LMFDB - Embedded newform 168.4.q.f.121.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,4,Mod(25,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("168.25");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	168.q (of order $3$, degree $2$, 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:	$9.91232088096$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 173x^{6} + 9457x^{4} + 168048x^{2} + 746496 $ x^8 + 173x^6 + 9457x^4 + 168048*x^2 + 746496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.4
Root			$4.63878i$ of defining polynomial
Character	$\chi$	$=$	168.121
Dual form			168.4.q.f.25.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{3} +(7.90648 - 13.6944i) q^{5} +(15.2050 - 10.5739i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})$	\(q+(-1.50000 - 2.59808i) q^{3} +(7.90648 - 13.6944i) q^{5} +(15.2050 - 10.5739i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(15.1240 + 26.1955i) q^{11} -61.6298 q^{13} -47.4389 q^{15} +(-28.0724 - 48.6228i) q^{17} +(69.7002 - 120.724i) q^{19} +(-50.2793 - 23.6429i) q^{21} +(4.07240 - 7.05360i) q^{23} +(-62.5247 - 108.296i) q^{25} +27.0000 q^{27} -0.217857 q^{29} +(-88.2903 - 152.923i) q^{31} +(45.3719 - 78.5864i) q^{33} +(-24.5855 - 291.826i) q^{35} +(-105.540 + 182.801i) q^{37} +(92.4447 + 160.119i) q^{39} -293.305 q^{41} +434.591 q^{43} +(71.1583 + 123.250i) q^{45} +(-241.698 + 418.633i) q^{47} +(119.385 - 321.553i) q^{49} +(-84.2172 + 145.868i) q^{51} +(-10.2536 - 17.7598i) q^{53} +478.309 q^{55} -418.201 q^{57} +(115.674 + 200.353i) q^{59} +(419.351 - 726.337i) q^{61} +(13.9930 + 166.094i) q^{63} +(-487.274 + 843.984i) q^{65} +(312.020 + 540.435i) q^{67} -24.4344 q^{69} +227.106 q^{71} +(-21.5247 - 37.2819i) q^{73} +(-187.574 + 324.888i) q^{75} +(506.949 + 238.383i) q^{77} +(154.530 - 267.654i) q^{79} +(-40.5000 - 70.1481i) q^{81} +1233.99 q^{83} -887.815 q^{85} +(0.326785 + 0.566009i) q^{87} +(-572.179 + 991.042i) q^{89} +(-937.082 + 651.668i) q^{91} +(-264.871 + 458.770i) q^{93} +(-1102.17 - 1909.01i) q^{95} +1688.12 q^{97} -272.231 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9}+O(q^{10}) $ 8 * q - 12 * q^3 - 4 * q^5 + 18 * q^7 - 36 * q^9	$ 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9} - 14 q^{11} + 44 q^{13} + 24 q^{15} - 96 q^{17} + 26 q^{19} - 36 q^{21} - 96 q^{23} - 110 q^{25} + 216 q^{27} - 152 q^{29} - 238 q^{31} - 42 q^{33} + 152 q^{35} - 562 q^{37} - 66 q^{39} + 856 q^{41} - 516 q^{43} - 36 q^{45} + 80 q^{47} + 156 q^{49} - 288 q^{51} + 2952 q^{55} - 156 q^{57} - 262 q^{59} + 276 q^{61} - 54 q^{63} - 2196 q^{65} - 150 q^{67} + 576 q^{69} - 1696 q^{71} + 218 q^{73} - 330 q^{75} - 764 q^{77} - 1762 q^{79} - 324 q^{81} + 6900 q^{83} + 2904 q^{85} + 228 q^{87} + 344 q^{89} - 2806 q^{91} - 714 q^{93} - 2004 q^{95} - 1244 q^{97} + 252 q^{99}+O(q^{100}) $ 8 * q - 12 * q^3 - 4 * q^5 + 18 * q^7 - 36 * q^9 - 14 * q^11 + 44 * q^13 + 24 * q^15 - 96 * q^17 + 26 * q^19 - 36 * q^21 - 96 * q^23 - 110 * q^25 + 216 * q^27 - 152 * q^29 - 238 * q^31 - 42 * q^33 + 152 * q^35 - 562 * q^37 - 66 * q^39 + 856 * q^41 - 516 * q^43 - 36 * q^45 + 80 * q^47 + 156 * q^49 - 288 * q^51 + 2952 * q^55 - 156 * q^57 - 262 * q^59 + 276 * q^61 - 54 * q^63 - 2196 * q^65 - 150 * q^67 + 576 * q^69 - 1696 * q^71 + 218 * q^73 - 330 * q^75 - 764 * q^77 - 1762 * q^79 - 324 * q^81 + 6900 * q^83 + 2904 * q^85 + 228 * q^87 + 344 * q^89 - 2806 * q^91 - 714 * q^93 - 2004 * q^95 - 1244 * q^97 + 252 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.50000	−	2.59808i	−0.288675	−	0.500000i
$3$	−1.50000	−	2.59808i	−0.288675	−	0.500000i
$4$		0			0
$5$	7.90648	−	13.6944i	0.707177	−	1.22487i	−0.258724	−	0.965951i	$-0.583302\pi$
$5$	7.90648	−	13.6944i	0.707177	−	1.22487i	0.965900	−	0.258915i	$-0.0833648\pi$
$6$		0			0
$7$	15.2050	−	10.5739i	0.820993	−	0.570938i
$7$	15.2050	−	10.5739i	0.820993	−	0.570938i
$8$		0			0
$9$	−4.50000	+	7.79423i	−0.166667	+	0.288675i
$10$		0			0
$11$	15.1240	+	26.1955i	0.414550	+	0.718021i	0.995381	−	0.0960028i	$-0.0306058\pi$
$11$	15.1240	+	26.1955i	0.414550	+	0.718021i	−0.580831	+	0.814024i	$0.697272\pi$
$12$		0			0
$13$	−61.6298			−1.31485			−0.657424	−	0.753521i	$-0.728354\pi$
$13$	−61.6298			−1.31485			−0.657424	+	0.753521i	$0.728354\pi$
$14$		0			0
$15$	−47.4389			−0.816577
$16$		0			0
$17$	−28.0724	−	48.6228i	−0.400503	−	0.693692i	0.593283	−	0.804994i	$-0.297831\pi$
$17$	−28.0724	−	48.6228i	−0.400503	−	0.693692i	−0.993787	+	0.111302i	$0.964498\pi$
$18$		0			0
$19$	69.7002	−	120.724i	0.841596	−	1.45769i	−0.0469481	−	0.998897i	$-0.514950\pi$
$19$	69.7002	−	120.724i	0.841596	−	1.45769i	0.888544	−	0.458790i	$-0.151717\pi$
$20$		0			0
$21$	−50.2793	−	23.6429i	−0.522469	−	0.245681i
$22$		0			0
$23$	4.07240	−	7.05360i	0.0369197	−	0.0639469i	−0.846975	−	0.531633i	$-0.821579\pi$
$23$	4.07240	−	7.05360i	0.0369197	−	0.0639469i	0.883895	+	0.467686i	$0.154912\pi$
$24$		0			0
$25$	−62.5247	−	108.296i	−0.500198	−	0.866368i
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	−0.217857			−0.00139500			−0.000697500	−	1.00000i	$-0.500222\pi$
$29$	−0.217857			−0.00139500			−0.000697500		1.00000i	$0.500222\pi$
$30$		0			0
$31$	−88.2903	−	152.923i	−0.511529	−	0.885994i	−0.999911	−	0.0133642i	$-0.995746\pi$
$31$	−88.2903	−	152.923i	−0.511529	−	0.885994i	0.488382	−	0.872630i	$-0.337587\pi$
$32$		0			0
$33$	45.3719	−	78.5864i	0.239340	−	0.414550i
$34$		0			0
$35$	−24.5855	−	291.826i	−0.118735	−	1.40936i
$36$		0			0
$37$	−105.540	+	182.801i	−0.468937	+	0.812223i	−0.999370	−	0.0355042i	$-0.988696\pi$
$37$	−105.540	+	182.801i	−0.468937	+	0.812223i	0.530432	+	0.847727i	$0.322030\pi$
$38$		0			0
$39$	92.4447	+	160.119i	0.379564	+	0.657424i
$40$		0			0
$41$	−293.305			−1.11723			−0.558616	−	0.829427i	$-0.688667\pi$
$41$	−293.305			−1.11723			−0.558616	+	0.829427i	$0.688667\pi$
$42$		0			0
$43$	434.591			1.54127			0.770634	−	0.637278i	$-0.219940\pi$
$43$	434.591			1.54127			0.770634	+	0.637278i	$0.219940\pi$
$44$		0			0
$45$	71.1583	+	123.250i	0.235726	+	0.408289i
$46$		0			0
$47$	−241.698	+	418.633i	−0.750112	+	1.29923i	0.197657	+	0.980271i	$0.436667\pi$
$47$	−241.698	+	418.633i	−0.750112	+	1.29923i	−0.947768	+	0.318960i	$0.896666\pi$
$48$		0			0
$49$	119.385	−	321.553i	0.348060	−	0.937472i
$50$		0			0
$51$	−84.2172	+	145.868i	−0.231231	+	0.400503i
$52$		0			0
$53$	−10.2536	−	17.7598i	−0.0265745	−	0.0460283i	0.852432	−	0.522838i	$-0.175127\pi$
$53$	−10.2536	−	17.7598i	−0.0265745	−	0.0460283i	−0.879007	+	0.476809i	$0.841793\pi$
$54$		0			0
$55$	478.309			1.17264
$56$		0			0
$57$	−418.201			−0.971792
$58$		0			0
$59$	115.674	+	200.353i	0.255245	+	0.442097i	0.964962	−	0.262390i	$-0.0845106\pi$
$59$	115.674	+	200.353i	0.255245	+	0.442097i	−0.709717	+	0.704487i	$0.751177\pi$
$60$		0			0
$61$	419.351	−	726.337i	0.880203	−	1.52456i	0.0290872	−	0.999577i	$-0.490740\pi$
$61$	419.351	−	726.337i	0.880203	−	1.52456i	0.851115	−	0.524979i	$-0.175927\pi$
$62$		0			0
$63$	13.9930	+	166.094i	0.0279833	+	0.332157i
$64$		0			0
$65$	−487.274	+	843.984i	−0.929830	+	1.61051i
$66$		0			0
$67$	312.020	+	540.435i	0.568945	+	0.985442i	0.996671	+	0.0815332i	$0.0259817\pi$
$67$	312.020	+	540.435i	0.568945	+	0.985442i	−0.427726	+	0.903909i	$0.640685\pi$
$68$		0			0
$69$	−24.4344			−0.0426312
$70$		0			0
$71$	227.106			0.379613			0.189806	−	0.981822i	$-0.439214\pi$
$71$	227.106			0.379613			0.189806	+	0.981822i	$0.439214\pi$
$72$		0			0
$73$	−21.5247	−	37.2819i	−0.0345107	−	0.0597742i	0.848254	−	0.529589i	$-0.177654\pi$
$73$	−21.5247	−	37.2819i	−0.0345107	−	0.0597742i	−0.882765	+	0.469815i	$0.844321\pi$
$74$		0			0
$75$	−187.574	+	324.888i	−0.288789	+	0.500198i
$76$		0			0
$77$	506.949	+	238.383i	0.750288	+	0.352809i
$78$		0			0
$79$	154.530	−	267.654i	0.220076	−	0.381183i	−0.734755	−	0.678333i	$-0.762703\pi$
$79$	154.530	−	267.654i	0.220076	−	0.381183i	0.954831	+	0.297150i	$0.0960362\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	1233.99			1.63190			0.815951	−	0.578122i	$-0.196214\pi$
$83$	1233.99			1.63190			0.815951	+	0.578122i	$0.196214\pi$
$84$		0			0
$85$	−887.815			−1.13291
$86$		0			0
$87$	0.326785	+	0.566009i	0.000402702	+	0.000697500i
$88$		0			0
$89$	−572.179	+	991.042i	−0.681470	+	1.18034i	0.293063	+	0.956093i	$0.405326\pi$
$89$	−572.179	+	991.042i	−0.681470	+	1.18034i	−0.974532	+	0.224247i	$0.928008\pi$
$90$		0			0
$91$	−937.082	+	651.668i	−1.07948	+	0.750696i
$92$		0			0
$93$	−264.871	+	458.770i	−0.295331	+	0.511529i
$94$		0			0
$95$	−1102.17	−	1909.01i	−1.19031	−	2.06169i
$96$		0			0
$97$	1688.12			1.76704			0.883520	−	0.468394i	$-0.155167\pi$
$97$	1688.12			1.76704			0.883520	+	0.468394i	$0.155167\pi$
$98$		0			0
$99$	−272.231			−0.276366
$100$		0			0
$101$	743.930	+	1288.52i	0.732909	+	1.26944i	0.955635	+	0.294554i	$0.0951712\pi$
$101$	743.930	+	1288.52i	0.732909	+	1.26944i	−0.222726	+	0.974881i	$0.571496\pi$
$102$		0			0
$103$	389.204	−	674.121i	0.372324	−	0.644884i	−0.617598	−	0.786494i	$-0.711894\pi$
$103$	389.204	−	674.121i	0.372324	−	0.644884i	0.989923	+	0.141609i	$0.0452276\pi$
$104$		0			0
$105$	−721.308	+	501.614i	−0.670405	+	0.466215i
$106$		0			0
$107$	−651.040	+	1127.63i	−0.588209	+	1.01881i	0.406258	+	0.913758i	$0.366833\pi$
$107$	−651.040	+	1127.63i	−0.588209	+	1.01881i	−0.994467	+	0.105050i	$0.966500\pi$
$108$		0			0
$109$	−234.541	−	406.237i	−0.206100	−	0.356976i	0.744382	−	0.667754i	$-0.232744\pi$
$109$	−234.541	−	406.237i	−0.206100	−	0.356976i	−0.950483	+	0.310777i	$0.899411\pi$
$110$		0			0
$111$	633.240			0.541482
$112$		0			0
$113$	1653.86			1.37684			0.688418	−	0.725314i	$-0.258306\pi$
$113$	1653.86			1.37684			0.688418	+	0.725314i	$0.258306\pi$
$114$		0			0
$115$	−64.3966	−	111.538i	−0.0522175	−	0.0904434i
$116$		0			0
$117$	277.334	−	480.357i	0.219141	−	0.379564i
$118$		0			0
$119$	−940.975	−	442.475i	−0.724866	−	0.340854i
$120$		0			0
$121$	208.032	−	360.321i	0.156297	−	0.270715i
$122$		0			0
$123$	439.957	+	762.028i	0.322517	+	0.558616i
$124$		0			0
$125$	−0.781685			−0.000559328
$126$		0			0
$127$	163.760			0.114420			0.0572100	−	0.998362i	$-0.481780\pi$
$127$	163.760			0.114420			0.0572100	+	0.998362i	$0.481780\pi$
$128$		0			0
$129$	−651.887	−	1129.10i	−0.444926	−	0.770634i
$130$		0			0
$131$	260.947	−	451.973i	0.174038	−	0.301443i	−0.765790	−	0.643091i	$-0.777652\pi$
$131$	260.947	−	451.973i	0.174038	−	0.301443i	0.939828	+	0.341648i	$0.110985\pi$
$132$		0			0
$133$	−216.736	−	2572.62i	−0.141304	−	1.67725i
$134$		0			0
$135$	213.475	−	369.749i	0.136096	−	0.235726i
$136$		0			0
$137$	−1338.44	−	2318.25i	−0.834679	−	1.44571i	−0.894291	−	0.447485i	$-0.852320\pi$
$137$	−1338.44	−	2318.25i	−0.834679	−	1.44571i	0.0596120	−	0.998222i	$-0.481014\pi$
$138$		0			0
$139$	853.692			0.520930			0.260465	−	0.965483i	$-0.416124\pi$
$139$	853.692			0.520930			0.260465	+	0.965483i	$0.416124\pi$
$140$		0			0
$141$	1450.19			0.866154
$142$		0			0
$143$	−932.087	−	1614.42i	−0.545070	−	0.944089i
$144$		0			0
$145$	−1.72248	+	2.98342i	−0.000986512	+	0.00170869i
$146$		0			0
$147$	−1014.50	+	172.159i	−0.569212	+	0.0965947i
$148$		0			0
$149$	−278.843	+	482.970i	−0.153313	+	0.265547i	−0.932444	−	0.361316i	$-0.882328\pi$
$149$	−278.843	+	482.970i	−0.153313	+	0.265547i	0.779130	+	0.626862i	$0.215661\pi$
$150$		0			0
$151$	−884.674	−	1532.30i	−0.476780	−	0.825807i	0.522866	−	0.852415i	$-0.324863\pi$
$151$	−884.674	−	1532.30i	−0.476780	−	0.825807i	−0.999646	+	0.0266077i	$0.991530\pi$
$152$		0			0
$153$	505.303			0.267002
$154$		0			0
$155$	−2792.26			−1.44697
$156$		0			0
$157$	1202.45	+	2082.70i	0.611247	+	1.05871i	0.991031	+	0.133635i	$0.0426651\pi$
$157$	1202.45	+	2082.70i	0.611247	+	1.05871i	−0.379784	+	0.925075i	$0.624002\pi$
$158$		0			0
$159$	−30.7609	+	53.2795i	−0.0153428	+	0.0265745i
$160$		0			0
$161$	−12.6633	−	150.311i	−0.00619881	−	0.0735788i
$162$		0			0
$163$	−1689.40	+	2926.12i	−0.811802	+	1.40608i	0.0997991	+	0.995008i	$0.468180\pi$
$163$	−1689.40	+	2926.12i	−0.811802	+	1.40608i	−0.911601	+	0.411075i	$0.865153\pi$
$164$		0			0
$165$	−717.463	−	1242.68i	−0.338512	−	0.586320i
$166$		0			0
$167$	−805.900			−0.373428			−0.186714	−	0.982414i	$-0.559784\pi$
$167$	−805.900			−0.373428			−0.186714	+	0.982414i	$0.559784\pi$
$168$		0			0
$169$	1601.23			0.728826
$170$		0			0
$171$	627.302	+	1086.52i	0.280532	+	0.485896i
$172$		0			0
$173$	420.252	−	727.898i	0.184689	−	0.319890i	−0.758783	−	0.651344i	$-0.774206\pi$
$173$	420.252	−	727.898i	0.184689	−	0.319890i	0.943472	+	0.331453i	$0.107539\pi$
$174$		0			0
$175$	−2095.80	−	985.511i	−0.905301	−	0.425701i
$176$		0			0
$177$	347.021	−	601.059i	0.147366	−	0.255245i
$178$		0			0
$179$	−802.305	−	1389.63i	−0.335012	−	0.580258i	0.648475	−	0.761236i	$-0.275407\pi$
$179$	−802.305	−	1389.63i	−0.335012	−	0.580258i	−0.983487	+	0.180978i	$0.942074\pi$
$180$		0			0
$181$	−3779.43			−1.55206			−0.776029	−	0.630697i	$-0.782769\pi$
$181$	−3779.43			−1.55206			−0.776029	+	0.630697i	$0.782769\pi$
$182$		0			0
$183$	−2516.10			−1.01637
$184$		0			0
$185$	1668.90	+	2890.62i	0.663243	+	1.14877i
$186$		0			0
$187$	849.132	−	1470.74i	0.332057	−	0.575140i
$188$		0			0
$189$	410.535	−	285.496i	0.158000	−	0.109877i
$190$		0			0
$191$	924.798	−	1601.80i	0.350346	−	0.606816i	−0.635964	−	0.771718i	$-0.719397\pi$
$191$	924.798	−	1601.80i	0.350346	−	0.606816i	0.986310	+	0.164902i	$0.0527308\pi$
$192$		0			0
$193$	176.501	+	305.709i	0.0658282	+	0.114018i	0.897061	−	0.441907i	$-0.145698\pi$
$193$	176.501	+	305.709i	0.0658282	+	0.114018i	−0.831233	+	0.555924i	$0.812364\pi$
$194$		0			0
$195$	2923.65			1.07368
$196$		0			0
$197$	4448.22			1.60874			0.804372	−	0.594125i	$-0.202502\pi$
$197$	4448.22			1.60874			0.804372	+	0.594125i	$0.202502\pi$
$198$		0			0
$199$	1120.08	+	1940.03i	0.398995	+	0.691080i	0.993602	−	0.112936i	$-0.0360256\pi$
$199$	1120.08	+	1940.03i	0.398995	+	0.691080i	−0.594607	+	0.804017i	$0.702692\pi$
$200$		0			0
$201$	936.060	−	1621.30i	0.328481	−	0.568945i
$202$		0			0
$203$	−3.31252	+	2.30360i	−0.00114529	+	0.000796458i
$204$		0			0
$205$	−2319.01	+	4016.64i	−0.790080	+	1.36846i
$206$		0			0
$207$	36.6516	+	63.4824i	0.0123066	+	0.0213156i
$208$		0			0
$209$	4216.58			1.39553
$210$		0			0
$211$	1224.34			0.399463			0.199732	−	0.979851i	$-0.435993\pi$
$211$	1224.34			0.399463			0.199732	+	0.979851i	$0.435993\pi$
$212$		0			0
$213$	−340.659	−	590.039i	−0.109585	−	0.189806i
$214$		0			0
$215$	3436.08	−	5951.47i	1.08995	−	1.88785i
$216$		0			0
$217$	−2959.45	−	1391.63i	−0.925810	−	0.435344i
$218$		0			0
$219$	−64.5741	+	111.846i	−0.0199247	+	0.0345107i
$220$		0			0
$221$	1730.10	+	2996.61i	0.526601	+	0.912100i
$222$		0			0
$223$	−3457.37			−1.03822			−0.519108	−	0.854708i	$-0.673736\pi$
$223$	−3457.37			−1.03822			−0.519108	+	0.854708i	$0.673736\pi$
$224$		0			0
$225$	1125.44			0.333465
$226$		0			0
$227$	2589.11	+	4484.47i	0.757027	+	1.31121i	0.944360	+	0.328913i	$0.106682\pi$
$227$	2589.11	+	4484.47i	0.757027	+	1.31121i	−0.187333	+	0.982296i	$0.559984\pi$
$228$		0			0
$229$	−1155.94	+	2002.15i	−0.333567	+	0.577755i	−0.983208	−	0.182486i	$-0.941586\pi$
$229$	−1155.94	+	2002.15i	−0.333567	+	0.577755i	0.649642	+	0.760241i	$0.274919\pi$
$230$		0			0
$231$	−141.086	−	1674.67i	−0.0401852	−	0.476991i
$232$		0			0
$233$	−2371.49	+	4107.54i	−0.666787	+	1.15491i	0.312011	+	0.950078i	$0.398997\pi$
$233$	−2371.49	+	4107.54i	−0.666787	+	1.15491i	−0.978798	+	0.204830i	$0.934336\pi$
$234$		0			0
$235$	3821.95	+	6619.82i	1.06092	+	1.83757i
$236$		0			0
$237$	−927.181			−0.254122
$238$		0			0
$239$	−1412.59			−0.382313			−0.191157	−	0.981560i	$-0.561224\pi$
$239$	−1412.59			−0.382313			−0.191157	+	0.981560i	$0.561224\pi$
$240$		0			0
$241$	940.577	+	1629.13i	0.251402	+	0.435441i	0.963912	−	0.266221i	$-0.0857750\pi$
$241$	940.577	+	1629.13i	0.251402	+	0.435441i	−0.712510	+	0.701662i	$0.752442\pi$
$242$		0			0
$243$	−121.500	+	210.444i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	−3459.57	−	4177.25i	−0.902138	−	1.08929i
$246$		0			0
$247$	−4295.61	+	7440.22i	−1.10657	+	1.91664i
$248$		0			0
$249$	−1850.98	−	3206.00i	−0.471089	−	0.815951i
$250$		0			0
$251$	4519.64			1.13656			0.568281	−	0.822834i	$-0.307609\pi$
$251$	4519.64			1.13656			0.568281	+	0.822834i	$0.307609\pi$
$252$		0			0
$253$	246.363			0.0612202
$254$		0			0
$255$	1331.72	+	2306.61i	0.327042	+	0.566453i
$256$		0			0
$257$	−3956.33	+	6852.56i	−0.960268	+	1.66323i	−0.238445	+	0.971156i	$0.576638\pi$
$257$	−3956.33	+	6852.56i	−0.960268	+	1.66323i	−0.721824	+	0.692077i	$0.756696\pi$
$258$		0			0
$259$	328.182	+	3895.46i	0.0787344	+	0.934564i
$260$		0			0
$261$	0.980356	−	1.69803i	0.000232500	−	0.000402702i
$262$		0			0
$263$	1715.58	+	2971.47i	0.402233	+	0.696688i	0.993995	−	0.109425i	$-0.0349009\pi$
$263$	1715.58	+	2971.47i	0.402233	+	0.696688i	−0.591762	+	0.806113i	$0.701568\pi$
$264$		0			0
$265$	−324.281			−0.0751714
$266$		0			0
$267$	3433.07			0.786894
$268$		0			0
$269$	−4149.60	−	7187.31i	−0.940540	−	1.62906i	−0.764443	−	0.644691i	$-0.776986\pi$
$269$	−4149.60	−	7187.31i	−0.940540	−	1.62906i	−0.176097	−	0.984373i	$-0.556347\pi$
$270$		0			0
$271$	2962.24	−	5130.75i	0.663998	−	1.15008i	−0.315558	−	0.948906i	$-0.602192\pi$
$271$	2962.24	−	5130.75i	0.663998	−	1.15008i	0.979556	−	0.201172i	$-0.0644749\pi$
$272$		0			0
$273$	3098.71	+	1457.11i	0.686968	+	0.323033i
$274$		0			0
$275$	1891.24	−	3275.73i	0.414714	−	0.718305i
$276$		0			0
$277$	−760.300	−	1316.88i	−0.164917	−	0.285645i	0.771709	−	0.635976i	$-0.219402\pi$
$277$	−760.300	−	1316.88i	−0.164917	−	0.285645i	−0.936626	+	0.350331i	$0.886069\pi$
$278$		0			0
$279$	1589.22			0.341019
$280$		0			0
$281$	−352.333			−0.0747986			−0.0373993	−	0.999300i	$-0.511907\pi$
$281$	−352.333			−0.0747986			−0.0373993	+	0.999300i	$0.511907\pi$
$282$		0			0
$283$	−2144.24	−	3713.94i	−0.450396	−	0.780109i	0.548014	−	0.836469i	$-0.315384\pi$
$283$	−2144.24	−	3713.94i	−0.450396	−	0.780109i	−0.998411	+	0.0563599i	$0.982051\pi$
$284$		0			0
$285$	−3306.50	+	5727.03i	−0.687229	+	1.19031i
$286$		0			0
$287$	−4459.70	+	3101.38i	−0.917239	+	0.637869i
$288$		0			0
$289$	880.381	−	1524.86i	0.179194	−	0.310373i
$290$		0			0
$291$	−2532.18	−	4385.87i	−0.510100	−	0.883520i
$292$		0			0
$293$	2661.99			0.530767			0.265384	−	0.964143i	$-0.414501\pi$
$293$	2661.99			0.530767			0.265384	+	0.964143i	$0.414501\pi$
$294$		0			0
$295$	3658.29			0.722013
$296$		0			0
$297$	408.347	+	707.278i	0.0797801	+	0.138183i
$298$		0			0
$299$	−250.981	+	434.712i	−0.0485438	+	0.0840804i
$300$		0			0
$301$	6607.96	−	4595.33i	1.26537	−	0.879968i
$302$		0			0
$303$	2231.79	−	3865.57i	0.423145	−	0.732909i
$304$		0			0
$305$	−6631.17	−	11485.5i	−1.24492	−	2.15626i
$306$		0			0
$307$	145.970			0.0271366			0.0135683	−	0.999908i	$-0.495681\pi$
$307$	145.970			0.0271366			0.0135683	+	0.999908i	$0.495681\pi$
$308$		0			0
$309$	−2335.22			−0.429923
$310$		0			0
$311$	−1708.26	−	2958.79i	−0.311467	−	0.539477i	0.667213	−	0.744867i	$-0.267487\pi$
$311$	−1708.26	−	2958.79i	−0.311467	−	0.539477i	−0.978680	+	0.205390i	$0.934154\pi$
$312$		0			0
$313$	−1841.18	+	3189.01i	−0.332490	+	0.575890i	−0.982999	−	0.183608i	$-0.941222\pi$
$313$	−1841.18	+	3189.01i	−0.332490	+	0.575890i	0.650509	+	0.759498i	$0.274555\pi$
$314$		0			0
$315$	2385.19	+	1121.59i	0.426637	+	0.200618i
$316$		0			0
$317$	688.389	−	1192.32i	0.121968	−	0.211254i	−0.798576	−	0.601894i	$-0.794413\pi$
$317$	688.389	−	1192.32i	0.121968	−	0.211254i	0.920544	+	0.390640i	$0.127746\pi$
$318$		0			0
$319$	−3.29486	−	5.70686i	−0.000578297	−	0.00100164i
$320$		0			0
$321$	3906.24			0.679205
$322$		0			0
$323$	−7826.61			−1.34825
$324$		0			0
$325$	3853.38	+	6674.26i	0.657684	+	1.13914i
$326$		0			0
$327$	−703.622	+	1218.71i	−0.118992	+	0.206100i
$328$		0			0
$329$	751.570	+	8921.01i	0.125943	+	1.49493i
$330$		0			0
$331$	−1087.46	+	1883.54i	−0.180581	+	0.312776i	−0.942079	−	0.335392i	$-0.891131\pi$
$331$	−1087.46	+	1883.54i	−0.180581	+	0.312776i	0.761497	+	0.648168i	$0.224465\pi$
$332$		0			0
$333$	−949.860	−	1645.21i	−0.156312	−	0.270741i
$334$		0			0
$335$	9867.92			1.60938
$336$		0			0
$337$	−6321.14			−1.02176			−0.510882	−	0.859651i	$-0.670681\pi$
$337$	−6321.14			−1.02176			−0.510882	+	0.859651i	$0.670681\pi$
$338$		0			0
$339$	−2480.80	−	4296.86i	−0.397458	−	0.688418i
$340$		0			0
$341$	2670.60	−	4625.61i	0.424108	−	0.734577i
$342$		0			0
$343$	−1584.83	−	6151.58i	−0.249483	−	0.968379i
$344$		0			0
$345$	−193.190	+	334.615i	−0.0301478	+	0.0522175i
$346$		0			0
$347$	5812.21	+	10067.0i	0.899181	+	1.55743i	0.828543	+	0.559925i	$0.189170\pi$
$347$	5812.21	+	10067.0i	0.899181	+	1.55743i	0.0706381	+	0.997502i	$0.477496\pi$
$348$		0			0
$349$	−9841.79			−1.50951			−0.754755	−	0.656007i	$-0.772244\pi$
$349$	−9841.79			−1.50951			−0.754755	+	0.656007i	$0.772244\pi$
$350$		0			0
$351$	−1664.00			−0.253043
$352$		0			0
$353$	−2265.94	−	3924.73i	−0.341654	−	0.591762i	0.643086	−	0.765794i	$-0.277654\pi$
$353$	−2265.94	−	3924.73i	−0.341654	−	0.591762i	−0.984740	+	0.174032i	$0.944320\pi$
$354$		0			0
$355$	1795.61	−	3110.08i	0.268453	−	0.464975i
$356$		0			0
$357$	261.877	+	3108.44i	0.0388236	+	0.460829i
$358$		0			0
$359$	1140.05	−	1974.63i	0.167603	−	0.290298i	−0.769973	−	0.638076i	$-0.779731\pi$
$359$	1140.05	−	1974.63i	0.167603	−	0.290298i	0.937577	+	0.347779i	$0.113064\pi$
$360$		0			0
$361$	−6286.75	−	10889.0i	−0.916569	−	1.58754i
$362$		0			0
$363$	−1248.19			−0.180476
$364$		0			0
$365$	−680.739			−0.0976205
$366$		0			0
$367$	−3168.76	−	5488.45i	−0.450702	−	0.780639i	0.547728	−	0.836657i	$-0.315493\pi$
$367$	−3168.76	−	5488.45i	−0.450702	−	0.780639i	−0.998430	+	0.0560176i	$0.982160\pi$
$368$		0			0
$369$	1319.87	−	2286.08i	0.186205	−	0.322517i
$370$		0			0
$371$	−343.698	−	161.617i	−0.0480968	−	0.0226166i
$372$		0			0
$373$	1812.48	−	3139.31i	0.251600	−	0.435783i	−0.712367	−	0.701807i	$-0.752377\pi$
$373$	1812.48	−	3139.31i	0.251600	−	0.435783i	0.963966	+	0.266024i	$0.0857100\pi$
$374$		0			0
$375$	1.17253	+	2.03088i	0.000161464	+	0.000279664i
$376$		0			0
$377$	13.4265			0.00183421
$378$		0			0
$379$	268.622			0.0364068			0.0182034	−	0.999834i	$-0.494205\pi$
$379$	268.622			0.0364068			0.0182034	+	0.999834i	$0.494205\pi$
$380$		0			0
$381$	−245.640	−	425.461i	−0.0330302	−	0.0572100i
$382$		0			0
$383$	1068.70	−	1851.03i	0.142579	−	0.246954i	−0.785888	−	0.618369i	$-0.787794\pi$
$383$	1068.70	−	1851.03i	0.142579	−	0.246954i	0.928467	+	0.371415i	$0.121127\pi$
$384$		0			0
$385$	7272.69	−	5057.60i	0.962729	−	0.669504i
$386$		0			0
$387$	−1955.66	+	3387.30i	−0.256878	+	0.444926i
$388$		0			0
$389$	−731.615	−	1267.19i	−0.0953583	−	0.165165i	0.814400	−	0.580304i	$-0.197066\pi$
$389$	−731.615	−	1267.19i	−0.0953583	−	0.165165i	−0.909758	+	0.415139i	$0.863733\pi$
$390$		0			0
$391$	−457.288			−0.0591459
$392$		0			0
$393$	−1565.68			−0.200962
$394$		0			0
$395$	−2443.58	−	4232.40i	−0.311265	−	0.539127i
$396$		0			0
$397$	618.718	−	1071.65i	0.0782180	−	0.135478i	−0.824263	−	0.566207i	$-0.808410\pi$
$397$	618.718	−	1071.65i	0.0782180	−	0.135478i	0.902481	+	0.430729i	$0.141744\pi$
$398$		0			0
$399$	−6358.76	+	4422.03i	−0.797835	+	0.554833i
$400$		0			0
$401$	3089.94	−	5351.94i	0.384799	−	0.666492i	−0.606942	−	0.794746i	$-0.707604\pi$
$401$	3089.94	−	5351.94i	0.384799	−	0.666492i	0.991741	+	0.128254i	$0.0409374\pi$
$402$		0			0
$403$	5441.31	+	9424.63i	0.672583	+	1.16495i
$404$		0			0
$405$	−1280.85			−0.157150
$406$		0			0
$407$	−6384.73			−0.777591
$408$		0			0
$409$	1127.49	+	1952.87i	0.136310	+	0.236096i	0.926097	−	0.377285i	$-0.123142\pi$
$409$	1127.49	+	1952.87i	0.136310	+	0.236096i	−0.789787	+	0.613381i	$0.789809\pi$
$410$		0			0
$411$	−4015.33	+	6954.76i	−0.481902	+	0.834679i
$412$		0			0
$413$	3877.34	+	1823.24i	0.461964	+	0.217230i
$414$		0			0
$415$	9756.50	−	16898.7i	1.15404	−	1.99886i
$416$		0			0
$417$	−1280.54	−	2217.96i	−0.150379	−	0.260465i
$418$		0			0
$419$	−1404.53			−0.163761			−0.0818806	−	0.996642i	$-0.526093\pi$
$419$	−1404.53			−0.163761			−0.0818806	+	0.996642i	$0.526093\pi$
$420$		0			0
$421$	13068.1			1.51283			0.756414	−	0.654094i	$-0.226950\pi$
$421$	13068.1			1.51283			0.756414	+	0.654094i	$0.226950\pi$
$422$		0			0
$423$	−2175.28	−	3767.70i	−0.250037	−	0.433077i
$424$		0			0
$425$	−3510.44	+	6080.26i	−0.400662	+	0.693966i
$426$		0			0
$427$	−1303.99	−	15478.1i	−0.147786	−	1.75419i
$428$		0			0
$429$	−2796.26	+	4843.26i	−0.314696	+	0.545070i
$430$		0			0
$431$	2681.39	+	4644.31i	0.299671	+	0.519046i	0.976061	−	0.217499i	$-0.0697897\pi$
$431$	2681.39	+	4644.31i	0.299671	+	0.519046i	−0.676390	+	0.736544i	$0.736456\pi$
$432$		0			0
$433$	−2495.82			−0.277001			−0.138501	−	0.990362i	$-0.544228\pi$
$433$	−2495.82			−0.277001			−0.138501	+	0.990362i	$0.544228\pi$
$434$		0			0
$435$	10.3349			0.00113913
$436$		0			0
$437$	−567.694	−	983.275i	−0.0621430	−	0.107635i
$438$		0			0
$439$	−1881.06	+	3258.09i	−0.204506	+	0.354215i	−0.949975	−	0.312325i	$-0.898892\pi$
$439$	−1881.06	+	3258.09i	−0.204506	+	0.354215i	0.745469	+	0.666540i	$0.232225\pi$
$440$		0			0
$441$	1969.03	+	2377.50i	0.212615	+	0.256722i
$442$		0			0
$443$	−5722.63	+	9911.88i	−0.613748	+	1.06304i	0.376855	+	0.926272i	$0.377005\pi$
$443$	−5722.63	+	9911.88i	−0.613748	+	1.06304i	−0.990603	+	0.136770i	$0.956328\pi$
$444$		0			0
$445$	9047.83	+	15671.3i	0.963839	+	1.66942i
$446$		0			0
$447$	1673.06			0.177031
$448$		0			0
$449$	−9420.02			−0.990107			−0.495054	−	0.868862i	$-0.664852\pi$
$449$	−9420.02			−0.990107			−0.495054	+	0.868862i	$0.664852\pi$
$450$		0			0
$451$	−4435.93	−	7683.25i	−0.463148	−	0.802196i
$452$		0			0
$453$	−2654.02	+	4596.90i	−0.275269	+	0.476780i
$454$		0			0
$455$	1515.20	+	17985.2i	0.156118	+	1.85310i
$456$		0			0
$457$	−5731.98	+	9928.08i	−0.586719	+	1.01623i	0.407939	+	0.913009i	$0.366247\pi$
$457$	−5731.98	+	9928.08i	−0.586719	+	1.01623i	−0.994659	+	0.103219i	$0.967086\pi$
$458$		0			0
$459$	−757.955	−	1312.82i	−0.0770769	−	0.133501i
$460$		0			0
$461$	−9751.10			−0.985149			−0.492575	−	0.870270i	$-0.663944\pi$
$461$	−9751.10			−0.985149			−0.492575	+	0.870270i	$0.663944\pi$
$462$		0			0
$463$	2182.03			0.219023			0.109512	−	0.993986i	$-0.465071\pi$
$463$	2182.03			0.219023			0.109512	+	0.993986i	$0.465071\pi$
$464$		0			0
$465$	4188.39	+	7254.50i	0.417703	+	0.723483i
$466$		0			0
$467$	−1598.17	+	2768.11i	−0.158361	+	0.274289i	−0.934278	−	0.356546i	$-0.883954\pi$
$467$	−1598.17	+	2768.11i	−0.158361	+	0.274289i	0.775917	+	0.630835i	$0.217288\pi$
$468$		0			0
$469$	10458.8	+	4918.04i	1.02973	+	0.484209i
$470$		0			0
$471$	3607.34	−	6248.10i	0.352904	−	0.611247i
$472$		0			0
$473$	6572.74	+	11384.3i	0.638932	+	1.10666i
$474$		0			0
$475$	−17432.0			−1.68386
$476$		0			0
$477$	184.566			0.0177163
$478$		0			0
$479$	6071.79	+	10516.7i	0.579180	+	1.00317i	0.995574	+	0.0939849i	$0.0299605\pi$
$479$	6071.79	+	10516.7i	0.579180	+	1.00317i	−0.416394	+	0.909184i	$0.636706\pi$
$480$		0			0
$481$	6504.41	−	11266.0i	0.616581	−	1.06795i
$482$		0			0
$483$	−371.525	+	258.367i	−0.0350000	+	0.0243398i
$484$		0			0
$485$	13347.1	−	23117.9i	1.24961	−	2.16439i
$486$		0			0
$487$	10151.2	+	17582.5i	0.944551	+	1.63601i	0.756647	+	0.653824i	$0.226836\pi$
$487$	10151.2	+	17582.5i	0.944551	+	1.63601i	0.187905	+	0.982187i	$0.439830\pi$
$488$		0			0
$489$	10136.4			0.937389
$490$		0			0
$491$	2562.41			0.235519			0.117760	−	0.993042i	$-0.462429\pi$
$491$	2562.41			0.235519			0.117760	+	0.993042i	$0.462429\pi$
$492$		0			0
$493$	6.11577	+	10.5928i	0.000558702	+	0.000967701i
$494$		0			0
$495$	−2152.39	+	3728.05i	−0.195440	+	0.338512i
$496$		0			0
$497$	3453.15	−	2401.40i	0.311660	−	0.216735i
$498$		0			0
$499$	5828.05	−	10094.5i	0.522844	−	0.905592i	−0.476803	−	0.879010i	$-0.658204\pi$
$499$	5828.05	−	10094.5i	0.522844	−	0.905592i	0.999647	−	0.0265820i	$-0.00846232\pi$
$500$		0			0
$501$	1208.85	+	2093.79i	0.107799	+	0.186714i
$502$		0			0
$503$	−18532.8			−1.64281			−0.821407	−	0.570342i	$-0.806811\pi$
$503$	−18532.8			−1.64281			−0.821407	+	0.570342i	$0.806811\pi$
$504$		0			0
$505$	23527.5			2.07318
$506$		0			0
$507$	−2401.85	−	4160.12i	−0.210394	−	0.364413i
$508$		0			0
$509$	−9366.05	+	16222.5i	−0.815605	+	1.41267i	0.0932881	+	0.995639i	$0.470262\pi$
$509$	−9366.05	+	16222.5i	−0.815605	+	1.41267i	−0.908893	+	0.417030i	$0.863071\pi$
$510$		0			0
$511$	−721.499	−	339.271i	−0.0624604	−	0.0293708i
$512$		0			0
$513$	1881.91	−	3259.56i	0.161965	−	0.280532i
$514$		0			0
$515$	−6154.46	−	10659.8i	−0.526598	−	0.912094i
$516$		0			0
$517$	−14621.7			−1.24383
$518$		0			0
$519$	−2521.51			−0.213260
$520$		0			0
$521$	1661.63	+	2878.02i	0.139726	+	0.242012i	0.927393	−	0.374089i	$-0.122045\pi$
$521$	1661.63	+	2878.02i	0.139726	+	0.242012i	−0.787667	+	0.616101i	$0.788711\pi$
$522$		0			0
$523$	11648.7	−	20176.2i	0.973925	−	1.68689i	0.290489	−	0.956878i	$-0.406182\pi$
$523$	11648.7	−	20176.2i	0.973925	−	1.68689i	0.683437	−	0.730010i	$-0.260484\pi$
$524$		0			0
$525$	583.270	+	6923.32i	0.0484876	+	0.575540i
$526$		0			0
$527$	−4957.04	+	8585.84i	−0.409738	+	0.709687i
$528$		0			0
$529$	6050.33	+	10479.5i	0.497274	+	0.861304i
$530$		0			0
$531$	−2082.13			−0.170163
$532$		0			0
$533$	18076.3			1.46899
$534$		0			0
$535$	10294.9	+	17831.2i	0.831936	+	1.44095i
$536$		0			0
$537$	−2406.92	+	4168.90i	−0.193419	+	0.335012i
$538$		0			0
$539$	10228.8	−	1735.82i	0.817413	−	0.138714i
$540$		0			0
$541$	−4525.01	+	7837.54i	−0.359603	+	0.622851i	−0.987894	−	0.155127i	$-0.950421\pi$
$541$	−4525.01	+	7837.54i	−0.359603	+	0.622851i	0.628292	+	0.777978i	$0.283755\pi$
$542$		0			0
$543$	5669.14	+	9819.24i	0.448041	+	0.776029i
$544$		0			0
$545$	−7417.56			−0.582997
$546$		0			0
$547$	−12316.7			−0.962749			−0.481374	−	0.876515i	$-0.659862\pi$
$547$	−12316.7			−0.962749			−0.481374	+	0.876515i	$0.659862\pi$
$548$		0			0
$549$	3774.16	+	6537.03i	0.293401	+	0.508185i
$550$		0			0
$551$	−15.1847	+	26.3006i	−0.00117403	+	0.00203347i
$552$		0			0
$553$	−480.518	−	5703.67i	−0.0369507	−	0.438598i
$554$		0			0
$555$	5006.70	−	8671.86i	0.382923	−	0.663243i
$556$		0			0
$557$	−12452.0	−	21567.6i	−0.947236	−	1.64066i	−0.751212	−	0.660061i	$-0.770530\pi$
$557$	−12452.0	−	21567.6i	−0.947236	−	1.64066i	−0.196024	−	0.980599i	$-0.562803\pi$
$558$		0			0
$559$	−26783.8			−2.02653
$560$		0			0
$561$	−5094.79			−0.383426
$562$		0			0
$563$	2073.30	+	3591.06i	0.155203	+	0.268819i	0.933133	−	0.359532i	$-0.117064\pi$
$563$	2073.30	+	3591.06i	0.155203	+	0.268819i	−0.777930	+	0.628351i	$0.783730\pi$
$564$		0			0
$565$	13076.2	−	22648.7i	0.973666	−	1.68644i
$566$		0			0
$567$	−1357.54	−	638.358i	−0.100549	−	0.0472814i
$568$		0			0
$569$	−3648.74	+	6319.80i	−0.268828	+	0.465624i	−0.968559	−	0.248782i	$-0.919970\pi$
$569$	−3648.74	+	6319.80i	−0.268828	+	0.465624i	0.699731	+	0.714406i	$0.253303\pi$
$570$		0			0
$571$	5782.45	+	10015.5i	0.423797	+	0.734037i	0.996307	−	0.0858600i	$-0.0273638\pi$
$571$	5782.45	+	10015.5i	0.423797	+	0.734037i	−0.572511	+	0.819897i	$0.694030\pi$
$572$		0			0
$573$	−5548.79			−0.404544
$574$		0			0
$575$	−1018.50			−0.0738687
$576$		0			0
$577$	−4736.72	−	8204.25i	−0.341755	−	0.591936i	0.643004	−	0.765863i	$-0.277688\pi$
$577$	−4736.72	−	8204.25i	−0.341755	−	0.591936i	−0.984759	+	0.173926i	$0.944354\pi$
$578$		0			0
$579$	529.504	−	917.128i	0.0380059	−	0.0658282i
$580$		0			0
$581$	18762.8	−	13048.1i	1.33978	−	0.931714i
$582$		0			0
$583$	310.152	−	537.198i	0.0220329	−	0.0381620i
$584$		0			0
$585$	−4385.47	−	7595.86i	−0.309943	−	0.536838i
$586$		0			0
$587$	7336.02			0.515826			0.257913	−	0.966168i	$-0.416965\pi$
$587$	7336.02			0.515826			0.257913	+	0.966168i	$0.416965\pi$
$588$		0			0
$589$	−24615.4			−1.72200
$590$		0			0
$591$	−6672.33	−	11556.8i	−0.464405	−	0.804372i
$592$		0			0
$593$	−9040.73	+	15659.0i	−0.626068	+	1.08438i	0.362265	+	0.932075i	$0.382004\pi$
$593$	−9040.73	+	15659.0i	−0.626068	+	1.08438i	−0.988333	+	0.152307i	$0.951330\pi$
$594$		0			0
$595$	−13499.2	+	9387.68i	−0.930109	+	0.646819i
$596$		0			0
$597$	3360.23	−	5820.08i	0.230360	−	0.398995i
$598$		0			0
$599$	96.2165	+	166.652i	0.00656310	+	0.0113676i	0.869288	−	0.494305i	$-0.164578\pi$
$599$	96.2165	+	166.652i	0.00656310	+	0.0113676i	−0.862725	+	0.505673i	$0.831244\pi$
$600$		0			0
$601$	−5055.76			−0.343143			−0.171571	−	0.985172i	$-0.554884\pi$
$601$	−5055.76			−0.343143			−0.171571	+	0.985172i	$0.554884\pi$
$602$		0			0
$603$	−5616.36			−0.379297
$604$		0			0
$605$	−3289.59	−	5697.74i	−0.221059	−	0.382886i
$606$		0			0
$607$	−10117.9	+	17524.7i	−0.676560	+	1.17184i	0.299451	+	0.954112i	$0.403197\pi$
$607$	−10117.9	+	17524.7i	−0.676560	+	1.17184i	−0.976010	+	0.217724i	$0.930137\pi$
$608$		0			0
$609$	10.9537	+	5.15077i	0.000728845	+	0.000342725i
$610$		0			0
$611$	14895.8	−	25800.2i	0.986283	−	1.70829i
$612$		0			0
$613$	1920.25	+	3325.98i	0.126523	+	0.219143i	0.922327	−	0.386410i	$-0.126285\pi$
$613$	1920.25	+	3325.98i	0.126523	+	0.219143i	−0.795805	+	0.605554i	$0.792952\pi$
$614$		0			0
$615$	13914.0			0.912306
$616$		0			0
$617$	−5720.89			−0.373281			−0.186641	−	0.982428i	$-0.559760\pi$
$617$	−5720.89			−0.373281			−0.186641	+	0.982428i	$0.559760\pi$
$618$		0			0
$619$	−11470.1	−	19866.8i	−0.744787	−	1.29001i	−0.950294	−	0.311354i	$-0.899218\pi$
$619$	−11470.1	−	19866.8i	−0.744787	−	1.29001i	0.205507	−	0.978656i	$-0.434116\pi$
$620$		0			0
$621$	109.955	−	190.447i	0.00710521	−	0.0123066i
$622$		0			0
$623$	1779.22	+	21119.0i	0.114419	+	1.35813i
$624$		0			0
$625$	7809.41	−	13526.3i	0.499802	−	0.865683i
$626$		0			0
$627$	−6324.86	−	10955.0i	−0.402856	−	0.697767i
$628$		0			0
$629$	11851.0			0.751244
$630$		0			0
$631$	7634.81			0.481675			0.240837	−	0.970565i	$-0.422578\pi$
$631$	7634.81			0.481675			0.240837	+	0.970565i	$0.422578\pi$
$632$		0			0
$633$	−1836.50	−	3180.92i	−0.115315	−	0.199732i
$634$		0			0
$635$	1294.76	−	2242.60i	0.0809152	−	0.140149i
$636$		0			0
$637$	−7357.65	+	19817.2i	−0.457646	+	1.23263i
$638$		0			0
$639$	−1021.98	+	1770.12i	−0.0632688	+	0.109585i
$640$		0			0
$641$	1389.80	+	2407.20i	0.0856376	+	0.148329i	0.905663	−	0.423999i	$-0.139374\pi$
$641$	1389.80	+	2407.20i	0.0856376	+	0.148329i	−0.820025	+	0.572328i	$0.806041\pi$
$642$		0			0
$643$	18304.5			1.12264			0.561322	−	0.827598i	$-0.310293\pi$
$643$	18304.5			1.12264			0.561322	+	0.827598i	$0.310293\pi$
$644$		0			0
$645$	−20616.5			−1.25856
$646$		0			0
$647$	−3276.27	−	5674.66i	−0.199078	−	0.344813i	0.749152	−	0.662398i	$-0.230461\pi$
$647$	−3276.27	−	5674.66i	−0.199078	−	0.344813i	−0.948230	+	0.317585i	$0.897128\pi$
$648$		0			0
$649$	−3498.89	+	6060.26i	−0.211623	+	0.366542i
$650$		0			0
$651$	823.628	+	9776.32i	0.0495861	+	0.588578i
$652$		0			0
$653$	15993.0	−	27700.8i	0.958432	−	1.66005i	0.232120	−	0.972687i	$-0.425434\pi$
$653$	15993.0	−	27700.8i	0.958432	−	1.66005i	0.726312	−	0.687365i	$-0.241233\pi$
$654$		0			0
$655$	−4126.34	−	7147.03i	−0.246152	−	0.426348i
$656$		0			0
$657$	387.445			0.0230071
$658$		0			0
$659$	517.327			0.0305799			0.0152900	−	0.999883i	$-0.495133\pi$
$659$	517.327			0.0305799			0.0152900	+	0.999883i	$0.495133\pi$
$660$		0			0
$661$	−9828.40	−	17023.3i	−0.578336	−	1.00171i	−0.995670	−	0.0929549i	$-0.970369\pi$
$661$	−9828.40	−	17023.3i	−0.578336	−	1.00171i	0.417334	−	0.908753i	$-0.362965\pi$
$662$		0			0
$663$	5190.29	−	8989.84i	0.304033	−	0.526601i
$664$		0			0
$665$	−36944.1	−	17372.3i	−2.15433	−	1.01303i
$666$		0			0
$667$	−0.887200	+	1.53668i	−5.15030e−5	+	8.92059e-5i
$668$		0			0
$669$	5186.05	+	8982.50i	0.299707	+	0.519108i
$670$		0			0
$671$	25369.0			1.45955
$672$		0			0
$673$	25836.1			1.47981			0.739904	−	0.672713i	$-0.234871\pi$
$673$	25836.1			1.47981			0.739904	+	0.672713i	$0.234871\pi$
$674$		0			0
$675$	−1688.17	−	2923.99i	−0.0962631	−	0.166733i
$676$		0			0
$677$	−13263.1	+	22972.4i	−0.752944	+	1.30414i	0.193447	+	0.981111i	$0.438033\pi$
$677$	−13263.1	+	22972.4i	−0.752944	+	1.30414i	−0.946390	+	0.323026i	$0.895300\pi$
$678$		0			0
$679$	25667.9	−	17850.1i	1.45073	−	1.00887i
$680$		0			0
$681$	7767.33	−	13453.4i	0.437070	−	0.757027i
$682$		0			0
$683$	−16939.4	−	29339.9i	−0.949002	−	1.64372i	−0.747532	−	0.664225i	$-0.768761\pi$
$683$	−16939.4	−	29339.9i	−0.949002	−	1.64372i	−0.201470	−	0.979495i	$-0.564572\pi$
$684$		0			0
$685$	−42329.5			−2.36106
$686$		0			0
$687$	6935.65			0.385170
$688$		0			0
$689$	631.930	+	1094.53i	0.0349414	+	0.0605203i
$690$		0			0
$691$	500.631	−	867.119i	0.0275614	−	0.0477377i	−0.851916	−	0.523679i	$-0.824559\pi$
$691$	500.631	−	867.119i	0.0275614	−	0.0477377i	0.879477	+	0.475941i	$0.157892\pi$
$692$		0			0
$693$	−4139.28	+	2878.55i	−0.226895	+	0.157788i
$694$		0			0
$695$	6749.70	−	11690.8i	0.368389	−	0.638069i
$696$		0			0
$697$	8233.76	+	14261.3i	0.447455	+	0.775015i
$698$		0			0
$699$	14228.9			0.769939
$700$		0			0
$701$	7713.38			0.415592			0.207796	−	0.978172i	$-0.433371\pi$
$701$	7713.38			0.415592			0.207796	+	0.978172i	$0.433371\pi$
$702$		0			0
$703$	14712.3	+	25482.5i	0.789312	+	1.36713i
$704$		0			0
$705$	11465.9	−	19859.5i	0.612524	−	1.06092i
$706$		0			0
$707$	24936.2	+	11725.8i	1.32648	+	0.623753i
$708$		0			0
$709$	−5017.56	+	8690.67i	−0.265781	+	0.460346i	−0.967768	−	0.251844i	$-0.918963\pi$
$709$	−5017.56	+	8690.67i	−0.265781	+	0.460346i	0.701987	+	0.712190i	$0.252296\pi$
$710$		0			0
$711$	1390.77	+	2408.89i	0.0733586	+	0.127061i
$712$		0			0
$713$	−1438.21			−0.0755421
$714$		0			0
$715$	−29478.1			−1.54184
$716$		0			0
$717$	2118.89	+	3670.02i	0.110364	+	0.191157i
$718$		0			0
$719$	10059.7	−	17423.9i	0.521785	−	0.903758i	−0.477894	−	0.878417i	$-0.658600\pi$
$719$	10059.7	−	17423.9i	0.521785	−	0.903758i	0.999679	−	0.0253401i	$-0.00806688\pi$
$720$		0			0
$721$	−1210.25	−	14365.4i	−0.0625131	−	0.742020i
$722$		0			0
$723$	2821.73	−	4887.38i	0.145147	−	0.251402i
$724$		0			0
$725$	13.6214	+	23.5930i	0.000697776	+	0.00120858i
$726$		0			0
$727$	−7567.09			−0.386035			−0.193018	−	0.981195i	$-0.561827\pi$
$727$	−7567.09			−0.386035			−0.193018	+	0.981195i	$0.561827\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−12200.0	−	21131.0i	−0.617283	−	1.06917i
$732$		0			0
$733$	−15400.0	+	26673.6i	−0.776006	+	1.34408i	0.158222	+	0.987404i	$0.449424\pi$
$733$	−15400.0	+	26673.6i	−0.776006	+	1.34408i	−0.934228	+	0.356678i	$0.883909\pi$
$734$		0			0
$735$	−5663.47	+	15254.1i	−0.284218	+	0.765518i
$736$		0			0
$737$	−9437.96	+	16347.0i	−0.471712	+	0.817029i
$738$		0			0
$739$	6659.66	+	11534.9i	0.331501	+	0.574177i	0.982806	−	0.184639i	$-0.0591115\pi$
$739$	6659.66	+	11534.9i	0.331501	+	0.574177i	−0.651305	+	0.758816i	$0.725778\pi$
$740$		0			0
$741$	25773.7			1.27776
$742$		0			0
$743$	−10747.7			−0.530682			−0.265341	−	0.964155i	$-0.585484\pi$
$743$	−10747.7			−0.530682			−0.265341	+	0.964155i	$0.585484\pi$
$744$		0			0
$745$	4409.33	+	7637.18i	0.216839	+	0.375577i
$746$		0			0
$747$	−5552.95	+	9617.99i	−0.271984	+	0.471089i
$748$		0			0
$749$	2024.44	+	24029.7i	0.0987601	+	1.17227i
$750$		0			0
$751$	−6106.70	+	10577.1i	−0.296720	+	0.513934i	−0.975383	−	0.220515i	$-0.929226\pi$
$751$	−6106.70	+	10577.1i	−0.296720	+	0.513934i	0.678664	+	0.734449i	$0.262559\pi$
$752$		0			0
$753$	−6779.46	−	11742.4i	−0.328097	−	0.568281i
$754$		0			0
$755$	−27978.6			−1.34867
$756$		0			0
$757$	−30173.2			−1.44870			−0.724349	−	0.689434i	$-0.757860\pi$
$757$	−30173.2			−1.44870			−0.724349	+	0.689434i	$0.757860\pi$
$758$		0			0
$759$	−369.545	−	640.070i	−0.0176728	−	0.0306101i
$760$		0			0
$761$	−7603.67	+	13169.9i	−0.362198	+	0.627346i	−0.988322	−	0.152378i	$-0.951307\pi$
$761$	−7603.67	+	13169.9i	−0.362198	+	0.627346i	0.626124	+	0.779723i	$0.284640\pi$
$762$		0			0
$763$	−7861.71	−	3696.82i	−0.373018	−	0.175405i
$764$		0			0
$765$	3995.17	−	6919.83i	0.188818	−	0.327042i
$766$		0			0
$767$	−7128.95	−	12347.7i	−0.335608	−	0.581290i
$768$		0			0
$769$	−9368.65			−0.439327			−0.219663	−	0.975576i	$-0.570496\pi$
$769$	−9368.65			−0.439327			−0.219663	+	0.975576i	$0.570496\pi$
$770$		0			0
$771$	23738.0			1.10882
$772$		0			0
$773$	−16258.2	−	28159.9i	−0.756488	−	1.31028i	−0.944631	−	0.328134i	$-0.893580\pi$
$773$	−16258.2	−	28159.9i	−0.756488	−	1.31028i	0.188143	−	0.982142i	$-0.439753\pi$
$774$		0			0
$775$	−11040.6	+	19123.0i	−0.511731	+	0.886345i
$776$		0			0
$777$	9628.42	−	6695.83i	0.444553	−	0.309152i
$778$		0			0
$779$	−20443.4	+	35409.0i	−0.940258	+	1.62857i
$780$		0			0
$781$	3434.74	+	5949.15i	0.157368	+	0.272570i
$782$		0			0
$783$	−5.88214			−0.000268468
$784$		0			0
$785$	38028.5			1.72904
$786$		0			0
$787$	20125.1	+	34857.8i	0.911542	+	1.57884i	0.811887	+	0.583815i	$0.198441\pi$
$787$	20125.1	+	34857.8i	0.911542	+	1.57884i	0.0996552	+	0.995022i	$0.468226\pi$
$788$		0			0
$789$	5146.74	−	8914.42i	0.232229	−	0.402233i
$790$		0			0
$791$	25147.0	−	17487.8i	1.13037	−	0.786087i
$792$		0			0
$793$	−25844.5	+	44764.0i	−1.15733	+	2.00456i
$794$		0			0
$795$	486.421	+	842.506i	0.0217001	+	0.0375857i
$796$		0			0
$797$	21387.5			0.950544			0.475272	−	0.879839i	$-0.342350\pi$
$797$	21387.5			0.950544			0.475272	+	0.879839i	$0.342350\pi$
$798$		0			0
$799$	27140.1			1.20169
$800$		0			0
$801$	−5149.61	−	8919.38i	−0.227157	−	0.393447i
$802$		0			0
$803$	651.078	−	1127.70i	0.0286128	−	0.0495587i
$804$		0			0
$805$	−2158.55	−	1015.02i	−0.0945078	−	0.0444405i
$806$		0			0
$807$	−12448.8	+	21561.9i	−0.543021	+	0.940540i
$808$		0			0
$809$	9817.06	+	17003.6i	0.426637	+	0.738957i	0.996572	−	0.0827330i	$-0.0263649\pi$
$809$	9817.06	+	17003.6i	0.426637	+	0.738957i	−0.569935	+	0.821690i	$0.693032\pi$
$810$		0			0
$811$	35238.1			1.52574			0.762872	−	0.646549i	$-0.223789\pi$
$811$	35238.1			1.52574			0.762872	+	0.646549i	$0.223789\pi$
$812$		0			0
$813$	−17773.5			−0.766719
$814$		0			0
$815$	26714.4	+	46270.6i	1.14818	+	1.98870i
$816$		0			0
$817$	30291.1	−	52465.7i	1.29713	−	2.24669i
$818$		0			0
$819$	−862.383	−	10236.3i	−0.0367938	−	0.436736i
$820$		0			0
$821$	3527.40	−	6109.63i	0.149948	−	0.259717i	−0.781260	−	0.624205i	$-0.785423\pi$
$821$	3527.40	−	6109.63i	0.149948	−	0.259717i	0.931208	+	0.364488i	$0.118756\pi$
$822$		0			0
$823$	−3094.16	−	5359.25i	−0.131052	−	0.226989i	0.793030	−	0.609182i	$-0.208502\pi$
$823$	−3094.16	−	5359.25i	−0.131052	−	0.226989i	−0.924082	+	0.382193i	$0.875169\pi$
$824$		0			0
$825$	−11347.5			−0.478870
$826$		0			0
$827$	−24000.0			−1.00915			−0.504573	−	0.863369i	$-0.668350\pi$
$827$	−24000.0			−1.00915			−0.504573	+	0.863369i	$0.668350\pi$
$828$		0			0
$829$	4354.43	+	7542.09i	0.182431	+	0.315980i	0.942708	−	0.333619i	$-0.108270\pi$
$829$	4354.43	+	7542.09i	0.182431	+	0.315980i	−0.760277	+	0.649599i	$0.774937\pi$
$830$		0			0
$831$	−2280.90	+	3950.63i	−0.0952149	+	0.164917i
$832$		0			0
$833$	−18986.2	+	3221.94i	−0.789716	+	0.134014i
$834$		0			0
$835$	−6371.83	+	11036.3i	−0.264079	+	0.457399i
$836$		0			0
$837$	−2383.84	−	4128.93i	−0.0984438	−	0.170510i
$838$		0			0
$839$	7820.46			0.321802			0.160901	−	0.986971i	$-0.448560\pi$
$839$	7820.46			0.321802			0.160901	+	0.986971i	$0.448560\pi$
$840$		0			0
$841$	−24389.0			−0.999998
$842$		0			0
$843$	528.499	+	915.388i	0.0215925	+	0.0373993i
$844$		0			0
$845$	12660.1	−	21927.9i	0.515409	−	0.892714i
$846$		0			0
$847$	−646.884	−	7678.39i	−0.0262422	−	0.311491i
$848$		0			0
$849$	−6432.73	+	11141.8i	−0.260036	+	0.450396i
$850$		0			0
$851$	859.602	+	1488.87i	0.0346261	+	0.0599741i
$852$		0			0
$853$	31176.8			1.25143			0.625717	−	0.780050i	$-0.284807\pi$
$853$	31176.8			1.25143			0.625717	+	0.780050i	$0.284807\pi$
$854$		0			0
$855$	19839.0			0.793543
$856$		0			0
$857$	−447.864	−	775.724i	−0.0178515	−	0.0309197i	0.856962	−	0.515380i	$-0.172349\pi$
$857$	−447.864	−	775.724i	−0.0178515	−	0.0309197i	−0.874813	+	0.484460i	$0.839016\pi$
$858$		0			0
$859$	−4557.51	+	7893.83i	−0.181025	+	0.313544i	−0.942230	−	0.334967i	$-0.891275\pi$
$859$	−4557.51	+	7893.83i	−0.181025	+	0.313544i	0.761205	+	0.648511i	$0.224608\pi$
$860$		0			0
$861$	14747.2	+	6934.57i	0.583719	+	0.274483i
$862$		0			0
$863$	10563.3	−	18296.2i	0.416662	−	0.721680i	−0.578939	−	0.815371i	$-0.696533\pi$
$863$	10563.3	−	18296.2i	0.416662	−	0.721680i	0.995601	+	0.0936906i	$0.0298664\pi$
$864$		0			0
$865$	−6645.42	−	11510.2i	−0.261215	−	0.452438i
$866$		0			0
$867$	−5282.29			−0.206916
$868$		0			0
$869$	9348.43			0.364930
$870$		0			0
$871$	−19229.7	−	33306.9i	−0.748076	−	1.29571i
$872$		0			0
$873$	−7596.55	+	13157.6i	−0.294507	+	0.510100i
$874$		0			0
$875$	−11.8855	+	8.26547i	−0.000459205	+	0.000319342i
$876$		0			0
$877$	9374.72	−	16237.5i	0.360960	−	0.625201i	−0.627159	−	0.778891i	$-0.715782\pi$
$877$	9374.72	−	16237.5i	0.360960	−	0.625201i	0.988119	+	0.153690i	$0.0491158\pi$
$878$		0			0
$879$	−3992.98	−	6916.04i	−0.153219	−	0.265384i
$880$		0			0
$881$	37000.3			1.41495			0.707475	−	0.706739i	$-0.249834\pi$
$881$	37000.3			1.41495			0.707475	+	0.706739i	$0.249834\pi$
$882$		0			0
$883$	21618.3			0.823913			0.411956	−	0.911204i	$-0.364846\pi$
$883$	21618.3			0.823913			0.411956	+	0.911204i	$0.364846\pi$
$884$		0			0
$885$	−5487.43	−	9504.51i	−0.208427	−	0.361006i
$886$		0			0
$887$	−4151.85	+	7191.21i	−0.157165	+	0.272218i	−0.933845	−	0.357677i	$-0.883569\pi$
$887$	−4151.85	+	7191.21i	−0.157165	+	0.272218i	0.776680	+	0.629895i	$0.216902\pi$
$888$		0			0
$889$	2489.97	−	1731.58i	0.0939381	−	0.0653267i
$890$		0			0
$891$	1225.04	−	2121.83i	0.0460611	−	0.0797801i
$892$		0			0
$893$	33692.8	+	58357.6i	1.26258	+	2.18686i
$894$		0			0
$895$	−25373.6			−0.947650
$896$		0			0
$897$	1505.89			0.0560536
$898$		0			0
$899$	19.2346	+	33.3154i	0.000713583	+	0.00123596i
$900$		0			0
$901$	−575.689	+	997.122i	−0.0212863	+	0.0368690i
$902$		0			0
$903$	−21851.0	−	10275.0i	−0.805265	−	0.378661i
$904$		0			0
$905$	−29881.9	+	51757.0i	−1.09758	+	1.90106i
$906$		0			0
$907$	17272.2	+	29916.4i	0.632321	+	1.09521i	0.987076	+	0.160253i	$0.0512310\pi$
$907$	17272.2	+	29916.4i	0.632321	+	1.09521i	−0.354755	+	0.934959i	$0.615436\pi$
$908$		0			0
$909$	−13390.7			−0.488606
$910$		0			0
$911$	−10505.3			−0.382059			−0.191030	−	0.981584i	$-0.561183\pi$
$911$	−10505.3			−0.382059			−0.191030	+	0.981584i	$0.561183\pi$
$912$		0			0
$913$	18662.8	+	32324.9i	0.676504	+	1.17174i
$914$		0			0
$915$	−19893.5	+	34456.6i	−0.718753	+	1.24492i
$916$		0			0
$917$	−811.426	−	9631.49i	−0.0292210	−	0.346848i
$918$		0			0
$919$	−16100.8	+	27887.4i	−0.577929	+	1.00100i	0.417787	+	0.908545i	$0.362806\pi$
$919$	−16100.8	+	27887.4i	−0.577929	+	1.00100i	−0.995717	+	0.0924578i	$0.970528\pi$
$920$		0			0
$921$	−218.955	−	379.241i	−0.00783366	−	0.0135683i
$922$		0			0
$923$	−13996.5			−0.499133
$924$		0			0
$925$	26395.4			0.938245
$926$		0			0
$927$	3502.83	+	6067.09i	0.124108	+	0.214961i
$928$		0			0
$929$	18517.4	−	32073.1i	0.653968	−	1.13271i	−0.328183	−	0.944614i	$-0.606436\pi$
$929$	18517.4	−	32073.1i	0.653968	−	1.13271i	0.982151	−	0.188092i	$-0.0602303\pi$
$930$		0			0
$931$	−30498.1	−	36825.0i	−1.07362	−	1.29634i
$932$		0			0
$933$	−5124.77	+	8876.36i	−0.179826	+	0.311467i
$934$		0			0
$935$	−13427.3	−	23256.7i	−0.469646	−	0.813451i
$936$		0			0
$937$	−7207.18			−0.251279			−0.125639	−	0.992076i	$-0.540098\pi$
$937$	−7207.18			−0.251279			−0.125639	+	0.992076i	$0.540098\pi$
$938$		0			0
$939$	11047.1			0.383927
$940$		0			0
$941$	10859.4	+	18809.1i	0.376203	+	0.651603i	0.990506	−	0.137467i	$-0.0438961\pi$
$941$	10859.4	+	18809.1i	0.376203	+	0.651603i	−0.614303	+	0.789070i	$0.710563\pi$
$942$		0			0
$943$	−1194.45	+	2068.85i	−0.0412479	+	0.0714434i
$944$		0			0
$945$	−663.810	−	7879.31i	−0.0228505	−	0.271232i
$946$		0			0
$947$	65.6714	−	113.746i	0.00225347	−	0.00390312i	−0.864896	−	0.501950i	$-0.832616\pi$
$947$	65.6714	−	113.746i	0.00225347	−	0.00390312i	0.867150	+	0.498047i	$0.165949\pi$
$948$		0			0
$949$	1326.56	+	2297.68i	0.0453763	+	0.0785940i
$950$		0			0
$951$	−4130.33			−0.140836
$952$		0			0
$953$	−39810.0			−1.35317			−0.676586	−	0.736363i	$-0.736541\pi$
$953$	−39810.0			−1.35317			−0.676586	+	0.736363i	$0.736541\pi$
$954$		0			0
$955$	−14623.8	−	25329.1i	−0.495512	−	0.858253i
$956$		0			0
$957$	−9.88458	+	17.1206i	−0.000333880	+	0.000578297i
$958$		0			0
$959$	−44864.1	−	21096.5i	−1.51067	−	0.710366i
$960$		0			0
$961$	−694.846	+	1203.51i	−0.0233240	+	0.0403984i
$962$		0			0
$963$	−5859.36	−	10148.7i	−0.196070	−	0.339603i
$964$		0			0
$965$	5582.01			0.186209
$966$		0			0
$967$	18858.5			0.627145			0.313573	−	0.949564i	$-0.398474\pi$
$967$	18858.5			0.627145			0.313573	+	0.949564i	$0.398474\pi$
$968$		0			0
$969$	11739.9	+	20334.1i	0.389206	+	0.674124i
$970$		0			0
$971$	24177.5	−	41876.7i	0.799067	−	1.38402i	−0.121158	−	0.992633i	$-0.538661\pi$
$971$	24177.5	−	41876.7i	0.799067	−	1.38402i	0.920225	−	0.391391i	$-0.128006\pi$
$972$		0			0
$973$	12980.4	−	9026.87i	0.427680	−	0.297418i
$974$		0			0
$975$	11560.2	−	20022.8i	0.379714	−	0.657684i
$976$		0			0
$977$	2542.86	+	4404.36i	0.0832685	+	0.144225i	0.904652	−	0.426151i	$-0.140131\pi$
$977$	2542.86	+	4404.36i	0.0832685	+	0.144225i	−0.821384	+	0.570376i	$0.806797\pi$
$978$		0			0
$979$	−34614.4			−1.13001
$980$		0			0
$981$	4221.73			0.137400
$982$		0			0
$983$	−24308.9	−	42104.2i	−0.788741	−	1.36614i	−0.926739	−	0.375707i	$-0.877400\pi$
$983$	−24308.9	−	42104.2i	−0.788741	−	1.36614i	0.137998	−	0.990433i	$-0.455933\pi$
$984$		0			0
$985$	35169.8	−	60915.8i	1.13767	−	1.97050i
$986$		0			0
$987$	22050.1	−	15334.1i	0.711107	−	0.494520i
$988$		0			0
$989$	1769.83	−	3065.43i	0.0569032	−	0.0985592i
$990$		0			0
$991$	−17500.0	−	30310.9i	−0.560955	−	0.971603i	−0.997413	−	0.0718786i	$-0.977101\pi$
$991$	−17500.0	−	30310.9i	−0.560955	−	0.971603i	0.436458	−	0.899725i	$-0.356233\pi$
$992$		0			0
$993$	6524.79			0.208517
$994$		0			0
$995$	35423.4			1.12864
$996$		0			0
$997$	−16403.9	−	28412.4i	−0.521081	−	0.902539i	−0.999699	−	0.0245157i	$-0.992196\pi$
$997$	−16403.9	−	28412.4i	−0.521081	−	0.902539i	0.478618	−	0.878023i	$-0.341138\pi$
$998$		0			0
$999$	−2849.58	+	4935.62i	−0.0902470	+	0.156312i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.4.q.f.121.4	yes	8
3.2	odd	2		504.4.s.j.289.1		8
4.3	odd	2		336.4.q.m.289.4		8
7.2	even	3		1176.4.a.bd.1.1		4
7.4	even	3	inner	168.4.q.f.25.4	✓	8
7.5	odd	6		1176.4.a.ba.1.4		4
21.11	odd	6		504.4.s.j.361.1		8
28.11	odd	6		336.4.q.m.193.4		8
28.19	even	6		2352.4.a.cp.1.4		4
28.23	odd	6		2352.4.a.cm.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.f.25.4	✓	8	7.4	even	3	inner
168.4.q.f.121.4	yes	8	1.1	even	1	trivial
336.4.q.m.193.4		8	28.11	odd	6
336.4.q.m.289.4		8	4.3	odd	2
504.4.s.j.289.1		8	3.2	odd	2
504.4.s.j.361.1		8	21.11	odd	6
1176.4.a.ba.1.4		4	7.5	odd	6
1176.4.a.bd.1.1		4	7.2	even	3
2352.4.a.cm.1.1		4	28.23	odd	6
2352.4.a.cp.1.4		4	28.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	168.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{3} +(7.90648 - 13.6944i) q^{5} +(15.2050 - 10.5739i) q^{7} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)	\(q+(-1.50000 - 2.59808i) q^{3} +(7.90648 - 13.6944i) q^{5} +(15.2050 - 10.5739i) q^{7} +(-4.50000 + 7.79423i) q^{9} +(15.1240 + 26.1955i) q^{11} -61.6298 q^{13} -47.4389 q^{15} +(-28.0724 - 48.6228i) q^{17} +(69.7002 - 120.724i) q^{19} +(-50.2793 - 23.6429i) q^{21} +(4.07240 - 7.05360i) q^{23} +(-62.5247 - 108.296i) q^{25} +27.0000 q^{27} -0.217857 q^{29} +(-88.2903 - 152.923i) q^{31} +(45.3719 - 78.5864i) q^{33} +(-24.5855 - 291.826i) q^{35} +(-105.540 + 182.801i) q^{37} +(92.4447 + 160.119i) q^{39} -293.305 q^{41} +434.591 q^{43} +(71.1583 + 123.250i) q^{45} +(-241.698 + 418.633i) q^{47} +(119.385 - 321.553i) q^{49} +(-84.2172 + 145.868i) q^{51} +(-10.2536 - 17.7598i) q^{53} +478.309 q^{55} -418.201 q^{57} +(115.674 + 200.353i) q^{59} +(419.351 - 726.337i) q^{61} +(13.9930 + 166.094i) q^{63} +(-487.274 + 843.984i) q^{65} +(312.020 + 540.435i) q^{67} -24.4344 q^{69} +227.106 q^{71} +(-21.5247 - 37.2819i) q^{73} +(-187.574 + 324.888i) q^{75} +(506.949 + 238.383i) q^{77} +(154.530 - 267.654i) q^{79} +(-40.5000 - 70.1481i) q^{81} +1233.99 q^{83} -887.815 q^{85} +(0.326785 + 0.566009i) q^{87} +(-572.179 + 991.042i) q^{89} +(-937.082 + 651.668i) q^{91} +(-264.871 + 458.770i) q^{93} +(-1102.17 - 1909.01i) q^{95} +1688.12 q^{97} -272.231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9}+O(q^{10}) \) 8 * q - 12 * q^3 - 4 * q^5 + 18 * q^7 - 36 * q^9	\( 8 q - 12 q^{3} - 4 q^{5} + 18 q^{7} - 36 q^{9} - 14 q^{11} + 44 q^{13} + 24 q^{15} - 96 q^{17} + 26 q^{19} - 36 q^{21} - 96 q^{23} - 110 q^{25} + 216 q^{27} - 152 q^{29} - 238 q^{31} - 42 q^{33} + 152 q^{35} - 562 q^{37} - 66 q^{39} + 856 q^{41} - 516 q^{43} - 36 q^{45} + 80 q^{47} + 156 q^{49} - 288 q^{51} + 2952 q^{55} - 156 q^{57} - 262 q^{59} + 276 q^{61} - 54 q^{63} - 2196 q^{65} - 150 q^{67} + 576 q^{69} - 1696 q^{71} + 218 q^{73} - 330 q^{75} - 764 q^{77} - 1762 q^{79} - 324 q^{81} + 6900 q^{83} + 2904 q^{85} + 228 q^{87} + 344 q^{89} - 2806 q^{91} - 714 q^{93} - 2004 q^{95} - 1244 q^{97} + 252 q^{99}+O(q^{100}) \) 8 * q - 12 * q^3 - 4 * q^5 + 18 * q^7 - 36 * q^9 - 14 * q^11 + 44 * q^13 + 24 * q^15 - 96 * q^17 + 26 * q^19 - 36 * q^21 - 96 * q^23 - 110 * q^25 + 216 * q^27 - 152 * q^29 - 238 * q^31 - 42 * q^33 + 152 * q^35 - 562 * q^37 - 66 * q^39 + 856 * q^41 - 516 * q^43 - 36 * q^45 + 80 * q^47 + 156 * q^49 - 288 * q^51 + 2952 * q^55 - 156 * q^57 - 262 * q^59 + 276 * q^61 - 54 * q^63 - 2196 * q^65 - 150 * q^67 + 576 * q^69 - 1696 * q^71 + 218 * q^73 - 330 * q^75 - 764 * q^77 - 1762 * q^79 - 324 * q^81 + 6900 * q^83 + 2904 * q^85 + 228 * q^87 + 344 * q^89 - 2806 * q^91 - 714 * q^93 - 2004 * q^95 - 1244 * q^97 + 252 * q^99

Introduction

L-functions

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