LMFDB - Embedded newform 2352.2.h.j.2255.4

Newspace parameters

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.h (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{11})$
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2255.4
Root			$-1.65831 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.j.2255.3

$q$-expansion

$f(q)$	$=$	$q+(1.65831 + 0.500000i) q^{3} -3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +O(q^{10})$	$q+(1.65831 + 0.500000i) q^{3} -3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +3.31662 q^{11} +4.00000 q^{13} +(1.65831 - 5.50000i) q^{15} +3.31662i q^{17} +7.00000i q^{19} +3.31662 q^{23} -6.00000 q^{25} +(3.31662 + 4.00000i) q^{27} -6.63325i q^{29} +3.00000i q^{31} +(5.50000 + 1.65831i) q^{33} -1.00000 q^{37} +(6.63325 + 2.00000i) q^{39} -6.63325i q^{41} +2.00000i q^{43} +(5.50000 - 8.29156i) q^{45} -9.94987 q^{47} +(-1.65831 + 5.50000i) q^{51} +3.31662i q^{53} -11.0000i q^{55} +(-3.50000 + 11.6082i) q^{57} +3.31662 q^{59} -3.00000 q^{61} -13.2665i q^{65} -9.00000i q^{67} +(5.50000 + 1.65831i) q^{69} +13.2665 q^{71} -7.00000 q^{73} +(-9.94987 - 3.00000i) q^{75} -9.00000i q^{79} +(3.50000 + 8.29156i) q^{81} +13.2665 q^{83} +11.0000 q^{85} +(3.31662 - 11.0000i) q^{87} +3.31662i q^{89} +(-1.50000 + 4.97494i) q^{93} +23.2164 q^{95} -8.00000 q^{97} +(8.29156 + 5.50000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{9}+O(q^{10}) $ 4 * q + 10 * q^9	$ 4 q + 10 q^{9} + 16 q^{13} - 24 q^{25} + 22 q^{33} - 4 q^{37} + 22 q^{45} - 14 q^{57} - 12 q^{61} + 22 q^{69} - 28 q^{73} + 14 q^{81} + 44 q^{85} - 6 q^{93} - 32 q^{97}+O(q^{100}) $ 4 * q + 10 * q^9 + 16 * q^13 - 24 * q^25 + 22 * q^33 - 4 * q^37 + 22 * q^45 - 14 * q^57 - 12 * q^61 + 22 * q^69 - 28 * q^73 + 14 * q^81 + 44 * q^85 - 6 * q^93 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	1.65831	+	0.500000i	0.957427	+	0.288675i
$3$	1.65831	+	0.500000i	0.957427	+	0.288675i
$4$		0			0
$5$		−	3.31662i		−	1.48324i	−0.670820	−	0.741620i	$-0.734058\pi$
$5$		−	3.31662i		−	1.48324i	0.670820	−	0.741620i	$-0.265942\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.50000	+	1.65831i	0.833333	+	0.552771i
$10$		0			0
$11$	3.31662			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$11$	3.31662			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$	1.65831	−	5.50000i	0.428174	−	1.42009i
$16$		0			0
$17$			3.31662i			0.804400i	0.915552	+	0.402200i	$0.131754\pi$
$17$			3.31662i			0.804400i	−0.915552	+	0.402200i	$0.868246\pi$
$18$		0			0
$19$			7.00000i			1.60591i	0.596040	+	0.802955i	$0.296740\pi$
$19$			7.00000i			1.60591i	−0.596040	+	0.802955i	$0.703260\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.31662			0.691564			0.345782	−	0.938315i	$-0.387614\pi$
$23$	3.31662			0.691564			0.345782	+	0.938315i	$0.387614\pi$
$24$		0			0
$25$	−6.00000			−1.20000
$26$		0			0
$27$	3.31662	+	4.00000i	0.638285	+	0.769800i
$28$		0			0
$29$		−	6.63325i		−	1.23176i	−0.787839	−	0.615882i	$-0.788800\pi$
$29$		−	6.63325i		−	1.23176i	0.787839	−	0.615882i	$-0.211200\pi$
$30$		0			0
$31$			3.00000i			0.538816i	0.963026	+	0.269408i	$0.0868280\pi$
$31$			3.00000i			0.538816i	−0.963026	+	0.269408i	$0.913172\pi$
$32$		0			0
$33$	5.50000	+	1.65831i	0.957427	+	0.288675i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$	6.63325	+	2.00000i	1.06217	+	0.320256i
$40$		0			0
$41$		−	6.63325i		−	1.03594i	−0.855399	−	0.517970i	$-0.826688\pi$
$41$		−	6.63325i		−	1.03594i	0.855399	−	0.517970i	$-0.173312\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$	5.50000	−	8.29156i	0.819892	−	1.23603i
$46$		0			0
$47$	−9.94987			−1.45134			−0.725669	−	0.688044i	$-0.758470\pi$
$47$	−9.94987			−1.45134			−0.725669	+	0.688044i	$0.758470\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−1.65831	+	5.50000i	−0.232210	+	0.770154i
$52$		0			0
$53$			3.31662i			0.455573i	0.973711	+	0.227787i	$0.0731489\pi$
$53$			3.31662i			0.455573i	−0.973711	+	0.227787i	$0.926851\pi$
$54$		0			0
$55$		−	11.0000i		−	1.48324i
$56$		0			0
$57$	−3.50000	+	11.6082i	−0.463586	+	1.53754i
$58$		0			0
$59$	3.31662			0.431788			0.215894	−	0.976417i	$-0.430733\pi$
$59$	3.31662			0.431788			0.215894	+	0.976417i	$0.430733\pi$
$60$		0			0
$61$	−3.00000			−0.384111			−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000			−0.384111			−0.192055	+	0.981384i	$0.561515\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	13.2665i		−	1.64551i
$66$		0			0
$67$		−	9.00000i		−	1.09952i	−0.835321	−	0.549762i	$-0.814718\pi$
$67$		−	9.00000i		−	1.09952i	0.835321	−	0.549762i	$-0.185282\pi$
$68$		0			0
$69$	5.50000	+	1.65831i	0.662122	+	0.199637i
$70$		0			0
$71$	13.2665			1.57444			0.787222	−	0.616670i	$-0.211519\pi$
$71$	13.2665			1.57444			0.787222	+	0.616670i	$0.211519\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$	−9.94987	−	3.00000i	−1.14891	−	0.346410i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	9.00000i		−	1.01258i	−0.862364	−	0.506290i	$-0.831017\pi$
$79$		−	9.00000i		−	1.01258i	0.862364	−	0.506290i	$-0.168983\pi$
$80$		0			0
$81$	3.50000	+	8.29156i	0.388889	+	0.921285i
$82$		0			0
$83$	13.2665			1.45619			0.728094	−	0.685478i	$-0.240407\pi$
$83$	13.2665			1.45619			0.728094	+	0.685478i	$0.240407\pi$
$84$		0			0
$85$	11.0000			1.19312
$86$		0			0
$87$	3.31662	−	11.0000i	0.355580	−	1.17932i
$88$		0			0
$89$			3.31662i			0.351562i	0.984429	+	0.175781i	$0.0562450\pi$
$89$			3.31662i			0.351562i	−0.984429	+	0.175781i	$0.943755\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−1.50000	+	4.97494i	−0.155543	+	0.515877i
$94$		0			0
$95$	23.2164			2.38195
$96$		0			0
$97$	−8.00000			−0.812277			−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000			−0.812277			−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$	8.29156	+	5.50000i	0.833333	+	0.552771i
$100$		0			0
$101$			3.31662i			0.330017i	0.986292	+	0.165008i	$0.0527651\pi$
$101$			3.31662i			0.330017i	−0.986292	+	0.165008i	$0.947235\pi$
$102$		0			0
$103$		−	11.0000i		−	1.08386i	−0.840423	−	0.541931i	$-0.817693\pi$
$103$		−	11.0000i		−	1.08386i	0.840423	−	0.541931i	$-0.182307\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.31662			−0.320630			−0.160315	−	0.987066i	$-0.551251\pi$
$107$	−3.31662			−0.320630			−0.160315	+	0.987066i	$0.551251\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$		0			0
$111$	−1.65831	−	0.500000i	−0.157400	−	0.0474579i
$112$		0			0
$113$		−	6.63325i		−	0.624004i	−0.950082	−	0.312002i	$-0.899000\pi$
$113$		−	6.63325i		−	0.624004i	0.950082	−	0.312002i	$-0.101000\pi$
$114$		0			0
$115$		−	11.0000i		−	1.02576i
$116$		0			0
$117$	10.0000	+	6.63325i	0.924500	+	0.613244i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$	3.31662	−	11.0000i	0.299050	−	0.991837i
$124$		0			0
$125$			3.31662i			0.296648i
$126$		0			0
$127$			2.00000i			0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$			2.00000i			0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$		0			0
$129$	−1.00000	+	3.31662i	−0.0880451	+	0.292013i
$130$		0			0
$131$	−16.5831			−1.44887			−0.724437	−	0.689341i	$-0.757900\pi$
$131$	−16.5831			−1.44887			−0.724437	+	0.689341i	$0.757900\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	13.2665	−	11.0000i	1.14180	−	0.946729i
$136$		0			0
$137$		−	16.5831i		−	1.41679i	−0.705815	−	0.708396i	$-0.749419\pi$
$137$		−	16.5831i		−	1.41679i	0.705815	−	0.708396i	$-0.250581\pi$
$138$		0			0
$139$			10.0000i			0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$			10.0000i			0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$		0			0
$141$	−16.5000	−	4.97494i	−1.38955	−	0.418965i
$142$		0			0
$143$	13.2665			1.10940
$144$		0			0
$145$	−22.0000			−1.82700
$146$		0			0
$147$		0			0
$148$		0			0
$149$			16.5831i			1.35854i	0.733887	+	0.679271i	$0.237704\pi$
$149$			16.5831i			1.35854i	−0.733887	+	0.679271i	$0.762296\pi$
$150$		0			0
$151$		−	7.00000i		−	0.569652i	−0.958579	−	0.284826i	$-0.908064\pi$
$151$		−	7.00000i		−	0.569652i	0.958579	−	0.284826i	$-0.0919358\pi$
$152$		0			0
$153$	−5.50000	+	8.29156i	−0.444649	+	0.670333i
$154$		0			0
$155$	9.94987			0.799193
$156$		0			0
$157$	−3.00000			−0.239426			−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000			−0.239426			−0.119713	+	0.992809i	$0.538197\pi$
$158$		0			0
$159$	−1.65831	+	5.50000i	−0.131513	+	0.436178i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			7.00000i			0.548282i	0.961689	+	0.274141i	$0.0883936\pi$
$163$			7.00000i			0.548282i	−0.961689	+	0.274141i	$0.911606\pi$
$164$		0			0
$165$	5.50000	−	18.2414i	0.428174	−	1.42009i
$166$		0			0
$167$	−6.63325			−0.513296			−0.256648	−	0.966505i	$-0.582618\pi$
$167$	−6.63325			−0.513296			−0.256648	+	0.966505i	$0.582618\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	−11.6082	+	17.5000i	−0.887700	+	1.33826i
$172$		0			0
$173$		−	9.94987i		−	0.756475i	−0.925709	−	0.378237i	$-0.876530\pi$
$173$		−	9.94987i		−	0.756475i	0.925709	−	0.378237i	$-0.123470\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.50000	+	1.65831i	0.413405	+	0.124646i
$178$		0			0
$179$	3.31662			0.247896			0.123948	−	0.992289i	$-0.460444\pi$
$179$	3.31662			0.247896			0.123948	+	0.992289i	$0.460444\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$	−4.97494	−	1.50000i	−0.367758	−	0.110883i
$184$		0			0
$185$			3.31662i			0.243843i
$186$		0			0
$187$			11.0000i			0.804400i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.94987			0.719948			0.359974	−	0.932962i	$-0.382786\pi$
$191$	9.94987			0.719948			0.359974	+	0.932962i	$0.382786\pi$
$192$		0			0
$193$	−3.00000			−0.215945			−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000			−0.215945			−0.107972	+	0.994154i	$0.534436\pi$
$194$		0			0
$195$	6.63325	−	22.0000i	0.475017	−	1.57545i
$196$		0			0
$197$			19.8997i			1.41780i	0.705310	+	0.708899i	$0.250808\pi$
$197$			19.8997i			1.41780i	−0.705310	+	0.708899i	$0.749192\pi$
$198$		0			0
$199$		−	1.00000i		−	0.0708881i	−0.999372	−	0.0354441i	$-0.988715\pi$
$199$		−	1.00000i		−	0.0708881i	0.999372	−	0.0354441i	$-0.0112846\pi$
$200$		0			0
$201$	4.50000	−	14.9248i	0.317406	−	1.05272i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−22.0000			−1.53655
$206$		0			0
$207$	8.29156	+	5.50000i	0.576303	+	0.382276i
$208$		0			0
$209$			23.2164i			1.60591i
$210$		0			0
$211$		−	26.0000i		−	1.78991i	−0.446153	−	0.894957i	$-0.647206\pi$
$211$		−	26.0000i		−	1.78991i	0.446153	−	0.894957i	$-0.352794\pi$
$212$		0			0
$213$	22.0000	+	6.63325i	1.50742	+	0.454503i
$214$		0			0
$215$	6.63325			0.452384
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−11.6082	−	3.50000i	−0.784409	−	0.236508i
$220$		0			0
$221$			13.2665i			0.892401i
$222$		0			0
$223$			14.0000i			0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$			14.0000i			0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$		0			0
$225$	−15.0000	−	9.94987i	−1.00000	−	0.663325i
$226$		0			0
$227$	−3.31662			−0.220132			−0.110066	−	0.993924i	$-0.535106\pi$
$227$	−3.31662			−0.220132			−0.110066	+	0.993924i	$0.535106\pi$
$228$		0			0
$229$	−29.0000			−1.91637			−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−29.0000			−1.91637			−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.5831i			1.08640i	0.839604	+	0.543198i	$0.182787\pi$
$233$			16.5831i			1.08640i	−0.839604	+	0.543198i	$0.817213\pi$
$234$		0			0
$235$			33.0000i			2.15268i
$236$		0			0
$237$	4.50000	−	14.9248i	0.292306	−	0.969471i
$238$		0			0
$239$	−6.63325			−0.429069			−0.214535	−	0.976716i	$-0.568823\pi$
$239$	−6.63325			−0.429069			−0.214535	+	0.976716i	$0.568823\pi$
$240$		0			0
$241$	−19.0000			−1.22390			−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000			−1.22390			−0.611949	+	0.790897i	$0.709614\pi$
$242$		0			0
$243$	1.65831	+	15.5000i	0.106381	+	0.994325i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			28.0000i			1.78160i
$248$		0			0
$249$	22.0000	+	6.63325i	1.39419	+	0.420365i
$250$		0			0
$251$	19.8997			1.25606			0.628031	−	0.778189i	$-0.283861\pi$
$251$	19.8997			1.25606			0.628031	+	0.778189i	$0.283861\pi$
$252$		0			0
$253$	11.0000			0.691564
$254$		0			0
$255$	18.2414	+	5.50000i	1.14232	+	0.344423i
$256$		0			0
$257$			23.2164i			1.44820i	0.689696	+	0.724099i	$0.257744\pi$
$257$			23.2164i			1.44820i	−0.689696	+	0.724099i	$0.742256\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	11.0000	−	16.5831i	0.680883	−	1.02647i
$262$		0			0
$263$	−3.31662			−0.204512			−0.102256	−	0.994758i	$-0.532606\pi$
$263$	−3.31662			−0.204512			−0.102256	+	0.994758i	$0.532606\pi$
$264$		0			0
$265$	11.0000			0.675725
$266$		0			0
$267$	−1.65831	+	5.50000i	−0.101487	+	0.336595i
$268$		0			0
$269$		−	16.5831i		−	1.01109i	−0.862800	−	0.505545i	$-0.831291\pi$
$269$		−	16.5831i		−	1.01109i	0.862800	−	0.505545i	$-0.168709\pi$
$270$		0			0
$271$			17.0000i			1.03268i	0.856385	+	0.516338i	$0.172705\pi$
$271$			17.0000i			1.03268i	−0.856385	+	0.516338i	$0.827295\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−19.8997			−1.20000
$276$		0			0
$277$	−13.0000			−0.781094			−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000			−0.781094			−0.390547	+	0.920583i	$0.627714\pi$
$278$		0			0
$279$	−4.97494	+	7.50000i	−0.297842	+	0.449013i
$280$		0			0
$281$			6.63325i			0.395706i	0.980232	+	0.197853i	$0.0633969\pi$
$281$			6.63325i			0.395706i	−0.980232	+	0.197853i	$0.936603\pi$
$282$		0			0
$283$			15.0000i			0.891657i	0.895118	+	0.445829i	$0.147091\pi$
$283$			15.0000i			0.891657i	−0.895118	+	0.445829i	$0.852909\pi$
$284$		0			0
$285$	38.5000	+	11.6082i	2.28054	+	0.687610i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.00000			0.352941
$290$		0			0
$291$	−13.2665	−	4.00000i	−0.777696	−	0.234484i
$292$		0			0
$293$		−	13.2665i		−	0.775037i	−0.921862	−	0.387519i	$-0.873332\pi$
$293$		−	13.2665i		−	0.775037i	0.921862	−	0.387519i	$-0.126668\pi$
$294$		0			0
$295$		−	11.0000i		−	0.640445i
$296$		0			0
$297$	11.0000	+	13.2665i	0.638285	+	0.769800i
$298$		0			0
$299$	13.2665			0.767221
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−1.65831	+	5.50000i	−0.0952676	+	0.315967i
$304$		0			0
$305$			9.94987i			0.569728i
$306$		0			0
$307$		−	6.00000i		−	0.342438i	−0.985233	−	0.171219i	$-0.945229\pi$
$307$		−	6.00000i		−	0.342438i	0.985233	−	0.171219i	$-0.0547706\pi$
$308$		0			0
$309$	5.50000	−	18.2414i	0.312884	−	1.03772i
$310$		0			0
$311$	−9.94987			−0.564206			−0.282103	−	0.959384i	$-0.591032\pi$
$311$	−9.94987			−0.564206			−0.282103	+	0.959384i	$0.591032\pi$
$312$		0			0
$313$	9.00000			0.508710			0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000			0.508710			0.254355	+	0.967111i	$0.418137\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	29.8496i		−	1.67652i	−0.545269	−	0.838261i	$-0.683573\pi$
$317$		−	29.8496i		−	1.67652i	0.545269	−	0.838261i	$-0.316427\pi$
$318$		0			0
$319$		−	22.0000i		−	1.23176i
$320$		0			0
$321$	−5.50000	−	1.65831i	−0.306980	−	0.0925580i
$322$		0			0
$323$	−23.2164			−1.29179
$324$		0			0
$325$	−24.0000			−1.33128
$326$		0			0
$327$	8.29156	+	2.50000i	0.458524	+	0.138250i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	17.0000i		−	0.934405i	−0.884150	−	0.467202i	$-0.845262\pi$
$331$		−	17.0000i		−	0.934405i	0.884150	−	0.467202i	$-0.154738\pi$
$332$		0			0
$333$	−2.50000	−	1.65831i	−0.136999	−	0.0908750i
$334$		0			0
$335$	−29.8496			−1.63086
$336$		0			0
$337$	28.0000			1.52526			0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000			1.52526			0.762629	+	0.646837i	$0.223908\pi$
$338$		0			0
$339$	3.31662	−	11.0000i	0.180134	−	0.597438i
$340$		0			0
$341$			9.94987i			0.538816i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	5.50000	−	18.2414i	0.296110	−	0.982086i
$346$		0			0
$347$	−29.8496			−1.60241			−0.801206	−	0.598389i	$-0.795808\pi$
$347$	−29.8496			−1.60241			−0.801206	+	0.598389i	$0.795808\pi$
$348$		0			0
$349$	8.00000			0.428230			0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000			0.428230			0.214115	+	0.976808i	$0.431313\pi$
$350$		0			0
$351$	13.2665	+	16.0000i	0.708113	+	0.854017i
$352$		0			0
$353$		−	36.4829i		−	1.94179i	−0.239511	−	0.970894i	$-0.576987\pi$
$353$		−	36.4829i		−	1.94179i	0.239511	−	0.970894i	$-0.423013\pi$
$354$		0			0
$355$		−	44.0000i		−	2.33528i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	36.4829			1.92549			0.962746	−	0.270407i	$-0.0871582\pi$
$359$	36.4829			1.92549			0.962746	+	0.270407i	$0.0871582\pi$
$360$		0			0
$361$	−30.0000			−1.57895
$362$		0			0
$363$		0			0
$364$		0			0
$365$			23.2164i			1.21520i
$366$		0			0
$367$			1.00000i			0.0521996i	0.999659	+	0.0260998i	$0.00830876\pi$
$367$			1.00000i			0.0521996i	−0.999659	+	0.0260998i	$0.991691\pi$
$368$		0			0
$369$	11.0000	−	16.5831i	0.572637	−	0.863283i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−13.0000			−0.673114			−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000			−0.673114			−0.336557	+	0.941663i	$0.609263\pi$
$374$		0			0
$375$	−1.65831	+	5.50000i	−0.0856349	+	0.284019i
$376$		0			0
$377$		−	26.5330i		−	1.36652i
$378$		0			0
$379$			24.0000i			1.23280i	0.787434	+	0.616399i	$0.211409\pi$
$379$			24.0000i			1.23280i	−0.787434	+	0.616399i	$0.788591\pi$
$380$		0			0
$381$	−1.00000	+	3.31662i	−0.0512316	+	0.169916i
$382$		0			0
$383$	16.5831			0.847358			0.423679	−	0.905812i	$-0.360738\pi$
$383$	16.5831			0.847358			0.423679	+	0.905812i	$0.360738\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.31662	+	5.00000i	−0.168594	+	0.254164i
$388$		0			0
$389$		−	3.31662i		−	0.168160i	−0.996459	−	0.0840798i	$-0.973205\pi$
$389$		−	3.31662i		−	0.168160i	0.996459	−	0.0840798i	$-0.0267951\pi$
$390$		0			0
$391$			11.0000i			0.556294i
$392$		0			0
$393$	−27.5000	−	8.29156i	−1.38719	−	0.418254i
$394$		0			0
$395$	−29.8496			−1.50190
$396$		0			0
$397$	−13.0000			−0.652451			−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000			−0.652451			−0.326226	+	0.945292i	$0.605777\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			16.5831i			0.828122i	0.910249	+	0.414061i	$0.135890\pi$
$401$			16.5831i			0.828122i	−0.910249	+	0.414061i	$0.864110\pi$
$402$		0			0
$403$			12.0000i			0.597763i
$404$		0			0
$405$	27.5000	−	11.6082i	1.36649	−	0.576815i
$406$		0			0
$407$	−3.31662			−0.164399
$408$		0			0
$409$	15.0000			0.741702			0.370851	−	0.928692i	$-0.379066\pi$
$409$	15.0000			0.741702			0.370851	+	0.928692i	$0.379066\pi$
$410$		0			0
$411$	8.29156	−	27.5000i	0.408993	−	1.35647i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	44.0000i		−	2.15988i
$416$		0			0
$417$	−5.00000	+	16.5831i	−0.244851	+	0.812079i
$418$		0			0
$419$	−26.5330			−1.29622			−0.648111	−	0.761546i	$-0.724441\pi$
$419$	−26.5330			−1.29622			−0.648111	+	0.761546i	$0.724441\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$		0			0
$423$	−24.8747	−	16.5000i	−1.20945	−	0.802257i
$424$		0			0
$425$		−	19.8997i		−	0.965280i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	22.0000	+	6.63325i	1.06217	+	0.320256i
$430$		0			0
$431$	−23.2164			−1.11829			−0.559147	−	0.829069i	$-0.688871\pi$
$431$	−23.2164			−1.11829			−0.559147	+	0.829069i	$0.688871\pi$
$432$		0			0
$433$	20.0000			0.961139			0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000			0.961139			0.480569	+	0.876957i	$0.340430\pi$
$434$		0			0
$435$	−36.4829	−	11.0000i	−1.74922	−	0.527410i
$436$		0			0
$437$			23.2164i			1.11059i
$438$		0			0
$439$		−	9.00000i		−	0.429547i	−0.976664	−	0.214773i	$-0.931099\pi$
$439$		−	9.00000i		−	0.429547i	0.976664	−	0.214773i	$-0.0689013\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−29.8496			−1.41820			−0.709099	−	0.705109i	$-0.750898\pi$
$443$	−29.8496			−1.41820			−0.709099	+	0.705109i	$0.750898\pi$
$444$		0			0
$445$	11.0000			0.521450
$446$		0			0
$447$	−8.29156	+	27.5000i	−0.392177	+	1.30071i
$448$		0			0
$449$		−	19.8997i		−	0.939127i	−0.882899	−	0.469564i	$-0.844411\pi$
$449$		−	19.8997i		−	0.939127i	0.882899	−	0.469564i	$-0.155589\pi$
$450$		0			0
$451$		−	22.0000i		−	1.03594i
$452$		0			0
$453$	3.50000	−	11.6082i	0.164444	−	0.545400i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.00000			0.140334			0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000			0.140334			0.0701670	+	0.997535i	$0.477647\pi$
$458$		0			0
$459$	−13.2665	+	11.0000i	−0.619227	+	0.513436i
$460$		0			0
$461$			33.1662i			1.54471i	0.635193	+	0.772353i	$0.280920\pi$
$461$			33.1662i			1.54471i	−0.635193	+	0.772353i	$0.719080\pi$
$462$		0			0
$463$			38.0000i			1.76601i	0.469364	+	0.883005i	$0.344483\pi$
$463$			38.0000i			1.76601i	−0.469364	+	0.883005i	$0.655517\pi$
$464$		0			0
$465$	16.5000	+	4.97494i	0.765169	+	0.230707i
$466$		0			0
$467$	−3.31662			−0.153475			−0.0767375	−	0.997051i	$-0.524450\pi$
$467$	−3.31662			−0.153475			−0.0767375	+	0.997051i	$0.524450\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−4.97494	−	1.50000i	−0.229233	−	0.0691164i
$472$		0			0
$473$			6.63325i			0.304997i
$474$		0			0
$475$		−	42.0000i		−	1.92709i
$476$		0			0
$477$	−5.50000	+	8.29156i	−0.251828	+	0.379645i
$478$		0			0
$479$	16.5831			0.757702			0.378851	−	0.925458i	$-0.376319\pi$
$479$	16.5831			0.757702			0.378851	+	0.925458i	$0.376319\pi$
$480$		0			0
$481$	−4.00000			−0.182384
$482$		0			0
$483$		0			0
$484$		0			0
$485$			26.5330i			1.20480i
$486$		0			0
$487$		−	17.0000i		−	0.770344i	−0.922845	−	0.385172i	$-0.874142\pi$
$487$		−	17.0000i		−	0.770344i	0.922845	−	0.385172i	$-0.125858\pi$
$488$		0			0
$489$	−3.50000	+	11.6082i	−0.158275	+	0.524940i
$490$		0			0
$491$	−6.63325			−0.299354			−0.149677	−	0.988735i	$-0.547823\pi$
$491$	−6.63325			−0.299354			−0.149677	+	0.988735i	$0.547823\pi$
$492$		0			0
$493$	22.0000			0.990830
$494$		0			0
$495$	18.2414	−	27.5000i	0.819892	−	1.23603i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			35.0000i			1.56682i	0.621508	+	0.783408i	$0.286520\pi$
$499$			35.0000i			1.56682i	−0.621508	+	0.783408i	$0.713480\pi$
$500$		0			0
$501$	−11.0000	−	3.31662i	−0.491444	−	0.148176i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	11.0000			0.489494
$506$		0			0
$507$	4.97494	+	1.50000i	0.220945	+	0.0666173i
$508$		0			0
$509$		−	16.5831i		−	0.735034i	−0.930017	−	0.367517i	$-0.880208\pi$
$509$		−	16.5831i		−	0.735034i	0.930017	−	0.367517i	$-0.119792\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−28.0000	+	23.2164i	−1.23623	+	1.02503i
$514$		0			0
$515$	−36.4829			−1.60763
$516$		0			0
$517$	−33.0000			−1.45134
$518$		0			0
$519$	4.97494	−	16.5000i	0.218376	−	0.724270i
$520$		0			0
$521$			23.2164i			1.01713i	0.861024	+	0.508564i	$0.169823\pi$
$521$			23.2164i			1.01713i	−0.861024	+	0.508564i	$0.830177\pi$
$522$		0			0
$523$			19.0000i			0.830812i	0.909636	+	0.415406i	$0.136360\pi$
$523$			19.0000i			0.830812i	−0.909636	+	0.415406i	$0.863640\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.94987			−0.433423
$528$		0			0
$529$	−12.0000			−0.521739
$530$		0			0
$531$	8.29156	+	5.50000i	0.359823	+	0.238680i
$532$		0			0
$533$		−	26.5330i		−	1.14927i
$534$		0			0
$535$			11.0000i			0.475571i
$536$		0			0
$537$	5.50000	+	1.65831i	0.237343	+	0.0715615i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	37.0000			1.59075			0.795377	−	0.606115i	$-0.207273\pi$
$541$	37.0000			1.59075			0.795377	+	0.606115i	$0.207273\pi$
$542$		0			0
$543$	−29.8496	−	9.00000i	−1.28097	−	0.386227i
$544$		0			0
$545$		−	16.5831i		−	0.710343i
$546$		0			0
$547$			40.0000i			1.71028i	0.518400	+	0.855138i	$0.326528\pi$
$547$			40.0000i			1.71028i	−0.518400	+	0.855138i	$0.673472\pi$
$548$		0			0
$549$	−7.50000	−	4.97494i	−0.320092	−	0.212325i
$550$		0			0
$551$	46.4327			1.97810
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.65831	+	5.50000i	−0.0703914	+	0.233462i
$556$		0			0
$557$		−	3.31662i		−	0.140530i	−0.997528	−	0.0702650i	$-0.977616\pi$
$557$		−	3.31662i		−	0.140530i	0.997528	−	0.0702650i	$-0.0223845\pi$
$558$		0			0
$559$			8.00000i			0.338364i
$560$		0			0
$561$	−5.50000	+	18.2414i	−0.232210	+	0.770154i
$562$		0			0
$563$	−9.94987			−0.419337			−0.209669	−	0.977773i	$-0.567239\pi$
$563$	−9.94987			−0.419337			−0.209669	+	0.977773i	$0.567239\pi$
$564$		0			0
$565$	−22.0000			−0.925547
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	29.8496i		−	1.25136i	−0.780079	−	0.625681i	$-0.784821\pi$
$569$		−	29.8496i		−	1.25136i	0.780079	−	0.625681i	$-0.215179\pi$
$570$		0			0
$571$			5.00000i			0.209243i	0.994512	+	0.104622i	$0.0333632\pi$
$571$			5.00000i			0.209243i	−0.994512	+	0.104622i	$0.966637\pi$
$572$		0			0
$573$	16.5000	+	4.97494i	0.689297	+	0.207831i
$574$		0			0
$575$	−19.8997			−0.829877
$576$		0			0
$577$	−35.0000			−1.45707			−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−35.0000			−1.45707			−0.728535	+	0.685009i	$0.759798\pi$
$578$		0			0
$579$	−4.97494	−	1.50000i	−0.206751	−	0.0623379i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			11.0000i			0.455573i
$584$		0			0
$585$	22.0000	−	33.1662i	0.909588	−	1.37126i
$586$		0			0
$587$	26.5330			1.09513			0.547567	−	0.836762i	$-0.315554\pi$
$587$	26.5330			1.09513			0.547567	+	0.836762i	$0.315554\pi$
$588$		0			0
$589$	−21.0000			−0.865290
$590$		0			0
$591$	−9.94987	+	33.0000i	−0.409283	+	1.35744i
$592$		0			0
$593$		−	3.31662i		−	0.136197i	−0.997679	−	0.0680987i	$-0.978307\pi$
$593$		−	3.31662i		−	0.136197i	0.997679	−	0.0680987i	$-0.0216933\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.500000	−	1.65831i	0.0204636	−	0.0678702i
$598$		0			0
$599$	9.94987			0.406541			0.203270	−	0.979123i	$-0.434843\pi$
$599$	9.94987			0.406541			0.203270	+	0.979123i	$0.434843\pi$
$600$		0			0
$601$	16.0000			0.652654			0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000			0.652654			0.326327	+	0.945257i	$0.394189\pi$
$602$		0			0
$603$	14.9248	−	22.5000i	0.607785	−	0.916271i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			7.00000i			0.284121i	0.989858	+	0.142061i	$0.0453728\pi$
$607$			7.00000i			0.284121i	−0.989858	+	0.142061i	$0.954627\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−39.7995			−1.61012
$612$		0			0
$613$	1.00000			0.0403896			0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000			0.0403896			0.0201948	+	0.999796i	$0.493571\pi$
$614$		0			0
$615$	−36.4829	−	11.0000i	−1.47113	−	0.443563i
$616$		0			0
$617$			6.63325i			0.267045i	0.991046	+	0.133522i	$0.0426288\pi$
$617$			6.63325i			0.267045i	−0.991046	+	0.133522i	$0.957371\pi$
$618$		0			0
$619$			25.0000i			1.00483i	0.864625	+	0.502417i	$0.167556\pi$
$619$			25.0000i			1.00483i	−0.864625	+	0.502417i	$0.832444\pi$
$620$		0			0
$621$	11.0000	+	13.2665i	0.441415	+	0.532366i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	−11.6082	+	38.5000i	−0.463586	+	1.53754i
$628$		0			0
$629$		−	3.31662i		−	0.132242i
$630$		0			0
$631$		−	18.0000i		−	0.716569i	−0.933613	−	0.358284i	$-0.883362\pi$
$631$		−	18.0000i		−	0.716569i	0.933613	−	0.358284i	$-0.116638\pi$
$632$		0			0
$633$	13.0000	−	43.1161i	0.516704	−	1.71371i
$634$		0			0
$635$	6.63325			0.263232
$636$		0			0
$637$		0			0
$638$		0			0
$639$	33.1662	+	22.0000i	1.31204	+	0.870307i
$640$		0			0
$641$		−	36.4829i		−	1.44099i	−0.693462	−	0.720493i	$-0.743915\pi$
$641$		−	36.4829i		−	1.44099i	0.693462	−	0.720493i	$-0.256085\pi$
$642$		0			0
$643$			2.00000i			0.0788723i	0.999222	+	0.0394362i	$0.0125562\pi$
$643$			2.00000i			0.0788723i	−0.999222	+	0.0394362i	$0.987444\pi$
$644$		0			0
$645$	11.0000	+	3.31662i	0.433125	+	0.130592i
$646$		0			0
$647$	3.31662			0.130390			0.0651950	−	0.997873i	$-0.479233\pi$
$647$	3.31662			0.130390			0.0651950	+	0.997873i	$0.479233\pi$
$648$		0			0
$649$	11.0000			0.431788
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	3.31662i		−	0.129790i	−0.997892	−	0.0648948i	$-0.979329\pi$
$653$		−	3.31662i		−	0.129790i	0.997892	−	0.0648948i	$-0.0206712\pi$
$654$		0			0
$655$			55.0000i			2.14903i
$656$		0			0
$657$	−17.5000	−	11.6082i	−0.682740	−	0.452878i
$658$		0			0
$659$	−26.5330			−1.03358			−0.516789	−	0.856113i	$-0.672873\pi$
$659$	−26.5330			−1.03358			−0.516789	+	0.856113i	$0.672873\pi$
$660$		0			0
$661$	−39.0000			−1.51692			−0.758462	−	0.651717i	$-0.774049\pi$
$661$	−39.0000			−1.51692			−0.758462	+	0.651717i	$0.774049\pi$
$662$		0			0
$663$	−6.63325	+	22.0000i	−0.257614	+	0.854409i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	22.0000i		−	0.851843i
$668$		0			0
$669$	−7.00000	+	23.2164i	−0.270636	+	0.897597i
$670$		0			0
$671$	−9.94987			−0.384111
$672$		0			0
$673$	40.0000			1.54189			0.770943	−	0.636904i	$-0.219785\pi$
$673$	40.0000			1.54189			0.770943	+	0.636904i	$0.219785\pi$
$674$		0			0
$675$	−19.8997	−	24.0000i	−0.765942	−	0.923760i
$676$		0			0
$677$			9.94987i			0.382405i	0.981551	+	0.191202i	$0.0612387\pi$
$677$			9.94987i			0.382405i	−0.981551	+	0.191202i	$0.938761\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−5.50000	−	1.65831i	−0.210760	−	0.0635467i
$682$		0			0
$683$	−23.2164			−0.888350			−0.444175	−	0.895940i	$-0.646503\pi$
$683$	−23.2164			−0.888350			−0.444175	+	0.895940i	$0.646503\pi$
$684$		0			0
$685$	−55.0000			−2.10144
$686$		0			0
$687$	−48.0911	−	14.5000i	−1.83479	−	0.553210i
$688$		0			0
$689$			13.2665i			0.505413i
$690$		0			0
$691$			33.0000i			1.25538i	0.778464	+	0.627690i	$0.215999\pi$
$691$			33.0000i			1.25538i	−0.778464	+	0.627690i	$0.784001\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	33.1662			1.25807
$696$		0			0
$697$	22.0000			0.833309
$698$		0			0
$699$	−8.29156	+	27.5000i	−0.313616	+	1.04015i
$700$		0			0
$701$		−	13.2665i		−	0.501069i	−0.968108	−	0.250534i	$-0.919394\pi$
$701$		−	13.2665i		−	0.501069i	0.968108	−	0.250534i	$-0.0806063\pi$
$702$		0			0
$703$		−	7.00000i		−	0.264010i
$704$		0			0
$705$	−16.5000	+	54.7243i	−0.621426	+	2.06104i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−33.0000			−1.23934			−0.619671	−	0.784862i	$-0.712734\pi$
$709$	−33.0000			−1.23934			−0.619671	+	0.784862i	$0.712734\pi$
$710$		0			0
$711$	14.9248	−	22.5000i	0.559724	−	0.843816i
$712$		0			0
$713$			9.94987i			0.372626i
$714$		0			0
$715$		−	44.0000i		−	1.64551i
$716$		0			0
$717$	−11.0000	−	3.31662i	−0.410803	−	0.123862i
$718$		0			0
$719$	23.2164			0.865825			0.432912	−	0.901436i	$-0.357486\pi$
$719$	23.2164			0.865825			0.432912	+	0.901436i	$0.357486\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−31.5079	−	9.50000i	−1.17179	−	0.353309i
$724$		0			0
$725$			39.7995i			1.47812i
$726$		0			0
$727$			10.0000i			0.370879i	0.982656	+	0.185440i	$0.0593710\pi$
$727$			10.0000i			0.370879i	−0.982656	+	0.185440i	$0.940629\pi$
$728$		0			0
$729$	−5.00000	+	26.5330i	−0.185185	+	0.982704i
$730$		0			0
$731$	−6.63325			−0.245340
$732$		0			0
$733$	−7.00000			−0.258551			−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000			−0.258551			−0.129275	+	0.991609i	$0.541265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	29.8496i		−	1.09952i
$738$		0			0
$739$		−	31.0000i		−	1.14035i	−0.821522	−	0.570177i	$-0.806875\pi$
$739$		−	31.0000i		−	1.14035i	0.821522	−	0.570177i	$-0.193125\pi$
$740$		0			0
$741$	−14.0000	+	46.4327i	−0.514303	+	1.70575i
$742$		0			0
$743$	13.2665			0.486701			0.243350	−	0.969938i	$-0.421754\pi$
$743$	13.2665			0.486701			0.243350	+	0.969938i	$0.421754\pi$
$744$		0			0
$745$	55.0000			2.01504
$746$		0			0
$747$	33.1662	+	22.0000i	1.21349	+	0.804938i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			5.00000i			0.182453i	0.995830	+	0.0912263i	$0.0290787\pi$
$751$			5.00000i			0.182453i	−0.995830	+	0.0912263i	$0.970921\pi$
$752$		0			0
$753$	33.0000	+	9.94987i	1.20259	+	0.362594i
$754$		0			0
$755$	−23.2164			−0.844930
$756$		0			0
$757$	−48.0000			−1.74459			−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000			−1.74459			−0.872295	+	0.488980i	$0.837369\pi$
$758$		0			0
$759$	18.2414	+	5.50000i	0.662122	+	0.199637i
$760$		0			0
$761$			23.2164i			0.841593i	0.907155	+	0.420796i	$0.138249\pi$
$761$			23.2164i			0.841593i	−0.907155	+	0.420796i	$0.861751\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	27.5000	+	18.2414i	0.994265	+	0.659521i
$766$		0			0
$767$	13.2665			0.479026
$768$		0			0
$769$	16.0000			0.576975			0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000			0.576975			0.288487	+	0.957484i	$0.406848\pi$
$770$		0			0
$771$	−11.6082	+	38.5000i	−0.418059	+	1.38654i
$772$		0			0
$773$			16.5831i			0.596454i	0.954495	+	0.298227i	$0.0963952\pi$
$773$			16.5831i			0.596454i	−0.954495	+	0.298227i	$0.903605\pi$
$774$		0			0
$775$		−	18.0000i		−	0.646579i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	46.4327			1.66363
$780$		0			0
$781$	44.0000			1.57444
$782$		0			0
$783$	26.5330	−	22.0000i	0.948212	−	0.786216i
$784$		0			0
$785$			9.94987i			0.355126i
$786$		0			0
$787$			45.0000i			1.60408i	0.597272	+	0.802038i	$0.296251\pi$
$787$			45.0000i			1.60408i	−0.597272	+	0.802038i	$0.703749\pi$
$788$		0			0
$789$	−5.50000	−	1.65831i	−0.195805	−	0.0590375i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−12.0000			−0.426132
$794$		0			0
$795$	18.2414	+	5.50000i	0.646957	+	0.195065i
$796$		0			0
$797$		−	6.63325i		−	0.234962i	−0.993075	−	0.117481i	$-0.962518\pi$
$797$		−	6.63325i		−	0.234962i	0.993075	−	0.117481i	$-0.0374819\pi$
$798$		0			0
$799$		−	33.0000i		−	1.16746i
$800$		0			0
$801$	−5.50000	+	8.29156i	−0.194333

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.65831 + 0.500000i) q^{3} -3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +O(q^{10})\)	\(q+(1.65831 + 0.500000i) q^{3} -3.31662i q^{5} +(2.50000 + 1.65831i) q^{9} +3.31662 q^{11} +4.00000 q^{13} +(1.65831 - 5.50000i) q^{15} +3.31662i q^{17} +7.00000i q^{19} +3.31662 q^{23} -6.00000 q^{25} +(3.31662 + 4.00000i) q^{27} -6.63325i q^{29} +3.00000i q^{31} +(5.50000 + 1.65831i) q^{33} -1.00000 q^{37} +(6.63325 + 2.00000i) q^{39} -6.63325i q^{41} +2.00000i q^{43} +(5.50000 - 8.29156i) q^{45} -9.94987 q^{47} +(-1.65831 + 5.50000i) q^{51} +3.31662i q^{53} -11.0000i q^{55} +(-3.50000 + 11.6082i) q^{57} +3.31662 q^{59} -3.00000 q^{61} -13.2665i q^{65} -9.00000i q^{67} +(5.50000 + 1.65831i) q^{69} +13.2665 q^{71} -7.00000 q^{73} +(-9.94987 - 3.00000i) q^{75} -9.00000i q^{79} +(3.50000 + 8.29156i) q^{81} +13.2665 q^{83} +11.0000 q^{85} +(3.31662 - 11.0000i) q^{87} +3.31662i q^{89} +(-1.50000 + 4.97494i) q^{93} +23.2164 q^{95} -8.00000 q^{97} +(8.29156 + 5.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{9}+O(q^{10}) \) 4 * q + 10 * q^9	\( 4 q + 10 q^{9} + 16 q^{13} - 24 q^{25} + 22 q^{33} - 4 q^{37} + 22 q^{45} - 14 q^{57} - 12 q^{61} + 22 q^{69} - 28 q^{73} + 14 q^{81} + 44 q^{85} - 6 q^{93} - 32 q^{97}+O(q^{100}) \) 4 * q + 10 * q^9 + 16 * q^13 - 24 * q^25 + 22 * q^33 - 4 * q^37 + 22 * q^45 - 14 * q^57 - 12 * q^61 + 22 * q^69 - 28 * q^73 + 14 * q^81 + 44 * q^85 - 6 * q^93 - 32 * q^97

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