LMFDB - Embedded newform 208.4.i.b.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(81,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("208.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.i (of order $3$, degree $2$, not 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:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	208.81
Dual form			208.4.i.b.113.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{3} +17.0000 q^{5} +(10.0000 - 17.3205i) q^{7} +(11.5000 - 19.9186i) q^{9} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{3} +17.0000 q^{5} +(10.0000 - 17.3205i) q^{7} +(11.5000 - 19.9186i) q^{9} +(-16.0000 - 27.7128i) q^{11} +(-45.5000 + 11.2583i) q^{13} +(17.0000 + 29.4449i) q^{15} +(6.50000 - 11.2583i) q^{17} +(15.0000 - 25.9808i) q^{19} +40.0000 q^{21} +(39.0000 + 67.5500i) q^{23} +164.000 q^{25} +100.000 q^{27} +(-98.5000 - 170.607i) q^{29} +74.0000 q^{31} +(32.0000 - 55.4256i) q^{33} +(170.000 - 294.449i) q^{35} +(113.500 + 196.588i) q^{37} +(-65.0000 - 67.5500i) q^{39} +(82.5000 + 142.894i) q^{41} +(-78.0000 + 135.100i) q^{43} +(195.500 - 338.616i) q^{45} +162.000 q^{47} +(-28.5000 - 49.3634i) q^{49} +26.0000 q^{51} +93.0000 q^{53} +(-272.000 - 471.118i) q^{55} +60.0000 q^{57} +(-432.000 + 748.246i) q^{59} +(-72.5000 + 125.574i) q^{61} +(-230.000 - 398.372i) q^{63} +(-773.500 + 191.392i) q^{65} +(431.000 + 746.514i) q^{67} +(-78.0000 + 135.100i) q^{69} +(327.000 - 566.381i) q^{71} +215.000 q^{73} +(164.000 + 284.056i) q^{75} -640.000 q^{77} +76.0000 q^{79} +(-210.500 - 364.597i) q^{81} -628.000 q^{83} +(110.500 - 191.392i) q^{85} +(197.000 - 341.214i) q^{87} +(133.000 + 230.363i) q^{89} +(-260.000 + 900.666i) q^{91} +(74.0000 + 128.172i) q^{93} +(255.000 - 441.673i) q^{95} +(-119.000 + 206.114i) q^{97} -736.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 34 q^{5} + 20 q^{7} + 23 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 34 * q^5 + 20 * q^7 + 23 * q^9	$ 2 q + 2 q^{3} + 34 q^{5} + 20 q^{7} + 23 q^{9} - 32 q^{11} - 91 q^{13} + 34 q^{15} + 13 q^{17} + 30 q^{19} + 80 q^{21} + 78 q^{23} + 328 q^{25} + 200 q^{27} - 197 q^{29} + 148 q^{31} + 64 q^{33} + 340 q^{35} + 227 q^{37} - 130 q^{39} + 165 q^{41} - 156 q^{43} + 391 q^{45} + 324 q^{47} - 57 q^{49} + 52 q^{51} + 186 q^{53} - 544 q^{55} + 120 q^{57} - 864 q^{59} - 145 q^{61} - 460 q^{63} - 1547 q^{65} + 862 q^{67} - 156 q^{69} + 654 q^{71} + 430 q^{73} + 328 q^{75} - 1280 q^{77} + 152 q^{79} - 421 q^{81} - 1256 q^{83} + 221 q^{85} + 394 q^{87} + 266 q^{89} - 520 q^{91} + 148 q^{93} + 510 q^{95} - 238 q^{97} - 1472 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 34 * q^5 + 20 * q^7 + 23 * q^9 - 32 * q^11 - 91 * q^13 + 34 * q^15 + 13 * q^17 + 30 * q^19 + 80 * q^21 + 78 * q^23 + 328 * q^25 + 200 * q^27 - 197 * q^29 + 148 * q^31 + 64 * q^33 + 340 * q^35 + 227 * q^37 - 130 * q^39 + 165 * q^41 - 156 * q^43 + 391 * q^45 + 324 * q^47 - 57 * q^49 + 52 * q^51 + 186 * q^53 - 544 * q^55 + 120 * q^57 - 864 * q^59 - 145 * q^61 - 460 * q^63 - 1547 * q^65 + 862 * q^67 - 156 * q^69 + 654 * q^71 + 430 * q^73 + 328 * q^75 - 1280 * q^77 + 152 * q^79 - 421 * q^81 - 1256 * q^83 + 221 * q^85 + 394 * q^87 + 266 * q^89 - 520 * q^91 + 148 * q^93 + 510 * q^95 - 238 * q^97 - 1472 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000	+	1.73205i	0.192450	+	0.333333i	0.946062	−	0.323987i	$-0.105023\pi$
$3$	1.00000	+	1.73205i	0.192450	+	0.333333i	−0.753612	+	0.657320i	$0.771690\pi$
$4$		0			0
$5$	17.0000			1.52053			0.760263	−	0.649615i	$-0.225070\pi$
$5$	17.0000			1.52053			0.760263	+	0.649615i	$0.225070\pi$
$6$		0			0
$7$	10.0000	−	17.3205i	0.539949	−	0.935220i	−0.458957	−	0.888459i	$-0.651777\pi$
$7$	10.0000	−	17.3205i	0.539949	−	0.935220i	0.998906	−	0.0467610i	$-0.0148899\pi$
$8$		0			0
$9$	11.5000	−	19.9186i	0.425926	−	0.737725i
$10$		0			0
$11$	−16.0000	−	27.7128i	−0.438562	−	0.759612i	0.559017	−	0.829156i	$-0.311179\pi$
$11$	−16.0000	−	27.7128i	−0.438562	−	0.759612i	−0.997579	+	0.0695447i	$0.977845\pi$
$12$		0			0
$13$	−45.5000	+	11.2583i	−0.970725	+	0.240192i
$13$	−45.5000	+	11.2583i	−0.970725	+	0.240192i
$14$		0			0
$15$	17.0000	+	29.4449i	0.292625	+	0.506842i
$16$		0			0
$17$	6.50000	−	11.2583i	0.0927342	−	0.160620i	−0.815927	−	0.578156i	$-0.803773\pi$
$17$	6.50000	−	11.2583i	0.0927342	−	0.160620i	0.908661	+	0.417535i	$0.137106\pi$
$18$		0			0
$19$	15.0000	−	25.9808i	0.181118	−	0.313705i	−0.761144	−	0.648583i	$-0.775362\pi$
$19$	15.0000	−	25.9808i	0.181118	−	0.313705i	0.942261	+	0.334878i	$0.108695\pi$
$20$		0			0
$21$	40.0000			0.415653
$22$		0			0
$23$	39.0000	+	67.5500i	0.353568	+	0.612398i	0.986872	−	0.161506i	$-0.0516350\pi$
$23$	39.0000	+	67.5500i	0.353568	+	0.612398i	−0.633304	+	0.773903i	$0.718302\pi$
$24$		0			0
$25$	164.000			1.31200
$26$		0			0
$27$	100.000			0.712778
$28$		0			0
$29$	−98.5000	−	170.607i	−0.630724	−	1.09245i	−0.987404	−	0.158219i	$-0.949425\pi$
$29$	−98.5000	−	170.607i	−0.630724	−	1.09245i	0.356680	−	0.934227i	$-0.383909\pi$
$30$		0			0
$31$	74.0000			0.428735			0.214368	−	0.976753i	$-0.431231\pi$
$31$	74.0000			0.428735			0.214368	+	0.976753i	$0.431231\pi$
$32$		0			0
$33$	32.0000	−	55.4256i	0.168803	−	0.292375i
$34$		0			0
$35$	170.000	−	294.449i	0.821007	−	1.42203i
$36$		0			0
$37$	113.500	+	196.588i	0.504305	+	0.873482i	0.999988	+	0.00497814i	$0.00158460\pi$
$37$	113.500	+	196.588i	0.504305	+	0.873482i	−0.495683	+	0.868504i	$0.665082\pi$
$38$		0			0
$39$	−65.0000	−	67.5500i	−0.266880	−	0.277350i
$40$		0			0
$41$	82.5000	+	142.894i	0.314252	+	0.544301i	0.979278	−	0.202520i	$-0.0649130\pi$
$41$	82.5000	+	142.894i	0.314252	+	0.544301i	−0.665026	+	0.746820i	$0.731580\pi$
$42$		0			0
$43$	−78.0000	+	135.100i	−0.276625	+	0.479129i	−0.970544	−	0.240924i	$-0.922549\pi$
$43$	−78.0000	+	135.100i	−0.276625	+	0.479129i	0.693919	+	0.720053i	$0.255883\pi$
$44$		0			0
$45$	195.500	−	338.616i	0.647632	−	1.12173i
$46$		0			0
$47$	162.000			0.502769			0.251384	−	0.967887i	$-0.419114\pi$
$47$	162.000			0.502769			0.251384	+	0.967887i	$0.419114\pi$
$48$		0			0
$49$	−28.5000	−	49.3634i	−0.0830904	−	0.143917i
$50$		0			0
$51$	26.0000			0.0713868
$52$		0			0
$53$	93.0000			0.241029			0.120514	−	0.992712i	$-0.461546\pi$
$53$	93.0000			0.241029			0.120514	+	0.992712i	$0.461546\pi$
$54$		0			0
$55$	−272.000	−	471.118i	−0.666845	−	1.15501i
$56$		0			0
$57$	60.0000			0.139424
$58$		0			0
$59$	−432.000	+	748.246i	−0.953248	+	1.65107i	−0.214919	+	0.976632i	$0.568949\pi$
$59$	−432.000	+	748.246i	−0.953248	+	1.65107i	−0.738328	+	0.674442i	$0.764384\pi$
$60$		0			0
$61$	−72.5000	+	125.574i	−0.152175	+	0.263575i	−0.932027	−	0.362389i	$-0.881961\pi$
$61$	−72.5000	+	125.574i	−0.152175	+	0.263575i	0.779852	+	0.625964i	$0.215294\pi$
$62$		0			0
$63$	−230.000	−	398.372i	−0.459957	−	0.796668i
$64$		0			0
$65$	−773.500	+	191.392i	−1.47601	+	0.365219i
$66$		0			0
$67$	431.000	+	746.514i	0.785896	+	1.36121i	0.928462	+	0.371427i	$0.121131\pi$
$67$	431.000	+	746.514i	0.785896	+	1.36121i	−0.142566	+	0.989785i	$0.545535\pi$
$68$		0			0
$69$	−78.0000	+	135.100i	−0.136088	+	0.235712i
$70$		0			0
$71$	327.000	−	566.381i	0.546588	−	0.946718i	−0.451917	−	0.892060i	$-0.649260\pi$
$71$	327.000	−	566.381i	0.546588	−	0.946718i	0.998505	−	0.0546585i	$-0.0174070\pi$
$72$		0			0
$73$	215.000			0.344710			0.172355	−	0.985035i	$-0.444862\pi$
$73$	215.000			0.344710			0.172355	+	0.985035i	$0.444862\pi$
$74$		0			0
$75$	164.000	+	284.056i	0.252495	+	0.437333i
$76$		0			0
$77$	−640.000			−0.947205
$78$		0			0
$79$	76.0000			0.108236			0.0541182	−	0.998535i	$-0.482765\pi$
$79$	76.0000			0.108236			0.0541182	+	0.998535i	$0.482765\pi$
$80$		0			0
$81$	−210.500	−	364.597i	−0.288752	−	0.500133i
$82$		0			0
$83$	−628.000			−0.830505			−0.415253	−	0.909706i	$-0.636307\pi$
$83$	−628.000			−0.830505			−0.415253	+	0.909706i	$0.636307\pi$
$84$		0			0
$85$	110.500	−	191.392i	0.141005	−	0.244227i
$86$		0			0
$87$	197.000	−	341.214i	0.242766	−	0.420483i
$88$		0			0
$89$	133.000	+	230.363i	0.158404	+	0.274364i	0.934293	−	0.356505i	$-0.116032\pi$
$89$	133.000	+	230.363i	0.158404	+	0.274364i	−0.775889	+	0.630869i	$0.782698\pi$
$90$		0			0
$91$	−260.000	+	900.666i	−0.299510	+	1.03753i
$92$		0			0
$93$	74.0000	+	128.172i	0.0825101	+	0.142912i
$94$		0			0
$95$	255.000	−	441.673i	0.275394	−	0.476997i
$96$		0			0
$97$	−119.000	+	206.114i	−0.124563	+	0.215750i	−0.921562	−	0.388231i	$-0.873086\pi$
$97$	−119.000	+	206.114i	−0.124563	+	0.215750i	0.796999	+	0.603981i	$0.206420\pi$
$98$		0			0
$99$	−736.000			−0.747180
$100$		0			0
$101$	409.500	+	709.275i	0.403433	+	0.698767i	0.994138	−	0.108121i	$-0.0344834\pi$
$101$	409.500	+	709.275i	0.403433	+	0.698767i	−0.590704	+	0.806888i	$0.701150\pi$
$102$		0			0
$103$	−1638.00			−1.56696			−0.783480	−	0.621417i	$-0.786557\pi$
$103$	−1638.00			−1.56696			−0.783480	+	0.621417i	$0.786557\pi$
$104$		0			0
$105$	680.000			0.632011
$106$		0			0
$107$	261.000	+	452.065i	0.235811	+	0.408437i	0.959508	−	0.281681i	$-0.0908919\pi$
$107$	261.000	+	452.065i	0.235811	+	0.408437i	−0.723697	+	0.690118i	$0.757559\pi$
$108$		0			0
$109$	−1634.00			−1.43586			−0.717930	−	0.696115i	$-0.754910\pi$
$109$	−1634.00			−1.43586			−0.717930	+	0.696115i	$0.754910\pi$
$110$		0			0
$111$	−227.000	+	393.176i	−0.194107	+	0.336203i
$112$		0			0
$113$	−163.500	+	283.190i	−0.136113	+	0.235755i	−0.926022	−	0.377469i	$-0.876794\pi$
$113$	−163.500	+	283.190i	−0.136113	+	0.235755i	0.789909	+	0.613224i	$0.210128\pi$
$114$		0			0
$115$	663.000	+	1148.35i	0.537609	+	0.931167i
$116$		0			0
$117$	−299.000	+	1035.77i	−0.236261	+	0.818433i
$118$		0			0
$119$	−130.000	−	225.167i	−0.100144	−	0.173454i
$120$		0			0
$121$	153.500	−	265.870i	0.115327	−	0.199752i
$122$		0			0
$123$	−165.000	+	285.788i	−0.120956	+	0.209501i
$124$		0			0
$125$	663.000			0.474404
$126$		0			0
$127$	−1079.00	−	1868.88i	−0.753904	−	1.30580i	−0.945918	−	0.324407i	$-0.894835\pi$
$127$	−1079.00	−	1868.88i	−0.753904	−	1.30580i	0.192014	−	0.981392i	$-0.438498\pi$
$128$		0			0
$129$	−312.000			−0.212946
$130$		0			0
$131$	−730.000			−0.486873			−0.243437	−	0.969917i	$-0.578275\pi$
$131$	−730.000			−0.486873			−0.243437	+	0.969917i	$0.578275\pi$
$132$		0			0
$133$	−300.000	−	519.615i	−0.195589	−	0.338770i
$134$		0			0
$135$	1700.00			1.08380
$136$		0			0
$137$	−835.500	+	1447.13i	−0.521033	+	0.902456i	0.478667	+	0.877996i	$0.341120\pi$
$137$	−835.500	+	1447.13i	−0.521033	+	0.902456i	−0.999701	+	0.0244601i	$0.992213\pi$
$138$		0			0
$139$	456.000	−	789.815i	0.278255	−	0.481951i	−0.692696	−	0.721229i	$-0.743577\pi$
$139$	456.000	−	789.815i	0.278255	−	0.481951i	0.970951	+	0.239278i	$0.0769107\pi$
$140$		0			0
$141$	162.000	+	280.592i	0.0967579	+	0.167590i
$142$		0			0
$143$	1040.00	+	1080.80i	0.608176	+	0.632035i
$144$		0			0
$145$	−1674.50	−	2900.32i	−0.959032	−	1.66109i
$146$		0			0
$147$	57.0000	−	98.7269i	0.0319815	−	0.0553936i
$148$		0			0
$149$	1057.50	−	1831.64i	0.581435	−	1.00707i	−0.413875	−	0.910334i	$-0.635825\pi$
$149$	1057.50	−	1831.64i	0.581435	−	1.00707i	0.995310	−	0.0967407i	$-0.0308418\pi$
$150$		0			0
$151$	−514.000			−0.277011			−0.138506	−	0.990362i	$-0.544230\pi$
$151$	−514.000			−0.277011			−0.138506	+	0.990362i	$0.544230\pi$
$152$		0			0
$153$	−149.500	−	258.942i	−0.0789958	−	0.136825i
$154$		0			0
$155$	1258.00			0.651903
$156$		0			0
$157$	2901.00			1.47468			0.737341	−	0.675521i	$-0.236081\pi$
$157$	2901.00			1.47468			0.737341	+	0.675521i	$0.236081\pi$
$158$		0			0
$159$	93.0000	+	161.081i	0.0463860	+	0.0803430i
$160$		0			0
$161$	1560.00			0.763635
$162$		0			0
$163$	1180.00	−	2043.82i	0.567023	−	0.982112i	−0.429835	−	0.902907i	$-0.641428\pi$
$163$	1180.00	−	2043.82i	0.567023	−	0.982112i	0.996858	−	0.0792052i	$-0.0252382\pi$
$164$		0			0
$165$	544.000	−	942.236i	0.256669	−	0.444563i
$166$		0			0
$167$	140.000	+	242.487i	0.0648714	+	0.112361i	0.896637	−	0.442767i	$-0.146003\pi$
$167$	140.000	+	242.487i	0.0648714	+	0.112361i	−0.831766	+	0.555127i	$0.812670\pi$
$168$		0			0
$169$	1943.50	−	1024.51i	0.884615	−	0.466321i
$170$		0			0
$171$	−345.000	−	597.558i	−0.154285	−	0.267230i
$172$		0			0
$173$	−663.000	+	1148.35i	−0.291370	+	0.504667i	−0.974134	−	0.225972i	$-0.927444\pi$
$173$	−663.000	+	1148.35i	−0.291370	+	0.504667i	0.682764	+	0.730639i	$0.260778\pi$
$174$		0			0
$175$	1640.00	−	2840.56i	0.708413	−	1.22701i
$176$		0			0
$177$	−1728.00			−0.733810
$178$		0			0
$179$	2132.00	+	3692.73i	0.890241	+	1.54194i	0.839586	+	0.543227i	$0.182798\pi$
$179$	2132.00	+	3692.73i	0.890241	+	1.54194i	0.0506550	+	0.998716i	$0.483869\pi$
$180$		0			0
$181$	−403.000			−0.165496			−0.0827479	−	0.996571i	$-0.526370\pi$
$181$	−403.000			−0.165496			−0.0827479	+	0.996571i	$0.526370\pi$
$182$		0			0
$183$	−290.000			−0.117144
$184$		0			0
$185$	1929.50	+	3341.99i	0.766809	+	1.32815i
$186$		0			0
$187$	−416.000			−0.162679
$188$		0			0
$189$	1000.00	−	1732.05i	0.384864	−	0.666604i
$190$		0			0
$191$	−623.000	+	1079.07i	−0.236014	+	0.408788i	−0.959567	−	0.281481i	$-0.909174\pi$
$191$	−623.000	+	1079.07i	−0.236014	+	0.408788i	0.723553	+	0.690269i	$0.242508\pi$
$192$		0			0
$193$	−133.500	−	231.229i	−0.0497904	−	0.0862394i	0.840056	−	0.542500i	$-0.182522\pi$
$193$	−133.500	−	231.229i	−0.0497904	−	0.0862394i	−0.889846	+	0.456260i	$0.849189\pi$
$194$		0			0
$195$	−1105.00	−	1148.35i	−0.405798	−	0.421718i
$196$		0			0
$197$	−639.000	−	1106.78i	−0.231101	−	0.400278i	0.727032	−	0.686604i	$-0.240899\pi$
$197$	−639.000	−	1106.78i	−0.231101	−	0.400278i	−0.958132	+	0.286326i	$0.907566\pi$
$198$		0			0
$199$	2119.00	−	3670.22i	0.754834	−	1.30741i	−0.190623	−	0.981663i	$-0.561051\pi$
$199$	2119.00	−	3670.22i	0.754834	−	1.30741i	0.945457	−	0.325747i	$-0.105616\pi$
$200$		0			0
$201$	−862.000	+	1493.03i	−0.302492	+	0.523931i
$202$		0			0
$203$	−3940.00			−1.36224
$204$		0			0
$205$	1402.50	+	2429.20i	0.477829	+	0.827623i
$206$		0			0
$207$	1794.00			0.602375
$208$		0			0
$209$	−960.000			−0.317725
$210$		0			0
$211$	1535.00	+	2658.70i	0.500823	+	0.867452i	1.00000		0.000951154i	$0.000302762\pi$
$211$	1535.00	+	2658.70i	0.500823	+	0.867452i	−0.499176	+	0.866501i	$0.666364\pi$
$212$		0			0
$213$	1308.00			0.420764
$214$		0			0
$215$	−1326.00	+	2296.70i	−0.420616	+	0.728528i
$216$		0			0
$217$	740.000	−	1281.72i	0.231495	−	0.400962i
$218$		0			0
$219$	215.000	+	372.391i	0.0663395	+	0.114903i
$220$		0			0
$221$	−169.000	+	585.433i	−0.0514397	+	0.178192i
$222$		0			0
$223$	−2689.00	−	4657.48i	−0.807483	−	1.39860i	−0.914602	−	0.404356i	$-0.867496\pi$
$223$	−2689.00	−	4657.48i	−0.807483	−	1.39860i	0.107119	−	0.994246i	$-0.465838\pi$
$224$		0			0
$225$	1886.00	−	3266.65i	0.558815	−	0.967896i
$226$		0			0
$227$	−1987.00	+	3441.58i	−0.580977	+	1.00628i	0.414387	+	0.910101i	$0.363996\pi$
$227$	−1987.00	+	3441.58i	−0.580977	+	1.00628i	−0.995364	+	0.0961811i	$0.969337\pi$
$228$		0			0
$229$	−6298.00			−1.81740			−0.908698	−	0.417455i	$-0.862922\pi$
$229$	−6298.00			−1.81740			−0.908698	+	0.417455i	$0.862922\pi$
$230$		0			0
$231$	−640.000	−	1108.51i	−0.182290	−	0.315735i
$232$		0			0
$233$	4030.00			1.13311			0.566554	−	0.824025i	$-0.308276\pi$
$233$	4030.00			1.13311			0.566554	+	0.824025i	$0.308276\pi$
$234$		0			0
$235$	2754.00			0.764473
$236$		0			0
$237$	76.0000	+	131.636i	0.0208301	+	0.0360788i
$238$		0			0
$239$	984.000			0.266317			0.133158	−	0.991095i	$-0.457488\pi$
$239$	984.000			0.266317			0.133158	+	0.991095i	$0.457488\pi$
$240$		0			0
$241$	−471.500	+	816.662i	−0.126025	+	0.218281i	−0.922133	−	0.386873i	$-0.873555\pi$
$241$	−471.500	+	816.662i	−0.126025	+	0.218281i	0.796108	+	0.605154i	$0.206889\pi$
$242$		0			0
$243$	1771.00	−	3067.46i	0.467530	−	0.809785i
$244$		0			0
$245$	−484.500	−	839.179i	−0.126341	−	0.218829i
$246$		0			0
$247$	−390.000	+	1351.00i	−0.100466	+	0.348024i
$248$		0			0
$249$	−628.000	−	1087.73i	−0.159831	−	0.276835i
$250$		0			0
$251$	−1365.00	+	2364.25i	−0.343259	+	0.594542i	−0.985036	−	0.172349i	$-0.944864\pi$
$251$	−1365.00	+	2364.25i	−0.343259	+	0.594542i	0.641777	+	0.766891i	$0.278198\pi$
$252$		0			0
$253$	1248.00	−	2161.60i	0.310123	−	0.537149i
$254$		0			0
$255$	442.000			0.108546
$256$		0			0
$257$	942.500	+	1632.46i	0.228761	+	0.396225i	0.957441	−	0.288629i	$-0.0931993\pi$
$257$	942.500	+	1632.46i	0.228761	+	0.396225i	−0.728680	+	0.684854i	$0.759866\pi$
$258$		0			0
$259$	4540.00			1.08920
$260$		0			0
$261$	−4531.00			−1.07457
$262$		0			0
$263$	2016.00	+	3491.81i	0.472669	+	0.818686i	0.999511	−	0.0312769i	$-0.00995738\pi$
$263$	2016.00	+	3491.81i	0.472669	+	0.818686i	−0.526842	+	0.849963i	$0.676624\pi$
$264$		0			0
$265$	1581.00			0.366491
$266$		0			0
$267$	−266.000	+	460.726i	−0.0609698	+	0.105603i
$268$		0			0
$269$	−2003.00	+	3469.30i	−0.453997	+	0.786345i	−0.998630	−	0.0523292i	$-0.983335\pi$
$269$	−2003.00	+	3469.30i	−0.453997	+	0.786345i	0.544633	+	0.838674i	$0.316669\pi$
$270$		0			0
$271$	−2148.00	−	3720.45i	−0.481482	−	0.833952i	0.518292	−	0.855204i	$-0.326568\pi$
$271$	−2148.00	−	3720.45i	−0.481482	−	0.833952i	−0.999774	+	0.0212520i	$0.993235\pi$
$272$		0			0
$273$	−1820.00	+	450.333i	−0.403485	+	0.0998367i
$274$		0			0
$275$	−2624.00	−	4544.90i	−0.575393	−	0.996610i
$276$		0			0
$277$	2775.50	−	4807.31i	0.602035	−	1.04275i	−0.390478	−	0.920612i	$-0.627690\pi$
$277$	2775.50	−	4807.31i	0.602035	−	1.04275i	0.992513	−	0.122142i	$-0.0389765\pi$
$278$		0			0
$279$	851.000	−	1473.98i	0.182609	−	0.316289i
$280$		0			0
$281$	−5557.00			−1.17973			−0.589863	−	0.807504i	$-0.700818\pi$
$281$	−5557.00			−1.17973			−0.589863	+	0.807504i	$0.700818\pi$
$282$		0			0
$283$	1560.00	+	2702.00i	0.327676	+	0.567552i	0.982050	−	0.188619i	$-0.0604012\pi$
$283$	1560.00	+	2702.00i	0.327676	+	0.567552i	−0.654374	+	0.756171i	$0.727068\pi$
$284$		0			0
$285$	1020.00			0.211999
$286$		0			0
$287$	3300.00			0.678721
$288$		0			0
$289$	2372.00	+	4108.42i	0.482801	+	0.836235i
$290$		0			0
$291$	−476.000			−0.0958887
$292$		0			0
$293$	−4150.50	+	7188.88i	−0.827559	+	1.43337i	0.0723887	+	0.997376i	$0.476938\pi$
$293$	−4150.50	+	7188.88i	−0.827559	+	1.43337i	−0.899948	+	0.435998i	$0.856396\pi$
$294$		0			0
$295$	−7344.00	+	12720.2i	−1.44944	+	2.51050i
$296$		0			0
$297$	−1600.00	−	2771.28i	−0.312597	−	0.541435i
$298$		0			0
$299$	−2535.00	−	2634.45i	−0.490310	−	0.509546i
$300$		0			0
$301$	1560.00	+	2702.00i	0.298727	+	0.517411i
$302$		0			0
$303$	−819.000	+	1418.55i	−0.155282	+	0.268956i
$304$		0			0
$305$	−1232.50	+	2134.75i	−0.231386	+	0.400772i
$306$		0			0
$307$	−8678.00			−1.61329			−0.806644	−	0.591037i	$-0.798719\pi$
$307$	−8678.00			−1.61329			−0.806644	+	0.591037i	$0.798719\pi$
$308$		0			0
$309$	−1638.00	−	2837.10i	−0.301562	−	0.522320i
$310$		0			0
$311$	−8658.00			−1.57862			−0.789309	−	0.613996i	$-0.789561\pi$
$311$	−8658.00			−1.57862			−0.789309	+	0.613996i	$0.789561\pi$
$312$		0			0
$313$	−5250.00			−0.948075			−0.474038	−	0.880505i	$-0.657204\pi$
$313$	−5250.00			−0.948075			−0.474038	+	0.880505i	$0.657204\pi$
$314$		0			0
$315$	−3910.00	−	6772.32i	−0.699376	−	1.21136i
$316$		0			0
$317$	6413.00			1.13625			0.568123	−	0.822944i	$-0.307670\pi$
$317$	6413.00			1.13625			0.568123	+	0.822944i	$0.307670\pi$
$318$		0			0
$319$	−3152.00	+	5459.42i	−0.553223	+	0.958210i
$320$		0			0
$321$	−522.000	+	904.131i	−0.0907639	+	0.157208i
$322$		0			0
$323$	−195.000	−	337.750i	−0.0335916	−	0.0581824i
$324$		0			0
$325$	−7462.00	+	1846.37i	−1.27359	+	0.315132i
$326$		0			0
$327$	−1634.00	−	2830.17i	−0.276332	−	0.478620i
$328$		0			0
$329$	1620.00	−	2805.92i	0.271470	−	0.470199i
$330$		0			0
$331$	1744.00	−	3020.70i	0.289604	−	0.501609i	−0.684111	−	0.729378i	$-0.739810\pi$
$331$	1744.00	−	3020.70i	0.289604	−	0.501609i	0.973715	+	0.227769i	$0.0731431\pi$
$332$		0			0
$333$	5221.00			0.859186
$334$		0			0
$335$	7327.00	+	12690.7i	1.19498	+	2.06976i
$336$		0			0
$337$	−1833.00			−0.296290			−0.148145	−	0.988966i	$-0.547330\pi$
$337$	−1833.00			−0.296290			−0.148145	+	0.988966i	$0.547330\pi$
$338$		0			0
$339$	−654.000			−0.104780
$340$		0			0
$341$	−1184.00	−	2050.75i	−0.188027	−	0.325672i
$342$		0			0
$343$	5720.00			0.900440
$344$		0			0
$345$	−1326.00	+	2296.70i	−0.206926	+	0.358406i
$346$		0			0
$347$	3615.00	−	6261.36i	0.559260	−	0.968667i	−0.438298	−	0.898830i	$-0.644419\pi$
$347$	3615.00	−	6261.36i	0.559260	−	0.968667i	0.997558	−	0.0698377i	$-0.0222481\pi$
$348$		0			0
$349$	2629.00	+	4553.56i	0.403230	+	0.698414i	0.994114	−	0.108342i	$-0.0345543\pi$
$349$	2629.00	+	4553.56i	0.403230	+	0.698414i	−0.590884	+	0.806757i	$0.701221\pi$
$350$		0			0
$351$	−4550.00	+	1125.83i	−0.691912	+	0.171204i
$352$		0			0
$353$	−1581.50	−	2739.24i	−0.238455	−	0.413017i	0.721816	−	0.692085i	$-0.243308\pi$
$353$	−1581.50	−	2739.24i	−0.238455	−	0.413017i	−0.960271	+	0.279068i	$0.909974\pi$
$354$		0			0
$355$	5559.00	−	9628.47i	0.831102	−	1.43951i
$356$		0			0
$357$	260.000	−	450.333i	0.0385453	−	0.0667624i
$358$		0			0
$359$	10068.0			1.48014			0.740068	−	0.672532i	$-0.234793\pi$
$359$	10068.0			1.48014			0.740068	+	0.672532i	$0.234793\pi$
$360$		0			0
$361$	2979.50	+	5160.65i	0.434393	+	0.752390i
$362$		0			0
$363$	614.000			0.0887786
$364$		0			0
$365$	3655.00			0.524141
$366$		0			0
$367$	3719.00	+	6441.50i	0.528965	+	0.916195i	0.999429	+	0.0337755i	$0.0107531\pi$
$367$	3719.00	+	6441.50i	0.528965	+	0.916195i	−0.470464	+	0.882419i	$0.655914\pi$
$368$		0			0
$369$	3795.00			0.535392
$370$		0			0
$371$	930.000	−	1610.81i	0.130143	−	0.225415i
$372$		0			0
$373$	4841.50	−	8385.72i	0.672073	−	1.16407i	−0.305242	−	0.952275i	$-0.598737\pi$
$373$	4841.50	−	8385.72i	0.672073	−	1.16407i	0.977315	−	0.211790i	$-0.0679294\pi$
$374$		0			0
$375$	663.000	+	1148.35i	0.0912991	+	0.158135i
$376$		0			0
$377$	6402.50	+	6653.67i	0.874657	+	0.908970i
$378$		0			0
$379$	−531.000	−	919.719i	−0.0719674	−	0.124651i	0.827796	−	0.561029i	$-0.189594\pi$
$379$	−531.000	−	919.719i	−0.0719674	−	0.124651i	−0.899763	+	0.436378i	$0.856261\pi$
$380$		0			0
$381$	2158.00	−	3737.77i	0.290178	−	0.502602i
$382$		0			0
$383$	−1766.00	+	3058.80i	−0.235609	+	0.408087i	−0.959450	−	0.281880i	$-0.909042\pi$
$383$	−1766.00	+	3058.80i	−0.235609	+	0.408087i	0.723840	+	0.689968i	$0.242375\pi$
$384$		0			0
$385$	−10880.0			−1.44025
$386$		0			0
$387$	1794.00	+	3107.30i	0.235644	+	0.408147i
$388$		0			0
$389$	−11063.0			−1.44194			−0.720972	−	0.692964i	$-0.756304\pi$
$389$	−11063.0			−1.44194			−0.720972	+	0.692964i	$0.756304\pi$
$390$		0			0
$391$	1014.00			0.131151
$392$		0			0
$393$	−730.000	−	1264.40i	−0.0936988	−	0.162291i
$394$		0			0
$395$	1292.00			0.164576
$396$		0			0
$397$	2993.00	−	5184.03i	0.378374	−	0.655362i	−0.612452	−	0.790508i	$-0.709817\pi$
$397$	2993.00	−	5184.03i	0.378374	−	0.655362i	0.990826	+	0.135145i	$0.0431501\pi$
$398$		0			0
$399$	600.000	−	1039.23i	0.0752821	−	0.130392i
$400$		0			0
$401$	−2967.50	−	5139.86i	−0.369551	−	0.640081i	0.619945	−	0.784646i	$-0.287155\pi$
$401$	−2967.50	−	5139.86i	−0.369551	−	0.640081i	−0.989495	+	0.144565i	$0.953822\pi$
$402$		0			0
$403$	−3367.00	+	833.116i	−0.416184	+	0.102979i
$404$		0			0
$405$	−3578.50	−	6198.14i	−0.439055	−	0.760465i
$406$		0			0
$407$	3632.00	−	6290.81i	0.442338	−	0.766152i
$408$		0			0
$409$	7544.50	−	13067.5i	0.912106	−	1.57981i	0.101023	−	0.994884i	$-0.467788\pi$
$409$	7544.50	−	13067.5i	0.912106	−	1.57981i	0.811083	−	0.584931i	$-0.198878\pi$
$410$		0			0
$411$	−3342.00			−0.401092
$412$		0			0
$413$	8640.00	+	14964.9i	1.02941	+	1.78299i
$414$		0			0
$415$	−10676.0			−1.26281
$416$		0			0
$417$	1824.00			0.214201
$418$		0			0
$419$	−5407.00	−	9365.20i	−0.630428	−	1.09193i	−0.987464	−	0.157843i	$-0.949546\pi$
$419$	−5407.00	−	9365.20i	−0.630428	−	1.09193i	0.357037	−	0.934090i	$-0.383787\pi$
$420$		0			0
$421$	−6535.00			−0.756524			−0.378262	−	0.925699i	$-0.623478\pi$
$421$	−6535.00			−0.756524			−0.378262	+	0.925699i	$0.623478\pi$
$422$		0			0
$423$	1863.00	−	3226.81i	0.214142	−	0.370905i
$424$		0			0
$425$	1066.00	−	1846.37i	0.121667	−	0.210734i
$426$		0			0
$427$	1450.00	+	2511.47i	0.164334	+	0.284634i
$428$		0			0
$429$	−832.000	+	2882.13i	−0.0936348	+	0.324361i
$430$		0			0
$431$	990.000	+	1714.73i	0.110642	+	0.191637i	0.916029	−	0.401112i	$-0.131376\pi$
$431$	990.000	+	1714.73i	0.110642	+	0.191637i	−0.805387	+	0.592749i	$0.798043\pi$
$432$		0			0
$433$	3464.50	−	6000.69i	0.384511	−	0.665993i	−0.607190	−	0.794556i	$-0.707703\pi$
$433$	3464.50	−	6000.69i	0.384511	−	0.665993i	0.991701	+	0.128564i	$0.0410368\pi$
$434$		0			0
$435$	3349.00	−	5800.64i	0.369132	−	0.639355i
$436$		0			0
$437$	2340.00			0.256150
$438$		0			0
$439$	−2288.00	−	3962.93i	−0.248748	−	0.430844i	0.714431	−	0.699706i	$-0.246686\pi$
$439$	−2288.00	−	3962.93i	−0.248748	−	0.430844i	−0.963179	+	0.268862i	$0.913352\pi$
$440$		0			0
$441$	−1311.00			−0.141561
$442$		0			0
$443$	8812.00			0.945081			0.472540	−	0.881309i	$-0.343337\pi$
$443$	8812.00			0.945081			0.472540	+	0.881309i	$0.343337\pi$
$444$		0			0
$445$	2261.00	+	3916.17i	0.240858	+	0.417178i
$446$		0			0
$447$	4230.00			0.447589
$448$		0			0
$449$	−959.000	+	1661.04i	−0.100797	+	0.174586i	−0.912013	−	0.410160i	$-0.865473\pi$
$449$	−959.000	+	1661.04i	−0.100797	+	0.174586i	0.811216	+	0.584747i	$0.198806\pi$
$450$		0			0
$451$	2640.00	−	4572.61i	0.275638	−	0.477419i
$452$		0			0
$453$	−514.000	−	890.274i	−0.0533109	−	0.0923371i
$454$		0			0
$455$	−4420.00	+	15311.3i	−0.455413	+	1.57760i
$456$		0			0
$457$	5880.50	+	10185.3i	0.601922	+	1.04256i	0.992530	+	0.122002i	$0.0389314\pi$
$457$	5880.50	+	10185.3i	0.601922	+	1.04256i	−0.390608	+	0.920557i	$0.627735\pi$
$458$		0			0
$459$	650.000	−	1125.83i	0.0660989	−	0.114487i
$460$		0			0
$461$	−450.500	+	780.289i	−0.0455138	+	0.0788323i	−0.887885	−	0.460066i	$-0.847826\pi$
$461$	−450.500	+	780.289i	−0.0455138	+	0.0788323i	0.842371	+	0.538898i	$0.181159\pi$
$462$		0			0
$463$	−1372.00			−0.137715			−0.0688577	−	0.997626i	$-0.521935\pi$
$463$	−1372.00			−0.137715			−0.0688577	+	0.997626i	$0.521935\pi$
$464$		0			0
$465$	1258.00	+	2178.92i	0.125459	+	0.217301i
$466$		0			0
$467$	6396.00			0.633772			0.316886	−	0.948464i	$-0.397363\pi$
$467$	6396.00			0.633772			0.316886	+	0.948464i	$0.397363\pi$
$468$		0			0
$469$	17240.0			1.69738
$470$		0			0
$471$	2901.00	+	5024.68i	0.283803	+	0.491561i
$472$		0			0
$473$	4992.00			0.485269
$474$		0			0
$475$	2460.00	−	4260.84i	0.237626	−	0.411581i
$476$		0			0
$477$	1069.50	−	1852.43i	0.102660	−	0.177813i
$478$		0			0
$479$	1635.00	+	2831.90i	0.155960	+	0.270131i	0.933408	−	0.358816i	$-0.116819\pi$
$479$	1635.00	+	2831.90i	0.155960	+	0.270131i	−0.777448	+	0.628947i	$0.783486\pi$
$480$		0			0
$481$	−7377.50	−	7666.92i	−0.699345	−	0.726781i
$482$		0			0
$483$	1560.00	+	2702.00i	0.146962	+	0.254545i
$484$		0			0
$485$	−2023.00	+	3503.94i	−0.189401	+	0.328053i
$486$		0			0
$487$	9960.00	−	17251.2i	0.926757	−	1.60519i	0.138046	−	0.990426i	$-0.455918\pi$
$487$	9960.00	−	17251.2i	0.926757	−	1.60519i	0.788711	−	0.614765i	$-0.210749\pi$
$488$		0			0
$489$	4720.00			0.436494
$490$		0			0
$491$	3276.00	+	5674.20i	0.301108	+	0.521534i	0.976387	−	0.216028i	$-0.0693103\pi$
$491$	3276.00	+	5674.20i	0.301108	+	0.521534i	−0.675280	+	0.737562i	$0.735977\pi$
$492$		0			0
$493$	−2561.00			−0.233959
$494$		0			0
$495$	−12512.0			−1.13611
$496$		0			0
$497$	−6540.00	−	11327.6i	−0.590260	−	1.02236i
$498$		0			0
$499$	−1746.00			−0.156637			−0.0783183	−	0.996928i	$-0.524955\pi$
$499$	−1746.00			−0.156637			−0.0783183	+	0.996928i	$0.524955\pi$
$500$		0			0
$501$	−280.000	+	484.974i	−0.0249690	+	0.0432476i
$502$		0			0
$503$	7346.00	−	12723.6i	0.651177	−	1.12787i	−0.331661	−	0.943399i	$-0.607609\pi$
$503$	7346.00	−	12723.6i	0.651177	−	1.12787i	0.982838	−	0.184473i	$-0.0590577\pi$
$504$		0			0
$505$	6961.50	+	12057.7i	0.613431	+	1.06249i
$506$		0			0
$507$	3718.00	+	2341.73i	0.325685	+	0.205128i
$508$		0			0
$509$	−4038.50	−	6994.89i	−0.351677	−	0.609122i	0.634867	−	0.772622i	$-0.281055\pi$
$509$	−4038.50	−	6994.89i	−0.351677	−	0.609122i	−0.986543	+	0.163500i	$0.947722\pi$
$510$		0			0
$511$	2150.00	−	3723.91i	0.186126	−	0.322380i
$512$		0			0
$513$	1500.00	−	2598.08i	0.129097	−	0.223602i
$514$		0			0
$515$	−27846.0			−2.38260
$516$		0			0
$517$	−2592.00	−	4489.48i	−0.220495	−	0.381909i
$518$		0			0
$519$	−2652.00			−0.224296
$520$		0			0
$521$	11247.0			0.945758			0.472879	−	0.881127i	$-0.343215\pi$
$521$	11247.0			0.945758			0.472879	+	0.881127i	$0.343215\pi$
$522$		0			0
$523$	1366.00	+	2365.98i	0.114208	+	0.197815i	0.917463	−	0.397821i	$-0.130233\pi$
$523$	1366.00	+	2365.98i	0.114208	+	0.197815i	−0.803255	+	0.595636i	$0.796900\pi$
$524$		0			0
$525$	6560.00			0.545337
$526$		0			0
$527$	481.000	−	833.116i	0.0397584	−	0.0688636i
$528$		0			0
$529$	3041.50	−	5268.03i	0.249979	−	0.432977i
$530$		0			0
$531$	9936.00	+	17209.7i	0.812026	+	1.40647i
$532$		0			0
$533$	−5362.50	−	5572.87i	−0.435789	−	0.452885i
$534$		0			0
$535$	4437.00	+	7685.11i	0.358557	+	0.621040i
$536$		0			0
$537$	−4264.00	+	7385.46i	−0.342654	+	0.593494i
$538$		0			0
$539$	−912.000	+	1579.63i	−0.0728806	+	0.126233i
$540$		0			0
$541$	−18375.0			−1.46026			−0.730132	−	0.683306i	$-0.760542\pi$
$541$	−18375.0			−1.46026			−0.730132	+	0.683306i	$0.760542\pi$
$542$		0			0
$543$	−403.000	−	698.016i	−0.0318497	−	0.0551653i
$544$		0			0
$545$	−27778.0			−2.18326
$546$		0			0
$547$	10346.0			0.808708			0.404354	−	0.914603i	$-0.367496\pi$
$547$	10346.0			0.808708			0.404354	+	0.914603i	$0.367496\pi$
$548$		0			0
$549$	1667.50	+	2888.19i	0.129631	+	0.224527i
$550$		0			0
$551$	−5910.00			−0.456941
$552$		0			0
$553$	760.000	−	1316.36i	0.0584421	−	0.101225i
$554$		0			0
$555$	−3859.00	+	6683.98i	−0.295145	+	0.511206i
$556$		0			0
$557$	−172.500	−	298.779i	−0.0131222	−	0.0227283i	0.859390	−	0.511321i	$-0.170844\pi$
$557$	−172.500	−	298.779i	−0.0131222	−	0.0227283i	−0.872512	+	0.488593i	$0.837510\pi$
$558$		0			0
$559$	2028.00	−	7025.20i	0.153444	−	0.531546i
$560$		0			0
$561$	−416.000	−	720.533i	−0.0313075	−	0.0542263i
$562$		0			0
$563$	−4290.00	+	7430.50i	−0.321140	+	0.556231i	−0.980724	−	0.195400i	$-0.937399\pi$
$563$	−4290.00	+	7430.50i	−0.321140	+	0.556231i	0.659583	+	0.751631i	$0.270733\pi$
$564$		0			0
$565$	−2779.50	+	4814.24i	−0.206964	+	0.358472i
$566$		0			0
$567$	−8420.00			−0.623645
$568$		0			0
$569$	9841.00	+	17045.1i	0.725055	+	1.25583i	0.958951	+	0.283570i	$0.0915189\pi$
$569$	9841.00	+	17045.1i	0.725055	+	1.25583i	−0.233897	+	0.972261i	$0.575148\pi$
$570$		0			0
$571$	−26624.0			−1.95128			−0.975639	−	0.219382i	$-0.929596\pi$
$571$	−26624.0			−1.95128			−0.975639	+	0.219382i	$0.929596\pi$
$572$		0			0
$573$	−2492.00			−0.181684
$574$		0			0
$575$	6396.00	+	11078.2i	0.463881	+	0.803466i
$576$		0			0
$577$	−14101.0			−1.01739			−0.508694	−	0.860948i	$-0.669871\pi$
$577$	−14101.0			−1.01739			−0.508694	+	0.860948i	$0.669871\pi$
$578$		0			0
$579$	267.000	−	462.458i	0.0191643	−	0.0331936i
$580$		0			0
$581$	−6280.00	+	10877.3i	−0.448431	+	0.776705i
$582$		0			0
$583$	−1488.00	−	2577.29i	−0.105706	−	0.183088i
$584$		0			0
$585$	−5083.00	+	17608.0i	−0.359241	+	1.24445i
$586$		0			0
$587$	704.000	+	1219.36i	0.0495012	+	0.0857386i	0.889714	−	0.456518i	$-0.150904\pi$
$587$	704.000	+	1219.36i	0.0495012	+	0.0857386i	−0.840213	+	0.542256i	$0.817570\pi$
$588$		0			0
$589$	1110.00	−	1922.58i	0.0776515	−	0.134496i
$590$		0			0
$591$	1278.00	−	2213.56i	0.0889508	−	0.154067i
$592$		0			0
$593$	−1241.00			−0.0859389			−0.0429694	−	0.999076i	$-0.513682\pi$
$593$	−1241.00			−0.0859389			−0.0429694	+	0.999076i	$0.513682\pi$
$594$		0			0
$595$	−2210.00	−	3827.83i	−0.152271	−	0.263741i
$596$		0			0
$597$	8476.00			0.581071
$598$		0			0
$599$	−11078.0			−0.755651			−0.377825	−	0.925877i	$-0.623328\pi$
$599$	−11078.0			−0.755651			−0.377825	+	0.925877i	$0.623328\pi$
$600$		0			0
$601$	6908.50	+	11965.9i	0.468891	+	0.812143i	0.999368	−	0.0355563i	$-0.0113203\pi$
$601$	6908.50	+	11965.9i	0.468891	+	0.812143i	−0.530477	+	0.847700i	$0.677987\pi$
$602$		0			0
$603$	19826.0			1.33893
$604$		0			0
$605$	2609.50	−	4519.79i	0.175357	−	0.303728i
$606$		0			0
$607$	4135.00	−	7162.03i	0.276498	−	0.478909i	−0.694014	−	0.719962i	$-0.744159\pi$
$607$	4135.00	−	7162.03i	0.276498	−	0.478909i	0.970512	+	0.241053i	$0.0774926\pi$
$608$		0			0
$609$	−3940.00	−	6824.28i	−0.262162	−	0.454078i
$610$		0			0
$611$	−7371.00	+	1823.85i	−0.488050	+	0.120761i
$612$		0			0
$613$	−11136.5	−	19289.0i	−0.733767	−	1.27092i	−0.955262	−	0.295760i	$-0.904427\pi$
$613$	−11136.5	−	19289.0i	−0.733767	−	1.27092i	0.221496	−	0.975161i	$-0.428906\pi$
$614$		0			0
$615$	−2805.00	+	4858.40i	−0.183916	+	0.318552i
$616$		0			0
$617$	9494.50	−	16445.0i	0.619504	−	1.07301i	−0.370072	−	0.929003i	$-0.620667\pi$
$617$	9494.50	−	16445.0i	0.619504	−	1.07301i	0.989576	−	0.144010i	$-0.0459997\pi$
$618$		0			0
$619$	−72.0000			−0.00467516			−0.00233758	−	0.999997i	$-0.500744\pi$
$619$	−72.0000			−0.00467516			−0.00233758	+	0.999997i	$0.500744\pi$
$620$		0			0
$621$	3900.00	+	6755.00i	0.252015	+	0.436504i
$622$		0			0
$623$	5320.00			0.342121
$624$		0			0
$625$	−9229.00			−0.590656
$626$		0			0
$627$	−960.000	−	1662.77i	−0.0611463	−	0.105908i
$628$		0			0
$629$	2951.00			0.187065
$630$		0			0
$631$	−11690.0	+	20247.7i	−0.737514	+	1.27741i	0.216097	+	0.976372i	$0.430667\pi$
$631$	−11690.0	+	20247.7i	−0.737514	+	1.27741i	−0.953611	+	0.301040i	$0.902666\pi$
$632$		0			0
$633$	−3070.00	+	5317.40i	−0.192767	+	0.333882i
$634$		0			0
$635$	−18343.0	−	31771.0i	−1.14633	−	1.98550i
$636$		0			0
$637$	1852.50	+	1925.17i	0.115226	+	0.119746i
$638$		0			0
$639$	−7521.00	−	13026.8i	−0.465612	−	0.806464i
$640$		0			0
$641$	−3191.50	+	5527.84i	−0.196656	+	0.340619i	−0.947442	−	0.319927i	$-0.896342\pi$
$641$	−3191.50	+	5527.84i	−0.196656	+	0.340619i	0.750786	+	0.660546i	$0.229675\pi$
$642$		0			0
$643$	−8552.00	+	14812.5i	−0.524507	+	0.908473i	0.475086	+	0.879939i	$0.342417\pi$
$643$	−8552.00	+	14812.5i	−0.524507	+	0.908473i	−0.999593	+	0.0285332i	$0.990916\pi$
$644$		0			0
$645$	−5304.00			−0.323790
$646$		0			0
$647$	3497.00	+	6056.98i	0.212490	+	0.368044i	0.952493	−	0.304560i	$-0.0985093\pi$
$647$	3497.00	+	6056.98i	0.212490	+	0.368044i	−0.740003	+	0.672604i	$0.765176\pi$
$648$		0			0
$649$	27648.0			1.67223
$650$		0			0
$651$	2960.00			0.178205
$652$		0			0
$653$	2625.00	+	4546.63i	0.157311	+	0.272471i	0.933898	−	0.357539i	$-0.116384\pi$
$653$	2625.00	+	4546.63i	0.157311	+	0.272471i	−0.776587	+	0.630010i	$0.783051\pi$
$654$		0			0
$655$	−12410.0			−0.740304
$656$		0			0
$657$	2472.50	−	4282.50i	0.146821	−	0.254301i
$658$		0			0
$659$	−2170.00	+	3758.55i	−0.128272	+	0.222173i	−0.923007	−	0.384783i	$-0.874276\pi$
$659$	−2170.00	+	3758.55i	−0.128272	+	0.222173i	0.794735	+	0.606956i	$0.207610\pi$
$660$		0			0
$661$	2089.50	+	3619.12i	0.122953	+	0.212961i	0.920931	−	0.389726i	$-0.127430\pi$
$661$	2089.50	+	3619.12i	0.122953	+	0.212961i	−0.797978	+	0.602687i	$0.794097\pi$
$662$		0			0
$663$	−1183.00	+	292.717i	−0.0692970	+	0.0171466i
$664$		0			0
$665$	−5100.00	−	8833.46i	−0.297398	−	0.515108i
$666$		0			0
$667$	7683.00	−	13307.3i	0.446007	−	0.772508i
$668$		0			0
$669$	5378.00	−	9314.97i	0.310800	−	0.538322i
$670$		0			0
$671$	4640.00			0.266953
$672$		0			0
$673$	−11433.5	−	19803.4i	−0.654872	−	1.13427i	−0.981926	−	0.189266i	$-0.939389\pi$
$673$	−11433.5	−	19803.4i	−0.654872	−	1.13427i	0.327054	−	0.945006i	$-0.393944\pi$
$674$		0			0
$675$	16400.0			0.935165
$676$		0			0
$677$	5410.00			0.307124			0.153562	−	0.988139i	$-0.450925\pi$
$677$	5410.00			0.307124			0.153562	+	0.988139i	$0.450925\pi$
$678$		0			0
$679$	2380.00	+	4122.28i	0.134515	+	0.232988i
$680$		0			0
$681$	−7948.00			−0.447236
$682$		0			0
$683$	−6789.00	+	11758.9i	−0.380342	+	0.658772i	−0.991111	−	0.133037i	$-0.957527\pi$
$683$	−6789.00	+	11758.9i	−0.380342	+	0.658772i	0.610769	+	0.791809i	$0.290861\pi$
$684$		0			0
$685$	−14203.5	+	24601.2i	−0.792245	+	1.37221i
$686$		0			0
$687$	−6298.00	−	10908.5i	−0.349758	−	0.605798i
$688$		0			0
$689$	−4231.50	+	1047.02i	−0.233973	+	0.0578933i
$690$		0			0
$691$	6372.00	+	11036.6i	0.350799	+	0.607602i	0.986390	−	0.164424i	$-0.0525766\pi$
$691$	6372.00	+	11036.6i	0.350799	+	0.607602i	−0.635590	+	0.772026i	$0.719243\pi$
$692$		0			0
$693$	−7360.00	+	12747.9i	−0.403439	+	0.698777i
$694$		0			0
$695$	7752.00	−	13426.9i	0.423094	−	0.732820i
$696$		0			0
$697$	2145.00			0.116568
$698$		0			0
$699$	4030.00	+	6980.16i	0.218067	+	0.377703i
$700$		0			0
$701$	16406.0			0.883946			0.441973	−	0.897028i	$-0.354279\pi$
$701$	16406.0			0.883946			0.441973	+	0.897028i	$0.354279\pi$
$702$		0			0
$703$	6810.00			0.365354
$704$		0			0
$705$	2754.00	+	4770.07i	0.147123	+	0.254824i
$706$		0			0
$707$	16380.0			0.871334
$708$		0			0
$709$	−354.500	+	614.012i	−0.0187779	+	0.0325243i	−0.875262	−	0.483650i	$-0.839311\pi$
$709$	−354.500	+	614.012i	−0.0187779	+	0.0325243i	0.856484	+	0.516174i	$0.172644\pi$
$710$		0			0
$711$	874.000	−	1513.81i	0.0461006	−	0.0798487i
$712$		0			0
$713$	2886.00	+	4998.70i	0.151587	+	0.262556i
$714$		0			0
$715$	17680.0	+	18373.6i	0.924748	+	0.961026i
$716$		0			0
$717$	984.000	+	1704.34i	0.0512527	+	0.0887722i
$718$		0			0
$719$	−3822.00	+	6619.90i	−0.198243	+	0.343367i	−0.947959	−	0.318393i	$-0.896857\pi$
$719$	−3822.00	+	6619.90i	−0.198243	+	0.343367i	0.749716	+	0.661760i	$0.230190\pi$
$720$		0			0
$721$	−16380.0	+	28371.0i	−0.846079	+	1.46545i
$722$		0			0
$723$	−1886.00			−0.0970140
$724$		0			0
$725$	−16154.0	−	27979.5i	−0.827510	−	1.43329i
$726$		0			0
$727$	15808.0			0.806446			0.403223	−	0.915102i	$-0.367890\pi$
$727$	15808.0			0.806446			0.403223	+	0.915102i	$0.367890\pi$
$728$		0			0
$729$	−4283.00			−0.217599
$730$		0			0
$731$	1014.00	+	1756.30i	0.0513053	+	0.0888633i
$732$		0			0
$733$	−2583.00			−0.130157			−0.0650786	−	0.997880i	$-0.520730\pi$
$733$	−2583.00			−0.130157			−0.0650786	+	0.997880i	$0.520730\pi$
$734$		0			0
$735$	969.000	−	1678.36i	0.0486287	−	0.0842274i
$736$		0			0
$737$	13792.0	−	23888.4i	0.689328	−	1.19395i
$738$		0			0
$739$	2038.00	+	3529.92i	0.101447	+	0.175711i	0.912281	−	0.409565i	$-0.134320\pi$
$739$	2038.00	+	3529.92i	0.101447	+	0.175711i	−0.810834	+	0.585276i	$0.800986\pi$
$740$		0			0
$741$	−2730.00	+	675.500i	−0.135343	+	0.0334887i
$742$		0			0
$743$	−17028.0	−	29493.4i	−0.840776	−	1.45627i	−0.889239	−	0.457442i	$-0.848766\pi$
$743$	−17028.0	−	29493.4i	−0.840776	−	1.45627i	0.0484632	−	0.998825i	$-0.484568\pi$
$744$		0			0
$745$	17977.5	−	31137.9i	0.884087	−	1.53128i
$746$		0			0
$747$	−7222.00	+	12508.9i	−0.353734	+	0.612685i
$748$		0			0
$749$	10440.0			0.509305
$750$		0			0
$751$	−182.000	−	315.233i	−0.00884324	−	0.0153169i	0.861570	−	0.507639i	$-0.169482\pi$
$751$	−182.000	−	315.233i	−0.00884324	−	0.0153169i	−0.870413	+	0.492322i	$0.836148\pi$
$752$		0			0
$753$	−5460.00			−0.264241
$754$		0			0
$755$	−8738.00			−0.421203
$756$		0			0
$757$	3457.00	+	5987.70i	0.165980	+	0.287486i	0.937003	−	0.349322i	$-0.113588\pi$
$757$	3457.00	+	5987.70i	0.165980	+	0.287486i	−0.771023	+	0.636807i	$0.780255\pi$
$758$		0			0
$759$	4992.00			0.238733
$760$		0			0
$761$	−6991.00	+	12108.8i	−0.333014	+	0.576797i	−0.983101	−	0.183062i	$-0.941399\pi$
$761$	−6991.00	+	12108.8i	−0.333014	+	0.576797i	0.650087	+	0.759859i	$0.274732\pi$
$762$		0			0
$763$	−16340.0	+	28301.7i	−0.775292	+	1.34284i
$764$		0			0
$765$	−2541.50	−	4402.01i	−0.120115	−	0.208046i
$766$		0			0
$767$	11232.0	−	38908.8i	0.528767	−	1.83170i
$768$		0			0
$769$	9033.00	+	15645.6i	0.423587	+	0.733674i	0.996287	−	0.0860907i	$-0.0274375\pi$
$769$	9033.00	+	15645.6i	0.423587	+	0.733674i	−0.572700	+	0.819765i	$0.694104\pi$
$770$		0			0
$771$	−1885.00	+	3264.92i	−0.0880501	+	0.152507i
$772$		0			0
$773$	−7217.00	+	12500.2i	−0.335805	+	0.581632i	−0.983639	−	0.180150i	$-0.942342\pi$
$773$	−7217.00	+	12500.2i	−0.335805	+	0.581632i	0.647834	+	0.761782i	$0.275675\pi$
$774$		0			0
$775$	12136.0			0.562501
$776$		0			0
$777$	4540.00	+	7863.51i	0.209616	+	0.363065i
$778$		0			0
$779$	4950.00			0.227666
$780$		0			0
$781$	−20928.0			−0.958851
$782$		0			0
$783$	−9850.00	−	17060.7i	−0.449566	−	0.778671i
$784$		0			0
$785$	49317.0			2.24229
$786$

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{3} +17.0000 q^{5} +(10.0000 - 17.3205i) q^{7} +(11.5000 - 19.9186i) q^{9} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{3} +17.0000 q^{5} +(10.0000 - 17.3205i) q^{7} +(11.5000 - 19.9186i) q^{9} +(-16.0000 - 27.7128i) q^{11} +(-45.5000 + 11.2583i) q^{13} +(17.0000 + 29.4449i) q^{15} +(6.50000 - 11.2583i) q^{17} +(15.0000 - 25.9808i) q^{19} +40.0000 q^{21} +(39.0000 + 67.5500i) q^{23} +164.000 q^{25} +100.000 q^{27} +(-98.5000 - 170.607i) q^{29} +74.0000 q^{31} +(32.0000 - 55.4256i) q^{33} +(170.000 - 294.449i) q^{35} +(113.500 + 196.588i) q^{37} +(-65.0000 - 67.5500i) q^{39} +(82.5000 + 142.894i) q^{41} +(-78.0000 + 135.100i) q^{43} +(195.500 - 338.616i) q^{45} +162.000 q^{47} +(-28.5000 - 49.3634i) q^{49} +26.0000 q^{51} +93.0000 q^{53} +(-272.000 - 471.118i) q^{55} +60.0000 q^{57} +(-432.000 + 748.246i) q^{59} +(-72.5000 + 125.574i) q^{61} +(-230.000 - 398.372i) q^{63} +(-773.500 + 191.392i) q^{65} +(431.000 + 746.514i) q^{67} +(-78.0000 + 135.100i) q^{69} +(327.000 - 566.381i) q^{71} +215.000 q^{73} +(164.000 + 284.056i) q^{75} -640.000 q^{77} +76.0000 q^{79} +(-210.500 - 364.597i) q^{81} -628.000 q^{83} +(110.500 - 191.392i) q^{85} +(197.000 - 341.214i) q^{87} +(133.000 + 230.363i) q^{89} +(-260.000 + 900.666i) q^{91} +(74.0000 + 128.172i) q^{93} +(255.000 - 441.673i) q^{95} +(-119.000 + 206.114i) q^{97} -736.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 34 q^{5} + 20 q^{7} + 23 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 34 * q^5 + 20 * q^7 + 23 * q^9	\( 2 q + 2 q^{3} + 34 q^{5} + 20 q^{7} + 23 q^{9} - 32 q^{11} - 91 q^{13} + 34 q^{15} + 13 q^{17} + 30 q^{19} + 80 q^{21} + 78 q^{23} + 328 q^{25} + 200 q^{27} - 197 q^{29} + 148 q^{31} + 64 q^{33} + 340 q^{35} + 227 q^{37} - 130 q^{39} + 165 q^{41} - 156 q^{43} + 391 q^{45} + 324 q^{47} - 57 q^{49} + 52 q^{51} + 186 q^{53} - 544 q^{55} + 120 q^{57} - 864 q^{59} - 145 q^{61} - 460 q^{63} - 1547 q^{65} + 862 q^{67} - 156 q^{69} + 654 q^{71} + 430 q^{73} + 328 q^{75} - 1280 q^{77} + 152 q^{79} - 421 q^{81} - 1256 q^{83} + 221 q^{85} + 394 q^{87} + 266 q^{89} - 520 q^{91} + 148 q^{93} + 510 q^{95} - 238 q^{97} - 1472 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 34 * q^5 + 20 * q^7 + 23 * q^9 - 32 * q^11 - 91 * q^13 + 34 * q^15 + 13 * q^17 + 30 * q^19 + 80 * q^21 + 78 * q^23 + 328 * q^25 + 200 * q^27 - 197 * q^29 + 148 * q^31 + 64 * q^33 + 340 * q^35 + 227 * q^37 - 130 * q^39 + 165 * q^41 - 156 * q^43 + 391 * q^45 + 324 * q^47 - 57 * q^49 + 52 * q^51 + 186 * q^53 - 544 * q^55 + 120 * q^57 - 864 * q^59 - 145 * q^61 - 460 * q^63 - 1547 * q^65 + 862 * q^67 - 156 * q^69 + 654 * q^71 + 430 * q^73 + 328 * q^75 - 1280 * q^77 + 152 * q^79 - 421 * q^81 - 1256 * q^83 + 221 * q^85 + 394 * q^87 + 266 * q^89 - 520 * q^91 + 148 * q^93 + 510 * q^95 - 238 * q^97 - 1472 * q^99

Introduction

L-functions

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