LMFDB - Embedded newform 2268.2.l.n.109.8

Newspace parameters

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.l (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$ x^{16} - 9x^{14} + 31x^{12} - 282x^{10} + 1695x^{8} - 3318x^{6} + 4606x^{4} - 4116x^{2} + 2401 $ x^16 - 9x^14 + 31x^12 - 282x^10 + 1695x^8 - 3318x^6 + 4606x^4 - 4116*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.8
Root			$1.04556 + 0.339889i$ of defining polynomial
Character	$\chi$	$=$	2268.109
Dual form			2268.2.l.n.541.8

$q$-expansion

$f(q)$	$=$	$q+3.67687 q^{5} +(-1.07542 - 2.41733i) q^{7} +O(q^{10})$	$q+3.67687 q^{5} +(-1.07542 - 2.41733i) q^{7} +0.603143 q^{11} +(2.62851 - 4.55271i) q^{13} +(-2.12557 + 3.68159i) q^{17} +(3.68426 + 6.38133i) q^{19} +1.15778 q^{23} +8.51937 q^{25} +(3.98826 + 6.90786i) q^{29} +(-1.57542 - 2.72871i) q^{31} +(-3.95419 - 8.88819i) q^{35} +(0.00266923 + 0.00462323i) q^{37} +(2.00937 - 3.48033i) q^{41} +(-3.66193 - 6.34264i) q^{43} +(6.10863 - 10.5805i) q^{47} +(-4.68693 + 5.19930i) q^{49} +(4.64928 - 8.05279i) q^{53} +2.21768 q^{55} +(3.30760 + 5.72894i) q^{59} +(0.969252 - 1.67879i) q^{61} +(9.66468 - 16.7397i) q^{65} +(-4.31544 - 7.47456i) q^{67} -1.13815 q^{71} +(-5.33511 + 9.24068i) q^{73} +(-0.648634 - 1.45799i) q^{77} +(-2.07275 + 3.59011i) q^{79} +(-6.24088 - 10.8095i) q^{83} +(-7.81544 + 13.5367i) q^{85} +(4.09464 + 7.09212i) q^{89} +(-13.8321 - 1.45787i) q^{91} +(13.5466 + 23.4633i) q^{95} +(6.77935 + 11.7422i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 10 q^{13} + 8 q^{19} - 8 q^{31} - 4 q^{37} - 10 q^{43} - 20 q^{49} - 32 q^{55} + 28 q^{61} + 18 q^{67} - 20 q^{79} - 38 q^{85} - 2 q^{91} + 42 q^{97}+O(q^{100}) $ 16 * q + 10 * q^13 + 8 * q^19 - 8 * q^31 - 4 * q^37 - 10 * q^43 - 20 * q^49 - 32 * q^55 + 28 * q^61 + 18 * q^67 - 20 * q^79 - 38 * q^85 - 2 * q^91 + 42 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	3.67687			1.64435			0.822173	−	0.569238i	$-0.192762\pi$
$5$	3.67687			1.64435			0.822173	+	0.569238i	$0.192762\pi$
$6$		0			0
$7$	−1.07542	−	2.41733i	−0.406472	−	0.913663i
$7$	−1.07542	−	2.41733i	−0.406472	−	0.913663i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.603143			0.181855			0.0909273	−	0.995858i	$-0.471017\pi$
$11$	0.603143			0.181855			0.0909273	+	0.995858i	$0.471017\pi$
$12$		0			0
$13$	2.62851	−	4.55271i	0.729017	−	1.26269i	−0.228282	−	0.973595i	$-0.573311\pi$
$13$	2.62851	−	4.55271i	0.729017	−	1.26269i	0.957299	−	0.289099i	$-0.0933558\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.12557	+	3.68159i	−0.515526	+	0.892918i	0.484311	+	0.874896i	$0.339070\pi$
$17$	−2.12557	+	3.68159i	−0.515526	+	0.892918i	−0.999838	+	0.0180219i	$0.994263\pi$
$18$		0			0
$19$	3.68426	+	6.38133i	0.845228	+	1.46398i	0.885423	+	0.464786i	$0.153869\pi$
$19$	3.68426	+	6.38133i	0.845228	+	1.46398i	−0.0401954	+	0.999192i	$0.512798\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.15778			0.241414			0.120707	−	0.992688i	$-0.461484\pi$
$23$	1.15778			0.241414			0.120707	+	0.992688i	$0.461484\pi$
$24$		0			0
$25$	8.51937			1.70387
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.98826	+	6.90786i	0.740601	+	1.28276i	0.952222	+	0.305406i	$0.0987922\pi$
$29$	3.98826	+	6.90786i	0.740601	+	1.28276i	−0.211622	+	0.977352i	$0.567874\pi$
$30$		0			0
$31$	−1.57542	−	2.72871i	−0.282954	−	0.490091i	0.689157	−	0.724612i	$-0.257981\pi$
$31$	−1.57542	−	2.72871i	−0.282954	−	0.490091i	−0.972111	+	0.234521i	$0.924648\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.95419	−	8.88819i	−0.668380	−	1.50238i
$36$		0			0
$37$	0.00266923	+	0.00462323i	0.000438818	+	0.000760055i	0.866245	−	0.499620i	$-0.166527\pi$
$37$	0.00266923	+	0.00462323i	0.000438818	+	0.000760055i	−0.865806	+	0.500380i	$0.833194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.00937	−	3.48033i	0.313811	−	0.543537i	−0.665373	−	0.746511i	$-0.731728\pi$
$41$	2.00937	−	3.48033i	0.313811	−	0.543537i	0.979184	+	0.202974i	$0.0650609\pi$
$42$		0			0
$43$	−3.66193	−	6.34264i	−0.558438	−	0.967244i	−0.997627	−	0.0688488i	$-0.978067\pi$
$43$	−3.66193	−	6.34264i	−0.558438	−	0.967244i	0.439189	−	0.898395i	$-0.355266\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.10863	−	10.5805i	0.891036	−	1.54332i	0.0524003	−	0.998626i	$-0.483313\pi$
$47$	6.10863	−	10.5805i	0.891036	−	1.54332i	0.838635	−	0.544693i	$-0.183354\pi$
$48$		0			0
$49$	−4.68693	+	5.19930i	−0.669562	+	0.742756i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.64928	−	8.05279i	0.638628	−	1.10614i	−0.347107	−	0.937826i	$-0.612836\pi$
$53$	4.64928	−	8.05279i	0.638628	−	1.10614i	0.985734	−	0.168310i	$-0.0538309\pi$
$54$		0			0
$55$	2.21768			0.299032
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.30760	+	5.72894i	0.430613	+	0.745844i	0.996926	−	0.0783462i	$-0.0249640\pi$
$59$	3.30760	+	5.72894i	0.430613	+	0.745844i	−0.566313	+	0.824190i	$0.691631\pi$
$60$		0			0
$61$	0.969252	−	1.67879i	0.124100	−	0.214948i	−0.797281	−	0.603609i	$-0.793729\pi$
$61$	0.969252	−	1.67879i	0.124100	−	0.214948i	0.921381	+	0.388661i	$0.127062\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.66468	−	16.7397i	1.19876	−	2.07631i
$66$		0			0
$67$	−4.31544	−	7.47456i	−0.527215	−	0.913163i	−0.999497	−	0.0317155i	$-0.989903\pi$
$67$	−4.31544	−	7.47456i	−0.527215	−	0.913163i	0.472282	−	0.881447i	$-0.343430\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.13815			−0.135074			−0.0675370	−	0.997717i	$-0.521514\pi$
$71$	−1.13815			−0.135074			−0.0675370	+	0.997717i	$0.521514\pi$
$72$		0			0
$73$	−5.33511	+	9.24068i	−0.624427	+	1.08154i	0.364224	+	0.931311i	$0.381334\pi$
$73$	−5.33511	+	9.24068i	−0.624427	+	1.08154i	−0.988651	+	0.150228i	$0.951999\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.648634	−	1.45799i	−0.0739187	−	0.166154i
$78$		0			0
$79$	−2.07275	+	3.59011i	−0.233203	+	0.403919i	−0.958749	−	0.284254i	$-0.908254\pi$
$79$	−2.07275	+	3.59011i	−0.233203	+	0.403919i	0.725546	+	0.688174i	$0.241587\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.24088	−	10.8095i	−0.685026	−	1.18650i	−0.973429	−	0.228990i	$-0.926458\pi$
$83$	−6.24088	−	10.8095i	−0.685026	−	1.18650i	0.288403	−	0.957509i	$-0.406876\pi$
$84$		0			0
$85$	−7.81544	+	13.5367i	−0.847703	+	1.46827i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.09464	+	7.09212i	0.434031	+	0.751764i	0.997216	−	0.0745672i	$-0.0237575\pi$
$89$	4.09464	+	7.09212i	0.434031	+	0.751764i	−0.563185	+	0.826331i	$0.690424\pi$
$90$		0			0
$91$	−13.8321	−	1.45787i	−1.45000	−	0.152827i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13.5466	+	23.4633i	1.38985	+	2.40729i
$96$		0			0
$97$	6.77935	+	11.7422i	0.688339	+	1.19224i	0.972375	+	0.233425i	$0.0749932\pi$
$97$	6.77935	+	11.7422i	0.688339	+	1.19224i	−0.284036	+	0.958814i	$0.591673\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15.3395			1.52634			0.763169	−	0.646200i	$-0.223643\pi$
$101$	15.3395			1.52634			0.763169	+	0.646200i	$0.223643\pi$
$102$		0			0
$103$	−4.96066			−0.488789			−0.244394	−	0.969676i	$-0.578589\pi$
$103$	−4.96066			−0.488789			−0.244394	+	0.969676i	$0.578589\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.41399	−	5.91320i	−0.330043	−	0.571651i	0.652477	−	0.757808i	$-0.273730\pi$
$107$	−3.41399	−	5.91320i	−0.330043	−	0.571651i	−0.982520	+	0.186158i	$0.940396\pi$
$108$		0			0
$109$	8.90194	−	15.4186i	0.852651	−	1.47684i	−0.0261554	−	0.999658i	$-0.508326\pi$
$109$	8.90194	−	15.4186i	0.852651	−	1.47684i	0.878807	−	0.477178i	$-0.158340\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.63946	−	8.03579i	0.436444	−	0.755943i	−0.560968	−	0.827837i	$-0.689571\pi$
$113$	4.63946	−	8.03579i	0.436444	−	0.755943i	0.997412	+	0.0718940i	$0.0229043\pi$
$114$		0			0
$115$	4.25702			0.396969
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.1855	+	1.17892i	1.02537	+	0.108072i
$120$		0			0
$121$	−10.6362			−0.966929
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.9403			1.15741
$126$		0			0
$127$	−14.9941			−1.33051			−0.665254	−	0.746617i	$-0.731677\pi$
$127$	−14.9941			−1.33051			−0.665254	+	0.746617i	$0.731677\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−22.8327			−1.99490			−0.997452	−	0.0713394i	$-0.977273\pi$
$131$	−22.8327			−1.99490			−0.997452	+	0.0713394i	$0.977273\pi$
$132$		0			0
$133$	11.4636	−	15.7687i	0.994022	−	1.36732i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.52557			−0.130338			−0.0651690	−	0.997874i	$-0.520759\pi$
$137$	−1.52557			−0.130338			−0.0651690	+	0.997874i	$0.520759\pi$
$138$		0			0
$139$	−3.31277	+	5.73789i	−0.280986	+	0.486681i	−0.971628	−	0.236515i	$-0.923995\pi$
$139$	−3.31277	+	5.73789i	−0.280986	+	0.486681i	0.690642	+	0.723197i	$0.257328\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.58537	−	2.74594i	0.132575	−	0.229627i
$144$		0			0
$145$	14.6643	+	25.3993i	1.21780	+	2.10930i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	17.7749			1.45618			0.728089	−	0.685483i	$-0.240409\pi$
$149$	17.7749			1.45618			0.728089	+	0.685483i	$0.240409\pi$
$150$		0			0
$151$	4.63622			0.377290			0.188645	−	0.982045i	$-0.439590\pi$
$151$	4.63622			0.377290			0.188645	+	0.982045i	$0.439590\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.79262	−	10.0331i	−0.465275	−	0.805880i
$156$		0			0
$157$	−1.70660	−	2.95592i	−0.136202	−	0.235908i	0.789854	−	0.613295i	$-0.210156\pi$
$157$	−1.70660	−	2.95592i	−0.136202	−	0.235908i	−0.926056	+	0.377387i	$0.876823\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.24511	−	2.79874i	−0.0981281	−	0.220571i
$162$		0			0
$163$	7.55012	+	13.0772i	0.591371	+	1.02428i	0.994048	+	0.108942i	$0.0347464\pi$
$163$	7.55012	+	13.0772i	0.591371	+	1.02428i	−0.402677	+	0.915342i	$0.631920\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.85782	−	4.94988i	0.221144	−	0.383033i	−0.734011	−	0.679137i	$-0.762354\pi$
$167$	2.85782	−	4.94988i	0.221144	−	0.383033i	0.955156	+	0.296104i	$0.0956875\pi$
$168$		0			0
$169$	−7.31811	−	12.6753i	−0.562931	−	0.975026i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.948157	−	1.64226i	0.0720871	−	0.124858i	−0.827729	−	0.561128i	$-0.810367\pi$
$173$	0.948157	−	1.64226i	0.0720871	−	0.124858i	0.899816	+	0.436270i	$0.143701\pi$
$174$		0			0
$175$	−9.16193	−	20.5941i	−0.692576	−	1.55677i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−7.29747	+	12.6396i	−0.545438	+	0.944727i	0.453141	+	0.891439i	$0.350304\pi$
$179$	−7.29747	+	12.6396i	−0.545438	+	0.944727i	−0.998579	+	0.0532881i	$0.983030\pi$
$180$		0			0
$181$	7.89857			0.587096			0.293548	−	0.955944i	$-0.405164\pi$
$181$	7.89857			0.587096			0.293548	+	0.955944i	$0.405164\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.00981439	+	0.0169990i	0.000721569	+	0.00124979i
$186$		0			0
$187$	−1.28202	+	2.22053i	−0.0937508	+	0.162381i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.23670	+	7.33818i	−0.306557	+	0.530972i	−0.977607	−	0.210440i	$-0.932510\pi$
$191$	−4.23670	+	7.33818i	−0.306557	+	0.530972i	0.671050	+	0.741412i	$0.265844\pi$
$192$		0			0
$193$	3.48300	+	6.03273i	0.250712	+	0.434246i	0.963722	−	0.266908i	$-0.0860020\pi$
$193$	3.48300	+	6.03273i	0.250712	+	0.434246i	−0.713010	+	0.701154i	$0.752669\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.2371			−1.15685			−0.578424	−	0.815736i	$-0.696332\pi$
$197$	−16.2371			−1.15685			−0.578424	+	0.815736i	$0.696332\pi$
$198$		0			0
$199$	−7.35153	+	12.7332i	−0.521136	+	0.902634i	0.478562	+	0.878054i	$0.341158\pi$
$199$	−7.35153	+	12.7332i	−0.521136	+	0.902634i	−0.999698	+	0.0245800i	$0.992175\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.4095	−	17.0698i	0.870976	−	1.19806i
$204$		0			0
$205$	7.38819	−	12.7967i	0.516014	−	0.893762i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.22214	+	3.84886i	0.153709	+	0.266231i
$210$		0			0
$211$	8.41053	−	14.5675i	0.579005	−	1.00287i	−0.416589	−	0.909095i	$-0.636775\pi$
$211$	8.41053	−	14.5675i	0.579005	−	1.00287i	0.995594	−	0.0937708i	$-0.0298921\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−13.4644	−	23.3211i	−0.918266	−	1.59048i
$216$		0			0
$217$	−4.90194	+	6.74283i	−0.332766	+	0.457733i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	11.1741	+	19.3542i	0.751655	+	1.30190i
$222$		0			0
$223$	3.45799	+	5.98942i	0.231564	+	0.401081i	0.958269	−	0.285869i	$-0.0922823\pi$
$223$	3.45799	+	5.98942i	0.231564	+	0.401081i	−0.726704	+	0.686950i	$0.758949\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.99634			0.464363			0.232182	−	0.972672i	$-0.425414\pi$
$227$	6.99634			0.464363			0.232182	+	0.972672i	$0.425414\pi$
$228$		0			0
$229$	−2.82700			−0.186813			−0.0934066	−	0.995628i	$-0.529776\pi$
$229$	−2.82700			−0.186813			−0.0934066	+	0.995628i	$0.529776\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.1679	+	19.3434i	0.731635	+	1.26723i	0.956184	+	0.292766i	$0.0945756\pi$
$233$	11.1679	+	19.3434i	0.731635	+	1.26723i	−0.224550	+	0.974463i	$0.572091\pi$
$234$		0			0
$235$	22.4606	−	38.9030i	1.46517	−	2.53775i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.954829	+	1.65381i	−0.0617628	+	0.106976i	−0.895253	−	0.445557i	$-0.853006\pi$
$239$	−0.954829	+	1.65381i	−0.0617628	+	0.106976i	0.833491	+	0.552534i	$0.186339\pi$
$240$		0			0
$241$	−19.6870			−1.26815			−0.634077	−	0.773270i	$-0.718620\pi$
$241$	−19.6870			−1.26815			−0.634077	+	0.773270i	$0.718620\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−17.2332	+	19.1171i	−1.10099	+	1.22135i
$246$		0			0
$247$	38.7365			2.46474
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.2990			0.776306			0.388153	−	0.921595i	$-0.373113\pi$
$251$	12.2990			0.776306			0.388153	+	0.921595i	$0.373113\pi$
$252$		0			0
$253$	0.698309			0.0439023
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.4242			−0.712622			−0.356311	−	0.934367i	$-0.615966\pi$
$257$	−11.4242			−0.712622			−0.356311	+	0.934367i	$0.615966\pi$
$258$		0			0
$259$	0.00830532	−	0.0114243i	0.000516067	−	0.000709873i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	19.3109			1.19076			0.595380	−	0.803445i	$-0.297002\pi$
$263$	19.3109			1.19076			0.595380	+	0.803445i	$0.297002\pi$
$264$		0			0
$265$	17.0948	−	29.6091i	1.05012	−	1.81887i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.00745	−	6.94110i	0.244338	−	0.423206i	−0.717607	−	0.696448i	$-0.754763\pi$
$269$	4.00745	−	6.94110i	0.244338	−	0.423206i	0.961945	+	0.273242i	$0.0880959\pi$
$270$		0			0
$271$	2.96658	+	5.13827i	0.180207	+	0.312128i	0.941951	−	0.335750i	$-0.108990\pi$
$271$	2.96658	+	5.13827i	0.180207	+	0.312128i	−0.761744	+	0.647878i	$0.775657\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.13840			0.309857
$276$		0			0
$277$	−12.0554			−0.724336			−0.362168	−	0.932113i	$-0.617963\pi$
$277$	−12.0554			−0.724336			−0.362168	+	0.932113i	$0.617963\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.73700	+	16.8650i	0.580861	+	1.00608i	0.995378	+	0.0960386i	$0.0306172\pi$
$281$	9.73700	+	16.8650i	0.580861	+	1.00608i	−0.414517	+	0.910042i	$0.636049\pi$
$282$		0			0
$283$	14.3518	+	24.8581i	0.853127	+	1.47766i	0.878372	+	0.477977i	$0.158630\pi$
$283$	14.3518	+	24.8581i	0.853127	+	1.47766i	−0.0252457	+	0.999681i	$0.508037\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.5740	−	1.11448i	−0.624165	−	0.0657854i
$288$		0			0
$289$	−0.536086	−	0.928529i	−0.0315345	−	0.0546193i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.3503	−	19.6593i	0.663090	−	1.14851i	−0.316709	−	0.948523i	$-0.602578\pi$
$293$	11.3503	−	19.6593i	0.663090	−	1.14851i	0.979799	−	0.199983i	$-0.0640888\pi$
$294$		0			0
$295$	12.1616	+	21.0646i	0.708077	+	1.22643i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.04324	−	5.27105i	0.175995	−	0.304833i
$300$		0			0
$301$	−11.3941	+	15.6731i	−0.656746	+	0.903382i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.56381	−	6.17271i	0.204063	−	0.353448i
$306$		0			0
$307$	−27.9486			−1.59511			−0.797555	−	0.603246i	$-0.793874\pi$
$307$	−27.9486			−1.59511			−0.797555	+	0.603246i	$0.793874\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.64908	−	13.2486i	−0.433739	−	0.751259i	0.563452	−	0.826149i	$-0.309473\pi$
$311$	−7.64908	−	13.2486i	−0.433739	−	0.751259i	−0.997192	+	0.0748898i	$0.976139\pi$
$312$		0			0
$313$	−1.67051	+	2.89341i	−0.0944230	+	0.163545i	−0.909368	−	0.415993i	$-0.863434\pi$
$313$	−1.67051	+	2.89341i	−0.0944230	+	0.163545i	0.814945	+	0.579539i	$0.196767\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.50722	+	7.80673i	−0.253151	+	0.438470i	−0.964392	−	0.264479i	$-0.914800\pi$
$317$	−4.50722	+	7.80673i	−0.253151	+	0.438470i	0.711241	+	0.702948i	$0.248133\pi$
$318$		0			0
$319$	2.40549	+	4.16643i	0.134682	+	0.233275i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−31.3246			−1.74295
$324$		0			0
$325$	22.3932	−	38.7862i	1.24215	−	2.15147i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−32.1458	−	3.38809i	−1.77226	−	0.186791i
$330$		0			0
$331$	−9.14760	+	15.8441i	−0.502797	+	0.870870i	0.497197	+	0.867637i	$0.334362\pi$
$331$	−9.14760	+	15.8441i	−0.502797	+	0.870870i	−0.999995	+	0.00323307i	$0.998971\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.8673	−	27.4830i	−0.866924	−	1.50156i
$336$		0			0
$337$	−0.868823	+	1.50485i	−0.0473278	+	0.0819741i	−0.888719	−	0.458453i	$-0.848404\pi$
$337$	−0.868823	+	1.50485i	−0.0473278	+	0.0819741i	0.841391	+	0.540427i	$0.181737\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.950206	−	1.64580i	−0.0514565	−	0.0891253i
$342$		0			0
$343$	17.6088	+	5.73840i	0.950787	+	0.309845i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.546358	−	0.946319i	−0.0293300	−	0.0508011i	0.850988	−	0.525185i	$-0.176004\pi$
$347$	−0.546358	−	0.946319i	−0.0293300	−	0.0508011i	−0.880318	+	0.474384i	$0.842671\pi$
$348$		0			0
$349$	−4.70096	−	8.14231i	−0.251637	−	0.435848i	0.712340	−	0.701835i	$-0.247636\pi$
$349$	−4.70096	−	8.14231i	−0.251637	−	0.435848i	−0.963977	+	0.265987i	$0.914302\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.2910			−1.02676			−0.513378	−	0.858163i	$-0.671606\pi$
$353$	−19.2910			−1.02676			−0.513378	+	0.858163i	$0.671606\pi$
$354$		0			0
$355$	−4.18484			−0.222108
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.94976	+	5.10914i	0.155682	+	0.269650i	0.933307	−	0.359079i	$-0.116909\pi$
$359$	2.94976	+	5.10914i	0.155682	+	0.269650i	−0.777625	+	0.628729i	$0.783576\pi$
$360$		0			0
$361$	−17.6476	+	30.5665i	−0.928820	+	1.60876i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−19.6165	+	33.9768i	−1.02677	+	1.77843i
$366$		0			0
$367$	10.9660			0.572421			0.286210	−	0.958167i	$-0.407604\pi$
$367$	10.9660			0.572421			0.286210	+	0.958167i	$0.407604\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−24.4662	−	2.57867i	−1.27022	−	0.133878i
$372$		0			0
$373$	−31.9381			−1.65369			−0.826847	−	0.562428i	$-0.809868\pi$
$373$	−31.9381			−1.65369			−0.826847	+	0.562428i	$0.809868\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	41.9326			2.15964
$378$		0			0
$379$	−14.4354			−0.741495			−0.370747	−	0.928734i	$-0.620898\pi$
$379$	−14.4354			−0.741495			−0.370747	+	0.928734i	$0.620898\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.7201			−0.905456			−0.452728	−	0.891649i	$-0.649549\pi$
$383$	−17.7201			−0.905456			−0.452728	+	0.891649i	$0.649549\pi$
$384$		0			0
$385$	−2.38494	−	5.36085i	−0.121548	−	0.273214i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.9750			−0.962072			−0.481036	−	0.876701i	$-0.659739\pi$
$389$	−18.9750			−0.962072			−0.481036	+	0.876701i	$0.659739\pi$
$390$		0			0
$391$	−2.46095	+	4.26249i	−0.124455	+	0.215563i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.62124	+	13.2004i	−0.383466	+	0.664183i
$396$		0			0
$397$	−10.5889	−	18.3405i	−0.531440	−	0.920482i	−0.999327	−	0.0366930i	$-0.988318\pi$
$397$	−10.5889	−	18.3405i	−0.531440	−	0.920482i	0.467886	−	0.883789i	$-0.345016\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.7808			−0.738116			−0.369058	−	0.929406i	$-0.620320\pi$
$401$	−14.7808			−0.738116			−0.369058	+	0.929406i	$0.620320\pi$
$402$		0			0
$403$	−16.5640			−0.825114
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.00160993	+	0.00278847i	7.98011e−5	+	0.000138219i
$408$		0			0
$409$	1.86575	+	3.23158i	0.0922554	+	0.159791i	0.908460	−	0.417972i	$-0.137259\pi$
$409$	1.86575	+	3.23158i	0.0922554	+	0.159791i	−0.816204	+	0.577763i	$0.803926\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	10.2916	−	14.1566i	0.506418	−	0.696600i
$414$		0			0
$415$	−22.9469	−	39.7452i	−1.12642	−	1.95102i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.1678	−	24.5393i	0.692142	−	1.19883i	−0.278993	−	0.960293i	$-0.590001\pi$
$419$	14.1678	−	24.5393i	0.692142	−	1.19883i	0.971135	−	0.238532i	$-0.0766661\pi$
$420$		0			0
$421$	−8.09776	−	14.0257i	−0.394661	−	0.683572i	0.598397	−	0.801200i	$-0.295805\pi$
$421$	−8.09776	−	14.0257i	−0.394661	−	0.683572i	−0.993058	+	0.117627i	$0.962471\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−18.1085	+	31.3649i	−0.878392	+	1.52142i
$426$		0			0
$427$	−5.10055	−	0.537585i	−0.246833	−	0.0260156i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13.5528	+	23.4741i	−0.652815	+	1.13071i	0.329622	+	0.944113i	$0.393079\pi$
$431$	−13.5528	+	23.4741i	−0.652815	+	1.13071i	−0.982437	+	0.186595i	$0.940255\pi$
$432$		0			0
$433$	−11.5028			−0.552789			−0.276394	−	0.961044i	$-0.589140\pi$
$433$	−11.5028			−0.552789			−0.276394	+	0.961044i	$0.589140\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.26558	+	7.38819i	0.204050	+	0.353425i
$438$		0			0
$439$	1.43357	−	2.48301i	0.0684205	−	0.118508i	−0.829786	−	0.558082i	$-0.811537\pi$
$439$	1.43357	−	2.48301i	0.0684205	−	0.118508i	0.898206	+	0.439574i	$0.144871\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.42151	+	7.65827i	−0.210072	+	0.363855i	−0.951737	−	0.306915i	$-0.900703\pi$
$443$	−4.42151	+	7.65827i	−0.210072	+	0.363855i	0.741665	+	0.670771i	$0.234037\pi$
$444$		0			0
$445$	15.0555	+	26.0768i	0.713697	+	1.23616i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	40.5033			1.91147			0.955735	−	0.294229i	$-0.0950629\pi$
$449$	40.5033			1.91147			0.955735	+	0.294229i	$0.0950629\pi$
$450$		0			0
$451$	1.21194	−	2.09914i	0.0570680	−	0.0988446i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−50.8590	−	5.36041i	−2.38431	−	0.251300i
$456$		0			0
$457$	−19.4049	+	33.6103i	−0.907723	+	1.57222i	−0.0905025	+	0.995896i	$0.528847\pi$
$457$	−19.4049	+	33.6103i	−0.907723	+	1.57222i	−0.817220	+	0.576326i	$0.804486\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.1328	+	27.9428i	0.751378	+	1.30142i	0.947155	+	0.320776i	$0.103944\pi$
$461$	16.1328	+	27.9428i	0.751378	+	1.30142i	−0.195777	+	0.980648i	$0.562723\pi$
$462$		0			0
$463$	−16.7430	+	28.9997i	−0.778112	+	1.34773i	0.154917	+	0.987927i	$0.450489\pi$
$463$	−16.7430	+	28.9997i	−0.778112	+	1.34773i	−0.933029	+	0.359802i	$0.882844\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.04280	+	8.73438i	0.233353	+	0.404179i	0.958793	−	0.284107i	$-0.0916970\pi$
$467$	5.04280	+	8.73438i	0.233353	+	0.404179i	−0.725440	+	0.688286i	$0.758364\pi$
$468$		0			0
$469$	−13.4275	+	18.4701i	−0.620026	+	0.852872i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.20867	−	3.82552i	−0.101555	−	0.175898i
$474$		0			0
$475$	31.3876	+	54.3649i	1.44016	+	2.49443i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.59519			−0.392724			−0.196362	−	0.980531i	$-0.562913\pi$
$479$	−8.59519			−0.392724			−0.196362	+	0.980531i	$0.562913\pi$
$480$		0			0
$481$	0.0280643			0.00127962
$482$		0			0
$483$		0			0
$484$		0			0
$485$	24.9268	+	43.1745i	1.13187	+	1.96045i
$486$		0			0
$487$	−0.298843	+	0.517612i	−0.0135419	+	0.0234552i	−0.872717	−	0.488227i	$-0.837644\pi$
$487$	−0.298843	+	0.517612i	−0.0135419	+	0.0234552i	0.859175	+	0.511682i	$0.170977\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.1521	+	29.7084i	−0.774066	+	1.34072i	0.161253	+	0.986913i	$0.448447\pi$
$491$	−17.1521	+	29.7084i	−0.774066	+	1.34072i	−0.935318	+	0.353808i	$0.884887\pi$
$492$		0			0
$493$	−33.9092			−1.52720
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.22400	+	2.75129i	0.0549038	+	0.123412i
$498$		0			0
$499$	30.0494			1.34520			0.672598	−	0.740008i	$-0.265178\pi$
$499$	30.0494			1.34520			0.672598	+	0.740008i	$0.265178\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−18.5650			−0.827773			−0.413887	−	0.910328i	$-0.635829\pi$
$503$	−18.5650			−0.827773			−0.413887	+	0.910328i	$0.635829\pi$
$504$		0			0
$505$	56.4013			2.50983
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.01815			0.311074			0.155537	−	0.987830i	$-0.450289\pi$
$509$	7.01815			0.311074			0.155537	+	0.987830i	$0.450289\pi$
$510$		0			0
$511$	28.0752	+	2.95906i	1.24198	+	0.130901i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−18.2397			−0.803738
$516$		0			0
$517$	3.68438	−	6.38154i	0.162039	−	0.280660i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.0742	+	26.1092i	−0.660411	+	1.14387i	0.320096	+	0.947385i	$0.396285\pi$
$521$	−15.0742	+	26.1092i	−0.660411	+	1.14387i	−0.980508	+	0.196481i	$0.937049\pi$
$522$		0			0
$523$	−14.1726	−	24.5476i	−0.619724	−	1.07339i	−0.989536	−	0.144287i	$-0.953911\pi$
$523$	−14.1726	−	24.5476i	−0.619724	−	1.07339i	0.369812	−	0.929107i	$-0.379422\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	13.3947			0.583481
$528$		0			0
$529$	−21.6595			−0.941719
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−10.5633	−	18.2962i	−0.457547	−	0.792495i
$534$		0			0
$535$	−12.5528	−	21.7421i	−0.542704	−	0.939991i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.82689	+	3.13592i	−0.121763	+	0.135074i
$540$		0			0
$541$	−14.5245	−	25.1572i	−0.624458	−	1.08159i	−0.988645	−	0.150268i	$-0.951986\pi$
$541$	−14.5245	−	25.1572i	−0.624458	−	1.08159i	0.364187	−	0.931326i	$-0.381347\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	32.7313	−	56.6922i	1.40205	−	2.42843i
$546$		0			0
$547$	−8.68455	−	15.0421i	−0.371324	−	0.643153i	0.618445	−	0.785828i	$-0.287763\pi$
$547$	−8.68455	−	15.0421i	−0.371324	−	0.643153i	−0.989770	+	0.142675i	$0.954430\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−29.3876	+	50.9008i	−1.25195	+	2.16844i
$552$		0			0
$553$	10.9076	+	1.14963i	0.463837	+	0.0488873i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.5927	+	33.9355i	−0.830169	+	1.43789i	0.0677355	+	0.997703i	$0.478423\pi$
$557$	−19.5927	+	33.9355i	−0.830169	+	1.43789i	−0.897904	+	0.440191i	$0.854911\pi$
$558$		0			0
$559$	−38.5016			−1.62844
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.2039	+	33.2622i	0.809349	+	1.40183i	0.913316	+	0.407252i	$0.133513\pi$
$563$	19.2039	+	33.2622i	0.809349	+	1.40183i	−0.103967	+	0.994581i	$0.533154\pi$
$564$		0			0
$565$	17.0587	−	29.5465i	0.717665	−	1.24303i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.54032	−	7.86407i	0.190340	−	0.329679i	−0.755023	−	0.655698i	$-0.772374\pi$
$569$	4.54032	−	7.86407i	0.190340	−	0.329679i	0.945363	+	0.326020i	$0.105708\pi$
$570$		0			0
$571$	18.5274	+	32.0904i	0.775347	+	1.34294i	0.934599	+	0.355702i	$0.115758\pi$
$571$	18.5274	+	32.0904i	0.775347	+	1.34294i	−0.159253	+	0.987238i	$0.550908\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	9.86358			0.411340
$576$		0			0
$577$	1.04241	−	1.80550i	0.0433960	−	0.0751641i	−0.843512	−	0.537111i	$-0.819516\pi$
$577$	1.04241	−	1.80550i	0.0433960	−	0.0751641i	0.886908	+	0.461947i	$0.152849\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−19.4186	+	26.7110i	−0.805617	+	1.10816i
$582$		0			0
$583$	2.80418	−	4.85699i	0.116137	−	0.201156i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.75919	−	8.24316i	−0.196433	−	0.340232i	0.750936	−	0.660374i	$-0.229602\pi$
$587$	−4.75919	−	8.24316i	−0.196433	−	0.340232i	−0.947369	+	0.320143i	$0.896269\pi$
$588$		0			0
$589$	11.6085	−	20.1066i	0.478322	−	0.828477i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7.63267	−	13.2202i	−0.313436	−	0.542887i	0.665668	−	0.746248i	$-0.268147\pi$
$593$	−7.63267	−	13.2202i	−0.313436	−	0.542887i	−0.979104	+	0.203361i	$0.934814\pi$
$594$		0			0
$595$	41.1276	+	4.33475i	1.68607	+	0.177707i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.0597	−	17.4240i	−0.411030	−	0.711924i	0.583973	−	0.811773i	$-0.301497\pi$
$599$	−10.0597	−	17.4240i	−0.411030	−	0.711924i	−0.995003	+	0.0998490i	$0.968164\pi$
$600$		0			0
$601$	−10.1529	−	17.5854i	−0.414146	−	0.717322i	0.581192	−	0.813766i	$-0.302586\pi$
$601$	−10.1529	−	17.5854i	−0.414146	−	0.717322i	−0.995338	+	0.0964440i	$0.969253\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−39.1080			−1.58997
$606$		0			0
$607$	−2.07298			−0.0841396			−0.0420698	−	0.999115i	$-0.513395\pi$
$607$	−2.07298			−0.0841396			−0.0420698	+	0.999115i	$0.513395\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−32.1132	−	55.6217i	−1.29916	−	2.25021i
$612$		0			0
$613$	−1.10053	+	1.90618i	−0.0444502	+	0.0769900i	−0.887395	−	0.461011i	$-0.847487\pi$
$613$	−1.10053	+	1.90618i	−0.0444502	+	0.0769900i	0.842944	+	0.538001i	$0.180820\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	21.1904	−	36.7029i	0.853095	−	1.47760i	−0.0253061	−	0.999680i	$-0.508056\pi$
$617$	21.1904	−	36.7029i	0.853095	−	1.47760i	0.878401	−	0.477924i	$-0.158611\pi$
$618$		0			0
$619$	13.8707			0.557511			0.278756	−	0.960362i	$-0.410078\pi$
$619$	13.8707			0.557511			0.278756	+	0.960362i	$0.410078\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.7405	−	17.5251i	0.510438	−	0.702129i
$624$		0			0
$625$	4.98282			0.199313
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.0226945			−0.000904889
$630$		0			0
$631$	45.1845			1.79876			0.899382	−	0.437163i	$-0.144017\pi$
$631$	45.1845			1.79876			0.899382	+	0.437163i	$0.144017\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−55.1312			−2.18782
$636$		0			0
$637$	11.3512	+	35.0046i	0.449753	+	1.38693i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	26.5128			1.04719			0.523597	−	0.851966i	$-0.324590\pi$
$641$	26.5128			1.04719			0.523597	+	0.851966i	$0.324590\pi$
$642$		0			0
$643$	24.3184	−	42.1207i	0.959024	−	1.66108i	0.234145	−	0.972202i	$-0.424771\pi$
$643$	24.3184	−	42.1207i	0.959024	−	1.66108i	0.724879	−	0.688876i	$-0.241896\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.7224	−	18.5717i	0.421540	−	0.730128i	−0.574551	−	0.818469i	$-0.694823\pi$
$647$	10.7224	−	18.5717i	0.421540	−	0.730128i	0.996090	+	0.0883409i	$0.0281565\pi$
$648$		0			0
$649$	1.99496	+	3.45537i	0.0783090	+	0.135635i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.48991			0.136571			0.0682854	−	0.997666i	$-0.478247\pi$
$653$	3.48991			0.136571			0.0682854	+	0.997666i	$0.478247\pi$
$654$		0			0
$655$	−83.9529			−3.28031
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.2754	+	31.6539i	0.711908	+	1.23306i	0.964140	+	0.265394i	$0.0855021\pi$
$659$	18.2754	+	31.6539i	0.711908	+	1.23306i	−0.252232	+	0.967667i	$0.581165\pi$
$660$		0			0
$661$	2.78748	+	4.82806i	0.108420	+	0.187790i	0.915131	−	0.403158i	$-0.132087\pi$
$661$	2.78748	+	4.82806i	0.108420	+	0.187790i	−0.806710	+	0.590947i	$0.798754\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	42.1502	−	57.9794i	1.63452	−	2.24835i
$666$		0			0
$667$	4.61753	+	7.99780i	0.178792	+	0.309676i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.584598	−	1.01255i	0.0225682	−	0.0390892i
$672$		0			0
$673$	1.25661	+	2.17652i	0.0484389	+	0.0838986i	0.889228	−	0.457464i	$-0.151242\pi$
$673$	1.25661	+	2.17652i	0.0484389	+	0.0838986i	−0.840789	+	0.541362i	$0.817909\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.8606	+	27.4714i	−0.609574	+	1.05581i	0.381737	+	0.924271i	$0.375326\pi$
$677$	−15.8606	+	27.4714i	−0.609574	+	1.05581i	−0.991311	+	0.131542i	$0.958007\pi$
$678$		0			0
$679$	21.0940	−	29.0157i	0.809514	−	1.11352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.4275	+	37.1136i	−0.819902	+	1.42011i	0.0858521	+	0.996308i	$0.472639\pi$
$683$	−21.4275	+	37.1136i	−0.819902	+	1.42011i	−0.905754	+	0.423804i	$0.860695\pi$
$684$		0			0
$685$	−5.60932			−0.214321
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−24.4413	−	42.3336i	−0.931141	−	1.61278i
$690$		0			0
$691$	7.38187	−	12.7858i	0.280820	−	0.486394i	−0.690767	−	0.723077i	$-0.742727\pi$
$691$	7.38187	−	12.7858i	0.280820	−	0.486394i	0.971587	+	0.236683i	$0.0760604\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.1806	+	21.0975i	−0.462037	+	0.800272i
$696$		0			0
$697$	8.54211	+	14.7954i	0.323556	+	0.560415i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−19.8352			−0.749167			−0.374583	−	0.927193i	$-0.622214\pi$
$701$	−19.8352			−0.749167			−0.374583	+	0.927193i	$0.622214\pi$
$702$		0			0
$703$	−0.0196683	+	0.0340664i	−0.000741802	+	0.00128484i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−16.4964	−	37.0806i	−0.620413	−	1.39456i
$708$		0			0
$709$	−5.10292	+	8.83852i	−0.191644	+	0.331937i	−0.945795	−	0.324763i	$-0.894715\pi$
$709$	−5.10292	+	8.83852i	−0.191644	+	0.331937i	0.754151	+	0.656701i	$0.228049\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.82400	−	3.15926i	−0.0683092	−	0.118315i
$714$		0			0
$715$	5.82919	−	10.0965i	0.217999	−	0.377586i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.2769	−	28.1924i	−0.607025	−	1.05140i	−0.991728	−	0.128358i	$-0.959029\pi$
$719$	−16.2769	−	28.1924i	−0.607025	−	1.05140i	0.384703	−	0.923040i	$-0.374304\pi$
$720$		0			0
$721$	5.33481	+	11.9915i	0.198679	+	0.446588i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	33.9774	+	58.8506i	1.26189	+	2.18566i
$726$		0			0
$727$	−7.65095	−	13.2518i	−0.283758	−	0.491483i	0.688549	−	0.725190i	$-0.258248\pi$
$727$	−7.65095	−	13.2518i	−0.283758	−	0.491483i	−0.972307	+	0.233706i	$0.924915\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	31.1347			1.15156
$732$		0			0
$733$	−8.69354			−0.321103			−0.160552	−	0.987027i	$-0.551327\pi$
$733$	−8.69354			−0.321103			−0.160552	+	0.987027i	$0.551327\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.60283	−	4.50823i	−0.0958764	−	0.166063i
$738$		0			0
$739$	−6.61922	+	11.4648i	−0.243492	+	0.421740i	−0.961707	−	0.274081i	$-0.911626\pi$
$739$	−6.61922	+	11.4648i	−0.243492	+	0.421740i	0.718215	+	0.695822i	$0.244960\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.7693	+	18.6530i	−0.395089	+	0.684314i	−0.993112	−	0.117165i	$-0.962619\pi$
$743$	−10.7693	+	18.6530i	−0.395089	+	0.684314i	0.598024	+	0.801478i	$0.295953\pi$
$744$		0			0
$745$	65.3561			2.39446
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−10.6227	+	14.6119i	−0.388143	+	0.533908i
$750$		0			0
$751$	39.6483			1.44679			0.723393	−	0.690437i	$-0.242582\pi$
$751$	39.6483			1.44679			0.723393	+	0.690437i	$0.242582\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17.0468			0.620395
$756$		0			0
$757$	13.0719			0.475108			0.237554	−	0.971374i	$-0.423654\pi$
$757$	13.0719			0.475108			0.237554	+	0.971374i	$0.423654\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−18.3429			−0.664930			−0.332465	−	0.943116i	$-0.607880\pi$
$761$	−18.3429			−0.664930			−0.332465	+	0.943116i	$0.607880\pi$
$762$		0			0
$763$	−46.8452	−	4.93737i	−1.69591	−	0.178745i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	34.7763			1.25570
$768$		0			0
$769$	7.46351	−	12.9272i	0.269141	−	0.466166i	−0.699499	−	0.714633i	$-0.746594\pi$
$769$	7.46351	−	12.9272i	0.269141	−	0.466166i	0.968640	+	0.248467i	$0.0799269\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.7818	+	25.6029i	−0.531666	+	0.920873i	0.467651	+	0.883913i	$0.345100\pi$
$773$	−14.7818	+	25.6029i	−0.531666	+	0.920873i	−0.999317	+	0.0369592i	$0.988233\pi$
$774$		0			0
$775$	−13.4216	−	23.2469i	−0.482118	−	0.835054i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	29.6122			1.06097
$780$		0			0
$781$	−0.686470			−0.0245638
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.27495	−	10.8685i	−0.223962	−	0.387914i
$786$		0			0
$787$	4.66430	+	8.07880i	0.166264	+	0.287978i	0.937104	−	0.349052i	$-0.113496\pi$
$787$	4.66430	+	8.07880i	0.166264	+	0.287978i	−0.770839	+	0.637030i	$0.780163\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−24.4145	−	2.57323i	−0.868080	−	0.0914935i
$792$		0			0
$793$	−5.09537	−	8.82545i	−0.180942	−	0.313401i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.9334	−	20.6692i	0.422701	−	0.732140i	−0.573502	−	0.819204i	$-0.694415\pi$
$797$	11.9334	−	20.6692i	0.422701	−	0.732140i	0.996203	+	0.0870647i	$0.0277487\pi$
$798$		0			0
$799$	25.9686	+	44.9790i	0.918705	+	1.59124i
$800$		0			0
$801$		0			0
$802$		0			0
$803$

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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.l (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+3.67687 q^{5} +(-1.07542 - 2.41733i) q^{7} +O(q^{10})\)	\(q+3.67687 q^{5} +(-1.07542 - 2.41733i) q^{7} +0.603143 q^{11} +(2.62851 - 4.55271i) q^{13} +(-2.12557 + 3.68159i) q^{17} +(3.68426 + 6.38133i) q^{19} +1.15778 q^{23} +8.51937 q^{25} +(3.98826 + 6.90786i) q^{29} +(-1.57542 - 2.72871i) q^{31} +(-3.95419 - 8.88819i) q^{35} +(0.00266923 + 0.00462323i) q^{37} +(2.00937 - 3.48033i) q^{41} +(-3.66193 - 6.34264i) q^{43} +(6.10863 - 10.5805i) q^{47} +(-4.68693 + 5.19930i) q^{49} +(4.64928 - 8.05279i) q^{53} +2.21768 q^{55} +(3.30760 + 5.72894i) q^{59} +(0.969252 - 1.67879i) q^{61} +(9.66468 - 16.7397i) q^{65} +(-4.31544 - 7.47456i) q^{67} -1.13815 q^{71} +(-5.33511 + 9.24068i) q^{73} +(-0.648634 - 1.45799i) q^{77} +(-2.07275 + 3.59011i) q^{79} +(-6.24088 - 10.8095i) q^{83} +(-7.81544 + 13.5367i) q^{85} +(4.09464 + 7.09212i) q^{89} +(-13.8321 - 1.45787i) q^{91} +(13.5466 + 23.4633i) q^{95} +(6.77935 + 11.7422i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 10 q^{13} + 8 q^{19} - 8 q^{31} - 4 q^{37} - 10 q^{43} - 20 q^{49} - 32 q^{55} + 28 q^{61} + 18 q^{67} - 20 q^{79} - 38 q^{85} - 2 q^{91} + 42 q^{97}+O(q^{100}) \) 16 * q + 10 * q^13 + 8 * q^19 - 8 * q^31 - 4 * q^37 - 10 * q^43 - 20 * q^49 - 32 * q^55 + 28 * q^61 + 18 * q^67 - 20 * q^79 - 38 * q^85 - 2 * q^91 + 42 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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