LMFDB - Embedded newform 162.3.d.c.107.4

Newspace parameters

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	162.d (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$4.41418028264$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			107.4
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	162.107
Dual form			162.3.d.c.53.4

$q$-expansion

$f(q)$	$=$	$q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(5.01910 - 2.89778i) q^{5} +(4.19615 - 7.26795i) q^{7} +2.82843i q^{8} +O(q^{10})$	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(5.01910 - 2.89778i) q^{5} +(4.19615 - 7.26795i) q^{7} +2.82843i q^{8} +8.19615 q^{10} +(-12.7279 - 7.34847i) q^{11} +(10.5981 + 18.3564i) q^{13} +(10.2784 - 5.93426i) q^{14} +(-2.00000 + 3.46410i) q^{16} +7.76457i q^{17} +24.3923 q^{19} +(10.0382 + 5.79555i) q^{20} +(-10.3923 - 18.0000i) q^{22} +(-12.7279 + 7.34847i) q^{23} +(4.29423 - 7.43782i) q^{25} +29.9759i q^{26} +16.7846 q^{28} +(-30.7387 - 17.7470i) q^{29} +(-4.00000 - 6.92820i) q^{31} +(-4.89898 + 2.82843i) q^{32} +(-5.49038 + 9.50962i) q^{34} -48.6381i q^{35} -60.5692 q^{37} +(29.8744 + 17.2480i) q^{38} +(8.19615 + 14.1962i) q^{40} +(-29.1301 + 16.8183i) q^{41} +(-4.58846 + 7.94744i) q^{43} -29.3939i q^{44} -20.7846 q^{46} +(14.6969 + 8.48528i) q^{47} +(-10.7154 - 18.5596i) q^{49} +(10.5187 - 6.07296i) q^{50} +(-21.1962 + 36.7128i) q^{52} +25.7605i q^{53} -85.1769 q^{55} +(20.5569 + 11.8685i) q^{56} +(-25.0981 - 43.4711i) q^{58} +(53.4083 - 30.8353i) q^{59} +(6.50000 - 11.2583i) q^{61} -11.3137i q^{62} -8.00000 q^{64} +(106.386 + 61.4217i) q^{65} +(-10.5885 - 18.3397i) q^{67} +(-13.4486 + 7.76457i) q^{68} +(34.3923 - 59.5692i) q^{70} -101.214i q^{71} +40.4115 q^{73} +(-74.1818 - 42.8289i) q^{74} +(24.3923 + 42.2487i) q^{76} +(-106.817 + 61.6706i) q^{77} +(-49.3731 + 85.5167i) q^{79} +23.1822i q^{80} -47.5692 q^{82} +(-89.6231 - 51.7439i) q^{83} +(22.5000 + 38.9711i) q^{85} +(-11.2394 + 6.48906i) q^{86} +(20.7846 - 36.0000i) q^{88} +134.130i q^{89} +177.885 q^{91} +(-25.4558 - 14.6969i) q^{92} +(12.0000 + 20.7846i) q^{94} +(122.427 - 70.6835i) q^{95} +(37.5692 - 65.0718i) q^{97} -30.3077i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^4 - 8 * q^7	$ 8 q + 8 q^{4} - 8 q^{7} + 24 q^{10} + 64 q^{13} - 16 q^{16} + 112 q^{19} - 28 q^{25} - 32 q^{28} - 32 q^{31} + 60 q^{34} - 152 q^{37} + 24 q^{40} + 88 q^{43} - 252 q^{49} - 128 q^{52} - 432 q^{55} - 180 q^{58} + 52 q^{61} - 64 q^{64} + 40 q^{67} + 192 q^{70} + 448 q^{73} + 112 q^{76} - 104 q^{79} - 48 q^{82} + 180 q^{85} + 176 q^{91} + 96 q^{94} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^4 - 8 * q^7 + 24 * q^10 + 64 * q^13 - 16 * q^16 + 112 * q^19 - 28 * q^25 - 32 * q^28 - 32 * q^31 + 60 * q^34 - 152 * q^37 + 24 * q^40 + 88 * q^43 - 252 * q^49 - 128 * q^52 - 432 * q^55 - 180 * q^58 + 52 * q^61 - 64 * q^64 + 40 * q^67 + 192 * q^70 + 448 * q^73 + 112 * q^76 - 104 * q^79 - 48 * q^82 + 180 * q^85 + 176 * q^91 + 96 * q^94 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$	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
$5$	5.01910	−	2.89778i	1.00382	−	0.579555i	0.0944434	−	0.995530i	$-0.469893\pi$
$5$	5.01910	−	2.89778i	1.00382	−	0.579555i	0.909376	+	0.415975i	$0.136560\pi$
$6$		0			0
$7$	4.19615	−	7.26795i	0.599450	−	1.03828i	−0.393452	−	0.919345i	$-0.628719\pi$
$7$	4.19615	−	7.26795i	0.599450	−	1.03828i	0.992902	−	0.118933i	$-0.0379475\pi$
$8$			2.82843i			0.353553i
$9$		0			0
$10$	8.19615			0.819615
$11$	−12.7279	−	7.34847i	−1.15708	−	0.668043i	−0.206480	−	0.978451i	$-0.566201\pi$
$11$	−12.7279	−	7.34847i	−1.15708	−	0.668043i	−0.950603	+	0.310408i	$0.899534\pi$
$12$		0			0
$13$	10.5981	+	18.3564i	0.815237	+	1.41203i	0.909158	+	0.416452i	$0.136726\pi$
$13$	10.5981	+	18.3564i	0.815237	+	1.41203i	−0.0939212	+	0.995580i	$0.529940\pi$
$14$	10.2784	−	5.93426i	0.734174	−	0.423875i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.125000	+	0.216506i
$17$			7.76457i			0.456739i	0.973574	+	0.228370i	$0.0733395\pi$
$17$			7.76457i			0.456739i	−0.973574	+	0.228370i	$0.926660\pi$
$18$		0			0
$19$	24.3923			1.28381			0.641903	−	0.766786i	$-0.278145\pi$
$19$	24.3923			1.28381			0.641903	+	0.766786i	$0.278145\pi$
$20$	10.0382	+	5.79555i	0.501910	+	0.289778i
$21$		0			0
$22$	−10.3923	−	18.0000i	−0.472377	−	0.818182i
$23$	−12.7279	+	7.34847i	−0.553388	+	0.319499i	−0.750487	−	0.660885i	$-0.770181\pi$
$23$	−12.7279	+	7.34847i	−0.553388	+	0.319499i	0.197099	+	0.980384i	$0.436848\pi$
$24$		0			0
$25$	4.29423	−	7.43782i	0.171769	−	0.297513i
$26$			29.9759i			1.15292i
$27$		0			0
$28$	16.7846			0.599450
$29$	−30.7387	−	17.7470i	−1.05996	−	0.611966i	−0.134536	−	0.990909i	$-0.542954\pi$
$29$	−30.7387	−	17.7470i	−1.05996	−	0.611966i	−0.925420	+	0.378942i	$0.876288\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.129032	−	0.223490i	0.794270	−	0.607565i	$-0.207854\pi$
$31$	−4.00000	−	6.92820i	−0.129032	−	0.223490i	−0.923302	+	0.384075i	$0.874520\pi$
$32$	−4.89898	+	2.82843i	−0.153093	+	0.0883883i
$33$		0			0
$34$	−5.49038	+	9.50962i	−0.161482	+	0.279695i
$35$		−	48.6381i		−	1.38966i
$36$		0			0
$37$	−60.5692			−1.63701			−0.818503	−	0.574502i	$-0.805196\pi$
$37$	−60.5692			−1.63701			−0.818503	+	0.574502i	$0.805196\pi$
$38$	29.8744	+	17.2480i	0.786167	+	0.453894i
$39$		0			0
$40$	8.19615	+	14.1962i	0.204904	+	0.354904i
$41$	−29.1301	+	16.8183i	−0.710490	+	0.410201i	−0.811242	−	0.584710i	$-0.801208\pi$
$41$	−29.1301	+	16.8183i	−0.710490	+	0.410201i	0.100753	+	0.994912i	$0.467875\pi$
$42$		0			0
$43$	−4.58846	+	7.94744i	−0.106708	+	0.184824i	−0.914435	−	0.404733i	$-0.867364\pi$
$43$	−4.58846	+	7.94744i	−0.106708	+	0.184824i	0.807727	+	0.589557i	$0.200698\pi$
$44$		−	29.3939i		−	0.668043i
$45$		0			0
$46$	−20.7846			−0.451839
$47$	14.6969	+	8.48528i	0.312701	+	0.180538i	0.648134	−	0.761526i	$-0.275549\pi$
$47$	14.6969	+	8.48528i	0.312701	+	0.180538i	−0.335434	+	0.942064i	$0.608883\pi$
$48$		0			0
$49$	−10.7154	−	18.5596i	−0.218681	−	0.378767i
$50$	10.5187	−	6.07296i	0.210373	−	0.121459i
$51$		0			0
$52$	−21.1962	+	36.7128i	−0.407618	+	0.706016i
$53$			25.7605i			0.486046i	0.970020	+	0.243023i	$0.0781391\pi$
$53$			25.7605i			0.486046i	−0.970020	+	0.243023i	$0.921861\pi$
$54$		0			0
$55$	−85.1769			−1.54867
$56$	20.5569	+	11.8685i	0.367087	+	0.211938i
$57$		0			0
$58$	−25.0981	−	43.4711i	−0.432725	−	0.749502i
$59$	53.4083	−	30.8353i	0.905225	−	0.522632i	0.0263336	−	0.999653i	$-0.491617\pi$
$59$	53.4083	−	30.8353i	0.905225	−	0.522632i	0.878892	+	0.477021i	$0.158283\pi$
$60$		0			0
$61$	6.50000	−	11.2583i	0.106557	−	0.184563i	−0.807816	−	0.589435i	$-0.799351\pi$
$61$	6.50000	−	11.2583i	0.106557	−	0.184563i	0.914373	+	0.404872i	$0.132684\pi$
$62$		−	11.3137i		−	0.182479i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	106.386	+	61.4217i	1.63670	+	0.944950i
$66$		0			0
$67$	−10.5885	−	18.3397i	−0.158037	−	0.273728i	0.776124	−	0.630580i	$-0.217183\pi$
$67$	−10.5885	−	18.3397i	−0.158037	−	0.273728i	−0.934161	+	0.356853i	$0.883850\pi$
$68$	−13.4486	+	7.76457i	−0.197774	+	0.114185i
$69$		0			0
$70$	34.3923	−	59.5692i	0.491319	−	0.850989i
$71$		−	101.214i		−	1.42555i	−0.701392	−	0.712776i	$-0.747438\pi$
$71$		−	101.214i		−	1.42555i	0.701392	−	0.712776i	$-0.252562\pi$
$72$		0			0
$73$	40.4115			0.553583			0.276791	−	0.960930i	$-0.410729\pi$
$73$	40.4115			0.553583			0.276791	+	0.960930i	$0.410729\pi$
$74$	−74.1818	−	42.8289i	−1.00246	−	0.578769i
$75$		0			0
$76$	24.3923	+	42.2487i	0.320951	+	0.555904i
$77$	−106.817	+	61.6706i	−1.38723	+	0.800917i
$78$		0			0
$79$	−49.3731	+	85.5167i	−0.624976	+	1.08249i	0.363570	+	0.931567i	$0.381558\pi$
$79$	−49.3731	+	85.5167i	−0.624976	+	1.08249i	−0.988546	+	0.150923i	$0.951776\pi$
$80$			23.1822i			0.289778i
$81$		0			0
$82$	−47.5692			−0.580112
$83$	−89.6231	−	51.7439i	−1.07980	−	0.623420i	−0.148954	−	0.988844i	$-0.547591\pi$
$83$	−89.6231	−	51.7439i	−1.07980	−	0.623420i	−0.930841	+	0.365424i	$0.880924\pi$
$84$		0			0
$85$	22.5000	+	38.9711i	0.264706	+	0.458484i
$86$	−11.2394	+	6.48906i	−0.130690	+	0.0754542i
$87$		0			0
$88$	20.7846	−	36.0000i	0.236189	−	0.409091i
$89$			134.130i			1.50708i	0.657403	+	0.753539i	$0.271655\pi$
$89$			134.130i			1.50708i	−0.657403	+	0.753539i	$0.728345\pi$
$90$		0			0
$91$	177.885			1.95478
$92$	−25.4558	−	14.6969i	−0.276694	−	0.159749i
$93$		0			0
$94$	12.0000	+	20.7846i	0.127660	+	0.221113i
$95$	122.427	−	70.6835i	1.28871	−	0.744037i
$96$		0			0
$97$	37.5692	−	65.0718i	0.387312	−	0.670843i	−0.604775	−	0.796396i	$-0.706737\pi$
$97$	37.5692	−	65.0718i	0.387312	−	0.670843i	0.992087	+	0.125553i	$0.0400705\pi$
$98$		−	30.3077i		−	0.309262i
$99$		0			0
$100$	17.1769			0.171769
$101$	25.1920	+	14.5446i	0.249426	+	0.144006i	0.619501	−	0.784995i	$-0.287335\pi$
$101$	25.1920	+	14.5446i	0.249426	+	0.144006i	−0.370075	+	0.929002i	$0.620668\pi$
$102$		0			0
$103$	−48.3538	−	83.7513i	−0.469455	−	0.813119i	0.529936	−	0.848038i	$-0.322216\pi$
$103$	−48.3538	−	83.7513i	−0.469455	−	0.813119i	−0.999390	+	0.0349186i	$0.988883\pi$
$104$	−51.9198	+	29.9759i	−0.499228	+	0.288230i
$105$		0			0
$106$	−18.2154	+	31.5500i	−0.171843	+	0.297641i
$107$			177.582i			1.65964i	0.558030	+	0.829821i	$0.311558\pi$
$107$			177.582i			1.65964i	−0.558030	+	0.829821i	$0.688442\pi$
$108$		0			0
$109$	61.9423			0.568278			0.284139	−	0.958783i	$-0.408292\pi$
$109$	61.9423			0.568278			0.284139	+	0.958783i	$0.408292\pi$
$110$	−104.320	−	60.2292i	−0.948364	−	0.547538i
$111$		0			0
$112$	16.7846	+	29.0718i	0.149863	+	0.259570i
$113$	95.0991	−	54.9055i	0.841585	−	0.485889i	−0.0162179	−	0.999868i	$-0.505163\pi$
$113$	95.0991	−	54.9055i	0.841585	−	0.485889i	0.857803	+	0.513979i	$0.171829\pi$
$114$		0			0
$115$	−42.5885	+	73.7654i	−0.370334	+	0.641438i
$116$		−	70.9881i		−	0.611966i
$117$		0			0
$118$	87.2154			0.739113
$119$	56.4325	+	32.5813i	0.474223	+	0.273793i
$120$		0			0
$121$	47.5000	+	82.2724i	0.392562	+	0.679937i
$122$	15.9217	−	9.19239i	0.130506	−	0.0753474i
$123$		0			0
$124$	8.00000	−	13.8564i	0.0645161	−	0.111745i
$125$			95.1140i			0.760912i
$126$		0			0
$127$	141.177			1.11163			0.555815	−	0.831306i	$-0.312406\pi$
$127$	141.177			1.11163			0.555815	+	0.831306i	$0.312406\pi$
$128$	−9.79796	−	5.65685i	−0.0765466	−	0.0441942i
$129$		0			0
$130$	86.8634	+	150.452i	0.668180	+	1.15732i
$131$	−46.0598	+	26.5927i	−0.351602	+	0.202997i	−0.665391	−	0.746495i	$-0.731735\pi$
$131$	−46.0598	+	26.5927i	−0.351602	+	0.202997i	0.313789	+	0.949493i	$0.398402\pi$
$132$		0			0
$133$	102.354	−	177.282i	0.769578	−	1.33295i
$134$		−	29.9487i		−	0.223498i
$135$		0			0
$136$	−21.9615			−0.161482
$137$	−1.41555	−	0.817267i	−0.0103325	−	0.00596545i	0.494825	−	0.868993i	$-0.335232\pi$
$137$	−1.41555	−	0.817267i	−0.0103325	−	0.00596545i	−0.505157	+	0.863027i	$0.668566\pi$
$138$		0			0
$139$	9.60770	+	16.6410i	0.0691201	+	0.119720i	0.898514	−	0.438944i	$-0.144647\pi$
$139$	9.60770	+	16.6410i	0.0691201	+	0.119720i	−0.829394	+	0.558664i	$0.811314\pi$
$140$	84.2436	−	48.6381i	0.601740	−	0.347415i
$141$		0			0
$142$	71.5692	−	123.962i	0.504009	−	0.872968i
$143$		−	311.519i		−	2.17845i
$144$		0			0
$145$	−205.708			−1.41867
$146$	49.4938	+	28.5753i	0.338999	+	0.195721i
$147$		0			0
$148$	−60.5692	−	104.909i	−0.409251	−	0.708844i
$149$	81.6504	−	47.1409i	0.547989	−	0.316382i	−0.200321	−	0.979730i	$-0.564199\pi$
$149$	81.6504	−	47.1409i	0.547989	−	0.316382i	0.748311	+	0.663348i	$0.230865\pi$
$150$		0			0
$151$	−16.0000	+	27.7128i	−0.105960	+	0.183529i	−0.914130	−	0.405421i	$-0.867125\pi$
$151$	−16.0000	+	27.7128i	−0.105960	+	0.183529i	0.808170	+	0.588949i	$0.200458\pi$
$152$			68.9919i			0.453894i
$153$		0			0
$154$	−174.431			−1.13267
$155$	−40.1528	−	23.1822i	−0.259050	−	0.149563i
$156$		0			0
$157$	−0.146171	−	0.253175i	−0.000931025	−	0.00161258i	0.865560	−	0.500806i	$-0.166963\pi$
$157$	−0.146171	−	0.253175i	−0.000931025	−	0.00161258i	−0.866491	+	0.499193i	$0.833630\pi$
$158$	−120.939	+	69.8241i	−0.765436	+	0.441924i
$159$		0			0
$160$	−16.3923	+	28.3923i	−0.102452	+	0.177452i
$161$			123.341i			0.766094i
$162$		0			0
$163$	28.7846			0.176593			0.0882963	−	0.996094i	$-0.471858\pi$
$163$	28.7846			0.176593			0.0882963	+	0.996094i	$0.471858\pi$
$164$	−58.2602	−	33.6365i	−0.355245	−	0.205101i
$165$		0			0
$166$	−73.1769	−	126.746i	−0.440825	−	0.763531i
$167$	48.5564	−	28.0341i	0.290757	−	0.167869i	−0.347526	−	0.937670i	$-0.612978\pi$
$167$	48.5564	−	28.0341i	0.290757	−	0.167869i	0.638283	+	0.769802i	$0.279645\pi$
$168$		0			0
$169$	−140.138	+	242.727i	−0.829222	+	1.43625i
$170$			63.6396i			0.374351i
$171$		0			0
$172$	−18.3538			−0.106708
$173$	191.350	+	110.476i	1.10607	+	0.638589i	0.937808	−	0.347154i	$-0.112852\pi$
$173$	191.350	+	110.476i	1.10607	+	0.638589i	0.168260	+	0.985743i	$0.446185\pi$
$174$		0			0
$175$	−36.0385	−	62.4205i	−0.205934	−	0.356688i
$176$	50.9117	−	29.3939i	0.289271	−	0.167011i
$177$		0			0
$178$	−94.8442	+	164.275i	−0.532833	+	0.922893i
$179$		−	248.347i		−	1.38741i	−0.720258	−	0.693706i	$-0.755977\pi$
$179$		−	248.347i		−	1.38741i	0.720258	−	0.693706i	$-0.244023\pi$
$180$		0			0
$181$	−158.277			−0.874458			−0.437229	−	0.899350i	$-0.644040\pi$
$181$	−158.277			−0.874458			−0.437229	+	0.899350i	$0.644040\pi$
$182$	217.863	+	125.783i	1.19705	+	0.691118i
$183$		0			0
$184$	−20.7846	−	36.0000i	−0.112960	−	0.195652i
$185$	−304.003	+	175.516i	−1.64326	+	0.948736i
$186$		0			0
$187$	57.0577	−	98.8269i	0.305121	−	0.528486i
$188$			33.9411i			0.180538i
$189$		0			0
$190$	199.923			1.05223
$191$	−29.9215	−	17.2752i	−0.156657	−	0.0904459i	0.419622	−	0.907699i	$-0.362163\pi$
$191$	−29.9215	−	17.2752i	−0.156657	−	0.0904459i	−0.576279	+	0.817253i	$0.695496\pi$
$192$		0			0
$193$	27.5000	+	47.6314i	0.142487	+	0.246795i	0.928433	−	0.371501i	$-0.121157\pi$
$193$	27.5000	+	47.6314i	0.142487	+	0.246795i	−0.785946	+	0.618296i	$0.787823\pi$
$194$	92.0254	−	53.1309i	0.474358	−	0.273871i
$195$		0			0
$196$	21.4308	−	37.1192i	0.109341	−	0.189384i
$197$			35.7170i			0.181305i	0.995883	+	0.0906524i	$0.0288952\pi$
$197$			35.7170i			0.181305i	−0.995883	+	0.0906524i	$0.971105\pi$
$198$		0			0
$199$	−375.138			−1.88512			−0.942559	−	0.334040i	$-0.891588\pi$
$199$	−375.138			−1.88512			−0.942559	+	0.334040i	$0.891588\pi$
$200$	21.0373	+	12.1459i	0.105187	+	0.0607296i
$201$		0			0
$202$	20.5692	+	35.6269i	0.101828	+	0.176371i
$203$	−257.969	+	148.938i	−1.27078	+	0.733687i
$204$		0			0
$205$	−97.4711	+	168.825i	−0.475469	+	0.823536i
$206$		−	136.765i		−	0.663909i
$207$		0			0
$208$	−84.7846			−0.407618
$209$	−310.463	−	179.246i	−1.48547	−	0.857637i
$210$		0			0
$211$	−153.727	−	266.263i	−0.728563	−	1.26191i	−0.957490	−	0.288465i	$-0.906855\pi$
$211$	−153.727	−	266.263i	−0.728563	−	1.26191i	0.228927	−	0.973444i	$-0.426478\pi$
$212$	−44.6184	+	25.7605i	−0.210464	+	0.121512i
$213$		0			0
$214$	−125.569	+	217.492i	−0.586772	+	1.01632i
$215$			53.1853i			0.247374i
$216$		0			0
$217$	−67.1384			−0.309394
$218$	75.8635	+	43.7998i	0.347998	+	0.200917i
$219$		0			0
$220$	−85.1769	−	147.531i	−0.387168	−	0.670594i
$221$	−142.530	+	82.2895i	−0.644930	+	0.372351i
$222$		0			0
$223$	169.296	−	293.229i	0.759175	−	1.31493i	−0.184096	−	0.982908i	$-0.558936\pi$
$223$	169.296	−	293.229i	0.759175	−	1.31493i	0.943272	−	0.332022i	$-0.107731\pi$
$224$			47.4740i			0.211938i
$225$		0			0
$226$	155.296			0.687151
$227$	−107.344	−	61.9752i	−0.472882	−	0.273019i	0.244563	−	0.969633i	$-0.421355\pi$
$227$	−107.344	−	61.9752i	−0.472882	−	0.273019i	−0.717445	+	0.696615i	$0.754689\pi$
$228$		0			0
$229$	−37.0289	−	64.1359i	−0.161698	−	0.280069i	0.773780	−	0.633455i	$-0.218364\pi$
$229$	−37.0289	−	64.1359i	−0.161698	−	0.280069i	−0.935478	+	0.353386i	$0.885030\pi$
$230$	−104.320	+	60.2292i	−0.453565	+	0.261866i
$231$		0			0
$232$	50.1962	−	86.9423i	0.216363	−	0.374751i
$233$		−	273.223i		−	1.17263i	−0.810082	−	0.586316i	$-0.800578\pi$
$233$		−	273.223i		−	1.17263i	0.810082	−	0.586316i	$-0.199422\pi$
$234$		0			0
$235$	98.3538			0.418527
$236$	106.817	+	61.6706i	0.452613	+	0.261316i
$237$		0			0
$238$	46.0770	+	79.8076i	0.193601	+	0.335326i
$239$	392.210	−	226.443i	1.64105	−	0.947459i	0.660585	−	0.750751i	$-0.270308\pi$
$239$	392.210	−	226.443i	1.64105	−	0.947459i	0.980462	−	0.196708i	$-0.0630252\pi$
$240$		0			0
$241$	191.344	−	331.418i	0.793959	−	1.37518i	−0.129538	−	0.991574i	$-0.541350\pi$
$241$	191.344	−	331.418i	0.793959	−	1.37518i	0.923498	−	0.383604i	$-0.125317\pi$
$242$			134.350i			0.555166i
$243$		0			0
$244$	26.0000			0.106557
$245$	−107.563	−	62.1016i	−0.439033	−	0.253476i
$246$		0			0
$247$	258.512	+	447.755i	1.04661	+	1.81277i
$248$	19.5959	−	11.3137i	0.0790158	−	0.0456198i
$249$		0			0
$250$	−67.2558	+	116.490i	−0.269023	+	0.465962i
$251$			73.9307i			0.294544i	0.989096	+	0.147272i	$0.0470493\pi$
$251$			73.9307i			0.294544i	−0.989096	+	0.147272i	$0.952951\pi$
$252$		0			0
$253$	216.000			0.853755
$254$	172.906	+	99.8272i	0.680731	+	0.393020i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.0312500	−	0.0541266i
$257$	138.398	−	79.9044i	0.538515	−	0.310912i	−0.205962	−	0.978560i	$-0.566032\pi$
$257$	138.398	−	79.9044i	0.538515	−	0.310912i	0.744477	+	0.667648i	$0.232699\pi$
$258$		0			0
$259$	−254.158	+	440.214i	−0.981304	+	1.69967i
$260$			245.687i			0.944950i
$261$		0			0
$262$	−75.2154			−0.287082
$263$	224.392	+	129.553i	0.853202	+	0.492596i	0.861730	−	0.507367i	$-0.169381\pi$
$263$	224.392	+	129.553i	0.853202	+	0.492596i	−0.00852798	+	0.999964i	$0.502715\pi$
$264$		0			0
$265$	74.6481	+	129.294i	0.281691	+	0.487903i
$266$	250.715	−	144.750i	0.942536	−	0.544174i
$267$		0			0
$268$	21.1769	−	36.6795i	0.0790183	−	0.136864i
$269$		−	24.8168i		−	0.0922556i	−0.998936	−	0.0461278i	$-0.985312\pi$
$269$		−	24.8168i		−	0.0922556i	0.998936	−	0.0461278i	$-0.0146881\pi$
$270$		0			0
$271$	98.1154			0.362050			0.181025	−	0.983479i	$-0.442059\pi$
$271$	98.1154			0.362050			0.181025	+	0.983479i	$0.442059\pi$
$272$	−26.8973	−	15.5291i	−0.0988870	−	0.0570924i
$273$		0			0
$274$	−1.15579	−	2.00189i	−0.00421821	−	0.00730616i
$275$	−109.313	+	63.1120i	−0.397503	+	0.229498i
$276$		0			0
$277$	0.707658	−	1.22570i	0.00255472	−	0.00442491i	−0.864745	−	0.502211i	$-0.832520\pi$
$277$	0.707658	−	1.22570i	0.00255472	−	0.00442491i	0.867300	+	0.497786i	$0.165853\pi$
$278$			27.1747i			0.0977506i
$279$		0			0
$280$	137.569			0.491319
$281$	−87.2230	−	50.3582i	−0.310402	−	0.179211i	0.336704	−	0.941610i	$-0.390688\pi$
$281$	−87.2230	−	50.3582i	−0.310402	−	0.179211i	−0.647106	+	0.762400i	$0.724021\pi$
$282$		0			0
$283$	23.6462	+	40.9564i	0.0835554	+	0.144722i	0.904775	−	0.425890i	$-0.140039\pi$
$283$	23.6462	+	40.9564i	0.0835554	+	0.144722i	−0.821219	+	0.570613i	$0.806706\pi$
$284$	175.308	−	101.214i	0.617282	−	0.356388i
$285$		0			0
$286$	220.277	−	381.531i	0.770199	−	1.33402i
$287$			282.288i			0.983582i
$288$		0			0
$289$	228.711			0.791389
$290$	−251.939	−	145.457i	−0.868757	−	0.501577i
$291$		0			0
$292$	40.4115	+	69.9948i	0.138396	+	0.239708i
$293$	288.920	−	166.808i	0.986074	−	0.569310i	0.0819755	−	0.996634i	$-0.473877\pi$
$293$	288.920	−	166.808i	0.986074	−	0.569310i	0.904098	+	0.427324i	$0.140544\pi$
$294$		0			0
$295$	178.708	−	309.531i	0.605789	−	1.04926i
$296$		−	171.316i		−	0.578769i
$297$		0			0
$298$	133.335			0.447432
$299$	−269.783	−	155.759i	−0.902284	−	0.520934i
$300$		0			0
$301$	38.5077	+	66.6973i	0.127933	+	0.221586i
$302$	−39.1918	+	22.6274i	−0.129774	+	0.0749252i
$303$		0			0
$304$	−48.7846	+	84.4974i	−0.160476	+	0.277952i
$305$		−	75.3422i		−	0.247024i
$306$		0			0
$307$	10.3538			0.0337258			0.0168629	−	0.999858i	$-0.494632\pi$
$307$	10.3538			0.0337258			0.0168629	+	0.999858i	$0.494632\pi$
$308$	−213.633	−	123.341i	−0.693614	−	0.400458i
$309$		0			0
$310$	−32.7846	−	56.7846i	−0.105757	−	0.183176i
$311$	8.26229	−	4.77024i	0.0265669	−	0.0153384i	−0.486658	−	0.873593i	$-0.661784\pi$
$311$	8.26229	−	4.77024i	0.0265669	−	0.0153384i	0.513225	+	0.858254i	$0.328451\pi$
$312$		0			0
$313$	239.638	−	415.066i	0.765618	−	1.32609i	−0.174301	−	0.984692i	$-0.555767\pi$
$313$	239.638	−	415.066i	0.765618	−	1.32609i	0.939919	−	0.341397i	$-0.110900\pi$
$314$		−	0.413434i		−	0.00131667i
$315$		0			0
$316$	−197.492			−0.624976
$317$	155.592	+	89.8311i	0.490827	+	0.283379i	0.724917	−	0.688836i	$-0.241878\pi$
$317$	155.592	+	89.8311i	0.490827	+	0.283379i	−0.234091	+	0.972215i	$0.575211\pi$
$318$		0			0
$319$	260.827	+	451.765i	0.817639	+	1.41619i
$320$	−40.1528	+	23.1822i	−0.125477	+	0.0724444i
$321$		0			0
$322$	−87.2154	+	151.061i	−0.270855	+	0.469135i
$323$			189.396i			0.586365i
$324$		0			0
$325$	182.042			0.560130
$326$	35.2538	+	20.3538i	0.108141	+	0.0624349i
$327$		0			0
$328$	−47.5692	−	82.3923i	−0.145028	−	0.251196i
$329$	123.341	−	71.2111i	0.374897	−	0.216447i
$330$		0			0
$331$	−147.727	+	255.870i	−0.446305	+	0.773023i	−0.998142	−	0.0609292i	$-0.980594\pi$
$331$	−147.727	+	255.870i	−0.446305	+	0.773023i	0.551837	+	0.833952i	$0.313927\pi$
$332$		−	206.976i		−	0.623420i
$333$		0			0
$334$	79.2923			0.237402
$335$	−106.289	−	61.3660i	−0.317281	−	0.183182i
$336$		0			0
$337$	−244.631	−	423.713i	−0.725907	−	1.25731i	−0.958600	−	0.284758i	$-0.908087\pi$
$337$	−244.631	−	423.713i	−0.725907	−	1.25731i	0.232692	−	0.972550i	$-0.425246\pi$
$338$	−343.268	+	198.186i	−1.01558	+	0.586348i
$339$		0			0
$340$	−45.0000	+	77.9423i	−0.132353	+	0.229242i
$341$			117.576i			0.344796i
$342$		0			0
$343$	231.369			0.674546
$344$	−22.4788	−	12.9781i	−0.0653452	−	0.0377271i
$345$		0			0
$346$	156.237	+	270.610i	0.451551	+	0.782109i
$347$	−577.363	+	333.341i	−1.66387	+	0.960637i	−0.693032	+	0.720907i	$0.743726\pi$
$347$	−577.363	+	333.341i	−1.66387	+	0.960637i	−0.970840	+	0.239730i	$0.922941\pi$
$348$		0			0
$349$	−255.985	+	443.378i	−0.733480	+	1.27042i	0.221907	+	0.975068i	$0.428772\pi$
$349$	−255.985	+	443.378i	−0.733480	+	1.27042i	−0.955387	+	0.295357i	$0.904561\pi$
$350$		−	101.932i		−	0.291235i
$351$		0			0
$352$	83.1384			0.236189
$353$	508.081	+	293.340i	1.43932	+	0.830993i	0.997802	−	0.0662588i	$-0.0211063\pi$
$353$	508.081	+	293.340i	1.43932	+	0.830993i	0.441519	+	0.897252i	$0.354440\pi$
$354$		0			0
$355$	−293.296	−	508.004i	−0.826186	−	1.43100i
$356$	−232.320	+	134.130i	−0.652584	+	0.376770i
$357$		0			0
$358$	175.608	−	304.161i	0.490524	−	0.849613i
$359$			534.573i			1.48906i	0.667589	+	0.744530i	$0.267326\pi$
$359$			534.573i			1.48906i	−0.667589	+	0.744530i	$0.732674\pi$
$360$		0			0
$361$	233.985			0.648157
$362$	−193.849	−	111.919i	−0.535494	−	0.309168i
$363$		0			0
$364$	177.885	+	308.105i	0.488694	+	0.846443i
$365$	202.829	−	117.104i	0.555697	−	0.320832i
$366$		0			0
$367$	−7.64617	+	13.2436i	−0.0208343	+	0.0360860i	−0.876255	−	0.481848i	$-0.839966\pi$
$367$	−7.64617	+	13.2436i	−0.0208343	+	0.0360860i	0.855420	+	0.517934i	$0.173299\pi$
$368$		−	58.7878i		−	0.159749i
$369$		0			0
$370$	−496.435			−1.34172
$371$	187.226	+	108.095i	0.504651	+	0.291361i
$372$		0			0
$373$	−326.492	−	565.501i	−0.875314	−	1.51609i	−0.856427	−	0.516267i	$-0.827321\pi$
$373$	−326.492	−	565.501i	−0.875314	−	1.51609i	−0.0188869	−	0.999822i	$-0.506012\pi$
$374$	139.762	−	80.6918i	0.373696	−	0.215753i
$375$		0			0
$376$	−24.0000	+	41.5692i	−0.0638298	+	0.110556i
$377$		−	752.337i		−	1.99559i
$378$		0			0
$379$	−655.215			−1.72880			−0.864400	−	0.502804i	$-0.832302\pi$
$379$	−655.215			−1.72880			−0.864400	+	0.502804i	$0.832302\pi$
$380$	244.855	+	141.367i	0.644355	+	0.372018i
$381$		0			0
$382$	−24.4308	−	42.3154i	−0.0639549	−	0.110773i
$383$	259.410	−	149.771i	0.677311	−	0.391046i	−0.121530	−	0.992588i	$-0.538780\pi$
$383$	259.410	−	149.771i	0.677311	−	0.391046i	0.798841	+	0.601542i	$0.205447\pi$
$384$		0			0
$385$	−357.415	+	619.061i	−0.928351	+	1.60795i
$386$			77.7817i			0.201507i
$387$		0			0
$388$	150.277			0.387312
$389$	−402.319	−	232.279i	−1.03424	−	0.597119i	−0.116043	−	0.993244i	$-0.537021\pi$
$389$	−402.319	−	232.279i	−1.03424	−	0.597119i	−0.918196	+	0.396126i	$0.870354\pi$
$390$		0			0
$391$	−57.0577	−	98.8269i	−0.145928	−	0.252754i
$392$	52.4945	−	30.3077i	0.133914	−	0.0773156i
$393$		0			0
$394$	−25.2558	+	43.7442i	−0.0641009	+	0.111026i
$395$			572.289i			1.44883i
$396$		0			0
$397$	−185.708			−0.467777			−0.233889	−	0.972263i	$-0.575145\pi$
$397$	−185.708			−0.467777			−0.233889	+	0.972263i	$0.575145\pi$
$398$	−459.449	−	265.263i	−1.15439	−	0.666490i
$399$		0			0
$400$	17.1769	+	29.7513i	0.0429423	+	0.0743782i
$401$	54.4376	−	31.4296i	0.135755	−	0.0783780i	−0.430585	−	0.902550i	$-0.641693\pi$
$401$	54.4376	−	31.4296i	0.135755	−	0.0783780i	0.566339	+	0.824172i	$0.308359\pi$
$402$		0			0
$403$	84.7846	−	146.851i	0.210384	−	0.364395i
$404$			58.1785i			0.144006i
$405$		0			0
$406$	−421.261			−1.03759
$407$	770.920	+	445.091i	1.89415	+	1.09359i
$408$		0			0
$409$	163.640	+	283.433i	0.400099	+	0.692991i	0.993737	−	0.111741i	$-0.0356425\pi$
$409$	163.640	+	283.433i	0.400099	+	0.692991i	−0.593639	+	0.804732i	$0.702309\pi$
$410$	−238.755	+	137.845i	−0.582328	+	0.336207i
$411$		0			0
$412$	96.7077	−	167.503i	0.234727	−	0.406560i
$413$		−	517.558i		−	1.25317i
$414$		0			0
$415$	−599.769			−1.44523
$416$	−103.840	−	59.9518i	−0.249614	−	0.144115i
$417$		0			0
$418$	−253.492	−	439.061i	−0.606441	−	1.05039i
$419$	−562.808	+	324.937i	−1.34322	+	0.775507i	−0.987278	−	0.159003i	$-0.949172\pi$
$419$	−562.808	+	324.937i	−1.34322	+	0.775507i	−0.355939	+	0.934509i	$0.615839\pi$
$420$		0			0
$421$	−1.65956	+	2.87445i	−0.00394196	+	0.00682767i	−0.867990	−	0.496582i	$-0.834588\pi$
$421$	−1.65956	+	2.87445i	−0.00394196	+	0.00682767i	0.864048	+	0.503410i	$0.167921\pi$
$422$		−	434.805i		−	1.03034i
$423$		0			0
$424$	−72.8616			−0.171843
$425$	57.7515	+	33.3428i	0.135886	+	0.0784538i
$426$		0			0
$427$	−54.5500	−	94.4833i	−0.127752	−	0.221272i
$428$	−307.581	+	177.582i	−0.718646	+	0.414910i
$429$		0			0
$430$	−37.6077	+	65.1384i	−0.0874598	+	0.151485i
$431$		−	803.502i		−	1.86427i	−0.362107	−	0.932136i	$-0.617943\pi$
$431$		−	803.502i		−	1.86427i	0.362107	−	0.932136i	$-0.382057\pi$
$432$		0			0
$433$	−93.1230			−0.215065			−0.107532	−	0.994202i	$-0.534295\pi$
$433$	−93.1230			−0.215065			−0.107532	+	0.994202i	$0.534295\pi$
$434$	−82.2275	−	47.4740i	−0.189464	−	0.109387i
$435$		0			0
$436$	61.9423	+	107.287i	0.142069	+	0.246072i
$437$	−310.463	+	179.246i	−0.710442	+	0.410174i
$438$		0			0
$439$	236.000	−	408.764i	0.537585	−	0.931125i	−0.461448	−	0.887167i	$-0.652670\pi$
$439$	236.000	−	408.764i	0.537585	−	0.931125i	0.999033	−	0.0439580i	$-0.0139968\pi$
$440$		−	240.917i		−	0.547538i
$441$		0			0
$442$	−232.750			−0.526584
$443$	493.365	+	284.844i	1.11369	+	0.642989i	0.939783	−	0.341773i	$-0.111027\pi$
$443$	493.365	+	284.844i	1.11369	+	0.642989i	0.173908	+	0.984762i	$0.444361\pi$
$444$		0			0
$445$	388.679	+	673.211i	0.873436	+	1.51283i
$446$	414.689	−	239.421i	0.929796	−	0.536818i
$447$		0			0
$448$	−33.5692	+	58.1436i	−0.0749313	+	0.129785i
$449$			559.115i			1.24524i	0.782523	+	0.622622i	$0.213933\pi$
$449$			559.115i			1.24524i	−0.782523	+	0.622622i	$0.786067\pi$
$450$		0			0
$451$	494.354			1.09613
$452$	190.198	+	109.811i	0.420792	+	0.242945i
$453$		0			0
$454$	−87.6462	−	151.808i	−0.193053	−	0.334378i
$455$	892.820	−	515.470i	1.96224	−	1.13290i
$456$		0			0
$457$	−302.148	+	523.336i	−0.661155	+	1.14515i	0.319157	+	0.947702i	$0.396600\pi$
$457$	−302.148	+	523.336i	−0.661155	+	1.14515i	−0.980312	+	0.197453i	$0.936733\pi$
$458$		−	104.733i		−	0.228676i
$459$		0			0
$460$	−170.354			−0.370334
$461$	−4.44669	−	2.56730i	−0.00964575	−	0.00556897i	0.495169	−	0.868796i	$-0.335106\pi$
$461$	−4.44669	−	2.56730i	−0.00964575	−	0.00556897i	−0.504815	+	0.863227i	$0.668439\pi$
$462$		0			0
$463$	80.7461	+	139.856i	0.174398	+	0.302066i	0.939953	−	0.341305i	$-0.110869\pi$
$463$	80.7461	+	139.856i	0.174398	+	0.302066i	−0.765555	+	0.643370i	$0.777536\pi$
$464$	122.955	−	70.9881i	0.264989	−	0.152992i
$465$		0			0
$466$	193.198	−	334.629i	0.414588	−	0.718088i
$467$			503.025i			1.07714i	0.842581	+	0.538570i	$0.181035\pi$
$467$			503.025i			1.07714i	−0.842581	+	0.538570i	$0.818965\pi$
$468$		0			0
$469$	−177.723			−0.378941
$470$	120.458	+	69.5467i	0.256294	+	0.147972i
$471$		0			0
$472$	87.2154	+	151.061i	0.184778	+	0.320046i
$473$	116.803	−	67.4363i	0.246941	−	0.142571i
$474$		0			0
$475$	104.746	−	181.426i	0.220518	−	0.381949i
$476$			130.325i			0.273793i
$477$		0			0
$478$	640.477			1.33991
$479$	185.153	+	106.898i	0.386541	+	0.223170i	0.680660	−	0.732599i	$-0.261693\pi$
$479$	185.153	+	106.898i	0.386541	+	0.223170i	−0.294119	+	0.955769i	$0.595026\pi$
$480$		0			0
$481$	−641.917	−	1111.83i	−1.33455	−	2.31150i
$482$	468.696	−	270.602i	0.972398	−	0.561414i
$483$		0			0
$484$	−95.0000	+	164.545i	−0.196281	+	0.339969i
$485$		−	435.469i		−	0.897874i
$486$		0			0
$487$	8.63071			0.0177222			0.00886110	−	0.999961i	$-0.497179\pi$
$487$	8.63071			0.0177222			0.00886110	+	0.999961i	$0.497179\pi$
$488$	31.8434	+	18.3848i	0.0652528	+	0.0376737i
$489$		0			0
$490$	−87.8250	−	152.117i	−0.179235	−	0.310443i
$491$	−798.873	+	461.229i	−1.62703	+	0.939367i	−0.642060	+	0.766654i	$0.721920\pi$
$491$	−798.873	+	461.229i	−1.62703	+	0.939367i	−0.984972	+	0.172713i	$0.944747\pi$
$492$		0			0
$493$	137.798	−	238.673i	0.279509	−	0.484124i
$494$			731.181i			1.48012i
$495$		0			0
$496$	32.0000			0.0645161
$497$	−735.619	−	424.710i	−1.48012	−	0.854547i
$498$		0			0
$499$	217.296	+	376.368i	0.435463	+	0.754244i	0.997333	−	0.0729811i	$-0.0232513\pi$
$499$	217.296	+	376.368i	0.435463	+	0.754244i	−0.561870	+	0.827225i	$0.689918\pi$
$500$	−164.742	+	95.1140i	−0.329485	+	0.190228i
$501$		0			0
$502$	−52.2769	+	90.5462i	−0.104137	+	0.180371i
$503$		−	144.087i		−	0.286454i	−0.989690	−	0.143227i	$-0.954252\pi$
$503$		−	144.087i		−	0.286454i	0.989690	−	0.143227i	$-0.0457480\pi$
$504$		0			0
$505$	168.588			0.333839
$506$	264.545	+	152.735i	0.522816	+	0.301848i
$507$		0			0
$508$	141.177	+	244.526i	0.277907	+	0.481350i
$509$	603.856	−	348.636i	1.18636	−	0.684943i	0.228880	−	0.973455i	$-0.426494\pi$
$509$	603.856	−	348.636i	1.18636	−	0.684943i	0.957477	+	0.288511i	$0.0931602\pi$
$510$		0			0
$511$	169.573	−	293.709i	0.331845	−	0.574773i
$512$		−	22.6274i		−	0.0441942i
$513$		0			0
$514$	226.004			0.439696
$515$	−485.385	−	280.237i	−0.942496	−	0.544150i
$516$		0			0
$517$	−124.708	−	216.000i	−0.241214	−	0.417795i
$518$	−622.557	+	359.433i	−1.20185	+	0.693887i
$519$		0			0
$520$	−173.727	+	300.904i	−0.334090	+	0.578661i
$521$			426.962i			0.819504i	0.912197	+	0.409752i	$0.134385\pi$
$521$			426.962i			0.819504i	−0.912197	+	0.409752i	$0.865615\pi$
$522$		0			0
$523$	179.762			0.343712			0.171856	−	0.985122i	$-0.445024\pi$
$523$	179.762			0.343712			0.171856	+	0.985122i	$0.445024\pi$
$524$	−92.1197	−	53.1853i	−0.175801	−	0.101499i
$525$		0			0
$526$	183.215	+	317.338i	0.348318	+	0.603305i
$527$	53.7945	−	31.0583i	0.102077	−	0.0589341i
$528$		0			0
$529$	−156.500	+	271.066i	−0.295841	+	0.512412i
$530$			211.137i			0.398371i
$531$		0			0
$532$	409.415			0.769578
$533$	−617.446	−	356.482i	−1.15843	−	0.668822i
$534$		0			0
$535$	514.592	+	891.300i	0.961855	+	1.66598i
$536$	51.8726	−	29.9487i	0.0967773	−	0.0558744i
$537$		0			0
$538$	17.5481	−	30.3942i	0.0326173	−	0.0564948i
$539$			314.967i			0.584354i
$540$		0			0
$541$	708.734			1.31005			0.655023	−	0.755609i	$-0.272659\pi$
$541$	708.734			1.31005			0.655023	+	0.755609i	$0.272659\pi$
$542$	120.166	+	69.3781i	0.221709	+	0.128004i
$543$		0			0
$544$	−21.9615	−	38.0385i	−0.0403704	−	0.0699237i
$545$	310.894	−	179.495i	0.570448	−	0.329349i
$546$		0			0
$547$	98.1154	−	169.941i	0.179370	−	0.310678i	−0.762295	−	0.647230i	$-0.775927\pi$
$547$	98.1154	−	169.941i	0.179370	−	0.310678i	0.941665	+	0.336552i	$0.109261\pi$
$548$		−	3.26907i		−	0.00596545i
$549$		0			0
$550$	−178.508			−0.324560
$551$	−749.789	−	432.891i	−1.36078	−	0.785646i
$552$		0			0
$553$	414.354	+	717.682i	0.749284	+	1.29780i
$554$	1.73340	−	1.00078i	0.00312888	−	0.00180646i
$555$		0			0
$556$	−19.2154	+	33.2820i	−0.0345601	+	0.0598598i
$557$		−	353.610i		−	0.634848i	−0.948284	−	0.317424i	$-0.897182\pi$
$557$		−	353.610i		−	0.634848i	0.948284	−	0.317424i	$-0.102818\pi$
$558$		0			0
$559$	−194.515			−0.347970
$560$	168.487	+	97.2761i	0.300870	+	0.173707i
$561$		0			0
$562$	−71.2173	−	123.352i	−0.126721	−	0.219487i
$563$	−323.050	+	186.513i	−0.573801	+	0.331284i	−0.758666	−	0.651480i	$-0.774149\pi$
$563$	−323.050	+	186.513i	−0.573801	+	0.331284i	0.184865	+	0.982764i	$0.440815\pi$
$564$		0			0
$565$	318.208	−	551.152i	0.563199	−	0.975490i
$566$			66.8815i			0.118165i
$567$		0			0
$568$	286.277			0.504009
$569$	52.0634	+	30.0588i	0.0914999	+	0.0528275i	0.545052	−	0.838402i	$-0.316510\pi$
$569$	52.0634	+	30.0588i	0.0914999	+	0.0528275i	−0.453552	+	0.891230i	$0.649843\pi$
$570$		0			0
$571$	−43.1000	−	74.6513i	−0.0754815	−	0.130738i	0.825814	−	0.563943i	$-0.190716\pi$
$571$	−43.1000	−	74.6513i	−0.0754815	−	0.130738i	−0.901296	+	0.433205i	$0.857383\pi$
$572$	539.566	−	311.519i	0.943297	−	0.544613i
$573$		0			0
$574$	−199.608	+	345.731i	−0.347749	+	0.602318i
$575$			126.224i			0.219520i
$576$		0			0
$577$	709.123			1.22898			0.614491	−	0.788924i	$-0.289361\pi$
$577$	709.123			1.22898			0.614491	+	0.788924i	$0.289361\pi$
$578$	280.113	+	161.723i	0.484625	+	0.279798i
$579$		0			0
$580$	−205.708	−	356.296i	−0.354668	−	0.614304i
$581$	−752.144	+	434.251i	−1.29457	+	0.747419i
$582$		0			0
$583$	189.300	−	327.877i	0.324700	−	0.562396i
$584$			114.301i			0.195721i
$585$		0			0
$586$	471.804			0.805126
$587$	−833.363	−	481.143i	−1.41970	−	0.819664i	−0.423427	−	0.905930i	$-0.639173\pi$
$587$	−833.363	−	481.143i	−1.41970	−	0.819664i	−0.996272	+	0.0862666i	$0.972506\pi$
$588$		0			0
$589$	−97.5692	−	168.995i	−0.165652	−	0.286918i
$590$	437.743	−	252.731i	0.741937	−	0.428357i
$591$		0			0
$592$	121.138	−	209.818i	0.204626	−	0.354422i
$593$			104.350i			0.175969i	0.996122	+	0.0879847i	$0.0280427\pi$
$593$			104.350i			0.175969i	−0.996122	+	0.0879847i	$0.971957\pi$
$594$		0			0
$595$	377.654			0.634712
$596$	163.301	+	94.2818i	0.273995	+	0.158191i
$597$		0			0
$598$	−220.277	−	381.531i	−0.368356	−	0.638011i
$599$	−247.738	+	143.031i	−0.413585	+	0.238784i	−0.692329	−	0.721582i	$-0.743415\pi$
$599$	−247.738	+	143.031i	−0.413585	+	0.238784i	0.278744	+	0.960366i	$0.410082\pi$
$600$		0			0
$601$	140.208	−	242.847i	0.233291	−	0.404071i	−0.725484	−	0.688239i	$-0.758384\pi$
$601$	140.208	−	242.847i	0.233291	−	0.404071i	0.958775	+	0.284168i	$0.0917173\pi$
$602$			108.916i			0.180924i
$603$		0			0
$604$	−64.0000			−0.105960
$605$	476.814	+	275.289i	0.788123	+	0.455023i
$606$		0			0
$607$	−368.865	−	638.894i	−0.607686	−	1.05254i	−0.991621	−	0.129183i	$-0.958765\pi$
$607$	−368.865	−	638.894i	−0.607686	−	1.05254i	0.383935	−	0.923360i	$-0.374569\pi$
$608$	−119.497	+	68.9919i	−0.196542	+	0.113473i
$609$		0			0
$610$	53.2750	−	92.2750i	0.0873361	−	0.151270i
$611$			359.711i			0.588724i
$612$		0			0
$613$	−679.415			−1.10834			−0.554172	−	0.832402i	$-0.686965\pi$
$613$	−679.415			−1.10834			−0.554172	+	0.832402i	$0.686965\pi$
$614$	12.6808	+	7.32126i	0.0206528	+	0.0119239i
$615$		0			0
$616$	−174.431	−	302.123i	−0.283167	−	0.490459i
$617$	409.326	−	236.325i	0.663414	−	0.383022i	−0.130163	−	0.991493i	$-0.541550\pi$
$617$	409.326	−	236.325i	0.663414	−	0.383022i	0.793576	+	0.608471i	$0.208217\pi$
$618$		0			0
$619$	−443.177	+	767.605i	−0.715956	+	1.24007i	0.246633	+	0.969109i	$0.420676\pi$
$619$	−443.177	+	767.605i	−0.715956	+	1.24007i	−0.962590	+	0.270964i	$0.912658\pi$
$620$		−	92.7289i		−	0.149563i
$621$		0			0
$622$	13.4923			0.0216917
$623$	974.850	+	562.830i	1.56477	+	0.903419i
$624$		0			0
$625$	382.975	+	663.332i	0.612760	+	1.06133i
$626$	586.992	−	338.900i	0.937687	−	0.541374i
$627$		0			0
$628$	0.292342	−	0.506351i	0.000465513	−	0.000806291i
$629$		−	470.294i		−	0.747685i
$630$		0			0
$631$	−729.108			−1.15548			−0.577740	−	0.816221i	$-0.696065\pi$
$631$	−729.108			−1.15548			−0.577740	+	0.816221i	$0.696065\pi$
$632$	−241.878	−	139.648i	−0.382718	−	0.220962i
$633$		0			0
$634$	127.040	+	220.040i	0.200379	+	0.347067i
$635$	708.581	−	409.099i	1.11588	−	0.644251i
$636$		0			0
$637$	227.125	−	393.392i	0.356554	−	0.617570i
$638$			737.730i			1.15632i
$639$		0			0
$640$	−65.5692			−0.102452
$641$	753.611	+	435.098i	1.17568	+	0.678780i	0.955011	−	0.296569i	$-0.0958425\pi$
$641$	753.611	+	435.098i	1.17568	+	0.678780i	0.220669	+	0.975349i	$0.429176\pi$
$642$		0			0
$643$	309.061	+	535.310i	0.480656	+	0.832520i	0.999754	−	0.0221949i	$-0.00706543\pi$
$643$	309.061	+	535.310i	0.480656	+	0.832520i	−0.519098	+	0.854715i	$0.673732\pi$
$644$	−213.633	+	123.341i	−0.331729	+	0.191524i
$645$		0			0
$646$	−133.923	+	231.962i	−0.207311	+	0.359074i
$647$		−	425.439i		−	0.657556i	−0.944407	−	0.328778i	$-0.893363\pi$
$647$		−	425.439i		−	0.657556i	0.944407	−	0.328778i	$-0.106637\pi$
$648$		0			0
$649$	−906.369			−1.39656
$650$	222.955	+	128.723i	0.343008	+	0.198036i
$651$		0			0
$652$	28.7846	+	49.8564i	0.0441482	+	0.0764669i
$653$	−163.513	+	94.4042i	−0.250403	+	0.144570i	−0.619949	−	0.784642i	$-0.712847\pi$
$653$	−163.513	+	94.4042i	−0.250403	+	0.144570i	0.369546	+	0.929212i	$0.379513\pi$
$654$		0			0
$655$	−154.119	+	266.942i	−0.235296	+	0.407545i
$656$		−	134.546i		−	0.205101i
$657$		0			0
$658$	201.415			0.306102
$659$	−571.984	−	330.235i	−0.867958	−	0.501116i	−0.00128859	−	0.999999i	$-0.500410\pi$
$659$	−571.984	−	330.235i	−0.867958	−	0.501116i	−0.866669	+	0.498884i	$0.833744\pi$
$660$		0			0
$661$	294.915	+	510.808i	0.446165	+	0.772781i	0.998133	−	0.0610847i	$-0.0194560\pi$
$661$	294.915	+	510.808i	0.446165	+	0.772781i	−0.551967	+	0.833866i	$0.686123\pi$
$662$	−361.856	+	208.917i	−0.546610	+	0.315585i
$663$		0			0
$664$	146.354	−	253.492i	0.220412	−	0.381765i
$665$		−	1186.39i		−	1.78405i
$666$		0			0
$667$	521.654			0.782090
$668$	97.1129	+	56.0682i	0.145379	+	0.0839344i
$669$		0			0
$670$	−86.7846	−	150.315i	−0.129529	−	0.224351i
$671$	−165.463	+	95.5301i	−0.246592	+	0.142370i
$672$		0			0
$673$	195.715	−	338.989i	0.290810	−	0.503698i	−0.683191	−	0.730240i	$-0.739408\pi$
$673$	195.715	−	338.989i	0.290810	−	0.503698i	0.974002	+	0.226541i	$0.0727418\pi$
$674$		−	691.920i		−	1.02659i
$675$		0			0
$676$	−560.554			−0.829222
$677$	−600.728	−	346.830i	−0.887338	−	0.512305i	−0.0142672	−	0.999898i	$-0.504542\pi$
$677$	−600.728	−	346.830i	−0.887338	−	0.512305i	−0.873071	+	0.487593i	$0.837875\pi$
$678$		0			0
$679$	−315.292	−	546.102i	−0.464348	−	0.804274i
$680$	−110.227	+	63.6396i	−0.162099	+	0.0935877i
$681$		0			0
$682$	−83.1384	+	144.000i	−0.121904	+	0.211144i
$683$		−	678.170i		−	0.992928i	−0.868057	−	0.496464i	$-0.834632\pi$
$683$		−	678.170i		−	0.992928i	0.868057	−	0.496464i	$-0.165368\pi$
$684$		0			0
$685$	−9.47303			−0.0138292
$686$	283.368	+	163.603i	0.413073	+	0.238488i
$687$		0			0
$688$	−18.3538	−	31.7898i	−0.0266771	−	0.0462061i
$689$	−472.869	+	273.011i	−0.686313	+	0.396243i
$690$		0			0
$691$	−337.888	+	585.240i	−0.488985	+	0.846946i	−0.999920	−	0.0126731i	$-0.995966\pi$
$691$	−337.888	+	585.240i	−0.488985	+	0.846946i	0.510935	+	0.859619i	$0.329299\pi$
$692$			441.904i			0.638589i
$693$		0			0
$694$	−942.831			−1.35855
$695$	96.4439	+	55.6819i	0.138768	+	0.0801179i
$696$		0			0
$697$	−130.587	−	226.183i	−0.187355	−	0.324509i
$698$	−627.031	+	362.017i	−0.898326	+	0.518649i
$699$		0			0
$700$	72.0770	−	124.841i	0.102967	−	0.178344i
$701$			797.260i			1.13732i	0.822573	+	0.568659i	$0.192538\pi$
$701$			797.260i			1.13732i	−0.822573	+	0.568659i	$0.807462\pi$
$702$		0			0
$703$	−1477.42			−2.10160
$704$	101.823	+	58.7878i	0.144635	+	0.0835053i
$705$		0			0
$706$	414.846	+	718.535i	0.587601	+	1.01775i
$707$	211.419	−	122.063i	0.299037	−	0.172649i
$708$		0			0
$709$	−86.7520	+	150.259i	−0.122358	+	0.211931i	−0.920697	−	0.390278i	$-0.872379\pi$
$709$	−86.7520	+	150.259i	−0.122358	+	0.211931i	0.798339	+	0.602208i	$0.205712\pi$
$710$		−	829.567i		−	1.16840i
$711$		0			0
$712$	−379.377			−0.532833
$713$	101.823	+	58.7878i	0.142810	+	0.0824513i
$714$		0			0
$715$	−902.711	−	1563.54i	−1.26253	−	2.18677i
$716$	430.149	−	248.347i	0.600767	−	0.346853i
$717$		0			0
$718$	−378.000	+	654.715i	−0.526462	+	0.911860i
$719$			188.177i			0.261721i	0.991401	+	0.130860i	$0.0417740\pi$
$719$			188.177i			0.261721i	−0.991401	+	0.130860i	$0.958226\pi$
$720$		0			0
$721$	−811.600			−1.12566
$722$	286.571	+	165.452i	0.396913	+	0.229158i
$723$		0			0
$724$	−158.277	−	274.144i	−0.218614	−	0.378651i
$725$	−263.998	+	152.420i	−0.364136	+	0.210234i
$726$		0			0
$727$	−355.888	+	616.417i	−0.489530	+	0.847891i	−0.999927	−	0.0120478i	$-0.996165\pi$
$727$	−355.888	+	616.417i	−0.489530	+	0.847891i	0.510397	+	0.859939i	$0.329498\pi$
$728$			503.134i			0.691118i
$729$		0			0
$730$	331.219			0.453725
$731$	−61.7085	−	35.6274i	−0.0844165	−	0.0487379i
$732$		0			0
$733$	−613.415	−	1062.47i	−0.836856	−	1.44948i	−0.892510	−	0.451027i	$-0.851058\pi$
$733$	−613.415	−	1062.47i	−0.836856	−	1.44948i	0.0556546	−	0.998450i	$-0.482275\pi$
$734$	−18.7292	+	10.8133i	−0.0255166	+	0.0147320i
$735$		0			0
$736$	41.5692	−	72.0000i	0.0564799	−	0.0978261i
$737$			311.236i			0.422301i
$738$		0			0
$739$	741.892			1.00391			0.501957	−	0.864893i	$-0.332614\pi$
$739$	741.892			1.00391			0.501957	+	0.864893i	$0.332614\pi$
$740$	−608.006	−	351.032i	−0.821629	−	0.474368i
$741$		0			0
$742$	152.869	+	264.777i	0.206023	+	0.356842i
$743$	677.218	−	390.992i	0.911464	−	0.526234i	0.0305622	−	0.999533i	$-0.490270\pi$
$743$	677.218	−	390.992i	0.911464	−	0.526234i	0.880902	+	0.473299i	$0.156937\pi$
$744$		0			0
$745$	273.208	−	473.210i	0.366722	−	0.635181i
$746$		−	923.460i		−	1.23788i
$747$		0			0
$748$	228.231			0.305121
$749$	1290.65	+	745.160i	1.72317	+	0.994873i
$750$		0			0
$751$	516.665	+	894.891i	0.687970	+	1.19160i	0.972494	+	0.232930i	$0.0748312\pi$
$751$	516.665	+	894.891i	0.687970	+	1.19160i	−0.284524	+	0.958669i	$0.591835\pi$
$752$	−58.7878	+	33.9411i	−0.0781752	+	0.0451345i
$753$		0			0
$754$	531.983	−	921.421i	0.705547	−	1.22204i
$755$			185.458i			0.245639i
$756$		0			0
$757$	473.877			0.625993			0.312997	−	0.949754i	$-0.398667\pi$
$757$	473.877			0.625993			0.312997	+	0.949754i	$0.398667\pi$
$758$	−802.472	−	463.307i	−1.05867	−	0.611223i
$759$		0			0
$760$	199.923	+	346.277i	0.263057	+	0.455627i
$761$	−814.458	+	470.227i	−1.07025	+	0.617907i	−0.928249	−	0.371960i	$-0.878686\pi$
$761$	−814.458	+	470.227i	−1.07025	+	0.617907i	−0.141998	+	0.989867i	$0.545353\pi$
$762$		0			0
$763$	259.919	−	450.193i	0.340654	−	0.590031i
$764$		−	69.1007i		−	0.0904459i
$765$		0			0
$766$	423.615			0.553023
$767$	1132.05	+	653.590i	1.47595	+	0.852138i
$768$		0			0
$769$	−490.408	−	849.411i	−0.637721	−	1.10457i	−0.985932	−	0.167149i	$-0.946544\pi$
$769$	−490.408	−	849.411i	−0.637721	−	1.10457i	0.348210	−	0.937416i	$-0.386789\pi$
$770$	−875.485	+	505.462i	−1.13699	+	0.656444i
$771$		0			0
$772$	−55.0000	+	95.2628i	−0.0712435	+	0.123397i
$773$			1042.20i			1.34825i	0.738618	+	0.674124i	$0.235479\pi$
$773$			1042.20i			1.34825i	−0.738618	+	0.674124i	$0.764521\pi$
$774$		0			0
$775$	−68.7077			−0.0886550
$776$	184.051	+	106.262i	0.237179	+	0.136935i
$777$		0			0
$778$	−328.492	−	568.965i	−0.422227	−	0.731318i
$779$	−710.550	+	410.236i	−0.912131	+	0.526619i
$780$		0			0
$781$	−743.769	+	1288.25i	−0.952329	+	1.64948i
$782$		−	161.384i		−	0.206373i
$783$		0			0
$784$	85.7231			0.109341
$785$	−1.46729	−	0.847142i	−0.00186916	−	0.00107916i
$786$		0			0
$787$	72.3501	+	125.314i

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	162.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(5.01910 - 2.89778i) q^{5} +(4.19615 - 7.26795i) q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(5.01910 - 2.89778i) q^{5} +(4.19615 - 7.26795i) q^{7} +2.82843i q^{8} +8.19615 q^{10} +(-12.7279 - 7.34847i) q^{11} +(10.5981 + 18.3564i) q^{13} +(10.2784 - 5.93426i) q^{14} +(-2.00000 + 3.46410i) q^{16} +7.76457i q^{17} +24.3923 q^{19} +(10.0382 + 5.79555i) q^{20} +(-10.3923 - 18.0000i) q^{22} +(-12.7279 + 7.34847i) q^{23} +(4.29423 - 7.43782i) q^{25} +29.9759i q^{26} +16.7846 q^{28} +(-30.7387 - 17.7470i) q^{29} +(-4.00000 - 6.92820i) q^{31} +(-4.89898 + 2.82843i) q^{32} +(-5.49038 + 9.50962i) q^{34} -48.6381i q^{35} -60.5692 q^{37} +(29.8744 + 17.2480i) q^{38} +(8.19615 + 14.1962i) q^{40} +(-29.1301 + 16.8183i) q^{41} +(-4.58846 + 7.94744i) q^{43} -29.3939i q^{44} -20.7846 q^{46} +(14.6969 + 8.48528i) q^{47} +(-10.7154 - 18.5596i) q^{49} +(10.5187 - 6.07296i) q^{50} +(-21.1962 + 36.7128i) q^{52} +25.7605i q^{53} -85.1769 q^{55} +(20.5569 + 11.8685i) q^{56} +(-25.0981 - 43.4711i) q^{58} +(53.4083 - 30.8353i) q^{59} +(6.50000 - 11.2583i) q^{61} -11.3137i q^{62} -8.00000 q^{64} +(106.386 + 61.4217i) q^{65} +(-10.5885 - 18.3397i) q^{67} +(-13.4486 + 7.76457i) q^{68} +(34.3923 - 59.5692i) q^{70} -101.214i q^{71} +40.4115 q^{73} +(-74.1818 - 42.8289i) q^{74} +(24.3923 + 42.2487i) q^{76} +(-106.817 + 61.6706i) q^{77} +(-49.3731 + 85.5167i) q^{79} +23.1822i q^{80} -47.5692 q^{82} +(-89.6231 - 51.7439i) q^{83} +(22.5000 + 38.9711i) q^{85} +(-11.2394 + 6.48906i) q^{86} +(20.7846 - 36.0000i) q^{88} +134.130i q^{89} +177.885 q^{91} +(-25.4558 - 14.6969i) q^{92} +(12.0000 + 20.7846i) q^{94} +(122.427 - 70.6835i) q^{95} +(37.5692 - 65.0718i) q^{97} -30.3077i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^4 - 8 * q^7	\( 8 q + 8 q^{4} - 8 q^{7} + 24 q^{10} + 64 q^{13} - 16 q^{16} + 112 q^{19} - 28 q^{25} - 32 q^{28} - 32 q^{31} + 60 q^{34} - 152 q^{37} + 24 q^{40} + 88 q^{43} - 252 q^{49} - 128 q^{52} - 432 q^{55} - 180 q^{58} + 52 q^{61} - 64 q^{64} + 40 q^{67} + 192 q^{70} + 448 q^{73} + 112 q^{76} - 104 q^{79} - 48 q^{82} + 180 q^{85} + 176 q^{91} + 96 q^{94} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^4 - 8 * q^7 + 24 * q^10 + 64 * q^13 - 16 * q^16 + 112 * q^19 - 28 * q^25 - 32 * q^28 - 32 * q^31 + 60 * q^34 - 152 * q^37 + 24 * q^40 + 88 * q^43 - 252 * q^49 - 128 * q^52 - 432 * q^55 - 180 * q^58 + 52 * q^61 - 64 * q^64 + 40 * q^67 + 192 * q^70 + 448 * q^73 + 112 * q^76 - 104 * q^79 - 48 * q^82 + 180 * q^85 + 176 * q^91 + 96 * q^94 - 32 * q^97

Introduction

L-functions

Modular forms

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