LMFDB - Embedded newform 165.6.c.b.34.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,6,Mod(34,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("165.34");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	165.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$26.4633302691$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			34.6
Character	$\chi$	$=$	165.34
Dual form			165.6.c.b.34.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.39855i q^{2} -9.00000i q^{3} -22.7385 q^{4} +(-50.7560 + 23.4272i) q^{5} -66.5869 q^{6} +150.852i q^{7} -68.5217i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q-7.39855i q^{2} -9.00000i q^{3} -22.7385 q^{4} +(-50.7560 + 23.4272i) q^{5} -66.5869 q^{6} +150.852i q^{7} -68.5217i q^{8} -81.0000 q^{9} +(173.327 + 375.520i) q^{10} +121.000 q^{11} +204.646i q^{12} +868.426i q^{13} +1116.08 q^{14} +(210.844 + 456.804i) q^{15} -1234.59 q^{16} -2317.87i q^{17} +599.282i q^{18} +2655.24 q^{19} +(1154.11 - 532.698i) q^{20} +1357.67 q^{21} -895.224i q^{22} +2537.58i q^{23} -616.695 q^{24} +(2027.34 - 2378.14i) q^{25} +6425.09 q^{26} +729.000i q^{27} -3430.14i q^{28} +819.917 q^{29} +(3379.68 - 1559.94i) q^{30} +7303.36 q^{31} +6941.50i q^{32} -1089.00i q^{33} -17148.9 q^{34} +(-3534.03 - 7656.63i) q^{35} +1841.82 q^{36} +2993.84i q^{37} -19644.9i q^{38} +7815.84 q^{39} +(1605.27 + 3477.89i) q^{40} -4567.72 q^{41} -10044.8i q^{42} +1022.10i q^{43} -2751.36 q^{44} +(4111.23 - 1897.60i) q^{45} +18774.4 q^{46} +24499.3i q^{47} +11111.3i q^{48} -5949.25 q^{49} +(-17594.8 - 14999.3i) q^{50} -20860.8 q^{51} -19746.7i q^{52} -13318.9i q^{53} +5393.54 q^{54} +(-6141.47 + 2834.69i) q^{55} +10336.6 q^{56} -23897.2i q^{57} -6066.20i q^{58} +29760.6 q^{59} +(-4794.28 - 10387.0i) q^{60} -17257.1 q^{61} -54034.2i q^{62} -12219.0i q^{63} +11850.0 q^{64} +(-20344.8 - 44077.8i) q^{65} -8057.02 q^{66} +18645.0i q^{67} +52704.9i q^{68} +22838.2 q^{69} +(-56647.9 + 26146.7i) q^{70} +47722.0 q^{71} +5550.26i q^{72} -19156.3i q^{73} +22150.1 q^{74} +(-21403.2 - 18246.0i) q^{75} -60376.2 q^{76} +18253.1i q^{77} -57825.8i q^{78} +3713.15 q^{79} +(62662.9 - 28923.0i) q^{80} +6561.00 q^{81} +33794.5i q^{82} -51919.0i q^{83} -30871.3 q^{84} +(54301.2 + 117646. i) q^{85} +7562.07 q^{86} -7379.26i q^{87} -8291.13i q^{88} +52966.5 q^{89} +(-14039.5 - 30417.2i) q^{90} -131004. q^{91} -57700.8i q^{92} -65730.2i q^{93} +181259. q^{94} +(-134769. + 62204.8i) q^{95} +62473.5 q^{96} +114526. i q^{97} +44015.8i q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 418 q^{4} - 98 q^{5} + 234 q^{6} - 2106 q^{9}+O(q^{10}) $ 26 * q - 418 * q^4 - 98 * q^5 + 234 * q^6 - 2106 * q^9	$ 26 q - 418 q^{4} - 98 q^{5} + 234 q^{6} - 2106 q^{9} + 236 q^{10} + 3146 q^{11} + 1220 q^{14} + 7002 q^{16} - 540 q^{19} + 4930 q^{20} + 5472 q^{21} - 7182 q^{24} + 218 q^{25} + 5304 q^{26} - 23904 q^{29} + 12114 q^{30} + 38192 q^{31} + 2604 q^{34} - 11988 q^{35} + 33858 q^{36} - 17748 q^{39} - 41096 q^{40} + 70368 q^{41} - 50578 q^{44} + 7938 q^{45} - 8240 q^{46} - 29114 q^{49} - 133876 q^{50} + 26568 q^{51} - 18954 q^{54} - 11858 q^{55} + 119604 q^{56} + 18384 q^{59} - 14148 q^{60} + 10876 q^{61} - 213114 q^{64} - 117068 q^{65} + 28314 q^{66} - 163512 q^{69} - 58660 q^{70} + 203400 q^{71} - 27352 q^{74} - 35352 q^{75} + 279932 q^{76} - 187908 q^{79} - 256654 q^{80} + 170586 q^{81} - 196560 q^{84} + 37396 q^{85} + 741860 q^{86} + 36836 q^{89} - 19116 q^{90} + 349072 q^{91} - 129040 q^{94} - 208284 q^{95} + 209898 q^{96} - 254826 q^{99}+O(q^{100}) $ 26 * q - 418 * q^4 - 98 * q^5 + 234 * q^6 - 2106 * q^9 + 236 * q^10 + 3146 * q^11 + 1220 * q^14 + 7002 * q^16 - 540 * q^19 + 4930 * q^20 + 5472 * q^21 - 7182 * q^24 + 218 * q^25 + 5304 * q^26 - 23904 * q^29 + 12114 * q^30 + 38192 * q^31 + 2604 * q^34 - 11988 * q^35 + 33858 * q^36 - 17748 * q^39 - 41096 * q^40 + 70368 * q^41 - 50578 * q^44 + 7938 * q^45 - 8240 * q^46 - 29114 * q^49 - 133876 * q^50 + 26568 * q^51 - 18954 * q^54 - 11858 * q^55 + 119604 * q^56 + 18384 * q^59 - 14148 * q^60 + 10876 * q^61 - 213114 * q^64 - 117068 * q^65 + 28314 * q^66 - 163512 * q^69 - 58660 * q^70 + 203400 * q^71 - 27352 * q^74 - 35352 * q^75 + 279932 * q^76 - 187908 * q^79 - 256654 * q^80 + 170586 * q^81 - 196560 * q^84 + 37396 * q^85 + 741860 * q^86 + 36836 * q^89 - 19116 * q^90 + 349072 * q^91 - 129040 * q^94 - 208284 * q^95 + 209898 * q^96 - 254826 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/165\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$67$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	7.39855i		−	1.30789i	−0.756542	−	0.653945i	$-0.773113\pi$
$2$		−	7.39855i		−	1.30789i	0.756542	−	0.653945i	$-0.226887\pi$
$3$		−	9.00000i		−	0.577350i
$3$		−	9.00000i		−	0.577350i
$4$	−22.7385			−0.710578
$5$	−50.7560	+	23.4272i	−0.907950	+	0.419078i
$5$	−50.7560	+	23.4272i	−0.907950	+	0.419078i
$6$	−66.5869			−0.755111
$7$			150.852i			1.16360i	0.813330	+	0.581802i	$0.197652\pi$
$7$			150.852i			1.16360i	−0.813330	+	0.581802i	$0.802348\pi$
$8$		−	68.5217i		−	0.378533i
$9$	−81.0000			−0.333333
$10$	173.327	+	375.520i	0.548108	+	1.18750i
$11$	121.000			0.301511
$11$	121.000			0.301511
$12$			204.646i			0.410252i
$13$			868.426i			1.42520i	0.701573	+	0.712598i	$0.252482\pi$
$13$			868.426i			1.42520i	−0.701573	+	0.712598i	$0.747518\pi$
$14$	1116.08			1.52187
$15$	210.844	+	456.804i	0.241955	+	0.524205i
$16$	−1234.59			−1.20566
$17$		−	2317.87i		−	1.94521i	−0.232458	−	0.972606i	$-0.574677\pi$
$17$		−	2317.87i		−	1.94521i	0.232458	−	0.972606i	$-0.425323\pi$
$18$			599.282i			0.435964i
$19$	2655.24			1.68741			0.843705	−	0.536808i	$-0.180370\pi$
$19$	2655.24			1.68741			0.843705	+	0.536808i	$0.180370\pi$
$20$	1154.11	−	532.698i	0.645169	−	0.297787i
$21$	1357.67			0.671807
$22$		−	895.224i		−	0.394344i
$23$			2537.58i			1.00023i	0.865959	+	0.500116i	$0.166709\pi$
$23$			2537.58i			1.00023i	−0.865959	+	0.500116i	$0.833291\pi$
$24$	−616.695			−0.218546
$25$	2027.34	−	2378.14i	0.648748	−	0.761004i
$26$	6425.09			1.86400
$27$			729.000i			0.192450i
$28$		−	3430.14i		−	0.826831i
$29$	819.917			0.181040			0.0905201	−	0.995895i	$-0.471147\pi$
$29$	819.917			0.181040			0.0905201	+	0.995895i	$0.471147\pi$
$30$	3379.68	−	1559.94i	0.685603	−	0.316450i
$31$	7303.36			1.36495			0.682477	−	0.730907i	$-0.260903\pi$
$31$	7303.36			1.36495			0.682477	+	0.730907i	$0.260903\pi$
$32$			6941.50i			1.19833i
$33$		−	1089.00i		−	0.174078i
$34$	−17148.9			−2.54413
$35$	−3534.03	−	7656.63i	−0.487641	−	1.05649i
$36$	1841.82			0.236859
$37$			2993.84i			0.359521i	0.983710	+	0.179761i	$0.0575323\pi$
$37$			2993.84i			0.359521i	−0.983710	+	0.179761i	$0.942468\pi$
$38$		−	19644.9i		−	2.20695i
$39$	7815.84			0.822837
$40$	1605.27	+	3477.89i	0.158635	+	0.343689i
$41$	−4567.72			−0.424365			−0.212182	−	0.977230i	$-0.568057\pi$
$41$	−4567.72			−0.424365			−0.212182	+	0.977230i	$0.568057\pi$
$42$		−	10044.8i		−	0.878650i
$43$			1022.10i			0.0842992i	0.999111	+	0.0421496i	$0.0134206\pi$
$43$			1022.10i			0.0842992i	−0.999111	+	0.0421496i	$0.986579\pi$
$44$	−2751.36			−0.214247
$45$	4111.23	−	1897.60i	0.302650	−	0.139693i
$46$	18774.4			1.30819
$47$			24499.3i			1.61774i	0.587987	+	0.808871i	$0.299921\pi$
$47$			24499.3i			1.61774i	−0.587987	+	0.808871i	$0.700079\pi$
$48$			11111.3i			0.696086i
$49$	−5949.25			−0.353974
$50$	−17594.8	−	14999.3i	−0.995309	−	0.848491i
$51$	−20860.8			−1.12307
$52$		−	19746.7i		−	1.01271i
$53$		−	13318.9i		−	0.651297i	−0.945491	−	0.325649i	$-0.894417\pi$
$53$		−	13318.9i		−	0.651297i	0.945491	−	0.325649i	$-0.105583\pi$
$54$	5393.54			0.251704
$55$	−6141.47	+	2834.69i	−0.273757	+	0.126357i
$56$	10336.6			0.440462
$57$		−	23897.2i		−	0.974226i
$58$		−	6066.20i		−	0.236781i
$59$	29760.6			1.11304			0.556522	−	0.830833i	$-0.312136\pi$
$59$	29760.6			1.11304			0.556522	+	0.830833i	$0.312136\pi$
$60$	−4794.28	−	10387.0i	−0.171928	−	0.372489i
$61$	−17257.1			−0.593805			−0.296903	−	0.954908i	$-0.595954\pi$
$61$	−17257.1			−0.593805			−0.296903	+	0.954908i	$0.595954\pi$
$62$		−	54034.2i		−	1.78521i
$63$		−	12219.0i		−	0.387868i
$64$	11850.0			0.361634
$65$	−20344.8	−	44077.8i	−0.597268	−	1.29401i
$66$	−8057.02			−0.227675
$67$			18645.0i			0.507429i	0.967279	+	0.253715i	$0.0816524\pi$
$67$			18645.0i			0.507429i	−0.967279	+	0.253715i	$0.918348\pi$
$68$			52704.9i			1.38223i
$69$	22838.2			0.577484
$70$	−56647.9	+	26146.7i	−1.38178	+	0.637781i
$71$	47722.0			1.12350			0.561750	−	0.827307i	$-0.310128\pi$
$71$	47722.0			1.12350			0.561750	+	0.827307i	$0.310128\pi$
$72$			5550.26i			0.126178i
$73$		−	19156.3i		−	0.420730i	−0.977623	−	0.210365i	$-0.932535\pi$
$73$		−	19156.3i		−	0.420730i	0.977623	−	0.210365i	$-0.0674652\pi$
$74$	22150.1			0.470214
$75$	−21403.2	−	18246.0i	−0.439366	−	0.374555i
$76$	−60376.2			−1.19904
$77$			18253.1i			0.350840i
$78$		−	57825.8i		−	1.07618i
$79$	3713.15			0.0669383			0.0334692	−	0.999440i	$-0.489344\pi$
$79$	3713.15			0.0669383			0.0334692	+	0.999440i	$0.489344\pi$
$80$	62662.9	−	28923.0i	1.09468	−	0.505264i
$81$	6561.00			0.111111
$82$			33794.5i			0.555023i
$83$		−	51919.0i		−	0.827239i	−0.910450	−	0.413619i	$-0.864264\pi$
$83$		−	51919.0i		−	0.827239i	0.910450	−	0.413619i	$-0.135736\pi$
$84$	−30871.3			−0.477371
$85$	54301.2	+	117646.i	0.815195	+	1.76616i
$86$	7562.07			0.110254
$87$		−	7379.26i		−	0.104524i
$88$		−	8291.13i		−	0.114132i
$89$	52966.5			0.708803			0.354402	−	0.935093i	$-0.384685\pi$
$89$	52966.5			0.708803			0.354402	+	0.935093i	$0.384685\pi$
$90$	−14039.5	−	30417.2i	−0.182703	−	0.395833i
$91$	−131004.			−1.65836
$92$		−	57700.8i		−	0.710742i
$93$		−	65730.2i		−	0.788057i
$94$	181259.			2.11583
$95$	−134769.	+	62204.8i	−1.53208	+	0.707156i
$96$	62473.5			0.691859
$97$			114526.i			1.23588i	0.786225	+	0.617940i	$0.212032\pi$
$97$			114526.i			1.23588i	−0.786225	+	0.617940i	$0.787968\pi$
$98$			44015.8i			0.462960i
$99$	−9801.00			−0.100504
$100$	−46098.6	+	54075.2i	−0.460986	+	0.540752i
$101$	154646.			1.50846			0.754232	−	0.656608i	$-0.228009\pi$
$101$	154646.			1.50846			0.754232	+	0.656608i	$0.228009\pi$
$102$			154340.i			1.46885i
$103$			117955.i			1.09552i	0.836634	+	0.547762i	$0.184520\pi$
$103$			117955.i			1.09552i	−0.836634	+	0.547762i	$0.815480\pi$
$104$	59506.1			0.539483
$105$	−68909.6	+	31806.2i	−0.609967	+	0.281539i
$106$	−98540.6			−0.851826
$107$			233086.i			1.96815i	0.177763	+	0.984073i	$0.443114\pi$
$107$			233086.i			1.96815i	−0.177763	+	0.984073i	$0.556886\pi$
$108$		−	16576.4i		−	0.136751i
$109$	18774.3			0.151355			0.0756775	−	0.997132i	$-0.475888\pi$
$109$	18774.3			0.151355			0.0756775	+	0.997132i	$0.475888\pi$
$110$	20972.6	+	45438.0i	0.165261	+	0.358045i
$111$	26944.6			0.207570
$112$		−	186240.i		−	1.40291i
$113$		−	165056.i		−	1.21600i	−0.793936	−	0.608001i	$-0.791971\pi$
$113$		−	165056.i		−	1.21600i	0.793936	−	0.608001i	$-0.208029\pi$
$114$	−176804.			−1.27418
$115$	−59448.3	−	128797.i	−0.419175	−	0.908160i
$116$	−18643.7			−0.128643
$117$		−	70342.5i		−	0.475065i
$118$		−	220186.i		−	1.45574i
$119$	349655.			2.26346
$120$	31301.0	−	14447.4i	0.198429	−	0.0915877i
$121$	14641.0			0.0909091
$122$			127678.i			0.776632i
$123$			41109.4i			0.245007i
$124$	−166067.			−0.969907
$125$	−47186.4	+	168199.i	−0.270111	+	0.962829i
$126$	−90402.8			−0.507289
$127$			203866.i			1.12159i	0.827953	+	0.560797i	$0.189505\pi$
$127$			203866.i			1.12159i	−0.827953	+	0.560797i	$0.810495\pi$
$128$			134455.i			0.725357i
$129$	9198.92			0.0486701
$130$	−326112.	+	150522.i	−1.69242	+	0.781161i
$131$	82311.6			0.419067			0.209533	−	0.977802i	$-0.432806\pi$
$131$	82311.6			0.419067			0.209533	+	0.977802i	$0.432806\pi$
$132$			24762.2i			0.123696i
$133$			400548.i			1.96348i
$134$	137946.			0.663662
$135$	−17078.4	−	37001.1i	−0.0806516	−	0.174735i
$136$	−158825.			−0.736326
$137$			202042.i			0.919687i	0.888000	+	0.459844i	$0.152095\pi$
$137$			202042.i			0.919687i	−0.888000	+	0.459844i	$0.847905\pi$
$138$		−	168970.i		−	0.755286i
$139$	200312.			0.879367			0.439683	−	0.898153i	$-0.355091\pi$
$139$	200312.			0.879367			0.439683	+	0.898153i	$0.355091\pi$
$140$	80358.5	+	174100.i	0.346507	+	0.750722i
$141$	220494.			0.934003
$142$		−	353074.i		−	1.46942i
$143$			105080.i			0.429713i
$144$	100002.			0.401886
$145$	−41615.7	+	19208.3i	−0.164376	+	0.0758699i
$146$	−141728.			−0.550269
$147$			53543.2i			0.204367i
$148$		−	68075.4i		−	0.255468i
$149$	−80425.4			−0.296775			−0.148388	−	0.988929i	$-0.547408\pi$
$149$	−80425.4			−0.296775			−0.148388	+	0.988929i	$0.547408\pi$
$150$	−134994.	+	158353.i	−0.489876	+	0.574642i
$151$	395216.			1.41056			0.705280	−	0.708928i	$-0.250821\pi$
$151$	395216.			1.41056			0.705280	+	0.708928i	$0.250821\pi$
$152$		−	181942.i		−	0.638739i
$153$			187748.i			0.648404i
$154$	135046.			0.458860
$155$	−370689.	+	171097.i	−1.23931	+	0.572022i
$156$	−177720.			−0.584690
$157$		−	510056.i		−	1.65146i	−0.564063	−	0.825732i	$-0.690763\pi$
$157$		−	510056.i		−	1.65146i	0.564063	−	0.825732i	$-0.309237\pi$
$158$		−	27471.9i		−	0.0875480i
$159$	−119870.			−0.376027
$160$	−162620.	−	352322.i	−0.502196	−	1.08803i
$161$	−382799.			−1.16387
$162$		−	48541.9i		−	0.145321i
$163$		−	307324.i		−	0.905998i	−0.891511	−	0.452999i	$-0.850354\pi$
$163$		−	307324.i		−	0.905998i	0.891511	−	0.452999i	$-0.149646\pi$
$164$	103863.			0.301544
$165$	25512.2	+	55273.2i	0.0729521	+	0.158054i
$166$	−384125.			−1.08194
$167$			214345.i			0.594733i	0.954763	+	0.297367i	$0.0961084\pi$
$167$			214345.i			0.594733i	−0.954763	+	0.297367i	$0.903892\pi$
$168$		−	93029.6i		−	0.254301i
$169$	−382871.			−1.03118
$170$	870408.	−	401750.i	2.30994	−	1.06619i
$171$	−215075.			−0.562470
$172$		−	23241.1i		−	0.0599011i
$173$		−	366833.i		−	0.931865i	−0.884820	−	0.465933i	$-0.845719\pi$
$173$		−	366833.i		−	0.931865i	0.884820	−	0.465933i	$-0.154281\pi$
$174$	−54595.8			−0.136705
$175$	358746.	+	305827.i	0.885507	+	0.754885i
$176$	−149386.			−0.363519
$177$		−	267846.i		−	0.642616i
$178$		−	391875.i		−	0.927037i
$179$	−678162.			−1.58198			−0.790990	−	0.611829i	$-0.790434\pi$
$179$	−678162.			−1.58198			−0.790990	+	0.611829i	$0.790434\pi$
$180$	−93483.2	+	43148.6i	−0.215056	+	0.0992625i
$181$	−270856.			−0.614529			−0.307265	−	0.951624i	$-0.599414\pi$
$181$	−270856.			−0.614529			−0.307265	+	0.951624i	$0.599414\pi$
$182$			969236.i			2.16896i
$183$			155314.i			0.342834i
$184$	173879.			0.378620
$185$	−70137.2	−	151955.i	−0.150667	−	0.326427i
$186$	−486308.			−1.03069
$187$		−	280462.i		−	0.586504i
$188$		−	557077.i		−	1.14953i
$189$	−109971.			−0.223936
$190$	460225.	+	997098.i	0.924882	+	2.00380i
$191$	81645.4			0.161938			0.0809689	−	0.996717i	$-0.474199\pi$
$191$	81645.4			0.161938			0.0809689	+	0.996717i	$0.474199\pi$
$192$		−	106650.i		−	0.208789i
$193$			170728.i			0.329922i	0.986300	+	0.164961i	$0.0527498\pi$
$193$			170728.i			0.329922i	−0.986300	+	0.164961i	$0.947250\pi$
$194$	847329.			1.61640
$195$	−396700.	+	183103.i	−0.747095	+	0.344833i
$196$	135277.			0.251526
$197$		−	173263.i		−	0.318082i	−0.987272	−	0.159041i	$-0.949160\pi$
$197$		−	173263.i		−	0.318082i	0.987272	−	0.159041i	$-0.0508402\pi$
$198$			72513.2i			0.131448i
$199$	−42534.0			−0.0761384			−0.0380692	−	0.999275i	$-0.512121\pi$
$199$	−42534.0			−0.0761384			−0.0380692	+	0.999275i	$0.512121\pi$
$200$	−162954.	−	138917.i	−0.288065	−	0.245572i
$201$	167805.			0.292964
$202$		−	1.14416e6i		−	1.97291i
$203$			123686.i			0.210659i
$204$	474344.			0.798028
$205$	231839.	−	107009.i	0.385302	−	0.177842i
$206$	872693.			1.43283
$207$		−	205544.i		−	0.333410i
$208$		−	1.07215e6i		−	1.71830i
$209$	321284.			0.508773
$210$	235320.	+	509831.i	0.368223	+	0.797771i
$211$	−625694.			−0.967511			−0.483755	−	0.875203i	$-0.660728\pi$
$211$	−625694.			−0.967511			−0.483755	+	0.875203i	$0.660728\pi$
$212$			302852.i			0.462797i
$213$		−	429498.i		−	0.648653i
$214$	1.72450e6			2.57412
$215$	−23945.0	−	51877.8i	−0.0353279	−	0.0765394i
$216$	49952.3			0.0728486
$217$			1.10172e6i			1.58827i
$218$		−	138902.i		−	0.197956i
$219$	−172406.			−0.242909
$220$	139648.	−	64456.5i	0.194526	−	0.0897863i
$221$	2.01290e6			2.77231
$222$		−	199351.i		−	0.271478i
$223$			42469.9i			0.0571899i	0.999591	+	0.0285950i	$0.00910330\pi$
$223$			42469.9i			0.0571899i	−0.999591	+	0.0285950i	$0.990897\pi$
$224$	−1.04714e6			−1.39439
$225$	−164214.	+	192629.i	−0.216249	+	0.253668i
$226$	−1.22117e6			−1.59040
$227$			21815.5i			0.0280996i	0.999901	+	0.0140498i	$0.00447233\pi$
$227$			21815.5i			0.0280996i	−0.999901	+	0.0140498i	$0.995528\pi$
$228$			543386.i			0.692263i
$229$	−58861.1			−0.0741720			−0.0370860	−	0.999312i	$-0.511808\pi$
$229$	−58861.1			−0.0741720			−0.0370860	+	0.999312i	$0.511808\pi$
$230$	−952914.	+	439831.i	−1.18777	+	0.548235i
$231$	164278.			0.202557
$232$		−	56182.1i		−	0.0685296i
$233$		−	418005.i		−	0.504419i	−0.967673	−	0.252210i	$-0.918843\pi$
$233$		−	418005.i		−	0.504419i	0.967673	−	0.252210i	$-0.0811572\pi$
$234$	−520432.			−0.621333
$235$	−573949.	−	1.24349e6i	−0.677959	−	1.46883i
$236$	−676712.			−0.790904
$237$		−	33418.3i		−	0.0386468i
$238$		−	2.58694e6i		−	2.96035i
$239$	1.29252e6			1.46367			0.731834	−	0.681483i	$-0.238665\pi$
$239$	1.29252e6			1.46367			0.731834	+	0.681483i	$0.238665\pi$
$240$	−260307.	−	563967.i	−0.291714	−	0.632012i
$241$	496148.			0.550261			0.275131	−	0.961407i	$-0.411279\pi$
$241$	496148.			0.550261			0.275131	+	0.961407i	$0.411279\pi$
$242$		−	108322.i		−	0.118899i
$243$		−	59049.0i		−	0.0641500i
$244$	392401.			0.421945
$245$	301960.	−	139374.i	0.321391	−	0.148343i
$246$	304150.			0.320443
$247$			2.30588e6i			2.40489i
$248$		−	500439.i		−	0.516680i
$249$	−467271.			−0.477606
$250$	1.24443e6	+	349111.i	1.25928	+	0.353276i
$251$	−1.58588e6			−1.58886			−0.794431	−	0.607354i	$-0.792231\pi$
$251$	−1.58588e6			−1.58886			−0.794431	+	0.607354i	$0.792231\pi$
$252$			277841.i			0.275610i
$253$			307047.i			0.301581i
$254$	1.50831e6			1.46692
$255$	1.05881e6	−	488710.i	1.01969	−	0.470653i
$256$	1.37397e6			1.31032
$257$			1.41881e6i			1.33995i	0.742381	+	0.669977i	$0.233696\pi$
$257$			1.41881e6i			1.33995i	−0.742381	+	0.669977i	$0.766304\pi$
$258$		−	68058.6i		−	0.0636552i
$259$	−451626.			−0.418340
$260$	462609.	+	1.00226e6i	0.424405	+	0.919493i
$261$	−66413.3			−0.0603467
$262$		−	608987.i		−	0.548093i
$263$		−	1.62046e6i		−	1.44461i	−0.691575	−	0.722304i	$-0.743083\pi$
$263$		−	1.62046e6i		−	1.44461i	0.691575	−	0.722304i	$-0.256917\pi$
$264$	−74620.1			−0.0658941
$265$	312024.	+	676014.i	0.272944	+	0.591346i
$266$	2.96347e6			2.56801
$267$		−	476698.i		−	0.409228i
$268$		−	423959.i		−	0.360568i
$269$	−1.25821e6			−1.06017			−0.530083	−	0.847946i	$-0.677839\pi$
$269$	−1.25821e6			−1.06017			−0.530083	+	0.847946i	$0.677839\pi$
$270$	−273754.	+	126355.i	−0.228534	+	0.105483i
$271$	−2.33387e6			−1.93043			−0.965214	−	0.261462i	$-0.915796\pi$
$271$	−2.33387e6			−1.93043			−0.965214	+	0.261462i	$0.915796\pi$
$272$			2.86163e6i			2.34526i
$273$			1.17903e6i			0.957457i
$274$	1.49482e6			1.20285
$275$	245308.	−	287754.i	0.195605	−	0.229451i
$276$	−519307.			−0.410347
$277$		−	952978.i		−	0.746248i	−0.927781	−	0.373124i	$-0.878287\pi$
$277$		−	952978.i		−	0.746248i	0.927781	−	0.373124i	$-0.121713\pi$
$278$		−	1.48202e6i		−	1.15012i
$279$	−591572.			−0.454985
$280$	−524645.	+	242158.i	−0.399918	+	0.184588i
$281$	−194035.			−0.146593			−0.0732966	−	0.997310i	$-0.523352\pi$
$281$	−194035.			−0.146593			−0.0732966	+	0.997310i	$0.523352\pi$
$282$		−	1.63133e6i		−	1.22157i
$283$			589829.i			0.437784i	0.975749	+	0.218892i	$0.0702442\pi$
$283$			589829.i			0.437784i	−0.975749	+	0.218892i	$0.929756\pi$
$284$	−1.08513e6			−0.798334
$285$	559843.	+	1.21293e6i	0.408276	+	0.884549i
$286$	777436.			0.562017
$287$		−	689048.i		−	0.493793i
$288$		−	562261.i		−	0.399445i
$289$	−3.95267e6			−2.78385
$290$	142114.	+	307896.i	0.0992296	+	0.214985i
$291$	1.03074e6			0.713535
$292$			435584.i			0.298961i
$293$			2.18674e6i			1.48809i	0.668130	+	0.744045i	$0.267095\pi$
$293$			2.18674e6i			1.48809i	−0.668130	+	0.744045i	$0.732905\pi$
$294$	396142.			0.267290
$295$	−1.51053e6	+	697207.i	−1.01059	+	0.466452i
$296$	205143.			0.136090
$297$			88209.0i			0.0580259i
$298$			595031.i			0.388150i
$299$	−2.20370e6			−1.42553
$300$	486677.	+	414887.i	0.312203	+	0.266150i
$301$	−154186.			−0.0980908
$302$		−	2.92402e6i		−	1.84486i
$303$		−	1.39181e6i		−	0.870913i
$304$	−3.27814e6			−2.03444
$305$	875902.	−	404286.i	0.539146	−	0.248851i
$306$	1.38906e6			0.848042
$307$			2.25874e6i			1.36780i	0.729578	+	0.683898i	$0.239717\pi$
$307$			2.25874e6i			1.36780i	−0.729578	+	0.683898i	$0.760283\pi$
$308$		−	415047.i		−	0.249299i
$309$	1.06159e6			0.632501
$310$	1.26587e6	+	2.74256e6i	0.748143	+	1.62088i
$311$	−3.00436e6			−1.76137			−0.880687	−	0.473699i	$-0.842918\pi$
$311$	−3.00436e6			−1.76137			−0.880687	+	0.473699i	$0.842918\pi$
$312$		−	535554.i		−	0.311471i
$313$		−	2.76965e6i		−	1.59795i	−0.601363	−	0.798976i	$-0.705376\pi$
$313$		−	2.76965e6i		−	1.59795i	0.601363	−	0.798976i	$-0.294624\pi$
$314$	−3.77367e6			−2.15993
$315$	286256.	+	620187.i	0.162547	+	0.352165i
$316$	−84431.4			−0.0475649
$317$			2.00320e6i			1.11963i	0.828616	+	0.559817i	$0.189129\pi$
$317$			2.00320e6i			1.11963i	−0.828616	+	0.559817i	$0.810871\pi$
$318$			886866.i			0.491802i
$319$	99210.0			0.0545857
$320$	−601459.	+	277612.i	−0.328346	+	0.151553i
$321$	2.09778e6			1.13631
$322$			2.83215e6i			1.52222i
$323$		−	6.15451e6i		−	3.28237i
$324$	−149187.			−0.0789531
$325$	2.06524e6	+	1.76059e6i	1.08458	+	0.924592i
$326$	−2.27375e6			−1.18495
$327$		−	168968.i		−	0.0873849i
$328$			312988.i			0.160636i
$329$	−3.69576e6			−1.88241
$330$	408942.	−	188753.i	0.206717	−	0.0954133i
$331$	1.09643e6			0.550059			0.275029	−	0.961436i	$-0.411312\pi$
$331$	1.09643e6			0.550059			0.275029	+	0.961436i	$0.411312\pi$
$332$			1.18056e6i			0.587817i
$333$		−	242501.i		−	0.119840i
$334$	1.58584e6			0.777846
$335$	−436800.	−	946345.i	−0.212652	−	0.460721i
$336$	−1.67616e6			−0.809969
$337$			594378.i			0.285094i	0.989788	+	0.142547i	$0.0455292\pi$
$337$			594378.i			0.285094i	−0.989788	+	0.142547i	$0.954471\pi$
$338$			2.83269e6i			1.34868i
$339$	−1.48550e6			−0.702060
$340$	−1.23473e6	−	2.67509e6i	−0.579260	−	1.25499i
$341$	883706.			0.411549
$342$			1.59124e6i			0.735649i
$343$			1.63791e6i			0.751718i
$344$	70036.2			0.0319100
$345$	−1.15918e6	+	535035.i	−0.524327	+	0.242011i
$346$	−2.71403e6			−1.21878
$347$		−	822560.i		−	0.366728i	−0.983045	−	0.183364i	$-0.941301\pi$
$347$		−	822560.i		−	0.366728i	0.983045	−	0.183364i	$-0.0586987\pi$
$348$			167793.i			0.0742722i
$349$	−956928.			−0.420548			−0.210274	−	0.977642i	$-0.567436\pi$
$349$	−956928.			−0.420548			−0.210274	+	0.977642i	$0.567436\pi$
$350$	2.26268e6	−	2.65420e6i	0.987307	−	1.15815i
$351$	−633083.			−0.274279
$352$			839921.i			0.361312i
$353$		−	218804.i		−	0.0934583i	−0.998908	−	0.0467291i	$-0.985120\pi$
$353$		−	218804.i		−	0.0934583i	0.998908	−	0.0467291i	$-0.0148798\pi$
$354$	−1.98167e6			−0.840472
$355$	−2.42218e6	+	1.11799e6i	−1.02008	+	0.470834i
$356$	−1.20438e6			−0.503660
$357$		−	3.14689e6i		−	1.30681i
$358$			5.01741e6i			2.06906i
$359$	2.72858e6			1.11738			0.558689	−	0.829378i	$-0.311305\pi$
$359$	2.72858e6			1.11738			0.558689	+	0.829378i	$0.311305\pi$
$360$	−130027.	−	281709.i	−0.0528782	−	0.114563i
$361$	4.57422e6			1.84735
$362$			2.00394e6i			0.803737i
$363$		−	131769.i		−	0.0524864i
$364$	2.97882e6			1.17840
$365$	448777.	+	972294.i	0.176319	+	0.382002i
$366$	1.14910e6			0.448389
$367$		−	1.94374e6i		−	0.753307i	−0.926354	−	0.376654i	$-0.877075\pi$
$367$		−	1.94374e6i		−	0.753307i	0.926354	−	0.376654i	$-0.122925\pi$
$368$		−	3.13288e6i		−	1.20594i
$369$	369985.			0.141455
$370$	−1.12425e6	+	518913.i	−0.426931	+	0.197056i
$371$	2.00918e6			0.757852
$372$			1.49461e6i			0.559976i
$373$			4.88061e6i			1.81636i	0.418580	+	0.908180i	$0.362528\pi$
$373$			4.88061e6i			1.81636i	−0.418580	+	0.908180i	$0.637472\pi$
$374$	−2.07501e6			−0.767083
$375$	1.51379e6	+	424678.i	0.555890	+	0.155949i
$376$	1.67873e6			0.612368
$377$			712038.i			0.258018i
$378$			813625.i			0.292883i
$379$	4.08830e6			1.46199			0.730995	−	0.682383i	$-0.239056\pi$
$379$	4.08830e6			1.46199			0.730995	+	0.682383i	$0.239056\pi$
$380$	3.06445e6	−	1.41444e6i	1.08866	−	0.502489i
$381$	1.83479e6			0.647553
$382$		−	604057.i		−	0.211797i
$383$		−	3.04558e6i		−	1.06090i	−0.847717	−	0.530448i	$-0.822024\pi$
$383$		−	3.04558e6i		−	1.06090i	0.847717	−	0.530448i	$-0.177976\pi$
$384$	1.21009e6			0.418785
$385$	−427617.	−	926452.i	−0.147029	−	0.318545i
$386$	1.26314e6			0.431502
$387$		−	82790.3i		−	0.0280997i
$388$		−	2.60416e6i		−	0.878189i
$389$	−2.13646e6			−0.715847			−0.357923	−	0.933751i	$-0.616515\pi$
$389$	−2.13646e6			−0.715847			−0.357923	+	0.933751i	$0.616515\pi$
$390$	1.35469e6	+	2.93501e6i	0.451004	+	0.977119i
$391$	5.88179e6			1.94566
$392$			407653.i			0.133991i
$393$		−	740805.i		−	0.241948i
$394$	−1.28189e6			−0.416017
$395$	−188464.	+	86988.6i	−0.0607767	+	0.0280524i
$396$	222860.			0.0714158
$397$		−	2.27058e6i		−	0.723038i	−0.932365	−	0.361519i	$-0.882258\pi$
$397$		−	2.27058e6i		−	0.723038i	0.932365	−	0.361519i	$-0.117742\pi$
$398$			314690.i			0.0995807i
$399$	3.60493e6			1.13361
$400$	−2.50293e6	+	2.93603e6i	−0.782167	+	0.917509i
$401$	981140.			0.304698			0.152349	−	0.988327i	$-0.451316\pi$
$401$	981140.			0.304698			0.152349	+	0.988327i	$0.451316\pi$
$402$		−	1.24151e6i		−	0.383165i
$403$			6.34243e6i			1.94533i
$404$	−3.51642e6			−1.07188
$405$	−333010.	+	153706.i	−0.100883	+	0.0465642i
$406$	915096.			0.275519
$407$			362255.i			0.108400i
$408$			1.42942e6i			0.425118i
$409$	1.98079e6			0.585505			0.292752	−	0.956188i	$-0.405429\pi$
$409$	1.98079e6			0.585505			0.292752	+	0.956188i	$0.405429\pi$
$410$	−791708.	−	1.71527e6i	−0.232598	−	0.503933i
$411$	1.81838e6			0.530982
$412$		−	2.68211e6i		−	0.778456i
$413$			4.48945e6i			1.29514i
$414$	−1.52073e6			−0.436064
$415$	1.21631e6	+	2.63520e6i	0.346677	+	0.751092i
$416$	−6.02818e6			−1.70786
$417$		−	1.80281e6i		−	0.507703i
$418$		−	2.37704e6i		−	0.665419i
$419$	−153079.			−0.0425972			−0.0212986	−	0.999773i	$-0.506780\pi$
$419$	−153079.			−0.0425972			−0.0212986	+	0.999773i	$0.506780\pi$
$420$	1.56690e6	−	723226.i	0.433429	−	0.200056i
$421$	848256.			0.233250			0.116625	−	0.993176i	$-0.462792\pi$
$421$	848256.			0.233250			0.116625	+	0.993176i	$0.462792\pi$
$422$			4.62923e6i			1.26540i
$423$		−	1.98444e6i		−	0.539247i
$424$	−912635.			−0.246537
$425$	−5.51222e6	−	4.69911e6i	−1.48031	−	1.26195i
$426$	−3.17766e6			−0.848367
$427$		−	2.60327e6i		−	0.690954i
$428$		−	5.30003e6i		−	1.39852i
$429$	945716.			0.248095
$430$	−383820.	+	177158.i	−0.100105	+	0.0462050i
$431$	−492966.			−0.127827			−0.0639137	−	0.997955i	$-0.520358\pi$
$431$	−492966.			−0.127827			−0.0639137	+	0.997955i	$0.520358\pi$
$432$		−	900018.i		−	0.232029i
$433$		−	5.12774e6i		−	1.31434i	−0.753744	−	0.657168i	$-0.771754\pi$
$433$		−	5.12774e6i		−	1.31434i	0.753744	−	0.657168i	$-0.228246\pi$
$434$	8.15116e6			2.07728
$435$	172875.	+	374541.i	0.0438035	+	0.0949023i
$436$	−426899.			−0.107550
$437$			6.73790e6i			1.68780i
$438$			1.27556e6i			0.317698i
$439$	3.98100e6			0.985895			0.492947	−	0.870059i	$-0.335919\pi$
$439$	3.98100e6			0.985895			0.492947	+	0.870059i	$0.335919\pi$
$440$	194238.	+	420824.i	0.0478301	+	0.103626i
$441$	481889.			0.117991
$442$		−	1.48925e7i		−	3.62588i
$443$			3.09112e6i			0.748353i	0.927357	+	0.374177i	$0.122075\pi$
$443$			3.09112e6i			0.748353i	−0.927357	+	0.374177i	$0.877925\pi$
$444$	−612679.			−0.147494
$445$	−2.68836e6	+	1.24085e6i	−0.643558	+	0.297044i
$446$	314216.			0.0747981
$447$			723829.i			0.171343i
$448$			1.78760e6i			0.420799i
$449$	1.51168e6			0.353870			0.176935	−	0.984223i	$-0.443382\pi$
$449$	1.51168e6			0.353870			0.176935	+	0.984223i	$0.443382\pi$
$450$	1.42517e6	+	1.21495e6i	0.331770	+	0.282830i
$451$	−552694.			−0.127951
$452$			3.75312e6i			0.864065i
$453$		−	3.55694e6i		−	0.814388i
$454$	161403.			0.0367512
$455$	6.64921e6	−	3.06904e6i	1.50571	−	0.694983i
$456$	−1.63748e6			−0.368776
$457$		−	5.58801e6i		−	1.25160i	−0.779982	−	0.625802i	$-0.784772\pi$
$457$		−	5.58801e6i		−	1.25160i	0.779982	−	0.625802i	$-0.215228\pi$
$458$			435487.i			0.0970088i
$459$	1.68973e6			0.374356
$460$	1.35177e6	+	2.92866e6i	0.297856	+	0.645319i
$461$	−819486.			−0.179593			−0.0897964	−	0.995960i	$-0.528622\pi$
$461$	−819486.			−0.179593			−0.0897964	+	0.995960i	$0.528622\pi$
$462$		−	1.21542e6i		−	0.264923i
$463$			987740.i			0.214136i	0.994252	+	0.107068i	$0.0341463\pi$
$463$			987740.i			0.214136i	−0.994252	+	0.107068i	$0.965854\pi$
$464$	−1.01226e6			−0.218272
$465$	1.53987e6	+	3.33620e6i	0.330257	+	0.715517i
$466$	−3.09263e6			−0.659725
$467$		−	3.26318e6i		−	0.692387i	−0.938163	−	0.346193i	$-0.887474\pi$
$467$		−	3.26318e6i		−	0.692387i	0.938163	−	0.346193i	$-0.112526\pi$
$468$			1.59948e6i			0.337571i
$469$	−2.81263e6			−0.590447
$470$	−9.19999e6	+	4.24639e6i	−1.92107	+	0.886697i
$471$	−4.59051e6			−0.953473
$472$		−	2.03925e6i		−	0.421323i
$473$			123674.i			0.0254172i
$474$	−247247.			−0.0505459
$475$	5.38307e6	−	6.31453e6i	1.09470	−	1.28412i
$476$	−7.95063e6			−1.60836
$477$			1.07883e6i			0.217099i
$478$		−	9.56277e6i		−	1.91432i
$479$	−5.61817e6			−1.11881			−0.559405	−	0.828895i	$-0.688970\pi$
$479$	−5.61817e6			−1.11881			−0.559405	+	0.828895i	$0.688970\pi$
$480$	−3.17090e6	+	1.46358e6i	−0.628174	+	0.289943i
$481$	−2.59993e6			−0.512388
$482$		−	3.67078e6i		−	0.719681i
$483$			3.44519e6i			0.671962i
$484$	−332914.			−0.0645980
$485$	−2.68303e6	−	5.81290e6i	−0.517930	−	1.12212i
$486$	−436877.			−0.0839012
$487$		−	4.31761e6i		−	0.824938i	−0.910972	−	0.412469i	$-0.864666\pi$
$487$		−	4.31761e6i		−	0.824938i	0.910972	−	0.412469i	$-0.135334\pi$
$488$			1.18249e6i			0.224775i
$489$	−2.76591e6			−0.523078
$490$	−1.03116e6	−	2.23406e6i	−0.194016	−	0.420344i
$491$	8.77622e6			1.64287			0.821436	−	0.570301i	$-0.193173\pi$
$491$	8.77622e6			1.64287			0.821436	+	0.570301i	$0.193173\pi$
$492$		−	934767.i		−	0.174097i
$493$		−	1.90046e6i		−	0.352162i
$494$	1.70602e7			3.14533
$495$	497459.	−	229610.i	0.0912524	−	0.0421189i
$496$	−9.01667e6			−1.64567
$497$			7.19895e6i			1.30731i
$498$			3.45712e6i			0.624657i
$499$	9.08110e6			1.63263			0.816313	−	0.577609i	$-0.196014\pi$
$499$	9.08110e6			1.63263			0.816313	+	0.577609i	$0.196014\pi$
$500$	1.07295e6	−	3.82460e6i	0.191935	−	0.684165i
$501$	1.92911e6			0.343370
$502$			1.17332e7i			2.07806i
$503$		−	4.35992e6i		−	0.768350i	−0.923260	−	0.384175i	$-0.874486\pi$
$503$		−	4.35992e6i		−	0.768350i	0.923260	−	0.384175i	$-0.125514\pi$
$504$	−837266.			−0.146821
$505$	−7.84921e6	+	3.62292e6i	−1.36961	+	0.632164i
$506$	2.27170e6			0.394435
$507$			3.44584e6i			0.595354i
$508$		−	4.63561e6i		−	0.796980i
$509$	−5.30215e6			−0.907105			−0.453552	−	0.891230i	$-0.649844\pi$
$509$	−5.30215e6			−0.907105			−0.453552	+	0.891230i	$0.649844\pi$
$510$	−3.61575e6	−	7.83367e6i	−0.615563	−	1.33364i
$511$	2.88975e6			0.489563
$512$		−	5.86284e6i		−	0.988400i
$513$			1.93567e6i			0.324742i
$514$	1.04971e7			1.75251
$515$	−2.76334e6	−	5.98690e6i	−0.459110	−	0.994682i
$516$	−209170.			−0.0345839
$517$			2.96442e6i			0.487767i
$518$			3.34138e6i			0.547143i
$519$	−3.30150e6			−0.538013
$520$	−3.02029e6	+	1.39406e6i	−0.489824	+	0.226085i
$521$	1.80284e6			0.290981			0.145490	−	0.989360i	$-0.453524\pi$
$521$	1.80284e6			0.290981			0.145490	+	0.989360i	$0.453524\pi$
$522$			491362.i			0.0789269i
$523$		−	1.48512e6i		−	0.237414i	−0.992929	−	0.118707i	$-0.962125\pi$
$523$		−	1.48512e6i		−	0.237414i	0.992929	−	0.118707i	$-0.0378750\pi$
$524$	−1.87164e6			−0.297779
$525$	2.75244e6	−	3.22871e6i	0.435833	−	0.511248i
$526$	−1.19891e7			−1.88939
$527$		−	1.69282e7i		−	2.65513i
$528$			1.34447e6i			0.209878i
$529$	−2976.47			−0.000462448
$530$	5.00152e6	−	2.30853e6i	0.773415	−	0.356981i
$531$	−2.41061e6			−0.371015
$532$		−	9.10786e6i		−	1.39520i
$533$		−	3.96672e6i		−	0.604803i
$534$	−3.52687e6			−0.535225
$535$	−5.46055e6	−	1.18305e7i	−0.824806	−	1.78698i
$536$	1.27759e6			0.192078
$537$			6.10346e6i			0.913356i
$538$			9.30896e6i			1.38658i
$539$	−719859.			−0.106727
$540$	388337.	+	841349.i	0.0573092	+	0.124163i
$541$	−2.18504e6			−0.320972			−0.160486	−	0.987038i	$-0.551306\pi$
$541$	−2.18504e6			−0.320972			−0.160486	+	0.987038i	$0.551306\pi$
$542$			1.72673e7i			2.52479i
$543$			2.43771e6i			0.354799i
$544$	1.60895e7			2.33102
$545$	−952906.	+	439828.i	−0.137423	+	0.0634295i
$546$	8.72313e6			1.25225
$547$			6.00853e6i			0.858618i	0.903158	+	0.429309i	$0.141243\pi$
$547$			6.00853e6i			0.858618i	−0.903158	+	0.429309i	$0.858757\pi$
$548$		−	4.59413e6i		−	0.653509i
$549$	1.39783e6			0.197935
$550$	−2.12897e6	−	1.81492e6i	−0.300097	−	0.255830i
$551$	2.17708e6			0.305489
$552$		−	1.56491e6i		−	0.218596i
$553$			560135.i			0.0778897i
$554$	−7.05065e6			−0.976011
$555$	−1.36760e6	+	631235.i	−0.188463	+	0.0869878i
$556$	−4.55479e6			−0.624858
$557$			8.40252e6i			1.14755i	0.819013	+	0.573775i	$0.194522\pi$
$557$			8.40252e6i			1.14755i	−0.819013	+	0.573775i	$0.805478\pi$
$558$			4.37677e6i			0.595071i
$559$	−887620.			−0.120143
$560$	4.36309e6	+	9.45281e6i	0.587927	+	1.27377i
$561$	−2.52416e6			−0.338618
$562$			1.43558e6i			0.191728i
$563$			1.68040e6i			0.223430i	0.993740	+	0.111715i	$0.0356343\pi$
$563$			1.68040e6i			0.223430i	−0.993740	+	0.111715i	$0.964366\pi$
$564$	−5.01369e6			−0.663682
$565$	3.86679e6	+	8.37757e6i	0.509600	+	1.10407i
$566$	4.36387e6			0.572573
$567$			989738.i			0.129289i
$568$		−	3.27000e6i		−	0.425281i
$569$	−1.43427e7			−1.85716			−0.928580	−	0.371131i	$-0.878970\pi$
$569$	−1.43427e7			−1.85716			−0.928580	+	0.371131i	$0.878970\pi$
$570$	8.97388e6	−	4.14203e6i	1.15689	−	0.533981i
$571$	9.83718e6			1.26264			0.631321	−	0.775521i	$-0.282513\pi$
$571$	9.83718e6			1.26264			0.631321	+	0.775521i	$0.282513\pi$
$572$		−	2.38935e6i		−	0.305344i
$573$		−	734809.i		−	0.0934949i
$574$	−5.09795e6			−0.645827
$575$	6.03471e6	+	5.14453e6i	0.761180	+	0.648898i
$576$	−959852.			−0.120545
$577$			4.88454e6i			0.610779i	0.952228	+	0.305390i	$0.0987867\pi$
$577$			4.88454e6i			0.610779i	−0.952228	+	0.305390i	$0.901213\pi$
$578$			2.92440e7i			3.64097i
$579$	1.53655e6			0.190481
$580$	946278.	−	436769.i	0.116802	−	0.0539115i
$581$	7.83207e6			0.962578
$582$		−	7.62596e6i		−	0.933226i
$583$		−	1.61159e6i		−	0.196374i
$584$	−1.31262e6			−0.159260
$585$	1.64793e6	+	3.57030e6i	0.199089	+	0.431336i
$586$	1.61787e7			1.94626
$587$			3.75376e6i			0.449646i	0.974400	+	0.224823i	$0.0721804\pi$
$587$			3.75376e6i			0.449646i	−0.974400	+	0.224823i	$0.927820\pi$
$588$		−	1.21749e6i		−	0.145219i
$589$	1.93922e7			2.30324
$590$	5.15832e6	+	1.11757e7i	0.610068	+	1.32174i
$591$	−1.55936e6			−0.183645
$592$		−	3.69617e6i		−	0.433459i
$593$		−	7.63599e6i		−	0.891720i	−0.895103	−	0.445860i	$-0.852898\pi$
$593$		−	7.63599e6i		−	0.891720i	0.895103	−	0.445860i	$-0.147102\pi$
$594$	652618.			0.0758915
$595$	−1.77471e7	+	8.19142e6i	−2.05511	+	0.948565i
$596$	1.82875e6			0.210882
$597$			382806.i			0.0439585i
$598$			1.63042e7i			1.86443i
$599$	1.06753e6			0.121567			0.0607834	−	0.998151i	$-0.480640\pi$
$599$	1.06753e6			0.121567			0.0607834	+	0.998151i	$0.480640\pi$
$600$	−1.25025e6	+	1.46659e6i	−0.141781	+	0.166314i
$601$	−4.83197e6			−0.545680			−0.272840	−	0.962059i	$-0.587963\pi$
$601$	−4.83197e6			−0.545680			−0.272840	+	0.962059i	$0.587963\pi$
$602$			1.14075e6i			0.128292i
$603$		−	1.51025e6i		−	0.169143i
$604$	−8.98661e6			−1.00231
$605$	−743118.	+	342997.i	−0.0825409	+	0.0380980i
$606$	−1.02974e7			−1.13906
$607$		−	8.95712e6i		−	0.986726i	−0.869824	−	0.493363i	$-0.835767\pi$
$607$		−	8.95712e6i		−	0.986726i	0.869824	−	0.493363i	$-0.164233\pi$
$608$			1.84314e7i			2.02208i
$609$	1.11317e6			0.121624
$610$	−2.99113e6	−	6.48040e6i	−0.325469	−	0.705144i
$611$	−2.12758e7			−2.30560
$612$		−	4.26910e6i		−	0.460742i
$613$			8.42569e6i			0.905637i	0.891603	+	0.452818i	$0.149581\pi$
$613$			8.42569e6i			0.905637i	−0.891603	+	0.452818i	$0.850419\pi$
$614$	1.67114e7			1.78893
$615$	−963077.	−	2.08655e6i	−0.102677	−	0.222454i
$616$	1.25073e6			0.132804
$617$		−	4.84242e6i		−	0.512094i	−0.966664	−	0.256047i	$-0.917580\pi$
$617$		−	4.84242e6i		−	0.512094i	0.966664	−	0.256047i	$-0.0824202\pi$
$618$		−	7.85424e6i		−	0.827243i
$619$	4.22402e6			0.443097			0.221549	−	0.975149i	$-0.428889\pi$
$619$	4.22402e6			0.443097			0.221549	+	0.975149i	$0.428889\pi$
$620$	8.42891e6	−	3.89049e6i	0.880627	−	0.406466i
$621$	−1.84990e6			−0.192495
$622$			2.22279e7i			2.30368i
$623$			7.99008e6i			0.824767i
$624$	−9.64938e6			−0.992059
$625$	−1.54544e6	−	9.64256e6i	−0.158253	−	0.987399i
$626$	−2.04914e7			−2.08995
$627$		−	2.89156e6i		−	0.293740i
$628$			1.15979e7i			1.17349i
$629$	6.93934e6			0.699345
$630$	4.58848e6	−	2.11788e6i	0.460593	−	0.212594i
$631$	1.18298e7			1.18278			0.591392	−	0.806384i	$-0.298579\pi$
$631$	1.18298e7			1.18278			0.591392	+	0.806384i	$0.298579\pi$
$632$		−	254431.i		−	0.0253383i
$633$			5.63125e6i			0.558593i
$634$	1.48208e7			1.46436
$635$	−4.77600e6	−	1.03474e7i	−0.470035	−	1.01835i
$636$	2.72567e6			0.267196
$637$		−	5.16648e6i		−	0.504483i
$638$		−	734010.i		−	0.0713921i
$639$	−3.86549e6			−0.374500
$640$	−3.14990e6	−	6.82439e6i	−0.303981	−	0.658588i
$641$	−2.19467e6			−0.210972			−0.105486	−	0.994421i	$-0.533640\pi$
$641$	−2.19467e6			−0.210972			−0.105486	+	0.994421i	$0.533640\pi$
$642$		−	1.55205e7i		−	1.48617i
$643$		−	467202.i		−	0.0445633i	−0.999752	−	0.0222816i	$-0.992907\pi$
$643$		−	467202.i		−	0.0445633i	0.999752	−	0.0222816i	$-0.00709305\pi$
$644$	8.70426e6			0.827022
$645$	−466900.	+	215505.i	−0.0441901	+	0.0203966i
$646$	−4.55345e7			−4.29298
$647$			944645.i			0.0887172i	0.999016	+	0.0443586i	$0.0141244\pi$
$647$			944645.i			0.0887172i	−0.999016	+	0.0443586i	$0.985876\pi$
$648$		−	449571.i		−	0.0420592i
$649$	3.60104e6			0.335595
$650$	1.30258e7	−	1.52797e7i	1.20927	−	1.41851i
$651$	9.91552e6			0.916986
$652$			6.98808e6i			0.643782i
$653$		−	917543.i		−	0.0842061i	−0.999113	−	0.0421030i	$-0.986594\pi$
$653$		−	917543.i		−	0.0842061i	0.999113	−	0.0421030i	$-0.0134058\pi$
$654$	−1.25012e6			−0.114290
$655$	−4.17781e6	+	1.92833e6i	−0.380492	+	0.175622i
$656$	5.63927e6			0.511638
$657$			1.55166e6i			0.140243i
$658$			2.73433e7i			2.46199i
$659$	5.17061e6			0.463798			0.231899	−	0.972740i	$-0.425506\pi$
$659$	5.17061e6			0.463798			0.231899	+	0.972740i	$0.425506\pi$
$660$	−580108.	−	1.25683e6i	−0.0518381	−	0.112310i
$661$	−1.78425e6			−0.158837			−0.0794185	−	0.996841i	$-0.525306\pi$
$661$	−1.78425e6			−0.158837			−0.0794185	+	0.996841i	$0.525306\pi$
$662$		−	8.11195e6i		−	0.719417i
$663$		−	1.81161e7i		−	1.60059i
$664$	−3.55758e6			−0.313137
$665$	−9.38370e6	−	2.03302e7i	−0.822849	−	1.78274i
$666$	−1.79416e6			−0.156738
$667$			2.08061e6i			0.181082i
$668$		−	4.87388e6i		−	0.422604i
$669$	382229.			0.0330186
$670$	−7.00158e6	+	3.23168e6i	−0.602572	+	0.278126i
$671$	−2.08811e6			−0.179039
$672$			9.42423e6i			0.805050i
$673$		−	9.10656e6i		−	0.775026i	−0.921864	−	0.387513i	$-0.873334\pi$
$673$		−	9.10656e6i		−	0.775026i	0.921864	−	0.387513i	$-0.126666\pi$
$674$	4.39753e6			0.372872
$675$	1.73366e6	+	1.47793e6i	0.146455	+	0.124852i
$676$	8.70591e6			0.732736
$677$			657297.i			0.0551175i	0.999620	+	0.0275588i	$0.00877334\pi$
$677$			657297.i			0.0551175i	−0.999620	+	0.0275588i	$0.991227\pi$
$678$			1.09906e7i			0.918217i
$679$	−1.72765e7			−1.43807
$680$	8.06129e6	−	3.72081e6i	0.668548	−	0.308578i
$681$	196339.			0.0162233
$682$		−	6.53814e6i		−	0.538262i
$683$		−	1.92866e7i		−	1.58199i	−0.611821	−	0.790997i	$-0.709563\pi$
$683$		−	1.92866e7i		−	1.58199i	0.611821	−	0.790997i	$-0.290437\pi$
$684$	4.89047e6			0.399678
$685$	−4.73327e6	−	1.02548e7i	−0.385420	−	0.835030i
$686$	1.21182e7			0.983165
$687$			529750.i			0.0428232i
$688$		−	1.26188e6i		−	0.101636i
$689$	1.15665e7			0.928226
$690$	3.95848e6	+	8.57622e6i	0.316523	+	0.685762i
$691$	7.09653e6			0.565394			0.282697	−	0.959209i	$-0.408771\pi$
$691$	7.09653e6			0.565394			0.282697	+	0.959209i	$0.408771\pi$
$692$			8.34123e6i			0.662163i
$693$		−	1.47850e6i		−	0.116947i
$694$	−6.08575e6			−0.479640
$695$	−1.01670e7	+	4.69274e6i	−0.798421	+	0.368523i
$696$	−505639.			−0.0395656
$697$			1.05874e7i			0.825480i
$698$			7.07987e6i			0.550031i
$699$	−3.76204e6			−0.291227
$700$	−8.15734e6	−	6.95405e6i	−0.629222	−	0.536405i
$701$	1.57310e7			1.20910			0.604550	−	0.796567i	$-0.293353\pi$
$701$	1.57310e7			1.20910			0.604550	+	0.796567i	$0.293353\pi$
$702$			4.68389e6i			0.358727i
$703$			7.94938e6i			0.606659i
$704$	1.43385e6			0.109037
$705$	−1.11914e7	+	5.16554e6i	−0.848029	+	0.391420i
$706$	−1.61883e6			−0.122233
$707$			2.33286e7i			1.75526i
$708$			6.09041e6i			0.456629i
$709$	−2.46071e7			−1.83842			−0.919210	−	0.393767i	$-0.871172\pi$
$709$	−2.46071e7			−1.83842			−0.919210	+	0.393767i	$0.871172\pi$
$710$	8.27152e6	+	1.79206e7i	0.615799	+	1.33416i
$711$	−300765.			−0.0223128
$712$		−	3.62935e6i		−	0.268305i
$713$			1.85329e7i			1.36527i
$714$	−2.32824e7			−1.70916
$715$	−2.46172e6	−	5.33342e6i	−0.180083	−	0.390158i
$716$	1.54204e7			1.12412
$717$		−	1.16327e7i		−	0.845049i
$718$		−	2.01875e7i		−	1.46141i
$719$	8.76856e6			0.632566			0.316283	−	0.948665i	$-0.397565\pi$
$719$	8.76856e6			0.632566			0.316283	+	0.948665i	$0.397565\pi$
$720$	−5.07570e6	+	2.34276e6i	−0.364892	+	0.168421i
$721$	−1.77937e7			−1.27476
$722$		−	3.38426e7i		−	2.41613i
$723$		−	4.46533e6i		−	0.317693i
$724$	6.15887e6			0.436671
$725$	1.66225e6	−	1.94988e6i	0.117449	−	0.137772i
$726$	−974899.			−0.0686465
$727$		−	1.62173e7i		−	1.13800i	−0.822338	−	0.568999i	$-0.807331\pi$
$727$		−	1.62173e7i		−	1.13800i	0.822338	−	0.568999i	$-0.192669\pi$
$728$			8.97659e6i			0.627745i
$729$	−531441.			−0.0370370
$730$	7.19356e6	−	3.32029e6i	0.499617	−	0.230605i
$731$	2.36910e6			0.163980
$732$		−	3.53161e6i		−	0.243610i
$733$			1.12288e7i			0.771919i	0.922516	+	0.385960i	$0.126130\pi$
$733$			1.12288e7i			0.771919i	−0.922516	+	0.385960i	$0.873870\pi$
$734$	−1.43808e7			−0.985243
$735$	−1.25437e6	−	2.71764e6i	−0.0856457	−	0.185555i
$736$	−1.76146e7			−1.19861
$737$			2.25605e6i			0.152996i
$738$		−	2.73735e6i		−	0.185008i
$739$	−8.22290e6			−0.553878			−0.276939	−	0.960888i	$-0.589320\pi$
$739$	−8.22290e6			−0.553878			−0.276939	+	0.960888i	$0.589320\pi$
$740$	1.59481e6	+	3.45523e6i	0.107061	+	0.231952i
$741$	2.07529e7			1.38846
$742$		−	1.48650e7i		−	0.991188i
$743$			2.24984e6i			0.149513i	0.997202	+	0.0747566i	$0.0238180\pi$
$743$			2.24984e6i			0.149513i	−0.997202	+	0.0747566i	$0.976182\pi$
$744$	−4.50395e6			−0.298305
$745$	4.08207e6	−	1.88414e6i	0.269457	−	0.124372i
$746$	3.61094e7			2.37560
$747$			4.20544e6i			0.275746i
$748$			6.37729e6i			0.416757i
$749$	−3.51615e7			−2.29014
$750$	3.14200e6	−	1.11999e7i	0.203964	−	0.727043i
$751$	−171752.			−0.0111122			−0.00555612	−	0.999985i	$-0.501769\pi$
$751$	−171752.			−0.0111122			−0.00555612	+	0.999985i	$0.501769\pi$
$752$		−	3.02467e7i		−	1.95044i
$753$			1.42729e7i			0.917330i
$754$	5.26804e6			0.337459
$755$	−2.00596e7	+	9.25878e6i	−1.28072	+	0.591135i
$756$	2.50057e6			0.159124
$757$			2.62239e7i			1.66325i	0.555339	+	0.831624i	$0.312589\pi$
$757$			2.62239e7i			1.66325i	−0.555339	+	0.831624i	$0.687411\pi$
$758$		−	3.02475e7i		−	1.91212i
$759$	2.76343e6			0.174118
$760$	4.26238e6	+	9.23463e6i	0.267681	+	0.579944i
$761$	−2.29774e7			−1.43827			−0.719134	−	0.694871i	$-0.755461\pi$
$761$	−2.29774e7			−1.43827			−0.719134	+	0.694871i	$0.755461\pi$
$762$		−	1.35748e7i		−	0.846928i
$763$			2.83213e6i			0.176117i
$764$	−1.85649e6			−0.115069
$765$	−4.39839e6	−	9.52931e6i	−0.271732	−	0.588719i
$766$	−2.25329e7			−1.38754
$767$			2.58449e7i			1.58631i
$768$		−	1.23657e7i		−	0.756515i
$769$	2.94748e6			0.179736			0.0898680	−	0.995954i	$-0.471355\pi$
$769$	2.94748e6			0.179736			0.0898680	+	0.995954i	$0.471355\pi$
$770$	−6.85440e6	+	3.16375e6i	−0.416622	+	0.192298i
$771$	1.27693e7			0.773623
$772$		−	3.88209e6i		−	0.234435i
$773$			1.27928e7i			0.770049i	0.922906	+	0.385024i	$0.125807\pi$
$773$			1.27928e7i			0.770049i	−0.922906	+	0.385024i	$0.874193\pi$
$774$	−612528.			−0.0367514
$775$	1.48064e7	−	1.73684e7i	0.885511	−	1.03874i
$776$	7.84754e6			0.467821
$777$			4.06463e6i			0.241529i
$778$			1.58067e7i			0.936249i
$779$	−1.21284e7			−0.716077
$780$	9.02037e6	−	4.16348e6i	0.530869	−	0.245031i
$781$	5.77437e6			0.338748
$782$		−	4.35167e7i		−	2.54471i
$783$			597720.i			0.0348412i
$784$	7.34490e6			0.426772
$785$	1.19492e7	+	2.58884e7i	0.692092	+	1.49945i
$786$	−5.48088e6			−0.316442
$787$		−	1.94779e6i		−	0.112100i	−0.998428	−	0.0560499i	$-0.982149\pi$
$787$		−	1.94779e6i		−	0.112100i	0.998428	−	0.0560499i	$-0.0178506\pi$
$788$			3.93973e6i			0.226022i
$789$	−1.45842e7			−0.834045
$790$	643589.	+	1.39436e6i	0.0366894	+	0.0794892i
$791$	2.48989e7			1.41495
$792$			671581.i			0.0380440i
$793$		−	1.49865e7i		−	0.846289i
$794$	−1.67990e7			−0.945655
$795$	6.08413e6	−	2.80822e6i	0.341414	−	0.157584i
$796$	967160.			0.0541023
$797$		−	2.72445e7i		−	1.51926i	−0.650355	−	0.759630i	$-0.725380\pi$
$797$		−	2.72445e7i		−	1.51926i	0.650355	−	0.759630i	$-0.274620\pi$
$798$		−	2.66713e7i		−	1.48264i
$799$	5.67862e7			3.14685
$800$	1.65078e7	+	1.40727e7i	0.911937	+	0.777417i
$801$	−4.29028e6			−0.236268
$802$		−	7.25901e6i		−	0.398512i
$803$		−	2.31791e6i		−	0.126855i
$804$	−3.81563e6			−0.208174
$805$	1.94293e7	−	8.96788e6i	1.05674	−	0.487753i
$806$	4.69247e7			2.54428
$807$			1.13239e7i			0.612087i
$808$		−	1.05966e7i		−	0.571003i
$809$	−1.56700e7			−0.841780			−0.420890	−	0.907112i	$-0.638282\pi$
$809$	−1.56700e7			−0.841780			−0.420890	+	0.907112i	$0.638282\pi$
$810$	1.13720e6	+	2.46379e6i	0.0609009	+	0.131944i
$811$	1.41672e7			0.756365			0.378182	−	0.925731i	$-0.376549\pi$
$811$	1.41672e7			0.756365			0.378182	+	0.925731i	$0.376549\pi$
$812$		−	2.81243e6i		−	0.149690i
$813$			2.10048e7i			1.11453i
$814$	2.68016e6			0.141775
$815$	7.19972e6	+	1.55985e7i	0.379683	+	0.822601i
$816$	2.57547e7			1.35404
$817$			2.71393e6i			0.142247i
$818$		−	1.46550e7i		−	0.765776i
$819$	1.06113e7			0.552788
$820$	−5.27166e6	+	2.43321e6i	−0.273787	+	0.126370i
$821$	6.51336e6			0.337246			0.168623	−	0.985681i	$-0.446068\pi$
$821$	6.51336e6			0.337246			0.168623	+	0.985681i	$0.446068\pi$
$822$		−	1.34533e7i		−	0.694466i
$823$		−	2.92352e7i		−	1.50455i	−0.658849	−	0.752275i	$-0.728956\pi$
$823$		−	2.92352e7i		−	1.50455i	0.658849	−	0.752275i	$-0.271044\pi$
$824$	8.08246e6			0.414692
$825$	−2.58979e6	−	2.20777e6i	−0.132474	−	0.112932i
$826$	3.32154e7			1.69390
$827$		−	2.56735e7i		−	1.30533i	−0.757646	−	0.652666i	$-0.773651\pi$
$827$		−	2.56735e7i		−	1.30533i	0.757646	−	0.652666i	$-0.226349\pi$
$828$			4.67376e6i			0.236914i
$829$	9.02846e6			0.456276			0.228138	−	0.973629i	$-0.426736\pi$
$829$	9.02846e6			0.456276			0.228138	+	0.973629i	$0.426736\pi$
$830$	1.94966e7	−	8.99895e6i	0.982346	−	0.453416i
$831$	−8.57680e6			−0.430847
$832$			1.02909e7i			0.515399i
$833$			1.37896e7i			0.688555i
$834$	−1.33382e7			−0.664019
$835$	−5.02150e6	−	1.08793e7i	−0.249240	−	0.539988i
$836$	−7.30552e6			−0.361523
$837$			5.32415e6i			0.262686i
$838$			1.13256e6i			0.0557124i
$839$	−3.00826e7			−1.47540			−0.737701	−	0.675128i	$-0.764088\pi$
$839$	−3.00826e7			−1.47540			−0.737701	+	0.675128i	$0.764088\pi$
$840$	2.17942e6	+	4.72181e6i	0.106572	+	0.230893i
$841$	−1.98389e7			−0.967224
$842$		−	6.27586e6i		−	0.305065i
$843$			1.74631e6i			0.0846356i
$844$	1.42273e7			0.687492
$845$	1.94330e7	−	8.96958e6i	0.936263	−	0.432146i
$846$	−1.46820e7			−0.705276
$847$			2.20862e6i			0.105782i
$848$			1.64434e7i			0.785241i
$849$	5.30846e6			0.252755
$850$	−3.47666e7	+	4.07824e7i	−1.65050	+	1.93609i
$851$	−7.59711e6			−0.359604
$852$			9.76614e6i			0.460919i
$853$		−	2.48619e7i		−	1.16994i	−0.811057	−	0.584968i	$-0.801107\pi$
$853$		−	2.48619e7i		−	1.16994i	0.811057	−	0.584968i	$-0.198893\pi$
$854$	−1.92604e7			−0.903692
$855$	1.09163e7	−	5.03859e6i	0.510695	−	0.235719i
$856$	1.59715e7			0.745008
$857$			8.27535e6i			0.384888i	0.981308	+	0.192444i	$0.0616414\pi$
$857$			8.27535e6i			0.384888i	−0.981308	+	0.192444i	$0.938359\pi$
$858$		−	6.99693e6i		−	0.324481i
$859$	−1.89682e7			−0.877088			−0.438544	−	0.898710i	$-0.644506\pi$
$859$	−1.89682e7			−0.877088			−0.438544	+	0.898710i	$0.644506\pi$
$860$	544472.	+	1.17962e6i	0.0251032	+	0.0543872i
$861$	−6.20143e6			−0.285091
$862$			3.64723e6i			0.167184i
$863$		−	1.70264e7i		−	0.778209i	−0.921194	−	0.389104i	$-0.872785\pi$
$863$		−	1.70264e7i		−	0.778209i	0.921194	−	0.389104i	$-0.127215\pi$
$864$	−5.06035e6			−0.230620
$865$	8.59385e6	+	1.86190e7i	0.390524	+	0.846087i
$866$	−3.79378e7			−1.71901
$867$			3.55741e7i			1.60726i
$868$		−	2.50515e7i		−	1.12859i
$869$	449291.			0.0201827
$870$	2.77106e6	−	1.27902e6i	0.124122	−	0.0572902i
$871$	−1.61918e7			−0.723186
$872$		−	1.28645e6i		−	0.0572928i
$873$		−	9.27664e6i		−	0.411960i
$874$	4.98506e7			2.20746
$875$	−2.53732e7	−	7.11816e6i	−1.12035	−	0.314302i
$876$	3.92026e6			0.172605
$877$			2.69746e7i			1.18428i	0.805834	+	0.592142i	$0.201717\pi$
$877$			2.69746e7i			1.18428i	−0.805834	+	0.592142i	$0.798283\pi$
$878$		−	2.94536e7i		−	1.28944i
$879$	1.96807e7			0.859149
$880$	7.58222e6	−	3.49968e6i	0.330057	−	0.152343i
$881$	2.13740e7			0.927781			0.463890	−	0.885893i	$-0.346453\pi$
$881$	2.13740e7			0.927781			0.463890	+	0.885893i	$0.346453\pi$
$882$		−	3.56528e6i		−	0.154320i
$883$			3.26501e6i			0.140923i	0.997514	+	0.0704616i	$0.0224472\pi$
$883$			3.26501e6i			0.140923i	−0.997514	+	0.0704616i	$0.977553\pi$
$884$	−4.57703e7			−1.96994
$885$	6.27487e6	+	1.35948e7i	0.269306	+	0.583464i
$886$	2.28698e7			0.978764
$887$		−	6.68667e6i		−	0.285365i	−0.989769	−	0.142682i	$-0.954427\pi$
$887$		−	6.68667e6i		−	0.285365i	0.989769	−	0.142682i	$-0.0455728\pi$
$888$		−	1.84629e6i		−	0.0785718i
$889$	−3.07536e7			−1.30509
$890$	9.18051e6	+	1.98900e7i	0.388501	+	0.841704i
$891$	793881.			0.0335013
$892$		−	965702.i		−	0.0406379i
$893$			6.50516e7i			2.72979i
$894$	5.35528e6			0.224098
$895$	3.44208e7	−	1.58874e7i	1.43636	−	0.662972i
$896$	−2.02828e7			−0.844029
$897$			1.98333e7i			0.823027i
$898$		−	1.11842e7i		−	0.462824i
$899$	5.98815e6			0.247112
$900$	3.73398e6	−	4.38009e6i	0.153662	−	0.180251i
$901$	−3.08715e7			−1.26691
$902$			4.08913e6i			0.167346i
$903$			1.38767e6i			0.0566328i
$904$	−1.13099e7			−0.460297
$905$	1.37476e7	−	6.34540e6i	0.557962	−	0.257536i
$906$	−2.63162e7			−1.06513
$907$			2.82586e7i			1.14060i	0.821437	+	0.570299i	$0.193173\pi$
$907$			2.82586e7i			1.14060i	−0.821437	+	0.570299i	$0.806827\pi$
$908$		−	496050.i		−	0.0199669i
$909$	−1.25263e7			−0.502822
$910$	−2.27065e7	−	4.91945e7i	−0.908962	−	1.96931i
$911$	−9.75713e6			−0.389517			−0.194758	−	0.980851i	$-0.562392\pi$
$911$	−9.75713e6			−0.389517			−0.194758	+	0.980851i	$0.562392\pi$
$912$			2.95033e7i			1.17458i
$913$		−	6.28219e6i		−	0.249422i
$914$	−4.13432e7			−1.63696
$915$	−3.63857e6	−	7.88312e6i	−0.143674	−	0.311276i
$916$	1.33841e6			0.0527050
$917$			1.24169e7i			0.487628i
$918$		−	1.25015e7i		−	0.489617i
$919$	3.69755e7			1.44419			0.722097	−	0.691792i	$-0.243179\pi$
$919$	3.69755e7			1.44419			0.722097	+	0.691792i	$0.243179\pi$
$920$	−8.82542e6	+	4.07350e6i	−0.343768	+	0.158671i
$921$	2.03287e7			0.789697
$922$			6.06300e6i			0.234888i
$923$			4.14431e7i			1.60121i
$924$	−3.73542e6			−0.143933
$925$	7.11976e6	+	6.06952e6i	0.273597	+	0.233238i
$926$	7.30784e6			0.280067
$927$		−	9.55433e6i		−	0.365175i
$928$			5.69145e6i			0.216947i
$929$	5.05419e7			1.92137			0.960687	−	0.277633i	$-0.0895500\pi$
$929$	5.05419e7			1.92137			0.960687	+	0.277633i	$0.0895500\pi$
$930$	2.46830e7	−	1.13928e7i	0.935818	−	0.431940i
$931$	−1.57967e7			−0.597299
$932$			9.50480e6i			0.358429i
$933$			2.70393e7i			1.01693i
$934$	−2.41428e7			−0.905566
$935$	6.57044e6	+	1.42351e7i	0.245791	+	0.532516i
$936$	−4.81999e6			−0.179828
$937$			1.08317e7i			0.403040i	0.979484	+	0.201520i	$0.0645880\pi$
$937$			1.08317e7i			0.403040i	−0.979484	+	0.201520i	$0.935412\pi$
$938$			2.08094e7i			0.772240i
$939$	−2.49268e7			−0.922578
$940$	1.30507e7	+	2.82750e7i	0.481743	+	1.04372i
$941$	1.45659e6			0.0536246			0.0268123	−	0.999640i	$-0.491464\pi$
$941$	1.45659e6			0.0536246			0.0268123	+	0.999640i	$0.491464\pi$
$942$			3.39631e7i			1.24704i
$943$		−	1.15909e7i		−	0.424463i
$944$	−3.67423e7			−1.34195
$945$	5.58168e6	−	2.57631e6i	0.203322	−	0.0938465i
$946$	915011.			0.0332429
$947$		−	4.32433e7i		−	1.56691i	−0.621449	−	0.783455i	$-0.713456\pi$
$947$		−	4.32433e7i		−	1.56691i	0.621449	−	0.783455i	$-0.286544\pi$
$948$			759883.i			0.0274616i
$949$	1.66358e7			0.599623
$950$	−4.67184e7	−	3.98269e7i	−1.67949	−	1.43175i
$951$	1.80288e7			0.646421
$952$		−	2.39590e7i		−	0.856792i
$953$		−	1.69154e7i		−	0.603322i	−0.953415	−	0.301661i	$-0.902459\pi$
$953$		−	1.69154e7i		−	0.603322i	0.953415	−	0.301661i	$-0.0975410\pi$
$954$	7.98179e6			0.283942
$955$	−4.14399e6	+	1.91272e6i	−0.147032	+	0.0678646i
$956$	−2.93900e7			−1.04005
$957$		−	892890.i		−	0.0315151i
$958$			4.15663e7i			1.46328i
$959$	−3.04784e7			−1.07015
$960$	2.49851e6	+	5.41313e6i	0.0874990	+	0.189570i
$961$	2.47099e7			0.863102
$962$			1.92357e7i			0.670147i
$963$		−	1.88800e7i		−	0.656049i
$964$	−1.12817e7			−0.391003
$965$	−3.99967e6	−	8.66546e6i	−0.138263	−	0.299553i
$966$	2.54894e7			0.878853
$967$			2.51066e7i			0.863421i	0.902012	+	0.431710i	$0.142090\pi$
$967$			2.51066e7i			0.863421i	−0.902012	+	0.431710i	$0.857910\pi$
$968$		−	1.00323e6i		−	0.0344121i
$969$	−5.53906e7			−1.89508
$970$	−4.30070e7	+	1.98505e7i	−1.46761	+	0.677395i
$971$	−3.10806e7			−1.05789			−0.528947	−	0.848655i	$-0.677413\pi$
$971$	−3.10806e7			−1.05789			−0.528947	+	0.848655i	$0.677413\pi$
$972$			1.34269e6i			0.0455836i
$973$			3.02174e7i			1.02323i
$974$	−3.19441e7			−1.07893
$975$	1.58453e7	−	1.85871e7i	0.533814	−	0.626182i
$976$	2.13055e7			0.715925
$977$		−	1.72964e7i		−	0.579722i	−0.957069	−	0.289861i	$-0.906391\pi$
$977$		−	1.72964e7i		−	0.579722i	0.957069	−	0.289861i	$-0.0936090\pi$
$978$			2.04637e7i			0.684129i
$979$	6.40894e6			0.213712
$980$	−6.86611e6	+	3.16915e6i	−0.228373	+	0.105409i
$981$	−1.52072e6			−0.0504517
$982$		−	6.49313e7i		−	2.14870i
$983$		−	2.02617e7i		−	0.668793i	−0.942433	−	0.334396i	$-0.891468\pi$
$983$		−	2.02617e7i		−	0.668793i	0.942433	−	0.334396i	$-0.108532\pi$
$984$	2.81689e6			0.0927432
$985$	4.05905e6	+	8.79411e6i	0.133301	+	0.288803i
$986$	−1.40607e7			−0.460589
$987$			3.32619e7i			1.08681i
$988$		−	5.24323e7i		−	1.70886i
$989$	−2.59367e6			−0.0843186
$990$	−1.69878e6	−	3.68048e6i	−0.0550869	−	0.119348i
$991$	1.64795e7			0.533041			0.266520	−	0.963829i	$-0.414126\pi$
$991$	1.64795e7			0.533041			0.266520	+	0.963829i	$0.414126\pi$
$992$			5.06962e7i			1.63567i
$993$		−	9.86783e6i		−	0.317577i
$994$	5.32618e7			1.70982
$995$	2.15886e6	−	996452.i	0.0691299	−	0.0319079i
$996$	1.06250e7			0.339377
$997$			364567.i			0.0116155i	0.999983	+	0.00580777i	$0.00184868\pi$
$997$			364567.i			0.0116155i	−0.999983	+	0.00580777i	$0.998151\pi$
$998$		−	6.71869e7i		−	2.13530i
$999$	−2.18251e6			−0.0691899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.6.c.b.34.6	✓	26
5.2	odd	4		825.6.a.v.1.11		13
5.3	odd	4		825.6.a.y.1.3		13
5.4	even	2	inner	165.6.c.b.34.21	yes	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.6	✓	26	1.1	even	1	trivial
165.6.c.b.34.21	yes	26	5.4	even	2	inner
825.6.a.v.1.11		13	5.2	odd	4
825.6.a.y.1.3		13	5.3	odd	4

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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	165.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.39855i q^{2} -9.00000i q^{3} -22.7385 q^{4} +(-50.7560 + 23.4272i) q^{5} -66.5869 q^{6} +150.852i q^{7} -68.5217i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q-7.39855i q^{2} -9.00000i q^{3} -22.7385 q^{4} +(-50.7560 + 23.4272i) q^{5} -66.5869 q^{6} +150.852i q^{7} -68.5217i q^{8} -81.0000 q^{9} +(173.327 + 375.520i) q^{10} +121.000 q^{11} +204.646i q^{12} +868.426i q^{13} +1116.08 q^{14} +(210.844 + 456.804i) q^{15} -1234.59 q^{16} -2317.87i q^{17} +599.282i q^{18} +2655.24 q^{19} +(1154.11 - 532.698i) q^{20} +1357.67 q^{21} -895.224i q^{22} +2537.58i q^{23} -616.695 q^{24} +(2027.34 - 2378.14i) q^{25} +6425.09 q^{26} +729.000i q^{27} -3430.14i q^{28} +819.917 q^{29} +(3379.68 - 1559.94i) q^{30} +7303.36 q^{31} +6941.50i q^{32} -1089.00i q^{33} -17148.9 q^{34} +(-3534.03 - 7656.63i) q^{35} +1841.82 q^{36} +2993.84i q^{37} -19644.9i q^{38} +7815.84 q^{39} +(1605.27 + 3477.89i) q^{40} -4567.72 q^{41} -10044.8i q^{42} +1022.10i q^{43} -2751.36 q^{44} +(4111.23 - 1897.60i) q^{45} +18774.4 q^{46} +24499.3i q^{47} +11111.3i q^{48} -5949.25 q^{49} +(-17594.8 - 14999.3i) q^{50} -20860.8 q^{51} -19746.7i q^{52} -13318.9i q^{53} +5393.54 q^{54} +(-6141.47 + 2834.69i) q^{55} +10336.6 q^{56} -23897.2i q^{57} -6066.20i q^{58} +29760.6 q^{59} +(-4794.28 - 10387.0i) q^{60} -17257.1 q^{61} -54034.2i q^{62} -12219.0i q^{63} +11850.0 q^{64} +(-20344.8 - 44077.8i) q^{65} -8057.02 q^{66} +18645.0i q^{67} +52704.9i q^{68} +22838.2 q^{69} +(-56647.9 + 26146.7i) q^{70} +47722.0 q^{71} +5550.26i q^{72} -19156.3i q^{73} +22150.1 q^{74} +(-21403.2 - 18246.0i) q^{75} -60376.2 q^{76} +18253.1i q^{77} -57825.8i q^{78} +3713.15 q^{79} +(62662.9 - 28923.0i) q^{80} +6561.00 q^{81} +33794.5i q^{82} -51919.0i q^{83} -30871.3 q^{84} +(54301.2 + 117646. i) q^{85} +7562.07 q^{86} -7379.26i q^{87} -8291.13i q^{88} +52966.5 q^{89} +(-14039.5 - 30417.2i) q^{90} -131004. q^{91} -57700.8i q^{92} -65730.2i q^{93} +181259. q^{94} +(-134769. + 62204.8i) q^{95} +62473.5 q^{96} +114526. i q^{97} +44015.8i q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 418 q^{4} - 98 q^{5} + 234 q^{6} - 2106 q^{9}+O(q^{10}) \) 26 * q - 418 * q^4 - 98 * q^5 + 234 * q^6 - 2106 * q^9	\( 26 q - 418 q^{4} - 98 q^{5} + 234 q^{6} - 2106 q^{9} + 236 q^{10} + 3146 q^{11} + 1220 q^{14} + 7002 q^{16} - 540 q^{19} + 4930 q^{20} + 5472 q^{21} - 7182 q^{24} + 218 q^{25} + 5304 q^{26} - 23904 q^{29} + 12114 q^{30} + 38192 q^{31} + 2604 q^{34} - 11988 q^{35} + 33858 q^{36} - 17748 q^{39} - 41096 q^{40} + 70368 q^{41} - 50578 q^{44} + 7938 q^{45} - 8240 q^{46} - 29114 q^{49} - 133876 q^{50} + 26568 q^{51} - 18954 q^{54} - 11858 q^{55} + 119604 q^{56} + 18384 q^{59} - 14148 q^{60} + 10876 q^{61} - 213114 q^{64} - 117068 q^{65} + 28314 q^{66} - 163512 q^{69} - 58660 q^{70} + 203400 q^{71} - 27352 q^{74} - 35352 q^{75} + 279932 q^{76} - 187908 q^{79} - 256654 q^{80} + 170586 q^{81} - 196560 q^{84} + 37396 q^{85} + 741860 q^{86} + 36836 q^{89} - 19116 q^{90} + 349072 q^{91} - 129040 q^{94} - 208284 q^{95} + 209898 q^{96} - 254826 q^{99}+O(q^{100}) \) 26 * q - 418 * q^4 - 98 * q^5 + 234 * q^6 - 2106 * q^9 + 236 * q^10 + 3146 * q^11 + 1220 * q^14 + 7002 * q^16 - 540 * q^19 + 4930 * q^20 + 5472 * q^21 - 7182 * q^24 + 218 * q^25 + 5304 * q^26 - 23904 * q^29 + 12114 * q^30 + 38192 * q^31 + 2604 * q^34 - 11988 * q^35 + 33858 * q^36 - 17748 * q^39 - 41096 * q^40 + 70368 * q^41 - 50578 * q^44 + 7938 * q^45 - 8240 * q^46 - 29114 * q^49 - 133876 * q^50 + 26568 * q^51 - 18954 * q^54 - 11858 * q^55 + 119604 * q^56 + 18384 * q^59 - 14148 * q^60 + 10876 * q^61 - 213114 * q^64 - 117068 * q^65 + 28314 * q^66 - 163512 * q^69 - 58660 * q^70 + 203400 * q^71 - 27352 * q^74 - 35352 * q^75 + 279932 * q^76 - 187908 * q^79 - 256654 * q^80 + 170586 * q^81 - 196560 * q^84 + 37396 * q^85 + 741860 * q^86 + 36836 * q^89 - 19116 * q^90 + 349072 * q^91 - 129040 * q^94 - 208284 * q^95 + 209898 * q^96 - 254826 * q^99

Introduction

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