LMFDB - Embedded newform 336.4.bc.f.17.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,Mod(17,336)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("336.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.12
Character	$\chi$	$=$	336.17
Dual form			336.4.bc.f.257.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.328966 + 5.18573i) q^{3} +(-7.91382 - 13.7071i) q^{5} +(-13.3812 + 12.8040i) q^{7} +(-26.7836 + 3.41186i) q^{9} +O(q^{10})$	\(q+(0.328966 + 5.18573i) q^{3} +(-7.91382 - 13.7071i) q^{5} +(-13.3812 + 12.8040i) q^{7} +(-26.7836 + 3.41186i) q^{9} +(4.95924 + 2.86322i) q^{11} +50.0739i q^{13} +(68.4781 - 45.5481i) q^{15} +(39.4108 - 68.2615i) q^{17} +(81.4814 - 47.0433i) q^{19} +(-70.8002 - 65.1793i) q^{21} +(96.5537 - 55.7453i) q^{23} +(-62.7571 + 108.698i) q^{25} +(-26.5038 - 137.770i) q^{27} -237.853i q^{29} +(-77.9496 - 45.0042i) q^{31} +(-13.2165 + 26.6592i) q^{33} +(281.403 + 82.0894i) q^{35} +(-27.2956 - 47.2774i) q^{37} +(-259.670 + 16.4726i) q^{39} +206.436 q^{41} +507.500 q^{43} +(258.727 + 340.125i) q^{45} +(-53.6291 - 92.8884i) q^{47} +(15.1138 - 342.667i) q^{49} +(366.950 + 181.918i) q^{51} +(-373.754 - 215.787i) q^{53} -90.6360i q^{55} +(270.758 + 407.065i) q^{57} +(-212.369 + 367.833i) q^{59} +(397.053 - 229.239i) q^{61} +(314.711 - 388.592i) q^{63} +(686.370 - 396.276i) q^{65} +(-476.436 + 825.211i) q^{67} +(320.843 + 482.363i) q^{69} +7.43938i q^{71} +(812.839 + 469.293i) q^{73} +(-584.325 - 289.683i) q^{75} +(-103.021 + 25.1849i) q^{77} +(-454.661 - 787.496i) q^{79} +(705.718 - 182.763i) q^{81} +199.553 q^{83} -1247.56 q^{85} +(1233.44 - 78.2455i) q^{87} +(-571.305 - 989.530i) q^{89} +(-641.148 - 670.050i) q^{91} +(207.737 - 419.030i) q^{93} +(-1289.66 - 744.584i) q^{95} -1269.59i q^{97} +(-142.595 - 59.7670i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 12 q^{7} + 14 q^{9}+O(q^{10}) $ 48 * q - 12 * q^7 + 14 * q^9	$ 48 q - 12 q^{7} + 14 q^{9} + 88 q^{15} + 270 q^{19} + 50 q^{21} - 438 q^{25} - 216 q^{31} - 372 q^{33} + 66 q^{37} - 242 q^{39} - 900 q^{43} - 294 q^{45} + 60 q^{49} + 138 q^{51} + 1384 q^{57} + 108 q^{61} - 1096 q^{63} - 6 q^{67} - 1206 q^{73} + 594 q^{75} + 588 q^{79} - 54 q^{81} - 240 q^{85} + 3522 q^{87} - 234 q^{91} - 608 q^{93} - 1988 q^{99}+O(q^{100}) $ 48 * q - 12 * q^7 + 14 * q^9 + 88 * q^15 + 270 * q^19 + 50 * q^21 - 438 * q^25 - 216 * q^31 - 372 * q^33 + 66 * q^37 - 242 * q^39 - 900 * q^43 - 294 * q^45 + 60 * q^49 + 138 * q^51 + 1384 * q^57 + 108 * q^61 - 1096 * q^63 - 6 * q^67 - 1206 * q^73 + 594 * q^75 + 588 * q^79 - 54 * q^81 - 240 * q^85 + 3522 * q^87 - 234 * q^91 - 608 * q^93 - 1988 * q^99

Display 100 coefficients 10 coefficients

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
$3$	0.328966	+	5.18573i	0.0633095	+	0.997994i
$3$	0.328966	+	5.18573i	0.0633095	+	0.997994i
$4$		0			0
$5$	−7.91382	−	13.7071i	−0.707833	−	1.22600i	−0.965659	−	0.259812i	$-0.916339\pi$
$5$	−7.91382	−	13.7071i	−0.707833	−	1.22600i	0.257826	−	0.966191i	$-0.416994\pi$
$6$		0			0
$7$	−13.3812	+	12.8040i	−0.722518	+	0.691352i
$7$	−13.3812	+	12.8040i	−0.722518	+	0.691352i
$8$		0			0
$9$	−26.7836	+	3.41186i	−0.991984	+	0.126365i
$10$		0			0
$11$	4.95924	+	2.86322i	0.135933	+	0.0784812i	0.566424	−	0.824114i	$-0.308326\pi$
$11$	4.95924	+	2.86322i	0.135933	+	0.0784812i	−0.430491	+	0.902595i	$0.641660\pi$
$12$		0			0
$13$			50.0739i			1.06831i	0.845387	+	0.534154i	$0.179370\pi$
$13$			50.0739i			1.06831i	−0.845387	+	0.534154i	$0.820630\pi$
$14$		0			0
$15$	68.4781	−	45.5481i	1.17873	−	0.784031i
$16$		0			0
$17$	39.4108	−	68.2615i	0.562266	−	0.973873i	−0.435032	−	0.900415i	$-0.643263\pi$
$17$	39.4108	−	68.2615i	0.562266	−	0.973873i	0.997298	−	0.0734583i	$-0.0234036\pi$
$18$		0			0
$19$	81.4814	−	47.0433i	0.983848	−	0.568025i	0.0804184	−	0.996761i	$-0.474374\pi$
$19$	81.4814	−	47.0433i	0.983848	−	0.568025i	0.903430	+	0.428736i	$0.141041\pi$
$20$		0			0
$21$	−70.8002	−	65.1793i	−0.735708	−	0.677299i
$22$		0			0
$23$	96.5537	−	55.7453i	0.875341	−	0.505378i	0.00622127	−	0.999981i	$-0.498020\pi$
$23$	96.5537	−	55.7453i	0.875341	−	0.505378i	0.869119	+	0.494603i	$0.164686\pi$
$24$		0			0
$25$	−62.7571	+	108.698i	−0.502057	+	0.869587i
$26$		0			0
$27$	−26.5038	−	137.770i	−0.188914	−	0.981994i
$28$		0			0
$29$		−	237.853i		−	1.52304i	−0.648141	−	0.761520i	$-0.724453\pi$
$29$		−	237.853i		−	1.52304i	0.648141	−	0.761520i	$-0.275547\pi$
$30$		0			0
$31$	−77.9496	−	45.0042i	−0.451618	−	0.260742i	0.256895	−	0.966439i	$-0.417300\pi$
$31$	−77.9496	−	45.0042i	−0.451618	−	0.260742i	−0.708513	+	0.705697i	$0.750634\pi$
$32$		0			0
$33$	−13.2165	+	26.6592i	−0.0697179	+	0.140629i
$34$		0			0
$35$	281.403	+	82.0894i	1.35902	+	0.396447i
$36$		0			0
$37$	−27.2956	−	47.2774i	−0.121280	−	0.210064i	0.798992	−	0.601341i	$-0.205367\pi$
$37$	−27.2956	−	47.2774i	−0.121280	−	0.210064i	−0.920273	+	0.391277i	$0.872033\pi$
$38$		0			0
$39$	−259.670	+	16.4726i	−1.06617	+	0.0676341i
$40$		0			0
$41$	206.436			0.786337			0.393169	−	0.919466i	$-0.371379\pi$
$41$	206.436			0.786337			0.393169	+	0.919466i	$0.371379\pi$
$42$		0			0
$43$	507.500			1.79984			0.899919	−	0.436056i	$-0.143625\pi$
$43$	507.500			1.79984			0.899919	+	0.436056i	$0.143625\pi$
$44$		0			0
$45$	258.727	+	340.125i	0.857083	+	1.12673i
$46$		0			0
$47$	−53.6291	−	92.8884i	−0.166439	−	0.288280i	0.770727	−	0.637166i	$-0.219893\pi$
$47$	−53.6291	−	92.8884i	−0.166439	−	0.288280i	−0.937165	+	0.348886i	$0.886560\pi$
$48$		0			0
$49$	15.1138	−	342.667i	0.0440637	−	0.999029i
$50$		0			0
$51$	366.950	+	181.918i	1.00752	+	0.499483i
$52$		0			0
$53$	−373.754	−	215.787i	−0.968661	−	0.559256i	−0.0698329	−	0.997559i	$-0.522247\pi$
$53$	−373.754	−	215.787i	−0.968661	−	0.559256i	−0.898828	+	0.438302i	$0.855580\pi$
$54$		0			0
$55$		−	90.6360i		−	0.222206i
$56$		0			0
$57$	270.758	+	407.065i	0.629172	+	0.945913i
$58$		0			0
$59$	−212.369	+	367.833i	−0.468611	+	0.811658i	−0.999356	−	0.0358736i	$-0.988579\pi$
$59$	−212.369	+	367.833i	−0.468611	+	0.811658i	0.530746	+	0.847531i	$0.321912\pi$
$60$		0			0
$61$	397.053	−	229.239i	0.833401	−	0.481164i	−0.0216149	−	0.999766i	$-0.506881\pi$
$61$	397.053	−	229.239i	0.833401	−	0.481164i	0.855016	+	0.518602i	$0.173547\pi$
$62$		0			0
$63$	314.711	−	388.592i	0.629363	−	0.777111i
$64$		0			0
$65$	686.370	−	396.276i	1.30975	−	0.756184i
$66$		0			0
$67$	−476.436	+	825.211i	−0.868745	+	1.50471i	−0.00546483	+	0.999985i	$0.501740\pi$
$67$	−476.436	+	825.211i	−0.868745	+	1.50471i	−0.863280	+	0.504725i	$0.831594\pi$
$68$		0			0
$69$	320.843	+	482.363i	0.559782	+	0.841589i
$70$		0			0
$71$			7.43938i			0.0124351i	0.999981	+	0.00621755i	$0.00197912\pi$
$71$			7.43938i			0.0124351i	−0.999981	+	0.00621755i	$0.998021\pi$
$72$		0			0
$73$	812.839	+	469.293i	1.30323	+	0.752419i	0.980956	−	0.194229i	$-0.0622204\pi$
$73$	812.839	+	469.293i	1.30323	+	0.752419i	0.322271	+	0.946647i	$0.395554\pi$
$74$		0			0
$75$	−584.325	−	289.683i	−0.899628	−	0.445996i
$76$		0			0
$77$	−103.021	+	25.1849i	−0.152472	+	0.0372738i
$78$		0			0
$79$	−454.661	−	787.496i	−0.647511	−	1.12152i	−0.983716	−	0.179732i	$-0.942477\pi$
$79$	−454.661	−	787.496i	−0.647511	−	1.12152i	0.336205	−	0.941789i	$-0.390856\pi$
$80$		0			0
$81$	705.718	−	182.763i	0.968064	−	0.250704i
$82$		0			0
$83$	199.553			0.263901			0.131950	−	0.991256i	$-0.457876\pi$
$83$	199.553			0.263901			0.131950	+	0.991256i	$0.457876\pi$
$84$		0			0
$85$	−1247.56			−1.59196
$86$		0			0
$87$	1233.44	−	78.2455i	1.51999	−	0.0964230i
$88$		0			0
$89$	−571.305	−	989.530i	−0.680430	−	1.17854i	−0.974850	−	0.222863i	$-0.928460\pi$
$89$	−571.305	−	989.530i	−0.680430	−	1.17854i	0.294420	−	0.955676i	$-0.404874\pi$
$90$		0			0
$91$	−641.148	−	670.050i	−0.738577	−	0.771872i
$92$		0			0
$93$	207.737	−	419.030i	0.231627	−	0.467219i
$94$		0			0
$95$	−1289.66	−	744.584i	−1.39280	−	0.804134i
$96$		0			0
$97$		−	1269.59i		−	1.32895i	−0.747313	−	0.664473i	$-0.768656\pi$
$97$		−	1269.59i		−	1.32895i	0.747313	−	0.664473i	$-0.231344\pi$
$98$		0			0
$99$	−142.595	−	59.7670i	−0.144761	−	0.0606748i
$100$		0			0
$101$	−738.334	+	1278.83i	−0.727396	+	1.25989i	0.230585	+	0.973052i	$0.425936\pi$
$101$	−738.334	+	1278.83i	−0.727396	+	1.25989i	−0.957980	+	0.286834i	$0.907397\pi$
$102$		0			0
$103$	−158.164	+	91.3159i	−0.151304	+	0.0873556i	−0.573741	−	0.819037i	$-0.694508\pi$
$103$	−158.164	+	91.3159i	−0.151304	+	0.0873556i	0.422436	+	0.906393i	$0.361175\pi$
$104$		0			0
$105$	−333.121	+	1486.28i	−0.309613	+	1.38140i
$106$		0			0
$107$	146.375	−	84.5097i	0.132249	−	0.0763538i	−0.432416	−	0.901674i	$-0.642339\pi$
$107$	146.375	−	84.5097i	0.132249	−	0.0763538i	0.564665	+	0.825320i	$0.309005\pi$
$108$		0			0
$109$	336.455	−	582.757i	0.295656	−	0.512092i	−0.679481	−	0.733693i	$-0.737795\pi$
$109$	336.455	−	582.757i	0.295656	−	0.512092i	0.975137	+	0.221601i	$0.0711283\pi$
$110$		0			0
$111$	236.189	−	157.100i	0.201964	−	0.134336i
$112$		0			0
$113$		−	987.845i		−	0.822378i	−0.911550	−	0.411189i	$-0.865114\pi$
$113$		−	987.845i		−	0.822378i	0.911550	−	0.411189i	$-0.134886\pi$
$114$		0			0
$115$	−1528.22	−	882.316i	−1.23919	−	0.715447i
$116$		0			0
$117$	−170.845	−	1341.16i	−0.134997	−	1.05974i
$118$		0			0
$119$	346.658	+	1418.04i	0.267042	+	1.09236i
$120$		0			0
$121$	−649.104	−	1124.28i	−0.487681	−	0.844689i
$122$		0			0
$123$	67.9103	+	1070.52i	0.0497827	+	0.784760i
$124$		0			0
$125$	8.13742			0.00582267
$126$		0			0
$127$	1858.55			1.29858			0.649291	−	0.760540i	$-0.275066\pi$
$127$	1858.55			1.29858			0.649291	+	0.760540i	$0.275066\pi$
$128$		0			0
$129$	166.950	+	2631.76i	0.113947	+	1.79623i
$130$		0			0
$131$	1075.06	+	1862.06i	0.717010	+	1.24190i	0.962179	+	0.272418i	$0.0878233\pi$
$131$	1075.06	+	1862.06i	0.717010	+	1.24190i	−0.245169	+	0.969480i	$0.578843\pi$
$132$		0			0
$133$	−487.976	+	1672.79i	−0.318142	+	1.09059i
$134$		0			0
$135$	−1678.68	+	1453.58i	−1.07021	+	0.926697i
$136$		0			0
$137$	45.8015	+	26.4435i	0.0285627	+	0.0164907i	0.514213	−	0.857662i	$-0.328084\pi$
$137$	45.8015	+	26.4435i	0.0285627	+	0.0164907i	−0.485651	+	0.874153i	$0.661417\pi$
$138$		0			0
$139$			1810.71i			1.10491i	0.833542	+	0.552456i	$0.186309\pi$
$139$			1810.71i			1.10491i	−0.833542	+	0.552456i	$0.813691\pi$
$140$		0			0
$141$	464.052	−	308.663i	0.277165	−	0.184356i
$142$		0			0
$143$	−143.373	+	248.329i	−0.0838421	+	0.145219i
$144$		0			0
$145$	−3260.28	+	1882.32i	−1.86725	+	1.07806i
$146$		0			0
$147$	1781.95	−	34.3494i	0.999814	−	0.0192727i
$148$		0			0
$149$	−440.769	+	254.478i	−0.242343	+	0.139917i	−0.616253	−	0.787548i	$-0.711350\pi$
$149$	−440.769	+	254.478i	−0.242343	+	0.139917i	0.373910	+	0.927465i	$0.378017\pi$
$150$		0			0
$151$	996.115	−	1725.32i	0.536839	−	0.929832i	−0.462233	−	0.886758i	$-0.652952\pi$
$151$	996.115	−	1725.32i	0.536839	−	0.929832i	0.999072	−	0.0430736i	$-0.0137150\pi$
$152$		0			0
$153$	−822.663	+	1962.75i	−0.434695	+	1.03712i
$154$		0			0
$155$			1424.62i			0.738247i
$156$		0			0
$157$	−1756.53	−	1014.14i	−0.892909	−	0.515521i	−0.0180159	−	0.999838i	$-0.505735\pi$
$157$	−1756.53	−	1014.14i	−0.892909	−	0.515521i	−0.874893	+	0.484317i	$0.839068\pi$
$158$		0			0
$159$	996.059	−	2009.17i	0.496809	−	1.00212i
$160$		0			0
$161$	−578.241	+	1982.22i	−0.283055	+	0.970313i
$162$		0			0
$163$	−806.781	−	1397.39i	−0.387681	−	0.671483i	0.604456	−	0.796638i	$-0.293390\pi$
$163$	−806.781	−	1397.39i	−0.387681	−	0.671483i	−0.992137	+	0.125155i	$0.960057\pi$
$164$		0			0
$165$	470.014	−	29.8162i	0.221761	−	0.0140678i
$166$		0			0
$167$	−942.060			−0.436520			−0.218260	−	0.975891i	$-0.570038\pi$
$167$	−942.060			−0.436520			−0.218260	+	0.975891i	$0.570038\pi$
$168$		0			0
$169$	−310.397			−0.141282
$170$		0			0
$171$	−2021.86	+	1537.99i	−0.904183	+	0.687796i
$172$		0			0
$173$	2151.67	+	3726.81i	0.945599	+	1.63783i	0.754548	+	0.656245i	$0.227856\pi$
$173$	2151.67	+	3726.81i	0.945599	+	1.63783i	0.191051	+	0.981580i	$0.438810\pi$
$174$		0			0
$175$	−552.012	−	2258.06i	−0.238447	−	0.975390i
$176$		0			0
$177$	−1977.35	−	980.281i	−0.839697	−	0.416285i
$178$		0			0
$179$	−981.393	−	566.608i	−0.409792	−	0.236594i	0.280908	−	0.959735i	$-0.409364\pi$
$179$	−981.393	−	566.608i	−0.409792	−	0.236594i	−0.690700	+	0.723141i	$0.742698\pi$
$180$		0			0
$181$		−	428.901i		−	0.176132i	−0.996115	−	0.0880661i	$-0.971931\pi$
$181$		−	428.901i		−	0.176132i	0.996115	−	0.0880661i	$-0.0280687\pi$
$182$		0			0
$183$	1319.39	+	1983.60i	0.532961	+	0.801267i
$184$		0			0
$185$	−432.025	+	748.290i	−0.171693	+	0.297380i
$186$		0			0
$187$	390.895	−	225.683i	0.152861	−	0.0882546i
$188$		0			0
$189$	2118.66	+	1504.17i	0.815397	+	0.578902i
$190$		0			0
$191$	2270.37	−	1310.80i	0.860095	−	0.496576i	−0.00394904	−	0.999992i	$-0.501257\pi$
$191$	2270.37	−	1310.80i	0.860095	−	0.496576i	0.864044	+	0.503416i	$0.167924\pi$
$192$		0			0
$193$	2369.95	−	4104.87i	0.883900	−	1.53096i	0.0369303	−	0.999318i	$-0.488242\pi$
$193$	2369.95	−	4104.87i	0.883900	−	1.53096i	0.846970	−	0.531641i	$-0.178425\pi$
$194$		0			0
$195$	2280.77	+	3428.97i	0.837587	+	1.25925i
$196$		0			0
$197$		−	3177.96i		−	1.14934i	−0.818385	−	0.574670i	$-0.805130\pi$
$197$		−	3177.96i		−	1.14934i	0.818385	−	0.574670i	$-0.194870\pi$
$198$		0			0
$199$	−4411.79	−	2547.15i	−1.57157	−	0.907349i	−0.995976	−	0.0896166i	$-0.971436\pi$
$199$	−4411.79	−	2547.15i	−1.57157	−	0.907349i	−0.575598	−	0.817733i	$-0.695231\pi$
$200$		0			0
$201$	−4436.05	−	2199.20i	−1.55669	−	0.771740i
$202$		0			0
$203$	3045.47	+	3182.76i	1.05296	+	1.10042i
$204$		0			0
$205$	−1633.69	−	2829.64i	−0.556596	−	0.964052i
$206$		0			0
$207$	−2395.86	+	1822.48i	−0.804462	+	0.611939i
$208$		0			0
$209$	538.781			0.178317
$210$		0			0
$211$	−2882.42			−0.940446			−0.470223	−	0.882548i	$-0.655827\pi$
$211$	−2882.42			−0.940446			−0.470223	+	0.882548i	$0.655827\pi$
$212$		0			0
$213$	−38.5786	+	2.44730i	−0.0124101	+	0.000787260i
$214$		0			0
$215$	−4016.27	−	6956.38i	−1.27399	−	2.20661i
$216$		0			0
$217$	1619.30	−	395.857i	0.506566	−	0.123837i
$218$		0			0
$219$	−2166.23	+	4369.54i	−0.668403	+	1.34825i
$220$		0			0
$221$	3418.12	+	1973.45i	1.04040	+	0.600673i
$222$		0			0
$223$			1677.02i			0.503596i	0.967780	+	0.251798i	$0.0810218\pi$
$223$			1677.02i			0.503596i	−0.967780	+	0.251798i	$0.918978\pi$
$224$		0			0
$225$	1309.99	−	3125.45i	0.388146	−	0.926059i
$226$		0			0
$227$	1985.74	−	3439.40i	0.580609	−	1.00564i	−0.414798	−	0.909913i	$-0.636148\pi$
$227$	1985.74	−	3439.40i	0.580609	−	1.00564i	0.995407	−	0.0957309i	$-0.0305188\pi$
$228$		0			0
$229$	870.768	−	502.738i	0.251275	−	0.145074i	−0.369073	−	0.929400i	$-0.620325\pi$
$229$	870.768	−	502.738i	0.251275	−	0.145074i	0.620348	+	0.784327i	$0.286991\pi$
$230$		0			0
$231$	−164.493	−	525.956i	−0.0468520	−	0.149807i
$232$		0			0
$233$	−533.387	+	307.951i	−0.149971	+	0.0865860i	−0.573108	−	0.819480i	$-0.694262\pi$
$233$	−533.387	+	307.951i	−0.149971	+	0.0865860i	0.423136	+	0.906066i	$0.360929\pi$
$234$		0			0
$235$	−848.823	+	1470.20i	−0.235622	+	0.408109i
$236$		0			0
$237$	3934.17	−	2616.81i	1.07828	−	0.717215i
$238$		0			0
$239$		−	4741.91i		−	1.28338i	−0.766963	−	0.641692i	$-0.778233\pi$
$239$		−	4741.91i		−	1.28338i	0.766963	−	0.641692i	$-0.221767\pi$
$240$		0			0
$241$	2931.82	+	1692.69i	0.783632	+	0.452430i	0.837716	−	0.546106i	$-0.183891\pi$
$241$	2931.82	+	1692.69i	0.783632	+	0.452430i	−0.0540841	+	0.998536i	$0.517224\pi$
$242$		0			0
$243$	1179.92	+	3599.54i	0.311489	+	0.950250i
$244$		0			0
$245$	−4816.59	+	2504.64i	−1.25600	+	0.653124i
$246$		0			0
$247$	2355.64	+	4080.09i	0.606826	+	1.05105i
$248$		0			0
$249$	65.6461	+	1034.83i	0.0167074	+	0.263371i
$250$		0			0
$251$	−1531.85			−0.385217			−0.192608	−	0.981276i	$-0.561695\pi$
$251$	−1531.85			−0.385217			−0.192608	+	0.981276i	$0.561695\pi$
$252$		0			0
$253$	638.444			0.158651
$254$		0			0
$255$	−410.405	−	6469.50i	−0.100786	−	1.58877i
$256$		0			0
$257$	1124.78	+	1948.17i	0.273003	+	0.472855i	0.969629	−	0.244579i	$-0.0786498\pi$
$257$	1124.78	+	1948.17i	0.273003	+	0.472855i	−0.696626	+	0.717434i	$0.745316\pi$
$258$		0			0
$259$	970.590	+	283.135i	0.232855	+	0.0679273i
$260$		0			0
$261$	811.520	+	6370.55i	0.192459	+	1.51083i
$262$		0			0
$263$	−5671.03	−	3274.17i	−1.32962	−	0.767658i	−0.344381	−	0.938830i	$-0.611911\pi$
$263$	−5671.03	−	3274.17i	−1.32962	−	0.767658i	−0.985241	+	0.171172i	$0.945245\pi$
$264$		0			0
$265$			6830.79i			1.58344i
$266$		0			0
$267$	4943.49	−	3288.16i	1.13310	−	0.753678i
$268$		0			0
$269$	2302.17	−	3987.48i	0.521806	−	0.903795i	−0.477872	−	0.878430i	$-0.658592\pi$
$269$	2302.17	−	3987.48i	0.521806	−	0.903795i	0.999678	−	0.0253656i	$-0.00807500\pi$
$270$		0			0
$271$	933.172	−	538.767i	0.209174	−	0.120767i	−0.391753	−	0.920070i	$-0.628131\pi$
$271$	933.172	−	538.767i	0.209174	−	0.120767i	0.600927	+	0.799304i	$0.294798\pi$
$272$		0			0
$273$	3263.78	−	3545.24i	0.723564	−	0.785963i
$274$		0			0
$275$	−622.455	+	359.374i	−0.136492	+	0.0788040i
$276$		0			0
$277$	2371.27	−	4107.16i	0.514353	−	0.890885i	−0.485508	−	0.874232i	$-0.661365\pi$
$277$	2371.27	−	4107.16i	0.514353	−	0.890885i	0.999861	−	0.0166534i	$-0.00530119\pi$
$278$		0			0
$279$	2241.32	+	939.420i	0.480946	+	0.201583i
$280$		0			0
$281$		−	1443.99i		−	0.306553i	−0.988183	−	0.153277i	$-0.951017\pi$
$281$		−	1443.99i		−	0.306553i	0.988183	−	0.153277i	$-0.0489826\pi$
$282$		0			0
$283$	16.3717	+	9.45219i	0.00343885	+	0.00198542i	0.501718	−	0.865031i	$-0.332701\pi$
$283$	16.3717	+	9.45219i	0.00343885	+	0.00198542i	−0.498280	+	0.867016i	$0.666035\pi$
$284$		0			0
$285$	3436.96	−	6932.76i	0.714343	−	1.44092i
$286$		0			0
$287$	−2762.36	+	2643.21i	−0.568143	+	0.543636i
$288$		0			0
$289$	−649.921	−	1125.70i	−0.132286	−	0.229126i
$290$		0			0
$291$	6583.77	−	417.653i	1.32628	−	0.0841349i
$292$		0			0
$293$	6714.84			1.33886			0.669428	−	0.742877i	$-0.266539\pi$
$293$	6714.84			1.33886			0.669428	+	0.742877i	$0.266539\pi$
$294$		0			0
$295$	6722.59			1.32679
$296$		0			0
$297$	263.026	−	759.120i	0.0513884	−	0.148312i
$298$		0			0
$299$	2791.39	+	4834.82i	0.539900	+	0.935133i
$300$		0			0
$301$	−6790.97	+	6498.05i	−1.30042	+	1.24432i
$302$		0			0
$303$	−6874.56	−	3408.11i	−1.30341	−	0.646174i
$304$		0			0
$305$	−6284.41	−	3628.31i	−1.17982	−	0.681168i
$306$		0			0
$307$			2266.12i			0.421285i	0.977563	+	0.210643i	$0.0675556\pi$
$307$			2266.12i			0.421285i	−0.977563	+	0.210643i	$0.932444\pi$
$308$		0			0
$309$	−525.570	−	790.155i	−0.0967593	−	0.145470i
$310$		0			0
$311$	−785.496	+	1360.52i	−0.143220	+	0.248064i	−0.928707	−	0.370813i	$-0.879079\pi$
$311$	−785.496	+	1360.52i	−0.143220	+	0.248064i	0.785487	+	0.618878i	$0.212412\pi$
$312$		0			0
$313$	5193.97	−	2998.74i	0.937958	−	0.541530i	0.0486382	−	0.998816i	$-0.484512\pi$
$313$	5193.97	−	2998.74i	0.937958	−	0.541530i	0.889320	+	0.457286i	$0.151179\pi$
$314$		0			0
$315$	−7817.05	−	1238.54i	−1.39823	−	0.221536i
$316$		0			0
$317$	−6213.65	+	3587.46i	−1.10093	+	0.635620i	−0.936464	−	0.350763i	$-0.885922\pi$
$317$	−6213.65	+	3587.46i	−1.10093	+	0.635620i	−0.164462	+	0.986383i	$0.552589\pi$
$318$		0			0
$319$	681.025	−	1179.57i	0.119530	−	0.207032i
$320$		0			0
$321$	486.397	+	731.260i	0.0845732	+	0.127149i
$322$		0			0
$323$		−	7416.06i		−	1.27752i
$324$		0			0
$325$	−5442.96	−	3142.49i	−0.928987	−	0.536351i
$326$		0			0
$327$	3132.70	+	1553.06i	0.529783	+	0.262643i
$328$		0			0
$329$	1906.97	+	556.291i	0.319558	+	0.0932198i
$330$		0			0
$331$	−1560.28	−	2702.49i	−0.259097	−	0.448768i	0.706904	−	0.707310i	$-0.250091\pi$
$331$	−1560.28	−	2702.49i	−0.259097	−	0.448768i	−0.966000	+	0.258542i	$0.916758\pi$
$332$		0			0
$333$	892.378	+	1173.13i	0.146853	+	0.193054i
$334$		0			0
$335$	15081.7			2.45971
$336$		0			0
$337$	3589.43			0.580203			0.290102	−	0.956996i	$-0.406311\pi$
$337$	3589.43			0.580203			0.290102	+	0.956996i	$0.406311\pi$
$338$		0			0
$339$	5122.70	−	324.968i	0.820728	−	0.0520644i
$340$		0			0
$341$	−257.714	−	446.373i	−0.0409266	−	0.0708870i
$342$		0			0
$343$	4185.27	+	4778.82i	0.658844	+	0.752279i
$344$		0			0
$345$	4072.72	−	8215.17i	0.635559	−	1.28200i
$346$		0			0
$347$	−664.055	−	383.393i	−0.102733	−	0.0593129i	0.447753	−	0.894157i	$-0.352224\pi$
$347$	−664.055	−	383.393i	−0.102733	−	0.0593129i	−0.550486	+	0.834844i	$0.685558\pi$
$348$		0			0
$349$			10054.2i			1.54209i	0.636778	+	0.771047i	$0.280267\pi$
$349$			10054.2i			1.54209i	−0.636778	+	0.771047i	$0.719733\pi$
$350$		0			0
$351$	6898.68	−	1327.15i	1.04907	−	0.201818i
$352$		0			0
$353$	−1041.77	+	1804.40i	−0.157076	+	0.272063i	−0.933813	−	0.357762i	$-0.883540\pi$
$353$	−1041.77	+	1804.40i	−0.157076	+	0.272063i	0.776737	+	0.629825i	$0.216873\pi$
$354$		0			0
$355$	101.973	−	58.8739i	0.0152455	−	0.00880198i
$356$		0			0
$357$	−7239.52	+	2264.16i	−1.07327	+	0.335664i
$358$		0			0
$359$	−5724.96	+	3305.31i	−0.841648	+	0.485926i	−0.857824	−	0.513943i	$-0.828184\pi$
$359$	−5724.96	+	3305.31i	−0.841648	+	0.485926i	0.0161758	+	0.999869i	$0.494851\pi$
$360$		0			0
$361$	996.645	−	1726.24i	0.145305	−	0.251675i
$362$		0			0
$363$	5616.68	−	3735.93i	0.812120	−	0.540180i
$364$		0			0
$365$		−	14855.6i		−	2.13035i
$366$		0			0
$367$	5556.00	+	3207.76i	0.790247	+	0.456250i	0.840050	−	0.542510i	$-0.182526\pi$
$367$	5556.00	+	3207.76i	0.790247	+	0.456250i	−0.0498022	+	0.998759i	$0.515859\pi$
$368$		0			0
$369$	−5529.08	+	704.329i	−0.780034	+	0.0993656i
$370$		0			0
$371$	7764.22	−	1898.06i	1.08652	−	0.265613i
$372$		0			0
$373$	2924.35	+	5065.13i	0.405944	+	0.703116i	0.994431	−	0.105391i	$-0.0336094\pi$
$373$	2924.35	+	5065.13i	0.405944	+	0.703116i	−0.588487	+	0.808507i	$0.700276\pi$
$374$		0			0
$375$	2.67694	+	42.1985i	0.000368630	+	0.00581099i
$376$		0			0
$377$	11910.2			1.62708
$378$		0			0
$379$	1147.80			0.155564			0.0777818	−	0.996970i	$-0.475216\pi$
$379$	1147.80			0.155564			0.0777818	+	0.996970i	$0.475216\pi$
$380$		0			0
$381$	611.401	+	9637.96i	0.0822127	+	1.29598i
$382$		0			0
$383$	2393.80	+	4146.19i	0.319367	+	0.553160i	0.980356	−	0.197235i	$-0.0631963\pi$
$383$	2393.80	+	4146.19i	0.319367	+	0.553160i	−0.660989	+	0.750396i	$0.729863\pi$
$384$		0			0
$385$	1160.51	+	1212.82i	0.153623	+	0.160548i
$386$		0			0
$387$	−13592.7	+	1731.52i	−1.78541	+	0.227437i
$388$		0			0
$389$	−5798.58	−	3347.81i	−0.755783	−	0.436351i	0.0719967	−	0.997405i	$-0.477063\pi$
$389$	−5798.58	−	3347.81i	−0.755783	−	0.436351i	−0.827780	+	0.561053i	$0.810396\pi$
$390$		0			0
$391$		−	8787.86i		−	1.13663i
$392$		0			0
$393$	−9302.46	+	6187.52i	−1.19401	+	0.794196i
$394$		0			0
$395$	−7196.21	+	12464.2i	−0.916659	+	1.58770i
$396$		0			0
$397$	1099.99	−	635.082i	0.139061	−	0.0802868i	−0.428855	−	0.903373i	$-0.641083\pi$
$397$	1099.99	−	635.082i	0.139061	−	0.0802868i	0.567916	+	0.823086i	$0.307750\pi$
$398$		0			0
$399$	−8835.14	−	1980.22i	−1.10855	−	0.248459i
$400$		0			0
$401$	−8683.23	+	5013.27i	−1.08135	+	0.624316i	−0.931259	−	0.364357i	$-0.881289\pi$
$401$	−8683.23	+	5013.27i	−1.08135	+	0.624316i	−0.150087	+	0.988673i	$0.547955\pi$
$402$		0			0
$403$	2253.54	−	3903.24i	0.278553	−	0.482467i
$404$		0			0
$405$	−8090.09	−	8227.02i	−0.992592	−	1.00939i
$406$		0			0
$407$		−	312.614i		−	0.0380729i
$408$		0			0
$409$	4280.55	+	2471.38i	0.517505	+	0.298782i	0.735913	−	0.677076i	$-0.236753\pi$
$409$	4280.55	+	2471.38i	0.517505	+	0.298782i	−0.218408	+	0.975858i	$0.570086\pi$
$410$		0			0
$411$	−122.062	+	246.213i	−0.0146493	+	0.0295494i
$412$		0			0
$413$	−1868.00	−	7641.23i	−0.222562	−	0.910412i
$414$		0			0
$415$	−1579.22	−	2735.30i	−0.186798	−	0.323543i
$416$		0			0
$417$	−9389.87	+	595.663i	−1.10270	+	0.0699515i
$418$		0			0
$419$	−9465.14			−1.10359			−0.551793	−	0.833981i	$-0.686056\pi$
$419$	−9465.14			−1.10359			−0.551793	+	0.833981i	$0.686056\pi$
$420$		0			0
$421$	5592.78			0.647448			0.323724	−	0.946152i	$-0.395065\pi$
$421$	5592.78			0.647448			0.323724	+	0.946152i	$0.395065\pi$
$422$		0			0
$423$	1753.30	+	2304.91i	0.201533	+	0.264937i
$424$		0			0
$425$	4946.61	+	8567.78i	0.564579	+	0.977879i
$426$		0			0
$427$	−2377.87	+	8151.37i	−0.269493	+	0.923823i
$428$		0			0
$429$	−1334.93	−	661.800i	−0.150235	−	0.0744802i
$430$		0			0
$431$	4799.54	+	2771.02i	0.536394	+	0.309687i	0.743616	−	0.668607i	$-0.233109\pi$
$431$	4799.54	+	2771.02i	0.536394	+	0.309687i	−0.207222	+	0.978294i	$0.566442\pi$
$432$		0			0
$433$		−	988.719i		−	0.109734i	−0.998494	−	0.0548670i	$-0.982527\pi$
$433$		−	988.719i		−	0.109734i	0.998494	−	0.0548670i	$-0.0174735\pi$
$434$		0			0
$435$	−10833.7	−	16287.7i	−1.19411	−	1.79526i
$436$		0			0
$437$	5244.89	−	9084.41i	0.574135	−	0.994431i
$438$		0			0
$439$	−7466.39	+	4310.72i	−0.811734	+	0.468655i	−0.847558	−	0.530703i	$-0.821928\pi$
$439$	−7466.39	+	4310.72i	−0.811734	+	0.468655i	0.0358237	+	0.999358i	$0.488595\pi$
$440$		0			0
$441$	764.328	+	9229.41i	0.0825319	+	0.996588i
$442$		0			0
$443$	8616.20	−	4974.56i	0.924081	−	0.533518i	0.0391463	−	0.999233i	$-0.487536\pi$
$443$	8616.20	−	4974.56i	0.924081	−	0.533518i	0.884935	+	0.465715i	$0.154203\pi$
$444$		0			0
$445$	−9042.42	+	15661.9i	−0.963262	+	1.66842i
$446$		0			0
$447$	−1464.65	−	2201.99i	−0.154979	−	0.232999i
$448$		0			0
$449$			2628.52i			0.276275i	0.990413	+	0.138137i	$0.0441116\pi$
$449$			2628.52i			0.276275i	−0.990413	+	0.138137i	$0.955888\pi$
$450$		0			0
$451$	1023.76	+	591.070i	0.106889	+	0.0617127i
$452$		0			0
$453$	9274.73	+	4598.01i	0.961954	+	0.476895i
$454$		0			0
$455$	−4110.54	+	14091.0i	−0.423527	+	1.45186i
$456$		0			0
$457$	−4366.94	−	7563.76i	−0.446995	−	0.774218i	0.551194	−	0.834377i	$-0.314172\pi$
$457$	−4366.94	−	7563.76i	−0.446995	−	0.774218i	−0.998189	+	0.0601593i	$0.980839\pi$
$458$		0			0
$459$	−10448.9	−	3620.43i	−1.06256	−	0.368164i
$460$		0			0
$461$	7852.73			0.793358			0.396679	−	0.917957i	$-0.370163\pi$
$461$	7852.73			0.793358			0.396679	+	0.917957i	$0.370163\pi$
$462$		0			0
$463$	−6173.31			−0.619650			−0.309825	−	0.950794i	$-0.600271\pi$
$463$	−6173.31			−0.619650			−0.309825	+	0.950794i	$0.600271\pi$
$464$		0			0
$465$	−7387.70	+	468.652i	−0.736766	+	0.0467381i
$466$		0			0
$467$	−5152.27	−	8923.99i	−0.510532	−	0.884267i	−0.999926	−	0.0122043i	$-0.996115\pi$
$467$	−5152.27	−	8923.99i	−0.510532	−	0.884267i	0.489394	−	0.872063i	$-0.337218\pi$
$468$		0			0
$469$	−4190.73	−	17142.6i	−0.412602	−	1.68779i
$470$		0			0
$471$	4681.19	−	9442.52i	0.457957	−	0.923755i
$472$		0			0
$473$	2516.82	+	1453.08i	0.244658	+	0.141253i
$474$		0			0
$475$			11809.2i			1.14072i
$476$		0			0
$477$	10746.7	+	4504.34i	1.03157	+	0.432368i
$478$		0			0
$479$	−4004.73	+	6936.40i	−0.382006	+	0.661654i	−0.991349	−	0.131253i	$-0.958100\pi$
$479$	−4004.73	+	6936.40i	−0.382006	+	0.661654i	0.609343	+	0.792907i	$0.291433\pi$
$480$		0			0
$481$	2367.37	−	1366.80i	0.224413	−	0.129565i
$482$		0			0
$483$	−10469.5	−	2346.52i	−0.986287	−	0.221057i
$484$		0			0
$485$	−17402.5	+	10047.3i	−1.62929	+	0.940672i
$486$		0			0
$487$	−822.529	+	1424.66i	−0.0765346	+	0.132562i	−0.901753	−	0.432253i	$-0.857719\pi$
$487$	−822.529	+	1424.66i	−0.0765346	+	0.132562i	0.825218	+	0.564814i	$0.191052\pi$
$488$		0			0
$489$	6981.06	−	4643.44i	0.645592	−	0.429414i
$490$		0			0
$491$			4333.80i			0.398334i	0.979966	+	0.199167i	$0.0638236\pi$
$491$			4333.80i			0.398334i	−0.979966	+	0.199167i	$0.936176\pi$
$492$		0			0
$493$	−16236.2	−	9373.97i	−1.48325	−	0.856354i
$494$		0			0
$495$	309.237	+	2427.55i	0.0280791	+	0.220425i
$496$		0			0
$497$	−95.2540	−	99.5479i	−0.00859703	−	0.00898458i
$498$		0			0
$499$	5349.68	+	9265.93i	0.479929	+	0.831262i	0.999735	−	0.0230225i	$-0.00732893\pi$
$499$	5349.68	+	9265.93i	0.479929	+	0.831262i	−0.519806	+	0.854285i	$0.673996\pi$
$500$		0			0
$501$	−309.906	−	4885.27i	−0.0276359	−	0.435644i
$502$		0			0
$503$	15884.8			1.40809			0.704044	−	0.710157i	$-0.251376\pi$
$503$	15884.8			1.40809			0.704044	+	0.710157i	$0.251376\pi$
$504$		0			0
$505$	23372.2			2.05950
$506$		0			0
$507$	−102.110	−	1609.64i	−0.00894452	−	0.140999i
$508$		0			0
$509$	3826.23	+	6627.22i	0.333192	+	0.577105i	0.983136	−	0.182877i	$-0.0585411\pi$
$509$	3826.23	+	6627.22i	0.333192	+	0.577105i	−0.649944	+	0.759982i	$0.725208\pi$
$510$		0			0
$511$	−16885.6	+	4127.90i	−1.46179	+	0.357354i
$512$		0			0
$513$	−8640.72	−	9978.86i	−0.743659	−	0.858825i
$514$		0			0
$515$	2503.36	+	1445.31i	0.214196	+	0.123666i
$516$		0			0
$517$		−	614.208i		−	0.0522492i
$518$		0			0
$519$	−18618.4	+	12384.0i	−1.57467	+	1.04739i
$520$		0			0
$521$	9407.14	−	16293.6i	0.791045	−	1.37013i	−0.134276	−	0.990944i	$-0.542871\pi$
$521$	9407.14	−	16293.6i	0.791045	−	1.37013i	0.925321	−	0.379186i	$-0.123796\pi$
$522$		0			0
$523$	14166.2	−	8178.84i	1.18440	−	0.683816i	0.227375	−	0.973807i	$-0.426986\pi$
$523$	14166.2	−	8178.84i	1.18440	−	0.683816i	0.957029	+	0.289991i	$0.0936524\pi$
$524$		0			0
$525$	11528.1	−	3605.41i	0.958338	−	0.299720i
$526$		0			0
$527$	−6144.11	+	3547.30i	−0.507859	+	0.293212i
$528$		0			0
$529$	131.575	−	227.894i	0.0108141	−	0.0187305i
$530$		0			0
$531$	4432.99	−	10576.5i	0.362289	−	0.864367i
$532$		0			0
$533$			10337.0i			0.840051i
$534$		0			0
$535$	−2316.77	−	1337.59i	−0.187220	−	0.108092i
$536$		0			0
$537$	2615.43	−	5275.63i	0.210175	−	0.423949i
$538$		0			0
$539$	1056.08	−	1656.09i	0.0843947	−	0.132343i
$540$		0			0
$541$	11184.7	+	19372.5i	0.888851	+	1.53953i	0.841235	+	0.540669i	$0.181829\pi$
$541$	11184.7	+	19372.5i	0.888851	+	1.53953i	0.0476155	+	0.998866i	$0.484838\pi$
$542$		0			0
$543$	2224.16	−	141.094i	0.175779	−	0.0111509i
$544$		0			0
$545$	−10650.6			−0.837102
$546$		0			0
$547$	1073.43			0.0839062			0.0419531	−	0.999120i	$-0.486642\pi$
$547$	1073.43			0.0839062			0.0419531	+	0.999120i	$0.486642\pi$
$548$		0			0
$549$	−9852.37	+	7494.52i	−0.765918	+	0.582620i
$550$		0			0
$551$	−11189.4	−	19380.6i	−0.865125	−	1.49844i
$552$		0			0
$553$	16167.0	+	4716.16i	1.24320	+	0.362661i
$554$		0			0
$555$	−4022.55	−	1994.20i	−0.307654	−	0.152521i
$556$		0			0
$557$	8875.39	+	5124.21i	0.675157	+	0.389802i	0.798028	−	0.602621i	$-0.205877\pi$
$557$	8875.39	+	5124.21i	0.675157	+	0.389802i	−0.122871	+	0.992423i	$0.539210\pi$
$558$		0			0
$559$			25412.5i			1.92278i
$560$		0			0
$561$	1298.92	+	1952.83i	0.0977551	+	0.146967i
$562$		0			0
$563$	−11480.8	+	19885.4i	−0.859429	+	1.48857i	0.0130451	+	0.999915i	$0.495847\pi$
$563$	−11480.8	+	19885.4i	−0.859429	+	1.48857i	−0.872474	+	0.488660i	$0.837486\pi$
$564$		0			0
$565$	−13540.5	+	7817.63i	−1.00824	+	0.582106i
$566$		0			0
$567$	−7103.26	+	11481.6i	−0.526118	+	0.850411i
$568$		0			0
$569$	−16065.5	+	9275.44i	−1.18366	+	0.683386i	−0.956858	−	0.290555i	$-0.906160\pi$
$569$	−16065.5	+	9275.44i	−1.18366	+	0.683386i	−0.226801	+	0.973941i	$0.572827\pi$
$570$		0			0
$571$	11632.5	−	20148.0i	0.852545	−	1.47665i	−0.0263595	−	0.999653i	$-0.508391\pi$
$571$	11632.5	−	20148.0i	0.852545	−	1.47665i	0.878904	−	0.476998i	$-0.158275\pi$
$572$		0			0
$573$	7544.32	+	11342.3i	0.550032	+	0.826932i
$574$		0			0
$575$			13993.6i			1.01491i
$576$		0			0
$577$	−7314.12	−	4222.81i	−0.527714	−	0.304676i	0.212371	−	0.977189i	$-0.431881\pi$
$577$	−7314.12	−	4222.81i	−0.527714	−	0.304676i	−0.740085	+	0.672513i	$0.765215\pi$
$578$		0			0
$579$	22066.4	+	10939.5i	1.58385	+	0.785202i
$580$		0			0
$581$	−2670.26	+	2555.08i	−0.190673	+	0.182448i
$582$		0			0
$583$	−1235.69	−	2140.28i	−0.0877822	−	0.152043i
$584$		0			0
$585$	−17031.4	+	12955.5i	−1.20370	+	0.915629i
$586$		0			0
$587$	−16734.6			−1.17668			−0.588339	−	0.808614i	$-0.700218\pi$
$587$	−16734.6			−1.17668			−0.588339	+	0.808614i	$0.700218\pi$
$588$		0			0
$589$	−8468.59			−0.592431
$590$		0			0
$591$	16480.0	−	1045.44i	1.14703	−	0.0727642i
$592$		0			0
$593$	−1219.96	−	2113.03i	−0.0844816	−	0.146326i	0.820689	−	0.571376i	$-0.193590\pi$
$593$	−1219.96	−	2113.03i	−0.0844816	−	0.146326i	−0.905170	+	0.425049i	$0.860257\pi$
$594$		0			0
$595$	16693.9	−	15973.8i	1.15022	−	1.10061i
$596$		0			0
$597$	11757.5	−	23716.3i	0.806033	−	1.62587i
$598$		0			0
$599$	3845.18	+	2220.01i	0.262287	+	0.151431i	0.625377	−	0.780323i	$-0.284945\pi$
$599$	3845.18	+	2220.01i	0.262287	+	0.151431i	−0.363091	+	0.931754i	$0.618278\pi$
$600$		0			0
$601$			15246.9i			1.03483i	0.855735	+	0.517415i	$0.173105\pi$
$601$			15246.9i			1.03483i	−0.855735	+	0.517415i	$0.826895\pi$
$602$		0			0
$603$	9945.15	−	23727.6i	0.671638	−	1.60243i
$604$		0			0
$605$	−10273.8	+	17794.7i	−0.690394	+	1.19580i
$606$		0			0
$607$	−13051.0	+	7535.02i	−0.872695	+	0.503850i	−0.868243	−	0.496140i	$-0.834750\pi$
$607$	−13051.0	+	7535.02i	−0.872695	+	0.503850i	−0.00445182	+	0.999990i	$0.501417\pi$
$608$		0			0
$609$	−15503.1	+	16840.0i	−1.03155	+	1.12051i
$610$		0			0
$611$	4651.29	−	2685.42i	0.307972	−	0.177808i
$612$		0			0
$613$	3211.01	−	5561.64i	0.211569	−	0.366448i	−0.740637	−	0.671905i	$-0.765476\pi$
$613$	3211.01	−	5561.64i	0.211569	−	0.366448i	0.952206	+	0.305458i	$0.0988094\pi$
$614$		0			0
$615$	14136.3	−	9402.75i	0.926881	−	0.616513i
$616$		0			0
$617$		−	7830.44i		−	0.510927i	−0.966819	−	0.255463i	$-0.917772\pi$
$617$		−	7830.44i		−	0.510927i	0.966819	−	0.255463i	$-0.0822280\pi$
$618$		0			0
$619$	107.096	+	61.8319i	0.00695405	+	0.00401492i	0.503473	−	0.864011i	$-0.332055\pi$
$619$	107.096	+	61.8319i	0.00695405	+	0.00401492i	−0.496519	+	0.868026i	$0.665389\pi$
$620$		0			0
$621$	−10239.1	−	11824.7i	−0.661642	−	0.764106i
$622$		0			0
$623$	20314.7	+	5926.10i	1.30641	+	0.381099i
$624$		0			0
$625$	7780.23	+	13475.8i	0.497935	+	0.862449i
$626$		0			0
$627$	177.241	+	2793.97i	0.0112892	+	0.177959i
$628$		0			0
$629$	−4302.97			−0.272767
$630$		0			0
$631$	3.00689			0.000189703			9.48514e−5	−	1.00000i	$-0.499970\pi$
$631$	3.00689			0.000189703			9.48514e−5		1.00000i	$0.499970\pi$
$632$		0			0
$633$	−948.219	−	14947.5i	−0.0595392	−	0.938560i
$634$		0			0
$635$	−14708.3	−	25475.5i	−0.919180	−	1.59207i
$636$		0			0
$637$	17158.7	+	756.810i	1.06727	+	0.0470736i
$638$		0			0
$639$	−25.3821	−	199.253i	−0.00157136	−	0.0123354i
$640$		0			0
$641$	−22462.2	−	12968.6i	−1.38410	−	0.799108i	−0.391454	−	0.920198i	$-0.628028\pi$
$641$	−22462.2	−	12968.6i	−1.38410	−	0.799108i	−0.992642	+	0.121089i	$0.961361\pi$
$642$		0			0
$643$			23754.6i			1.45691i	0.685096	+	0.728453i	$0.259760\pi$
$643$			23754.6i			1.45691i	−0.685096	+	0.728453i	$0.740240\pi$
$644$		0			0
$645$	34752.7	−	23115.7i	2.12153	−	1.41113i
$646$		0			0
$647$	−680.871	+	1179.30i	−0.0413722	+	0.0716587i	−0.885970	−	0.463743i	$-0.846506\pi$
$647$	−680.871	+	1179.30i	−0.0413722	+	0.0716587i	0.844598	+	0.535401i	$0.179840\pi$
$648$		0			0
$649$	−2106.37	+	1216.12i	−0.127400	+	0.0735542i
$650$		0			0
$651$	2585.50	+	8267.00i	0.155659	+	0.497710i
$652$		0			0
$653$	26870.9	−	15513.9i	1.61032	−	0.929719i	0.621024	−	0.783791i	$-0.286717\pi$
$653$	26870.9	−	15513.9i	1.61032	−	0.929719i	0.989295	−	0.145927i	$-0.0466166\pi$
$654$		0			0
$655$	17015.6	−	29472.0i	1.01505	−	1.75811i
$656$		0			0
$657$	−23371.9	−	9796.04i	−1.38786	−	0.581705i
$658$		0			0
$659$		−	21216.5i		−	1.25414i	−0.778963	−	0.627070i	$-0.784254\pi$
$659$		−	21216.5i		−	1.25414i	0.778963	−	0.627070i	$-0.215746\pi$
$660$		0			0
$661$	−15965.0	−	9217.37i	−0.939432	−	0.542381i	−0.0496500	−	0.998767i	$-0.515811\pi$
$661$	−15965.0	−	9217.37i	−0.939432	−	0.542381i	−0.889782	+	0.456385i	$0.849144\pi$
$662$		0			0
$663$	−9109.34	+	18374.6i	−0.533601	+	1.07634i
$664$		0			0
$665$	26790.9	−	6549.37i	1.56226	−	0.381915i
$666$		0			0
$667$	−13259.2	−	22965.6i	−0.769711	−	1.33318i
$668$		0			0
$669$	−8696.59	+	551.684i	−0.502586	+	0.0318824i
$670$		0			0
$671$	2625.44			0.151049
$672$		0			0
$673$	−10525.7			−0.602878			−0.301439	−	0.953485i	$-0.597467\pi$
$673$	−10525.7			−0.602878			−0.301439	+	0.953485i	$0.597467\pi$
$674$		0			0
$675$	16638.7	+	5765.11i	0.948775	+	0.328739i
$676$		0			0
$677$	−10231.2	−	17721.0i	−0.580824	−	1.00602i	−0.995382	−	0.0959933i	$-0.969397\pi$
$677$	−10231.2	−	17721.0i	−0.580824	−	1.00602i	0.414558	−	0.910023i	$-0.363936\pi$
$678$		0			0
$679$	16255.9	+	16988.7i	0.918769	+	0.960186i
$680$		0			0
$681$	18489.1	+	9166.07i	1.04038	+	0.515777i
$682$		0			0
$683$	17893.6	+	10330.9i	1.00246	+	0.578770i	0.908975	−	0.416851i	$-0.136866\pi$
$683$	17893.6	+	10330.9i	1.00246	+	0.578770i	0.0934838	+	0.995621i	$0.470200\pi$
$684$		0			0
$685$		−	837.077i		−	0.0466906i
$686$		0			0
$687$	2893.52	+	4350.18i	0.160691	+	0.241586i
$688$		0			0
$689$	10805.3	−	18715.3i	0.597458	−	1.03483i
$690$		0			0
$691$	−13873.9	+	8010.09i	−0.763802	+	0.440981i	−0.830659	−	0.556781i	$-0.812036\pi$
$691$	−13873.9	+	8010.09i	−0.763802	+	0.440981i	0.0668569	+	0.997763i	$0.478703\pi$
$692$		0			0
$693$	2673.35	−	1026.04i	0.146540	−	0.0562422i
$694$		0			0
$695$	24819.7	−	14329.7i	1.35463	−	0.782094i
$696$		0			0
$697$	8135.79	−	14091.6i	0.442131	−	0.765793i
$698$		0			0
$699$	−1772.42	−	2664.69i	−0.0959070	−	0.144189i
$700$		0			0
$701$			5541.82i			0.298590i	0.988793	+	0.149295i	$0.0477004\pi$
$701$			5541.82i			0.298590i	−0.988793	+	0.149295i	$0.952300\pi$
$702$		0			0
$703$	−4448.17	−	2568.15i	−0.238643	−	0.137781i
$704$		0			0
$705$	−7903.31	−	3918.12i	−0.422207	−	0.209312i
$706$		0			0
$707$	−6494.39	−	26566.0i	−0.345469	−	1.41318i
$708$		0			0
$709$	−11560.2	−	20022.8i	−0.612343	−	1.06061i	−0.990845	−	0.135008i	$-0.956894\pi$
$709$	−11560.2	−	20022.8i	−0.612343	−	1.06061i	0.378502	−	0.925601i	$-0.376439\pi$
$710$		0			0
$711$	14864.3	+	19540.7i	0.784041	+	1.03071i
$712$		0			0
$713$	−10035.1			−0.527093
$714$		0			0
$715$	4538.50			0.237385
$716$		0			0
$717$	24590.3	−	1559.93i	1.28081	−	0.0812504i
$718$		0			0
$719$	−4614.48	−	7992.52i	−0.239348	−	0.414563i	0.721179	−	0.692748i	$-0.243600\pi$
$719$	−4614.48	−	7992.52i	−0.239348	−	0.414563i	−0.960527	+	0.278186i	$0.910267\pi$
$720$		0			0
$721$	947.212	−	3247.05i	0.0489265	−	0.167720i
$722$		0			0
$723$	−7813.35	+	15760.5i	−0.401911	+	0.810703i
$724$		0			0
$725$	25854.2	+	14926.9i	1.32442	+	0.764652i
$726$		0			0
$727$		−	14795.0i		−	0.754767i	−0.926057	−	0.377384i	$-0.876824\pi$
$727$		−	14795.0i		−	0.754767i	0.926057	−	0.377384i	$-0.123176\pi$
$728$		0			0
$729$	−18278.1	+	7302.87i	−0.928623	+	0.371024i
$730$		0			0
$731$	20001.0	−	34642.7i	1.01199	−	1.75281i
$732$		0			0
$733$	−15930.4	+	9197.42i	−0.802732	+	0.463457i	−0.844426	−	0.535673i	$-0.820058\pi$
$733$	−15930.4	+	9197.42i	−0.802732	+	0.463457i	0.0416936	+	0.999130i	$0.486725\pi$
$734$		0			0
$735$	−14572.9	−	24153.6i	−0.731330	−	1.21213i
$736$		0			0
$737$	−4725.52	+	2728.28i	−0.236183	+	0.136360i
$738$		0			0
$739$	2188.73	−	3790.99i	0.108949	−	0.188706i	−0.806395	−	0.591377i	$-0.798585\pi$
$739$	2188.73	−	3790.99i	0.108949	−	0.188706i	0.915345	+	0.402671i	$0.131918\pi$
$740$		0			0
$741$	−20383.3	+	13557.9i	−1.01053	+	0.672150i
$742$		0			0
$743$		−	611.994i		−	0.0302179i	−0.999886	−	0.0151089i	$-0.995190\pi$
$743$		−	611.994i		−	0.0302179i	0.999886	−	0.0151089i	$-0.00480951\pi$
$744$		0			0
$745$	6976.33	+	4027.78i	0.343078	+	0.198076i
$746$		0			0
$747$	−5344.73	+	680.845i	−0.261785	+	0.0333478i
$748$		0			0
$749$	−876.612	+	3005.03i	−0.0427646	+	0.146597i
$750$		0			0
$751$	3213.32	+	5565.63i	0.156133	+	0.270430i	0.933471	−	0.358653i	$-0.116764\pi$
$751$	3213.32	+	5565.63i	0.156133	+	0.270430i	−0.777338	+	0.629083i	$0.783431\pi$
$752$		0			0
$753$	−503.926	−	7943.75i	−0.0243879	−	0.384444i
$754$		0			0
$755$	−31532.3			−1.51997
$756$		0			0
$757$	−18664.6			−0.896138			−0.448069	−	0.893999i	$-0.647888\pi$
$757$	−18664.6			−0.896138			−0.448069	+	0.893999i	$0.647888\pi$
$758$		0			0
$759$	210.026	+	3310.80i	0.0100441	+	0.158332i
$760$		0			0
$761$	−603.686	−	1045.61i	−0.0287564	−	0.0498075i	0.851289	−	0.524697i	$-0.175821\pi$
$761$	−603.686	−	1045.61i	−0.0287564	−	0.0498075i	−0.880045	+	0.474889i	$0.842488\pi$
$762$		0			0
$763$	2959.46	+	12106.0i	0.140419	+	0.574398i
$764$		0			0
$765$	33414.1	−	4256.49i	1.57920	−	0.201168i
$766$		0			0
$767$	−18418.8	−	10634.1i	−0.867100	−	0.500621i
$768$		0			0
$769$		−	10700.4i		−	0.501776i	−0.968016	−	0.250888i	$-0.919277\pi$
$769$		−	10700.4i		−	0.501776i	0.968016	−	0.250888i	$-0.0807227\pi$
$770$		0			0
$771$	−9732.68	+	6473.68i	−0.454623	+	0.302391i
$772$		0			0
$773$	−14178.3	+	24557.6i	−0.659715	+	1.14266i	0.320975	+	0.947088i	$0.395989\pi$
$773$	−14178.3	+	24557.6i	−0.659715	+	1.14266i	−0.980689	+	0.195571i	$0.937344\pi$
$774$		0			0
$775$	9783.77	−	5648.66i	0.453475	−	0.261814i
$776$		0			0
$777$	−1148.97	+	5126.36i	−0.0530491	+	0.236689i
$778$		0			0
$779$	16820.7	−	9711.42i	0.773637	−	0.446659i
$780$		0			0
$781$	−21.3006	+	36.8937i	−0.000975921	+	0.00169034i
$782$		0			0
$783$	−32769.0	+	6304.02i	−1.49562	+	0.287723i
$784$		0			0
$785$			32102.7i			1.45961i
$786$		0			0
$787$	4035.13	+	2329.69i	0.182766	+	0.105520i	0.588592	−	0.808430i	$-0.299683\pi$
$787$	4035.13	+	2329.69i	0.182766	+	0.105520i	−0.405825	+	0.913951i	$0.633016\pi$
$788$		0			0
$789$	15113.4	−	30485.5i	0.681940	−	1.37556i
$790$		0			0
$791$	12648.4	+	13218.6i	0.568553	+	0.594182i
$792$		0			0
$793$	11478.9	+	19882.0i	0.514032	+	0.890329i
$794$		0			0
$795$	−35422.6	+	2247.10i	−1.58027	+	0.100247i
$796$		0			0
$797$	28669.9			1.27420			0.637101	−	0.770780i	$-0.280133\pi$
$797$	28669.9			1.27420			0.637101	+	0.770780i	$0.280133\pi$
$798$		0			0
$799$	−8454.27			−0.374331
$800$		0			0
$801$	18677.7	+	24553.9i	0.823902	+	1.08311i
$802$		0			0
$803$	2687.38	+	4654.67i	0.118101	+	0.204558i
$804$		0			0
$805$	31746.6	−	7760.86i	1.38996	−	0.339794i
$806$		0			0
$807$	21435.3	+	10626.7i	0.935017	+	0.463541i
$808$		0			0
$809$	38166.0	+	22035.2i	1.65865	+	0.957620i	0.973340	+	0.229366i	$0.0736651\pi$
$809$	38166.0	+	22035.2i	1.65865	+	0.957620i	0.685307	+	0.728255i	$0.259668\pi$
$810$		0			0
$811$		−	41979.6i		−	1.81764i	−0.417192	−	0.908819i	$-0.636986\pi$
$811$		−	41979.6i		−	1.81764i	0.417192	−	0.908819i	$-0.363014\pi$
$812$		0			0
$813$	3100.88	+	4661.94i	0.133767	+	0.201109i
$814$		0			0
$815$	−12769.4	+	22117.3i	−0.548827	+	0.950596i
$816$		0			0
$817$	41351.8	−	23874.5i	1.77077	−	1.02235i
$818$		0			0
$819$	19458.3	+	15758.8i	0.830194	+	0.672354i
$820$		0			0
$821$	−20435.5	+	11798.4i	−0.868702	+	0.501545i	−0.866917	−	0.498453i	$-0.833902\pi$
$821$	−20435.5	+	11798.4i	−0.868702	+	0.501545i	−0.00178534	+	0.999998i	$0.500568\pi$
$822$		0			0
$823$	−16447.0	+	28487.0i	−0.696606	+	1.20656i	0.273031	+	0.962005i	$0.411974\pi$
$823$	−16447.0	+	28487.0i	−0.696606	+	1.20656i	−0.969636	+	0.244551i	$0.921359\pi$
$824$		0			0
$825$	−2068.38	−	3109.66i	−0.0872872	−	0.131230i
$826$		0			0
$827$			16838.7i			0.708029i	0.935240	+	0.354015i	$0.115184\pi$
$827$			16838.7i			0.708029i	−0.935240	+	0.354015i	$0.884816\pi$
$828$		0			0
$829$	8630.34	+	4982.73i	0.361573	+	0.208754i	0.669771	−	0.742568i	$-0.266392\pi$
$829$	8630.34	+	4982.73i	0.361573	+	0.208754i	−0.308197	+	0.951322i	$0.599726\pi$
$830$		0			0
$831$	22078.7	+	10945.6i	0.921662	+	0.456920i
$832$		0			0
$833$	−22795.3	−	14536.5i	−0.948152	−	0.604632i
$834$		0			0
$835$	7455.29	+	12912.9i	0.308983	+	0.535175i
$836$		0			0
$837$	−4134.26	+	11931.9i	−0.170730	+	0.492744i
$838$		0			0
$839$	14935.3			0.614572			0.307286	−	0.951617i	$-0.400579\pi$
$839$	14935.3			0.614572			0.307286	+	0.951617i	$0.400579\pi$
$840$		0			0
$841$	−32185.0			−1.31965
$842$		0			0
$843$	7488.16	−	475.025i	0.305938	−	0.0194077i
$844$		0			0
$845$	2456.43	+	4254.66i	0.100004	+	0.173213i
$846$		0			0
$847$	23081.1	+	6733.10i	0.936336	+	0.273143i
$848$		0			0
$849$	−43.6308	+	88.0085i	−0.00176373	+	0.00355765i
$850$		0			0
$851$	−5270.99	−	3043.21i	−0.212323	−	0.122585i
$852$		0			0
$853$			16347.1i			0.656170i	0.944648	+	0.328085i	$0.106403\pi$
$853$			16347.1i			0.656170i	−0.944648	+	0.328085i	$0.893597\pi$
$854$		0			0
$855$	37082.1	+	15542.5i	1.48325	+	0.621687i
$856$		0			0
$857$	17946.5	−	31084.3i	0.715335	−	1.23900i	−0.247496	−	0.968889i	$-0.579608\pi$
$857$	17946.5	−	31084.3i	0.715335	−	1.23900i	0.962830	−	0.270107i	$-0.0870591\pi$
$858$		0			0
$859$	−41465.7	+	23940.2i	−1.64702	+	0.950909i	−0.668776	+	0.743464i	$0.733181\pi$
$859$	−41465.7	+	23940.2i	−1.64702	+	0.950909i	−0.978247	+	0.207445i	$0.933485\pi$
$860$		0			0
$861$	−14615.7	−	13455.3i	−0.578515	−	0.532586i
$862$		0			0
$863$	−8269.47	+	4774.38i	−0.326183	+	0.188322i	−0.654145	−	0.756369i	$-0.726971\pi$
$863$	−8269.47	+	4774.38i	−0.326183	+	0.188322i	0.327962	+	0.944691i	$0.393638\pi$
$864$		0			0
$865$	34055.9	−	58986.5i	1.33865	−	2.31862i
$866$		0			0
$867$	5623.75	−	3740.63i	0.220291	−	0.146526i
$868$		0			0
$869$		−	5207.17i		−	0.203270i
$870$		0			0
$871$	−41321.6	−	23857.0i	−1.60749	−	0.928087i
$872$		0			0
$873$	4331.67	+	34004.2i	0.167932	+	1.31829i
$874$		0			0
$875$	−108.889	+	104.192i	−0.00420698	+	0.00402551i
$876$		0			0
$877$	−18554.2	−	32136.9i	−0.714404	−	1.23738i	−0.963189	−	0.268825i	$-0.913365\pi$
$877$	−18554.2	−	32136.9i	−0.714404	−	1.23738i	0.248785	−	0.968559i	$-0.419969\pi$
$878$		0			0
$879$	2208.95	+	34821.3i	0.0847624	+	1.33617i
$880$		0			0
$881$	24632.1			0.941973			0.470986	−	0.882140i	$-0.343898\pi$
$881$	24632.1			0.941973			0.470986	+	0.882140i	$0.343898\pi$
$882$		0			0
$883$	−37746.3			−1.43858			−0.719289	−	0.694711i	$-0.755532\pi$
$883$	−37746.3			−1.43858			−0.719289	+	0.694711i	$0.755532\pi$
$884$		0			0
$885$	2211.50	+	34861.5i	0.0839987	+	1.32413i
$886$		0			0
$887$	−7916.98	−	13712.6i	−0.299691	−	0.519081i	0.676374	−	0.736559i	$-0.263550\pi$
$887$	−7916.98	−	13712.6i	−0.299691	−	0.519081i	−0.976065	+	0.217478i	$0.930217\pi$
$888$		0			0
$889$	−24869.7	+	23797.0i	−0.938249	+	0.897778i
$890$		0			0
$891$	4023.12	+	1114.26i	0.151268	+	0.0418957i
$892$		0			0
$893$	−8739.55	−	5045.78i	−0.327501	−	0.189083i
$894$		0			0
$895$			17936.1i			0.669875i
$896$		0			0
$897$	−24153.8	+	16065.9i	−0.899077	+	0.598019i
$898$		0			0
$899$	−10704.4	+	18540.5i	−0.397120	+	0.687832i
$900$		0			0
$901$	−29459.9	+	17008.7i	−1.08929	+	0.628902i
$902$		0			0
$903$	−35931.1	−	33078.5i	−1.32416	−	1.21903i
$904$		0			0
$905$	−5879.00	+	3394.24i	−0.215939	+	0.124672i
$906$		0			0
$907$	10609.8	−	18376.6i	0.388414	−	0.672752i	−0.603823	−	0.797119i	$-0.706357\pi$
$907$	10609.8	−	18376.6i	0.388414	−	0.672752i	0.992236	+	0.124367i	$0.0396899\pi$
$908$		0			0
$909$	15412.0	−	36770.8i	0.562359	−	1.34170i
$910$		0			0
$911$		−	2011.38i		−	0.0731503i	−0.999331	−	0.0365752i	$-0.988355\pi$
$911$		−	2011.38i		−	0.0731503i	0.999331	−	0.0365752i	$-0.0116448\pi$
$912$		0			0
$913$	989.630	+	571.363i	0.0358729	+	0.0207112i
$914$		0			0
$915$	16748.1	−	33782.9i	0.605108	−	1.22058i
$916$		0			0
$917$	−38227.4	−	11151.5i	−1.37664	−	0.401587i
$918$		0			0
$919$	−4632.59	−	8023.89i	−0.166284	−	0.288013i	0.770826	−	0.637045i	$-0.219844\pi$
$919$	−4632.59	−	8023.89i	−0.166284	−	0.288013i	−0.937111	+	0.349033i	$0.886510\pi$
$920$		0			0
$921$	−11751.5	+	745.478i	−0.420440	+	0.0266714i
$922$		0			0
$923$	−372.519			−0.0132845
$924$		0			0
$925$	6851.98			0.243558
$926$		0			0
$927$	3924.63	−	2985.40i	0.139053	−	0.105775i
$928$		0			0
$929$	23761.9	+	41156.8i	0.839185	+	1.45351i	0.890577	+	0.454833i	$0.150301\pi$
$929$	23761.9	+	41156.8i	0.839185	+	1.45351i	−0.0513919	+	0.998679i	$0.516366\pi$
$930$		0			0
$931$	−14888.7	−	28632.0i	−0.524121	−	1.00792i
$932$		0			0
$933$	−7313.68	−	3625.80i	−0.256634	−	0.127228i
$934$		0			0
$935$	−6186.95	−	3572.04i	−0.216401	−	0.124939i
$936$		0			0
$937$		−	12685.2i		−	0.442272i	−0.975243	−	0.221136i	$-0.929024\pi$
$937$		−	12685.2i		−	0.442272i	0.975243	−	0.221136i	$-0.0709765\pi$
$938$		0			0
$939$	17259.3	+	25948.1i	0.599825	+	0.901792i
$940$		0			0
$941$	−7710.52	+	13355.0i	−0.267116	+	0.462658i	−0.968116	−	0.250503i	$-0.919404\pi$
$941$	−7710.52	+	13355.0i	−0.267116	+	0.462658i	0.701000	+	0.713161i	$0.252737\pi$
$942$		0			0
$943$	19932.1	−	11507.8i	0.688313	−	0.397398i
$944$		0			0
$945$	3851.19	−	40944.6i	0.132570	−	1.40945i
$946$		0			0
$947$	−3351.40	+	1934.93i	−0.115001	+	0.0663958i	−0.556397	−	0.830917i	$-0.687817\pi$
$947$	−3351.40	+	1934.93i	−0.115001	+	0.0663958i	0.441396	+	0.897312i	$0.354483\pi$
$948$		0			0
$949$	−23499.3	+	40702.0i	−0.803815	+	1.39225i
$950$		0			0
$951$	−20647.6	−	31042.2i	−0.704044	−	1.05848i
$952$		0			0
$953$		−	1032.39i		−	0.0350918i	−0.999846	−	0.0175459i	$-0.994415\pi$
$953$		−	1032.39i		−	0.0350918i	0.999846	−	0.0175459i	$-0.00558532\pi$
$954$		0			0
$955$	−35934.6	−	20746.8i	−1.21761	−	0.702986i
$956$		0			0
$957$	6340.96	+	3143.57i	0.214184	+	0.106183i
$958$		0			0
$959$	−951.463	+	232.597i	−0.0320379	+	0.00783208i
$960$		0			0
$961$	−10844.7	−	18783.6i	−0.364027	−	0.630514i
$962$		0			0
$963$	−3632.11	+	2762.88i	−0.121540	+	0.0924533i
$964$		0			0
$965$	−75021.4			−2.50262
$966$		0			0
$967$	6010.36			0.199876			0.0999380	−	0.994994i	$-0.468136\pi$
$967$	6010.36			0.199876			0.0999380	+	0.994994i	$0.468136\pi$
$968$		0			0
$969$	38457.7	−	2439.63i	1.27496	−	0.0808795i
$970$		0			0
$971$	20647.3	+	35762.1i	0.682392	+	1.18194i	0.974249	+	0.225475i	$0.0723935\pi$
$971$	20647.3	+	35762.1i	0.682392	+	1.18194i	−0.291857	+	0.956462i	$0.594273\pi$
$972$		0			0
$973$	−23184.4	−	24229.6i	−0.763883	−	0.798318i
$974$		0			0
$975$	14505.6	−	29259.5i	0.476461	−	0.961080i
$976$		0			0
$977$	−45450.5	−	26240.8i	−1.48832	−	0.859282i	−0.488409	−	0.872615i	$-0.662423\pi$
$977$	−45450.5	−	26240.8i	−1.48832	−	0.859282i	−0.999911	+	0.0133324i	$0.995756\pi$
$978$		0			0
$979$		−	6543.09i		−	0.213604i
$980$		0			0
$981$	−7023.18	+	16756.3i	−0.228576	+	0.545348i
$982$		0			0
$983$	14605.8	−	25298.0i	0.473910	−	0.820837i	−0.525644	−	0.850705i	$-0.676175\pi$
$983$	14605.8	−	25298.0i	0.473910	−	0.820837i	0.999554	+	0.0298683i	$0.00950880\pi$
$984$		0			0
$985$	−43560.7	+	25149.8i	−1.40910	+	0.813541i
$986$		0			0
$987$	−2257.44	+	10072.0i	−0.0728017	+	0.324819i
$988$		0			0
$989$	49001.0	−	28290.7i	1.57547	−	0.909599i
$990$		0			0
$991$	−9881.78	+	17115.7i	−0.316756	+	0.548637i	−0.979809	−	0.199935i	$-0.935927\pi$
$991$	−9881.78	+	17115.7i	−0.316756	+	0.548637i	0.663053	+	0.748572i	$0.269260\pi$
$992$		0			0
$993$	13501.1	−	8980.24i	0.431465	−	0.286988i
$994$		0			0
$995$			80630.6i			2.56901i
$996$		0			0
$997$	−24678.2	−	14248.0i	−0.783917	−	0.452595i	0.0538994	−	0.998546i	$-0.482835\pi$
$997$	−24678.2	−	14248.0i	−0.783917	−	0.452595i	−0.837817	+	0.545951i	$0.816168\pi$
$998$		0			0
$999$	−5789.97	+	5013.55i	−0.183370	+	0.158781i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.bc.f.17.12		48
3.2	odd	2	inner	336.4.bc.f.17.20		48
4.3	odd	2		168.4.u.a.17.13	yes	48
7.5	odd	6	inner	336.4.bc.f.257.20		48
12.11	even	2		168.4.u.a.17.5	✓	48
21.5	even	6	inner	336.4.bc.f.257.12		48
28.19	even	6		168.4.u.a.89.5	yes	48
84.47	odd	6		168.4.u.a.89.13	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.u.a.17.5	✓	48	12.11	even	2
168.4.u.a.17.13	yes	48	4.3	odd	2
168.4.u.a.89.5	yes	48	28.19	even	6
168.4.u.a.89.13	yes	48	84.47	odd	6
336.4.bc.f.17.12		48	1.1	even	1	trivial
336.4.bc.f.17.20		48	3.2	odd	2	inner
336.4.bc.f.257.12		48	21.5	even	6	inner
336.4.bc.f.257.20		48	7.5	odd	6	inner

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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(0.328966 + 5.18573i) q^{3} +(-7.91382 - 13.7071i) q^{5} +(-13.3812 + 12.8040i) q^{7} +(-26.7836 + 3.41186i) q^{9} +O(q^{10})\)	\(q+(0.328966 + 5.18573i) q^{3} +(-7.91382 - 13.7071i) q^{5} +(-13.3812 + 12.8040i) q^{7} +(-26.7836 + 3.41186i) q^{9} +(4.95924 + 2.86322i) q^{11} +50.0739i q^{13} +(68.4781 - 45.5481i) q^{15} +(39.4108 - 68.2615i) q^{17} +(81.4814 - 47.0433i) q^{19} +(-70.8002 - 65.1793i) q^{21} +(96.5537 - 55.7453i) q^{23} +(-62.7571 + 108.698i) q^{25} +(-26.5038 - 137.770i) q^{27} -237.853i q^{29} +(-77.9496 - 45.0042i) q^{31} +(-13.2165 + 26.6592i) q^{33} +(281.403 + 82.0894i) q^{35} +(-27.2956 - 47.2774i) q^{37} +(-259.670 + 16.4726i) q^{39} +206.436 q^{41} +507.500 q^{43} +(258.727 + 340.125i) q^{45} +(-53.6291 - 92.8884i) q^{47} +(15.1138 - 342.667i) q^{49} +(366.950 + 181.918i) q^{51} +(-373.754 - 215.787i) q^{53} -90.6360i q^{55} +(270.758 + 407.065i) q^{57} +(-212.369 + 367.833i) q^{59} +(397.053 - 229.239i) q^{61} +(314.711 - 388.592i) q^{63} +(686.370 - 396.276i) q^{65} +(-476.436 + 825.211i) q^{67} +(320.843 + 482.363i) q^{69} +7.43938i q^{71} +(812.839 + 469.293i) q^{73} +(-584.325 - 289.683i) q^{75} +(-103.021 + 25.1849i) q^{77} +(-454.661 - 787.496i) q^{79} +(705.718 - 182.763i) q^{81} +199.553 q^{83} -1247.56 q^{85} +(1233.44 - 78.2455i) q^{87} +(-571.305 - 989.530i) q^{89} +(-641.148 - 670.050i) q^{91} +(207.737 - 419.030i) q^{93} +(-1289.66 - 744.584i) q^{95} -1269.59i q^{97} +(-142.595 - 59.7670i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{7} + 14 q^{9}+O(q^{10}) \) 48 * q - 12 * q^7 + 14 * q^9	\( 48 q - 12 q^{7} + 14 q^{9} + 88 q^{15} + 270 q^{19} + 50 q^{21} - 438 q^{25} - 216 q^{31} - 372 q^{33} + 66 q^{37} - 242 q^{39} - 900 q^{43} - 294 q^{45} + 60 q^{49} + 138 q^{51} + 1384 q^{57} + 108 q^{61} - 1096 q^{63} - 6 q^{67} - 1206 q^{73} + 594 q^{75} + 588 q^{79} - 54 q^{81} - 240 q^{85} + 3522 q^{87} - 234 q^{91} - 608 q^{93} - 1988 q^{99}+O(q^{100}) \) 48 * q - 12 * q^7 + 14 * q^9 + 88 * q^15 + 270 * q^19 + 50 * q^21 - 438 * q^25 - 216 * q^31 - 372 * q^33 + 66 * q^37 - 242 * q^39 - 900 * q^43 - 294 * q^45 + 60 * q^49 + 138 * q^51 + 1384 * q^57 + 108 * q^61 - 1096 * q^63 - 6 * q^67 - 1206 * q^73 + 594 * q^75 + 588 * q^79 - 54 * q^81 - 240 * q^85 + 3522 * q^87 - 234 * q^91 - 608 * q^93 - 1988 * q^99

Introduction

L-functions

Modular forms

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Database

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