LMFDB - Embedded newform 384.3.i.b.353.3

Newspace parameters

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	384.i (of order $4$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.629407744.1
Defining polynomial:	$ x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 2x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			353.3
Root			$-0.767178 - 1.18804i$ of defining polynomial
Character	$\chi$	$=$	384.353
Dual form			384.3.i.b.161.3

$q$-expansion

$f(q)$	$=$	$q+(1.13234 + 2.77809i) q^{3} +(-6.28651 + 6.28651i) q^{5} -1.64575i q^{7} +(-6.43560 + 6.29150i) q^{9} +O(q^{10})$	\(q+(1.13234 + 2.77809i) q^{3} +(-6.28651 + 6.28651i) q^{5} -1.64575i q^{7} +(-6.43560 + 6.29150i) q^{9} +(-4.75216 + 4.75216i) q^{11} +(9.35425 - 9.35425i) q^{13} +(-24.5830 - 10.3460i) q^{15} -11.4859i q^{17} +(-8.58301 + 8.58301i) q^{19} +(4.57205 - 1.86355i) q^{21} -16.2381 q^{23} -54.0405i q^{25} +(-24.7657 - 10.7546i) q^{27} +(10.7405 + 10.7405i) q^{29} -6.35425 q^{31} +(-18.5830 - 7.82087i) q^{33} +(10.3460 + 10.3460i) q^{35} +(-27.2288 - 27.2288i) q^{37} +(36.5792 + 15.3948i) q^{39} +1.98162 q^{41} +(-19.4170 - 19.4170i) q^{43} +(0.905893 - 80.0091i) q^{45} +74.9474i q^{47} +46.2915 q^{49} +(31.9090 - 13.0060i) q^{51} +(-4.00671 + 4.00671i) q^{53} -59.7490i q^{55} +(-33.5633 - 14.1255i) q^{57} +(-27.9694 + 27.9694i) q^{59} +(-39.2288 + 39.2288i) q^{61} +(10.3542 + 10.5914i) q^{63} +117.611i q^{65} +(-68.6863 + 68.6863i) q^{67} +(-18.3871 - 45.1110i) q^{69} -40.6822 q^{71} +59.0405i q^{73} +(150.130 - 61.1923i) q^{75} +(7.82087 + 7.82087i) q^{77} -17.3948 q^{79} +(1.83399 - 80.9792i) q^{81} +(-75.1400 - 75.1400i) q^{83} +(72.2065 + 72.2065i) q^{85} +(-17.6762 + 42.0000i) q^{87} +78.8051 q^{89} +(-15.3948 - 15.3948i) q^{91} +(-7.19518 - 17.6527i) q^{93} -107.914i q^{95} -38.8340 q^{97} +(0.684791 - 60.4812i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3}+O(q^{10}) $ 8 * q + 4 * q^3	$ 8 q + 4 q^{3} + 96 q^{13} - 112 q^{15} + 16 q^{19} + 32 q^{21} - 68 q^{27} - 72 q^{31} - 64 q^{33} - 112 q^{37} - 240 q^{43} + 112 q^{45} + 328 q^{49} - 32 q^{51} - 208 q^{61} + 104 q^{63} - 232 q^{67} + 324 q^{75} + 136 q^{79} + 184 q^{81} + 112 q^{85} + 152 q^{91} - 64 q^{93} - 480 q^{97} + 160 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 + 96 * q^13 - 112 * q^15 + 16 * q^19 + 32 * q^21 - 68 * q^27 - 72 * q^31 - 64 * q^33 - 112 * q^37 - 240 * q^43 + 112 * q^45 + 328 * q^49 - 32 * q^51 - 208 * q^61 + 104 * q^63 - 232 * q^67 + 324 * q^75 + 136 * q^79 + 184 * q^81 + 112 * q^85 + 152 * q^91 - 64 * q^93 - 480 * q^97 + 160 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.13234	+	2.77809i	0.377447	+	0.926031i
$3$	1.13234	+	2.77809i	0.377447	+	0.926031i
$4$		0			0
$5$	−6.28651	+	6.28651i	−1.25730	+	1.25730i	−0.304927	+	0.952376i	$0.598632\pi$
$5$	−6.28651	+	6.28651i	−1.25730	+	1.25730i	−0.952376	+	0.304927i	$0.901368\pi$
$6$		0			0
$7$		−	1.64575i		−	0.235107i	−0.993067	−	0.117554i	$-0.962495\pi$
$7$		−	1.64575i		−	0.235107i	0.993067	−	0.117554i	$-0.0375052\pi$
$8$		0			0
$9$	−6.43560	+	6.29150i	−0.715067	+	0.699056i
$10$		0			0
$11$	−4.75216	+	4.75216i	−0.432014	+	0.432014i	−0.889313	−	0.457299i	$-0.848817\pi$
$11$	−4.75216	+	4.75216i	−0.432014	+	0.432014i	0.457299	+	0.889313i	$0.348817\pi$
$12$		0			0
$13$	9.35425	−	9.35425i	0.719558	−	0.719558i	−0.248957	−	0.968515i	$-0.580088\pi$
$13$	9.35425	−	9.35425i	0.719558	−	0.719558i	0.968515	+	0.248957i	$0.0800878\pi$
$14$		0			0
$15$	−24.5830	−	10.3460i	−1.63887	−	0.689736i
$16$		0			0
$17$		−	11.4859i		−	0.675644i	−0.941210	−	0.337822i	$-0.890310\pi$
$17$		−	11.4859i		−	0.675644i	0.941210	−	0.337822i	$-0.109690\pi$
$18$		0			0
$19$	−8.58301	+	8.58301i	−0.451737	+	0.451737i	−0.895931	−	0.444194i	$-0.853490\pi$
$19$	−8.58301	+	8.58301i	−0.451737	+	0.451737i	0.444194	+	0.895931i	$0.353490\pi$
$20$		0			0
$21$	4.57205	−	1.86355i	0.217717	−	0.0887406i
$22$		0			0
$23$	−16.2381			−0.706004			−0.353002	−	0.935623i	$-0.614839\pi$
$23$	−16.2381			−0.706004			−0.353002	+	0.935623i	$0.614839\pi$
$24$		0			0
$25$		−	54.0405i		−	2.16162i
$26$		0			0
$27$	−24.7657	−	10.7546i	−0.917248	−	0.398318i
$28$		0			0
$29$	10.7405	+	10.7405i	0.370362	+	0.370362i	0.867609	−	0.497247i	$-0.165656\pi$
$29$	10.7405	+	10.7405i	0.370362	+	0.370362i	−0.497247	+	0.867609i	$0.665656\pi$
$30$		0			0
$31$	−6.35425			−0.204976			−0.102488	−	0.994734i	$-0.532680\pi$
$31$	−6.35425			−0.204976			−0.102488	+	0.994734i	$0.532680\pi$
$32$		0			0
$33$	−18.5830	−	7.82087i	−0.563121	−	0.236996i
$34$		0			0
$35$	10.3460	+	10.3460i	0.295601	+	0.295601i
$36$		0			0
$37$	−27.2288	−	27.2288i	−0.735912	−	0.735912i	0.235872	−	0.971784i	$-0.424205\pi$
$37$	−27.2288	−	27.2288i	−0.735912	−	0.735912i	−0.971784	+	0.235872i	$0.924205\pi$
$38$		0			0
$39$	36.5792	+	15.3948i	0.937928	+	0.394738i
$40$		0			0
$41$	1.98162			0.0483323			0.0241662	−	0.999708i	$-0.492307\pi$
$41$	1.98162			0.0483323			0.0241662	+	0.999708i	$0.492307\pi$
$42$		0			0
$43$	−19.4170	−	19.4170i	−0.451558	−	0.451558i	0.444313	−	0.895871i	$-0.353448\pi$
$43$	−19.4170	−	19.4170i	−0.451558	−	0.451558i	−0.895871	+	0.444313i	$0.853448\pi$
$44$		0			0
$45$	0.905893	−	80.0091i	0.0201310	−	1.77798i
$46$		0			0
$47$			74.9474i			1.59463i	0.603566	+	0.797313i	$0.293746\pi$
$47$			74.9474i			1.59463i	−0.603566	+	0.797313i	$0.706254\pi$
$48$		0			0
$49$	46.2915			0.944725
$50$		0			0
$51$	31.9090	−	13.0060i	0.625667	−	0.255020i
$52$		0			0
$53$	−4.00671	+	4.00671i	−0.0755983	+	0.0755983i	−0.743895	−	0.668297i	$-0.767024\pi$
$53$	−4.00671	+	4.00671i	−0.0755983	+	0.0755983i	0.668297	+	0.743895i	$0.267024\pi$
$54$		0			0
$55$		−	59.7490i		−	1.08635i
$56$		0			0
$57$	−33.5633	−	14.1255i	−0.588830	−	0.247816i
$58$		0			0
$59$	−27.9694	+	27.9694i	−0.474058	+	0.474058i	−0.903225	−	0.429167i	$-0.858807\pi$
$59$	−27.9694	+	27.9694i	−0.474058	+	0.474058i	0.429167	+	0.903225i	$0.358807\pi$
$60$		0			0
$61$	−39.2288	+	39.2288i	−0.643094	+	0.643094i	−0.951315	−	0.308221i	$-0.900267\pi$
$61$	−39.2288	+	39.2288i	−0.643094	+	0.643094i	0.308221	+	0.951315i	$0.400267\pi$
$62$		0			0
$63$	10.3542	+	10.5914i	0.164353	+	0.168118i
$64$		0			0
$65$			117.611i			1.80940i
$66$		0			0
$67$	−68.6863	+	68.6863i	−1.02517	+	1.02517i	−0.0254932	+	0.999675i	$0.508116\pi$
$67$	−68.6863	+	68.6863i	−1.02517	+	1.02517i	−0.999675	+	0.0254932i	$0.991884\pi$
$68$		0			0
$69$	−18.3871	−	45.1110i	−0.266479	−	0.653782i
$70$		0			0
$71$	−40.6822			−0.572988			−0.286494	−	0.958082i	$-0.592490\pi$
$71$	−40.6822			−0.572988			−0.286494	+	0.958082i	$0.592490\pi$
$72$		0			0
$73$			59.0405i			0.808774i	0.914588	+	0.404387i	$0.132515\pi$
$73$			59.0405i			0.808774i	−0.914588	+	0.404387i	$0.867485\pi$
$74$		0			0
$75$	150.130	−	61.1923i	2.00173	−	0.815898i
$76$		0			0
$77$	7.82087	+	7.82087i	0.101570	+	0.101570i
$78$		0			0
$79$	−17.3948			−0.220187			−0.110093	−	0.993921i	$-0.535115\pi$
$79$	−17.3948			−0.220187			−0.110093	+	0.993921i	$0.535115\pi$
$80$		0			0
$81$	1.83399	−	80.9792i	0.0226418	−	0.999744i
$82$		0			0
$83$	−75.1400	−	75.1400i	−0.905301	−	0.905301i	0.0905874	−	0.995889i	$-0.471126\pi$
$83$	−75.1400	−	75.1400i	−0.905301	−	0.905301i	−0.995889	+	0.0905874i	$0.971126\pi$
$84$		0			0
$85$	72.2065	+	72.2065i	0.849489	+	0.849489i
$86$		0			0
$87$	−17.6762	+	42.0000i	−0.203174	+	0.482759i
$88$		0			0
$89$	78.8051			0.885450			0.442725	−	0.896657i	$-0.354012\pi$
$89$	78.8051			0.885450			0.442725	+	0.896657i	$0.354012\pi$
$90$		0			0
$91$	−15.3948	−	15.3948i	−0.169173	−	0.169173i
$92$		0			0
$93$	−7.19518	−	17.6527i	−0.0773675	−	0.189814i
$94$		0			0
$95$		−	107.914i		−	1.13594i
$96$		0			0
$97$	−38.8340			−0.400350			−0.200175	−	0.979760i	$-0.564151\pi$
$97$	−38.8340			−0.400350			−0.200175	+	0.979760i	$0.564151\pi$
$98$		0			0
$99$	0.684791	−	60.4812i	0.00691708	−	0.610921i
$100$		0			0
$101$	41.5332	−	41.5332i	0.411220	−	0.411220i	−0.470943	−	0.882164i	$-0.656086\pi$
$101$	41.5332	−	41.5332i	0.411220	−	0.411220i	0.882164	+	0.470943i	$0.156086\pi$
$102$		0			0
$103$			98.8118i			0.959337i	0.877450	+	0.479669i	$0.159243\pi$
$103$			98.8118i			0.959337i	−0.877450	+	0.479669i	$0.840757\pi$
$104$		0			0
$105$	−17.0270	+	40.4575i	−0.162162	+	0.385310i
$106$		0			0
$107$	−98.8480	+	98.8480i	−0.923813	+	0.923813i	−0.997296	−	0.0734837i	$-0.976588\pi$
$107$	−98.8480	+	98.8480i	−0.923813	+	0.923813i	0.0734837	+	0.997296i	$0.476588\pi$
$108$		0			0
$109$	−68.8523	+	68.8523i	−0.631672	+	0.631672i	−0.948487	−	0.316815i	$-0.897387\pi$
$109$	−68.8523	+	68.8523i	−0.631672	+	0.631672i	0.316815	+	0.948487i	$0.397387\pi$
$110$		0			0
$111$	44.8118	−	106.476i	0.403710	−	0.959246i
$112$		0			0
$113$		−	8.31160i		−	0.0735540i	−0.999323	−	0.0367770i	$-0.988291\pi$
$113$		−	8.31160i		−	0.0735540i	0.999323	−	0.0367770i	$-0.0117091\pi$
$114$		0			0
$115$	102.081	−	102.081i	0.887661	−	0.887661i
$116$		0			0
$117$	−1.34796	+	119.053i	−0.0115210	+	1.01754i
$118$		0			0
$119$	−18.9030			−0.158849
$120$		0			0
$121$			75.8340i			0.626727i
$122$		0			0
$123$	2.24388	+	5.50514i	0.0182429	+	0.0447572i
$124$		0			0
$125$	182.564	+	182.564i	1.46051	+	1.46051i
$126$		0			0
$127$	195.933			1.54278			0.771391	−	0.636361i	$-0.219561\pi$
$127$	195.933			1.54278			0.771391	+	0.636361i	$0.219561\pi$
$128$		0			0
$129$	31.9555	−	75.9289i	0.247717	−	0.588596i
$130$		0			0
$131$	142.127	+	142.127i	1.08494	+	1.08494i	0.996041	+	0.0888967i	$0.0283341\pi$
$131$	142.127	+	142.127i	1.08494	+	1.08494i	0.0888967	+	0.996041i	$0.471666\pi$
$132$		0			0
$133$	14.1255	+	14.1255i	0.106207	+	0.106207i
$134$		0			0
$135$	223.299	−	88.0810i	1.65406	−	0.652452i
$136$		0			0
$137$	−50.4847			−0.368501			−0.184251	−	0.982879i	$-0.558986\pi$
$137$	−50.4847			−0.368501			−0.184251	+	0.982879i	$0.558986\pi$
$138$		0			0
$139$	171.727	+	171.727i	1.23544	+	1.23544i	0.961843	+	0.273601i	$0.0882149\pi$
$139$	171.727	+	171.727i	1.23544	+	1.23544i	0.273601	+	0.961843i	$0.411785\pi$
$140$		0			0
$141$	−208.211	+	84.8661i	−1.47667	+	0.601887i
$142$		0			0
$143$			88.9057i			0.621718i
$144$		0			0
$145$	−135.041			−0.931314
$146$		0			0
$147$	52.4178	+	128.602i	0.356584	+	0.874844i
$148$		0			0
$149$	−84.4952	+	84.4952i	−0.567082	+	0.567082i	−0.931310	−	0.364228i	$-0.881333\pi$
$149$	−84.4952	+	84.4952i	−0.567082	+	0.567082i	0.364228	+	0.931310i	$0.381333\pi$
$150$		0			0
$151$		−	30.1033i		−	0.199359i	−0.995020	−	0.0996797i	$-0.968218\pi$
$151$		−	30.1033i		−	0.199359i	0.995020	−	0.0996797i	$-0.0317818\pi$
$152$		0			0
$153$	72.2638	+	73.9190i	0.472313	+	0.483130i
$154$		0			0
$155$	39.9461	−	39.9461i	0.257717	−	0.257717i
$156$		0			0
$157$	181.265	−	181.265i	1.15456	−	1.15456i	0.168928	−	0.985628i	$-0.445970\pi$
$157$	181.265	−	181.265i	1.15456	−	1.15456i	0.985628	−	0.168928i	$-0.0540305\pi$
$158$		0			0
$159$	−15.6680	−	6.59405i	−0.0985407	−	0.0414720i
$160$		0			0
$161$			26.7239i			0.165987i
$162$		0			0
$163$	200.081	−	200.081i	1.22749	−	1.22749i	0.262581	−	0.964910i	$-0.415426\pi$
$163$	200.081	−	200.081i	1.22749	−	1.22749i	0.964910	−	0.262581i	$-0.0845737\pi$
$164$		0			0
$165$	165.988	−	67.6563i	1.00599	−	0.410038i
$166$		0			0
$167$	−172.656			−1.03387			−0.516933	−	0.856026i	$-0.672926\pi$
$167$	−172.656			−1.03387			−0.516933	+	0.856026i	$0.672926\pi$
$168$		0			0
$169$		−	6.00394i		−	0.0355263i
$170$		0			0
$171$	1.23682	−	109.237i	0.00723287	−	0.638812i
$172$		0			0
$173$	−40.8313	−	40.8313i	−0.236019	−	0.236019i	0.579181	−	0.815199i	$-0.303373\pi$
$173$	−40.8313	−	40.8313i	−0.236019	−	0.236019i	−0.815199	+	0.579181i	$0.803373\pi$
$174$		0			0
$175$	−88.9373			−0.508213
$176$		0			0
$177$	−109.373	−	46.0307i	−0.617924	−	0.260060i
$178$		0			0
$179$	−152.613	−	152.613i	−0.852584	−	0.852584i	0.137866	−	0.990451i	$-0.455976\pi$
$179$	−152.613	−	152.613i	−0.852584	−	0.852584i	−0.990451	+	0.137866i	$0.955976\pi$
$180$		0			0
$181$	−166.601	−	166.601i	−0.920449	−	0.920449i	0.0766118	−	0.997061i	$-0.475590\pi$
$181$	−166.601	−	166.601i	−0.920449	−	0.920449i	−0.997061	+	0.0766118i	$0.975590\pi$
$182$		0			0
$183$	−153.402	−	64.5608i	−0.838260	−	0.352791i
$184$		0			0
$185$	342.348			1.85053
$186$		0			0
$187$	54.5830	+	54.5830i	0.291888	+	0.291888i
$188$		0			0
$189$	−17.6994	+	40.7582i	−0.0936474	+	0.215652i
$190$		0			0
$191$		−	14.3434i		−	0.0750963i	−0.999295	−	0.0375482i	$-0.988045\pi$
$191$		−	14.3434i		−	0.0750963i	0.999295	−	0.0375482i	$-0.0119548\pi$
$192$		0			0
$193$	207.373			1.07447			0.537235	−	0.843433i	$-0.319469\pi$
$193$	207.373			1.07447			0.537235	+	0.843433i	$0.319469\pi$
$194$		0			0
$195$	−326.735	+	133.176i	−1.67556	+	0.682954i
$196$		0			0
$197$	97.2608	−	97.2608i	0.493710	−	0.493710i	−0.415763	−	0.909473i	$-0.636486\pi$
$197$	97.2608	−	97.2608i	0.493710	−	0.493710i	0.909473	+	0.415763i	$0.136486\pi$
$198$		0			0
$199$			82.7673i			0.415916i	0.978138	+	0.207958i	$0.0666818\pi$
$199$			82.7673i			0.415916i	−0.978138	+	0.207958i	$0.933318\pi$
$200$		0			0
$201$	−268.593	−	113.041i	−1.33628	−	0.562391i
$202$		0			0
$203$	17.6762	−	17.6762i	0.0870748	−	0.0870748i
$204$		0			0
$205$	−12.4575	+	12.4575i	−0.0607684	+	0.0607684i
$206$		0			0
$207$	104.502	−	102.162i	0.504840	−	0.493536i
$208$		0			0
$209$		−	81.5756i		−	0.390314i
$210$		0			0
$211$	−201.646	+	201.646i	−0.955667	+	0.955667i	−0.999058	−	0.0433911i	$-0.986184\pi$
$211$	−201.646	+	201.646i	−0.955667	+	0.955667i	0.0433911	+	0.999058i	$0.486184\pi$
$212$		0			0
$213$	−46.0661	−	113.019i	−0.216273	−	0.530605i
$214$		0			0
$215$	244.130			1.13549
$216$		0			0
$217$			10.4575i			0.0481913i
$218$		0			0
$219$	−164.020	+	66.8541i	−0.748950	+	0.305270i
$220$		0			0
$221$	−107.442	−	107.442i	−0.486164	−	0.486164i
$222$		0			0
$223$	−233.261			−1.04602			−0.523008	−	0.852328i	$-0.675190\pi$
$223$	−233.261			−1.04602			−0.523008	+	0.852328i	$0.675190\pi$
$224$		0			0
$225$	339.996	+	347.783i	1.51109	+	1.54570i
$226$		0			0
$227$	−94.3599	−	94.3599i	−0.415682	−	0.415682i	0.468030	−	0.883712i	$-0.344964\pi$
$227$	−94.3599	−	94.3599i	−0.415682	−	0.415682i	−0.883712	+	0.468030i	$0.844964\pi$
$228$		0			0
$229$	138.063	+	138.063i	0.602894	+	0.602894i	0.941080	−	0.338185i	$-0.109813\pi$
$229$	138.063	+	138.063i	0.602894	+	0.602894i	−0.338185	+	0.941080i	$0.609813\pi$
$230$		0			0
$231$	−12.8712	+	30.5830i	−0.0557195	+	0.132394i
$232$		0			0
$233$	−396.796			−1.70299			−0.851493	−	0.524366i	$-0.824303\pi$
$233$	−396.796			−1.70299			−0.851493	+	0.524366i	$0.824303\pi$
$234$		0			0
$235$	−471.158	−	471.158i	−2.00493	−	2.00493i
$236$		0			0
$237$	−19.6968	−	48.3243i	−0.0831090	−	0.203900i
$238$		0			0
$239$			284.813i			1.19168i	0.803102	+	0.595842i	$0.203182\pi$
$239$			284.813i			1.19168i	−0.803102	+	0.595842i	$0.796818\pi$
$240$		0			0
$241$	−266.531			−1.10594			−0.552968	−	0.833202i	$-0.686505\pi$
$241$	−266.531			−1.10594			−0.552968	+	0.833202i	$0.686505\pi$
$242$		0			0
$243$	227.045	−	86.6012i	0.934340	−	0.356383i
$244$		0			0
$245$	−291.012	+	291.012i	−1.18780	+	1.18780i
$246$		0			0
$247$			160.575i			0.650102i
$248$		0			0
$249$	123.662	−	293.830i	0.496633	−	1.18004i
$250$		0			0
$251$	153.945	−	153.945i	0.613327	−	0.613327i	−0.330485	−	0.943811i	$-0.607212\pi$
$251$	153.945	−	153.945i	0.613327	−	0.613327i	0.943811	+	0.330485i	$0.107212\pi$
$252$		0			0
$253$	77.1660	−	77.1660i	0.305004	−	0.305004i
$254$		0			0
$255$	−118.834	+	282.359i	−0.466016	+	1.10729i
$256$		0			0
$257$			240.167i			0.934503i	0.884125	+	0.467251i	$0.154756\pi$
$257$			240.167i			0.934503i	−0.884125	+	0.467251i	$0.845244\pi$
$258$		0			0
$259$	−44.8118	+	44.8118i	−0.173018	+	0.173018i
$260$		0			0
$261$	−136.695	−	1.54772i	−0.523737	−	0.00592995i
$262$		0			0
$263$	−140.707			−0.535009			−0.267505	−	0.963557i	$-0.586199\pi$
$263$	−140.707			−0.535009			−0.267505	+	0.963557i	$0.586199\pi$
$264$		0			0
$265$		−	50.3765i		−	0.190100i
$266$		0			0
$267$	89.2343	+	218.928i	0.334211	+	0.819954i
$268$		0			0
$269$	−229.830	−	229.830i	−0.854388	−	0.854388i	0.136282	−	0.990670i	$-0.456485\pi$
$269$	−229.830	−	229.830i	−0.854388	−	0.854388i	−0.990670	+	0.136282i	$0.956485\pi$
$270$		0			0
$271$	−228.731			−0.844025			−0.422012	−	0.906590i	$-0.638676\pi$
$271$	−228.731			−0.844025			−0.422012	+	0.906590i	$0.638676\pi$
$272$		0			0
$273$	25.3360	−	60.2002i	0.0928057	−	0.220514i
$274$		0			0
$275$	256.809	+	256.809i	0.933851	+	0.933851i
$276$		0			0
$277$	103.265	+	103.265i	0.372799	+	0.372799i	0.868496	−	0.495697i	$-0.165087\pi$
$277$	103.265	+	103.265i	0.372799	+	0.372799i	−0.495697	+	0.868496i	$0.665087\pi$
$278$		0			0
$279$	40.8934	−	39.9778i	0.146571	−	0.143290i
$280$		0			0
$281$	−283.552			−1.00908			−0.504540	−	0.863388i	$-0.668338\pi$
$281$	−283.552			−1.00908			−0.504540	+	0.863388i	$0.668338\pi$
$282$		0			0
$283$	23.4758	+	23.4758i	0.0829534	+	0.0829534i	0.747366	−	0.664413i	$-0.231318\pi$
$283$	23.4758	+	23.4758i	0.0829534	+	0.0829534i	−0.664413	+	0.747366i	$0.731318\pi$
$284$		0			0
$285$	299.796	−	122.196i	1.05192	−	0.428758i
$286$		0			0
$287$		−	3.26126i		−	0.0113633i
$288$		0			0
$289$	157.073			0.543506
$290$		0			0
$291$	−43.9734	−	107.884i	−0.151111	−	0.370737i
$292$		0			0
$293$	381.409	−	381.409i	1.30174	−	1.30174i	0.374516	−	0.927220i	$-0.377809\pi$
$293$	381.409	−	381.409i	1.30174	−	1.30174i	0.927220	−	0.374516i	$-0.122191\pi$
$294$		0			0
$295$		−	351.660i		−	1.19207i
$296$		0			0
$297$	168.798	−	66.5830i	0.568343	−	0.224185i
$298$		0			0
$299$	−151.895	+	151.895i	−0.508011	+	0.508011i
$300$		0			0
$301$	−31.9555	+	31.9555i	−0.106165	+	0.106165i
$302$		0			0
$303$	162.413	+	68.3534i	0.536017	+	0.225589i
$304$		0			0
$305$		−	493.224i		−	1.61713i
$306$		0			0
$307$	−209.055	+	209.055i	−0.680960	+	0.680960i	−0.960217	−	0.279256i	$-0.909912\pi$
$307$	−209.055	+	209.055i	−0.680960	+	0.680960i	0.279256	+	0.960217i	$0.409912\pi$
$308$		0			0
$309$	−274.508	+	111.889i	−0.888376	+	0.362099i
$310$		0			0
$311$	111.176			0.357478			0.178739	−	0.983897i	$-0.442798\pi$
$311$	111.176			0.357478			0.178739	+	0.983897i	$0.442798\pi$
$312$		0			0
$313$		−	282.280i		−	0.901852i	−0.892561	−	0.450926i	$-0.851094\pi$
$313$		−	282.280i		−	0.901852i	0.892561	−	0.450926i	$-0.148906\pi$
$314$		0			0
$315$	−131.675	−	1.49088i	−0.418016	−	0.00473294i
$316$		0			0
$317$	206.983	+	206.983i	0.652943	+	0.652943i	0.953701	−	0.300758i	$-0.0972395\pi$
$317$	206.983	+	206.983i	0.652943	+	0.652943i	−0.300758	+	0.953701i	$0.597240\pi$
$318$		0			0
$319$	−102.081			−0.320003
$320$		0			0
$321$	−386.539	−	162.679i	−1.20417	−	0.506789i
$322$		0			0
$323$	98.5839	+	98.5839i	0.305213	+	0.305213i
$324$		0			0
$325$	−505.508	−	505.508i	−1.55541	−	1.55541i
$326$		0			0
$327$	−269.242	−	113.314i	−0.823371	−	0.346525i
$328$		0			0
$329$	123.345			0.374908
$330$		0			0
$331$	127.431	+	127.431i	0.384989	+	0.384989i	0.872896	−	0.487907i	$-0.162239\pi$
$331$	127.431	+	127.431i	0.384989	+	0.384989i	−0.487907	+	0.872896i	$0.662239\pi$
$332$		0			0
$333$	346.543	+	3.92369i	1.04067	+	0.0117829i
$334$		0			0
$335$		−	863.594i		−	2.57789i
$336$		0			0
$337$	68.9595			0.204628			0.102314	−	0.994752i	$-0.467375\pi$
$337$	68.9595			0.204628			0.102314	+	0.994752i	$0.467375\pi$
$338$		0			0
$339$	23.0904	−	9.41157i	0.0681133	−	0.0277627i
$340$		0			0
$341$	30.1964	−	30.1964i	0.0885525	−	0.0885525i
$342$		0			0
$343$		−	156.826i		−	0.457219i
$344$		0			0
$345$	399.181	+	168.000i	1.15705	+	0.486957i
$346$		0			0
$347$	54.0628	−	54.0628i	0.155801	−	0.155801i	−0.624902	−	0.780703i	$-0.714861\pi$
$347$	54.0628	−	54.0628i	0.155801	−	0.155801i	0.780703	+	0.624902i	$0.214861\pi$
$348$		0			0
$349$	−0.107201	+	0.107201i	−0.000307168	+	0.000307168i	−0.707260	−	0.706953i	$-0.750069\pi$
$349$	−0.107201	+	0.107201i	−0.000307168	+	0.000307168i	0.706953	+	0.707260i	$0.250069\pi$
$350$		0			0
$351$	−332.265	+	131.063i	−0.946625	+	0.373400i
$352$		0			0
$353$		−	194.223i		−	0.550208i	−0.961414	−	0.275104i	$-0.911288\pi$
$353$		−	194.223i		−	0.550208i	0.961414	−	0.275104i	$-0.0887123\pi$
$354$		0			0
$355$	255.749	−	255.749i	0.720420	−	0.720420i
$356$		0			0
$357$	−21.4047	−	52.5143i	−0.0599570	−	0.147099i
$358$		0			0
$359$	437.689			1.21919			0.609595	−	0.792713i	$-0.291332\pi$
$359$	437.689			1.21919			0.609595	+	0.792713i	$0.291332\pi$
$360$		0			0
$361$			213.664i			0.591867i
$362$		0			0
$363$	−210.674	+	85.8700i	−0.580369	+	0.236556i
$364$		0			0
$365$	−371.159	−	371.159i	−1.01687	−	1.01687i
$366$		0			0
$367$	246.678			0.672148			0.336074	−	0.941836i	$-0.390901\pi$
$367$	246.678			0.672148			0.336074	+	0.941836i	$0.390901\pi$
$368$		0			0
$369$	−12.7530	+	12.4674i	−0.0345608	+	0.0337870i
$370$		0			0
$371$	6.59405	+	6.59405i	0.0177737	+	0.0177737i
$372$		0			0
$373$	349.678	+	349.678i	0.937476	+	0.937476i	0.998157	−	0.0606816i	$-0.0193274\pi$
$373$	349.678	+	349.678i	0.937476	+	0.937476i	−0.0606816	+	0.998157i	$0.519327\pi$
$374$		0			0
$375$	−300.454	+	713.903i	−0.801212	+	1.90374i
$376$		0			0
$377$	200.938			0.532993
$378$		0			0
$379$	235.668	+	235.668i	0.621815	+	0.621815i	0.945995	−	0.324180i	$-0.105088\pi$
$379$	235.668	+	235.668i	0.621815	+	0.621815i	−0.324180	+	0.945995i	$0.605088\pi$
$380$		0			0
$381$	221.864	+	544.321i	0.582319	+	1.42866i
$382$		0			0
$383$			64.2130i			0.167658i	0.996480	+	0.0838290i	$0.0267149\pi$
$383$			64.2130i			0.167658i	−0.996480	+	0.0838290i	$0.973285\pi$
$384$		0			0
$385$	−98.3320			−0.255408
$386$		0			0
$387$	247.122	+	2.79801i	0.638559	+	0.00723000i
$388$		0			0
$389$	−273.321	+	273.321i	−0.702624	+	0.702624i	−0.964973	−	0.262349i	$-0.915503\pi$
$389$	−273.321	+	273.321i	−0.702624	+	0.702624i	0.262349	+	0.964973i	$0.415503\pi$
$390$		0			0
$391$			186.510i			0.477007i
$392$		0			0
$393$	−233.905	+	555.778i	−0.595179	+	1.41419i
$394$		0			0
$395$	109.352	−	109.352i	0.276842	−	0.276842i
$396$		0			0
$397$	−141.678	+	141.678i	−0.356873	+	0.356873i	−0.862659	−	0.505786i	$-0.831202\pi$
$397$	−141.678	+	141.678i	−0.356873	+	0.356873i	0.505786	+	0.862659i	$0.331202\pi$
$398$		0			0
$399$	−23.2470	+	55.2368i	−0.0582633	+	0.138438i
$400$		0			0
$401$		−	194.801i		−	0.485788i	−0.970053	−	0.242894i	$-0.921903\pi$
$401$		−	194.801i		−	0.485788i	0.970053	−	0.242894i	$-0.0780967\pi$
$402$		0			0
$403$	−59.4392	+	59.4392i	−0.147492	+	0.147492i
$404$		0			0
$405$	497.548	+	520.607i	1.22851	+	1.28545i
$406$		0			0
$407$	258.791			0.635849
$408$		0			0
$409$		−	420.826i		−	1.02891i	−0.857516	−	0.514457i	$-0.827993\pi$
$409$		−	420.826i		−	1.02891i	0.857516	−	0.514457i	$-0.172007\pi$
$410$		0			0
$411$	−57.1659	−	140.251i	−0.139090	−	0.341244i
$412$		0			0
$413$	46.0307	+	46.0307i	0.111454	+	0.111454i
$414$		0			0
$415$	944.737			2.27648
$416$		0			0
$417$	−282.620	+	671.526i	−0.677745	+	1.61038i
$418$		0			0
$419$	−186.421	−	186.421i	−0.444919	−	0.444919i	0.448742	−	0.893661i	$-0.351872\pi$
$419$	−186.421	−	186.421i	−0.444919	−	0.444919i	−0.893661	+	0.448742i	$0.851872\pi$
$420$		0			0
$421$	186.889	+	186.889i	0.443917	+	0.443917i	0.893326	−	0.449409i	$-0.148366\pi$
$421$	186.889	+	186.889i	0.443917	+	0.443917i	−0.449409	+	0.893326i	$0.648366\pi$
$422$		0			0
$423$	−471.532	−	482.332i	−1.11473	−	1.14026i
$424$		0			0
$425$	−620.706			−1.46049
$426$		0			0
$427$	64.5608	+	64.5608i	0.151196	+	0.151196i
$428$		0			0
$429$	−246.988	+	100.672i	−0.575731	+	0.234666i
$430$		0			0
$431$			128.395i			0.297901i	0.988845	+	0.148950i	$0.0475895\pi$
$431$			128.395i			0.297901i	−0.988845	+	0.148950i	$0.952411\pi$
$432$		0			0
$433$	684.737			1.58138			0.790690	−	0.612217i	$-0.209722\pi$
$433$	684.737			1.58138			0.790690	+	0.612217i	$0.209722\pi$
$434$		0			0
$435$	−152.912	−	375.155i	−0.351522	−	0.862426i
$436$		0			0
$437$	139.372	−	139.372i	0.318928	−	0.318928i
$438$		0			0
$439$		−	239.107i		−	0.544663i	−0.962203	−	0.272332i	$-0.912205\pi$
$439$		−	239.107i		−	0.544663i	0.962203	−	0.272332i	$-0.0877948\pi$
$440$		0			0
$441$	−297.914	+	291.243i	−0.675541	+	0.660415i
$442$		0			0
$443$	310.189	−	310.189i	0.700200	−	0.700200i	−0.264253	−	0.964453i	$-0.585125\pi$
$443$	310.189	−	310.189i	0.700200	−	0.700200i	0.964453	+	0.264253i	$0.0851255\pi$
$444$		0			0
$445$	−495.409	+	495.409i	−1.11328	+	1.11328i
$446$		0			0
$447$	−330.413	−	139.058i	−0.739179	−	0.311092i
$448$		0			0
$449$			545.902i			1.21582i	0.794007	+	0.607908i	$0.207991\pi$
$449$			545.902i			1.21582i	−0.794007	+	0.607908i	$0.792009\pi$
$450$		0			0
$451$	−9.41699	+	9.41699i	−0.0208803	+	0.0208803i
$452$		0			0
$453$	83.6297	−	34.0872i	0.184613	−	0.0752477i
$454$		0			0
$455$	193.559			0.425404
$456$		0			0
$457$			289.579i			0.633652i	0.948484	+	0.316826i	$0.102617\pi$
$457$			289.579i			0.633652i	−0.948484	+	0.316826i	$0.897383\pi$
$458$		0			0
$459$	−123.526	+	284.457i	−0.269121	+	0.619732i
$460$		0			0
$461$	−160.511	−	160.511i	−0.348180	−	0.348180i	0.511251	−	0.859431i	$-0.329182\pi$
$461$	−160.511	−	160.511i	−0.348180	−	0.348180i	−0.859431	+	0.511251i	$0.829182\pi$
$462$		0			0
$463$	197.573			0.426723			0.213361	−	0.976973i	$-0.431559\pi$
$463$	197.573			0.426723			0.213361	+	0.976973i	$0.431559\pi$
$464$		0			0
$465$	156.207	+	65.7413i	0.335928	+	0.141379i
$466$		0			0
$467$	52.7645	+	52.7645i	0.112986	+	0.112986i	0.761339	−	0.648353i	$-0.224542\pi$
$467$	52.7645	+	52.7645i	0.112986	+	0.112986i	−0.648353	+	0.761339i	$0.724542\pi$
$468$		0			0
$469$	113.041	+	113.041i	0.241025	+	0.241025i
$470$		0			0
$471$	708.826	+	298.318i	1.50494	+	0.633371i
$472$		0			0
$473$	184.545			0.390159
$474$		0			0
$475$	463.830	+	463.830i	0.976484	+	0.976484i
$476$		0			0
$477$	0.577371	−	50.9938i	0.00121042	−	0.106905i
$478$		0			0
$479$		−	175.985i		−	0.367401i	−0.982982	−	0.183700i	$-0.941192\pi$
$479$		−	175.985i		−	0.367401i	0.982982	−	0.183700i	$-0.0588076\pi$
$480$		0			0
$481$	−509.409			−1.05906
$482$		0			0
$483$	−74.2414	+	30.2606i	−0.153709	+	0.0626513i
$484$		0			0
$485$	244.130	−	244.130i	0.503362	−	0.503362i
$486$		0			0
$487$			965.217i			1.98196i	0.133991	+	0.990982i	$0.457221\pi$
$487$			965.217i			1.98196i	−0.133991	+	0.990982i	$0.542779\pi$
$488$		0			0
$489$	782.404	+	329.284i	1.60001	+	0.673382i
$490$		0			0
$491$	600.614	−	600.614i	1.22325	−	1.22325i	0.256775	−	0.966471i	$-0.417340\pi$
$491$	600.614	−	600.614i	1.22325	−	1.22325i	0.966471	−	0.256775i	$-0.0826598\pi$
$492$		0			0
$493$	123.365	−	123.365i	0.250233	−	0.250233i
$494$		0			0
$495$	375.911	+	384.521i	0.759416	+	0.776810i
$496$		0			0
$497$			66.9527i			0.134714i
$498$		0			0
$499$	−51.6092	+	51.6092i	−0.103425	+	0.103425i	−0.756926	−	0.653501i	$-0.773300\pi$
$499$	−51.6092	+	51.6092i	−0.103425	+	0.103425i	0.653501	+	0.756926i	$0.273300\pi$
$500$		0			0
$501$	−195.505	−	479.653i	−0.390230	−	0.957391i
$502$		0			0
$503$	−847.530			−1.68495			−0.842475	−	0.538735i	$-0.818902\pi$
$503$	−847.530			−1.68495			−0.842475	+	0.538735i	$0.818902\pi$
$504$		0			0
$505$			522.199i			1.03406i
$506$		0			0
$507$	16.6795	−	6.79851i	0.0328984	−	0.0134093i
$508$		0			0
$509$	128.457	+	128.457i	0.252372	+	0.252372i	0.821942	−	0.569570i	$-0.192890\pi$
$509$	128.457	+	128.457i	0.252372	+	0.252372i	−0.569570	+	0.821942i	$0.692890\pi$
$510$		0			0
$511$	97.1660			0.190149
$512$		0			0
$513$	304.871	−	120.257i	0.594290	−	0.234420i
$514$		0			0
$515$	−621.182	−	621.182i	−1.20618	−	1.20618i
$516$		0			0
$517$	−356.162	−	356.162i	−0.688901	−	0.688901i
$518$		0			0
$519$	67.1981	−	159.668i	0.129476	−	0.307645i
$520$		0			0
$521$	676.366			1.29821			0.649103	−	0.760700i	$-0.275144\pi$
$521$	676.366			1.29821			0.649103	+	0.760700i	$0.275144\pi$
$522$		0			0
$523$	−600.494	−	600.494i	−1.14817	−	1.14817i	−0.986912	−	0.161260i	$-0.948444\pi$
$523$	−600.494	−	600.494i	−1.14817	−	1.14817i	−0.161260	−	0.986912i	$-0.551556\pi$
$524$		0			0
$525$	−100.707	−	247.076i	−0.191824	−	0.470621i
$526$		0			0
$527$			72.9845i			0.138491i
$528$		0			0
$529$	−265.324			−0.501558
$530$		0			0
$531$	4.03042	−	355.970i	0.00759025	−	0.670376i
$532$		0			0
$533$	18.5366	−	18.5366i	0.0347779	−	0.0347779i
$534$		0			0
$535$		−	1242.82i		−	2.32302i
$536$		0			0
$537$	251.162	−	596.782i	0.467714	−	1.11133i
$538$		0			0
$539$	−219.985	+	219.985i	−0.408135	+	0.408135i
$540$		0			0
$541$	−43.4797	+	43.4797i	−0.0803692	+	0.0803692i	−0.746149	−	0.665779i	$-0.768099\pi$
$541$	−43.4797	+	43.4797i	−0.0803692	+	0.0803692i	0.665779	+	0.746149i	$0.268099\pi$
$542$		0			0
$543$	274.184	−	651.484i	0.504943	−	1.19979i
$544$		0			0
$545$		−	865.682i		−	1.58841i
$546$		0			0
$547$	−125.498	+	125.498i	−0.229430	+	0.229430i	−0.812454	−	0.583025i	$-0.801869\pi$
$547$	−125.498	+	125.498i	−0.229430	+	0.229430i	0.583025	+	0.812454i	$0.301869\pi$
$548$		0			0
$549$	5.65291	−	499.269i	0.0102967	−	0.909414i
$550$		0			0
$551$	−184.371			−0.334612
$552$		0			0
$553$			28.6275i			0.0517676i
$554$		0			0
$555$	387.655	+	951.074i	0.698477	+	1.71365i
$556$		0			0
$557$	184.272	+	184.272i	0.330829	+	0.330829i	0.852901	−	0.522072i	$-0.174841\pi$
$557$	184.272	+	184.272i	0.330829	+	0.330829i	−0.522072	+	0.852901i	$0.674841\pi$
$558$		0			0
$559$	−363.263			−0.649844
$560$		0			0
$561$	−89.8301	+	213.443i	−0.160125	+	0.380469i
$562$		0			0
$563$	523.489	+	523.489i	0.929820	+	0.929820i	0.997694	−	0.0678736i	$-0.0216215\pi$
$563$	523.489	+	523.489i	0.929820	+	0.929820i	−0.0678736	+	0.997694i	$0.521621\pi$
$564$		0			0
$565$	52.2510	+	52.2510i	0.0924796	+	0.0924796i
$566$		0			0
$567$	−133.272	−	3.01829i	−0.235047	−	0.00532326i
$568$		0			0
$569$	−52.6214			−0.0924805			−0.0462403	−	0.998930i	$-0.514724\pi$
$569$	−52.6214			−0.0924805			−0.0462403	+	0.998930i	$0.514724\pi$
$570$		0			0
$571$	−114.561	−	114.561i	−0.200632	−	0.200632i	0.599639	−	0.800271i	$-0.295311\pi$
$571$	−114.561	−	114.561i	−0.200632	−	0.200632i	−0.800271	+	0.599639i	$0.795311\pi$
$572$		0			0
$573$	39.8473	−	16.2416i	0.0695415	−	0.0283449i
$574$		0			0
$575$			877.515i			1.52611i
$576$		0			0
$577$	496.442			0.860384			0.430192	−	0.902737i	$-0.358446\pi$
$577$	496.442			0.860384			0.430192	+	0.902737i	$0.358446\pi$
$578$		0			0
$579$	234.817	+	576.100i	0.405555	+	0.994992i
$580$		0			0
$581$	−123.662	+	123.662i	−0.212843	+	0.212843i
$582$		0			0
$583$		−	38.0810i		−	0.0653191i
$584$		0			0
$585$	−739.951	−	756.899i	−1.26487	−	1.29384i
$586$		0			0
$587$	115.260	−	115.260i	0.196354	−	0.196354i	−0.602081	−	0.798435i	$-0.705662\pi$
$587$	115.260	−	115.260i	0.196354	−	0.196354i	0.798435	+	0.602081i	$0.205662\pi$
$588$		0			0
$589$	54.5385	−	54.5385i	0.0925952	−	0.0925952i
$590$		0			0
$591$	380.332	+	160.067i	0.643540	+	0.270841i
$592$		0			0
$593$		−	227.756i		−	0.384074i	−0.981388	−	0.192037i	$-0.938491\pi$
$593$		−	227.756i		−	0.384074i	0.981388	−	0.192037i	$-0.0615094\pi$
$594$		0			0
$595$	118.834	−	118.834i	0.199721	−	0.199721i
$596$		0			0
$597$	−229.935	+	93.7209i	−0.385151	+	0.156986i
$598$		0			0
$599$	760.308			1.26930			0.634648	−	0.772802i	$-0.281145\pi$
$599$	760.308			1.26930			0.634648	+	0.772802i	$0.281145\pi$
$600$		0			0
$601$			85.7856i			0.142738i	0.997450	+	0.0713690i	$0.0227368\pi$
$601$			85.7856i			0.142738i	−0.997450	+	0.0713690i	$0.977263\pi$
$602$		0			0
$603$	9.89776	−	874.177i	0.0164142	−	1.44971i
$604$		0			0
$605$	−476.731	−	476.731i	−0.787986	−	0.787986i
$606$		0			0
$607$	−685.217			−1.12886			−0.564429	−	0.825482i	$-0.690904\pi$
$607$	−685.217			−1.12886			−0.564429	+	0.825482i	$0.690904\pi$
$608$		0			0
$609$	69.1216	+	29.0906i	0.113500	+	0.0477678i
$610$		0			0
$611$	701.077	+	701.077i	1.14743	+	1.14743i
$612$		0			0
$613$	544.727	+	544.727i	0.888624	+	0.888624i	0.994391	−	0.105767i	$-0.0337296\pi$
$613$	544.727	+	544.727i	0.888624	+	0.888624i	−0.105767	+	0.994391i	$0.533730\pi$
$614$		0			0
$615$	−48.7143	−	20.5020i	−0.0792102	−	0.0333365i
$616$		0			0
$617$	−383.577			−0.621681			−0.310840	−	0.950462i	$-0.600610\pi$
$617$	−383.577			−0.621681			−0.310840	+	0.950462i	$0.600610\pi$
$618$		0			0
$619$	81.7634	+	81.7634i	0.132089	+	0.132089i	0.770060	−	0.637971i	$-0.220226\pi$
$619$	81.7634	+	81.7634i	0.132089	+	0.132089i	−0.637971	+	0.770060i	$0.720226\pi$
$620$		0			0
$621$	402.148	+	174.634i	0.647581	+	0.281214i
$622$		0			0
$623$		−	129.694i		−	0.208176i
$624$		0			0
$625$	−944.365			−1.51098
$626$		0			0
$627$	226.625	−	92.3715i	0.361443	−	0.147323i
$628$		0			0
$629$	−312.748	+	312.748i	−0.497214	+	0.497214i
$630$		0			0
$631$			944.242i			1.49642i	0.663461	+	0.748211i	$0.269087\pi$
$631$			944.242i			1.49642i	−0.663461	+	0.748211i	$0.730913\pi$
$632$		0			0
$633$	−788.523	−	331.859i	−1.24569	−	0.524263i
$634$		0			0
$635$	−1231.74	+	1231.74i	−1.93974	+	1.93974i
$636$		0			0
$637$	433.022	−	433.022i	0.679784	−	0.679784i
$638$		0			0
$639$	261.814	−	255.952i	0.409725	−	0.400551i
$640$		0			0
$641$			1102.48i			1.71994i	0.510344	+	0.859970i	$0.329518\pi$
$641$			1102.48i			1.71994i	−0.510344	+	0.859970i	$0.670482\pi$
$642$		0			0
$643$	794.664	−	794.664i	1.23587	−	1.23587i	0.274195	−	0.961674i	$-0.411588\pi$
$643$	794.664	−	794.664i	1.23587	−	1.23587i	0.961674	−	0.274195i	$-0.0884115\pi$
$644$		0			0
$645$	276.439	+	678.217i	0.428588	+	1.05150i
$646$		0			0
$647$	768.446			1.18771			0.593853	−	0.804574i	$-0.297606\pi$
$647$	768.446			1.18771			0.593853	+	0.804574i	$0.297606\pi$
$648$		0			0
$649$		−	265.830i		−	0.409599i
$650$		0			0
$651$	−29.0519	+	11.8415i	−0.0446266	+	0.0181897i
$652$		0			0
$653$	829.478	+	829.478i	1.27026	+	1.27026i	0.945953	+	0.324305i	$0.105130\pi$
$653$	829.478	+	829.478i	1.27026	+	1.27026i	0.324305	+	0.945953i	$0.394870\pi$
$654$		0			0
$655$	−1786.96			−2.72819
$656$		0			0
$657$	−371.454	−	379.961i	−0.565378	−	0.578328i
$658$		0			0
$659$	−653.956	−	653.956i	−0.992346	−	0.992346i	0.00762509	−	0.999971i	$-0.497573\pi$
$659$	−653.956	−	653.956i	−0.992346	−	0.992346i	−0.999971	+	0.00762509i	$0.997573\pi$
$660$		0			0
$661$	−734.342	−	734.342i	−1.11096	−	1.11096i	−0.993021	−	0.117936i	$-0.962372\pi$
$661$	−734.342	−	734.342i	−1.11096	−	1.11096i	−0.117936	−	0.993021i	$-0.537628\pi$
$662$		0			0
$663$	176.823	−	420.146i	0.266702	−	0.633705i
$664$		0			0
$665$	−177.600			−0.267068
$666$		0			0
$667$	−174.405	−	174.405i	−0.261477	−	0.261477i
$668$		0			0
$669$	−264.132	−	648.022i	−0.394816	−	0.968643i
$670$		0			0
$671$		−	372.842i		−	0.555652i
$672$		0			0
$673$	514.259			0.764129			0.382065	−	0.924136i	$-0.375213\pi$
$673$	514.259			0.764129			0.382065	+	0.924136i	$0.375213\pi$
$674$		0			0
$675$	−581.183	+	1338.35i	−0.861012	+	1.98274i
$676$		0			0
$677$	−662.519	+	662.519i	−0.978610	+	0.978610i	−0.999776	−	0.0211661i	$-0.993262\pi$
$677$	−662.519	+	662.519i	−0.978610	+	0.978610i	0.0211661	+	0.999776i	$0.493262\pi$
$678$		0			0
$679$			63.9111i			0.0941253i
$680$		0			0
$681$	155.293	−	368.988i	0.228037	−	0.541833i
$682$		0			0
$683$	−280.446	+	280.446i	−0.410608	+	0.410608i	−0.881950	−	0.471342i	$-0.843770\pi$
$683$	−280.446	+	280.446i	−0.410608	+	0.410608i	0.471342	+	0.881950i	$0.343770\pi$
$684$		0			0
$685$	317.373	−	317.373i	0.463318	−	0.463318i
$686$		0			0
$687$	−227.217	+	539.885i	−0.330738	+	0.785859i
$688$		0			0
$689$			74.9595i			0.108795i
$690$		0			0
$691$	−631.830	+	631.830i	−0.914371	+	0.914371i	−0.996612	−	0.0822418i	$-0.973792\pi$
$691$	−631.830	+	631.830i	−0.914371	+	0.914371i	0.0822418	+	0.996612i	$0.473792\pi$
$692$		0			0
$693$	−99.5370	−	1.12700i	−0.143632	−	0.00162626i
$694$		0			0
$695$	−2159.13			−3.10666
$696$		0			0
$697$		−	22.7608i		−	0.0326554i
$698$		0			0
$699$	−449.309	−	1102.34i	−0.642788	−	1.57702i
$700$		0			0
$701$	−160.480	−	160.480i	−0.228930	−	0.228930i	0.583315	−	0.812246i	$-0.301755\pi$
$701$	−160.480	−	160.480i	−0.228930	−	0.228930i	−0.812246	+	0.583315i	$0.801755\pi$
$702$		0			0
$703$	467.409			0.664878
$704$		0			0
$705$	775.409	−	1842.43i	1.09987	−	2.61338i
$706$		0			0
$707$	−68.3534	−	68.3534i	−0.0966809	−	0.0966809i
$708$		0			0
$709$	410.261	+	410.261i	0.578648	+	0.578648i	0.934531	−	0.355883i	$-0.115820\pi$
$709$	410.261	+	410.261i	0.578648	+	0.578648i	−0.355883	+	0.934531i	$0.615820\pi$
$710$		0			0
$711$	111.946	−	109.439i	0.157448	−	0.153923i
$712$		0			0
$713$	103.181			0.144714
$714$		0			0
$715$	−558.907	−	558.907i	−0.781688	−	0.781688i
$716$		0			0
$717$	−791.236	+	322.505i	−1.10354	+	0.449798i
$718$		0			0
$719$		−	1069.18i		−	1.48704i	−0.668716	−	0.743518i	$-0.733156\pi$
$719$		−	1069.18i		−	1.48704i	0.668716	−	0.743518i	$-0.266844\pi$
$720$		0			0
$721$	162.620			0.225547
$722$		0			0
$723$	−301.804	−	740.447i	−0.417433	−	1.02413i
$724$		0			0
$725$	580.422	−	580.422i	0.800582	−	0.800582i
$726$		0			0
$727$			148.864i			0.204765i	0.994745	+	0.102382i	$0.0326466\pi$
$727$			148.864i			0.204765i	−0.994745	+	0.102382i	$0.967353\pi$
$728$		0			0
$729$	497.678	+	532.689i	0.682686	+	0.730712i
$730$		0			0
$731$	−223.022	+	223.022i	−0.305092	+	0.305092i
$732$		0			0
$733$	690.136	−	690.136i	0.941522	−	0.941522i	−0.0568598	−	0.998382i	$-0.518109\pi$
$733$	690.136	−	690.136i	0.941522	−	0.941522i	0.998382	+	0.0568598i	$0.0181088\pi$
$734$		0			0
$735$	−1137.98	−	478.934i	−1.54828	−	0.651610i
$736$		0			0
$737$		−	652.816i		−	0.885775i
$738$		0			0
$739$	−535.593	+	535.593i	−0.724754	+	0.724754i	−0.969570	−	0.244815i	$-0.921273\pi$
$739$	−535.593	+	535.593i	−0.724754	+	0.724754i	0.244815	+	0.969570i	$0.421273\pi$
$740$		0			0
$741$	−446.093	+	181.826i	−0.602014	+	0.245379i
$742$		0			0
$743$	−20.5116			−0.0276065			−0.0138032	−	0.999905i	$-0.504394\pi$
$743$	−20.5116			−0.0276065			−0.0138032	+	0.999905i	$0.504394\pi$
$744$		0			0
$745$		−	1062.36i		−	1.42599i
$746$		0			0
$747$	956.315	+	10.8278i	1.28021	+	0.0144950i
$748$		0			0
$749$	162.679	+	162.679i	0.217195	+	0.217195i
$750$		0			0
$751$	15.8000			0.0210385			0.0105193	−	0.999945i	$-0.496652\pi$
$751$	15.8000			0.0210385			0.0105193	+	0.999945i	$0.496652\pi$
$752$		0			0
$753$	601.992	+	253.355i	0.799458	+	0.336461i
$754$		0			0
$755$	189.245	+	189.245i	0.250655	+	0.250655i
$756$		0			0
$757$	−810.497	−	810.497i	−1.07067	−	1.07067i	−0.997305	−	0.0733640i	$-0.976627\pi$
$757$	−810.497	−	810.497i	−1.07067	−	1.07067i	−0.0733640	−	0.997305i	$-0.523373\pi$
$758$		0			0
$759$	301.753	+	126.996i	0.397566	+	0.167320i
$760$		0			0
$761$	212.194			0.278836			0.139418	−	0.990234i	$-0.455477\pi$
$761$	212.194			0.278836			0.139418	+	0.990234i	$0.455477\pi$
$762$		0			0
$763$	113.314	+	113.314i	0.148511	+	0.148511i
$764$		0			0
$765$	−918.980	−	10.4050i	−1.20128	−	0.0136014i
$766$		0			0
$767$			523.266i			0.682224i
$768$		0			0
$769$	−883.681			−1.14913			−0.574565	−	0.818459i	$-0.694829\pi$
$769$	−883.681			−1.14913			−0.574565	+	0.818459i	$0.694829\pi$
$770$		0			0
$771$	−667.207	+	271.951i	−0.865378	+	0.352725i
$772$		0			0
$773$	515.805	−	515.805i	0.667277	−	0.667277i	−0.289808	−	0.957085i	$-0.593591\pi$
$773$	515.805	−	515.805i	0.667277	−	0.667277i	0.957085	+	0.289808i	$0.0935914\pi$
$774$		0			0
$775$			343.387i			0.443080i
$776$		0			0
$777$	−175.233	−	73.7490i	−0.225526	−	0.0949151i
$778$		0			0
$779$	−17.0083	+	17.0083i	−0.0218335	+	0.0218335i
$780$		0			0
$781$	193.328	−	193.328i	0.247539	−	0.247539i
$782$		0			0
$783$	−150.486	−	381.505i	−0.192192	−	0.487235i
$784$		0			0
$785$			2279.05i			2.90325i
$786$		0			0
$787$	279.150	−	279.150i	0.354702	−	0.354702i	−0.507154	−	0.861856i	$-0.669302\pi$
$787$	279.150	−	279.150i	0.354702	−	0.354702i	0.861856	+	0.507154i	$0.169302\pi$
$788$		0			0
$789$	−159.329	−	390.898i	−0.201938	−	0.495435i
$790$		0			0
$791$	−13.6788			−0.0172931
$792$		0			0
$793$			733.911i			0.925487i

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.i (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.13234 + 2.77809i) q^{3} +(-6.28651 + 6.28651i) q^{5} -1.64575i q^{7} +(-6.43560 + 6.29150i) q^{9} +O(q^{10})\)	\(q+(1.13234 + 2.77809i) q^{3} +(-6.28651 + 6.28651i) q^{5} -1.64575i q^{7} +(-6.43560 + 6.29150i) q^{9} +(-4.75216 + 4.75216i) q^{11} +(9.35425 - 9.35425i) q^{13} +(-24.5830 - 10.3460i) q^{15} -11.4859i q^{17} +(-8.58301 + 8.58301i) q^{19} +(4.57205 - 1.86355i) q^{21} -16.2381 q^{23} -54.0405i q^{25} +(-24.7657 - 10.7546i) q^{27} +(10.7405 + 10.7405i) q^{29} -6.35425 q^{31} +(-18.5830 - 7.82087i) q^{33} +(10.3460 + 10.3460i) q^{35} +(-27.2288 - 27.2288i) q^{37} +(36.5792 + 15.3948i) q^{39} +1.98162 q^{41} +(-19.4170 - 19.4170i) q^{43} +(0.905893 - 80.0091i) q^{45} +74.9474i q^{47} +46.2915 q^{49} +(31.9090 - 13.0060i) q^{51} +(-4.00671 + 4.00671i) q^{53} -59.7490i q^{55} +(-33.5633 - 14.1255i) q^{57} +(-27.9694 + 27.9694i) q^{59} +(-39.2288 + 39.2288i) q^{61} +(10.3542 + 10.5914i) q^{63} +117.611i q^{65} +(-68.6863 + 68.6863i) q^{67} +(-18.3871 - 45.1110i) q^{69} -40.6822 q^{71} +59.0405i q^{73} +(150.130 - 61.1923i) q^{75} +(7.82087 + 7.82087i) q^{77} -17.3948 q^{79} +(1.83399 - 80.9792i) q^{81} +(-75.1400 - 75.1400i) q^{83} +(72.2065 + 72.2065i) q^{85} +(-17.6762 + 42.0000i) q^{87} +78.8051 q^{89} +(-15.3948 - 15.3948i) q^{91} +(-7.19518 - 17.6527i) q^{93} -107.914i q^{95} -38.8340 q^{97} +(0.684791 - 60.4812i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3}+O(q^{10}) \) 8 * q + 4 * q^3	\( 8 q + 4 q^{3} + 96 q^{13} - 112 q^{15} + 16 q^{19} + 32 q^{21} - 68 q^{27} - 72 q^{31} - 64 q^{33} - 112 q^{37} - 240 q^{43} + 112 q^{45} + 328 q^{49} - 32 q^{51} - 208 q^{61} + 104 q^{63} - 232 q^{67} + 324 q^{75} + 136 q^{79} + 184 q^{81} + 112 q^{85} + 152 q^{91} - 64 q^{93} - 480 q^{97} + 160 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 + 96 * q^13 - 112 * q^15 + 16 * q^19 + 32 * q^21 - 68 * q^27 - 72 * q^31 - 64 * q^33 - 112 * q^37 - 240 * q^43 + 112 * q^45 + 328 * q^49 - 32 * q^51 - 208 * q^61 + 104 * q^63 - 232 * q^67 + 324 * q^75 + 136 * q^79 + 184 * q^81 + 112 * q^85 + 152 * q^91 - 64 * q^93 - 480 * q^97 + 160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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