LMFDB - Embedded newform 750.3.f.b.193.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [750,3,Mod(193,750)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(750, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("750.193"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	750.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,-16,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$20.4360198270$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.6879707136000000000000.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 21x^{12} + 86x^{8} + 36x^{4} + 1 $ x^16 + 21x^12 + 86x^8 + 36*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.8
Root			$1.40647 + 1.40647i$ of defining polynomial
Character	$\chi$	$=$	750.193
Dual form			750.3.f.b.307.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} -2.44949 q^{6} +(4.02347 - 4.02347i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} -0.880368 q^{11} +(2.44949 - 2.44949i) q^{12} +(-16.1259 - 16.1259i) q^{13} +8.04693i q^{14} -4.00000 q^{16} +(-23.0372 + 23.0372i) q^{17} +(-3.00000 - 3.00000i) q^{18} +21.7203i q^{19} +9.85544 q^{21} +(0.880368 - 0.880368i) q^{22} +(17.6357 + 17.6357i) q^{23} +4.89898i q^{24} +32.2518 q^{26} +(-3.67423 + 3.67423i) q^{27} +(-8.04693 - 8.04693i) q^{28} +18.3656i q^{29} +43.1540 q^{31} +(4.00000 - 4.00000i) q^{32} +(-1.07823 - 1.07823i) q^{33} -46.0744i q^{34} +6.00000 q^{36} +(-49.8860 + 49.8860i) q^{37} +(-21.7203 - 21.7203i) q^{38} -39.5002i q^{39} +42.0242 q^{41} +(-9.85544 + 9.85544i) q^{42} +(32.3173 + 32.3173i) q^{43} +1.76074i q^{44} -35.2713 q^{46} +(-11.4478 + 11.4478i) q^{47} +(-4.89898 - 4.89898i) q^{48} +16.6234i q^{49} -56.4294 q^{51} +(-32.2518 + 32.2518i) q^{52} +(28.3583 + 28.3583i) q^{53} -7.34847i q^{54} +16.0939 q^{56} +(-26.6018 + 26.6018i) q^{57} +(-18.3656 - 18.3656i) q^{58} +73.9494i q^{59} -20.3308 q^{61} +(-43.1540 + 43.1540i) q^{62} +(12.0704 + 12.0704i) q^{63} +8.00000i q^{64} +2.15645 q^{66} +(10.5235 - 10.5235i) q^{67} +(46.0744 + 46.0744i) q^{68} +43.1984i q^{69} +38.9877 q^{71} +(-6.00000 + 6.00000i) q^{72} +(-66.8385 - 66.8385i) q^{73} -99.7721i q^{74} +43.4405 q^{76} +(-3.54213 + 3.54213i) q^{77} +(39.5002 + 39.5002i) q^{78} +0.0459774i q^{79} -9.00000 q^{81} +(-42.0242 + 42.0242i) q^{82} +(-3.58119 - 3.58119i) q^{83} -19.7109i q^{84} -64.6347 q^{86} +(-22.4932 + 22.4932i) q^{87} +(-1.76074 - 1.76074i) q^{88} -63.8377i q^{89} -129.764 q^{91} +(35.2713 - 35.2713i) q^{92} +(52.8527 + 52.8527i) q^{93} -22.8957i q^{94} +9.79796 q^{96} +(-63.0888 + 63.0888i) q^{97} +(-16.6234 - 16.6234i) q^{98} -2.64110i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{2} + 24 q^{7} + 32 q^{8} - 24 q^{11} - 48 q^{13} - 64 q^{16} - 16 q^{17} - 48 q^{18} - 48 q^{21} + 24 q^{22} + 104 q^{23} + 96 q^{26} - 48 q^{28} + 200 q^{31} + 64 q^{32} - 48 q^{33} + 96 q^{36}+ \cdots - 224 q^{98}+O(q^{100}) $ 16 * q - 16 * q^2 + 24 * q^7 + 32 * q^8 - 24 * q^11 - 48 * q^13 - 64 * q^16 - 16 * q^17 - 48 * q^18 - 48 * q^21 + 24 * q^22 + 104 * q^23 + 96 * q^26 - 48 * q^28 + 200 * q^31 + 64 * q^32 - 48 * q^33 + 96 * q^36 - 128 * q^37 + 56 * q^38 - 192 * q^41 + 48 * q^42 + 88 * q^43 - 208 * q^46 + 16 * q^47 - 48 * q^51 - 96 * q^52 - 160 * q^53 + 96 * q^56 + 48 * q^57 + 24 * q^58 - 24 * q^61 - 200 * q^62 + 72 * q^63 + 96 * q^66 + 264 * q^67 + 32 * q^68 + 176 * q^71 - 96 * q^72 + 32 * q^73 - 112 * q^76 + 32 * q^77 + 96 * q^78 - 144 * q^81 + 192 * q^82 - 288 * q^83 - 176 * q^86 - 168 * q^87 - 48 * q^88 + 32 * q^91 + 208 * q^92 + 144 * q^93 - 48 * q^97 - 224 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$4$		−	2.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$	−2.44949			−0.408248
$7$	4.02347	−	4.02347i	0.574781	−	0.574781i	−0.358680	−	0.933461i	$-0.616773\pi$
$7$	4.02347	−	4.02347i	0.574781	−	0.574781i	0.933461	+	0.358680i	$0.116773\pi$
$8$	2.00000	+	2.00000i	0.250000	+	0.250000i
$9$			3.00000i			0.333333i
$10$		0			0
$11$	−0.880368			−0.0800335			−0.0400167	−	0.999199i	$-0.512741\pi$
$11$	−0.880368			−0.0800335			−0.0400167	+	0.999199i	$0.512741\pi$
$12$	2.44949	−	2.44949i	0.204124	−	0.204124i
$13$	−16.1259	−	16.1259i	−1.24045	−	1.24045i	−0.959813	−	0.280641i	$-0.909453\pi$
$13$	−16.1259	−	16.1259i	−1.24045	−	1.24045i	−0.280641	−	0.959813i	$-0.590547\pi$
$14$			8.04693i			0.574781i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	−23.0372	+	23.0372i	−1.35513	+	1.35513i	−0.475314	+	0.879816i	$0.657666\pi$
$17$	−23.0372	+	23.0372i	−1.35513	+	1.35513i	−0.879816	+	0.475314i	$0.842334\pi$
$18$	−3.00000	−	3.00000i	−0.166667	−	0.166667i
$19$			21.7203i			1.14317i	0.820542	+	0.571586i	$0.193672\pi$
$19$			21.7203i			1.14317i	−0.820542	+	0.571586i	$0.806328\pi$
$20$		0			0
$21$	9.85544			0.469307
$22$	0.880368	−	0.880368i	0.0400167	−	0.0400167i
$23$	17.6357	+	17.6357i	0.766768	+	0.766768i	0.977536	−	0.210768i	$-0.0675965\pi$
$23$	17.6357	+	17.6357i	0.766768	+	0.766768i	−0.210768	+	0.977536i	$0.567597\pi$
$24$			4.89898i			0.204124i
$25$		0			0
$26$	32.2518			1.24045
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$	−8.04693	−	8.04693i	−0.287390	−	0.287390i
$29$			18.3656i			0.633296i	0.948543	+	0.316648i	$0.102557\pi$
$29$			18.3656i			0.633296i	−0.948543	+	0.316648i	$0.897443\pi$
$30$		0			0
$31$	43.1540			1.39207			0.696033	−	0.718010i	$-0.254947\pi$
$31$	43.1540			1.39207			0.696033	+	0.718010i	$0.254947\pi$
$32$	4.00000	−	4.00000i	0.125000	−	0.125000i
$33$	−1.07823	−	1.07823i	−0.0326735	−	0.0326735i
$34$		−	46.0744i		−	1.35513i
$35$		0			0
$36$	6.00000			0.166667
$37$	−49.8860	+	49.8860i	−1.34827	+	1.34827i	−0.460732	+	0.887539i	$0.652413\pi$
$37$	−49.8860	+	49.8860i	−1.34827	+	1.34827i	−0.887539	+	0.460732i	$0.847587\pi$
$38$	−21.7203	−	21.7203i	−0.571586	−	0.571586i
$39$		−	39.5002i		−	1.01283i
$40$		0			0
$41$	42.0242			1.02498			0.512490	−	0.858693i	$-0.328723\pi$
$41$	42.0242			1.02498			0.512490	+	0.858693i	$0.328723\pi$
$42$	−9.85544	+	9.85544i	−0.234653	+	0.234653i
$43$	32.3173	+	32.3173i	0.751566	+	0.751566i	0.974771	−	0.223206i	$-0.0716521\pi$
$43$	32.3173	+	32.3173i	0.751566	+	0.751566i	−0.223206	+	0.974771i	$0.571652\pi$
$44$			1.76074i			0.0400167i
$45$		0			0
$46$	−35.2713			−0.766768
$47$	−11.4478	+	11.4478i	−0.243571	+	0.243571i	−0.818326	−	0.574755i	$-0.805097\pi$
$47$	−11.4478	+	11.4478i	−0.243571	+	0.243571i	0.574755	+	0.818326i	$0.305097\pi$
$48$	−4.89898	−	4.89898i	−0.102062	−	0.102062i
$49$			16.6234i			0.339254i
$50$		0			0
$51$	−56.4294			−1.10646
$52$	−32.2518	+	32.2518i	−0.620227	+	0.620227i
$53$	28.3583	+	28.3583i	0.535062	+	0.535062i	0.922074	−	0.387013i	$-0.126493\pi$
$53$	28.3583	+	28.3583i	0.535062	+	0.535062i	−0.387013	+	0.922074i	$0.626493\pi$
$54$		−	7.34847i		−	0.136083i
$55$		0			0
$56$	16.0939			0.287390
$57$	−26.6018	+	26.6018i	−0.466698	+	0.466698i
$58$	−18.3656	−	18.3656i	−0.316648	−	0.316648i
$59$			73.9494i			1.25338i	0.779269	+	0.626689i	$0.215591\pi$
$59$			73.9494i			1.25338i	−0.779269	+	0.626689i	$0.784409\pi$
$60$		0			0
$61$	−20.3308			−0.333291			−0.166646	−	0.986017i	$-0.553294\pi$
$61$	−20.3308			−0.333291			−0.166646	+	0.986017i	$0.553294\pi$
$62$	−43.1540	+	43.1540i	−0.696033	+	0.696033i
$63$	12.0704	+	12.0704i	0.191594	+	0.191594i
$64$			8.00000i			0.125000i
$65$		0			0
$66$	2.15645			0.0326735
$67$	10.5235	−	10.5235i	0.157067	−	0.157067i	−0.624198	−	0.781266i	$-0.714574\pi$
$67$	10.5235	−	10.5235i	0.157067	−	0.157067i	0.781266	+	0.624198i	$0.214574\pi$
$68$	46.0744	+	46.0744i	0.677565	+	0.677565i
$69$			43.1984i			0.626063i
$70$		0			0
$71$	38.9877			0.549122			0.274561	−	0.961570i	$-0.411467\pi$
$71$	38.9877			0.549122			0.274561	+	0.961570i	$0.411467\pi$
$72$	−6.00000	+	6.00000i	−0.0833333	+	0.0833333i
$73$	−66.8385	−	66.8385i	−0.915596	−	0.915596i	0.0811094	−	0.996705i	$-0.474154\pi$
$73$	−66.8385	−	66.8385i	−0.915596	−	0.915596i	−0.996705	+	0.0811094i	$0.974154\pi$
$74$		−	99.7721i		−	1.34827i
$75$		0			0
$76$	43.4405			0.571586
$77$	−3.54213	+	3.54213i	−0.0460017	+	0.0460017i
$78$	39.5002	+	39.5002i	0.506413	+	0.506413i
$79$			0.0459774i			0.000581992i	1.00000		0.000290996i	$9.26270e-5\pi$
$79$			0.0459774i			0.000581992i	−1.00000		0.000290996i	$0.999907\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$	−42.0242	+	42.0242i	−0.512490	+	0.512490i
$83$	−3.58119	−	3.58119i	−0.0431469	−	0.0431469i	0.685204	−	0.728351i	$-0.259713\pi$
$83$	−3.58119	−	3.58119i	−0.0431469	−	0.0431469i	−0.728351	+	0.685204i	$0.759713\pi$
$84$		−	19.7109i		−	0.234653i
$85$		0			0
$86$	−64.6347			−0.751566
$87$	−22.4932	+	22.4932i	−0.258542	+	0.258542i
$88$	−1.76074	−	1.76074i	−0.0200084	−	0.0200084i
$89$		−	63.8377i		−	0.717278i	−0.933476	−	0.358639i	$-0.883241\pi$
$89$		−	63.8377i		−	0.717278i	0.933476	−	0.358639i	$-0.116759\pi$
$90$		0			0
$91$	−129.764			−1.42598
$92$	35.2713	−	35.2713i	0.383384	−	0.383384i
$93$	52.8527	+	52.8527i	0.568309	+	0.568309i
$94$		−	22.8957i		−	0.243571i
$95$		0			0
$96$	9.79796			0.102062
$97$	−63.0888	+	63.0888i	−0.650400	+	0.650400i	−0.953089	−	0.302689i	$-0.902116\pi$
$97$	−63.0888	+	63.0888i	−0.650400	+	0.650400i	0.302689	+	0.953089i	$0.402116\pi$
$98$	−16.6234	−	16.6234i	−0.169627	−	0.169627i
$99$		−	2.64110i		−	0.0266778i
$100$		0			0
$101$	84.2261			0.833922			0.416961	−	0.908924i	$-0.363095\pi$
$101$	84.2261			0.833922			0.416961	+	0.908924i	$0.363095\pi$
$102$	56.4294	−	56.4294i	0.553230	−	0.553230i
$103$	−74.7312	−	74.7312i	−0.725546	−	0.725546i	0.244183	−	0.969729i	$-0.421480\pi$
$103$	−74.7312	−	74.7312i	−0.725546	−	0.725546i	−0.969729	+	0.244183i	$0.921480\pi$
$104$		−	64.5036i		−	0.620227i
$105$		0			0
$106$	−56.7166			−0.535062
$107$	−131.877	+	131.877i	−1.23249	+	1.23249i	−0.269491	+	0.963003i	$0.586856\pi$
$107$	−131.877	+	131.877i	−1.23249	+	1.23249i	−0.963003	+	0.269491i	$0.913144\pi$
$108$	7.34847	+	7.34847i	0.0680414	+	0.0680414i
$109$		−	77.0255i		−	0.706656i	−0.935500	−	0.353328i	$-0.885050\pi$
$109$		−	77.0255i		−	0.706656i	0.935500	−	0.353328i	$-0.114950\pi$
$110$		0			0
$111$	−122.195			−1.10086
$112$	−16.0939	+	16.0939i	−0.143695	+	0.143695i
$113$	−63.9083	−	63.9083i	−0.565560	−	0.565560i	0.365321	−	0.930881i	$-0.380959\pi$
$113$	−63.9083	−	63.9083i	−0.565560	−	0.565560i	−0.930881	+	0.365321i	$0.880959\pi$
$114$		−	53.2035i		−	0.466698i
$115$		0			0
$116$	36.7312			0.316648
$117$	48.3777	−	48.3777i	0.413485	−	0.413485i
$118$	−73.9494	−	73.9494i	−0.626689	−	0.626689i
$119$			185.379i			1.55781i
$120$		0			0
$121$	−120.225			−0.993595
$122$	20.3308	−	20.3308i	0.166646	−	0.166646i
$123$	51.4689	+	51.4689i	0.418446	+	0.418446i
$124$		−	86.3081i		−	0.696033i
$125$		0			0
$126$	−24.1408			−0.191594
$127$	61.1374	−	61.1374i	0.481397	−	0.481397i	−0.424181	−	0.905578i	$-0.639438\pi$
$127$	61.1374	−	61.1374i	0.481397	−	0.481397i	0.905578	+	0.424181i	$0.139438\pi$
$128$	−8.00000	−	8.00000i	−0.0625000	−	0.0625000i
$129$			79.1610i			0.613651i
$130$		0			0
$131$	135.535			1.03462			0.517308	−	0.855799i	$-0.326934\pi$
$131$	135.535			1.03462			0.517308	+	0.855799i	$0.326934\pi$
$132$	−2.15645	+	2.15645i	−0.0163368	+	0.0163368i
$133$	87.3907	+	87.3907i	0.657073	+	0.657073i
$134$			21.0470i			0.157067i
$135$		0			0
$136$	−92.1489			−0.677565
$137$	−87.9906	+	87.9906i	−0.642267	+	0.642267i	−0.951112	−	0.308845i	$-0.900058\pi$
$137$	−87.9906	+	87.9906i	−0.642267	+	0.642267i	0.308845	+	0.951112i	$0.400058\pi$
$138$	−43.1984	−	43.1984i	−0.313032	−	0.313032i
$139$			108.094i			0.777653i	0.921311	+	0.388827i	$0.127120\pi$
$139$			108.094i			0.777653i	−0.921311	+	0.388827i	$0.872880\pi$
$140$		0			0
$141$	−28.0414			−0.198875
$142$	−38.9877	+	38.9877i	−0.274561	+	0.274561i
$143$	14.1967	+	14.1967i	0.0992778	+	0.0992778i
$144$		−	12.0000i		−	0.0833333i
$145$		0			0
$146$	133.677			0.915596
$147$	−20.3595	+	20.3595i	−0.138500	+	0.138500i
$148$	99.7721	+	99.7721i	0.674136	+	0.674136i
$149$			77.0332i			0.517001i	0.966011	+	0.258501i	$0.0832284\pi$
$149$			77.0332i			0.517001i	−0.966011	+	0.258501i	$0.916772\pi$
$150$		0			0
$151$	111.285			0.736986			0.368493	−	0.929630i	$-0.379874\pi$
$151$	111.285			0.736986			0.368493	+	0.929630i	$0.379874\pi$
$152$	−43.4405	+	43.4405i	−0.285793	+	0.285793i
$153$	−69.1116	−	69.1116i	−0.451710	−	0.451710i
$154$		−	7.08426i		−	0.0460017i
$155$		0			0
$156$	−79.0005			−0.506413
$157$	133.008	−	133.008i	0.847183	−	0.847183i	−0.142597	−	0.989781i	$-0.545545\pi$
$157$	133.008	−	133.008i	0.847183	−	0.847183i	0.989781	+	0.142597i	$0.0455454\pi$
$158$	−0.0459774	−	0.0459774i	−0.000290996	−	0.000290996i
$159$			69.4633i			0.436876i
$160$		0			0
$161$	141.913			0.881447
$162$	9.00000	−	9.00000i	0.0555556	−	0.0555556i
$163$	−164.580	−	164.580i	−1.00969	−	1.00969i	−0.999953	−	0.00974130i	$-0.996899\pi$
$163$	−164.580	−	164.580i	−1.00969	−	1.00969i	−0.00974130	−	0.999953i	$-0.503101\pi$
$164$		−	84.0484i		−	0.512490i
$165$		0			0
$166$	7.16239			0.0431469
$167$	138.034	−	138.034i	0.826551	−	0.826551i	−0.160487	−	0.987038i	$-0.551306\pi$
$167$	138.034	−	138.034i	0.826551	−	0.826551i	0.987038	+	0.160487i	$0.0513064\pi$
$168$	19.7109	+	19.7109i	0.117327	+	0.117327i
$169$			351.089i			2.07745i
$170$		0			0
$171$	−65.1608			−0.381057
$172$	64.6347	−	64.6347i	0.375783	−	0.375783i
$173$	44.2455	+	44.2455i	0.255754	+	0.255754i	0.823325	−	0.567571i	$-0.192117\pi$
$173$	44.2455	+	44.2455i	0.255754	+	0.255754i	−0.567571	+	0.823325i	$0.692117\pi$
$174$		−	44.9863i		−	0.258542i
$175$		0			0
$176$	3.52147			0.0200084
$177$	−90.5691	+	90.5691i	−0.511690	+	0.511690i
$178$	63.8377	+	63.8377i	0.358639	+	0.358639i
$179$		−	104.668i		−	0.584737i	−0.956306	−	0.292369i	$-0.905557\pi$
$179$		−	104.668i		−	0.584737i	0.956306	−	0.292369i	$-0.0944434\pi$
$180$		0			0
$181$	−200.119			−1.10563			−0.552815	−	0.833304i	$-0.686446\pi$
$181$	−200.119			−1.10563			−0.552815	+	0.833304i	$0.686446\pi$
$182$	129.764	−	129.764i	0.712989	−	0.712989i
$183$	−24.9000	−	24.9000i	−0.136066	−	0.136066i
$184$			70.5426i			0.383384i
$185$		0			0
$186$	−105.705			−0.568309
$187$	20.2812	−	20.2812i	0.108456	−	0.108456i
$188$	22.8957	+	22.8957i	0.121786	+	0.121786i
$189$			29.5663i			0.156436i
$190$		0			0
$191$	−327.224			−1.71321			−0.856607	−	0.515969i	$-0.827432\pi$
$191$	−327.224			−1.71321			−0.856607	+	0.515969i	$0.827432\pi$
$192$	−9.79796	+	9.79796i	−0.0510310	+	0.0510310i
$193$	−23.2601	−	23.2601i	−0.120519	−	0.120519i	0.644275	−	0.764794i	$-0.277159\pi$
$193$	−23.2601	−	23.2601i	−0.120519	−	0.120519i	−0.764794	+	0.644275i	$0.777159\pi$
$194$		−	126.178i		−	0.650400i
$195$		0			0
$196$	33.2469			0.169627
$197$	164.292	−	164.292i	0.833969	−	0.833969i	−0.154088	−	0.988057i	$-0.549244\pi$
$197$	164.292	−	164.292i	0.833969	−	0.833969i	0.988057	+	0.154088i	$0.0492440\pi$
$198$	2.64110	+	2.64110i	0.0133389	+	0.0133389i
$199$		−	53.3274i		−	0.267977i	−0.990983	−	0.133988i	$-0.957222\pi$
$199$		−	53.3274i		−	0.267977i	0.990983	−	0.133988i	$-0.0427785\pi$
$200$		0			0
$201$	25.7772			0.128245
$202$	−84.2261	+	84.2261i	−0.416961	+	0.416961i
$203$	73.8933	+	73.8933i	0.364006	+	0.364006i
$204$			112.859i			0.553230i
$205$		0			0
$206$	149.462			0.725546
$207$	−52.9070	+	52.9070i	−0.255589	+	0.255589i
$208$	64.5036	+	64.5036i	0.310113	+	0.310113i
$209$		−	19.1218i		−	0.0914920i
$210$		0			0
$211$	−399.659			−1.89412			−0.947059	−	0.321058i	$-0.895961\pi$
$211$	−399.659			−1.89412			−0.947059	+	0.321058i	$0.895961\pi$
$212$	56.7166	−	56.7166i	0.267531	−	0.267531i
$213$	47.7500	+	47.7500i	0.224178	+	0.224178i
$214$		−	263.754i		−	1.23249i
$215$		0			0
$216$	−14.6969			−0.0680414
$217$	173.629	−	173.629i	0.800133	−	0.800133i
$218$	77.0255	+	77.0255i	0.353328	+	0.353328i
$219$		−	163.720i		−	0.747581i
$220$		0			0
$221$	742.992			3.36195
$222$	122.195	−	122.195i	0.550429	−	0.550429i
$223$	−18.2069	−	18.2069i	−0.0816452	−	0.0816452i	0.665105	−	0.746750i	$-0.268387\pi$
$223$	−18.2069	−	18.2069i	−0.0816452	−	0.0816452i	−0.746750	+	0.665105i	$0.768387\pi$
$224$		−	32.1877i		−	0.143695i
$225$		0			0
$226$	127.817			0.565560
$227$	222.960	−	222.960i	0.982203	−	0.982203i	−0.0176417	−	0.999844i	$-0.505616\pi$
$227$	222.960	−	222.960i	0.982203	−	0.982203i	0.999844	+	0.0176417i	$0.00561580\pi$
$228$	53.2035	+	53.2035i	0.233349	+	0.233349i
$229$			76.8571i			0.335621i	0.985819	+	0.167810i	$0.0536696\pi$
$229$			76.8571i			0.335621i	−0.985819	+	0.167810i	$0.946330\pi$
$230$		0			0
$231$	−8.67641			−0.0375602
$232$	−36.7312	+	36.7312i	−0.158324	+	0.158324i
$233$	−58.7236	−	58.7236i	−0.252033	−	0.252033i	0.569771	−	0.821804i	$-0.307032\pi$
$233$	−58.7236	−	58.7236i	−0.252033	−	0.252033i	−0.821804	+	0.569771i	$0.807032\pi$
$234$			96.7554i			0.413485i
$235$		0			0
$236$	147.899			0.626689
$237$	−0.0563106	+	0.0563106i	−0.000237597	+	0.000237597i
$238$	−185.379	−	185.379i	−0.778903	−	0.778903i
$239$		−	146.681i		−	0.613729i	−0.951753	−	0.306865i	$-0.900720\pi$
$239$		−	146.681i		−	0.613729i	0.951753	−	0.306865i	$-0.0992799\pi$
$240$		0			0
$241$	191.834			0.795990			0.397995	−	0.917388i	$-0.369706\pi$
$241$	191.834			0.795990			0.397995	+	0.917388i	$0.369706\pi$
$242$	120.225	−	120.225i	0.496797	−	0.496797i
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$			40.6615i			0.166646i
$245$		0			0
$246$	−102.938			−0.418446
$247$	350.259	−	350.259i	1.41805	−	1.41805i
$248$	86.3081	+	86.3081i	0.348017	+	0.348017i
$249$		−	8.77210i		−	0.0352293i
$250$		0			0
$251$	397.176			1.58237			0.791187	−	0.611574i	$-0.209463\pi$
$251$	397.176			1.58237			0.791187	+	0.611574i	$0.209463\pi$
$252$	24.1408	−	24.1408i	0.0957968	−	0.0957968i
$253$	−15.5259	−	15.5259i	−0.0613671	−	0.0613671i
$254$			122.275i			0.481397i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−222.079	+	222.079i	−0.864120	+	0.864120i	−0.991814	−	0.127693i	$-0.959243\pi$
$257$	−222.079	+	222.079i	−0.864120	+	0.864120i	0.127693	+	0.991814i	$0.459243\pi$
$258$	−79.1610	−	79.1610i	−0.306825	−	0.306825i
$259$			401.430i			1.54992i
$260$		0			0
$261$	−55.0967			−0.211099
$262$	−135.535	+	135.535i	−0.517308	+	0.517308i
$263$	148.964	+	148.964i	0.566402	+	0.566402i	0.931118	−	0.364717i	$-0.118834\pi$
$263$	148.964	+	148.964i	0.566402	+	0.566402i	−0.364717	+	0.931118i	$0.618834\pi$
$264$		−	4.31290i		−	0.0163368i
$265$		0			0
$266$	−174.781			−0.657073
$267$	78.1849	−	78.1849i	0.292827	−	0.292827i
$268$	−21.0470	−	21.0470i	−0.0785337	−	0.0785337i
$269$			399.749i			1.48606i	0.669261	+	0.743028i	$0.266611\pi$
$269$			399.749i			1.48606i	−0.669261	+	0.743028i	$0.733389\pi$
$270$		0			0
$271$	146.491			0.540558			0.270279	−	0.962782i	$-0.412884\pi$
$271$	146.491			0.540558			0.270279	+	0.962782i	$0.412884\pi$
$272$	92.1489	−	92.1489i	0.338783	−	0.338783i
$273$	−158.928	−	158.928i	−0.582153	−	0.582153i
$274$		−	175.981i		−	0.642267i
$275$		0			0
$276$	86.3967			0.313032
$277$	179.944	−	179.944i	0.649616	−	0.649616i	−0.303284	−	0.952900i	$-0.598083\pi$
$277$	179.944	−	179.944i	0.649616	−	0.649616i	0.952900	+	0.303284i	$0.0980831\pi$
$278$	−108.094	−	108.094i	−0.388827	−	0.388827i
$279$			129.462i			0.464022i
$280$		0			0
$281$	286.333			1.01898			0.509489	−	0.860477i	$-0.329835\pi$
$281$	286.333			1.01898			0.509489	+	0.860477i	$0.329835\pi$
$282$	28.0414	−	28.0414i	0.0994375	−	0.0994375i
$283$	222.713	+	222.713i	0.786972	+	0.786972i	0.980997	−	0.194025i	$-0.0621542\pi$
$283$	222.713	+	222.713i	0.786972	+	0.786972i	−0.194025	+	0.980997i	$0.562154\pi$
$284$		−	77.9754i		−	0.274561i
$285$		0			0
$286$	−28.3935			−0.0992778
$287$	169.083	−	169.083i	0.589139	−	0.589139i
$288$	12.0000	+	12.0000i	0.0416667	+	0.0416667i
$289$		−	772.426i		−	2.67276i
$290$		0			0
$291$	−154.535			−0.531050
$292$	−133.677	+	133.677i	−0.457798	+	0.457798i
$293$	100.223	+	100.223i	0.342059	+	0.342059i	0.857141	−	0.515082i	$-0.172238\pi$
$293$	100.223	+	100.223i	0.342059	+	0.342059i	−0.515082	+	0.857141i	$0.672238\pi$
$294$		−	40.7189i		−	0.138500i
$295$		0			0
$296$	−199.544			−0.674136
$297$	3.23468	−	3.23468i	0.0108912	−	0.0108912i
$298$	−77.0332	−	77.0332i	−0.258501	−	0.258501i
$299$		−	568.782i		−	1.90228i
$300$		0			0
$301$	260.055			0.863971
$302$	−111.285	+	111.285i	−0.368493	+	0.368493i
$303$	103.156	+	103.156i	0.340447	+	0.340447i
$304$		−	86.8810i		−	0.285793i
$305$		0			0
$306$	138.223			0.451710
$307$	171.837	−	171.837i	0.559730	−	0.559730i	−0.369501	−	0.929230i	$-0.620471\pi$
$307$	171.837	−	171.837i	0.559730	−	0.559730i	0.929230	+	0.369501i	$0.120471\pi$
$308$	7.08426	+	7.08426i	0.0230008	+	0.0230008i
$309$		−	183.053i		−	0.592406i
$310$		0			0
$311$	−140.504			−0.451781			−0.225890	−	0.974153i	$-0.572529\pi$
$311$	−140.504			−0.451781			−0.225890	+	0.974153i	$0.572529\pi$
$312$	79.0005	−	79.0005i	0.253207	−	0.253207i
$313$	391.527	+	391.527i	1.25089	+	1.25089i	0.955323	+	0.295563i	$0.0955071\pi$
$313$	391.527	+	391.527i	1.25089	+	1.25089i	0.295563	+	0.955323i	$0.404493\pi$
$314$			266.016i			0.847183i
$315$		0			0
$316$	0.0919548			0.000290996
$317$	−193.032	+	193.032i	−0.608934	+	0.608934i	−0.942667	−	0.333734i	$-0.891691\pi$
$317$	−193.032	+	193.032i	−0.608934	+	0.608934i	0.333734	+	0.942667i	$0.391691\pi$
$318$	−69.4633	−	69.4633i	−0.218438	−	0.218438i
$319$		−	16.1685i		−	0.0506849i
$320$		0			0
$321$	−323.031			−1.00633
$322$	−141.913	+	141.913i	−0.440724	+	0.440724i
$323$	−500.374	−	500.374i	−1.54915	−	1.54915i
$324$			18.0000i			0.0555556i
$325$		0			0
$326$	329.160			1.00969
$327$	94.3366	−	94.3366i	0.288491	−	0.288491i
$328$	84.0484	+	84.0484i	0.256245	+	0.256245i
$329$			92.1200i			0.280000i
$330$		0			0
$331$	−339.875			−1.02681			−0.513406	−	0.858146i	$-0.671617\pi$
$331$	−339.875			−1.02681			−0.513406	+	0.858146i	$0.671617\pi$
$332$	−7.16239	+	7.16239i	−0.0215735	+	0.0215735i
$333$	−149.658	−	149.658i	−0.449424	−	0.449424i
$334$			276.068i			0.826551i
$335$		0			0
$336$	−39.4218			−0.117327
$337$	251.241	−	251.241i	0.745522	−	0.745522i	−0.228113	−	0.973635i	$-0.573256\pi$
$337$	251.241	−	251.241i	0.745522	−	0.745522i	0.973635	+	0.228113i	$0.0732555\pi$
$338$	−351.089	−	351.089i	−1.03873	−	1.03873i
$339$		−	156.543i		−	0.461778i
$340$		0			0
$341$	−37.9914			−0.111412
$342$	65.1608	−	65.1608i	0.190529	−	0.190529i
$343$	264.034	+	264.034i	0.769778	+	0.769778i
$344$			129.269i			0.375783i
$345$		0			0
$346$	−88.4909			−0.255754
$347$	−414.443	+	414.443i	−1.19436	+	1.19436i	−0.218531	+	0.975830i	$0.570127\pi$
$347$	−414.443	+	414.443i	−1.19436	+	1.19436i	−0.975830	+	0.218531i	$0.929873\pi$
$348$	44.9863	+	44.9863i	0.129271	+	0.129271i
$349$		−	311.540i		−	0.892665i	−0.894867	−	0.446332i	$-0.852730\pi$
$349$		−	311.540i		−	0.892665i	0.894867	−	0.446332i	$-0.147270\pi$
$350$		0			0
$351$	118.501			0.337609
$352$	−3.52147	+	3.52147i	−0.0100042	+	0.0100042i
$353$	−241.357	−	241.357i	−0.683732	−	0.683732i	0.277107	−	0.960839i	$-0.410624\pi$
$353$	−241.357	−	241.357i	−0.683732	−	0.683732i	−0.960839	+	0.277107i	$0.910624\pi$
$354$		−	181.138i		−	0.511690i
$355$		0			0
$356$	−127.675			−0.358639
$357$	−227.042	+	227.042i	−0.635972	+	0.635972i
$358$	104.668	+	104.668i	0.292369	+	0.292369i
$359$			65.5040i			0.182462i	0.995830	+	0.0912312i	$0.0290802\pi$
$359$			65.5040i			0.182462i	−0.995830	+	0.0912312i	$0.970920\pi$
$360$		0			0
$361$	−110.770			−0.306841
$362$	200.119	−	200.119i	0.552815	−	0.552815i
$363$	−147.245	−	147.245i	−0.405633	−	0.405633i
$364$			259.528i			0.712989i
$365$		0			0
$366$	49.8000			0.136066
$367$	−177.298	+	177.298i	−0.483102	+	0.483102i	−0.906121	−	0.423019i	$-0.860970\pi$
$367$	−177.298	+	177.298i	−0.483102	+	0.483102i	0.423019	+	0.906121i	$0.360970\pi$
$368$	−70.5426	−	70.5426i	−0.191692	−	0.191692i
$369$			126.073i			0.341660i
$370$		0			0
$371$	228.197			0.615087
$372$	105.705	−	105.705i	0.284154	−	0.284154i
$373$	63.9535	+	63.9535i	0.171457	+	0.171457i	0.787619	−	0.616162i	$-0.211313\pi$
$373$	63.9535	+	63.9535i	0.171457	+	0.171457i	−0.616162	+	0.787619i	$0.711313\pi$
$374$			40.5625i			0.108456i
$375$		0			0
$376$	−45.7914			−0.121786
$377$	296.162	−	296.162i	0.785574	−	0.785574i
$378$	−29.5663	−	29.5663i	−0.0782178	−	0.0782178i
$379$			344.443i			0.908820i	0.890793	+	0.454410i	$0.150150\pi$
$379$			344.443i			0.908820i	−0.890793	+	0.454410i	$0.849850\pi$
$380$		0			0
$381$	149.755			0.393059
$382$	327.224	−	327.224i	0.856607	−	0.856607i
$383$	−22.6888	−	22.6888i	−0.0592396	−	0.0592396i	0.676866	−	0.736106i	$-0.263337\pi$
$383$	−22.6888	−	22.6888i	−0.0592396	−	0.0592396i	−0.736106	+	0.676866i	$0.763337\pi$
$384$		−	19.5959i		−	0.0510310i
$385$		0			0
$386$	46.5203			0.120519
$387$	−96.9520	+	96.9520i	−0.250522	+	0.250522i
$388$	126.178	+	126.178i	0.325200	+	0.325200i
$389$			234.054i			0.601680i	0.953675	+	0.300840i	$0.0972671\pi$
$389$			234.054i			0.601680i	−0.953675	+	0.300840i	$0.902733\pi$
$390$		0			0
$391$	−812.553			−2.07814
$392$	−33.2469	+	33.2469i	−0.0848135	+	0.0848135i
$393$	165.996	+	165.996i	0.422381	+	0.422381i
$394$			328.584i			0.833969i
$395$		0			0
$396$	−5.28221			−0.0133389
$397$	−495.008	+	495.008i	−1.24687	+	1.24687i	−0.289779	+	0.957094i	$0.593582\pi$
$397$	−495.008	+	495.008i	−1.24687	+	1.24687i	−0.957094	+	0.289779i	$0.906418\pi$
$398$	53.3274	+	53.3274i	0.133988	+	0.133988i
$399$			214.063i			0.536498i
$400$		0			0
$401$	−46.2845			−0.115423			−0.0577114	−	0.998333i	$-0.518380\pi$
$401$	−46.2845			−0.115423			−0.0577114	+	0.998333i	$0.518380\pi$
$402$	−25.7772	+	25.7772i	−0.0641225	+	0.0641225i
$403$	−695.898	−	695.898i	−1.72679	−	1.72679i
$404$		−	168.452i		−	0.416961i
$405$		0			0
$406$	−147.787			−0.364006
$407$	43.9181	−	43.9181i	0.107907	−	0.107907i
$408$	−112.859	−	112.859i	−0.276615	−	0.276615i
$409$		−	239.436i		−	0.585419i	−0.956201	−	0.292710i	$-0.905443\pi$
$409$		−	239.436i		−	0.585419i	0.956201	−	0.292710i	$-0.0945569\pi$
$410$		0			0
$411$	−215.532			−0.524409
$412$	−149.462	+	149.462i	−0.362773	+	0.362773i
$413$	297.533	+	297.533i	0.720418	+	0.720418i
$414$		−	105.814i		−	0.255589i
$415$		0			0
$416$	−129.007			−0.310113
$417$	−132.387	+	132.387i	−0.317476	+	0.317476i
$418$	19.1218	+	19.1218i	0.0457460	+	0.0457460i
$419$			150.579i			0.359378i	0.983723	+	0.179689i	$0.0575091\pi$
$419$			150.579i			0.359378i	−0.983723	+	0.179689i	$0.942491\pi$
$420$		0			0
$421$	485.057			1.15215			0.576077	−	0.817396i	$-0.304583\pi$
$421$	485.057			1.15215			0.576077	+	0.817396i	$0.304583\pi$
$422$	399.659	−	399.659i	0.947059	−	0.947059i
$423$	−34.3435	−	34.3435i	−0.0811903	−	0.0811903i
$424$			113.433i			0.267531i
$425$		0			0
$426$	−95.4999			−0.224178
$427$	−81.8001	+	81.8001i	−0.191569	+	0.191569i
$428$	263.754	+	263.754i	0.616247	+	0.616247i
$429$			34.7747i			0.0810600i
$430$		0			0
$431$	−790.360			−1.83378			−0.916891	−	0.399138i	$-0.869310\pi$
$431$	−790.360			−1.83378			−0.916891	+	0.399138i	$0.869310\pi$
$432$	14.6969	−	14.6969i	0.0340207	−	0.0340207i
$433$	552.690	+	552.690i	1.27642	+	1.27642i	0.942657	+	0.333763i	$0.108319\pi$
$433$	552.690	+	552.690i	1.27642	+	1.27642i	0.333763	+	0.942657i	$0.391681\pi$
$434$			347.258i			0.800133i
$435$		0			0
$436$	−154.051			−0.353328
$437$	−383.051	+	383.051i	−0.876547	+	0.876547i
$438$	163.720	+	163.720i	0.373790	+	0.373790i
$439$		−	649.354i		−	1.47917i	−0.673066	−	0.739583i	$-0.735023\pi$
$439$		−	649.354i		−	1.47917i	0.673066	−	0.739583i	$-0.264977\pi$
$440$		0			0
$441$	−49.8703			−0.113085
$442$	−742.992	+	742.992i	−1.68098	+	1.68098i
$443$	−55.3639	−	55.3639i	−0.124975	−	0.124975i	0.641853	−	0.766828i	$-0.278166\pi$
$443$	−55.3639	−	55.3639i	−0.124975	−	0.124975i	−0.766828	+	0.641853i	$0.778166\pi$
$444$			244.391i			0.550429i
$445$		0			0
$446$	36.4138			0.0816452
$447$	−94.3460	+	94.3460i	−0.211065	+	0.211065i
$448$	32.1877	+	32.1877i	0.0718476	+	0.0718476i
$449$		−	835.180i		−	1.86009i	−0.367446	−	0.930045i	$-0.619768\pi$
$449$		−	835.180i		−	1.86009i	0.367446	−	0.930045i	$-0.380232\pi$
$450$		0			0
$451$	−36.9967			−0.0820327
$452$	−127.817	+	127.817i	−0.282780	+	0.282780i
$453$	136.296	+	136.296i	0.300873	+	0.300873i
$454$			445.920i			0.982203i
$455$		0			0
$456$	−106.407			−0.233349
$457$	268.706	−	268.706i	0.587978	−	0.587978i	−0.349105	−	0.937083i	$-0.613514\pi$
$457$	268.706	−	268.706i	0.587978	−	0.587978i	0.937083	+	0.349105i	$0.113514\pi$
$458$	−76.8571	−	76.8571i	−0.167810	−	0.167810i
$459$		−	169.288i		−	0.368820i
$460$		0			0
$461$	−337.848			−0.732860			−0.366430	−	0.930446i	$-0.619420\pi$
$461$	−337.848			−0.732860			−0.366430	+	0.930446i	$0.619420\pi$
$462$	8.67641	−	8.67641i	0.0187801	−	0.0187801i
$463$	127.722	+	127.722i	0.275857	+	0.275857i	0.831452	−	0.555596i	$-0.187510\pi$
$463$	127.722	+	127.722i	0.275857	+	0.275857i	−0.555596	+	0.831452i	$0.687510\pi$
$464$		−	73.4623i		−	0.158324i
$465$		0			0
$466$	117.447			0.252033
$467$	−48.8594	+	48.8594i	−0.104624	+	0.104624i	−0.757481	−	0.652857i	$-0.773570\pi$
$467$	−48.8594	+	48.8594i	−0.104624	+	0.104624i	0.652857	+	0.757481i	$0.273570\pi$
$468$	−96.7554	−	96.7554i	−0.206742	−	0.206742i
$469$		−	84.6820i		−	0.180559i
$470$		0			0
$471$	325.801			0.691722
$472$	−147.899	+	147.899i	−0.313345	+	0.313345i
$473$	−28.4511	−	28.4511i	−0.0601504	−	0.0601504i
$474$		−	0.112621i		−	0.000237597i
$475$		0			0
$476$	370.758			0.778903
$477$	−85.0748	+	85.0748i	−0.178354	+	0.178354i
$478$	146.681	+	146.681i	0.306865	+	0.306865i
$479$			11.1370i			0.0232506i	0.999932	+	0.0116253i	$0.00370053\pi$
$479$			11.1370i			0.0232506i	−0.999932	+	0.0116253i	$0.996299\pi$
$480$		0			0
$481$	1608.91			3.34494
$482$	−191.834	+	191.834i	−0.397995	+	0.397995i
$483$	173.807	+	173.807i	0.359849	+	0.359849i
$484$			240.450i			0.496797i
$485$		0			0
$486$	22.0454			0.0453609
$487$	368.776	−	368.776i	0.757240	−	0.757240i	−0.218579	−	0.975819i	$-0.570142\pi$
$487$	368.776	−	368.776i	0.757240	−	0.757240i	0.975819	+	0.218579i	$0.0701421\pi$
$488$	−40.6615	−	40.6615i	−0.0833228	−	0.0833228i
$489$		−	403.137i		−	0.824412i
$490$		0			0
$491$	−535.323			−1.09027			−0.545136	−	0.838348i	$-0.683522\pi$
$491$	−535.323			−1.09027			−0.545136	+	0.838348i	$0.683522\pi$
$492$	102.938	−	102.938i	0.209223	−	0.209223i
$493$	−423.092	−	423.092i	−0.858198	−	0.858198i
$494$			700.517i			1.41805i
$495$		0			0
$496$	−172.616			−0.348017
$497$	156.866	−	156.866i	0.315625	−	0.315625i
$498$	8.77210	+	8.77210i	0.0176147	+	0.0176147i
$499$			224.560i			0.450020i	0.974356	+	0.225010i	$0.0722415\pi$
$499$			224.560i			0.450020i	−0.974356	+	0.225010i	$0.927758\pi$
$500$		0			0
$501$	338.113			0.674876
$502$	−397.176	+	397.176i	−0.791187	+	0.791187i
$503$	527.137	+	527.137i	1.04799	+	1.04799i	0.998789	+	0.0491977i	$0.0156664\pi$
$503$	527.137	+	527.137i	1.04799	+	1.04799i	0.0491977	+	0.998789i	$0.484334\pi$
$504$			48.2816i			0.0957968i
$505$		0			0
$506$	31.0517			0.0613671
$507$	−429.995	+	429.995i	−0.848116	+	0.848116i
$508$	−122.275	−	122.275i	−0.240698	−	0.240698i
$509$			672.701i			1.32161i	0.750556	+	0.660807i	$0.229786\pi$
$509$			672.701i			1.32161i	−0.750556	+	0.660807i	$0.770214\pi$
$510$		0			0
$511$	−537.845			−1.05253
$512$	−16.0000	+	16.0000i	−0.0312500	+	0.0312500i
$513$	−79.8053	−	79.8053i	−0.155566	−	0.155566i
$514$		−	444.158i		−	0.864120i
$515$		0			0
$516$	158.322			0.306825
$517$	10.0783	−	10.0783i	0.0194938	−	0.0194938i
$518$	−401.430	−	401.430i	−0.774961	−	0.774961i
$519$			108.379i			0.208822i
$520$		0			0
$521$	−478.786			−0.918975			−0.459487	−	0.888184i	$-0.651967\pi$
$521$	−478.786			−0.918975			−0.459487	+	0.888184i	$0.651967\pi$
$522$	55.0967	−	55.0967i	0.105549	−	0.105549i
$523$	371.133	+	371.133i	0.709623	+	0.709623i	0.966456	−	0.256833i	$-0.0826791\pi$
$523$	371.133	+	371.133i	0.709623	+	0.709623i	−0.256833	+	0.966456i	$0.582679\pi$
$524$		−	271.070i		−	0.517308i
$525$		0			0
$526$	−297.927			−0.566402
$527$	−994.149	+	994.149i	−1.88643	+	1.88643i
$528$	4.31290	+	4.31290i	0.00816838	+	0.00816838i
$529$			93.0331i			0.175866i
$530$		0			0
$531$	−221.848			−0.417793
$532$	174.781	−	174.781i	0.328537	−	0.328537i
$533$	−677.678	−	677.678i	−1.27144	−	1.27144i
$534$			156.370i			0.292827i
$535$		0			0
$536$	42.0941			0.0785337
$537$	128.192	−	128.192i	0.238718	−	0.238718i
$538$	−399.749	−	399.749i	−0.743028	−	0.743028i
$539$		−	14.6347i		−	0.0271517i
$540$		0			0
$541$	2.49427			0.00461049			0.00230524	−	0.999997i	$-0.499266\pi$
$541$	2.49427			0.00461049			0.00230524	+	0.999997i	$0.499266\pi$
$542$	−146.491	+	146.491i	−0.270279	+	0.270279i
$543$	−245.095	−	245.095i	−0.451371	−	0.451371i
$544$			184.298i			0.338783i
$545$		0			0
$546$	317.856			0.582153
$547$	−476.872	+	476.872i	−0.871796	+	0.871796i	−0.992668	−	0.120872i	$-0.961431\pi$
$547$	−476.872	+	476.872i	−0.871796	+	0.871796i	0.120872	+	0.992668i	$0.461431\pi$
$548$	175.981	+	175.981i	0.321134	+	0.321134i
$549$		−	60.9923i		−	0.111097i
$550$		0			0
$551$	−398.905			−0.723966
$552$	−86.3967	+	86.3967i	−0.156516	+	0.156516i
$553$	0.184988	+	0.184988i	0.000334518	+	0.000334518i
$554$			359.887i			0.649616i
$555$		0			0
$556$	216.188			0.388827
$557$	−480.526	+	480.526i	−0.862704	+	0.862704i	−0.991651	−	0.128947i	$-0.958840\pi$
$557$	−480.526	+	480.526i	−0.862704	+	0.862704i	0.128947	+	0.991651i	$0.458840\pi$
$558$	−129.462	−	129.462i	−0.232011	−	0.232011i
$559$		−	1042.29i		−	1.86457i
$560$		0			0
$561$	49.6787			0.0885538
$562$	−286.333	+	286.333i	−0.509489	+	0.509489i
$563$	528.527	+	528.527i	0.938770	+	0.938770i	0.998231	−	0.0594608i	$-0.0189381\pi$
$563$	528.527	+	528.527i	0.938770	+	0.938770i	−0.0594608	+	0.998231i	$0.518938\pi$
$564$			56.0827i			0.0994375i
$565$		0			0
$566$	−445.426			−0.786972
$567$	−36.2112	+	36.2112i	−0.0638645	+	0.0638645i
$568$	77.9754	+	77.9754i	0.137281	+	0.137281i
$569$			153.089i			0.269050i	0.990910	+	0.134525i	$0.0429509\pi$
$569$			153.089i			0.269050i	−0.990910	+	0.134525i	$0.957049\pi$
$570$		0			0
$571$	−462.664			−0.810269			−0.405135	−	0.914257i	$-0.632775\pi$
$571$	−462.664			−0.810269			−0.405135	+	0.914257i	$0.632775\pi$
$572$	28.3935	−	28.3935i	0.0496389	−	0.0496389i
$573$	−400.766	−	400.766i	−0.699417	−	0.699417i
$574$			338.166i			0.589139i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$	444.484	−	444.484i	0.770336	−	0.770336i	−0.207829	−	0.978165i	$-0.566640\pi$
$577$	444.484	−	444.484i	0.770336	−	0.770336i	0.978165	+	0.207829i	$0.0666398\pi$
$578$	772.426	+	772.426i	1.33638	+	1.33638i
$579$		−	56.9754i		−	0.0984032i
$580$		0			0
$581$	−28.8176			−0.0496000
$582$	154.535	−	154.535i	0.265525	−	0.265525i
$583$	−24.9657	−	24.9657i	−0.0428229	−	0.0428229i
$584$		−	267.354i		−	0.457798i
$585$		0			0
$586$	−200.447			−0.342059
$587$	−102.115	+	102.115i	−0.173961	+	0.173961i	−0.788717	−	0.614756i	$-0.789254\pi$
$587$	−102.115	+	102.115i	−0.173961	+	0.173961i	0.614756	+	0.788717i	$0.289254\pi$
$588$	40.7189	+	40.7189i	0.0692499	+	0.0692499i
$589$			937.317i			1.59137i
$590$		0			0
$591$	402.431			0.680933
$592$	199.544	−	199.544i	0.337068	−	0.337068i
$593$	−38.6110	−	38.6110i	−0.0651112	−	0.0651112i	0.673801	−	0.738913i	$-0.264660\pi$
$593$	−38.6110	−	38.6110i	−0.0651112	−	0.0651112i	−0.738913	+	0.673801i	$0.764660\pi$
$594$			6.46936i			0.0108912i
$595$		0			0
$596$	154.066			0.258501
$597$	65.3124	−	65.3124i	0.109401	−	0.109401i
$598$	568.782	+	568.782i	0.951140	+	0.951140i
$599$		−	458.928i		−	0.766157i	−0.923716	−	0.383079i	$-0.874864\pi$
$599$		−	458.928i		−	0.766157i	0.923716	−	0.383079i	$-0.125136\pi$
$600$		0			0
$601$	796.357			1.32505			0.662527	−	0.749038i	$-0.269484\pi$
$601$	796.357			1.32505			0.662527	+	0.749038i	$0.269484\pi$
$602$	−260.055	+	260.055i	−0.431986	+	0.431986i
$603$	31.5706	+	31.5706i	0.0523558	+	0.0523558i
$604$		−	222.570i		−	0.368493i
$605$		0			0
$606$	−206.311			−0.340447
$607$	−292.246	+	292.246i	−0.481460	+	0.481460i	−0.905598	−	0.424138i	$-0.860577\pi$
$607$	−292.246	+	292.246i	−0.481460	+	0.481460i	0.424138	+	0.905598i	$0.360577\pi$
$608$	86.8810	+	86.8810i	0.142896	+	0.142896i
$609$			181.001i			0.297210i
$610$		0			0
$611$	369.213			0.604277
$612$	−138.223	+	138.223i	−0.225855	+	0.225855i
$613$	−105.765	−	105.765i	−0.172536	−	0.172536i	0.615557	−	0.788093i	$-0.288931\pi$
$613$	−105.765	−	105.765i	−0.172536	−	0.172536i	−0.788093	+	0.615557i	$0.788931\pi$
$614$			343.674i			0.559730i
$615$		0			0
$616$	−14.1685			−0.0230008
$617$	156.187	−	156.187i	0.253139	−	0.253139i	−0.569117	−	0.822256i	$-0.692715\pi$
$617$	156.187	−	156.187i	0.253139	−	0.253139i	0.822256	+	0.569117i	$0.192715\pi$
$618$	183.053	+	183.053i	0.296203	+	0.296203i
$619$		−	526.395i		−	0.850396i	−0.905100	−	0.425198i	$-0.860205\pi$
$619$		−	526.395i		−	0.850396i	0.905100	−	0.425198i	$-0.139795\pi$
$620$		0			0
$621$	−129.595			−0.208688
$622$	140.504	−	140.504i	0.225890	−	0.225890i
$623$	−256.849	−	256.849i	−0.412278	−	0.412278i
$624$			158.001i			0.253207i
$625$		0			0
$626$	−783.055			−1.25089
$627$	23.4194	−	23.4194i	0.0373514	−	0.0373514i
$628$	−266.016	−	266.016i	−0.423592	−	0.423592i
$629$		−	2298.47i		−	3.65417i
$630$		0			0
$631$	−77.2365			−0.122403			−0.0612017	−	0.998125i	$-0.519493\pi$
$631$	−77.2365			−0.122403			−0.0612017	+	0.998125i	$0.519493\pi$
$632$	−0.0919548	+	0.0919548i	−0.000145498	+	0.000145498i
$633$	−489.480	−	489.480i	−0.773271	−	0.773271i
$634$		−	386.064i		−	0.608934i
$635$		0			0
$636$	138.927			0.218438
$637$	268.068	−	268.068i	0.420829	−	0.420829i
$638$	16.1685	+	16.1685i	0.0253424	+	0.0253424i
$639$			116.963i			0.183041i
$640$		0			0
$641$	533.896			0.832911			0.416455	−	0.909156i	$-0.363272\pi$
$641$	533.896			0.832911			0.416455	+	0.909156i	$0.363272\pi$
$642$	323.031	−	323.031i	0.503164	−	0.503164i
$643$	−666.085	−	666.085i	−1.03590	−	1.03590i	−0.999331	−	0.0365703i	$-0.988357\pi$
$643$	−666.085	−	666.085i	−1.03590	−	1.03590i	−0.0365703	−	0.999331i	$-0.511643\pi$
$644$		−	283.826i		−	0.440724i
$645$		0			0
$646$	1000.75			1.54915
$647$	−574.511	+	574.511i	−0.887962	+	0.887962i	−0.994327	−	0.106365i	$-0.966079\pi$
$647$	−574.511	+	574.511i	−0.887962	+	0.887962i	0.106365	+	0.994327i	$0.466079\pi$
$648$	−18.0000	−	18.0000i	−0.0277778	−	0.0277778i
$649$		−	65.1026i		−	0.100312i
$650$		0			0
$651$	425.302			0.653306
$652$	−329.160	+	329.160i	−0.504847	+	0.504847i
$653$	−882.196	−	882.196i	−1.35099	−	1.35099i	−0.884556	−	0.466434i	$-0.845539\pi$
$653$	−882.196	−	882.196i	−1.35099	−	1.35099i	−0.466434	−	0.884556i	$-0.654461\pi$
$654$			188.673i			0.288491i
$655$		0			0
$656$	−168.097			−0.256245
$657$	200.515	−	200.515i	0.305199	−	0.305199i
$658$	−92.1200	−	92.1200i	−0.140000	−	0.140000i
$659$		−	566.500i		−	0.859636i	−0.902916	−	0.429818i	$-0.858578\pi$
$659$		−	566.500i		−	0.859636i	0.902916	−	0.429818i	$-0.141422\pi$
$660$		0			0
$661$	237.420			0.359184			0.179592	−	0.983741i	$-0.442522\pi$
$661$	237.420			0.359184			0.179592	+	0.983741i	$0.442522\pi$
$662$	339.875	−	339.875i	0.513406	−	0.513406i
$663$	909.975	+	909.975i	1.37251	+	1.37251i
$664$		−	14.3248i		−	0.0215735i
$665$		0			0
$666$	299.316			0.449424
$667$	−323.889	+	323.889i	−0.485591	+	0.485591i
$668$	−276.068	−	276.068i	−0.413276	−	0.413276i
$669$		−	44.5976i		−	0.0666630i
$670$		0			0
$671$	17.8986			0.0266744
$672$	39.4218	−	39.4218i	0.0586633	−	0.0586633i
$673$	46.1772	+	46.1772i	0.0686139	+	0.0686139i	0.740581	−	0.671967i	$-0.234550\pi$
$673$	46.1772	+	46.1772i	0.0686139	+	0.0686139i	−0.671967	+	0.740581i	$0.734550\pi$
$674$			502.482i			0.745522i
$675$		0			0
$676$	702.179			1.03873
$677$	746.476	−	746.476i	1.10262	−	1.10262i	0.108531	−	0.994093i	$-0.465385\pi$
$677$	746.476	−	746.476i	1.10262	−	1.10262i	0.994093	−	0.108531i	$-0.0346145\pi$
$678$	156.543	+	156.543i	0.230889	+	0.230889i
$679$			507.672i			0.747675i
$680$		0			0
$681$	546.138			0.801965
$682$	37.9914	−	37.9914i	0.0557059	−	0.0557059i
$683$	367.617	+	367.617i	0.538238	+	0.538238i	0.923011	−	0.384773i	$-0.125720\pi$
$683$	367.617	+	367.617i	0.538238	+	0.538238i	−0.384773	+	0.923011i	$0.625720\pi$
$684$			130.322i			0.190529i
$685$		0			0
$686$	−528.067			−0.769778
$687$	−94.1304	+	94.1304i	−0.137017	+	0.137017i
$688$	−129.269	−	129.269i	−0.187891	−	0.187891i
$689$		−	914.606i		−	1.32744i
$690$		0			0
$691$	867.808			1.25587			0.627937	−	0.778264i	$-0.283900\pi$
$691$	867.808			1.25587			0.627937	+	0.778264i	$0.283900\pi$
$692$	88.4909	−	88.4909i	0.127877	−	0.127877i
$693$	−10.6264	−	10.6264i	−0.0153339	−	0.0153339i
$694$		−	828.887i		−	1.19436i
$695$		0			0
$696$	−89.9726			−0.129271
$697$	−968.120	+	968.120i	−1.38898	+	1.38898i
$698$	311.540	+	311.540i	0.446332	+	0.446332i
$699$		−	143.843i		−	0.205784i
$700$		0			0
$701$	−73.7516			−0.105209			−0.0526046	−	0.998615i	$-0.516752\pi$
$701$	−73.7516			−0.105209			−0.0526046	+	0.998615i	$0.516752\pi$
$702$	−118.501	+	118.501i	−0.168804	+	0.168804i
$703$	−1083.54	−	1083.54i	−1.54131	−	1.54131i
$704$		−	7.04294i		−	0.0100042i
$705$		0			0
$706$	482.715			0.683732
$707$	338.881	−	338.881i	0.479323	−	0.479323i
$708$	181.138	+	181.138i	0.255845	+	0.255845i
$709$			1222.00i			1.72356i	0.507286	+	0.861778i	$0.330649\pi$
$709$			1222.00i			1.72356i	−0.507286	+	0.861778i	$0.669351\pi$
$710$		0			0
$711$	−0.137932			−0.000193997
$712$	127.675	−	127.675i	0.179319	−	0.179319i
$713$	761.050	+	761.050i	1.06739	+	1.06739i
$714$		−	454.084i		−	0.635972i
$715$		0			0
$716$	−209.336			−0.292369
$717$	179.647	−	179.647i	0.250554	−	0.250554i
$718$	−65.5040	−	65.5040i	−0.0912312	−	0.0912312i
$719$			1271.29i			1.76814i	0.467354	+	0.884070i	$0.345207\pi$
$719$			1271.29i			1.76814i	−0.467354	+	0.884070i	$0.654793\pi$
$720$		0			0
$721$	−601.357			−0.834060
$722$	110.770	−	110.770i	0.153420	−	0.153420i
$723$	234.947	+	234.947i	0.324962	+	0.324962i
$724$			400.238i			0.552815i
$725$		0			0
$726$	294.490			0.405633
$727$	1004.06	−	1004.06i	1.38110	−	1.38110i	0.538434	−	0.842668i	$-0.319016\pi$
$727$	1004.06	−	1004.06i	1.38110	−	1.38110i	0.842668	−	0.538434i	$-0.180984\pi$
$728$	−259.528	−	259.528i	−0.356495	−	0.356495i
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−1489.00			−2.03694
$732$	−49.8000	+	49.8000i	−0.0680328	+	0.0680328i
$733$	249.129	+	249.129i	0.339876	+	0.339876i	0.856321	−	0.516444i	$-0.172745\pi$
$733$	249.129	+	249.129i	0.339876	+	0.339876i	−0.516444	+	0.856321i	$0.672745\pi$
$734$		−	354.597i		−	0.483102i
$735$		0			0
$736$	141.085			0.191692
$737$	−9.26457	+	9.26457i	−0.0125706	+	0.0125706i
$738$	−126.073	−	126.073i	−0.170830	−	0.170830i
$739$			147.407i			0.199468i	0.995014	+	0.0997340i	$0.0317992\pi$
$739$			147.407i			0.199468i	−0.995014	+	0.0997340i	$0.968201\pi$
$740$		0			0
$741$	857.955			1.15783
$742$	−228.197	+	228.197i	−0.307543	+	0.307543i
$743$	593.793	+	593.793i	0.799183	+	0.799183i	0.982967	−	0.183784i	$-0.0588345\pi$
$743$	593.793	+	593.793i	0.799183	+	0.799183i	−0.183784	+	0.982967i	$0.558835\pi$
$744$			211.411i			0.284154i
$745$		0			0
$746$	−127.907			−0.171457
$747$	10.7436	−	10.7436i	0.0143823	−	0.0143823i
$748$	−40.5625	−	40.5625i	−0.0542279	−	0.0542279i
$749$			1061.20i			1.41683i
$750$		0			0
$751$	410.932			0.547180			0.273590	−	0.961846i	$-0.411789\pi$
$751$	410.932			0.547180			0.273590	+	0.961846i	$0.411789\pi$
$752$	45.7914	−	45.7914i	0.0608928	−	0.0608928i
$753$	486.439	+	486.439i	0.646002	+	0.646002i
$754$			592.323i			0.785574i
$755$		0			0
$756$	59.1326			0.0782178
$757$	853.737	−	853.737i	1.12779	−	1.12779i	0.137254	−	0.990536i	$-0.456172\pi$
$757$	853.737	−	853.737i	1.12779	−	1.12779i	0.990536	−	0.137254i	$-0.0438276\pi$
$758$	−344.443	−	344.443i	−0.454410	−	0.454410i
$759$		−	38.0305i		−	0.0501060i
$760$		0			0
$761$	−179.153			−0.235418			−0.117709	−	0.993048i	$-0.537555\pi$
$761$	−179.153			−0.235418			−0.117709	+	0.993048i	$0.537555\pi$
$762$	−149.755	+	149.755i	−0.196529	+	0.196529i
$763$	−309.909	−	309.909i	−0.406172	−	0.406172i
$764$			654.448i			0.856607i
$765$		0			0
$766$	45.3775			0.0592396
$767$	1192.50	−	1192.50i	1.55476	−	1.55476i
$768$	19.5959	+	19.5959i	0.0255155	+	0.0255155i
$769$		−	189.333i		−	0.246207i	−0.992394	−	0.123104i	$-0.960715\pi$
$769$		−	189.333i		−	0.246207i	0.992394	−	0.123104i	$-0.0392848\pi$
$770$		0			0
$771$	−543.980			−0.705551
$772$	−46.5203	+	46.5203i	−0.0602594	+	0.0602594i
$773$	349.358	+	349.358i	0.451950	+	0.451950i	0.896002	−	0.444051i	$-0.146459\pi$
$773$	349.358	+	349.358i	0.451950	+	0.451950i	−0.444051	+	0.896002i	$0.646459\pi$
$774$		−	193.904i		−	0.250522i
$775$		0			0
$776$	−252.355			−0.325200
$777$	−491.649	+	491.649i	−0.632753	+	0.632753i
$778$	−234.054	−	234.054i	−0.300840	−	0.300840i
$779$			912.776i			1.17173i
$780$		0			0
$781$	−34.3235			−0.0439481
$782$	812.553	−	812.553i	1.03907	−	1.03907i
$783$	−67.4795	−	67.4795i	−0.0861807	−	0.0861807i
$784$		−	66.4938i		−	0.0848135i
$785$		0			0
$786$	−331.991			−0.422381
$787$	511.653	−	511.653i	0.650131	−	0.650131i	−0.302893	−	0.953025i	$-0.597953\pi$
$787$	511.653	−	511.653i	0.650131	−	0.650131i	0.953025	+	0.302893i	$0.0979525\pi$
$788$	−328.584	−	328.584i	−0.416984	−	0.416984i
$789$			364.885i			0.462465i
$790$		0			0
$791$	−514.266			−0.650146
$792$	5.28221	−	5.28221i	0.00666945	−	0.00666945i
$793$	327.852	+	327.852i	0.413432	+	0.413432i
$794$		−	990.017i		−	1.24687i
$795$		0			0
$796$	−106.655			−0.133988
$797$	−80.9097	+	80.9097i	−0.101518	+	0.101518i	−0.756041	−	0.654524i	$-0.772869\pi$
$797$	−80.9097	+	80.9097i	−0.101518	+	0.101518i	0.654524	+	0.756041i	$0.272869\pi$
$798$	−214.063	−	214.063i	−0.268249	−	0.268249i
$799$		−	527.453i		−	0.660141i
$800$		0			0
$801$	191.513			0.239093
$802$	46.2845	−	46.2845i	0.0577114	−	0.0577114i
$803$	58.8425	+	58.8425i	0.0732783	+	0.0732783i
$804$		−	51.5545i		−	0.0641225i
$805$		0			0
$806$	1391.80			1.72679
$807$	−489.590	+	489.590i	−0.606680	+	0.606680i
$808$	168.452	+	168.452i	0.208481	+	0.208481i
$809$			256.018i			0.316462i	0.987402	+	0.158231i	$0.0505791\pi$
$809$			256.018i			0.316462i	−0.987402	+	0.158231i	$0.949421\pi$
$810$		0			0
$811$	497.936			0.613978			0.306989	−	0.951713i	$-0.400678\pi$
$811$	497.936			0.613978			0.306989	+	0.951713i	$0.400678\pi$
$812$	147.787	−	147.787i	0.182003	−	0.182003i
$813$	179.414	+	179.414i	0.220682	+	0.220682i
$814$			87.8361i			0.107907i
$815$		0			0
$816$	225.718			0.276615
$817$	−701.941	+	701.941i	−0.859169	+	0.859169i
$818$	239.436	+	239.436i	0.292710	+	0.292710i
$819$		−	389.292i		−	0.475326i
$820$		0			0
$821$	976.995			1.19001			0.595003	−	0.803724i	$-0.297151\pi$
$821$	976.995			1.19001			0.595003	+	0.803724i	$0.297151\pi$
$822$	215.532	−	215.532i	0.262205	−	0.262205i
$823$	49.6700	+	49.6700i	0.0603524	+	0.0603524i	0.736639	−	0.676286i	$-0.236412\pi$
$823$	49.6700	+	49.6700i	0.0603524	+	0.0603524i	−0.676286	+	0.736639i	$0.736412\pi$
$824$		−	298.925i		−	0.362773i
$825$		0			0
$826$	−595.065			−0.720418
$827$	−150.970	+	150.970i	−0.182551	+	0.182551i	−0.792467	−	0.609915i	$-0.791203\pi$
$827$	−150.970	+	150.970i	−0.182551	+	0.182551i	0.609915	+	0.792467i	$0.291203\pi$
$828$	105.814	+	105.814i	0.127795	+	0.127795i
$829$		−	808.898i		−	0.975752i	−0.872913	−	0.487876i	$-0.837772\pi$
$829$		−	808.898i		−	0.975752i	0.872913	−	0.487876i	$-0.162228\pi$
$830$		0			0
$831$	440.770			0.530409
$832$	129.007	−	129.007i	0.155057	−	0.155057i
$833$	−382.958	−	382.958i	−0.459733	−	0.459733i
$834$		−	264.775i		−	0.317476i
$835$		0			0
$836$	−38.2436			−0.0457460
$837$	−158.558	+	158.558i	−0.189436	+	0.189436i
$838$	−150.579	−	150.579i	−0.179689	−	0.179689i
$839$			288.815i			0.344237i	0.985076	+	0.172119i	$0.0550613\pi$
$839$			288.815i			0.344237i	−0.985076	+	0.172119i	$0.944939\pi$
$840$		0			0
$841$	503.705			0.598936
$842$	−485.057	+	485.057i	−0.576077	+	0.576077i
$843$	350.684	+	350.684i	0.415996	+	0.415996i
$844$			799.318i			0.947059i
$845$		0			0
$846$	68.6870			0.0811903
$847$	−483.721	+	483.721i	−0.571099	+	0.571099i
$848$	−113.433	−	113.433i	−0.133765	−	0.133765i
$849$			545.533i			0.642560i
$850$		0			0
$851$	−1759.55			−2.06762
$852$	95.4999	−	95.4999i	0.112089	−	0.112089i
$853$	586.627	+	586.627i	0.687723	+	0.687723i	0.961728	−	0.274005i	$-0.0883486\pi$
$853$	586.627	+	586.627i	0.687723	+	0.687723i	−0.274005	+	0.961728i	$0.588349\pi$
$854$		−	163.600i		−	0.191569i
$855$		0			0
$856$	−527.507			−0.616247
$857$	−293.219	+	293.219i	−0.342146	+	0.342146i	−0.857174	−	0.515028i	$-0.827782\pi$
$857$	−293.219	+	293.219i	−0.342146	+	0.342146i	0.515028	+	0.857174i	$0.327782\pi$
$858$	−34.7747	−	34.7747i	−0.0405300	−	0.0405300i
$859$			311.960i			0.363167i	0.983376	+	0.181583i	$0.0581222\pi$
$859$			311.960i			0.363167i	−0.983376	+	0.181583i	$0.941878\pi$
$860$		0			0
$861$	414.167			0.481030
$862$	790.360	−	790.360i	0.916891	−	0.916891i
$863$	−590.699	−	590.699i	−0.684471	−	0.684471i	0.276533	−	0.961004i	$-0.410814\pi$
$863$	−590.699	−	590.699i	−0.684471	−	0.684471i	−0.961004	+	0.276533i	$0.910814\pi$
$864$			29.3939i			0.0340207i
$865$		0			0
$866$	−1105.38			−1.27642
$867$	946.025	−	946.025i	1.09115	−	1.09115i
$868$	−347.258	−	347.258i	−0.400066	−	0.400066i
$869$		−	0.0404770i		−	4.65789e-5i
$870$		0			0
$871$	−339.402			−0.389670
$872$	154.051	−	154.051i	0.176664	−	0.176664i
$873$	−189.267	−	189.267i	−0.216800	−	0.216800i
$874$		−	766.102i		−	0.876547i
$875$		0			0
$876$	−327.440			−0.373790
$877$	881.008	−	881.008i	1.00457	−	1.00457i	0.00458097	−	0.999990i	$-0.498542\pi$
$877$	881.008	−	881.008i	1.00457	−	1.00457i	0.999990	−	0.00458097i	$-0.00145817\pi$
$878$	649.354	+	649.354i	0.739583	+	0.739583i
$879$			245.496i			0.279290i
$880$		0			0
$881$	−705.494			−0.800787			−0.400394	−	0.916343i	$-0.631127\pi$
$881$	−705.494			−0.800787			−0.400394	+	0.916343i	$0.631127\pi$
$882$	49.8703	−	49.8703i	0.0565423	−	0.0565423i
$883$	−295.341	−	295.341i	−0.334475	−	0.334475i	0.519808	−	0.854283i	$-0.326003\pi$
$883$	−295.341	−	295.341i	−0.334475	−	0.334475i	−0.854283	+	0.519808i	$0.826003\pi$
$884$		−	1485.98i		−	1.68098i
$885$		0			0
$886$	110.728			0.124975
$887$	157.844	−	157.844i	0.177953	−	0.177953i	−0.612510	−	0.790463i	$-0.709840\pi$
$887$	157.844	−	157.844i	0.177953	−	0.177953i	0.790463	+	0.612510i	$0.209840\pi$
$888$	−244.391	−	244.391i	−0.275215	−	0.275215i
$889$		−	491.968i		−	0.553395i
$890$		0			0
$891$	7.92331			0.00889261
$892$	−36.4138	+	36.4138i	−0.0408226	+	0.0408226i
$893$	−248.650	−	248.650i	−0.278443	−	0.278443i
$894$		−	188.692i		−	0.211065i
$895$		0			0
$896$	−64.3755			−0.0718476
$897$	696.613	−	696.613i	0.776603	−	0.776603i
$898$	835.180	+	835.180i	0.930045	+	0.930045i
$899$			792.549i			0.881590i
$900$		0			0
$901$	−1306.59			−1.45016
$902$	36.9967	−	36.9967i	0.0410164	−	0.0410164i
$903$	318.502	+	318.502i	0.352715	+	0.352715i
$904$		−	255.633i		−	0.282780i
$905$		0			0
$906$	−272.591			−0.300873
$907$	1061.72	−	1061.72i	1.17059	−	1.17059i	0.188516	−	0.982070i	$-0.439632\pi$
$907$	1061.72	−	1061.72i	1.17059	−	1.17059i	0.982070	−	0.188516i	$-0.0603676\pi$
$908$	−445.920	−	445.920i	−0.491101	−	0.491101i
$909$			252.678i			0.277974i
$910$		0			0
$911$	−938.162			−1.02982			−0.514908	−	0.857246i	$-0.672174\pi$
$911$	−938.162			−1.02982			−0.514908	+	0.857246i	$0.672174\pi$
$912$	106.407	−	106.407i	0.116674	−	0.116674i
$913$	3.15277	+	3.15277i	0.00345320	+	0.00345320i
$914$			537.412i			0.587978i
$915$		0			0
$916$	153.714			0.167810
$917$	545.320	−	545.320i	0.594678	−	0.594678i
$918$	169.288	+	169.288i	0.184410	+	0.184410i
$919$		−	552.208i		−	0.600880i	−0.953801	−	0.300440i	$-0.902867\pi$
$919$		−	552.208i		−	0.600880i	0.953801	−	0.300440i	$-0.0971335\pi$
$920$		0			0
$921$	420.913			0.457017
$922$	337.848	−	337.848i	0.366430	−	0.366430i
$923$	−628.711	−	628.711i	−0.681161	−	0.681161i
$924$			17.3528i			0.0187801i
$925$		0			0
$926$	−255.443			−0.275857
$927$	224.194	−	224.194i	0.241849	−	0.241849i
$928$	73.4623	+	73.4623i	0.0791620	+	0.0791620i
$929$			773.378i			0.832484i	0.909254	+	0.416242i	$0.136653\pi$
$929$			773.378i			0.832484i	−0.909254	+	0.416242i	$0.863347\pi$
$930$		0			0
$931$	−361.065			−0.387825
$932$	−117.447	+	117.447i	−0.126016	+	0.126016i
$933$	−172.081	−	172.081i	−0.184439	−	0.184439i
$934$		−	97.7188i		−	0.104624i
$935$		0			0
$936$	193.511			0.206742
$937$	−271.711	+	271.711i	−0.289980	+	0.289980i	−0.837072	−	0.547093i	$-0.815735\pi$
$937$	−271.711	+	271.711i	−0.289980	+	0.289980i	0.547093	+	0.837072i	$0.315735\pi$
$938$	84.6820	+	84.6820i	0.0902793	+	0.0902793i
$939$			959.042i			1.02134i
$940$		0			0
$941$	−919.761			−0.977429			−0.488714	−	0.872444i	$-0.662534\pi$
$941$	−919.761			−0.977429			−0.488714	+	0.872444i	$0.662534\pi$
$942$	−325.801	+	325.801i	−0.345861	+	0.345861i
$943$	741.124	+	741.124i	0.785922	+	0.785922i
$944$		−	295.797i		−	0.313345i
$945$		0			0
$946$	56.9023			0.0601504
$947$	1053.78	−	1053.78i	1.11276	−	1.11276i	0.119984	−	0.992776i	$-0.461716\pi$
$947$	1053.78	−	1053.78i	1.11276	−	1.11276i	0.992776	−	0.119984i	$-0.0382843\pi$
$948$	0.112621	+	0.112621i	0.000118799	+	0.000118799i
$949$			2155.66i			2.27151i
$950$		0			0
$951$	−472.830			−0.497192
$952$	−370.758	+	370.758i	−0.389451	+	0.389451i
$953$	926.983	+	926.983i	0.972700	+	0.972700i	0.999637	−	0.0269370i	$-0.00857537\pi$
$953$	926.983	+	926.983i	0.972700	+	0.972700i	−0.0269370	+	0.999637i	$0.508575\pi$
$954$		−	170.150i		−	0.178354i
$955$		0			0
$956$	−293.363			−0.306865
$957$	19.8022	−	19.8022i	0.0206920	−	0.0206920i
$958$	−11.1370	−	11.1370i	−0.0116253	−	0.0116253i
$959$			708.055i			0.738326i
$960$		0			0
$961$	901.272			0.937848
$962$	−1608.91	+	1608.91i	−1.67247	+	1.67247i
$963$	−395.631	−	395.631i	−0.410831	−	0.410831i
$964$		−	383.667i		−	0.397995i
$965$		0			0
$966$	−347.614			−0.359849
$967$	32.5615	−	32.5615i	0.0336727	−	0.0336727i	−0.690070	−	0.723743i	$-0.742420\pi$
$967$	32.5615	−	32.5615i	0.0336727	−	0.0336727i	0.723743	+	0.690070i	$0.242420\pi$
$968$	−240.450	−	240.450i	−0.248399	−	0.248399i
$969$		−	1225.66i		−	1.26487i
$970$		0			0
$971$	−942.036			−0.970171			−0.485085	−	0.874467i	$-0.661212\pi$
$971$	−942.036			−0.970171			−0.485085	+	0.874467i	$0.661212\pi$
$972$	−22.0454	+	22.0454i	−0.0226805	+	0.0226805i
$973$	434.912	+	434.912i	0.446980	+	0.446980i
$974$			737.552i			0.757240i
$975$		0			0
$976$	81.3230			0.0833228
$977$	89.4543	−	89.4543i	0.0915602	−	0.0915602i	−0.659843	−	0.751403i	$-0.729377\pi$
$977$	89.4543	−	89.4543i	0.0915602	−	0.0915602i	0.751403	+	0.659843i	$0.229377\pi$
$978$	403.137	+	403.137i	0.412206	+	0.412206i
$979$			56.2007i			0.0574062i
$980$		0			0
$981$	231.076			0.235552
$982$	535.323	−	535.323i	0.545136	−	0.545136i
$983$	454.713	+	454.713i	0.462576	+	0.462576i	0.899499	−	0.436923i	$-0.143932\pi$
$983$	454.713	+	454.713i	0.462576	+	0.462576i	−0.436923	+	0.899499i	$0.643932\pi$
$984$			205.876i			0.209223i
$985$		0			0
$986$	846.184			0.858198
$987$	−112.823	+	112.823i	−0.114309	+	0.114309i
$988$	−700.517	−	700.517i	−0.709026	−	0.709026i
$989$			1139.88i			1.15255i
$990$		0			0
$991$	1488.35			1.50187			0.750935	−	0.660376i	$-0.229603\pi$
$991$	1488.35			1.50187			0.750935	+	0.660376i	$0.229603\pi$
$992$	172.616	−	172.616i	0.174008	−	0.174008i
$993$	−416.260	−	416.260i	−0.419195	−	0.419195i
$994$			313.731i			0.315625i
$995$		0			0
$996$	−17.5442			−0.0176147
$997$	496.448	−	496.448i	0.497942	−	0.497942i	−0.412855	−	0.910797i	$-0.635468\pi$
$997$	496.448	−	496.448i	0.497942	−	0.497942i	0.910797	+	0.412855i	$0.135468\pi$
$998$	−224.560	−	224.560i	−0.225010	−	0.225010i
$999$		−	366.586i		−	0.366953i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.3.f.b.193.8	✓	16
5.2	odd	4	inner	750.3.f.b.307.8	yes	16
5.3	odd	4		750.3.f.c.307.1	yes	16
5.4	even	2		750.3.f.c.193.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.3.f.b.193.8	✓	16	1.1	even	1	trivial
750.3.f.b.307.8	yes	16	5.2	odd	4	inner
750.3.f.c.193.1	yes	16	5.4	even	2
750.3.f.c.307.1	yes	16	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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	750.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} -2.44949 q^{6} +(4.02347 - 4.02347i) q^{7} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} -0.880368 q^{11} +(2.44949 - 2.44949i) q^{12} +(-16.1259 - 16.1259i) q^{13} +8.04693i q^{14} -4.00000 q^{16} +(-23.0372 + 23.0372i) q^{17} +(-3.00000 - 3.00000i) q^{18} +21.7203i q^{19} +9.85544 q^{21} +(0.880368 - 0.880368i) q^{22} +(17.6357 + 17.6357i) q^{23} +4.89898i q^{24} +32.2518 q^{26} +(-3.67423 + 3.67423i) q^{27} +(-8.04693 - 8.04693i) q^{28} +18.3656i q^{29} +43.1540 q^{31} +(4.00000 - 4.00000i) q^{32} +(-1.07823 - 1.07823i) q^{33} -46.0744i q^{34} +6.00000 q^{36} +(-49.8860 + 49.8860i) q^{37} +(-21.7203 - 21.7203i) q^{38} -39.5002i q^{39} +42.0242 q^{41} +(-9.85544 + 9.85544i) q^{42} +(32.3173 + 32.3173i) q^{43} +1.76074i q^{44} -35.2713 q^{46} +(-11.4478 + 11.4478i) q^{47} +(-4.89898 - 4.89898i) q^{48} +16.6234i q^{49} -56.4294 q^{51} +(-32.2518 + 32.2518i) q^{52} +(28.3583 + 28.3583i) q^{53} -7.34847i q^{54} +16.0939 q^{56} +(-26.6018 + 26.6018i) q^{57} +(-18.3656 - 18.3656i) q^{58} +73.9494i q^{59} -20.3308 q^{61} +(-43.1540 + 43.1540i) q^{62} +(12.0704 + 12.0704i) q^{63} +8.00000i q^{64} +2.15645 q^{66} +(10.5235 - 10.5235i) q^{67} +(46.0744 + 46.0744i) q^{68} +43.1984i q^{69} +38.9877 q^{71} +(-6.00000 + 6.00000i) q^{72} +(-66.8385 - 66.8385i) q^{73} -99.7721i q^{74} +43.4405 q^{76} +(-3.54213 + 3.54213i) q^{77} +(39.5002 + 39.5002i) q^{78} +0.0459774i q^{79} -9.00000 q^{81} +(-42.0242 + 42.0242i) q^{82} +(-3.58119 - 3.58119i) q^{83} -19.7109i q^{84} -64.6347 q^{86} +(-22.4932 + 22.4932i) q^{87} +(-1.76074 - 1.76074i) q^{88} -63.8377i q^{89} -129.764 q^{91} +(35.2713 - 35.2713i) q^{92} +(52.8527 + 52.8527i) q^{93} -22.8957i q^{94} +9.79796 q^{96} +(-63.0888 + 63.0888i) q^{97} +(-16.6234 - 16.6234i) q^{98} -2.64110i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{2} + 24 q^{7} + 32 q^{8} - 24 q^{11} - 48 q^{13} - 64 q^{16} - 16 q^{17} - 48 q^{18} - 48 q^{21} + 24 q^{22} + 104 q^{23} + 96 q^{26} - 48 q^{28} + 200 q^{31} + 64 q^{32} - 48 q^{33} + 96 q^{36}+ \cdots - 224 q^{98}+O(q^{100}) \) 16 * q - 16 * q^2 + 24 * q^7 + 32 * q^8 - 24 * q^11 - 48 * q^13 - 64 * q^16 - 16 * q^17 - 48 * q^18 - 48 * q^21 + 24 * q^22 + 104 * q^23 + 96 * q^26 - 48 * q^28 + 200 * q^31 + 64 * q^32 - 48 * q^33 + 96 * q^36 - 128 * q^37 + 56 * q^38 - 192 * q^41 + 48 * q^42 + 88 * q^43 - 208 * q^46 + 16 * q^47 - 48 * q^51 - 96 * q^52 - 160 * q^53 + 96 * q^56 + 48 * q^57 + 24 * q^58 - 24 * q^61 - 200 * q^62 + 72 * q^63 + 96 * q^66 + 264 * q^67 + 32 * q^68 + 176 * q^71 - 96 * q^72 + 32 * q^73 - 112 * q^76 + 32 * q^77 + 96 * q^78 - 144 * q^81 + 192 * q^82 - 288 * q^83 - 176 * q^86 - 168 * q^87 - 48 * q^88 + 32 * q^91 + 208 * q^92 + 144 * q^93 - 48 * q^97 - 224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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