LMFDB - Embedded newform 448.3.s.d.129.2

Newspace parameters

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.s (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			129.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	448.129
Dual form			448.3.s.d.257.2

$q$-expansion

$f(q)$	$=$	$q+(3.62132 + 2.09077i) q^{3} +(-2.74264 + 1.58346i) q^{5} +(-2.24264 + 6.63103i) q^{7} +(4.24264 + 7.34847i) q^{9} +O(q^{10})$	\(q+(3.62132 + 2.09077i) q^{3} +(-2.74264 + 1.58346i) q^{5} +(-2.24264 + 6.63103i) q^{7} +(4.24264 + 7.34847i) q^{9} +(-6.62132 + 11.4685i) q^{11} -5.49333i q^{13} -13.2426 q^{15} +(-11.7426 - 6.77962i) q^{17} +(0.621320 - 0.358719i) q^{19} +(-21.9853 + 19.3242i) q^{21} +(1.13604 + 1.96768i) q^{23} +(-7.48528 + 12.9649i) q^{25} -2.15232i q^{27} -20.4853 q^{29} +(21.3198 + 12.3090i) q^{31} +(-47.9558 + 27.6873i) q^{33} +(-4.34924 - 21.7377i) q^{35} +(32.4706 + 56.2407i) q^{37} +(11.4853 - 19.8931i) q^{39} +21.0308i q^{41} -6.48528 q^{43} +(-23.2721 - 13.4361i) q^{45} +(41.3787 - 23.8900i) q^{47} +(-38.9411 - 29.7420i) q^{49} +(-28.3492 - 49.1023i) q^{51} +(11.0147 - 19.0781i) q^{53} -41.9385i q^{55} +3.00000 q^{57} +(72.5330 + 41.8770i) q^{59} +(-57.3823 + 33.1297i) q^{61} +(-58.2426 + 11.6531i) q^{63} +(8.69848 + 15.0662i) q^{65} +(46.3198 - 80.2283i) q^{67} +9.50079i q^{69} -48.4264 q^{71} +(113.441 + 65.4953i) q^{73} +(-54.2132 + 31.3000i) q^{75} +(-61.1985 - 69.6258i) q^{77} +(38.1066 + 66.0026i) q^{79} +(42.6838 - 73.9305i) q^{81} +107.981i q^{83} +42.9411 q^{85} +(-74.1838 - 42.8300i) q^{87} +(-145.412 + 83.9535i) q^{89} +(36.4264 + 12.3196i) q^{91} +(51.4706 + 89.1496i) q^{93} +(-1.13604 + 1.96768i) q^{95} -25.5816i q^{97} -112.368 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{3} + 6 q^{5} + 8 q^{7}+O(q^{10}) $ 4 * q + 6 * q^3 + 6 * q^5 + 8 * q^7	$ 4 q + 6 q^{3} + 6 q^{5} + 8 q^{7} - 18 q^{11} - 36 q^{15} - 30 q^{17} - 6 q^{19} - 54 q^{21} + 30 q^{23} + 4 q^{25} - 48 q^{29} - 42 q^{31} - 90 q^{33} + 42 q^{35} + 62 q^{37} + 12 q^{39} + 8 q^{43} - 144 q^{45} + 174 q^{47} - 20 q^{49} - 54 q^{51} + 78 q^{53} + 12 q^{57} + 78 q^{59} + 42 q^{61} - 216 q^{63} - 84 q^{65} + 58 q^{67} - 24 q^{71} + 318 q^{73} - 132 q^{75} - 126 q^{77} + 110 q^{79} + 18 q^{81} + 36 q^{85} - 144 q^{87} - 378 q^{89} - 24 q^{91} + 138 q^{93} - 30 q^{95} - 144 q^{99}+O(q^{100}) $ 4 * q + 6 * q^3 + 6 * q^5 + 8 * q^7 - 18 * q^11 - 36 * q^15 - 30 * q^17 - 6 * q^19 - 54 * q^21 + 30 * q^23 + 4 * q^25 - 48 * q^29 - 42 * q^31 - 90 * q^33 + 42 * q^35 + 62 * q^37 + 12 * q^39 + 8 * q^43 - 144 * q^45 + 174 * q^47 - 20 * q^49 - 54 * q^51 + 78 * q^53 + 12 * q^57 + 78 * q^59 + 42 * q^61 - 216 * q^63 - 84 * q^65 + 58 * q^67 - 24 * q^71 + 318 * q^73 - 132 * q^75 - 126 * q^77 + 110 * q^79 + 18 * q^81 + 36 * q^85 - 144 * q^87 - 378 * q^89 - 24 * q^91 + 138 * q^93 - 30 * q^95 - 144 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$1$

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
$3$	3.62132	+	2.09077i	1.20711	+	0.696923i	0.962126	−	0.272605i	$-0.0878851\pi$
$3$	3.62132	+	2.09077i	1.20711	+	0.696923i	0.244981	+	0.969528i	$0.421218\pi$
$4$		0			0
$5$	−2.74264	+	1.58346i	−0.548528	+	0.316693i	−0.748528	−	0.663103i	$-0.769239\pi$
$5$	−2.74264	+	1.58346i	−0.548528	+	0.316693i	0.200000	+	0.979796i	$0.435906\pi$
$6$		0			0
$7$	−2.24264	+	6.63103i	−0.320377	+	0.947290i
$7$	−2.24264	+	6.63103i	−0.320377	+	0.947290i
$8$		0			0
$9$	4.24264	+	7.34847i	0.471405	+	0.816497i
$10$		0			0
$11$	−6.62132	+	11.4685i	−0.601938	+	1.04259i	0.390589	+	0.920565i	$0.372271\pi$
$11$	−6.62132	+	11.4685i	−0.601938	+	1.04259i	−0.992527	+	0.122022i	$0.961062\pi$
$12$		0			0
$13$		−	5.49333i		−	0.422563i	−0.977425	−	0.211282i	$-0.932236\pi$
$13$		−	5.49333i		−	0.422563i	0.977425	−	0.211282i	$-0.0677638\pi$
$14$		0			0
$15$	−13.2426			−0.882843
$16$		0			0
$17$	−11.7426	−	6.77962i	−0.690744	−	0.398801i	0.113147	−	0.993578i	$-0.463907\pi$
$17$	−11.7426	−	6.77962i	−0.690744	−	0.398801i	−0.803891	+	0.594777i	$0.797240\pi$
$18$		0			0
$19$	0.621320	−	0.358719i	0.0327011	−	0.0188800i	−0.483560	−	0.875311i	$-0.660657\pi$
$19$	0.621320	−	0.358719i	0.0327011	−	0.0188800i	0.516261	+	0.856431i	$0.327323\pi$
$20$		0			0
$21$	−21.9853	+	19.3242i	−1.04692	+	0.920202i
$22$		0			0
$23$	1.13604	+	1.96768i	0.0493930	+	0.0855512i	0.889665	−	0.456614i	$-0.150938\pi$
$23$	1.13604	+	1.96768i	0.0493930	+	0.0855512i	−0.840272	+	0.542165i	$0.817605\pi$
$24$		0			0
$25$	−7.48528	+	12.9649i	−0.299411	+	0.518596i
$26$		0			0
$27$		−	2.15232i		−	0.0797154i
$28$		0			0
$29$	−20.4853			−0.706389			−0.353195	−	0.935550i	$-0.614905\pi$
$29$	−20.4853			−0.706389			−0.353195	+	0.935550i	$0.614905\pi$
$30$		0			0
$31$	21.3198	+	12.3090i	0.687736	+	0.397064i	0.802763	−	0.596298i	$-0.203362\pi$
$31$	21.3198	+	12.3090i	0.687736	+	0.397064i	−0.115028	+	0.993362i	$0.536696\pi$
$32$		0			0
$33$	−47.9558	+	27.6873i	−1.45321	+	0.839010i
$34$		0			0
$35$	−4.34924	−	21.7377i	−0.124264	−	0.621076i
$36$		0			0
$37$	32.4706	+	56.2407i	0.877583	+	1.52002i	0.853986	+	0.520296i	$0.174179\pi$
$37$	32.4706	+	56.2407i	0.877583	+	1.52002i	0.0235970	+	0.999722i	$0.492488\pi$
$38$		0			0
$39$	11.4853	−	19.8931i	0.294494	−	0.510079i
$40$		0			0
$41$			21.0308i			0.512946i	0.966551	+	0.256473i	$0.0825605\pi$
$41$			21.0308i			0.512946i	−0.966551	+	0.256473i	$0.917439\pi$
$42$		0			0
$43$	−6.48528			−0.150820			−0.0754102	−	0.997153i	$-0.524027\pi$
$43$	−6.48528			−0.150820			−0.0754102	+	0.997153i	$0.524027\pi$
$44$		0			0
$45$	−23.2721	−	13.4361i	−0.517157	−	0.298581i
$46$		0			0
$47$	41.3787	−	23.8900i	0.880397	−	0.508298i	0.00960801	−	0.999954i	$-0.496942\pi$
$47$	41.3787	−	23.8900i	0.880397	−	0.508298i	0.870789	+	0.491656i	$0.163608\pi$
$48$		0			0
$49$	−38.9411	−	29.7420i	−0.794717	−	0.606980i
$50$		0			0
$51$	−28.3492	−	49.1023i	−0.555867	−	0.962791i
$52$		0			0
$53$	11.0147	−	19.0781i	0.207825	−	0.359963i	−0.743204	−	0.669065i	$-0.766695\pi$
$53$	11.0147	−	19.0781i	0.207825	−	0.359963i	0.951029	+	0.309101i	$0.100028\pi$
$54$		0			0
$55$		−	41.9385i		−	0.762518i
$56$		0			0
$57$	3.00000			0.0526316
$58$		0			0
$59$	72.5330	+	41.8770i	1.22937	+	0.709779i	0.966899	−	0.255160i	$-0.0821282\pi$
$59$	72.5330	+	41.8770i	1.22937	+	0.709779i	0.262474	+	0.964939i	$0.415462\pi$
$60$		0			0
$61$	−57.3823	+	33.1297i	−0.940693	+	0.543109i	−0.890177	−	0.455614i	$-0.849420\pi$
$61$	−57.3823	+	33.1297i	−0.940693	+	0.543109i	−0.0505153	+	0.998723i	$0.516086\pi$
$62$		0			0
$63$	−58.2426	+	11.6531i	−0.924486	+	0.184970i
$64$		0			0
$65$	8.69848	+	15.0662i	0.133823	+	0.231788i
$66$		0			0
$67$	46.3198	−	80.2283i	0.691340	−	1.19744i	−0.280058	−	0.959983i	$-0.590354\pi$
$67$	46.3198	−	80.2283i	0.691340	−	1.19744i	0.971399	−	0.237454i	$-0.0763128\pi$
$68$		0			0
$69$			9.50079i			0.137693i
$70$		0			0
$71$	−48.4264			−0.682062			−0.341031	−	0.940052i	$-0.610776\pi$
$71$	−48.4264			−0.682062			−0.341031	+	0.940052i	$0.610776\pi$
$72$		0			0
$73$	113.441	+	65.4953i	1.55399	+	0.897195i	0.997811	+	0.0661316i	$0.0210657\pi$
$73$	113.441	+	65.4953i	1.55399	+	0.897195i	0.556177	+	0.831064i	$0.312268\pi$
$74$		0			0
$75$	−54.2132	+	31.3000i	−0.722843	+	0.417333i
$76$		0			0
$77$	−61.1985	−	69.6258i	−0.794786	−	0.904231i
$78$		0			0
$79$	38.1066	+	66.0026i	0.482362	+	0.835476i	0.999795	−	0.0202482i	$-0.00644564\pi$
$79$	38.1066	+	66.0026i	0.482362	+	0.835476i	−0.517433	+	0.855724i	$0.673112\pi$
$80$		0			0
$81$	42.6838	−	73.9305i	0.526960	−	0.912722i
$82$		0			0
$83$			107.981i			1.30098i	0.759514	+	0.650491i	$0.225437\pi$
$83$			107.981i			1.30098i	−0.759514	+	0.650491i	$0.774563\pi$
$84$		0			0
$85$	42.9411			0.505190
$86$		0			0
$87$	−74.1838	−	42.8300i	−0.852687	−	0.492299i
$88$		0			0
$89$	−145.412	+	83.9535i	−1.63384	+	0.943297i	−0.650945	+	0.759125i	$0.725627\pi$
$89$	−145.412	+	83.9535i	−1.63384	+	0.943297i	−0.982894	+	0.184173i	$0.941039\pi$
$90$		0			0
$91$	36.4264	+	12.3196i	0.400290	+	0.135380i
$92$		0			0
$93$	51.4706	+	89.1496i	0.553447	+	0.958598i
$94$		0			0
$95$	−1.13604	+	1.96768i	−0.0119583	+	0.0207124i
$96$		0			0
$97$		−	25.5816i		−	0.263728i	−0.991268	−	0.131864i	$-0.957904\pi$
$97$		−	25.5816i		−	0.263728i	0.991268	−	0.131864i	$-0.0420962\pi$
$98$		0			0
$99$	−112.368			−1.13503
$100$		0			0
$101$	−24.6838	−	14.2512i	−0.244394	−	0.141101i	0.372801	−	0.927911i	$-0.378397\pi$
$101$	−24.6838	−	14.2512i	−0.244394	−	0.141101i	−0.617194	+	0.786811i	$0.711731\pi$
$102$		0			0
$103$	48.9228	−	28.2456i	0.474979	−	0.274229i	−0.243343	−	0.969940i	$-0.578244\pi$
$103$	48.9228	−	28.2456i	0.474979	−	0.274229i	0.718322	+	0.695711i	$0.244911\pi$
$104$		0			0
$105$	29.6985	−	87.8124i	0.282843	−	0.836308i
$106$		0			0
$107$	23.8051	+	41.2316i	0.222477	+	0.385342i	0.955560	−	0.294798i	$-0.0952523\pi$
$107$	23.8051	+	41.2316i	0.222477	+	0.385342i	−0.733082	+	0.680140i	$0.761919\pi$
$108$		0			0
$109$	37.6543	−	65.2192i	0.345453	−	0.598341i	−0.639983	−	0.768389i	$-0.721059\pi$
$109$	37.6543	−	65.2192i	0.345453	−	0.598341i	0.985436	+	0.170047i	$0.0543921\pi$
$110$		0			0
$111$			271.554i			2.44643i
$112$		0			0
$113$	85.4558			0.756246			0.378123	−	0.925755i	$-0.376570\pi$
$113$	85.4558			0.756246			0.378123	+	0.925755i	$0.376570\pi$
$114$		0			0
$115$	−6.23149	−	3.59775i	−0.0541869	−	0.0312848i
$116$		0			0
$117$	40.3675	−	23.3062i	0.345022	−	0.199198i
$118$		0			0
$119$	71.2904	−	62.6616i	0.599079	−	0.526568i
$120$		0			0
$121$	−27.1838	−	47.0837i	−0.224659	−	0.389121i
$122$		0			0
$123$	−43.9706	+	76.1592i	−0.357484	+	0.619181i
$124$		0			0
$125$		−	126.584i		−	1.01267i
$126$		0			0
$127$	−60.6619			−0.477653			−0.238826	−	0.971062i	$-0.576763\pi$
$127$	−60.6619			−0.477653			−0.238826	+	0.971062i	$0.576763\pi$
$128$		0			0
$129$	−23.4853	−	13.5592i	−0.182056	−	0.105110i
$130$		0			0
$131$	115.136	−	66.4738i	0.878901	−	0.507434i	0.00860515	−	0.999963i	$-0.497261\pi$
$131$	115.136	−	66.4738i	0.878901	−	0.507434i	0.870296	+	0.492529i	$0.163928\pi$
$132$		0			0
$133$	0.985281	+	4.92447i	0.00740813	+	0.0370261i
$134$		0			0
$135$	3.40812	+	5.90303i	0.0252453	+	0.0437262i
$136$		0			0
$137$	58.7132	−	101.694i	0.428564	−	0.742294i	−0.568182	−	0.822903i	$-0.692353\pi$
$137$	58.7132	−	101.694i	0.428564	−	0.742294i	0.996746	+	0.0806089i	$0.0256865\pi$
$138$		0			0
$139$			68.5857i			0.493422i	0.969089	+	0.246711i	$0.0793499\pi$
$139$			68.5857i			0.493422i	−0.969089	+	0.246711i	$0.920650\pi$
$140$		0			0
$141$	199.794			1.41698
$142$		0			0
$143$	63.0000	+	36.3731i	0.440559	+	0.254357i
$144$		0			0
$145$	56.1838	−	32.4377i	0.387474	−	0.223708i
$146$		0			0
$147$	−78.8345	−	189.122i	−0.536289	−	1.28655i
$148$		0			0
$149$	−13.1985	−	22.8604i	−0.0885804	−	0.153426i	0.818331	−	0.574747i	$-0.194900\pi$
$149$	−13.1985	−	22.8604i	−0.0885804	−	0.153426i	−0.906911	+	0.421322i	$0.861566\pi$
$150$		0			0
$151$	67.1066	−	116.232i	0.444415	−	0.769749i	−0.553597	−	0.832785i	$-0.686745\pi$
$151$	67.1066	−	116.232i	0.444415	−	0.769749i	0.998011	+	0.0630363i	$0.0200784\pi$
$152$		0			0
$153$		−	115.054i		−	0.751986i
$154$		0			0
$155$	−77.9634			−0.502990
$156$		0			0
$157$	196.323	+	113.347i	1.25047	+	0.721958i	0.971202	−	0.238256i	$-0.0765759\pi$
$157$	196.323	+	113.347i	1.25047	+	0.721958i	0.279265	+	0.960214i	$0.409909\pi$
$158$		0			0
$159$	79.7756	−	46.0585i	0.501734	−	0.289676i
$160$		0			0
$161$	−15.5955	+	3.12032i	−0.0968662	+	0.0193808i
$162$		0			0
$163$	−45.9889	−	79.6550i	−0.282140	−	0.488681i	0.689771	−	0.724027i	$-0.257711\pi$
$163$	−45.9889	−	79.6550i	−0.282140	−	0.488681i	−0.971912	+	0.235346i	$0.924378\pi$
$164$		0			0
$165$	87.6838	−	151.873i	0.531417	−	0.920441i
$166$		0			0
$167$		−	203.482i		−	1.21845i	−0.792996	−	0.609227i	$-0.791480\pi$
$167$		−	203.482i		−	1.21845i	0.792996	−	0.609227i	$-0.208520\pi$
$168$		0			0
$169$	138.823			0.821440
$170$		0			0
$171$	5.27208	+	3.04384i	0.0308309	+	0.0178002i
$172$		0			0
$173$	−61.3234	+	35.4051i	−0.354470	+	0.204654i	−0.666652	−	0.745369i	$-0.732273\pi$
$173$	−61.3234	+	35.4051i	−0.354470	+	0.204654i	0.312182	+	0.950022i	$0.398940\pi$
$174$		0			0
$175$	−69.1838	−	78.7107i	−0.395336	−	0.449775i
$176$		0			0
$177$	175.110	+	303.300i	0.989323	+	1.71356i
$178$		0			0
$179$	54.4081	−	94.2376i	0.303956	−	0.526467i	−0.673072	−	0.739577i	$-0.735026\pi$
$179$	54.4081	−	94.2376i	0.303956	−	0.526467i	0.977028	+	0.213109i	$0.0683591\pi$
$180$		0			0
$181$		−	99.6607i		−	0.550611i	−0.961357	−	0.275306i	$-0.911221\pi$
$181$		−	99.6607i		−	0.550611i	0.961357	−	0.275306i	$-0.0887791\pi$
$182$		0			0
$183$	−277.066			−1.51402
$184$		0			0
$185$	−178.110	−	102.832i	−0.962758	−	0.555848i
$186$		0			0
$187$	155.504	−	89.7800i	0.831570	−	0.480107i
$188$		0			0
$189$	14.2721	+	4.82687i	0.0755136	+	0.0255390i
$190$		0			0
$191$	−34.9523	−	60.5391i	−0.182996	−	0.316959i	0.759903	−	0.650036i	$-0.225246\pi$
$191$	−34.9523	−	60.5391i	−0.182996	−	0.316959i	−0.942899	+	0.333077i	$0.891913\pi$
$192$		0			0
$193$	16.1690	−	28.0056i	0.0837774	−	0.145107i	−0.821092	−	0.570796i	$-0.806635\pi$
$193$	16.1690	−	28.0056i	0.0837774	−	0.145107i	0.904870	+	0.425689i	$0.139968\pi$
$194$		0			0
$195$			72.7461i			0.373057i
$196$		0			0
$197$	−277.103			−1.40661			−0.703306	−	0.710887i	$-0.748294\pi$
$197$	−277.103			−1.40661			−0.703306	+	0.710887i	$0.748294\pi$
$198$		0			0
$199$	145.011	+	83.7222i	0.728699	+	0.420715i	0.817946	−	0.575295i	$-0.195113\pi$
$199$	145.011	+	83.7222i	0.728699	+	0.420715i	−0.0892469	+	0.996010i	$0.528446\pi$
$200$		0			0
$201$	335.478	−	193.688i	1.66904	−	0.963623i
$202$		0			0
$203$	45.9411	−	135.839i	0.226311	−	0.669155i
$204$		0			0
$205$	−33.3015	−	57.6799i	−0.162446	−	0.281365i
$206$		0			0
$207$	−9.63961	+	16.6963i	−0.0465682	+	0.0806584i
$208$		0			0
$209$			9.50079i			0.0454583i
$210$		0			0
$211$	128.073			0.606982			0.303491	−	0.952834i	$-0.401848\pi$
$211$	128.073			0.606982			0.303491	+	0.952834i	$0.401848\pi$
$212$		0			0
$213$	−175.368	−	101.248i	−0.823322	−	0.475345i
$214$		0			0
$215$	17.7868	−	10.2692i	0.0827293	−	0.0477638i
$216$		0			0
$217$	−129.434	+	113.768i	−0.596470	+	0.524275i
$218$		0			0
$219$	273.871	+	474.359i	1.25055	+	2.16602i
$220$		0			0
$221$	−37.2426	+	64.5061i	−0.168519	+	0.291883i
$222$		0			0
$223$			417.169i			1.87071i	0.353705	+	0.935357i	$0.384922\pi$
$223$			417.169i			1.87071i	−0.353705	+	0.935357i	$0.615078\pi$
$224$		0			0
$225$	−127.029			−0.564575
$226$		0			0
$227$	−201.143	−	116.130i	−0.886093	−	0.511586i	−0.0134307	−	0.999910i	$-0.504275\pi$
$227$	−201.143	−	116.130i	−0.886093	−	0.511586i	−0.872663	+	0.488324i	$0.837609\pi$
$228$		0			0
$229$	72.4188	−	41.8110i	0.316239	−	0.182581i	−0.333476	−	0.942759i	$-0.608222\pi$
$229$	72.4188	−	41.8110i	0.316239	−	0.182581i	0.649715	+	0.760178i	$0.274888\pi$
$230$		0			0
$231$	−76.0477	−	380.089i	−0.329211	−	1.64541i
$232$		0			0
$233$	−109.537	−	189.723i	−0.470114	−	0.814261i	0.529302	−	0.848434i	$-0.322454\pi$
$233$	−109.537	−	189.723i	−0.470114	−	0.814261i	−0.999416	+	0.0341721i	$0.989121\pi$
$234$		0			0
$235$	−75.6579	+	131.043i	−0.321949	+	0.557631i
$236$		0			0
$237$			318.689i			1.34468i
$238$		0			0
$239$	193.103			0.807961			0.403980	−	0.914768i	$-0.367626\pi$
$239$	193.103			0.807961			0.403980	+	0.914768i	$0.367626\pi$
$240$		0			0
$241$	42.8970	+	24.7666i	0.177996	+	0.102766i	0.586351	−	0.810057i	$-0.300564\pi$
$241$	42.8970	+	24.7666i	0.177996	+	0.102766i	−0.408355	+	0.912823i	$0.633897\pi$
$242$		0			0
$243$	292.368	−	168.798i	1.20316	−	0.694644i
$244$		0			0
$245$	153.897	+	19.9098i	0.628151	+	0.0812646i
$246$		0			0
$247$	−1.97056	−	3.41311i	−0.00797799	−	0.0138183i
$248$		0			0
$249$	−225.765	+	391.036i	−0.906685	+	1.57042i
$250$		0			0
$251$			162.524i			0.647507i	0.946141	+	0.323754i	$0.104945\pi$
$251$			162.524i			0.647507i	−0.946141	+	0.323754i	$0.895055\pi$
$252$		0			0
$253$	−30.0883			−0.118926
$254$		0			0
$255$	155.504	+	89.7800i	0.609818	+	0.352079i
$256$		0			0
$257$	−85.8747	+	49.5798i	−0.334143	+	0.192917i	−0.657679	−	0.753298i	$-0.728462\pi$
$257$	−85.8747	+	49.5798i	−0.334143	+	0.192917i	0.323536	+	0.946216i	$0.395128\pi$
$258$		0			0
$259$	−445.753	+	89.1857i	−1.72106	+	0.344346i
$260$		0			0
$261$	−86.9117	−	150.535i	−0.332995	−	0.576764i
$262$		0			0
$263$	−217.173	+	376.154i	−0.825751	+	1.43024i	0.0755923	+	0.997139i	$0.475915\pi$
$263$	−217.173	+	376.154i	−0.825751	+	1.43024i	−0.901344	+	0.433105i	$0.857418\pi$
$264$		0			0
$265$			69.7657i			0.263267i
$266$		0			0
$267$	−702.110			−2.62962
$268$		0			0
$269$	79.1619	+	45.7041i	0.294282	+	0.169904i	0.639871	−	0.768482i	$-0.278988\pi$
$269$	79.1619	+	45.7041i	0.294282	+	0.169904i	−0.345589	+	0.938386i	$0.612321\pi$
$270$		0			0
$271$	14.8051	−	8.54772i	0.0546313	−	0.0315414i	−0.472436	−	0.881365i	$-0.656625\pi$
$271$	14.8051	−	8.54772i	0.0546313	−	0.0315414i	0.527067	+	0.849824i	$0.323292\pi$
$272$		0			0
$273$	106.154	+	120.772i	0.388844	+	0.442389i
$274$		0			0
$275$	−99.1249	−	171.689i	−0.360454	−	0.624325i
$276$		0			0
$277$	−200.206	+	346.766i	−0.722764	+	1.25186i	0.237124	+	0.971479i	$0.423795\pi$
$277$	−200.206	+	346.766i	−0.722764	+	1.25186i	−0.959888	+	0.280385i	$0.909538\pi$
$278$		0			0
$279$			208.891i			0.748712i
$280$		0			0
$281$	−538.690			−1.91705			−0.958524	−	0.285012i	$-0.908002\pi$
$281$	−538.690			−1.91705			−0.958524	+	0.285012i	$0.908002\pi$
$282$		0			0
$283$	267.783	+	154.604i	0.946229	+	0.546306i	0.891907	−	0.452218i	$-0.149367\pi$
$283$	267.783	+	154.604i	0.946229	+	0.546306i	0.0543215	+	0.998523i	$0.482700\pi$
$284$		0			0
$285$	−8.22792	+	4.75039i	−0.0288699	+	0.0166680i
$286$		0			0
$287$	−139.456	−	47.1645i	−0.485909	−	0.164336i
$288$		0			0
$289$	−52.5736	−	91.0601i	−0.181916	−	0.315087i
$290$		0			0
$291$	53.4853	−	92.6392i	0.183798	−	0.318348i
$292$		0			0
$293$			327.391i			1.11738i	0.829378	+	0.558688i	$0.188695\pi$
$293$			327.391i			1.11738i	−0.829378	+	0.558688i	$0.811305\pi$
$294$		0			0
$295$	−265.243			−0.899128
$296$		0			0
$297$	24.6838	+	14.2512i	0.0831103	+	0.0479838i
$298$		0			0
$299$	10.8091	−	6.24063i	0.0361508	−	0.0208717i
$300$		0			0
$301$	14.5442	−	43.0041i	0.0483195	−	0.142871i
$302$		0			0
$303$	−59.5919	−	103.216i	−0.196673	−	0.340647i
$304$		0			0
$305$	104.919	−	181.725i	0.343998	−	0.595821i
$306$		0			0
$307$			256.140i			0.834331i	0.908831	+	0.417165i	$0.136976\pi$
$307$			256.140i			0.834331i	−0.908831	+	0.417165i	$0.863024\pi$
$308$		0			0
$309$	236.220			0.764467
$310$		0			0
$311$	187.349	+	108.166i	0.602409	+	0.347801i	0.769989	−	0.638057i	$-0.220262\pi$
$311$	187.349	+	108.166i	0.602409	+	0.347801i	−0.167580	+	0.985859i	$0.553595\pi$
$312$		0			0
$313$	135.809	−	78.4092i	0.433893	−	0.250509i	−0.267110	−	0.963666i	$-0.586069\pi$
$313$	135.809	−	78.4092i	0.433893	−	0.250509i	0.701004	+	0.713157i	$0.252736\pi$
$314$		0			0
$315$	141.286	−	124.185i	0.448528	−	0.394239i
$316$		0			0
$317$	−224.015	−	388.005i	−0.706671	−	1.22399i	−0.966085	−	0.258224i	$-0.916863\pi$
$317$	−224.015	−	388.005i	−0.706671	−	1.22399i	0.259414	−	0.965766i	$-0.416471\pi$
$318$		0			0
$319$	135.640	−	234.935i	0.425203	−	0.736472i
$320$		0			0
$321$			199.084i			0.620199i
$322$		0			0
$323$	−9.72792			−0.0301174
$324$		0			0
$325$	71.2203	+	41.1191i	0.219140	+	0.126520i
$326$		0			0
$327$	272.717	−	157.453i	0.833996	−	0.481508i
$328$		0			0
$329$	65.6177	+	327.960i	0.199446	+	0.996839i
$330$		0			0
$331$	−27.5036	−	47.6376i	−0.0830924	−	0.143920i	0.821484	−	0.570231i	$-0.193146\pi$
$331$	−27.5036	−	47.6376i	−0.0830924	−	0.143920i	−0.904577	+	0.426311i	$0.859813\pi$
$332$		0			0
$333$	−275.522	+	477.218i	−0.827393	+	1.43309i
$334$		0			0
$335$			293.383i			0.875770i
$336$		0			0
$337$	−111.632			−0.331254			−0.165627	−	0.986189i	$-0.552965\pi$
$337$	−111.632			−0.331254			−0.165627	+	0.986189i	$0.552965\pi$
$338$		0			0
$339$	309.463	+	178.669i	0.912870	+	0.527046i
$340$		0			0
$341$	−282.331	+	163.004i	−0.827949	+	0.478016i
$342$		0			0
$343$	284.551	−	191.519i	0.829596	−	0.558365i
$344$		0			0
$345$	−15.0442	−	26.0572i	−0.0436062	−	0.0755282i
$346$		0			0
$347$	188.628	−	326.714i	0.543598	−	0.941539i	−0.455096	−	0.890442i	$-0.650395\pi$
$347$	188.628	−	326.714i	0.543598	−	0.941539i	0.998694	−	0.0510967i	$-0.0162717\pi$
$348$		0			0
$349$		−	204.034i		−	0.584624i	−0.956323	−	0.292312i	$-0.905575\pi$
$349$		−	204.034i		−	0.584624i	0.956323	−	0.292312i	$-0.0944246\pi$
$350$		0			0
$351$	−11.8234			−0.0336848
$352$		0			0
$353$	−361.198	−	208.538i	−1.02323	−	0.590759i	−0.108189	−	0.994130i	$-0.534505\pi$
$353$	−361.198	−	208.538i	−1.02323	−	0.590759i	−0.915036	+	0.403371i	$0.867838\pi$
$354$		0			0
$355$	132.816	−	76.6815i	0.374130	−	0.216004i
$356$		0			0
$357$	389.176	−	77.8658i	1.09013	−	0.218112i
$358$		0			0
$359$	89.4153	+	154.872i	0.249068	+	0.431398i	0.963267	−	0.268544i	$-0.0865425\pi$
$359$	89.4153	+	154.872i	0.249068	+	0.431398i	−0.714200	+	0.699942i	$0.753209\pi$
$360$		0			0
$361$	−180.243	+	312.189i	−0.499287	+	0.864791i
$362$		0			0
$363$		−	227.340i		−	0.626281i
$364$		0			0
$365$	−414.838			−1.13654
$366$		0			0
$367$	−544.724	−	314.497i	−1.48426	−	0.856939i	−0.484422	−	0.874835i	$-0.660970\pi$
$367$	−544.724	−	314.497i	−1.48426	−	0.856939i	−0.999840	+	0.0178960i	$0.994303\pi$
$368$		0			0
$369$	−154.544	+	89.2261i	−0.418819	+	0.241805i
$370$		0			0
$371$	101.805	+	115.824i	0.274407	+	0.312194i
$372$		0			0
$373$	−127.779	−	221.320i	−0.342572	−	0.593351i	0.642338	−	0.766422i	$-0.277965\pi$
$373$	−127.779	−	221.320i	−0.342572	−	0.593351i	−0.984910	+	0.173070i	$0.944631\pi$
$374$		0			0
$375$	264.658	−	458.401i	0.705754	−	1.22240i
$376$		0			0
$377$			112.532i			0.298494i
$378$		0			0
$379$	−219.750			−0.579816			−0.289908	−	0.957055i	$-0.593625\pi$
$379$	−219.750			−0.579816			−0.289908	+	0.957055i	$0.593625\pi$
$380$		0			0
$381$	−219.676	−	126.830i	−0.576578	−	0.332887i
$382$		0			0
$383$	−14.7534	+	8.51785i	−0.0385205	+	0.0222398i	−0.519137	−	0.854691i	$-0.673746\pi$
$383$	−14.7534	+	8.51785i	−0.0385205	+	0.0222398i	0.480616	+	0.876931i	$0.340413\pi$
$384$		0			0
$385$	278.095	+	94.0530i	0.722326	+	0.244293i
$386$		0			0
$387$	−27.5147	−	47.6569i	−0.0710975	−	0.123144i
$388$		0			0
$389$	−76.1102	+	131.827i	−0.195656	+	0.338886i	−0.947115	−	0.320893i	$-0.896017\pi$
$389$	−76.1102	+	131.827i	−0.195656	+	0.338886i	0.751459	+	0.659779i	$0.229350\pi$
$390$		0			0
$391$		−	30.8076i		−	0.0787919i
$392$		0			0
$393$	555.926			1.41457
$394$		0			0
$395$	−209.025	−	120.681i	−0.529178	−	0.305521i
$396$		0			0
$397$	322.786	−	186.361i	0.813064	−	0.469423i	−0.0349549	−	0.999389i	$-0.511129\pi$
$397$	322.786	−	186.361i	0.813064	−	0.469423i	0.848019	+	0.529966i	$0.177795\pi$
$398$		0			0
$399$	−6.72792	+	19.8931i	−0.0168620	+	0.0498574i
$400$		0			0
$401$	−325.786	−	564.279i	−0.812435	−	1.40718i	−0.911155	−	0.412063i	$-0.864808\pi$
$401$	−325.786	−	564.279i	−0.812435	−	1.40718i	0.0987205	−	0.995115i	$-0.468525\pi$
$402$		0			0
$403$	67.6173	−	117.117i	0.167785	−	0.290612i
$404$		0			0
$405$			270.353i			0.667538i
$406$		0			0
$407$	−859.992			−2.11300
$408$		0			0
$409$	462.081	+	266.782i	1.12978	+	0.652280i	0.943880	−	0.330289i	$-0.107146\pi$
$409$	462.081	+	266.782i	1.12978	+	0.652280i	0.185902	+	0.982568i	$0.440479\pi$
$410$		0			0
$411$	425.239	−	245.512i	1.03464	−	0.597352i
$412$		0			0
$413$	−440.353	+	387.054i	−1.06623	+	0.937176i
$414$		0			0
$415$	−170.985	−	296.154i	−0.412012	−	0.713625i
$416$		0			0
$417$	−143.397	+	248.371i	−0.343878	+	0.595614i
$418$		0			0
$419$			534.252i			1.27507i	0.770423	+	0.637533i	$0.220045\pi$
$419$			534.252i			1.27507i	−0.770423	+	0.637533i	$0.779955\pi$
$420$		0			0
$421$	−157.220			−0.373445			−0.186723	−	0.982413i	$-0.559787\pi$
$421$	−157.220			−0.373445			−0.186723	+	0.982413i	$0.559787\pi$
$422$		0			0
$423$	351.110	+	202.713i	0.830047	+	0.479228i
$424$		0			0
$425$	175.794	−	101.495i	0.413633	−	0.238811i
$426$		0			0
$427$	−90.9960	−	454.801i	−0.213105	−	1.06511i
$428$		0			0
$429$	152.095	+	263.437i	0.354535	+	0.614072i
$430$		0			0
$431$	114.268	−	197.918i	0.265123	−	0.459207i	−0.702473	−	0.711711i	$-0.747921\pi$
$431$	114.268	−	197.918i	0.265123	−	0.459207i	0.967596	+	0.252504i	$0.0812541\pi$
$432$		0			0
$433$			47.5549i			0.109827i	0.998491	+	0.0549133i	$0.0174882\pi$
$433$			47.5549i			0.109827i	−0.998491	+	0.0549133i	$0.982512\pi$
$434$		0			0
$435$	271.279			0.623630
$436$		0			0
$437$	1.41169	+	0.815039i	0.00323041	+	0.00186508i
$438$		0			0
$439$	−63.9594	+	36.9270i	−0.145693	+	0.0841161i	−0.571075	−	0.820898i	$-0.693473\pi$
$439$	−63.9594	+	36.9270i	−0.145693	+	0.0841161i	0.425381	+	0.905014i	$0.360140\pi$
$440$		0			0
$441$	53.3452	−	412.342i	0.120964	−	0.935017i
$442$		0			0
$443$	117.320	+	203.204i	0.264830	+	0.458699i	0.967519	−	0.252798i	$-0.0813507\pi$
$443$	117.320	+	203.204i	0.264830	+	0.458699i	−0.702689	+	0.711497i	$0.748017\pi$
$444$		0			0
$445$	265.875	−	460.508i	0.597471	−	1.03485i
$446$		0			0
$447$		−	110.380i		−	0.246935i
$448$		0			0
$449$	−255.161			−0.568288			−0.284144	−	0.958782i	$-0.591709\pi$
$449$	−255.161			−0.568288			−0.284144	+	0.958782i	$0.591709\pi$
$450$		0			0
$451$	−241.191	−	139.252i	−0.534791	−	0.308762i
$452$		0			0
$453$	486.029	−	280.609i	1.07291	−	0.619446i
$454$		0			0
$455$	−119.412	+	23.8918i	−0.262444	+	0.0525095i
$456$		0			0
$457$	72.8675	+	126.210i	0.159448	+	0.276171i	0.934670	−	0.355518i	$-0.115695\pi$
$457$	72.8675	+	126.210i	0.159448	+	0.276171i	−0.775222	+	0.631689i	$0.782362\pi$
$458$		0			0
$459$	−14.5919	+	25.2739i	−0.0317906	+	0.0550629i
$460$		0			0
$461$		−	888.329i		−	1.92696i	−0.267777	−	0.963481i	$-0.586289\pi$
$461$		−	888.329i		−	1.92696i	0.267777	−	0.963481i	$-0.413711\pi$
$462$		0			0
$463$	234.014			0.505430			0.252715	−	0.967541i	$-0.418676\pi$
$463$	234.014			0.505430			0.252715	+	0.967541i	$0.418676\pi$
$464$		0			0
$465$	−282.331	−	163.004i	−0.607162	−	0.350545i
$466$		0			0
$467$	−681.231	+	393.309i	−1.45874	+	0.842204i	−0.998950	−	0.0458237i	$-0.985409\pi$
$467$	−681.231	+	393.309i	−1.45874	+	0.842204i	−0.459790	+	0.888028i	$0.652075\pi$
$468$		0			0
$469$	428.117	+	487.071i	0.912830	+	1.03853i
$470$		0			0
$471$	473.967	+	820.934i	1.00630	+	1.74296i
$472$		0			0
$473$	42.9411	−	74.3762i	0.0907846	−	0.157244i
$474$		0			0
$475$			10.7405i			0.0226115i
$476$		0			0
$477$	186.926			0.391878
$478$		0			0
$479$	638.202	+	368.466i	1.33236	+	0.769240i	0.985661	−	0.168735i	$-0.0539682\pi$
$479$	638.202	+	368.466i	1.33236	+	0.769240i	0.346702	+	0.937975i	$0.387302\pi$
$480$		0			0
$481$	308.948	−	178.371i	0.642304	−	0.370834i
$482$		0			0
$483$	−63.0000	−	21.3068i	−0.130435	−	0.0441136i
$484$		0			0
$485$	40.5076	+	70.1612i	0.0835208	+	0.144662i
$486$		0			0
$487$	−135.349	+	234.432i	−0.277925	+	0.481379i	−0.970869	−	0.239612i	$-0.922980\pi$
$487$	−135.349	+	234.432i	−0.277925	+	0.481379i	0.692944	+	0.720991i	$0.256313\pi$
$488$		0			0
$489$		−	384.609i		−	0.786520i
$490$		0			0
$491$	−760.161			−1.54819			−0.774094	−	0.633070i	$-0.781794\pi$
$491$	−760.161			−1.54819			−0.774094	+	0.633070i	$0.781794\pi$
$492$		0			0
$493$	240.551	+	138.882i	0.487934	+	0.281709i
$494$		0			0
$495$	308.184	−	177.930i	0.622593	−	0.359455i
$496$		0			0
$497$	108.603	−	321.117i	0.218517	−	0.646111i
$498$		0			0
$499$	62.7462	+	108.680i	0.125744	+	0.217795i	0.922023	−	0.387134i	$-0.126535\pi$
$499$	62.7462	+	108.680i	0.125744	+	0.217795i	−0.796280	+	0.604929i	$0.793202\pi$
$500$		0			0
$501$	425.434	−	736.873i	0.849169	−	1.47080i
$502$		0			0
$503$			117.083i			0.232770i	0.993204	+	0.116385i	$0.0371306\pi$
$503$			117.083i			0.232770i	−0.993204	+	0.116385i	$0.962869\pi$
$504$		0			0
$505$	90.2649			0.178742
$506$		0			0
$507$	502.724	+	290.248i	0.991566	+	0.572481i
$508$		0			0
$509$	574.110	−	331.463i	1.12792	−	0.651204i	0.184507	−	0.982831i	$-0.440931\pi$
$509$	574.110	−	331.463i	1.12792	−	0.651204i	0.943410	+	0.331627i	$0.107598\pi$
$510$		0			0
$511$	−688.709	+	605.349i	−1.34777	+	1.18464i
$512$		0			0
$513$	−0.772078	−	1.33728i	−0.00150503	−	0.00260678i
$514$		0			0
$515$	−89.4518	+	154.935i	−0.173693	+	0.300845i
$516$		0			0
$517$			632.733i			1.22386i
$518$		0			0
$519$	−296.095			−0.570511
$520$		0			0
$521$	−40.8229	−	23.5691i	−0.0783550	−	0.0452383i	0.460311	−	0.887758i	$-0.347738\pi$
$521$	−40.8229	−	23.5691i	−0.0783550	−	0.0452383i	−0.538666	+	0.842520i	$0.681071\pi$
$522$		0			0
$523$	−432.554	+	249.735i	−0.827064	+	0.477506i	−0.852846	−	0.522162i	$-0.825126\pi$
$523$	−432.554	+	249.735i	−0.827064	+	0.477506i	0.0257824	+	0.999668i	$0.491792\pi$
$524$		0			0
$525$	−85.9706	−	429.684i	−0.163753	−	0.818446i
$526$		0			0
$527$	−166.901	−	289.080i	−0.316699	−	0.548539i
$528$		0			0
$529$	261.919	−	453.657i	0.495121	−	0.857574i
$530$		0			0
$531$			710.675i			1.33837i
$532$		0			0
$533$	115.529			0.216752
$534$		0			0
$535$	−130.578	−	75.3890i	−0.244070	−	0.140914i
$536$		0			0
$537$	394.058	−	227.510i	0.733815	−	0.423668i
$538$		0			0
$539$	598.937	−	249.663i	1.11120	−	0.463197i
$540$		0			0
$541$	249.405	+	431.981i	0.461007	+	0.798487i	0.999011	−	0.0444550i	$-0.0141551\pi$
$541$	249.405	+	431.981i	0.461007	+	0.798487i	−0.538005	+	0.842942i	$0.680822\pi$
$542$		0			0
$543$	208.368	−	360.903i	0.383734	−	0.664647i
$544$		0			0
$545$			238.497i			0.437609i
$546$		0			0
$547$	279.897			0.511694			0.255847	−	0.966717i	$-0.417646\pi$
$547$	279.897			0.511694			0.255847	+	0.966717i	$0.417646\pi$
$548$		0			0
$549$	−486.905	−	281.114i	−0.886894	−	0.512048i
$550$		0			0
$551$	−12.7279	+	7.34847i	−0.0230997	+	0.0133366i
$552$		0			0
$553$	−523.124	+	104.666i	−0.945976	+	0.189269i
$554$		0			0
$555$	−429.996	−	744.775i	−0.774768	−	1.34194i
$556$		0			0
$557$	−130.890	+	226.708i	−0.234991	+	0.407016i	−0.959270	−	0.282491i	$-0.908839\pi$
$557$	−130.890	+	226.708i	−0.234991	+	0.407016i	0.724279	+	0.689507i	$0.242173\pi$
$558$		0			0
$559$			35.6258i			0.0637312i
$560$		0			0
$561$	750.838			1.33839
$562$		0			0
$563$	420.076	+	242.531i	0.746139	+	0.430784i	0.824297	−	0.566157i	$-0.191571\pi$
$563$	420.076	+	242.531i	0.746139	+	0.430784i	−0.0781581	+	0.996941i	$0.524904\pi$
$564$		0			0
$565$	−234.375	+	135.316i	−0.414822	+	0.239498i
$566$		0			0
$567$	394.511	+	448.837i	0.695786	+	0.791599i
$568$		0			0
$569$	227.000	+	393.175i	0.398945	+	0.690993i	0.993596	−	0.112991i	$-0.0360432\pi$
$569$	227.000	+	393.175i	0.398945	+	0.690993i	−0.594651	+	0.803984i	$0.702710\pi$
$570$		0			0
$571$	−115.769	+	200.517i	−0.202747	+	0.351168i	−0.949413	−	0.314032i	$-0.898320\pi$
$571$	−115.769	+	200.517i	−0.202747	+	0.351168i	0.746666	+	0.665200i	$0.231654\pi$
$572$		0			0
$573$		−	292.309i		−	0.510137i
$574$		0			0
$575$	−34.0143			−0.0591553
$576$		0			0
$577$	564.014	+	325.634i	0.977494	+	0.564356i	0.901513	−	0.432753i	$-0.142458\pi$
$577$	564.014	+	325.634i	0.977494	+	0.564356i	0.0759812	+	0.997109i	$0.475791\pi$
$578$		0			0
$579$	117.107	−	67.6115i	0.202257	−	0.116773i
$580$		0			0
$581$	−716.029	−	242.164i	−1.23241	−	0.416805i
$582$		0			0
$583$	145.864	+	252.644i	0.250195	+	0.433351i
$584$		0			0
$585$	−73.8091	+	127.841i	−0.126169	+	0.218532i
$586$		0			0
$587$		−	823.029i		−	1.40209i	−0.713116	−	0.701046i	$-0.752717\pi$
$587$		−	823.029i		−	1.40209i	0.713116	−	0.701046i	$-0.247283\pi$
$588$		0			0
$589$	17.6619			0.0299863
$590$		0			0
$591$	−1003.48	−	579.358i	−1.69793	−	0.980301i
$592$		0			0
$593$	−700.110	+	404.209i	−1.18062	+	0.681634i	−0.956159	−	0.292848i	$-0.905397\pi$
$593$	−700.110	+	404.209i	−1.18062	+	0.681634i	−0.224465	+	0.974482i	$0.572064\pi$
$594$		0			0
$595$	−96.3015	+	284.744i	−0.161851	+	0.478561i
$596$		0			0
$597$	350.088	+	606.370i	0.586412	+	1.01570i
$598$		0			0
$599$	−265.422	+	459.725i	−0.443109	+	0.767488i	−0.997918	−	0.0644900i	$-0.979458\pi$
$599$	−265.422	+	459.725i	−0.443109	+	0.767488i	0.554809	+	0.831978i	$0.312791\pi$
$600$		0			0
$601$		−	936.503i		−	1.55824i	−0.626874	−	0.779121i	$-0.715666\pi$
$601$		−	936.503i		−	1.55824i	0.626874	−	0.779121i	$-0.284334\pi$
$602$		0			0
$603$	786.073			1.30360
$604$		0			0
$605$	149.111	+	86.0890i	0.246464	+	0.142296i
$606$		0			0
$607$	521.452	−	301.060i	0.859064	−	0.495981i	−0.00463474	−	0.999989i	$-0.501475\pi$
$607$	521.452	−	301.060i	0.859064	−	0.495981i	0.863699	+	0.504008i	$0.168142\pi$
$608$		0			0
$609$	450.375	−	395.862i	0.739531	−	0.650020i
$610$		0			0
$611$	−131.235	−	227.307i	−0.214788	−	0.372024i
$612$		0			0
$613$	548.448	−	949.940i	0.894695	−	1.54966i	0.0605142	−	0.998167i	$-0.480726\pi$
$613$	548.448	−	949.940i	0.894695	−	1.54966i	0.834181	−	0.551491i	$-0.185941\pi$
$614$		0			0
$615$		−	278.503i		−	0.452851i
$616$		0			0
$617$	−432.956			−0.701712			−0.350856	−	0.936429i	$-0.614109\pi$
$617$	−432.956			−0.701712			−0.350856	+	0.936429i	$0.614109\pi$
$618$		0			0
$619$	−194.951	−	112.555i	−0.314946	−	0.181834i	0.334192	−	0.942505i	$-0.391537\pi$
$619$	−194.951	−	112.555i	−0.314946	−	0.181834i	−0.649137	+	0.760671i	$0.724870\pi$
$620$		0			0
$621$	4.23506	−	2.44512i	0.00681975	−	0.00393738i
$622$		0			0
$623$	−230.592	−	1152.51i	−0.370131	−	1.84993i
$624$		0			0
$625$	13.3091	+	23.0520i	0.0212945	+	0.0368832i
$626$		0			0
$627$	−19.8640	+	34.4054i	−0.0316810	+	0.0548730i
$628$		0			0
$629$		−	880.552i		−	1.39992i
$630$		0			0
$631$	750.514			1.18940			0.594702	−	0.803946i	$-0.297270\pi$
$631$	750.514			1.18940			0.594702	+	0.803946i	$0.297270\pi$
$632$		0			0
$633$	463.794	+	267.772i	0.732692	+	0.423020i
$634$		0			0
$635$	166.374	−	96.0560i	0.262006	−	0.151269i
$636$		0			0
$637$	−163.383	+	213.916i	−0.256488	+	0.335818i
$638$		0			0
$639$	−205.456	−	355.860i	−0.321527	−	0.556901i
$640$		0			0
$641$	580.926	−	1006.19i	0.906281	−	1.56973i	0.0870937	−	0.996200i	$-0.472242\pi$
$641$	580.926	−	1006.19i	0.906281	−	1.56973i	0.819188	−	0.573525i	$-0.194425\pi$
$642$		0			0
$643$		−	121.957i		−	0.189669i	−0.995493	−	0.0948347i	$-0.969768\pi$
$643$		−	121.957i		−	0.189669i	0.995493	−	0.0948347i	$-0.0302322\pi$
$644$		0			0
$645$	85.8823			0.133151
$646$		0			0
$647$	−137.504	−	79.3877i	−0.212525	−	0.122701i	0.389959	−	0.920832i	$-0.372489\pi$
$647$	−137.504	−	79.3877i	−0.212525	−	0.122701i	−0.602484	+	0.798131i	$0.705822\pi$
$648$		0			0
$649$	−960.529	+	554.561i	−1.48001	+	0.854486i
$650$		0			0
$651$	−706.584	+	141.372i	−1.08538	+	0.217162i
$652$		0			0
$653$	−195.471	−	338.565i	−0.299342	−	0.518476i	0.676643	−	0.736311i	$-0.263434\pi$
$653$	−195.471	−	338.565i	−0.299342	−	0.518476i	−0.975986	+	0.217835i	$0.930101\pi$
$654$		0			0
$655$	−210.518	+	364.628i	−0.321401	+	0.556683i
$656$		0			0
$657$			1111.49i			1.69177i
$658$		0			0
$659$	331.955			0.503726			0.251863	−	0.967763i	$-0.418957\pi$
$659$	331.955			0.503726			0.251863	+	0.967763i	$0.418957\pi$
$660$		0			0
$661$	−561.029	−	323.910i	−0.848758	−	0.490031i	0.0114736	−	0.999934i	$-0.496348\pi$
$661$	−561.029	−	323.910i	−0.848758	−	0.490031i	−0.860232	+	0.509904i	$0.829681\pi$
$662$		0			0
$663$	−269.735	+	155.732i	−0.406840	+	0.234889i
$664$		0			0
$665$	−10.5000	−	11.9459i	−0.0157895	−	0.0179638i
$666$		0			0
$667$	−23.2721	−	40.3084i	−0.0348907	−	0.0604324i
$668$		0			0
$669$	−872.205	+	1510.70i	−1.30374	+	2.25815i
$670$		0			0
$671$		−	877.448i		−	1.30767i
$672$		0			0
$673$	100.956			0.150009			0.0750047	−	0.997183i	$-0.476103\pi$
$673$	100.956			0.150009			0.0750047	+	0.997183i	$0.476103\pi$
$674$		0			0
$675$	27.9045	+	16.1107i	0.0413401	+	0.0238677i
$676$		0			0
$677$	−643.610	+	371.588i	−0.950679	+	0.548875i	−0.893292	−	0.449477i	$-0.851610\pi$
$677$	−643.610	+	371.588i	−0.950679	+	0.548875i	−0.0573873	+	0.998352i	$0.518277\pi$
$678$		0			0
$679$	169.632	+	57.3704i	0.249827	+	0.0844924i
$680$		0			0
$681$	−485.603	−	841.088i	−0.713073	−	1.23508i
$682$		0			0
$683$	2.21721	−	3.84032i	0.00324628	−	0.00562272i	−0.864398	−	0.502809i	$-0.832300\pi$
$683$	2.21721	−	3.84032i	0.00324628	−	0.00562272i	0.867644	+	0.497186i	$0.165633\pi$
$684$		0			0
$685$			371.881i			0.542892i
$686$		0			0
$687$	349.669			0.508980
$688$		0			0
$689$	−104.802	−	60.5074i	−0.152107	−	0.0878192i
$690$		0			0
$691$	846.253	−	488.584i	1.22468	−	0.707069i	0.258767	−	0.965940i	$-0.416684\pi$
$691$	846.253	−	488.584i	1.22468	−	0.707069i	0.965912	+	0.258871i	$0.0833506\pi$
$692$		0			0
$693$	252.000	−	745.113i	0.363636	−	1.07520i
$694$		0			0
$695$	−108.603	−	188.106i	−0.156263	−	0.270656i
$696$		0			0
$697$	142.581	−	246.957i	0.204563	−	0.354314i
$698$		0			0
$699$		−	916.063i		−	1.31053i
$700$		0			0
$701$	840.177			1.19854			0.599270	−	0.800547i	$-0.295458\pi$
$701$	840.177			1.19854			0.599270	+	0.800547i	$0.295458\pi$
$702$		0			0
$703$	40.3492	+	23.2956i	0.0573958	+	0.0331375i
$704$		0			0
$705$	−547.963	+	316.367i	−0.777252	+	0.448747i
$706$		0			0
$707$	149.857	−	131.719i	0.211962	−	0.186306i
$708$		0			0
$709$	341.279	+	591.112i	0.481352	+	0.833727i	0.999771	−	0.0214003i	$-0.00681244\pi$
$709$	341.279	+	591.112i	0.481352	+	0.833727i	−0.518419	+	0.855127i	$0.673479\pi$
$710$		0			0
$711$	−323.345	+	560.050i	−0.454775	+	0.787694i
$712$		0			0
$713$			55.9340i			0.0784488i
$714$		0			0
$715$	−230.382			−0.322212
$716$		0			0
$717$	699.286	+	403.733i	0.975295	+	0.563087i
$718$		0			0
$719$	119.187	−	68.8126i	0.165768	−	0.0957060i	−0.414821	−	0.909903i	$-0.636156\pi$
$719$	119.187	−	68.8126i	0.165768	−	0.0957060i	0.580589	+	0.814197i	$0.302822\pi$
$720$		0			0
$721$	77.5812	+	387.754i	0.107602	+	0.537800i
$722$		0			0
$723$	103.562	+	179.375i	0.143240	+	0.248099i
$724$		0			0
$725$	153.338	−	265.589i	0.211501	−	0.366330i
$726$		0			0
$727$			264.137i			0.363325i	0.983361	+	0.181662i	$0.0581478\pi$
$727$			264.137i			0.363325i	−0.983361	+	0.181662i	$0.941852\pi$
$728$		0			0
$729$	643.368			0.882534
$730$		0			0
$731$	76.1543	+	43.9677i	0.104178	+	0.0601474i
$732$		0			0
$733$	−501.705	+	289.660i	−0.684455	+	0.395170i	−0.801531	−	0.597953i	$-0.795981\pi$
$733$	−501.705	+	289.660i	−0.684455	+	0.395170i	0.117077	+	0.993123i	$0.462648\pi$
$734$		0			0
$735$	515.683	+	393.863i	0.701610	+	0.535868i
$736$		0			0
$737$	613.397	+	1062.43i	0.832288	+	1.44157i
$738$		0			0
$739$	−99.0477	+	171.556i	−0.134029	+	0.232146i	−0.925226	−	0.379416i	$-0.876125\pi$
$739$	−99.0477	+	171.556i	−0.134029	+	0.232146i	0.791197	+	0.611562i	$0.209458\pi$
$740$		0			0
$741$		−	16.4800i		−	0.0222402i
$742$		0			0
$743$	976.690			1.31452			0.657261	−	0.753663i	$-0.271715\pi$
$743$	976.690			1.31452			0.657261	+	0.753663i	$0.271715\pi$
$744$		0			0
$745$	72.3974	+	41.7987i	0.0971777	+	0.0561056i
$746$		0			0
$747$	−793.499	+	458.127i	−1.06225	+	0.613289i
$748$		0			0
$749$	−326.794	+	65.3845i	−0.436308	+	0.0872958i
$750$		0			0
$751$	417.665	+	723.417i	0.556145	+	0.963272i	0.997813	+	0.0660933i	$0.0210535\pi$
$751$	417.665	+	723.417i	0.556145	+	0.963272i	−0.441668	+	0.897178i	$0.645613\pi$
$752$		0			0
$753$	−339.801	+	588.553i	−0.451263	+	0.781611i
$754$		0			0
$755$			425.044i			0.562972i
$756$		0			0
$757$	−104.221			−0.137677			−0.0688383	−	0.997628i	$-0.521929\pi$
$757$	−104.221			−0.137677			−0.0688383	+	0.997628i	$0.521929\pi$
$758$		0			0
$759$	−108.959	−	62.9077i	−0.143557	−	0.0828824i
$760$		0			0
$761$	−473.785	+	273.540i	−0.622583	+	0.359448i	−0.777874	−	0.628420i	$-0.783702\pi$
$761$	−473.785	+	273.540i	−0.622583	+	0.359448i	0.155291	+	0.987869i	$0.450368\pi$
$762$		0			0
$763$	348.025	+	395.950i	0.456128	+	0.518939i
$764$		0			0
$765$	182.184	+	315.552i	0.238149	+	0.412486i
$766$		0			0
$767$	230.044	−	398.447i	0.299927	−	0.519488i
$768$		0			0
$769$		−	341.205i		−	0.443700i	−0.975081	−	0.221850i	$-0.928790\pi$
$769$		−	341.205i		−	0.443700i	0.975081	−	0.221850i	$-0.0712095\pi$
$770$		0			0
$771$	−414.640			−0.537795
$772$		0			0
$773$	425.213	+	245.497i	0.550081	+	0.317590i	0.749155	−	0.662395i	$-0.230460\pi$
$773$	425.213	+	245.497i	0.550081	+	0.317590i	−0.199074	+	0.979985i	$0.563793\pi$
$774$		0			0
$775$	−319.169	+	184.273i	−0.411832	+	0.237771i
$776$		0			0
$777$	−1800.68	−	608.998i	−2.31748	−	0.783781i
$778$		0			0
$779$	7.54416	+	13.0669i	0.00968441	+	0.0167739i
$780$		0			0
$781$	320.647	−	555.376i	0.410559	−	0.711109i
$782$		0			0
$783$			44.0908i			0.0563101i
$784$		0			0
$785$	−717.926			−0.914555
$786$		0			0
$787$	−260.202	−	150.228i	−0.330625	−	0.190887i	0.325493	−	0.945544i	$-0.394470\pi$
$787$	−260.202	−	150.228i	−0.330625	−	0.190887i	−0.656119	+	0.754658i	$0.727803\pi$
$788$		0			0
$789$	−1572.90	+	908.116i	−1.99354	+	1.15097i
$790$		0			0
$791$	−191.647	+	566.660i	−0.242284	+	0.716385i
$792$		0			0

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.62132 + 2.09077i) q^{3} +(-2.74264 + 1.58346i) q^{5} +(-2.24264 + 6.63103i) q^{7} +(4.24264 + 7.34847i) q^{9} +O(q^{10})\)	\(q+(3.62132 + 2.09077i) q^{3} +(-2.74264 + 1.58346i) q^{5} +(-2.24264 + 6.63103i) q^{7} +(4.24264 + 7.34847i) q^{9} +(-6.62132 + 11.4685i) q^{11} -5.49333i q^{13} -13.2426 q^{15} +(-11.7426 - 6.77962i) q^{17} +(0.621320 - 0.358719i) q^{19} +(-21.9853 + 19.3242i) q^{21} +(1.13604 + 1.96768i) q^{23} +(-7.48528 + 12.9649i) q^{25} -2.15232i q^{27} -20.4853 q^{29} +(21.3198 + 12.3090i) q^{31} +(-47.9558 + 27.6873i) q^{33} +(-4.34924 - 21.7377i) q^{35} +(32.4706 + 56.2407i) q^{37} +(11.4853 - 19.8931i) q^{39} +21.0308i q^{41} -6.48528 q^{43} +(-23.2721 - 13.4361i) q^{45} +(41.3787 - 23.8900i) q^{47} +(-38.9411 - 29.7420i) q^{49} +(-28.3492 - 49.1023i) q^{51} +(11.0147 - 19.0781i) q^{53} -41.9385i q^{55} +3.00000 q^{57} +(72.5330 + 41.8770i) q^{59} +(-57.3823 + 33.1297i) q^{61} +(-58.2426 + 11.6531i) q^{63} +(8.69848 + 15.0662i) q^{65} +(46.3198 - 80.2283i) q^{67} +9.50079i q^{69} -48.4264 q^{71} +(113.441 + 65.4953i) q^{73} +(-54.2132 + 31.3000i) q^{75} +(-61.1985 - 69.6258i) q^{77} +(38.1066 + 66.0026i) q^{79} +(42.6838 - 73.9305i) q^{81} +107.981i q^{83} +42.9411 q^{85} +(-74.1838 - 42.8300i) q^{87} +(-145.412 + 83.9535i) q^{89} +(36.4264 + 12.3196i) q^{91} +(51.4706 + 89.1496i) q^{93} +(-1.13604 + 1.96768i) q^{95} -25.5816i q^{97} -112.368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{3} + 6 q^{5} + 8 q^{7}+O(q^{10}) \) 4 * q + 6 * q^3 + 6 * q^5 + 8 * q^7	\( 4 q + 6 q^{3} + 6 q^{5} + 8 q^{7} - 18 q^{11} - 36 q^{15} - 30 q^{17} - 6 q^{19} - 54 q^{21} + 30 q^{23} + 4 q^{25} - 48 q^{29} - 42 q^{31} - 90 q^{33} + 42 q^{35} + 62 q^{37} + 12 q^{39} + 8 q^{43} - 144 q^{45} + 174 q^{47} - 20 q^{49} - 54 q^{51} + 78 q^{53} + 12 q^{57} + 78 q^{59} + 42 q^{61} - 216 q^{63} - 84 q^{65} + 58 q^{67} - 24 q^{71} + 318 q^{73} - 132 q^{75} - 126 q^{77} + 110 q^{79} + 18 q^{81} + 36 q^{85} - 144 q^{87} - 378 q^{89} - 24 q^{91} + 138 q^{93} - 30 q^{95} - 144 q^{99}+O(q^{100}) \) 4 * q + 6 * q^3 + 6 * q^5 + 8 * q^7 - 18 * q^11 - 36 * q^15 - 30 * q^17 - 6 * q^19 - 54 * q^21 + 30 * q^23 + 4 * q^25 - 48 * q^29 - 42 * q^31 - 90 * q^33 + 42 * q^35 + 62 * q^37 + 12 * q^39 + 8 * q^43 - 144 * q^45 + 174 * q^47 - 20 * q^49 - 54 * q^51 + 78 * q^53 + 12 * q^57 + 78 * q^59 + 42 * q^61 - 216 * q^63 - 84 * q^65 + 58 * q^67 - 24 * q^71 + 318 * q^73 - 132 * q^75 - 126 * q^77 + 110 * q^79 + 18 * q^81 + 36 * q^85 - 144 * q^87 - 378 * q^89 - 24 * q^91 + 138 * q^93 - 30 * q^95 - 144 * q^99

Introduction

L-functions

Modular forms

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