.721001
$115$	−35302.5	−0.248921
$116$	−29652.9	−0.204608
$117$	62154.0	0.419763
$118$	53874.9	0.356190
$119$	321886.	2.08370
$120$	117833.	0.746986
$121$	156740.	0.973230
$122$	−213311.	−1.29752
$123$	−145870.	−0.869369
$124$	−23482.6	−0.137149
$125$	−119890.	−0.686288
$126$	−73074.4	−0.410052
$127$	−49454.2	−0.272078	−0.136039	−	0.990703i	$-0.543437\pi$
$127$	−49454.2	−0.272078	−0.136039	+	0.990703i	$0.543437\pi$
$128$	83222.6	0.448969
$129$	−63530.6	−0.336131
$130$	249311.	1.29385
$131$	96091.5	0.489223	0.244612	−	0.969621i	$-0.421340\pi$
$131$	96091.5	0.489223	0.244612	+	0.969621i	$0.421340\pi$
$132$	−42092.5	−0.210266
$133$	423079.	2.07392
$134$	168082.	0.808648
$135$	−48649.4	−0.229743
$136$	340802.	1.57999
$137$	−257144.	−1.17051	−0.585255	−	0.810850i	$-0.699005\pi$
$137$	−257144.	−1.17051	−0.585255	+	0.810850i	$0.699005\pi$
$138$	23179.6	0.103611
$139$	127180.	0.558319	0.279159	−	0.960245i	$-0.409944\pi$
$139$	127180.	0.558319	0.279159	+	0.960245i	$0.409944\pi$
$140$	102592.	0.442379
$141$	84024.1	0.355923
$142$	−285359.	−1.18760
$143$	−432568.	−1.76895
$144$	−55864.3	−0.224506
$145$	238520.	0.942118
$146$	−305237.	−1.18510
$147$	−157759.	−0.602145
$148$	−37295.4	−0.139959
$149$	623.566	0.00230100	0.00115050	−	0.999999i	$-0.499634\pi$
$149$	623.566	0.00230100	0.00115050	+	0.999999i	$0.499634\pi$
$150$	−58210.9	−0.211240
$151$	−333623.	−1.19073	−0.595366	−	0.803455i	$-0.702993\pi$
$151$	−333623.	−1.19073	−0.595366	+	0.803455i	$0.702993\pi$
$152$	447942.	1.57258
$153$	−140706.	−0.485942
$154$	508570.	1.72802
$155$	188888.	0.631502
$156$	57295.2	0.188498
$157$	−574134.	−1.85893	−0.929467	−	0.368904i	$-0.879733\pi$
$157$	−574134.	−1.85893	−0.929467	+	0.368904i	$0.879733\pi$
$158$	−279999.	−0.892305
$159$	180118.	0.565022
$160$	194879.	0.601818
$161$	98023.3	0.298033
$162$	31943.1	0.0956289
$163$	239894.	0.707214	0.353607	−	0.935394i	$-0.384955\pi$
$163$	239894.	0.707214	0.353607	+	0.935394i	$0.384955\pi$
$164$	−134467.	−0.390397
$165$	338581.	0.968173
$166$	268652.	0.756694
$167$	309931.	0.859951	0.429975	−	0.902841i	$-0.358522\pi$
$167$	309931.	0.859951	0.429975	+	0.902841i	$0.358522\pi$
$168$	−327182.	−0.894368
$169$	217508.	0.585811
$170$	−564397.	−1.49783
$171$	−184941.	−0.483663
$172$	−58564.2	−0.150942
$173$	−81138.3	−0.206115	−0.103058	−	0.994675i	$-0.532863\pi$
$173$	−81138.3	−0.206115	−0.103058	+	0.994675i	$0.532863\pi$
$174$	−156612.	−0.392150
$175$	−246166.	−0.607622
$176$	388795.	0.946103
$177$	−99591.5	−0.238940
$178$	−397701.	−0.940820
$179$	−118862.	−0.277275	−0.138637	−	0.990343i	$-0.544272\pi$
$179$	−118862.	−0.277275	−0.138637	+	0.990343i	$0.544272\pi$
$180$	−44846.3	−0.103168
$181$	−541106.	−1.22768	−0.613841	−	0.789430i	$-0.710376\pi$
$181$	−541106.	−1.22768	−0.613841	+	0.789430i	$0.710376\pi$
$182$	−692252.	−1.54912
$183$	394320.	0.870405
$184$	103784.	0.225988
$185$	299994.	0.644442
$186$	−124023.	−0.262858
$187$	979262.	2.04783
$188$	77455.6	0.159830
$189$	135083.	0.275072
$190$	−741831.	−1.49081
$191$	268445.	0.532441	0.266220	−	0.963912i	$-0.414225\pi$
$191$	268445.	0.532441	0.266220	+	0.963912i	$0.414225\pi$
$192$	−326586.	−0.639358
$193$	−602619.	−1.16453	−0.582264	−	0.813000i	$-0.697833\pi$
$193$	−602619.	−1.16453	−0.582264	+	0.813000i	$0.697833\pi$
$194$	−834956.	−1.59279
$195$	−460868.	−0.867940
$196$	−145426.	−0.270398
$197$	482982.	0.886676	0.443338	−	0.896354i	$-0.353794\pi$
$197$	482982.	0.886676	0.443338	+	0.896354i	$0.353794\pi$
$198$	−222312.	−0.402995
$199$	573505.	1.02661	0.513304	−	0.858207i	$-0.328422\pi$
$199$	573505.	1.02661	0.513304	+	0.858207i	$0.328422\pi$
$200$	−260633.	−0.460738
$201$	−310711.	−0.542459
$202$	−209753.	−0.361684
$203$	−662291.	−1.12800
$204$	−129707.	−0.218216
$205$	1.08162e6	1.79759
$206$	201491.	0.330816
$207$	−42849.0	−0.0695048
$208$	−529217.	−0.848155
$209$	1.28712e6	2.03823
$210$	541842.	0.847860
$211$	−1.23893e6	−1.91577	−0.957883	−	0.287160i	$-0.907289\pi$
$211$	−1.23893e6	−1.91577	−0.957883	+	0.287160i	$0.907289\pi$
$212$	166038.	0.253727
$213$	527506.	0.796670
$214$	804469.	1.20081
$215$	471075.	0.695015
$216$	143021.	0.208577
$217$	−524479.	−0.756099
$218$	664148.	0.946507
$219$	564252.	0.794993
$220$	312113.	0.434766
$221$	−1.33295e6	−1.83583
$222$	−196976.	−0.268244
$223$	−308404.	−0.415296	−0.207648	−	0.978204i	$-0.566581\pi$
$223$	−308404.	−0.415296	−0.207648	+	0.978204i	$0.566581\pi$
$224$	−541114.	−0.720558
$225$	107607.	0.141705
$226$	360978.	0.470120
$227$	764147.	0.984266	0.492133	−	0.870520i	$-0.336217\pi$
$227$	764147.	0.984266	0.492133	+	0.870520i	$0.336217\pi$
$228$	−170484.	−0.217193
$229$	1.01846e6	1.28338	0.641692	−	0.766962i	$-0.278233\pi$
$229$	1.01846e6	1.28338	0.641692	+	0.766962i	$0.278233\pi$
$230$	−171875.	−0.214236
$231$	−940127.	−1.15920
$232$	−701211.	−0.855321
$233$	−120752.	−0.145715	−0.0728577	−	0.997342i	$-0.523212\pi$
$233$	−120752.	−0.145715	−0.0728577	+	0.997342i	$0.523212\pi$
$234$	302605.	0.361274
$235$	−623033.	−0.735938
$236$	−91806.1	−0.107298
$237$	517597.	0.598578
$238$	1.56714e6	1.79335
$239$	−317622.	−0.359680	−0.179840	−	0.983696i	$-0.557558\pi$
$239$	−317622.	−0.359680	−0.179840	+	0.983696i	$0.557558\pi$
$240$	414230.	0.464209
$241$	−66852.0	−0.0741432	−0.0370716	−	0.999313i	$-0.511803\pi$
$241$	−66852.0	−0.0741432	−0.0370716	+	0.999313i	$0.511803\pi$
$242$	763107.	0.837620
$243$	−59049.0	−0.0641500
$244$	363495.	0.390862
$245$	1.16977e6	1.24505
$246$	−710189.	−0.748231
$247$	−1.75199e6	−1.82722
$248$	−555300.	−0.573322
$249$	−496622.	−0.507607
$250$	−583698.	−0.590661
$251$	−101729.	−0.101920	−0.0509600	−	0.998701i	$-0.516228\pi$
$251$	−101729.	−0.101920	−0.0509600	+	0.998701i	$0.516228\pi$
$252$	124523.	0.123523
$253$	298213.	0.292904
$254$	−240774.	−0.234167
$255$	1.04333e6	1.00478
$256$	−756014.	−0.720991
$257$	2.07807e6	1.96258	0.981291	−	0.192533i	$-0.0616701\pi$
$257$	2.07807e6	1.96258	0.981291	+	0.192533i	$0.0616701\pi$
$258$	−309307.	−0.289295
$259$	−832985.	−0.771592
$260$	−424840.	−0.389755
$261$	289508.	0.263063
$262$	467834.	0.421055
$263$	−437706.	−0.390205	−0.195102	−	0.980783i	$-0.562504\pi$
$263$	−437706.	−0.390205	−0.195102	+	0.980783i	$0.562504\pi$
$264$	−995375.	−0.878975
$265$	−1.33557e6	−1.16829
$266$	2.05982e6	1.78494
$267$	735177.	0.631123
$268$	−286422.	−0.243596
$269$	1.43024e6	1.20511	0.602556	−	0.798076i	$-0.294149\pi$
$269$	1.43024e6	1.20511	0.602556	+	0.798076i	$0.294149\pi$
$270$	−236856.	−0.197731
$271$	442343.	0.365877	0.182939	−	0.983124i	$-0.441439\pi$
$271$	442343.	0.365877	0.182939	+	0.983124i	$0.441439\pi$
$272$	1.19806e6	0.981874
$273$	1.27968e6	1.03919
$274$	−1.25194e6	−1.00741
$275$	−748903.	−0.597165
$276$	−39499.3	−0.0312117
$277$	−1.01001e6	−0.790909	−0.395455	−	0.918486i	$-0.629413\pi$
$277$	−1.01001e6	−0.790909	−0.395455	+	0.918486i	$0.629413\pi$
$278$	619193.	0.480523
$279$	229266.	0.176331
$280$	2.42603e6	1.84928
$281$	−1.44669e6	−1.09298	−0.546488	−	0.837467i	$-0.684036\pi$
$281$	−1.44669e6	−1.09298	−0.546488	+	0.837467i	$0.684036\pi$
$282$	409082.	0.306329
$283$	647359.	0.480484	0.240242	−	0.970713i	$-0.422773\pi$
$283$	647359.	0.480484	0.240242	+	0.970713i	$0.422773\pi$
$284$	486269.	0.357751
$285$	1.37133e6	1.00007
$286$	−2.10602e6	−1.52246
$287$	−3.00329e6	−2.15225
$288$	236538.	0.168042
$289$	1.59771e6	1.12526
$290$	1.16127e6	0.810844
$291$	1.54347e6	1.06848
$292$	520143.	0.356998
$293$	−1.04046e6	−0.708040	−0.354020	−	0.935238i	$-0.615185\pi$
$293$	−1.04046e6	−0.708040	−0.354020	+	0.935238i	$0.615185\pi$
$294$	−768070.	−0.518242
$295$	738465.	0.494054
$296$	−881936.	−0.585070
$297$	410959.	0.270338
$298$	3035.91	0.00198038
$299$	−405919.	−0.262580
$300$	99194.9	0.0636336
$301$	−1.30802e6	−0.832143
$302$	−1.62429e6	−1.02482
$303$	387743.	0.242626
$304$	1.57470e6	0.977269
$305$	−2.92386e6	−1.79973
$306$	−685047.	−0.418231
$307$	792650.	0.479994	0.239997	−	0.970774i	$-0.422854\pi$
$307$	792650.	0.479994	0.239997	+	0.970774i	$0.422854\pi$
$308$	−866634.	−0.520546
$309$	−372469.	−0.221919
$310$	919625.	0.543509
$311$	−2.27129e6	−1.33159	−0.665797	−	0.746133i	$-0.731908\pi$
$311$	−2.27129e6	−1.33159	−0.665797	+	0.746133i	$0.731908\pi$
$312$	1.35488e6	0.787977
$313$	1.32722e6	0.765741	0.382870	−	0.923802i	$-0.374936\pi$
$313$	1.32722e6	0.765741	0.382870	+	0.923802i	$0.374936\pi$
$314$	−2.79525e6	−1.59991
$315$	−1.00163e6	−0.568764
$316$	477134.	0.268796
$317$	1.54042e6	0.860977	0.430489	−	0.902596i	$-0.358341\pi$
$317$	1.54042e6	0.860977	0.430489	+	0.902596i	$0.358341\pi$
$318$	876930.	0.486292
$319$	−2.01487e6	−1.10859
$320$	2.42161e6	1.32199
$321$	−1.48712e6	−0.805532
$322$	477239.	0.256505
$323$	3.96622e6	2.11529
$324$	−54432.9	−0.0288071
$325$	1.01939e6	0.535342
$326$	1.16796e6	0.608671
$327$	−1.22772e6	−0.634938
$328$	−3.17979e6	−1.63197
$329$	1.72995e6	0.881140
$330$	1.64843e6	0.833268
$331$	−2.90735e6	−1.45857	−0.729286	−	0.684209i	$-0.760148\pi$
$331$	−2.90735e6	−1.45857	−0.729286	+	0.684209i	$0.760148\pi$
$332$	−457799.	−0.227945
$333$	364123.	0.179944
$334$	1.50894e6	0.740126
$335$	2.30390e6	1.12164
$336$	−1.15018e6	−0.555799
$337$	−2.54377e6	−1.22012	−0.610061	−	0.792355i	$-0.708855\pi$
$337$	−2.54377e6	−1.22012	−0.610061	+	0.792355i	$0.708855\pi$
$338$	1.05896e6	0.504185
$339$	−667292.	−0.315367
$340$	961766.	0.451203
$341$	−1.59560e6	−0.743086
$342$	−900410.	−0.416270
$343$	−133745.	−0.0613820
$344$	−1.38489e6	−0.630984
$345$	317723.	0.143714
$346$	−395032.	−0.177395
$347$	1.06956e6	0.476851	0.238426	−	0.971161i	$-0.423369\pi$
$347$	1.06956e6	0.476851	0.238426	+	0.971161i	$0.423369\pi$
$348$	266876.	0.118130
$349$	−3.46366e6	−1.52220	−0.761099	−	0.648635i	$-0.775340\pi$
$349$	−3.46366e6	−1.52220	−0.761099	+	0.648635i	$0.775340\pi$
$350$	−1.19849e6	−0.522956
$351$	−559386.	−0.242350
$352$	−1.64621e6	−0.708156
$353$	−1.80510e6	−0.771018	−0.385509	−	0.922704i	$-0.625974\pi$
$353$	−1.80510e6	−0.771018	−0.385509	+	0.922704i	$0.625974\pi$
$354$	−484874.	−0.205646
$355$	−3.91142e6	−1.64727
$356$	677706.	0.283411
$357$	−2.89697e6	−1.20302
$358$	−578695.	−0.238640
$359$	−2.44266e6	−1.00029	−0.500146	−	0.865941i	$-0.666720\pi$
$359$	−2.44266e6	−1.00029	−0.500146	+	0.865941i	$0.666720\pi$
$360$	−1.06049e6	−0.431273
$361$	2.73701e6	1.10537
$362$	−2.63444e6	−1.05662
$363$	−1.41066e6	−0.561895
$364$	1.17964e6	0.466655
$365$	−4.18389e6	−1.64380
$366$	1.91980e6	0.749123
$367$	−1.73440e6	−0.672179	−0.336090	−	0.941830i	$-0.609105\pi$
$367$	−1.73440e6	−0.672179	−0.336090	+	0.941830i	$0.609105\pi$
$368$	364842.	0.140438
$369$	1.31283e6	0.501930
$370$	1.46056e6	0.554646
$371$	3.70842e6	1.39880
$372$	211343.	0.0791828
$373$	670423.	0.249503	0.124752	−	0.992188i	$-0.460187\pi$
$373$	670423.	0.249503	0.124752	+	0.992188i	$0.460187\pi$
$374$	4.76766e6	1.76249
$375$	1.07901e6	0.396229
$376$	1.83162e6	0.668136
$377$	2.74258e6	0.993817
$378$	657670.	0.236744
$379$	−1.58515e6	−0.566857	−0.283428	−	0.958993i	$-0.591472\pi$
$379$	−1.58515e6	−0.566857	−0.283428	+	0.958993i	$0.591472\pi$
$380$	1.26412e6	0.449087
$381$	445088.	0.157085
$382$	1.30696e6	0.458251
$383$	1.13803e6	0.396421	0.198210	−	0.980160i	$-0.436487\pi$
$383$	1.13803e6	0.396421	0.198210	+	0.980160i	$0.436487\pi$
$384$	−749003.	−0.259212
$385$	6.97098e6	2.39686
$386$	−2.93393e6	−1.00226
$387$	571776.	0.194065
$388$	1.42282e6	0.479810
$389$	5.06039e6	1.69555	0.847774	−	0.530358i	$-0.177942\pi$
$389$	5.06039e6	1.69555	0.847774	+	0.530358i	$0.177942\pi$
$390$	−2.24380e6	−0.747002
$391$	918933.	0.303978
$392$	−3.43894e6	−1.13034
$393$	−864824.	−0.282453
$394$	2.35146e6	0.763127
$395$	−3.83795e6	−1.23767
$396$	378832.	0.121397
$397$	2.94086e6	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$397$	2.94086e6	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$398$	2.79218e6	0.883560
$399$	−3.80771e6	−1.19738
$400$	−916231.	−0.286322
$401$	3.32124e6	1.03143	0.515715	−	0.856760i	$-0.327526\pi$
$401$	3.32124e6	1.03143	0.515715	+	0.856760i	$0.327526\pi$
$402$	−1.51274e6	−0.466873
$403$	2.17189e6	0.666156
$404$	357431.	0.108953
$405$	437844.	0.132642
$406$	−3.22445e6	−0.970824
$407$	−2.53416e6	−0.758312
$408$	−3.06722e6	−0.912208
$409$	−1.11681e6	−0.330120	−0.165060	−	0.986283i	$-0.552782\pi$
$409$	−1.11681e6	−0.330120	−0.165060	+	0.986283i	$0.552782\pi$
$410$	5.26600e6	1.54711
$411$	2.31430e6	0.675794
$412$	−343352.	−0.0996544
$413$	−2.05047e6	−0.591532
$414$	−208616.	−0.0598200
$415$	3.68242e6	1.04957
$416$	2.24078e6	0.634843
$417$	−1.14462e6	−0.322345
$418$	6.26651e6	1.75422
$419$	707320.	0.196825	0.0984126	−	0.995146i	$-0.468624\pi$
$419$	707320.	0.196825	0.0984126	+	0.995146i	$0.468624\pi$
$420$	−923331.	−0.255408
$421$	5.82295e6	1.60117	0.800586	−	0.599218i	$-0.204522\pi$
$421$	5.82295e6	1.60117	0.800586	+	0.599218i	$0.204522\pi$
$422$	−6.03192e6	−1.64882
$423$	−756217.	−0.205492
$424$	3.92635e6	1.06066
$425$	−2.30772e6	−0.619742
$426$	2.56823e6	0.685662
$427$	8.11858e6	2.15482
$428$	−1.37086e6	−0.361730
$429$	3.89311e6	1.02130
$430$	2.29349e6	0.598172
$431$	−3.42318e6	−0.887640	−0.443820	−	0.896116i	$-0.646377\pi$
$431$	−3.42318e6	−0.887640	−0.443820	+	0.896116i	$0.646377\pi$
$432$	502779.	0.129619
$433$	1.45426e6	0.372754	0.186377	−	0.982478i	$-0.440325\pi$
$433$	1.45426e6	0.372754	0.186377	+	0.982478i	$0.440325\pi$
$434$	−2.55349e6	−0.650744
$435$	−2.14668e6	−0.543932
$436$	−1.13175e6	−0.285124
$437$	1.20783e6	0.302552
$438$	2.74714e6	0.684219
$439$	−3.79123e6	−0.938899	−0.469450	−	0.882959i	$-0.655548\pi$
$439$	−3.79123e6	−0.938899	−0.469450	+	0.882959i	$0.655548\pi$
$440$	7.38064e6	1.81745
$441$	1.41983e6	0.347648
$442$	−6.48962e6	−1.58002
$443$	−672461.	−0.162801	−0.0814006	−	0.996681i	$-0.525939\pi$
$443$	−672461.	−0.162801	−0.0814006	+	0.996681i	$0.525939\pi$
$444$	335658.	0.0808054
$445$	−5.45129e6	−1.30497
$446$	−1.50150e6	−0.357429
$447$	−5612.10	−0.00132848
$448$	−6.72401e6	−1.58283
$449$	4.85755e6	1.13711	0.568554	−	0.822646i	$-0.307503\pi$
$449$	4.85755e6	1.13711	0.568554	+	0.822646i	$0.307503\pi$
$450$	523898.	0.121960
$451$	−9.13682e6	−2.11521
$452$	−615127.	−0.141618
$453$	3.00261e6	0.687470
$454$	3.72035e6	0.847118
$455$	−9.48871e6	−2.14872
$456$	−4.03148e6	−0.907930
$457$	698808.	0.156519	0.0782596	−	0.996933i	$-0.475064\pi$
$457$	698808.	0.156519	0.0782596	+	0.996933i	$0.475064\pi$
$458$	4.95852e6	1.10456
$459$	1.26636e6	0.280559
$460$	292885.	0.0645361
$461$	422691.	0.0926341	0.0463170	−	0.998927i	$-0.485252\pi$
$461$	422691.	0.0926341	0.0463170	+	0.998927i	$0.485252\pi$
$462$	−4.57713e6	−0.997673
$463$	4.13790e6	0.897073	0.448536	−	0.893765i	$-0.351945\pi$
$463$	4.13790e6	0.897073	0.448536	+	0.893765i	$0.351945\pi$
$464$	−2.46505e6	−0.531533
$465$	−1.69999e6	−0.364598
$466$	−587898.	−0.125411
$467$	6.02577e6	1.27856	0.639279	−	0.768975i	$-0.279233\pi$
$467$	6.02577e6	1.27856	0.639279	+	0.768975i	$0.279233\pi$
$468$	−515657.	−0.108829
$469$	−6.39717e6	−1.34294
$470$	−3.03332e6	−0.633393
$471$	5.16721e6	1.07326
$472$	−2.17097e6	−0.448537
$473$	−3.97934e6	−0.817821
$474$	2.51999e6	0.515172
$475$	−3.03322e6	−0.616836
$476$	−2.67050e6	−0.540227
$477$	−1.62107e6	−0.326216
$478$	−1.54639e6	−0.309562
$479$	−2.25772e6	−0.449604	−0.224802	−	0.974404i	$-0.572174\pi$
$479$	−2.25772e6	−0.449604	−0.224802	+	0.974404i	$0.572174\pi$
$480$	−1.75391e6	−0.347460
$481$	3.44943e6	0.679806
$482$	−325478.	−0.0638122
$483$	−882209.	−0.172069
$484$	−1.30038e6	−0.252323
$485$	−1.14448e7	−2.20929
$486$	−287488.	−0.0552114
$487$	5.05901e6	0.966592	0.483296	−	0.875457i	$-0.339439\pi$
$487$	5.05901e6	0.966592	0.483296	+	0.875457i	$0.339439\pi$
$488$	8.59567e6	1.63392
$489$	−2.15905e6	−0.408310
$490$	5.69519e6	1.07156
$491$	−5.75970e6	−1.07819	−0.539096	−	0.842244i	$-0.681234\pi$
$491$	−5.75970e6	−1.07819	−0.539096	+	0.842244i	$0.681234\pi$
$492$	1.21020e6	0.225396
$493$	−6.20874e6	−1.15050
$494$	−8.52981e6	−1.57261
$495$	−3.04723e6	−0.558975
$496$	−1.95211e6	−0.356287
$497$	1.08607e7	1.97227
$498$	−2.41787e6	−0.436877
$499$	−956632.	−0.171986	−0.0859930	−	0.996296i	$-0.527406\pi$
$499$	−956632.	−0.171986	−0.0859930	+	0.996296i	$0.527406\pi$
$500$	994657.	0.177930
$501$	−2.78938e6	−0.496493
$502$	−495280.	−0.0877186
$503$	4.84338e6	0.853548	0.426774	−	0.904358i	$-0.359650\pi$
$503$	4.84338e6	0.853548	0.426774	+	0.904358i	$0.359650\pi$
$504$	2.94464e6	0.516364
$505$	−2.87509e6	−0.501675
$506$	1.45189e6	0.252091
$507$	−1.95757e6	−0.338218
$508$	410294.	0.0705400
$509$	2.65927e6	0.454955	0.227478	−	0.973783i	$-0.426952\pi$
$509$	2.65927e6	0.454955	0.227478	+	0.973783i	$0.426952\pi$
$510$	5.07958e6	0.864773
$511$	1.16173e7	1.96812
$512$	−6.34388e6	−1.06950
$513$	1.66447e6	0.279243
$514$	1.01174e7	1.68912
$515$	2.76183e6	0.458859
$516$	527078.	0.0871466
$517$	5.26298e6	0.865975
$518$	−4.05549e6	−0.664078
$519$	730244.	0.119001
$520$	−1.00463e7	−1.62929
$521$	1.09007e6	0.175938	0.0879691	−	0.996123i	$-0.471962\pi$
$521$	1.09007e6	0.175938	0.0879691	+	0.996123i	$0.471962\pi$
$522$	1.40951e6	0.226408
$523$	−1.03436e7	−1.65354	−0.826772	−	0.562537i	$-0.809825\pi$
$523$	−1.03436e7	−1.65354	−0.826772	+	0.562537i	$0.809825\pi$
$524$	−797217.	−0.126838
$525$	2.21550e6	0.350811
$526$	−2.13103e6	−0.335834
$527$	−4.91680e6	−0.771181
$528$	−3.49915e6	−0.546233
$529$	279841.	0.0434783
$530$	−6.50238e6	−1.00550
$531$	896324.	0.137952
$532$	−3.51005e6	−0.537693
$533$	1.24368e7	1.89623
$534$	3.57931e6	0.543182
$535$	1.10269e7	1.66559
$536$	−6.77311e6	−1.01830
$537$	1.06976e6	0.160085
$538$	6.96330e6	1.03719
$539$	−9.88148e6	−1.46504
$540$	403616.	0.0595641
$541$	−1.07591e7	−1.58045	−0.790227	−	0.612814i	$-0.790037\pi$
$541$	−1.07591e7	−1.58045	−0.790227	+	0.612814i	$0.790037\pi$
$542$	2.15360e6	0.314896
$543$	4.86995e6	0.708802
$544$	−5.07275e6	−0.734931
$545$	9.10349e6	1.31285
$546$	6.23027e6	0.894387
$547$	−1.31666e7	−1.88150	−0.940749	−	0.339103i	$-0.889876\pi$
$547$	−1.31666e7	−1.88150	−0.940749	+	0.339103i	$0.889876\pi$
$548$	2.13338e6	0.303470
$549$	−3.54888e6	−0.502528
$550$	−3.64613e6	−0.513956
$551$	−8.16064e6	−1.14510
$552$	−934053.	−0.130474
$553$	1.06567e7	1.48187
$554$	−4.91737e6	−0.680704
$555$	−2.69995e6	−0.372069
$556$	−1.05514e6	−0.144752
$557$	2.57299e6	0.351399	0.175699	−	0.984444i	$-0.443781\pi$
$557$	2.57299e6	0.351399	0.175699	+	0.984444i	$0.443781\pi$
$558$	1.11621e6	0.151761
$559$	5.41658e6	0.733154
$560$	8.52850e6	1.14922
$561$	−8.81336e6	−1.18232
$562$	−7.04341e6	−0.940681
$563$	5.77557e6	0.767933	0.383967	−	0.923347i	$-0.374558\pi$
$563$	5.77557e6	0.767933	0.383967	+	0.923347i	$0.374558\pi$
$564$	−697100.	−0.0922779
$565$	4.94793e6	0.652082
$566$	3.15175e6	0.413533
$567$	−1.21575e6	−0.158813
$568$	1.14990e7	1.49550
$569$	4.07120e6	0.527160	0.263580	−	0.964638i	$-0.415097\pi$
$569$	4.07120e6	0.527160	0.263580	+	0.964638i	$0.415097\pi$
$570$	6.67648e6	0.860717
$571$	−5.03144e6	−0.645806	−0.322903	−	0.946432i	$-0.604659\pi$
$571$	−5.03144e6	−0.645806	−0.322903	+	0.946432i	$0.604659\pi$
$572$	3.58878e6	0.458623
$573$	−2.41600e6	−0.307405
$574$	−1.46219e7	−1.85236
$575$	−702766.	−0.0886424
$576$	2.93927e6	0.369134
$577$	−2.49388e6	−0.311843	−0.155921	−	0.987769i	$-0.549835\pi$
$577$	−2.49388e6	−0.311843	−0.155921	+	0.987769i	$0.549835\pi$
$578$	7.77865e6	0.968466
$579$	5.42357e6	0.672340
$580$	−1.97887e6	−0.244257
$581$	−1.02249e7	−1.25666
$582$	7.51461e6	0.919600
$583$	1.12820e7	1.37472
$584$	1.23000e7	1.49236
$585$	4.14781e6	0.501106
$586$	−5.06563e6	−0.609382
$587$	5.36774e6	0.642979	0.321489	−	0.946913i	$-0.395817\pi$
$587$	5.36774e6	0.642979	0.321489	+	0.946913i	$0.395817\pi$
$588$	1.30884e6	0.156114
$589$	−6.46253e6	−0.767564
$590$	3.59531e6	0.425213
$591$	−4.34683e6	−0.511923
$592$	−3.10037e6	−0.363587
$593$	1.07373e7	1.25389	0.626943	−	0.779065i	$-0.284306\pi$
$593$	1.07373e7	1.25389	0.626943	+	0.779065i	$0.284306\pi$
$594$	2.00081e6	0.232669
$595$	2.14809e7	2.48748
$596$	−5173.38	−0.000596566 0
$597$	−5.16154e6	−0.592712
$598$	−1.97627e6	−0.225992
$599$	−3.52730e6	−0.401675	−0.200838	−	0.979625i	$-0.564366\pi$
$599$	−3.52730e6	−0.401675	−0.200838	+	0.979625i	$0.564366\pi$
$600$	2.34569e6	0.266007
$601$	−2.73536e6	−0.308907	−0.154454	−	0.988000i	$-0.549362\pi$
$601$	−2.73536e6	−0.308907	−0.154454	+	0.988000i	$0.549362\pi$
$602$	−6.36826e6	−0.716192
$603$	2.79640e6	0.313189
$604$	2.76789e6	0.308714
$605$	1.04599e7	1.16182
$606$	1.88778e6	0.208818
$607$	3.19959e6	0.352470	0.176235	−	0.984348i	$-0.443608\pi$
$607$	3.19959e6	0.352470	0.176235	+	0.984348i	$0.443608\pi$
$608$	−6.66751e6	−0.731484
$609$	5.96062e6	0.651251
$610$	−1.42352e7	−1.54895
$611$	−7.16383e6	−0.776323
$612$	1.16736e6	0.125987
$613$	−1.31336e7	−1.41167	−0.705833	−	0.708378i	$-0.749427\pi$
$613$	−1.31336e7	−1.41167	−0.705833	+	0.708378i	$0.749427\pi$
$614$	3.85912e6	0.413112
$615$	−9.73457e6	−1.03784
$616$	−2.04936e7	−2.17603
$617$	5.00019e6	0.528778	0.264389	−	0.964416i	$-0.414830\pi$
$617$	5.00019e6	0.528778	0.264389	+	0.964416i	$0.414830\pi$
$618$	−1.81341e6	−0.190997
$619$	−8.11569e6	−0.851332	−0.425666	−	0.904880i	$-0.639960\pi$
$619$	−8.11569e6	−0.851332	−0.425666	+	0.904880i	$0.639960\pi$
$620$	−1.56710e6	−0.163726
$621$	385641.	0.0401286
$622$	−1.10581e7	−1.14605
$623$	1.51364e7	1.56244
$624$	4.76295e6	0.489683
$625$	−1.21523e7	−1.24439
$626$	6.46174e6	0.659043
$627$	−1.15841e7	−1.17677
$628$	4.76327e6	0.481954
$629$	−7.80893e6	−0.786983
$630$	−4.87658e6	−0.489512
$631$	−121821.	−0.0121801	−0.00609003	−	0.999981i	$-0.501939\pi$
$631$	−121821.	−0.0121801	−0.00609003	+	0.999981i	$0.501939\pi$
$632$	1.12829e7	1.12365
$633$	1.11504e7	1.10607
$634$	7.49975e6	0.741009
$635$	−3.30030e6	−0.324802
$636$	−1.49434e6	−0.146490
$637$	1.34504e7	1.31337
$638$	−9.80964e6	−0.954116
$639$	−4.74755e6	−0.459957
$640$	5.55381e6	0.535971
$641$	4.06055e6	0.390337	0.195169	−	0.980770i	$-0.437475\pi$
$641$	4.06055e6	0.390337	0.195169	+	0.980770i	$0.437475\pi$
$642$	−7.24022e6	−0.693289
$643$	1.62068e7	1.54586	0.772928	−	0.634494i	$-0.218792\pi$
$643$	1.62068e7	1.54586	0.772928	+	0.634494i	$0.218792\pi$
$644$	−813244.	−0.0772691
$645$	−4.23968e6	−0.401267
$646$	1.93101e7	1.82055
$647$	4.33831e6	0.407436	0.203718	−	0.979030i	$-0.434697\pi$
$647$	4.33831e6	0.407436	0.203718	+	0.979030i	$0.434697\pi$
$648$	−1.28719e6	−0.120422
$649$	−6.23807e6	−0.581351
$650$	4.96302e6	0.460747
$651$	4.72031e6	0.436534
$652$	−1.99027e6	−0.183355
$653$	3.28611e6	0.301578	0.150789	−	0.988566i	$-0.451819\pi$
$653$	3.28611e6	0.301578	0.150789	+	0.988566i	$0.451819\pi$
$654$	−5.97733e6	−0.546466
$655$	6.41261e6	0.584025
$656$	−1.11783e7	−1.01418
$657$	−5.07827e6	−0.458989
$658$	8.42251e6	0.758362
$659$	−2.08664e7	−1.87169	−0.935847	−	0.352408i	$-0.885363\pi$
$659$	−2.08664e7	−1.87169	−0.935847	+	0.352408i	$0.885363\pi$
$660$	−2.80902e6	−0.251012
$661$	−1.40139e7	−1.24754	−0.623769	−	0.781609i	$-0.714399\pi$
$661$	−1.40139e7	−1.24754	−0.623769	+	0.781609i	$0.714399\pi$
$662$	−1.41548e7	−1.25534
$663$	1.19965e7	1.05992
$664$	−1.08257e7	−0.952877
$665$	2.82340e7	2.47581
$666$	1.77278e6	0.154871
$667$	−1.89074e6	−0.164557
$668$	−2.57132e6	−0.222954
$669$	2.77563e6	0.239771
$670$	1.12169e7	0.965349
$671$	2.46989e7	2.11773
$672$	4.87003e6	0.416014
$673$	6.69255e6	0.569579	0.284790	−	0.958590i	$-0.408076\pi$
$673$	6.69255e6	0.569579	0.284790	+	0.958590i	$0.408076\pi$
$674$	−1.23847e7	−1.05011
$675$	−968463.	−0.0818132
$676$	−1.80454e6	−0.151880
$677$	−1.24923e7	−1.04754	−0.523769	−	0.851860i	$-0.675475\pi$
$677$	−1.24923e7	−1.04754	−0.523769	+	0.851860i	$0.675475\pi$
$678$	−3.24880e6	−0.271424
$679$	3.17783e7	2.64518
$680$	2.27432e7	1.88616
$681$	−6.87732e6	−0.568266
$682$	−7.76840e6	−0.639545
$683$	975374.	0.0800054	0.0400027	−	0.999200i	$-0.487263\pi$
$683$	975374.	0.0800054	0.0400027	+	0.999200i	$0.487263\pi$
$684$	1.53435e6	0.125396
$685$	−1.71603e7	−1.39733
$686$	−651153.	−0.0528290
$687$	−9.16617e6	−0.740962
$688$	−4.86845e6	−0.392120
$689$	−1.53568e7	−1.23240
$690$	1.54687e6	0.123689
$691$	5.56198e6	0.443133	0.221566	−	0.975145i	$-0.428883\pi$
$691$	5.56198e6	0.443133	0.221566	+	0.975145i	$0.428883\pi$
$692$	673158.	0.0534382
$693$	8.46114e6	0.669262
$694$	5.20731e6	0.410407
$695$	8.48729e6	0.666510
$696$	6.31090e6	0.493820
$697$	−2.81548e7	−2.19518
$698$	−1.68633e7	−1.31010
$699$	1.08677e6	0.0841288
$700$	2.04230e6	0.157534
$701$	−9.57353e6	−0.735829	−0.367915	−	0.929860i	$-0.619928\pi$
$701$	−9.57353e6	−0.735829	−0.367915	+	0.929860i	$0.619928\pi$
$702$	−2.72344e6	−0.208581
$703$	−1.02639e7	−0.783292
$704$	−2.04562e7	−1.55559
$705$	5.60730e6	0.424894
$706$	−8.78836e6	−0.663584
$707$	7.98316e6	0.600656
$708$	826255.	0.0619485
$709$	1.85648e7	1.38700	0.693498	−	0.720458i	$-0.256068\pi$
$709$	1.85648e7	1.38700	0.693498	+	0.720458i	$0.256068\pi$
$710$	−1.90433e7	−1.41774
$711$	−4.65837e6	−0.345589
$712$	1.60259e7	1.18474
$713$	−1.49730e6	−0.110303
$714$	−1.41043e7	−1.03539
$715$	−2.88672e7	−2.11173
$716$	986131.	0.0718873
$717$	2.85860e6	0.207661
$718$	−1.18924e7	−0.860911
$719$	−2.42441e7	−1.74897	−0.874487	−	0.485050i	$-0.838801\pi$
$719$	−2.42441e7	−1.74897	−0.874487	+	0.485050i	$0.838801\pi$
$720$	−3.72807e6	−0.268011
$721$	−7.66869e6	−0.549393
$722$	1.33255e7	0.951350
$723$	601668.	0.0428066
$724$	4.48925e6	0.318293
$725$	4.74822e6	0.335495
$726$	−6.86797e6	−0.483600
$727$	1.23980e7	0.869990	0.434995	−	0.900433i	$-0.356750\pi$
$727$	1.23980e7	0.869990	0.434995	+	0.900433i	$0.356750\pi$
$728$	2.78953e7	1.95075
$729$	531441.	0.0370370
$730$	−2.03698e7	−1.41475
$731$	−1.22622e7	−0.848742
$732$	−3.27145e6	−0.225664
$733$	2.06634e7	1.42050	0.710252	−	0.703947i	$-0.248581\pi$
$733$	2.06634e7	1.42050	0.710252	+	0.703947i	$0.248581\pi$
$734$	−8.44417e6	−0.578518
$735$	−1.05280e7	−0.718829
$736$	−1.54480e6	−0.105118
$737$	−1.94619e7	−1.31983
$738$	6.39170e6	0.431992
$739$	1.86671e7	1.25738	0.628688	−	0.777658i	$-0.283592\pi$
$739$	1.86671e7	1.25738	0.628688	+	0.777658i	$0.283592\pi$
$740$	−2.48889e6	−0.167080
$741$	1.57679e7	1.05494
$742$	1.80549e7	1.20389
$743$	8.53060e6	0.566902	0.283451	−	0.958987i	$-0.408521\pi$
$743$	8.53060e6	0.566902	0.283451	+	0.958987i	$0.408521\pi$
$744$	4.99770e6	0.331008
$745$	41613.3	0.00274689
$746$	3.26404e6	0.214738
$747$	4.46960e6	0.293067
$748$	−8.12438e6	−0.530929
$749$	−3.06179e7	−1.99421
$750$	5.25329e6	0.341018
$751$	9.57195e6	0.619299	0.309650	−	0.950851i	$-0.399788\pi$
$751$	9.57195e6	0.619299	0.309650	+	0.950851i	$0.399788\pi$
$752$	6.43889e6	0.415208
$753$	915560.	0.0588436
$754$	1.33526e7	0.855339
$755$	−2.22642e7	−1.42147
$756$	−1.12071e6	−0.0713162
$757$	3.09107e7	1.96051	0.980255	−	0.197738i	$-0.0633596\pi$
$757$	3.09107e7	1.96051	0.980255	+	0.197738i	$0.0633596\pi$
$758$	−7.71752e6	−0.487871
$759$	−2.68391e6	−0.169108
$760$	2.98932e7	1.87732
$761$	−3.87764e6	−0.242720	−0.121360	−	0.992609i	$-0.538726\pi$
$761$	−3.87764e6	−0.242720	−0.121360	+	0.992609i	$0.538726\pi$
$762$	2.16697e6	0.135196
$763$	−2.52774e7	−1.57188
$764$	−2.22713e6	−0.138043
$765$	−9.38995e6	−0.580109
$766$	5.54064e6	0.341184
$767$	8.49110e6	0.521165
$768$	6.80413e6	0.416265
$769$	−2.77315e7	−1.69105	−0.845527	−	0.533933i	$-0.820714\pi$
$769$	−2.77315e7	−1.69105	−0.845527	+	0.533933i	$0.820714\pi$
$770$	3.39391e7	2.06288
$771$	−1.87026e7	−1.13310
$772$	4.99959e6	0.301920
$773$	−2.27867e7	−1.37162	−0.685810	−	0.727781i	$-0.740552\pi$
$773$	−2.27867e7	−1.37162	−0.685810	+	0.727781i	$0.740552\pi$
$774$	2.78376e6	0.167024
$775$	3.76019e6	0.224882
$776$	3.36458e7	2.00575
$777$	7.49686e6	0.445479
$778$	2.46372e7	1.45929
$779$	−3.70061e7	−2.18489
$780$	3.82356e6	0.225025
$781$	3.30412e7	1.93833
$782$	4.47395e6	0.261622
$783$	−2.60557e6	−0.151879
$784$	−1.20893e7	−0.702443
$785$	−3.83145e7	−2.21916
$786$	−4.21051e6	−0.243096
$787$	1.70385e7	0.980604	0.490302	−	0.871553i	$-0.336886\pi$
$787$	1.70385e7	0.980604	0.490302	+	0.871553i	$0.336886\pi$
$788$	−4.00703e6	−0.229883
$789$	3.93935e6	0.225285
$790$	−1.86855e7	−1.06522
$791$	−1.37387e7	−0.780739
$792$	8.95837e6	0.507477
$793$	−3.36194e7	−1.89849
$794$	1.43180e7	0.805992
$795$	1.20201e7	0.674513
$796$	−4.75804e6	−0.266162
$797$	−2.04743e7	−1.14173	−0.570864	−	0.821045i	$-0.693392\pi$
$797$	−2.04743e7	−1.14173	−0.570864	+	0.821045i	$0.693392\pi$
$798$	−1.85384e7	−1.03054
$799$	1.62177e7	0.898716
$800$	3.87945e6	0.214312
$801$	−6.61660e6	−0.364379
$802$	1.61699e7	0.887710
$803$	3.53428e7	1.93425
$804$	2.57780e6	0.140640
$805$	6.54152e6	0.355786
$806$	1.05741e7	0.573334
$807$	−1.28721e7	−0.695772
$808$	8.45230e6	0.455456
$809$	8.90982e6	0.478628	0.239314	−	0.970942i	$-0.423078\pi$
$809$	8.90982e6	0.478628	0.239314	+	0.970942i	$0.423078\pi$
$810$	2.13170e6	0.114160
$811$	4.16300e6	0.222256	0.111128	−	0.993806i	$-0.464554\pi$
$811$	4.16300e6	0.222256	0.111128	+	0.993806i	$0.464554\pi$
$812$	5.49466e6	0.292449
$813$	−3.98108e6	−0.211239
$814$	−1.23379e7	−0.652649
$815$	1.60092e7	0.844259
$816$	−1.07825e7	−0.566885
$817$	−1.61172e7	−0.844761
$818$	−5.43735e6	−0.284122
$819$	−1.15171e7	−0.599975
$820$	−8.97358e6	−0.466048
$821$	8.55489e6	0.442952	0.221476	−	0.975166i	$-0.428913\pi$
$821$	8.55489e6	0.442952	0.221476	+	0.975166i	$0.428913\pi$
$822$	1.12674e7	0.581629
$823$	2.62720e7	1.35205	0.676025	−	0.736878i	$-0.263701\pi$
$823$	2.62720e7	1.35205	0.676025	+	0.736878i	$0.263701\pi$
$824$	−8.11935e6	−0.416585
$825$	6.74013e6	0.344773
$826$	−9.98298e6	−0.509108
$827$	777655.	0.0395388	0.0197694	−	0.999805i	$-0.493707\pi$
$827$	777655.	0.0395388	0.0197694	+	0.999805i	$0.493707\pi$
$828$	355494.	0.0180201
$829$	−1.04189e6	−0.0526546	−0.0263273	−	0.999653i	$-0.508381\pi$
$829$	−1.04189e6	−0.0526546	−0.0263273	+	0.999653i	$0.508381\pi$
$830$	1.79283e7	0.903327
$831$	9.09010e6	0.456632
$832$	2.78445e7	1.39454
$833$	−3.04495e7	−1.52043
$834$	−5.57274e6	−0.277430
$835$	2.06830e7	1.02659
$836$	−1.06785e7	−0.528439
$837$	−2.06339e6	−0.101805
$838$	3.44368e6	0.169400
$839$	2.90427e7	1.42440	0.712201	−	0.701976i	$-0.247699\pi$
$839$	2.90427e7	1.42440	0.712201	+	0.701976i	$0.247699\pi$
$840$	−2.18343e7	−1.06768
$841$	−7.73644e6	−0.377182
$842$	2.83498e7	1.37806
$843$	1.30202e7	0.631030
$844$	1.02787e7	0.496688
$845$	1.45152e7	0.699331
$846$	−3.68174e6	−0.176859
$847$	−2.90437e7	−1.39105
$848$	1.38027e7	0.659137
$849$	−5.82623e6	−0.277408
$850$	−1.12354e7	−0.533388
$851$	−2.37804e6	−0.112563
$852$	−4.37642e6	−0.206547
$853$	−1.29014e7	−0.607103	−0.303552	−	0.952815i	$-0.598172\pi$
$853$	−1.29014e7	−0.607103	−0.303552	+	0.952815i	$0.598172\pi$
$854$	3.95263e7	1.85456
$855$	−1.23419e7	−0.577388
$856$	−3.24172e7	−1.51214
$857$	2.64342e7	1.22946	0.614729	−	0.788738i	$-0.289265\pi$
$857$	2.64342e7	1.22946	0.614729	+	0.788738i	$0.289265\pi$
$858$	1.89541e7	0.878994
$859$	3.34613e7	1.54725	0.773623	−	0.633646i	$-0.218442\pi$
$859$	3.34613e7	1.54725	0.773623	+	0.633646i	$0.218442\pi$
$860$	−3.90825e6	−0.180192
$861$	2.70296e7	1.24260
$862$	−1.66662e7	−0.763956
$863$	−1.30469e7	−0.596320	−0.298160	−	0.954516i	$-0.596373\pi$
$863$	−1.30469e7	−0.596320	−0.298160	+	0.954516i	$0.596373\pi$
$864$	−2.12884e6	−0.0970194
$865$	−5.41471e6	−0.246057
$866$	7.08025e6	0.320814
$867$	−1.43794e7	−0.649669
$868$	4.35130e6	0.196029
$869$	3.24205e7	1.45637
$870$	−1.04514e7	−0.468141
$871$	2.64910e7	1.18319
$872$	−2.67628e7	−1.19190
$873$	−1.38913e7	−0.616888
$874$	5.88046e6	0.260395
$875$	2.22154e7	0.980923
$876$	−4.68128e6	−0.206113
$877$	−3.46134e7	−1.51966	−0.759828	−	0.650124i	$-0.774717\pi$
$877$	−3.46134e7	−1.51966	−0.759828	+	0.650124i	$0.774717\pi$
$878$	−1.84581e7	−0.808073
$879$	9.36417e6	0.408787
$880$	2.59460e7	1.12944
$881$	−2.51488e7	−1.09163	−0.545817	−	0.837905i	$-0.683781\pi$
$881$	−2.51488e7	−1.09163	−0.545817	+	0.837905i	$0.683781\pi$
$882$	6.91263e6	0.299207
$883$	8.32506e6	0.359323	0.179662	−	0.983728i	$-0.442500\pi$
$883$	8.32506e6	0.359323	0.179662	+	0.983728i	$0.442500\pi$
$884$	1.10587e7	0.475963
$885$	−6.64618e6	−0.285242
$886$	−3.27396e6	−0.140117
$887$	3.90242e7	1.66543	0.832713	−	0.553705i	$-0.186786\pi$
$887$	3.90242e7	1.66543	0.832713	+	0.553705i	$0.186786\pi$
$888$	7.93742e6	0.337790
$889$	9.16383e6	0.388886
$890$	−2.65403e7	−1.12313
$891$	−3.69863e6	−0.156080
$892$	2.55865e6	0.107671
$893$	2.13162e7	0.894501
$894$	−27323.2	−0.00114337
$895$	−7.93219e6	−0.331006
$896$	−1.54211e7	−0.641719
$897$	3.65327e6	0.151601
$898$	2.36496e7	0.978664
$899$	1.01165e7	0.417475
$900$	−892754.	−0.0367389
$901$	3.47651e7	1.42670
$902$	−4.44838e7	−1.82048
$903$	1.17722e7	0.480438
$904$	−1.45461e7	−0.592006
$905$	−3.61104e7	−1.46558
$906$	1.46186e7	0.591678
$907$	−3.55617e7	−1.43537	−0.717685	−	0.696368i	$-0.754798\pi$
$907$	−3.55617e7	−1.43537	−0.717685	+	0.696368i	$0.754798\pi$
$908$	−6.33970e6	−0.255184
$909$	−3.48968e6	−0.140080
$910$	−4.61970e7	−1.84931
$911$	1.54307e6	0.0616013	0.0308007	−	0.999526i	$-0.490194\pi$
$911$	1.54307e6	0.0616013	0.0308007	+	0.999526i	$0.490194\pi$
$912$	−1.41723e7	−0.564227
$913$	−3.11067e7	−1.23503
$914$	3.40224e6	0.134710
$915$	2.63147e7	1.03907
$916$	−8.44962e6	−0.332735
$917$	−1.78057e7	−0.699254
$918$	6.16542e6	0.241466
$919$	5.41531e6	0.211512	0.105756	−	0.994392i	$-0.466274\pi$
$919$	5.41531e6	0.211512	0.105756	+	0.994392i	$0.466274\pi$
$920$	6.92594e6	0.269780
$921$	−7.13385e6	−0.277125
$922$	2.05793e6	0.0797265
$923$	−4.49748e7	−1.73766
$924$	7.79970e6	0.300537
$925$	5.97199e6	0.229490
$926$	2.01459e7	0.772075
$927$	3.35222e6	0.128125
$928$	1.04374e7	0.397851
$929$	−2.37496e7	−0.902851	−0.451426	−	0.892309i	$-0.649084\pi$
$929$	−2.37496e7	−0.902851	−0.451426	+	0.892309i	$0.649084\pi$
$930$	−8.27663e6	−0.313795
$931$	−4.00221e7	−1.51330
$932$	1.00181e6	0.0377787
$933$	2.04416e7	0.768796
$934$	2.93372e7	1.10040
$935$	6.53505e7	2.44467
$936$	−1.21939e7	−0.454939
$937$	−1.35178e7	−0.502987	−0.251493	−	0.967859i	$-0.580922\pi$
$937$	−1.35178e7	−0.502987	−0.251493	+	0.967859i	$0.580922\pi$
$938$	−3.11455e7	−1.15581
$939$	−1.19450e7	−0.442101
$940$	5.16895e6	0.190802
$941$	−3.13260e7	−1.15327	−0.576635	−	0.817002i	$-0.695634\pi$
$941$	−3.13260e7	−1.15327	−0.576635	+	0.817002i	$0.695634\pi$
$942$	2.51572e7	0.923709
$943$	−8.57393e6	−0.313979
$944$	−7.63184e6	−0.278740
$945$	9.01469e6	0.328376
$946$	−1.93739e7	−0.703866
$947$	−2.57698e7	−0.933761	−0.466880	−	0.884320i	$-0.654622\pi$
$947$	−2.57698e7	−0.933761	−0.466880	+	0.884320i	$0.654622\pi$
$948$	−4.29421e6	−0.155190
$949$	−4.81078e7	−1.73400
$950$	−1.47676e7	−0.530886
$951$	−1.38638e7	−0.497086
$952$	−6.31503e7	−2.25831
$953$	1.18796e7	0.423710	0.211855	−	0.977301i	$-0.432050\pi$
$953$	1.18796e7	0.423710	0.211855	+	0.977301i	$0.432050\pi$
$954$	−7.89237e6	−0.280761
$955$	1.79145e7	0.635618
$956$	2.63513e6	0.0932520
$957$	1.81338e7	0.640042
$958$	−1.09920e7	−0.386957
$959$	4.76485e7	1.67303
$960$	−2.17945e7	−0.763254
$961$	−2.06177e7	−0.720166
$962$	1.67940e7	0.585082
$963$	1.33841e7	0.465074
$964$	554633.	0.0192226
$965$	−4.02154e7	−1.39019
$966$	−4.29515e6	−0.148093
$967$	−8.80618e6	−0.302846	−0.151423	−	0.988469i	$-0.548386\pi$
$967$	−8.80618e6	−0.302846	−0.151423	+	0.988469i	$0.548386\pi$
$968$	−3.07505e7	−1.05478
$969$	−3.56960e7	−1.22126
$970$	−5.57203e7	−1.90145
$971$	1.00214e7	0.341099	0.170550	−	0.985349i	$-0.445446\pi$
$971$	1.00214e7	0.341099	0.170550	+	0.985349i	$0.445446\pi$
$972$	489896.	0.0166318
$973$	−2.35664e7	−0.798014
$974$	2.46305e7	0.831908
$975$	−9.17449e6	−0.309080
$976$	3.02173e7	1.01539
$977$	3.76716e7	1.26264	0.631318	−	0.775524i	$-0.282514\pi$
$977$	3.76716e7	1.26264	0.631318	+	0.775524i	$0.282514\pi$
$978$	−1.05116e7	−0.351417
$979$	4.60490e7	1.53555
$980$	−9.70494e6	−0.322796
$981$	1.10495e7	0.366582
$982$	−2.80419e7	−0.927958
$983$	−4.31076e7	−1.42288	−0.711442	−	0.702744i	$-0.751958\pi$
$983$	−4.31076e7	−1.42288	−0.711442	+	0.702744i	$0.751958\pi$
$984$	2.86181e7	0.942220
$985$	3.22315e7	1.05850
$986$	−3.02281e7	−0.990189
$987$	−1.55696e7	−0.508726
$988$	1.45353e7	0.473731
$989$	−3.73419e6	−0.121396
$990$	−1.48358e7	−0.481088
$991$	3.25349e6	0.105236	0.0526181	−	0.998615i	$-0.483243\pi$
$991$	3.25349e6	0.105236	0.0526181	+	0.998615i	$0.483243\pi$
$992$	8.26551e6	0.266680
$993$	2.61662e7	0.842107
$994$	5.28768e7	1.69746
$995$	3.82725e7	1.22554
$996$	4.12019e6	0.131604
$997$	1.82210e7	0.580544	0.290272	−	0.956944i	$-0.406254\pi$
$997$	1.82210e7	0.580544	0.290272	+	0.956944i	$0.406254\pi$
$998$	−4.65749e6	−0.148022
$999$	−3.27711e6	−0.103891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.d.1.3	✓	4
3.2	odd	2		207.6.a.e.1.2		4
4.3	odd	2		1104.6.a.o.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.d.1.3	✓	4	1.1	even	1	trivial
207.6.a.e.1.2		4	3.2	odd	2
1104.6.a.o.1.4		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)

\(f(q)\)	\(=\)	\(q+4.86863 q^{2} -9.00000 q^{3} -8.29644 q^{4} +66.7344 q^{5} -43.8177 q^{6} -185.299 q^{7} -196.188 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+4.86863 q^{2} -9.00000 q^{3} -8.29644 q^{4} +66.7344 q^{5} -43.8177 q^{6} -185.299 q^{7} -196.188 q^{8} +81.0000 q^{9} +324.905 q^{10} -563.729 q^{11} +74.6679 q^{12} +767.333 q^{13} -902.153 q^{14} -600.610 q^{15} -689.683 q^{16} -1737.11 q^{17} +394.359 q^{18} -2283.22 q^{19} -553.658 q^{20} +1667.69 q^{21} -2744.59 q^{22} -529.000 q^{23} +1765.70 q^{24} +1328.48 q^{25} +3735.86 q^{26} -729.000 q^{27} +1537.32 q^{28} +3574.17 q^{29} -2924.15 q^{30} +2830.44 q^{31} +2920.22 q^{32} +5073.56 q^{33} -8457.37 q^{34} -12365.8 q^{35} -672.011 q^{36} +4495.35 q^{37} -11116.2 q^{38} -6906.00 q^{39} -13092.5 q^{40} +16207.8 q^{41} +8119.38 q^{42} +7058.96 q^{43} +4676.94 q^{44} +5405.49 q^{45} -2575.51 q^{46} -9336.01 q^{47} +6207.15 q^{48} +17528.8 q^{49} +6467.88 q^{50} +15634.0 q^{51} -6366.13 q^{52} -20013.2 q^{53} -3549.23 q^{54} -37620.1 q^{55} +36353.6 q^{56} +20549.0 q^{57} +17401.3 q^{58} +11065.7 q^{59} +4982.92 q^{60} -43813.3 q^{61} +13780.4 q^{62} -15009.2 q^{63} +36287.3 q^{64} +51207.5 q^{65} +24701.3 q^{66} +34523.5 q^{67} +14411.9 q^{68} +4761.00 q^{69} -60204.7 q^{70} -58611.8 q^{71} -15891.3 q^{72} -62694.7 q^{73} +21886.2 q^{74} -11956.3 q^{75} +18942.6 q^{76} +104459. q^{77} -33622.8 q^{78} -57510.8 q^{79} -46025.6 q^{80} +6561.00 q^{81} +78909.8 q^{82} +55180.2 q^{83} -13835.9 q^{84} -115925. q^{85} +34367.5 q^{86} -32167.6 q^{87} +110597. q^{88} -81686.4 q^{89} +26317.3 q^{90} -142186. q^{91} +4388.81 q^{92} -25474.0 q^{93} -45453.6 q^{94} -152370. q^{95} -26282.0 q^{96} -171497. q^{97} +85341.1 q^{98} -45662.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 36 q^{3} + 26 q^{4} + 22 q^{5} - 36 q^{6} - 62 q^{7} + 72 q^{8} + 324 q^{9}+O(q^{10}) \) 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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