LMFDB - Embedded newform 336.5.bh.d.241.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,5,Mod(145,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.145"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	336.bh (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,18,0,-66] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$34.7323075962$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			241.1
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	336.241
Dual form			336.5.bh.d.145.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.50000 + 2.59808i) q^{3} +(-20.7426 + 11.9758i) q^{5} +(47.1985 - 13.1645i) q^{7} +(13.5000 + 23.3827i) q^{9} +(48.9853 - 84.8450i) q^{11} -104.211i q^{13} -124.456 q^{15} +(93.2498 + 53.8378i) q^{17} +(-33.2498 + 19.1968i) q^{19} +(246.595 + 63.3852i) q^{21} +(-510.749 - 884.644i) q^{23} +(-25.6619 + 44.4477i) q^{25} +140.296i q^{27} +621.603 q^{29} +(1315.61 + 759.568i) q^{31} +(440.868 - 254.535i) q^{33} +(-821.367 + 838.304i) q^{35} +(281.118 + 486.910i) q^{37} +(270.749 - 468.952i) q^{39} -1023.20i q^{41} +3382.41 q^{43} +(-560.051 - 323.346i) q^{45} +(3416.50 - 1972.52i) q^{47} +(2054.39 - 1242.69i) q^{49} +(279.749 + 484.540i) q^{51} +(-1095.30 + 1897.12i) q^{53} +2346.55i q^{55} -199.499 q^{57} +(-2541.67 - 1467.44i) q^{59} +(576.207 - 332.673i) q^{61} +(945.000 + 925.907i) q^{63} +(1248.01 + 2161.62i) q^{65} +(-2962.69 + 5131.53i) q^{67} -5307.86i q^{69} +4494.41 q^{71} +(7767.32 + 4484.46i) q^{73} +(-230.957 + 133.343i) q^{75} +(1195.09 - 4649.42i) q^{77} +(-5223.41 - 9047.22i) q^{79} +(-364.500 + 631.333i) q^{81} -1269.28i q^{83} -2579.00 q^{85} +(2797.21 + 1614.97i) q^{87} +(-3121.46 + 1802.18i) q^{89} +(-1371.89 - 4918.62i) q^{91} +(3946.83 + 6836.11i) q^{93} +(459.792 - 796.383i) q^{95} -1950.53i q^{97} +2645.21 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 18 q^{3} - 66 q^{5} + 70 q^{7} + 54 q^{9} + 162 q^{11} - 396 q^{15} - 204 q^{17} + 444 q^{19} + 630 q^{21} - 312 q^{23} - 476 q^{25} + 2724 q^{29} + 3786 q^{31} + 1458 q^{33} - 672 q^{35} + 1396 q^{37}+ \cdots + 8748 q^{99}+O(q^{100}) $ 4 * q + 18 * q^3 - 66 * q^5 + 70 * q^7 + 54 * q^9 + 162 * q^11 - 396 * q^15 - 204 * q^17 + 444 * q^19 + 630 * q^21 - 312 * q^23 - 476 * q^25 + 2724 * q^29 + 3786 * q^31 + 1458 * q^33 - 672 * q^35 + 1396 * q^37 - 648 * q^39 + 632 * q^43 - 1782 * q^45 + 7896 * q^47 - 98 * q^49 - 612 * q^51 - 1038 * q^53 + 2664 * q^57 + 966 * q^59 + 5088 * q^61 + 3780 * q^63 - 744 * q^65 - 14600 * q^67 + 9696 * q^71 + 22584 * q^73 - 4284 * q^75 - 3654 * q^77 - 3974 * q^79 - 1458 * q^81 + 1224 * q^85 + 12258 * q^87 - 33156 * q^89 + 18984 * q^91 + 11358 * q^93 - 3252 * q^95 + 8748 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	4.50000	+	2.59808i	0.500000	+	0.288675i
$3$	4.50000	+	2.59808i	0.500000	+	0.288675i
$4$		0			0
$5$	−20.7426	+	11.9758i	−0.829706	+	0.479031i	−0.853752	−	0.520680i	$-0.825678\pi$
$5$	−20.7426	+	11.9758i	−0.829706	+	0.479031i	0.0240462	+	0.999711i	$0.492345\pi$
$6$		0			0
$7$	47.1985	−	13.1645i	0.963234	−	0.268662i
$7$	47.1985	−	13.1645i	0.963234	−	0.268662i
$8$		0			0
$9$	13.5000	+	23.3827i	0.166667	+	0.288675i
$10$		0			0
$11$	48.9853	−	84.8450i	0.404837	−	0.701198i	−0.589465	−	0.807794i	$-0.700662\pi$
$11$	48.9853	−	84.8450i	0.404837	−	0.701198i	0.994302	+	0.106595i	$0.0339949\pi$
$12$		0			0
$13$		−	104.211i		−	0.616636i	−0.951283	−	0.308318i	$-0.900234\pi$
$13$		−	104.211i		−	0.616636i	0.951283	−	0.308318i	$-0.0997661\pi$
$14$		0			0
$15$	−124.456			−0.553137
$16$		0			0
$17$	93.2498	+	53.8378i	0.322664	+	0.186290i	0.652579	−	0.757721i	$-0.273687\pi$
$17$	93.2498	+	53.8378i	0.322664	+	0.186290i	−0.329916	+	0.944010i	$0.607020\pi$
$18$		0			0
$19$	−33.2498	+	19.1968i	−0.0921047	+	0.0531767i	−0.545345	−	0.838212i	$-0.683601\pi$
$19$	−33.2498	+	19.1968i	−0.0921047	+	0.0531767i	0.453240	+	0.891388i	$0.350268\pi$
$20$		0			0
$21$	246.595	+	63.3852i	0.559173	+	0.143731i
$22$		0			0
$23$	−510.749	−	884.644i	−0.965500	−	1.67229i	−0.708266	−	0.705945i	$-0.750522\pi$
$23$	−510.749	−	884.644i	−0.965500	−	1.67229i	−0.257233	−	0.966349i	$-0.582811\pi$
$24$		0			0
$25$	−25.6619	+	44.4477i	−0.0410590	+	0.0711164i
$26$		0			0
$27$			140.296i			0.192450i
$28$		0			0
$29$	621.603			0.739124			0.369562	−	0.929206i	$-0.379508\pi$
$29$	621.603			0.739124			0.369562	+	0.929206i	$0.379508\pi$
$30$		0			0
$31$	1315.61	+	759.568i	1.36900	+	0.790393i	0.990800	−	0.135331i	$-0.0432097\pi$
$31$	1315.61	+	759.568i	1.36900	+	0.790393i	0.378200	+	0.925724i	$0.376543\pi$
$32$		0			0
$33$	440.868	−	254.535i	0.404837	−	0.233733i
$34$		0			0
$35$	−821.367	+	838.304i	−0.670503	+	0.684330i
$36$		0			0
$37$	281.118	+	486.910i	0.205345	+	0.355669i	0.950243	−	0.311511i	$-0.100835\pi$
$37$	281.118	+	486.910i	0.205345	+	0.355669i	−0.744897	+	0.667179i	$0.767502\pi$
$38$		0			0
$39$	270.749	−	468.952i	0.178007	−	0.308318i
$40$		0			0
$41$		−	1023.20i		−	0.608684i	−0.952563	−	0.304342i	$-0.901563\pi$
$41$		−	1023.20i		−	0.608684i	0.952563	−	0.304342i	$-0.0984365\pi$
$42$		0			0
$43$	3382.41			1.82932			0.914658	−	0.404228i	$-0.132460\pi$
$43$	3382.41			1.82932			0.914658	+	0.404228i	$0.132460\pi$
$44$		0			0
$45$	−560.051	−	323.346i	−0.276569	−	0.159677i
$46$		0			0
$47$	3416.50	−	1972.52i	1.54663	−	0.892945i	0.548231	−	0.836327i	$-0.315302\pi$
$47$	3416.50	−	1972.52i	1.54663	−	0.892945i	0.998396	−	0.0566179i	$-0.0180317\pi$
$48$		0			0
$49$	2054.39	−	1242.69i	0.855641	−	0.517570i
$50$		0			0
$51$	279.749	+	484.540i	0.107555	+	0.186290i
$52$		0			0
$53$	−1095.30	+	1897.12i	−0.389925	+	0.675370i	−0.992439	−	0.122738i	$-0.960833\pi$
$53$	−1095.30	+	1897.12i	−0.389925	+	0.675370i	0.602514	+	0.798108i	$0.294166\pi$
$54$		0			0
$55$			2346.55i			0.775718i
$56$		0			0
$57$	−199.499			−0.0614031
$58$		0			0
$59$	−2541.67	−	1467.44i	−0.730156	−	0.421556i	0.0883234	−	0.996092i	$-0.471849\pi$
$59$	−2541.67	−	1467.44i	−0.730156	−	0.421556i	−0.818479	+	0.574536i	$0.805182\pi$
$60$		0			0
$61$	576.207	−	332.673i	0.154853	−	0.0894043i	−0.420571	−	0.907259i	$-0.638170\pi$
$61$	576.207	−	332.673i	0.154853	−	0.0894043i	0.575424	+	0.817855i	$0.304837\pi$
$62$		0			0
$63$	945.000	+	925.907i	0.238095	+	0.233285i
$64$		0			0
$65$	1248.01	+	2161.62i	0.295388	+	0.511626i
$66$		0			0
$67$	−2962.69	+	5131.53i	−0.659989	+	1.14314i	0.320629	+	0.947205i	$0.396106\pi$
$67$	−2962.69	+	5131.53i	−0.659989	+	1.14314i	−0.980618	+	0.195930i	$0.937227\pi$
$68$		0			0
$69$		−	5307.86i		−	1.11486i
$70$		0			0
$71$	4494.41			0.891571			0.445785	−	0.895140i	$-0.352924\pi$
$71$	4494.41			0.891571			0.445785	+	0.895140i	$0.352924\pi$
$72$		0			0
$73$	7767.32	+	4484.46i	1.45756	+	0.841521i	0.998891	−	0.0470879i	$-0.0149941\pi$
$73$	7767.32	+	4484.46i	1.45756	+	0.841521i	0.458666	+	0.888609i	$0.348327\pi$
$74$		0			0
$75$	−230.957	+	133.343i	−0.0410590	+	0.0237055i
$76$		0			0
$77$	1195.09	−	4649.42i	0.201567	−	0.784183i
$78$		0			0
$79$	−5223.41	−	9047.22i	−0.836951	−	1.44964i	−0.892432	−	0.451182i	$-0.851002\pi$
$79$	−5223.41	−	9047.22i	−0.836951	−	1.44964i	0.0554805	−	0.998460i	$-0.482331\pi$
$80$		0			0
$81$	−364.500	+	631.333i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		−	1269.28i		−	0.184247i	−0.995748	−	0.0921234i	$-0.970635\pi$
$83$		−	1269.28i		−	0.184247i	0.995748	−	0.0921234i	$-0.0293654\pi$
$84$		0			0
$85$	−2579.00			−0.356954
$86$		0			0
$87$	2797.21	+	1614.97i	0.369562	+	0.213367i
$88$		0			0
$89$	−3121.46	+	1802.18i	−0.394074	+	0.227519i	−0.683924	−	0.729553i	$-0.739728\pi$
$89$	−3121.46	+	1802.18i	−0.394074	+	0.227519i	0.289850	+	0.957072i	$0.406395\pi$
$90$		0			0
$91$	−1371.89	−	4918.62i	−0.165667	−	0.593965i
$92$		0			0
$93$	3946.83	+	6836.11i	0.456334	+	0.790393i
$94$		0			0
$95$	459.792	−	796.383i	0.0509465	−	0.0882419i
$96$		0			0
$97$		−	1950.53i		−	0.207304i	−0.994614	−	0.103652i	$-0.966947\pi$
$97$		−	1950.53i		−	0.207304i	0.994614	−	0.103652i	$-0.0330529\pi$
$98$		0			0
$99$	2645.21			0.269891
$100$		0			0
$101$	9808.81	+	5663.12i	0.961554	+	0.555153i	0.896651	−	0.442738i	$-0.145993\pi$
$101$	9808.81	+	5663.12i	0.961554	+	0.555153i	0.0649030	+	0.997892i	$0.479326\pi$
$102$		0			0
$103$	14141.2	−	8164.43i	1.33294	−	0.769576i	0.347195	−	0.937793i	$-0.387134\pi$
$103$	14141.2	−	8164.43i	1.33294	−	0.769576i	0.985750	+	0.168217i	$0.0538010\pi$
$104$		0			0
$105$	−5874.13	+	1638.39i	−0.532801	+	0.148607i
$106$		0			0
$107$	2198.23	+	3807.45i	0.192002	+	0.332557i	0.945914	−	0.324419i	$-0.105169\pi$
$107$	2198.23	+	3807.45i	0.192002	+	0.332557i	−0.753912	+	0.656976i	$0.771835\pi$
$108$		0			0
$109$	7348.89	−	12728.6i	0.618541	−	1.07134i	−0.371211	−	0.928548i	$-0.621057\pi$
$109$	7348.89	−	12728.6i	0.618541	−	1.07134i	0.989752	−	0.142796i	$-0.0456093\pi$
$110$		0			0
$111$			2921.46i			0.237112i
$112$		0			0
$113$	2124.36			0.166368			0.0831842	−	0.996534i	$-0.473491\pi$
$113$	2124.36			0.166368			0.0831842	+	0.996534i	$0.473491\pi$
$114$		0			0
$115$	21188.6	+	12233.2i	1.60216	+	0.925008i
$116$		0			0
$117$	2436.74	−	1406.85i	0.178007	−	0.102773i
$118$		0			0
$119$	5109.99	+	1313.48i	0.360850	+	0.0927533i
$120$		0			0
$121$	2521.38	+	4367.17i	0.172214	+	0.298283i
$122$		0			0
$123$	2658.35	−	4604.39i	0.175712	−	0.304342i
$124$		0			0
$125$		−	16199.0i		−	1.03674i
$126$		0			0
$127$	−8498.07			−0.526881			−0.263441	−	0.964676i	$-0.584857\pi$
$127$	−8498.07			−0.526881			−0.263441	+	0.964676i	$0.584857\pi$
$128$		0			0
$129$	15220.8	+	8787.75i	0.914658	+	0.528078i
$130$		0			0
$131$	−12533.4	+	7236.14i	−0.730340	+	0.421662i	−0.818547	−	0.574440i	$-0.805220\pi$
$131$	−12533.4	+	7236.14i	−0.730340	+	0.421662i	0.0882063	+	0.996102i	$0.471887\pi$
$132$		0			0
$133$	−1316.62	+	1343.77i	−0.0744318	+	0.0759666i
$134$		0			0
$135$	−1680.15	−	2910.11i	−0.0921895	−	0.159677i
$136$		0			0
$137$	−6427.59	+	11132.9i	−0.342458	+	0.593154i	−0.984888	−	0.173190i	$-0.944593\pi$
$137$	−6427.59	+	11132.9i	−0.342458	+	0.593154i	0.642431	+	0.766344i	$0.277926\pi$
$138$		0			0
$139$		−	5009.46i		−	0.259275i	−0.991561	−	0.129638i	$-0.958619\pi$
$139$		−	5009.46i		−	0.259275i	0.991561	−	0.129638i	$-0.0413814\pi$
$140$		0			0
$141$	20499.0			1.03108
$142$		0			0
$143$	−8841.82	−	5104.83i	−0.432384	−	0.249637i
$144$		0			0
$145$	−12893.7	+	7444.17i	−0.613255	+	0.354063i
$146$		0			0
$147$	12473.4	−	254.611i	0.577230	−	0.0117827i
$148$		0			0
$149$	−12699.5	−	21996.1i	−0.572022	−	0.990771i	−0.996358	−	0.0852664i	$-0.972826\pi$
$149$	−12699.5	−	21996.1i	−0.572022	−	0.990771i	0.424336	−	0.905505i	$-0.360507\pi$
$150$		0			0
$151$	−9154.39	+	15855.9i	−0.401491	+	0.695402i	−0.993906	−	0.110230i	$-0.964841\pi$
$151$	−9154.39	+	15855.9i	−0.401491	+	0.695402i	0.592415	+	0.805633i	$0.298174\pi$
$152$		0			0
$153$			2907.24i			0.124193i
$154$		0			0
$155$	−36385.6			−1.51449
$156$		0			0
$157$	−30522.6	−	17622.2i	−1.23829	−	0.714926i	−0.269544	−	0.962988i	$-0.586873\pi$
$157$	−30522.6	−	17622.2i	−1.23829	−	0.714926i	−0.968744	+	0.248062i	$0.920206\pi$
$158$		0			0
$159$	−9857.70	+	5691.35i	−0.389925	+	0.225123i
$160$		0			0
$161$	−35752.5	−	35030.1i	−1.37929	−	1.35142i
$162$		0			0
$163$	72.9693	+	126.386i	0.00274641	+	0.00475691i	0.867395	−	0.497620i	$-0.165792\pi$
$163$	72.9693	+	126.386i	0.00274641	+	0.00475691i	−0.864649	+	0.502377i	$0.832459\pi$
$164$		0			0
$165$	−6096.50	+	10559.5i	−0.223930	+	0.387859i
$166$		0			0
$167$		−	31314.1i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$167$		−	31314.1i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$168$		0			0
$169$	17701.0			0.619760
$170$		0			0
$171$	−897.744	−	518.313i	−0.0307016	−	0.0177256i
$172$		0			0
$173$	−34741.3	+	20057.9i	−1.16079	+	0.670183i	−0.951494	−	0.307669i	$-0.900451\pi$
$173$	−34741.3	+	20057.9i	−1.16079	+	0.670183i	−0.209298	+	0.977852i	$0.567118\pi$
$174$		0			0
$175$	−626.073	+	2435.69i	−0.0204432	+	0.0795327i
$176$		0			0
$177$	−7625.02	−	13206.9i	−0.243385	−	0.421556i
$178$		0			0
$179$	−25077.2	+	43435.0i	−0.782661	+	1.35561i	0.147726	+	0.989028i	$0.452805\pi$
$179$	−25077.2	+	43435.0i	−0.782661	+	1.35561i	−0.930387	+	0.366580i	$0.880529\pi$
$180$		0			0
$181$			63974.3i			1.95276i	0.216069	+	0.976378i	$0.430677\pi$
$181$			63974.3i			1.95276i	−0.216069	+	0.976378i	$0.569323\pi$
$182$		0			0
$183$	3457.24			0.103235
$184$		0			0
$185$	−11662.2	−	6733.20i	−0.340752	−	0.196733i
$186$		0			0
$187$	9135.73	−	5274.52i	0.261252	−	0.150834i
$188$		0			0
$189$	1846.92	+	6621.76i	0.0517041	+	0.185375i
$190$		0			0
$191$	−21384.4	−	37039.0i	−0.586180	−	1.01529i	−0.994727	−	0.102557i	$-0.967298\pi$
$191$	−21384.4	−	37039.0i	−0.586180	−	1.01529i	0.408547	−	0.912737i	$-0.366036\pi$
$192$		0			0
$193$	4386.29	−	7597.28i	0.117756	−	0.203959i	−0.801122	−	0.598501i	$-0.795763\pi$
$193$	4386.29	−	7597.28i	0.117756	−	0.203959i	0.918878	+	0.394542i	$0.129097\pi$
$194$		0			0
$195$			12969.7i			0.341084i
$196$		0			0
$197$	−32825.0			−0.845808			−0.422904	−	0.906174i	$-0.638989\pi$
$197$	−32825.0			−0.845808			−0.422904	+	0.906174i	$0.638989\pi$
$198$		0			0
$199$	22918.7	+	13232.1i	0.578740	+	0.334136i	0.760632	−	0.649183i	$-0.224889\pi$
$199$	22918.7	+	13232.1i	0.578740	+	0.334136i	−0.181893	+	0.983318i	$0.558222\pi$
$200$		0			0
$201$	−26664.2	+	15394.6i	−0.659989	+	0.381045i
$202$		0			0
$203$	29338.7	−	8183.07i	0.711949	−	0.198575i
$204$		0			0
$205$	12253.6	+	21223.8i	0.291578	+	0.505028i
$206$		0			0
$207$	13790.2	−	23885.4i	0.321833	−	0.557432i
$208$		0			0
$209$			3761.44i			0.0861115i
$210$		0			0
$211$	−67056.3			−1.50617			−0.753087	−	0.657921i	$-0.771436\pi$
$211$	−67056.3			−1.50617			−0.753087	+	0.657921i	$0.771436\pi$
$212$		0			0
$213$	20224.8	+	11676.8i	0.445785	+	0.257374i
$214$		0			0
$215$	−70160.1	+	40506.9i	−1.51779	+	0.876299i
$216$		0			0
$217$	72094.1	+	18531.1i	1.53102	+	0.393535i
$218$		0			0
$219$	23302.0	+	40360.2i	0.485852	+	0.841521i
$220$		0			0
$221$	5610.52	−	9717.70i	0.114873	−	0.198966i
$222$		0			0
$223$		−	56260.1i		−	1.13133i	−0.824634	−	0.565666i	$-0.808619\pi$
$223$		−	56260.1i		−	1.13133i	0.824634	−	0.565666i	$-0.191381\pi$
$224$		0			0
$225$	−1385.74			−0.0273727
$226$		0			0
$227$	−52955.8	−	30574.1i	−1.02769	−	0.593337i	−0.111368	−	0.993779i	$-0.535523\pi$
$227$	−52955.8	−	30574.1i	−1.02769	−	0.593337i	−0.916322	+	0.400442i	$0.868857\pi$
$228$		0			0
$229$	29274.9	−	16901.9i	0.558244	−	0.322303i	−0.194196	−	0.980963i	$-0.562210\pi$
$229$	29274.9	−	16901.9i	0.558244	−	0.322303i	0.752441	+	0.658660i	$0.228877\pi$
$230$		0			0
$231$	17457.5	−	17817.4i	0.327158	−	0.333904i
$232$		0			0
$233$	−4343.63	−	7523.39i	−0.0800094	−	0.138580i	0.823244	−	0.567687i	$-0.192162\pi$
$233$	−4343.63	−	7523.39i	−0.0800094	−	0.138580i	−0.903254	+	0.429107i	$0.858828\pi$
$234$		0			0
$235$	−47244.8	+	81830.4i	−0.855496	+	1.48176i
$236$		0			0
$237$		−	54283.3i		−	0.966428i
$238$		0			0
$239$	15715.0			0.275118			0.137559	−	0.990494i	$-0.456074\pi$
$239$	15715.0			0.275118			0.137559	+	0.990494i	$0.456074\pi$
$240$		0			0
$241$	−4744.80	−	2739.41i	−0.0816928	−	0.0471654i	0.458597	−	0.888644i	$-0.348352\pi$
$241$	−4744.80	−	2739.41i	−0.0816928	−	0.0471654i	−0.540290	+	0.841479i	$0.681685\pi$
$242$		0			0
$243$	−3280.50	+	1894.00i	−0.0555556	+	0.0320750i
$244$		0			0
$245$	−27731.4	+	50379.5i	−0.461998	+	0.839309i
$246$		0			0
$247$	2000.52	+	3465.01i	0.0327906	+	0.0567950i
$248$		0			0
$249$	3297.68	−	5711.74i	0.0531875	−	0.0921234i
$250$		0			0
$251$			24350.9i			0.386516i	0.981148	+	0.193258i	$0.0619054\pi$
$251$			24350.9i			0.386516i	−0.981148	+	0.193258i	$0.938095\pi$
$252$		0			0
$253$	−100077.			−1.56348
$254$		0			0
$255$	−11605.5	−	6700.43i	−0.178477	−	0.103044i
$256$		0			0
$257$	10051.7	−	5803.37i	0.152186	−	0.0878647i	−0.421973	−	0.906608i	$-0.638662\pi$
$257$	10051.7	−	5803.37i	0.152186	−	0.0878647i	0.574159	+	0.818744i	$0.305329\pi$
$258$		0			0
$259$	19678.2	+	19280.7i	0.293350	+	0.287424i
$260$		0			0
$261$	8391.64	+	14534.7i	0.123187	+	0.213367i
$262$		0			0
$263$	60333.8	−	104501.i	0.872266	−	1.51081i	0.0126200	−	0.999920i	$-0.495983\pi$
$263$	60333.8	−	104501.i	0.872266	−	1.51081i	0.859646	−	0.510889i	$-0.170684\pi$
$264$		0			0
$265$		−	52468.3i		−	0.747145i
$266$		0			0
$267$	−18728.8			−0.262716
$268$		0			0
$269$	−23596.3	−	13623.3i	−0.326091	−	0.188269i	0.328013	−	0.944673i	$-0.393621\pi$
$269$	−23596.3	−	13623.3i	−0.326091	−	0.188269i	−0.654104	+	0.756404i	$0.726954\pi$
$270$		0			0
$271$	84041.9	−	48521.6i	1.14435	−	0.660689i	0.196843	−	0.980435i	$-0.436931\pi$
$271$	84041.9	−	48521.6i	1.14435	−	0.660689i	0.947503	+	0.319746i	$0.103598\pi$
$272$		0			0
$273$	6605.46	−	25698.1i	0.0886294	−	0.344806i
$274$		0			0
$275$	2514.11	+	4354.57i	0.0332444	+	0.0575811i
$276$		0			0
$277$	−14876.4	+	25766.7i	−0.193882	+	0.335814i	−0.946534	−	0.322605i	$-0.895441\pi$
$277$	−14876.4	+	25766.7i	−0.193882	+	0.335814i	0.752651	+	0.658419i	$0.228775\pi$
$278$		0			0
$279$			41016.7i			0.526929i
$280$		0			0
$281$	−122275.			−1.54855			−0.774276	−	0.632849i	$-0.781886\pi$
$281$	−122275.			−1.54855			−0.774276	+	0.632849i	$0.781886\pi$
$282$		0			0
$283$	52568.4	+	30350.4i	0.656374	+	0.378958i	0.790894	−	0.611953i	$-0.209616\pi$
$283$	52568.4	+	30350.4i	0.656374	+	0.378958i	−0.134520	+	0.990911i	$0.542949\pi$
$284$		0			0
$285$	4138.13	−	2389.15i	0.0509465	−	0.0294140i
$286$		0			0
$287$	−13469.8	−	48293.4i	−0.163531	−	0.586305i
$288$		0			0
$289$	−35963.5	−	62290.6i	−0.430592	−	0.745807i
$290$		0			0
$291$	5067.62	−	8777.37i	0.0598436	−	0.103652i
$292$		0			0
$293$		−	29557.4i		−	0.344296i	−0.985071	−	0.172148i	$-0.944929\pi$
$293$		−	29557.4i		−	0.344296i	0.985071	−	0.172148i	$-0.0550707\pi$
$294$		0			0
$295$	70294.7			0.807752
$296$		0			0
$297$	11903.4	+	6872.44i	0.134946	+	0.0779109i
$298$		0			0
$299$	−92190.0	+	53225.9i	−1.03120	+	0.595362i
$300$		0			0
$301$	159644.	−	44527.6i	1.76206	−	0.491469i
$302$		0			0
$303$	29426.4	+	50968.1i	0.320518	+	0.555153i
$304$		0			0
$305$	−7968.04	+	13801.0i	−0.0856548	+	0.148358i
$306$		0			0
$307$			107567.i			1.14131i	0.821191	+	0.570654i	$0.193310\pi$
$307$			107567.i			1.14131i	−0.821191	+	0.570654i	$0.806690\pi$
$308$		0			0
$309$	84847.3			0.888630
$310$		0			0
$311$	−37561.0	−	21685.9i	−0.388344	−	0.224211i	0.293098	−	0.956082i	$-0.405314\pi$
$311$	−37561.0	−	21685.9i	−0.388344	−	0.224211i	−0.681442	+	0.731872i	$0.738647\pi$
$312$		0			0
$313$	−93311.1	+	53873.2i	−0.952455	+	0.549900i	−0.893843	−	0.448381i	$-0.852001\pi$
$313$	−93311.1	+	53873.2i	−0.952455	+	0.549900i	−0.0586125	+	0.998281i	$0.518668\pi$
$314$		0			0
$315$	−30690.2	−	7888.66i	−0.309300	−	0.0795027i
$316$		0			0
$317$	77996.9	+	135095.i	0.776174	+	1.34437i	0.934133	+	0.356926i	$0.116175\pi$
$317$	77996.9	+	135095.i	0.776174	+	1.34437i	−0.157959	+	0.987446i	$0.550491\pi$
$318$		0			0
$319$	30449.4	−	52739.9i	0.299225	−	0.518272i
$320$		0			0
$321$			22844.7i			0.221705i
$322$		0			0
$323$	−4134.05			−0.0396251
$324$		0			0
$325$	4631.96	+	2674.26i	0.0438529	+	0.0253185i
$326$		0			0
$327$	66140.0	−	38185.9i	0.618541	−	0.357115i
$328$		0			0
$329$	135286.	−	138076.i	1.24986	−	1.27564i
$330$		0			0
$331$	20109.4	+	34830.4i	0.183545	+	0.317909i	0.943085	−	0.332551i	$-0.107909\pi$
$331$	20109.4	+	34830.4i	0.183545	+	0.317909i	−0.759540	+	0.650460i	$0.774576\pi$
$332$		0			0
$333$	−7590.18	+	13146.6i	−0.0684484	+	0.118556i
$334$		0			0
$335$		−	141922.i		−	1.26462i
$336$		0			0
$337$	90476.4			0.796665			0.398332	−	0.917241i	$-0.369589\pi$
$337$	90476.4			0.796665			0.398332	+	0.917241i	$0.369589\pi$
$338$		0			0
$339$	9559.61	+	5519.25i	0.0831842	+	0.0480264i
$340$		0			0
$341$	128891.	−	74415.3i	1.10844	−	0.639961i
$342$		0			0
$343$	80605.0	−	85697.8i	0.685131	−	0.728420i
$344$		0			0
$345$	63565.7	+	110099.i	0.534054	+	0.925008i
$346$		0			0
$347$	16037.2	−	27777.3i	0.133190	−	0.230691i	−0.791715	−	0.610891i	$-0.790811\pi$
$347$	16037.2	−	27777.3i	0.133190	−	0.230691i	0.924904	+	0.380200i	$0.124145\pi$
$348$		0			0
$349$		−	116783.i		−	0.958805i	−0.877595	−	0.479402i	$-0.840853\pi$
$349$		−	116783.i		−	0.958805i	0.877595	−	0.479402i	$-0.159147\pi$
$350$		0			0
$351$	14620.5			0.118672
$352$		0			0
$353$	107957.	+	62329.1i	0.866367	+	0.500197i	0.866139	−	0.499803i	$-0.166594\pi$
$353$	107957.	+	62329.1i	0.866367	+	0.500197i	0.000227639		1.00000i	$0.499928\pi$
$354$		0			0
$355$	−93225.9	+	53824.0i	−0.739741	+	0.427090i
$356$		0			0
$357$	19582.5	+	19186.8i	0.153649	+	0.150545i
$358$		0			0
$359$	49721.0	+	86119.4i	0.385790	+	0.668208i	0.991878	−	0.127189i	$-0.0405955\pi$
$359$	49721.0	+	86119.4i	0.385790	+	0.668208i	−0.606088	+	0.795397i	$0.707262\pi$
$360$		0			0
$361$	−64423.5	+	111585.i	−0.494344	+	0.856230i
$362$		0			0
$363$			26203.0i			0.198856i
$364$		0			0
$365$	−214820.			−1.61246
$366$		0			0
$367$	−77002.3	−	44457.3i	−0.571704	−	0.330074i	0.186126	−	0.982526i	$-0.440407\pi$
$367$	−77002.3	−	44457.3i	−0.571704	−	0.330074i	−0.757830	+	0.652452i	$0.773740\pi$
$368$		0			0
$369$	23925.1	−	13813.2i	0.175712	−	0.101447i
$370$		0			0
$371$	−26722.0	+	103960.i	−0.194143	+	0.755298i
$372$		0			0
$373$	45676.4	+	79113.9i	0.328303	+	0.568637i	0.982175	−	0.187968i	$-0.0601900\pi$
$373$	45676.4	+	79113.9i	0.328303	+	0.568637i	−0.653872	+	0.756605i	$0.726857\pi$
$374$		0			0
$375$	42086.2	−	72895.5i	0.299280	−	0.518368i
$376$		0			0
$377$		−	64778.2i		−	0.455770i
$378$		0			0
$379$	−6328.51			−0.0440578			−0.0220289	−	0.999757i	$-0.507013\pi$
$379$	−6328.51			−0.0440578			−0.0220289	+	0.999757i	$0.507013\pi$
$380$		0			0
$381$	−38241.3	−	22078.6i	−0.263441	−	0.152097i
$382$		0			0
$383$	11635.9	−	6717.98i	0.0793235	−	0.0457974i	−0.459814	−	0.888015i	$-0.652084\pi$
$383$	11635.9	−	6717.98i	0.0793235	−	0.0457974i	0.539137	+	0.842218i	$0.318750\pi$
$384$		0			0
$385$	30891.0	+	110753.i	0.208406	+	0.747198i
$386$		0			0
$387$	45662.5	+	79089.8i	0.304886	+	0.528078i
$388$		0			0
$389$	−44148.1	+	76466.8i	−0.291752	+	0.505328i	−0.974224	−	0.225583i	$-0.927571\pi$
$389$	−44148.1	+	76466.8i	−0.291752	+	0.505328i	0.682472	+	0.730911i	$0.260905\pi$
$390$		0			0
$391$		−	109990.i		−	0.719451i
$392$		0			0
$393$	−75200.2			−0.486894
$394$		0			0
$395$	216695.	+	125109.i	1.38885	+	0.801851i
$396$		0			0
$397$	187570.	−	108294.i	1.19010	−	0.687103i	0.231769	−	0.972771i	$-0.425549\pi$
$397$	187570.	−	108294.i	1.19010	−	0.687103i	0.958329	+	0.285668i	$0.0922155\pi$
$398$		0			0
$399$	−9416.04	+	2626.29i	−0.0591456	+	0.0164967i
$400$		0			0
$401$	2583.42	+	4474.61i	0.0160659	+	0.0278270i	0.873947	−	0.486022i	$-0.161553\pi$
$401$	2583.42	+	4474.61i	0.0160659	+	0.0278270i	−0.857881	+	0.513849i	$0.828219\pi$
$402$		0			0
$403$	79155.7	−	137102.i	0.487385	−	0.844175i
$404$		0			0
$405$		−	17460.7i		−	0.106451i
$406$		0			0
$407$	55082.5			0.332526
$408$		0			0
$409$	33697.7	+	19455.3i	0.201443	+	0.116303i	0.597329	−	0.801997i	$-0.296229\pi$
$409$	33697.7	+	19455.3i	0.201443	+	0.116303i	−0.395885	+	0.918300i	$0.629562\pi$
$410$		0			0
$411$	−57848.3	+	33398.7i	−0.342458	+	0.197718i
$412$		0			0
$413$	−139281.	−	35801.0i	−0.816567	−	0.209891i
$414$		0			0
$415$	15200.6	+	26328.1i	0.0882599	+	0.152871i
$416$		0			0
$417$	13015.0	−	22542.6i	0.0748463	−	0.129638i
$418$		0			0
$419$		−	263425.i		−	1.50048i	−0.661167	−	0.750239i	$-0.729938\pi$
$419$		−	263425.i		−	1.50048i	0.661167	−	0.750239i	$-0.270062\pi$
$420$		0			0
$421$	−16375.1			−0.0923889			−0.0461944	−	0.998932i	$-0.514709\pi$
$421$	−16375.1			−0.0923889			−0.0461944	+	0.998932i	$0.514709\pi$
$422$		0			0
$423$	92245.4	+	53257.9i	0.515542	+	0.297648i
$424$		0			0
$425$	−4785.93	+	2763.16i	−0.0264965	+	0.0152978i
$426$		0			0
$427$	22816.6	−	23287.1i	0.125140	−	0.127720i
$428$		0			0
$429$	−26525.5	−	45943.5i	−0.144128	−	0.249637i
$430$		0			0
$431$	−125208.	+	216867.i	−0.674028	+	1.16745i	0.302724	+	0.953078i	$0.402104\pi$
$431$	−125208.	+	216867.i	−0.674028	+	1.16745i	−0.976752	+	0.214373i	$0.931229\pi$
$432$		0			0
$433$		−	91495.1i		−	0.488002i	−0.969775	−	0.244001i	$-0.921540\pi$
$433$		−	91495.1i		−	0.488002i	0.969775	−	0.244001i	$-0.0784601\pi$
$434$		0			0
$435$	−77362.1			−0.408837
$436$		0			0
$437$	33964.6	+	19609.5i	0.177854	+	0.102684i
$438$		0			0
$439$	−7672.44	+	4429.69i	−0.0398111	+	0.0229850i	−0.519773	−	0.854304i	$-0.673984\pi$
$439$	−7672.44	+	4429.69i	−0.0398111	+	0.0229850i	0.479962	+	0.877289i	$0.340650\pi$
$440$		0			0
$441$	56791.6	+	31261.0i	0.292016	+	0.160741i
$442$		0			0
$443$	150105.	+	259989.i	0.764868	+	1.32479i	0.940316	+	0.340302i	$0.110529\pi$
$443$	150105.	+	259989.i	0.764868	+	1.32479i	−0.175448	+	0.984489i	$0.556137\pi$
$444$		0			0
$445$	43164.9	−	74763.9i	0.217977	−	0.377548i
$446$		0			0
$447$		−	131977.i		−	0.660514i
$448$		0			0
$449$	116866.			0.579691			0.289846	−	0.957073i	$-0.406396\pi$
$449$	116866.			0.579691			0.289846	+	0.957073i	$0.406396\pi$
$450$		0			0
$451$	−86813.2	−	50121.6i	−0.426808	−	0.246418i
$452$		0			0
$453$	−82389.5	+	47567.6i	−0.401491	+	0.231801i
$454$		0			0
$455$	87360.9	+	85595.8i	0.421982	+	0.413456i
$456$		0			0
$457$	−36872.9	−	63865.7i	−0.176553	−	0.305798i	0.764145	−	0.645045i	$-0.223161\pi$
$457$	−36872.9	−	63865.7i	−0.176553	−	0.305798i	−0.940698	+	0.339246i	$0.889828\pi$
$458$		0			0
$459$	−7553.23	+	13082.6i	−0.0358515	+	0.0620966i
$460$		0			0
$461$		−	88219.1i		−	0.415108i	−0.978224	−	0.207554i	$-0.933450\pi$
$461$		−	88219.1i		−	0.415108i	0.978224	−	0.207554i	$-0.0665502\pi$
$462$		0			0
$463$	−297880.			−1.38957			−0.694784	−	0.719219i	$-0.744500\pi$
$463$	−297880.			−1.38957			−0.694784	+	0.719219i	$0.744500\pi$
$464$		0			0
$465$	−163735.	−	94532.6i	−0.757245	−	0.437196i
$466$		0			0
$467$	162353.	−	93734.8i	0.744436	−	0.429801i	−0.0792437	−	0.996855i	$-0.525251\pi$
$467$	162353.	−	93734.8i	0.744436	−	0.429801i	0.823680	+	0.567055i	$0.191917\pi$
$468$		0			0
$469$	−72280.7	+	281203.i	−0.328607	+	1.27842i
$470$		0			0
$471$	−91567.7	−	158600.i	−0.412763	−	0.714926i
$472$		0			0
$473$	165688.	−	286980.i	0.740575	−	1.28271i
$474$		0			0
$475$		−	1970.50i		−	0.00873353i
$476$		0			0
$477$	−59146.2			−0.259950
$478$		0			0
$479$	104347.	+	60244.5i	0.454786	+	0.262571i	0.709849	−	0.704354i	$-0.248763\pi$
$479$	104347.	+	60244.5i	0.454786	+	0.262571i	−0.255063	+	0.966924i	$0.582096\pi$
$480$		0			0
$481$	50741.6	−	29295.7i	0.219318	−	0.126623i
$482$		0			0
$483$	−69875.2	−	250523.i	−0.299522	−	1.07387i
$484$		0			0
$485$	23359.1	+	40459.1i	0.0993052	+	0.172002i
$486$		0			0
$487$	−133697.	+	231570.i	−0.563721	+	0.976393i	0.433447	+	0.901179i	$0.357297\pi$
$487$	−133697.	+	231570.i	−0.563721	+	0.976393i	−0.997167	+	0.0752136i	$0.976036\pi$
$488$		0			0
$489$			758.319i			0.00317128i
$490$		0			0
$491$	−178364.			−0.739851			−0.369926	−	0.929061i	$-0.620617\pi$
$491$	−178364.			−0.739851			−0.369926	+	0.929061i	$0.620617\pi$
$492$		0			0
$493$	57964.3	+	33465.7i	0.238488	+	0.137691i
$494$		0			0
$495$	−54868.5	+	31678.4i	−0.223930	+	0.129286i
$496$		0			0
$497$	212129.	−	59166.5i	0.858792	−	0.239532i
$498$		0			0
$499$	131873.	+	228411.i	0.529610	+	0.917311i	0.999403	+	0.0345349i	$0.0109950\pi$
$499$	131873.	+	228411.i	0.529610	+	0.917311i	−0.469794	+	0.882776i	$0.655672\pi$
$500$		0			0
$501$	81356.3	−	140913.i	0.324128	−	0.561405i
$502$		0			0
$503$		−	480056.i		−	1.89739i	−0.316197	−	0.948694i	$-0.602406\pi$
$503$		−	480056.i		−	1.89739i	0.316197	−	0.948694i	$-0.397594\pi$
$504$		0			0
$505$	−271281.			−1.06374
$506$		0			0
$507$	79654.4	+	45988.5i	0.309880	+	0.178909i
$508$		0			0
$509$	−52192.6	+	30133.4i	−0.201453	+	0.116309i	−0.597333	−	0.801993i	$-0.703773\pi$
$509$	−52192.6	+	30133.4i	−0.201453	+	0.116309i	0.395880	+	0.918302i	$0.370440\pi$
$510$		0			0
$511$	425641.	+	109407.i	1.63005	+	0.418991i
$512$		0			0
$513$	−2693.23	−	4664.82i	−0.0102339	−	0.0177256i
$514$		0			0
$515$	−195551.	+	338704.i	−0.737301	+	1.27704i
$516$		0			0
$517$		−	386497.i		−	1.44599i
$518$		0			0
$519$	−208448.			−0.773861
$520$		0			0
$521$	248823.	+	143658.i	0.916674	+	0.529242i	0.882573	−	0.470176i	$-0.155810\pi$
$521$	248823.	+	143658.i	0.916674	+	0.529242i	0.0341018	+	0.999418i	$0.489143\pi$
$522$		0			0
$523$	239407.	−	138222.i	0.875254	−	0.505328i	0.00616355	−	0.999981i	$-0.498038\pi$
$523$	239407.	−	138222.i	0.875254	−	0.505328i	0.869091	+	0.494653i	$0.164705\pi$
$524$		0			0
$525$	−9145.44	+	9334.02i	−0.0331807	+	0.0338649i
$526$		0			0
$527$	81786.9	+	141659.i	0.294484	+	0.510062i
$528$		0			0
$529$	−381809.	+	661313.i	−1.36438	+	2.36317i
$530$		0			0
$531$		−	79241.5i		−	0.281037i
$532$		0			0
$533$	−106629.			−0.375336
$534$		0			0
$535$	−91194.3	−	52651.0i	−0.318610	−	0.183950i
$536$		0			0
$537$	−225695.	+	130305.i	−0.782661	+	0.451869i
$538$		0			0
$539$	−4800.56	−	235178.i	−0.0165240	−	0.809505i
$540$		0			0
$541$	29832.0	+	51670.6i	0.101927	+	0.176542i	0.912478	−	0.409125i	$-0.134166\pi$
$541$	29832.0	+	51670.6i	0.101927	+	0.176542i	−0.810552	+	0.585667i	$0.800833\pi$
$542$		0			0
$543$	−166210.	+	287884.i	−0.563712	+	0.976378i
$544$		0			0
$545$			352034.i			1.18520i
$546$		0			0
$547$	−489585.			−1.63627			−0.818133	−	0.575030i	$-0.804990\pi$
$547$	−489585.			−1.63627			−0.818133	+	0.575030i	$0.804990\pi$
$548$		0			0
$549$	15557.6	+	8982.18i	0.0516176	+	0.0298014i
$550$		0			0
$551$	−20668.2	+	11932.8i	−0.0680767	+	0.0393041i
$552$		0			0
$553$	−365639.	−	358251.i	−1.19564	−	1.17149i
$554$		0			0
$555$	−34986.7	−	60598.8i	−0.113584	−	0.196733i
$556$		0			0
$557$	−254267.	+	440404.i	−0.819558	+	1.41952i	0.0864495	+	0.996256i	$0.472448\pi$
$557$	−254267.	+	440404.i	−0.819558	+	1.41952i	−0.906008	+	0.423261i	$0.860885\pi$
$558$		0			0
$559$		−	352486.i		−	1.12802i
$560$		0			0
$561$	54814.4			0.174168
$562$		0			0
$563$	−103765.	−	59908.5i	−0.327365	−	0.189004i	0.327306	−	0.944919i	$-0.393859\pi$
$563$	−103765.	−	59908.5i	−0.327365	−	0.189004i	−0.654671	+	0.755914i	$0.727193\pi$
$564$		0			0
$565$	−44064.8	+	25440.8i	−0.138037	+	0.0796956i
$566$		0			0
$567$	−8892.70	+	34596.4i	−0.0276610	+	0.107613i
$568$		0			0
$569$	48829.7	+	84575.5i	0.150820	+	0.261228i	0.931529	−	0.363667i	$-0.118475\pi$
$569$	48829.7	+	84575.5i	0.150820	+	0.261228i	−0.780709	+	0.624895i	$0.785142\pi$
$570$		0			0
$571$	23825.7	−	41267.3i	0.0730757	−	0.126571i	−0.827172	−	0.561949i	$-0.810052\pi$
$571$	23825.7	−	41267.3i	0.0730757	−	0.126571i	0.900248	+	0.435378i	$0.143385\pi$
$572$		0			0
$573$		−	222234.i		−	0.676863i
$574$		0			0
$575$	52427.2			0.158570
$576$		0			0
$577$	−113845.	−	65728.4i	−0.341949	−	0.197425i	0.319184	−	0.947693i	$-0.396591\pi$
$577$	−113845.	−	65728.4i	−0.341949	−	0.197425i	−0.661134	+	0.750268i	$0.729924\pi$
$578$		0			0
$579$	39476.6	−	22791.8i	0.117756	−	0.0679864i
$580$		0			0
$581$	−16709.3	−	59907.9i	−0.0495002	−	0.177473i
$582$		0			0
$583$	107307.	+	185861.i	0.315712	+	0.546830i
$584$		0			0
$585$	−33696.3	+	58363.8i	−0.0984625	+	0.170542i
$586$		0			0
$587$			135493.i			0.393224i	0.980481	+	0.196612i	$0.0629940\pi$
$587$			135493.i			0.393224i	−0.980481	+	0.196612i	$0.937006\pi$
$588$		0			0
$589$	−58325.0			−0.168122
$590$		0			0
$591$	−147712.	−	85281.8i	−0.422904	−	0.244164i
$592$		0			0
$593$	162993.	−	94104.0i	0.463510	−	0.267608i	−0.250009	−	0.968244i	$-0.580433\pi$
$593$	162993.	−	94104.0i	0.463510	−	0.267608i	0.713519	+	0.700636i	$0.247100\pi$
$594$		0			0
$595$	−121725.	+	33951.1i	−0.343831	+	0.0959003i
$596$		0			0
$597$	68756.0	+	119089.i	0.192913	+	0.334136i
$598$		0			0
$599$	−39202.4	+	67900.5i	−0.109259	+	0.189243i	−0.915470	−	0.402385i	$-0.868181\pi$
$599$	−39202.4	+	67900.5i	−0.109259	+	0.189243i	0.806211	+	0.591628i	$0.201515\pi$
$600$		0			0
$601$		−	254898.i		−	0.705695i	−0.935681	−	0.352848i	$-0.885213\pi$
$601$		−	254898.i		−	0.705695i	0.935681	−	0.352848i	$-0.114787\pi$
$602$		0			0
$603$	−159985.			−0.439993
$604$		0			0
$605$	−104600.	−	60391.0i	−0.285774	−	0.164992i
$606$		0			0
$607$	−489896.	+	282842.i	−1.32962	+	0.767655i	−0.985241	−	0.171176i	$-0.945243\pi$
$607$	−489896.	+	282842.i	−1.32962	+	0.767655i	−0.344378	+	0.938831i	$0.611910\pi$
$608$		0			0
$609$	153284.	+	39400.4i	0.413298	+	0.106235i
$610$		0			0
$611$	−205559.	−	356038.i	−0.550622	−	0.953705i
$612$		0			0
$613$	207111.	−	358726.i	0.551164	−	0.954645i	−0.447026	−	0.894521i	$-0.647517\pi$
$613$	207111.	−	358726.i	0.551164	−	0.954645i	0.998191	−	0.0601241i	$-0.0191496\pi$
$614$		0			0
$615$			127343.i			0.336686i
$616$		0			0
$617$	22896.0			0.0601435			0.0300717	−	0.999548i	$-0.490426\pi$
$617$	22896.0			0.0601435			0.0300717	+	0.999548i	$0.490426\pi$
$618$		0			0
$619$	−139564.	−	80577.4i	−0.364244	−	0.210296i	0.306697	−	0.951807i	$-0.400776\pi$
$619$	−139564.	−	80577.4i	−0.364244	−	0.210296i	−0.670941	+	0.741511i	$0.734110\pi$
$620$		0			0
$621$	124112.	−	71656.1i	0.321833	−	0.185811i
$622$		0			0
$623$	−123604.	+	126152.i	−0.318460	+	0.325027i
$624$		0			0
$625$	177957.	+	308230.i	0.455569	+	0.789069i
$626$		0			0
$627$	−9772.50	+	16926.5i	−0.0248583	+	0.0430558i
$628$		0			0
$629$			60539.0i			0.153015i
$630$		0			0
$631$	−489038.			−1.22824			−0.614121	−	0.789212i	$-0.710489\pi$
$631$	−489038.			−1.22824			−0.614121	+	0.789212i	$0.710489\pi$
$632$		0			0
$633$	−301754.	−	174217.i	−0.753087	−	0.434795i
$634$		0			0
$635$	176272.	−	101771.i	0.437156	−	0.252392i
$636$		0			0
$637$	−129502.	−	214091.i	−0.319152	−	0.527619i
$638$		0			0
$639$	60674.5	+	105091.i	0.148595	+	0.257374i
$640$		0			0
$641$	−59669.8	+	103351.i	−0.145224	+	0.251536i	−0.929457	−	0.368932i	$-0.879724\pi$
$641$	−59669.8	+	103351.i	−0.145224	+	0.251536i	0.784232	+	0.620467i	$0.213057\pi$
$642$		0			0
$643$			324224.i			0.784194i	0.919924	+	0.392097i	$0.128250\pi$
$643$			324224.i			0.784194i	−0.919924	+	0.392097i	$0.871750\pi$
$644$		0			0
$645$	−420960.			−1.01186
$646$		0			0
$647$	399048.	+	230391.i	0.953272	+	0.550372i	0.894096	−	0.447876i	$-0.147819\pi$
$647$	399048.	+	230391.i	0.953272	+	0.550372i	0.0591759	+	0.998248i	$0.481153\pi$
$648$		0			0
$649$	−249009.	+	143765.i	−0.591188	+	0.341323i
$650$		0			0
$651$	276278.	+	270696.i	0.651905	+	0.638734i
$652$		0			0
$653$	279562.	+	484216.i	0.655620	+	1.13557i	0.981738	+	0.190238i	$0.0609260\pi$
$653$	279562.	+	484216.i	0.655620	+	1.13557i	−0.326118	+	0.945329i	$0.605741\pi$
$654$		0			0
$655$	173317.	−	300193.i	0.403978	−	0.699711i
$656$		0			0
$657$			242161.i			0.561014i
$658$		0			0
$659$	184372.			0.424546			0.212273	−	0.977210i	$-0.431913\pi$
$659$	184372.			0.424546			0.212273	+	0.977210i	$0.431913\pi$
$660$		0			0
$661$	68612.1	+	39613.2i	0.157035	+	0.0906645i	0.576458	−	0.817126i	$-0.304434\pi$
$661$	68612.1	+	39613.2i	0.157035	+	0.0906645i	−0.419423	+	0.907791i	$0.637768\pi$
$662$		0			0
$663$	50494.6	−	29153.1i	0.114873	−	0.0663220i
$664$		0			0
$665$	11217.5	−	43641.0i	0.0253661	−	0.0986851i
$666$		0			0
$667$	−317483.	−	549897.i	−0.713624	−	1.23603i
$668$		0			0
$669$	146168.	−	253170.i	0.326588	−	0.565666i
$670$		0			0
$671$		−	65184.4i		−	0.144777i
$672$		0			0
$673$	−482896.			−1.06616			−0.533082	−	0.846064i	$-0.678966\pi$
$673$	−482896.			−1.06616			−0.533082	+	0.846064i	$0.678966\pi$
$674$		0			0
$675$	−6235.84	−	3600.27i	−0.0136863	−	0.00790182i
$676$		0			0
$677$	−329375.	+	190165.i	−0.718642	+	0.414908i	−0.814253	−	0.580511i	$-0.802853\pi$
$677$	−329375.	+	190165.i	−0.718642	+	0.414908i	0.0956105	+	0.995419i	$0.469520\pi$
$678$		0			0
$679$	−25677.6	−	92061.9i	−0.0556949	−	0.199683i
$680$		0			0
$681$	−158868.	−	275167.i	−0.342563	−	0.593337i
$682$		0			0
$683$	−171034.	+	296240.i	−0.366642	+	0.635042i	−0.989038	−	0.147660i	$-0.952826\pi$
$683$	−171034.	+	296240.i	−0.366642	+	0.635042i	0.622396	+	0.782702i	$0.286159\pi$
$684$		0			0
$685$		−	307901.i		−	0.656191i
$686$		0			0
$687$	175649.			0.372163
$688$		0			0
$689$	197701.	+	114143.i	0.416458	+	0.240442i
$690$		0			0
$691$	−63274.0	+	36531.3i	−0.132516	+	0.0765084i	−0.564793	−	0.825233i	$-0.691044\pi$
$691$	−63274.0	+	36531.3i	−0.132516	+	0.0765084i	0.432276	+	0.901741i	$0.357711\pi$
$692$		0			0
$693$	124850.	−	34822.7i	0.259969	−	0.0725097i
$694$		0			0
$695$	59992.1	+	103909.i	0.124201	+	0.215122i
$696$		0			0
$697$	55086.7	−	95413.0i	0.113392	−	0.196400i
$698$		0			0
$699$		−	45140.3i		−	0.0923869i
$700$		0			0
$701$	−288287.			−0.586664			−0.293332	−	0.956011i	$-0.594764\pi$
$701$	−288287.			−0.586664			−0.293332	+	0.956011i	$0.594764\pi$
$702$		0			0
$703$	−18694.2	−	10793.1i	−0.0378265	−	0.0218392i
$704$		0			0
$705$	−425203.	+	245491.i	−0.855496	+	0.493921i
$706$		0			0
$707$	537513.	+	138163.i	1.07535	+	0.276409i
$708$		0			0
$709$	155256.	+	268911.i	0.308856	+	0.534954i	0.978112	−	0.208078i	$-0.0667206\pi$
$709$	155256.	+	268911.i	0.308856	+	0.534954i	−0.669257	+	0.743031i	$0.733387\pi$
$710$		0			0
$711$	141032.	−	244275.i	0.278984	−	0.483214i
$712$		0			0
$713$		−	1.55179e6i		−	3.05250i
$714$		0			0
$715$	244537.			0.478335
$716$		0			0
$717$	70717.6	+	40828.8i	0.137559	+	0.0794198i
$718$		0			0
$719$	−8733.60	+	5042.35i	−0.0168941	+	0.00975383i	−0.508423	−	0.861107i	$-0.669771\pi$
$719$	−8733.60	+	5042.35i	−0.0168941	+	0.00975383i	0.491529	+	0.870861i	$0.336438\pi$
$720$		0			0
$721$	559963.	−	571510.i	1.07718	−	1.09939i
$722$		0			0
$723$	−14234.4	−	24654.7i	−0.0272309	−	0.0471654i
$724$		0			0
$725$	−15951.5	+	27628.8i	−0.0303477	+	0.0525638i
$726$		0			0
$727$			638552.i			1.20817i	0.796921	+	0.604084i	$0.206461\pi$
$727$			638552.i			1.20817i	−0.796921	+	0.604084i	$0.793539\pi$
$728$		0			0
$729$	−19683.0			−0.0370370
$730$		0			0
$731$	315409.	+	182101.i	0.590254	+	0.340783i
$732$		0			0
$733$	−715410.	+	413042.i	−1.33152	+	0.768752i	−0.985532	−	0.169488i	$-0.945789\pi$
$733$	−715410.	+	413042.i	−1.33152	+	0.768752i	−0.345985	+	0.938240i	$0.612455\pi$
$734$		0			0
$735$	−255681.	+	154659.i	−0.473287	+	0.286287i
$736$		0			0
$737$	290257.	+	502739.i	0.534376	+	0.925567i
$738$		0			0
$739$	384995.	−	666831.i	0.704962	−	1.22103i	−0.261743	−	0.965138i	$-0.584297\pi$
$739$	384995.	−	666831.i	0.704962	−	1.22103i	0.966705	−	0.255893i	$-0.0823695\pi$
$740$		0			0
$741$			20790.1i			0.0378634i
$742$		0			0
$743$	611777.			1.10819			0.554097	−	0.832452i	$-0.313064\pi$
$743$	611777.			1.10819			0.554097	+	0.832452i	$0.313064\pi$
$744$		0			0
$745$	526841.	+	304172.i	0.949220	+	0.548032i
$746$		0			0
$747$	29679.1	−	17135.2i	0.0531875	−	0.0307078i
$748$		0			0
$749$	153876.	+	150767.i	0.274289	+	0.268747i
$750$		0			0
$751$	185311.	+	320968.i	0.328565	+	0.569091i	0.982227	−	0.187695i	$-0.0601018\pi$
$751$	185311.	+	320968.i	0.328565	+	0.569091i	−0.653663	+	0.756786i	$0.726768\pi$
$752$		0			0
$753$	−63265.5	+	109579.i	−0.111578	+	0.193258i
$754$		0			0
$755$		−	438523.i		−	0.769305i
$756$		0			0
$757$	298778.			0.521384			0.260692	−	0.965422i	$-0.416049\pi$
$757$	298778.			0.521384			0.260692	+	0.965422i	$0.416049\pi$
$758$		0			0
$759$	−450346.	−	260007.i	−0.781740	−	0.451338i
$760$		0			0
$761$	695541.	−	401571.i	1.20103	−	0.693414i	0.240245	−	0.970712i	$-0.422772\pi$
$761$	695541.	−	401571.i	1.20103	−	0.693414i	0.960784	+	0.277298i	$0.0894389\pi$
$762$		0			0
$763$	179291.	−	697517.i	0.307970	−	1.19813i
$764$		0			0
$765$	−34816.4	−	60303.8i	−0.0594924	−	0.103044i
$766$		0			0
$767$	−152924.	+	264871.i	−0.259946	+	0.450240i
$768$		0			0
$769$			452541.i			0.765253i	0.923903	+	0.382626i	$0.124980\pi$
$769$			452541.i			0.765253i	−0.923903	+	0.382626i	$0.875020\pi$
$770$		0			0
$771$	60310.4			0.101457
$772$		0			0
$773$	908057.	+	524267.i	1.51969	+	0.877392i	0.999731	+	0.0231996i	$0.00738531\pi$
$773$	908057.	+	524267.i	1.51969	+	0.877392i	0.519957	+	0.854193i	$0.325948\pi$
$774$		0			0
$775$	−67522.1	+	38983.9i	−0.112420	+	0.0649056i
$776$		0			0
$777$	38459.5	+	137889.i	0.0637032	+	0.228395i
$778$		0			0
$779$	19642.1	+	34021.1i	0.0323678	+	0.0560626i
$780$		0			0
$781$	220160.	−	381328.i	0.360941	−	0.625168i
$782$		0			0
$783$			87208.5i			0.142244i
$784$		0			0
$785$	844159.			1.36989
$786$		0			0
$787$	−169712.	−	97983.5i	−0.274009	−	0.158199i	0.356699	−	0.934219i	$-0.383902\pi$
$787$	−169712.	−	97983.5i	−0.274009	−	0.158199i	−0.630708	+	0.776020i	$0.717235\pi$
$788$		0			0
$789$	543004.	−	313504.i	0.872266	−	0.503603i
$790$		0			0
$791$	100266.	−	27966.0i	0.160252	−	0.0446970i
$792$		0			0
$793$	−34668.4	−	60047.4i	−0.0551299	−	0.0954877i
$794$		0			0
$795$	136317.	−	236107.i	0.215682	−	0.373572i
$796$		0			0
$797$			66453.8i			0.104617i	0.998631	+	0.0523086i	$0.0166579\pi$
$797$			66453.8i			0.104617i	−0.998631	+	0.0523086i	$0.983342\pi$
$798$		0			0
$799$	424784.			0.665387
$800$		0			0
$801$	−84279.5	−	48658.8i	−0.131358	−	0.0758397i
$802$		0			0
$803$	760969.	−	439346.i	1.18015	−	0.681358i
$804$		0			0
$805$	1.16111e6	+	298454.i	1.79177	+	0.460559i
$806$		0			0
$807$	−70788.8	−	122610.i	−0.108697	−	0.188269i
$808$		0			0
$809$	−41152.3	+	71277.9i	−0.0628777	+	0.108907i	−0.895751	−	0.444557i	$-0.853361\pi$
$809$	−41152.3	+	71277.9i	−0.0628777	+	0.108907i	0.832873	+	0.553464i	$0.186694\pi$
$810$		0			0
$811$		−	469097.i		−	0.713215i	−0.934254	−	0.356608i	$-0.883933\pi$
$811$		−	469097.i		−	0.713215i	0.934254	−	0.356608i	$-0.116067\pi$
$812$		0			0
$813$	504252.			0.762898
$814$		0			0
$815$	−3027.15	−	1747.73i	−0.00455742	−	0.00263123i
$816$		0			0
$817$	−112464.	+	64931.3i	−0.168489	+	0.0972769i
$818$		0			0
$819$	96490.1	−	98479.8i	0.143852	−	0.146818i
$820$		0			0
$821$	2689.76	+	4658.80i	0.00399050	+	0.00691174i	0.868014	−	0.496540i	$-0.165396\pi$
$821$	2689.76	+	4658.80i	0.00399050	+	0.00691174i	−0.864023	+	0.503452i	$0.832063\pi$
$822$		0			0
$823$	−484903.	+	839877.i	−0.715905	+	1.23998i	0.246704	+	0.969091i	$0.420652\pi$
$823$	−484903.	+	839877.i	−0.715905	+	1.23998i	−0.962609	+	0.270893i	$0.912681\pi$
$824$		0			0
$825$			26127.4i			0.0383874i
$826$		0			0
$827$	−838747.			−1.22637			−0.613183	−	0.789941i	$-0.710111\pi$
$827$	−838747.			−1.22637			−0.613183	+	0.789941i	$0.710111\pi$
$828$		0			0
$829$	−765309.	−	441852.i	−1.11360	−	0.642935i	−0.173838	−	0.984774i	$-0.555617\pi$
$829$	−765309.	−	441852.i	−1.11360	−	0.642935i	−0.939759	+	0.341839i	$0.888950\pi$
$830$		0			0
$831$	−133888.	+	77300.0i	−0.193882	+	0.111938i
$832$		0			0
$833$	258475.	−	5276.10i	0.372502	−	0.00760367i
$834$		0			0
$835$	375010.	+	649536.i	0.537861	+	0.931602i
$836$		0			0
$837$	−106564.	+	184575.i	−0.152111	+	0.263464i
$838$		0			0
$839$			1.12626e6i			1.59998i	0.600011	+	0.799992i	$0.295163\pi$
$839$			1.12626e6i			1.59998i	−0.600011	+	0.799992i	$0.704837\pi$
$840$		0			0
$841$	−320891.			−0.453696
$842$		0			0
$843$	−550238.	−	317680.i	−0.774276	−	0.447028i
$844$		0			0
$845$	−367165.	+	211983.i	−0.514218	+	0.296884i
$846$		0			0
$847$	176497.	+	172931.i	0.246020	+	0.241049i
$848$		0			0
$849$	157705.	+	273153.i	0.218791	+	0.378958i
$850$		0			0
$851$	287161.	−	497378.i	0.396522	−	0.686796i
$852$		0			0
$853$			805486.i			1.10703i	0.832839	+	0.553516i	$0.186714\pi$
$853$			805486.i			1.10703i	−0.832839	+	0.553516i	$0.813286\pi$
$854$		0			0
$855$	24828.8			0.0339643
$856$		0			0
$857$	−849118.	−	490238.i	−1.15613	−	0.667491i	−0.205756	−	0.978603i	$-0.565965\pi$
$857$	−849118.	−	490238.i	−1.15613	−	0.667491i	−0.950373	+	0.311112i	$0.899299\pi$
$858$		0			0
$859$	375665.	−	216890.i	0.509113	−	0.293936i	−0.223356	−	0.974737i	$-0.571701\pi$
$859$	375665.	−	216890.i	0.509113	−	0.293936i	0.732469	+	0.680800i	$0.238368\pi$
$860$		0			0
$861$	64855.6	−	252316.i	0.0874865	−	0.340360i
$862$		0			0
$863$	−151578.	−	262542.i	−0.203524	−	0.352514i	0.746137	−	0.665792i	$-0.231906\pi$
$863$	−151578.	−	262542.i	−0.203524	−	0.352514i	−0.949661	+	0.313278i	$0.898573\pi$
$864$		0			0
$865$	480418.	−	832108.i	0.642077	−	1.11211i
$866$		0			0
$867$		−	373744.i		−	0.497205i
$868$		0			0
$869$	−1.02348e6			−1.35532
$870$		0			0
$871$	534765.	+	308747.i	0.704898	+	0.406973i
$872$		0			0
$873$	45608.6	−	26332.1i	0.0598436	−	0.0345507i
$874$		0			0
$875$	−213251.	−	764568.i	−0.278532	−	0.998619i
$876$		0			0
$877$	−177500.	−	307439.i	−0.230781	−	0.399724i	0.727257	−	0.686365i	$-0.240795\pi$
$877$	−177500.	−	307439.i	−0.230781	−	0.399724i	−0.958038	+	0.286641i	$0.907461\pi$
$878$		0			0
$879$	76792.5	−	133008.i	0.0993896	−	0.172148i
$880$		0			0
$881$		−	320002.i		−	0.412288i	−0.978522	−	0.206144i	$-0.933909\pi$
$881$		−	320002.i		−	0.412288i	0.978522	−	0.206144i	$-0.0660915\pi$
$882$		0			0
$883$	1.08605e6			1.39293			0.696467	−	0.717589i	$-0.254754\pi$
$883$	1.08605e6			1.39293			0.696467	+	0.717589i	$0.254754\pi$
$884$		0			0
$885$	316326.	+	182631.i	0.403876	+	0.233178i
$886$		0			0
$887$	−1.11218e6	+	642117.i	−1.41360	+	0.816144i	−0.995726	−	0.0923591i	$-0.970559\pi$
$887$	−1.11218e6	+	642117.i	−1.41360	+	0.816144i	−0.417878	+	0.908503i	$0.637226\pi$
$888$		0			0
$889$	−401096.	+	111872.i	−0.507510	+	0.141553i
$890$		0			0
$891$	35710.3	+	61852.0i	0.0449819	+	0.0779109i
$892$		0			0
$893$	−75731.9	+	131171.i	−0.0949677	+	0.164489i
$894$		0			0
$895$		−	1.20128e6i		−	1.49967i
$896$		0			0
$897$	−553140.			−0.687465
$898$		0			0
$899$	817787.	+	472150.i	1.01186	+	0.584198i
$900$		0			0
$901$	−204273.	+	117937.i	−0.251629	+	0.145278i
$902$		0			0
$903$	834086.	+	214394.i	1.02291	+	0.262929i
$904$		0			0
$905$	−766141.	−	1.32699e6i	−0.935430	−	1.62021i
$906$		0			0
$907$	24217.8	−	41946.4i	0.0294388	−	0.0509894i	−0.850931	−	0.525278i	$-0.823961\pi$
$907$	24217.8	−	41946.4i	0.0294388	−	0.0509894i	0.880369	+	0.474289i	$0.157295\pi$
$908$		0			0
$909$			305808.i			0.370102i
$910$		0			0
$911$	−727675.			−0.876801			−0.438400	−	0.898780i	$-0.644455\pi$
$911$	−727675.			−0.876801			−0.438400	+	0.898780i	$0.644455\pi$
$912$		0			0
$913$	−107692.	−	62175.8i	−0.129194	−	0.0745899i
$914$		0			0
$915$	−71712.3	+	41403.1i	−0.0856548	+	0.0494528i
$916$		0			0
$917$	−496296.	+	506530.i	−0.590204	+	0.602375i
$918$		0			0
$919$	43858.0	+	75964.2i	0.0519299	+	0.0899452i	0.890822	−	0.454353i	$-0.150129\pi$
$919$	43858.0	+	75964.2i	0.0519299	+	0.0899452i	−0.838892	+	0.544298i	$0.816796\pi$
$920$		0			0
$921$	−279468.	+	484052.i	−0.329467	+	0.570654i
$922$		0			0
$923$		−	468369.i		−	0.549775i
$924$		0			0
$925$	−28856.1			−0.0337251
$926$		0			0
$927$	381813.	+	220440.i	0.444315	+	0.256525i
$928$		0			0
$929$	781079.	−	450956.i	0.905032	−	0.522520i	0.0262024	−	0.999657i	$-0.491659\pi$
$929$	781079.	−	450956.i	0.905032	−	0.522520i	0.878829	+	0.477136i	$0.158325\pi$
$930$		0			0
$931$	−44452.6	+	80756.7i	−0.0512859	+	0.0931707i
$932$		0			0
$933$	−112683.	−	195173.i	−0.129448	−	0.224211i
$934$		0			0
$935$	−126333.	+	218815.i	−0.144508	+	0.250296i
$936$		0			0
$937$			1.24186e6i			1.41447i	0.706978	+	0.707236i	$0.250058\pi$
$937$			1.24186e6i			1.41447i	−0.706978	+	0.707236i	$0.749942\pi$
$938$		0			0
$939$	−559867.			−0.634970
$940$		0			0
$941$	369139.	+	213122.i	0.416879	+	0.240685i	0.693741	−	0.720224i	$-0.255961\pi$
$941$	369139.	+	213122.i	0.416879	+	0.240685i	−0.276862	+	0.960910i	$0.589294\pi$
$942$		0			0
$943$	−905165.	+	522598.i	−1.01790	+	0.587684i
$944$		0			0
$945$	−117611.	−	115235.i	−0.131699	−	0.129038i
$946$		0			0
$947$	13665.2	+	23668.9i	0.0152376	+	0.0263923i	0.873544	−	0.486746i	$-0.161816\pi$
$947$	13665.2	+	23668.9i	0.0152376	+	0.0263923i	−0.858306	+	0.513138i	$0.828483\pi$
$948$		0			0
$949$	467333.	−	809444.i	0.518912	−	0.898782i
$950$		0			0
$951$			810568.i			0.896248i
$952$		0			0
$953$	1.30093e6			1.43242			0.716208	−	0.697887i	$-0.245876\pi$
$953$	1.30093e6			1.43242			0.716208	+	0.697887i	$0.245876\pi$
$954$		0			0
$955$	887140.	+	512190.i	0.972714	+	0.561597i
$956$		0			0
$957$	274045.	−	158220.i	0.299225	−	0.172757i
$958$		0			0
$959$	−156814.	+	610072.i	−0.170509	+	0.663352i
$960$		0			0
$961$	692125.	+	1.19880e6i	0.749442	+	1.29807i
$962$		0			0
$963$	−59352.3	+	102801.i	−0.0640007	+	0.110852i
$964$		0			0
$965$			210117.i			0.225635i
$966$		0			0
$967$	−463079.			−0.495225			−0.247612	−	0.968859i	$-0.579646\pi$
$967$	−463079.			−0.495225			−0.247612	+	0.968859i	$0.579646\pi$
$968$		0			0
$969$	−18603.2	−	10740.6i	−0.0198125	−	0.0114388i
$970$		0			0
$971$	−54723.1	+	31594.4i	−0.0580406	+	0.0335098i	−0.528739	−	0.848784i	$-0.677335\pi$
$971$	−54723.1	+	31594.4i	−0.0580406	+	0.0335098i	0.470699	+	0.882294i	$0.344002\pi$
$972$		0			0
$973$	−65946.8	−	236439.i	−0.0696575	−	0.249743i
$974$		0			0
$975$	13895.9	+	24068.4i	0.0146176	+	0.0253185i
$976$		0			0
$977$	−719497.	+	1.24621e6i	−0.753772	+	1.30557i	0.192210	+	0.981354i	$0.438434\pi$
$977$	−719497.	+	1.24621e6i	−0.753772	+	1.30557i	−0.945982	+	0.324218i	$0.894899\pi$
$978$		0			0
$979$			353121.i			0.368432i
$980$		0			0
$981$	396840.			0.412361
$982$		0			0
$983$	−346926.	−	200298.i	−0.359030	−	0.207286i	0.309625	−	0.950859i	$-0.399796\pi$
$983$	−346926.	−	200298.i	−0.359030	−	0.207286i	−0.668655	+	0.743573i	$0.733130\pi$
$984$		0			0
$985$	680877.	−	393104.i	0.701772	−	0.405168i
$986$		0			0
$987$	967521.	−	269858.i	0.993176	−	0.277014i
$988$		0			0
$989$	−1.72756e6	−	2.99223e6i	−1.76620	−	3.05916i
$990$		0			0
$991$	99274.5	−	171949.i	0.101086	−	0.175086i	−0.811046	−	0.584982i	$-0.801102\pi$
$991$	99274.5	−	171949.i	0.101086	−	0.175086i	0.912132	+	0.409896i	$0.134435\pi$
$992$		0			0
$993$			208983.i			0.211939i
$994$		0			0
$995$	−633858.			−0.640245
$996$		0			0
$997$	792228.	+	457393.i	0.797003	+	0.460150i	0.842422	−	0.538818i	$-0.181129\pi$
$997$	792228.	+	457393.i	0.797003	+	0.460150i	−0.0454194	+	0.998968i	$0.514462\pi$
$998$		0			0
$999$	−68311.6	+	39439.7i	−0.0684484	+	0.0395187i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.5.bh.d.241.1		4
4.3	odd	2		42.5.g.a.31.2	yes	4
7.5	odd	6	inner	336.5.bh.d.145.1		4
12.11	even	2		126.5.n.b.73.1		4
28.3	even	6		294.5.c.a.97.1		4
28.11	odd	6		294.5.c.a.97.2		4
28.19	even	6		42.5.g.a.19.2	✓	4
28.23	odd	6		294.5.g.c.19.2		4
28.27	even	2		294.5.g.c.31.2		4
84.11	even	6		882.5.c.a.685.4		4
84.47	odd	6		126.5.n.b.19.1		4
84.59	odd	6		882.5.c.a.685.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.5.g.a.19.2	✓	4	28.19	even	6
42.5.g.a.31.2	yes	4	4.3	odd	2
126.5.n.b.19.1		4	84.47	odd	6
126.5.n.b.73.1		4	12.11	even	2
294.5.c.a.97.1		4	28.3	even	6
294.5.c.a.97.2		4	28.11	odd	6
294.5.g.c.19.2		4	28.23	odd	6
294.5.g.c.31.2		4	28.27	even	2
336.5.bh.d.145.1		4	7.5	odd	6	inner
336.5.bh.d.241.1		4	1.1	even	1	trivial
882.5.c.a.685.3		4	84.59	odd	6
882.5.c.a.685.4		4	84.11	even	6

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	336.bh (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 + 2.59808i) q^{3} +(-20.7426 + 11.9758i) q^{5} +(47.1985 - 13.1645i) q^{7} +(13.5000 + 23.3827i) q^{9} +(48.9853 - 84.8450i) q^{11} -104.211i q^{13} -124.456 q^{15} +(93.2498 + 53.8378i) q^{17} +(-33.2498 + 19.1968i) q^{19} +(246.595 + 63.3852i) q^{21} +(-510.749 - 884.644i) q^{23} +(-25.6619 + 44.4477i) q^{25} +140.296i q^{27} +621.603 q^{29} +(1315.61 + 759.568i) q^{31} +(440.868 - 254.535i) q^{33} +(-821.367 + 838.304i) q^{35} +(281.118 + 486.910i) q^{37} +(270.749 - 468.952i) q^{39} -1023.20i q^{41} +3382.41 q^{43} +(-560.051 - 323.346i) q^{45} +(3416.50 - 1972.52i) q^{47} +(2054.39 - 1242.69i) q^{49} +(279.749 + 484.540i) q^{51} +(-1095.30 + 1897.12i) q^{53} +2346.55i q^{55} -199.499 q^{57} +(-2541.67 - 1467.44i) q^{59} +(576.207 - 332.673i) q^{61} +(945.000 + 925.907i) q^{63} +(1248.01 + 2161.62i) q^{65} +(-2962.69 + 5131.53i) q^{67} -5307.86i q^{69} +4494.41 q^{71} +(7767.32 + 4484.46i) q^{73} +(-230.957 + 133.343i) q^{75} +(1195.09 - 4649.42i) q^{77} +(-5223.41 - 9047.22i) q^{79} +(-364.500 + 631.333i) q^{81} -1269.28i q^{83} -2579.00 q^{85} +(2797.21 + 1614.97i) q^{87} +(-3121.46 + 1802.18i) q^{89} +(-1371.89 - 4918.62i) q^{91} +(3946.83 + 6836.11i) q^{93} +(459.792 - 796.383i) q^{95} -1950.53i q^{97} +2645.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 18 q^{3} - 66 q^{5} + 70 q^{7} + 54 q^{9} + 162 q^{11} - 396 q^{15} - 204 q^{17} + 444 q^{19} + 630 q^{21} - 312 q^{23} - 476 q^{25} + 2724 q^{29} + 3786 q^{31} + 1458 q^{33} - 672 q^{35} + 1396 q^{37}+ \cdots + 8748 q^{99}+O(q^{100}) \) 4 * q + 18 * q^3 - 66 * q^5 + 70 * q^7 + 54 * q^9 + 162 * q^11 - 396 * q^15 - 204 * q^17 + 444 * q^19 + 630 * q^21 - 312 * q^23 - 476 * q^25 + 2724 * q^29 + 3786 * q^31 + 1458 * q^33 - 672 * q^35 + 1396 * q^37 - 648 * q^39 + 632 * q^43 - 1782 * q^45 + 7896 * q^47 - 98 * q^49 - 612 * q^51 - 1038 * q^53 + 2664 * q^57 + 966 * q^59 + 5088 * q^61 + 3780 * q^63 - 744 * q^65 - 14600 * q^67 + 9696 * q^71 + 22584 * q^73 - 4284 * q^75 - 3654 * q^77 - 3974 * q^79 - 1458 * q^81 + 1224 * q^85 + 12258 * q^87 - 33156 * q^89 + 18984 * q^91 + 11358 * q^93 - 3252 * q^95 + 8748 * q^99

Introduction

L-functions

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