LMFDB - Embedded newform 117.4.q.b.82.1

Newspace parameters

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.q (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			82.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	117.82
Dual form			117.4.q.b.10.1

$q$-expansion

$f(q)$	$=$	$q+(-4.00000 - 6.92820i) q^{4} +5.19615i q^{5} +(9.00000 - 5.19615i) q^{7} +O(q^{10})$	\(q+(-4.00000 - 6.92820i) q^{4} +5.19615i q^{5} +(9.00000 - 5.19615i) q^{7} +(-45.0000 - 25.9808i) q^{11} +(-32.5000 - 33.7750i) q^{13} +(-32.0000 + 55.4256i) q^{16} +(-58.5000 - 101.325i) q^{17} +(-21.0000 + 12.1244i) q^{19} +(36.0000 - 20.7846i) q^{20} +(9.00000 - 15.5885i) q^{23} +98.0000 q^{25} +(-72.0000 - 41.5692i) q^{28} +(-49.5000 + 85.7365i) q^{29} -193.990i q^{31} +(27.0000 + 46.7654i) q^{35} +(97.5000 + 56.2917i) q^{37} +(31.5000 + 18.1865i) q^{41} +(41.0000 + 71.0141i) q^{43} +415.692i q^{44} -72.7461i q^{47} +(-117.500 + 203.516i) q^{49} +(-104.000 + 360.267i) q^{52} +261.000 q^{53} +(135.000 - 233.827i) q^{55} +(684.000 - 394.908i) q^{59} +(359.500 + 622.672i) q^{61} +512.000 q^{64} +(175.500 - 168.875i) q^{65} +(-609.000 - 351.606i) q^{67} +(-468.000 + 810.600i) q^{68} +(405.000 - 233.827i) q^{71} -684.160i q^{73} +(168.000 + 96.9948i) q^{76} -540.000 q^{77} -440.000 q^{79} +(-288.000 - 166.277i) q^{80} +1195.12i q^{83} +(526.500 - 303.975i) q^{85} +(-1314.00 - 758.638i) q^{89} +(-468.000 - 135.100i) q^{91} -144.000 q^{92} +(-63.0000 - 109.119i) q^{95} +(-1002.00 + 578.505i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 18 q^{7}+O(q^{10}) $ 2 * q - 8 * q^4 + 18 * q^7	$ 2 q - 8 q^{4} + 18 q^{7} - 90 q^{11} - 65 q^{13} - 64 q^{16} - 117 q^{17} - 42 q^{19} + 72 q^{20} + 18 q^{23} + 196 q^{25} - 144 q^{28} - 99 q^{29} + 54 q^{35} + 195 q^{37} + 63 q^{41} + 82 q^{43} - 235 q^{49} - 208 q^{52} + 522 q^{53} + 270 q^{55} + 1368 q^{59} + 719 q^{61} + 1024 q^{64} + 351 q^{65} - 1218 q^{67} - 936 q^{68} + 810 q^{71} + 336 q^{76} - 1080 q^{77} - 880 q^{79} - 576 q^{80} + 1053 q^{85} - 2628 q^{89} - 936 q^{91} - 288 q^{92} - 126 q^{95} - 2004 q^{97}+O(q^{100}) $ 2 * q - 8 * q^4 + 18 * q^7 - 90 * q^11 - 65 * q^13 - 64 * q^16 - 117 * q^17 - 42 * q^19 + 72 * q^20 + 18 * q^23 + 196 * q^25 - 144 * q^28 - 99 * q^29 + 54 * q^35 + 195 * q^37 + 63 * q^41 + 82 * q^43 - 235 * q^49 - 208 * q^52 + 522 * q^53 + 270 * q^55 + 1368 * q^59 + 719 * q^61 + 1024 * q^64 + 351 * q^65 - 1218 * q^67 - 936 * q^68 + 810 * q^71 + 336 * q^76 - 1080 * q^77 - 880 * q^79 - 576 * q^80 + 1053 * q^85 - 2628 * q^89 - 936 * q^91 - 288 * q^92 - 126 * q^95 - 2004 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$

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		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$		0			0
$3$		0			0
$4$	−4.00000	−	6.92820i	−0.500000	−	0.866025i
$5$			5.19615i			0.464758i	0.972625	+	0.232379i	$0.0746510\pi$
$5$			5.19615i			0.464758i	−0.972625	+	0.232379i	$0.925349\pi$
$6$		0			0
$7$	9.00000	−	5.19615i	0.485954	−	0.280566i	−0.236940	−	0.971524i	$-0.576145\pi$
$7$	9.00000	−	5.19615i	0.485954	−	0.280566i	0.722895	+	0.690958i	$0.242811\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−45.0000	−	25.9808i	−1.23346	−	0.712136i	−0.265707	−	0.964054i	$-0.585605\pi$
$11$	−45.0000	−	25.9808i	−1.23346	−	0.712136i	−0.967749	+	0.251918i	$0.918939\pi$
$12$		0			0
$13$	−32.5000	−	33.7750i	−0.693375	−	0.720577i
$13$	−32.5000	−	33.7750i	−0.693375	−	0.720577i
$14$		0			0
$15$		0			0
$16$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$17$	−58.5000	−	101.325i	−0.834608	−	1.44558i	−0.894349	−	0.447369i	$-0.852361\pi$
$17$	−58.5000	−	101.325i	−0.834608	−	1.44558i	0.0597414	−	0.998214i	$-0.480972\pi$
$18$		0			0
$19$	−21.0000	+	12.1244i	−0.253565	+	0.146396i	−0.621395	−	0.783497i	$-0.713434\pi$
$19$	−21.0000	+	12.1244i	−0.253565	+	0.146396i	0.367831	+	0.929893i	$0.380101\pi$
$20$	36.0000	−	20.7846i	0.402492	−	0.232379i
$21$		0			0
$22$		0			0
$23$	9.00000	−	15.5885i	0.0815926	−	0.141323i	−0.822342	−	0.568994i	$-0.807333\pi$
$23$	9.00000	−	15.5885i	0.0815926	−	0.141323i	0.903934	+	0.427672i	$0.140666\pi$
$24$		0			0
$25$	98.0000			0.784000
$26$		0			0
$27$		0			0
$28$	−72.0000	−	41.5692i	−0.485954	−	0.280566i
$29$	−49.5000	+	85.7365i	−0.316963	+	0.548996i	−0.979853	−	0.199721i	$-0.935996\pi$
$29$	−49.5000	+	85.7365i	−0.316963	+	0.548996i	0.662890	+	0.748717i	$0.269330\pi$
$30$		0			0
$31$		−	193.990i		−	1.12392i	−0.827164	−	0.561961i	$-0.810047\pi$
$31$		−	193.990i		−	1.12392i	0.827164	−	0.561961i	$-0.189953\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	27.0000	+	46.7654i	0.130395	+	0.225851i
$36$		0			0
$37$	97.5000	+	56.2917i	0.433214	+	0.250116i	0.700715	−	0.713442i	$-0.252865\pi$
$37$	97.5000	+	56.2917i	0.433214	+	0.250116i	−0.267501	+	0.963558i	$0.586198\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	31.5000	+	18.1865i	0.119987	+	0.0692746i	0.558792	−	0.829308i	$-0.311265\pi$
$41$	31.5000	+	18.1865i	0.119987	+	0.0692746i	−0.438805	+	0.898582i	$0.644598\pi$
$42$		0			0
$43$	41.0000	+	71.0141i	0.145406	+	0.251850i	0.929524	−	0.368761i	$-0.120218\pi$
$43$	41.0000	+	71.0141i	0.145406	+	0.251850i	−0.784119	+	0.620611i	$0.786885\pi$
$44$			415.692i			1.42427i
$45$		0			0
$46$		0			0
$47$		−	72.7461i		−	0.225768i	−0.993608	−	0.112884i	$-0.963991\pi$
$47$		−	72.7461i		−	0.225768i	0.993608	−	0.112884i	$-0.0360089\pi$
$48$		0			0
$49$	−117.500	+	203.516i	−0.342566	+	0.593341i
$50$		0			0
$51$		0			0
$52$	−104.000	+	360.267i	−0.277350	+	0.960769i
$53$	261.000			0.676436			0.338218	−	0.941068i	$-0.390176\pi$
$53$	261.000			0.676436			0.338218	+	0.941068i	$0.390176\pi$
$54$		0			0
$55$	135.000	−	233.827i	0.330971	−	0.573258i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	684.000	−	394.908i	1.50931	−	0.871400i	0.509368	−	0.860549i	$-0.329879\pi$
$59$	684.000	−	394.908i	1.50931	−	0.871400i	0.999941	−	0.0108508i	$-0.00345397\pi$
$60$		0			0
$61$	359.500	+	622.672i	0.754578	+	1.30697i	0.945584	+	0.325379i	$0.105492\pi$
$61$	359.500	+	622.672i	0.754578	+	1.30697i	−0.191006	+	0.981589i	$0.561175\pi$
$62$		0			0
$63$		0			0
$64$	512.000			1.00000
$65$	175.500	−	168.875i	0.334894	−	0.322252i
$66$		0			0
$67$	−609.000	−	351.606i	−1.11047	−	0.641128i	−0.171516	−	0.985181i	$-0.554866\pi$
$67$	−609.000	−	351.606i	−1.11047	−	0.641128i	−0.938950	+	0.344054i	$0.888200\pi$
$68$	−468.000	+	810.600i	−0.834608	+	1.44558i
$69$		0			0
$70$		0			0
$71$	405.000	−	233.827i	0.676967	−	0.390847i	−0.121744	−	0.992561i	$-0.538849\pi$
$71$	405.000	−	233.827i	0.676967	−	0.390847i	0.798711	+	0.601714i	$0.205515\pi$
$72$		0			0
$73$		−	684.160i		−	1.09692i	−0.836178	−	0.548458i	$-0.815215\pi$
$73$		−	684.160i		−	1.09692i	0.836178	−	0.548458i	$-0.184785\pi$
$74$		0			0
$75$		0			0
$76$	168.000	+	96.9948i	0.253565	+	0.146396i
$77$	−540.000			−0.799204
$78$		0			0
$79$	−440.000			−0.626631			−0.313316	−	0.949649i	$-0.601440\pi$
$79$	−440.000			−0.626631			−0.313316	+	0.949649i	$0.601440\pi$
$80$	−288.000	−	166.277i	−0.402492	−	0.232379i
$81$		0			0
$82$		0			0
$83$			1195.12i			1.58049i	0.612789	+	0.790247i	$0.290048\pi$
$83$			1195.12i			1.58049i	−0.612789	+	0.790247i	$0.709952\pi$
$84$		0			0
$85$	526.500	−	303.975i	0.671846	−	0.387891i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1314.00	−	758.638i	−1.56499	−	0.903545i	−0.996740	−	0.0806862i	$-0.974289\pi$
$89$	−1314.00	−	758.638i	−1.56499	−	0.903545i	−0.568246	−	0.822859i	$-0.692378\pi$
$90$		0			0
$91$	−468.000	−	135.100i	−0.539118	−	0.155630i
$92$	−144.000			−0.163185
$93$		0			0
$94$		0			0
$95$	−63.0000	−	109.119i	−0.0680386	−	0.117846i
$96$		0			0
$97$	−1002.00	+	578.505i	−1.04884	+	0.605549i	−0.922325	−	0.386415i	$-0.873713\pi$
$97$	−1002.00	+	578.505i	−1.04884	+	0.605549i	−0.126517	+	0.991964i	$0.540380\pi$
$98$		0			0
$99$		0			0
$100$	−392.000	−	678.964i	−0.392000	−	0.678964i
$101$	787.500	−	1363.99i	0.775833	−	1.34378i	−0.158491	−	0.987360i	$-0.550663\pi$
$101$	787.500	−	1363.99i	0.775833	−	1.34378i	0.934325	−	0.356423i	$-0.116004\pi$
$102$		0			0
$103$	794.000			0.759565			0.379782	−	0.925076i	$-0.375999\pi$
$103$	794.000			0.759565			0.379782	+	0.925076i	$0.375999\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	225.000	−	389.711i	0.203286	−	0.352101i	−0.746299	−	0.665610i	$-0.768171\pi$
$107$	225.000	−	389.711i	0.203286	−	0.352101i	0.949585	+	0.313509i	$0.101505\pi$
$108$		0			0
$109$		−	595.825i		−	0.523576i	−0.965125	−	0.261788i	$-0.915688\pi$
$109$		−	595.825i		−	0.523576i	0.965125	−	0.261788i	$-0.0843120\pi$
$110$		0			0
$111$		0			0
$112$			665.108i			0.561132i
$113$	−850.500	−	1473.11i	−0.708038	−	1.22636i	−0.965584	−	0.260092i	$-0.916247\pi$
$113$	−850.500	−	1473.11i	−0.708038	−	1.22636i	0.257546	−	0.966266i	$-0.417086\pi$
$114$		0			0
$115$	81.0000	+	46.7654i	0.0656808	+	0.0379208i
$116$	792.000			0.633925
$117$		0			0
$118$		0			0
$119$	−1053.00	−	607.950i	−0.811163	−	0.468325i
$120$		0			0
$121$	684.500	+	1185.59i	0.514275	+	0.890750i
$122$		0			0
$123$		0			0
$124$	−1344.00	+	775.959i	−0.973345	+	0.561961i
$125$			1158.74i			0.829128i
$126$		0			0
$127$	832.000	−	1441.07i	0.581323	−	1.00688i	−0.414000	−	0.910277i	$-0.635868\pi$
$127$	832.000	−	1441.07i	0.581323	−	1.00688i	0.995323	−	0.0966044i	$-0.0307982\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1476.00			0.984418			0.492209	−	0.870477i	$-0.336190\pi$
$131$	1476.00			0.984418			0.492209	+	0.870477i	$0.336190\pi$
$132$		0			0
$133$	−126.000	+	218.238i	−0.0821473	+	0.142283i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−877.500	+	506.625i	−0.547225	+	0.315941i	−0.748002	−	0.663696i	$-0.768987\pi$
$137$	−877.500	+	506.625i	−0.547225	+	0.315941i	0.200777	+	0.979637i	$0.435653\pi$
$138$		0			0
$139$	−562.000	−	973.413i	−0.342937	−	0.593984i	0.642040	−	0.766671i	$-0.278088\pi$
$139$	−562.000	−	973.413i	−0.342937	−	0.593984i	−0.984977	+	0.172687i	$0.944755\pi$
$140$	216.000	−	374.123i	0.130395	−	0.225851i
$141$		0			0
$142$		0			0
$143$	585.000	+	2364.25i	0.342099	+	1.38258i
$144$		0			0
$145$	−445.500	−	257.210i	−0.255150	−	0.147311i
$146$		0			0
$147$		0			0
$148$		−	900.666i		−	0.500232i
$149$	−2830.50	+	1634.19i	−1.55627	+	0.898510i	−0.558657	+	0.829399i	$0.688683\pi$
$149$	−2830.50	+	1634.19i	−1.55627	+	0.898510i	−0.997609	+	0.0691115i	$0.977984\pi$
$150$		0			0
$151$			1638.52i			0.883052i	0.897248	+	0.441526i	$0.145563\pi$
$151$			1638.52i			0.883052i	−0.897248	+	0.441526i	$0.854437\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1008.00			0.522352
$156$		0			0
$157$	1259.00			0.639995			0.319997	−	0.947418i	$-0.396318\pi$
$157$	1259.00			0.639995			0.319997	+	0.947418i	$0.396318\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	187.061i		−	0.0915684i
$162$		0			0
$163$	2556.00	−	1475.71i	1.22823	−	0.709118i	0.261570	−	0.965185i	$-0.415760\pi$
$163$	2556.00	−	1475.71i	1.22823	−	0.709118i	0.966659	+	0.256066i	$0.0824264\pi$
$164$		−	290.985i		−	0.138549i
$165$		0			0
$166$		0			0
$167$	2718.00	+	1569.24i	1.25943	+	0.727133i	0.972964	−	0.230956i	$-0.0741855\pi$
$167$	2718.00	+	1569.24i	1.25943	+	0.727133i	0.286468	+	0.958090i	$0.407519\pi$
$168$		0			0
$169$	−84.5000	+	2195.37i	−0.0384615	+	0.999260i
$170$		0			0
$171$		0			0
$172$	328.000	−	568.113i	0.145406	−	0.251850i
$173$	2133.00	+	3694.46i	0.937393	+	1.62361i	0.770310	+	0.637669i	$0.220101\pi$
$173$	2133.00	+	3694.46i	0.937393	+	1.62361i	0.167083	+	0.985943i	$0.446565\pi$
$174$		0			0
$175$	882.000	−	509.223i	0.380988	−	0.219964i
$176$	2880.00	−	1662.77i	1.23346	−	0.712136i
$177$		0			0
$178$		0			0
$179$	1503.00	−	2603.27i	0.627595	−	1.08703i	−0.360438	−	0.932783i	$-0.617373\pi$
$179$	1503.00	−	2603.27i	0.627595	−	1.08703i	0.988033	−	0.154243i	$-0.0492939\pi$
$180$		0			0
$181$	−1873.00			−0.769166			−0.384583	−	0.923090i	$-0.625655\pi$
$181$	−1873.00			−0.769166			−0.384583	+	0.923090i	$0.625655\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−292.500	+	506.625i	−0.116243	+	0.201339i
$186$		0			0
$187$			6079.50i			2.37742i
$188$	−504.000	+	290.985i	−0.195521	+	0.112884i
$189$		0			0
$190$		0			0
$191$	−1368.00	−	2369.45i	−0.518246	−	0.897629i	−0.999775	−	0.0211985i	$-0.993252\pi$
$191$	−1368.00	−	2369.45i	−0.518246	−	0.897629i	0.481529	−	0.876430i	$-0.340082\pi$
$192$		0			0
$193$	−2254.50	−	1301.64i	−0.840842	−	0.485460i	0.0167085	−	0.999860i	$-0.494681\pi$
$193$	−2254.50	−	1301.64i	−0.840842	−	0.485460i	−0.857550	+	0.514400i	$0.828015\pi$
$194$		0			0
$195$		0			0
$196$	1880.00			0.685131
$197$	−3222.00	−	1860.22i	−1.16527	−	0.672768i	−0.212708	−	0.977116i	$-0.568228\pi$
$197$	−3222.00	−	1860.22i	−1.16527	−	0.672768i	−0.952561	+	0.304347i	$0.901562\pi$
$198$		0			0
$199$	−599.000	−	1037.50i	−0.213377	−	0.369579i	0.739392	−	0.673275i	$-0.235113\pi$
$199$	−599.000	−	1037.50i	−0.213377	−	0.369579i	−0.952769	+	0.303695i	$0.901780\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1028.84i			0.355716i
$204$		0			0
$205$	−94.5000	+	163.679i	−0.0321959	+	0.0557650i
$206$		0			0
$207$		0			0
$208$	2912.00	−	720.533i	0.970725	−	0.240192i
$209$	1260.00			0.417014
$210$		0			0
$211$	−1196.00	+	2071.53i	−0.390218	+	0.675878i	−0.992478	−	0.122422i	$-0.960934\pi$
$211$	−1196.00	+	2071.53i	−0.390218	+	0.675878i	0.602260	+	0.798300i	$0.294267\pi$
$212$	−1044.00	−	1808.26i	−0.338218	−	0.585811i
$213$		0			0
$214$		0			0
$215$	−369.000	+	213.042i	−0.117049	+	0.0675784i
$216$		0			0
$217$	−1008.00	−	1745.91i	−0.315334	−	0.546175i
$218$		0			0
$219$		0			0
$220$	−2160.00			−0.661942
$221$	−1521.00	+	5268.90i	−0.462957	+	1.60373i
$222$		0			0
$223$	−1764.00	−	1018.45i	−0.529714	−	0.305830i	0.211186	−	0.977446i	$-0.432267\pi$
$223$	−1764.00	−	1018.45i	−0.529714	−	0.305830i	−0.740900	+	0.671615i	$0.765601\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1863.00	+	1075.60i	−0.544721	+	0.314495i	−0.746990	−	0.664835i	$-0.768502\pi$
$227$	−1863.00	+	1075.60i	−0.544721	+	0.314495i	0.202269	+	0.979330i	$0.435168\pi$
$228$		0			0
$229$		−	3471.03i		−	1.00162i	−0.865556	−	0.500812i	$-0.833035\pi$
$229$		−	3471.03i		−	1.00162i	0.865556	−	0.500812i	$-0.166965\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1854.00			0.521286			0.260643	−	0.965435i	$-0.416065\pi$
$233$	1854.00			0.521286			0.260643	+	0.965435i	$0.416065\pi$
$234$		0			0
$235$	378.000			0.104928
$236$	−5472.00	−	3159.26i	−1.50931	−	0.871400i
$237$		0			0
$238$		0			0
$239$		−	4458.30i		−	1.20662i	−0.797505	−	0.603312i	$-0.793847\pi$
$239$		−	4458.30i		−	1.20662i	0.797505	−	0.603312i	$-0.206153\pi$
$240$		0			0
$241$	−361.500	+	208.712i	−0.0966235	+	0.0557856i	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−361.500	+	208.712i	−0.0966235	+	0.0557856i	0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$		0			0
$244$	2876.00	−	4981.38i	0.754578	−	1.30697i
$245$	−1057.50	−	610.548i	−0.275760	−	0.159210i
$246$		0			0
$247$	1092.00	+	315.233i	0.281305	+	0.0812057i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2052.00	+	3554.17i	0.516020	+	0.893773i	0.999827	+	0.0185985i	$0.00592043\pi$
$251$	2052.00	+	3554.17i	0.516020	+	0.893773i	−0.483807	+	0.875175i	$0.660746\pi$
$252$		0			0
$253$	−810.000	+	467.654i	−0.201282	+	0.116210i
$254$		0			0
$255$		0			0
$256$	−2048.00	−	3547.24i	−0.500000	−	0.866025i
$257$	994.500	−	1722.52i	0.241382	−	0.418086i	−0.719726	−	0.694258i	$-0.755733\pi$
$257$	994.500	−	1722.52i	0.241382	−	0.418086i	0.961108	+	0.276172i	$0.0890660\pi$
$258$		0			0
$259$	1170.00			0.280696
$260$	−1872.00	−	540.400i	−0.446525	−	0.128901i
$261$		0			0
$262$		0			0
$263$	369.000	−	639.127i	0.0865153	−	0.149849i	−0.819521	−	0.573050i	$-0.805760\pi$
$263$	369.000	−	639.127i	0.0865153	−	0.149849i	0.906036	+	0.423201i	$0.139094\pi$
$264$		0			0
$265$			1356.20i			0.314379i
$266$		0			0
$267$		0			0
$268$			5625.70i			1.28226i
$269$	−1053.00	−	1823.85i	−0.238671	−	0.413391i	0.721662	−	0.692246i	$-0.243378\pi$
$269$	−1053.00	−	1823.85i	−0.238671	−	0.413391i	−0.960333	+	0.278855i	$0.910045\pi$
$270$		0			0
$271$	594.000	+	342.946i	0.133147	+	0.0768727i	0.565094	−	0.825026i	$-0.308840\pi$
$271$	594.000	+	342.946i	0.133147	+	0.0768727i	−0.431947	+	0.901899i	$0.642173\pi$
$272$	7488.00			1.66922
$273$		0			0
$274$		0			0
$275$	−4410.00	−	2546.11i	−0.967029	−	0.558315i
$276$		0			0
$277$	−1832.50	−	3173.98i	−0.397488	−	0.688470i	0.595927	−	0.803039i	$-0.296785\pi$
$277$	−1832.50	−	3173.98i	−0.397488	−	0.688470i	−0.993415	+	0.114569i	$0.963451\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1719.93i		−	0.365132i	−0.983194	−	0.182566i	$-0.941560\pi$
$281$		−	1719.93i		−	0.365132i	0.983194	−	0.182566i	$-0.0584404\pi$
$282$		0			0
$283$	913.000	−	1581.36i	0.191775	−	0.332163i	−0.754064	−	0.656801i	$-0.771909\pi$
$283$	913.000	−	1581.36i	0.191775	−	0.332163i	0.945838	+	0.324638i	$0.105242\pi$
$284$	−3240.00	−	1870.61i	−0.676967	−	0.390847i
$285$		0			0
$286$		0			0
$287$	378.000			0.0777444
$288$		0			0
$289$	−4388.00	+	7600.24i	−0.893141	+	1.54696i
$290$		0			0
$291$		0			0
$292$	−4740.00	+	2736.64i	−0.949957	+	0.548458i
$293$	436.500	−	252.013i	0.0870328	−	0.0502484i	−0.455852	−	0.890056i	$-0.650665\pi$
$293$	436.500	−	252.013i	0.0870328	−	0.0502484i	0.542885	+	0.839807i	$0.317332\pi$
$294$		0			0
$295$	2052.00	+	3554.17i	0.404990	+	0.701463i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−819.000	+	202.650i	−0.158408	+	0.0391958i
$300$		0			0
$301$	738.000	+	426.084i	0.141321	+	0.0815917i
$302$		0			0
$303$		0			0
$304$		−	1551.92i		−	0.292791i
$305$	−3235.50	+	1868.02i	−0.607424	+	0.350696i
$306$		0			0
$307$		−	1950.29i		−	0.362570i	−0.983431	−	0.181285i	$-0.941974\pi$
$307$		−	1950.29i		−	0.362570i	0.983431	−	0.181285i	$-0.0580256\pi$
$308$	2160.00	+	3741.23i	0.399602	+	0.692131i
$309$		0			0
$310$		0			0
$311$	3798.00			0.692491			0.346246	−	0.938144i	$-0.387456\pi$
$311$	3798.00			0.692491			0.346246	+	0.938144i	$0.387456\pi$
$312$		0			0
$313$	1378.00			0.248847			0.124424	−	0.992229i	$-0.460292\pi$
$313$	1378.00			0.248847			0.124424	+	0.992229i	$0.460292\pi$
$314$		0			0
$315$		0			0
$316$	1760.00	+	3048.41i	0.313316	+	0.542679i
$317$			7103.14i			1.25852i	0.777193	+	0.629262i	$0.216643\pi$
$317$			7103.14i			1.25852i	−0.777193	+	0.629262i	$0.783357\pi$
$318$		0			0
$319$	4455.00	−	2572.10i	0.781919	−	0.451441i
$320$			2660.43i			0.464758i
$321$		0			0
$322$		0			0
$323$	2457.00	+	1418.55i	0.423254	+	0.244366i
$324$		0			0
$325$	−3185.00	−	3309.95i	−0.543606	−	0.564932i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−378.000	−	654.715i	−0.0633429	−	0.109713i
$330$		0			0
$331$	8724.00	−	5036.80i	1.44868	−	0.836398i	0.450281	−	0.892887i	$-0.351324\pi$
$331$	8724.00	−	5036.80i	1.44868	−	0.836398i	0.998403	+	0.0564889i	$0.0179906\pi$
$332$	8280.00	−	4780.46i	1.36875	−	0.790247i
$333$		0			0
$334$		0			0
$335$	1827.00	−	3164.46i	0.297969	−	0.516098i
$336$		0			0
$337$	9001.00			1.45494			0.727471	−	0.686138i	$-0.240695\pi$
$337$	9001.00			1.45494			0.727471	+	0.686138i	$0.240695\pi$
$338$		0			0
$339$		0			0
$340$	−4212.00	−	2431.80i	−0.671846	−	0.387891i
$341$	−5040.00	+	8729.54i	−0.800385	+	1.38631i
$342$		0			0
$343$			6006.75i			0.945581i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1647.00	+	2852.69i	0.254800	+	0.441327i	0.964841	−	0.262834i	$-0.0846570\pi$
$347$	1647.00	+	2852.69i	0.254800	+	0.441327i	−0.710041	+	0.704160i	$0.751324\pi$
$348$		0			0
$349$	−9132.00	−	5272.36i	−1.40064	−	0.808662i	−0.406185	−	0.913791i	$-0.633141\pi$
$349$	−9132.00	−	5272.36i	−1.40064	−	0.808662i	−0.994459	+	0.105129i	$0.966475\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2146.50	+	1239.28i	0.323645	+	0.186856i	0.653016	−	0.757344i	$-0.273503\pi$
$353$	2146.50	+	1239.28i	0.323645	+	0.186856i	−0.329371	+	0.944201i	$0.606837\pi$
$354$		0			0
$355$	1215.00	+	2104.44i	0.181649	+	0.314626i
$356$			12138.2i			1.80709i
$357$		0			0
$358$		0			0
$359$		−	5414.39i		−	0.795991i	−0.917387	−	0.397995i	$-0.869706\pi$
$359$		−	5414.39i		−	0.795991i	0.917387	−	0.397995i	$-0.130294\pi$
$360$		0			0
$361$	−3135.50	+	5430.85i	−0.457137	+	0.791784i
$362$		0			0
$363$		0			0
$364$	936.000	+	3782.80i	0.134779	+	0.544705i
$365$	3555.00			0.509801
$366$		0			0
$367$	4973.00	−	8613.49i	0.707326	−	1.22512i	−0.258520	−	0.966006i	$-0.583235\pi$
$367$	4973.00	−	8613.49i	0.707326	−	1.22512i	0.965846	−	0.259118i	$-0.0834318\pi$
$368$	576.000	+	997.661i	0.0815926	+	0.141323i
$369$		0			0
$370$		0			0
$371$	2349.00	−	1356.20i	0.328717	−	0.189785i
$372$		0			0
$373$	3650.50	+	6322.85i	0.506745	+	0.877707i	0.999970	+	0.00780555i	$0.00248461\pi$
$373$	3650.50	+	6322.85i	0.506745	+	0.877707i	−0.493225	+	0.869902i	$0.664182\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4504.50	−	1114.57i	0.615368	−	0.152264i
$378$		0			0
$379$	−2964.00	−	1711.27i	−0.401716	−	0.231931i	0.285508	−	0.958376i	$-0.407838\pi$
$379$	−2964.00	−	1711.27i	−0.401716	−	0.231931i	−0.687224	+	0.726445i	$0.741171\pi$
$380$	−504.000	+	872.954i	−0.0680386	+	0.117846i
$381$		0			0
$382$		0			0
$383$	5004.00	−	2889.06i	0.667604	−	0.385442i	−0.127564	−	0.991830i	$-0.540716\pi$
$383$	5004.00	−	2889.06i	0.667604	−	0.385442i	0.795168	+	0.606389i	$0.207382\pi$
$384$		0			0
$385$		−	2805.92i		−	0.371436i
$386$		0			0
$387$		0			0
$388$	8016.00	+	4628.04i	1.04884	+	0.605549i
$389$	−9153.00			−1.19300			−0.596498	−	0.802614i	$-0.703442\pi$
$389$	−9153.00			−1.19300			−0.596498	+	0.802614i	$0.703442\pi$
$390$		0			0
$391$	−2106.00			−0.272391
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	2286.31i		−	0.291232i
$396$		0			0
$397$	1752.00	−	1011.52i	0.221487	−	0.127876i	−0.385152	−	0.922853i	$-0.625851\pi$
$397$	1752.00	−	1011.52i	0.221487	−	0.127876i	0.606639	+	0.794978i	$0.292518\pi$
$398$		0			0
$399$		0			0
$400$	−3136.00	+	5431.71i	−0.392000	+	0.678964i
$401$	−7195.50	−	4154.32i	−0.896075	−	0.517349i	−0.0201504	−	0.999797i	$-0.506414\pi$
$401$	−7195.50	−	4154.32i	−0.896075	−	0.517349i	−0.875925	+	0.482448i	$0.839748\pi$
$402$		0			0
$403$	−6552.00	+	6304.66i	−0.809872	+	0.779300i
$404$	−12600.0			−1.55167
$405$		0			0
$406$		0			0
$407$	−2925.00	−	5066.25i	−0.356233	−	0.617014i
$408$		0			0
$409$	−9022.50	+	5209.14i	−1.09079	+	0.629769i	−0.933787	−	0.357829i	$-0.883517\pi$
$409$	−9022.50	+	5209.14i	−1.09079	+	0.629769i	−0.157005	+	0.987598i	$0.550184\pi$
$410$		0			0
$411$		0			0
$412$	−3176.00	−	5500.99i	−0.379782	−	0.657802i
$413$	4104.00	−	7108.34i	0.488970	−	0.846921i
$414$		0			0
$415$	−6210.00			−0.734547
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2088.00	−	3616.52i	0.243450	−	0.421667i	−0.718245	−	0.695790i	$-0.755054\pi$
$419$	2088.00	−	3616.52i	0.243450	−	0.421667i	0.961695	+	0.274123i	$0.0883875\pi$
$420$		0			0
$421$		−	14471.3i		−	1.67527i	−0.546233	−	0.837633i	$-0.683939\pi$
$421$		−	14471.3i		−	1.67527i	0.546233	−	0.837633i	$-0.316061\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5733.00	−	9929.85i	−0.654333	−	1.13334i
$426$		0			0
$427$	6471.00	+	3736.03i	0.733381	+	0.423418i
$428$	−3600.00			−0.406571
$429$		0			0
$430$		0			0
$431$	5697.00	+	3289.16i	0.636693	+	0.367595i	0.783340	−	0.621594i	$-0.213515\pi$
$431$	5697.00	+	3289.16i	0.636693	+	0.367595i	−0.146646	+	0.989189i	$0.546848\pi$
$432$		0			0
$433$	3302.50	+	5720.10i	0.366531	+	0.634851i	0.989021	−	0.147778i	$-0.0472120\pi$
$433$	3302.50	+	5720.10i	0.366531	+	0.634851i	−0.622489	+	0.782628i	$0.713879\pi$
$434$		0			0
$435$		0			0
$436$	−4128.00	+	2383.30i	−0.453430	+	0.261788i
$437$			436.477i			0.0477792i
$438$		0			0
$439$	4271.00	−	7397.59i	0.464336	−	0.804254i	−0.534835	−	0.844957i	$-0.679626\pi$
$439$	4271.00	−	7397.59i	0.464336	−	0.804254i	0.999171	+	0.0407023i	$0.0129595\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14328.0			1.53667			0.768334	−	0.640049i	$-0.221086\pi$
$443$	14328.0			1.53667			0.768334	+	0.640049i	$0.221086\pi$
$444$		0			0
$445$	3942.00	−	6827.74i	0.419930	−	0.727340i
$446$		0			0
$447$		0			0
$448$	4608.00	−	2660.43i	0.485954	−	0.280566i
$449$	−2610.00	+	1506.88i	−0.274329	+	0.158384i	−0.630853	−	0.775902i	$-0.717295\pi$
$449$	−2610.00	+	1506.88i	−0.274329	+	0.158384i	0.356525	+	0.934286i	$0.383962\pi$
$450$		0			0
$451$	−945.000	−	1636.79i	−0.0986659	−	0.170894i
$452$	−6804.00	+	11784.9i	−0.708038	+	1.22636i
$453$		0			0
$454$		0			0
$455$	702.000	−	2431.80i	0.0723303	−	0.250559i
$456$		0			0
$457$	−2500.50	−	1443.66i	−0.255948	−	0.147772i	0.366536	−	0.930404i	$-0.380543\pi$
$457$	−2500.50	−	1443.66i	−0.255948	−	0.147772i	−0.622485	+	0.782632i	$0.713877\pi$
$458$		0			0
$459$		0			0
$460$		−	748.246i		−	0.0758416i
$461$	−3118.50	+	1800.47i	−0.315061	+	0.181900i	−0.649189	−	0.760627i	$-0.724892\pi$
$461$	−3118.50	+	1800.47i	−0.315061	+	0.181900i	0.334128	+	0.942528i	$0.391558\pi$
$462$		0			0
$463$		−	2677.75i		−	0.268781i	−0.990928	−	0.134391i	$-0.957092\pi$
$463$		−	2677.75i		−	0.268781i	0.990928	−	0.134391i	$-0.0429077\pi$
$464$	−3168.00	−	5487.14i	−0.316963	−	0.548996i
$465$		0			0
$466$		0			0
$467$	−13878.0			−1.37515			−0.687577	−	0.726111i	$-0.741326\pi$
$467$	−13878.0			−1.37515			−0.687577	+	0.726111i	$0.741326\pi$
$468$		0			0
$469$	−7308.00			−0.719514
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	4260.84i		−	0.414194i
$474$		0			0
$475$	−2058.00	+	1188.19i	−0.198795	+	0.114774i
$476$			9727.20i			0.936650i
$477$		0			0
$478$		0			0
$479$	−954.000	−	550.792i	−0.0910008	−	0.0525393i	0.453809	−	0.891099i	$-0.350065\pi$
$479$	−954.000	−	550.792i	−0.0910008	−	0.0525393i	−0.544810	+	0.838560i	$0.683398\pi$
$480$		0			0
$481$	−1267.50	−	5122.54i	−0.120152	−	0.485588i
$482$		0			0
$483$		0			0
$484$	5476.00	−	9484.71i	0.514275	−	0.890750i
$485$	−3006.00	−	5206.54i	−0.281434	−	0.487458i
$486$		0			0
$487$	−14829.0	+	8561.53i	−1.37981	+	0.796632i	−0.992136	−	0.125166i	$-0.960054\pi$
$487$	−14829.0	+	8561.53i	−1.37981	+	0.796632i	−0.387671	+	0.921798i	$0.626720\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	225.000	−	389.711i	0.0206805	−	0.0358196i	−0.855500	−	0.517803i	$-0.826750\pi$
$491$	225.000	−	389.711i	0.0206805	−	0.0358196i	0.876180	+	0.481983i	$0.160083\pi$
$492$		0			0
$493$	11583.0			1.05816
$494$		0			0
$495$		0			0
$496$	10752.0	+	6207.67i	0.973345	+	0.561961i
$497$	2430.00	−	4208.88i	0.219317	−	0.379868i
$498$		0			0
$499$		−	13219.0i		−	1.18590i	−0.805239	−	0.592950i	$-0.797963\pi$
$499$		−	13219.0i		−	1.18590i	0.805239	−	0.592950i	$-0.202037\pi$
$500$	8028.00	−	4634.97i	0.718046	−	0.414564i
$501$		0			0
$502$		0			0
$503$	2673.00	+	4629.77i	0.236945	+	0.410400i	0.959836	−	0.280561i	$-0.0905206\pi$
$503$	2673.00	+	4629.77i	0.236945	+	0.410400i	−0.722891	+	0.690962i	$0.757187\pi$
$504$		0			0
$505$	7087.50	+	4091.97i	0.624534	+	0.360575i
$506$		0			0
$507$		0			0
$508$	−13312.0			−1.16265
$509$	5080.50	+	2933.23i	0.442415	+	0.255428i	0.704621	−	0.709583i	$-0.251117\pi$
$509$	5080.50	+	2933.23i	0.442415	+	0.255428i	−0.262207	+	0.965012i	$0.584450\pi$
$510$		0			0
$511$	−3555.00	−	6157.44i	−0.307757	−	0.533051i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			4125.75i			0.353014i
$516$		0			0
$517$	−1890.00	+	3273.58i	−0.160778	+	0.278475i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9657.00			0.812055			0.406028	−	0.913861i	$-0.366914\pi$
$521$	9657.00			0.812055			0.406028	+	0.913861i	$0.366914\pi$
$522$		0			0
$523$	−10813.0	+	18728.7i	−0.904053	+	1.56586i	−0.0818685	+	0.996643i	$0.526089\pi$
$523$	−10813.0	+	18728.7i	−0.904053	+	1.56586i	−0.822184	+	0.569222i	$0.807245\pi$
$524$	−5904.00	−	10226.0i	−0.492209	−	0.852531i
$525$		0			0
$526$		0			0
$527$	−19656.0	+	11348.4i	−1.62472	+	0.938034i
$528$		0			0
$529$	5921.50	+	10256.3i	0.486685	+	0.842964i
$530$		0			0
$531$		0			0
$532$	2016.00			0.164295
$533$	−409.500	−	1654.97i	−0.0332785	−	0.134493i
$534$		0			0
$535$	2025.00	+	1169.13i	0.163642	+	0.0944787i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10575.0	−	6105.48i	0.845079	−	0.487906i
$540$		0			0
$541$			5371.09i			0.426841i	0.976960	+	0.213421i	$0.0684605\pi$
$541$			5371.09i			0.426841i	−0.976960	+	0.213421i	$0.931540\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3096.00			0.243336
$546$		0			0
$547$	16946.0			1.32460			0.662302	−	0.749237i	$-0.269579\pi$
$547$	16946.0			1.32460			0.662302	+	0.749237i	$0.269579\pi$
$548$	7020.00	+	4053.00i	0.547225	+	0.315941i
$549$		0			0
$550$		0			0
$551$		−	2400.62i		−	0.185608i
$552$		0			0
$553$	−3960.00	+	2286.31i	−0.304514	+	0.175811i
$554$		0			0
$555$		0			0
$556$	−4496.00	+	7787.30i	−0.342937	+	0.593984i
$557$	−3343.50	−	1930.37i	−0.254342	−	0.146845i	0.367409	−	0.930060i	$-0.380245\pi$
$557$	−3343.50	−	1930.37i	−0.254342	−	0.146845i	−0.621751	+	0.783215i	$0.713578\pi$
$558$		0			0
$559$	1066.00	−	3692.73i	0.0806565	−	0.279402i
$560$	−3456.00			−0.260790
$561$		0			0
$562$		0			0
$563$	10836.0	+	18768.5i	0.811160	+	1.40497i	0.912053	+	0.410073i	$0.134497\pi$
$563$	10836.0	+	18768.5i	0.811160	+	1.40497i	−0.100893	+	0.994897i	$0.532170\pi$
$564$		0			0
$565$	7654.50	−	4419.33i	0.569960	−	0.329066i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−693.000	+	1200.31i	−0.0510581	+	0.0884353i	−0.890425	−	0.455130i	$-0.849593\pi$
$569$	−693.000	+	1200.31i	−0.0510581	+	0.0884353i	0.839367	+	0.543565i	$0.182926\pi$
$570$		0			0
$571$	1162.00			0.0851632			0.0425816	−	0.999093i	$-0.486442\pi$
$571$	1162.00			0.0851632			0.0425816	+	0.999093i	$0.486442\pi$
$572$	14040.0	−	13510.0i	1.02630	−	0.987555i
$573$		0			0
$574$		0			0
$575$	882.000	−	1527.67i	0.0639686	−	0.110797i
$576$		0			0
$577$			8045.38i			0.580474i	0.956955	+	0.290237i	$0.0937341\pi$
$577$			8045.38i			0.580474i	−0.956955	+	0.290237i	$0.906266\pi$
$578$		0			0
$579$		0			0
$580$			4115.35i			0.294622i
$581$	6210.00	+	10756.0i	0.443432	+	0.768047i
$582$		0			0
$583$	−11745.0	−	6780.98i	−0.834354	−	0.481714i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23922.0	+	13811.4i	1.68206	+	0.971135i	0.960293	+	0.278995i	$0.0900013\pi$
$587$	23922.0	+	13811.4i	1.68206	+	0.971135i	0.721763	+	0.692140i	$0.243332\pi$
$588$		0			0
$589$	2352.00	+	4073.78i	0.164537	+	0.284987i
$590$		0			0
$591$		0			0
$592$	−6240.00	+	3602.67i	−0.433214	+	0.250116i
$593$		−	275.396i		−	0.0190711i	−0.999955	−	0.00953555i	$-0.996965\pi$
$593$		−	275.396i		−	0.0190711i	0.999955	−	0.00953555i	$-0.00303531\pi$
$594$		0			0
$595$	3159.00	−	5471.55i	0.217658	−	0.376994i
$596$	22644.0	+	13073.5i	1.55627	+	0.898510i
$597$		0			0
$598$		0			0
$599$	−22356.0			−1.52494			−0.762472	−	0.647021i	$-0.776014\pi$
$599$	−22356.0			−1.52494			−0.762472	+	0.647021i	$0.776014\pi$
$600$		0			0
$601$	9041.50	−	15660.3i	0.613661	−	1.06289i	−0.376956	−	0.926231i	$-0.623029\pi$
$601$	9041.50	−	15660.3i	0.613661	−	1.06289i	0.990618	−	0.136662i	$-0.0436373\pi$
$602$		0			0
$603$		0			0
$604$	11352.0	−	6554.08i	0.764746	−	0.441526i
$605$	−6160.50	+	3556.77i	−0.413983	+	0.239013i
$606$		0			0
$607$	−2740.00	−	4745.82i	−0.183218	−	0.317342i	0.759757	−	0.650207i	$-0.225318\pi$
$607$	−2740.00	−	4745.82i	−0.183218	−	0.317342i	−0.942975	+	0.332865i	$0.891985\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2457.00	+	2364.25i	−0.162683	+	0.156542i
$612$		0			0
$613$	15361.5	+	8868.97i	1.01215	+	0.584362i	0.911819	−	0.410592i	$-0.134678\pi$
$613$	15361.5	+	8868.97i	1.01215	+	0.584362i	0.100326	+	0.994955i	$0.468011\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8545.50	+	4933.75i	−0.557583	+	0.321921i	−0.752175	−	0.658963i	$-0.770995\pi$
$617$	−8545.50	+	4933.75i	−0.557583	+	0.321921i	0.194592	+	0.980884i	$0.437662\pi$
$618$		0			0
$619$		−	4115.35i		−	0.267221i	−0.991034	−	0.133611i	$-0.957343\pi$
$619$		−	4115.35i		−	0.267221i	0.991034	−	0.133611i	$-0.0426572\pi$
$620$	−4032.00	−	6983.63i	−0.261176	−	0.452370i
$621$		0			0
$622$		0			0
$623$	−15768.0			−1.01402
$624$		0			0
$625$	6229.00			0.398656
$626$		0			0
$627$		0			0
$628$	−5036.00	−	8722.61i	−0.319997	−	0.554252i
$629$		−	13172.2i		−	0.834995i
$630$		0			0
$631$	−10968.0	+	6332.38i	−0.691964	+	0.399506i	−0.804347	−	0.594159i	$-0.797485\pi$
$631$	−10968.0	+	6332.38i	−0.691964	+	0.399506i	0.112383	+	0.993665i	$0.464151\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7488.00	+	4323.20i	0.467956	+	0.270175i
$636$		0			0
$637$	10692.5	−	2645.71i	0.665074	−	0.164563i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1894.50	−	3281.37i	−0.116737	−	0.202194i	0.801736	−	0.597678i	$-0.203910\pi$
$641$	−1894.50	−	3281.37i	−0.116737	−	0.202194i	−0.918473	+	0.395484i	$0.870577\pi$
$642$		0			0
$643$	14646.0	−	8455.87i	0.898261	−	0.518611i	0.0216255	−	0.999766i	$-0.493116\pi$
$643$	14646.0	−	8455.87i	0.898261	−	0.518611i	0.876636	+	0.481155i	$0.159783\pi$
$644$	−1296.00	+	748.246i	−0.0793006	+	0.0457842i
$645$		0			0
$646$		0			0
$647$	−13896.0	+	24068.6i	−0.844371	+	1.46249i	0.0417951	+	0.999126i	$0.486692\pi$
$647$	−13896.0	+	24068.6i	−0.844371	+	1.46249i	−0.886166	+	0.463368i	$0.846641\pi$
$648$		0			0
$649$	−41040.0			−2.48222
$650$		0			0
$651$		0			0
$652$	−20448.0	−	11805.7i	−1.22823	−	0.709118i
$653$	−297.000	+	514.419i	−0.0177986	+	0.0308281i	−0.874788	−	0.484507i	$-0.838999\pi$
$653$	−297.000	+	514.419i	−0.0177986	+	0.0308281i	0.856989	+	0.515335i	$0.172332\pi$
$654$		0			0
$655$			7669.52i			0.457516i
$656$	−2016.00	+	1163.94i	−0.119987	+	0.0692746i
$657$		0			0
$658$		0			0
$659$	8874.00	+	15370.2i	0.524555	+	0.908556i	0.999591	+	0.0285901i	$0.00910174\pi$
$659$	8874.00	+	15370.2i	0.524555	+	0.908556i	−0.475036	+	0.879966i	$0.657565\pi$
$660$		0			0
$661$	13675.5	+	7895.55i	0.804713	+	0.464601i	0.845117	−	0.534582i	$-0.179531\pi$
$661$	13675.5	+	7895.55i	0.804713	+	0.464601i	−0.0404035	+	0.999183i	$0.512864\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1134.00	−	654.715i	−0.0661273	−	0.0381786i
$666$		0			0
$667$	891.000	+	1543.26i	0.0517236	+	0.0895879i
$668$		−	25107.8i		−	1.45427i
$669$		0			0
$670$		0			0
$671$		−	37360.3i		−	2.14945i
$672$		0			0
$673$	−10466.5	+	18128.5i	−0.599486	+	1.03834i	0.393411	+	0.919363i	$0.371295\pi$
$673$	−10466.5	+	18128.5i	−0.599486	+	1.03834i	−0.992897	+	0.118977i	$0.962038\pi$
$674$		0			0
$675$		0			0
$676$	15548.0	−	8196.06i	0.884615	−	0.466321i
$677$	−3402.00			−0.193131			−0.0965653	−	0.995327i	$-0.530786\pi$
$677$	−3402.00			−0.193131			−0.0965653	+	0.995327i	$0.530786\pi$
$678$		0			0
$679$	−6012.00	+	10413.1i	−0.339793	+	0.588539i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21636.0	+	12491.6i	−1.21212	+	0.699818i	−0.963221	−	0.268711i	$-0.913402\pi$
$683$	−21636.0	+	12491.6i	−1.21212	+	0.699818i	−0.248900	+	0.968529i	$0.580069\pi$
$684$		0			0
$685$	−2632.50	−	4559.62i	−0.146836	−	0.254327i
$686$		0			0
$687$		0			0
$688$	−5248.00			−0.290811
$689$	−8482.50	−	8815.27i	−0.469024	−	0.487424i
$690$		0			0
$691$	12009.0	+	6933.40i	0.661134	+	0.381706i	0.792709	−	0.609600i	$-0.208670\pi$
$691$	12009.0	+	6933.40i	0.661134	+	0.381706i	−0.131575	+	0.991306i	$0.542003\pi$
$692$	17064.0	−	29555.7i	0.937393	−	1.62361i
$693$		0			0
$694$		0			0
$695$	5058.00	−	2920.24i	0.276059	−	0.159383i
$696$		0			0
$697$		−	4255.65i		−	0.231269i
$698$		0			0
$699$		0			0
$700$	−7056.00	−	4073.78i	−0.380988	−	0.219964i
$701$	21906.0			1.18028			0.590141	−	0.807300i	$-0.299072\pi$
$701$	21906.0			1.18028			0.590141	+	0.807300i	$0.299072\pi$
$702$		0			0
$703$	−2730.00			−0.146464
$704$	−23040.0	−	13302.2i	−1.23346	−	0.712136i
$705$		0			0
$706$		0			0
$707$		−	16367.9i		−	0.870690i
$708$		0			0
$709$	−11308.5	+	6528.97i	−0.599012	+	0.345840i	−0.768653	−	0.639666i	$-0.779073\pi$
$709$	−11308.5	+	6528.97i	−0.599012	+	0.345840i	0.169641	+	0.985506i	$0.445739\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3024.00	−	1745.91i	−0.158835	−	0.0917037i
$714$		0			0
$715$	−12285.0	+	3039.75i	−0.642564	+	0.158993i
$716$	−24048.0			−1.25519
$717$		0			0
$718$		0			0
$719$	−7110.00	−	12314.9i	−0.368788	−	0.638759i	0.620589	−	0.784136i	$-0.286894\pi$
$719$	−7110.00	−	12314.9i	−0.368788	−	0.638759i	−0.989376	+	0.145377i	$0.953560\pi$
$720$		0			0
$721$	7146.00	−	4125.75i	0.369114	−	0.213108i
$722$		0			0
$723$		0			0
$724$	7492.00	+	12976.5i	0.384583	+	0.666117i
$725$	−4851.00	+	8402.18i	−0.248499	+	0.430413i
$726$		0			0
$727$	−5282.00			−0.269462			−0.134731	−	0.990882i	$-0.543017\pi$
$727$	−5282.00			−0.269462			−0.134731	+	0.990882i	$0.543017\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4797.00	−	8308.65i	0.242713	−	0.420392i
$732$		0			0
$733$		−	11419.4i		−	0.575424i	−0.957717	−	0.287712i	$-0.907105\pi$
$733$		−	11419.4i		−	0.575424i	0.957717	−	0.287712i	$-0.0928945\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18270.0	+	31644.6i	0.913140	+	1.58160i
$738$		0			0
$739$	−17784.0	−	10267.6i	−0.885244	−	0.511096i	−0.0128599	−	0.999917i	$-0.504094\pi$
$739$	−17784.0	−	10267.6i	−0.885244	−	0.511096i	−0.872384	+	0.488822i	$0.837427\pi$
$740$	4680.00			0.232487
$741$		0			0
$742$		0			0
$743$	18036.0	+	10413.1i	0.890547	+	0.514158i	0.874121	−	0.485707i	$-0.161438\pi$
$743$	18036.0	+	10413.1i	0.890547	+	0.514158i	0.0164258	+	0.999865i	$0.494771\pi$
$744$		0			0
$745$	−8491.50	−	14707.7i	−0.417590	−	0.723287i
$746$		0			0
$747$		0			0
$748$	42120.0	−	24318.0i	2.05890	−	1.18871i
$749$		−	4676.54i		−	0.228140i
$750$		0			0
$751$	2417.00	−	4186.37i	0.117440	−	0.203412i	−0.801312	−	0.598246i	$-0.795864\pi$
$751$	2417.00	−	4186.37i	0.117440	−	0.203412i	0.918753	+	0.394834i	$0.129198\pi$
$752$	4032.00	+	2327.88i	0.195521	+	0.112884i
$753$		0			0
$754$		0			0
$755$	−8514.00			−0.410406
$756$		0			0
$757$	−4523.00	+	7834.07i	−0.217161	+	0.376135i	−0.953939	−	0.300001i	$-0.903013\pi$
$757$	−4523.00	+	7834.07i	−0.217161	+	0.376135i	0.736778	+	0.676135i	$0.236346\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10422.0	−	6017.14i	0.496448	−	0.286625i	−0.230797	−	0.973002i	$-0.574133\pi$
$761$	10422.0	−	6017.14i	0.496448	−	0.286625i	0.727246	+	0.686377i	$0.240800\pi$
$762$		0			0
$763$	−3096.00	−	5362.43i	−0.146897	−	0.254434i
$764$	−10944.0	+	18955.6i	−0.518246	+	0.897629i
$765$		0			0
$766$		0			0
$767$	−35568.0	−	10267.6i	−1.67443	−	0.483366i
$768$		0			0
$769$	32514.0	+	18772.0i	1.52469	+	0.880279i	0.999572	+	0.0292479i	$0.00931121\pi$
$769$	32514.0	+	18772.0i	1.52469	+	0.880279i	0.525115	+	0.851031i	$0.324022\pi$
$770$		0			0
$771$		0			0
$772$			20826.2i			0.970920i
$773$	13608.0	−	7856.58i	0.633177	−	0.365565i	−0.148804	−	0.988867i	$-0.547543\pi$
$773$	13608.0	−	7856.58i	0.633177	−	0.365565i	0.781981	+	0.623302i	$0.214209\pi$
$774$		0			0
$775$		−	19011.0i		−	0.881155i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−882.000			−0.0405660
$780$		0			0
$781$	−24300.0			−1.11334
$782$		0			0
$783$		0			0
$784$	−7520.00	−	13025.0i	−0.342566	−	0.593341i
$785$			6541.96i			0.297443i
$786$		0			0
$787$	−3252.00	+	1877.54i	−0.147295	+	0.0850409i	−0.571836	−	0.820368i	$-0.693769\pi$
$787$	−3252.00	+	1877.54i	−0.147295	+	0.0850409i	0.424541	+	0.905409i	$0.360435\pi$
$788$			29763.6i			1.34554i
$789$		0			0
$790$		0			0
$791$	−15309.0	−	8838.66i	−0.688148	−	0.397303i
$792$		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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.00000 - 6.92820i) q^{4} +5.19615i q^{5} +(9.00000 - 5.19615i) q^{7} +O(q^{10})\)	\(q+(-4.00000 - 6.92820i) q^{4} +5.19615i q^{5} +(9.00000 - 5.19615i) q^{7} +(-45.0000 - 25.9808i) q^{11} +(-32.5000 - 33.7750i) q^{13} +(-32.0000 + 55.4256i) q^{16} +(-58.5000 - 101.325i) q^{17} +(-21.0000 + 12.1244i) q^{19} +(36.0000 - 20.7846i) q^{20} +(9.00000 - 15.5885i) q^{23} +98.0000 q^{25} +(-72.0000 - 41.5692i) q^{28} +(-49.5000 + 85.7365i) q^{29} -193.990i q^{31} +(27.0000 + 46.7654i) q^{35} +(97.5000 + 56.2917i) q^{37} +(31.5000 + 18.1865i) q^{41} +(41.0000 + 71.0141i) q^{43} +415.692i q^{44} -72.7461i q^{47} +(-117.500 + 203.516i) q^{49} +(-104.000 + 360.267i) q^{52} +261.000 q^{53} +(135.000 - 233.827i) q^{55} +(684.000 - 394.908i) q^{59} +(359.500 + 622.672i) q^{61} +512.000 q^{64} +(175.500 - 168.875i) q^{65} +(-609.000 - 351.606i) q^{67} +(-468.000 + 810.600i) q^{68} +(405.000 - 233.827i) q^{71} -684.160i q^{73} +(168.000 + 96.9948i) q^{76} -540.000 q^{77} -440.000 q^{79} +(-288.000 - 166.277i) q^{80} +1195.12i q^{83} +(526.500 - 303.975i) q^{85} +(-1314.00 - 758.638i) q^{89} +(-468.000 - 135.100i) q^{91} -144.000 q^{92} +(-63.0000 - 109.119i) q^{95} +(-1002.00 + 578.505i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 18 q^{7}+O(q^{10}) \) 2 * q - 8 * q^4 + 18 * q^7	\( 2 q - 8 q^{4} + 18 q^{7} - 90 q^{11} - 65 q^{13} - 64 q^{16} - 117 q^{17} - 42 q^{19} + 72 q^{20} + 18 q^{23} + 196 q^{25} - 144 q^{28} - 99 q^{29} + 54 q^{35} + 195 q^{37} + 63 q^{41} + 82 q^{43} - 235 q^{49} - 208 q^{52} + 522 q^{53} + 270 q^{55} + 1368 q^{59} + 719 q^{61} + 1024 q^{64} + 351 q^{65} - 1218 q^{67} - 936 q^{68} + 810 q^{71} + 336 q^{76} - 1080 q^{77} - 880 q^{79} - 576 q^{80} + 1053 q^{85} - 2628 q^{89} - 936 q^{91} - 288 q^{92} - 126 q^{95} - 2004 q^{97}+O(q^{100}) \) 2 * q - 8 * q^4 + 18 * q^7 - 90 * q^11 - 65 * q^13 - 64 * q^16 - 117 * q^17 - 42 * q^19 + 72 * q^20 + 18 * q^23 + 196 * q^25 - 144 * q^28 - 99 * q^29 + 54 * q^35 + 195 * q^37 + 63 * q^41 + 82 * q^43 - 235 * q^49 - 208 * q^52 + 522 * q^53 + 270 * q^55 + 1368 * q^59 + 719 * q^61 + 1024 * q^64 + 351 * q^65 - 1218 * q^67 - 936 * q^68 + 810 * q^71 + 336 * q^76 - 1080 * q^77 - 880 * q^79 - 576 * q^80 + 1053 * q^85 - 2628 * q^89 - 936 * q^91 - 288 * q^92 - 126 * q^95 - 2004 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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