LMFDB - Embedded newform 69.6.a.b.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:	$3$
Coefficient field:	3.3.5333.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 8 $ x^3 - x^2 - 11*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.20733$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.49429 q^{2} +9.00000 q^{3} -19.7900 q^{4} -59.5955 q^{5} +31.4486 q^{6} +96.8268 q^{7} -180.969 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+3.49429 q^{2} +9.00000 q^{3} -19.7900 q^{4} -59.5955 q^{5} +31.4486 q^{6} +96.8268 q^{7} -180.969 q^{8} +81.0000 q^{9} -208.244 q^{10} -731.078 q^{11} -178.110 q^{12} -631.879 q^{13} +338.341 q^{14} -536.360 q^{15} +0.920920 q^{16} +75.7854 q^{17} +283.037 q^{18} +2.35318 q^{19} +1179.39 q^{20} +871.441 q^{21} -2554.60 q^{22} +529.000 q^{23} -1628.72 q^{24} +426.625 q^{25} -2207.97 q^{26} +729.000 q^{27} -1916.20 q^{28} -2425.82 q^{29} -1874.19 q^{30} -349.683 q^{31} +5794.23 q^{32} -6579.70 q^{33} +264.816 q^{34} -5770.44 q^{35} -1602.99 q^{36} +8314.43 q^{37} +8.22270 q^{38} -5686.91 q^{39} +10784.9 q^{40} +4198.56 q^{41} +3045.07 q^{42} +9372.62 q^{43} +14468.0 q^{44} -4827.24 q^{45} +1848.48 q^{46} -2436.94 q^{47} +8.28828 q^{48} -7431.57 q^{49} +1490.75 q^{50} +682.069 q^{51} +12504.8 q^{52} -34979.0 q^{53} +2547.34 q^{54} +43569.0 q^{55} -17522.6 q^{56} +21.1787 q^{57} -8476.51 q^{58} +15086.9 q^{59} +10614.5 q^{60} +987.986 q^{61} -1221.89 q^{62} +7842.97 q^{63} +20217.2 q^{64} +37657.1 q^{65} -22991.4 q^{66} +44337.8 q^{67} -1499.79 q^{68} +4761.00 q^{69} -20163.6 q^{70} -33934.4 q^{71} -14658.5 q^{72} -42483.0 q^{73} +29053.0 q^{74} +3839.62 q^{75} -46.5694 q^{76} -70788.0 q^{77} -19871.7 q^{78} +51187.8 q^{79} -54.8827 q^{80} +6561.00 q^{81} +14671.0 q^{82} -88496.5 q^{83} -17245.8 q^{84} -4516.47 q^{85} +32750.6 q^{86} -21832.4 q^{87} +132303. q^{88} -34083.9 q^{89} -16867.8 q^{90} -61182.8 q^{91} -10468.9 q^{92} -3147.15 q^{93} -8515.36 q^{94} -140.239 q^{95} +52148.0 q^{96} -156803. q^{97} -25968.0 q^{98} -59217.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{2} + 27 q^{3} + 22 q^{4} - 56 q^{5} - 72 q^{6} - 114 q^{7} - 510 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q - 8 * q^2 + 27 * q^3 + 22 * q^4 - 56 * q^5 - 72 * q^6 - 114 * q^7 - 510 * q^8 + 243 * q^9	\( 3 q - 8 q^{2} + 27 q^{3} + 22 q^{4} - 56 q^{5} - 72 q^{6} - 114 q^{7} - 510 q^{8} + 243 q^{9} + 282 q^{10} - 376 q^{11} + 198 q^{12} - 858 q^{13} + 588 q^{14} - 504 q^{15} + 2738 q^{16} - 2548 q^{17} - 648 q^{18} - 2846 q^{19} - 4618 q^{20} - 1026 q^{21} - 5050 q^{22} + 1587 q^{23} - 4590 q^{24} + 753 q^{25} - 7788 q^{26} + 2187 q^{27} + 4736 q^{28} - 16370 q^{29} + 2538 q^{30} - 14756 q^{31} - 3878 q^{32} - 3384 q^{33} + 16520 q^{34} - 18520 q^{35} + 1782 q^{36} + 15874 q^{37} + 12438 q^{38} - 7722 q^{39} + 38270 q^{40} + 12606 q^{41} + 5292 q^{42} + 3154 q^{43} + 27114 q^{44} - 4536 q^{45} - 4232 q^{46} + 29928 q^{47} + 24642 q^{48} + 4471 q^{49} + 1452 q^{50} - 22932 q^{51} + 86856 q^{52} - 44084 q^{53} - 5832 q^{54} + 38360 q^{55} - 35704 q^{56} - 25614 q^{57} + 73316 q^{58} - 29300 q^{59} - 41562 q^{60} + 54010 q^{61} + 99908 q^{62} - 9234 q^{63} - 1582 q^{64} - 51216 q^{65} - 45450 q^{66} + 43390 q^{67} - 69840 q^{68} + 14283 q^{69} - 2476 q^{70} + 23424 q^{71} - 41310 q^{72} - 91402 q^{73} - 2294 q^{74} + 6777 q^{75} - 14274 q^{76} - 97208 q^{77} - 70092 q^{78} - 49398 q^{79} - 52626 q^{80} + 19683 q^{81} + 40152 q^{82} - 103936 q^{83} + 42624 q^{84} + 5888 q^{85} + 133634 q^{86} - 147330 q^{87} + 48898 q^{88} + 96112 q^{89} + 22842 q^{90} + 129228 q^{91} + 11638 q^{92} - 132804 q^{93} - 133688 q^{94} - 55928 q^{95} - 34902 q^{96} - 135318 q^{97} + 108440 q^{98} - 30456 q^{99}+O(q^{100}) \) 3 * q - 8 * q^2 + 27 * q^3 + 22 * q^4 - 56 * q^5 - 72 * q^6 - 114 * q^7 - 510 * q^8 + 243 * q^9 + 282 * q^10 - 376 * q^11 + 198 * q^12 - 858 * q^13 + 588 * q^14 - 504 * q^15 + 2738 * q^16 - 2548 * q^17 - 648 * q^18 - 2846 * q^19 - 4618 * q^20 - 1026 * q^21 - 5050 * q^22 + 1587 * q^23 - 4590 * q^24 + 753 * q^25 - 7788 * q^26 + 2187 * q^27 + 4736 * q^28 - 16370 * q^29 + 2538 * q^30 - 14756 * q^31 - 3878 * q^32 - 3384 * q^33 + 16520 * q^34 - 18520 * q^35 + 1782 * q^36 + 15874 * q^37 + 12438 * q^38 - 7722 * q^39 + 38270 * q^40 + 12606 * q^41 + 5292 * q^42 + 3154 * q^43 + 27114 * q^44 - 4536 * q^45 - 4232 * q^46 + 29928 * q^47 + 24642 * q^48 + 4471 * q^49 + 1452 * q^50 - 22932 * q^51 + 86856 * q^52 - 44084 * q^53 - 5832 * q^54 + 38360 * q^55 - 35704 * q^56 - 25614 * q^57 + 73316 * q^58 - 29300 * q^59 - 41562 * q^60 + 54010 * q^61 + 99908 * q^62 - 9234 * q^63 - 1582 * q^64 - 51216 * q^65 - 45450 * q^66 + 43390 * q^67 - 69840 * q^68 + 14283 * q^69 - 2476 * q^70 + 23424 * q^71 - 41310 * q^72 - 91402 * q^73 - 2294 * q^74 + 6777 * q^75 - 14274 * q^76 - 97208 * q^77 - 70092 * q^78 - 49398 * q^79 - 52626 * q^80 + 19683 * q^81 + 40152 * q^82 - 103936 * q^83 + 42624 * q^84 + 5888 * q^85 + 133634 * q^86 - 147330 * q^87 + 48898 * q^88 + 96112 * q^89 + 22842 * q^90 + 129228 * q^91 + 11638 * q^92 - 132804 * q^93 - 133688 * q^94 - 55928 * q^95 - 34902 * q^96 - 135318 * q^97 + 108440 * q^98 - 30456 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.49429	0.617709	0.308854	−	0.951109i	$-0.400054\pi$
$2$	3.49429	0.617709	0.308854	+	0.951109i	$0.400054\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−19.7900	−0.618436
$5$	−59.5955	−1.06608	−0.533038	−	0.846091i	$-0.678950\pi$
$5$	−59.5955	−1.06608	−0.533038	+	0.846091i	$0.678950\pi$
$6$	31.4486	0.356634
$7$	96.8268	0.746879	0.373440	−	0.927654i	$-0.378178\pi$
$7$	96.8268	0.746879	0.373440	+	0.927654i	$0.378178\pi$
$8$	−180.969	−0.999722
$9$	81.0000	0.333333
$10$	−208.244	−0.658525
$11$	−731.078	−1.82172	−0.910861	−	0.412713i	$-0.864581\pi$
$11$	−731.078	−1.82172	−0.910861	+	0.412713i	$0.864581\pi$
$12$	−178.110	−0.357054
$13$	−631.879	−1.03699	−0.518496	−	0.855080i	$-0.673508\pi$
$13$	−631.879	−1.03699	−0.518496	+	0.855080i	$0.673508\pi$
$14$	338.341	0.461354
$15$	−536.360	−0.615500
$16$	0.920920	0.000899336 0
$17$	75.7854	0.0636009	0.0318005	−	0.999494i	$-0.489876\pi$
$17$	75.7854	0.0636009	0.0318005	+	0.999494i	$0.489876\pi$
$18$	283.037	0.205903
$19$	2.35318	0.00149545	0.000747725	−	1.00000i	$-0.499762\pi$
$19$	2.35318	0.00149545	0.000747725		1.00000i	$0.499762\pi$
$20$	1179.39	0.659300
$21$	871.441	0.431211
$22$	−2554.60	−1.12529
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	−1628.72	−0.577190
$25$	426.625	0.136520
$26$	−2207.97	−0.640559
$27$	729.000	0.192450
$28$	−1916.20	−0.461897
$29$	−2425.82	−0.535628	−0.267814	−	0.963471i	$-0.586301\pi$
$29$	−2425.82	−0.535628	−0.267814	+	0.963471i	$0.586301\pi$
$30$	−1874.19	−0.380200
$31$	−349.683	−0.0653537	−0.0326769	−	0.999466i	$-0.510403\pi$
$31$	−349.683	−0.0653537	−0.0326769	+	0.999466i	$0.510403\pi$
$32$	5794.23	1.00028
$33$	−6579.70	−1.05177
$34$	264.816	0.0392868
$35$	−5770.44	−0.796231
$36$	−1602.99	−0.206145
$37$	8314.43	0.998454	0.499227	−	0.866471i	$-0.333617\pi$
$37$	8314.43	0.998454	0.499227	+	0.866471i	$0.333617\pi$
$38$	8.22270	0.000923753 0
$39$	−5686.91	−0.598707
$40$	10784.9	1.06578
$41$	4198.56	0.390068	0.195034	−	0.980796i	$-0.437518\pi$
$41$	4198.56	0.390068	0.195034	+	0.980796i	$0.437518\pi$
$42$	3045.07	0.266363
$43$	9372.62	0.773019	0.386509	−	0.922285i	$-0.373681\pi$
$43$	9372.62	0.773019	0.386509	+	0.922285i	$0.373681\pi$
$44$	14468.0	1.12662
$45$	−4827.24	−0.355359
$46$	1848.48	0.128801
$47$	−2436.94	−0.160916	−0.0804581	−	0.996758i	$-0.525638\pi$
$47$	−2436.94	−0.160916	−0.0804581	+	0.996758i	$0.525638\pi$
$48$	8.28828	0.000519232 0
$49$	−7431.57	−0.442171
$50$	1490.75	0.0843296
$51$	682.069	0.0367200
$52$	12504.8	0.641313
$53$	−34979.0	−1.71048	−0.855241	−	0.518231i	$-0.826591\pi$
$53$	−34979.0	−1.71048	−0.855241	+	0.518231i	$0.826591\pi$
$54$	2547.34	0.118878
$55$	43569.0	1.94210
$56$	−17522.6	−0.746672
$57$	21.1787	0.000863399 0
$58$	−8476.51	−0.330862
$59$	15086.9	0.564249	0.282125	−	0.959378i	$-0.408961\pi$
$59$	15086.9	0.564249	0.282125	+	0.959378i	$0.408961\pi$
$60$	10614.5	0.380647
$61$	987.986	0.0339959	0.0169979	−	0.999856i	$-0.494589\pi$
$61$	987.986	0.0339959	0.0169979	+	0.999856i	$0.494589\pi$
$62$	−1221.89	−0.0403696
$63$	7842.97	0.248960
$64$	20217.2	0.616981
$65$	37657.1	1.10551
$66$	−22991.4	−0.649689
$67$	44337.8	1.20667	0.603333	−	0.797490i	$-0.293839\pi$
$67$	44337.8	1.20667	0.603333	+	0.797490i	$0.293839\pi$
$68$	−1499.79	−0.0393331
$69$	4761.00	0.120386
$70$	−20163.6	−0.491839
$71$	−33934.4	−0.798904	−0.399452	−	0.916754i	$-0.630800\pi$
$71$	−33934.4	−0.798904	−0.399452	+	0.916754i	$0.630800\pi$
$72$	−14658.5	−0.333241
$73$	−42483.0	−0.933058	−0.466529	−	0.884506i	$-0.654496\pi$
$73$	−42483.0	−0.933058	−0.466529	+	0.884506i	$0.654496\pi$
$74$	29053.0	0.616753
$75$	3839.62	0.0788198
$76$	−46.5694	−0.000924841 0
$77$	−70788.0	−1.36061
$78$	−19871.7	−0.369827
$79$	51187.8	0.922782	0.461391	−	0.887197i	$-0.347351\pi$
$79$	51187.8	0.922782	0.461391	+	0.887197i	$0.347351\pi$
$80$	−54.8827	−0.000958761 0
$81$	6561.00	0.111111
$82$	14671.0	0.240949
$83$	−88496.5	−1.41004	−0.705019	−	0.709188i	$-0.749062\pi$
$83$	−88496.5	−1.41004	−0.705019	+	0.709188i	$0.749062\pi$
$84$	−17245.8	−0.266676
$85$	−4516.47	−0.0678035
$86$	32750.6	0.477500
$87$	−21832.4	−0.309245
$88$	132303.	1.82122
$89$	−34083.9	−0.456115	−0.228058	−	0.973648i	$-0.573237\pi$
$89$	−34083.9	−0.456115	−0.228058	+	0.973648i	$0.573237\pi$
$90$	−16867.8	−0.219508
$91$	−61182.8	−0.774508
$92$	−10468.9	−0.128953
$93$	−3147.15	−0.0377320
$94$	−8515.36	−0.0993993
$95$	−140.239	−0.00159427
$96$	52148.0	0.577510
$97$	−156803.	−1.69209	−0.846047	−	0.533108i	$-0.821024\pi$
$97$	−156803.	−1.69209	−0.846047	+	0.533108i	$0.821024\pi$
$98$	−25968.0	−0.273133
$99$	−59217.3	−0.607241
$100$	−8442.89	−0.0844289
$101$	−148400.	−1.44754	−0.723772	−	0.690039i	$-0.757593\pi$
$101$	−148400.	−1.44754	−0.723772	+	0.690039i	$0.757593\pi$
$102$	2383.34	0.0226823
$103$	155116.	1.44067	0.720335	−	0.693626i	$-0.243988\pi$
$103$	155116.	1.44067	0.720335	+	0.693626i	$0.243988\pi$
$104$	114350.	1.03670
$105$	−51934.0	−0.459704
$106$	−122227.	−1.05658
$107$	−177527.	−1.49901	−0.749504	−	0.662000i	$-0.769708\pi$
$107$	−177527.	−1.49901	−0.749504	+	0.662000i	$0.769708\pi$
$108$	−14426.9	−0.119018
$109$	−169314.	−1.36498	−0.682492	−	0.730893i	$-0.739104\pi$
$109$	−169314.	−1.36498	−0.682492	+	0.730893i	$0.739104\pi$
$110$	152243.	1.19965
$111$	74829.8	0.576458
$112$	89.1697	0.000671696 0
$113$	81710.2	0.601977	0.300989	−	0.953628i	$-0.402683\pi$
$113$	81710.2	0.601977	0.300989	+	0.953628i	$0.402683\pi$
$114$	74.0043	0.000533329 0
$115$	−31526.0	−0.222292
$116$	48006.9	0.331252
$117$	−51182.2	−0.345664
$118$	52718.1	0.348542
$119$	7338.06	0.0475022
$120$	97064.5	0.615329
$121$	373424.	2.31867
$122$	3452.31	0.0209996
$123$	37787.0	0.225206
$124$	6920.21	0.0404171
$125$	160811.	0.920536
$126$	27405.6	0.153785
$127$	−317307.	−1.74570	−0.872851	−	0.487986i	$-0.837732\pi$
$127$	−317307.	−1.74570	−0.872851	+	0.487986i	$0.837732\pi$
$128$	−114770.	−0.619163
$129$	84353.6	0.446303
$130$	131585.	0.682885
$131$	−154610.	−0.787151	−0.393576	−	0.919292i	$-0.628762\pi$
$131$	−154610.	−0.787151	−0.393576	+	0.919292i	$0.628762\pi$
$132$	130212.	0.650454
$133$	227.851	0.00111692
$134$	154929.	0.745368
$135$	−43445.1	−0.205167
$136$	−13714.8	−0.0635832
$137$	217676.	0.990852	0.495426	−	0.868650i	$-0.335012\pi$
$137$	217676.	0.990852	0.495426	+	0.868650i	$0.335012\pi$
$138$	16636.3	0.0743634
$139$	236914.	1.04005	0.520023	−	0.854152i	$-0.325923\pi$
$139$	236914.	1.04005	0.520023	+	0.854152i	$0.325923\pi$
$140$	114197.	0.492418
$141$	−21932.4	−0.0929050
$142$	−118577.	−0.493490
$143$	461953.	1.88911
$144$	74.5945	0.000299779 0
$145$	144568.	0.571021
$146$	−148448.	−0.576358
$147$	−66884.1	−0.255288
$148$	−164542.	−0.617480
$149$	434197.	1.60221	0.801107	−	0.598521i	$-0.204245\pi$
$149$	434197.	1.60221	0.801107	+	0.598521i	$0.204245\pi$
$150$	13416.8	0.0486877
$151$	−135760.	−0.484541	−0.242270	−	0.970209i	$-0.577892\pi$
$151$	−135760.	−0.484541	−0.242270	+	0.970209i	$0.577892\pi$
$152$	−425.853	−0.00149503
$153$	6138.62	0.0212003
$154$	−247354.	−0.840459
$155$	20839.5	0.0696721
$156$	112544.	0.370262
$157$	−355399.	−1.15071	−0.575356	−	0.817903i	$-0.695136\pi$
$157$	−355399.	−1.15071	−0.575356	+	0.817903i	$0.695136\pi$
$158$	178865.	0.570010
$159$	−314811.	−0.987547
$160$	−345310.	−1.06637
$161$	51221.4	0.155735
$162$	22926.0	0.0686343
$163$	−514344.	−1.51630	−0.758148	−	0.652082i	$-0.773896\pi$
$163$	−514344.	−1.51630	−0.758148	+	0.652082i	$0.773896\pi$
$164$	−83089.3	−0.241232
$165$	392121.	1.12127
$166$	−309232.	−0.870993
$167$	178285.	0.494680	0.247340	−	0.968929i	$-0.420444\pi$
$167$	178285.	0.494680	0.247340	+	0.968929i	$0.420444\pi$
$168$	−157704.	−0.431091
$169$	27977.5	0.0753515
$170$	−15781.8	−0.0418828
$171$	190.608	0.000498483 0
$172$	−185484.	−0.478063
$173$	−535921.	−1.36140	−0.680699	−	0.732563i	$-0.738324\pi$
$173$	−535921.	−1.36140	−0.680699	+	0.732563i	$0.738324\pi$
$174$	−76288.6	−0.191023
$175$	41308.7	0.101964
$176$	−673.265	−0.00163834
$177$	135782.	0.325769
$178$	−119099.	−0.281746
$179$	−58891.2	−0.137378	−0.0686891	−	0.997638i	$-0.521882\pi$
$179$	−58891.2	−0.137378	−0.0686891	+	0.997638i	$0.521882\pi$
$180$	95530.8	0.219767
$181$	829741.	1.88255	0.941274	−	0.337644i	$-0.109630\pi$
$181$	829741.	1.88255	0.941274	+	0.337644i	$0.109630\pi$
$182$	−213790.	−0.478420
$183$	8891.88	0.0196275
$184$	−95732.6	−0.208456
$185$	−495502.	−1.06443
$186$	−10997.0	−0.0233074
$187$	−55405.1	−0.115863
$188$	48226.9	0.0995164
$189$	70586.7	0.143737
$190$	−490.036	−0.000984791 0
$191$	149335.	0.296195	0.148097	−	0.988973i	$-0.452685\pi$
$191$	149335.	0.296195	0.148097	+	0.988973i	$0.452685\pi$
$192$	181955.	0.356214
$193$	913030.	1.76438	0.882189	−	0.470895i	$-0.156069\pi$
$193$	913030.	1.76438	0.882189	+	0.470895i	$0.156069\pi$
$194$	−547914.	−1.04522
$195$	338914.	0.638268
$196$	147070.	0.273455
$197$	−919205.	−1.68751	−0.843756	−	0.536727i	$-0.819661\pi$
$197$	−919205.	−1.68751	−0.843756	+	0.536727i	$0.819661\pi$
$198$	−206922.	−0.375098
$199$	493730.	0.883806	0.441903	−	0.897063i	$-0.354304\pi$
$199$	493730.	0.883806	0.441903	+	0.897063i	$0.354304\pi$
$200$	−77205.9	−0.136482
$201$	399040.	0.696669
$202$	−518554.	−0.894160
$203$	−234884.	−0.400050
$204$	−13498.1	−0.0227090
$205$	−250215.	−0.415843
$206$	542021.	0.889914
$207$	42849.0	0.0695048
$208$	−581.910	−0.000932604 0
$209$	−1720.36	−0.00272430
$210$	−181472.	−0.283963
$211$	−133387.	−0.206257	−0.103129	−	0.994668i	$-0.532885\pi$
$211$	−133387.	−0.206257	−0.103129	+	0.994668i	$0.532885\pi$
$212$	692234.	1.05782
$213$	−305410.	−0.461247
$214$	−620329.	−0.925950
$215$	−558566.	−0.824098
$216$	−131926.	−0.192397
$217$	−33858.7	−0.0488114
$218$	−591633.	−0.843162
$219$	−382347.	−0.538701
$220$	−862228.	−1.20106
$221$	−47887.2	−0.0659536
$222$	261477.	0.356083
$223$	362735.	0.488457	0.244229	−	0.969718i	$-0.421465\pi$
$223$	362735.	0.488457	0.244229	+	0.969718i	$0.421465\pi$
$224$	561036.	0.747087
$225$	34556.6	0.0455067
$226$	285519.	0.371846
$227$	−190962.	−0.245970	−0.122985	−	0.992409i	$-0.539247\pi$
$227$	−190962.	−0.245970	−0.122985	+	0.992409i	$0.539247\pi$
$228$	−419.125	−0.000533957 0
$229$	745968.	0.940008	0.470004	−	0.882664i	$-0.344252\pi$
$229$	745968.	0.940008	0.470004	+	0.882664i	$0.344252\pi$
$230$	−110161.	−0.137312
$231$	−637092.	−0.785547
$232$	438998.	0.535479
$233$	636804.	0.768450	0.384225	−	0.923239i	$-0.374469\pi$
$233$	636804.	0.768450	0.384225	+	0.923239i	$0.374469\pi$
$234$	−178845.	−0.213520
$235$	145231.	0.171549
$236$	−298570.	−0.348952
$237$	460691.	0.532768
$238$	25641.3	0.0293425
$239$	1.44028e6	1.63099	0.815494	−	0.578765i	$-0.196465\pi$
$239$	1.44028e6	1.63099	0.815494	+	0.578765i	$0.196465\pi$
$240$	−493.944	−0.000553541 0
$241$	1.13226e6	1.25575	0.627876	−	0.778313i	$-0.283924\pi$
$241$	1.13226e6	1.25575	0.627876	+	0.778313i	$0.283924\pi$
$242$	1.30485e6	1.43226
$243$	59049.0	0.0641500
$244$	−19552.2	−0.0210243
$245$	442888.	0.471388
$246$	132039.	0.139112
$247$	−1486.93	−0.00155077
$248$	63281.8	0.0653356
$249$	−796469.	−0.814086
$250$	561920.	0.568623
$251$	−622529.	−0.623700	−0.311850	−	0.950131i	$-0.600949\pi$
$251$	−622529.	−0.623700	−0.311850	+	0.950131i	$0.600949\pi$
$252$	−155212.	−0.153966
$253$	−386740.	−0.379855
$254$	−1.10876e6	−1.07834
$255$	−40648.2	−0.0391463
$256$	−1.04799e6	−0.999443
$257$	−2.10304e6	−1.98616	−0.993082	−	0.117419i	$-0.962538\pi$
$257$	−2.10304e6	−1.98616	−0.993082	+	0.117419i	$0.962538\pi$
$258$	294756.	0.275685
$259$	805059.	0.745725
$260$	−745233.	−0.683689
$261$	−196491.	−0.178543
$262$	−540250.	−0.486230
$263$	838283.	0.747310	0.373655	−	0.927568i	$-0.378104\pi$
$263$	838283.	0.747310	0.373655	+	0.927568i	$0.378104\pi$
$264$	1.19072e6	1.05148
$265$	2.08459e6	1.82350
$266$	796.178	0.000689932 0
$267$	−306755.	−0.263338
$268$	−877443.	−0.746246
$269$	897532.	0.756257	0.378128	−	0.925753i	$-0.376568\pi$
$269$	897532.	0.756257	0.378128	+	0.925753i	$0.376568\pi$
$270$	−151810.	−0.126733
$271$	−1.00088e6	−0.827864	−0.413932	−	0.910308i	$-0.635845\pi$
$271$	−1.00088e6	−0.827864	−0.413932	+	0.910308i	$0.635845\pi$
$272$	69.7923	5.71986e−5 0
$273$	−550645.	−0.447162
$274$	760622.	0.612058
$275$	−311896.	−0.248701
$276$	−94220.0	−0.0744510
$277$	2.09974e6	1.64424	0.822120	−	0.569314i	$-0.192791\pi$
$277$	2.09974e6	1.64424	0.822120	+	0.569314i	$0.192791\pi$
$278$	827844.	0.642446
$279$	−28324.3	−0.0217846
$280$	1.04427e6	0.796009
$281$	503227.	0.380188	0.190094	−	0.981766i	$-0.439121\pi$
$281$	503227.	0.380188	0.190094	+	0.981766i	$0.439121\pi$
$282$	−76638.3	−0.0573882
$283$	−1.61771e6	−1.20070	−0.600349	−	0.799738i	$-0.704972\pi$
$283$	−1.61771e6	−1.20070	−0.600349	+	0.799738i	$0.704972\pi$
$284$	671561.	0.494071
$285$	−1262.15	−0.000920449 0
$286$	1.61420e6	1.16692
$287$	406533.	0.291334
$288$	469332.	0.333426
$289$	−1.41411e6	−0.995955
$290$	505162.	0.352725
$291$	−1.41123e6	−0.976932
$292$	840738.	0.577037
$293$	1.04521e6	0.711270	0.355635	−	0.934625i	$-0.384265\pi$
$293$	1.04521e6	0.711270	0.355635	+	0.934625i	$0.384265\pi$
$294$	−233712.	−0.157693
$295$	−899113.	−0.601533
$296$	−1.50465e6	−0.998176
$297$	−532956.	−0.350591
$298$	1.51721e6	0.989702
$299$	−334264.	−0.216228
$300$	−75986.0	−0.0487450
$301$	907521.	0.577352
$302$	−474385.	−0.299305
$303$	−1.33560e6	−0.835740
$304$	2.16710	1.34491e−6 0
$305$	−58879.6	−0.0362422
$306$	21450.1	0.0130956
$307$	−697539.	−0.422399	−0.211199	−	0.977443i	$-0.567737\pi$
$307$	−697539.	−0.422399	−0.211199	+	0.977443i	$0.567737\pi$
$308$	1.40089e6	0.841448
$309$	1.39605e6	0.831771
$310$	72819.4	0.0430371
$311$	−2.68619e6	−1.57484	−0.787419	−	0.616419i	$-0.788583\pi$
$311$	−2.68619e6	−1.57484	−0.787419	+	0.616419i	$0.788583\pi$
$312$	1.02915e6	0.598541
$313$	1.62222e6	0.935944	0.467972	−	0.883743i	$-0.344985\pi$
$313$	1.62222e6	0.935944	0.467972	+	0.883743i	$0.344985\pi$
$314$	−1.24186e6	−0.710805
$315$	−467406.	−0.265410
$316$	−1.01301e6	−0.570682
$317$	648171.	0.362277	0.181139	−	0.983458i	$-0.442022\pi$
$317$	648171.	0.362277	0.181139	+	0.983458i	$0.442022\pi$
$318$	−1.10004e6	−0.610016
$319$	1.77346e6	0.975766
$320$	−1.20486e6	−0.657749
$321$	−1.59774e6	−0.865452
$322$	178982.	0.0961989
$323$	178.337	9.51120e−5 0
$324$	−129842.	−0.0687151
$325$	−269575.	−0.141570
$326$	−1.79726e6	−0.936630
$327$	−1.52383e6	−0.788074
$328$	−759809.	−0.389960
$329$	−235961.	−0.120185
$330$	1.37018e6	0.692618
$331$	−1.66550e6	−0.835552	−0.417776	−	0.908550i	$-0.637190\pi$
$331$	−1.66550e6	−0.835552	−0.417776	+	0.908550i	$0.637190\pi$
$332$	1.75134e6	0.872019
$333$	673468.	0.332818
$334$	622980.	0.305568
$335$	−2.64233e6	−1.28640
$336$	802.528	0.000387804 0
$337$	−2.11943e6	−1.01659	−0.508294	−	0.861184i	$-0.669724\pi$
$337$	−2.11943e6	−1.01659	−0.508294	+	0.861184i	$0.669724\pi$
$338$	97761.3	0.0465452
$339$	735392.	0.347552
$340$	89380.8	0.0419321
$341$	255646.	0.119056
$342$	666.039	0.000307918 0
$343$	−2.34694e6	−1.07713
$344$	−1.69615e6	−0.772804
$345$	−283734.	−0.128341
$346$	−1.87266e6	−0.840948
$347$	−791717.	−0.352977	−0.176488	−	0.984303i	$-0.556474\pi$
$347$	−791717.	−0.352977	−0.176488	+	0.984303i	$0.556474\pi$
$348$	432062.	0.191248
$349$	743781.	0.326875	0.163437	−	0.986554i	$-0.447742\pi$
$349$	743781.	0.326875	0.163437	+	0.986554i	$0.447742\pi$
$350$	144345.	0.0629840
$351$	−460639.	−0.199569
$352$	−4.23603e6	−1.82223
$353$	−646287.	−0.276051	−0.138025	−	0.990429i	$-0.544076\pi$
$353$	−646287.	−0.276051	−0.138025	+	0.990429i	$0.544076\pi$
$354$	474463.	0.201231
$355$	2.02234e6	0.851693
$356$	674519.	0.282078
$357$	66042.5	0.0274254
$358$	−205783.	−0.0848597
$359$	1.89862e6	0.777503	0.388752	−	0.921343i	$-0.372906\pi$
$359$	1.89862e6	0.777503	0.388752	+	0.921343i	$0.372906\pi$
$360$	873580.	0.355260
$361$	−2.47609e6	−0.999998
$362$	2.89935e6	1.16287
$363$	3.36082e6	1.33869
$364$	1.21080e6	0.478983
$365$	2.53180e6	0.994711
$366$	31070.8	0.0121241
$367$	1.19082e6	0.461509	0.230754	−	0.973012i	$-0.425881\pi$
$367$	1.19082e6	0.461509	0.230754	+	0.973012i	$0.425881\pi$
$368$	487.167	0.000187525 0
$369$	340083.	0.130023
$370$	−1.73143e6	−0.657507
$371$	−3.38691e6	−1.27752
$372$	62281.9	0.0233348
$373$	−254301.	−0.0946402	−0.0473201	−	0.998880i	$-0.515068\pi$
$373$	−254301.	−0.0946402	−0.0473201	+	0.998880i	$0.515068\pi$
$374$	−193601.	−0.0715697
$375$	1.44730e6	0.531472
$376$	441010.	0.160871
$377$	1.53282e6	0.555442
$378$	246650.	0.0887876
$379$	−5.16696e6	−1.84772	−0.923861	−	0.382728i	$-0.874985\pi$
$379$	−5.16696e6	−1.84772	−0.923861	+	0.382728i	$0.874985\pi$
$380$	2775.33	0.000985951 0
$381$	−2.85576e6	−1.00788
$382$	521818.	0.182962
$383$	5.10331e6	1.77769	0.888843	−	0.458212i	$-0.151510\pi$
$383$	5.10331e6	1.77769	0.888843	+	0.458212i	$0.151510\pi$
$384$	−1.03293e6	−0.357474
$385$	4.21865e6	1.45051
$386$	3.19039e6	1.08987
$387$	759183.	0.257673
$388$	3.10312e6	1.04645
$389$	−1.00400e6	−0.336403	−0.168201	−	0.985753i	$-0.553796\pi$
$389$	−1.00400e6	−0.336403	−0.168201	+	0.985753i	$0.553796\pi$
$390$	1.18426e6	0.394264
$391$	40090.5	0.0132617
$392$	1.34488e6	0.442048
$393$	−1.39149e6	−0.454462
$394$	−3.21197e6	−1.04239
$395$	−3.05057e6	−0.983757
$396$	1.17191e6	0.375540
$397$	3.62003e6	1.15275	0.576377	−	0.817184i	$-0.304466\pi$
$397$	3.62003e6	1.15275	0.576377	+	0.817184i	$0.304466\pi$
$398$	1.72523e6	0.545934
$399$	2050.66	0.000644855 0
$400$	392.887	0.000122777 0
$401$	1.18825e6	0.369016	0.184508	−	0.982831i	$-0.440931\pi$
$401$	1.18825e6	0.369016	0.184508	+	0.982831i	$0.440931\pi$
$402$	1.39436e6	0.430338
$403$	220957.	0.0677713
$404$	2.93684e6	0.895213
$405$	−391006.	−0.118453
$406$	−820753.	−0.247114
$407$	−6.07850e6	−1.81891
$408$	−123433.	−0.0367098
$409$	−64602.3	−0.0190959	−0.00954794	−	0.999954i	$-0.503039\pi$
$409$	−64602.3	−0.0190959	−0.00954794	+	0.999954i	$0.503039\pi$
$410$	−874324.	−0.256870
$411$	1.95908e6	0.572068
$412$	−3.06975e6	−0.890962
$413$	1.46082e6	0.421426
$414$	149727.	0.0429337
$415$	5.27399e6	1.50321
$416$	−3.66125e6	−1.03728
$417$	2.13222e6	0.600471
$418$	−6011.44	−0.00168282
$419$	−2.92057e6	−0.812704	−0.406352	−	0.913717i	$-0.633199\pi$
$419$	−2.92057e6	−0.812704	−0.406352	+	0.913717i	$0.633199\pi$
$420$	1.02777e6	0.284298
$421$	−2.99592e6	−0.823807	−0.411904	−	0.911227i	$-0.635136\pi$
$421$	−2.99592e6	−0.823807	−0.411904	+	0.911227i	$0.635136\pi$
$422$	−466094.	−0.127407
$423$	−197392.	−0.0536387
$424$	6.33012e6	1.71001
$425$	32331.9	0.00868280
$426$	−1.06719e6	−0.284916
$427$	95663.6	0.0253908
$428$	3.51324e6	0.927040
$429$	4.15757e6	1.09068
$430$	−1.95179e6	−0.509052
$431$	5.82982e6	1.51169	0.755844	−	0.654752i	$-0.227227\pi$
$431$	5.82982e6	1.51169	0.755844	+	0.654752i	$0.227227\pi$
$432$	671.351	0.000173077 0
$433$	−2.98624e6	−0.765429	−0.382714	−	0.923867i	$-0.625011\pi$
$433$	−2.98624e6	−0.765429	−0.382714	+	0.923867i	$0.625011\pi$
$434$	−118312.	−0.0301512
$435$	1.30111e6	0.329679
$436$	3.35072e6	0.844155
$437$	1244.83	0.000311823 0
$438$	−1.33603e6	−0.332760
$439$	−5.68591e6	−1.40812	−0.704058	−	0.710142i	$-0.748631\pi$
$439$	−5.68591e6	−1.40812	−0.704058	+	0.710142i	$0.748631\pi$
$440$	−7.88464e6	−1.94156
$441$	−601957.	−0.147390
$442$	−167332.	−0.0407401
$443$	−68926.8	−0.0166870	−0.00834352	−	0.999965i	$-0.502656\pi$
$443$	−68926.8	−0.0166870	−0.00834352	+	0.999965i	$0.502656\pi$
$444$	−1.48088e6	−0.356502
$445$	2.03125e6	0.486254
$446$	1.26750e6	0.301724
$447$	3.90777e6	0.925039
$448$	1.95757e6	0.460810
$449$	4.63362e6	1.08469	0.542343	−	0.840157i	$-0.317537\pi$
$449$	4.63362e6	1.08469	0.542343	+	0.840157i	$0.317537\pi$
$450$	120751.	0.0281099
$451$	−3.06948e6	−0.710596
$452$	−1.61704e6	−0.372284
$453$	−1.22184e6	−0.279750
$454$	−667275.	−0.151937
$455$	3.64622e6	0.825685
$456$	−3832.68	−0.000863159 0
$457$	−4.33421e6	−0.970777	−0.485388	−	0.874299i	$-0.661322\pi$
$457$	−4.33421e6	−0.970777	−0.485388	+	0.874299i	$0.661322\pi$
$458$	2.60663e6	0.580651
$459$	55247.6	0.0122400
$460$	623899.	0.137474
$461$	2.08958e6	0.457938	0.228969	−	0.973434i	$-0.426465\pi$
$461$	2.08958e6	0.457938	0.228969	+	0.973434i	$0.426465\pi$
$462$	−2.22618e6	−0.485239
$463$	−1.24932e6	−0.270846	−0.135423	−	0.990788i	$-0.543239\pi$
$463$	−1.24932e6	−0.270846	−0.135423	+	0.990788i	$0.543239\pi$
$464$	−2233.99	−0.000481710 0
$465$	187556.	0.0402252
$466$	2.22517e6	0.474678
$467$	−157302.	−0.0333765	−0.0166883	−	0.999861i	$-0.505312\pi$
$467$	−157302.	−0.0333765	−0.0166883	+	0.999861i	$0.505312\pi$
$468$	1.01289e6	0.213771
$469$	4.29309e6	0.901234
$470$	507477.	0.105967
$471$	−3.19859e6	−0.664364
$472$	−2.73027e6	−0.564092
$473$	−6.85212e6	−1.40823
$474$	1.60979e6	0.329096
$475$	1003.93	0.000204159 0
$476$	−145220.	−0.0293771
$477$	−2.83330e6	−0.570160
$478$	5.03274e6	1.00748
$479$	2.74855e6	0.547350	0.273675	−	0.961822i	$-0.411761\pi$
$479$	2.74855e6	0.547350	0.273675	+	0.961822i	$0.411761\pi$
$480$	−3.10779e6	−0.615671
$481$	−5.25371e6	−1.03539
$482$	3.95645e6	0.775689
$483$	460992.	0.0899137
$484$	−7.39005e6	−1.43395
$485$	9.34475e6	1.80390
$486$	206334.	0.0396260
$487$	−1.01838e7	−1.94576	−0.972878	−	0.231319i	$-0.925696\pi$
$487$	−1.01838e7	−1.94576	−0.972878	+	0.231319i	$0.925696\pi$
$488$	−178795.	−0.0339864
$489$	−4.62909e6	−0.875434
$490$	1.54758e6	0.291181
$491$	−8.81893e6	−1.65087	−0.825434	−	0.564499i	$-0.809069\pi$
$491$	−8.81893e6	−1.65087	−0.825434	+	0.564499i	$0.809069\pi$
$492$	−747804.	−0.139276
$493$	−183842.	−0.0340664
$494$	−5195.75	−0.000957924 0
$495$	3.52909e6	0.647365
$496$	−322.030	−5.87750e−5 0
$497$	−3.28576e6	−0.596685
$498$	−2.78309e6	−0.502868
$499$	−7.27705e6	−1.30829	−0.654145	−	0.756369i	$-0.726971\pi$
$499$	−7.27705e6	−1.30829	−0.654145	+	0.756369i	$0.726971\pi$
$500$	−3.18244e6	−0.569293
$501$	1.60457e6	0.285604
$502$	−2.17530e6	−0.385265
$503$	−2.82184e6	−0.497293	−0.248646	−	0.968594i	$-0.579986\pi$
$503$	−2.82184e6	−0.497293	−0.248646	+	0.968594i	$0.579986\pi$
$504$	−1.41933e6	−0.248891
$505$	8.84400e6	1.54319
$506$	−1.35138e6	−0.234640
$507$	251797.	0.0435042
$508$	6.27949e6	1.07961
$509$	−4.78063e6	−0.817881	−0.408941	−	0.912561i	$-0.634102\pi$
$509$	−4.78063e6	−0.817881	−0.408941	+	0.912561i	$0.634102\pi$
$510$	−142037.	−0.0241810
$511$	−4.11350e6	−0.696882
$512$	10669.1	0.00179867
$513$	1715.47	0.000287800 0
$514$	−7.34864e6	−1.22687
$515$	−9.24424e6	−1.53587
$516$	−1.66935e6	−0.276010
$517$	1.78159e6	0.293145
$518$	2.81311e6	0.460640
$519$	−4.82329e6	−0.786004
$520$	−6.81477e6	−1.10521
$521$	7.11976e6	1.14914	0.574568	−	0.818457i	$-0.305170\pi$
$521$	7.11976e6	1.14914	0.574568	+	0.818457i	$0.305170\pi$
$522$	−686597.	−0.110287
$523$	4.97980e6	0.796082	0.398041	−	0.917368i	$-0.369690\pi$
$523$	4.97980e6	0.796082	0.398041	+	0.917368i	$0.369690\pi$
$524$	3.05972e6	0.486803
$525$	371779.	0.0588689
$526$	2.92920e6	0.461620
$527$	−26500.9	−0.00415656
$528$	−6059.38	−0.000945896 0
$529$	279841.	0.0434783
$530$	7.28417e6	1.12639
$531$	1.22204e6	0.188083
$532$	−4509.17	−0.000690744 0
$533$	−2.65298e6	−0.404498
$534$	−1.07189e6	−0.162666
$535$	1.05798e7	1.59806
$536$	−8.02376e6	−1.20633
$537$	−530021.	−0.0793153
$538$	3.13624e6	0.467146
$539$	5.43306e6	0.805513
$540$	859777.	0.126882
$541$	2.34995e6	0.345196	0.172598	−	0.984992i	$-0.444784\pi$
$541$	2.34995e6	0.345196	0.172598	+	0.984992i	$0.444784\pi$
$542$	−3.49737e6	−0.511379
$543$	7.46767e6	1.08689
$544$	439118.	0.0636186
$545$	1.00904e7	1.45518
$546$	−1.92411e6	−0.276216
$547$	1.27753e7	1.82559	0.912795	−	0.408419i	$-0.133920\pi$
$547$	1.27753e7	1.82559	0.912795	+	0.408419i	$0.133920\pi$
$548$	−4.30779e6	−0.612778
$549$	80026.9	0.0113320
$550$	−1.08986e6	−0.153625
$551$	−5708.40	−0.000801005 0
$552$	−861593.	−0.120352
$553$	4.95636e6	0.689207
$554$	7.33708e6	1.01566
$555$	−4.45952e6	−0.614548
$556$	−4.68851e6	−0.643203
$557$	−6.13111e6	−0.837339	−0.418670	−	0.908139i	$-0.637503\pi$
$557$	−6.13111e6	−0.837339	−0.418670	+	0.908139i	$0.637503\pi$
$558$	−98973.4	−0.0134565
$559$	−5.92236e6	−0.801614
$560$	−5314.12	−0.000716079 0
$561$	−498646.	−0.0668937
$562$	1.75842e6	0.234845
$563$	1.30613e7	1.73667	0.868334	−	0.495979i	$-0.165191\pi$
$563$	1.30613e7	1.73667	0.868334	+	0.495979i	$0.165191\pi$
$564$	434042.	0.0574558
$565$	−4.86956e6	−0.641754
$566$	−5.65274e6	−0.741682
$567$	635281.	0.0829866
$568$	6.14108e6	0.798682
$569$	−5.03952e6	−0.652542	−0.326271	−	0.945276i	$-0.605792\pi$
$569$	−5.03952e6	−0.652542	−0.326271	+	0.945276i	$0.605792\pi$
$570$	−4410.33	−0.000568569 0
$571$	−7.67025e6	−0.984508	−0.492254	−	0.870452i	$-0.663827\pi$
$571$	−7.67025e6	−0.984508	−0.492254	+	0.870452i	$0.663827\pi$
$572$	−9.14202e6	−1.16829
$573$	1.34401e6	0.171008
$574$	1.42054e6	0.179960
$575$	225685.	0.0284664
$576$	1.63759e6	0.205660
$577$	1.25985e7	1.57536	0.787681	−	0.616083i	$-0.211281\pi$
$577$	1.25985e7	1.57536	0.787681	+	0.616083i	$0.211281\pi$
$578$	−4.94132e6	−0.615210
$579$	8.21727e6	1.01866
$580$	−2.86099e6	−0.353140
$581$	−8.56883e6	−1.05313
$582$	−4.93123e6	−0.603459
$583$	2.55724e7	3.11602
$584$	7.68811e6	0.932798
$585$	3.05023e6	0.368504
$586$	3.65226e6	0.439357
$587$	5.11295e6	0.612457	0.306229	−	0.951958i	$-0.400933\pi$
$587$	5.11295e6	0.612457	0.306229	+	0.951958i	$0.400933\pi$
$588$	1.32363e6	0.157879
$589$	−822.869	−9.77333e−5 0
$590$	−3.14176e6	−0.371572
$591$	−8.27285e6	−0.974286
$592$	7656.92	0.000897945 0
$593$	5.45932e6	0.637532	0.318766	−	0.947833i	$-0.396732\pi$
$593$	5.45932e6	0.637532	0.318766	+	0.947833i	$0.396732\pi$
$594$	−1.86230e6	−0.216563
$595$	−437315.	−0.0506410
$596$	−8.59273e6	−0.990867
$597$	4.44357e6	0.510265
$598$	−1.16801e6	−0.133566
$599$	−22496.6	−0.00256183	−0.00128091	−	0.999999i	$-0.500408\pi$
$599$	−22496.6	−0.00256183	−0.00128091	+	0.999999i	$0.500408\pi$
$600$	−694853.	−0.0787979
$601$	1.00912e6	0.113962	0.0569808	−	0.998375i	$-0.481853\pi$
$601$	1.00912e6	0.113962	0.0569808	+	0.998375i	$0.481853\pi$
$602$	3.17114e6	0.356635
$603$	3.59136e6	0.402222
$604$	2.68669e6	0.299657
$605$	−2.22544e7	−2.47188
$606$	−4.66698e6	−0.516244
$607$	−68503.9	−0.00754646	−0.00377323	−	0.999993i	$-0.501201\pi$
$607$	−68503.9	−0.00754646	−0.00377323	+	0.999993i	$0.501201\pi$
$608$	13634.9	0.00149587
$609$	−2.11396e6	−0.230969
$610$	−205742.	−0.0223871
$611$	1.53985e6	0.166869
$612$	−121483.	−0.0131110
$613$	7.67884e6	0.825362	0.412681	−	0.910876i	$-0.364593\pi$
$613$	7.67884e6	0.825362	0.412681	+	0.910876i	$0.364593\pi$
$614$	−2.43740e6	−0.260919
$615$	−2.25194e6	−0.240087
$616$	1.28104e7	1.36023
$617$	−1.85528e6	−0.196199	−0.0980995	−	0.995177i	$-0.531276\pi$
$617$	−1.85528e6	−0.196199	−0.0980995	+	0.995177i	$0.531276\pi$
$618$	4.87819e6	0.513792
$619$	−2.88941e6	−0.303098	−0.151549	−	0.988450i	$-0.548426\pi$
$619$	−2.88941e6	−0.303098	−0.151549	+	0.988450i	$0.548426\pi$
$620$	−412414.	−0.0430878
$621$	385641.	0.0401286
$622$	−9.38632e6	−0.972791
$623$	−3.30024e6	−0.340663
$624$	−5237.19	−0.000538439 0
$625$	−1.09168e7	−1.11788
$626$	5.66851e6	0.578140
$627$	−15483.3	−0.00157287
$628$	7.03332e6	0.711642
$629$	630112.	0.0635026
$630$	−1.63325e6	−0.163946
$631$	−1.22946e7	−1.22925	−0.614625	−	0.788819i	$-0.710693\pi$
$631$	−1.22946e7	−1.22925	−0.614625	+	0.788819i	$0.710693\pi$
$632$	−9.26341e6	−0.922525
$633$	−1.20049e6	−0.119083
$634$	2.26489e6	0.223782
$635$	1.89101e7	1.86105
$636$	6.23010e6	0.610735
$637$	4.69585e6	0.458528
$638$	6.19699e6	0.602739
$639$	−2.74869e6	−0.266301
$640$	6.83980e6	0.660076
$641$	−7.87250e6	−0.756776	−0.378388	−	0.925647i	$-0.623522\pi$
$641$	−7.87250e6	−0.756776	−0.378388	+	0.925647i	$0.623522\pi$
$642$	−5.58296e6	−0.534597
$643$	3.14575e6	0.300052	0.150026	−	0.988682i	$-0.452064\pi$
$643$	3.14575e6	0.300052	0.150026	+	0.988682i	$0.452064\pi$
$644$	−1.01367e6	−0.0963122
$645$	−5.02710e6	−0.475793
$646$	623.161	5.87515e−5 0
$647$	4.14059e6	0.388867	0.194434	−	0.980916i	$-0.437713\pi$
$647$	4.14059e6	0.388867	0.194434	+	0.980916i	$0.437713\pi$
$648$	−1.18734e6	−0.111080
$649$	−1.10297e7	−1.02791
$650$	−941973.	−0.0874490
$651$	−304728.	−0.0281813
$652$	1.01788e7	0.937733
$653$	1.09894e7	1.00854	0.504268	−	0.863547i	$-0.331762\pi$
$653$	1.09894e7	1.00854	0.504268	+	0.863547i	$0.331762\pi$
$654$	−5.32470e6	−0.486800
$655$	9.21404e6	0.839164
$656$	3866.54	0.000350803 0
$657$	−3.44113e6	−0.311019
$658$	−824515.	−0.0742393
$659$	1.78286e7	1.59920	0.799602	−	0.600530i	$-0.205044\pi$
$659$	1.78286e7	1.59920	0.799602	+	0.600530i	$0.205044\pi$
$660$	−7.76005e6	−0.693434
$661$	4.39680e6	0.391411	0.195705	−	0.980663i	$-0.437300\pi$
$661$	4.39680e6	0.391411	0.195705	+	0.980663i	$0.437300\pi$
$662$	−5.81972e6	−0.516128
$663$	−430985.	−0.0380783
$664$	1.60151e7	1.40965
$665$	−13578.9	−0.00119072
$666$	2.35329e6	0.205584
$667$	−1.28326e6	−0.111686
$668$	−3.52826e6	−0.305928
$669$	3.26461e6	0.282011
$670$	−9.23307e6	−0.794619
$671$	−722295.	−0.0619311
$672$	5.04933e6	0.431331
$673$	−1.98954e7	−1.69322	−0.846612	−	0.532210i	$-0.821362\pi$
$673$	−1.98954e7	−1.69322	−0.846612	+	0.532210i	$0.821362\pi$
$674$	−7.40591e6	−0.627955
$675$	311010.	0.0262733
$676$	−553673.	−0.0466001
$677$	−796059.	−0.0667535	−0.0333767	−	0.999443i	$-0.510626\pi$
$677$	−796059.	−0.0667535	−0.0333767	+	0.999443i	$0.510626\pi$
$678$	2.56967e6	0.214686
$679$	−1.51827e7	−1.26379
$680$	817341.	0.0677846
$681$	−1.71865e6	−0.142011
$682$	893300.	0.0735421
$683$	−9.97449e6	−0.818162	−0.409081	−	0.912498i	$-0.634151\pi$
$683$	−9.97449e6	−0.818162	−0.409081	+	0.912498i	$0.634151\pi$
$684$	−3772.12	−0.000308280 0
$685$	−1.29725e7	−1.05632
$686$	−8.20089e6	−0.665351
$687$	6.71371e6	0.542714
$688$	8631.44	0.000695204 0
$689$	2.21025e7	1.77375
$690$	−991449.	−0.0792771
$691$	−6.67183e6	−0.531557	−0.265779	−	0.964034i	$-0.585629\pi$
$691$	−6.67183e6	−0.531557	−0.265779	+	0.964034i	$0.585629\pi$
$692$	1.06058e7	0.841938
$693$	−5.73383e6	−0.453536
$694$	−2.76649e6	−0.218037
$695$	−1.41190e7	−1.10877
$696$	3.95098e6	0.309159
$697$	318190.	0.0248087
$698$	2.59898e6	0.201913
$699$	5.73123e6	0.443665
$700$	−817498.	−0.0630582
$701$	5.64132e6	0.433597	0.216798	−	0.976216i	$-0.430439\pi$
$701$	5.64132e6	0.433597	0.216798	+	0.976216i	$0.430439\pi$
$702$	−1.60961e6	−0.123276
$703$	19565.4	0.00149314
$704$	−1.47804e7	−1.12397
$705$	1.30708e6	0.0990439
$706$	−2.25831e6	−0.170519
$707$	−1.43691e7	−1.08114
$708$	−2.68713e6	−0.201468
$709$	−1.86950e6	−0.139672	−0.0698361	−	0.997558i	$-0.522248\pi$
$709$	−1.86950e6	−0.139672	−0.0698361	+	0.997558i	$0.522248\pi$
$710$	7.06663e6	0.526098
$711$	4.14622e6	0.307594
$712$	6.16813e6	0.455988
$713$	−184982.	−0.0136272
$714$	230772.	0.0169409
$715$	−2.75303e7	−2.01394
$716$	1.16545e6	0.0849596
$717$	1.29625e7	0.941652
$718$	6.63433e6	0.480270
$719$	1.87348e7	1.35153	0.675766	−	0.737116i	$-0.263813\pi$
$719$	1.87348e7	1.35153	0.675766	+	0.737116i	$0.263813\pi$
$720$	−4445.50	−0.000319587 0
$721$	1.50194e7	1.07601
$722$	−8.65218e6	−0.617707
$723$	1.01904e7	0.725009
$724$	−1.64205e7	−1.16424
$725$	−1.03492e6	−0.0731240
$726$	1.17437e7	0.826918
$727$	7.88165e6	0.553071	0.276536	−	0.961004i	$-0.410814\pi$
$727$	7.88165e6	0.553071	0.276536	+	0.961004i	$0.410814\pi$
$728$	1.10722e7	0.774292
$729$	531441.	0.0370370
$730$	8.84683e6	0.614442
$731$	710308.	0.0491647
$732$	−175970.	−0.0121384
$733$	196627.	0.0135171	0.00675854	−	0.999977i	$-0.497849\pi$
$733$	196627.	0.0135171	0.00675854	+	0.999977i	$0.497849\pi$
$734$	4.16106e6	0.285078
$735$	3.98599e6	0.272156
$736$	3.06515e6	0.208572
$737$	−3.24144e7	−2.19821
$738$	1.18835e6	0.0803162
$739$	−1.89161e7	−1.27415	−0.637074	−	0.770803i	$-0.719855\pi$
$739$	−1.89161e7	−1.27415	−0.637074	+	0.770803i	$0.719855\pi$
$740$	9.80597e6	0.658281
$741$	−13382.3	−0.000895337 0
$742$	−1.18348e7	−0.789137
$743$	−5.29482e6	−0.351867	−0.175934	−	0.984402i	$-0.556294\pi$
$743$	−5.29482e6	−0.351867	−0.175934	+	0.984402i	$0.556294\pi$
$744$	569536.	0.0377215
$745$	−2.58762e7	−1.70808
$746$	−888600.	−0.0584601
$747$	−7.16822e6	−0.470013
$748$	1.09646e6	0.0716540
$749$	−1.71893e7	−1.11958
$750$	5.05728e6	0.328295
$751$	−1.23104e7	−0.796478	−0.398239	−	0.917282i	$-0.630379\pi$
$751$	−1.23104e7	−0.796478	−0.398239	+	0.917282i	$0.630379\pi$
$752$	−2244.23	−0.000144718 0
$753$	−5.60276e6	−0.360093
$754$	5.35612e6	0.343101
$755$	8.09070e6	0.516558
$756$	−1.39691e6	−0.0888922
$757$	−7.33051e6	−0.464938	−0.232469	−	0.972604i	$-0.574680\pi$
$757$	−7.33051e6	−0.464938	−0.232469	+	0.972604i	$0.574680\pi$
$758$	−1.80548e7	−1.14135
$759$	−3.48066e6	−0.219310
$760$	25379.0	0.00159382
$761$	3.11913e6	0.195242	0.0976208	−	0.995224i	$-0.468877\pi$
$761$	3.11913e6	0.195242	0.0976208	+	0.995224i	$0.468877\pi$
$762$	−9.97886e6	−0.622577
$763$	−1.63942e7	−1.01948
$764$	−2.95533e6	−0.183178
$765$	−365834.	−0.0226012
$766$	1.78324e7	1.09809
$767$	−9.53311e6	−0.585122
$768$	−9.43193e6	−0.577029
$769$	−7.07903e6	−0.431676	−0.215838	−	0.976429i	$-0.569248\pi$
$769$	−7.07903e6	−0.431676	−0.215838	+	0.976429i	$0.569248\pi$
$770$	1.47412e7	0.895993
$771$	−1.89274e7	−1.14671
$772$	−1.80688e7	−1.09116
$773$	1.91702e7	1.15393	0.576964	−	0.816770i	$-0.304237\pi$
$773$	1.91702e7	1.15393	0.576964	+	0.816770i	$0.304237\pi$
$774$	2.65280e6	0.159167
$775$	−149184.	−0.00892209
$776$	2.83765e7	1.69162
$777$	7.24553e6	0.430544
$778$	−3.50826e6	−0.207799
$779$	9879.99	0.000583328 0
$780$	−6.70710e6	−0.394728
$781$	2.48087e7	1.45538
$782$	140088.	0.00819187
$783$	−1.76842e6	−0.103082
$784$	−6843.88	−0.000397660 0
$785$	2.11802e7	1.22675
$786$	−4.86225e6	−0.280725
$787$	8.38766e6	0.482730	0.241365	−	0.970434i	$-0.422405\pi$
$787$	8.38766e6	0.482730	0.241365	+	0.970434i	$0.422405\pi$
$788$	1.81910e7	1.04362
$789$	7.54454e6	0.431460
$790$	−1.06596e7	−0.607675
$791$	7.91173e6	0.449604
$792$	1.07165e7	0.607072
$793$	−624287.	−0.0352534
$794$	1.26494e7	0.712066
$795$	1.87613e7	1.05280
$796$	−9.77089e6	−0.546577
$797$	−1.36638e7	−0.761950	−0.380975	−	0.924585i	$-0.624412\pi$
$797$	−1.36638e7	−0.761950	−0.380975	+	0.924585i	$0.624412\pi$
$798$	7165.60	0.000398332 0
$799$	−184684.	−0.0102344
$800$	2.47196e6	0.136558
$801$	−2.76080e6	−0.152038
$802$	4.15207e6	0.227944
$803$	3.10584e7	1.69977
$804$	−7.89698e6	−0.430845
$805$	−3.05256e6	−0.166026
$806$	772088.	0.0418629
$807$	8.07779e6	0.436625
$808$	2.68559e7	1.44714
$809$	5.16198e6	0.277297	0.138648	−	0.990342i	$-0.455724\pi$
$809$	5.16198e6	0.277297	0.138648	+	0.990342i	$0.455724\pi$
$810$	−1.36629e6	−0.0731694
$811$	1.20275e7	0.642131	0.321065	−	0.947057i	$-0.395959\pi$
$811$	1.20275e7	0.642131	0.321065	+	0.947057i	$0.395959\pi$
$812$	4.64835e6	0.247405
$813$	−9.00793e6	−0.477968
$814$	−2.12400e7	−1.12355
$815$	3.06526e7	1.61649
$816$	628.131	3.30236e−5 0
$817$	22055.5	0.00115601
$818$	−225739.	−0.0117957
$819$	−4.95580e6	−0.258169
$820$	4.95175e6	0.257172
$821$	−4.25894e6	−0.220518	−0.110259	−	0.993903i	$-0.535168\pi$
$821$	−4.25894e6	−0.220518	−0.110259	+	0.993903i	$0.535168\pi$
$822$	6.84560e6	0.353372
$823$	1.63473e7	0.841290	0.420645	−	0.907225i	$-0.361804\pi$
$823$	1.63473e7	0.841290	0.420645	+	0.907225i	$0.361804\pi$
$824$	−2.80712e7	−1.44027
$825$	−2.80707e6	−0.143588
$826$	5.10452e6	0.260318
$827$	−1.62566e7	−0.826542	−0.413271	−	0.910608i	$-0.635614\pi$
$827$	−1.62566e7	−0.826542	−0.413271	+	0.910608i	$0.635614\pi$
$828$	−847980.	−0.0429843
$829$	3.87989e7	1.96080	0.980400	−	0.197018i	$-0.0631256\pi$
$829$	3.87989e7	1.96080	0.980400	+	0.197018i	$0.0631256\pi$
$830$	1.84289e7	0.928545
$831$	1.88976e7	0.949302
$832$	−1.27748e7	−0.639804
$833$	−563205.	−0.0281225
$834$	7.45060e6	0.370916
$835$	−1.06250e7	−0.527367
$836$	34045.9	0.00168480
$837$	−254919.	−0.0125773
$838$	−1.02053e7	−0.502014
$839$	−2.96779e7	−1.45555	−0.727776	−	0.685815i	$-0.759446\pi$
$839$	−2.96779e7	−1.45555	−0.727776	+	0.685815i	$0.759446\pi$
$840$	9.39844e6	0.459576
$841$	−1.46265e7	−0.713102
$842$	−1.04686e7	−0.508873
$843$	4.52904e6	0.219502
$844$	2.63973e6	0.127557
$845$	−1.66733e6	−0.0803305
$846$	−689744.	−0.0331331
$847$	3.61575e7	1.73177
$848$	−32212.9	−0.00153830
$849$	−1.45594e7	−0.693224
$850$	112977.	0.00536344
$851$	4.39833e6	0.208192
$852$	6.04405e6	0.285252
$853$	−3.20704e7	−1.50915	−0.754573	−	0.656216i	$-0.772156\pi$
$853$	−3.20704e7	−1.50915	−0.754573	+	0.656216i	$0.772156\pi$
$854$	334276.	0.0156841
$855$	−11359.4	−0.000531422 0
$856$	3.21268e7	1.49859
$857$	−1.10540e7	−0.514123	−0.257061	−	0.966395i	$-0.582754\pi$
$857$	−1.10540e7	−0.514123	−0.257061	+	0.966395i	$0.582754\pi$
$858$	1.45278e7	0.673721
$859$	1.49699e6	0.0692206	0.0346103	−	0.999401i	$-0.488981\pi$
$859$	1.49699e6	0.0692206	0.0346103	+	0.999401i	$0.488981\pi$
$860$	1.10540e7	0.509652
$861$	3.65880e6	0.168202
$862$	2.03711e7	0.933783
$863$	7.26672e6	0.332132	0.166066	−	0.986115i	$-0.446893\pi$
$863$	7.26672e6	0.332132	0.166066	+	0.986115i	$0.446893\pi$
$864$	4.22399e6	0.192503
$865$	3.19385e7	1.45136
$866$	−1.04348e7	−0.472812
$867$	−1.27270e7	−0.575015
$868$	670062.	0.0301867
$869$	−3.74223e7	−1.68105
$870$	4.54646e6	0.203646
$871$	−2.80161e7	−1.25130
$872$	3.06406e7	1.36460
$873$	−1.27010e7	−0.564032
$874$	4349.81	0.000192616 0
$875$	1.55708e7	0.687529
$876$	7.56664e6	0.333152
$877$	−3.36708e6	−0.147827	−0.0739137	−	0.997265i	$-0.523549\pi$
$877$	−3.36708e6	−0.147827	−0.0739137	+	0.997265i	$0.523549\pi$
$878$	−1.98682e7	−0.869806
$879$	9.40688e6	0.410652
$880$	40123.6	0.00174660
$881$	−1.40164e7	−0.608409	−0.304204	−	0.952607i	$-0.598391\pi$
$881$	−1.40164e7	−0.608409	−0.304204	+	0.952607i	$0.598391\pi$
$882$	−2.10341e6	−0.0910443
$883$	−4.00921e7	−1.73044	−0.865220	−	0.501392i	$-0.832821\pi$
$883$	−4.00921e7	−1.73044	−0.865220	+	0.501392i	$0.832821\pi$
$884$	947685.	0.0407881
$885$	−8.09202e6	−0.347295
$886$	−240850.	−0.0103077
$887$	3.57779e7	1.52688	0.763442	−	0.645877i	$-0.223508\pi$
$887$	3.57779e7	1.52688	0.763442	+	0.645877i	$0.223508\pi$
$888$	−1.35419e7	−0.576297
$889$	−3.07238e7	−1.30383
$890$	7.09777e6	0.300363
$891$	−4.79660e6	−0.202414
$892$	−7.17850e6	−0.302080
$893$	−5734.56	−0.000240642 0
$894$	1.36549e7	0.571405
$895$	3.50965e6	0.146456
$896$	−1.11129e7	−0.462440
$897$	−3.00837e6	−0.124839
$898$	1.61912e7	0.670020
$899$	848268.	0.0350053
$900$	−683874.	−0.0281430
$901$	−2.65090e6	−0.108788
$902$	−1.07256e7	−0.438941
$903$	8.16769e6	0.333334
$904$	−1.47870e7	−0.601810
$905$	−4.94488e7	−2.00694
$906$	−4.26947e6	−0.172804
$907$	2.64880e6	0.106913	0.0534565	−	0.998570i	$-0.482976\pi$
$907$	2.64880e6	0.106913	0.0534565	+	0.998570i	$0.482976\pi$
$908$	3.77912e6	0.152116
$909$	−1.20204e7	−0.482515
$910$	1.27409e7	0.510033
$911$	−3.49769e7	−1.39632	−0.698161	−	0.715941i	$-0.745998\pi$
$911$	−3.49769e7	−1.39632	−0.698161	+	0.715941i	$0.745998\pi$
$912$	19.5039	7.76486e−7 0
$913$	6.46979e7	2.56870
$914$	−1.51450e7	−0.599657
$915$	−529916.	−0.0209245
$916$	−1.47627e7	−0.581335
$917$	−1.49704e7	−0.587907
$918$	193051.	0.00756075
$919$	−2.60166e7	−1.01616	−0.508079	−	0.861310i	$-0.669644\pi$
$919$	−2.60166e7	−1.01616	−0.508079	+	0.861310i	$0.669644\pi$
$920$	5.70523e6	0.222231
$921$	−6.27785e6	−0.243872
$922$	7.30160e6	0.282872
$923$	2.14424e7	0.828457
$924$	1.26080e7	0.485810
$925$	3.54714e6	0.136309
$926$	−4.36550e6	−0.167304
$927$	1.25644e7	0.480223
$928$	−1.40557e7	−0.535777
$929$	4.76233e7	1.81042	0.905212	−	0.424960i	$-0.139712\pi$
$929$	4.76233e7	1.81042	0.905212	+	0.424960i	$0.139712\pi$
$930$	655374.	0.0248475
$931$	−17487.9	−0.000661245 0
$932$	−1.26023e7	−0.475237
$933$	−2.41757e7	−0.909233
$934$	−549657.	−0.0206170
$935$	3.30189e6	0.123519
$936$	9.26238e6	0.345568
$937$	−5.81079e6	−0.216215	−0.108108	−	0.994139i	$-0.534479\pi$
$937$	−5.81079e6	−0.216215	−0.108108	+	0.994139i	$0.534479\pi$
$938$	1.50013e7	0.556700
$939$	1.46000e7	0.540367
$940$	−2.87411e6	−0.106092
$941$	−6.47870e6	−0.238514	−0.119257	−	0.992863i	$-0.538051\pi$
$941$	−6.47870e6	−0.238514	−0.119257	+	0.992863i	$0.538051\pi$
$942$	−1.11768e7	−0.410383
$943$	2.22104e6	0.0813349
$944$	13893.9	0.000507450 0
$945$	−4.20665e6	−0.153235
$946$	−2.39433e7	−0.869873
$947$	−4.16257e6	−0.150830	−0.0754149	−	0.997152i	$-0.524028\pi$
$947$	−4.16257e6	−0.150830	−0.0754149	+	0.997152i	$0.524028\pi$
$948$	−9.11705e6	−0.329483
$949$	2.68441e7	0.967573
$950$	3508.01	0.000126111 0
$951$	5.83354e6	0.209161
$952$	−1.32796e6	−0.0474890
$953$	−2.93375e7	−1.04638	−0.523191	−	0.852216i	$-0.675259\pi$
$953$	−2.93375e7	−1.04638	−0.523191	+	0.852216i	$0.675259\pi$
$954$	−9.90037e6	−0.352193
$955$	−8.89968e6	−0.315766
$956$	−2.85030e7	−1.00866
$957$	1.59612e7	0.563359
$958$	9.60422e6	0.338103
$959$	2.10768e7	0.740047
$960$	−1.08437e7	−0.379751
$961$	−2.85069e7	−0.995729
$962$	−1.83580e7	−0.639568
$963$	−1.43796e7	−0.499669
$964$	−2.24074e7	−0.776603
$965$	−5.44125e7	−1.88096
$966$	1.61084e6	0.0555405
$967$	4.34493e6	0.149423	0.0747113	−	0.997205i	$-0.476196\pi$
$967$	4.34493e6	0.149423	0.0747113	+	0.997205i	$0.476196\pi$
$968$	−6.75783e7	−2.31803
$969$	1605.03	5.49129e−5 0
$970$	3.26532e7	1.11429
$971$	−1.84656e7	−0.628514	−0.314257	−	0.949338i	$-0.601755\pi$
$971$	−1.84656e7	−0.628514	−0.314257	+	0.949338i	$0.601755\pi$
$972$	−1.16858e6	−0.0396727
$973$	2.29396e7	0.776790
$974$	−3.55852e7	−1.20191
$975$	−2.42618e6	−0.0817355
$976$	909.857	3.05737e−5 0
$977$	4.05887e7	1.36041	0.680203	−	0.733023i	$-0.261891\pi$
$977$	4.05887e7	1.36041	0.680203	+	0.733023i	$0.261891\pi$
$978$	−1.61754e7	−0.540763
$979$	2.49180e7	0.830915
$980$	−8.76474e6	−0.291524
$981$	−1.37145e7	−0.454995
$982$	−3.08159e7	−1.01975
$983$	5.29637e7	1.74822	0.874108	−	0.485732i	$-0.161447\pi$
$983$	5.29637e7	1.74822	0.874108	+	0.485732i	$0.161447\pi$
$984$	−6.83828e6	−0.225143
$985$	5.47805e7	1.79902
$986$	−642396.	−0.0210431
$987$	−2.12365e6	−0.0693888
$988$	29426.2	0.000959052 0
$989$	4.95812e6	0.161186
$990$	1.23316e7	0.399883
$991$	1.26953e7	0.410636	0.205318	−	0.978695i	$-0.434177\pi$
$991$	1.26953e7	0.410636	0.205318	+	0.978695i	$0.434177\pi$
$992$	−2.02614e6	−0.0653719
$993$	−1.49895e7	−0.482406
$994$	−1.14814e7	−0.368577
$995$	−2.94241e7	−0.942205
$996$	1.57621e7	0.503460
$997$	4.65074e7	1.48178	0.740890	−	0.671626i	$-0.234404\pi$
$997$	4.65074e7	1.48178	0.740890	+	0.671626i	$0.234404\pi$
$998$	−2.54281e7	−0.808142
$999$	6.06122e6	0.192153

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.b.1.3	✓	3
3.2	odd	2		207.6.a.c.1.1		3
4.3	odd	2		1104.6.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.b.1.3	✓	3	1.1	even	1	trivial
207.6.a.c.1.1		3	3.2	odd	2
1104.6.a.i.1.1		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+3.49429 q^{2} +9.00000 q^{3} -19.7900 q^{4} -59.5955 q^{5} +31.4486 q^{6} +96.8268 q^{7} -180.969 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+3.49429 q^{2} +9.00000 q^{3} -19.7900 q^{4} -59.5955 q^{5} +31.4486 q^{6} +96.8268 q^{7} -180.969 q^{8} +81.0000 q^{9} -208.244 q^{10} -731.078 q^{11} -178.110 q^{12} -631.879 q^{13} +338.341 q^{14} -536.360 q^{15} +0.920920 q^{16} +75.7854 q^{17} +283.037 q^{18} +2.35318 q^{19} +1179.39 q^{20} +871.441 q^{21} -2554.60 q^{22} +529.000 q^{23} -1628.72 q^{24} +426.625 q^{25} -2207.97 q^{26} +729.000 q^{27} -1916.20 q^{28} -2425.82 q^{29} -1874.19 q^{30} -349.683 q^{31} +5794.23 q^{32} -6579.70 q^{33} +264.816 q^{34} -5770.44 q^{35} -1602.99 q^{36} +8314.43 q^{37} +8.22270 q^{38} -5686.91 q^{39} +10784.9 q^{40} +4198.56 q^{41} +3045.07 q^{42} +9372.62 q^{43} +14468.0 q^{44} -4827.24 q^{45} +1848.48 q^{46} -2436.94 q^{47} +8.28828 q^{48} -7431.57 q^{49} +1490.75 q^{50} +682.069 q^{51} +12504.8 q^{52} -34979.0 q^{53} +2547.34 q^{54} +43569.0 q^{55} -17522.6 q^{56} +21.1787 q^{57} -8476.51 q^{58} +15086.9 q^{59} +10614.5 q^{60} +987.986 q^{61} -1221.89 q^{62} +7842.97 q^{63} +20217.2 q^{64} +37657.1 q^{65} -22991.4 q^{66} +44337.8 q^{67} -1499.79 q^{68} +4761.00 q^{69} -20163.6 q^{70} -33934.4 q^{71} -14658.5 q^{72} -42483.0 q^{73} +29053.0 q^{74} +3839.62 q^{75} -46.5694 q^{76} -70788.0 q^{77} -19871.7 q^{78} +51187.8 q^{79} -54.8827 q^{80} +6561.00 q^{81} +14671.0 q^{82} -88496.5 q^{83} -17245.8 q^{84} -4516.47 q^{85} +32750.6 q^{86} -21832.4 q^{87} +132303. q^{88} -34083.9 q^{89} -16867.8 q^{90} -61182.8 q^{91} -10468.9 q^{92} -3147.15 q^{93} -8515.36 q^{94} -140.239 q^{95} +52148.0 q^{96} -156803. q^{97} -25968.0 q^{98} -59217.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{2} + 27 q^{3} + 22 q^{4} - 56 q^{5} - 72 q^{6} - 114 q^{7} - 510 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q - 8 * q^2 + 27 * q^3 + 22 * q^4 - 56 * q^5 - 72 * q^6 - 114 * q^7 - 510 * q^8 + 243 * q^9	\( 3 q - 8 q^{2} + 27 q^{3} + 22 q^{4} - 56 q^{5} - 72 q^{6} - 114 q^{7} - 510 q^{8} + 243 q^{9} + 282 q^{10} - 376 q^{11} + 198 q^{12} - 858 q^{13} + 588 q^{14} - 504 q^{15} + 2738 q^{16} - 2548 q^{17} - 648 q^{18} - 2846 q^{19} - 4618 q^{20} - 1026 q^{21} - 5050 q^{22} + 1587 q^{23} - 4590 q^{24} + 753 q^{25} - 7788 q^{26} + 2187 q^{27} + 4736 q^{28} - 16370 q^{29} + 2538 q^{30} - 14756 q^{31} - 3878 q^{32} - 3384 q^{33} + 16520 q^{34} - 18520 q^{35} + 1782 q^{36} + 15874 q^{37} + 12438 q^{38} - 7722 q^{39} + 38270 q^{40} + 12606 q^{41} + 5292 q^{42} + 3154 q^{43} + 27114 q^{44} - 4536 q^{45} - 4232 q^{46} + 29928 q^{47} + 24642 q^{48} + 4471 q^{49} + 1452 q^{50} - 22932 q^{51} + 86856 q^{52} - 44084 q^{53} - 5832 q^{54} + 38360 q^{55} - 35704 q^{56} - 25614 q^{57} + 73316 q^{58} - 29300 q^{59} - 41562 q^{60} + 54010 q^{61} + 99908 q^{62} - 9234 q^{63} - 1582 q^{64} - 51216 q^{65} - 45450 q^{66} + 43390 q^{67} - 69840 q^{68} + 14283 q^{69} - 2476 q^{70} + 23424 q^{71} - 41310 q^{72} - 91402 q^{73} - 2294 q^{74} + 6777 q^{75} - 14274 q^{76} - 97208 q^{77} - 70092 q^{78} - 49398 q^{79} - 52626 q^{80} + 19683 q^{81} + 40152 q^{82} - 103936 q^{83} + 42624 q^{84} + 5888 q^{85} + 133634 q^{86} - 147330 q^{87} + 48898 q^{88} + 96112 q^{89} + 22842 q^{90} + 129228 q^{91} + 11638 q^{92} - 132804 q^{93} - 133688 q^{94} - 55928 q^{95} - 34902 q^{96} - 135318 q^{97} + 108440 q^{98} - 30456 q^{99}+O(q^{100}) \) 3 * q - 8 * q^2 + 27 * q^3 + 22 * q^4 - 56 * q^5 - 72 * q^6 - 114 * q^7 - 510 * q^8 + 243 * q^9 + 282 * q^10 - 376 * q^11 + 198 * q^12 - 858 * q^13 + 588 * q^14 - 504 * q^15 + 2738 * q^16 - 2548 * q^17 - 648 * q^18 - 2846 * q^19 - 4618 * q^20 - 1026 * q^21 - 5050 * q^22 + 1587 * q^23 - 4590 * q^24 + 753 * q^25 - 7788 * q^26 + 2187 * q^27 + 4736 * q^28 - 16370 * q^29 + 2538 * q^30 - 14756 * q^31 - 3878 * q^32 - 3384 * q^33 + 16520 * q^34 - 18520 * q^35 + 1782 * q^36 + 15874 * q^37 + 12438 * q^38 - 7722 * q^39 + 38270 * q^40 + 12606 * q^41 + 5292 * q^42 + 3154 * q^43 + 27114 * q^44 - 4536 * q^45 - 4232 * q^46 + 29928 * q^47 + 24642 * q^48 + 4471 * q^49 + 1452 * q^50 - 22932 * q^51 + 86856 * q^52 - 44084 * q^53 - 5832 * q^54 + 38360 * q^55 - 35704 * q^56 - 25614 * q^57 + 73316 * q^58 - 29300 * q^59 - 41562 * q^60 + 54010 * q^61 + 99908 * q^62 - 9234 * q^63 - 1582 * q^64 - 51216 * q^65 - 45450 * q^66 + 43390 * q^67 - 69840 * q^68 + 14283 * q^69 - 2476 * q^70 + 23424 * q^71 - 41310 * q^72 - 91402 * q^73 - 2294 * q^74 + 6777 * q^75 - 14274 * q^76 - 97208 * q^77 - 70092 * q^78 - 49398 * q^79 - 52626 * q^80 + 19683 * q^81 + 40152 * q^82 - 103936 * q^83 + 42624 * q^84 + 5888 * q^85 + 133634 * q^86 - 147330 * q^87 + 48898 * q^88 + 96112 * q^89 + 22842 * q^90 + 129228 * q^91 + 11638 * q^92 - 132804 * q^93 - 133688 * q^94 - 55928 * q^95 - 34902 * q^96 - 135318 * q^97 + 108440 * q^98 - 30456 * q^99

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