LMFDB - Embedded newform 1900.2.i.e.201.6

Newspace parameters

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.i (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
Defining polynomial:	$x^{12} - 3 x^{11} + 15 x^{10} - 16 x^{9} + 79 x^{8} - 65 x^{7} + 298 x^{6} + 13 x^{5} + 233 x^{4} + 14 x^{3} + 145 x^{2} + 33 x + 9$
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			201.6
Root			$1.42323 - 2.46510i$ of defining polynomial
Character	$\chi$	$=$	1900.201
Dual form			1900.2.i.e.501.6

$q$-expansion

$f(q)$	$=$	$q+(0.923228 - 1.59908i) q^{3} +1.72830 q^{7} +(-0.204702 - 0.354554i) q^{9} +O(q^{10})$	\(q+(0.923228 - 1.59908i) q^{3} +1.72830 q^{7} +(-0.204702 - 0.354554i) q^{9} +2.66366 q^{11} +(-0.544662 - 0.943382i) q^{13} +(-0.660249 + 1.14358i) q^{17} +(2.41510 + 3.62868i) q^{19} +(1.59561 - 2.76368i) q^{21} +(-0.704108 - 1.21955i) q^{23} +4.78343 q^{27} +(0.0446618 + 0.0773564i) q^{29} +8.90368 q^{31} +(2.45917 - 4.25940i) q^{33} +3.32732 q^{37} -2.01139 q^{39} +(-0.296903 + 0.514251i) q^{41} +(1.31334 - 2.27477i) q^{43} +(-3.51092 - 6.08109i) q^{47} -4.01299 q^{49} +(1.21912 + 2.11158i) q^{51} +(0.275627 + 0.477400i) q^{53} +(8.03223 - 0.511833i) q^{57} +(-4.07268 + 7.05408i) q^{59} +(-0.207276 - 0.359013i) q^{61} +(-0.353785 - 0.612774i) q^{63} +(-2.11271 - 3.65932i) q^{67} -2.60021 q^{69} +(1.36672 - 2.36723i) q^{71} +(4.53139 - 7.84860i) q^{73} +4.60359 q^{77} +(-5.86470 + 10.1580i) q^{79} +(5.03030 - 8.71273i) q^{81} +7.76556 q^{83} +0.164932 q^{87} +(-0.132475 - 0.229454i) q^{89} +(-0.941337 - 1.63044i) q^{91} +(8.22013 - 14.2377i) q^{93} +(-0.843674 + 1.46129i) q^{97} +(-0.545256 - 0.944410i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} - 3 q^{9} + O(q^{10}) $	$ 12 q - 3 q^{3} - 3 q^{9} + 2 q^{11} - 7 q^{13} + q^{17} + q^{21} + 2 q^{23} + 24 q^{27} + q^{29} + 2 q^{31} - 10 q^{33} - 20 q^{37} + 36 q^{39} - 7 q^{41} - 19 q^{43} + 14 q^{47} + 8 q^{49} + 11 q^{51} - 6 q^{53} + 28 q^{57} - 5 q^{61} + 11 q^{63} - 14 q^{67} - 14 q^{69} + 8 q^{71} + 9 q^{73} - 2 q^{77} + q^{79} + 2 q^{81} + 26 q^{83} - 30 q^{87} - 8 q^{89} + 3 q^{91} - 9 q^{93} + 11 q^{97} - 18 q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	0.923228	−	1.59908i	0.533026	−	0.923228i	−0.466230	−	0.884664i	$-0.654388\pi$
$3$	0.923228	−	1.59908i	0.533026	−	0.923228i	0.999256	−	0.0385649i	$-0.0122786\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.72830			0.653234			0.326617	−	0.945157i	$-0.394091\pi$
$7$	1.72830			0.653234			0.326617	+	0.945157i	$0.394091\pi$
$8$		0			0
$9$	−0.204702	−	0.354554i	−0.0682339	−	0.118185i
$10$		0			0
$11$	2.66366			0.803123			0.401562	−	0.915832i	$-0.368468\pi$
$11$	2.66366			0.803123			0.401562	+	0.915832i	$0.368468\pi$
$12$		0			0
$13$	−0.544662	−	0.943382i	−0.151062	−	0.261647i	0.780556	−	0.625086i	$-0.214936\pi$
$13$	−0.544662	−	0.943382i	−0.151062	−	0.261647i	−0.931618	+	0.363439i	$0.881603\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.660249	+	1.14358i	−0.160134	+	0.277360i	−0.934917	−	0.354868i	$-0.884526\pi$
$17$	−0.660249	+	1.14358i	−0.160134	+	0.277360i	0.774783	+	0.632228i	$0.217859\pi$
$18$		0			0
$19$	2.41510	+	3.62868i	0.554061	+	0.832476i
$19$	2.41510	+	3.62868i	0.554061	+	0.832476i
$20$		0			0
$21$	1.59561	−	2.76368i	0.348191	−	0.603085i
$22$		0			0
$23$	−0.704108	−	1.21955i	−0.146817	−	0.254294i	0.783233	−	0.621729i	$-0.213569\pi$
$23$	−0.704108	−	1.21955i	−0.146817	−	0.254294i	−0.930049	+	0.367435i	$0.880236\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.78343			0.920571
$28$		0			0
$29$	0.0446618	+	0.0773564i	0.00829348	+	0.0143647i	0.870142	−	0.492800i	$-0.164027\pi$
$29$	0.0446618	+	0.0773564i	0.00829348	+	0.0143647i	−0.861849	+	0.507165i	$0.830693\pi$
$30$		0			0
$31$	8.90368			1.59915			0.799574	−	0.600568i	$-0.205059\pi$
$31$	8.90368			1.59915			0.799574	+	0.600568i	$0.205059\pi$
$32$		0			0
$33$	2.45917	−	4.25940i	0.428086	−	0.741466i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.32732			0.547008			0.273504	−	0.961871i	$-0.411817\pi$
$37$	3.32732			0.547008			0.273504	+	0.961871i	$0.411817\pi$
$38$		0			0
$39$	−2.01139			−0.322080
$40$		0			0
$41$	−0.296903	+	0.514251i	−0.0463685	+	0.0803126i	−0.888278	−	0.459306i	$-0.848098\pi$
$41$	−0.296903	+	0.514251i	−0.0463685	+	0.0803126i	0.841910	+	0.539618i	$0.181432\pi$
$42$		0			0
$43$	1.31334	−	2.27477i	0.200282	−	0.346898i	−0.748337	−	0.663318i	$-0.769148\pi$
$43$	1.31334	−	2.27477i	0.200282	−	0.346898i	0.948619	+	0.316420i	$0.102481\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.51092	−	6.08109i	−0.512120	−	0.887018i	−0.999901	−	0.0140521i	$-0.995527\pi$
$47$	−3.51092	−	6.08109i	−0.512120	−	0.887018i	0.487781	−	0.872966i	$-0.337806\pi$
$48$		0			0
$49$	−4.01299			−0.573285
$50$		0			0
$51$	1.21912	+	2.11158i	0.170711	+	0.295680i
$52$		0			0
$53$	0.275627	+	0.477400i	0.0378602	+	0.0655759i	0.884335	−	0.466854i	$-0.154612\pi$
$53$	0.275627	+	0.477400i	0.0378602	+	0.0655759i	−0.846474	+	0.532429i	$0.821279\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	8.03223	−	0.511833i	1.06389	−	0.0677939i
$58$		0			0
$59$	−4.07268	+	7.05408i	−0.530217	+	0.918364i	0.469161	+	0.883113i	$0.344556\pi$
$59$	−4.07268	+	7.05408i	−0.530217	+	0.918364i	−0.999378	+	0.0352509i	$0.988777\pi$
$60$		0			0
$61$	−0.207276	−	0.359013i	−0.0265390	−	0.0459669i	0.852451	−	0.522807i	$-0.175115\pi$
$61$	−0.207276	−	0.359013i	−0.0265390	−	0.0459669i	−0.878990	+	0.476840i	$0.841782\pi$
$62$		0			0
$63$	−0.353785	−	0.612774i	−0.0445727	−	0.0772022i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2.11271	−	3.65932i	−0.258108	−	0.447057i	0.707627	−	0.706586i	$-0.249766\pi$
$67$	−2.11271	−	3.65932i	−0.258108	−	0.447057i	−0.965735	+	0.259529i	$0.916433\pi$
$68$		0			0
$69$	−2.60021			−0.313028
$70$		0			0
$71$	1.36672	−	2.36723i	0.162200	−	0.280939i	−0.773457	−	0.633848i	$-0.781474\pi$
$71$	1.36672	−	2.36723i	0.162200	−	0.280939i	0.935657	+	0.352910i	$0.114808\pi$
$72$		0			0
$73$	4.53139	−	7.84860i	0.530359	−	0.918609i	−0.469013	−	0.883191i	$-0.655390\pi$
$73$	4.53139	−	7.84860i	0.530359	−	0.918609i	0.999373	−	0.0354182i	$-0.0112763\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.60359			0.524628
$78$		0			0
$79$	−5.86470	+	10.1580i	−0.659831	+	1.14286i	0.320829	+	0.947137i	$0.396039\pi$
$79$	−5.86470	+	10.1580i	−0.659831	+	1.14286i	−0.980659	+	0.195723i	$0.937295\pi$
$80$		0			0
$81$	5.03030	−	8.71273i	0.558922	−	0.968082i
$82$		0			0
$83$	7.76556			0.852381			0.426190	−	0.904633i	$-0.359855\pi$
$83$	7.76556			0.852381			0.426190	+	0.904633i	$0.359855\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.164932			0.0176826
$88$		0			0
$89$	−0.132475	−	0.229454i	−0.0140423	−	0.0243221i	0.858919	−	0.512112i	$-0.171137\pi$
$89$	−0.132475	−	0.229454i	−0.0140423	−	0.0243221i	−0.872961	+	0.487790i	$0.837803\pi$
$90$		0			0
$91$	−0.941337	−	1.63044i	−0.0986789	−	0.170917i
$92$		0			0
$93$	8.22013	−	14.2377i	0.852387	−	1.47638i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.843674	+	1.46129i	−0.0856621	+	0.148371i	−0.905673	−	0.423977i	$-0.860634\pi$
$97$	−0.843674	+	1.46129i	−0.0856621	+	0.148371i	0.820011	+	0.572348i	$0.193967\pi$
$98$		0			0
$99$	−0.545256	−	0.944410i	−0.0548002	−	0.0949168i
$100$		0			0
$101$	−4.80562	−	8.32358i	−0.478177	−	0.828227i	0.521510	−	0.853245i	$-0.325369\pi$
$101$	−4.80562	−	8.32358i	−0.478177	−	0.828227i	−0.999687	+	0.0250179i	$0.992036\pi$
$102$		0			0
$103$	5.41108			0.533169			0.266585	−	0.963812i	$-0.414105\pi$
$103$	5.41108			0.533169			0.266585	+	0.963812i	$0.414105\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.99613			−0.676341			−0.338171	−	0.941085i	$-0.609808\pi$
$107$	−6.99613			−0.676341			−0.338171	+	0.941085i	$0.609808\pi$
$108$		0			0
$109$	−2.05095	+	3.55235i	−0.196445	+	0.340253i	−0.947373	−	0.320131i	$-0.896273\pi$
$109$	−2.05095	+	3.55235i	−0.196445	+	0.340253i	0.750928	+	0.660384i	$0.229607\pi$
$110$		0			0
$111$	3.07187	−	5.32064i	0.291569	−	0.505013i
$112$		0			0
$113$	12.9414			1.21742			0.608710	−	0.793393i	$-0.291687\pi$
$113$	12.9414			1.21742			0.608710	+	0.793393i	$0.291687\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.222986	+	0.386224i	−0.0206151	+	0.0357064i
$118$		0			0
$119$	−1.14110	+	1.97645i	−0.104605	+	0.181181i
$120$		0			0
$121$	−3.90492			−0.354993
$122$		0			0
$123$	0.548219	+	0.949543i	0.0494312	+	0.0856174i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−5.90009	−	10.2192i	−0.523548	−	0.906812i	−0.999624	−	0.0274077i	$-0.991275\pi$
$127$	−5.90009	−	10.2192i	−0.523548	−	0.906812i	0.476076	−	0.879404i	$-0.342059\pi$
$128$		0			0
$129$	−2.42502	−	4.20026i	−0.213511	−	0.369812i
$130$		0			0
$131$	−2.25635	+	3.90811i	−0.197138	+	0.341453i	−0.947599	−	0.319461i	$-0.896498\pi$
$131$	−2.25635	+	3.90811i	−0.197138	+	0.341453i	0.750461	+	0.660915i	$0.229831\pi$
$132$		0			0
$133$	4.17400	+	6.27143i	0.361932	+	0.543802i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.63770	+	2.83658i	0.139918	+	0.242345i	0.927465	−	0.373909i	$-0.121983\pi$
$137$	1.63770	+	2.83658i	0.139918	+	0.242345i	−0.787547	+	0.616254i	$0.788649\pi$
$138$		0			0
$139$	−3.84162	−	6.65388i	−0.325842	−	0.564375i	0.655840	−	0.754900i	$-0.272314\pi$
$139$	−3.84162	−	6.65388i	−0.325842	−	0.564375i	−0.981682	+	0.190525i	$0.938981\pi$
$140$		0			0
$141$	−12.9655			−1.09189
$142$		0			0
$143$	−1.45079	−	2.51285i	−0.121321	−	0.210135i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−3.70491	+	6.41709i	−0.305576	+	0.529273i
$148$		0			0
$149$	1.15941	−	2.00816i	0.0949828	−	0.164515i	−0.814619	−	0.579997i	$-0.803054\pi$
$149$	1.15941	−	2.00816i	0.0949828	−	0.164515i	0.909601	+	0.415482i	$0.136387\pi$
$150$		0			0
$151$	3.18280			0.259012			0.129506	−	0.991579i	$-0.458661\pi$
$151$	3.18280			0.259012			0.129506	+	0.991579i	$0.458661\pi$
$152$		0			0
$153$	0.540616			0.0437062
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.981487	−	1.69999i	0.0783312	−	0.135674i	−0.824199	−	0.566300i	$-0.808374\pi$
$157$	0.981487	−	1.69999i	0.0783312	−	0.135674i	0.902530	+	0.430627i	$0.141708\pi$
$158$		0			0
$159$	1.01787			0.0807220
$160$		0			0
$161$	−1.21691	−	2.10774i	−0.0959057	−	0.166114i
$162$		0			0
$163$	1.63601			0.128142			0.0640711	−	0.997945i	$-0.479592\pi$
$163$	1.63601			0.128142			0.0640711	+	0.997945i	$0.479592\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.5824	−	21.7934i	−0.973658	−	1.68642i	−0.684295	−	0.729205i	$-0.739890\pi$
$167$	−12.5824	−	21.7934i	−0.973658	−	1.68642i	−0.289363	−	0.957219i	$-0.593443\pi$
$168$		0			0
$169$	5.90669	−	10.2307i	0.454361	−	0.786976i
$170$		0			0
$171$	0.792187	−	1.59908i	0.0605800	−	0.122285i
$172$		0			0
$173$	7.01277	−	12.1465i	0.533171	−	0.923479i	−0.466079	−	0.884743i	$-0.654334\pi$
$173$	7.01277	−	12.1465i	0.533171	−	0.923479i	0.999249	−	0.0387355i	$-0.0123330\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.52002	+	13.0251i	0.565240	+	0.979024i
$178$		0			0
$179$	−23.7004			−1.77145			−0.885727	−	0.464207i	$-0.846339\pi$
$179$	−23.7004			−1.77145			−0.885727	+	0.464207i	$0.846339\pi$
$180$		0			0
$181$	−3.91555	−	6.78193i	−0.291040	−	0.504097i	0.683016	−	0.730404i	$-0.260668\pi$
$181$	−3.91555	−	6.78193i	−0.291040	−	0.504097i	−0.974056	+	0.226307i	$0.927335\pi$
$182$		0			0
$183$	−0.765453			−0.0565839
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.75868	+	3.04612i	−0.128607	+	0.222754i
$188$		0			0
$189$	8.26717			0.601348
$190$		0			0
$191$	6.03698			0.436820			0.218410	−	0.975857i	$-0.429913\pi$
$191$	6.03698			0.436820			0.218410	+	0.975857i	$0.429913\pi$
$192$		0			0
$193$	−7.51822	+	13.0219i	−0.541173	+	0.937339i	0.457664	+	0.889125i	$0.348686\pi$
$193$	−7.51822	+	13.0219i	−0.541173	+	0.937339i	−0.998837	+	0.0482141i	$0.984647\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.45409			0.103599			0.0517996	−	0.998657i	$-0.483504\pi$
$197$	1.45409			0.103599			0.0517996	+	0.998657i	$0.483504\pi$
$198$		0			0
$199$	12.0696	+	20.9051i	0.855590	+	1.48193i	0.876096	+	0.482136i	$0.160139\pi$
$199$	12.0696	+	20.9051i	0.855590	+	1.48193i	−0.0205062	+	0.999790i	$0.506528\pi$
$200$		0			0
$201$	−7.80205			−0.550314
$202$		0			0
$203$	0.0771887	+	0.133695i	0.00541759	+	0.00938354i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.288264	+	0.499288i	−0.0200357	+	0.0347029i
$208$		0			0
$209$	6.43300	+	9.66556i	0.444980	+	0.668581i
$210$		0			0
$211$	−7.63568	+	13.2254i	−0.525662	+	0.910473i	0.473891	+	0.880583i	$0.342849\pi$
$211$	−7.63568	+	13.2254i	−0.525662	+	0.910473i	−0.999553	+	0.0298898i	$0.990484\pi$
$212$		0			0
$213$	−2.52359	−	4.37099i	−0.172914	−	0.299496i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	15.3882			1.04462
$218$		0			0
$219$	−8.36702	−	14.4921i	−0.565391	−	0.979286i
$220$		0			0
$221$	1.43845			0.0967605
$222$		0			0
$223$	−1.72194	+	2.98248i	−0.115309	+	0.199722i	−0.917903	−	0.396804i	$-0.870119\pi$
$223$	−1.72194	+	2.98248i	−0.115309	+	0.199722i	0.802594	+	0.596526i	$0.203453\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−24.9955			−1.65901			−0.829504	−	0.558501i	$-0.811377\pi$
$227$	−24.9955			−1.65901			−0.829504	+	0.558501i	$0.811377\pi$
$228$		0			0
$229$	5.35855			0.354103			0.177051	−	0.984202i	$-0.443344\pi$
$229$	5.35855			0.354103			0.177051	+	0.984202i	$0.443344\pi$
$230$		0			0
$231$	4.25017	−	7.36150i	0.279640	−	0.484351i
$232$		0			0
$233$	−6.50466	+	11.2664i	−0.426135	+	0.738087i	−0.996526	−	0.0832864i	$-0.973458\pi$
$233$	−6.50466	+	11.2664i	−0.426135	+	0.738087i	0.570391	+	0.821373i	$0.306792\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	10.8289	+	18.7562i	0.703414	+	1.21835i
$238$		0			0
$239$	−8.48487			−0.548841			−0.274420	−	0.961610i	$-0.588486\pi$
$239$	−8.48487			−0.548841			−0.274420	+	0.961610i	$0.588486\pi$
$240$		0			0
$241$	0.0131746	+	0.0228190i	0.000848648	+	0.00146990i	0.866449	−	0.499265i	$-0.166397\pi$
$241$	0.0131746	+	0.0228190i	0.000848648	+	0.00146990i	−0.865601	+	0.500735i	$0.833063\pi$
$242$		0			0
$243$	−2.11309	−	3.65999i	−0.135555	−	0.234788i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.10782	−	4.25476i	0.134117	−	0.270724i
$248$		0			0
$249$	7.16939	−	12.4177i	0.454341	−	0.786942i
$250$		0			0
$251$	9.82041	+	17.0094i	0.619859	+	1.07363i	0.989511	+	0.144457i	$0.0461434\pi$
$251$	9.82041	+	17.0094i	0.619859	+	1.07363i	−0.369653	+	0.929170i	$0.620523\pi$
$252$		0			0
$253$	−1.87550	−	3.24847i	−0.117912	−	0.204229i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.76869	−	8.25962i	−0.297463	−	0.515221i	0.678092	−	0.734977i	$-0.262807\pi$
$257$	−4.76869	−	8.25962i	−0.297463	−	0.515221i	−0.975555	+	0.219756i	$0.929474\pi$
$258$		0			0
$259$	5.75059			0.357324
$260$		0			0
$261$	0.0182847	−	0.0316700i	0.00113179	−	0.00196032i
$262$		0			0
$263$	−5.64831	+	9.78317i	−0.348290	+	0.603256i	−0.985946	−	0.167065i	$-0.946571\pi$
$263$	−5.64831	+	9.78317i	−0.348290	+	0.603256i	0.637656	+	0.770321i	$0.279904\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.489220			−0.0299398
$268$		0			0
$269$	−8.18113	+	14.1701i	−0.498812	+	0.863968i	−0.999999	−	0.00137116i	$-0.999564\pi$
$269$	−8.18113	+	14.1701i	−0.498812	+	0.863968i	0.501187	+	0.865339i	$0.332897\pi$
$270$		0			0
$271$	2.25370	−	3.90352i	0.136903	−	0.237122i	−0.789420	−	0.613853i	$-0.789619\pi$
$271$	2.25370	−	3.90352i	0.136903	−	0.237122i	0.926323	+	0.376731i	$0.122952\pi$
$272$		0			0
$273$	−3.47628			−0.210394
$274$		0			0
$275$		0			0
$276$		0			0
$277$	26.7914			1.60974			0.804871	−	0.593450i	$-0.202235\pi$
$277$	26.7914			1.60974			0.804871	+	0.593450i	$0.202235\pi$
$278$		0			0
$279$	−1.82260	−	3.15683i	−0.109116	−	0.188995i
$280$		0			0
$281$	9.65081	+	16.7157i	0.575719	+	0.997174i	0.995963	+	0.0897633i	$0.0286111\pi$
$281$	9.65081	+	16.7157i	0.575719	+	0.997174i	−0.420244	+	0.907411i	$0.638056\pi$
$282$		0			0
$283$	1.15440	−	1.99948i	0.0686221	−	0.118857i	−0.829673	−	0.558250i	$-0.811473\pi$
$283$	1.15440	−	1.99948i	0.0686221	−	0.118857i	0.898295	+	0.439393i	$0.144806\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.513136	+	0.888778i	−0.0302895	+	0.0524629i
$288$		0			0
$289$	7.62814	+	13.2123i	0.448714	+	0.777196i
$290$		0			0
$291$	1.55781	+	2.69820i	0.0913203	+	0.158171i
$292$		0			0
$293$	−20.5156			−1.19854			−0.599268	−	0.800549i	$-0.704541\pi$
$293$	−20.5156			−1.19854			−0.599268	+	0.800549i	$0.704541\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	12.7414			0.739332
$298$		0			0
$299$	−0.767001	+	1.32849i	−0.0443568	+	0.0768283i
$300$		0			0
$301$	2.26983	−	3.93147i	0.130831	−	0.226606i
$302$		0			0
$303$	−17.7468			−1.01952
$304$		0			0
$305$		0			0
$306$		0			0
$307$	2.03655	−	3.52742i	0.116232	−	0.201320i	−0.802039	−	0.597271i	$-0.796252\pi$
$307$	2.03655	−	3.52742i	0.116232	−	0.201320i	0.918272	+	0.395951i	$0.129585\pi$
$308$		0			0
$309$	4.99566	−	8.65274i	0.284193	−	0.492237i
$310$		0			0
$311$	11.1999			0.635087			0.317543	−	0.948244i	$-0.397142\pi$
$311$	11.1999			0.635087			0.317543	+	0.948244i	$0.397142\pi$
$312$		0			0
$313$	13.0281	+	22.5654i	0.736393	+	1.27547i	0.954110	+	0.299458i	$0.0968057\pi$
$313$	13.0281	+	22.5654i	0.736393	+	1.27547i	−0.217717	+	0.976012i	$0.569861\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.9315	−	24.1300i	−0.782469	−	1.35528i	−0.930499	−	0.366294i	$-0.880627\pi$
$317$	−13.9315	−	24.1300i	−0.782469	−	1.35528i	0.148030	−	0.988983i	$-0.452707\pi$
$318$		0			0
$319$	0.118964	+	0.206051i	0.00666069	+	0.0115367i
$320$		0			0
$321$	−6.45902	+	11.1874i	−0.360508	+	0.624417i
$322$		0			0
$323$	−5.74426	+	0.366038i	−0.319619	+	0.0203669i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	3.78699	+	6.55926i	0.209421	+	0.362728i
$328$		0			0
$329$	−6.06791	−	10.5099i	−0.334534	−	0.579431i
$330$		0			0
$331$	−20.9379			−1.15085			−0.575426	−	0.817854i	$-0.695164\pi$
$331$	−20.9379			−1.15085			−0.575426	+	0.817854i	$0.695164\pi$
$332$		0			0
$333$	−0.681108	−	1.17971i	−0.0373245	−	0.0646479i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.75456	+	13.4313i	−0.422418	+	0.731649i	−0.996175	−	0.0873761i	$-0.972152\pi$
$337$	−7.75456	+	13.4313i	−0.422418	+	0.731649i	0.573758	+	0.819025i	$0.305485\pi$
$338$		0			0
$339$	11.9478	−	20.6942i	0.648917	−	1.12396i
$340$		0			0
$341$	23.7164			1.28431
$342$		0			0
$343$	−19.0337			−1.02772
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.5770	+	30.4443i	−0.943584	+	1.63434i	−0.185023	+	0.982734i	$0.559236\pi$
$347$	−17.5770	+	30.4443i	−0.943584	+	1.63434i	−0.758561	+	0.651602i	$0.774097\pi$
$348$		0			0
$349$	29.5120			1.57974			0.789870	−	0.613274i	$-0.210148\pi$
$349$	29.5120			1.57974			0.789870	+	0.613274i	$0.210148\pi$
$350$		0			0
$351$	−2.60535	−	4.51260i	−0.139063	−	0.240865i
$352$		0			0
$353$	−31.0179			−1.65092			−0.825458	−	0.564463i	$-0.809083\pi$
$353$	−31.0179			−1.65092			−0.825458	+	0.564463i	$0.809083\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	2.10700	+	3.64943i	0.111514	+	0.193148i
$358$		0			0
$359$	16.5956	−	28.7444i	0.875882	−	1.51707i	0.0200625	−	0.999799i	$-0.493613\pi$
$359$	16.5956	−	28.7444i	0.875882	−	1.51707i	0.855820	−	0.517274i	$-0.173053\pi$
$360$		0			0
$361$	−7.33460	+	17.5272i	−0.386032	+	0.922485i
$362$		0			0
$363$	−3.60513	+	6.24428i	−0.189220	+	0.327739i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.46462	+	12.9291i	0.389650	+	0.674894i	0.992402	−	0.123035i	$-0.0392626\pi$
$367$	7.46462	+	12.9291i	0.389650	+	0.674894i	−0.602752	+	0.797928i	$0.705929\pi$
$368$		0			0
$369$	0.243106			0.0126556
$370$		0			0
$371$	0.476365	+	0.825088i	0.0247316	+	0.0428364i
$372$		0			0
$373$	−0.129504			−0.00670546			−0.00335273	−	0.999994i	$-0.501067\pi$
$373$	−0.129504			−0.00670546			−0.00335273	+	0.999994i	$0.501067\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.0486511	−	0.0842662i	0.00250566	−	0.00433993i
$378$		0			0
$379$	9.91054			0.509070			0.254535	−	0.967064i	$-0.418078\pi$
$379$	9.91054			0.509070			0.254535	+	0.967064i	$0.418078\pi$
$380$		0			0
$381$	−21.7885			−1.11626
$382$		0			0
$383$	−18.1930	+	31.5112i	−0.929619	+	1.61015i	−0.145659	+	0.989335i	$0.546530\pi$
$383$	−18.1930	+	31.5112i	−0.929619	+	1.61015i	−0.783960	+	0.620812i	$0.786803\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.07537			−0.0546641
$388$		0			0
$389$	11.3433	+	19.6472i	0.575129	+	0.996153i	0.996028	+	0.0890454i	$0.0283816\pi$
$389$	11.3433	+	19.6472i	0.575129	+	0.996153i	−0.420898	+	0.907108i	$0.638285\pi$
$390$		0			0
$391$	1.85954			0.0940412
$392$		0			0
$393$	4.16625	+	7.21616i	0.210160	+	0.364007i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−9.67149	+	16.7515i	−0.485398	+	0.840734i	−0.999859	−	0.0167797i	$-0.994659\pi$
$397$	−9.67149	+	16.7515i	−0.485398	+	0.840734i	0.514461	+	0.857514i	$0.327992\pi$
$398$		0			0
$399$	13.8821	−	0.884599i	0.694973	−	0.0442853i
$400$		0			0
$401$	−1.11006	+	1.92268i	−0.0554337	+	0.0960140i	−0.892411	−	0.451224i	$-0.850987\pi$
$401$	−1.11006	+	1.92268i	−0.0554337	+	0.0960140i	0.836977	+	0.547238i	$0.184321\pi$
$402$		0			0
$403$	−4.84949	−	8.39957i	−0.241570	−	0.418412i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.86284			0.439315
$408$		0			0
$409$	−5.19101	−	8.99110i	−0.256679	−	0.444581i	0.708671	−	0.705539i	$-0.249295\pi$
$409$	−5.19101	−	8.99110i	−0.256679	−	0.444581i	−0.965350	+	0.260958i	$0.915962\pi$
$410$		0			0
$411$	6.04788			0.298320
$412$		0			0
$413$	−7.03879	+	12.1915i	−0.346356	+	0.599907i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−14.1868			−0.694730
$418$		0			0
$419$	−15.0958			−0.737479			−0.368740	−	0.929533i	$-0.620211\pi$
$419$	−15.0958			−0.737479			−0.368740	+	0.929533i	$0.620211\pi$
$420$		0			0
$421$	−6.47485	+	11.2148i	−0.315565	+	0.546574i	−0.979557	−	0.201165i	$-0.935527\pi$
$421$	−6.47485	+	11.2148i	−0.315565	+	0.546574i	0.663993	+	0.747739i	$0.268861\pi$
$422$		0			0
$423$	−1.43738	+	2.48962i	−0.0698879	+	0.121049i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.358235	−	0.620480i	−0.0173362	−	0.0300272i
$428$		0			0
$429$	−5.35765			−0.258670
$430$		0			0
$431$	15.0654	+	26.0940i	0.725674	+	1.25690i	0.958696	+	0.284433i	$0.0918053\pi$
$431$	15.0654	+	26.0940i	0.725674	+	1.25690i	−0.233022	+	0.972472i	$0.574861\pi$
$432$		0			0
$433$	12.9281	+	22.3922i	0.621287	+	1.07610i	0.989246	+	0.146259i	$0.0467232\pi$
$433$	12.9281	+	22.3922i	0.621287	+	1.07610i	−0.367959	+	0.929842i	$0.619943\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.72487	−	5.50031i	0.130348	−	0.263116i
$438$		0			0
$439$	4.35016	−	7.53469i	0.207622	−	0.359611i	−0.743343	−	0.668910i	$-0.766761\pi$
$439$	4.35016	−	7.53469i	0.207622	−	0.359611i	0.950965	+	0.309299i	$0.100094\pi$
$440$		0			0
$441$	0.821467	+	1.42282i	0.0391175	+	0.0677534i
$442$		0			0
$443$	−15.0822	−	26.1231i	−0.716575	−	1.24114i	−0.962349	−	0.271817i	$-0.912375\pi$
$443$	−15.0822	−	26.1231i	−0.716575	−	1.24114i	0.245774	−	0.969327i	$-0.420958\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−2.14081	−	3.70798i	−0.101257	−	0.175382i
$448$		0			0
$449$	23.7821			1.12235			0.561173	−	0.827699i	$-0.310350\pi$
$449$	23.7821			1.12235			0.561173	+	0.827699i	$0.310350\pi$
$450$		0			0
$451$	−0.790849	+	1.36979i	−0.0372396	+	0.0645009i
$452$		0			0
$453$	2.93845	−	5.08954i	0.138060	−	0.239128i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	26.5258			1.24082			0.620412	−	0.784276i	$-0.286966\pi$
$457$	26.5258			1.24082			0.620412	+	0.784276i	$0.286966\pi$
$458$		0			0
$459$	−3.15825	+	5.47025i	−0.147414	+	0.255329i
$460$		0			0
$461$	7.34402	−	12.7202i	0.342045	−	0.592439i	−0.642767	−	0.766061i	$-0.722214\pi$
$461$	7.34402	−	12.7202i	0.342045	−	0.592439i	0.984812	+	0.173622i	$0.0555472\pi$
$462$		0			0
$463$	9.22167			0.428567			0.214284	−	0.976771i	$-0.431258\pi$
$463$	9.22167			0.428567			0.214284	+	0.976771i	$0.431258\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−31.3069			−1.44871			−0.724355	−	0.689427i	$-0.757862\pi$
$467$	−31.3069			−1.44871			−0.724355	+	0.689427i	$0.757862\pi$
$468$		0			0
$469$	−3.65139	−	6.32439i	−0.168605	−	0.292033i
$470$		0			0
$471$	−1.81227	−	3.13895i	−0.0835051	−	0.144635i
$472$		0			0
$473$	3.49828	−	6.05920i	0.160851	−	0.278602i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.112843	−	0.195449i	0.00516671	−	0.00894900i
$478$		0			0
$479$	−16.0062	−	27.7236i	−0.731344	−	1.26672i	−0.956309	−	0.292357i	$-0.905560\pi$
$479$	−16.0062	−	27.7236i	−0.731344	−	1.26672i	0.224966	−	0.974367i	$-0.427773\pi$
$480$		0			0
$481$	−1.81226	−	3.13893i	−0.0826321	−	0.143123i
$482$		0			0
$483$	−4.49393			−0.204481
$484$		0			0
$485$		0			0
$486$		0			0
$487$	2.19899			0.0996456			0.0498228	−	0.998758i	$-0.484134\pi$
$487$	2.19899			0.0996456			0.0498228	+	0.998758i	$0.484134\pi$
$488$		0			0
$489$	1.51041	−	2.61611i	0.0683031	−	0.118304i
$490$		0			0
$491$	−6.78283	+	11.7482i	−0.306105	+	0.530189i	−0.977507	−	0.210904i	$-0.932359\pi$
$491$	−6.78283	+	11.7482i	−0.306105	+	0.530189i	0.671402	+	0.741094i	$0.265693\pi$
$492$		0			0
$493$	−0.117951			−0.00531227
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.36210	−	4.09128i	0.105955	−	0.183519i
$498$		0			0
$499$	−6.20880	+	10.7540i	−0.277944	+	0.481413i	−0.970874	−	0.239592i	$-0.922986\pi$
$499$	−6.20880	+	10.7540i	−0.277944	+	0.481413i	0.692930	+	0.721005i	$0.256320\pi$
$500$		0			0
$501$	−46.4658			−2.07594
$502$		0			0
$503$	8.43957	+	14.6178i	0.376302	+	0.651774i	0.990521	−	0.137361i	$-0.0438622\pi$
$503$	8.43957	+	14.6178i	0.376302	+	0.651774i	−0.614219	+	0.789136i	$0.710529\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−10.9064	−	18.8905i	−0.484372	−	0.838957i
$508$		0			0
$509$	12.4081	+	21.4915i	0.549981	+	0.952594i	0.998275	+	0.0587085i	$0.0186983\pi$
$509$	12.4081	+	21.4915i	0.549981	+	0.952594i	−0.448295	+	0.893886i	$0.647968\pi$
$510$		0			0
$511$	7.83159	−	13.5647i	0.346449	−	0.600067i
$512$		0			0
$513$	11.5524	+	17.3575i	0.510053	+	0.766353i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−9.35189	−	16.1979i	−0.411296	−	0.712385i
$518$		0			0
$519$	−12.9488	−	22.4279i	−0.568388	−	0.984477i
$520$		0			0
$521$	−32.6549			−1.43064			−0.715318	−	0.698799i	$-0.753718\pi$
$521$	−32.6549			−1.43064			−0.715318	+	0.698799i	$0.753718\pi$
$522$		0			0
$523$	−1.72603	−	2.98957i	−0.0754741	−	0.130725i	0.825818	−	0.563936i	$-0.190714\pi$
$523$	−1.72603	−	2.98957i	−0.0754741	−	0.130725i	−0.901292	+	0.433211i	$0.857380\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.87864	+	10.1821i	−0.256078	+	0.443539i
$528$		0			0
$529$	10.5085	−	18.2012i	0.456890	−	0.791356i
$530$		0			0
$531$	3.33474			0.144715
$532$		0			0
$533$	0.646847			0.0280181
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−21.8809	+	37.8989i	−0.944231	+	1.63546i
$538$		0			0
$539$	−10.6892			−0.460418
$540$		0			0
$541$	−3.69787	−	6.40490i	−0.158984	−	0.275368i	0.775519	−	0.631325i	$-0.217488\pi$
$541$	−3.69787	−	6.40490i	−0.158984	−	0.275368i	−0.934503	+	0.355956i	$0.884155\pi$
$542$		0			0
$543$	−14.4598			−0.620529
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−3.74778	−	6.49134i	−0.160243	−	0.277550i	0.774713	−	0.632314i	$-0.217895\pi$
$547$	−3.74778	−	6.49134i	−0.160243	−	0.277550i	−0.934956	+	0.354764i	$0.884561\pi$
$548$		0			0
$549$	−0.0848596	+	0.146981i	−0.00362172	+	0.00627300i
$550$		0			0
$551$	−0.172839	+	0.348887i	−0.00736319	+	0.0148631i
$552$		0			0
$553$	−10.1359	+	17.5560i	−0.431024	+	0.746556i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.1744	−	22.8187i	−0.558217	−	0.966859i	−0.997645	−	0.0685820i	$-0.978153\pi$
$557$	−13.1744	−	22.8187i	−0.558217	−	0.966859i	0.439429	−	0.898277i	$-0.355181\pi$
$558$		0			0
$559$	−2.86130			−0.121020
$560$		0			0
$561$	3.24732	+	5.62453i	0.137102	+	0.237468i
$562$		0			0
$563$	−10.8731			−0.458247			−0.229123	−	0.973397i	$-0.573586\pi$
$563$	−10.8731			−0.458247			−0.229123	+	0.973397i	$0.573586\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	8.69385	−	15.0582i	0.365107	−	0.632384i
$568$		0			0
$569$	−18.8171			−0.788853			−0.394427	−	0.918927i	$-0.629057\pi$
$569$	−18.8171			−0.788853			−0.394427	+	0.918927i	$0.629057\pi$
$570$		0			0
$571$	−29.0661			−1.21638			−0.608189	−	0.793792i	$-0.708104\pi$
$571$	−29.0661			−1.21638			−0.608189	+	0.793792i	$0.708104\pi$
$572$		0			0
$573$	5.57351	−	9.65360i	0.232837	−	0.403285i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	17.6665			0.735468			0.367734	−	0.929931i	$-0.380134\pi$
$577$	17.6665			0.735468			0.367734	+	0.929931i	$0.380134\pi$
$578$		0			0
$579$	13.8821	+	24.0444i	0.576919	+	0.999253i
$580$		0			0
$581$	13.4212			0.556804
$582$		0			0
$583$	0.734176	+	1.27163i	0.0304065	+	0.0526655i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.2815	+	26.4684i	−0.630736	+	1.09247i	0.356666	+	0.934232i	$0.383913\pi$
$587$	−15.2815	+	26.4684i	−0.630736	+	1.09247i	−0.987402	+	0.158234i	$0.949420\pi$
$588$		0			0
$589$	21.5032	+	32.3086i	0.886026	+	1.33125i
$590$		0			0
$591$	1.34245	−	2.32520i	0.0552211	−	0.0956458i
$592$		0			0
$593$	11.0606	+	19.1576i	0.454206	+	0.786707i	0.998642	−	0.0520949i	$-0.0165898\pi$
$593$	11.0606	+	19.1576i	0.454206	+	0.786707i	−0.544437	+	0.838802i	$0.683257\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	44.5719			1.82421
$598$		0			0
$599$	−14.0909	−	24.4062i	−0.575740	−	0.997210i	−0.995961	−	0.0897885i	$-0.971381\pi$
$599$	−14.0909	−	24.4062i	−0.575740	−	0.997210i	0.420221	−	0.907422i	$-0.361952\pi$
$600$		0			0
$601$	−32.7927			−1.33764			−0.668821	−	0.743424i	$-0.733201\pi$
$601$	−32.7927			−1.33764			−0.668821	+	0.743424i	$0.733201\pi$
$602$		0			0
$603$	−0.864950	+	1.49814i	−0.0352235	+	0.0610089i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−38.2403			−1.55213			−0.776063	−	0.630656i	$-0.782786\pi$
$607$	−38.2403			−1.55213			−0.776063	+	0.630656i	$0.782786\pi$
$608$		0			0
$609$	0.285051			0.0115509
$610$		0			0
$611$	−3.82453	+	6.62427i	−0.154724	+	0.267989i
$612$		0			0
$613$	−3.72203	+	6.44674i	−0.150331	+	0.260381i	−0.931349	−	0.364127i	$-0.881367\pi$
$613$	−3.72203	+	6.44674i	−0.150331	+	0.260381i	0.781018	+	0.624509i	$0.214701\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.23294	−	7.33167i	−0.170412	−	0.295162i	0.768152	−	0.640267i	$-0.221176\pi$
$617$	−4.23294	−	7.33167i	−0.170412	−	0.295162i	−0.938564	+	0.345106i	$0.887843\pi$
$618$		0			0
$619$	−35.1940			−1.41457			−0.707283	−	0.706931i	$-0.750079\pi$
$619$	−35.1940			−1.41457			−0.707283	+	0.706931i	$0.750079\pi$
$620$		0			0
$621$	−3.36805	−	5.83363i	−0.135155	−	0.234095i
$622$		0			0
$623$	−0.228956	−	0.396564i	−0.00917294	−	0.0158880i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	21.3951	−	1.36335i	0.854439	−	0.0544469i
$628$		0			0
$629$	−2.19686	+	3.80507i	−0.0875944	+	0.151718i
$630$		0			0
$631$	−5.23316	−	9.06410i	−0.208329	−	0.360836i	0.742859	−	0.669447i	$-0.233469\pi$
$631$	−5.23316	−	9.06410i	−0.208329	−	0.360836i	−0.951188	+	0.308611i	$0.900136\pi$
$632$		0			0
$633$	14.0990	+	24.4201i	0.560383	+	0.970612i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.18572	+	3.78579i	0.0866016	+	0.149998i
$638$		0			0
$639$	−1.11908			−0.0442702
$640$		0			0
$641$	2.85836	−	4.95082i	0.112898	−	0.195546i	−0.804039	−	0.594576i	$-0.797320\pi$
$641$	2.85836	−	4.95082i	0.112898	−	0.195546i	0.916938	+	0.399030i	$0.130653\pi$
$642$		0			0
$643$	17.5249	−	30.3540i	0.691114	−	1.19705i	−0.280359	−	0.959895i	$-0.590453\pi$
$643$	17.5249	−	30.3540i	0.691114	−	1.19705i	0.971473	−	0.237150i	$-0.0762133\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.40523			0.330444			0.165222	−	0.986256i	$-0.447166\pi$
$647$	8.40523			0.330444			0.165222	+	0.986256i	$0.447166\pi$
$648$		0			0
$649$	−10.8482	+	18.7897i	−0.425830	+	0.737559i
$650$		0			0
$651$	14.2068	−	24.6069i	0.556809	−	0.964421i
$652$		0			0
$653$	−0.586740			−0.0229609			−0.0114805	−	0.999934i	$-0.503654\pi$
$653$	−0.586740			−0.0229609			−0.0114805	+	0.999934i	$0.503654\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−3.71034			−0.144754
$658$		0			0
$659$	−4.16313	−	7.21076i	−0.162173	−	0.280891i	0.773475	−	0.633827i	$-0.218517\pi$
$659$	−4.16313	−	7.21076i	−0.162173	−	0.280891i	−0.935648	+	0.352936i	$0.885183\pi$
$660$		0			0
$661$	0.564657	+	0.978014i	0.0219626	+	0.0380403i	0.876798	−	0.480859i	$-0.159675\pi$
$661$	0.564657	+	0.978014i	0.0219626	+	0.0380403i	−0.854835	+	0.518900i	$0.826342\pi$
$662$		0			0
$663$	1.32802	−	2.30019i	0.0515759	−	0.0893321i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.0628934	−	0.108935i	0.00243524	−	0.00421796i
$668$		0			0
$669$	3.17948	+	5.50703i	0.122926	+	0.212914i
$670$		0			0
$671$	−0.552113	−	0.956288i	−0.0213141	−	0.0369171i
$672$		0			0
$673$	9.90978			0.381994			0.190997	−	0.981591i	$-0.438828\pi$
$673$	9.90978			0.381994			0.190997	+	0.981591i	$0.438828\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−32.9039			−1.26460			−0.632301	−	0.774723i	$-0.717889\pi$
$677$	−32.9039			−1.26460			−0.632301	+	0.774723i	$0.717889\pi$
$678$		0			0
$679$	−1.45812	+	2.52553i	−0.0559574	+	0.0969211i
$680$		0			0
$681$	−23.0765	+	39.9697i	−0.884295	+	1.53164i
$682$		0			0
$683$	3.22470			0.123390			0.0616949	−	0.998095i	$-0.480349\pi$
$683$	3.22470			0.123390			0.0616949	+	0.998095i	$0.480349\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	4.94716	−	8.56874i	0.188746	−	0.326918i
$688$		0			0
$689$	0.300247	−	0.520043i	0.0114385	−	0.0198120i
$690$		0			0
$691$	−8.98569			−0.341832			−0.170916	−	0.985286i	$-0.554673\pi$
$691$	−8.98569			−0.341832			−0.170916	+	0.985286i	$0.554673\pi$
$692$		0			0
$693$	−0.942363	−	1.63222i	−0.0357974	−	0.0620029i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.392060	−	0.679067i	−0.0148503	−	0.0257215i
$698$		0			0
$699$	12.0106	+	20.8029i	0.454282	+	0.786839i
$700$		0			0
$701$	7.62340	−	13.2041i	0.287932	−	0.498713i	−0.685384	−	0.728182i	$-0.740366\pi$
$701$	7.62340	−	13.2041i	0.287932	−	0.498713i	0.973316	+	0.229469i	$0.0736990\pi$
$702$		0			0
$703$	8.03580	+	12.0738i	0.303076	+	0.455371i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.30554	−	14.3856i	−0.312362	−	0.541027i
$708$		0			0
$709$	−17.6736	−	30.6116i	−0.663746	−	1.14964i	−0.979624	−	0.200841i	$-0.935632\pi$
$709$	−17.6736	−	30.6116i	−0.663746	−	1.14964i	0.315878	−	0.948800i	$-0.397701\pi$
$710$		0			0
$711$	4.80206			0.180091
$712$		0			0
$713$	−6.26915	−	10.8585i	−0.234781	−	0.406653i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−7.83347	+	13.5680i	−0.292546	+	0.506705i
$718$		0			0
$719$	14.9566	−	25.9056i	0.557787	−	0.966115i	−0.439894	−	0.898050i	$-0.644984\pi$
$719$	14.9566	−	25.9056i	0.557787	−	0.966115i	0.997681	−	0.0680651i	$-0.0216826\pi$
$720$		0			0
$721$	9.35194			0.348284
$722$		0			0
$723$	0.0486525			0.00180941
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−19.1230	+	33.1220i	−0.709232	+	1.22843i	0.255911	+	0.966700i	$0.417625\pi$
$727$	−19.1230	+	33.1220i	−0.709232	+	1.22843i	−0.965142	+	0.261725i	$0.915709\pi$
$728$		0			0
$729$	22.3783			0.828827
$730$		0			0
$731$	1.73426	+	3.00382i	0.0641438	+	0.111100i
$732$		0			0
$733$	1.55553			0.0574549			0.0287274	−	0.999587i	$-0.490855\pi$
$733$	1.55553			0.0574549			0.0287274	+	0.999587i	$0.490855\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.62754	−	9.74718i	−0.207293	−	0.359042i
$738$		0			0
$739$	13.5945	−	23.5464i	0.500083	−	0.866169i	−0.499917	−	0.866073i	$-0.666636\pi$
$739$	13.5945	−	23.5464i	0.500083	−	0.866169i	1.00000		9.57964e-5i	$-3.04929e-5\pi$
$740$		0			0
$741$	−4.85770	−	7.29868i	−0.178452	−	0.268124i
$742$		0			0
$743$	9.54725	−	16.5363i	0.350255	−	0.606659i	−0.636039	−	0.771657i	$-0.719428\pi$
$743$	9.54725	−	16.5363i	0.350255	−	0.606659i	0.986294	+	0.164998i	$0.0527617\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.58962	−	2.75331i	−0.0581613	−	0.100738i
$748$		0			0
$749$	−12.0914			−0.441809
$750$		0			0
$751$	−1.08150	−	1.87321i	−0.0394644	−	0.0683544i	0.845619	−	0.533788i	$-0.179232\pi$
$751$	−1.08150	−	1.87321i	−0.0394644	−	0.0683544i	−0.885083	+	0.465433i	$0.845899\pi$
$752$		0			0
$753$	36.2659			1.32160
$754$		0			0
$755$		0			0
$756$		0			0
$757$	24.1677	−	41.8598i	0.878392	−	1.52142i	0.0252862	−	0.999680i	$-0.491950\pi$
$757$	24.1677	−	41.8598i	0.878392	−	1.52142i	0.853105	−	0.521739i	$-0.174716\pi$
$758$		0			0
$759$	−6.92607			−0.251401
$760$		0			0
$761$	49.4377			1.79211			0.896057	−	0.443939i	$-0.146419\pi$
$761$	49.4377			1.79211			0.896057	+	0.443939i	$0.146419\pi$
$762$		0			0
$763$	−3.54465	+	6.13951i	−0.128325	+	0.222265i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.87293			0.320383
$768$		0			0
$769$	10.7781	+	18.6682i	0.388667	+	0.673192i	0.992271	−	0.124093i	$-0.0396022\pi$
$769$	10.7781	+	18.6682i	0.388667	+	0.673192i	−0.603603	+	0.797285i	$0.706269\pi$
$770$		0			0
$771$	−17.6104			−0.634222
$772$		0			0
$773$	−22.9699	−	39.7850i	−0.826169	−	1.43097i	−0.901022	−	0.433772i	$-0.857182\pi$
$773$	−22.9699	−	39.7850i	−0.826169	−	1.43097i	0.0748533	−	0.997195i	$-0.476151\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	5.30911	−	9.19564i	0.190463	−	0.329892i
$778$		0			0
$779$	−2.58310	+	0.164602i	−0.0925493	+	0.00589746i
$780$		0			0
$781$	3.64048	−	6.30550i	0.130267	−	0.225629i
$782$		0			0
$783$	0.213636	+	0.370029i	0.00763473	+	0.0132237i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	43.7925			1.56103			0.780517	−	0.625135i	$-0.214956\pi$
$787$	43.7925			1.56103			0.780517	+	0.625135i	$0.214956\pi$
$788$		0			0
$789$	10.4294	+	18.0642i	0.371295	+	0.643102i
$790$		0			0
$791$	22.3665			0.795261
$792$		0			0
$793$	−0.225791	+	0.391081i	−0.00801807	+	0.0138877i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−44.8269			−1.58785			−0.793925	−	0.608016i	$-0.791966\pi$
$797$	−44.8269			−1.58785			−0.793925	+	0.608016i	$0.791966\pi$
$798$		0			0
$799$	9.27232			0.328031
$800$		0			0
$801$	−0.0542358	+	0.0939392i	−0.00191633	+

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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.923228 - 1.59908i) q^{3} +1.72830 q^{7} +(-0.204702 - 0.354554i) q^{9} +O(q^{10})\)	\(q+(0.923228 - 1.59908i) q^{3} +1.72830 q^{7} +(-0.204702 - 0.354554i) q^{9} +2.66366 q^{11} +(-0.544662 - 0.943382i) q^{13} +(-0.660249 + 1.14358i) q^{17} +(2.41510 + 3.62868i) q^{19} +(1.59561 - 2.76368i) q^{21} +(-0.704108 - 1.21955i) q^{23} +4.78343 q^{27} +(0.0446618 + 0.0773564i) q^{29} +8.90368 q^{31} +(2.45917 - 4.25940i) q^{33} +3.32732 q^{37} -2.01139 q^{39} +(-0.296903 + 0.514251i) q^{41} +(1.31334 - 2.27477i) q^{43} +(-3.51092 - 6.08109i) q^{47} -4.01299 q^{49} +(1.21912 + 2.11158i) q^{51} +(0.275627 + 0.477400i) q^{53} +(8.03223 - 0.511833i) q^{57} +(-4.07268 + 7.05408i) q^{59} +(-0.207276 - 0.359013i) q^{61} +(-0.353785 - 0.612774i) q^{63} +(-2.11271 - 3.65932i) q^{67} -2.60021 q^{69} +(1.36672 - 2.36723i) q^{71} +(4.53139 - 7.84860i) q^{73} +4.60359 q^{77} +(-5.86470 + 10.1580i) q^{79} +(5.03030 - 8.71273i) q^{81} +7.76556 q^{83} +0.164932 q^{87} +(-0.132475 - 0.229454i) q^{89} +(-0.941337 - 1.63044i) q^{91} +(8.22013 - 14.2377i) q^{93} +(-0.843674 + 1.46129i) q^{97} +(-0.545256 - 0.944410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} - 3 q^{9} + O(q^{10}) \)	\( 12 q - 3 q^{3} - 3 q^{9} + 2 q^{11} - 7 q^{13} + q^{17} + q^{21} + 2 q^{23} + 24 q^{27} + q^{29} + 2 q^{31} - 10 q^{33} - 20 q^{37} + 36 q^{39} - 7 q^{41} - 19 q^{43} + 14 q^{47} + 8 q^{49} + 11 q^{51} - 6 q^{53} + 28 q^{57} - 5 q^{61} + 11 q^{63} - 14 q^{67} - 14 q^{69} + 8 q^{71} + 9 q^{73} - 2 q^{77} + q^{79} + 2 q^{81} + 26 q^{83} - 30 q^{87} - 8 q^{89} + 3 q^{91} - 9 q^{93} + 11 q^{97} - 18 q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more