LMFDB - Embedded newform 138.6.a.g.1.1

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			1.1
Root			$-27.1335$ 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)

$f(q)$	$=$	$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -73.1526 q^{5} +36.0000 q^{6} +90.1716 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -73.1526 q^{5} +36.0000 q^{6} +90.1716 q^{7} -64.0000 q^{8} +81.0000 q^{9} +292.610 q^{10} +481.155 q^{11} -144.000 q^{12} -57.0787 q^{13} -360.686 q^{14} +658.373 q^{15} +256.000 q^{16} -1085.40 q^{17} -324.000 q^{18} +2246.34 q^{19} -1170.44 q^{20} -811.544 q^{21} -1924.62 q^{22} -529.000 q^{23} +576.000 q^{24} +2226.30 q^{25} +228.315 q^{26} -729.000 q^{27} +1442.75 q^{28} -4472.51 q^{29} -2633.49 q^{30} +5874.35 q^{31} -1024.00 q^{32} -4330.39 q^{33} +4341.59 q^{34} -6596.28 q^{35} +1296.00 q^{36} -10539.5 q^{37} -8985.36 q^{38} +513.708 q^{39} +4681.76 q^{40} -4019.01 q^{41} +3246.18 q^{42} -21160.9 q^{43} +7698.48 q^{44} -5925.36 q^{45} +2116.00 q^{46} +2871.45 q^{47} -2304.00 q^{48} -8676.08 q^{49} -8905.18 q^{50} +9768.57 q^{51} -913.259 q^{52} -12032.2 q^{53} +2916.00 q^{54} -35197.7 q^{55} -5770.98 q^{56} -20217.1 q^{57} +17890.0 q^{58} -36043.3 q^{59} +10534.0 q^{60} -7609.04 q^{61} -23497.4 q^{62} +7303.90 q^{63} +4096.00 q^{64} +4175.45 q^{65} +17321.6 q^{66} +38427.2 q^{67} -17366.3 q^{68} +4761.00 q^{69} +26385.1 q^{70} -17529.1 q^{71} -5184.00 q^{72} -9465.69 q^{73} +42158.1 q^{74} -20036.7 q^{75} +35941.4 q^{76} +43386.5 q^{77} -2054.83 q^{78} +79632.8 q^{79} -18727.1 q^{80} +6561.00 q^{81} +16076.0 q^{82} +63720.0 q^{83} -12984.7 q^{84} +79399.5 q^{85} +84643.5 q^{86} +40252.6 q^{87} -30793.9 q^{88} -94203.0 q^{89} +23701.4 q^{90} -5146.88 q^{91} -8464.00 q^{92} -52869.2 q^{93} -11485.8 q^{94} -164325. q^{95} +9216.00 q^{96} -123274. q^{97} +34704.3 q^{98} +38973.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−73.1526	−1.30859	−0.654296	−	0.756238i	$-0.727035\pi$
$5$	−73.1526	−1.30859	−0.654296	+	0.756238i	$0.727035\pi$
$6$	36.0000	0.408248
$7$	90.1716	0.695544	0.347772	−	0.937579i	$-0.386938\pi$
$7$	90.1716	0.695544	0.347772	+	0.937579i	$0.386938\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	292.610	0.925315
$11$	481.155	1.19896	0.599478	−	0.800391i	$-0.295375\pi$
$11$	481.155	1.19896	0.599478	+	0.800391i	$0.295375\pi$
$12$	−144.000	−0.288675
$13$	−57.0787	−0.0936733	−0.0468366	−	0.998903i	$-0.514914\pi$
$13$	−57.0787	−0.0936733	−0.0468366	+	0.998903i	$0.514914\pi$
$14$	−360.686	−0.491824
$15$	658.373	0.755516
$16$	256.000	0.250000
$17$	−1085.40	−0.910891	−0.455445	−	0.890264i	$-0.650520\pi$
$17$	−1085.40	−0.910891	−0.455445	+	0.890264i	$0.650520\pi$
$18$	−324.000	−0.235702
$19$	2246.34	1.42755	0.713775	−	0.700375i	$-0.246984\pi$
$19$	2246.34	1.42755	0.713775	+	0.700375i	$0.246984\pi$
$20$	−1170.44	−0.654296
$21$	−811.544	−0.401573
$22$	−1924.62	−0.847790
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	576.000	0.204124
$25$	2226.30	0.712415
$26$	228.315	0.0662370
$27$	−729.000	−0.192450
$28$	1442.75	0.347772
$29$	−4472.51	−0.987544	−0.493772	−	0.869591i	$-0.664382\pi$
$29$	−4472.51	−0.987544	−0.493772	+	0.869591i	$0.664382\pi$
$30$	−2633.49	−0.534231
$31$	5874.35	1.09788	0.548941	−	0.835861i	$-0.315031\pi$
$31$	5874.35	1.09788	0.548941	+	0.835861i	$0.315031\pi$
$32$	−1024.00	−0.176777
$33$	−4330.39	−0.692218
$34$	4341.59	0.644097
$35$	−6596.28	−0.910184
$36$	1296.00	0.166667
$37$	−10539.5	−1.26566	−0.632829	−	0.774292i	$-0.718106\pi$
$37$	−10539.5	−1.26566	−0.632829	+	0.774292i	$0.718106\pi$
$38$	−8985.36	−1.00943
$39$	513.708	0.0540823
$40$	4681.76	0.462657
$41$	−4019.01	−0.373387	−0.186694	−	0.982418i	$-0.559777\pi$
$41$	−4019.01	−0.373387	−0.186694	+	0.982418i	$0.559777\pi$
$42$	3246.18	0.283955
$43$	−21160.9	−1.74527	−0.872635	−	0.488373i	$-0.837591\pi$
$43$	−21160.9	−1.74527	−0.872635	+	0.488373i	$0.837591\pi$
$44$	7698.48	0.599478
$45$	−5925.36	−0.436198
$46$	2116.00	0.147442
$47$	2871.45	0.189608	0.0948041	−	0.995496i	$-0.469778\pi$
$47$	2871.45	0.189608	0.0948041	+	0.995496i	$0.469778\pi$
$48$	−2304.00	−0.144338
$49$	−8676.08	−0.516218
$50$	−8905.18	−0.503753
$51$	9768.57	0.525903
$52$	−913.259	−0.0468366
$53$	−12032.2	−0.588374	−0.294187	−	0.955748i	$-0.595049\pi$
$53$	−12032.2	−0.588374	−0.294187	+	0.955748i	$0.595049\pi$
$54$	2916.00	0.136083
$55$	−35197.7	−1.56894
$56$	−5770.98	−0.245912
$57$	−20217.1	−0.824197
$58$	17890.0	0.698299
$59$	−36043.3	−1.34801	−0.674006	−	0.738725i	$-0.735428\pi$
$59$	−36043.3	−1.34801	−0.674006	+	0.738725i	$0.735428\pi$
$60$	10534.0	0.377758
$61$	−7609.04	−0.261821	−0.130911	−	0.991394i	$-0.541790\pi$
$61$	−7609.04	−0.261821	−0.130911	+	0.991394i	$0.541790\pi$
$62$	−23497.4	−0.776320
$63$	7303.90	0.231848
$64$	4096.00	0.125000
$65$	4175.45	0.122580
$66$	17321.6	0.489472
$67$	38427.2	1.04581	0.522904	−	0.852392i	$-0.324849\pi$
$67$	38427.2	1.04581	0.522904	+	0.852392i	$0.324849\pi$
$68$	−17366.3	−0.455445
$69$	4761.00	0.120386
$70$	26385.1	0.643597
$71$	−17529.1	−0.412679	−0.206340	−	0.978480i	$-0.566155\pi$
$71$	−17529.1	−0.412679	−0.206340	+	0.978480i	$0.566155\pi$
$72$	−5184.00	−0.117851
$73$	−9465.69	−0.207896	−0.103948	−	0.994583i	$-0.533147\pi$
$73$	−9465.69	−0.207896	−0.103948	+	0.994583i	$0.533147\pi$
$74$	42158.1	0.894955
$75$	−20036.7	−0.411313
$76$	35941.4	0.713775
$77$	43386.5	0.833927
$78$	−2054.83	−0.0382420
$79$	79632.8	1.43557	0.717784	−	0.696265i	$-0.245156\pi$
$79$	79632.8	1.43557	0.717784	+	0.696265i	$0.245156\pi$
$80$	−18727.1	−0.327148
$81$	6561.00	0.111111
$82$	16076.0	0.264025
$83$	63720.0	1.01527	0.507634	−	0.861573i	$-0.330520\pi$
$83$	63720.0	1.01527	0.507634	+	0.861573i	$0.330520\pi$
$84$	−12984.7	−0.200786
$85$	79399.5	1.19198
$86$	84643.5	1.23409
$87$	40252.6	0.570159
$88$	−30793.9	−0.423895
$89$	−94203.0	−1.26064	−0.630318	−	0.776337i	$-0.717075\pi$
$89$	−94203.0	−1.26064	−0.630318	+	0.776337i	$0.717075\pi$
$90$	23701.4	0.308438
$91$	−5146.88	−0.0651539
$92$	−8464.00	−0.104257
$93$	−52869.2	−0.633863
$94$	−11485.8	−0.134073
$95$	−164325.	−1.86808
$96$	9216.00	0.102062
$97$	−123274.	−1.33028	−0.665141	−	0.746718i	$-0.731629\pi$
$97$	−123274.	−1.33028	−0.665141	+	0.746718i	$0.731629\pi$
$98$	34704.3	0.365021
$99$	38973.5	0.399652
$100$	35620.7	0.356207
$101$	−19121.6	−0.186518	−0.0932592	−	0.995642i	$-0.529729\pi$
$101$	−19121.6	−0.186518	−0.0932592	+	0.995642i	$0.529729\pi$
$102$	−39074.3	−0.371870
$103$	85424.2	0.793392	0.396696	−	0.917950i	$-0.370157\pi$
$103$	85424.2	0.793392	0.396696	+	0.917950i	$0.370157\pi$
$104$	3653.04	0.0331185
$105$	59366.5	0.525495
$106$	48128.6	0.416043
$107$	−204868.	−1.72988	−0.864939	−	0.501876i	$-0.832643\pi$
$107$	−204868.	−1.72988	−0.864939	+	0.501876i	$0.832643\pi$
$108$	−11664.0	−0.0962250
$109$	168038.	1.35470	0.677348	−	0.735663i	$-0.263129\pi$
$109$	168038.	1.35470	0.677348	+	0.735663i	$0.263129\pi$
$110$	140791.	1.10941
$111$	94855.6	0.730728
$112$	23083.9	0.173886
$113$	68458.0	0.504346	0.252173	−	0.967682i	$-0.418855\pi$
$113$	68458.0	0.504346	0.252173	+	0.967682i	$0.418855\pi$
$114$	80868.2	0.582795
$115$	38697.7	0.272860
$116$	−71560.2	−0.493772
$117$	−4623.37	−0.0312244
$118$	144173.	0.953189
$119$	−97872.0	−0.633565
$120$	−42135.9	−0.267115
$121$	70459.1	0.437495
$122$	30436.1	0.185136
$123$	36171.1	0.215575
$124$	93989.7	0.548941
$125$	65742.5	0.376332
$126$	−29215.6	−0.163941
$127$	−240503.	−1.32316	−0.661579	−	0.749876i	$-0.730113\pi$
$127$	−240503.	−1.32316	−0.661579	+	0.749876i	$0.730113\pi$
$128$	−16384.0	−0.0883883
$129$	190448.	1.00763
$130$	−16701.8	−0.0866773
$131$	−4927.51	−0.0250870	−0.0125435	−	0.999921i	$-0.503993\pi$
$131$	−4927.51	−0.0250870	−0.0125435	+	0.999921i	$0.503993\pi$
$132$	−69286.3	−0.346109
$133$	202556.	0.992924
$134$	−153709.	−0.739498
$135$	53328.2	0.251839
$136$	69465.4	0.322048
$137$	155495.	0.707805	0.353903	−	0.935282i	$-0.384854\pi$
$137$	155495.	0.707805	0.353903	+	0.935282i	$0.384854\pi$
$138$	−19044.0	−0.0851257
$139$	−177025.	−0.777138	−0.388569	−	0.921420i	$-0.627031\pi$
$139$	−177025.	−0.777138	−0.388569	+	0.921420i	$0.627031\pi$
$140$	−105541.	−0.455092
$141$	−25843.1	−0.109470
$142$	70116.2	0.291808
$143$	−27463.7	−0.112310
$144$	20736.0	0.0833333
$145$	327176.	1.29229
$146$	37862.8	0.147004
$147$	78084.7	0.298039
$148$	−168632.	−0.632829
$149$	−165548.	−0.610882	−0.305441	−	0.952211i	$-0.598804\pi$
$149$	−165548.	−0.610882	−0.305441	+	0.952211i	$0.598804\pi$
$150$	80146.6	0.290842
$151$	−143317.	−0.511513	−0.255756	−	0.966741i	$-0.582324\pi$
$151$	−143317.	−0.511513	−0.255756	+	0.966741i	$0.582324\pi$
$152$	−143766.	−0.504715
$153$	−87917.1	−0.303630
$154$	−173546.	−0.589675
$155$	−429724.	−1.43668
$156$	8219.33	0.0270411
$157$	−543347.	−1.75925	−0.879626	−	0.475666i	$-0.842207\pi$
$157$	−543347.	−1.75925	−0.879626	+	0.475666i	$0.842207\pi$
$158$	−318531.	−1.01510
$159$	108289.	0.339698
$160$	74908.2	0.231329
$161$	−47700.8	−0.145031
$162$	−26244.0	−0.0785674
$163$	−249178.	−0.734583	−0.367291	−	0.930106i	$-0.619715\pi$
$163$	−249178.	−0.734583	−0.367291	+	0.930106i	$0.619715\pi$
$164$	−64304.1	−0.186694
$165$	316779.	0.905831
$166$	−254880.	−0.717903
$167$	224312.	0.622388	0.311194	−	0.950346i	$-0.399271\pi$
$167$	224312.	0.622388	0.311194	+	0.950346i	$0.399271\pi$
$168$	51938.8	0.141977
$169$	−368035.	−0.991225
$170$	−317598.	−0.842860
$171$	181954.	0.475850
$172$	−338574.	−0.872635
$173$	35179.1	0.0893655	0.0446828	−	0.999001i	$-0.485772\pi$
$173$	35179.1	0.0893655	0.0446828	+	0.999001i	$0.485772\pi$
$174$	−161010.	−0.403163
$175$	200749.	0.495516
$176$	123176.	0.299739
$177$	324389.	0.778276
$178$	376812.	0.891404
$179$	−565587.	−1.31937	−0.659685	−	0.751542i	$-0.729310\pi$
$179$	−565587.	−1.31937	−0.659685	+	0.751542i	$0.729310\pi$
$180$	−94805.7	−0.218099
$181$	−527974.	−1.19789	−0.598944	−	0.800791i	$-0.704413\pi$
$181$	−527974.	−1.19789	−0.598944	+	0.800791i	$0.704413\pi$
$182$	20587.5	0.0460708
$183$	68481.3	0.151163
$184$	33856.0	0.0737210
$185$	770992.	1.65623
$186$	211477.	0.448209
$187$	−522244.	−1.09212
$188$	45943.3	0.0948041
$189$	−65735.1	−0.133858
$190$	657302.	1.32093
$191$	−367744.	−0.729393	−0.364697	−	0.931126i	$-0.618827\pi$
$191$	−367744.	−0.729393	−0.364697	+	0.931126i	$0.618827\pi$
$192$	−36864.0	−0.0721688
$193$	139116.	0.268834	0.134417	−	0.990925i	$-0.457084\pi$
$193$	139116.	0.268834	0.134417	+	0.990925i	$0.457084\pi$
$194$	493098.	0.940651
$195$	−37579.1	−0.0707717
$196$	−138817.	−0.258109
$197$	166240.	0.305190	0.152595	−	0.988289i	$-0.451237\pi$
$197$	166240.	0.305190	0.152595	+	0.988289i	$0.451237\pi$
$198$	−155894.	−0.282597
$199$	−269311.	−0.482083	−0.241041	−	0.970515i	$-0.577489\pi$
$199$	−269311.	−0.482083	−0.241041	+	0.970515i	$0.577489\pi$
$200$	−142483.	−0.251877
$201$	−345845.	−0.603797
$202$	76486.6	0.131888
$203$	−403293.	−0.686880
$204$	156297.	0.262951
$205$	294001.	0.488612
$206$	−341697.	−0.561013
$207$	−42849.0	−0.0695048
$208$	−14612.1	−0.0234183
$209$	1.08084e6	1.71157
$210$	−237466.	−0.371581
$211$	83967.5	0.129839	0.0649195	−	0.997891i	$-0.479321\pi$
$211$	83967.5	0.129839	0.0649195	+	0.997891i	$0.479321\pi$
$212$	−192514.	−0.294187
$213$	157762.	0.238261
$214$	819474.	1.22321
$215$	1.54797e6	2.28385
$216$	46656.0	0.0680414
$217$	529700.	0.763626
$218$	−672153.	−0.957915
$219$	85191.2	0.120029
$220$	−563163.	−0.784472
$221$	61953.0	0.0853261
$222$	−379423.	−0.516703
$223$	312135.	0.420320	0.210160	−	0.977667i	$-0.432601\pi$
$223$	312135.	0.420320	0.210160	+	0.977667i	$0.432601\pi$
$224$	−92335.7	−0.122956
$225$	180330.	0.237472
$226$	−273832.	−0.356626
$227$	−1.09930e6	−1.41596	−0.707982	−	0.706230i	$-0.750394\pi$
$227$	−1.09930e6	−1.41596	−0.707982	+	0.706230i	$0.750394\pi$
$228$	−323473.	−0.412098
$229$	1.47358e6	1.85689	0.928445	−	0.371469i	$-0.121146\pi$
$229$	1.47358e6	1.85689	0.928445	+	0.371469i	$0.121146\pi$
$230$	−154791.	−0.192941
$231$	−390479.	−0.481468
$232$	286241.	0.349149
$233$	−1.32062e6	−1.59364	−0.796819	−	0.604218i	$-0.793485\pi$
$233$	−1.32062e6	−1.59364	−0.796819	+	0.604218i	$0.793485\pi$
$234$	18493.5	0.0220790
$235$	−210054.	−0.248120
$236$	−576692.	−0.674006
$237$	−716695.	−0.828826
$238$	391488.	0.447998
$239$	1.21476e6	1.37562	0.687808	−	0.725892i	$-0.258573\pi$
$239$	1.21476e6	1.37562	0.687808	+	0.725892i	$0.258573\pi$
$240$	168543.	0.188879
$241$	−903951.	−1.00254	−0.501271	−	0.865290i	$-0.667134\pi$
$241$	−903951.	−1.00254	−0.501271	+	0.865290i	$0.667134\pi$
$242$	−281836.	−0.309356
$243$	−59049.0	−0.0641500
$244$	−121745.	−0.130911
$245$	634677.	0.675519
$246$	−144684.	−0.152435
$247$	−128218.	−0.133723
$248$	−375959.	−0.388160
$249$	−573480.	−0.586165
$250$	−262970.	−0.266107
$251$	1.30065e6	1.30309	0.651545	−	0.758610i	$-0.274121\pi$
$251$	1.30065e6	1.30309	0.651545	+	0.758610i	$0.274121\pi$
$252$	116862.	0.115924
$253$	−254531.	−0.250000
$254$	962013.	0.935614
$255$	−714596.	−0.688193
$256$	65536.0	0.0625000
$257$	1.60440e6	1.51524	0.757619	−	0.652697i	$-0.226363\pi$
$257$	1.60440e6	1.51524	0.757619	+	0.652697i	$0.226363\pi$
$258$	−761792.	−0.712504
$259$	−950365.	−0.880321
$260$	66807.2	0.0612901
$261$	−362273.	−0.329181
$262$	19710.0	0.0177392
$263$	−731881.	−0.652456	−0.326228	−	0.945291i	$-0.605778\pi$
$263$	−731881.	−0.652456	−0.326228	+	0.945291i	$0.605778\pi$
$264$	277145.	0.244736
$265$	880183.	0.769942
$266$	−810224.	−0.702104
$267$	847827.	0.727828
$268$	614835.	0.522904
$269$	−787223.	−0.663311	−0.331655	−	0.943401i	$-0.607607\pi$
$269$	−787223.	−0.663311	−0.331655	+	0.943401i	$0.607607\pi$
$270$	−213313.	−0.178077
$271$	2.32012e6	1.91906	0.959529	−	0.281610i	$-0.0908685\pi$
$271$	2.32012e6	1.91906	0.959529	+	0.281610i	$0.0908685\pi$
$272$	−277862.	−0.227723
$273$	46321.9	0.0376166
$274$	−621978.	−0.500494
$275$	1.07119e6	0.854154
$276$	76176.0	0.0601929
$277$	−1.14853e6	−0.899383	−0.449691	−	0.893184i	$-0.648466\pi$
$277$	−1.14853e6	−0.899383	−0.449691	+	0.893184i	$0.648466\pi$
$278$	708101.	0.549520
$279$	475823.	0.365961
$280$	422162.	0.321799
$281$	952010.	0.719243	0.359622	−	0.933098i	$-0.382906\pi$
$281$	952010.	0.719243	0.359622	+	0.933098i	$0.382906\pi$
$282$	103372.	0.0774072
$283$	−1.87476e6	−1.39149	−0.695743	−	0.718291i	$-0.744925\pi$
$283$	−1.87476e6	−1.39149	−0.695743	+	0.718291i	$0.744925\pi$
$284$	−280465.	−0.206340
$285$	1.47893e6	1.07854
$286$	109855.	0.0794153
$287$	−362401.	−0.259707
$288$	−82944.0	−0.0589256
$289$	−241771.	−0.170278
$290$	−1.30870e6	−0.913789
$291$	1.10947e6	0.768039
$292$	−151451.	−0.103948
$293$	405698.	0.276080	0.138040	−	0.990427i	$-0.455920\pi$
$293$	405698.	0.276080	0.138040	+	0.990427i	$0.455920\pi$
$294$	−312339.	−0.210745
$295$	2.63666e6	1.76400
$296$	674529.	0.447478
$297$	−350762.	−0.230739
$298$	662190.	0.431959
$299$	30194.6	0.0195322
$300$	−320587.	−0.205656
$301$	−1.90811e6	−1.21391
$302$	573269.	0.361694
$303$	172095.	0.107686
$304$	575063.	0.356888
$305$	556620.	0.342617
$306$	351669.	0.214699
$307$	624995.	0.378469	0.189235	−	0.981932i	$-0.439399\pi$
$307$	624995.	0.378469	0.189235	+	0.981932i	$0.439399\pi$
$308$	694184.	0.416963
$309$	−768818.	−0.458065
$310$	1.71890e6	1.01589
$311$	1.34802e6	0.790308	0.395154	−	0.918615i	$-0.370691\pi$
$311$	1.34802e6	0.790308	0.395154	+	0.918615i	$0.370691\pi$
$312$	−32877.3	−0.0191210
$313$	776161.	0.447807	0.223904	−	0.974611i	$-0.428120\pi$
$313$	776161.	0.447807	0.223904	+	0.974611i	$0.428120\pi$
$314$	2.17339e6	1.24398
$315$	−534299.	−0.303395
$316$	1.27412e6	0.717784
$317$	1.31829e6	0.736821	0.368411	−	0.929663i	$-0.379902\pi$
$317$	1.31829e6	0.736821	0.368411	+	0.929663i	$0.379902\pi$
$318$	−433158.	−0.240203
$319$	−2.15197e6	−1.18402
$320$	−299633.	−0.163574
$321$	1.84382e6	0.998746
$322$	190803.	0.102552
$323$	−2.43817e6	−1.30034
$324$	104976.	0.0555556
$325$	−127074.	−0.0667342
$326$	996712.	0.519428
$327$	−1.51234e6	−0.782134
$328$	257217.	0.132012
$329$	258924.	0.131881
$330$	−1.26712e6	−0.640519
$331$	−2.95244e6	−1.48119	−0.740596	−	0.671950i	$-0.765457\pi$
$331$	−2.95244e6	−1.48119	−0.740596	+	0.671950i	$0.765457\pi$
$332$	1.01952e6	0.507634
$333$	−853701.	−0.421886
$334$	−897247.	−0.440094
$335$	−2.81105e6	−1.36854
$336$	−207755.	−0.100393
$337$	−2.30114e6	−1.10374	−0.551871	−	0.833930i	$-0.686086\pi$
$337$	−2.30114e6	−1.10374	−0.551871	+	0.833930i	$0.686086\pi$
$338$	1.47214e6	0.700902
$339$	−616122.	−0.291184
$340$	1.27039e6	0.595992
$341$	2.82647e6	1.31631
$342$	−727814.	−0.336477
$343$	−2.29785e6	−1.05460
$344$	1.35430e6	0.617046
$345$	−348279.	−0.157536
$346$	−140717.	−0.0631910
$347$	−1.09062e6	−0.486239	−0.243119	−	0.969996i	$-0.578171\pi$
$347$	−1.09062e6	−0.486239	−0.243119	+	0.969996i	$0.578171\pi$
$348$	644041.	0.285079
$349$	−2.85065e6	−1.25280	−0.626398	−	0.779504i	$-0.715471\pi$
$349$	−2.85065e6	−1.25280	−0.626398	+	0.779504i	$0.715471\pi$
$350$	−802995.	−0.350383
$351$	41610.4	0.0180274
$352$	−492703.	−0.211947
$353$	1.98999e6	0.849993	0.424996	−	0.905195i	$-0.360275\pi$
$353$	1.98999e6	0.849993	0.424996	+	0.905195i	$0.360275\pi$
$354$	−1.29756e6	−0.550324
$355$	1.28230e6	0.540029
$356$	−1.50725e6	−0.630318
$357$	880848.	0.365789
$358$	2.26235e6	0.932936
$359$	176892.	0.0724391	0.0362196	−	0.999344i	$-0.488468\pi$
$359$	176892.	0.0724391	0.0362196	+	0.999344i	$0.488468\pi$
$360$	379223.	0.154219
$361$	2.56994e6	1.03790
$362$	2.11189e6	0.847034
$363$	−634132.	−0.252588
$364$	−82350.1	−0.0325769
$365$	692440.	0.272051
$366$	−273925.	−0.106888
$367$	−2.81552e6	−1.09117	−0.545587	−	0.838054i	$-0.683693\pi$
$367$	−2.81552e6	−1.09117	−0.545587	+	0.838054i	$0.683693\pi$
$368$	−135424.	−0.0521286
$369$	−325540.	−0.124462
$370$	−3.08397e6	−1.17113
$371$	−1.08496e6	−0.409240
$372$	−845907.	−0.316931
$373$	−2.42214e6	−0.901422	−0.450711	−	0.892670i	$-0.648829\pi$
$373$	−2.42214e6	−0.901422	−0.450711	+	0.892670i	$0.648829\pi$
$374$	2.08898e6	0.772244
$375$	−591683.	−0.217275
$376$	−183773.	−0.0670366
$377$	255285.	0.0925065
$378$	262940.	0.0946516
$379$	3.47574e6	1.24294	0.621469	−	0.783438i	$-0.286536\pi$
$379$	3.47574e6	1.24294	0.621469	+	0.783438i	$0.286536\pi$
$380$	−2.62921e6	−0.934041
$381$	2.16453e6	0.763925
$382$	1.47097e6	0.515759
$383$	2.71374e6	0.945304	0.472652	−	0.881249i	$-0.343297\pi$
$383$	2.71374e6	0.945304	0.472652	+	0.881249i	$0.343297\pi$
$384$	147456.	0.0510310
$385$	−3.17383e6	−1.09127
$386$	−556465.	−0.190094
$387$	−1.71403e6	−0.581757
$388$	−1.97239e6	−0.665141
$389$	−1.69768e6	−0.568829	−0.284414	−	0.958701i	$-0.591799\pi$
$389$	−1.69768e6	−0.568829	−0.284414	+	0.958701i	$0.591799\pi$
$390$	150316.	0.0500431
$391$	574175.	0.189934
$392$	555269.	0.182511
$393$	44347.6	0.0144840
$394$	−664961.	−0.215802
$395$	−5.82534e6	−1.87857
$396$	623577.	0.199826
$397$	706475.	0.224968	0.112484	−	0.993654i	$-0.464119\pi$
$397$	706475.	0.224968	0.112484	+	0.993654i	$0.464119\pi$
$398$	1.07724e6	0.340884
$399$	−1.82300e6	−0.573265
$400$	569932.	0.178104
$401$	3.05420e6	0.948500	0.474250	−	0.880390i	$-0.342719\pi$
$401$	3.05420e6	0.948500	0.474250	+	0.880390i	$0.342719\pi$
$402$	1.38338e6	0.426949
$403$	−335301.	−0.102842
$404$	−305946.	−0.0932592
$405$	−479954.	−0.145399
$406$	1.61317e6	0.485698
$407$	−5.07114e6	−1.51747
$408$	−625189.	−0.185935
$409$	720250.	0.212900	0.106450	−	0.994318i	$-0.466052\pi$
$409$	720250.	0.212900	0.106450	+	0.994318i	$0.466052\pi$
$410$	−1.17600e6	−0.345501
$411$	−1.39945e6	−0.408652
$412$	1.36679e6	0.396696
$413$	−3.25008e6	−0.937603
$414$	171396.	0.0491473
$415$	−4.66128e6	−1.32857
$416$	58448.6	0.0165593
$417$	1.59323e6	0.448681
$418$	−4.32335e6	−1.21026
$419$	216286.	0.0601857	0.0300929	−	0.999547i	$-0.490420\pi$
$419$	216286.	0.0601857	0.0300929	+	0.999547i	$0.490420\pi$
$420$	949865.	0.262747
$421$	−4.81267e6	−1.32337	−0.661685	−	0.749782i	$-0.730158\pi$
$421$	−4.81267e6	−1.32337	−0.661685	+	0.749782i	$0.730158\pi$
$422$	−335870.	−0.0918100
$423$	232588.	0.0632027
$424$	770058.	0.208022
$425$	−2.41641e6	−0.648932
$426$	−631046.	−0.168476
$427$	−686119.	−0.182108
$428$	−3.27790e6	−0.864939
$429$	247173.	0.0648423
$430$	−6.19189e6	−1.61492
$431$	6.01799e6	1.56048	0.780240	−	0.625481i	$-0.215097\pi$
$431$	6.01799e6	1.56048	0.780240	+	0.625481i	$0.215097\pi$
$432$	−186624.	−0.0481125
$433$	5.56276e6	1.42584	0.712919	−	0.701246i	$-0.247373\pi$
$433$	5.56276e6	1.42584	0.712919	+	0.701246i	$0.247373\pi$
$434$	−2.11880e6	−0.539965
$435$	−2.94458e6	−0.746105
$436$	2.68861e6	0.677348
$437$	−1.18831e6	−0.297665
$438$	−340765.	−0.0848730
$439$	−3.56797e6	−0.883607	−0.441804	−	0.897112i	$-0.645661\pi$
$439$	−3.56797e6	−0.883607	−0.441804	+	0.897112i	$0.645661\pi$
$440$	2.25265e6	0.554706
$441$	−702763.	−0.172073
$442$	−247812.	−0.0603347
$443$	−3.30895e6	−0.801090	−0.400545	−	0.916277i	$-0.631179\pi$
$443$	−3.30895e6	−0.801090	−0.400545	+	0.916277i	$0.631179\pi$
$444$	1.51769e6	0.365364
$445$	6.89119e6	1.64966
$446$	−1.24854e6	−0.297211
$447$	1.48993e6	0.352693
$448$	369343.	0.0869430
$449$	6.07797e6	1.42280	0.711398	−	0.702789i	$-0.248062\pi$
$449$	6.07797e6	1.42280	0.711398	+	0.702789i	$0.248062\pi$
$450$	−721320.	−0.167918
$451$	−1.93377e6	−0.447675
$452$	1.09533e6	0.252173
$453$	1.28986e6	0.295322
$454$	4.39721e6	1.00124
$455$	376507.	0.0852599
$456$	1.29389e6	0.291397
$457$	797304.	0.178580	0.0892902	−	0.996006i	$-0.471540\pi$
$457$	797304.	0.178580	0.0892902	+	0.996006i	$0.471540\pi$
$458$	−5.89434e6	−1.31302
$459$	791254.	0.175301
$460$	619163.	0.136430
$461$	1.35264e6	0.296435	0.148217	−	0.988955i	$-0.452646\pi$
$461$	1.35264e6	0.296435	0.148217	+	0.988955i	$0.452646\pi$
$462$	1.56191e6	0.340449
$463$	−3.41112e6	−0.739511	−0.369755	−	0.929129i	$-0.620558\pi$
$463$	−3.41112e6	−0.739511	−0.369755	+	0.929129i	$0.620558\pi$
$464$	−1.14496e6	−0.246886
$465$	3.86752e6	0.829468
$466$	5.28250e6	1.12687
$467$	−2.64231e6	−0.560650	−0.280325	−	0.959905i	$-0.590442\pi$
$467$	−2.64231e6	−0.560650	−0.280325	+	0.959905i	$0.590442\pi$
$468$	−73974.0	−0.0156122
$469$	3.46504e6	0.727405
$470$	840217.	0.175447
$471$	4.89012e6	1.01570
$472$	2.30677e6	0.476595
$473$	−1.01817e7	−2.09250
$474$	2.86678e6	0.586069
$475$	5.00102e6	1.01701
$476$	−1.56595e6	−0.316782
$477$	−974604.	−0.196125
$478$	−4.85906e6	−0.972708
$479$	9.33738e6	1.85946	0.929729	−	0.368245i	$-0.120041\pi$
$479$	9.33738e6	1.85946	0.929729	+	0.368245i	$0.120041\pi$
$480$	−674174.	−0.133558
$481$	601582.	0.118558
$482$	3.61581e6	0.708904
$483$	429307.	0.0837337
$484$	1.12735e6	0.218748
$485$	9.01784e6	1.74080
$486$	236196.	0.0453609
$487$	−3.73933e6	−0.714449	−0.357224	−	0.934019i	$-0.616277\pi$
$487$	−3.73933e6	−0.714449	−0.357224	+	0.934019i	$0.616277\pi$
$488$	486978.	0.0925678
$489$	2.24260e6	0.424111
$490$	−2.53871e6	−0.477664
$491$	3.60405e6	0.674663	0.337331	−	0.941386i	$-0.390476\pi$
$491$	3.60405e6	0.674663	0.337331	+	0.941386i	$0.390476\pi$
$492$	578737.	0.107788
$493$	4.85445e6	0.899544
$494$	512873.	0.0945567
$495$	−2.85101e6	−0.522982
$496$	1.50383e6	0.274471
$497$	−1.58062e6	−0.287037
$498$	2.29392e6	0.414481
$499$	5.20906e6	0.936501	0.468250	−	0.883596i	$-0.344885\pi$
$499$	5.20906e6	0.936501	0.468250	+	0.883596i	$0.344885\pi$
$500$	1.05188e6	0.188166
$501$	−2.01881e6	−0.359336
$502$	−5.20258e6	−0.921424
$503$	1.04910e7	1.84883	0.924415	−	0.381388i	$-0.124554\pi$
$503$	1.04910e7	1.84883	0.924415	+	0.381388i	$0.124554\pi$
$504$	−467450.	−0.0819707
$505$	1.39880e6	0.244077
$506$	1.01812e6	0.176776
$507$	3.31232e6	0.572284
$508$	−3.84805e6	−0.661579
$509$	−188826.	−0.0323048	−0.0161524	−	0.999870i	$-0.505142\pi$
$509$	−188826.	−0.0323048	−0.0161524	+	0.999870i	$0.505142\pi$
$510$	2.85838e6	0.486626
$511$	−853537.	−0.144601
$512$	−262144.	−0.0441942
$513$	−1.63758e6	−0.274732
$514$	−6.41761e6	−1.07143
$515$	−6.24900e6	−1.03823
$516$	3.04717e6	0.503816
$517$	1.38161e6	0.227332
$518$	3.80146e6	0.622481
$519$	−316612.	−0.0515952
$520$	−267229.	−0.0433386
$521$	9.11989e6	1.47196	0.735979	−	0.677005i	$-0.236722\pi$
$521$	9.11989e6	1.47196	0.735979	+	0.677005i	$0.236722\pi$
$522$	1.44909e6	0.232766
$523$	3.59196e6	0.574219	0.287109	−	0.957898i	$-0.407306\pi$
$523$	3.59196e6	0.574219	0.287109	+	0.957898i	$0.407306\pi$
$524$	−78840.2	−0.0125435
$525$	−1.80674e6	−0.286086
$526$	2.92752e6	0.461356
$527$	−6.37601e6	−1.00005
$528$	−1.10858e6	−0.173054
$529$	279841.	0.0434783
$530$	−3.52073e6	−0.544431
$531$	−2.91950e6	−0.449338
$532$	3.24090e6	0.496462
$533$	229400.	0.0349764
$534$	−3.39131e6	−0.514652
$535$	1.49867e7	2.26371
$536$	−2.45934e6	−0.369749
$537$	5.09028e6	0.761739
$538$	3.14889e6	0.469032
$539$	−4.17454e6	−0.618923
$540$	853251.	0.125919
$541$	2.24924e6	0.330402	0.165201	−	0.986260i	$-0.447173\pi$
$541$	2.24924e6	0.330402	0.165201	+	0.986260i	$0.447173\pi$
$542$	−9.28050e6	−1.35698
$543$	4.75176e6	0.691600
$544$	1.11145e6	0.161024
$545$	−1.22924e7	−1.77274
$546$	−185288.	−0.0265990
$547$	−2.43062e6	−0.347335	−0.173668	−	0.984804i	$-0.555562\pi$
$547$	−2.43062e6	−0.347335	−0.173668	+	0.984804i	$0.555562\pi$
$548$	2.48791e6	0.353903
$549$	−616332.	−0.0872738
$550$	−4.28477e6	−0.603978
$551$	−1.00468e7	−1.40977
$552$	−304704.	−0.0425628
$553$	7.18061e6	0.998502
$554$	4.59414e6	0.635960
$555$	−6.93893e6	−0.956225
$556$	−2.83240e6	−0.388569
$557$	−1.72797e6	−0.235993	−0.117997	−	0.993014i	$-0.537647\pi$
$557$	−1.72797e6	−0.235993	−0.117997	+	0.993014i	$0.537647\pi$
$558$	−1.90329e6	−0.258773
$559$	1.20784e6	0.163485
$560$	−1.68865e6	−0.227546
$561$	4.70020e6	0.630534
$562$	−3.80804e6	−0.508582
$563$	−1.07764e7	−1.43286	−0.716428	−	0.697661i	$-0.754224\pi$
$563$	−1.07764e7	−1.43286	−0.716428	+	0.697661i	$0.754224\pi$
$564$	−413489.	−0.0547352
$565$	−5.00788e6	−0.659983
$566$	7.49903e6	0.983929
$567$	591616.	0.0772827
$568$	1.12186e6	0.145904
$569$	−9.82612e6	−1.27234	−0.636168	−	0.771551i	$-0.719481\pi$
$569$	−9.82612e6	−1.27234	−0.636168	+	0.771551i	$0.719481\pi$
$570$	−5.91572e6	−0.762641
$571$	1.03649e7	1.33038	0.665191	−	0.746673i	$-0.268350\pi$
$571$	1.03649e7	1.33038	0.665191	+	0.746673i	$0.268350\pi$
$572$	−439419.	−0.0561551
$573$	3.30969e6	0.421115
$574$	1.44960e6	0.183641
$575$	−1.17771e6	−0.148549
$576$	331776.	0.0416667
$577$	7.23020e6	0.904089	0.452044	−	0.891995i	$-0.350695\pi$
$577$	7.23020e6	0.904089	0.452044	+	0.891995i	$0.350695\pi$
$578$	967083.	0.120405
$579$	−1.25205e6	−0.155211
$580$	5.23481e6	0.646146
$581$	5.74573e6	0.706163
$582$	−4.43788e6	−0.543085
$583$	−5.78933e6	−0.705435
$584$	605804.	0.0735022
$585$	338212.	0.0408601
$586$	−1.62279e6	−0.195218
$587$	1.63852e7	1.96271	0.981354	−	0.192210i	$-0.0615655\pi$
$587$	1.63852e7	1.96271	0.981354	+	0.192210i	$0.0615655\pi$
$588$	1.24936e6	0.149019
$589$	1.31958e7	1.56728
$590$	−1.05466e7	−1.24734
$591$	−1.49616e6	−0.176202
$592$	−2.69812e6	−0.316414
$593$	−4.80444e6	−0.561056	−0.280528	−	0.959846i	$-0.590510\pi$
$593$	−4.80444e6	−0.561056	−0.280528	+	0.959846i	$0.590510\pi$
$594$	1.40305e6	0.163157
$595$	7.15958e6	0.829078
$596$	−2.64876e6	−0.305441
$597$	2.42380e6	0.278331
$598$	−120779.	−0.0138114
$599$	−1.58237e7	−1.80195	−0.900973	−	0.433876i	$-0.857146\pi$
$599$	−1.58237e7	−1.80195	−0.900973	+	0.433876i	$0.857146\pi$
$600$	1.28235e6	0.145421
$601$	805030.	0.0909130	0.0454565	−	0.998966i	$-0.485526\pi$
$601$	805030.	0.0909130	0.0454565	+	0.998966i	$0.485526\pi$
$602$	7.63244e6	0.858366
$603$	3.11260e6	0.348603
$604$	−2.29308e6	−0.255756
$605$	−5.15426e6	−0.572503
$606$	−688379.	−0.0761458
$607$	1.16520e7	1.28360	0.641801	−	0.766871i	$-0.278188\pi$
$607$	1.16520e7	1.28360	0.641801	+	0.766871i	$0.278188\pi$
$608$	−2.30025e6	−0.252358
$609$	3.62964e6	0.396571
$610$	−2.22648e6	−0.242267
$611$	−163899.	−0.0177612
$612$	−1.40667e6	−0.151815
$613$	−2.45088e6	−0.263433	−0.131716	−	0.991287i	$-0.542049\pi$
$613$	−2.45088e6	−0.263433	−0.131716	+	0.991287i	$0.542049\pi$
$614$	−2.49998e6	−0.267618
$615$	−2.64601e6	−0.282100
$616$	−2.77674e6	−0.294838
$617$	−1.63832e7	−1.73255	−0.866275	−	0.499567i	$-0.833492\pi$
$617$	−1.63832e7	−1.73255	−0.866275	+	0.499567i	$0.833492\pi$
$618$	3.07527e6	0.323901
$619$	−7.56892e6	−0.793976	−0.396988	−	0.917824i	$-0.629944\pi$
$619$	−7.56892e6	−0.793976	−0.396988	+	0.917824i	$0.629944\pi$
$620$	−6.87558e6	−0.718341
$621$	385641.	0.0401286
$622$	−5.39209e6	−0.558832
$623$	−8.49443e6	−0.876828
$624$	131509.	0.0135206
$625$	−1.17664e7	−1.20488
$626$	−3.10465e6	−0.316648
$627$	−9.72754e6	−0.988175
$628$	−8.69355e6	−0.879626
$629$	1.14396e7	1.15288
$630$	2.13720e6	0.214532
$631$	1.43986e7	1.43961	0.719807	−	0.694174i	$-0.244230\pi$
$631$	1.43986e7	1.43961	0.719807	+	0.694174i	$0.244230\pi$
$632$	−5.09650e6	−0.507550
$633$	−755707.	−0.0749625
$634$	−5.27315e6	−0.521011
$635$	1.75934e7	1.73147
$636$	1.73263e6	0.169849
$637$	495219.	0.0483559
$638$	8.60788e6	0.837230
$639$	−1.41985e6	−0.137560
$640$	1.19853e6	0.115664
$641$	−1.20691e7	−1.16019	−0.580097	−	0.814548i	$-0.696985\pi$
$641$	−1.20691e7	−1.16019	−0.580097	+	0.814548i	$0.696985\pi$
$642$	−7.37527e6	−0.706220
$643$	3.88907e6	0.370953	0.185476	−	0.982649i	$-0.440617\pi$
$643$	3.88907e6	0.370953	0.185476	+	0.982649i	$0.440617\pi$
$644$	−763213.	−0.0725155
$645$	−1.39318e7	−1.31858
$646$	9.75268e6	0.919481
$647$	1.99120e7	1.87005	0.935026	−	0.354578i	$-0.115376\pi$
$647$	1.99120e7	1.87005	0.935026	+	0.354578i	$0.115376\pi$
$648$	−419904.	−0.0392837
$649$	−1.73424e7	−1.61621
$650$	508296.	0.0471882
$651$	−4.76730e6	−0.440880
$652$	−3.98685e6	−0.367291
$653$	1.34173e7	1.23135	0.615676	−	0.787999i	$-0.288883\pi$
$653$	1.34173e7	1.23135	0.615676	+	0.787999i	$0.288883\pi$
$654$	6.04938e6	0.553052
$655$	360460.	0.0328287
$656$	−1.02887e6	−0.0933468
$657$	−766721.	−0.0692985
$658$	−1.03569e6	−0.0932539
$659$	−1.85192e7	−1.66115	−0.830574	−	0.556908i	$-0.811988\pi$
$659$	−1.85192e7	−1.66115	−0.830574	+	0.556908i	$0.811988\pi$
$660$	5.06847e6	0.452915
$661$	−1.17733e7	−1.04808	−0.524039	−	0.851694i	$-0.675575\pi$
$661$	−1.17733e7	−1.04808	−0.524039	+	0.851694i	$0.675575\pi$
$662$	1.18098e7	1.04736
$663$	−557577.	−0.0492630
$664$	−4.07808e6	−0.358951
$665$	−1.48175e7	−1.29933
$666$	3.41480e6	0.298318
$667$	2.36596e6	0.205917
$668$	3.58899e6	0.311194
$669$	−2.80922e6	−0.242672
$670$	1.12442e7	0.967701
$671$	−3.66112e6	−0.313912
$672$	831022.	0.0709887
$673$	1.04932e7	0.893041	0.446521	−	0.894773i	$-0.352663\pi$
$673$	1.04932e7	0.893041	0.446521	+	0.894773i	$0.352663\pi$
$674$	9.20454e6	0.780463
$675$	−1.62297e6	−0.137104
$676$	−5.88856e6	−0.495613
$677$	−1.30808e7	−1.09689	−0.548444	−	0.836187i	$-0.684780\pi$
$677$	−1.30808e7	−1.09689	−0.548444	+	0.836187i	$0.684780\pi$
$678$	2.46449e6	0.205898
$679$	−1.11159e7	−0.925270
$680$	−5.08157e6	−0.421430
$681$	9.89372e6	0.817508
$682$	−1.13059e7	−0.930774
$683$	−3.14143e6	−0.257677	−0.128839	−	0.991666i	$-0.541125\pi$
$683$	−3.14143e6	−0.257677	−0.128839	+	0.991666i	$0.541125\pi$
$684$	2.91126e6	0.237925
$685$	−1.13748e7	−0.926229
$686$	9.19140e6	0.745713
$687$	−1.32623e7	−1.07208
$688$	−5.41719e6	−0.436318
$689$	686780.	0.0551149
$690$	1.39312e6	0.111395
$691$	2.47309e7	1.97036	0.985180	−	0.171525i	$-0.0548696\pi$
$691$	2.47309e7	1.97036	0.985180	+	0.171525i	$0.0548696\pi$
$692$	562866.	0.0446828
$693$	3.51431e6	0.277976
$694$	4.36248e6	0.343823
$695$	1.29498e7	1.01696
$696$	−2.57617e6	−0.201582
$697$	4.36222e6	0.340115
$698$	1.14026e7	0.885860
$699$	1.18856e7	0.920087
$700$	3.21198e6	0.247758
$701$	3.56709e6	0.274169	0.137085	−	0.990559i	$-0.456227\pi$
$701$	3.56709e6	0.274169	0.137085	+	0.990559i	$0.456227\pi$
$702$	−166441.	−0.0127473
$703$	−2.36753e7	−1.80679
$704$	1.97081e6	0.149869
$705$	1.89049e6	0.143252
$706$	−7.95998e6	−0.601036
$707$	−1.72423e6	−0.129732
$708$	5.19023e6	0.389138
$709$	−1.83365e7	−1.36994	−0.684969	−	0.728572i	$-0.740184\pi$
$709$	−1.83365e7	−1.36994	−0.684969	+	0.728572i	$0.740184\pi$
$710$	−5.12918e6	−0.381858
$711$	6.45025e6	0.478523
$712$	6.02899e6	0.445702
$713$	−3.10753e6	−0.228924
$714$	−3.52339e6	−0.258652
$715$	2.00904e6	0.146968
$716$	−9.04939e6	−0.659685
$717$	−1.09329e7	−0.794213
$718$	−707570.	−0.0512222
$719$	2.22328e6	0.160388	0.0801938	−	0.996779i	$-0.474446\pi$
$719$	2.22328e6	0.160388	0.0801938	+	0.996779i	$0.474446\pi$
$720$	−1.51689e6	−0.109049
$721$	7.70284e6	0.551839
$722$	−1.02798e7	−0.733906
$723$	8.13556e6	0.578818
$724$	−8.44758e6	−0.598944
$725$	−9.95713e6	−0.703541
$726$	2.53653e6	0.178607
$727$	1.04368e7	0.732374	0.366187	−	0.930541i	$-0.380663\pi$
$727$	1.04368e7	0.732374	0.366187	+	0.930541i	$0.380663\pi$
$728$	329400.	0.0230354
$729$	531441.	0.0370370
$730$	−2.76976e6	−0.192369
$731$	2.29680e7	1.58975
$732$	1.09570e6	0.0755813
$733$	−1.65631e7	−1.13863	−0.569313	−	0.822121i	$-0.692791\pi$
$733$	−1.65631e7	−1.13863	−0.569313	+	0.822121i	$0.692791\pi$
$734$	1.12621e7	0.771576
$735$	−5.71210e6	−0.390011
$736$	541696.	0.0368605
$737$	1.84894e7	1.25388
$738$	1.30216e6	0.0880082
$739$	−2.06873e7	−1.39346	−0.696728	−	0.717335i	$-0.745362\pi$
$739$	−2.06873e7	−1.39346	−0.696728	+	0.717335i	$0.745362\pi$
$740$	1.23359e7	0.828115
$741$	1.15396e6	0.0772052
$742$	4.33983e6	0.289377
$743$	2.80214e6	0.186216	0.0931080	−	0.995656i	$-0.470320\pi$
$743$	2.80214e6	0.186216	0.0931080	+	0.995656i	$0.470320\pi$
$744$	3.38363e6	0.224104
$745$	1.21102e7	0.799395
$746$	9.68858e6	0.637401
$747$	5.16132e6	0.338423
$748$	−8.35590e6	−0.546059
$749$	−1.84733e7	−1.20321
$750$	2.36673e6	0.153637
$751$	1.79496e7	1.16133	0.580664	−	0.814143i	$-0.302793\pi$
$751$	1.79496e7	1.16133	0.580664	+	0.814143i	$0.302793\pi$
$752$	735092.	0.0474021
$753$	−1.17058e7	−0.752339
$754$	−1.02114e6	−0.0654119
$755$	1.04840e7	0.669362
$756$	−1.05176e6	−0.0669288
$757$	−3.00311e7	−1.90472	−0.952361	−	0.304973i	$-0.901353\pi$
$757$	−3.00311e7	−1.90472	−0.952361	+	0.304973i	$0.901353\pi$
$758$	−1.39030e7	−0.878890
$759$	2.29078e6	0.144337
$760$	1.05168e7	0.660467
$761$	−5.60169e6	−0.350637	−0.175318	−	0.984512i	$-0.556095\pi$
$761$	−5.60169e6	−0.350637	−0.175318	+	0.984512i	$0.556095\pi$
$762$	−8.65812e6	−0.540177
$763$	1.51523e7	0.942251
$764$	−5.88390e6	−0.364697
$765$	6.43136e6	0.397328
$766$	−1.08550e7	−0.668431
$767$	2.05730e6	0.126273
$768$	−589824.	−0.0360844
$769$	−3.06498e6	−0.186901	−0.0934506	−	0.995624i	$-0.529790\pi$
$769$	−3.06498e6	−0.186901	−0.0934506	+	0.995624i	$0.529790\pi$
$770$	1.26953e7	0.771645
$771$	−1.44396e7	−0.874823
$772$	2.22586e6	0.134417
$773$	−2.27700e6	−0.137061	−0.0685304	−	0.997649i	$-0.521831\pi$
$773$	−2.27700e6	−0.137061	−0.0685304	+	0.997649i	$0.521831\pi$
$774$	6.85613e6	0.411364
$775$	1.30781e7	0.782148
$776$	7.88956e6	0.470326
$777$	8.55329e6	0.508254
$778$	6.79072e6	0.402223
$779$	−9.02806e6	−0.533029
$780$	−601265.	−0.0353858
$781$	−8.43419e6	−0.494784
$782$	−2.29670e6	−0.134303
$783$	3.26046e6	0.190053
$784$	−2.22108e6	−0.129055
$785$	3.97472e7	2.30214
$786$	−177390.	−0.0102417
$787$	−1.07214e6	−0.0617041	−0.0308521	−	0.999524i	$-0.509822\pi$
$787$	−1.07214e6	−0.0617041	−0.0308521	+	0.999524i	$0.509822\pi$
$788$	2.65984e6	0.152595
$789$	6.58693e6	0.376695
$790$	2.33014e7	1.32835
$791$	6.17297e6	0.350795
$792$	−2.49431e6	−0.141298
$793$	434314.	0.0245257
$794$	−2.82590e6	−0.159076
$795$	−7.92165e6	−0.444526
$796$	−4.30898e6	−0.241041
$797$	2.24366e7	1.25116	0.625579	−	0.780161i	$-0.284863\pi$
$797$	2.24366e7	1.25116	0.625579	+	0.780161i	$0.284863\pi$
$798$	7.29202e6	0.405360
$799$	−3.11667e6	−0.172712
$800$	−2.27973e6	−0.125938
$801$	−7.63044e6	−0.420212
$802$	−1.22168e7	−0.670691
$803$	−4.55447e6	−0.249258
$804$	−5.53352e6	−0.301899
$805$	3.48943e6	0.189786
$806$	1.34120e6	0.0727205
$807$	7.08501e6	0.382963
$808$	1.22378e6	0.0659442
$809$	−1.26281e7	−0.678371	−0.339185	−	0.940720i	$-0.610151\pi$
$809$	−1.26281e7	−0.678371	−0.339185	+	0.940720i	$0.610151\pi$
$810$	1.91982e6	0.102813
$811$	1.85726e7	0.991565	0.495783	−	0.868447i	$-0.334881\pi$
$811$	1.85726e7	0.991565	0.495783	+	0.868447i	$0.334881\pi$
$812$	−6.45270e6	−0.343440
$813$	−2.08811e7	−1.10797
$814$	2.02846e7	1.07301
$815$	1.82280e7	0.961269
$816$	2.50075e6	0.131476
$817$	−4.75345e7	−2.49146
$818$	−2.88100e6	−0.150543
$819$	−416897.	−0.0217180
$820$	4.70401e6	0.244306
$821$	6.23383e6	0.322773	0.161387	−	0.986891i	$-0.448403\pi$
$821$	6.23383e6	0.322773	0.161387	+	0.986891i	$0.448403\pi$
$822$	5.59781e6	0.288960
$823$	9.80588e6	0.504646	0.252323	−	0.967643i	$-0.418805\pi$
$823$	9.80588e6	0.504646	0.252323	+	0.967643i	$0.418805\pi$
$824$	−5.46715e6	−0.280506
$825$	−9.64074e6	−0.493146
$826$	1.30003e7	0.662985
$827$	−2.84744e7	−1.44774	−0.723871	−	0.689935i	$-0.757639\pi$
$827$	−2.84744e7	−1.44774	−0.723871	+	0.689935i	$0.757639\pi$
$828$	−685584.	−0.0347524
$829$	−1.56570e7	−0.791264	−0.395632	−	0.918409i	$-0.629474\pi$
$829$	−1.56570e7	−0.791264	−0.395632	+	0.918409i	$0.629474\pi$
$830$	1.86451e7	0.939442
$831$	1.03368e7	0.519259
$832$	−233794.	−0.0117092
$833$	9.41699e6	0.470218
$834$	−6.37291e6	−0.317265
$835$	−1.64090e7	−0.814452
$836$	1.72934e7	0.855785
$837$	−4.28240e6	−0.211288
$838$	−865144.	−0.0425577
$839$	−3.50245e7	−1.71778	−0.858890	−	0.512160i	$-0.828845\pi$
$839$	−3.50245e7	−1.71778	−0.858890	+	0.512160i	$0.828845\pi$
$840$	−3.79946e6	−0.185791
$841$	−507802.	−0.0247574
$842$	1.92507e7	0.935764
$843$	−8.56809e6	−0.415255
$844$	1.34348e6	0.0649195
$845$	2.69227e7	1.29711
$846$	−930351.	−0.0446911
$847$	6.35341e6	0.304297
$848$	−3.08023e6	−0.147094
$849$	1.68728e7	0.803375
$850$	9.66566e6	0.458864
$851$	5.57540e6	0.263908
$852$	2.52418e6	0.119130
$853$	−2.41446e7	−1.13618	−0.568090	−	0.822967i	$-0.692317\pi$
$853$	−2.41446e7	−1.13618	−0.568090	+	0.822967i	$0.692317\pi$
$854$	2.74448e6	0.128770
$855$	−1.33104e7	−0.622694
$856$	1.31116e7	0.611605
$857$	2.30752e7	1.07323	0.536615	−	0.843827i	$-0.319703\pi$
$857$	2.30752e7	1.07323	0.536615	+	0.843827i	$0.319703\pi$
$858$	−988693.	−0.0458504
$859$	1.11770e7	0.516822	0.258411	−	0.966035i	$-0.416801\pi$
$859$	1.11770e7	0.516822	0.258411	+	0.966035i	$0.416801\pi$
$860$	2.47676e7	1.14192
$861$	3.26160e6	0.149942
$862$	−2.40719e7	−1.10343
$863$	−1.36487e7	−0.623826	−0.311913	−	0.950111i	$-0.600970\pi$
$863$	−1.36487e7	−0.623826	−0.311913	+	0.950111i	$0.600970\pi$
$864$	746496.	0.0340207
$865$	−2.57344e6	−0.116943
$866$	−2.22510e7	−1.00822
$867$	2.17594e6	0.0983102
$868$	8.47520e6	0.381813
$869$	3.83157e7	1.72118
$870$	1.17783e7	0.527576
$871$	−2.19337e6	−0.0979642
$872$	−1.07544e7	−0.478957
$873$	−9.98523e6	−0.443427
$874$	4.75325e6	0.210481
$875$	5.92811e6	0.261756
$876$	1.36306e6	0.0600143
$877$	2.05105e7	0.900486	0.450243	−	0.892906i	$-0.351337\pi$
$877$	2.05105e7	0.900486	0.450243	+	0.892906i	$0.351337\pi$
$878$	1.42719e7	0.624805
$879$	−3.65129e6	−0.159395
$880$	−9.01061e6	−0.392236
$881$	2.20938e7	0.959028	0.479514	−	0.877534i	$-0.340813\pi$
$881$	2.20938e7	0.959028	0.479514	+	0.877534i	$0.340813\pi$
$882$	2.81105e6	0.121674
$883$	−1.17759e6	−0.0508268	−0.0254134	−	0.999677i	$-0.508090\pi$
$883$	−1.17759e6	−0.0508268	−0.0254134	+	0.999677i	$0.508090\pi$
$884$	991249.	0.0426631
$885$	−2.37299e7	−1.01845
$886$	1.32358e7	0.566456
$887$	−2.35414e7	−1.00467	−0.502336	−	0.864673i	$-0.667526\pi$
$887$	−2.35414e7	−1.00467	−0.502336	+	0.864673i	$0.667526\pi$
$888$	−6.07076e6	−0.258351
$889$	−2.16866e7	−0.920315
$890$	−2.75648e7	−1.16648
$891$	3.15686e6	0.133217
$892$	4.99416e6	0.210160
$893$	6.45026e6	0.270675
$894$	−5.95971e6	−0.249391
$895$	4.13741e7	1.72652
$896$	−1.47737e6	−0.0614780
$897$	−271752.	−0.0112769
$898$	−2.43119e7	−1.00607
$899$	−2.62731e7	−1.08421
$900$	2.88528e6	0.118736
$901$	1.30597e7	0.535945
$902$	7.73506e6	0.316554
$903$	1.71730e7	0.700853
$904$	−4.38131e6	−0.178313
$905$	3.86226e7	1.56755
$906$	−5.15942e6	−0.208824
$907$	−2.63830e7	−1.06489	−0.532446	−	0.846464i	$-0.678727\pi$
$907$	−2.63830e7	−1.06489	−0.532446	+	0.846464i	$0.678727\pi$
$908$	−1.75888e7	−0.707982
$909$	−1.54885e6	−0.0621728
$910$	−1.50603e6	−0.0602879
$911$	2.24004e7	0.894251	0.447126	−	0.894471i	$-0.352448\pi$
$911$	2.24004e7	0.894251	0.447126	+	0.894471i	$0.352448\pi$
$912$	−5.17557e6	−0.206049
$913$	3.06592e7	1.21726
$914$	−3.18922e6	−0.126275
$915$	−5.00958e6	−0.197810
$916$	2.35774e7	0.928445
$917$	−444321.	−0.0174491
$918$	−3.16502e6	−0.123957
$919$	2.00873e6	0.0784570	0.0392285	−	0.999230i	$-0.487510\pi$
$919$	2.00873e6	0.0784570	0.0392285	+	0.999230i	$0.487510\pi$
$920$	−2.47665e6	−0.0964707
$921$	−5.62496e6	−0.218509
$922$	−5.41055e6	−0.209611
$923$	1.00054e6	0.0386570
$924$	−6.24766e6	−0.240734
$925$	−2.34641e7	−0.901673
$926$	1.36445e7	0.522913
$927$	6.91936e6	0.264464
$928$	4.57985e6	0.174575
$929$	−3.10083e7	−1.17880	−0.589399	−	0.807842i	$-0.700635\pi$
$929$	−3.10083e7	−1.17880	−0.589399	+	0.807842i	$0.700635\pi$
$930$	−1.54701e7	−0.586523
$931$	−1.94894e7	−0.736928
$932$	−2.11300e7	−0.796819
$933$	−1.21322e7	−0.456284
$934$	1.05692e7	0.396439
$935$	3.82035e7	1.42914
$936$	295896.	0.0110395
$937$	−1.31851e7	−0.490608	−0.245304	−	0.969446i	$-0.578888\pi$
$937$	−1.31851e7	−0.490608	−0.245304	+	0.969446i	$0.578888\pi$
$938$	−1.38602e7	−0.514353
$939$	−6.98545e6	−0.258542
$940$	−3.36087e6	−0.124060
$941$	−2.53246e7	−0.932327	−0.466164	−	0.884699i	$-0.654364\pi$
$941$	−2.53246e7	−0.932327	−0.466164	+	0.884699i	$0.654364\pi$
$942$	−1.95605e7	−0.718212
$943$	2.12606e6	0.0778566
$944$	−9.22708e6	−0.337003
$945$	4.80869e6	0.175165
$946$	4.07267e7	1.47962
$947$	3.30648e7	1.19810	0.599048	−	0.800713i	$-0.295546\pi$
$947$	3.30648e7	1.19810	0.599048	+	0.800713i	$0.295546\pi$
$948$	−1.14671e7	−0.414413
$949$	540290.	0.0194743
$950$	−2.00041e7	−0.719133
$951$	−1.18646e7	−0.425404
$952$	6.26381e6	0.223999
$953$	−1.65258e6	−0.0589429	−0.0294714	−	0.999566i	$-0.509382\pi$
$953$	−1.65258e6	−0.0589429	−0.0294714	+	0.999566i	$0.509382\pi$
$954$	3.89842e6	0.138681
$955$	2.69014e7	0.954478
$956$	1.94362e7	0.687808
$957$	1.93677e7	0.683595
$958$	−3.73495e7	−1.31484
$959$	1.40212e7	0.492310
$960$	2.69670e6	0.0944395
$961$	5.87889e6	0.205346
$962$	−2.40633e6	−0.0838334
$963$	−1.65943e7	−0.576626
$964$	−1.44632e7	−0.501271
$965$	−1.01767e7	−0.351794
$966$	−1.71723e6	−0.0592087
$967$	3.55675e7	1.22317	0.611585	−	0.791179i	$-0.290532\pi$
$967$	3.55675e7	1.22317	0.611585	+	0.791179i	$0.290532\pi$
$968$	−4.50938e6	−0.154678
$969$	2.19435e7	0.750753
$970$	−3.60714e7	−1.23093
$971$	−3.62629e7	−1.23428	−0.617141	−	0.786853i	$-0.711709\pi$
$971$	−3.62629e7	−1.23428	−0.617141	+	0.786853i	$0.711709\pi$
$972$	−944784.	−0.0320750
$973$	−1.59627e7	−0.540534
$974$	1.49573e7	0.505191
$975$	1.14367e6	0.0385290
$976$	−1.94791e6	−0.0654553
$977$	−1.64779e7	−0.552289	−0.276145	−	0.961116i	$-0.589057\pi$
$977$	−1.64779e7	−0.552289	−0.276145	+	0.961116i	$0.589057\pi$
$978$	−8.97041e6	−0.299892
$979$	−4.53262e7	−1.51145
$980$	1.01548e7	0.337760
$981$	1.36111e7	0.451565
$982$	−1.44162e7	−0.477059
$983$	−4.25070e7	−1.40306	−0.701530	−	0.712640i	$-0.747499\pi$
$983$	−4.25070e7	−1.40306	−0.701530	+	0.712640i	$0.747499\pi$
$984$	−2.31495e6	−0.0762173
$985$	−1.21609e7	−0.399370
$986$	−1.94178e7	−0.636074
$987$	−2.33031e6	−0.0761415
$988$	−2.05149e6	−0.0668617
$989$	1.11941e7	0.363914
$990$	1.14041e7	0.369804
$991$	4.79987e7	1.55255	0.776274	−	0.630395i	$-0.217107\pi$
$991$	4.79987e7	1.55255	0.776274	+	0.630395i	$0.217107\pi$
$992$	−6.01534e6	−0.194080
$993$	2.65720e7	0.855167
$994$	6.32249e6	0.202966
$995$	1.97008e7	0.630850
$996$	−9.17568e6	−0.293083
$997$	−1.98378e7	−0.632057	−0.316028	−	0.948750i	$-0.602350\pi$
$997$	−1.98378e7	−0.632057	−0.316028	+	0.948750i	$0.602350\pi$
$998$	−2.08362e7	−0.662206
$999$	7.68331e6	0.243576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.6.a.g.1.1	✓	3
3.2	odd	2		414.6.a.n.1.3		3
4.3	odd	2		1104.6.a.k.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.6.a.g.1.1	✓	3	1.1	even	1	trivial
414.6.a.n.1.3		3	3.2	odd	2
1104.6.a.k.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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -73.1526 q^{5} +36.0000 q^{6} +90.1716 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -73.1526 q^{5} +36.0000 q^{6} +90.1716 q^{7} -64.0000 q^{8} +81.0000 q^{9} +292.610 q^{10} +481.155 q^{11} -144.000 q^{12} -57.0787 q^{13} -360.686 q^{14} +658.373 q^{15} +256.000 q^{16} -1085.40 q^{17} -324.000 q^{18} +2246.34 q^{19} -1170.44 q^{20} -811.544 q^{21} -1924.62 q^{22} -529.000 q^{23} +576.000 q^{24} +2226.30 q^{25} +228.315 q^{26} -729.000 q^{27} +1442.75 q^{28} -4472.51 q^{29} -2633.49 q^{30} +5874.35 q^{31} -1024.00 q^{32} -4330.39 q^{33} +4341.59 q^{34} -6596.28 q^{35} +1296.00 q^{36} -10539.5 q^{37} -8985.36 q^{38} +513.708 q^{39} +4681.76 q^{40} -4019.01 q^{41} +3246.18 q^{42} -21160.9 q^{43} +7698.48 q^{44} -5925.36 q^{45} +2116.00 q^{46} +2871.45 q^{47} -2304.00 q^{48} -8676.08 q^{49} -8905.18 q^{50} +9768.57 q^{51} -913.259 q^{52} -12032.2 q^{53} +2916.00 q^{54} -35197.7 q^{55} -5770.98 q^{56} -20217.1 q^{57} +17890.0 q^{58} -36043.3 q^{59} +10534.0 q^{60} -7609.04 q^{61} -23497.4 q^{62} +7303.90 q^{63} +4096.00 q^{64} +4175.45 q^{65} +17321.6 q^{66} +38427.2 q^{67} -17366.3 q^{68} +4761.00 q^{69} +26385.1 q^{70} -17529.1 q^{71} -5184.00 q^{72} -9465.69 q^{73} +42158.1 q^{74} -20036.7 q^{75} +35941.4 q^{76} +43386.5 q^{77} -2054.83 q^{78} +79632.8 q^{79} -18727.1 q^{80} +6561.00 q^{81} +16076.0 q^{82} +63720.0 q^{83} -12984.7 q^{84} +79399.5 q^{85} +84643.5 q^{86} +40252.6 q^{87} -30793.9 q^{88} -94203.0 q^{89} +23701.4 q^{90} -5146.88 q^{91} -8464.00 q^{92} -52869.2 q^{93} -11485.8 q^{94} -164325. q^{95} +9216.00 q^{96} -123274. q^{97} +34704.3 q^{98} +38973.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 12 q^{2} - 27 q^{3} + 48 q^{4} - 18 q^{5} + 108 q^{6} - 50 q^{7} - 192 q^{8} + 243 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more