LMFDB - Embedded newform 680.2.bd.b.361.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [680,2,Mod(81,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.81"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 680 = 2^{3} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	680.bd (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.42982733745$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 38 x^{18} + 597 x^{16} + 5004 x^{14} + 24072 x^{12} + 66452 x^{10} + 99328 x^{8} + 70784 x^{6} + \cdots + 16 $ x^20 + 38x^18 + 597x^16 + 5004x^14 + 24072x^12 + 66452x^10 + 99328x^8 + 70784x^6 + 19972x^4 + 1104*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			361.6
Root			$-0.155688i$ of defining polynomial
Character	$\chi$	$=$	680.361
Dual form			680.2.bd.b.81.6

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.110088 - 0.110088i) q^{3} +(-0.707107 + 0.707107i) q^{5} +(-1.59814 - 1.59814i) q^{7} +2.97576i q^{9} +(2.29246 + 2.29246i) q^{11} -6.35384 q^{13} +0.155688i q^{15} +(-3.14352 - 2.66801i) q^{17} +3.74959i q^{19} -0.351873 q^{21} +(1.61226 + 1.61226i) q^{23} -1.00000i q^{25} +(0.657861 + 0.657861i) q^{27} +(-3.11528 + 3.11528i) q^{29} +(-5.15119 + 5.15119i) q^{31} +0.504746 q^{33} +2.26011 q^{35} +(1.52587 - 1.52587i) q^{37} +(-0.699484 + 0.699484i) q^{39} +(1.63254 + 1.63254i) q^{41} +10.1998i q^{43} +(-2.10418 - 2.10418i) q^{45} -6.71856 q^{47} -1.89191i q^{49} +(-0.639782 + 0.0523473i) q^{51} +2.23744i q^{53} -3.24203 q^{55} +(0.412786 + 0.412786i) q^{57} -5.91043i q^{59} +(-5.35172 - 5.35172i) q^{61} +(4.75568 - 4.75568i) q^{63} +(4.49284 - 4.49284i) q^{65} +9.42323 q^{67} +0.354982 q^{69} +(3.13126 - 3.13126i) q^{71} +(-5.11930 + 5.11930i) q^{73} +(-0.110088 - 0.110088i) q^{75} -7.32733i q^{77} +(4.51283 + 4.51283i) q^{79} -8.78244 q^{81} +2.99110i q^{83} +(4.10937 - 0.336231i) q^{85} +0.685912i q^{87} -3.95093 q^{89} +(10.1543 + 10.1543i) q^{91} +1.13417i q^{93} +(-2.65136 - 2.65136i) q^{95} +(1.15110 - 1.15110i) q^{97} +(-6.82181 + 6.82181i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} + 4 q^{7} + 12 q^{11} + 4 q^{13} + 4 q^{17} + 16 q^{21} - 4 q^{23} - 16 q^{27} - 4 q^{29} - 4 q^{31} + 16 q^{33} - 8 q^{35} - 8 q^{37} - 8 q^{39} - 8 q^{41} + 4 q^{47} - 4 q^{51} - 8 q^{55}+ \cdots + 52 q^{99}+O(q^{100}) $ 20 * q - 4 * q^3 + 4 * q^7 + 12 * q^11 + 4 * q^13 + 4 * q^17 + 16 * q^21 - 4 * q^23 - 16 * q^27 - 4 * q^29 - 4 * q^31 + 16 * q^33 - 8 * q^35 - 8 * q^37 - 8 * q^39 - 8 * q^41 + 4 * q^47 - 4 * q^51 - 8 * q^55 - 24 * q^57 - 24 * q^61 + 60 * q^63 - 4 * q^65 - 56 * q^67 + 24 * q^69 - 40 * q^71 - 20 * q^73 + 4 * q^75 - 8 * q^79 + 4 * q^81 - 12 * q^85 + 28 * q^89 - 4 * q^95 + 24 * q^97 + 52 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/680\mathbb{Z}\right)^\times$.

$n$	$137$	$241$	$341$	$511$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	0.110088	−	0.110088i	0.0635595	−	0.0635595i	−0.674613	−	0.738172i	$-0.735689\pi$
$3$	0.110088	−	0.110088i	0.0635595	−	0.0635595i	0.738172	+	0.674613i	$0.235689\pi$
$4$		0			0
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$6$		0			0
$7$	−1.59814	−	1.59814i	−0.604039	−	0.604039i	0.337343	−	0.941382i	$-0.390472\pi$
$7$	−1.59814	−	1.59814i	−0.604039	−	0.604039i	−0.941382	+	0.337343i	$0.890472\pi$
$8$		0			0
$9$			2.97576i			0.991920i
$10$		0			0
$11$	2.29246	+	2.29246i	0.691202	+	0.691202i	0.962496	−	0.271294i	$-0.0874517\pi$
$11$	2.29246	+	2.29246i	0.691202	+	0.691202i	−0.271294	+	0.962496i	$0.587452\pi$
$12$		0			0
$13$	−6.35384			−1.76224			−0.881119	−	0.472894i	$-0.843209\pi$
$13$	−6.35384			−1.76224			−0.881119	+	0.472894i	$0.843209\pi$
$14$		0			0
$15$			0.155688i			0.0401986i
$16$		0			0
$17$	−3.14352	−	2.66801i	−0.762415	−	0.647089i
$17$	−3.14352	−	2.66801i	−0.762415	−	0.647089i
$18$		0			0
$19$			3.74959i			0.860214i	0.902778	+	0.430107i	$0.141524\pi$
$19$			3.74959i			0.860214i	−0.902778	+	0.430107i	$0.858476\pi$
$20$		0			0
$21$	−0.351873			−0.0767849
$22$		0			0
$23$	1.61226	+	1.61226i	0.336180	+	0.336180i	0.854927	−	0.518748i	$-0.173602\pi$
$23$	1.61226	+	1.61226i	0.336180	+	0.336180i	−0.518748	+	0.854927i	$0.673602\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$	0.657861	+	0.657861i	0.126606	+	0.126606i
$28$		0			0
$29$	−3.11528	+	3.11528i	−0.578493	+	0.578493i	−0.934488	−	0.355995i	$-0.884142\pi$
$29$	−3.11528	+	3.11528i	−0.578493	+	0.578493i	0.355995	+	0.934488i	$0.384142\pi$
$30$		0			0
$31$	−5.15119	+	5.15119i	−0.925182	+	0.925182i	−0.997390	−	0.0722081i	$-0.976995\pi$
$31$	−5.15119	+	5.15119i	−0.925182	+	0.925182i	0.0722081	+	0.997390i	$0.476995\pi$
$32$		0			0
$33$	0.504746			0.0878650
$34$		0			0
$35$	2.26011			0.382028
$36$		0			0
$37$	1.52587	−	1.52587i	0.250851	−	0.250851i	−0.570468	−	0.821320i	$-0.693238\pi$
$37$	1.52587	−	1.52587i	0.250851	−	0.250851i	0.821320	+	0.570468i	$0.193238\pi$
$38$		0			0
$39$	−0.699484	+	0.699484i	−0.112007	+	0.112007i
$40$		0			0
$41$	1.63254	+	1.63254i	0.254960	+	0.254960i	0.823000	−	0.568041i	$-0.192298\pi$
$41$	1.63254	+	1.63254i	0.254960	+	0.254960i	−0.568041	+	0.823000i	$0.692298\pi$
$42$		0			0
$43$			10.1998i			1.55545i	0.628605	+	0.777725i	$0.283626\pi$
$43$			10.1998i			1.55545i	−0.628605	+	0.777725i	$0.716374\pi$
$44$		0			0
$45$	−2.10418	−	2.10418i	−0.313673	−	0.313673i
$46$		0			0
$47$	−6.71856			−0.980002			−0.490001	−	0.871722i	$-0.663004\pi$
$47$	−6.71856			−0.980002			−0.490001	+	0.871722i	$0.663004\pi$
$48$		0			0
$49$		−	1.89191i		−	0.270273i
$50$		0			0
$51$	−0.639782	+	0.0523473i	−0.0895874	+	0.00733008i
$52$		0			0
$53$			2.23744i			0.307336i	0.988123	+	0.153668i	$0.0491087\pi$
$53$			2.23744i			0.307336i	−0.988123	+	0.153668i	$0.950891\pi$
$54$		0			0
$55$	−3.24203			−0.437155
$56$		0			0
$57$	0.412786	+	0.412786i	0.0546748	+	0.0546748i
$58$		0			0
$59$		−	5.91043i		−	0.769473i	−0.923026	−	0.384737i	$-0.874292\pi$
$59$		−	5.91043i		−	0.769473i	0.923026	−	0.384737i	$-0.125708\pi$
$60$		0			0
$61$	−5.35172	−	5.35172i	−0.685218	−	0.685218i	0.275953	−	0.961171i	$-0.411007\pi$
$61$	−5.35172	−	5.35172i	−0.685218	−	0.685218i	−0.961171	+	0.275953i	$0.911007\pi$
$62$		0			0
$63$	4.75568	−	4.75568i	0.599159	−	0.599159i
$64$		0			0
$65$	4.49284	−	4.49284i	0.557269	−	0.557269i
$66$		0			0
$67$	9.42323			1.15123			0.575615	−	0.817721i	$-0.304763\pi$
$67$	9.42323			1.15123			0.575615	+	0.817721i	$0.304763\pi$
$68$		0			0
$69$	0.354982			0.0427348
$70$		0			0
$71$	3.13126	−	3.13126i	0.371612	−	0.371612i	−0.496452	−	0.868064i	$-0.665364\pi$
$71$	3.13126	−	3.13126i	0.371612	−	0.371612i	0.868064	+	0.496452i	$0.165364\pi$
$72$		0			0
$73$	−5.11930	+	5.11930i	−0.599168	+	0.599168i	−0.940091	−	0.340923i	$-0.889261\pi$
$73$	−5.11930	+	5.11930i	−0.599168	+	0.599168i	0.340923	+	0.940091i	$0.389261\pi$
$74$		0			0
$75$	−0.110088	−	0.110088i	−0.0127119	−	0.0127119i
$76$		0			0
$77$		−	7.32733i		−	0.835027i
$78$		0			0
$79$	4.51283	+	4.51283i	0.507734	+	0.507734i	0.913830	−	0.406097i	$-0.133110\pi$
$79$	4.51283	+	4.51283i	0.507734	+	0.507734i	−0.406097	+	0.913830i	$0.633110\pi$
$80$		0			0
$81$	−8.78244			−0.975826
$82$		0			0
$83$			2.99110i			0.328316i	0.986434	+	0.164158i	$0.0524906\pi$
$83$			2.99110i			0.328316i	−0.986434	+	0.164158i	$0.947509\pi$
$84$		0			0
$85$	4.10937	−	0.336231i	0.445724	−	0.0364694i
$86$		0			0
$87$			0.685912i			0.0735375i
$88$		0			0
$89$	−3.95093			−0.418798			−0.209399	−	0.977830i	$-0.567151\pi$
$89$	−3.95093			−0.418798			−0.209399	+	0.977830i	$0.567151\pi$
$90$		0			0
$91$	10.1543	+	10.1543i	1.06446	+	1.06446i
$92$		0			0
$93$			1.13417i			0.117608i
$94$		0			0
$95$	−2.65136	−	2.65136i	−0.272024	−	0.272024i
$96$		0			0
$97$	1.15110	−	1.15110i	0.116876	−	0.116876i	−0.646250	−	0.763126i	$-0.723664\pi$
$97$	1.15110	−	1.15110i	0.116876	−	0.116876i	0.763126	+	0.646250i	$0.223664\pi$
$98$		0			0
$99$	−6.82181	+	6.82181i	−0.685617	+	0.685617i
$100$		0			0
$101$	−7.30222			−0.726598			−0.363299	−	0.931673i	$-0.618350\pi$
$101$	−7.30222			−0.726598			−0.363299	+	0.931673i	$0.618350\pi$
$102$		0			0
$103$	16.1560			1.59190			0.795949	−	0.605364i	$-0.206973\pi$
$103$	16.1560			1.59190			0.795949	+	0.605364i	$0.206973\pi$
$104$		0			0
$105$	0.248812	−	0.248812i	0.0242815	−	0.0242815i
$106$		0			0
$107$	4.84617	−	4.84617i	0.468497	−	0.468497i	−0.432931	−	0.901427i	$-0.642520\pi$
$107$	4.84617	−	4.84617i	0.468497	−	0.468497i	0.901427	+	0.432931i	$0.142520\pi$
$108$		0			0
$109$	11.0534	+	11.0534i	1.05873	+	1.05873i	0.998164	+	0.0605625i	$0.0192894\pi$
$109$	11.0534	+	11.0534i	1.05873	+	1.05873i	0.0605625	+	0.998164i	$0.480711\pi$
$110$		0			0
$111$		−	0.335960i		−	0.0318879i
$112$		0			0
$113$	−4.34740	−	4.34740i	−0.408969	−	0.408969i	0.472410	−	0.881379i	$-0.343384\pi$
$113$	−4.34740	−	4.34740i	−0.408969	−	0.408969i	−0.881379	+	0.472410i	$0.843384\pi$
$114$		0			0
$115$	−2.28008			−0.212619
$116$		0			0
$117$		−	18.9075i		−	1.74800i
$118$		0			0
$119$	0.759919	+	9.28763i	0.0696616	+	0.851396i
$120$		0			0
$121$		−	0.489271i		−	0.0444792i
$122$		0			0
$123$	0.359447			0.0324102
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$		−	16.0773i		−	1.42663i	−0.700845	−	0.713313i	$-0.747194\pi$
$127$		−	16.0773i		−	1.42663i	0.700845	−	0.713313i	$-0.252806\pi$
$128$		0			0
$129$	1.12288	+	1.12288i	0.0988637	+	0.0988637i
$130$		0			0
$131$	11.5325	−	11.5325i	1.00760	−	1.00760i	0.00763061	−	0.999971i	$-0.497571\pi$
$131$	11.5325	−	11.5325i	1.00760	−	1.00760i	0.999971	−	0.00763061i	$-0.00242892\pi$
$132$		0			0
$133$	5.99236	−	5.99236i	0.519603	−	0.519603i
$134$		0			0
$135$	−0.930357			−0.0800724
$136$		0			0
$137$	−14.6889			−1.25495			−0.627477	−	0.778635i	$-0.715912\pi$
$137$	−14.6889			−1.25495			−0.627477	+	0.778635i	$0.715912\pi$
$138$		0			0
$139$	−11.0894	+	11.0894i	−0.940587	+	0.940587i	−0.998331	−	0.0577442i	$-0.981609\pi$
$139$	−11.0894	+	11.0894i	−0.940587	+	0.940587i	0.0577442	+	0.998331i	$0.481609\pi$
$140$		0			0
$141$	−0.739635	+	0.739635i	−0.0622885	+	0.0622885i
$142$		0			0
$143$	−14.5659	−	14.5659i	−1.21806	−	1.21806i
$144$		0			0
$145$		−	4.40567i		−	0.365871i
$146$		0			0
$147$	−0.208277	−	0.208277i	−0.0171784	−	0.0171784i
$148$		0			0
$149$	15.7780			1.29259			0.646294	−	0.763089i	$-0.276318\pi$
$149$	15.7780			1.29259			0.646294	+	0.763089i	$0.276318\pi$
$150$		0			0
$151$		−	20.9005i		−	1.70085i	−0.526093	−	0.850427i	$-0.676343\pi$
$151$		−	20.9005i		−	1.70085i	0.526093	−	0.850427i	$-0.323657\pi$
$152$		0			0
$153$	7.93937	−	9.35436i	0.641860	−	0.756255i
$154$		0			0
$155$		−	7.28489i		−	0.585136i
$156$		0			0
$157$	12.2692			0.979189			0.489595	−	0.871950i	$-0.337145\pi$
$157$	12.2692			0.979189			0.489595	+	0.871950i	$0.337145\pi$
$158$		0			0
$159$	0.246316	+	0.246316i	0.0195342	+	0.0195342i
$160$		0			0
$161$		−	5.15323i		−	0.406132i
$162$		0			0
$163$	12.2783	+	12.2783i	0.961708	+	0.961708i	0.999293	−	0.0375855i	$-0.0119667\pi$
$163$	12.2783	+	12.2783i	0.961708	+	0.961708i	−0.0375855	+	0.999293i	$0.511967\pi$
$164$		0			0
$165$	−0.356909	+	0.356909i	−0.0277853	+	0.0277853i
$166$		0			0
$167$	6.54441	−	6.54441i	0.506422	−	0.506422i	−0.407004	−	0.913426i	$-0.633427\pi$
$167$	6.54441	−	6.54441i	0.506422	−	0.506422i	0.913426	+	0.407004i	$0.133427\pi$
$168$		0			0
$169$	27.3713			2.10548
$170$		0			0
$171$	−11.1579			−0.853264
$172$		0			0
$173$	−15.5882	+	15.5882i	−1.18515	+	1.18515i	−0.206756	+	0.978392i	$0.566291\pi$
$173$	−15.5882	+	15.5882i	−1.18515	+	1.18515i	−0.978392	+	0.206756i	$0.933709\pi$
$174$		0			0
$175$	−1.59814	+	1.59814i	−0.120808	+	0.120808i
$176$		0			0
$177$	−0.650670	−	0.650670i	−0.0489073	−	0.0489073i
$178$		0			0
$179$		−	1.00295i		−	0.0749643i	−0.999297	−	0.0374822i	$-0.988066\pi$
$179$		−	1.00295i		−	0.0749643i	0.999297	−	0.0374822i	$-0.0119337\pi$
$180$		0			0
$181$	11.6481	+	11.6481i	0.865797	+	0.865797i	0.992004	−	0.126207i	$-0.0402802\pi$
$181$	11.6481	+	11.6481i	0.865797	+	0.865797i	−0.126207	+	0.992004i	$0.540280\pi$
$182$		0			0
$183$	−1.17832			−0.0871043
$184$		0			0
$185$			2.15790i			0.158652i
$186$		0			0
$187$	−1.09007	−	13.3227i	−0.0797138	−	0.974252i
$188$		0			0
$189$		−	2.10271i		−	0.152949i
$190$		0			0
$191$	−4.58530			−0.331781			−0.165890	−	0.986144i	$-0.553050\pi$
$191$	−4.58530			−0.331781			−0.165890	+	0.986144i	$0.553050\pi$
$192$		0			0
$193$	2.85427	+	2.85427i	0.205455	+	0.205455i	0.802332	−	0.596877i	$-0.203592\pi$
$193$	2.85427	+	2.85427i	0.205455	+	0.205455i	−0.596877	+	0.802332i	$0.703592\pi$
$194$		0			0
$195$		−	0.989219i		−	0.0708395i
$196$		0			0
$197$	−12.3384	−	12.3384i	−0.879074	−	0.879074i	0.114365	−	0.993439i	$-0.463517\pi$
$197$	−12.3384	−	12.3384i	−0.879074	−	0.879074i	−0.993439	+	0.114365i	$0.963517\pi$
$198$		0			0
$199$	−12.4256	+	12.4256i	−0.880829	+	0.880829i	−0.993619	−	0.112790i	$-0.964021\pi$
$199$	−12.4256	+	12.4256i	−0.880829	+	0.880829i	0.112790	+	0.993619i	$0.464021\pi$
$200$		0			0
$201$	1.03739	−	1.03739i	0.0731716	−	0.0731716i
$202$		0			0
$203$	9.95730			0.698865
$204$		0			0
$205$	−2.30876			−0.161251
$206$		0			0
$207$	−4.79770	+	4.79770i	−0.333464	+	0.333464i
$208$		0			0
$209$	−8.59577	+	8.59577i	−0.594582	+	0.594582i
$210$		0			0
$211$	10.6791	+	10.6791i	0.735179	+	0.735179i	0.971641	−	0.236462i	$-0.0759879\pi$
$211$	10.6791	+	10.6791i	0.735179	+	0.735179i	−0.236462	+	0.971641i	$0.575988\pi$
$212$		0			0
$213$		−	0.689430i		−	0.0472390i
$214$		0			0
$215$	−7.21233	−	7.21233i	−0.491876	−	0.491876i
$216$		0			0
$217$	16.4646			1.11769
$218$		0			0
$219$			1.12715i			0.0761657i
$220$		0			0
$221$	19.9734	+	16.9521i	1.34356	+	1.14032i
$222$		0			0
$223$			2.95689i			0.198008i	0.995087	+	0.0990039i	$0.0315656\pi$
$223$			2.95689i			0.198008i	−0.995087	+	0.0990039i	$0.968434\pi$
$224$		0			0
$225$	2.97576			0.198384
$226$		0			0
$227$	9.41235	+	9.41235i	0.624719	+	0.624719i	0.946735	−	0.322015i	$-0.104360\pi$
$227$	9.41235	+	9.41235i	0.624719	+	0.624719i	−0.322015	+	0.946735i	$0.604360\pi$
$228$		0			0
$229$		−	18.0840i		−	1.19502i	−0.801860	−	0.597511i	$-0.796156\pi$
$229$		−	18.0840i		−	1.19502i	0.801860	−	0.597511i	$-0.203844\pi$
$230$		0			0
$231$	−0.806653	−	0.806653i	−0.0530739	−	0.0530739i
$232$		0			0
$233$	−12.1221	+	12.1221i	−0.794145	+	0.794145i	−0.982165	−	0.188020i	$-0.939793\pi$
$233$	−12.1221	+	12.1221i	−0.794145	+	0.794145i	0.188020	+	0.982165i	$0.439793\pi$
$234$		0			0
$235$	4.75074	−	4.75074i	0.309904	−	0.309904i
$236$		0			0
$237$	0.993621			0.0645426
$238$		0			0
$239$	−9.17681			−0.593599			−0.296799	−	0.954940i	$-0.595919\pi$
$239$	−9.17681			−0.593599			−0.296799	+	0.954940i	$0.595919\pi$
$240$		0			0
$241$	−4.11535	+	4.11535i	−0.265093	+	0.265093i	−0.827119	−	0.562027i	$-0.810022\pi$
$241$	−4.11535	+	4.11535i	−0.265093	+	0.265093i	0.562027	+	0.827119i	$0.310022\pi$
$242$		0			0
$243$	−2.94043	+	2.94043i	−0.188629	+	0.188629i
$244$		0			0
$245$	1.33778	+	1.33778i	0.0854677	+	0.0854677i
$246$		0			0
$247$		−	23.8243i		−	1.51590i
$248$		0			0
$249$	0.329285	+	0.329285i	0.0208676	+	0.0208676i
$250$		0			0
$251$	21.5093			1.35765			0.678826	−	0.734299i	$-0.262489\pi$
$251$	21.5093			1.35765			0.678826	+	0.734299i	$0.262489\pi$
$252$		0			0
$253$			7.39208i			0.464736i
$254$		0			0
$255$	0.415379	−	0.489409i	0.0260120	−	0.0306480i
$256$		0			0
$257$		−	10.6671i		−	0.665398i	−0.943033	−	0.332699i	$-0.892041\pi$
$257$		−	10.6671i		−	0.665398i	0.943033	−	0.332699i	$-0.107959\pi$
$258$		0			0
$259$	−4.87709			−0.303048
$260$		0			0
$261$	−9.27033	−	9.27033i	−0.573819	−	0.573819i
$262$		0			0
$263$		−	4.92413i		−	0.303635i	−0.988409	−	0.151817i	$-0.951487\pi$
$263$		−	4.92413i		−	0.303635i	0.988409	−	0.151817i	$-0.0485126\pi$
$264$		0			0
$265$	−1.58211	−	1.58211i	−0.0971883	−	0.0971883i
$266$		0			0
$267$	−0.434951	+	0.434951i	−0.0266186	+	0.0266186i
$268$		0			0
$269$	7.15963	−	7.15963i	0.436530	−	0.436530i	−0.454312	−	0.890843i	$-0.650115\pi$
$269$	7.15963	−	7.15963i	0.436530	−	0.436530i	0.890843	+	0.454312i	$0.150115\pi$
$270$		0			0
$271$	−18.1444			−1.10219			−0.551096	−	0.834442i	$-0.685790\pi$
$271$	−18.1444			−1.10219			−0.551096	+	0.834442i	$0.685790\pi$
$272$		0			0
$273$	2.23574			0.135313
$274$		0			0
$275$	2.29246	−	2.29246i	0.138240	−	0.138240i
$276$		0			0
$277$	−18.8451	+	18.8451i	−1.13229	+	1.13229i	−0.142497	+	0.989795i	$0.545513\pi$
$277$	−18.8451	+	18.8451i	−1.13229	+	1.13229i	−0.989795	+	0.142497i	$0.954487\pi$
$278$		0			0
$279$	−15.3287	−	15.3287i	−0.917706	−	0.917706i
$280$		0			0
$281$			20.6326i			1.23084i	0.788201	+	0.615418i	$0.211013\pi$
$281$			20.6326i			1.23084i	−0.788201	+	0.615418i	$0.788987\pi$
$282$		0			0
$283$	9.58317	+	9.58317i	0.569660	+	0.569660i	0.932033	−	0.362373i	$-0.118033\pi$
$283$	9.58317	+	9.58317i	0.569660	+	0.569660i	−0.362373	+	0.932033i	$0.618033\pi$
$284$		0			0
$285$	−0.583767			−0.0345794
$286$		0			0
$287$		−	5.21805i		−	0.308012i
$288$		0			0
$289$	2.76340	+	16.7739i	0.162553	+	0.986700i
$290$		0			0
$291$		−	0.253445i		−	0.0148572i
$292$		0			0
$293$	−28.8138			−1.68332			−0.841660	−	0.540008i	$-0.818421\pi$
$293$	−28.8138			−1.68332			−0.841660	+	0.540008i	$0.818421\pi$
$294$		0			0
$295$	4.17931	+	4.17931i	0.243329	+	0.243329i
$296$		0			0
$297$			3.01624i			0.175020i
$298$		0			0
$299$	−10.2441	−	10.2441i	−0.592429	−	0.592429i
$300$		0			0
$301$	16.3006	−	16.3006i	0.939553	−	0.939553i
$302$		0			0
$303$	−0.803889	+	0.803889i	−0.0461822	+	0.0461822i
$304$		0			0
$305$	7.56848			0.433370
$306$		0			0
$307$	−29.1337			−1.66275			−0.831375	−	0.555712i	$-0.812446\pi$
$307$	−29.1337			−1.66275			−0.831375	+	0.555712i	$0.812446\pi$
$308$		0			0
$309$	1.77859	−	1.77859i	0.101180	−	0.101180i
$310$		0			0
$311$	−20.3810	+	20.3810i	−1.15570	+	1.15570i	−0.170313	+	0.985390i	$0.554478\pi$
$311$	−20.3810	+	20.3810i	−1.15570	+	1.15570i	−0.985390	+	0.170313i	$0.945522\pi$
$312$		0			0
$313$	−17.4936	−	17.4936i	−0.988796	−	0.988796i	0.0111421	−	0.999938i	$-0.496453\pi$
$313$	−17.4936	−	17.4936i	−0.988796	−	0.988796i	−0.999938	+	0.0111421i	$0.996453\pi$
$314$		0			0
$315$			6.72554i			0.378941i
$316$		0			0
$317$	12.3914	+	12.3914i	0.695970	+	0.695970i	0.963539	−	0.267569i	$-0.0862203\pi$
$317$	12.3914	+	12.3914i	0.695970	+	0.695970i	−0.267569	+	0.963539i	$0.586220\pi$
$318$		0			0
$319$	−14.2833			−0.799711
$320$		0			0
$321$		−	1.06701i		−	0.0595549i
$322$		0			0
$323$	10.0040	−	11.7869i	0.556635	−	0.655840i
$324$		0			0
$325$			6.35384i			0.352448i
$326$		0			0
$327$	2.43371			0.134584
$328$		0			0
$329$	10.7372	+	10.7372i	0.591960	+	0.591960i
$330$		0			0
$331$			15.2629i			0.838923i	0.907773	+	0.419461i	$0.137781\pi$
$331$			15.2629i			0.838923i	−0.907773	+	0.419461i	$0.862219\pi$
$332$		0			0
$333$	4.54062	+	4.54062i	0.248824	+	0.248824i
$334$		0			0
$335$	−6.66323	+	6.66323i	−0.364051	+	0.364051i
$336$		0			0
$337$	−15.5078	+	15.5078i	−0.844766	+	0.844766i	−0.989474	−	0.144709i	$-0.953776\pi$
$337$	−15.5078	+	15.5078i	−0.844766	+	0.844766i	0.144709	+	0.989474i	$0.453776\pi$
$338$		0			0
$339$	−0.957196			−0.0519878
$340$		0			0
$341$	−23.6178			−1.27897
$342$		0			0
$343$	−14.2105	+	14.2105i	−0.767295	+	0.767295i
$344$		0			0
$345$	−0.251010	+	0.251010i	−0.0135139	+	0.0135139i
$346$		0			0
$347$	−6.88455	−	6.88455i	−0.369582	−	0.369582i	0.497743	−	0.867325i	$-0.334162\pi$
$347$	−6.88455	−	6.88455i	−0.369582	−	0.369582i	−0.867325	+	0.497743i	$0.834162\pi$
$348$		0			0
$349$			36.1613i			1.93567i	0.251589	+	0.967834i	$0.419047\pi$
$349$			36.1613i			1.93567i	−0.251589	+	0.967834i	$0.580953\pi$
$350$		0			0
$351$	−4.17995	−	4.17995i	−0.223109	−	0.223109i
$352$		0			0
$353$	24.0430			1.27968			0.639839	−	0.768509i	$-0.279001\pi$
$353$	24.0430			1.27968			0.639839	+	0.768509i	$0.279001\pi$
$354$		0			0
$355$			4.42827i			0.235028i
$356$		0			0
$357$	1.10612	+	0.938801i	0.0585420	+	0.0496866i
$358$		0			0
$359$		−	8.62910i		−	0.455426i	−0.973728	−	0.227713i	$-0.926875\pi$
$359$		−	8.62910i		−	0.455426i	0.973728	−	0.227713i	$-0.0731248\pi$
$360$		0			0
$361$	4.94060			0.260032
$362$		0			0
$363$	−0.0538630	−	0.0538630i	−0.00282708	−	0.00282708i
$364$		0			0
$365$		−	7.23978i		−	0.378947i
$366$		0			0
$367$	13.9153	+	13.9153i	0.726373	+	0.726373i	0.969895	−	0.243522i	$-0.0783030\pi$
$367$	13.9153	+	13.9153i	0.726373	+	0.726373i	−0.243522	+	0.969895i	$0.578303\pi$
$368$		0			0
$369$	−4.85805	+	4.85805i	−0.252900	+	0.252900i
$370$		0			0
$371$	3.57574	−	3.57574i	0.185643	−	0.185643i
$372$		0			0
$373$	17.7649			0.919830			0.459915	−	0.887963i	$-0.347880\pi$
$373$	17.7649			0.919830			0.459915	+	0.887963i	$0.347880\pi$
$374$		0			0
$375$	0.155688			0.00803971
$376$		0			0
$377$	19.7940	−	19.7940i	1.01944	−	1.01944i
$378$		0			0
$379$	−4.83105	+	4.83105i	−0.248154	+	0.248154i	−0.820213	−	0.572059i	$-0.806145\pi$
$379$	−4.83105	+	4.83105i	−0.248154	+	0.248154i	0.572059	+	0.820213i	$0.306145\pi$
$380$		0			0
$381$	−1.76992	−	1.76992i	−0.0906757	−	0.0906757i
$382$		0			0
$383$			20.5719i			1.05118i	0.850739	+	0.525588i	$0.176155\pi$
$383$			20.5719i			1.05118i	−0.850739	+	0.525588i	$0.823845\pi$
$384$		0			0
$385$	5.18120	+	5.18120i	0.264059	+	0.264059i
$386$		0			0
$387$	−30.3521			−1.54288
$388$		0			0
$389$			15.0964i			0.765418i	0.923869	+	0.382709i	$0.125009\pi$
$389$			15.0964i			0.765418i	−0.923869	+	0.382709i	$0.874991\pi$
$390$		0			0
$391$	−0.766634	−	9.36971i	−0.0387704	−	0.473846i
$392$		0			0
$393$		−	2.53919i		−	0.128085i
$394$		0			0
$395$	−6.38211			−0.321119
$396$		0			0
$397$	14.1646	+	14.1646i	0.710899	+	0.710899i	0.966723	−	0.255824i	$-0.0823468\pi$
$397$	14.1646	+	14.1646i	0.710899	+	0.710899i	−0.255824	+	0.966723i	$0.582347\pi$
$398$		0			0
$399$		−	1.31938i		−	0.0660515i
$400$		0			0
$401$	−21.5575	−	21.5575i	−1.07653	−	1.07653i	−0.996818	−	0.0797099i	$-0.974601\pi$
$401$	−21.5575	−	21.5575i	−1.07653	−	1.07653i	−0.0797099	−	0.996818i	$-0.525399\pi$
$402$		0			0
$403$	32.7299	−	32.7299i	1.63039	−	1.63039i
$404$		0			0
$405$	6.21012	−	6.21012i	0.308583	−	0.308583i
$406$		0			0
$407$	6.99598			0.346778
$408$		0			0
$409$	18.4810			0.913829			0.456914	−	0.889511i	$-0.348955\pi$
$409$	18.4810			0.913829			0.456914	+	0.889511i	$0.348955\pi$
$410$		0			0
$411$	−1.61707	+	1.61707i	−0.0797642	+	0.0797642i
$412$		0			0
$413$	−9.44569	+	9.44569i	−0.464792	+	0.464792i
$414$		0			0
$415$	−2.11503	−	2.11503i	−0.103822	−	0.103822i
$416$		0			0
$417$			2.44162i			0.119567i
$418$		0			0
$419$	14.1681	+	14.1681i	0.692158	+	0.692158i	0.962706	−	0.270549i	$-0.0872051\pi$
$419$	14.1681	+	14.1681i	0.692158	+	0.692158i	−0.270549	+	0.962706i	$0.587205\pi$
$420$		0			0
$421$	−23.9933			−1.16936			−0.584681	−	0.811263i	$-0.698780\pi$
$421$	−23.9933			−1.16936			−0.584681	+	0.811263i	$0.698780\pi$
$422$		0			0
$423$		−	19.9928i		−	0.972084i
$424$		0			0
$425$	−2.66801	+	3.14352i	−0.129418	+	0.152483i
$426$		0			0
$427$			17.1056i			0.827798i
$428$		0			0
$429$	−3.20707			−0.154839
$430$		0			0
$431$	−0.287583	−	0.287583i	−0.0138524	−	0.0138524i	0.700147	−	0.713999i	$-0.253118\pi$
$431$	−0.287583	−	0.287583i	−0.0138524	−	0.0138524i	−0.713999	+	0.700147i	$0.753118\pi$
$432$		0			0
$433$			6.37599i			0.306410i	0.988194	+	0.153205i	$0.0489595\pi$
$433$			6.37599i			0.306410i	−0.988194	+	0.153205i	$0.951040\pi$
$434$		0			0
$435$	−0.485013	−	0.485013i	−0.0232546	−	0.0232546i
$436$		0			0
$437$	−6.04531	+	6.04531i	−0.289187	+	0.289187i
$438$		0			0
$439$	−13.9079	+	13.9079i	−0.663790	+	0.663790i	−0.956271	−	0.292481i	$-0.905519\pi$
$439$	−13.9079	+	13.9079i	−0.663790	+	0.663790i	0.292481	+	0.956271i	$0.405519\pi$
$440$		0			0
$441$	5.62987			0.268089
$442$		0			0
$443$	−30.2115			−1.43539			−0.717697	−	0.696356i	$-0.754804\pi$
$443$	−30.2115			−1.43539			−0.717697	+	0.696356i	$0.754804\pi$
$444$		0			0
$445$	2.79373	−	2.79373i	0.132436	−	0.132436i
$446$		0			0
$447$	1.73698	−	1.73698i	0.0821562	−	0.0821562i
$448$		0			0
$449$	−12.5317	−	12.5317i	−0.591406	−	0.591406i	0.346605	−	0.938011i	$-0.387334\pi$
$449$	−12.5317	−	12.5317i	−0.591406	−	0.591406i	−0.938011	+	0.346605i	$0.887334\pi$
$450$		0			0
$451$			7.48506i			0.352458i
$452$		0			0
$453$	−2.30090	−	2.30090i	−0.108105	−	0.108105i
$454$		0			0
$455$	−14.3604			−0.673225
$456$		0			0
$457$			9.06827i			0.424196i	0.977248	+	0.212098i	$0.0680296\pi$
$457$			9.06827i			0.424196i	−0.977248	+	0.212098i	$0.931970\pi$
$458$		0			0
$459$	−0.312815	−	3.82318i	−0.0146009	−	0.178451i
$460$		0			0
$461$		−	6.13947i		−	0.285944i	−0.989727	−	0.142972i	$-0.954334\pi$
$461$		−	6.13947i		−	0.285944i	0.989727	−	0.142972i	$-0.0456659\pi$
$462$		0			0
$463$	23.8176			1.10690			0.553450	−	0.832883i	$-0.313311\pi$
$463$	23.8176			1.10690			0.553450	+	0.832883i	$0.313311\pi$
$464$		0			0
$465$	−0.801981	−	0.801981i	−0.0371910	−	0.0371910i
$466$		0			0
$467$			8.03489i			0.371810i	0.982568	+	0.185905i	$0.0595217\pi$
$467$			8.03489i			0.371810i	−0.982568	+	0.185905i	$0.940478\pi$
$468$		0			0
$469$	−15.0596	−	15.0596i	−0.695388	−	0.695388i
$470$		0			0
$471$	1.35070	−	1.35070i	0.0622368	−	0.0622368i
$472$		0			0
$473$	−23.3825	+	23.3825i	−1.07513	+	1.07513i
$474$		0			0
$475$	3.74959			0.172043
$476$		0			0
$477$	−6.65809			−0.304853
$478$		0			0
$479$	17.2392	−	17.2392i	0.787679	−	0.787679i	−0.193434	−	0.981113i	$-0.561963\pi$
$479$	17.2392	−	17.2392i	0.787679	−	0.787679i	0.981113	+	0.193434i	$0.0619626\pi$
$480$		0			0
$481$	−9.69512	+	9.69512i	−0.442059	+	0.442059i
$482$		0			0
$483$	−0.567311	−	0.567311i	−0.0258135	−	0.0258135i
$484$		0			0
$485$			1.62790i			0.0739192i
$486$		0			0
$487$	14.6286	+	14.6286i	0.662887	+	0.662887i	0.956059	−	0.293173i	$-0.0947111\pi$
$487$	14.6286	+	14.6286i	0.662887	+	0.662887i	−0.293173	+	0.956059i	$0.594711\pi$
$488$		0			0
$489$	2.70339			0.122251
$490$		0			0
$491$		−	31.4474i		−	1.41920i	−0.704604	−	0.709601i	$-0.748875\pi$
$491$		−	31.4474i		−	1.41920i	0.704604	−	0.709601i	$-0.251125\pi$
$492$		0			0
$493$	18.1046	−	1.48132i	0.815388	−	0.0667155i
$494$		0			0
$495$		−	9.64749i		−	0.433623i
$496$		0			0
$497$	−10.0084			−0.448937
$498$		0			0
$499$	5.22038	+	5.22038i	0.233696	+	0.233696i	0.814234	−	0.580537i	$-0.197157\pi$
$499$	5.22038	+	5.22038i	0.233696	+	0.233696i	−0.580537	+	0.814234i	$0.697157\pi$
$500$		0			0
$501$		−	1.44093i		−	0.0643759i
$502$		0			0
$503$	−13.8503	−	13.8503i	−0.617553	−	0.617553i	0.327350	−	0.944903i	$-0.393844\pi$
$503$	−13.8503	−	13.8503i	−0.617553	−	0.617553i	−0.944903	+	0.327350i	$0.893844\pi$
$504$		0			0
$505$	5.16345	−	5.16345i	0.229770	−	0.229770i
$506$		0			0
$507$	3.01326	−	3.01326i	0.133824	−	0.133824i
$508$		0			0
$509$	36.3104			1.60943			0.804715	−	0.593661i	$-0.202318\pi$
$509$	36.3104			1.60943			0.804715	+	0.593661i	$0.202318\pi$
$510$		0			0
$511$	16.3627			0.723843
$512$		0			0
$513$	−2.46671	+	2.46671i	−0.108908	+	0.108908i
$514$		0			0
$515$	−11.4240	+	11.4240i	−0.503402	+	0.503402i
$516$		0			0
$517$	−15.4020	−	15.4020i	−0.677380	−	0.677380i
$518$		0			0
$519$			3.43216i			0.150655i
$520$		0			0
$521$	13.5155	+	13.5155i	0.592125	+	0.592125i	0.938205	−	0.346080i	$-0.112488\pi$
$521$	13.5155	+	13.5155i	0.592125	+	0.592125i	−0.346080	+	0.938205i	$0.612488\pi$
$522$		0			0
$523$	−11.6507			−0.509449			−0.254724	−	0.967014i	$-0.581985\pi$
$523$	−11.6507			−0.509449			−0.254724	+	0.967014i	$0.581985\pi$
$524$		0			0
$525$			0.351873i			0.0153570i
$526$		0			0
$527$	29.9363	−	2.44941i	1.30405	−	0.106698i
$528$		0			0
$529$		−	17.8012i		−	0.773966i
$530$		0			0
$531$	17.5880			0.763256
$532$		0			0
$533$	−10.3729	−	10.3729i	−0.449300	−	0.449300i
$534$		0			0
$535$			6.85352i			0.296303i
$536$		0			0
$537$	−0.110414	−	0.110414i	−0.00476470	−	0.00476470i
$538$		0			0
$539$	4.33712	−	4.33712i	0.186813	−	0.186813i
$540$		0			0
$541$	6.17032	−	6.17032i	0.265283	−	0.265283i	−0.561913	−	0.827196i	$-0.689935\pi$
$541$	6.17032	−	6.17032i	0.265283	−	0.265283i	0.827196	+	0.561913i	$0.189935\pi$
$542$		0			0
$543$	2.56464			0.110059
$544$		0			0
$545$	−15.6319			−0.669598
$546$		0			0
$547$	−18.6105	+	18.6105i	−0.795729	+	0.795729i	−0.982419	−	0.186690i	$-0.940224\pi$
$547$	−18.6105	+	18.6105i	−0.795729	+	0.795729i	0.186690	+	0.982419i	$0.440224\pi$
$548$		0			0
$549$	15.9255	−	15.9255i	0.679682	−	0.679682i
$550$		0			0
$551$	−11.6810	−	11.6810i	−0.497628	−	0.497628i
$552$		0			0
$553$		−	14.4243i		−	0.613382i
$554$		0			0
$555$	0.237560	+	0.237560i	0.0100839	+	0.0100839i
$556$		0			0
$557$	3.77947			0.160141			0.0800707	−	0.996789i	$-0.474485\pi$
$557$	3.77947			0.160141			0.0800707	+	0.996789i	$0.474485\pi$
$558$		0			0
$559$		−	64.8077i		−	2.74107i
$560$		0			0
$561$	−1.58668	−	1.34667i	−0.0669895	−	0.0568564i
$562$		0			0
$563$			12.0518i			0.507922i	0.967214	+	0.253961i	$0.0817336\pi$
$563$			12.0518i			0.507922i	−0.967214	+	0.253961i	$0.918266\pi$
$564$		0			0
$565$	6.14815			0.258655
$566$		0			0
$567$	14.0355	+	14.0355i	0.589438	+	0.589438i
$568$		0			0
$569$		−	45.8883i		−	1.92374i	−0.273510	−	0.961869i	$-0.588185\pi$
$569$		−	45.8883i		−	1.92374i	0.273510	−	0.961869i	$-0.411815\pi$
$570$		0			0
$571$	29.6260	+	29.6260i	1.23981	+	1.23981i	0.960079	+	0.279730i	$0.0902450\pi$
$571$	29.6260	+	29.6260i	1.23981	+	1.23981i	0.279730	+	0.960079i	$0.409755\pi$
$572$		0			0
$573$	−0.504788	+	0.504788i	−0.0210878	+	0.0210878i
$574$		0			0
$575$	1.61226	−	1.61226i	0.0672359	−	0.0672359i
$576$		0			0
$577$	35.9945			1.49847			0.749236	−	0.662303i	$-0.230421\pi$
$577$	35.9945			1.49847			0.749236	+	0.662303i	$0.230421\pi$
$578$		0			0
$579$	0.628444			0.0261173
$580$		0			0
$581$	4.78019	−	4.78019i	0.198316	−	0.198316i
$582$		0			0
$583$	−5.12924	+	5.12924i	−0.212432	+	0.212432i
$584$		0			0
$585$	13.3696	+	13.3696i	0.552766	+	0.552766i
$586$		0			0
$587$			13.6745i			0.564408i	0.959354	+	0.282204i	$0.0910655\pi$
$587$			13.6745i			0.564408i	−0.959354	+	0.282204i	$0.908935\pi$
$588$		0			0
$589$	−19.3148	−	19.3148i	−0.795854	−	0.795854i
$590$		0			0
$591$	−2.71662			−0.111747
$592$		0			0
$593$		−	16.1481i		−	0.663122i	−0.943434	−	0.331561i	$-0.892425\pi$
$593$		−	16.1481i		−	0.663122i	0.943434	−	0.331561i	$-0.107575\pi$
$594$		0			0
$595$	−7.10469	−	6.03000i	−0.291264	−	0.247206i
$596$		0			0
$597$			2.73583i			0.111970i
$598$		0			0
$599$	−28.7857			−1.17615			−0.588076	−	0.808806i	$-0.700114\pi$
$599$	−28.7857			−1.17615			−0.588076	+	0.808806i	$0.700114\pi$
$600$		0			0
$601$	−0.605266	−	0.605266i	−0.0246893	−	0.0246893i	0.694654	−	0.719344i	$-0.255557\pi$
$601$	−0.605266	−	0.605266i	−0.0246893	−	0.0246893i	−0.719344	+	0.694654i	$0.755557\pi$
$602$		0			0
$603$			28.0413i			1.14193i
$604$		0			0
$605$	0.345967	+	0.345967i	0.0140656	+	0.0140656i
$606$		0			0
$607$	−5.16467	+	5.16467i	−0.209627	+	0.209627i	−0.804109	−	0.594482i	$-0.797357\pi$
$607$	−5.16467	+	5.16467i	−0.209627	+	0.209627i	0.594482	+	0.804109i	$0.297357\pi$
$608$		0			0
$609$	1.09618	−	1.09618i	0.0444195	−	0.0444195i
$610$		0			0
$611$	42.6886			1.72700
$612$		0			0
$613$	−8.72541			−0.352416			−0.176208	−	0.984353i	$-0.556383\pi$
$613$	−8.72541			−0.352416			−0.176208	+	0.984353i	$0.556383\pi$
$614$		0			0
$615$	−0.254167	+	0.254167i	−0.0102490	+	0.0102490i
$616$		0			0
$617$	−17.7377	+	17.7377i	−0.714093	+	0.714093i	−0.967389	−	0.253296i	$-0.918485\pi$
$617$	−17.7377	+	17.7377i	−0.714093	+	0.714093i	0.253296	+	0.967389i	$0.418485\pi$
$618$		0			0
$619$	20.1175	+	20.1175i	0.808591	+	0.808591i	0.984421	−	0.175830i	$-0.0562609\pi$
$619$	20.1175	+	20.1175i	0.808591	+	0.808591i	−0.175830	+	0.984421i	$0.556261\pi$
$620$		0			0
$621$			2.12129i			0.0851244i
$622$		0			0
$623$	6.31413	+	6.31413i	0.252970	+	0.252970i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$			1.89259i			0.0755827i
$628$		0			0
$629$	−8.86763	+	0.725554i	−0.353575	+	0.0289297i
$630$		0			0
$631$			5.58672i			0.222404i	0.993798	+	0.111202i	$0.0354700\pi$
$631$			5.58672i			0.222404i	−0.993798	+	0.111202i	$0.964530\pi$
$632$		0			0
$633$	2.35128			0.0934552
$634$		0			0
$635$	11.3683	+	11.3683i	0.451139	+	0.451139i
$636$		0			0
$637$			12.0209i			0.476285i
$638$		0			0
$639$	9.31788	+	9.31788i	0.368610	+	0.368610i
$640$		0			0
$641$	−9.53481	+	9.53481i	−0.376603	+	0.376603i	−0.869875	−	0.493272i	$-0.835801\pi$
$641$	−9.53481	+	9.53481i	−0.376603	+	0.376603i	0.493272	+	0.869875i	$0.335801\pi$
$642$		0			0
$643$	17.4747	−	17.4747i	0.689135	−	0.689135i	−0.272906	−	0.962041i	$-0.587985\pi$
$643$	17.4747	−	17.4747i	0.689135	−	0.689135i	0.962041	+	0.272906i	$0.0879848\pi$
$644$		0			0
$645$	−1.58799			−0.0625269
$646$		0			0
$647$	35.8120			1.40791			0.703957	−	0.710243i	$-0.251415\pi$
$647$	35.8120			1.40791			0.703957	+	0.710243i	$0.251415\pi$
$648$		0			0
$649$	13.5494	−	13.5494i	0.531861	−	0.531861i
$650$		0			0
$651$	1.81256	−	1.81256i	0.0710400	−	0.0710400i
$652$		0			0
$653$	−14.6559	−	14.6559i	−0.573529	−	0.573529i	0.359584	−	0.933113i	$-0.382919\pi$
$653$	−14.6559	−	14.6559i	−0.573529	−	0.573529i	−0.933113	+	0.359584i	$0.882919\pi$
$654$		0			0
$655$			16.3095i			0.637263i
$656$		0			0
$657$	−15.2338	−	15.2338i	−0.594327	−	0.594327i
$658$		0			0
$659$	−32.6942			−1.27359			−0.636793	−	0.771035i	$-0.719739\pi$
$659$	−32.6942			−1.27359			−0.636793	+	0.771035i	$0.719739\pi$
$660$		0			0
$661$		−	0.329952i		−	0.0128336i	−0.999979	−	0.00641682i	$-0.997957\pi$
$661$		−	0.329952i		−	0.0128336i	0.999979	−	0.00641682i	$-0.00204255\pi$
$662$		0			0
$663$	4.06507	−	0.332606i	0.157874	−	0.0129174i
$664$		0			0
$665$			8.47447i			0.328626i
$666$		0			0
$667$	−10.0453			−0.388955
$668$		0			0
$669$	0.325519	+	0.325519i	0.0125853	+	0.0125853i
$670$		0			0
$671$		−	24.5372i		−	0.947248i
$672$		0			0
$673$	−21.5369	−	21.5369i	−0.830187	−	0.830187i	0.157355	−	0.987542i	$-0.449703\pi$
$673$	−21.5369	−	21.5369i	−0.830187	−	0.830187i	−0.987542	+	0.157355i	$0.949703\pi$
$674$		0			0
$675$	0.657861	−	0.657861i	0.0253211	−	0.0253211i
$676$		0			0
$677$	6.09948	−	6.09948i	0.234422	−	0.234422i	−0.580114	−	0.814536i	$-0.696992\pi$
$677$	6.09948	−	6.09948i	0.234422	−	0.234422i	0.814536	+	0.580114i	$0.196992\pi$
$678$		0			0
$679$	−3.67923			−0.141196
$680$		0			0
$681$	2.07238			0.0794137
$682$		0			0
$683$	24.3481	−	24.3481i	0.931653	−	0.931653i	−0.0661559	−	0.997809i	$-0.521073\pi$
$683$	24.3481	−	24.3481i	0.931653	−	0.931653i	0.997809	+	0.0661559i	$0.0210735\pi$
$684$		0			0
$685$	10.3866	−	10.3866i	0.396851	−	0.396851i
$686$		0			0
$687$	−1.99083	−	1.99083i	−0.0759551	−	0.0759551i
$688$		0			0
$689$		−	14.2164i		−	0.541600i
$690$		0			0
$691$	−17.5503	−	17.5503i	−0.667643	−	0.667643i	0.289527	−	0.957170i	$-0.406502\pi$
$691$	−17.5503	−	17.5503i	−0.667643	−	0.667643i	−0.957170	+	0.289527i	$0.906502\pi$
$692$		0			0
$693$	21.8044			0.828280
$694$		0			0
$695$		−	15.6827i		−	0.594880i
$696$		0			0
$697$	−0.776277	−	9.48755i	−0.0294036	−	0.359367i
$698$		0			0
$699$			2.66900i			0.100951i
$700$		0			0
$701$	17.8489			0.674142			0.337071	−	0.941479i	$-0.390564\pi$
$701$	17.8489			0.674142			0.337071	+	0.941479i	$0.390564\pi$
$702$		0			0
$703$	5.72137	+	5.72137i	0.215786	+	0.215786i
$704$		0			0
$705$		−	1.04600i		−	0.0393947i
$706$		0			0
$707$	11.6700	+	11.6700i	0.438894	+	0.438894i
$708$		0			0
$709$	4.23484	−	4.23484i	0.159043	−	0.159043i	−0.623100	−	0.782142i	$-0.714127\pi$
$709$	4.23484	−	4.23484i	0.159043	−	0.159043i	0.782142	+	0.623100i	$0.214127\pi$
$710$		0			0
$711$	−13.4291	+	13.4291i	−0.503631	+	0.503631i
$712$		0			0
$713$	−16.6101			−0.622055
$714$		0			0
$715$	20.5993			0.770371
$716$		0			0
$717$	−1.01026	+	1.01026i	−0.0377288	+	0.0377288i
$718$		0			0
$719$	30.4445	−	30.4445i	1.13539	−	1.13539i	0.146121	−	0.989267i	$-0.453321\pi$
$719$	30.4445	−	30.4445i	1.13539	−	1.13539i	0.989267	−	0.146121i	$-0.0466787\pi$
$720$		0			0
$721$	−25.8195	−	25.8195i	−0.961569	−	0.961569i
$722$		0			0
$723$			0.906103i			0.0336983i
$724$		0			0
$725$	3.11528	+	3.11528i	0.115699	+	0.115699i
$726$		0			0
$727$	3.48872			0.129390			0.0646948	−	0.997905i	$-0.479393\pi$
$727$	3.48872			0.129390			0.0646948	+	0.997905i	$0.479393\pi$
$728$		0			0
$729$		−	25.6999i		−	0.951848i
$730$		0			0
$731$	27.2131	−	32.0631i	1.00651	−	1.18590i
$732$		0			0
$733$		−	31.1271i		−	1.14971i	−0.818257	−	0.574853i	$-0.805059\pi$
$733$		−	31.1271i		−	1.14971i	0.818257	−	0.574853i	$-0.194941\pi$
$734$		0			0
$735$	0.294548			0.0108646
$736$		0			0
$737$	21.6024	+	21.6024i	0.795733	+	0.795733i
$738$		0			0
$739$		−	45.9291i		−	1.68953i	−0.535139	−	0.844764i	$-0.679741\pi$
$739$		−	45.9291i		−	1.68953i	0.535139	−	0.844764i	$-0.320259\pi$
$740$		0			0
$741$	−2.62277	−	2.62277i	−0.0963500	−	0.0963500i
$742$		0			0
$743$	31.8533	−	31.8533i	1.16858	−	1.16858i	0.186043	−	0.982542i	$-0.440434\pi$
$743$	31.8533	−	31.8533i	1.16858	−	1.16858i	0.982542	−	0.186043i	$-0.0595662\pi$
$744$		0			0
$745$	−11.1568	+	11.1568i	−0.408752	+	0.408752i
$746$		0			0
$747$	−8.90079			−0.325663
$748$		0			0
$749$	−15.4897			−0.565981
$750$		0			0
$751$	−37.5622	+	37.5622i	−1.37066	+	1.37066i	−0.511205	+	0.859459i	$0.670801\pi$
$751$	−37.5622	+	37.5622i	−1.37066	+	1.37066i	−0.859459	+	0.511205i	$0.829199\pi$
$752$		0			0
$753$	2.36792	−	2.36792i	0.0862917	−	0.0862917i
$754$		0			0
$755$	14.7789	+	14.7789i	0.537857	+	0.537857i
$756$		0			0
$757$		−	1.41070i		−	0.0512729i	−0.999671	−	0.0256365i	$-0.991839\pi$
$757$		−	1.41070i		−	0.0512729i	0.999671	−	0.0256365i	$-0.00816124\pi$
$758$		0			0
$759$	0.813782	+	0.813782i	0.0295384	+	0.0295384i
$760$		0			0
$761$	18.6971			0.677768			0.338884	−	0.940828i	$-0.389951\pi$
$761$	18.6971			0.677768			0.338884	+	0.940828i	$0.389951\pi$
$762$		0			0
$763$		−	35.3298i		−	1.27903i
$764$		0			0
$765$	1.00054	+	12.2285i	0.0361747	+	0.442123i
$766$		0			0
$767$			37.5540i			1.35599i
$768$		0			0
$769$	−17.7284			−0.639301			−0.319651	−	0.947535i	$-0.603566\pi$
$769$	−17.7284			−0.639301			−0.319651	+	0.947535i	$0.603566\pi$
$770$		0			0
$771$	−1.17433	−	1.17433i	−0.0422924	−	0.0422924i
$772$		0			0
$773$		−	20.3379i		−	0.731503i	−0.930713	−	0.365751i	$-0.880812\pi$
$773$		−	20.3379i		−	0.731503i	0.930713	−	0.365751i	$-0.119188\pi$
$774$		0			0
$775$	5.15119	+	5.15119i	0.185036	+	0.185036i
$776$		0			0
$777$	−0.536911	+	0.536911i	−0.0192616	+	0.0192616i
$778$		0			0
$779$	−6.12135	+	6.12135i	−0.219320	+	0.219320i
$780$		0			0
$781$	14.3566			0.513718
$782$		0			0
$783$	−4.09885			−0.146481
$784$		0			0
$785$	−8.67564	+	8.67564i	−0.309647	+	0.309647i
$786$		0			0
$787$	−16.0350	+	16.0350i	−0.571585	+	0.571585i	−0.932571	−	0.360986i	$-0.882440\pi$
$787$	−16.0350	+	16.0350i	−0.571585	+	0.571585i	0.360986	+	0.932571i	$0.382440\pi$
$788$		0			0
$789$	−0.542089	−	0.542089i	−0.0192989	−	0.0192989i
$790$		0			0
$791$			13.8955i			0.494067i
$792$		0			0
$793$	34.0040	+	34.0040i	1.20752	+	1.20752i
$794$		0			0
$795$	−0.348344			−0.0123545
$796$		0			0
$797$			26.2980i			0.931522i	0.884910	+	0.465761i	$0.154219\pi$
$797$			26.2980i			0.931522i	−0.884910	+	0.465761i	$0.845781\pi$
$798$		0			0
$799$	21.1199	+	17.9252i	0.747168	+	0.634148i
$800$		0			0
$801$		−	11.7570i		−	0.415414i
$802$		0			0
$803$	−23.4716			−0.828293
$804$		0			0
$805$	3.64389	+	3.64389i	0.128430	+	0.128430i
$806$		0			0
$807$		−	1.57638i		−	0.0554913i
$808$		0			0
$809$	26.4838	+	26.4838i	0.931120	+	0.931120i	0.997776	−	0.0666563i	$-0.0212331\pi$
$809$	26.4838	+	26.4838i	0.931120	+	0.931120i	−0.0666563	+	0.997776i	$0.521233\pi$
$810$		0			0
$811$	1.14490	−	1.14490i	0.0402029	−	0.0402029i	−0.686720	−	0.726922i	$-0.740950\pi$
$811$	1.14490	−	1.14490i	0.0402029	−	0.0402029i	0.726922	+	0.686720i	$0.240950\pi$
$812$		0			0
$813$	−1.99748	+	1.99748i	−0.0700549	+	0.0700549i
$814$		0			0
$815$	−17.3641			−0.608237
$816$		0			0
$817$	−38.2449			−1.33802
$818$		0			0
$819$	−30.2168	+	30.2168i	−1.05586	+	1.05586i
$820$		0			0
$821$	−16.1479	+	16.1479i	−0.563566	+	0.563566i	−0.930319	−	0.366753i	$-0.880470\pi$
$821$	−16.1479	+	16.1479i	−0.563566	+	0.563566i	0.366753	+	0.930319i	$0.380470\pi$
$822$		0			0
$823$	25.3910	+	25.3910i	0.885073	+	0.885073i	0.994045	−	0.108972i	$-0.0347558\pi$
$823$	25.3910	+	25.3910i	0.885073	+	0.885073i	−0.108972	+	0.994045i	$0.534756\pi$
$824$		0			0
$825$		−	0.504746i		−	0.0175730i
$826$		0			0
$827$	−21.1096	−	21.1096i	−0.734053	−	0.734053i	0.237367	−	0.971420i	$-0.423716\pi$
$827$	−21.1096	−	21.1096i	−0.734053	−	0.734053i	−0.971420	+	0.237367i	$0.923716\pi$
$828$		0			0
$829$	−31.8056			−1.10466			−0.552328	−	0.833627i	$-0.686260\pi$
$829$	−31.8056			−1.10466			−0.552328	+	0.833627i	$0.686260\pi$
$830$		0			0
$831$			4.14925i			0.143936i
$832$		0			0
$833$	−5.04764	+	5.94725i	−0.174890	+	0.206060i
$834$		0			0
$835$			9.25520i			0.320289i
$836$		0			0
$837$	−6.77754			−0.234266
$838$		0			0
$839$	27.6685	+	27.6685i	0.955224	+	0.955224i	0.999040	−	0.0438159i	$-0.0139515\pi$
$839$	27.6685	+	27.6685i	0.955224	+	0.955224i	−0.0438159	+	0.999040i	$0.513951\pi$
$840$		0			0
$841$			9.59005i			0.330691i
$842$		0			0
$843$	2.27140	+	2.27140i	0.0782313	+	0.0782313i
$844$		0			0
$845$	−19.3544	+	19.3544i	−0.665812	+	0.665812i
$846$		0			0
$847$	−0.781923	+	0.781923i	−0.0268672	+	0.0268672i
$848$		0			0
$849$	2.10999			0.0724146
$850$		0			0
$851$	4.92019			0.168662
$852$		0			0
$853$	1.24673	−	1.24673i	0.0426872	−	0.0426872i	−0.685441	−	0.728128i	$-0.740391\pi$
$853$	1.24673	−	1.24673i	0.0426872	−	0.0426872i	0.728128	+	0.685441i	$0.240391\pi$
$854$		0			0
$855$	7.88981	−	7.88981i	0.269826	−	0.269826i
$856$		0			0
$857$	38.6780	+	38.6780i	1.32122	+	1.32122i	0.912792	+	0.408425i	$0.133922\pi$
$857$	38.6780	+	38.6780i	1.32122	+	1.32122i	0.408425	+	0.912792i	$0.366078\pi$
$858$		0			0
$859$		−	21.1269i		−	0.720842i	−0.932790	−	0.360421i	$-0.882633\pi$
$859$		−	21.1269i		−	0.720842i	0.932790	−	0.360421i	$-0.117367\pi$
$860$		0			0
$861$	−0.574446	−	0.574446i	−0.0195771	−	0.0195771i
$862$		0			0
$863$	7.08367			0.241131			0.120565	−	0.992705i	$-0.461529\pi$
$863$	7.08367			0.241131			0.120565	+	0.992705i	$0.461529\pi$
$864$		0			0
$865$		−	22.0450i		−	0.749554i
$866$		0			0
$867$	2.15083	+	1.54239i	0.0730459	+	0.0523824i
$868$		0			0
$869$			20.6910i			0.701893i
$870$		0			0
$871$	−59.8737			−2.02874
$872$		0			0
$873$	3.42540	+	3.42540i	0.115932	+	0.115932i
$874$		0			0
$875$		−	2.26011i		−	0.0764056i
$876$		0			0
$877$	−8.26296	−	8.26296i	−0.279020	−	0.279020i	0.553698	−	0.832718i	$-0.313216\pi$
$877$	−8.26296	−	8.26296i	−0.279020	−	0.279020i	−0.832718	+	0.553698i	$0.813216\pi$
$878$		0			0
$879$	−3.17206	+	3.17206i	−0.106991	+	0.106991i
$880$		0			0
$881$	35.3498	−	35.3498i	1.19096	−	1.19096i	0.214167	−	0.976797i	$-0.431296\pi$
$881$	35.3498	−	35.3498i	1.19096	−	1.19096i	0.976797	−	0.214167i	$-0.0687036\pi$
$882$		0			0
$883$	−41.3693			−1.39219			−0.696093	−	0.717951i	$-0.745080\pi$
$883$	−41.3693			−1.39219			−0.696093	+	0.717951i	$0.745080\pi$
$884$		0			0
$885$	0.920186			0.0309317
$886$		0			0
$887$	−5.69731	+	5.69731i	−0.191297	+	0.191297i	−0.796256	−	0.604959i	$-0.793189\pi$
$887$	−5.69731	+	5.69731i	−0.191297	+	0.191297i	0.604959	+	0.796256i	$0.293189\pi$
$888$		0			0
$889$	−25.6937	+	25.6937i	−0.861739	+	0.861739i
$890$		0			0
$891$	−20.1334	−	20.1334i	−0.674493	−	0.674493i
$892$		0			0
$893$		−	25.1918i		−	0.843012i
$894$		0			0
$895$	0.709196	+	0.709196i	0.0237058	+	0.0237058i
$896$		0			0
$897$	−2.25550			−0.0753090
$898$		0			0
$899$		−	32.0948i		−	1.07042i
$900$		0			0
$901$	5.96953	−	7.03344i	0.198874	−	0.234318i
$902$		0			0
$903$		−	3.58902i		−	0.119435i
$904$		0			0
$905$	−16.4729			−0.547578
$906$		0			0
$907$	1.94441	+	1.94441i	0.0645629	+	0.0645629i	0.738651	−	0.674088i	$-0.235463\pi$
$907$	1.94441	+	1.94441i	0.0645629	+	0.0645629i	−0.674088	+	0.738651i	$0.735463\pi$
$908$		0			0
$909$		−	21.7297i		−	0.720727i
$910$		0			0
$911$	17.9083	+	17.9083i	0.593329	+	0.593329i	0.938529	−	0.345200i	$-0.112189\pi$
$911$	17.9083	+	17.9083i	0.593329	+	0.593329i	−0.345200	+	0.938529i	$0.612189\pi$
$912$		0			0
$913$	−6.85697	+	6.85697i	−0.226932	+	0.226932i
$914$		0			0
$915$	0.833201	−	0.833201i	0.0275448	−	0.0275448i
$916$		0			0
$917$	−36.8611			−1.21726
$918$		0			0
$919$	54.7178			1.80497			0.902486	−	0.430719i	$-0.141740\pi$
$919$	54.7178			1.80497			0.902486	+	0.430719i	$0.141740\pi$
$920$		0			0
$921$	−3.20728	+	3.20728i	−0.105684	+	0.105684i
$922$		0			0
$923$	−19.8955	+	19.8955i	−0.654869	+	0.654869i
$924$		0			0
$925$	−1.52587	−	1.52587i	−0.0501702	−	0.0501702i
$926$		0			0
$927$			48.0764i			1.57904i
$928$		0			0
$929$	−38.8266	−	38.8266i	−1.27386	−	1.27386i	−0.944046	−	0.329814i	$-0.893014\pi$
$929$	−38.8266	−	38.8266i	−1.27386	−	1.27386i	−0.329814	−	0.944046i	$-0.606986\pi$
$930$		0			0
$931$	7.09387			0.232492
$932$		0			0
$933$			4.48743i			0.146912i
$934$		0			0
$935$	10.1914	+	8.64977i	0.333293	+	0.282878i
$936$		0			0
$937$		−	16.2957i		−	0.532358i	−0.963924	−	0.266179i	$-0.914239\pi$
$937$		−	16.2957i		−	0.532358i	0.963924	−	0.266179i	$-0.0857612\pi$
$938$		0			0
$939$	−3.85168			−0.125695
$940$		0			0
$941$	11.1227	+	11.1227i	0.362588	+	0.362588i	0.864765	−	0.502177i	$-0.167467\pi$
$941$	11.1227	+	11.1227i	0.362588	+	0.362588i	−0.502177	+	0.864765i	$0.667467\pi$
$942$		0			0
$943$			5.26416i			0.171425i
$944$		0			0
$945$	1.48684	+	1.48684i	0.0483669	+	0.0483669i
$946$		0			0
$947$	15.1051	−	15.1051i	0.490849	−	0.490849i	−0.417725	−	0.908574i	$-0.637172\pi$
$947$	15.1051	−	15.1051i	0.490849	−	0.490849i	0.908574	+	0.417725i	$0.137172\pi$
$948$		0			0
$949$	32.5272	−	32.5272i	1.05588	−	1.05588i
$950$		0			0
$951$	2.72829			0.0884710
$952$		0			0
$953$	−43.2594			−1.40131			−0.700655	−	0.713500i	$-0.747109\pi$
$953$	−43.2594			−1.40131			−0.700655	+	0.713500i	$0.747109\pi$
$954$		0			0
$955$	3.24230	−	3.24230i	0.104918	−	0.104918i
$956$		0			0
$957$	−1.57242	+	1.57242i	−0.0508293	+	0.0508293i
$958$		0			0
$959$	23.4748	+	23.4748i	0.758041	+	0.758041i
$960$		0			0
$961$		−	22.0696i		−	0.711922i
$962$		0			0
$963$	14.4210	+	14.4210i	0.464711	+	0.464711i
$964$		0			0
$965$	−4.03655			−0.129941
$966$		0			0
$967$			4.94557i			0.159039i	0.996833	+	0.0795194i	$0.0253386\pi$
$967$			4.94557i			0.159039i	−0.996833	+	0.0795194i	$0.974661\pi$
$968$		0			0
$969$	−0.196281	−	2.39892i	−0.00630544	−	0.0770643i
$970$		0			0
$971$		−	2.33246i		−	0.0748521i	−0.999299	−	0.0374260i	$-0.988084\pi$
$971$		−	2.33246i		−	0.0748521i	0.999299	−	0.0374260i	$-0.0119159\pi$
$972$		0			0
$973$	35.4447			1.13630
$974$		0			0
$975$	0.699484	+	0.699484i	0.0224014	+	0.0224014i
$976$		0			0
$977$		−	22.9940i		−	0.735642i	−0.929897	−	0.367821i	$-0.880104\pi$
$977$		−	22.9940i		−	0.735642i	0.929897	−	0.367821i	$-0.119896\pi$
$978$		0			0
$979$	−9.05734	−	9.05734i	−0.289474	−	0.289474i
$980$		0			0
$981$	−32.8924	+	32.8924i	−1.05017	+	1.05017i
$982$		0			0
$983$	−27.8553	+	27.8553i	−0.888445	+	0.888445i	−0.994374	−	0.105929i	$-0.966218\pi$
$983$	−27.8553	+	27.8553i	−0.888445	+	0.888445i	0.105929	+	0.994374i	$0.466218\pi$
$984$		0			0
$985$	17.4491			0.555975
$986$		0			0
$987$	2.36408			0.0752494
$988$		0			0
$989$	−16.4447	+	16.4447i	−0.522911	+	0.522911i
$990$		0			0
$991$	−2.49805	+	2.49805i	−0.0793531	+	0.0793531i	−0.745669	−	0.666316i	$-0.767870\pi$
$991$	−2.49805	+	2.49805i	−0.0793531	+	0.0793531i	0.666316	+	0.745669i	$0.267870\pi$
$992$		0			0
$993$	1.68026	+	1.68026i	0.0533215	+	0.0533215i
$994$		0			0
$995$		−	17.5725i		−	0.557085i
$996$		0			0
$997$	22.4811	+	22.4811i	0.711983	+	0.711983i	0.966950	−	0.254967i	$-0.0820645\pi$
$997$	22.4811	+	22.4811i	0.711983	+	0.711983i	−0.254967	+	0.966950i	$0.582065\pi$
$998$		0			0
$999$	2.00762			0.0635183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	680.2.bd.b.361.6	yes	20
4.3	odd	2		1360.2.bt.f.1041.5		20
17.13	even	4	inner	680.2.bd.b.81.6	✓	20
68.47	odd	4		1360.2.bt.f.81.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.bd.b.81.6	✓	20	17.13	even	4	inner
680.2.bd.b.361.6	yes	20	1.1	even	1	trivial
1360.2.bt.f.81.5		20	68.47	odd	4
1360.2.bt.f.1041.5		20	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.bd (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.110088 - 0.110088i) q^{3} +(-0.707107 + 0.707107i) q^{5} +(-1.59814 - 1.59814i) q^{7} +2.97576i q^{9} +(2.29246 + 2.29246i) q^{11} -6.35384 q^{13} +0.155688i q^{15} +(-3.14352 - 2.66801i) q^{17} +3.74959i q^{19} -0.351873 q^{21} +(1.61226 + 1.61226i) q^{23} -1.00000i q^{25} +(0.657861 + 0.657861i) q^{27} +(-3.11528 + 3.11528i) q^{29} +(-5.15119 + 5.15119i) q^{31} +0.504746 q^{33} +2.26011 q^{35} +(1.52587 - 1.52587i) q^{37} +(-0.699484 + 0.699484i) q^{39} +(1.63254 + 1.63254i) q^{41} +10.1998i q^{43} +(-2.10418 - 2.10418i) q^{45} -6.71856 q^{47} -1.89191i q^{49} +(-0.639782 + 0.0523473i) q^{51} +2.23744i q^{53} -3.24203 q^{55} +(0.412786 + 0.412786i) q^{57} -5.91043i q^{59} +(-5.35172 - 5.35172i) q^{61} +(4.75568 - 4.75568i) q^{63} +(4.49284 - 4.49284i) q^{65} +9.42323 q^{67} +0.354982 q^{69} +(3.13126 - 3.13126i) q^{71} +(-5.11930 + 5.11930i) q^{73} +(-0.110088 - 0.110088i) q^{75} -7.32733i q^{77} +(4.51283 + 4.51283i) q^{79} -8.78244 q^{81} +2.99110i q^{83} +(4.10937 - 0.336231i) q^{85} +0.685912i q^{87} -3.95093 q^{89} +(10.1543 + 10.1543i) q^{91} +1.13417i q^{93} +(-2.65136 - 2.65136i) q^{95} +(1.15110 - 1.15110i) q^{97} +(-6.82181 + 6.82181i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 4 q^{7} + 12 q^{11} + 4 q^{13} + 4 q^{17} + 16 q^{21} - 4 q^{23} - 16 q^{27} - 4 q^{29} - 4 q^{31} + 16 q^{33} - 8 q^{35} - 8 q^{37} - 8 q^{39} - 8 q^{41} + 4 q^{47} - 4 q^{51} - 8 q^{55}+ \cdots + 52 q^{99}+O(q^{100}) \) 20 * q - 4 * q^3 + 4 * q^7 + 12 * q^11 + 4 * q^13 + 4 * q^17 + 16 * q^21 - 4 * q^23 - 16 * q^27 - 4 * q^29 - 4 * q^31 + 16 * q^33 - 8 * q^35 - 8 * q^37 - 8 * q^39 - 8 * q^41 + 4 * q^47 - 4 * q^51 - 8 * q^55 - 24 * q^57 - 24 * q^61 + 60 * q^63 - 4 * q^65 - 56 * q^67 + 24 * q^69 - 40 * q^71 - 20 * q^73 + 4 * q^75 - 8 * q^79 + 4 * q^81 - 12 * q^85 + 28 * q^89 - 4 * q^95 + 24 * q^97 + 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more