LMFDB - Embedded newform 648.2.n.n.109.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(109,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("648.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.n (of order $6$, 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:	$5.17430605098$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			109.3
Root			$0.228425 + 1.39564i$ of defining polynomial
Character	$\chi$	$=$	648.109
Dual form			648.2.n.n.541.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.228425 + 1.39564i) q^{2} +(-1.89564 + 0.637600i) q^{4} +(-0.866025 - 0.500000i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-1.32288 - 2.50000i) q^{8} +O(q^{10})$	\(q+(0.228425 + 1.39564i) q^{2} +(-1.89564 + 0.637600i) q^{4} +(-0.866025 - 0.500000i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-1.32288 - 2.50000i) q^{8} +(0.500000 - 1.32288i) q^{10} +(2.59808 - 1.50000i) q^{11} +(-4.58258 - 2.64575i) q^{13} +(1.09445 - 0.895644i) q^{14} +(3.18693 - 2.41733i) q^{16} +5.29150 q^{17} -5.29150i q^{19} +(1.96048 + 0.395644i) q^{20} +(2.68693 + 3.28335i) q^{22} +(-2.64575 + 4.58258i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(2.64575 - 7.00000i) q^{26} +(1.50000 + 1.32288i) q^{28} +(5.19615 - 3.00000i) q^{29} +(3.50000 - 6.06218i) q^{31} +(4.10170 + 3.89564i) q^{32} +(1.20871 + 7.38505i) q^{34} +1.00000i q^{35} +5.29150i q^{37} +(7.38505 - 1.20871i) q^{38} +(-0.104356 + 2.82650i) q^{40} +(2.64575 - 4.58258i) q^{41} +(-9.16515 + 5.29150i) q^{43} +(-3.96863 + 4.50000i) q^{44} +(-7.00000 - 2.64575i) q^{46} +(3.00000 - 5.19615i) q^{49} +(4.37780 - 3.58258i) q^{50} +(10.3739 + 2.09355i) q^{52} -9.00000i q^{53} -3.00000 q^{55} +(-1.50363 + 2.39564i) q^{56} +(5.37386 + 6.56670i) q^{58} +(3.46410 + 2.00000i) q^{59} +(9.26013 + 3.50000i) q^{62} +(-4.50000 + 6.61438i) q^{64} +(2.64575 + 4.58258i) q^{65} +(-10.0308 + 3.37386i) q^{68} +(-1.39564 + 0.228425i) q^{70} -15.8745 q^{71} +3.00000 q^{73} +(-7.38505 + 1.20871i) q^{74} +(3.37386 + 10.0308i) q^{76} +(-2.59808 - 1.50000i) q^{77} +(-2.00000 - 3.46410i) q^{79} +(-3.96863 + 0.500000i) q^{80} +(7.00000 + 2.64575i) q^{82} +(-6.06218 + 3.50000i) q^{83} +(-4.58258 - 2.64575i) q^{85} +(-9.47860 - 11.5826i) q^{86} +(-7.18693 - 4.51088i) q^{88} +10.5830 q^{89} +5.29150i q^{91} +(2.09355 - 10.3739i) q^{92} +(-2.64575 + 4.58258i) q^{95} +(-3.50000 - 6.06218i) q^{97} +(7.93725 + 3.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{4} - 4 q^{7}+O(q^{10}) $ 8 * q - 6 * q^4 - 4 * q^7	$ 8 q - 6 q^{4} - 4 q^{7} + 4 q^{10} - 2 q^{16} - 6 q^{22} - 16 q^{25} + 12 q^{28} + 28 q^{31} + 28 q^{34} - 10 q^{40} - 56 q^{46} + 24 q^{49} + 28 q^{52} - 24 q^{55} - 12 q^{58} - 36 q^{64} - 2 q^{70} + 24 q^{73} - 28 q^{76} - 16 q^{79} + 56 q^{82} - 30 q^{88} - 28 q^{97}+O(q^{100}) $ 8 * q - 6 * q^4 - 4 * q^7 + 4 * q^10 - 2 * q^16 - 6 * q^22 - 16 * q^25 + 12 * q^28 + 28 * q^31 + 28 * q^34 - 10 * q^40 - 56 * q^46 + 24 * q^49 + 28 * q^52 - 24 * q^55 - 12 * q^58 - 36 * q^64 - 2 * q^70 + 24 * q^73 - 28 * q^76 - 16 * q^79 + 56 * q^82 - 30 * q^88 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	0.228425	+	1.39564i	0.161521	+	0.986869i
$2$	0.228425	+	1.39564i	0.161521	+	0.986869i
$3$		0			0
$3$		0			0
$4$	−1.89564	+	0.637600i	−0.947822	+	0.318800i
$5$	−0.866025	−	0.500000i	−0.387298	−	0.223607i	0.293691	−	0.955901i	$-0.405116\pi$
$5$	−0.866025	−	0.500000i	−0.387298	−	0.223607i	−0.680989	+	0.732294i	$0.738450\pi$
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i	0.755929	−	0.654654i	$-0.227186\pi$
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i	−0.944911	+	0.327327i	$0.893852\pi$
$8$	−1.32288	−	2.50000i	−0.467707	−	0.883883i
$9$		0			0
$10$	0.500000	−	1.32288i	0.158114	−	0.418330i
$11$	2.59808	−	1.50000i	0.783349	−	0.452267i	−0.0542666	−	0.998526i	$-0.517282\pi$
$11$	2.59808	−	1.50000i	0.783349	−	0.452267i	0.837616	+	0.546259i	$0.183949\pi$
$12$		0			0
$13$	−4.58258	−	2.64575i	−1.27098	−	0.733799i	−0.295806	−	0.955248i	$-0.595588\pi$
$13$	−4.58258	−	2.64575i	−1.27098	−	0.733799i	−0.975172	+	0.221449i	$0.928921\pi$
$14$	1.09445	−	0.895644i	0.292504	−	0.239371i
$15$		0			0
$16$	3.18693	−	2.41733i	0.796733	−	0.604332i
$17$	5.29150			1.28338			0.641689	−	0.766965i	$-0.278234\pi$
$17$	5.29150			1.28338			0.641689	+	0.766965i	$0.278234\pi$
$18$		0			0
$19$		−	5.29150i		−	1.21395i	−0.794719	−	0.606977i	$-0.792382\pi$
$19$		−	5.29150i		−	1.21395i	0.794719	−	0.606977i	$-0.207618\pi$
$20$	1.96048	+	0.395644i	0.438376	+	0.0884687i
$21$		0			0
$22$	2.68693	+	3.28335i	0.572856	+	0.700013i
$23$	−2.64575	+	4.58258i	−0.551677	+	0.955533i	0.446476	+	0.894795i	$0.352679\pi$
$23$	−2.64575	+	4.58258i	−0.551677	+	0.955533i	−0.998154	+	0.0607377i	$0.980655\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$	2.64575	−	7.00000i	0.518875	−	1.37281i
$27$		0			0
$28$	1.50000	+	1.32288i	0.283473	+	0.250000i
$29$	5.19615	−	3.00000i	0.964901	−	0.557086i	0.0672232	−	0.997738i	$-0.478586\pi$
$29$	5.19615	−	3.00000i	0.964901	−	0.557086i	0.897678	+	0.440652i	$0.145253\pi$
$30$		0			0
$31$	3.50000	−	6.06218i	0.628619	−	1.08880i	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	3.50000	−	6.06218i	0.628619	−	1.08880i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$	4.10170	+	3.89564i	0.725085	+	0.688659i
$33$		0			0
$34$	1.20871	+	7.38505i	0.207292	+	1.26653i
$35$			1.00000i			0.169031i
$36$		0			0
$37$			5.29150i			0.869918i	0.900450	+	0.434959i	$0.143237\pi$
$37$			5.29150i			0.869918i	−0.900450	+	0.434959i	$0.856763\pi$
$38$	7.38505	−	1.20871i	1.19801	−	0.196079i
$39$		0			0
$40$	−0.104356	+	2.82650i	−0.0165001	+	0.446909i
$41$	2.64575	−	4.58258i	0.413197	−	0.715678i	−0.582040	−	0.813160i	$-0.697745\pi$
$41$	2.64575	−	4.58258i	0.413197	−	0.715678i	0.995237	+	0.0974818i	$0.0310788\pi$
$42$		0			0
$43$	−9.16515	+	5.29150i	−1.39767	+	0.806947i	−0.994148	−	0.108022i	$-0.965548\pi$
$43$	−9.16515	+	5.29150i	−1.39767	+	0.806947i	−0.403524	+	0.914969i	$0.632215\pi$
$44$	−3.96863	+	4.50000i	−0.598293	+	0.678401i
$45$		0			0
$46$	−7.00000	−	2.64575i	−1.03209	−	0.390095i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	3.00000	−	5.19615i	0.428571	−	0.742307i
$50$	4.37780	−	3.58258i	0.619115	−	0.506653i
$51$		0			0
$52$	10.3739	+	2.09355i	1.43860	+	0.290323i
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$	−1.50363	+	2.39564i	−0.200930	+	0.320131i
$57$		0			0
$58$	5.37386	+	6.56670i	0.705623	+	0.862250i
$59$	3.46410	+	2.00000i	0.450988	+	0.260378i	0.708247	−	0.705965i	$-0.249486\pi$
$59$	3.46410	+	2.00000i	0.450988	+	0.260378i	−0.257260	+	0.966342i	$0.582820\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$	9.26013	+	3.50000i	1.17604	+	0.444500i
$63$		0			0
$64$	−4.50000	+	6.61438i	−0.562500	+	0.826797i
$65$	2.64575	+	4.58258i	0.328165	+	0.568399i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	−10.0308	+	3.37386i	−1.21641	+	0.409141i
$69$		0			0
$70$	−1.39564	+	0.228425i	−0.166811	+	0.0273020i
$71$	−15.8745			−1.88396			−0.941979	−	0.335673i	$-0.891036\pi$
$71$	−15.8745			−1.88396			−0.941979	+	0.335673i	$0.891036\pi$
$72$		0			0
$73$	3.00000			0.351123			0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000			0.351123			0.175562	+	0.984468i	$0.443826\pi$
$74$	−7.38505	+	1.20871i	−0.858495	+	0.140510i
$75$		0			0
$76$	3.37386	+	10.0308i	0.387009	+	1.15061i
$77$	−2.59808	−	1.50000i	−0.296078	−	0.170941i
$78$		0			0
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	−0.956325	+	0.292306i	$0.905577\pi$
$80$	−3.96863	+	0.500000i	−0.443706	+	0.0559017i
$81$		0			0
$82$	7.00000	+	2.64575i	0.773021	+	0.292174i
$83$	−6.06218	+	3.50000i	−0.665410	+	0.384175i	−0.794335	−	0.607479i	$-0.792181\pi$
$83$	−6.06218	+	3.50000i	−0.665410	+	0.384175i	0.128925	+	0.991654i	$0.458847\pi$
$84$		0			0
$85$	−4.58258	−	2.64575i	−0.497050	−	0.286972i
$86$	−9.47860	−	11.5826i	−1.02210	−	1.24898i
$87$		0			0
$88$	−7.18693	−	4.51088i	−0.766130	−	0.480861i
$89$	10.5830			1.12180			0.560898	−	0.827885i	$-0.310456\pi$
$89$	10.5830			1.12180			0.560898	+	0.827885i	$0.310456\pi$
$90$		0			0
$91$			5.29150i			0.554700i
$92$	2.09355	−	10.3739i	0.218268	−	1.08155i
$93$		0			0
$94$		0			0
$95$	−2.64575	+	4.58258i	−0.271448	+	0.470162i
$96$		0			0
$97$	−3.50000	−	6.06218i	−0.355371	−	0.615521i	0.631810	−	0.775123i	$-0.282312\pi$
$97$	−3.50000	−	6.06218i	−0.355371	−	0.615521i	−0.987181	+	0.159602i	$0.948979\pi$
$98$	7.93725	+	3.00000i	0.801784	+	0.303046i
$99$		0			0
$100$	6.00000	+	5.29150i	0.600000	+	0.529150i
$101$	−14.7224	+	8.50000i	−1.46494	+	0.845782i	−0.999233	−	0.0391591i	$-0.987532\pi$
$101$	−14.7224	+	8.50000i	−1.46494	+	0.845782i	−0.465704	+	0.884941i	$0.654199\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$	−0.552200	+	14.9564i	−0.0541477	+	1.46660i
$105$		0			0
$106$	12.5608	−	2.05583i	1.22001	−	0.199679i
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$		0			0
$109$		−	5.29150i		−	0.506834i	−0.967357	−	0.253417i	$-0.918446\pi$
$109$		−	5.29150i		−	0.506834i	0.967357	−	0.253417i	$-0.0815545\pi$
$110$	−0.685275	−	4.18693i	−0.0653384	−	0.399208i
$111$		0			0
$112$	−3.68693	−	1.55130i	−0.348382	−	0.146584i
$113$	7.93725	−	13.7477i	0.746674	−	1.29328i	−0.202735	−	0.979234i	$-0.564983\pi$
$113$	7.93725	−	13.7477i	0.746674	−	1.29328i	0.949409	−	0.314044i	$-0.101684\pi$
$114$		0			0
$115$	4.58258	−	2.64575i	0.427327	−	0.246718i
$116$	−7.93725	+	9.00000i	−0.736956	+	0.835629i
$117$		0			0
$118$	−2.00000	+	5.29150i	−0.184115	+	0.487122i
$119$	−2.64575	−	4.58258i	−0.242536	−	0.420084i
$120$		0			0
$121$	−1.00000	+	1.73205i	−0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$	−2.76951	+	13.7233i	−0.248709	+	1.23239i
$125$			9.00000i			0.804984i
$126$		0			0
$127$	13.0000			1.15356			0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000			1.15356			0.576782	+	0.816898i	$0.304308\pi$
$128$	−10.2592	−	4.76951i	−0.906796	−	0.421569i
$129$		0			0
$130$	−5.79129	+	4.73930i	−0.507930	+	0.415664i
$131$	6.06218	+	3.50000i	0.529655	+	0.305796i	0.740876	−	0.671642i	$-0.234411\pi$
$131$	6.06218	+	3.50000i	0.529655	+	0.305796i	−0.211221	+	0.977438i	$0.567744\pi$
$132$		0			0
$133$	−4.58258	+	2.64575i	−0.397360	+	0.229416i
$134$		0			0
$135$		0			0
$136$	−7.00000	−	13.2288i	−0.600245	−	1.13436i
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$	−9.16515	−	5.29150i	−0.777378	−	0.448819i	0.0581222	−	0.998309i	$-0.481489\pi$
$139$	−9.16515	−	5.29150i	−0.777378	−	0.448819i	−0.835500	+	0.549490i	$0.814822\pi$
$140$	−0.637600	−	1.89564i	−0.0538871	−	0.160211i
$141$		0			0
$142$	−3.62614	−	22.1552i	−0.304299	−	1.85922i
$143$	−15.8745			−1.32749
$144$		0			0
$145$	−6.00000			−0.498273
$146$	0.685275	+	4.18693i	0.0567138	+	0.346513i
$147$		0			0
$148$	−3.37386	−	10.0308i	−0.277330	−	0.824527i
$149$	2.59808	+	1.50000i	0.212843	+	0.122885i	0.602632	−	0.798019i	$-0.294119\pi$
$149$	2.59808	+	1.50000i	0.212843	+	0.122885i	−0.389789	+	0.920904i	$0.627452\pi$
$150$		0			0
$151$	−1.50000	−	2.59808i	−0.122068	−	0.211428i	0.798515	−	0.601975i	$-0.205619\pi$
$151$	−1.50000	−	2.59808i	−0.122068	−	0.211428i	−0.920583	+	0.390547i	$0.872286\pi$
$152$	−13.2288	+	7.00000i	−1.07299	+	0.567775i
$153$		0			0
$154$	1.50000	−	3.96863i	0.120873	−	0.319801i
$155$	−6.06218	+	3.50000i	−0.486926	+	0.281127i
$156$		0			0
$157$	9.16515	+	5.29150i	0.731459	+	0.422308i	0.818956	−	0.573857i	$-0.194553\pi$
$157$	9.16515	+	5.29150i	0.731459	+	0.422308i	−0.0874969	+	0.996165i	$0.527887\pi$
$158$	4.37780	−	3.58258i	0.348279	−	0.285014i
$159$		0			0
$160$	−1.60436	−	5.42458i	−0.126835	−	0.428851i
$161$	5.29150			0.417029
$162$		0			0
$163$			5.29150i			0.414462i	0.978292	+	0.207231i	$0.0664452\pi$
$163$			5.29150i			0.414462i	−0.978292	+	0.207231i	$0.933555\pi$
$164$	−2.09355	+	10.3739i	−0.163479	+	0.810063i
$165$		0			0
$166$	−6.26951	−	7.66115i	−0.486608	−	0.594621i
$167$	−5.29150	+	9.16515i	−0.409469	+	0.709221i	−0.994830	−	0.101552i	$-0.967619\pi$
$167$	−5.29150	+	9.16515i	−0.409469	+	0.709221i	0.585361	+	0.810772i	$0.300953\pi$
$168$		0			0
$169$	7.50000	+	12.9904i	0.576923	+	0.999260i
$170$	2.64575	−	7.00000i	0.202920	−	0.536875i
$171$		0			0
$172$	14.0000	−	15.8745i	1.06749	−	1.21042i
$173$	12.9904	−	7.50000i	0.987640	−	0.570214i	0.0830722	−	0.996544i	$-0.473527\pi$
$173$	12.9904	−	7.50000i	0.987640	−	0.570214i	0.904568	+	0.426329i	$0.140193\pi$
$174$		0			0
$175$	−2.00000	+	3.46410i	−0.151186	+	0.261861i
$176$	4.65390	−	11.0608i	0.350801	−	0.833739i
$177$		0			0
$178$	2.41742	+	14.7701i	0.181194	+	1.10707i
$179$			23.0000i			1.71910i	0.511051	+	0.859550i	$0.329256\pi$
$179$			23.0000i			1.71910i	−0.511051	+	0.859550i	$0.670744\pi$
$180$		0			0
$181$		−	21.1660i		−	1.57326i	−0.617426	−	0.786629i	$-0.711825\pi$
$181$		−	21.1660i		−	1.57326i	0.617426	−	0.786629i	$-0.288175\pi$
$182$	−7.38505	+	1.20871i	−0.547417	+	0.0895957i
$183$		0			0
$184$	14.9564	+	0.552200i	1.10260	+	0.0407088i
$185$	2.64575	−	4.58258i	0.194520	−	0.336918i
$186$		0			0
$187$	13.7477	−	7.93725i	1.00533	−	0.580429i
$188$		0			0
$189$		0			0
$190$	−7.00000	−	2.64575i	−0.507833	−	0.191943i
$191$	−2.64575	−	4.58258i	−0.191440	−	0.331584i	0.754288	−	0.656544i	$-0.227982\pi$
$191$	−2.64575	−	4.58258i	−0.191440	−	0.331584i	−0.945728	+	0.324960i	$0.894649\pi$
$192$		0			0
$193$	1.50000	−	2.59808i	0.107972	−	0.187014i	−0.806976	−	0.590584i	$-0.798898\pi$
$193$	1.50000	−	2.59808i	0.107972	−	0.187014i	0.914949	+	0.403570i	$0.132231\pi$
$194$	7.66115	−	6.26951i	0.550039	−	0.450124i
$195$		0			0
$196$	−2.37386	+	11.7629i	−0.169562	+	0.840204i
$197$		−	13.0000i		−	0.926212i	−0.886303	−	0.463106i	$-0.846735\pi$
$197$		−	13.0000i		−	0.926212i	0.886303	−	0.463106i	$-0.153265\pi$
$198$		0			0
$199$	−17.0000			−1.20510			−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000			−1.20510			−0.602549	+	0.798082i	$0.705848\pi$
$200$	−6.01450	+	9.58258i	−0.425290	+	0.677590i
$201$		0			0
$202$	−15.2259	−	18.6057i	−1.07129	−	1.30909i
$203$	−5.19615	−	3.00000i	−0.364698	−	0.210559i
$204$		0			0
$205$	−4.58258	+	2.64575i	−0.320061	+	0.184787i
$206$		0			0
$207$		0			0
$208$	−21.0000	+	2.64575i	−1.45609	+	0.183450i
$209$	−7.93725	−	13.7477i	−0.549031	−	0.950950i
$210$		0			0
$211$	−4.58258	−	2.64575i	−0.315478	−	0.182141i	0.333897	−	0.942609i	$-0.391636\pi$
$211$	−4.58258	−	2.64575i	−0.315478	−	0.182141i	−0.649375	+	0.760468i	$0.724969\pi$
$212$	5.73840	+	17.0608i	0.394115	+	1.17174i
$213$		0			0
$214$	−4.18693	+	0.685275i	−0.286213	+	0.0468445i
$215$	10.5830			0.721755
$216$		0			0
$217$	−7.00000			−0.475191
$218$	7.38505	−	1.20871i	0.500179	−	0.0818643i
$219$		0			0
$220$	5.68693	−	1.91280i	0.383413	−	0.128961i
$221$	−24.2487	−	14.0000i	−1.63114	−	0.941742i
$222$		0			0
$223$	8.00000	+	13.8564i	0.535720	+	0.927894i	0.999128	+	0.0417488i	$0.0132929\pi$
$223$	8.00000	+	13.8564i	0.535720	+	0.927894i	−0.463409	+	0.886145i	$0.653374\pi$
$224$	1.32288	−	5.50000i	0.0883883	−	0.367484i
$225$		0			0
$226$	21.0000	+	7.93725i	1.39690	+	0.527978i
$227$	3.46410	−	2.00000i	0.229920	−	0.132745i	−0.380615	−	0.924734i	$-0.624288\pi$
$227$	3.46410	−	2.00000i	0.229920	−	0.132745i	0.610535	+	0.791989i	$0.290954\pi$
$228$		0			0
$229$	18.3303	+	10.5830i	1.21130	+	0.699345i	0.963043	−	0.269349i	$-0.0868086\pi$
$229$	18.3303	+	10.5830i	1.21130	+	0.699345i	0.248258	+	0.968694i	$0.420142\pi$
$230$	4.73930	+	5.79129i	0.312500	+	0.381866i
$231$		0			0
$232$	−14.3739	−	9.02175i	−0.943690	−	0.592307i
$233$	10.5830			0.693316			0.346658	−	0.937992i	$-0.387316\pi$
$233$	10.5830			0.693316			0.346658	+	0.937992i	$0.387316\pi$
$234$		0			0
$235$		0			0
$236$	−7.84190	−	1.58258i	−0.510464	−	0.103017i
$237$		0			0
$238$	5.79129	−	4.73930i	0.375393	−	0.307203i
$239$	−7.93725	+	13.7477i	−0.513418	+	0.889267i	0.486461	+	0.873703i	$0.338288\pi$
$239$	−7.93725	+	13.7477i	−0.513418	+	0.889267i	−0.999879	+	0.0155640i	$0.995046\pi$
$240$		0			0
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	0.980917	−	0.194429i	$-0.0622852\pi$
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	−0.658838	+	0.752285i	$0.728952\pi$
$242$	−2.64575	−	1.00000i	−0.170075	−	0.0642824i
$243$		0			0
$244$		0			0
$245$	−5.19615	+	3.00000i	−0.331970	+	0.191663i
$246$		0			0
$247$	−14.0000	+	24.2487i	−0.890799	+	1.54291i
$248$	−19.7855	−	0.730493i	−1.25638	−	0.0463863i
$249$		0			0
$250$	−12.5608	+	2.05583i	−0.794414	+	0.130022i
$251$		−	12.0000i		−	0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$		−	12.0000i		−	0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$		0			0
$253$			15.8745i			0.998022i
$254$	2.96953	+	18.1434i	0.186325	+	1.13842i
$255$		0			0
$256$	4.31307	−	15.4077i	0.269567	−	0.962982i
$257$	−7.93725	+	13.7477i	−0.495112	+	0.857560i	−0.999984	−	0.00563467i	$-0.998206\pi$
$257$	−7.93725	+	13.7477i	−0.495112	+	0.857560i	0.504872	+	0.863194i	$0.331540\pi$
$258$		0			0
$259$	4.58258	−	2.64575i	0.284747	−	0.164399i
$260$	−7.93725	−	7.00000i	−0.492248	−	0.434122i
$261$		0			0
$262$	−3.50000	+	9.26013i	−0.216231	+	0.572093i
$263$	2.64575	+	4.58258i	0.163144	+	0.282574i	0.935995	−	0.352014i	$-0.114503\pi$
$263$	2.64575	+	4.58258i	0.163144	+	0.282574i	−0.772851	+	0.634588i	$0.781170\pi$
$264$		0			0
$265$	−4.50000	+	7.79423i	−0.276433	+	0.478796i
$266$	−4.73930	−	5.79129i	−0.290585	−	0.355087i
$267$		0			0
$268$		0			0
$269$			18.0000i			1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$			18.0000i			1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$		0			0
$271$	15.0000			0.911185			0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000			0.911185			0.455593	+	0.890188i	$0.349427\pi$
$272$	16.8637	−	12.7913i	1.02251	−	0.775586i
$273$		0			0
$274$		0			0
$275$	−10.3923	−	6.00000i	−0.626680	−	0.361814i
$276$		0			0
$277$	9.16515	−	5.29150i	0.550681	−	0.317936i	−0.198716	−	0.980057i	$-0.563677\pi$
$277$	9.16515	−	5.29150i	0.550681	−	0.317936i	0.749396	+	0.662122i	$0.230344\pi$
$278$	5.29150	−	14.0000i	0.317363	−	0.839664i
$279$		0			0
$280$	2.50000	−	1.32288i	0.149404	−	0.0790569i
$281$	7.93725	+	13.7477i	0.473497	+	0.820121i	0.999540	−	0.0303374i	$-0.00965819\pi$
$281$	7.93725	+	13.7477i	0.473497	+	0.820121i	−0.526043	+	0.850458i	$0.676325\pi$
$282$		0			0
$283$	22.9129	+	13.2288i	1.36203	+	0.786368i	0.989894	−	0.141810i	$-0.0452924\pi$
$283$	22.9129	+	13.2288i	1.36203	+	0.786368i	0.372135	+	0.928178i	$0.378626\pi$
$284$	30.0924	−	10.1216i	1.78566	−	0.600606i
$285$		0			0
$286$	−3.62614	−	22.1552i	−0.214418	−	1.31006i
$287$	−5.29150			−0.312348
$288$		0			0
$289$	11.0000			0.647059
$290$	−1.37055	−	8.37386i	−0.0804815	−	0.491730i
$291$		0			0
$292$	−5.68693	+	1.91280i	−0.332803	+	0.111938i
$293$	25.9808	+	15.0000i	1.51781	+	0.876309i	0.999781	+	0.0209480i	$0.00666844\pi$
$293$	25.9808	+	15.0000i	1.51781	+	0.876309i	0.518032	+	0.855361i	$0.326665\pi$
$294$		0			0
$295$	−2.00000	−	3.46410i	−0.116445	−	0.201688i
$296$	13.2288	−	7.00000i	0.768906	−	0.406867i
$297$		0			0
$298$	−1.50000	+	3.96863i	−0.0868927	+	0.229896i
$299$	24.2487	−	14.0000i	1.40234	−	0.809641i
$300$		0			0
$301$	9.16515	+	5.29150i	0.528271	+	0.304997i
$302$	3.28335	−	2.68693i	0.188936	−	0.154616i
$303$		0			0
$304$	−12.7913	−	16.8637i	−0.733631	−	0.967197i
$305$		0			0
$306$		0			0
$307$			15.8745i			0.906006i	0.891509	+	0.453003i	$0.149647\pi$
$307$			15.8745i			0.906006i	−0.891509	+	0.453003i	$0.850353\pi$
$308$	5.88143	+	1.18693i	0.335125	+	0.0676317i
$309$		0			0
$310$	−6.26951	−	7.66115i	−0.356084	−	0.435124i
$311$	13.2288	−	22.9129i	0.750134	−	1.29927i	−0.197623	−	0.980278i	$-0.563322\pi$
$311$	13.2288	−	22.9129i	0.750134	−	1.29927i	0.947757	−	0.318992i	$-0.103344\pi$
$312$		0			0
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	0.879810	−	0.475325i	$-0.157669\pi$
$313$	0.500000	+	0.866025i	0.0282617	+	0.0489506i	−0.851549	+	0.524276i	$0.824336\pi$
$314$	−5.29150	+	14.0000i	−0.298617	+	0.790066i
$315$		0			0
$316$	6.00000	+	5.29150i	0.337526	+	0.297670i
$317$	11.2583	−	6.50000i	0.632331	−	0.365076i	−0.149323	−	0.988788i	$-0.547710\pi$
$317$	11.2583	−	6.50000i	0.632331	−	0.365076i	0.781654	+	0.623712i	$0.214376\pi$
$318$		0			0
$319$	9.00000	−	15.5885i	0.503903	−	0.872786i
$320$	7.20430	−	3.47822i	0.402733	−	0.194438i
$321$		0			0
$322$	1.20871	+	7.38505i	0.0673589	+	0.411553i
$323$		−	28.0000i		−	1.55796i
$324$		0			0
$325$			21.1660i			1.17408i
$326$	−7.38505	+	1.20871i	−0.409020	+	0.0669444i
$327$		0			0
$328$	−14.9564	−	0.552200i	−0.825831	−	0.0304902i
$329$		0			0
$330$		0			0
$331$	18.3303	−	10.5830i	1.00752	−	0.581695i	0.0970586	−	0.995279i	$-0.469057\pi$
$331$	18.3303	−	10.5830i	1.00752	−	0.581695i	0.910466	+	0.413584i	$0.135723\pi$
$332$	9.26013	−	10.5000i	0.508216	−	0.576262i
$333$		0			0
$334$	−14.0000	−	5.29150i	−0.766046	−	0.289538i
$335$		0			0
$336$		0			0
$337$	−17.0000	+	29.4449i	−0.926049	+	1.60396i	−0.136184	+	0.990684i	$0.543484\pi$
$337$	−17.0000	+	29.4449i	−0.926049	+	1.60396i	−0.789865	+	0.613280i	$0.789850\pi$
$338$	−16.4168	+	13.4347i	−0.892954	+	0.730749i
$339$		0			0
$340$	10.3739	+	2.09355i	0.562602	+	0.113539i
$341$		−	21.0000i		−	1.13721i
$342$		0			0
$343$	−13.0000			−0.701934
$344$	25.3531	+	15.9129i	1.36695	+	0.857965i
$345$		0			0
$346$	13.4347	+	16.4168i	0.722252	+	0.882570i
$347$	19.9186	+	11.5000i	1.06929	+	0.617352i	0.927986	−	0.372615i	$-0.121539\pi$
$347$	19.9186	+	11.5000i	1.06929	+	0.617352i	0.141299	+	0.989967i	$0.454872\pi$
$348$		0			0
$349$	4.58258	−	2.64575i	0.245300	−	0.141624i	−0.372310	−	0.928108i	$-0.621434\pi$
$349$	4.58258	−	2.64575i	0.245300	−	0.141624i	0.617610	+	0.786484i	$0.288101\pi$
$350$	−5.29150	−	2.00000i	−0.282843	−	0.106904i
$351$		0			0
$352$	16.5000	+	3.96863i	0.879453	+	0.211529i
$353$	7.93725	+	13.7477i	0.422457	+	0.731718i	0.996179	−	0.0873325i	$-0.0278343\pi$
$353$	7.93725	+	13.7477i	0.422457	+	0.731718i	−0.573722	+	0.819050i	$0.694501\pi$
$354$		0			0
$355$	13.7477	+	7.93725i	0.729654	+	0.421266i
$356$	−20.0616	+	6.74773i	−1.06326	+	0.357629i
$357$		0			0
$358$	−32.0998	+	5.25378i	−1.69653	+	0.277671i
$359$	21.1660			1.11710			0.558550	−	0.829471i	$-0.311358\pi$
$359$	21.1660			1.11710			0.558550	+	0.829471i	$0.311358\pi$
$360$		0			0
$361$	−9.00000			−0.473684
$362$	29.5402	−	4.83485i	1.55260	−	0.254114i
$363$		0			0
$364$	−3.37386	−	10.0308i	−0.176838	−	0.525757i
$365$	−2.59808	−	1.50000i	−0.135990	−	0.0785136i
$366$		0			0
$367$	1.50000	+	2.59808i	0.0782994	+	0.135618i	0.902516	−	0.430656i	$-0.141718\pi$
$367$	1.50000	+	2.59808i	0.0782994	+	0.135618i	−0.824217	+	0.566274i	$0.808384\pi$
$368$	2.64575	+	21.0000i	0.137919	+	1.09470i
$369$		0			0
$370$	7.00000	+	2.64575i	0.363913	+	0.137546i
$371$	−7.79423	+	4.50000i	−0.404656	+	0.233628i
$372$		0			0
$373$	−22.9129	−	13.2288i	−1.18638	−	0.684959i	−0.228901	−	0.973450i	$-0.573513\pi$
$373$	−22.9129	−	13.2288i	−1.18638	−	0.684959i	−0.957483	+	0.288491i	$0.906847\pi$
$374$	14.2179	+	17.3739i	0.735190	+	0.898381i
$375$		0			0
$376$		0			0
$377$	−31.7490			−1.63516
$378$		0			0
$379$		−	21.1660i		−	1.08722i	−0.839336	−	0.543612i	$-0.817056\pi$
$379$		−	21.1660i		−	1.08722i	0.839336	−	0.543612i	$-0.182944\pi$
$380$	2.09355	−	10.3739i	0.107397	−	0.532168i
$381$		0			0
$382$	5.79129	−	4.73930i	0.296308	−	0.242484i
$383$	7.93725	−	13.7477i	0.405575	−	0.702476i	−0.588813	−	0.808269i	$-0.700405\pi$
$383$	7.93725	−	13.7477i	0.405575	−	0.702476i	0.994388	+	0.105793i	$0.0337381\pi$
$384$		0			0
$385$	1.50000	+	2.59808i	0.0764471	+	0.132410i
$386$	3.96863	+	1.50000i	0.201998	+	0.0763480i
$387$		0			0
$388$	10.5000	+	9.26013i	0.533057	+	0.470112i
$389$	4.33013	−	2.50000i	0.219546	−	0.126755i	−0.386194	−	0.922418i	$-0.626210\pi$
$389$	4.33013	−	2.50000i	0.219546	−	0.126755i	0.605740	+	0.795663i	$0.292877\pi$
$390$		0			0
$391$	−14.0000	+	24.2487i	−0.708010	+	1.22631i
$392$	−16.9590	−	0.626136i	−0.856559	−	0.0316247i
$393$		0			0
$394$	18.1434	−	2.96953i	0.914050	−	0.149603i
$395$			4.00000i			0.201262i
$396$		0			0
$397$		−	5.29150i		−	0.265573i	−0.991145	−	0.132786i	$-0.957608\pi$
$397$		−	5.29150i		−	0.265573i	0.991145	−	0.132786i	$-0.0423924\pi$
$398$	−3.88323	−	23.7259i	−0.194649	−	1.18927i
$399$		0			0
$400$	−14.7477	−	6.20520i	−0.737386	−	0.310260i
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$	−32.0780	+	18.5203i	−1.59792	+	0.922560i
$404$	22.4889	−	25.5000i	1.11886	−	1.26867i
$405$		0			0
$406$	3.00000	−	7.93725i	0.148888	−	0.393919i
$407$	7.93725	+	13.7477i	0.393435	+	0.681450i
$408$		0			0
$409$	5.50000	−	9.52628i	0.271957	−	0.471044i	−0.697406	−	0.716677i	$-0.745662\pi$
$409$	5.50000	−	9.52628i	0.271957	−	0.471044i	0.969363	+	0.245633i	$0.0789957\pi$
$410$	−4.73930	−	5.79129i	−0.234057	−	0.286011i
$411$		0			0
$412$		0			0
$413$		−	4.00000i		−	0.196827i
$414$		0			0
$415$	7.00000			0.343616
$416$	−8.48945	−	28.7042i	−0.416230	−	1.40734i
$417$		0			0
$418$	17.3739	−	14.2179i	0.849783	−	0.695421i
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	9.16515	−	5.29150i	0.446682	−	0.257892i	−0.259746	−	0.965677i	$-0.583639\pi$
$421$	9.16515	−	5.29150i	0.446682	−	0.257892i	0.706428	+	0.707785i	$0.250305\pi$
$422$	2.64575	−	7.00000i	0.128793	−	0.340755i
$423$		0			0
$424$	−22.5000	+	11.9059i	−1.09270	+	0.578201i
$425$	−10.5830	−	18.3303i	−0.513351	−	0.889150i
$426$		0			0
$427$		0			0
$428$	−1.91280	−	5.68693i	−0.0924587	−	0.274888i
$429$		0			0
$430$	2.41742	+	14.7701i	0.116579	+	0.712278i
$431$	31.7490			1.52930			0.764648	−	0.644448i	$-0.222913\pi$
$431$	31.7490			1.52930			0.764648	+	0.644448i	$0.222913\pi$
$432$		0			0
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$	−1.59898	−	9.76951i	−0.0767533	−	0.468951i
$435$		0			0
$436$	3.37386	+	10.0308i	0.161579	+	0.480388i
$437$	24.2487	+	14.0000i	1.15997	+	0.669711i
$438$		0			0
$439$	10.5000	+	18.1865i	0.501138	+	0.867996i	0.999999	+	0.00131415i	$0.000418308\pi$
$439$	10.5000	+	18.1865i	0.501138	+	0.867996i	−0.498861	+	0.866682i	$0.666248\pi$
$440$	3.96863	+	7.50000i	0.189197	+	0.357548i
$441$		0			0
$442$	14.0000	−	37.0405i	0.665912	−	1.76184i
$443$	−20.7846	+	12.0000i	−0.987507	+	0.570137i	−0.904528	−	0.426414i	$-0.859777\pi$
$443$	−20.7846	+	12.0000i	−0.987507	+	0.570137i	−0.0829786	+	0.996551i	$0.526443\pi$
$444$		0			0
$445$	−9.16515	−	5.29150i	−0.434470	−	0.250841i
$446$	−17.5112	+	14.3303i	−0.829180	+	0.678560i
$447$		0			0
$448$	7.97822	+	0.589925i	0.376935	+	0.0278713i
$449$	−15.8745			−0.749164			−0.374582	−	0.927194i	$-0.622214\pi$
$449$	−15.8745			−0.749164			−0.374582	+	0.927194i	$0.622214\pi$
$450$		0			0
$451$		−	15.8745i		−	0.747501i
$452$	−6.28065	+	31.1216i	−0.295417	+	1.46384i
$453$		0			0
$454$	3.58258	+	4.37780i	0.168139	+	0.205460i
$455$	2.64575	−	4.58258i	0.124035	−	0.214834i
$456$		0			0
$457$	−8.50000	−	14.7224i	−0.397613	−	0.688686i	0.595818	−	0.803120i	$-0.296828\pi$
$457$	−8.50000	−	14.7224i	−0.397613	−	0.688686i	−0.993431	+	0.114433i	$0.963495\pi$
$458$	−10.5830	+	28.0000i	−0.494511	+	1.30835i
$459$		0			0
$460$	−7.00000	+	7.93725i	−0.326377	+	0.370076i
$461$	−7.79423	+	4.50000i	−0.363013	+	0.209586i	−0.670402	−	0.741998i	$-0.733878\pi$
$461$	−7.79423	+	4.50000i	−0.363013	+	0.209586i	0.307388	+	0.951584i	$0.400545\pi$
$462$		0			0
$463$	20.5000	−	35.5070i	0.952716	−	1.65015i	0.213205	−	0.977007i	$-0.431610\pi$
$463$	20.5000	−	35.5070i	0.952716	−	1.65015i	0.739511	−	0.673145i	$-0.235057\pi$
$464$	9.30780	−	22.1216i	0.432104	−	1.02697i
$465$		0			0
$466$	2.41742	+	14.7701i	0.111985	+	0.684212i
$467$			1.00000i			0.0462745i	0.999732	+	0.0231372i	$0.00736547\pi$
$467$			1.00000i			0.0462745i	−0.999732	+	0.0231372i	$0.992635\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.417424	−	11.3060i	0.0192135	−	0.520401i
$473$	−15.8745	+	27.4955i	−0.729911	+	1.26424i
$474$		0			0
$475$	−18.3303	+	10.5830i	−0.841052	+	0.485582i
$476$	7.93725	+	7.00000i	0.363803	+	0.320844i
$477$		0			0
$478$	−21.0000	−	7.93725i	−0.960518	−	0.363042i
$479$	10.5830	+	18.3303i	0.483550	+	0.837533i	0.999822	−	0.0188920i	$-0.00601385\pi$
$479$	10.5830	+	18.3303i	0.483550	+	0.837533i	−0.516272	+	0.856425i	$0.672681\pi$
$480$		0			0
$481$	14.0000	−	24.2487i	0.638345	−	1.10565i
$482$	−10.9445	+	8.95644i	−0.498509	+	0.407955i
$483$		0			0
$484$	0.791288	−	3.92095i	0.0359676	−	0.178225i
$485$			7.00000i			0.317854i
$486$		0			0
$487$	−40.0000			−1.81257			−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000			−1.81257			−0.906287	+	0.422664i	$0.861095\pi$
$488$		0			0
$489$		0			0
$490$	−5.37386	−	6.56670i	−0.242766	−	0.296653i
$491$	−25.1147	−	14.5000i	−1.13341	−	0.654376i	−0.188621	−	0.982050i	$-0.560402\pi$
$491$	−25.1147	−	14.5000i	−1.13341	−	0.654376i	−0.944791	+	0.327674i	$0.893735\pi$
$492$		0			0
$493$	27.4955	−	15.8745i	1.23833	−	0.714952i
$494$	−37.0405	−	14.0000i	−1.66653	−	0.629890i
$495$		0			0
$496$	−3.50000	−	27.7804i	−0.157155	−	1.24738i
$497$	7.93725	+	13.7477i	0.356034	+	0.616670i
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−5.73840	−	17.0608i	−0.256629	−	0.762982i
$501$		0			0
$502$	16.7477	−	2.74110i	0.747488	−	0.122341i
$503$	−15.8745			−0.707809			−0.353905	−	0.935282i	$-0.615146\pi$
$503$	−15.8745			−0.707809			−0.353905	+	0.935282i	$0.615146\pi$
$504$		0			0
$505$	17.0000			0.756490
$506$	−22.1552	+	3.62614i	−0.984917	+	0.161201i
$507$		0			0
$508$	−24.6434	+	8.28880i	−1.09337	+	0.367756i
$509$	−18.1865	−	10.5000i	−0.806104	−	0.465404i	0.0394971	−	0.999220i	$-0.487424\pi$
$509$	−18.1865	−	10.5000i	−0.806104	−	0.465404i	−0.845601	+	0.533815i	$0.820758\pi$
$510$		0			0
$511$	−1.50000	−	2.59808i	−0.0663561	−	0.114932i
$512$	22.4889	+	2.50000i	0.993878	+	0.110485i
$513$		0			0
$514$	−21.0000	−	7.93725i	−0.926270	−	0.350097i
$515$		0			0
$516$		0			0
$517$		0			0
$518$	4.73930	+	5.79129i	0.208233	+	0.254455i
$519$		0			0
$520$	7.95644	−	12.6766i	0.348913	−	0.555904i
$521$	−15.8745			−0.695475			−0.347737	−	0.937592i	$-0.613050\pi$
$521$	−15.8745			−0.695475			−0.347737	+	0.937592i	$0.613050\pi$
$522$		0			0
$523$		−	15.8745i		−	0.694144i	−0.937839	−	0.347072i	$-0.887176\pi$
$523$		−	15.8745i		−	0.694144i	0.937839	−	0.347072i	$-0.112824\pi$
$524$	−13.7233	−	2.76951i	−0.599507	−	0.120986i
$525$		0			0
$526$	−5.79129	+	4.73930i	−0.252512	+	0.206643i
$527$	18.5203	−	32.0780i	0.806755	−	1.39734i
$528$		0			0
$529$	−2.50000	−	4.33013i	−0.108696	−	0.188266i
$530$	−11.9059	−	4.50000i	−0.517158	−	0.195468i
$531$		0			0
$532$	7.00000	−	7.93725i	0.303488	−	0.344124i
$533$	−24.2487	+	14.0000i	−1.05033	+	0.606407i
$534$		0			0
$535$	1.50000	−	2.59808i	0.0648507	−	0.112325i
$536$		0			0
$537$		0			0
$538$	−25.1216	+	4.11165i	−1.08307	+	0.177266i
$539$		−	18.0000i		−	0.775315i
$540$		0			0
$541$		−	10.5830i		−	0.454999i	−0.973778	−	0.227499i	$-0.926945\pi$
$541$		−	10.5830i		−	0.454999i	0.973778	−	0.227499i	$-0.0730550\pi$
$542$	3.42638	+	20.9347i	0.147175	+	0.899221i
$543$		0			0
$544$	21.7042	+	20.6138i	0.930558	+	0.883810i
$545$	−2.64575	+	4.58258i	−0.113332	+	0.196296i
$546$		0			0
$547$	−4.58258	+	2.64575i	−0.195937	+	0.113124i	−0.594759	−	0.803904i	$-0.702752\pi$
$547$	−4.58258	+	2.64575i	−0.195937	+	0.113124i	0.398822	+	0.917028i	$0.369419\pi$
$548$		0			0
$549$		0			0
$550$	6.00000	−	15.8745i	0.255841	−	0.676891i
$551$	−15.8745	−	27.4955i	−0.676277	−	1.17135i
$552$		0			0
$553$	−2.00000	+	3.46410i	−0.0850487	+	0.147309i
$554$	9.47860	+	11.5826i	0.402707	+	0.492096i
$555$		0			0
$556$	20.7477	+	4.18710i	0.879900	+	0.177573i
$557$			39.0000i			1.65248i	0.563316	+	0.826242i	$0.309525\pi$
$557$			39.0000i			1.65248i	−0.563316	+	0.826242i	$0.690475\pi$
$558$		0			0
$559$	56.0000			2.36855
$560$	2.41733	+	3.18693i	0.102151	+	0.134672i
$561$		0			0
$562$	−17.3739	+	14.2179i	−0.732872	+	0.599746i
$563$	−2.59808	−	1.50000i	−0.109496	−	0.0632175i	0.444252	−	0.895902i	$-0.353470\pi$
$563$	−2.59808	−	1.50000i	−0.109496	−	0.0632175i	−0.553748	+	0.832684i	$0.686803\pi$
$564$		0			0
$565$	−13.7477	+	7.93725i	−0.578371	+	0.333923i
$566$	−13.2288	+	35.0000i	−0.556046	+	1.47116i
$567$		0			0
$568$	21.0000	+	39.6863i	0.881140	+	1.66520i
$569$	−15.8745	−	27.4955i	−0.665494	−	1.15267i	−0.979151	−	0.203133i	$-0.934888\pi$
$569$	−15.8745	−	27.4955i	−0.665494	−	1.15267i	0.313657	−	0.949536i	$-0.398446\pi$
$570$		0			0
$571$	−22.9129	−	13.2288i	−0.958874	−	0.553606i	−0.0630478	−	0.998011i	$-0.520082\pi$
$571$	−22.9129	−	13.2288i	−0.958874	−	0.553606i	−0.895826	+	0.444404i	$0.853415\pi$
$572$	30.0924	−	10.1216i	1.25823	−	0.423205i
$573$		0			0
$574$	−1.20871	−	7.38505i	−0.0504507	−	0.308246i
$575$	21.1660			0.882684
$576$		0			0
$577$	−18.0000			−0.749350			−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000			−0.749350			−0.374675	+	0.927156i	$0.622246\pi$
$578$	2.51268	+	15.3521i	0.104514	+	0.638562i
$579$		0			0
$580$	11.3739	−	3.82560i	0.472274	−	0.158849i
$581$	6.06218	+	3.50000i	0.251502	+	0.145204i
$582$		0			0
$583$	−13.5000	−	23.3827i	−0.559113	−	0.968412i
$584$	−3.96863	−	7.50000i	−0.164223	−	0.310352i
$585$		0			0
$586$	−15.0000	+	39.6863i	−0.619644	+	1.63942i
$587$	19.9186	−	11.5000i	0.822128	−	0.474656i	−0.0290218	−	0.999579i	$-0.509239\pi$
$587$	19.9186	−	11.5000i	0.822128	−	0.474656i	0.851150	+	0.524923i	$0.175906\pi$
$588$		0			0
$589$	−32.0780	−	18.5203i	−1.32175	−	0.763114i
$590$	4.37780	−	3.58258i	0.180231	−	0.147492i
$591$		0			0
$592$	12.7913	+	16.8637i	0.525719	+	0.693092i
$593$	−31.7490			−1.30378			−0.651888	−	0.758315i	$-0.726023\pi$
$593$	−31.7490			−1.30378			−0.651888	+	0.758315i	$0.726023\pi$
$594$		0			0
$595$			5.29150i			0.216930i
$596$	−5.88143	−	1.18693i	−0.240913	−	0.0486186i
$597$		0			0
$598$	25.0780	+	30.6446i	1.02552	+	1.25315i
$599$	−10.5830	+	18.3303i	−0.432410	+	0.748956i	−0.997080	−	0.0763606i	$-0.975670\pi$
$599$	−10.5830	+	18.3303i	−0.432410	+	0.748956i	0.564670	+	0.825317i	$0.309003\pi$
$600$		0			0
$601$	−2.50000	−	4.33013i	−0.101977	−	0.176630i	0.810522	−	0.585708i	$-0.199184\pi$
$601$	−2.50000	−	4.33013i	−0.101977	−	0.176630i	−0.912499	+	0.409079i	$0.865850\pi$
$602$	−5.29150	+	14.0000i	−0.215666	+	0.570597i
$603$		0			0
$604$	4.50000	+	3.96863i	0.183102	+	0.161481i
$605$	1.73205	−	1.00000i	0.0704179	−	0.0406558i
$606$		0			0
$607$	4.00000	−	6.92820i	0.162355	−	0.281207i	−0.773358	−	0.633970i	$-0.781424\pi$
$607$	4.00000	−	6.92820i	0.162355	−	0.281207i	0.935713	+	0.352763i	$0.114758\pi$
$608$	20.6138	−	21.7042i	0.836000	−	0.880220i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			10.5830i			0.427444i	0.976895	+	0.213722i	$0.0685586\pi$
$613$			10.5830i			0.427444i	−0.976895	+	0.213722i	$0.931441\pi$
$614$	−22.1552	+	3.62614i	−0.894110	+	0.146339i
$615$		0			0
$616$	−0.313068	+	8.47950i	−0.0126139	+	0.341649i
$617$	−2.64575	+	4.58258i	−0.106514	+	0.184488i	−0.914356	−	0.404912i	$-0.867302\pi$
$617$	−2.64575	+	4.58258i	−0.106514	+	0.184488i	0.807842	+	0.589399i	$0.200636\pi$
$618$		0			0
$619$	−13.7477	+	7.93725i	−0.552568	+	0.319025i	−0.750157	−	0.661260i	$-0.770022\pi$
$619$	−13.7477	+	7.93725i	−0.552568	+	0.319025i	0.197589	+	0.980285i	$0.436689\pi$
$620$	9.26013	−	10.5000i	0.371896	−	0.421690i
$621$		0			0
$622$	35.0000	+	13.2288i	1.40337	+	0.530425i
$623$	−5.29150	−	9.16515i	−0.212000	−	0.367194i
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$	−1.09445	+	0.895644i	−0.0437430	+	0.0357971i
$627$		0			0
$628$	−20.7477	−	4.18710i	−0.827924	−	0.167084i
$629$			28.0000i			1.11643i
$630$		0			0
$631$	−23.0000			−0.915616			−0.457808	−	0.889051i	$-0.651365\pi$
$631$	−23.0000			−0.915616			−0.457808	+	0.889051i	$0.651365\pi$
$632$	−6.01450	+	9.58258i	−0.239244	+	0.381174i
$633$		0			0
$634$	11.6434	+	14.2279i	0.462417	+	0.565060i
$635$	−11.2583	−	6.50000i	−0.446773	−	0.257945i
$636$		0			0
$637$	−27.4955	+	15.8745i	−1.08941	+	0.628971i
$638$	23.8118	+	9.00000i	0.942717	+	0.356313i
$639$		0			0
$640$	6.50000	+	9.26013i	0.256935	+	0.366039i
$641$	−18.5203	−	32.0780i	−0.731506	−	1.26701i	−0.956239	−	0.292586i	$-0.905484\pi$
$641$	−18.5203	−	32.0780i	−0.731506	−	1.26701i	0.224733	−	0.974420i	$-0.427849\pi$
$642$		0			0
$643$	−18.3303	−	10.5830i	−0.722877	−	0.417353i	0.0929339	−	0.995672i	$-0.470375\pi$
$643$	−18.3303	−	10.5830i	−0.722877	−	0.417353i	−0.815811	+	0.578319i	$0.803709\pi$
$644$	−10.0308	+	3.37386i	−0.395269	+	0.132949i
$645$		0			0
$646$	39.0780	−	6.39590i	1.53750	−	0.251643i
$647$	31.7490			1.24818			0.624091	−	0.781351i	$-0.285469\pi$
$647$	31.7490			1.24818			0.624091	+	0.781351i	$0.285469\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$	−29.5402	+	4.83485i	−1.15866	+	0.189638i
$651$		0			0
$652$	−3.37386	−	10.0308i	−0.132131	−	0.392837i
$653$	−2.59808	−	1.50000i	−0.101671	−	0.0586995i	0.448303	−	0.893882i	$-0.352029\pi$
$653$	−2.59808	−	1.50000i	−0.101671	−	0.0586995i	−0.549973	+	0.835182i	$0.685362\pi$
$654$		0			0
$655$	−3.50000	−	6.06218i	−0.136756	−	0.236869i
$656$	−2.64575	−	21.0000i	−0.103299	−	0.819912i
$657$		0			0
$658$		0			0
$659$	12.9904	−	7.50000i	0.506033	−	0.292159i	−0.225168	−	0.974320i	$-0.572293\pi$
$659$	12.9904	−	7.50000i	0.506033	−	0.292159i	0.731202	+	0.682161i	$0.238960\pi$
$660$		0			0
$661$	22.9129	+	13.2288i	0.891208	+	0.514539i	0.874337	−	0.485319i	$-0.161296\pi$
$661$	22.9129	+	13.2288i	0.891208	+	0.514539i	0.0168704	+	0.999858i	$0.494630\pi$
$662$	18.9572	+	23.1652i	0.736793	+	0.900339i
$663$		0			0
$664$	16.7695	+	10.5254i	0.650783	+	0.408464i
$665$	5.29150			0.205196
$666$		0			0
$667$			31.7490i			1.22933i
$668$	4.18710	−	20.7477i	0.162004	−	0.802754i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−14.5000	−	25.1147i	−0.558934	−	0.968102i	−0.997586	−	0.0694449i	$-0.977877\pi$
$673$	−14.5000	−	25.1147i	−0.558934	−	0.968102i	0.438652	−	0.898657i	$-0.355456\pi$
$674$	−44.9778	−	17.0000i	−1.73248	−	0.654816i
$675$		0			0
$676$	−22.5000	−	19.8431i	−0.865385	−	0.763197i
$677$	5.19615	−	3.00000i	0.199704	−	0.115299i	−0.396813	−	0.917899i	$-0.629884\pi$
$677$	5.19615	−	3.00000i	0.199704	−	0.115299i	0.596518	+	0.802600i	$0.296551\pi$
$678$		0			0
$679$	−3.50000	+	6.06218i	−0.134318	+	0.232645i
$680$	−0.552200	+	14.9564i	−0.0211759	+	0.573553i
$681$		0			0
$682$	29.3085	−	4.79693i	1.12228	−	0.183684i
$683$			44.0000i			1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$			44.0000i			1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$		0			0
$685$		0			0
$686$	−2.96953	−	18.1434i	−0.113377	−	0.692717i
$687$		0			0
$688$	−16.4174	+	39.0188i	−0.625908	+	1.48758i
$689$	−23.8118	+	41.2432i	−0.907156	+	1.57124i
$690$		0			0
$691$	36.6606	−	21.1660i	1.39464	−	0.805193i	0.400811	−	0.916161i	$-0.368728\pi$
$691$	36.6606	−	21.1660i	1.39464	−	0.805193i	0.993824	+	0.110968i	$0.0353950\pi$
$692$	−19.8431	+	22.5000i	−0.754323	+	0.855322i
$693$		0			0
$694$	−11.5000	+	30.4261i	−0.436534	+	1.15496i
$695$	5.29150	+	9.16515i	0.200718	+	0.347654i
$696$		0			0
$697$	14.0000	−	24.2487i	0.530288	−	0.918485i
$698$	4.73930	+	5.79129i	0.179385	+	0.219203i
$699$		0			0
$700$	1.58258	−	7.84190i	0.0598157	−	0.296396i
$701$			1.00000i			0.0377695i	0.999822	+	0.0188847i	$0.00601156\pi$
$701$			1.00000i			0.0377695i	−0.999822	+	0.0188847i	$0.993988\pi$
$702$		0			0
$703$	28.0000			1.05604
$704$	−1.76978	+	23.9347i	−0.0667009	+	0.902071i
$705$		0			0
$706$	−17.3739	+	14.2179i	−0.653874	+	0.535098i
$707$	14.7224	+	8.50000i	0.553694	+	0.319675i
$708$		0			0
$709$	−18.3303	+	10.5830i	−0.688409	+	0.397453i	−0.803016	−	0.595958i	$-0.796773\pi$
$709$	−18.3303	+	10.5830i	−0.688409	+	0.397453i	0.114607	+	0.993411i	$0.463439\pi$
$710$	−7.93725	+	21.0000i	−0.297880	+	0.788116i
$711$		0			0
$712$	−14.0000	−	26.4575i	−0.524672	−	0.991537i
$713$	18.5203	+	32.0780i	0.693589	+	1.20133i
$714$		0			0
$715$	13.7477	+	7.93725i	0.514136	+	0.296836i
$716$	−14.6648	−	43.5998i	−0.548050	−	1.62940i
$717$		0			0
$718$	4.83485	+	29.5402i	0.180435	+	1.10243i
$719$	21.1660			0.789359			0.394679	−	0.918819i	$-0.370856\pi$
$719$	21.1660			0.789359			0.394679	+	0.918819i	$0.370856\pi$
$720$		0			0
$721$		0			0
$722$	−2.05583	−	12.5608i	−0.0765099	−	0.467464i
$723$		0			0
$724$	13.4955	+	40.1232i	0.501555	+	1.49117i
$725$	−20.7846	−	12.0000i	−0.771921	−	0.445669i
$726$		0			0
$727$	−16.5000	−	28.5788i	−0.611951	−	1.05993i	−0.990911	−	0.134517i	$-0.957052\pi$
$727$	−16.5000	−	28.5788i	−0.611951	−	1.05993i	0.378960	−	0.925413i	$-0.376282\pi$
$728$	13.2288	−	7.00000i	0.490290	−	0.259437i
$729$		0			0
$730$	1.50000	−	3.96863i	0.0555175	−	0.146885i
$731$	−48.4974	+	28.0000i	−1.79374	+	1.03562i
$732$		0			0
$733$	−4.58258	−	2.64575i	−0.169261	−	0.0977231i	0.412976	−	0.910742i	$-0.364489\pi$
$733$	−4.58258	−	2.64575i	−0.169261	−	0.0977231i	−0.582237	+	0.813019i	$0.697823\pi$
$734$	−3.28335	+	2.68693i	−0.121191	+	0.0991765i
$735$		0			0
$736$	−28.7042	+	8.48945i	−1.05805	+	0.312925i
$737$		0			0
$738$		0			0
$739$			5.29150i			0.194651i	0.995253	+	0.0973255i	$0.0310288\pi$
$739$			5.29150i			0.194651i	−0.995253	+	0.0973255i	$0.968971\pi$
$740$	−2.09355	+	10.3739i	−0.0769605	+	0.381351i
$741$		0			0
$742$	−8.06080	−	9.85005i	−0.295921	−	0.361607i
$743$	18.5203	−	32.0780i	0.679442	−	1.17683i	−0.295707	−	0.955279i	$-0.595555\pi$
$743$	18.5203	−	32.0780i	0.679442	−	1.17683i	0.975149	−	0.221550i	$-0.0711116\pi$
$744$		0			0
$745$	−1.50000	−	2.59808i	−0.0549557	−	0.0951861i
$746$	13.2288	−	35.0000i	0.484339	−	1.28144i
$747$		0			0
$748$	−21.0000	+	23.8118i	−0.767836	+	0.870644i
$749$	2.59808	−	1.50000i	0.0949316	−	0.0548088i
$750$		0			0
$751$	15.5000	−	26.8468i	0.565603	−	0.979653i	−0.431390	−	0.902165i	$-0.641977\pi$
$751$	15.5000	−	26.8468i	0.565603	−	0.979653i	0.996993	−	0.0774878i	$-0.0246899\pi$
$752$		0			0
$753$		0			0
$754$	−7.25227	−	44.3103i	−0.264112	−	1.61369i
$755$			3.00000i			0.109181i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	29.5402	−	4.83485i	1.07295	−	0.175610i
$759$		0			0
$760$	14.9564	+	0.552200i	0.542527	+	0.0200304i
$761$	5.29150	−	9.16515i	0.191817	−	0.332236i	−0.754036	−	0.656834i	$-0.771895\pi$
$761$	5.29150	−	9.16515i	0.191817	−	0.332236i	0.945852	+	0.324597i	$0.105229\pi$
$762$		0			0
$763$	−4.58258	+	2.64575i	−0.165900	+	0.0957826i
$764$	7.93725	+	7.00000i	0.287160	+	0.253251i
$765$		0			0
$766$	21.0000	+	7.93725i	0.758761	+	0.286785i
$767$	−10.5830	−	18.3303i	−0.382130	−	0.661869i
$768$		0			0
$769$	17.5000	−	30.3109i	0.631066	−	1.09304i	−0.356268	−	0.934384i	$-0.615951\pi$
$769$	17.5000	−	30.3109i	0.631066	−	1.09304i	0.987334	−	0.158655i	$-0.0507157\pi$
$770$	−3.28335	+	2.68693i	−0.118324	+	0.0968303i
$771$		0			0
$772$	−1.18693	+	5.88143i	−0.0427186	+	0.211677i
$773$		−	14.0000i		−	0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$		−	14.0000i		−	0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$		0			0
$775$	−28.0000			−1.00579
$776$	−10.5254	+	16.7695i	−0.377839	+	0.601990i
$777$		0			0
$778$	4.47822	+	5.47225i	0.160552	+	0.196190i
$779$	−24.2487	−	14.0000i	−0.868800	−	0.501602i
$780$		0			0
$781$	−41.2432	+	23.8118i	−1.47580	+	0.852052i
$782$	−37.0405	−	14.0000i	−1.32457	−	0.500639i
$783$		0			0
$784$	−3.00000	−	23.8118i	−0.107143	−	0.850420i
$785$	−5.29150	−	9.16515i	−0.188862	−	0.327118i
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	8.28880	+	24.6434i	0.295276	+	0.877884i
$789$		0			0
$790$	−5.58258	+	0.913701i	−0.198619	+	0.0325080i
$791$	−15.8745			−0.564433
$792$		0			0
$793$		0			0
$794$	7.38505	−	1.20871i	0.262086	−	0.0428956i
$795$		0			0
$796$	32.2259	−	10.8392i	1.14222	−	0.384185i
$797$	42.4352	+	24.5000i	1.50313	+	0.867835i	0.999993	+	0.00362965i	$0.00115536\pi$
$797$	42.4352	+	24.5000i	1.50313	+	0.867835i	0.503140	+	0.864205i	$0.332178\pi$
$798$		0			0
$799$		0			0
$800$	5.29150	−	22.0000i	0.187083	−	0.777817i
$801$		0			0
$802$		0			0
$803$	7.79423	−	4.50000i	0.275052	−	0.158802i
$804$		0			0
$805$	−4.58258	−	2.64575i	−0.161515	−	0.0932505i
$806$	−33.1751	−	40.5390i	−1.16854	−	1.42793i
$807$		0			0
$808$	40.7259	+	25.5616i	1.43273	+	0.899255i
$809$	−31.7490			−1.11624			−0.558118	−	0.829762i	$-0.688476\pi$
$809$	−31.7490			−1.11624			−0.558118	+	0.829762i	$0.688476\pi$
$810$		0			0
$811$			37.0405i			1.30067i	0.759648	+	0.650334i	$0.225371\pi$
$811$			37.0405i			1.30067i	−0.759648	+	0.650334i	$0.774629\pi$
$812$	11.7629	+	2.37386i	0.412795	+	0.0833063i
$813$		0			0
$814$	−17.3739	+	14.2179i	−0.608954	+	0.498337i
$815$	2.64575	−	4.58258i	0.0926766	−	0.160521i
$816$		0			0
$817$	28.0000	+	48.4974i	0.979596	+	1.69671i
$818$	14.5516	+	5.50000i	0.508786	+	0.192303i
$819$		0			0
$820$	7.00000	−	7.93725i	0.244451	−	0.277181i
$821$	8.66025	−	5.00000i	0.302245	−	0.174501i	−0.341206	−	0.939989i	$-0.610835\pi$
$821$	8.66025	−	5.00000i	0.302245	−	0.174501i	0.643451	+	0.765487i	$0.277502\pi$
$822$		0			0
$823$	−11.5000	+	19.9186i	−0.400865	+	0.694318i	−0.993831	−	0.110910i	$-0.964624\pi$
$823$	−11.5000	+	19.9186i	−0.400865	+	0.694318i	0.592966	+	0.805228i	$0.297957\pi$
$824$		0			0
$825$		0			0
$826$	5.58258	−	0.913701i	0.194243	−	0.0317917i
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$		−	21.1660i		−	0.735126i	−0.929999	−	0.367563i	$-0.880192\pi$
$829$		−	21.1660i		−	0.735126i	0.929999	−	0.367563i	$-0.119808\pi$
$830$	1.59898	+	9.76951i	0.0555013	+	0.339105i
$831$		0			0
$832$	38.1216	−	18.4050i	1.32163	−	0.638079i
$833$	15.8745	−	27.4955i	0.550019	−	0.952661i
$834$		0			0
$835$	9.16515	−	5.29150i	0.317173	−	0.183120i
$836$	23.8118	+	21.0000i	0.823547	+	0.726300i
$837$		0			0
$838$		0			0
$839$	18.5203	+	32.0780i	0.639390	+	1.10746i	0.985567	+	0.169287i	$0.0541465\pi$
$839$	18.5203	+	32.0780i	0.639390	+	1.10746i	−0.346176	+	0.938169i	$0.612520\pi$
$840$		0			0
$841$	3.50000	−	6.06218i	0.120690	−	0.209041i
$842$	9.47860	+	11.5826i	0.326654	+	0.399162i
$843$		0			0
$844$	10.3739	+	2.09355i	0.357083	+	0.0720630i
$845$		−	15.0000i		−	0.516016i
$846$		0			0
$847$	2.00000			0.0687208
$848$	−21.7559	−	28.6824i	−0.747102	−	0.984957i
$849$		0			0
$850$	23.1652	−	18.9572i	0.794558	−	0.650227i
$851$	−24.2487	−	14.0000i	−0.831235	−	0.479914i
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	7.50000	−	3.96863i	0.256345	−	0.135645i
$857$	7.93725	+	13.7477i	0.271131	+	0.469613i	0.969152	−	0.246464i	$-0.0792689\pi$
$857$	7.93725	+	13.7477i	0.271131	+	0.469613i	−0.698020	+	0.716078i	$0.745936\pi$
$858$		0			0
$859$	−45.8258	−	26.4575i	−1.56355	−	0.902719i	−0.996893	−	0.0787681i	$-0.974901\pi$
$859$	−45.8258	−	26.4575i	−1.56355	−	0.902719i	−0.566662	−	0.823951i	$-0.691765\pi$
$860$	−20.0616	+	6.74773i	−0.684095	+	0.230096i
$861$		0			0
$862$	7.25227	+	44.3103i	0.247013	+	1.50922i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−15.0000			−0.510015
$866$	1.14213	+	6.97822i	0.0388110	+	0.237130i
$867$		0			0
$868$	13.2695	−	4.46320i	0.450396	−	0.151491i
$869$	−10.3923	−	6.00000i	−0.352535	−	0.203536i
$870$		0			0
$871$		0			0
$872$	−13.2288	+	7.00000i	−0.447982	+	0.237050i
$873$		0			0
$874$	−14.0000	+	37.0405i	−0.473557	+	1.25291i
$875$	7.79423	−	4.50000i	0.263493	−	0.152128i
$876$		0			0
$877$	45.8258	+	26.4575i	1.54743	+	0.893407i	0.998337	+	0.0576426i	$0.0183584\pi$
$877$	45.8258	+	26.4575i	1.54743	+	0.893407i	0.549089	+	0.835764i	$0.314975\pi$
$878$	−22.9835	+	18.8085i	−0.775654	+	0.634757i
$879$		0			0
$880$	−9.56080	+	7.25198i	−0.322294	+	0.244464i
$881$	26.4575			0.891376			0.445688	−	0.895188i	$-0.352959\pi$
$881$	26.4575			0.891376			0.445688	+	0.895188i	$0.352959\pi$
$882$		0			0
$883$		−	58.2065i		−	1.95881i	−0.201916	−	0.979403i	$-0.564717\pi$
$883$		−	58.2065i		−	1.95881i	0.201916	−	0.979403i	$-0.435283\pi$
$884$	54.8933	+	11.0780i	1.84626	+	0.372594i
$885$		0			0
$886$	−21.4955	−	26.2668i	−0.722154	−	0.882451i
$887$	−7.93725	+	13.7477i	−0.266507	+	0.461603i	−0.967957	−	0.251115i	$-0.919203\pi$
$887$	−7.93725	+	13.7477i	−0.266507	+	0.461603i	0.701450	+	0.712718i	$0.252536\pi$
$888$		0			0
$889$	−6.50000	−	11.2583i	−0.218003	−	0.377592i
$890$	5.29150	−	14.0000i	0.177372	−	0.469281i
$891$		0			0
$892$	−24.0000	−	21.1660i	−0.803579	−	0.708690i
$893$		0			0
$894$		0			0
$895$	11.5000	−	19.9186i	0.384403	−	0.665805i
$896$	0.999100	+	11.2695i	0.0333776	+	0.376488i
$897$		0			0
$898$	−3.62614	−	22.1552i	−0.121006	−	0.739327i
$899$		−	42.0000i		−	1.40078i
$900$		0			0
$901$		−	47.6235i		−	1.58657i
$902$	22.1552	−	3.62614i	0.737686	−	0.120737i
$903$		0			0
$904$	−44.8693	−	1.65660i	−1.49233	−	0.0550977i
$905$	−10.5830	+	18.3303i	−0.351791	+	0.609320i
$906$		0			0
$907$	−4.58258	+	2.64575i	−0.152162	+	0.0878507i	−0.574148	−	0.818752i	$-0.694667\pi$
$907$	−4.58258	+	2.64575i	−0.152162	+	0.0878507i	0.421986	+	0.906602i	$0.361333\pi$
$908$	−5.29150	+	6.00000i	−0.175605	+	0.199117i
$909$		0			0
$910$	7.00000	+	2.64575i	0.232048	+	0.0877058i
$911$	15.8745	+	27.4955i	0.525946	+	0.910965i	0.999543	+	0.0302235i	$0.00962191\pi$
$911$	15.8745	+	27.4955i	0.525946	+	0.910965i	−0.473597	+	0.880742i	$0.657045\pi$
$912$		0			0
$913$	−10.5000	+	18.1865i	−0.347499	+	0.601886i
$914$	18.6057	−	15.2259i	0.615421	−	0.503630i
$915$		0			0
$916$	−41.4955	−	8.37420i	−1.37105	−	0.276691i
$917$		−	7.00000i		−	0.231160i
$918$		0			0
$919$	3.00000			0.0989609			0.0494804	−	0.998775i	$-0.484243\pi$
$919$	3.00000			0.0989609			0.0494804	+	0.998775i	$0.484243\pi$
$920$	−12.6766	−	7.95644i	−0.417934	−	0.262316i
$921$		0			0
$922$	−8.06080	−	9.85005i	−0.265468	−	0.324394i
$923$	72.7461	+	42.0000i	2.39447	+	1.38245i
$924$		0			0
$925$	18.3303	−	10.5830i	0.602697	−	0.347967i
$926$	54.2379	+	20.5000i	1.78237	+	0.673672i
$927$		0			0
$928$	33.0000	+	7.93725i	1.08328	+	0.260553i
$929$	5.29150	+	9.16515i	0.173609	+	0.300699i	0.939679	−	0.342058i	$-0.111124\pi$
$929$	5.29150	+	9.16515i	0.173609	+	0.300699i	−0.766070	+	0.642757i	$0.777791\pi$
$930$		0			0
$931$	−27.4955	−	15.8745i	−0.901127	−	0.520266i
$932$	−20.0616	+	6.74773i	−0.657140	+	0.221029i
$933$		0			0
$934$	−1.39564	+	0.228425i	−0.0456669	+	0.00747430i
$935$	−15.8745			−0.519152
$936$		0			0
$937$	−19.0000			−0.620703			−0.310351	−	0.950622i	$-0.600447\pi$
$937$	−19.0000			−0.620703			−0.310351	+	0.950622i	$0.600447\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.1865	+	10.5000i	0.592864	+	0.342290i	0.766229	−	0.642567i	$-0.222131\pi$
$941$	18.1865	+	10.5000i	0.592864	+	0.342290i	−0.173365	+	0.984858i	$0.555464\pi$
$942$		0			0
$943$	14.0000	+	24.2487i	0.455903	+	0.789647i
$944$	15.8745	−	2.00000i	0.516671	−	0.0650945i
$945$		0			0
$946$	−42.0000	−	15.8745i	−1.36554	−	0.516125i
$947$	33.7750	−	19.5000i	1.09754	−	0.633665i	0.161966	−	0.986796i	$-0.448217\pi$
$947$	33.7750	−	19.5000i	1.09754	−	0.633665i	0.935574	+	0.353131i	$0.114883\pi$
$948$		0			0
$949$	−13.7477	−	7.93725i	−0.446270	−	0.257654i
$950$	−18.9572	−	23.1652i	−0.615053	−	0.751577i
$951$		0			0
$952$	−7.95644	+	12.6766i	−0.257870	+	0.410850i
$953$	31.7490			1.02845			0.514226	−	0.857655i	$-0.328079\pi$
$953$	31.7490			1.02845			0.514226	+	0.857655i	$0.328079\pi$
$954$		0			0
$955$			5.29150i			0.171229i
$956$	6.28065	−	31.1216i	0.203131	−	1.00654i
$957$		0			0
$958$	−23.1652	+	18.9572i	−0.748432	+	0.612480i
$959$		0			0
$960$		0			0
$961$	−9.00000	−	15.5885i	−0.290323	−	0.502853i
$962$	37.0405	+	14.0000i	1.19423	+	0.451378i
$963$		0			0
$964$	−15.0000	−	13.2288i	−0.483117	−	0.426070i
$965$	−2.59808	+	1.50000i	−0.0836350	+	0.0482867i
$966$		0			0
$967$	−20.5000	+	35.5070i	−0.659236	+	1.14183i	0.321578	+	0.946883i	$0.395787\pi$
$967$	−20.5000	+	35.5070i	−0.659236	+	1.14183i	−0.980814	+	0.194946i	$0.937547\pi$
$968$	5.65300	+	0.208712i	0.181694	+	0.00670826i
$969$		0			0
$970$	−9.76951	+	1.59898i	−0.313680	+	0.0513400i
$971$			15.0000i			0.481373i	0.970603	+	0.240686i	$0.0773725\pi$
$971$			15.0000i			0.481373i	−0.970603	+	0.240686i	$0.922627\pi$
$972$		0			0
$973$			10.5830i			0.339276i
$974$	−9.13701	−	55.8258i	−0.292769	−	1.78877i
$975$		0			0
$976$		0			0
$977$	5.29150	−	9.16515i	0.169290	−	0.293219i	−0.768880	−	0.639393i	$-0.779186\pi$
$977$	5.29150	−	9.16515i	0.169290	−	0.293219i	0.938170	+	0.346174i	$0.112519\pi$
$978$		0			0
$979$	27.4955	−	15.8745i	0.878759	−	0.507351i
$980$	7.93725	−	9.00000i	0.253546	−	0.287494i
$981$		0			0
$982$	14.5000	−	38.3634i	0.462714	−	1.22423i
$983$	−21.1660	−	36.6606i	−0.675091	−	1.16929i	−0.976442	−	0.215779i	$-0.930771\pi$
$983$	−21.1660	−	36.6606i	−0.675091	−	1.16929i	0.301351	−	0.953513i	$-0.402562\pi$
$984$		0			0
$985$	−6.50000	+	11.2583i	−0.207107	+	0.358720i
$986$	28.4358	+	34.7477i	0.905581	+	1.10659i
$987$		0			0
$988$	11.0780	−	54.8933i	0.352439	−	1.74639i
$989$		−	56.0000i		−	1.78070i
$990$		0			0
$991$	47.0000			1.49300			0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000			1.49300			0.746502	+	0.665383i	$0.231732\pi$
$992$	37.9720	−	11.2305i	1.20561	−	0.356568i
$993$		0			0
$994$	−17.3739	+	14.2179i	−0.551065	+	0.450965i
$995$	14.7224	+	8.50000i	0.466732	+	0.269468i
$996$		0			0
$997$	13.7477	−	7.93725i	0.435395	−	0.251375i	−0.266247	−	0.963905i	$-0.585784\pi$
$997$	13.7477	−	7.93725i	0.435395	−	0.251375i	0.701642	+	0.712529i	$0.252450\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.n.n.109.3		8
3.2	odd	2	inner	648.2.n.n.109.2		8
4.3	odd	2		2592.2.r.p.433.1		8
8.3	odd	2		2592.2.r.p.433.4		8
8.5	even	2	inner	648.2.n.n.109.4		8
9.2	odd	6	inner	648.2.n.n.541.1		8
9.4	even	3		216.2.d.b.109.1	✓	4
9.5	odd	6		216.2.d.b.109.4	yes	4
9.7	even	3	inner	648.2.n.n.541.4		8
12.11	even	2		2592.2.r.p.433.3		8
24.5	odd	2	inner	648.2.n.n.109.1		8
24.11	even	2		2592.2.r.p.433.2		8
36.7	odd	6		2592.2.r.p.2161.4		8
36.11	even	6		2592.2.r.p.2161.2		8
36.23	even	6		864.2.d.a.433.2		4
36.31	odd	6		864.2.d.a.433.4		4
72.5	odd	6		216.2.d.b.109.3	yes	4
72.11	even	6		2592.2.r.p.2161.3		8
72.13	even	6		216.2.d.b.109.2	yes	4
72.29	odd	6	inner	648.2.n.n.541.2		8
72.43	odd	6		2592.2.r.p.2161.1		8
72.59	even	6		864.2.d.a.433.3		4
72.61	even	6	inner	648.2.n.n.541.3		8
72.67	odd	6		864.2.d.a.433.1		4
144.5	odd	12		6912.2.a.bu.1.2		2
144.13	even	12		6912.2.a.bu.1.1		2
144.59	even	12		6912.2.a.bv.1.2		2
144.67	odd	12		6912.2.a.bv.1.1		2
144.77	odd	12		6912.2.a.bc.1.1		2
144.85	even	12		6912.2.a.bc.1.2		2
144.131	even	12		6912.2.a.bd.1.1		2
144.139	odd	12		6912.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.d.b.109.1	✓	4	9.4	even	3
216.2.d.b.109.2	yes	4	72.13	even	6
216.2.d.b.109.3	yes	4	72.5	odd	6
216.2.d.b.109.4	yes	4	9.5	odd	6
648.2.n.n.109.1		8	24.5	odd	2	inner
648.2.n.n.109.2		8	3.2	odd	2	inner
648.2.n.n.109.3		8	1.1	even	1	trivial
648.2.n.n.109.4		8	8.5	even	2	inner
648.2.n.n.541.1		8	9.2	odd	6	inner
648.2.n.n.541.2		8	72.29	odd	6	inner
648.2.n.n.541.3		8	72.61	even	6	inner
648.2.n.n.541.4		8	9.7	even	3	inner
864.2.d.a.433.1		4	72.67	odd	6
864.2.d.a.433.2		4	36.23	even	6
864.2.d.a.433.3		4	72.59	even	6
864.2.d.a.433.4		4	36.31	odd	6
2592.2.r.p.433.1		8	4.3	odd	2
2592.2.r.p.433.2		8	24.11	even	2
2592.2.r.p.433.3		8	12.11	even	2
2592.2.r.p.433.4		8	8.3	odd	2
2592.2.r.p.2161.1		8	72.43	odd	6
2592.2.r.p.2161.2		8	36.11	even	6
2592.2.r.p.2161.3		8	72.11	even	6
2592.2.r.p.2161.4		8	36.7	odd	6
6912.2.a.bc.1.1		2	144.77	odd	12
6912.2.a.bc.1.2		2	144.85	even	12
6912.2.a.bd.1.1		2	144.131	even	12
6912.2.a.bd.1.2		2	144.139	odd	12
6912.2.a.bu.1.1		2	144.13	even	12
6912.2.a.bu.1.2		2	144.5	odd	12
6912.2.a.bv.1.1		2	144.67	odd	12
6912.2.a.bv.1.2		2	144.59	even	12

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.n (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.228425 + 1.39564i) q^{2} +(-1.89564 + 0.637600i) q^{4} +(-0.866025 - 0.500000i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-1.32288 - 2.50000i) q^{8} +O(q^{10})\)	\(q+(0.228425 + 1.39564i) q^{2} +(-1.89564 + 0.637600i) q^{4} +(-0.866025 - 0.500000i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-1.32288 - 2.50000i) q^{8} +(0.500000 - 1.32288i) q^{10} +(2.59808 - 1.50000i) q^{11} +(-4.58258 - 2.64575i) q^{13} +(1.09445 - 0.895644i) q^{14} +(3.18693 - 2.41733i) q^{16} +5.29150 q^{17} -5.29150i q^{19} +(1.96048 + 0.395644i) q^{20} +(2.68693 + 3.28335i) q^{22} +(-2.64575 + 4.58258i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(2.64575 - 7.00000i) q^{26} +(1.50000 + 1.32288i) q^{28} +(5.19615 - 3.00000i) q^{29} +(3.50000 - 6.06218i) q^{31} +(4.10170 + 3.89564i) q^{32} +(1.20871 + 7.38505i) q^{34} +1.00000i q^{35} +5.29150i q^{37} +(7.38505 - 1.20871i) q^{38} +(-0.104356 + 2.82650i) q^{40} +(2.64575 - 4.58258i) q^{41} +(-9.16515 + 5.29150i) q^{43} +(-3.96863 + 4.50000i) q^{44} +(-7.00000 - 2.64575i) q^{46} +(3.00000 - 5.19615i) q^{49} +(4.37780 - 3.58258i) q^{50} +(10.3739 + 2.09355i) q^{52} -9.00000i q^{53} -3.00000 q^{55} +(-1.50363 + 2.39564i) q^{56} +(5.37386 + 6.56670i) q^{58} +(3.46410 + 2.00000i) q^{59} +(9.26013 + 3.50000i) q^{62} +(-4.50000 + 6.61438i) q^{64} +(2.64575 + 4.58258i) q^{65} +(-10.0308 + 3.37386i) q^{68} +(-1.39564 + 0.228425i) q^{70} -15.8745 q^{71} +3.00000 q^{73} +(-7.38505 + 1.20871i) q^{74} +(3.37386 + 10.0308i) q^{76} +(-2.59808 - 1.50000i) q^{77} +(-2.00000 - 3.46410i) q^{79} +(-3.96863 + 0.500000i) q^{80} +(7.00000 + 2.64575i) q^{82} +(-6.06218 + 3.50000i) q^{83} +(-4.58258 - 2.64575i) q^{85} +(-9.47860 - 11.5826i) q^{86} +(-7.18693 - 4.51088i) q^{88} +10.5830 q^{89} +5.29150i q^{91} +(2.09355 - 10.3739i) q^{92} +(-2.64575 + 4.58258i) q^{95} +(-3.50000 - 6.06218i) q^{97} +(7.93725 + 3.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{4} - 4 q^{7}+O(q^{10}) \) 8 * q - 6 * q^4 - 4 * q^7	\( 8 q - 6 q^{4} - 4 q^{7} + 4 q^{10} - 2 q^{16} - 6 q^{22} - 16 q^{25} + 12 q^{28} + 28 q^{31} + 28 q^{34} - 10 q^{40} - 56 q^{46} + 24 q^{49} + 28 q^{52} - 24 q^{55} - 12 q^{58} - 36 q^{64} - 2 q^{70} + 24 q^{73} - 28 q^{76} - 16 q^{79} + 56 q^{82} - 30 q^{88} - 28 q^{97}+O(q^{100}) \) 8 * q - 6 * q^4 - 4 * q^7 + 4 * q^10 - 2 * q^16 - 6 * q^22 - 16 * q^25 + 12 * q^28 + 28 * q^31 + 28 * q^34 - 10 * q^40 - 56 * q^46 + 24 * q^49 + 28 * q^52 - 24 * q^55 - 12 * q^58 - 36 * q^64 - 2 * q^70 + 24 * q^73 - 28 * q^76 - 16 * q^79 + 56 * q^82 - 30 * q^88 - 28 * q^97

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