LMFDB - Embedded newform 168.5.z.a.73.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,5,Mod(73,168)]

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, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("168.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	168.z (of order $6$, 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:	$17.3661537981$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 99 x^{14} - 1810 x^{13} + 14212 x^{12} - 199882 x^{11} + 1800935 x^{10} + \cdots + 41390114348800 $ x^16 - 6x^15 + 99x^14 - 1810x^13 + 14212x^12 - 199882x^11 + 1800935x^10 - 9073270x^9 + 62016647x^8 - 13497916x^7 - 872612906x^6 + 8208467240x^5 + 11144877252x^4 - 185824445600x^3 + 3750935164160x^2 - 13970712492800*x + 41390114348800
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{36}\cdot 7^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			73.2
Root			$-5.43765 - 1.98832i$ of defining polynomial
Character	$\chi$	$=$	168.73
Dual form			168.5.z.a.145.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 - 2.59808i) q^{3} +(-31.0314 + 17.9160i) q^{5} +(-41.3748 + 26.2512i) q^{7} +(13.5000 + 23.3827i) q^{9} +O(q^{10})$	\(q+(-4.50000 - 2.59808i) q^{3} +(-31.0314 + 17.9160i) q^{5} +(-41.3748 + 26.2512i) q^{7} +(13.5000 + 23.3827i) q^{9} +(21.0862 - 36.5224i) q^{11} -22.7779i q^{13} +186.188 q^{15} +(111.418 + 64.3274i) q^{17} +(338.683 - 195.539i) q^{19} +(254.389 - 10.6353i) q^{21} +(-331.549 - 574.260i) q^{23} +(329.465 - 570.651i) q^{25} -140.296i q^{27} -604.215 q^{29} +(1100.33 + 635.273i) q^{31} +(-189.776 + 109.567i) q^{33} +(813.603 - 1555.88i) q^{35} +(-1069.76 - 1852.88i) q^{37} +(-59.1788 + 102.501i) q^{39} +244.193i q^{41} +3480.28 q^{43} +(-837.848 - 483.732i) q^{45} +(114.489 - 66.1002i) q^{47} +(1022.75 - 2172.27i) q^{49} +(-334.255 - 578.946i) q^{51} +(481.312 - 833.657i) q^{53} +1511.12i q^{55} -2032.10 q^{57} +(5542.71 + 3200.09i) q^{59} +(-524.016 + 302.541i) q^{61} +(-1172.38 - 613.064i) q^{63} +(408.089 + 706.831i) q^{65} +(1596.09 - 2764.51i) q^{67} +3445.56i q^{69} -6484.24 q^{71} +(526.915 + 304.214i) q^{73} +(-2965.19 + 1711.95i) q^{75} +(86.3170 + 2064.64i) q^{77} +(596.115 + 1032.50i) q^{79} +(-364.500 + 631.333i) q^{81} -4573.10i q^{83} -4609.95 q^{85} +(2718.97 + 1569.80i) q^{87} +(-9458.27 + 5460.73i) q^{89} +(597.947 + 942.433i) q^{91} +(-3300.98 - 5717.46i) q^{93} +(-7006.54 + 12135.7i) q^{95} +9241.16i q^{97} +1138.66 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) $ 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9	$ 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9} - 108 q^{11} - 72 q^{15} - 168 q^{17} + 948 q^{19} - 252 q^{21} - 768 q^{23} + 1908 q^{25} - 1608 q^{29} - 3216 q^{31} + 972 q^{33} + 696 q^{35} - 1820 q^{37} - 1188 q^{39} + 2888 q^{43} + 324 q^{45} + 744 q^{47} - 3784 q^{49} + 504 q^{51} + 4476 q^{53} - 5688 q^{57} - 4668 q^{59} + 17760 q^{61} + 1188 q^{63} + 8760 q^{65} + 1580 q^{67} + 48 q^{71} + 588 q^{73} - 17172 q^{75} - 17508 q^{77} - 3824 q^{79} - 5832 q^{81} + 11440 q^{85} + 7236 q^{87} - 360 q^{89} + 25860 q^{91} + 9648 q^{93} - 21792 q^{95} - 5832 q^{99}+O(q^{100}) $ 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9 - 108 * q^11 - 72 * q^15 - 168 * q^17 + 948 * q^19 - 252 * q^21 - 768 * q^23 + 1908 * q^25 - 1608 * q^29 - 3216 * q^31 + 972 * q^33 + 696 * q^35 - 1820 * q^37 - 1188 * q^39 + 2888 * q^43 + 324 * q^45 + 744 * q^47 - 3784 * q^49 + 504 * q^51 + 4476 * q^53 - 5688 * q^57 - 4668 * q^59 + 17760 * q^61 + 1188 * q^63 + 8760 * q^65 + 1580 * q^67 + 48 * q^71 + 588 * q^73 - 17172 * q^75 - 17508 * q^77 - 3824 * q^79 - 5832 * q^81 + 11440 * q^85 + 7236 * q^87 - 360 * q^89 + 25860 * q^91 + 9648 * q^93 - 21792 * q^95 - 5832 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$	$1$

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
$3$	−4.50000	−	2.59808i	−0.500000	−	0.288675i
$3$	−4.50000	−	2.59808i	−0.500000	−	0.288675i
$4$		0			0
$5$	−31.0314	+	17.9160i	−1.24126	+	0.716640i	−0.969350	−	0.245684i	$-0.920987\pi$
$5$	−31.0314	+	17.9160i	−1.24126	+	0.716640i	−0.271906	+	0.962324i	$0.587654\pi$
$6$		0			0
$7$	−41.3748	+	26.2512i	−0.844384	+	0.535738i
$7$	−41.3748	+	26.2512i	−0.844384	+	0.535738i
$8$		0			0
$9$	13.5000	+	23.3827i	0.166667	+	0.288675i
$10$		0			0
$11$	21.0862	−	36.5224i	0.174266	−	0.301838i	−0.765641	−	0.643268i	$-0.777578\pi$
$11$	21.0862	−	36.5224i	0.174266	−	0.301838i	0.939907	+	0.341430i	$0.110911\pi$
$12$		0			0
$13$		−	22.7779i		−	0.134781i	−0.997727	−	0.0673903i	$-0.978533\pi$
$13$		−	22.7779i		−	0.134781i	0.997727	−	0.0673903i	$-0.0214673\pi$
$14$		0			0
$15$	186.188			0.827504
$16$		0			0
$17$	111.418	+	64.3274i	0.385530	+	0.222586i	0.680222	−	0.733006i	$-0.261884\pi$
$17$	111.418	+	64.3274i	0.385530	+	0.222586i	−0.294691	+	0.955592i	$0.595217\pi$
$18$		0			0
$19$	338.683	−	195.539i	0.938180	−	0.541659i	0.0487908	−	0.998809i	$-0.484463\pi$
$19$	338.683	−	195.539i	0.938180	−	0.541659i	0.889389	+	0.457150i	$0.151130\pi$
$20$		0			0
$21$	254.389	−	10.6353i	0.576846	−	0.0241163i
$22$		0			0
$23$	−331.549	−	574.260i	−0.626746	−	1.08556i	−0.988200	−	0.153167i	$-0.951053\pi$
$23$	−331.549	−	574.260i	−0.626746	−	1.08556i	0.361454	−	0.932390i	$-0.382281\pi$
$24$		0			0
$25$	329.465	−	570.651i	0.527145	−	0.913042i
$26$		0			0
$27$		−	140.296i		−	0.192450i
$28$		0			0
$29$	−604.215			−0.718448			−0.359224	−	0.933251i	$-0.616959\pi$
$29$	−604.215			−0.718448			−0.359224	+	0.933251i	$0.616959\pi$
$30$		0			0
$31$	1100.33	+	635.273i	1.14498	+	0.661054i	0.947659	−	0.319285i	$-0.103443\pi$
$31$	1100.33	+	635.273i	1.14498	+	0.661054i	0.197321	+	0.980339i	$0.436776\pi$
$32$		0			0
$33$	−189.776	+	109.567i	−0.174266	+	0.100613i
$34$		0			0
$35$	813.603	−	1555.88i	0.664166	−	1.27011i
$36$		0			0
$37$	−1069.76	−	1852.88i	−0.781417	−	1.35345i	−0.931116	−	0.364722i	$-0.881164\pi$
$37$	−1069.76	−	1852.88i	−0.781417	−	1.35345i	0.149700	−	0.988732i	$-0.452169\pi$
$38$		0			0
$39$	−59.1788	+	102.501i	−0.0389078	+	0.0673903i
$40$		0			0
$41$			244.193i			0.145267i	0.997359	+	0.0726333i	$0.0231403\pi$
$41$			244.193i			0.145267i	−0.997359	+	0.0726333i	$0.976860\pi$
$42$		0			0
$43$	3480.28			1.88225			0.941124	−	0.338062i	$-0.109771\pi$
$43$	3480.28			1.88225			0.941124	+	0.338062i	$0.109771\pi$
$44$		0			0
$45$	−837.848	−	483.732i	−0.413752	−	0.238880i
$46$		0			0
$47$	114.489	−	66.1002i	0.0518284	−	0.0299231i	−0.473862	−	0.880599i	$-0.657140\pi$
$47$	114.489	−	66.1002i	0.0518284	−	0.0299231i	0.525690	+	0.850676i	$0.323807\pi$
$48$		0			0
$49$	1022.75	−	2172.27i	0.425969	−	0.904738i
$50$		0			0
$51$	−334.255	−	578.946i	−0.128510	−	0.222586i
$52$		0			0
$53$	481.312	−	833.657i	0.171346	−	0.296781i	−0.767544	−	0.640996i	$-0.778522\pi$
$53$	481.312	−	833.657i	0.171346	−	0.296781i	0.938891	+	0.344215i	$0.111855\pi$
$54$		0			0
$55$			1511.12i			0.499544i
$56$		0			0
$57$	−2032.10			−0.625453
$58$		0			0
$59$	5542.71	+	3200.09i	1.59228	+	0.919301i	0.992916	+	0.118822i	$0.0379119\pi$
$59$	5542.71	+	3200.09i	1.59228	+	0.919301i	0.599361	+	0.800479i	$0.295421\pi$
$60$		0			0
$61$	−524.016	+	302.541i	−0.140827	+	0.0813063i	−0.568758	−	0.822505i	$-0.692576\pi$
$61$	−524.016	+	302.541i	−0.140827	+	0.0813063i	0.427931	+	0.903811i	$0.359243\pi$
$62$		0			0
$63$	−1172.38	−	613.064i	−0.295385	−	0.154463i
$64$		0			0
$65$	408.089	+	706.831i	0.0965892	+	0.167297i
$66$		0			0
$67$	1596.09	−	2764.51i	0.355556	−	0.615841i	−0.631657	−	0.775248i	$-0.717625\pi$
$67$	1596.09	−	2764.51i	0.355556	−	0.615841i	0.987213	+	0.159407i	$0.0509582\pi$
$68$		0			0
$69$			3445.56i			0.723705i
$70$		0			0
$71$	−6484.24			−1.28630			−0.643150	−	0.765740i	$-0.722373\pi$
$71$	−6484.24			−1.28630			−0.643150	+	0.765740i	$0.722373\pi$
$72$		0			0
$73$	526.915	+	304.214i	0.0988769	+	0.0570866i	0.548623	−	0.836070i	$-0.315152\pi$
$73$	526.915	+	304.214i	0.0988769	+	0.0570866i	−0.449746	+	0.893156i	$0.648486\pi$
$74$		0			0
$75$	−2965.19	+	1711.95i	−0.527145	+	0.304347i
$76$		0			0
$77$	86.3170	+	2064.64i	0.0145584	+	0.348228i
$78$		0			0
$79$	596.115	+	1032.50i	0.0955160	+	0.165439i	0.909824	−	0.414995i	$-0.136217\pi$
$79$	596.115	+	1032.50i	0.0955160	+	0.165439i	−0.814308	+	0.580433i	$0.802883\pi$
$80$		0			0
$81$	−364.500	+	631.333i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		−	4573.10i		−	0.663826i	−0.943310	−	0.331913i	$-0.892306\pi$
$83$		−	4573.10i		−	0.663826i	0.943310	−	0.331913i	$-0.107694\pi$
$84$		0			0
$85$	−4609.95			−0.638056
$86$		0			0
$87$	2718.97	+	1569.80i	0.359224	+	0.207398i
$88$		0			0
$89$	−9458.27	+	5460.73i	−1.19407	+	0.689399i	−0.959228	−	0.282633i	$-0.908792\pi$
$89$	−9458.27	+	5460.73i	−1.19407	+	0.689399i	−0.234847	+	0.972032i	$0.575459\pi$
$90$		0			0
$91$	597.947	+	942.433i	0.0722071	+	0.113807i
$92$		0			0
$93$	−3300.98	−	5717.46i	−0.381660	−	0.661054i
$94$		0			0
$95$	−7006.54	+	12135.7i	−0.776348	+	1.34467i
$96$		0			0
$97$			9241.16i			0.982161i	0.871114	+	0.491081i	$0.163398\pi$
$97$			9241.16i			0.982161i	−0.871114	+	0.491081i	$0.836602\pi$
$98$		0			0
$99$	1138.66			0.116177
$100$		0			0
$101$	−5517.47	−	3185.52i	−0.540876	−	0.312275i	0.204558	−	0.978854i	$-0.434424\pi$
$101$	−5517.47	−	3185.52i	−0.540876	−	0.312275i	−0.745434	+	0.666580i	$0.767758\pi$
$102$		0			0
$103$	13222.0	−	7633.74i	1.24630	−	0.719553i	0.275932	−	0.961177i	$-0.411013\pi$
$103$	13222.0	−	7633.74i	1.24630	−	0.719553i	0.970370	+	0.241624i	$0.0776801\pi$
$104$		0			0
$105$	−7703.51	+	4887.66i	−0.698731	+	0.443326i
$106$		0			0
$107$	321.963	+	557.655i	0.0281215	+	0.0487078i	0.879744	−	0.475448i	$-0.157714\pi$
$107$	321.963	+	557.655i	0.0281215	+	0.0487078i	−0.851622	+	0.524156i	$0.824381\pi$
$108$		0			0
$109$	10814.7	−	18731.7i	0.910255	−	1.57661i	0.0965516	−	0.995328i	$-0.469219\pi$
$109$	10814.7	−	18731.7i	0.910255	−	1.57661i	0.813704	−	0.581280i	$-0.197448\pi$
$110$		0			0
$111$			11117.3i			0.902302i
$112$		0			0
$113$	−21093.9			−1.65196			−0.825980	−	0.563699i	$-0.809378\pi$
$113$	−21093.9			−1.65196			−0.825980	+	0.563699i	$0.809378\pi$
$114$		0			0
$115$	20576.9	+	11880.1i	1.55591	+	0.898303i
$116$		0			0
$117$	532.609	−	307.502i	0.0389078	−	0.0224634i
$118$		0			0
$119$	−6298.58	+	263.326i	−0.444783	+	0.0185951i
$120$		0			0
$121$	6431.24	+	11139.2i	0.439263	+	0.760825i
$122$		0			0
$123$	634.432	−	1098.87i	0.0419348	−	0.0726333i
$124$		0			0
$125$			1215.81i			0.0778121i
$126$		0			0
$127$	2459.47			0.152487			0.0762436	−	0.997089i	$-0.475707\pi$
$127$	2459.47			0.152487			0.0762436	+	0.997089i	$0.475707\pi$
$128$		0			0
$129$	−15661.2	−	9042.02i	−0.941124	−	0.543358i
$130$		0			0
$131$	−4575.85	+	2641.87i	−0.266642	+	0.153946i	−0.627361	−	0.778729i	$-0.715865\pi$
$131$	−4575.85	+	2641.87i	−0.266642	+	0.153946i	0.360719	+	0.932675i	$0.382532\pi$
$132$		0			0
$133$	−8879.83	+	16981.2i	−0.501997	+	0.959987i
$134$		0			0
$135$	2513.54	+	4353.59i	0.137917	+	0.238880i
$136$		0			0
$137$	3070.83	−	5318.83i	0.163612	−	0.283384i	−0.772550	−	0.634954i	$-0.781019\pi$
$137$	3070.83	−	5318.83i	0.163612	−	0.283384i	0.936161	+	0.351571i	$0.114352\pi$
$138$		0			0
$139$		−	15814.8i		−	0.818528i	−0.912416	−	0.409264i	$-0.865786\pi$
$139$		−	15814.8i		−	0.818528i	0.912416	−	0.409264i	$-0.134214\pi$
$140$		0			0
$141$	−686.933			−0.0345522
$142$		0			0
$143$	−831.904	−	480.300i	−0.0406819	−	0.0234877i
$144$		0			0
$145$	18749.6	−	10825.1i	0.891778	−	0.514868i
$146$		0			0
$147$	−10246.1	+	7118.05i	−0.474160	+	0.329402i
$148$		0			0
$149$	18574.4	+	32171.8i	0.836647	+	1.44912i	0.892682	+	0.450687i	$0.148821\pi$
$149$	18574.4	+	32171.8i	0.836647	+	1.44912i	−0.0560350	+	0.998429i	$0.517846\pi$
$150$		0			0
$151$	1288.66	−	2232.02i	0.0565177	−	0.0978915i	−0.836382	−	0.548147i	$-0.815334\pi$
$151$	1288.66	−	2232.02i	0.0565177	−	0.0978915i	0.892900	+	0.450255i	$0.148667\pi$
$152$		0			0
$153$			3473.68i			0.148391i
$154$		0			0
$155$	−45526.2			−1.89495
$156$		0			0
$157$	−12641.9	−	7298.82i	−0.512878	−	0.296110i	0.221138	−	0.975243i	$-0.429023\pi$
$157$	−12641.9	−	7298.82i	−0.512878	−	0.296110i	−0.734016	+	0.679132i	$0.762356\pi$
$158$		0			0
$159$	−4331.81	+	2500.97i	−0.171346	+	0.0989269i
$160$		0			0
$161$	28792.8	+	15056.3i	1.11079	+	0.580855i
$162$		0			0
$163$	−18687.7	−	32368.1i	−0.703365	−	1.21826i	−0.967278	−	0.253718i	$-0.918347\pi$
$163$	−18687.7	−	32368.1i	−0.703365	−	1.21826i	0.263913	−	0.964546i	$-0.414987\pi$
$164$		0			0
$165$	3926.01	−	6800.05i	0.144206	−	0.249772i
$166$		0			0
$167$		−	46771.2i		−	1.67705i	−0.544865	−	0.838524i	$-0.683419\pi$
$167$		−	46771.2i		−	1.67705i	0.544865	−	0.838524i	$-0.316581\pi$
$168$		0			0
$169$	28042.2			0.981834
$170$		0			0
$171$	9144.44	+	5279.55i	0.312727	+	0.180553i
$172$		0			0
$173$	42135.5	−	24327.0i	1.40785	−	0.812822i	0.412668	−	0.910881i	$-0.364597\pi$
$173$	42135.5	−	24327.0i	1.40785	−	0.812822i	0.995181	+	0.0980592i	$0.0312635\pi$
$174$		0			0
$175$	1348.68	+	32259.4i	0.0440384	+	1.05337i
$176$		0			0
$177$	−16628.1	−	28800.8i	−0.530759	−	0.919301i
$178$		0			0
$179$	26442.3	−	45799.3i	0.825263	−	1.42940i	−0.0764547	−	0.997073i	$-0.524360\pi$
$179$	26442.3	−	45799.3i	0.825263	−	1.42940i	0.901718	−	0.432325i	$-0.142307\pi$
$180$		0			0
$181$		−	34634.2i		−	1.05718i	−0.848878	−	0.528589i	$-0.822721\pi$
$181$		−	34634.2i		−	1.05718i	0.848878	−	0.528589i	$-0.177279\pi$
$182$		0			0
$183$	3144.10			0.0938844
$184$		0			0
$185$	66392.3	+	38331.6i	1.93988	+	1.11999i
$186$		0			0
$187$	4698.78	−	2712.84i	0.134370	−	0.0775784i
$188$		0			0
$189$	3682.94	+	5804.73i	0.103103	+	0.162502i
$190$		0			0
$191$	−7227.43	−	12518.3i	−0.198115	−	0.343145i	0.749802	−	0.661662i	$-0.230149\pi$
$191$	−7227.43	−	12518.3i	−0.198115	−	0.343145i	−0.947917	+	0.318517i	$0.896815\pi$
$192$		0			0
$193$	34230.2	−	59288.4i	0.918956	−	1.59168i	0.117953	−	0.993019i	$-0.462367\pi$
$193$	34230.2	−	59288.4i	0.918956	−	1.59168i	0.801003	−	0.598660i	$-0.204300\pi$
$194$		0			0
$195$		−	4240.99i		−	0.111532i
$196$		0			0
$197$	3031.37			0.0781099			0.0390549	−	0.999237i	$-0.487565\pi$
$197$	3031.37			0.0781099			0.0390549	+	0.999237i	$0.487565\pi$
$198$		0			0
$199$	10625.4	+	6134.58i	0.268312	+	0.154910i	0.628120	−	0.778116i	$-0.283825\pi$
$199$	10625.4	+	6134.58i	0.268312	+	0.154910i	−0.359809	+	0.933026i	$0.617158\pi$
$200$		0			0
$201$	−14364.8	+	8293.53i	−0.355556	+	0.205280i
$202$		0			0
$203$	24999.3	−	15861.3i	0.606646	−	0.384900i
$204$		0			0
$205$	−4374.96	−	7577.65i	−0.104104	−	0.180313i
$206$		0			0
$207$	8951.82	−	15505.0i	0.208915	−	0.361852i
$208$		0			0
$209$		−	16492.7i		−	0.377571i
$210$		0			0
$211$	−13520.5			−0.303687			−0.151844	−	0.988405i	$-0.548521\pi$
$211$	−13520.5			−0.303687			−0.151844	+	0.988405i	$0.548521\pi$
$212$		0			0
$213$	29179.1	+	16846.6i	0.643150	+	0.371323i
$214$		0			0
$215$	−107998.	+	62352.6i	−2.33635	+	1.34889i
$216$		0			0
$217$	−62202.4	+	2600.51i	−1.32095	+	0.0552254i
$218$		0			0
$219$	−1580.74	−	2737.93i	−0.0329590	−	0.0570866i
$220$		0			0
$221$	1465.24	−	2537.88i	0.0300003	−	0.0519620i
$222$		0			0
$223$			69011.1i			1.38774i	0.720099	+	0.693872i	$0.244097\pi$
$223$			69011.1i			1.38774i	−0.720099	+	0.693872i	$0.755903\pi$
$224$		0			0
$225$	17791.1			0.351430
$226$		0			0
$227$	11677.0	+	6741.75i	0.226611	+	0.130834i	0.609008	−	0.793164i	$-0.291568\pi$
$227$	11677.0	+	6741.75i	0.226611	+	0.130834i	−0.382397	+	0.923998i	$0.624901\pi$
$228$		0			0
$229$	−39740.5	+	22944.2i	−0.757813	+	0.437524i	−0.828510	−	0.559974i	$-0.810811\pi$
$229$	−39740.5	+	22944.2i	−0.757813	+	0.437524i	0.0706970	+	0.997498i	$0.477478\pi$
$230$		0			0
$231$	4975.68	−	9515.16i	0.0932456	−	0.178317i
$232$		0			0
$233$	−29095.4	−	50394.7i	−0.535936	−	0.928268i	−0.999117	−	0.0420044i	$-0.986626\pi$
$233$	−29095.4	−	50394.7i	−0.535936	−	0.928268i	0.463182	−	0.886263i	$-0.346708\pi$
$234$		0			0
$235$	−2368.50	+	4102.36i	−0.0428882	+	0.0742845i
$236$		0			0
$237$		−	6195.01i		−	0.110292i
$238$		0			0
$239$	−66901.8			−1.17123			−0.585615	−	0.810590i	$-0.699147\pi$
$239$	−66901.8			−1.17123			−0.585615	+	0.810590i	$0.699147\pi$
$240$		0			0
$241$	53146.0	+	30683.9i	0.915033	+	0.528294i	0.882047	−	0.471162i	$-0.156165\pi$
$241$	53146.0	+	30683.9i	0.915033	+	0.528294i	0.0329856	+	0.999456i	$0.489498\pi$
$242$		0			0
$243$	3280.50	−	1894.00i	0.0555556	−	0.0320750i
$244$		0			0
$245$	7181.01	+	85732.4i	0.119634	+	1.42828i
$246$		0			0
$247$	−4453.97	−	7714.50i	−0.0730051	−	0.126449i
$248$		0			0
$249$	−11881.3	+	20578.9i	−0.191630	+	0.331913i
$250$		0			0
$251$		−	84016.1i		−	1.33357i	−0.745251	−	0.666784i	$-0.767670\pi$
$251$		−	84016.1i		−	1.33357i	0.745251	−	0.666784i	$-0.232330\pi$
$252$		0			0
$253$	−27964.4			−0.436883
$254$		0			0
$255$	20744.8	+	11977.0i	0.319028	+	0.184191i
$256$		0			0
$257$	−30370.2	+	17534.2i	−0.459813	+	0.265473i	−0.711966	−	0.702214i	$-0.752195\pi$
$257$	−30370.2	+	17534.2i	−0.459813	+	0.265473i	0.252153	+	0.967688i	$0.418862\pi$
$258$		0			0
$259$	92901.3	+	48580.1i	1.38491	+	0.724200i
$260$		0			0
$261$	−8156.90	−	14128.2i	−0.119741	−	0.207398i
$262$		0			0
$263$	−228.845	+	396.370i	−0.00330848	+	0.00573046i	−0.867675	−	0.497132i	$-0.834386\pi$
$263$	−228.845	+	396.370i	−0.00330848	+	0.00573046i	0.864366	+	0.502862i	$0.167720\pi$
$264$		0			0
$265$			34492.7i			0.491174i
$266$		0			0
$267$	56749.6			0.796050
$268$		0			0
$269$	43296.6	+	24997.3i	0.598342	+	0.345453i	0.768389	−	0.639983i	$-0.221059\pi$
$269$	43296.6	+	24997.3i	0.598342	+	0.345453i	−0.170047	+	0.985436i	$0.554392\pi$
$270$		0			0
$271$	−75498.8	+	43589.3i	−1.02802	+	0.593528i	−0.916417	−	0.400225i	$-0.868932\pi$
$271$	−75498.8	+	43589.3i	−1.02802	+	0.593528i	−0.111603	+	0.993753i	$0.535599\pi$
$272$		0			0
$273$	−242.250	−	5794.46i	−0.00325041	−	0.0777477i
$274$		0			0
$275$	−13894.4	−	24065.7i	−0.183727	−	0.318225i
$276$		0			0
$277$	−8440.28	+	14619.0i	−0.110001	+	0.190528i	−0.915770	−	0.401702i	$-0.868419\pi$
$277$	−8440.28	+	14619.0i	−0.110001	+	0.190528i	0.805769	+	0.592229i	$0.201752\pi$
$278$		0			0
$279$			34304.8i			0.440703i
$280$		0			0
$281$	92330.2			1.16931			0.584657	−	0.811281i	$-0.301229\pi$
$281$	92330.2			1.16931			0.584657	+	0.811281i	$0.301229\pi$
$282$		0			0
$283$	−21570.6	−	12453.8i	−0.269333	−	0.155499i	0.359252	−	0.933241i	$-0.383032\pi$
$283$	−21570.6	−	12453.8i	−0.269333	−	0.155499i	−0.628584	+	0.777741i	$0.716365\pi$
$284$		0			0
$285$	63058.9	−	36407.1i	0.776348	−	0.448225i
$286$		0			0
$287$	−6410.35	−	10103.4i	−0.0778248	−	0.122661i
$288$		0			0
$289$	−33484.5	−	57996.8i	−0.400911	−	0.694398i
$290$		0			0
$291$	24009.2	−	41585.2i	0.283526	−	0.491081i
$292$		0			0
$293$			119112.i			1.38746i	0.720235	+	0.693730i	$0.244034\pi$
$293$			119112.i			1.38746i	−0.720235	+	0.693730i	$0.755966\pi$
$294$		0			0
$295$	−229331.			−2.63523
$296$		0			0
$297$	−5123.95	−	2958.31i	−0.0580887	−	0.0335375i
$298$		0			0
$299$	−13080.4	+	7552.00i	−0.146312	+	0.0844733i
$300$		0			0
$301$	−143996.	+	91361.3i	−1.58934	+	1.00839i
$302$		0			0
$303$	16552.4	+	28669.6i	0.180292	+	0.312275i
$304$		0			0
$305$	10840.6	−	18776.5i	0.116535	−	0.201844i
$306$		0			0
$307$		−	104503.i		−	1.10880i	−0.832252	−	0.554398i	$-0.812948\pi$
$307$		−	104503.i		−	1.10880i	0.832252	−	0.554398i	$-0.187052\pi$
$308$		0			0
$309$	−79332.1			−0.830868
$310$		0			0
$311$	110013.	+	63515.8i	1.13742	+	0.656691i	0.945791	−	0.324777i	$-0.105289\pi$
$311$	110013.	+	63515.8i	1.13742	+	0.656691i	0.191631	+	0.981467i	$0.438622\pi$
$312$		0			0
$313$	−163258.	+	94257.0i	−1.66642	+	0.962111i	−0.696883	+	0.717185i	$0.745430\pi$
$313$	−163258.	+	94257.0i	−1.66642	+	0.962111i	−0.969542	+	0.244926i	$0.921236\pi$
$314$		0			0
$315$	47364.3	−	1980.17i	0.477343	−	0.0199564i
$316$		0			0
$317$	−62987.3	−	109097.i	−0.626808	−	1.08566i	−0.988188	−	0.153245i	$-0.951028\pi$
$317$	−62987.3	−	109097.i	−0.626808	−	1.08566i	0.361380	−	0.932419i	$-0.382306\pi$
$318$		0			0
$319$	−12740.6	+	22067.4i	−0.125201	+	0.216855i
$320$		0			0
$321$		−	3345.93i		−	0.0324719i
$322$		0			0
$323$	50314.0			0.482262
$324$		0			0
$325$	−12998.2	−	7504.54i	−0.123060	−	0.0710489i
$326$		0			0
$327$	−97332.7	+	56195.0i	−0.910255	+	0.525536i
$328$		0			0
$329$	−3001.75	+	5740.35i	−0.0277321	+	0.0530330i
$330$		0			0
$331$	47987.1	+	83116.1i	0.437994	+	0.758629i	0.997535	−	0.0701745i	$-0.0223556\pi$
$331$	47987.1	+	83116.1i	0.437994	+	0.758629i	−0.559540	+	0.828803i	$0.689022\pi$
$332$		0			0
$333$	28883.5	−	50027.7i	0.260472	−	0.451151i
$334$		0			0
$335$			114382.i			1.01922i
$336$		0			0
$337$	−110570.			−0.973593			−0.486797	−	0.873515i	$-0.661835\pi$
$337$	−110570.			−0.973593			−0.486797	+	0.873515i	$0.661835\pi$
$338$		0			0
$339$	94922.4	+	54803.5i	0.825980	+	0.476880i
$340$		0			0
$341$	46403.4	−	26791.0i	0.399062	−	0.230399i
$342$		0			0
$343$	14708.5	+	116726.i	0.125021	+	0.992154i
$344$		0			0
$345$	−61730.6	−	106920.i	−0.518635	−	0.898303i
$346$		0			0
$347$	94317.8	−	163363.i	0.783312	−	1.35674i	−0.146690	−	0.989183i	$-0.546862\pi$
$347$	94317.8	−	163363.i	0.783312	−	1.35674i	0.930002	−	0.367554i	$-0.119805\pi$
$348$		0			0
$349$		−	137640.i		−	1.13004i	−0.825076	−	0.565022i	$-0.808868\pi$
$349$		−	137640.i		−	1.13004i	0.825076	−	0.565022i	$-0.191132\pi$
$350$		0			0
$351$	−3195.66			−0.0259386
$352$		0			0
$353$	71165.2	+	41087.2i	0.571108	+	0.329729i	0.757592	−	0.652729i	$-0.226376\pi$
$353$	71165.2	+	41087.2i	0.571108	+	0.329729i	−0.186484	+	0.982458i	$0.559709\pi$
$354$		0			0
$355$	201215.	−	116172.i	1.59663	−	0.921814i
$356$		0			0
$357$	29027.7	+	15179.2i	0.227760	+	0.119100i
$358$		0			0
$359$	21237.9	+	36785.1i	0.164787	+	0.285419i	0.936580	−	0.350455i	$-0.113973\pi$
$359$	21237.9	+	36785.1i	0.164787	+	0.285419i	−0.771793	+	0.635874i	$0.780640\pi$
$360$		0			0
$361$	11310.3	−	19590.0i	0.0867880	−	0.150321i
$362$		0			0
$363$		−	66835.4i		−	0.507217i
$364$		0			0
$365$	−21801.2			−0.163642
$366$		0			0
$367$	−9927.06	−	5731.39i	−0.0737035	−	0.0425528i	0.462695	−	0.886517i	$-0.346882\pi$
$367$	−9927.06	−	5731.39i	−0.0737035	−	0.0425528i	−0.536399	+	0.843965i	$0.680216\pi$
$368$		0			0
$369$	−5709.89	+	3296.61i	−0.0419348	+	0.0242111i
$370$		0			0
$371$	1970.26	+	47127.4i	0.0143145	+	0.342394i
$372$		0			0
$373$	114052.	+	197544.i	0.819756	+	1.41986i	0.905862	+	0.423573i	$0.139224\pi$
$373$	114052.	+	197544.i	0.819756	+	1.41986i	−0.0861056	+	0.996286i	$0.527442\pi$
$374$		0			0
$375$	3158.78	−	5471.17i	0.0224624	−	0.0389061i
$376$		0			0
$377$			13762.8i			0.0968329i
$378$		0			0
$379$	−90556.7			−0.630437			−0.315219	−	0.949019i	$-0.602078\pi$
$379$	−90556.7			−0.630437			−0.315219	+	0.949019i	$0.602078\pi$
$380$		0			0
$381$	−11067.6	−	6389.88i	−0.0762436	−	0.0440193i
$382$		0			0
$383$	176241.	−	101753.i	1.20146	−	0.693662i	0.240578	−	0.970630i	$-0.422663\pi$
$383$	176241.	−	101753.i	1.20146	−	0.693662i	0.960879	+	0.276968i	$0.0893296\pi$
$384$		0			0
$385$	−39668.7	−	62522.4i	−0.267625	−	0.421807i
$386$		0			0
$387$	46983.7	+	81378.2i	0.313708	+	0.543358i
$388$		0			0
$389$	63925.1	−	110722.i	0.422447	−	0.731700i	−0.573731	−	0.819044i	$-0.694505\pi$
$389$	63925.1	−	110722.i	0.422447	−	0.731700i	0.996178	+	0.0873441i	$0.0278379\pi$
$390$		0			0
$391$		−	85310.7i		−	0.558020i
$392$		0			0
$393$	27455.1			0.177761
$394$		0			0
$395$	−36996.6	−	21360.0i	−0.237120	−	0.136901i
$396$		0			0
$397$	189719.	−	109535.i	1.20373	−	0.694977i	0.242351	−	0.970189i	$-0.422081\pi$
$397$	189719.	−	109535.i	1.20373	−	0.694977i	0.961384	+	0.275212i	$0.0887480\pi$
$398$		0			0
$399$	84077.7	−	53344.9i	0.528123	−	0.335079i
$400$		0			0
$401$	−39090.9	−	67707.4i	−0.243101	−	0.421064i	0.718495	−	0.695532i	$-0.244831\pi$
$401$	−39090.9	−	67707.4i	−0.243101	−	0.421064i	−0.961596	+	0.274469i	$0.911498\pi$
$402$		0			0
$403$	14470.2	−	25063.1i	0.0890973	−	0.154321i
$404$		0			0
$405$		−	26121.5i		−	0.159253i
$406$		0			0
$407$	−90228.7			−0.544698
$408$		0			0
$409$	−96786.9	−	55880.0i	−0.578589	−	0.334048i	0.181983	−	0.983302i	$-0.441748\pi$
$409$	−96786.9	−	55880.0i	−0.578589	−	0.334048i	−0.760572	+	0.649253i	$0.775082\pi$
$410$		0			0
$411$	−27637.4	+	15956.5i	−0.163612	+	0.0944612i
$412$		0			0
$413$	−313335.	+	13099.6i	−1.83700	+	0.0767997i
$414$		0			0
$415$	81931.6	+	141910.i	0.475724	+	0.823978i
$416$		0			0
$417$	−41088.0	+	71166.5i	−0.236289	+	0.409264i
$418$		0			0
$419$		−	104053.i		−	0.592687i	−0.955081	−	0.296343i	$-0.904233\pi$
$419$		−	104053.i		−	0.592687i	0.955081	−	0.296343i	$-0.0957673\pi$
$420$		0			0
$421$	100894.			0.569250			0.284625	−	0.958639i	$-0.408131\pi$
$421$	100894.			0.569250			0.284625	+	0.958639i	$0.408131\pi$
$422$		0			0
$423$	3091.20	+	1784.70i	0.0172761	+	0.00997437i
$424$		0			0
$425$	73416.9	−	42387.3i	0.406461	−	0.234670i
$426$		0			0
$427$	13739.0	−	26273.6i	0.0753529	−	0.144100i
$428$		0			0
$429$	2495.71	+	4322.70i	0.0135606	+	0.0234877i
$430$		0			0
$431$	−29456.3	+	51019.9i	−0.158571	+	0.274653i	−0.934354	−	0.356347i	$-0.884022\pi$
$431$	−29456.3	+	51019.9i	−0.158571	+	0.274653i	0.775783	+	0.631000i	$0.217355\pi$
$432$		0			0
$433$			195037.i			1.04026i	0.854087	+	0.520130i	$0.174116\pi$
$433$			195037.i			1.04026i	−0.854087	+	0.520130i	$0.825884\pi$
$434$		0			0
$435$	−112498.			−0.594519
$436$		0			0
$437$	−224580.	−	129661.i	−1.17600	−	0.678965i
$438$		0			0
$439$	294023.	−	169754.i	1.52564	−	0.880828i	0.526101	−	0.850422i	$-0.323653\pi$
$439$	294023.	−	169754.i	1.52564	−	0.880828i	0.999538	−	0.0304055i	$-0.00967985\pi$
$440$		0			0
$441$	64600.8	−	5411.01i	0.332170	−	0.0278228i
$442$		0			0
$443$	−17099.0	−	29616.4i	−0.0871293	−	0.150912i	0.819167	−	0.573555i	$-0.194436\pi$
$443$	−17099.0	−	29616.4i	−0.0871293	−	0.150912i	−0.906297	+	0.422642i	$0.861103\pi$
$444$		0			0
$445$	195669.	−	338908.i	0.988102	−	1.71144i
$446$		0			0
$447$		−	193031.i		−	0.966077i
$448$		0			0
$449$	149675.			0.742433			0.371217	−	0.928546i	$-0.378941\pi$
$449$	149675.			0.742433			0.371217	+	0.928546i	$0.378941\pi$
$450$		0			0
$451$	8918.51	+	5149.11i	0.0438469	+	0.0253150i
$452$		0			0
$453$	−11597.9	+	6696.07i	−0.0565177	+	0.0326305i
$454$		0			0
$455$	−35439.8	−	18532.2i	−0.171186	−	0.0895167i
$456$		0			0
$457$	−73251.9	−	126876.i	−0.350741	−	0.607501i	0.635638	−	0.771987i	$-0.280737\pi$
$457$	−73251.9	−	126876.i	−0.350741	−	0.607501i	−0.986379	+	0.164486i	$0.947404\pi$
$458$		0			0
$459$	9024.88	−	15631.5i	0.0428367	−	0.0741953i
$460$		0			0
$461$			360521.i			1.69640i	0.529673	+	0.848202i	$0.322315\pi$
$461$			360521.i			1.69640i	−0.529673	+	0.848202i	$0.677685\pi$
$462$		0			0
$463$	−128447.			−0.599186			−0.299593	−	0.954067i	$-0.596851\pi$
$463$	−128447.			−0.599186			−0.299593	+	0.954067i	$0.596851\pi$
$464$		0			0
$465$	204868.	+	118281.i	0.947476	+	0.547025i
$466$		0			0
$467$	−154958.	+	89465.2i	−0.710527	+	0.410223i	−0.811256	−	0.584691i	$-0.801216\pi$
$467$	−154958.	+	89465.2i	−0.710527	+	0.410223i	0.100729	+	0.994914i	$0.467883\pi$
$468$		0			0
$469$	6533.64	+	156280.i	0.0297036	+	0.710491i
$470$		0			0
$471$	37925.8	+	65689.4i	0.170959	+	0.296110i
$472$		0			0
$473$	73385.8	−	127108.i	0.328012	−	0.568134i
$474$		0			0
$475$		−	257693.i		−	1.14213i
$476$		0			0
$477$	25990.8			0.114231
$478$		0			0
$479$	185371.	+	107024.i	0.807925	+	0.466456i	0.846235	−	0.532810i	$-0.178864\pi$
$479$	185371.	+	107024.i	0.807925	+	0.466456i	−0.0383100	+	0.999266i	$0.512197\pi$
$480$		0			0
$481$	−42204.7	+	24366.9i	−0.182419	+	0.105320i
$482$		0			0
$483$	−90449.9	−	142559.i	−0.387716	−	0.611085i
$484$		0			0
$485$	−165564.	−	286766.i	−0.703856	−	1.21911i
$486$		0			0
$487$	−5595.57	+	9691.80i	−0.0235932	+	0.0408645i	−0.877581	−	0.479429i	$-0.840844\pi$
$487$	−5595.57	+	9691.80i	−0.0235932	+	0.0408645i	0.853988	+	0.520293i	$0.174177\pi$
$488$		0			0
$489$			194208.i			0.812176i
$490$		0			0
$491$	273731.			1.13543			0.567715	−	0.823225i	$-0.307827\pi$
$491$	273731.			1.13543			0.567715	+	0.823225i	$0.307827\pi$
$492$		0			0
$493$	−67320.6	−	38867.5i	−0.276984	−	0.159917i
$494$		0			0
$495$	−35334.1	+	20400.1i	−0.144206	+	0.0832574i
$496$		0			0
$497$	268284.	−	170219.i	1.08613	−	0.689120i
$498$		0			0
$499$	153737.	+	266281.i	0.617417	+	1.06940i	0.989955	+	0.141380i	$0.0451540\pi$
$499$	153737.	+	266281.i	0.617417	+	1.06940i	−0.372539	+	0.928017i	$0.621513\pi$
$500$		0			0
$501$	−121515.	+	210470.i	−0.484122	+	0.838524i
$502$		0			0
$503$		−	172978.i		−	0.683684i	−0.939757	−	0.341842i	$-0.888949\pi$
$503$		−	172978.i		−	0.683684i	0.939757	−	0.341842i	$-0.111051\pi$
$504$		0			0
$505$	228287.			0.895154
$506$		0			0
$507$	−126190.	−	72855.7i	−0.490917	−	0.283431i
$508$		0			0
$509$	−368421.	+	212708.i	−1.42203	+	0.821010i	−0.996472	−	0.0839203i	$-0.973256\pi$
$509$	−368421.	+	212708.i	−1.42203	+	0.821010i	−0.425559	+	0.904931i	$0.639923\pi$
$510$		0			0
$511$	−29787.0	+	1245.31i	−0.114074	+	0.00476909i
$512$		0			0
$513$	−27433.3	−	47515.9i	−0.104242	−	0.180553i
$514$		0			0
$515$	−273532.	+	473771.i	−1.03132	+	1.78630i
$516$		0			0
$517$		−	5575.21i		−	0.0208583i
$518$		0			0
$519$	−252813.			−0.938566
$520$		0			0
$521$	−98727.1	−	57000.1i	−0.363715	−	0.209991i	0.306994	−	0.951711i	$-0.400677\pi$
$521$	−98727.1	−	57000.1i	−0.363715	−	0.209991i	−0.670709	+	0.741721i	$0.734010\pi$
$522$		0			0
$523$	−48047.0	+	27739.9i	−0.175656	+	0.101415i	−0.585250	−	0.810853i	$-0.699004\pi$
$523$	−48047.0	+	27739.9i	−0.175656	+	0.101415i	0.409594	+	0.912268i	$0.365670\pi$
$524$		0			0
$525$	77743.4	−	148671.i	0.282062	−	0.539398i
$526$		0			0
$527$	81730.9	+	141562.i	0.294283	+	0.509713i
$528$		0			0
$529$	−79928.8	+	138441.i	−0.285622	+	0.494712i
$530$		0			0
$531$			172805.i			0.612867i
$532$		0			0
$533$	5562.21			0.0195791
$534$		0			0
$535$	−19981.9	−	11536.6i	−0.0698119	−	0.0403059i
$536$		0			0
$537$	−237980.	+	137398.i	−0.825263	+	0.476466i
$538$		0			0
$539$	−57770.7	−	83158.4i	−0.198852	−	0.286239i
$540$		0			0
$541$	196179.	+	339792.i	0.670283	+	1.16096i	0.977824	+	0.209429i	$0.0671606\pi$
$541$	196179.	+	339792.i	0.670283	+	1.16096i	−0.307541	+	0.951535i	$0.599506\pi$
$542$		0			0
$543$	−89982.3	+	155854.i	−0.305181	+	0.528589i
$544$		0			0
$545$			775027.i			2.60930i
$546$		0			0
$547$	−313473.			−1.04767			−0.523836	−	0.851819i	$-0.675500\pi$
$547$	−313473.			−1.04767			−0.523836	+	0.851819i	$0.675500\pi$
$548$		0			0
$549$	−14148.4	−	8168.60i	−0.0469422	−	0.0271021i
$550$		0			0
$551$	−204637.	+	118147.i	−0.674034	+	0.389154i
$552$		0			0
$553$	−51768.5	−	27070.9i	−0.169284	−	0.0885221i
$554$		0			0
$555$	−199177.	−	344984.i	−0.646626	−	1.11999i
$556$		0			0
$557$	51331.9	−	88909.4i	0.165454	−	0.286574i	−0.771363	−	0.636396i	$-0.780424\pi$
$557$	51331.9	−	88909.4i	0.165454	−	0.286574i	0.936816	+	0.349822i	$0.113758\pi$
$558$		0			0
$559$		−	79273.5i		−	0.253691i
$560$		0			0
$561$	−28192.7			−0.0895798
$562$		0			0
$563$	364209.	+	210276.i	1.14904	+	0.663397i	0.948653	−	0.316320i	$-0.102447\pi$
$563$	364209.	+	210276.i	1.14904	+	0.663397i	0.200385	+	0.979717i	$0.435781\pi$
$564$		0			0
$565$	654573.	−	377918.i	2.05051	−	1.18386i
$566$		0			0
$567$	−1492.09	−	35689.8i	−0.00464119	−	0.111014i
$568$		0			0
$569$	−236703.	−	409981.i	−0.731103	−	1.26631i	−0.956412	−	0.292020i	$-0.905673\pi$
$569$	−236703.	−	409981.i	−0.731103	−	1.26631i	0.225310	−	0.974287i	$-0.427661\pi$
$570$		0			0
$571$	−213031.	+	368981.i	−0.653388	+	1.13170i	0.328908	+	0.944362i	$0.393319\pi$
$571$	−213031.	+	368981.i	−0.653388	+	1.13170i	−0.982295	+	0.187339i	$0.940014\pi$
$572$		0			0
$573$			75109.7i			0.228763i
$574$		0			0
$575$	−436936.			−1.32154
$576$		0			0
$577$	−140040.	−	80852.1i	−0.420630	−	0.242851i	0.274717	−	0.961525i	$-0.411416\pi$
$577$	−140040.	−	80852.1i	−0.420630	−	0.242851i	−0.695347	+	0.718674i	$0.744749\pi$
$578$		0			0
$579$	−308072.	+	177865.i	−0.918956	+	0.530560i
$580$		0			0
$581$	120049.	+	189211.i	0.355637	+	0.560524i
$582$		0			0
$583$	−20298.1	−	35157.3i	−0.0597198	−	0.103438i
$584$		0			0
$585$	−11018.4	+	19084.4i	−0.0321964	+	0.0557658i
$586$		0			0
$587$		−	68383.2i		−	0.198460i	−0.995065	−	0.0992300i	$-0.968362\pi$
$587$		−	68383.2i		−	0.198460i	0.995065	−	0.0992300i	$-0.0316380\pi$
$588$		0			0
$589$	496882.			1.43226
$590$		0			0
$591$	−13641.1	−	7875.72i	−0.0390549	−	0.0225484i
$592$		0			0
$593$	−182097.	+	105134.i	−0.517837	+	0.298974i	−0.736049	−	0.676928i	$-0.763311\pi$
$593$	−182097.	+	105134.i	−0.517837	+	0.298974i	0.218212	+	0.975901i	$0.429978\pi$
$594$		0			0
$595$	190736.	−	121017.i	0.538764	−	0.341831i
$596$		0			0
$597$	−31876.2	−	55211.2i	−0.0894372	−	0.154910i
$598$		0			0
$599$	−232560.	+	402806.i	−0.648159	+	1.12265i	0.335403	+	0.942075i	$0.391128\pi$
$599$	−232560.	+	402806.i	−0.648159	+	1.12265i	−0.983562	+	0.180570i	$0.942206\pi$
$600$		0			0
$601$			17745.4i			0.0491288i	0.999698	+	0.0245644i	$0.00781989\pi$
$601$			17745.4i			0.0491288i	−0.999698	+	0.0245644i	$0.992180\pi$
$602$		0			0
$603$	86188.9			0.237037
$604$		0			0
$605$	−399141.	−	230444.i	−1.09047	−	0.629586i
$606$		0			0
$607$	1576.06	−	909.938i	0.00427755	−	0.00246965i	−0.497860	−	0.867258i	$-0.665881\pi$
$607$	1576.06	−	909.938i	0.00427755	−	0.00246965i	0.502137	+	0.864788i	$0.332547\pi$
$608$		0			0
$609$	−153706.	+	6426.00i	−0.414434	+	0.0173263i
$610$		0			0
$611$	−1505.62	−	2607.82i	−0.00403306	−	0.00698546i
$612$		0			0
$613$	−108295.	+	187573.i	−0.288196	+	0.499171i	−0.973379	−	0.229200i	$-0.926389\pi$
$613$	−108295.	+	187573.i	−0.288196	+	0.499171i	0.685183	+	0.728371i	$0.259722\pi$
$614$		0			0
$615$			45465.9i			0.120209i
$616$		0			0
$617$	702974.			1.84658			0.923292	−	0.384098i	$-0.125488\pi$
$617$	702974.			1.84658			0.923292	+	0.384098i	$0.125488\pi$
$618$		0			0
$619$	−518344.	−	299266.i	−1.35281	−	0.781045i	−0.364168	−	0.931333i	$-0.618647\pi$
$619$	−518344.	−	299266.i	−1.35281	−	0.781045i	−0.988642	+	0.150288i	$0.951980\pi$
$620$		0			0
$621$	−80566.4	+	46515.0i	−0.208915	+	0.120617i
$622$		0			0
$623$	247984.	−	474227.i	0.638920	−	1.22183i
$624$		0			0
$625$	184133.	+	318928.i	0.471382	+	0.816457i
$626$		0			0
$627$	−42849.2	+	74217.1i	−0.108995	+	0.188786i
$628$		0			0
$629$		−	275259.i		−	0.695730i
$630$		0			0
$631$	446883.			1.12237			0.561183	−	0.827692i	$-0.310346\pi$
$631$	446883.			1.12237			0.561183	+	0.827692i	$0.310346\pi$
$632$		0			0
$633$	60842.1	+	35127.2i	0.151844	+	0.0876670i
$634$		0			0
$635$	−76320.7	+	44063.8i	−0.189276	+	0.109278i
$636$		0			0
$637$	−49479.9	−	23296.2i	−0.121941	−	0.0574124i
$638$		0			0
$639$	−87537.3	−	151619.i	−0.214383	−	0.371323i
$640$		0			0
$641$	−93993.5	+	162802.i	−0.228761	+	0.396226i	−0.957441	−	0.288629i	$-0.906801\pi$
$641$	−93993.5	+	162802.i	−0.228761	+	0.396226i	0.728680	+	0.684854i	$0.240134\pi$
$642$		0			0
$643$		−	610836.i		−	1.47742i	−0.674025	−	0.738708i	$-0.735436\pi$
$643$		−	610836.i		−	1.47742i	0.674025	−	0.738708i	$-0.264564\pi$
$644$		0			0
$645$	647987.			1.55757
$646$		0			0
$647$	−479220.	−	276678.i	−1.14479	−	0.660946i	−0.197179	−	0.980367i	$-0.563178\pi$
$647$	−479220.	−	276678.i	−1.14479	−	0.660946i	−0.947613	+	0.319421i	$0.896511\pi$
$648$		0			0
$649$	233750.	−	134955.i	0.554960	−	0.320406i
$650$		0			0
$651$	286667.	+	149904.i	0.676420	+	0.353714i
$652$		0			0
$653$	68934.6	+	119398.i	0.161663	+	0.280009i	0.935465	−	0.353419i	$-0.114981\pi$
$653$	68934.6	+	119398.i	0.161663	+	0.280009i	−0.773802	+	0.633427i	$0.781648\pi$
$654$		0			0
$655$	94663.3	−	163962.i	0.220648	−	0.382173i
$656$		0			0
$657$			16427.6i			0.0380577i
$658$		0			0
$659$	208675.			0.480506			0.240253	−	0.970710i	$-0.422770\pi$
$659$	208675.			0.480506			0.240253	+	0.970710i	$0.422770\pi$
$660$		0			0
$661$	335103.	+	193472.i	0.766965	+	0.442807i	0.831791	−	0.555089i	$-0.187316\pi$
$661$	335103.	+	193472.i	0.766965	+	0.442807i	−0.0648259	+	0.997897i	$0.520649\pi$
$662$		0			0
$663$	−13187.2	+	7613.63i	−0.0300003	+	0.0173207i
$664$		0			0
$665$	−28681.5	−	686042.i	−0.0648572	−	1.55134i
$666$		0			0
$667$	200327.	+	346976.i	0.450285	+	0.779916i
$668$		0			0
$669$	179296.	−	310550.i	0.400607	−	0.693872i
$670$		0			0
$671$			25517.8i			0.0566758i
$672$		0			0
$673$	360814.			0.796624			0.398312	−	0.917250i	$-0.369596\pi$
$673$	360814.			0.796624			0.398312	+	0.917250i	$0.369596\pi$
$674$		0			0
$675$	−80060.1	−	46222.7i	−0.175715	−	0.101449i
$676$		0			0
$677$	−247314.	+	142787.i	−0.539600	+	0.311538i	−0.744917	−	0.667157i	$-0.767511\pi$
$677$	−247314.	+	142787.i	−0.539600	+	0.311538i	0.205317	+	0.978696i	$0.434178\pi$
$678$		0			0
$679$	−242591.	−	382351.i	−0.526181	−	0.829322i
$680$		0			0
$681$	−35031.1	−	60675.7i	−0.0755371	−	0.130834i
$682$		0			0
$683$	−142256.	+	246395.i	−0.304951	+	0.528191i	−0.977250	−	0.212089i	$-0.931973\pi$
$683$	−142256.	+	246395.i	−0.304951	+	0.528191i	0.672299	+	0.740279i	$0.265307\pi$
$684$		0			0
$685$			220068.i			0.469002i
$686$		0			0
$687$	238443.			0.505209
$688$		0			0
$689$	−18989.0	−	10963.3i	−0.0400003	−	0.0230942i
$690$		0			0
$691$	51567.9	−	29772.8i	0.108000	−	0.0623538i	−0.445027	−	0.895517i	$-0.646806\pi$
$691$	51567.9	−	29772.8i	0.108000	−	0.0623538i	0.553027	+	0.833163i	$0.313473\pi$
$692$		0			0
$693$	−47111.7	+	29891.0i	−0.0980984	+	0.0622407i
$694$		0			0
$695$	283337.	+	490755.i	0.586590	+	1.01600i
$696$		0			0
$697$	−15708.3	+	27207.6i	−0.0323343	+	0.0560047i
$698$		0			0
$699$			302368.i			0.618845i
$700$		0			0
$701$	−750279.			−1.52682			−0.763408	−	0.645916i	$-0.776476\pi$
$701$	−750279.			−1.52682			−0.763408	+	0.645916i	$0.776476\pi$
$702$		0			0
$703$	−724619.	−	418359.i	−1.46622	−	0.846522i
$704$		0			0
$705$	21316.5	−	12307.1i	0.0428882	−	0.0247615i
$706$		0			0
$707$	311908.	−	13040.0i	0.624004	−	0.0260879i
$708$		0			0
$709$	−143489.	−	248530.i	−0.285447	−	0.494409i	0.687270	−	0.726402i	$-0.258809\pi$
$709$	−143489.	−	248530.i	−0.285447	−	0.494409i	−0.972718	+	0.231993i	$0.925475\pi$
$710$		0			0
$711$	−16095.1	+	27877.6i	−0.0318387	+	0.0551462i
$712$		0			0
$713$		−	842497.i		−	1.65725i
$714$		0			0
$715$	34420.2			0.0673289
$716$		0			0
$717$	301058.	+	173816.i	0.585615	+	0.338105i
$718$		0			0
$719$	455133.	−	262771.i	0.880401	−	0.508300i	0.00961017	−	0.999954i	$-0.496941\pi$
$719$	455133.	−	262771.i	0.880401	−	0.508300i	0.870790	+	0.491654i	$0.163608\pi$
$720$		0			0
$721$	−346664.	+	662938.i	−0.666866	+	1.27527i
$722$		0			0
$723$	−159438.	−	276155.i	−0.305011	−	0.528294i
$724$		0			0
$725$	−199068.	+	344796.i	−0.378726	+	0.655973i
$726$		0			0
$727$		−	120256.i		−	0.227530i	−0.993508	−	0.113765i	$-0.963709\pi$
$727$		−	120256.i		−	0.227530i	0.993508	−	0.113765i	$-0.0362911\pi$
$728$		0			0
$729$	−19683.0			−0.0370370
$730$		0			0
$731$	387766.	+	223877.i	0.725663	+	0.418962i
$732$		0			0
$733$	321060.	−	185364.i	0.597556	−	0.344999i	−0.170523	−	0.985354i	$-0.554546\pi$
$733$	321060.	−	185364.i	0.597556	−	0.344999i	0.768080	+	0.640354i	$0.221212\pi$
$734$		0			0
$735$	190425.	−	404452.i	0.352491	−	0.748674i
$736$		0			0
$737$	−67311.0	−	116586.i	−0.123923	−	0.214641i
$738$		0			0
$739$	262961.	−	455462.i	0.481507	−	0.833995i	−0.518267	−	0.855219i	$-0.673423\pi$
$739$	262961.	−	455462.i	0.481507	−	0.833995i	0.999775	+	0.0212235i	$0.00675617\pi$
$740$		0			0
$741$			46287.0i			0.0842990i
$742$		0			0
$743$	601996.			1.09048			0.545238	−	0.838281i	$-0.316439\pi$
$743$	601996.			1.09048			0.545238	+	0.838281i	$0.316439\pi$
$744$		0			0
$745$	−1.15278e6	−	665558.i	−2.07699	−	1.19915i
$746$		0			0
$747$	106931.	−	61736.8i	0.191630	−	0.110638i
$748$		0			0
$749$	−27960.2	−	14621.0i	−0.0498399	−	0.0260624i
$750$		0			0
$751$	482086.	+	834998.i	0.854762	+	1.48049i	0.876866	+	0.480735i	$0.159630\pi$
$751$	482086.	+	834998.i	0.854762	+	1.48049i	−0.0221042	+	0.999756i	$0.507037\pi$
$752$		0			0
$753$	−218280.	+	378073.i	−0.384968	+	0.666784i
$754$		0			0
$755$			92350.5i			0.162011i
$756$		0			0
$757$	−101611.			−0.177317			−0.0886583	−	0.996062i	$-0.528258\pi$
$757$	−101611.			−0.177317			−0.0886583	+	0.996062i	$0.528258\pi$
$758$		0			0
$759$	125840.	+	72653.7i	0.218441	+	0.126117i
$760$		0			0
$761$	616909.	−	356172.i	1.06525	−	0.615022i	0.138370	−	0.990381i	$-0.455814\pi$
$761$	616909.	−	356172.i	1.06525	−	0.615022i	0.926880	+	0.375358i	$0.122480\pi$
$762$		0			0
$763$	44270.4	+	1.05892e6i	0.0760440	+	1.81892i
$764$		0			0
$765$	−62234.4	−	107793.i	−0.106343	−	0.184191i
$766$		0			0
$767$	72891.4	−	126252.i	0.123904	−	0.214608i
$768$		0			0
$769$			164666.i			0.278453i	0.990261	+	0.139226i	$0.0444616\pi$
$769$			164666.i			0.278453i	−0.990261	+	0.139226i	$0.955538\pi$
$770$		0			0
$771$	182221.			0.306542
$772$		0			0
$773$	−396268.	−	228785.i	−0.663177	−	0.382886i	0.130309	−	0.991473i	$-0.458403\pi$
$773$	−396268.	−	228785.i	−0.663177	−	0.382886i	−0.793487	+	0.608588i	$0.791736\pi$
$774$		0			0
$775$	725039.	−	418601.i	1.20714	−	0.696943i
$776$		0			0
$777$	−291841.	−	459975.i	−0.483398	−	0.761890i
$778$		0			0
$779$	47749.2	+	82704.1i	0.0786849	+	0.136286i
$780$		0			0
$781$	−136728.	+	236820.i	−0.224159	+	0.388254i
$782$		0			0
$783$			84769.0i			0.138265i
$784$		0			0
$785$	523062.			0.848817
$786$		0			0
$787$	−746239.	−	430841.i	−1.20484	−	0.695613i	−0.243210	−	0.969974i	$-0.578200\pi$
$787$	−746239.	−	430841.i	−1.20484	−	0.695613i	−0.961627	+	0.274361i	$0.911534\pi$
$788$		0			0
$789$	2059.60	−	1189.11i	0.00330848	−	0.00191015i
$790$		0			0
$791$	872755.	−	553739.i	1.39489	−	0.885018i
$792$		0			0
$793$	6891.25	+	11936.0i	0.0109585	+	0.0189807i
$794$		0			0
$795$	89614.7	−	155217.i	0.141790	−	0.245587i
$796$		0			0
$797$		−	492535.i		−	0.775390i	−0.921788	−	0.387695i	$-0.873271\pi$
$797$		−	492535.i		−	0.775390i	0.921788	−	0.387695i	$-0.126729\pi$
$798$		0			0
$799$	17008.2			0.0266419
$800$		0			0
$801$	−255373.	−	147440.i	−0.398025	−	0.229800i
$802$		0			0
$803$	22221.3	−	12829.5i	0.0344618	−	0.0198965i
$804$		0			0
$805$	−1.16323e6	+	48631.3i	−1.79504	+	0.0750455i
$806$		0			0
$807$	−129890.	−	224976.i	−0.199447	−	0.345453i
$808$		0			0
$809$	81016.1	−	140324.i	0.123787	−	0.214405i	−0.797471	−	0.603357i	$-0.793829\pi$
$809$	81016.1	−	140324.i	0.123787	−	0.214405i	0.921258	+	0.388952i	$0.127163\pi$
$810$		0			0
$811$		−	202170.i		−	0.307379i	−0.988119	−	0.153690i	$-0.950884\pi$
$811$		−	202170.i		−	0.307379i	0.988119	−	0.153690i	$-0.0491156\pi$
$812$		0			0
$813$	452993.			0.685347
$814$		0			0
$815$	1.15981e6	+	669617.i	1.74611	+	1.00812i
$816$		0			0
$817$	1.17871e6	−	680529.i	1.76589	−	1.01954i
$818$		0			0
$819$	−13964.3	+	26704.5i	−0.0208186	+	0.0398122i
$820$		0			0
$821$	564790.	+	978245.i	0.837917	+	1.45131i	0.891633	+	0.452758i	$0.149560\pi$
$821$	564790.	+	978245.i	0.837917	+	1.45131i	−0.0537165	+	0.998556i	$0.517107\pi$
$822$		0			0
$823$	354598.	−	614183.i	0.523525	−	0.906771i	−0.476100	−	0.879391i	$-0.657950\pi$
$823$	354598.	−	614183.i	0.523525	−	0.906771i	0.999625	−	0.0273804i	$-0.00871655\pi$
$824$		0			0
$825$			144394.i			0.212150i
$826$		0			0
$827$	−273000.			−0.399164			−0.199582	−	0.979881i	$-0.563958\pi$
$827$	−273000.			−0.399164			−0.199582	+	0.979881i	$0.563958\pi$
$828$		0			0
$829$	−1.00278e6	−	578956.i	−1.45914	−	0.842435i	−0.460171	−	0.887830i	$-0.652212\pi$
$829$	−1.00278e6	−	578956.i	−1.45914	−	0.842435i	−0.998969	+	0.0453951i	$0.985545\pi$
$830$		0			0
$831$	75962.5	−	43857.0i	0.110001	−	0.0635092i
$832$		0			0
$833$	253690.	−	176240.i	0.365606	−	0.253989i
$834$		0			0
$835$	837952.	+	1.45138e6i	1.20184	+	2.08165i
$836$		0			0
$837$	89126.4	−	154371.i	0.127220	−	0.220351i
$838$		0			0
$839$		−	928206.i		−	1.31862i	−0.751870	−	0.659311i	$-0.770848\pi$
$839$		−	928206.i		−	1.31862i	0.751870	−	0.659311i	$-0.229152\pi$
$840$		0			0
$841$	−342205.			−0.483832
$842$		0			0
$843$	−415486.	−	239881.i	−0.584657	−	0.337552i
$844$		0			0
$845$	−870188.	+	502403.i	−1.21871	+	0.703621i
$846$		0			0
$847$	−558510.	−	292057.i	−0.778509	−	0.407099i
$848$		0			0
$849$	64711.8	+	112084.i	0.0897776	+	0.155499i
$850$		0			0
$851$	−709355.	+	1.22864e6i	−0.979500	+	1.69654i
$852$		0			0
$853$			459810.i			0.631946i	0.948768	+	0.315973i	$0.102331\pi$
$853$			459810.i			0.631946i	−0.948768	+	0.315973i	$0.897669\pi$
$854$		0			0
$855$	−378353.			−0.517565
$856$		0			0
$857$	−347830.	−	200820.i	−0.473593	−	0.273429i	0.244149	−	0.969738i	$-0.421491\pi$
$857$	−347830.	−	200820.i	−0.473593	−	0.273429i	−0.717743	+	0.696308i	$0.754825\pi$
$858$		0			0
$859$	−1.26172e6	+	728454.i	−1.70992	+	0.987224i	−0.775291	+	0.631605i	$0.782397\pi$
$859$	−1.26172e6	+	728454.i	−1.70992	+	0.987224i	−0.934631	+	0.355619i	$0.884270\pi$
$860$		0			0
$861$	2597.07	+	62120.1i	0.00350329	+	0.0837965i
$862$		0			0
$863$	−545774.	−	945308.i	−0.732809	−	1.26926i	−0.955678	−	0.294415i	$-0.904875\pi$
$863$	−545774.	−	945308.i	−0.732809	−	1.26926i	0.222868	−	0.974849i	$-0.428458\pi$
$864$		0			0
$865$	−871683.	+	1.50980e6i	−1.16500	+	2.01784i
$866$		0			0
$867$			347981.i			0.462932i
$868$		0			0
$869$	50279.2			0.0665808
$870$		0			0
$871$	−62969.8	−	36355.7i	−0.0830035	−	0.0479221i
$872$		0			0
$873$	−216083.	+	124756.i	−0.283526	+	0.163694i
$874$		0			0
$875$	−31916.6	−	50304.1i	−0.0416869	−	0.0657033i
$876$		0			0
$877$	−377043.	−	653057.i	−0.490220	−	0.849087i	0.509716	−	0.860343i	$-0.329750\pi$
$877$	−377043.	−	653057.i	−0.490220	−	0.849087i	−0.999937	+	0.0112559i	$0.996417\pi$
$878$		0			0
$879$	309462.	−	536005.i	0.400525	−	0.693730i
$880$		0			0
$881$			551998.i			0.711190i	0.934640	+	0.355595i	$0.115722\pi$
$881$			551998.i			0.711190i	−0.934640	+	0.355595i	$0.884278\pi$
$882$		0			0
$883$	−824248.			−1.05715			−0.528575	−	0.848887i	$-0.677273\pi$
$883$	−824248.			−1.05715			−0.528575	+	0.848887i	$0.677273\pi$
$884$		0			0
$885$	1.03199e6	+	595819.i	1.31762	+	0.760726i
$886$		0			0
$887$	−31832.5	+	18378.5i	−0.0404598	+	0.0233595i	−0.520093	−	0.854109i	$-0.674103\pi$
$887$	−31832.5	+	18378.5i	−0.0404598	+	0.0233595i	0.479634	+	0.877469i	$0.340770\pi$
$888$		0			0
$889$	−101760.	+	64563.9i	−0.128758	+	0.0816932i
$890$		0			0
$891$	15371.8	+	26624.8i	0.0193629	+	0.0335375i
$892$		0			0
$893$	25850.3	−	44774.0i	0.0324162	−	0.0561465i
$894$		0			0
$895$			1.89496e6i			2.36567i
$896$		0			0
$897$	78482.7			0.0975414
$898$		0			0
$899$	−664833.	−	383842.i	−0.822609	−	0.474933i
$900$		0			0
$901$	107254.	−	61923.0i	0.132118	−	0.0762786i
$902$		0			0
$903$	885345.	−	37013.8i	1.08577	−	0.0453929i
$904$		0			0
$905$	620506.	+	1.07475e6i	0.757616	+	1.31223i
$906$		0			0
$907$	−730093.	+	1.26456e6i	−0.887490	+	1.53718i	−0.0446579	+	0.999002i	$0.514220\pi$
$907$	−730093.	+	1.26456e6i	−0.887490	+	1.53718i	−0.842832	+	0.538176i	$0.819114\pi$
$908$		0			0
$909$		−	172018.i		−	0.208183i
$910$		0			0
$911$	−351572.			−0.423621			−0.211811	−	0.977311i	$-0.567936\pi$
$911$	−351572.			−0.423621			−0.211811	+	0.977311i	$0.567936\pi$
$912$		0			0
$913$	−167020.	−	96429.3i	−0.200368	−	0.115682i
$914$		0			0
$915$	−97565.7	+	56329.6i	−0.116535	+	0.0672813i
$916$		0			0
$917$	119973.	−	229428.i	0.142674	−	0.272840i
$918$		0			0
$919$	−273275.	−	473327.i	−0.323571	−	0.560441i	0.657651	−	0.753322i	$-0.271550\pi$
$919$	−273275.	−	473327.i	−0.323571	−	0.560441i	−0.981222	+	0.192882i	$0.938217\pi$
$920$		0			0
$921$	−271507.	+	470263.i	−0.320082	+	0.554398i
$922$		0			0
$923$			147698.i			0.173368i
$924$		0			0
$925$	−1.40980e6			−1.64768
$926$		0			0
$927$	356995.	+	206111.i	0.415434	+	0.239851i
$928$		0			0
$929$	−31052.0	+	17927.9i	−0.0359798	+	0.0207729i	−0.517882	−	0.855452i	$-0.673279\pi$
$929$	−31052.0	+	17927.9i	−0.0359798	+	0.0207729i	0.481902	+	0.876225i	$0.339946\pi$
$930$		0			0
$931$	−78375.0	−	935700.i	−0.0904228	−	1.07954i
$932$		0			0
$933$	−330038.	−	571642.i	−0.379140	−	0.656691i
$934$		0			0
$935$	−97206.4	+	168366.i	−0.111192	+	0.192589i
$936$		0			0
$937$		−	1.27743e6i		−	1.45498i	−0.686119	−	0.727489i	$-0.740687\pi$
$937$		−	1.27743e6i		−	1.45498i	0.686119	−	0.727489i	$-0.259313\pi$
$938$		0			0
$939$	979548.			1.11095
$940$		0			0
$941$	1.03433e6	+	597168.i	1.16809	+	0.674399i	0.953230	−	0.302245i	$-0.0977359\pi$
$941$	1.03433e6	+	597168.i	1.16809	+	0.674399i	0.214864	+	0.976644i	$0.431069\pi$
$942$		0			0
$943$	140230.	−	80961.9i	0.157695	−	0.0910453i
$944$		0			0
$945$	−218284.	−	114145.i	−0.244432	−	0.127819i
$946$		0			0
$947$	−427985.	−	741292.i	−0.477231	−	0.826589i	0.522428	−	0.852683i	$-0.325026\pi$
$947$	−427985.	−	741292.i	−0.477231	−	0.826589i	−0.999659	+	0.0260943i	$0.991693\pi$
$948$		0			0
$949$	6929.38	−	12002.0i	0.00769417	−	0.0133267i
$950$		0			0
$951$			654584.i			0.723776i
$952$		0			0
$953$	−271093.			−0.298491			−0.149246	−	0.988800i	$-0.547685\pi$
$953$	−271093.			−0.298491			−0.149246	+	0.988800i	$0.547685\pi$
$954$		0			0
$955$	448555.	+	258973.i	0.491823	+	0.283954i
$956$		0			0
$957$	114665.	−	66202.1i	0.125201	−	0.0722850i
$958$		0			0
$959$	12570.5	+	300678.i	0.0136683	+	0.326938i
$960$		0			0
$961$	345384.	+	598222.i	0.373986	+	0.647762i
$962$		0			0
$963$	−8692.99	+	15056.7i	−0.00937382	+	0.0162359i
$964$		0			0
$965$			2.45307e6i			2.63424i
$966$		0			0
$967$	1.14518e6			1.22467			0.612336	−	0.790598i	$-0.290230\pi$
$967$	1.14518e6			1.22467			0.612336	+	0.790598i	$0.290230\pi$
$968$		0			0
$969$	−226413.	−	130720.i	−0.241131	−	0.139217i
$970$		0			0
$971$	−459920.	+	265535.i	−0.487802	+	0.281633i	−0.723662	−	0.690154i	$-0.757543\pi$
$971$	−459920.	+	265535.i	−0.487802	+	0.281633i	0.235860	+	0.971787i	$0.424209\pi$
$972$		0			0
$973$	415156.	+	654334.i	0.438517	+	0.691152i
$974$		0			0
$975$	38994.7	+	67540.9i	0.0410201	+	0.0710489i
$976$		0			0
$977$	756793.	−	1.31080e6i	0.792845	−	1.37325i	−0.131354	−	0.991336i	$-0.541932\pi$
$977$	756793.	−	1.31080e6i	0.792845	−	1.37325i	0.924199	−	0.381912i	$-0.124734\pi$
$978$		0			0
$979$			460585.i			0.480556i
$980$		0			0
$981$	583996.			0.606837
$982$		0			0
$983$	480929.	+	277664.i	0.497707	+	0.287351i	0.727766	−	0.685825i	$-0.240559\pi$
$983$	480929.	+	277664.i	0.497707	+	0.287351i	−0.230059	+	0.973177i	$0.573892\pi$
$984$		0			0
$985$	−94067.5	+	54309.9i	−0.0969544	+	0.0559766i
$986$		0			0
$987$	28421.7	−	18032.8i	0.0291754	−	0.0185109i
$988$		0			0
$989$	−1.15388e6	−	1.99858e6i	−1.17969	−	2.04329i
$990$		0			0
$991$	584486.	−	1.01236e6i	0.595151	−	1.03083i	−0.398375	−	0.917223i	$-0.630426\pi$
$991$	584486.	−	1.01236e6i	0.595151	−	1.03083i	0.993526	−	0.113609i	$-0.0362411\pi$
$992$		0			0
$993$		−	498697.i		−	0.505752i
$994$		0			0
$995$	−439628.			−0.444058
$996$		0			0
$997$	880818.	+	508541.i	0.886127	+	0.511606i	0.872674	−	0.488304i	$-0.162384\pi$
$997$	880818.	+	508541.i	0.886127	+	0.511606i	0.0134533	+	0.999909i	$0.495718\pi$
$998$		0			0
$999$	−259952.	+	150083.i	−0.260472	+	0.150384i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.5.z.a.73.2	✓	16
3.2	odd	2		504.5.by.a.73.7		16
4.3	odd	2		336.5.bh.i.241.2		16
7.3	odd	6		1176.5.f.b.97.7		16
7.4	even	3		1176.5.f.b.97.10		16
7.5	odd	6	inner	168.5.z.a.145.2	yes	16
21.5	even	6		504.5.by.a.145.7		16
28.19	even	6		336.5.bh.i.145.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.5.z.a.73.2	✓	16	1.1	even	1	trivial
168.5.z.a.145.2	yes	16	7.5	odd	6	inner
336.5.bh.i.145.2		16	28.19	even	6
336.5.bh.i.241.2		16	4.3	odd	2
504.5.by.a.73.7		16	3.2	odd	2
504.5.by.a.145.7		16	21.5	even	6
1176.5.f.b.97.7		16	7.3	odd	6
1176.5.f.b.97.10		16	7.4	even	3

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	168.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 2.59808i) q^{3} +(-31.0314 + 17.9160i) q^{5} +(-41.3748 + 26.2512i) q^{7} +(13.5000 + 23.3827i) q^{9} +O(q^{10})\)	\(q+(-4.50000 - 2.59808i) q^{3} +(-31.0314 + 17.9160i) q^{5} +(-41.3748 + 26.2512i) q^{7} +(13.5000 + 23.3827i) q^{9} +(21.0862 - 36.5224i) q^{11} -22.7779i q^{13} +186.188 q^{15} +(111.418 + 64.3274i) q^{17} +(338.683 - 195.539i) q^{19} +(254.389 - 10.6353i) q^{21} +(-331.549 - 574.260i) q^{23} +(329.465 - 570.651i) q^{25} -140.296i q^{27} -604.215 q^{29} +(1100.33 + 635.273i) q^{31} +(-189.776 + 109.567i) q^{33} +(813.603 - 1555.88i) q^{35} +(-1069.76 - 1852.88i) q^{37} +(-59.1788 + 102.501i) q^{39} +244.193i q^{41} +3480.28 q^{43} +(-837.848 - 483.732i) q^{45} +(114.489 - 66.1002i) q^{47} +(1022.75 - 2172.27i) q^{49} +(-334.255 - 578.946i) q^{51} +(481.312 - 833.657i) q^{53} +1511.12i q^{55} -2032.10 q^{57} +(5542.71 + 3200.09i) q^{59} +(-524.016 + 302.541i) q^{61} +(-1172.38 - 613.064i) q^{63} +(408.089 + 706.831i) q^{65} +(1596.09 - 2764.51i) q^{67} +3445.56i q^{69} -6484.24 q^{71} +(526.915 + 304.214i) q^{73} +(-2965.19 + 1711.95i) q^{75} +(86.3170 + 2064.64i) q^{77} +(596.115 + 1032.50i) q^{79} +(-364.500 + 631.333i) q^{81} -4573.10i q^{83} -4609.95 q^{85} +(2718.97 + 1569.80i) q^{87} +(-9458.27 + 5460.73i) q^{89} +(597.947 + 942.433i) q^{91} +(-3300.98 - 5717.46i) q^{93} +(-7006.54 + 12135.7i) q^{95} +9241.16i q^{97} +1138.66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) \) 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9	\( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9} - 108 q^{11} - 72 q^{15} - 168 q^{17} + 948 q^{19} - 252 q^{21} - 768 q^{23} + 1908 q^{25} - 1608 q^{29} - 3216 q^{31} + 972 q^{33} + 696 q^{35} - 1820 q^{37} - 1188 q^{39} + 2888 q^{43} + 324 q^{45} + 744 q^{47} - 3784 q^{49} + 504 q^{51} + 4476 q^{53} - 5688 q^{57} - 4668 q^{59} + 17760 q^{61} + 1188 q^{63} + 8760 q^{65} + 1580 q^{67} + 48 q^{71} + 588 q^{73} - 17172 q^{75} - 17508 q^{77} - 3824 q^{79} - 5832 q^{81} + 11440 q^{85} + 7236 q^{87} - 360 q^{89} + 25860 q^{91} + 9648 q^{93} - 21792 q^{95} - 5832 q^{99}+O(q^{100}) \) 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9 - 108 * q^11 - 72 * q^15 - 168 * q^17 + 948 * q^19 - 252 * q^21 - 768 * q^23 + 1908 * q^25 - 1608 * q^29 - 3216 * q^31 + 972 * q^33 + 696 * q^35 - 1820 * q^37 - 1188 * q^39 + 2888 * q^43 + 324 * q^45 + 744 * q^47 - 3784 * q^49 + 504 * q^51 + 4476 * q^53 - 5688 * q^57 - 4668 * q^59 + 17760 * q^61 + 1188 * q^63 + 8760 * q^65 + 1580 * q^67 + 48 * q^71 + 588 * q^73 - 17172 * q^75 - 17508 * q^77 - 3824 * q^79 - 5832 * q^81 + 11440 * q^85 + 7236 * q^87 - 360 * q^89 + 25860 * q^91 + 9648 * q^93 - 21792 * q^95 - 5832 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more