LMFDB - Embedded newform 1360.2.bt.a.81.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1360,2,Mod(81,1360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1360.81"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1360, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1360 = 2^{4} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1360.bt (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$10.8596546749$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			81.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1360.81
Dual form			1360.2.bt.a.1041.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.70711 - 1.70711i) q^{3} +(0.707107 + 0.707107i) q^{5} +(0.414214 - 0.414214i) q^{7} +2.82843i q^{9} +(-1.00000 + 1.00000i) q^{11} +1.00000 q^{13} -2.41421i q^{15} +(-4.12132 - 0.121320i) q^{17} +2.41421i q^{19} -1.41421 q^{21} +(-2.24264 + 2.24264i) q^{23} +1.00000i q^{25} +(-0.292893 + 0.292893i) q^{27} +(-6.94975 - 6.94975i) q^{29} +(3.70711 + 3.70711i) q^{31} +3.41421 q^{33} +0.585786 q^{35} +(1.58579 + 1.58579i) q^{37} +(-1.70711 - 1.70711i) q^{39} +(-6.65685 + 6.65685i) q^{41} -10.2426i q^{43} +(-2.00000 + 2.00000i) q^{45} -3.24264 q^{47} +6.65685i q^{49} +(6.82843 + 7.24264i) q^{51} -3.48528i q^{53} -1.41421 q^{55} +(4.12132 - 4.12132i) q^{57} +10.8995i q^{59} +(-5.77817 + 5.77817i) q^{61} +(1.17157 + 1.17157i) q^{63} +(0.707107 + 0.707107i) q^{65} -8.82843 q^{67} +7.65685 q^{69} +(8.29289 + 8.29289i) q^{71} +(5.53553 + 5.53553i) q^{73} +(1.70711 - 1.70711i) q^{75} +0.828427i q^{77} +(-8.24264 + 8.24264i) q^{79} +9.48528 q^{81} -9.89949i q^{83} +(-2.82843 - 3.00000i) q^{85} +23.7279i q^{87} +14.6569 q^{89} +(0.414214 - 0.414214i) q^{91} -12.6569i q^{93} +(-1.70711 + 1.70711i) q^{95} +(10.1213 + 10.1213i) q^{97} +(-2.82843 - 2.82843i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{7} - 4 q^{11} + 4 q^{13} - 8 q^{17} + 8 q^{23} - 4 q^{27} - 8 q^{29} + 12 q^{31} + 8 q^{33} + 8 q^{35} + 12 q^{37} - 4 q^{39} - 4 q^{41} - 8 q^{45} + 4 q^{47} + 16 q^{51} + 8 q^{57}+ \cdots + 32 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^7 - 4 * q^11 + 4 * q^13 - 8 * q^17 + 8 * q^23 - 4 * q^27 - 8 * q^29 + 12 * q^31 + 8 * q^33 + 8 * q^35 + 12 * q^37 - 4 * q^39 - 4 * q^41 - 8 * q^45 + 4 * q^47 + 16 * q^51 + 8 * q^57 + 8 * q^61 + 16 * q^63 - 24 * q^67 + 8 * q^69 + 36 * q^71 + 8 * q^73 + 4 * q^75 - 16 * q^79 + 4 * q^81 + 36 * q^89 - 4 * q^91 - 4 * q^95 + 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times$.

$n$	$241$	$341$	$511$	$817$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.70711	−	1.70711i	−0.985599	−	0.985599i	0.0142992	−	0.999898i	$-0.495448\pi$
$3$	−1.70711	−	1.70711i	−0.985599	−	0.985599i	−0.999898	+	0.0142992i	$0.995448\pi$
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$	0.414214	−	0.414214i	0.156558	−	0.156558i	−0.624482	−	0.781040i	$-0.714690\pi$
$7$	0.414214	−	0.414214i	0.156558	−	0.156558i	0.781040	+	0.624482i	$0.214690\pi$
$8$		0			0
$9$			2.82843i			0.942809i
$10$		0			0
$11$	−1.00000	+	1.00000i	−0.301511	+	0.301511i	−0.841605	−	0.540094i	$-0.818389\pi$
$11$	−1.00000	+	1.00000i	−0.301511	+	0.301511i	0.540094	+	0.841605i	$0.318389\pi$
$12$		0			0
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		0			0
$15$		−	2.41421i		−	0.623347i
$16$		0			0
$17$	−4.12132	−	0.121320i	−0.999567	−	0.0294245i
$17$	−4.12132	−	0.121320i	−0.999567	−	0.0294245i
$18$		0			0
$19$			2.41421i			0.553859i	0.960890	+	0.276929i	$0.0893168\pi$
$19$			2.41421i			0.553859i	−0.960890	+	0.276929i	$0.910683\pi$
$20$		0			0
$21$	−1.41421			−0.308607
$22$		0			0
$23$	−2.24264	+	2.24264i	−0.467623	+	0.467623i	−0.901144	−	0.433521i	$-0.857271\pi$
$23$	−2.24264	+	2.24264i	−0.467623	+	0.467623i	0.433521	+	0.901144i	$0.357271\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$	−0.292893	+	0.292893i	−0.0563673	+	0.0563673i
$28$		0			0
$29$	−6.94975	−	6.94975i	−1.29054	−	1.29054i	−0.934455	−	0.356080i	$-0.884113\pi$
$29$	−6.94975	−	6.94975i	−1.29054	−	1.29054i	−0.356080	−	0.934455i	$-0.615887\pi$
$30$		0			0
$31$	3.70711	+	3.70711i	0.665816	+	0.665816i	0.956745	−	0.290929i	$-0.0939642\pi$
$31$	3.70711	+	3.70711i	0.665816	+	0.665816i	−0.290929	+	0.956745i	$0.593964\pi$
$32$		0			0
$33$	3.41421			0.594338
$34$		0			0
$35$	0.585786			0.0990160
$36$		0			0
$37$	1.58579	+	1.58579i	0.260702	+	0.260702i	0.825339	−	0.564637i	$-0.190984\pi$
$37$	1.58579	+	1.58579i	0.260702	+	0.260702i	−0.564637	+	0.825339i	$0.690984\pi$
$38$		0			0
$39$	−1.70711	−	1.70711i	−0.273356	−	0.273356i
$40$		0			0
$41$	−6.65685	+	6.65685i	−1.03963	+	1.03963i	−0.0404442	+	0.999182i	$0.512877\pi$
$41$	−6.65685	+	6.65685i	−1.03963	+	1.03963i	−0.999182	+	0.0404442i	$0.987123\pi$
$42$		0			0
$43$		−	10.2426i		−	1.56199i	−0.624538	−	0.780994i	$-0.714713\pi$
$43$		−	10.2426i		−	1.56199i	0.624538	−	0.780994i	$-0.285287\pi$
$44$		0			0
$45$	−2.00000	+	2.00000i	−0.298142	+	0.298142i
$46$		0			0
$47$	−3.24264			−0.472988			−0.236494	−	0.971633i	$-0.575998\pi$
$47$	−3.24264			−0.472988			−0.236494	+	0.971633i	$0.575998\pi$
$48$		0			0
$49$			6.65685i			0.950979i
$50$		0			0
$51$	6.82843	+	7.24264i	0.956171	+	1.01417i
$52$		0			0
$53$		−	3.48528i		−	0.478740i	−0.970928	−	0.239370i	$-0.923059\pi$
$53$		−	3.48528i		−	0.478740i	0.970928	−	0.239370i	$-0.0769409\pi$
$54$		0			0
$55$	−1.41421			−0.190693
$56$		0			0
$57$	4.12132	−	4.12132i	0.545882	−	0.545882i
$58$		0			0
$59$			10.8995i			1.41899i	0.704709	+	0.709497i	$0.251078\pi$
$59$			10.8995i			1.41899i	−0.704709	+	0.709497i	$0.748922\pi$
$60$		0			0
$61$	−5.77817	+	5.77817i	−0.739819	+	0.739819i	−0.972543	−	0.232723i	$-0.925236\pi$
$61$	−5.77817	+	5.77817i	−0.739819	+	0.739819i	0.232723	+	0.972543i	$0.425236\pi$
$62$		0			0
$63$	1.17157	+	1.17157i	0.147604	+	0.147604i
$64$		0			0
$65$	0.707107	+	0.707107i	0.0877058	+	0.0877058i
$66$		0			0
$67$	−8.82843			−1.07856			−0.539282	−	0.842125i	$-0.681304\pi$
$67$	−8.82843			−1.07856			−0.539282	+	0.842125i	$0.681304\pi$
$68$		0			0
$69$	7.65685			0.921777
$70$		0			0
$71$	8.29289	+	8.29289i	0.984185	+	0.984185i	0.999877	−	0.0156915i	$-0.00499497\pi$
$71$	8.29289	+	8.29289i	0.984185	+	0.984185i	−0.0156915	+	0.999877i	$0.504995\pi$
$72$		0			0
$73$	5.53553	+	5.53553i	0.647885	+	0.647885i	0.952482	−	0.304596i	$-0.0985215\pi$
$73$	5.53553	+	5.53553i	0.647885	+	0.647885i	−0.304596	+	0.952482i	$0.598522\pi$
$74$		0			0
$75$	1.70711	−	1.70711i	0.197120	−	0.197120i
$76$		0			0
$77$			0.828427i			0.0944080i
$78$		0			0
$79$	−8.24264	+	8.24264i	−0.927370	+	0.927370i	−0.997535	−	0.0701658i	$-0.977647\pi$
$79$	−8.24264	+	8.24264i	−0.927370	+	0.927370i	0.0701658	+	0.997535i	$0.477647\pi$
$80$		0			0
$81$	9.48528			1.05392
$82$		0			0
$83$		−	9.89949i		−	1.08661i	−0.839535	−	0.543305i	$-0.817173\pi$
$83$		−	9.89949i		−	1.08661i	0.839535	−	0.543305i	$-0.182827\pi$
$84$		0			0
$85$	−2.82843	−	3.00000i	−0.306786	−	0.325396i
$86$		0			0
$87$			23.7279i			2.54390i
$88$		0			0
$89$	14.6569			1.55362			0.776812	−	0.629733i	$-0.216836\pi$
$89$	14.6569			1.55362			0.776812	+	0.629733i	$0.216836\pi$
$90$		0			0
$91$	0.414214	−	0.414214i	0.0434214	−	0.0434214i
$92$		0			0
$93$		−	12.6569i		−	1.31245i
$94$		0			0
$95$	−1.70711	+	1.70711i	−0.175145	+	0.175145i
$96$		0			0
$97$	10.1213	+	10.1213i	1.02766	+	1.02766i	0.999606	+	0.0280581i	$0.00893234\pi$
$97$	10.1213	+	10.1213i	1.02766	+	1.02766i	0.0280581	+	0.999606i	$0.491068\pi$
$98$		0			0
$99$	−2.82843	−	2.82843i	−0.284268	−	0.284268i
$100$		0			0
$101$	−4.34315			−0.432159			−0.216080	−	0.976376i	$-0.569327\pi$
$101$	−4.34315			−0.432159			−0.216080	+	0.976376i	$0.569327\pi$
$102$		0			0
$103$	2.34315			0.230877			0.115439	−	0.993315i	$-0.463173\pi$
$103$	2.34315			0.230877			0.115439	+	0.993315i	$0.463173\pi$
$104$		0			0
$105$	−1.00000	−	1.00000i	−0.0975900	−	0.0975900i
$106$		0			0
$107$	8.82843	+	8.82843i	0.853476	+	0.853476i	0.990560	−	0.137083i	$-0.0437728\pi$
$107$	8.82843	+	8.82843i	0.853476	+	0.853476i	−0.137083	+	0.990560i	$0.543773\pi$
$108$		0			0
$109$	−8.36396	+	8.36396i	−0.801122	+	0.801122i	−0.983271	−	0.182149i	$-0.941695\pi$
$109$	−8.36396	+	8.36396i	−0.801122	+	0.801122i	0.182149	+	0.983271i	$0.441695\pi$
$110$		0			0
$111$		−	5.41421i		−	0.513894i
$112$		0			0
$113$	2.46447	−	2.46447i	0.231837	−	0.231837i	−0.581622	−	0.813459i	$-0.697582\pi$
$113$	2.46447	−	2.46447i	0.231837	−	0.231837i	0.813459	+	0.581622i	$0.197582\pi$
$114$		0			0
$115$	−3.17157			−0.295751
$116$		0			0
$117$			2.82843i			0.261488i
$118$		0			0
$119$	−1.75736	+	1.65685i	−0.161097	+	0.151884i
$120$		0			0
$121$			9.00000i			0.818182i
$122$		0			0
$123$	22.7279			2.04931
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$			7.72792i			0.685742i	0.939382	+	0.342871i	$0.111399\pi$
$127$			7.72792i			0.685742i	−0.939382	+	0.342871i	$0.888601\pi$
$128$		0			0
$129$	−17.4853	+	17.4853i	−1.53949	+	1.53949i
$130$		0			0
$131$	−14.0711	−	14.0711i	−1.22939	−	1.22939i	−0.964193	−	0.265202i	$-0.914561\pi$
$131$	−14.0711	−	14.0711i	−1.22939	−	1.22939i	−0.265202	−	0.964193i	$-0.585439\pi$
$132$		0			0
$133$	1.00000	+	1.00000i	0.0867110	+	0.0867110i
$134$		0			0
$135$	−0.414214			−0.0356498
$136$		0			0
$137$	−16.7279			−1.42916			−0.714581	−	0.699552i	$-0.753383\pi$
$137$	−16.7279			−1.42916			−0.714581	+	0.699552i	$0.753383\pi$
$138$		0			0
$139$	0.656854	+	0.656854i	0.0557137	+	0.0557137i	0.734415	−	0.678701i	$-0.237457\pi$
$139$	0.656854	+	0.656854i	0.0557137	+	0.0557137i	−0.678701	+	0.734415i	$0.737457\pi$
$140$		0			0
$141$	5.53553	+	5.53553i	0.466176	+	0.466176i
$142$		0			0
$143$	−1.00000	+	1.00000i	−0.0836242	+	0.0836242i
$144$		0			0
$145$		−	9.82843i		−	0.816206i
$146$		0			0
$147$	11.3640	−	11.3640i	0.937284	−	0.937284i
$148$		0			0
$149$	−1.75736			−0.143968			−0.0719842	−	0.997406i	$-0.522933\pi$
$149$	−1.75736			−0.143968			−0.0719842	+	0.997406i	$0.522933\pi$
$150$		0			0
$151$		−	4.82843i		−	0.392932i	−0.980511	−	0.196466i	$-0.937053\pi$
$151$		−	4.82843i		−	0.392932i	0.980511	−	0.196466i	$-0.0629465\pi$
$152$		0			0
$153$	0.343146	−	11.6569i	0.0277417	−	0.942401i
$154$		0			0
$155$			5.24264i			0.421099i
$156$		0			0
$157$	2.82843			0.225733			0.112867	−	0.993610i	$-0.463997\pi$
$157$	2.82843			0.225733			0.112867	+	0.993610i	$0.463997\pi$
$158$		0			0
$159$	−5.94975	+	5.94975i	−0.471846	+	0.471846i
$160$		0			0
$161$			1.85786i			0.146420i
$162$		0			0
$163$	−0.343146	+	0.343146i	−0.0268772	+	0.0268772i	−0.720418	−	0.693540i	$-0.756050\pi$
$163$	−0.343146	+	0.343146i	−0.0268772	+	0.0268772i	0.693540	+	0.720418i	$0.256050\pi$
$164$		0			0
$165$	2.41421	+	2.41421i	0.187946	+	0.187946i
$166$		0			0
$167$	−10.7279	−	10.7279i	−0.830152	−	0.830152i	0.157386	−	0.987537i	$-0.449693\pi$
$167$	−10.7279	−	10.7279i	−0.830152	−	0.830152i	−0.987537	+	0.157386i	$0.949693\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$	−6.82843			−0.522183
$172$		0			0
$173$	0.928932	+	0.928932i	0.0706254	+	0.0706254i	0.741537	−	0.670912i	$-0.234097\pi$
$173$	0.928932	+	0.928932i	0.0706254	+	0.0706254i	−0.670912	+	0.741537i	$0.734097\pi$
$174$		0			0
$175$	0.414214	+	0.414214i	0.0313116	+	0.0313116i
$176$		0			0
$177$	18.6066	−	18.6066i	1.39856	−	1.39856i
$178$		0			0
$179$		−	6.82843i		−	0.510381i	−0.966891	−	0.255190i	$-0.917862\pi$
$179$		−	6.82843i		−	0.510381i	0.966891	−	0.255190i	$-0.0821381\pi$
$180$		0			0
$181$	6.82843	−	6.82843i	0.507553	−	0.507553i	−0.406222	−	0.913775i	$-0.633154\pi$
$181$	6.82843	−	6.82843i	0.507553	−	0.507553i	0.913775	+	0.406222i	$0.133154\pi$
$182$		0			0
$183$	19.7279			1.45833
$184$		0			0
$185$			2.24264i			0.164882i
$186$		0			0
$187$	4.24264	−	4.00000i	0.310253	−	0.292509i
$188$		0			0
$189$			0.242641i			0.0176495i
$190$		0			0
$191$	−0.242641			−0.0175569			−0.00877843	−	0.999961i	$-0.502794\pi$
$191$	−0.242641			−0.0175569			−0.00877843	+	0.999961i	$0.502794\pi$
$192$		0			0
$193$	2.82843	−	2.82843i	0.203595	−	0.203595i	−0.597944	−	0.801538i	$-0.704015\pi$
$193$	2.82843	−	2.82843i	0.203595	−	0.203595i	0.801538	+	0.597944i	$0.204015\pi$
$194$		0			0
$195$		−	2.41421i		−	0.172885i
$196$		0			0
$197$	−18.0711	+	18.0711i	−1.28751	+	1.28751i	−0.351216	+	0.936295i	$0.614232\pi$
$197$	−18.0711	+	18.0711i	−1.28751	+	1.28751i	−0.936295	+	0.351216i	$0.885768\pi$
$198$		0			0
$199$	8.05025	+	8.05025i	0.570667	+	0.570667i	0.932315	−	0.361648i	$-0.117786\pi$
$199$	8.05025	+	8.05025i	0.570667	+	0.570667i	−0.361648	+	0.932315i	$0.617786\pi$
$200$		0			0
$201$	15.0711	+	15.0711i	1.06303	+	1.06303i
$202$		0			0
$203$	−5.75736			−0.404087
$204$		0			0
$205$	−9.41421			−0.657517
$206$		0			0
$207$	−6.34315	−	6.34315i	−0.440879	−	0.440879i
$208$		0			0
$209$	−2.41421	−	2.41421i	−0.166995	−	0.166995i
$210$		0			0
$211$	−3.41421	+	3.41421i	−0.235044	+	0.235044i	−0.814794	−	0.579750i	$-0.803150\pi$
$211$	−3.41421	+	3.41421i	−0.235044	+	0.235044i	0.579750	+	0.814794i	$0.303150\pi$
$212$		0			0
$213$		−	28.3137i		−	1.94002i
$214$		0			0
$215$	7.24264	−	7.24264i	0.493944	−	0.493944i
$216$		0			0
$217$	3.07107			0.208478
$218$		0			0
$219$		−	18.8995i		−	1.27711i
$220$		0			0
$221$	−4.12132	−	0.121320i	−0.277230	−	0.00816089i
$222$		0			0
$223$		−	25.3848i		−	1.69989i	−0.526871	−	0.849945i	$-0.676635\pi$
$223$		−	25.3848i		−	1.69989i	0.526871	−	0.849945i	$-0.323365\pi$
$224$		0			0
$225$	−2.82843			−0.188562
$226$		0			0
$227$	14.5355	−	14.5355i	0.964757	−	0.964757i	−0.0346425	−	0.999400i	$-0.511029\pi$
$227$	14.5355	−	14.5355i	0.964757	−	0.964757i	0.999400	+	0.0346425i	$0.0110293\pi$
$228$		0			0
$229$		−	16.2426i		−	1.07334i	−0.843791	−	0.536672i	$-0.819681\pi$
$229$		−	16.2426i		−	1.07334i	0.843791	−	0.536672i	$-0.180319\pi$
$230$		0			0
$231$	1.41421	−	1.41421i	0.0930484	−	0.0930484i
$232$		0			0
$233$	−8.36396	−	8.36396i	−0.547941	−	0.547941i	0.377904	−	0.925845i	$-0.376645\pi$
$233$	−8.36396	−	8.36396i	−0.547941	−	0.547941i	−0.925845	+	0.377904i	$0.876645\pi$
$234$		0			0
$235$	−2.29289	−	2.29289i	−0.149572	−	0.149572i
$236$		0			0
$237$	28.1421			1.82803
$238$		0			0
$239$	−27.4142			−1.77328			−0.886639	−	0.462462i	$-0.846966\pi$
$239$	−27.4142			−1.77328			−0.886639	+	0.462462i	$0.846966\pi$
$240$		0			0
$241$	−19.8284	−	19.8284i	−1.27726	−	1.27726i	−0.942194	−	0.335067i	$-0.891241\pi$
$241$	−19.8284	−	19.8284i	−1.27726	−	1.27726i	−0.335067	−	0.942194i	$-0.608759\pi$
$242$		0			0
$243$	−15.3137	−	15.3137i	−0.982375	−	0.982375i
$244$		0			0
$245$	−4.70711	+	4.70711i	−0.300726	+	0.300726i
$246$		0			0
$247$			2.41421i			0.153613i
$248$		0			0
$249$	−16.8995	+	16.8995i	−1.07096	+	1.07096i
$250$		0			0
$251$	2.14214			0.135210			0.0676052	−	0.997712i	$-0.478464\pi$
$251$	2.14214			0.135210			0.0676052	+	0.997712i	$0.478464\pi$
$252$		0			0
$253$		−	4.48528i		−	0.281987i
$254$		0			0
$255$	−0.292893	+	9.94975i	−0.0183417	+	0.623077i
$256$		0			0
$257$		−	22.5858i		−	1.40886i	−0.709772	−	0.704431i	$-0.751202\pi$
$257$		−	22.5858i		−	1.40886i	0.709772	−	0.704431i	$-0.248798\pi$
$258$		0			0
$259$	1.31371			0.0816299
$260$		0			0
$261$	19.6569	−	19.6569i	1.21673	−	1.21673i
$262$		0			0
$263$			29.7279i			1.83310i	0.399918	+	0.916551i	$0.369039\pi$
$263$			29.7279i			1.83310i	−0.399918	+	0.916551i	$0.630961\pi$
$264$		0			0
$265$	2.46447	−	2.46447i	0.151391	−	0.151391i
$266$		0			0
$267$	−25.0208	−	25.0208i	−1.53125	−	1.53125i
$268$		0			0
$269$	−2.46447	−	2.46447i	−0.150261	−	0.150261i	0.627974	−	0.778235i	$-0.283885\pi$
$269$	−2.46447	−	2.46447i	−0.150261	−	0.150261i	−0.778235	+	0.627974i	$0.783885\pi$
$270$		0			0
$271$	6.34315			0.385319			0.192659	−	0.981266i	$-0.438289\pi$
$271$	6.34315			0.385319			0.192659	+	0.981266i	$0.438289\pi$
$272$		0			0
$273$	−1.41421			−0.0855921
$274$		0			0
$275$	−1.00000	−	1.00000i	−0.0603023	−	0.0603023i
$276$		0			0
$277$	5.89949	+	5.89949i	0.354466	+	0.354466i	0.861768	−	0.507302i	$-0.169357\pi$
$277$	5.89949	+	5.89949i	0.354466	+	0.354466i	−0.507302	+	0.861768i	$0.669357\pi$
$278$		0			0
$279$	−10.4853	+	10.4853i	−0.627737	+	0.627737i
$280$		0			0
$281$		−	1.48528i		−	0.0886045i	−0.999018	−	0.0443022i	$-0.985894\pi$
$281$		−	1.48528i		−	0.0886045i	0.999018	−	0.0443022i	$-0.0141065\pi$
$282$		0			0
$283$	−6.77817	+	6.77817i	−0.402921	+	0.402921i	−0.879261	−	0.476340i	$-0.841963\pi$
$283$	−6.77817	+	6.77817i	−0.402921	+	0.402921i	0.476340	+	0.879261i	$0.341963\pi$
$284$		0			0
$285$	5.82843			0.345246
$286$		0			0
$287$			5.51472i			0.325524i
$288$		0			0
$289$	16.9706	+	1.00000i	0.998268	+	0.0588235i
$290$		0			0
$291$		−	34.5563i		−	2.02573i
$292$		0			0
$293$	5.48528			0.320454			0.160227	−	0.987080i	$-0.448777\pi$
$293$	5.48528			0.320454			0.160227	+	0.987080i	$0.448777\pi$
$294$		0			0
$295$	−7.70711	+	7.70711i	−0.448725	+	0.448725i
$296$		0			0
$297$		−	0.585786i		−	0.0339908i
$298$		0			0
$299$	−2.24264	+	2.24264i	−0.129695	+	0.129695i
$300$		0			0
$301$	−4.24264	−	4.24264i	−0.244542	−	0.244542i
$302$		0			0
$303$	7.41421	+	7.41421i	0.425935	+	0.425935i
$304$		0			0
$305$	−8.17157			−0.467903
$306$		0			0
$307$	−9.89949			−0.564994			−0.282497	−	0.959268i	$-0.591163\pi$
$307$	−9.89949			−0.564994			−0.282497	+	0.959268i	$0.591163\pi$
$308$		0			0
$309$	−4.00000	−	4.00000i	−0.227552	−	0.227552i
$310$		0			0
$311$	2.58579	+	2.58579i	0.146626	+	0.146626i	0.776609	−	0.629983i	$-0.216938\pi$
$311$	2.58579	+	2.58579i	0.146626	+	0.146626i	−0.629983	+	0.776609i	$0.716938\pi$
$312$		0			0
$313$	6.48528	−	6.48528i	0.366570	−	0.366570i	−0.499655	−	0.866225i	$-0.666540\pi$
$313$	6.48528	−	6.48528i	0.366570	−	0.366570i	0.866225	+	0.499655i	$0.166540\pi$
$314$		0			0
$315$			1.65685i			0.0933532i
$316$		0			0
$317$	−24.4853	+	24.4853i	−1.37523	+	1.37523i	−0.522735	+	0.852495i	$0.675088\pi$
$317$	−24.4853	+	24.4853i	−1.37523	+	1.37523i	−0.852495	+	0.522735i	$0.824912\pi$
$318$		0			0
$319$	13.8995			0.778222
$320$		0			0
$321$		−	30.1421i		−	1.68237i
$322$		0			0
$323$	0.292893	−	9.94975i	0.0162970	−	0.553619i
$324$		0			0
$325$			1.00000i			0.0554700i
$326$		0			0
$327$	28.5563			1.57917
$328$		0			0
$329$	−1.34315	+	1.34315i	−0.0740500	+	0.0740500i
$330$		0			0
$331$			8.55635i			0.470299i	0.971959	+	0.235150i	$0.0755581\pi$
$331$			8.55635i			0.470299i	−0.971959	+	0.235150i	$0.924442\pi$
$332$		0			0
$333$	−4.48528	+	4.48528i	−0.245792	+	0.245792i
$334$		0			0
$335$	−6.24264	−	6.24264i	−0.341072	−	0.341072i
$336$		0			0
$337$	−21.4350	−	21.4350i	−1.16764	−	1.16764i	−0.982761	−	0.184879i	$-0.940811\pi$
$337$	−21.4350	−	21.4350i	−1.16764	−	1.16764i	−0.184879	−	0.982761i	$-0.559189\pi$
$338$		0			0
$339$	−8.41421			−0.456997
$340$		0			0
$341$	−7.41421			−0.401502
$342$		0			0
$343$	5.65685	+	5.65685i	0.305441	+	0.305441i
$344$		0			0
$345$	5.41421	+	5.41421i	0.291491	+	0.291491i
$346$		0			0
$347$	10.4350	−	10.4350i	0.560182	−	0.560182i	−0.369177	−	0.929359i	$-0.620360\pi$
$347$	10.4350	−	10.4350i	0.560182	−	0.560182i	0.929359	+	0.369177i	$0.120360\pi$
$348$		0			0
$349$			8.97056i			0.480183i	0.970750	+	0.240092i	$0.0771775\pi$
$349$			8.97056i			0.480183i	−0.970750	+	0.240092i	$0.922823\pi$
$350$		0			0
$351$	−0.292893	+	0.292893i	−0.0156335	+	0.0156335i
$352$		0			0
$353$	5.31371			0.282820			0.141410	−	0.989951i	$-0.454836\pi$
$353$	5.31371			0.282820			0.141410	+	0.989951i	$0.454836\pi$
$354$		0			0
$355$			11.7279i			0.622453i
$356$		0			0
$357$	5.82843	+	0.171573i	0.308473	+	0.00908060i
$358$		0			0
$359$			12.3848i			0.653643i	0.945086	+	0.326822i	$0.105978\pi$
$359$			12.3848i			0.653643i	−0.945086	+	0.326822i	$0.894022\pi$
$360$		0			0
$361$	13.1716			0.693241
$362$		0			0
$363$	15.3640	−	15.3640i	0.806399	−	0.806399i
$364$		0			0
$365$			7.82843i			0.409759i
$366$		0			0
$367$	−5.55635	+	5.55635i	−0.290039	+	0.290039i	−0.837096	−	0.547057i	$-0.815748\pi$
$367$	−5.55635	+	5.55635i	−0.290039	+	0.290039i	0.547057	+	0.837096i	$0.315748\pi$
$368$		0			0
$369$	−18.8284	−	18.8284i	−0.980169	−	0.980169i
$370$		0			0
$371$	−1.44365	−	1.44365i	−0.0749506	−	0.0749506i
$372$		0			0
$373$	24.2843			1.25739			0.628696	−	0.777651i	$-0.283589\pi$
$373$	24.2843			1.25739			0.628696	+	0.777651i	$0.283589\pi$
$374$		0			0
$375$	2.41421			0.124669
$376$		0			0
$377$	−6.94975	−	6.94975i	−0.357930	−	0.357930i
$378$		0			0
$379$	4.00000	+	4.00000i	0.205466	+	0.205466i	0.802337	−	0.596871i	$-0.203590\pi$
$379$	4.00000	+	4.00000i	0.205466	+	0.205466i	−0.596871	+	0.802337i	$0.703590\pi$
$380$		0			0
$381$	13.1924	−	13.1924i	0.675867	−	0.675867i
$382$		0			0
$383$		−	22.2132i		−	1.13504i	−0.823359	−	0.567521i	$-0.807903\pi$
$383$		−	22.2132i		−	1.13504i	0.823359	−	0.567521i	$-0.192097\pi$
$384$		0			0
$385$	−0.585786	+	0.585786i	−0.0298544	+	0.0298544i
$386$		0			0
$387$	28.9706			1.47266
$388$		0			0
$389$		−	10.6274i		−	0.538831i	−0.963024	−	0.269416i	$-0.913169\pi$
$389$		−	10.6274i		−	0.538831i	0.963024	−	0.269416i	$-0.0868306\pi$
$390$		0			0
$391$	9.51472	−	8.97056i	0.481180	−	0.453661i
$392$		0			0
$393$			48.0416i			2.42338i
$394$		0			0
$395$	−11.6569			−0.586520
$396$		0			0
$397$	−16.4853	+	16.4853i	−0.827373	+	0.827373i	−0.987153	−	0.159780i	$-0.948921\pi$
$397$	−16.4853	+	16.4853i	−0.827373	+	0.827373i	0.159780	+	0.987153i	$0.448921\pi$
$398$		0			0
$399$		−	3.41421i		−	0.170924i
$400$		0			0
$401$	14.8284	−	14.8284i	0.740496	−	0.740496i	−0.232177	−	0.972674i	$-0.574585\pi$
$401$	14.8284	−	14.8284i	0.740496	−	0.740496i	0.972674	+	0.232177i	$0.0745849\pi$
$402$		0			0
$403$	3.70711	+	3.70711i	0.184664	+	0.184664i
$404$		0			0
$405$	6.70711	+	6.70711i	0.333279	+	0.333279i
$406$		0			0
$407$	−3.17157			−0.157209
$408$		0			0
$409$	11.4853			0.567911			0.283955	−	0.958838i	$-0.408353\pi$
$409$	11.4853			0.567911			0.283955	+	0.958838i	$0.408353\pi$
$410$		0			0
$411$	28.5563	+	28.5563i	1.40858	+	1.40858i
$412$		0			0
$413$	4.51472	+	4.51472i	0.222155	+	0.222155i
$414$		0			0
$415$	7.00000	−	7.00000i	0.343616	−	0.343616i
$416$		0			0
$417$		−	2.24264i		−	0.109823i
$418$		0			0
$419$	−9.75736	+	9.75736i	−0.476678	+	0.476678i	−0.904068	−	0.427389i	$-0.859433\pi$
$419$	−9.75736	+	9.75736i	−0.476678	+	0.476678i	0.427389	+	0.904068i	$0.359433\pi$
$420$		0			0
$421$	22.9289			1.11749			0.558744	−	0.829340i	$-0.311283\pi$
$421$	22.9289			1.11749			0.558744	+	0.829340i	$0.311283\pi$
$422$		0			0
$423$		−	9.17157i		−	0.445937i
$424$		0			0
$425$	0.121320	−	4.12132i	0.00588490	−	0.199913i
$426$		0			0
$427$			4.78680i			0.231649i
$428$		0			0
$429$	3.41421			0.164840
$430$		0			0
$431$	−16.0000	+	16.0000i	−0.770693	+	0.770693i	−0.978228	−	0.207535i	$-0.933456\pi$
$431$	−16.0000	+	16.0000i	−0.770693	+	0.770693i	0.207535	+	0.978228i	$0.433456\pi$
$432$		0			0
$433$		−	0.928932i		−	0.0446416i	−0.999751	−	0.0223208i	$-0.992894\pi$
$433$		−	0.928932i		−	0.0446416i	0.999751	−	0.0223208i	$-0.00710553\pi$
$434$		0			0
$435$	−16.7782	+	16.7782i	−0.804452	+	0.804452i
$436$		0			0
$437$	−5.41421	−	5.41421i	−0.258997	−	0.258997i
$438$		0			0
$439$	22.6274	+	22.6274i	1.07995	+	1.07995i	0.996513	+	0.0834344i	$0.0265889\pi$
$439$	22.6274	+	22.6274i	1.07995	+	1.07995i	0.0834344	+	0.996513i	$0.473411\pi$
$440$		0			0
$441$	−18.8284			−0.896592
$442$		0			0
$443$	20.0000			0.950229			0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000			0.950229			0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$	10.3640	+	10.3640i	0.491299	+	0.491299i
$446$		0			0
$447$	3.00000	+	3.00000i	0.141895	+	0.141895i
$448$		0			0
$449$	−29.3848	+	29.3848i	−1.38675	+	1.38675i	−0.554709	+	0.832045i	$0.687170\pi$
$449$	−29.3848	+	29.3848i	−1.38675	+	1.38675i	−0.832045	+	0.554709i	$0.812830\pi$
$450$		0			0
$451$		−	13.3137i		−	0.626918i
$452$		0			0
$453$	−8.24264	+	8.24264i	−0.387273	+	0.387273i
$454$		0			0
$455$	0.585786			0.0274621
$456$		0			0
$457$		−	27.5563i		−	1.28903i	−0.764591	−	0.644516i	$-0.777059\pi$
$457$		−	27.5563i		−	1.28903i	0.764591	−	0.644516i	$-0.222941\pi$
$458$		0			0
$459$	1.24264	−	1.17157i	0.0580015	−	0.0546843i
$460$		0			0
$461$		−	6.97056i		−	0.324651i	−0.986737	−	0.162326i	$-0.948100\pi$
$461$		−	6.97056i		−	0.324651i	0.986737	−	0.162326i	$-0.0518995\pi$
$462$		0			0
$463$	1.10051			0.0511448			0.0255724	−	0.999673i	$-0.491859\pi$
$463$	1.10051			0.0511448			0.0255724	+	0.999673i	$0.491859\pi$
$464$		0			0
$465$	8.94975	−	8.94975i	0.415035	−	0.415035i
$466$		0			0
$467$		−	25.2132i		−	1.16673i	−0.812211	−	0.583364i	$-0.801736\pi$
$467$		−	25.2132i		−	1.16673i	0.812211	−	0.583364i	$-0.198264\pi$
$468$		0			0
$469$	−3.65685	+	3.65685i	−0.168858	+	0.168858i
$470$		0			0
$471$	−4.82843	−	4.82843i	−0.222482	−	0.222482i
$472$		0			0
$473$	10.2426	+	10.2426i	0.470957	+	0.470957i
$474$		0			0
$475$	−2.41421			−0.110772
$476$		0			0
$477$	9.85786			0.451361
$478$		0			0
$479$	−19.0208	−	19.0208i	−0.869083	−	0.869083i	0.123288	−	0.992371i	$-0.460656\pi$
$479$	−19.0208	−	19.0208i	−0.869083	−	0.869083i	−0.992371	+	0.123288i	$0.960656\pi$
$480$		0			0
$481$	1.58579	+	1.58579i	0.0723056	+	0.0723056i
$482$		0			0
$483$	3.17157	−	3.17157i	0.144312	−	0.144312i
$484$		0			0
$485$			14.3137i			0.649952i
$486$		0			0
$487$	−14.4142	+	14.4142i	−0.653170	+	0.653170i	−0.953755	−	0.300585i	$-0.902818\pi$
$487$	−14.4142	+	14.4142i	−0.653170	+	0.653170i	0.300585	+	0.953755i	$0.402818\pi$
$488$		0			0
$489$	1.17157			0.0529804
$490$		0			0
$491$			8.27208i			0.373314i	0.982425	+	0.186657i	$0.0597652\pi$
$491$			8.27208i			0.373314i	−0.982425	+	0.186657i	$0.940235\pi$
$492$		0			0
$493$	27.7990	+	29.4853i	1.25200	+	1.32795i
$494$		0			0
$495$		−	4.00000i		−	0.179787i
$496$		0			0
$497$	6.87006			0.308164
$498$		0			0
$499$	19.2426	−	19.2426i	0.861419	−	0.861419i	−0.130084	−	0.991503i	$-0.541525\pi$
$499$	19.2426	−	19.2426i	0.861419	−	0.861419i	0.991503	+	0.130084i	$0.0415247\pi$
$500$		0			0
$501$			36.6274i			1.63639i
$502$		0			0
$503$	−26.1421	+	26.1421i	−1.16562	+	1.16562i	−0.182395	+	0.983225i	$0.558385\pi$
$503$	−26.1421	+	26.1421i	−1.16562	+	1.16562i	−0.983225	+	0.182395i	$0.941615\pi$
$504$		0			0
$505$	−3.07107	−	3.07107i	−0.136661	−	0.136661i
$506$		0			0
$507$	20.4853	+	20.4853i	0.909783	+	0.909783i
$508$		0			0
$509$	32.0000			1.41838			0.709188	−	0.705020i	$-0.249062\pi$
$509$	32.0000			1.41838			0.709188	+	0.705020i	$0.249062\pi$
$510$		0			0
$511$	4.58579			0.202863
$512$		0			0
$513$	−0.707107	−	0.707107i	−0.0312195	−	0.0312195i
$514$		0			0
$515$	1.65685	+	1.65685i	0.0730097	+	0.0730097i
$516$		0			0
$517$	3.24264	−	3.24264i	0.142611	−	0.142611i
$518$		0			0
$519$		−	3.17157i		−	0.139217i
$520$		0			0
$521$	4.68629	−	4.68629i	0.205310	−	0.205310i	−0.596960	−	0.802271i	$-0.703625\pi$
$521$	4.68629	−	4.68629i	0.205310	−	0.205310i	0.802271	+	0.596960i	$0.203625\pi$
$522$		0			0
$523$	−4.82843			−0.211132			−0.105566	−	0.994412i	$-0.533665\pi$
$523$	−4.82843			−0.211132			−0.105566	+	0.994412i	$0.533665\pi$
$524$		0			0
$525$		−	1.41421i		−	0.0617213i
$526$		0			0
$527$	−14.8284	−	15.7279i	−0.645936	−	0.685119i
$528$		0			0
$529$			12.9411i			0.562658i
$530$		0			0
$531$	−30.8284			−1.33784
$532$		0			0
$533$	−6.65685	+	6.65685i	−0.288340	+	0.288340i
$534$		0			0
$535$			12.4853i			0.539786i
$536$		0			0
$537$	−11.6569	+	11.6569i	−0.503030	+	0.503030i
$538$		0			0
$539$	−6.65685	−	6.65685i	−0.286731	−	0.286731i
$540$		0			0
$541$	−11.3137	−	11.3137i	−0.486414	−	0.486414i	0.420758	−	0.907173i	$-0.361764\pi$
$541$	−11.3137	−	11.3137i	−0.486414	−	0.486414i	−0.907173	+	0.420758i	$0.861764\pi$
$542$		0			0
$543$	−23.3137			−1.00049
$544$		0			0
$545$	−11.8284			−0.506674
$546$		0			0
$547$	−11.4645	−	11.4645i	−0.490185	−	0.490185i	0.418179	−	0.908364i	$-0.362668\pi$
$547$	−11.4645	−	11.4645i	−0.490185	−	0.490185i	−0.908364	+	0.418179i	$0.862668\pi$
$548$		0			0
$549$	−16.3431	−	16.3431i	−0.697508	−	0.697508i
$550$		0			0
$551$	16.7782	−	16.7782i	0.714774	−	0.714774i
$552$		0			0
$553$			6.82843i			0.290374i
$554$		0			0
$555$	3.82843	−	3.82843i	0.162508	−	0.162508i
$556$		0			0
$557$	7.82843			0.331701			0.165851	−	0.986151i	$-0.446963\pi$
$557$	7.82843			0.331701			0.165851	+	0.986151i	$0.446963\pi$
$558$		0			0
$559$		−	10.2426i		−	0.433218i
$560$		0			0
$561$	−14.0711	−	0.414214i	−0.594081	−	0.0174881i
$562$		0			0
$563$			32.1421i			1.35463i	0.735693	+	0.677315i	$0.236856\pi$
$563$			32.1421i			1.35463i	−0.735693	+	0.677315i	$0.763144\pi$
$564$		0			0
$565$	3.48528			0.146627
$566$		0			0
$567$	3.92893	−	3.92893i	0.165000	−	0.165000i
$568$		0			0
$569$			25.9706i			1.08874i	0.838844	+	0.544371i	$0.183232\pi$
$569$			25.9706i			1.08874i	−0.838844	+	0.544371i	$0.816768\pi$
$570$		0			0
$571$	−23.0416	+	23.0416i	−0.964262	+	0.964262i	−0.999383	−	0.0351208i	$-0.988818\pi$
$571$	−23.0416	+	23.0416i	−0.964262	+	0.964262i	0.0351208	+	0.999383i	$0.488818\pi$
$572$		0			0
$573$	0.414214	+	0.414214i	0.0173040	+	0.0173040i
$574$		0			0
$575$	−2.24264	−	2.24264i	−0.0935246	−	0.0935246i
$576$		0			0
$577$	18.3431			0.763635			0.381818	−	0.924238i	$-0.375298\pi$
$577$	18.3431			0.763635			0.381818	+	0.924238i	$0.375298\pi$
$578$		0			0
$579$	−9.65685			−0.401325
$580$		0			0
$581$	−4.10051	−	4.10051i	−0.170118	−	0.170118i
$582$		0			0
$583$	3.48528	+	3.48528i	0.144346	+	0.144346i
$584$		0			0
$585$	−2.00000	+	2.00000i	−0.0826898	+	0.0826898i
$586$		0			0
$587$			3.17157i			0.130905i	0.997856	+	0.0654524i	$0.0208491\pi$
$587$			3.17157i			0.130905i	−0.997856	+	0.0654524i	$0.979151\pi$
$588$		0			0
$589$	−8.94975	+	8.94975i	−0.368768	+	0.368768i
$590$		0			0
$591$	61.6985			2.53794
$592$		0			0
$593$		−	14.2843i		−	0.586585i	−0.956023	−	0.293292i	$-0.905249\pi$
$593$		−	14.2843i		−	0.586585i	0.956023	−	0.293292i	$-0.0947509\pi$
$594$		0			0
$595$	−2.41421	−	0.0710678i	−0.0989731	−	0.00291350i
$596$		0			0
$597$		−	27.4853i		−	1.12490i
$598$		0			0
$599$	−42.7696			−1.74752			−0.873758	−	0.486360i	$-0.838324\pi$
$599$	−42.7696			−1.74752			−0.873758	+	0.486360i	$0.838324\pi$
$600$		0			0
$601$	13.1716	−	13.1716i	0.537280	−	0.537280i	−0.385449	−	0.922729i	$-0.625954\pi$
$601$	13.1716	−	13.1716i	0.537280	−	0.537280i	0.922729	+	0.385449i	$0.125954\pi$
$602$		0			0
$603$		−	24.9706i		−	1.01688i
$604$		0			0
$605$	−6.36396	+	6.36396i	−0.258732	+	0.258732i
$606$		0			0
$607$	4.07107	+	4.07107i	0.165240	+	0.165240i	0.784883	−	0.619644i	$-0.212723\pi$
$607$	4.07107	+	4.07107i	0.165240	+	0.165240i	−0.619644	+	0.784883i	$0.712723\pi$
$608$		0			0
$609$	9.82843	+	9.82843i	0.398268	+	0.398268i
$610$		0			0
$611$	−3.24264			−0.131183
$612$		0			0
$613$	−23.2843			−0.940443			−0.470221	−	0.882548i	$-0.655826\pi$
$613$	−23.2843			−0.940443			−0.470221	+	0.882548i	$0.655826\pi$
$614$		0			0
$615$	16.0711	+	16.0711i	0.648048	+	0.648048i
$616$		0			0
$617$	0.564971	+	0.564971i	0.0227449	+	0.0227449i	0.718388	−	0.695643i	$-0.244880\pi$
$617$	0.564971	+	0.564971i	0.0227449	+	0.0227449i	−0.695643	+	0.718388i	$0.744880\pi$
$618$		0			0
$619$	10.2426	−	10.2426i	0.411686	−	0.411686i	−0.470639	−	0.882326i	$-0.655977\pi$
$619$	10.2426	−	10.2426i	0.411686	−	0.411686i	0.882326	+	0.470639i	$0.155977\pi$
$620$		0			0
$621$		−	1.31371i		−	0.0527173i
$622$		0			0
$623$	6.07107	−	6.07107i	0.243232	−	0.243232i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$			8.24264i			0.329179i
$628$		0			0
$629$	−6.34315	−	6.72792i	−0.252918	−	0.268260i
$630$		0			0
$631$		−	49.6569i		−	1.97681i	−0.151847	−	0.988404i	$-0.548522\pi$
$631$		−	49.6569i		−	1.97681i	0.151847	−	0.988404i	$-0.451478\pi$
$632$		0			0
$633$	11.6569			0.463318
$634$		0			0
$635$	−5.46447	+	5.46447i	−0.216851	+	0.216851i
$636$		0			0
$637$			6.65685i			0.263754i
$638$		0			0
$639$	−23.4558	+	23.4558i	−0.927899	+	0.927899i
$640$		0			0
$641$	−11.0711	−	11.0711i	−0.437281	−	0.437281i	0.453815	−	0.891096i	$-0.350063\pi$
$641$	−11.0711	−	11.0711i	−0.437281	−	0.437281i	−0.891096	+	0.453815i	$0.850063\pi$
$642$		0			0
$643$	18.3848	+	18.3848i	0.725025	+	0.725025i	0.969624	−	0.244599i	$-0.0786565\pi$
$643$	18.3848	+	18.3848i	0.725025	+	0.725025i	−0.244599	+	0.969624i	$0.578656\pi$
$644$		0			0
$645$	−24.7279			−0.973661
$646$		0			0
$647$	−48.2132			−1.89546			−0.947728	−	0.319078i	$-0.896627\pi$
$647$	−48.2132			−1.89546			−0.947728	+	0.319078i	$0.896627\pi$
$648$		0			0
$649$	−10.8995	−	10.8995i	−0.427843	−	0.427843i
$650$		0			0
$651$	−5.24264	−	5.24264i	−0.205475	−	0.205475i
$652$		0			0
$653$	6.51472	−	6.51472i	0.254941	−	0.254941i	−0.568052	−	0.822993i	$-0.692303\pi$
$653$	6.51472	−	6.51472i	0.254941	−	0.254941i	0.822993	+	0.568052i	$0.192303\pi$
$654$		0			0
$655$		−	19.8995i		−	0.777538i
$656$		0			0
$657$	−15.6569	+	15.6569i	−0.610832	+	0.610832i
$658$		0			0
$659$	10.7574			0.419047			0.209524	−	0.977804i	$-0.432809\pi$
$659$	10.7574			0.419047			0.209524	+	0.977804i	$0.432809\pi$
$660$		0			0
$661$		−	11.3137i		−	0.440052i	−0.975494	−	0.220026i	$-0.929386\pi$
$661$		−	11.3137i		−	0.440052i	0.975494	−	0.220026i	$-0.0706143\pi$
$662$		0			0
$663$	6.82843	+	7.24264i	0.265194	+	0.281281i
$664$		0			0
$665$			1.41421i			0.0548408i
$666$		0			0
$667$	31.1716			1.20697
$668$		0			0
$669$	−43.3345	+	43.3345i	−1.67541	+	1.67541i
$670$		0			0
$671$		−	11.5563i		−	0.446128i
$672$		0			0
$673$	−8.46447	+	8.46447i	−0.326281	+	0.326281i	−0.851170	−	0.524889i	$-0.824107\pi$
$673$	−8.46447	+	8.46447i	−0.326281	+	0.326281i	0.524889	+	0.851170i	$0.324107\pi$
$674$		0			0
$675$	−0.292893	−	0.292893i	−0.0112735	−	0.0112735i
$676$		0			0
$677$	−31.8284	−	31.8284i	−1.22327	−	1.22327i	−0.966464	−	0.256802i	$-0.917331\pi$
$677$	−31.8284	−	31.8284i	−1.22327	−	1.22327i	−0.256802	−	0.966464i	$-0.582669\pi$
$678$		0			0
$679$	8.38478			0.321778
$680$		0			0
$681$	−49.6274			−1.90173
$682$		0			0
$683$	−8.19239	−	8.19239i	−0.313473	−	0.313473i	0.532780	−	0.846253i	$-0.321147\pi$
$683$	−8.19239	−	8.19239i	−0.313473	−	0.313473i	−0.846253	+	0.532780i	$0.821147\pi$
$684$		0			0
$685$	−11.8284	−	11.8284i	−0.451941	−	0.451941i
$686$		0			0
$687$	−27.7279	+	27.7279i	−1.05789	+	1.05789i
$688$		0			0
$689$		−	3.48528i		−	0.132779i
$690$		0			0
$691$	23.8284	−	23.8284i	0.906476	−	0.906476i	−0.0895098	−	0.995986i	$-0.528530\pi$
$691$	23.8284	−	23.8284i	0.906476	−	0.906476i	0.995986	+	0.0895098i	$0.0285301\pi$
$692$		0			0
$693$	−2.34315			−0.0890087
$694$		0			0
$695$			0.928932i			0.0352364i
$696$		0			0
$697$	28.2426	−	26.6274i	1.06977	−	1.00859i
$698$		0			0
$699$			28.5563i			1.08010i
$700$		0			0
$701$	8.97056			0.338813			0.169407	−	0.985546i	$-0.445815\pi$
$701$	8.97056			0.338813			0.169407	+	0.985546i	$0.445815\pi$
$702$		0			0
$703$	−3.82843	+	3.82843i	−0.144392	+	0.144392i
$704$		0			0
$705$			7.82843i			0.294836i
$706$		0			0
$707$	−1.79899	+	1.79899i	−0.0676580	+	0.0676580i
$708$		0			0
$709$	−16.5061	−	16.5061i	−0.619899	−	0.619899i	0.325606	−	0.945506i	$-0.394432\pi$
$709$	−16.5061	−	16.5061i	−0.619899	−	0.619899i	−0.945506	+	0.325606i	$0.894432\pi$
$710$		0			0
$711$	−23.3137	−	23.3137i	−0.874332	−	0.874332i
$712$		0			0
$713$	−16.6274			−0.622702
$714$		0			0
$715$	−1.41421			−0.0528886
$716$		0			0
$717$	46.7990	+	46.7990i	1.74774	+	1.74774i
$718$		0			0
$719$	36.6777	+	36.6777i	1.36785	+	1.36785i	0.863504	+	0.504343i	$0.168265\pi$
$719$	36.6777	+	36.6777i	1.36785	+	1.36785i	0.504343	+	0.863504i	$0.331735\pi$
$720$		0			0
$721$	0.970563	−	0.970563i	0.0361456	−	0.0361456i
$722$		0			0
$723$			67.6985i			2.51773i
$724$		0			0
$725$	6.94975	−	6.94975i	0.258107	−	0.258107i
$726$		0			0
$727$	22.2132			0.823842			0.411921	−	0.911220i	$-0.364858\pi$
$727$	22.2132			0.823842			0.411921	+	0.911220i	$0.364858\pi$
$728$		0			0
$729$			23.8284i			0.882534i
$730$		0			0
$731$	−1.24264	+	42.2132i	−0.0459607	+	1.56131i
$732$		0			0
$733$		−	10.8284i		−	0.399957i	−0.979800	−	0.199979i	$-0.935913\pi$
$733$		−	10.8284i		−	0.399957i	0.979800	−	0.199979i	$-0.0640872\pi$
$734$		0			0
$735$	16.0711			0.592790
$736$		0			0
$737$	8.82843	−	8.82843i	0.325199	−	0.325199i
$738$		0			0
$739$		−	5.87006i		−	0.215934i	−0.994155	−	0.107967i	$-0.965566\pi$
$739$		−	5.87006i		−	0.215934i	0.994155	−	0.107967i	$-0.0344340\pi$
$740$		0			0
$741$	4.12132	−	4.12132i	0.151400	−	0.151400i
$742$		0			0
$743$	−4.55635	−	4.55635i	−0.167156	−	0.167156i	0.618572	−	0.785728i	$-0.287712\pi$
$743$	−4.55635	−	4.55635i	−0.167156	−	0.167156i	−0.785728	+	0.618572i	$0.787712\pi$
$744$		0			0
$745$	−1.24264	−	1.24264i	−0.0455268	−	0.0455268i
$746$		0			0
$747$	28.0000			1.02447
$748$		0			0
$749$	7.31371			0.267237
$750$		0			0
$751$	2.15076	+	2.15076i	0.0784823	+	0.0784823i	0.745258	−	0.666776i	$-0.232326\pi$
$751$	2.15076	+	2.15076i	0.0784823	+	0.0784823i	−0.666776	+	0.745258i	$0.732326\pi$
$752$		0			0
$753$	−3.65685	−	3.65685i	−0.133263	−	0.133263i
$754$		0			0
$755$	3.41421	−	3.41421i	0.124256	−	0.124256i
$756$		0			0
$757$			39.8284i			1.44759i	0.690016	+	0.723794i	$0.257604\pi$
$757$			39.8284i			1.44759i	−0.690016	+	0.723794i	$0.742396\pi$
$758$		0			0
$759$	−7.65685	+	7.65685i	−0.277926	+	0.277926i
$760$		0			0
$761$	43.1127			1.56283			0.781417	−	0.624009i	$-0.214497\pi$
$761$	43.1127			1.56283			0.781417	+	0.624009i	$0.214497\pi$
$762$		0			0
$763$			6.92893i			0.250844i
$764$		0			0
$765$	8.48528	−	8.00000i	0.306786	−	0.289241i
$766$		0			0
$767$			10.8995i			0.393558i
$768$		0			0
$769$	23.1421			0.834527			0.417263	−	0.908786i	$-0.362989\pi$
$769$	23.1421			0.834527			0.417263	+	0.908786i	$0.362989\pi$
$770$		0			0
$771$	−38.5563	+	38.5563i	−1.38857	+	1.38857i
$772$		0			0
$773$		−	23.5147i		−	0.845766i	−0.906184	−	0.422883i	$-0.861018\pi$
$773$		−	23.5147i		−	0.845766i	0.906184	−	0.422883i	$-0.138982\pi$
$774$		0			0
$775$	−3.70711	+	3.70711i	−0.133163	+	0.133163i
$776$		0			0
$777$	−2.24264	−	2.24264i	−0.0804543	−	0.0804543i
$778$		0			0
$779$	−16.0711	−	16.0711i	−0.575806	−	0.575806i
$780$		0			0
$781$	−16.5858			−0.593486
$782$		0			0
$783$	4.07107			0.145488
$784$		0			0
$785$	2.00000	+	2.00000i	0.0713831	+	0.0713831i
$786$		0			0
$787$	0.393398	+	0.393398i	0.0140231	+	0.0140231i	0.714084	−	0.700060i	$-0.246844\pi$
$787$	0.393398	+	0.393398i	0.0140231	+	0.0140231i	−0.700060	+	0.714084i	$0.746844\pi$
$788$		0			0
$789$	50.7487	−	50.7487i	1.80670	−	1.80670i
$790$		0			0
$791$		−	2.04163i		−	0.0725920i
$792$		0			0
$793$	−5.77817	+	5.77817i	−0.205189	+	0.205189i
$794$		0			0
$795$	−8.41421			−0.298421
$796$		0			0
$797$			26.6274i			0.943192i	0.881815	+	0.471596i	$0.156322\pi$
$797$			26.6274i			0.943192i	−0.881815	+	0.471596i	$0.843678\pi$
$798$		0			0
$799$	13.3640	+	0.393398i	0.472783	+	0.0139174i
$800$		0			0
$801$			41.4558i			1.46477i
$802$		0			0
$803$	−11.0711			−0.390689
$804$		0			0
$805$	−1.31371	+	1.31371i	−0.0463021	+	0.0463021i
$806$		0			0
$807$			8.41421i			0.296194i
$808$		0			0
$809$	25.8284	−	25.8284i	0.908079	−	0.908079i	−0.0880380	−	0.996117i	$-0.528060\pi$
$809$	25.8284	−	25.8284i	0.908079	−	0.908079i	0.996117	+	0.0880380i	$0.0280597\pi$
$810$		0			0
$811$	−36.0711	−	36.0711i	−1.26663	−	1.26663i	−0.947820	−	0.318807i	$-0.896718\pi$
$811$	−36.0711	−	36.0711i	−1.26663	−	1.26663i	−0.318807	−	0.947820i	$-0.603282\pi$
$812$		0			0
$813$	−10.8284	−	10.8284i	−0.379770	−	0.379770i
$814$		0			0
$815$	−0.485281			−0.0169987
$816$		0			0
$817$	24.7279			0.865120
$818$		0			0
$819$	1.17157	+	1.17157i	0.0409381	+	0.0409381i
$820$		0			0
$821$	−3.92031	−	3.92031i	−0.136820	−	0.136820i	0.635380	−	0.772200i	$-0.280844\pi$
$821$	−3.92031	−	3.92031i	−0.136820	−	0.136820i	−0.772200	+	0.635380i	$0.780844\pi$
$822$		0			0
$823$	−8.58579	+	8.58579i	−0.299282	+	0.299282i	−0.840732	−	0.541451i	$-0.817875\pi$
$823$	−8.58579	+	8.58579i	−0.299282	+	0.299282i	0.541451	+	0.840732i	$0.317875\pi$
$824$		0			0
$825$			3.41421i			0.118868i
$826$		0			0
$827$	−10.1421	+	10.1421i	−0.352677	+	0.352677i	−0.861105	−	0.508428i	$-0.830227\pi$
$827$	−10.1421	+	10.1421i	−0.352677	+	0.352677i	0.508428	+	0.861105i	$0.330227\pi$
$828$		0			0
$829$	−11.3137			−0.392941			−0.196471	−	0.980510i	$-0.562948\pi$
$829$	−11.3137			−0.392941			−0.196471	+	0.980510i	$0.562948\pi$
$830$		0			0
$831$		−	20.1421i		−	0.698723i
$832$		0			0
$833$	0.807612	−	27.4350i	0.0279821	−	0.950567i
$834$		0			0
$835$		−	15.1716i		−	0.525034i
$836$		0			0
$837$	−2.17157			−0.0750605
$838$		0			0
$839$	8.87868	−	8.87868i	0.306526	−	0.306526i	−0.537034	−	0.843560i	$-0.680455\pi$
$839$	8.87868	−	8.87868i	0.306526	−	0.306526i	0.843560	+	0.537034i	$0.180455\pi$
$840$		0			0
$841$			67.5980i			2.33096i
$842$		0			0
$843$	−2.53553	+	2.53553i	−0.0873284	+	0.0873284i
$844$		0			0
$845$	−8.48528	−	8.48528i	−0.291903	−	0.291903i
$846$		0			0
$847$	3.72792	+	3.72792i	0.128093	+	0.128093i
$848$		0			0
$849$	23.1421			0.794236
$850$		0			0
$851$	−7.11270			−0.243820
$852$		0			0
$853$	−0.514719	−	0.514719i	−0.0176236	−	0.0176236i	0.698240	−	0.715864i	$-0.253967\pi$
$853$	−0.514719	−	0.514719i	−0.0176236	−	0.0176236i	−0.715864	+	0.698240i	$0.753967\pi$
$854$		0			0
$855$	−4.82843	−	4.82843i	−0.165129	−	0.165129i
$856$		0			0
$857$	−15.4350	+	15.4350i	−0.527251	+	0.527251i	−0.919752	−	0.392501i	$-0.871610\pi$
$857$	−15.4350	+	15.4350i	−0.527251	+	0.527251i	0.392501	+	0.919752i	$0.371610\pi$
$858$		0			0
$859$			47.8701i			1.63331i	0.577129	+	0.816653i	$0.304173\pi$
$859$			47.8701i			1.63331i	−0.577129	+	0.816653i	$0.695827\pi$
$860$		0			0
$861$	9.41421	−	9.41421i	0.320836	−	0.320836i
$862$		0			0
$863$	1.45584			0.0495575			0.0247788	−	0.999693i	$-0.492112\pi$
$863$	1.45584			0.0495575			0.0247788	+	0.999693i	$0.492112\pi$
$864$		0			0
$865$			1.31371i			0.0446674i
$866$		0			0
$867$	−27.2635	−	30.6777i	−0.925916	−	1.04187i
$868$		0			0
$869$		−	16.4853i		−	0.559225i
$870$		0			0
$871$	−8.82843			−0.299140
$872$		0			0
$873$	−28.6274	+	28.6274i	−0.968891	+	0.968891i
$874$		0			0
$875$			0.585786i			0.0198032i
$876$		0			0
$877$	−34.2132	+	34.2132i	−1.15530	+	1.15530i	−0.169823	+	0.985475i	$0.554320\pi$
$877$	−34.2132	+	34.2132i	−1.15530	+	1.15530i	−0.985475	+	0.169823i	$0.945680\pi$
$878$		0			0
$879$	−9.36396	−	9.36396i	−0.315839	−	0.315839i
$880$		0			0
$881$	28.0416	+	28.0416i	0.944747	+	0.944747i	0.998551	−	0.0538049i	$-0.0171349\pi$
$881$	28.0416	+	28.0416i	0.944747	+	0.944747i	−0.0538049	+	0.998551i	$0.517135\pi$
$882$		0			0
$883$	55.4975			1.86764			0.933819	−	0.357745i	$-0.116454\pi$
$883$	55.4975			1.86764			0.933819	+	0.357745i	$0.116454\pi$
$884$		0			0
$885$	26.3137			0.884526
$886$		0			0
$887$	−36.8995	−	36.8995i	−1.23896	−	1.23896i	−0.960425	−	0.278539i	$-0.910150\pi$
$887$	−36.8995	−	36.8995i	−1.23896	−	1.23896i	−0.278539	−	0.960425i	$-0.589850\pi$
$888$		0			0
$889$	3.20101	+	3.20101i	0.107358	+	0.107358i
$890$		0			0
$891$	−9.48528	+	9.48528i	−0.317769	+	0.317769i
$892$		0			0
$893$		−	7.82843i		−	0.261968i
$894$		0			0
$895$	4.82843	−	4.82843i	0.161397	−	0.161397i
$896$		0			0
$897$	7.65685			0.255655
$898$		0			0
$899$		−	51.5269i		−	1.71852i
$900$		0			0
$901$	−0.422836	+	14.3640i	−0.0140867	+	0.478533i
$902$		0			0
$903$			14.4853i			0.482040i
$904$		0			0
$905$	9.65685			0.321005
$906$		0			0
$907$	−2.97918	+	2.97918i	−0.0989222	+	0.0989222i	−0.754836	−	0.655914i	$-0.772284\pi$
$907$	−2.97918	+	2.97918i	−0.0989222	+	0.0989222i	0.655914	+	0.754836i	$0.272284\pi$
$908$		0			0
$909$		−	12.2843i		−	0.407444i
$910$		0			0
$911$	3.31371	−	3.31371i	0.109788	−	0.109788i	−0.650079	−	0.759867i	$-0.725264\pi$
$911$	3.31371	−	3.31371i	0.109788	−	0.109788i	0.759867	+	0.650079i	$0.225264\pi$
$912$		0			0
$913$	9.89949	+	9.89949i	0.327625	+	0.327625i
$914$		0			0
$915$	13.9497	+	13.9497i	0.461164	+	0.461164i
$916$		0			0
$917$	−11.6569			−0.384943
$918$		0			0
$919$	43.0122			1.41884			0.709421	−	0.704785i	$-0.248957\pi$
$919$	43.0122			1.41884			0.709421	+	0.704785i	$0.248957\pi$
$920$		0			0
$921$	16.8995	+	16.8995i	0.556857	+	0.556857i
$922$		0			0
$923$	8.29289	+	8.29289i	0.272964	+	0.272964i
$924$		0			0
$925$	−1.58579	+	1.58579i	−0.0521403	+	0.0521403i
$926$		0			0
$927$			6.62742i			0.217673i
$928$		0			0
$929$	13.4853	−	13.4853i	0.442438	−	0.442438i	−0.450393	−	0.892831i	$-0.648716\pi$
$929$	13.4853	−	13.4853i	0.442438	−	0.442438i	0.892831	+	0.450393i	$0.148716\pi$
$930$		0			0
$931$	−16.0711			−0.526708
$932$		0			0
$933$		−	8.82843i		−	0.289030i
$934$		0			0
$935$	5.82843	+	0.171573i	0.190610	+	0.00561103i
$936$		0			0
$937$			45.8406i			1.49755i	0.662826	+	0.748774i	$0.269357\pi$
$937$			45.8406i			1.49755i	−0.662826	+	0.748774i	$0.730643\pi$
$938$		0			0
$939$	−22.1421			−0.722581
$940$		0			0
$941$	−38.4056	+	38.4056i	−1.25199	+	1.25199i	−0.297158	+	0.954828i	$0.596039\pi$
$941$	−38.4056	+	38.4056i	−1.25199	+	1.25199i	−0.954828	+	0.297158i	$0.903961\pi$
$942$		0			0
$943$		−	29.8579i		−	0.972306i
$944$		0			0
$945$	−0.171573	+	0.171573i	−0.00558127	+	0.00558127i
$946$		0			0
$947$	14.7782	+	14.7782i	0.480226	+	0.480226i	0.905204	−	0.424978i	$-0.139718\pi$
$947$	14.7782	+	14.7782i	0.480226	+	0.480226i	−0.424978	+	0.905204i	$0.639718\pi$
$948$		0			0
$949$	5.53553	+	5.53553i	0.179691	+	0.179691i
$950$		0			0
$951$	83.5980			2.71085
$952$		0			0
$953$	−20.5858			−0.666839			−0.333420	−	0.942779i	$-0.608203\pi$
$953$	−20.5858			−0.666839			−0.333420	+	0.942779i	$0.608203\pi$
$954$		0			0
$955$	−0.171573	−	0.171573i	−0.00555197	−	0.00555197i
$956$		0			0
$957$	−23.7279	−	23.7279i	−0.767015	−	0.767015i
$958$		0			0
$959$	−6.92893	+	6.92893i	−0.223747	+	0.223747i
$960$		0			0
$961$		−	3.51472i		−	0.113378i
$962$		0			0
$963$	−24.9706	+	24.9706i	−0.804665	+	0.804665i
$964$		0			0
$965$	4.00000			0.128765
$966$		0			0
$967$			33.5980i			1.08044i	0.841524	+	0.540219i	$0.181659\pi$
$967$			33.5980i			1.08044i	−0.841524	+	0.540219i	$0.818341\pi$
$968$		0			0
$969$	−17.4853	+	16.4853i	−0.561708	+	0.529584i
$970$		0			0
$971$			22.0122i			0.706405i	0.935547	+	0.353202i	$0.114907\pi$
$971$			22.0122i			0.706405i	−0.935547	+	0.353202i	$0.885093\pi$
$972$		0			0
$973$	0.544156			0.0174448
$974$		0			0
$975$	1.70711	−	1.70711i	0.0546712	−	0.0546712i
$976$		0			0
$977$		−	34.2426i		−	1.09552i	−0.836636	−	0.547760i	$-0.815481\pi$
$977$		−	34.2426i		−	1.09552i	0.836636	−	0.547760i	$-0.184519\pi$
$978$		0			0
$979$	−14.6569	+	14.6569i	−0.468435	+	0.468435i
$980$		0			0
$981$	−23.6569	−	23.6569i	−0.755305	−	0.755305i
$982$		0			0
$983$	8.07107	+	8.07107i	0.257427	+	0.257427i	0.824007	−	0.566580i	$-0.191734\pi$
$983$	8.07107	+	8.07107i	0.257427	+	0.257427i	−0.566580	+	0.824007i	$0.691734\pi$
$984$		0			0
$985$	−25.5563			−0.814293
$986$		0			0
$987$	4.58579			0.145967
$988$		0			0
$989$	22.9706	+	22.9706i	0.730421	+	0.730421i
$990$		0			0
$991$	−22.0919	−	22.0919i	−0.701772	−	0.701772i	0.263019	−	0.964791i	$-0.415282\pi$
$991$	−22.0919	−	22.0919i	−0.701772	−	0.701772i	−0.964791	+	0.263019i	$0.915282\pi$
$992$		0			0
$993$	14.6066	−	14.6066i	0.463526	−	0.463526i
$994$		0			0
$995$			11.3848i			0.360922i
$996$		0			0
$997$	33.2843	−	33.2843i	1.05412	−	1.05412i	0.0556745	−	0.998449i	$-0.482269\pi$
$997$	33.2843	−	33.2843i	1.05412	−	1.05412i	0.998449	−	0.0556745i	$-0.0177309\pi$
$998$		0			0
$999$	−0.928932			−0.0293901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1360.2.bt.a.81.1		4
4.3	odd	2		170.2.h.a.81.2	yes	4
12.11	even	2		1530.2.q.c.1441.1		4
17.4	even	4	inner	1360.2.bt.a.1041.1		4
20.3	even	4		850.2.g.e.149.1		4
20.7	even	4		850.2.g.h.149.2		4
20.19	odd	2		850.2.h.g.251.1		4
68.15	odd	8		2890.2.a.t.1.1		2
68.19	odd	8		2890.2.a.v.1.2		2
68.43	odd	8		2890.2.b.j.2311.4		4
68.55	odd	4		170.2.h.a.21.2	✓	4
68.59	odd	8		2890.2.b.j.2311.1		4
204.191	even	4		1530.2.q.c.361.1		4
340.123	even	4		850.2.g.h.599.2		4
340.259	odd	4		850.2.h.g.701.1		4
340.327	even	4		850.2.g.e.599.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.h.a.21.2	✓	4	68.55	odd	4
170.2.h.a.81.2	yes	4	4.3	odd	2
850.2.g.e.149.1		4	20.3	even	4
850.2.g.e.599.1		4	340.327	even	4
850.2.g.h.149.2		4	20.7	even	4
850.2.g.h.599.2		4	340.123	even	4
850.2.h.g.251.1		4	20.19	odd	2
850.2.h.g.701.1		4	340.259	odd	4
1360.2.bt.a.81.1		4	1.1	even	1	trivial
1360.2.bt.a.1041.1		4	17.4	even	4	inner
1530.2.q.c.361.1		4	204.191	even	4
1530.2.q.c.1441.1		4	12.11	even	2
2890.2.a.t.1.1		2	68.15	odd	8
2890.2.a.v.1.2		2	68.19	odd	8
2890.2.b.j.2311.1		4	68.59	odd	8
2890.2.b.j.2311.4		4	68.43	odd	8

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:	\( N \)	\(=\)	\( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1360.bt (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.70711 - 1.70711i) q^{3} +(0.707107 + 0.707107i) q^{5} +(0.414214 - 0.414214i) q^{7} +2.82843i q^{9} +(-1.00000 + 1.00000i) q^{11} +1.00000 q^{13} -2.41421i q^{15} +(-4.12132 - 0.121320i) q^{17} +2.41421i q^{19} -1.41421 q^{21} +(-2.24264 + 2.24264i) q^{23} +1.00000i q^{25} +(-0.292893 + 0.292893i) q^{27} +(-6.94975 - 6.94975i) q^{29} +(3.70711 + 3.70711i) q^{31} +3.41421 q^{33} +0.585786 q^{35} +(1.58579 + 1.58579i) q^{37} +(-1.70711 - 1.70711i) q^{39} +(-6.65685 + 6.65685i) q^{41} -10.2426i q^{43} +(-2.00000 + 2.00000i) q^{45} -3.24264 q^{47} +6.65685i q^{49} +(6.82843 + 7.24264i) q^{51} -3.48528i q^{53} -1.41421 q^{55} +(4.12132 - 4.12132i) q^{57} +10.8995i q^{59} +(-5.77817 + 5.77817i) q^{61} +(1.17157 + 1.17157i) q^{63} +(0.707107 + 0.707107i) q^{65} -8.82843 q^{67} +7.65685 q^{69} +(8.29289 + 8.29289i) q^{71} +(5.53553 + 5.53553i) q^{73} +(1.70711 - 1.70711i) q^{75} +0.828427i q^{77} +(-8.24264 + 8.24264i) q^{79} +9.48528 q^{81} -9.89949i q^{83} +(-2.82843 - 3.00000i) q^{85} +23.7279i q^{87} +14.6569 q^{89} +(0.414214 - 0.414214i) q^{91} -12.6569i q^{93} +(-1.70711 + 1.70711i) q^{95} +(10.1213 + 10.1213i) q^{97} +(-2.82843 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{7} - 4 q^{11} + 4 q^{13} - 8 q^{17} + 8 q^{23} - 4 q^{27} - 8 q^{29} + 12 q^{31} + 8 q^{33} + 8 q^{35} + 12 q^{37} - 4 q^{39} - 4 q^{41} - 8 q^{45} + 4 q^{47} + 16 q^{51} + 8 q^{57}+ \cdots + 32 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^7 - 4 * q^11 + 4 * q^13 - 8 * q^17 + 8 * q^23 - 4 * q^27 - 8 * q^29 + 12 * q^31 + 8 * q^33 + 8 * q^35 + 12 * q^37 - 4 * q^39 - 4 * q^41 - 8 * q^45 + 4 * q^47 + 16 * q^51 + 8 * q^57 + 8 * q^61 + 16 * q^63 - 24 * q^67 + 8 * q^69 + 36 * q^71 + 8 * q^73 + 4 * q^75 - 16 * q^79 + 4 * q^81 + 36 * q^89 - 4 * q^91 - 4 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more