LMFDB - Embedded newform 138.6.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,6,Mod(1,138)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, 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("138.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	138.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:	$22.1329671342$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 1383x - 16813 $ x^3 - x^2 - 1383*x - 16813
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$42.6589$ of defining polynomial
Character	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -24.6530 q^{5} +36.0000 q^{6} +62.9650 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -24.6530 q^{5} +36.0000 q^{6} +62.9650 q^{7} -64.0000 q^{8} +81.0000 q^{9} +98.6122 q^{10} -614.231 q^{11} -144.000 q^{12} +1123.48 q^{13} -251.860 q^{14} +221.877 q^{15} +256.000 q^{16} +1423.58 q^{17} -324.000 q^{18} -326.515 q^{19} -394.449 q^{20} -566.685 q^{21} +2456.92 q^{22} -529.000 q^{23} +576.000 q^{24} -2517.23 q^{25} -4493.92 q^{26} -729.000 q^{27} +1007.44 q^{28} +2130.56 q^{29} -887.509 q^{30} -6278.98 q^{31} -1024.00 q^{32} +5528.08 q^{33} -5694.32 q^{34} -1552.28 q^{35} +1296.00 q^{36} +6116.03 q^{37} +1306.06 q^{38} -10111.3 q^{39} +1577.79 q^{40} -19704.5 q^{41} +2266.74 q^{42} +920.714 q^{43} -9827.70 q^{44} -1996.90 q^{45} +2116.00 q^{46} -25873.5 q^{47} -2304.00 q^{48} -12842.4 q^{49} +10068.9 q^{50} -12812.2 q^{51} +17975.7 q^{52} -36328.3 q^{53} +2916.00 q^{54} +15142.7 q^{55} -4029.76 q^{56} +2938.63 q^{57} -8522.26 q^{58} +22090.3 q^{59} +3550.04 q^{60} -37622.3 q^{61} +25115.9 q^{62} +5100.17 q^{63} +4096.00 q^{64} -27697.2 q^{65} -22112.3 q^{66} -11706.1 q^{67} +22777.3 q^{68} +4761.00 q^{69} +6209.12 q^{70} +38068.4 q^{71} -5184.00 q^{72} +50101.7 q^{73} -24464.1 q^{74} +22655.0 q^{75} -5224.24 q^{76} -38675.1 q^{77} +40445.2 q^{78} +20644.2 q^{79} -6311.18 q^{80} +6561.00 q^{81} +78818.2 q^{82} -66247.5 q^{83} -9066.96 q^{84} -35095.5 q^{85} -3682.86 q^{86} -19175.1 q^{87} +39310.8 q^{88} +63637.6 q^{89} +7987.58 q^{90} +70739.9 q^{91} -8464.00 q^{92} +56510.8 q^{93} +103494. q^{94} +8049.58 q^{95} +9216.00 q^{96} +120727. q^{97} +51369.6 q^{98} -49752.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 12 q^{2} - 27 q^{3} + 48 q^{4} - 18 q^{5} + 108 q^{6} - 50 q^{7} - 192 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q - 12 * q^2 - 27 * q^3 + 48 * q^4 - 18 * q^5 + 108 * q^6 - 50 * q^7 - 192 * q^8 + 243 * q^9	\( 3 q - 12 q^{2} - 27 q^{3} + 48 q^{4} - 18 q^{5} + 108 q^{6} - 50 q^{7} - 192 q^{8} + 243 q^{9} + 72 q^{10} + 6 q^{11} - 432 q^{12} + 790 q^{13} + 200 q^{14} + 162 q^{15} + 768 q^{16} + 1148 q^{17} - 972 q^{18} + 2692 q^{19} - 288 q^{20} + 450 q^{21} - 24 q^{22} - 1587 q^{23} + 1728 q^{24} + 2953 q^{25} - 3160 q^{26} - 2187 q^{27} - 800 q^{28} - 5910 q^{29} - 648 q^{30} - 4224 q^{31} - 3072 q^{32} - 54 q^{33} - 4592 q^{34} - 24360 q^{35} + 3888 q^{36} - 10368 q^{37} - 10768 q^{38} - 7110 q^{39} + 1152 q^{40} - 37786 q^{41} - 1800 q^{42} - 28616 q^{43} + 96 q^{44} - 1458 q^{45} + 6348 q^{46} - 43348 q^{47} - 6912 q^{48} + 2939 q^{49} - 11812 q^{50} - 10332 q^{51} + 12640 q^{52} - 41566 q^{53} + 8748 q^{54} - 8956 q^{55} + 3200 q^{56} - 24228 q^{57} + 23640 q^{58} - 46560 q^{59} + 2592 q^{60} + 7924 q^{61} + 16896 q^{62} - 4050 q^{63} + 12288 q^{64} - 45580 q^{65} + 216 q^{66} - 17016 q^{67} + 18368 q^{68} + 14283 q^{69} + 97440 q^{70} - 38280 q^{71} - 15552 q^{72} - 11710 q^{73} + 41472 q^{74} - 26577 q^{75} + 43072 q^{76} - 23540 q^{77} + 28440 q^{78} + 72930 q^{79} - 4608 q^{80} + 19683 q^{81} + 151144 q^{82} - 50102 q^{83} + 7200 q^{84} + 108932 q^{85} + 114464 q^{86} + 53190 q^{87} - 384 q^{88} + 46212 q^{89} + 5832 q^{90} + 121740 q^{91} - 25392 q^{92} + 38016 q^{93} + 173392 q^{94} - 94652 q^{95} + 27648 q^{96} - 101642 q^{97} - 11756 q^{98} + 486 q^{99}+O(q^{100}) \) 3 * q - 12 * q^2 - 27 * q^3 + 48 * q^4 - 18 * q^5 + 108 * q^6 - 50 * q^7 - 192 * q^8 + 243 * q^9 + 72 * q^10 + 6 * q^11 - 432 * q^12 + 790 * q^13 + 200 * q^14 + 162 * q^15 + 768 * q^16 + 1148 * q^17 - 972 * q^18 + 2692 * q^19 - 288 * q^20 + 450 * q^21 - 24 * q^22 - 1587 * q^23 + 1728 * q^24 + 2953 * q^25 - 3160 * q^26 - 2187 * q^27 - 800 * q^28 - 5910 * q^29 - 648 * q^30 - 4224 * q^31 - 3072 * q^32 - 54 * q^33 - 4592 * q^34 - 24360 * q^35 + 3888 * q^36 - 10368 * q^37 - 10768 * q^38 - 7110 * q^39 + 1152 * q^40 - 37786 * q^41 - 1800 * q^42 - 28616 * q^43 + 96 * q^44 - 1458 * q^45 + 6348 * q^46 - 43348 * q^47 - 6912 * q^48 + 2939 * q^49 - 11812 * q^50 - 10332 * q^51 + 12640 * q^52 - 41566 * q^53 + 8748 * q^54 - 8956 * q^55 + 3200 * q^56 - 24228 * q^57 + 23640 * q^58 - 46560 * q^59 + 2592 * q^60 + 7924 * q^61 + 16896 * q^62 - 4050 * q^63 + 12288 * q^64 - 45580 * q^65 + 216 * q^66 - 17016 * q^67 + 18368 * q^68 + 14283 * q^69 + 97440 * q^70 - 38280 * q^71 - 15552 * q^72 - 11710 * q^73 + 41472 * q^74 - 26577 * q^75 + 43072 * q^76 - 23540 * q^77 + 28440 * q^78 + 72930 * q^79 - 4608 * q^80 + 19683 * q^81 + 151144 * q^82 - 50102 * q^83 + 7200 * q^84 + 108932 * q^85 + 114464 * q^86 + 53190 * q^87 - 384 * q^88 + 46212 * q^89 + 5832 * q^90 + 121740 * q^91 - 25392 * q^92 + 38016 * q^93 + 173392 * q^94 - 94652 * q^95 + 27648 * q^96 - 101642 * q^97 - 11756 * q^98 + 486 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.0000	0.500000
$5$	−24.6530	−0.441007	−0.220503	−	0.975386i	$-0.570770\pi$
$5$	−24.6530	−0.441007	−0.220503	+	0.975386i	$0.570770\pi$
$6$	36.0000	0.408248
$7$	62.9650	0.485684	0.242842	−	0.970066i	$-0.421920\pi$
$7$	62.9650	0.485684	0.242842	+	0.970066i	$0.421920\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	98.6122	0.311839
$11$	−614.231	−1.53056	−0.765279	−	0.643698i	$-0.777399\pi$
$11$	−614.231	−1.53056	−0.765279	+	0.643698i	$0.777399\pi$
$12$	−144.000	−0.288675
$13$	1123.48	1.84377	0.921885	−	0.387465i	$-0.126649\pi$
$13$	1123.48	1.84377	0.921885	+	0.387465i	$0.126649\pi$
$14$	−251.860	−0.343431
$15$	221.877	0.254615
$16$	256.000	0.250000
$17$	1423.58	1.19470	0.597350	−	0.801980i	$-0.296220\pi$
$17$	1423.58	1.19470	0.597350	+	0.801980i	$0.296220\pi$
$18$	−324.000	−0.235702
$19$	−326.515	−0.207500	−0.103750	−	0.994603i	$-0.533084\pi$
$19$	−326.515	−0.207500	−0.103750	+	0.994603i	$0.533084\pi$
$20$	−394.449	−0.220503
$21$	−566.685	−0.280410
$22$	2456.92	1.08227
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	576.000	0.204124
$25$	−2517.23	−0.805513
$26$	−4493.92	−1.30374
$27$	−729.000	−0.192450
$28$	1007.44	0.242842
$29$	2130.56	0.470435	0.235217	−	0.971943i	$-0.424420\pi$
$29$	2130.56	0.470435	0.235217	+	0.971943i	$0.424420\pi$
$30$	−887.509	−0.180040
$31$	−6278.98	−1.17350	−0.586752	−	0.809767i	$-0.699594\pi$
$31$	−6278.98	−1.17350	−0.586752	+	0.809767i	$0.699594\pi$
$32$	−1024.00	−0.176777
$33$	5528.08	0.883669
$34$	−5694.32	−0.844781
$35$	−1552.28	−0.214190
$36$	1296.00	0.166667
$37$	6116.03	0.734455	0.367228	−	0.930131i	$-0.380307\pi$
$37$	6116.03	0.734455	0.367228	+	0.930131i	$0.380307\pi$
$38$	1306.06	0.146725
$39$	−10111.3	−1.06450
$40$	1577.79	0.155920
$41$	−19704.5	−1.83066	−0.915328	−	0.402709i	$-0.868069\pi$
$41$	−19704.5	−1.83066	−0.915328	+	0.402709i	$0.868069\pi$
$42$	2266.74	0.198280
$43$	920.714	0.0759370	0.0379685	−	0.999279i	$-0.487911\pi$
$43$	920.714	0.0759370	0.0379685	+	0.999279i	$0.487911\pi$
$44$	−9827.70	−0.765279
$45$	−1996.90	−0.147002
$46$	2116.00	0.147442
$47$	−25873.5	−1.70848	−0.854242	−	0.519875i	$-0.825978\pi$
$47$	−25873.5	−1.70848	−0.854242	+	0.519875i	$0.825978\pi$
$48$	−2304.00	−0.144338
$49$	−12842.4	−0.764111
$50$	10068.9	0.569584
$51$	−12812.2	−0.689761
$52$	17975.7	0.921885
$53$	−36328.3	−1.77646	−0.888229	−	0.459401i	$-0.848064\pi$
$53$	−36328.3	−1.77646	−0.888229	+	0.459401i	$0.848064\pi$
$54$	2916.00	0.136083
$55$	15142.7	0.674987
$56$	−4029.76	−0.171715
$57$	2938.63	0.119800
$58$	−8522.26	−0.332648
$59$	22090.3	0.826174	0.413087	−	0.910692i	$-0.364451\pi$
$59$	22090.3	0.826174	0.413087	+	0.910692i	$0.364451\pi$
$60$	3550.04	0.127308
$61$	−37622.3	−1.29456	−0.647278	−	0.762254i	$-0.724093\pi$
$61$	−37622.3	−1.29456	−0.647278	+	0.762254i	$0.724093\pi$
$62$	25115.9	0.829793
$63$	5100.17	0.161895
$64$	4096.00	0.125000
$65$	−27697.2	−0.813115
$66$	−22112.3	−0.624848
$67$	−11706.1	−0.318584	−0.159292	−	0.987231i	$-0.550921\pi$
$67$	−11706.1	−0.318584	−0.159292	+	0.987231i	$0.550921\pi$
$68$	22777.3	0.597350
$69$	4761.00	0.120386
$70$	6209.12	0.151455
$71$	38068.4	0.896229	0.448114	−	0.893976i	$-0.352096\pi$
$71$	38068.4	0.896229	0.448114	+	0.893976i	$0.352096\pi$
$72$	−5184.00	−0.117851
$73$	50101.7	1.10039	0.550193	−	0.835038i	$-0.314554\pi$
$73$	50101.7	1.10039	0.550193	+	0.835038i	$0.314554\pi$
$74$	−24464.1	−0.519338
$75$	22655.0	0.465063
$76$	−5224.24	−0.103750
$77$	−38675.1	−0.743369
$78$	40445.2	0.752716
$79$	20644.2	0.372160	0.186080	−	0.982535i	$-0.440422\pi$
$79$	20644.2	0.372160	0.186080	+	0.982535i	$0.440422\pi$
$80$	−6311.18	−0.110252
$81$	6561.00	0.111111
$82$	78818.2	1.29447
$83$	−66247.5	−1.05554	−0.527770	−	0.849388i	$-0.676972\pi$
$83$	−66247.5	−1.05554	−0.527770	+	0.849388i	$0.676972\pi$
$84$	−9066.96	−0.140205
$85$	−35095.5	−0.526871
$86$	−3682.86	−0.0536956
$87$	−19175.1	−0.271606
$88$	39310.8	0.541134
$89$	63637.6	0.851606	0.425803	−	0.904816i	$-0.359992\pi$
$89$	63637.6	0.851606	0.425803	+	0.904816i	$0.359992\pi$
$90$	7987.58	0.103946
$91$	70739.9	0.895490
$92$	−8464.00	−0.104257
$93$	56510.8	0.677523
$94$	103494.	1.20808
$95$	8049.58	0.0915091
$96$	9216.00	0.102062
$97$	120727.	1.30279	0.651396	−	0.758738i	$-0.274184\pi$
$97$	120727.	1.30279	0.651396	+	0.758738i	$0.274184\pi$
$98$	51369.6	0.540308
$99$	−49752.7	−0.510186
$100$	−40275.6	−0.402756
$101$	−111073.	−1.08344	−0.541720	−	0.840559i	$-0.682227\pi$
$101$	−111073.	−1.08344	−0.541720	+	0.840559i	$0.682227\pi$
$102$	51248.8	0.487735
$103$	−173554.	−1.61191	−0.805956	−	0.591975i	$-0.798348\pi$
$103$	−173554.	−1.61191	−0.805956	+	0.591975i	$0.798348\pi$
$104$	−71902.7	−0.651871
$105$	13970.5	0.123663
$106$	145313.	1.25615
$107$	−164472.	−1.38878	−0.694391	−	0.719598i	$-0.744326\pi$
$107$	−164472.	−1.38878	−0.694391	+	0.719598i	$0.744326\pi$
$108$	−11664.0	−0.0962250
$109$	−32848.2	−0.264817	−0.132408	−	0.991195i	$-0.542271\pi$
$109$	−32848.2	−0.264817	−0.132408	+	0.991195i	$0.542271\pi$
$110$	−60570.6	−0.477288
$111$	−55044.3	−0.424038
$112$	16119.0	0.121421
$113$	151033.	1.11269	0.556346	−	0.830951i	$-0.312203\pi$
$113$	151033.	1.11269	0.556346	+	0.830951i	$0.312203\pi$
$114$	−11754.5	−0.0847117
$115$	13041.5	0.0919563
$116$	34089.0	0.235217
$117$	91001.8	0.614590
$118$	−88361.2	−0.584193
$119$	89635.7	0.580248
$120$	−14200.1	−0.0900202
$121$	216229.	1.34261
$122$	150489.	0.915389
$123$	177341.	1.05693
$124$	−100464.	−0.586752
$125$	139098.	0.796244
$126$	−20400.7	−0.114477
$127$	284508.	1.56525	0.782627	−	0.622491i	$-0.213879\pi$
$127$	284508.	1.56525	0.782627	+	0.622491i	$0.213879\pi$
$128$	−16384.0	−0.0883883
$129$	−8286.42	−0.0438423
$130$	110789.	0.574959
$131$	−115857.	−0.589855	−0.294927	−	0.955520i	$-0.595295\pi$
$131$	−115857.	−0.589855	−0.294927	+	0.955520i	$0.595295\pi$
$132$	88449.3	0.441834
$133$	−20559.0	−0.100780
$134$	46824.3	0.225273
$135$	17972.1	0.0848718
$136$	−91109.0	−0.422391
$137$	−137796.	−0.627240	−0.313620	−	0.949548i	$-0.601542\pi$
$137$	−137796.	−0.627240	−0.313620	+	0.949548i	$0.601542\pi$
$138$	−19044.0	−0.0851257
$139$	−54810.8	−0.240619	−0.120309	−	0.992736i	$-0.538389\pi$
$139$	−54810.8	−0.240619	−0.120309	+	0.992736i	$0.538389\pi$
$140$	−24836.5	−0.107095
$141$	232862.	0.986394
$142$	−152274.	−0.633729
$143$	−690076.	−2.82200
$144$	20736.0	0.0833333
$145$	−52524.9	−0.207465
$146$	−200407.	−0.778090
$147$	115582.	0.441159
$148$	97856.5	0.367228
$149$	−206539.	−0.762142	−0.381071	−	0.924546i	$-0.624445\pi$
$149$	−206539.	−0.762142	−0.381071	+	0.924546i	$0.624445\pi$
$150$	−90620.2	−0.328849
$151$	−423510.	−1.51155	−0.755773	−	0.654834i	$-0.772739\pi$
$151$	−423510.	−1.51155	−0.755773	+	0.654834i	$0.772739\pi$
$152$	20896.9	0.0733624
$153$	115310.	0.398234
$154$	154700.	0.525641
$155$	154796.	0.517523
$156$	−161781.	−0.532250
$157$	−350924.	−1.13623	−0.568113	−	0.822951i	$-0.692326\pi$
$157$	−350924.	−1.13623	−0.568113	+	0.822951i	$0.692326\pi$
$158$	−82576.7	−0.263157
$159$	326954.	1.02564
$160$	25244.7	0.0779598
$161$	−33308.5	−0.101272
$162$	−26244.0	−0.0785674
$163$	−587393.	−1.73165	−0.865824	−	0.500349i	$-0.833205\pi$
$163$	−587393.	−1.73165	−0.865824	+	0.500349i	$0.833205\pi$
$164$	−315273.	−0.915328
$165$	−136284.	−0.389704
$166$	264990.	0.746379
$167$	−570758.	−1.58366	−0.791828	−	0.610745i	$-0.790870\pi$
$167$	−570758.	−1.58366	−0.791828	+	0.610745i	$0.790870\pi$
$168$	36267.8	0.0991399
$169$	890912.	2.39948
$170$	140382.	0.372554
$171$	−26447.7	−0.0691668
$172$	14731.4	0.0379685
$173$	522262.	1.32670	0.663350	−	0.748309i	$-0.269134\pi$
$173$	522262.	1.32670	0.663350	+	0.748309i	$0.269134\pi$
$174$	76700.3	0.192054
$175$	−158497.	−0.391225
$176$	−157243.	−0.382640
$177$	−198813.	−0.476992
$178$	−254551.	−0.602177
$179$	630105.	1.46987	0.734937	−	0.678135i	$-0.237211\pi$
$179$	630105.	1.46987	0.734937	+	0.678135i	$0.237211\pi$
$180$	−31950.3	−0.0735012
$181$	−518974.	−1.17747	−0.588734	−	0.808327i	$-0.700373\pi$
$181$	−518974.	−1.17747	−0.588734	+	0.808327i	$0.700373\pi$
$182$	−282959.	−0.633207
$183$	338601.	0.747412
$184$	33856.0	0.0737210
$185$	−150779.	−0.323900
$186$	−226043.	−0.479081
$187$	−874406.	−1.82856
$188$	−413976.	−0.854242
$189$	−45901.5	−0.0934700
$190$	−32198.3	−0.0647067
$191$	675832.	1.34047	0.670233	−	0.742151i	$-0.266194\pi$
$191$	675832.	1.34047	0.670233	+	0.742151i	$0.266194\pi$
$192$	−36864.0	−0.0721688
$193$	−145152.	−0.280499	−0.140249	−	0.990116i	$-0.544790\pi$
$193$	−145152.	−0.280499	−0.140249	+	0.990116i	$0.544790\pi$
$194$	−482908.	−0.921213
$195$	249275.	0.469452
$196$	−205479.	−0.382055
$197$	−481154.	−0.883321	−0.441660	−	0.897182i	$-0.645610\pi$
$197$	−481154.	−0.883321	−0.441660	+	0.897182i	$0.645610\pi$
$198$	199011.	0.360756
$199$	571805.	1.02356	0.511782	−	0.859115i	$-0.328986\pi$
$199$	571805.	1.02356	0.511782	+	0.859115i	$0.328986\pi$
$200$	161103.	0.284792
$201$	105355.	0.183935
$202$	444291.	0.766107
$203$	134151.	0.228483
$204$	−204995.	−0.344880
$205$	485777.	0.807332
$206$	694216.	1.13979
$207$	−42849.0	−0.0695048
$208$	287611.	0.460942
$209$	200556.	0.317591
$210$	−55882.0	−0.0874428
$211$	−48648.3	−0.0752249	−0.0376125	−	0.999292i	$-0.511975\pi$
$211$	−48648.3	−0.0752249	−0.0376125	+	0.999292i	$0.511975\pi$
$212$	−581252.	−0.888229
$213$	−342616.	−0.517438
$214$	657890.	0.982017
$215$	−22698.4	−0.0334888
$216$	46656.0	0.0680414
$217$	−395356.	−0.569953
$218$	131393.	0.187254
$219$	−450915.	−0.635308
$220$	242283.	0.337494
$221$	1.59936e6	2.20275
$222$	220177.	0.299840
$223$	162891.	0.219348	0.109674	−	0.993968i	$-0.465019\pi$
$223$	162891.	0.219348	0.109674	+	0.993968i	$0.465019\pi$
$224$	−64476.2	−0.0858577
$225$	−203895.	−0.268504
$226$	−604131.	−0.786792
$227$	−580710.	−0.747988	−0.373994	−	0.927431i	$-0.622012\pi$
$227$	−580710.	−0.747988	−0.373994	+	0.927431i	$0.622012\pi$
$228$	47018.1	0.0599002
$229$	−673938.	−0.849241	−0.424621	−	0.905371i	$-0.639592\pi$
$229$	−673938.	−0.849241	−0.424621	+	0.905371i	$0.639592\pi$
$230$	−52165.8	−0.0650229
$231$	348076.	0.429184
$232$	−136356.	−0.166324
$233$	526937.	0.635871	0.317936	−	0.948112i	$-0.397010\pi$
$233$	526937.	0.635871	0.317936	+	0.948112i	$0.397010\pi$
$234$	−364007.	−0.434581
$235$	637861.	0.753453
$236$	353445.	0.413087
$237$	−185798.	−0.214867
$238$	−358543.	−0.410297
$239$	−22953.2	−0.0259925	−0.0129962	−	0.999916i	$-0.504137\pi$
$239$	−22953.2	−0.0259925	−0.0129962	+	0.999916i	$0.504137\pi$
$240$	56800.6	0.0636539
$241$	741679.	0.822571	0.411286	−	0.911507i	$-0.365080\pi$
$241$	741679.	0.822571	0.411286	+	0.911507i	$0.365080\pi$
$242$	−864915.	−0.949369
$243$	−59049.0	−0.0641500
$244$	−601957.	−0.647278
$245$	316604.	0.336978
$246$	−709364.	−0.747362
$247$	−366832.	−0.382583
$248$	401854.	0.414896
$249$	596228.	0.609416
$250$	−556392.	−0.563029
$251$	−1.46839e6	−1.47115	−0.735575	−	0.677443i	$-0.763088\pi$
$251$	−1.46839e6	−1.47115	−0.735575	+	0.677443i	$0.763088\pi$
$252$	81602.7	0.0809474
$253$	324928.	0.319144
$254$	−1.13803e6	−1.10680
$255$	315860.	0.304189
$256$	65536.0	0.0625000
$257$	527652.	0.498328	0.249164	−	0.968461i	$-0.419844\pi$
$257$	527652.	0.498328	0.249164	+	0.968461i	$0.419844\pi$
$258$	33145.7	0.0310012
$259$	385096.	0.356714
$260$	−443155.	−0.406558
$261$	172576.	0.156812
$262$	463429.	0.417090
$263$	740956.	0.660546	0.330273	−	0.943885i	$-0.392859\pi$
$263$	740956.	0.660546	0.330273	+	0.943885i	$0.392859\pi$
$264$	−353797.	−0.312424
$265$	895602.	0.783430
$266$	82236.0	0.0712620
$267$	−572739.	−0.491675
$268$	−187297.	−0.159292
$269$	1.23653e6	1.04189	0.520947	−	0.853589i	$-0.325579\pi$
$269$	1.23653e6	1.04189	0.520947	+	0.853589i	$0.325579\pi$
$270$	−71888.3	−0.0600134
$271$	1.08298e6	0.895770	0.447885	−	0.894091i	$-0.352177\pi$
$271$	1.08298e6	0.895770	0.447885	+	0.894091i	$0.352177\pi$
$272$	364436.	0.298675
$273$	−636659.	−0.517011
$274$	551183.	0.443526
$275$	1.54616e6	1.23288
$276$	76176.0	0.0601929
$277$	−1.01473e6	−0.794605	−0.397302	−	0.917688i	$-0.630054\pi$
$277$	−1.01473e6	−0.794605	−0.397302	+	0.917688i	$0.630054\pi$
$278$	219243.	0.170143
$279$	−508597.	−0.391168
$280$	99345.8	0.0757277
$281$	−805939.	−0.608887	−0.304443	−	0.952530i	$-0.598470\pi$
$281$	−805939.	−0.608887	−0.304443	+	0.952530i	$0.598470\pi$
$282$	−931447.	−0.697486
$283$	−1.22359e6	−0.908178	−0.454089	−	0.890956i	$-0.650035\pi$
$283$	−1.22359e6	−0.908178	−0.454089	+	0.890956i	$0.650035\pi$
$284$	609095.	0.448114
$285$	−72446.2	−0.0528328
$286$	2.76030e6	1.99545
$287$	−1.24070e6	−0.889121
$288$	−82944.0	−0.0589256
$289$	606720.	0.427310
$290$	210100.	0.146700
$291$	−1.08654e6	−0.752167
$292$	801627.	0.550193
$293$	1.65539e6	1.12650	0.563251	−	0.826286i	$-0.309550\pi$
$293$	1.65539e6	1.12650	0.563251	+	0.826286i	$0.309550\pi$
$294$	−462327.	−0.311947
$295$	−544593.	−0.364348
$296$	−391426.	−0.259669
$297$	447774.	0.294556
$298$	826155.	0.538916
$299$	−594320.	−0.384452
$300$	362481.	0.232532
$301$	57972.8	0.0368814
$302$	1.69404e6	1.06882
$303$	999656.	0.625524
$304$	−83587.8	−0.0518751
$305$	927504.	0.570908
$306$	−461240.	−0.281594
$307$	32809.9	0.0198682	0.00993410	−	0.999951i	$-0.496838\pi$
$307$	32809.9	0.0198682	0.00993410	+	0.999951i	$0.496838\pi$
$308$	−618801.	−0.371684
$309$	1.56199e6	0.930638
$310$	−619183.	−0.365944
$311$	1.60728e6	0.942302	0.471151	−	0.882053i	$-0.343839\pi$
$311$	1.60728e6	0.942302	0.471151	+	0.882053i	$0.343839\pi$
$312$	647124.	0.376358
$313$	−2.48058e6	−1.43117	−0.715587	−	0.698523i	$-0.753841\pi$
$313$	−2.48058e6	−1.43117	−0.715587	+	0.698523i	$0.753841\pi$
$314$	1.40370e6	0.803432
$315$	−125735.	−0.0713967
$316$	330307.	0.186080
$317$	771301.	0.431098	0.215549	−	0.976493i	$-0.430846\pi$
$317$	771301.	0.431098	0.215549	+	0.976493i	$0.430846\pi$
$318$	−1.30782e6	−0.725236
$319$	−1.30866e6	−0.720028
$320$	−100979.	−0.0551259
$321$	1.48025e6	0.801813
$322$	133234.	0.0716103
$323$	−464820.	−0.247901
$324$	104976.	0.0555556
$325$	−2.82805e6	−1.48518
$326$	2.34957e6	1.22446
$327$	295634.	0.152892
$328$	1.26109e6	0.647235
$329$	−1.62913e6	−0.829784
$330$	545136.	0.275562
$331$	954600.	0.478907	0.239454	−	0.970908i	$-0.423032\pi$
$331$	954600.	0.478907	0.239454	+	0.970908i	$0.423032\pi$
$332$	−1.05996e6	−0.527770
$333$	495399.	0.244818
$334$	2.28303e6	1.11981
$335$	288590.	0.140498
$336$	−145071.	−0.0701025
$337$	2.42580e6	1.16354	0.581769	−	0.813354i	$-0.302361\pi$
$337$	2.42580e6	1.16354	0.581769	+	0.813354i	$0.302361\pi$
$338$	−3.56365e6	−1.69669
$339$	−1.35929e6	−0.642413
$340$	−561529.	−0.263436
$341$	3.85674e6	1.79612
$342$	105791.	0.0489083
$343$	−1.86688e6	−0.856801
$344$	−58925.7	−0.0268478
$345$	−117373.	−0.0530910
$346$	−2.08905e6	−0.938119
$347$	383446.	0.170954	0.0854772	−	0.996340i	$-0.472759\pi$
$347$	383446.	0.170954	0.0854772	+	0.996340i	$0.472759\pi$
$348$	−306801.	−0.135803
$349$	−2.57020e6	−1.12955	−0.564773	−	0.825246i	$-0.691036\pi$
$349$	−2.57020e6	−1.12955	−0.564773	+	0.825246i	$0.691036\pi$
$350$	633989.	0.276638
$351$	−819016.	−0.354834
$352$	628973.	0.270567
$353$	−999241.	−0.426809	−0.213404	−	0.976964i	$-0.568455\pi$
$353$	−999241.	−0.426809	−0.213404	+	0.976964i	$0.568455\pi$
$354$	795251.	0.337284
$355$	−938502.	−0.395243
$356$	1.01820e6	0.425803
$357$	−806721.	−0.335006
$358$	−2.52042e6	−1.03936
$359$	838307.	0.343295	0.171647	−	0.985158i	$-0.445091\pi$
$359$	838307.	0.343295	0.171647	+	0.985158i	$0.445091\pi$
$360$	127801.	0.0519732
$361$	−2.36949e6	−0.956944
$362$	2.07590e6	0.832595
$363$	−1.94606e6	−0.775157
$364$	1.13184e6	0.447745
$365$	−1.23516e6	−0.485278
$366$	−1.35440e6	−0.528500
$367$	2.14837e6	0.832614	0.416307	−	0.909224i	$-0.363324\pi$
$367$	2.14837e6	0.832614	0.416307	+	0.909224i	$0.363324\pi$
$368$	−135424.	−0.0521286
$369$	−1.59607e6	−0.610219
$370$	603115.	0.229032
$371$	−2.28741e6	−0.862798
$372$	904172.	0.338761
$373$	−2.28650e6	−0.850941	−0.425470	−	0.904972i	$-0.639891\pi$
$373$	−2.28650e6	−0.850941	−0.425470	+	0.904972i	$0.639891\pi$
$374$	3.49763e6	1.29299
$375$	−1.25188e6	−0.459712
$376$	1.65591e6	0.604040
$377$	2.39364e6	0.867374
$378$	183606.	0.0660933
$379$	−3.62369e6	−1.29585	−0.647923	−	0.761706i	$-0.724362\pi$
$379$	−3.62369e6	−1.29585	−0.647923	+	0.761706i	$0.724362\pi$
$380$	128793.	0.0457545
$381$	−2.56057e6	−0.903700
$382$	−2.70333e6	−0.947852
$383$	−51051.6	−0.0177833	−0.00889164	−	0.999960i	$-0.502830\pi$
$383$	−51051.6	−0.0177833	−0.00889164	+	0.999960i	$0.502830\pi$
$384$	147456.	0.0510310
$385$	953458.	0.327831
$386$	580609.	0.198342
$387$	74577.8	0.0253123
$388$	1.93163e6	0.651396
$389$	−2.05966e6	−0.690116	−0.345058	−	0.938581i	$-0.612141\pi$
$389$	−2.05966e6	−0.690116	−0.345058	+	0.938581i	$0.612141\pi$
$390$	−997098.	−0.331953
$391$	−753073.	−0.249112
$392$	821914.	0.270154
$393$	1.04272e6	0.340553
$394$	1.92462e6	0.624602
$395$	−508942.	−0.164125
$396$	−796043.	−0.255093
$397$	4.91836e6	1.56619	0.783094	−	0.621904i	$-0.213641\pi$
$397$	4.91836e6	1.56619	0.783094	+	0.621904i	$0.213641\pi$
$398$	−2.28722e6	−0.723769
$399$	185031.	0.0581852
$400$	−644410.	−0.201378
$401$	−25160.4	−0.00781371	−0.00390685	−	0.999992i	$-0.501244\pi$
$401$	−25160.4	−0.00781371	−0.00390685	+	0.999992i	$0.501244\pi$
$402$	−421419.	−0.130062
$403$	−7.05430e6	−2.16367
$404$	−1.77717e6	−0.541720
$405$	−161749.	−0.0490008
$406$	−536604.	−0.161562
$407$	−3.75666e6	−1.12413
$408$	819981.	0.243867
$409$	−185434.	−0.0548126	−0.0274063	−	0.999624i	$-0.508725\pi$
$409$	−185434.	−0.0548126	−0.0274063	+	0.999624i	$0.508725\pi$
$410$	−1.94311e6	−0.570870
$411$	1.24016e6	0.362137
$412$	−2.77686e6	−0.805956
$413$	1.39092e6	0.401260
$414$	171396.	0.0491473
$415$	1.63320e6	0.465500
$416$	−1.15044e6	−0.325935
$417$	493297.	0.138921
$418$	−802222.	−0.224571
$419$	305449.	0.0849969	0.0424984	−	0.999097i	$-0.486468\pi$
$419$	305449.	0.0849969	0.0424984	+	0.999097i	$0.486468\pi$
$420$	223528.	0.0618314
$421$	−5.01698e6	−1.37955	−0.689775	−	0.724024i	$-0.742290\pi$
$421$	−5.01698e6	−1.37955	−0.689775	+	0.724024i	$0.742290\pi$
$422$	194593.	0.0531920
$423$	−2.09576e6	−0.569495
$424$	2.32501e6	0.628073
$425$	−3.58347e6	−0.962347
$426$	1.37046e6	0.365884
$427$	−2.36889e6	−0.628745
$428$	−2.63156e6	−0.694391
$429$	6.21068e6	1.62928
$430$	90793.6	0.0236801
$431$	−4.75913e6	−1.23406	−0.617028	−	0.786942i	$-0.711663\pi$
$431$	−4.75913e6	−1.23406	−0.617028	+	0.786942i	$0.711663\pi$
$432$	−186624.	−0.0481125
$433$	4.58785e6	1.17595	0.587976	−	0.808878i	$-0.299925\pi$
$433$	4.58785e6	1.17595	0.587976	+	0.808878i	$0.299925\pi$
$434$	1.58142e6	0.403017
$435$	472724.	0.119780
$436$	−525572.	−0.132408
$437$	172726.	0.0432668
$438$	1.80366e6	0.449231
$439$	−6.62970e6	−1.64185	−0.820923	−	0.571039i	$-0.806540\pi$
$439$	−6.62970e6	−1.64185	−0.820923	+	0.571039i	$0.806540\pi$
$440$	−969130.	−0.238644
$441$	−1.04023e6	−0.254704
$442$	−6.39744e6	−1.55758
$443$	−918838.	−0.222448	−0.111224	−	0.993795i	$-0.535477\pi$
$443$	−918838.	−0.222448	−0.111224	+	0.993795i	$0.535477\pi$
$444$	−880709.	−0.212019
$445$	−1.56886e6	−0.375564
$446$	−651563.	−0.155103
$447$	1.85885e6	0.440023
$448$	257905.	0.0607106
$449$	8.16971e6	1.91245	0.956227	−	0.292627i	$-0.0945296\pi$
$449$	8.16971e6	1.91245	0.956227	+	0.292627i	$0.0945296\pi$
$450$	815582.	0.189861
$451$	1.21031e7	2.80193
$452$	2.41652e6	0.556346
$453$	3.81159e6	0.872691
$454$	2.32284e6	0.528907
$455$	−1.74395e6	−0.394917
$456$	−188073.	−0.0423558
$457$	59641.9	0.0133586	0.00667930	−	0.999978i	$-0.497874\pi$
$457$	59641.9	0.0133586	0.00667930	+	0.999978i	$0.497874\pi$
$458$	2.69575e6	0.600504
$459$	−1.03779e6	−0.229920
$460$	208663.	0.0459782
$461$	1.59900e6	0.350426	0.175213	−	0.984531i	$-0.443939\pi$
$461$	1.59900e6	0.350426	0.175213	+	0.984531i	$0.443939\pi$
$462$	−1.39230e6	−0.303479
$463$	2.17474e6	0.471470	0.235735	−	0.971817i	$-0.424250\pi$
$463$	2.17474e6	0.471470	0.235735	+	0.971817i	$0.424250\pi$
$464$	545424.	0.117609
$465$	−1.39316e6	−0.298792
$466$	−2.10775e6	−0.449629
$467$	4.43242e6	0.940479	0.470240	−	0.882539i	$-0.344168\pi$
$467$	4.43242e6	0.940479	0.470240	+	0.882539i	$0.344168\pi$
$468$	1.45603e6	0.307295
$469$	−737074.	−0.154731
$470$	−2.55144e6	−0.532772
$471$	3.15832e6	0.656000
$472$	−1.41378e6	−0.292097
$473$	−565531.	−0.116226
$474$	743190.	0.151934
$475$	821912.	0.167144
$476$	1.43417e6	0.290124
$477$	−2.94259e6	−0.592152
$478$	91812.7	0.0183795
$479$	−5.69579e6	−1.13427	−0.567133	−	0.823626i	$-0.691947\pi$
$479$	−5.69579e6	−1.13427	−0.567133	+	0.823626i	$0.691947\pi$
$480$	−227202.	−0.0450101
$481$	6.87123e6	1.35417
$482$	−2.96672e6	−0.581646
$483$	299776.	0.0584695
$484$	3.45966e6	0.671305
$485$	−2.97629e6	−0.574540
$486$	236196.	0.0453609
$487$	−6.11927e6	−1.16917	−0.584585	−	0.811333i	$-0.698743\pi$
$487$	−6.11927e6	−1.16917	−0.584585	+	0.811333i	$0.698743\pi$
$488$	2.40783e6	0.457694
$489$	5.28653e6	0.999767
$490$	−1.26642e6	−0.238279
$491$	5.80932e6	1.08748	0.543740	−	0.839254i	$-0.317008\pi$
$491$	5.80932e6	1.08748	0.543740	+	0.839254i	$0.317008\pi$
$492$	2.83745e6	0.528465
$493$	3.03303e6	0.562029
$494$	1.46733e6	0.270527
$495$	1.22656e6	0.224996
$496$	−1.60742e6	−0.293376
$497$	2.39698e6	0.435284
$498$	−2.38491e6	−0.430922
$499$	4.23911e6	0.762120	0.381060	−	0.924550i	$-0.375559\pi$
$499$	4.23911e6	0.762120	0.381060	+	0.924550i	$0.375559\pi$
$500$	2.22557e6	0.398122
$501$	5.13682e6	0.914324
$502$	5.87356e6	1.04026
$503$	2.69629e6	0.475167	0.237584	−	0.971367i	$-0.423645\pi$
$503$	2.69629e6	0.475167	0.237584	+	0.971367i	$0.423645\pi$
$504$	−326411.	−0.0572385
$505$	2.73828e6	0.477804
$506$	−1.29971e6	−0.225669
$507$	−8.01821e6	−1.38534
$508$	4.55212e6	0.782627
$509$	389811.	0.0666898	0.0333449	−	0.999444i	$-0.489384\pi$
$509$	389811.	0.0666898	0.0333449	+	0.999444i	$0.489384\pi$
$510$	−1.26344e6	−0.215094
$511$	3.15465e6	0.534440
$512$	−262144.	−0.0441942
$513$	238029.	0.0399335
$514$	−2.11061e6	−0.352371
$515$	4.27863e6	0.710865
$516$	−132583.	−0.0219211
$517$	1.58923e7	2.61494
$518$	−1.54038e6	−0.252235
$519$	−4.70035e6	−0.765971
$520$	1.77262e6	0.287480
$521$	3.41346e6	0.550936	0.275468	−	0.961310i	$-0.411167\pi$
$521$	3.41346e6	0.550936	0.275468	+	0.961310i	$0.411167\pi$
$522$	−690303.	−0.110883
$523$	1.58263e6	0.253003	0.126502	−	0.991966i	$-0.459625\pi$
$523$	1.58263e6	0.253003	0.126502	+	0.991966i	$0.459625\pi$
$524$	−1.85372e6	−0.294927
$525$	1.42648e6	0.225874
$526$	−2.96382e6	−0.467077
$527$	−8.93862e6	−1.40199
$528$	1.41519e6	0.220917
$529$	279841.	0.0434783
$530$	−3.58241e6	−0.553969
$531$	1.78931e6	0.275391
$532$	−328944.	−0.0503898
$533$	−2.21376e7	−3.37531
$534$	2.29095e6	0.347667
$535$	4.05475e6	0.612462
$536$	749189.	0.112637
$537$	−5.67095e6	−0.848633
$538$	−4.94612e6	−0.736731
$539$	7.88821e6	1.16952
$540$	287553.	0.0424359
$541$	5.85776e6	0.860475	0.430238	−	0.902716i	$-0.358430\pi$
$541$	5.85776e6	0.860475	0.430238	+	0.902716i	$0.358430\pi$
$542$	−4.33191e6	−0.633405
$543$	4.67076e6	0.679811
$544$	−1.45774e6	−0.211195
$545$	809809.	0.116786
$546$	2.54664e6	0.365582
$547$	−4.41901e6	−0.631476	−0.315738	−	0.948846i	$-0.602252\pi$
$547$	−4.41901e6	−0.631476	−0.315738	+	0.948846i	$0.602252\pi$
$548$	−2.20473e6	−0.313620
$549$	−3.04741e6	−0.431518
$550$	−6.18464e6	−0.871781
$551$	−695661.	−0.0976154
$552$	−304704.	−0.0425628
$553$	1.29986e6	0.180752
$554$	4.05892e6	0.561870
$555$	1.35701e6	0.187004
$556$	−876973.	−0.120309
$557$	4.99496e6	0.682172	0.341086	−	0.940032i	$-0.389205\pi$
$557$	4.99496e6	0.682172	0.341086	+	0.940032i	$0.389205\pi$
$558$	2.03439e6	0.276598
$559$	1.03440e6	0.140010
$560$	−397383.	−0.0535476
$561$	7.86966e6	1.05572
$562$	3.22376e6	0.430548
$563$	−1.71595e6	−0.228157	−0.114078	−	0.993472i	$-0.536391\pi$
$563$	−1.71595e6	−0.228157	−0.114078	+	0.993472i	$0.536391\pi$
$564$	3.72579e6	0.493197
$565$	−3.72341e6	−0.490705
$566$	4.89437e6	0.642179
$567$	413113.	0.0539649
$568$	−2.43638e6	−0.316865
$569$	−731339.	−0.0946975	−0.0473487	−	0.998878i	$-0.515077\pi$
$569$	−731339.	−0.0946975	−0.0473487	+	0.998878i	$0.515077\pi$
$570$	289785.	0.0373584
$571$	8.74520e6	1.12248	0.561241	−	0.827652i	$-0.310324\pi$
$571$	8.74520e6	1.12248	0.561241	+	0.827652i	$0.310324\pi$
$572$	−1.10412e7	−1.41100
$573$	−6.08249e6	−0.773918
$574$	4.96279e6	0.628704
$575$	1.33161e6	0.167961
$576$	331776.	0.0416667
$577$	−8.82854e6	−1.10395	−0.551975	−	0.833861i	$-0.686126\pi$
$577$	−8.82854e6	−1.10395	−0.551975	+	0.833861i	$0.686126\pi$
$578$	−2.42688e6	−0.302154
$579$	1.30637e6	0.161946
$580$	−840398.	−0.103733
$581$	−4.17128e6	−0.512659
$582$	4.34617e6	0.531863
$583$	2.23139e7	2.71897
$584$	−3.20651e6	−0.389045
$585$	−2.24347e6	−0.271038
$586$	−6.62157e6	−0.796557
$587$	−8.53872e6	−1.02282	−0.511408	−	0.859338i	$-0.670876\pi$
$587$	−8.53872e6	−1.02282	−0.511408	+	0.859338i	$0.670876\pi$
$588$	1.84931e6	0.220580
$589$	2.05018e6	0.243502
$590$	2.17837e6	0.257633
$591$	4.33038e6	0.509985
$592$	1.56570e6	0.183614
$593$	3.60062e6	0.420475	0.210237	−	0.977650i	$-0.432576\pi$
$593$	3.60062e6	0.420475	0.210237	+	0.977650i	$0.432576\pi$
$594$	−1.79110e6	−0.208283
$595$	−2.20979e6	−0.255893
$596$	−3.30462e6	−0.381071
$597$	−5.14624e6	−0.590955
$598$	2.37728e6	0.271849
$599$	−3.30268e6	−0.376097	−0.188048	−	0.982160i	$-0.560216\pi$
$599$	−3.30268e6	−0.376097	−0.188048	+	0.982160i	$0.560216\pi$
$600$	−1.44992e6	−0.164425
$601$	7.97390e6	0.900502	0.450251	−	0.892902i	$-0.351335\pi$
$601$	7.97390e6	0.900502	0.450251	+	0.892902i	$0.351335\pi$
$602$	−231891.	−0.0260791
$603$	−948193.	−0.106195
$604$	−6.77616e6	−0.755773
$605$	−5.33070e6	−0.592101
$606$	−3.99862e6	−0.442312
$607$	9.94310e6	1.09534	0.547671	−	0.836694i	$-0.315514\pi$
$607$	9.94310e6	1.09534	0.547671	+	0.836694i	$0.315514\pi$
$608$	334351.	0.0366812
$609$	−1.20736e6	−0.131915
$610$	−3.71001e6	−0.403693
$611$	−2.90684e7	−3.15005
$612$	1.84496e6	0.199117
$613$	−1.06848e7	−1.14845	−0.574227	−	0.818696i	$-0.694697\pi$
$613$	−1.06848e7	−1.14845	−0.574227	+	0.818696i	$0.694697\pi$
$614$	−131239.	−0.0140489
$615$	−4.37199e6	−0.466113
$616$	2.47520e6	0.262821
$617$	4.42104e6	0.467532	0.233766	−	0.972293i	$-0.424895\pi$
$617$	4.42104e6	0.467532	0.233766	+	0.972293i	$0.424895\pi$
$618$	−6.24794e6	−0.658061
$619$	−1.62693e7	−1.70664	−0.853320	−	0.521387i	$-0.825415\pi$
$619$	−1.62693e7	−1.70664	−0.853320	+	0.521387i	$0.825415\pi$
$620$	2.47673e6	0.258762
$621$	385641.	0.0401286
$622$	−6.42911e6	−0.666308
$623$	4.00694e6	0.413612
$624$	−2.58850e6	−0.266125
$625$	4.43715e6	0.454364
$626$	9.92233e6	1.01199
$627$	−1.80500e6	−0.183362
$628$	−5.61479e6	−0.568113
$629$	8.70665e6	0.877455
$630$	502938.	0.0504851
$631$	−38051.9	−0.00380455	−0.00190227	−	0.999998i	$-0.500606\pi$
$631$	−38051.9	−0.00380455	−0.00190227	+	0.999998i	$0.500606\pi$
$632$	−1.32123e6	−0.131578
$633$	437835.	0.0434311
$634$	−3.08520e6	−0.304832
$635$	−7.01398e6	−0.690288
$636$	5.23127e6	0.512819
$637$	−1.44282e7	−1.40884
$638$	5.23463e6	0.509137
$639$	3.08354e6	0.298743
$640$	403915.	0.0389799
$641$	−1.86773e6	−0.179544	−0.0897718	−	0.995962i	$-0.528614\pi$
$641$	−1.86773e6	−0.179544	−0.0897718	+	0.995962i	$0.528614\pi$
$642$	−5.92101e6	−0.566968
$643$	6.54234e6	0.624031	0.312015	−	0.950077i	$-0.398996\pi$
$643$	6.54234e6	0.624031	0.312015	+	0.950077i	$0.398996\pi$
$644$	−532936.	−0.0506361
$645$	204286.	0.0193347
$646$	1.85928e6	0.175292
$647$	−1.89181e6	−0.177671	−0.0888357	−	0.996046i	$-0.528315\pi$
$647$	−1.89181e6	−0.177671	−0.0888357	+	0.996046i	$0.528315\pi$
$648$	−419904.	−0.0392837
$649$	−1.35685e7	−1.26451
$650$	1.13122e7	1.05018
$651$	3.55820e6	0.329062
$652$	−9.39828e6	−0.865824
$653$	−1.59554e6	−0.146428	−0.0732139	−	0.997316i	$-0.523326\pi$
$653$	−1.59554e6	−0.146428	−0.0732139	+	0.997316i	$0.523326\pi$
$654$	−1.18254e6	−0.108111
$655$	2.85623e6	0.260130
$656$	−5.04436e6	−0.457664
$657$	4.05824e6	0.366795
$658$	6.51651e6	0.586746
$659$	5.41003e6	0.485273	0.242636	−	0.970117i	$-0.421988\pi$
$659$	5.41003e6	0.485273	0.242636	+	0.970117i	$0.421988\pi$
$660$	−2.18054e6	−0.194852
$661$	1.79886e7	1.60138	0.800691	−	0.599078i	$-0.204466\pi$
$661$	1.79886e7	1.60138	0.800691	+	0.599078i	$0.204466\pi$
$662$	−3.81840e6	−0.338639
$663$	−1.43942e7	−1.27176
$664$	4.23984e6	0.373189
$665$	506842.	0.0444445
$666$	−1.98159e6	−0.173113
$667$	−1.12707e6	−0.0980925
$668$	−9.13213e6	−0.791828
$669$	−1.46602e6	−0.126641
$670$	−1.15436e6	−0.0993470
$671$	2.31088e7	1.98139
$672$	580286.	0.0495700
$673$	1.21561e7	1.03456	0.517282	−	0.855815i	$-0.326944\pi$
$673$	1.21561e7	1.03456	0.517282	+	0.855815i	$0.326944\pi$
$674$	−9.70320e6	−0.822745
$675$	1.83506e6	0.155021
$676$	1.42546e7	1.19974
$677$	1.89587e7	1.58978	0.794891	−	0.606753i	$-0.207528\pi$
$677$	1.89587e7	1.58978	0.794891	+	0.606753i	$0.207528\pi$
$678$	5.43718e6	0.454254
$679$	7.60158e6	0.632746
$680$	2.24611e6	0.186277
$681$	5.22639e6	0.431851
$682$	−1.54270e7	−1.27005
$683$	−1.87970e7	−1.54183	−0.770917	−	0.636936i	$-0.780202\pi$
$683$	−1.87970e7	−1.54183	−0.770917	+	0.636936i	$0.780202\pi$
$684$	−423163.	−0.0345834
$685$	3.39708e6	0.276617
$686$	7.46750e6	0.605850
$687$	6.06544e6	0.490310
$688$	235703.	0.0189843
$689$	−4.08140e7	−3.27538
$690$	469492.	0.0375410
$691$	3.09571e6	0.246641	0.123321	−	0.992367i	$-0.460646\pi$
$691$	3.09571e6	0.246641	0.123321	+	0.992367i	$0.460646\pi$
$692$	8.35619e6	0.663350
$693$	−3.13268e6	−0.247790
$694$	−1.53378e6	−0.120883
$695$	1.35125e6	0.106114
$696$	1.22720e6	0.0960271
$697$	−2.80510e7	−2.18709
$698$	1.02808e7	0.798710
$699$	−4.74244e6	−0.367120
$700$	−2.53596e6	−0.195613
$701$	−9.71754e6	−0.746898	−0.373449	−	0.927651i	$-0.621825\pi$
$701$	−9.71754e6	−0.746898	−0.373449	+	0.927651i	$0.621825\pi$
$702$	3.27606e6	0.250905
$703$	−1.99697e6	−0.152400
$704$	−2.51589e6	−0.191320
$705$	−5.74075e6	−0.435007
$706$	3.99696e6	0.301799
$707$	−6.99370e6	−0.526210
$708$	−3.18100e6	−0.238496
$709$	−1.95429e7	−1.46007	−0.730033	−	0.683411i	$-0.760496\pi$
$709$	−1.95429e7	−1.46007	−0.730033	+	0.683411i	$0.760496\pi$
$710$	3.75401e6	0.279479
$711$	1.67218e6	0.124053
$712$	−4.07281e6	−0.301088
$713$	3.32158e6	0.244692
$714$	3.22688e6	0.236885
$715$	1.70125e7	1.24452
$716$	1.00817e7	0.734937
$717$	206579.	0.0150068
$718$	−3.35323e6	−0.242746
$719$	8.02925e6	0.579232	0.289616	−	0.957143i	$-0.406472\pi$
$719$	8.02925e6	0.579232	0.289616	+	0.957143i	$0.406472\pi$
$720$	−511205.	−0.0367506
$721$	−1.09278e7	−0.782881
$722$	9.47795e6	0.676661
$723$	−6.67511e6	−0.474912
$724$	−8.30358e6	−0.588734
$725$	−5.36312e6	−0.378941
$726$	7.78424e6	0.548118
$727$	8.55901e6	0.600603	0.300302	−	0.953844i	$-0.402913\pi$
$727$	8.55901e6	0.600603	0.300302	+	0.953844i	$0.402913\pi$
$728$	−4.52735e6	−0.316604
$729$	531441.	0.0370370
$730$	4.94063e6	0.343143
$731$	1.31071e6	0.0907220
$732$	5.41761e6	0.373706
$733$	−1.51462e7	−1.04122	−0.520611	−	0.853794i	$-0.674296\pi$
$733$	−1.51462e7	−1.04122	−0.520611	+	0.853794i	$0.674296\pi$
$734$	−8.59347e6	−0.588747
$735$	−2.84944e6	−0.194554
$736$	541696.	0.0368605
$737$	7.19024e6	0.487612
$738$	6.38427e6	0.431490
$739$	1.47295e7	0.992146	0.496073	−	0.868281i	$-0.334775\pi$
$739$	1.47295e7	0.992146	0.496073	+	0.868281i	$0.334775\pi$
$740$	−2.41246e6	−0.161950
$741$	3.30149e6	0.220884
$742$	9.14964e6	0.610090
$743$	6.32849e6	0.420560	0.210280	−	0.977641i	$-0.432562\pi$
$743$	6.32849e6	0.420560	0.210280	+	0.977641i	$0.432562\pi$
$744$	−3.61669e6	−0.239540
$745$	5.09181e6	0.336110
$746$	9.14600e6	0.601706
$747$	−5.36605e6	−0.351846
$748$	−1.39905e7	−0.914280
$749$	−1.03560e7	−0.674509
$750$	5.00753e6	0.325065
$751$	2.35372e6	0.152284	0.0761420	−	0.997097i	$-0.475740\pi$
$751$	2.35372e6	0.152284	0.0761420	+	0.997097i	$0.475740\pi$
$752$	−6.62362e6	−0.427121
$753$	1.32155e7	0.849369
$754$	−9.57458e6	−0.613326
$755$	1.04408e7	0.666602
$756$	−734424.	−0.0467350
$757$	−4.00595e6	−0.254077	−0.127039	−	0.991898i	$-0.540547\pi$
$757$	−4.00595e6	−0.254077	−0.127039	+	0.991898i	$0.540547\pi$
$758$	1.44948e7	0.916301
$759$	−2.92435e6	−0.184258
$760$	−515173.	−0.0323533
$761$	1.10008e7	0.688590	0.344295	−	0.938862i	$-0.388118\pi$
$761$	1.10008e7	0.688590	0.344295	+	0.938862i	$0.388118\pi$
$762$	1.02423e7	0.639012
$763$	−2.06829e6	−0.128617
$764$	1.08133e7	0.670233
$765$	−2.84274e6	−0.175624
$766$	204206.	0.0125747
$767$	2.48180e7	1.52327
$768$	−589824.	−0.0360844
$769$	−1.86532e7	−1.13746	−0.568732	−	0.822523i	$-0.692566\pi$
$769$	−1.86532e7	−1.13746	−0.568732	+	0.822523i	$0.692566\pi$
$770$	−3.81383e6	−0.231811
$771$	−4.74887e6	−0.287710
$772$	−2.32244e6	−0.140249
$773$	2.20769e7	1.32889	0.664444	−	0.747338i	$-0.268668\pi$
$773$	2.20769e7	1.32889	0.664444	+	0.747338i	$0.268668\pi$
$774$	−298311.	−0.0178985
$775$	1.58056e7	0.945273
$776$	−7.72653e6	−0.460607
$777$	−3.46586e6	−0.205949
$778$	8.23866e6	0.487986
$779$	6.43383e6	0.379862
$780$	3.98839e6	0.234726
$781$	−2.33828e7	−1.37173
$782$	3.01229e6	0.176149
$783$	−1.55318e6	−0.0905353
$784$	−3.28766e6	−0.191028
$785$	8.65135e6	0.501083
$786$	−4.17086e6	−0.240807
$787$	−1.48726e6	−0.0855951	−0.0427976	−	0.999084i	$-0.513627\pi$
$787$	−1.48726e6	−0.0855951	−0.0427976	+	0.999084i	$0.513627\pi$
$788$	−7.69846e6	−0.441660
$789$	−6.66860e6	−0.381366
$790$	2.03577e6	0.116054
$791$	9.50977e6	0.540417
$792$	3.18417e6	0.180378
$793$	−4.22678e7	−2.38686
$794$	−1.96734e7	−1.10746
$795$	−8.06042e6	−0.452314
$796$	9.14888e6	0.511782
$797$	1.83345e7	1.02241	0.511203	−	0.859460i	$-0.329200\pi$
$797$	1.83345e7	1.02241	0.511203	+	0.859460i	$0.329200\pi$
$798$	−740124.	−0.0411431
$799$	−3.68330e7	−2.04113
$800$	2.57764e6	0.142396
$801$	5.15465e6	0.283869
$802$	100642.	0.00552512
$803$	−3.07740e7	−1.68421
$804$	1.68568e6	0.0919674
$805$	821156.	0.0446617
$806$	2.82172e7	1.52995
$807$	−1.11288e7	−0.601538
$808$	7.10866e6	0.383054
$809$	6.80583e6	0.365603	0.182802	−	0.983150i	$-0.441483\pi$
$809$	6.80583e6	0.365603	0.182802	+	0.983150i	$0.441483\pi$
$810$	646994.	0.0346488
$811$	2.95810e7	1.57929	0.789644	−	0.613566i	$-0.210265\pi$
$811$	2.95810e7	1.57929	0.789644	+	0.613566i	$0.210265\pi$
$812$	2.14642e6	0.114241
$813$	−9.74680e6	−0.517173
$814$	1.50266e7	0.794878
$815$	1.44810e7	0.763669
$816$	−3.27993e6	−0.172440
$817$	−300627.	−0.0157570
$818$	741735.	0.0387584
$819$	5.72993e6	0.298497
$820$	7.77243e6	0.403666
$821$	8.28541e6	0.428999	0.214500	−	0.976724i	$-0.431188\pi$
$821$	8.28541e6	0.428999	0.214500	+	0.976724i	$0.431188\pi$
$822$	−4.96064e6	−0.256070
$823$	1.47148e7	0.757278	0.378639	−	0.925544i	$-0.376392\pi$
$823$	1.47148e7	0.757278	0.378639	+	0.925544i	$0.376392\pi$
$824$	1.11075e7	0.569897
$825$	−1.39154e7	−0.711806
$826$	−5.56366e6	−0.283734
$827$	8.48341e6	0.431327	0.215664	−	0.976468i	$-0.430809\pi$
$827$	8.48341e6	0.431327	0.215664	+	0.976468i	$0.430809\pi$
$828$	−685584.	−0.0347524
$829$	3.29360e7	1.66450	0.832251	−	0.554399i	$-0.187052\pi$
$829$	3.29360e7	1.66450	0.832251	+	0.554399i	$0.187052\pi$
$830$	−6.53281e6	−0.329158
$831$	9.13257e6	0.458765
$832$	4.60177e6	0.230471
$833$	−1.82822e7	−0.912884
$834$	−1.97319e6	−0.0982321
$835$	1.40709e7	0.698403
$836$	3.20889e6	0.158796
$837$	4.57737e6	0.225841
$838$	−1.22179e6	−0.0601019
$839$	−6.46559e6	−0.317105	−0.158553	−	0.987351i	$-0.550683\pi$
$839$	−6.46559e6	−0.317105	−0.158553	+	0.987351i	$0.550683\pi$
$840$	−894113.	−0.0437214
$841$	−1.59718e7	−0.778691
$842$	2.00679e7	0.975489
$843$	7.25345e6	0.351541
$844$	−778373.	−0.0376125
$845$	−2.19637e7	−1.05819
$846$	8.38302e6	0.402694
$847$	1.36148e7	0.652085
$848$	−9.30003e6	−0.444114
$849$	1.10123e7	0.524337
$850$	1.43339e7	0.680482
$851$	−3.23538e6	−0.153145
$852$	−5.48185e6	−0.258719
$853$	1.56915e6	0.0738401	0.0369201	−	0.999318i	$-0.488245\pi$
$853$	1.56915e6	0.0738401	0.0369201	+	0.999318i	$0.488245\pi$
$854$	9.47555e6	0.444590
$855$	652016.	0.0305030
$856$	1.05262e7	0.491008
$857$	1.95888e7	0.911080	0.455540	−	0.890215i	$-0.349446\pi$
$857$	1.95888e7	0.911080	0.455540	+	0.890215i	$0.349446\pi$
$858$	−2.48427e7	−1.15208
$859$	−7.75837e6	−0.358747	−0.179373	−	0.983781i	$-0.557407\pi$
$859$	−7.75837e6	−0.358747	−0.179373	+	0.983781i	$0.557407\pi$
$860$	−363174.	−0.0167444
$861$	1.11663e7	0.513334
$862$	1.90365e7	0.872609
$863$	7.14209e6	0.326436	0.163218	−	0.986590i	$-0.447813\pi$
$863$	7.14209e6	0.326436	0.163218	+	0.986590i	$0.447813\pi$
$864$	746496.	0.0340207
$865$	−1.28753e7	−0.585084
$866$	−1.83514e7	−0.831524
$867$	−5.46048e6	−0.246708
$868$	−6.32569e6	−0.284976
$869$	−1.26803e7	−0.569613
$870$	−1.89090e6	−0.0846973
$871$	−1.31515e7	−0.587396
$872$	2.10229e6	0.0936269
$873$	9.77889e6	0.434264
$874$	−690905.	−0.0305943
$875$	8.75831e6	0.386723
$876$	−7.21464e6	−0.317654
$877$	−3.29064e6	−0.144471	−0.0722357	−	0.997388i	$-0.523013\pi$
$877$	−3.29064e6	−0.144471	−0.0722357	+	0.997388i	$0.523013\pi$
$878$	2.65188e7	1.16096
$879$	−1.48985e7	−0.650386
$880$	3.87652e6	0.168747
$881$	1.65827e7	0.719808	0.359904	−	0.932989i	$-0.382809\pi$
$881$	1.65827e7	0.719808	0.359904	+	0.932989i	$0.382809\pi$
$882$	4.16094e6	0.180103
$883$	1.73068e7	0.746991	0.373496	−	0.927632i	$-0.378159\pi$
$883$	1.73068e7	0.746991	0.373496	+	0.927632i	$0.378159\pi$
$884$	2.55898e7	1.10138
$885$	4.90134e6	0.210357
$886$	3.67535e6	0.157295
$887$	−2.48291e7	−1.05962	−0.529812	−	0.848115i	$-0.677737\pi$
$887$	−2.48291e7	−1.05962	−0.529812	+	0.848115i	$0.677737\pi$
$888$	3.52283e6	0.149920
$889$	1.79140e7	0.760219
$890$	6.27544e6	0.265564
$891$	−4.02997e6	−0.170062
$892$	2.60625e6	0.109674
$893$	8.44809e6	0.354511
$894$	−7.43540e6	−0.311143
$895$	−1.55340e7	−0.648225
$896$	−1.03162e6	−0.0429288
$897$	5.34888e6	0.221964
$898$	−3.26788e7	−1.35231
$899$	−1.33778e7	−0.552057
$900$	−3.26233e6	−0.134252
$901$	−5.17161e7	−2.12234
$902$	−4.84126e7	−1.98126
$903$	−521755.	−0.0212935
$904$	−9.66609e6	−0.393396
$905$	1.27943e7	0.519271
$906$	−1.52464e7	−0.617086
$907$	−2.19445e7	−0.885743	−0.442871	−	0.896585i	$-0.646040\pi$
$907$	−2.19445e7	−0.885743	−0.442871	+	0.896585i	$0.646040\pi$
$908$	−9.29136e6	−0.373994
$909$	−8.99690e6	−0.361146
$910$	6.97581e6	0.279249
$911$	2.99103e7	1.19405	0.597027	−	0.802221i	$-0.296348\pi$
$911$	2.99103e7	1.19405	0.597027	+	0.802221i	$0.296348\pi$
$912$	752290.	0.0299501
$913$	4.06913e7	1.61556
$914$	−238567.	−0.00944595
$915$	−8.34753e6	−0.329614
$916$	−1.07830e7	−0.424621
$917$	−7.29496e6	−0.286483
$918$	4.15116e6	0.162578
$919$	−7.27607e6	−0.284190	−0.142095	−	0.989853i	$-0.545384\pi$
$919$	−7.27607e6	−0.284190	−0.142095	+	0.989853i	$0.545384\pi$
$920$	−834653.	−0.0325115
$921$	−295289.	−0.0114709
$922$	−6.39601e6	−0.247789
$923$	4.27691e7	1.65244
$924$	5.56921e6	0.214592
$925$	−1.53954e7	−0.591613
$926$	−8.69895e6	−0.333380
$927$	−1.40579e7	−0.537304
$928$	−2.18170e6	−0.0831619
$929$	−1.27760e7	−0.485688	−0.242844	−	0.970065i	$-0.578080\pi$
$929$	−1.27760e7	−0.485688	−0.242844	+	0.970065i	$0.578080\pi$
$930$	5.57265e6	0.211278
$931$	4.19324e6	0.158553
$932$	8.43100e6	0.317936
$933$	−1.44655e7	−0.544038
$934$	−1.77297e7	−0.665019
$935$	2.15568e7	0.806408
$936$	−5.82411e6	−0.217290
$937$	−4.59979e7	−1.71155	−0.855774	−	0.517349i	$-0.826919\pi$
$937$	−4.59979e7	−1.71155	−0.855774	+	0.517349i	$0.826919\pi$
$938$	2.94829e6	0.109412
$939$	2.23252e7	0.826289
$940$	1.02058e7	0.376727
$941$	4.91509e7	1.80950	0.904748	−	0.425948i	$-0.140059\pi$
$941$	4.91509e7	1.80950	0.904748	+	0.425948i	$0.140059\pi$
$942$	−1.26333e7	−0.463862
$943$	1.04237e7	0.381718
$944$	5.65512e6	0.206544
$945$	1.13161e6	0.0412209
$946$	2.26212e6	0.0821842
$947$	−1.34393e7	−0.486970	−0.243485	−	0.969905i	$-0.578291\pi$
$947$	−1.34393e7	−0.486970	−0.243485	+	0.969905i	$0.578291\pi$
$948$	−2.97276e6	−0.107433
$949$	5.62882e7	2.02886
$950$	−3.28765e6	−0.118189
$951$	−6.94171e6	−0.248894
$952$	−5.73668e6	−0.205149
$953$	−3.53308e7	−1.26015	−0.630074	−	0.776535i	$-0.716975\pi$
$953$	−3.53308e7	−1.26015	−0.630074	+	0.776535i	$0.716975\pi$
$954$	1.17704e7	0.418715
$955$	−1.66613e7	−0.591155
$956$	−367251.	−0.0129962
$957$	1.17779e7	0.415709
$958$	2.27831e7	0.802047
$959$	−8.67630e6	−0.304641
$960$	908810.	0.0318269
$961$	1.07964e7	0.377112
$962$	−2.74849e7	−0.957540
$963$	−1.33223e7	−0.462927
$964$	1.18669e7	0.411286
$965$	3.57844e6	0.123702
$966$	−1.19911e6	−0.0413442
$967$	2.34640e7	0.806930	0.403465	−	0.914995i	$-0.367806\pi$
$967$	2.34640e7	0.806930	0.403465	+	0.914995i	$0.367806\pi$
$968$	−1.38386e7	−0.474685
$969$	4.18338e6	0.143126
$970$	1.19052e7	0.406261
$971$	3.86147e7	1.31433	0.657164	−	0.753747i	$-0.271756\pi$
$971$	3.86147e7	1.31433	0.657164	+	0.753747i	$0.271756\pi$
$972$	−944784.	−0.0320750
$973$	−3.45116e6	−0.116865
$974$	2.44771e7	0.826728
$975$	2.54525e7	0.857469
$976$	−9.63131e6	−0.323639
$977$	−2.51154e7	−0.841790	−0.420895	−	0.907109i	$-0.638284\pi$
$977$	−2.51154e7	−0.841790	−0.420895	+	0.907109i	$0.638284\pi$
$978$	−2.11461e7	−0.706942
$979$	−3.90882e7	−1.30343
$980$	5.06567e6	0.168489
$981$	−2.66071e6	−0.0882723
$982$	−2.32373e7	−0.768965
$983$	1.63217e7	0.538742	0.269371	−	0.963037i	$-0.413184\pi$
$983$	1.63217e7	0.538742	0.269371	+	0.963037i	$0.413184\pi$
$984$	−1.13498e7	−0.373681
$985$	1.18619e7	0.389550
$986$	−1.21321e7	−0.397415
$987$	1.46621e7	0.479076
$988$	−5.86932e6	−0.191291
$989$	−487058.	−0.0158340
$990$	−4.90622e6	−0.159096
$991$	8.45877e6	0.273605	0.136802	−	0.990598i	$-0.456318\pi$
$991$	8.45877e6	0.273605	0.136802	+	0.990598i	$0.456318\pi$
$992$	6.42967e6	0.207448
$993$	−8.59140e6	−0.276497
$994$	−9.58791e6	−0.307793
$995$	−1.40967e7	−0.451399
$996$	9.53964e6	0.304708
$997$	4.36472e7	1.39065	0.695326	−	0.718695i	$-0.255260\pi$
$997$	4.36472e7	1.39065	0.695326	+	0.718695i	$0.255260\pi$
$998$	−1.69564e7	−0.538900
$999$	−4.45859e6	−0.141346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.6.a.g.1.2	✓	3
3.2	odd	2		414.6.a.n.1.2		3
4.3	odd	2		1104.6.a.k.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.6.a.g.1.2	✓	3	1.1	even	1	trivial
414.6.a.n.1.2		3	3.2	odd	2
1104.6.a.k.1.2		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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -24.6530 q^{5} +36.0000 q^{6} +62.9650 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -24.6530 q^{5} +36.0000 q^{6} +62.9650 q^{7} -64.0000 q^{8} +81.0000 q^{9} +98.6122 q^{10} -614.231 q^{11} -144.000 q^{12} +1123.48 q^{13} -251.860 q^{14} +221.877 q^{15} +256.000 q^{16} +1423.58 q^{17} -324.000 q^{18} -326.515 q^{19} -394.449 q^{20} -566.685 q^{21} +2456.92 q^{22} -529.000 q^{23} +576.000 q^{24} -2517.23 q^{25} -4493.92 q^{26} -729.000 q^{27} +1007.44 q^{28} +2130.56 q^{29} -887.509 q^{30} -6278.98 q^{31} -1024.00 q^{32} +5528.08 q^{33} -5694.32 q^{34} -1552.28 q^{35} +1296.00 q^{36} +6116.03 q^{37} +1306.06 q^{38} -10111.3 q^{39} +1577.79 q^{40} -19704.5 q^{41} +2266.74 q^{42} +920.714 q^{43} -9827.70 q^{44} -1996.90 q^{45} +2116.00 q^{46} -25873.5 q^{47} -2304.00 q^{48} -12842.4 q^{49} +10068.9 q^{50} -12812.2 q^{51} +17975.7 q^{52} -36328.3 q^{53} +2916.00 q^{54} +15142.7 q^{55} -4029.76 q^{56} +2938.63 q^{57} -8522.26 q^{58} +22090.3 q^{59} +3550.04 q^{60} -37622.3 q^{61} +25115.9 q^{62} +5100.17 q^{63} +4096.00 q^{64} -27697.2 q^{65} -22112.3 q^{66} -11706.1 q^{67} +22777.3 q^{68} +4761.00 q^{69} +6209.12 q^{70} +38068.4 q^{71} -5184.00 q^{72} +50101.7 q^{73} -24464.1 q^{74} +22655.0 q^{75} -5224.24 q^{76} -38675.1 q^{77} +40445.2 q^{78} +20644.2 q^{79} -6311.18 q^{80} +6561.00 q^{81} +78818.2 q^{82} -66247.5 q^{83} -9066.96 q^{84} -35095.5 q^{85} -3682.86 q^{86} -19175.1 q^{87} +39310.8 q^{88} +63637.6 q^{89} +7987.58 q^{90} +70739.9 q^{91} -8464.00 q^{92} +56510.8 q^{93} +103494. q^{94} +8049.58 q^{95} +9216.00 q^{96} +120727. q^{97} +51369.6 q^{98} -49752.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 12 q^{2} - 27 q^{3} + 48 q^{4} - 18 q^{5} + 108 q^{6} - 50 q^{7} - 192 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q - 12 * q^2 - 27 * q^3 + 48 * q^4 - 18 * q^5 + 108 * q^6 - 50 * q^7 - 192 * q^8 + 243 * q^9	\( 3 q - 12 q^{2} - 27 q^{3} + 48 q^{4} - 18 q^{5} + 108 q^{6} - 50 q^{7} - 192 q^{8} + 243 q^{9} + 72 q^{10} + 6 q^{11} - 432 q^{12} + 790 q^{13} + 200 q^{14} + 162 q^{15} + 768 q^{16} + 1148 q^{17} - 972 q^{18} + 2692 q^{19} - 288 q^{20} + 450 q^{21} - 24 q^{22} - 1587 q^{23} + 1728 q^{24} + 2953 q^{25} - 3160 q^{26} - 2187 q^{27} - 800 q^{28} - 5910 q^{29} - 648 q^{30} - 4224 q^{31} - 3072 q^{32} - 54 q^{33} - 4592 q^{34} - 24360 q^{35} + 3888 q^{36} - 10368 q^{37} - 10768 q^{38} - 7110 q^{39} + 1152 q^{40} - 37786 q^{41} - 1800 q^{42} - 28616 q^{43} + 96 q^{44} - 1458 q^{45} + 6348 q^{46} - 43348 q^{47} - 6912 q^{48} + 2939 q^{49} - 11812 q^{50} - 10332 q^{51} + 12640 q^{52} - 41566 q^{53} + 8748 q^{54} - 8956 q^{55} + 3200 q^{56} - 24228 q^{57} + 23640 q^{58} - 46560 q^{59} + 2592 q^{60} + 7924 q^{61} + 16896 q^{62} - 4050 q^{63} + 12288 q^{64} - 45580 q^{65} + 216 q^{66} - 17016 q^{67} + 18368 q^{68} + 14283 q^{69} + 97440 q^{70} - 38280 q^{71} - 15552 q^{72} - 11710 q^{73} + 41472 q^{74} - 26577 q^{75} + 43072 q^{76} - 23540 q^{77} + 28440 q^{78} + 72930 q^{79} - 4608 q^{80} + 19683 q^{81} + 151144 q^{82} - 50102 q^{83} + 7200 q^{84} + 108932 q^{85} + 114464 q^{86} + 53190 q^{87} - 384 q^{88} + 46212 q^{89} + 5832 q^{90} + 121740 q^{91} - 25392 q^{92} + 38016 q^{93} + 173392 q^{94} - 94652 q^{95} + 27648 q^{96} - 101642 q^{97} - 11756 q^{98} + 486 q^{99}+O(q^{100}) \) 3 * q - 12 * q^2 - 27 * q^3 + 48 * q^4 - 18 * q^5 + 108 * q^6 - 50 * q^7 - 192 * q^8 + 243 * q^9 + 72 * q^10 + 6 * q^11 - 432 * q^12 + 790 * q^13 + 200 * q^14 + 162 * q^15 + 768 * q^16 + 1148 * q^17 - 972 * q^18 + 2692 * q^19 - 288 * q^20 + 450 * q^21 - 24 * q^22 - 1587 * q^23 + 1728 * q^24 + 2953 * q^25 - 3160 * q^26 - 2187 * q^27 - 800 * q^28 - 5910 * q^29 - 648 * q^30 - 4224 * q^31 - 3072 * q^32 - 54 * q^33 - 4592 * q^34 - 24360 * q^35 + 3888 * q^36 - 10368 * q^37 - 10768 * q^38 - 7110 * q^39 + 1152 * q^40 - 37786 * q^41 - 1800 * q^42 - 28616 * q^43 + 96 * q^44 - 1458 * q^45 + 6348 * q^46 - 43348 * q^47 - 6912 * q^48 + 2939 * q^49 - 11812 * q^50 - 10332 * q^51 + 12640 * q^52 - 41566 * q^53 + 8748 * q^54 - 8956 * q^55 + 3200 * q^56 - 24228 * q^57 + 23640 * q^58 - 46560 * q^59 + 2592 * q^60 + 7924 * q^61 + 16896 * q^62 - 4050 * q^63 + 12288 * q^64 - 45580 * q^65 + 216 * q^66 - 17016 * q^67 + 18368 * q^68 + 14283 * q^69 + 97440 * q^70 - 38280 * q^71 - 15552 * q^72 - 11710 * q^73 + 41472 * q^74 - 26577 * q^75 + 43072 * q^76 - 23540 * q^77 + 28440 * q^78 + 72930 * q^79 - 4608 * q^80 + 19683 * q^81 + 151144 * q^82 - 50102 * q^83 + 7200 * q^84 + 108932 * q^85 + 114464 * q^86 + 53190 * q^87 - 384 * q^88 + 46212 * q^89 + 5832 * q^90 + 121740 * q^91 - 25392 * q^92 + 38016 * q^93 + 173392 * q^94 - 94652 * q^95 + 27648 * q^96 - 101642 * q^97 - 11756 * q^98 + 486 * q^99

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