LMFDB - Embedded newform 378.3.i.a.179.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [378,3,Mod(179,378)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("378.179"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	378.i (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$10.2997539928$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			179.6
Character	$\chi$	$=$	378.179
Dual form			378.3.i.a.359.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +2.45915i q^{5} +(1.16622 - 6.90217i) q^{7} -2.82843i q^{8} +(1.73888 - 3.01183i) q^{10} -11.2668i q^{11} +(-9.55569 + 16.5509i) q^{13} +(-6.30889 + 7.62875i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(8.32907 + 4.80879i) q^{17} +(-8.82426 - 15.2841i) q^{19} +(-4.25936 + 2.45915i) q^{20} +(-7.96680 + 13.7989i) q^{22} -32.3395i q^{23} +18.9526 q^{25} +(23.4066 - 13.5138i) q^{26} +(13.1211 - 4.88221i) q^{28} +(25.7644 - 14.8751i) q^{29} +(-9.69972 - 16.8004i) q^{31} +(4.89898 - 2.82843i) q^{32} +(-6.80066 - 11.7791i) q^{34} +(16.9734 + 2.86791i) q^{35} +(-15.3594 - 26.6032i) q^{37} +24.9588i q^{38} +6.95551 q^{40} +(-10.3804 - 5.99315i) q^{41} +(-30.0344 - 52.0211i) q^{43} +(19.5146 - 11.2668i) q^{44} +(-22.8675 + 39.6076i) q^{46} +(-36.2320 - 20.9185i) q^{47} +(-46.2799 - 16.0989i) q^{49} +(-23.2121 - 13.4015i) q^{50} -38.2228 q^{52} +(20.2918 + 11.7155i) q^{53} +27.7066 q^{55} +(-19.5223 - 3.29857i) q^{56} -42.0731 q^{58} +(-4.21468 + 2.43335i) q^{59} +(16.6932 - 28.9135i) q^{61} +27.4350i q^{62} -8.00000 q^{64} +(-40.7012 - 23.4988i) q^{65} +(54.1318 + 93.7590i) q^{67} +19.2352i q^{68} +(-18.7602 - 15.5145i) q^{70} -133.605i q^{71} +(60.3429 - 104.517i) q^{73} +43.4429i q^{74} +(17.6485 - 30.5681i) q^{76} +(-77.7651 - 13.1395i) q^{77} +(-13.0333 + 22.5743i) q^{79} +(-8.51873 - 4.91829i) q^{80} +(8.47559 + 14.6801i) q^{82} +(59.7033 - 34.4697i) q^{83} +(-11.8255 + 20.4824i) q^{85} +84.9501i q^{86} -31.8672 q^{88} +(-128.675 + 74.2905i) q^{89} +(103.093 + 85.2571i) q^{91} +(56.0136 - 32.3395i) q^{92} +(29.5833 + 51.2398i) q^{94} +(37.5857 - 21.7001i) q^{95} +(11.3016 + 19.5750i) q^{97} +(45.2974 + 52.4419i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 32 q^{4} + 2 q^{7} + 10 q^{13} - 36 q^{14} - 64 q^{16} - 54 q^{17} + 28 q^{19} - 160 q^{25} - 72 q^{26} - 4 q^{28} - 36 q^{29} - 8 q^{31} - 90 q^{35} + 22 q^{37} - 72 q^{41} + 16 q^{43} + 72 q^{44}+ \cdots + 288 q^{98}+O(q^{100}) $ 32 * q + 32 * q^4 + 2 * q^7 + 10 * q^13 - 36 * q^14 - 64 * q^16 - 54 * q^17 + 28 * q^19 - 160 * q^25 - 72 * q^26 - 4 * q^28 - 36 * q^29 - 8 * q^31 - 90 * q^35 + 22 * q^37 - 72 * q^41 + 16 * q^43 + 72 * q^44 - 12 * q^46 + 108 * q^47 + 74 * q^49 + 288 * q^50 + 40 * q^52 - 72 * q^53 - 24 * q^55 + 48 * q^58 + 90 * q^59 - 62 * q^61 - 256 * q^64 - 378 * q^65 + 70 * q^67 - 108 * q^70 + 196 * q^73 - 56 * q^76 - 630 * q^77 - 38 * q^79 + 60 * q^85 - 486 * q^89 - 122 * q^91 - 252 * q^92 + 168 * q^94 + 72 * q^95 - 38 * q^97 + 288 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	−1.22474	−	0.707107i	−0.612372	−	0.353553i
$2$	−1.22474	−	0.707107i	−0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	1.00000	+	1.73205i	0.250000	+	0.433013i
$5$			2.45915i			0.491829i	0.969292	+	0.245915i	$0.0790883\pi$
$5$			2.45915i			0.491829i	−0.969292	+	0.245915i	$0.920912\pi$
$6$		0			0
$7$	1.16622	−	6.90217i	0.166603	−	0.986024i
$7$	1.16622	−	6.90217i	0.166603	−	0.986024i
$8$		−	2.82843i		−	0.353553i
$9$		0			0
$10$	1.73888	−	3.01183i	0.173888	−	0.301183i
$11$		−	11.2668i		−	1.02425i	−0.858911	−	0.512125i	$-0.828858\pi$
$11$		−	11.2668i		−	1.02425i	0.858911	−	0.512125i	$-0.171142\pi$
$12$		0			0
$13$	−9.55569	+	16.5509i	−0.735053	+	1.27315i	0.219646	+	0.975580i	$0.429510\pi$
$13$	−9.55569	+	16.5509i	−0.735053	+	1.27315i	−0.954700	+	0.297570i	$0.903824\pi$
$14$	−6.30889	+	7.62875i	−0.450635	+	0.544911i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.125000	+	0.216506i
$17$	8.32907	+	4.80879i	0.489945	+	0.282870i	0.724552	−	0.689220i	$-0.242047\pi$
$17$	8.32907	+	4.80879i	0.489945	+	0.282870i	−0.234606	+	0.972090i	$0.575380\pi$
$18$		0			0
$19$	−8.82426	−	15.2841i	−0.464435	−	0.804424i	0.534741	−	0.845016i	$-0.320409\pi$
$19$	−8.82426	−	15.2841i	−0.464435	−	0.804424i	−0.999176	+	0.0405916i	$0.987076\pi$
$20$	−4.25936	+	2.45915i	−0.212968	+	0.122957i
$21$		0			0
$22$	−7.96680	+	13.7989i	−0.362127	+	0.627223i
$23$		−	32.3395i		−	1.40606i	−0.711158	−	0.703032i	$-0.751829\pi$
$23$		−	32.3395i		−	1.40606i	0.711158	−	0.703032i	$-0.248171\pi$
$24$		0			0
$25$	18.9526			0.758104
$26$	23.4066	−	13.5138i	0.900253	−	0.519761i
$27$		0			0
$28$	13.1211	−	4.88221i	0.468612	−	0.174365i
$29$	25.7644	−	14.8751i	0.888428	−	0.512934i	0.0149997	−	0.999887i	$-0.495225\pi$
$29$	25.7644	−	14.8751i	0.888428	−	0.512934i	0.873428	+	0.486954i	$0.161892\pi$
$30$		0			0
$31$	−9.69972	−	16.8004i	−0.312894	−	0.541949i	0.666093	−	0.745868i	$-0.267965\pi$
$31$	−9.69972	−	16.8004i	−0.312894	−	0.541949i	−0.978988	+	0.203920i	$0.934632\pi$
$32$	4.89898	−	2.82843i	0.153093	−	0.0883883i
$33$		0			0
$34$	−6.80066	−	11.7791i	−0.200019	−	0.346444i
$35$	16.9734	+	2.86791i	0.484955	+	0.0819402i
$36$		0			0
$37$	−15.3594	−	26.6032i	−0.415118	−	0.719006i	0.580322	−	0.814387i	$-0.302927\pi$
$37$	−15.3594	−	26.6032i	−0.415118	−	0.719006i	−0.995441	+	0.0953806i	$0.969593\pi$
$38$			24.9588i			0.656810i
$39$		0			0
$40$	6.95551			0.173888
$41$	−10.3804	−	5.99315i	−0.253181	−	0.146174i	0.368039	−	0.929810i	$-0.380029\pi$
$41$	−10.3804	−	5.99315i	−0.253181	−	0.146174i	−0.621220	+	0.783636i	$0.713363\pi$
$42$		0			0
$43$	−30.0344	−	52.0211i	−0.698475	−	1.20979i	−0.968995	−	0.247079i	$-0.920529\pi$
$43$	−30.0344	−	52.0211i	−0.698475	−	1.20979i	0.270521	−	0.962714i	$-0.412804\pi$
$44$	19.5146	−	11.2668i	0.443514	−	0.256063i
$45$		0			0
$46$	−22.8675	+	39.6076i	−0.497119	+	0.861035i
$47$	−36.2320	−	20.9185i	−0.770893	−	0.445075i	0.0622998	−	0.998057i	$-0.480156\pi$
$47$	−36.2320	−	20.9185i	−0.770893	−	0.445075i	−0.833193	+	0.552982i	$0.813490\pi$
$48$		0			0
$49$	−46.2799	−	16.0989i	−0.944487	−	0.328549i
$50$	−23.2121	−	13.4015i	−0.464242	−	0.268030i
$51$		0			0
$52$	−38.2228			−0.735053
$53$	20.2918	+	11.7155i	0.382865	+	0.221047i	0.679064	−	0.734079i	$-0.262386\pi$
$53$	20.2918	+	11.7155i	0.382865	+	0.221047i	−0.296199	+	0.955126i	$0.595719\pi$
$54$		0			0
$55$	27.7066			0.503756
$56$	−19.5223	−	3.29857i	−0.348612	−	0.0589030i
$57$		0			0
$58$	−42.0731			−0.725398
$59$	−4.21468	+	2.43335i	−0.0714353	+	0.0412432i	−0.535292	−	0.844667i	$-0.679799\pi$
$59$	−4.21468	+	2.43335i	−0.0714353	+	0.0412432i	0.463857	+	0.885910i	$0.346465\pi$
$60$		0			0
$61$	16.6932	−	28.9135i	0.273659	−	0.473991i	−0.696137	−	0.717909i	$-0.745099\pi$
$61$	16.6932	−	28.9135i	0.273659	−	0.473991i	0.969796	+	0.243918i	$0.0784327\pi$
$62$			27.4350i			0.442499i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	−40.7012	−	23.4988i	−0.626172	−	0.361521i
$66$		0			0
$67$	54.1318	+	93.7590i	0.807937	+	1.39939i	0.914290	+	0.405059i	$0.132749\pi$
$67$	54.1318	+	93.7590i	0.807937	+	1.39939i	−0.106354	+	0.994328i	$0.533918\pi$
$68$			19.2352i			0.282870i
$69$		0			0
$70$	−18.7602	−	15.5145i	−0.268003	−	0.221636i
$71$		−	133.605i		−	1.88175i	−0.338748	−	0.940877i	$-0.610003\pi$
$71$		−	133.605i		−	1.88175i	0.338748	−	0.940877i	$-0.389997\pi$
$72$		0			0
$73$	60.3429	−	104.517i	0.826615	−	1.43174i	−0.0740630	−	0.997254i	$-0.523597\pi$
$73$	60.3429	−	104.517i	0.826615	−	1.43174i	0.900678	−	0.434486i	$-0.143070\pi$
$74$			43.4429i			0.587066i
$75$		0			0
$76$	17.6485	−	30.5681i	0.232217	−	0.402212i
$77$	−77.7651	−	13.1395i	−1.00994	−	0.170643i
$78$		0			0
$79$	−13.0333	+	22.5743i	−0.164979	+	0.285751i	−0.936648	−	0.350273i	$-0.886089\pi$
$79$	−13.0333	+	22.5743i	−0.164979	+	0.285751i	0.771669	+	0.636024i	$0.219422\pi$
$80$	−8.51873	−	4.91829i	−0.106484	−	0.0614786i
$81$		0			0
$82$	8.47559	+	14.6801i	0.103361	+	0.179026i
$83$	59.7033	−	34.4697i	0.719317	−	0.415298i	−0.0951842	−	0.995460i	$-0.530344\pi$
$83$	59.7033	−	34.4697i	0.719317	−	0.415298i	0.814501	+	0.580162i	$0.197011\pi$
$84$		0			0
$85$	−11.8255	+	20.4824i	−0.139124	+	0.240969i
$86$			84.9501i			0.987792i
$87$		0			0
$88$	−31.8672			−0.362127
$89$	−128.675	+	74.2905i	−1.44579	+	0.834725i	−0.998227	−	0.0595295i	$-0.981040\pi$
$89$	−128.675	+	74.2905i	−1.44579	+	0.834725i	−0.447559	+	0.894254i	$0.647707\pi$
$90$		0			0
$91$	103.093	+	85.2571i	1.13289	+	0.936891i
$92$	56.0136	−	32.3395i	0.608843	−	0.351516i
$93$		0			0
$94$	29.5833	+	51.2398i	0.314716	+	0.545104i
$95$	37.5857	−	21.7001i	0.395639	−	0.228422i
$96$		0			0
$97$	11.3016	+	19.5750i	0.116512	+	0.201804i	0.918383	−	0.395693i	$-0.129495\pi$
$97$	11.3016	+	19.5750i	0.116512	+	0.201804i	−0.801871	+	0.597497i	$0.796162\pi$
$98$	45.2974	+	52.4419i	0.462218	+	0.535121i
$99$		0			0
$100$	18.9526	+	32.8269i	0.189526	+	0.328269i
$101$			69.5252i			0.688368i	0.938902	+	0.344184i	$0.111844\pi$
$101$			69.5252i			0.688368i	−0.938902	+	0.344184i	$0.888156\pi$
$102$		0			0
$103$	91.1054			0.884519			0.442259	−	0.896887i	$-0.354177\pi$
$103$	91.1054			0.884519			0.442259	+	0.896887i	$0.354177\pi$
$104$	46.8132	+	27.0276i	0.450126	+	0.259881i
$105$		0			0
$106$	−16.5682	−	28.6970i	−0.156304	−	0.270726i
$107$	−136.016	+	78.5290i	−1.27118	+	0.733916i	−0.975210	−	0.221282i	$-0.928976\pi$
$107$	−136.016	+	78.5290i	−1.27118	+	0.733916i	−0.295969	+	0.955197i	$0.595643\pi$
$108$		0			0
$109$	26.9553	−	46.6880i	0.247296	−	0.428330i	−0.715478	−	0.698635i	$-0.753791\pi$
$109$	26.9553	−	46.6880i	0.247296	−	0.428330i	0.962775	+	0.270305i	$0.0871245\pi$
$110$	−33.9335	−	19.5915i	−0.308486	−	0.178105i
$111$		0			0
$112$	21.5774	+	17.8442i	0.192655	+	0.159324i
$113$	112.172	+	64.7627i	0.992675	+	0.573121i	0.906073	−	0.423122i	$-0.139066\pi$
$113$	112.172	+	64.7627i	0.992675	+	0.573121i	0.0866020	+	0.996243i	$0.472399\pi$
$114$		0			0
$115$	79.5275			0.691543
$116$	51.5288	+	29.7502i	0.444214	+	0.256467i
$117$		0			0
$118$	6.88255			0.0583267
$119$	42.9046	−	51.8805i	0.360543	−	0.435971i
$120$		0			0
$121$	−5.93985			−0.0490896
$122$	−40.8898	+	23.6077i	−0.335162	+	0.193506i
$123$		0			0
$124$	19.3994	−	33.6008i	0.156447	−	0.270974i
$125$			108.086i			0.864687i
$126$		0			0
$127$	69.5844			0.547908			0.273954	−	0.961743i	$-0.411668\pi$
$127$	69.5844			0.547908			0.273954	+	0.961743i	$0.411668\pi$
$128$	9.79796	+	5.65685i	0.0765466	+	0.0441942i
$129$		0			0
$130$	33.2324	+	57.5602i	0.255634	+	0.442771i
$131$		−	1.80136i		−	0.0137508i	−0.999976	−	0.00687541i	$-0.997811\pi$
$131$		−	1.80136i		−	0.0137508i	0.999976	−	0.00687541i	$-0.00218853\pi$
$132$		0			0
$133$	−115.784	+	43.0819i	−0.870558	+	0.323924i
$134$		−	153.108i		−	1.14260i
$135$		0			0
$136$	13.6013	−	23.5582i	0.100010	−	0.173222i
$137$			45.1078i			0.329254i	0.986356	+	0.164627i	$0.0526420\pi$
$137$			45.1078i			0.329254i	−0.986356	+	0.164627i	$0.947358\pi$
$138$		0			0
$139$	−127.215	+	220.344i	−0.915219	+	1.58521i	−0.108638	+	0.994081i	$0.534649\pi$
$139$	−127.215	+	220.344i	−0.915219	+	1.58521i	−0.806581	+	0.591124i	$0.798684\pi$
$140$	12.0061	+	32.2668i	0.0857577	+	0.230477i
$141$		0			0
$142$	−94.4727	+	163.631i	−0.665301	+	1.15233i
$143$	186.476	+	107.662i	1.30402	+	0.752879i
$144$		0			0
$145$	36.5800	+	63.3584i	0.252276	+	0.436955i
$146$	−147.809	+	85.3378i	−1.01239	+	0.584505i
$147$		0			0
$148$	30.7188	−	53.2065i	0.207559	−	0.359503i
$149$		−	33.6374i		−	0.225754i	−0.993609	−	0.112877i	$-0.963993\pi$
$149$		−	33.6374i		−	0.225754i	0.993609	−	0.112877i	$-0.0360066\pi$
$150$		0			0
$151$	−174.677			−1.15680			−0.578402	−	0.815752i	$-0.696324\pi$
$151$	−174.677			−1.15680			−0.578402	+	0.815752i	$0.696324\pi$
$152$	−43.2299	+	24.9588i	−0.284407	+	0.164202i
$153$		0			0
$154$	85.9513	+	71.0808i	0.558125	+	0.461563i
$155$	41.3147	−	23.8530i	0.266546	−	0.153890i
$156$		0			0
$157$	−26.0441	−	45.1098i	−0.165886	−	0.287323i	0.771083	−	0.636734i	$-0.219715\pi$
$157$	−26.0441	−	45.1098i	−0.165886	−	0.287323i	−0.936970	+	0.349411i	$0.886382\pi$
$158$	31.9250	−	18.4319i	0.202057	−	0.116657i
$159$		0			0
$160$	6.95551	+	12.0473i	0.0434720	+	0.0752956i
$161$	−223.212	−	37.7150i	−1.38641	−	0.234254i
$162$		0			0
$163$	68.5686	+	118.764i	0.420666	+	0.728615i	0.996005	−	0.0893001i	$-0.0284630\pi$
$163$	68.5686	+	118.764i	0.420666	+	0.728615i	−0.575339	+	0.817915i	$0.695130\pi$
$164$		−	23.9726i		−	0.146174i
$165$		0			0
$166$	−97.4951			−0.587320
$167$	99.7282	+	57.5781i	0.597175	+	0.344779i	0.767929	−	0.640535i	$-0.221287\pi$
$167$	99.7282	+	57.5781i	0.597175	+	0.344779i	−0.170755	+	0.985314i	$0.554621\pi$
$168$		0			0
$169$	−98.1226	−	169.953i	−0.580607	−	1.00564i
$170$	28.9665	−	16.7238i	0.170391	−	0.0983753i
$171$		0			0
$172$	60.0688	−	104.042i	0.349237	−	0.604897i
$173$	217.116	+	125.352i	1.25501	+	0.724578i	0.972099	−	0.234569i	$-0.0753679\pi$
$173$	217.116	+	125.352i	1.25501	+	0.724578i	0.282907	+	0.959147i	$0.408701\pi$
$174$		0			0
$175$	22.1029	−	130.814i	0.126302	−	0.747509i
$176$	39.0292	+	22.5335i	0.221757	+	0.128031i
$177$		0			0
$178$	210.125			1.18048
$179$	21.9209	+	12.6560i	0.122463	+	0.0707041i	0.559980	−	0.828506i	$-0.310809\pi$
$179$	21.9209	+	12.6560i	0.122463	+	0.0707041i	−0.437517	+	0.899210i	$0.644142\pi$
$180$		0			0
$181$	−35.6698			−0.197070			−0.0985352	−	0.995134i	$-0.531416\pi$
$181$	−35.6698			−0.197070			−0.0985352	+	0.995134i	$0.531416\pi$
$182$	−65.9772	−	177.316i	−0.362512	−	0.974265i
$183$		0			0
$184$	−91.4698			−0.497119
$185$	65.4212	−	37.7710i	0.353628	−	0.204167i
$186$		0			0
$187$	54.1795	−	93.8416i	0.289730	−	0.501827i
$188$		−	83.6742i		−	0.445075i
$189$		0			0
$190$	−61.3772			−0.323038
$191$	30.3226	+	17.5068i	0.158757	+	0.0916585i	0.577274	−	0.816551i	$-0.304117\pi$
$191$	30.3226	+	17.5068i	0.158757	+	0.0916585i	−0.418517	+	0.908209i	$0.637450\pi$
$192$		0			0
$193$	69.6945	+	120.714i	0.361111	+	0.625463i	0.988144	−	0.153530i	$-0.0490641\pi$
$193$	69.6945	+	120.714i	0.361111	+	0.625463i	−0.627033	+	0.778993i	$0.715731\pi$
$194$		−	31.9658i		−	0.164772i
$195$		0			0
$196$	−18.3957	−	96.2580i	−0.0938558	−	0.491112i
$197$		−	85.6801i		−	0.434924i	−0.976069	−	0.217462i	$-0.930222\pi$
$197$		−	85.6801i		−	0.434924i	0.976069	−	0.217462i	$-0.0697779\pi$
$198$		0			0
$199$	−61.4480	+	106.431i	−0.308784	+	0.534830i	−0.978097	−	0.208151i	$-0.933255\pi$
$199$	−61.4480	+	106.431i	−0.308784	+	0.534830i	0.669313	+	0.742981i	$0.266589\pi$
$200$		−	53.6061i		−	0.268030i
$201$		0			0
$202$	49.1617	−	85.1506i	0.243375	−	0.421538i
$203$	−72.6234	−	195.178i	−0.357751	−	0.961467i
$204$		0			0
$205$	14.7380	−	25.5270i	0.0718928	−	0.124522i
$206$	−111.581	−	64.4213i	−0.541655	−	0.312725i
$207$		0			0
$208$	−38.2228	−	66.2038i	−0.183763	−	0.318287i
$209$	−172.202	+	99.4208i	−0.823932	+	0.475697i
$210$		0			0
$211$	135.356	−	234.444i	0.641499	−	1.11111i	−0.343600	−	0.939116i	$-0.611646\pi$
$211$	135.356	−	234.444i	0.641499	−	1.11111i	0.985098	−	0.171992i	$-0.0550203\pi$
$212$			46.8620i			0.221047i
$213$		0			0
$214$	222.114			1.03791
$215$	127.928	−	73.8590i	0.595012	−	0.343530i
$216$		0			0
$217$	−127.271	+	47.3561i	−0.586504	+	0.218231i
$218$	−66.0268	+	38.1206i	−0.302875	+	0.174865i
$219$		0			0
$220$	27.7066	+	47.9892i	0.125939	+	0.218133i
$221$	−159.180	+	91.9027i	−0.720272	+	0.415849i
$222$		0			0
$223$	35.6354	+	61.7223i	0.159800	+	0.276781i	0.934796	−	0.355184i	$-0.115582\pi$
$223$	35.6354	+	61.7223i	0.159800	+	0.276781i	−0.774997	+	0.631965i	$0.782248\pi$
$224$	−13.8090	−	37.1122i	−0.0616473	−	0.165679i
$225$		0			0
$226$	−91.5883	−	158.636i	−0.405258	−	0.701927i
$227$			244.104i			1.07535i	0.843152	+	0.537675i	$0.180697\pi$
$227$			244.104i			1.07535i	−0.843152	+	0.537675i	$0.819303\pi$
$228$		0			0
$229$	−446.853			−1.95132			−0.975662	−	0.219279i	$-0.929629\pi$
$229$	−446.853			−1.95132			−0.975662	+	0.219279i	$0.929629\pi$
$230$	−97.4008	−	56.2344i	−0.423482	−	0.244497i
$231$		0			0
$232$	−42.0731	−	72.8727i	−0.181350	−	0.314107i
$233$	−230.950	+	133.339i	−0.991200	+	0.572269i	−0.905633	−	0.424063i	$-0.860604\pi$
$233$	−230.950	+	133.339i	−0.991200	+	0.572269i	−0.0855670	+	0.996332i	$0.527270\pi$
$234$		0			0
$235$	51.4418	−	89.0997i	0.218901	−	0.379148i
$236$	−8.42937	−	4.86670i	−0.0357177	−	0.0206216i
$237$		0			0
$238$	−89.2323	+	33.2023i	−0.374926	+	0.139505i
$239$	−314.587	−	181.627i	−1.31627	−	0.759946i	−0.333140	−	0.942878i	$-0.608108\pi$
$239$	−314.587	−	181.627i	−1.31627	−	0.759946i	−0.983126	+	0.182931i	$0.941441\pi$
$240$		0			0
$241$	326.755			1.35583			0.677915	−	0.735141i	$-0.262884\pi$
$241$	326.755			1.35583			0.677915	+	0.735141i	$0.262884\pi$
$242$	7.27480	+	4.20011i	0.0300611	+	0.0173558i
$243$		0			0
$244$	66.7728			0.273659
$245$	39.5895	−	113.809i	0.161590	−	0.464526i
$246$		0			0
$247$	337.288			1.36554
$248$	−47.5187	+	27.4350i	−0.191608	+	0.110625i
$249$		0			0
$250$	76.4282	−	132.378i	0.305713	−	0.529510i
$251$		−	239.251i		−	0.953191i	−0.879123	−	0.476595i	$-0.841871\pi$
$251$		−	239.251i		−	0.953191i	0.879123	−	0.476595i	$-0.158129\pi$
$252$		0			0
$253$	−364.361			−1.44016
$254$	−85.2231	−	49.2036i	−0.335524	−	0.193715i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.0312500	−	0.0541266i
$257$		−	335.542i		−	1.30561i	−0.757525	−	0.652806i	$-0.773592\pi$
$257$		−	335.542i		−	1.30561i	0.757525	−	0.652806i	$-0.226408\pi$
$258$		0			0
$259$	−201.532	+	74.9878i	−0.778117	+	0.289528i
$260$		−	93.9954i		−	0.361521i
$261$		0			0
$262$	−1.27375	+	2.20620i	−0.00486165	+	0.00842063i
$263$			265.889i			1.01099i	0.862831	+	0.505493i	$0.168689\pi$
$263$			265.889i			1.01099i	−0.862831	+	0.505493i	$0.831311\pi$
$264$		0			0
$265$	−28.8101	+	49.9006i	−0.108717	+	0.188304i
$266$	172.270	+	29.1074i	0.647630	+	0.109426i
$267$		0			0
$268$	−108.264	+	187.518i	−0.403968	+	0.699694i
$269$	283.798	+	163.851i	1.05501	+	0.609111i	0.924048	−	0.382277i	$-0.124860\pi$
$269$	283.798	+	163.851i	1.05501	+	0.609111i	0.130963	+	0.991387i	$0.458193\pi$
$270$		0			0
$271$	−190.337	−	329.674i	−0.702351	−	1.21651i	−0.967639	−	0.252339i	$-0.918800\pi$
$271$	−190.337	−	329.674i	−0.702351	−	1.21651i	0.265288	−	0.964169i	$-0.414533\pi$
$272$	−33.3163	+	19.2352i	−0.122486	+	0.0707175i
$273$		0			0
$274$	31.8960	−	55.2456i	0.116409	−	0.201626i
$275$		−	213.534i		−	0.776489i
$276$		0			0
$277$	−203.417			−0.734357			−0.367178	−	0.930151i	$-0.619676\pi$
$277$	−203.417			−0.734357			−0.367178	+	0.930151i	$0.619676\pi$
$278$	311.613	−	179.910i	1.12091	−	0.647157i
$279$		0			0
$280$	8.11166	−	48.0081i	0.0289702	−	0.171458i
$281$	309.847	−	178.890i	1.10266	−	0.636620i	0.165740	−	0.986170i	$-0.446999\pi$
$281$	309.847	−	178.890i	1.10266	−	0.636620i	0.936918	+	0.349550i	$0.113665\pi$
$282$		0			0
$283$	50.6119	+	87.6624i	0.178841	+	0.309761i	0.941484	−	0.337059i	$-0.109432\pi$
$283$	50.6119	+	87.6624i	0.178841	+	0.309761i	−0.762643	+	0.646820i	$0.776099\pi$
$284$	231.410	−	133.605i	0.814824	−	0.470439i
$285$		0			0
$286$	−152.257	−	263.716i	−0.532366	−	0.922085i
$287$	−53.4716	+	64.6582i	−0.186312	+	0.225290i
$288$		0			0
$289$	−98.2511	−	170.176i	−0.339969	−	0.588844i
$290$		−	103.464i		−	0.356772i
$291$		0			0
$292$	241.372			0.826615
$293$	47.8628	+	27.6336i	0.163354	+	0.0943126i	0.579448	−	0.815009i	$-0.303268\pi$
$293$	47.8628	+	27.6336i	0.163354	+	0.0943126i	−0.416094	+	0.909322i	$0.636601\pi$
$294$		0			0
$295$	−5.98396	−	10.3645i	−0.0202846	−	0.0351340i
$296$	−75.2453	+	43.4429i	−0.254207	+	0.146767i
$297$		0			0
$298$	−23.7852	+	41.1972i	−0.0798162	+	0.138246i
$299$	535.249	+	309.026i	1.79013	+	1.03353i
$300$		0			0
$301$	−394.085	+	146.634i	−1.30925	+	0.487158i
$302$	213.935	+	123.516i	0.708395	+	0.408992i
$303$		0			0
$304$	70.5941			0.232217
$305$	71.1024	+	41.0510i	0.233123	+	0.134593i
$306$		0			0
$307$	311.039			1.01316			0.506578	−	0.862194i	$-0.330910\pi$
$307$	311.039			1.01316			0.506578	+	0.862194i	$0.330910\pi$
$308$	−55.0067	−	147.833i	−0.178593	−	0.479976i
$309$		0			0
$310$	−67.4665			−0.217634
$311$	36.6055	−	21.1342i	0.117702	−	0.0679555i	−0.439993	−	0.898001i	$-0.645019\pi$
$311$	36.6055	−	21.1342i	0.117702	−	0.0679555i	0.557695	+	0.830046i	$0.311686\pi$
$312$		0			0
$313$	−63.7340	+	110.390i	−0.203623	+	0.352685i	−0.949693	−	0.313182i	$-0.898605\pi$
$313$	−63.7340	+	110.390i	−0.203623	+	0.352685i	0.746070	+	0.665867i	$0.231938\pi$
$314$			73.6639i			0.234598i
$315$		0			0
$316$	−52.1332			−0.164979
$317$	−492.304	−	284.232i	−1.55301	−	0.896630i	−0.997895	−	0.0648560i	$-0.979341\pi$
$317$	−492.304	−	284.232i	−1.55301	−	0.896630i	−0.555114	−	0.831774i	$-0.687325\pi$
$318$		0			0
$319$	−167.594	−	290.281i	−0.525373	−	0.909973i
$320$		−	19.6732i		−	0.0614786i
$321$		0			0
$322$	246.710	+	204.026i	0.766180	+	0.633622i
$323$		−	169.736i		−	0.525499i
$324$		0			0
$325$	−181.105	+	313.684i	−0.557247	+	0.965180i
$326$		−	193.941i		−	0.594912i
$327$		0			0
$328$	−16.9512	+	29.3603i	−0.0516804	+	0.0895131i
$329$	−186.638	+	225.684i	−0.567288	+	0.685969i
$330$		0			0
$331$	−198.102	+	343.122i	−0.598495	+	1.03662i	0.394549	+	0.918875i	$0.370901\pi$
$331$	−198.102	+	343.122i	−0.598495	+	1.03662i	−0.993043	+	0.117748i	$0.962432\pi$
$332$	119.407	+	68.9395i	0.359659	+	0.207649i
$333$		0			0
$334$	−81.4277	−	141.037i	−0.243795	−	0.422266i
$335$	−230.567	+	133.118i	−0.688260	+	0.397367i
$336$		0			0
$337$	169.043	−	292.792i	0.501612	−	0.868818i	−0.498386	−	0.866955i	$-0.666074\pi$
$337$	169.043	−	292.792i	0.501612	−	0.868818i	0.999998	−	0.00186256i	$-0.000592871\pi$
$338$			277.533i			0.821103i
$339$		0			0
$340$	−47.3021			−0.139124
$341$	−189.286	+	109.284i	−0.555091	+	0.320482i
$342$		0			0
$343$	−165.090	+	300.657i	−0.481312	+	0.876550i
$344$	−147.138	+	84.9501i	−0.427727	+	0.246948i
$345$		0			0
$346$	−177.275	−	307.049i	−0.512354	−	0.887424i
$347$	94.4271	−	54.5175i	0.272124	−	0.157111i	−0.357728	−	0.933826i	$-0.616449\pi$
$347$	94.4271	−	54.5175i	0.272124	−	0.157111i	0.629853	+	0.776715i	$0.283115\pi$
$348$		0			0
$349$	307.589	+	532.760i	0.881344	+	1.52653i	0.849847	+	0.527030i	$0.176694\pi$
$349$	307.589	+	532.760i	0.881344	+	1.52653i	0.0314975	+	0.999504i	$0.489972\pi$
$350$	−119.570	+	144.585i	−0.341628	+	0.413099i
$351$		0			0
$352$	−31.8672	−	55.1956i	−0.0905318	−	0.156806i
$353$		−	318.694i		−	0.902816i	−0.892318	−	0.451408i	$-0.850922\pi$
$353$		−	318.694i		−	0.902816i	0.892318	−	0.451408i	$-0.149078\pi$
$354$		0			0
$355$	328.553			0.925502
$356$	−257.350	−	148.581i	−0.722893	−	0.417362i
$357$		0			0
$358$	−17.8983	−	31.0008i	−0.0499954	−	0.0865945i
$359$	566.210	−	326.902i	1.57719	−	0.910590i	0.581938	−	0.813233i	$-0.302295\pi$
$359$	566.210	−	326.902i	1.57719	−	0.910590i	0.995250	−	0.0973563i	$-0.0310386\pi$
$360$		0			0
$361$	24.7650	−	42.8942i	0.0686011	−	0.118821i
$362$	43.6864	+	25.2223i	0.120681	+	0.0696749i
$363$		0			0
$364$	−44.5762	+	263.820i	−0.122462	+	0.724780i
$365$	257.023	+	148.392i	0.704171	+	0.406554i
$366$		0			0
$367$	639.707			1.74307			0.871536	−	0.490332i	$-0.163124\pi$
$367$	639.707			1.74307			0.871536	+	0.490332i	$0.163124\pi$
$368$	112.027	+	64.6789i	0.304422	+	0.175758i
$369$		0			0
$370$	−106.832			−0.288736
$371$	104.527	−	126.395i	0.281744	−	0.340687i
$372$		0			0
$373$	436.699			1.17078			0.585388	−	0.810753i	$-0.300942\pi$
$373$	436.699			1.17078			0.585388	+	0.810753i	$0.300942\pi$
$374$	−132.712	+	76.6214i	−0.354845	+	0.204870i
$375$		0			0
$376$	−59.1666	+	102.480i	−0.157358	+	0.272552i
$377$			568.567i			1.50814i
$378$		0			0
$379$	−351.261			−0.926810			−0.463405	−	0.886146i	$-0.653373\pi$
$379$	−351.261			−0.926810			−0.463405	+	0.886146i	$0.653373\pi$
$380$	75.1715	+	43.4003i	0.197820	+	0.114211i
$381$		0			0
$382$	−24.7583	−	42.8827i	−0.0648124	−	0.112258i
$383$			276.521i			0.721986i	0.932569	+	0.360993i	$0.117562\pi$
$383$			276.521i			0.721986i	−0.932569	+	0.360993i	$0.882438\pi$
$384$		0			0
$385$	32.3120	−	191.236i	0.0839273	−	0.496716i
$386$		−	197.126i		−	0.510689i
$387$		0			0
$388$	−22.6033	+	39.1500i	−0.0582558	+	0.100902i
$389$		−	57.6095i		−	0.148096i	−0.997255	−	0.0740482i	$-0.976408\pi$
$389$		−	57.6095i		−	0.148096i	0.997255	−	0.0740482i	$-0.0235919\pi$
$390$		0			0
$391$	155.514	−	269.358i	0.397733	−	0.688894i
$392$	−45.5346	+	130.899i	−0.116160	+	0.333927i
$393$		0			0
$394$	−60.5850	+	104.936i	−0.153769	+	0.266336i
$395$	−55.5136	−	32.0508i	−0.140541	−	0.0811413i
$396$		0			0
$397$	134.380	+	232.752i	0.338488	+	0.586278i	0.984149	−	0.177346i	$-0.0567512\pi$
$397$	134.380	+	232.752i	0.338488	+	0.586278i	−0.645661	+	0.763625i	$0.723418\pi$
$398$	150.516	−	86.9006i	0.378182	−	0.218343i
$399$		0			0
$400$	−37.9052	+	65.6537i	−0.0947630	+	0.164134i
$401$			46.8624i			0.116864i	0.998291	+	0.0584319i	$0.0186101\pi$
$401$			46.8624i			0.116864i	−0.998291	+	0.0584319i	$0.981390\pi$
$402$		0			0
$403$	370.750			0.919976
$404$	−120.421	+	69.5252i	−0.298072	+	0.172092i
$405$		0			0
$406$	−49.0665	+	290.396i	−0.120853	+	0.715260i
$407$	−299.732	+	173.050i	−0.736443	+	0.425185i
$408$		0			0
$409$	−44.5923	−	77.2361i	−0.109028	−	0.188841i	0.806349	−	0.591440i	$-0.201440\pi$
$409$	−44.5923	−	77.2361i	−0.109028	−	0.188841i	−0.915377	+	0.402599i	$0.868107\pi$
$410$	−36.1006	+	20.8427i	−0.0880503	+	0.0508359i
$411$		0			0
$412$	91.1054	+	157.799i	0.221130	+	0.383008i
$413$	11.8801	+	31.9283i	0.0287654	+	0.0773082i
$414$		0			0
$415$	84.7661	+	146.819i	0.204256	+	0.353781i
$416$			108.110i			0.259881i
$417$		0			0
$418$	281.204			0.672738
$419$	559.214	+	322.862i	1.33464	+	0.770554i	0.986007	−	0.166706i	$-0.0533130\pi$
$419$	559.214	+	322.862i	1.33464	+	0.770554i	0.348632	+	0.937260i	$0.386646\pi$
$420$		0			0
$421$	−8.15020	−	14.1166i	−0.0193592	−	0.0335310i	0.856184	−	0.516672i	$-0.172829\pi$
$421$	−8.15020	−	14.1166i	−0.0193592	−	0.0335310i	−0.875543	+	0.483141i	$0.839496\pi$
$422$	−331.554	+	191.423i	−0.785672	+	0.453608i
$423$		0			0
$424$	33.1364	−	57.3940i	0.0781520	−	0.135363i
$425$	157.858	+	91.1391i	0.371430	+	0.214445i
$426$		0			0
$427$	−180.098	−	148.939i	−0.421774	−	0.348803i
$428$	−272.032	−	157.058i	−0.635590	−	0.366958i
$429$		0			0
$430$	−208.905			−0.485825
$431$	−675.352	−	389.915i	−1.56694	−	0.904674i	−0.996523	−	0.0833199i	$-0.973448\pi$
$431$	−675.352	−	389.915i	−1.56694	−	0.904674i	−0.570419	−	0.821354i	$-0.693219\pi$
$432$		0			0
$433$	−266.616			−0.615742			−0.307871	−	0.951428i	$-0.599616\pi$
$433$	−266.616			−0.615742			−0.307871	+	0.951428i	$0.599616\pi$
$434$	189.361	+	31.9952i	0.436315	+	0.0737217i
$435$		0			0
$436$	107.821			0.247296
$437$	−494.278	+	285.372i	−1.13107	+	0.653025i
$438$		0			0
$439$	21.4115	−	37.0858i	0.0487734	−	0.0844780i	−0.840608	−	0.541644i	$-0.817802\pi$
$439$	21.4115	−	37.0858i	0.0487734	−	0.0844780i	0.889381	+	0.457166i	$0.151135\pi$
$440$		−	78.3661i		−	0.178105i
$441$		0			0
$442$	259.940			0.588100
$443$	−398.451	−	230.046i	−0.899437	−	0.519290i	−0.0224196	−	0.999749i	$-0.507137\pi$
$443$	−398.451	−	230.046i	−0.899437	−	0.519290i	−0.877018	+	0.480458i	$0.840470\pi$
$444$		0			0
$445$	−182.691	−	316.430i	−0.410542	−	0.711080i
$446$		−	100.792i		−	0.225991i
$447$		0			0
$448$	−9.32977	+	55.2173i	−0.0208254	+	0.123253i
$449$			568.150i			1.26537i	0.774410	+	0.632684i	$0.218047\pi$
$449$			568.150i			1.26537i	−0.774410	+	0.632684i	$0.781953\pi$
$450$		0			0
$451$	−67.5233	+	116.954i	−0.149719	+	0.259321i
$452$			259.051i			0.573121i
$453$		0			0
$454$	172.608	−	298.965i	0.380193	−	0.658514i
$455$	−209.660	+	253.522i	−0.460790	+	0.557190i
$456$		0			0
$457$	−53.3761	+	92.4500i	−0.116797	+	0.202298i	−0.918497	−	0.395429i	$-0.870596\pi$
$457$	−53.3761	+	92.4500i	−0.116797	+	0.202298i	0.801700	+	0.597727i	$0.203929\pi$
$458$	547.281	+	315.973i	1.19494	+	0.689897i
$459$		0			0
$460$	79.5275	+	137.746i	0.172886	+	0.299447i
$461$	687.776	−	397.088i	1.49192	−	0.861362i	0.491965	−	0.870615i	$-0.336279\pi$
$461$	687.776	−	397.088i	1.49192	−	0.861362i	0.999957	+	0.00925333i	$0.00294547\pi$
$462$		0			0
$463$	−33.1409	+	57.4018i	−0.0715787	+	0.123978i	−0.899593	−	0.436728i	$-0.856137\pi$
$463$	−33.1409	+	57.4018i	−0.0715787	+	0.123978i	0.828015	+	0.560706i	$0.189470\pi$
$464$			119.001i			0.256467i
$465$		0			0
$466$	377.139			0.809311
$467$	−68.5040	+	39.5508i	−0.146690	+	0.0846912i	−0.571548	−	0.820568i	$-0.693657\pi$
$467$	−68.5040	+	39.5508i	−0.146690	+	0.0846912i	0.424859	+	0.905260i	$0.360324\pi$
$468$		0			0
$469$	710.270	−	264.283i	1.51443	−	0.563503i
$470$	−126.006	+	72.7496i	−0.268098	+	0.154786i
$471$		0			0
$472$	6.88255	+	11.9209i	0.0145817	+	0.0252562i
$473$	−586.109	+	338.390i	−1.23913	+	0.715413i
$474$		0			0
$475$	−167.243	−	289.673i	−0.352090	−	0.609837i
$476$	132.764	+	22.4324i	0.278917	+	0.0471270i
$477$		0			0
$478$	256.860	+	444.894i	0.537363	+	0.930740i
$479$		−	370.623i		−	0.773744i	−0.922133	−	0.386872i	$-0.873555\pi$
$479$		−	370.623i		−	0.773744i	0.922133	−	0.386872i	$-0.126445\pi$
$480$		0			0
$481$	587.078			1.22054
$482$	−400.191	−	231.051i	−0.830273	−	0.479358i
$483$		0			0
$484$	−5.93985	−	10.2881i	−0.0122724	−	0.0212564i
$485$	−48.1378	+	27.7924i	−0.0992532	+	0.0573038i
$486$		0			0
$487$	4.05330	−	7.02052i	0.00832299	−	0.0144158i	−0.861834	−	0.507191i	$-0.830684\pi$
$487$	4.05330	−	7.02052i	0.00832299	−	0.0144158i	0.870157	+	0.492775i	$0.164017\pi$
$488$	−81.7796	−	47.2155i	−0.167581	−	0.0967530i
$489$		0			0
$490$	−128.962	+	111.393i	−0.263188	+	0.227332i
$491$	535.748	+	309.314i	1.09114	+	0.629968i	0.933879	−	0.357590i	$-0.116401\pi$
$491$	535.748	+	309.314i	1.09114	+	0.629968i	0.157257	+	0.987558i	$0.449735\pi$
$492$		0			0
$493$	286.125			0.580375
$494$	−413.091	−	238.498i	−0.836217	−	0.482790i
$495$		0			0
$496$	77.5978			0.156447
$497$	−922.161	−	155.812i	−1.85546	−	0.313506i
$498$		0			0
$499$	−49.7159			−0.0996311			−0.0498156	−	0.998758i	$-0.515863\pi$
$499$	−49.7159			−0.0996311			−0.0498156	+	0.998758i	$0.515863\pi$
$500$	−187.210	+	108.086i	−0.374420	+	0.216172i
$501$		0			0
$502$	−169.176	+	293.021i	−0.337004	+	0.583708i
$503$		−	330.210i		−	0.656480i	−0.944594	−	0.328240i	$-0.893544\pi$
$503$		−	330.210i		−	0.656480i	0.944594	−	0.328240i	$-0.106456\pi$
$504$		0			0
$505$	−170.972			−0.338559
$506$	446.249	+	257.642i	0.881915	+	0.509174i
$507$		0			0
$508$	69.5844	+	120.524i	0.136977	+	0.237251i
$509$		−	787.097i		−	1.54636i	−0.634187	−	0.773180i	$-0.718665\pi$
$509$		−	787.097i		−	1.54636i	0.634187	−	0.773180i	$-0.281335\pi$
$510$		0			0
$511$	−651.021	−	538.387i	−1.27401	−	1.05359i
$512$			22.6274i			0.0441942i
$513$		0			0
$514$	−237.264	+	410.954i	−0.461604	+	0.799521i
$515$			224.042i			0.435032i
$516$		0			0
$517$	−235.684	+	408.217i	−0.455869	+	0.789588i
$518$	299.850	+	50.6640i	0.578861	+	0.0978070i
$519$		0			0
$520$	−66.4648	+	115.120i	−0.127817	+	0.221385i
$521$	32.2385	+	18.6129i	0.0618782	+	0.0357254i	0.530620	−	0.847610i	$-0.321959\pi$
$521$	32.2385	+	18.6129i	0.0618782	+	0.0357254i	−0.468742	+	0.883335i	$0.655293\pi$
$522$		0			0
$523$	91.4338	+	158.368i	0.174826	+	0.302807i	0.940101	−	0.340896i	$-0.110730\pi$
$523$	91.4338	+	158.368i	0.174826	+	0.302807i	−0.765275	+	0.643703i	$0.777397\pi$
$524$	3.12004	−	1.80136i	0.00595428	−	0.00343771i
$525$		0			0
$526$	188.012	−	325.646i	0.357437	−	0.619099i
$527$		−	186.576i		−	0.354034i
$528$		0			0
$529$	−516.841			−0.977015
$530$	70.5701	−	40.7437i	0.133151	−	0.0768749i
$531$		0			0
$532$	−190.404	−	157.462i	−0.357903	−	0.295982i
$533$	198.384	−	114.537i	0.372204	−	0.214892i
$534$		0			0
$535$	−193.114	−	334.484i	−0.360961	−	0.625203i
$536$	265.190	−	153.108i	0.494758	−	0.285649i
$537$		0			0
$538$	−231.720	−	401.351i	−0.430706	−	0.746005i
$539$	−181.382	+	521.424i	−0.336517	+	0.967391i
$540$		0			0
$541$	−458.958	−	794.939i	−0.848351	−	1.46939i	−0.882679	−	0.469977i	$-0.844262\pi$
$541$	−458.958	−	794.939i	−0.848351	−	1.46939i	0.0343272	−	0.999411i	$-0.489071\pi$
$542$			538.355i			0.993275i
$543$		0			0
$544$	54.4053			0.100010
$545$	114.813	+	66.2870i	0.210665	+	0.121628i
$546$		0			0
$547$	531.064	+	919.829i	0.970866	+	1.68159i	0.692952	+	0.720984i	$0.256310\pi$
$547$	531.064	+	919.829i	0.970866	+	1.68159i	0.277914	+	0.960606i	$0.410357\pi$
$548$	−78.1290	+	45.1078i	−0.142571	+	0.0823135i
$549$		0			0
$550$	−150.992	+	261.525i	−0.274530	+	0.475500i
$551$	−454.703	−	262.523i	−0.825233	−	0.476448i
$552$		0			0
$553$	140.612	+	116.285i	0.254272	+	0.210280i
$554$	249.134	+	143.837i	0.449700	+	0.259634i
$555$		0			0
$556$	−508.862			−0.915219
$557$	93.5858	+	54.0318i	0.168018	+	0.0970050i	0.581651	−	0.813439i	$-0.302407\pi$
$557$	93.5858	+	54.0318i	0.168018	+	0.0970050i	−0.413633	+	0.910444i	$0.635740\pi$
$558$		0			0
$559$	1148.00			2.05366
$560$	−43.8816	+	53.0619i	−0.0783600	+	0.0947534i
$561$		0			0
$562$	−505.978			−0.900316
$563$	935.789	−	540.278i	1.66215	−	0.959641i	0.690460	−	0.723370i	$-0.257408\pi$
$563$	935.789	−	540.278i	1.66215	−	0.959641i	0.971687	−	0.236271i	$-0.0759253\pi$
$564$		0			0
$565$	−159.261	+	275.848i	−0.281878	+	0.488226i
$566$		−	143.152i		−	0.252919i
$567$		0			0
$568$	−377.891			−0.665301
$569$	605.640	+	349.666i	1.06439	+	0.614528i	0.926644	−	0.375940i	$-0.122680\pi$
$569$	605.640	+	349.666i	1.06439	+	0.614528i	0.137749	+	0.990467i	$0.456013\pi$
$570$		0			0
$571$	112.622	+	195.067i	0.197237	+	0.341624i	0.947631	−	0.319366i	$-0.103470\pi$
$571$	112.622	+	195.067i	0.197237	+	0.341624i	−0.750395	+	0.660990i	$0.770137\pi$
$572$			430.647i			0.752879i
$573$		0			0
$574$	111.209	−	41.3796i	0.193744	−	0.0720900i
$575$		−	612.917i		−	1.06594i
$576$		0			0
$577$	−7.39884	+	12.8152i	−0.0128230	+	0.0222100i	−0.872366	−	0.488854i	$-0.837415\pi$
$577$	−7.39884	+	12.8152i	−0.0128230	+	0.0222100i	0.859543	+	0.511064i	$0.170748\pi$
$578$			277.896i			0.480789i
$579$		0			0
$580$	−73.1600	+	126.717i	−0.126138	+	0.218477i
$581$	−168.289	−	452.282i	−0.289653	−	0.778454i
$582$		0			0
$583$	131.996	−	228.623i	0.226408	−	0.392150i
$584$	−295.619	−	170.676i	−0.506197	−	0.292253i
$585$		0			0
$586$	−39.0798	−	67.6882i	−0.0666891	−	0.115509i
$587$	−97.5653	+	56.3294i	−0.166210	+	0.0959615i	−0.580798	−	0.814048i	$-0.697259\pi$
$587$	−97.5653	+	56.3294i	−0.166210	+	0.0959615i	0.414587	+	0.910009i	$0.363926\pi$
$588$		0			0
$589$	−171.186	+	296.502i	−0.290638	+	0.503399i
$590$			16.9252i			0.0286868i
$591$		0			0
$592$	122.875			0.207559
$593$	46.3761	−	26.7752i	0.0782059	−	0.0451522i	−0.460387	−	0.887718i	$-0.652289\pi$
$593$	46.3761	−	26.7752i	0.0782059	−	0.0451522i	0.538593	+	0.842566i	$0.318956\pi$
$594$		0			0
$595$	127.582	+	105.509i	0.214423	+	0.177326i
$596$	58.2617	−	33.6374i	0.0977545	−	0.0564386i
$597$		0			0
$598$	−437.029	−	756.956i	−0.730818	−	1.26581i
$599$	28.7901	−	16.6220i	0.0480637	−	0.0277496i	−0.475776	−	0.879567i	$-0.657833\pi$
$599$	28.7901	−	16.6220i	0.0480637	−	0.0277496i	0.523839	+	0.851817i	$0.324499\pi$
$600$		0			0
$601$	−433.346	−	750.577i	−0.721041	−	1.24888i	−0.960583	−	0.277994i	$-0.910331\pi$
$601$	−433.346	−	750.577i	−0.721041	−	1.24888i	0.239542	−	0.970886i	$-0.423003\pi$
$602$	586.340	+	99.0706i	0.973987	+	0.164569i
$603$		0			0
$604$	−174.677	−	302.550i	−0.289201	−	0.500911i
$605$		−	14.6069i		−	0.0241437i
$606$		0			0
$607$	−247.487			−0.407721			−0.203861	−	0.979000i	$-0.565349\pi$
$607$	−247.487			−0.407721			−0.203861	+	0.979000i	$0.565349\pi$
$608$	−86.4597	−	49.9175i	−0.142203	−	0.0821012i
$609$		0			0
$610$	−58.0549	−	100.554i	−0.0951719	−	0.164843i
$611$	692.444	−	399.783i	1.13330	−	0.654309i
$612$		0			0
$613$	250.537	−	433.943i	0.408706	−	0.707900i	−0.586039	−	0.810283i	$-0.699313\pi$
$613$	250.537	−	433.943i	0.408706	−	0.707900i	0.994745	+	0.102383i	$0.0326467\pi$
$614$	−380.943	−	219.938i	−0.620429	−	0.358205i
$615$		0			0
$616$	−37.1642	+	219.953i	−0.0603315	+	0.357066i
$617$	−212.273	−	122.556i	−0.344041	−	0.198632i	0.318017	−	0.948085i	$-0.396983\pi$
$617$	−212.273	−	122.556i	−0.344041	−	0.198632i	−0.662058	+	0.749453i	$0.730317\pi$
$618$		0			0
$619$	439.570			0.710129			0.355065	−	0.934842i	$-0.384459\pi$
$619$	439.570			0.710129			0.355065	+	0.934842i	$0.384459\pi$
$620$	82.6293	+	47.7061i	0.133273	+	0.0769452i
$621$		0			0
$622$	−59.7765			−0.0961036
$623$	362.702	+	974.775i	0.582187	+	1.56465i
$624$		0			0
$625$	208.016			0.332826
$626$	156.116	−	90.1334i	0.249386	−	0.143983i
$627$		0			0
$628$	52.0883	−	90.2195i	0.0829431	−	0.143662i
$629$		−	295.440i		−	0.469698i
$630$		0			0
$631$	−548.428			−0.869142			−0.434571	−	0.900638i	$-0.643100\pi$
$631$	−548.428			−0.869142			−0.434571	+	0.900638i	$0.643100\pi$
$632$	63.8499	+	36.8638i	0.101028	+	0.0583287i
$633$		0			0
$634$	401.964	+	696.223i	0.634013	+	1.09814i
$635$			171.118i			0.269477i
$636$		0			0
$637$	708.688	−	612.139i	1.11254	−	0.960972i
$638$			474.027i			0.742990i
$639$		0			0
$640$	−13.9110	+	24.0946i	−0.0217360	+	0.0376478i
$641$		−	919.200i		−	1.43401i	−0.697068	−	0.717005i	$-0.745513\pi$
$641$		−	919.200i		−	1.43401i	0.697068	−	0.717005i	$-0.254487\pi$
$642$		0			0
$643$	−636.355	+	1102.20i	−0.989666	+	1.71415i	−0.370653	+	0.928772i	$0.620866\pi$
$643$	−636.355	+	1102.20i	−0.989666	+	1.71415i	−0.619013	+	0.785380i	$0.712467\pi$
$644$	−157.888	−	424.330i	−0.245168	−	0.658898i
$645$		0			0
$646$	−120.021	+	207.883i	−0.185792	+	0.321801i
$647$	−205.822	−	118.831i	−0.318117	−	0.183665i	0.332436	−	0.943126i	$-0.392129\pi$
$647$	−205.822	−	118.831i	−0.318117	−	0.183665i	−0.650553	+	0.759461i	$0.725463\pi$
$648$		0			0
$649$	27.4160	+	47.4858i	0.0422434	+	0.0731677i
$650$	443.616	−	256.122i	0.682485	−	0.394033i
$651$		0			0
$652$	−137.137	+	237.529i	−0.210333	+	0.364308i
$653$			62.5079i			0.0957242i	0.998854	+	0.0478621i	$0.0152408\pi$
$653$			62.5079i			0.0957242i	−0.998854	+	0.0478621i	$0.984759\pi$
$654$		0			0
$655$	4.42980			0.00676306
$656$	41.5217	−	23.9726i	0.0632953	−	0.0365436i
$657$		0			0
$658$	388.166	−	144.432i	0.589918	−	0.219502i
$659$	−304.697	+	175.917i	−0.462363	+	0.266945i	−0.713037	−	0.701126i	$-0.752681\pi$
$659$	−304.697	+	175.917i	−0.462363	+	0.266945i	0.250675	+	0.968071i	$0.419348\pi$
$660$		0			0
$661$	−142.614	−	247.014i	−0.215754	−	0.373697i	0.737751	−	0.675073i	$-0.235888\pi$
$661$	−142.614	−	247.014i	−0.215754	−	0.373697i	−0.953506	+	0.301375i	$0.902554\pi$
$662$	485.248	−	280.158i	0.733003	−	0.423200i
$663$		0			0
$664$	−97.4951	−	168.866i	−0.146830	−	0.254317i
$665$	−105.945	−	284.730i	−0.159315	−	0.428166i
$666$		0			0
$667$	−481.052	−	833.207i	−0.721218	−	1.24919i
$668$			230.312i			0.344779i
$669$		0			0
$670$	376.514			0.561962
$671$	−325.761	−	188.078i	−0.485486	−	0.280295i
$672$		0			0
$673$	597.026	+	1034.08i	0.887111	+	1.53652i	0.843275	+	0.537483i	$0.180625\pi$
$673$	597.026	+	1034.08i	0.887111	+	1.53652i	0.0438362	+	0.999039i	$0.486042\pi$
$674$	−414.070	+	239.063i	−0.614347	+	0.354693i
$675$		0			0
$676$	196.245	−	339.907i	0.290304	−	0.502821i
$677$	245.949	+	141.999i	0.363293	+	0.209747i	0.670524	−	0.741888i	$-0.266069\pi$
$677$	245.949	+	141.999i	0.363293	+	0.209747i	−0.307232	+	0.951635i	$0.599403\pi$
$678$		0			0
$679$	148.290	−	55.1770i	0.218395	−	0.0812622i
$680$	57.9330	+	33.4476i	0.0851955	+	0.0491877i
$681$		0			0
$682$	309.103			0.453230
$683$	36.6557	+	21.1632i	0.0536687	+	0.0309856i	0.526594	−	0.850117i	$-0.323469\pi$
$683$	36.6557	+	21.1632i	0.0536687	+	0.0309856i	−0.472926	+	0.881102i	$0.656802\pi$
$684$		0			0
$685$	−110.927			−0.161937
$686$	414.789	−	251.491i	0.604649	−	0.366605i
$687$		0			0
$688$	240.275			0.349237
$689$	−387.805	+	223.900i	−0.562853	+	0.324963i
$690$		0			0
$691$	67.0552	−	116.143i	0.0970407	−	0.168080i	−0.813418	−	0.581680i	$-0.802396\pi$
$691$	67.0552	−	116.143i	0.0970407	−	0.168080i	0.910458	+	0.413600i	$0.135729\pi$
$692$			501.408i			0.724578i
$693$		0			0
$694$	−154.199			−0.222189
$695$	−541.857	−	312.841i	−0.779650	−	0.450131i
$696$		0			0
$697$	−57.6396	−	99.8347i	−0.0826967	−	0.143235i
$698$		−	869.994i		−	1.24641i
$699$		0			0
$700$	248.680	−	92.5307i	0.355256	−	0.132187i
$701$		−	516.988i		−	0.737501i	−0.929528	−	0.368751i	$-0.879786\pi$
$701$		−	516.988i		−	0.737501i	0.929528	−	0.368751i	$-0.120214\pi$
$702$		0			0
$703$	−271.070	+	469.507i	−0.385591	+	0.667863i
$704$			90.1341i			0.128031i
$705$		0			0
$706$	−225.351	+	390.319i	−0.319194	+	0.552859i
$707$	479.874	+	81.0817i	0.678747	+	0.114684i
$708$		0			0
$709$	261.577	−	453.065i	0.368938	−	0.639019i	−0.620462	−	0.784237i	$-0.713055\pi$
$709$	261.577	−	453.065i	0.368938	−	0.639019i	0.989400	+	0.145217i	$0.0463882\pi$
$710$	−402.394	−	232.322i	−0.566752	−	0.327214i
$711$		0			0
$712$	210.125	+	363.948i	0.295120	+	0.511162i
$713$	−543.316	+	313.684i	−0.762014	+	0.439949i
$714$		0			0
$715$	−264.756	+	458.571i	−0.370288	+	0.641357i
$716$			50.6242i			0.0707041i
$717$		0			0
$718$	−924.618			−1.28777
$719$	−1081.87	+	624.616i	−1.50468	+	0.868729i	−0.504697	+	0.863296i	$0.668396\pi$
$719$	−1081.87	+	624.616i	−1.50468	+	0.868729i	−0.999985	+	0.00543238i	$0.998271\pi$
$720$		0			0
$721$	106.249	−	628.825i	0.147363	−	0.872157i
$722$	−60.6616	+	35.0230i	−0.0840188	+	0.0485083i
$723$		0			0
$724$	−35.6698	−	61.7818i	−0.0492676	−	0.0853340i
$725$	488.302	−	281.922i	0.673521	−	0.388857i
$726$		0			0
$727$	−429.100	−	743.222i	−0.590233	−	1.02231i	−0.994201	−	0.107541i	$-0.965702\pi$
$727$	−429.100	−	743.222i	−0.590233	−	1.02231i	0.403968	−	0.914773i	$-0.367631\pi$
$728$	241.143	−	291.592i	0.331241	−	0.400539i
$729$		0			0
$730$	−209.858	−	363.485i	−0.287477	−	0.497924i
$731$		−	577.717i		−	0.790310i
$732$		0			0
$733$	−466.133			−0.635925			−0.317963	−	0.948103i	$-0.602999\pi$
$733$	−466.133			−0.635925			−0.317963	+	0.948103i	$0.602999\pi$
$734$	−783.478	−	452.341i	−1.06741	−	0.616269i
$735$		0			0
$736$	−91.4698	−	158.430i	−0.124280	−	0.215259i
$737$	1056.36	−	609.890i	1.43332	−	0.827530i
$738$		0			0
$739$	−58.9593	+	102.121i	−0.0797825	+	0.138187i	−0.903156	−	0.429312i	$-0.858756\pi$
$739$	−58.9593	+	102.121i	−0.0797825	+	0.138187i	0.823373	+	0.567500i	$0.192089\pi$
$740$	130.842	+	75.5419i	0.176814	+	0.102084i
$741$		0			0
$742$	−217.394	+	80.8896i	−0.292984	+	0.109016i
$743$	−271.243	−	156.602i	−0.365064	−	0.210770i	0.306236	−	0.951956i	$-0.400930\pi$
$743$	−271.243	−	156.602i	−0.365064	−	0.210770i	−0.671300	+	0.741186i	$0.734264\pi$
$744$		0			0
$745$	82.7192			0.111033
$746$	−534.845	−	308.793i	−0.716951	−	0.413932i
$747$		0			0
$748$	216.718			0.289730
$749$	383.395	+	1030.39i	0.511876	+	1.37569i
$750$		0			0
$751$	551.023			0.733719			0.366860	−	0.930276i	$-0.380433\pi$
$751$	551.023			0.733719			0.366860	+	0.930276i	$0.380433\pi$
$752$	144.928	−	83.6742i	0.192723	−	0.111269i
$753$		0			0
$754$	402.038	−	696.350i	0.533206	−	0.923540i
$755$		−	429.557i		−	0.568950i
$756$		0			0
$757$	1111.61			1.46844			0.734220	−	0.678912i	$-0.237548\pi$
$757$	1111.61			1.46844			0.734220	+	0.678912i	$0.237548\pi$
$758$	430.205	+	248.379i	0.567553	+	0.327677i
$759$		0			0
$760$	−61.3772	−	106.308i	−0.0807595	−	0.139880i
$761$			637.688i			0.837960i	0.907995	+	0.418980i	$0.137612\pi$
$761$			637.688i			0.837960i	−0.907995	+	0.418980i	$0.862388\pi$
$762$		0			0
$763$	−290.812	−	240.499i	−0.381143	−	0.315201i
$764$			70.0271i			0.0916585i
$765$		0			0
$766$	195.530	−	338.667i	0.255260	−	0.442124i
$767$		−	93.0093i		−	0.121264i
$768$		0			0
$769$	−121.673	+	210.745i	−0.158223	+	0.274050i	−0.934228	−	0.356677i	$-0.883910\pi$
$769$	−121.673	+	210.745i	−0.158223	+	0.274050i	0.776005	+	0.630727i	$0.217243\pi$
$770$	−174.798	+	211.367i	−0.227010	+	0.274502i
$771$		0			0
$772$	−139.389	+	241.429i	−0.180556	+	0.312732i
$773$	−154.913	−	89.4393i	−0.200405	−	0.115704i	0.396439	−	0.918061i	$-0.370246\pi$
$773$	−154.913	−	89.4393i	−0.200405	−	0.115704i	−0.596845	+	0.802357i	$0.703579\pi$
$774$		0			0
$775$	−183.835	−	318.412i	−0.237206	−	0.410854i
$776$	55.3665	−	31.9658i	0.0713485	−	0.0411931i
$777$		0			0
$778$	−40.7361	+	70.5569i	−0.0523600	+	0.0906901i
$779$			211.540i			0.271554i
$780$		0			0
$781$	−1505.29			−1.92739
$782$	−380.929	+	219.930i	−0.487122	+	0.281240i
$783$		0			0
$784$	148.328	−	128.120i	0.189194	−	0.163419i
$785$	110.931	−	64.0463i	0.141314	−	0.0815876i
$786$		0			0
$787$	92.3278	+	159.916i	0.117316	+	0.203198i	0.918703	−	0.394949i	$-0.129238\pi$
$787$	92.3278	+	159.916i	0.117316	+	0.203198i	−0.801387	+	0.598146i	$0.795904\pi$
$788$	148.402	−	85.6801i	0.188328	−	0.108731i
$789$		0			0
$790$	45.3267	+	78.5081i	0.0573755	+	0.0993773i
$791$	577.821	−	698.704i	0.730494	−	0.883318i
$792$		0			0
$793$	319.030	+	552.576i	0.402308	+	0.696818i
$794$		−	380.083i		−	0.478694i
$795$		0			0
$796$	−245.792			−0.308784
$797$	337.135	+	194.645i	0.423004	+	0.244222i	0.696362	−	0.717691i	$-0.254801\pi$
$797$	337.135	+	194.645i	0.423004	+	0.244222i	−0.273357	+	0.961913i	$0.588134\pi$
$798$		0			0
$799$	−201.186	−	348.464i	−0.251797	−	0.436125i
$800$	92.8484	−	53.6061i	0.116061	−	0.0670076i
$801$		0			0
$802$	33.1367	−	57.3945i	0.0413176	−	0.0715642i
$803$	−1177.57	−	679.869i	−1.46646	−	0.846661i
$804$		0			0
$805$	92.7466	−	548.912i	0.115213	−	0.681878i
$806$	−454.075	−	262.160i	−0.563368	−	0.325261i
$807$		0			0
$808$	196.647			0.243375
$809$	1250.90	+	722.207i	1.54623	+	0.892715i	0.998425	+	0.0561081i	$0.0178691\pi$
$809$	1250.90	+	722.207i	1.54623	+	0.892715i	0.547803	+	0.836607i	$0.315464\pi$
$810$		0			0
$811$	−61.6267			−0.0759885			−0.0379943	−	0.999278i	$-0.512097\pi$
$811$	−61.6267			−0.0759885			−0.0379943	+	0.999278i	$0.512097\pi$
$812$	265.435	−	320.965i	0.326890	−	0.395277i
$813$		0			0
$814$	489.461			0.601303
$815$	−292.059	+	168.620i	−0.358354	+	0.206896i
$816$		0			0
$817$	−530.063	+	918.095i	−0.648791	+	1.12374i
$818$			126.126i			0.154188i
$819$		0			0
$820$	58.9521			0.0718928
$821$	181.227	+	104.632i	0.220740	+	0.127444i	0.606293	−	0.795241i	$-0.292656\pi$
$821$	181.227	+	104.632i	0.220740	+	0.127444i	−0.385553	+	0.922686i	$0.625989\pi$
$822$		0			0
$823$	−116.003	−	200.923i	−0.140951	−	0.244135i	0.786904	−	0.617076i	$-0.211683\pi$
$823$	−116.003	−	200.923i	−0.140951	−	0.244135i	−0.927855	+	0.372941i	$0.878349\pi$
$824$		−	257.685i		−	0.312725i
$825$		0			0
$826$	8.02657	−	47.5045i	0.00971740	−	0.0575115i
$827$			1375.87i			1.66369i	0.555009	+	0.831844i	$0.312715\pi$
$827$			1375.87i			1.66369i	−0.555009	+	0.831844i	$0.687285\pi$
$828$		0			0
$829$	−517.892	+	897.015i	−0.624719	+	1.08204i	0.363876	+	0.931447i	$0.381453\pi$
$829$	−517.892	+	897.015i	−0.624719	+	1.08204i	−0.988595	+	0.150597i	$0.951880\pi$
$830$		−	239.755i		−	0.288861i
$831$		0			0
$832$	76.4456	−	132.408i	0.0918817	−	0.159144i
$833$	−308.052	−	356.639i	−0.369810	−	0.428138i
$834$		0			0
$835$	−141.593	+	245.246i	−0.169572	+	0.293708i
$836$	−344.404	−	198.842i	−0.411966	−	0.237849i
$837$		0			0
$838$	−456.596	−	790.847i	−0.544864	−	0.943732i
$839$	−560.099	+	323.373i	−0.667579	+	0.385427i	−0.795159	−	0.606402i	$-0.792612\pi$
$839$	−560.099	+	323.373i	−0.667579	+	0.385427i	0.127580	+	0.991828i	$0.459279\pi$
$840$		0			0
$841$	22.0362	−	38.1678i	0.0262024	−	0.0453838i
$842$			23.0523i			0.0273780i
$843$		0			0
$844$	541.425			0.641499
$845$	417.940	−	241.298i	0.494604	−	0.285559i
$846$		0			0
$847$	−6.92717	+	40.9978i	−0.00817848	+	0.0484036i
$848$	−81.1674	+	46.8620i	−0.0957163	+	0.0552618i
$849$		0			0
$850$	−128.890	−	223.244i	−0.151635	−	0.262640i
$851$	−860.334	+	496.714i	−1.01097	+	0.583683i
$852$		0			0
$853$	−387.624	−	671.385i	−0.454424	−	0.787086i	0.544230	−	0.838936i	$-0.316822\pi$
$853$	−387.624	−	671.385i	−0.454424	−	0.787086i	−0.998655	+	0.0518495i	$0.983488\pi$
$854$	115.258	+	309.760i	0.134963	+	0.362717i
$855$		0			0
$856$	222.114	+	384.712i	0.259478	+	0.449430i
$857$		−	115.567i		−	0.134850i	−0.997724	−	0.0674251i	$-0.978522\pi$
$857$		−	115.567i		−	0.134850i	0.997724	−	0.0674251i	$-0.0214784\pi$
$858$		0			0
$859$	−710.100			−0.826659			−0.413330	−	0.910582i	$-0.635634\pi$
$859$	−710.100			−0.826659			−0.413330	+	0.910582i	$0.635634\pi$
$860$	255.855	+	147.718i	0.297506	+	0.171765i
$861$		0			0
$862$	551.422	+	955.092i	0.639701	+	1.10799i
$863$	543.437	−	313.753i	0.629707	−	0.363561i	−0.150932	−	0.988544i	$-0.548227\pi$
$863$	543.437	−	313.753i	0.629707	−	0.363561i	0.780638	+	0.624983i	$0.214894\pi$
$864$		0			0
$865$	−308.259	+	533.920i	−0.356369	+	0.617249i
$866$	326.537	+	188.526i	0.377063	+	0.217698i
$867$		0			0
$868$	−209.294	−	173.084i	−0.241123	−	0.199406i
$869$	254.340	+	146.843i	0.292681	+	0.168979i
$870$		0			0
$871$	−2069.07			−2.37551
$872$	−132.054	−	76.2411i	−0.151438	−	0.0874325i
$873$		0			0
$874$	807.153			0.923516
$875$	746.027	+	126.052i	0.852602	+	0.144059i
$876$		0			0
$877$	1174.92			1.33970			0.669850	−	0.742497i	$-0.266358\pi$
$877$	1174.92			1.33970			0.669850	+	0.742497i	$0.266358\pi$
$878$	−52.4473	+	30.2804i	−0.0597349	+	0.0344880i
$879$		0			0
$880$	−55.4132	+	95.9785i	−0.0629695	+	0.109066i
$881$		−	125.159i		−	0.142065i	−0.997474	−	0.0710323i	$-0.977371\pi$
$881$		−	125.159i		−	0.142065i	0.997474	−	0.0710323i	$-0.0226294\pi$
$882$		0			0
$883$	660.599			0.748130			0.374065	−	0.927403i	$-0.377964\pi$
$883$	660.599			0.748130			0.374065	+	0.927403i	$0.377964\pi$
$884$	−318.360	−	183.805i	−0.360136	−	0.207925i
$885$		0			0
$886$	325.334	+	563.494i	0.367194	+	0.635998i
$887$			428.773i			0.483397i	0.970351	+	0.241698i	$0.0777045\pi$
$887$			428.773i			0.483397i	−0.970351	+	0.241698i	$0.922296\pi$
$888$		0			0
$889$	81.1507	−	480.283i	0.0912832	−	0.540251i
$890$			516.729i			0.580594i
$891$		0			0
$892$	−71.2707	+	123.445i	−0.0798999	+	0.138391i
$893$			738.363i			0.826834i
$894$		0			0
$895$	−31.1230	+	53.9067i	−0.0347744	+	0.0602309i
$896$	50.4711	−	61.0300i	0.0563294	−	0.0681139i
$897$		0			0
$898$	401.743	−	695.839i	0.447375	−	0.774876i
$899$	−499.815	−	288.568i	−0.555968	−	0.320988i
$900$		0			0
$901$	112.675	+	195.159i	0.125055	+	0.216602i
$902$	165.398	−	95.4924i	0.183368	−	0.105867i
$903$		0			0
$904$	183.177	−	317.271i	0.202629	−	0.350964i
$905$		−	87.7171i		−	0.0969250i
$906$		0			0
$907$	377.272			0.415956			0.207978	−	0.978134i	$-0.433312\pi$
$907$	377.272			0.415956			0.207978	+	0.978134i	$0.433312\pi$
$908$	−422.801	+	244.104i	−0.465640	+	0.268837i
$909$		0			0
$910$	436.046	−	162.248i	0.479172	−	0.178294i
$911$	−1546.16	+	892.675i	−1.69721	+	0.979884i	−0.748821	+	0.662772i	$0.769380\pi$
$911$	−1546.16	+	892.675i	−1.69721	+	0.979884i	−0.948388	+	0.317112i	$0.897287\pi$
$912$		0			0
$913$	−388.362	−	672.663i	−0.425369	−	0.736761i
$914$	130.744	−	75.4851i	0.143046	−	0.0825877i
$915$		0			0
$916$	−446.853	−	773.973i	−0.487831	−	0.844948i
$917$	−12.4333	−	2.10078i	−0.0135586	−	0.00229093i
$918$		0			0
$919$	47.0807	+	81.5461i	0.0512303	+	0.0887335i	0.890503	−	0.454977i	$-0.150352\pi$
$919$	47.0807	+	81.5461i	0.0512303	+	0.0887335i	−0.839273	+	0.543710i	$0.817019\pi$
$920$		−	224.938i		−	0.244497i
$921$		0			0
$922$	−1123.13			−1.21815
$923$	2211.28	+	1276.68i	2.39576	+	1.38319i
$924$		0			0
$925$	−291.100	−	504.200i	−0.314703	−	0.545082i
$926$	81.1784	−	46.8684i	0.0876657	−	0.0506138i
$927$		0			0
$928$	84.1462	−	145.745i	0.0906748	−	0.157053i
$929$	562.464	+	324.739i	0.605451	+	0.349557i	0.771183	−	0.636614i	$-0.219665\pi$
$929$	562.464	+	324.739i	0.605451	+	0.349557i	−0.165732	+	0.986171i	$0.552999\pi$
$930$		0			0
$931$	162.329	+	849.405i	0.174360	+	0.912358i
$932$	−461.899	−	266.678i	−0.495600	−	0.286135i
$933$		0			0
$934$	111.867			0.119771
$935$	230.770	+	133.235i	0.246813	+	0.142498i
$936$		0			0
$937$	683.065			0.728991			0.364496	−	0.931205i	$-0.381241\pi$
$937$	683.065			0.728991			0.364496	+	0.931205i	$0.381241\pi$
$938$	−1056.78	−	178.557i	−1.12663	−	0.190360i
$939$		0			0
$940$	205.767			0.218901
$941$	−660.406	+	381.286i	−0.701813	+	0.405192i	−0.808022	−	0.589152i	$-0.799462\pi$
$941$	−660.406	+	381.286i	−0.701813	+	0.405192i	0.106209	+	0.994344i	$0.466129\pi$
$942$		0			0
$943$	−193.815	+	335.698i	−0.205530	+	0.355989i
$944$		−	19.4668i		−	0.0206216i
$945$		0			0
$946$	957.113			1.01175
$947$	1133.04	+	654.162i	1.19645	+	0.690773i	0.959763	−	0.280812i	$-0.0906039\pi$
$947$	1133.04	+	654.162i	1.19645	+	0.690773i	0.236691	+	0.971585i	$0.423937\pi$
$948$		0			0
$949$	1153.24	+	1997.47i	1.21521	+	2.10481i
$950$			473.034i			0.497930i
$951$		0			0
$952$	−146.740	−	121.353i	−0.154139	−	0.127471i
$953$			703.289i			0.737974i	0.929435	+	0.368987i	$0.120295\pi$
$953$			703.289i			0.737974i	−0.929435	+	0.368987i	$0.879705\pi$
$954$		0			0
$955$	−43.0517	+	74.5678i	−0.0450803	+	0.0780814i
$956$		−	726.508i		−	0.759946i
$957$		0			0
$958$	−262.070	+	453.919i	−0.273560	+	0.473820i
$959$	311.342	+	52.6057i	0.324652	+	0.0548547i
$960$		0			0
$961$	292.331	−	506.332i	0.304194	−	0.526880i
$962$	−719.021	−	415.127i	−0.747423	−	0.431525i
$963$		0			0
$964$	326.755	+	565.956i	0.338957	+	0.587091i
$965$	−296.854	+	171.389i	−0.307621	+	0.177605i
$966$		0			0
$967$	−104.996	+	181.858i	−0.108579	+	0.188064i	−0.915195	−	0.403012i	$-0.867963\pi$
$967$	−104.996	+	181.858i	−0.108579	+	0.188064i	0.806616	+	0.591076i	$0.201297\pi$
$968$			16.8004i			0.0173558i
$969$		0			0
$970$	78.6087			0.0810399
$971$	−752.182	+	434.272i	−0.774647	+	0.447243i	−0.834530	−	0.550963i	$-0.814261\pi$
$971$	−752.182	+	434.272i	−0.774647	+	0.447243i	0.0598830	+	0.998205i	$0.480927\pi$
$972$		0			0
$973$	1372.49	+	1135.03i	1.41057	+	1.16653i
$974$	−9.92851	+	5.73223i	−0.0101935	+	0.00588524i
$975$		0			0
$976$	66.7728	+	115.654i	0.0684147	+	0.118498i
$977$	−393.671	+	227.286i	−0.402938	+	0.232636i	−0.687751	−	0.725947i	$-0.741402\pi$
$977$	−393.671	+	227.286i	−0.402938	+	0.232636i	0.284813	+	0.958583i	$0.408069\pi$
$978$		0			0
$979$	837.013	+	1449.75i	0.854968	+	1.48085i
$980$	236.712	−	45.2378i	0.241543	−	0.0461610i
$981$		0			0
$982$	−437.436	−	757.662i	−0.445454	−	0.771550i
$983$		−	1129.10i		−	1.14863i	−0.818636	−	0.574313i	$-0.805269\pi$
$983$		−	1129.10i		−	1.14863i	0.818636	−	0.574313i	$-0.194731\pi$
$984$		0			0
$985$	210.700			0.213908
$986$	−350.430	−	202.321i	−0.355405	−	0.205193i
$987$		0			0
$988$	337.288	+	584.199i	0.341384	+	0.591295i
$989$	−1682.34	+	971.297i	−1.70105	+	0.982100i
$990$		0			0
$991$	79.7669	−	138.160i	0.0804914	−	0.139415i	−0.822970	−	0.568085i	$-0.807684\pi$
$991$	79.7669	−	138.160i	0.0804914	−	0.139415i	0.903461	+	0.428670i	$0.141018\pi$
$992$	−95.0375	−	54.8699i	−0.0958039	−	0.0553124i
$993$		0			0
$994$	1019.24	+	842.897i	1.02539	+	0.847985i
$995$	−261.730	−	151.110i	−0.263045	−	0.151869i
$996$		0			0
$997$	381.181			0.382328			0.191164	−	0.981558i	$-0.438774\pi$
$997$	381.181			0.382328			0.191164	+	0.981558i	$0.438774\pi$
$998$	60.8893	+	35.1545i	0.0610114	+	0.0352249i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.3.i.a.179.6		32
3.2	odd	2		126.3.i.a.95.16	yes	32
7.2	even	3		378.3.r.a.233.14		32
9.2	odd	6		378.3.r.a.305.6		32
9.7	even	3		126.3.r.a.11.12	yes	32
21.2	odd	6		126.3.r.a.23.4	yes	32
63.2	odd	6	inner	378.3.i.a.359.3		32
63.16	even	3		126.3.i.a.65.16	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.i.a.65.16	✓	32	63.16	even	3
126.3.i.a.95.16	yes	32	3.2	odd	2
126.3.r.a.11.12	yes	32	9.7	even	3
126.3.r.a.23.4	yes	32	21.2	odd	6
378.3.i.a.179.6		32	1.1	even	1	trivial
378.3.i.a.359.3		32	63.2	odd	6	inner
378.3.r.a.233.14		32	7.2	even	3
378.3.r.a.305.6		32	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +2.45915i q^{5} +(1.16622 - 6.90217i) q^{7} -2.82843i q^{8} +(1.73888 - 3.01183i) q^{10} -11.2668i q^{11} +(-9.55569 + 16.5509i) q^{13} +(-6.30889 + 7.62875i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(8.32907 + 4.80879i) q^{17} +(-8.82426 - 15.2841i) q^{19} +(-4.25936 + 2.45915i) q^{20} +(-7.96680 + 13.7989i) q^{22} -32.3395i q^{23} +18.9526 q^{25} +(23.4066 - 13.5138i) q^{26} +(13.1211 - 4.88221i) q^{28} +(25.7644 - 14.8751i) q^{29} +(-9.69972 - 16.8004i) q^{31} +(4.89898 - 2.82843i) q^{32} +(-6.80066 - 11.7791i) q^{34} +(16.9734 + 2.86791i) q^{35} +(-15.3594 - 26.6032i) q^{37} +24.9588i q^{38} +6.95551 q^{40} +(-10.3804 - 5.99315i) q^{41} +(-30.0344 - 52.0211i) q^{43} +(19.5146 - 11.2668i) q^{44} +(-22.8675 + 39.6076i) q^{46} +(-36.2320 - 20.9185i) q^{47} +(-46.2799 - 16.0989i) q^{49} +(-23.2121 - 13.4015i) q^{50} -38.2228 q^{52} +(20.2918 + 11.7155i) q^{53} +27.7066 q^{55} +(-19.5223 - 3.29857i) q^{56} -42.0731 q^{58} +(-4.21468 + 2.43335i) q^{59} +(16.6932 - 28.9135i) q^{61} +27.4350i q^{62} -8.00000 q^{64} +(-40.7012 - 23.4988i) q^{65} +(54.1318 + 93.7590i) q^{67} +19.2352i q^{68} +(-18.7602 - 15.5145i) q^{70} -133.605i q^{71} +(60.3429 - 104.517i) q^{73} +43.4429i q^{74} +(17.6485 - 30.5681i) q^{76} +(-77.7651 - 13.1395i) q^{77} +(-13.0333 + 22.5743i) q^{79} +(-8.51873 - 4.91829i) q^{80} +(8.47559 + 14.6801i) q^{82} +(59.7033 - 34.4697i) q^{83} +(-11.8255 + 20.4824i) q^{85} +84.9501i q^{86} -31.8672 q^{88} +(-128.675 + 74.2905i) q^{89} +(103.093 + 85.2571i) q^{91} +(56.0136 - 32.3395i) q^{92} +(29.5833 + 51.2398i) q^{94} +(37.5857 - 21.7001i) q^{95} +(11.3016 + 19.5750i) q^{97} +(45.2974 + 52.4419i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{4} + 2 q^{7} + 10 q^{13} - 36 q^{14} - 64 q^{16} - 54 q^{17} + 28 q^{19} - 160 q^{25} - 72 q^{26} - 4 q^{28} - 36 q^{29} - 8 q^{31} - 90 q^{35} + 22 q^{37} - 72 q^{41} + 16 q^{43} + 72 q^{44}+ \cdots + 288 q^{98}+O(q^{100}) \) 32 * q + 32 * q^4 + 2 * q^7 + 10 * q^13 - 36 * q^14 - 64 * q^16 - 54 * q^17 + 28 * q^19 - 160 * q^25 - 72 * q^26 - 4 * q^28 - 36 * q^29 - 8 * q^31 - 90 * q^35 + 22 * q^37 - 72 * q^41 + 16 * q^43 + 72 * q^44 - 12 * q^46 + 108 * q^47 + 74 * q^49 + 288 * q^50 + 40 * q^52 - 72 * q^53 - 24 * q^55 + 48 * q^58 + 90 * q^59 - 62 * q^61 - 256 * q^64 - 378 * q^65 + 70 * q^67 - 108 * q^70 + 196 * q^73 - 56 * q^76 - 630 * q^77 - 38 * q^79 + 60 * q^85 - 486 * q^89 - 122 * q^91 - 252 * q^92 + 168 * q^94 + 72 * q^95 - 38 * q^97 + 288 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more