LMFDB - Embedded newform 138.5.b.a.91.9

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			91.9
Root			$0.707107 + 1.47108i$ of defining polynomial
Character	$\chi$	$=$	138.91
Dual form			138.5.b.a.91.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{2} -5.19615 q^{3} +8.00000 q^{4} -19.9722i q^{5} -14.6969 q^{6} +58.9751i q^{7} +22.6274 q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+2.82843 q^{2} -5.19615 q^{3} +8.00000 q^{4} -19.9722i q^{5} -14.6969 q^{6} +58.9751i q^{7} +22.6274 q^{8} +27.0000 q^{9} -56.4898i q^{10} -168.627i q^{11} -41.5692 q^{12} +288.481 q^{13} +166.807i q^{14} +103.778i q^{15} +64.0000 q^{16} -311.477i q^{17} +76.3675 q^{18} -118.278i q^{19} -159.777i q^{20} -306.444i q^{21} -476.949i q^{22} +(-383.722 - 364.140i) q^{23} -117.576 q^{24} +226.112 q^{25} +815.949 q^{26} -140.296 q^{27} +471.801i q^{28} +991.638 q^{29} +293.530i q^{30} +827.898 q^{31} +181.019 q^{32} +876.212i q^{33} -880.991i q^{34} +1177.86 q^{35} +216.000 q^{36} -80.7914i q^{37} -334.541i q^{38} -1498.99 q^{39} -451.919i q^{40} +187.684 q^{41} -866.754i q^{42} -331.510i q^{43} -1349.02i q^{44} -539.249i q^{45} +(-1085.33 - 1029.94i) q^{46} -1896.65 q^{47} -332.554 q^{48} -1077.07 q^{49} +639.542 q^{50} +1618.48i q^{51} +2307.85 q^{52} +4190.84i q^{53} -396.817 q^{54} -3367.85 q^{55} +1334.46i q^{56} +614.591i q^{57} +2804.78 q^{58} -4961.39 q^{59} +830.228i q^{60} +5565.38i q^{61} +2341.65 q^{62} +1592.33i q^{63} +512.000 q^{64} -5761.60i q^{65} +2478.30i q^{66} -6833.36i q^{67} -2491.82i q^{68} +(1993.88 + 1892.13i) q^{69} +3331.50 q^{70} +1911.46 q^{71} +610.940 q^{72} -6606.73 q^{73} -228.512i q^{74} -1174.91 q^{75} -946.225i q^{76} +9944.80 q^{77} -4239.79 q^{78} +11343.7i q^{79} -1278.22i q^{80} +729.000 q^{81} +530.850 q^{82} -3573.28i q^{83} -2451.55i q^{84} -6220.88 q^{85} -937.652i q^{86} -5152.70 q^{87} -3815.59i q^{88} -3641.31i q^{89} -1525.23i q^{90} +17013.2i q^{91} +(-3069.78 - 2913.12i) q^{92} -4301.88 q^{93} -5364.55 q^{94} -2362.27 q^{95} -940.604 q^{96} -9509.93i q^{97} -3046.41 q^{98} -4552.93i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	19.9722i		−	0.798887i	−0.916758	−	0.399443i	$-0.869203\pi$
$5$		−	19.9722i		−	0.798887i	0.916758	−	0.399443i	$-0.130797\pi$
$6$	−14.6969			−0.408248
$7$			58.9751i			1.20357i	0.798656	+	0.601787i	$0.205544\pi$
$7$			58.9751i			1.20357i	−0.798656	+	0.601787i	$0.794456\pi$
$8$	22.6274			0.353553
$9$	27.0000			0.333333
$10$		−	56.4898i		−	0.564898i
$11$		−	168.627i		−	1.39361i	−0.717260	−	0.696806i	$-0.754604\pi$
$11$		−	168.627i		−	1.39361i	0.717260	−	0.696806i	$-0.245396\pi$
$12$	−41.5692			−0.288675
$13$	288.481			1.70699			0.853495	−	0.521101i	$-0.174478\pi$
$13$	288.481			1.70699			0.853495	+	0.521101i	$0.174478\pi$
$14$			166.807i			0.851056i
$15$			103.778i			0.461238i
$16$	64.0000			0.250000
$17$		−	311.477i		−	1.07778i	−0.842377	−	0.538888i	$-0.818845\pi$
$17$		−	311.477i		−	1.07778i	0.842377	−	0.538888i	$-0.181155\pi$
$18$	76.3675			0.235702
$19$		−	118.278i		−	0.327640i	−0.986490	−	0.163820i	$-0.947618\pi$
$19$		−	118.278i		−	0.327640i	0.986490	−	0.163820i	$-0.0523816\pi$
$20$		−	159.777i		−	0.399443i
$21$		−	306.444i		−	0.694884i
$22$		−	476.949i		−	0.985432i
$23$	−383.722	−	364.140i	−0.725373	−	0.688356i
$23$	−383.722	−	364.140i	−0.725373	−	0.688356i
$24$	−117.576			−0.204124
$25$	226.112			0.361780
$26$	815.949			1.20702
$27$	−140.296			−0.192450
$28$			471.801i			0.601787i
$29$	991.638			1.17912			0.589559	−	0.807725i	$-0.299302\pi$
$29$	991.638			1.17912			0.589559	+	0.807725i	$0.299302\pi$
$30$			293.530i			0.326144i
$31$	827.898			0.861496			0.430748	−	0.902472i	$-0.358250\pi$
$31$	827.898			0.861496			0.430748	+	0.902472i	$0.358250\pi$
$32$	181.019			0.176777
$33$			876.212i			0.804602i
$34$		−	880.991i		−	0.762103i
$35$	1177.86			0.961520
$36$	216.000			0.166667
$37$		−	80.7914i		−	0.0590149i	−0.999565	−	0.0295074i	$-0.990606\pi$
$37$		−	80.7914i		−	0.0590149i	0.999565	−	0.0295074i	$-0.00939387\pi$
$38$		−	334.541i		−	0.231677i
$39$	−1498.99			−0.985531
$40$		−	451.919i		−	0.282449i
$41$	187.684			0.111650			0.0558251	−	0.998441i	$-0.482221\pi$
$41$	187.684			0.111650			0.0558251	+	0.998441i	$0.482221\pi$
$42$		−	866.754i		−	0.491357i
$43$		−	331.510i		−	0.179292i	−0.995974	−	0.0896458i	$-0.971426\pi$
$43$		−	331.510i		−	0.179292i	0.995974	−	0.0896458i	$-0.0285735\pi$
$44$		−	1349.02i		−	0.696806i
$45$		−	539.249i		−	0.266296i
$46$	−1085.33	−	1029.94i	−0.512916	−	0.486741i
$47$	−1896.65			−0.858603			−0.429302	−	0.903161i	$-0.641240\pi$
$47$	−1896.65			−0.858603			−0.429302	+	0.903161i	$0.641240\pi$
$48$	−332.554			−0.144338
$49$	−1077.07			−0.448591
$50$	639.542			0.255817
$51$			1618.48i			0.622254i
$52$	2307.85			0.853495
$53$			4190.84i			1.49193i	0.665984	+	0.745966i	$0.268012\pi$
$53$			4190.84i			1.49193i	−0.665984	+	0.745966i	$0.731988\pi$
$54$	−396.817			−0.136083
$55$	−3367.85			−1.11334
$56$			1334.46i			0.425528i
$57$			614.591i			0.189163i
$58$	2804.78			0.833762
$59$	−4961.39			−1.42528			−0.712638	−	0.701532i	$-0.752500\pi$
$59$	−4961.39			−1.42528			−0.712638	+	0.701532i	$0.752500\pi$
$60$			830.228i			0.230619i
$61$			5565.38i			1.49567i	0.663886	+	0.747834i	$0.268906\pi$
$61$			5565.38i			1.49567i	−0.663886	+	0.747834i	$0.731094\pi$
$62$	2341.65			0.609170
$63$			1592.33i			0.401191i
$64$	512.000			0.125000
$65$		−	5761.60i		−	1.36369i
$66$			2478.30i			0.568940i
$67$		−	6833.36i		−	1.52225i	−0.648608	−	0.761123i	$-0.724649\pi$
$67$		−	6833.36i		−	1.52225i	0.648608	−	0.761123i	$-0.275351\pi$
$68$		−	2491.82i		−	0.538888i
$69$	1993.88	+	1892.13i	0.418794	+	0.397422i
$70$	3331.50			0.679897
$71$	1911.46			0.379183			0.189592	−	0.981863i	$-0.439284\pi$
$71$	1911.46			0.379183			0.189592	+	0.981863i	$0.439284\pi$
$72$	610.940			0.117851
$73$	−6606.73			−1.23977			−0.619885	−	0.784693i	$-0.712821\pi$
$73$	−6606.73			−1.23977			−0.619885	+	0.784693i	$0.712821\pi$
$74$		−	228.512i		−	0.0417298i
$75$	−1174.91			−0.208874
$76$		−	946.225i		−	0.163820i
$77$	9944.80			1.67732
$78$	−4239.79			−0.696876
$79$			11343.7i			1.81761i	0.417226	+	0.908803i	$0.363002\pi$
$79$			11343.7i			1.81761i	−0.417226	+	0.908803i	$0.636998\pi$
$80$		−	1278.22i		−	0.199722i
$81$	729.000			0.111111
$82$	530.850			0.0789486
$83$		−	3573.28i		−	0.518694i	−0.965784	−	0.259347i	$-0.916493\pi$
$83$		−	3573.28i		−	0.518694i	0.965784	−	0.259347i	$-0.0835073\pi$
$84$		−	2451.55i		−	0.347442i
$85$	−6220.88			−0.861021
$86$		−	937.652i		−	0.126778i
$87$	−5152.70			−0.680764
$88$		−	3815.59i		−	0.492716i
$89$		−	3641.31i		−	0.459703i	−0.973226	−	0.229851i	$-0.926176\pi$
$89$		−	3641.31i		−	0.459703i	0.973226	−	0.229851i	$-0.0738240\pi$
$90$		−	1525.23i		−	0.188299i
$91$			17013.2i			2.05449i
$92$	−3069.78	−	2913.12i	−0.362687	−	0.344178i
$93$	−4301.88			−0.497385
$94$	−5364.55			−0.607124
$95$	−2362.27			−0.261747
$96$	−940.604			−0.102062
$97$		−	9509.93i		−	1.01073i	−0.862907	−	0.505363i	$-0.831358\pi$
$97$		−	9509.93i		−	1.01073i	0.862907	−	0.505363i	$-0.168642\pi$
$98$	−3046.41			−0.317202
$99$		−	4552.93i		−	0.464537i
$100$	1808.90			0.180890
$101$	7894.84			0.773928			0.386964	−	0.922095i	$-0.373524\pi$
$101$	7894.84			0.773928			0.386964	+	0.922095i	$0.373524\pi$
$102$			4577.76i			0.440000i
$103$			9239.82i			0.870942i	0.900203	+	0.435471i	$0.143418\pi$
$103$			9239.82i			0.870942i	−0.900203	+	0.435471i	$0.856582\pi$
$104$	6527.59			0.603512
$105$	−6120.35			−0.555134
$106$			11853.5i			1.05496i
$107$			18393.4i			1.60655i	0.595607	+	0.803276i	$0.296911\pi$
$107$			18393.4i			1.60655i	−0.595607	+	0.803276i	$0.703089\pi$
$108$	−1122.37			−0.0962250
$109$		−	1577.05i		−	0.132737i	−0.997795	−	0.0663687i	$-0.978859\pi$
$109$		−	1577.05i		−	0.132737i	0.997795	−	0.0663687i	$-0.0211414\pi$
$110$	−9525.71			−0.787249
$111$			419.804i			0.0340723i
$112$			3774.41i			0.300894i
$113$		−	12500.1i		−	0.978942i	−0.872020	−	0.489471i	$-0.837190\pi$
$113$		−	12500.1i		−	0.978942i	0.872020	−	0.489471i	$-0.162810\pi$
$114$			1738.33i			0.133759i
$115$	−7272.67	+	7663.77i	−0.549918	+	0.579491i
$116$	7933.10			0.589559
$117$	7789.00			0.568997
$118$	−14032.9			−1.00782
$119$	18369.4			1.29718
$120$			2348.24i			0.163072i
$121$	−13794.1			−0.942153
$122$			15741.3i			1.05760i
$123$	−975.234			−0.0644612
$124$	6623.18			0.430748
$125$		−	16998.6i		−	1.08791i
$126$			4503.79i			0.283685i
$127$	−26262.9			−1.62830			−0.814152	−	0.580652i	$-0.802798\pi$
$127$	−26262.9			−1.62830			−0.814152	+	0.580652i	$0.802798\pi$
$128$	1448.15			0.0883883
$129$			1722.58i			0.103514i
$130$		−	16296.3i		−	0.964276i
$131$	18959.4			1.10479			0.552397	−	0.833581i	$-0.313713\pi$
$131$	18959.4			1.10479			0.552397	+	0.833581i	$0.313713\pi$
$132$			7009.69i			0.402301i
$133$	6975.47			0.394339
$134$		−	19327.7i		−	1.07639i
$135$			2802.02i			0.153746i
$136$		−	7047.93i		−	0.381051i
$137$		−	10213.0i		−	0.544140i	−0.962277	−	0.272070i	$-0.912292\pi$
$137$		−	10213.0i		−	0.544140i	0.962277	−	0.272070i	$-0.0877082\pi$
$138$	5639.55	+	5351.75i	0.296132	+	0.281020i
$139$	19685.0			1.01884			0.509420	−	0.860518i	$-0.329860\pi$
$139$	19685.0			1.01884			0.509420	+	0.860518i	$0.329860\pi$
$140$	9422.90			0.480760
$141$	9855.30			0.495715
$142$	5406.43			0.268123
$143$		−	48645.8i		−	2.37888i
$144$	1728.00			0.0833333
$145$		−	19805.2i		−	0.941982i
$146$	−18686.7			−0.876650
$147$	5596.61			0.258994
$148$		−	646.331i		−	0.0295074i
$149$			18845.6i			0.848864i	0.905460	+	0.424432i	$0.139526\pi$
$149$			18845.6i			0.848864i	−0.905460	+	0.424432i	$0.860474\pi$
$150$	−3323.16			−0.147696
$151$	−6343.55			−0.278214			−0.139107	−	0.990277i	$-0.544423\pi$
$151$	−6343.55			−0.278214			−0.139107	+	0.990277i	$0.544423\pi$
$152$		−	2676.33i		−	0.115838i
$153$		−	8409.89i		−	0.359259i
$154$	28128.1			1.18604
$155$		−	16534.9i		−	0.688238i
$156$	−11991.9			−0.492766
$157$			26439.6i			1.07264i	0.844014	+	0.536321i	$0.180186\pi$
$157$			26439.6i			1.07264i	−0.844014	+	0.536321i	$0.819814\pi$
$158$			32084.8i			1.28524i
$159$		−	21776.2i		−	0.861367i
$160$		−	3615.35i		−	0.141225i
$161$	21475.2	−	22630.1i	0.828487	−	0.873041i
$162$	2061.92			0.0785674
$163$	11346.9			0.427072			0.213536	−	0.976935i	$-0.431502\pi$
$163$	11346.9			0.427072			0.213536	+	0.976935i	$0.431502\pi$
$164$	1501.47			0.0558251
$165$	17499.9			0.642786
$166$		−	10106.8i		−	0.366772i
$167$	6625.97			0.237584			0.118792	−	0.992919i	$-0.462098\pi$
$167$	6625.97			0.237584			0.118792	+	0.992919i	$0.462098\pi$
$168$		−	6934.03i		−	0.245679i
$169$	54660.5			1.91382
$170$	−17595.3			−0.608834
$171$		−	3193.51i		−	0.109213i
$172$		−	2652.08i		−	0.0896458i
$173$	27706.0			0.925725			0.462863	−	0.886430i	$-0.346822\pi$
$173$	27706.0			0.925725			0.462863	+	0.886430i	$0.346822\pi$
$174$	−14574.0			−0.481373
$175$			13335.0i			0.435429i
$176$		−	10792.1i		−	0.348403i
$177$	25780.1			0.822884
$178$		−	10299.2i		−	0.325059i
$179$	−22956.0			−0.716457			−0.358228	−	0.933634i	$-0.616619\pi$
$179$	−22956.0			−0.716457			−0.358228	+	0.933634i	$0.616619\pi$
$180$		−	4313.99i		−	0.133148i
$181$		−	19018.6i		−	0.580526i	−0.956947	−	0.290263i	$-0.906257\pi$
$181$		−	19018.6i		−	0.580526i	0.956947	−	0.290263i	$-0.0937427\pi$
$182$			48120.7i			1.45274i
$183$		−	28918.6i		−	0.863524i
$184$	−8682.65	−	8239.55i	−0.256458	−	0.243371i
$185$	−1613.58			−0.0471462
$186$	−12167.6			−0.351704
$187$	−52523.5			−1.50200
$188$	−15173.2			−0.429302
$189$		−	8273.98i		−	0.231628i
$190$	−6681.51			−0.185083
$191$			70888.2i			1.94315i	0.236726	+	0.971576i	$0.423926\pi$
$191$			70888.2i			1.94315i	−0.236726	+	0.971576i	$0.576074\pi$
$192$	−2660.43			−0.0721688
$193$	23342.4			0.626658			0.313329	−	0.949645i	$-0.398556\pi$
$193$	23342.4			0.626658			0.313329	+	0.949645i	$0.398556\pi$
$194$		−	26898.1i		−	0.714692i
$195$			29938.2i			0.787328i
$196$	−8616.54			−0.224296
$197$	−51130.3			−1.31749			−0.658743	−	0.752368i	$-0.728911\pi$
$197$	−51130.3			−1.31749			−0.658743	+	0.752368i	$0.728911\pi$
$198$		−	12877.6i		−	0.328477i
$199$		−	68616.0i		−	1.73268i	−0.499452	−	0.866342i	$-0.666465\pi$
$199$		−	68616.0i		−	1.73268i	0.499452	−	0.866342i	$-0.333535\pi$
$200$	5116.34			0.127908
$201$			35507.2i			0.878869i
$202$	22330.0			0.547250
$203$			58482.0i			1.41916i
$204$			12947.9i			0.311127i
$205$		−	3748.45i		−	0.0891958i
$206$			26134.2i			0.615849i
$207$	−10360.5	−	9831.79i	−0.241791	−	0.229452i
$208$	18462.8			0.426748
$209$	−19944.9			−0.456603
$210$	−17311.0			−0.392539
$211$	−59489.3			−1.33621			−0.668104	−	0.744068i	$-0.732894\pi$
$211$	−59489.3			−1.33621			−0.668104	+	0.744068i	$0.732894\pi$
$212$			33526.7i			0.745966i
$213$	−9932.25			−0.218922
$214$			52024.4i			1.13600i
$215$	−6620.98			−0.143234
$216$	−3174.54			−0.0680414
$217$			48825.4i			1.03687i
$218$		−	4460.58i		−	0.0938595i
$219$	34329.6			0.715781
$220$	−26942.8			−0.556669
$221$		−	89855.4i		−	1.83975i
$222$			1187.39i			0.0240927i
$223$	59731.0			1.20113			0.600565	−	0.799576i	$-0.294942\pi$
$223$	59731.0			1.20113			0.600565	+	0.799576i	$0.294942\pi$
$224$			10675.6i			0.212764i
$225$	6105.03			0.120593
$226$		−	35355.7i		−	0.692217i
$227$			36166.2i			0.701860i	0.936402	+	0.350930i	$0.114135\pi$
$227$			36166.2i			0.701860i	−0.936402	+	0.350930i	$0.885865\pi$
$228$			4916.73i			0.0945816i
$229$			45527.5i			0.868165i	0.900873	+	0.434083i	$0.142927\pi$
$229$			45527.5i			0.868165i	−0.900873	+	0.434083i	$0.857073\pi$
$230$	−20570.2	+	21676.4i	−0.388851	+	0.409762i
$231$	−51674.7			−0.968398
$232$	22438.2			0.416881
$233$	−34923.6			−0.643291			−0.321645	−	0.946860i	$-0.604236\pi$
$233$	−34923.6			−0.643291			−0.321645	+	0.946860i	$0.604236\pi$
$234$	22030.6			0.402342
$235$			37880.3i			0.685927i
$236$	−39691.1			−0.712638
$237$		−	58943.5i		−	1.04939i
$238$	51956.6			0.917247
$239$	−87697.7			−1.53530			−0.767649	−	0.640871i	$-0.778573\pi$
$239$	−87697.7			−1.53530			−0.767649	+	0.640871i	$0.778573\pi$
$240$			6641.82i			0.115309i
$241$		−	78064.1i		−	1.34406i	−0.740526	−	0.672028i	$-0.765423\pi$
$241$		−	78064.1i		−	1.34406i	0.740526	−	0.672028i	$-0.234577\pi$
$242$	−39015.5			−0.666203
$243$	−3788.00			−0.0641500
$244$			44523.0i			0.747834i
$245$			21511.4i			0.358374i
$246$	−2758.38			−0.0455810
$247$		−	34121.0i		−	0.559279i
$248$	18733.2			0.304585
$249$			18567.3i			0.299468i
$250$		−	48079.2i		−	0.769267i
$251$			51539.5i			0.818075i	0.912518	+	0.409037i	$0.134135\pi$
$251$			51539.5i			0.818075i	−0.912518	+	0.409037i	$0.865865\pi$
$252$			12738.6i			0.200596i
$253$	−61403.9	+	64706.0i	−0.959301	+	1.01089i
$254$	−74282.7			−1.15138
$255$	32324.6			0.497111
$256$	4096.00			0.0625000
$257$	−44584.2			−0.675016			−0.337508	−	0.941323i	$-0.609584\pi$
$257$	−44584.2			−0.675016			−0.337508	+	0.941323i	$0.609584\pi$
$258$			4872.18i			0.0731955i
$259$	4764.68			0.0710288
$260$		−	46092.8i		−	0.681846i
$261$	26774.2			0.393039
$262$	53625.1			0.781207
$263$			100390.i			1.45138i	0.688023	+	0.725688i	$0.258479\pi$
$263$			100390.i			1.45138i	−0.688023	+	0.725688i	$0.741521\pi$
$264$			19826.4i			0.284470i
$265$	83700.1			1.19188
$266$	19729.6			0.278840
$267$			18920.8i			0.265410i
$268$		−	54666.9i		−	0.761123i
$269$	89798.2			1.24097			0.620487	−	0.784216i	$-0.286935\pi$
$269$	89798.2			1.24097			0.620487	+	0.784216i	$0.286935\pi$
$270$			7925.30i			0.108715i
$271$	−9946.27			−0.135432			−0.0677161	−	0.997705i	$-0.521571\pi$
$271$	−9946.27			−0.135432			−0.0677161	+	0.997705i	$0.521571\pi$
$272$		−	19934.5i		−	0.269444i
$273$		−	88403.4i		−	1.18616i
$274$		−	28886.6i		−	0.384765i
$275$		−	38128.6i		−	0.504180i
$276$	15951.0	+	15137.0i	0.209397	+	0.198711i
$277$	−137747.			−1.79524			−0.897620	−	0.440770i	$-0.854705\pi$
$277$	−137747.			−1.79524			−0.897620	+	0.440770i	$0.854705\pi$
$278$	55677.6			0.720428
$279$	22353.2			0.287165
$280$	26652.0			0.339949
$281$			144672.i			1.83220i	0.400952	+	0.916099i	$0.368679\pi$
$281$			144672.i			1.83220i	−0.400952	+	0.916099i	$0.631321\pi$
$282$	27875.0			0.350523
$283$			11368.5i			0.141948i	0.997478	+	0.0709740i	$0.0226107\pi$
$283$			11368.5i			0.141948i	−0.997478	+	0.0709740i	$0.977389\pi$
$284$	15291.7			0.189592
$285$	12274.7			0.151120
$286$		−	137591.i		−	1.68212i
$287$			11068.7i			0.134379i
$288$	4887.52			0.0589256
$289$	−13497.1			−0.161601
$290$		−	56017.5i		−	0.666082i
$291$			49415.0i			0.583543i
$292$	−52853.9			−0.619885
$293$			16897.0i			0.196823i	0.995146	+	0.0984113i	$0.0313761\pi$
$293$			16897.0i			0.196823i	−0.995146	+	0.0984113i	$0.968624\pi$
$294$	15829.6			0.183137
$295$			99089.7i			1.13863i
$296$		−	1828.10i		−	0.0208649i
$297$			23657.7i			0.268201i
$298$			53303.5i			0.600237i
$299$	−110697.	−	105048.i	−1.23821	−	1.17502i
$300$	−9399.31			−0.104437
$301$	19550.9			0.215791
$302$	−17942.3			−0.196727
$303$	−41022.8			−0.446828
$304$		−	7569.80i		−	0.0819100i
$305$	111153.			1.19487
$306$		−	23786.8i		−	0.254034i
$307$	−61670.7			−0.654338			−0.327169	−	0.944966i	$-0.606095\pi$
$307$	−61670.7			−0.654338			−0.327169	+	0.944966i	$0.606095\pi$
$308$	79558.4			0.838658
$309$		−	48011.5i		−	0.502839i
$310$		−	46767.8i		−	0.486658i
$311$	94648.6			0.978574			0.489287	−	0.872123i	$-0.337257\pi$
$311$	94648.6			0.978574			0.489287	+	0.872123i	$0.337257\pi$
$312$	−33918.3			−0.348438
$313$		−	64314.1i		−	0.656474i	−0.944595	−	0.328237i	$-0.893546\pi$
$313$		−	64314.1i		−	0.656474i	0.944595	−	0.328237i	$-0.106454\pi$
$314$			74782.4i			0.758473i
$315$	31802.3			0.320507
$316$			90749.4i			0.908803i
$317$	−41158.9			−0.409586			−0.204793	−	0.978805i	$-0.565652\pi$
$317$	−41158.9			−0.409586			−0.204793	+	0.978805i	$0.565652\pi$
$318$		−	61592.5i		−	0.609079i
$319$		−	167217.i		−	1.64323i
$320$		−	10225.8i		−	0.0998609i
$321$		−	95575.0i		−	0.927543i
$322$	60741.1	−	64007.6i	0.585829	−	0.617333i
$323$	−36840.9			−0.353123
$324$	5832.00			0.0555556
$325$	65229.2			0.617554
$326$	32093.8			0.301986
$327$			8194.61i			0.0766360i
$328$	4246.80			0.0394743
$329$		−	111855.i		−	1.03339i
$330$	49497.1			0.454518
$331$	114900.			1.04873			0.524366	−	0.851493i	$-0.324302\pi$
$331$	114900.			1.04873			0.524366	+	0.851493i	$0.324302\pi$
$332$		−	28586.3i		−	0.259347i
$333$		−	2181.37i		−	0.0196716i
$334$	18741.1			0.167997
$335$	−136477.			−1.21610
$336$		−	19612.4i		−	0.173721i
$337$			93546.8i			0.823701i	0.911252	+	0.411850i	$0.135117\pi$
$337$			93546.8i			0.823701i	−0.911252	+	0.411850i	$0.864883\pi$
$338$	154603.			1.35327
$339$			64952.5i			0.565192i
$340$	−49767.0			−0.430511
$341$		−	139606.i		−	1.20059i
$342$		−	9032.60i		−	0.0772255i
$343$			78079.1i			0.663661i
$344$		−	7501.22i		−	0.0633892i
$345$	37789.9	−	39822.1i	0.317496	−	0.334569i
$346$	78364.5			0.654587
$347$	−30832.5			−0.256064			−0.128032	−	0.991770i	$-0.540866\pi$
$347$	−30832.5			−0.256064			−0.128032	+	0.991770i	$0.540866\pi$
$348$	−41221.6			−0.340382
$349$	33514.4			0.275157			0.137578	−	0.990491i	$-0.456068\pi$
$349$	33514.4			0.275157			0.137578	+	0.990491i	$0.456068\pi$
$350$			37717.1i			0.307895i
$351$	−40472.8			−0.328510
$352$		−	30524.7i		−	0.246358i
$353$	111286.			0.893082			0.446541	−	0.894763i	$-0.352656\pi$
$353$	111286.			0.893082			0.446541	+	0.894763i	$0.352656\pi$
$354$	72917.2			0.581867
$355$		−	38176.1i		−	0.302925i
$356$		−	29130.4i		−	0.229851i
$357$	−95450.3			−0.748929
$358$	−64929.4			−0.506611
$359$			90598.3i			0.702961i	0.936195	+	0.351480i	$0.114322\pi$
$359$			90598.3i			0.702961i	−0.936195	+	0.351480i	$0.885678\pi$
$360$		−	12201.8i		−	0.0941497i
$361$	116331.			0.892652
$362$		−	53792.7i		−	0.410494i
$363$	71676.1			0.543952
$364$			136106.i			1.02725i
$365$			131951.i			0.990436i
$366$		−	81794.1i		−	0.610604i
$367$			120673.i			0.895936i	0.894049	+	0.447968i	$0.147852\pi$
$367$			120673.i			0.895936i	−0.894049	+	0.447968i	$0.852148\pi$
$368$	−24558.2	−	23305.0i	−0.181343	−	0.172089i
$369$	5067.46			0.0372167
$370$	−4563.89			−0.0333374
$371$	−247155.			−1.79565
$372$	−34415.1			−0.248693
$373$			67722.9i			0.486763i	0.969931	+	0.243382i	$0.0782567\pi$
$373$			67722.9i			0.486763i	−0.969931	+	0.243382i	$0.921743\pi$
$374$	−148559.			−1.06208
$375$			88327.1i			0.628104i
$376$	−42916.4			−0.303562
$377$	286069.			2.01274
$378$		−	23402.4i		−	0.163786i
$379$		−	82059.4i		−	0.571281i	−0.958337	−	0.285641i	$-0.907794\pi$
$379$		−	82059.4i		−	0.571281i	0.958337	−	0.285641i	$-0.0922064\pi$
$380$	−18898.2			−0.130874
$381$	136466.			0.940101
$382$			200502.i			1.37402i
$383$		−	137181.i		−	0.935181i	−0.883945	−	0.467590i	$-0.845122\pi$
$383$		−	137181.i		−	0.935181i	0.883945	−	0.467590i	$-0.154878\pi$
$384$	−7524.83			−0.0510310
$385$		−	198619.i		−	1.33999i
$386$	66022.3			0.443114
$387$		−	8950.78i		−	0.0597639i
$388$		−	76079.4i		−	0.505363i
$389$		−	8522.96i		−	0.0563237i	−0.999603	−	0.0281619i	$-0.991035\pi$
$389$		−	8522.96i		−	0.0563237i	0.999603	−	0.0281619i	$-0.00896538\pi$
$390$			84677.9i			0.556725i
$391$	−113421.	+	119521.i	−0.741893	+	0.781790i
$392$	−24371.3			−0.158601
$393$	−98515.7			−0.637853
$394$	−144618.			−0.931603
$395$	226558.			1.45206
$396$		−	36423.4i		−	0.232269i
$397$	−187463.			−1.18942			−0.594708	−	0.803942i	$-0.702732\pi$
$397$	−187463.			−1.18942			−0.594708	+	0.803942i	$0.702732\pi$
$398$		−	194075.i		−	1.22519i
$399$	−36245.6			−0.227672
$400$	14471.2			0.0904449
$401$			113907.i			0.708374i	0.935175	+	0.354187i	$0.115242\pi$
$401$			113907.i			0.708374i	−0.935175	+	0.354187i	$0.884758\pi$
$402$			100429.i			0.621454i
$403$	238833.			1.47057
$404$	63158.7			0.386964
$405$		−	14559.7i		−	0.0887652i
$406$			165412.i			1.00349i
$407$	−13623.6			−0.0822438
$408$			36622.1i			0.220000i
$409$	−260370.			−1.55649			−0.778243	−	0.627964i	$-0.783889\pi$
$409$	−260370.			−1.55649			−0.778243	+	0.627964i	$0.783889\pi$
$410$		−	10602.2i		−	0.0630710i
$411$			53068.1i			0.314159i
$412$			73918.6i			0.435471i
$413$		−	292598.i		−	1.71543i
$414$	−29303.9	−	27808.5i	−0.170972	−	0.162247i
$415$	−71366.2			−0.414378
$416$	52220.7			0.301756
$417$	−102286.			−0.588227
$418$	−56412.6			−0.322867
$419$		−	93509.6i		−	0.532633i	−0.963886	−	0.266317i	$-0.914193\pi$
$419$		−	93509.6i		−	0.532633i	0.963886	−	0.266317i	$-0.0858067\pi$
$420$	−48962.8			−0.277567
$421$		−	41315.4i		−	0.233103i	−0.993185	−	0.116552i	$-0.962816\pi$
$421$		−	41315.4i		−	0.233103i	0.993185	−	0.116552i	$-0.0371840\pi$
$422$	−168261.			−0.944841
$423$	−51209.7			−0.286201
$424$			94827.8i			0.527478i
$425$		−	70428.8i		−	0.389917i
$426$	−28092.6			−0.154801
$427$	−328219.			−1.80015
$428$			147147.i			0.803276i
$429$			252771.i			1.37345i
$430$	−18727.0			−0.101282
$431$		−	106474.i		−	0.573178i	−0.958054	−	0.286589i	$-0.907479\pi$
$431$		−	106474.i		−	0.573178i	0.958054	−	0.286589i	$-0.0925215\pi$
$432$	−8978.95			−0.0481125
$433$		−	147335.i		−	0.785832i	−0.919574	−	0.392916i	$-0.871466\pi$
$433$		−	147335.i		−	0.785832i	0.919574	−	0.392916i	$-0.128534\pi$
$434$			138099.i			0.733181i
$435$			102911.i			0.543853i
$436$		−	12616.4i		−	0.0663687i
$437$	−43069.8	+	45386.0i	−0.225533	+	0.237661i
$438$	97098.7			0.506134
$439$	277220.			1.43845			0.719227	−	0.694775i	$-0.244496\pi$
$439$	277220.			1.43845			0.719227	+	0.694775i	$0.244496\pi$
$440$	−76205.7			−0.393624
$441$	−29080.8			−0.149530
$442$		−	254149.i		−	1.30090i
$443$	17965.1			0.0915425			0.0457712	−	0.998952i	$-0.485425\pi$
$443$	17965.1			0.0915425			0.0457712	+	0.998952i	$0.485425\pi$
$444$			3358.43i			0.0170361i
$445$	−72724.8			−0.367251
$446$	168945.			0.849327
$447$		−	97924.7i		−	0.490092i
$448$			30195.3i			0.150447i
$449$	186125.			0.923235			0.461617	−	0.887079i	$-0.347269\pi$
$449$	186125.			0.923235			0.461617	+	0.887079i	$0.347269\pi$
$450$	17267.6			0.0852723
$451$		−	31648.6i		−	0.155597i
$452$		−	100001.i		−	0.489471i
$453$	32962.1			0.160627
$454$			102293.i			0.496290i
$455$	339791.			1.64131
$456$			13906.6i			0.0668793i
$457$			356339.i			1.70621i	0.521743	+	0.853103i	$0.325282\pi$
$457$			356339.i			1.70621i	−0.521743	+	0.853103i	$0.674718\pi$
$458$			128771.i			0.613886i
$459$			43699.1i			0.207418i
$460$	−58181.4	+	61310.2i	−0.274959	+	0.289746i
$461$	−107546.			−0.506046			−0.253023	−	0.967460i	$-0.581425\pi$
$461$	−107546.			−0.506046			−0.253023	+	0.967460i	$0.581425\pi$
$462$	−146158.			−0.684761
$463$	71300.2			0.332605			0.166303	−	0.986075i	$-0.446817\pi$
$463$	71300.2			0.332605			0.166303	+	0.986075i	$0.446817\pi$
$464$	63464.8			0.294779
$465$			85918.0i			0.397354i
$466$	−98778.9			−0.454875
$467$			297142.i			1.36248i	0.732059	+	0.681241i	$0.238559\pi$
$467$			297142.i			1.36248i	−0.732059	+	0.681241i	$0.761441\pi$
$468$	62312.0			0.284498
$469$	402998.			1.83214
$470$			107142.i			0.485024i
$471$		−	137384.i		−	0.619291i
$472$	−112263.			−0.503911
$473$	−55901.6			−0.249863
$474$		−	166717.i		−	0.742034i
$475$		−	26744.1i		−	0.118534i
$476$	146955.			0.648592
$477$			113153.i			0.497311i
$478$	−248047.			−1.08562
$479$		−	9251.37i		−	0.0403213i	−0.999797	−	0.0201607i	$-0.993582\pi$
$479$		−	9251.37i		−	0.0403213i	0.999797	−	0.0201607i	$-0.00641778\pi$
$480$			18785.9i			0.0815361i
$481$		−	23306.8i		−	0.100738i
$482$		−	220799.i		−	0.950391i
$483$	−111589.	+	117589.i	−0.478327	+	0.504050i
$484$	−110353.			−0.471077
$485$	−189934.			−0.807456
$486$	−10714.1			−0.0453609
$487$	404796.			1.70678			0.853391	−	0.521271i	$-0.174542\pi$
$487$	404796.			1.70678			0.853391	+	0.521271i	$0.174542\pi$
$488$			125930.i			0.528799i
$489$	−58960.1			−0.246570
$490$			60843.4i			0.253409i
$491$	−156227.			−0.648029			−0.324014	−	0.946052i	$-0.605033\pi$
$491$	−156227.			−0.648029			−0.324014	+	0.946052i	$0.605033\pi$
$492$	−7801.87			−0.0322306
$493$		−	308873.i		−	1.27082i
$494$		−	96508.8i		−	0.395470i
$495$	−90931.9			−0.371113
$496$	52985.5			0.215374
$497$			112729.i			0.456375i
$498$			52516.3i			0.211756i
$499$	−272853.			−1.09579			−0.547896	−	0.836547i	$-0.684571\pi$
$499$	−272853.			−1.09579			−0.547896	+	0.836547i	$0.684571\pi$
$500$		−	135988.i		−	0.543954i
$501$	−34429.6			−0.137169
$502$			145776.i			0.578466i
$503$		−	25089.1i		−	0.0991631i	−0.998770	−	0.0495815i	$-0.984211\pi$
$503$		−	25089.1i		−	0.0991631i	0.998770	−	0.0495815i	$-0.0157888\pi$
$504$			36030.3i			0.141843i
$505$		−	157677.i		−	0.618281i
$506$	−173676.	+	183016.i	−0.678328	+	0.714806i
$507$	−284024.			−1.10494
$508$	−210103.			−0.814152
$509$	−208154.			−0.803432			−0.401716	−	0.915764i	$-0.631586\pi$
$509$	−208154.			−0.803432			−0.401716	+	0.915764i	$0.631586\pi$
$510$	91427.9			0.351510
$511$		−	389633.i		−	1.49216i
$512$	11585.2			0.0441942
$513$			16594.0i			0.0630544i
$514$	−126103.			−0.477309
$515$	184539.			0.695784
$516$			13780.6i			0.0517570i
$517$			319827.i			1.19656i
$518$	13476.6			0.0502249
$519$	−143965.			−0.534468
$520$		−	130370.i		−	0.482138i
$521$		−	318213.i		−	1.17231i	−0.810200	−	0.586154i	$-0.800641\pi$
$521$		−	318213.i		−	1.17231i	0.810200	−	0.586154i	$-0.199359\pi$
$522$	75728.9			0.277921
$523$		−	266854.i		−	0.975596i	−0.872957	−	0.487798i	$-0.837800\pi$
$523$		−	266854.i		−	0.975596i	0.872957	−	0.487798i	$-0.162200\pi$
$524$	151675.			0.552397
$525$		−	69290.7i		−	0.251395i
$526$			283947.i			1.02628i
$527$		−	257871.i		−	0.928500i
$528$			56077.5i			0.201151i
$529$	14644.8	+	279458.i	0.0523326	+	0.998630i
$530$	236740.			0.842790
$531$	−133957.			−0.475092
$532$	55803.7			0.197170
$533$	54143.3			0.190586
$534$			53516.0i			0.187673i
$535$	367356.			1.28345
$536$		−	154621.i		−	0.538195i
$537$	119283.			0.413647
$538$	253988.			0.877502
$539$			181623.i			0.625162i
$540$			22416.1i			0.0768729i
$541$	385497.			1.31712			0.658562	−	0.752526i	$-0.271165\pi$
$541$	385497.			1.31712			0.658562	+	0.752526i	$0.271165\pi$
$542$	−28132.3			−0.0957650
$543$			98823.6i			0.335167i
$544$		−	56383.4i		−	0.190526i
$545$	−31497.2			−0.106042
$546$		−	250042.i		−	0.838742i
$547$	366081.			1.22350			0.611748	−	0.791053i	$-0.290467\pi$
$547$	366081.			1.22350			0.611748	+	0.791053i	$0.290467\pi$
$548$		−	81703.7i		−	0.272070i
$549$			150265.i			0.498556i
$550$		−	107844.i		−	0.356509i
$551$		−	117289.i		−	0.386326i
$552$	45116.4	+	42814.0i	0.148066	+	0.140510i
$553$	−668995.			−2.18762
$554$	−389607.			−1.26943
$555$	8384.40			0.0272199
$556$	157480.			0.509420
$557$		−	28811.7i		−	0.0928665i	−0.998921	−	0.0464333i	$-0.985215\pi$
$557$		−	28811.7i		−	0.0928665i	0.998921	−	0.0464333i	$-0.0147855\pi$
$558$	63224.5			0.203057
$559$		−	95634.5i		−	0.306049i
$560$	75383.2			0.240380
$561$	272920.			0.867181
$562$			409195.i			1.29556i
$563$			137155.i			0.432708i	0.976315	+	0.216354i	$0.0694166\pi$
$563$			137155.i			0.432708i	−0.976315	+	0.216354i	$0.930583\pi$
$564$	78842.4			0.247857
$565$	−249654.			−0.782064
$566$			32154.9i			0.100372i
$567$			42992.9i			0.133730i
$568$	43251.5			0.134062
$569$			451318.i			1.39399i	0.717078	+	0.696993i	$0.245479\pi$
$569$			451318.i			1.39399i	−0.717078	+	0.696993i	$0.754521\pi$
$570$	34718.1			0.106858
$571$			143304.i			0.439528i	0.975553	+	0.219764i	$0.0705287\pi$
$571$			143304.i			0.439528i	−0.975553	+	0.219764i	$0.929471\pi$
$572$		−	389166.i		−	1.18944i
$573$		−	368346.i		−	1.12188i
$574$			31307.0i			0.0950205i
$575$	−86764.4	−	82336.6i	−0.262425	−	0.249033i
$576$	13824.0			0.0416667
$577$	−467347.			−1.40374			−0.701871	−	0.712304i	$-0.747652\pi$
$577$	−467347.			−1.40374			−0.701871	+	0.712304i	$0.747652\pi$
$578$	−38175.6			−0.114269
$579$	−121291.			−0.361801
$580$		−	158441.i		−	0.470991i
$581$	210735.			0.624287
$582$			139767.i			0.412627i
$583$	706688.			2.07917
$584$	−149493.			−0.438325
$585$		−	155563.i		−	0.454564i
$586$			47792.0i			0.139175i
$587$	13978.2			0.0405672			0.0202836	−	0.999794i	$-0.493543\pi$
$587$	13978.2			0.0405672			0.0202836	+	0.999794i	$0.493543\pi$
$588$	44772.9			0.129497
$589$		−	97922.2i		−	0.282261i
$590$			280268.i			0.805136i
$591$	265681.			0.760651
$592$		−	5170.65i		−	0.0147537i
$593$	485533.			1.38073			0.690365	−	0.723461i	$-0.257450\pi$
$593$	485533.			1.38073			0.690365	+	0.723461i	$0.257450\pi$
$594$			66914.1i			0.189647i
$595$		−	366877.i		−	1.03630i
$596$			150765.i			0.424432i
$597$			356539.i			1.00037i
$598$	−313098.	−	297120.i	−0.875543	−	0.830862i
$599$	−473979.			−1.32101			−0.660503	−	0.750823i	$-0.729657\pi$
$599$	−473979.			−1.32101			−0.660503	+	0.750823i	$0.729657\pi$
$600$	−26585.3			−0.0738480
$601$	275818.			0.763615			0.381807	−	0.924242i	$-0.375302\pi$
$601$	275818.			0.763615			0.381807	+	0.924242i	$0.375302\pi$
$602$	55298.2			0.152587
$603$		−	184501.i		−	0.507415i
$604$	−50748.4			−0.139107
$605$			275497.i			0.752674i
$606$	−116030.			−0.315955
$607$	351708.			0.954564			0.477282	−	0.878750i	$-0.341622\pi$
$607$	351708.			0.954564			0.477282	+	0.878750i	$0.341622\pi$
$608$		−	21410.6i		−	0.0579191i
$609$		−	303881.i		−	0.819350i
$610$	314387.			0.844900
$611$	−547149.			−1.46563
$612$		−	67279.1i		−	0.179629i
$613$			178842.i			0.475935i	0.971273	+	0.237968i	$0.0764812\pi$
$613$			178842.i			0.475935i	−0.971273	+	0.237968i	$0.923519\pi$
$614$	−174431.			−0.462687
$615$			19477.5i			0.0514972i
$616$	225025.			0.593021
$617$		−	223536.i		−	0.587189i	−0.955930	−	0.293594i	$-0.905148\pi$
$617$		−	223536.i		−	0.587189i	0.955930	−	0.293594i	$-0.0948515\pi$
$618$		−	135797.i		−	0.355561i
$619$		−	332661.i		−	0.868200i	−0.900865	−	0.434100i	$-0.857066\pi$
$619$		−	332661.i		−	0.868200i	0.900865	−	0.434100i	$-0.142934\pi$
$620$		−	132279.i		−	0.344119i
$621$	53834.8	+	51087.5i	0.139598	+	0.132474i
$622$	267707.			0.691956
$623$	214747.			0.553287
$624$	−95935.6			−0.246383
$625$	−198178.			−0.507336
$626$		−	181908.i		−	0.464197i
$627$	103637.			0.263620
$628$			211517.i			0.536321i
$629$	−25164.7			−0.0636048
$630$	89950.4			0.226632
$631$		−	664444.i		−	1.66878i	−0.551173	−	0.834391i	$-0.685820\pi$
$631$		−	664444.i		−	1.66878i	0.551173	−	0.834391i	$-0.314180\pi$
$632$			256678.i			0.642620i
$633$	309115.			0.771459
$634$	−116415.			−0.289621
$635$			524527.i			1.30083i
$636$		−	174210.i		−	0.430684i
$637$	−310714.			−0.765741
$638$		−	472961.i		−	1.16194i
$639$	51609.5			0.126394
$640$		−	28922.8i		−	0.0706123i
$641$		−	338234.i		−	0.823191i	−0.911367	−	0.411596i	$-0.864972\pi$
$641$		−	338234.i		−	0.823191i	0.911367	−	0.411596i	$-0.135028\pi$
$642$		−	270327.i		−	0.655872i
$643$			262721.i			0.635438i	0.948185	+	0.317719i	$0.102917\pi$
$643$			262721.i			0.635438i	−0.948185	+	0.317719i	$0.897083\pi$
$644$	171802.	−	181041.i	0.414244	−	0.436520i
$645$	34403.6			0.0826960
$646$	−104202.			−0.249695
$647$	348233.			0.831881			0.415941	−	0.909392i	$-0.363452\pi$
$647$	348233.			0.831881			0.415941	+	0.909392i	$0.363452\pi$
$648$	16495.4			0.0392837
$649$			836624.i			1.98628i
$650$	184496.			0.436677
$651$		−	253704.i		−	0.598640i
$652$	90775.1			0.213536
$653$	−592841.			−1.39031			−0.695155	−	0.718859i	$-0.744664\pi$
$653$	−592841.			−1.39031			−0.695155	+	0.718859i	$0.744664\pi$
$654$			23177.9i			0.0541898i
$655$		−	378659.i		−	0.882605i
$656$	12011.8			0.0279125
$657$	−178382.			−0.413257
$658$		−	316375.i		−	0.730719i
$659$		−	377415.i		−	0.869058i	−0.900658	−	0.434529i	$-0.856915\pi$
$659$		−	377415.i		−	0.869058i	0.900658	−	0.434529i	$-0.143085\pi$
$660$	139999.			0.321393
$661$			547932.i			1.25408i	0.778988	+	0.627038i	$0.215733\pi$
$661$			547932.i			1.25408i	−0.778988	+	0.627038i	$0.784267\pi$
$662$	324986.			0.741565
$663$			466902.i			1.06218i
$664$		−	80854.2i		−	0.183386i
$665$		−	139315.i		−	0.315032i
$666$		−	6169.84i		−	0.0139099i
$667$	−380514.	−	361095.i	−0.855300	−	0.811652i
$668$	53007.8			0.118792
$669$	−310371.			−0.693473
$670$	−386015.			−0.859914
$671$	938473.			2.08438
$672$		−	55472.3i		−	0.122839i
$673$	−776783.			−1.71502			−0.857511	−	0.514466i	$-0.827990\pi$
$673$	−776783.			−1.71502			−0.857511	+	0.514466i	$0.827990\pi$
$674$			264590.i			0.582444i
$675$	−31722.7			−0.0696245
$676$	437284.			0.956908
$677$			377606.i			0.823876i	0.911212	+	0.411938i	$0.135148\pi$
$677$			377606.i			0.823876i	−0.911212	+	0.411938i	$0.864852\pi$
$678$			183713.i			0.399651i
$679$	560849.			1.21648
$680$	−140762.			−0.304417
$681$		−	187925.i		−	0.405219i
$682$		−	394865.i		−	0.848946i
$683$	−67277.0			−0.144220			−0.0721099	−	0.997397i	$-0.522973\pi$
$683$	−67277.0			−0.144220			−0.0721099	+	0.997397i	$0.522973\pi$
$684$		−	25548.1i		−	0.0546067i
$685$	−203975.			−0.434706
$686$			220841.i			0.469279i
$687$		−	236568.i		−	0.501235i
$688$		−	21216.7i		−	0.0448229i
$689$			1.20898e6i			2.54671i
$690$	106886.	−	112634.i	0.224503	−	0.236576i
$691$	540961.			1.13295			0.566474	−	0.824080i	$-0.308307\pi$
$691$	540961.			1.13295			0.566474	+	0.824080i	$0.308307\pi$
$692$	221648.			0.462863
$693$	268510.			0.559105
$694$	−87207.4			−0.181065
$695$		−	393152.i		−	0.813938i
$696$	−116592.			−0.240686
$697$		−	58459.3i		−	0.120334i
$698$	94792.9			0.194565
$699$	181468.			0.371404
$700$			106680.i			0.217714i
$701$			356596.i			0.725671i	0.931853	+	0.362836i	$0.118191\pi$
$701$			356596.i			0.725671i	−0.931853	+	0.362836i	$0.881809\pi$
$702$	−114474.			−0.232292
$703$	−9555.85			−0.0193356
$704$		−	86337.0i		−	0.174201i
$705$		−	196832.i		−	0.396020i
$706$	314764.			0.631504
$707$			465600.i			0.931480i
$708$	206241.			0.411442
$709$		−	608246.i		−	1.21000i	−0.796224	−	0.605002i	$-0.793172\pi$
$709$		−	608246.i		−	1.21000i	0.796224	−	0.605002i	$-0.206828\pi$
$710$		−	107978.i		−	0.214200i
$711$			306279.i			0.605868i
$712$		−	82393.3i		−	0.162529i
$713$	−317683.	−	301471.i	−0.624906	−	0.593016i
$714$	−269974.			−0.529573
$715$	−971561.			−1.90046
$716$	−183648.			−0.358228
$717$	455691.			0.886405
$718$			256251.i			0.497068i
$719$	−770910.			−1.49123			−0.745617	−	0.666374i	$-0.767845\pi$
$719$	−770910.			−1.49123			−0.745617	+	0.666374i	$0.767845\pi$
$720$		−	34511.9i		−	0.0665739i
$721$	−544920.			−1.04824
$722$	329035.			0.631200
$723$			405633.i			0.775991i
$724$		−	152149.i		−	0.290263i
$725$	224222.			0.426581
$726$	202731.			0.384632
$727$			53283.8i			0.100815i	0.998729	+	0.0504076i	$0.0160520\pi$
$727$			53283.8i			0.100815i	−0.998729	+	0.0504076i	$0.983948\pi$
$728$			384966.i			0.726372i
$729$	19683.0			0.0370370
$730$			373213.i			0.700344i
$731$	−103258.			−0.193236
$732$		−	231349.i		−	0.431762i
$733$			865807.i			1.61144i	0.592299	+	0.805718i	$0.298220\pi$
$733$			865807.i			1.61144i	−0.592299	+	0.805718i	$0.701780\pi$
$734$			341314.i			0.633523i
$735$		−	111776.i		−	0.206907i
$736$	−69461.2	−	65916.4i	−0.128229	−	0.121685i
$737$	−1.15229e6			−2.12142
$738$	14333.0			0.0263162
$739$	227470.			0.416520			0.208260	−	0.978074i	$-0.433220\pi$
$739$	227470.			0.416520			0.208260	+	0.978074i	$0.433220\pi$
$740$	−12908.6			−0.0235731
$741$			177298.i			0.322900i
$742$	−699060.			−1.26972
$743$			101640.i			0.184114i	0.995754	+	0.0920568i	$0.0293441\pi$
$743$			101640.i			0.184114i	−0.995754	+	0.0920568i	$0.970656\pi$
$744$	−97340.5			−0.175852
$745$	376388.			0.678146
$746$			191549.i			0.344194i
$747$		−	96478.7i		−	0.172898i
$748$	−420188.			−0.751001
$749$	−1.08475e6			−1.93360
$750$			249827.i			0.444137i
$751$			699200.i			1.23971i	0.784715	+	0.619857i	$0.212809\pi$
$751$			699200.i			1.23971i	−0.784715	+	0.619857i	$0.787191\pi$
$752$	−121386.			−0.214651
$753$		−	267807.i		−	0.472316i
$754$	809126.			1.42322
$755$			126695.i			0.222261i
$756$		−	66191.9i		−	0.115814i
$757$			420292.i			0.733430i	0.930333	+	0.366715i	$0.119518\pi$
$757$			420292.i			0.733430i	−0.930333	+	0.366715i	$0.880482\pi$
$758$		−	232099.i		−	0.403957i
$759$	319064.	−	336222.i	0.553852	−	0.583637i
$760$	−53452.1			−0.0925417
$761$	−557730.			−0.963063			−0.481531	−	0.876429i	$-0.659919\pi$
$761$	−557730.			−0.963063			−0.481531	+	0.876429i	$0.659919\pi$
$762$	385984.			0.664752
$763$	93007.0			0.159759
$764$			567105.i			0.971576i
$765$	−167964.			−0.287007
$766$		−	388006.i		−	0.661273i
$767$	−1.43127e6			−2.43293
$768$	−21283.4			−0.0360844
$769$			521694.i			0.882192i	0.897460	+	0.441096i	$0.145410\pi$
$769$			521694.i			0.882192i	−0.897460	+	0.441096i	$0.854590\pi$
$770$		−	561780.i		−	0.947513i
$771$	231666.			0.389721
$772$	186739.			0.313329
$773$		−	769723.i		−	1.28818i	−0.764951	−	0.644089i	$-0.777237\pi$
$773$		−	769723.i		−	1.28818i	0.764951	−	0.644089i	$-0.222763\pi$
$774$		−	25316.6i		−	0.0422594i
$775$	187198.			0.311672
$776$		−	215185.i		−	0.357346i
$777$	−24758.0			−0.0410085
$778$		−	24106.6i		−	0.0398269i
$779$		−	22198.9i		−	0.0365811i
$780$			239505.i			0.393664i
$781$		−	322324.i		−	0.528434i
$782$	−320804.	+	338056.i	−0.524598	+	0.552809i
$783$	−139123.			−0.226921
$784$	−68932.4			−0.112148
$785$	528056.			0.856920
$786$	−278644.			−0.451030
$787$		−	65163.1i		−	0.105209i	−0.998615	−	0.0526044i	$-0.983248\pi$
$787$		−	65163.1i		−	0.105209i	0.998615	−	0.0526044i	$-0.0167522\pi$
$788$	−409042.			−0.658743
$789$		−	521643.i		−	0.837953i
$790$	640802.			1.02676
$791$	737196.			1.17823
$792$		−	103021.i		−	0.164239i
$793$			1.60551e6i			2.55309i
$794$	−530225.			−0.841044
$795$	−434919.			−0.688135
$796$		−	548928.i		−	0.866342i
$797$		−	322367.i		−	0.507498i	−0.967270	−	0.253749i	$-0.918336\pi$
$797$		−	322367.i		−	0.507498i	0.967270	−	0.253749i	$-0.0816637\pi$
$798$	−102518.			−0.160988
$799$			590765.i			0.925382i
$800$	40930.7			0.0639542
$801$		−	98315.3i		−	0.153234i
$802$			322178.i			0.500896i
$803$			1.11407e6i			1.72776i
$804$			284057.i			0.439434i
$805$	−451972.	−	428907.i	−0.697461	−	0.661868i
$806$	675522.			1.03985
$807$	−466605.			−0.716477
$808$	178640.			0.273625
$809$	498598.			0.761822			0.380911	−	0.924612i	$-0.375610\pi$
$809$	498598.			0.761822			0.380911	+	0.924612i	$0.375610\pi$
$810$		−	41181.1i		−	0.0627665i
$811$	−572320.			−0.870156			−0.435078	−	0.900393i	$-0.643279\pi$
$811$	−572320.			−0.870156			−0.435078	+	0.900393i	$0.643279\pi$
$812$			467856.i			0.709578i
$813$	51682.4			0.0781918
$814$	−38533.4			−0.0581552
$815$		−	226622.i		−	0.341182i
$816$			103583.i			0.155564i
$817$	−39210.4			−0.0587431
$818$	−736439.			−1.10060
$819$			459357.i			0.684830i
$820$		−	29987.6i		−	0.0445979i
$821$	710376.			1.05391			0.526953	−	0.849894i	$-0.323334\pi$
$821$	710376.			1.05391			0.526953	+	0.849894i	$0.323334\pi$
$822$			150099.i			0.222144i
$823$	−219640.			−0.324273			−0.162137	−	0.986768i	$-0.551839\pi$
$823$	−219640.			−0.324273			−0.162137	+	0.986768i	$0.551839\pi$
$824$			209073.i			0.307925i
$825$			198122.i			0.291089i
$826$		−	827593.i		−	1.21299i
$827$			771580.i			1.12816i	0.825721	+	0.564079i	$0.190769\pi$
$827$			771580.i			1.12816i	−0.825721	+	0.564079i	$0.809231\pi$
$828$	−82884.0	−	78654.3i	−0.120896	−	0.114726i
$829$	1.08700e6			1.58169			0.790844	−	0.612018i	$-0.209642\pi$
$829$	1.08700e6			1.58169			0.790844	+	0.612018i	$0.209642\pi$
$830$	−201854.			−0.293009
$831$	715754.			1.03648
$832$	147702.			0.213374
$833$			335482.i			0.483481i
$834$	−289309.			−0.415940
$835$		−	132335.i		−	0.189802i
$836$	−159559.			−0.228302
$837$	−116151.			−0.165795
$838$		−	264485.i		−	0.376629i
$839$		−	331804.i		−	0.471365i	−0.971830	−	0.235683i	$-0.924267\pi$
$839$		−	331804.i		−	0.471365i	0.971830	−	0.235683i	$-0.0757326\pi$
$840$	−138488.			−0.196269
$841$	276065.			0.390318
$842$		−	116858.i		−	0.164829i
$843$		−	751739.i		−	1.05782i
$844$	−475914.			−0.668104
$845$		−	1.09169e6i		−	1.52892i
$846$	−144843.			−0.202375
$847$		−	813507.i		−	1.13395i
$848$			268214.i			0.372983i
$849$		−	59072.3i		−	0.0819537i
$850$		−	199203.i		−	0.275713i
$851$	−29419.4	+	31001.5i	−0.0406232	+	0.0428078i
$852$	−79458.0			−0.109461
$853$	126462.			0.173805			0.0869024	−	0.996217i	$-0.472303\pi$
$853$	126462.			0.173805			0.0869024	+	0.996217i	$0.472303\pi$
$854$	−928344.			−1.27290
$855$	−63781.3			−0.0872491
$856$			416195.i			0.568002i
$857$	−80673.1			−0.109842			−0.0549208	−	0.998491i	$-0.517491\pi$
$857$	−80673.1			−0.109842			−0.0549208	+	0.998491i	$0.517491\pi$
$858$			714944.i			0.971174i
$859$	672529.			0.911433			0.455717	−	0.890125i	$-0.349383\pi$
$859$	672529.			0.911433			0.455717	+	0.890125i	$0.349383\pi$
$860$	−52967.8			−0.0716169
$861$		−	57514.6i		−	0.0775839i
$862$		−	301154.i		−	0.405298i
$863$	993649.			1.33417			0.667086	−	0.744981i	$-0.267542\pi$
$863$	993649.			1.33417			0.667086	+	0.744981i	$0.267542\pi$
$864$	−25396.3			−0.0340207
$865$		−	553350.i		−	0.739550i
$866$		−	416726.i		−	0.555667i
$867$	70133.0			0.0933006
$868$			390603.i			0.518437i
$869$	1.91285e6			2.53304
$870$			291075.i			0.384562i
$871$		−	1.97130e6i		−	2.59846i
$872$		−	35684.6i		−	0.0469298i
$873$		−	256768.i		−	0.336909i
$874$	−121820.	+	128371.i	−0.159476	+	0.168052i
$875$	1.00249e6			1.30938
$876$	274637.			0.357891
$877$	−416237.			−0.541180			−0.270590	−	0.962695i	$-0.587219\pi$
$877$	−416237.			−0.541180			−0.270590	+	0.962695i	$0.587219\pi$
$878$	784098.			1.01714
$879$		−	87799.5i		−	0.113636i
$880$	−215542.			−0.278335
$881$		−	636344.i		−	0.819861i	−0.912117	−	0.409930i	$-0.865553\pi$
$881$		−	636344.i		−	0.819861i	0.912117	−	0.409930i	$-0.134447\pi$
$882$	−82253.0			−0.105734
$883$	125554.			0.161031			0.0805153	−	0.996753i	$-0.474343\pi$
$883$	125554.			0.161031			0.0805153	+	0.996753i	$0.474343\pi$
$884$		−	718843.i		−	0.919877i
$885$		−	514885.i		−	0.657391i
$886$	50813.0			0.0647303
$887$	−299033.			−0.380077			−0.190039	−	0.981777i	$-0.560861\pi$
$887$	−299033.			−0.380077			−0.190039	+	0.981777i	$0.560861\pi$
$888$			9499.09i			0.0120464i
$889$		−	1.54886e6i		−	1.95978i
$890$	−205697.			−0.259685
$891$		−	122929.i		−	0.154846i
$892$	477848.			0.600565
$893$			224333.i			0.281313i
$894$		−	276973.i		−	0.346547i
$895$			458481.i			0.572368i
$896$			85405.1i			0.106382i
$897$	575197.	+	545844.i	0.714878	+	0.678396i
$898$	526441.			0.652826
$899$	820975.			1.01581
$900$	48840.3			0.0602966
$901$	1.30535e6			1.60797
$902$		−	89515.7i		−	0.110024i
$903$	−101589.			−0.124587
$904$		−	282845.i		−	0.346108i
$905$	−379843.			−0.463774
$906$	93230.8			0.113580
$907$			237752.i			0.289008i	0.989504	+	0.144504i	$0.0461587\pi$
$907$			237752.i			0.289008i	−0.989504	+	0.144504i	$0.953841\pi$
$908$			289329.i			0.350930i
$909$	213161.			0.257976
$910$	961075.			1.16058
$911$		−	1.18670e6i		−	1.42990i	−0.699176	−	0.714950i	$-0.746450\pi$
$911$		−	1.18670e6i		−	1.42990i	0.699176	−	0.714950i	$-0.253550\pi$
$912$			39333.8i			0.0472908i
$913$	−602552.			−0.722858
$914$			1.00788e6i			1.20647i
$915$	−577567.			−0.689858
$916$			364220.i			0.434083i
$917$			1.11813e6i			1.32970i
$918$			123600.i			0.146667i
$919$		−	1.42655e6i		−	1.68910i	−0.535477	−	0.844550i	$-0.679868\pi$
$919$		−	1.42655e6i		−	1.68910i	0.535477	−	0.844550i	$-0.320132\pi$
$920$	−164562.	+	173411.i	−0.194426	+	0.204881i
$921$	320450.			0.377782
$922$	−304185.			−0.357829
$923$	551421.			0.647262
$924$	−413398.			−0.484199
$925$		−	18267.9i		−	0.0213504i
$926$	201667.			0.235187
$927$			249475.i			0.290314i
$928$	179506.			0.208441
$929$	194050.			0.224845			0.112423	−	0.993660i	$-0.464139\pi$
$929$	194050.			0.224845			0.112423	+	0.993660i	$0.464139\pi$
$930$			243013.i			0.280972i
$931$			127394.i			0.146977i
$932$	−279389.			−0.321645
$933$	−491809.			−0.564980
$934$			840445.i			0.963420i
$935$			1.04901e6i			1.19993i
$936$	176245.			0.201171
$937$		−	330519.i		−	0.376459i	−0.982125	−	0.188229i	$-0.939725\pi$
$937$		−	330519.i		−	0.376459i	0.982125	−	0.188229i	$-0.0602748\pi$
$938$	1.13985e6			1.29552
$939$			334186.i			0.379015i
$940$			303042.i			0.342963i
$941$		−	1.47447e6i		−	1.66517i	−0.553900	−	0.832583i	$-0.686861\pi$
$941$		−	1.47447e6i		−	1.66517i	0.553900	−	0.832583i	$-0.313139\pi$
$942$		−	388581.i		−	0.437905i
$943$	−72018.5	−	68343.2i	−0.0809880	−	0.0768550i
$944$	−317529.			−0.356319
$945$	−165249.			−0.185045
$946$	−158114.			−0.176680
$947$	−494341.			−0.551222			−0.275611	−	0.961269i	$-0.588880\pi$
$947$	−494341.			−0.551222			−0.275611	+	0.961269i	$0.588880\pi$
$948$		−	471548.i		−	0.524697i
$949$	−1.90592e6			−2.11627
$950$		−	75643.8i		−	0.0838159i
$951$	213868.			0.236475
$952$	415653.			0.458624
$953$		−	1.71443e6i		−	1.88771i	−0.330367	−	0.943853i	$-0.607172\pi$
$953$		−	1.71443e6i		−	1.88771i	0.330367	−	0.943853i	$-0.392828\pi$
$954$			320044.i			0.351652i
$955$	1.41579e6			1.55236
$956$	−701582.			−0.767649
$957$			868885.i			0.948720i
$958$		−	26166.8i		−	0.0285115i
$959$	602311.			0.654913
$960$			53134.6i			0.0576547i
$961$	−238106.			−0.257824
$962$		−	65921.6i		−	0.0712324i
$963$			496622.i			0.535517i
$964$		−	624513.i		−	0.672028i
$965$		−	466198.i		−	0.500629i
$966$	−315620.	+	332593.i	−0.338229	+	0.356417i
$967$	−1.20027e6			−1.28359			−0.641795	−	0.766877i	$-0.721810\pi$
$967$	−1.20027e6			−1.28359			−0.641795	+	0.766877i	$0.721810\pi$
$968$	−312124.			−0.333101
$969$	191431.			0.203875
$970$	−537214.			−0.570958
$971$		−	1.61911e6i		−	1.71727i	−0.512589	−	0.858634i	$-0.671313\pi$
$971$		−	1.61911e6i		−	1.71727i	0.512589	−	0.858634i	$-0.328687\pi$
$972$	−30304.0			−0.0320750
$973$			1.16093e6i			1.22625i
$974$	1.14494e6			1.20688
$975$	−338941.			−0.356545
$976$			356184.i			0.373917i
$977$			109801.i			0.115032i	0.998345	+	0.0575158i	$0.0183179\pi$
$977$			109801.i			0.115032i	−0.998345	+	0.0575158i	$0.981682\pi$
$978$	−166764.			−0.174352
$979$	−614022.			−0.640647
$980$			172091.i			0.179187i
$981$		−	42580.4i		−	0.0442458i
$982$	−441878.			−0.458225
$983$		−	1.29774e6i		−	1.34301i	−0.741000	−	0.671505i	$-0.765648\pi$
$983$		−	1.29774e6i		−	1.34301i	0.741000	−	0.671505i	$-0.234352\pi$
$984$	−22067.0			−0.0227905
$985$			1.02118e6i			1.05252i
$986$		−	873624.i		−	0.898609i
$987$			581218.i			0.596630i
$988$		−	272968.i		−	0.279639i
$989$	−120716.	+	127208.i	−0.123416	+	0.130053i
$990$	−257194.			−0.262416
$991$	−569407.			−0.579797			−0.289898	−	0.957057i	$-0.593621\pi$
$991$	−569407.			−0.579797			−0.289898	+	0.957057i	$0.593621\pi$
$992$	149866.			0.152292
$993$	−597038.			−0.605485
$994$			318845.i			0.322706i
$995$	−1.37041e6			−1.38422
$996$			148539.i			0.149734i
$997$	1.02559e6			1.03178			0.515888	−	0.856656i	$-0.327462\pi$
$997$	1.02559e6			1.03178			0.515888	+	0.856656i	$0.327462\pi$
$998$	−771745.			−0.774842
$999$			11334.7i			0.0113574i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.b.a.91.9	✓	16
3.2	odd	2		414.5.b.b.91.7		16
4.3	odd	2		1104.5.c.a.1057.11		16
23.22	odd	2	inner	138.5.b.a.91.12	yes	16
69.68	even	2		414.5.b.b.91.2		16
92.91	even	2		1104.5.c.a.1057.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.b.a.91.9	✓	16	1.1	even	1	trivial
138.5.b.a.91.12	yes	16	23.22	odd	2	inner
414.5.b.b.91.2		16	69.68	even	2
414.5.b.b.91.7		16	3.2	odd	2
1104.5.c.a.1057.11		16	4.3	odd	2
1104.5.c.a.1057.14		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)

\(f(q)\)	\(=\)	\(q+2.82843 q^{2} -5.19615 q^{3} +8.00000 q^{4} -19.9722i q^{5} -14.6969 q^{6} +58.9751i q^{7} +22.6274 q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+2.82843 q^{2} -5.19615 q^{3} +8.00000 q^{4} -19.9722i q^{5} -14.6969 q^{6} +58.9751i q^{7} +22.6274 q^{8} +27.0000 q^{9} -56.4898i q^{10} -168.627i q^{11} -41.5692 q^{12} +288.481 q^{13} +166.807i q^{14} +103.778i q^{15} +64.0000 q^{16} -311.477i q^{17} +76.3675 q^{18} -118.278i q^{19} -159.777i q^{20} -306.444i q^{21} -476.949i q^{22} +(-383.722 - 364.140i) q^{23} -117.576 q^{24} +226.112 q^{25} +815.949 q^{26} -140.296 q^{27} +471.801i q^{28} +991.638 q^{29} +293.530i q^{30} +827.898 q^{31} +181.019 q^{32} +876.212i q^{33} -880.991i q^{34} +1177.86 q^{35} +216.000 q^{36} -80.7914i q^{37} -334.541i q^{38} -1498.99 q^{39} -451.919i q^{40} +187.684 q^{41} -866.754i q^{42} -331.510i q^{43} -1349.02i q^{44} -539.249i q^{45} +(-1085.33 - 1029.94i) q^{46} -1896.65 q^{47} -332.554 q^{48} -1077.07 q^{49} +639.542 q^{50} +1618.48i q^{51} +2307.85 q^{52} +4190.84i q^{53} -396.817 q^{54} -3367.85 q^{55} +1334.46i q^{56} +614.591i q^{57} +2804.78 q^{58} -4961.39 q^{59} +830.228i q^{60} +5565.38i q^{61} +2341.65 q^{62} +1592.33i q^{63} +512.000 q^{64} -5761.60i q^{65} +2478.30i q^{66} -6833.36i q^{67} -2491.82i q^{68} +(1993.88 + 1892.13i) q^{69} +3331.50 q^{70} +1911.46 q^{71} +610.940 q^{72} -6606.73 q^{73} -228.512i q^{74} -1174.91 q^{75} -946.225i q^{76} +9944.80 q^{77} -4239.79 q^{78} +11343.7i q^{79} -1278.22i q^{80} +729.000 q^{81} +530.850 q^{82} -3573.28i q^{83} -2451.55i q^{84} -6220.88 q^{85} -937.652i q^{86} -5152.70 q^{87} -3815.59i q^{88} -3641.31i q^{89} -1525.23i q^{90} +17013.2i q^{91} +(-3069.78 - 2913.12i) q^{92} -4301.88 q^{93} -5364.55 q^{94} -2362.27 q^{95} -940.604 q^{96} -9509.93i q^{97} -3046.41 q^{98} -4552.93i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) \) 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

L-functions

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