LMFDB - Embedded newform 672.2.q.c.193.1

Newspace parameters

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.q (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			193.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	672.193
Dual form			672.2.q.c.289.1

$q$-expansion

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(0.500000 - 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 - 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +5.00000 q^{13} +(1.00000 - 1.73205i) q^{17} +(-1.50000 - 2.59808i) q^{19} +(2.00000 + 1.73205i) q^{21} +(-1.00000 - 1.73205i) q^{23} +(2.50000 - 4.33013i) q^{25} +1.00000 q^{27} +8.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(2.50000 + 4.33013i) q^{37} +(-2.50000 + 4.33013i) q^{39} +2.00000 q^{41} +7.00000 q^{43} +(4.00000 + 6.92820i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(1.00000 + 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} +3.00000 q^{57} +(-5.00000 + 8.66025i) q^{59} +(1.00000 + 1.73205i) q^{61} +(-2.50000 + 0.866025i) q^{63} +(5.50000 - 9.52628i) q^{67} +2.00000 q^{69} +12.0000 q^{71} +(1.50000 - 2.59808i) q^{73} +(2.50000 + 4.33013i) q^{75} +(4.00000 + 3.46410i) q^{77} +(-8.50000 - 14.7224i) q^{79} +(-0.500000 + 0.866025i) q^{81} -16.0000 q^{83} +(-4.00000 + 6.92820i) q^{87} +(-6.00000 - 10.3923i) q^{89} +(2.50000 - 12.9904i) q^{91} +(0.500000 + 0.866025i) q^{93} -14.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q - q^{3} + q^{7} - q^{9} + O(q^{10}) $	$ 2q - q^{3} + q^{7} - q^{9} - 2q^{11} + 10q^{13} + 2q^{17} - 3q^{19} + 4q^{21} - 2q^{23} + 5q^{25} + 2q^{27} + 16q^{29} + q^{31} - 2q^{33} + 5q^{37} - 5q^{39} + 4q^{41} + 14q^{43} + 8q^{47} - 13q^{49} + 2q^{51} + 2q^{53} + 6q^{57} - 10q^{59} + 2q^{61} - 5q^{63} + 11q^{67} + 4q^{69} + 24q^{71} + 3q^{73} + 5q^{75} + 8q^{77} - 17q^{79} - q^{81} - 32q^{83} - 8q^{87} - 12q^{89} + 5q^{91} + q^{93} - 28q^{97} + 4q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	5.00000			1.38675			0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000			1.38675			0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	−0.718900	−	0.695113i	$-0.755354\pi$
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	0.961436	+	0.275029i	$0.0886875\pi$
$18$		0			0
$19$	−1.50000	−	2.59808i	−0.344124	−	0.596040i	0.641071	−	0.767482i	$-0.278491\pi$
$19$	−1.50000	−	2.59808i	−0.344124	−	0.596040i	−0.985194	+	0.171442i	$0.945157\pi$
$20$		0			0
$21$	2.00000	+	1.73205i	0.436436	+	0.377964i
$22$		0			0
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	0.742732	−	0.669588i	$-0.233529\pi$
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	−0.951247	+	0.308431i	$0.900196\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	8.00000			1.48556			0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000			1.48556			0.742781	+	0.669534i	$0.233506\pi$
$30$		0			0
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	−0.817625	−	0.575751i	$-0.804710\pi$
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	0.907428	+	0.420208i	$0.138043\pi$
$32$		0			0
$33$	−1.00000	−	1.73205i	−0.174078	−	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.50000	+	4.33013i	0.410997	+	0.711868i	0.994999	−	0.0998840i	$-0.0318472\pi$
$37$	2.50000	+	4.33013i	0.410997	+	0.711868i	−0.584002	+	0.811752i	$0.698514\pi$
$38$		0			0
$39$	−2.50000	+	4.33013i	−0.400320	+	0.693375i
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	7.00000			1.06749			0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000			1.06749			0.533745	+	0.845645i	$0.320784\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00000	+	6.92820i	0.583460	+	1.01058i	0.995066	+	0.0992202i	$0.0316348\pi$
$47$	4.00000	+	6.92820i	0.583460	+	1.01058i	−0.411606	+	0.911362i	$0.635032\pi$
$48$		0			0
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$		0			0
$51$	1.00000	+	1.73205i	0.140028	+	0.242536i
$52$		0			0
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	−0.789136	−	0.614218i	$-0.789471\pi$
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	0.926497	+	0.376303i	$0.122805\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.00000			0.397360
$58$		0			0
$59$	−5.00000	+	8.66025i	−0.650945	+	1.12747i	0.331949	+	0.943297i	$0.392294\pi$
$59$	−5.00000	+	8.66025i	−0.650945	+	1.12747i	−0.982894	+	0.184172i	$0.941040\pi$
$60$		0			0
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	0.922916	−	0.385002i	$-0.125799\pi$
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	−0.794879	+	0.606768i	$0.792466\pi$
$62$		0			0
$63$	−2.50000	+	0.866025i	−0.314970	+	0.109109i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$	2.00000			0.240772
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	1.50000	−	2.59808i	0.175562	−	0.304082i	−0.764794	−	0.644275i	$-0.777159\pi$
$73$	1.50000	−	2.59808i	0.175562	−	0.304082i	0.940356	+	0.340193i	$0.110493\pi$
$74$		0			0
$75$	2.50000	+	4.33013i	0.288675	+	0.500000i
$76$		0			0
$77$	4.00000	+	3.46410i	0.455842	+	0.394771i
$78$		0			0
$79$	−8.50000	−	14.7224i	−0.956325	−	1.65640i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−8.50000	−	14.7224i	−0.956325	−	1.65640i	−0.225018	−	0.974355i	$-0.572244\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.00000	+	6.92820i	−0.428845	+	0.742781i
$88$		0			0
$89$	−6.00000	−	10.3923i	−0.635999	−	1.10158i	−0.986303	−	0.164946i	$-0.947255\pi$
$89$	−6.00000	−	10.3923i	−0.635999	−	1.10158i	0.350304	−	0.936636i	$-0.386078\pi$
$90$		0			0
$91$	2.50000	−	12.9904i	0.262071	−	1.36176i
$92$		0			0
$93$	0.500000	+	0.866025i	0.0518476	+	0.0898027i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.0623905	+	0.998052i	$0.519872\pi$
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.833143	+	0.553058i	$0.813461\pi$
$102$		0			0
$103$	0.500000	+	0.866025i	0.0492665	+	0.0853320i	0.889607	−	0.456727i	$-0.150978\pi$
$103$	0.500000	+	0.866025i	0.0492665	+	0.0853320i	−0.840341	+	0.542059i	$0.817645\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.00000	−	1.73205i	−0.0966736	−	0.167444i	0.813632	−	0.581380i	$-0.197487\pi$
$107$	−1.00000	−	1.73205i	−0.0966736	−	0.167444i	−0.910306	+	0.413936i	$0.864154\pi$
$108$		0			0
$109$	−0.500000	+	0.866025i	−0.0478913	+	0.0829502i	−0.888977	−	0.457951i	$-0.848583\pi$
$109$	−0.500000	+	0.866025i	−0.0478913	+	0.0829502i	0.841086	+	0.540901i	$0.181917\pi$
$110$		0			0
$111$	−5.00000			−0.474579
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.50000	−	4.33013i	−0.231125	−	0.400320i
$118$		0			0
$119$	−4.00000	−	3.46410i	−0.366679	−	0.317554i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$	−1.00000	+	1.73205i	−0.0901670	+	0.156174i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	11.0000			0.976092			0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000			0.976092			0.488046	+	0.872818i	$0.337710\pi$
$128$		0			0
$129$	−3.50000	+	6.06218i	−0.308158	+	0.533745i
$130$		0			0
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	0.966803	−	0.255524i	$-0.0822479\pi$
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	−0.704692	+	0.709514i	$0.748915\pi$
$132$		0			0
$133$	−7.50000	+	2.59808i	−0.650332	+	0.225282i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.00000	+	3.46410i	−0.170872	+	0.295958i	−0.938725	−	0.344668i	$-0.887992\pi$
$137$	−2.00000	+	3.46410i	−0.170872	+	0.295958i	0.767853	+	0.640626i	$0.221325\pi$
$138$		0			0
$139$	−1.00000			−0.0848189			−0.0424094	−	0.999100i	$-0.513503\pi$
$139$	−1.00000			−0.0848189			−0.0424094	+	0.999100i	$0.513503\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$	−5.00000	+	8.66025i	−0.418121	+	0.724207i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.50000	−	4.33013i	0.453632	−	0.357143i
$148$		0			0
$149$	−6.00000	−	10.3923i	−0.491539	−	0.851371i	0.508413	−	0.861113i	$-0.330232\pi$
$149$	−6.00000	−	10.3923i	−0.491539	−	0.851371i	−0.999953	+	0.00974235i	$0.996899\pi$
$150$		0			0
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	−0.981617	−	0.190864i	$-0.938871\pi$
$151$	−4.00000	+	6.92820i	−0.325515	+	0.563809i	0.656101	+	0.754673i	$0.272204\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3.00000	−	5.19615i	0.239426	−	0.414698i	−0.721124	−	0.692806i	$-0.756374\pi$
$157$	3.00000	−	5.19615i	0.239426	−	0.414698i	0.960550	+	0.278108i	$0.0897074\pi$
$158$		0			0
$159$	1.00000	+	1.73205i	0.0793052	+	0.137361i
$160$		0			0
$161$	−5.00000	+	1.73205i	−0.394055	+	0.136505i
$162$		0			0
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	0.777007	−	0.629492i	$-0.216737\pi$
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	−0.933659	+	0.358162i	$0.883403\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	20.0000			1.54765			0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000			1.54765			0.773823	+	0.633402i	$0.218342\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	−1.50000	+	2.59808i	−0.114708	+	0.198680i
$172$		0			0
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	0.957241	−	0.289292i	$-0.0934200\pi$
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	−0.729155	+	0.684349i	$0.760087\pi$
$174$		0			0
$175$	−10.0000	−	8.66025i	−0.755929	−	0.654654i
$176$		0			0
$177$	−5.00000	−	8.66025i	−0.375823	−	0.650945i
$178$		0			0
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	−0.998286	−	0.0585225i	$-0.981361\pi$
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	0.549825	+	0.835280i	$0.314694\pi$
$180$		0			0
$181$	−15.0000			−1.11494			−0.557471	−	0.830197i	$-0.688228\pi$
$181$	−15.0000			−1.11494			−0.557471	+	0.830197i	$0.688228\pi$
$182$		0			0
$183$	−2.00000			−0.147844
$184$		0			0
$185$		0			0
$186$		0			0
$187$	2.00000	+	3.46410i	0.146254	+	0.253320i
$188$		0			0
$189$	0.500000	−	2.59808i	0.0363696	−	0.188982i
$190$		0			0
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	0.563081	−	0.826402i	$-0.309616\pi$
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	−0.997225	+	0.0744412i	$0.976283\pi$
$192$		0			0
$193$	−11.5000	+	19.9186i	−0.827788	+	1.43377i	0.0719816	+	0.997406i	$0.477068\pi$
$193$	−11.5000	+	19.9186i	−0.827788	+	1.43377i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	−2.00000	+	3.46410i	−0.141776	+	0.245564i	−0.928166	−	0.372168i	$-0.878615\pi$
$199$	−2.00000	+	3.46410i	−0.141776	+	0.245564i	0.786389	+	0.617731i	$0.211948\pi$
$200$		0			0
$201$	5.50000	+	9.52628i	0.387940	+	0.671932i
$202$		0			0
$203$	4.00000	−	20.7846i	0.280745	−	1.45879i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.00000	+	1.73205i	−0.0695048	+	0.120386i
$208$		0			0
$209$	6.00000			0.415029
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$	−6.00000	+	10.3923i	−0.411113	+	0.712069i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.00000	−	1.73205i	−0.135769	−	0.117579i
$218$		0			0
$219$	1.50000	+	2.59808i	0.101361	+	0.175562i
$220$		0			0
$221$	5.00000	−	8.66025i	0.336336	−	0.582552i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−5.00000			−0.333333
$226$		0			0
$227$	−13.0000	+	22.5167i	−0.862840	+	1.49448i	0.00633544	+	0.999980i	$0.497983\pi$
$227$	−13.0000	+	22.5167i	−0.862840	+	1.49448i	−0.869176	+	0.494503i	$0.835350\pi$
$228$		0			0
$229$	−6.50000	−	11.2583i	−0.429532	−	0.743971i	0.567300	−	0.823511i	$-0.307988\pi$
$229$	−6.50000	−	11.2583i	−0.429532	−	0.743971i	−0.996832	+	0.0795401i	$0.974655\pi$
$230$		0			0
$231$	−5.00000	+	1.73205i	−0.328976	+	0.113961i
$232$		0			0
$233$	6.00000	+	10.3923i	0.393073	+	0.680823i	0.992853	−	0.119342i	$-0.0380786\pi$
$233$	6.00000	+	10.3923i	0.393073	+	0.680823i	−0.599780	+	0.800165i	$0.704745\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	17.0000			1.10427
$238$		0			0
$239$	30.0000			1.94054			0.970269	−	0.242028i	$-0.0778125\pi$
$239$	30.0000			1.94054			0.970269	+	0.242028i	$0.0778125\pi$
$240$		0			0
$241$	11.0000	−	19.0526i	0.708572	−	1.22728i	−0.256814	−	0.966461i	$-0.582673\pi$
$241$	11.0000	−	19.0526i	0.708572	−	1.22728i	0.965387	−	0.260822i	$-0.0839937\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−7.50000	−	12.9904i	−0.477214	−	0.826558i
$248$		0			0
$249$	8.00000	−	13.8564i	0.506979	−	0.878114i
$250$		0			0
$251$	−22.0000			−1.38863			−0.694314	−	0.719672i	$-0.744292\pi$
$251$	−22.0000			−1.38863			−0.694314	+	0.719672i	$0.744292\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.00000	+	13.8564i	0.499026	+	0.864339i	0.999999	−	0.00112398i	$-0.000357774\pi$
$257$	8.00000	+	13.8564i	0.499026	+	0.864339i	−0.500973	+	0.865463i	$0.667024\pi$
$258$		0			0
$259$	12.5000	−	4.33013i	0.776712	−	0.269061i
$260$		0			0
$261$	−4.00000	−	6.92820i	−0.247594	−	0.428845i
$262$		0			0
$263$	−4.00000	+	6.92820i	−0.246651	+	0.427211i	−0.962594	−	0.270947i	$-0.912663\pi$
$263$	−4.00000	+	6.92820i	−0.246651	+	0.427211i	0.715944	+	0.698158i	$0.245997\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	12.0000			0.734388
$268$		0			0
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	−0.449622	−	0.893219i	$-0.648441\pi$
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	0.998361	−	0.0572259i	$-0.0182255\pi$
$270$		0			0
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	0.999870	−	0.0161307i	$-0.00513477\pi$
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	−0.513905	+	0.857847i	$0.671801\pi$
$272$		0			0
$273$	10.0000	+	8.66025i	0.605228	+	0.524142i
$274$		0			0
$275$	5.00000	+	8.66025i	0.301511	+	0.522233i
$276$		0			0
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	0.489207	+	0.872167i	$0.337286\pi$
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	−0.999923	+	0.0124177i	$0.996047\pi$
$278$		0			0
$279$	−1.00000			−0.0598684
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$	−15.5000	+	26.8468i	−0.921379	+	1.59588i	−0.124096	+	0.992270i	$0.539603\pi$
$283$	−15.5000	+	26.8468i	−0.921379	+	1.59588i	−0.797283	+	0.603606i	$0.793730\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.00000	−	5.19615i	0.0590281	−	0.306719i
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$	7.00000	−	12.1244i	0.410347	−	0.710742i
$292$		0			0
$293$	−4.00000			−0.233682			−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000			−0.233682			−0.116841	+	0.993151i	$0.537277\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.00000	+	1.73205i	−0.0580259	+	0.100504i
$298$		0			0
$299$	−5.00000	−	8.66025i	−0.289157	−	0.500835i
$300$		0			0
$301$	3.50000	−	18.1865i	0.201737	−	1.04825i
$302$		0			0
$303$	−9.00000	−	15.5885i	−0.517036	−	0.895533i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−23.0000			−1.31268			−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000			−1.31268			−0.656340	+	0.754466i	$0.727896\pi$
$308$		0			0
$309$	−1.00000			−0.0568880
$310$		0			0
$311$	15.0000	−	25.9808i	0.850572	−	1.47323i	−0.0301210	−	0.999546i	$-0.509589\pi$
$311$	15.0000	−	25.9808i	0.850572	−	1.47323i	0.880693	−	0.473688i	$-0.157077\pi$
$312$		0			0
$313$	8.50000	+	14.7224i	0.480448	+	0.832161i	0.999748	−	0.0224310i	$-0.00714060\pi$
$313$	8.50000	+	14.7224i	0.480448	+	0.832161i	−0.519300	+	0.854592i	$0.673807\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.0000	−	17.3205i	−0.561656	−	0.972817i	−0.997352	−	0.0727229i	$-0.976831\pi$
$317$	−10.0000	−	17.3205i	−0.561656	−	0.972817i	0.435696	−	0.900094i	$-0.356502\pi$
$318$		0			0
$319$	−8.00000	+	13.8564i	−0.447914	+	0.775810i
$320$		0			0
$321$	2.00000			0.111629
$322$		0			0
$323$	−6.00000			−0.333849
$324$		0			0
$325$	12.5000	−	21.6506i	0.693375	−	1.20096i
$326$		0			0
$327$	−0.500000	−	0.866025i	−0.0276501	−	0.0478913i
$328$		0			0
$329$	20.0000	−	6.92820i	1.10264	−	0.381964i
$330$		0			0
$331$	7.50000	+	12.9904i	0.412237	+	0.714016i	0.995134	−	0.0985303i	$-0.0314141\pi$
$331$	7.50000	+	12.9904i	0.412237	+	0.714016i	−0.582897	+	0.812546i	$0.698081\pi$
$332$		0			0
$333$	2.50000	−	4.33013i	0.136999	−	0.237289i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.00000			0.490261			0.245131	−	0.969490i	$-0.421169\pi$
$337$	9.00000			0.490261			0.245131	+	0.969490i	$0.421169\pi$
$338$		0			0
$339$	6.00000	−	10.3923i	0.325875	−	0.564433i
$340$		0			0
$341$	1.00000	+	1.73205i	0.0541530	+	0.0937958i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.0000	−	25.9808i	0.805242	−	1.39472i	−0.110885	−	0.993833i	$-0.535369\pi$
$347$	15.0000	−	25.9808i	0.805242	−	1.39472i	0.916127	−	0.400887i	$-0.131298\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$	5.00000			0.266880
$352$		0			0
$353$	−12.0000	+	20.7846i	−0.638696	+	1.10625i	0.347024	+	0.937856i	$0.387192\pi$
$353$	−12.0000	+	20.7846i	−0.638696	+	1.10625i	−0.985719	+	0.168397i	$0.946141\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	5.00000	−	1.73205i	0.264628	−	0.0916698i
$358$		0			0
$359$	−16.0000	−	27.7128i	−0.844448	−	1.46263i	−0.886100	−	0.463494i	$-0.846596\pi$
$359$	−16.0000	−	27.7128i	−0.844448	−	1.46263i	0.0416523	−	0.999132i	$-0.486738\pi$
$360$		0			0
$361$	5.00000	−	8.66025i	0.263158	−	0.455803i
$362$		0			0
$363$	−7.00000			−0.367405
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	−0.997960	−	0.0638362i	$-0.979666\pi$
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	0.554264	+	0.832341i	$0.313000\pi$
$368$		0			0
$369$	−1.00000	−	1.73205i	−0.0520579	−	0.0901670i
$370$		0			0
$371$	−4.00000	−	3.46410i	−0.207670	−	0.179847i
$372$		0			0
$373$	10.5000	+	18.1865i	0.543669	+	0.941663i	0.998689	+	0.0511818i	$0.0162988\pi$
$373$	10.5000	+	18.1865i	0.543669	+	0.941663i	−0.455020	+	0.890481i	$0.650368\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	40.0000			2.06010
$378$		0			0
$379$	1.00000			0.0513665			0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000			0.0513665			0.0256833	+	0.999670i	$0.491824\pi$
$380$		0			0
$381$	−5.50000	+	9.52628i	−0.281774	+	0.488046i
$382$		0			0
$383$	11.0000	+	19.0526i	0.562074	+	0.973540i	0.997315	+	0.0732266i	$0.0233296\pi$
$383$	11.0000	+	19.0526i	0.562074	+	0.973540i	−0.435242	+	0.900314i	$0.643337\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.50000	−	6.06218i	−0.177915	−	0.308158i
$388$		0			0
$389$	12.0000	−	20.7846i	0.608424	−	1.05382i	−0.383076	−	0.923717i	$-0.625135\pi$
$389$	12.0000	−	20.7846i	0.608424	−	1.05382i	0.991500	−	0.130105i	$-0.0415314\pi$
$390$		0			0
$391$	−4.00000			−0.202289
$392$		0			0
$393$	−6.00000			−0.302660
$394$		0			0
$395$		0			0
$396$		0			0
$397$	6.50000	+	11.2583i	0.326226	+	0.565039i	0.981760	−	0.190126i	$-0.0608897\pi$
$397$	6.50000	+	11.2583i	0.326226	+	0.565039i	−0.655534	+	0.755166i	$0.727556\pi$
$398$		0			0
$399$	1.50000	−	7.79423i	0.0750939	−	0.390199i
$400$		0			0
$401$	−5.00000	−	8.66025i	−0.249688	−	0.432472i	0.713751	−	0.700399i	$-0.246995\pi$
$401$	−5.00000	−	8.66025i	−0.249688	−	0.432472i	−0.963439	+	0.267927i	$0.913661\pi$
$402$		0			0
$403$	2.50000	−	4.33013i	0.124534	−	0.215699i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.0000			−0.495682
$408$		0			0
$409$	17.5000	−	30.3109i	0.865319	−	1.49878i	−0.00141047	−	0.999999i	$-0.500449\pi$
$409$	17.5000	−	30.3109i	0.865319	−	1.49878i	0.866730	−	0.498778i	$-0.166218\pi$
$410$		0			0
$411$	−2.00000	−	3.46410i	−0.0986527	−	0.170872i
$412$		0			0
$413$	20.0000	+	17.3205i	0.984136	+	0.852286i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.500000	−	0.866025i	0.0244851	−	0.0424094i
$418$		0			0
$419$	18.0000			0.879358			0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000			0.879358			0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$	9.00000			0.438633			0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000			0.438633			0.219317	+	0.975654i	$0.429617\pi$
$422$		0			0
$423$	4.00000	−	6.92820i	0.194487	−	0.336861i
$424$		0			0
$425$	−5.00000	−	8.66025i	−0.242536	−	0.420084i
$426$		0			0
$427$	5.00000	−	1.73205i	0.241967	−	0.0838198i
$428$		0			0
$429$	−5.00000	−	8.66025i	−0.241402	−	0.418121i
$430$		0			0
$431$	−10.0000	+	17.3205i	−0.481683	+	0.834300i	−0.999779	−	0.0210230i	$-0.993308\pi$
$431$	−10.0000	+	17.3205i	−0.481683	+	0.834300i	0.518096	+	0.855323i	$0.326641\pi$
$432$		0			0
$433$	−17.0000			−0.816968			−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000			−0.816968			−0.408484	+	0.912766i	$0.633942\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.00000	+	5.19615i	−0.143509	+	0.248566i
$438$		0			0
$439$	12.0000	+	20.7846i	0.572729	+	0.991995i	0.996284	+	0.0861252i	$0.0274485\pi$
$439$	12.0000	+	20.7846i	0.572729	+	0.991995i	−0.423556	+	0.905870i	$0.639218\pi$
$440$		0			0
$441$	1.00000	+	6.92820i	0.0476190	+	0.329914i
$442$		0			0
$443$	−3.00000	−	5.19615i	−0.142534	−	0.246877i	0.785916	−	0.618333i	$-0.212192\pi$
$443$	−3.00000	−	5.19615i	−0.142534	−	0.246877i	−0.928450	+	0.371457i	$0.878858\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	12.0000			0.567581
$448$		0			0
$449$	−24.0000			−1.13263			−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000			−1.13263			−0.566315	+	0.824189i	$0.691631\pi$
$450$		0			0
$451$	−2.00000	+	3.46410i	−0.0941763	+	0.163118i
$452$		0			0
$453$	−4.00000	−	6.92820i	−0.187936	−	0.325515i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.5000	+	19.9186i	0.537947	+	0.931752i	0.999014	+	0.0443868i	$0.0141334\pi$
$457$	11.5000	+	19.9186i	0.537947	+	0.931752i	−0.461067	+	0.887365i	$0.652533\pi$
$458$		0			0
$459$	1.00000	−	1.73205i	0.0466760	−	0.0808452i
$460$		0			0
$461$	−20.0000			−0.931493			−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000			−0.931493			−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$	9.00000			0.418265			0.209133	−	0.977887i	$-0.432936\pi$
$463$	9.00000			0.418265			0.209133	+	0.977887i	$0.432936\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.00000	−	10.3923i	−0.277647	−	0.480899i	0.693153	−	0.720791i	$-0.256221\pi$
$467$	−6.00000	−	10.3923i	−0.277647	−	0.480899i	−0.970799	+	0.239892i	$0.922888\pi$
$468$		0			0
$469$	−22.0000	−	19.0526i	−1.01587	−	0.879765i
$470$		0			0
$471$	3.00000	+	5.19615i	0.138233	+	0.239426i
$472$		0			0
$473$	−7.00000	+	12.1244i	−0.321860	+	0.557478i
$474$		0			0
$475$	−15.0000			−0.688247
$476$		0			0
$477$	−2.00000			−0.0915737
$478$		0			0
$479$	−9.00000	+	15.5885i	−0.411220	+	0.712255i	−0.995023	−	0.0996406i	$-0.968231\pi$
$479$	−9.00000	+	15.5885i	−0.411220	+	0.712255i	0.583803	+	0.811895i	$0.301564\pi$
$480$		0			0
$481$	12.5000	+	21.6506i	0.569951	+	0.987184i
$482$		0			0
$483$	1.00000	−	5.19615i	0.0455016	−	0.236433i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−10.5000	+	18.1865i	−0.475800	+	0.824110i	−0.999616	−	0.0277214i	$-0.991175\pi$
$487$	−10.5000	+	18.1865i	−0.475800	+	0.824110i	0.523815	+	0.851832i	$0.324508\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	32.0000			1.44414			0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000			1.44414			0.722070	+	0.691820i	$0.243191\pi$
$492$		0			0
$493$	8.00000	−	13.8564i	0.360302	−	0.624061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.00000	−	31.1769i	0.269137	−	1.39848i
$498$		0			0
$499$	20.5000	+	35.5070i	0.917706	+	1.58951i	0.802890	+	0.596127i	$0.203294\pi$
$499$	20.5000	+	35.5070i	0.917706	+	1.58951i	0.114816	+	0.993387i	$0.463372\pi$
$500$		0			0
$501$	−10.0000	+	17.3205i	−0.446767	+	0.773823i
$502$		0			0
$503$	14.0000			0.624229			0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000			0.624229			0.312115	+	0.950044i	$0.398963\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−6.00000	+	10.3923i	−0.266469	+	0.461538i
$508$		0			0
$509$	2.00000	+	3.46410i	0.0886484	+	0.153544i	0.906940	−	0.421260i	$-0.138412\pi$
$509$	2.00000	+	3.46410i	0.0886484	+	0.153544i	−0.818292	+	0.574803i	$0.805079\pi$
$510$		0			0
$511$	−6.00000	−	5.19615i	−0.265424	−	0.229864i
$512$		0			0
$513$	−1.50000	−	2.59808i	−0.0662266	−	0.114708i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−16.0000			−0.703679
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	−16.0000	+	27.7128i	−0.700973	+	1.21412i	0.267153	+	0.963654i	$0.413917\pi$
$521$	−16.0000	+	27.7128i	−0.700973	+	1.21412i	−0.968125	+	0.250466i	$0.919416\pi$
$522$		0			0
$523$	5.50000	+	9.52628i	0.240498	+	0.416555i	0.960856	−	0.277047i	$-0.0893559\pi$
$523$	5.50000	+	9.52628i	0.240498	+	0.416555i	−0.720358	+	0.693602i	$0.756023\pi$
$524$		0			0
$525$	12.5000	−	4.33013i	0.545545	−	0.188982i
$526$		0			0
$527$	−1.00000	−	1.73205i	−0.0435607	−	0.0754493i
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$		0			0
$531$	10.0000			0.433963
$532$		0			0
$533$	10.0000			0.433148
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−6.00000	−	10.3923i	−0.258919	−	0.448461i
$538$		0			0
$539$	11.0000	−	8.66025i	0.473804	−	0.373024i
$540$		0			0
$541$	13.5000	+	23.3827i	0.580410	+	1.00530i	0.995431	+	0.0954880i	$0.0304412\pi$
$541$	13.5000	+	23.3827i	0.580410	+	1.00530i	−0.415020	+	0.909812i	$0.636226\pi$
$542$		0			0
$543$	7.50000	−	12.9904i	0.321856	−	0.557471i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−44.0000			−1.88130			−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000			−1.88130			−0.940652	+	0.339372i	$0.889785\pi$
$548$		0			0
$549$	1.00000	−	1.73205i	0.0426790	−	0.0739221i
$550$		0			0
$551$	−12.0000	−	20.7846i	−0.511217	−	0.885454i
$552$		0			0
$553$	−42.5000	+	14.7224i	−1.80728	+	0.626061i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.00000	+	15.5885i	−0.381342	+	0.660504i	−0.991254	−	0.131965i	$-0.957871\pi$
$557$	−9.00000	+	15.5885i	−0.381342	+	0.660504i	0.609912	+	0.792469i	$0.291205\pi$
$558$		0			0
$559$	35.0000			1.48034
$560$		0			0
$561$	−4.00000			−0.168880
$562$		0			0
$563$	10.0000	−	17.3205i	0.421450	−	0.729972i	−0.574632	−	0.818412i	$-0.694855\pi$
$563$	10.0000	−	17.3205i	0.421450	−	0.729972i	0.996082	+	0.0884397i	$0.0281881\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	2.00000	+	1.73205i	0.0839921	+	0.0727393i
$568$		0			0
$569$	−21.0000	−	36.3731i	−0.880366	−	1.52484i	−0.850935	−	0.525271i	$-0.823964\pi$
$569$	−21.0000	−	36.3731i	−0.880366	−	1.52484i	−0.0294311	−	0.999567i	$-0.509370\pi$
$570$		0			0
$571$	−11.5000	+	19.9186i	−0.481260	+	0.833567i	−0.999769	−	0.0215055i	$-0.993154\pi$
$571$	−11.5000	+	19.9186i	−0.481260	+	0.833567i	0.518509	+	0.855072i	$0.326487\pi$
$572$		0			0
$573$	12.0000			0.501307
$574$		0			0
$575$	−10.0000			−0.417029
$576$		0			0
$577$	−3.50000	+	6.06218i	−0.145707	+	0.252372i	−0.929636	−	0.368478i	$-0.879879\pi$
$577$	−3.50000	+	6.06218i	−0.145707	+	0.252372i	0.783930	+	0.620850i	$0.213212\pi$
$578$		0			0
$579$	−11.5000	−	19.9186i	−0.477924	−	0.827788i
$580$		0			0
$581$	−8.00000	+	41.5692i	−0.331896	+	1.72458i
$582$		0			0
$583$	2.00000	+	3.46410i	0.0828315	+	0.143468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−24.0000			−0.990586			−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000			−0.990586			−0.495293	+	0.868726i	$0.664939\pi$
$588$		0			0
$589$	−3.00000			−0.123613
$590$		0			0
$591$	9.00000	−	15.5885i	0.370211	−	0.641223i
$592$		0			0
$593$	−15.0000	−	25.9808i	−0.615976	−	1.06690i	−0.990212	−	0.139569i	$-0.955428\pi$
$593$	−15.0000	−	25.9808i	−0.615976	−	1.06690i	0.374236	−	0.927333i	$-0.377905\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.00000	−	3.46410i	−0.0818546	−	0.141776i
$598$		0			0
$599$	18.0000	−	31.1769i	0.735460	−	1.27385i	−0.219061	−	0.975711i	$-0.570299\pi$
$599$	18.0000	−	31.1769i	0.735460	−	1.27385i	0.954521	−	0.298143i	$-0.0963673\pi$
$600$		0			0
$601$	−37.0000			−1.50926			−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000			−1.50926			−0.754631	+	0.656150i	$0.772184\pi$
$602$		0			0
$603$	−11.0000			−0.447955
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−16.5000	−	28.5788i	−0.669714	−	1.15998i	−0.977984	−	0.208680i	$-0.933083\pi$
$607$	−16.5000	−	28.5788i	−0.669714	−	1.15998i	0.308270	−	0.951299i	$-0.400250\pi$
$608$		0			0
$609$	16.0000	+	13.8564i	0.648353	+	0.561490i
$610$		0			0
$611$	20.0000	+	34.6410i	0.809113	+	1.40143i
$612$		0			0
$613$	−13.0000	+	22.5167i	−0.525065	+	0.909439i	0.474509	+	0.880251i	$0.342626\pi$
$613$	−13.0000	+	22.5167i	−0.525065	+	0.909439i	−0.999574	+	0.0291886i	$0.990708\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.0000			−0.563619			−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000			−0.563619			−0.281809	+	0.959470i	$0.590935\pi$
$618$		0			0
$619$	−2.50000	+	4.33013i	−0.100483	+	0.174042i	−0.911884	−	0.410448i	$-0.865372\pi$
$619$	−2.50000	+	4.33013i	−0.100483	+	0.174042i	0.811400	+	0.584491i	$0.198706\pi$
$620$		0			0
$621$	−1.00000	−	1.73205i	−0.0401286	−	0.0695048i
$622$		0			0
$623$	−30.0000	+	10.3923i	−1.20192	+	0.416359i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$	−3.00000	+	5.19615i	−0.119808	+	0.207514i
$628$		0			0
$629$	10.0000			0.398726
$630$		0			0
$631$	−4.00000			−0.159237			−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000			−0.159237			−0.0796187	+	0.996825i	$0.525370\pi$
$632$		0			0
$633$	−10.0000	+	17.3205i	−0.397464	+	0.688428i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−32.5000	−	12.9904i	−1.28770	−	0.514698i
$638$		0			0
$639$	−6.00000	−	10.3923i	−0.237356	−	0.411113i
$640$		0			0
$641$	−13.0000	+	22.5167i	−0.513469	+	0.889355i	0.486409	+	0.873731i	$0.338307\pi$
$641$	−13.0000	+	22.5167i	−0.513469	+	0.889355i	−0.999878	+	0.0156233i	$0.995027\pi$
$642$		0			0
$643$	11.0000			0.433798			0.216899	−	0.976194i	$-0.430406\pi$
$643$	11.0000			0.433798			0.216899	+	0.976194i	$0.430406\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	−0.633090	−	0.774078i	$-0.718214\pi$
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	0.986916	+	0.161233i	$0.0515470\pi$
$648$		0			0
$649$	−10.0000	−	17.3205i	−0.392534	−	0.679889i
$650$		0			0
$651$	2.50000	−	0.866025i	0.0979827	−	0.0339422i
$652$		0			0
$653$	7.00000	+	12.1244i	0.273931	+	0.474463i	0.969865	−	0.243643i	$-0.0783426\pi$
$653$	7.00000	+	12.1244i	0.273931	+	0.474463i	−0.695934	+	0.718106i	$0.745009\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−3.00000			−0.117041
$658$		0			0
$659$	−42.0000			−1.63609			−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000			−1.63609			−0.818044	+	0.575156i	$0.804941\pi$
$660$		0			0
$661$	−1.50000	+	2.59808i	−0.0583432	+	0.101053i	−0.893722	−	0.448622i	$-0.851915\pi$
$661$	−1.50000	+	2.59808i	−0.0583432	+	0.101053i	0.835379	+	0.549675i	$0.185248\pi$
$662$		0			0
$663$	5.00000	+	8.66025i	0.194184	+	0.336336i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.00000	−	13.8564i	−0.309761	−	0.536522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.00000			−0.154418
$672$		0			0
$673$	−21.0000			−0.809491			−0.404745	−	0.914429i	$-0.632640\pi$
$673$	−21.0000			−0.809491			−0.404745	+	0.914429i	$0.632640\pi$
$674$		0			0
$675$	2.50000	−	4.33013i	0.0962250	−	0.166667i
$676$		0			0
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	0.957984	−	0.286820i	$-0.0925982\pi$
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	−0.727386	+	0.686229i	$0.759265\pi$
$678$		0			0
$679$	−7.00000	+	36.3731i	−0.268635	+	1.39587i
$680$		0			0
$681$	−13.0000	−	22.5167i	−0.498161	−	0.862840i
$682$		0			0
$683$	15.0000	−	25.9808i	0.573959	−	0.994126i	−0.422195	−	0.906505i	$-0.638740\pi$
$683$	15.0000	−	25.9808i	0.573959	−	0.994126i	0.996154	−	0.0876211i	$-0.0279265\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	13.0000			0.495981
$688$		0			0
$689$	5.00000	−	8.66025i	0.190485	−	0.329929i
$690$		0			0
$691$	−2.50000	−	4.33013i	−0.0951045	−	0.164726i	0.814548	−	0.580097i	$-0.196985\pi$
$691$	−2.50000	−	4.33013i	−0.0951045	−	0.164726i	−0.909652	+	0.415371i	$0.863652\pi$
$692$		0			0
$693$	1.00000	−	5.19615i	0.0379869	−	0.197386i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.00000	−	3.46410i	0.0757554	−	0.131212i
$698$		0			0
$699$	−12.0000			−0.453882
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$	7.50000	−	12.9904i	0.282868	−	0.489942i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	36.0000	+	31.1769i	1.35392	+	1.17253i
$708$		0			0
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	0.995228	−	0.0975728i	$-0.0311079\pi$
$709$	11.0000	+	19.0526i	0.413114	+	0.715534i	−0.582115	+	0.813107i	$0.697775\pi$
$710$		0			0
$711$	−8.50000	+	14.7224i	−0.318775	+	0.552134i
$712$		0			0
$713$	−2.00000			−0.0749006
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−15.0000	+	25.9808i	−0.560185	+	0.970269i
$718$		0			0
$719$	3.00000	+	5.19615i	0.111881	+	0.193784i	0.916529	−	0.399969i	$-0.130979\pi$
$719$	3.00000	+	5.19615i	0.111881	+	0.193784i	−0.804648	+	0.593753i	$0.797646\pi$
$720$		0			0
$721$	2.50000	−	0.866025i	0.0931049	−	0.0322525i
$722$		0			0
$723$	11.0000	+	19.0526i	0.409094	+	0.708572i
$724$		0			0
$725$	20.0000	−	34.6410i	0.742781	−	1.28654i
$726$		0			0
$727$	13.0000			0.482143			0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000			0.482143			0.241072	+	0.970507i	$0.422501\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	7.00000	−	12.1244i	0.258904	−	0.448435i
$732$		0			0
$733$	9.50000	+	16.4545i	0.350891	+	0.607760i	0.986406	−	0.164328i	$-0.0525456\pi$
$733$	9.50000	+	16.4545i	0.350891	+	0.607760i	−0.635515	+	0.772088i	$0.719212\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11.0000	+	19.0526i	0.405190	+	0.701810i
$738$		0			0
$739$	8.50000	−	14.7224i	0.312678	−	0.541573i	−0.666264	−	0.745716i	$-0.732107\pi$
$739$	8.50000	−	14.7224i	0.312678	−	0.541573i	0.978941	+	0.204143i	$0.0654407\pi$
$740$		0			0
$741$	15.0000			0.551039
$742$		0			0
$743$	−14.0000			−0.513610			−0.256805	−	0.966463i	$-0.582670\pi$
$743$	−14.0000			−0.513610			−0.256805	+	0.966463i	$0.582670\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	8.00000	+	13.8564i	0.292705	+	0.506979i
$748$		0			0
$749$	−5.00000	+	1.73205i	−0.182696	+	0.0632878i
$750$		0			0
$751$	−13.5000	−	23.3827i	−0.492622	−	0.853246i	0.507342	−	0.861745i	$-0.330628\pi$
$751$	−13.5000	−	23.3827i	−0.492622	−	0.853246i	−0.999964	+	0.00849853i	$0.997295\pi$
$752$		0			0
$753$	11.0000	−	19.0526i	0.400862	−	0.694314i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$	−2.00000	+	3.46410i	−0.0725954	+	0.125739i
$760$		0			0
$761$	−15.0000	−	25.9808i	−0.543750	−	0.941802i	−0.998684	−	0.0512772i	$-0.983671\pi$
$761$	−15.0000	−	25.9808i	−0.543750	−	0.941802i	0.454935	−	0.890525i	$-0.349663\pi$
$762$		0			0
$763$	2.00000	+	1.73205i	0.0724049	+	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−25.0000	+	43.3013i	−0.902698	+	1.56352i
$768$		0			0
$769$	−9.00000			−0.324548			−0.162274	−	0.986746i	$-0.551883\pi$
$769$	−9.00000			−0.324548			−0.162274	+	0.986746i	$0.551883\pi$
$770$		0			0
$771$	−16.0000			−0.576226
$772$		0			0
$773$	−9.00000	+	15.5885i	−0.323708	+	0.560678i	−0.981250	−	0.192740i	$-0.938263\pi$
$773$	−9.00000	+	15.5885i	−0.323708	+	0.560678i	0.657542	+	0.753418i	$0.271596\pi$
$774$		0			0
$775$	−2.50000	−	4.33013i	−0.0898027	−	0.155543i
$776$		0			0
$777$	−2.50000	+	12.9904i	−0.0896870	+	0.466027i
$778$		0			0
$779$	−3.00000	−	5.19615i	−0.107486	−	0.186171i
$780$		0			0
$781$	−12.0000	+	20.7846i	−0.429394	+	0.743732i
$782$		0			0
$783$	8.00000			0.285897
$784$		0			0
$785$		0			0
$786$		0			0
$787$	20.0000	−	34.6410i	0.712923	−	1.23482i	−0.250832	−	0.968031i	$-0.580704\pi$
$787$	20.0000	−	34.6410i	0.712923	−	1.23482i	0.963755	−	0.266788i	$-0.0859624\pi$
$788$		0			0
$789$	−4.00000	−	6.92820i	−0.142404	−	0.246651i
$790$		0			0
$791$	−6.00000	+	31.1769i	−0.213335	+	1.10852i
$792$		0			0
$793$	5.00000	+	8.66025i	0.177555	+	0.307535i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−34.0000			−1.20434			−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000			−1.20434			−0.602171	+	0.798367i	$0.705697\pi$
$798$		0			0
$799$	16.0000

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 - 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(0.500000 - 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +5.00000 q^{13} +(1.00000 - 1.73205i) q^{17} +(-1.50000 - 2.59808i) q^{19} +(2.00000 + 1.73205i) q^{21} +(-1.00000 - 1.73205i) q^{23} +(2.50000 - 4.33013i) q^{25} +1.00000 q^{27} +8.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(-1.00000 - 1.73205i) q^{33} +(2.50000 + 4.33013i) q^{37} +(-2.50000 + 4.33013i) q^{39} +2.00000 q^{41} +7.00000 q^{43} +(4.00000 + 6.92820i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(1.00000 + 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} +3.00000 q^{57} +(-5.00000 + 8.66025i) q^{59} +(1.00000 + 1.73205i) q^{61} +(-2.50000 + 0.866025i) q^{63} +(5.50000 - 9.52628i) q^{67} +2.00000 q^{69} +12.0000 q^{71} +(1.50000 - 2.59808i) q^{73} +(2.50000 + 4.33013i) q^{75} +(4.00000 + 3.46410i) q^{77} +(-8.50000 - 14.7224i) q^{79} +(-0.500000 + 0.866025i) q^{81} -16.0000 q^{83} +(-4.00000 + 6.92820i) q^{87} +(-6.00000 - 10.3923i) q^{89} +(2.50000 - 12.9904i) q^{91} +(0.500000 + 0.866025i) q^{93} -14.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q - q^{3} + q^{7} - q^{9} + O(q^{10}) \)	\( 2q - q^{3} + q^{7} - q^{9} - 2q^{11} + 10q^{13} + 2q^{17} - 3q^{19} + 4q^{21} - 2q^{23} + 5q^{25} + 2q^{27} + 16q^{29} + q^{31} - 2q^{33} + 5q^{37} - 5q^{39} + 4q^{41} + 14q^{43} + 8q^{47} - 13q^{49} + 2q^{51} + 2q^{53} + 6q^{57} - 10q^{59} + 2q^{61} - 5q^{63} + 11q^{67} + 4q^{69} + 24q^{71} + 3q^{73} + 5q^{75} + 8q^{77} - 17q^{79} - q^{81} - 32q^{83} - 8q^{87} - 12q^{89} + 5q^{91} + q^{93} - 28q^{97} + 4q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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