LMFDB - Embedded newform 69.5.d.a.22.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,5,Mod(22,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, 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("69.22");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	69.d (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:	$7.13252745279$
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} + 5598 x^{14} + 11369517 x^{12} + 11272666128 x^{10} + 5958872960073 x^{8} + \cdots + 13\!\cdots\!52 $ x^16 + 5598x^14 + 11369517x^12 + 11272666128x^10 + 5958872960073x^8 + 1661250441139200x^6 + 215215956313867368x^4 + 8236967436040727664*x^2 + 13861812915684254052
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.11
Root			$50.5339i$ of defining polynomial
Character	$\chi$	$=$	69.22
Dual form			69.5.d.a.22.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.93471 q^{2} -5.19615 q^{3} -0.518045 q^{4} -11.5477i q^{5} -20.4454 q^{6} -67.1686i q^{7} -64.9937 q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+3.93471 q^{2} -5.19615 q^{3} -0.518045 q^{4} -11.5477i q^{5} -20.4454 q^{6} -67.1686i q^{7} -64.9937 q^{8} +27.0000 q^{9} -45.4369i q^{10} -106.647i q^{11} +2.69184 q^{12} -34.3967 q^{13} -264.289i q^{14} +60.0037i q^{15} -247.443 q^{16} -210.225i q^{17} +106.237 q^{18} +177.938i q^{19} +5.98224i q^{20} +349.018i q^{21} -419.626i q^{22} +(-211.819 + 484.741i) q^{23} +337.717 q^{24} +491.650 q^{25} -135.341 q^{26} -140.296 q^{27} +34.7964i q^{28} +233.806 q^{29} +236.097i q^{30} -359.135 q^{31} +66.2834 q^{32} +554.156i q^{33} -827.173i q^{34} -775.644 q^{35} -13.9872 q^{36} -817.931i q^{37} +700.136i q^{38} +178.730 q^{39} +750.529i q^{40} +1982.50 q^{41} +1373.29i q^{42} -2563.43i q^{43} +55.2481i q^{44} -311.788i q^{45} +(-833.446 + 1907.32i) q^{46} -238.461 q^{47} +1285.75 q^{48} -2110.62 q^{49} +1934.50 q^{50} +1092.36i q^{51} +17.8190 q^{52} +4421.31i q^{53} -552.025 q^{54} -1231.53 q^{55} +4365.54i q^{56} -924.594i q^{57} +919.961 q^{58} +3206.60 q^{59} -31.0846i q^{60} -3353.19i q^{61} -1413.09 q^{62} -1813.55i q^{63} +4219.89 q^{64} +397.203i q^{65} +2180.44i q^{66} -7681.83i q^{67} +108.906i q^{68} +(1100.64 - 2518.79i) q^{69} -3051.94 q^{70} -889.199 q^{71} -1754.83 q^{72} -6763.88 q^{73} -3218.32i q^{74} -2554.69 q^{75} -92.1801i q^{76} -7163.35 q^{77} +703.253 q^{78} -8091.81i q^{79} +2857.40i q^{80} +729.000 q^{81} +7800.55 q^{82} -4538.20i q^{83} -180.807i q^{84} -2427.61 q^{85} -10086.4i q^{86} -1214.89 q^{87} +6931.41i q^{88} -5556.68i q^{89} -1226.80i q^{90} +2310.38i q^{91} +(109.732 - 251.118i) q^{92} +1866.12 q^{93} -938.275 q^{94} +2054.78 q^{95} -344.419 q^{96} +3517.90i q^{97} -8304.70 q^{98} -2879.48i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9}+O(q^{10}) $ 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9	$ 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9} + 104 q^{13} + 680 q^{16} + 324 q^{18} - 732 q^{23} - 1764 q^{24} - 2984 q^{25} + 1800 q^{26} - 3528 q^{29} - 400 q^{31} + 5244 q^{32} + 912 q^{35} + 3888 q^{36} + 2016 q^{39} + 1008 q^{41} - 1168 q^{46} - 8664 q^{47} - 2016 q^{48} + 7240 q^{49} - 18852 q^{50} - 20952 q^{52} - 972 q^{54} + 6816 q^{55} - 13352 q^{58} + 20112 q^{59} + 4248 q^{62} - 896 q^{64} - 10044 q^{69} - 10680 q^{70} + 40368 q^{71} + 10044 q^{72} - 9568 q^{73} + 7560 q^{75} + 2952 q^{77} - 6912 q^{78} + 11664 q^{81} + 71800 q^{82} + 42744 q^{85} + 8352 q^{87} - 9876 q^{92} - 10008 q^{93} + 73720 q^{94} + 33312 q^{95} - 24948 q^{96} - 59052 q^{98}+O(q^{100}) $ 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9 + 104 * q^13 + 680 * q^16 + 324 * q^18 - 732 * q^23 - 1764 * q^24 - 2984 * q^25 + 1800 * q^26 - 3528 * q^29 - 400 * q^31 + 5244 * q^32 + 912 * q^35 + 3888 * q^36 + 2016 * q^39 + 1008 * q^41 - 1168 * q^46 - 8664 * q^47 - 2016 * q^48 + 7240 * q^49 - 18852 * q^50 - 20952 * q^52 - 972 * q^54 + 6816 * q^55 - 13352 * q^58 + 20112 * q^59 + 4248 * q^62 - 896 * q^64 - 10044 * q^69 - 10680 * q^70 + 40368 * q^71 + 10044 * q^72 - 9568 * q^73 + 7560 * q^75 + 2952 * q^77 - 6912 * q^78 + 11664 * q^81 + 71800 * q^82 + 42744 * q^85 + 8352 * q^87 - 9876 * q^92 - 10008 * q^93 + 73720 * q^94 + 33312 * q^95 - 24948 * q^96 - 59052 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.93471			0.983678			0.491839	−	0.870686i	$-0.336325\pi$
$2$	3.93471			0.983678			0.491839	+	0.870686i	$0.336325\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$	−0.518045			−0.0323778
$5$		−	11.5477i		−	0.461909i	−0.972965	−	0.230954i	$-0.925815\pi$
$5$		−	11.5477i		−	0.461909i	0.972965	−	0.230954i	$-0.0741848\pi$
$6$	−20.4454			−0.567927
$7$		−	67.1686i		−	1.37079i	−0.728172	−	0.685394i	$-0.759630\pi$
$7$		−	67.1686i		−	1.37079i	0.728172	−	0.685394i	$-0.240370\pi$
$8$	−64.9937			−1.01553
$9$	27.0000			0.333333
$10$		−	45.4369i		−	0.454369i
$11$		−	106.647i		−	0.881382i	−0.897659	−	0.440691i	$-0.854733\pi$
$11$		−	106.647i		−	0.881382i	0.897659	−	0.440691i	$-0.145267\pi$
$12$	2.69184			0.0186933
$13$	−34.3967			−0.203531			−0.101765	−	0.994808i	$-0.532449\pi$
$13$	−34.3967			−0.203531			−0.101765	+	0.994808i	$0.532449\pi$
$14$		−	264.289i		−	1.34841i
$15$			60.0037i			0.266683i
$16$	−247.443			−0.966574
$17$		−	210.225i		−	0.727421i	−0.931512	−	0.363710i	$-0.881510\pi$
$17$		−	210.225i		−	0.727421i	0.931512	−	0.363710i	$-0.118490\pi$
$18$	106.237			0.327893
$19$			177.938i			0.492904i	0.969155	+	0.246452i	$0.0792647\pi$
$19$			177.938i			0.492904i	−0.969155	+	0.246452i	$0.920735\pi$
$20$			5.98224i			0.0149556i
$21$			349.018i			0.791425i
$22$		−	419.626i		−	0.866996i
$23$	−211.819	+	484.741i	−0.400413	+	0.916335i
$23$	−211.819	+	484.741i	−0.400413	+	0.916335i
$24$	337.717			0.586315
$25$	491.650			0.786640
$26$	−135.341			−0.200209
$27$	−140.296			−0.192450
$28$			34.7964i			0.0443831i
$29$	233.806			0.278010			0.139005	−	0.990292i	$-0.455610\pi$
$29$	233.806			0.278010			0.139005	+	0.990292i	$0.455610\pi$
$30$			236.097i			0.262330i
$31$	−359.135			−0.373709			−0.186855	−	0.982388i	$-0.559829\pi$
$31$	−359.135			−0.373709			−0.186855	+	0.982388i	$0.559829\pi$
$32$	66.2834			0.0647299
$33$			554.156i			0.508866i
$34$		−	827.173i		−	0.715548i
$35$	−775.644			−0.633179
$36$	−13.9872			−0.0107926
$37$		−	817.931i		−	0.597466i	−0.954337	−	0.298733i	$-0.903436\pi$
$37$		−	817.931i		−	0.597466i	0.954337	−	0.298733i	$-0.0965640\pi$
$38$			700.136i			0.484859i
$39$	178.730			0.117509
$40$			750.529i			0.469081i
$41$	1982.50			1.17936			0.589678	−	0.807639i	$-0.299255\pi$
$41$	1982.50			1.17936			0.589678	+	0.807639i	$0.299255\pi$
$42$			1373.29i			0.778507i
$43$		−	2563.43i		−	1.38639i	−0.720751	−	0.693194i	$-0.756203\pi$
$43$		−	2563.43i		−	1.38639i	0.720751	−	0.693194i	$-0.243797\pi$
$44$			55.2481i			0.0285372i
$45$		−	311.788i		−	0.153970i
$46$	−833.446	+	1907.32i	−0.393878	+	0.901378i
$47$	−238.461			−0.107950			−0.0539749	−	0.998542i	$-0.517189\pi$
$47$	−238.461			−0.107950			−0.0539749	+	0.998542i	$0.517189\pi$
$48$	1285.75			0.558052
$49$	−2110.62			−0.879061
$50$	1934.50			0.773801
$51$			1092.36i			0.419977i
$52$	17.8190			0.00658988
$53$			4421.31i			1.57398i	0.616965	+	0.786991i	$0.288362\pi$
$53$			4421.31i			1.57398i	−0.616965	+	0.786991i	$0.711638\pi$
$54$	−552.025			−0.189309
$55$	−1231.53			−0.407118
$56$			4365.54i			1.39207i
$57$		−	924.594i		−	0.284578i
$58$	919.961			0.273472
$59$	3206.60			0.921172			0.460586	−	0.887615i	$-0.347639\pi$
$59$	3206.60			0.921172			0.460586	+	0.887615i	$0.347639\pi$
$60$		−	31.0846i		−	0.00863462i
$61$		−	3353.19i		−	0.901153i	−0.892738	−	0.450577i	$-0.851218\pi$
$61$		−	3353.19i		−	0.901153i	0.892738	−	0.450577i	$-0.148782\pi$
$62$	−1413.09			−0.367610
$63$		−	1813.55i		−	0.456929i
$64$	4219.89			1.03025
$65$			397.203i			0.0940126i
$66$			2180.44i			0.500561i
$67$		−	7681.83i		−	1.71126i	−0.517591	−	0.855628i	$-0.673171\pi$
$67$		−	7681.83i		−	1.71126i	0.517591	−	0.855628i	$-0.326829\pi$
$68$			108.906i			0.0235523i
$69$	1100.64	−	2518.79i	0.231179	−	0.529046i
$70$	−3051.94			−0.622844
$71$	−889.199			−0.176393			−0.0881967	−	0.996103i	$-0.528110\pi$
$71$	−889.199			−0.176393			−0.0881967	+	0.996103i	$0.528110\pi$
$72$	−1754.83			−0.338509
$73$	−6763.88			−1.26926			−0.634629	−	0.772817i	$-0.718847\pi$
$73$	−6763.88			−1.26926			−0.634629	+	0.772817i	$0.718847\pi$
$74$		−	3218.32i		−	0.587714i
$75$	−2554.69			−0.454167
$76$		−	92.1801i		−	0.0159592i
$77$	−7163.35			−1.20819
$78$	703.253			0.115591
$79$		−	8091.81i		−	1.29656i	−0.761403	−	0.648278i	$-0.775489\pi$
$79$		−	8091.81i		−	1.29656i	0.761403	−	0.648278i	$-0.224511\pi$
$80$			2857.40i			0.446469i
$81$	729.000			0.111111
$82$	7800.55			1.16011
$83$		−	4538.20i		−	0.658761i	−0.944197	−	0.329381i	$-0.893160\pi$
$83$		−	4538.20i		−	0.658761i	0.944197	−	0.329381i	$-0.106840\pi$
$84$		−	180.807i		−	0.0256246i
$85$	−2427.61			−0.336002
$86$		−	10086.4i		−	1.36376i
$87$	−1214.89			−0.160509
$88$			6931.41i			0.895068i
$89$		−	5556.68i		−	0.701512i	−0.936467	−	0.350756i	$-0.885925\pi$
$89$		−	5556.68i		−	0.701512i	0.936467	−	0.350756i	$-0.114075\pi$
$90$		−	1226.80i		−	0.151456i
$91$			2310.38i			0.278998i
$92$	109.732	−	251.118i	0.0129645	−	0.0296689i
$93$	1866.12			0.215761
$94$	−938.275			−0.106188
$95$	2054.78			0.227677
$96$	−344.419			−0.0373718
$97$			3517.90i			0.373887i	0.982371	+	0.186943i	$0.0598581\pi$
$97$			3517.90i			0.373887i	−0.982371	+	0.186943i	$0.940142\pi$
$98$	−8304.70			−0.864713
$99$		−	2879.48i		−	0.293794i
$100$	−254.697			−0.0254697
$101$	15869.4			1.55567			0.777835	−	0.628469i	$-0.216318\pi$
$101$	15869.4			1.55567			0.777835	+	0.628469i	$0.216318\pi$
$102$			4298.12i			0.413122i
$103$		−	1219.14i		−	0.114915i	−0.998348	−	0.0574576i	$-0.981701\pi$
$103$		−	1219.14i		−	0.114915i	0.998348	−	0.0574576i	$-0.0182994\pi$
$104$	2235.57			0.206691
$105$	4030.37			0.365566
$106$			17396.6i			1.54829i
$107$			518.772i			0.0453115i	0.999743	+	0.0226558i	$0.00721217\pi$
$107$			518.772i			0.0453115i	−0.999743	+	0.0226558i	$0.992788\pi$
$108$	72.6797			0.00623111
$109$			15880.7i			1.33665i	0.743870	+	0.668324i	$0.232988\pi$
$109$			15880.7i			1.33665i	−0.743870	+	0.668324i	$0.767012\pi$
$110$	−4845.73			−0.400473
$111$			4250.10i			0.344947i
$112$			16620.4i			1.32497i
$113$			21986.0i			1.72183i	0.508752	+	0.860913i	$0.330107\pi$
$113$			21986.0i			1.72183i	−0.508752	+	0.860913i	$0.669893\pi$
$114$		−	3638.01i		−	0.279933i
$115$	5597.65	+	2446.02i	0.423263	+	0.184954i
$116$	−121.122			−0.00900136
$117$	−928.711			−0.0678436
$118$	12617.0			0.906136
$119$	−14120.5			−0.997140
$120$		−	3899.86i		−	0.270824i
$121$	3267.36			0.223165
$122$		−	13193.8i		−	0.886444i
$123$	−10301.4			−0.680901
$124$	186.048			0.0120999
$125$		−	12894.8i		−	0.825265i
$126$		−	7135.81i		−	0.449471i
$127$	26516.3			1.64401			0.822006	−	0.569479i	$-0.192855\pi$
$127$	26516.3			1.64401			0.822006	+	0.569479i	$0.192855\pi$
$128$	15543.5			0.948702
$129$			13320.0i			0.800431i
$130$			1562.88i			0.0924781i
$131$	−13523.7			−0.788049			−0.394025	−	0.919100i	$-0.628918\pi$
$131$	−13523.7			−0.788049			−0.394025	+	0.919100i	$0.628918\pi$
$132$		−	287.078i		−	0.0164760i
$133$	11951.9			0.675667
$134$		−	30225.8i		−	1.68332i
$135$			1620.10i			0.0888944i
$136$			13663.3i			0.738716i
$137$			16378.7i			0.872645i	0.899790	+	0.436323i	$0.143719\pi$
$137$			16378.7i			0.872645i	−0.899790	+	0.436323i	$0.856281\pi$
$138$	4330.71	−	9910.70i	0.227405	−	0.520411i
$139$	−35236.5			−1.82374			−0.911869	−	0.410481i	$-0.865361\pi$
$139$	−35236.5			−1.82374			−0.911869	+	0.410481i	$0.865361\pi$
$140$	401.819			0.0205010
$141$	1239.08			0.0623248
$142$	−3498.74			−0.173514
$143$			3668.31i			0.179388i
$144$	−6680.96			−0.322191
$145$		−	2699.93i		−	0.128415i
$146$	−26613.9			−1.24854
$147$	10967.1			0.507526
$148$			423.725i			0.0193447i
$149$			8092.91i			0.364529i	0.983250	+	0.182265i	$0.0583427\pi$
$149$			8092.91i			0.364529i	−0.983250	+	0.182265i	$0.941657\pi$
$150$	−10052.0			−0.446754
$151$	33345.5			1.46246			0.731229	−	0.682132i	$-0.238947\pi$
$151$	33345.5			1.46246			0.731229	+	0.682132i	$0.238947\pi$
$152$		−	11564.9i		−	0.500557i
$153$		−	5676.06i		−	0.242474i
$154$	−28185.7			−1.18847
$155$			4147.19i			0.172620i
$156$	−92.5905			−0.00380467
$157$		−	18911.0i		−	0.767212i	−0.923497	−	0.383606i	$-0.874682\pi$
$157$		−	18911.0i		−	0.767212i	0.923497	−	0.383606i	$-0.125318\pi$
$158$		−	31838.9i		−	1.27539i
$159$		−	22973.8i		−	0.908739i
$160$		−	765.422i		−	0.0298993i
$161$	32559.4	+	14227.6i	1.25610	+	0.548882i
$162$	2868.40			0.109298
$163$	−17331.8			−0.652332			−0.326166	−	0.945312i	$-0.605757\pi$
$163$	−17331.8			−0.652332			−0.326166	+	0.945312i	$0.605757\pi$
$164$	−1027.02			−0.0381850
$165$	6399.23			0.235050
$166$		−	17856.5i		−	0.648009i
$167$	−23360.5			−0.837623			−0.418811	−	0.908073i	$-0.637553\pi$
$167$	−23360.5			−0.837623			−0.418811	+	0.908073i	$0.637553\pi$
$168$		−	22684.0i		−	0.803714i
$169$	−27377.9			−0.958575
$170$	−9551.96			−0.330518
$171$			4804.33i			0.164301i
$172$			1327.97i			0.0448882i
$173$	41758.7			1.39526			0.697629	−	0.716459i	$-0.254238\pi$
$173$	41758.7			1.39526			0.697629	+	0.716459i	$0.254238\pi$
$174$	−4780.26			−0.157889
$175$		−	33023.5i		−	1.07832i
$176$			26389.1i			0.851921i
$177$	−16662.0			−0.531839
$178$		−	21863.9i		−	0.690062i
$179$	29144.1			0.909588			0.454794	−	0.890597i	$-0.349713\pi$
$179$	29144.1			0.909588			0.454794	+	0.890597i	$0.349713\pi$
$180$			161.520i			0.00498520i
$181$			43559.9i			1.32963i	0.747010	+	0.664813i	$0.231489\pi$
$181$			43559.9i			1.32963i	−0.747010	+	0.664813i	$0.768511\pi$
$182$			9090.67i			0.274444i
$183$			17423.7i			0.520281i
$184$	13766.9	−	31505.1i	0.406631	−	0.930563i
$185$	−9445.24			−0.275975
$186$	7342.64			0.212240
$187$	−22419.9			−0.641136
$188$	123.534			0.00349518
$189$			9423.50i			0.263808i
$190$	8084.97			0.223960
$191$			43781.6i			1.20012i	0.799955	+	0.600060i	$0.204857\pi$
$191$			43781.6i			1.20012i	−0.799955	+	0.600060i	$0.795143\pi$
$192$	−21927.2			−0.594814
$193$	−18997.2			−0.510005			−0.255003	−	0.966940i	$-0.582076\pi$
$193$	−18997.2			−0.510005			−0.255003	+	0.966940i	$0.582076\pi$
$194$			13841.9i			0.367784i
$195$		−	2063.93i		−	0.0542782i
$196$	1093.40			0.0284621
$197$	10244.5			0.263972			0.131986	−	0.991252i	$-0.457865\pi$
$197$	10244.5			0.263972			0.131986	+	0.991252i	$0.457865\pi$
$198$		−	11329.9i		−	0.288999i
$199$		−	28173.2i		−	0.711427i	−0.934595	−	0.355713i	$-0.884238\pi$
$199$		−	28173.2i		−	0.711427i	0.934595	−	0.355713i	$-0.115762\pi$
$200$	−31954.2			−0.798855
$201$			39915.9i			0.987994i
$202$	62441.4			1.53028
$203$		−	15704.5i		−	0.381093i
$204$		−	565.891i		−	0.0135979i
$205$		−	22893.3i		−	0.544754i
$206$		−	4796.95i		−	0.113040i
$207$	−5719.11	+	13088.0i	−0.133471	+	0.305445i
$208$	8511.22			0.196727
$209$	18976.6			0.434437
$210$	15858.3			0.359599
$211$	8055.93			0.180947			0.0904734	−	0.995899i	$-0.471162\pi$
$211$	8055.93			0.180947			0.0904734	+	0.995899i	$0.471162\pi$
$212$		−	2290.44i		−	0.0509621i
$213$	4620.42			0.101841
$214$			2041.22i			0.0445719i
$215$	−29601.8			−0.640384
$216$	9118.37			0.195438
$217$			24122.6i			0.512276i
$218$			62486.1i			1.31483i
$219$	35146.1			0.732807
$220$	637.989			0.0131816
$221$			7231.03i			0.148052i
$222$			16722.9i			0.339317i
$223$	−55411.7			−1.11427			−0.557137	−	0.830421i	$-0.688100\pi$
$223$	−55411.7			−1.11427			−0.557137	+	0.830421i	$0.688100\pi$
$224$		−	4452.17i		−	0.0887310i
$225$	13274.6			0.262213
$226$			86508.6i			1.69372i
$227$		−	18323.7i		−	0.355599i	−0.984067	−	0.177800i	$-0.943102\pi$
$227$		−	18323.7i		−	0.355599i	0.984067	−	0.177800i	$-0.0568979\pi$
$228$			478.982i			0.00921402i
$229$		−	99721.6i		−	1.90160i	−0.309812	−	0.950798i	$-0.600266\pi$
$229$		−	99721.6i		−	1.90160i	0.309812	−	0.950798i	$-0.399734\pi$
$230$	22025.1	+	9624.39i	0.416354	+	0.181936i
$231$	37221.9			0.697548
$232$	−15196.0			−0.282327
$233$	−9345.15			−0.172137			−0.0860685	−	0.996289i	$-0.527430\pi$
$233$	−9345.15			−0.172137			−0.0860685	+	0.996289i	$0.527430\pi$
$234$	−3654.21			−0.0667362
$235$			2753.68i			0.0498629i
$236$	−1661.16			−0.0298255
$237$			42046.3i			0.748567i
$238$	−55560.1			−0.980864
$239$	11561.5			0.202405			0.101202	−	0.994866i	$-0.467731\pi$
$239$	11561.5			0.202405			0.101202	+	0.994866i	$0.467731\pi$
$240$		−	14847.5i		−	0.257769i
$241$		−	4972.45i		−	0.0856123i	−0.999083	−	0.0428062i	$-0.986370\pi$
$241$		−	4972.45i		−	0.0856123i	0.999083	−	0.0428062i	$-0.0136298\pi$
$242$	12856.1			0.219522
$243$	−3788.00			−0.0641500
$244$			1737.10i			0.0291774i
$245$			24372.9i			0.406046i
$246$	−40532.9			−0.669787
$247$		−	6120.49i		−	0.100321i
$248$	23341.5			0.379512
$249$			23581.2i			0.380336i
$250$		−	50737.2i		−	0.811795i
$251$		−	77812.7i		−	1.23510i	−0.786531	−	0.617551i	$-0.788125\pi$
$251$		−	77812.7i		−	1.23510i	0.786531	−	0.617551i	$-0.211875\pi$
$252$			939.502i			0.0147944i
$253$	51696.3	+	22589.9i	0.807641	+	0.352917i
$254$	104334.			1.61718
$255$	12614.3			0.193991
$256$	−6358.99			−0.0970305
$257$	−50269.2			−0.761090			−0.380545	−	0.924763i	$-0.624264\pi$
$257$	−50269.2			−0.761090			−0.380545	+	0.924763i	$0.624264\pi$
$258$			52410.3i			0.787367i
$259$	−54939.3			−0.819000
$260$		−	205.769i		−	0.00304392i
$261$	6312.77			0.0926700
$262$	−53211.9			−0.775187
$263$		−	27622.3i		−	0.399345i	−0.979863	−	0.199672i	$-0.936012\pi$
$263$		−	27622.3i		−	0.399345i	0.979863	−	0.199672i	$-0.0639878\pi$
$264$		−	36016.6i		−	0.516768i
$265$	51056.1			0.727036
$266$	47027.2			0.664639
$267$			28873.3i			0.405018i
$268$			3979.53i			0.0554067i
$269$	−130511.			−1.80361			−0.901803	−	0.432147i	$-0.857756\pi$
$269$	−130511.			−1.80361			−0.901803	+	0.432147i	$0.857756\pi$
$270$			6374.63i			0.0874434i
$271$	78992.5			1.07559			0.537796	−	0.843075i	$-0.319257\pi$
$271$	78992.5			1.07559			0.537796	+	0.843075i	$0.319257\pi$
$272$			52018.6i			0.703106i
$273$		−	12005.1i		−	0.161079i
$274$			64445.4i			0.858402i
$275$		−	52433.2i		−	0.693331i
$276$	−570.182	+	1304.85i	−0.00748507	+	0.0171294i
$277$	22244.1			0.289905			0.144953	−	0.989439i	$-0.453697\pi$
$277$	22244.1			0.289905			0.144953	+	0.989439i	$0.453697\pi$
$278$	−138645.			−1.79397
$279$	−9696.64			−0.124570
$280$	50412.0			0.643011
$281$			96292.7i			1.21950i	0.792595	+	0.609748i	$0.208729\pi$
$281$			96292.7i			1.21950i	−0.792595	+	0.609748i	$0.791271\pi$
$282$	4875.42			0.0613076
$283$		−	48448.5i		−	0.604934i	−0.953160	−	0.302467i	$-0.902190\pi$
$283$		−	48448.5i		−	0.604934i	0.953160	−	0.302467i	$-0.0978101\pi$
$284$	460.645			0.00571124
$285$	−10677.0			−0.131449
$286$			14433.8i			0.176460i
$287$		−	133162.i		−	1.61665i
$288$	1789.65			0.0215766
$289$	39326.6			0.470859
$290$		−	10623.4i		−	0.126319i
$291$		−	18279.5i		−	0.215864i
$292$	3503.99			0.0410958
$293$		−	55068.5i		−	0.641458i	−0.947171	−	0.320729i	$-0.896072\pi$
$293$		−	55068.5i		−	0.641458i	0.947171	−	0.320729i	$-0.103928\pi$
$294$	43152.5			0.499242
$295$		−	37028.9i		−	0.425497i
$296$			53160.4i			0.606743i
$297$			14962.2i			0.169622i
$298$			31843.3i			0.358579i
$299$	7285.86	−	16673.5i	0.0814964	−	0.186502i
$300$	1323.44			0.0147049
$301$	−172182.			−1.90044
$302$	131205.			1.43859
$303$	−82459.7			−0.898166
$304$		−	44029.6i		−	0.476428i
$305$	−38721.7			−0.416250
$306$		−	22333.7i		−	0.238516i
$307$	29990.2			0.318201			0.159101	−	0.987262i	$-0.449141\pi$
$307$	29990.2			0.318201			0.159101	+	0.987262i	$0.449141\pi$
$308$	3710.94			0.0391185
$309$			6334.82i			0.0663464i
$310$			16318.0i			0.169802i
$311$	47727.5			0.493455			0.246728	−	0.969085i	$-0.420645\pi$
$311$	47727.5			0.493455			0.246728	+	0.969085i	$0.420645\pi$
$312$	−11616.4			−0.119333
$313$		−	33601.0i		−	0.342976i	−0.985186	−	0.171488i	$-0.945143\pi$
$313$		−	33601.0i		−	0.342976i	0.985186	−	0.171488i	$-0.0548574\pi$
$314$		−	74409.4i		−	0.754690i
$315$	−20942.4			−0.211060
$316$			4191.92i			0.0419797i
$317$	59945.9			0.596542			0.298271	−	0.954481i	$-0.403590\pi$
$317$	59945.9			0.596542			0.298271	+	0.954481i	$0.403590\pi$
$318$		−	90395.4i		−	0.893906i
$319$		−	24934.8i		−	0.245033i
$320$		−	48730.1i		−	0.475880i
$321$		−	2695.62i		−	0.0261606i
$322$	128112.	+	55981.4i	1.23560	+	0.539923i
$323$	37407.0			0.358548
$324$	−377.655			−0.00359754
$325$	−16911.1			−0.160105
$326$	−68195.7			−0.641685
$327$		−	82518.7i		−	0.771714i
$328$	−128850.			−1.19767
$329$			16017.1i			0.147976i
$330$	25179.1			0.231213
$331$	−171157.			−1.56221			−0.781104	−	0.624401i	$-0.785343\pi$
$331$	−171157.			−1.56221			−0.781104	+	0.624401i	$0.785343\pi$
$332$			2350.99i			0.0213292i
$333$		−	22084.1i		−	0.199155i
$334$	−91916.7			−0.823951
$335$	−88707.6			−0.790444
$336$		−	86362.1i		−	0.764971i
$337$			81678.7i			0.719199i	0.933107	+	0.359599i	$0.117087\pi$
$337$			81678.7i			0.719199i	−0.933107	+	0.359599i	$0.882913\pi$
$338$	−107724.			−0.942929
$339$		−	114243.i		−	0.994097i
$340$	1257.61			0.0108790
$341$			38300.7i			0.329381i
$342$			18903.7i			0.161620i
$343$		−	19504.1i		−	0.165782i
$344$			166607.i			1.40791i
$345$	−29086.3	−	12709.9i	−0.244371	−	0.106783i
$346$	164308.			1.37248
$347$	−127948.			−1.06261			−0.531306	−	0.847180i	$-0.678298\pi$
$347$	−127948.			−1.06261			−0.531306	+	0.847180i	$0.678298\pi$
$348$	629.370			0.00519694
$349$	205824.			1.68984			0.844919	−	0.534894i	$-0.179648\pi$
$349$	205824.			1.68984			0.844919	+	0.534894i	$0.179648\pi$
$350$		−	129938.i		−	1.06072i
$351$	4825.72			0.0391695
$352$		−	7068.95i		−	0.0570518i
$353$	159071.			1.27656			0.638281	−	0.769803i	$-0.279646\pi$
$353$	159071.			1.27656			0.638281	+	0.769803i	$0.279646\pi$
$354$	−65560.1			−0.523158
$355$			10268.2i			0.0814777i
$356$			2878.61i			0.0227134i
$357$	73372.3			0.575699
$358$	114674.			0.894742
$359$			129066.i			1.00143i	0.865611	+	0.500717i	$0.166930\pi$
$359$			129066.i			1.00143i	−0.865611	+	0.500717i	$0.833070\pi$
$360$			20264.3i			0.156360i
$361$	98659.0			0.757046
$362$			171396.i			1.30792i
$363$	−16977.7			−0.128844
$364$		−	1196.88i		−	0.00903333i
$365$			78107.3i			0.586281i
$366$			68557.2i			0.511789i
$367$			256298.i			1.90289i	0.307820	+	0.951445i	$0.400401\pi$
$367$			256298.i			1.90289i	−0.307820	+	0.951445i	$0.599599\pi$
$368$	52413.0	−	119946.i	0.387029	−	0.885705i
$369$	53527.4			0.393118
$370$	−37164.3			−0.271470
$371$	296974.			2.15760
$372$	−966.734			−0.00698588
$373$		−	62967.8i		−	0.452586i	−0.974059	−	0.226293i	$-0.927339\pi$
$373$		−	62967.8i		−	0.452586i	0.974059	−	0.226293i	$-0.0726607\pi$
$374$	−88215.8			−0.630671
$375$			67003.1i			0.476467i
$376$	15498.5			0.109626
$377$	−8042.17			−0.0565836
$378$			37078.7i			0.259502i
$379$			220972.i			1.53836i	0.639031	+	0.769181i	$0.279336\pi$
$379$			220972.i			1.53836i	−0.639031	+	0.769181i	$0.720664\pi$
$380$	−1064.47			−0.00737167
$381$	−137783.			−0.949171
$382$			172268.i			1.18053i
$383$		−	136454.i		−	0.930228i	−0.885251	−	0.465114i	$-0.846013\pi$
$383$		−	136454.i		−	0.930228i	0.885251	−	0.465114i	$-0.153987\pi$
$384$	−80766.5			−0.547733
$385$			82720.4i			0.558073i
$386$	−74748.5			−0.501681
$387$		−	69212.6i		−	0.462129i
$388$		−	1822.43i		−	0.0121056i
$389$			260760.i			1.72323i	0.507566	+	0.861613i	$0.330545\pi$
$389$			260760.i			1.72323i	−0.507566	+	0.861613i	$0.669455\pi$
$390$		−	8120.96i		−	0.0533923i
$391$	101904.	+	44529.5i	0.666561	+	0.291269i
$392$	137177.			0.892710
$393$	70271.3			0.454980
$394$	40309.1			0.259663
$395$	−93441.9			−0.598891
$396$			1491.70i			0.00951241i
$397$	−170714.			−1.08315			−0.541573	−	0.840654i	$-0.682171\pi$
$397$	−170714.			−1.08315			−0.541573	+	0.840654i	$0.682171\pi$
$398$		−	110853.i		−	0.699815i
$399$	−62103.7			−0.390096
$400$	−121655.			−0.760346
$401$			67483.8i			0.419673i	0.977737	+	0.209836i	$0.0672932\pi$
$401$			67483.8i			0.419673i	−0.977737	+	0.209836i	$0.932707\pi$
$402$			157058.i			0.971868i
$403$	12353.0			0.0760613
$404$	−8221.06			−0.0503692
$405$		−	8418.29i		−	0.0513232i
$406$		−	61792.5i		−	0.374873i
$407$	−87230.2			−0.526596
$408$		−	70996.5i		−	0.426498i
$409$	282547.			1.68905			0.844527	−	0.535513i	$-0.179882\pi$
$409$	282547.			1.68905			0.844527	+	0.535513i	$0.179882\pi$
$410$		−	90078.5i		−	0.535863i
$411$		−	85106.1i		−	0.503822i
$412$			631.567i			0.00372071i
$413$		−	215383.i		−	1.26273i
$414$	−22503.0	+	51497.5i	−0.131293	+	0.300459i
$415$	−52405.9			−0.304287
$416$	−2279.93			−0.0131745
$417$	183094.			1.05294
$418$	74667.6			0.427346
$419$		−	2860.17i		−	0.0162916i	−0.999967	−	0.00814581i	$-0.997407\pi$
$419$		−	2860.17i		−	0.0162916i	0.999967	−	0.00814581i	$-0.00259292\pi$
$420$	−2087.91			−0.0118362
$421$			126512.i			0.713784i	0.934146	+	0.356892i	$0.116164\pi$
$421$			126512.i			0.713784i	−0.934146	+	0.356892i	$0.883836\pi$
$422$	31697.8			0.177993
$423$	−6438.45			−0.0359833
$424$		−	287358.i		−	1.59842i
$425$		−	103357.i		−	0.572219i
$426$	18180.0			0.100179
$427$	−225229.			−1.23529
$428$		−	268.747i		−	0.00146709i
$429$		−	19061.1i		−	0.103570i
$430$	−116474.			−0.629932
$431$		−	103422.i		−	0.556746i	−0.960473	−	0.278373i	$-0.910205\pi$
$431$		−	103422.i		−	0.556746i	0.960473	−	0.278373i	$-0.0897951\pi$
$432$	34715.3			0.186017
$433$		−	140033.i		−	0.746885i	−0.927653	−	0.373442i	$-0.878177\pi$
$433$		−	140033.i		−	0.746885i	0.927653	−	0.373442i	$-0.121823\pi$
$434$			94915.4i			0.503915i
$435$			14029.3i			0.0741406i
$436$		−	8226.93i		−	0.0432778i
$437$	−86254.0	−	37690.7i	−0.451665	−	0.197365i
$438$	138290.			0.720846
$439$	−30777.9			−0.159702			−0.0798509	−	0.996807i	$-0.525444\pi$
$439$	−30777.9			−0.159702			−0.0798509	+	0.996807i	$0.525444\pi$
$440$	80041.9			0.413440
$441$	−56986.9			−0.293020
$442$			28452.0i			0.145636i
$443$	−240292.			−1.22443			−0.612213	−	0.790693i	$-0.709721\pi$
$443$	−240292.			−1.22443			−0.612213	+	0.790693i	$0.709721\pi$
$444$		−	2201.74i		−	0.0111686i
$445$	−64166.9			−0.324035
$446$	−218029.			−1.09609
$447$		−	42052.0i		−	0.210461i
$448$		−	283444.i		−	1.41225i
$449$	187901.			0.932043			0.466022	−	0.884773i	$-0.345687\pi$
$449$	187901.			0.932043			0.466022	+	0.884773i	$0.345687\pi$
$450$	52231.6			0.257934
$451$		−	211428.i		−	1.03946i
$452$		−	11389.7i		−	0.0557490i
$453$	−173268.			−0.844351
$454$		−	72098.4i		−	0.349795i
$455$	26679.6			0.128871
$456$			60092.9i			0.288997i
$457$		−	77806.0i		−	0.372547i	−0.982498	−	0.186273i	$-0.940359\pi$
$457$		−	77806.0i		−	0.372547i	0.982498	−	0.186273i	$-0.0596410\pi$
$458$		−	392376.i		−	1.87056i
$459$			29493.7i			0.139992i
$460$	−2899.84	−	1267.15i	−0.0137043	−	0.00598842i
$461$	52422.9			0.246672			0.123336	−	0.992365i	$-0.460641\pi$
$461$	52422.9			0.246672			0.123336	+	0.992365i	$0.460641\pi$
$462$	146457.			0.686163
$463$	212488.			0.991226			0.495613	−	0.868544i	$-0.334943\pi$
$463$	212488.			0.991226			0.495613	+	0.868544i	$0.334943\pi$
$464$	−57853.8			−0.268717
$465$		−	21549.4i		−	0.0996620i
$466$	−36770.5			−0.169327
$467$		−	418429.i		−	1.91862i	−0.282361	−	0.959308i	$-0.591118\pi$
$467$		−	418429.i		−	1.91862i	0.282361	−	0.959308i	$-0.408882\pi$
$468$	481.114			0.00219663
$469$	−515978.			−2.34577
$470$			10834.9i			0.0490491i
$471$			98264.5i			0.442950i
$472$	−208409.			−0.935475
$473$	−273383.			−1.22194
$474$			165440.i			0.736349i
$475$			87483.4i			0.387738i
$476$	7315.06			0.0322852
$477$			119375.i			0.524660i
$478$	45491.4			0.199101
$479$			425969.i			1.85655i	0.371891	+	0.928276i	$0.378709\pi$
$479$			425969.i			1.85655i	−0.371891	+	0.928276i	$0.621291\pi$
$480$			3977.25i			0.0172624i
$481$			28134.1i			0.121603i
$482$		−	19565.2i		−	0.0842150i
$483$	−169184.	−	73928.6i	−0.725210	−	0.316897i
$484$	−1692.64			−0.00722560
$485$	40623.7			0.172702
$486$	−14904.7			−0.0631030
$487$	455530.			1.92070			0.960350	−	0.278798i	$-0.0899361\pi$
$487$	455530.			1.92070			0.960350	+	0.278798i	$0.0899361\pi$
$488$			217936.i			0.915146i
$489$	90058.8			0.376624
$490$			95900.3i			0.399418i
$491$	175464.			0.727823			0.363911	−	0.931434i	$-0.381441\pi$
$491$	175464.			0.727823			0.363911	+	0.931434i	$0.381441\pi$
$492$	5336.57			0.0220461
$493$		−	49151.9i		−	0.202230i
$494$		−	24082.4i		−	0.0986836i
$495$	−33251.4			−0.135706
$496$	88865.3			0.361218
$497$			59726.3i			0.241798i
$498$			92785.2i			0.374128i
$499$	342274.			1.37459			0.687295	−	0.726378i	$-0.258798\pi$
$499$	342274.			1.37459			0.687295	+	0.726378i	$0.258798\pi$
$500$			6680.07i			0.0267203i
$501$	121385.			0.483602
$502$		−	306170.i		−	1.21494i
$503$			92211.2i			0.364458i	0.983256	+	0.182229i	$0.0583313\pi$
$503$			92211.2i			0.364458i	−0.983256	+	0.182229i	$0.941669\pi$
$504$			117870.i			0.464024i
$505$		−	183255.i		−	0.718577i
$506$	203410.	+	88884.7i	0.794459	+	0.347157i
$507$	142260.			0.553434
$508$	−13736.6			−0.0532295
$509$	400394.			1.54544			0.772719	−	0.634748i	$-0.218896\pi$
$509$	400394.			1.54544			0.772719	+	0.634748i	$0.218896\pi$
$510$	49633.4			0.190824
$511$			454320.i			1.73988i
$512$	−273717.			−1.04415
$513$		−	24964.0i		−	0.0948594i
$514$	−197795.			−0.748667
$515$	−14078.2			−0.0530803
$516$		−	6900.35i		−	0.0259162i
$517$			25431.2i			0.0951450i
$518$	−216170.			−0.805632
$519$	−216984.			−0.805553
$520$		−	25815.7i		−	0.0954724i
$521$			406550.i			1.49775i	0.662712	+	0.748874i	$0.269405\pi$
$521$			406550.i			1.49775i	−0.662712	+	0.748874i	$0.730595\pi$
$522$	24838.9			0.0911575
$523$		−	239071.i		−	0.874026i	−0.899455	−	0.437013i	$-0.856036\pi$
$523$		−	239071.i		−	0.874026i	0.899455	−	0.437013i	$-0.143964\pi$
$524$	7005.89			0.0255153
$525$			171595.i			0.622567i
$526$		−	108686.i		−	0.392826i
$527$			75498.9i			0.271844i
$528$		−	137122.i		−	0.491857i
$529$	−190107.	−	205354.i	−0.679338	−	0.733825i
$530$	200891.			0.715169
$531$	86578.2			0.307057
$532$	−6191.61			−0.0218766
$533$	−68191.3			−0.240035
$534$			113608.i			0.398407i
$535$	5990.63			0.0209298
$536$			499271.i			1.73783i
$537$	−151437.			−0.525151
$538$	−513522.			−1.77417
$539$			225092.i			0.774789i
$540$		−	839.285i		−	0.00287821i
$541$	65179.7			0.222699			0.111349	−	0.993781i	$-0.464483\pi$
$541$	65179.7			0.222699			0.111349	+	0.993781i	$0.464483\pi$
$542$	310813.			1.05804
$543$		−	226344.i		−	0.767660i
$544$		−	13934.4i		−	0.0470859i
$545$	183386.			0.617410
$546$		−	47236.5i		−	0.158450i
$547$	245413.			0.820206			0.410103	−	0.912039i	$-0.365493\pi$
$547$	245413.			0.820206			0.410103	+	0.912039i	$0.365493\pi$
$548$		−	8484.90i		−	0.0282544i
$549$		−	90536.2i		−	0.300384i
$550$		−	206309.i		−	0.682014i
$551$			41603.1i			0.137032i
$552$	−71534.9	+	163705.i	−0.234768	+	0.537261i
$553$	−543516.			−1.77730
$554$	87524.3			0.285173
$555$	49078.9			0.159334
$556$	18254.1			0.0590487
$557$		−	425630.i		−	1.37190i	−0.727650	−	0.685949i	$-0.759387\pi$
$557$		−	425630.i		−	1.37190i	0.727650	−	0.685949i	$-0.240613\pi$
$558$	−38153.5			−0.122537
$559$			88173.5i			0.282172i
$560$	191928.			0.612014
$561$	116497.			0.370160
$562$			378884.i			1.19959i
$563$			8177.89i			0.0258003i	0.999917	+	0.0129001i	$0.00410636\pi$
$563$			8177.89i			0.0258003i	−0.999917	+	0.0129001i	$0.995894\pi$
$564$	−641.899			−0.00201794
$565$	253888.			0.795326
$566$		−	190631.i		−	0.595060i
$567$		−	48965.9i		−	0.152310i
$568$	57792.4			0.179132
$569$			540842.i			1.67050i	0.549871	+	0.835249i	$0.314677\pi$
$569$			540842.i			1.67050i	−0.549871	+	0.835249i	$0.685323\pi$
$570$	−42010.7			−0.129304
$571$			440456.i			1.35092i	0.737396	+	0.675460i	$0.236055\pi$
$571$			440456.i			1.35092i	−0.737396	+	0.675460i	$0.763945\pi$
$572$		−	1900.35i		−	0.00580821i
$573$		−	227496.i		−	0.692890i
$574$		−	523952.i		−	1.59026i
$575$	−104141.	+	238323.i	−0.314981	+	0.720826i
$576$	113937.			0.343416
$577$	−297085.			−0.892336			−0.446168	−	0.894949i	$-0.647212\pi$
$577$	−297085.			−0.892336			−0.446168	+	0.894949i	$0.647212\pi$
$578$	154739.			0.463174
$579$	98712.3			0.294452
$580$			1398.69i			0.00415781i
$581$	−304825.			−0.903022
$582$		−	71924.7i		−	0.212340i
$583$	471521.			1.38728
$584$	439610.			1.28897
$585$			10724.5i			0.0313375i
$586$		−	216679.i		−	0.630988i
$587$	153326.			0.444978			0.222489	−	0.974935i	$-0.428582\pi$
$587$	153326.			0.444978			0.222489	+	0.974935i	$0.428582\pi$
$588$	−5681.47			−0.0164326
$589$		−	63903.8i		−	0.184203i
$590$		−	145698.i		−	0.418552i
$591$	−53231.9			−0.152404
$592$			202391.i			0.577495i
$593$	−424013.			−1.20578			−0.602892	−	0.797823i	$-0.705985\pi$
$593$	−424013.			−1.20578			−0.602892	+	0.797823i	$0.705985\pi$
$594$			58871.9i			0.166854i
$595$			163060.i			0.460588i
$596$		−	4192.49i		−	0.0118027i
$597$			146392.i			0.410742i
$598$	28667.8	−	65605.4i	0.0801662	−	0.183458i
$599$	119251.			0.332360			0.166180	−	0.986095i	$-0.446857\pi$
$599$	119251.			0.332360			0.166180	+	0.986095i	$0.446857\pi$
$600$	166039.			0.461219
$601$	−150101.			−0.415561			−0.207780	−	0.978175i	$-0.566624\pi$
$601$	−150101.			−0.415561			−0.207780	+	0.978175i	$0.566624\pi$
$602$	−677487.			−1.86942
$603$		−	207409.i		−	0.570419i
$604$	−17274.5			−0.0473512
$605$		−	37730.5i		−	0.103082i
$606$	−324455.			−0.883506
$607$	−147608.			−0.400619			−0.200310	−	0.979733i	$-0.564195\pi$
$607$	−147608.			−0.400619			−0.200310	+	0.979733i	$0.564195\pi$
$608$			11794.4i			0.0319056i
$609$			81602.8i			0.220024i
$610$	−152359.			−0.409456
$611$	8202.27			0.0219711
$612$			2940.46i			0.00785077i
$613$		−	255821.i		−	0.680793i	−0.940282	−	0.340396i	$-0.889439\pi$
$613$		−	255821.i		−	0.680793i	0.940282	−	0.340396i	$-0.110561\pi$
$614$	118003.			0.313008
$615$			118957.i			0.314514i
$616$	465573.			1.22695
$617$		−	495699.i		−	1.30211i	−0.759030	−	0.651055i	$-0.774327\pi$
$617$		−	495699.i		−	1.30211i	0.759030	−	0.651055i	$-0.225673\pi$
$618$			24925.7i			0.0652634i
$619$		−	38021.9i		−	0.0992322i	−0.998768	−	0.0496161i	$-0.984200\pi$
$619$		−	38021.9i		−	0.0992322i	0.998768	−	0.0496161i	$-0.0157998\pi$
$620$		−	2148.43i		−	0.00558905i
$621$	29717.3	−	68007.3i	0.0770596	−	0.176349i
$622$	187794.			0.485401
$623$	−373234.			−0.961625
$624$	−44225.6			−0.113581
$625$	158376.			0.405443
$626$		−	132210.i		−	0.337378i
$627$	−98605.5			−0.250822
$628$			9796.76i			0.0248407i
$629$	−171949.			−0.434609
$630$	−82402.3			−0.207615
$631$			518382.i			1.30194i	0.759104	+	0.650970i	$0.225638\pi$
$631$			518382.i			1.30194i	−0.759104	+	0.650970i	$0.774362\pi$
$632$			525917.i			1.31669i
$633$	−41859.8			−0.104470
$634$	235870.			0.586806
$635$		−	306202.i		−	0.759383i
$636$			11901.5i			0.0294230i
$637$	72598.5			0.178916
$638$		−	98111.3i		−	0.241034i
$639$	−24008.4			−0.0587978
$640$		−	179492.i		−	0.438213i
$641$		−	577465.i		−	1.40543i	−0.711471	−	0.702716i	$-0.751971\pi$
$641$		−	577465.i		−	1.40543i	0.711471	−	0.702716i	$-0.248029\pi$
$642$		−	10606.5i		−	0.0257336i
$643$			585729.i			1.41669i	0.705867	+	0.708345i	$0.250558\pi$
$643$			585729.i			1.41669i	−0.705867	+	0.708345i	$0.749442\pi$
$644$	−16867.2	−	7370.53i	−0.0406698	−	0.0177716i
$645$	153815.			0.369726
$646$	147186.			0.352696
$647$	356259.			0.851054			0.425527	−	0.904946i	$-0.360089\pi$
$647$	356259.			0.851054			0.425527	+	0.904946i	$0.360089\pi$
$648$	−47380.4			−0.112836
$649$		−	341975.i		−	0.811905i
$650$	−66540.5			−0.157492
$651$		−	125345.i		−	0.295763i
$652$	8978.66			0.0211211
$653$	780402.			1.83017			0.915086	−	0.403259i	$-0.132123\pi$
$653$	780402.			1.83017			0.915086	+	0.403259i	$0.132123\pi$
$654$		−	324687.i		−	0.759118i
$655$			156168.i			0.364007i
$656$	−490555.			−1.13993
$657$	−182625.			−0.423086
$658$			63022.7i			0.145561i
$659$		−	231275.i		−	0.532546i	−0.963898	−	0.266273i	$-0.914208\pi$
$659$		−	231275.i		−	0.532546i	0.963898	−	0.266273i	$-0.0857923\pi$
$660$	−3315.09			−0.00761040
$661$		−	157495.i		−	0.360467i	−0.983624	−	0.180233i	$-0.942315\pi$
$661$		−	157495.i		−	0.360467i	0.983624	−	0.180233i	$-0.0576853\pi$
$662$	−673453.			−1.53671
$663$		−	37573.5i		−	0.0854781i
$664$			294955.i			0.668990i
$665$		−	138017.i		−	0.312096i
$666$		−	86894.7i		−	0.195905i
$667$	−49524.6	+	113336.i	−0.111319	+	0.254750i
$668$	12101.8			0.0271204
$669$	287928.			0.643326
$670$	−349039.			−0.777542
$671$	−357609.			−0.794261
$672$			23134.1i			0.0512289i
$673$	23575.4			0.0520510			0.0260255	−	0.999661i	$-0.491715\pi$
$673$	23575.4			0.0520510			0.0260255	+	0.999661i	$0.491715\pi$
$674$			321382.i			0.707460i
$675$	−68976.6			−0.151389
$676$	14183.0			0.0310366
$677$		−	783873.i		−	1.71029i	−0.518393	−	0.855143i	$-0.673470\pi$
$677$		−	783873.i		−	1.71029i	0.518393	−	0.855143i	$-0.326530\pi$
$678$		−	449512.i		−	0.977871i
$679$	236293.			0.512520
$680$	157780.			0.341219
$681$			95212.6i			0.205305i
$682$			150702.i			0.324005i
$683$	−376520.			−0.807136			−0.403568	−	0.914950i	$-0.632230\pi$
$683$	−376520.			−0.807136			−0.403568	+	0.914950i	$0.632230\pi$
$684$		−	2488.86i		−	0.00531972i
$685$	189136.			0.403082
$686$		−	76743.0i		−	0.163076i
$687$			518169.i			1.09789i
$688$			634303.i			1.34005i
$689$		−	152079.i		−	0.320354i
$690$	−114446.	−	50009.8i	−0.240382	−	0.105041i
$691$	−803264.			−1.68230			−0.841148	−	0.540805i	$-0.818120\pi$
$691$	−803264.			−1.68230			−0.841148	+	0.540805i	$0.818120\pi$
$692$	−21632.9			−0.0451754
$693$	−193410.			−0.402730
$694$	−503439.			−1.04527
$695$			406901.i			0.842401i
$696$	78960.5			0.163001
$697$		−	416769.i		−	0.857888i
$698$	809858.			1.66226
$699$	48558.8			0.0993834
$700$			17107.7i			0.0349136i
$701$		−	232131.i		−	0.472387i	−0.971706	−	0.236193i	$-0.924100\pi$
$701$		−	232131.i		−	0.472387i	0.971706	−	0.236193i	$-0.0758998\pi$
$702$	18987.8			0.0385302
$703$	145541.			0.294493
$704$		−	450040.i		−	0.908042i
$705$		−	14308.5i		−	0.0287884i
$706$	625899.			1.25573
$707$		−	1.06592e6i		−	2.13249i
$708$	8631.66			0.0172198
$709$		−	163365.i		−	0.324987i	−0.986710	−	0.162494i	$-0.948046\pi$
$709$		−	163365.i		−	0.324987i	0.986710	−	0.162494i	$-0.0519537\pi$
$710$			40402.5i			0.0801478i
$711$		−	218479.i		−	0.432186i
$712$			361149.i			0.712405i
$713$	76071.4	−	174087.i	0.149638	−	0.342443i
$714$	288699.			0.566302
$715$	42360.6			0.0828611
$716$	−15098.0			−0.0294505
$717$	−60075.6			−0.116858
$718$			507837.i			0.985089i
$719$	185602.			0.359026			0.179513	−	0.983756i	$-0.442548\pi$
$719$	185602.			0.359026			0.179513	+	0.983756i	$0.442548\pi$
$720$			77149.8i			0.148823i
$721$	−81887.7			−0.157524
$722$	388195.			0.744689
$723$			25837.6i			0.0494283i
$724$		−	22566.0i		−	0.0430504i
$725$	114951.			0.218694
$726$	−66802.3			−0.126741
$727$			965256.i			1.82631i	0.407615	+	0.913154i	$0.366360\pi$
$727$			965256.i			1.82631i	−0.407615	+	0.913154i	$0.633640\pi$
$728$		−	150160.i		−	0.283330i
$729$	19683.0			0.0370370
$730$			307330.i			0.576712i
$731$	−538896.			−1.00849
$732$		−	9026.26i		−	0.0168456i
$733$		−	581367.i		−	1.08204i	−0.841011	−	0.541018i	$-0.818039\pi$
$733$		−	581367.i		−	1.08204i	0.841011	−	0.541018i	$-0.181961\pi$
$734$			1.00846e6i			1.87183i
$735$		−	126645.i		−	0.234431i
$736$	−14040.1	+	32130.3i	−0.0259187	+	0.0593143i
$737$	−819246.			−1.50827
$738$	210615.			0.386702
$739$	−546933.			−1.00149			−0.500743	−	0.865596i	$-0.666940\pi$
$739$	−546933.			−1.00149			−0.500743	+	0.865596i	$0.666940\pi$
$740$	4893.06			0.00893546
$741$			31803.0i			0.0579204i
$742$	1.16851e6			2.12238
$743$		−	270012.i		−	0.489109i	−0.969636	−	0.244555i	$-0.921358\pi$
$743$		−	270012.i		−	0.489109i	0.969636	−	0.244555i	$-0.0786418\pi$
$744$	−121286.			−0.219111
$745$	93454.6			0.168379
$746$		−	247760.i		−	0.445199i
$747$		−	122532.i		−	0.219587i
$748$	11614.5			0.0207586
$749$	34845.2			0.0621125
$750$			263638.i			0.468690i
$751$			524790.i			0.930477i	0.885185	+	0.465238i	$0.154031\pi$
$751$			524790.i			0.930477i	−0.885185	+	0.465238i	$0.845969\pi$
$752$	59005.5			0.104341
$753$			404327.i			0.713087i
$754$	−31643.6			−0.0556600
$755$		−	385064.i		−	0.675522i
$756$		−	4881.80i		−	0.00854154i
$757$		−	280320.i		−	0.489174i	−0.969627	−	0.244587i	$-0.921348\pi$
$757$		−	280320.i		−	0.489174i	0.969627	−	0.244587i	$-0.0786523\pi$
$758$			869461.i			1.51325i
$759$	−268622.	−	117381.i	−0.466292	−	0.203757i
$760$	−133548.			−0.231212
$761$	−641718.			−1.10809			−0.554045	−	0.832487i	$-0.686916\pi$
$761$	−641718.			−1.10809			−0.554045	+	0.832487i	$0.686916\pi$
$762$	−542135.			−0.933678
$763$	1.06669e6			1.83226
$764$		−	22680.8i		−	0.0388573i
$765$	−65545.6			−0.112001
$766$		−	536908.i		−	0.915045i
$767$	−110296.			−0.187487
$768$	33042.3			0.0560206
$769$		−	500311.i		−	0.846034i	−0.906122	−	0.423017i	$-0.860971\pi$
$769$		−	500311.i		−	0.846034i	0.906122	−	0.423017i	$-0.139029\pi$
$770$			325481.i			0.548964i
$771$	261206.			0.439415
$772$	9841.40			0.0165129
$773$			960312.i			1.60714i	0.595211	+	0.803569i	$0.297068\pi$
$773$			960312.i			1.60714i	−0.595211	+	0.803569i	$0.702932\pi$
$774$		−	272332.i		−	0.454586i
$775$	−176569.			−0.293975
$776$		−	228641.i		−	0.379692i
$777$	285473.			0.472850
$778$			1.02602e6i			1.69510i
$779$			352762.i			0.581309i
$780$			1069.21i			0.00175741i
$781$			94830.7i			0.155470i
$782$	400965.	+	175211.i	0.655681	+	0.286515i
$783$	−32802.1			−0.0535031
$784$	522259.			0.849677
$785$	−218379.			−0.354382
$786$	276497.			0.447554
$787$		−	955534.i		−	1.54275i	−0.636378	−	0.771377i	$-0.719568\pi$
$787$		−	955534.i		−	1.54275i	0.636378	−	0.771377i	$-0.280432\pi$
$788$	−5307.10			−0.00854683
$789$			143530.i			0.230562i
$790$	−367667.			−0.589116
$791$	1.47677e6			2.36026
$792$			187148.i			0.298356i
$793$			115339.i			0.183412i
$794$	−671709.			−1.06547
$795$	−265295.			−0.419754
$796$			14595.0i			0.0230345i
$797$			756938.i			1.19164i	0.803120	+	0.595818i	$0.203172\pi$
$797$			756938.i			1.19164i	−0.803120	+	0.595818i	$0.796828\pi$
$798$	−244360.			−0.383729
$799$			50130.4i			0.0785249i
$800$	32588.3			0.0509192
$801$		−	150030.i		−	0.233837i
$802$			265529.i			0.412823i
$803$			721349.i			1.11870i
$804$		−	20678.3i		−	0.0319891i
$805$	164296.	−	375987.i	0.253533	−	0.580204i
$806$	48605.7			0.0748199
$807$	678154.			1.04131
$808$	−1.03141e6			−1.57982
$809$	27315.7			0.0417364			0.0208682	−	0.999782i	$-0.493357\pi$
$809$	27315.7			0.0417364			0.0208682	+	0.999782i	$0.493357\pi$
$810$		−	33123.5i		−	0.0504855i
$811$	643247.			0.977993			0.488997	−	0.872286i	$-0.337363\pi$
$811$	643247.			0.977993			0.488997	+	0.872286i	$0.337363\pi$
$812$			8135.62i			0.0123390i
$813$	−410457.			−0.620993
$814$	−343225.			−0.518001
$815$			200143.i			0.301318i
$816$		−	270297.i		−	0.405938i
$817$	456132.			0.683356
$818$	1.11174e6			1.66149
$819$			62380.2i			0.0929992i
$820$			11859.8i			0.0176380i
$821$	593789.			0.880940			0.440470	−	0.897767i	$-0.354812\pi$
$821$	593789.			0.880940			0.440470	+	0.897767i	$0.354812\pi$
$822$		−	334868.i		−	0.495599i
$823$	−436521.			−0.644474			−0.322237	−	0.946659i	$-0.604435\pi$
$823$	−436521.			−0.644474			−0.322237	+	0.946659i	$0.604435\pi$
$824$			79236.2i			0.116700i
$825$			272451.i			0.400295i
$826$		−	847469.i		−	1.24212i
$827$		−	213829.i		−	0.312648i	−0.987706	−	0.156324i	$-0.950036\pi$
$827$		−	213829.i		−	0.312648i	0.987706	−	0.156324i	$-0.0499645\pi$
$828$	2962.75	−	6780.18i	0.00432151	−	0.00988964i
$829$	−889288.			−1.29400			−0.646999	−	0.762491i	$-0.723976\pi$
$829$	−889288.			−1.29400			−0.646999	+	0.762491i	$0.723976\pi$
$830$	−206202.			−0.299321
$831$	−115584.			−0.167377
$832$	−145150.			−0.209687
$833$			443705.i			0.639447i
$834$	720422.			1.03575
$835$			269760.i			0.386905i
$836$	−9830.75			−0.0140661
$837$	50385.2			0.0719204
$838$		−	11254.0i		−	0.0160257i
$839$			401532.i			0.570422i	0.958465	+	0.285211i	$0.0920637\pi$
$839$			401532.i			0.570422i	−0.958465	+	0.285211i	$0.907936\pi$
$840$	−261949.			−0.371242
$841$	−652616.			−0.922710
$842$			497788.i			0.702134i
$843$		−	500351.i		−	0.704077i
$844$	−4173.34			−0.00585866
$845$			316152.i			0.442774i
$846$	−25333.4			−0.0353959
$847$		−	219464.i		−	0.305912i
$848$		−	1.09402e6i		−	1.52137i
$849$			251746.i			0.349259i
$850$		−	406680.i		−	0.562879i
$851$	396485.	+	173253.i	0.547479	+	0.239234i
$852$	−2393.58			−0.00329738
$853$	554765.			0.762449			0.381225	−	0.924482i	$-0.375502\pi$
$853$	554765.			0.762449			0.381225	+	0.924482i	$0.375502\pi$
$854$	−886212.			−1.21513
$855$	55479.1			0.0758922
$856$		−	33716.9i		−	0.0460151i
$857$	774321.			1.05429			0.527144	−	0.849776i	$-0.323263\pi$
$857$	774321.			1.05429			0.527144	+	0.849776i	$0.323263\pi$
$858$		−	75000.0i		−	0.101879i
$859$	−1.19815e6			−1.62377			−0.811887	−	0.583814i	$-0.801560\pi$
$859$	−1.19815e6			−1.62377			−0.811887	+	0.583814i	$0.801560\pi$
$860$	15335.1			0.0207343
$861$			691928.i			0.933371i
$862$		−	406935.i		−	0.547659i
$863$	1.29705e6			1.74155			0.870776	−	0.491680i	$-0.163617\pi$
$863$	1.29705e6			1.74155			0.870776	+	0.491680i	$0.163617\pi$
$864$	−9299.31			−0.0124573
$865$		−	482217.i		−	0.644482i
$866$		−	550988.i		−	0.734694i
$867$	−204347.			−0.271851
$868$		−	12496.6i		−	0.0165864i
$869$	−862969.			−1.14276
$870$			55201.1i			0.0729305i
$871$			264229.i			0.348293i
$872$		−	1.03215e6i		−	1.35740i
$873$			94983.3i			0.124629i
$874$	−339385.	−	148302.i	−0.444293	−	0.194144i
$875$	−866123.			−1.13126
$876$	−18207.3			−0.0237267
$877$	−957051.			−1.24433			−0.622165	−	0.782886i	$-0.713747\pi$
$877$	−957051.			−1.24433			−0.622165	+	0.782886i	$0.713747\pi$
$878$	−121102.			−0.157095
$879$			286144.i			0.370346i
$880$	304734.			0.393510
$881$			1.13453e6i			1.46172i	0.682530	+	0.730858i	$0.260880\pi$
$881$			1.13453e6i			1.46172i	−0.682530	+	0.730858i	$0.739120\pi$
$882$	−224227.			−0.288238
$883$	688460.			0.882994			0.441497	−	0.897263i	$-0.354448\pi$
$883$	688460.			0.882994			0.441497	+	0.897263i	$0.354448\pi$
$884$		−	3746.00i		−	0.00479362i
$885$			192408.i			0.245661i
$886$	−945481.			−1.20444
$887$	−957189.			−1.21661			−0.608304	−	0.793704i	$-0.708150\pi$
$887$	−957189.			−1.21661			−0.608304	+	0.793704i	$0.708150\pi$
$888$		−	276230.i		−	0.350303i
$889$		−	1.78106e6i		−	2.25359i
$890$	−252478.			−0.318746
$891$		−	77745.9i		−	0.0979314i
$892$	28705.8			0.0360778
$893$		−	42431.3i		−	0.0532089i
$894$		−	165462.i		−	0.207026i
$895$		−	336548.i		−	0.420147i
$896$		−	1.04404e6i		−	1.30047i
$897$	−37858.5	+	86638.0i	−0.0470520	+	0.107677i
$898$	739336.			0.916830
$899$	−83968.0			−0.103895
$900$	−6876.82			−0.00848990
$901$	929469.			1.14495
$902$		−	831907.i		−	1.02250i
$903$	894685.			1.09722
$904$		−	1.42895e6i		−	1.74856i
$905$	503017.			0.614166
$906$	−681761.			−0.830569
$907$		−	796894.i		−	0.968693i	−0.874876	−	0.484346i	$-0.839057\pi$
$907$		−	796894.i		−	0.968693i	0.874876	−	0.484346i	$-0.160943\pi$
$908$			9492.49i			0.0115135i
$909$	428473.			0.518556
$910$	104977.			0.126768
$911$		−	847677.i		−	1.02140i	−0.859760	−	0.510698i	$-0.829387\pi$
$911$		−	847677.i		−	1.02140i	0.859760	−	0.510698i	$-0.170613\pi$
$912$			228784.i			0.275066i
$913$	−483987.			−0.580620
$914$		−	306144.i		−	0.366466i
$915$	201204.			0.240322
$916$			51660.3i			0.0615695i
$917$			908369.i			1.08025i
$918$			116049.i			0.137707i
$919$			395433.i			0.468212i	0.972211	+	0.234106i	$0.0752162\pi$
$919$			395433.i			0.468212i	−0.972211	+	0.234106i	$0.924784\pi$
$920$	−363812.	−	158976.i	−0.429835	−	0.187826i
$921$	−155833.			−0.183714
$922$	206269.			0.242646
$923$	30585.5			0.0359015
$924$	−19282.6			−0.0225851
$925$		−	402136.i		−	0.469991i
$926$	836079.			0.975047
$927$		−	32916.7i		−	0.0383051i
$928$	15497.5			0.0179956
$929$	1.21526e6			1.40812			0.704058	−	0.710142i	$-0.251369\pi$
$929$	1.21526e6			1.40812			0.704058	+	0.710142i	$0.251369\pi$
$930$		−	84790.7i		−	0.0980353i
$931$		−	375561.i		−	0.433292i
$932$	4841.21			0.00557342
$933$	−247999.			−0.284897
$934$		−	1.64640e6i		−	1.88730i
$935$			258898.i			0.296146i
$936$	60360.4			0.0688970
$937$			402700.i			0.458673i	0.973347	+	0.229336i	$0.0736555\pi$
$937$			402700.i			0.458673i	−0.973347	+	0.229336i	$0.926344\pi$
$938$	−2.03022e6			−2.30748
$939$			174596.i			0.198017i
$940$		−	1426.53i		−	0.00161445i
$941$		−	347426.i		−	0.392358i	−0.980568	−	0.196179i	$-0.937147\pi$
$941$		−	347426.i		−	0.392358i	0.980568	−	0.196179i	$-0.0628534\pi$
$942$			386643.i			0.435720i
$943$	−419930.	+	960997.i	−0.472230	+	1.08068i
$944$	−793450.			−0.890381
$945$	108820.			0.121855
$946$	−1.07568e6			−1.20199
$947$	−1.25264e6			−1.39678			−0.698389	−	0.715718i	$-0.746099\pi$
$947$	−1.25264e6			−1.39678			−0.698389	+	0.715718i	$0.746099\pi$
$948$		−	21781.9i		−	0.0242370i
$949$	232655.			0.258333
$950$			344222.i			0.381409i
$951$	−311488.			−0.344414
$952$	917744.			1.01262
$953$			683312.i			0.752373i	0.926544	+	0.376186i	$0.122765\pi$
$953$			683312.i			0.752373i	−0.926544	+	0.376186i	$0.877235\pi$
$954$			469708.i			0.516097i
$955$	505577.			0.554346
$956$	−5989.40			−0.00655342
$957$			129565.i			0.141470i
$958$			1.67607e6i			1.82625i
$959$	1.10013e6			1.19621
$960$			253209.i			0.274750i
$961$	−794543.			−0.860341
$962$			110700.i			0.119618i
$963$			14006.8i			0.0151038i
$964$			2575.95i			0.00277194i
$965$			219374.i			0.235576i
$966$	−665688.	−	290888.i	−0.713373	−	0.311725i
$967$	502022.			0.536871			0.268436	−	0.963298i	$-0.413493\pi$
$967$	502022.			0.536871			0.268436	+	0.963298i	$0.413493\pi$
$968$	−212358.			−0.226630
$969$	−194372.			−0.207008
$970$	159843.			0.169883
$971$			871880.i			0.924737i	0.886688	+	0.462369i	$0.153000\pi$
$971$			871880.i			0.924737i	−0.886688	+	0.462369i	$0.847000\pi$
$972$	1962.35			0.00207704
$973$			2.36678e6i			2.49996i
$974$	1.79238e6			1.88935
$975$	87872.9			0.0924370
$976$			829723.i			0.871031i
$977$		−	732847.i		−	0.767758i	−0.923383	−	0.383879i	$-0.874588\pi$
$977$		−	732847.i		−	0.767758i	0.923383	−	0.383879i	$-0.125412\pi$
$978$	354355.			0.370477
$979$	−592605.			−0.618301
$980$		−	12626.3i		−	0.0131469i
$981$			428779.i			0.445550i
$982$	690401.			0.715943
$983$			736382.i			0.762073i	0.924560	+	0.381036i	$0.124433\pi$
$983$			736382.i			0.762073i	−0.924560	+	0.381036i	$0.875567\pi$
$984$	669523.			0.691474
$985$		−	118300.i		−	0.121931i
$986$		−	193398.i		−	0.198929i
$987$		−	83227.3i		−	0.0854341i
$988$			3170.69i			0.00324818i
$989$	1.24260e6	+	542983.i	1.27039	+	0.555128i
$990$	−130835.			−0.133491
$991$	647860.			0.659681			0.329840	−	0.944037i	$-0.393005\pi$
$991$	647860.			0.659681			0.329840	+	0.944037i	$0.393005\pi$
$992$	−23804.7			−0.0241902
$993$	889358.			0.901941
$994$			235006.i			0.237851i
$995$	−325336.			−0.328614
$996$		−	12216.1i		−	0.0123144i
$997$	−740784.			−0.745249			−0.372624	−	0.927982i	$-0.621542\pi$
$997$	−740784.			−0.745249			−0.372624	+	0.927982i	$0.621542\pi$
$998$	1.34675e6			1.35215
$999$			114753.i			0.114982i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.d.a.22.11	✓	16
3.2	odd	2		207.5.d.c.91.6		16
4.3	odd	2		1104.5.c.c.1057.11		16
23.22	odd	2	inner	69.5.d.a.22.12	yes	16
69.68	even	2		207.5.d.c.91.5		16
92.91	even	2		1104.5.c.c.1057.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.d.a.22.11	✓	16	1.1	even	1	trivial
69.5.d.a.22.12	yes	16	23.22	odd	2	inner
207.5.d.c.91.5		16	69.68	even	2
207.5.d.c.91.6		16	3.2	odd	2
1104.5.c.c.1057.11		16	4.3	odd	2
1104.5.c.c.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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.93471 q^{2} -5.19615 q^{3} -0.518045 q^{4} -11.5477i q^{5} -20.4454 q^{6} -67.1686i q^{7} -64.9937 q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+3.93471 q^{2} -5.19615 q^{3} -0.518045 q^{4} -11.5477i q^{5} -20.4454 q^{6} -67.1686i q^{7} -64.9937 q^{8} +27.0000 q^{9} -45.4369i q^{10} -106.647i q^{11} +2.69184 q^{12} -34.3967 q^{13} -264.289i q^{14} +60.0037i q^{15} -247.443 q^{16} -210.225i q^{17} +106.237 q^{18} +177.938i q^{19} +5.98224i q^{20} +349.018i q^{21} -419.626i q^{22} +(-211.819 + 484.741i) q^{23} +337.717 q^{24} +491.650 q^{25} -135.341 q^{26} -140.296 q^{27} +34.7964i q^{28} +233.806 q^{29} +236.097i q^{30} -359.135 q^{31} +66.2834 q^{32} +554.156i q^{33} -827.173i q^{34} -775.644 q^{35} -13.9872 q^{36} -817.931i q^{37} +700.136i q^{38} +178.730 q^{39} +750.529i q^{40} +1982.50 q^{41} +1373.29i q^{42} -2563.43i q^{43} +55.2481i q^{44} -311.788i q^{45} +(-833.446 + 1907.32i) q^{46} -238.461 q^{47} +1285.75 q^{48} -2110.62 q^{49} +1934.50 q^{50} +1092.36i q^{51} +17.8190 q^{52} +4421.31i q^{53} -552.025 q^{54} -1231.53 q^{55} +4365.54i q^{56} -924.594i q^{57} +919.961 q^{58} +3206.60 q^{59} -31.0846i q^{60} -3353.19i q^{61} -1413.09 q^{62} -1813.55i q^{63} +4219.89 q^{64} +397.203i q^{65} +2180.44i q^{66} -7681.83i q^{67} +108.906i q^{68} +(1100.64 - 2518.79i) q^{69} -3051.94 q^{70} -889.199 q^{71} -1754.83 q^{72} -6763.88 q^{73} -3218.32i q^{74} -2554.69 q^{75} -92.1801i q^{76} -7163.35 q^{77} +703.253 q^{78} -8091.81i q^{79} +2857.40i q^{80} +729.000 q^{81} +7800.55 q^{82} -4538.20i q^{83} -180.807i q^{84} -2427.61 q^{85} -10086.4i q^{86} -1214.89 q^{87} +6931.41i q^{88} -5556.68i q^{89} -1226.80i q^{90} +2310.38i q^{91} +(109.732 - 251.118i) q^{92} +1866.12 q^{93} -938.275 q^{94} +2054.78 q^{95} -344.419 q^{96} +3517.90i q^{97} -8304.70 q^{98} -2879.48i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9}+O(q^{10}) \) 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9	\( 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9} + 104 q^{13} + 680 q^{16} + 324 q^{18} - 732 q^{23} - 1764 q^{24} - 2984 q^{25} + 1800 q^{26} - 3528 q^{29} - 400 q^{31} + 5244 q^{32} + 912 q^{35} + 3888 q^{36} + 2016 q^{39} + 1008 q^{41} - 1168 q^{46} - 8664 q^{47} - 2016 q^{48} + 7240 q^{49} - 18852 q^{50} - 20952 q^{52} - 972 q^{54} + 6816 q^{55} - 13352 q^{58} + 20112 q^{59} + 4248 q^{62} - 896 q^{64} - 10044 q^{69} - 10680 q^{70} + 40368 q^{71} + 10044 q^{72} - 9568 q^{73} + 7560 q^{75} + 2952 q^{77} - 6912 q^{78} + 11664 q^{81} + 71800 q^{82} + 42744 q^{85} + 8352 q^{87} - 9876 q^{92} - 10008 q^{93} + 73720 q^{94} + 33312 q^{95} - 24948 q^{96} - 59052 q^{98}+O(q^{100}) \) 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9 + 104 * q^13 + 680 * q^16 + 324 * q^18 - 732 * q^23 - 1764 * q^24 - 2984 * q^25 + 1800 * q^26 - 3528 * q^29 - 400 * q^31 + 5244 * q^32 + 912 * q^35 + 3888 * q^36 + 2016 * q^39 + 1008 * q^41 - 1168 * q^46 - 8664 * q^47 - 2016 * q^48 + 7240 * q^49 - 18852 * q^50 - 20952 * q^52 - 972 * q^54 + 6816 * q^55 - 13352 * q^58 + 20112 * q^59 + 4248 * q^62 - 896 * q^64 - 10044 * q^69 - 10680 * q^70 + 40368 * q^71 + 10044 * q^72 - 9568 * q^73 + 7560 * q^75 + 2952 * q^77 - 6912 * q^78 + 11664 * q^81 + 71800 * q^82 + 42744 * q^85 + 8352 * q^87 - 9876 * q^92 - 10008 * q^93 + 73720 * q^94 + 33312 * q^95 - 24948 * q^96 - 59052 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more