LMFDB - Embedded newform 69.6.a.c.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:	$0$
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} - 39x^{2} - 30x + 20 $ x^4 - 39x^2 - 30x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.76108$ 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)

$f(q)$	$=$	$q-4.76108 q^{2} -9.00000 q^{3} -9.33211 q^{4} -97.1410 q^{5} +42.8497 q^{6} -211.220 q^{7} +196.786 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-4.76108 q^{2} -9.00000 q^{3} -9.33211 q^{4} -97.1410 q^{5} +42.8497 q^{6} -211.220 q^{7} +196.786 q^{8} +81.0000 q^{9} +462.496 q^{10} -633.540 q^{11} +83.9890 q^{12} -59.3594 q^{13} +1005.63 q^{14} +874.269 q^{15} -638.284 q^{16} -1809.15 q^{17} -385.648 q^{18} +1986.94 q^{19} +906.530 q^{20} +1900.98 q^{21} +3016.33 q^{22} +529.000 q^{23} -1771.07 q^{24} +6311.37 q^{25} +282.615 q^{26} -729.000 q^{27} +1971.12 q^{28} -2631.22 q^{29} -4162.46 q^{30} -4916.81 q^{31} -3258.21 q^{32} +5701.86 q^{33} +8613.53 q^{34} +20518.1 q^{35} -755.901 q^{36} -4871.13 q^{37} -9459.96 q^{38} +534.235 q^{39} -19115.9 q^{40} -13838.6 q^{41} -9050.70 q^{42} -1272.49 q^{43} +5912.26 q^{44} -7868.42 q^{45} -2518.61 q^{46} -17922.1 q^{47} +5744.56 q^{48} +27806.7 q^{49} -30048.9 q^{50} +16282.4 q^{51} +553.948 q^{52} -19517.0 q^{53} +3470.83 q^{54} +61542.6 q^{55} -41565.0 q^{56} -17882.4 q^{57} +12527.5 q^{58} -17569.0 q^{59} -8158.77 q^{60} -20453.8 q^{61} +23409.3 q^{62} -17108.8 q^{63} +35937.7 q^{64} +5766.23 q^{65} -27147.0 q^{66} -11300.7 q^{67} +16883.2 q^{68} -4761.00 q^{69} -97688.2 q^{70} +33263.2 q^{71} +15939.6 q^{72} +22571.9 q^{73} +23191.9 q^{74} -56802.3 q^{75} -18542.3 q^{76} +133816. q^{77} -2543.53 q^{78} -9902.06 q^{79} +62003.6 q^{80} +6561.00 q^{81} +65886.5 q^{82} -24218.1 q^{83} -17740.1 q^{84} +175743. q^{85} +6058.43 q^{86} +23681.0 q^{87} -124671. q^{88} +40463.9 q^{89} +37462.2 q^{90} +12537.9 q^{91} -4936.69 q^{92} +44251.3 q^{93} +85328.5 q^{94} -193013. q^{95} +29323.9 q^{96} -18327.4 q^{97} -132390. q^{98} -51316.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 36 q^{3} - 46 q^{4} - 122 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 - 46 * q^4 - 122 * q^5 - 36 * q^6 + 62 * q^7 + 72 * q^8 + 324 * q^9	\( 4 q + 4 q^{2} - 36 q^{3} - 46 q^{4} - 122 q^{5} - 36 q^{6} + 62 q^{7} + 72 q^{8} + 324 q^{9} + 642 q^{10} + 32 q^{11} + 414 q^{12} + 1364 q^{13} + 2754 q^{14} + 1098 q^{15} + 18 q^{16} + 278 q^{17} + 324 q^{18} + 2862 q^{19} + 3830 q^{20} - 558 q^{21} + 3176 q^{22} + 2116 q^{23} - 648 q^{24} + 5944 q^{25} + 6996 q^{26} - 2916 q^{27} + 4738 q^{28} - 5180 q^{29} - 5778 q^{30} - 1788 q^{31} - 7352 q^{32} - 288 q^{33} + 15818 q^{34} + 11768 q^{35} - 3726 q^{36} + 3348 q^{37} + 1050 q^{38} - 12276 q^{39} - 13462 q^{40} - 17664 q^{41} - 24786 q^{42} + 25398 q^{43} - 16848 q^{44} - 9882 q^{45} + 2116 q^{46} - 26040 q^{47} - 162 q^{48} + 55720 q^{49} - 35256 q^{50} - 2502 q^{51} - 2752 q^{52} - 32006 q^{53} - 2916 q^{54} + 34904 q^{55} - 68542 q^{56} - 25758 q^{57} - 40804 q^{58} - 61136 q^{59} - 34470 q^{60} + 35844 q^{61} - 47524 q^{62} + 5022 q^{63} - 35142 q^{64} - 48036 q^{65} - 28584 q^{66} + 73458 q^{67} + 17910 q^{68} - 19044 q^{69} - 59104 q^{70} + 24432 q^{71} + 5832 q^{72} + 122512 q^{73} - 20828 q^{74} - 53496 q^{75} + 56834 q^{76} + 159496 q^{77} - 62964 q^{78} + 90170 q^{79} - 36546 q^{80} + 26244 q^{81} + 84144 q^{82} + 28592 q^{83} - 42642 q^{84} + 355124 q^{85} + 103778 q^{86} + 46620 q^{87} - 150776 q^{88} - 27926 q^{89} + 52002 q^{90} + 334180 q^{91} - 24334 q^{92} + 16092 q^{93} + 113632 q^{94} + 113392 q^{95} + 66168 q^{96} + 16580 q^{97} + 49688 q^{98} + 2592 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 36 * q^3 - 46 * q^4 - 122 * q^5 - 36 * q^6 + 62 * q^7 + 72 * q^8 + 324 * q^9 + 642 * q^10 + 32 * q^11 + 414 * q^12 + 1364 * q^13 + 2754 * q^14 + 1098 * q^15 + 18 * q^16 + 278 * q^17 + 324 * q^18 + 2862 * q^19 + 3830 * q^20 - 558 * q^21 + 3176 * q^22 + 2116 * q^23 - 648 * q^24 + 5944 * q^25 + 6996 * q^26 - 2916 * q^27 + 4738 * q^28 - 5180 * q^29 - 5778 * q^30 - 1788 * q^31 - 7352 * q^32 - 288 * q^33 + 15818 * q^34 + 11768 * q^35 - 3726 * q^36 + 3348 * q^37 + 1050 * q^38 - 12276 * q^39 - 13462 * q^40 - 17664 * q^41 - 24786 * q^42 + 25398 * q^43 - 16848 * q^44 - 9882 * q^45 + 2116 * q^46 - 26040 * q^47 - 162 * q^48 + 55720 * q^49 - 35256 * q^50 - 2502 * q^51 - 2752 * q^52 - 32006 * q^53 - 2916 * q^54 + 34904 * q^55 - 68542 * q^56 - 25758 * q^57 - 40804 * q^58 - 61136 * q^59 - 34470 * q^60 + 35844 * q^61 - 47524 * q^62 + 5022 * q^63 - 35142 * q^64 - 48036 * q^65 - 28584 * q^66 + 73458 * q^67 + 17910 * q^68 - 19044 * q^69 - 59104 * q^70 + 24432 * q^71 + 5832 * q^72 + 122512 * q^73 - 20828 * q^74 - 53496 * q^75 + 56834 * q^76 + 159496 * q^77 - 62964 * q^78 + 90170 * q^79 - 36546 * q^80 + 26244 * q^81 + 84144 * q^82 + 28592 * q^83 - 42642 * q^84 + 355124 * q^85 + 103778 * q^86 + 46620 * q^87 - 150776 * q^88 - 27926 * q^89 + 52002 * q^90 + 334180 * q^91 - 24334 * q^92 + 16092 * q^93 + 113632 * q^94 + 113392 * q^95 + 66168 * q^96 + 16580 * q^97 + 49688 * q^98 + 2592 * 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 $

$2$	−4.76108	−0.841648	−0.420824	−	0.907142i	$-0.638259\pi$
$2$	−4.76108	−0.841648	−0.420824	+	0.907142i	$0.638259\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−9.33211	−0.291628
$5$	−97.1410	−1.73771	−0.868855	−	0.495066i	$-0.835144\pi$
$5$	−97.1410	−1.73771	−0.868855	+	0.495066i	$0.835144\pi$
$6$	42.8497	0.485926
$7$	−211.220	−1.62926	−0.814628	−	0.579985i	$-0.803059\pi$
$7$	−211.220	−1.62926	−0.814628	+	0.579985i	$0.803059\pi$
$8$	196.786	1.08710
$9$	81.0000	0.333333
$10$	462.496	1.46254
$11$	−633.540	−1.57867	−0.789336	−	0.613961i	$-0.789575\pi$
$11$	−633.540	−1.57867	−0.789336	+	0.613961i	$0.789575\pi$
$12$	83.9890	0.168372
$13$	−59.3594	−0.0974162	−0.0487081	−	0.998813i	$-0.515510\pi$
$13$	−59.3594	−0.0974162	−0.0487081	+	0.998813i	$0.515510\pi$
$14$	1005.63	1.37126
$15$	874.269	1.00327
$16$	−638.284	−0.623325
$17$	−1809.15	−1.51828	−0.759142	−	0.650925i	$-0.774381\pi$
$17$	−1809.15	−1.51828	−0.759142	+	0.650925i	$0.774381\pi$
$18$	−385.648	−0.280549
$19$	1986.94	1.26270	0.631350	−	0.775498i	$-0.282501\pi$
$19$	1986.94	1.26270	0.631350	+	0.775498i	$0.282501\pi$
$20$	906.530	0.506766
$21$	1900.98	0.940651
$22$	3016.33	1.32869
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	−1771.07	−0.627636
$25$	6311.37	2.01964
$26$	282.615	0.0819902
$27$	−729.000	−0.192450
$28$	1971.12	0.475137
$29$	−2631.22	−0.580982	−0.290491	−	0.956878i	$-0.593819\pi$
$29$	−2631.22	−0.580982	−0.290491	+	0.956878i	$0.593819\pi$
$30$	−4162.46	−0.844398
$31$	−4916.81	−0.918923	−0.459462	−	0.888198i	$-0.651958\pi$
$31$	−4916.81	−0.918923	−0.459462	+	0.888198i	$0.651958\pi$
$32$	−3258.21	−0.562477
$33$	5701.86	0.911447
$34$	8613.53	1.27786
$35$	20518.1	2.83117
$36$	−755.901	−0.0972095
$37$	−4871.13	−0.584959	−0.292480	−	0.956272i	$-0.594480\pi$
$37$	−4871.13	−0.584959	−0.292480	+	0.956272i	$0.594480\pi$
$38$	−9459.96	−1.06275
$39$	534.235	0.0562433
$40$	−19115.9	−1.88906
$41$	−13838.6	−1.28567	−0.642837	−	0.766003i	$-0.722243\pi$
$41$	−13838.6	−1.28567	−0.642837	+	0.766003i	$0.722243\pi$
$42$	−9050.70	−0.791697
$43$	−1272.49	−0.104950	−0.0524751	−	0.998622i	$-0.516711\pi$
$43$	−1272.49	−0.104950	−0.0524751	+	0.998622i	$0.516711\pi$
$44$	5912.26	0.460386
$45$	−7868.42	−0.579237
$46$	−2518.61	−0.175496
$47$	−17922.1	−1.18343	−0.591717	−	0.806146i	$-0.701550\pi$
$47$	−17922.1	−1.18343	−0.591717	+	0.806146i	$0.701550\pi$
$48$	5744.56	0.359877
$49$	27806.7	1.65447
$50$	−30048.9	−1.69982
$51$	16282.4	0.876582
$52$	553.948	0.0284093
$53$	−19517.0	−0.954383	−0.477192	−	0.878799i	$-0.658345\pi$
$53$	−19517.0	−0.954383	−0.477192	+	0.878799i	$0.658345\pi$
$54$	3470.83	0.161975
$55$	61542.6	2.74328
$56$	−41565.0	−1.77116
$57$	−17882.4	−0.729020
$58$	12527.5	0.488982
$59$	−17569.0	−0.657078	−0.328539	−	0.944490i	$-0.606556\pi$
$59$	−17569.0	−0.657078	−0.328539	+	0.944490i	$0.606556\pi$
$60$	−8158.77	−0.292581
$61$	−20453.8	−0.703801	−0.351901	−	0.936037i	$-0.614464\pi$
$61$	−20453.8	−0.703801	−0.351901	+	0.936037i	$0.614464\pi$
$62$	23409.3	0.773410
$63$	−17108.8	−0.543085
$64$	35937.7	1.09673
$65$	5766.23	0.169281
$66$	−27147.0	−0.767118
$67$	−11300.7	−0.307552	−0.153776	−	0.988106i	$-0.549143\pi$
$67$	−11300.7	−0.307552	−0.153776	+	0.988106i	$0.549143\pi$
$68$	16883.2	0.442775
$69$	−4761.00	−0.120386
$70$	−97688.2	−2.38285
$71$	33263.2	0.783102	0.391551	−	0.920156i	$-0.371939\pi$
$71$	33263.2	0.783102	0.391551	+	0.920156i	$0.371939\pi$
$72$	15939.6	0.362366
$73$	22571.9	0.495749	0.247874	−	0.968792i	$-0.420268\pi$
$73$	22571.9	0.495749	0.247874	+	0.968792i	$0.420268\pi$
$74$	23191.9	0.492330
$75$	−56802.3	−1.16604
$76$	−18542.3	−0.368239
$77$	133816.	2.57206
$78$	−2543.53	−0.0473370
$79$	−9902.06	−0.178508	−0.0892540	−	0.996009i	$-0.528448\pi$
$79$	−9902.06	−0.178508	−0.0892540	+	0.996009i	$0.528448\pi$
$80$	62003.6	1.08316
$81$	6561.00	0.111111
$82$	65886.5	1.08209
$83$	−24218.1	−0.385873	−0.192937	−	0.981211i	$-0.561801\pi$
$83$	−24218.1	−0.385873	−0.192937	+	0.981211i	$0.561801\pi$
$84$	−17740.1	−0.274320
$85$	175743.	2.63834
$86$	6058.43	0.0883311
$87$	23681.0	0.335430
$88$	−124671.	−1.71617
$89$	40463.9	0.541493	0.270746	−	0.962651i	$-0.412730\pi$
$89$	40463.9	0.541493	0.270746	+	0.962651i	$0.412730\pi$
$90$	37462.2	0.487514
$91$	12537.9	0.158716
$92$	−4936.69	−0.0608087
$93$	44251.3	0.530541
$94$	85328.5	0.996035
$95$	−193013.	−2.19421
$96$	29323.9	0.324746
$97$	−18327.4	−0.197776	−0.0988878	−	0.995099i	$-0.531528\pi$
$97$	−18327.4	−0.197776	−0.0988878	+	0.995099i	$0.531528\pi$
$98$	−132390.	−1.39248
$99$	−51316.7	−0.526224
$100$	−58898.4	−0.588984
$101$	135506.	1.32177	0.660886	−	0.750486i	$-0.270181\pi$
$101$	135506.	1.32177	0.660886	+	0.750486i	$0.270181\pi$
$102$	−77521.8	−0.737774
$103$	13708.4	0.127319	0.0636595	−	0.997972i	$-0.479723\pi$
$103$	13708.4	0.127319	0.0636595	+	0.997972i	$0.479723\pi$
$104$	−11681.1	−0.105901
$105$	−184663.	−1.63458
$106$	92921.9	0.803255
$107$	193324.	1.63240	0.816200	−	0.577770i	$-0.196077\pi$
$107$	193324.	1.63240	0.816200	+	0.577770i	$0.196077\pi$
$108$	6803.11	0.0561239
$109$	−116451.	−0.938804	−0.469402	−	0.882984i	$-0.655531\pi$
$109$	−116451.	−0.938804	−0.469402	+	0.882984i	$0.655531\pi$
$110$	−293010.	−2.30887
$111$	43840.2	0.337727
$112$	134818.	1.01555
$113$	−179696.	−1.32386	−0.661929	−	0.749567i	$-0.730262\pi$
$113$	−179696.	−1.32386	−0.661929	+	0.749567i	$0.730262\pi$
$114$	85139.7	0.613578
$115$	−51387.6	−0.362338
$116$	24554.9	0.169431
$117$	−4808.11	−0.0324721
$118$	83647.4	0.553028
$119$	382129.	2.47367
$120$	172043.	1.09065
$121$	240321.	1.49221
$122$	97382.3	0.592353
$123$	124547.	0.742284
$124$	45884.2	0.267984
$125$	−309527.	−1.77183
$126$	81456.3	0.457086
$127$	−246258.	−1.35482	−0.677408	−	0.735607i	$-0.736897\pi$
$127$	−246258.	−1.35482	−0.677408	+	0.735607i	$0.736897\pi$
$128$	−66839.5	−0.360586
$129$	11452.4	0.0605930
$130$	−27453.5	−0.142475
$131$	68004.2	0.346224	0.173112	−	0.984902i	$-0.444618\pi$
$131$	68004.2	0.346224	0.173112	+	0.984902i	$0.444618\pi$
$132$	−53210.3	−0.265804
$133$	−419680.	−2.05726
$134$	53803.6	0.258851
$135$	70815.8	0.334423
$136$	−356015.	−1.65052
$137$	−378451.	−1.72269	−0.861346	−	0.508018i	$-0.830378\pi$
$137$	−378451.	−1.72269	−0.861346	+	0.508018i	$0.830378\pi$
$138$	22667.5	0.101323
$139$	−112126.	−0.492231	−0.246115	−	0.969241i	$-0.579154\pi$
$139$	−112126.	−0.492231	−0.246115	+	0.969241i	$0.579154\pi$
$140$	−191477.	−0.825651
$141$	161299.	0.683256
$142$	−158369.	−0.659096
$143$	37606.5	0.153788
$144$	−51701.0	−0.207775
$145$	255599.	1.00958
$146$	−107467.	−0.417246
$147$	−250260.	−0.955210
$148$	45458.0	0.170591
$149$	290306.	1.07125	0.535625	−	0.844456i	$-0.320076\pi$
$149$	290306.	1.07125	0.535625	+	0.844456i	$0.320076\pi$
$150$	270440.	0.981394
$151$	125549.	0.448095	0.224048	−	0.974578i	$-0.428073\pi$
$151$	125549.	0.448095	0.224048	+	0.974578i	$0.428073\pi$
$152$	391000.	1.37268
$153$	−146541.	−0.506095
$154$	−637109.	−2.16477
$155$	477624.	1.59682
$156$	−4985.54	−0.0164021
$157$	−165073.	−0.534473	−0.267237	−	0.963631i	$-0.586111\pi$
$157$	−165073.	−0.534473	−0.267237	+	0.963631i	$0.586111\pi$
$158$	47144.5	0.150241
$159$	175653.	0.551013
$160$	316506.	0.977422
$161$	−111735.	−0.339723
$162$	−31237.5	−0.0935165
$163$	330262.	0.973620	0.486810	−	0.873508i	$-0.338160\pi$
$163$	330262.	0.973620	0.486810	+	0.873508i	$0.338160\pi$
$164$	129143.	0.374939
$165$	−553884.	−1.58383
$166$	115304.	0.324769
$167$	−21121.5	−0.0586048	−0.0293024	−	0.999571i	$-0.509329\pi$
$167$	−21121.5	−0.0586048	−0.0293024	+	0.999571i	$0.509329\pi$
$168$	374085.	1.02258
$169$	−367769.	−0.990510
$170$	−836726.	−2.22055
$171$	160942.	0.420900
$172$	11875.0	0.0306065
$173$	−519214.	−1.31896	−0.659480	−	0.751722i	$-0.729223\pi$
$173$	−519214.	−1.31896	−0.659480	+	0.751722i	$0.729223\pi$
$174$	−112747.	−0.282314
$175$	−1.33308e6	−3.29050
$176$	404378.	0.984025
$177$	158121.	0.379364
$178$	−192652.	−0.455746
$179$	420339.	0.980544	0.490272	−	0.871570i	$-0.336898\pi$
$179$	420339.	0.980544	0.490272	+	0.871570i	$0.336898\pi$
$180$	73428.9	0.168922
$181$	79336.5	0.180002	0.0900008	−	0.995942i	$-0.471313\pi$
$181$	79336.5	0.180002	0.0900008	+	0.995942i	$0.471313\pi$
$182$	−59693.8	−0.133583
$183$	184084.	0.406340
$184$	104100.	0.226675
$185$	473187.	1.01649
$186$	−210684.	−0.446529
$187$	1.14617e6	2.39687
$188$	167251.	0.345123
$189$	153979.	0.313550
$190$	918950.	1.84675
$191$	409572.	0.812357	0.406179	−	0.913794i	$-0.366861\pi$
$191$	409572.	0.812357	0.406179	+	0.913794i	$0.366861\pi$
$192$	−323439.	−0.633199
$193$	424832.	0.820963	0.410482	−	0.911869i	$-0.365361\pi$
$193$	424832.	0.820963	0.410482	+	0.911869i	$0.365361\pi$
$194$	87258.5	0.166457
$195$	−51896.1	−0.0977345
$196$	−259495.	−0.482491
$197$	−716052.	−1.31456	−0.657278	−	0.753648i	$-0.728292\pi$
$197$	−716052.	−1.31456	−0.657278	+	0.753648i	$0.728292\pi$
$198$	244323.	0.442896
$199$	−636722.	−1.13977	−0.569885	−	0.821724i	$-0.693012\pi$
$199$	−636722.	−1.13977	−0.569885	+	0.821724i	$0.693012\pi$
$200$	1.24199e6	2.19554
$201$	101706.	0.177565
$202$	−645157.	−1.11247
$203$	555766.	0.946568
$204$	−151949.	−0.255636
$205$	1.34429e6	2.23413
$206$	−65266.7	−0.107158
$207$	42849.0	0.0695048
$208$	37888.2	0.0607219
$209$	−1.25880e6	−1.99339
$210$	879194.	1.37574
$211$	694318.	1.07362	0.536812	−	0.843702i	$-0.319628\pi$
$211$	694318.	1.07362	0.536812	+	0.843702i	$0.319628\pi$
$212$	182134.	0.278325
$213$	−299369.	−0.452124
$214$	−920432.	−1.37391
$215$	123611.	0.182373
$216$	−143457.	−0.209212
$217$	1.03853e6	1.49716
$218$	554430.	0.790143
$219$	−203147.	−0.286221
$220$	−574323.	−0.800017
$221$	107390.	0.147906
$222$	−208727.	−0.284247
$223$	−555721.	−0.748333	−0.374166	−	0.927362i	$-0.622071\pi$
$223$	−555721.	−0.748333	−0.374166	+	0.927362i	$0.622071\pi$
$224$	688198.	0.916418
$225$	511221.	0.673212
$226$	855545.	1.11422
$227$	−513115.	−0.660922	−0.330461	−	0.943820i	$-0.607204\pi$
$227$	−513115.	−0.660922	−0.330461	+	0.943820i	$0.607204\pi$
$228$	166881.	0.212603
$229$	−1.29876e6	−1.63659	−0.818296	−	0.574798i	$-0.805081\pi$
$229$	−1.29876e6	−1.63659	−0.818296	+	0.574798i	$0.805081\pi$
$230$	244660.	0.304961
$231$	−1.20434e6	−1.48498
$232$	−517786.	−0.631583
$233$	−522889.	−0.630986	−0.315493	−	0.948928i	$-0.602170\pi$
$233$	−522889.	−0.630986	−0.315493	+	0.948928i	$0.602170\pi$
$234$	22891.8	0.0273301
$235$	1.74097e6	2.05646
$236$	163956.	0.191623
$237$	89118.5	0.103062
$238$	−1.81935e6	−2.08196
$239$	1.02363e6	1.15917	0.579587	−	0.814910i	$-0.303214\pi$
$239$	1.02363e6	1.15917	0.579587	+	0.814910i	$0.303214\pi$
$240$	−558032.	−0.625361
$241$	−1.66479e6	−1.84636	−0.923180	−	0.384367i	$-0.874420\pi$
$241$	−1.66479e6	−1.84636	−0.923180	+	0.384367i	$0.874420\pi$
$242$	−1.14419e6	−1.25591
$243$	−59049.0	−0.0641500
$244$	190877.	0.205248
$245$	−2.70117e6	−2.87499
$246$	−592978.	−0.624742
$247$	−117943.	−0.123007
$248$	−967557.	−0.998958
$249$	217963.	0.222784
$250$	1.47368e6	1.49126
$251$	1.05735e6	1.05934	0.529671	−	0.848203i	$-0.322315\pi$
$251$	1.05735e6	1.05934	0.529671	+	0.848203i	$0.322315\pi$
$252$	159661.	0.158379
$253$	−335142.	−0.329176
$254$	1.17245e6	1.14028
$255$	−1.58169e6	−1.52325
$256$	−831778.	−0.793246
$257$	−1.06720e6	−1.00789	−0.503943	−	0.863737i	$-0.668118\pi$
$257$	−1.06720e6	−1.00789	−0.503943	+	0.863737i	$0.668118\pi$
$258$	−54525.9	−0.0509980
$259$	1.02888e6	0.953048
$260$	−53811.1	−0.0493672
$261$	−213129.	−0.193661
$262$	−323773.	−0.291399
$263$	−1.72834e6	−1.54077	−0.770387	−	0.637577i	$-0.779937\pi$
$263$	−1.72834e6	−1.54077	−0.770387	+	0.637577i	$0.779937\pi$
$264$	1.12204e6	0.990831
$265$	1.89590e6	1.65844
$266$	1.99813e6	1.73149
$267$	−364175.	−0.312631
$268$	105459.	0.0896909
$269$	447676.	0.377210	0.188605	−	0.982053i	$-0.439603\pi$
$269$	447676.	0.377210	0.188605	+	0.982053i	$0.439603\pi$
$270$	−337160.	−0.281466
$271$	−1.72413e6	−1.42609	−0.713047	−	0.701117i	$-0.752685\pi$
$271$	−1.72413e6	−1.42609	−0.713047	+	0.701117i	$0.752685\pi$
$272$	1.15475e6	0.946384
$273$	−112841.	−0.0916346
$274$	1.80183e6	1.44990
$275$	−3.99850e6	−3.18835
$276$	44430.2	0.0351079
$277$	−114905.	−0.0899784	−0.0449892	−	0.998987i	$-0.514325\pi$
$277$	−114905.	−0.0899784	−0.0449892	+	0.998987i	$0.514325\pi$
$278$	533840.	0.414285
$279$	−398262.	−0.306308
$280$	4.03766e6	3.07776
$281$	−2.02575e6	−1.53046	−0.765228	−	0.643760i	$-0.777374\pi$
$281$	−2.02575e6	−1.53046	−0.765228	+	0.643760i	$0.777374\pi$
$282$	−767957.	−0.575061
$283$	−1.29230e6	−0.959177	−0.479589	−	0.877493i	$-0.659214\pi$
$283$	−1.29230e6	−0.959177	−0.479589	+	0.877493i	$0.659214\pi$
$284$	−310416.	−0.228375
$285$	1.73712e6	1.26683
$286$	−179048.	−0.129436
$287$	2.92297e6	2.09469
$288$	−263915.	−0.187492
$289$	1.85318e6	1.30519
$290$	−1.21693e6	−0.849710
$291$	164947.	0.114186
$292$	−210644.	−0.144574
$293$	−838415.	−0.570545	−0.285272	−	0.958446i	$-0.592084\pi$
$293$	−838415.	−0.570545	−0.285272	+	0.958446i	$0.592084\pi$
$294$	1.19151e6	0.803951
$295$	1.70667e6	1.14181
$296$	−958569.	−0.635908
$297$	461850.	0.303816
$298$	−1.38217e6	−0.901616
$299$	−31401.1	−0.0203127
$300$	530085.	0.340050
$301$	268775.	0.170991
$302$	−597748.	−0.377139
$303$	−1.21956e6	−0.763125
$304$	−1.26823e6	−0.787071
$305$	1.98690e6	1.22300
$306$	697696.	0.425954
$307$	1.15370e6	0.698632	0.349316	−	0.937005i	$-0.386414\pi$
$307$	1.15370e6	0.698632	0.349316	+	0.937005i	$0.386414\pi$
$308$	−1.24878e6	−0.750086
$309$	−123375.	−0.0735076
$310$	−2.27401e6	−1.34396
$311$	1.46590e6	0.859414	0.429707	−	0.902968i	$-0.358617\pi$
$311$	1.46590e6	0.859414	0.429707	+	0.902968i	$0.358617\pi$
$312$	105130.	0.0611419
$313$	2.94425e6	1.69869	0.849343	−	0.527841i	$-0.176998\pi$
$313$	2.94425e6	1.69869	0.849343	+	0.527841i	$0.176998\pi$
$314$	785924.	0.449838
$315$	1.66196e6	0.943724
$316$	92407.1	0.0520580
$317$	−561175.	−0.313653	−0.156827	−	0.987626i	$-0.550126\pi$
$317$	−561175.	−0.313653	−0.156827	+	0.987626i	$0.550126\pi$
$318$	−836297.	−0.463759
$319$	1.66698e6	0.917180
$320$	−3.49102e6	−1.90580
$321$	−1.73992e6	−0.942466
$322$	531980.	0.285927
$323$	−3.59467e6	−1.91714
$324$	−61228.0	−0.0324032
$325$	−374639.	−0.196745
$326$	−1.57240e6	−0.819445
$327$	1.04805e6	0.542019
$328$	−2.72323e6	−1.39765
$329$	3.78550e6	1.92812
$330$	2.63709e6	1.33303
$331$	3.55649e6	1.78423	0.892117	−	0.451805i	$-0.149220\pi$
$331$	3.55649e6	1.78423	0.892117	+	0.451805i	$0.149220\pi$
$332$	226006.	0.112532
$333$	−394562.	−0.194986
$334$	100561.	0.0493246
$335$	1.09776e6	0.534436
$336$	−1.21336e6	−0.586331
$337$	−1.16442e6	−0.558516	−0.279258	−	0.960216i	$-0.590088\pi$
$337$	−1.16442e6	−0.558516	−0.279258	+	0.960216i	$0.590088\pi$
$338$	1.75098e6	0.833661
$339$	1.61726e6	0.764330
$340$	−1.64005e6	−0.769415
$341$	3.11499e6	1.45068
$342$	−766257.	−0.354249
$343$	−2.32335e6	−1.06630
$344$	−250408.	−0.114091
$345$	462488.	0.209196
$346$	2.47202e6	1.11010
$347$	2.09076e6	0.932139	0.466069	−	0.884748i	$-0.345670\pi$
$347$	2.09076e6	0.932139	0.466069	+	0.884748i	$0.345670\pi$
$348$	−220994.	−0.0978209
$349$	2.28656e6	1.00489	0.502445	−	0.864609i	$-0.332434\pi$
$349$	2.28656e6	1.00489	0.502445	+	0.864609i	$0.332434\pi$
$350$	6.34692e6	2.76945
$351$	43273.0	0.0187478
$352$	2.06421e6	0.887966
$353$	1.42523e6	0.608764	0.304382	−	0.952550i	$-0.401550\pi$
$353$	1.42523e6	0.608764	0.304382	+	0.952550i	$0.401550\pi$
$354$	−752826.	−0.319291
$355$	−3.23122e6	−1.36080
$356$	−377613.	−0.157915
$357$	−3.43916e6	−1.42818
$358$	−2.00127e6	−0.825273
$359$	−1.73659e6	−0.711152	−0.355576	−	0.934647i	$-0.615715\pi$
$359$	−1.73659e6	−0.711152	−0.355576	+	0.934647i	$0.615715\pi$
$360$	−1.54839e6	−0.629686
$361$	1.47182e6	0.594409
$362$	−377727.	−0.151498
$363$	−2.16289e6	−0.861526
$364$	−117005.	−0.0462860
$365$	−2.19266e6	−0.861467
$366$	−876441.	−0.341995
$367$	1.37269e6	0.531994	0.265997	−	0.963974i	$-0.414299\pi$
$367$	1.37269e6	0.531994	0.265997	+	0.963974i	$0.414299\pi$
$368$	−337652.	−0.129972
$369$	−1.12092e6	−0.428558
$370$	−2.25288e6	−0.855527
$371$	4.12237e6	1.55493
$372$	−412958.	−0.154721
$373$	−539618.	−0.200824	−0.100412	−	0.994946i	$-0.532016\pi$
$373$	−539618.	−0.200824	−0.100412	+	0.994946i	$0.532016\pi$
$374$	−5.45701e6	−2.01732
$375$	2.78574e6	1.02297
$376$	−3.52681e6	−1.28651
$377$	156188.	0.0565970
$378$	−733107.	−0.263899
$379$	2.91919e6	1.04391	0.521956	−	0.852972i	$-0.325203\pi$
$379$	2.91919e6	1.04391	0.521956	+	0.852972i	$0.325203\pi$
$380$	1.80122e6	0.639893
$381$	2.21632e6	0.782204
$382$	−1.95001e6	−0.683719
$383$	−2.78248e6	−0.969247	−0.484623	−	0.874723i	$-0.661043\pi$
$383$	−2.78248e6	−0.969247	−0.484623	+	0.874723i	$0.661043\pi$
$384$	601556.	0.208184
$385$	−1.29990e7	−4.46950
$386$	−2.02266e6	−0.690962
$387$	−103072.	−0.0349834
$388$	171034.	0.0576770
$389$	1.77472e6	0.594642	0.297321	−	0.954778i	$-0.403907\pi$
$389$	1.77472e6	0.594642	0.297321	+	0.954778i	$0.403907\pi$
$390$	247081.	0.0822581
$391$	−957042.	−0.316584
$392$	5.47196e6	1.79857
$393$	−612037.	−0.199893
$394$	3.40918e6	1.10639
$395$	961896.	0.310195
$396$	478893.	0.153462
$397$	−470654.	−0.149874	−0.0749369	−	0.997188i	$-0.523876\pi$
$397$	−470654.	−0.149874	−0.0749369	+	0.997188i	$0.523876\pi$
$398$	3.03149e6	0.959285
$399$	3.77712e6	1.18776
$400$	−4.02845e6	−1.25889
$401$	2.32453e6	0.721894	0.360947	−	0.932586i	$-0.382453\pi$
$401$	2.32453e6	0.721894	0.360947	+	0.932586i	$0.382453\pi$
$402$	−484232.	−0.149447
$403$	291859.	0.0895180
$404$	−1.26456e6	−0.385466
$405$	−637342.	−0.193079
$406$	−2.64605e6	−0.796677
$407$	3.08606e6	0.923459
$408$	3.20414e6	0.952929
$409$	−3.54081e6	−1.04663	−0.523317	−	0.852138i	$-0.675305\pi$
$409$	−3.54081e6	−1.04663	−0.523317	+	0.852138i	$0.675305\pi$
$410$	−6.40027e6	−1.88035
$411$	3.40606e6	0.994597
$412$	−127928.	−0.0371298
$413$	3.71091e6	1.07055
$414$	−204008.	−0.0584986
$415$	2.35257e6	0.670536
$416$	193406.	0.0547943
$417$	1.00913e6	0.284190
$418$	5.99326e6	1.67773
$419$	1.84820e6	0.514297	0.257149	−	0.966372i	$-0.417217\pi$
$419$	1.84820e6	0.514297	0.257149	+	0.966372i	$0.417217\pi$
$420$	1.72329e6	0.476690
$421$	1.97062e6	0.541874	0.270937	−	0.962597i	$-0.412667\pi$
$421$	1.97062e6	0.541874	0.270937	+	0.962597i	$0.412667\pi$
$422$	−3.30571e6	−0.903614
$423$	−1.45169e6	−0.394478
$424$	−3.84066e6	−1.03751
$425$	−1.14182e7	−3.06638
$426$	1.42532e6	0.380529
$427$	4.32025e6	1.14667
$428$	−1.80412e6	−0.476054
$429$	−338459.	−0.0887897
$430$	−588522.	−0.153494
$431$	−3.45702e6	−0.896415	−0.448208	−	0.893929i	$-0.647937\pi$
$431$	−3.45702e6	−0.896415	−0.448208	+	0.893929i	$0.647937\pi$
$432$	465309.	0.119959
$433$	−4.90614e6	−1.25753	−0.628767	−	0.777594i	$-0.716440\pi$
$433$	−4.90614e6	−1.25753	−0.628767	+	0.777594i	$0.716440\pi$
$434$	−4.94451e6	−1.26008
$435$	−2.30040e6	−0.582880
$436$	1.08673e6	0.273782
$437$	1.05109e6	0.263291
$438$	967201.	0.240897
$439$	400905.	0.0992842	0.0496421	−	0.998767i	$-0.484192\pi$
$439$	400905.	0.0992842	0.0496421	+	0.998767i	$0.484192\pi$
$440$	1.21107e7	2.98221
$441$	2.25234e6	0.551491
$442$	−511294.	−0.124484
$443$	−5.31050e6	−1.28566	−0.642829	−	0.766009i	$-0.722240\pi$
$443$	−5.31050e6	−1.28566	−0.642829	+	0.766009i	$0.722240\pi$
$444$	−409122.	−0.0984906
$445$	−3.93070e6	−0.940958
$446$	2.64583e6	0.629833
$447$	−2.61276e6	−0.618486
$448$	−7.59075e6	−1.78686
$449$	−501557.	−0.117410	−0.0587049	−	0.998275i	$-0.518697\pi$
$449$	−501557.	−0.117410	−0.0587049	+	0.998275i	$0.518697\pi$
$450$	−2.43396e6	−0.566608
$451$	8.76727e6	2.02966
$452$	1.67694e6	0.386075
$453$	−1.12994e6	−0.258708
$454$	2.44298e6	0.556264
$455$	−1.21794e6	−0.275802
$456$	−3.51900e6	−0.792515
$457$	−5.15796e6	−1.15528	−0.577641	−	0.816291i	$-0.696027\pi$
$457$	−5.15796e6	−1.15528	−0.577641	+	0.816291i	$0.696027\pi$
$458$	6.18350e6	1.37743
$459$	1.31887e6	0.292194
$460$	479554.	0.105668
$461$	−3.35669e6	−0.735628	−0.367814	−	0.929899i	$-0.619894\pi$
$461$	−3.35669e6	−0.735628	−0.367814	+	0.929899i	$0.619894\pi$
$462$	5.73398e6	1.24983
$463$	1.83888e6	0.398658	0.199329	−	0.979933i	$-0.436124\pi$
$463$	1.83888e6	0.398658	0.199329	+	0.979933i	$0.436124\pi$
$464$	1.67947e6	0.362140
$465$	−4.29861e6	−0.921926
$466$	2.48952e6	0.531068
$467$	4.47719e6	0.949977	0.474988	−	0.879992i	$-0.342452\pi$
$467$	4.47719e6	0.949977	0.474988	+	0.879992i	$0.342452\pi$
$468$	44869.8	0.00946977
$469$	2.38693e6	0.501081
$470$	−8.28889e6	−1.73082
$471$	1.48565e6	0.308578
$472$	−3.45732e6	−0.714307
$473$	806173.	0.165682
$474$	−424301.	−0.0867417
$475$	1.25403e7	2.55019
$476$	−3.56607e6	−0.721393
$477$	−1.58087e6	−0.318128
$478$	−4.87359e6	−0.975617
$479$	−1.13426e6	−0.225878	−0.112939	−	0.993602i	$-0.536026\pi$
$479$	−1.13426e6	−0.225878	−0.112939	+	0.993602i	$0.536026\pi$
$480$	−2.84855e6	−0.564315
$481$	289148.	0.0569845
$482$	7.92620e6	1.55399
$483$	1.00562e6	0.196139
$484$	−2.24270e6	−0.435170
$485$	1.78035e6	0.343677
$486$	281137.	0.0539918
$487$	−4.84753e6	−0.926185	−0.463093	−	0.886310i	$-0.653260\pi$
$487$	−4.84753e6	−0.926185	−0.463093	+	0.886310i	$0.653260\pi$
$488$	−4.02502e6	−0.765100
$489$	−2.97236e6	−0.562120
$490$	1.28605e7	2.41973
$491$	5.80829e6	1.08729	0.543644	−	0.839316i	$-0.317044\pi$
$491$	5.80829e6	1.08729	0.543644	+	0.839316i	$0.317044\pi$
$492$	−1.16229e6	−0.216471
$493$	4.76029e6	0.882096
$494$	561538.	0.103529
$495$	4.98495e6	0.914425
$496$	3.13832e6	0.572787
$497$	−7.02584e6	−1.27587
$498$	−1.03774e6	−0.187506
$499$	−3.67933e6	−0.661481	−0.330741	−	0.943722i	$-0.607299\pi$
$499$	−3.67933e6	−0.661481	−0.330741	+	0.943722i	$0.607299\pi$
$500$	2.88854e6	0.516717
$501$	190093.	0.0338355
$502$	−5.03415e6	−0.891593
$503$	8.28397e6	1.45988	0.729942	−	0.683509i	$-0.239547\pi$
$503$	8.28397e6	1.45988	0.729942	+	0.683509i	$0.239547\pi$
$504$	−3.36676e6	−0.590386
$505$	−1.31632e7	−2.29686
$506$	1.59564e6	0.277050
$507$	3.30993e6	0.571871
$508$	2.29810e6	0.395103
$509$	−1.50326e6	−0.257182	−0.128591	−	0.991698i	$-0.541045\pi$
$509$	−1.50326e6	−0.257182	−0.128591	+	0.991698i	$0.541045\pi$
$510$	7.53054e6	1.28204
$511$	−4.76763e6	−0.807701
$512$	6.09903e6	1.02822
$513$	−1.44848e6	−0.243007
$514$	5.08100e6	0.848285
$515$	−1.33165e6	−0.221243
$516$	−106875.	−0.0176706
$517$	1.13544e7	1.86825
$518$	−4.89858e6	−0.802131
$519$	4.67293e6	0.761502
$520$	1.13471e6	0.184025
$521$	−7.04086e6	−1.13640	−0.568200	−	0.822891i	$-0.692360\pi$
$521$	−7.04086e6	−1.13640	−0.568200	+	0.822891i	$0.692360\pi$
$522$	1.01472e6	0.162994
$523$	−1.68119e6	−0.268759	−0.134380	−	0.990930i	$-0.542904\pi$
$523$	−1.68119e6	−0.268759	−0.134380	+	0.990930i	$0.542904\pi$
$524$	−634622.	−0.100969
$525$	1.19978e7	1.89977
$526$	8.22875e6	1.29679
$527$	8.89527e6	1.39519
$528$	−3.63941e6	−0.568127
$529$	279841.	0.0434783
$530$	−9.02652e6	−1.39582
$531$	−1.42309e6	−0.219026
$532$	3.91650e6	0.599955
$533$	821448.	0.125245
$534$	1.73387e6	0.263125
$535$	−1.87797e7	−2.83664
$536$	−2.22382e6	−0.334339
$537$	−3.78305e6	−0.566117
$538$	−2.13142e6	−0.317478
$539$	−1.76166e7	−2.61187
$540$	−660860.	−0.0975271
$541$	−2.09033e6	−0.307059	−0.153529	−	0.988144i	$-0.549064\pi$
$541$	−2.09033e6	−0.307059	−0.153529	+	0.988144i	$0.549064\pi$
$542$	8.20874e6	1.20027
$543$	−714028.	−0.103924
$544$	5.89461e6	0.854000
$545$	1.13121e7	1.63137
$546$	537244.	0.0771241
$547$	−9.24304e6	−1.32083	−0.660414	−	0.750901i	$-0.729619\pi$
$547$	−9.24304e6	−1.32083	−0.660414	+	0.750901i	$0.729619\pi$
$548$	3.53174e6	0.502386
$549$	−1.65676e6	−0.234600
$550$	1.90372e7	2.68347
$551$	−5.22807e6	−0.733605
$552$	−936896.	−0.130871
$553$	2.09151e6	0.290835
$554$	547071.	0.0757302
$555$	−4.25868e6	−0.586871
$556$	1.04637e6	0.143548
$557$	−1.67077e6	−0.228181	−0.114091	−	0.993470i	$-0.536395\pi$
$557$	−1.67077e6	−0.228181	−0.114091	+	0.993470i	$0.536395\pi$
$558$	1.89616e6	0.257803
$559$	75534.3	0.0102238
$560$	−1.30964e7	−1.76474
$561$	−1.03155e7	−1.38384
$562$	9.64477e6	1.28810
$563$	−1.12609e7	−1.49728	−0.748639	−	0.662978i	$-0.769292\pi$
$563$	−1.12609e7	−1.49728	−0.748639	+	0.662978i	$0.769292\pi$
$564$	−1.50526e6	−0.199257
$565$	1.74558e7	2.30048
$566$	6.15277e6	0.807290
$567$	−1.38581e6	−0.181028
$568$	6.54572e6	0.851307
$569$	1.05116e7	1.36110	0.680549	−	0.732703i	$-0.261741\pi$
$569$	1.05116e7	1.36110	0.680549	+	0.732703i	$0.261741\pi$
$570$	−8.27055e6	−1.06622
$571$	1.48085e7	1.90073	0.950363	−	0.311142i	$-0.100711\pi$
$571$	1.48085e7	1.90073	0.950363	+	0.311142i	$0.100711\pi$
$572$	−350948.	−0.0448490
$573$	−3.68615e6	−0.469015
$574$	−1.39165e7	−1.76299
$575$	3.33871e6	0.421124
$576$	2.91095e6	0.365577
$577$	−3.06552e6	−0.383323	−0.191661	−	0.981461i	$-0.561387\pi$
$577$	−3.06552e6	−0.383323	−0.191661	+	0.981461i	$0.561387\pi$
$578$	−8.82315e6	−1.09851
$579$	−3.82349e6	−0.473983
$580$	−2.38528e6	−0.294422
$581$	5.11533e6	0.628686
$582$	−785326.	−0.0961042
$583$	1.23648e7	1.50666
$584$	4.44183e6	0.538927
$585$	467065.	0.0564270
$586$	3.99176e6	0.480198
$587$	4.49808e6	0.538805	0.269403	−	0.963028i	$-0.413174\pi$
$587$	4.49808e6	0.538805	0.269403	+	0.963028i	$0.413174\pi$
$588$	2.33546e6	0.278566
$589$	−9.76939e6	−1.16032
$590$	−8.12559e6	−0.961003
$591$	6.44447e6	0.758959
$592$	3.10917e6	0.364620
$593$	2.42205e6	0.282844	0.141422	−	0.989949i	$-0.454833\pi$
$593$	2.42205e6	0.282844	0.141422	+	0.989949i	$0.454833\pi$
$594$	−2.19891e6	−0.255706
$595$	−3.71204e7	−4.29853
$596$	−2.70917e6	−0.312407
$597$	5.73050e6	0.658046
$598$	149503.	0.0170961
$599$	2.86365e6	0.326102	0.163051	−	0.986618i	$-0.447866\pi$
$599$	2.86365e6	0.326102	0.163051	+	0.986618i	$0.447866\pi$
$600$	−1.11779e7	−1.26760
$601$	3.17881e6	0.358987	0.179493	−	0.983759i	$-0.442554\pi$
$601$	3.17881e6	0.358987	0.179493	+	0.983759i	$0.442554\pi$
$602$	−1.27966e6	−0.143914
$603$	−915357.	−0.102517
$604$	−1.17164e6	−0.130677
$605$	−2.33450e7	−2.59302
$606$	5.80641e6	0.642283
$607$	−1.62063e7	−1.78531	−0.892653	−	0.450745i	$-0.851158\pi$
$607$	−1.62063e7	−1.78531	−0.892653	+	0.450745i	$0.851158\pi$
$608$	−6.47386e6	−0.710239
$609$	−5.00189e6	−0.546501
$610$	−9.45981e6	−1.02934
$611$	1.06384e6	0.115286
$612$	1.36754e6	0.147592
$613$	5.85294e6	0.629105	0.314552	−	0.949240i	$-0.398146\pi$
$613$	5.85294e6	0.629105	0.314552	+	0.949240i	$0.398146\pi$
$614$	−5.49288e6	−0.588002
$615$	−1.20986e7	−1.28988
$616$	2.63330e7	2.79608
$617$	6.26326e6	0.662350	0.331175	−	0.943569i	$-0.392555\pi$
$617$	6.26326e6	0.662350	0.331175	+	0.943569i	$0.392555\pi$
$618$	587400.	0.0618676
$619$	−4.36108e6	−0.457475	−0.228737	−	0.973488i	$-0.573460\pi$
$619$	−4.36108e6	−0.457475	−0.228737	+	0.973488i	$0.573460\pi$
$620$	−4.45724e6	−0.465679
$621$	−385641.	−0.0401286
$622$	−6.97925e6	−0.723324
$623$	−8.54677e6	−0.882230
$624$	−340994.	−0.0350578
$625$	1.03447e7	1.05930
$626$	−1.40178e7	−1.42970
$627$	1.13292e7	1.15088
$628$	1.54048e6	0.155868
$629$	8.81263e6	0.888135
$630$	−7.91274e6	−0.794284
$631$	−1.48998e7	−1.48973	−0.744866	−	0.667214i	$-0.767487\pi$
$631$	−1.48998e7	−1.48973	−0.744866	+	0.667214i	$0.767487\pi$
$632$	−1.94858e6	−0.194056
$633$	−6.24887e6	−0.619858
$634$	2.67180e6	0.263986
$635$	2.39217e7	2.35428
$636$	−1.63921e6	−0.160691
$637$	−1.65059e6	−0.161172
$638$	−7.93664e6	−0.771943
$639$	2.69432e6	0.261034
$640$	6.49286e6	0.626594
$641$	826180.	0.0794199	0.0397100	−	0.999211i	$-0.487357\pi$
$641$	826180.	0.0794199	0.0397100	+	0.999211i	$0.487357\pi$
$642$	8.28388e6	0.793225
$643$	−1.37326e7	−1.30986	−0.654929	−	0.755691i	$-0.727301\pi$
$643$	−1.37326e7	−1.30986	−0.654929	+	0.755691i	$0.727301\pi$
$644$	1.04272e6	0.0990729
$645$	−1.11250e6	−0.105293
$646$	1.71145e7	1.61355
$647$	2.70016e6	0.253588	0.126794	−	0.991929i	$-0.459531\pi$
$647$	2.70016e6	0.253588	0.126794	+	0.991929i	$0.459531\pi$
$648$	1.29111e6	0.120789
$649$	1.11306e7	1.03731
$650$	1.78369e6	0.165590
$651$	−9.34674e6	−0.864386
$652$	−3.08204e6	−0.283935
$653$	−1.01591e7	−0.932339	−0.466169	−	0.884696i	$-0.654366\pi$
$653$	−1.01591e7	−0.932339	−0.466169	+	0.884696i	$0.654366\pi$
$654$	−4.98987e6	−0.456189
$655$	−6.60599e6	−0.601637
$656$	8.83293e6	0.801392
$657$	1.82833e6	0.165250
$658$	−1.80230e7	−1.62279
$659$	1.53744e7	1.37906	0.689531	−	0.724256i	$-0.257817\pi$
$659$	1.53744e7	1.37906	0.689531	+	0.724256i	$0.257817\pi$
$660$	5.16890e6	0.461890
$661$	−1.47641e7	−1.31433	−0.657163	−	0.753748i	$-0.728244\pi$
$661$	−1.47641e7	−1.31433	−0.657163	+	0.753748i	$0.728244\pi$
$662$	−1.69327e7	−1.50170
$663$	−966513.	−0.0853933
$664$	−4.76577e6	−0.419481
$665$	4.07681e7	3.57492
$666$	1.87854e6	0.164110
$667$	−1.39192e6	−0.121143
$668$	197108.	0.0170908
$669$	5.00149e6	0.432050
$670$	−5.22653e6	−0.449807
$671$	1.29583e7	1.11107
$672$	−6.19379e6	−0.529094
$673$	1.79584e7	1.52837	0.764187	−	0.644995i	$-0.223141\pi$
$673$	1.79584e7	1.52837	0.764187	+	0.644995i	$0.223141\pi$
$674$	5.54391e6	0.470074
$675$	−4.60099e6	−0.388679
$676$	3.43206e6	0.288861
$677$	7.66782e6	0.642984	0.321492	−	0.946912i	$-0.395816\pi$
$677$	7.66782e6	0.642984	0.321492	+	0.946912i	$0.395816\pi$
$678$	−7.69991e6	−0.643297
$679$	3.87112e6	0.322227
$680$	3.45837e7	2.86813
$681$	4.61804e6	0.381584
$682$	−1.48307e7	−1.22096
$683$	−3.48360e6	−0.285744	−0.142872	−	0.989741i	$-0.545634\pi$
$683$	−3.48360e6	−0.285744	−0.142872	+	0.989741i	$0.545634\pi$
$684$	−1.50193e6	−0.122746
$685$	3.67631e7	2.99354
$686$	1.10617e7	0.897451
$687$	1.16888e7	0.944886
$688$	812211.	0.0654180
$689$	1.15852e6	0.0929724
$690$	−2.20194e6	−0.176069
$691$	−1.24991e7	−0.995829	−0.497914	−	0.867226i	$-0.665901\pi$
$691$	−1.24991e7	−0.995829	−0.497914	+	0.867226i	$0.665901\pi$
$692$	4.84537e6	0.384646
$693$	1.08391e7	0.857353
$694$	−9.95428e6	−0.784533
$695$	1.08920e7	0.855355
$696$	4.66008e6	0.364645
$697$	2.50361e7	1.95202
$698$	−1.08865e7	−0.845765
$699$	4.70600e6	0.364300
$700$	1.24405e7	0.959604
$701$	−2.20409e7	−1.69408	−0.847042	−	0.531525i	$-0.821619\pi$
$701$	−2.20409e7	−1.69408	−0.847042	+	0.531525i	$0.821619\pi$
$702$	−206026.	−0.0157790
$703$	−9.67863e6	−0.738628
$704$	−2.27680e7	−1.73138
$705$	−1.56687e7	−1.18730
$706$	−6.78564e6	−0.512365
$707$	−2.86216e7	−2.15350
$708$	−1.47560e6	−0.110633
$709$	4.99549e6	0.373218	0.186609	−	0.982434i	$-0.440250\pi$
$709$	4.99549e6	0.373218	0.186609	+	0.982434i	$0.440250\pi$
$710$	1.53841e7	1.14532
$711$	−802067.	−0.0595027
$712$	7.96271e6	0.588655
$713$	−2.60099e6	−0.191609
$714$	1.63741e7	1.20202
$715$	−3.65313e6	−0.267239
$716$	−3.92265e6	−0.285954
$717$	−9.21268e6	−0.669250
$718$	8.26806e6	0.598539
$719$	2.06128e6	0.148701	0.0743507	−	0.997232i	$-0.476312\pi$
$719$	2.06128e6	0.148701	0.0743507	+	0.997232i	$0.476312\pi$
$720$	5.02229e6	0.361052
$721$	−2.89548e6	−0.207435
$722$	−7.00744e6	−0.500284
$723$	1.49831e7	1.06600
$724$	−740377.	−0.0524936
$725$	−1.66066e7	−1.17337
$726$	1.02977e7	0.725102
$727$	1.74867e7	1.22708	0.613539	−	0.789665i	$-0.289746\pi$
$727$	1.74867e7	1.22708	0.613539	+	0.789665i	$0.289746\pi$
$728$	2.46727e6	0.172539
$729$	531441.	0.0370370
$730$	1.04394e7	0.725052
$731$	2.30213e6	0.159344
$732$	−1.71790e6	−0.118500
$733$	8.68392e6	0.596975	0.298487	−	0.954414i	$-0.403518\pi$
$733$	8.68392e6	0.596975	0.298487	+	0.954414i	$0.403518\pi$
$734$	−6.53549e6	−0.447752
$735$	2.43105e7	1.65988
$736$	−1.72359e6	−0.117285
$737$	7.15944e6	0.485524
$738$	5.33680e6	0.360695
$739$	1.15780e7	0.779871	0.389936	−	0.920842i	$-0.372497\pi$
$739$	1.15780e7	0.779871	0.389936	+	0.920842i	$0.372497\pi$
$740$	−4.41583e6	−0.296437
$741$	1.06149e6	0.0710183
$742$	−1.96269e7	−1.30871
$743$	−6.91849e6	−0.459768	−0.229884	−	0.973218i	$-0.573835\pi$
$743$	−6.91849e6	−0.459768	−0.229884	+	0.973218i	$0.573835\pi$
$744$	8.70801e6	0.576749
$745$	−2.82006e7	−1.86152
$746$	2.56917e6	0.169023
$747$	−1.96166e6	−0.128624
$748$	−1.06962e7	−0.698997
$749$	−4.08338e7	−2.65960
$750$	−1.32631e7	−0.860980
$751$	3.10339e6	0.200787	0.100394	−	0.994948i	$-0.467990\pi$
$751$	3.10339e6	0.200787	0.100394	+	0.994948i	$0.467990\pi$
$752$	1.14394e7	0.737663
$753$	−9.51618e6	−0.611611
$754$	−743623.	−0.0476348
$755$	−1.21959e7	−0.778660
$756$	−1.43695e6	−0.0914402
$757$	−9.03023e6	−0.572742	−0.286371	−	0.958119i	$-0.592449\pi$
$757$	−9.03023e6	−0.572742	−0.286371	+	0.958119i	$0.592449\pi$
$758$	−1.38985e7	−0.878606
$759$	3.01628e6	0.190050
$760$	−3.79821e7	−2.38531
$761$	9.70856e6	0.607706	0.303853	−	0.952719i	$-0.401727\pi$
$761$	9.70856e6	0.607706	0.303853	+	0.952719i	$0.401727\pi$
$762$	−1.05521e7	−0.658340
$763$	2.45966e7	1.52955
$764$	−3.82217e6	−0.236906
$765$	1.42352e7	0.879446
$766$	1.32476e7	0.815765
$767$	1.04288e6	0.0640100
$768$	7.48601e6	0.457981
$769$	−1.31499e7	−0.801874	−0.400937	−	0.916106i	$-0.631315\pi$
$769$	−1.31499e7	−0.801874	−0.400937	+	0.916106i	$0.631315\pi$
$770$	6.18893e7	3.76174
$771$	9.60476e6	0.581903
$772$	−3.96458e6	−0.239416
$773$	−1.58606e7	−0.954706	−0.477353	−	0.878712i	$-0.658404\pi$
$773$	−1.58606e7	−0.954706	−0.477353	+	0.878712i	$0.658404\pi$
$774$	490733.	0.0294437
$775$	−3.10318e7	−1.85589
$776$	−3.60658e6	−0.215001
$777$	−9.25991e6	−0.550243
$778$	−8.44958e6	−0.500479
$779$	−2.74963e7	−1.62342
$780$	484300.	0.0285022
$781$	−2.10736e7	−1.23626
$782$	4.55656e6	0.266453
$783$	1.91816e6	0.111810
$784$	−1.77486e7	−1.03127
$785$	1.60353e7	0.928760
$786$	2.91396e6	0.168239
$787$	3.25773e7	1.87490	0.937449	−	0.348122i	$-0.113180\pi$
$787$	3.25773e7	1.87490	0.937449	+	0.348122i	$0.113180\pi$
$788$	6.68228e6	0.383362
$789$	1.55550e7	0.889566
$790$	−4.57966e6	−0.261075
$791$	3.79552e7	2.15690
$792$	−1.00984e7	−0.572056
$793$	1.21413e6	0.0685616
$794$	2.24082e6	0.126141
$795$	−1.70631e7	−0.957502
$796$	5.94196e6	0.332389
$797$	2.89765e7	1.61585	0.807923	−	0.589288i	$-0.200592\pi$
$797$	2.89765e7	1.61585	0.807923	+	0.589288i	$0.200592\pi$
$798$	−1.79832e7	−0.999675
$799$	3.24238e7	1.79679
$800$	−2.05638e7	−1.13600
$801$	3.27758e6	0.180498
$802$	−1.10673e7	−0.607581
$803$	−1.43002e7	−0.782625
$804$	−949135.	−0.0517831
$805$	1.08541e7	0.590340
$806$	−1.38956e6	−0.0753427
$807$	−4.02909e6	−0.217782
$808$	2.66657e7	1.43689
$809$	−1.73592e7	−0.932521	−0.466261	−	0.884647i	$-0.654399\pi$
$809$	−1.73592e7	−0.932521	−0.466261	+	0.884647i	$0.654399\pi$
$810$	3.03444e6	0.162505
$811$	−2.90389e7	−1.55034	−0.775172	−	0.631750i	$-0.782337\pi$
$811$	−2.90389e7	−1.55034	−0.775172	+	0.631750i	$0.782337\pi$
$812$	−5.18647e6	−0.276046
$813$	1.55172e7	0.823355
$814$	−1.46930e7	−0.777228
$815$	−3.20820e7	−1.69187
$816$	−1.03928e7	−0.546395
$817$	−2.52836e6	−0.132521
$818$	1.68581e7	0.880897
$819$	1.01557e6	0.0529053
$820$	−1.25451e7	−0.651536
$821$	2.46510e7	1.27637	0.638186	−	0.769882i	$-0.279685\pi$
$821$	2.46510e7	1.27637	0.638186	+	0.769882i	$0.279685\pi$
$822$	−1.62165e7	−0.837101
$823$	−2.84702e7	−1.46518	−0.732589	−	0.680671i	$-0.761688\pi$
$823$	−2.84702e7	−1.46518	−0.732589	+	0.680671i	$0.761688\pi$
$824$	2.69761e6	0.138408
$825$	3.59865e7	1.84079
$826$	−1.76680e7	−0.901024
$827$	2.38791e7	1.21410	0.607050	−	0.794664i	$-0.292353\pi$
$827$	2.38791e7	1.21410	0.607050	+	0.794664i	$0.292353\pi$
$828$	−399872.	−0.0202696
$829$	−9.49660e6	−0.479934	−0.239967	−	0.970781i	$-0.577137\pi$
$829$	−9.49660e6	−0.479934	−0.239967	+	0.970781i	$0.577137\pi$
$830$	−1.12008e7	−0.564355
$831$	1.03414e6	0.0519491
$832$	−2.13324e6	−0.106839
$833$	−5.03066e7	−2.51196
$834$	−4.80456e6	−0.239188
$835$	2.05176e6	0.101838
$836$	1.17473e7	0.581329
$837$	3.58436e6	0.176847
$838$	−8.79944e6	−0.432857
$839$	−2.25952e7	−1.10818	−0.554091	−	0.832456i	$-0.686934\pi$
$839$	−2.25952e7	−1.10818	−0.554091	+	0.832456i	$0.686934\pi$
$840$	−3.63389e7	−1.77695
$841$	−1.35878e7	−0.662460
$842$	−9.38229e6	−0.456067
$843$	1.82318e7	0.883609
$844$	−6.47945e6	−0.313099
$845$	3.57255e7	1.72122
$846$	6.91161e6	0.332012
$847$	−5.07606e7	−2.43119
$848$	1.24574e7	0.594890
$849$	1.16307e7	0.553781
$850$	5.43631e7	2.58082
$851$	−2.57683e6	−0.121972
$852$	2.79374e6	0.131852
$853$	−687989.	−0.0323749	−0.0161875	−	0.999869i	$-0.505153\pi$
$853$	−687989.	−0.0323749	−0.0161875	+	0.999869i	$0.505153\pi$
$854$	−2.05691e7	−0.965094
$855$	−1.56340e7	−0.731402
$856$	3.80434e7	1.77458
$857$	1.52420e7	0.708906	0.354453	−	0.935074i	$-0.384667\pi$
$857$	1.52420e7	0.708906	0.354453	+	0.935074i	$0.384667\pi$
$858$	1.61143e6	0.0747297
$859$	−2.56393e7	−1.18556	−0.592780	−	0.805364i	$-0.701970\pi$
$859$	−2.56393e7	−1.18556	−0.592780	+	0.805364i	$0.701970\pi$
$860$	−1.15355e6	−0.0531852
$861$	−2.63068e7	−1.20937
$862$	1.64592e7	0.754466
$863$	4.25337e7	1.94405	0.972023	−	0.234888i	$-0.0754722\pi$
$863$	4.25337e7	1.94405	0.972023	+	0.234888i	$0.0754722\pi$
$864$	2.37524e6	0.108249
$865$	5.04370e7	2.29197
$866$	2.33585e7	1.05840
$867$	−1.66786e7	−0.753551
$868$	−9.69164e6	−0.436614
$869$	6.27335e6	0.281806
$870$	1.09524e7	0.490580
$871$	670803.	0.0299605
$872$	−2.29158e7	−1.02057
$873$	−1.48452e6	−0.0659252
$874$	−5.00432e6	−0.221598
$875$	6.53781e7	2.88677
$876$	1.89579e6	0.0834700
$877$	−2.00396e7	−0.879811	−0.439906	−	0.898044i	$-0.644988\pi$
$877$	−2.00396e7	−0.879811	−0.439906	+	0.898044i	$0.644988\pi$
$878$	−1.90874e6	−0.0835624
$879$	7.54573e6	0.329404
$880$	−3.92817e7	−1.70995
$881$	1.60786e7	0.697925	0.348962	−	0.937137i	$-0.386534\pi$
$881$	1.60786e7	0.697925	0.348962	+	0.937137i	$0.386534\pi$
$882$	−1.07236e7	−0.464161
$883$	1.82621e7	0.788222	0.394111	−	0.919063i	$-0.371053\pi$
$883$	1.82621e7	0.788222	0.394111	+	0.919063i	$0.371053\pi$
$884$	−1.00218e6	−0.0431334
$885$	−1.53600e7	−0.659225
$886$	2.52837e7	1.08207
$887$	−2.37902e7	−1.01529	−0.507645	−	0.861567i	$-0.669484\pi$
$887$	−2.37902e7	−1.01529	−0.507645	+	0.861567i	$0.669484\pi$
$888$	8.62712e6	0.367141
$889$	5.20145e7	2.20734
$890$	1.87144e7	0.791955
$891$	−4.15665e6	−0.175408
$892$	5.18605e6	0.218235
$893$	−3.56100e7	−1.49432
$894$	1.24395e7	0.520548
$895$	−4.08321e7	−1.70390
$896$	1.41178e7	0.587486
$897$	282610.	0.0117275
$898$	2.38795e6	0.0988178
$899$	1.29372e7	0.533878
$900$	−4.77077e6	−0.196328
$901$	3.53092e7	1.44903
$902$	−4.17417e7	−1.70826
$903$	−2.41897e6	−0.0987215
$904$	−3.53615e7	−1.43916
$905$	−7.70682e6	−0.312791
$906$	5.37973e6	0.217741
$907$	7.37144e6	0.297532	0.148766	−	0.988872i	$-0.452470\pi$
$907$	7.37144e6	0.297532	0.148766	+	0.988872i	$0.452470\pi$
$908$	4.78845e6	0.192744
$909$	1.09760e7	0.440591
$910$	5.79871e6	0.232128
$911$	−4.34077e7	−1.73289	−0.866445	−	0.499273i	$-0.833601\pi$
$911$	−4.34077e7	−1.73289	−0.866445	+	0.499273i	$0.833601\pi$
$912$	1.14141e7	0.454416
$913$	1.53431e7	0.609167
$914$	2.45575e7	0.972341
$915$	−1.78821e7	−0.706101
$916$	1.21202e7	0.477276
$917$	−1.43638e7	−0.564087
$918$	−6.27926e6	−0.245925
$919$	1.26700e6	0.0494865	0.0247433	−	0.999694i	$-0.492123\pi$
$919$	1.26700e6	0.0494865	0.0247433	+	0.999694i	$0.492123\pi$
$920$	−1.01123e7	−0.393896
$921$	−1.03833e7	−0.403355
$922$	1.59815e7	0.619140
$923$	−1.97448e6	−0.0762868
$924$	1.12391e7	0.433062
$925$	−3.07435e7	−1.18141
$926$	−8.75506e6	−0.335530
$927$	1.11038e6	0.0424397
$928$	8.57308e6	0.326789
$929$	632559.	0.0240470	0.0120235	−	0.999928i	$-0.496173\pi$
$929$	632559.	0.0240470	0.0120235	+	0.999928i	$0.496173\pi$
$930$	2.04660e7	0.775937
$931$	5.52502e7	2.08910
$932$	4.87966e6	0.184013
$933$	−1.31931e7	−0.496183
$934$	−2.13163e7	−0.799546
$935$	−1.11340e8	−4.16507
$936$	−946167.	−0.0353003
$937$	−3.21230e6	−0.119527	−0.0597637	−	0.998213i	$-0.519035\pi$
$937$	−3.21230e6	−0.119527	−0.0597637	+	0.998213i	$0.519035\pi$
$938$	−1.13644e7	−0.421734
$939$	−2.64982e7	−0.980737
$940$	−1.62469e7	−0.599724
$941$	−2.08925e7	−0.769161	−0.384580	−	0.923091i	$-0.625654\pi$
$941$	−2.08925e7	−0.769161	−0.384580	+	0.923091i	$0.625654\pi$
$942$	−7.07332e6	−0.259714
$943$	−7.32059e6	−0.268082
$944$	1.12140e7	0.409573
$945$	−1.49577e7	−0.544860
$946$	−3.83825e6	−0.139446
$947$	4.64335e7	1.68250	0.841252	−	0.540643i	$-0.181819\pi$
$947$	4.64335e7	1.68250	0.841252	+	0.540643i	$0.181819\pi$
$948$	−831664.	−0.0300557
$949$	−1.33986e6	−0.0482939
$950$	−5.97053e7	−2.14637
$951$	5.05057e6	0.181088
$952$	7.51974e7	2.68912
$953$	3.54643e7	1.26491	0.632454	−	0.774598i	$-0.282048\pi$
$953$	3.54643e7	1.26491	0.632454	+	0.774598i	$0.282048\pi$
$954$	7.52667e6	0.267752
$955$	−3.97862e7	−1.41164
$956$	−9.55264e6	−0.338048
$957$	−1.50029e7	−0.529534
$958$	5.40030e6	0.190110
$959$	7.99362e7	2.80671
$960$	3.14192e7	1.10032
$961$	−4.45412e6	−0.155580
$962$	−1.37666e6	−0.0479609
$963$	1.56592e7	0.544133
$964$	1.55360e7	0.538451
$965$	−4.12686e7	−1.42660
$966$	−4.78782e6	−0.165080
$967$	4.27394e7	1.46981	0.734906	−	0.678169i	$-0.237226\pi$
$967$	4.27394e7	1.46981	0.734906	+	0.678169i	$0.237226\pi$
$968$	4.72918e7	1.62217
$969$	3.23521e7	1.10686
$970$	−8.47637e6	−0.289255
$971$	1.05226e7	0.358157	0.179079	−	0.983835i	$-0.442688\pi$
$971$	1.05226e7	0.358157	0.179079	+	0.983835i	$0.442688\pi$
$972$	551052.	0.0187080
$973$	2.36832e7	0.801970
$974$	2.30795e7	0.779522
$975$	3.37175e6	0.113591
$976$	1.30554e7	0.438696
$977$	5.68345e7	1.90492	0.952458	−	0.304670i	$-0.0985461\pi$
$977$	5.68345e7	1.90492	0.952458	+	0.304670i	$0.0985461\pi$
$978$	1.41516e7	0.473107
$979$	−2.56355e7	−0.854840
$980$	2.52076e7	0.838430
$981$	−9.43249e6	−0.312935
$982$	−2.76537e7	−0.915114
$983$	1.82745e7	0.603201	0.301600	−	0.953434i	$-0.402479\pi$
$983$	1.82745e7	0.603201	0.301600	+	0.953434i	$0.402479\pi$
$984$	2.45090e7	0.806935
$985$	6.95580e7	2.28432
$986$	−2.26641e7	−0.742414
$987$	−3.40695e7	−1.11320
$988$	1.10066e6	0.0358724
$989$	−673147.	−0.0218836
$990$	−2.37338e7	−0.769624
$991$	−1.33296e7	−0.431155	−0.215577	−	0.976487i	$-0.569163\pi$
$991$	−1.33296e7	−0.431155	−0.215577	+	0.976487i	$0.569163\pi$
$992$	1.60200e7	0.516873
$993$	−3.20084e7	−1.03013
$994$	3.34506e7	1.07384
$995$	6.18518e7	1.98059
$996$	−2.03405e6	−0.0649701
$997$	−1.11514e7	−0.355296	−0.177648	−	0.984094i	$-0.556849\pi$
$997$	−1.11514e7	−0.355296	−0.177648	+	0.984094i	$0.556849\pi$
$998$	1.75176e7	0.556735
$999$	3.55106e6	0.112576

Twists

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

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

\(f(q)\)	\(=\)	\(q-4.76108 q^{2} -9.00000 q^{3} -9.33211 q^{4} -97.1410 q^{5} +42.8497 q^{6} -211.220 q^{7} +196.786 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-4.76108 q^{2} -9.00000 q^{3} -9.33211 q^{4} -97.1410 q^{5} +42.8497 q^{6} -211.220 q^{7} +196.786 q^{8} +81.0000 q^{9} +462.496 q^{10} -633.540 q^{11} +83.9890 q^{12} -59.3594 q^{13} +1005.63 q^{14} +874.269 q^{15} -638.284 q^{16} -1809.15 q^{17} -385.648 q^{18} +1986.94 q^{19} +906.530 q^{20} +1900.98 q^{21} +3016.33 q^{22} +529.000 q^{23} -1771.07 q^{24} +6311.37 q^{25} +282.615 q^{26} -729.000 q^{27} +1971.12 q^{28} -2631.22 q^{29} -4162.46 q^{30} -4916.81 q^{31} -3258.21 q^{32} +5701.86 q^{33} +8613.53 q^{34} +20518.1 q^{35} -755.901 q^{36} -4871.13 q^{37} -9459.96 q^{38} +534.235 q^{39} -19115.9 q^{40} -13838.6 q^{41} -9050.70 q^{42} -1272.49 q^{43} +5912.26 q^{44} -7868.42 q^{45} -2518.61 q^{46} -17922.1 q^{47} +5744.56 q^{48} +27806.7 q^{49} -30048.9 q^{50} +16282.4 q^{51} +553.948 q^{52} -19517.0 q^{53} +3470.83 q^{54} +61542.6 q^{55} -41565.0 q^{56} -17882.4 q^{57} +12527.5 q^{58} -17569.0 q^{59} -8158.77 q^{60} -20453.8 q^{61} +23409.3 q^{62} -17108.8 q^{63} +35937.7 q^{64} +5766.23 q^{65} -27147.0 q^{66} -11300.7 q^{67} +16883.2 q^{68} -4761.00 q^{69} -97688.2 q^{70} +33263.2 q^{71} +15939.6 q^{72} +22571.9 q^{73} +23191.9 q^{74} -56802.3 q^{75} -18542.3 q^{76} +133816. q^{77} -2543.53 q^{78} -9902.06 q^{79} +62003.6 q^{80} +6561.00 q^{81} +65886.5 q^{82} -24218.1 q^{83} -17740.1 q^{84} +175743. q^{85} +6058.43 q^{86} +23681.0 q^{87} -124671. q^{88} +40463.9 q^{89} +37462.2 q^{90} +12537.9 q^{91} -4936.69 q^{92} +44251.3 q^{93} +85328.5 q^{94} -193013. q^{95} +29323.9 q^{96} -18327.4 q^{97} -132390. q^{98} -51316.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 36 q^{3} - 46 q^{4} - 122 q^{5} - 36 q^{6} + 62 q^{7} + 72 q^{8} + 324 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 36 * q^3 - 46 * q^4 - 122 * q^5 - 36 * q^6 + 62 * q^7 + 72 * q^8 + 324 * q^9	\( 4 q + 4 q^{2} - 36 q^{3} - 46 q^{4} - 122 q^{5} - 36 q^{6} + 62 q^{7} + 72 q^{8} + 324 q^{9} + 642 q^{10} + 32 q^{11} + 414 q^{12} + 1364 q^{13} + 2754 q^{14} + 1098 q^{15} + 18 q^{16} + 278 q^{17} + 324 q^{18} + 2862 q^{19} + 3830 q^{20} - 558 q^{21} + 3176 q^{22} + 2116 q^{23} - 648 q^{24} + 5944 q^{25} + 6996 q^{26} - 2916 q^{27} + 4738 q^{28} - 5180 q^{29} - 5778 q^{30} - 1788 q^{31} - 7352 q^{32} - 288 q^{33} + 15818 q^{34} + 11768 q^{35} - 3726 q^{36} + 3348 q^{37} + 1050 q^{38} - 12276 q^{39} - 13462 q^{40} - 17664 q^{41} - 24786 q^{42} + 25398 q^{43} - 16848 q^{44} - 9882 q^{45} + 2116 q^{46} - 26040 q^{47} - 162 q^{48} + 55720 q^{49} - 35256 q^{50} - 2502 q^{51} - 2752 q^{52} - 32006 q^{53} - 2916 q^{54} + 34904 q^{55} - 68542 q^{56} - 25758 q^{57} - 40804 q^{58} - 61136 q^{59} - 34470 q^{60} + 35844 q^{61} - 47524 q^{62} + 5022 q^{63} - 35142 q^{64} - 48036 q^{65} - 28584 q^{66} + 73458 q^{67} + 17910 q^{68} - 19044 q^{69} - 59104 q^{70} + 24432 q^{71} + 5832 q^{72} + 122512 q^{73} - 20828 q^{74} - 53496 q^{75} + 56834 q^{76} + 159496 q^{77} - 62964 q^{78} + 90170 q^{79} - 36546 q^{80} + 26244 q^{81} + 84144 q^{82} + 28592 q^{83} - 42642 q^{84} + 355124 q^{85} + 103778 q^{86} + 46620 q^{87} - 150776 q^{88} - 27926 q^{89} + 52002 q^{90} + 334180 q^{91} - 24334 q^{92} + 16092 q^{93} + 113632 q^{94} + 113392 q^{95} + 66168 q^{96} + 16580 q^{97} + 49688 q^{98} + 2592 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 36 * q^3 - 46 * q^4 - 122 * q^5 - 36 * q^6 + 62 * q^7 + 72 * q^8 + 324 * q^9 + 642 * q^10 + 32 * q^11 + 414 * q^12 + 1364 * q^13 + 2754 * q^14 + 1098 * q^15 + 18 * q^16 + 278 * q^17 + 324 * q^18 + 2862 * q^19 + 3830 * q^20 - 558 * q^21 + 3176 * q^22 + 2116 * q^23 - 648 * q^24 + 5944 * q^25 + 6996 * q^26 - 2916 * q^27 + 4738 * q^28 - 5180 * q^29 - 5778 * q^30 - 1788 * q^31 - 7352 * q^32 - 288 * q^33 + 15818 * q^34 + 11768 * q^35 - 3726 * q^36 + 3348 * q^37 + 1050 * q^38 - 12276 * q^39 - 13462 * q^40 - 17664 * q^41 - 24786 * q^42 + 25398 * q^43 - 16848 * q^44 - 9882 * q^45 + 2116 * q^46 - 26040 * q^47 - 162 * q^48 + 55720 * q^49 - 35256 * q^50 - 2502 * q^51 - 2752 * q^52 - 32006 * q^53 - 2916 * q^54 + 34904 * q^55 - 68542 * q^56 - 25758 * q^57 - 40804 * q^58 - 61136 * q^59 - 34470 * q^60 + 35844 * q^61 - 47524 * q^62 + 5022 * q^63 - 35142 * q^64 - 48036 * q^65 - 28584 * q^66 + 73458 * q^67 + 17910 * q^68 - 19044 * q^69 - 59104 * q^70 + 24432 * q^71 + 5832 * q^72 + 122512 * q^73 - 20828 * q^74 - 53496 * q^75 + 56834 * q^76 + 159496 * q^77 - 62964 * q^78 + 90170 * q^79 - 36546 * q^80 + 26244 * q^81 + 84144 * q^82 + 28592 * q^83 - 42642 * q^84 + 355124 * q^85 + 103778 * q^86 + 46620 * q^87 - 150776 * q^88 - 27926 * q^89 + 52002 * q^90 + 334180 * q^91 - 24334 * q^92 + 16092 * q^93 + 113632 * q^94 + 113392 * q^95 + 66168 * q^96 + 16580 * q^97 + 49688 * q^98 + 2592 * q^99

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