LMFDB - Embedded newform 108.4.i.a.49.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(13,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 8]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.i (of order $9$, degree $6$, 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:	$6.37220628062$
Analytic rank:	$0$
Dimension:	$54$
Relative dimension:	$9$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			49.8
Character	$\chi$	$=$	108.49
Dual form			108.4.i.a.97.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.05072 + 1.22075i) q^{3} +(11.1609 - 4.06223i) q^{5} +(-4.28132 - 24.2806i) q^{7} +(24.0196 + 12.3313i) q^{9} +O(q^{10})$	\(q+(5.05072 + 1.22075i) q^{3} +(11.1609 - 4.06223i) q^{5} +(-4.28132 - 24.2806i) q^{7} +(24.0196 + 12.3313i) q^{9} +(-9.00480 - 3.27748i) q^{11} +(6.18439 + 5.18932i) q^{13} +(61.3295 - 6.89258i) q^{15} +(-16.8131 + 29.1212i) q^{17} +(57.3341 + 99.3056i) q^{19} +(8.01668 - 127.861i) q^{21} +(29.4732 - 167.151i) q^{23} +(12.3082 - 10.3278i) q^{25} +(106.263 + 91.6038i) q^{27} +(-1.33025 + 1.11621i) q^{29} +(-26.8769 + 152.426i) q^{31} +(-41.4798 - 27.5462i) q^{33} +(-146.417 - 253.601i) q^{35} +(-143.047 + 247.765i) q^{37} +(24.9008 + 33.7594i) q^{39} +(93.7251 + 78.6447i) q^{41} +(-328.750 - 119.655i) q^{43} +(318.172 + 40.0553i) q^{45} +(-40.1073 - 227.460i) q^{47} +(-248.901 + 90.5926i) q^{49} +(-120.468 + 126.559i) q^{51} -647.947 q^{53} -113.816 q^{55} +(168.352 + 571.555i) q^{57} +(-824.149 + 299.966i) q^{59} +(126.231 + 715.889i) q^{61} +(196.576 - 636.002i) q^{63} +(90.1035 + 32.7950i) q^{65} +(-210.269 - 176.437i) q^{67} +(352.910 - 808.253i) q^{69} +(392.428 - 679.706i) q^{71} +(176.253 + 305.280i) q^{73} +(74.7732 - 37.1378i) q^{75} +(-41.0266 + 232.674i) q^{77} +(820.969 - 688.875i) q^{79} +(424.878 + 592.385i) q^{81} +(1012.90 - 849.924i) q^{83} +(-69.3525 + 393.318i) q^{85} +(-8.08135 + 4.01378i) q^{87} +(15.8047 + 27.3746i) q^{89} +(99.5222 - 172.377i) q^{91} +(-321.821 + 737.052i) q^{93} +(1043.30 + 875.434i) q^{95} +(64.2391 + 23.3811i) q^{97} +(-175.876 - 189.765i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) $ 54 * q + 12 * q^5 - 48 * q^9	$ 54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100}) $ 54 * q + 12 * q^5 - 48 * q^9 - 87 * q^11 + 234 * q^15 + 204 * q^17 - 12 * q^21 + 96 * q^23 - 216 * q^25 + 27 * q^27 + 318 * q^29 - 54 * q^31 + 63 * q^33 + 6 * q^35 + 66 * q^39 + 867 * q^41 - 513 * q^43 - 306 * q^45 - 1548 * q^47 + 594 * q^49 - 1368 * q^51 - 1068 * q^53 - 1269 * q^57 - 1218 * q^59 - 54 * q^61 + 30 * q^63 + 96 * q^65 - 2997 * q^67 + 1476 * q^69 - 120 * q^71 - 216 * q^73 + 732 * q^75 + 3480 * q^77 + 2808 * q^79 + 3348 * q^81 + 4464 * q^83 + 2160 * q^85 + 4824 * q^87 + 4029 * q^89 + 270 * q^91 + 1164 * q^93 - 1650 * q^95 - 3483 * q^97 - 5076 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{7}{9}\right)$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	5.05072	+	1.22075i	0.972012	+	0.234933i
$3$	5.05072	+	1.22075i	0.972012	+	0.234933i
$4$		0			0
$5$	11.1609	−	4.06223i	0.998261	−	0.363337i	0.209347	−	0.977841i	$-0.432866\pi$
$5$	11.1609	−	4.06223i	0.998261	−	0.363337i	0.788914	+	0.614504i	$0.210644\pi$
$6$		0			0
$7$	−4.28132	−	24.2806i	−0.231169	−	1.31103i	−0.850533	−	0.525922i	$-0.823720\pi$
$7$	−4.28132	−	24.2806i	−0.231169	−	1.31103i	0.619363	−	0.785105i	$-0.287391\pi$
$8$		0			0
$9$	24.0196	+	12.3313i	0.889613	+	0.456715i
$10$		0			0
$11$	−9.00480	−	3.27748i	−0.246823	−	0.0898361i	0.215646	−	0.976472i	$-0.430814\pi$
$11$	−9.00480	−	3.27748i	−0.246823	−	0.0898361i	−0.462469	+	0.886635i	$0.653036\pi$
$12$		0			0
$13$	6.18439	+	5.18932i	0.131942	+	0.110712i	0.706370	−	0.707842i	$-0.250331\pi$
$13$	6.18439	+	5.18932i	0.131942	+	0.110712i	−0.574429	+	0.818555i	$0.694776\pi$
$14$		0			0
$15$	61.3295	−	6.89258i	1.05568	−	0.118644i
$16$		0			0
$17$	−16.8131	+	29.1212i	−0.239870	+	0.415467i	−0.960677	−	0.277669i	$-0.910438\pi$
$17$	−16.8131	+	29.1212i	−0.239870	+	0.415467i	0.720807	+	0.693136i	$0.243771\pi$
$18$		0			0
$19$	57.3341	+	99.3056i	0.692281	+	1.19907i	0.971089	+	0.238719i	$0.0767276\pi$
$19$	57.3341	+	99.3056i	0.692281	+	1.19907i	−0.278807	+	0.960347i	$0.589939\pi$
$20$		0			0
$21$	8.01668	−	127.861i	0.0833040	−	1.32864i
$22$		0			0
$23$	29.4732	−	167.151i	0.267199	−	1.51536i	−0.495499	−	0.868609i	$-0.665015\pi$
$23$	29.4732	−	167.151i	0.267199	−	1.51536i	0.762698	−	0.646755i	$-0.223874\pi$
$24$		0			0
$25$	12.3082	−	10.3278i	0.0984660	−	0.0826228i
$26$		0			0
$27$	106.263	+	91.6038i	0.757417	+	0.652932i
$28$		0			0
$29$	−1.33025	+	1.11621i	−0.00851799	+	0.00714744i	−0.647037	−	0.762459i	$-0.723992\pi$
$29$	−1.33025	+	1.11621i	−0.00851799	+	0.00714744i	0.638519	+	0.769606i	$0.279548\pi$
$30$		0			0
$31$	−26.8769	+	152.426i	−0.155717	+	0.883115i	0.802410	+	0.596773i	$0.203551\pi$
$31$	−26.8769	+	152.426i	−0.155717	+	0.883115i	−0.958127	+	0.286342i	$0.907561\pi$
$32$		0			0
$33$	−41.4798	−	27.5462i	−0.218809	−	0.145309i
$34$		0			0
$35$	−146.417	−	253.601i	−0.707112	−	1.22475i
$36$		0			0
$37$	−143.047	+	247.765i	−0.635589	+	1.10087i	0.350801	+	0.936450i	$0.385909\pi$
$37$	−143.047	+	247.765i	−0.635589	+	1.10087i	−0.986390	+	0.164422i	$0.947424\pi$
$38$		0			0
$39$	24.9008	+	33.7594i	0.102239	+	0.138611i
$40$		0			0
$41$	93.7251	+	78.6447i	0.357010	+	0.299567i	0.803598	−	0.595173i	$-0.202916\pi$
$41$	93.7251	+	78.6447i	0.357010	+	0.299567i	−0.446588	+	0.894740i	$0.647361\pi$
$42$		0			0
$43$	−328.750	−	119.655i	−1.16590	−	0.424354i	−0.314701	−	0.949191i	$-0.601904\pi$
$43$	−328.750	−	119.655i	−1.16590	−	0.424354i	−0.851203	+	0.524837i	$0.824126\pi$
$44$		0			0
$45$	318.172	+	40.0553i	1.05401	+	0.132691i
$46$		0			0
$47$	−40.1073	−	227.460i	−0.124474	−	0.705925i	−0.981619	−	0.190850i	$-0.938875\pi$
$47$	−40.1073	−	227.460i	−0.124474	−	0.705925i	0.857146	−	0.515074i	$-0.172236\pi$
$48$		0			0
$49$	−248.901	+	90.5926i	−0.725659	+	0.264118i
$50$		0			0
$51$	−120.468	+	126.559i	−0.330763	+	0.347485i
$52$		0			0
$53$	−647.947			−1.67929			−0.839645	−	0.543136i	$-0.817237\pi$
$53$	−647.947			−1.67929			−0.839645	+	0.543136i	$0.817237\pi$
$54$		0			0
$55$	−113.816			−0.279034
$56$		0			0
$57$	168.352	+	571.555i	0.391205	+	1.32815i
$58$		0			0
$59$	−824.149	+	299.966i	−1.81856	+	0.661902i	−0.822970	+	0.568085i	$0.807685\pi$
$59$	−824.149	+	299.966i	−1.81856	+	0.661902i	−0.995589	+	0.0938171i	$0.970093\pi$
$60$		0			0
$61$	126.231	+	715.889i	0.264954	+	1.50263i	0.769167	+	0.639048i	$0.220671\pi$
$61$	126.231	+	715.889i	0.264954	+	1.50263i	−0.504214	+	0.863579i	$0.668218\pi$
$62$		0			0
$63$	196.576	−	636.002i	0.393114	−	1.27189i
$64$		0			0
$65$	90.1035	+	32.7950i	0.171938	+	0.0625803i
$66$		0			0
$67$	−210.269	−	176.437i	−0.383410	−	0.321719i	0.430629	−	0.902529i	$-0.358292\pi$
$67$	−210.269	−	176.437i	−0.383410	−	0.321719i	−0.814039	+	0.580810i	$0.802736\pi$
$68$		0			0
$69$	352.910	−	808.253i	0.615730	−	1.41018i
$70$		0			0
$71$	392.428	−	679.706i	0.655953	−	1.13614i	−0.325701	−	0.945473i	$-0.605600\pi$
$71$	392.428	−	679.706i	0.655953	−	1.13614i	0.981654	−	0.190671i	$-0.0610664\pi$
$72$		0			0
$73$	176.253	+	305.280i	0.282588	+	0.489456i	0.972021	−	0.234893i	$-0.0754739\pi$
$73$	176.253	+	305.280i	0.282588	+	0.489456i	−0.689434	+	0.724349i	$0.742141\pi$
$74$		0			0
$75$	74.7732	−	37.1378i	0.115121	−	0.0571774i
$76$		0			0
$77$	−41.0266	+	232.674i	−0.0607197	+	0.344359i
$78$		0			0
$79$	820.969	−	688.875i	1.16919	−	0.981070i	0.169203	−	0.985581i	$-0.445880\pi$
$79$	820.969	−	688.875i	1.16919	−	0.981070i	0.999990	+	0.00451148i	$0.00143605\pi$
$80$		0			0
$81$	424.878	+	592.385i	0.582823	+	0.812599i
$82$		0			0
$83$	1012.90	−	849.924i	1.33952	−	1.12399i	0.357772	−	0.933809i	$-0.383536\pi$
$83$	1012.90	−	849.924i	1.33952	−	1.12399i	0.981749	−	0.190182i	$-0.0609080\pi$
$84$		0			0
$85$	−69.3525	+	393.318i	−0.0884981	+	0.501897i
$86$		0			0
$87$	−8.08135	+	4.01378i	−0.00995875	+	0.00494624i
$88$		0			0
$89$	15.8047	+	27.3746i	0.0188236	+	0.0326034i	0.875284	−	0.483610i	$-0.160675\pi$
$89$	15.8047	+	27.3746i	0.0188236	+	0.0326034i	−0.856460	+	0.516213i	$0.827341\pi$
$90$		0			0
$91$	99.5222	−	172.377i	0.114646	−	0.198572i
$92$		0			0
$93$	−321.821	+	737.052i	−0.358831	+	0.821815i
$94$		0			0
$95$	1043.30	+	875.434i	1.12674	+	0.945449i
$96$		0			0
$97$	64.2391	+	23.3811i	0.0672422	+	0.0244742i	0.375422	−	0.926854i	$-0.377498\pi$
$97$	64.2391	+	23.3811i	0.0672422	+	0.0244742i	−0.308180	+	0.951328i	$0.599720\pi$
$98$		0			0
$99$	−175.876	−	189.765i	−0.178547	−	0.192647i
$100$		0			0
$101$	−111.400	−	631.783i	−0.109750	−	0.622423i	−0.989216	−	0.146462i	$-0.953211\pi$
$101$	−111.400	−	631.783i	−0.109750	−	0.622423i	0.879466	−	0.475961i	$-0.157900\pi$
$102$		0			0
$103$	−1465.14	+	533.267i	−1.40160	+	0.510140i	−0.928652	−	0.370952i	$-0.879031\pi$
$103$	−1465.14	+	533.267i	−1.40160	+	0.510140i	−0.472945	+	0.881092i	$0.656809\pi$
$104$		0			0
$105$	−429.927	−	1459.61i	−0.399586	−	1.35660i
$106$		0			0
$107$	−84.9771			−0.0767762			−0.0383881	−	0.999263i	$-0.512222\pi$
$107$	−84.9771			−0.0767762			−0.0383881	+	0.999263i	$0.512222\pi$
$108$		0			0
$109$	2199.34			1.93265			0.966324	−	0.257329i	$-0.0828424\pi$
$109$	2199.34			1.93265			0.966324	+	0.257329i	$0.0828424\pi$
$110$		0			0
$111$	−1024.95	+	1076.77i	−0.876431	+	0.920740i
$112$		0			0
$113$	184.357	−	67.1006i	0.153477	−	0.0558610i	−0.264139	−	0.964485i	$-0.585088\pi$
$113$	184.357	−	67.1006i	0.153477	−	0.0558610i	0.417616	+	0.908624i	$0.362866\pi$
$114$		0			0
$115$	−350.058	−	1985.28i	−0.283853	−	1.60981i
$116$		0			0
$117$	84.5552	+	200.907i	0.0668131	+	0.158751i
$118$		0			0
$119$	779.061	+	283.555i	0.600138	+	0.218432i
$120$		0			0
$121$	−949.261	−	796.524i	−0.713194	−	0.598440i
$122$		0			0
$123$	377.374	+	511.627i	0.276640	+	0.375056i
$124$		0			0
$125$	−646.907	+	1120.47i	−0.462889	+	0.801747i
$126$		0			0
$127$	697.879	+	1208.76i	0.487612	+	0.844569i	0.999899	−	0.0142455i	$-0.00453464\pi$
$127$	697.879	+	1208.76i	0.487612	+	0.844569i	−0.512286	+	0.858815i	$0.671201\pi$
$128$		0			0
$129$	−1514.35	−	1005.66i	−1.03358	−	0.686386i
$130$		0			0
$131$	−436.059	+	2473.01i	−0.290829	+	1.64937i	0.392858	+	0.919599i	$0.371487\pi$
$131$	−436.059	+	2473.01i	−0.290829	+	1.64937i	−0.683687	+	0.729775i	$0.739625\pi$
$132$		0			0
$133$	2165.73	−	1817.26i	1.41197	−	1.18479i
$134$		0			0
$135$	1558.10	+	590.716i	0.993334	+	0.376598i
$136$		0			0
$137$	−320.673	+	269.077i	−0.199978	+	0.167801i	−0.737277	−	0.675590i	$-0.763889\pi$
$137$	−320.673	+	269.077i	−0.199978	+	0.167801i	0.537300	+	0.843391i	$0.319444\pi$
$138$		0			0
$139$	128.625	−	729.470i	0.0784881	−	0.445128i	−0.920085	−	0.391720i	$-0.871880\pi$
$139$	128.625	−	729.470i	0.0784881	−	0.445128i	0.998573	−	0.0534085i	$-0.0170086\pi$
$140$		0			0
$141$	75.1002	−	1197.80i	0.0448552	−	0.715410i
$142$		0			0
$143$	−38.6813	−	66.9980i	−0.0226202	−	0.0391794i
$144$		0			0
$145$	−10.3125	+	17.8617i	−0.00590624	+	0.0102299i
$146$		0			0
$147$	−1367.72	+	153.713i	−0.767399	+	0.0862449i
$148$		0			0
$149$	165.624	+	138.975i	0.0910633	+	0.0764112i	0.687183	−	0.726485i	$-0.258847\pi$
$149$	165.624	+	138.975i	0.0910633	+	0.0764112i	−0.596120	+	0.802896i	$0.703292\pi$
$150$		0			0
$151$	−1498.00	−	545.227i	−0.807320	−	0.293841i	−0.0948039	−	0.995496i	$-0.530222\pi$
$151$	−1498.00	−	545.227i	−0.807320	−	0.293841i	−0.712516	+	0.701655i	$0.752445\pi$
$152$		0			0
$153$	−762.947	+	492.151i	−0.403141	+	0.260052i
$154$		0			0
$155$	319.221	+	1810.39i	0.165422	+	0.938156i
$156$		0			0
$157$	−340.759	+	124.026i	−0.173220	+	0.0630469i	−0.427174	−	0.904169i	$-0.640491\pi$
$157$	−340.759	+	124.026i	−0.173220	+	0.0630469i	0.253954	+	0.967216i	$0.418269\pi$
$158$		0			0
$159$	−3272.60	−	790.979i	−1.63229	−	0.394520i
$160$		0			0
$161$	−4184.70			−2.04845
$162$		0			0
$163$	−1635.53			−0.785916			−0.392958	−	0.919557i	$-0.628548\pi$
$163$	−1635.53			−0.785916			−0.392958	+	0.919557i	$0.628548\pi$
$164$		0			0
$165$	−574.850	−	138.940i	−0.271225	−	0.0655543i
$166$		0			0
$167$	1583.68	−	576.413i	0.733826	−	0.267091i	0.0520427	−	0.998645i	$-0.483427\pi$
$167$	1583.68	−	576.413i	0.733826	−	0.267091i	0.681784	+	0.731554i	$0.261205\pi$
$168$		0			0
$169$	−370.187	−	2099.44i	−0.168497	−	0.955593i
$170$		0			0
$171$	152.572	+	3092.28i	0.0682309	+	1.38288i
$172$		0			0
$173$	3073.84	+	1118.79i	1.35087	+	0.491675i	0.913218	−	0.407471i	$-0.133589\pi$
$173$	3073.84	+	1118.79i	1.35087	+	0.491675i	0.437648	+	0.899146i	$0.355811\pi$
$174$		0			0
$175$	−303.461	−	254.634i	−0.131083	−	0.109992i
$176$		0			0
$177$	−4528.73	+	508.965i	−1.92316	+	0.216137i
$178$		0			0
$179$	1379.57	−	2389.48i	0.576054	−	0.997755i	−0.419872	−	0.907583i	$-0.637925\pi$
$179$	1379.57	−	2389.48i	0.576054	−	0.997755i	0.995926	−	0.0901720i	$-0.0287417\pi$
$180$		0			0
$181$	1160.13	+	2009.40i	0.476418	+	0.825180i	0.999635	−	0.0270195i	$-0.00860161\pi$
$181$	1160.13	+	2009.40i	0.476418	+	0.825180i	−0.523217	+	0.852199i	$0.675268\pi$
$182$		0			0
$183$	−236.364	+	3769.85i	−0.0954784	+	1.52282i
$184$		0			0
$185$	−590.055	+	3346.37i	−0.234496	+	1.32989i
$186$		0			0
$187$	246.843	−	207.126i	0.0965292	−	0.0809976i
$188$		0			0
$189$	1769.25	−	2972.30i	0.680919	−	1.14393i
$190$		0			0
$191$	3603.66	−	3023.83i	1.36519	−	1.14553i	0.390851	−	0.920454i	$-0.372181\pi$
$191$	3603.66	−	3023.83i	1.36519	−	1.14553i	0.974341	−	0.225078i	$-0.0722636\pi$
$192$		0			0
$193$	−37.9402	+	215.169i	−0.0141502	+	0.0802499i	−0.991065	−	0.133379i	$-0.957417\pi$
$193$	−37.9402	+	215.169i	−0.0141502	+	0.0802499i	0.976915	+	0.213629i	$0.0685284\pi$
$194$		0			0
$195$	415.053	+	275.632i	0.152423	+	0.101223i
$196$		0			0
$197$	113.711	+	196.953i	0.0411246	+	0.0712299i	0.885855	−	0.463962i	$-0.153573\pi$
$197$	113.711	+	196.953i	0.0411246	+	0.0712299i	−0.844730	+	0.535192i	$0.820239\pi$
$198$		0			0
$199$	−501.833	+	869.199i	−0.178764	+	0.309628i	−0.941457	−	0.337132i	$-0.890543\pi$
$199$	−501.833	+	869.199i	−0.178764	+	0.309628i	0.762694	+	0.646760i	$0.223876\pi$
$200$		0			0
$201$	−846.627	−	1147.82i	−0.297097	−	0.402791i
$202$		0			0
$203$	32.7975	+	27.5204i	0.0113396	+	0.00951504i
$204$		0			0
$205$	1365.53	+	497.012i	0.465233	+	0.169331i
$206$		0			0
$207$	2769.12	−	3651.45i	0.929793	−	1.22605i
$208$		0			0
$209$	−190.810	−	1082.14i	−0.0631513	−	0.358149i
$210$		0			0
$211$	1881.22	−	684.709i	0.613785	−	0.223400i	−0.0163734	−	0.999866i	$-0.505212\pi$
$211$	1881.22	−	684.709i	0.613785	−	0.223400i	0.630159	+	0.776466i	$0.282990\pi$
$212$		0			0
$213$	2811.79	−	2953.95i	0.904512	−	0.950240i
$214$		0			0
$215$	−4155.21			−1.31806
$216$		0			0
$217$	3816.06			1.19378
$218$		0			0
$219$	517.537	+	1757.04i	0.159689	+	0.542146i
$220$		0			0
$221$	−255.098	+	92.8481i	−0.0776460	+	0.0282608i
$222$		0			0
$223$	−6.76080	−	38.3424i	−0.00203021	−	0.0115139i	0.983776	−	0.179402i	$-0.0574163\pi$
$223$	−6.76080	−	38.3424i	−0.00203021	−	0.0115139i	−0.985806	+	0.167888i	$0.946305\pi$
$224$		0			0
$225$	422.994	−	96.2935i	0.125332	−	0.0285314i
$226$		0			0
$227$	414.300	+	150.793i	0.121137	+	0.0440902i	0.401877	−	0.915694i	$-0.368358\pi$
$227$	414.300	+	150.793i	0.121137	+	0.0440902i	−0.280741	+	0.959784i	$0.590580\pi$
$228$		0			0
$229$	266.737	+	223.819i	0.0769715	+	0.0645867i	0.680462	−	0.732784i	$-0.261779\pi$
$229$	266.737	+	223.819i	0.0769715	+	0.0645867i	−0.603490	+	0.797370i	$0.706224\pi$
$230$		0			0
$231$	−491.250	+	1125.09i	−0.139921	+	0.320455i
$232$		0			0
$233$	2660.33	−	4607.83i	0.748001	−	1.29558i	−0.200779	−	0.979637i	$-0.564347\pi$
$233$	2660.33	−	4607.83i	0.748001	−	1.29558i	0.948780	−	0.315938i	$-0.102319\pi$
$234$		0			0
$235$	−1371.63	−	2375.73i	−0.380746	−	0.659471i
$236$		0			0
$237$	4987.43	−	2477.12i	1.36695	−	0.678929i
$238$		0			0
$239$	−266.898	+	1513.65i	−0.0722350	+	0.409665i	0.927153	+	0.374683i	$0.122249\pi$
$239$	−266.898	+	1513.65i	−0.0722350	+	0.409665i	−0.999388	+	0.0349821i	$0.988863\pi$
$240$		0			0
$241$	−2059.17	+	1727.85i	−0.550384	+	0.461827i	−0.875071	−	0.483995i	$-0.839185\pi$
$241$	−2059.17	+	1727.85i	−0.550384	+	0.461827i	0.324687	+	0.945822i	$0.394741\pi$
$242$		0			0
$243$	1422.79	+	3510.64i	0.375604	+	0.926780i
$244$		0			0
$245$	−2409.95	+	2022.19i	−0.628433	+	0.527318i
$246$		0			0
$247$	−160.752	+	911.669i	−0.0414105	+	0.234851i
$248$		0			0
$249$	6153.42	−	3056.23i	1.56609	−	0.777835i
$250$		0			0
$251$	1559.04	+	2700.34i	0.392055	+	0.679060i	0.992720	−	0.120441i	$-0.0384309\pi$
$251$	1559.04	+	2700.34i	0.392055	+	0.679060i	−0.600665	+	0.799501i	$0.705098\pi$
$252$		0			0
$253$	−813.234	+	1408.56i	−0.202085	+	0.350022i
$254$		0			0
$255$	−830.421	+	1901.88i	−0.203933	+	0.467059i
$256$		0			0
$257$	−2743.88	−	2302.39i	−0.665985	−	0.558828i	0.245889	−	0.969298i	$-0.420920\pi$
$257$	−2743.88	−	2302.39i	−0.665985	−	0.558828i	−0.911874	+	0.410470i	$0.865365\pi$
$258$		0			0
$259$	6628.29	+	2412.50i	1.59020	+	0.578786i
$260$		0			0
$261$	−45.7164	+	10.4072i	−0.0108421	+	0.00246816i
$262$		0			0
$263$	−653.310	−	3705.11i	−0.153174	−	0.868694i	−0.960436	−	0.278501i	$-0.910163\pi$
$263$	−653.310	−	3705.11i	−0.153174	−	0.868694i	0.807262	−	0.590194i	$-0.200949\pi$
$264$		0			0
$265$	−7231.66	+	2632.11i	−1.67637	+	0.610148i
$266$		0			0
$267$	46.4078	+	157.555i	0.0106371	+	0.0361131i
$268$		0			0
$269$	−1101.88			−0.249751			−0.124875	−	0.992172i	$-0.539853\pi$
$269$	−1101.88			−0.249751			−0.124875	+	0.992172i	$0.539853\pi$
$270$		0			0
$271$	−2560.66			−0.573981			−0.286991	−	0.957933i	$-0.592655\pi$
$271$	−2560.66			−0.573981			−0.286991	+	0.957933i	$0.592655\pi$
$272$		0			0
$273$	713.088	−	749.139i	0.158088	−	0.166080i
$274$		0			0
$275$	−144.683	+	52.6602i	−0.0317261	+	0.0115474i
$276$		0			0
$277$	−861.916	−	4888.17i	−0.186958	−	1.06029i	−0.923413	−	0.383807i	$-0.874613\pi$
$277$	−861.916	−	4888.17i	−0.186958	−	1.06029i	0.736455	−	0.676487i	$-0.236498\pi$
$278$		0			0
$279$	−2525.18	+	3329.78i	−0.541860	+	0.714512i
$280$		0			0
$281$	−7661.08	−	2788.41i	−1.62641	−	0.591966i	−0.641824	−	0.766852i	$-0.721822\pi$
$281$	−7661.08	−	2788.41i	−1.62641	−	0.591966i	−0.984589	+	0.174886i	$0.944044\pi$
$282$		0			0
$283$	−3237.19	−	2716.33i	−0.679969	−	0.570562i	0.236028	−	0.971746i	$-0.424154\pi$
$283$	−3237.19	−	2716.33i	−0.679969	−	0.570562i	−0.915997	+	0.401184i	$0.868599\pi$
$284$		0			0
$285$	4200.74	+	5695.18i	0.873090	+	1.18370i
$286$		0			0
$287$	1508.27	−	2612.40i	0.310210	−	0.537300i
$288$		0			0
$289$	1891.14	+	3275.54i	0.384925	+	0.666710i
$290$		0			0
$291$	295.911	+	196.511i	0.0596104	+	0.0395866i
$292$		0			0
$293$	1152.32	−	6535.14i	0.229759	−	1.30303i	−0.623616	−	0.781731i	$-0.714337\pi$
$293$	1152.32	−	6535.14i	0.229759	−	1.30303i	0.853375	−	0.521297i	$-0.174552\pi$
$294$		0			0
$295$	−7979.71	+	6695.77i	−1.57490	+	1.32150i
$296$		0			0
$297$	−656.645	−	1173.15i	−0.128291	−	0.229202i
$298$		0			0
$299$	1049.67	−	880.780i	0.203024	−	0.170357i
$300$		0			0
$301$	−1497.81	+	8494.51i	−0.286819	+	1.62663i
$302$		0			0
$303$	208.595	−	3326.95i	0.0395494	−	0.630786i
$304$		0			0
$305$	4316.96	+	7477.19i	0.810453	+	1.40375i
$306$		0			0
$307$	1427.01	−	2471.65i	0.265289	−	0.459494i	−0.702350	−	0.711831i	$-0.747866\pi$
$307$	1427.01	−	2471.65i	0.265289	−	0.459494i	0.967639	+	0.252337i	$0.0811993\pi$
$308$		0			0
$309$	−8051.00	+	904.819i	−1.48222	+	0.166580i
$310$		0			0
$311$	85.2222	+	71.5099i	0.0155386	+	0.0130384i	0.650524	−	0.759486i	$-0.274549\pi$
$311$	85.2222	+	71.5099i	0.0155386	+	0.0130384i	−0.634985	+	0.772524i	$0.718994\pi$
$312$		0			0
$313$	−1134.60	−	412.959i	−0.204892	−	0.0745745i	0.237536	−	0.971379i	$-0.423660\pi$
$313$	−1134.60	−	412.959i	−0.204892	−	0.0745745i	−0.442427	+	0.896804i	$0.645883\pi$
$314$		0			0
$315$	−389.630	−	7896.89i	−0.0696926	−	1.41251i
$316$		0			0
$317$	−897.165	−	5088.08i	−0.158958	−	0.901498i	−0.955077	−	0.296357i	$-0.904228\pi$
$317$	−897.165	−	5088.08i	−0.158958	−	0.901498i	0.796119	−	0.605140i	$-0.206883\pi$
$318$		0			0
$319$	15.6370	−	5.69141i	0.00274453	−	0.000998928i
$320$		0			0
$321$	−429.196	−	103.736i	−0.0746273	−	0.0180372i
$322$		0			0
$323$	−3855.86			−0.664229
$324$		0			0
$325$	129.713			0.0221391
$326$		0			0
$327$	11108.3	+	2684.84i	1.87856	+	0.454043i
$328$		0			0
$329$	−5351.14	+	1947.66i	−0.896712	+	0.326376i
$330$		0			0
$331$	1643.91	+	9323.07i	0.272983	+	1.54816i	0.745297	+	0.666733i	$0.232308\pi$
$331$	1643.91	+	9323.07i	0.272983	+	1.54816i	−0.472314	+	0.881430i	$0.656581\pi$
$332$		0			0
$333$	−6491.19	+	4187.24i	−1.06821	+	0.689067i
$334$		0			0
$335$	−3063.52	−	1115.03i	−0.499636	−	0.181853i
$336$		0			0
$337$	5945.60	+	4988.95i	0.961061	+	0.806426i	0.981125	−	0.193373i	$-0.0619427\pi$
$337$	5945.60	+	4988.95i	0.961061	+	0.806426i	−0.0200644	+	0.999799i	$0.506387\pi$
$338$		0			0
$339$	1013.05	−	113.853i	0.162305	−	0.0182408i
$340$		0			0
$341$	741.595	−	1284.48i	0.117770	−	0.203984i
$342$		0			0
$343$	−963.091	−	1668.12i	−0.151609	−	0.262595i
$344$		0			0
$345$	655.477	−	10454.4i	0.102289	−	1.63144i
$346$		0			0
$347$	1749.24	−	9920.45i	0.270618	−	1.53475i	−0.481930	−	0.876210i	$-0.660064\pi$
$347$	1749.24	−	9920.45i	0.270618	−	1.53475i	0.752547	−	0.658538i	$-0.228825\pi$
$348$		0			0
$349$	2871.58	−	2409.54i	0.440436	−	0.369570i	−0.395437	−	0.918493i	$-0.629407\pi$
$349$	2871.58	−	2409.54i	0.440436	−	0.369570i	0.835872	+	0.548924i	$0.184962\pi$
$350$		0			0
$351$	181.808	+	1117.94i	0.0276473	+	0.170004i
$352$		0			0
$353$	−6741.74	+	5656.99i	−1.01651	+	0.852950i	−0.989184	−	0.146677i	$-0.953142\pi$
$353$	−6741.74	+	5656.99i	−1.01651	+	0.852950i	−0.0273217	+	0.999627i	$0.508698\pi$
$354$		0			0
$355$	1618.73	−	9180.26i	0.242009	−	1.37250i
$356$		0			0
$357$	3588.67	+	2383.20i	0.532024	+	0.353311i
$358$		0			0
$359$	−569.401	−	986.232i	−0.0837099	−	0.144990i	0.821131	−	0.570740i	$-0.193344\pi$
$359$	−569.401	−	986.232i	−0.0837099	−	0.144990i	−0.904841	+	0.425750i	$0.860010\pi$
$360$		0			0
$361$	−3144.90	+	5447.12i	−0.458507	+	0.794157i
$362$		0			0
$363$	−3822.10	−	5181.83i	−0.552639	−	0.749244i
$364$		0			0
$365$	3207.26	+	2691.21i	0.459934	+	0.385930i
$366$		0			0
$367$	−6133.18	−	2232.29i	−0.872341	−	0.317506i	−0.133226	−	0.991086i	$-0.542534\pi$
$367$	−6133.18	−	2232.29i	−0.872341	−	0.317506i	−0.739115	+	0.673579i	$0.764756\pi$
$368$		0			0
$369$	1281.44	+	3044.76i	0.180784	+	0.429550i
$370$		0			0
$371$	2774.07	+	15732.5i	0.388200	+	2.20159i
$372$		0			0
$373$	11369.5	−	4138.17i	1.57826	−	0.574440i	0.603436	−	0.797411i	$-0.293798\pi$
$373$	11369.5	−	4138.17i	1.57826	−	0.574440i	0.974825	+	0.222972i	$0.0715757\pi$
$374$		0			0
$375$	−4635.16	+	4869.50i	−0.638290	+	0.670559i
$376$		0			0
$377$	−14.0192			−0.00191518
$378$		0			0
$379$	−13220.9			−1.79185			−0.895924	−	0.444208i	$-0.853485\pi$
$379$	−13220.9			−1.79185			−0.895924	+	0.444208i	$0.853485\pi$
$380$		0			0
$381$	2049.20	+	6957.05i	0.275548	+	0.935487i
$382$		0			0
$383$	−3370.04	+	1226.60i	−0.449611	+	0.163645i	−0.556894	−	0.830583i	$-0.688007\pi$
$383$	−3370.04	+	1226.60i	−0.449611	+	0.163645i	0.107283	+	0.994229i	$0.465785\pi$
$384$		0			0
$385$	487.280	+	2763.50i	0.0645042	+	0.365821i
$386$		0			0
$387$	−6420.92	−	6927.97i	−0.843394	−	0.909997i
$388$		0			0
$389$	4592.47	+	1671.52i	0.598580	+	0.217865i	0.623499	−	0.781824i	$-0.285711\pi$
$389$	4592.47	+	1671.52i	0.598580	+	0.217865i	−0.0249191	+	0.999689i	$0.507933\pi$
$390$		0			0
$391$	4372.10	+	3668.63i	0.565490	+	0.474502i
$392$		0			0
$393$	−5221.33	+	11958.2i	−0.670182	+	1.53489i
$394$		0			0
$395$	6364.38	−	11023.4i	0.810701	−	1.40417i
$396$		0			0
$397$	−7094.60	−	12288.2i	−0.896896	−	1.55347i	−0.831440	−	0.555614i	$-0.812483\pi$
$397$	−7094.60	−	12288.2i	−0.896896	−	1.55347i	−0.0654562	−	0.997855i	$-0.520850\pi$
$398$		0			0
$399$	13156.9	−	6534.68i	1.65080	−	0.819907i
$400$		0			0
$401$	617.602	−	3502.60i	0.0769117	−	0.436188i	−0.921899	−	0.387430i	$-0.873363\pi$
$401$	617.602	−	3502.60i	0.0769117	−	0.436188i	0.998811	−	0.0487575i	$-0.0155261\pi$
$402$		0			0
$403$	−957.205	+	803.190i	−0.118317	+	0.0992798i
$404$		0			0
$405$	7148.42	+	4885.59i	0.877057	+	0.599425i
$406$		0			0
$407$	2100.15	−	1762.24i	0.255776	−	0.214621i
$408$		0			0
$409$	2004.03	−	11365.4i	0.242281	−	1.37404i	−0.584444	−	0.811434i	$-0.698687\pi$
$409$	2004.03	−	11365.4i	0.242281	−	1.37404i	0.826724	−	0.562607i	$-0.190202\pi$
$410$		0			0
$411$	−1948.11	+	967.571i	−0.233803	+	0.116123i
$412$		0			0
$413$	10811.8	+	18726.5i	1.28817	+	2.23117i
$414$		0			0
$415$	7852.28	−	13600.5i	0.928803	−	1.60873i
$416$		0			0
$417$	1540.15	−	3527.33i	0.180867	−	0.414230i
$418$		0			0
$419$	−1394.94	−	1170.50i	−0.162643	−	0.136474i	0.557834	−	0.829953i	$-0.311633\pi$
$419$	−1394.94	−	1170.50i	−0.162643	−	0.136474i	−0.720477	+	0.693479i	$0.756077\pi$
$420$		0			0
$421$	683.841	+	248.898i	0.0791647	+	0.0288136i	0.381299	−	0.924452i	$-0.375477\pi$
$421$	683.841	+	248.898i	0.0791647	+	0.0288136i	−0.302134	+	0.953265i	$0.597699\pi$
$422$		0			0
$423$	1841.52	−	5958.07i	0.211673	−	0.684849i
$424$		0			0
$425$	93.8191	+	532.074i	0.0107080	+	0.0607280i
$426$		0			0
$427$	16841.8	−	6129.90i	1.90873	−	0.694723i
$428$		0			0
$429$	−113.581	−	385.608i	−0.0127826	−	0.0433971i
$430$		0			0
$431$	−10589.8			−1.18350			−0.591752	−	0.806120i	$-0.701564\pi$
$431$	−10589.8			−1.18350			−0.591752	+	0.806120i	$0.701564\pi$
$432$		0			0
$433$	8248.97			0.915520			0.457760	−	0.889076i	$-0.348652\pi$
$433$	8248.97			0.915520			0.457760	+	0.889076i	$0.348652\pi$
$434$		0			0
$435$	−73.8901	+	77.6257i	−0.00814428	+	0.00855602i
$436$		0			0
$437$	18288.8	−	6656.59i	2.00200	−	0.728668i
$438$		0			0
$439$	755.154	+	4282.69i	0.0820992	+	0.465608i	0.997945	+	0.0640773i	$0.0204104\pi$
$439$	755.154	+	4282.69i	0.0820992	+	0.465608i	−0.915846	+	0.401530i	$0.868478\pi$
$440$		0			0
$441$	−7095.62	−	893.281i	−0.766183	−	0.0964563i
$442$		0			0
$443$	9558.38	+	3478.97i	1.02513	+	0.373117i	0.799224	−	0.601033i	$-0.205244\pi$
$443$	9558.38	+	3478.97i	1.02513	+	0.373117i	0.225905	+	0.974149i	$0.427466\pi$
$444$		0			0
$445$	287.597	+	241.322i	0.0306368	+	0.0257074i
$446$		0			0
$447$	666.867	+	904.108i	0.0705631	+	0.0956663i
$448$		0			0
$449$	3202.24	−	5546.45i	0.336577	−	0.582969i	−0.647209	−	0.762312i	$-0.724064\pi$
$449$	3202.24	−	5546.45i	0.336577	−	0.582969i	0.983787	+	0.179343i	$0.0573973\pi$
$450$		0			0
$451$	−586.220	−	1015.36i	−0.0612062	−	0.106012i
$452$		0			0
$453$	−6900.39	−	4582.47i	−0.715692	−	0.475283i
$454$		0			0
$455$	410.519	−	2328.17i	0.0422976	−	0.239882i
$456$		0			0
$457$	−1976.70	+	1658.65i	−0.202333	+	0.169778i	−0.738324	−	0.674446i	$-0.764383\pi$
$457$	−1976.70	+	1658.65i	−0.202333	+	0.169778i	0.535991	+	0.844224i	$0.319938\pi$
$458$		0			0
$459$	−4454.22	+	1554.35i	−0.452953	+	0.158063i
$460$		0			0
$461$	−8257.79	+	6929.11i	−0.834282	+	0.700045i	−0.956270	−	0.292486i	$-0.905517\pi$
$461$	−8257.79	+	6929.11i	−0.834282	+	0.700045i	0.121988	+	0.992532i	$0.461073\pi$
$462$		0			0
$463$	−169.398	+	960.704i	−0.0170034	+	0.0964313i	−0.992128	−	0.125224i	$-0.960035\pi$
$463$	−169.398	+	960.704i	−0.0170034	+	0.0964313i	0.975125	+	0.221656i	$0.0711461\pi$
$464$		0			0
$465$	−597.735	+	9533.48i	−0.0596114	+	0.950762i
$466$		0			0
$467$	8841.29	+	15313.6i	0.876073	+	1.51740i	0.855616	+	0.517612i	$0.173179\pi$
$467$	8841.29	+	15313.6i	0.876073	+	1.51740i	0.0204571	+	0.999791i	$0.493488\pi$
$468$		0			0
$469$	−3383.76	+	5860.84i	−0.333150	+	0.577033i
$470$		0			0
$471$	−1872.48	+	210.441i	−0.183184	+	0.0205873i
$472$		0			0
$473$	2568.16	+	2154.94i	0.249649	+	0.209481i
$474$		0			0
$475$	1731.29	+	630.140i	0.167236	+	0.0608690i
$476$		0			0
$477$	−15563.4	−	7990.03i	−1.49392	−	0.766957i
$478$		0			0
$479$	616.357	+	3495.54i	0.0587935	+	0.333434i	0.999990	−	0.00442458i	$-0.00140839\pi$
$479$	616.357	+	3495.54i	0.0587935	+	0.333434i	−0.941197	+	0.337859i	$0.890297\pi$
$480$		0			0
$481$	−2170.39	+	789.957i	−0.205740	+	0.0748834i
$482$		0			0
$483$	−21135.7	−	5108.46i	−1.99112	−	0.481248i
$484$		0			0
$485$	811.945			0.0760176
$486$		0			0
$487$	−2435.85			−0.226651			−0.113325	−	0.993558i	$-0.536150\pi$
$487$	−2435.85			−0.226651			−0.113325	+	0.993558i	$0.536150\pi$
$488$		0			0
$489$	−8260.58	−	1996.56i	−0.763919	−	0.184637i
$490$		0			0
$491$	−16681.1	+	6071.43i	−1.53321	+	0.558044i	−0.964406	−	0.264427i	$-0.914817\pi$
$491$	−16681.1	+	6071.43i	−1.53321	+	0.558044i	−0.568808	+	0.822471i	$0.692595\pi$
$492$		0			0
$493$	−10.1398	−	57.5056i	−0.000926315	−	0.00525339i
$494$		0			0
$495$	−2733.80	−	1403.49i	−0.248233	−	0.127439i
$496$		0			0
$497$	−18183.7	−	6618.34i	−1.64115	−	0.597330i
$498$		0			0
$499$	7703.84	+	6464.29i	0.691125	+	0.579923i	0.919233	−	0.393713i	$-0.128810\pi$
$499$	7703.84	+	6464.29i	0.691125	+	0.579923i	−0.228108	+	0.973636i	$0.573254\pi$
$500$		0			0
$501$	8702.39	−	978.026i	0.776036	−	0.0872156i
$502$		0			0
$503$	5162.70	−	8942.06i	0.457641	−	0.792658i	−0.541195	−	0.840897i	$-0.682028\pi$
$503$	5162.70	−	8942.06i	0.457641	−	0.792658i	0.998836	+	0.0482397i	$0.0153611\pi$
$504$		0			0
$505$	−3809.78	−	6598.73i	−0.335709	−	0.581464i
$506$		0			0
$507$	693.169	−	11055.6i	0.0607193	−	0.968433i
$508$		0			0
$509$	1494.33	−	8474.74i	0.130127	−	0.737989i	−0.848002	−	0.529993i	$-0.822195\pi$
$509$	1494.33	−	8474.74i	0.130127	−	0.737989i	0.978130	−	0.207996i	$-0.0666942\pi$
$510$		0			0
$511$	6657.76	−	5586.53i	0.576364	−	0.483627i
$512$		0			0
$513$	−3004.29	+	15804.5i	−0.258563	+	1.36021i
$514$		0			0
$515$	−14186.0	+	11903.5i	−1.21381	+	1.01850i
$516$		0			0
$517$	−384.337	+	2179.68i	−0.0326946	+	0.185421i
$518$		0			0
$519$	14159.4	+	9403.07i	1.19755	+	0.795277i
$520$		0			0
$521$	11243.8	+	19474.8i	0.945486	+	1.63763i	0.754774	+	0.655984i	$0.227746\pi$
$521$	11243.8	+	19474.8i	0.945486	+	1.63763i	0.190712	+	0.981646i	$0.438920\pi$
$522$		0			0
$523$	2521.73	−	4367.76i	0.210837	−	0.365180i	−0.741140	−	0.671351i	$-0.765715\pi$
$523$	2521.73	−	4367.76i	0.210837	−	0.365180i	0.951977	+	0.306171i	$0.0990479\pi$
$524$		0			0
$525$	−1221.85	−	1656.54i	−0.101573	−	0.137709i
$526$		0			0
$527$	−3986.95	−	3345.45i	−0.329553	−	0.276528i
$528$		0			0
$529$	−15637.5	−	5691.58i	−1.28524	−	0.467789i
$530$		0			0
$531$	−23494.7	−	2957.79i	−1.92011	−	0.241727i
$532$		0			0
$533$	171.520	+	972.739i	0.0139388	+	0.0790506i
$534$		0			0
$535$	−948.421	+	345.197i	−0.0766426	+	0.0278956i
$536$		0			0
$537$	9884.76	−	10384.5i	0.794337	−	0.834496i
$538$		0			0
$539$	2538.22			0.202837
$540$		0			0
$541$	−8015.37			−0.636983			−0.318491	−	0.947926i	$-0.603176\pi$
$541$	−8015.37			−0.636983			−0.318491	+	0.947926i	$0.603176\pi$
$542$		0			0
$543$	3406.51	+	11565.1i	0.269222	+	0.914011i
$544$		0			0
$545$	24546.6	−	8934.23i	1.92929	−	0.702203i
$546$		0			0
$547$	2843.34	+	16125.4i	0.222253	+	1.26046i	0.867867	+	0.496797i	$0.165491\pi$
$547$	2843.34	+	16125.4i	0.222253	+	1.26046i	−0.645614	+	0.763664i	$0.723398\pi$
$548$		0			0
$549$	−5795.85	+	18751.9i	−0.450566	+	1.45776i
$550$		0			0
$551$	−187.115	−	68.1043i	−0.0144671	−	0.00526559i
$552$		0			0
$553$	−20241.1	−	16984.3i	−1.55649	−	1.30605i
$554$		0			0
$555$	−7065.27	+	16181.2i	−0.540367	+	1.23758i
$556$		0			0
$557$	−4603.16	+	7972.91i	−0.350166	+	0.606504i	−0.986278	−	0.165092i	$-0.947208\pi$
$557$	−4603.16	+	7972.91i	−0.350166	+	0.606504i	0.636113	+	0.771596i	$0.280541\pi$
$558$		0			0
$559$	−1412.19	−	2445.98i	−0.106850	−	0.185070i
$560$		0			0
$561$	1499.58	−	744.803i	0.112857	−	0.0560527i
$562$		0			0
$563$	−4038.40	+	22902.9i	−0.302306	+	1.71446i	0.333615	+	0.942709i	$0.391732\pi$
$563$	−4038.40	+	22902.9i	−0.302306	+	1.71446i	−0.635921	+	0.771754i	$0.719380\pi$
$564$		0			0
$565$	1785.01	−	1497.80i	0.132913	−	0.111528i
$566$		0			0
$567$	12564.4	−	12852.5i	0.930608	−	0.951944i
$568$		0			0
$569$	13470.6	−	11303.1i	0.992470	−	0.832781i	0.00654627	−	0.999979i	$-0.497916\pi$
$569$	13470.6	−	11303.1i	0.992470	−	0.832781i	0.985923	+	0.167198i	$0.0534718\pi$
$570$		0			0
$571$	398.268	−	2258.69i	0.0291891	−	0.165540i	−0.966729	−	0.255804i	$-0.917660\pi$
$571$	398.268	−	2258.69i	0.0291891	−	0.165540i	0.995918	+	0.0902640i	$0.0287711\pi$
$572$		0			0
$573$	21892.4	−	10873.4i	1.59611	−	0.792742i
$574$		0			0
$575$	−1363.54	−	2361.73i	−0.0988935	−	0.171288i
$576$		0			0
$577$	10473.0	−	18139.7i	0.755625	−	1.30878i	−0.189438	−	0.981893i	$-0.560667\pi$
$577$	10473.0	−	18139.7i	0.755625	−	1.30878i	0.945063	−	0.326888i	$-0.106000\pi$
$578$		0			0
$579$	−454.293	+	1040.45i	−0.0326075	+	0.0746795i
$580$		0			0
$581$	−24973.2	−	20955.0i	−1.78324	−	1.49632i
$582$		0			0
$583$	5834.63	+	2123.63i	0.414487	+	0.150861i
$584$		0			0
$585$	1759.84	+	1898.81i	0.124377	+	0.134199i
$586$		0			0
$587$	−2212.95	−	12550.2i	−0.155601	−	0.882459i	−0.958234	−	0.285986i	$-0.907679\pi$
$587$	−2212.95	−	12550.2i	−0.155601	−	0.882459i	0.802633	−	0.596474i	$-0.203432\pi$
$588$		0			0
$589$	−16677.7	+	6070.20i	−1.16671	+	0.424649i
$590$		0			0
$591$	333.891	+	1133.56i	0.0232393	+	0.0788978i
$592$		0			0
$593$	9461.47			0.655204			0.327602	−	0.944816i	$-0.393759\pi$
$593$	9461.47			0.655204			0.327602	+	0.944816i	$0.393759\pi$
$594$		0			0
$595$	9846.89			0.678459
$596$		0			0
$597$	−3595.69	+	3777.47i	−0.246502	+	0.258964i
$598$		0			0
$599$	−11410.1	+	4152.94i	−0.778305	+	0.283280i	−0.700466	−	0.713686i	$-0.747024\pi$
$599$	−11410.1	+	4152.94i	−0.778305	+	0.283280i	−0.0778394	+	0.996966i	$0.524802\pi$
$600$		0			0
$601$	−459.651	−	2606.81i	−0.0311973	−	0.176928i	0.965228	−	0.261410i	$-0.0841876\pi$
$601$	−459.651	−	2606.81i	−0.0311973	−	0.176928i	−0.996425	+	0.0844819i	$0.973076\pi$
$602$		0			0
$603$	−2874.88	−	6830.83i	−0.194153	−	0.461315i
$604$		0			0
$605$	−13830.3	−	5033.80i	−0.929389	−	0.338270i
$606$		0			0
$607$	13255.2	+	11122.4i	0.886345	+	0.743732i	0.967474	−	0.252972i	$-0.0814079\pi$
$607$	13255.2	+	11122.4i	0.886345	+	0.743732i	−0.0811282	+	0.996704i	$0.525852\pi$
$608$		0			0
$609$	132.056	+	179.035i	0.00878681	+	0.0119128i
$610$		0			0
$611$	932.323	−	1614.83i	0.0617312	−	0.106922i
$612$		0			0
$613$	1505.84	+	2608.19i	0.0992175	+	0.171850i	0.911361	−	0.411608i	$-0.135033\pi$
$613$	1505.84	+	2608.19i	0.0992175	+	0.171850i	−0.812143	+	0.583458i	$0.801699\pi$
$614$		0			0
$615$	6290.18	+	4177.23i	0.412430	+	0.273890i
$616$		0			0
$617$	−863.521	+	4897.27i	−0.0563437	+	0.319541i	−0.999933	−	0.0115772i	$-0.996315\pi$
$617$	−863.521	+	4897.27i	−0.0563437	+	0.319541i	0.943589	+	0.331118i	$0.107426\pi$
$618$		0			0
$619$	−2683.55	+	2251.77i	−0.174250	+	0.146213i	−0.725742	−	0.687967i	$-0.758503\pi$
$619$	−2683.55	+	2251.77i	−0.174250	+	0.146213i	0.551492	+	0.834181i	$0.314059\pi$
$620$		0			0
$621$	18443.5	−	15062.0i	1.19181	−	0.973299i
$622$		0			0
$623$	597.005	−	500.947i	0.0383925	−	0.0322151i
$624$		0			0
$625$	−3017.18	+	17111.3i	−0.193100	+	1.09512i
$626$		0			0
$627$	357.288	−	5698.51i	0.0227571	−	0.362961i
$628$		0			0
$629$	−4810.14	−	8331.40i	−0.304917	−	0.528132i
$630$		0			0
$631$	−15195.1	+	26318.6i	−0.958646	+	1.66042i	−0.232852	+	0.972512i	$0.574806\pi$
$631$	−15195.1	+	26318.6i	−0.958646	+	1.66042i	−0.725794	+	0.687912i	$0.758527\pi$
$632$		0			0
$633$	10337.4	−	1161.78i	0.649090	−	0.0729486i
$634$		0			0
$635$	12699.2	+	10655.9i	0.793628	+	0.665933i
$636$		0			0
$637$	−2009.41	−	731.367i	−0.124986	−	0.0454911i
$638$		0			0
$639$	17807.6	−	11487.1i	1.10244	−	0.711145i
$640$		0			0
$641$	367.411	+	2083.69i	0.0226394	+	0.128394i	0.994033	−	0.109082i	$-0.0347910\pi$
$641$	367.411	+	2083.69i	0.0226394	+	0.128394i	−0.971393	+	0.237476i	$0.923680\pi$
$642$		0			0
$643$	16121.2	−	5867.63i	0.988735	−	0.359870i	0.203505	−	0.979074i	$-0.434767\pi$
$643$	16121.2	−	5867.63i	0.988735	−	0.359870i	0.785230	+	0.619204i	$0.212544\pi$
$644$		0			0
$645$	−20986.8	−	5072.46i	−1.28117	−	0.309656i
$646$		0			0
$647$	−24282.8			−1.47551			−0.737756	−	0.675068i	$-0.764114\pi$
$647$	−24282.8			−1.47551			−0.737756	+	0.675068i	$0.764114\pi$
$648$		0			0
$649$	8404.43			0.508325
$650$		0			0
$651$	19273.9	+	4658.45i	1.16037	+	0.280459i
$652$		0			0
$653$	15070.7	−	5485.28i	0.903156	−	0.328722i	0.151639	−	0.988436i	$-0.451545\pi$
$653$	15070.7	−	5485.28i	0.903156	−	0.328722i	0.751517	+	0.659714i	$0.229323\pi$
$654$		0			0
$655$	5179.15	+	29372.4i	0.308956	+	1.75217i
$656$		0			0
$657$	469.029	+	9506.12i	0.0278517	+	0.564488i
$658$		0			0
$659$	12869.4	+	4684.06i	0.760727	+	0.276882i	0.693112	−	0.720829i	$-0.256239\pi$
$659$	12869.4	+	4684.06i	0.760727	+	0.276882i	0.0676147	+	0.997712i	$0.478461\pi$
$660$		0			0
$661$	2109.37	+	1769.97i	0.124123	+	0.104151i	0.702736	−	0.711451i	$-0.251962\pi$
$661$	2109.37	+	1769.97i	0.124123	+	0.104151i	−0.578613	+	0.815602i	$0.696406\pi$
$662$		0			0
$663$	−1401.77	+	157.540i	−0.0821122	+	0.00922826i
$664$		0			0
$665$	16789.3	−	29080.0i	0.979041	−	1.69575i
$666$		0			0
$667$	147.369	+	255.251i	0.00855497	+	0.0148176i
$668$		0			0
$669$	12.6595	−	201.910i	0.000731604	−	0.0116686i
$670$		0			0
$671$	1209.63	−	6860.16i	0.0695936	−	0.394685i
$672$		0			0
$673$	7306.81	−	6131.14i	0.418509	−	0.351171i	−0.409086	−	0.912496i	$-0.634153\pi$
$673$	7306.81	−	6131.14i	0.418509	−	0.351171i	0.827596	+	0.561325i	$0.189708\pi$
$674$		0			0
$675$	2253.98	+	30.0176i	0.128527	+	0.00171167i
$676$		0			0
$677$	8203.11	−	6883.23i	0.465689	−	0.390759i	−0.379530	−	0.925179i	$-0.623914\pi$
$677$	8203.11	−	6883.23i	0.465689	−	0.390759i	0.845219	+	0.534420i	$0.179470\pi$
$678$		0			0
$679$	292.679	−	1659.86i	0.0165419	−	0.0938140i
$680$		0			0
$681$	1908.43	+	1267.37i	0.107388	+	0.0713152i
$682$		0			0
$683$	−9802.23	−	16978.0i	−0.549153	−	0.951162i	−0.998333	−	0.0577201i	$-0.981617\pi$
$683$	−9802.23	−	16978.0i	−0.549153	−	0.951162i	0.449179	−	0.893442i	$-0.351716\pi$
$684$		0			0
$685$	−2485.95	+	4305.79i	−0.138662	+	0.240169i
$686$		0			0
$687$	1073.99	+	1456.06i	0.0596436	+	0.0808622i
$688$		0			0
$689$	−4007.15	−	3362.40i	−0.221568	−	0.185918i
$690$		0			0
$691$	5541.06	+	2016.78i	0.305053	+	0.111030i	0.490011	−	0.871716i	$-0.336993\pi$
$691$	5541.06	+	2016.78i	0.305053	+	0.111030i	−0.184958	+	0.982746i	$0.559215\pi$
$692$		0			0
$693$	−3854.61	+	5082.80i	−0.211291	+	0.278614i
$694$		0			0
$695$	−1527.70	−	8664.04i	−0.0833800	−	0.472872i
$696$		0			0
$697$	−3866.04	+	1407.12i	−0.210096	+	0.0764686i
$698$		0			0
$699$	19061.6	−	20025.3i	1.03144	−	1.08358i
$700$		0			0
$701$	−26136.5			−1.40822			−0.704109	−	0.710092i	$-0.748654\pi$
$701$	−26136.5			−1.40822			−0.704109	+	0.710092i	$0.748654\pi$
$702$		0			0
$703$	−32805.9			−1.76002
$704$		0			0
$705$	−4027.55	−	13673.6i	−0.215158	−	0.730463i
$706$		0			0
$707$	−14863.1	+	5409.73i	−0.790643	+	0.287770i
$708$		0			0
$709$	−3375.05	−	19140.9i	−0.178777	−	1.01389i	−0.933694	−	0.358073i	$-0.883434\pi$
$709$	−3375.05	−	19140.9i	−0.178777	−	1.01389i	0.754917	−	0.655820i	$-0.227677\pi$
$710$		0			0
$711$	28214.0	−	6422.85i	1.48820	−	0.338784i
$712$		0			0
$713$	24686.0	+	8984.98i	1.29663	+	0.471936i
$714$		0			0
$715$	−703.879	−	590.625i	−0.0368162	−	0.0308925i
$716$		0			0
$717$	−3195.81	+	7319.22i	−0.166457	+	0.381229i
$718$		0			0
$719$	11662.7	−	20200.4i	0.604930	−	1.04777i	−0.387133	−	0.922024i	$-0.626534\pi$
$719$	11662.7	−	20200.4i	0.604930	−	1.04777i	0.992063	−	0.125745i	$-0.0401322\pi$
$720$		0			0
$721$	19220.8	+	33291.3i	0.992813	+	1.71960i
$722$		0			0
$723$	−12509.5	+	6213.14i	−0.643478	+	0.319598i
$724$		0			0
$725$	−4.84498	+	27.4773i	−0.000248191	+	0.00140756i
$726$		0			0
$727$	−21572.4	+	18101.4i	−1.10052	+	0.923444i	−0.997460	−	0.0712312i	$-0.977307\pi$
$727$	−21572.4	+	18101.4i	−1.10052	+	0.923444i	−0.103058	+	0.994675i	$0.532863\pi$
$728$		0			0
$729$	2900.50	+	19468.1i	0.147361	+	0.989083i
$730$		0			0
$731$	9011.82	−	7561.81i	0.455970	−	0.382604i
$732$		0			0
$733$	2490.57	−	14124.7i	0.125500	−	0.711745i	−0.855510	−	0.517786i	$-0.826756\pi$
$733$	2490.57	−	14124.7i	0.125500	−	0.711745i	0.981010	−	0.193958i	$-0.0621327\pi$
$734$		0			0
$735$	−14640.6	+	7271.57i	−0.734728	+	0.364919i
$736$		0			0
$737$	1315.17	+	2277.93i	0.0657323	+	0.113852i
$738$		0			0
$739$	−1865.54	+	3231.22i	−0.0928622	+	0.160842i	−0.908714	−	0.417419i	$-0.862935\pi$
$739$	−1865.54	+	3231.22i	−0.0928622	+	0.160842i	0.815852	+	0.578260i	$0.196268\pi$
$740$		0			0
$741$	−1924.83	+	4408.35i	−0.0954256	+	0.218549i
$742$		0			0
$743$	−3841.35	−	3223.28i	−0.189671	−	0.159153i	0.543007	−	0.839728i	$-0.317286\pi$
$743$	−3841.35	−	3223.28i	−0.189671	−	0.159153i	−0.732678	+	0.680575i	$0.761730\pi$
$744$		0			0
$745$	2413.06	+	878.282i	0.118668	+	0.0431916i
$746$		0			0
$747$	34810.1	−	7924.42i	1.70500	−	0.388138i
$748$		0			0
$749$	363.814	+	2063.29i	0.0177483	+	0.100656i
$750$		0			0
$751$	−11741.0	+	4273.37i	−0.570485	+	0.207640i	−0.611125	−	0.791534i	$-0.709283\pi$
$751$	−11741.0	+	4273.37i	−0.570485	+	0.207640i	0.0406395	+	0.999174i	$0.487060\pi$
$752$		0			0
$753$	4577.85	+	15541.9i	0.221549	+	0.752160i
$754$		0			0
$755$	−18933.8			−0.912679
$756$		0			0
$757$	5318.20			0.255341			0.127671	−	0.991817i	$-0.459250\pi$
$757$	5318.20			0.255341			0.127671	+	0.991817i	$0.459250\pi$
$758$		0			0
$759$	−5826.92	+	6121.50i	−0.278661	+	0.292749i
$760$		0			0
$761$	−263.705	+	95.9809i	−0.0125615	+	0.00457202i	−0.348293	−	0.937386i	$-0.613239\pi$
$761$	−263.705	+	95.9809i	−0.0125615	+	0.00457202i	0.335732	+	0.941958i	$0.391016\pi$
$762$		0			0
$763$	−9416.07	−	53401.2i	−0.446769	−	2.53375i
$764$		0			0
$765$	−6515.94	+	8592.11i	−0.307953	+	0.406076i
$766$		0			0
$767$	−6653.47	−	2421.67i	−0.313224	−	0.114004i
$768$		0			0
$769$	−4855.57	−	4074.30i	−0.227693	−	0.191057i	0.521803	−	0.853066i	$-0.325260\pi$
$769$	−4855.57	−	4074.30i	−0.227693	−	0.191057i	−0.749496	+	0.662009i	$0.769704\pi$
$770$		0			0
$771$	−11047.9	−	14978.3i	−0.516058	−	0.699649i
$772$		0			0
$773$	−16590.1	+	28734.9i	−0.771934	+	1.33703i	0.164568	+	0.986366i	$0.447377\pi$
$773$	−16590.1	+	28734.9i	−0.771934	+	1.33703i	−0.936502	+	0.350663i	$0.885956\pi$
$774$		0			0
$775$	1243.43	+	2153.68i	0.0576325	+	0.0998225i
$776$		0			0
$777$	30532.6	+	20276.3i	1.40972	+	0.936177i
$778$		0			0
$779$	−2436.21	+	13816.4i	−0.112049	+	0.635463i
$780$		0			0
$781$	−5761.46	+	4834.44i	−0.263971	+	0.221498i
$782$		0			0
$783$	−243.606	−	3.24425i	−0.0111185	−	0.000148072i
$784$		0			0
$785$	−3299.35	+	2768.49i	−0.150011	+	0.125875i
$786$		0			0
$787$	−3540.75	+	20080.6i	−0.160374	+	0.909524i	0.793333	+	0.608787i	$0.208344\pi$
$787$	−3540.75	+	20080.6i	−0.160374	+	0.909524i	−0.953707	+	0.300737i	$0.902767\pi$
$788$		0			0
$789$	1223.31	−	19511.0i	0.0551977	−	0.880367i
$790$		0			0
$791$	−2418.53	−	4189.02i	−0.108714	−	0.188299i
$792$		0			0
$793$	−2934.32	+	5082.39i	−0.131401	+	0.227593i
$794$		0			0
$795$	−39738.3	+	4466.02i	−1.77279	+	0.199237i
$796$		0			0
$797$	12799.7	+	10740.3i	0.568871	+	0.477339i	0.881271	−	0.472611i	$-0.156689\pi$
$797$	12799.7	+	10740.3i	0.568871	+	0.477339i	−0.312400	+	0.949951i	$0.601133\pi$
$798$		0			0
$799$	7298.24	+	2656.34i	0.323146	+	0.117615i
$800$		0			0
$801$	42.0581	+	852.418i	0.00185524	+	0.0376014i
$802$		0			0
$803$	−586.578	−	3326.65i	−0.0257782	−	0.146195i
$804$		0			0
$805$	−46705.0	+	16999.2i	−2.04489	+	0.744278i
$806$		0			0
$807$	−5565.30	−	1345.12i	−0.242760	−	0.0586746i
$808$		0			0
$809$	−35383.0			−1.53770			−0.768851	−	0.639428i	$-0.779171\pi$
$809$	−35383.0			−1.53770			−0.768851	+	0.639428i	$0.779171\pi$
$810$		0			0
$811$	31405.6			1.35980			0.679901	−	0.733304i	$-0.262023\pi$
$811$	31405.6			1.35980			0.679901	+	0.733304i	$0.262023\pi$
$812$		0			0
$813$	−12933.2	−	3125.91i	−0.557916	−	0.134847i
$814$		0			0
$815$	−18253.9	+	6643.89i	−0.784549	+	0.285552i
$816$		0			0
$817$	−6966.15	−	39507.0i	−0.298304	−	1.69177i
$818$		0			0
$819$	4516.12	−	2913.19i	0.192681	−	0.124292i
$820$		0			0
$821$	−21704.2	−	7899.68i	−0.922633	−	0.335811i	−0.163347	−	0.986569i	$-0.552229\pi$
$821$	−21704.2	−	7899.68i	−0.922633	−	0.335811i	−0.759285	+	0.650758i	$0.774451\pi$
$822$		0			0
$823$	−10126.1	−	8496.77i	−0.428885	−	0.359877i	0.402646	−	0.915356i	$-0.368091\pi$
$823$	−10126.1	−	8496.77i	−0.428885	−	0.359877i	−0.831531	+	0.555479i	$0.812535\pi$
$824$		0			0
$825$	−795.036	+	89.3509i	−0.0335510	+	0.00377067i
$826$		0			0
$827$	−20082.7	+	34784.3i	−0.844432	+	1.46260i	0.0416807	+	0.999131i	$0.486729\pi$
$827$	−20082.7	+	34784.3i	−0.844432	+	1.46260i	−0.886113	+	0.463469i	$0.846605\pi$
$828$		0			0
$829$	9303.77	+	16114.6i	0.389787	+	0.675130i	0.992421	−	0.122887i	$-0.0392154\pi$
$829$	9303.77	+	16114.6i	0.389787	+	0.675130i	−0.602634	+	0.798018i	$0.705882\pi$
$830$		0			0
$831$	1613.92	−	25740.9i	0.0673722	−	1.07454i
$832$		0			0
$833$	1546.64	−	8771.45i	0.0643313	−	0.364841i
$834$		0			0
$835$	15333.8	−	12866.6i	0.635506	−	0.533253i
$836$		0			0
$837$	−16818.8	+	13735.2i	−0.694556	+	0.567213i
$838$		0			0
$839$	5456.36	−	4578.43i	0.224523	−	0.188397i	−0.523586	−	0.851973i	$-0.675406\pi$
$839$	5456.36	−	4578.43i	0.224523	−	0.188397i	0.748109	+	0.663576i	$0.230962\pi$
$840$		0			0
$841$	−4234.58	+	24015.5i	−0.173627	+	0.984686i
$842$		0			0
$843$	−35290.0	−	23435.7i	−1.44182	−	0.957495i
$844$		0			0
$845$	−12660.0	−	21927.8i	−0.515406	−	0.892709i
$846$		0			0
$847$	−15276.0	+	26458.7i	−0.619703	+	1.07336i
$848$		0			0
$849$	−13034.2	−	17671.2i	−0.526894	−	0.714340i
$850$		0			0
$851$	37198.0	+	31212.8i	1.49839	+	1.25730i
$852$		0			0
$853$	5664.58	+	2061.74i	0.227376	+	0.0827581i	0.453196	−	0.891411i	$-0.350284\pi$
$853$	5664.58	+	2061.74i	0.227376	+	0.0827581i	−0.225820	+	0.974169i	$0.572506\pi$
$854$		0			0
$855$	14264.4	+	33892.8i	0.570564	+	1.35568i
$856$		0			0
$857$	2996.65	+	16994.8i	0.119444	+	0.677400i	0.984454	+	0.175645i	$0.0562011\pi$
$857$	2996.65	+	16994.8i	0.119444	+	0.677400i	−0.865010	+	0.501755i	$0.832688\pi$
$858$		0			0
$859$	18113.2	−	6592.66i	0.719458	−	0.261861i	0.0437621	−	0.999042i	$-0.486066\pi$
$859$	18113.2	−	6592.66i	0.719458	−	0.261861i	0.675695	+	0.737181i	$0.263843\pi$
$860$		0			0
$861$	10806.9	−	11353.3i	0.427757	−	0.449383i
$862$		0			0
$863$	38434.6			1.51602			0.758011	−	0.652241i	$-0.226171\pi$
$863$	38434.6			1.51602			0.758011	+	0.652241i	$0.226171\pi$
$864$		0			0
$865$	38851.6			1.52716
$866$		0			0
$867$	5552.99	+	18852.5i	0.217520	+	0.738481i
$868$		0			0
$869$	−9650.44	+	3512.47i	−0.376719	+	0.137115i
$870$		0			0
$871$	−384.800	−	2182.31i	−0.0149695	−	0.0848963i
$872$		0			0
$873$	1254.67	+	1353.76i	0.0486418	+	0.0524830i
$874$		0			0
$875$	29975.4	+	10910.1i	1.15812	+	0.421520i
$876$		0			0
$877$	−7769.12	−	6519.07i	−0.299139	−	0.251007i	0.480847	−	0.876805i	$-0.340329\pi$
$877$	−7769.12	−	6519.07i	−0.299139	−	0.251007i	−0.779986	+	0.625797i	$0.784774\pi$
$878$		0			0
$879$	13797.8	−	31600.5i	0.529452	−	1.21258i
$880$		0			0
$881$	12949.0	−	22428.4i	0.495192	−	0.857698i	−0.504793	−	0.863241i	$-0.668431\pi$
$881$	12949.0	−	22428.4i	0.495192	−	0.857698i	0.999985	+	0.00554305i	$0.00176442\pi$
$882$		0			0
$883$	−11843.7	−	20513.9i	−0.451385	−	0.781822i	0.547087	−	0.837075i	$-0.315737\pi$
$883$	−11843.7	−	20513.9i	−0.451385	−	0.781822i	−0.998472	+	0.0552538i	$0.982403\pi$
$884$		0			0
$885$	−48477.1	+	24077.2i	−1.84129	+	0.914517i
$886$		0			0
$887$	1105.81	−	6271.37i	0.0418596	−	0.237398i	−0.956698	−	0.291081i	$-0.905985\pi$
$887$	1105.81	−	6271.37i	0.0418596	−	0.237398i	0.998558	+	0.0536833i	$0.0170961\pi$
$888$		0			0
$889$	26361.6	−	22120.0i	0.994532	−	0.834511i
$890$		0			0
$891$	−1884.41	−	6726.84i	−0.0708532	−	0.252927i
$892$		0			0
$893$	20288.5	−	17024.1i	0.760280	−	0.637951i
$894$		0			0
$895$	5690.58	−	32272.9i	0.212531	−	1.20532i
$896$		0			0
$897$	6376.81	−	3167.19i	0.237364	−	0.117892i
$898$		0			0
$899$	−134.387	−	232.766i	−0.00498561	−	0.00863534i
$900$		0			0
$901$	10894.0	−	18869.0i	0.402811	−	0.697688i
$902$		0			0
$903$	−17934.7	+	41074.9i	−0.660939	+	1.51372i
$904$		0			0
$905$	21110.7	+	17714.0i	0.775408	+	0.650644i
$906$		0			0
$907$	17551.1	+	6388.10i	0.642532	+	0.233862i	0.642677	−	0.766138i	$-0.277824\pi$
$907$	17551.1	+	6388.10i	0.642532	+	0.233862i	−0.000144890		1.00000i	$0.500046\pi$
$908$		0			0
$909$	5114.92	−	16548.9i	0.186635	−	0.603840i
$910$		0			0
$911$	−6259.33	−	35498.4i	−0.227641	−	1.29102i	−0.857572	−	0.514364i	$-0.828028\pi$
$911$	−6259.33	−	35498.4i	−0.227641	−	1.29102i	0.629931	−	0.776651i	$-0.283083\pi$
$912$		0			0
$913$	−11906.6	+	4333.64i	−0.431599	+	0.157089i
$914$		0			0
$915$	12676.0	+	43035.1i	0.457984	+	1.55486i
$916$		0			0
$917$	61913.0			2.22961
$918$		0			0
$919$	7124.80			0.255740			0.127870	−	0.991791i	$-0.459186\pi$
$919$	7124.80			0.255740			0.127870	+	0.991791i	$0.459186\pi$
$920$		0			0
$921$	10224.7	−	10741.6i	0.365814	−	0.384308i
$922$		0			0
$923$	5954.14	−	2167.13i	0.212332	−	0.0772827i
$924$		0			0
$925$	798.217	+	4526.92i	0.0283732	+	0.160913i
$926$		0			0
$927$	−41767.9	−	5258.24i	−1.47987	−	0.186304i
$928$		0			0
$929$	12267.6	+	4465.05i	0.433249	+	0.157690i	0.549433	−	0.835538i	$-0.314844\pi$
$929$	12267.6	+	4465.05i	0.433249	+	0.157690i	−0.116184	+	0.993228i	$0.537066\pi$
$930$		0			0
$931$	−23266.9	−	19523.2i	−0.819056	−	0.687269i
$932$		0			0
$933$	343.138	+	465.211i	0.0120406	+	0.0163240i
$934$		0			0
$935$	1913.60	−	3314.45i	0.0669319	−	0.115929i
$936$		0			0
$937$	16553.1	+	28670.9i	0.577126	+	0.999612i	0.995807	+	0.0914784i	$0.0291592\pi$
$937$	16553.1	+	28670.9i	0.577126	+	0.999612i	−0.418681	+	0.908133i	$0.637507\pi$
$938$		0			0
$939$	−5226.41	−	3470.79i	−0.181637	−	0.120623i
$940$		0			0
$941$	−6249.23	+	35441.1i	−0.216492	+	1.22779i	0.661807	+	0.749675i	$0.269790\pi$
$941$	−6249.23	+	35441.1i	−0.216492	+	1.22779i	−0.878299	+	0.478113i	$0.841321\pi$
$942$		0			0
$943$	15907.9	−	13348.3i	0.549345	−	0.460956i
$944$		0			0
$945$	7672.19	−	40360.6i	0.264102	−	1.38935i
$946$		0			0
$947$	8533.82	−	7160.73i	0.292832	−	0.245715i	−0.484521	−	0.874779i	$-0.661006\pi$
$947$	8533.82	−	7160.73i	0.292832	−	0.245715i	0.777353	+	0.629064i	$0.216562\pi$
$948$		0			0
$949$	−494.174	+	2802.60i	−0.0169037	+	0.0958655i
$950$		0			0
$951$	1679.92	−	26793.7i	0.0572821	−	0.913611i
$952$		0			0
$953$	10825.3	+	18750.0i	0.367961	+	0.637328i	0.989247	−	0.146257i	$-0.0467226\pi$
$953$	10825.3	+	18750.0i	0.367961	+	0.637328i	−0.621285	+	0.783584i	$0.713389\pi$
$954$		0			0
$955$	27936.6	−	48387.5i	0.946603	−	1.63956i
$956$		0			0
$957$	85.9260	−	9.65688i	0.00290240	−	0.000326189i
$958$		0			0
$959$	7906.24	+	6634.12i	0.266221	+	0.223386i
$960$		0			0
$961$	5483.00	+	1995.65i	0.184049	+	0.0669883i
$962$		0			0
$963$	−2041.11	−	1047.88i	−0.0683011	−	0.0350648i
$964$		0			0
$965$	450.622	+	2555.61i	0.0150322	+	0.0852517i
$966$		0			0
$967$	−24009.7	+	8738.82i	−0.798449	+	0.290612i	−0.708844	−	0.705366i	$-0.750783\pi$
$967$	−24009.7	+	8738.82i	−0.798449	+	0.290612i	−0.0896056	+	0.995977i	$0.528561\pi$
$968$		0			0
$969$	−19474.9	−	4707.04i	−0.645638	−	0.156049i
$970$		0			0
$971$	−16218.0			−0.536005			−0.268003	−	0.963418i	$-0.586364\pi$
$971$	−16218.0			−0.536005			−0.268003	+	0.963418i	$0.586364\pi$
$972$		0			0
$973$	−18262.6			−0.601719
$974$		0			0
$975$	655.146	+	158.347i	0.0215195	+	0.00520120i
$976$		0			0
$977$	−41442.9	+	15084.0i	−1.35709	+	0.493940i	−0.915153	−	0.403107i	$-0.867930\pi$
$977$	−41442.9	+	15084.0i	−1.35709	+	0.493940i	−0.441936	+	0.897047i	$0.645708\pi$
$978$		0			0
$979$	−52.5988	−	298.302i	−0.00171712	−	0.00973829i
$980$		0			0
$981$	52827.2	+	27120.7i	1.71931	+	0.882669i
$982$		0			0
$983$	47663.2	+	17348.0i	1.54651	+	0.562883i	0.967595	−	0.252506i	$-0.0812546\pi$
$983$	47663.2	+	17348.0i	1.54651	+	0.562883i	0.578914	+	0.815389i	$0.303477\pi$
$984$		0			0
$985$	2069.18	+	1736.25i	0.0669335	+	0.0561639i
$986$		0			0
$987$	−29404.7	+	3304.68i	−0.948291	+	0.106575i
$988$		0			0
$989$	−29689.8	+	51424.2i	−0.954580	+	1.65338i
$990$		0			0
$991$	−8577.07	−	14855.9i	−0.274934	−	0.476199i	0.695185	−	0.718831i	$-0.255323\pi$
$991$	−8577.07	−	14855.9i	−0.274934	−	0.476199i	−0.970118	+	0.242632i	$0.921989\pi$
$992$		0			0
$993$	−3078.19	+	49095.0i	−0.0983719	+	1.56897i
$994$		0			0
$995$	−2070.01	+	11739.6i	−0.0659535	+	0.374041i
$996$		0			0
$997$	−5448.01	+	4571.43i	−0.173060	+	0.145214i	−0.725203	−	0.688535i	$-0.758254\pi$
$997$	−5448.01	+	4571.43i	−0.173060	+	0.145214i	0.552144	+	0.833749i	$0.313810\pi$
$998$		0			0
$999$	−37896.7	+	13224.5i	−1.20020	+	0.418823i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.i.a.49.8	✓	54
3.2	odd	2		324.4.i.a.37.3		54
27.11	odd	18		324.4.i.a.289.3		54
27.16	even	9	inner	108.4.i.a.97.8	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.8	✓	54	1.1	even	1	trivial
108.4.i.a.97.8	yes	54	27.16	even	9	inner
324.4.i.a.37.3		54	3.2	odd	2
324.4.i.a.289.3		54	27.11	odd	18

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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(5.05072 + 1.22075i) q^{3} +(11.1609 - 4.06223i) q^{5} +(-4.28132 - 24.2806i) q^{7} +(24.0196 + 12.3313i) q^{9} +O(q^{10})\)	\(q+(5.05072 + 1.22075i) q^{3} +(11.1609 - 4.06223i) q^{5} +(-4.28132 - 24.2806i) q^{7} +(24.0196 + 12.3313i) q^{9} +(-9.00480 - 3.27748i) q^{11} +(6.18439 + 5.18932i) q^{13} +(61.3295 - 6.89258i) q^{15} +(-16.8131 + 29.1212i) q^{17} +(57.3341 + 99.3056i) q^{19} +(8.01668 - 127.861i) q^{21} +(29.4732 - 167.151i) q^{23} +(12.3082 - 10.3278i) q^{25} +(106.263 + 91.6038i) q^{27} +(-1.33025 + 1.11621i) q^{29} +(-26.8769 + 152.426i) q^{31} +(-41.4798 - 27.5462i) q^{33} +(-146.417 - 253.601i) q^{35} +(-143.047 + 247.765i) q^{37} +(24.9008 + 33.7594i) q^{39} +(93.7251 + 78.6447i) q^{41} +(-328.750 - 119.655i) q^{43} +(318.172 + 40.0553i) q^{45} +(-40.1073 - 227.460i) q^{47} +(-248.901 + 90.5926i) q^{49} +(-120.468 + 126.559i) q^{51} -647.947 q^{53} -113.816 q^{55} +(168.352 + 571.555i) q^{57} +(-824.149 + 299.966i) q^{59} +(126.231 + 715.889i) q^{61} +(196.576 - 636.002i) q^{63} +(90.1035 + 32.7950i) q^{65} +(-210.269 - 176.437i) q^{67} +(352.910 - 808.253i) q^{69} +(392.428 - 679.706i) q^{71} +(176.253 + 305.280i) q^{73} +(74.7732 - 37.1378i) q^{75} +(-41.0266 + 232.674i) q^{77} +(820.969 - 688.875i) q^{79} +(424.878 + 592.385i) q^{81} +(1012.90 - 849.924i) q^{83} +(-69.3525 + 393.318i) q^{85} +(-8.08135 + 4.01378i) q^{87} +(15.8047 + 27.3746i) q^{89} +(99.5222 - 172.377i) q^{91} +(-321.821 + 737.052i) q^{93} +(1043.30 + 875.434i) q^{95} +(64.2391 + 23.3811i) q^{97} +(-175.876 - 189.765i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) 54 * q + 12 * q^5 - 48 * q^9	\( 54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100}) \) 54 * q + 12 * q^5 - 48 * q^9 - 87 * q^11 + 234 * q^15 + 204 * q^17 - 12 * q^21 + 96 * q^23 - 216 * q^25 + 27 * q^27 + 318 * q^29 - 54 * q^31 + 63 * q^33 + 6 * q^35 + 66 * q^39 + 867 * q^41 - 513 * q^43 - 306 * q^45 - 1548 * q^47 + 594 * q^49 - 1368 * q^51 - 1068 * q^53 - 1269 * q^57 - 1218 * q^59 - 54 * q^61 + 30 * q^63 + 96 * q^65 - 2997 * q^67 + 1476 * q^69 - 120 * q^71 - 216 * q^73 + 732 * q^75 + 3480 * q^77 + 2808 * q^79 + 3348 * q^81 + 4464 * q^83 + 2160 * q^85 + 4824 * q^87 + 4029 * q^89 + 270 * q^91 + 1164 * q^93 - 1650 * q^95 - 3483 * q^97 - 5076 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more