LMFDB - Embedded newform 162.5.d.e.53.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,5,Mod(53,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.53"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	162.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,32,0,0,52,0,0,240] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.7459340196$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.3
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	162.53
Dual form			162.5.d.e.107.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.44949 - 1.41421i) q^{2} +(4.00000 - 6.92820i) q^{4} +(-0.360355 - 0.208051i) q^{5} +(-1.29423 - 2.24167i) q^{7} -22.6274i q^{8} -1.17691 q^{10} +(47.0443 - 27.1611i) q^{11} +(114.442 - 198.220i) q^{13} +(-6.34040 - 3.66063i) q^{14} +(-32.0000 - 55.4256i) q^{16} -380.263i q^{17} -298.242 q^{19} +(-2.88284 + 1.66441i) q^{20} +(76.8231 - 133.061i) q^{22} +(547.178 + 315.913i) q^{23} +(-312.413 - 541.116i) q^{25} -647.383i q^{26} -20.7077 q^{28} +(-568.722 + 328.352i) q^{29} +(772.296 - 1337.66i) q^{31} +(-156.767 - 90.5097i) q^{32} +(-537.773 - 931.450i) q^{34} +1.07706i q^{35} -187.842 q^{37} +(-730.541 + 421.778i) q^{38} +(-4.70766 + 8.15390i) q^{40} +(1848.89 + 1067.46i) q^{41} +(-1152.49 - 1996.16i) q^{43} -434.577i q^{44} +1787.08 q^{46} +(-1736.82 + 1002.76i) q^{47} +(1197.15 - 2073.52i) q^{49} +(-1530.51 - 883.639i) q^{50} +(-915.538 - 1585.76i) q^{52} +3293.71i q^{53} -22.6036 q^{55} +(-50.7232 + 29.2851i) q^{56} +(-928.719 + 1608.59i) q^{58} +(2780.93 + 1605.57i) q^{59} +(911.109 + 1578.09i) q^{61} -4368.77i q^{62} -512.000 q^{64} +(-82.4797 + 47.6197i) q^{65} +(-1049.04 + 1817.00i) q^{67} +(-2634.54 - 1521.05i) q^{68} +(1.52320 + 2.63825i) q^{70} +2181.30i q^{71} -5527.36 q^{73} +(-460.117 + 265.649i) q^{74} +(-1192.97 + 2066.28i) q^{76} +(-121.772 - 70.3052i) q^{77} +(5513.17 + 9549.08i) q^{79} +26.6305i q^{80} +6038.44 q^{82} +(10443.1 - 6029.35i) q^{83} +(-79.1141 + 137.030i) q^{85} +(-5646.01 - 3259.72i) q^{86} +(-614.585 - 1064.49i) q^{88} +2905.35i q^{89} -592.458 q^{91} +(4377.43 - 2527.31i) q^{92} +(-2836.22 + 4912.48i) q^{94} +(107.473 + 62.0496i) q^{95} +(6446.38 + 11165.5i) q^{97} -6772.10i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{4} + 52 q^{7} + 240 q^{10} + 292 q^{13} - 256 q^{16} + 1480 q^{19} + 864 q^{22} - 1564 q^{25} + 832 q^{28} + 5056 q^{31} + 312 q^{34} + 8848 q^{37} + 960 q^{40} - 428 q^{43} + 4320 q^{46} + 7956 q^{49}+ \cdots + 3808 q^{97}+O(q^{100}) $ 8 * q + 32 * q^4 + 52 * q^7 + 240 * q^10 + 292 * q^13 - 256 * q^16 + 1480 * q^19 + 864 * q^22 - 1564 * q^25 + 832 * q^28 + 5056 * q^31 + 312 * q^34 + 8848 * q^37 + 960 * q^40 - 428 * q^43 + 4320 * q^46 + 7956 * q^49 - 2336 * q^52 + 8424 * q^55 - 72 * q^58 - 4496 * q^61 - 4096 * q^64 - 16436 * q^67 + 3504 * q^70 - 41600 * q^73 + 5920 * q^76 + 16732 * q^79 - 3072 * q^82 - 19152 * q^85 - 6912 * q^88 - 11848 * q^91 - 29424 * q^94 + 3808 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{5}{6}\right)$

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 $

$2$	2.44949	−	1.41421i	0.612372	−	0.353553i
$2$	2.44949	−	1.41421i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	4.00000	−	6.92820i	0.250000	−	0.433013i
$5$	−0.360355	−	0.208051i	−0.0144142	−	0.00832204i	0.492776	−	0.870156i	$-0.335982\pi$
$5$	−0.360355	−	0.208051i	−0.0144142	−	0.00832204i	−0.507190	+	0.861834i	$0.669316\pi$
$6$		0			0
$7$	−1.29423	−	2.24167i	−0.0264128	−	0.0457484i	0.852517	−	0.522700i	$-0.175075\pi$
$7$	−1.29423	−	2.24167i	−0.0264128	−	0.0457484i	−0.878930	+	0.476951i	$0.841742\pi$
$8$		−	22.6274i		−	0.353553i
$9$		0			0
$10$	−1.17691			−0.0117691
$11$	47.0443	−	27.1611i	0.388796	−	0.224472i	−0.292842	−	0.956161i	$-0.594601\pi$
$11$	47.0443	−	27.1611i	0.388796	−	0.224472i	0.681638	+	0.731689i	$0.261268\pi$
$12$		0			0
$13$	114.442	−	198.220i	0.677173	−	1.17290i	−0.298655	−	0.954361i	$-0.596538\pi$
$13$	114.442	−	198.220i	0.677173	−	1.17290i	0.975829	−	0.218537i	$-0.0701286\pi$
$14$	−6.34040	−	3.66063i	−0.0323490	−	0.0186767i
$15$		0			0
$16$	−32.0000	−	55.4256i	−0.125000	−	0.216506i
$17$		−	380.263i		−	1.31579i	−0.753110	−	0.657894i	$-0.771447\pi$
$17$		−	380.263i		−	1.31579i	0.753110	−	0.657894i	$-0.228553\pi$
$18$		0			0
$19$	−298.242			−0.826156			−0.413078	−	0.910696i	$-0.635546\pi$
$19$	−298.242			−0.826156			−0.413078	+	0.910696i	$0.635546\pi$
$20$	−2.88284	+	1.66441i	−0.00720710	+	0.00416102i
$21$		0			0
$22$	76.8231	−	133.061i	0.158725	−	0.274920i
$23$	547.178	+	315.913i	1.03436	+	0.597190i	0.918232	−	0.396044i	$-0.129617\pi$
$23$	547.178	+	315.913i	1.03436	+	0.597190i	0.116132	+	0.993234i	$0.462951\pi$
$24$		0			0
$25$	−312.413	−	541.116i	−0.499861	−	0.865785i
$26$		−	647.383i		−	0.957668i
$27$		0			0
$28$	−20.7077			−0.0264128
$29$	−568.722	+	328.352i	−0.676245	+	0.390430i	−0.798439	−	0.602076i	$-0.794340\pi$
$29$	−568.722	+	328.352i	−0.676245	+	0.390430i	0.122194	+	0.992506i	$0.461007\pi$
$30$		0			0
$31$	772.296	−	1337.66i	0.803638	−	1.39194i	−0.113569	−	0.993530i	$-0.536228\pi$
$31$	772.296	−	1337.66i	0.803638	−	1.39194i	0.917207	−	0.398412i	$-0.130438\pi$
$32$	−156.767	−	90.5097i	−0.153093	−	0.0883883i
$33$		0			0
$34$	−537.773	−	931.450i	−0.465201	−	0.805753i
$35$			1.07706i			0.000879235i
$36$		0			0
$37$	−187.842			−0.137211			−0.0686055	−	0.997644i	$-0.521855\pi$
$37$	−187.842			−0.137211			−0.0686055	+	0.997644i	$0.521855\pi$
$38$	−730.541	+	421.778i	−0.505915	+	0.292090i
$39$		0			0
$40$	−4.70766	+	8.15390i	−0.00294229	+	0.00509619i
$41$	1848.89	+	1067.46i	1.09987	+	0.635013i	0.936188	−	0.351500i	$-0.114328\pi$
$41$	1848.89	+	1067.46i	1.09987	+	0.635013i	0.163686	+	0.986512i	$0.447662\pi$
$42$		0			0
$43$	−1152.49	−	1996.16i	−0.623302	−	1.07959i	−0.988867	−	0.148805i	$-0.952457\pi$
$43$	−1152.49	−	1996.16i	−0.623302	−	1.07959i	0.365564	−	0.930786i	$-0.380876\pi$
$44$		−	434.577i		−	0.224472i
$45$		0			0
$46$	1787.08			0.844554
$47$	−1736.82	+	1002.76i	−0.786249	+	0.453941i	−0.838641	−	0.544685i	$-0.816649\pi$
$47$	−1736.82	+	1002.76i	−0.786249	+	0.453941i	0.0523911	+	0.998627i	$0.483316\pi$
$48$		0			0
$49$	1197.15	−	2073.52i	0.498605	−	0.863609i
$50$	−1530.51	−	883.639i	−0.612203	−	0.353455i
$51$		0			0
$52$	−915.538	−	1585.76i	−0.338587	−	0.586449i
$53$			3293.71i			1.17256i	0.810110	+	0.586278i	$0.199407\pi$
$53$			3293.71i			1.17256i	−0.810110	+	0.586278i	$0.800593\pi$
$54$		0			0
$55$	−22.6036			−0.00747225
$56$	−50.7232	+	29.2851i	−0.0161745	+	0.00933835i
$57$		0			0
$58$	−928.719	+	1608.59i	−0.276076	+	0.478177i
$59$	2780.93	+	1605.57i	0.798888	+	0.461238i	0.843082	−	0.537785i	$-0.180739\pi$
$59$	2780.93	+	1605.57i	0.798888	+	0.461238i	−0.0441943	+	0.999023i	$0.514072\pi$
$60$		0			0
$61$	911.109	+	1578.09i	0.244856	+	0.424103i	0.962091	−	0.272728i	$-0.0879260\pi$
$61$	911.109	+	1578.09i	0.244856	+	0.424103i	−0.717235	+	0.696831i	$0.754593\pi$
$62$		−	4368.77i		−	1.13652i
$63$		0			0
$64$	−512.000			−0.125000
$65$	−82.4797	+	47.6197i	−0.0195218	+	0.0112709i
$66$		0			0
$67$	−1049.04	+	1817.00i	−0.233692	+	0.404767i	−0.958892	−	0.283772i	$-0.908414\pi$
$67$	−1049.04	+	1817.00i	−0.233692	+	0.404767i	0.725200	+	0.688539i	$0.241747\pi$
$68$	−2634.54	−	1521.05i	−0.569753	−	0.328947i
$69$		0			0
$70$	1.52320	+	2.63825i	0.000310856	+	0.000538419i
$71$			2181.30i			0.432713i	0.976314	+	0.216356i	$0.0694173\pi$
$71$			2181.30i			0.432713i	−0.976314	+	0.216356i	$0.930583\pi$
$72$		0			0
$73$	−5527.36			−1.03722			−0.518611	−	0.855010i	$-0.673551\pi$
$73$	−5527.36			−1.03722			−0.518611	+	0.855010i	$0.673551\pi$
$74$	−460.117	+	265.649i	−0.0840243	+	0.0485114i
$75$		0			0
$76$	−1192.97	+	2066.28i	−0.206539	+	0.357736i
$77$	−121.772	−	70.3052i	−0.0205384	−	0.0118579i
$78$		0			0
$79$	5513.17	+	9549.08i	0.883379	+	1.53006i	0.847561	+	0.530698i	$0.178070\pi$
$79$	5513.17	+	9549.08i	0.883379	+	1.53006i	0.0358180	+	0.999358i	$0.488596\pi$
$80$			26.6305i			0.00416102i
$81$		0			0
$82$	6038.44			0.898043
$83$	10443.1	−	6029.35i	1.51591	−	0.875214i	0.516089	−	0.856535i	$-0.327387\pi$
$83$	10443.1	−	6029.35i	1.51591	−	0.875214i	0.999826	−	0.0186785i	$-0.00594589\pi$
$84$		0			0
$85$	−79.1141	+	137.030i	−0.0109500	+	0.0189660i
$86$	−5646.01	−	3259.72i	−0.763386	−	0.440741i
$87$		0			0
$88$	−614.585	−	1064.49i	−0.0793627	−	0.137460i
$89$			2905.35i			0.366791i	0.983039	+	0.183395i	$0.0587089\pi$
$89$			2905.35i			0.366791i	−0.983039	+	0.183395i	$0.941291\pi$
$90$		0			0
$91$	−592.458			−0.0715443
$92$	4377.43	−	2527.31i	0.517182	−	0.298595i
$93$		0			0
$94$	−2836.22	+	4912.48i	−0.320985	+	0.555962i
$95$	107.473	+	62.0496i	0.0119084	+	0.00687530i
$96$		0			0
$97$	6446.38	+	11165.5i	0.685129	+	1.18668i	0.973396	+	0.229128i	$0.0735875\pi$
$97$	6446.38	+	11165.5i	0.685129	+	1.18668i	−0.288267	+	0.957550i	$0.593079\pi$
$98$		−	6772.10i		−	0.705134i
$99$		0			0
$100$	−4998.61			−0.499861
$101$	3121.12	−	1801.98i	0.305962	−	0.176647i	−0.339156	−	0.940730i	$-0.610141\pi$
$101$	3121.12	−	1801.98i	0.305962	−	0.176647i	0.645118	+	0.764083i	$0.276808\pi$
$102$		0			0
$103$	−1607.06	+	2783.51i	−0.151481	+	0.262373i	−0.931772	−	0.363044i	$-0.881738\pi$
$103$	−1607.06	+	2783.51i	−0.151481	+	0.262373i	0.780291	+	0.625416i	$0.215071\pi$
$104$	−4485.20	−	2589.53i	−0.414682	−	0.239417i
$105$		0			0
$106$	4658.01	+	8067.91i	0.414561	+	0.718041i
$107$			13043.6i			1.13928i	0.821895	+	0.569639i	$0.192917\pi$
$107$			13043.6i			1.13928i	−0.821895	+	0.569639i	$0.807083\pi$
$108$		0			0
$109$	3214.31			0.270542			0.135271	−	0.990809i	$-0.456810\pi$
$109$	3214.31			0.270542			0.135271	+	0.990809i	$0.456810\pi$
$110$	−55.3672	+	31.9662i	−0.00457580	+	0.00264184i
$111$		0			0
$112$	−82.8306	+	143.467i	−0.00660321	+	0.0114371i
$113$	−9741.28	−	5624.13i	−0.762885	−	0.440452i	0.0674456	−	0.997723i	$-0.478515\pi$
$113$	−9741.28	−	5624.13i	−0.762885	−	0.440452i	−0.830331	+	0.557271i	$0.811848\pi$
$114$		0			0
$115$	−131.452	−	227.682i	−0.00993968	−	0.0172160i
$116$			5253.63i			0.390430i
$117$		0			0
$118$	9082.47			0.652289
$119$	−852.424	+	492.147i	−0.0601952	+	0.0347537i
$120$		0			0
$121$	−5845.05	+	10123.9i	−0.399225	+	0.691478i
$122$	4463.51	+	2577.01i	0.299886	+	0.173139i
$123$		0			0
$124$	−6178.37	−	10701.2i	−0.401819	−	0.695971i
$125$			520.056i			0.0332836i
$126$		0			0
$127$	−1395.11			−0.0864973			−0.0432487	−	0.999064i	$-0.513771\pi$
$127$	−1395.11			−0.0864973			−0.0432487	+	0.999064i	$0.513771\pi$
$128$	−1254.14	+	724.077i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	−134.689	+	233.288i	−0.00796975	+	0.0138040i
$131$	14510.4	+	8377.57i	0.845543	+	0.488175i	0.859145	−	0.511733i	$-0.170996\pi$
$131$	14510.4	+	8377.57i	0.845543	+	0.488175i	−0.0136013	+	0.999907i	$0.504330\pi$
$132$		0			0
$133$	385.994	+	668.560i	0.0218211	+	0.0377953i
$134$			5934.29i			0.330491i
$135$		0			0
$136$	−8604.37			−0.465201
$137$	−15867.6	+	9161.18i	−0.845417	+	0.488102i	−0.859102	−	0.511804i	$-0.828977\pi$
$137$	−15867.6	+	9161.18i	−0.845417	+	0.488102i	0.0136847	+	0.999906i	$0.495644\pi$
$138$		0			0
$139$	−4140.02	+	7170.72i	−0.214275	+	0.371136i	−0.953048	−	0.302819i	$-0.902072\pi$
$139$	−4140.02	+	7170.72i	−0.214275	+	0.371136i	0.738773	+	0.673955i	$0.235406\pi$
$140$	7.46211	+	4.30825i	0.000380720	+	0.000219809i
$141$		0			0
$142$	3084.83	+	5343.08i	0.152987	+	0.264981i
$143$		−	12433.5i		−	0.608025i
$144$		0			0
$145$	273.256			0.0129967
$146$	−13539.2	+	7816.86i	−0.635166	+	0.366713i
$147$		0			0
$148$	−751.368	+	1301.41i	−0.0343028	+	0.0594141i
$149$	−10055.1	−	5805.30i	−0.452911	−	0.261488i	0.256148	−	0.966638i	$-0.417547\pi$
$149$	−10055.1	−	5805.30i	−0.452911	−	0.261488i	−0.709059	+	0.705149i	$0.750880\pi$
$150$		0			0
$151$	−21954.7	−	38026.6i	−0.962881	−	1.66776i	−0.715203	−	0.698916i	$-0.753666\pi$
$151$	−21954.7	−	38026.6i	−0.962881	−	1.66776i	−0.247678	−	0.968843i	$-0.579667\pi$
$152$			6748.45i			0.292090i
$153$		0			0
$154$	−397.707			−0.0167695
$155$	−556.602	+	321.354i	−0.0231676	+	0.0133758i
$156$		0			0
$157$	18240.8	−	31594.0i	0.740022	−	1.28175i	−0.212463	−	0.977169i	$-0.568149\pi$
$157$	18240.8	−	31594.0i	0.740022	−	1.28175i	0.952485	−	0.304586i	$-0.0985181\pi$
$158$	27008.9	+	15593.6i	1.08191	+	0.624643i
$159$		0			0
$160$	37.6613	+	65.2312i	0.00147114	+	0.00254809i
$161$		−	1635.46i		−	0.0630939i
$162$		0			0
$163$	34529.1			1.29960			0.649800	−	0.760105i	$-0.274853\pi$
$163$	34529.1			1.29960			0.649800	+	0.760105i	$0.274853\pi$
$164$	14791.1	−	8539.65i	0.549937	−	0.317506i
$165$		0			0
$166$	17053.6	−	29537.6i	0.618869	−	1.07191i
$167$	−2914.01	−	1682.40i	−0.104486	−	0.0603250i	0.446846	−	0.894611i	$-0.352547\pi$
$167$	−2914.01	−	1682.40i	−0.104486	−	0.0603250i	−0.551332	+	0.834286i	$0.685880\pi$
$168$		0			0
$169$	−11913.6	−	20634.9i	−0.417127	−	0.722486i
$170$			447.537i			0.0154857i
$171$		0			0
$172$	−18439.8			−0.623302
$173$	−38526.6	+	22243.4i	−1.28727	+	0.743204i	−0.978166	−	0.207825i	$-0.933362\pi$
$173$	−38526.6	+	22243.4i	−1.28727	+	0.743204i	−0.309102	+	0.951029i	$0.600028\pi$
$174$		0			0
$175$	−808.669	+	1400.66i	−0.0264055	+	0.0457357i
$176$	−3010.84	−	1738.31i	−0.0971990	−	0.0561179i
$177$		0			0
$178$	4108.79	+	7116.63i	0.129680	+	0.224613i
$179$		−	57987.0i		−	1.80977i	−0.425652	−	0.904887i	$-0.639955\pi$
$179$		−	57987.0i		−	1.80977i	0.425652	−	0.904887i	$-0.360045\pi$
$180$		0			0
$181$	−33114.4			−1.01079			−0.505393	−	0.862889i	$-0.668653\pi$
$181$	−33114.4			−1.01079			−0.505393	+	0.862889i	$0.668653\pi$
$182$	−1451.22	+	837.862i	−0.0438117	+	0.0252947i
$183$		0			0
$184$	7148.31	−	12381.2i	0.211139	−	0.365703i
$185$	67.6898	+	39.0807i	0.00197779	+	0.00114188i
$186$		0			0
$187$	−10328.3	−	17889.2i	−0.295357	−	0.511574i
$188$			16044.1i			0.453941i
$189$		0			0
$190$	351.006			0.00972315
$191$	53380.7	−	30819.4i	1.46325	−	0.844807i	0.464088	−	0.885789i	$-0.346382\pi$
$191$	53380.7	−	30819.4i	1.46325	−	0.844807i	0.999160	+	0.0409825i	$0.0130488\pi$
$192$		0			0
$193$	18839.4	−	32630.8i	0.505769	−	0.876017i	−0.494209	−	0.869343i	$-0.664542\pi$
$193$	18839.4	−	32630.8i	0.505769	−	0.876017i	0.999978	−	0.00667399i	$-0.00212441\pi$
$194$	31580.7	+	18233.1i	0.839108	+	0.484459i
$195$		0			0
$196$	−9577.20	−	16588.2i	−0.249302	−	0.431804i
$197$		−	53748.5i		−	1.38495i	−0.721442	−	0.692475i	$-0.756520\pi$
$197$		−	53748.5i		−	1.38495i	0.721442	−	0.692475i	$-0.243480\pi$
$198$		0			0
$199$	16462.2			0.415702			0.207851	−	0.978161i	$-0.433353\pi$
$199$	16462.2			0.415702			0.207851	+	0.978161i	$0.433353\pi$
$200$	−12244.1	+	7069.11i	−0.306101	+	0.176728i
$201$		0			0
$202$	5096.77	−	8827.86i	0.124908	−	0.216348i
$203$	1472.11	+	849.924i	0.0357231	+	0.0206247i
$204$		0			0
$205$	−444.171	−	769.326i	−0.0105692	−	0.0183064i
$206$			9090.91i			0.214226i
$207$		0			0
$208$	−14648.6			−0.338587
$209$	−14030.6	+	8100.57i	−0.321206	+	0.185448i
$210$		0			0
$211$	−4283.80	+	7419.76i	−0.0962197	+	0.166657i	−0.910117	−	0.414351i	$-0.864008\pi$
$211$	−4283.80	+	7419.76i	−0.0962197	+	0.166657i	0.813897	+	0.581009i	$0.197342\pi$
$212$	22819.5	+	13174.8i	0.507731	+	0.293139i
$213$		0			0
$214$	18446.4	+	31950.1i	0.402795	+	0.697662i
$215$			959.104i			0.0207486i
$216$		0			0
$217$	−3998.11			−0.0849054
$218$	7873.41	−	4545.72i	0.165672	−	0.0956510i
$219$		0			0
$220$	−90.4142	+	156.602i	−0.00186806	+	0.00323558i
$221$	−75375.7	−	43518.2i	−1.54329	−	0.891017i
$222$		0			0
$223$	−4162.64	−	7209.90i	−0.0837065	−	0.144984i	0.821133	−	0.570737i	$-0.193342\pi$
$223$	−4162.64	−	7209.90i	−0.0837065	−	0.144984i	−0.904839	+	0.425753i	$0.860009\pi$
$224$			468.561i			0.00933835i
$225$		0			0
$226$	−31814.9			−0.622893
$227$	−26569.1	+	15339.7i	−0.515615	+	0.297690i	−0.735139	−	0.677917i	$-0.762883\pi$
$227$	−26569.1	+	15339.7i	−0.515615	+	0.297690i	0.219524	+	0.975607i	$0.429550\pi$
$228$		0			0
$229$	−25223.1	+	43687.6i	−0.480980	+	0.833081i	−0.999762	−	0.0218252i	$-0.993052\pi$
$229$	−25223.1	+	43687.6i	−0.480980	+	0.833081i	0.518782	+	0.854907i	$0.326386\pi$
$230$	−643.982	−	371.803i	−0.0121736	−	0.00702842i
$231$		0			0
$232$	7429.75	+	12868.7i	0.138038	+	0.239089i
$233$		−	7603.00i		−	0.140047i	−0.997545	−	0.0700234i	$-0.977693\pi$
$233$		−	7603.00i		−	0.140047i	0.997545	−	0.0700234i	$-0.0223074\pi$
$234$		0			0
$235$	834.498			0.0151109
$236$	22247.4	−	12844.6i	0.399444	−	0.230619i
$237$		0			0
$238$	−1392.00	+	2411.02i	−0.0245746	+	0.0425644i
$239$	86873.5	+	50156.5i	1.52087	+	0.878074i	0.999697	+	0.0246224i	$0.00783833\pi$
$239$	86873.5	+	50156.5i	1.52087	+	0.878074i	0.521172	+	0.853452i	$0.325495\pi$
$240$		0			0
$241$	18969.7	+	32856.4i	0.326607	+	0.565700i	0.981836	−	0.189730i	$-0.0607613\pi$
$241$	18969.7	+	32856.4i	0.326607	+	0.565700i	−0.655229	+	0.755430i	$0.727428\pi$
$242$			33064.6i			0.564589i
$243$		0			0
$244$	14577.7			0.244856
$245$	−862.798	+	498.137i	−0.0143740	+	0.00829882i
$246$		0			0
$247$	−34131.5	+	59117.5i	−0.559451	+	0.968997i
$248$	−30267.7	−	17475.1i	−0.492126	−	0.284129i
$249$		0			0
$250$	735.470	+	1273.87i	0.0117675	+	0.0203819i
$251$			58448.0i			0.927732i	0.885905	+	0.463866i	$0.153538\pi$
$251$			58448.0i			0.927732i	−0.885905	+	0.463866i	$0.846462\pi$
$252$		0			0
$253$	34322.2			0.536209
$254$	−3417.32	+	1972.99i	−0.0529686	+	0.0305814i
$255$		0			0
$256$	−2048.00	+	3547.24i	−0.0312500	+	0.0541266i
$257$	−102349.	−	59091.3i	−1.54959	−	0.894658i	−0.998172	−	0.0604307i	$-0.980753\pi$
$257$	−102349.	−	59091.3i	−1.54959	−	0.894658i	−0.551421	−	0.834227i	$-0.685914\pi$
$258$		0			0
$259$	243.110	+	421.080i	0.00362413	+	0.00627718i
$260$			761.915i			0.0112709i
$261$		0			0
$262$	47390.7			0.690383
$263$	92101.7	−	53175.0i	1.33155	−	0.768769i	0.346010	−	0.938231i	$-0.387536\pi$
$263$	92101.7	−	53175.0i	1.33155	−	0.768769i	0.985537	+	0.169462i	$0.0542030\pi$
$264$		0			0
$265$	685.260	−	1186.90i	0.00975806	−	0.0169015i
$266$	1890.97	+	1091.75i	0.0267253	+	0.0154299i
$267$		0			0
$268$	8392.36	+	14536.0i	0.116846	+	0.202383i
$269$			137219.i			1.89631i	0.317813	+	0.948154i	$0.397052\pi$
$269$			137219.i			1.89631i	−0.317813	+	0.948154i	$0.602948\pi$
$270$		0			0
$271$	60463.3			0.823291			0.411646	−	0.911344i	$-0.364954\pi$
$271$	60463.3			0.823291			0.411646	+	0.911344i	$0.364954\pi$
$272$	−21076.3	+	12168.4i	−0.284877	+	0.164474i
$273$		0			0
$274$	−25911.7	+	44880.5i	−0.345140	+	0.597800i
$275$	−29394.6	−	16971.0i	−0.388688	−	0.224409i
$276$		0			0
$277$	59621.4	+	103267.i	0.777039	+	1.34587i	0.933641	+	0.358209i	$0.116612\pi$
$277$	59621.4	+	103267.i	0.777039	+	1.34587i	−0.156603	+	0.987662i	$0.550054\pi$
$278$			23419.5i			0.303031i
$279$		0			0
$280$	24.3711			0.000310856
$281$	−89797.9	+	51844.8i	−1.13724	+	0.656588i	−0.945746	−	0.324906i	$-0.894667\pi$
$281$	−89797.9	+	51844.8i	−1.13724	+	0.656588i	−0.191497	+	0.981493i	$0.561334\pi$
$282$		0			0
$283$	−33820.9	+	58579.5i	−0.422291	+	0.731429i	−0.996163	−	0.0875154i	$-0.972107\pi$
$283$	−33820.9	+	58579.5i	−0.422291	+	0.731429i	0.573872	+	0.818945i	$0.305441\pi$
$284$	15112.5	+	8725.22i	0.187370	+	0.108178i
$285$		0			0
$286$	−17583.6	−	30455.7i	−0.214969	−	0.372338i
$287$		−	5526.13i		−	0.0670899i
$288$		0			0
$289$	−61078.9			−0.731299
$290$	669.337	−	386.442i	0.00795882	−	0.00459503i
$291$		0			0
$292$	−22109.4	+	38294.7i	−0.259306	+	0.449130i
$293$	42689.8	+	24647.0i	0.497267	+	0.287097i	0.727584	−	0.686019i	$-0.240643\pi$
$293$	42689.8	+	24647.0i	0.497267	+	0.287097i	−0.230317	+	0.973116i	$0.573976\pi$
$294$		0			0
$295$	−668.081	−	1157.15i	−0.00767689	−	0.0132968i
$296$			4250.38i			0.0485114i
$297$		0			0
$298$	−32839.7			−0.369800
$299$	125241.	−	72307.7i	1.40089	−	0.808802i
$300$		0			0
$301$	−2983.16	+	5166.99i	−0.0329264	+	0.0570301i
$302$	−107555.	−	62097.1i	−1.17928	−	0.680860i
$303$		0			0
$304$	9543.75	+	16530.3i	0.103269	+	0.178868i
$305$		−	758.229i		−	0.00815081i
$306$		0			0
$307$	19565.2			0.207591			0.103795	−	0.994599i	$-0.466901\pi$
$307$	19565.2			0.207591			0.103795	+	0.994599i	$0.466901\pi$
$308$	−974.178	+	562.442i	−0.0102692	+	0.00592893i
$309$		0			0
$310$	−908.927	+	1574.31i	−0.00945813	+	0.0163820i
$311$	43177.8	+	24928.7i	0.446416	+	0.257738i	0.706315	−	0.707897i	$-0.250356\pi$
$311$	43177.8	+	24928.7i	0.446416	+	0.257738i	−0.259900	+	0.965636i	$0.583689\pi$
$312$		0			0
$313$	53091.8	+	91957.7i	0.541925	+	0.938641i	0.998794	+	0.0491071i	$0.0156376\pi$
$313$	53091.8	+	91957.7i	0.541925	+	0.938641i	−0.456869	+	0.889534i	$0.651029\pi$
$314$		−	103186.i		−	1.04655i
$315$		0			0
$316$	88210.7			0.883379
$317$	−75738.7	+	43727.7i	−0.753701	+	0.435149i	−0.827030	−	0.562158i	$-0.809971\pi$
$317$	−75738.7	+	43727.7i	−0.753701	+	0.435149i	0.0733287	+	0.997308i	$0.476638\pi$
$318$		0			0
$319$	−17836.8	+	30894.2i	−0.175281	+	0.303596i
$320$	184.502	+	106.522i	0.00180178	+	0.00104026i
$321$		0			0
$322$	−2312.89	−	4006.04i	−0.0223071	−	0.0386370i
$323$			113410.i			1.08705i
$324$		0			0
$325$	−143013.			−1.35397
$326$	84578.6	−	48831.5i	0.795839	−	0.459478i
$327$		0			0
$328$	24153.8	−	41835.6i	0.224511	−	0.388864i
$329$	4495.70	+	2595.59i	0.0415341	+	0.0239797i
$330$		0			0
$331$	−10937.7	−	18944.6i	−0.0998319	−	0.172914i	0.811783	−	0.583959i	$-0.198497\pi$
$331$	−10937.7	−	18944.6i	−0.0998319	−	0.172914i	−0.911615	+	0.411045i	$0.865164\pi$
$332$		−	96469.5i		−	0.875214i
$333$		0			0
$334$	−9517.11			−0.0853124
$335$	756.057	−	436.510i	0.00673697	−	0.00388959i
$336$		0			0
$337$	94021.5	−	162850.i	0.827880	−	1.43393i	−0.0718186	−	0.997418i	$-0.522880\pi$
$337$	94021.5	−	162850.i	0.827880	−	1.43393i	0.899698	−	0.436512i	$-0.143786\pi$
$338$	−58364.4	−	33696.7i	−0.510875	−	0.294954i
$339$		0			0
$340$	632.913	+	1096.24i	0.00547502	+	0.00948302i
$341$		−	83905.5i		−	0.721576i
$342$		0			0
$343$	−12412.4			−0.105504
$344$	−45168.1	+	26077.8i	−0.381693	+	0.220371i
$345$		0			0
$346$	−62913.7	+	108970.i	−0.525525	+	0.910236i
$347$	162975.	+	94093.6i	1.35351	+	0.781450i	0.988739	−	0.149648i	$-0.0478140\pi$
$347$	162975.	+	94093.6i	1.35351	+	0.781450i	0.364771	+	0.931097i	$0.381147\pi$
$348$		0			0
$349$	−39328.3	−	68118.6i	−0.322890	−	0.559262i	0.658193	−	0.752849i	$-0.271321\pi$
$349$	−39328.3	−	68118.6i	−0.322890	−	0.559262i	−0.981083	+	0.193587i	$0.937988\pi$
$350$			4574.52i			0.0373430i
$351$		0			0
$352$	−9833.35			−0.0793627
$353$	100346.	−	57934.8i	0.805287	−	0.464933i	−0.0400293	−	0.999199i	$-0.512745\pi$
$353$	100346.	−	57934.8i	0.805287	−	0.464933i	0.845317	+	0.534266i	$0.179412\pi$
$354$		0			0
$355$	453.823	−	786.044i	0.00360105	−	0.00623721i
$356$	20128.9	+	11621.4i	0.158825	+	0.0916977i
$357$		0			0
$358$	−82005.9	−	142038.i	−0.639852	−	1.10826i
$359$		−	187042.i		−	1.45128i	−0.688075	−	0.725640i	$-0.741544\pi$
$359$		−	187042.i		−	1.45128i	0.688075	−	0.725640i	$-0.258456\pi$
$360$		0			0
$361$	−41372.6			−0.317467
$362$	−81113.3	+	46830.8i	−0.618978	+	0.357367i
$363$		0			0
$364$	−2369.83	+	4104.67i	−0.0178861	+	0.0309796i
$365$	1991.81	+	1149.97i	0.0149507	+	0.00863181i
$366$		0			0
$367$	30070.0	+	52082.8i	0.223255	+	0.386689i	0.955795	−	0.294035i	$-0.0949983\pi$
$367$	30070.0	+	52082.8i	0.223255	+	0.386689i	−0.732539	+	0.680725i	$0.761665\pi$
$368$		−	40436.9i		−	0.298595i
$369$		0			0
$370$	221.074			0.00161486
$371$	7383.41	−	4262.81i	0.0536425	−	0.0309705i
$372$		0			0
$373$	121324.	−	210138.i	0.872022	−	1.51039i	0.0121200	−	0.999927i	$-0.496142\pi$
$373$	121324.	−	210138.i	0.872022	−	1.51039i	0.859902	−	0.510459i	$-0.170525\pi$
$374$	−50598.3	−	29213.0i	−0.361737	−	0.208849i
$375$		0			0
$376$	22689.8	+	39299.9i	0.160492	+	0.277981i
$377$			150309.i			1.05756i
$378$		0			0
$379$	339.729			0.00236513			0.00118256	−	0.999999i	$-0.499624\pi$
$379$	339.729			0.00236513			0.00118256	+	0.999999i	$0.499624\pi$
$380$	859.784	−	496.397i	0.00595419	−	0.00343765i
$381$		0			0
$382$	87170.4	−	150984.i	0.597368	−	1.03467i
$383$	36136.1	+	20863.2i	0.246345	+	0.142227i	0.618090	−	0.786108i	$-0.287907\pi$
$383$	36136.1	+	20863.2i	0.246345	+	0.142227i	−0.371744	+	0.928335i	$0.621240\pi$
$384$		0			0
$385$	29.2542	+	50.6697i	0.000197363	+	0.000341843i
$386$		−	106572.i		−	0.715265i
$387$		0			0
$388$	103142.			0.685129
$389$	−152182.	+	87862.2i	−1.00569	+	0.580635i	−0.909927	−	0.414769i	$-0.863862\pi$
$389$	−152182.	+	87862.2i	−1.00569	+	0.580635i	−0.0957624	+	0.995404i	$0.530529\pi$
$390$		0			0
$391$	120130.	−	208072.i	0.785776	−	1.36100i
$392$	−46918.5	−	27088.4i	−0.305332	−	0.176283i
$393$		0			0
$394$	−76011.9	−	131656.i	−0.489654	−	0.848105i
$395$		−	4588.08i		−	0.0294061i
$396$		0			0
$397$	−13497.3			−0.0856379			−0.0428190	−	0.999083i	$-0.513634\pi$
$397$	−13497.3			−0.0856379			−0.0428190	+	0.999083i	$0.513634\pi$
$398$	40324.0	−	23281.1i	0.254564	−	0.146973i
$399$		0			0
$400$	−19994.5	+	34631.4i	−0.124965	+	0.216446i
$401$	59305.0	+	34239.8i	0.368810	+	0.212933i	0.672939	−	0.739698i	$-0.265032\pi$
$401$	59305.0	+	34239.8i	0.368810	+	0.212933i	−0.304128	+	0.952631i	$0.598365\pi$
$402$		0			0
$403$	−176767.	−	306169.i	−1.08840	−	1.88517i
$404$		−	28831.7i		−	0.176647i
$405$		0			0
$406$	4807.90			0.0291678
$407$	−8836.90	+	5101.99i	−0.0533471	+	0.0308000i
$408$		0			0
$409$	−89877.3	+	155672.i	−0.537283	+	0.930602i	0.461766	+	0.887002i	$0.347216\pi$
$409$	−89877.3	+	155672.i	−0.537283	+	0.930602i	−0.999049	+	0.0436002i	$0.986117\pi$
$410$	−2175.98	−	1256.30i	−0.0129446	−	0.00747356i
$411$		0			0
$412$	12856.5	+	22268.1i	0.0757405	+	0.131186i
$413$		−	8311.90i		−	0.0487304i
$414$		0			0
$415$	−5017.65			−0.0291343
$416$	−35881.6	+	20716.3i	−0.207341	+	0.119708i
$417$		0			0
$418$	−22911.9	+	39684.5i	−0.131132	+	0.227127i
$419$	251540.	+	145227.i	1.43278	+	0.827214i	0.997332	−	0.0730012i	$-0.0232577\pi$
$419$	251540.	+	145227.i	1.43278	+	0.827214i	0.435445	+	0.900215i	$0.356591\pi$
$420$		0			0
$421$	−19653.9	−	34041.6i	−0.110888	−	0.192064i	0.805240	−	0.592948i	$-0.202036\pi$
$421$	−19653.9	−	34041.6i	−0.110888	−	0.192064i	−0.916129	+	0.400884i	$0.868703\pi$
$422$			24232.8i			0.136075i
$423$		0			0
$424$	74528.1			0.414561
$425$	−205766.	+	118799.i	−1.13919	+	0.657712i
$426$		0			0
$427$	2358.37	−	4084.81i	0.0129347	−	0.0224035i
$428$	90368.6	+	52174.4i	0.493322	+	0.284819i
$429$		0			0
$430$	1356.38	+	2349.32i	0.00733574	+	0.0127059i
$431$			144937.i			0.780235i	0.920765	+	0.390117i	$0.127566\pi$
$431$			144937.i			0.780235i	−0.920765	+	0.390117i	$0.872434\pi$
$432$		0			0
$433$	34162.1			0.182208			0.0911042	−	0.995841i	$-0.470960\pi$
$433$	34162.1			0.182208			0.0911042	+	0.995841i	$0.470960\pi$
$434$	−9793.33	+	5654.18i	−0.0519937	+	0.0300186i
$435$		0			0
$436$	12857.2	−	22269.4i	0.0676355	−	0.117148i
$437$	−163192.	−	94218.7i	−0.854545	−	0.493372i
$438$		0			0
$439$	10986.2	+	19028.7i	0.0570058	+	0.0987369i	0.893120	−	0.449818i	$-0.148511\pi$
$439$	10986.2	+	19028.7i	0.0570058	+	0.0987369i	−0.836114	+	0.548555i	$0.815178\pi$
$440$			511.460i			0.00264184i
$441$		0			0
$442$	−246176.			−1.26009
$443$	−55479.7	+	32031.2i	−0.282701	+	0.163217i	−0.634645	−	0.772803i	$-0.718854\pi$
$443$	−55479.7	+	32031.2i	−0.282701	+	0.163217i	0.351945	+	0.936021i	$0.385520\pi$
$444$		0			0
$445$	604.461	−	1046.96i	0.00305245	−	0.00528700i
$446$	−20392.7	−	11773.7i	−0.102519	−	0.0591894i
$447$		0			0
$448$	662.645	+	1147.73i	0.00330160	+	0.00571855i
$449$		−	162975.i		−	0.808403i	−0.914670	−	0.404202i	$-0.867549\pi$
$449$		−	162975.i		−	0.808403i	0.914670	−	0.404202i	$-0.132451\pi$
$450$		0			0
$451$	115973.			0.570169
$452$	−77930.2	+	44993.0i	−0.381443	+	0.220226i
$453$		0			0
$454$	−43387.2	+	75148.8i	−0.210499	+	0.364595i
$455$	213.495	+	123.262i	0.00103125	+	0.000595394i
$456$		0			0
$457$	−74178.2	−	128480.i	−0.355176	−	0.615183i	0.631972	−	0.774991i	$-0.282246\pi$
$457$	−74178.2	−	128480.i	−0.355176	−	0.615183i	−0.987148	+	0.159808i	$0.948912\pi$
$458$			142683.i			0.680208i
$459$		0			0
$460$	−2103.24			−0.00993968
$461$	51581.1	−	29780.4i	0.242711	−	0.140129i	−0.373711	−	0.927545i	$-0.621915\pi$
$461$	51581.1	−	29780.4i	0.242711	−	0.140129i	0.616422	+	0.787416i	$0.288582\pi$
$462$		0			0
$463$	−33888.7	+	58696.9i	−0.158086	+	0.273812i	−0.934178	−	0.356806i	$-0.883866\pi$
$463$	−33888.7	+	58696.9i	−0.158086	+	0.273812i	0.776093	+	0.630619i	$0.217199\pi$
$464$	36398.2	+	21014.5i	0.169061	+	0.0976075i
$465$		0			0
$466$	−10752.3	−	18623.5i	−0.0495140	−	0.0857607i
$467$			57234.8i			0.262438i	0.991353	+	0.131219i	$0.0418891\pi$
$467$			57234.8i			0.262438i	−0.991353	+	0.131219i	$0.958111\pi$
$468$		0			0
$469$	5430.81			0.0246899
$470$	2044.09	−	1180.16i	0.00925348	−	0.00534250i
$471$		0			0
$472$	36329.9	−	62925.2i	0.163072	−	0.282450i
$473$	−108436.	−	62605.5i	−0.484675	−	0.279827i
$474$		0			0
$475$	93174.9	+	161384.i	0.412963	+	0.715274i
$476$			7874.35i			0.0347537i
$477$		0			0
$478$	283728.			1.24178
$479$	164946.	−	95231.6i	0.718903	−	0.415059i	−0.0954455	−	0.995435i	$-0.530428\pi$
$479$	164946.	−	95231.6i	0.718903	−	0.415059i	0.814349	+	0.580376i	$0.197094\pi$
$480$		0			0
$481$	−21497.1	+	37234.0i	−0.0929157	+	0.160935i
$482$	92932.0	+	53654.3i	0.400010	+	0.230946i
$483$		0			0
$484$	46760.4	+	80991.4i	0.199613	+	0.345739i
$485$		−	5364.70i		−	0.0228067i
$486$		0			0
$487$	−167195.			−0.704962			−0.352481	−	0.935819i	$-0.614662\pi$
$487$	−167195.			−0.704962			−0.352481	+	0.935819i	$0.614662\pi$
$488$	35708.0	−	20616.0i	0.149943	−	0.0865697i
$489$		0			0
$490$	−1408.94	+	2440.36i	−0.00586815	+	0.0101639i
$491$	−214569.	−	123881.i	−0.890028	−	0.513858i	−0.0160763	−	0.999871i	$-0.505117\pi$
$491$	−214569.	−	123881.i	−0.890028	−	0.513858i	−0.873952	+	0.486013i	$0.838451\pi$
$492$		0			0
$493$	124860.	+	216264.i	0.513723	+	0.889795i
$494$			193077.i			0.791183i
$495$		0			0
$496$	−98853.9			−0.401819
$497$	4889.76	−	2823.11i	0.0197959	−	0.0114292i
$498$		0			0
$499$	−205515.	+	355963.i	−0.825359	+	1.42956i	0.0762861	+	0.997086i	$0.475694\pi$
$499$	−205515.	+	355963.i	−0.825359	+	1.42956i	−0.901645	+	0.432477i	$0.857640\pi$
$500$	3603.05	+	2080.22i	0.0144122	+	0.00832089i
$501$		0			0
$502$	82658.0	+	143168.i	0.328003	+	0.568118i
$503$		−	108804.i		−	0.430039i	−0.976610	−	0.215020i	$-0.931018\pi$
$503$		−	108804.i		−	0.430039i	0.976610	−	0.215020i	$-0.0689816\pi$
$504$		0			0
$505$	−1499.61			−0.00588026
$506$	84071.8	−	48538.9i	0.328359	−	0.189578i
$507$		0			0
$508$	−5580.46	+	9665.64i	−0.0216243	+	0.0374544i
$509$	68677.5	+	39651.0i	0.265081	+	0.153045i	0.626650	−	0.779301i	$-0.284425\pi$
$509$	68677.5	+	39651.0i	0.265081	+	0.153045i	−0.361569	+	0.932345i	$0.617759\pi$
$510$		0			0
$511$	7153.66	+	12390.5i	0.0273960	+	0.0474512i
$512$			11585.2i			0.0441942i
$513$		0			0
$514$	−334271.			−1.26524
$515$	1158.23	−	668.702i	0.00436695	−	0.00252126i
$516$		0			0
$517$	−54471.9	+	94348.0i	−0.203794	+	0.352981i
$518$	1190.99	+	687.620i	0.00443864	+	0.00256265i
$519$		0			0
$520$	1077.51	+	1866.30i	0.00398488	+	0.00690201i
$521$			55028.4i			0.202727i	0.994849	+	0.101364i	$0.0323205\pi$
$521$			55028.4i			0.202727i	−0.994849	+	0.101364i	$0.967679\pi$
$522$		0			0
$523$	473394.			1.73069			0.865346	−	0.501176i	$-0.167099\pi$
$523$	473394.			1.73069			0.865346	+	0.501176i	$0.167099\pi$
$524$	116083.	−	67020.5i	0.422772	−	0.244087i
$525$		0			0
$526$	150402.	−	260503.i	0.543602	−	0.941546i
$527$	−508661.	−	293676.i	−1.83150	−	1.05742i
$528$		0			0
$529$	59682.2	+	103373.i	0.213272	+	0.369397i
$530$		−	3876.41i		−	0.0138000i
$531$		0			0
$532$	6175.90			0.0218211
$533$	423182.	−	244324.i	1.48961	−	0.860027i
$534$		0			0
$535$	2713.73	−	4700.32i	0.00948112	−	0.0164218i
$536$	41114.0	+	23737.2i	0.143107	+	0.0826227i
$537$		0			0
$538$	194057.	+	336116.i	0.670446	+	1.16125i
$539$		−	130063.i		−	0.447690i
$540$		0			0
$541$	239116.			0.816984			0.408492	−	0.912762i	$-0.366055\pi$
$541$	239116.			0.816984			0.408492	+	0.912762i	$0.366055\pi$
$542$	148104.	−	85508.0i	0.504161	−	0.291077i
$543$		0			0
$544$	−34417.5	+	59612.8i	−0.116300	+	0.201438i
$545$	−1158.29	−	668.740i	−0.00389964	−	0.00225146i
$546$		0			0
$547$	91487.1	+	158460.i	0.305763	+	0.529597i	0.977431	−	0.211255i	$-0.0677551\pi$
$547$	91487.1	+	158460.i	0.305763	+	0.529597i	−0.671668	+	0.740853i	$0.734422\pi$
$548$			146579.i			0.488102i
$549$		0			0
$550$	−96002.3			−0.317363
$551$	169617.	−	97928.3i	0.558683	−	0.322556i
$552$		0			0
$553$	14270.6	−	24717.4i	0.0466651	−	0.0808263i
$554$	292084.	+	168635.i	0.951674	+	0.549449i
$555$		0			0
$556$	33120.1	+	57365.8i	0.107138	+	0.185568i
$557$			63739.1i			0.205445i	0.994710	+	0.102722i	$0.0327553\pi$
$557$			63739.1i			0.205445i	−0.994710	+	0.102722i	$0.967245\pi$
$558$		0			0
$559$	−527573.			−1.68834
$560$	59.6969	−	34.4660i	0.000190360	−	0.000109904i
$561$		0			0
$562$	−146639.	+	253987.i	−0.464278	+	0.804152i
$563$	−306343.	−	176867.i	−0.966476	−	0.557995i	−0.0683162	−	0.997664i	$-0.521763\pi$
$563$	−306343.	−	176867.i	−0.966476	−	0.557995i	−0.898160	+	0.439668i	$0.855096\pi$
$564$		0			0
$565$	2340.21	+	4053.37i	0.00733092	+	0.0126975i
$566$			191320.i			0.597210i
$567$		0			0
$568$	49357.3			0.152987
$569$	76108.2	−	43941.1i	0.235075	−	0.135721i	−0.377836	−	0.925873i	$-0.623332\pi$
$569$	76108.2	−	43941.1i	0.235075	−	0.135721i	0.612911	+	0.790152i	$0.289998\pi$
$570$		0			0
$571$	−100166.	+	173493.i	−0.307220	+	0.532120i	−0.977753	−	0.209759i	$-0.932732\pi$
$571$	−100166.	+	173493.i	−0.307220	+	0.532120i	0.670533	+	0.741879i	$0.266065\pi$
$572$	−86141.8	−	49734.0i	−0.263282	−	0.152006i
$573$		0			0
$574$	−7815.13	−	13536.2i	−0.0237199	−	0.0410840i
$575$		−	394782.i		−	1.19405i
$576$		0			0
$577$	−3625.36			−0.0108893			−0.00544464	−	0.999985i	$-0.501733\pi$
$577$	−3625.36			−0.0108893			−0.00544464	+	0.999985i	$0.501733\pi$
$578$	−149612.	+	86378.5i	−0.447828	+	0.258553i
$579$		0			0
$580$	1093.02	−	1893.17i	0.00324918	−	0.00562774i
$581$	−27031.6	−	15606.7i	−0.0800792	−	0.0462337i
$582$		0			0
$583$	89460.6	+	154950.i	0.263205	+	0.455885i
$584$			125070.i			0.366713i
$585$		0			0
$586$	139424.			0.406016
$587$	−439182.	+	253562.i	−1.27458	+	0.735880i	−0.975847	−	0.218456i	$-0.929898\pi$
$587$	−439182.	+	253562.i	−1.27458	+	0.735880i	−0.298735	+	0.954336i	$0.596565\pi$
$588$		0			0
$589$	−230331.	+	398945.i	−0.663930	+	1.14996i
$590$	−3272.92	−	1889.62i	−0.00940223	−	0.00542838i
$591$		0			0
$592$	6010.94	+	10411.3i	0.0171514	+	0.0297071i
$593$		−	255264.i		−	0.725906i	−0.931807	−	0.362953i	$-0.881769\pi$
$593$		−	255264.i		−	0.725906i	0.931807	−	0.362953i	$-0.118231\pi$
$594$		0			0
$595$	409.567			0.00115689
$596$	−80440.6	+	46442.4i	−0.226455	+	0.130744i
$597$		0			0
$598$	204517.	−	354234.i	0.571910	−	0.990576i
$599$	248668.	+	143569.i	0.693053	+	0.400134i	0.804755	−	0.593608i	$-0.202297\pi$
$599$	248668.	+	143569.i	0.693053	+	0.400134i	−0.111702	+	0.993742i	$0.535630\pi$
$600$		0			0
$601$	−3352.40	−	5806.52i	−0.00928125	−	0.0160756i	0.861347	−	0.508016i	$-0.169621\pi$
$601$	−3352.40	−	5806.52i	−0.00928125	−	0.0160756i	−0.870629	+	0.491941i	$0.836288\pi$
$602$			16875.3i			0.0465649i
$603$		0			0
$604$	−351274.			−0.962881
$605$	4212.59	−	2432.14i	0.0115090	−	0.00664474i
$606$		0			0
$607$	84488.0	−	146338.i	0.229307	−	0.397172i	−0.728296	−	0.685263i	$-0.759687\pi$
$607$	84488.0	−	146338.i	0.229307	−	0.397172i	0.957603	+	0.288091i	$0.0930206\pi$
$608$	46754.6	+	26993.8i	0.126479	+	0.0730225i
$609$		0			0
$610$	−1072.30	−	1857.27i	−0.00288175	−	0.00499133i
$611$			459031.i			1.22959i
$612$		0			0
$613$	110387.			0.293763			0.146882	−	0.989154i	$-0.453076\pi$
$613$	110387.			0.293763			0.146882	+	0.989154i	$0.453076\pi$
$614$	47924.8	−	27669.4i	0.127123	−	0.0733945i
$615$		0			0
$616$	−1590.83	+	2755.39i	−0.00419239	+	0.00726143i
$617$	−116125.	−	67045.0i	−0.305040	−	0.176115i	0.339665	−	0.940547i	$-0.389686\pi$
$617$	−116125.	−	67045.0i	−0.305040	−	0.176115i	−0.644705	+	0.764432i	$0.723020\pi$
$618$		0			0
$619$	−179601.	−	311077.i	−0.468734	−	0.811871i	0.530627	−	0.847605i	$-0.321956\pi$
$619$	−179601.	−	311077.i	−0.468734	−	0.811871i	−0.999361	+	0.0357339i	$0.988623\pi$
$620$			5141.66i			0.0133758i
$621$		0			0
$622$	141018.			0.364497
$623$	6512.84	−	3760.19i	0.0167801	−	0.00968799i
$624$		0			0
$625$	−195150.	+	338010.i	−0.499585	+	0.865306i
$626$	260096.	+	150166.i	0.663720	+	0.383199i
$627$		0			0
$628$	−145926.	−	252752.i	−0.370011	−	0.640877i
$629$			71429.3i			0.180541i
$630$		0			0
$631$	431012.			1.08251			0.541253	−	0.840860i	$-0.317950\pi$
$631$	431012.			1.08251			0.541253	+	0.840860i	$0.317950\pi$
$632$	216071.	−	124749.i	0.540957	−	0.312322i
$633$		0			0
$634$	−123681.	+	214221.i	−0.307697	+	0.532947i
$635$	502.737	+	290.255i	0.00124679	+	0.000719834i
$636$		0			0
$637$	−274009.	−	474598.i	−0.675284	−	1.16963i
$638$			100900.i			0.247885i
$639$		0			0
$640$	602.580			0.00147114
$641$	55097.4	−	31810.5i	0.134096	−	0.0774203i	−0.431451	−	0.902136i	$-0.641998\pi$
$641$	55097.4	−	31810.5i	0.134096	−	0.0774203i	0.565547	+	0.824716i	$0.308665\pi$
$642$		0			0
$643$	398579.	−	690359.i	0.964034	−	1.66976i	0.251846	−	0.967767i	$-0.418962\pi$
$643$	398579.	−	690359.i	0.964034	−	1.66976i	0.712188	−	0.701989i	$-0.247704\pi$
$644$	−11330.8	−	6541.83i	−0.0273205	−	0.0157735i
$645$		0			0
$646$	160387.	+	277798.i	0.384329	+	0.665677i
$647$		−	185467.i		−	0.443054i	−0.975154	−	0.221527i	$-0.928896\pi$
$647$		−	185467.i		−	0.443054i	0.975154	−	0.221527i	$-0.0711042\pi$
$648$		0			0
$649$	174436.			0.414139
$650$	−350309.	+	202251.i	−0.829135	+	0.478701i
$651$		0			0
$652$	138116.	−	239224.i	0.324900	−	0.562743i
$653$	92645.7	+	53489.0i	0.217270	+	0.125441i	0.604685	−	0.796464i	$-0.293299\pi$
$653$	92645.7	+	53489.0i	0.217270	+	0.125441i	−0.387416	+	0.921905i	$0.626632\pi$
$654$		0			0
$655$	−3485.92	−	6037.80i	−0.00812522	−	0.0140733i
$656$		−	136634.i		−	0.317506i
$657$		0			0
$658$	14682.9			0.0339125
$659$	−659558.	+	380796.i	−1.51874	+	0.876843i	−0.518980	+	0.854786i	$0.673688\pi$
$659$	−659558.	+	380796.i	−1.51874	+	0.876843i	−0.999757	+	0.0220565i	$0.992979\pi$
$660$		0			0
$661$	54379.4	−	94187.9i	0.124461	−	0.215572i	−0.797061	−	0.603898i	$-0.793613\pi$
$661$	54379.4	−	94187.9i	0.124461	−	0.215572i	0.921522	+	0.388326i	$0.126947\pi$
$662$	−53583.5	−	30936.4i	−0.122269	−	0.0705918i
$663$		0			0
$664$	−136429.	−	236301.i	−0.309435	−	0.535957i
$665$		−	321.225i		−	0.000726385i
$666$		0			0
$667$	−414923.			−0.932644
$668$	−23312.1	+	13459.2i	−0.0522430	+	0.0301625i
$669$		0			0
$670$	1234.64	−	2138.45i	0.00275036	−	0.00476376i
$671$	85725.1	+	49493.4i	0.190398	+	0.109926i
$672$		0			0
$673$	−277175.	−	480081.i	−0.611961	−	1.05995i	−0.990910	−	0.134529i	$-0.957048\pi$
$673$	−277175.	−	480081.i	−0.611961	−	1.05995i	0.378949	−	0.925418i	$-0.376286\pi$
$674$		−	531866.i		−	1.17080i
$675$		0			0
$676$	−190617.			−0.417127
$677$	−333771.	+	192703.i	−0.728235	+	0.420446i	−0.817776	−	0.575537i	$-0.804793\pi$
$677$	−333771.	+	192703.i	−0.728235	+	0.420446i	0.0895414	+	0.995983i	$0.471460\pi$
$678$		0			0
$679$	16686.2	−	28901.3i	0.0361924	−	0.0626871i
$680$	3100.63	+	1790.15i	0.00670551	+	0.00387143i
$681$		0			0
$682$	−118660.	−	205526.i	−0.255115	−	0.441873i
$683$			731269.i			1.56760i	0.621012	+	0.783801i	$0.286722\pi$
$683$			731269.i			1.56760i	−0.621012	+	0.783801i	$0.713278\pi$
$684$		0			0
$685$	7623.98			0.0162480
$686$	−30404.1	+	17553.8i	−0.0646077	+	0.0373013i
$687$		0			0
$688$	−73759.1	+	127755.i	−0.155826	+	0.269898i
$689$	652878.	+	376940.i	1.37529	+	0.794023i
$690$		0			0
$691$	−25691.9	−	44499.7i	−0.0538072	−	0.0931969i	0.837867	−	0.545874i	$-0.183802\pi$
$691$	−25691.9	−	44499.7i	−0.0538072	−	0.0931969i	−0.891674	+	0.452677i	$0.850469\pi$
$692$			355894.i			0.743204i
$693$		0			0
$694$	532274.			1.10514
$695$	2983.75	−	1722.67i	0.00617722	−	0.00356642i
$696$		0			0
$697$	405914.	−	703064.i	0.835542	−	1.44720i
$698$	−192669.	−	111237.i	−0.395458	−	0.228318i
$699$		0			0
$700$	6469.35	+	11205.2i	0.0132028	+	0.0228678i
$701$		−	538501.i		−	1.09585i	−0.836528	−	0.547924i	$-0.815418\pi$
$701$		−	538501.i		−	1.09585i	0.836528	−	0.547924i	$-0.184582\pi$
$702$		0			0
$703$	56022.4			0.113358
$704$	−24086.7	+	13906.5i	−0.0485995	+	0.0280589i
$705$		0			0
$706$	163864.	−	283821.i	0.328757	−	0.569424i
$707$	−8078.88	−	4664.35i	−0.0161626	−	0.00933151i
$708$		0			0
$709$	−161752.	−	280163.i	−0.321779	−	0.557337i	0.659077	−	0.752076i	$-0.270947\pi$
$709$	−161752.	−	280163.i	−0.321779	−	0.557337i	−0.980855	+	0.194739i	$0.937614\pi$
$710$		−	2567.21i		−	0.00509266i
$711$		0			0
$712$	65740.6			0.129680
$713$	845167.	−	487958.i	1.66251	−	0.959849i
$714$		0			0
$715$	−2586.80	+	4480.47i	−0.00506001	+	0.00876419i
$716$	−401745.	−	231948.i	−0.783655	−	0.452443i
$717$		0			0
$718$	−264518.	−	458158.i	−0.513105	−	0.888724i
$719$		−	759369.i		−	1.46891i	−0.678658	−	0.734455i	$-0.737438\pi$
$719$		−	759369.i		−	1.46891i	0.678658	−	0.734455i	$-0.262562\pi$
$720$		0			0
$721$	8319.62			0.0160042
$722$	−101342.	+	58509.7i	−0.194408	+	0.112242i
$723$		0			0
$724$	−132457.	+	229423.i	−0.252697	+	0.437683i
$725$	355353.	+	205163.i	0.676057	+	0.390322i
$726$		0			0
$727$	243697.	+	422096.i	0.461086	+	0.798624i	0.999015	−	0.0443658i	$-0.0141267\pi$
$727$	243697.	+	422096.i	0.461086	+	0.798624i	−0.537930	+	0.842990i	$0.680793\pi$
$728$			13405.8i			0.0252947i
$729$		0			0
$730$	6505.23			0.0122072
$731$	−759067.	+	438248.i	−1.42051	+	0.820134i
$732$		0			0
$733$	230033.	−	398429.i	0.428137	−	0.741555i	−0.568571	−	0.822634i	$-0.692503\pi$
$733$	230033.	−	398429.i	0.428137	−	0.741555i	0.996708	+	0.0810797i	$0.0258368\pi$
$734$	147312.	+	85050.9i	0.273431	+	0.157865i
$735$		0			0
$736$	−57186.5	−	99049.8i	−0.105569	−	0.182851i
$737$			113973.i			0.209829i
$738$		0			0
$739$	−870177.			−1.59338			−0.796689	−	0.604390i	$-0.793417\pi$
$739$	−870177.			−1.59338			−0.796689	+	0.604390i	$0.793417\pi$
$740$	541.518	−	312.646i	0.000988894	−	0.000570938i
$741$		0			0
$742$	12057.1	−	20883.4i	0.0218995	−	0.0379310i
$743$	−193682.	−	111822.i	−0.350842	−	0.202559i	0.314214	−	0.949352i	$-0.398259\pi$
$743$	−193682.	−	111822.i	−0.350842	−	0.202559i	−0.665056	+	0.746794i	$0.731592\pi$
$744$		0			0
$745$	2415.60	+	4183.94i	0.00435223	+	0.00753829i
$746$		−	686309.i		−	1.23323i
$747$		0			0
$748$	−165253.			−0.295357
$749$	29239.4	−	16881.4i	0.0521201	−	0.0300915i
$750$		0			0
$751$	358800.	−	621459.i	0.636169	−	1.10188i	−0.350098	−	0.936713i	$-0.613852\pi$
$751$	358800.	−	621459.i	0.636169	−	1.10188i	0.986266	−	0.165163i	$-0.0528151\pi$
$752$	111157.	+	64176.4i	0.196562	+	0.113485i
$753$		0			0
$754$	212569.	+	368181.i	0.373902	+	0.647618i
$755$			18270.8i			0.0320525i
$756$		0			0
$757$	248013.			0.432795			0.216398	−	0.976305i	$-0.430569\pi$
$757$	248013.			0.432795			0.216398	+	0.976305i	$0.430569\pi$
$758$	832.163	−	480.450i	0.00144834	−	0.000836199i
$759$		0			0
$760$	1404.02	−	2431.84i	0.00243079	−	0.00421025i
$761$	752960.	+	434722.i	1.30018	+	0.750658i	0.980434	−	0.196846i	$-0.0630700\pi$
$761$	752960.	+	434722.i	1.30018	+	0.750658i	0.319743	+	0.947504i	$0.396403\pi$
$762$		0			0
$763$	−4160.05	−	7205.42i	−0.00714578	−	0.0123768i
$764$		−	493110.i		−	0.844807i
$765$		0			0
$766$	118020.			0.201140
$767$	636512.	−	367490.i	1.08197	−	0.624676i
$768$		0			0
$769$	−474853.	+	822469.i	−0.802983	+	1.39081i	0.114662	+	0.993405i	$0.463421\pi$
$769$	−474853.	+	822469.i	−0.802983	+	1.39081i	−0.917645	+	0.397402i	$0.869912\pi$
$770$	143.316	+	82.7433i	0.000241720	+	0.000139557i
$771$		0			0
$772$	−150715.	−	261046.i	−0.252884	−	0.438009i
$773$			608953.i			1.01912i	0.860435	+	0.509560i	$0.170192\pi$
$773$			608953.i			1.01912i	−0.860435	+	0.509560i	$0.829808\pi$
$774$		0			0
$775$	−965103.			−1.60683
$776$	252645.	−	145865.i	0.419554	−	0.242230i
$777$		0			0
$778$	−248512.	+	430435.i	−0.410571	+	0.711129i
$779$	−551417.	−	318360.i	−0.908667	−	0.524619i
$780$		0			0
$781$	59246.6	+	102618.i	0.0971317	+	0.168237i
$782$		−	679559.i		−	1.11125i
$783$		0			0
$784$	−153235.			−0.249302
$785$	−13146.3	+	7590.03i	−0.0213336	+	0.0123170i
$786$		0			0
$787$	−75627.2	+	130990.i	−0.122104	+	0.211490i	−0.920597	−	0.390514i	$-0.872297\pi$
$787$	−75627.2	+	130990.i	−0.122104	+	0.211490i	0.798493	+	0.602004i	$0.205631\pi$
$788$	−372381.	−	214994.i	−0.599701	−	0.346238i
$789$		0			0
$790$	−6488.53	−	11238.5i	−0.0103966	−	0.0180075i
$791$			29115.6i			0.0465343i
$792$		0			0
$793$	417078.			0.663240
$794$	−33061.5	+	19088.1i	−0.0524423	+	0.0302776i
$795$		0			0
$796$	65848.8	−	114054.i	0.103925	−	0.180004i
$797$	774339.	+	447065.i	1.21903	+	0.703807i	0.964711	−	0.263313i	$-0.0848151\pi$
$797$	774339.	+	447065.i	1.21903	+	0.703807i	0.254320	+	0.967120i	$0.418148\pi$
$798$		0			0
$799$	381311.	+	660450.i	0.597291	+	1.03454i
$800$			113106.i			0.176728i
$801$		0			0
$802$	193689.			0.301132
$803$	−260031.	+	150129.i	−0.403268	+	0.232827i
$804$		0			0
$805$	−340.259	+	589.345i	−0.000525070	+	0.000909448i
$806$	−865976.	−	499972.i	−1.33302	−	0.769618i
$807$		0			0
$808$	−40774.1	−	70622.9i	−0.0624542	−	0.108174i
$809$		−	61045.4i		−	0.0932730i	−0.998912	−	0.0466365i	$-0.985150\pi$
$809$		−	61045.4i		−	0.0932730i	0.998912	−	0.0466365i	$-0.0148502\pi$
$810$		0			0
$811$	716958.			1.09006			0.545032	−	0.838415i	$-0.316517\pi$
$811$	716958.			1.09006			0.545032	+	0.838415i	$0.316517\pi$
$812$	11776.9	−	6799.40i	0.0178615	−	0.0103124i
$813$		0			0
$814$	−14430.6	+	24994.5i	−0.0217789	+	0.0377221i
$815$	−12442.7	−	7183.81i	−0.0187327	−	0.0108153i
$816$		0			0
$817$	343720.	+	595341.i	0.514945	+	0.891911i
$818$			508423.i			0.759834i
$819$		0			0
$820$	−7106.73			−0.0105692
$821$	858337.	−	495561.i	1.27342	−	0.735209i	0.297789	−	0.954632i	$-0.403751\pi$
$821$	858337.	−	495561.i	1.27342	−	0.735209i	0.975630	+	0.219423i	$0.0704174\pi$
$822$		0			0
$823$	488314.	−	845785.i	0.720941	−	1.24871i	−0.239682	−	0.970852i	$-0.577043\pi$
$823$	488314.	−	845785.i	0.720941	−	1.24871i	0.960623	−	0.277855i	$-0.0896236\pi$
$824$	62983.7	+	36363.7i	0.0927628	+	0.0535566i
$825$		0			0
$826$	−11754.8	−	20359.9i	−0.0172288	−	0.0298412i
$827$			997499.i			1.45848i	0.684256	+	0.729242i	$0.260127\pi$
$827$			997499.i			1.45848i	−0.684256	+	0.729242i	$0.739873\pi$
$828$		0			0
$829$	−1.02815e6			−1.49606			−0.748029	−	0.663666i	$-0.768999\pi$
$829$	−1.02815e6			−1.49606			−0.748029	+	0.663666i	$0.768999\pi$
$830$	−12290.7	+	7096.03i	−0.0178410	+	0.0103005i
$831$		0			0
$832$	−58594.5	+	101489.i	−0.0846467	+	0.146612i
$833$	−788484.	−	455232.i	−1.13633	−	0.656058i
$834$		0			0
$835$	700.052	+	1212.53i	0.00100405	+	0.00173907i
$836$			129609.i			0.185448i
$837$		0			0
$838$	821525.			1.16986
$839$	−369159.	+	213134.i	−0.524432	+	0.302781i	−0.738746	−	0.673984i	$-0.764582\pi$
$839$	−369159.	+	213134.i	−0.524432	+	0.302781i	0.214314	+	0.976765i	$0.431248\pi$
$840$		0			0
$841$	−138011.	+	239042.i	−0.195129	+	0.337973i
$842$	−96284.2	−	55589.7i	−0.135810	−	0.0784098i
$843$		0			0
$844$	34270.4	+	59358.1i	0.0481099	+	0.0833287i
$845$			9914.53i			0.0138854i
$846$		0			0
$847$	30259.3			0.0421786
$848$	182556.	−	105399.i	0.253866	−	0.146569i
$849$		0			0
$850$	−336015.	+	581995.i	−0.465073	+	0.805529i
$851$	−102783.	−	59341.8i	−0.141926	−	0.0819411i
$852$		0			0
$853$	254338.	+	440527.i	0.349553	+	0.605444i	0.986170	−	0.165736i	$-0.0530000\pi$
$853$	254338.	+	440527.i	0.349553	+	0.605444i	−0.636617	+	0.771180i	$0.719667\pi$
$854$		−	13340.9i		−	0.0182924i
$855$		0			0
$856$	295143.			0.402795
$857$	402015.	−	232103.i	0.547369	−	0.316024i	−0.200691	−	0.979655i	$-0.564319\pi$
$857$	402015.	−	232103.i	0.547369	−	0.316024i	0.748060	+	0.663631i	$0.230985\pi$
$858$		0			0
$859$	−537730.	+	931376.i	−0.728749	+	1.26223i	0.228663	+	0.973506i	$0.426564\pi$
$859$	−537730.	+	931376.i	−0.728749	+	1.26223i	−0.957412	+	0.288725i	$0.906769\pi$
$860$	6644.87	+	3836.42i	0.00898441	+	0.00518715i
$861$		0			0
$862$	204972.	+	355022.i	0.275855	+	0.477794i
$863$		−	353636.i		−	0.474827i	−0.971409	−	0.237413i	$-0.923700\pi$
$863$		−	353636.i		−	0.474827i	0.971409	−	0.237413i	$-0.0762996\pi$
$864$		0			0
$865$	18511.0			0.0247399
$866$	83679.6	−	48312.4i	0.111579	−	0.0644204i
$867$		0			0
$868$	−15992.4	+	27699.7i	−0.0212264	+	0.0367651i
$869$	518727.	+	299487.i	0.686909	+	0.396587i
$870$		0			0
$871$	240110.	+	415883.i	0.316500	+	0.548195i
$872$		−	72731.5i		−	0.0956510i
$873$		0			0
$874$	−532982.			−0.697733
$875$	1165.79	−	673.071i	0.00152267	−	0.000879113i
$876$		0			0
$877$	−593602.	+	1.02815e6i	−0.771785	+	1.33677i	0.164799	+	0.986327i	$0.447303\pi$
$877$	−593602.	+	1.02815e6i	−0.771785	+	1.33677i	−0.936584	+	0.350444i	$0.886031\pi$
$878$	53821.2	+	31073.7i	0.0698175	+	0.0403092i
$879$		0			0
$880$	723.314	+	1252.82i	0.000934031	+	0.00161779i
$881$			1.51991e6i			1.95824i	0.203294	+	0.979118i	$0.434835\pi$
$881$			1.51991e6i			1.95824i	−0.203294	+	0.979118i	$0.565165\pi$
$882$		0			0
$883$	794738.			1.01930			0.509650	−	0.860382i	$-0.329775\pi$
$883$	794738.			1.01930			0.509650	+	0.860382i	$0.329775\pi$
$884$	−603005.	+	348145.i	−0.771643	+	0.445508i
$885$		0			0
$886$	−90598.0	+	156920.i	−0.115412	+	0.199900i
$887$	−709913.	−	409868.i	−0.902314	−	0.520951i	−0.0243637	−	0.999703i	$-0.507756\pi$
$887$	−709913.	−	409868.i	−0.902314	−	0.520951i	−0.877950	+	0.478752i	$0.841089\pi$
$888$		0			0
$889$	1805.60	+	3127.39i	0.00228464	+	0.00395711i
$890$		−	3419.35i		−	0.00431682i
$891$		0			0
$892$	−66602.2			−0.0837065
$893$	517994.	−	299064.i	0.649564	−	0.375026i
$894$		0			0
$895$	−12064.2	+	20895.9i	−0.0150610	+	0.0260864i
$896$	3246.28	+	1874.24i	0.00404362	+	0.00233459i
$897$		0			0
$898$	−230481.	−	399205.i	−0.285814	−	0.495044i
$899$			1.01434e6i			1.25506i
$900$		0			0
$901$	1.25248e6			1.54284
$902$	284075.	−	164011.i	0.349156	−	0.201585i
$903$		0			0
$904$	−127260.	+	220420.i	−0.155723	+	0.269721i
$905$	11932.9	+	6889.48i	0.0145697	+	0.00841181i
$906$		0			0
$907$	−149614.	−	259140.i	−0.181869	−	0.315007i	0.760648	−	0.649165i	$-0.224881\pi$
$907$	−149614.	−	259140.i	−0.181869	−	0.315007i	−0.942517	+	0.334158i	$0.891548\pi$
$908$			245435.i			0.297690i
$909$		0			0
$910$	697.272			0.000842015
$911$	−249530.	+	144066.i	−0.300667	+	0.173590i	−0.642743	−	0.766082i	$-0.722204\pi$
$911$	−249530.	+	144066.i	−0.300667	+	0.173590i	0.342075	+	0.939673i	$0.388870\pi$
$912$		0			0
$913$	327527.	−	567293.i	0.392921	−	0.680559i
$914$	−363397.	−	209808.i	−0.435000	−	0.251147i
$915$		0			0
$916$	201784.	+	349501.i	0.240490	+	0.416541i
$917$		−	43369.9i		−	0.0515763i
$918$		0			0
$919$	−1.26206e6			−1.49434			−0.747172	−	0.664631i	$-0.768589\pi$
$919$	−1.26206e6			−1.49434			−0.747172	+	0.664631i	$0.768589\pi$
$920$	−5151.86	+	2974.43i	−0.00608679	+	0.00351421i
$921$		0			0
$922$	84231.7	−	145893.i	0.0990863	−	0.171622i
$923$	432378.	+	249633.i	0.507528	+	0.293021i
$924$		0			0
$925$	58684.3	+	101644.i	0.0685865	+	0.118795i
$926$			191703.i			0.223567i
$927$		0			0
$928$	118876.			0.138038
$929$	−643193.	+	371348.i	−0.745263	+	0.430278i	−0.823980	−	0.566619i	$-0.808251\pi$
$929$	−643193.	+	371348.i	−0.745263	+	0.430278i	0.0787166	+	0.996897i	$0.474918\pi$
$930$		0			0
$931$	−357041.	+	618412.i	−0.411925	+	0.713475i
$932$	−52675.1	−	30412.0i	−0.0606420	−	0.0350117i
$933$		0			0
$934$	80942.3	+	140196.i	0.0927858	+	0.160710i
$935$			8595.29i			0.00983190i
$936$		0			0
$937$	359577.			0.409555			0.204778	−	0.978809i	$-0.434353\pi$
$937$	359577.			0.409555			0.204778	+	0.978809i	$0.434353\pi$
$938$	13302.7	−	7680.33i	0.0151194	−	0.00872920i
$939$		0			0
$940$	3337.99	−	5781.57i	0.00377772	−	0.00654320i
$941$	−1.06711e6	−	616095.i	−1.20512	−	0.695774i	−0.243428	−	0.969919i	$-0.578272\pi$
$941$	−1.06711e6	−	616095.i	−1.20512	−	0.695774i	−0.961688	+	0.274145i	$0.911605\pi$
$942$		0			0
$943$	674448.	+	1.16818e6i	0.758446	+	1.31367i
$944$		−	205513.i		−	0.230619i
$945$		0			0
$946$	−354150.			−0.395736
$947$	−773051.	+	446321.i	−0.862002	+	0.497677i	−0.864682	−	0.502319i	$-0.832480\pi$
$947$	−773051.	+	446321.i	−0.862002	+	0.497677i	0.00268003	+	0.999996i	$0.499147\pi$
$948$		0			0
$949$	−632563.	+	1.09563e6i	−0.702379	+	1.21656i
$950$	456462.	+	263538.i	0.505775	+	0.292009i
$951$		0			0
$952$	11136.0	+	19288.1i	0.0122873	+	0.0212822i
$953$		−	737750.i		−	0.812313i	−0.913803	−	0.406157i	$-0.866869\pi$
$953$		−	737750.i		−	0.812313i	0.913803	−	0.406157i	$-0.133131\pi$
$954$		0			0
$955$	−25648.0			−0.0281221
$956$	694988.	−	401252.i	0.760434	−	0.439037i
$957$		0			0
$958$	269356.	−	466538.i	0.293491	−	0.508342i
$959$	41072.7	+	23713.3i	0.0446597	+	0.0257843i
$960$		0			0
$961$	−731122.	−	1.26634e6i	−0.791668	−	1.37121i
$962$			121606.i			0.131403i
$963$		0			0
$964$	303515.			0.326607
$965$	−13577.7	+	7839.11i	−0.0145805	+	0.00841806i
$966$		0			0
$967$	−24080.7	+	41708.9i	−0.0257523	+	0.0446042i	−0.878614	−	0.477532i	$-0.841531\pi$
$967$	−24080.7	+	41708.9i	−0.0257523	+	0.0446042i	0.852862	+	0.522136i	$0.174865\pi$
$968$	229078.	+	132258.i	0.244474	+	0.141147i
$969$		0			0
$970$	−7586.84	−	13140.8i	−0.00806338	−	0.0139662i
$971$		−	80929.3i		−	0.0858356i	−0.999079	−	0.0429178i	$-0.986335\pi$
$971$		−	80929.3i		−	0.0858356i	0.999079	−	0.0429178i	$-0.0136654\pi$
$972$		0			0
$973$	21432.5			0.0226385
$974$	−409543.	+	236450.i	−0.431699	+	0.249242i
$975$		0			0
$976$	58311.0	−	100998.i	0.0612140	−	0.106026i
$977$	525681.	+	303502.i	0.550723	+	0.317960i	0.749414	−	0.662102i	$-0.230335\pi$
$977$	525681.	+	303502.i	0.550723	+	0.317960i	−0.198690	+	0.980062i	$0.563669\pi$
$978$		0			0
$979$	78912.4	+	136680.i	0.0823341	+	0.142607i
$980$			7970.19i			0.00829882i
$981$		0			0
$982$	−700779.			−0.726705
$983$	1.31392e6	−	758595.i	1.35976	−	0.785060i	0.370172	−	0.928963i	$-0.379299\pi$
$983$	1.31392e6	−	758595.i	1.35976	−	0.785060i	0.989592	+	0.143904i	$0.0459655\pi$
$984$		0			0
$985$	−11182.4	+	19368.6i	−0.0115256	+	0.0199630i
$986$	611686.	+	353157.i	0.629180	+	0.363257i
$987$		0			0
$988$	273052.	+	472940.i	0.279725	+	0.484498i
$989$		−	1.45634e6i		−	1.48892i
$990$		0			0
$991$	−1.27300e6			−1.29623			−0.648116	−	0.761542i	$-0.724443\pi$
$991$	−1.27300e6			−1.29623			−0.648116	+	0.761542i	$0.724443\pi$
$992$	−242142.	+	139801.i	−0.246063	+	0.142064i
$993$		0			0
$994$	7984.95	−	13830.3i	0.00808164	−	0.0139978i
$995$	−5932.24	−	3424.98i	−0.00599201	−	0.00345949i
$996$		0			0
$997$	−596309.	−	1.03284e6i	−0.599903	−	1.03906i	−0.992835	−	0.119494i	$-0.961873\pi$
$997$	−596309.	−	1.03284e6i	−0.599903	−	1.03906i	0.392932	−	0.919567i	$-0.371461\pi$
$998$			1.16257e6i			1.16723i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.5.d.e.53.3		8
3.2	odd	2	inner	162.5.d.e.53.2		8
9.2	odd	6	inner	162.5.d.e.107.3		8
9.4	even	3		162.5.b.b.161.4	yes	4
9.5	odd	6		162.5.b.b.161.1	✓	4
9.7	even	3	inner	162.5.d.e.107.2		8
36.23	even	6		1296.5.e.a.161.2		4
36.31	odd	6		1296.5.e.a.161.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.5.b.b.161.1	✓	4	9.5	odd	6
162.5.b.b.161.4	yes	4	9.4	even	3
162.5.d.e.53.2		8	3.2	odd	2	inner
162.5.d.e.53.3		8	1.1	even	1	trivial
162.5.d.e.107.2		8	9.7	even	3	inner
162.5.d.e.107.3		8	9.2	odd	6	inner
1296.5.e.a.161.2		4	36.23	even	6
1296.5.e.a.161.3		4	36.31	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	162.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.44949 - 1.41421i) q^{2} +(4.00000 - 6.92820i) q^{4} +(-0.360355 - 0.208051i) q^{5} +(-1.29423 - 2.24167i) q^{7} -22.6274i q^{8} -1.17691 q^{10} +(47.0443 - 27.1611i) q^{11} +(114.442 - 198.220i) q^{13} +(-6.34040 - 3.66063i) q^{14} +(-32.0000 - 55.4256i) q^{16} -380.263i q^{17} -298.242 q^{19} +(-2.88284 + 1.66441i) q^{20} +(76.8231 - 133.061i) q^{22} +(547.178 + 315.913i) q^{23} +(-312.413 - 541.116i) q^{25} -647.383i q^{26} -20.7077 q^{28} +(-568.722 + 328.352i) q^{29} +(772.296 - 1337.66i) q^{31} +(-156.767 - 90.5097i) q^{32} +(-537.773 - 931.450i) q^{34} +1.07706i q^{35} -187.842 q^{37} +(-730.541 + 421.778i) q^{38} +(-4.70766 + 8.15390i) q^{40} +(1848.89 + 1067.46i) q^{41} +(-1152.49 - 1996.16i) q^{43} -434.577i q^{44} +1787.08 q^{46} +(-1736.82 + 1002.76i) q^{47} +(1197.15 - 2073.52i) q^{49} +(-1530.51 - 883.639i) q^{50} +(-915.538 - 1585.76i) q^{52} +3293.71i q^{53} -22.6036 q^{55} +(-50.7232 + 29.2851i) q^{56} +(-928.719 + 1608.59i) q^{58} +(2780.93 + 1605.57i) q^{59} +(911.109 + 1578.09i) q^{61} -4368.77i q^{62} -512.000 q^{64} +(-82.4797 + 47.6197i) q^{65} +(-1049.04 + 1817.00i) q^{67} +(-2634.54 - 1521.05i) q^{68} +(1.52320 + 2.63825i) q^{70} +2181.30i q^{71} -5527.36 q^{73} +(-460.117 + 265.649i) q^{74} +(-1192.97 + 2066.28i) q^{76} +(-121.772 - 70.3052i) q^{77} +(5513.17 + 9549.08i) q^{79} +26.6305i q^{80} +6038.44 q^{82} +(10443.1 - 6029.35i) q^{83} +(-79.1141 + 137.030i) q^{85} +(-5646.01 - 3259.72i) q^{86} +(-614.585 - 1064.49i) q^{88} +2905.35i q^{89} -592.458 q^{91} +(4377.43 - 2527.31i) q^{92} +(-2836.22 + 4912.48i) q^{94} +(107.473 + 62.0496i) q^{95} +(6446.38 + 11165.5i) q^{97} -6772.10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{4} + 52 q^{7} + 240 q^{10} + 292 q^{13} - 256 q^{16} + 1480 q^{19} + 864 q^{22} - 1564 q^{25} + 832 q^{28} + 5056 q^{31} + 312 q^{34} + 8848 q^{37} + 960 q^{40} - 428 q^{43} + 4320 q^{46} + 7956 q^{49}+ \cdots + 3808 q^{97}+O(q^{100}) \) 8 * q + 32 * q^4 + 52 * q^7 + 240 * q^10 + 292 * q^13 - 256 * q^16 + 1480 * q^19 + 864 * q^22 - 1564 * q^25 + 832 * q^28 + 5056 * q^31 + 312 * q^34 + 8848 * q^37 + 960 * q^40 - 428 * q^43 + 4320 * q^46 + 7956 * q^49 - 2336 * q^52 + 8424 * q^55 - 72 * q^58 - 4496 * q^61 - 4096 * q^64 - 16436 * q^67 + 3504 * q^70 - 41600 * q^73 + 5920 * q^76 + 16732 * q^79 - 3072 * q^82 - 19152 * q^85 - 6912 * q^88 - 11848 * q^91 - 29424 * q^94 + 3808 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more