LMFDB - Embedded newform 1365.2.f.d.274.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1365,2,Mod(274,1365)] 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("1365.274"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1365, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1365 = 3 \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1365.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-32,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$10.8995798759$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 32x^{14} + 416x^{12} + 2802x^{10} + 10280x^{8} + 19568x^{6} + 16273x^{4} + 4072x^{2} + 144 $ x^16 + 32x^14 + 416x^12 + 2802x^10 + 10280x^8 + 19568x^6 + 16273x^4 + 4072*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.13
Root			$2.21394i$ of defining polynomial
Character	$\chi$	$=$	1365.274
Dual form			1365.2.f.d.274.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.21394i q^{2} -1.00000i q^{3} -2.90155 q^{4} +(-1.43370 + 1.71595i) q^{5} +2.21394 q^{6} -1.00000i q^{7} -1.99598i q^{8} -1.00000 q^{9} +(-3.79903 - 3.17414i) q^{10} +1.31239 q^{11} +2.90155i q^{12} -1.00000i q^{13} +2.21394 q^{14} +(1.71595 + 1.43370i) q^{15} -1.38411 q^{16} +0.961936i q^{17} -2.21394i q^{18} -6.74102 q^{19} +(4.15996 - 4.97893i) q^{20} -1.00000 q^{21} +2.90557i q^{22} -3.62489i q^{23} -1.99598 q^{24} +(-0.888995 - 4.92033i) q^{25} +2.21394 q^{26} +1.00000i q^{27} +2.90155i q^{28} -1.40802 q^{29} +(-3.17414 + 3.79903i) q^{30} -1.90547 q^{31} -7.05630i q^{32} -1.31239i q^{33} -2.12967 q^{34} +(1.71595 + 1.43370i) q^{35} +2.90155 q^{36} -0.468408i q^{37} -14.9242i q^{38} -1.00000 q^{39} +(3.42501 + 2.86164i) q^{40} +1.05570 q^{41} -2.21394i q^{42} -5.08537i q^{43} -3.80798 q^{44} +(1.43370 - 1.71595i) q^{45} +8.02530 q^{46} -0.376207i q^{47} +1.38411i q^{48} -1.00000 q^{49} +(10.8933 - 1.96819i) q^{50} +0.961936 q^{51} +2.90155i q^{52} -11.6841i q^{53} -2.21394 q^{54} +(-1.88158 + 2.25201i) q^{55} -1.99598 q^{56} +6.74102i q^{57} -3.11728i q^{58} +2.84906 q^{59} +(-4.97893 - 4.15996i) q^{60} +1.02391 q^{61} -4.21860i q^{62} +1.00000i q^{63} +12.8540 q^{64} +(1.71595 + 1.43370i) q^{65} +2.90557 q^{66} -7.67831i q^{67} -2.79111i q^{68} -3.62489 q^{69} +(-3.17414 + 3.79903i) q^{70} -2.90557 q^{71} +1.99598i q^{72} +2.39384i q^{73} +1.03703 q^{74} +(-4.92033 + 0.888995i) q^{75} +19.5594 q^{76} -1.31239i q^{77} -2.21394i q^{78} -8.28161 q^{79} +(1.98440 - 2.37507i) q^{80} +1.00000 q^{81} +2.33726i q^{82} -2.94509i q^{83} +2.90155 q^{84} +(-1.65064 - 1.37913i) q^{85} +11.2587 q^{86} +1.40802i q^{87} -2.61951i q^{88} +10.3109 q^{89} +(3.79903 + 3.17414i) q^{90} -1.00000 q^{91} +10.5178i q^{92} +1.90547i q^{93} +0.832902 q^{94} +(9.66462 - 11.5673i) q^{95} -7.05630 q^{96} +12.6919i q^{97} -2.21394i q^{98} -1.31239 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} - 2 q^{5} + 4 q^{6} - 16 q^{9} - 2 q^{10} + 4 q^{11} + 4 q^{14} + 64 q^{16} - 4 q^{19} - 2 q^{20} - 16 q^{21} - 24 q^{24} + 2 q^{25} + 4 q^{26} + 24 q^{29} + 2 q^{30} + 4 q^{31} - 4 q^{34}+ \cdots - 4 q^{99}+O(q^{100}) $ 16 * q - 32 * q^4 - 2 * q^5 + 4 * q^6 - 16 * q^9 - 2 * q^10 + 4 * q^11 + 4 * q^14 + 64 * q^16 - 4 * q^19 - 2 * q^20 - 16 * q^21 - 24 * q^24 + 2 * q^25 + 4 * q^26 + 24 * q^29 + 2 * q^30 + 4 * q^31 - 4 * q^34 + 32 * q^36 - 16 * q^39 - 74 * q^40 + 20 * q^41 + 96 * q^44 + 2 * q^45 - 16 * q^49 - 48 * q^50 + 8 * q^51 - 4 * q^54 - 4 * q^55 - 24 * q^56 - 4 * q^59 + 10 * q^60 + 56 * q^61 - 44 * q^64 + 40 * q^66 - 52 * q^69 + 2 * q^70 - 40 * q^71 - 16 * q^75 + 40 * q^76 + 24 * q^79 - 54 * q^80 + 16 * q^81 + 32 * q^84 + 28 * q^85 - 60 * q^86 + 28 * q^89 + 2 * q^90 - 16 * q^91 - 16 * q^94 + 50 * q^95 + 52 * q^96 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1365\mathbb{Z}\right)^\times$.

$n$	$106$	$547$	$911$	$976$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$			2.21394i			1.56550i	0.622339	+	0.782748i	$0.286182\pi$
$2$			2.21394i			1.56550i	−0.622339	+	0.782748i	$0.713818\pi$
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−2.90155			−1.45077
$5$	−1.43370	+	1.71595i	−0.641171	+	0.767398i
$5$	−1.43370	+	1.71595i	−0.641171	+	0.767398i
$6$	2.21394			0.903839
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		−	1.99598i		−	0.705686i
$9$	−1.00000			−0.333333
$10$	−3.79903	−	3.17414i	−1.20136	−	1.00375i
$11$	1.31239			0.395702			0.197851	−	0.980232i	$-0.436604\pi$
$11$	1.31239			0.395702			0.197851	+	0.980232i	$0.436604\pi$
$12$			2.90155i			0.837605i
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$	2.21394			0.591702
$15$	1.71595	+	1.43370i	0.443057	+	0.370180i
$16$	−1.38411			−0.346027
$17$			0.961936i			0.233304i	0.993173	+	0.116652i	$0.0372162\pi$
$17$			0.961936i			0.233304i	−0.993173	+	0.116652i	$0.962784\pi$
$18$		−	2.21394i		−	0.521832i
$19$	−6.74102			−1.54650			−0.773248	−	0.634103i	$-0.781369\pi$
$19$	−6.74102			−1.54650			−0.773248	+	0.634103i	$0.781369\pi$
$20$	4.15996	−	4.97893i	0.930195	−	1.11332i
$21$	−1.00000			−0.218218
$22$			2.90557i			0.619469i
$23$		−	3.62489i		−	0.755842i	−0.925838	−	0.377921i	$-0.876639\pi$
$23$		−	3.62489i		−	0.755842i	0.925838	−	0.377921i	$-0.123361\pi$
$24$	−1.99598			−0.407428
$25$	−0.888995	−	4.92033i	−0.177799	−	0.984067i
$26$	2.21394			0.434190
$27$			1.00000i			0.192450i
$28$			2.90155i			0.548341i
$29$	−1.40802			−0.261463			−0.130732	−	0.991418i	$-0.541733\pi$
$29$	−1.40802			−0.261463			−0.130732	+	0.991418i	$0.541733\pi$
$30$	−3.17414	+	3.79903i	−0.579516	+	0.693604i
$31$	−1.90547			−0.342232			−0.171116	−	0.985251i	$-0.554737\pi$
$31$	−1.90547			−0.342232			−0.171116	+	0.985251i	$0.554737\pi$
$32$		−	7.05630i		−	1.24739i
$33$		−	1.31239i		−	0.228459i
$34$	−2.12967			−0.365236
$35$	1.71595	+	1.43370i	0.290049	+	0.242340i
$36$	2.90155			0.483592
$37$		−	0.468408i		−	0.0770058i	−0.999258	−	0.0385029i	$-0.987741\pi$
$37$		−	0.468408i		−	0.0770058i	0.999258	−	0.0385029i	$-0.0122589\pi$
$38$		−	14.9242i		−	2.42103i
$39$	−1.00000			−0.160128
$40$	3.42501	+	2.86164i	0.541542	+	0.452465i
$41$	1.05570			0.164873			0.0824363	−	0.996596i	$-0.473730\pi$
$41$	1.05570			0.164873			0.0824363	+	0.996596i	$0.473730\pi$
$42$		−	2.21394i		−	0.341619i
$43$		−	5.08537i		−	0.775511i	−0.921762	−	0.387756i	$-0.873250\pi$
$43$		−	5.08537i		−	0.775511i	0.921762	−	0.387756i	$-0.126750\pi$
$44$	−3.80798			−0.574074
$45$	1.43370	−	1.71595i	0.213724	−	0.255799i
$46$	8.02530			1.18327
$47$		−	0.376207i		−	0.0548755i	−0.999624	−	0.0274377i	$-0.991265\pi$
$47$		−	0.376207i		−	0.0548755i	0.999624	−	0.0274377i	$-0.00873480\pi$
$48$			1.38411i			0.199779i
$49$	−1.00000			−0.142857
$50$	10.8933	−	1.96819i	1.54055	−	0.278343i
$51$	0.961936			0.134698
$52$			2.90155i			0.402373i
$53$		−	11.6841i		−	1.60493i	−0.596697	−	0.802466i	$-0.703521\pi$
$53$		−	11.6841i		−	1.60493i	0.596697	−	0.802466i	$-0.296479\pi$
$54$	−2.21394			−0.301280
$55$	−1.88158	+	2.25201i	−0.253713	+	0.303661i
$56$	−1.99598			−0.266724
$57$			6.74102i			0.892870i
$58$		−	3.11728i		−	0.409319i
$59$	2.84906			0.370917			0.185458	−	0.982652i	$-0.440623\pi$
$59$	2.84906			0.370917			0.185458	+	0.982652i	$0.440623\pi$
$60$	−4.97893	−	4.15996i	−0.642777	−	0.537048i
$61$	1.02391			0.131099			0.0655494	−	0.997849i	$-0.479120\pi$
$61$	1.02391			0.131099			0.0655494	+	0.997849i	$0.479120\pi$
$62$		−	4.21860i		−	0.535763i
$63$			1.00000i			0.125988i
$64$	12.8540			1.60676
$65$	1.71595	+	1.43370i	0.212838	+	0.177829i
$66$	2.90557			0.357651
$67$		−	7.67831i		−	0.938054i	−0.883184	−	0.469027i	$-0.844605\pi$
$67$		−	7.67831i		−	0.938054i	0.883184	−	0.469027i	$-0.155395\pi$
$68$		−	2.79111i		−	0.338471i
$69$	−3.62489			−0.436385
$70$	−3.17414	+	3.79903i	−0.379382	+	0.454070i
$71$	−2.90557			−0.344828			−0.172414	−	0.985025i	$-0.555157\pi$
$71$	−2.90557			−0.344828			−0.172414	+	0.985025i	$0.555157\pi$
$72$			1.99598i			0.235229i
$73$			2.39384i			0.280178i	0.990139	+	0.140089i	$0.0447389\pi$
$73$			2.39384i			0.280178i	−0.990139	+	0.140089i	$0.955261\pi$
$74$	1.03703			0.120552
$75$	−4.92033	+	0.888995i	−0.568151	+	0.102652i
$76$	19.5594			2.24362
$77$		−	1.31239i		−	0.149561i
$78$		−	2.21394i		−	0.250680i
$79$	−8.28161			−0.931754			−0.465877	−	0.884849i	$-0.654261\pi$
$79$	−8.28161			−0.931754			−0.465877	+	0.884849i	$0.654261\pi$
$80$	1.98440	−	2.37507i	0.221863	−	0.265540i
$81$	1.00000			0.111111
$82$			2.33726i			0.258107i
$83$		−	2.94509i		−	0.323265i	−0.986851	−	0.161633i	$-0.948324\pi$
$83$		−	2.94509i		−	0.323265i	0.986851	−	0.161633i	$-0.0516760\pi$
$84$	2.90155			0.316585
$85$	−1.65064	−	1.37913i	−0.179037	−	0.149588i
$86$	11.2587			1.21406
$87$			1.40802i			0.150956i
$88$		−	2.61951i		−	0.279241i
$89$	10.3109			1.09296			0.546479	−	0.837473i	$-0.315968\pi$
$89$	10.3109			1.09296			0.546479	+	0.837473i	$0.315968\pi$
$90$	3.79903	+	3.17414i	0.400453	+	0.334583i
$91$	−1.00000			−0.104828
$92$			10.5178i			1.09656i
$93$			1.90547i			0.197588i
$94$	0.832902			0.0859073
$95$	9.66462	−	11.5673i	0.991569	−	1.18678i
$96$	−7.05630			−0.720181
$97$			12.6919i			1.28866i	0.764746	+	0.644332i	$0.222864\pi$
$97$			12.6919i			1.28866i	−0.764746	+	0.644332i	$0.777136\pi$
$98$		−	2.21394i		−	0.223642i
$99$	−1.31239			−0.131901
$100$	2.57946	+	14.2766i	0.257946	+	1.42766i
$101$	2.20823			0.219727			0.109863	−	0.993947i	$-0.464959\pi$
$101$	2.20823			0.219727			0.109863	+	0.993947i	$0.464959\pi$
$102$			2.12967i			0.210869i
$103$		−	12.3920i		−	1.22102i	−0.792009	−	0.610509i	$-0.790965\pi$
$103$		−	12.3920i		−	1.22102i	0.792009	−	0.610509i	$-0.209035\pi$
$104$	−1.99598			−0.195722
$105$	1.43370	−	1.71595i	0.139915	−	0.167460i
$106$	25.8679			2.51251
$107$			17.5241i			1.69412i	0.531497	+	0.847060i	$0.321630\pi$
$107$			17.5241i			1.69412i	−0.531497	+	0.847060i	$0.678370\pi$
$108$		−	2.90155i		−	0.279202i
$109$	−2.29539			−0.219859			−0.109929	−	0.993939i	$-0.535062\pi$
$109$	−2.29539			−0.219859			−0.109929	+	0.993939i	$0.535062\pi$
$110$	−4.98582	−	4.16572i	−0.475379	−	0.397186i
$111$	−0.468408			−0.0444593
$112$			1.38411i			0.130786i
$113$		−	18.6247i		−	1.75207i	−0.482250	−	0.876034i	$-0.660180\pi$
$113$		−	18.6247i		−	1.75207i	0.482250	−	0.876034i	$-0.339820\pi$
$114$	−14.9242			−1.39778
$115$	6.22014	+	5.19701i	0.580031	+	0.484624i
$116$	4.08545			0.379324
$117$			1.00000i			0.0924500i
$118$			6.30767i			0.580668i
$119$	0.961936			0.0881805
$120$	2.86164	−	3.42501i	0.261231	−	0.312659i
$121$	−9.27762			−0.843420
$122$			2.26689i			0.205234i
$123$		−	1.05570i		−	0.0951893i
$124$	5.52881			0.496502
$125$	9.71762	+	5.52882i	0.869170	+	0.494513i
$126$	−2.21394			−0.197234
$127$		−	19.8495i		−	1.76136i	−0.473713	−	0.880680i	$-0.657086\pi$
$127$		−	19.8495i		−	1.76136i	0.473713	−	0.880680i	$-0.342914\pi$
$128$			14.3455i			1.26798i
$129$	−5.08537			−0.447742
$130$	−3.17414	+	3.79903i	−0.278390	+	0.333197i
$131$	−15.4353			−1.34859			−0.674293	−	0.738464i	$-0.735552\pi$
$131$	−15.4353			−1.34859			−0.674293	+	0.738464i	$0.735552\pi$
$132$			3.80798i			0.331442i
$133$			6.74102i			0.584521i
$134$	16.9993			1.46852
$135$	−1.71595	−	1.43370i	−0.147686	−	0.123393i
$136$	1.92001			0.164639
$137$		−	17.4660i		−	1.49222i	−0.665824	−	0.746109i	$-0.731920\pi$
$137$		−	17.4660i		−	1.49222i	0.665824	−	0.746109i	$-0.268080\pi$
$138$		−	8.02530i		−	0.683159i
$139$	−16.8610			−1.43013			−0.715065	−	0.699058i	$-0.753603\pi$
$139$	−16.8610			−1.43013			−0.715065	+	0.699058i	$0.753603\pi$
$140$	−4.97893	−	4.15996i	−0.420796	−	0.351581i
$141$	−0.376207			−0.0316824
$142$		−	6.43277i		−	0.539826i
$143$		−	1.31239i		−	0.109748i
$144$	1.38411			0.115342
$145$	2.01868	−	2.41610i	0.167643	−	0.200646i
$146$	−5.29984			−0.438618
$147$			1.00000i			0.0824786i
$148$			1.35911i			0.111718i
$149$	1.90560			0.156113			0.0780563	−	0.996949i	$-0.475129\pi$
$149$	1.90560			0.156113			0.0780563	+	0.996949i	$0.475129\pi$
$150$	−1.96819	−	10.8933i	−0.160702	−	0.889438i
$151$	−7.73417			−0.629398			−0.314699	−	0.949192i	$-0.601904\pi$
$151$	−7.73417			−0.629398			−0.314699	+	0.949192i	$0.601904\pi$
$152$			13.4550i			1.09134i
$153$		−	0.961936i		−	0.0777679i
$154$	2.90557			0.234137
$155$	2.73188	−	3.26970i	0.219429	−	0.262628i
$156$	2.90155			0.232310
$157$			17.8129i			1.42162i	0.703383	+	0.710811i	$0.251672\pi$
$157$			17.8129i			1.42162i	−0.703383	+	0.710811i	$0.748328\pi$
$158$		−	18.3350i		−	1.45866i
$159$	−11.6841			−0.926608
$160$	12.1083	+	10.1166i	0.957244	+	0.799790i
$161$	−3.62489			−0.285681
$162$			2.21394i			0.173944i
$163$		−	3.84000i		−	0.300772i	−0.988627	−	0.150386i	$-0.951948\pi$
$163$		−	3.84000i		−	0.300772i	0.988627	−	0.150386i	$-0.0480516\pi$
$164$	−3.06317			−0.239193
$165$	2.25201	+	1.88158i	0.175319	+	0.146481i
$166$	6.52026			0.506070
$167$			6.14901i			0.475825i	0.971287	+	0.237912i	$0.0764631\pi$
$167$			6.14901i			0.475825i	−0.971287	+	0.237912i	$0.923537\pi$
$168$			1.99598i			0.153993i
$169$	−1.00000			−0.0769231
$170$	3.05332	−	3.65442i	0.234179	−	0.280281i
$171$	6.74102			0.515499
$172$			14.7554i			1.12509i
$173$			17.9828i			1.36721i	0.729852	+	0.683605i	$0.239589\pi$
$173$			17.9828i			1.36721i	−0.729852	+	0.683605i	$0.760411\pi$
$174$	−3.11728			−0.236321
$175$	−4.92033	+	0.888995i	−0.371942	+	0.0672017i
$176$	−1.81650			−0.136924
$177$		−	2.84906i		−	0.214149i
$178$			22.8279i			1.71102i
$179$	−10.7729			−0.805206			−0.402603	−	0.915375i	$-0.631894\pi$
$179$	−10.7729			−0.805206			−0.402603	+	0.915375i	$0.631894\pi$
$180$	−4.15996	+	4.97893i	−0.310065	+	0.371107i
$181$	−0.731579			−0.0543778			−0.0271889	−	0.999630i	$-0.508656\pi$
$181$	−0.731579			−0.0543778			−0.0271889	+	0.999630i	$0.508656\pi$
$182$		−	2.21394i		−	0.164108i
$183$		−	1.02391i		−	0.0756899i
$184$	−7.23521			−0.533387
$185$	0.803767	+	0.671558i	0.0590941	+	0.0493739i
$186$	−4.21860			−0.309323
$187$			1.26244i			0.0923187i
$188$			1.09158i			0.0796120i
$189$	1.00000			0.0727393
$190$	25.6093	+	21.3969i	1.85790	+	1.55230i
$191$	−3.41082			−0.246798			−0.123399	−	0.992357i	$-0.539380\pi$
$191$	−3.41082			−0.246798			−0.123399	+	0.992357i	$0.539380\pi$
$192$		−	12.8540i		−	0.927661i
$193$			22.5412i			1.62255i	0.584665	+	0.811275i	$0.301226\pi$
$193$			22.5412i			1.62255i	−0.584665	+	0.811275i	$0.698774\pi$
$194$	−28.0991			−2.01740
$195$	1.43370	−	1.71595i	0.102670	−	0.122882i
$196$	2.90155			0.207254
$197$			8.50143i			0.605702i	0.953038	+	0.302851i	$0.0979385\pi$
$197$			8.50143i			0.605702i	−0.953038	+	0.302851i	$0.902062\pi$
$198$		−	2.90557i		−	0.206490i
$199$	−5.25603			−0.372590			−0.186295	−	0.982494i	$-0.559648\pi$
$199$	−5.25603			−0.372590			−0.186295	+	0.982494i	$0.559648\pi$
$200$	−9.82089	+	1.77442i	−0.694442	+	0.125470i
$201$	−7.67831			−0.541586
$202$			4.88889i			0.343981i
$203$			1.40802i			0.0988238i
$204$	−2.79111			−0.195416
$205$	−1.51356	+	1.81153i	−0.105712	+	0.126523i
$206$	27.4352			1.91150
$207$			3.62489i			0.251947i
$208$			1.38411i			0.0959706i
$209$	−8.84688			−0.611952
$210$	3.79903	+	3.17414i	0.262158	+	0.219036i
$211$	−7.55756			−0.520284			−0.260142	−	0.965570i	$-0.583769\pi$
$211$	−7.55756			−0.520284			−0.260142	+	0.965570i	$0.583769\pi$
$212$			33.9020i			2.32840i
$213$			2.90557i			0.199086i
$214$	−38.7974			−2.65214
$215$	8.72626	+	7.29090i	0.595126	+	0.497236i
$216$	1.99598			0.135809
$217$			1.90547i			0.129352i
$218$		−	5.08187i		−	0.344188i
$219$	2.39384			0.161761
$220$	5.45951	−	6.53431i	0.368080	−	0.440543i
$221$	0.961936			0.0647068
$222$		−	1.03703i		−	0.0696009i
$223$			17.3850i			1.16419i	0.813122	+	0.582093i	$0.197766\pi$
$223$			17.3850i			1.16419i	−0.813122	+	0.582093i	$0.802234\pi$
$224$	−7.05630			−0.471469
$225$	0.888995	+	4.92033i	0.0592663	+	0.328022i
$226$	41.2341			2.74285
$227$			7.21108i			0.478616i	0.970944	+	0.239308i	$0.0769206\pi$
$227$			7.21108i			0.478616i	−0.970944	+	0.239308i	$0.923079\pi$
$228$		−	19.5594i		−	1.29535i
$229$	−9.36239			−0.618684			−0.309342	−	0.950951i	$-0.600109\pi$
$229$	−9.36239			−0.618684			−0.309342	+	0.950951i	$0.600109\pi$
$230$	−11.5059	+	13.7710i	−0.758676	+	0.908036i
$231$	−1.31239			−0.0863492
$232$			2.81039i			0.184511i
$233$			3.92008i			0.256813i	0.991722	+	0.128407i	$0.0409862\pi$
$233$			3.92008i			0.256813i	−0.991722	+	0.128407i	$0.959014\pi$
$234$	−2.21394			−0.144730
$235$	0.645554	+	0.539369i	0.0421113	+	0.0351846i
$236$	−8.26670			−0.538117
$237$			8.28161i			0.537948i
$238$			2.12967i			0.138046i
$239$	−0.661837			−0.0428107			−0.0214053	−	0.999771i	$-0.506814\pi$
$239$	−0.661837			−0.0428107			−0.0214053	+	0.999771i	$0.506814\pi$
$240$	−2.37507	−	1.98440i	−0.153310	−	0.128092i
$241$	−6.92832			−0.446292			−0.223146	−	0.974785i	$-0.571633\pi$
$241$	−6.92832			−0.446292			−0.223146	+	0.974785i	$0.571633\pi$
$242$		−	20.5401i		−	1.32037i
$243$		−	1.00000i		−	0.0641500i
$244$	−2.97094			−0.190195
$245$	1.43370	−	1.71595i	0.0915959	−	0.109628i
$246$	2.33726			0.149018
$247$			6.74102i			0.428921i
$248$			3.80328i			0.241509i
$249$	−2.94509			−0.186637
$250$	−12.2405	+	21.5143i	−0.774157	+	1.36068i
$251$	−31.3623			−1.97957			−0.989785	−	0.142567i	$-0.954464\pi$
$251$	−31.3623			−1.97957			−0.989785	+	0.142567i	$0.954464\pi$
$252$		−	2.90155i		−	0.182780i
$253$		−	4.75728i		−	0.299088i
$254$	43.9457			2.75740
$255$	−1.37913	+	1.65064i	−0.0863645	+	0.103367i
$256$	−6.05213			−0.378258
$257$			7.87232i			0.491062i	0.969389	+	0.245531i	$0.0789623\pi$
$257$			7.87232i			0.491062i	−0.969389	+	0.245531i	$0.921038\pi$
$258$		−	11.2587i		−	0.700937i
$259$	−0.468408			−0.0291055
$260$	−4.97893	−	4.15996i	−0.308780	−	0.257990i
$261$	1.40802			0.0871544
$262$		−	34.1728i		−	2.11121i
$263$			23.6347i			1.45738i	0.684846	+	0.728688i	$0.259870\pi$
$263$			23.6347i			1.45738i	−0.684846	+	0.728688i	$0.740130\pi$
$264$	−2.61951			−0.161220
$265$	20.0494	+	16.7515i	1.23162	+	1.02904i
$266$	−14.9242			−0.915064
$267$		−	10.3109i		−	0.631020i
$268$			22.2790i			1.36091i
$269$	−9.64340			−0.587968			−0.293984	−	0.955810i	$-0.594981\pi$
$269$	−9.64340			−0.587968			−0.293984	+	0.955810i	$0.594981\pi$
$270$	3.17414	−	3.79903i	0.193172	−	0.231201i
$271$	11.5739			0.703067			0.351533	−	0.936175i	$-0.385660\pi$
$271$	11.5739			0.703067			0.351533	+	0.936175i	$0.385660\pi$
$272$		−	1.33142i		−	0.0807294i
$273$			1.00000i			0.0605228i
$274$	38.6687			2.33606
$275$	−1.16671	−	6.45742i	−0.0703554	−	0.389397i
$276$	10.5178			0.633097
$277$			3.95685i			0.237744i	0.992910	+	0.118872i	$0.0379278\pi$
$277$			3.95685i			0.237744i	−0.992910	+	0.118872i	$0.962072\pi$
$278$		−	37.3293i		−	2.23886i
$279$	1.90547			0.114077
$280$	2.86164	−	3.42501i	0.171016	−	0.204684i
$281$	−8.33607			−0.497288			−0.248644	−	0.968595i	$-0.579985\pi$
$281$	−8.33607			−0.497288			−0.248644	+	0.968595i	$0.579985\pi$
$282$		−	0.832902i		−	0.0495986i
$283$		−	12.8317i		−	0.762763i	−0.924418	−	0.381382i	$-0.875448\pi$
$283$		−	12.8317i		−	0.762763i	0.924418	−	0.381382i	$-0.124552\pi$
$284$	8.43065			0.500267
$285$	−11.5673	−	9.66462i	−0.685187	−	0.572483i
$286$	2.90557			0.171810
$287$		−	1.05570i		−	0.0623160i
$288$			7.05630i			0.415797i
$289$	16.0747			0.945569
$290$	5.34911	+	4.46926i	0.314111	+	0.262444i
$291$	12.6919			0.744010
$292$		−	6.94586i		−	0.406475i
$293$		−	24.8191i		−	1.44995i	−0.688777	−	0.724973i	$-0.741852\pi$
$293$		−	24.8191i		−	1.44995i	0.688777	−	0.724973i	$-0.258148\pi$
$294$	−2.21394			−0.129120
$295$	−4.08471	+	4.88886i	−0.237821	+	0.284641i
$296$	−0.934934			−0.0543419
$297$			1.31239i			0.0761529i
$298$			4.21889i			0.244393i
$299$	−3.62489			−0.209633
$300$	14.2766	−	2.57946i	0.824260	−	0.148925i
$301$	−5.08537			−0.293116
$302$		−	17.1230i		−	0.985319i
$303$		−	2.20823i		−	0.126859i
$304$	9.33030			0.535130
$305$	−1.46799	+	1.75699i	−0.0840567	+	0.100605i
$306$	2.12967			0.121745
$307$		−	28.2873i		−	1.61444i	−0.590249	−	0.807221i	$-0.700970\pi$
$307$		−	28.2873i		−	1.61444i	0.590249	−	0.807221i	$-0.299030\pi$
$308$			3.80798i			0.216980i
$309$	−12.3920			−0.704956
$310$	7.23893	+	6.04822i	0.411143	+	0.343516i
$311$	−5.62324			−0.318865			−0.159432	−	0.987209i	$-0.550966\pi$
$311$	−5.62324			−0.318865			−0.159432	+	0.987209i	$0.550966\pi$
$312$			1.99598i			0.113000i
$313$		−	11.2246i		−	0.634450i	−0.948350	−	0.317225i	$-0.897249\pi$
$313$		−	11.2246i		−	0.634450i	0.948350	−	0.317225i	$-0.102751\pi$
$314$	−39.4367			−2.22554
$315$	−1.71595	−	1.43370i	−0.0966830	−	0.0807800i
$316$	24.0295			1.35177
$317$		−	4.83778i		−	0.271717i	−0.990728	−	0.135859i	$-0.956621\pi$
$317$		−	4.83778i		−	0.271717i	0.990728	−	0.135859i	$-0.0433793\pi$
$318$		−	25.8679i		−	1.45060i
$319$	−1.84788			−0.103461
$320$	−18.4289	+	22.0569i	−1.03021	+	1.23302i
$321$	17.5241			0.978101
$322$		−	8.02530i		−	0.447233i
$323$		−	6.48443i		−	0.360803i
$324$	−2.90155			−0.161197
$325$	−4.92033	+	0.888995i	−0.272931	+	0.0493126i
$326$	8.50154			0.470857
$327$			2.29539i			0.126936i
$328$		−	2.10716i		−	0.116348i
$329$	−0.376207			−0.0207410
$330$	−4.16572	+	4.98582i	−0.229315	+	0.274460i
$331$	−29.9041			−1.64368			−0.821839	−	0.569720i	$-0.807052\pi$
$331$	−29.9041			−1.64368			−0.821839	+	0.569720i	$0.807052\pi$
$332$			8.54532i			0.468985i
$333$			0.468408i			0.0256686i
$334$	−13.6136			−0.744901
$335$	13.1756	+	11.0084i	0.719861	+	0.601453i
$336$	1.38411			0.0755093
$337$		−	32.8948i		−	1.79190i	−0.444160	−	0.895948i	$-0.646498\pi$
$337$		−	32.8948i		−	1.79190i	0.444160	−	0.895948i	$-0.353502\pi$
$338$		−	2.21394i		−	0.120423i
$339$	−18.6247			−1.01156
$340$	4.78941	+	4.00161i	0.259742	+	0.217018i
$341$	−2.50073			−0.135422
$342$			14.9242i			0.807011i
$343$			1.00000i			0.0539949i
$344$	−10.1503			−0.547267
$345$	5.19701	−	6.22014i	0.279798	−	0.334881i
$346$	−39.8130			−2.14036
$347$		−	35.6847i		−	1.91565i	−0.287347	−	0.957827i	$-0.592773\pi$
$347$		−	35.6847i		−	1.91565i	0.287347	−	0.957827i	$-0.407227\pi$
$348$		−	4.08545i		−	0.219003i
$349$	−16.2559			−0.870158			−0.435079	−	0.900392i	$-0.643280\pi$
$349$	−16.2559			−0.870158			−0.435079	+	0.900392i	$0.643280\pi$
$350$	−1.96819	−	10.8933i	−0.105204	−	0.582274i
$351$	1.00000			0.0533761
$352$		−	9.26065i		−	0.493594i
$353$			26.7866i			1.42571i	0.701313	+	0.712854i	$0.252598\pi$
$353$			26.7866i			1.42571i	−0.701313	+	0.712854i	$0.747402\pi$
$354$	6.30767			0.335249
$355$	4.16572	−	4.98582i	0.221093	−	0.264620i
$356$	−29.9177			−1.58564
$357$		−	0.961936i		−	0.0509110i
$358$		−	23.8506i		−	1.26055i
$359$	23.1586			1.22227			0.611133	−	0.791528i	$-0.290714\pi$
$359$	23.1586			1.22227			0.611133	+	0.791528i	$0.290714\pi$
$360$	−3.42501	−	2.86164i	−0.180514	−	0.150822i
$361$	26.4414			1.39165
$362$		−	1.61967i		−	0.0851282i
$363$			9.27762i			0.486949i
$364$	2.90155			0.152083
$365$	−4.10772	−	3.43206i	−0.215008	−	0.179642i
$366$	2.26689			0.118492
$367$		−	16.3496i		−	0.853445i	−0.904383	−	0.426722i	$-0.859668\pi$
$367$		−	16.3496i		−	0.853445i	0.904383	−	0.426722i	$-0.140332\pi$
$368$			5.01724i			0.261542i
$369$	−1.05570			−0.0549576
$370$	−1.48679	+	1.77949i	−0.0772946	+	0.0925115i
$371$	−11.6841			−0.606608
$372$		−	5.52881i		−	0.286656i
$373$			28.5777i			1.47970i	0.672773	+	0.739849i	$0.265103\pi$
$373$			28.5777i			1.47970i	−0.672773	+	0.739849i	$0.734897\pi$
$374$	−2.79497			−0.144525
$375$	5.52882	−	9.71762i	0.285507	−	0.501816i
$376$	−0.750903			−0.0387249
$377$			1.40802i			0.0725168i
$378$			2.21394i			0.113873i
$379$	19.8566			1.01997			0.509983	−	0.860185i	$-0.329652\pi$
$379$	19.8566			1.01997			0.509983	+	0.860185i	$0.329652\pi$
$380$	−28.0424	+	33.5630i	−1.43854	+	1.72175i
$381$	−19.8495			−1.01692
$382$		−	7.55137i		−	0.386361i
$383$		−	15.7082i		−	0.802652i	−0.915935	−	0.401326i	$-0.868550\pi$
$383$		−	15.7082i		−	0.802652i	0.915935	−	0.401326i	$-0.131450\pi$
$384$	14.3455			0.732067
$385$	2.25201	+	1.88158i	0.114773	+	0.0958944i
$386$	−49.9049			−2.54009
$387$			5.08537i			0.258504i
$388$		−	36.8261i		−	1.86956i
$389$	18.0392			0.914626			0.457313	−	0.889306i	$-0.348812\pi$
$389$	18.0392			0.914626			0.457313	+	0.889306i	$0.348812\pi$
$390$	3.79903	+	3.17414i	0.192371	+	0.160729i
$391$	3.48691			0.176341
$392$			1.99598i			0.100812i
$393$			15.4353i			0.778607i
$394$	−18.8217			−0.948224
$395$	11.8734	−	14.2109i	0.597414	−	0.715026i
$396$	3.80798			0.191358
$397$			0.0326359i			0.00163795i	1.00000		0.000818974i	$0.000260688\pi$
$397$			0.0326359i			0.00163795i	−1.00000		0.000818974i	$0.999739\pi$
$398$		−	11.6366i		−	0.583288i
$399$	6.74102			0.337473
$400$	1.23047	+	6.81028i	0.0615233	+	0.340514i
$401$	−11.9135			−0.594933			−0.297466	−	0.954732i	$-0.596142\pi$
$401$	−11.9135			−0.594933			−0.297466	+	0.954732i	$0.596142\pi$
$402$		−	16.9993i		−	0.847850i
$403$			1.90547i			0.0949182i
$404$	−6.40728			−0.318774
$405$	−1.43370	+	1.71595i	−0.0712412	+	0.0852664i
$406$	−3.11728			−0.154708
$407$		−	0.614736i		−	0.0304713i
$408$		−	1.92001i		−	0.0950545i
$409$	−16.5244			−0.817081			−0.408541	−	0.912740i	$-0.633962\pi$
$409$	−16.5244			−0.817081			−0.408541	+	0.912740i	$0.633962\pi$
$410$	−4.01063	−	3.35094i	−0.198071	−	0.165491i
$411$	−17.4660			−0.861532
$412$			35.9560i			1.77142i
$413$		−	2.84906i		−	0.140193i
$414$	−8.02530			−0.394422
$415$	5.05363	+	4.22238i	0.248073	+	0.207268i
$416$	−7.05630			−0.345964
$417$			16.8610i			0.825686i
$418$		−	19.5865i		−	0.958007i
$419$	−8.69657			−0.424855			−0.212428	−	0.977177i	$-0.568137\pi$
$419$	−8.69657			−0.424855			−0.212428	+	0.977177i	$0.568137\pi$
$420$	−4.15996	+	4.97893i	−0.202985	+	0.242947i
$421$	3.58737			0.174838			0.0874188	−	0.996172i	$-0.472138\pi$
$421$	3.58737			0.174838			0.0874188	+	0.996172i	$0.472138\pi$
$422$		−	16.7320i		−	0.814502i
$423$			0.376207i			0.0182918i
$424$	−23.3212			−1.13258
$425$	4.73305	−	0.855156i	0.229586	−	0.0414812i
$426$	−6.43277			−0.311669
$427$		−	1.02391i		−	0.0495507i
$428$		−	50.8471i		−	2.45779i
$429$	−1.31239			−0.0633630
$430$	−16.1417	+	19.3194i	−0.778420	+	0.931666i
$431$	19.7370			0.950697			0.475348	−	0.879798i	$-0.342322\pi$
$431$	19.7370			0.950697			0.475348	+	0.879798i	$0.342322\pi$
$432$		−	1.38411i		−	0.0665929i
$433$			6.28972i			0.302265i	0.988514	+	0.151132i	$0.0482919\pi$
$433$			6.28972i			0.302265i	−0.988514	+	0.151132i	$0.951708\pi$
$434$	−4.21860			−0.202499
$435$	−2.41610	−	2.01868i	−0.115843	−	0.0967885i
$436$	6.66020			0.318966
$437$			24.4355i			1.16891i
$438$			5.29984i			0.253236i
$439$	−5.22802			−0.249520			−0.124760	−	0.992187i	$-0.539816\pi$
$439$	−5.22802			−0.249520			−0.124760	+	0.992187i	$0.539816\pi$
$440$	4.49497	+	3.75560i	0.214289	+	0.179041i
$441$	1.00000			0.0476190
$442$			2.12967i			0.101298i
$443$			19.9130i			0.946097i	0.881036	+	0.473049i	$0.156846\pi$
$443$			19.9130i			0.946097i	−0.881036	+	0.473049i	$0.843154\pi$
$444$	1.35911			0.0645005
$445$	−14.7828	+	17.6931i	−0.700773	+	0.838734i
$446$	−38.4894			−1.82253
$447$		−	1.90560i		−	0.0901316i
$448$		−	12.8540i		−	0.607296i
$449$	6.76117			0.319079			0.159540	−	0.987192i	$-0.448999\pi$
$449$	6.76117			0.319079			0.159540	+	0.987192i	$0.448999\pi$
$450$	−10.8933	+	1.96819i	−0.513517	+	0.0927811i
$451$	1.38550			0.0652404
$452$			54.0406i			2.54186i
$453$			7.73417i			0.363383i
$454$	−15.9649			−0.749271
$455$	1.43370	−	1.71595i	0.0672130	−	0.0804452i
$456$	13.4550			0.630086
$457$			26.5031i			1.23976i	0.784696	+	0.619881i	$0.212819\pi$
$457$			26.5031i			1.23976i	−0.784696	+	0.619881i	$0.787181\pi$
$458$		−	20.7278i		−	0.968547i
$459$	−0.961936			−0.0448993
$460$	−18.0481	−	15.0794i	−0.841495	−	0.703080i
$461$	9.98485			0.465041			0.232520	−	0.972592i	$-0.425303\pi$
$461$	9.98485			0.465041			0.232520	+	0.972592i	$0.425303\pi$
$462$		−	2.90557i		−	0.135179i
$463$		−	28.2047i		−	1.31078i	−0.755289	−	0.655392i	$-0.772503\pi$
$463$		−	28.2047i		−	1.31078i	0.755289	−	0.655392i	$-0.227497\pi$
$464$	1.94885			0.0904733
$465$	−3.26970	−	2.73188i	−0.151629	−	0.126688i
$466$	−8.67885			−0.402040
$467$			8.20146i			0.379518i	0.981831	+	0.189759i	$0.0607707\pi$
$467$			8.20146i			0.379518i	−0.981831	+	0.189759i	$0.939229\pi$
$468$		−	2.90155i		−	0.134124i
$469$	−7.67831			−0.354551
$470$	−1.19413	+	1.42922i	−0.0550813	+	0.0659251i
$471$	17.8129			0.820774
$472$		−	5.68668i		−	0.261751i
$473$		−	6.67401i		−	0.306871i
$474$	−18.3350			−0.842156
$475$	5.99273	+	33.1681i	0.274966	+	1.52186i
$476$	−2.79111			−0.127930
$477$			11.6841i			0.534978i
$478$		−	1.46527i		−	0.0670199i
$479$	25.1767			1.15035			0.575177	−	0.818029i	$-0.304933\pi$
$479$	25.1767			1.15035			0.575177	+	0.818029i	$0.304933\pi$
$480$	10.1166	−	12.1083i	0.461759	−	0.552665i
$481$	−0.468408			−0.0213576
$482$		−	15.3389i		−	0.698668i
$483$			3.62489i			0.164938i
$484$	26.9195			1.22361
$485$	−21.7787	−	18.1964i	−0.988918	−	0.826254i
$486$	2.21394			0.100427
$487$			13.7244i			0.621913i	0.950424	+	0.310957i	$0.100649\pi$
$487$			13.7244i			0.621913i	−0.950424	+	0.310957i	$0.899351\pi$
$488$		−	2.04371i		−	0.0925145i
$489$	−3.84000			−0.173651
$490$	3.79903	+	3.17414i	0.171623	+	0.143393i
$491$	5.45181			0.246037			0.123018	−	0.992404i	$-0.460743\pi$
$491$	5.45181			0.246037			0.123018	+	0.992404i	$0.460743\pi$
$492$			3.06317i			0.138098i
$493$		−	1.35443i		−	0.0610003i
$494$	−14.9242			−0.671474
$495$	1.88158	−	2.25201i	0.0845709	−	0.101220i
$496$	2.63738			0.118422
$497$			2.90557i			0.130333i
$498$		−	6.52026i		−	0.292180i
$499$	36.7954			1.64719			0.823595	−	0.567179i	$-0.191965\pi$
$499$	36.7954			1.64719			0.823595	+	0.567179i	$0.191965\pi$
$500$	−28.1962	−	16.0421i	−1.26097	−	0.717427i
$501$	6.14901			0.274717
$502$		−	69.4344i		−	3.09901i
$503$			7.17093i			0.319736i	0.987138	+	0.159868i	$0.0511069\pi$
$503$			7.17093i			0.319736i	−0.987138	+	0.159868i	$0.948893\pi$
$504$	1.99598			0.0889081
$505$	−3.16594	+	3.78922i	−0.140883	+	0.168618i
$506$	10.5324			0.468221
$507$			1.00000i			0.0444116i
$508$			57.5943i			2.55534i
$509$	24.2551			1.07509			0.537545	−	0.843235i	$-0.319352\pi$
$509$	24.2551			1.07509			0.537545	+	0.843235i	$0.319352\pi$
$510$	−3.65442	−	3.05332i	−0.161820	−	0.135203i
$511$	2.39384			0.105897
$512$			15.2920i			0.675817i
$513$		−	6.74102i		−	0.297623i
$514$	−17.4289			−0.768755
$515$	21.2641	+	17.7664i	0.937007	+	0.782882i
$516$	14.7554			0.649572
$517$		−	0.493732i		−	0.0217143i
$518$		−	1.03703i		−	0.0455645i
$519$	17.9828			0.789359
$520$	2.86164	−	3.42501i	0.125491	−	0.150197i
$521$	−6.95695			−0.304789			−0.152395	−	0.988320i	$-0.548698\pi$
$521$	−6.95695			−0.304789			−0.152395	+	0.988320i	$0.548698\pi$
$522$			3.11728i			0.136440i
$523$		−	21.7172i		−	0.949626i	−0.880087	−	0.474813i	$-0.842516\pi$
$523$		−	21.7172i		−	0.949626i	0.880087	−	0.474813i	$-0.157484\pi$
$524$	44.7862			1.95650
$525$	0.888995	+	4.92033i	0.0387989	+	0.214741i
$526$	−52.3258			−2.28152
$527$		−	1.83294i		−	0.0798441i
$528$			1.81650i			0.0790528i
$529$	9.86018			0.428703
$530$	−37.0869	+	44.3882i	−1.61095	+	1.92810i
$531$	−2.84906			−0.123639
$532$		−	19.5594i		−	0.848008i
$533$		−	1.05570i		−	0.0457275i
$534$	22.8279			0.987858
$535$	−30.0706	−	25.1244i	−1.30006	−	1.08622i
$536$	−15.3258			−0.661972
$537$			10.7729i			0.464886i
$538$		−	21.3499i		−	0.920461i
$539$	−1.31239			−0.0565288
$540$	4.97893	+	4.15996i	0.214259	+	0.179016i
$541$	−29.3660			−1.26254			−0.631272	−	0.775562i	$-0.717467\pi$
$541$	−29.3660			−1.26254			−0.631272	+	0.775562i	$0.717467\pi$
$542$			25.6240i			1.10065i
$543$			0.731579i			0.0313950i
$544$	6.78771			0.291021
$545$	3.29091	−	3.93879i	0.140967	−	0.168719i
$546$	−2.21394			−0.0947481
$547$			20.7560i			0.887462i	0.896160	+	0.443731i	$0.146346\pi$
$547$			20.7560i			0.887462i	−0.896160	+	0.443731i	$0.853654\pi$
$548$			50.6784i			2.16487i
$549$	−1.02391			−0.0436996
$550$	14.2964	−	2.58304i	0.609599	−	0.110141i
$551$	9.49151			0.404352
$552$			7.23521i			0.307951i
$553$			8.28161i			0.352170i
$554$	−8.76024			−0.372187
$555$	0.671558	−	0.803767i	0.0285060	−	0.0341180i
$556$	48.9230			2.07480
$557$			7.26865i			0.307983i	0.988072	+	0.153991i	$0.0492128\pi$
$557$			7.26865i			0.307983i	−0.988072	+	0.153991i	$0.950787\pi$
$558$			4.21860i			0.178588i
$559$	−5.08537			−0.215088
$560$	−2.37507	−	1.98440i	−0.100365	−	0.0838562i
$561$	1.26244			0.0533002
$562$		−	18.4556i		−	0.778503i
$563$			4.89336i			0.206230i	0.994669	+	0.103115i	$0.0328810\pi$
$563$			4.89336i			0.206230i	−0.994669	+	0.103115i	$0.967119\pi$
$564$	1.09158			0.0459640
$565$	31.9592	+	26.7023i	1.34453	+	1.12338i
$566$	28.4086			1.19410
$567$		−	1.00000i		−	0.0419961i
$568$			5.79946i			0.243340i
$569$	15.0651			0.631561			0.315781	−	0.948832i	$-0.397734\pi$
$569$	15.0651			0.631561			0.315781	+	0.948832i	$0.397734\pi$
$570$	21.3969	−	25.6093i	0.896219	−	1.07266i
$571$	8.10710			0.339272			0.169636	−	0.985507i	$-0.445741\pi$
$571$	8.10710			0.339272			0.169636	+	0.985507i	$0.445741\pi$
$572$			3.80798i			0.159220i
$573$			3.41082i			0.142489i
$574$	2.33726			0.0975554
$575$	−17.8357	+	3.22251i	−0.743799	+	0.134388i
$576$	−12.8540			−0.535585
$577$		−	36.3014i		−	1.51125i	−0.655006	−	0.755624i	$-0.727334\pi$
$577$		−	36.3014i		−	1.51125i	0.655006	−	0.755624i	$-0.272666\pi$
$578$			35.5884i			1.48028i
$579$	22.5412			0.936779
$580$	−5.85731	+	7.01044i	−0.243212	+	0.291093i
$581$	−2.94509			−0.122183
$582$			28.0991i			1.16474i
$583$		−	15.3341i		−	0.635075i
$584$	4.77807			0.197718
$585$	−1.71595	−	1.43370i	−0.0709460	−	0.0592763i
$586$	54.9481			2.26988
$587$			6.22753i			0.257038i	0.991707	+	0.128519i	$0.0410223\pi$
$587$			6.22753i			0.257038i	−0.991707	+	0.128519i	$0.958978\pi$
$588$		−	2.90155i		−	0.119658i
$589$	12.8448			0.529261
$590$	−10.8237	−	9.04332i	−0.445604	−	0.372308i
$591$	8.50143			0.349702
$592$			0.648328i			0.0266461i
$593$		−	1.33243i		−	0.0547165i	−0.999626	−	0.0273583i	$-0.991291\pi$
$593$		−	1.33243i		−	0.0547165i	0.999626	−	0.0273583i	$-0.00870950\pi$
$594$	−2.90557			−0.119217
$595$	−1.37913	+	1.65064i	−0.0565388	+	0.0676695i
$596$	−5.52918			−0.226484
$597$			5.25603i			0.215115i
$598$		−	8.02530i		−	0.328179i
$599$	38.0056			1.55287			0.776433	−	0.630200i	$-0.217027\pi$
$599$	38.0056			1.55287			0.776433	+	0.630200i	$0.217027\pi$
$600$	1.77442	+	9.82089i	0.0724403	+	0.400936i
$601$	−25.8801			−1.05567			−0.527837	−	0.849346i	$-0.676997\pi$
$601$	−25.8801			−1.05567			−0.527837	+	0.849346i	$0.676997\pi$
$602$		−	11.2587i		−	0.458871i
$603$			7.67831i			0.312685i
$604$	22.4411			0.913115
$605$	13.3013	−	15.9200i	0.540777	−	0.647239i
$606$	4.88889			0.198598
$607$		−	27.6938i		−	1.12406i	−0.827118	−	0.562029i	$-0.810021\pi$
$607$		−	27.6938i		−	1.12406i	0.827118	−	0.562029i	$-0.189979\pi$
$608$			47.5667i			1.92908i
$609$	1.40802			0.0570559
$610$	−3.88988	−	3.25004i	−0.157496	−	0.131590i
$611$	−0.376207			−0.0152197
$612$			2.79111i			0.112824i
$613$		−	29.3432i		−	1.18516i	−0.805511	−	0.592581i	$-0.798109\pi$
$613$		−	29.3432i		−	1.18516i	0.805511	−	0.592581i	$-0.201891\pi$
$614$	62.6266			2.52740
$615$	1.81153	+	1.51356i	0.0730481	+	0.0610326i
$616$	−2.61951			−0.105543
$617$			30.4475i			1.22577i	0.790172	+	0.612885i	$0.209991\pi$
$617$			30.4475i			1.22577i	−0.790172	+	0.612885i	$0.790009\pi$
$618$		−	27.4352i		−	1.10360i
$619$	−39.5946			−1.59144			−0.795722	−	0.605663i	$-0.792908\pi$
$619$	−39.5946			−1.59144			−0.795722	+	0.605663i	$0.792908\pi$
$620$	−7.92667	+	9.48719i	−0.318343	+	0.381015i
$621$	3.62489			0.145462
$622$		−	12.4495i		−	0.499181i
$623$		−	10.3109i		−	0.413099i
$624$	1.38411			0.0554087
$625$	−23.4194	+	8.74830i	−0.936775	+	0.349932i
$626$	24.8506			0.993229
$627$			8.84688i			0.353310i
$628$		−	51.6850i		−	2.06245i
$629$	0.450579			0.0179657
$630$	3.17414	−	3.79903i	0.126461	−	0.151357i
$631$	−24.0453			−0.957230			−0.478615	−	0.878025i	$-0.658861\pi$
$631$	−24.0453			−0.957230			−0.478615	+	0.878025i	$0.658861\pi$
$632$			16.5299i			0.657526i
$633$			7.55756i			0.300386i
$634$	10.7106			0.425372
$635$	34.0608	+	28.4583i	1.35166	+	1.12933i
$636$	33.9020			1.34430
$637$			1.00000i			0.0396214i
$638$		−	4.09110i		−	0.161968i
$639$	2.90557			0.114943
$640$	−24.6163	−	20.5672i	−0.973044	−	0.812991i
$641$	−24.4052			−0.963949			−0.481975	−	0.876185i	$-0.660080\pi$
$641$	−24.4052			−0.963949			−0.481975	+	0.876185i	$0.660080\pi$
$642$			38.7974i			1.53121i
$643$			12.7923i			0.504478i	0.967665	+	0.252239i	$0.0811669\pi$
$643$			12.7923i			0.504478i	−0.967665	+	0.252239i	$0.918833\pi$
$644$	10.5178			0.414459
$645$	7.29090	−	8.72626i	0.287079	−	0.343596i
$646$	14.3562			0.564836
$647$			30.1091i			1.18371i	0.806044	+	0.591855i	$0.201604\pi$
$647$			30.1091i			1.18371i	−0.806044	+	0.591855i	$0.798396\pi$
$648$		−	1.99598i		−	0.0784095i
$649$	3.73910			0.146772
$650$	−1.96819	−	10.8933i	−0.0771986	−	0.427272i
$651$	1.90547			0.0746812
$652$			11.1419i			0.436352i
$653$			47.1317i			1.84440i	0.386708	+	0.922202i	$0.373612\pi$
$653$			47.1317i			1.84440i	−0.386708	+	0.922202i	$0.626388\pi$
$654$	−5.08187			−0.198717
$655$	22.1296	−	26.4862i	0.864675	−	1.03490i
$656$	−1.46120			−0.0570504
$657$		−	2.39384i		−	0.0933927i
$658$		−	0.832902i		−	0.0324699i
$659$	14.3200			0.557828			0.278914	−	0.960316i	$-0.410026\pi$
$659$	14.3200			0.557828			0.278914	+	0.960316i	$0.410026\pi$
$660$	−6.53431	−	5.45951i	−0.254348	−	0.212511i
$661$	0.151327			0.00588593			0.00294297	−	0.999996i	$-0.499063\pi$
$661$	0.151327			0.00588593			0.00294297	+	0.999996i	$0.499063\pi$
$662$		−	66.2060i		−	2.57317i
$663$		−	0.961936i		−	0.0373585i
$664$	−5.87834			−0.228124
$665$	−11.5673	−	9.66462i	−0.448560	−	0.374778i
$666$	−1.03703			−0.0401841
$667$			5.10392i			0.197625i
$668$		−	17.8417i		−	0.690314i
$669$	17.3850			0.672143
$670$	−24.3720	+	29.1701i	−0.941572	+	1.12694i
$671$	1.34378			0.0518760
$672$			7.05630i			0.272203i
$673$		−	20.1434i		−	0.776470i	−0.921560	−	0.388235i	$-0.873085\pi$
$673$		−	20.1434i		−	0.776470i	0.921560	−	0.388235i	$-0.126915\pi$
$674$	72.8273			2.80520
$675$	4.92033	−	0.888995i	0.189384	−	0.0342174i
$676$	2.90155			0.111598
$677$			29.2800i			1.12532i	0.826688	+	0.562661i	$0.190222\pi$
$677$			29.2800i			1.12532i	−0.826688	+	0.562661i	$0.809778\pi$
$678$		−	41.2341i		−	1.58359i
$679$	12.6919			0.487069
$680$	−2.75272	+	3.29464i	−0.105562	+	0.126344i
$681$	7.21108			0.276329
$682$		−	5.53647i		−	0.212002i
$683$		−	41.9206i		−	1.60405i	−0.597293	−	0.802023i	$-0.703757\pi$
$683$		−	41.9206i		−	1.60405i	0.597293	−	0.802023i	$-0.296243\pi$
$684$	−19.5594			−0.747873
$685$	29.9708	+	25.0410i	1.14512	+	0.956767i
$686$	−2.21394			−0.0845288
$687$			9.36239i			0.357197i
$688$			7.03870i			0.268348i
$689$	−11.6841			−0.445128
$690$	13.7710	+	11.5059i	0.524255	+	0.438022i
$691$	37.3266			1.41997			0.709985	−	0.704217i	$-0.248702\pi$
$691$	37.3266			1.41997			0.709985	+	0.704217i	$0.248702\pi$
$692$		−	52.1781i		−	1.98351i
$693$			1.31239i			0.0498537i
$694$	79.0039			2.99895
$695$	24.1736	−	28.9327i	0.916959	−	1.09748i
$696$	2.81039			0.106527
$697$			1.01552i			0.0384654i
$698$		−	35.9897i		−	1.36223i
$699$	3.92008			0.148271
$700$	14.2766	−	2.57946i	0.539605	−	0.0974945i
$701$	−37.3462			−1.41055			−0.705273	−	0.708936i	$-0.749175\pi$
$701$	−37.3462			−1.41055			−0.705273	+	0.708936i	$0.749175\pi$
$702$			2.21394i			0.0835599i
$703$			3.15755i			0.119089i
$704$	16.8696			0.635796
$705$	0.539369	−	0.645554i	0.0203138	−	0.0243130i
$706$	−59.3041			−2.23194
$707$		−	2.20823i		−	0.0830490i
$708$			8.26670i			0.310682i
$709$	12.3937			0.465457			0.232729	−	0.972542i	$-0.425235\pi$
$709$	12.3937			0.465457			0.232729	+	0.972542i	$0.425235\pi$
$710$	11.0383	+	9.22267i	0.414261	+	0.346121i
$711$	8.28161			0.310585
$712$		−	20.5805i		−	0.771285i
$713$			6.90711i			0.258673i
$714$	2.12967			0.0797010
$715$	2.25201	+	1.88158i	0.0842203	+	0.0703672i
$716$	31.2582			1.16817
$717$			0.661837i			0.0247167i
$718$			51.2719i			1.91345i
$719$	23.9746			0.894102			0.447051	−	0.894508i	$-0.352474\pi$
$719$	23.9746			0.894102			0.447051	+	0.894508i	$0.352474\pi$
$720$	−1.98440	+	2.37507i	−0.0739542	+	0.0885135i
$721$	−12.3920			−0.461502
$722$			58.5397i			2.17862i
$723$			6.92832i			0.257667i
$724$	2.12271			0.0788900
$725$	1.25172	+	6.92794i	0.0464879	+	0.257297i
$726$	−20.5401			−0.762316
$727$			20.0330i			0.742983i	0.928436	+	0.371492i	$0.121154\pi$
$727$			20.0330i			0.742983i	−0.928436	+	0.371492i	$0.878846\pi$
$728$			1.99598i			0.0739760i
$729$	−1.00000			−0.0370370
$730$	7.59839	−	9.09427i	0.281229	−	0.336594i
$731$	4.89180			0.180930
$732$			2.97094i			0.109809i
$733$		−	34.4602i		−	1.27282i	−0.771352	−	0.636408i	$-0.780419\pi$
$733$		−	34.4602i		−	1.27282i	0.771352	−	0.636408i	$-0.219581\pi$
$734$	36.1972			1.33606
$735$	−1.71595	−	1.43370i	−0.0632939	−	0.0528829i
$736$	−25.5783			−0.942829
$737$		−	10.0770i		−	0.371190i
$738$		−	2.33726i		−	0.0860358i
$739$	−1.55512			−0.0572060			−0.0286030	−	0.999591i	$-0.509106\pi$
$739$	−1.55512			−0.0572060			−0.0286030	+	0.999591i	$0.509106\pi$
$740$	−2.33217	−	1.94856i	−0.0857322	−	0.0716304i
$741$	6.74102			0.247638
$742$		−	25.8679i		−	0.949641i
$743$		−	2.56709i		−	0.0941774i	−0.998891	−	0.0470887i	$-0.985006\pi$
$743$		−	2.56709i		−	0.0941774i	0.998891	−	0.0470887i	$-0.0149943\pi$
$744$	3.80328			0.139435
$745$	−2.73206	+	3.26992i	−0.100095	+	0.119800i
$746$	−63.2695			−2.31646
$747$			2.94509i			0.107755i
$748$		−	3.66303i		−	0.133934i
$749$	17.5241			0.640317
$750$	21.5143	+	12.2405i	0.785590	+	0.446960i
$751$	33.1988			1.21144			0.605720	−	0.795678i	$-0.292885\pi$
$751$	33.1988			1.21144			0.605720	+	0.795678i	$0.292885\pi$
$752$			0.520712i			0.0189884i
$753$			31.3623i			1.14291i
$754$	−3.11728			−0.113525
$755$	11.0885	−	13.2715i	0.403552	−	0.482999i
$756$	−2.90155			−0.105528
$757$			5.34377i			0.194222i	0.995274	+	0.0971112i	$0.0309603\pi$
$757$			5.34377i			0.194222i	−0.995274	+	0.0971112i	$0.969040\pi$
$758$			43.9614i			1.59675i
$759$	−4.75728			−0.172678
$760$	−23.0881	−	19.2904i	−0.837493	−	0.699736i
$761$	30.9794			1.12300			0.561501	−	0.827476i	$-0.310224\pi$
$761$	30.9794			1.12300			0.561501	+	0.827476i	$0.310224\pi$
$762$		−	43.9457i		−	1.59199i
$763$			2.29539i			0.0830988i
$764$	9.89666			0.358049
$765$	1.65064	+	1.37913i	0.0596789	+	0.0498625i
$766$	34.7771			1.25655
$767$		−	2.84906i		−	0.102874i
$768$			6.05213i			0.218387i
$769$	−20.9410			−0.755152			−0.377576	−	0.925979i	$-0.623242\pi$
$769$	−20.9410			−0.755152			−0.377576	+	0.925979i	$0.623242\pi$
$770$	−4.16572	+	4.98582i	−0.150122	+	0.179677i
$771$	7.87232			0.283515
$772$		−	65.4043i		−	2.35395i
$773$			20.3744i			0.732815i	0.930455	+	0.366407i	$0.119412\pi$
$773$			20.3744i			0.732815i	−0.930455	+	0.366407i	$0.880588\pi$
$774$	−11.2587			−0.404686
$775$	1.69395	+	9.37554i	0.0608486	+	0.336779i
$776$	25.3327			0.909392
$777$			0.468408i			0.0168040i
$778$			39.9379i			1.43184i
$779$	−7.11650			−0.254975
$780$	−4.15996	+	4.97893i	−0.148950	+	0.178274i
$781$	−3.81325			−0.136449
$782$			7.71983i			0.276060i
$783$		−	1.40802i		−	0.0503186i
$784$	1.38411			0.0494324
$785$	−30.5661	−	25.5384i	−1.09095	−	0.911503i
$786$	−34.1728			−1.21890
$787$			21.0425i			0.750085i	0.927008	+	0.375043i	$0.122372\pi$
$787$			21.0425i			0.750085i	−0.927008	+	0.375043i	$0.877628\pi$
$788$		−	24.6673i		−	0.878737i
$789$	23.6347			0.841416
$790$	31.4621	+	26.2870i	1.11937	+	0.935248i
$791$	−18.6247			−0.662219
$792$			2.61951i			0.0930804i
$793$		−	1.02391i		−	0.0363602i
$794$	−0.0722541			−0.00256420
$795$	16.7515	−	20.0494i	0.594115	−	0.711077i
$796$	15.2506			0.540544
$797$			7.69130i			0.272440i	0.990679	+	0.136220i	$0.0434954\pi$
$797$			7.69130i			0.272440i	−0.990679	+	0.136220i	$0.956505\pi$
$798$			14.9242i			0.528313i
$799$	0.361887			0.0128027
$800$	−34.7194	+	6.27302i	−1.22751	+	0.221785i
$801$	−10.3109			−0.364319
$802$		−	26.3759i		−	0.931364i
$803$			3.14167i			0.110867i
$804$	22.2790			0.785719
$805$	5.19701	−	6.22014i	0.183171	−	0.219231i
$806$	−4.21860			−0.148594
$807$			9.64340i			0.339464i
$808$		−	4.40758i		−	0.155058i
$809$	−49.2709			−1.73227			−0.866137	−	0.499807i	$-0.833404\pi$
$809$	−49.2709			−1.73227			−0.866137	+	0.499807i	$0.833404\pi$
$810$	−3.79903	−	3.17414i	−0.133484	−	0.111528i
$811$	−47.2650			−1.65970			−0.829849	−	0.557988i	$-0.811573\pi$
$811$	−47.2650			−1.65970			−0.829849	+	0.557988i	$0.811573\pi$
$812$		−	4.08545i		−	0.143371i
$813$		−	11.5739i		−	0.405916i
$814$	1.36099			0.0477027
$815$	6.58926	+	5.50541i	0.230812	+	0.192846i
$816$	−1.33142			−0.0466091
$817$			34.2806i			1.19933i
$818$		−	36.5842i		−	1.27914i
$819$	1.00000			0.0349428
$820$	4.39167	−	5.25625i	0.153364	−	0.183556i
$821$	29.6834			1.03596			0.517979	−	0.855394i	$-0.326685\pi$
$821$	29.6834			1.03596			0.517979	+	0.855394i	$0.326685\pi$
$822$		−	38.6687i		−	1.34872i
$823$		−	49.6019i		−	1.72902i	−0.502620	−	0.864508i	$-0.667630\pi$
$823$		−	49.6019i		−	1.72902i	0.502620	−	0.864508i	$-0.332370\pi$
$824$	−24.7342			−0.861656
$825$	−6.45742	+	1.16671i	−0.224819	+	0.0406197i
$826$	6.30767			0.219472
$827$			7.59316i			0.264040i	0.991247	+	0.132020i	$0.0421463\pi$
$827$			7.59316i			0.264040i	−0.991247	+	0.132020i	$0.957854\pi$
$828$		−	10.5178i		−	0.365519i
$829$	−3.19718			−0.111043			−0.0555214	−	0.998457i	$-0.517682\pi$
$829$	−3.19718			−0.111043			−0.0555214	+	0.998457i	$0.517682\pi$
$830$	−9.34811	+	11.1885i	−0.324478	+	0.388357i
$831$	3.95685			0.137262
$832$		−	12.8540i		−	0.445634i
$833$		−	0.961936i		−	0.0333291i
$834$	−37.3293			−1.29261
$835$	−10.5514	−	8.81585i	−0.365147	−	0.305085i
$836$	25.6697			0.887804
$837$		−	1.90547i		−	0.0658626i
$838$		−	19.2537i		−	0.665109i
$839$	46.6361			1.61006			0.805028	−	0.593236i	$-0.202150\pi$
$839$	46.6361			1.61006			0.805028	+	0.593236i	$0.202150\pi$
$840$	−3.42501	−	2.86164i	−0.118174	−	0.0987361i
$841$	−27.0175			−0.931637
$842$			7.94223i			0.273707i
$843$			8.33607i			0.287110i
$844$	21.9286			0.754815
$845$	1.43370	−	1.71595i	0.0493209	−	0.0590306i
$846$	−0.832902			−0.0286358
$847$			9.27762i			0.318783i
$848$			16.1720i			0.555350i
$849$	−12.8317			−0.440382
$850$	1.89327	+	10.4787i	0.0649386	+	0.359416i
$851$	−1.69793			−0.0582042
$852$		−	8.43065i		−	0.288829i
$853$		−	8.46888i		−	0.289969i	−0.989434	−	0.144984i	$-0.953687\pi$
$853$		−	8.46888i		−	0.289969i	0.989434	−	0.144984i	$-0.0463132\pi$
$854$	2.26689			0.0775713
$855$	−9.66462	+	11.5673i	−0.330523	+	0.395593i
$856$	34.9778			1.19552
$857$		−	12.0676i		−	0.412220i	−0.978529	−	0.206110i	$-0.933919\pi$
$857$		−	12.0676i		−	0.412220i	0.978529	−	0.206110i	$-0.0660806\pi$
$858$		−	2.90557i		−	0.0991945i
$859$	−11.1054			−0.378911			−0.189455	−	0.981889i	$-0.560672\pi$
$859$	−11.1054			−0.378911			−0.189455	+	0.981889i	$0.560672\pi$
$860$	−25.3197	−	21.1549i	−0.863394	−	0.721377i
$861$	−1.05570			−0.0359782
$862$			43.6966i			1.48831i
$863$		−	52.4943i		−	1.78693i	−0.449134	−	0.893464i	$-0.648268\pi$
$863$		−	52.4943i		−	1.78693i	0.449134	−	0.893464i	$-0.351732\pi$
$864$	7.05630			0.240060
$865$	−30.8577	−	25.7820i	−1.04919	−	0.876616i
$866$	−13.9251			−0.473194
$867$		−	16.0747i		−	0.545925i
$868$		−	5.52881i		−	0.187660i
$869$	−10.8687			−0.368697
$870$	4.46926	−	5.34911i	0.151522	−	0.181352i
$871$	−7.67831			−0.260169
$872$			4.58156i			0.155151i
$873$		−	12.6919i		−	0.429555i
$874$	−54.0987			−1.82992
$875$	5.52882	−	9.71762i	0.186908	−	0.328516i
$876$	−6.94586			−0.234679
$877$			17.8109i			0.601432i	0.953714	+	0.300716i	$0.0972256\pi$
$877$			17.8109i			0.601432i	−0.953714	+	0.300716i	$0.902774\pi$
$878$		−	11.5746i		−	0.390622i
$879$	−24.8191			−0.837127
$880$	2.60431	−	3.11702i	0.0877914	−	0.105075i
$881$	5.08831			0.171429			0.0857147	−	0.996320i	$-0.472683\pi$
$881$	5.08831			0.171429			0.0857147	+	0.996320i	$0.472683\pi$
$882$			2.21394i			0.0745474i
$883$			13.4240i			0.451754i	0.974156	+	0.225877i	$0.0725247\pi$
$883$			13.4240i			0.451754i	−0.974156	+	0.225877i	$0.927475\pi$
$884$	−2.79111			−0.0938750
$885$	4.88886	+	4.08471i	0.164337	+	0.137306i
$886$	−44.0864			−1.48111
$887$		−	26.2023i		−	0.879786i	−0.898050	−	0.439893i	$-0.855016\pi$
$887$		−	26.2023i		−	0.879786i	0.898050	−	0.439893i	$-0.144984\pi$
$888$			0.934934i			0.0313743i
$889$	−19.8495			−0.665731
$890$	−39.1716	−	32.7284i	−1.31303	−	1.09706i
$891$	1.31239			0.0439669
$892$		−	50.4435i		−	1.68897i
$893$			2.53602i			0.0848647i
$894$	4.21889			0.141101
$895$	15.4452	−	18.4858i	0.516275	−	0.617913i
$896$	14.3455			0.479251
$897$			3.62489i			0.121032i
$898$			14.9689i			0.499517i
$899$	2.68294			0.0894811
$900$	−2.57946	−	14.2766i	−0.0859821	−	0.475886i
$901$	11.2393			0.374437
$902$			3.06741i			0.102134i
$903$			5.08537i			0.169230i
$904$	−37.1746			−1.23641
$905$	1.04887	−	1.25536i	0.0348655	−	0.0417294i
$906$	−17.1230			−0.568874
$907$			17.6095i			0.584714i	0.956309	+	0.292357i	$0.0944396\pi$
$907$			17.6095i			0.584714i	−0.956309	+	0.292357i	$0.905560\pi$
$908$		−	20.9233i		−	0.694364i
$909$	−2.20823			−0.0732423
$910$	3.79903	+	3.17414i	0.125936	+	0.105222i
$911$	32.1496			1.06516			0.532582	−	0.846378i	$-0.321222\pi$
$911$	32.1496			1.06516			0.532582	+	0.846378i	$0.321222\pi$
$912$		−	9.33030i		−	0.308957i
$913$		−	3.86512i		−	0.127917i
$914$	−58.6763			−1.94084
$915$	1.75699	+	1.46799i	0.0580843	+	0.0485302i
$916$	27.1654			0.897571
$917$			15.4353i			0.509718i
$918$		−	2.12967i		−	0.0702897i
$919$	41.7527			1.37729			0.688647	−	0.725096i	$-0.258205\pi$
$919$	41.7527			1.37729			0.688647	+	0.725096i	$0.258205\pi$
$920$	10.3731	−	12.4153i	0.341992	−	0.409320i
$921$	−28.2873			−0.932099
$922$			22.1059i			0.728019i
$923$			2.90557i			0.0956380i
$924$	3.80798			0.125273
$925$	−2.30472	+	0.416412i	−0.0757789	+	0.0136916i
$926$	62.4437			2.05203
$927$			12.3920i			0.407006i
$928$			9.93543i			0.326146i
$929$	54.1717			1.77732			0.888658	−	0.458570i	$-0.151638\pi$
$929$	54.1717			1.77732			0.888658	+	0.458570i	$0.151638\pi$
$930$	6.04822	−	7.23893i	0.198329	−	0.237374i
$931$	6.74102			0.220928
$932$		−	11.3743i		−	0.372578i
$933$			5.62324i			0.184097i
$934$	−18.1576			−0.594134
$935$	−2.16629	−	1.80996i	−0.0708452	−	0.0591921i
$936$	1.99598			0.0652407
$937$		−	11.6548i		−	0.380745i	−0.981712	−	0.190372i	$-0.939031\pi$
$937$		−	11.6548i		−	0.380745i	0.981712	−	0.190372i	$-0.0609695\pi$
$938$		−	16.9993i		−	0.555048i
$939$	−11.2246			−0.366300
$940$	−1.87311	−	1.56501i	−0.0610941	−	0.0510449i
$941$	−1.09707			−0.0357634			−0.0178817	−	0.999840i	$-0.505692\pi$
$941$	−1.09707			−0.0357634			−0.0178817	+	0.999840i	$0.505692\pi$
$942$			39.4367i			1.28492i
$943$		−	3.82680i		−	0.124618i
$944$	−3.94341			−0.128347
$945$	−1.43370	+	1.71595i	−0.0466383	+	0.0558200i
$946$	14.7759			0.480405
$947$			8.91437i			0.289678i	0.989455	+	0.144839i	$0.0462665\pi$
$947$			8.91437i			0.289678i	−0.989455	+	0.144839i	$0.953734\pi$
$948$		−	24.0295i		−	0.780442i
$949$	2.39384			0.0777074
$950$	−73.4323	+	13.2676i	−2.38246	+	0.430457i
$951$	−4.83778			−0.156876
$952$		−	1.92001i		−	0.0622278i
$953$		−	17.4179i		−	0.564222i	−0.959382	−	0.282111i	$-0.908965\pi$
$953$		−	17.4179i		−	0.564222i	0.959382	−	0.282111i	$-0.0910347\pi$
$954$	−25.8679			−0.837505
$955$	4.89010	−	5.85281i	0.158240	−	0.189392i
$956$	1.92035			0.0621086
$957$			1.84788i			0.0597335i
$958$			55.7398i			1.80087i
$959$	−17.4660			−0.564005
$960$	22.0569	+	18.4289i	0.711885	+	0.594789i
$961$	−27.3692			−0.882877
$962$		−	1.03703i		−	0.0334352i
$963$		−	17.5241i		−	0.564707i
$964$	20.1029			0.647469
$965$	−38.6796	−	32.3173i	−1.24514	−	1.04033i
$966$	−8.02530			−0.258210
$967$			40.0216i			1.28701i	0.765443	+	0.643503i	$0.222520\pi$
$967$			40.0216i			1.28701i	−0.765443	+	0.643503i	$0.777480\pi$
$968$			18.5180i			0.595190i
$969$	−6.48443			−0.208310
$970$	40.2857	−	48.2167i	1.29350	−	1.54815i
$971$	37.8074			1.21330			0.606648	−	0.794970i	$-0.292514\pi$
$971$	37.8074			1.21330			0.606648	+	0.794970i	$0.292514\pi$
$972$			2.90155i			0.0930673i
$973$			16.8610i			0.540539i
$974$	−30.3851			−0.973602
$975$	0.888995	+	4.92033i	0.0284706	+	0.157577i
$976$	−1.41721			−0.0453637
$977$		−	41.2256i		−	1.31892i	−0.751738	−	0.659462i	$-0.770784\pi$
$977$		−	41.2256i		−	1.31892i	0.751738	−	0.659462i	$-0.229216\pi$
$978$		−	8.50154i		−	0.271849i
$979$	13.5320			0.432486
$980$	−4.15996	+	4.97893i	−0.132885	+	0.159046i
$981$	2.29539			0.0732863
$982$			12.0700i			0.385169i
$983$			51.4552i			1.64117i	0.571527	+	0.820583i	$0.306351\pi$
$983$			51.4552i			1.64117i	−0.571527	+	0.820583i	$0.693649\pi$
$984$	−2.10716			−0.0671737
$985$	−14.5881	−	12.1885i	−0.464814	−	0.388359i
$986$	2.99863			0.0954957
$987$			0.376207i			0.0119748i
$988$		−	19.5594i		−	0.622268i
$989$	−18.4339			−0.586164
$990$	4.98582	+	4.16572i	0.158460	+	0.132395i
$991$	−39.2567			−1.24703			−0.623515	−	0.781811i	$-0.714296\pi$
$991$	−39.2567			−1.24703			−0.623515	+	0.781811i	$0.714296\pi$
$992$			13.4456i			0.426897i
$993$			29.9041i			0.948978i
$994$	−6.43277			−0.204035
$995$	7.53558	−	9.01910i	0.238894	−	0.285925i
$996$	8.54532			0.270769
$997$		−	9.99230i		−	0.316459i	−0.987402	−	0.158230i	$-0.949421\pi$
$997$		−	9.99230i		−	0.316459i	0.987402	−	0.158230i	$-0.0505786\pi$
$998$			81.4630i			2.57867i
$999$	0.468408			0.0148198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1365.2.f.d.274.13	yes	16
5.2	odd	4		6825.2.a.bv.1.3		8
5.3	odd	4		6825.2.a.ca.1.6		8
5.4	even	2	inner	1365.2.f.d.274.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1365.2.f.d.274.4	✓	16	5.4	even	2	inner
1365.2.f.d.274.13	yes	16	1.1	even	1	trivial
6825.2.a.bv.1.3		8	5.2	odd	4
6825.2.a.ca.1.6		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 1365 = 3 \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1365.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.21394i q^{2} -1.00000i q^{3} -2.90155 q^{4} +(-1.43370 + 1.71595i) q^{5} +2.21394 q^{6} -1.00000i q^{7} -1.99598i q^{8} -1.00000 q^{9} +(-3.79903 - 3.17414i) q^{10} +1.31239 q^{11} +2.90155i q^{12} -1.00000i q^{13} +2.21394 q^{14} +(1.71595 + 1.43370i) q^{15} -1.38411 q^{16} +0.961936i q^{17} -2.21394i q^{18} -6.74102 q^{19} +(4.15996 - 4.97893i) q^{20} -1.00000 q^{21} +2.90557i q^{22} -3.62489i q^{23} -1.99598 q^{24} +(-0.888995 - 4.92033i) q^{25} +2.21394 q^{26} +1.00000i q^{27} +2.90155i q^{28} -1.40802 q^{29} +(-3.17414 + 3.79903i) q^{30} -1.90547 q^{31} -7.05630i q^{32} -1.31239i q^{33} -2.12967 q^{34} +(1.71595 + 1.43370i) q^{35} +2.90155 q^{36} -0.468408i q^{37} -14.9242i q^{38} -1.00000 q^{39} +(3.42501 + 2.86164i) q^{40} +1.05570 q^{41} -2.21394i q^{42} -5.08537i q^{43} -3.80798 q^{44} +(1.43370 - 1.71595i) q^{45} +8.02530 q^{46} -0.376207i q^{47} +1.38411i q^{48} -1.00000 q^{49} +(10.8933 - 1.96819i) q^{50} +0.961936 q^{51} +2.90155i q^{52} -11.6841i q^{53} -2.21394 q^{54} +(-1.88158 + 2.25201i) q^{55} -1.99598 q^{56} +6.74102i q^{57} -3.11728i q^{58} +2.84906 q^{59} +(-4.97893 - 4.15996i) q^{60} +1.02391 q^{61} -4.21860i q^{62} +1.00000i q^{63} +12.8540 q^{64} +(1.71595 + 1.43370i) q^{65} +2.90557 q^{66} -7.67831i q^{67} -2.79111i q^{68} -3.62489 q^{69} +(-3.17414 + 3.79903i) q^{70} -2.90557 q^{71} +1.99598i q^{72} +2.39384i q^{73} +1.03703 q^{74} +(-4.92033 + 0.888995i) q^{75} +19.5594 q^{76} -1.31239i q^{77} -2.21394i q^{78} -8.28161 q^{79} +(1.98440 - 2.37507i) q^{80} +1.00000 q^{81} +2.33726i q^{82} -2.94509i q^{83} +2.90155 q^{84} +(-1.65064 - 1.37913i) q^{85} +11.2587 q^{86} +1.40802i q^{87} -2.61951i q^{88} +10.3109 q^{89} +(3.79903 + 3.17414i) q^{90} -1.00000 q^{91} +10.5178i q^{92} +1.90547i q^{93} +0.832902 q^{94} +(9.66462 - 11.5673i) q^{95} -7.05630 q^{96} +12.6919i q^{97} -2.21394i q^{98} -1.31239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 2 q^{5} + 4 q^{6} - 16 q^{9} - 2 q^{10} + 4 q^{11} + 4 q^{14} + 64 q^{16} - 4 q^{19} - 2 q^{20} - 16 q^{21} - 24 q^{24} + 2 q^{25} + 4 q^{26} + 24 q^{29} + 2 q^{30} + 4 q^{31} - 4 q^{34}+ \cdots - 4 q^{99}+O(q^{100}) \) 16 * q - 32 * q^4 - 2 * q^5 + 4 * q^6 - 16 * q^9 - 2 * q^10 + 4 * q^11 + 4 * q^14 + 64 * q^16 - 4 * q^19 - 2 * q^20 - 16 * q^21 - 24 * q^24 + 2 * q^25 + 4 * q^26 + 24 * q^29 + 2 * q^30 + 4 * q^31 - 4 * q^34 + 32 * q^36 - 16 * q^39 - 74 * q^40 + 20 * q^41 + 96 * q^44 + 2 * q^45 - 16 * q^49 - 48 * q^50 + 8 * q^51 - 4 * q^54 - 4 * q^55 - 24 * q^56 - 4 * q^59 + 10 * q^60 + 56 * q^61 - 44 * q^64 + 40 * q^66 - 52 * q^69 + 2 * q^70 - 40 * q^71 - 16 * q^75 + 40 * q^76 + 24 * q^79 - 54 * q^80 + 16 * q^81 + 32 * q^84 + 28 * q^85 - 60 * q^86 + 28 * q^89 + 2 * q^90 - 16 * q^91 - 16 * q^94 + 50 * q^95 + 52 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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