LMFDB - Embedded newform 1152.2.i.l.769.3

Newspace parameters

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.i (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
Defining polynomial:	$ x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} - 216 x^{3} + 243 x^{2} - 486 x + 729 $ x^12 - 2x^11 + 3x^10 - 8x^9 + 22x^8 - 42x^7 + 51x^6 - 126x^5 + 198x^4 - 216x^3 + 243x^2 - 486*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			769.3
Root			$-1.28252 - 1.16410i$ of defining polynomial
Character	$\chi$	$=$	1152.769
Dual form			1152.2.i.l.385.3

$q$-expansion

$f(q)$	$=$	$q+(0.366879 + 1.69275i) q^{3} +(1.05471 - 1.82681i) q^{5} +(1.43914 + 2.49267i) q^{7} +(-2.73080 + 1.24207i) q^{9} +O(q^{10})$	\(q+(0.366879 + 1.69275i) q^{3} +(1.05471 - 1.82681i) q^{5} +(1.43914 + 2.49267i) q^{7} +(-2.73080 + 1.24207i) q^{9} +(-1.21325 - 2.10141i) q^{11} +(-3.30008 + 5.71590i) q^{13} +(3.47929 + 1.11514i) q^{15} -7.56848 q^{17} +6.25779 q^{19} +(-3.69147 + 3.35061i) q^{21} +(-2.63611 + 4.56587i) q^{23} +(0.275172 + 0.476612i) q^{25} +(-3.10438 - 4.16687i) q^{27} +(1.57821 + 2.73353i) q^{29} +(-1.79039 + 3.10104i) q^{31} +(3.11204 - 2.82469i) q^{33} +6.07151 q^{35} +6.70957 q^{37} +(-10.8863 - 3.48916i) q^{39} +(-1.74537 + 3.02306i) q^{41} +(3.12570 + 5.41388i) q^{43} +(-0.611180 + 6.29868i) q^{45} +(1.32972 + 2.30314i) q^{47} +(-0.642255 + 1.11242i) q^{49} +(-2.77671 - 12.8115i) q^{51} -0.953009 q^{53} -5.11850 q^{55} +(2.29585 + 10.5929i) q^{57} +(4.84757 - 8.39624i) q^{59} +(-2.57821 - 4.46558i) q^{61} +(-7.02607 - 5.01946i) q^{63} +(6.96125 + 12.0572i) q^{65} +(0.949546 - 1.64466i) q^{67} +(-8.69601 - 2.78715i) q^{69} +5.82491 q^{71} -5.01222 q^{73} +(-0.705829 + 0.640656i) q^{75} +(3.49207 - 6.04844i) q^{77} +(6.49996 + 11.2583i) q^{79} +(5.91454 - 6.78368i) q^{81} +(1.54502 + 2.67606i) q^{83} +(-7.98256 + 13.8262i) q^{85} +(-4.04818 + 3.67438i) q^{87} -2.95301 q^{89} -18.9971 q^{91} +(-5.90614 - 1.89297i) q^{93} +(6.60015 - 11.4318i) q^{95} +(-5.51242 - 9.54779i) q^{97} +(5.92323 + 4.23159i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 2 q^{5} - 6 q^{7} - 2 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 2 * q^5 - 6 * q^7 - 2 * q^9	$ 12 q + 4 q^{3} + 2 q^{5} - 6 q^{7} - 2 q^{9} + 4 q^{11} - 10 q^{13} - 4 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} - 8 q^{23} - 14 q^{25} - 14 q^{27} + 2 q^{29} - 8 q^{31} - 10 q^{33} + 8 q^{35} - 22 q^{39} - 2 q^{41} - 2 q^{43} - 10 q^{45} + 14 q^{47} - 18 q^{49} - 38 q^{51} - 24 q^{53} + 16 q^{55} - 38 q^{57} + 6 q^{59} - 14 q^{61} + 16 q^{63} - 8 q^{65} + 4 q^{67} + 50 q^{69} + 28 q^{71} + 60 q^{73} + 50 q^{75} - 2 q^{77} - 16 q^{79} + 22 q^{81} + 24 q^{83} - 16 q^{85} + 36 q^{87} - 48 q^{89} - 52 q^{91} - 42 q^{93} + 20 q^{95} - 14 q^{97} + 68 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 + 2 * q^5 - 6 * q^7 - 2 * q^9 + 4 * q^11 - 10 * q^13 - 4 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 - 8 * q^23 - 14 * q^25 - 14 * q^27 + 2 * q^29 - 8 * q^31 - 10 * q^33 + 8 * q^35 - 22 * q^39 - 2 * q^41 - 2 * q^43 - 10 * q^45 + 14 * q^47 - 18 * q^49 - 38 * q^51 - 24 * q^53 + 16 * q^55 - 38 * q^57 + 6 * q^59 - 14 * q^61 + 16 * q^63 - 8 * q^65 + 4 * q^67 + 50 * q^69 + 28 * q^71 + 60 * q^73 + 50 * q^75 - 2 * q^77 - 16 * q^79 + 22 * q^81 + 24 * q^83 - 16 * q^85 + 36 * q^87 - 48 * q^89 - 52 * q^91 - 42 * q^93 + 20 * q^95 - 14 * q^97 + 68 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.366879	+	1.69275i	0.211818	+	0.977309i
$3$	0.366879	+	1.69275i	0.211818	+	0.977309i
$4$		0			0
$5$	1.05471	−	1.82681i	0.471681	−	0.816975i	−0.527794	−	0.849372i	$-0.676981\pi$
$5$	1.05471	−	1.82681i	0.471681	−	0.816975i	0.999475	+	0.0323971i	$0.0103141\pi$
$6$		0			0
$7$	1.43914	+	2.49267i	0.543944	+	0.942139i	0.998673	+	0.0515089i	$0.0164031\pi$
$7$	1.43914	+	2.49267i	0.543944	+	0.942139i	−0.454728	+	0.890630i	$0.650264\pi$
$8$		0			0
$9$	−2.73080	+	1.24207i	−0.910267	+	0.414022i
$10$		0			0
$11$	−1.21325	−	2.10141i	−0.365808	−	0.633598i	0.623097	−	0.782144i	$-0.285874\pi$
$11$	−1.21325	−	2.10141i	−0.365808	−	0.633598i	−0.988905	+	0.148546i	$0.952541\pi$
$12$		0			0
$13$	−3.30008	+	5.71590i	−0.915277	+	1.58531i	−0.108781	+	0.994066i	$0.534695\pi$
$13$	−3.30008	+	5.71590i	−0.915277	+	1.58531i	−0.806496	+	0.591240i	$0.798639\pi$
$14$		0			0
$15$	3.47929	+	1.11514i	0.898348	+	0.287928i
$16$		0			0
$17$	−7.56848			−1.83563			−0.917813	−	0.397013i	$-0.870047\pi$
$17$	−7.56848			−1.83563			−0.917813	+	0.397013i	$0.870047\pi$
$18$		0			0
$19$	6.25779			1.43563			0.717817	−	0.696231i	$-0.245141\pi$
$19$	6.25779			1.43563			0.717817	+	0.696231i	$0.245141\pi$
$20$		0			0
$21$	−3.69147	+	3.35061i	−0.805544	+	0.731163i
$22$		0			0
$23$	−2.63611	+	4.56587i	−0.549666	+	0.952050i	0.448631	+	0.893717i	$0.351912\pi$
$23$	−2.63611	+	4.56587i	−0.549666	+	0.952050i	−0.998297	+	0.0583329i	$0.981422\pi$
$24$		0			0
$25$	0.275172	+	0.476612i	0.0550344	+	0.0953223i
$26$		0			0
$27$	−3.10438	−	4.16687i	−0.597438	−	0.801915i
$28$		0			0
$29$	1.57821	+	2.73353i	0.293065	+	0.507604i	0.974533	−	0.224244i	$-0.0719913\pi$
$29$	1.57821	+	2.73353i	0.293065	+	0.507604i	−0.681468	+	0.731848i	$0.738658\pi$
$30$		0			0
$31$	−1.79039	+	3.10104i	−0.321563	+	0.556964i	−0.980811	−	0.194962i	$-0.937542\pi$
$31$	−1.79039	+	3.10104i	−0.321563	+	0.556964i	0.659248	+	0.751926i	$0.270875\pi$
$32$		0			0
$33$	3.11204	−	2.82469i	0.541737	−	0.491715i
$34$		0			0
$35$	6.07151			1.02627
$36$		0			0
$37$	6.70957			1.10305			0.551524	−	0.834159i	$-0.314047\pi$
$37$	6.70957			1.10305			0.551524	+	0.834159i	$0.314047\pi$
$38$		0			0
$39$	−10.8863	−	3.48916i	−1.74321	−	0.558713i
$40$		0			0
$41$	−1.74537	+	3.02306i	−0.272580	+	0.472123i	−0.969522	−	0.245005i	$-0.921210\pi$
$41$	−1.74537	+	3.02306i	−0.272580	+	0.472123i	0.696941	+	0.717128i	$0.254544\pi$
$42$		0			0
$43$	3.12570	+	5.41388i	0.476665	+	0.825608i	0.999642	−	0.0267383i	$-0.00851207\pi$
$43$	3.12570	+	5.41388i	0.476665	+	0.825608i	−0.522977	+	0.852347i	$0.675179\pi$
$44$		0			0
$45$	−0.611180	+	6.29868i	−0.0911093	+	0.938952i
$46$		0			0
$47$	1.32972	+	2.30314i	0.193960	+	0.335948i	0.946559	−	0.322531i	$-0.104534\pi$
$47$	1.32972	+	2.30314i	0.193960	+	0.335948i	−0.752599	+	0.658479i	$0.771200\pi$
$48$		0			0
$49$	−0.642255	+	1.11242i	−0.0917508	+	0.158917i
$50$		0			0
$51$	−2.77671	−	12.8115i	−0.388818	−	1.79397i
$52$		0			0
$53$	−0.953009			−0.130906			−0.0654529	−	0.997856i	$-0.520849\pi$
$53$	−0.953009			−0.130906			−0.0654529	+	0.997856i	$0.520849\pi$
$54$		0			0
$55$	−5.11850			−0.690178
$56$		0			0
$57$	2.29585	+	10.5929i	0.304093	+	1.40306i
$58$		0			0
$59$	4.84757	−	8.39624i	0.631100	−	1.09310i	−0.356227	−	0.934400i	$-0.615937\pi$
$59$	4.84757	−	8.39624i	0.631100	−	1.09310i	0.987327	−	0.158698i	$-0.0507297\pi$
$60$		0			0
$61$	−2.57821	−	4.46558i	−0.330105	−	0.571759i	0.652427	−	0.757852i	$-0.273751\pi$
$61$	−2.57821	−	4.46558i	−0.330105	−	0.571759i	−0.982532	+	0.186092i	$0.940418\pi$
$62$		0			0
$63$	−7.02607	−	5.01946i	−0.885201	−	0.632393i
$64$		0			0
$65$	6.96125	+	12.0572i	0.863437	+	1.49552i
$66$		0			0
$67$	0.949546	−	1.64466i	0.116005	−	0.200927i	−0.802176	−	0.597088i	$-0.796324\pi$
$67$	0.949546	−	1.64466i	0.116005	−	0.200927i	0.918181	+	0.396161i	$0.129658\pi$
$68$		0			0
$69$	−8.69601	−	2.78715i	−1.04688	−	0.335533i
$70$		0			0
$71$	5.82491			0.691290			0.345645	−	0.938365i	$-0.387660\pi$
$71$	5.82491			0.691290			0.345645	+	0.938365i	$0.387660\pi$
$72$		0			0
$73$	−5.01222			−0.586636			−0.293318	−	0.956015i	$-0.594760\pi$
$73$	−5.01222			−0.586636			−0.293318	+	0.956015i	$0.594760\pi$
$74$		0			0
$75$	−0.705829	+	0.640656i	−0.0815021	+	0.0739765i
$76$		0			0
$77$	3.49207	−	6.04844i	0.397958	−	0.689284i
$78$		0			0
$79$	6.49996	+	11.2583i	0.731303	+	1.26665i	0.956327	+	0.292301i	$0.0944207\pi$
$79$	6.49996	+	11.2583i	0.731303	+	1.26665i	−0.225024	+	0.974353i	$0.572246\pi$
$80$		0			0
$81$	5.91454	−	6.78368i	0.657171	−	0.753742i
$82$		0			0
$83$	1.54502	+	2.67606i	0.169588	+	0.293735i	0.938275	−	0.345890i	$-0.112423\pi$
$83$	1.54502	+	2.67606i	0.169588	+	0.293735i	−0.768687	+	0.639625i	$0.779090\pi$
$84$		0			0
$85$	−7.98256	+	13.8262i	−0.865830	+	1.49966i
$86$		0			0
$87$	−4.04818	+	3.67438i	−0.434010	+	0.393935i
$88$		0			0
$89$	−2.95301			−0.313018			−0.156509	−	0.987677i	$-0.550024\pi$
$89$	−2.95301			−0.313018			−0.156509	+	0.987677i	$0.550024\pi$
$90$		0			0
$91$	−18.9971			−1.99144
$92$		0			0
$93$	−5.90614	−	1.89297i	−0.612439	−	0.196292i
$94$		0			0
$95$	6.60015	−	11.4318i	0.677161	−	1.17288i
$96$		0			0
$97$	−5.51242	−	9.54779i	−0.559702	−	0.969432i	−0.997521	−	0.0703686i	$-0.977582\pi$
$97$	−5.51242	−	9.54779i	−0.559702	−	0.969432i	0.437820	−	0.899063i	$-0.355751\pi$
$98$		0			0
$99$	5.92323	+	4.23159i	0.595307	+	0.425290i
$100$		0			0
$101$	−1.82357	−	3.15852i	−0.181452	−	0.314284i	0.760923	−	0.648842i	$-0.224746\pi$
$101$	−1.82357	−	3.15852i	−0.181452	−	0.314284i	−0.942375	+	0.334558i	$0.891413\pi$
$102$		0			0
$103$	−5.43914	+	9.42087i	−0.535935	+	0.928266i	0.463183	+	0.886263i	$0.346707\pi$
$103$	−5.43914	+	9.42087i	−0.535935	+	0.928266i	−0.999117	+	0.0420031i	$0.986626\pi$
$104$		0			0
$105$	2.22751	+	10.2775i	0.217382	+	1.00299i
$106$		0			0
$107$	−5.94922			−0.575133			−0.287567	−	0.957761i	$-0.592846\pi$
$107$	−5.94922			−0.575133			−0.287567	+	0.957761i	$0.592846\pi$
$108$		0			0
$109$	−9.87113			−0.945483			−0.472741	−	0.881201i	$-0.656735\pi$
$109$	−9.87113			−0.945483			−0.472741	+	0.881201i	$0.656735\pi$
$110$		0			0
$111$	2.46160	+	11.3576i	0.233645	+	1.07802i
$112$		0			0
$113$	−7.51518	+	13.0167i	−0.706969	+	1.22451i	0.259008	+	0.965875i	$0.416604\pi$
$113$	−7.51518	+	13.0167i	−0.706969	+	1.22451i	−0.965976	+	0.258630i	$0.916729\pi$
$114$		0			0
$115$	5.56066	+	9.63134i	0.518534	+	0.898128i
$116$		0			0
$117$	1.91232	−	19.7079i	0.176794	−	1.82200i
$118$		0			0
$119$	−10.8921	−	18.8657i	−0.998478	−	1.72942i
$120$		0			0
$121$	2.55606	−	4.42722i	0.232369	−	0.402475i
$122$		0			0
$123$	−5.75763	−	1.84537i	−0.519148	−	0.166391i
$124$		0			0
$125$	11.7080			1.04720
$126$		0			0
$127$	8.07789			0.716797			0.358398	−	0.933569i	$-0.383323\pi$
$127$	8.07789			0.716797			0.358398	+	0.933569i	$0.383323\pi$
$128$		0			0
$129$	−8.01758	+	7.27727i	−0.705909	+	0.640728i
$130$		0			0
$131$	11.1329	−	19.2827i	0.972684	−	1.68474i	0.285309	−	0.958435i	$-0.407904\pi$
$131$	11.1329	−	19.2827i	0.972684	−	1.68474i	0.687375	−	0.726303i	$-0.258763\pi$
$132$		0			0
$133$	9.00584	+	15.5986i	0.780905	+	1.35257i
$134$		0			0
$135$	−10.8863	+	1.27628i	−0.936945	+	0.109844i
$136$		0			0
$137$	−4.62365	−	8.00839i	−0.395025	−	0.684203i	0.598079	−	0.801437i	$-0.295931\pi$
$137$	−4.62365	−	8.00839i	−0.395025	−	0.684203i	−0.993104	+	0.117234i	$0.962597\pi$
$138$		0			0
$139$	4.20256	−	7.27905i	0.356456	−	0.617401i	−0.630910	−	0.775856i	$-0.717318\pi$
$139$	4.20256	−	7.27905i	0.356456	−	0.617401i	0.987366	+	0.158456i	$0.0506515\pi$
$140$		0			0
$141$	−3.41080	+	3.09586i	−0.287241	+	0.260718i
$142$		0			0
$143$	16.0152			1.33926
$144$		0			0
$145$	6.65820			0.552934
$146$		0			0
$147$	−2.11868	−	0.679055i	−0.174745	−	0.0560075i
$148$		0			0
$149$	−6.17836	+	10.7012i	−0.506151	+	0.876679i	0.493824	+	0.869562i	$0.335599\pi$
$149$	−6.17836	+	10.7012i	−0.506151	+	0.876679i	−0.999975	+	0.00711709i	$0.997735\pi$
$150$		0			0
$151$	−4.91424	−	8.51171i	−0.399915	−	0.692673i	0.593800	−	0.804613i	$-0.297627\pi$
$151$	−4.91424	−	8.51171i	−0.399915	−	0.692673i	−0.993715	+	0.111940i	$0.964294\pi$
$152$		0			0
$153$	20.6680	−	9.40056i	1.67091	−	0.759990i
$154$		0			0
$155$	3.77668	+	6.54141i	0.303350	+	0.525418i
$156$		0			0
$157$	0.108083	−	0.187206i	0.00862598	−	0.0149406i	−0.861680	−	0.507452i	$-0.830588\pi$
$157$	0.108083	−	0.187206i	0.00862598	−	0.0149406i	0.870306	+	0.492511i	$0.163921\pi$
$158$		0			0
$159$	−0.349639	−	1.61321i	−0.0277282	−	0.127935i
$160$		0			0
$161$	−15.1749			−1.19595
$162$		0			0
$163$	−5.78116			−0.452815			−0.226408	−	0.974033i	$-0.572698\pi$
$163$	−5.78116			−0.452815			−0.226408	+	0.974033i	$0.572698\pi$
$164$		0			0
$165$	−1.87787	−	8.66434i	−0.146192	−	0.674518i
$166$		0			0
$167$	0.818646	−	1.41794i	0.0633487	−	0.109723i	−0.832612	−	0.553857i	$-0.813155\pi$
$167$	0.818646	−	1.41794i	0.0633487	−	0.109723i	0.895960	+	0.444134i	$0.146489\pi$
$168$		0			0
$169$	−15.2810	−	26.4675i	−1.17546	−	2.03596i
$170$		0			0
$171$	−17.0888	+	7.77259i	−1.30681	+	0.594385i
$172$		0			0
$173$	2.99228	+	5.18278i	0.227499	+	0.394040i	0.957066	−	0.289869i	$-0.0936119\pi$
$173$	2.99228	+	5.18278i	0.227499	+	0.394040i	−0.729567	+	0.683909i	$0.760279\pi$
$174$		0			0
$175$	−0.792022	+	1.37182i	−0.0598713	+	0.103700i
$176$		0			0
$177$	15.9912	+	5.12532i	1.20197	+	0.385243i
$178$		0			0
$179$	9.51313			0.711045			0.355522	−	0.934668i	$-0.384303\pi$
$179$	9.51313			0.711045			0.355522	+	0.934668i	$0.384303\pi$
$180$		0			0
$181$	23.7526			1.76551			0.882757	−	0.469830i	$-0.155685\pi$
$181$	23.7526			1.76551			0.882757	+	0.469830i	$0.155685\pi$
$182$		0			0
$183$	6.61322	−	6.00258i	0.488864	−	0.443724i
$184$		0			0
$185$	7.07666	−	12.2571i	0.520286	−	0.901162i
$186$		0			0
$187$	9.18244	+	15.9045i	0.671487	+	1.16305i
$188$		0			0
$189$	5.91898	−	13.7349i	0.430542	−	0.999067i
$190$		0			0
$191$	4.14164	+	7.17352i	0.299678	+	0.519058i	0.976062	−	0.217491i	$-0.0697873\pi$
$191$	4.14164	+	7.17352i	0.299678	+	0.519058i	−0.676384	+	0.736549i	$0.736454\pi$
$192$		0			0
$193$	5.00315	−	8.66572i	0.360135	−	0.623772i	−0.627848	−	0.778336i	$-0.716064\pi$
$193$	5.00315	−	8.66572i	0.360135	−	0.623772i	0.987983	+	0.154564i	$0.0493973\pi$
$194$		0			0
$195$	−17.8559	+	16.2072i	−1.27869	+	1.16062i
$196$		0			0
$197$	−0.556259			−0.0396318			−0.0198159	−	0.999804i	$-0.506308\pi$
$197$	−0.556259			−0.0396318			−0.0198159	+	0.999804i	$0.506308\pi$
$198$		0			0
$199$	21.5526			1.52782			0.763912	−	0.645320i	$-0.223276\pi$
$199$	21.5526			1.52782			0.763912	+	0.645320i	$0.223276\pi$
$200$		0			0
$201$	3.13237	+	1.00395i	0.220940	+	0.0708133i
$202$		0			0
$203$	−4.54252	+	7.86788i	−0.318823	+	0.552217i
$204$		0			0
$205$	3.68171	+	6.37691i	0.257142	+	0.445383i
$206$		0			0
$207$	1.52756	−	15.7427i	0.106173	−	1.09419i
$208$		0			0
$209$	−7.59225	−	13.1502i	−0.525167	−	0.909615i
$210$		0			0
$211$	−8.32984	+	14.4277i	−0.573450	+	0.993245i	0.422758	+	0.906243i	$0.361062\pi$
$211$	−8.32984	+	14.4277i	−0.573450	+	0.993245i	−0.996208	+	0.0870022i	$0.972271\pi$
$212$		0			0
$213$	2.13704	+	9.86011i	0.146427	+	0.675604i
$214$		0			0
$215$	13.1868			0.899335
$216$		0			0
$217$	−10.3065			−0.699650
$218$		0			0
$219$	−1.83888	−	8.48443i	−0.124260	−	0.573325i
$220$		0			0
$221$	24.9766	−	43.2607i	1.68011	−	2.91003i
$222$		0			0
$223$	−4.49251	−	7.78126i	−0.300841	−	0.521072i	0.675486	−	0.737373i	$-0.263934\pi$
$223$	−4.49251	−	7.78126i	−0.300841	−	0.521072i	−0.976327	+	0.216301i	$0.930601\pi$
$224$		0			0
$225$	−1.34342	−	0.959749i	−0.0895615	−	0.0639833i
$226$		0			0
$227$	5.32448	+	9.22226i	0.353398	+	0.612103i	0.986842	−	0.161685i	$-0.0516929\pi$
$227$	5.32448	+	9.22226i	0.353398	+	0.612103i	−0.633445	+	0.773788i	$0.718360\pi$
$228$		0			0
$229$	5.50786	−	9.53988i	0.363969	−	0.630413i	−0.624641	−	0.780912i	$-0.714755\pi$
$229$	5.50786	−	9.53988i	0.363969	−	0.630413i	0.988610	+	0.150499i	$0.0480880\pi$
$230$		0			0
$231$	11.5197	+	3.69215i	0.757938	+	0.242926i
$232$		0			0
$233$	27.2595			1.78583			0.892915	−	0.450225i	$-0.148656\pi$
$233$	27.2595			1.78583			0.892915	+	0.450225i	$0.148656\pi$
$234$		0			0
$235$	5.60988			0.365948
$236$		0			0
$237$	−16.6727	+	15.1332i	−1.08301	+	0.983009i
$238$		0			0
$239$	5.82295	−	10.0856i	0.376655	−	0.652386i	−0.613918	−	0.789370i	$-0.710407\pi$
$239$	5.82295	−	10.0856i	0.376655	−	0.652386i	0.990573	+	0.136984i	$0.0437407\pi$
$240$		0			0
$241$	9.83554	+	17.0356i	0.633563	+	1.09736i	0.986818	+	0.161835i	$0.0517414\pi$
$241$	9.83554	+	17.0356i	0.633563	+	1.09736i	−0.353255	+	0.935527i	$0.614925\pi$
$242$		0			0
$243$	13.6530	+	7.52304i	0.875839	+	0.482603i
$244$		0			0
$245$	1.35479	+	2.34656i	0.0865542	+	0.149916i
$246$		0			0
$247$	−20.6512	+	35.7689i	−1.31400	+	2.27592i
$248$		0			0
$249$	−3.96306	+	3.59712i	−0.251149	+	0.227958i
$250$		0			0
$251$	23.3802			1.47575			0.737873	−	0.674940i	$-0.235830\pi$
$251$	23.3802			1.47575			0.737873	+	0.674940i	$0.235830\pi$
$252$		0			0
$253$	12.7930			0.804289
$254$		0			0
$255$	−26.3329	−	8.43993i	−1.64903	−	0.528529i
$256$		0			0
$257$	−1.89037	+	3.27421i	−0.117918	+	0.204240i	−0.918942	−	0.394392i	$-0.870955\pi$
$257$	−1.89037	+	3.27421i	−0.117918	+	0.204240i	0.801024	+	0.598632i	$0.204289\pi$
$258$		0			0
$259$	9.65603	+	16.7247i	0.599996	+	1.03922i
$260$		0			0
$261$	−7.70500	−	5.50449i	−0.476927	−	0.340720i
$262$		0			0
$263$	10.3638	+	17.9507i	0.639060	+	1.10689i	0.985639	+	0.168864i	$0.0540100\pi$
$263$	10.3638	+	17.9507i	0.639060	+	1.10689i	−0.346579	+	0.938021i	$0.612657\pi$
$264$		0			0
$265$	−1.00515	+	1.74097i	−0.0617458	+	0.106947i
$266$		0			0
$267$	−1.08340	−	4.99870i	−0.0663028	−	0.305916i
$268$		0			0
$269$	−0.965772			−0.0588842			−0.0294421	−	0.999566i	$-0.509373\pi$
$269$	−0.965772			−0.0588842			−0.0294421	+	0.999566i	$0.509373\pi$
$270$		0			0
$271$	4.28219			0.260124			0.130062	−	0.991506i	$-0.458482\pi$
$271$	4.28219			0.260124			0.130062	+	0.991506i	$0.458482\pi$
$272$		0			0
$273$	−6.96963	−	32.1573i	−0.421821	−	1.94625i
$274$		0			0
$275$	0.667703	−	1.15650i	0.0402640	−	0.0697393i
$276$		0			0
$277$	−4.83619	−	8.37653i	−0.290579	−	0.503297i	0.683368	−	0.730074i	$-0.260514\pi$
$277$	−4.83619	−	8.37653i	−0.290579	−	0.503297i	−0.973947	+	0.226777i	$0.927181\pi$
$278$		0			0
$279$	1.03749	−	10.6921i	0.0621128	−	0.640120i
$280$		0			0
$281$	7.34848	+	12.7279i	0.438373	+	0.759285i	0.997564	−	0.0697540i	$-0.0222214\pi$
$281$	7.34848	+	12.7279i	0.438373	+	0.759285i	−0.559191	+	0.829039i	$0.688888\pi$
$282$		0			0
$283$	−7.90174	+	13.6862i	−0.469710	+	0.813561i	−0.999400	−	0.0346299i	$-0.988975\pi$
$283$	−7.90174	+	13.6862i	−0.469710	+	0.813561i	0.529690	+	0.848191i	$0.322308\pi$
$284$		0			0
$285$	21.7726	+	6.97832i	1.28970	+	0.413360i
$286$		0			0
$287$	−10.0473			−0.593074
$288$		0			0
$289$	40.2819			2.36952
$290$		0			0
$291$	14.1396	−	12.8340i	0.828880	−	0.752344i
$292$		0			0
$293$	2.99866	−	5.19384i	0.175184	−	0.303427i	−0.765041	−	0.643981i	$-0.777281\pi$
$293$	2.99866	−	5.19384i	0.175184	−	0.303427i	0.940225	+	0.340554i	$0.110615\pi$
$294$		0			0
$295$	−10.2256	−	17.7112i	−0.595356	−	1.03119i
$296$		0			0
$297$	−4.98991	+	11.5790i	−0.289544	+	0.671883i
$298$		0			0
$299$	−17.3987	−	30.1355i	−1.00619	−	1.74278i
$300$		0			0
$301$	−8.99666	+	15.5827i	−0.518559	+	0.898170i
$302$		0			0
$303$	4.67755	−	4.24564i	0.268718	−	0.243906i
$304$		0			0
$305$	−10.8770			−0.622818
$306$		0			0
$307$	−11.7568			−0.670994			−0.335497	−	0.942041i	$-0.608904\pi$
$307$	−11.7568			−0.670994			−0.335497	+	0.942041i	$0.608904\pi$
$308$		0			0
$309$	−17.9427	−	5.75079i	−1.02072	−	0.327151i
$310$		0			0
$311$	16.1797	−	28.0240i	0.917465	−	1.58910i	0.114213	−	0.993456i	$-0.463565\pi$
$311$	16.1797	−	28.0240i	0.917465	−	1.58910i	0.803252	−	0.595640i	$-0.203101\pi$
$312$		0			0
$313$	−14.6062	−	25.2987i	−0.825591	−	1.42997i	−0.901466	−	0.432849i	$-0.857508\pi$
$313$	−14.6062	−	25.2987i	−0.825591	−	1.42997i	0.0758750	−	0.997117i	$-0.475825\pi$
$314$		0			0
$315$	−16.5801	+	7.54122i	−0.934182	+	0.424900i
$316$		0			0
$317$	−13.7011	−	23.7310i	−0.769530	−	1.33286i	−0.937818	−	0.347127i	$-0.887157\pi$
$317$	−13.7011	−	23.7310i	−0.769530	−	1.33286i	0.168288	−	0.985738i	$-0.446176\pi$
$318$		0			0
$319$	3.82951	−	6.63291i	0.214411	−	0.371371i
$320$		0			0
$321$	−2.18264	−	10.0705i	−0.121823	−	0.562083i
$322$		0			0
$323$	−47.3619			−2.63529
$324$		0			0
$325$	−3.63235			−0.201487
$326$		0			0
$327$	−3.62151	−	16.7093i	−0.200270	−	0.924029i
$328$		0			0
$329$	−3.82731	+	6.62910i	−0.211007	+	0.365474i
$330$		0			0
$331$	−1.08930	−	1.88673i	−0.0598735	−	0.103704i	0.834535	−	0.550955i	$-0.185736\pi$
$331$	−1.08930	−	1.88673i	−0.0598735	−	0.103704i	−0.894408	+	0.447251i	$0.852403\pi$
$332$		0			0
$333$	−18.3225	+	8.33374i	−1.00407	+	0.456686i
$334$		0			0
$335$	−2.00299	−	3.46928i	−0.109435	−	0.189547i
$336$		0			0
$337$	−0.715700	+	1.23963i	−0.0389866	+	0.0675268i	−0.884860	−	0.465856i	$-0.845746\pi$
$337$	−0.715700	+	1.23963i	−0.0389866	+	0.0675268i	0.845874	+	0.533383i	$0.179080\pi$
$338$		0			0
$339$	−24.7911	−	7.94577i	−1.34647	−	0.431555i
$340$		0			0
$341$	8.68874			0.470522
$342$		0			0
$343$	16.4508			0.888259
$344$		0			0
$345$	−14.2634	+	12.9463i	−0.767914	+	0.697007i
$346$		0			0
$347$	−13.1223	+	22.7285i	−0.704441	+	1.22013i	0.262452	+	0.964945i	$0.415469\pi$
$347$	−13.1223	+	22.7285i	−0.704441	+	1.22013i	−0.966893	+	0.255183i	$0.917864\pi$
$348$		0			0
$349$	−17.1673	−	29.7346i	−0.918944	−	1.59166i	−0.801022	−	0.598635i	$-0.795710\pi$
$349$	−17.1673	−	29.7346i	−0.918944	−	1.59166i	−0.117923	−	0.993023i	$-0.537623\pi$
$350$		0			0
$351$	34.0621	−	3.99334i	1.81810	−	0.213149i
$352$		0			0
$353$	−2.28933	−	3.96523i	−0.121849	−	0.211048i	0.798648	−	0.601798i	$-0.205549\pi$
$353$	−2.28933	−	3.96523i	−0.121849	−	0.211048i	−0.920497	+	0.390750i	$0.872216\pi$
$354$		0			0
$355$	6.14359	−	10.6410i	0.326068	−	0.564766i
$356$		0			0
$357$	27.9388	−	25.3590i	1.47868	−	1.34214i
$358$		0			0
$359$	−13.0591			−0.689231			−0.344616	−	0.938744i	$-0.611991\pi$
$359$	−13.0591			−0.689231			−0.344616	+	0.938744i	$0.611991\pi$
$360$		0			0
$361$	20.1599			1.06105
$362$		0			0
$363$	8.43195	+	2.70251i	0.442562	+	0.141845i
$364$		0			0
$365$	−5.28644	+	9.15639i	−0.276705	+	0.479267i
$366$		0			0
$367$	−1.70626	−	2.95533i	−0.0890660	−	0.154267i	0.818051	−	0.575146i	$-0.195055\pi$
$367$	−1.70626	−	2.95533i	−0.0890660	−	0.154267i	−0.907117	+	0.420879i	$0.861722\pi$
$368$		0			0
$369$	1.01140	−	10.4232i	0.0526513	−	0.542612i
$370$		0			0
$371$	−1.37151	−	2.37553i	−0.0712055	−	0.123332i
$372$		0			0
$373$	−6.42179	+	11.1229i	−0.332508	+	0.575921i	−0.983003	−	0.183590i	$-0.941228\pi$
$373$	−6.42179	+	11.1229i	−0.332508	+	0.575921i	0.650495	+	0.759511i	$0.274561\pi$
$374$		0			0
$375$	4.29542	+	19.8187i	0.221815	+	1.02343i
$376$		0			0
$377$	−20.8328			−1.07294
$378$		0			0
$379$	12.1642			0.624833			0.312417	−	0.949945i	$-0.398862\pi$
$379$	12.1642			0.624833			0.312417	+	0.949945i	$0.398862\pi$
$380$		0			0
$381$	2.96361	+	13.6738i	0.151830	+	0.700532i
$382$		0			0
$383$	−8.84072	+	15.3126i	−0.451740	+	0.782436i	−0.998494	−	0.0548568i	$-0.982530\pi$
$383$	−8.84072	+	15.3126i	−0.451740	+	0.782436i	0.546755	+	0.837293i	$0.315863\pi$
$384$		0			0
$385$	−7.36625	−	12.7587i	−0.375419	−	0.650244i
$386$		0			0
$387$	−15.2601	−	10.9019i	−0.775713	−	0.554174i
$388$		0			0
$389$	3.77658	+	6.54123i	0.191480	+	0.331654i	0.945741	−	0.324921i	$-0.105338\pi$
$389$	3.77658	+	6.54123i	0.191480	+	0.331654i	−0.754261	+	0.656575i	$0.772005\pi$
$390$		0			0
$391$	19.9513	−	34.5567i	1.00898	−	1.74761i
$392$		0			0
$393$	36.7252	+	11.7708i	1.85254	+	0.593756i
$394$		0			0
$395$	27.4223			1.37977
$396$		0			0
$397$	−7.97075			−0.400040			−0.200020	−	0.979792i	$-0.564101\pi$
$397$	−7.97075			−0.400040			−0.200020	+	0.979792i	$0.564101\pi$
$398$		0			0
$399$	−23.1004	+	20.9674i	−1.15647	+	1.04968i
$400$		0			0
$401$	−3.53771	+	6.12750i	−0.176665	+	0.305993i	−0.940736	−	0.339139i	$-0.889864\pi$
$401$	−3.53771	+	6.12750i	−0.176665	+	0.305993i	0.764071	+	0.645132i	$0.223198\pi$
$402$		0			0
$403$	−11.8168	−	20.4674i	−0.588639	−	1.01955i
$404$		0			0
$405$	−6.15438	−	17.9596i	−0.305813	−	0.892418i
$406$		0			0
$407$	−8.14038	−	14.0995i	−0.403503	−	0.698889i
$408$		0			0
$409$	−16.0499	+	27.7993i	−0.793619	+	1.37459i	0.130094	+	0.991502i	$0.458472\pi$
$409$	−16.0499	+	27.7993i	−0.793619	+	1.37459i	−0.923713	+	0.383086i	$0.874861\pi$
$410$		0			0
$411$	11.8599	−	10.7648i	0.585005	−	0.530988i
$412$		0			0
$413$	27.9054			1.37313
$414$		0			0
$415$	6.51820			0.319966
$416$		0			0
$417$	13.8634	+	4.44335i	0.678895	+	0.217592i
$418$		0			0
$419$	−5.63571	+	9.76133i	−0.275322	+	0.476872i	−0.970216	−	0.242240i	$-0.922118\pi$
$419$	−5.63571	+	9.76133i	−0.275322	+	0.476872i	0.694894	+	0.719112i	$0.255451\pi$
$420$		0			0
$421$	−4.82872	−	8.36359i	−0.235337	−	0.407616i	0.724033	−	0.689765i	$-0.242286\pi$
$421$	−4.82872	−	8.36359i	−0.235337	−	0.407616i	−0.959371	+	0.282149i	$0.908953\pi$
$422$		0			0
$423$	−6.49186	−	4.63782i	−0.315645	−	0.225499i
$424$		0			0
$425$	−2.08263	−	3.60723i	−0.101023	−	0.174976i
$426$		0			0
$427$	7.42081	−	12.8532i	0.359118	−	0.622011i
$428$		0			0
$429$	5.87565	+	27.1098i	0.283679	+	1.30887i
$430$		0			0
$431$	−40.1842			−1.93560			−0.967802	−	0.251711i	$-0.919007\pi$
$431$	−40.1842			−1.93560			−0.967802	+	0.251711i	$0.919007\pi$
$432$		0			0
$433$	16.1510			0.776168			0.388084	−	0.921624i	$-0.373137\pi$
$433$	16.1510			0.776168			0.388084	+	0.921624i	$0.373137\pi$
$434$		0			0
$435$	2.44275	+	11.2707i	0.117121	+	0.540387i
$436$		0			0
$437$	−16.4962	+	28.5723i	−0.789120	+	1.36680i
$438$		0			0
$439$	9.25383	+	16.0281i	0.441661	+	0.764980i	0.997813	−	0.0661011i	$-0.0210560\pi$
$439$	9.25383	+	16.0281i	0.441661	+	0.764980i	−0.556152	+	0.831081i	$0.687723\pi$
$440$		0			0
$441$	0.372172	−	3.83552i	0.0177225	−	0.182644i
$442$		0			0
$443$	1.72586	+	2.98927i	0.0819979	+	0.142025i	0.904108	−	0.427304i	$-0.140537\pi$
$443$	1.72586	+	2.98927i	0.0819979	+	0.142025i	−0.822110	+	0.569329i	$0.807203\pi$
$444$		0			0
$445$	−3.11457	+	5.39459i	−0.147645	+	0.255728i
$446$		0			0
$447$	−20.3812	−	6.53236i	−0.963998	−	0.308970i
$448$		0			0
$449$	9.83790			0.464279			0.232140	−	0.972682i	$-0.425427\pi$
$449$	9.83790			0.464279			0.232140	+	0.972682i	$0.425427\pi$
$450$		0			0
$451$	8.47025			0.398848
$452$		0			0
$453$	12.6053	−	11.4413i	0.592247	−	0.537561i
$454$		0			0
$455$	−20.0364	+	34.7041i	−0.939323	+	1.62696i
$456$		0			0
$457$	0.0111990	+	0.0193973i	0.000523869	+	0.000907367i	0.866287	−	0.499546i	$-0.166500\pi$
$457$	0.0111990	+	0.0193973i	0.000523869	+	0.000907367i	−0.865763	+	0.500454i	$0.833167\pi$
$458$		0			0
$459$	23.4954	+	31.5369i	1.09667	+	1.47202i
$460$		0			0
$461$	7.50986	+	13.0075i	0.349769	+	0.605818i	0.986208	−	0.165509i	$-0.0529267\pi$
$461$	7.50986	+	13.0075i	0.349769	+	0.605818i	−0.636439	+	0.771327i	$0.719593\pi$
$462$		0			0
$463$	−16.2691	+	28.1790i	−0.756091	+	1.30959i	0.188738	+	0.982027i	$0.439560\pi$
$463$	−16.2691	+	28.1790i	−0.756091	+	1.30959i	−0.944830	+	0.327561i	$0.893773\pi$
$464$		0			0
$465$	−9.68738	+	8.79288i	−0.449241	+	0.407760i
$466$		0			0
$467$	10.2983			0.476550			0.238275	−	0.971198i	$-0.423418\pi$
$467$	10.2983			0.476550			0.238275	+	0.971198i	$0.423418\pi$
$468$		0			0
$469$	5.46612			0.252402
$470$		0			0
$471$	0.356545	+	0.114276i	0.0164287	+	0.00526556i
$472$		0			0
$473$	7.58450	−	13.1367i	0.348736	−	0.604028i
$474$		0			0
$475$	1.72197	+	2.98253i	0.0790093	+	0.136848i
$476$		0			0
$477$	2.60248	−	1.18370i	0.119159	−	0.0541980i
$478$		0			0
$479$	3.79020	+	6.56481i	0.173178	+	0.299954i	0.939529	−	0.342468i	$-0.111263\pi$
$479$	3.79020	+	6.56481i	0.173178	+	0.299954i	−0.766351	+	0.642422i	$0.777930\pi$
$480$		0			0
$481$	−22.1421	+	38.3513i	−1.00959	+	1.74867i
$482$		0			0
$483$	−5.56736	−	25.6873i	−0.253323	−	1.16881i
$484$		0			0
$485$	−23.2560			−1.05600
$486$		0			0
$487$	20.6214			0.934444			0.467222	−	0.884140i	$-0.345255\pi$
$487$	20.6214			0.934444			0.467222	+	0.884140i	$0.345255\pi$
$488$		0			0
$489$	−2.12098	−	9.78605i	−0.0959142	−	0.442541i
$490$		0			0
$491$	−3.98229	+	6.89752i	−0.179718	+	0.311281i	−0.941784	−	0.336219i	$-0.890852\pi$
$491$	−3.98229	+	6.89752i	−0.179718	+	0.311281i	0.762066	+	0.647499i	$0.224185\pi$
$492$		0			0
$493$	−11.9446	−	20.6887i	−0.537959	−	0.931772i
$494$		0			0
$495$	13.9776	−	6.35752i	0.628246	−	0.285749i
$496$		0			0
$497$	8.38287	+	14.5196i	0.376023	+	0.651291i
$498$		0			0
$499$	10.5911	−	18.3444i	0.474125	−	0.821208i	−0.525436	−	0.850833i	$-0.676098\pi$
$499$	10.5911	−	18.3444i	0.474125	−	0.821208i	0.999561	+	0.0296248i	$0.00943125\pi$
$500$		0			0
$501$	2.70055	+	0.865552i	0.120652	+	0.0386700i
$502$		0			0
$503$	19.0876			0.851073			0.425536	−	0.904941i	$-0.360085\pi$
$503$	19.0876			0.851073			0.425536	+	0.904941i	$0.360085\pi$
$504$		0			0
$505$	−7.69336			−0.342350
$506$		0			0
$507$	39.1965	−	35.5773i	1.74078	−	1.58004i
$508$		0			0
$509$	−18.3596	+	31.7998i	−0.813776	+	1.40950i	0.0964283	+	0.995340i	$0.469258\pi$
$509$	−18.3596	+	31.7998i	−0.813776	+	1.40950i	−0.910204	+	0.414161i	$0.864075\pi$
$510$		0			0
$511$	−7.21330	−	12.4938i	−0.319097	−	0.552693i
$512$		0			0
$513$	−19.4266	−	26.0754i	−0.857703	−	1.15126i
$514$		0			0
$515$	11.4734	+	19.8726i	0.505580	+	0.875690i
$516$		0			0
$517$	3.22656	−	5.58857i	0.141904	−	0.245785i
$518$		0			0
$519$	−7.67535	+	6.96663i	−0.336910	+	0.305801i
$520$		0			0
$521$	20.5620			0.900839			0.450419	−	0.892817i	$-0.351275\pi$
$521$	20.5620			0.900839			0.450419	+	0.892817i	$0.351275\pi$
$522$		0			0
$523$	−16.2754			−0.711672			−0.355836	−	0.934548i	$-0.615804\pi$
$523$	−16.2754			−0.711672			−0.355836	+	0.934548i	$0.615804\pi$
$524$		0			0
$525$	−2.61273	−	0.837403i	−0.114029	−	0.0365472i
$526$		0			0
$527$	13.5505	−	23.4702i	0.590270	−	1.02238i
$528$		0			0
$529$	−2.39812	−	4.15367i	−0.104266	−	0.180594i
$530$		0			0
$531$	−2.80905	+	28.9495i	−0.121903	+	1.25630i
$532$		0			0
$533$	−11.5197	−	19.9527i	−0.498973	−	0.864246i
$534$		0			0
$535$	−6.27471	+	10.8681i	−0.271279	+	0.469870i
$536$		0			0
$537$	3.49017	+	16.1033i	0.150612	+	0.694911i
$538$		0			0
$539$	3.11686			0.134253
$540$		0			0
$541$	3.19402			0.137322			0.0686609	−	0.997640i	$-0.478127\pi$
$541$	3.19402			0.137322			0.0686609	+	0.997640i	$0.478127\pi$
$542$		0			0
$543$	8.71431	+	40.2071i	0.373967	+	1.72545i
$544$		0			0
$545$	−10.4112	+	18.0327i	−0.445966	+	0.772436i
$546$		0			0
$547$	−7.43936	−	12.8854i	−0.318084	−	0.550938i	0.662004	−	0.749500i	$-0.269706\pi$
$547$	−7.43936	−	12.8854i	−0.318084	−	0.550938i	−0.980088	+	0.198562i	$0.936373\pi$
$548$		0			0
$549$	12.5871	+	8.99231i	0.537205	+	0.383782i
$550$		0			0
$551$	9.87608	+	17.1059i	0.420735	+	0.728734i
$552$		0			0
$553$	−18.7087	+	32.4045i	−0.795576	+	1.37798i
$554$		0			0
$555$	23.3445	+	7.48213i	0.990920	+	0.317599i
$556$		0			0
$557$	5.76686			0.244350			0.122175	−	0.992509i	$-0.461013\pi$
$557$	5.76686			0.244350			0.122175	+	0.992509i	$0.461013\pi$
$558$		0			0
$559$	−41.2602			−1.74512
$560$		0			0
$561$	−23.5534	+	21.3786i	−0.994426	+	0.902604i
$562$		0			0
$563$	4.76738	−	8.25734i	0.200921	−	0.348005i	−0.747904	−	0.663806i	$-0.768940\pi$
$563$	4.76738	−	8.25734i	0.200921	−	0.348005i	0.948825	+	0.315801i	$0.102273\pi$
$564$		0			0
$565$	15.8527	+	27.4576i	0.666927	+	1.15515i
$566$		0			0
$567$	25.4213	+	4.98030i	1.06759	+	0.209153i
$568$		0			0
$569$	−4.27200	−	7.39933i	−0.179092	−	0.310196i	0.762478	−	0.647014i	$-0.223983\pi$
$569$	−4.27200	−	7.39933i	−0.179092	−	0.310196i	−0.941570	+	0.336818i	$0.890649\pi$
$570$		0			0
$571$	18.1444	−	31.4270i	0.759318	−	1.31518i	−0.183881	−	0.982949i	$-0.558866\pi$
$571$	18.1444	−	31.4270i	0.759318	−	1.31518i	0.943199	−	0.332229i	$-0.107801\pi$
$572$		0			0
$573$	−10.6235	+	9.64256i	−0.443803	+	0.402824i
$574$		0			0
$575$	−2.90153			−0.121002
$576$		0			0
$577$	−6.11652			−0.254634			−0.127317	−	0.991862i	$-0.540637\pi$
$577$	−6.11652			−0.254634			−0.127317	+	0.991862i	$0.540637\pi$
$578$		0			0
$579$	16.5044	+	5.28982i	0.685901	+	0.219837i
$580$		0			0
$581$	−4.44701	+	7.70245i	−0.184493	+	0.319551i
$582$		0			0
$583$	1.15624	+	2.00266i	0.0478864	+	0.0829417i
$584$		0			0
$585$	−33.9857	−	24.2796i	−1.40514	−	1.00384i
$586$		0			0
$587$	1.92500	+	3.33419i	0.0794532	+	0.137617i	0.903014	−	0.429611i	$-0.141349\pi$
$587$	1.92500	+	3.33419i	0.0794532	+	0.137617i	−0.823561	+	0.567228i	$0.808016\pi$
$588$		0			0
$589$	−11.2039	+	19.4057i	−0.461647	+	0.799597i
$590$		0			0
$591$	−0.204079	−	0.941606i	−0.00839471	−	0.0387325i
$592$		0			0
$593$	16.6462			0.683579			0.341789	−	0.939777i	$-0.388967\pi$
$593$	16.6462			0.683579			0.341789	+	0.939777i	$0.388967\pi$
$594$		0			0
$595$	−45.9521			−1.88385
$596$		0			0
$597$	7.90720	+	36.4832i	0.323620	+	1.49316i
$598$		0			0
$599$	−17.8791	+	30.9675i	−0.730520	+	1.26530i	0.226141	+	0.974095i	$0.427389\pi$
$599$	−17.8791	+	30.9675i	−0.730520	+	1.26530i	−0.956661	+	0.291203i	$0.905944\pi$
$600$		0			0
$601$	6.77202	+	11.7295i	0.276236	+	0.478455i	0.970446	−	0.241317i	$-0.0775794\pi$
$601$	6.77202	+	11.7295i	0.276236	+	0.478455i	−0.694210	+	0.719773i	$0.744246\pi$
$602$		0			0
$603$	−0.550239	+	5.67064i	−0.0224075	+	0.230926i
$604$		0			0
$605$	−5.39181	−	9.33888i	−0.219208	−	0.379679i
$606$		0			0
$607$	−14.0131	+	24.2714i	−0.568775	+	0.985147i	0.427913	+	0.903820i	$0.359249\pi$
$607$	−14.0131	+	24.2714i	−0.568775	+	0.985147i	−0.996688	+	0.0813266i	$0.974084\pi$
$608$		0			0
$609$	−14.9849	−	4.80279i	−0.607219	−	0.194619i
$610$		0			0
$611$	−17.5527			−0.710107
$612$		0			0
$613$	37.6283			1.51979			0.759897	−	0.650043i	$-0.225249\pi$
$613$	37.6283			1.51979			0.759897	+	0.650043i	$0.225249\pi$
$614$		0			0
$615$	−9.44377	+	8.57177i	−0.380810	+	0.345647i
$616$		0			0
$617$	5.57241	−	9.65169i	0.224337	−	0.388562i	−0.731784	−	0.681537i	$-0.761312\pi$
$617$	5.57241	−	9.65169i	0.224337	−	0.388562i	0.956120	+	0.292975i	$0.0946451\pi$
$618$		0			0
$619$	9.54119	+	16.5258i	0.383493	+	0.664229i	0.991559	−	0.129658i	$-0.0413878\pi$
$619$	9.54119	+	16.5258i	0.383493	+	0.664229i	−0.608066	+	0.793886i	$0.708054\pi$
$620$		0			0
$621$	27.2089	−	3.18989i	1.09185	−	0.128006i
$622$		0			0
$623$	−4.24980	−	7.36086i	−0.170265	−	0.294907i
$624$		0			0
$625$	10.9727	−	19.0053i	0.438908	−	0.760211i
$626$		0			0
$627$	19.4745	−	17.6763i	0.777736	−	0.705923i
$628$		0			0
$629$	−50.7813			−2.02478
$630$		0			0
$631$	−29.7049			−1.18253			−0.591267	−	0.806476i	$-0.701372\pi$
$631$	−29.7049			−1.18253			−0.591267	+	0.806476i	$0.701372\pi$
$632$		0			0
$633$	−27.4785	−	8.80712i	−1.09217	−	0.350051i
$634$		0			0
$635$	8.51984	−	14.7568i	0.338099	−	0.585605i
$636$		0			0
$637$	−4.23898	−	7.34214i	−0.167955	−	0.290906i
$638$		0			0
$639$	−15.9067	+	7.23493i	−0.629258	+	0.286209i
$640$		0			0
$641$	−9.60138	−	16.6301i	−0.379232	−	0.656848i	0.611719	−	0.791075i	$-0.290478\pi$
$641$	−9.60138	−	16.6301i	−0.379232	−	0.656848i	−0.990951	+	0.134227i	$0.957145\pi$
$642$		0			0
$643$	3.52181	−	6.09995i	0.138887	−	0.240559i	−0.788189	−	0.615434i	$-0.788981\pi$
$643$	3.52181	−	6.09995i	0.138887	−	0.240559i	0.927075	+	0.374875i	$0.122314\pi$
$644$		0			0
$645$	4.83797	+	22.3220i	0.190495	+	0.878929i
$646$		0			0
$647$	36.8474			1.44862			0.724311	−	0.689473i	$-0.242158\pi$
$647$	36.8474			1.44862			0.724311	+	0.689473i	$0.242158\pi$
$648$		0			0
$649$	−23.5252			−0.923446
$650$		0			0
$651$	−3.78123	−	17.4463i	−0.148198	−	0.683774i
$652$		0			0
$653$	−20.6354	+	35.7416i	−0.807526	+	1.39868i	0.107046	+	0.994254i	$0.465861\pi$
$653$	−20.6354	+	35.7416i	−0.807526	+	1.39868i	−0.914572	+	0.404422i	$0.867473\pi$
$654$		0			0
$655$	−23.4839	−	40.6754i	−0.917593	−	1.58932i
$656$		0			0
$657$	13.6874	−	6.22552i	0.533995	−	0.242881i
$658$		0			0
$659$	−7.79606	−	13.5032i	−0.303691	−	0.526009i	0.673278	−	0.739390i	$-0.264886\pi$
$659$	−7.79606	−	13.5032i	−0.303691	−	0.526009i	−0.976969	+	0.213381i	$0.931552\pi$
$660$		0			0
$661$	0.273228	−	0.473246i	0.0106274	−	0.0184071i	−0.860663	−	0.509175i	$-0.829950\pi$
$661$	0.273228	−	0.473246i	0.0106274	−	0.0184071i	0.871290	+	0.490768i	$0.163284\pi$
$662$		0			0
$663$	82.3929	+	26.4076i	3.19987	+	1.02559i
$664$		0			0
$665$	37.9942			1.47335
$666$		0			0
$667$	−16.6413			−0.644353
$668$		0			0
$669$	11.5235	−	10.4595i	0.445525	−	0.404387i
$670$		0			0
$671$	−6.25601	+	10.8357i	−0.241510	+	0.418308i
$672$		0			0
$673$	7.71994	+	13.3713i	0.297582	+	0.515427i	0.975582	−	0.219635i	$-0.0704866\pi$
$673$	7.71994	+	13.3713i	0.297582	+	0.515427i	−0.678000	+	0.735061i	$0.737153\pi$
$674$		0			0
$675$	1.13174	−	2.62619i	0.0435607	−	0.101082i
$676$		0			0
$677$	20.7394	+	35.9216i	0.797079	+	1.38058i	0.921511	+	0.388353i	$0.126956\pi$
$677$	20.7394	+	35.9216i	0.797079	+	1.38058i	−0.124432	+	0.992228i	$0.539711\pi$
$678$		0			0
$679$	15.8663	−	27.4812i	0.608893	−	1.05463i
$680$		0			0
$681$	−13.6575	+	12.3965i	−0.523358	+	0.475033i
$682$		0			0
$683$	7.83506			0.299800			0.149900	−	0.988701i	$-0.452105\pi$
$683$	7.83506			0.299800			0.149900	+	0.988701i	$0.452105\pi$
$684$		0			0
$685$	−19.5064			−0.745303
$686$		0			0
$687$	18.1693	+	5.82344i	0.693204	+	0.222178i
$688$		0			0
$689$	3.14500	−	5.44730i	0.119815	−	0.207526i
$690$		0			0
$691$	11.8366	+	20.5016i	0.450286	+	0.779918i	0.998404	−	0.0564831i	$-0.0179887\pi$
$691$	11.8366	+	20.5016i	0.450286	+	0.779918i	−0.548118	+	0.836401i	$0.684655\pi$
$692$		0			0
$693$	−2.02357	+	20.8545i	−0.0768691	+	0.792196i
$694$		0			0
$695$	−8.86497	−	15.3546i	−0.336267	−	0.582432i
$696$		0			0
$697$	13.2098	−	22.8800i	0.500356	−	0.866642i
$698$		0			0
$699$	10.0009	+	46.1435i	0.378270	+	1.74531i
$700$		0			0
$701$	7.14909			0.270017			0.135009	−	0.990844i	$-0.456894\pi$
$701$	7.14909			0.270017			0.135009	+	0.990844i	$0.456894\pi$
$702$		0			0
$703$	41.9871			1.58357
$704$		0			0
$705$	2.05815	+	9.49612i	0.0775143	+	0.357645i
$706$		0			0
$707$	5.24876	−	9.09111i	0.197400	−	0.341906i
$708$		0			0
$709$	−19.3826	−	33.5716i	−0.727928	−	1.26081i	−0.957757	−	0.287577i	$-0.907150\pi$
$709$	−19.3826	−	33.5716i	−0.727928	−	1.26081i	0.229830	−	0.973231i	$-0.426183\pi$
$710$		0			0
$711$	−31.7336	−	22.6707i	−1.19010	−	0.850217i
$712$		0			0
$713$	−9.43931	−	16.3494i	−0.353505	−	0.612289i
$714$		0			0
$715$	16.8914	−	29.2568i	0.631704	−	1.09414i
$716$		0			0
$717$	19.2088	+	6.15659i	0.717365	+	0.229922i
$718$		0			0
$719$	−38.3429			−1.42995			−0.714975	−	0.699151i	$-0.753562\pi$
$719$	−38.3429			−1.42995			−0.714975	+	0.699151i	$0.753562\pi$
$720$		0			0
$721$	−31.3108			−1.16607
$722$		0			0
$723$	−25.2286	+	22.8991i	−0.938263	+	0.851627i
$724$		0			0
$725$	−0.868556	+	1.50438i	−0.0322574	+	0.0558714i
$726$		0			0
$727$	−21.9734	−	38.0591i	−0.814950	−	1.41154i	−0.909364	−	0.416002i	$-0.863431\pi$
$727$	−21.9734	−	38.0591i	−0.814950	−	1.41154i	0.0944136	−	0.995533i	$-0.469902\pi$
$728$		0			0
$729$	−7.72564	+	25.8711i	−0.286135	+	0.958189i
$730$		0			0
$731$	−23.6568	−	40.9748i	−0.874979	−	1.51551i
$732$		0			0
$733$	24.7222	−	42.8202i	0.913137	−	1.58160i	0.103531	−	0.994626i	$-0.466986\pi$
$733$	24.7222	−	42.8202i	0.913137	−	1.58160i	0.809606	−	0.586973i	$-0.199681\pi$
$734$		0			0
$735$	−3.47510	+	3.15422i	−0.128181	+	0.116345i
$736$		0			0
$737$	−4.60814			−0.169743
$738$		0			0
$739$	9.25073			0.340294			0.170147	−	0.985419i	$-0.445576\pi$
$739$	9.25073			0.340294			0.170147	+	0.985419i	$0.445576\pi$
$740$		0			0
$741$	−68.1242	−	21.8344i	−2.50261	−	0.802107i
$742$		0			0
$743$	−20.6473	+	35.7621i	−0.757475	+	1.31198i	0.186660	+	0.982425i	$0.440234\pi$
$743$	−20.6473	+	35.7621i	−0.757475	+	1.31198i	−0.944135	+	0.329560i	$0.893100\pi$
$744$		0			0
$745$	13.0328	+	22.5734i	0.477483	+	0.827025i
$746$		0			0
$747$	−7.54299	−	5.38875i	−0.275984	−	0.197164i
$748$		0			0
$749$	−8.56178	−	14.8294i	−0.312840	−	0.541856i
$750$		0			0
$751$	10.2101	−	17.6845i	0.372573	−	0.645315i	−0.617388	−	0.786659i	$-0.711809\pi$
$751$	10.2101	−	17.6845i	0.372573	−	0.645315i	0.989961	+	0.141344i	$0.0451423\pi$
$752$		0			0
$753$	8.57770	+	39.5768i	0.312589	+	1.44226i
$754$		0			0
$755$	−20.7324			−0.754529
$756$		0			0
$757$	−7.74944			−0.281658			−0.140829	−	0.990034i	$-0.544977\pi$
$757$	−7.74944			−0.281658			−0.140829	+	0.990034i	$0.544977\pi$
$758$		0			0
$759$	4.69348	+	21.6554i	0.170363	+	0.786039i
$760$		0			0
$761$	13.3416	−	23.1084i	0.483634	−	0.837679i	−0.516189	−	0.856475i	$-0.672650\pi$
$761$	13.3416	−	23.1084i	0.483634	−	0.837679i	0.999823	+	0.0187955i	$0.00598315\pi$
$762$		0			0
$763$	−14.2059	−	24.6054i	−0.514290	−	0.890776i
$764$		0			0
$765$	4.62570	−	47.6714i	0.167243	−	1.72356i
$766$		0			0
$767$	31.9947	+	55.4165i	1.15526	+	2.00097i
$768$		0			0
$769$	−24.5226	+	42.4744i	−0.884307	+	1.53166i	−0.0378010	+	0.999285i	$0.512035\pi$
$769$	−24.5226	+	42.4744i	−0.884307	+	1.53166i	−0.846506	+	0.532379i	$0.821298\pi$
$770$		0			0
$771$	−6.23596	−	1.99868i	−0.224583	−	0.0719807i
$772$		0			0
$773$	−12.1406			−0.436665			−0.218333	−	0.975874i	$-0.570062\pi$
$773$	−12.1406			−0.436665			−0.218333	+	0.975874i	$0.570062\pi$
$774$		0			0
$775$	−1.97066			−0.0707881
$776$		0			0
$777$	−24.7682	+	22.4812i	−0.888553	+	0.806508i
$778$		0			0
$779$	−10.9221	+	18.9177i	−0.391326	+	0.677796i
$780$		0			0
$781$	−7.06706	−	12.2405i	−0.252879	−	0.438000i
$782$		0			0
$783$	6.49093	−	15.0621i	0.231967	−	0.538276i
$784$		0			0
$785$	−0.227993	−	0.394895i	−0.00813742	−	0.0140944i
$786$		0			0
$787$	24.2553	−	42.0114i	0.864608	−	1.49754i	−0.00282812	−	0.999996i	$-0.500900\pi$
$787$	24.2553	−	42.0114i	0.864608	−	1.49754i	0.867436	−	0.497549i	$-0.165766\pi$
$788$		0			0
$789$	−26.5837	+	24.1291i	−0.946405	+	0.859017i
$790$		0			0
$791$	−43.2616			−1.53821
$792$		0			0
$793$	34.0331			1.20855
$794$		0

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.366879 + 1.69275i) q^{3} +(1.05471 - 1.82681i) q^{5} +(1.43914 + 2.49267i) q^{7} +(-2.73080 + 1.24207i) q^{9} +O(q^{10})\)	\(q+(0.366879 + 1.69275i) q^{3} +(1.05471 - 1.82681i) q^{5} +(1.43914 + 2.49267i) q^{7} +(-2.73080 + 1.24207i) q^{9} +(-1.21325 - 2.10141i) q^{11} +(-3.30008 + 5.71590i) q^{13} +(3.47929 + 1.11514i) q^{15} -7.56848 q^{17} +6.25779 q^{19} +(-3.69147 + 3.35061i) q^{21} +(-2.63611 + 4.56587i) q^{23} +(0.275172 + 0.476612i) q^{25} +(-3.10438 - 4.16687i) q^{27} +(1.57821 + 2.73353i) q^{29} +(-1.79039 + 3.10104i) q^{31} +(3.11204 - 2.82469i) q^{33} +6.07151 q^{35} +6.70957 q^{37} +(-10.8863 - 3.48916i) q^{39} +(-1.74537 + 3.02306i) q^{41} +(3.12570 + 5.41388i) q^{43} +(-0.611180 + 6.29868i) q^{45} +(1.32972 + 2.30314i) q^{47} +(-0.642255 + 1.11242i) q^{49} +(-2.77671 - 12.8115i) q^{51} -0.953009 q^{53} -5.11850 q^{55} +(2.29585 + 10.5929i) q^{57} +(4.84757 - 8.39624i) q^{59} +(-2.57821 - 4.46558i) q^{61} +(-7.02607 - 5.01946i) q^{63} +(6.96125 + 12.0572i) q^{65} +(0.949546 - 1.64466i) q^{67} +(-8.69601 - 2.78715i) q^{69} +5.82491 q^{71} -5.01222 q^{73} +(-0.705829 + 0.640656i) q^{75} +(3.49207 - 6.04844i) q^{77} +(6.49996 + 11.2583i) q^{79} +(5.91454 - 6.78368i) q^{81} +(1.54502 + 2.67606i) q^{83} +(-7.98256 + 13.8262i) q^{85} +(-4.04818 + 3.67438i) q^{87} -2.95301 q^{89} -18.9971 q^{91} +(-5.90614 - 1.89297i) q^{93} +(6.60015 - 11.4318i) q^{95} +(-5.51242 - 9.54779i) q^{97} +(5.92323 + 4.23159i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 2 q^{5} - 6 q^{7} - 2 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 2 * q^5 - 6 * q^7 - 2 * q^9	\( 12 q + 4 q^{3} + 2 q^{5} - 6 q^{7} - 2 q^{9} + 4 q^{11} - 10 q^{13} - 4 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} - 8 q^{23} - 14 q^{25} - 14 q^{27} + 2 q^{29} - 8 q^{31} - 10 q^{33} + 8 q^{35} - 22 q^{39} - 2 q^{41} - 2 q^{43} - 10 q^{45} + 14 q^{47} - 18 q^{49} - 38 q^{51} - 24 q^{53} + 16 q^{55} - 38 q^{57} + 6 q^{59} - 14 q^{61} + 16 q^{63} - 8 q^{65} + 4 q^{67} + 50 q^{69} + 28 q^{71} + 60 q^{73} + 50 q^{75} - 2 q^{77} - 16 q^{79} + 22 q^{81} + 24 q^{83} - 16 q^{85} + 36 q^{87} - 48 q^{89} - 52 q^{91} - 42 q^{93} + 20 q^{95} - 14 q^{97} + 68 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 + 2 * q^5 - 6 * q^7 - 2 * q^9 + 4 * q^11 - 10 * q^13 - 4 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 - 8 * q^23 - 14 * q^25 - 14 * q^27 + 2 * q^29 - 8 * q^31 - 10 * q^33 + 8 * q^35 - 22 * q^39 - 2 * q^41 - 2 * q^43 - 10 * q^45 + 14 * q^47 - 18 * q^49 - 38 * q^51 - 24 * q^53 + 16 * q^55 - 38 * q^57 + 6 * q^59 - 14 * q^61 + 16 * q^63 - 8 * q^65 + 4 * q^67 + 50 * q^69 + 28 * q^71 + 60 * q^73 + 50 * q^75 - 2 * q^77 - 16 * q^79 + 22 * q^81 + 24 * q^83 - 16 * q^85 + 36 * q^87 - 48 * q^89 - 52 * q^91 - 42 * q^93 + 20 * q^95 - 14 * q^97 + 68 * q^99

Introduction

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