LMFDB - Embedded newform 1368.2.g.b.685.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1368,2,Mod(685,1368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1368.685"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,2,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 4x^12 + 4x^11 - 10x^10 + 24x^9 - 40x^8 + 48x^7 - 40x^6 + 32x^5 + 64x^4 - 128x^3 + 192x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.7
Root			$1.33852 + 0.456455i$ of defining polynomial
Character	$\chi$	$=$	1368.685
Dual form			1368.2.g.b.685.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.456455 - 1.33852i) q^{2} +(-1.58330 + 1.22195i) q^{4} -3.36827i q^{5} +4.47116 q^{7} +(2.35832 + 1.56152i) q^{8} +(-4.50852 + 1.53746i) q^{10} -0.608709i q^{11} -1.03922i q^{13} +(-2.04088 - 5.98476i) q^{14} +(1.01366 - 3.86943i) q^{16} +3.06367 q^{17} -1.00000i q^{19} +(4.11587 + 5.33298i) q^{20} +(-0.814773 + 0.277848i) q^{22} +8.50224 q^{23} -6.34525 q^{25} +(-1.39102 + 0.474357i) q^{26} +(-7.07918 + 5.46355i) q^{28} +7.27343i q^{29} +4.02054 q^{31} +(-5.64202 + 0.409404i) q^{32} +(-1.39843 - 4.10080i) q^{34} -15.0601i q^{35} +4.31265i q^{37} +(-1.33852 + 0.456455i) q^{38} +(5.25962 - 7.94345i) q^{40} +4.15770 q^{41} -6.27910i q^{43} +(0.743814 + 0.963768i) q^{44} +(-3.88089 - 11.3805i) q^{46} -4.73522 q^{47} +12.9913 q^{49} +(2.89632 + 8.49328i) q^{50} +(1.26988 + 1.64539i) q^{52} -6.98132i q^{53} -2.05030 q^{55} +(10.5444 + 6.98180i) q^{56} +(9.73567 - 3.31999i) q^{58} +2.64652i q^{59} +5.11145i q^{61} +(-1.83520 - 5.38159i) q^{62} +(3.12332 + 7.36511i) q^{64} -3.50037 q^{65} -2.62178i q^{67} +(-4.85071 + 3.74366i) q^{68} +(-20.1583 + 6.87425i) q^{70} -12.0085 q^{71} -12.5175 q^{73} +(5.77259 - 1.96853i) q^{74} +(1.22195 + 1.58330i) q^{76} -2.72164i q^{77} -0.913307 q^{79} +(-13.0333 - 3.41430i) q^{80} +(-1.89780 - 5.56518i) q^{82} -0.887809i q^{83} -10.3193i q^{85} +(-8.40473 + 2.86613i) q^{86} +(0.950510 - 1.43553i) q^{88} +7.61792 q^{89} -4.64652i q^{91} +(-13.4616 + 10.3893i) q^{92} +(2.16141 + 6.33820i) q^{94} -3.36827 q^{95} -5.93426 q^{97} +(-5.92994 - 17.3892i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{4} - 8 q^{7} + 12 q^{8} - 8 q^{10} - 4 q^{14} + 2 q^{16} + 8 q^{17} - 8 q^{20} + 20 q^{22} - 24 q^{25} + 10 q^{26} - 14 q^{28} + 16 q^{31} + 20 q^{32} - 2 q^{38} + 28 q^{40} - 16 q^{41} + 28 q^{44}+ \cdots + 48 q^{98}+O(q^{100}) $ 16 * q + 2 * q^4 - 8 * q^7 + 12 * q^8 - 8 * q^10 - 4 * q^14 + 2 * q^16 + 8 * q^17 - 8 * q^20 + 20 * q^22 - 24 * q^25 + 10 * q^26 - 14 * q^28 + 16 * q^31 + 20 * q^32 - 2 * q^38 + 28 * q^40 - 16 * q^41 + 28 * q^44 - 48 * q^46 - 24 * q^47 + 24 * q^49 - 12 * q^50 + 8 * q^52 + 16 * q^55 + 48 * q^56 + 38 * q^58 + 16 * q^62 + 14 * q^64 - 16 * q^65 + 26 * q^68 - 32 * q^70 - 48 * q^71 + 20 * q^74 - 4 * q^76 - 48 * q^79 - 4 * q^80 - 12 * q^82 - 48 * q^86 + 40 * q^88 + 16 * q^89 - 62 * q^92 - 36 * q^94 - 16 * q^95 + 32 * q^97 + 48 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	−0.456455	−	1.33852i	−0.322762	−	0.946480i
$2$	−0.456455	−	1.33852i	−0.322762	−	0.946480i
$3$		0			0
$3$		0			0
$4$	−1.58330	+	1.22195i	−0.791649	+	0.610976i
$5$		−	3.36827i		−	1.50634i	−0.657828	−	0.753168i	$-0.728525\pi$
$5$		−	3.36827i		−	1.50634i	0.657828	−	0.753168i	$-0.271475\pi$
$6$		0			0
$7$	4.47116			1.68994			0.844970	−	0.534813i	$-0.179618\pi$
$7$	4.47116			1.68994			0.844970	+	0.534813i	$0.179618\pi$
$8$	2.35832	+	1.56152i	0.833791	+	0.552080i
$9$		0			0
$10$	−4.50852	+	1.53746i	−1.42572	+	0.486189i
$11$		−	0.608709i		−	0.183533i	−0.995781	−	0.0917664i	$-0.970749\pi$
$11$		−	0.608709i		−	0.183533i	0.995781	−	0.0917664i	$-0.0292513\pi$
$12$		0			0
$13$		−	1.03922i		−	0.288228i	−0.989561	−	0.144114i	$-0.953967\pi$
$13$		−	1.03922i		−	0.288228i	0.989561	−	0.144114i	$-0.0460331\pi$
$14$	−2.04088	−	5.98476i	−0.545449	−	1.59950i
$15$		0			0
$16$	1.01366	−	3.86943i	0.253416	−	0.967357i
$17$	3.06367			0.743050			0.371525	−	0.928423i	$-0.378835\pi$
$17$	3.06367			0.743050			0.371525	+	0.928423i	$0.378835\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$	4.11587	+	5.33298i	0.920336	+	1.19249i
$21$		0			0
$22$	−0.814773	+	0.277848i	−0.173710	+	0.0592375i
$23$	8.50224			1.77284			0.886419	−	0.462883i	$-0.153185\pi$
$23$	8.50224			1.77284			0.886419	+	0.462883i	$0.153185\pi$
$24$		0			0
$25$	−6.34525			−1.26905
$26$	−1.39102	+	0.474357i	−0.272802	+	0.0930290i
$27$		0			0
$28$	−7.07918	+	5.46355i	−1.33784	+	1.03251i
$29$			7.27343i			1.35064i	0.737524	+	0.675321i	$0.235995\pi$
$29$			7.27343i			1.35064i	−0.737524	+	0.675321i	$0.764005\pi$
$30$		0			0
$31$	4.02054			0.722111			0.361055	−	0.932544i	$-0.382417\pi$
$31$	4.02054			0.722111			0.361055	+	0.932544i	$0.382417\pi$
$32$	−5.64202	+	0.409404i	−0.997378	+	0.0723731i
$33$		0			0
$34$	−1.39843	−	4.10080i	−0.239828	−	0.703282i
$35$		−	15.0601i		−	2.54562i
$36$		0			0
$37$			4.31265i			0.708995i	0.935057	+	0.354498i	$0.115348\pi$
$37$			4.31265i			0.708995i	−0.935057	+	0.354498i	$0.884652\pi$
$38$	−1.33852	+	0.456455i	−0.217137	+	0.0740468i
$39$		0			0
$40$	5.25962	−	7.94345i	0.831618	−	1.25597i
$41$	4.15770			0.649323			0.324661	−	0.945830i	$-0.394750\pi$
$41$	4.15770			0.649323			0.324661	+	0.945830i	$0.394750\pi$
$42$		0			0
$43$		−	6.27910i		−	0.957554i	−0.877937	−	0.478777i	$-0.841080\pi$
$43$		−	6.27910i		−	0.957554i	0.877937	−	0.478777i	$-0.158920\pi$
$44$	0.743814	+	0.963768i	0.112134	+	0.145294i
$45$		0			0
$46$	−3.88089	−	11.3805i	−0.572205	−	1.67796i
$47$	−4.73522			−0.690702			−0.345351	−	0.938474i	$-0.612240\pi$
$47$	−4.73522			−0.690702			−0.345351	+	0.938474i	$0.612240\pi$
$48$		0			0
$49$	12.9913			1.85590
$50$	2.89632	+	8.49328i	0.409602	+	1.20113i
$51$		0			0
$52$	1.26988	+	1.64539i	0.176100	+	0.228175i
$53$		−	6.98132i		−	0.958958i	−0.877553	−	0.479479i	$-0.840826\pi$
$53$		−	6.98132i		−	0.958958i	0.877553	−	0.479479i	$-0.159174\pi$
$54$		0			0
$55$	−2.05030			−0.276462
$56$	10.5444	+	6.98180i	1.40906	+	0.932982i
$57$		0			0
$58$	9.73567	−	3.31999i	1.27836	−	0.435936i
$59$			2.64652i			0.344547i	0.985049	+	0.172274i	$0.0551113\pi$
$59$			2.64652i			0.344547i	−0.985049	+	0.172274i	$0.944889\pi$
$60$		0			0
$61$			5.11145i			0.654454i	0.944946	+	0.327227i	$0.106114\pi$
$61$			5.11145i			0.654454i	−0.944946	+	0.327227i	$0.893886\pi$
$62$	−1.83520	−	5.38159i	−0.233070	−	0.683463i
$63$		0			0
$64$	3.12332	+	7.36511i	0.390416	+	0.920639i
$65$	−3.50037			−0.434168
$66$		0			0
$67$		−	2.62178i		−	0.320301i	−0.987093	−	0.160150i	$-0.948802\pi$
$67$		−	2.62178i		−	0.320301i	0.987093	−	0.160150i	$-0.0511979\pi$
$68$	−4.85071	+	3.74366i	−0.588235	+	0.453986i
$69$		0			0
$70$	−20.1583	+	6.87425i	−2.40938	+	0.821630i
$71$	−12.0085			−1.42514			−0.712572	−	0.701599i	$-0.752470\pi$
$71$	−12.0085			−1.42514			−0.712572	+	0.701599i	$0.752470\pi$
$72$		0			0
$73$	−12.5175			−1.46507			−0.732533	−	0.680731i	$-0.761662\pi$
$73$	−12.5175			−1.46507			−0.732533	+	0.680731i	$0.761662\pi$
$74$	5.77259	−	1.96853i	0.671050	−	0.228837i
$75$		0			0
$76$	1.22195	+	1.58330i	0.140168	+	0.181617i
$77$		−	2.72164i		−	0.310159i
$78$		0			0
$79$	−0.913307			−0.102755			−0.0513775	−	0.998679i	$-0.516361\pi$
$79$	−0.913307			−0.102755			−0.0513775	+	0.998679i	$0.516361\pi$
$80$	−13.0333	−	3.41430i	−1.45717	−	0.381730i
$81$		0			0
$82$	−1.89780	−	5.56518i	−0.209577	−	0.614571i
$83$		−	0.887809i		−	0.0974497i	−0.998812	−	0.0487249i	$-0.984484\pi$
$83$		−	0.887809i		−	0.0974497i	0.998812	−	0.0487249i	$-0.0155158\pi$
$84$		0			0
$85$		−	10.3193i		−	1.11928i
$86$	−8.40473	+	2.86613i	−0.906306	+	0.309062i
$87$		0			0
$88$	0.950510	−	1.43553i	0.101325	−	0.153028i
$89$	7.61792			0.807498			0.403749	−	0.914870i	$-0.367707\pi$
$89$	7.61792			0.807498			0.403749	+	0.914870i	$0.367707\pi$
$90$		0			0
$91$		−	4.64652i		−	0.487087i
$92$	−13.4616	+	10.3893i	−1.40347	+	1.08316i
$93$		0			0
$94$	2.16141	+	6.33820i	0.222933	+	0.653736i
$95$	−3.36827			−0.345577
$96$		0			0
$97$	−5.93426			−0.602533			−0.301267	−	0.953540i	$-0.597409\pi$
$97$	−5.93426			−0.602533			−0.301267	+	0.953540i	$0.597409\pi$
$98$	−5.92994	−	17.3892i	−0.599014	−	1.75657i
$99$		0			0
$100$	10.0464	−	7.75360i	1.00464	−	0.775360i
$101$		−	16.5793i		−	1.64970i	−0.565353	−	0.824849i	$-0.691260\pi$
$101$		−	16.5793i		−	1.64970i	0.565353	−	0.824849i	$-0.308740\pi$
$102$		0			0
$103$	−11.6631			−1.14920			−0.574600	−	0.818434i	$-0.694842\pi$
$103$	−11.6631			−1.14920			−0.574600	+	0.818434i	$0.694842\pi$
$104$	1.62276	−	2.45081i	0.159125	−	0.240322i
$105$		0			0
$106$	−9.34467	+	3.18666i	−0.907635	+	0.309516i
$107$			10.3213i			0.997801i	0.866659	+	0.498900i	$0.166263\pi$
$107$			10.3213i			0.997801i	−0.866659	+	0.498900i	$0.833737\pi$
$108$		0			0
$109$		−	1.85592i		−	0.177765i	−0.996042	−	0.0888824i	$-0.971670\pi$
$109$		−	1.85592i		−	0.177765i	0.996042	−	0.0888824i	$-0.0283295\pi$
$110$	0.935868	+	2.74438i	0.0892316	+	0.261666i
$111$		0			0
$112$	4.53226	−	17.3008i	0.428258	−	1.63478i
$113$	−18.4343			−1.73415			−0.867077	−	0.498174i	$-0.834004\pi$
$113$	−18.4343			−1.73415			−0.867077	+	0.498174i	$0.834004\pi$
$114$		0			0
$115$		−	28.6378i		−	2.67049i
$116$	−8.88779	−	11.5160i	−0.825210	−	1.06923i
$117$		0			0
$118$	3.54243	−	1.20802i	0.326107	−	0.111207i
$119$	13.6982			1.25571
$120$		0			0
$121$	10.6295			0.966316
$122$	6.84181	−	2.33315i	0.619428	−	0.211233i
$123$		0			0
$124$	−6.36571	+	4.91291i	−0.571658	+	0.441192i
$125$			4.53117i			0.405281i
$126$		0			0
$127$	−9.29686			−0.824963			−0.412481	−	0.910966i	$-0.635338\pi$
$127$	−9.29686			−0.824963			−0.412481	+	0.910966i	$0.635338\pi$
$128$	8.43273	−	7.54249i	0.745355	−	0.666668i
$129$		0			0
$130$	1.59776	+	4.68534i	0.140133	+	0.410931i
$131$		−	8.15016i		−	0.712083i	−0.934470	−	0.356042i	$-0.884126\pi$
$131$		−	8.15016i		−	0.712083i	0.934470	−	0.356042i	$-0.115874\pi$
$132$		0			0
$133$		−	4.47116i		−	0.387699i
$134$	−3.50931	+	1.19672i	−0.303158	+	0.103381i
$135$		0			0
$136$	7.22511	+	4.78398i	0.619548	+	0.410223i
$137$	−11.0756			−0.946254			−0.473127	−	0.880994i	$-0.656875\pi$
$137$	−11.0756			−0.946254			−0.473127	+	0.880994i	$0.656875\pi$
$138$		0			0
$139$			12.2091i			1.03557i	0.855512	+	0.517783i	$0.173242\pi$
$139$			12.2091i			1.03557i	−0.855512	+	0.517783i	$0.826758\pi$
$140$	18.4027	+	23.8446i	1.55531	+	2.01524i
$141$		0			0
$142$	5.48133	+	16.0736i	0.459983	+	1.34887i
$143$	−0.632582			−0.0528992
$144$		0			0
$145$	24.4989			2.03452
$146$	5.71369	+	16.7550i	0.472868	+	1.38666i
$147$		0			0
$148$	−5.26985	−	6.82821i	−0.433179	−	0.561275i
$149$			4.31060i			0.353138i	0.984288	+	0.176569i	$0.0564998\pi$
$149$			4.31060i			0.353138i	−0.984288	+	0.176569i	$0.943500\pi$
$150$		0			0
$151$	−9.89022			−0.804855			−0.402428	−	0.915452i	$-0.631833\pi$
$151$	−9.89022			−0.804855			−0.402428	+	0.915452i	$0.631833\pi$
$152$	1.56152	−	2.35832i	0.126656	−	0.191285i
$153$		0			0
$154$	−3.64298	+	1.24230i	−0.293560	+	0.100108i
$155$		−	13.5423i		−	1.08774i
$156$		0			0
$157$			7.39359i			0.590073i	0.955486	+	0.295036i	$0.0953318\pi$
$157$			7.39359i			0.590073i	−0.955486	+	0.295036i	$0.904668\pi$
$158$	0.416883	+	1.22248i	0.0331655	+	0.0972556i
$159$		0			0
$160$	1.37898	+	19.0039i	0.109018	+	1.50239i
$161$	38.0149			2.99599
$162$		0			0
$163$		−	12.5566i		−	0.983509i	−0.870734	−	0.491755i	$-0.836356\pi$
$163$		−	12.5566i		−	0.983509i	0.870734	−	0.491755i	$-0.163644\pi$
$164$	−6.58287	+	5.08051i	−0.514036	+	0.396721i
$165$		0			0
$166$	−1.18836	+	0.405245i	−0.0922342	+	0.0314531i
$167$	0.00624861			0.000483532			0.000241766	−	1.00000i	$-0.499923\pi$
$167$	0.00624861			0.000483532			0.000241766		1.00000i	$0.499923\pi$
$168$		0			0
$169$	11.9200			0.916925
$170$	−13.8126	+	4.71029i	−1.05938	+	0.361262i
$171$		0			0
$172$	7.67276	+	9.94169i	0.585043	+	0.758046i
$173$			4.45345i			0.338589i	0.985565	+	0.169295i	$0.0541489\pi$
$173$			4.45345i			0.338589i	−0.985565	+	0.169295i	$0.945851\pi$
$174$		0			0
$175$	−28.3707			−2.14462
$176$	−2.35536	−	0.617027i	−0.177542	−	0.0465102i
$177$		0			0
$178$	−3.47724	−	10.1968i	−0.260630	−	0.764281i
$179$		−	9.80470i		−	0.732838i	−0.930450	−	0.366419i	$-0.880584\pi$
$179$		−	9.80470i		−	0.732838i	0.930450	−	0.366419i	$-0.119416\pi$
$180$		0			0
$181$			17.9475i			1.33403i	0.745046	+	0.667013i	$0.232427\pi$
$181$			17.9475i			1.33403i	−0.745046	+	0.667013i	$0.767573\pi$
$182$	−6.21948	+	2.12093i	−0.461019	+	0.157213i
$183$		0			0
$184$	20.0510	+	13.2764i	1.47818	+	0.978749i
$185$	14.5262			1.06799
$186$		0			0
$187$		−	1.86489i		−	0.136374i
$188$	7.49726	−	5.78621i	0.546794	−	0.422003i
$189$		0			0
$190$	1.53746	+	4.50852i	0.111539	+	0.327082i
$191$	−3.85328			−0.278814			−0.139407	−	0.990235i	$-0.544520\pi$
$191$	−3.85328			−0.278814			−0.139407	+	0.990235i	$0.544520\pi$
$192$		0			0
$193$	19.5610			1.40803			0.704017	−	0.710183i	$-0.251388\pi$
$193$	19.5610			1.40803			0.704017	+	0.710183i	$0.251388\pi$
$194$	2.70872	+	7.94316i	0.194475	+	0.570285i
$195$		0			0
$196$	−20.5691	+	15.8747i	−1.46922	+	1.13391i
$197$			3.44831i			0.245682i	0.992426	+	0.122841i	$0.0392006\pi$
$197$			3.44831i			0.245682i	−0.992426	+	0.122841i	$0.960799\pi$
$198$		0			0
$199$	−7.64829			−0.542173			−0.271087	−	0.962555i	$-0.587383\pi$
$199$	−7.64829			−0.542173			−0.271087	+	0.962555i	$0.587383\pi$
$200$	−14.9641	−	9.90822i	−1.05812	−	0.700617i
$201$		0			0
$202$	−22.1918	+	7.56768i	−1.56141	+	0.532460i
$203$			32.5207i			2.28251i
$204$		0			0
$205$		−	14.0042i		−	0.978099i
$206$	5.32368	+	15.6114i	0.370919	+	1.08770i
$207$		0			0
$208$	−4.02118	−	1.05342i	−0.278819	−	0.0730415i
$209$	−0.608709			−0.0421053
$210$		0			0
$211$			14.4648i			0.995801i	0.867234	+	0.497900i	$0.165895\pi$
$211$			14.4648i			0.995801i	−0.867234	+	0.497900i	$0.834105\pi$
$212$	8.53084	+	11.0535i	0.585901	+	0.759158i
$213$		0			0
$214$	13.8154	−	4.71122i	0.944398	−	0.322052i
$215$	−21.1497			−1.44240
$216$		0			0
$217$	17.9765			1.22032
$218$	−2.48419	+	0.847143i	−0.168251	+	0.0573758i
$219$		0			0
$220$	3.24623	−	2.50537i	0.218861	−	0.168912i
$221$		−	3.18383i		−	0.214167i
$222$		0			0
$223$	4.74734			0.317906			0.158953	−	0.987286i	$-0.449188\pi$
$223$	4.74734			0.317906			0.158953	+	0.987286i	$0.449188\pi$
$224$	−25.2264	+	1.83051i	−1.68551	+	0.122306i
$225$		0			0
$226$	8.41443	+	24.6748i	0.559719	+	1.64134i
$227$		−	2.67149i		−	0.177313i	−0.996062	−	0.0886566i	$-0.971743\pi$
$227$		−	2.67149i		−	0.177313i	0.996062	−	0.0886566i	$-0.0282574\pi$
$228$		0			0
$229$			7.87392i			0.520323i	0.965565	+	0.260162i	$0.0837758\pi$
$229$			7.87392i			0.520323i	−0.965565	+	0.260162i	$0.916224\pi$
$230$	−38.3325	+	13.0719i	−2.52757	+	0.861934i
$231$		0			0
$232$	−11.3576	+	17.1531i	−0.745662	+	1.12615i
$233$	10.2320			0.670321			0.335160	−	0.942161i	$-0.391209\pi$
$233$	10.2320			0.670321			0.335160	+	0.942161i	$0.391209\pi$
$234$		0			0
$235$			15.9495i			1.04043i
$236$	−3.23392	−	4.19023i	−0.210510	−	0.272760i
$237$		0			0
$238$	−6.25260	−	18.3354i	−0.405296	−	1.18850i
$239$	15.1051			0.977067			0.488533	−	0.872545i	$-0.337532\pi$
$239$	15.1051			0.977067			0.488533	+	0.872545i	$0.337532\pi$
$240$		0			0
$241$	6.17274			0.397621			0.198811	−	0.980038i	$-0.436292\pi$
$241$	6.17274			0.397621			0.198811	+	0.980038i	$0.436292\pi$
$242$	−4.85187	−	14.2278i	−0.311890	−	0.914599i
$243$		0			0
$244$	−6.24595	−	8.09295i	−0.399856	−	0.518098i
$245$		−	43.7582i		−	2.79561i
$246$		0			0
$247$	−1.03922			−0.0661239
$248$	9.48171	+	6.27815i	0.602089	+	0.398663i
$249$		0			0
$250$	6.06509	−	2.06828i	0.383590	−	0.130809i
$251$			6.78315i			0.428149i	0.976817	+	0.214074i	$0.0686735\pi$
$251$			6.78315i			0.428149i	−0.976817	+	0.214074i	$0.931327\pi$
$252$		0			0
$253$		−	5.17539i		−	0.325374i
$254$	4.24360	+	12.4441i	0.266267	+	0.780811i
$255$		0			0
$256$	−13.9450	−	7.84461i	−0.871560	−	0.490288i
$257$	26.6333			1.66134			0.830669	−	0.556766i	$-0.187958\pi$
$257$	26.6333			1.66134			0.830669	+	0.556766i	$0.187958\pi$
$258$		0			0
$259$			19.2826i			1.19816i
$260$	5.54213	−	4.27729i	0.343708	−	0.265266i
$261$		0			0
$262$	−10.9092	+	3.72018i	−0.673972	+	0.229834i
$263$	−14.6785			−0.905117			−0.452559	−	0.891735i	$-0.649489\pi$
$263$	−14.6785			−0.905117			−0.452559	+	0.891735i	$0.649489\pi$
$264$		0			0
$265$	−23.5150			−1.44451
$266$	−5.98476	+	2.04088i	−0.366949	+	0.125135i
$267$		0			0
$268$	3.20369	+	4.15105i	0.195696	+	0.253566i
$269$			11.6590i			0.710862i	0.934702	+	0.355431i	$0.115666\pi$
$269$			11.6590i			0.710862i	−0.934702	+	0.355431i	$0.884334\pi$
$270$		0			0
$271$	0.125029			0.00759496			0.00379748	−	0.999993i	$-0.498791\pi$
$271$	0.125029			0.00759496			0.00379748	+	0.999993i	$0.498791\pi$
$272$	3.10554	−	11.8547i	0.188301	−	0.718795i
$273$		0			0
$274$	5.05552	+	14.8250i	0.305415	+	0.895610i
$275$			3.86241i			0.232912i
$276$		0			0
$277$			19.8203i			1.19088i	0.803398	+	0.595442i	$0.203023\pi$
$277$			19.8203i			1.19088i	−0.803398	+	0.595442i	$0.796977\pi$
$278$	16.3422	−	5.57292i	0.980142	−	0.334242i
$279$		0			0
$280$	23.5166	−	35.5165i	1.40539	−	2.12252i
$281$	−9.24019			−0.551223			−0.275612	−	0.961269i	$-0.588880\pi$
$281$	−9.24019			−0.551223			−0.275612	+	0.961269i	$0.588880\pi$
$282$		0			0
$283$		−	30.3491i		−	1.80407i	−0.431664	−	0.902035i	$-0.642073\pi$
$283$		−	30.3491i		−	1.80407i	0.431664	−	0.902035i	$-0.357927\pi$
$284$	19.0130	−	14.6738i	1.12821	−	0.870729i
$285$		0			0
$286$	0.288745	+	0.846727i	0.0170739	+	0.0500680i
$287$	18.5897			1.09732
$288$		0			0
$289$	−7.61391			−0.447877
$290$	−11.1826	−	32.7924i	−0.656667	−	1.92563i
$291$		0			0
$292$	19.8190	−	15.2958i	1.15982	−	0.895121i
$293$			19.5780i			1.14376i	0.820337	+	0.571880i	$0.193786\pi$
$293$			19.5780i			1.14376i	−0.820337	+	0.571880i	$0.806214\pi$
$294$		0			0
$295$	8.91419			0.519004
$296$	−6.73428	+	10.1706i	−0.391422	+	0.591154i
$297$		0			0
$298$	5.76984	−	1.96759i	0.334238	−	0.113980i
$299$		−	8.83569i		−	0.510981i
$300$		0			0
$301$		−	28.0749i		−	1.61821i
$302$	4.51444	+	13.2383i	0.259777	+	0.761779i
$303$		0			0
$304$	−3.86943	−	1.01366i	−0.221927	−	0.0581377i
$305$	17.2168			0.985829
$306$		0			0
$307$			1.54809i			0.0883541i	0.999024	+	0.0441770i	$0.0140666\pi$
$307$			1.54809i			0.0883541i	−0.999024	+	0.0441770i	$0.985933\pi$
$308$	3.32571	+	4.30916i	0.189500	+	0.245537i
$309$		0			0
$310$	−18.1267	+	6.18144i	−1.02953	+	0.351082i
$311$	3.34671			0.189774			0.0948872	−	0.995488i	$-0.469751\pi$
$311$	3.34671			0.189774			0.0948872	+	0.995488i	$0.469751\pi$
$312$		0			0
$313$	6.40232			0.361881			0.180940	−	0.983494i	$-0.442086\pi$
$313$	6.40232			0.361881			0.180940	+	0.983494i	$0.442086\pi$
$314$	9.89651	−	3.37484i	0.558492	−	0.190453i
$315$		0			0
$316$	1.44604	−	1.11602i	0.0813459	−	0.0627809i
$317$		−	9.23830i		−	0.518875i	−0.965760	−	0.259437i	$-0.916463\pi$
$317$		−	9.23830i		−	0.518875i	0.965760	−	0.259437i	$-0.0835371\pi$
$318$		0			0
$319$	4.42741			0.247887
$320$	24.8077	−	10.5202i	1.38679	−	0.588097i
$321$		0			0
$322$	−17.3521	−	50.8839i	−0.966993	−	2.83565i
$323$		−	3.06367i		−	0.170467i
$324$		0			0
$325$			6.59411i			0.365775i
$326$	−16.8073	+	5.73152i	−0.930872	+	0.317440i
$327$		0			0
$328$	9.80517	+	6.49232i	0.541400	+	0.358478i
$329$	−21.1719			−1.16725
$330$		0			0
$331$			4.65778i			0.256015i	0.991773	+	0.128007i	$0.0408582\pi$
$331$			4.65778i			0.256015i	−0.991773	+	0.128007i	$0.959142\pi$
$332$	1.08486	+	1.40567i	0.0595395	+	0.0771460i
$333$		0			0
$334$	−0.00285221	−	0.00836392i	−0.000156066	−	0.000457654i
$335$	−8.83085			−0.482481
$336$		0			0
$337$	−9.34371			−0.508984			−0.254492	−	0.967075i	$-0.581908\pi$
$337$	−9.34371			−0.508984			−0.254492	+	0.967075i	$0.581908\pi$
$338$	−5.44095	−	15.9552i	−0.295949	−	0.867851i
$339$		0			0
$340$	12.6097	+	16.3385i	0.683855	+	0.886079i
$341$		−	2.44734i		−	0.132531i
$342$		0			0
$343$	26.7881			1.44642
$344$	9.80493	−	14.8081i	0.528646	−	0.798400i
$345$		0			0
$346$	5.96105	−	2.03280i	0.320468	−	0.109284i
$347$			30.9185i			1.65979i	0.557917	+	0.829897i	$0.311601\pi$
$347$			30.9185i			1.65979i	−0.557917	+	0.829897i	$0.688399\pi$
$348$		0			0
$349$			25.2445i			1.35131i	0.737220	+	0.675653i	$0.236138\pi$
$349$			25.2445i			1.35131i	−0.737220	+	0.675653i	$0.763862\pi$
$350$	12.9499	+	37.9748i	0.692202	+	2.02984i
$351$		0			0
$352$	0.249208	+	3.43435i	0.0132828	+	0.183051i
$353$	2.15727			0.114820			0.0574099	−	0.998351i	$-0.481716\pi$
$353$	2.15727			0.114820			0.0574099	+	0.998351i	$0.481716\pi$
$354$		0			0
$355$			40.4478i			2.14675i
$356$	−12.0614	+	9.30874i	−0.639255	+	0.493362i
$357$		0			0
$358$	−13.1238	+	4.47540i	−0.693616	+	0.236532i
$359$	−32.0925			−1.69378			−0.846889	−	0.531770i	$-0.821527\pi$
$359$	−32.0925			−1.69378			−0.846889	+	0.531770i	$0.821527\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	24.0231	−	8.19221i	1.26263	−	0.430573i
$363$		0			0
$364$	5.67782	+	7.35682i	0.297599	+	0.385602i
$365$			42.1624i			2.20688i
$366$		0			0
$367$	10.3206			0.538729			0.269365	−	0.963038i	$-0.413186\pi$
$367$	10.3206			0.538729			0.269365	+	0.963038i	$0.413186\pi$
$368$	8.61842	−	32.8988i	0.449266	−	1.71497i
$369$		0			0
$370$	−6.63054	−	19.4437i	−0.344706	−	1.01083i
$371$		−	31.2146i		−	1.62058i
$372$		0			0
$373$		−	20.3031i		−	1.05125i	−0.850715	−	0.525627i	$-0.823831\pi$
$373$		−	20.3031i		−	1.05125i	0.850715	−	0.525627i	$-0.176169\pi$
$374$	−2.49620	+	0.851236i	−0.129075	+	0.0440164i
$375$		0			0
$376$	−11.1671	−	7.39412i	−0.575901	−	0.381323i
$377$	7.55869			0.389292
$378$		0			0
$379$			15.4355i			0.792867i	0.918063	+	0.396433i	$0.129752\pi$
$379$			15.4355i			0.792867i	−0.918063	+	0.396433i	$0.870248\pi$
$380$	5.33298	−	4.11587i	0.273576	−	0.211140i
$381$		0			0
$382$	1.75885	+	5.15771i	0.0899906	+	0.263892i
$383$	37.7837			1.93066			0.965328	−	0.261041i	$-0.0840658\pi$
$383$	37.7837			1.93066			0.965328	+	0.261041i	$0.0840658\pi$
$384$		0			0
$385$	−9.16722			−0.467205
$386$	−8.92873	−	26.1829i	−0.454460	−	1.33268i
$387$		0			0
$388$	9.39571	−	7.25139i	0.476995	−	0.368133i
$389$			13.7943i			0.699399i	0.936862	+	0.349700i	$0.113716\pi$
$389$			13.7943i			0.699399i	−0.936862	+	0.349700i	$0.886284\pi$
$390$		0			0
$391$	26.0481			1.31731
$392$	30.6376	+	20.2861i	1.54743	+	1.02460i
$393$		0			0
$394$	4.61566	−	1.57400i	0.232533	−	0.0792970i
$395$			3.07627i			0.154784i
$396$		0			0
$397$			33.1665i			1.66458i	0.554343	+	0.832288i	$0.312970\pi$
$397$			33.1665i			1.66458i	−0.554343	+	0.832288i	$0.687030\pi$
$398$	3.49110	+	10.2374i	0.174993	+	0.513156i
$399$		0			0
$400$	−6.43196	+	24.5525i	−0.321598	+	1.22763i
$401$	8.24620			0.411796			0.205898	−	0.978573i	$-0.433989\pi$
$401$	8.24620			0.411796			0.205898	+	0.978573i	$0.433989\pi$
$402$		0			0
$403$		−	4.17822i		−	0.208132i
$404$	20.2591	+	26.2499i	1.00793	+	1.30598i
$405$		0			0
$406$	43.5298	−	14.8442i	2.16035	−	0.736707i
$407$	2.62515			0.130124
$408$		0			0
$409$	30.4291			1.50462			0.752312	−	0.658807i	$-0.228939\pi$
$409$	30.4291			1.50462			0.752312	+	0.658807i	$0.228939\pi$
$410$	−18.7450	+	6.39231i	−0.925751	+	0.315694i
$411$		0			0
$412$	18.4662	−	14.2518i	0.909763	−	0.702134i
$413$			11.8330i			0.582264i
$414$		0			0
$415$	−2.99038			−0.146792
$416$	0.425460	+	5.86329i	0.0208599	+	0.287472i
$417$		0			0
$418$	0.277848	+	0.814773i	0.0135900	+	0.0398518i
$419$			5.11734i			0.249999i	0.992157	+	0.124999i	$0.0398929\pi$
$419$			5.11734i			0.249999i	−0.992157	+	0.124999i	$0.960107\pi$
$420$		0			0
$421$		−	28.8854i		−	1.40779i	−0.710305	−	0.703894i	$-0.751443\pi$
$421$		−	28.8854i		−	1.40779i	0.710305	−	0.703894i	$-0.248557\pi$
$422$	19.3616	−	6.60255i	0.942506	−	0.321407i
$423$		0			0
$424$	10.9015	−	16.4642i	0.529422	−	0.799571i
$425$	−19.4398			−0.942967
$426$		0			0
$427$			22.8541i			1.10599i
$428$	−12.6122	−	16.3417i	−0.609632	−	0.789908i
$429$		0			0
$430$	9.65389	+	28.3094i	0.465552	+	1.36520i
$431$	19.3963			0.934288			0.467144	−	0.884181i	$-0.345283\pi$
$431$	19.3963			0.934288			0.467144	+	0.884181i	$0.345283\pi$
$432$		0			0
$433$	30.5791			1.46954			0.734770	−	0.678317i	$-0.237290\pi$
$433$	30.5791			1.46954			0.734770	+	0.678317i	$0.237290\pi$
$434$	−8.20546	−	24.0620i	−0.393875	−	1.15501i
$435$		0			0
$436$	2.26784	+	2.93847i	0.108610	+	0.140727i
$437$		−	8.50224i		−	0.406717i
$438$		0			0
$439$	−21.4056			−1.02163			−0.510817	−	0.859690i	$-0.670657\pi$
$439$	−21.4056			−1.02163			−0.510817	+	0.859690i	$0.670657\pi$
$440$	−4.83525	−	3.20158i	−0.230512	−	0.152629i
$441$		0			0
$442$	−4.26163	+	1.45327i	−0.202705	+	0.0691252i
$443$		−	8.83665i		−	0.419842i	−0.977718	−	0.209921i	$-0.932679\pi$
$443$		−	8.83665i		−	0.419842i	0.977718	−	0.209921i	$-0.0673207\pi$
$444$		0			0
$445$		−	25.6592i		−	1.21636i
$446$	−2.16695	−	6.35444i	−0.102608	−	0.300891i
$447$		0			0
$448$	13.9649	+	32.9306i	0.659779	+	1.55582i
$449$	−17.9257			−0.845968			−0.422984	−	0.906137i	$-0.639017\pi$
$449$	−17.9257			−0.845968			−0.422984	+	0.906137i	$0.639017\pi$
$450$		0			0
$451$		−	2.53083i		−	0.119172i
$452$	29.1870	−	22.5258i	1.37284	−	1.05953i
$453$		0			0
$454$	−3.57586	+	1.21942i	−0.167823	+	0.0572300i
$455$	−15.6507			−0.733718
$456$		0			0
$457$	−11.8630			−0.554930			−0.277465	−	0.960736i	$-0.589494\pi$
$457$	−11.8630			−0.554930			−0.277465	+	0.960736i	$0.589494\pi$
$458$	10.5394	−	3.59409i	0.492475	−	0.167941i
$459$		0			0
$460$	34.9941	+	45.3422i	1.63161	+	2.11409i
$461$			7.03036i			0.327436i	0.986507	+	0.163718i	$0.0523488\pi$
$461$			7.03036i			0.327436i	−0.986507	+	0.163718i	$0.947651\pi$
$462$		0			0
$463$	−0.523381			−0.0243236			−0.0121618	−	0.999926i	$-0.503871\pi$
$463$	−0.523381			−0.0243236			−0.0121618	+	0.999926i	$0.503871\pi$
$464$	28.1440	+	7.37282i	1.30655	+	0.342275i
$465$		0			0
$466$	−4.67045	−	13.6958i	−0.216354	−	0.634445i
$467$		−	15.1010i		−	0.698789i	−0.936976	−	0.349395i	$-0.886387\pi$
$467$		−	15.1010i		−	0.698789i	0.936976	−	0.349395i	$-0.113613\pi$
$468$		0			0
$469$		−	11.7224i		−	0.541290i
$470$	21.3488	−	7.28022i	0.984746	−	0.335812i
$471$		0			0
$472$	−4.13258	+	6.24133i	−0.190218	+	0.287280i
$473$	−3.82215			−0.175742
$474$		0			0
$475$			6.34525i			0.291140i
$476$	−21.6883	+	16.7385i	−0.994081	+	0.767209i
$477$		0			0
$478$	−6.89479	−	20.2185i	−0.315360	−	0.924774i
$479$	−42.8653			−1.95857			−0.979284	−	0.202493i	$-0.935096\pi$
$479$	−42.8653			−1.95857			−0.979284	+	0.202493i	$0.935096\pi$
$480$		0			0
$481$	4.48179			0.204352
$482$	−2.81758	−	8.26237i	−0.128337	−	0.376340i
$483$		0			0
$484$	−16.8296	+	12.9887i	−0.764983	+	0.590396i
$485$			19.9882i			0.907618i
$486$		0			0
$487$	−26.1098			−1.18315			−0.591573	−	0.806251i	$-0.701493\pi$
$487$	−26.1098			−1.18315			−0.591573	+	0.806251i	$0.701493\pi$
$488$	−7.98162	+	12.0544i	−0.361311	+	0.545678i
$489$		0			0
$490$	−58.5715	+	19.9736i	−2.64599	+	0.902317i
$491$		−	31.5875i		−	1.42552i	−0.701407	−	0.712761i	$-0.747444\pi$
$491$		−	31.5875i		−	1.42552i	0.701407	−	0.712761i	$-0.252556\pi$
$492$		0			0
$493$			22.2834i			1.00359i
$494$	0.474357	+	1.39102i	0.0213423	+	0.0625850i
$495$		0			0
$496$	4.07548	−	15.5572i	0.182995	−	0.698539i
$497$	−53.6919			−2.40841
$498$		0			0
$499$		−	16.1119i		−	0.721268i	−0.932707	−	0.360634i	$-0.882560\pi$
$499$		−	16.1119i		−	0.721268i	0.932707	−	0.360634i	$-0.117440\pi$
$500$	−5.53688	−	7.17420i	−0.247617	−	0.320840i
$501$		0			0
$502$	9.07941	−	3.09620i	0.405234	−	0.138190i
$503$	−6.81093			−0.303684			−0.151842	−	0.988405i	$-0.548521\pi$
$503$	−6.81093			−0.303684			−0.151842	+	0.988405i	$0.548521\pi$
$504$		0			0
$505$	−55.8435			−2.48500
$506$	−6.92739	+	2.36233i	−0.307960	+	0.105018i
$507$		0			0
$508$	14.7197	−	11.3603i	0.653081	−	0.504033i
$509$			15.5299i			0.688350i	0.938905	+	0.344175i	$0.111841\pi$
$509$			15.5299i			0.688350i	−0.938905	+	0.344175i	$0.888159\pi$
$510$		0			0
$511$	−55.9679			−2.47588
$512$	−4.13496	+	22.2464i	−0.182741	+	0.983161i
$513$		0			0
$514$	−12.1569	−	35.6493i	−0.536217	−	1.57242i
$515$			39.2845i			1.73108i
$516$		0			0
$517$			2.88237i			0.126766i
$518$	25.8102	−	8.80162i	1.13403	−	0.386721i
$519$		0			0
$520$	−8.25499	−	5.46589i	−0.362005	−	0.239695i
$521$	4.33054			0.189724			0.0948622	−	0.995490i	$-0.469759\pi$
$521$	4.33054			0.189724			0.0948622	+	0.995490i	$0.469759\pi$
$522$		0			0
$523$		−	5.17275i		−	0.226189i	−0.993584	−	0.113094i	$-0.963924\pi$
$523$		−	5.17275i		−	0.226189i	0.993584	−	0.113094i	$-0.0360762\pi$
$524$	9.95911	+	12.9041i	0.435066	+	0.563720i
$525$		0			0
$526$	6.70009	+	19.6476i	0.292138	+	0.856675i
$527$	12.3176			0.536564
$528$		0			0
$529$	49.2880			2.14296
$530$	10.7335	+	31.4754i	0.466235	+	1.36720i
$531$		0			0
$532$	5.46355	+	7.07918i	0.236875	+	0.306922i
$533$		−	4.32076i		−	0.187153i
$534$		0			0
$535$	34.7650			1.50302
$536$	4.09395	−	6.18298i	0.176832	−	0.267064i
$537$		0			0
$538$	15.6059	−	5.32181i	0.672817	−	0.229440i
$539$		−	7.90792i		−	0.340618i
$540$		0			0
$541$		−	18.2195i		−	0.783316i	−0.920111	−	0.391658i	$-0.871902\pi$
$541$		−	18.2195i		−	0.783316i	0.920111	−	0.391658i	$-0.128098\pi$
$542$	−0.0570700	−	0.167354i	−0.00245137	−	0.00718848i
$543$		0			0
$544$	−17.2853	+	1.25428i	−0.741101	+	0.0537768i
$545$	−6.25124			−0.267774
$546$		0			0
$547$		−	39.0195i		−	1.66835i	−0.551499	−	0.834176i	$-0.685944\pi$
$547$		−	39.0195i		−	1.66835i	0.551499	−	0.834176i	$-0.314056\pi$
$548$	17.5360	−	13.5339i	0.749101	−	0.578139i
$549$		0			0
$550$	5.16994	−	1.76302i	0.220447	−	0.0751753i
$551$	7.27343			0.309859
$552$		0			0
$553$	−4.08354			−0.173650
$554$	26.5299	−	9.04706i	1.12715	−	0.384373i
$555$		0			0
$556$	−14.9190	−	19.3307i	−0.632706	−	0.819805i
$557$			17.3116i			0.733516i	0.930316	+	0.366758i	$0.119532\pi$
$557$			17.3116i			0.733516i	−0.930316	+	0.366758i	$0.880468\pi$
$558$		0			0
$559$	−6.52536			−0.275993
$560$	−58.2740	−	15.2659i	−2.46252	−	0.645101i
$561$		0			0
$562$	4.21773	+	12.3682i	0.177914	+	0.521722i
$563$			1.53556i			0.0647162i	0.999476	+	0.0323581i	$0.0103017\pi$
$563$			1.53556i			0.0647162i	−0.999476	+	0.0323581i	$0.989698\pi$
$564$		0			0
$565$			62.0917i			2.61222i
$566$	−40.6231	+	13.8530i	−1.70752	+	0.582286i
$567$		0			0
$568$	−28.3198	−	18.7515i	−1.18827	−	0.786793i
$569$	−25.4671			−1.06764			−0.533818	−	0.845600i	$-0.679243\pi$
$569$	−25.4671			−1.06764			−0.533818	+	0.845600i	$0.679243\pi$
$570$		0			0
$571$		−	31.1021i		−	1.30158i	−0.759256	−	0.650792i	$-0.774437\pi$
$571$		−	31.1021i		−	1.30158i	0.759256	−	0.650792i	$-0.225563\pi$
$572$	1.00157	−	0.772985i	0.0418776	−	0.0323201i
$573$		0			0
$574$	−8.48537	−	24.8828i	−0.354173	−	1.03859i
$575$	−53.9488			−2.24982
$576$		0			0
$577$	6.93064			0.288526			0.144263	−	0.989539i	$-0.453919\pi$
$577$	6.93064			0.288526			0.144263	+	0.989539i	$0.453919\pi$
$578$	3.47541	+	10.1914i	0.144558	+	0.423907i
$579$		0			0
$580$	−38.7890	+	29.9365i	−1.61063	+	1.24304i
$581$		−	3.96954i		−	0.164684i
$582$		0			0
$583$	−4.24960			−0.176000
$584$	−29.5203	−	19.5463i	−1.22156	−	0.808834i
$585$		0			0
$586$	26.2057	−	8.93649i	1.08255	−	0.369163i
$587$		−	14.2948i		−	0.590009i	−0.955496	−	0.295005i	$-0.904679\pi$
$587$		−	14.2948i		−	0.590009i	0.955496	−	0.295005i	$-0.0953212\pi$
$588$		0			0
$589$		−	4.02054i		−	0.165664i
$590$	−4.06893	−	11.9319i	−0.167515	−	0.491227i
$591$		0			0
$592$	16.6875	+	4.37158i	0.685852	+	0.179671i
$593$	21.2046			0.870770			0.435385	−	0.900244i	$-0.356612\pi$
$593$	21.2046			0.870770			0.435385	+	0.900244i	$0.356612\pi$
$594$		0			0
$595$		−	46.1392i		−	1.89152i
$596$	−5.26734	−	6.82496i	−0.215759	−	0.279561i
$597$		0			0
$598$	−11.8268	+	4.03309i	−0.483633	+	0.164925i
$599$	−6.52684			−0.266679			−0.133340	−	0.991070i	$-0.542570\pi$
$599$	−6.52684			−0.266679			−0.133340	+	0.991070i	$0.542570\pi$
$600$		0			0
$601$	−37.6306			−1.53498			−0.767492	−	0.641059i	$-0.778495\pi$
$601$	−37.6306			−1.53498			−0.767492	+	0.641059i	$0.778495\pi$
$602$	−37.5789	+	12.8149i	−1.53160	+	0.522297i
$603$		0			0
$604$	15.6592	−	12.0854i	0.637163	−	0.491747i
$605$		−	35.8029i		−	1.45560i
$606$		0			0
$607$	−24.8911			−1.01030			−0.505150	−	0.863032i	$-0.668563\pi$
$607$	−24.8911			−1.01030			−0.505150	+	0.863032i	$0.668563\pi$
$608$	0.409404	+	5.64202i	0.0166035	+	0.228814i
$609$		0			0
$610$	−7.85867	−	23.0451i	−0.318188	−	0.933067i
$611$			4.92093i			0.199079i
$612$		0			0
$613$			0.491774i			0.0198626i	0.999951	+	0.00993128i	$0.00316128\pi$
$613$			0.491774i			0.0198626i	−0.999951	+	0.00993128i	$0.996839\pi$
$614$	2.07216	−	0.706633i	0.0836254	−	0.0285174i
$615$		0			0
$616$	4.24989	−	6.41849i	0.171233	−	0.258608i
$617$	−14.2755			−0.574711			−0.287356	−	0.957824i	$-0.592776\pi$
$617$	−14.2755			−0.574711			−0.287356	+	0.957824i	$0.592776\pi$
$618$		0			0
$619$		−	17.6251i		−	0.708411i	−0.935168	−	0.354206i	$-0.884751\pi$
$619$		−	17.6251i		−	0.708411i	0.935168	−	0.354206i	$-0.115249\pi$
$620$	16.5480	+	21.4415i	0.664584	+	0.861110i
$621$		0			0
$622$	−1.52762	−	4.47965i	−0.0612520	−	0.179618i
$623$	34.0610			1.36462
$624$		0			0
$625$	−16.4640			−0.658561
$626$	−2.92237	−	8.56967i	−0.116801	−	0.342513i
$627$		0			0
$628$	−9.03462	−	11.7063i	−0.360520	−	0.467130i
$629$			13.2125i			0.526819i
$630$		0			0
$631$	−0.532668			−0.0212052			−0.0106026	−	0.999944i	$-0.503375\pi$
$631$	−0.532668			−0.0212052			−0.0106026	+	0.999944i	$0.503375\pi$
$632$	−2.15387	−	1.42615i	−0.0856763	−	0.0567290i
$633$		0			0
$634$	−12.3657	+	4.21687i	−0.491105	+	0.167473i
$635$			31.3143i			1.24267i
$636$		0			0
$637$		−	13.5008i		−	0.534921i
$638$	−2.02091	−	5.92619i	−0.0800086	−	0.234620i
$639$		0			0
$640$	−25.4051	−	28.4037i	−1.00423	−	1.12276i
$641$	−16.5745			−0.654654			−0.327327	−	0.944911i	$-0.606148\pi$
$641$	−16.5745			−0.654654			−0.327327	+	0.944911i	$0.606148\pi$
$642$		0			0
$643$			9.17617i			0.361873i	0.983495	+	0.180936i	$0.0579128\pi$
$643$			9.17617i			0.361873i	−0.983495	+	0.180936i	$0.942087\pi$
$644$	−60.1889	+	46.4524i	−2.37177	+	1.83048i
$645$		0			0
$646$	−4.10080	+	1.39843i	−0.161344	+	0.0550204i
$647$	20.1025			0.790311			0.395155	−	0.918614i	$-0.370691\pi$
$647$	20.1025			0.790311			0.395155	+	0.918614i	$0.370691\pi$
$648$		0			0
$649$	1.61096			0.0632357
$650$	8.82638	−	3.00991i	0.346199	−	0.118058i
$651$		0			0
$652$	15.3436	+	19.8809i	0.600901	+	0.778594i
$653$		−	42.6830i		−	1.67031i	−0.550012	−	0.835157i	$-0.685377\pi$
$653$		−	42.6830i		−	1.67031i	0.550012	−	0.835157i	$-0.314623\pi$
$654$		0			0
$655$	−27.4520			−1.07264
$656$	4.21451	−	16.0879i	0.164549	−	0.628127i
$657$		0			0
$658$	9.66403	+	28.3391i	0.376743	+	1.10477i
$659$			34.7270i			1.35277i	0.736548	+	0.676385i	$0.236454\pi$
$659$			34.7270i			1.35277i	−0.736548	+	0.676385i	$0.763546\pi$
$660$		0			0
$661$		−	0.0670508i		−	0.00260797i	−0.999999	−	0.00130399i	$-0.999585\pi$
$661$		−	0.0670508i		−	0.00260797i	0.999999	−	0.00130399i	$-0.000415072\pi$
$662$	6.23456	−	2.12607i	0.242313	−	0.0826320i
$663$		0			0
$664$	1.38633	−	2.09374i	0.0538000	−	0.0812527i
$665$	−15.0601			−0.584005
$666$		0			0
$667$			61.8404i			2.39447i
$668$	−0.00989342	+	0.00763551i	−0.000382788	+	0.000295427i
$669$		0			0
$670$	4.03089	+	11.8203i	0.155727	+	0.456659i
$671$	3.11139			0.120114
$672$		0			0
$673$	−26.4049			−1.01783			−0.508917	−	0.860816i	$-0.669954\pi$
$673$	−26.4049			−1.01783			−0.508917	+	0.860816i	$0.669954\pi$
$674$	4.26498	+	12.5068i	0.164281	+	0.481743i
$675$		0			0
$676$	−18.8729	+	14.5657i	−0.725883	+	0.560219i
$677$			17.9627i			0.690361i	0.938536	+	0.345181i	$0.112182\pi$
$677$			17.9627i			0.690361i	−0.938536	+	0.345181i	$0.887818\pi$
$678$		0			0
$679$	−26.5331			−1.01825
$680$	16.1137	−	24.3361i	0.617934	−	0.933248i
$681$		0			0
$682$	−3.27583	+	1.11710i	−0.125438	+	0.0427760i
$683$		−	32.3268i		−	1.23695i	−0.785805	−	0.618475i	$-0.787751\pi$
$683$		−	32.3268i		−	1.23695i	0.785805	−	0.618475i	$-0.212249\pi$
$684$		0			0
$685$			37.3057i			1.42538i
$686$	−12.2275	−	35.8565i	−0.466850	−	1.36901i
$687$		0			0
$688$	−24.2965	−	6.36490i	−0.926297	−	0.242660i
$689$	−7.25512			−0.276398
$690$		0			0
$691$			1.75007i			0.0665757i	0.999446	+	0.0332878i	$0.0105978\pi$
$691$			1.75007i			0.0665757i	−0.999446	+	0.0332878i	$0.989402\pi$
$692$	−5.44190	−	7.05113i	−0.206870	−	0.268044i
$693$		0			0
$694$	41.3852	−	14.1129i	1.57096	−	0.535719i
$695$	41.1237			1.55991
$696$		0			0
$697$	12.7378			0.482479
$698$	33.7903	−	11.5230i	1.27898	−	0.436150i
$699$		0			0
$700$	44.9192	−	34.6676i	1.69779	−	1.31031i
$701$			27.7248i			1.04715i	0.851979	+	0.523576i	$0.175403\pi$
$701$			27.7248i			1.04715i	−0.851979	+	0.523576i	$0.824597\pi$
$702$		0			0
$703$	4.31265			0.162655
$704$	4.48321	−	1.90120i	0.168967	−	0.0716540i
$705$		0			0
$706$	−0.984696	−	2.88756i	−0.0370595	−	0.108675i
$707$		−	74.1286i		−	2.78789i
$708$		0			0
$709$			50.2818i			1.88837i	0.329412	+	0.944186i	$0.393150\pi$
$709$			50.2818i			1.88837i	−0.329412	+	0.944186i	$0.606850\pi$
$710$	54.1404	−	18.4626i	2.03185	−	0.692889i
$711$		0			0
$712$	17.9655	+	11.8955i	0.673285	+	0.445804i
$713$	34.1836			1.28019
$714$		0			0
$715$			2.13071i			0.0796840i
$716$	11.9809	+	15.5238i	0.447746	+	0.580150i
$717$		0			0
$718$	14.6488	+	42.9566i	0.546687	+	1.60313i
$719$	13.1202			0.489300			0.244650	−	0.969611i	$-0.421327\pi$
$719$	13.1202			0.489300			0.244650	+	0.969611i	$0.421327\pi$
$720$		0			0
$721$	−52.1477			−1.94208
$722$	0.456455	+	1.33852i	0.0169875	+	0.0498147i
$723$		0			0
$724$	−21.9310	−	28.4162i	−0.815058	−	1.05608i
$725$		−	46.1518i		−	1.71403i
$726$		0			0
$727$	32.5922			1.20878			0.604389	−	0.796690i	$-0.293417\pi$
$727$	32.5922			1.20878			0.604389	+	0.796690i	$0.293417\pi$
$728$	7.25562	−	10.9580i	0.268911	−	0.406129i
$729$		0			0
$730$	56.4355	−	19.2453i	2.08877	−	0.712299i
$731$		−	19.2371i		−	0.711510i
$732$		0			0
$733$		−	8.28509i		−	0.306017i	−0.988225	−	0.153008i	$-0.951104\pi$
$733$		−	8.28509i		−	0.306017i	0.988225	−	0.153008i	$-0.0488961\pi$
$734$	−4.71087	−	13.8143i	−0.173882	−	0.509897i
$735$		0			0
$736$	−47.9698	+	3.48085i	−1.76819	+	0.128306i
$737$	−1.59590			−0.0587857
$738$		0			0
$739$			48.5291i			1.78517i	0.450876	+	0.892587i	$0.351112\pi$
$739$			48.5291i			1.78517i	−0.450876	+	0.892587i	$0.648888\pi$
$740$	−22.9993	+	17.7503i	−0.845470	+	0.652514i
$741$		0			0
$742$	−41.7816	+	14.2481i	−1.53385	+	0.523063i
$743$	−3.81374			−0.139913			−0.0699563	−	0.997550i	$-0.522286\pi$
$743$	−3.81374			−0.139913			−0.0699563	+	0.997550i	$0.522286\pi$
$744$		0			0
$745$	14.5193			0.531945
$746$	−27.1762	+	9.26743i	−0.994990	+	0.339305i
$747$		0			0
$748$	2.27880	+	2.95267i	0.0833212	+	0.107960i
$749$			46.1483i			1.68622i
$750$		0			0
$751$	46.7083			1.70441			0.852205	−	0.523208i	$-0.175265\pi$
$751$	46.7083			1.70441			0.852205	+	0.523208i	$0.175265\pi$
$752$	−4.79992	+	18.3226i	−0.175035	+	0.668156i
$753$		0			0
$754$	−3.45020	−	10.1175i	−0.125649	−	0.368457i
$755$			33.3130i			1.21238i
$756$		0			0
$757$		−	38.8886i		−	1.41343i	−0.707498	−	0.706716i	$-0.750176\pi$
$757$		−	38.8886i		−	1.41343i	0.707498	−	0.706716i	$-0.249824\pi$
$758$	20.6608	−	7.04560i	0.750433	−	0.255908i
$759$		0			0
$760$	−7.94345	−	5.25962i	−0.288139	−	0.190786i
$761$	−36.7676			−1.33283			−0.666413	−	0.745583i	$-0.732171\pi$
$761$	−36.7676			−1.33283			−0.666413	+	0.745583i	$0.732171\pi$
$762$		0			0
$763$		−	8.29812i		−	0.300412i
$764$	6.10089	−	4.70853i	0.220723	−	0.170349i
$765$		0			0
$766$	−17.2465	−	50.5744i	−0.623143	−	1.82733i
$767$	2.75031			0.0993080
$768$		0			0
$769$	52.3408			1.88746			0.943730	−	0.330718i	$-0.107291\pi$
$769$	52.3408			1.88746			0.943730	+	0.330718i	$0.107291\pi$
$770$	4.18442	+	12.2705i	0.150796	+	0.442200i
$771$		0			0
$772$	−30.9709	+	23.9026i	−1.11467	+	0.860275i
$773$			13.8632i			0.498626i	0.968423	+	0.249313i	$0.0802048\pi$
$773$			13.8632i			0.498626i	−0.968423	+	0.249313i	$0.919795\pi$
$774$		0			0
$775$	−25.5113			−0.916395
$776$	−13.9949	−	9.26646i	−0.502387	−	0.332646i
$777$		0			0
$778$	18.4640	−	6.29648i	0.661967	−	0.225740i
$779$		−	4.15770i		−	0.148965i
$780$		0			0
$781$			7.30967i			0.261561i
$782$	−11.8898	−	34.8660i	−0.425177	−	1.24680i
$783$		0			0
$784$	13.1688	−	50.2689i	0.470315	−	1.79532i
$785$	24.9036			0.888848
$786$		0			0
$787$			7.53623i			0.268638i	0.990938	+	0.134319i	$0.0428846\pi$
$787$			7.53623i			0.268638i	−0.990938	+	0.134319i	$0.957115\pi$
$788$	−4.21368	−	5.45971i	−0.150106	−	0.194494i
$789$		0			0
$790$	4.11766	−	1.40418i	0.146500	−	0.0499584i
$791$	−82.4228			−2.93062
$792$		0			0
$793$	5.31192			0.188632
$794$	44.3941	−	15.1390i	1.57549	−	0.537263i
$795$		0			0
$796$	12.1095	−	9.34585i	0.429211	−	0.331255i
$797$			44.5030i			1.57638i	0.615433	+	0.788189i	$0.288981\pi$
$797$			44.5030i			1.57638i	−0.615433	+	0.788189i	$0.711019\pi$
$798$		0			0
$799$	−14.5071			−0.513226
$800$	35.8000	−	2.59777i	1.26572	−	0.0918451i
$801$		0			0
$802$	−3.76402	−	11.0377i	−0.132912	−	0.389756i
$803$			7.61954i			0.268888i
$804$		0			0
$805$		−	128.044i		−	4.51297i
$806$	−5.59266	+	1.90717i	−0.196993	+	0.0671772i
$807$		0			0
$808$	25.8888	−	39.0992i	0.910765	−	1.37550i
$809$	3.45607			0.121509			0.0607544	−	0.998153i	$-0.480649\pi$
$809$	3.45607			0.121509			0.0607544	+	0.998153i	$0.480649\pi$
$810$		0			0
$811$			14.9470i			0.524860i	0.964951	+	0.262430i	$0.0845239\pi$
$811$			14.9470i			0.524860i	−0.964951	+	0.262430i	$0.915476\pi$
$812$	−39.7387	−	51.4900i	−1.39456	−	1.80694i
$813$		0			0
$814$	−1.19826	−	3.51383i	−0.0419991	−	0.123160i
$815$	−42.2941			−1.48150
$816$		0			0
$817$	−6.27910			−0.219678
$818$	−13.8895	−	40.7302i	−0.485636	−	1.42410i
$819$		0			0
$820$	17.1125	+	22.1729i	0.597595	+	0.774311i
$821$			3.72440i			0.129982i	0.997886	+	0.0649912i	$0.0207019\pi$
$821$			3.72440i			0.129982i	−0.997886	+	0.0649912i	$0.979298\pi$
$822$		0			0
$823$	15.0687			0.525262			0.262631	−	0.964896i	$-0.415410\pi$
$823$	15.0687			0.525262			0.262631	+	0.964896i	$0.415410\pi$
$824$	−27.5053	−	18.2122i	−0.958193	−	0.634450i
$825$		0			0
$826$	15.8388	−	5.40124i	0.551102	−	0.187933i
$827$			37.8202i			1.31514i	0.753395	+	0.657568i	$0.228415\pi$
$827$			37.8202i			1.31514i	−0.753395	+	0.657568i	$0.771585\pi$
$828$		0			0
$829$			38.8523i			1.34940i	0.738094	+	0.674698i	$0.235726\pi$
$829$			38.8523i			1.34940i	−0.738094	+	0.674698i	$0.764274\pi$
$830$	1.36497	+	4.00270i	0.0473790	+	0.138936i
$831$		0			0
$832$	7.65396	−	3.24582i	0.265353	−	0.112529i
$833$	39.8011			1.37903
$834$		0			0
$835$		−	0.0210470i		−	0.000728362i
$836$	0.963768	−	0.743814i	0.0333326	−	0.0257253i
$837$		0			0
$838$	6.84969	−	2.33584i	0.236619	−	0.0806901i
$839$	15.8272			0.546417			0.273209	−	0.961955i	$-0.411915\pi$
$839$	15.8272			0.546417			0.273209	+	0.961955i	$0.411915\pi$
$840$		0			0
$841$	−23.9028			−0.824235
$842$	−38.6638	+	13.1849i	−1.33244	+	0.454381i
$843$		0			0
$844$	−17.6753	−	22.9022i	−0.608411	−	0.788325i
$845$		−	40.1499i		−	1.38120i
$846$		0			0
$847$	47.5261			1.63302
$848$	−27.0137	−	7.07672i	−0.927655	−	0.243016i
$849$		0			0
$850$	8.87338	+	26.0206i	0.304354	+	0.892500i
$851$			36.6672i			1.25693i
$852$		0			0
$853$			6.18601i			0.211805i	0.994377	+	0.105902i	$0.0337731\pi$
$853$			6.18601i			0.211805i	−0.994377	+	0.105902i	$0.966227\pi$
$854$	30.5908	−	10.4319i	1.04680	−	0.356972i
$855$		0			0
$856$	−16.1169	+	24.3410i	−0.550866	+	0.831957i
$857$	−12.2246			−0.417583			−0.208792	−	0.977960i	$-0.566953\pi$
$857$	−12.2246			−0.417583			−0.208792	+	0.977960i	$0.566953\pi$
$858$		0			0
$859$		−	44.9215i		−	1.53270i	−0.642423	−	0.766350i	$-0.722071\pi$
$859$		−	44.9215i		−	1.53270i	0.642423	−	0.766350i	$-0.277929\pi$
$860$	33.4863	−	25.8439i	1.14187	−	0.881271i
$861$		0			0
$862$	−8.85355	−	25.9625i	−0.301553	−	0.884285i
$863$	−21.2268			−0.722568			−0.361284	−	0.932456i	$-0.617661\pi$
$863$	−21.2268			−0.722568			−0.361284	+	0.932456i	$0.617661\pi$
$864$		0			0
$865$	15.0004			0.510029
$866$	−13.9580	−	40.9309i	−0.474312	−	1.39089i
$867$		0			0
$868$	−28.4621	+	21.9664i	−0.966068	+	0.745589i
$869$			0.555938i			0.0188589i
$870$		0			0
$871$	−2.72460			−0.0923196
$872$	2.89805	−	4.37685i	0.0981404	−	0.148219i
$873$		0			0
$874$	−11.3805	+	3.88089i	−0.384950	+	0.131273i
$875$			20.2596i			0.684900i
$876$		0			0
$877$			15.4462i			0.521582i	0.965395	+	0.260791i	$0.0839834\pi$
$877$			15.4462i			0.521582i	−0.965395	+	0.260791i	$0.916017\pi$
$878$	9.77069	+	28.6519i	0.329745	+	0.966956i
$879$		0			0
$880$	−2.07832	+	7.93348i	−0.0700600	+	0.267438i
$881$	25.4049			0.855912			0.427956	−	0.903800i	$-0.359234\pi$
$881$	25.4049			0.855912			0.427956	+	0.903800i	$0.359234\pi$
$882$		0			0
$883$		−	29.9928i		−	1.00934i	−0.863313	−	0.504669i	$-0.831615\pi$
$883$		−	29.9928i		−	1.00934i	0.863313	−	0.504669i	$-0.168385\pi$
$884$	3.89048	+	5.04095i	0.130851	+	0.169545i
$885$		0			0
$886$	−11.8281	+	4.03353i	−0.397372	+	0.135509i
$887$	27.5162			0.923903			0.461952	−	0.886905i	$-0.347149\pi$
$887$	27.5162			0.923903			0.461952	+	0.886905i	$0.347149\pi$
$888$		0			0
$889$	−41.5678			−1.39414
$890$	−34.3455	+	11.7123i	−1.15126	+	0.392597i
$891$		0			0
$892$	−7.51646	+	5.80103i	−0.251670	+	0.194233i
$893$			4.73522i			0.158458i
$894$		0			0
$895$	−33.0249			−1.10390
$896$	37.7041	−	33.7237i	1.25961	−	1.12663i
$897$		0			0
$898$	8.18229	+	23.9940i	0.273046	+	0.800692i
$899$			29.2431i			0.975313i
$900$		0			0
$901$		−	21.3885i		−	0.712554i
$902$	−3.38758	+	1.15521i	−0.112794	+	0.0384642i
$903$		0			0
$904$	−43.4739	−	28.7855i	−1.44592	−	0.957391i
$905$	60.4520			2.00949
$906$		0			0
$907$		−	26.8589i		−	0.891837i	−0.895074	−	0.445918i	$-0.852877\pi$
$907$		−	26.8589i		−	0.891837i	0.895074	−	0.445918i	$-0.147123\pi$
$908$	3.26444	+	4.22977i	0.108334	+	0.140370i
$909$		0			0
$910$	7.14385	+	20.9489i	0.236816	+	0.694449i
$911$	34.5330			1.14413			0.572065	−	0.820208i	$-0.306142\pi$
$911$	34.5330			1.14413			0.572065	+	0.820208i	$0.306142\pi$
$912$		0			0
$913$	−0.540418			−0.0178852
$914$	5.41494	+	15.8790i	0.179110	+	0.525230i
$915$		0			0
$916$	−9.62155	−	12.4668i	−0.317905	−	0.411913i
$917$		−	36.4407i		−	1.20338i
$918$		0			0
$919$	−20.0300			−0.660729			−0.330365	−	0.943853i	$-0.607172\pi$
$919$	−20.0300			−0.660729			−0.330365	+	0.943853i	$0.607172\pi$
$920$	44.7185	−	67.5371i	1.47432	−	2.22663i
$921$		0			0
$922$	9.41031	−	3.20904i	0.309912	−	0.105684i
$923$			12.4794i			0.410766i
$924$		0			0
$925$		−	27.3649i		−	0.899751i
$926$	0.238900	+	0.700559i	0.00785073	+	0.0230218i
$927$		0			0
$928$	−2.97777	−	41.0368i	−0.0977502	−	1.34710i
$929$	−2.91198			−0.0955389			−0.0477695	−	0.998858i	$-0.515211\pi$
$929$	−2.91198			−0.0955389			−0.0477695	+	0.998858i	$0.515211\pi$
$930$		0			0
$931$		−	12.9913i		−	0.425773i
$932$	−16.2003	+	12.5030i	−0.530659	+	0.409550i
$933$		0			0
$934$	−20.2130	+	6.89291i	−0.661390	+	0.225543i
$935$	−6.28144			−0.205425
$936$		0			0
$937$	4.03574			0.131842			0.0659210	−	0.997825i	$-0.479001\pi$
$937$	4.03574			0.131842			0.0659210	+	0.997825i	$0.479001\pi$
$938$	−15.6907	+	5.35074i	−0.512320	+	0.174708i
$939$		0			0
$940$	−19.4895	−	25.2528i	−0.635678	−	0.823655i
$941$			51.2751i			1.67152i	0.549096	+	0.835760i	$0.314972\pi$
$941$			51.2751i			1.67152i	−0.549096	+	0.835760i	$0.685028\pi$
$942$		0			0
$943$	35.3497			1.15114
$944$	10.2405	+	2.68268i	0.333300	+	0.0873139i
$945$		0			0
$946$	1.74464	+	5.11604i	0.0567230	+	0.166337i
$947$			37.7009i			1.22511i	0.790426	+	0.612557i	$0.209859\pi$
$947$			37.7009i			1.22511i	−0.790426	+	0.612557i	$0.790141\pi$
$948$		0			0
$949$			13.0085i			0.422272i
$950$	8.49328	−	2.89632i	0.275558	−	0.0939691i
$951$		0			0
$952$	32.3047	+	21.3899i	1.04700	+	0.693252i
$953$	−41.4066			−1.34129			−0.670645	−	0.741779i	$-0.733983\pi$
$953$	−41.4066			−1.34129			−0.670645	+	0.741779i	$0.733983\pi$
$954$		0			0
$955$			12.9789i			0.419987i
$956$	−23.9159	+	18.4577i	−0.773494	+	0.596964i
$957$		0			0
$958$	19.5661	+	57.3763i	0.632152	+	1.85375i
$959$	−49.5209			−1.59911
$960$		0			0
$961$	−14.8352			−0.478556
$962$	−2.04573	−	5.99899i	−0.0659571	−	0.193415i
$963$		0			0
$964$	−9.77329	+	7.54279i	−0.314776	+	0.242937i
$965$		−	65.8869i		−	2.12097i
$966$		0			0
$967$	29.9529			0.963220			0.481610	−	0.876386i	$-0.340052\pi$
$967$	29.9529			0.963220			0.481610	+	0.876386i	$0.340052\pi$
$968$	25.0677	+	16.5981i	0.805706	+	0.533483i
$969$		0			0
$970$	26.7547	−	9.12371i	0.859042	−	0.292945i
$971$		−	53.5633i		−	1.71893i	−0.511195	−	0.859465i	$-0.670797\pi$
$971$		−	53.5633i		−	1.71893i	0.511195	−	0.859465i	$-0.329203\pi$
$972$		0			0
$973$			54.5890i			1.75004i
$974$	11.9179	+	34.9486i	0.381875	+	1.11982i
$975$		0			0
$976$	19.7784	+	5.18130i	0.633091	+	0.165849i
$977$	26.2092			0.838506			0.419253	−	0.907869i	$-0.362292\pi$
$977$	26.2092			0.838506			0.419253	+	0.907869i	$0.362292\pi$
$978$		0			0
$979$		−	4.63710i		−	0.148202i
$980$	53.4704	+	69.2823i	1.70805	+	2.21314i
$981$		0			0
$982$	−42.2806	+	14.4182i	−1.34923	+	0.460105i
$983$	−6.77344			−0.216039			−0.108020	−	0.994149i	$-0.534451\pi$
$983$	−6.77344			−0.216039			−0.108020	+	0.994149i	$0.534451\pi$
$984$		0			0
$985$	11.6149			0.370080
$986$	29.8269	−	10.1714i	0.949882	−	0.323922i
$987$		0			0
$988$	1.64539	−	1.26988i	0.0523469	−	0.0404001i
$989$		−	53.3864i		−	1.69759i
$990$		0			0
$991$	−8.71348			−0.276793			−0.138396	−	0.990377i	$-0.544195\pi$
$991$	−8.71348			−0.276793			−0.138396	+	0.990377i	$0.544195\pi$
$992$	−22.6840	+	1.64603i	−0.720217	+	0.0522614i
$993$		0			0
$994$	24.5079	+	71.8679i	0.777344	+	2.27951i
$995$			25.7615i			0.816695i
$996$		0			0
$997$		−	58.7540i		−	1.86076i	−0.366599	−	0.930379i	$-0.619478\pi$
$997$		−	58.7540i		−	1.86076i	0.366599	−	0.930379i	$-0.380522\pi$
$998$	−21.5662	+	7.35436i	−0.682666	+	0.232798i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.g.b.685.7		16
3.2	odd	2		152.2.c.b.77.10	yes	16
4.3	odd	2		5472.2.g.b.2737.2		16
8.3	odd	2		5472.2.g.b.2737.15		16
8.5	even	2	inner	1368.2.g.b.685.8		16
12.11	even	2		608.2.c.b.305.13		16
24.5	odd	2		152.2.c.b.77.9	✓	16
24.11	even	2		608.2.c.b.305.4		16
48.5	odd	4		4864.2.a.bo.1.2		8
48.11	even	4		4864.2.a.bn.1.7		8
48.29	odd	4		4864.2.a.bq.1.7		8
48.35	even	4		4864.2.a.bp.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.9	✓	16	24.5	odd	2
152.2.c.b.77.10	yes	16	3.2	odd	2
608.2.c.b.305.4		16	24.11	even	2
608.2.c.b.305.13		16	12.11	even	2
1368.2.g.b.685.7		16	1.1	even	1	trivial
1368.2.g.b.685.8		16	8.5	even	2	inner
4864.2.a.bn.1.7		8	48.11	even	4
4864.2.a.bo.1.2		8	48.5	odd	4
4864.2.a.bp.1.2		8	48.35	even	4
4864.2.a.bq.1.7		8	48.29	odd	4
5472.2.g.b.2737.2		16	4.3	odd	2
5472.2.g.b.2737.15		16	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.456455 - 1.33852i) q^{2} +(-1.58330 + 1.22195i) q^{4} -3.36827i q^{5} +4.47116 q^{7} +(2.35832 + 1.56152i) q^{8} +(-4.50852 + 1.53746i) q^{10} -0.608709i q^{11} -1.03922i q^{13} +(-2.04088 - 5.98476i) q^{14} +(1.01366 - 3.86943i) q^{16} +3.06367 q^{17} -1.00000i q^{19} +(4.11587 + 5.33298i) q^{20} +(-0.814773 + 0.277848i) q^{22} +8.50224 q^{23} -6.34525 q^{25} +(-1.39102 + 0.474357i) q^{26} +(-7.07918 + 5.46355i) q^{28} +7.27343i q^{29} +4.02054 q^{31} +(-5.64202 + 0.409404i) q^{32} +(-1.39843 - 4.10080i) q^{34} -15.0601i q^{35} +4.31265i q^{37} +(-1.33852 + 0.456455i) q^{38} +(5.25962 - 7.94345i) q^{40} +4.15770 q^{41} -6.27910i q^{43} +(0.743814 + 0.963768i) q^{44} +(-3.88089 - 11.3805i) q^{46} -4.73522 q^{47} +12.9913 q^{49} +(2.89632 + 8.49328i) q^{50} +(1.26988 + 1.64539i) q^{52} -6.98132i q^{53} -2.05030 q^{55} +(10.5444 + 6.98180i) q^{56} +(9.73567 - 3.31999i) q^{58} +2.64652i q^{59} +5.11145i q^{61} +(-1.83520 - 5.38159i) q^{62} +(3.12332 + 7.36511i) q^{64} -3.50037 q^{65} -2.62178i q^{67} +(-4.85071 + 3.74366i) q^{68} +(-20.1583 + 6.87425i) q^{70} -12.0085 q^{71} -12.5175 q^{73} +(5.77259 - 1.96853i) q^{74} +(1.22195 + 1.58330i) q^{76} -2.72164i q^{77} -0.913307 q^{79} +(-13.0333 - 3.41430i) q^{80} +(-1.89780 - 5.56518i) q^{82} -0.887809i q^{83} -10.3193i q^{85} +(-8.40473 + 2.86613i) q^{86} +(0.950510 - 1.43553i) q^{88} +7.61792 q^{89} -4.64652i q^{91} +(-13.4616 + 10.3893i) q^{92} +(2.16141 + 6.33820i) q^{94} -3.36827 q^{95} -5.93426 q^{97} +(-5.92994 - 17.3892i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{4} - 8 q^{7} + 12 q^{8} - 8 q^{10} - 4 q^{14} + 2 q^{16} + 8 q^{17} - 8 q^{20} + 20 q^{22} - 24 q^{25} + 10 q^{26} - 14 q^{28} + 16 q^{31} + 20 q^{32} - 2 q^{38} + 28 q^{40} - 16 q^{41} + 28 q^{44}+ \cdots + 48 q^{98}+O(q^{100}) \) 16 * q + 2 * q^4 - 8 * q^7 + 12 * q^8 - 8 * q^10 - 4 * q^14 + 2 * q^16 + 8 * q^17 - 8 * q^20 + 20 * q^22 - 24 * q^25 + 10 * q^26 - 14 * q^28 + 16 * q^31 + 20 * q^32 - 2 * q^38 + 28 * q^40 - 16 * q^41 + 28 * q^44 - 48 * q^46 - 24 * q^47 + 24 * q^49 - 12 * q^50 + 8 * q^52 + 16 * q^55 + 48 * q^56 + 38 * q^58 + 16 * q^62 + 14 * q^64 - 16 * q^65 + 26 * q^68 - 32 * q^70 - 48 * q^71 + 20 * q^74 - 4 * q^76 - 48 * q^79 - 4 * q^80 - 12 * q^82 - 48 * q^86 + 40 * q^88 + 16 * q^89 - 62 * q^92 - 36 * q^94 - 16 * q^95 + 32 * q^97 + 48 * q^98

Introduction

L-functions

Modular forms

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