LMFDB - Embedded newform 980.2.i.k.961.2

Newspace parameters

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.i (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	980.961
Dual form			980.2.i.k.361.2

$q$-expansion

$f(q)$	$=$	$q+(0.207107 + 0.358719i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.41421 - 2.44949i) q^{9} +O(q^{10})$	\(q+(0.207107 + 0.358719i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.41421 - 2.44949i) q^{9} +(1.91421 + 3.31552i) q^{11} +3.58579 q^{13} +0.414214 q^{15} +(-3.20711 - 5.55487i) q^{17} +(-1.82843 + 3.16693i) q^{19} +(-0.292893 + 0.507306i) q^{23} +(-0.500000 - 0.866025i) q^{25} +2.41421 q^{27} +6.65685 q^{29} +(-2.29289 - 3.97141i) q^{31} +(-0.792893 + 1.37333i) q^{33} +(1.70711 - 2.95680i) q^{37} +(0.742641 + 1.28629i) q^{39} +0.585786 q^{41} +11.6569 q^{43} +(-1.41421 - 2.44949i) q^{45} +(4.44975 - 7.70719i) q^{47} +(1.32843 - 2.30090i) q^{51} +(1.87868 + 3.25397i) q^{53} +3.82843 q^{55} -1.51472 q^{57} +(-1.70711 - 2.95680i) q^{59} +(-2.58579 + 4.47871i) q^{61} +(1.79289 - 3.10538i) q^{65} +(5.53553 + 9.58783i) q^{67} -0.242641 q^{69} +6.48528 q^{71} +(-2.58579 - 4.47871i) q^{73} +(0.207107 - 0.358719i) q^{75} +(-6.57107 + 11.3814i) q^{79} +(-3.74264 - 6.48244i) q^{81} -8.00000 q^{83} -6.41421 q^{85} +(1.37868 + 2.38794i) q^{87} +(-8.48528 + 14.6969i) q^{89} +(0.949747 - 1.64501i) q^{93} +(1.82843 + 3.16693i) q^{95} +15.7279 q^{97} +10.8284 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 2 q^{5}+O(q^{10}) $ 4 * q - 2 * q^3 + 2 * q^5	$ 4 q - 2 q^{3} + 2 q^{5} + 2 q^{11} + 20 q^{13} - 4 q^{15} - 10 q^{17} + 4 q^{19} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 4 q^{29} - 12 q^{31} - 6 q^{33} + 4 q^{37} - 14 q^{39} + 8 q^{41} + 24 q^{43} - 2 q^{47} - 6 q^{51} + 16 q^{53} + 4 q^{55} - 40 q^{57} - 4 q^{59} - 16 q^{61} + 10 q^{65} + 8 q^{67} + 16 q^{69} - 8 q^{71} - 16 q^{73} - 2 q^{75} + 2 q^{79} + 2 q^{81} - 32 q^{83} - 20 q^{85} + 14 q^{87} - 16 q^{93} - 4 q^{95} + 12 q^{97} + 32 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 2 * q^5 + 2 * q^11 + 20 * q^13 - 4 * q^15 - 10 * q^17 + 4 * q^19 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 4 * q^29 - 12 * q^31 - 6 * q^33 + 4 * q^37 - 14 * q^39 + 8 * q^41 + 24 * q^43 - 2 * q^47 - 6 * q^51 + 16 * q^53 + 4 * q^55 - 40 * q^57 - 4 * q^59 - 16 * q^61 + 10 * q^65 + 8 * q^67 + 16 * q^69 - 8 * q^71 - 16 * q^73 - 2 * q^75 + 2 * q^79 + 2 * q^81 - 32 * q^83 - 20 * q^85 + 14 * q^87 - 16 * q^93 - 4 * q^95 + 12 * q^97 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$3$	0.207107	+	0.358719i	0.119573	+	0.207107i	0.919599	−	0.392859i	$-0.128514\pi$
$3$	0.207107	+	0.358719i	0.119573	+	0.207107i	−0.800025	+	0.599966i	$0.795181\pi$
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.41421	−	2.44949i	0.471405	−	0.816497i
$10$		0			0
$11$	1.91421	+	3.31552i	0.577157	+	0.999665i	0.995804	+	0.0915161i	$0.0291713\pi$
$11$	1.91421	+	3.31552i	0.577157	+	0.999665i	−0.418646	+	0.908149i	$0.637495\pi$
$12$		0			0
$13$	3.58579			0.994518			0.497259	−	0.867602i	$-0.334340\pi$
$13$	3.58579			0.994518			0.497259	+	0.867602i	$0.334340\pi$
$14$		0			0
$15$	0.414214			0.106949
$16$		0			0
$17$	−3.20711	−	5.55487i	−0.777838	−	1.34725i	−0.933186	−	0.359395i	$-0.882983\pi$
$17$	−3.20711	−	5.55487i	−0.777838	−	1.34725i	0.155348	−	0.987860i	$-0.450350\pi$
$18$		0			0
$19$	−1.82843	+	3.16693i	−0.419470	+	0.726543i	−0.995886	−	0.0906130i	$-0.971117\pi$
$19$	−1.82843	+	3.16693i	−0.419470	+	0.726543i	0.576416	+	0.817156i	$0.304451\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.292893	+	0.507306i	−0.0610725	+	0.105781i	−0.894945	−	0.446176i	$-0.852785\pi$
$23$	−0.292893	+	0.507306i	−0.0610725	+	0.105781i	0.833873	+	0.551957i	$0.186119\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	2.41421			0.464616
$28$		0			0
$29$	6.65685			1.23615			0.618073	−	0.786120i	$-0.287913\pi$
$29$	6.65685			1.23615			0.618073	+	0.786120i	$0.287913\pi$
$30$		0			0
$31$	−2.29289	−	3.97141i	−0.411816	−	0.713286i	0.583273	−	0.812276i	$-0.301772\pi$
$31$	−2.29289	−	3.97141i	−0.411816	−	0.713286i	−0.995088	+	0.0989906i	$0.968439\pi$
$32$		0			0
$33$	−0.792893	+	1.37333i	−0.138025	+	0.239066i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.70711	−	2.95680i	0.280647	−	0.486094i	−0.690898	−	0.722953i	$-0.742784\pi$
$37$	1.70711	−	2.95680i	0.280647	−	0.486094i	0.971544	+	0.236858i	$0.0761178\pi$
$38$		0			0
$39$	0.742641	+	1.28629i	0.118918	+	0.205971i
$40$		0			0
$41$	0.585786			0.0914845			0.0457422	−	0.998953i	$-0.485435\pi$
$41$	0.585786			0.0914845			0.0457422	+	0.998953i	$0.485435\pi$
$42$		0			0
$43$	11.6569			1.77765			0.888827	−	0.458243i	$-0.151521\pi$
$43$	11.6569			1.77765			0.888827	+	0.458243i	$0.151521\pi$
$44$		0			0
$45$	−1.41421	−	2.44949i	−0.210819	−	0.365148i
$46$		0			0
$47$	4.44975	−	7.70719i	0.649062	−	1.12421i	−0.334285	−	0.942472i	$-0.608495\pi$
$47$	4.44975	−	7.70719i	0.649062	−	1.12421i	0.983347	−	0.181737i	$-0.0581719\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.32843	−	2.30090i	0.186017	−	0.322191i
$52$		0			0
$53$	1.87868	+	3.25397i	0.258056	+	0.446967i	0.965721	−	0.259581i	$-0.0835846\pi$
$53$	1.87868	+	3.25397i	0.258056	+	0.446967i	−0.707665	+	0.706548i	$0.750251\pi$
$54$		0			0
$55$	3.82843			0.516225
$56$		0			0
$57$	−1.51472			−0.200629
$58$		0			0
$59$	−1.70711	−	2.95680i	−0.222246	−	0.384942i	0.733243	−	0.679966i	$-0.238006\pi$
$59$	−1.70711	−	2.95680i	−0.222246	−	0.384942i	−0.955490	+	0.295024i	$0.904672\pi$
$60$		0			0
$61$	−2.58579	+	4.47871i	−0.331076	+	0.573441i	−0.982723	−	0.185082i	$-0.940745\pi$
$61$	−2.58579	+	4.47871i	−0.331076	+	0.573441i	0.651647	+	0.758522i	$0.274078\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.79289	−	3.10538i	0.222381	−	0.385175i
$66$		0			0
$67$	5.53553	+	9.58783i	0.676273	+	1.17134i	0.976095	+	0.217344i	$0.0697394\pi$
$67$	5.53553	+	9.58783i	0.676273	+	1.17134i	−0.299822	+	0.953995i	$0.596927\pi$
$68$		0			0
$69$	−0.242641			−0.0292105
$70$		0			0
$71$	6.48528			0.769661			0.384831	−	0.922987i	$-0.374260\pi$
$71$	6.48528			0.769661			0.384831	+	0.922987i	$0.374260\pi$
$72$		0			0
$73$	−2.58579	−	4.47871i	−0.302643	−	0.524194i	0.674090	−	0.738649i	$-0.264536\pi$
$73$	−2.58579	−	4.47871i	−0.302643	−	0.524194i	−0.976734	+	0.214455i	$0.931202\pi$
$74$		0			0
$75$	0.207107	−	0.358719i	0.0239146	−	0.0414214i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.57107	+	11.3814i	−0.739303	+	1.28051i	0.213507	+	0.976942i	$0.431511\pi$
$79$	−6.57107	+	11.3814i	−0.739303	+	1.28051i	−0.952810	+	0.303569i	$0.901822\pi$
$80$		0			0
$81$	−3.74264	−	6.48244i	−0.415849	−	0.720272i
$82$		0			0
$83$	−8.00000			−0.878114			−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000			−0.878114			−0.439057	+	0.898459i	$0.644687\pi$
$84$		0			0
$85$	−6.41421			−0.695719
$86$		0			0
$87$	1.37868	+	2.38794i	0.147810	+	0.256014i
$88$		0			0
$89$	−8.48528	+	14.6969i	−0.899438	+	1.55787i	−0.0712241	+	0.997460i	$0.522691\pi$
$89$	−8.48528	+	14.6969i	−0.899438	+	1.55787i	−0.828214	+	0.560412i	$0.810643\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	0.949747	−	1.64501i	0.0984842	−	0.170580i
$94$		0			0
$95$	1.82843	+	3.16693i	0.187593	+	0.324920i
$96$		0			0
$97$	15.7279			1.59693			0.798464	−	0.602042i	$-0.205646\pi$
$97$	15.7279			1.59693			0.798464	+	0.602042i	$0.205646\pi$
$98$		0			0
$99$	10.8284			1.08830
$100$		0			0
$101$	2.41421	+	4.18154i	0.240223	+	0.416079i	0.960778	−	0.277319i	$-0.0894460\pi$
$101$	2.41421	+	4.18154i	0.240223	+	0.416079i	−0.720555	+	0.693398i	$0.756113\pi$
$102$		0			0
$103$	−0.792893	+	1.37333i	−0.0781261	+	0.135318i	−0.902441	−	0.430813i	$-0.858227\pi$
$103$	−0.792893	+	1.37333i	−0.0781261	+	0.135318i	0.824315	+	0.566131i	$0.191560\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.41421	−	14.5738i	0.813433	−	1.40891i	−0.0970151	−	0.995283i	$-0.530930\pi$
$107$	8.41421	−	14.5738i	0.813433	−	1.40891i	0.910448	−	0.413624i	$-0.135737\pi$
$108$		0			0
$109$	−4.50000	−	7.79423i	−0.431022	−	0.746552i	0.565940	−	0.824447i	$-0.308513\pi$
$109$	−4.50000	−	7.79423i	−0.431022	−	0.746552i	−0.996962	+	0.0778949i	$0.975180\pi$
$110$		0			0
$111$	1.41421			0.134231
$112$		0			0
$113$	−5.07107			−0.477046			−0.238523	−	0.971137i	$-0.576663\pi$
$113$	−5.07107			−0.477046			−0.238523	+	0.971137i	$0.576663\pi$
$114$		0			0
$115$	0.292893	+	0.507306i	0.0273124	+	0.0473065i
$116$		0			0
$117$	5.07107	−	8.78335i	0.468820	−	0.812021i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.82843	+	3.16693i	−0.166221	+	0.287903i
$122$		0			0
$123$	0.121320	+	0.210133i	0.0109391	+	0.0189471i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−21.8995			−1.94327			−0.971633	−	0.236494i	$-0.924002\pi$
$127$	−21.8995			−1.94327			−0.971633	+	0.236494i	$0.924002\pi$
$128$		0			0
$129$	2.41421	+	4.18154i	0.212560	+	0.368164i
$130$		0			0
$131$	5.87868	−	10.1822i	0.513623	−	0.889620i	−0.486253	−	0.873818i	$-0.661636\pi$
$131$	5.87868	−	10.1822i	0.513623	−	0.889620i	0.999875	−	0.0158021i	$-0.00503019\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.20711	−	2.09077i	0.103891	−	0.179945i
$136$		0			0
$137$	−6.48528	−	11.2328i	−0.554075	−	0.959686i	−0.997975	−	0.0636096i	$-0.979739\pi$
$137$	−6.48528	−	11.2328i	−0.554075	−	0.959686i	0.443900	−	0.896076i	$-0.353595\pi$
$138$		0			0
$139$	−13.8995			−1.17894			−0.589470	−	0.807790i	$-0.700663\pi$
$139$	−13.8995			−1.17894			−0.589470	+	0.807790i	$0.700663\pi$
$140$		0			0
$141$	3.68629			0.310442
$142$		0			0
$143$	6.86396	+	11.8887i	0.573993	+	0.994185i
$144$		0			0
$145$	3.32843	−	5.76500i	0.276411	−	0.478758i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.75736	+	3.04384i	−0.143968	+	0.249361i	−0.928988	−	0.370111i	$-0.879320\pi$
$149$	−1.75736	+	3.04384i	−0.143968	+	0.249361i	0.785019	+	0.619472i	$0.212653\pi$
$150$		0			0
$151$	5.91421	+	10.2437i	0.481292	+	0.833622i	0.999770	−	0.0214692i	$-0.00683439\pi$
$151$	5.91421	+	10.2437i	0.481292	+	0.833622i	−0.518478	+	0.855091i	$0.673501\pi$
$152$		0			0
$153$	−18.1421			−1.46670
$154$		0			0
$155$	−4.58579			−0.368339
$156$		0			0
$157$	−5.24264	−	9.08052i	−0.418408	−	0.724704i	0.577371	−	0.816482i	$-0.304079\pi$
$157$	−5.24264	−	9.08052i	−0.418408	−	0.724704i	−0.995780	+	0.0917773i	$0.970745\pi$
$158$		0			0
$159$	−0.778175	+	1.34784i	−0.0617133	+	0.106891i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.7782	+	18.6683i	−0.844212	+	1.46222i	0.0420922	+	0.999114i	$0.486598\pi$
$163$	−10.7782	+	18.6683i	−0.844212	+	1.46222i	−0.886304	+	0.463104i	$0.846736\pi$
$164$		0			0
$165$	0.792893	+	1.37333i	0.0617267	+	0.106914i
$166$		0			0
$167$	2.41421			0.186817			0.0934087	−	0.995628i	$-0.470224\pi$
$167$	2.41421			0.186817			0.0934087	+	0.995628i	$0.470224\pi$
$168$		0			0
$169$	−0.142136			−0.0109335
$170$		0			0
$171$	5.17157	+	8.95743i	0.395480	+	0.684992i
$172$		0			0
$173$	−9.27817	+	16.0703i	−0.705407	+	1.22180i	0.261138	+	0.965301i	$0.415902\pi$
$173$	−9.27817	+	16.0703i	−0.705407	+	1.22180i	−0.966545	+	0.256499i	$0.917431\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.707107	−	1.22474i	0.0531494	−	0.0920575i
$178$		0			0
$179$	−3.24264	−	5.61642i	−0.242366	−	0.419791i	0.719022	−	0.694988i	$-0.244590\pi$
$179$	−3.24264	−	5.61642i	−0.242366	−	0.419791i	−0.961388	+	0.275197i	$0.911257\pi$
$180$		0			0
$181$	1.75736			0.130623			0.0653117	−	0.997865i	$-0.479196\pi$
$181$	1.75736			0.130623			0.0653117	+	0.997865i	$0.479196\pi$
$182$		0			0
$183$	−2.14214			−0.158351
$184$		0			0
$185$	−1.70711	−	2.95680i	−0.125509	−	0.217388i
$186$		0			0
$187$	12.2782	−	21.2664i	0.897869	−	1.55515i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.32843	+	14.4253i	−0.602624	+	1.04378i	0.389798	+	0.920900i	$0.372545\pi$
$191$	−8.32843	+	14.4253i	−0.602624	+	1.04378i	−0.992422	+	0.122875i	$0.960789\pi$
$192$		0			0
$193$	−2.82843	−	4.89898i	−0.203595	−	0.352636i	0.746089	−	0.665846i	$-0.231929\pi$
$193$	−2.82843	−	4.89898i	−0.203595	−	0.352636i	−0.949684	+	0.313210i	$0.898596\pi$
$194$		0			0
$195$	1.48528			0.106363
$196$		0			0
$197$	−27.5563			−1.96331			−0.981654	−	0.190669i	$-0.938934\pi$
$197$	−27.5563			−1.96331			−0.981654	+	0.190669i	$0.938934\pi$
$198$		0			0
$199$	13.3640	+	23.1471i	0.947346	+	1.64085i	0.750984	+	0.660320i	$0.229579\pi$
$199$	13.3640	+	23.1471i	0.947346	+	1.64085i	0.196362	+	0.980532i	$0.437087\pi$
$200$		0			0
$201$	−2.29289	+	3.97141i	−0.161728	+	0.280121i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.292893	−	0.507306i	0.0204565	−	0.0354318i
$206$		0			0
$207$	0.828427	+	1.43488i	0.0575797	+	0.0997309i
$208$		0			0
$209$	−14.0000			−0.968400
$210$		0			0
$211$	−24.3137			−1.67382			−0.836912	−	0.547337i	$-0.815642\pi$
$211$	−24.3137			−1.67382			−0.836912	+	0.547337i	$0.815642\pi$
$212$		0			0
$213$	1.34315	+	2.32640i	0.0920308	+	0.159402i
$214$		0			0
$215$	5.82843	−	10.0951i	0.397495	−	0.688482i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	1.07107	−	1.85514i	0.0723761	−	0.125359i
$220$		0			0
$221$	−11.5000	−	19.9186i	−0.773574	−	1.33987i
$222$		0			0
$223$	24.0711			1.61192			0.805959	−	0.591971i	$-0.201650\pi$
$223$	24.0711			1.61192			0.805959	+	0.591971i	$0.201650\pi$
$224$		0			0
$225$	−2.82843			−0.188562
$226$		0			0
$227$	−3.86396	−	6.69258i	−0.256460	−	0.444202i	0.708831	−	0.705378i	$-0.249223\pi$
$227$	−3.86396	−	6.69258i	−0.256460	−	0.444202i	−0.965291	+	0.261176i	$0.915890\pi$
$228$		0			0
$229$	−11.9497	+	20.6976i	−0.789662	+	1.36773i	0.136513	+	0.990638i	$0.456411\pi$
$229$	−11.9497	+	20.6976i	−0.789662	+	1.36773i	−0.926174	+	0.377096i	$0.876923\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.58579	−	7.94282i	0.300425	−	0.520351i	−0.675807	−	0.737078i	$-0.736205\pi$
$233$	4.58579	−	7.94282i	0.300425	−	0.520351i	0.976232	+	0.216727i	$0.0695382\pi$
$234$		0			0
$235$	−4.44975	−	7.70719i	−0.290270	−	0.502762i
$236$		0			0
$237$	−5.44365			−0.353603
$238$		0			0
$239$	14.1716			0.916683			0.458341	−	0.888776i	$-0.348444\pi$
$239$	14.1716			0.916683			0.458341	+	0.888776i	$0.348444\pi$
$240$		0			0
$241$	1.77817	+	3.07989i	0.114542	+	0.198393i	0.917597	−	0.397513i	$-0.130127\pi$
$241$	1.77817	+	3.07989i	0.114542	+	0.198393i	−0.803054	+	0.595906i	$0.796793\pi$
$242$		0			0
$243$	5.17157	−	8.95743i	0.331757	−	0.574619i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.55635	+	11.3559i	−0.417171	+	0.722561i
$248$		0			0
$249$	−1.65685	−	2.86976i	−0.104999	−	0.181863i
$250$		0			0
$251$	−27.0711			−1.70871			−0.854355	−	0.519689i	$-0.826048\pi$
$251$	−27.0711			−1.70871			−0.854355	+	0.519689i	$0.826048\pi$
$252$		0			0
$253$	−2.24264			−0.140994
$254$		0			0
$255$	−1.32843	−	2.30090i	−0.0831893	−	0.144088i
$256$		0			0
$257$	12.8995	−	22.3426i	0.804648	−	1.39369i	−0.111880	−	0.993722i	$-0.535687\pi$
$257$	12.8995	−	22.3426i	0.804648	−	1.39369i	0.916528	−	0.399970i	$-0.130980\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	9.41421	−	16.3059i	0.582725	−	1.00931i
$262$		0			0
$263$	−5.00000	−	8.66025i	−0.308313	−	0.534014i	0.669680	−	0.742650i	$-0.266431\pi$
$263$	−5.00000	−	8.66025i	−0.308313	−	0.534014i	−0.977993	+	0.208635i	$0.933098\pi$
$264$		0			0
$265$	3.75736			0.230813
$266$		0			0
$267$	−7.02944			−0.430195
$268$		0			0
$269$	8.36396	+	14.4868i	0.509960	+	0.883276i	0.999933	+	0.0115389i	$0.00367303\pi$
$269$	8.36396	+	14.4868i	0.509960	+	0.883276i	−0.489974	+	0.871737i	$0.662994\pi$
$270$		0			0
$271$	−14.4853	+	25.0892i	−0.879918	+	1.52406i	−0.0284886	+	0.999594i	$0.509069\pi$
$271$	−14.4853	+	25.0892i	−0.879918	+	1.52406i	−0.851430	+	0.524469i	$0.824264\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.91421	−	3.31552i	0.115431	−	0.199933i
$276$		0			0
$277$	3.70711	+	6.42090i	0.222738	+	0.385794i	0.955639	−	0.294542i	$-0.0951671\pi$
$277$	3.70711	+	6.42090i	0.222738	+	0.385794i	−0.732900	+	0.680336i	$0.761834\pi$
$278$		0			0
$279$	−12.9706			−0.776527
$280$		0			0
$281$	−1.00000			−0.0596550			−0.0298275	−	0.999555i	$-0.509496\pi$
$281$	−1.00000			−0.0596550			−0.0298275	+	0.999555i	$0.509496\pi$
$282$		0			0
$283$	9.37868	+	16.2443i	0.557505	+	0.965626i	0.997704	+	0.0677263i	$0.0215745\pi$
$283$	9.37868	+	16.2443i	0.557505	+	0.965626i	−0.440199	+	0.897900i	$0.645092\pi$
$284$		0			0
$285$	−0.757359	+	1.31178i	−0.0448621	+	0.0777034i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−12.0711	+	20.9077i	−0.710063	+	1.22986i
$290$		0			0
$291$	3.25736	+	5.64191i	0.190950	+	0.330735i
$292$		0			0
$293$	−14.4142			−0.842087			−0.421044	−	0.907040i	$-0.638336\pi$
$293$	−14.4142			−0.842087			−0.421044	+	0.907040i	$0.638336\pi$
$294$		0			0
$295$	−3.41421			−0.198783
$296$		0			0
$297$	4.62132	+	8.00436i	0.268156	+	0.464460i
$298$		0			0
$299$	−1.05025	+	1.81909i	−0.0607377	+	0.105201i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−1.00000	+	1.73205i	−0.0574485	+	0.0995037i
$304$		0			0
$305$	2.58579	+	4.47871i	0.148062	+	0.256450i
$306$		0			0
$307$	3.58579			0.204652			0.102326	−	0.994751i	$-0.467372\pi$
$307$	3.58579			0.204652			0.102326	+	0.994751i	$0.467372\pi$
$308$		0			0
$309$	−0.656854			−0.0373671
$310$		0			0
$311$	−3.48528	−	6.03668i	−0.197632	−	0.342309i	0.750128	−	0.661293i	$-0.229992\pi$
$311$	−3.48528	−	6.03668i	−0.197632	−	0.342309i	−0.947760	+	0.318984i	$0.896658\pi$
$312$		0			0
$313$	−8.27817	+	14.3382i	−0.467910	+	0.810444i	−0.999328	−	0.0366660i	$-0.988326\pi$
$313$	−8.27817	+	14.3382i	−0.467910	+	0.810444i	0.531417	+	0.847110i	$0.321660\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.48528	−	12.9649i	0.420415	−	0.728181i	−0.575565	−	0.817756i	$-0.695218\pi$
$317$	7.48528	−	12.9649i	0.420415	−	0.728181i	0.995980	+	0.0895756i	$0.0285511\pi$
$318$		0			0
$319$	12.7426	+	22.0709i	0.713451	+	1.23573i
$320$		0			0
$321$	6.97056			0.389059
$322$		0			0
$323$	23.4558			1.30512
$324$		0			0
$325$	−1.79289	−	3.10538i	−0.0994518	−	0.172256i
$326$		0			0
$327$	1.86396	−	3.22848i	0.103077	−	0.178535i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.41421	+	4.18154i	−0.132697	+	0.229838i	−0.924715	−	0.380659i	$-0.875697\pi$
$331$	−2.41421	+	4.18154i	−0.132697	+	0.229838i	0.792018	+	0.610497i	$0.209030\pi$
$332$		0			0
$333$	−4.82843	−	8.36308i	−0.264596	−	0.458294i
$334$		0			0
$335$	11.0711			0.604877
$336$		0			0
$337$	−18.7279			−1.02017			−0.510087	−	0.860123i	$-0.670387\pi$
$337$	−18.7279			−1.02017			−0.510087	+	0.860123i	$0.670387\pi$
$338$		0			0
$339$	−1.05025	−	1.81909i	−0.0570419	−	0.0987994i
$340$		0			0
$341$	8.77817	−	15.2042i	0.475365	−	0.823356i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.121320	+	0.210133i	−0.00653167	+	0.0113132i
$346$		0			0
$347$	3.70711	+	6.42090i	0.199008	+	0.344692i	0.948207	−	0.317653i	$-0.102895\pi$
$347$	3.70711	+	6.42090i	0.199008	+	0.344692i	−0.749199	+	0.662345i	$0.769561\pi$
$348$		0			0
$349$	18.0000			0.963518			0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000			0.963518			0.481759	+	0.876304i	$0.339998\pi$
$350$		0			0
$351$	8.65685			0.462069
$352$		0			0
$353$	1.03553	+	1.79360i	0.0551159	+	0.0954636i	0.892267	−	0.451508i	$-0.149114\pi$
$353$	1.03553	+	1.79360i	0.0551159	+	0.0954636i	−0.837151	+	0.546972i	$0.815781\pi$
$354$		0			0
$355$	3.24264	−	5.61642i	0.172101	−	0.298089i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5.65685	+	9.79796i	−0.298557	+	0.517116i	−0.975806	−	0.218638i	$-0.929839\pi$
$359$	−5.65685	+	9.79796i	−0.298557	+	0.517116i	0.677249	+	0.735754i	$0.263172\pi$
$360$		0			0
$361$	2.81371	+	4.87349i	0.148090	+	0.256499i
$362$		0			0
$363$	−1.51472			−0.0795021
$364$		0			0
$365$	−5.17157			−0.270692
$366$		0			0
$367$	−9.86396	−	17.0849i	−0.514895	−	0.891824i	−0.999851	−	0.0172850i	$-0.994498\pi$
$367$	−9.86396	−	17.0849i	−0.514895	−	0.891824i	0.484956	−	0.874539i	$-0.338836\pi$
$368$		0			0
$369$	0.828427	−	1.43488i	0.0431262	−	0.0746968i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−8.24264	+	14.2767i	−0.426788	+	0.739218i	−0.996586	−	0.0825669i	$-0.973688\pi$
$373$	−8.24264	+	14.2767i	−0.426788	+	0.739218i	0.569798	+	0.821785i	$0.307022\pi$
$374$		0			0
$375$	−0.207107	−	0.358719i	−0.0106949	−	0.0185242i
$376$		0			0
$377$	23.8701			1.22937
$378$		0			0
$379$	14.0000			0.719132			0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000			0.719132			0.359566	+	0.933120i	$0.382925\pi$
$380$		0			0
$381$	−4.53553	−	7.85578i	−0.232362	−	0.402464i
$382$		0			0
$383$	1.75736	−	3.04384i	0.0897969	−	0.155533i	−0.817628	−	0.575746i	$-0.804712\pi$
$383$	1.75736	−	3.04384i	0.0897969	−	0.155533i	0.907425	+	0.420214i	$0.138045\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	16.4853	−	28.5533i	0.837994	−	1.45145i
$388$		0			0
$389$	−6.57107	−	11.3814i	−0.333166	−	0.577061i	0.649965	−	0.759965i	$-0.274784\pi$
$389$	−6.57107	−	11.3814i	−0.333166	−	0.577061i	−0.983131	+	0.182903i	$0.941450\pi$
$390$		0			0
$391$	3.75736			0.190018
$392$		0			0
$393$	4.87006			0.245662
$394$		0			0
$395$	6.57107	+	11.3814i	0.330626	+	0.572662i
$396$		0			0
$397$	−7.20711	+	12.4831i	−0.361714	+	0.626508i	−0.988243	−	0.152891i	$-0.951142\pi$
$397$	−7.20711	+	12.4831i	−0.361714	+	0.626508i	0.626529	+	0.779398i	$0.284475\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.257359	+	0.445759i	−0.0128519	+	0.0222602i	−0.872380	−	0.488829i	$-0.837424\pi$
$401$	−0.257359	+	0.445759i	−0.0128519	+	0.0222602i	0.859528	+	0.511089i	$0.170758\pi$
$402$		0			0
$403$	−8.22183	−	14.2406i	−0.409558	−	0.709376i
$404$		0			0
$405$	−7.48528			−0.371947
$406$		0			0
$407$	13.0711			0.647909
$408$		0			0
$409$	−16.4142	−	28.4303i	−0.811631	−	1.40579i	−0.911722	−	0.410807i	$-0.865247\pi$
$409$	−16.4142	−	28.4303i	−0.811631	−	1.40579i	0.100092	−	0.994978i	$-0.468086\pi$
$410$		0			0
$411$	2.68629	−	4.65279i	0.132505	−	0.229505i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.00000	+	6.92820i	−0.196352	+	0.340092i
$416$		0			0
$417$	−2.87868	−	4.98602i	−0.140970	−	0.244166i
$418$		0			0
$419$	−9.55635			−0.466858			−0.233429	−	0.972374i	$-0.574995\pi$
$419$	−9.55635			−0.466858			−0.233429	+	0.972374i	$0.574995\pi$
$420$		0			0
$421$	20.3137			0.990030			0.495015	−	0.868885i	$-0.335163\pi$
$421$	20.3137			0.990030			0.495015	+	0.868885i	$0.335163\pi$
$422$		0			0
$423$	−12.5858	−	21.7992i	−0.611942	−	1.05991i
$424$		0			0
$425$	−3.20711	+	5.55487i	−0.155568	+	0.269451i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−2.84315	+	4.92447i	−0.137268	+	0.237756i
$430$		0			0
$431$	4.91421	+	8.51167i	0.236709	+	0.409993i	0.959768	−	0.280794i	$-0.0905978\pi$
$431$	4.91421	+	8.51167i	0.236709	+	0.409993i	−0.723059	+	0.690787i	$0.757264\pi$
$432$		0			0
$433$	−22.2843			−1.07091			−0.535457	−	0.844563i	$-0.679861\pi$
$433$	−22.2843			−1.07091			−0.535457	+	0.844563i	$0.679861\pi$
$434$		0			0
$435$	2.75736			0.132205
$436$		0			0
$437$	−1.07107	−	1.85514i	−0.0512361	−	0.0887436i
$438$		0			0
$439$	−12.1213	+	20.9947i	−0.578519	+	1.00202i	0.417130	+	0.908847i	$0.363036\pi$
$439$	−12.1213	+	20.9947i	−0.578519	+	1.00202i	−0.995649	+	0.0931778i	$0.970298\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	17.7279	−	30.7057i	0.842279	−	1.45887i	−0.0456844	−	0.998956i	$-0.514547\pi$
$443$	17.7279	−	30.7057i	0.842279	−	1.45887i	0.887963	−	0.459914i	$-0.152120\pi$
$444$		0			0
$445$	8.48528	+	14.6969i	0.402241	+	0.696702i
$446$		0			0
$447$	−1.45584			−0.0688591
$448$		0			0
$449$	23.8284			1.12453			0.562267	−	0.826956i	$-0.309930\pi$
$449$	23.8284			1.12453			0.562267	+	0.826956i	$0.309930\pi$
$450$		0			0
$451$	1.12132	+	1.94218i	0.0528009	+	0.0914539i
$452$		0			0
$453$	−2.44975	+	4.24309i	−0.115099	+	0.199358i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.53553	−	13.0519i	0.352497	−	0.610543i	−0.634189	−	0.773178i	$-0.718666\pi$
$457$	7.53553	−	13.0519i	0.352497	−	0.610543i	0.986686	+	0.162635i	$0.0519992\pi$
$458$		0			0
$459$	−7.74264	−	13.4106i	−0.361396	−	0.625955i
$460$		0			0
$461$	22.3431			1.04062			0.520312	−	0.853976i	$-0.325816\pi$
$461$	22.3431			1.04062			0.520312	+	0.853976i	$0.325816\pi$
$462$		0			0
$463$	−13.4558			−0.625346			−0.312673	−	0.949861i	$-0.601224\pi$
$463$	−13.4558			−0.625346			−0.312673	+	0.949861i	$0.601224\pi$
$464$		0			0
$465$	−0.949747	−	1.64501i	−0.0440435	−	0.0762856i
$466$		0			0
$467$	−0.449747	+	0.778985i	−0.0208118	+	0.0360471i	−0.876244	−	0.481868i	$-0.839958\pi$
$467$	−0.449747	+	0.778985i	−0.0208118	+	0.0360471i	0.855432	+	0.517915i	$0.173292\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	2.17157	−	3.76127i	0.100061	−	0.173310i
$472$		0			0
$473$	22.3137	+	38.6485i	1.02599	+	1.77706i
$474$		0			0
$475$	3.65685			0.167788
$476$		0			0
$477$	10.6274			0.486596
$478$		0			0
$479$	−4.70711	−	8.15295i	−0.215073	−	0.372518i	0.738222	−	0.674558i	$-0.235666\pi$
$479$	−4.70711	−	8.15295i	−0.215073	−	0.372518i	−0.953295	+	0.302040i	$0.902332\pi$
$480$		0			0
$481$	6.12132	−	10.6024i	0.279108	−	0.483430i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.86396	−	13.6208i	0.357084	−	0.618488i
$486$		0			0
$487$	3.70711	+	6.42090i	0.167985	+	0.290959i	0.937711	−	0.347415i	$-0.112941\pi$
$487$	3.70711	+	6.42090i	0.167985	+	0.290959i	−0.769726	+	0.638374i	$0.779607\pi$
$488$		0			0
$489$	−8.92893			−0.403780
$490$		0			0
$491$	17.2843			0.780028			0.390014	−	0.920809i	$-0.372470\pi$
$491$	17.2843			0.780028			0.390014	+	0.920809i	$0.372470\pi$
$492$		0			0
$493$	−21.3492	−	36.9780i	−0.961522	−	1.66540i
$494$		0			0
$495$	5.41421	−	9.37769i	0.243351	−	0.421496i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.81371	−	11.8017i	0.305023	−	0.528316i	−0.672243	−	0.740331i	$-0.734669\pi$
$499$	6.81371	−	11.8017i	0.305023	−	0.528316i	0.977267	+	0.212014i	$0.0680024\pi$
$500$		0			0
$501$	0.500000	+	0.866025i	0.0223384	+	0.0386912i
$502$		0			0
$503$	39.0416			1.74078			0.870390	−	0.492363i	$-0.163867\pi$
$503$	39.0416			1.74078			0.870390	+	0.492363i	$0.163867\pi$
$504$		0			0
$505$	4.82843			0.214862
$506$		0			0
$507$	−0.0294373	−	0.0509868i	−0.00130735	−	0.00226440i
$508$		0			0
$509$	17.5355	−	30.3724i	0.777249	−	1.34623i	−0.156273	−	0.987714i	$-0.549948\pi$
$509$	17.5355	−	30.3724i	0.777249	−	1.34623i	0.933522	−	0.358521i	$-0.116719\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−4.41421	+	7.64564i	−0.194892	+	0.337563i
$514$		0			0
$515$	0.792893	+	1.37333i	0.0349390	+	0.0605162i
$516$		0			0
$517$	34.0711			1.49844
$518$		0			0
$519$	−7.68629			−0.337391
$520$		0			0
$521$	−3.34315	−	5.79050i	−0.146466	−	0.253686i	0.783453	−	0.621451i	$-0.213457\pi$
$521$	−3.34315	−	5.79050i	−0.146466	−	0.253686i	−0.929919	+	0.367765i	$0.880123\pi$
$522$		0			0
$523$	−0.928932	+	1.60896i	−0.0406194	+	0.0703548i	−0.885620	−	0.464410i	$-0.846266\pi$
$523$	−0.928932	+	1.60896i	−0.0406194	+	0.0703548i	0.845001	+	0.534765i	$0.179600\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.7071	+	25.4735i	−0.640652	+	1.10964i
$528$		0			0
$529$	11.3284	+	19.6214i	0.492540	+	0.853105i
$530$		0			0
$531$	−9.65685			−0.419072
$532$		0			0
$533$	2.10051			0.0909830
$534$		0			0
$535$	−8.41421	−	14.5738i	−0.363778	−	0.630082i
$536$		0			0
$537$	1.34315	−	2.32640i	0.0579610	−	0.100391i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7.98528	+	13.8309i	−0.343314	+	0.594637i	−0.985046	−	0.172292i	$-0.944883\pi$
$541$	−7.98528	+	13.8309i	−0.343314	+	0.594637i	0.641732	+	0.766929i	$0.278216\pi$
$542$		0			0
$543$	0.363961	+	0.630399i	0.0156191	+	0.0270530i
$544$		0			0
$545$	−9.00000			−0.385518
$546$		0			0
$547$	7.51472			0.321306			0.160653	−	0.987011i	$-0.448640\pi$
$547$	7.51472			0.321306			0.160653	+	0.987011i	$0.448640\pi$
$548$		0			0
$549$	7.31371	+	12.6677i	0.312141	+	0.540645i
$550$		0			0
$551$	−12.1716	+	21.0818i	−0.518526	+	0.898114i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.707107	−	1.22474i	0.0300150	−	0.0519875i
$556$		0			0
$557$	3.89949	+	6.75412i	0.165227	+	0.286181i	0.936736	−	0.350037i	$-0.113831\pi$
$557$	3.89949	+	6.75412i	0.165227	+	0.286181i	−0.771509	+	0.636218i	$0.780498\pi$
$558$		0			0
$559$	41.7990			1.76791
$560$		0			0
$561$	10.1716			0.429444
$562$		0			0
$563$	10.3137	+	17.8639i	0.434671	+	0.752872i	0.997269	−	0.0738585i	$-0.0235313\pi$
$563$	10.3137	+	17.8639i	0.434671	+	0.752872i	−0.562598	+	0.826731i	$0.690198\pi$
$564$		0			0
$565$	−2.53553	+	4.39167i	−0.106671	+	0.184759i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.8995	−	27.5387i	0.666542	−	1.15448i	−0.312323	−	0.949976i	$-0.601107\pi$
$569$	15.8995	−	27.5387i	0.666542	−	1.15448i	0.978865	−	0.204508i	$-0.0655594\pi$
$570$		0			0
$571$	−2.41421	−	4.18154i	−0.101032	−	0.174992i	0.811078	−	0.584938i	$-0.198881\pi$
$571$	−2.41421	−	4.18154i	−0.101032	−	0.174992i	−0.912110	+	0.409946i	$0.865548\pi$
$572$		0			0
$573$	−6.89949			−0.288231
$574$		0			0
$575$	0.585786			0.0244290
$576$		0			0
$577$	7.03553	+	12.1859i	0.292893	+	0.507306i	0.974493	−	0.224420i	$-0.0720487\pi$
$577$	7.03553	+	12.1859i	0.292893	+	0.507306i	−0.681599	+	0.731725i	$0.738715\pi$
$578$		0			0
$579$	1.17157	−	2.02922i	0.0486889	−	0.0843317i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−7.19239	+	12.4576i	−0.297878	+	0.515940i
$584$		0			0
$585$	−5.07107	−	8.78335i	−0.209663	−	0.363147i
$586$		0			0
$587$	−30.8284			−1.27243			−0.636213	−	0.771514i	$-0.719500\pi$
$587$	−30.8284			−1.27243			−0.636213	+	0.771514i	$0.719500\pi$
$588$		0			0
$589$	16.7696			0.690977
$590$		0			0
$591$	−5.70711	−	9.88500i	−0.234759	−	0.406615i
$592$		0			0
$593$	−8.86396	+	15.3528i	−0.363999	+	0.630465i	−0.988615	−	0.150467i	$-0.951922\pi$
$593$	−8.86396	+	15.3528i	−0.363999	+	0.630465i	0.624616	+	0.780932i	$0.285256\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5.53553	+	9.58783i	−0.226554	+	0.392404i
$598$		0			0
$599$	6.39949	+	11.0843i	0.261476	+	0.452890i	0.966634	−	0.256160i	$-0.0824573\pi$
$599$	6.39949	+	11.0843i	0.261476	+	0.452890i	−0.705158	+	0.709050i	$0.749124\pi$
$600$		0			0
$601$	39.6569			1.61764			0.808818	−	0.588058i	$-0.200108\pi$
$601$	39.6569			1.61764			0.808818	+	0.588058i	$0.200108\pi$
$602$		0			0
$603$	31.3137			1.27519
$604$		0			0
$605$	1.82843	+	3.16693i	0.0743361	+	0.128754i
$606$		0			0
$607$	−6.96447	+	12.0628i	−0.282679	+	0.489614i	−0.972044	−	0.234800i	$-0.924556\pi$
$607$	−6.96447	+	12.0628i	−0.282679	+	0.489614i	0.689365	+	0.724414i	$0.257890\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.9558	−	27.6363i	0.645504	−	1.11805i
$612$		0			0
$613$	20.6569	+	35.7787i	0.834322	+	1.44509i	0.894581	+	0.446906i	$0.147474\pi$
$613$	20.6569	+	35.7787i	0.834322	+	1.44509i	−0.0602584	+	0.998183i	$0.519192\pi$
$614$		0			0
$615$	0.242641			0.00978422
$616$		0			0
$617$	40.8701			1.64537			0.822683	−	0.568500i	$-0.192476\pi$
$617$	40.8701			1.64537			0.822683	+	0.568500i	$0.192476\pi$
$618$		0			0
$619$	−5.43503	−	9.41375i	−0.218452	−	0.378370i	0.735883	−	0.677109i	$-0.236767\pi$
$619$	−5.43503	−	9.41375i	−0.218452	−	0.378370i	−0.954335	+	0.298739i	$0.903434\pi$
$620$		0			0
$621$	−0.707107	+	1.22474i	−0.0283752	+	0.0491473i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−2.89949	−	5.02207i	−0.115795	−	0.200562i
$628$		0			0
$629$	−21.8995			−0.873190
$630$		0			0
$631$	−39.4853			−1.57188			−0.785942	−	0.618300i	$-0.787822\pi$
$631$	−39.4853			−1.57188			−0.785942	+	0.618300i	$0.787822\pi$
$632$		0			0
$633$	−5.03553	−	8.72180i	−0.200145	−	0.346660i
$634$		0			0
$635$	−10.9497	+	18.9655i	−0.434527	+	0.752624i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	9.17157	−	15.8856i	0.362822	−	0.628426i
$640$		0			0
$641$	−19.6569	−	34.0467i	−0.776399	−	1.34476i	−0.934005	−	0.357261i	$-0.883711\pi$
$641$	−19.6569	−	34.0467i	−0.776399	−	1.34476i	0.157606	−	0.987502i	$-0.449622\pi$
$642$		0			0
$643$	4.21320			0.166153			0.0830763	−	0.996543i	$-0.473525\pi$
$643$	4.21320			0.166153			0.0830763	+	0.996543i	$0.473525\pi$
$644$		0			0
$645$	4.82843			0.190119
$646$		0			0
$647$	−17.5563	−	30.4085i	−0.690211	−	1.19548i	−0.971768	−	0.235937i	$-0.924184\pi$
$647$	−17.5563	−	30.4085i	−0.690211	−	1.19548i	0.281557	−	0.959544i	$-0.409149\pi$
$648$		0			0
$649$	6.53553	−	11.3199i	0.256542	−	0.444344i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.3431	−	23.1110i	0.522158	−	0.904404i	−0.477510	−	0.878626i	$-0.658461\pi$
$653$	13.3431	−	23.1110i	0.522158	−	0.904404i	0.999668	−	0.0257773i	$-0.00820608\pi$
$654$		0			0
$655$	−5.87868	−	10.1822i	−0.229699	−	0.397850i
$656$		0			0
$657$	−14.6274			−0.570670
$658$		0			0
$659$	−20.6569			−0.804677			−0.402338	−	0.915491i	$-0.631802\pi$
$659$	−20.6569			−0.804677			−0.402338	+	0.915491i	$0.631802\pi$
$660$		0			0
$661$	17.0711	+	29.5680i	0.663988	+	1.15006i	0.979559	+	0.201158i	$0.0644706\pi$
$661$	17.0711	+	29.5680i	0.663988	+	1.15006i	−0.315571	+	0.948902i	$0.602196\pi$
$662$		0			0
$663$	4.76346	−	8.25055i	0.184997	−	0.320425i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.94975	+	3.37706i	−0.0754945	+	0.130760i
$668$		0			0
$669$	4.98528	+	8.63476i	0.192742	+	0.333839i
$670$		0			0
$671$	−19.7990			−0.764332
$672$		0			0
$673$	−27.5147			−1.06061			−0.530307	−	0.847806i	$-0.677923\pi$
$673$	−27.5147			−1.06061			−0.530307	+	0.847806i	$0.677923\pi$
$674$		0			0
$675$	−1.20711	−	2.09077i	−0.0464616	−	0.0804738i
$676$		0			0
$677$	5.13604	−	8.89588i	0.197394	−	0.341896i	−0.750289	−	0.661110i	$-0.770086\pi$
$677$	5.13604	−	8.89588i	0.197394	−	0.341896i	0.947683	+	0.319214i	$0.103419\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.60051	−	2.77216i	0.0613315	−	0.106229i
$682$		0			0
$683$	−17.5563	−	30.4085i	−0.671775	−	1.16355i	−0.977400	−	0.211397i	$-0.932199\pi$
$683$	−17.5563	−	30.4085i	−0.671775	−	1.16355i	0.305625	−	0.952152i	$-0.401135\pi$
$684$		0			0
$685$	−12.9706			−0.495580
$686$		0			0
$687$	−9.89949			−0.377689
$688$		0			0
$689$	6.73654	+	11.6680i	0.256642	+	0.444517i
$690$		0			0
$691$	−1.75736	+	3.04384i	−0.0668531	+	0.115793i	−0.897515	−	0.440985i	$-0.854629\pi$
$691$	−1.75736	+	3.04384i	−0.0668531	+	0.115793i	0.830661	+	0.556778i	$0.187963\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.94975	+	12.0373i	−0.263619	+	0.456601i
$696$		0			0
$697$	−1.87868	−	3.25397i	−0.0711601	−	0.123253i
$698$		0			0
$699$	3.79899			0.143691
$700$		0			0
$701$	0.514719			0.0194407			0.00972033	−	0.999953i	$-0.496906\pi$
$701$	0.514719			0.0194407			0.00972033	+	0.999953i	$0.496906\pi$
$702$		0			0
$703$	6.24264	+	10.8126i	0.235446	+	0.407804i
$704$		0			0
$705$	1.84315	−	3.19242i	0.0694169	−	0.120234i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.81371	+	3.14144i	−0.0681153	+	0.117979i	−0.898072	−	0.439849i	$-0.855032\pi$
$709$	−1.81371	+	3.14144i	−0.0681153	+	0.117979i	0.829956	+	0.557828i	$0.188365\pi$
$710$		0			0
$711$	18.5858	+	32.1915i	0.697021	+	1.20728i
$712$		0			0
$713$	2.68629			0.100602
$714$		0			0
$715$	13.7279			0.513395
$716$		0			0
$717$	2.93503	+	5.08362i	0.109611	+	0.189851i
$718$		0			0
$719$	−7.87868	+	13.6463i	−0.293825	+	0.508920i	−0.974711	−	0.223470i	$-0.928262\pi$
$719$	−7.87868	+	13.6463i	−0.293825	+	0.508920i	0.680886	+	0.732390i	$0.261595\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−0.736544	+	1.27573i	−0.0273924	+	0.0474450i
$724$		0			0
$725$	−3.32843	−	5.76500i	−0.123615	−	0.214107i
$726$		0			0
$727$	−18.0000			−0.667583			−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000			−0.667583			−0.333792	+	0.942647i	$0.608328\pi$
$728$		0			0
$729$	−18.1716			−0.673021
$730$		0			0
$731$	−37.3848	−	64.7523i	−1.38273	−	2.39495i
$732$		0			0
$733$	15.3492	−	26.5857i	0.566937	−	0.981964i	−0.429929	−	0.902863i	$-0.641462\pi$
$733$	15.3492	−	26.5857i	0.566937	−	0.981964i	0.996867	−	0.0791015i	$-0.0252051\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.1924	+	36.7063i	−0.780632	+	1.35209i
$738$		0			0
$739$	−10.3284	−	17.8894i	−0.379937	−	0.658071i	0.611115	−	0.791541i	$-0.290721\pi$
$739$	−10.3284	−	17.8894i	−0.379937	−	0.658071i	−0.991053	+	0.133471i	$0.957388\pi$
$740$		0			0
$741$	−5.43146			−0.199530
$742$		0			0
$743$	−8.92893			−0.327571			−0.163785	−	0.986496i	$-0.552370\pi$
$743$	−8.92893			−0.327571			−0.163785	+	0.986496i	$0.552370\pi$
$744$		0			0
$745$	1.75736	+	3.04384i	0.0643847	+	0.111518i
$746$		0			0
$747$	−11.3137	+	19.5959i	−0.413947	+	0.716977i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.571068	+	0.989118i	−0.0208386	+	0.0360934i	−0.876257	−	0.481845i	$-0.839967\pi$
$751$	−0.571068	+	0.989118i	−0.0208386	+	0.0360934i	0.855418	+	0.517938i	$0.173300\pi$
$752$		0			0
$753$	−5.60660	−	9.71092i	−0.204316	−	0.353886i
$754$		0			0
$755$	11.8284			0.430481
$756$		0			0
$757$	37.1716			1.35102			0.675512	−	0.737349i	$-0.263923\pi$
$757$	37.1716			1.35102			0.675512	+	0.737349i	$0.263923\pi$
$758$		0			0
$759$	−0.464466	−	0.804479i	−0.0168591	−	0.0292007i
$760$		0			0
$761$	9.43503	−	16.3419i	0.342020	−	0.592395i	−0.642788	−	0.766044i	$-0.722222\pi$
$761$	9.43503	−	16.3419i	0.342020	−	0.592395i	0.984808	+	0.173649i	$0.0555558\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−9.07107	+	15.7116i	−0.327965	+	0.568052i
$766$		0			0
$767$	−6.12132	−	10.6024i	−0.221028	−	0.382832i
$768$		0			0
$769$	28.1421			1.01483			0.507416	−	0.861701i	$-0.330601\pi$
$769$	28.1421			1.01483			0.507416	+	0.861701i	$0.330601\pi$
$770$		0			0
$771$	10.6863			0.384857
$772$		0			0
$773$	21.8640	+	37.8695i	0.786392	+	1.36207i	0.928164	+	0.372172i	$0.121387\pi$
$773$	21.8640	+	37.8695i	0.786392	+	1.36207i	−0.141772	+	0.989899i	$0.545280\pi$
$774$		0			0
$775$	−2.29289	+	3.97141i	−0.0823632	+	0.142657i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.07107	+	1.85514i	−0.0383750	+	0.0664674i
$780$		0			0
$781$	12.4142	+	21.5020i	0.444215	+	0.769404i
$782$		0			0
$783$	16.0711			0.574333
$784$		0			0
$785$	−10.4853			−0.374236
$786$		0			0
$787$	21.8640	+	37.8695i	0.779366	+	1.34990i	0.932308	+	0.361666i	$0.117792\pi$
$787$	21.8640	+	37.8695i	0.779366	+	1.34990i	−0.152942	+	0.988235i	$0.548875\pi$
$788$		0			0
$789$	2.07107	−	3.58719i	0.0737320	−	0.127708i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−9.27208	+	16.0597i	−0.329261	+	0.570297i
$794$		0			0
$795$	0.778175	+	1.34784i	0.0275990	+	0.0478029i
$796$		0			0
$797$	−35.3848			−1.25339			−0.626697	−	0.779263i	$-0.715593\pi$
$797$	−35.3848			−1.25339			−0.626697	+

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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.207107 + 0.358719i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.41421 - 2.44949i) q^{9} +O(q^{10})\)	\(q+(0.207107 + 0.358719i) q^{3} +(0.500000 - 0.866025i) q^{5} +(1.41421 - 2.44949i) q^{9} +(1.91421 + 3.31552i) q^{11} +3.58579 q^{13} +0.414214 q^{15} +(-3.20711 - 5.55487i) q^{17} +(-1.82843 + 3.16693i) q^{19} +(-0.292893 + 0.507306i) q^{23} +(-0.500000 - 0.866025i) q^{25} +2.41421 q^{27} +6.65685 q^{29} +(-2.29289 - 3.97141i) q^{31} +(-0.792893 + 1.37333i) q^{33} +(1.70711 - 2.95680i) q^{37} +(0.742641 + 1.28629i) q^{39} +0.585786 q^{41} +11.6569 q^{43} +(-1.41421 - 2.44949i) q^{45} +(4.44975 - 7.70719i) q^{47} +(1.32843 - 2.30090i) q^{51} +(1.87868 + 3.25397i) q^{53} +3.82843 q^{55} -1.51472 q^{57} +(-1.70711 - 2.95680i) q^{59} +(-2.58579 + 4.47871i) q^{61} +(1.79289 - 3.10538i) q^{65} +(5.53553 + 9.58783i) q^{67} -0.242641 q^{69} +6.48528 q^{71} +(-2.58579 - 4.47871i) q^{73} +(0.207107 - 0.358719i) q^{75} +(-6.57107 + 11.3814i) q^{79} +(-3.74264 - 6.48244i) q^{81} -8.00000 q^{83} -6.41421 q^{85} +(1.37868 + 2.38794i) q^{87} +(-8.48528 + 14.6969i) q^{89} +(0.949747 - 1.64501i) q^{93} +(1.82843 + 3.16693i) q^{95} +15.7279 q^{97} +10.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 2 q^{5}+O(q^{10}) \) 4 * q - 2 * q^3 + 2 * q^5	\( 4 q - 2 q^{3} + 2 q^{5} + 2 q^{11} + 20 q^{13} - 4 q^{15} - 10 q^{17} + 4 q^{19} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 4 q^{29} - 12 q^{31} - 6 q^{33} + 4 q^{37} - 14 q^{39} + 8 q^{41} + 24 q^{43} - 2 q^{47} - 6 q^{51} + 16 q^{53} + 4 q^{55} - 40 q^{57} - 4 q^{59} - 16 q^{61} + 10 q^{65} + 8 q^{67} + 16 q^{69} - 8 q^{71} - 16 q^{73} - 2 q^{75} + 2 q^{79} + 2 q^{81} - 32 q^{83} - 20 q^{85} + 14 q^{87} - 16 q^{93} - 4 q^{95} + 12 q^{97} + 32 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 2 * q^5 + 2 * q^11 + 20 * q^13 - 4 * q^15 - 10 * q^17 + 4 * q^19 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 4 * q^29 - 12 * q^31 - 6 * q^33 + 4 * q^37 - 14 * q^39 + 8 * q^41 + 24 * q^43 - 2 * q^47 - 6 * q^51 + 16 * q^53 + 4 * q^55 - 40 * q^57 - 4 * q^59 - 16 * q^61 + 10 * q^65 + 8 * q^67 + 16 * q^69 - 8 * q^71 - 16 * q^73 - 2 * q^75 + 2 * q^79 + 2 * q^81 - 32 * q^83 - 20 * q^85 + 14 * q^87 - 16 * q^93 - 4 * q^95 + 12 * q^97 + 32 * q^99

Introduction

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