LMFDB - Embedded newform 138.5.b.a.91.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,5,Mod(91,138)]

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, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("138.91");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	138.b (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:	$14.2650549056$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1428 x^{14} - 600 x^{13} + 788282 x^{12} - 529464 x^{11} + 213396724 x^{10} + \cdots + 274129967370817 $ x^16 + 1428x^14 - 600x^13 + 788282x^12 - 529464x^11 + 213396724x^10 - 175079484x^9 + 29130946113x^8 - 26553654912x^7 + 1790717926120x^6 - 1672125302460x^5 + 32715955056348x^4 - 11255739266544x^3 + 140972261417692x^2 - 70961983286748x + 274129967370817
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.13
Root			$0.707107 - 14.6559i$ of defining polynomial
Character	$\chi$	$=$	138.91
Dual form			138.5.b.a.91.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -33.4782i q^{5} +14.6969 q^{6} -19.6774i q^{7} +22.6274 q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -33.4782i q^{5} +14.6969 q^{6} -19.6774i q^{7} +22.6274 q^{8} +27.0000 q^{9} -94.6906i q^{10} -54.0138i q^{11} +41.5692 q^{12} -159.682 q^{13} -55.6561i q^{14} -173.958i q^{15} +64.0000 q^{16} -77.2301i q^{17} +76.3675 q^{18} -379.657i q^{19} -267.826i q^{20} -102.247i q^{21} -152.774i q^{22} +(142.258 + 509.513i) q^{23} +117.576 q^{24} -495.789 q^{25} -451.648 q^{26} +140.296 q^{27} -157.419i q^{28} +1624.58 q^{29} -492.027i q^{30} +651.991 q^{31} +181.019 q^{32} -280.664i q^{33} -218.440i q^{34} -658.763 q^{35} +216.000 q^{36} +974.907i q^{37} -1073.83i q^{38} -829.730 q^{39} -757.525i q^{40} -171.454 q^{41} -289.197i q^{42} -531.633i q^{43} -432.110i q^{44} -903.911i q^{45} +(402.367 + 1441.12i) q^{46} -2572.53 q^{47} +332.554 q^{48} +2013.80 q^{49} -1402.30 q^{50} -401.299i q^{51} -1277.45 q^{52} +2079.78i q^{53} +396.817 q^{54} -1808.28 q^{55} -445.248i q^{56} -1972.76i q^{57} +4595.01 q^{58} +8.57641 q^{59} -1391.66i q^{60} +1060.41i q^{61} +1844.11 q^{62} -531.289i q^{63} +512.000 q^{64} +5345.85i q^{65} -793.838i q^{66} +4746.67i q^{67} -617.840i q^{68} +(739.195 + 2647.51i) q^{69} -1863.26 q^{70} +1386.31 q^{71} +610.940 q^{72} -2617.74 q^{73} +2757.45i q^{74} -2576.20 q^{75} -3037.26i q^{76} -1062.85 q^{77} -2346.83 q^{78} -3921.65i q^{79} -2142.60i q^{80} +729.000 q^{81} -484.944 q^{82} +10557.1i q^{83} -817.974i q^{84} -2585.52 q^{85} -1503.68i q^{86} +8441.58 q^{87} -1222.19i q^{88} +11318.8i q^{89} -2556.65i q^{90} +3142.12i q^{91} +(1138.07 + 4076.10i) q^{92} +3387.85 q^{93} -7276.22 q^{94} -12710.2 q^{95} +940.604 q^{96} -2774.76i q^{97} +5695.89 q^{98} -1458.37i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) $ 16 * q + 128 * q^4 + 432 * q^9	$ 16 q + 128 q^{4} + 432 q^{9} - 208 q^{13} + 1024 q^{16} + 840 q^{23} + 1056 q^{25} + 1920 q^{26} + 3600 q^{29} + 224 q^{31} - 3264 q^{35} + 3456 q^{36} - 2016 q^{39} - 6144 q^{41} + 1280 q^{46} + 8880 q^{47} - 13888 q^{49} + 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} - 18240 q^{59} + 8192 q^{64} + 10584 q^{69} + 19584 q^{70} - 30048 q^{71} + 9536 q^{73} - 4176 q^{75} + 14160 q^{77} + 6912 q^{78} + 11664 q^{81} - 19584 q^{82} - 32496 q^{85} - 8064 q^{87} + 6720 q^{92} - 11952 q^{93} - 21248 q^{94} - 20064 q^{95} + 21504 q^{98}+O(q^{100}) $ 16 * q + 128 * q^4 + 432 * q^9 - 208 * q^13 + 1024 * q^16 + 840 * q^23 + 1056 * q^25 + 1920 * q^26 + 3600 * q^29 + 224 * q^31 - 3264 * q^35 + 3456 * q^36 - 2016 * q^39 - 6144 * q^41 + 1280 * q^46 + 8880 * q^47 - 13888 * q^49 + 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 - 18240 * q^59 + 8192 * q^64 + 10584 * q^69 + 19584 * q^70 - 30048 * q^71 + 9536 * q^73 - 4176 * q^75 + 14160 * q^77 + 6912 * q^78 + 11664 * q^81 - 19584 * q^82 - 32496 * q^85 - 8064 * q^87 + 6720 * q^92 - 11952 * q^93 - 21248 * q^94 - 20064 * q^95 + 21504 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.82843			0.707107
$2$	2.82843			0.707107
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	8.00000			0.500000
$5$		−	33.4782i		−	1.33913i	−0.742755	−	0.669564i	$-0.766481\pi$
$5$		−	33.4782i		−	1.33913i	0.742755	−	0.669564i	$-0.233519\pi$
$6$	14.6969			0.408248
$7$		−	19.6774i		−	0.401579i	−0.979634	−	0.200790i	$-0.935649\pi$
$7$		−	19.6774i		−	0.401579i	0.979634	−	0.200790i	$-0.0643508\pi$
$8$	22.6274			0.353553
$9$	27.0000			0.333333
$10$		−	94.6906i		−	0.946906i
$11$		−	54.0138i		−	0.446395i	−0.974773	−	0.223198i	$-0.928350\pi$
$11$		−	54.0138i		−	0.446395i	0.974773	−	0.223198i	$-0.0716495\pi$
$12$	41.5692			0.288675
$13$	−159.682			−0.944862			−0.472431	−	0.881368i	$-0.656623\pi$
$13$	−159.682			−0.944862			−0.472431	+	0.881368i	$0.656623\pi$
$14$		−	55.6561i		−	0.283959i
$15$		−	173.958i		−	0.773146i
$16$	64.0000			0.250000
$17$		−	77.2301i		−	0.267232i	−0.991033	−	0.133616i	$-0.957341\pi$
$17$		−	77.2301i		−	0.267232i	0.991033	−	0.133616i	$-0.0426589\pi$
$18$	76.3675			0.235702
$19$		−	379.657i		−	1.05168i	−0.850583	−	0.525841i	$-0.823751\pi$
$19$		−	379.657i		−	1.05168i	0.850583	−	0.525841i	$-0.176249\pi$
$20$		−	267.826i		−	0.669564i
$21$		−	102.247i		−	0.231852i
$22$		−	152.774i		−	0.315649i
$23$	142.258	+	509.513i	0.268919	+	0.963163i
$23$	142.258	+	509.513i	0.268919	+	0.963163i
$24$	117.576			0.204124
$25$	−495.789			−0.793263
$26$	−451.648			−0.668118
$27$	140.296			0.192450
$28$		−	157.419i		−	0.200790i
$29$	1624.58			1.93173			0.965863	−	0.259052i	$-0.0834100\pi$
$29$	1624.58			1.93173			0.965863	+	0.259052i	$0.0834100\pi$
$30$		−	492.027i		−	0.546697i
$31$	651.991			0.678451			0.339225	−	0.940705i	$-0.389835\pi$
$31$	651.991			0.678451			0.339225	+	0.940705i	$0.389835\pi$
$32$	181.019			0.176777
$33$		−	280.664i		−	0.257726i
$34$		−	218.440i		−	0.188962i
$35$	−658.763			−0.537766
$36$	216.000			0.166667
$37$			974.907i			0.712131i	0.934461	+	0.356065i	$0.115882\pi$
$37$			974.907i			0.712131i	−0.934461	+	0.356065i	$0.884118\pi$
$38$		−	1073.83i		−	0.743652i
$39$	−829.730			−0.545516
$40$		−	757.525i		−	0.473453i
$41$	−171.454			−0.101995			−0.0509975	−	0.998699i	$-0.516240\pi$
$41$	−171.454			−0.101995			−0.0509975	+	0.998699i	$0.516240\pi$
$42$		−	289.197i		−	0.163944i
$43$		−	531.633i		−	0.287524i	−0.989612	−	0.143762i	$-0.954080\pi$
$43$		−	531.633i		−	0.287524i	0.989612	−	0.143762i	$-0.0459201\pi$
$44$		−	432.110i		−	0.223198i
$45$		−	903.911i		−	0.446376i
$46$	402.367	+	1441.12i	0.190155	+	0.681059i
$47$	−2572.53			−1.16457			−0.582284	−	0.812985i	$-0.697841\pi$
$47$	−2572.53			−1.16457			−0.582284	+	0.812985i	$0.697841\pi$
$48$	332.554			0.144338
$49$	2013.80			0.838734
$50$	−1402.30			−0.560921
$51$		−	401.299i		−	0.154286i
$52$	−1277.45			−0.472431
$53$			2079.78i			0.740397i	0.928953	+	0.370199i	$0.120710\pi$
$53$			2079.78i			0.740397i	−0.928953	+	0.370199i	$0.879290\pi$
$54$	396.817			0.136083
$55$	−1808.28			−0.597780
$56$		−	445.248i		−	0.141980i
$57$		−	1972.76i		−	0.607189i
$58$	4595.01			1.36594
$59$	8.57641			0.00246378			0.00123189	−	0.999999i	$-0.499608\pi$
$59$	8.57641			0.00246378			0.00123189	+	0.999999i	$0.499608\pi$
$60$		−	1391.66i		−	0.386573i
$61$			1060.41i			0.284981i	0.989796	+	0.142490i	$0.0455110\pi$
$61$			1060.41i			0.284981i	−0.989796	+	0.142490i	$0.954489\pi$
$62$	1844.11			0.479737
$63$		−	531.289i		−	0.133860i
$64$	512.000			0.125000
$65$			5345.85i			1.26529i
$66$		−	793.838i		−	0.182240i
$67$			4746.67i			1.05740i	0.848809	+	0.528700i	$0.177320\pi$
$67$			4746.67i			1.05740i	−0.848809	+	0.528700i	$0.822680\pi$
$68$		−	617.840i		−	0.133616i
$69$	739.195	+	2647.51i	0.155261	+	0.556082i
$70$	−1863.26			−0.380258
$71$	1386.31			0.275007			0.137503	−	0.990501i	$-0.456092\pi$
$71$	1386.31			0.275007			0.137503	+	0.990501i	$0.456092\pi$
$72$	610.940			0.117851
$73$	−2617.74			−0.491225			−0.245613	−	0.969368i	$-0.578989\pi$
$73$	−2617.74			−0.491225			−0.245613	+	0.969368i	$0.578989\pi$
$74$			2757.45i			0.503553i
$75$	−2576.20			−0.457990
$76$		−	3037.26i		−	0.525841i
$77$	−1062.85			−0.179263
$78$	−2346.83			−0.385738
$79$		−	3921.65i		−	0.628368i	−0.949362	−	0.314184i	$-0.898269\pi$
$79$		−	3921.65i		−	0.628368i	0.949362	−	0.314184i	$-0.101731\pi$
$80$		−	2142.60i		−	0.334782i
$81$	729.000			0.111111
$82$	−484.944			−0.0721214
$83$			10557.1i			1.53245i	0.642571	+	0.766226i	$0.277868\pi$
$83$			10557.1i			1.53245i	−0.642571	+	0.766226i	$0.722132\pi$
$84$		−	817.974i		−	0.115926i
$85$	−2585.52			−0.357858
$86$		−	1503.68i		−	0.203311i
$87$	8441.58			1.11528
$88$		−	1222.19i		−	0.157825i
$89$			11318.8i			1.42896i	0.699655	+	0.714481i	$0.253337\pi$
$89$			11318.8i			1.42896i	−0.699655	+	0.714481i	$0.746663\pi$
$90$		−	2556.65i		−	0.315635i
$91$			3142.12i			0.379437i
$92$	1138.07	+	4076.10i	0.134460	+	0.481581i
$93$	3387.85			0.391704
$94$	−7276.22			−0.823474
$95$	−12710.2			−1.40834
$96$	940.604			0.102062
$97$		−	2774.76i		−	0.294905i	−0.989069	−	0.147453i	$-0.952893\pi$
$97$		−	2774.76i		−	0.294905i	0.989069	−	0.147453i	$-0.0471074\pi$
$98$	5695.89			0.593075
$99$		−	1458.37i		−	0.148798i
$100$	−3966.31			−0.396631
$101$	−3202.26			−0.313916			−0.156958	−	0.987605i	$-0.550169\pi$
$101$	−3202.26			−0.313916			−0.156958	+	0.987605i	$0.550169\pi$
$102$		−	1135.05i		−	0.109097i
$103$		−	14013.2i		−	1.32087i	−0.750881	−	0.660437i	$-0.770371\pi$
$103$		−	14013.2i		−	1.32087i	0.750881	−	0.660437i	$-0.229629\pi$
$104$	−3613.18			−0.334059
$105$	−3423.03			−0.310479
$106$			5882.49i			0.523540i
$107$			19115.3i			1.66961i	0.550548	+	0.834804i	$0.314419\pi$
$107$			19115.3i			1.66961i	−0.550548	+	0.834804i	$0.685581\pi$
$108$	1122.37			0.0962250
$109$		−	11486.1i		−	0.966764i	−0.875409	−	0.483382i	$-0.839408\pi$
$109$		−	11486.1i		−	0.966764i	0.875409	−	0.483382i	$-0.160592\pi$
$110$	−5114.60			−0.422694
$111$			5065.77i			0.411149i
$112$		−	1259.35i		−	0.100395i
$113$			16746.6i			1.31150i	0.754977	+	0.655751i	$0.227648\pi$
$113$			16746.6i			1.31150i	−0.754977	+	0.655751i	$0.772352\pi$
$114$		−	5579.80i		−	0.429348i
$115$	17057.6	−	4762.55i	1.28980	−	0.360117i
$116$	12996.7			0.965863
$117$	−4311.40			−0.314954
$118$	24.2578			0.00174215
$119$	−1519.69			−0.107315
$120$		−	3936.22i		−	0.273348i
$121$	11723.5			0.800731
$122$			2999.30i			0.201512i
$123$	−890.899			−0.0588869
$124$	5215.93			0.339225
$125$		−	4325.75i		−	0.276848i
$126$		−	1502.71i		−	0.0946531i
$127$	6576.24			0.407728			0.203864	−	0.978999i	$-0.434650\pi$
$127$	6576.24			0.407728			0.203864	+	0.978999i	$0.434650\pi$
$128$	1448.15			0.0883883
$129$		−	2762.44i		−	0.166002i
$130$			15120.4i			0.894695i
$131$	−7041.13			−0.410298			−0.205149	−	0.978731i	$-0.565768\pi$
$131$	−7041.13			−0.410298			−0.205149	+	0.978731i	$0.565768\pi$
$132$		−	2245.31i		−	0.128863i
$133$	−7470.66			−0.422334
$134$			13425.6i			0.747695i
$135$		−	4696.86i		−	0.257715i
$136$		−	1747.52i		−	0.0944808i
$137$		−	26091.9i		−	1.39016i	−0.718932	−	0.695080i	$-0.755369\pi$
$137$		−	26091.9i		−	1.39016i	0.718932	−	0.695080i	$-0.244631\pi$
$138$	2090.76	+	7488.28i	0.109786	+	0.393210i
$139$	1401.19			0.0725216			0.0362608	−	0.999342i	$-0.488455\pi$
$139$	1401.19			0.0725216			0.0362608	+	0.999342i	$0.488455\pi$
$140$	−5270.11			−0.268883
$141$	−13367.3			−0.672364
$142$	3921.07			0.194459
$143$			8625.01i			0.421782i
$144$	1728.00			0.0833333
$145$		−	54388.1i		−	2.58683i
$146$	−7404.08			−0.347349
$147$	10464.0			0.484243
$148$			7799.26i			0.356065i
$149$		−	16105.3i		−	0.725433i	−0.931899	−	0.362717i	$-0.881849\pi$
$149$		−	16105.3i		−	0.725433i	0.931899	−	0.362717i	$-0.118151\pi$
$150$	−7286.58			−0.323848
$151$	−5562.22			−0.243946			−0.121973	−	0.992533i	$-0.538922\pi$
$151$	−5562.22			−0.243946			−0.121973	+	0.992533i	$0.538922\pi$
$152$		−	8590.67i		−	0.371826i
$153$		−	2085.21i		−	0.0890773i
$154$	−3006.20			−0.126758
$155$		−	21827.5i		−	0.908532i
$156$	−6637.84			−0.272758
$157$		−	41229.4i		−	1.67266i	−0.548227	−	0.836330i	$-0.684697\pi$
$157$		−	41229.4i		−	1.67266i	0.548227	−	0.836330i	$-0.315303\pi$
$158$		−	11092.1i		−	0.444323i
$159$			10806.8i			0.427468i
$160$		−	6060.20i		−	0.236727i
$161$	10025.9	−	2799.27i	0.386786	−	0.107992i
$162$	2061.92			0.0785674
$163$	38706.4			1.45683			0.728413	−	0.685138i	$-0.240258\pi$
$163$	38706.4			1.45683			0.728413	+	0.685138i	$0.240258\pi$
$164$	−1371.63			−0.0509975
$165$	−9396.12			−0.345128
$166$			29859.9i			1.08361i
$167$	−38699.6			−1.38763			−0.693815	−	0.720153i	$-0.744071\pi$
$167$	−38699.6			−1.38763			−0.693815	+	0.720153i	$0.744071\pi$
$168$		−	2313.58i		−	0.0819720i
$169$	−3062.78			−0.107236
$170$	−7312.96			−0.253044
$171$		−	10250.7i		−	0.350561i
$172$		−	4253.06i		−	0.143762i
$173$	30765.0			1.02793			0.513966	−	0.857811i	$-0.328176\pi$
$173$	30765.0			1.02793			0.513966	+	0.857811i	$0.328176\pi$
$174$	23876.4			0.788624
$175$			9755.83i			0.318558i
$176$		−	3456.88i		−	0.111599i
$177$	44.5643			0.00142246
$178$			32014.4i			1.01043i
$179$	39384.6			1.22920			0.614598	−	0.788841i	$-0.289318\pi$
$179$	39384.6			1.22920			0.614598	+	0.788841i	$0.289318\pi$
$180$		−	7231.29i		−	0.223188i
$181$		−	32728.1i		−	0.998997i	−0.866315	−	0.499499i	$-0.833518\pi$
$181$		−	32728.1i		−	0.998997i	0.866315	−	0.499499i	$-0.166482\pi$
$182$			8887.25i			0.268302i
$183$			5510.07i			0.164534i
$184$	3218.94	+	11529.0i	0.0950773	+	0.340529i
$185$	32638.1			0.953634
$186$	9582.27			0.276976
$187$	−4171.49			−0.119291
$188$	−20580.3			−0.582284
$189$		−	2760.66i		−	0.0772840i
$190$	−35950.0			−0.995845
$191$		−	22931.5i		−	0.628588i	−0.949326	−	0.314294i	$-0.898232\pi$
$191$		−	22931.5i		−	0.628588i	0.949326	−	0.314294i	$-0.101768\pi$
$192$	2660.43			0.0721688
$193$	−25528.5			−0.685347			−0.342674	−	0.939454i	$-0.611333\pi$
$193$	−25528.5			−0.685347			−0.342674	+	0.939454i	$0.611333\pi$
$194$		−	7848.21i		−	0.208529i
$195$			27777.9i			0.730516i
$196$	16110.4			0.419367
$197$	47220.8			1.21675			0.608374	−	0.793651i	$-0.291822\pi$
$197$	47220.8			1.21675			0.608374	+	0.793651i	$0.291822\pi$
$198$		−	4124.90i		−	0.105216i
$199$			43028.7i			1.08655i	0.839553	+	0.543277i	$0.182817\pi$
$199$			43028.7i			1.08655i	−0.839553	+	0.543277i	$0.817183\pi$
$200$	−11218.4			−0.280461
$201$			24664.4i			0.610490i
$202$	−9057.36			−0.221972
$203$		−	31967.5i		−	0.775742i
$204$		−	3210.39i		−	0.0771432i
$205$			5739.96i			0.136584i
$206$		−	39635.2i		−	0.933999i
$207$	3840.97	+	13756.9i	0.0896397	+	0.321054i
$208$	−10219.6			−0.236215
$209$	−20506.7			−0.469466
$210$	−9681.80			−0.219542
$211$	−37106.5			−0.833462			−0.416731	−	0.909030i	$-0.636824\pi$
$211$	−37106.5			−0.833462			−0.416731	+	0.909030i	$0.636824\pi$
$212$			16638.2i			0.370199i
$213$	7203.47			0.158775
$214$			54066.3i			1.18059i
$215$	−17798.1			−0.385032
$216$	3174.54			0.0680414
$217$		−	12829.5i		−	0.272452i
$218$		−	32487.7i		−	0.683606i
$219$	−13602.2			−0.283609
$220$	−14466.3			−0.298890
$221$			12332.2i			0.252497i
$222$			14328.1i			0.290726i
$223$	−79480.0			−1.59826			−0.799131	−	0.601157i	$-0.794707\pi$
$223$	−79480.0			−1.59826			−0.799131	+	0.601157i	$0.794707\pi$
$224$		−	3561.99i		−	0.0709899i
$225$	−13386.3			−0.264421
$226$			47366.5i			0.927372i
$227$			74663.7i			1.44896i	0.689293	+	0.724482i	$0.257921\pi$
$227$			74663.7i			1.44896i	−0.689293	+	0.724482i	$0.742079\pi$
$228$		−	15782.1i		−	0.303595i
$229$			32658.3i			0.622763i	0.950285	+	0.311381i	$0.100792\pi$
$229$			32658.3i			0.622763i	−0.950285	+	0.311381i	$0.899208\pi$
$230$	48246.1	−	13470.5i	0.912025	−	0.254641i
$231$	−5522.73			−0.103498
$232$	36760.1			0.682969
$233$	19403.8			0.357417			0.178708	−	0.983902i	$-0.442808\pi$
$233$	19403.8			0.357417			0.178708	+	0.983902i	$0.442808\pi$
$234$	−12194.5			−0.222706
$235$			86123.7i			1.55951i
$236$	68.6113			0.00123189
$237$		−	20377.5i		−	0.362789i
$238$	−4298.32			−0.0758831
$239$	−74488.3			−1.30404			−0.652022	−	0.758200i	$-0.726079\pi$
$239$	−74488.3			−1.30404			−0.652022	+	0.758200i	$0.726079\pi$
$240$		−	11133.3i		−	0.193286i
$241$			30092.6i			0.518114i	0.965862	+	0.259057i	$0.0834117\pi$
$241$			30092.6i			0.518114i	−0.965862	+	0.259057i	$0.916588\pi$
$242$	33159.1			0.566203
$243$	3788.00			0.0641500
$244$			8483.31i			0.142490i
$245$		−	67418.4i		−	1.12317i
$246$	−2519.84			−0.0416393
$247$			60624.3i			0.993694i
$248$	14752.9			0.239869
$249$			54856.1i			0.884761i
$250$		−	12235.1i		−	0.195761i
$251$			63697.7i			1.01106i	0.862809	+	0.505530i	$0.168703\pi$
$251$			63697.7i			1.01106i	−0.862809	+	0.505530i	$0.831297\pi$
$252$		−	4250.32i		−	0.0669299i
$253$	27520.7	−	7683.91i	0.429951	−	0.120044i
$254$	18600.4			0.288307
$255$	−13434.8			−0.206609
$256$	4096.00			0.0625000
$257$	−70085.0			−1.06111			−0.530553	−	0.847652i	$-0.678016\pi$
$257$	−70085.0			−1.06111			−0.530553	+	0.847652i	$0.678016\pi$
$258$		−	7813.37i		−	0.117381i
$259$	19183.6			0.285977
$260$			42766.8i			0.632645i
$261$	43863.7			0.643909
$262$	−19915.3			−0.290125
$263$			23606.9i			0.341293i	0.985332	+	0.170647i	$0.0545857\pi$
$263$			23606.9i			0.341293i	−0.985332	+	0.170647i	$0.945414\pi$
$264$		−	6350.70i		−	0.0911200i
$265$	69627.1			0.991486
$266$	−21130.2			−0.298635
$267$			58814.2i			0.825011i
$268$			37973.3i			0.528700i
$269$	−12740.0			−0.176062			−0.0880308	−	0.996118i	$-0.528057\pi$
$269$	−12740.0			−0.176062			−0.0880308	+	0.996118i	$0.528057\pi$
$270$		−	13284.7i		−	0.182232i
$271$	455.015			0.00619565			0.00309783	−	0.999995i	$-0.499014\pi$
$271$	455.015			0.00619565			0.00309783	+	0.999995i	$0.499014\pi$
$272$		−	4942.72i		−	0.0668080i
$273$			16326.9i			0.219068i
$274$		−	73799.1i		−	0.982992i
$275$			26779.5i			0.354109i
$276$	5913.56	+	21180.1i	0.0776303	+	0.278041i
$277$	46365.1			0.604271			0.302136	−	0.953265i	$-0.402300\pi$
$277$	46365.1			0.604271			0.302136	+	0.953265i	$0.402300\pi$
$278$	3963.16			0.0512805
$279$	17603.8			0.226150
$280$	−14906.1			−0.190129
$281$		−	117828.i		−	1.49223i	−0.665819	−	0.746113i	$-0.731918\pi$
$281$		−	117828.i		−	1.49223i	0.665819	−	0.746113i	$-0.268082\pi$
$282$	−37808.3			−0.475433
$283$			32448.1i			0.405151i	0.979267	+	0.202575i	$0.0649311\pi$
$283$			32448.1i			0.405151i	−0.979267	+	0.202575i	$0.935069\pi$
$284$	11090.5			0.137503
$285$	−66044.3			−0.813104
$286$			24395.2i			0.298245i
$287$			3373.76i			0.0409591i
$288$	4887.52			0.0589256
$289$	77556.5			0.928587
$290$		−	153833.i		−	1.82916i
$291$		−	14418.1i		−	0.170264i
$292$	−20941.9			−0.245613
$293$		−	88859.5i		−	1.03507i	−0.855663	−	0.517534i	$-0.826850\pi$
$293$		−	88859.5i		−	1.03507i	0.855663	−	0.517534i	$-0.173150\pi$
$294$	29596.7			0.342412
$295$		−	287.123i		−	0.00329931i
$296$			22059.6i			0.251776i
$297$		−	7577.93i		−	0.0859088i
$298$		−	45552.8i		−	0.512959i
$299$	−22716.0	−	81359.9i	−0.254091	−	0.910056i
$300$	−20609.6			−0.228995
$301$	−10461.1			−0.115464
$302$	−15732.3			−0.172496
$303$	−16639.4			−0.181240
$304$		−	24298.1i		−	0.262921i
$305$	35500.7			0.381626
$306$		−	5897.87i		−	0.0629872i
$307$	−20919.2			−0.221957			−0.110978	−	0.993823i	$-0.535398\pi$
$307$	−20919.2			−0.221957			−0.110978	+	0.993823i	$0.535398\pi$
$308$	−8502.80			−0.0896315
$309$		−	72814.5i		−	0.762607i
$310$		−	61737.4i		−	0.642429i
$311$	29482.6			0.304821			0.152411	−	0.988317i	$-0.451296\pi$
$311$	29482.6			0.304821			0.152411	+	0.988317i	$0.451296\pi$
$312$	−18774.6			−0.192869
$313$		−	80312.8i		−	0.819777i	−0.912136	−	0.409889i	$-0.865568\pi$
$313$		−	80312.8i		−	0.819777i	0.912136	−	0.409889i	$-0.134432\pi$
$314$		−	116614.i		−	1.18275i
$315$	−17786.6			−0.179255
$316$		−	31373.2i		−	0.314184i
$317$	−24797.0			−0.246763			−0.123381	−	0.992359i	$-0.539374\pi$
$317$	−24797.0			−0.246763			−0.123381	+	0.992359i	$0.539374\pi$
$318$			30566.3i			0.302266i
$319$		−	87749.9i		−	0.862314i
$320$		−	17140.8i		−	0.167391i
$321$			99326.2i			0.963948i
$322$	28357.5	−	7917.53i	0.273499	−	0.0763621i
$323$	−29321.0			−0.281043
$324$	5832.00			0.0555556
$325$	79168.4			0.749523
$326$	109478.			1.03013
$327$		−	59683.7i		−	0.558162i
$328$	−3879.55			−0.0360607
$329$			50620.7i			0.467667i
$330$	−26576.2			−0.244043
$331$	−127053.			−1.15966			−0.579830	−	0.814738i	$-0.696881\pi$
$331$	−127053.			−1.15966			−0.579830	+	0.814738i	$0.696881\pi$
$332$			84456.5i			0.766226i
$333$			26322.5i			0.237377i
$334$	−109459.			−0.981203
$335$	158910.			1.41599
$336$		−	6543.79i		−	0.0579630i
$337$		−	70632.6i		−	0.621935i	−0.950420	−	0.310968i	$-0.899347\pi$
$337$		−	70632.6i		−	0.621935i	0.950420	−	0.310968i	$-0.100653\pi$
$338$	−8662.85			−0.0758276
$339$			87017.7i			0.757196i
$340$	−20684.2			−0.178929
$341$		−	35216.5i		−	0.302857i
$342$		−	28993.5i		−	0.247884i
$343$		−	86871.7i		−	0.738398i
$344$		−	12029.5i		−	0.101655i
$345$	88633.8	−	24746.9i	0.744665	−	0.207914i
$346$	87016.5			0.726857
$347$	191595.			1.59120			0.795602	−	0.605819i	$-0.207155\pi$
$347$	191595.			1.59120			0.795602	+	0.605819i	$0.207155\pi$
$348$	67532.6			0.557642
$349$	−145705.			−1.19625			−0.598126	−	0.801402i	$-0.704088\pi$
$349$	−145705.			−1.19625			−0.598126	+	0.801402i	$0.704088\pi$
$350$			27593.7i			0.225254i
$351$	−22402.7			−0.181839
$352$		−	9777.54i		−	0.0789123i
$353$	−68772.2			−0.551904			−0.275952	−	0.961171i	$-0.588993\pi$
$353$	−68772.2			−0.551904			−0.275952	+	0.961171i	$0.588993\pi$
$354$	126.047			0.00100583
$355$		−	46411.1i		−	0.368269i
$356$			90550.4i			0.714481i
$357$	−7896.52			−0.0619583
$358$	111397.			0.869172
$359$			133581.i			1.03647i	0.855239	+	0.518234i	$0.173410\pi$
$359$			133581.i			1.03647i	−0.855239	+	0.518234i	$0.826590\pi$
$360$		−	20453.2i		−	0.157818i
$361$	−13818.7			−0.106036
$362$		−	92569.2i		−	0.706398i
$363$	60917.1			0.462302
$364$			25136.9i			0.189718i
$365$			87637.1i			0.657813i
$366$			15584.8i			0.116343i
$367$			194975.i			1.44760i	0.690012	+	0.723798i	$0.257605\pi$
$367$			194975.i			1.44760i	−0.690012	+	0.723798i	$0.742395\pi$
$368$	9104.53	+	32608.8i	0.0672298	+	0.240791i
$369$	−4629.25			−0.0339983
$370$	92314.6			0.674321
$371$	40924.5			0.297328
$372$	27102.8			0.195852
$373$		−	245451.i		−	1.76419i	−0.471067	−	0.882097i	$-0.656131\pi$
$373$		−	245451.i		−	1.76419i	0.471067	−	0.882097i	$-0.343869\pi$
$374$	−11798.8			−0.0843515
$375$		−	22477.2i		−	0.159838i
$376$	−58209.7			−0.411737
$377$	−259416.			−1.82521
$378$		−	7808.33i		−	0.0546480i
$379$			211323.i			1.47119i	0.677422	+	0.735594i	$0.263097\pi$
$379$			211323.i			1.47119i	−0.677422	+	0.735594i	$0.736903\pi$
$380$	−101682.			−0.704168
$381$	34171.2			0.235402
$382$		−	64860.1i		−	0.444479i
$383$		−	172464.i		−	1.17571i	−0.808965	−	0.587857i	$-0.799972\pi$
$383$		−	172464.i		−	1.17571i	0.808965	−	0.587857i	$-0.200028\pi$
$384$	7524.83			0.0510310
$385$			35582.3i			0.240056i
$386$	−72205.5			−0.484614
$387$		−	14354.1i		−	0.0958415i
$388$		−	22198.1i		−	0.147453i
$389$			140755.i			0.930177i	0.885264	+	0.465088i	$0.153977\pi$
$389$			140755.i			0.930177i	−0.885264	+	0.465088i	$0.846023\pi$
$390$			78567.7i			0.516553i
$391$	39349.7	−	10986.6i	0.257388	−	0.0718638i
$392$	45567.1			0.296537
$393$	−36586.8			−0.236886
$394$	133560.			0.860370
$395$	−131290.			−0.841465
$396$		−	11667.0i		−	0.0743992i
$397$	−302137.			−1.91700			−0.958502	−	0.285085i	$-0.907978\pi$
$397$	−302137.			−1.91700			−0.958502	+	0.285085i	$0.907978\pi$
$398$			121703.i			0.768310i
$399$	−38818.7			−0.243835
$400$	−31730.5			−0.198316
$401$			251342.i			1.56306i	0.623865	+	0.781532i	$0.285561\pi$
$401$			251342.i			1.56306i	−0.623865	+	0.781532i	$0.714439\pi$
$402$			69761.5i			0.431682i
$403$	−104111.			−0.641042
$404$	−25618.1			−0.156958
$405$		−	24405.6i		−	0.148792i
$406$		−	90417.8i		−	0.548532i
$407$	52658.4			0.317892
$408$		−	9080.36i		−	0.0545485i
$409$	118085.			0.705909			0.352954	−	0.935641i	$-0.385177\pi$
$409$	118085.			0.705909			0.352954	+	0.935641i	$0.385177\pi$
$410$			16235.1i			0.0965797i
$411$		−	135578.i		−	0.802610i
$412$		−	112105.i		−	0.660437i
$413$		−	168.761i		−	0.000989402i
$414$	10863.9	+	38910.3i	0.0633849	+	0.227020i
$415$	353431.			2.05215
$416$	−28905.5			−0.167030
$417$	7280.79			0.0418703
$418$	−58001.8			−0.331963
$419$			102383.i			0.583176i	0.956544	+	0.291588i	$0.0941837\pi$
$419$			102383.i			0.583176i	−0.956544	+	0.291588i	$0.905816\pi$
$420$	−27384.3			−0.155240
$421$		−	13174.0i		−	0.0743283i	−0.999309	−	0.0371641i	$-0.988168\pi$
$421$		−	13174.0i		−	0.0743283i	0.999309	−	0.0371641i	$-0.0118324\pi$
$422$	−104953.			−0.589346
$423$	−69458.4			−0.388189
$424$			47059.9i			0.261770i
$425$			38289.8i			0.211985i
$426$	20374.5			0.112271
$427$	20866.2			0.114442
$428$			152923.i			0.834804i
$429$			44816.9i			0.243516i
$430$	−50340.6			−0.272259
$431$		−	83944.7i		−	0.451896i	−0.974139	−	0.225948i	$-0.927452\pi$
$431$		−	83944.7i		−	0.451896i	0.974139	−	0.225948i	$-0.0725480\pi$
$432$	8978.95			0.0481125
$433$		−	57692.2i		−	0.307710i	−0.988093	−	0.153855i	$-0.950831\pi$
$433$		−	57692.2i		−	0.307710i	0.988093	−	0.153855i	$-0.0491689\pi$
$434$		−	36287.3i		−	0.192652i
$435$		−	282609.i		−	1.49351i
$436$		−	91889.0i		−	0.483382i
$437$	193440.	−	54009.4i	1.01294	−	0.282818i
$438$	−38472.7			−0.200542
$439$	−182534.			−0.947140			−0.473570	−	0.880756i	$-0.657035\pi$
$439$	−182534.			−0.947140			−0.473570	+	0.880756i	$0.657035\pi$
$440$	−40916.8			−0.211347
$441$	54372.6			0.279578
$442$			34880.8i			0.178543i
$443$	216357.			1.10246			0.551230	−	0.834353i	$-0.314159\pi$
$443$	216357.			1.10246			0.551230	+	0.834353i	$0.314159\pi$
$444$			40526.1i			0.205574i
$445$	378933.			1.91356
$446$	−224803.			−1.13014
$447$		−	83685.8i		−	0.418829i
$448$		−	10074.8i		−	0.0501974i
$449$	−368789.			−1.82930			−0.914651	−	0.404244i	$-0.867534\pi$
$449$	−368789.			−1.82930			−0.914651	+	0.404244i	$0.867534\pi$
$450$	−37862.2			−0.186974
$451$			9260.86i			0.0455301i
$452$			133973.i			0.655751i
$453$	−28902.1			−0.140842
$454$			211181.i			1.02457i
$455$	105192.			0.508114
$456$		−	44638.4i		−	0.214674i
$457$			154736.i			0.740897i	0.928853	+	0.370448i	$0.120796\pi$
$457$			154736.i			0.740897i	−0.928853	+	0.370448i	$0.879204\pi$
$458$			92371.6i			0.440360i
$459$		−	10835.1i		−	0.0514288i
$460$	136461.	−	38100.4i	0.644899	−	0.180059i
$461$	138326.			0.650880			0.325440	−	0.945563i	$-0.394488\pi$
$461$	138326.			0.650880			0.325440	+	0.945563i	$0.394488\pi$
$462$	−15620.6			−0.0731838
$463$	48659.5			0.226989			0.113495	−	0.993539i	$-0.463795\pi$
$463$	48659.5			0.226989			0.113495	+	0.993539i	$0.463795\pi$
$464$	103973.			0.482932
$465$		−	113419.i		−	0.524541i
$466$	54882.2			0.252732
$467$			273257.i			1.25296i	0.779437	+	0.626480i	$0.215505\pi$
$467$			273257.i			1.25296i	−0.779437	+	0.626480i	$0.784495\pi$
$468$	−34491.2			−0.157477
$469$	93402.0			0.424630
$470$			243595.i			1.10274i
$471$		−	214234.i		−	0.965710i
$472$	194.062			0.000871077
$473$	−28715.5			−0.128350
$474$		−	57636.2i		−	0.256530i
$475$			188230.i			0.834260i
$476$	−12157.5			−0.0536574
$477$			56153.9i			0.246799i
$478$	−210685.			−0.922098
$479$		−	206455.i		−	0.899816i	−0.893075	−	0.449908i	$-0.851457\pi$
$479$		−	206455.i		−	0.899816i	0.893075	−	0.449908i	$-0.148543\pi$
$480$		−	31489.7i		−	0.136674i
$481$		−	155675.i		−	0.672865i
$482$			85114.6i			0.366362i
$483$	52096.0	−	14545.4i	0.223311	−	0.0623494i
$484$	93788.1			0.400366
$485$	−92894.0			−0.394916
$486$	10714.1			0.0453609
$487$	−119940.			−0.505716			−0.252858	−	0.967503i	$-0.581371\pi$
$487$	−119940.			−0.505716			−0.252858	+	0.967503i	$0.581371\pi$
$488$			23994.4i			0.100756i
$489$	201124.			0.841099
$490$		−	190688.i		−	0.794202i
$491$	−103912.			−0.431024			−0.215512	−	0.976501i	$-0.569142\pi$
$491$	−103912.			−0.431024			−0.215512	+	0.976501i	$0.569142\pi$
$492$	−7127.19			−0.0294434
$493$		−	125467.i		−	0.516219i
$494$			171471.i			0.702648i
$495$	−48823.7			−0.199260
$496$	41727.4			0.169613
$497$		−	27278.9i		−	0.110437i
$498$			155156.i			0.625621i
$499$	−454701.			−1.82610			−0.913050	−	0.407848i	$-0.866279\pi$
$499$	−454701.			−1.82610			−0.913050	+	0.407848i	$0.866279\pi$
$500$		−	34606.0i		−	0.138424i
$501$	−201089.			−0.801149
$502$			180164.i			0.714927i
$503$			154319.i			0.609934i	0.952363	+	0.304967i	$0.0986454\pi$
$503$			154319.i			0.609934i	−0.952363	+	0.304967i	$0.901355\pi$
$504$		−	12021.7i		−	0.0473266i
$505$			107206.i			0.420374i
$506$	77840.4	−	21733.4i	0.304021	−	0.0848841i
$507$	−15914.7			−0.0619130
$508$	52610.0			0.203864
$509$	384796.			1.48523			0.742617	−	0.669717i	$-0.233585\pi$
$509$	384796.			1.48523			0.742617	+	0.669717i	$0.233585\pi$
$510$	−37999.3			−0.146095
$511$			51510.2i			0.197266i
$512$	11585.2			0.0441942
$513$		−	53264.5i		−	0.202396i
$514$	−198230.			−0.750315
$515$	−469135.			−1.76882
$516$		−	22099.6i		−	0.0830012i
$517$			138952.i			0.519858i
$518$	54259.5			0.202216
$519$	159859.			0.593477
$520$			120963.i			0.447348i
$521$		−	45060.9i		−	0.166006i	−0.996549	−	0.0830031i	$-0.973549\pi$
$521$		−	45060.9i		−	0.166006i	0.996549	−	0.0830031i	$-0.0264511\pi$
$522$	124065.			0.455312
$523$			152841.i			0.558774i	0.960179	+	0.279387i	$0.0901313\pi$
$523$			152841.i			0.558774i	−0.960179	+	0.279387i	$0.909869\pi$
$524$	−56329.0			−0.205149
$525$			50692.8i			0.183919i
$526$			66770.5i			0.241331i
$527$		−	50353.3i		−	0.181304i
$528$		−	17962.5i		−	0.0644316i
$529$	−239366.	+	144965.i	−0.855365	+	0.518026i
$530$	196935.			0.701086
$531$	231.563			0.000821259
$532$	−59765.3			−0.211167
$533$	27378.0			0.0963712
$534$			166352.i			0.583371i
$535$	639947.			2.23582
$536$			107405.i			0.373847i
$537$	204649.			0.709676
$538$	−36034.1			−0.124494
$539$		−	108773.i		−	0.374407i
$540$		−	37574.9i		−	0.128858i
$541$	−538523.			−1.83997			−0.919983	−	0.391958i	$-0.871798\pi$
$541$	−538523.			−1.83997			−0.919983	+	0.391958i	$0.871798\pi$
$542$	1286.98			0.00438099
$543$		−	170060.i		−	0.576771i
$544$		−	13980.1i		−	0.0472404i
$545$	−384535.			−1.29462
$546$			46179.5i			0.154904i
$547$	−392350.			−1.31129			−0.655645	−	0.755069i	$-0.727603\pi$
$547$	−392350.			−1.31129			−0.655645	+	0.755069i	$0.727603\pi$
$548$		−	208735.i		−	0.695080i
$549$			28631.2i			0.0949936i
$550$			75743.8i			0.250393i
$551$		−	616785.i		−	2.03156i
$552$	16726.1	+	59906.3i	0.0548929	+	0.196605i
$553$	−77167.7			−0.252340
$554$	131140.			0.427284
$555$	169593.			0.550581
$556$	11209.5			0.0362608
$557$			558524.i			1.80025i	0.435637	+	0.900123i	$0.356523\pi$
$557$			558524.i			1.80025i	−0.435637	+	0.900123i	$0.643477\pi$
$558$	49791.0			0.159912
$559$			84892.0i			0.271671i
$560$	−42160.8			−0.134441
$561$	−21675.7			−0.0688727
$562$		−	333267.i		−	1.05516i
$563$		−	417048.i		−	1.31574i	−0.753132	−	0.657869i	$-0.771458\pi$
$563$		−	417048.i		−	1.31574i	0.753132	−	0.657869i	$-0.228542\pi$
$564$	−106938.			−0.336182
$565$	560645.			1.75627
$566$			91777.1i			0.286485i
$567$		−	14344.8i		−	0.0446199i
$568$	31368.6			0.0972296
$569$		−	56795.6i		−	0.175425i	−0.996146	−	0.0877123i	$-0.972044\pi$
$569$		−	56795.6i		−	0.175425i	0.996146	−	0.0877123i	$-0.0279556\pi$
$570$	−186802.			−0.574951
$571$		−	281453.i		−	0.863243i	−0.902055	−	0.431621i	$-0.857942\pi$
$571$		−	281453.i		−	0.863243i	0.902055	−	0.431621i	$-0.142058\pi$
$572$			69000.1i			0.210891i
$573$		−	119156.i		−	0.362915i
$574$			9542.43i			0.0289625i
$575$	−70530.1	−	252611.i	−0.213324	−	0.764041i
$576$	13824.0			0.0416667
$577$	491209.			1.47542			0.737709	−	0.675119i	$-0.235908\pi$
$577$	491209.			1.47542			0.737709	+	0.675119i	$0.235908\pi$
$578$	219363.			0.656610
$579$	−132650.			−0.395685
$580$		−	435105.i		−	1.29341i
$581$	207735.			0.615401
$582$		−	40780.5i		−	0.120395i
$583$	112337.			0.330510
$584$	−59232.7			−0.173674
$585$			144338.i			0.421763i
$586$		−	251333.i		−	0.731903i
$587$	−169969.			−0.493281			−0.246640	−	0.969107i	$-0.579327\pi$
$587$	−169969.			−0.493281			−0.246640	+	0.969107i	$0.579327\pi$
$588$	83712.1			0.242122
$589$		−	247533.i		−	0.713515i
$590$		−	812.106i		−	0.00233297i
$591$	245366.			0.702490
$592$			62394.1i			0.178033i
$593$	401812.			1.14265			0.571325	−	0.820724i	$-0.306430\pi$
$593$	401812.			1.14265			0.571325	+	0.820724i	$0.306430\pi$
$594$		−	21433.6i		−	0.0607467i
$595$			50876.3i			0.143708i
$596$		−	128843.i		−	0.362717i
$597$			223583.i			0.627323i
$598$	−64250.6	−	230120.i	−0.179670	−	0.643506i
$599$	182719.			0.509250			0.254625	−	0.967040i	$-0.418048\pi$
$599$	182719.			0.509250			0.254625	+	0.967040i	$0.418048\pi$
$600$	−58292.7			−0.161924
$601$	281509.			0.779369			0.389685	−	0.920948i	$-0.372584\pi$
$601$	281509.			0.779369			0.389685	+	0.920948i	$0.372584\pi$
$602$	−29588.6			−0.0816453
$603$			128160.i			0.352467i
$604$	−44497.7			−0.121973
$605$		−	392482.i		−	1.07228i
$606$	−47063.4			−0.128156
$607$	−422711.			−1.14727			−0.573635	−	0.819111i	$-0.694467\pi$
$607$	−422711.			−1.14727			−0.573635	+	0.819111i	$0.694467\pi$
$608$		−	68725.3i		−	0.185913i
$609$		−	166108.i		−	0.447875i
$610$	100411.			0.269850
$611$	410786.			1.10036
$612$		−	16681.7i		−	0.0445387i
$613$			359369.i			0.956355i	0.878263	+	0.478177i	$0.158702\pi$
$613$			359369.i			0.956355i	−0.878263	+	0.478177i	$0.841298\pi$
$614$	−59168.5			−0.156947
$615$			29825.7i			0.0788570i
$616$	−24049.6			−0.0633791
$617$			343800.i			0.903099i	0.892246	+	0.451550i	$0.149129\pi$
$617$			343800.i			0.903099i	−0.892246	+	0.451550i	$0.850871\pi$
$618$		−	205951.i		−	0.539245i
$619$			625411.i			1.63224i	0.577882	+	0.816120i	$0.303879\pi$
$619$			625411.i			1.63224i	−0.577882	+	0.816120i	$0.696121\pi$
$620$		−	174620.i		−	0.454266i
$621$	19958.3	+	71482.7i	0.0517535	+	0.185361i
$622$	83389.4			0.215541
$623$	222724.			0.573841
$624$	−53102.7			−0.136379
$625$	−454686.			−1.16400
$626$		−	227159.i		−	0.579670i
$627$	−106556.			−0.271046
$628$		−	329835.i		−	0.836330i
$629$	75292.1			0.190304
$630$	−50308.1			−0.126753
$631$		−	384826.i		−	0.966508i	−0.875480	−	0.483254i	$-0.839455\pi$
$631$		−	384826.i		−	0.966508i	0.875480	−	0.483254i	$-0.160545\pi$
$632$		−	88736.7i		−	0.222162i
$633$	−192811.			−0.481199
$634$	−70136.4			−0.174488
$635$		−	220161.i		−	0.546000i
$636$			86454.6i			0.213734i
$637$	−321567.			−0.792488
$638$		−	248194.i		−	0.609748i
$639$	37430.3			0.0916689
$640$		−	48481.6i		−	0.118363i
$641$		−	284276.i		−	0.691870i	−0.938258	−	0.345935i	$-0.887562\pi$
$641$		−	284276.i		−	0.691870i	0.938258	−	0.345935i	$-0.112438\pi$
$642$			280937.i			0.681614i
$643$		−	416229.i		−	1.00672i	−0.864076	−	0.503361i	$-0.832096\pi$
$643$		−	416229.i		−	1.00672i	0.864076	−	0.503361i	$-0.167904\pi$
$644$	80207.1	−	22394.2i	0.193393	−	0.0539962i
$645$	−92481.7			−0.222298
$646$	−82932.2			−0.198728
$647$	−501651.			−1.19838			−0.599188	−	0.800609i	$-0.704510\pi$
$647$	−501651.			−1.19838			−0.599188	+	0.800609i	$0.704510\pi$
$648$	16495.4			0.0392837
$649$		−	463.245i		−	0.00109982i
$650$	223922.			0.529993
$651$		−	66663.9i		−	0.157300i
$652$	309651.			0.728413
$653$	−243366.			−0.570734			−0.285367	−	0.958418i	$-0.592116\pi$
$653$	−243366.			−0.570734			−0.285367	+	0.958418i	$0.592116\pi$
$654$		−	168811.i		−	0.394680i
$655$			235724.i			0.549442i
$656$	−10973.0			−0.0254988
$657$	−70678.9			−0.163742
$658$			143177.i			0.330690i
$659$		−	91222.5i		−	0.210054i	−0.994469	−	0.105027i	$-0.966507\pi$
$659$		−	91222.5i		−	0.210054i	0.994469	−	0.105027i	$-0.0334929\pi$
$660$	−75169.0			−0.172564
$661$		−	748739.i		−	1.71367i	−0.515589	−	0.856836i	$-0.672427\pi$
$661$		−	748739.i		−	1.71367i	0.515589	−	0.856836i	$-0.327573\pi$
$662$	−359362.			−0.820003
$663$			64080.1i			0.145779i
$664$			238879.i			0.541803i
$665$			250104.i			0.565559i
$666$			74451.2i			0.167851i
$667$	231110.	+	827746.i	0.519478	+	1.86057i
$668$	−309597.			−0.693815
$669$	−412990.			−0.922757
$670$	449465.			1.00126
$671$	57277.0			0.127214
$672$		−	18508.6i		−	0.0409860i
$673$	616620.			1.36141			0.680703	−	0.732560i	$-0.261675\pi$
$673$	616620.			1.36141			0.680703	+	0.732560i	$0.261675\pi$
$674$		−	199779.i		−	0.439775i
$675$	−69557.3			−0.152663
$676$	−24502.2			−0.0536182
$677$			823346.i			1.79641i	0.439578	+	0.898205i	$0.355128\pi$
$677$			823346.i			1.79641i	−0.439578	+	0.898205i	$0.644872\pi$
$678$			246123.i			0.535419i
$679$	−54600.1			−0.118428
$680$	−58503.7			−0.126522
$681$			387964.i			0.836560i
$682$		−	99607.4i		−	0.214152i
$683$	−354815.			−0.760606			−0.380303	−	0.924862i	$-0.624180\pi$
$683$	−354815.			−0.760606			−0.380303	+	0.924862i	$0.624180\pi$
$684$		−	82006.0i		−	0.175280i
$685$	−873511.			−1.86160
$686$		−	245710.i		−	0.522126i
$687$			169697.i			0.359552i
$688$		−	34024.5i		−	0.0718811i
$689$		−	332102.i		−	0.699573i
$690$	250694.	−	69994.9i	0.526558	−	0.147017i
$691$	567858.			1.18928			0.594639	−	0.803993i	$-0.297295\pi$
$691$	567858.			1.18928			0.594639	+	0.803993i	$0.297295\pi$
$692$	246120.			0.513966
$693$	−28697.0			−0.0597543
$694$	541913.			1.12515
$695$		−	46909.3i		−	0.0971156i
$696$	191011.			0.394312
$697$			13241.4i			0.0272563i
$698$	−412115.			−0.845878
$699$	100825.			0.206355
$700$			78046.7i			0.159279i
$701$		−	190169.i		−	0.386994i	−0.981101	−	0.193497i	$-0.938017\pi$
$701$		−	190169.i		−	0.386994i	0.981101	−	0.193497i	$-0.0619830\pi$
$702$	−63364.4			−0.128579
$703$	370131.			0.748935
$704$		−	27655.1i		−	0.0557994i
$705$			447512.i			0.900381i
$706$	−194517.			−0.390255
$707$			63012.1i			0.126062i
$708$	356.515			0.000711231
$709$			814163.i			1.61964i	0.586677	+	0.809821i	$0.300436\pi$
$709$			814163.i			1.61964i	−0.586677	+	0.809821i	$0.699564\pi$
$710$		−	131270.i		−	0.260406i
$711$		−	105884.i		−	0.209456i
$712$			256115.i			0.505214i
$713$	92751.1	+	332198.i	0.182448	+	0.653458i
$714$	−22334.7			−0.0438111
$715$	288750.			0.564819
$716$	315077.			0.614598
$717$	−387052.			−0.752890
$718$			377824.i			0.732894i
$719$	976490.			1.88890			0.944452	−	0.328649i	$-0.106593\pi$
$719$	976490.			1.88890			0.944452	+	0.328649i	$0.106593\pi$
$720$		−	57850.3i		−	0.111594i
$721$	−275742.			−0.530436
$722$	−39085.2			−0.0749787
$723$			156366.i			0.299133i
$724$		−	261825.i		−	0.499499i
$725$	−805450.			−1.53237
$726$	172300.			0.326897
$727$			364607.i			0.689853i	0.938630	+	0.344926i	$0.112096\pi$
$727$			364607.i			0.689853i	−0.938630	+	0.344926i	$0.887904\pi$
$728$			71098.0i			0.134151i
$729$	19683.0			0.0370370
$730$			247875.i			0.465144i
$731$	−41058.0			−0.0768357
$732$			44080.6i			0.0822669i
$733$		−	801046.i		−	1.49090i	−0.666560	−	0.745451i	$-0.732234\pi$
$733$		−	801046.i		−	1.49090i	0.666560	−	0.745451i	$-0.267766\pi$
$734$			551473.i			1.02360i
$735$		−	350316.i		−	0.648464i
$736$	25751.5	+	92231.7i	0.0475386	+	0.170265i
$737$	256386.			0.472018
$738$	−13093.5			−0.0240405
$739$	−197657.			−0.361929			−0.180965	−	0.983490i	$-0.557922\pi$
$739$	−197657.			−0.361929			−0.180965	+	0.983490i	$0.557922\pi$
$740$	261105.			0.476817
$741$			315013.i			0.573710i
$742$	115752.			0.210243
$743$		−	179668.i		−	0.325456i	−0.986671	−	0.162728i	$-0.947971\pi$
$743$		−	179668.i		−	0.325456i	0.986671	−	0.162728i	$-0.0520293\pi$
$744$	76658.2			0.138488
$745$	−539178.			−0.971448
$746$		−	694239.i		−	1.24747i
$747$			285041.i			0.510817i
$748$	−33371.9			−0.0596455
$749$	376140.			0.670480
$750$		−	63575.2i		−	0.113023i
$751$		−	454544.i		−	0.805928i	−0.915216	−	0.402964i	$-0.867980\pi$
$751$		−	454544.i		−	0.805928i	0.915216	−	0.402964i	$-0.132020\pi$
$752$	−164642.			−0.291142
$753$			330983.i			0.583735i
$754$	−733739.			−1.29062
$755$			186213.i			0.326675i
$756$		−	22085.3i		−	0.0386420i
$757$			461212.i			0.804838i	0.915456	+	0.402419i	$0.131831\pi$
$757$			461212.i			0.804838i	−0.915456	+	0.402419i	$0.868169\pi$
$758$			597712.i			1.04029i
$759$	143002.	−	39926.8i	0.248232	−	0.0693076i
$760$	−287600.			−0.497922
$761$	408299.			0.705032			0.352516	−	0.935806i	$-0.385326\pi$
$761$	408299.			0.705032			0.352516	+	0.935806i	$0.385326\pi$
$762$	96650.7			0.166454
$763$	−226017.			−0.388233
$764$		−	183452.i		−	0.314294i
$765$	−69809.1			−0.119286
$766$		−	487803.i		−	0.831355i
$767$	−1369.50			−0.00232793
$768$	21283.4			0.0360844
$769$			56576.3i			0.0956714i	0.998855	+	0.0478357i	$0.0152324\pi$
$769$			56576.3i			0.0956714i	−0.998855	+	0.0478357i	$0.984768\pi$
$770$			100642.i			0.169745i
$771$	−364172.			−0.612630
$772$	−204228.			−0.342674
$773$			714995.i			1.19659i	0.801277	+	0.598293i	$0.204154\pi$
$773$			714995.i			1.19659i	−0.801277	+	0.598293i	$0.795846\pi$
$774$		−	40599.5i		−	0.0677702i
$775$	−323250.			−0.538190
$776$		−	62785.7i		−	0.104265i
$777$	99681.0			0.165109
$778$			398116.i			0.657734i
$779$			65093.6i			0.107266i
$780$			222223.i			0.365258i
$781$		−	74879.8i		−	0.122762i
$782$	111298.	−	31074.8i	0.182001	−	0.0508154i
$783$	227923.			0.371761
$784$	128883.			0.209684
$785$	−1.38028e6			−2.23990
$786$	−103483.			−0.167504
$787$			361359.i			0.583431i	0.956505	+	0.291715i	$0.0942260\pi$
$787$			361359.i			0.583431i	−0.956505	+	0.291715i	$0.905774\pi$
$788$	377766.			0.608374
$789$			122665.i			0.197046i
$790$	−371343.			−0.595006
$791$	329529.			0.526672
$792$		−	32999.2i		−	0.0526082i
$793$		−	169329.i		−	0.269267i
$794$	−854573.			−1.35553
$795$	361793.			0.572435
$796$			344229.i			0.543277i
$797$			388399.i			0.611451i	0.952120	+	0.305726i	$0.0988990\pi$
$797$			388399.i			0.611451i	−0.952120	+	0.305726i	$0.901101\pi$
$798$	−109796.			−0.172417
$799$			198677.i			0.311210i
$800$	−89747.4			−0.140230
$801$			305608.i			0.476320i
$802$			710903.i			1.10525i
$803$			141394.i			0.219280i
$804$			197315.i			0.305245i
$805$	−93714.5	−	335648.i	−0.144616	−	0.517956i
$806$	−294470.			−0.453285
$807$	−66198.9			−0.101649
$808$	−72458.9			−0.110986
$809$	170967.			0.261225			0.130612	−	0.991434i	$-0.458306\pi$
$809$	170967.			0.261225			0.130612	+	0.991434i	$0.458306\pi$
$810$		−	69029.5i		−	0.105212i
$811$	−476324.			−0.724203			−0.362102	−	0.932139i	$-0.617941\pi$
$811$	−476324.			−0.724203			−0.362102	+	0.932139i	$0.617941\pi$
$812$		−	255740.i		−	0.387871i
$813$	2364.33			0.00357706
$814$	148941.			0.224783
$815$		−	1.29582e6i		−	1.95088i
$816$		−	25683.1i		−	0.0385716i
$817$	−201838.			−0.302384
$818$	333995.			0.499153
$819$			84837.2i			0.126479i
$820$			45919.7i			0.0682922i
$821$	1.06850e6			1.58521			0.792605	−	0.609736i	$-0.208724\pi$
$821$	1.06850e6			1.58521			0.792605	+	0.609736i	$0.208724\pi$
$822$		−	383472.i		−	0.567531i
$823$	−663222.			−0.979172			−0.489586	−	0.871955i	$-0.662852\pi$
$823$	−663222.			−0.979172			−0.489586	+	0.871955i	$0.662852\pi$
$824$		−	317082.i		−	0.467000i
$825$			139150.i			0.204445i
$826$		−	477.329i		−	0.000699613i
$827$			285779.i			0.417850i	0.977932	+	0.208925i	$0.0669964\pi$
$827$			285779.i			0.417850i	−0.977932	+	0.208925i	$0.933004\pi$
$828$	30727.8	+	110055.i	0.0448199	+	0.160527i
$829$	−3536.66			−0.00514618			−0.00257309	−	0.999997i	$-0.500819\pi$
$829$	−3536.66			−0.00514618			−0.00257309	+	0.999997i	$0.500819\pi$
$830$	999654.			1.45109
$831$	240920.			0.348876
$832$	−81757.0			−0.118108
$833$		−	155526.i		−	0.224137i
$834$	20593.2			0.0296068
$835$			1.29559e6i			1.85821i
$836$	−164054.			−0.234733
$837$	91471.8			0.130568
$838$			289583.i			0.412368i
$839$		−	1.26209e6i		−	1.79294i	−0.443104	−	0.896470i	$-0.646123\pi$
$839$		−	1.26209e6i		−	1.79294i	0.443104	−	0.896470i	$-0.353877\pi$
$840$	−77454.4			−0.109771
$841$	1.93199e6			2.73157
$842$		−	37261.7i		−	0.0525580i
$843$		−	612251.i		−	0.861537i
$844$	−296852.			−0.416731
$845$			102536.i			0.143603i
$846$	−196458.			−0.274491
$847$		−	230688.i		−	0.321557i
$848$			133106.i			0.185099i
$849$			168605.i			0.233914i
$850$			108300.i			0.149896i
$851$	−496728.	+	138689.i	−0.685898	+	0.191506i
$852$	57627.8			0.0793876
$853$	−39888.6			−0.0548214			−0.0274107	−	0.999624i	$-0.508726\pi$
$853$	−39888.6			−0.0548214			−0.0274107	+	0.999624i	$0.508726\pi$
$854$	59018.4			0.0809230
$855$	−343176.			−0.469446
$856$			432531.i			0.590295i
$857$	1.06525e6			1.45040			0.725202	−	0.688536i	$-0.241746\pi$
$857$	1.06525e6			1.45040			0.725202	+	0.688536i	$0.241746\pi$
$858$			126761.i			0.172192i
$859$	292913.			0.396965			0.198483	−	0.980104i	$-0.436399\pi$
$859$	292913.			0.396965			0.198483	+	0.980104i	$0.436399\pi$
$860$	−142385.			−0.192516
$861$			17530.6i			0.0236477i
$862$		−	237432.i		−	0.319539i
$863$	−1.27218e6			−1.70816			−0.854080	−	0.520141i	$-0.825879\pi$
$863$	−1.27218e6			−1.70816			−0.854080	+	0.520141i	$0.825879\pi$
$864$	25396.3			0.0340207
$865$		−	1.02996e6i		−	1.37653i
$866$		−	163178.i		−	0.217584i
$867$	402995.			0.536120
$868$		−	102636.i		−	0.136226i
$869$	−211823.			−0.280501
$870$		−	799338.i		−	1.05607i
$871$		−	757956.i		−	0.999097i
$872$		−	259901.i		−	0.341803i
$873$		−	74918.6i		−	0.0983017i
$874$	547132.	−	152762.i	0.716258	−	0.199982i
$875$	−85119.4			−0.111176
$876$	−108817.			−0.141804
$877$	145686.			0.189417			0.0947083	−	0.995505i	$-0.469808\pi$
$877$	145686.			0.189417			0.0947083	+	0.995505i	$0.469808\pi$
$878$	−516284.			−0.669729
$879$		−	461728.i		−	0.597597i
$880$	−115730.			−0.149445
$881$		−	781737.i		−	1.00718i	−0.863941	−	0.503592i	$-0.832011\pi$
$881$		−	781737.i		−	1.00718i	0.863941	−	0.503592i	$-0.167989\pi$
$882$	153789.			0.197692
$883$	1.11076e6			1.42462			0.712312	−	0.701863i	$-0.247648\pi$
$883$	1.11076e6			1.42462			0.712312	+	0.701863i	$0.247648\pi$
$884$			98657.8i			0.126249i
$885$		−	1491.93i		−	0.00190486i
$886$	611949.			0.779557
$887$	275114.			0.349676			0.174838	−	0.984597i	$-0.444060\pi$
$887$	275114.			0.349676			0.174838	+	0.984597i	$0.444060\pi$
$888$			114625.i			0.145363i
$889$		−	129403.i		−	0.163735i
$890$	1.07178e6			1.35309
$891$		−	39376.1i		−	0.0495995i
$892$	−635840.			−0.799131
$893$			976681.i			1.22476i
$894$		−	236699.i		−	0.296157i
$895$		−	1.31853e6i		−	1.64605i
$896$		−	28495.9i		−	0.0354949i
$897$	−118036.	−	422758.i	−0.146700	−	0.525421i
$898$	−1.04309e6			−1.29351
$899$	1.05921e6			1.31058
$900$	−107090.			−0.132210
$901$	160621.			0.197858
$902$			26193.7i			0.0321946i
$903$	−54357.7			−0.0666631
$904$			378932.i			0.463686i
$905$	−1.09568e6			−1.33778
$906$	−81747.6			−0.0995906
$907$			772272.i			0.938762i	0.882996	+	0.469381i	$0.155523\pi$
$907$			772272.i			0.938762i	−0.882996	+	0.469381i	$0.844477\pi$
$908$			597310.i			0.724482i
$909$	−86461.0			−0.104639
$910$	297529.			0.359291
$911$		−	1.42645e6i		−	1.71878i	−0.511321	−	0.859390i	$-0.670844\pi$
$911$		−	1.42645e6i		−	1.71878i	0.511321	−	0.859390i	$-0.329156\pi$
$912$		−	126256.i		−	0.151797i
$913$	570227.			0.684079
$914$			437658.i			0.523893i
$915$	184467.			0.220332
$916$			261266.i			0.311381i
$917$			138551.i			0.164767i
$918$		−	30646.2i		−	0.0363657i
$919$		−	369970.i		−	0.438062i	−0.975718	−	0.219031i	$-0.929710\pi$
$919$		−	369970.i		−	0.438062i	0.975718	−	0.219031i	$-0.0702896\pi$
$920$	385969.	−	107764.i	0.456012	−	0.127321i
$921$	−108699.			−0.128147
$922$	391244.			0.460241
$923$	−221368.			−0.259843
$924$	−44181.9			−0.0517488
$925$		−	483348.i		−	0.564907i
$926$	137630.			0.160506
$927$		−	378355.i		−	0.440292i
$928$	294081.			0.341484
$929$	−1.25590e6			−1.45520			−0.727599	−	0.686002i	$-0.759364\pi$
$929$	−1.25590e6			−1.45520			−0.727599	+	0.686002i	$0.759364\pi$
$930$		−	320797.i		−	0.370907i
$931$		−	764554.i		−	0.882082i
$932$	155230.			0.178708
$933$	153196.			0.175989
$934$			772887.i			0.885977i
$935$			139654.i			0.159746i
$936$	−97555.9			−0.111353
$937$		−	1.39349e6i		−	1.58717i	−0.608457	−	0.793587i	$-0.708211\pi$
$937$		−	1.39349e6i		−	1.58717i	0.608457	−	0.793587i	$-0.291789\pi$
$938$	264181.			0.300259
$939$		−	417317.i		−	0.473299i
$940$			688990.i			0.779753i
$941$		−	1.37875e6i		−	1.55706i	−0.627605	−	0.778532i	$-0.715965\pi$
$941$		−	1.37875e6i		−	1.55706i	0.627605	−	0.778532i	$-0.284035\pi$
$942$		−	605946.i		−	0.682860i
$943$	−24390.7	−	87357.9i	−0.0274284	−	0.0982378i
$944$	548.890			0.000615945
$945$	−92421.9			−0.103493
$946$	−81219.7			−0.0907568
$947$	−1.30227e6			−1.45212			−0.726059	−	0.687632i	$-0.758650\pi$
$947$	−1.30227e6			−1.45212			−0.726059	+	0.687632i	$0.758650\pi$
$948$		−	163020.i		−	0.181394i
$949$	418005.			0.464140
$950$			532395.i			0.589911i
$951$	−128849.			−0.142469
$952$	−34386.6			−0.0379415
$953$			161841.i			0.178198i	0.996023	+	0.0890991i	$0.0283988\pi$
$953$			161841.i			0.178198i	−0.996023	+	0.0890991i	$0.971601\pi$
$954$			158827.i			0.174513i
$955$	−767705.			−0.841759
$956$	−595906.			−0.652022
$957$		−	455962.i		−	0.497857i
$958$		−	583942.i		−	0.636266i
$959$	−513421.			−0.558260
$960$		−	89066.4i		−	0.0966432i
$961$	−498429.			−0.539705
$962$		−	440315.i		−	0.475787i
$963$			516114.i			0.556536i
$964$			240740.i			0.259057i
$965$			854648.i			0.917768i
$966$	147350.	−	41140.7i	0.157905	−	0.0440877i
$967$	−234245.			−0.250506			−0.125253	−	0.992125i	$-0.539974\pi$
$967$	−234245.			−0.250506			−0.125253	+	0.992125i	$0.539974\pi$
$968$	265273.			0.283101
$969$	−152356.			−0.162260
$970$	−262744.			−0.279248
$971$		−	27276.6i		−	0.0289302i	−0.999895	−	0.0144651i	$-0.995395\pi$
$971$		−	27276.6i		−	0.0289302i	0.999895	−	0.0144651i	$-0.00460454\pi$
$972$	30304.0			0.0320750
$973$		−	27571.7i		−	0.0291232i
$974$	−339242.			−0.357595
$975$	411371.			0.432738
$976$			67866.5i			0.0712452i
$977$			1.49726e6i			1.56858i	0.620392	+	0.784292i	$0.286973\pi$
$977$			1.49726e6i			1.56858i	−0.620392	+	0.784292i	$0.713027\pi$
$978$	568866.			0.594747
$979$	611372.			0.637881
$980$		−	539347.i		−	0.561586i
$981$		−	310125.i		−	0.322255i
$982$	−293907.			−0.304780
$983$			1.82952e6i			1.89335i	0.322190	+	0.946675i	$0.395581\pi$
$983$			1.82952e6i			1.89335i	−0.322190	+	0.946675i	$0.604419\pi$
$984$	−20158.7			−0.0208196
$985$		−	1.58087e6i		−	1.62938i
$986$		−	354873.i		−	0.365022i
$987$			263033.i			0.270007i
$988$			484994.i			0.496847i
$989$	270874.	−	75629.1i	0.276933	−	0.0773208i
$990$	−138094.			−0.140898
$991$	362117.			0.368724			0.184362	−	0.982858i	$-0.440978\pi$
$991$	362117.			0.368724			0.184362	+	0.982858i	$0.440978\pi$
$992$	118023.			0.119934
$993$	−660189.			−0.669530
$994$		−	77156.5i		−	0.0780908i
$995$	1.44052e6			1.45504
$996$			438849.i			0.442381i
$997$	1.39824e6			1.40667			0.703335	−	0.710859i	$-0.251694\pi$
$997$	1.39824e6			1.40667			0.703335	+	0.710859i	$0.251694\pi$
$998$	−1.28609e6			−1.29125
$999$			136776.i			0.137050i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.b.a.91.13	✓	16
3.2	odd	2		414.5.b.b.91.8		16
4.3	odd	2		1104.5.c.a.1057.2		16
23.22	odd	2	inner	138.5.b.a.91.16	yes	16
69.68	even	2		414.5.b.b.91.1		16
92.91	even	2		1104.5.c.a.1057.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.b.a.91.13	✓	16	1.1	even	1	trivial
138.5.b.a.91.16	yes	16	23.22	odd	2	inner
414.5.b.b.91.1		16	69.68	even	2
414.5.b.b.91.8		16	3.2	odd	2
1104.5.c.a.1057.2		16	4.3	odd	2
1104.5.c.a.1057.7		16	92.91	even	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -33.4782i q^{5} +14.6969 q^{6} -19.6774i q^{7} +22.6274 q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -33.4782i q^{5} +14.6969 q^{6} -19.6774i q^{7} +22.6274 q^{8} +27.0000 q^{9} -94.6906i q^{10} -54.0138i q^{11} +41.5692 q^{12} -159.682 q^{13} -55.6561i q^{14} -173.958i q^{15} +64.0000 q^{16} -77.2301i q^{17} +76.3675 q^{18} -379.657i q^{19} -267.826i q^{20} -102.247i q^{21} -152.774i q^{22} +(142.258 + 509.513i) q^{23} +117.576 q^{24} -495.789 q^{25} -451.648 q^{26} +140.296 q^{27} -157.419i q^{28} +1624.58 q^{29} -492.027i q^{30} +651.991 q^{31} +181.019 q^{32} -280.664i q^{33} -218.440i q^{34} -658.763 q^{35} +216.000 q^{36} +974.907i q^{37} -1073.83i q^{38} -829.730 q^{39} -757.525i q^{40} -171.454 q^{41} -289.197i q^{42} -531.633i q^{43} -432.110i q^{44} -903.911i q^{45} +(402.367 + 1441.12i) q^{46} -2572.53 q^{47} +332.554 q^{48} +2013.80 q^{49} -1402.30 q^{50} -401.299i q^{51} -1277.45 q^{52} +2079.78i q^{53} +396.817 q^{54} -1808.28 q^{55} -445.248i q^{56} -1972.76i q^{57} +4595.01 q^{58} +8.57641 q^{59} -1391.66i q^{60} +1060.41i q^{61} +1844.11 q^{62} -531.289i q^{63} +512.000 q^{64} +5345.85i q^{65} -793.838i q^{66} +4746.67i q^{67} -617.840i q^{68} +(739.195 + 2647.51i) q^{69} -1863.26 q^{70} +1386.31 q^{71} +610.940 q^{72} -2617.74 q^{73} +2757.45i q^{74} -2576.20 q^{75} -3037.26i q^{76} -1062.85 q^{77} -2346.83 q^{78} -3921.65i q^{79} -2142.60i q^{80} +729.000 q^{81} -484.944 q^{82} +10557.1i q^{83} -817.974i q^{84} -2585.52 q^{85} -1503.68i q^{86} +8441.58 q^{87} -1222.19i q^{88} +11318.8i q^{89} -2556.65i q^{90} +3142.12i q^{91} +(1138.07 + 4076.10i) q^{92} +3387.85 q^{93} -7276.22 q^{94} -12710.2 q^{95} +940.604 q^{96} -2774.76i q^{97} +5695.89 q^{98} -1458.37i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) \) 16 * q + 128 * q^4 + 432 * q^9	\( 16 q + 128 q^{4} + 432 q^{9} - 208 q^{13} + 1024 q^{16} + 840 q^{23} + 1056 q^{25} + 1920 q^{26} + 3600 q^{29} + 224 q^{31} - 3264 q^{35} + 3456 q^{36} - 2016 q^{39} - 6144 q^{41} + 1280 q^{46} + 8880 q^{47} - 13888 q^{49} + 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} - 18240 q^{59} + 8192 q^{64} + 10584 q^{69} + 19584 q^{70} - 30048 q^{71} + 9536 q^{73} - 4176 q^{75} + 14160 q^{77} + 6912 q^{78} + 11664 q^{81} - 19584 q^{82} - 32496 q^{85} - 8064 q^{87} + 6720 q^{92} - 11952 q^{93} - 21248 q^{94} - 20064 q^{95} + 21504 q^{98}+O(q^{100}) \) 16 * q + 128 * q^4 + 432 * q^9 - 208 * q^13 + 1024 * q^16 + 840 * q^23 + 1056 * q^25 + 1920 * q^26 + 3600 * q^29 + 224 * q^31 - 3264 * q^35 + 3456 * q^36 - 2016 * q^39 - 6144 * q^41 + 1280 * q^46 + 8880 * q^47 - 13888 * q^49 + 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 - 18240 * q^59 + 8192 * q^64 + 10584 * q^69 + 19584 * q^70 - 30048 * q^71 + 9536 * q^73 - 4176 * q^75 + 14160 * q^77 + 6912 * q^78 + 11664 * q^81 - 19584 * q^82 - 32496 * q^85 - 8064 * q^87 + 6720 * q^92 - 11952 * q^93 - 21248 * q^94 - 20064 * q^95 + 21504 * q^98

Introduction

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