LMFDB - Embedded newform 75.5.f.e.43.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,5,Mod(7,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.7"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	75.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$7.75274723129$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676 $ x^8 - 60x^5 + 1973x^4 - 3300x^3 + 1800x^2 + 31560*x + 276676
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.1
Root			$3.80336 + 3.80336i$ of defining polynomial
Character	$\chi$	$=$	75.43
Dual form			75.5.f.e.7.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-3.80336 + 3.80336i) q^{2} +(3.67423 + 3.67423i) q^{3} -12.9311i q^{4} -27.9489 q^{6} +(-16.6149 + 16.6149i) q^{7} +(-11.6720 - 11.6720i) q^{8} +27.0000i q^{9} -215.278 q^{11} +(47.5121 - 47.5121i) q^{12} +(29.9347 + 29.9347i) q^{13} -126.385i q^{14} +295.684 q^{16} +(3.97694 - 3.97694i) q^{17} +(-102.691 - 102.691i) q^{18} -604.349i q^{19} -122.094 q^{21} +(818.782 - 818.782i) q^{22} +(-376.379 - 376.379i) q^{23} -85.7710i q^{24} -227.705 q^{26} +(-99.2043 + 99.2043i) q^{27} +(214.850 + 214.850i) q^{28} +624.166i q^{29} +263.888 q^{31} +(-937.842 + 937.842i) q^{32} +(-790.983 - 790.983i) q^{33} +30.2515i q^{34} +349.141 q^{36} +(-979.613 + 979.613i) q^{37} +(2298.56 + 2298.56i) q^{38} +219.974i q^{39} -1579.25 q^{41} +(464.368 - 464.368i) q^{42} +(30.8409 + 30.8409i) q^{43} +2783.80i q^{44} +2863.01 q^{46} +(-1785.16 + 1785.16i) q^{47} +(1086.41 + 1086.41i) q^{48} +1848.89i q^{49} +29.2245 q^{51} +(387.089 - 387.089i) q^{52} +(-707.912 - 707.912i) q^{53} -754.620i q^{54} +387.857 q^{56} +(2220.52 - 2220.52i) q^{57} +(-2373.93 - 2373.93i) q^{58} -1366.84i q^{59} -2983.71 q^{61} +(-1003.66 + 1003.66i) q^{62} +(-448.602 - 448.602i) q^{63} -2402.96i q^{64} +6016.80 q^{66} +(986.003 - 986.003i) q^{67} +(-51.4265 - 51.4265i) q^{68} -2765.81i q^{69} +68.0686 q^{71} +(315.143 - 315.143i) q^{72} +(-2476.90 - 2476.90i) q^{73} -7451.65i q^{74} -7814.93 q^{76} +(3576.83 - 3576.83i) q^{77} +(-836.641 - 836.641i) q^{78} +6047.25i q^{79} -729.000 q^{81} +(6006.45 - 6006.45i) q^{82} +(5158.95 + 5158.95i) q^{83} +1578.82i q^{84} -234.598 q^{86} +(-2293.33 + 2293.33i) q^{87} +(2512.72 + 2512.72i) q^{88} +7028.50i q^{89} -994.722 q^{91} +(-4867.01 + 4867.01i) q^{92} +(969.586 + 969.586i) q^{93} -13579.2i q^{94} -6891.70 q^{96} +(-10869.3 + 10869.3i) q^{97} +(-7032.00 - 7032.00i) q^{98} -5812.52i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{6} - 20 q^{7} - 180 q^{8} - 288 q^{11} + 360 q^{12} + 340 q^{13} + 620 q^{16} - 900 q^{17} + 792 q^{21} + 1100 q^{22} + 1560 q^{23} - 3024 q^{26} - 3580 q^{28} - 512 q^{31} - 4980 q^{32} - 2700 q^{33}+ \cdots - 46440 q^{98}+O(q^{100}) $ 8 * q + 36 * q^6 - 20 * q^7 - 180 * q^8 - 288 * q^11 + 360 * q^12 + 340 * q^13 + 620 * q^16 - 900 * q^17 + 792 * q^21 + 1100 * q^22 + 1560 * q^23 - 3024 * q^26 - 3580 * q^28 - 512 * q^31 - 4980 * q^32 - 2700 * q^33 + 2484 * q^36 + 3820 * q^37 + 7680 * q^38 - 2712 * q^41 + 7380 * q^42 + 1240 * q^43 + 13528 * q^46 - 4800 * q^47 - 3600 * q^48 + 6264 * q^51 + 1240 * q^52 - 1020 * q^53 - 30720 * q^56 + 5400 * q^57 - 2340 * q^58 - 4760 * q^61 - 28680 * q^62 - 540 * q^63 + 10008 * q^66 + 8920 * q^67 + 1920 * q^68 + 7536 * q^71 + 4860 * q^72 - 11600 * q^73 + 4344 * q^76 + 360 * q^77 + 4680 * q^78 - 5832 * q^81 + 27200 * q^82 + 32400 * q^83 + 14592 * q^86 - 10620 * q^87 + 14340 * q^88 + 16528 * q^91 + 31800 * q^92 - 14040 * q^93 - 4068 * q^96 - 58640 * q^97 - 46440 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$	−3.80336	+	3.80336i	−0.950841	+	0.950841i	−0.998847	−	0.0480062i	$-0.984713\pi$
$2$	−3.80336	+	3.80336i	−0.950841	+	0.950841i	0.0480062	+	0.998847i	$0.484713\pi$
$3$	3.67423	+	3.67423i	0.408248	+	0.408248i
$3$	3.67423	+	3.67423i	0.408248	+	0.408248i
$4$		−	12.9311i		−	0.808197i
$5$		0			0
$5$		0			0
$6$	−27.9489			−0.776358
$7$	−16.6149	+	16.6149i	−0.339079	+	0.339079i	−0.856021	−	0.516941i	$-0.827071\pi$
$7$	−16.6149	+	16.6149i	−0.339079	+	0.339079i	0.516941	+	0.856021i	$0.327071\pi$
$8$	−11.6720	−	11.6720i	−0.182374	−	0.182374i
$9$			27.0000i			0.333333i
$10$		0			0
$11$	−215.278			−1.77916			−0.889580	−	0.456779i	$-0.849003\pi$
$11$	−215.278			−1.77916			−0.889580	+	0.456779i	$0.849003\pi$
$12$	47.5121	−	47.5121i	0.329945	−	0.329945i
$13$	29.9347	+	29.9347i	0.177128	+	0.177128i	0.790103	−	0.612975i	$-0.210027\pi$
$13$	29.9347	+	29.9347i	0.177128	+	0.177128i	−0.612975	+	0.790103i	$0.710027\pi$
$14$		−	126.385i		−	0.644821i
$15$		0			0
$16$	295.684			1.15501
$17$	3.97694	−	3.97694i	0.0137611	−	0.0137611i	−0.700193	−	0.713954i	$-0.746903\pi$
$17$	3.97694	−	3.97694i	0.0137611	−	0.0137611i	0.713954	+	0.700193i	$0.246903\pi$
$18$	−102.691	−	102.691i	−0.316947	−	0.316947i
$19$		−	604.349i		−	1.67410i	−0.547128	−	0.837049i	$-0.684279\pi$
$19$		−	604.349i		−	1.67410i	0.547128	−	0.837049i	$-0.315721\pi$
$20$		0			0
$21$	−122.094			−0.276857
$22$	818.782	−	818.782i	1.69170	−	1.69170i
$23$	−376.379	−	376.379i	−0.711491	−	0.711491i	0.255356	−	0.966847i	$-0.417807\pi$
$23$	−376.379	−	376.379i	−0.711491	−	0.711491i	−0.966847	+	0.255356i	$0.917807\pi$
$24$		−	85.7710i		−	0.148908i
$25$		0			0
$26$	−227.705			−0.336841
$27$	−99.2043	+	99.2043i	−0.136083	+	0.136083i
$28$	214.850	+	214.850i	0.274043	+	0.274043i
$29$			624.166i			0.742171i	0.928599	+	0.371086i	$0.121014\pi$
$29$			624.166i			0.742171i	−0.928599	+	0.371086i	$0.878986\pi$
$30$		0			0
$31$	263.888			0.274597			0.137299	−	0.990530i	$-0.456158\pi$
$31$	263.888			0.274597			0.137299	+	0.990530i	$0.456158\pi$
$32$	−937.842	+	937.842i	−0.915861	+	0.915861i
$33$	−790.983	−	790.983i	−0.726339	−	0.726339i
$34$			30.2515i			0.0261691i
$35$		0			0
$36$	349.141			0.269399
$37$	−979.613	+	979.613i	−0.715568	+	0.715568i	−0.967694	−	0.252126i	$-0.918870\pi$
$37$	−979.613	+	979.613i	−0.715568	+	0.715568i	0.252126	+	0.967694i	$0.418870\pi$
$38$	2298.56	+	2298.56i	1.59180	+	1.59180i
$39$			219.974i			0.144625i
$40$		0			0
$41$	−1579.25			−0.939468			−0.469734	−	0.882808i	$-0.655650\pi$
$41$	−1579.25			−0.939468			−0.469734	+	0.882808i	$0.655650\pi$
$42$	464.368	−	464.368i	0.263247	−	0.263247i
$43$	30.8409	+	30.8409i	0.0166797	+	0.0166797i	0.715397	−	0.698718i	$-0.246246\pi$
$43$	30.8409	+	30.8409i	0.0166797	+	0.0166797i	−0.698718	+	0.715397i	$0.746246\pi$
$44$			2783.80i			1.43791i
$45$		0			0
$46$	2863.01			1.35303
$47$	−1785.16	+	1785.16i	−0.808129	+	0.808129i	−0.984351	−	0.176221i	$-0.943613\pi$
$47$	−1785.16	+	1785.16i	−0.808129	+	0.808129i	0.176221	+	0.984351i	$0.443613\pi$
$48$	1086.41	+	1086.41i	0.471533	+	0.471533i
$49$			1848.89i			0.770050i
$50$		0			0
$51$	29.2245			0.0112359
$52$	387.089	−	387.089i	0.143154	−	0.143154i
$53$	−707.912	−	707.912i	−0.252016	−	0.252016i	0.569781	−	0.821797i	$-0.307028\pi$
$53$	−707.912	−	707.912i	−0.252016	−	0.252016i	−0.821797	+	0.569781i	$0.807028\pi$
$54$		−	754.620i		−	0.258786i
$55$		0			0
$56$	387.857			0.123679
$57$	2220.52	−	2220.52i	0.683448	−	0.683448i
$58$	−2373.93	−	2373.93i	−0.705687	−	0.705687i
$59$		−	1366.84i		−	0.392657i	−0.980538	−	0.196329i	$-0.937098\pi$
$59$		−	1366.84i		−	0.392657i	0.980538	−	0.196329i	$-0.0629020\pi$
$60$		0			0
$61$	−2983.71			−0.801856			−0.400928	−	0.916110i	$-0.631312\pi$
$61$	−2983.71			−0.801856			−0.400928	+	0.916110i	$0.631312\pi$
$62$	−1003.66	+	1003.66i	−0.261098	+	0.261098i
$63$	−448.602	−	448.602i	−0.113026	−	0.113026i
$64$		−	2402.96i		−	0.586661i
$65$		0			0
$66$	6016.80			1.38127
$67$	986.003	−	986.003i	0.219649	−	0.219649i	−0.588702	−	0.808350i	$-0.700361\pi$
$67$	986.003	−	986.003i	0.219649	−	0.219649i	0.808350	+	0.588702i	$0.200361\pi$
$68$	−51.4265	−	51.4265i	−0.0111216	−	0.0111216i
$69$		−	2765.81i		−	0.580930i
$70$		0			0
$71$	68.0686			0.0135030			0.00675150	−	0.999977i	$-0.497851\pi$
$71$	68.0686			0.0135030			0.00675150	+	0.999977i	$0.497851\pi$
$72$	315.143	−	315.143i	0.0607914	−	0.0607914i
$73$	−2476.90	−	2476.90i	−0.464797	−	0.464797i	0.435427	−	0.900224i	$-0.356597\pi$
$73$	−2476.90	−	2476.90i	−0.464797	−	0.464797i	−0.900224	+	0.435427i	$0.856597\pi$
$74$		−	7451.65i		−	1.36078i
$75$		0			0
$76$	−7814.93			−1.35300
$77$	3576.83	−	3576.83i	0.603277	−	0.603277i
$78$	−836.641	−	836.641i	−0.137515	−	0.137515i
$79$			6047.25i			0.968955i	0.874804	+	0.484478i	$0.160990\pi$
$79$			6047.25i			0.968955i	−0.874804	+	0.484478i	$0.839010\pi$
$80$		0			0
$81$	−729.000			−0.111111
$82$	6006.45	−	6006.45i	0.893285	−	0.893285i
$83$	5158.95	+	5158.95i	0.748868	+	0.748868i	0.974267	−	0.225398i	$-0.0723684\pi$
$83$	5158.95	+	5158.95i	0.748868	+	0.748868i	−0.225398	+	0.974267i	$0.572368\pi$
$84$			1578.82i			0.223755i
$85$		0			0
$86$	−234.598			−0.0317196
$87$	−2293.33	+	2293.33i	−0.302990	+	0.302990i
$88$	2512.72	+	2512.72i	0.324473	+	0.324473i
$89$			7028.50i			0.887325i	0.896194	+	0.443662i	$0.146321\pi$
$89$			7028.50i			0.887325i	−0.896194	+	0.443662i	$0.853679\pi$
$90$		0			0
$91$	−994.722			−0.120121
$92$	−4867.01	+	4867.01i	−0.575025	+	0.575025i
$93$	969.586	+	969.586i	0.112104	+	0.112104i
$94$		−	13579.2i		−	1.53681i
$95$		0			0
$96$	−6891.70			−0.747797
$97$	−10869.3	+	10869.3i	−1.15520	+	1.15520i	−0.169703	+	0.985495i	$0.554281\pi$
$97$	−10869.3	+	10869.3i	−1.15520	+	1.15520i	−0.985495	+	0.169703i	$0.945719\pi$
$98$	−7032.00	−	7032.00i	−0.732195	−	0.732195i
$99$		−	5812.52i		−	0.593054i
$100$		0			0
$101$	−4043.94			−0.396426			−0.198213	−	0.980159i	$-0.563514\pi$
$101$	−4043.94			−0.396426			−0.198213	+	0.980159i	$0.563514\pi$
$102$	−111.151	+	111.151i	−0.0106835	+	0.0106835i
$103$	119.716	+	119.716i	0.0112844	+	0.0112844i	0.712726	−	0.701442i	$-0.247460\pi$
$103$	119.716	+	119.716i	0.0112844	+	0.0112844i	−0.701442	+	0.712726i	$0.747460\pi$
$104$		−	698.792i		−	0.0646073i
$105$		0			0
$106$	5384.89			0.479254
$107$	8330.41	−	8330.41i	0.727610	−	0.727610i	−0.242533	−	0.970143i	$-0.577978\pi$
$107$	8330.41	−	8330.41i	0.727610	−	0.727610i	0.970143	+	0.242533i	$0.0779783\pi$
$108$	1282.83	+	1282.83i	0.109982	+	0.109982i
$109$			8380.72i			0.705389i	0.935739	+	0.352694i	$0.114734\pi$
$109$			8380.72i			0.705389i	−0.935739	+	0.352694i	$0.885266\pi$
$110$		0			0
$111$	−7198.66			−0.584259
$112$	−4912.75	+	4912.75i	−0.391642	+	0.391642i
$113$	−968.067	−	968.067i	−0.0758139	−	0.0758139i	0.668183	−	0.743997i	$-0.267072\pi$
$113$	−968.067	−	968.067i	−0.0758139	−	0.0758139i	−0.743997	+	0.668183i	$0.767072\pi$
$114$			16890.9i			1.29970i
$115$		0			0
$116$	8071.19			0.599821
$117$	−808.236	+	808.236i	−0.0590427	+	0.0590427i
$118$	5198.59	+	5198.59i	0.373355	+	0.373355i
$119$			132.153i			0.00933218i
$120$		0			0
$121$	31703.8			2.16541
$122$	11348.1	−	11348.1i	0.762437	−	0.762437i
$123$	−5802.52	−	5802.52i	−0.383536	−	0.383536i
$124$		−	3412.37i		−	0.221929i
$125$		0			0
$126$	3412.39			0.214940
$127$	12023.3	−	12023.3i	0.745446	−	0.745446i	−0.228174	−	0.973620i	$-0.573276\pi$
$127$	12023.3	−	12023.3i	0.745446	−	0.745446i	0.973620	+	0.228174i	$0.0732757\pi$
$128$	−5866.12	−	5866.12i	−0.358039	−	0.358039i
$129$			226.633i			0.0136190i
$130$		0			0
$131$	4866.47			0.283577			0.141789	−	0.989897i	$-0.454715\pi$
$131$	4866.47			0.283577			0.141789	+	0.989897i	$0.454715\pi$
$132$	−10228.3	+	10228.3i	−0.587025	+	0.587025i
$133$	10041.2	+	10041.2i	0.567652	+	0.567652i
$134$			7500.26i			0.417702i
$135$		0			0
$136$	−92.8374			−0.00501933
$137$	−10167.9	+	10167.9i	−0.541736	+	0.541736i	−0.924038	−	0.382301i	$-0.875132\pi$
$137$	−10167.9	+	10167.9i	−0.541736	+	0.541736i	0.382301	+	0.924038i	$0.375132\pi$
$138$	10519.4	+	10519.4i	0.552372	+	0.552372i
$139$			4195.51i			0.217148i	0.994088	+	0.108574i	$0.0346284\pi$
$139$			4195.51i			0.217148i	−0.994088	+	0.108574i	$0.965372\pi$
$140$		0			0
$141$	−13118.2			−0.659835
$142$	−258.890	+	258.890i	−0.0128392	+	0.0128392i
$143$	−6444.29	−	6444.29i	−0.315139	−	0.315139i
$144$			7983.46i			0.385005i
$145$		0			0
$146$	18841.1			0.883895
$147$	−6793.26	+	6793.26i	−0.314372	+	0.314372i
$148$	12667.5	+	12667.5i	0.578320	+	0.578320i
$149$		−	24880.4i		−	1.12069i	−0.828259	−	0.560345i	$-0.810669\pi$
$149$		−	24880.4i		−	1.12069i	0.828259	−	0.560345i	$-0.189331\pi$
$150$		0			0
$151$	−37791.8			−1.65746			−0.828732	−	0.559645i	$-0.810937\pi$
$151$	−37791.8			−1.65746			−0.828732	+	0.559645i	$0.810937\pi$
$152$	−7053.94	+	7053.94i	−0.305312	+	0.305312i
$153$	107.377	+	107.377i	0.00458702	+	0.00458702i
$154$			27208.0i			1.14724i
$155$		0			0
$156$	2844.52			0.116885
$157$	−18627.8	+	18627.8i	−0.755723	+	0.755723i	−0.975541	−	0.219818i	$-0.929454\pi$
$157$	−18627.8	+	18627.8i	−0.755723	+	0.755723i	0.219818	+	0.975541i	$0.429454\pi$
$158$	−22999.9	−	22999.9i	−0.921322	−	0.921322i
$159$		−	5202.07i		−	0.205770i
$160$		0			0
$161$	12507.0			0.482504
$162$	2772.65	−	2772.65i	0.105649	−	0.105649i
$163$	280.411	+	280.411i	0.0105541	+	0.0105541i	0.712364	−	0.701810i	$-0.247624\pi$
$163$	280.411	+	280.411i	0.0105541	+	0.0105541i	−0.701810	+	0.712364i	$0.747624\pi$
$164$			20421.5i			0.759275i
$165$		0			0
$166$	−39242.8			−1.42411
$167$	25855.1	−	25855.1i	0.927073	−	0.927073i	−0.0704428	−	0.997516i	$-0.522441\pi$
$167$	25855.1	−	25855.1i	0.927073	−	0.927073i	0.997516	+	0.0704428i	$0.0224412\pi$
$168$	1425.08	+	1425.08i	0.0504916	+	0.0504916i
$169$		−	26768.8i		−	0.937251i
$170$		0			0
$171$	16317.4			0.558033
$172$	398.808	−	398.808i	0.0134805	−	0.0134805i
$173$	31991.2	+	31991.2i	1.06890	+	1.06890i	0.997443	+	0.0714600i	$0.0227658\pi$
$173$	31991.2	+	31991.2i	1.06890	+	1.06890i	0.0714600	+	0.997443i	$0.477234\pi$
$174$		−	17444.8i		−	0.576191i
$175$		0			0
$176$	−63654.3			−2.05496
$177$	5022.09	−	5022.09i	0.160302	−	0.160302i
$178$	−26731.9	−	26731.9i	−0.843705	−	0.843705i
$179$		−	17987.8i		−	0.561400i	−0.959796	−	0.280700i	$-0.909433\pi$
$179$		−	17987.8i		−	0.561400i	0.959796	−	0.280700i	$-0.0905666\pi$
$180$		0			0
$181$	23072.1			0.704255			0.352127	−	0.935952i	$-0.385458\pi$
$181$	23072.1			0.704255			0.352127	+	0.935952i	$0.385458\pi$
$182$	3783.29	−	3783.29i	0.114216	−	0.114216i
$183$	−10962.8	−	10962.8i	−0.327356	−	0.327356i
$184$			8786.16i			0.259515i
$185$		0			0
$186$	−7375.38			−0.213186
$187$	−856.150	+	856.150i	−0.0244831	+	0.0244831i
$188$	23084.1	+	23084.1i	0.653128	+	0.653128i
$189$		−	3296.54i		−	0.0922857i
$190$		0			0
$191$	51892.0			1.42244			0.711220	−	0.702970i	$-0.248143\pi$
$191$	51892.0			1.42244			0.711220	+	0.702970i	$0.248143\pi$
$192$	8829.06	−	8829.06i	0.239503	−	0.239503i
$193$	−50542.0	−	50542.0i	−1.35687	−	1.35687i	−0.877751	−	0.479118i	$-0.840957\pi$
$193$	−50542.0	−	50542.0i	−1.35687	−	1.35687i	−0.479118	−	0.877751i	$-0.659043\pi$
$194$		−	82679.5i		−	2.19682i
$195$		0			0
$196$	23908.3			0.622352
$197$	22327.7	−	22327.7i	0.575323	−	0.575323i	−0.358288	−	0.933611i	$-0.616640\pi$
$197$	22327.7	−	22327.7i	0.575323	−	0.575323i	0.933611	+	0.358288i	$0.116640\pi$
$198$	22107.1	+	22107.1i	0.563900	+	0.563900i
$199$		−	57027.8i		−	1.44006i	−0.693943	−	0.720030i	$-0.744128\pi$
$199$		−	57027.8i		−	1.44006i	0.693943	−	0.720030i	$-0.255872\pi$
$200$		0			0
$201$	7245.61			0.179342
$202$	15380.6	−	15380.6i	0.376938	−	0.376938i
$203$	−10370.5	−	10370.5i	−0.251655	−	0.251655i
$204$		−	377.906i		−	0.00908078i
$205$		0			0
$206$	−910.647			−0.0214593
$207$	10162.2	−	10162.2i	0.237164	−	0.237164i
$208$	8851.19	+	8851.19i	0.204586	+	0.204586i
$209$			130103.i			2.97849i
$210$		0			0
$211$	−50695.3			−1.13868			−0.569341	−	0.822101i	$-0.692802\pi$
$211$	−50695.3			−1.13868			−0.569341	+	0.822101i	$0.692802\pi$
$212$	−9154.12	+	9154.12i	−0.203678	+	0.203678i
$213$	250.100	+	250.100i	0.00551258	+	0.00551258i
$214$			63367.1i			1.38368i
$215$		0			0
$216$	2315.82			0.0496360
$217$	−4384.47	+	4384.47i	−0.0931102	+	0.0931102i
$218$	−31874.9	−	31874.9i	−0.670712	−	0.670712i
$219$		−	18201.4i		−	0.379505i
$220$		0			0
$221$	238.097			0.00487494
$222$	27379.1	−	27379.1i	0.555537	−	0.555537i
$223$	18991.4	+	18991.4i	0.381898	+	0.381898i	0.871786	−	0.489888i	$-0.162962\pi$
$223$	18991.4	+	18991.4i	0.381898	+	0.381898i	−0.489888	+	0.871786i	$0.662962\pi$
$224$		−	31164.3i		−	0.621099i
$225$		0			0
$226$	7363.82			0.144174
$227$	−24016.1	+	24016.1i	−0.466069	+	0.466069i	−0.900638	−	0.434570i	$-0.856900\pi$
$227$	−24016.1	+	24016.1i	−0.466069	+	0.466069i	0.434570	+	0.900638i	$0.356900\pi$
$228$	−28713.9	−	28713.9i	−0.552360	−	0.552360i
$229$			4953.50i			0.0944586i	0.998884	+	0.0472293i	$0.0150391\pi$
$229$			4953.50i			0.0944586i	−0.998884	+	0.0472293i	$0.984961\pi$
$230$		0			0
$231$	26284.2			0.492573
$232$	7285.24	−	7285.24i	0.135353	−	0.135353i
$233$	60878.5	+	60878.5i	1.12138	+	1.12138i	0.991534	+	0.129844i	$0.0414475\pi$
$233$	60878.5	+	60878.5i	1.12138	+	1.12138i	0.129844	+	0.991534i	$0.458552\pi$
$234$		−	6148.03i		−	0.112280i
$235$		0			0
$236$	−17674.8			−0.317345
$237$	−22219.0	+	22219.0i	−0.395574	+	0.395574i
$238$	−502.626	−	502.626i	−0.00887342	−	0.00887342i
$239$		−	40609.0i		−	0.710929i	−0.934690	−	0.355464i	$-0.884323\pi$
$239$		−	40609.0i		−	0.710929i	0.934690	−	0.355464i	$-0.115677\pi$
$240$		0			0
$241$	24443.6			0.420854			0.210427	−	0.977610i	$-0.432515\pi$
$241$	24443.6			0.420854			0.210427	+	0.977610i	$0.432515\pi$
$242$	−120581.	+	120581.i	−2.05896	+	2.05896i
$243$	−2678.52	−	2678.52i	−0.0453609	−	0.0453609i
$244$			38582.7i			0.648057i
$245$		0			0
$246$	44138.2			0.729364
$247$	18091.0	−	18091.0i	0.296530	−	0.296530i
$248$	−3080.09	−	3080.09i	−0.0500795	−	0.0500795i
$249$			37910.4i			0.611448i
$250$		0			0
$251$	−71951.0			−1.14206			−0.571030	−	0.820929i	$-0.693456\pi$
$251$	−71951.0			−1.14206			−0.571030	+	0.820929i	$0.693456\pi$
$252$	−5800.94	+	5800.94i	−0.0913476	+	0.0913476i
$253$	81026.3	+	81026.3i	1.26586	+	1.26586i
$254$			91457.9i			1.41760i
$255$		0			0
$256$	83069.4			1.26754
$257$	−33507.3	+	33507.3i	−0.507310	+	0.507310i	−0.913700	−	0.406390i	$-0.866787\pi$
$257$	−33507.3	+	33507.3i	−0.507310	+	0.507310i	0.406390	+	0.913700i	$0.366787\pi$
$258$	−861.968	−	861.968i	−0.0129495	−	0.0129495i
$259$		−	32552.3i		−	0.485269i
$260$		0			0
$261$	−16852.5			−0.247390
$262$	−18508.9	+	18508.9i	−0.269637	+	0.269637i
$263$	18051.0	+	18051.0i	0.260969	+	0.260969i	0.825448	−	0.564479i	$-0.190923\pi$
$263$	18051.0	+	18051.0i	0.260969	+	0.260969i	−0.564479	+	0.825448i	$0.690923\pi$
$264$			18464.6i			0.264931i
$265$		0			0
$266$	−76380.6			−1.07949
$267$	−25824.4	+	25824.4i	−0.362249	+	0.362249i
$268$	−12750.2	−	12750.2i	−0.177519	−	0.177519i
$269$			90057.7i			1.24456i	0.782794	+	0.622281i	$0.213794\pi$
$269$			90057.7i			1.24456i	−0.782794	+	0.622281i	$0.786206\pi$
$270$		0			0
$271$	28019.1			0.381518			0.190759	−	0.981637i	$-0.438905\pi$
$271$	28019.1			0.381518			0.190759	+	0.981637i	$0.438905\pi$
$272$	1175.92	−	1175.92i	0.0158942	−	0.0158942i
$273$	−3654.84	−	3654.84i	−0.0490392	−	0.0490392i
$274$		−	77344.1i		−	1.03021i
$275$		0			0
$276$	−35765.1			−0.469506
$277$	77139.7	−	77139.7i	1.00535	−	1.00535i	0.00536651	−	0.999986i	$-0.498292\pi$
$277$	77139.7	−	77139.7i	1.00535	−	1.00535i	0.999986	−	0.00536651i	$-0.00170822\pi$
$278$	−15957.1	−	15957.1i	−0.206473	−	0.206473i
$279$			7124.97i			0.0915324i
$280$		0			0
$281$	59704.3			0.756125			0.378062	−	0.925780i	$-0.376591\pi$
$281$	59704.3			0.756125			0.378062	+	0.925780i	$0.376591\pi$
$282$	49893.2	−	49893.2i	0.627398	−	0.627398i
$283$	−71516.3	−	71516.3i	−0.892960	−	0.892960i	0.101841	−	0.994801i	$-0.467527\pi$
$283$	−71516.3	−	71516.3i	−0.892960	−	0.892960i	−0.994801	+	0.101841i	$0.967527\pi$
$284$		−	880.205i		−	0.0109131i
$285$		0			0
$286$	49019.9			0.599295
$287$	26239.0	−	26239.0i	0.318554	−	0.318554i
$288$	−25321.7	−	25321.7i	−0.305287	−	0.305287i
$289$			83489.4i			0.999621i
$290$		0			0
$291$	−79872.5			−0.943216
$292$	−32029.2	+	32029.2i	−0.375647	+	0.375647i
$293$	−2048.52	−	2048.52i	−0.0238619	−	0.0238619i	0.695075	−	0.718937i	$-0.255371\pi$
$293$	−2048.52	−	2048.52i	−0.0238619	−	0.0238619i	−0.718937	+	0.695075i	$0.755371\pi$
$294$		−	51674.5i		−	0.597835i
$295$		0			0
$296$	22868.0			0.261003
$297$	21356.6	−	21356.6i	0.242113	−	0.242113i
$298$	94629.4	+	94629.4i	1.06560	+	1.06560i
$299$		−	22533.5i		−	0.252050i
$300$		0			0
$301$	−1024.83			−0.0113115
$302$	143736.	−	143736.i	1.57598	−	1.57598i
$303$	−14858.4	−	14858.4i	−0.161840	−	0.161840i
$304$		−	178696.i		−	1.93361i
$305$		0			0
$306$	−816.791			−0.00872305
$307$	28977.5	−	28977.5i	0.307457	−	0.307457i	−0.536465	−	0.843922i	$-0.680241\pi$
$307$	28977.5	−	28977.5i	0.307457	−	0.307457i	0.843922	+	0.536465i	$0.180241\pi$
$308$	−46252.5	−	46252.5i	−0.487566	−	0.487566i
$309$			879.729i			0.00921366i
$310$		0			0
$311$	−20134.0			−0.208166			−0.104083	−	0.994569i	$-0.533191\pi$
$311$	−20134.0			−0.208166			−0.104083	+	0.994569i	$0.533191\pi$
$312$	2567.53	−	2567.53i	0.0263758	−	0.0263758i
$313$	−115839.	−	115839.i	−1.18240	−	1.18240i	−0.979120	−	0.203283i	$-0.934839\pi$
$313$	−115839.	−	115839.i	−1.18240	−	1.18240i	−0.203283	−	0.979120i	$-0.565161\pi$
$314$		−	141697.i		−	1.43715i
$315$		0			0
$316$	78197.9			0.783106
$317$	−56563.1	+	56563.1i	−0.562879	+	0.562879i	−0.930124	−	0.367245i	$-0.880301\pi$
$317$	−56563.1	+	56563.1i	−0.562879	+	0.562879i	0.367245	+	0.930124i	$0.380301\pi$
$318$	19785.4	+	19785.4i	0.195655	+	0.195655i
$319$		−	134370.i		−	1.32044i
$320$		0			0
$321$	61215.7			0.594091
$322$	−47568.6	+	47568.6i	−0.458785	+	0.458785i
$323$	−2403.46	−	2403.46i	−0.0230373	−	0.0230373i
$324$			9426.81i			0.0897996i
$325$		0			0
$326$	−2133.01			−0.0200705
$327$	−30792.7	+	30792.7i	−0.287974	+	0.287974i
$328$	18432.9	+	18432.9i	0.171335	+	0.171335i
$329$		−	59320.4i		−	0.548040i
$330$		0			0
$331$	−76249.8			−0.695958			−0.347979	−	0.937502i	$-0.613132\pi$
$331$	−76249.8			−0.695958			−0.347979	+	0.937502i	$0.613132\pi$
$332$	66711.2	−	66711.2i	0.605233	−	0.605233i
$333$	−26449.6	−	26449.6i	−0.238523	−	0.238523i
$334$			196673.i			1.76300i
$335$		0			0
$336$	−36101.2			−0.319774
$337$	−88861.9	+	88861.9i	−0.782448	+	0.782448i	−0.980243	−	0.197795i	$-0.936622\pi$
$337$	−88861.9	+	88861.9i	−0.782448	+	0.782448i	0.197795	+	0.980243i	$0.436622\pi$
$338$	101812.	+	101812.i	0.891177	+	0.891177i
$339$		−	7113.81i		−	0.0619018i
$340$		0			0
$341$	−56809.4			−0.488552
$342$	−62061.1	+	62061.1i	−0.530600	+	0.530600i
$343$	−70611.5	−	70611.5i	−0.600188	−	0.600188i
$344$		−	719.946i		−	0.00608392i
$345$		0			0
$346$	−243348.			−2.03271
$347$	41523.5	−	41523.5i	0.344854	−	0.344854i	−0.513334	−	0.858189i	$-0.671590\pi$
$347$	41523.5	−	41523.5i	0.344854	−	0.344854i	0.858189	+	0.513334i	$0.171590\pi$
$348$	29655.4	+	29655.4i	0.244876	+	0.244876i
$349$			94995.3i			0.779922i	0.920831	+	0.389961i	$0.127511\pi$
$349$			94995.3i			0.779922i	−0.920831	+	0.389961i	$0.872489\pi$
$350$		0			0
$351$	−5939.30			−0.0482082
$352$	201897.	−	201897.i	1.62946	−	1.62946i
$353$	−54491.3	−	54491.3i	−0.437298	−	0.437298i	0.453804	−	0.891102i	$-0.350067\pi$
$353$	−54491.3	−	54491.3i	−0.437298	−	0.437298i	−0.891102	+	0.453804i	$0.850067\pi$
$354$			38201.7i			0.304843i
$355$		0			0
$356$	90886.6			0.717133
$357$	−485.561	+	485.561i	−0.00380985	+	0.00380985i
$358$	68414.2	+	68414.2i	0.533802	+	0.533802i
$359$			60457.0i			0.469092i	0.972105	+	0.234546i	$0.0753603\pi$
$359$			60457.0i			0.469092i	−0.972105	+	0.234546i	$0.924640\pi$
$360$		0			0
$361$	−234917.			−1.80260
$362$	−87751.6	+	87751.6i	−0.669634	+	0.669634i
$363$	116487.	+	116487.i	0.884026	+	0.884026i
$364$			12862.9i			0.0970814i
$365$		0			0
$366$	83391.3			0.622527
$367$	−84336.6	+	84336.6i	−0.626158	+	0.626158i	−0.947099	−	0.320941i	$-0.896001\pi$
$367$	−84336.6	+	84336.6i	−0.626158	+	0.626158i	0.320941	+	0.947099i	$0.396001\pi$
$368$	−111289.	−	111289.i	−0.821783	−	0.821783i
$369$		−	42639.7i		−	0.313156i
$370$		0			0
$371$	23523.8			0.170907
$372$	12537.9	−	12537.9i	0.0906020	−	0.0906020i
$373$	67604.9	+	67604.9i	0.485915	+	0.485915i	0.907015	−	0.421099i	$-0.138356\pi$
$373$	67604.9	+	67604.9i	0.485915	+	0.485915i	−0.421099	+	0.907015i	$0.638356\pi$
$374$		−	6512.50i		−	0.0465591i
$375$		0			0
$376$	41672.6			0.294764
$377$	−18684.2	+	18684.2i	−0.131459	+	0.131459i
$378$	12537.9	+	12537.9i	0.0877490	+	0.0877490i
$379$			128651.i			0.895645i	0.894123	+	0.447822i	$0.147800\pi$
$379$			128651.i			0.895645i	−0.894123	+	0.447822i	$0.852200\pi$
$380$		0			0
$381$	88352.8			0.608654
$382$	−197364.	+	197364.i	−1.35251	+	1.35251i
$383$	−147476.	−	147476.i	−1.00536	−	1.00536i	−0.999986	−	0.00537910i	$-0.998288\pi$
$383$	−147476.	−	147476.i	−1.00536	−	1.00536i	−0.00537910	−	0.999986i	$-0.501712\pi$
$384$		−	43107.0i		−	0.292338i
$385$		0			0
$386$	384459.			2.58033
$387$	−832.703	+	832.703i	−0.00555992	+	0.00555992i
$388$	140552.	+	140552.i	0.933628	+	0.933628i
$389$		−	175663.i		−	1.16086i	−0.814309	−	0.580432i	$-0.802884\pi$
$389$		−	175663.i		−	1.16086i	0.814309	−	0.580432i	$-0.197116\pi$
$390$		0			0
$391$	−2993.68			−0.0195817
$392$	21580.2	−	21580.2i	0.140437	−	0.140437i
$393$	17880.5	+	17880.5i	0.115770	+	0.115770i
$394$			169841.i			1.09408i
$395$		0			0
$396$	−75162.5			−0.479304
$397$	−59199.8	+	59199.8i	−0.375612	+	0.375612i	−0.869516	−	0.493904i	$-0.835569\pi$
$397$	−59199.8	+	59199.8i	−0.375612	+	0.375612i	0.493904	+	0.869516i	$0.335569\pi$
$398$	216897.	+	216897.i	1.36927	+	1.36927i
$399$			73787.4i			0.463486i
$400$		0			0
$401$	81929.2			0.509507			0.254753	−	0.967006i	$-0.418006\pi$
$401$	81929.2			0.509507			0.254753	+	0.967006i	$0.418006\pi$
$402$	−27557.7	+	27557.7i	−0.170526	+	0.170526i
$403$	7899.39	+	7899.39i	0.0486389	+	0.0486389i
$404$			52292.8i			0.320390i
$405$		0			0
$406$	78885.2			0.478568
$407$	210890.	−	210890.i	1.27311	−	1.27311i
$408$	−341.107	−	341.107i	−0.00204913	−	0.00204913i
$409$		−	107251.i		−	0.641145i	−0.947224	−	0.320573i	$-0.896125\pi$
$409$		−	107251.i		−	0.641145i	0.947224	−	0.320573i	$-0.103875\pi$
$410$		0			0
$411$	−74718.1			−0.442326
$412$	1548.07	−	1548.07i	0.00912000	−	0.00912000i
$413$	22709.9	+	22709.9i	0.133142	+	0.133142i
$414$			77301.3i			0.451010i
$415$		0			0
$416$	−56147.9			−0.324449
$417$	−15415.3	+	15415.3i	−0.0886502	+	0.0886502i
$418$	−494830.	−	494830.i	−2.83207	−	2.83207i
$419$			25226.4i			0.143690i	0.997416	+	0.0718451i	$0.0228887\pi$
$419$			25226.4i			0.143690i	−0.997416	+	0.0718451i	$0.977111\pi$
$420$		0			0
$421$	165855.			0.935762			0.467881	−	0.883792i	$-0.345018\pi$
$421$	165855.			0.935762			0.467881	+	0.883792i	$0.345018\pi$
$422$	192813.	−	192813.i	1.08271	−	1.08271i
$423$	−48199.3	−	48199.3i	−0.269376	−	0.269376i
$424$			16525.4i			0.0919224i
$425$		0			0
$426$	−1902.44			−0.0104832
$427$	49573.9	−	49573.9i	0.271893	−	0.271893i
$428$	−107722.	−	107722.i	−0.588052	−	0.588052i
$429$		−	47355.6i		−	0.257310i
$430$		0			0
$431$	−181899.			−0.979210			−0.489605	−	0.871944i	$-0.662859\pi$
$431$	−181899.			−0.979210			−0.489605	+	0.871944i	$0.662859\pi$
$432$	−29333.1	+	29333.1i	−0.157178	+	0.157178i
$433$	178342.	+	178342.i	0.951214	+	0.951214i	0.998864	−	0.0476501i	$-0.0151732\pi$
$433$	178342.	+	178342.i	0.951214	+	0.951214i	−0.0476501	+	0.998864i	$0.515173\pi$
$434$		−	33351.5i		−	0.177066i
$435$		0			0
$436$	108372.			0.570093
$437$	−227464.	+	227464.i	−1.19111	+	1.19111i
$438$	69226.7	+	69226.7i	0.360849	+	0.360849i
$439$		−	24924.4i		−	0.129329i	−0.997907	−	0.0646645i	$-0.979402\pi$
$439$		−	24924.4i		−	0.129329i	0.997907	−	0.0646645i	$-0.0205977\pi$
$440$		0			0
$441$	−49920.1			−0.256683
$442$	−905.569	+	905.569i	−0.00463529	+	0.00463529i
$443$	75101.4	+	75101.4i	0.382684	+	0.382684i	0.872068	−	0.489384i	$-0.162778\pi$
$443$	75101.4	+	75101.4i	0.382684	+	0.382684i	−0.489384	+	0.872068i	$0.662778\pi$
$444$			93086.9i			0.472196i
$445$		0			0
$446$	−144462.			−0.726248
$447$	91416.6	−	91416.6i	0.457520	−	0.457520i
$448$	39925.0	+	39925.0i	0.198925	+	0.198925i
$449$			226867.i			1.12533i	0.826687	+	0.562663i	$0.190223\pi$
$449$			226867.i			1.12533i	−0.826687	+	0.562663i	$0.809777\pi$
$450$		0			0
$451$	339978.			1.67147
$452$	−12518.2	+	12518.2i	−0.0612725	+	0.0612725i
$453$	−138856.	−	138856.i	−0.676657	−	0.676657i
$454$		−	182684.i		−	0.886314i
$455$		0			0
$456$	−51835.7			−0.249287
$457$	3272.88	−	3272.88i	0.0156710	−	0.0156710i	−0.699228	−	0.714899i	$-0.746473\pi$
$457$	3272.88	−	3272.88i	0.0156710	−	0.0156710i	0.714899	+	0.699228i	$0.246473\pi$
$458$	−18840.0	−	18840.0i	−0.0898151	−	0.0898151i
$459$			789.060i			0.00374528i
$460$		0			0
$461$	94982.0			0.446930			0.223465	−	0.974712i	$-0.428263\pi$
$461$	94982.0			0.446930			0.223465	+	0.974712i	$0.428263\pi$
$462$	−99968.4	+	99968.4i	−0.468359	+	0.468359i
$463$	216080.	+	216080.i	1.00798	+	1.00798i	0.999968	+	0.00801234i	$0.00255044\pi$
$463$	216080.	+	216080.i	1.00798	+	1.00798i	0.00801234	+	0.999968i	$0.497450\pi$
$464$			184556.i			0.857219i
$465$		0			0
$466$	−463086.			−2.13250
$467$	−7732.59	+	7732.59i	−0.0354561	+	0.0354561i	−0.724613	−	0.689156i	$-0.757981\pi$
$467$	−7732.59	+	7732.59i	−0.0354561	+	0.0354561i	0.689156	+	0.724613i	$0.257981\pi$
$468$	10451.4	+	10451.4i	0.0477181	+	0.0477181i
$469$			32764.7i			0.148957i
$470$		0			0
$471$	−136886.			−0.617045
$472$	−15953.7	+	15953.7i	−0.0716106	+	0.0716106i
$473$	−6639.37	−	6639.37i	−0.0296759	−	0.0296759i
$474$		−	169014.i		−	0.752256i
$475$		0			0
$476$	1708.89			0.00754224
$477$	19113.6	−	19113.6i	0.0840052	−	0.0840052i
$478$	154451.	+	154451.i	0.675980	+	0.675980i
$479$		−	320749.i		−	1.39796i	−0.715142	−	0.698980i	$-0.753638\pi$
$479$		−	320749.i		−	1.39796i	0.715142	−	0.698980i	$-0.246362\pi$
$480$		0			0
$481$	−58648.8			−0.253495
$482$	−92967.9	+	92967.9i	−0.400165	+	0.400165i
$483$	45953.6	+	45953.6i	0.196981	+	0.196981i
$484$		−	409967.i		−	1.75008i
$485$		0			0
$486$	20374.7			0.0862620
$487$	−5262.88	+	5262.88i	−0.0221904	+	0.0221904i	−0.718115	−	0.695925i	$-0.754995\pi$
$487$	−5262.88	+	5262.88i	−0.0221904	+	0.0221904i	0.695925	+	0.718115i	$0.254995\pi$
$488$	34825.7	+	34825.7i	0.146238	+	0.146238i
$489$			2060.59i			0.00861737i
$490$		0			0
$491$	191704.			0.795183			0.397592	−	0.917562i	$-0.369846\pi$
$491$	191704.			0.795183			0.397592	+	0.917562i	$0.369846\pi$
$492$	−75033.3	+	75033.3i	−0.309973	+	0.309973i
$493$	2482.27	+	2482.27i	0.0102131	+	0.0102131i
$494$			137613.i			0.563905i
$495$		0			0
$496$	78027.4			0.317164
$497$	−1130.95	+	1130.95i	−0.00457859	+	0.00457859i
$498$	−144187.	−	144187.i	−0.581390	−	0.581390i
$499$			261730.i			1.05112i	0.850756	+	0.525561i	$0.176145\pi$
$499$			261730.i			1.05112i	−0.850756	+	0.525561i	$0.823855\pi$
$500$		0			0
$501$	189996.			0.756952
$502$	273656.	−	273656.i	1.08592	−	1.08592i
$503$	−206538.	−	206538.i	−0.816328	−	0.816328i	0.169245	−	0.985574i	$-0.445867\pi$
$503$	−206538.	−	206538.i	−0.816328	−	0.816328i	−0.985574	+	0.169245i	$0.945867\pi$
$504$			10472.1i			0.0412263i
$505$		0			0
$506$	−616345.			−2.40726
$507$	98355.0	−	98355.0i	0.382631	−	0.382631i
$508$	−155475.	−	155475.i	−0.602467	−	0.602467i
$509$		−	44082.3i		−	0.170149i	−0.996375	−	0.0850743i	$-0.972887\pi$
$509$		−	44082.3i		−	0.170149i	0.996375	−	0.0850743i	$-0.0271128\pi$
$510$		0			0
$511$	82306.9			0.315206
$512$	−222085.	+	222085.i	−0.847188	+	0.847188i
$513$	59954.1	+	59954.1i	0.227816	+	0.227816i
$514$		−	254881.i		−	0.964741i
$515$		0			0
$516$	2930.63			0.0110068
$517$	384306.	−	384306.i	1.43779	−	1.43779i
$518$	123808.	+	123808.i	0.461414	+	0.461414i
$519$			235086.i			0.872756i
$520$		0			0
$521$	−507981.			−1.87142			−0.935712	−	0.352764i	$-0.885242\pi$
$521$	−507981.			−1.87142			−0.935712	+	0.352764i	$0.885242\pi$
$522$	64096.1	−	64096.1i	0.235229	−	0.235229i
$523$	119838.	+	119838.i	0.438119	+	0.438119i	0.891378	−	0.453260i	$-0.149739\pi$
$523$	119838.	+	119838.i	0.438119	+	0.438119i	−0.453260	+	0.891378i	$0.649739\pi$
$524$		−	62929.0i		−	0.229186i
$525$		0			0
$526$	−137309.			−0.496280
$527$	1049.47	−	1049.47i	0.00377875	−	0.00377875i
$528$	−233881.	−	233881.i	−0.838933	−	0.838933i
$529$			3481.11i			0.0124396i
$530$		0			0
$531$	36904.7			0.130886
$532$	129844.	−	129844.i	0.458775	−	0.458775i
$533$	−47274.2	−	47274.2i	−0.166406	−	0.166406i
$534$		−	196439.i		−	0.688882i
$535$		0			0
$536$	−23017.2			−0.0801166
$537$	66091.5	−	66091.5i	0.229191	−	0.229191i
$538$	−342522.	−	342522.i	−1.18338	−	1.18338i
$539$		−	398026.i		−	1.37004i
$540$		0			0
$541$	509491.			1.74077			0.870387	−	0.492368i	$-0.163869\pi$
$541$	509491.			1.74077			0.870387	+	0.492368i	$0.163869\pi$
$542$	−106567.	+	106567.i	−0.362763	+	0.362763i
$543$	84772.3	+	84772.3i	0.287511	+	0.287511i
$544$			7459.49i			0.0252064i
$545$		0			0
$546$	27801.4			0.0932569
$547$	−376658.	+	376658.i	−1.25885	+	1.25885i	−0.307204	+	0.951644i	$0.599393\pi$
$547$	−376658.	+	376658.i	−1.25885	+	1.25885i	−0.951644	+	0.307204i	$0.900607\pi$
$548$	131482.	+	131482.i	0.437830	+	0.437830i
$549$		−	80560.0i		−	0.267285i
$550$		0			0
$551$	377214.			1.24247
$552$	−32282.4	+	32282.4i	−0.105947	+	0.105947i
$553$	−100474.	−	100474.i	−0.328553	−	0.328553i
$554$			586780.i			1.91186i
$555$		0			0
$556$	54252.8			0.175498
$557$	76108.6	−	76108.6i	0.245315	−	0.245315i	−0.573730	−	0.819045i	$-0.694504\pi$
$557$	76108.6	−	76108.6i	0.245315	−	0.245315i	0.819045	+	0.573730i	$0.194504\pi$
$558$	−27098.9	−	27098.9i	−0.0870327	−	0.0870327i
$559$			1846.42i			0.00590891i
$560$		0			0
$561$	−6291.39			−0.0199904
$562$	−227077.	+	227077.i	−0.718954	+	0.718954i
$563$	242876.	+	242876.i	0.766244	+	0.766244i	0.977443	−	0.211199i	$-0.0677368\pi$
$563$	242876.	+	242876.i	0.766244	+	0.766244i	−0.211199	+	0.977443i	$0.567737\pi$
$564$			169633.i			0.533276i
$565$		0			0
$566$	544005.			1.69813
$567$	12112.3	−	12112.3i	0.0376755	−	0.0376755i
$568$	−794.494	−	794.494i	−0.00246260	−	0.00246260i
$569$		−	285859.i		−	0.882934i	−0.897278	−	0.441467i	$-0.854458\pi$
$569$		−	285859.i		−	0.882934i	0.897278	−	0.441467i	$-0.145542\pi$
$570$		0			0
$571$	−28753.2			−0.0881890			−0.0440945	−	0.999027i	$-0.514040\pi$
$571$	−28753.2			−0.0881890			−0.0440945	+	0.999027i	$0.514040\pi$
$572$	−83332.0	+	83332.0i	−0.254695	+	0.254695i
$573$	190663.	+	190663.i	0.580708	+	0.580708i
$574$			199593.i			0.605789i
$575$		0			0
$576$	64880.0			0.195554
$577$	429650.	−	429650.i	1.29052	−	1.29052i	0.356048	−	0.934467i	$-0.384124\pi$
$577$	429650.	−	429650.i	1.29052	−	1.29052i	0.934467	−	0.356048i	$-0.115876\pi$
$578$	−317540.	−	317540.i	−0.950481	−	0.950481i
$579$		−	371406.i		−	1.10788i
$580$		0			0
$581$	−171431.			−0.507852
$582$	303784.	−	303784.i	0.896848	−	0.896848i
$583$	152398.	+	152398.i	0.448376	+	0.448376i
$584$			57820.6i			0.169534i
$585$		0			0
$586$	15582.6			0.0453778
$587$	116615.	−	116615.i	0.338436	−	0.338436i	−0.517343	−	0.855778i	$-0.673079\pi$
$587$	116615.	−	116615.i	0.338436	−	0.338436i	0.855778	+	0.517343i	$0.173079\pi$
$588$	87844.6	+	87844.6i	0.254074	+	0.254074i
$589$		−	159480.i		−	0.459703i
$590$		0			0
$591$	164074.			0.469749
$592$	−289656.	+	289656.i	−0.826492	+	0.826492i
$593$	80973.9	+	80973.9i	0.230269	+	0.230269i	0.812805	−	0.582536i	$-0.197939\pi$
$593$	80973.9	+	80973.9i	0.230269	+	0.230269i	−0.582536	+	0.812805i	$0.697939\pi$
$594$			162453.i			0.460422i
$595$		0			0
$596$	−321733.			−0.905738
$597$	209533.	−	209533.i	0.587902	−	0.587902i
$598$	85703.3	+	85703.3i	0.239660	+	0.239660i
$599$			683966.i			1.90626i	0.302569	+	0.953128i	$0.402156\pi$
$599$			683966.i			1.90626i	−0.302569	+	0.953128i	$0.597844\pi$
$600$		0			0
$601$	−211089.			−0.584408			−0.292204	−	0.956356i	$-0.594389\pi$
$601$	−211089.			−0.584408			−0.292204	+	0.956356i	$0.594389\pi$
$602$	3897.82	−	3897.82i	0.0107555	−	0.0107555i
$603$	26622.1	+	26622.1i	0.0732162	+	0.0732162i
$604$			488692.i			1.33956i
$605$		0			0
$606$	113024.			0.307768
$607$	−502579.	+	502579.i	−1.36404	+	1.36404i	−0.495340	+	0.868699i	$0.664956\pi$
$607$	−502579.	+	502579.i	−1.36404	+	1.36404i	−0.868699	+	0.495340i	$0.835044\pi$
$608$	566784.	+	566784.i	1.53324	+	1.53324i
$609$		−	76207.0i		−	0.205475i
$610$		0			0
$611$	−106876.			−0.286285
$612$	1388.51	−	1388.51i	0.00370721	−	0.00370721i
$613$	−402276.	−	402276.i	−1.07054	−	1.07054i	−0.997316	−	0.0732238i	$-0.976671\pi$
$613$	−402276.	−	402276.i	−1.07054	−	1.07054i	−0.0732238	−	0.997316i	$-0.523329\pi$
$614$			220424.i			0.584685i
$615$		0			0
$616$	−83497.2			−0.220044
$617$	−439415.	+	439415.i	−1.15426	+	1.15426i	−0.168574	+	0.985689i	$0.553916\pi$
$617$	−439415.	+	439415.i	−1.15426	+	1.15426i	−0.985689	+	0.168574i	$0.946084\pi$
$618$	−3345.93	−	3345.93i	−0.00876072	−	0.00876072i
$619$			555813.i			1.45060i	0.688434	+	0.725299i	$0.258299\pi$
$619$			555813.i			1.45060i	−0.688434	+	0.725299i	$0.741701\pi$
$620$		0			0
$621$	74676.8			0.193643
$622$	76576.9	−	76576.9i	0.197932	−	0.197932i
$623$	−116778.	−	116778.i	−0.300874	−	0.300874i
$624$			65042.7i			0.167043i
$625$		0			0
$626$	881154.			2.24855
$627$	−478030.	+	478030.i	−1.21596	+	1.21596i
$628$	240879.	+	240879.i	0.610773	+	0.610773i
$629$			7791.73i			0.0196939i
$630$		0			0
$631$	−448846.			−1.12730			−0.563648	−	0.826015i	$-0.690603\pi$
$631$	−448846.			−1.12730			−0.563648	+	0.826015i	$0.690603\pi$
$632$	70583.2	−	70583.2i	0.176713	−	0.176713i
$633$	−186266.	−	186266.i	−0.464865	−	0.464865i
$634$		−	430260.i		−	1.07042i
$635$		0			0
$636$	−67268.7			−0.166303
$637$	−55345.9	+	55345.9i	−0.136398	+	0.136398i
$638$	511056.	+	511056.i	1.25553	+	1.25553i
$639$			1837.85i			0.00450100i
$640$		0			0
$641$	−579776.			−1.41106			−0.705528	−	0.708682i	$-0.749290\pi$
$641$	−579776.			−1.41106			−0.705528	+	0.708682i	$0.749290\pi$
$642$	−232826.	+	232826.i	−0.564886	+	0.564886i
$643$	−485382.	−	485382.i	−1.17398	−	1.17398i	−0.981252	−	0.192731i	$-0.938266\pi$
$643$	−485382.	−	485382.i	−1.17398	−	1.17398i	−0.192731	−	0.981252i	$-0.561734\pi$
$644$		−	161730.i		−	0.389958i
$645$		0			0
$646$	18282.5			0.0438097
$647$	−227180.	+	227180.i	−0.542703	+	0.542703i	−0.924320	−	0.381618i	$-0.875367\pi$
$647$	−227180.	+	227180.i	−0.542703	+	0.542703i	0.381618	+	0.924320i	$0.375367\pi$
$648$	8508.86	+	8508.86i	0.0202638	+	0.0202638i
$649$			294251.i			0.698601i
$650$		0			0
$651$	−32219.1			−0.0760242
$652$	3626.04	−	3626.04i	0.00852977	−	0.00852977i
$653$	−51532.9	−	51532.9i	−0.120853	−	0.120853i	0.644093	−	0.764947i	$-0.277235\pi$
$653$	−51532.9	−	51532.9i	−0.120853	−	0.120853i	−0.764947	+	0.644093i	$0.777235\pi$
$654$		−	234232.i		−	0.547634i
$655$		0			0
$656$	−466958.			−1.08510
$657$	66876.3	−	66876.3i	0.154932	−	0.154932i
$658$	225617.	+	225617.i	0.521099	+	0.521099i
$659$		−	447720.i		−	1.03095i	−0.856906	−	0.515473i	$-0.827616\pi$
$659$		−	447720.i		−	1.03095i	0.856906	−	0.515473i	$-0.172384\pi$
$660$		0			0
$661$	−99640.8			−0.228052			−0.114026	−	0.993478i	$-0.536375\pi$
$661$	−99640.8			−0.228052			−0.114026	+	0.993478i	$0.536375\pi$
$662$	290006.	−	290006.i	0.661745	−	0.661745i
$663$	874.824	+	874.824i	0.00199019	+	0.00199019i
$664$		−	120430.i		−	0.273149i
$665$		0			0
$666$	201195.			0.453594
$667$	234923.	−	234923.i	0.528048	−	0.528048i
$668$	−334337.	−	334337.i	−0.749257	−	0.749257i
$669$			139558.i			0.311818i
$670$		0			0
$671$	642327.			1.42663
$672$	114505.	−	114505.i	0.253563	−	0.253563i
$673$	170652.	+	170652.i	0.376774	+	0.376774i	0.869937	−	0.493163i	$-0.164159\pi$
$673$	170652.	+	170652.i	0.376774	+	0.376774i	−0.493163	+	0.869937i	$0.664159\pi$
$674$		−	675948.i		−	1.48797i
$675$		0			0
$676$	−346152.			−0.757483
$677$	178335.	−	178335.i	0.389099	−	0.389099i	−0.485267	−	0.874366i	$-0.661278\pi$
$677$	178335.	−	178335.i	0.389099	−	0.389099i	0.874366	+	0.485267i	$0.161278\pi$
$678$	27056.4	+	27056.4i	0.0588587	+	0.0588587i
$679$		−	361183.i		−	0.783408i
$680$		0			0
$681$	−176481.			−0.380543
$682$	216067.	−	216067.i	0.464536	−	0.464536i
$683$	−621359.	−	621359.i	−1.33199	−	1.33199i	−0.903588	−	0.428402i	$-0.859077\pi$
$683$	−621359.	−	621359.i	−1.33199	−	1.33199i	−0.428402	−	0.903588i	$-0.640923\pi$
$684$		−	211003.i		−	0.451000i
$685$		0			0
$686$	537122.			1.14137
$687$	−18200.3	+	18200.3i	−0.0385626	+	0.0385626i
$688$	9119.14	+	9119.14i	0.0192654	+	0.0192654i
$689$		−	42382.2i		−	0.0892782i
$690$		0			0
$691$	−142980.			−0.299446			−0.149723	−	0.988728i	$-0.547838\pi$
$691$	−142980.			−0.299446			−0.149723	+	0.988728i	$0.547838\pi$
$692$	413683.	−	413683.i	0.863884	−	0.863884i
$693$	96574.3	+	96574.3i	0.201092	+	0.201092i
$694$			315858.i			0.655803i
$695$		0			0
$696$	53535.4			0.110515
$697$	−6280.57	+	6280.57i	−0.0129281	+	0.0129281i
$698$	−361301.	−	361301.i	−0.741582	−	0.741582i
$699$			447364.i			0.915601i
$700$		0			0
$701$	248096.			0.504876			0.252438	−	0.967613i	$-0.418768\pi$
$701$	248096.			0.504876			0.252438	+	0.967613i	$0.418768\pi$
$702$	22589.3	−	22589.3i	0.0458383	−	0.0458383i
$703$	592028.	+	592028.i	1.19793	+	1.19793i
$704$			517306.i			1.04376i
$705$		0			0
$706$	414500.			0.831602
$707$	67189.6	−	67189.6i	0.134420	−	0.134420i
$708$	−64941.4	−	64941.4i	−0.129555	−	0.129555i
$709$		−	669146.i		−	1.33115i	−0.746329	−	0.665577i	$-0.768186\pi$
$709$		−	669146.i		−	1.33115i	0.746329	−	0.665577i	$-0.231814\pi$
$710$		0			0
$711$	−163276.			−0.322985
$712$	82036.4	−	82036.4i	0.161825	−	0.161825i
$713$	−99321.8	−	99321.8i	−0.195373	−	0.195373i
$714$		−	3693.53i		−	0.00724511i
$715$		0			0
$716$	−232603.			−0.453722
$717$	149207.	−	149207.i	0.290235	−	0.290235i
$718$	−229940.	−	229940.i	−0.446031	−	0.446031i
$719$		−	244719.i		−	0.473380i	−0.971585	−	0.236690i	$-0.923937\pi$
$719$		−	244719.i		−	0.473380i	0.971585	−	0.236690i	$-0.0760626\pi$
$720$		0			0
$721$	−3978.14			−0.00765260
$722$	893475.	−	893475.i	1.71399	−	1.71399i
$723$	89811.5	+	89811.5i	0.171813	+	0.171813i
$724$		−	298349.i		−	0.569177i
$725$		0			0
$726$	−886086.			−1.68114
$727$	515647.	−	515647.i	0.975626	−	0.975626i	−0.0240837	−	0.999710i	$-0.507667\pi$
$727$	515647.	−	515647.i	0.975626	−	0.975626i	0.999710	+	0.0240837i	$0.00766684\pi$
$728$	11610.4	+	11610.4i	0.0219070	+	0.0219070i
$729$		−	19683.0i		−	0.0370370i
$730$		0			0
$731$	245.305			0.000459062
$732$	−141762.	+	141762.i	−0.264568	+	0.264568i
$733$	−98009.3	−	98009.3i	−0.182414	−	0.182414i	0.609993	−	0.792407i	$-0.291172\pi$
$733$	−98009.3	−	98009.3i	−0.182414	−	0.182414i	−0.792407	+	0.609993i	$0.791172\pi$
$734$		−	641526.i		−	1.19075i
$735$		0			0
$736$	705967.			1.30325
$737$	−212265.	+	212265.i	−0.390790	+	0.390790i
$738$	162174.	+	162174.i	0.297762	+	0.297762i
$739$			31323.0i			0.0573554i	0.999589	+	0.0286777i	$0.00912965\pi$
$739$			31323.0i			0.0573554i	−0.999589	+	0.0286777i	$0.990870\pi$
$740$		0			0
$741$	132941.			0.242116
$742$	−89469.4	+	89469.4i	−0.162505	+	0.162505i
$743$	−46804.2	−	46804.2i	−0.0847828	−	0.0847828i	0.663444	−	0.748226i	$-0.269094\pi$
$743$	−46804.2	−	46804.2i	−0.0847828	−	0.0847828i	−0.748226	+	0.663444i	$0.769094\pi$
$744$		−	22633.9i		−	0.0408897i
$745$		0			0
$746$	−514252.			−0.924056
$747$	−139292.	+	139292.i	−0.249623	+	0.249623i
$748$	11071.0	+	11071.0i	0.0197872	+	0.0197872i
$749$			276818.i			0.493435i
$750$		0			0
$751$	−186.753			−0.000331121			−0.000165561	−	1.00000i	$-0.500053\pi$
$751$	−186.753			−0.000331121			−0.000165561		1.00000i	$0.500053\pi$
$752$	−527842.	+	527842.i	−0.933401	+	0.933401i
$753$	−264365.	−	264365.i	−0.466244	−	0.466244i
$754$		−	142126.i		−	0.249994i
$755$		0			0
$756$	−42628.0			−0.0745850
$757$	−596186.	+	596186.i	−1.04037	+	1.04037i	−0.0412248	+	0.999150i	$0.513126\pi$
$757$	−596186.	+	596186.i	−1.04037	+	1.04037i	−0.999150	+	0.0412248i	$0.986874\pi$
$758$	−489308.	−	489308.i	−0.851616	−	0.851616i
$759$			595419.i			1.03357i
$760$		0			0
$761$	−1.04441e6			−1.80344			−0.901722	−	0.432315i	$-0.857697\pi$
$761$	−1.04441e6			−1.80344			−0.901722	+	0.432315i	$0.857697\pi$
$762$	−336038.	+	336038.i	−0.578733	+	0.578733i
$763$	−139245.	−	139245.i	−0.239183	−	0.239183i
$764$		−	671023.i		−	1.14961i
$765$		0			0
$766$	1.12181e6			1.91188
$767$	40915.9	−	40915.9i	0.0695507	−	0.0695507i
$768$	305216.	+	305216.i	0.517470	+	0.517470i
$769$		−	577158.i		−	0.975982i	−0.872849	−	0.487991i	$-0.837730\pi$
$769$		−	577158.i		−	0.975982i	0.872849	−	0.487991i	$-0.162270\pi$
$770$		0			0
$771$	−246227.			−0.414217
$772$	−653566.	+	653566.i	−1.09662	+	1.09662i
$773$	414104.	+	414104.i	0.693027	+	0.693027i	0.962897	−	0.269870i	$-0.0869807\pi$
$773$	414104.	+	414104.i	0.693027	+	0.693027i	−0.269870	+	0.962897i	$0.586981\pi$
$774$		−	6334.14i		−	0.0105732i
$775$		0			0
$776$	253731.			0.421357
$777$	119605.	−	119605.i	0.198110	−	0.198110i
$778$	668111.	+	668111.i	1.10380	+	1.10380i
$779$			954417.i			1.57276i
$780$		0			0
$781$	−14653.7			−0.0240240
$782$	11386.0	−	11386.0i	0.0186191	−	0.0186191i
$783$	−61920.0	−	61920.0i	−0.100997	−	0.100997i
$784$			546687.i			0.889419i
$785$		0			0
$786$	−136012.			−0.220157
$787$	337250.	−	337250.i	0.544505	−	0.544505i	−0.380341	−	0.924846i	$-0.624193\pi$
$787$	337250.	−	337250.i	0.544505	−	0.544505i	0.924846	+	0.380341i	$0.124193\pi$
$788$	−288723.	−	288723.i	−0.464974	−	0.464974i
$789$			132647.i			0.213080i
$790$		0			0
$791$	32168.7			0.0514138
$792$	−67843.5	+	67843.5i	−0.108158	+	0.108158i
$793$	−89316.2	−	89316.2i	−0.142031	−	0.142031i
$794$		−	450317.i		−	0.714294i
$795$		0			0
$796$	−737435.			−1.16385
$797$	365276.	−	365276.i	0.575048	−	0.575048i	−0.358487	−	0.933535i	$-0.616707\pi$
$797$	365276.	−	365276.i	0.575048	−	0.575048i	0.933535	+	0.358487i	$0.116707\pi$
$798$	−280640.	−	280640.i	−0.440701	−	0.440701i
$799$			14198.9i			0.0222414i
$800$		0			0
$801$	−189770.			−0.295775
$802$	−311606.	+	311606.i	−0.484460	+	0.484460i
$803$	533223.	+	533223.i	0.826948	+	0.826948i
$804$		−	93694.1i		−	0.144944i
$805$		0			0
$806$	−60088.5			−0.0924957
$807$	−330893.	+	330893.i	−0.508090	+	0.508090i
$808$	47200.7	+	47200.7i	0.0722979	+	0.0722979i
$809$			1.04908e6i			1.60292i	0.598051	+	0.801458i	$0.295942\pi$
$809$			1.04908e6i			1.60292i	−0.598051	+	0.801458i	$0.704058\pi$
$810$		0			0
$811$	−882064.			−1.34109			−0.670546	−	0.741868i	$-0.733940\pi$
$811$	−882064.			−1.34109			−0.670546	+	0.741868i	$0.733940\pi$
$812$	−134102.	+	134102.i	−0.203387	+	0.203387i
$813$	102949.	+	102949.i	0.155754	+	0.155754i
$814$			1.60418e6i			2.42105i
$815$		0			0
$816$	8641.20			0.0129776
$817$	18638.6	−	18638.6i	0.0279235	−	0.0279235i
$818$	407916.	+	407916.i	0.609627	+	0.609627i
$819$		−	26857.5i		−	0.0400403i
$820$		0			0
$821$	1.11452e6			1.65348			0.826742	−	0.562581i	$-0.190191\pi$
$821$	1.11452e6			1.65348			0.826742	+	0.562581i	$0.190191\pi$
$822$	284180.	−	284180.i	0.420582	−	0.420582i
$823$	258025.	+	258025.i	0.380945	+	0.380945i	0.871443	−	0.490497i	$-0.163185\pi$
$823$	258025.	+	258025.i	0.380945	+	0.380945i	−0.490497	+	0.871443i	$0.663185\pi$
$824$		−	2794.64i		−	0.00411596i
$825$		0			0
$826$	−172748.			−0.253194
$827$	55607.8	−	55607.8i	0.0813063	−	0.0813063i	−0.665284	−	0.746590i	$-0.731690\pi$
$827$	55607.8	−	55607.8i	0.0813063	−	0.0813063i	0.746590	+	0.665284i	$0.231690\pi$
$828$	−131409.	−	131409.i	−0.191675	−	0.191675i
$829$		−	1.03445e6i		−	1.50522i	−0.658468	−	0.752609i	$-0.728795\pi$
$829$		−	1.03445e6i		−	1.50522i	0.658468	−	0.752609i	$-0.271205\pi$
$830$		0			0
$831$	566858.			0.820867
$832$	71931.9	−	71931.9i	0.103914	−	0.103914i
$833$	7352.94	+	7352.94i	0.0105967	+	0.0105967i
$834$		−	117260.i		−	0.168584i
$835$		0			0
$836$	1.68239e6			2.40720
$837$	−26178.8	+	26178.8i	−0.0373679	+	0.0373679i
$838$	−95945.1	−	95945.1i	−0.136626	−	0.136626i
$839$			426954.i			0.606537i	0.952905	+	0.303268i	$0.0980778\pi$
$839$			426954.i			0.606537i	−0.952905	+	0.303268i	$0.901922\pi$
$840$		0			0
$841$	317698.			0.449182
$842$	−630808.	+	630808.i	−0.889761	+	0.889761i
$843$	219368.	+	219368.i	0.308687	+	0.308687i
$844$			655548.i			0.920279i
$845$		0			0
$846$	366639.			0.512268
$847$	−526755.	+	526755.i	−0.734247	+	0.734247i
$848$	−209318.	−	209318.i	−0.291082	−	0.291082i
$849$		−	525535.i		−	0.729099i
$850$		0			0
$851$	737411.			1.01824
$852$	3234.08	−	3234.08i	0.00445525	−	0.00445525i
$853$	63890.0	+	63890.0i	0.0878081	+	0.0878081i	0.749647	−	0.661838i	$-0.230224\pi$
$853$	63890.0	+	63890.0i	0.0878081	+	0.0878081i	−0.661838	+	0.749647i	$0.730224\pi$
$854$			377095.i			0.517053i
$855$		0			0
$856$	−194464.			−0.265395
$857$	104328.	−	104328.i	0.142049	−	0.142049i	−0.632506	−	0.774555i	$-0.717974\pi$
$857$	104328.	−	104328.i	0.142049	−	0.142049i	0.774555	+	0.632506i	$0.217974\pi$
$858$	180111.	+	180111.i	0.244661	+	0.244661i
$859$			1.25585e6i			1.70196i	0.525196	+	0.850982i	$0.323992\pi$
$859$			1.25585e6i			1.70196i	−0.525196	+	0.850982i	$0.676008\pi$
$860$		0			0
$861$	192817.			0.260099
$862$	691828.	−	691828.i	0.931073	−	0.931073i
$863$	−618705.	−	618705.i	−0.830734	−	0.830734i	0.156883	−	0.987617i	$-0.449855\pi$
$863$	−618705.	−	618705.i	−0.830734	−	0.830734i	−0.987617	+	0.156883i	$0.949855\pi$
$864$		−	186076.i		−	0.249266i
$865$		0			0
$866$	−1.35660e6			−1.80891
$867$	−306760.	+	306760.i	−0.408094	+	0.408094i
$868$	56696.2	+	56696.2i	0.0752514	+	0.0752514i
$869$		−	1.30184e6i		−	1.72393i
$870$		0			0
$871$	59031.3			0.0778119
$872$	97819.4	−	97819.4i	0.128645	−	0.128645i
$873$	−293470.	−	293470.i	−0.385066	−	0.385066i
$874$		−	1.73026e6i		−	2.26510i
$875$		0			0
$876$	−235365.			−0.306715
$877$	446207.	−	446207.i	0.580146	−	0.580146i	−0.354797	−	0.934943i	$-0.615450\pi$
$877$	446207.	−	446207.i	0.580146	−	0.580146i	0.934943	+	0.354797i	$0.115450\pi$
$878$	94796.6	+	94796.6i	0.122971	+	0.122971i
$879$		−	15053.5i		−	0.0194832i
$880$		0			0
$881$	−226795.			−0.292201			−0.146101	−	0.989270i	$-0.546672\pi$
$881$	−226795.			−0.292201			−0.146101	+	0.989270i	$0.546672\pi$
$882$	189864.	−	189864.i	0.244065	−	0.244065i
$883$	−235051.	−	235051.i	−0.301467	−	0.301467i	0.540121	−	0.841588i	$-0.318379\pi$
$883$	−235051.	−	235051.i	−0.301467	−	0.301467i	−0.841588	+	0.540121i	$0.818379\pi$
$884$		−	3078.87i		−	0.00393991i
$885$		0			0
$886$	−571276.			−0.727744
$887$	789601.	−	789601.i	1.00360	−	1.00360i	0.00360590	−	0.999993i	$-0.498852\pi$
$887$	789601.	−	789601.i	1.00360	−	1.00360i	0.999993	−	0.00360590i	$-0.00114779\pi$
$888$	84022.4	+	84022.4i	0.106554	+	0.106554i
$889$			399531.i			0.505531i
$890$		0			0
$891$	156938.			0.197685
$892$	245581.	−	245581.i	0.308649	−	0.308649i
$893$	1.07886e6	+	1.07886e6i	1.35289	+	1.35289i
$894$			695381.i			0.870057i
$895$		0			0
$896$	194930.			0.242808
$897$	82793.5	−	82793.5i	0.102899	−	0.102899i
$898$	−862857.	−	862857.i	−1.07001	−	1.07001i
$899$			164710.i			0.203798i
$900$		0			0
$901$	−5630.65			−0.00693600
$902$	−1.29306e6	+	1.29306e6i	−1.58930	+	1.58930i
$903$	−3765.48	−	3765.48i	−0.00461791	−	0.00461791i
$904$			22598.5i			0.0276530i
$905$		0			0
$906$	1.05624e6			1.28679
$907$	36080.2	−	36080.2i	0.0438585	−	0.0438585i	−0.684837	−	0.728696i	$-0.740127\pi$
$907$	36080.2	−	36080.2i	0.0438585	−	0.0438585i	0.728696	+	0.684837i	$0.240127\pi$
$908$	310555.	+	310555.i	0.376675	+	0.376675i
$909$		−	109186.i		−	0.132142i
$910$		0			0
$911$	216456.			0.260815			0.130407	−	0.991460i	$-0.458371\pi$
$911$	216456.			0.260815			0.130407	+	0.991460i	$0.458371\pi$
$912$	656572.	−	656572.i	0.789392	−	0.789392i
$913$	−1.11061e6	−	1.11061e6i	−1.33236	−	1.33236i
$914$			24895.9i			0.0298013i
$915$		0			0
$916$	64054.5			0.0763411
$917$	−80855.8	+	80855.8i	−0.0961552	+	0.0961552i
$918$	−3001.08	−	3001.08i	−0.00356117	−	0.00356117i
$919$			358123.i			0.424034i	0.977266	+	0.212017i	$0.0680032\pi$
$919$			358123.i			0.424034i	−0.977266	+	0.212017i	$0.931997\pi$
$920$		0			0
$921$	212940.			0.251037
$922$	−361251.	+	361251.i	−0.424959	+	0.424959i
$923$	2037.61	+	2037.61i	0.00239176	+	0.00239176i
$924$		−	339885.i		−	0.398096i
$925$		0			0
$926$	−1.64366e6			−1.91686
$927$	−3232.33	+	3232.33i	−0.00376146	+	0.00376146i
$928$	−585369.	−	585369.i	−0.679726	−	0.679726i
$929$		−	12991.3i		−	0.0150530i	−0.999972	−	0.00752649i	$-0.997604\pi$
$929$		−	12991.3i		−	0.0150530i	0.999972	−	0.00752649i	$-0.00239578\pi$
$930$		0			0
$931$	1.11738e6			1.28914
$932$	787229.	−	787229.i	0.906294	−	0.906294i
$933$	−73977.0	−	73977.0i	−0.0849832	−	0.0849832i
$934$		−	58819.7i		−	0.0674262i
$935$		0			0
$936$	18867.4			0.0215358
$937$	−253354.	+	253354.i	−0.288568	+	0.288568i	−0.836514	−	0.547946i	$-0.815410\pi$
$937$	−253354.	+	253354.i	−0.288568	+	0.288568i	0.547946	+	0.836514i	$0.315410\pi$
$938$	−124616.	−	124616.i	−0.141634	−	0.141634i
$939$		−	851238.i		−	0.965428i
$940$		0			0
$941$	−1.24657e6			−1.40779			−0.703893	−	0.710306i	$-0.748557\pi$
$941$	−1.24657e6			−1.40779			−0.703893	+	0.710306i	$0.748557\pi$
$942$	520627.	−	520627.i	0.586712	−	0.586712i
$943$	594395.	+	594395.i	0.668424	+	0.668424i
$944$		−	404153.i		−	0.453525i
$945$		0			0
$946$	50503.9			0.0564342
$947$	−550159.	+	550159.i	−0.613462	+	0.613462i	−0.943847	−	0.330384i	$-0.892822\pi$
$947$	−550159.	+	550159.i	−0.613462	+	0.613462i	0.330384	+	0.943847i	$0.392822\pi$
$948$	287317.	+	287317.i	0.319702	+	0.319702i
$949$		−	148290.i		−	0.164657i
$950$		0			0
$951$	−415652.			−0.459589
$952$	1542.48	−	1542.48i	0.00170195	−	0.00170195i
$953$	−457502.	−	457502.i	−0.503741	−	0.503741i	0.408858	−	0.912598i	$-0.365927\pi$
$953$	−457502.	−	457502.i	−0.503741	−	0.503741i	−0.912598	+	0.408858i	$0.865927\pi$
$954$			145392.i			0.159751i
$955$		0			0
$956$	−525121.			−0.574570
$957$	493705.	−	493705.i	0.539068	−	0.539068i
$958$	1.21993e6	+	1.21993e6i	1.32924	+	1.32924i
$959$		−	337875.i		−	0.367383i
$960$		0			0
$961$	−853884.			−0.924596
$962$	223063.	−	223063.i	0.241033	−	0.241033i
$963$	224921.	+	224921.i	0.242537	+	0.242537i
$964$		−	316084.i		−	0.340133i
$965$		0			0
$966$	−349557.			−0.374596
$967$	−72229.7	+	72229.7i	−0.0772437	+	0.0772437i	−0.744673	−	0.667429i	$-0.767395\pi$
$967$	−72229.7	+	72229.7i	−0.0772437	+	0.0772437i	0.667429	+	0.744673i	$0.267395\pi$
$968$	−370045.	−	370045.i	−0.394916	−	0.394916i
$969$		−	17661.8i		−	0.0188099i
$970$		0			0
$971$	864636.			0.917054			0.458527	−	0.888680i	$-0.348377\pi$
$971$	864636.			0.917054			0.458527	+	0.888680i	$0.348377\pi$
$972$	−34636.3	+	34636.3i	−0.0366606	+	0.0366606i
$973$	−69708.0	−	69708.0i	−0.0736303	−	0.0736303i
$974$		−	40033.3i		−	0.0421992i
$975$		0			0
$976$	−882233.			−0.926155
$977$	−838896.	+	838896.i	−0.878859	+	0.878859i	−0.993417	−	0.114558i	$-0.963455\pi$
$977$	−838896.	+	838896.i	−0.878859	+	0.878859i	0.114558	+	0.993417i	$0.463455\pi$
$978$	−7837.19	−	7837.19i	−0.00819374	−	0.00819374i
$979$		−	1.51308e6i		−	1.57869i
$980$		0			0
$981$	−226280.			−0.235130
$982$	−729119.	+	729119.i	−0.756093	+	0.756093i
$983$	1.04189e6	+	1.04189e6i	1.07823	+	1.07823i	0.996668	+	0.0815669i	$0.0259924\pi$
$983$	1.04189e6	+	1.04189e6i	1.07823	+	1.07823i	0.0815669	+	0.996668i	$0.474008\pi$
$984$			135454.i			0.139894i
$985$		0			0
$986$	−18882.0			−0.0194220
$987$	217957.	−	217957.i	0.223736	−	0.223736i
$988$	−233937.	−	233937.i	−0.239654	−	0.239654i
$989$		−	23215.7i		−	0.0237350i
$990$		0			0
$991$	−1.02008e6			−1.03869			−0.519347	−	0.854564i	$-0.673825\pi$
$991$	−1.02008e6			−1.03869			−0.519347	+	0.854564i	$0.673825\pi$
$992$	−247485.	+	247485.i	−0.251493	+	0.251493i
$993$	−280160.	−	280160.i	−0.284124	−	0.284124i
$994$		−	8602.85i		−	0.00870702i
$995$		0			0
$996$	490225.			0.494171
$997$	398909.	−	398909.i	0.401313	−	0.401313i	−0.477382	−	0.878696i	$-0.658414\pi$
$997$	398909.	−	398909.i	0.401313	−	0.401313i	0.878696	+	0.477382i	$0.158414\pi$
$998$	−995456.	−	995456.i	−0.999450	−	0.999450i
$999$		−	194364.i		−	0.194753i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.5.f.e.43.1		8
3.2	odd	2		225.5.g.m.118.4		8
5.2	odd	4	inner	75.5.f.e.7.1		8
5.3	odd	4		15.5.f.a.7.4	✓	8
5.4	even	2		15.5.f.a.13.4	yes	8
15.2	even	4		225.5.g.m.82.4		8
15.8	even	4		45.5.g.e.37.1		8
15.14	odd	2		45.5.g.e.28.1		8
20.3	even	4		240.5.bg.c.97.4		8
20.19	odd	2		240.5.bg.c.193.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.5.f.a.7.4	✓	8	5.3	odd	4
15.5.f.a.13.4	yes	8	5.4	even	2
45.5.g.e.28.1		8	15.14	odd	2
45.5.g.e.37.1		8	15.8	even	4
75.5.f.e.7.1		8	5.2	odd	4	inner
75.5.f.e.43.1		8	1.1	even	1	trivial
225.5.g.m.82.4		8	15.2	even	4
225.5.g.m.118.4		8	3.2	odd	2
240.5.bg.c.97.4		8	20.3	even	4
240.5.bg.c.193.4		8	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	75.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.80336 + 3.80336i) q^{2} +(3.67423 + 3.67423i) q^{3} -12.9311i q^{4} -27.9489 q^{6} +(-16.6149 + 16.6149i) q^{7} +(-11.6720 - 11.6720i) q^{8} +27.0000i q^{9} -215.278 q^{11} +(47.5121 - 47.5121i) q^{12} +(29.9347 + 29.9347i) q^{13} -126.385i q^{14} +295.684 q^{16} +(3.97694 - 3.97694i) q^{17} +(-102.691 - 102.691i) q^{18} -604.349i q^{19} -122.094 q^{21} +(818.782 - 818.782i) q^{22} +(-376.379 - 376.379i) q^{23} -85.7710i q^{24} -227.705 q^{26} +(-99.2043 + 99.2043i) q^{27} +(214.850 + 214.850i) q^{28} +624.166i q^{29} +263.888 q^{31} +(-937.842 + 937.842i) q^{32} +(-790.983 - 790.983i) q^{33} +30.2515i q^{34} +349.141 q^{36} +(-979.613 + 979.613i) q^{37} +(2298.56 + 2298.56i) q^{38} +219.974i q^{39} -1579.25 q^{41} +(464.368 - 464.368i) q^{42} +(30.8409 + 30.8409i) q^{43} +2783.80i q^{44} +2863.01 q^{46} +(-1785.16 + 1785.16i) q^{47} +(1086.41 + 1086.41i) q^{48} +1848.89i q^{49} +29.2245 q^{51} +(387.089 - 387.089i) q^{52} +(-707.912 - 707.912i) q^{53} -754.620i q^{54} +387.857 q^{56} +(2220.52 - 2220.52i) q^{57} +(-2373.93 - 2373.93i) q^{58} -1366.84i q^{59} -2983.71 q^{61} +(-1003.66 + 1003.66i) q^{62} +(-448.602 - 448.602i) q^{63} -2402.96i q^{64} +6016.80 q^{66} +(986.003 - 986.003i) q^{67} +(-51.4265 - 51.4265i) q^{68} -2765.81i q^{69} +68.0686 q^{71} +(315.143 - 315.143i) q^{72} +(-2476.90 - 2476.90i) q^{73} -7451.65i q^{74} -7814.93 q^{76} +(3576.83 - 3576.83i) q^{77} +(-836.641 - 836.641i) q^{78} +6047.25i q^{79} -729.000 q^{81} +(6006.45 - 6006.45i) q^{82} +(5158.95 + 5158.95i) q^{83} +1578.82i q^{84} -234.598 q^{86} +(-2293.33 + 2293.33i) q^{87} +(2512.72 + 2512.72i) q^{88} +7028.50i q^{89} -994.722 q^{91} +(-4867.01 + 4867.01i) q^{92} +(969.586 + 969.586i) q^{93} -13579.2i q^{94} -6891.70 q^{96} +(-10869.3 + 10869.3i) q^{97} +(-7032.00 - 7032.00i) q^{98} -5812.52i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{6} - 20 q^{7} - 180 q^{8} - 288 q^{11} + 360 q^{12} + 340 q^{13} + 620 q^{16} - 900 q^{17} + 792 q^{21} + 1100 q^{22} + 1560 q^{23} - 3024 q^{26} - 3580 q^{28} - 512 q^{31} - 4980 q^{32} - 2700 q^{33}+ \cdots - 46440 q^{98}+O(q^{100}) \) 8 * q + 36 * q^6 - 20 * q^7 - 180 * q^8 - 288 * q^11 + 360 * q^12 + 340 * q^13 + 620 * q^16 - 900 * q^17 + 792 * q^21 + 1100 * q^22 + 1560 * q^23 - 3024 * q^26 - 3580 * q^28 - 512 * q^31 - 4980 * q^32 - 2700 * q^33 + 2484 * q^36 + 3820 * q^37 + 7680 * q^38 - 2712 * q^41 + 7380 * q^42 + 1240 * q^43 + 13528 * q^46 - 4800 * q^47 - 3600 * q^48 + 6264 * q^51 + 1240 * q^52 - 1020 * q^53 - 30720 * q^56 + 5400 * q^57 - 2340 * q^58 - 4760 * q^61 - 28680 * q^62 - 540 * q^63 + 10008 * q^66 + 8920 * q^67 + 1920 * q^68 + 7536 * q^71 + 4860 * q^72 - 11600 * q^73 + 4344 * q^76 + 360 * q^77 + 4680 * q^78 - 5832 * q^81 + 27200 * q^82 + 32400 * q^83 + 14592 * q^86 - 10620 * q^87 + 14340 * q^88 + 16528 * q^91 + 31800 * q^92 - 14040 * q^93 - 4068 * q^96 - 58640 * q^97 - 46440 * q^98

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