LMFDB - Embedded newform 135.4.e.c.91.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [135,4,Mod(46,135)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("135.46"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$7.96525785077$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + \cdots + 82944 $ x^14 - 2x^13 + 48x^12 - 60x^11 + 1605x^10 - 1800x^9 + 23232x^8 - 2346x^7 + 209529x^6 - 55412x^5 + 765088x^4 + 276096x^3 + 1572480x^2 - 345600*x + 82944
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{13} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			91.3
Root			$1.09722 - 1.90044i$ of defining polynomial
Character	$\chi$	$=$	135.91
Dual form			135.4.e.c.46.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.09722 - 1.90044i) q^{2} +(1.59221 - 2.75778i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-1.38302 - 2.39547i) q^{7} -24.5436 q^{8} +10.9722 q^{10} +(-26.3295 - 45.6040i) q^{11} +(-10.2267 + 17.7132i) q^{13} +(-3.03497 + 5.25672i) q^{14} +(14.1921 + 24.5814i) q^{16} -3.66084 q^{17} -95.6705 q^{19} +(7.96104 + 13.7889i) q^{20} +(-57.7786 + 100.075i) q^{22} +(-44.9206 + 77.8047i) q^{23} +(-12.5000 - 21.6506i) q^{25} +44.8840 q^{26} -8.80825 q^{28} +(-113.890 - 197.264i) q^{29} +(-139.569 + 241.741i) q^{31} +(-67.0306 + 116.100i) q^{32} +(4.01675 + 6.95722i) q^{34} +13.8302 q^{35} +273.725 q^{37} +(104.972 + 181.816i) q^{38} +(61.3589 - 106.277i) q^{40} +(32.4323 - 56.1744i) q^{41} +(-209.381 - 362.658i) q^{43} -167.688 q^{44} +197.151 q^{46} +(69.3544 + 120.125i) q^{47} +(167.674 - 290.421i) q^{49} +(-27.4306 + 47.5111i) q^{50} +(32.5661 + 56.4062i) q^{52} +197.063 q^{53} +263.295 q^{55} +(33.9444 + 58.7934i) q^{56} +(-249.926 + 432.884i) q^{58} +(370.552 - 641.814i) q^{59} +(-244.234 - 423.026i) q^{61} +612.554 q^{62} +521.263 q^{64} +(-51.1336 - 88.5660i) q^{65} +(205.734 - 356.341i) q^{67} +(-5.82882 + 10.0958i) q^{68} +(-15.1748 - 26.2836i) q^{70} +310.343 q^{71} -51.0260 q^{73} +(-300.338 - 520.200i) q^{74} +(-152.327 + 263.839i) q^{76} +(-72.8286 + 126.143i) q^{77} +(-603.999 - 1046.16i) q^{79} -141.921 q^{80} -142.342 q^{82} +(452.611 + 783.945i) q^{83} +(9.15210 - 15.8519i) q^{85} +(-459.475 + 795.833i) q^{86} +(646.220 + 1119.29i) q^{88} -663.633 q^{89} +56.5752 q^{91} +(143.046 + 247.763i) q^{92} +(152.194 - 263.608i) q^{94} +(239.176 - 414.265i) q^{95} +(362.668 + 628.159i) q^{97} -735.905 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8} + 20 q^{10} - 23 q^{11} - 96 q^{13} + 21 q^{14} - 324 q^{16} + 322 q^{17} + 558 q^{19} - 180 q^{20} - 311 q^{22} - 96 q^{23} - 175 q^{25} - 716 q^{26}+ \cdots + 2558 q^{98}+O(q^{100}) $ 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8 + 20 * q^10 - 23 * q^11 - 96 * q^13 + 21 * q^14 - 324 * q^16 + 322 * q^17 + 558 * q^19 - 180 * q^20 - 311 * q^22 - 96 * q^23 - 175 * q^25 - 716 * q^26 + 674 * q^28 + 296 * q^29 - 244 * q^31 + 314 * q^32 - 125 * q^34 + 220 * q^35 + 808 * q^37 - 305 * q^38 - 90 * q^40 + 47 * q^41 - 525 * q^43 + 110 * q^44 + 1434 * q^46 - 164 * q^47 - 1225 * q^49 - 50 * q^50 - 1682 * q^52 + 1012 * q^53 + 230 * q^55 + 981 * q^56 - 1183 * q^58 + 85 * q^59 - 828 * q^61 - 1572 * q^62 + 4472 * q^64 - 480 * q^65 - 1093 * q^67 - 2473 * q^68 + 105 * q^70 + 656 * q^71 + 4170 * q^73 + 1316 * q^74 - 2789 * q^76 - 24 * q^77 - 2110 * q^79 + 3240 * q^80 - 124 * q^82 - 1290 * q^83 - 805 * q^85 + 2569 * q^86 - 2271 * q^88 - 6096 * q^89 + 6676 * q^91 - 2763 * q^92 + 517 * q^94 - 1395 * q^95 - 1787 * q^97 + 2558 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	−1.09722	−	1.90044i	−0.387927	−	0.671909i	0.604244	−	0.796799i	$-0.293475\pi$
$2$	−1.09722	−	1.90044i	−0.387927	−	0.671909i	−0.992170	+	0.124891i	$0.960142\pi$
$3$		0			0
$3$		0			0
$4$	1.59221	−	2.75778i	0.199026	−	0.344723i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	−1.38302	−	2.39547i	−0.0746763	−	0.129343i	0.826269	−	0.563275i	$-0.190459\pi$
$7$	−1.38302	−	2.39547i	−0.0746763	−	0.129343i	−0.900945	+	0.433932i	$0.857126\pi$
$8$	−24.5436			−1.08468
$9$		0			0
$10$	10.9722			0.346972
$11$	−26.3295	−	45.6040i	−0.721694	−	1.25001i	−0.960320	−	0.278900i	$-0.910030\pi$
$11$	−26.3295	−	45.6040i	−0.721694	−	1.25001i	0.238626	−	0.971112i	$-0.423303\pi$
$12$		0			0
$13$	−10.2267	+	17.7132i	−0.218183	+	0.377905i	−0.954253	−	0.299002i	$-0.903346\pi$
$13$	−10.2267	+	17.7132i	−0.218183	+	0.377905i	0.736069	+	0.676906i	$0.236680\pi$
$14$	−3.03497	+	5.25672i	−0.0579378	+	0.100351i
$15$		0			0
$16$	14.1921	+	24.5814i	0.221751	+	0.384085i
$17$	−3.66084			−0.0522285			−0.0261142	−	0.999659i	$-0.508313\pi$
$17$	−3.66084			−0.0522285			−0.0261142	+	0.999659i	$0.508313\pi$
$18$		0			0
$19$	−95.6705			−1.15517			−0.577587	−	0.816329i	$-0.696006\pi$
$19$	−95.6705			−1.15517			−0.577587	+	0.816329i	$0.696006\pi$
$20$	7.96104	+	13.7889i	0.0890071	+	0.154165i
$21$		0			0
$22$	−57.7786	+	100.075i	−0.559929	+	0.969825i
$23$	−44.9206	+	77.8047i	−0.407243	+	0.705365i	−0.994580	−	0.103977i	$-0.966843\pi$
$23$	−44.9206	+	77.8047i	−0.407243	+	0.705365i	0.587337	+	0.809343i	$0.300176\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	44.8840			0.338556
$27$		0			0
$28$	−8.80825			−0.0594501
$29$	−113.890	−	197.264i	−0.729272	−	1.26314i	−0.957191	−	0.289456i	$-0.906526\pi$
$29$	−113.890	−	197.264i	−0.729272	−	1.26314i	0.227919	−	0.973680i	$-0.426808\pi$
$30$		0			0
$31$	−139.569	+	241.741i	−0.808625	+	1.40058i	0.105191	+	0.994452i	$0.466455\pi$
$31$	−139.569	+	241.741i	−0.808625	+	1.40058i	−0.913816	+	0.406128i	$0.866879\pi$
$32$	−67.0306	+	116.100i	−0.370295	+	0.641370i
$33$		0			0
$34$	4.01675	+	6.95722i	0.0202608	+	0.0350928i
$35$	13.8302			0.0667925
$36$		0			0
$37$	273.725			1.21622			0.608110	−	0.793852i	$-0.291928\pi$
$37$	273.725			1.21622			0.608110	+	0.793852i	$0.291928\pi$
$38$	104.972	+	181.816i	0.448123	+	0.776172i
$39$		0			0
$40$	61.3589	−	106.277i	0.242542	−	0.420096i
$41$	32.4323	−	56.1744i	0.123539	−	0.213975i	−0.797622	−	0.603157i	$-0.793909\pi$
$41$	32.4323	−	56.1744i	0.123539	−	0.213975i	0.921161	+	0.389182i	$0.127242\pi$
$42$		0			0
$43$	−209.381	−	362.658i	−0.742565	−	1.28616i	−0.951324	−	0.308193i	$-0.900276\pi$
$43$	−209.381	−	362.658i	−0.742565	−	1.28616i	0.208759	−	0.977967i	$-0.433057\pi$
$44$	−167.688			−0.574544
$45$		0			0
$46$	197.151			0.631921
$47$	69.3544	+	120.125i	0.215242	+	0.372810i	0.953347	−	0.301875i	$-0.0976126\pi$
$47$	69.3544	+	120.125i	0.215242	+	0.372810i	−0.738105	+	0.674685i	$0.764279\pi$
$48$		0			0
$49$	167.674	−	290.421i	0.488847	−	0.846708i
$50$	−27.4306	+	47.5111i	−0.0775853	+	0.134382i
$51$		0			0
$52$	32.5661	+	56.4062i	0.0868483	+	0.150426i
$53$	197.063			0.510730			0.255365	−	0.966845i	$-0.417804\pi$
$53$	197.063			0.510730			0.255365	+	0.966845i	$0.417804\pi$
$54$		0			0
$55$	263.295			0.645503
$56$	33.9444	+	58.7934i	0.0810001	+	0.140296i
$57$		0			0
$58$	−249.926	+	432.884i	−0.565808	+	0.980008i
$59$	370.552	−	641.814i	0.817656	−	1.41622i	−0.0897490	−	0.995964i	$-0.528606\pi$
$59$	370.552	−	641.814i	0.817656	−	1.41622i	0.907405	−	0.420257i	$-0.138060\pi$
$60$		0			0
$61$	−244.234	−	423.026i	−0.512639	−	0.887916i	−0.999893	−	0.0146560i	$-0.995335\pi$
$61$	−244.234	−	423.026i	−0.512639	−	0.887916i	0.487254	−	0.873260i	$-0.337999\pi$
$62$	612.554			1.25475
$63$		0			0
$64$	521.263			1.01809
$65$	−51.1336	−	88.5660i	−0.0975746	−	0.169004i
$66$		0			0
$67$	205.734	−	356.341i	0.375140	−	0.649762i	−0.615208	−	0.788365i	$-0.710928\pi$
$67$	205.734	−	356.341i	0.375140	−	0.649762i	0.990348	+	0.138603i	$0.0442613\pi$
$68$	−5.82882	+	10.0958i	−0.0103948	+	0.0180044i
$69$		0			0
$70$	−15.1748	−	26.2836i	−0.0259106	−	0.0448785i
$71$	310.343			0.518746			0.259373	−	0.965777i	$-0.416484\pi$
$71$	310.343			0.518746			0.259373	+	0.965777i	$0.416484\pi$
$72$		0			0
$73$	−51.0260			−0.0818101			−0.0409051	−	0.999163i	$-0.513024\pi$
$73$	−51.0260			−0.0818101			−0.0409051	+	0.999163i	$0.513024\pi$
$74$	−300.338	−	520.200i	−0.471804	−	0.817189i
$75$		0			0
$76$	−152.327	+	263.839i	−0.229910	+	0.398215i
$77$	−72.8286	+	126.143i	−0.107787	+	0.186692i
$78$		0			0
$79$	−603.999	−	1046.16i	−0.860193	−	1.48990i	−0.871742	−	0.489964i	$-0.837010\pi$
$79$	−603.999	−	1046.16i	−0.860193	−	1.48990i	0.0115496	−	0.999933i	$-0.496324\pi$
$80$	−141.921			−0.198340
$81$		0			0
$82$	−142.342			−0.191695
$83$	452.611	+	783.945i	0.598560	+	1.03674i	0.993034	+	0.117829i	$0.0375935\pi$
$83$	452.611	+	783.945i	0.598560	+	1.03674i	−0.394474	+	0.918907i	$0.629073\pi$
$84$		0			0
$85$	9.15210	−	15.8519i	0.0116786	−	0.0202280i
$86$	−459.475	+	795.833i	−0.576121	+	0.997871i
$87$		0			0
$88$	646.220	+	1119.29i	0.782810	+	1.35587i
$89$	−663.633			−0.790393			−0.395197	−	0.918597i	$-0.629324\pi$
$89$	−663.633			−0.790393			−0.395197	+	0.918597i	$0.629324\pi$
$90$		0			0
$91$	56.5752			0.0651725
$92$	143.046	+	247.763i	0.162104	+	0.280772i
$93$		0			0
$94$	152.194	−	263.608i	0.166996	−	0.289246i
$95$	239.176	−	414.265i	0.258305	−	0.447397i
$96$		0			0
$97$	362.668	+	628.159i	0.379622	+	0.657525i	0.991007	−	0.133809i	$-0.0427207\pi$
$97$	362.668	+	628.159i	0.379622	+	0.657525i	−0.611385	+	0.791333i	$0.709387\pi$
$98$	−735.905			−0.758547
$99$		0			0
$100$	−79.6104			−0.0796104
$101$	488.891	+	846.784i	0.481648	+	0.834239i	0.999778	−	0.0210629i	$-0.00670503\pi$
$101$	488.891	+	846.784i	0.481648	+	0.834239i	−0.518130	+	0.855302i	$0.673372\pi$
$102$		0			0
$103$	−793.852	+	1374.99i	−0.759423	+	1.31536i	0.183722	+	0.982978i	$0.441185\pi$
$103$	−793.852	+	1374.99i	−0.759423	+	1.31536i	−0.943145	+	0.332381i	$0.892148\pi$
$104$	251.000	−	434.745i	0.236660	−	0.409907i
$105$		0			0
$106$	−216.222	−	374.507i	−0.198126	−	0.343164i
$107$	−897.731			−0.811093			−0.405546	−	0.914074i	$-0.632919\pi$
$107$	−897.731			−0.811093			−0.405546	+	0.914074i	$0.632919\pi$
$108$		0			0
$109$	855.492			0.751754			0.375877	−	0.926669i	$-0.377341\pi$
$109$	855.492			0.751754			0.375877	+	0.926669i	$0.377341\pi$
$110$	−288.893	−	500.377i	−0.250408	−	0.433719i
$111$		0			0
$112$	39.2560	−	67.9934i	0.0331191	−	0.0573640i
$113$	455.121	−	788.292i	0.378886	−	0.656250i	−0.612014	−	0.790847i	$-0.709641\pi$
$113$	455.121	−	788.292i	0.378886	−	0.656250i	0.990900	+	0.134597i	$0.0429738\pi$
$114$		0			0
$115$	−224.603	−	389.024i	−0.182125	−	0.315449i
$116$	−725.348			−0.580576
$117$		0			0
$118$	−1626.31			−1.26876
$119$	5.06303	+	8.76943i	0.00390023	+	0.00675539i
$120$		0			0
$121$	−720.984	+	1248.78i	−0.541686	+	0.938227i
$122$	−535.958	+	928.306i	−0.397732	+	0.688893i
$123$		0			0
$124$	444.447	+	769.804i	0.321875	+	0.557504i
$125$	125.000			0.0894427
$126$		0			0
$127$	−2038.25			−1.42414			−0.712068	−	0.702111i	$-0.752241\pi$
$127$	−2038.25			−1.42414			−0.712068	+	0.702111i	$0.752241\pi$
$128$	−35.6968	−	61.8287i	−0.0246499	−	0.0426948i
$129$		0			0
$130$	−112.210	+	194.353i	−0.0757035	+	0.131122i
$131$	117.823	−	204.075i	0.0785817	−	0.136108i	−0.824057	−	0.566508i	$-0.808294\pi$
$131$	117.823	−	204.075i	0.0785817	−	0.136108i	0.902638	+	0.430400i	$0.141628\pi$
$132$		0			0
$133$	132.315	+	229.176i	0.0862642	+	0.149414i
$134$	−902.942			−0.582107
$135$		0			0
$136$	89.8501			0.0566513
$137$	−1486.45	−	2574.60i	−0.926975	−	1.60557i	−0.788354	−	0.615222i	$-0.789066\pi$
$137$	−1486.45	−	2574.60i	−0.926975	−	1.60557i	−0.138621	−	0.990345i	$-0.544267\pi$
$138$		0			0
$139$	1047.32	−	1814.00i	0.639080	−	1.10692i	−0.346555	−	0.938030i	$-0.612648\pi$
$139$	1047.32	−	1814.00i	0.639080	−	1.10692i	0.985635	−	0.168890i	$-0.0540182\pi$
$140$	22.0206	−	38.1408i	0.0132934	−	0.0230249i
$141$		0			0
$142$	−340.515	−	589.790i	−0.201235	−	0.348550i
$143$	1077.06			0.629847
$144$		0			0
$145$	1138.90			0.652281
$146$	55.9868	+	96.9720i	0.0317363	+	0.0549689i
$147$		0			0
$148$	435.828	−	754.876i	0.242060	−	0.419259i
$149$	136.969	−	237.237i	0.0753081	−	0.130437i	−0.825912	−	0.563799i	$-0.809339\pi$
$149$	136.969	−	237.237i	0.0753081	−	0.130437i	0.901220	+	0.433361i	$0.142673\pi$
$150$		0			0
$151$	468.422	+	811.331i	0.252448	+	0.437253i	0.964199	−	0.265179i	$-0.0854310\pi$
$151$	468.422	+	811.331i	0.252448	+	0.437253i	−0.711751	+	0.702432i	$0.752098\pi$
$152$	2348.10			1.25300
$153$		0			0
$154$	319.637			0.167254
$155$	−697.846	−	1208.71i	−0.361628	−	0.626358i
$156$		0			0
$157$	−199.218	+	345.055i	−0.101269	+	0.175404i	−0.912208	−	0.409728i	$-0.865624\pi$
$157$	−199.218	+	345.055i	−0.101269	+	0.175404i	0.810938	+	0.585131i	$0.198957\pi$
$158$	−1325.44	+	2295.73i	−0.667383	+	1.15594i
$159$		0			0
$160$	−335.153	−	580.502i	−0.165601	−	0.286829i
$161$	248.505			0.121646
$162$		0			0
$163$	478.154			0.229766			0.114883	−	0.993379i	$-0.463351\pi$
$163$	478.154			0.229766			0.114883	+	0.993379i	$0.463351\pi$
$164$	−103.278	−	178.883i	−0.0491747	−	0.0851732i
$165$		0			0
$166$	993.229	−	1720.32i	0.464395	−	0.804355i
$167$	−170.181	+	294.763i	−0.0788564	+	0.136583i	−0.902757	−	0.430151i	$-0.858460\pi$
$167$	−170.181	+	294.763i	−0.0788564	+	0.136583i	0.823900	+	0.566735i	$0.191794\pi$
$168$		0			0
$169$	889.328	+	1540.36i	0.404792	+	0.701120i
$170$	−40.1675			−0.0181218
$171$		0			0
$172$	−1333.51			−0.591159
$173$	1888.03	+	3270.17i	0.829738	+	1.43715i	0.898244	+	0.439497i	$0.144843\pi$
$173$	1888.03	+	3270.17i	0.829738	+	1.43715i	−0.0685065	+	0.997651i	$0.521823\pi$
$174$		0			0
$175$	−34.5756	+	59.8867i	−0.0149353	+	0.0258686i
$176$	747.341	−	1294.43i	0.320073	−	0.554384i
$177$		0			0
$178$	728.153	+	1261.20i	0.306615	+	0.531072i
$179$	−186.652			−0.0779385			−0.0389693	−	0.999240i	$-0.512407\pi$
$179$	−186.652			−0.0779385			−0.0389693	+	0.999240i	$0.512407\pi$
$180$		0			0
$181$	1438.75			0.590837			0.295418	−	0.955368i	$-0.404541\pi$
$181$	1438.75			0.590837			0.295418	+	0.955368i	$0.404541\pi$
$182$	−62.0756	−	107.518i	−0.0252821	−	0.0437900i
$183$		0			0
$184$	1102.51	−	1909.61i	0.441729	−	0.765098i
$185$	−684.313	+	1185.27i	−0.271955	+	0.471040i
$186$		0			0
$187$	96.3880	+	166.949i	0.0376930	+	0.0652862i
$188$	441.706			0.171355
$189$		0			0
$190$	−1049.72			−0.400813
$191$	−195.218	−	338.127i	−0.0739552	−	0.128094i	0.826676	−	0.562678i	$-0.190229\pi$
$191$	−195.218	−	338.127i	−0.0739552	−	0.128094i	−0.900631	+	0.434584i	$0.856896\pi$
$192$		0			0
$193$	1957.52	−	3390.53i	0.730081	−	1.26454i	−0.226767	−	0.973949i	$-0.572816\pi$
$193$	1957.52	−	3390.53i	0.730081	−	1.26454i	0.956848	−	0.290588i	$-0.0938510\pi$
$194$	795.854	−	1378.46i	0.294531	−	0.510143i
$195$		0			0
$196$	−533.945	−	924.820i	−0.194586	−	0.337034i
$197$	892.680			0.322847			0.161423	−	0.986885i	$-0.448392\pi$
$197$	892.680			0.322847			0.161423	+	0.986885i	$0.448392\pi$
$198$		0			0
$199$	−2770.50			−0.986913			−0.493457	−	0.869770i	$-0.664267\pi$
$199$	−2770.50			−0.986913			−0.493457	+	0.869770i	$0.664267\pi$
$200$	306.795	+	531.384i	0.108468	+	0.187873i
$201$		0			0
$202$	1072.84	−	1858.22i	0.373688	−	0.647247i
$203$	−315.026	+	545.641i	−0.108919	+	0.188653i
$204$		0			0
$205$	162.162	+	280.872i	0.0552481	+	0.0956925i
$206$	3484.13			1.17840
$207$		0			0
$208$	−580.554			−0.193530
$209$	2518.96	+	4362.96i	0.833683	+	1.44398i
$210$		0			0
$211$	−2291.25	+	3968.57i	−0.747566	+	1.29482i	0.201420	+	0.979505i	$0.435444\pi$
$211$	−2291.25	+	3968.57i	−0.747566	+	1.29482i	−0.948986	+	0.315318i	$0.897889\pi$
$212$	313.765	−	543.458i	0.101649	−	0.176060i
$213$		0			0
$214$	985.010	+	1706.09i	0.314644	+	0.544980i
$215$	2093.81			0.664170
$216$		0			0
$217$	772.111			0.241541
$218$	−938.664	−	1625.81i	−0.291626	−	0.505110i
$219$		0			0
$220$	419.220	−	726.111i	0.128472	−	0.222520i
$221$	37.4384	−	64.8452i	0.0113954	−	0.0197374i
$222$		0			0
$223$	−2047.67	−	3546.66i	−0.614897	−	1.06503i	−0.990403	−	0.138212i	$-0.955864\pi$
$223$	−2047.67	−	3546.66i	−0.614897	−	1.06503i	0.375506	−	0.926820i	$-0.377469\pi$
$224$	370.820			0.110609
$225$		0			0
$226$	−1997.47			−0.587920
$227$	−1934.68	−	3350.96i	−0.565678	−	0.979784i	−0.996986	−	0.0775789i	$-0.975281\pi$
$227$	−1934.68	−	3350.96i	−0.565678	−	0.979784i	0.431308	−	0.902205i	$-0.358052\pi$
$228$		0			0
$229$	−618.006	+	1070.42i	−0.178336	+	0.308887i	−0.941311	−	0.337541i	$-0.890405\pi$
$229$	−618.006	+	1070.42i	−0.178336	+	0.308887i	0.762975	+	0.646429i	$0.223738\pi$
$230$	−492.878	+	853.690i	−0.141302	+	0.244742i
$231$		0			0
$232$	2795.27	+	4841.56i	0.791029	+	1.37010i
$233$	−2207.05			−0.620552			−0.310276	−	0.950647i	$-0.600421\pi$
$233$	−2207.05			−0.620552			−0.310276	+	0.950647i	$0.600421\pi$
$234$		0			0
$235$	−693.544			−0.192518
$236$	−1179.99	−	2043.80i	−0.325470	−	0.563730i
$237$		0			0
$238$	11.1105	−	19.2440i	0.00302600	−	0.00524119i
$239$	438.324	−	759.199i	0.118631	−	0.205475i	−0.800594	−	0.599207i	$-0.795483\pi$
$239$	438.324	−	759.199i	0.118631	−	0.205475i	0.919225	+	0.393732i	$0.128816\pi$
$240$		0			0
$241$	238.931	+	413.840i	0.0638626	+	0.110613i	0.896189	−	0.443673i	$-0.146325\pi$
$241$	238.931	+	413.840i	0.0638626	+	0.110613i	−0.832326	+	0.554286i	$0.812991\pi$
$242$	3164.32			0.840537
$243$		0			0
$244$	−1555.49			−0.408114
$245$	838.372	+	1452.10i	0.218619	+	0.378659i
$246$		0			0
$247$	978.396	−	1694.63i	0.252040	−	0.436546i
$248$	3425.53	−	5933.19i	0.877102	−	1.51919i
$249$		0			0
$250$	−137.153	−	237.556i	−0.0346972	−	0.0600973i
$251$	−6892.28			−1.73322			−0.866608	−	0.498990i	$-0.833704\pi$
$251$	−6892.28			−1.73322			−0.866608	+	0.498990i	$0.833704\pi$
$252$		0			0
$253$	4730.94			1.17562
$254$	2236.41	+	3873.57i	0.552460	+	0.956889i
$255$		0			0
$256$	2006.72	−	3475.74i	0.489921	−	0.848568i
$257$	−3628.35	+	6284.49i	−0.880664	+	1.52535i	−0.0300589	+	0.999548i	$0.509569\pi$
$257$	−3628.35	+	6284.49i	−0.880664	+	1.52535i	−0.850605	+	0.525806i	$0.823764\pi$
$258$		0			0
$259$	−378.569	−	655.701i	−0.0908229	−	0.157310i
$260$	−325.661			−0.0776795
$261$		0			0
$262$	−517.110			−0.121936
$263$	−3158.78	−	5471.16i	−0.740603	−	1.28276i	−0.952221	−	0.305409i	$-0.901207\pi$
$263$	−3158.78	−	5471.16i	−0.740603	−	1.28276i	0.211618	−	0.977352i	$-0.432127\pi$
$264$		0			0
$265$	−492.658	+	853.308i	−0.114203	+	0.197805i
$266$	290.357	−	502.913i	0.0669283	−	0.115923i
$267$		0			0
$268$	−655.142	−	1134.74i	−0.149325	−	0.258639i
$269$	−5746.22			−1.30243			−0.651214	−	0.758894i	$-0.725740\pi$
$269$	−5746.22			−1.30243			−0.651214	+	0.758894i	$0.725740\pi$
$270$		0			0
$271$	4925.20			1.10400			0.552001	−	0.833844i	$-0.313865\pi$
$271$	4925.20			1.10400			0.552001	+	0.833844i	$0.313865\pi$
$272$	−51.9550	−	89.9886i	−0.0115817	−	0.0200602i
$273$		0			0
$274$	−3261.92	+	5649.81i	−0.719196	+	1.24568i
$275$	−658.237	+	1140.10i	−0.144339	+	0.250002i
$276$		0			0
$277$	−1162.94	−	2014.27i	−0.252254	−	0.436917i	0.711892	−	0.702289i	$-0.247839\pi$
$277$	−1162.94	−	2014.27i	−0.252254	−	0.436917i	−0.964146	+	0.265372i	$0.914505\pi$
$278$	−4596.55			−0.991665
$279$		0			0
$280$	−339.444			−0.0724487
$281$	1641.71	+	2843.52i	0.348527	+	0.603667i	0.985988	−	0.166816i	$-0.0533485\pi$
$281$	1641.71	+	2843.52i	0.348527	+	0.603667i	−0.637461	+	0.770483i	$0.720015\pi$
$282$		0			0
$283$	−1007.11	+	1744.36i	−0.211541	+	0.366400i	−0.952197	−	0.305484i	$-0.901182\pi$
$283$	−1007.11	+	1744.36i	−0.211541	+	0.366400i	0.740656	+	0.671885i	$0.234515\pi$
$284$	494.131	−	855.860i	0.103244	−	0.178824i
$285$		0			0
$286$	−1181.77	−	2046.89i	−0.244334	−	0.423199i
$287$	−179.419			−0.0369016
$288$		0			0
$289$	−4899.60			−0.997272
$290$	−1249.63	−	2164.42i	−0.253037	−	0.438273i
$291$		0			0
$292$	−81.2440	+	140.719i	−0.0162823	+	0.0282018i
$293$	240.348	−	416.295i	0.0479225	−	0.0830042i	−0.841069	−	0.540928i	$-0.818073\pi$
$293$	240.348	−	416.295i	0.0479225	−	0.0830042i	0.888992	+	0.457923i	$0.151407\pi$
$294$		0			0
$295$	1852.76	+	3209.07i	0.365667	+	0.633354i
$296$	−6718.20			−1.31921
$297$		0			0
$298$	−601.140			−0.116856
$299$	−918.781	−	1591.37i	−0.177707	−	0.307798i
$300$		0			0
$301$	−579.158	+	1003.13i	−0.110904	+	0.192091i
$302$	1027.93	−	1780.42i	0.195863	−	0.339244i
$303$		0			0
$304$	−1357.76	−	2351.72i	−0.256162	−	0.443685i
$305$	2442.34			0.458518
$306$		0			0
$307$	3222.21			0.599026			0.299513	−	0.954092i	$-0.403176\pi$
$307$	3222.21			0.599026			0.299513	+	0.954092i	$0.403176\pi$
$308$	231.917	+	401.691i	0.0429048	+	0.0743133i
$309$		0			0
$310$	−1531.38	+	2652.44i	−0.280570	+	0.485962i
$311$	1207.00	−	2090.59i	0.220074	−	0.381179i	−0.734757	−	0.678331i	$-0.762704\pi$
$311$	1207.00	−	2090.59i	0.220074	−	0.381179i	0.954830	+	0.297152i	$0.0960369\pi$
$312$		0			0
$313$	1253.47	+	2171.07i	0.226359	+	0.392065i	0.956726	−	0.290990i	$-0.0939846\pi$
$313$	1253.47	+	2171.07i	0.226359	+	0.392065i	−0.730368	+	0.683054i	$0.760651\pi$
$314$	874.344			0.157140
$315$		0			0
$316$	−3846.77			−0.684803
$317$	1853.56	+	3210.46i	0.328411	+	0.568825i	0.982197	−	0.187855i	$-0.0601535\pi$
$317$	1853.56	+	3210.46i	0.328411	+	0.568825i	−0.653786	+	0.756680i	$0.726820\pi$
$318$		0			0
$319$	−5997.34	+	10387.7i	−1.05262	+	1.82320i
$320$	−1303.16	+	2257.13i	−0.227652	+	0.394305i
$321$		0			0
$322$	−272.665	−	472.270i	−0.0471895	−	0.0817347i
$323$	350.234			0.0603330
$324$		0			0
$325$	511.336			0.0872733
$326$	−524.641	−	908.704i	−0.0891324	−	0.154382i
$327$		0			0
$328$	−796.005	+	1378.72i	−0.134000	+	0.232095i
$329$	191.838	−	332.272i	0.0321470	−	0.0556802i
$330$		0			0
$331$	−1276.72	−	2211.35i	−0.212009	−	0.367211i	0.740334	−	0.672239i	$-0.234667\pi$
$331$	−1276.72	−	2211.35i	−0.212009	−	0.367211i	−0.952343	+	0.305029i	$0.901334\pi$
$332$	2882.60			0.476516
$333$		0			0
$334$	746.906			0.122362
$335$	1028.67	+	1781.71i	0.167768	+	0.290582i
$336$		0			0
$337$	−1776.44	+	3076.89i	−0.287148	+	0.497356i	−0.973128	−	0.230265i	$-0.926041\pi$
$337$	−1776.44	+	3076.89i	−0.287148	+	0.497356i	0.685979	+	0.727621i	$0.259374\pi$
$338$	1951.58	−	3380.24i	0.314059	−	0.543966i
$339$		0			0
$340$	−29.1441	−	50.4790i	−0.00464871	−	0.00805179i
$341$	14699.2			2.33432
$342$		0			0
$343$	−1876.35			−0.295374
$344$	5138.95	+	8900.93i	0.805447	+	1.39508i
$345$		0			0
$346$	4143.19	−	7176.21i	0.643754	−	1.11502i
$347$	−3884.01	+	6727.30i	−0.600878	+	1.04075i	0.391811	+	0.920046i	$0.371849\pi$
$347$	−3884.01	+	6727.30i	−0.600878	+	1.04075i	−0.992688	+	0.120705i	$0.961485\pi$
$348$		0			0
$349$	−348.003	−	602.760i	−0.0533759	−	0.0924498i	0.838103	−	0.545512i	$-0.183665\pi$
$349$	−348.003	−	602.760i	−0.0533759	−	0.0924498i	−0.891479	+	0.453062i	$0.850331\pi$
$350$	151.748			0.0231751
$351$		0			0
$352$	7059.52			1.06896
$353$	−2725.94	−	4721.46i	−0.411011	−	0.711892i	0.583990	−	0.811761i	$-0.301491\pi$
$353$	−2725.94	−	4721.46i	−0.411011	−	0.711892i	−0.995001	+	0.0998692i	$0.968158\pi$
$354$		0			0
$355$	−775.858	+	1343.83i	−0.115995	+	0.200909i
$356$	−1056.64	+	1830.16i	−0.157309	+	0.272467i
$357$		0			0
$358$	204.798	+	354.721i	0.0302344	+	0.0523675i
$359$	−4036.41			−0.593408			−0.296704	−	0.954969i	$-0.595887\pi$
$359$	−4036.41			−0.593408			−0.296704	+	0.954969i	$0.595887\pi$
$360$		0			0
$361$	2293.85			0.334429
$362$	−1578.63	−	2734.26i	−0.229201	−	0.396988i
$363$		0			0
$364$	90.0795	−	156.022i	0.0129710	−	0.0224665i
$365$	127.565	−	220.949i	0.0182933	−	0.0316849i
$366$		0			0
$367$	−5619.96	−	9734.06i	−0.799345	−	1.38451i	−0.920043	−	0.391817i	$-0.871847\pi$
$367$	−5619.96	−	9734.06i	−0.799345	−	1.38451i	0.120698	−	0.992689i	$-0.461487\pi$
$368$	−2550.07			−0.361227
$369$		0			0
$370$	3003.38			0.421995
$371$	−272.543	−	472.059i	−0.0381394	−	0.0660595i
$372$		0			0
$373$	3160.36	−	5473.90i	0.438706	−	0.759860i	−0.558884	−	0.829246i	$-0.688770\pi$
$373$	3160.36	−	5473.90i	0.438706	−	0.759860i	0.997590	+	0.0693853i	$0.0221038\pi$
$374$	211.518	−	366.360i	0.0292442	−	0.0506525i
$375$		0			0
$376$	−1702.20	−	2948.30i	−0.233469	−	0.404381i
$377$	4658.90			0.636460
$378$		0			0
$379$	9325.49			1.26390			0.631950	−	0.775009i	$-0.282255\pi$
$379$	9325.49			1.26390			0.631950	+	0.775009i	$0.282255\pi$
$380$	−761.637	−	1319.19i	−0.102819	−	0.178087i
$381$		0			0
$382$	−428.394	+	742.000i	−0.0573784	+	0.0993823i
$383$	5574.01	−	9654.47i	0.743652	−	1.28804i	−0.207170	−	0.978305i	$-0.566425\pi$
$383$	5574.01	−	9654.47i	0.743652	−	1.28804i	0.950822	−	0.309738i	$-0.100241\pi$
$384$		0			0
$385$	−364.143	−	630.715i	−0.0482038	−	0.0834914i
$386$	−8591.35			−1.13287
$387$		0			0
$388$	2309.77			0.302219
$389$	3285.41	+	5690.49i	0.428218	+	0.741695i	0.996715	−	0.0809902i	$-0.0258082\pi$
$389$	3285.41	+	5690.49i	0.428218	+	0.741695i	−0.568497	+	0.822685i	$0.692475\pi$
$390$		0			0
$391$	164.447	−	284.831i	0.0212697	−	0.0368402i
$392$	−4115.33	+	7127.96i	−0.530244	+	0.918409i
$393$		0			0
$394$	−979.469	−	1696.49i	−0.125241	−	0.216924i
$395$	6039.99			0.769380
$396$		0			0
$397$	3969.33			0.501800			0.250900	−	0.968013i	$-0.419273\pi$
$397$	3969.33			0.501800			0.250900	+	0.968013i	$0.419273\pi$
$398$	3039.86	+	5265.19i	0.382850	+	0.663115i
$399$		0			0
$400$	354.802	−	614.535i	0.0443503	−	0.0768169i
$401$	2187.35	−	3788.60i	0.272396	−	0.471804i	−0.697079	−	0.716995i	$-0.745517\pi$
$401$	2187.35	−	3788.60i	0.272396	−	0.471804i	0.969475	+	0.245190i	$0.0788505\pi$
$402$		0			0
$403$	−2854.67	−	4944.44i	−0.352857	−	0.611166i
$404$	3113.66			0.383442
$405$		0			0
$406$	1382.61			0.169010
$407$	−7207.05	−	12483.0i	−0.877740	−	1.52029i
$408$		0			0
$409$	4436.16	−	7683.66i	0.536318	−	0.928930i	−0.462780	−	0.886473i	$-0.653148\pi$
$409$	4436.16	−	7683.66i	0.536318	−	0.928930i	0.999098	−	0.0424573i	$-0.0135186\pi$
$410$	355.855	−	616.358i	0.0428644	−	0.0742433i
$411$		0			0
$412$	2527.95	+	4378.55i	0.302290	+	0.523581i
$413$	−2049.93			−0.244238
$414$		0			0
$415$	−4526.11			−0.535368
$416$	−1371.01	−	2374.65i	−0.161584	−	0.279872i
$417$		0			0
$418$	5527.71	−	9574.27i	0.646816	−	1.12032i
$419$	−507.087	+	878.300i	−0.0591236	+	0.102405i	−0.894072	−	0.447923i	$-0.852164\pi$
$419$	−507.087	+	878.300i	−0.0591236	+	0.102405i	0.834949	+	0.550328i	$0.185497\pi$
$420$		0			0
$421$	7446.57	+	12897.8i	0.862051	+	1.49312i	0.869946	+	0.493147i	$0.164154\pi$
$421$	7446.57	+	12897.8i	0.862051	+	1.49312i	−0.00789511	+	0.999969i	$0.502513\pi$
$422$	10056.1			1.16000
$423$		0			0
$424$	−4836.63			−0.553980
$425$	45.7605	+	79.2595i	0.00522285	+	0.00904624i
$426$		0			0
$427$	−675.563	+	1170.11i	−0.0765639	+	0.132613i
$428$	−1429.37	+	2475.75i	−0.161429	+	0.279602i
$429$		0			0
$430$	−2297.37	−	3979.17i	−0.257649	−	0.446261i
$431$	4363.90			0.487707			0.243853	−	0.969812i	$-0.421588\pi$
$431$	4363.90			0.487707			0.243853	+	0.969812i	$0.421588\pi$
$432$		0			0
$433$	9301.59			1.03235			0.516173	−	0.856484i	$-0.327356\pi$
$433$	9301.59			1.03235			0.516173	+	0.856484i	$0.327356\pi$
$434$	−847.177	−	1467.35i	−0.0937000	−	0.162293i
$435$		0			0
$436$	1362.12	−	2359.26i	0.149619	−	0.259147i
$437$	4297.57	−	7443.62i	0.470437	−	0.814820i
$438$		0			0
$439$	−760.076	−	1316.49i	−0.0826343	−	0.143127i	0.821746	−	0.569853i	$-0.193000\pi$
$439$	−760.076	−	1316.49i	−0.0826343	−	0.143127i	−0.904381	+	0.426727i	$0.859667\pi$
$440$	−6462.20			−0.700166
$441$		0			0
$442$	−164.313			−0.0176823
$443$	2231.24	+	3864.62i	0.239299	+	0.414478i	0.960513	−	0.278234i	$-0.0897490\pi$
$443$	2231.24	+	3864.62i	0.239299	+	0.414478i	−0.721214	+	0.692712i	$0.756416\pi$
$444$		0			0
$445$	1659.08	−	2873.62i	0.176737	−	0.306118i
$446$	−4493.49	+	7782.96i	−0.477070	+	0.826309i
$447$		0			0
$448$	−720.919	−	1248.67i	−0.0760273	−	0.131683i
$449$	5371.66			0.564598			0.282299	−	0.959326i	$-0.408903\pi$
$449$	5371.66			0.564598			0.282299	+	0.959326i	$0.408903\pi$
$450$		0			0
$451$	−3415.71			−0.356628
$452$	−1449.29	−	2510.25i	−0.150816	−	0.261222i
$453$		0			0
$454$	−4245.54	+	7353.49i	−0.438883	+	0.760168i
$455$	−141.438	+	244.978i	−0.0145730	+	0.0252412i
$456$		0			0
$457$	−7762.71	−	13445.4i	−0.794583	−	1.37626i	−0.923104	−	0.384551i	$-0.874356\pi$
$457$	−7762.71	−	13445.4i	−0.794583	−	1.37626i	0.128520	−	0.991707i	$-0.458977\pi$
$458$	2712.36			0.276725
$459$		0			0
$460$	−1430.46			−0.144990
$461$	−28.2090	−	48.8594i	−0.00284994	−	0.00493624i	0.864597	−	0.502466i	$-0.167574\pi$
$461$	−28.2090	−	48.8594i	−0.00284994	−	0.00493624i	−0.867447	+	0.497530i	$0.834240\pi$
$462$		0			0
$463$	−6686.89	+	11582.0i	−0.671201	+	1.16255i	0.306363	+	0.951915i	$0.400888\pi$
$463$	−6686.89	+	11582.0i	−0.671201	+	1.16255i	−0.977564	+	0.210640i	$0.932445\pi$
$464$	3232.68	−	5599.17i	0.323434	−	0.560204i
$465$		0			0
$466$	2421.62	+	4194.37i	0.240729	+	0.416954i
$467$	−2677.46			−0.265306			−0.132653	−	0.991163i	$-0.542350\pi$
$467$	−2677.46			−0.265306			−0.132653	+	0.991163i	$0.542350\pi$
$468$		0			0
$469$	−1138.14			−0.112056
$470$	760.972	+	1318.04i	0.0746830	+	0.129355i
$471$		0			0
$472$	−9094.66	+	15752.4i	−0.886898	+	1.53615i
$473$	−11025.8	+	19097.2i	−1.07181	+	1.85643i
$474$		0			0
$475$	1195.88	+	2071.33i	0.115517	+	0.200082i
$476$	32.2456			0.00310499
$477$		0			0
$478$	−1923.75			−0.184080
$479$	−1734.96	−	3005.05i	−0.165496	−	0.286647i	0.771335	−	0.636429i	$-0.219589\pi$
$479$	−1734.96	−	3005.05i	−0.165496	−	0.286647i	−0.936831	+	0.349782i	$0.886256\pi$
$480$		0			0
$481$	−2799.31	+	4848.55i	−0.265359	+	0.459616i
$482$	524.320	−	908.149i	0.0495480	−	0.0858196i
$483$		0			0
$484$	2295.91	+	3976.64i	0.215619	+	0.373463i
$485$	−3626.68			−0.339544
$486$		0			0
$487$	−14040.6			−1.30645			−0.653224	−	0.757165i	$-0.726584\pi$
$487$	−14040.6			−1.30645			−0.653224	+	0.757165i	$0.726584\pi$
$488$	5994.38	+	10382.6i	0.556051	+	0.963108i
$489$		0			0
$490$	1839.76	−	3186.56i	0.169616	−	0.293784i
$491$	4907.72	−	8500.42i	0.451084	−	0.781301i	−0.547369	−	0.836891i	$-0.684371\pi$
$491$	4907.72	−	8500.42i	0.451084	−	0.781301i	0.998454	+	0.0555902i	$0.0177040\pi$
$492$		0			0
$493$	416.934	+	722.151i	0.0380888	+	0.0659717i
$494$	−4294.07			−0.391092
$495$		0			0
$496$	−7923.12			−0.717255
$497$	−429.212	−	743.418i	−0.0387380	−	0.0670962i
$498$		0			0
$499$	6026.00	−	10437.3i	0.540603	−	0.936351i	−0.458267	−	0.888815i	$-0.651530\pi$
$499$	6026.00	−	10437.3i	0.540603	−	0.936351i	0.998869	−	0.0475367i	$-0.0151371\pi$
$500$	199.026	−	344.723i	0.0178014	−	0.0308330i
$501$		0			0
$502$	7562.36	+	13098.4i	0.672360	+	1.16456i
$503$	4695.09			0.416191			0.208095	−	0.978109i	$-0.433274\pi$
$503$	4695.09			0.416191			0.208095	+	0.978109i	$0.433274\pi$
$504$		0			0
$505$	−4888.91			−0.430799
$506$	−5190.89	−	8990.89i	−0.456054	−	0.789909i
$507$		0			0
$508$	−3245.31	+	5621.05i	−0.283440	+	0.490932i
$509$	−409.907	+	709.979i	−0.0356951	+	0.0618257i	−0.883321	−	0.468769i	$-0.844698\pi$
$509$	−409.907	+	709.979i	−0.0356951	+	0.0618257i	0.847626	+	0.530594i	$0.178031\pi$
$510$		0			0
$511$	70.5702	+	122.231i	0.00610928	+	0.0105816i
$512$	−9378.41			−0.809514
$513$		0			0
$514$	15924.4			1.36653
$515$	−3969.26	−	6874.96i	−0.339624	−	0.588247i
$516$		0			0
$517$	3652.13	−	6325.68i	0.310678	−	0.538110i
$518$	−830.748	+	1438.90i	−0.0704652	+	0.122049i
$519$		0			0
$520$	1255.00	+	2173.73i	0.105837	+	0.183316i
$521$	−3282.80			−0.276050			−0.138025	−	0.990429i	$-0.544075\pi$
$521$	−3282.80			−0.276050			−0.138025	+	0.990429i	$0.544075\pi$
$522$		0			0
$523$	−10768.1			−0.900300			−0.450150	−	0.892953i	$-0.648629\pi$
$523$	−10768.1			−0.900300			−0.450150	+	0.892953i	$0.648629\pi$
$524$	−375.196	−	649.859i	−0.0312796	−	0.0541779i
$525$		0			0
$526$	−6931.76	+	12006.2i	−0.574599	+	0.995235i
$527$	510.941	−	884.975i	0.0422333	−	0.0731502i
$528$		0			0
$529$	2047.78	+	3546.87i	0.168306	+	0.291515i
$530$	2162.22			0.177209
$531$		0			0
$532$	842.690			0.0686752
$533$	663.353	+	1148.96i	0.0539081	+	0.0933716i
$534$		0			0
$535$	2244.33	−	3887.29i	0.181366	−	0.314135i
$536$	−5049.44	+	8745.89i	−0.406908	+	0.704785i
$537$		0			0
$538$	6304.88	+	10920.4i	0.505247	+	0.875113i
$539$	−17659.1			−1.41119
$540$		0			0
$541$	−16037.9			−1.27453			−0.637266	−	0.770644i	$-0.719935\pi$
$541$	−16037.9			−1.27453			−0.637266	+	0.770644i	$0.719935\pi$
$542$	−5404.04	−	9360.06i	−0.428272	−	0.741788i
$543$		0			0
$544$	245.388	−	425.025i	0.0193399	−	0.0334978i
$545$	−2138.73	+	3704.39i	−0.168097	+	0.291153i
$546$		0			0
$547$	−1024.56	−	1774.59i	−0.0800862	−	0.138713i	0.823201	−	0.567751i	$-0.192186\pi$
$547$	−1024.56	−	1774.59i	−0.0800862	−	0.138713i	−0.903287	+	0.429037i	$0.858853\pi$
$548$	−9466.92			−0.737968
$549$		0			0
$550$	2888.93			0.223972
$551$	10895.9	+	18872.3i	0.842437	+	1.45914i
$552$		0			0
$553$	−1670.69	+	2893.72i	−0.128472	+	0.222520i
$554$	−2552.01	+	4420.21i	−0.195712	+	0.338983i
$555$		0			0
$556$	−3335.09	−	5776.54i	−0.254387	−	0.440611i
$557$	−3644.07			−0.277207			−0.138603	−	0.990348i	$-0.544261\pi$
$557$	−3644.07			−0.277207			−0.138603	+	0.990348i	$0.544261\pi$
$558$		0			0
$559$	8565.12			0.648061
$560$	196.280	+	339.967i	0.0148113	+	0.0256540i
$561$		0			0
$562$	3602.64	−	6239.96i	0.270406	−	0.468357i
$563$	175.368	−	303.746i	0.0131277	−	0.0227378i	−0.859387	−	0.511326i	$-0.829155\pi$
$563$	175.368	−	303.746i	0.0131277	−	0.0227378i	0.872515	+	0.488588i	$0.162488\pi$
$564$		0			0
$565$	2275.60	+	3941.46i	0.169443	+	0.293484i
$566$	4420.07			0.328250
$567$		0			0
$568$	−7616.93			−0.562675
$569$	−9724.34	−	16843.1i	−0.716460	−	1.24095i	−0.962394	−	0.271658i	$-0.912428\pi$
$569$	−9724.34	−	16843.1i	−0.716460	−	1.24095i	0.245934	−	0.969287i	$-0.420905\pi$
$570$		0			0
$571$	7572.97	−	13116.8i	0.555024	−	0.961330i	−0.442877	−	0.896582i	$-0.646042\pi$
$571$	7572.97	−	13116.8i	0.555024	−	0.961330i	0.997902	−	0.0647480i	$-0.0206244\pi$
$572$	1714.90	−	2970.29i	0.125356	−	0.217123i
$573$		0			0
$574$	196.862	+	340.975i	0.0143151	+	0.0247945i
$575$	2246.03			0.162897
$576$		0			0
$577$	−6365.11			−0.459243			−0.229621	−	0.973280i	$-0.573749\pi$
$577$	−6365.11			−0.459243			−0.229621	+	0.973280i	$0.573749\pi$
$578$	5375.95	+	9311.41i	0.386868	+	0.670076i
$579$		0			0
$580$	1813.37	−	3140.85i	0.129821	−	0.224856i
$581$	1251.94	−	2168.43i	0.0893965	−	0.154839i
$582$		0			0
$583$	−5188.57	−	8986.87i	−0.368591	−	0.638419i
$584$	1252.36			0.0887381
$585$		0			0
$586$	−1054.86			−0.0743617
$587$	5142.18	+	8906.52i	0.361568	+	0.626254i	0.988219	−	0.153046i	$-0.0489081\pi$
$587$	5142.18	+	8906.52i	0.361568	+	0.626254i	−0.626651	+	0.779300i	$0.715575\pi$
$588$		0			0
$589$	13352.7	−	23127.5i	0.934103	−	1.61791i
$590$	4065.77	−	7042.13i	0.283704	−	0.491389i
$591$		0			0
$592$	3884.73	+	6728.56i	0.269699	+	0.467132i
$593$	−666.566			−0.0461595			−0.0230798	−	0.999734i	$-0.507347\pi$
$593$	−666.566			−0.0461595			−0.0230798	+	0.999734i	$0.507347\pi$
$594$		0			0
$595$	−50.6303			−0.00348847
$596$	−436.165	−	755.460i	−0.0299765	−	0.0519209i
$597$		0			0
$598$	−2016.21	+	3492.18i	−0.137875	+	0.238806i
$599$	12606.6	−	21835.3i	0.859922	−	1.48943i	−0.0120807	−	0.999927i	$-0.503845\pi$
$599$	12606.6	−	21835.3i	0.859922	−	1.48943i	0.872003	−	0.489501i	$-0.162821\pi$
$600$		0			0
$601$	10309.3	+	17856.3i	0.699712	+	1.21194i	0.968566	+	0.248755i	$0.0800215\pi$
$601$	10309.3	+	17856.3i	0.699712	+	1.21194i	−0.268855	+	0.963181i	$0.586645\pi$
$602$	2541.86			0.172090
$603$		0			0
$604$	2983.30			0.200975
$605$	−3604.92	−	6243.90i	−0.242249	−	0.419588i
$606$		0			0
$607$	2541.76	−	4402.45i	0.169962	−	0.294382i	−0.768444	−	0.639916i	$-0.778969\pi$
$607$	2541.76	−	4402.45i	0.169962	−	0.294382i	0.938406	+	0.345534i	$0.112302\pi$
$608$	6412.85	−	11107.4i	0.427755	−	0.740894i
$609$		0			0
$610$	−2679.79	−	4641.53i	−0.177871	−	0.308082i
$611$	−2837.07			−0.187849
$612$		0			0
$613$	2625.18			0.172969			0.0864845	−	0.996253i	$-0.472437\pi$
$613$	2625.18			0.172969			0.0864845	+	0.996253i	$0.472437\pi$
$614$	−3535.47	−	6123.62i	−0.232378	−	0.402491i
$615$		0			0
$616$	1787.48	−	3096.00i	0.116915	−	0.202502i
$617$	−815.659	+	1412.76i	−0.0532208	+	0.0921811i	−0.891408	−	0.453201i	$-0.850282\pi$
$617$	−815.659	+	1412.76i	−0.0532208	+	0.0921811i	0.838188	+	0.545382i	$0.183615\pi$
$618$		0			0
$619$	−1592.33	−	2758.00i	−0.103395	−	0.179085i	0.809687	−	0.586863i	$-0.199637\pi$
$619$	−1592.33	−	2758.00i	−0.103395	−	0.179085i	−0.913081	+	0.407778i	$0.866304\pi$
$620$	−4444.47			−0.287894
$621$		0			0
$622$	−5297.40			−0.341489
$623$	917.821	+	1589.71i	0.0590236	+	0.102232i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	2750.67	−	4764.29i	0.175621	−	0.304184i
$627$		0			0
$628$	634.392	+	1098.80i	0.0403105	+	0.0698199i
$629$	−1002.06			−0.0635214
$630$		0			0
$631$	10436.7			0.658447			0.329223	−	0.944252i	$-0.393213\pi$
$631$	10436.7			0.658447			0.329223	+	0.944252i	$0.393213\pi$
$632$	14824.3	+	25676.4i	0.933037	+	1.61607i
$633$		0			0
$634$	4067.54	−	7045.18i	0.254799	−	0.441325i
$635$	5095.62	−	8825.87i	0.318446	−	0.551565i
$636$		0			0
$637$	3429.52	+	5940.11i	0.213316	+	0.369475i
$638$	26321.7			1.63336
$639$		0			0
$640$	356.968			0.0220475
$641$	4541.73	+	7866.50i	0.279856	+	0.484724i	0.971349	−	0.237659i	$-0.0763801\pi$
$641$	4541.73	+	7866.50i	0.279856	+	0.484724i	−0.691493	+	0.722383i	$0.743047\pi$
$642$		0			0
$643$	−8593.81	+	14884.9i	−0.527071	+	0.912914i	0.472431	+	0.881368i	$0.343377\pi$
$643$	−8593.81	+	14884.9i	−0.527071	+	0.912914i	−0.999502	+	0.0315465i	$0.989957\pi$
$644$	395.672	−	685.323i	0.0242106	−	0.0419340i
$645$		0			0
$646$	−384.285	−	665.601i	−0.0234048	−	0.0405383i
$647$	−2359.17			−0.143352			−0.0716758	−	0.997428i	$-0.522835\pi$
$647$	−2359.17			−0.143352			−0.0716758	+	0.997428i	$0.522835\pi$
$648$		0			0
$649$	−39025.7			−2.36039
$650$	−561.049	−	971.766i	−0.0338556	−	0.0586397i
$651$		0			0
$652$	761.320	−	1318.64i	0.0457294	−	0.0792057i
$653$	2718.06	−	4707.81i	0.162888	−	0.282130i	−0.773015	−	0.634387i	$-0.781252\pi$
$653$	2718.06	−	4707.81i	0.162888	−	0.282130i	0.935903	+	0.352257i	$0.114586\pi$
$654$		0			0
$655$	589.113	+	1020.37i	0.0351428	+	0.0608691i
$656$	1841.13			0.109579
$657$		0			0
$658$	−841.954			−0.0498826
$659$	1544.45	+	2675.07i	0.0912950	+	0.158128i	0.908056	−	0.418848i	$-0.137566\pi$
$659$	1544.45	+	2675.07i	0.0912950	+	0.158128i	−0.816761	+	0.576976i	$0.804233\pi$
$660$		0			0
$661$	9526.94	−	16501.2i	0.560598	−	0.970984i	−0.436846	−	0.899536i	$-0.643905\pi$
$661$	9526.94	−	16501.2i	0.560598	−	0.970984i	0.997444	−	0.0714479i	$-0.0227620\pi$
$662$	−2801.70	+	4852.68i	−0.164488	+	0.284902i
$663$		0			0
$664$	−11108.7	−	19240.8i	−0.649248	−	1.12453i
$665$	−1323.15			−0.0771570
$666$		0			0
$667$	20464.1			1.18796
$668$	541.928	+	938.646i	0.0313889	+	0.0543672i
$669$		0			0
$670$	2257.36	−	3909.86i	0.130163	−	0.225449i
$671$	−12861.1	+	22276.1i	−0.739937	+	1.28161i
$672$		0			0
$673$	14618.7	+	25320.4i	0.837312	+	1.45027i	0.892134	+	0.451771i	$0.149208\pi$
$673$	14618.7	+	25320.4i	0.837312	+	1.45027i	−0.0548215	+	0.998496i	$0.517459\pi$
$674$	7796.61			0.445570
$675$		0			0
$676$	5663.98			0.322257
$677$	7235.62	+	12532.5i	0.410764	+	0.711465i	0.994974	−	0.100138i	$-0.0319285\pi$
$677$	7235.62	+	12532.5i	0.410764	+	0.711465i	−0.584209	+	0.811603i	$0.698595\pi$
$678$		0			0
$679$	1003.16	−	1737.52i	0.0566975	−	0.0982030i
$680$	−224.625	+	389.062i	−0.0126676	+	0.0219410i
$681$		0			0
$682$	−16128.2	−	27934.9i	−0.905545	−	1.56845i
$683$	−5782.94			−0.323980			−0.161990	−	0.986792i	$-0.551791\pi$
$683$	−5782.94			−0.323980			−0.161990	+	0.986792i	$0.551791\pi$
$684$		0			0
$685$	14864.5			0.829112
$686$	2058.77	+	3565.89i	0.114583	+	0.198464i
$687$		0			0
$688$	5943.10	−	10293.8i	0.329329	−	0.570415i
$689$	−2015.31	+	3490.62i	−0.111433	+	0.193007i
$690$		0			0
$691$	−5527.56	−	9574.01i	−0.304310	−	0.527080i	0.672797	−	0.739827i	$-0.265093\pi$
$691$	−5527.56	−	9574.01i	−0.304310	−	0.527080i	−0.977107	+	0.212746i	$0.931759\pi$
$692$	12024.6			0.660557
$693$		0			0
$694$	17046.5			0.932386
$695$	5236.58	+	9070.02i	0.285805	+	0.495029i
$696$		0			0
$697$	−118.730	+	205.646i	−0.00645223	+	0.0111756i
$698$	−763.674	+	1322.72i	−0.0414119	+	0.0717275i
$699$		0			0
$700$	110.103	+	190.704i	0.00594501	+	0.0102971i
$701$	13554.2			0.730294			0.365147	−	0.930950i	$-0.381019\pi$
$701$	13554.2			0.730294			0.365147	+	0.930950i	$0.381019\pi$
$702$		0			0
$703$	−26187.4			−1.40495
$704$	−13724.6	−	23771.7i	−0.734751	−	1.27263i
$705$		0			0
$706$	−5981.91	+	10361.0i	−0.318884	+	0.552324i
$707$	1352.30	−	2342.25i	0.0719354	−	0.124596i
$708$		0			0
$709$	4694.58	+	8131.26i	0.248672	+	0.430713i	0.963158	−	0.268937i	$-0.0866724\pi$
$709$	4694.58	+	8131.26i	0.248672	+	0.430713i	−0.714485	+	0.699650i	$0.753339\pi$
$710$	3405.15			0.179990
$711$		0			0
$712$	16287.9			0.857326
$713$	−12539.1	−	21718.3i	−0.658614	−	1.14075i
$714$		0			0
$715$	−2692.64	+	4663.80i	−0.140838	+	0.243939i
$716$	−297.188	+	514.745i	−0.0155118	+	0.0268672i
$717$		0			0
$718$	4428.84	+	7670.97i	0.230199	+	0.398716i
$719$	26301.8			1.36424			0.682122	−	0.731238i	$-0.261057\pi$
$719$	26301.8			1.36424			0.682122	+	0.731238i	$0.261057\pi$
$720$		0			0
$721$	4391.67			0.226844
$722$	−2516.86	−	4359.33i	−0.129734	−	0.224705i
$723$		0			0
$724$	2290.79	−	3967.76i	0.117592	−	0.203675i
$725$	−2847.26	+	4931.59i	−0.145854	+	0.252627i
$726$		0			0
$727$	12769.3	+	22117.1i	0.651427	+	1.12830i	0.982777	+	0.184797i	$0.0591626\pi$
$727$	12769.3	+	22117.1i	0.651427	+	1.12830i	−0.331350	+	0.943508i	$0.607504\pi$
$728$	−1388.56			−0.0706915
$729$		0			0
$730$	−559.868			−0.0283858
$731$	766.510	+	1327.63i	0.0387830	+	0.0671742i
$732$		0			0
$733$	−3491.06	+	6046.70i	−0.175914	+	0.304693i	−0.940477	−	0.339856i	$-0.889622\pi$
$733$	−3491.06	+	6046.70i	−0.175914	+	0.304693i	0.764563	+	0.644549i	$0.222955\pi$
$734$	−12332.7	+	21360.8i	−0.620174	+	1.07417i
$735$		0			0
$736$	−6022.10	−	10430.6i	−0.301600	−	0.522387i
$737$	−21667.5			−1.08295
$738$		0			0
$739$	8863.91			0.441224			0.220612	−	0.975362i	$-0.429195\pi$
$739$	8863.91			0.441224			0.220612	+	0.975362i	$0.429195\pi$
$740$	2179.14	+	3774.38i	0.108252	+	0.187499i
$741$		0			0
$742$	−598.081	+	1035.91i	−0.0295906	+	0.0512524i
$743$	−19472.4	+	33727.3i	−0.961473	+	1.66532i	−0.242668	+	0.970110i	$0.578022\pi$
$743$	−19472.4	+	33727.3i	−0.961473	+	1.66532i	−0.718806	+	0.695211i	$0.755311\pi$
$744$		0			0
$745$	684.843	+	1186.18i	0.0336788	+	0.0583334i
$746$	−13870.5			−0.680742
$747$		0			0
$748$	613.879			0.0300075
$749$	1241.58	+	2150.49i	0.0605694	+	0.104909i
$750$		0			0
$751$	−8795.16	+	15233.7i	−0.427350	+	0.740192i	−0.996637	−	0.0819470i	$-0.973886\pi$
$751$	−8795.16	+	15233.7i	−0.427350	+	0.740192i	0.569287	+	0.822139i	$0.307220\pi$
$752$	−1968.57	+	3409.66i	−0.0954604	+	0.165342i
$753$		0			0
$754$	−5111.84	−	8853.97i	−0.246900	−	0.427643i
$755$	−4684.22			−0.225796
$756$		0			0
$757$	−4075.85			−0.195693			−0.0978463	−	0.995202i	$-0.531195\pi$
$757$	−4075.85			−0.195693			−0.0978463	+	0.995202i	$0.531195\pi$
$758$	−10232.1	−	17722.6i	−0.490300	−	0.849225i
$759$		0			0
$760$	−5870.24	+	10167.6i	−0.280179	+	0.485284i
$761$	20268.5	−	35106.1i	0.965483	−	1.67227i	0.257171	−	0.966366i	$-0.417210\pi$
$761$	20268.5	−	35106.1i	0.965483	−	1.67227i	0.708312	−	0.705899i	$-0.249457\pi$
$762$		0			0
$763$	−1183.17	−	2049.30i	−0.0561382	−	0.0972343i
$764$	−1243.31			−0.0588761
$765$		0			0
$766$	−24463.7			−1.15393
$767$	7579.06	+	13127.3i	0.356798	+	0.617992i
$768$		0			0
$769$	−12016.3	+	20812.9i	−0.563485	+	0.975985i	0.433704	+	0.901056i	$0.357206\pi$
$769$	−12016.3	+	20812.9i	−0.563485	+	0.975985i	−0.997189	+	0.0749293i	$0.976127\pi$
$770$	−799.092	+	1384.07i	−0.0373991	+	0.0647771i
$771$		0			0
$772$	−6233.57	−	10796.9i	−0.290610	−	0.503351i
$773$	20881.5			0.971611			0.485805	−	0.874067i	$-0.338526\pi$
$773$	20881.5			0.971611			0.485805	+	0.874067i	$0.338526\pi$
$774$		0			0
$775$	6978.46			0.323450
$776$	−8901.16	−	15417.3i	−0.411770	−	0.713206i
$777$		0			0
$778$	7209.64	−	12487.5i	0.332234	−	0.575446i
$779$	−3102.82	+	5374.24i	−0.142709	+	0.247179i
$780$		0			0
$781$	−8171.18	−	14152.9i	−0.374376	−	0.648439i
$782$	−721.740			−0.0330043
$783$		0			0
$784$	9518.60			0.433610
$785$	−996.089	−	1725.28i	−0.0452891	−	0.0784430i
$786$		0			0
$787$	2568.20	−	4448.25i	0.116323	−	0.201478i	−0.801985	−	0.597345i	$-0.796223\pi$
$787$	2568.20	−	4448.25i	0.116323	−	0.201478i	0.918308	+	0.395867i	$0.129556\pi$
$788$	1421.33	−	2461.82i	0.0642549	−	0.111293i
$789$		0			0
$790$	−6627.21	−	11478.7i	−0.298463	−	0.516953i
$791$	−2517.77			−0.113175
$792$		0			0
$793$	9990.86			0.447397
$794$	−4355.23	−	7543.48i	−0.194662	−	0.337164i
$795$		0			0
$796$	−4411.22	+	7640.45i	−0.196421	+	0.340212i
$797$	10998.0	−	19049.1i	0.488794	−	0.846615i	−0.511123	−	0.859507i	$-0.670770\pi$
$797$	10998.0	−	19049.1i	0.488794	−	0.846615i	0.999917	+	0.0128922i	$0.00410382\pi$
$798$		0			0
$799$	−253.895	−	439.760i	−0.0112418	−	0.0194713i
$800$	3351.53			0.148118
$801$		0			0
$802$	−9600.03			−0.422679
$803$	1343.49	+	2326.99i	0.0590419	+	0.102264i
$804$		0			0
$805$	−621.262	+	1076.06i	−0.0272008	+	0.0471131i
$806$	−6264.42	+	10850.3i	−0.273765	+	0.474175i
$807$		0			0
$808$	−11999.1	−	20783.1i	−0.522435	−	0.904885i
$809$	−31094.2			−1.35132			−0.675658	−	0.737215i	$-0.736140\pi$
$809$	−31094.2			−1.35132			−0.675658	+	0.737215i	$0.736140\pi$
$810$		0			0
$811$	19130.6			0.828320			0.414160	−	0.910204i	$-0.364075\pi$
$811$	19130.6			0.828320			0.414160	+	0.910204i	$0.364075\pi$
$812$	1003.17	+	1737.55i	0.0433553	+	0.0750936i
$813$		0			0
$814$	−15815.5	+	27393.2i	−0.680997	+	1.17952i
$815$	−1195.38	+	2070.47i	−0.0513773	+	0.0889881i
$816$		0			0
$817$	20031.6	+	34695.7i	0.857792	+	1.48574i
$818$	−19469.8			−0.832208
$819$		0			0
$820$	1032.78			0.0439832
$821$	−1778.81	−	3080.99i	−0.0756162	−	0.130971i	0.825738	−	0.564054i	$-0.190759\pi$
$821$	−1778.81	−	3080.99i	−0.0756162	−	0.130971i	−0.901354	+	0.433083i	$0.857426\pi$
$822$		0			0
$823$	−3074.69	+	5325.52i	−0.130227	+	0.225560i	−0.923764	−	0.382962i	$-0.874904\pi$
$823$	−3074.69	+	5325.52i	−0.130227	+	0.225560i	0.793537	+	0.608522i	$0.208237\pi$
$824$	19484.0	−	33747.2i	0.823733	−	1.42675i
$825$		0			0
$826$	2249.23	+	3895.77i	0.0947464	+	0.164106i
$827$	−21152.8			−0.889425			−0.444713	−	0.895673i	$-0.646694\pi$
$827$	−21152.8			−0.889425			−0.444713	+	0.895673i	$0.646694\pi$
$828$		0			0
$829$	−17402.4			−0.729083			−0.364541	−	0.931187i	$-0.618774\pi$
$829$	−17402.4			−0.729083			−0.364541	+	0.931187i	$0.618774\pi$
$830$	4966.14	+	8601.61i	0.207684	+	0.359719i
$831$		0			0
$832$	−5330.81	+	9233.24i	−0.222131	+	0.384742i
$833$	−613.829	+	1063.18i	−0.0255317	+	0.0442222i
$834$		0			0
$835$	−850.906	−	1473.81i	−0.0352657	−	0.0610819i
$836$	16042.8			0.663698
$837$		0			0
$838$	2225.55			0.0917425
$839$	−9037.12	−	15652.8i	−0.371867	−	0.644092i	0.617986	−	0.786189i	$-0.287949\pi$
$839$	−9037.12	−	15652.8i	−0.371867	−	0.644092i	−0.989853	+	0.142097i	$0.954615\pi$
$840$		0			0
$841$	−13747.5	+	23811.3i	−0.563675	+	0.976314i
$842$	16341.1	−	28303.6i	0.668825	−	1.15844i
$843$		0			0
$844$	7296.31	+	12637.6i	0.297570	+	0.515407i
$845$	−8893.28			−0.362057
$846$		0			0
$847$	3988.55			0.161804
$848$	2796.74	+	4844.09i	0.113255	+	0.196164i
$849$		0			0
$850$	100.419	−	173.931i	0.00405216	−	0.00701855i
$851$	−12295.9	+	21297.1i	−0.495297	+	0.857880i
$852$		0			0
$853$	−20396.1	−	35327.2i	−0.818699	−	1.41803i	−0.906641	−	0.421903i	$-0.861362\pi$
$853$	−20396.1	−	35327.2i	−0.818699	−	1.41803i	0.0879416	−	0.996126i	$-0.471971\pi$
$854$	2964.97			0.118805
$855$		0			0
$856$	22033.5			0.879778
$857$	−5528.93	−	9576.39i	−0.220379	−	0.381707i	0.734544	−	0.678561i	$-0.237396\pi$
$857$	−5528.93	−	9576.39i	−0.220379	−	0.381707i	−0.954923	+	0.296854i	$0.904063\pi$
$858$		0			0
$859$	876.815	−	1518.69i	0.0348272	−	0.0603224i	−0.848086	−	0.529858i	$-0.822245\pi$
$859$	876.815	−	1518.69i	0.0348272	−	0.0603224i	0.882914	+	0.469535i	$0.155579\pi$
$860$	3333.78	−	5774.27i	0.132187	−	0.228955i
$861$		0			0
$862$	−4788.17	−	8293.35i	−0.189194	−	0.327694i
$863$	19186.5			0.756796			0.378398	−	0.925643i	$-0.376475\pi$
$863$	19186.5			0.756796			0.378398	+	0.925643i	$0.376475\pi$
$864$		0			0
$865$	−18880.3			−0.742140
$866$	−10205.9	−	17677.2i	−0.400475	−	0.693642i
$867$		0			0
$868$	1229.36	−	2129.32i	0.0480728	−	0.0832646i
$869$	−31806.0	+	55089.6i	−1.24159	+	2.15050i
$870$		0			0
$871$	4207.97	+	7288.41i	0.163699	+	0.283534i
$872$	−20996.8			−0.815415
$873$		0			0
$874$	−18861.6			−0.729980
$875$	−172.878	−	299.434i	−0.00667925	−	0.0115688i
$876$		0			0
$877$	4257.30	−	7373.86i	0.163921	−	0.283920i	−0.772350	−	0.635197i	$-0.780919\pi$
$877$	4257.30	−	7373.86i	0.163921	−	0.283920i	0.936272	+	0.351277i	$0.114252\pi$
$878$	−1667.94	+	2888.96i	−0.0641121	+	0.111045i
$879$		0			0
$880$	3736.70	+	6472.16i	0.143141	+	0.247928i
$881$	−41177.0			−1.57467			−0.787337	−	0.616522i	$-0.788541\pi$
$881$	−41177.0			−1.57467			−0.787337	+	0.616522i	$0.788541\pi$
$882$		0			0
$883$	32540.4			1.24017			0.620086	−	0.784533i	$-0.287098\pi$
$883$	32540.4			1.24017			0.620086	+	0.784533i	$0.287098\pi$
$884$	−119.219	−	206.494i	−0.00453595	−	0.00785650i
$885$		0			0
$886$	4896.34	−	8480.70i	0.185661	−	0.321574i
$887$	1083.05	−	1875.90i	0.0409980	−	0.0710106i	−0.844798	−	0.535085i	$-0.820280\pi$
$887$	1083.05	−	1875.90i	0.0409980	−	0.0710106i	0.885796	+	0.464074i	$0.153613\pi$
$888$		0			0
$889$	2818.95	+	4882.56i	0.106349	+	0.184202i
$890$	−7281.53			−0.274244
$891$		0			0
$892$	−13041.2			−0.489522
$893$	−6635.17	−	11492.4i	−0.248642	−	0.430661i
$894$		0			0
$895$	466.629	−	808.225i	0.0174276	−	0.0301855i
$896$	−98.7391	+	171.021i	−0.00368152	+	0.00637658i
$897$		0			0
$898$	−5893.91	−	10208.5i	−0.219023	−	0.379358i
$899$	63582.3			2.35883
$900$		0			0
$901$	−721.416			−0.0266747
$902$	3747.79	+	6491.36i	0.138346	+	0.239622i
$903$		0			0
$904$	−11170.3	+	19347.5i	−0.410971	+	0.711823i
$905$	−3596.88	+	6229.97i	−0.132115	+	0.228830i
$906$		0			0
$907$	353.031	+	611.468i	0.0129241	+	0.0223853i	0.872415	−	0.488766i	$-0.162553\pi$
$907$	353.031	+	611.468i	0.0129241	+	0.0223853i	−0.859491	+	0.511151i	$0.829219\pi$
$908$	−12321.6			−0.450339
$909$		0			0
$910$	620.756			0.0226130
$911$	−247.743	−	429.103i	−0.00900997	−	0.0156057i	0.861485	−	0.507783i	$-0.169535\pi$
$911$	−247.743	−	429.103i	−0.00900997	−	0.0156057i	−0.870495	+	0.492177i	$0.836201\pi$
$912$		0			0
$913$	23834.0	−	41281.7i	0.863955	−	1.49641i
$914$	−17034.8	+	29505.2i	−0.616480	+	1.06777i
$915$		0			0
$916$	1967.99	+	3408.66i	0.0709871	+	0.122953i
$917$	−651.806			−0.0234728
$918$		0			0
$919$	20473.6			0.734889			0.367445	−	0.930045i	$-0.380233\pi$
$919$	20473.6			0.734889			0.367445	+	0.930045i	$0.380233\pi$
$920$	5512.56	+	9548.03i	0.197547	+	0.342162i
$921$		0			0
$922$	−61.9030	+	107.219i	−0.00221114	+	0.00382980i
$923$	−3173.80	+	5497.17i	−0.113182	+	0.196037i
$924$		0			0
$925$	−3421.57	−	5926.33i	−0.121622	−	0.210656i
$926$	29348.0			1.04151
$927$		0			0
$928$	30536.5			1.08018
$929$	24857.7	+	43054.7i	0.877883	+	1.52054i	0.853660	+	0.520831i	$0.174378\pi$
$929$	24857.7	+	43054.7i	0.877883	+	1.52054i	0.0242231	+	0.999707i	$0.492289\pi$
$930$		0			0
$931$	−16041.5	+	27784.7i	−0.564704	+	0.978095i
$932$	−3514.08	+	6086.57i	−0.123506	+	0.213919i
$933$		0			0
$934$	2937.77	+	5088.36i	0.102919	+	0.178261i
$935$	−963.880			−0.0337136
$936$		0			0
$937$	−31524.9			−1.09912			−0.549560	−	0.835454i	$-0.685204\pi$
$937$	−31524.9			−1.09912			−0.549560	+	0.835454i	$0.685204\pi$
$938$	1248.79	+	2162.97i	0.0434696	+	0.0752916i
$939$		0			0
$940$	−1104.27	+	1912.64i	−0.0383161	+	0.0663655i
$941$	18764.2	−	32500.6i	0.650050	−	1.12592i	−0.333060	−	0.942905i	$-0.608081\pi$
$941$	18764.2	−	32500.6i	0.650050	−	1.12592i	0.983110	−	0.183014i	$-0.0585854\pi$
$942$		0			0
$943$	2913.76	+	5046.78i	0.100620	+	0.174280i
$944$	21035.6			0.725265
$945$		0			0
$946$	48390.9			1.66313
$947$	12931.7	+	22398.3i	0.443741	+	0.768582i	0.997964	−	0.0637862i	$-0.0203176\pi$
$947$	12931.7	+	22398.3i	0.443741	+	0.768582i	−0.554222	+	0.832369i	$0.686984\pi$
$948$		0			0
$949$	521.829	−	903.834i	0.0178496	−	0.0309164i
$950$	2624.29	−	4545.41i	0.0896246	−	0.155234i
$951$		0			0
$952$	−124.265	−	215.233i	−0.00423051	−	0.00732746i
$953$	−1060.03			−0.0360312			−0.0180156	−	0.999838i	$-0.505735\pi$
$953$	−1060.03			−0.0360312			−0.0180156	+	0.999838i	$0.505735\pi$
$954$		0			0
$955$	1952.18			0.0661476
$956$	−1395.81	−	2417.61i	−0.0472213	−	0.0817897i
$957$		0			0
$958$	−3807.28	+	6594.40i	−0.128400	+	0.222396i
$959$	−4111.58	+	7121.46i	−0.138446	+	0.239796i
$960$		0			0
$961$	−24063.7	−	41679.5i	−0.807750	−	1.39906i
$962$	12285.9			0.411759
$963$		0			0
$964$	1521.71			0.0508412
$965$	9787.62	+	16952.6i	0.326502	+	0.565518i
$966$		0			0
$967$	2592.51	−	4490.35i	0.0862145	−	0.149328i	−0.819694	−	0.572802i	$-0.805856\pi$
$967$	2592.51	−	4490.35i	0.0862145	−	0.149328i	0.905908	+	0.423474i	$0.139190\pi$
$968$	17695.5	−	30649.5i	0.587557	−	1.01768i
$969$		0			0
$970$	3979.27	+	6892.30i	0.131718	+	0.228143i
$971$	−28314.9			−0.935805			−0.467903	−	0.883780i	$-0.654990\pi$
$971$	−28314.9			−0.935805			−0.467903	+	0.883780i	$0.654990\pi$
$972$		0			0
$973$	−5793.85			−0.190897
$974$	15405.6	+	26683.4i	0.506806	+	0.877813i
$975$		0			0
$976$	6932.38	−	12007.2i	0.227357	−	0.393793i
$977$	22448.3	−	38881.6i	0.735093	−	1.27322i	−0.219590	−	0.975592i	$-0.570472\pi$
$977$	22448.3	−	38881.6i	0.735093	−	1.27322i	0.954683	−	0.297626i	$-0.0961948\pi$
$978$		0			0
$979$	17473.1	+	30264.3i	0.570422	+	0.988001i
$980$	5339.45			0.174043
$981$		0			0
$982$	−21539.4			−0.699950
$983$	−7286.40	−	12620.4i	−0.236419	−	0.409490i	0.723265	−	0.690571i	$-0.242640\pi$
$983$	−7286.40	−	12620.4i	−0.236419	−	0.409490i	−0.959684	+	0.281081i	$0.909307\pi$
$984$		0			0
$985$	−2231.70	+	3865.42i	−0.0721908	+	0.125038i
$986$	914.938	−	1584.72i	0.0295513	−	0.0511843i
$987$		0			0
$988$	−3115.62	−	5396.41i	−0.100325	−	0.173768i
$989$	37622.0			1.20962
$990$		0			0
$991$	10602.1			0.339844			0.169922	−	0.985457i	$-0.445648\pi$
$991$	10602.1			0.339844			0.169922	+	0.985457i	$0.445648\pi$
$992$	−18710.8	−	32408.1i	−0.598860	−	1.03726i
$993$		0			0
$994$	−941.882	+	1631.39i	−0.0300550	+	0.0520568i
$995$	6926.26	−	11996.6i	0.220681	−	0.382230i
$996$		0			0
$997$	−13948.5	−	24159.5i	−0.443082	−	0.767441i	0.554834	−	0.831961i	$-0.312782\pi$
$997$	−13948.5	−	24159.5i	−0.443082	−	0.767441i	−0.997916	+	0.0645202i	$0.979448\pi$
$998$	−26447.4			−0.838857
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.e.c.91.3		14
3.2	odd	2		45.4.e.c.31.5	yes	14
9.2	odd	6		45.4.e.c.16.5	✓	14
9.4	even	3		405.4.a.n.1.5		7
9.5	odd	6		405.4.a.m.1.3		7
9.7	even	3	inner	135.4.e.c.46.3		14
15.2	even	4		225.4.k.d.49.5		28
15.8	even	4		225.4.k.d.49.10		28
15.14	odd	2		225.4.e.d.76.3		14
45.2	even	12		225.4.k.d.124.10		28
45.4	even	6		2025.4.a.ba.1.3		7
45.14	odd	6		2025.4.a.bb.1.5		7
45.29	odd	6		225.4.e.d.151.3		14
45.38	even	12		225.4.k.d.124.5		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.5	✓	14	9.2	odd	6
45.4.e.c.31.5	yes	14	3.2	odd	2
135.4.e.c.46.3		14	9.7	even	3	inner
135.4.e.c.91.3		14	1.1	even	1	trivial
225.4.e.d.76.3		14	15.14	odd	2
225.4.e.d.151.3		14	45.29	odd	6
225.4.k.d.49.5		28	15.2	even	4
225.4.k.d.49.10		28	15.8	even	4
225.4.k.d.124.5		28	45.38	even	12
225.4.k.d.124.10		28	45.2	even	12
405.4.a.m.1.3		7	9.5	odd	6
405.4.a.n.1.5		7	9.4	even	3
2025.4.a.ba.1.3		7	45.4	even	6
2025.4.a.bb.1.5		7	45.14	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.09722 - 1.90044i) q^{2} +(1.59221 - 2.75778i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-1.38302 - 2.39547i) q^{7} -24.5436 q^{8} +10.9722 q^{10} +(-26.3295 - 45.6040i) q^{11} +(-10.2267 + 17.7132i) q^{13} +(-3.03497 + 5.25672i) q^{14} +(14.1921 + 24.5814i) q^{16} -3.66084 q^{17} -95.6705 q^{19} +(7.96104 + 13.7889i) q^{20} +(-57.7786 + 100.075i) q^{22} +(-44.9206 + 77.8047i) q^{23} +(-12.5000 - 21.6506i) q^{25} +44.8840 q^{26} -8.80825 q^{28} +(-113.890 - 197.264i) q^{29} +(-139.569 + 241.741i) q^{31} +(-67.0306 + 116.100i) q^{32} +(4.01675 + 6.95722i) q^{34} +13.8302 q^{35} +273.725 q^{37} +(104.972 + 181.816i) q^{38} +(61.3589 - 106.277i) q^{40} +(32.4323 - 56.1744i) q^{41} +(-209.381 - 362.658i) q^{43} -167.688 q^{44} +197.151 q^{46} +(69.3544 + 120.125i) q^{47} +(167.674 - 290.421i) q^{49} +(-27.4306 + 47.5111i) q^{50} +(32.5661 + 56.4062i) q^{52} +197.063 q^{53} +263.295 q^{55} +(33.9444 + 58.7934i) q^{56} +(-249.926 + 432.884i) q^{58} +(370.552 - 641.814i) q^{59} +(-244.234 - 423.026i) q^{61} +612.554 q^{62} +521.263 q^{64} +(-51.1336 - 88.5660i) q^{65} +(205.734 - 356.341i) q^{67} +(-5.82882 + 10.0958i) q^{68} +(-15.1748 - 26.2836i) q^{70} +310.343 q^{71} -51.0260 q^{73} +(-300.338 - 520.200i) q^{74} +(-152.327 + 263.839i) q^{76} +(-72.8286 + 126.143i) q^{77} +(-603.999 - 1046.16i) q^{79} -141.921 q^{80} -142.342 q^{82} +(452.611 + 783.945i) q^{83} +(9.15210 - 15.8519i) q^{85} +(-459.475 + 795.833i) q^{86} +(646.220 + 1119.29i) q^{88} -663.633 q^{89} +56.5752 q^{91} +(143.046 + 247.763i) q^{92} +(152.194 - 263.608i) q^{94} +(239.176 - 414.265i) q^{95} +(362.668 + 628.159i) q^{97} -735.905 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{2} - 36 q^{4} - 35 q^{5} - 22 q^{7} + 36 q^{8} + 20 q^{10} - 23 q^{11} - 96 q^{13} + 21 q^{14} - 324 q^{16} + 322 q^{17} + 558 q^{19} - 180 q^{20} - 311 q^{22} - 96 q^{23} - 175 q^{25} - 716 q^{26}+ \cdots + 2558 q^{98}+O(q^{100}) \) 14 * q - 2 * q^2 - 36 * q^4 - 35 * q^5 - 22 * q^7 + 36 * q^8 + 20 * q^10 - 23 * q^11 - 96 * q^13 + 21 * q^14 - 324 * q^16 + 322 * q^17 + 558 * q^19 - 180 * q^20 - 311 * q^22 - 96 * q^23 - 175 * q^25 - 716 * q^26 + 674 * q^28 + 296 * q^29 - 244 * q^31 + 314 * q^32 - 125 * q^34 + 220 * q^35 + 808 * q^37 - 305 * q^38 - 90 * q^40 + 47 * q^41 - 525 * q^43 + 110 * q^44 + 1434 * q^46 - 164 * q^47 - 1225 * q^49 - 50 * q^50 - 1682 * q^52 + 1012 * q^53 + 230 * q^55 + 981 * q^56 - 1183 * q^58 + 85 * q^59 - 828 * q^61 - 1572 * q^62 + 4472 * q^64 - 480 * q^65 - 1093 * q^67 - 2473 * q^68 + 105 * q^70 + 656 * q^71 + 4170 * q^73 + 1316 * q^74 - 2789 * q^76 - 24 * q^77 - 2110 * q^79 + 3240 * q^80 - 124 * q^82 - 1290 * q^83 - 805 * q^85 + 2569 * q^86 - 2271 * q^88 - 6096 * q^89 + 6676 * q^91 - 2763 * q^92 + 517 * q^94 - 1395 * q^95 - 1787 * q^97 + 2558 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more