LMFDB - Embedded newform 2352.2.q.bb.1537.1

Newspace parameters

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.q (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
Defining polynomial:	$x^{4} + 2 x^{2} + 4$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1537.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2352.1537
Dual form			2352.2.q.bb.961.1

$q$-expansion

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(0.292893 + 0.507306i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(0.292893 + 0.507306i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} -5.41421 q^{13} -0.585786 q^{15} +(3.12132 - 5.40629i) q^{17} +(-1.41421 - 2.44949i) q^{19} +(1.82843 + 3.16693i) q^{23} +(2.32843 - 4.03295i) q^{25} +1.00000 q^{27} -1.17157 q^{29} +(3.41421 - 5.91359i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(2.00000 + 3.46410i) q^{37} +(2.70711 - 4.68885i) q^{39} +2.24264 q^{41} +5.65685 q^{43} +(0.292893 - 0.507306i) q^{45} +(-1.41421 - 2.44949i) q^{47} +(3.12132 + 5.40629i) q^{51} +(1.00000 - 1.73205i) q^{53} -1.17157 q^{55} +2.82843 q^{57} +(3.41421 - 5.91359i) q^{59} +(1.87868 + 3.25397i) q^{61} +(-1.58579 - 2.74666i) q^{65} +(2.82843 - 4.89898i) q^{67} -3.65685 q^{69} +13.3137 q^{71} +(-2.94975 + 5.10911i) q^{73} +(2.32843 + 4.03295i) q^{75} +(1.17157 + 2.02922i) q^{79} +(-0.500000 + 0.866025i) q^{81} -15.3137 q^{83} +3.65685 q^{85} +(0.585786 - 1.01461i) q^{87} +(-2.87868 - 4.98602i) q^{89} +(3.41421 + 5.91359i) q^{93} +(0.828427 - 1.43488i) q^{95} -5.41421 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4q - 2q^{3} + 4q^{5} - 2q^{9} + O(q^{10}) $	$ 4q - 2q^{3} + 4q^{5} - 2q^{9} - 4q^{11} - 16q^{13} - 8q^{15} + 4q^{17} - 4q^{23} - 2q^{25} + 4q^{27} - 16q^{29} + 8q^{31} - 4q^{33} + 8q^{37} + 8q^{39} - 8q^{41} + 4q^{45} + 4q^{51} + 4q^{53} - 16q^{55} + 8q^{59} + 16q^{61} - 12q^{65} + 8q^{69} + 8q^{71} + 8q^{73} - 2q^{75} + 16q^{79} - 2q^{81} - 16q^{83} - 8q^{85} + 8q^{87} - 20q^{89} + 8q^{93} - 8q^{95} - 16q^{97} + 8q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$	0.292893	+	0.507306i	0.130986	+	0.226874i	0.924057	−	0.382255i	$-0.124852\pi$
$5$	0.292893	+	0.507306i	0.130986	+	0.226874i	−0.793071	+	0.609129i	$0.791519\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	−5.41421			−1.50163			−0.750816	−	0.660511i	$-0.770340\pi$
$13$	−5.41421			−1.50163			−0.750816	+	0.660511i	$0.770340\pi$
$14$		0			0
$15$	−0.585786			−0.151249
$16$		0			0
$17$	3.12132	−	5.40629i	0.757031	−	1.31122i	−0.187327	−	0.982298i	$-0.559982\pi$
$17$	3.12132	−	5.40629i	0.757031	−	1.31122i	0.944358	−	0.328919i	$-0.106684\pi$
$18$		0			0
$19$	−1.41421	−	2.44949i	−0.324443	−	0.561951i	0.656957	−	0.753928i	$-0.271843\pi$
$19$	−1.41421	−	2.44949i	−0.324443	−	0.561951i	−0.981399	+	0.191977i	$0.938510\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.82843	+	3.16693i	0.381253	+	0.660350i	0.991242	−	0.132060i	$-0.0421592\pi$
$23$	1.82843	+	3.16693i	0.381253	+	0.660350i	−0.609988	+	0.792410i	$0.708826\pi$
$24$		0			0
$25$	2.32843	−	4.03295i	0.465685	−	0.806591i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−1.17157			−0.217556			−0.108778	−	0.994066i	$-0.534694\pi$
$29$	−1.17157			−0.217556			−0.108778	+	0.994066i	$0.534694\pi$
$30$		0			0
$31$	3.41421	−	5.91359i	0.613211	−	1.06211i	−0.377485	−	0.926016i	$-0.623211\pi$
$31$	3.41421	−	5.91359i	0.613211	−	1.06211i	0.990696	−	0.136097i	$-0.0434557\pi$
$32$		0			0
$33$	−1.00000	−	1.73205i	−0.174078	−	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000	+	3.46410i	0.328798	+	0.569495i	0.982274	−	0.187453i	$-0.0600231\pi$
$37$	2.00000	+	3.46410i	0.328798	+	0.569495i	−0.653476	+	0.756948i	$0.726690\pi$
$38$		0			0
$39$	2.70711	−	4.68885i	0.433484	−	0.750816i
$40$		0			0
$41$	2.24264			0.350242			0.175121	−	0.984547i	$-0.443968\pi$
$41$	2.24264			0.350242			0.175121	+	0.984547i	$0.443968\pi$
$42$		0			0
$43$	5.65685			0.862662			0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685			0.862662			0.431331	+	0.902194i	$0.358044\pi$
$44$		0			0
$45$	0.292893	−	0.507306i	0.0436619	−	0.0756247i
$46$		0			0
$47$	−1.41421	−	2.44949i	−0.206284	−	0.357295i	0.744257	−	0.667893i	$-0.232804\pi$
$47$	−1.41421	−	2.44949i	−0.206284	−	0.357295i	−0.950541	+	0.310599i	$0.899470\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	3.12132	+	5.40629i	0.437072	+	0.757031i
$52$		0			0
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	−0.789136	−	0.614218i	$-0.789471\pi$
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	0.926497	+	0.376303i	$0.122805\pi$
$54$		0			0
$55$	−1.17157			−0.157975
$56$		0			0
$57$	2.82843			0.374634
$58$		0			0
$59$	3.41421	−	5.91359i	0.444493	−	0.769884i	−0.553524	−	0.832833i	$-0.686717\pi$
$59$	3.41421	−	5.91359i	0.444493	−	0.769884i	0.998017	+	0.0629492i	$0.0200506\pi$
$60$		0			0
$61$	1.87868	+	3.25397i	0.240540	+	0.416628i	0.960868	−	0.277006i	$-0.0893421\pi$
$61$	1.87868	+	3.25397i	0.240540	+	0.416628i	−0.720328	+	0.693634i	$0.756009\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.58579	−	2.74666i	−0.196693	−	0.340682i
$66$		0			0
$67$	2.82843	−	4.89898i	0.345547	−	0.598506i	−0.639906	−	0.768453i	$-0.721027\pi$
$67$	2.82843	−	4.89898i	0.345547	−	0.598506i	0.985453	+	0.169948i	$0.0543599\pi$
$68$		0			0
$69$	−3.65685			−0.440234
$70$		0			0
$71$	13.3137			1.58005			0.790023	−	0.613077i	$-0.210068\pi$
$71$	13.3137			1.58005			0.790023	+	0.613077i	$0.210068\pi$
$72$		0			0
$73$	−2.94975	+	5.10911i	−0.345242	+	0.597976i	−0.985398	−	0.170269i	$-0.945536\pi$
$73$	−2.94975	+	5.10911i	−0.345242	+	0.597976i	0.640156	+	0.768245i	$0.278870\pi$
$74$		0			0
$75$	2.32843	+	4.03295i	0.268864	+	0.465685i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.17157	+	2.02922i	0.131812	+	0.228306i	0.924375	−	0.381485i	$-0.124587\pi$
$79$	1.17157	+	2.02922i	0.131812	+	0.228306i	−0.792563	+	0.609790i	$0.791254\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−15.3137			−1.68090			−0.840449	−	0.541891i	$-0.817709\pi$
$83$	−15.3137			−1.68090			−0.840449	+	0.541891i	$0.817709\pi$
$84$		0			0
$85$	3.65685			0.396642
$86$		0			0
$87$	0.585786	−	1.01461i	0.0628029	−	0.108778i
$88$		0			0
$89$	−2.87868	−	4.98602i	−0.305139	−	0.528517i	0.672153	−	0.740412i	$-0.265370\pi$
$89$	−2.87868	−	4.98602i	−0.305139	−	0.528517i	−0.977292	+	0.211895i	$0.932036\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	3.41421	+	5.91359i	0.354037	+	0.613211i
$94$		0			0
$95$	0.828427	−	1.43488i	0.0849948	−	0.147215i
$96$		0			0
$97$	−5.41421			−0.549730			−0.274865	−	0.961483i	$-0.588633\pi$
$97$	−5.41421			−0.549730			−0.274865	+	0.961483i	$0.588633\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	8.53553	−	14.7840i	0.849317	−	1.47106i	−0.0325010	−	0.999472i	$-0.510347\pi$
$101$	8.53553	−	14.7840i	0.849317	−	1.47106i	0.881818	−	0.471589i	$-0.156319\pi$
$102$		0			0
$103$	6.24264	+	10.8126i	0.615106	+	1.06539i	0.990366	+	0.138475i	$0.0442200\pi$
$103$	6.24264	+	10.8126i	0.615106	+	1.06539i	−0.375260	+	0.926919i	$0.622447\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.82843	−	10.0951i	−0.563455	−	0.975933i	−0.997192	−	0.0748933i	$-0.976138\pi$
$107$	−5.82843	−	10.0951i	−0.563455	−	0.975933i	0.433736	−	0.901040i	$-0.357195\pi$
$108$		0			0
$109$	−2.82843	+	4.89898i	−0.270914	+	0.469237i	−0.969096	−	0.246683i	$-0.920659\pi$
$109$	−2.82843	+	4.89898i	−0.270914	+	0.469237i	0.698182	+	0.715920i	$0.253993\pi$
$110$		0			0
$111$	−4.00000			−0.379663
$112$		0			0
$113$	17.3137			1.62874			0.814368	−	0.580348i	$-0.197084\pi$
$113$	17.3137			1.62874			0.814368	+	0.580348i	$0.197084\pi$
$114$		0			0
$115$	−1.07107	+	1.85514i	−0.0998776	+	0.172993i
$116$		0			0
$117$	2.70711	+	4.68885i	0.250272	+	0.433484i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$	−1.12132	+	1.94218i	−0.101106	+	0.175121i
$124$		0			0
$125$	5.65685			0.505964
$126$		0			0
$127$	−9.65685			−0.856907			−0.428454	−	0.903564i	$-0.640941\pi$
$127$	−9.65685			−0.856907			−0.428454	+	0.903564i	$0.640941\pi$
$128$		0			0
$129$	−2.82843	+	4.89898i	−0.249029	+	0.431331i
$130$		0			0
$131$	−3.65685	−	6.33386i	−0.319501	−	0.553392i	0.660883	−	0.750489i	$-0.270182\pi$
$131$	−3.65685	−	6.33386i	−0.319501	−	0.553392i	−0.980384	+	0.197097i	$0.936849\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	0.292893	+	0.507306i	0.0252082	+	0.0436619i
$136$		0			0
$137$	7.07107	−	12.2474i	0.604122	−	1.04637i	−0.388067	−	0.921631i	$-0.626857\pi$
$137$	7.07107	−	12.2474i	0.604122	−	1.04637i	0.992190	−	0.124739i	$-0.0398094\pi$
$138$		0			0
$139$	−6.34315			−0.538019			−0.269009	−	0.963138i	$-0.586696\pi$
$139$	−6.34315			−0.538019			−0.269009	+	0.963138i	$0.586696\pi$
$140$		0			0
$141$	2.82843			0.238197
$142$		0			0
$143$	5.41421	−	9.37769i	0.452759	−	0.784202i
$144$		0			0
$145$	−0.343146	−	0.594346i	−0.0284967	−	0.0493577i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.65685	+	4.60181i	0.217658	+	0.376995i	0.954092	−	0.299515i	$-0.0968249\pi$
$149$	2.65685	+	4.60181i	0.217658	+	0.376995i	−0.736434	+	0.676510i	$0.763492\pi$
$150$		0			0
$151$	6.00000	−	10.3923i	0.488273	−	0.845714i	−0.511636	−	0.859202i	$-0.670960\pi$
$151$	6.00000	−	10.3923i	0.488273	−	0.845714i	0.999909	+	0.0134886i	$0.00429367\pi$
$152$		0			0
$153$	−6.24264			−0.504688
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	10.1213	−	17.5306i	0.807769	−	1.39910i	−0.106636	−	0.994298i	$-0.534008\pi$
$157$	10.1213	−	17.5306i	0.807769	−	1.39910i	0.914406	−	0.404799i	$-0.132659\pi$
$158$		0			0
$159$	1.00000	+	1.73205i	0.0793052	+	0.137361i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.65685	+	9.79796i	0.443079	+	0.767435i	0.997916	−	0.0645236i	$-0.0205528\pi$
$163$	5.65685	+	9.79796i	0.443079	+	0.767435i	−0.554837	+	0.831959i	$0.687219\pi$
$164$		0			0
$165$	0.585786	−	1.01461i	0.0456034	−	0.0789874i
$166$		0			0
$167$	19.7990			1.53209			0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990			1.53209			0.766046	+	0.642786i	$0.222221\pi$
$168$		0			0
$169$	16.3137			1.25490
$170$		0			0
$171$	−1.41421	+	2.44949i	−0.108148	+	0.187317i
$172$		0			0
$173$	3.46447	+	6.00063i	0.263398	+	0.456220i	0.967143	−	0.254234i	$-0.0818233\pi$
$173$	3.46447	+	6.00063i	0.263398	+	0.456220i	−0.703744	+	0.710453i	$0.748490\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	3.41421	+	5.91359i	0.256628	+	0.444493i
$178$		0			0
$179$	−4.17157	+	7.22538i	−0.311798	+	0.540050i	−0.978752	−	0.205049i	$-0.934265\pi$
$179$	−4.17157	+	7.22538i	−0.311798	+	0.540050i	0.666954	+	0.745099i	$0.267598\pi$
$180$		0			0
$181$	5.41421			0.402435			0.201218	−	0.979547i	$-0.435510\pi$
$181$	5.41421			0.402435			0.201218	+	0.979547i	$0.435510\pi$
$182$		0			0
$183$	−3.75736			−0.277752
$184$		0			0
$185$	−1.17157	+	2.02922i	−0.0861358	+	0.149191i
$186$		0			0
$187$	6.24264	+	10.8126i	0.456507	+	0.790693i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	−0.982828	−	0.184525i	$-0.940925\pi$
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	0.331611	−	0.943416i	$-0.392408\pi$
$192$		0			0
$193$	8.65685	−	14.9941i	0.623134	−	1.07930i	−0.365765	−	0.930707i	$-0.619192\pi$
$193$	8.65685	−	14.9941i	0.623134	−	1.07930i	0.988899	−	0.148592i	$-0.0474742\pi$
$194$		0			0
$195$	3.17157			0.227121
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	−5.17157	+	8.95743i	−0.366603	+	0.634975i	−0.989032	−	0.147701i	$-0.952813\pi$
$199$	−5.17157	+	8.95743i	−0.366603	+	0.634975i	0.622429	+	0.782676i	$0.286146\pi$
$200$		0			0
$201$	2.82843	+	4.89898i	0.199502	+	0.345547i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.656854	+	1.13770i	0.0458767	+	0.0794608i
$206$		0			0
$207$	1.82843	−	3.16693i	0.127084	−	0.220117i
$208$		0			0
$209$	5.65685			0.391293
$210$		0			0
$211$	20.9706			1.44367			0.721837	−	0.692064i	$-0.243298\pi$
$211$	20.9706			1.44367			0.721837	+	0.692064i	$0.243298\pi$
$212$		0			0
$213$	−6.65685	+	11.5300i	−0.456120	+	0.790023i
$214$		0			0
$215$	1.65685	+	2.86976i	0.112997	+	0.195716i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−2.94975	−	5.10911i	−0.199325	−	0.345242i
$220$		0			0
$221$	−16.8995	+	29.2708i	−1.13678	+	1.96897i
$222$		0			0
$223$	−8.97056			−0.600713			−0.300357	−	0.953827i	$-0.597106\pi$
$223$	−8.97056			−0.600713			−0.300357	+	0.953827i	$0.597106\pi$
$224$		0			0
$225$	−4.65685			−0.310457
$226$		0			0
$227$	7.89949	−	13.6823i	0.524308	−	0.908128i	−0.475292	−	0.879828i	$-0.657657\pi$
$227$	7.89949	−	13.6823i	0.524308	−	0.908128i	0.999599	−	0.0282996i	$-0.00900924\pi$
$228$		0			0
$229$	4.12132	+	7.13834i	0.272345	+	0.471715i	0.969462	−	0.245243i	$-0.0788676\pi$
$229$	4.12132	+	7.13834i	0.272345	+	0.471715i	−0.697117	+	0.716957i	$0.745534\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.0711	−	19.1757i	−0.725290	−	1.25624i	−0.958855	−	0.283898i	$-0.908372\pi$
$233$	−11.0711	−	19.1757i	−0.725290	−	1.25624i	0.233565	−	0.972341i	$-0.424961\pi$
$234$		0			0
$235$	0.828427	−	1.43488i	0.0540406	−	0.0936011i
$236$		0			0
$237$	−2.34315			−0.152204
$238$		0			0
$239$	4.34315			0.280935			0.140467	−	0.990085i	$-0.455139\pi$
$239$	4.34315			0.280935			0.140467	+	0.990085i	$0.455139\pi$
$240$		0			0
$241$	−3.87868	+	6.71807i	−0.249848	+	0.432749i	−0.963483	−	0.267768i	$-0.913714\pi$
$241$	−3.87868	+	6.71807i	−0.249848	+	0.432749i	0.713636	+	0.700517i	$0.247047\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	7.65685	+	13.2621i	0.487194	+	0.843845i
$248$		0			0
$249$	7.65685	−	13.2621i	0.485233	−	0.840449i
$250$		0			0
$251$	4.48528			0.283108			0.141554	−	0.989931i	$-0.454790\pi$
$251$	4.48528			0.283108			0.141554	+	0.989931i	$0.454790\pi$
$252$		0			0
$253$	−7.31371			−0.459809
$254$		0			0
$255$	−1.82843	+	3.16693i	−0.114501	+	0.198321i
$256$		0			0
$257$	−9.60660	−	16.6391i	−0.599243	−	1.03792i	−0.992933	−	0.118677i	$-0.962135\pi$
$257$	−9.60660	−	16.6391i	−0.599243	−	1.03792i	0.393690	−	0.919243i	$-0.371198\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.585786	+	1.01461i	0.0362593	+	0.0628029i
$262$		0			0
$263$	−8.65685	+	14.9941i	−0.533805	+	0.924577i	0.465416	+	0.885092i	$0.345905\pi$
$263$	−8.65685	+	14.9941i	−0.533805	+	0.924577i	−0.999220	+	0.0394843i	$0.987428\pi$
$264$		0			0
$265$	1.17157			0.0719691
$266$		0			0
$267$	5.75736			0.352345
$268$		0			0
$269$	5.36396	−	9.29065i	0.327046	−	0.566461i	−0.654878	−	0.755735i	$-0.727280\pi$
$269$	5.36396	−	9.29065i	0.327046	−	0.566461i	0.981924	+	0.189274i	$0.0606133\pi$
$270$		0			0
$271$	−9.07107	−	15.7116i	−0.551028	−	0.954409i	−0.998201	−	0.0599610i	$-0.980902\pi$
$271$	−9.07107	−	15.7116i	−0.551028	−	0.954409i	0.447173	−	0.894448i	$-0.352431\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.65685	+	8.06591i	0.280819	+	0.486393i
$276$		0			0
$277$	−6.65685	+	11.5300i	−0.399972	+	0.692771i	−0.993722	−	0.111878i	$-0.964313\pi$
$277$	−6.65685	+	11.5300i	−0.399972	+	0.692771i	0.593750	+	0.804649i	$0.297647\pi$
$278$		0			0
$279$	−6.82843			−0.408807
$280$		0			0
$281$	−16.4853			−0.983429			−0.491715	−	0.870756i	$-0.663630\pi$
$281$	−16.4853			−0.983429			−0.491715	+	0.870756i	$0.663630\pi$
$282$		0			0
$283$	−4.24264	+	7.34847i	−0.252199	+	0.436821i	−0.964131	−	0.265427i	$-0.914487\pi$
$283$	−4.24264	+	7.34847i	−0.252199	+	0.436821i	0.711932	+	0.702248i	$0.247820\pi$
$284$		0			0
$285$	0.828427	+	1.43488i	0.0490718	+	0.0849948i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−10.9853	−	19.0271i	−0.646193	−	1.11924i
$290$		0			0
$291$	2.70711	−	4.68885i	0.158693	−	0.274865i
$292$		0			0
$293$	−19.4142			−1.13419			−0.567095	−	0.823652i	$-0.691933\pi$
$293$	−19.4142			−1.13419			−0.567095	+	0.823652i	$0.691933\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$	−1.00000	+	1.73205i	−0.0580259	+	0.100504i
$298$		0			0
$299$	−9.89949	−	17.1464i	−0.572503	−	0.991604i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	8.53553	+	14.7840i	0.490354	+	0.849317i
$304$		0			0
$305$	−1.10051	+	1.90613i	−0.0630147	+	0.109145i
$306$		0			0
$307$	1.85786			0.106034			0.0530170	−	0.998594i	$-0.483116\pi$
$307$	1.85786			0.106034			0.0530170	+	0.998594i	$0.483116\pi$
$308$		0			0
$309$	−12.4853			−0.710263
$310$		0			0
$311$	11.0711	−	19.1757i	0.627783	−	1.08735i	−0.360213	−	0.932870i	$-0.617296\pi$
$311$	11.0711	−	19.1757i	0.627783	−	1.08735i	0.987996	−	0.154481i	$-0.0493707\pi$
$312$		0			0
$313$	−8.94975	−	15.5014i	−0.505870	−	0.876192i	−0.999977	−	0.00679098i	$-0.997838\pi$
$313$	−8.94975	−	15.5014i	−0.505870	−	0.876192i	0.494107	−	0.869401i	$-0.335495\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.00000	−	8.66025i	−0.280828	−	0.486408i	0.690761	−	0.723083i	$-0.257276\pi$
$317$	−5.00000	−	8.66025i	−0.280828	−	0.486408i	−0.971589	+	0.236675i	$0.923942\pi$
$318$		0			0
$319$	1.17157	−	2.02922i	0.0655955	−	0.113615i
$320$		0			0
$321$	11.6569			0.650622
$322$		0			0
$323$	−17.6569			−0.982454
$324$		0			0
$325$	−12.6066	+	21.8353i	−0.699288	+	1.21120i
$326$		0			0
$327$	−2.82843	−	4.89898i	−0.156412	−	0.270914i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.00000	−	3.46410i	−0.109930	−	0.190404i	0.805812	−	0.592172i	$-0.201729\pi$
$331$	−2.00000	−	3.46410i	−0.109930	−	0.190404i	−0.915742	+	0.401768i	$0.868396\pi$
$332$		0			0
$333$	2.00000	−	3.46410i	0.109599	−	0.189832i
$334$		0			0
$335$	3.31371			0.181047
$336$		0			0
$337$	−18.3431			−0.999215			−0.499607	−	0.866252i	$-0.666522\pi$
$337$	−18.3431			−0.999215			−0.499607	+	0.866252i	$0.666522\pi$
$338$		0			0
$339$	−8.65685	+	14.9941i	−0.470176	+	0.814368i
$340$		0			0
$341$	6.82843	+	11.8272i	0.369780	+	0.640478i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1.07107	−	1.85514i	−0.0576644	−	0.0998776i
$346$		0			0
$347$	5.34315	−	9.25460i	0.286835	−	0.496813i	−0.686217	−	0.727396i	$-0.740730\pi$
$347$	5.34315	−	9.25460i	0.286835	−	0.496813i	0.973053	+	0.230584i	$0.0740635\pi$
$348$		0			0
$349$	−9.89949			−0.529908			−0.264954	−	0.964261i	$-0.585357\pi$
$349$	−9.89949			−0.529908			−0.264954	+	0.964261i	$0.585357\pi$
$350$		0			0
$351$	−5.41421			−0.288989
$352$		0			0
$353$	5.36396	−	9.29065i	0.285495	−	0.494492i	−0.687234	−	0.726436i	$-0.741175\pi$
$353$	5.36396	−	9.29065i	0.285495	−	0.494492i	0.972729	+	0.231944i	$0.0745087\pi$
$354$		0			0
$355$	3.89949	+	6.75412i	0.206964	+	0.358472i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5.82843	−	10.0951i	−0.307613	−	0.532801i	0.670227	−	0.742156i	$-0.266197\pi$
$359$	−5.82843	−	10.0951i	−0.307613	−	0.532801i	−0.977840	+	0.209355i	$0.932863\pi$
$360$		0			0
$361$	5.50000	−	9.52628i	0.289474	−	0.501383i
$362$		0			0
$363$	−7.00000			−0.367405
$364$		0			0
$365$	−3.45584			−0.180887
$366$		0			0
$367$	−9.65685	+	16.7262i	−0.504084	+	0.873099i	0.495905	+	0.868377i	$0.334836\pi$
$367$	−9.65685	+	16.7262i	−0.504084	+	0.873099i	−0.999989	+	0.00472187i	$0.998497\pi$
$368$		0			0
$369$	−1.12132	−	1.94218i	−0.0583736	−	0.101106i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	16.6569	+	28.8505i	0.862459	+	1.49382i	0.869548	+	0.493848i	$0.164410\pi$
$373$	16.6569	+	28.8505i	0.862459	+	1.49382i	−0.00708885	+	0.999975i	$0.502256\pi$
$374$		0			0
$375$	−2.82843	+	4.89898i	−0.146059	+	0.252982i
$376$		0			0
$377$	6.34315			0.326689
$378$		0			0
$379$	−31.3137			−1.60848			−0.804239	−	0.594307i	$-0.797427\pi$
$379$	−31.3137			−1.60848			−0.804239	+	0.594307i	$0.797427\pi$
$380$		0			0
$381$	4.82843	−	8.36308i	0.247368	−	0.428454i
$382$		0			0
$383$	14.8284	+	25.6836i	0.757697	+	1.31237i	0.944022	+	0.329882i	$0.107009\pi$
$383$	14.8284	+	25.6836i	0.757697	+	1.31237i	−0.186325	+	0.982488i	$0.559658\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.82843	−	4.89898i	−0.143777	−	0.249029i
$388$		0			0
$389$	−5.07107	+	8.78335i	−0.257113	+	0.445333i	−0.965467	−	0.260524i	$-0.916105\pi$
$389$	−5.07107	+	8.78335i	−0.257113	+	0.445333i	0.708354	+	0.705857i	$0.249438\pi$
$390$		0			0
$391$	22.8284			1.15448
$392$		0			0
$393$	7.31371			0.368928
$394$		0			0
$395$	−0.686292	+	1.18869i	−0.0345311	+	0.0598096i
$396$		0			0
$397$	17.1924	+	29.7781i	0.862861	+	1.49452i	0.869155	+	0.494539i	$0.164663\pi$
$397$	17.1924	+	29.7781i	0.862861	+	1.49452i	−0.00629405	+	0.999980i	$0.502003\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.0711	−	19.1757i	−0.552863	−	0.957586i	−0.998066	−	0.0621570i	$-0.980202\pi$
$401$	−11.0711	−	19.1757i	−0.552863	−	0.957586i	0.445204	−	0.895429i	$-0.353131\pi$
$402$		0			0
$403$	−18.4853	+	32.0174i	−0.920817	+	1.59490i
$404$		0			0
$405$	−0.585786			−0.0291080
$406$		0			0
$407$	−8.00000			−0.396545
$408$		0			0
$409$	−9.29289	+	16.0958i	−0.459504	+	0.795884i	−0.998935	−	0.0461457i	$-0.985306\pi$
$409$	−9.29289	+	16.0958i	−0.459504	+	0.795884i	0.539431	+	0.842030i	$0.318639\pi$
$410$		0			0
$411$	7.07107	+	12.2474i	0.348790	+	0.604122i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.48528	−	7.76874i	−0.220174	−	0.381352i
$416$		0			0
$417$	3.17157	−	5.49333i	0.155313	−	0.269009i
$418$		0			0
$419$	38.8284			1.89689			0.948446	−	0.316938i	$-0.102655\pi$
$419$	38.8284			1.89689			0.948446	+	0.316938i	$0.102655\pi$
$420$		0			0
$421$	−28.6274			−1.39521			−0.697607	−	0.716480i	$-0.745752\pi$
$421$	−28.6274			−1.39521			−0.697607	+	0.716480i	$0.745752\pi$
$422$		0			0
$423$	−1.41421	+	2.44949i	−0.0687614	+	0.119098i
$424$		0			0
$425$	−14.5355	−	25.1763i	−0.705077	−	1.22123i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	5.41421	+	9.37769i	0.261401	+	0.452759i
$430$		0			0
$431$	3.48528	−	6.03668i	0.167880	−	0.290777i	−0.769794	−	0.638292i	$-0.779641\pi$
$431$	3.48528	−	6.03668i	0.167880	−	0.290777i	0.937674	+	0.347515i	$0.112975\pi$
$432$		0			0
$433$	11.7574			0.565023			0.282511	−	0.959264i	$-0.408833\pi$
$433$	11.7574			0.565023			0.282511	+	0.959264i	$0.408833\pi$
$434$		0			0
$435$	0.686292			0.0329052
$436$		0			0
$437$	5.17157	−	8.95743i	0.247390	−	0.428492i
$438$		0			0
$439$	−17.6569	−	30.5826i	−0.842716	−	1.45963i	−0.887590	−	0.460634i	$-0.847622\pi$
$439$	−17.6569	−	30.5826i	−0.842716	−	1.45963i	0.0448746	−	0.998993i	$-0.485711\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.514719	+	0.891519i	0.0244550	+	0.0423573i	0.877994	−	0.478672i	$-0.158882\pi$
$443$	0.514719	+	0.891519i	0.0244550	+	0.0423573i	−0.853539	+	0.521029i	$0.825548\pi$
$444$		0			0
$445$	1.68629	−	2.92074i	0.0799379	−	0.138456i
$446$		0			0
$447$	−5.31371			−0.251330
$448$		0			0
$449$	17.3137			0.817084			0.408542	−	0.912739i	$-0.366037\pi$
$449$	17.3137			0.817084			0.408542	+	0.912739i	$0.366037\pi$
$450$		0			0
$451$	−2.24264	+	3.88437i	−0.105602	+	0.182908i
$452$		0			0
$453$	6.00000	+	10.3923i	0.281905	+	0.488273i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	9.00000	+	15.5885i	0.421002	+	0.729197i	0.996038	−	0.0889312i	$-0.0283451\pi$
$457$	9.00000	+	15.5885i	0.421002	+	0.729197i	−0.575036	+	0.818128i	$0.695012\pi$
$458$		0			0
$459$	3.12132	−	5.40629i	0.145691	−	0.252344i
$460$		0			0
$461$	19.4142			0.904210			0.452105	−	0.891965i	$-0.350673\pi$
$461$	19.4142			0.904210			0.452105	+	0.891965i	$0.350673\pi$
$462$		0			0
$463$	−18.6274			−0.865689			−0.432845	−	0.901468i	$-0.642490\pi$
$463$	−18.6274			−0.865689			−0.432845	+	0.901468i	$0.642490\pi$
$464$		0			0
$465$	−2.00000	+	3.46410i	−0.0927478	+	0.160644i
$466$		0			0
$467$	−19.8995	−	34.4669i	−0.920839	−	1.59494i	−0.798120	−	0.602498i	$-0.794172\pi$
$467$	−19.8995	−	34.4669i	−0.920839	−	1.59494i	−0.122718	−	0.992442i	$-0.539161\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	10.1213	+	17.5306i	0.466366	+	0.807769i
$472$		0			0
$473$	−5.65685	+	9.79796i	−0.260102	+	0.450511i
$474$		0			0
$475$	−13.1716			−0.604353
$476$		0			0
$477$	−2.00000			−0.0915737
$478$		0			0
$479$	−15.0711	+	26.1039i	−0.688615	+	1.19272i	0.283671	+	0.958922i	$0.408447\pi$
$479$	−15.0711	+	26.1039i	−0.688615	+	1.19272i	−0.972286	+	0.233794i	$0.924886\pi$
$480$		0			0
$481$	−10.8284	−	18.7554i	−0.493734	−	0.855172i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.58579	−	2.74666i	−0.0720069	−	0.124720i
$486$		0			0
$487$	−9.31371	+	16.1318i	−0.422044	+	0.731002i	−0.996139	−	0.0877864i	$-0.972021\pi$
$487$	−9.31371	+	16.1318i	−0.422044	+	0.731002i	0.574095	+	0.818789i	$0.305354\pi$
$488$		0			0
$489$	−11.3137			−0.511624
$490$		0			0
$491$	−38.9706			−1.75872			−0.879358	−	0.476160i	$-0.842028\pi$
$491$	−38.9706			−1.75872			−0.879358	+	0.476160i	$0.842028\pi$
$492$		0			0
$493$	−3.65685	+	6.33386i	−0.164696	+	0.285263i
$494$		0			0
$495$	0.585786	+	1.01461i	0.0263291	+	0.0456034i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−9.65685	−	16.7262i	−0.432300	−	0.748766i	0.564771	−	0.825248i	$-0.308965\pi$
$499$	−9.65685	−	16.7262i	−0.432300	−	0.748766i	−0.997071	+	0.0764820i	$0.975631\pi$
$500$		0			0
$501$	−9.89949	+	17.1464i	−0.442277	+	0.766046i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	10.0000			0.444994
$506$		0			0
$507$	−8.15685	+	14.1281i	−0.362259	+	0.627450i
$508$		0			0
$509$	−12.7782	−	22.1324i	−0.566383	−	0.981003i	−0.996920	−	0.0784305i	$-0.975009\pi$
$509$	−12.7782	−	22.1324i	−0.566383	−	0.981003i	0.430537	−	0.902573i	$-0.358324\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−1.41421	−	2.44949i	−0.0624391	−	0.108148i
$514$		0			0
$515$	−3.65685	+	6.33386i	−0.161140	+	0.279103i
$516$		0			0
$517$	5.65685			0.248788
$518$		0			0
$519$	−6.92893			−0.304146
$520$		0			0
$521$	−16.2929	+	28.2201i	−0.713805	+	1.23635i	0.249614	+	0.968345i	$0.419696\pi$
$521$	−16.2929	+	28.2201i	−0.713805	+	1.23635i	−0.963419	+	0.268000i	$0.913637\pi$
$522$		0			0
$523$	7.17157	+	12.4215i	0.313591	+	0.543156i	0.979137	−	0.203201i	$-0.0651346\pi$
$523$	7.17157	+	12.4215i	0.313591	+	0.543156i	−0.665546	+	0.746357i	$0.731801\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.3137	−	36.9164i	−0.928440	−	1.60810i
$528$		0			0
$529$	4.81371	−	8.33759i	0.209292	−	0.362504i
$530$		0			0
$531$	−6.82843			−0.296328
$532$		0			0
$533$	−12.1421			−0.525934
$534$		0			0
$535$	3.41421	−	5.91359i	0.147609	−	0.255667i
$536$		0			0
$537$	−4.17157	−	7.22538i	−0.180017	−	0.311798i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2.65685	+	4.60181i	0.114227	+	0.197847i	0.917471	−	0.397804i	$-0.130228\pi$
$541$	2.65685	+	4.60181i	0.114227	+	0.197847i	−0.803243	+	0.595651i	$0.796894\pi$
$542$		0			0
$543$	−2.70711	+	4.68885i	−0.116173	+	0.201218i
$544$		0			0
$545$	−3.31371			−0.141944
$546$		0			0
$547$	3.02944			0.129529			0.0647647	−	0.997901i	$-0.479370\pi$
$547$	3.02944			0.129529			0.0647647	+	0.997901i	$0.479370\pi$
$548$		0			0
$549$	1.87868	−	3.25397i	0.0801801	−	0.138876i
$550$		0			0
$551$	1.65685	+	2.86976i	0.0705844	+	0.122256i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.17157	−	2.02922i	−0.0497305	−	0.0861358i
$556$		0			0
$557$	−13.0000	+	22.5167i	−0.550828	+	0.954062i	0.447387	+	0.894340i	$0.352355\pi$
$557$	−13.0000	+	22.5167i	−0.550828	+	0.954062i	−0.998215	+	0.0597213i	$0.980979\pi$
$558$		0			0
$559$	−30.6274			−1.29540
$560$		0			0
$561$	−12.4853			−0.527129
$562$		0			0
$563$	3.41421	−	5.91359i	0.143892	−	0.249228i	−0.785067	−	0.619411i	$-0.787372\pi$
$563$	3.41421	−	5.91359i	0.143892	−	0.249228i	0.928959	+	0.370183i	$0.120705\pi$
$564$		0			0
$565$	5.07107	+	8.78335i	0.213341	+	0.369518i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.242641	−	0.420266i	−0.0101720	−	0.0176185i	0.860895	−	0.508783i	$-0.169905\pi$
$569$	−0.242641	−	0.420266i	−0.0101720	−	0.0176185i	−0.871067	+	0.491165i	$0.836571\pi$
$570$		0			0
$571$	16.8284	−	29.1477i	0.704248	−	1.21979i	−0.262715	−	0.964874i	$-0.584618\pi$
$571$	16.8284	−	29.1477i	0.704248	−	1.21979i	0.966962	−	0.254919i	$-0.0820489\pi$
$572$		0			0
$573$	18.0000			0.751961
$574$		0			0
$575$	17.0294			0.710177
$576$		0			0
$577$	−7.05025	+	12.2114i	−0.293506	+	0.508367i	−0.974636	−	0.223795i	$-0.928155\pi$
$577$	−7.05025	+	12.2114i	−0.293506	+	0.508367i	0.681130	+	0.732162i	$0.261489\pi$
$578$		0			0
$579$	8.65685	+	14.9941i	0.359767	+	0.623134i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.00000	+	3.46410i	0.0828315	+	0.143468i
$584$		0			0
$585$	−1.58579	+	2.74666i	−0.0655642	+	0.113561i
$586$		0			0
$587$	−17.1716			−0.708747			−0.354373	−	0.935104i	$-0.615306\pi$
$587$	−17.1716			−0.708747			−0.354373	+	0.935104i	$0.615306\pi$
$588$		0			0
$589$	−19.3137			−0.795807
$590$		0			0
$591$	−1.00000	+	1.73205i	−0.0411345	+	0.0712470i
$592$		0			0
$593$	−10.5355	−	18.2481i	−0.432643	−	0.749359i	0.564457	−	0.825462i	$-0.309085\pi$
$593$	−10.5355	−	18.2481i	−0.432643	−	0.749359i	−0.997100	+	0.0761034i	$0.975752\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5.17157	−	8.95743i	−0.211658	−	0.366603i
$598$		0			0
$599$	−1.00000	+	1.73205i	−0.0408589	+	0.0707697i	−0.885732	−	0.464198i	$-0.846343\pi$
$599$	−1.00000	+	1.73205i	−0.0408589	+	0.0707697i	0.844873	+	0.534967i	$0.179676\pi$
$600$		0			0
$601$	0.928932			0.0378919			0.0189460	−	0.999821i	$-0.493969\pi$
$601$	0.928932			0.0378919			0.0189460	+	0.999821i	$0.493969\pi$
$602$		0			0
$603$	−5.65685			−0.230365
$604$		0			0
$605$	−2.05025	+	3.55114i	−0.0833546	+	0.144374i
$606$		0			0
$607$	14.8284	+	25.6836i	0.601867	+	1.04246i	0.992538	+	0.121934i	$0.0389097\pi$
$607$	14.8284	+	25.6836i	0.601867	+	1.04246i	−0.390671	+	0.920530i	$0.627757\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.65685	+	13.2621i	0.309763	+	0.536526i
$612$		0			0
$613$	−13.6569	+	23.6544i	−0.551595	+	0.955391i	0.446565	+	0.894751i	$0.352647\pi$
$613$	−13.6569	+	23.6544i	−0.551595	+	0.955391i	−0.998160	+	0.0606394i	$0.980686\pi$
$614$		0			0
$615$	−1.31371			−0.0529738
$616$		0			0
$617$	−7.51472			−0.302531			−0.151266	−	0.988493i	$-0.548335\pi$
$617$	−7.51472			−0.302531			−0.151266	+	0.988493i	$0.548335\pi$
$618$		0			0
$619$	2.48528	−	4.30463i	0.0998919	−	0.173018i	−0.811748	−	0.584008i	$-0.801484\pi$
$619$	2.48528	−	4.30463i	0.0998919	−	0.173018i	0.911640	+	0.410990i	$0.134817\pi$
$620$		0			0
$621$	1.82843	+	3.16693i	0.0733723	+	0.127084i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−9.98528	−	17.2950i	−0.399411	−	0.691801i
$626$		0			0
$627$	−2.82843	+	4.89898i	−0.112956	+	0.195646i
$628$		0			0
$629$	24.9706			0.995642
$630$		0			0
$631$	−0.686292			−0.0273208			−0.0136604	−	0.999907i	$-0.504348\pi$
$631$	−0.686292			−0.0273208			−0.0136604	+	0.999907i	$0.504348\pi$
$632$		0			0
$633$	−10.4853	+	18.1610i	−0.416753	+	0.721837i
$634$		0			0
$635$	−2.82843	−	4.89898i	−0.112243	−	0.194410i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−6.65685	−	11.5300i	−0.263341	−	0.456120i
$640$		0			0
$641$	2.58579	−	4.47871i	0.102132	−	0.176899i	−0.810431	−	0.585835i	$-0.800767\pi$
$641$	2.58579	−	4.47871i	0.102132	−	0.176899i	0.912563	+	0.408936i	$0.134100\pi$
$642$		0			0
$643$	50.4264			1.98862			0.994312	−	0.106510i	$-0.0339675\pi$
$643$	50.4264			1.98862			0.994312	+	0.106510i	$0.0339675\pi$
$644$		0			0
$645$	−3.31371			−0.130477
$646$		0			0
$647$	−10.5858	+	18.3351i	−0.416170	+	0.720828i	−0.995551	−	0.0942294i	$-0.969961\pi$
$647$	−10.5858	+	18.3351i	−0.416170	+	0.720828i	0.579380	+	0.815057i	$0.303295\pi$
$648$		0			0
$649$	6.82843	+	11.8272i	0.268039	+	0.464258i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.75736	−	16.9002i	−0.381835	−	0.661358i	0.609490	−	0.792794i	$-0.291374\pi$
$653$	−9.75736	−	16.9002i	−0.381835	−	0.661358i	−0.991325	+	0.131436i	$0.958041\pi$
$654$		0			0
$655$	2.14214	−	3.71029i	0.0837002	−	0.144973i
$656$		0			0
$657$	5.89949			0.230161
$658$		0			0
$659$	13.3137			0.518628			0.259314	−	0.965793i	$-0.416503\pi$
$659$	13.3137			0.518628			0.259314	+	0.965793i	$0.416503\pi$
$660$		0			0
$661$	3.77817	−	6.54399i	0.146954	−	0.254532i	−0.783146	−	0.621838i	$-0.786386\pi$
$661$	3.77817	−	6.54399i	0.146954	−	0.254532i	0.930100	+	0.367306i	$0.119720\pi$
$662$		0			0
$663$	−16.8995	−	29.2708i	−0.656322	−	1.13678i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.14214	−	3.71029i	−0.0829438	−	0.143663i
$668$		0			0
$669$	4.48528	−	7.76874i	0.173411	−	0.300357i
$670$		0			0
$671$	−7.51472			−0.290102
$672$		0			0
$673$	0.686292			0.0264546			0.0132273	−	0.999913i	$-0.495789\pi$
$673$	0.686292			0.0264546			0.0132273	+	0.999913i	$0.495789\pi$
$674$		0			0
$675$	2.32843	−	4.03295i	0.0896212	−	0.155228i
$676$		0			0
$677$	14.2929	+	24.7560i	0.549321	+	0.951451i	0.998321	+	0.0579196i	$0.0184467\pi$
$677$	14.2929	+	24.7560i	0.549321	+	0.951451i	−0.449001	+	0.893531i	$0.648220\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	7.89949	+	13.6823i	0.302709	+	0.524308i
$682$		0			0
$683$	−4.17157	+	7.22538i	−0.159621	+	0.276471i	−0.934732	−	0.355354i	$-0.884360\pi$
$683$	−4.17157	+	7.22538i	−0.159621	+	0.276471i	0.775111	+	0.631825i	$0.217694\pi$
$684$		0			0
$685$	8.28427			0.316526
$686$		0			0
$687$	−8.24264			−0.314476
$688$		0			0
$689$	−5.41421	+	9.37769i	−0.206265	+	0.357262i
$690$		0			0
$691$	11.6569	+	20.1903i	0.443448	+	0.768074i	0.997943	−	0.0641132i	$-0.0204219\pi$
$691$	11.6569	+	20.1903i	0.443448	+	0.768074i	−0.554495	+	0.832187i	$0.687089\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.85786	−	3.21792i	−0.0704728	−	0.122062i
$696$		0			0
$697$	7.00000	−	12.1244i	0.265144	−	0.459243i
$698$		0			0
$699$	22.1421			0.837492
$700$		0			0
$701$	−22.8284			−0.862218			−0.431109	−	0.902300i	$-0.641878\pi$
$701$	−22.8284			−0.862218			−0.431109	+	0.902300i	$0.641878\pi$
$702$		0			0
$703$	5.65685	−	9.79796i	0.213352	−	0.369537i
$704$		0			0
$705$	0.828427	+	1.43488i	0.0312004	+	0.0540406i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−10.1421	−	17.5667i	−0.380896	−	0.659731i	0.610295	−	0.792174i	$-0.291051\pi$
$709$	−10.1421	−	17.5667i	−0.380896	−	0.659731i	−0.991191	+	0.132443i	$0.957718\pi$
$710$		0			0
$711$	1.17157	−	2.02922i	0.0439374	−	0.0761018i
$712$		0			0
$713$	24.9706			0.935155
$714$		0			0
$715$	6.34315			0.237220
$716$		0			0
$717$	−2.17157	+	3.76127i	−0.0810989	+	0.140467i
$718$		0			0
$719$	12.9706	+	22.4657i	0.483720	+	0.837828i	0.999825	−	0.0186972i	$-0.00595184\pi$
$719$	12.9706	+	22.4657i	0.483720	+	0.837828i	−0.516105	+	0.856525i	$0.672619\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−3.87868	−	6.71807i	−0.144250	−	0.249848i
$724$		0			0
$725$	−2.72792	+	4.72490i	−0.101312	+	0.175478i
$726$		0			0
$727$	−4.48528			−0.166350			−0.0831749	−	0.996535i	$-0.526506\pi$
$727$	−4.48528			−0.166350			−0.0831749	+	0.996535i	$0.526506\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	17.6569	−	30.5826i	0.653062	−	1.13114i
$732$		0			0
$733$	−4.84924	−	8.39913i	−0.179111	−	0.310229i	0.762465	−	0.647029i	$-0.223989\pi$
$733$	−4.84924	−	8.39913i	−0.179111	−	0.310229i	−0.941576	+	0.336800i	$0.890655\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.65685	+	9.79796i	0.208373	+	0.360912i
$738$		0			0
$739$	13.6569	−	23.6544i	0.502376	−	0.870140i	−0.497621	−	0.867395i	$-0.665793\pi$
$739$	13.6569	−	23.6544i	0.502376	−	0.870140i	0.999996	−	0.00274517i	$-0.000873816\pi$
$740$		0			0
$741$	−15.3137			−0.562563
$742$		0			0
$743$	17.0294			0.624749			0.312375	−	0.949959i	$-0.398876\pi$
$743$	17.0294			0.624749			0.312375	+	0.949959i	$0.398876\pi$
$744$		0			0
$745$	−1.55635	+	2.69568i	−0.0570202	+	0.0987619i
$746$		0			0
$747$	7.65685	+	13.2621i	0.280150	+	0.485233i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.17157	+	2.02922i	0.0427513	+	0.0740474i	0.886609	−	0.462519i	$-0.153054\pi$
$751$	1.17157	+	2.02922i	0.0427513	+	0.0740474i	−0.843858	+	0.536567i	$0.819721\pi$
$752$		0			0
$753$	−2.24264	+	3.88437i	−0.0817264	+	0.141554i
$754$		0			0
$755$	7.02944			0.255827
$756$		0			0
$757$	37.6569			1.36866			0.684331	−	0.729172i	$-0.260094\pi$
$757$	37.6569			1.36866			0.684331	+	0.729172i	$0.260094\pi$
$758$		0			0
$759$	3.65685	−	6.33386i	0.132735	−	0.229904i
$760$		0			0
$761$	23.2635	+	40.2935i	0.843300	+	1.46064i	0.887090	+	0.461597i	$0.152723\pi$
$761$	23.2635	+	40.2935i	0.843300	+	1.46064i	−0.0437901	+	0.999041i	$0.513943\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.82843	−	3.16693i	−0.0661069	−	0.114501i
$766$		0			0
$767$	−18.4853	+	32.0174i	−0.667465	+	1.15608i
$768$		0			0
$769$	29.6985			1.07095			0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985			1.07095			0.535477	+	0.844550i	$0.320132\pi$
$770$		0			0
$771$	19.2132			0.691947
$772$		0			0
$773$	10.7782	−	18.6683i	0.387664	−	0.671454i	−0.604471	−	0.796627i	$-0.706615\pi$
$773$	10.7782	−	18.6683i	0.387664	−	0.671454i	0.992135	+	0.125174i	$0.0399488\pi$
$774$		0			0
$775$	−15.8995	−	27.5387i	−0.571127	−	0.989220i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.17157	−	5.49333i	−0.113633	−	0.196819i
$780$		0			0
$781$	−13.3137	+	23.0600i	−0.476402	+	0.825152i
$782$		0			0
$783$	−1.17157			−0.0418686
$784$		0			0
$785$	11.8579			0.423225
$786$		0			0
$787$	−23.6569	+	40.9749i	−0.843276	+	1.46060i	0.0438344	+	0.999039i	$0.486043\pi$
$787$	−23.6569	+	40.9749i	−0.843276	+	1.46060i	−0.887110	+	0.461558i	$0.847291\pi$
$788$		0			0
$789$	−8.65685	−	14.9941i	−0.308192	−	0.533805i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−10.1716	−	17.6177i	−0.361203	−	0.625622i
$794$		0			0
$795$	−0.585786	+	1.01461i	−0.0207757	+	0.0359846i
$796$		0			0
$797$	−28.3848			−1.00544			−0.502720	−	0.864449i	$-0.667667\pi$
$797$	−28.3848			−1.00544			−0.502720	+	0.864449i	$0.667667\pi$
$798$		0			0
$799$	−17.6569			−0.624655

Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.q (of order \(3\), degree \(2\), not minimal)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

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