LMFDB - Embedded newform 256.2.g.a.97.1

Newspace parameters

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	256.g (of order $8$, degree $4$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$2.04417029174$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			97.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	256.97
Dual form			256.2.g.a.161.1

$q$-expansion

$f(q)$	$=$	$q+(-0.707107 + 0.292893i) q^{3} +(-1.12132 + 2.70711i) q^{5} +(-1.00000 - 1.00000i) q^{7} +(-1.70711 + 1.70711i) q^{9} +O(q^{10})$	\(q+(-0.707107 + 0.292893i) q^{3} +(-1.12132 + 2.70711i) q^{5} +(-1.00000 - 1.00000i) q^{7} +(-1.70711 + 1.70711i) q^{9} +(-4.12132 - 1.70711i) q^{11} +(-0.292893 - 0.707107i) q^{13} -2.24264i q^{15} +2.82843i q^{17} +(1.53553 + 3.70711i) q^{19} +(1.00000 + 0.414214i) q^{21} +(-5.82843 + 5.82843i) q^{23} +(-2.53553 - 2.53553i) q^{25} +(1.58579 - 3.82843i) q^{27} +(3.12132 - 1.29289i) q^{29} +4.00000 q^{31} +3.41421 q^{33} +(3.82843 - 1.58579i) q^{35} +(-0.292893 + 0.707107i) q^{37} +(0.414214 + 0.414214i) q^{39} +(-0.171573 + 0.171573i) q^{41} +(4.70711 + 1.94975i) q^{43} +(-2.70711 - 6.53553i) q^{45} -0.343146i q^{47} -5.00000i q^{49} +(-0.828427 - 2.00000i) q^{51} +(1.12132 + 0.464466i) q^{53} +(9.24264 - 9.24264i) q^{55} +(-2.17157 - 2.17157i) q^{57} +(-1.87868 + 4.53553i) q^{59} +(-1.70711 + 0.707107i) q^{61} +3.41421 q^{63} +2.24264 q^{65} +(-5.53553 + 2.29289i) q^{67} +(2.41421 - 5.82843i) q^{69} +(5.82843 + 5.82843i) q^{71} +(7.00000 - 7.00000i) q^{73} +(2.53553 + 1.05025i) q^{75} +(2.41421 + 5.82843i) q^{77} +6.00000i q^{79} -4.07107i q^{81} +(1.87868 + 4.53553i) q^{83} +(-7.65685 - 3.17157i) q^{85} +(-1.82843 + 1.82843i) q^{87} +(8.65685 + 8.65685i) q^{89} +(-0.414214 + 1.00000i) q^{91} +(-2.82843 + 1.17157i) q^{93} -11.7574 q^{95} -18.4853 q^{97} +(9.94975 - 4.12132i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 - 4 * q^7 - 4 * q^9	$ 4 q + 4 q^{5} - 4 q^{7} - 4 q^{9} - 8 q^{11} - 4 q^{13} - 8 q^{19} + 4 q^{21} - 12 q^{23} + 4 q^{25} + 12 q^{27} + 4 q^{29} + 16 q^{31} + 8 q^{33} + 4 q^{35} - 4 q^{37} - 4 q^{39} - 12 q^{41} + 16 q^{43} - 8 q^{45} + 8 q^{51} - 4 q^{53} + 20 q^{55} - 20 q^{57} - 16 q^{59} - 4 q^{61} + 8 q^{63} - 8 q^{65} - 8 q^{67} + 4 q^{69} + 12 q^{71} + 28 q^{73} - 4 q^{75} + 4 q^{77} + 16 q^{83} - 8 q^{85} + 4 q^{87} + 12 q^{89} + 4 q^{91} - 64 q^{95} - 40 q^{97} + 20 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^7 - 4 * q^9 - 8 * q^11 - 4 * q^13 - 8 * q^19 + 4 * q^21 - 12 * q^23 + 4 * q^25 + 12 * q^27 + 4 * q^29 + 16 * q^31 + 8 * q^33 + 4 * q^35 - 4 * q^37 - 4 * q^39 - 12 * q^41 + 16 * q^43 - 8 * q^45 + 8 * q^51 - 4 * q^53 + 20 * q^55 - 20 * q^57 - 16 * q^59 - 4 * q^61 + 8 * q^63 - 8 * q^65 - 8 * q^67 + 4 * q^69 + 12 * q^71 + 28 * q^73 - 4 * q^75 + 4 * q^77 + 16 * q^83 - 8 * q^85 + 4 * q^87 + 12 * q^89 + 4 * q^91 - 64 * q^95 - 40 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$e\left(\frac{5}{8}\right)$	$1$

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
$3$	−0.707107	+	0.292893i	−0.408248	+	0.169102i	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−0.707107	+	0.292893i	−0.408248	+	0.169102i	0.169102	+	0.985599i	$0.445913\pi$
$4$		0			0
$5$	−1.12132	+	2.70711i	−0.501470	+	1.21065i	0.447214	+	0.894427i	$0.352416\pi$
$5$	−1.12132	+	2.70711i	−0.501470	+	1.21065i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$	−1.00000	−	1.00000i	−0.377964	−	0.377964i	0.492403	−	0.870367i	$-0.336119\pi$
$7$	−1.00000	−	1.00000i	−0.377964	−	0.377964i	−0.870367	+	0.492403i	$0.836119\pi$
$8$		0			0
$9$	−1.70711	+	1.70711i	−0.569036	+	0.569036i
$10$		0			0
$11$	−4.12132	−	1.70711i	−1.24262	−	0.514712i	−0.338091	−	0.941113i	$-0.609781\pi$
$11$	−4.12132	−	1.70711i	−1.24262	−	0.514712i	−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$	−0.292893	−	0.707107i	−0.0812340	−	0.196116i	0.878044	−	0.478580i	$-0.158848\pi$
$13$	−0.292893	−	0.707107i	−0.0812340	−	0.196116i	−0.959278	+	0.282464i	$0.908848\pi$
$14$		0			0
$15$		−	2.24264i		−	0.579047i
$16$		0			0
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$		0			0
$19$	1.53553	+	3.70711i	0.352276	+	0.850469i	0.996339	+	0.0854961i	$0.0272475\pi$
$19$	1.53553	+	3.70711i	0.352276	+	0.850469i	−0.644063	+	0.764973i	$0.722752\pi$
$20$		0			0
$21$	1.00000	+	0.414214i	0.218218	+	0.0903888i
$22$		0			0
$23$	−5.82843	+	5.82843i	−1.21531	+	1.21531i	−0.246055	+	0.969256i	$0.579134\pi$
$23$	−5.82843	+	5.82843i	−1.21531	+	1.21531i	−0.969256	+	0.246055i	$0.920866\pi$
$24$		0			0
$25$	−2.53553	−	2.53553i	−0.507107	−	0.507107i
$26$		0			0
$27$	1.58579	−	3.82843i	0.305185	−	0.736781i
$28$		0			0
$29$	3.12132	−	1.29289i	0.579615	−	0.240084i	−0.0735609	−	0.997291i	$-0.523436\pi$
$29$	3.12132	−	1.29289i	0.579615	−	0.240084i	0.653176	+	0.757206i	$0.273436\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$	3.41421			0.594338
$34$		0			0
$35$	3.82843	−	1.58579i	0.647122	−	0.268047i
$36$		0			0
$37$	−0.292893	+	0.707107i	−0.0481513	+	0.116248i	−0.946125	−	0.323802i	$-0.895039\pi$
$37$	−0.292893	+	0.707107i	−0.0481513	+	0.116248i	0.897974	+	0.440049i	$0.145039\pi$
$38$		0			0
$39$	0.414214	+	0.414214i	0.0663273	+	0.0663273i
$40$		0			0
$41$	−0.171573	+	0.171573i	−0.0267952	+	0.0267952i	−0.720377	−	0.693582i	$-0.756031\pi$
$41$	−0.171573	+	0.171573i	−0.0267952	+	0.0267952i	0.693582	+	0.720377i	$0.256031\pi$
$42$		0			0
$43$	4.70711	+	1.94975i	0.717827	+	0.297334i	0.711539	−	0.702647i	$-0.247998\pi$
$43$	4.70711	+	1.94975i	0.717827	+	0.297334i	0.00628798	+	0.999980i	$0.497998\pi$
$44$		0			0
$45$	−2.70711	−	6.53553i	−0.403552	−	0.974260i
$46$		0			0
$47$		−	0.343146i		−	0.0500530i	−0.999687	−	0.0250265i	$-0.992033\pi$
$47$		−	0.343146i		−	0.0500530i	0.999687	−	0.0250265i	$-0.00796701\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$	−0.828427	−	2.00000i	−0.116003	−	0.280056i
$52$		0			0
$53$	1.12132	+	0.464466i	0.154025	+	0.0637993i	0.458364	−	0.888764i	$-0.348436\pi$
$53$	1.12132	+	0.464466i	0.154025	+	0.0637993i	−0.304339	+	0.952564i	$0.598436\pi$
$54$		0			0
$55$	9.24264	−	9.24264i	1.24628	−	1.24628i
$56$		0			0
$57$	−2.17157	−	2.17157i	−0.287632	−	0.287632i
$58$		0			0
$59$	−1.87868	+	4.53553i	−0.244583	+	0.590476i	−0.997727	−	0.0673793i	$-0.978536\pi$
$59$	−1.87868	+	4.53553i	−0.244583	+	0.590476i	0.753144	+	0.657855i	$0.228536\pi$
$60$		0			0
$61$	−1.70711	+	0.707107i	−0.218573	+	0.0905357i	−0.489283	−	0.872125i	$-0.662741\pi$
$61$	−1.70711	+	0.707107i	−0.218573	+	0.0905357i	0.270710	+	0.962661i	$0.412741\pi$
$62$		0			0
$63$	3.41421			0.430150
$64$		0			0
$65$	2.24264			0.278165
$66$		0			0
$67$	−5.53553	+	2.29289i	−0.676273	+	0.280121i	−0.694268	−	0.719717i	$-0.744272\pi$
$67$	−5.53553	+	2.29289i	−0.676273	+	0.280121i	0.0179949	+	0.999838i	$0.494272\pi$
$68$		0			0
$69$	2.41421	−	5.82843i	0.290637	−	0.701660i
$70$		0			0
$71$	5.82843	+	5.82843i	0.691707	+	0.691707i	0.962607	−	0.270900i	$-0.0873214\pi$
$71$	5.82843	+	5.82843i	0.691707	+	0.691707i	−0.270900	+	0.962607i	$0.587321\pi$
$72$		0			0
$73$	7.00000	−	7.00000i	0.819288	−	0.819288i	−0.166717	−	0.986005i	$-0.553317\pi$
$73$	7.00000	−	7.00000i	0.819288	−	0.819288i	0.986005	+	0.166717i	$0.0533166\pi$
$74$		0			0
$75$	2.53553	+	1.05025i	0.292778	+	0.121273i
$76$		0			0
$77$	2.41421	+	5.82843i	0.275125	+	0.664211i
$78$		0			0
$79$			6.00000i			0.675053i	0.941316	+	0.337526i	$0.109590\pi$
$79$			6.00000i			0.675053i	−0.941316	+	0.337526i	$0.890410\pi$
$80$		0			0
$81$		−	4.07107i		−	0.452341i
$82$		0			0
$83$	1.87868	+	4.53553i	0.206212	+	0.497840i	0.992821	−	0.119612i	$-0.0381651\pi$
$83$	1.87868	+	4.53553i	0.206212	+	0.497840i	−0.786609	+	0.617452i	$0.788165\pi$
$84$		0			0
$85$	−7.65685	−	3.17157i	−0.830502	−	0.344005i
$86$		0			0
$87$	−1.82843	+	1.82843i	−0.196028	+	0.196028i
$88$		0			0
$89$	8.65685	+	8.65685i	0.917625	+	0.917625i	0.996856	−	0.0792315i	$-0.0252466\pi$
$89$	8.65685	+	8.65685i	0.917625	+	0.917625i	−0.0792315	+	0.996856i	$0.525247\pi$
$90$		0			0
$91$	−0.414214	+	1.00000i	−0.0434214	+	0.104828i
$92$		0			0
$93$	−2.82843	+	1.17157i	−0.293294	+	0.121486i
$94$		0			0
$95$	−11.7574			−1.20628
$96$		0			0
$97$	−18.4853			−1.87690			−0.938448	−	0.345421i	$-0.887736\pi$
$97$	−18.4853			−1.87690			−0.938448	+	0.345421i	$0.887736\pi$
$98$		0			0
$99$	9.94975	−	4.12132i	0.999987	−	0.414208i
$100$		0			0
$101$	1.36396	−	3.29289i	0.135719	−	0.327655i	−0.841379	−	0.540446i	$-0.818255\pi$
$101$	1.36396	−	3.29289i	0.135719	−	0.327655i	0.977098	+	0.212791i	$0.0682554\pi$
$102$		0			0
$103$	−9.48528	−	9.48528i	−0.934613	−	0.934613i	0.0633771	−	0.997990i	$-0.479813\pi$
$103$	−9.48528	−	9.48528i	−0.934613	−	0.934613i	−0.997990	+	0.0633771i	$0.979813\pi$
$104$		0			0
$105$	−2.24264	+	2.24264i	−0.218859	+	0.218859i
$106$		0			0
$107$	−4.12132	−	1.70711i	−0.398423	−	0.165032i	0.174470	−	0.984663i	$-0.444179\pi$
$107$	−4.12132	−	1.70711i	−0.398423	−	0.165032i	−0.572893	+	0.819630i	$0.694179\pi$
$108$		0			0
$109$	5.70711	+	13.7782i	0.546642	+	1.31971i	0.919962	+	0.392007i	$0.128219\pi$
$109$	5.70711	+	13.7782i	0.546642	+	1.31971i	−0.373320	+	0.927702i	$0.621781\pi$
$110$		0			0
$111$		−	0.585786i		−	0.0556004i
$112$		0			0
$113$			6.34315i			0.596713i	0.954455	+	0.298356i	$0.0964384\pi$
$113$			6.34315i			0.596713i	−0.954455	+	0.298356i	$0.903562\pi$
$114$		0			0
$115$	−9.24264	−	22.3137i	−0.861881	−	2.08076i
$116$		0			0
$117$	1.70711	+	0.707107i	0.157822	+	0.0653720i
$118$		0			0
$119$	2.82843	−	2.82843i	0.259281	−	0.259281i
$120$		0			0
$121$	6.29289	+	6.29289i	0.572081	+	0.572081i
$122$		0			0
$123$	0.0710678	−	0.171573i	0.00640797	−	0.0154702i
$124$		0			0
$125$	−3.82843	+	1.58579i	−0.342425	+	0.141837i
$126$		0			0
$127$	−12.9706			−1.15095			−0.575476	−	0.817819i	$-0.695183\pi$
$127$	−12.9706			−1.15095			−0.575476	+	0.817819i	$0.695183\pi$
$128$		0			0
$129$	−3.89949			−0.343331
$130$		0			0
$131$	−16.3640	+	6.77817i	−1.42973	+	0.592212i	−0.957284	−	0.289150i	$-0.906627\pi$
$131$	−16.3640	+	6.77817i	−1.42973	+	0.592212i	−0.472442	+	0.881362i	$0.656627\pi$
$132$		0			0
$133$	2.17157	−	5.24264i	0.188299	−	0.454595i
$134$		0			0
$135$	8.58579	+	8.58579i	0.738947	+	0.738947i
$136$		0			0
$137$	−8.65685	+	8.65685i	−0.739605	+	0.739605i	−0.972502	−	0.232897i	$-0.925180\pi$
$137$	−8.65685	+	8.65685i	−0.739605	+	0.739605i	0.232897	+	0.972502i	$0.425180\pi$
$138$		0			0
$139$	13.1924	+	5.46447i	1.11896	+	0.463490i	0.864016	−	0.503465i	$-0.167942\pi$
$139$	13.1924	+	5.46447i	1.11896	+	0.463490i	0.254948	+	0.966955i	$0.417942\pi$
$140$		0			0
$141$	0.100505	+	0.242641i	0.00846405	+	0.0204340i
$142$		0			0
$143$			3.41421i			0.285511i
$144$		0			0
$145$			9.89949i			0.822108i
$146$		0			0
$147$	1.46447	+	3.53553i	0.120787	+	0.291606i
$148$		0			0
$149$	15.6066	+	6.46447i	1.27854	+	0.529590i	0.915551	−	0.402203i	$-0.131755\pi$
$149$	15.6066	+	6.46447i	1.27854	+	0.529590i	0.362992	+	0.931792i	$0.381755\pi$
$150$		0			0
$151$	1.48528	−	1.48528i	0.120870	−	0.120870i	−0.644084	−	0.764955i	$-0.722761\pi$
$151$	1.48528	−	1.48528i	0.120870	−	0.120870i	0.764955	+	0.644084i	$0.222761\pi$
$152$		0			0
$153$	−4.82843	−	4.82843i	−0.390355	−	0.390355i
$154$		0			0
$155$	−4.48528	+	10.8284i	−0.360266	+	0.869760i
$156$		0			0
$157$	−1.70711	+	0.707107i	−0.136242	+	0.0564333i	−0.449763	−	0.893148i	$-0.648491\pi$
$157$	−1.70711	+	0.707107i	−0.136242	+	0.0564333i	0.313521	+	0.949581i	$0.398491\pi$
$158$		0			0
$159$	−0.928932			−0.0736691
$160$		0			0
$161$	11.6569			0.918689
$162$		0			0
$163$	0.464466	−	0.192388i	0.0363798	−	0.0150690i	−0.364419	−	0.931235i	$-0.618733\pi$
$163$	0.464466	−	0.192388i	0.0363798	−	0.0150690i	0.400799	+	0.916166i	$0.368733\pi$
$164$		0			0
$165$	−3.82843	+	9.24264i	−0.298043	+	0.719539i
$166$		0			0
$167$	−14.6569	−	14.6569i	−1.13418	−	1.13418i	−0.989475	−	0.144707i	$-0.953776\pi$
$167$	−14.6569	−	14.6569i	−1.13418	−	1.13418i	−0.144707	−	0.989475i	$-0.546224\pi$
$168$		0			0
$169$	8.77817	−	8.77817i	0.675244	−	0.675244i
$170$		0			0
$171$	−8.94975	−	3.70711i	−0.684404	−	0.283490i
$172$		0			0
$173$	−3.12132	−	7.53553i	−0.237310	−	0.572916i	0.759693	−	0.650282i	$-0.225349\pi$
$173$	−3.12132	−	7.53553i	−0.237310	−	0.572916i	−0.997003	+	0.0773656i	$0.975349\pi$
$174$		0			0
$175$			5.07107i			0.383337i
$176$		0			0
$177$		−	3.75736i		−	0.282420i
$178$		0			0
$179$	−1.63604	−	3.94975i	−0.122283	−	0.295218i	0.850870	−	0.525377i	$-0.176076\pi$
$179$	−1.63604	−	3.94975i	−0.122283	−	0.295218i	−0.973153	+	0.230159i	$0.926076\pi$
$180$		0			0
$181$	−16.1924	−	6.70711i	−1.20357	−	0.498535i	−0.311420	−	0.950272i	$-0.600804\pi$
$181$	−16.1924	−	6.70711i	−1.20357	−	0.498535i	−0.892151	+	0.451737i	$0.850804\pi$
$182$		0			0
$183$	1.00000	−	1.00000i	0.0739221	−	0.0739221i
$184$		0			0
$185$	−1.58579	−	1.58579i	−0.116589	−	0.116589i
$186$		0			0
$187$	4.82843	−	11.6569i	0.353090	−	0.852434i
$188$		0			0
$189$	−5.41421	+	2.24264i	−0.393826	+	0.163128i
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$	−1.51472			−0.109032			−0.0545159	−	0.998513i	$-0.517362\pi$
$193$	−1.51472			−0.109032			−0.0545159	+	0.998513i	$0.517362\pi$
$194$		0			0
$195$	−1.58579	+	0.656854i	−0.113561	+	0.0470383i
$196$		0			0
$197$	−4.63604	+	11.1924i	−0.330304	+	0.797425i	0.668264	+	0.743924i	$0.267038\pi$
$197$	−4.63604	+	11.1924i	−0.330304	+	0.797425i	−0.998568	+	0.0535002i	$0.982962\pi$
$198$		0			0
$199$	15.9706	+	15.9706i	1.13212	+	1.13212i	0.989824	+	0.142300i	$0.0454496\pi$
$199$	15.9706	+	15.9706i	1.13212	+	1.13212i	0.142300	+	0.989824i	$0.454550\pi$
$200$		0			0
$201$	3.24264	−	3.24264i	0.228718	−	0.228718i
$202$		0			0
$203$	−4.41421	−	1.82843i	−0.309817	−	0.128330i
$204$		0			0
$205$	−0.272078	−	0.656854i	−0.0190027	−	0.0458767i
$206$		0			0
$207$		−	19.8995i		−	1.38311i
$208$		0			0
$209$		−	17.8995i		−	1.23813i
$210$		0			0
$211$	7.53553	+	18.1924i	0.518768	+	1.25242i	0.938661	+	0.344842i	$0.112068\pi$
$211$	7.53553	+	18.1924i	0.518768	+	1.25242i	−0.419893	+	0.907574i	$0.637932\pi$
$212$		0			0
$213$	−5.82843	−	2.41421i	−0.399357	−	0.165419i
$214$		0			0
$215$	−10.5563	+	10.5563i	−0.719937	+	0.719937i
$216$		0			0
$217$	−4.00000	−	4.00000i	−0.271538	−	0.271538i
$218$		0			0
$219$	−2.89949	+	7.00000i	−0.195930	+	0.473016i
$220$		0			0
$221$	2.00000	−	0.828427i	0.134535	−	0.0557260i
$222$		0			0
$223$	20.9706			1.40429			0.702146	−	0.712033i	$-0.252225\pi$
$223$	20.9706			1.40429			0.702146	+	0.712033i	$0.252225\pi$
$224$		0			0
$225$	8.65685			0.577124
$226$		0			0
$227$	18.6066	−	7.70711i	1.23496	−	0.511539i	0.332826	−	0.942988i	$-0.391998\pi$
$227$	18.6066	−	7.70711i	1.23496	−	0.511539i	0.902137	+	0.431449i	$0.141998\pi$
$228$		0			0
$229$	9.22183	−	22.2635i	0.609395	−	1.47121i	−0.254264	−	0.967135i	$-0.581833\pi$
$229$	9.22183	−	22.2635i	0.609395	−	1.47121i	0.863659	−	0.504076i	$-0.168167\pi$
$230$		0			0
$231$	−3.41421	−	3.41421i	−0.224639	−	0.224639i
$232$		0			0
$233$	−2.65685	+	2.65685i	−0.174056	+	0.174056i	−0.788759	−	0.614703i	$-0.789276\pi$
$233$	−2.65685	+	2.65685i	−0.174056	+	0.174056i	0.614703	+	0.788759i	$0.289276\pi$
$234$		0			0
$235$	0.928932	+	0.384776i	0.0605969	+	0.0251000i
$236$		0			0
$237$	−1.75736	−	4.24264i	−0.114153	−	0.275589i
$238$		0			0
$239$		−	5.31371i		−	0.343715i	−0.985122	−	0.171858i	$-0.945023\pi$
$239$		−	5.31371i		−	0.343715i	0.985122	−	0.171858i	$-0.0549769\pi$
$240$		0			0
$241$			8.48528i			0.546585i	0.961931	+	0.273293i	$0.0881127\pi$
$241$			8.48528i			0.546585i	−0.961931	+	0.273293i	$0.911887\pi$
$242$		0			0
$243$	5.94975	+	14.3640i	0.381676	+	0.921449i
$244$		0			0
$245$	13.5355	+	5.60660i	0.864754	+	0.358193i
$246$		0			0
$247$	2.17157	−	2.17157i	0.138174	−	0.138174i
$248$		0			0
$249$	−2.65685	−	2.65685i	−0.168371	−	0.168371i
$250$		0			0
$251$	6.60660	−	15.9497i	0.417005	−	1.00674i	−0.566205	−	0.824264i	$-0.691589\pi$
$251$	6.60660	−	15.9497i	0.417005	−	1.00674i	0.983210	−	0.182475i	$-0.0584109\pi$
$252$		0			0
$253$	33.9706	−	14.0711i	2.13571	−	0.884640i
$254$		0			0
$255$	6.34315			0.397223
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$	1.00000	−	0.414214i	0.0621370	−	0.0257380i
$260$		0			0
$261$	−3.12132	+	7.53553i	−0.193205	+	0.466438i
$262$		0			0
$263$	5.82843	+	5.82843i	0.359396	+	0.359396i	0.863590	−	0.504194i	$-0.168210\pi$
$263$	5.82843	+	5.82843i	0.359396	+	0.359396i	−0.504194	+	0.863590i	$0.668210\pi$
$264$		0			0
$265$	−2.51472	+	2.51472i	−0.154478	+	0.154478i
$266$		0			0
$267$	−8.65685	−	3.58579i	−0.529791	−	0.219447i
$268$		0			0
$269$	−9.12132	−	22.0208i	−0.556137	−	1.34263i	−0.912803	−	0.408401i	$-0.866087\pi$
$269$	−9.12132	−	22.0208i	−0.556137	−	1.34263i	0.356666	−	0.934232i	$-0.383913\pi$
$270$		0			0
$271$			18.0000i			1.09342i	0.837321	+	0.546711i	$0.184120\pi$
$271$			18.0000i			1.09342i	−0.837321	+	0.546711i	$0.815880\pi$
$272$		0			0
$273$		−	0.828427i		−	0.0501387i
$274$		0			0
$275$	6.12132	+	14.7782i	0.369130	+	0.891157i
$276$		0			0
$277$	−1.70711	−	0.707107i	−0.102570	−	0.0424859i	0.330808	−	0.943698i	$-0.392679\pi$
$277$	−1.70711	−	0.707107i	−0.102570	−	0.0424859i	−0.433378	+	0.901212i	$0.642679\pi$
$278$		0			0
$279$	−6.82843	+	6.82843i	−0.408807	+	0.408807i
$280$		0			0
$281$	−11.8284	−	11.8284i	−0.705625	−	0.705625i	0.259987	−	0.965612i	$-0.416282\pi$
$281$	−11.8284	−	11.8284i	−0.705625	−	0.705625i	−0.965612	+	0.259987i	$0.916282\pi$
$282$		0			0
$283$	5.77817	−	13.9497i	0.343477	−	0.829226i	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	5.77817	−	13.9497i	0.343477	−	0.829226i	0.997359	−	0.0726300i	$-0.0231392\pi$
$284$		0			0
$285$	8.31371	−	3.44365i	0.492462	−	0.203984i
$286$		0			0
$287$	0.343146			0.0202553
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$	13.0711	−	5.41421i	0.766240	−	0.317387i
$292$		0			0
$293$	−9.60660	+	23.1924i	−0.561224	+	1.35491i	0.347565	+	0.937656i	$0.387009\pi$
$293$	−9.60660	+	23.1924i	−0.561224	+	1.35491i	−0.908788	+	0.417258i	$0.862991\pi$
$294$		0			0
$295$	−10.1716	−	10.1716i	−0.592212	−	0.592212i
$296$		0			0
$297$	−13.0711	+	13.0711i	−0.758460	+	0.758460i
$298$		0			0
$299$	5.82843	+	2.41421i	0.337067	+	0.139618i
$300$		0			0
$301$	−2.75736	−	6.65685i	−0.158932	−	0.383695i
$302$		0			0
$303$			2.72792i			0.156715i
$304$		0			0
$305$		−	5.41421i		−	0.310017i
$306$		0			0
$307$	−6.94975	−	16.7782i	−0.396643	−	0.957581i	−0.988456	−	0.151506i	$-0.951588\pi$
$307$	−6.94975	−	16.7782i	−0.396643	−	0.957581i	0.591813	−	0.806075i	$-0.298412\pi$
$308$		0			0
$309$	9.48528	+	3.92893i	0.539599	+	0.223509i
$310$		0			0
$311$	2.65685	−	2.65685i	0.150656	−	0.150656i	−0.627755	−	0.778411i	$-0.716026\pi$
$311$	2.65685	−	2.65685i	0.150656	−	0.150656i	0.778411	+	0.627755i	$0.216026\pi$
$312$		0			0
$313$	−7.48528	−	7.48528i	−0.423093	−	0.423093i	0.463174	−	0.886267i	$-0.346710\pi$
$313$	−7.48528	−	7.48528i	−0.423093	−	0.423093i	−0.886267	+	0.463174i	$0.846710\pi$
$314$		0			0
$315$	−3.82843	+	9.24264i	−0.215707	+	0.520764i
$316$		0			0
$317$	−17.3640	+	7.19239i	−0.975257	+	0.403965i	−0.812667	−	0.582729i	$-0.801985\pi$
$317$	−17.3640	+	7.19239i	−0.975257	+	0.403965i	−0.162591	+	0.986694i	$0.551985\pi$
$318$		0			0
$319$	−15.0711			−0.843818
$320$		0			0
$321$	3.41421			0.190563
$322$		0			0
$323$	−10.4853	+	4.34315i	−0.583417	+	0.241659i
$324$		0			0
$325$	−1.05025	+	2.53553i	−0.0582575	+	0.140646i
$326$		0			0
$327$	−8.07107	−	8.07107i	−0.446331	−	0.446331i
$328$		0			0
$329$	−0.343146	+	0.343146i	−0.0189182	+	0.0189182i
$330$		0			0
$331$	−1.29289	−	0.535534i	−0.0710638	−	0.0294356i	0.346868	−	0.937914i	$-0.387245\pi$
$331$	−1.29289	−	0.535534i	−0.0710638	−	0.0294356i	−0.417932	+	0.908478i	$0.637245\pi$
$332$		0			0
$333$	−0.707107	−	1.70711i	−0.0387492	−	0.0935489i
$334$		0			0
$335$		−	17.5563i		−	0.959206i
$336$		0			0
$337$			16.9706i			0.924445i	0.886764	+	0.462223i	$0.152948\pi$
$337$			16.9706i			0.924445i	−0.886764	+	0.462223i	$0.847052\pi$
$338$		0			0
$339$	−1.85786	−	4.48528i	−0.100905	−	0.243607i
$340$		0			0
$341$	−16.4853	−	6.82843i	−0.892728	−	0.369780i
$342$		0			0
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$		0			0
$345$	13.0711	+	13.0711i	0.703723	+	0.703723i
$346$		0			0
$347$	1.63604	−	3.94975i	0.0878272	−	0.212034i	−0.873863	−	0.486172i	$-0.838393\pi$
$347$	1.63604	−	3.94975i	0.0878272	−	0.212034i	0.961690	+	0.274139i	$0.0883927\pi$
$348$		0			0
$349$	−24.6777	+	10.2218i	−1.32097	+	0.547162i	−0.928065	−	0.372419i	$-0.878528\pi$
$349$	−24.6777	+	10.2218i	−1.32097	+	0.547162i	−0.392901	+	0.919581i	$0.628528\pi$
$350$		0			0
$351$	−3.17157			−0.169286
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$	−22.3137	+	9.24264i	−1.18429	+	0.490548i
$356$		0			0
$357$	−1.17157	+	2.82843i	−0.0620062	+	0.149696i
$358$		0			0
$359$	17.8284	+	17.8284i	0.940948	+	0.940948i	0.998351	−	0.0574027i	$-0.0182819\pi$
$359$	17.8284	+	17.8284i	0.940948	+	0.940948i	−0.0574027	+	0.998351i	$0.518282\pi$
$360$		0			0
$361$	2.05025	−	2.05025i	0.107908	−	0.107908i
$362$		0			0
$363$	−6.29289	−	2.60660i	−0.330291	−	0.136811i
$364$		0			0
$365$	11.1005	+	26.7990i	0.581027	+	1.40272i
$366$		0			0
$367$		−	6.00000i		−	0.313197i	−0.987662	−	0.156599i	$-0.949947\pi$
$367$		−	6.00000i		−	0.313197i	0.987662	−	0.156599i	$-0.0500529\pi$
$368$		0			0
$369$		−	0.585786i		−	0.0304948i
$370$		0			0
$371$	−0.656854	−	1.58579i	−0.0341022	−	0.0823299i
$372$		0			0
$373$	10.2929	+	4.26346i	0.532946	+	0.220753i	0.632893	−	0.774239i	$-0.281867\pi$
$373$	10.2929	+	4.26346i	0.532946	+	0.220753i	−0.0999471	+	0.994993i	$0.531867\pi$
$374$		0			0
$375$	2.24264	−	2.24264i	0.115809	−	0.115809i
$376$		0			0
$377$	−1.82843	−	1.82843i	−0.0941688	−	0.0941688i
$378$		0			0
$379$	−13.6777	+	33.0208i	−0.702575	+	1.69617i	0.0151948	+	0.999885i	$0.495163\pi$
$379$	−13.6777	+	33.0208i	−0.702575	+	1.69617i	−0.717769	+	0.696281i	$0.754837\pi$
$380$		0			0
$381$	9.17157	−	3.79899i	0.469874	−	0.194628i
$382$		0			0
$383$	16.9706			0.867155			0.433578	−	0.901116i	$-0.357251\pi$
$383$	16.9706			0.867155			0.433578	+	0.901116i	$0.357251\pi$
$384$		0			0
$385$	−18.4853			−0.942097
$386$		0			0
$387$	−11.3640	+	4.70711i	−0.577663	+	0.239276i
$388$		0			0
$389$	8.39340	−	20.2635i	0.425562	−	1.02740i	−0.555117	−	0.831773i	$-0.687326\pi$
$389$	8.39340	−	20.2635i	0.425562	−	1.02740i	0.980679	−	0.195625i	$-0.0626737\pi$
$390$		0			0
$391$	−16.4853	−	16.4853i	−0.833697	−	0.833697i
$392$		0			0
$393$	9.58579	−	9.58579i	0.483539	−	0.483539i
$394$		0			0
$395$	−16.2426	−	6.72792i	−0.817256	−	0.338518i
$396$		0			0
$397$	9.22183	+	22.2635i	0.462830	+	1.11737i	0.967230	+	0.253901i	$0.0817137\pi$
$397$	9.22183	+	22.2635i	0.462830	+	1.11737i	−0.504400	+	0.863470i	$0.668286\pi$
$398$		0			0
$399$			4.34315i			0.217429i
$400$		0			0
$401$			2.82843i			0.141245i	0.997503	+	0.0706225i	$0.0224986\pi$
$401$			2.82843i			0.141245i	−0.997503	+	0.0706225i	$0.977501\pi$
$402$		0			0
$403$	−1.17157	−	2.82843i	−0.0583602	−	0.140894i
$404$		0			0
$405$	11.0208	+	4.56497i	0.547629	+	0.226835i
$406$		0			0
$407$	2.41421	−	2.41421i	0.119668	−	0.119668i
$408$		0			0
$409$	21.4853	+	21.4853i	1.06238	+	1.06238i	0.997920	+	0.0644584i	$0.0205320\pi$
$409$	21.4853	+	21.4853i	1.06238	+	1.06238i	0.0644584	+	0.997920i	$0.479468\pi$
$410$		0			0
$411$	3.58579	−	8.65685i	0.176874	−	0.427011i
$412$		0			0
$413$	6.41421	−	2.65685i	0.315623	−	0.130735i
$414$		0			0
$415$	−14.3848			−0.706121
$416$		0			0
$417$	−10.9289			−0.535192
$418$		0			0
$419$	12.6066	−	5.22183i	0.615873	−	0.255103i	−0.0528644	−	0.998602i	$-0.516835\pi$
$419$	12.6066	−	5.22183i	0.615873	−	0.255103i	0.668737	+	0.743499i	$0.266835\pi$
$420$		0			0
$421$	−6.29289	+	15.1924i	−0.306697	+	0.740432i	0.693111	+	0.720831i	$0.256240\pi$
$421$	−6.29289	+	15.1924i	−0.306697	+	0.740432i	−0.999808	+	0.0196009i	$0.993760\pi$
$422$		0			0
$423$	0.585786	+	0.585786i	0.0284819	+	0.0284819i
$424$		0			0
$425$	7.17157	−	7.17157i	0.347872	−	0.347872i
$426$		0			0
$427$	2.41421	+	1.00000i	0.116832	+	0.0483934i
$428$		0			0
$429$	−1.00000	−	2.41421i	−0.0482805	−	0.116559i
$430$		0			0
$431$		−	12.3431i		−	0.594548i	−0.954792	−	0.297274i	$-0.903922\pi$
$431$		−	12.3431i		−	0.594548i	0.954792	−	0.297274i	$-0.0960775\pi$
$432$		0			0
$433$			15.5147i			0.745590i	0.927914	+	0.372795i	$0.121600\pi$
$433$			15.5147i			0.745590i	−0.927914	+	0.372795i	$0.878400\pi$
$434$		0			0
$435$	−2.89949	−	7.00000i	−0.139020	−	0.335624i
$436$		0			0
$437$	−30.5563	−	12.6569i	−1.46171	−	0.605459i
$438$		0			0
$439$	17.0000	−	17.0000i	0.811366	−	0.811366i	−0.173473	−	0.984839i	$-0.555499\pi$
$439$	17.0000	−	17.0000i	0.811366	−	0.811366i	0.984839	+	0.173473i	$0.0554989\pi$
$440$		0			0
$441$	8.53553	+	8.53553i	0.406454	+	0.406454i
$442$		0			0
$443$	0.606602	−	1.46447i	0.0288205	−	0.0695789i	−0.908814	−	0.417201i	$-0.863011\pi$
$443$	0.606602	−	1.46447i	0.0288205	−	0.0695789i	0.937635	+	0.347623i	$0.113011\pi$
$444$		0			0
$445$	−33.1421	+	13.7279i	−1.57109	+	0.650766i
$446$		0			0
$447$	−12.9289			−0.611518
$448$		0			0
$449$	−19.4558			−0.918178			−0.459089	−	0.888390i	$-0.651824\pi$
$449$	−19.4558			−0.918178			−0.459089	+	0.888390i	$0.651824\pi$
$450$		0			0
$451$	1.00000	−	0.414214i	0.0470882	−	0.0195046i
$452$		0			0
$453$	−0.615224	+	1.48528i	−0.0289057	+	0.0697846i
$454$		0			0
$455$	−2.24264	−	2.24264i	−0.105137	−	0.105137i
$456$		0			0
$457$	−7.48528	+	7.48528i	−0.350147	+	0.350147i	−0.860164	−	0.510017i	$-0.829639\pi$
$457$	−7.48528	+	7.48528i	−0.350147	+	0.350147i	0.510017	+	0.860164i	$0.329639\pi$
$458$		0			0
$459$	10.8284	+	4.48528i	0.505428	+	0.209355i
$460$		0			0
$461$	−0.636039	−	1.53553i	−0.0296233	−	0.0715169i	0.908376	−	0.418155i	$-0.137323\pi$
$461$	−0.636039	−	1.53553i	−0.0296233	−	0.0715169i	−0.937999	+	0.346638i	$0.887323\pi$
$462$		0			0
$463$			22.9706i			1.06753i	0.845632	+	0.533766i	$0.179224\pi$
$463$			22.9706i			1.06753i	−0.845632	+	0.533766i	$0.820776\pi$
$464$		0			0
$465$		−	8.97056i		−	0.416000i
$466$		0			0
$467$	−9.09188	−	21.9497i	−0.420722	−	1.01571i	−0.982135	−	0.188177i	$-0.939742\pi$
$467$	−9.09188	−	21.9497i	−0.420722	−	1.01571i	0.561413	−	0.827536i	$-0.310258\pi$
$468$		0			0
$469$	7.82843	+	3.24264i	0.361483	+	0.149731i
$470$		0			0
$471$	1.00000	−	1.00000i	0.0460776	−	0.0460776i
$472$		0			0
$473$	−16.0711	−	16.0711i	−0.738948	−	0.738948i
$474$		0			0
$475$	5.50610	−	13.2929i	0.252637	−	0.609920i
$476$		0			0
$477$	−2.70711	+	1.12132i	−0.123950	+	0.0513417i
$478$		0			0
$479$	−28.9706			−1.32370			−0.661849	−	0.749637i	$-0.730228\pi$
$479$	−28.9706			−1.32370			−0.661849	+	0.749637i	$0.730228\pi$
$480$		0			0
$481$	0.585786			0.0267096
$482$		0			0
$483$	−8.24264	+	3.41421i	−0.375053	+	0.155352i
$484$		0			0
$485$	20.7279	−	50.0416i	0.941206	−	2.27227i
$486$		0			0
$487$	11.0000	+	11.0000i	0.498458	+	0.498458i	0.910958	−	0.412500i	$-0.135344\pi$
$487$	11.0000	+	11.0000i	0.498458	+	0.498458i	−0.412500	+	0.910958i	$0.635344\pi$
$488$		0			0
$489$	−0.272078	+	0.272078i	−0.0123038	+	0.0123038i
$490$		0			0
$491$	39.3345	+	16.2929i	1.77514	+	0.735288i	0.993800	+	0.111186i	$0.0354648\pi$
$491$	39.3345	+	16.2929i	1.77514	+	0.735288i	0.781343	+	0.624102i	$0.214535\pi$
$492$		0			0
$493$	3.65685	+	8.82843i	0.164696	+	0.397612i
$494$		0			0
$495$			31.5563i			1.41835i
$496$		0			0
$497$		−	11.6569i		−	0.522881i
$498$		0			0
$499$	−0.949747	−	2.29289i	−0.0425165	−	0.102644i	0.901195	−	0.433415i	$-0.142691\pi$
$499$	−0.949747	−	2.29289i	−0.0425165	−	0.102644i	−0.943711	+	0.330771i	$0.892691\pi$
$500$		0			0
$501$	14.6569	+	6.07107i	0.654820	+	0.271235i
$502$		0			0
$503$	11.1421	−	11.1421i	0.496803	−	0.496803i	−0.413638	−	0.910441i	$-0.635742\pi$
$503$	11.1421	−	11.1421i	0.496803	−	0.496803i	0.910441	+	0.413638i	$0.135742\pi$
$504$		0			0
$505$	7.38478	+	7.38478i	0.328618	+	0.328618i
$506$		0			0
$507$	−3.63604	+	8.77817i	−0.161482	+	0.389852i
$508$		0			0
$509$	26.0919	−	10.8076i	1.15650	−	0.479039i	0.279793	−	0.960060i	$-0.409734\pi$
$509$	26.0919	−	10.8076i	1.15650	−	0.479039i	0.876709	+	0.481021i	$0.159734\pi$
$510$		0			0
$511$	−14.0000			−0.619324
$512$		0			0
$513$	16.6274			0.734118
$514$		0			0
$515$	36.3137	−	15.0416i	1.60017	−	0.662813i
$516$		0			0
$517$	−0.585786	+	1.41421i	−0.0257629	+	0.0621970i
$518$		0			0
$519$	4.41421	+	4.41421i	0.193762	+	0.193762i
$520$		0			0
$521$	3.34315	−	3.34315i	0.146466	−	0.146466i	−0.630071	−	0.776537i	$-0.716974\pi$
$521$	3.34315	−	3.34315i	0.146466	−	0.146466i	0.776537	+	0.630071i	$0.216974\pi$
$522$		0			0
$523$	19.1924	+	7.94975i	0.839225	+	0.347618i	0.760548	−	0.649282i	$-0.224930\pi$
$523$	19.1924	+	7.94975i	0.839225	+	0.347618i	0.0786768	+	0.996900i	$0.474930\pi$
$524$		0			0
$525$	−1.48528	−	3.58579i	−0.0648230	−	0.156497i
$526$		0			0
$527$			11.3137i			0.492833i
$528$		0			0
$529$		−	44.9411i		−	1.95396i
$530$		0			0
$531$	−4.53553	−	10.9497i	−0.196825	−	0.475179i
$532$		0			0
$533$	0.171573	+	0.0710678i	0.00743165	+	0.00307829i
$534$		0			0
$535$	9.24264	−	9.24264i	0.399594	−	0.399594i
$536$		0			0
$537$	2.31371	+	2.31371i	0.0998439	+	0.0998439i
$538$		0			0
$539$	−8.53553	+	20.6066i	−0.367651	+	0.887589i
$540$		0			0
$541$	27.2635	−	11.2929i	1.17215	−	0.485519i	0.290246	−	0.956952i	$-0.406263\pi$
$541$	27.2635	−	11.2929i	1.17215	−	0.485519i	0.881902	+	0.471433i	$0.156263\pi$
$542$		0			0
$543$	13.4142			0.575659
$544$		0			0
$545$	−43.6985			−1.87184
$546$		0			0
$547$	−17.5355	+	7.26346i	−0.749765	+	0.310563i	−0.724646	−	0.689122i	$-0.757997\pi$
$547$	−17.5355	+	7.26346i	−0.749765	+	0.310563i	−0.0251195	+	0.999684i	$0.507997\pi$
$548$		0			0
$549$	1.70711	−	4.12132i	0.0728575	−	0.175894i
$550$		0			0
$551$	9.58579	+	9.58579i	0.408368	+	0.408368i
$552$		0			0
$553$	6.00000	−	6.00000i	0.255146	−	0.255146i
$554$		0			0
$555$	1.58579	+	0.656854i	0.0673129	+	0.0278819i
$556$		0			0
$557$	−15.1213	−	36.5061i	−0.640711	−	1.54681i	−0.825722	−	0.564077i	$-0.809232\pi$
$557$	−15.1213	−	36.5061i	−0.640711	−	1.54681i	0.185012	−	0.982736i	$-0.440768\pi$
$558$		0			0
$559$		−	3.89949i		−	0.164931i
$560$		0			0
$561$			9.65685i			0.407713i
$562$		0			0
$563$	7.87868	+	19.0208i	0.332047	+	0.801632i	0.998430	+	0.0560220i	$0.0178417\pi$
$563$	7.87868	+	19.0208i	0.332047	+	0.801632i	−0.666383	+	0.745610i	$0.732158\pi$
$564$		0			0
$565$	−17.1716	−	7.11270i	−0.722414	−	0.299233i
$566$		0			0
$567$	−4.07107	+	4.07107i	−0.170969	+	0.170969i
$568$		0			0
$569$	14.6569	+	14.6569i	0.614447	+	0.614447i	0.944102	−	0.329654i	$-0.106932\pi$
$569$	14.6569	+	14.6569i	0.614447	+	0.614447i	−0.329654	+	0.944102i	$0.606932\pi$
$570$		0			0
$571$	−2.70711	+	6.53553i	−0.113289	+	0.273504i	−0.970347	−	0.241716i	$-0.922290\pi$
$571$	−2.70711	+	6.53553i	−0.113289	+	0.273504i	0.857058	+	0.515220i	$0.172290\pi$
$572$		0			0
$573$	−8.48528	+	3.51472i	−0.354478	+	0.146829i
$574$		0			0
$575$	29.5563			1.23258
$576$		0			0
$577$	18.9706			0.789755			0.394877	−	0.918734i	$-0.370787\pi$
$577$	18.9706			0.789755			0.394877	+	0.918734i	$0.370787\pi$
$578$		0			0
$579$	1.07107	−	0.443651i	0.0445121	−	0.0184375i
$580$		0			0
$581$	2.65685	−	6.41421i	0.110225	−	0.266106i
$582$		0			0
$583$	−3.82843	−	3.82843i	−0.158557	−	0.158557i
$584$		0			0
$585$	−3.82843	+	3.82843i	−0.158286	+	0.158286i
$586$		0			0
$587$	−12.6066	−	5.22183i	−0.520330	−	0.215528i	0.107032	−	0.994256i	$-0.465865\pi$
$587$	−12.6066	−	5.22183i	−0.520330	−	0.215528i	−0.627362	+	0.778728i	$0.715865\pi$
$588$		0			0
$589$	6.14214	+	14.8284i	0.253082	+	0.610995i
$590$		0			0
$591$		−	9.27208i		−	0.381402i
$592$		0			0
$593$			28.2843i			1.16150i	0.814083	+	0.580748i	$0.197240\pi$
$593$			28.2843i			1.16150i	−0.814083	+	0.580748i	$0.802760\pi$
$594$		0			0
$595$	4.48528	+	10.8284i	0.183879	+	0.443922i
$596$		0			0
$597$	−15.9706	−	6.61522i	−0.653632	−	0.270743i
$598$		0			0
$599$	26.6569	−	26.6569i	1.08917	−	1.08917i	0.0935555	−	0.995614i	$-0.470177\pi$
$599$	26.6569	−	26.6569i	1.08917	−	1.08917i	0.995614	−	0.0935555i	$-0.0298232\pi$
$600$		0			0
$601$	−21.9706	−	21.9706i	−0.896198	−	0.896198i	0.0988995	−	0.995097i	$-0.468468\pi$
$601$	−21.9706	−	21.9706i	−0.896198	−	0.896198i	−0.995097	+	0.0988995i	$0.968468\pi$
$602$		0			0
$603$	5.53553	−	13.3640i	0.225424	−	0.544223i
$604$		0			0
$605$	−24.0919	+	9.97918i	−0.979474	+	0.405712i
$606$		0			0
$607$	32.9706			1.33823			0.669117	−	0.743157i	$-0.266673\pi$
$607$	32.9706			1.33823			0.669117	+	0.743157i	$0.266673\pi$
$608$		0			0
$609$	3.65685			0.148183
$610$		0			0
$611$	−0.242641	+	0.100505i	−0.00981619	+	0.00406600i
$612$		0			0
$613$	−1.32233	+	3.19239i	−0.0534084	+	0.128939i	−0.948332	−	0.317281i	$-0.897230\pi$
$613$	−1.32233	+	3.19239i	−0.0534084	+	0.128939i	0.894923	+	0.446220i	$0.147230\pi$
$614$		0			0
$615$	0.384776	+	0.384776i	0.0155157	+	0.0155157i
$616$		0			0
$617$	22.7990	−	22.7990i	0.917853	−	0.917853i	−0.0790202	−	0.996873i	$-0.525179\pi$
$617$	22.7990	−	22.7990i	0.917853	−	0.917853i	0.996873	+	0.0790202i	$0.0251792\pi$
$618$		0			0
$619$	−21.7782	−	9.02082i	−0.875339	−	0.362577i	−0.100651	−	0.994922i	$-0.532093\pi$
$619$	−21.7782	−	9.02082i	−0.875339	−	0.362577i	−0.774687	+	0.632345i	$0.782093\pi$
$620$		0			0
$621$	13.0711	+	31.5563i	0.524524	+	1.26631i
$622$		0			0
$623$		−	17.3137i		−	0.693659i
$624$		0			0
$625$		−	30.0711i		−	1.20284i
$626$		0			0
$627$	5.24264	+	12.6569i	0.209371	+	0.505466i
$628$		0			0
$629$	−2.00000	−	0.828427i	−0.0797452	−	0.0330316i
$630$		0			0
$631$	−32.4558	+	32.4558i	−1.29205	+	1.29205i	−0.358528	+	0.933519i	$0.616721\pi$
$631$	−32.4558	+	32.4558i	−1.29205	+	1.29205i	−0.933519	+	0.358528i	$0.883279\pi$
$632$		0			0
$633$	−10.6569	−	10.6569i	−0.423572	−	0.423572i
$634$		0			0
$635$	14.5442	−	35.1127i	0.577167	−	1.39340i
$636$		0			0
$637$	−3.53553	+	1.46447i	−0.140083	+	0.0580243i
$638$		0			0
$639$	−19.8995			−0.787212
$640$		0			0
$641$	7.45584			0.294488			0.147244	−	0.989100i	$-0.452960\pi$
$641$	7.45584			0.294488			0.147244	+	0.989100i	$0.452960\pi$
$642$		0			0
$643$	11.4350	−	4.73654i	0.450954	−	0.186791i	−0.145635	−	0.989338i	$-0.546522\pi$
$643$	11.4350	−	4.73654i	0.450954	−	0.186791i	0.596588	+	0.802547i	$0.296522\pi$
$644$		0			0
$645$	4.37258	−	10.5563i	0.172170	−	0.415656i
$646$		0			0
$647$	−6.17157	−	6.17157i	−0.242630	−	0.242630i	0.575308	−	0.817937i	$-0.304882\pi$
$647$	−6.17157	−	6.17157i	−0.242630	−	0.242630i	−0.817937	+	0.575308i	$0.804882\pi$
$648$		0			0
$649$	15.4853	−	15.4853i	0.607850	−	0.607850i
$650$		0			0
$651$	4.00000	+	1.65685i	0.156772	+	0.0649372i
$652$		0			0
$653$	−2.09188	−	5.05025i	−0.0818617	−	0.197632i	0.877649	−	0.479304i	$-0.159111\pi$
$653$	−2.09188	−	5.05025i	−0.0818617	−	0.197632i	−0.959511	+	0.281672i	$0.909111\pi$
$654$		0			0
$655$		−	51.8995i		−	2.02788i
$656$		0			0
$657$			23.8995i			0.932408i
$658$		0			0
$659$	−10.1213	−	24.4350i	−0.394271	−	0.951854i	−0.988998	−	0.147926i	$-0.952740\pi$
$659$	−10.1213	−	24.4350i	−0.394271	−	0.951854i	0.594728	−	0.803927i	$-0.297260\pi$
$660$		0			0
$661$	41.7487	+	17.2929i	1.62384	+	0.672616i	0.994521	−	0.104534i	$-0.0333350\pi$
$661$	41.7487	+	17.2929i	1.62384	+	0.672616i	0.629316	+	0.777149i	$0.283335\pi$
$662$		0			0
$663$	−1.17157	+	1.17157i	−0.0455001	+	0.0455001i
$664$		0			0
$665$	11.7574	+	11.7574i	0.455931	+	0.455931i
$666$		0			0
$667$	−10.6569	+	25.7279i	−0.412635	+	0.996189i
$668$		0			0
$669$	−14.8284	+	6.14214i	−0.573300	+	0.237469i
$670$		0			0
$671$	8.24264			0.318204
$672$		0			0
$673$	22.4853			0.866744			0.433372	−	0.901215i	$-0.357324\pi$
$673$	22.4853			0.866744			0.433372	+	0.901215i	$0.357324\pi$
$674$		0			0
$675$	−13.7279	+	5.68629i	−0.528388	+	0.218865i
$676$		0			0
$677$	−15.6066	+	37.6777i	−0.599810	+	1.44807i	0.273964	+	0.961740i	$0.411665\pi$
$677$	−15.6066	+	37.6777i	−0.599810	+	1.44807i	−0.873775	+	0.486331i	$0.838335\pi$
$678$		0			0
$679$	18.4853	+	18.4853i	0.709400	+	0.709400i
$680$		0			0
$681$	−10.8995	+	10.8995i	−0.417670	+	0.417670i
$682$		0			0
$683$	−10.1213	−	4.19239i	−0.387282	−	0.160417i	0.180543	−	0.983567i	$-0.442215\pi$
$683$	−10.1213	−	4.19239i	−0.387282	−	0.160417i	−0.567824	+	0.823150i	$0.692215\pi$
$684$		0			0
$685$	−13.7279	−	33.1421i	−0.524517	−	1.26630i
$686$		0			0
$687$			18.4437i			0.703669i
$688$		0			0
$689$		−	0.928932i		−	0.0353895i
$690$		0			0
$691$	12.5061	+	30.1924i	0.475754	+	1.14857i	0.961582	+	0.274518i	$0.0885183\pi$
$691$	12.5061	+	30.1924i	0.475754	+	1.14857i	−0.485828	+	0.874055i	$0.661482\pi$
$692$		0			0
$693$	−14.0711	−	5.82843i	−0.534516	−	0.221404i
$694$		0			0
$695$	−29.5858	+	29.5858i	−1.12225	+	1.12225i
$696$		0			0
$697$	−0.485281	−	0.485281i	−0.0183813	−	0.0183813i
$698$		0			0
$699$	1.10051	−	2.65685i	0.0416249	−	0.100491i
$700$		0			0
$701$	−2.87868	+	1.19239i	−0.108726	+	0.0450359i	−0.436383	−	0.899761i	$-0.643741\pi$
$701$	−2.87868	+	1.19239i	−0.108726	+	0.0450359i	0.327657	+	0.944797i	$0.393741\pi$
$702$		0			0
$703$	−3.07107			−0.115828
$704$		0			0
$705$	−0.769553			−0.0289830
$706$		0			0
$707$	−4.65685	+	1.92893i	−0.175139	+	0.0725450i
$708$		0			0
$709$	−8.77817	+	21.1924i	−0.329671	+	0.795897i	0.668945	+	0.743312i	$0.266746\pi$
$709$	−8.77817	+	21.1924i	−0.329671	+	0.795897i	−0.998616	+	0.0525851i	$0.983254\pi$
$710$		0			0
$711$	−10.2426	−	10.2426i	−0.384129	−	0.384129i
$712$		0			0
$713$	−23.3137	+	23.3137i	−0.873105	+	0.873105i
$714$		0			0
$715$	−9.24264	−	3.82843i	−0.345655	−	0.143175i
$716$		0			0
$717$	1.55635	+	3.75736i	0.0581229	+	0.140321i
$718$		0			0
$719$			35.6569i			1.32978i	0.746943	+	0.664888i	$0.231521\pi$
$719$			35.6569i			1.32978i	−0.746943	+	0.664888i	$0.768479\pi$
$720$		0			0
$721$			18.9706i			0.706501i
$722$		0			0
$723$	−2.48528	−	6.00000i	−0.0924286	−	0.223142i
$724$		0			0
$725$	−11.1924	−	4.63604i	−0.415675	−	0.172178i
$726$		0			0
$727$	9.97056	−	9.97056i	0.369788	−	0.369788i	−0.497612	−	0.867400i	$-0.665790\pi$
$727$	9.97056	−	9.97056i	0.369788	−	0.369788i	0.867400	+	0.497612i	$0.165790\pi$
$728$		0			0
$729$	0.221825	+	0.221825i	0.00821576	+	0.00821576i
$730$		0			0
$731$	−5.51472	+	13.3137i	−0.203969	+	0.492425i
$732$		0			0
$733$	33.2635	−	13.7782i	1.22861	−	0.508908i	0.328475	−	0.944513i	$-0.393465\pi$
$733$	33.2635	−	13.7782i	1.22861	−	0.508908i	0.900138	+	0.435604i	$0.143465\pi$
$734$		0			0
$735$	−11.2132			−0.413605
$736$		0			0
$737$	26.7279			0.984536
$738$		0			0
$739$	0.464466	−	0.192388i	0.0170857	−	0.00707711i	−0.374124	−	0.927379i	$-0.622057\pi$
$739$	0.464466	−	0.192388i	0.0170857	−	0.00707711i	0.391210	+	0.920301i	$0.372057\pi$
$740$		0			0
$741$	−0.899495	+	2.17157i	−0.0330438	+	0.0797747i
$742$		0			0
$743$	−31.6274	−	31.6274i	−1.16030	−	1.16030i	−0.984410	−	0.175887i	$-0.943721\pi$
$743$	−31.6274	−	31.6274i	−1.16030	−	1.16030i	−0.175887	−	0.984410i	$-0.556279\pi$
$744$		0			0
$745$	−35.0000	+	35.0000i	−1.28230	+	1.28230i
$746$		0			0
$747$	−10.9497	−	4.53553i	−0.400630	−	0.165947i
$748$		0			0
$749$	2.41421	+	5.82843i	0.0882134	+	0.212966i
$750$		0			0
$751$		−	10.9706i		−	0.400322i	−0.979763	−	0.200161i	$-0.935854\pi$
$751$		−	10.9706i		−	0.400322i	0.979763	−	0.200161i	$-0.0641464\pi$
$752$		0			0
$753$			13.2132i			0.481516i
$754$		0			0
$755$	2.35534	+	5.68629i	0.0857196	+	0.206945i
$756$		0			0
$757$	33.2635	+	13.7782i	1.20898	+	0.500776i	0.893890	−	0.448285i	$-0.147965\pi$
$757$	33.2635	+	13.7782i	1.20898	+	0.500776i	0.315090	+	0.949062i	$0.397965\pi$
$758$		0			0
$759$	−19.8995	+	19.8995i	−0.722306	+	0.722306i
$760$		0			0
$761$	−29.8284	−	29.8284i	−1.08128	−	1.08128i	−0.996390	−	0.0848892i	$-0.972946\pi$
$761$	−29.8284	−	29.8284i	−1.08128	−	1.08128i	−0.0848892	−	0.996390i	$-0.527054\pi$
$762$		0			0
$763$	8.07107	−	19.4853i	0.292192	−	0.705415i
$764$		0			0
$765$	18.4853	−	7.65685i	0.668337	−	0.276834i
$766$		0			0
$767$	3.75736			0.135670
$768$		0			0
$769$	5.51472			0.198866			0.0994329	−	0.995044i	$-0.468297\pi$
$769$	5.51472			0.198866			0.0994329	+	0.995044i	$0.468297\pi$
$770$		0			0
$771$	−4.24264	+	1.75736i	−0.152795	+	0.0632897i
$772$		0			0
$773$	−12.0919	+	29.1924i	−0.434915	+	1.04998i	0.542766	+	0.839884i	$0.317377\pi$
$773$	−12.0919	+	29.1924i	−0.434915	+	1.04998i	−0.977681	+	0.210094i	$0.932623\pi$
$774$		0			0
$775$	−10.1421	−	10.1421i	−0.364316	−	0.364316i
$776$		0			0
$777$	−0.585786	+	0.585786i	−0.0210150	+	0.0210150i
$778$		0			0
$779$	−0.899495	−	0.372583i	−0.0322278	−	0.0133492i
$780$		0			0
$781$	−14.0711	−	33.9706i	−0.503502	−	1.21556i
$782$		0			0
$783$		−	14.0000i		−	0.500319i
$784$		0			0
$785$		−	5.41421i		−	0.193242i
$786$		0			0
$787$	−0.949747	−	2.29289i	−0.0338548	−	0.0817328i	0.906048	−	0.423175i	$-0.139085\pi$
$787$	−0.949747	−	2.29289i	−0.0338548	−	0.0817328i	−0.939903	+	0.341442i	$0.889085\pi$
$788$		0			0
$789$	−5.82843	−	2.41421i	−0.207498	−	0.0859483i
$790$		0			0
$791$	6.34315	−	6.34315i	0.225536	−	0.225536i
$792$		0			0
$793$	1.00000	+	1.00000i	0.0355110	+

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	256.g (of order \(8\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.292893i) q^{3} +(-1.12132 + 2.70711i) q^{5} +(-1.00000 - 1.00000i) q^{7} +(-1.70711 + 1.70711i) q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.292893i) q^{3} +(-1.12132 + 2.70711i) q^{5} +(-1.00000 - 1.00000i) q^{7} +(-1.70711 + 1.70711i) q^{9} +(-4.12132 - 1.70711i) q^{11} +(-0.292893 - 0.707107i) q^{13} -2.24264i q^{15} +2.82843i q^{17} +(1.53553 + 3.70711i) q^{19} +(1.00000 + 0.414214i) q^{21} +(-5.82843 + 5.82843i) q^{23} +(-2.53553 - 2.53553i) q^{25} +(1.58579 - 3.82843i) q^{27} +(3.12132 - 1.29289i) q^{29} +4.00000 q^{31} +3.41421 q^{33} +(3.82843 - 1.58579i) q^{35} +(-0.292893 + 0.707107i) q^{37} +(0.414214 + 0.414214i) q^{39} +(-0.171573 + 0.171573i) q^{41} +(4.70711 + 1.94975i) q^{43} +(-2.70711 - 6.53553i) q^{45} -0.343146i q^{47} -5.00000i q^{49} +(-0.828427 - 2.00000i) q^{51} +(1.12132 + 0.464466i) q^{53} +(9.24264 - 9.24264i) q^{55} +(-2.17157 - 2.17157i) q^{57} +(-1.87868 + 4.53553i) q^{59} +(-1.70711 + 0.707107i) q^{61} +3.41421 q^{63} +2.24264 q^{65} +(-5.53553 + 2.29289i) q^{67} +(2.41421 - 5.82843i) q^{69} +(5.82843 + 5.82843i) q^{71} +(7.00000 - 7.00000i) q^{73} +(2.53553 + 1.05025i) q^{75} +(2.41421 + 5.82843i) q^{77} +6.00000i q^{79} -4.07107i q^{81} +(1.87868 + 4.53553i) q^{83} +(-7.65685 - 3.17157i) q^{85} +(-1.82843 + 1.82843i) q^{87} +(8.65685 + 8.65685i) q^{89} +(-0.414214 + 1.00000i) q^{91} +(-2.82843 + 1.17157i) q^{93} -11.7574 q^{95} -18.4853 q^{97} +(9.94975 - 4.12132i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 - 4 * q^7 - 4 * q^9	\( 4 q + 4 q^{5} - 4 q^{7} - 4 q^{9} - 8 q^{11} - 4 q^{13} - 8 q^{19} + 4 q^{21} - 12 q^{23} + 4 q^{25} + 12 q^{27} + 4 q^{29} + 16 q^{31} + 8 q^{33} + 4 q^{35} - 4 q^{37} - 4 q^{39} - 12 q^{41} + 16 q^{43} - 8 q^{45} + 8 q^{51} - 4 q^{53} + 20 q^{55} - 20 q^{57} - 16 q^{59} - 4 q^{61} + 8 q^{63} - 8 q^{65} - 8 q^{67} + 4 q^{69} + 12 q^{71} + 28 q^{73} - 4 q^{75} + 4 q^{77} + 16 q^{83} - 8 q^{85} + 4 q^{87} + 12 q^{89} + 4 q^{91} - 64 q^{95} - 40 q^{97} + 20 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^7 - 4 * q^9 - 8 * q^11 - 4 * q^13 - 8 * q^19 + 4 * q^21 - 12 * q^23 + 4 * q^25 + 12 * q^27 + 4 * q^29 + 16 * q^31 + 8 * q^33 + 4 * q^35 - 4 * q^37 - 4 * q^39 - 12 * q^41 + 16 * q^43 - 8 * q^45 + 8 * q^51 - 4 * q^53 + 20 * q^55 - 20 * q^57 - 16 * q^59 - 4 * q^61 + 8 * q^63 - 8 * q^65 - 8 * q^67 + 4 * q^69 + 12 * q^71 + 28 * q^73 - 4 * q^75 + 4 * q^77 + 16 * q^83 - 8 * q^85 + 4 * q^87 + 12 * q^89 + 4 * q^91 - 64 * q^95 - 40 * q^97 + 20 * q^99

Introduction

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