LMFDB - Embedded Newform 20.3.f.a.13.1

Newspace parameters

Level:	$ N $	=	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	=	$ 3 $
Character orbit:	$[\chi]$	=	20.f (of order $4$ and degree $2$)

Newform invariants

Self dual:	No
Analytic conductor:	$0.544960528721$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.1
Root			$-1.00000i$
Character	$\chi$	=	20.13
Dual form			20.3.f.a.17.1

$q$-expansion

$f(q)$	$=$	$q+(1.00000 + 1.00000i) q^{3} +(-3.00000 - 4.00000i) q^{5} +(-7.00000 + 7.00000i) q^{7} -7.00000i q^{9} +O(q^{10})$	\(q+(1.00000 + 1.00000i) q^{3} +(-3.00000 - 4.00000i) q^{5} +(-7.00000 + 7.00000i) q^{7} -7.00000i q^{9} +10.0000 q^{11} +(9.00000 + 9.00000i) q^{13} +(1.00000 - 7.00000i) q^{15} +(1.00000 - 1.00000i) q^{17} +8.00000i q^{19} -14.0000 q^{21} +(-23.0000 - 23.0000i) q^{23} +(-7.00000 + 24.0000i) q^{25} +(16.0000 - 16.0000i) q^{27} +8.00000i q^{29} -14.0000 q^{31} +(10.0000 + 10.0000i) q^{33} +(49.0000 + 7.00000i) q^{35} +(33.0000 - 33.0000i) q^{37} +18.0000i q^{39} -14.0000 q^{41} +(-15.0000 - 15.0000i) q^{43} +(-28.0000 + 21.0000i) q^{45} +(-39.0000 + 39.0000i) q^{47} -49.0000i q^{49} +2.00000 q^{51} +(-7.00000 - 7.00000i) q^{53} +(-30.0000 - 40.0000i) q^{55} +(-8.00000 + 8.00000i) q^{57} +56.0000i q^{59} +42.0000 q^{61} +(49.0000 + 49.0000i) q^{63} +(9.00000 - 63.0000i) q^{65} +(-7.00000 + 7.00000i) q^{67} -46.0000i q^{69} +98.0000 q^{71} +(49.0000 + 49.0000i) q^{73} +(-31.0000 + 17.0000i) q^{75} +(-70.0000 + 70.0000i) q^{77} -96.0000i q^{79} -31.0000 q^{81} +(-63.0000 - 63.0000i) q^{83} +(-7.00000 - 1.00000i) q^{85} +(-8.00000 + 8.00000i) q^{87} +112.000i q^{89} -126.000 q^{91} +(-14.0000 - 14.0000i) q^{93} +(32.0000 - 24.0000i) q^{95} +(33.0000 - 33.0000i) q^{97} -70.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q + 2q^{3} - 6q^{5} - 14q^{7} + O(q^{10}) $	$ 2q + 2q^{3} - 6q^{5} - 14q^{7} + 20q^{11} + 18q^{13} + 2q^{15} + 2q^{17} - 28q^{21} - 46q^{23} - 14q^{25} + 32q^{27} - 28q^{31} + 20q^{33} + 98q^{35} + 66q^{37} - 28q^{41} - 30q^{43} - 56q^{45} - 78q^{47} + 4q^{51} - 14q^{53} - 60q^{55} - 16q^{57} + 84q^{61} + 98q^{63} + 18q^{65} - 14q^{67} + 196q^{71} + 98q^{73} - 62q^{75} - 140q^{77} - 62q^{81} - 126q^{83} - 14q^{85} - 16q^{87} - 252q^{91} - 28q^{93} + 64q^{95} + 66q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character Values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000	+	1.00000i	0.333333	+	0.333333i	0.853851	−	0.520518i	$-0.174261\pi$
$3$	1.00000	+	1.00000i	0.333333	+	0.333333i	−0.520518	+	0.853851i	$0.674261\pi$
$4$		0			0
$5$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$5$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$6$		0			0
$7$	−7.00000	+	7.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$7$	−7.00000	+	7.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$8$		0			0
$9$		−	7.00000i		−	0.777778i
$10$		0			0
$11$	10.0000			0.909091			0.454545	−	0.890724i	$-0.349802\pi$
$11$	10.0000			0.909091			0.454545	+	0.890724i	$0.349802\pi$
$12$		0			0
$13$	9.00000	+	9.00000i	0.692308	+	0.692308i	0.962739	−	0.270432i	$-0.0871664\pi$
$13$	9.00000	+	9.00000i	0.692308	+	0.692308i	−0.270432	+	0.962739i	$0.587166\pi$
$14$		0			0
$15$	1.00000	−	7.00000i	0.0666667	−	0.466667i
$16$		0			0
$17$	1.00000	−	1.00000i	0.0588235	−	0.0588235i	−0.677083	−	0.735907i	$-0.736756\pi$
$17$	1.00000	−	1.00000i	0.0588235	−	0.0588235i	0.735907	+	0.677083i	$0.236756\pi$
$18$		0			0
$19$			8.00000i			0.421053i	0.977588	+	0.210526i	$0.0675178\pi$
$19$			8.00000i			0.421053i	−0.977588	+	0.210526i	$0.932482\pi$
$20$		0			0
$21$	−14.0000			−0.666667
$22$		0			0
$23$	−23.0000	−	23.0000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$23$	−23.0000	−	23.0000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$24$		0			0
$25$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$26$		0			0
$27$	16.0000	−	16.0000i	0.592593	−	0.592593i
$28$		0			0
$29$			8.00000i			0.275862i	0.990442	+	0.137931i	$0.0440452\pi$
$29$			8.00000i			0.275862i	−0.990442	+	0.137931i	$0.955955\pi$
$30$		0			0
$31$	−14.0000			−0.451613			−0.225806	−	0.974172i	$-0.572502\pi$
$31$	−14.0000			−0.451613			−0.225806	+	0.974172i	$0.572502\pi$
$32$		0			0
$33$	10.0000	+	10.0000i	0.303030	+	0.303030i
$34$		0			0
$35$	49.0000	+	7.00000i	1.40000	+	0.200000i
$36$		0			0
$37$	33.0000	−	33.0000i	0.891892	−	0.891892i	−0.102809	−	0.994701i	$-0.532783\pi$
$37$	33.0000	−	33.0000i	0.891892	−	0.891892i	0.994701	+	0.102809i	$0.0327831\pi$
$38$		0			0
$39$			18.0000i			0.461538i
$40$		0			0
$41$	−14.0000			−0.341463			−0.170732	−	0.985318i	$-0.554613\pi$
$41$	−14.0000			−0.341463			−0.170732	+	0.985318i	$0.554613\pi$
$42$		0			0
$43$	−15.0000	−	15.0000i	−0.348837	−	0.348837i	0.510839	−	0.859676i	$-0.329335\pi$
$43$	−15.0000	−	15.0000i	−0.348837	−	0.348837i	−0.859676	+	0.510839i	$0.829335\pi$
$44$		0			0
$45$	−28.0000	+	21.0000i	−0.622222	+	0.466667i
$46$		0			0
$47$	−39.0000	+	39.0000i	−0.829787	+	0.829787i	−0.987487	−	0.157700i	$-0.949592\pi$
$47$	−39.0000	+	39.0000i	−0.829787	+	0.829787i	0.157700	+	0.987487i	$0.449592\pi$
$48$		0			0
$49$		−	49.0000i		−	1.00000i
$50$		0			0
$51$	2.00000			0.0392157
$52$		0			0
$53$	−7.00000	−	7.00000i	−0.132075	−	0.132075i	0.637979	−	0.770054i	$-0.279771\pi$
$53$	−7.00000	−	7.00000i	−0.132075	−	0.132075i	−0.770054	+	0.637979i	$0.779771\pi$
$54$		0			0
$55$	−30.0000	−	40.0000i	−0.545455	−	0.727273i
$56$		0			0
$57$	−8.00000	+	8.00000i	−0.140351	+	0.140351i
$58$		0			0
$59$			56.0000i			0.949153i	0.880214	+	0.474576i	$0.157399\pi$
$59$			56.0000i			0.949153i	−0.880214	+	0.474576i	$0.842601\pi$
$60$		0			0
$61$	42.0000			0.688525			0.344262	−	0.938874i	$-0.388129\pi$
$61$	42.0000			0.688525			0.344262	+	0.938874i	$0.388129\pi$
$62$		0			0
$63$	49.0000	+	49.0000i	0.777778	+	0.777778i
$64$		0			0
$65$	9.00000	−	63.0000i	0.138462	−	0.969231i
$66$		0			0
$67$	−7.00000	+	7.00000i	−0.104478	+	0.104478i	−0.757413	−	0.652936i	$-0.773537\pi$
$67$	−7.00000	+	7.00000i	−0.104478	+	0.104478i	0.652936	+	0.757413i	$0.273537\pi$
$68$		0			0
$69$		−	46.0000i		−	0.666667i
$70$		0			0
$71$	98.0000			1.38028			0.690141	−	0.723675i	$-0.257549\pi$
$71$	98.0000			1.38028			0.690141	+	0.723675i	$0.257549\pi$
$72$		0			0
$73$	49.0000	+	49.0000i	0.671233	+	0.671233i	0.958000	−	0.286767i	$-0.0925807\pi$
$73$	49.0000	+	49.0000i	0.671233	+	0.671233i	−0.286767	+	0.958000i	$0.592581\pi$
$74$		0			0
$75$	−31.0000	+	17.0000i	−0.413333	+	0.226667i
$76$		0			0
$77$	−70.0000	+	70.0000i	−0.909091	+	0.909091i
$78$		0			0
$79$		−	96.0000i		−	1.21519i	−0.794247	−	0.607595i	$-0.792134\pi$
$79$		−	96.0000i		−	1.21519i	0.794247	−	0.607595i	$-0.207866\pi$
$80$		0			0
$81$	−31.0000			−0.382716
$82$		0			0
$83$	−63.0000	−	63.0000i	−0.759036	−	0.759036i	0.217111	−	0.976147i	$-0.430337\pi$
$83$	−63.0000	−	63.0000i	−0.759036	−	0.759036i	−0.976147	+	0.217111i	$0.930337\pi$
$84$		0			0
$85$	−7.00000	−	1.00000i	−0.0823529	−	0.0117647i
$86$		0			0
$87$	−8.00000	+	8.00000i	−0.0919540	+	0.0919540i
$88$		0			0
$89$			112.000i			1.25843i	0.777233	+	0.629213i	$0.216623\pi$
$89$			112.000i			1.25843i	−0.777233	+	0.629213i	$0.783377\pi$
$90$		0			0
$91$	−126.000			−1.38462
$92$		0			0
$93$	−14.0000	−	14.0000i	−0.150538	−	0.150538i
$94$		0			0
$95$	32.0000	−	24.0000i	0.336842	−	0.252632i
$96$		0			0
$97$	33.0000	−	33.0000i	0.340206	−	0.340206i	−0.516239	−	0.856445i	$-0.672668\pi$
$97$	33.0000	−	33.0000i	0.340206	−	0.340206i	0.856445	+	0.516239i	$0.172668\pi$
$98$		0			0
$99$		−	70.0000i		−	0.707071i
$100$		0			0
$101$	26.0000			0.257426			0.128713	−	0.991682i	$-0.458915\pi$
$101$	26.0000			0.257426			0.128713	+	0.991682i	$0.458915\pi$
$102$		0			0
$103$	73.0000	+	73.0000i	0.708738	+	0.708738i	0.966270	−	0.257532i	$-0.0829093\pi$
$103$	73.0000	+	73.0000i	0.708738	+	0.708738i	−0.257532	+	0.966270i	$0.582909\pi$
$104$		0			0
$105$	42.0000	+	56.0000i	0.400000	+	0.533333i
$106$		0			0
$107$	121.000	−	121.000i	1.13084	−	1.13084i	0.140804	−	0.990038i	$-0.455031\pi$
$107$	121.000	−	121.000i	1.13084	−	1.13084i	0.990038	−	0.140804i	$-0.0449686\pi$
$108$		0			0
$109$			136.000i			1.24771i	0.781542	+	0.623853i	$0.214434\pi$
$109$			136.000i			1.24771i	−0.781542	+	0.623853i	$0.785566\pi$
$110$		0			0
$111$	66.0000			0.594595
$112$		0			0
$113$	−127.000	−	127.000i	−1.12389	−	1.12389i	−0.991150	−	0.132743i	$-0.957621\pi$
$113$	−127.000	−	127.000i	−1.12389	−	1.12389i	−0.132743	−	0.991150i	$-0.542379\pi$
$114$		0			0
$115$	−23.0000	+	161.000i	−0.200000	+	1.40000i
$116$		0			0
$117$	63.0000	−	63.0000i	0.538462	−	0.538462i
$118$		0			0
$119$			14.0000i			0.117647i
$120$		0			0
$121$	−21.0000			−0.173554
$122$		0			0
$123$	−14.0000	−	14.0000i	−0.113821	−	0.113821i
$124$		0			0
$125$	117.000	−	44.0000i	0.936000	−	0.352000i
$126$		0			0
$127$	−7.00000	+	7.00000i	−0.0551181	+	0.0551181i	−0.734129	−	0.679010i	$-0.762409\pi$
$127$	−7.00000	+	7.00000i	−0.0551181	+	0.0551181i	0.679010	+	0.734129i	$0.262409\pi$
$128$		0			0
$129$		−	30.0000i		−	0.232558i
$130$		0			0
$131$	−230.000			−1.75573			−0.877863	−	0.478913i	$-0.841031\pi$
$131$	−230.000			−1.75573			−0.877863	+	0.478913i	$0.841031\pi$
$132$		0			0
$133$	−56.0000	−	56.0000i	−0.421053	−	0.421053i
$134$		0			0
$135$	−112.000	−	16.0000i	−0.829630	−	0.118519i
$136$		0			0
$137$	−63.0000	+	63.0000i	−0.459854	+	0.459854i	−0.898607	−	0.438753i	$-0.855420\pi$
$137$	−63.0000	+	63.0000i	−0.459854	+	0.459854i	0.438753	+	0.898607i	$0.355420\pi$
$138$		0			0
$139$			88.0000i			0.633094i	0.948577	+	0.316547i	$0.102523\pi$
$139$			88.0000i			0.633094i	−0.948577	+	0.316547i	$0.897477\pi$
$140$		0			0
$141$	−78.0000			−0.553191
$142$		0			0
$143$	90.0000	+	90.0000i	0.629371	+	0.629371i
$144$		0			0
$145$	32.0000	−	24.0000i	0.220690	−	0.165517i
$146$		0			0
$147$	49.0000	−	49.0000i	0.333333	−	0.333333i
$148$		0			0
$149$		−	168.000i		−	1.12752i	−0.825940	−	0.563758i	$-0.809355\pi$
$149$		−	168.000i		−	1.12752i	0.825940	−	0.563758i	$-0.190645\pi$
$150$		0			0
$151$	130.000			0.860927			0.430464	−	0.902608i	$-0.358350\pi$
$151$	130.000			0.860927			0.430464	+	0.902608i	$0.358350\pi$
$152$		0			0
$153$	−7.00000	−	7.00000i	−0.0457516	−	0.0457516i
$154$		0			0
$155$	42.0000	+	56.0000i	0.270968	+	0.361290i
$156$		0			0
$157$	−63.0000	+	63.0000i	−0.401274	+	0.401274i	−0.878682	−	0.477408i	$-0.841576\pi$
$157$	−63.0000	+	63.0000i	−0.401274	+	0.401274i	0.477408	+	0.878682i	$0.341576\pi$
$158$		0			0
$159$		−	14.0000i		−	0.0880503i
$160$		0			0
$161$	322.000			2.00000
$162$		0			0
$163$	65.0000	+	65.0000i	0.398773	+	0.398773i	0.877800	−	0.479027i	$-0.159010\pi$
$163$	65.0000	+	65.0000i	0.398773	+	0.398773i	−0.479027	+	0.877800i	$0.659010\pi$
$164$		0			0
$165$	10.0000	−	70.0000i	0.0606061	−	0.424242i
$166$		0			0
$167$	−103.000	+	103.000i	−0.616766	+	0.616766i	−0.944701	−	0.327934i	$-0.893648\pi$
$167$	−103.000	+	103.000i	−0.616766	+	0.616766i	0.327934	+	0.944701i	$0.393648\pi$
$168$		0			0
$169$		−	7.00000i		−	0.0414201i
$170$		0			0
$171$	56.0000			0.327485
$172$		0			0
$173$	73.0000	+	73.0000i	0.421965	+	0.421965i	0.885880	−	0.463915i	$-0.153556\pi$
$173$	73.0000	+	73.0000i	0.421965	+	0.421965i	−0.463915	+	0.885880i	$0.653556\pi$
$174$		0			0
$175$	−119.000	−	217.000i	−0.680000	−	1.24000i
$176$		0			0
$177$	−56.0000	+	56.0000i	−0.316384	+	0.316384i
$178$		0			0
$179$		−	56.0000i		−	0.312849i	−0.987690	−	0.156425i	$-0.950003\pi$
$179$		−	56.0000i		−	0.312849i	0.987690	−	0.156425i	$-0.0499968\pi$
$180$		0			0
$181$	−70.0000			−0.386740			−0.193370	−	0.981126i	$-0.561942\pi$
$181$	−70.0000			−0.386740			−0.193370	+	0.981126i	$0.561942\pi$
$182$		0			0
$183$	42.0000	+	42.0000i	0.229508	+	0.229508i
$184$		0			0
$185$	−231.000	−	33.0000i	−1.24865	−	0.178378i
$186$		0			0
$187$	10.0000	−	10.0000i	0.0534759	−	0.0534759i
$188$		0			0
$189$			224.000i			1.18519i
$190$		0			0
$191$	−142.000			−0.743455			−0.371728	−	0.928342i	$-0.621235\pi$
$191$	−142.000			−0.743455			−0.371728	+	0.928342i	$0.621235\pi$
$192$		0			0
$193$	−63.0000	−	63.0000i	−0.326425	−	0.326425i	0.524800	−	0.851225i	$-0.324140\pi$
$193$	−63.0000	−	63.0000i	−0.326425	−	0.326425i	−0.851225	+	0.524800i	$0.824140\pi$
$194$		0			0
$195$	72.0000	−	54.0000i	0.369231	−	0.276923i
$196$		0			0
$197$	−63.0000	+	63.0000i	−0.319797	+	0.319797i	−0.848689	−	0.528892i	$-0.822608\pi$
$197$	−63.0000	+	63.0000i	−0.319797	+	0.319797i	0.528892	+	0.848689i	$0.322608\pi$
$198$		0			0
$199$		−	336.000i		−	1.68844i	−0.535995	−	0.844221i	$-0.680063\pi$
$199$		−	336.000i		−	1.68844i	0.535995	−	0.844221i	$-0.319937\pi$
$200$		0			0
$201$	−14.0000			−0.0696517
$202$		0			0
$203$	−56.0000	−	56.0000i	−0.275862	−	0.275862i
$204$		0			0
$205$	42.0000	+	56.0000i	0.204878	+	0.273171i
$206$		0			0
$207$	−161.000	+	161.000i	−0.777778	+	0.777778i
$208$		0			0
$209$			80.0000i			0.382775i
$210$		0			0
$211$	314.000			1.48815			0.744076	−	0.668095i	$-0.232890\pi$
$211$	314.000			1.48815			0.744076	+	0.668095i	$0.232890\pi$
$212$		0			0
$213$	98.0000	+	98.0000i	0.460094	+	0.460094i
$214$		0			0
$215$	−15.0000	+	105.000i	−0.0697674	+	0.488372i
$216$		0			0
$217$	98.0000	−	98.0000i	0.451613	−	0.451613i
$218$		0			0
$219$			98.0000i			0.447489i
$220$		0			0
$221$	18.0000			0.0814480
$222$		0			0
$223$	−135.000	−	135.000i	−0.605381	−	0.605381i	0.336354	−	0.941736i	$-0.390806\pi$
$223$	−135.000	−	135.000i	−0.605381	−	0.605381i	−0.941736	+	0.336354i	$0.890806\pi$
$224$		0			0
$225$	168.000	+	49.0000i	0.746667	+	0.217778i
$226$		0			0
$227$	281.000	−	281.000i	1.23789	−	1.23789i	0.277022	−	0.960864i	$-0.410653\pi$
$227$	281.000	−	281.000i	1.23789	−	1.23789i	0.960864	−	0.277022i	$-0.0893474\pi$
$228$		0			0
$229$		−	168.000i		−	0.733624i	−0.930295	−	0.366812i	$-0.880449\pi$
$229$		−	168.000i		−	0.733624i	0.930295	−	0.366812i	$-0.119551\pi$
$230$		0			0
$231$	−140.000			−0.606061
$232$		0			0
$233$	273.000	+	273.000i	1.17167	+	1.17167i	0.981811	+	0.189863i	$0.0608045\pi$
$233$	273.000	+	273.000i	1.17167	+	1.17167i	0.189863	+	0.981811i	$0.439196\pi$
$234$		0			0
$235$	273.000	+	39.0000i	1.16170	+	0.165957i
$236$		0			0
$237$	96.0000	−	96.0000i	0.405063	−	0.405063i
$238$		0			0
$239$			288.000i			1.20502i	0.798111	+	0.602510i	$0.205833\pi$
$239$			288.000i			1.20502i	−0.798111	+	0.602510i	$0.794167\pi$
$240$		0			0
$241$	−446.000			−1.85062			−0.925311	−	0.379209i	$-0.876196\pi$
$241$	−446.000			−1.85062			−0.925311	+	0.379209i	$0.876196\pi$
$242$		0			0
$243$	−175.000	−	175.000i	−0.720165	−	0.720165i
$244$		0			0
$245$	−196.000	+	147.000i	−0.800000	+	0.600000i
$246$		0			0
$247$	−72.0000	+	72.0000i	−0.291498	+	0.291498i
$248$		0			0
$249$		−	126.000i		−	0.506024i
$250$		0			0
$251$	−150.000			−0.597610			−0.298805	−	0.954314i	$-0.596588\pi$
$251$	−150.000			−0.597610			−0.298805	+	0.954314i	$0.596588\pi$
$252$		0			0
$253$	−230.000	−	230.000i	−0.909091	−	0.909091i
$254$		0			0
$255$	−6.00000	−	8.00000i	−0.0235294	−	0.0313725i
$256$		0			0
$257$	161.000	−	161.000i	0.626459	−	0.626459i	−0.320716	−	0.947175i	$-0.603924\pi$
$257$	161.000	−	161.000i	0.626459	−	0.626459i	0.947175	+	0.320716i	$0.103924\pi$
$258$		0			0
$259$			462.000i			1.78378i
$260$		0			0
$261$	56.0000			0.214559
$262$		0			0
$263$	−151.000	−	151.000i	−0.574144	−	0.574144i	0.359139	−	0.933284i	$-0.383070\pi$
$263$	−151.000	−	151.000i	−0.574144	−	0.574144i	−0.933284	+	0.359139i	$0.883070\pi$
$264$		0			0
$265$	−7.00000	+	49.0000i	−0.0264151	+	0.184906i
$266$		0			0
$267$	−112.000	+	112.000i	−0.419476	+	0.419476i
$268$		0			0
$269$		−	376.000i		−	1.39777i	−0.715234	−	0.698885i	$-0.753680\pi$
$269$		−	376.000i		−	1.39777i	0.715234	−	0.698885i	$-0.246320\pi$
$270$		0			0
$271$	210.000			0.774908			0.387454	−	0.921889i	$-0.373355\pi$
$271$	210.000			0.774908			0.387454	+	0.921889i	$0.373355\pi$
$272$		0			0
$273$	−126.000	−	126.000i	−0.461538	−	0.461538i
$274$		0			0
$275$	−70.0000	+	240.000i	−0.254545	+	0.872727i
$276$		0			0
$277$	129.000	−	129.000i	0.465704	−	0.465704i	−0.434816	−	0.900520i	$-0.643186\pi$
$277$	129.000	−	129.000i	0.465704	−	0.465704i	0.900520	+	0.434816i	$0.143186\pi$
$278$		0			0
$279$			98.0000i			0.351254i
$280$		0			0
$281$	−174.000			−0.619217			−0.309609	−	0.950864i	$-0.600198\pi$
$281$	−174.000			−0.619217			−0.309609	+	0.950864i	$0.600198\pi$
$282$		0			0
$283$	113.000	+	113.000i	0.399293	+	0.399293i	0.877984	−	0.478690i	$-0.158888\pi$
$283$	113.000	+	113.000i	0.399293	+	0.399293i	−0.478690	+	0.877984i	$0.658888\pi$
$284$		0			0
$285$	56.0000	+	8.00000i	0.196491	+	0.0280702i
$286$		0			0
$287$	98.0000	−	98.0000i	0.341463	−	0.341463i
$288$		0			0
$289$			287.000i			0.993080i
$290$		0			0
$291$	66.0000			0.226804
$292$		0			0
$293$	345.000	+	345.000i	1.17747	+	1.17747i	0.980386	+	0.197089i	$0.0631487\pi$
$293$	345.000	+	345.000i	1.17747	+	1.17747i	0.197089	+	0.980386i	$0.436851\pi$
$294$		0			0
$295$	224.000	−	168.000i	0.759322	−	0.569492i
$296$		0			0
$297$	160.000	−	160.000i	0.538721	−	0.538721i
$298$		0			0
$299$		−	414.000i		−	1.38462i
$300$		0			0
$301$	210.000			0.697674
$302$		0			0
$303$	26.0000	+	26.0000i	0.0858086	+	0.0858086i
$304$		0			0
$305$	−126.000	−	168.000i	−0.413115	−	0.550820i
$306$		0			0
$307$	−327.000	+	327.000i	−1.06515	+	1.06515i	−0.0674220	+	0.997725i	$0.521477\pi$
$307$	−327.000	+	327.000i	−1.06515	+	1.06515i	−0.997725	+	0.0674220i	$0.978523\pi$
$308$		0			0
$309$			146.000i			0.472492i
$310$		0			0
$311$	2.00000			0.00643087			0.00321543	−	0.999995i	$-0.498976\pi$
$311$	2.00000			0.00643087			0.00321543	+	0.999995i	$0.498976\pi$
$312$		0			0
$313$	81.0000	+	81.0000i	0.258786	+	0.258786i	0.824560	−	0.565774i	$-0.191423\pi$
$313$	81.0000	+	81.0000i	0.258786	+	0.258786i	−0.565774	+	0.824560i	$0.691423\pi$
$314$		0			0
$315$	49.0000	−	343.000i	0.155556	−	1.08889i
$316$		0			0
$317$	−159.000	+	159.000i	−0.501577	+	0.501577i	−0.911928	−	0.410351i	$-0.865406\pi$
$317$	−159.000	+	159.000i	−0.501577	+	0.501577i	0.410351	+	0.911928i	$0.365406\pi$
$318$		0			0
$319$			80.0000i			0.250784i
$320$		0			0
$321$	242.000			0.753894
$322$		0			0
$323$	8.00000	+	8.00000i	0.0247678	+	0.0247678i
$324$		0			0
$325$	−279.000	+	153.000i	−0.858462	+	0.470769i
$326$		0			0
$327$	−136.000	+	136.000i	−0.415902	+	0.415902i
$328$		0			0
$329$		−	546.000i		−	1.65957i
$330$		0			0
$331$	−182.000			−0.549849			−0.274924	−	0.961466i	$-0.588653\pi$
$331$	−182.000			−0.549849			−0.274924	+	0.961466i	$0.588653\pi$
$332$		0			0
$333$	−231.000	−	231.000i	−0.693694	−	0.693694i
$334$		0			0
$335$	49.0000	+	7.00000i	0.146269	+	0.0208955i
$336$		0			0
$337$	−447.000	+	447.000i	−1.32641	+	1.32641i	−0.417930	+	0.908479i	$0.637244\pi$
$337$	−447.000	+	447.000i	−1.32641	+	1.32641i	−0.908479	+	0.417930i	$0.862756\pi$
$338$		0			0
$339$		−	254.000i		−	0.749263i
$340$		0			0
$341$	−140.000			−0.410557
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−184.000	+	138.000i	−0.533333	+	0.400000i
$346$		0			0
$347$	25.0000	−	25.0000i	0.0720461	−	0.0720461i	−0.670166	−	0.742212i	$-0.733777\pi$
$347$	25.0000	−	25.0000i	0.0720461	−	0.0720461i	0.742212	+	0.670166i	$0.233777\pi$
$348$		0			0
$349$			200.000i			0.573066i	0.958070	+	0.286533i	$0.0925028\pi$
$349$			200.000i			0.573066i	−0.958070	+	0.286533i	$0.907497\pi$
$350$		0			0
$351$	288.000			0.820513
$352$		0			0
$353$	321.000	+	321.000i	0.909348	+	0.909348i	0.996220	−	0.0868711i	$-0.0276868\pi$
$353$	321.000	+	321.000i	0.909348	+	0.909348i	−0.0868711	+	0.996220i	$0.527687\pi$
$354$		0			0
$355$	−294.000	−	392.000i	−0.828169	−	1.10423i
$356$		0			0
$357$	−14.0000	+	14.0000i	−0.0392157	+	0.0392157i
$358$		0			0
$359$			112.000i			0.311978i	0.987759	+	0.155989i	$0.0498564\pi$
$359$			112.000i			0.311978i	−0.987759	+	0.155989i	$0.950144\pi$
$360$		0			0
$361$	297.000			0.822715
$362$		0			0
$363$	−21.0000	−	21.0000i	−0.0578512	−	0.0578512i
$364$		0			0
$365$	49.0000	−	343.000i	0.134247	−	0.939726i
$366$		0			0
$367$	377.000	−	377.000i	1.02725	−	1.02725i	0.0276297	−	0.999618i	$-0.491204\pi$
$367$	377.000	−	377.000i	1.02725	−	1.02725i	0.999618	−	0.0276297i	$-0.00879594\pi$
$368$		0			0
$369$			98.0000i			0.265583i
$370$		0			0
$371$	98.0000			0.264151
$372$		0			0
$373$	217.000	+	217.000i	0.581769	+	0.581769i	0.935389	−	0.353620i	$-0.115049\pi$
$373$	217.000	+	217.000i	0.581769	+	0.581769i	−0.353620	+	0.935389i	$0.615049\pi$
$374$		0			0
$375$	161.000	+	73.0000i	0.429333	+	0.194667i
$376$		0			0
$377$	−72.0000	+	72.0000i	−0.190981	+	0.190981i
$378$		0			0
$379$			56.0000i			0.147757i	0.997267	+	0.0738786i	$0.0235377\pi$
$379$			56.0000i			0.147757i	−0.997267	+	0.0738786i	$0.976462\pi$
$380$		0			0
$381$	−14.0000			−0.0367454
$382$		0			0
$383$	57.0000	+	57.0000i	0.148825	+	0.148825i	0.777593	−	0.628768i	$-0.216440\pi$
$383$	57.0000	+	57.0000i	0.148825	+	0.148825i	−0.628768	+	0.777593i	$0.716440\pi$
$384$		0			0
$385$	490.000	+	70.0000i	1.27273	+	0.181818i
$386$		0			0
$387$	−105.000	+	105.000i	−0.271318	+	0.271318i
$388$		0			0
$389$			312.000i			0.802057i	0.916066	+	0.401028i	$0.131347\pi$
$389$			312.000i			0.802057i	−0.916066	+	0.401028i	$0.868653\pi$
$390$		0			0
$391$	−46.0000			−0.117647
$392$		0			0
$393$	−230.000	−	230.000i	−0.585242	−	0.585242i
$394$		0			0
$395$	−384.000	+	288.000i	−0.972152	+	0.729114i
$396$		0			0
$397$	193.000	−	193.000i	0.486146	−	0.486146i	−0.420942	−	0.907088i	$-0.638300\pi$
$397$	193.000	−	193.000i	0.486146	−	0.486146i	0.907088	+	0.420942i	$0.138300\pi$
$398$		0			0
$399$		−	112.000i		−	0.280702i
$400$		0			0
$401$	−30.0000			−0.0748130			−0.0374065	−	0.999300i	$-0.511910\pi$
$401$	−30.0000			−0.0748130			−0.0374065	+	0.999300i	$0.511910\pi$
$402$		0			0
$403$	−126.000	−	126.000i	−0.312655	−	0.312655i
$404$		0			0
$405$	93.0000	+	124.000i	0.229630	+	0.306173i
$406$		0			0
$407$	330.000	−	330.000i	0.810811	−	0.810811i
$408$		0			0
$409$		−	432.000i		−	1.05623i	−0.849171	−	0.528117i	$-0.822898\pi$
$409$		−	432.000i		−	1.05623i	0.849171	−	0.528117i	$-0.177102\pi$
$410$		0			0
$411$	−126.000			−0.306569
$412$		0			0
$413$	−392.000	−	392.000i	−0.949153	−	0.949153i
$414$		0			0
$415$	−63.0000	+	441.000i	−0.151807	+	1.06265i
$416$		0			0
$417$	−88.0000	+	88.0000i	−0.211031	+	0.211031i
$418$		0			0
$419$			168.000i			0.400955i	0.979698	+	0.200477i	$0.0642493\pi$
$419$			168.000i			0.400955i	−0.979698	+	0.200477i	$0.935751\pi$
$420$		0			0
$421$	−454.000			−1.07838			−0.539192	−	0.842183i	$-0.681270\pi$
$421$	−454.000			−1.07838			−0.539192	+	0.842183i	$0.681270\pi$
$422$		0			0
$423$	273.000	+	273.000i	0.645390	+	0.645390i
$424$		0			0
$425$	17.0000	+	31.0000i	0.0400000	+	0.0729412i
$426$		0			0
$427$	−294.000	+	294.000i	−0.688525	+	0.688525i
$428$		0			0
$429$			180.000i			0.419580i
$430$		0			0
$431$	−494.000			−1.14617			−0.573086	−	0.819495i	$-0.694254\pi$
$431$	−494.000			−1.14617			−0.573086	+	0.819495i	$0.694254\pi$
$432$		0			0
$433$	−511.000	−	511.000i	−1.18014	−	1.18014i	−0.979708	−	0.200431i	$-0.935766\pi$
$433$	−511.000	−	511.000i	−1.18014	−	1.18014i	−0.200431	−	0.979708i	$-0.564234\pi$
$434$		0			0
$435$	56.0000	+	8.00000i	0.128736	+	0.0183908i
$436$		0			0
$437$	184.000	−	184.000i	0.421053	−	0.421053i
$438$		0			0
$439$		−	176.000i		−	0.400911i	−0.979703	−	0.200456i	$-0.935758\pi$
$439$		−	176.000i		−	0.400911i	0.979703	−	0.200456i	$-0.0642422\pi$
$440$		0			0
$441$	−343.000			−0.777778
$442$		0			0
$443$	177.000	+	177.000i	0.399549	+	0.399549i	0.878074	−	0.478525i	$-0.158828\pi$
$443$	177.000	+	177.000i	0.399549	+	0.399549i	−0.478525	+	0.878074i	$0.658828\pi$
$444$		0			0
$445$	448.000	−	336.000i	1.00674	−	0.755056i
$446$		0			0
$447$	168.000	−	168.000i	0.375839	−	0.375839i
$448$		0			0
$449$			608.000i			1.35412i	0.735928	+	0.677060i	$0.236746\pi$
$449$			608.000i			1.35412i	−0.735928	+	0.677060i	$0.763254\pi$
$450$		0			0
$451$	−140.000			−0.310421
$452$		0			0
$453$	130.000	+	130.000i	0.286976	+	0.286976i
$454$		0			0
$455$	378.000	+	504.000i	0.830769	+	1.10769i
$456$		0			0
$457$	609.000	−	609.000i	1.33260	−	1.33260i	0.429571	−	0.903033i	$-0.358665\pi$
$457$	609.000	−	609.000i	1.33260	−	1.33260i	0.903033	−	0.429571i	$-0.141335\pi$
$458$		0			0
$459$		−	32.0000i		−	0.0697168i
$460$		0			0
$461$	490.000			1.06291			0.531453	−	0.847088i	$-0.321646\pi$
$461$	490.000			1.06291			0.531453	+	0.847088i	$0.321646\pi$
$462$		0			0
$463$	−7.00000	−	7.00000i	−0.0151188	−	0.0151188i	0.699507	−	0.714626i	$-0.253403\pi$
$463$	−7.00000	−	7.00000i	−0.0151188	−	0.0151188i	−0.714626	+	0.699507i	$0.753403\pi$
$464$		0			0
$465$	−14.0000	+	98.0000i	−0.0301075	+	0.210753i
$466$		0			0
$467$	25.0000	−	25.0000i	0.0535332	−	0.0535332i	−0.679833	−	0.733367i	$-0.737948\pi$
$467$	25.0000	−	25.0000i	0.0535332	−	0.0535332i	0.733367	+	0.679833i	$0.237948\pi$
$468$		0			0
$469$		−	98.0000i		−	0.208955i
$470$		0			0
$471$	−126.000			−0.267516
$472$		0			0
$473$	−150.000	−	150.000i	−0.317125	−	0.317125i
$474$		0			0
$475$	−192.000	−	56.0000i	−0.404211	−	0.117895i
$476$		0			0
$477$	−49.0000	+	49.0000i	−0.102725	+	0.102725i
$478$		0			0
$479$			128.000i			0.267223i	0.991034	+	0.133612i	$0.0426575\pi$
$479$			128.000i			0.267223i	−0.991034	+	0.133612i	$0.957343\pi$
$480$		0			0
$481$	594.000			1.23493
$482$		0			0
$483$	322.000	+	322.000i	0.666667	+	0.666667i
$484$		0			0
$485$	−231.000	−	33.0000i	−0.476289	−	0.0680412i
$486$		0			0
$487$	249.000	−	249.000i	0.511294	−	0.511294i	−0.403629	−	0.914923i	$-0.632251\pi$
$487$	249.000	−	249.000i	0.511294	−	0.511294i	0.914923	+	0.403629i	$0.132251\pi$
$488$		0			0
$489$			130.000i			0.265849i
$490$		0			0
$491$	650.000			1.32383			0.661914	−	0.749579i	$-0.269744\pi$
$491$	650.000			1.32383			0.661914	+	0.749579i	$0.269744\pi$
$492$		0			0
$493$	8.00000	+	8.00000i	0.0162272	+	0.0162272i
$494$		0			0
$495$	−280.000	+	210.000i	−0.565657	+	0.424242i
$496$		0			0
$497$	−686.000	+	686.000i	−1.38028	+	1.38028i
$498$		0			0
$499$		−	632.000i		−	1.26653i	−0.773934	−	0.633267i	$-0.781714\pi$
$499$		−	632.000i		−	1.26653i	0.773934	−	0.633267i	$-0.218286\pi$
$500$		0			0
$501$	−206.000			−0.411178
$502$		0			0
$503$	−471.000	−	471.000i	−0.936382	−	0.936382i	0.0617123	−	0.998094i	$-0.480344\pi$
$503$	−471.000	−	471.000i	−0.936382	−	0.936382i	−0.998094	+	0.0617123i	$0.980344\pi$
$504$		0			0
$505$	−78.0000	−	104.000i	−0.154455	−	0.205941i
$506$		0			0
$507$	7.00000	−	7.00000i	0.0138067	−	0.0138067i
$508$		0			0
$509$			8.00000i			0.0157171i	0.999969	+	0.00785855i	$0.00250148\pi$
$509$			8.00000i			0.0157171i	−0.999969	+	0.00785855i	$0.997499\pi$
$510$		0			0
$511$	−686.000			−1.34247
$512$		0			0
$513$	128.000	+	128.000i	0.249513	+	0.249513i
$514$		0			0
$515$	73.0000	−	511.000i	0.141748	−	0.992233i
$516$		0			0
$517$	−390.000	+	390.000i	−0.754352	+	0.754352i
$518$		0			0
$519$			146.000i			0.281310i
$520$		0			0
$521$	−366.000			−0.702495			−0.351248	−	0.936283i	$-0.614242\pi$
$521$	−366.000			−0.702495			−0.351248	+	0.936283i	$0.614242\pi$
$522$		0			0
$523$	273.000	+	273.000i	0.521989	+	0.521989i	0.918172	−	0.396183i	$-0.129665\pi$
$523$	273.000	+	273.000i	0.521989	+	0.521989i	−0.396183	+	0.918172i	$0.629665\pi$
$524$		0			0
$525$	98.0000	−	336.000i	0.186667	−	0.640000i
$526$		0			0
$527$	−14.0000	+	14.0000i	−0.0265655	+	0.0265655i
$528$		0			0
$529$			529.000i			1.00000i
$530$		0			0
$531$	392.000			0.738230
$532$		0			0
$533$	−126.000	−	126.000i	−0.236398	−	0.236398i
$534$		0			0
$535$	−847.000	−	121.000i	−1.58318	−	0.226168i
$536$		0			0
$537$	56.0000	−	56.0000i	0.104283	−	0.104283i
$538$		0			0
$539$		−	490.000i		−	0.909091i
$540$		0			0
$541$	394.000			0.728281			0.364140	−	0.931344i	$-0.381363\pi$
$541$	394.000			0.728281			0.364140	+	0.931344i	$0.381363\pi$
$542$		0			0
$543$	−70.0000	−	70.0000i	−0.128913	−	0.128913i
$544$		0			0
$545$	544.000	−	408.000i	0.998165	−	0.748624i
$546$		0			0
$547$	−231.000	+	231.000i	−0.422303	+	0.422303i	−0.885996	−	0.463693i	$-0.846524\pi$
$547$	−231.000	+	231.000i	−0.422303	+	0.422303i	0.463693	+	0.885996i	$0.346524\pi$
$548$		0			0
$549$		−	294.000i		−	0.535519i
$550$		0			0
$551$	−64.0000			−0.116152
$552$		0			0
$553$	672.000	+	672.000i	1.21519	+	1.21519i
$554$		0			0
$555$	−198.000	−	264.000i	−0.356757	−	0.475676i
$556$		0			0
$557$	−735.000	+	735.000i	−1.31957	+	1.31957i	−0.405453	+	0.914116i	$0.632886\pi$
$557$	−735.000	+	735.000i	−1.31957	+	1.31957i	−0.914116	+	0.405453i	$0.867114\pi$
$558$		0			0
$559$		−	270.000i		−	0.483005i
$560$		0			0
$561$	20.0000			0.0356506
$562$		0			0
$563$	609.000	+	609.000i	1.08171	+	1.08171i	0.996351	+	0.0853545i	$0.0272023\pi$
$563$	609.000	+	609.000i	1.08171	+	1.08171i	0.0853545	+	0.996351i	$0.472798\pi$
$564$		0			0
$565$	−127.000	+	889.000i	−0.224779	+	1.57345i
$566$		0			0
$567$	217.000	−	217.000i	0.382716	−	0.382716i
$568$		0			0
$569$		−	560.000i		−	0.984183i	−0.870544	−	0.492091i	$-0.836233\pi$
$569$		−	560.000i		−	0.984183i	0.870544	−	0.492091i	$-0.163767\pi$
$570$		0			0
$571$	938.000			1.64273			0.821366	−	0.570401i	$-0.193212\pi$
$571$	938.000			1.64273			0.821366	+	0.570401i	$0.193212\pi$
$572$		0			0
$573$	−142.000	−	142.000i	−0.247818	−	0.247818i
$574$		0			0
$575$	713.000	−	391.000i	1.24000	−	0.680000i
$576$		0			0
$577$	97.0000	−	97.0000i	0.168111	−	0.168111i	−0.618038	−	0.786149i	$-0.712072\pi$
$577$	97.0000	−	97.0000i	0.168111	−	0.168111i	0.786149	+	0.618038i	$0.212072\pi$
$578$		0			0
$579$		−	126.000i		−	0.217617i
$580$		0			0
$581$	882.000			1.51807
$582$		0			0
$583$	−70.0000	−	70.0000i	−0.120069	−	0.120069i
$584$		0			0
$585$	−441.000	−	63.0000i	−0.753846	−	0.107692i
$586$		0			0
$587$	−39.0000	+	39.0000i	−0.0664395	+	0.0664395i	−0.739546	−	0.673106i	$-0.764960\pi$
$587$	−39.0000	+	39.0000i	−0.0664395	+	0.0664395i	0.673106	+	0.739546i	$0.264960\pi$
$588$		0			0
$589$		−	112.000i		−	0.190153i
$590$		0			0
$591$	−126.000			−0.213198
$592$		0			0
$593$	−479.000	−	479.000i	−0.807757	−	0.807757i	0.176537	−	0.984294i	$-0.443511\pi$
$593$	−479.000	−	479.000i	−0.807757	−	0.807757i	−0.984294	+	0.176537i	$0.943511\pi$
$594$		0			0
$595$	56.0000	−	42.0000i	0.0941176	−	0.0705882i
$596$		0			0
$597$	336.000	−	336.000i	0.562814	−	0.562814i
$598$		0			0
$599$			1040.00i			1.73623i	0.496366	+	0.868114i	$0.334668\pi$
$599$			1040.00i			1.73623i	−0.496366	+	0.868114i	$0.665332\pi$
$600$		0			0
$601$	−430.000			−0.715474			−0.357737	−	0.933822i	$-0.616452\pi$
$601$	−430.000			−0.715474			−0.357737	+	0.933822i	$0.616452\pi$
$602$		0			0
$603$	49.0000	+	49.0000i	0.0812604	+	0.0812604i
$604$		0			0
$605$	63.0000	+	84.0000i	0.104132	+	0.138843i
$606$		0			0
$607$	−423.000	+	423.000i	−0.696870	+	0.696870i	−0.963734	−	0.266864i	$-0.914012\pi$
$607$	−423.000	+	423.000i	−0.696870	+	0.696870i	0.266864	+	0.963734i	$0.414012\pi$
$608$		0			0
$609$		−	112.000i		−	0.183908i
$610$		0			0
$611$	−702.000			−1.14894
$612$		0			0
$613$	249.000	+	249.000i	0.406199	+	0.406199i	0.880411	−	0.474212i	$-0.157267\pi$
$613$	249.000	+	249.000i	0.406199	+	0.406199i	−0.474212	+	0.880411i	$0.657267\pi$
$614$		0			0
$615$	−14.0000	+	98.0000i	−0.0227642	+	0.159350i
$616$		0			0
$617$	321.000	−	321.000i	0.520259	−	0.520259i	−0.397390	−	0.917650i	$-0.630084\pi$
$617$	321.000	−	321.000i	0.520259	−	0.520259i	0.917650	+	0.397390i	$0.130084\pi$
$618$		0			0
$619$			600.000i			0.969305i	0.874707	+	0.484653i	$0.161054\pi$
$619$			600.000i			0.969305i	−0.874707	+	0.484653i	$0.838946\pi$
$620$		0			0
$621$	−736.000			−1.18519
$622$		0			0
$623$	−784.000	−	784.000i	−1.25843	−	1.25843i
$624$		0			0
$625$	−527.000	−	336.000i	−0.843200	−	0.537600i
$626$		0			0
$627$	−80.0000	+	80.0000i	−0.127592	+	0.127592i
$628$		0			0
$629$		−	66.0000i		−	0.104928i
$630$		0			0
$631$	−638.000			−1.01109			−0.505547	−	0.862799i	$-0.668709\pi$
$631$	−638.000			−1.01109			−0.505547	+	0.862799i	$0.668709\pi$
$632$		0			0
$633$	314.000	+	314.000i	0.496051	+	0.496051i
$634$		0			0
$635$	49.0000	+	7.00000i	0.0771654	+	0.0110236i
$636$		0			0
$637$	441.000	−	441.000i	0.692308	−	0.692308i
$638$		0			0
$639$		−	686.000i		−	1.07355i
$640$		0			0
$641$	482.000			0.751950			0.375975	−	0.926630i	$-0.377308\pi$
$641$	482.000			0.751950			0.375975	+	0.926630i	$0.377308\pi$
$642$		0			0
$643$	33.0000	+	33.0000i	0.0513219	+	0.0513219i	0.732302	−	0.680980i	$-0.238446\pi$
$643$	33.0000	+	33.0000i	0.0513219	+	0.0513219i	−0.680980	+	0.732302i	$0.738446\pi$
$644$		0			0
$645$	−120.000	+	90.0000i	−0.186047	+	0.139535i
$646$		0			0
$647$	−7.00000	+	7.00000i	−0.0108192	+	0.0108192i	−0.712496	−	0.701677i	$-0.752435\pi$
$647$	−7.00000	+	7.00000i	−0.0108192	+	0.0108192i	0.701677	+	0.712496i	$0.252435\pi$
$648$		0			0
$649$			560.000i			0.862866i
$650$		0			0
$651$	196.000			0.301075
$652$		0			0
$653$	−471.000	−	471.000i	−0.721286	−	0.721286i	0.247581	−	0.968867i	$-0.420364\pi$
$653$	−471.000	−	471.000i	−0.721286	−	0.721286i	−0.968867	+	0.247581i	$0.920364\pi$
$654$		0			0
$655$	690.000	+	920.000i	1.05344	+	1.40458i
$656$		0			0
$657$	343.000	−	343.000i	0.522070	−	0.522070i
$658$		0			0
$659$			328.000i			0.497724i	0.968539	+	0.248862i	$0.0800565\pi$
$659$			328.000i			0.497724i	−0.968539	+	0.248862i	$0.919943\pi$
$660$		0			0
$661$	−742.000			−1.12254			−0.561271	−	0.827632i	$-0.689687\pi$
$661$	−742.000			−1.12254			−0.561271	+	0.827632i	$0.689687\pi$
$662$		0			0
$663$	18.0000	+	18.0000i	0.0271493	+	0.0271493i
$664$		0			0
$665$	−56.0000	+	392.000i	−0.0842105	+	0.589474i
$666$		0			0
$667$	184.000	−	184.000i	0.275862	−	0.275862i
$668$		0			0
$669$		−	270.000i		−	0.403587i
$670$		0			0
$671$	420.000			0.625931
$672$		0			0
$673$	−287.000	−	287.000i	−0.426449	−	0.426449i	0.460968	−	0.887417i	$-0.347502\pi$
$673$	−287.000	−	287.000i	−0.426449	−	0.426449i	−0.887417	+	0.460968i	$0.847502\pi$
$674$		0			0
$675$	272.000	+	496.000i	0.402963	+	0.734815i
$676$		0			0
$677$	577.000	−	577.000i	0.852290	−	0.852290i	−0.138125	−	0.990415i	$-0.544108\pi$
$677$	577.000	−	577.000i	0.852290	−	0.852290i	0.990415	+	0.138125i	$0.0441077\pi$
$678$		0			0
$679$			462.000i			0.680412i
$680$		0			0
$681$	562.000			0.825257
$682$		0			0
$683$	−399.000	−	399.000i	−0.584187	−	0.584187i	0.351864	−	0.936051i	$-0.385548\pi$
$683$	−399.000	−	399.000i	−0.584187	−	0.584187i	−0.936051	+	0.351864i	$0.885548\pi$
$684$		0			0
$685$	441.000	+	63.0000i	0.643796	+	0.0919708i
$686$		0			0
$687$	168.000	−	168.000i	0.244541	−	0.244541i
$688$		0			0
$689$		−	126.000i		−	0.182874i
$690$		0			0
$691$	378.000			0.547033			0.273517	−	0.961867i	$-0.411813\pi$
$691$	378.000			0.547033			0.273517	+	0.961867i	$0.411813\pi$
$692$		0			0
$693$	490.000	+	490.000i	0.707071	+	0.707071i
$694$		0			0
$695$	352.000	−	264.000i	0.506475	−	0.379856i
$696$		0			0
$697$	−14.0000	+	14.0000i	−0.0200861	+	0.0200861i
$698$		0			0
$699$			546.000i			0.781116i
$700$		0			0
$701$	−470.000			−0.670471			−0.335235	−	0.942134i	$-0.608816\pi$
$701$	−470.000			−0.670471			−0.335235	+	0.942134i	$0.608816\pi$
$702$		0			0
$703$	264.000	+	264.000i	0.375533	+	0.375533i
$704$		0			0
$705$	234.000	+	312.000i	0.331915	+	0.442553i
$706$		0			0
$707$	−182.000	+	182.000i	−0.257426	+	0.257426i
$708$		0			0
$709$		−	1192.00i		−	1.68124i	−0.541624	−	0.840621i	$-0.682190\pi$
$709$		−	1192.00i		−	1.68124i	0.541624	−	0.840621i	$-0.317810\pi$
$710$		0			0
$711$	−672.000			−0.945148
$712$		0			0
$713$	322.000	+	322.000i	0.451613	+	0.451613i
$714$		0			0
$715$	90.0000	−	630.000i	0.125874	−	0.881119i
$716$		0			0
$717$	−288.000	+	288.000i	−0.401674	+	0.401674i
$718$		0			0
$719$			608.000i			0.845619i	0.906219	+	0.422809i	$0.138956\pi$
$719$			608.000i			0.845619i	−0.906219	+	0.422809i	$0.861044\pi$
$720$		0			0
$721$	−1022.00			−1.41748
$722$		0			0
$723$	−446.000	−	446.000i	−0.616874	−	0.616874i
$724$		0			0
$725$	−192.000	−	56.0000i	−0.264828	−	0.0772414i
$726$		0			0
$727$	441.000	−	441.000i	0.606602	−	0.606602i	−0.335454	−	0.942057i	$-0.608890\pi$
$727$	441.000	−	441.000i	0.606602	−	0.606602i	0.942057	+	0.335454i	$0.108890\pi$
$728$		0			0
$729$		−	71.0000i		−	0.0973937i
$730$		0			0
$731$	−30.0000			−0.0410397
$732$		0			0
$733$	361.000	+	361.000i	0.492497	+	0.492497i	0.909092	−	0.416595i	$-0.136777\pi$
$733$	361.000	+	361.000i	0.492497	+	0.492497i	−0.416595	+	0.909092i	$0.636777\pi$
$734$		0			0
$735$	−343.000	−	49.0000i	−0.466667	−	0.0666667i
$736$		0			0
$737$	−70.0000	+	70.0000i	−0.0949796	+	0.0949796i
$738$		0			0
$739$		−	920.000i		−	1.24493i	−0.782649	−	0.622463i	$-0.786132\pi$
$739$		−	920.000i		−	1.24493i	0.782649	−	0.622463i	$-0.213868\pi$
$740$		0			0
$741$	−144.000			−0.194332
$742$		0			0
$743$	−343.000	−	343.000i	−0.461642	−	0.461642i	0.437551	−	0.899193i	$-0.355846\pi$
$743$	−343.000	−	343.000i	−0.461642	−	0.461642i	−0.899193	+	0.437551i	$0.855846\pi$
$744$		0			0
$745$	−672.000	+	504.000i	−0.902013	+	0.676510i
$746$		0			0
$747$	−441.000	+	441.000i	−0.590361	+	0.590361i
$748$		0			0
$749$			1694.00i			2.26168i
$750$		0			0
$751$	786.000			1.04660			0.523302	−	0.852147i	$-0.324700\pi$
$751$	786.000			1.04660			0.523302	+	0.852147i	$0.324700\pi$
$752$		0			0
$753$	−150.000	−	150.000i	−0.199203	−	0.199203i
$754$		0			0
$755$	−390.000	−	520.000i	−0.516556	−	0.688742i
$756$		0			0
$757$	−63.0000	+	63.0000i	−0.0832232	+	0.0832232i	−0.747493	−	0.664270i	$-0.768743\pi$
$757$	−63.0000	+	63.0000i	−0.0832232	+	0.0832232i	0.664270	+	0.747493i	$0.268743\pi$
$758$		0			0
$759$		−	460.000i		−	0.606061i
$760$		0			0
$761$	−398.000			−0.522996			−0.261498	−	0.965204i	$-0.584216\pi$
$761$	−398.000			−0.522996			−0.261498	+	0.965204i	$0.584216\pi$
$762$		0			0
$763$	−952.000	−	952.000i	−1.24771	−	1.24771i
$764$		0			0
$765$	−7.00000	+	49.0000i	−0.00915033	+	0.0640523i
$766$		0			0
$767$	−504.000	+	504.000i	−0.657106	+	0.657106i
$768$		0			0
$769$		−	704.000i		−	0.915475i	−0.889087	−	0.457737i	$-0.848660\pi$
$769$		−	704.000i		−	0.915475i	0.889087	−	0.457737i	$-0.151340\pi$
$770$		0			0
$771$	322.000			0.417639
$772$		0			0
$773$	825.000	+	825.000i	1.06727	+	1.06727i	0.997568	+	0.0697026i	$0.0222050\pi$
$773$	825.000	+	825.000i	1.06727	+	1.06727i	0.0697026	+	0.997568i	$0.477795\pi$
$774$		0			0
$775$	98.0000	−	336.000i	0.126452	−	0.433548i
$776$		0			0
$777$	−462.000	+	462.000i	−0.594595	+	0.594595i
$778$		0			0
$779$		−	112.000i		−	0.143774i
$780$		0			0
$781$	980.000			1.25480
$782$		0			0
$783$	128.000	+	128.000i	0.163474	+	0.163474i
$784$		0			0
$785$	441.000	+	63.0000i	0.561783	+	0.0802548i
$786$		0			0
$787$	−71.0000	+	71.0000i	−0.0902160	+	0.0902160i	−0.750775	−	0.660559i	$-0.770320\pi$
$787$	−71.0000	+	71.0000i	−0.0902160	+	0.0902160i	0.660559	+	0.750775i	$0.270320\pi$
$788$		0			0
$789$		−	302.000i		−	0.382763i
$790$		0			0
$791$	1778.00			2.24779
$792$		0			0
$793$	378.000	+	378.000i	0.476671	+	0.476671i
$794$		0			0
$795$	−56.0000	+	42.0000i	−0.0704403	+	0.0528302i
$796$		0			0
$797$	33.0000	−	33.0000i	0.0414053	−	0.0414053i	−0.686101	−	0.727506i	$-0.740679\pi$
$797$	33.0000	−	33.0000i	0.0414053	−	0.0414053i	0.727506	+	0.686101i	$0.240679\pi$
$798$		0			0
$799$			78.0000i			0.0976220i
$800$		0			0
$801$	784.000			0.978777
$802$		0			0
$803$	490.000	+	490.000i	0.610212	+	0.610212i
$804$		0			0
$805$	−966.000	−	1288.00i	−1.20000	−	1.60000i
$806$		0			0
$807$	376.000	−	376.000i	0.465923	−	0.465923i
$808$		0			0
$809$		−	368.000i		−

Introduction	Features
Universe	News

Level:	\( N \)	=	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	=	\( 3 \)
Character orbit:	\([\chi]\)	=	20.f (of order \(4\) and degree \(2\))

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{3} +(-3.00000 - 4.00000i) q^{5} +(-7.00000 + 7.00000i) q^{7} -7.00000i q^{9} +O(q^{10})\)	\(q+(1.00000 + 1.00000i) q^{3} +(-3.00000 - 4.00000i) q^{5} +(-7.00000 + 7.00000i) q^{7} -7.00000i q^{9} +10.0000 q^{11} +(9.00000 + 9.00000i) q^{13} +(1.00000 - 7.00000i) q^{15} +(1.00000 - 1.00000i) q^{17} +8.00000i q^{19} -14.0000 q^{21} +(-23.0000 - 23.0000i) q^{23} +(-7.00000 + 24.0000i) q^{25} +(16.0000 - 16.0000i) q^{27} +8.00000i q^{29} -14.0000 q^{31} +(10.0000 + 10.0000i) q^{33} +(49.0000 + 7.00000i) q^{35} +(33.0000 - 33.0000i) q^{37} +18.0000i q^{39} -14.0000 q^{41} +(-15.0000 - 15.0000i) q^{43} +(-28.0000 + 21.0000i) q^{45} +(-39.0000 + 39.0000i) q^{47} -49.0000i q^{49} +2.00000 q^{51} +(-7.00000 - 7.00000i) q^{53} +(-30.0000 - 40.0000i) q^{55} +(-8.00000 + 8.00000i) q^{57} +56.0000i q^{59} +42.0000 q^{61} +(49.0000 + 49.0000i) q^{63} +(9.00000 - 63.0000i) q^{65} +(-7.00000 + 7.00000i) q^{67} -46.0000i q^{69} +98.0000 q^{71} +(49.0000 + 49.0000i) q^{73} +(-31.0000 + 17.0000i) q^{75} +(-70.0000 + 70.0000i) q^{77} -96.0000i q^{79} -31.0000 q^{81} +(-63.0000 - 63.0000i) q^{83} +(-7.00000 - 1.00000i) q^{85} +(-8.00000 + 8.00000i) q^{87} +112.000i q^{89} -126.000 q^{91} +(-14.0000 - 14.0000i) q^{93} +(32.0000 - 24.0000i) q^{95} +(33.0000 - 33.0000i) q^{97} -70.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q + 2q^{3} - 6q^{5} - 14q^{7} + O(q^{10}) \)	\( 2q + 2q^{3} - 6q^{5} - 14q^{7} + 20q^{11} + 18q^{13} + 2q^{15} + 2q^{17} - 28q^{21} - 46q^{23} - 14q^{25} + 32q^{27} - 28q^{31} + 20q^{33} + 98q^{35} + 66q^{37} - 28q^{41} - 30q^{43} - 56q^{45} - 78q^{47} + 4q^{51} - 14q^{53} - 60q^{55} - 16q^{57} + 84q^{61} + 98q^{63} + 18q^{65} - 14q^{67} + 196q^{71} + 98q^{73} - 62q^{75} - 140q^{77} - 62q^{81} - 126q^{83} - 14q^{85} - 16q^{87} - 252q^{91} - 28q^{93} + 64q^{95} + 66q^{97} + O(q^{100}) \)

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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