LMFDB - Embedded newform 684.3.b.a.683.9

Newspace parameters

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			683.9
Character	$\chi$	$=$	684.683
Dual form			684.3.b.a.683.11

$q$-expansion

$f(q)$	$=$	$q+(-1.88680 - 0.663313i) q^{2} +(3.12003 + 2.50308i) q^{4} +0.647124i q^{5} -4.63116i q^{7} +(-4.22656 - 6.79236i) q^{8} +O(q^{10})$	\(q+(-1.88680 - 0.663313i) q^{2} +(3.12003 + 2.50308i) q^{4} +0.647124i q^{5} -4.63116i q^{7} +(-4.22656 - 6.79236i) q^{8} +(0.429246 - 1.22099i) q^{10} -14.8893 q^{11} -22.0174i q^{13} +(-3.07191 + 8.73808i) q^{14} +(3.46921 + 15.6194i) q^{16} +18.9443i q^{17} +(1.76418 + 18.9179i) q^{19} +(-1.61980 + 2.01905i) q^{20} +(28.0932 + 9.87628i) q^{22} +13.7179 q^{23} +24.5812 q^{25} +(-14.6044 + 41.5424i) q^{26} +(11.5922 - 14.4494i) q^{28} -21.8371 q^{29} -19.3751 q^{31} +(3.81482 - 31.7718i) q^{32} +(12.5660 - 35.7442i) q^{34} +2.99694 q^{35} +31.3197i q^{37} +(9.21984 - 36.8645i) q^{38} +(4.39550 - 2.73511i) q^{40} -38.1704 q^{41} +32.1401i q^{43} +(-46.4552 - 37.2691i) q^{44} +(-25.8830 - 9.09926i) q^{46} -75.4122 q^{47} +27.5523 q^{49} +(-46.3799 - 16.3050i) q^{50} +(55.1112 - 68.6949i) q^{52} +60.7547 q^{53} -9.63524i q^{55} +(-31.4565 + 19.5739i) q^{56} +(41.2022 + 14.4848i) q^{58} +113.793i q^{59} -106.204 q^{61} +(36.5569 + 12.8517i) q^{62} +(-28.2724 + 57.4166i) q^{64} +14.2480 q^{65} -25.0449 q^{67} +(-47.4191 + 59.1069i) q^{68} +(-5.65462 - 1.98791i) q^{70} -67.2579i q^{71} +99.5142 q^{73} +(20.7747 - 59.0940i) q^{74} +(-41.8487 + 63.4404i) q^{76} +68.9549i q^{77} -84.4233 q^{79} +(-10.1077 + 2.24501i) q^{80} +(72.0199 + 25.3189i) q^{82} -92.6388 q^{83} -12.2593 q^{85} +(21.3189 - 60.6420i) q^{86} +(62.9306 + 101.134i) q^{88} +120.417 q^{89} -101.966 q^{91} +(42.8003 + 34.3370i) q^{92} +(142.288 + 50.0219i) q^{94} +(-12.2422 + 1.14164i) q^{95} +140.014i q^{97} +(-51.9858 - 18.2758i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 8 q^{4}+O(q^{10}) $ 80 * q - 8 * q^4	$ 80 q - 8 q^{4} - 56 q^{16} - 400 q^{25} - 464 q^{49} - 272 q^{58} - 352 q^{61} - 200 q^{64} + 480 q^{73} + 152 q^{76} + 32 q^{82} + 704 q^{85}+O(q^{100}) $ 80 * q - 8 * q^4 - 56 * q^16 - 400 * q^25 - 464 * q^49 - 272 * q^58 - 352 * q^61 - 200 * q^64 + 480 * q^73 + 152 * q^76 + 32 * q^82 + 704 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.88680	−	0.663313i	−0.943400	−	0.331656i
$2$	−1.88680	−	0.663313i	−0.943400	−	0.331656i
$3$		0			0
$3$		0			0
$4$	3.12003	+	2.50308i	0.780008	+	0.625769i
$5$			0.647124i			0.129425i	0.997904	+	0.0647124i	$0.0206130\pi$
$5$			0.647124i			0.129425i	−0.997904	+	0.0647124i	$0.979387\pi$
$6$		0			0
$7$		−	4.63116i		−	0.661595i	−0.943702	−	0.330797i	$-0.892682\pi$
$7$		−	4.63116i		−	0.661595i	0.943702	−	0.330797i	$-0.107318\pi$
$8$	−4.22656	−	6.79236i	−0.528320	−	0.849046i
$9$		0			0
$10$	0.429246	−	1.22099i	0.0429246	−	0.122099i
$11$	−14.8893			−1.35357			−0.676787	−	0.736179i	$-0.736628\pi$
$11$	−14.8893			−1.35357			−0.676787	+	0.736179i	$0.736628\pi$
$12$		0			0
$13$		−	22.0174i		−	1.69364i	−0.531877	−	0.846822i	$-0.678513\pi$
$13$		−	22.0174i		−	1.69364i	0.531877	−	0.846822i	$-0.321487\pi$
$14$	−3.07191	+	8.73808i	−0.219422	+	0.624149i
$15$		0			0
$16$	3.46921	+	15.6194i	0.216826	+	0.976210i
$17$			18.9443i			1.11437i	0.830388	+	0.557186i	$0.188119\pi$
$17$			18.9443i			1.11437i	−0.830388	+	0.557186i	$0.811881\pi$
$18$		0			0
$19$	1.76418	+	18.9179i	0.0928516	+	0.995680i
$19$	1.76418	+	18.9179i	0.0928516	+	0.995680i
$20$	−1.61980	+	2.01905i	−0.0809901	+	0.100952i
$21$		0			0
$22$	28.0932	+	9.87628i	1.27696	+	0.448922i
$23$	13.7179			0.596431			0.298215	−	0.954499i	$-0.403609\pi$
$23$	13.7179			0.596431			0.298215	+	0.954499i	$0.403609\pi$
$24$		0			0
$25$	24.5812			0.983249
$26$	−14.6044	+	41.5424i	−0.561707	+	1.59778i
$27$		0			0
$28$	11.5922	−	14.4494i	0.414006	−	0.516049i
$29$	−21.8371			−0.753002			−0.376501	−	0.926416i	$-0.622873\pi$
$29$	−21.8371			−0.753002			−0.376501	+	0.926416i	$0.622873\pi$
$30$		0			0
$31$	−19.3751			−0.625003			−0.312501	−	0.949917i	$-0.601167\pi$
$31$	−19.3751			−0.625003			−0.312501	+	0.949917i	$0.601167\pi$
$32$	3.81482	−	31.7718i	0.119213	−	0.992869i
$33$		0			0
$34$	12.5660	−	35.7442i	0.369589	−	1.05130i
$35$	2.99694			0.0856268
$36$		0			0
$37$			31.3197i			0.846478i	0.906018	+	0.423239i	$0.139107\pi$
$37$			31.3197i			0.846478i	−0.906018	+	0.423239i	$0.860893\pi$
$38$	9.21984	−	36.8645i	0.242627	−	0.970120i
$39$		0			0
$40$	4.39550	−	2.73511i	0.109888	−	0.0683777i
$41$	−38.1704			−0.930985			−0.465492	−	0.885052i	$-0.654123\pi$
$41$	−38.1704			−0.930985			−0.465492	+	0.885052i	$0.654123\pi$
$42$		0			0
$43$			32.1401i			0.747444i	0.927541	+	0.373722i	$0.121919\pi$
$43$			32.1401i			0.747444i	−0.927541	+	0.373722i	$0.878081\pi$
$44$	−46.4552	−	37.2691i	−1.05580	−	0.847026i
$45$		0			0
$46$	−25.8830	−	9.09926i	−0.562673	−	0.197810i
$47$	−75.4122			−1.60451			−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−75.4122			−1.60451			−0.802257	+	0.596978i	$0.796368\pi$
$48$		0			0
$49$	27.5523			0.562292
$50$	−46.3799	−	16.3050i	−0.927598	−	0.326101i
$51$		0			0
$52$	55.1112	−	68.6949i	1.05983	−	1.32106i
$53$	60.7547			1.14632			0.573158	−	0.819445i	$-0.305718\pi$
$53$	60.7547			1.14632			0.573158	+	0.819445i	$0.305718\pi$
$54$		0			0
$55$		−	9.63524i		−	0.175186i
$56$	−31.4565	+	19.5739i	−0.561724	+	0.349533i
$57$		0			0
$58$	41.2022	+	14.4848i	0.710383	+	0.249738i
$59$			113.793i			1.92870i	0.264634	+	0.964349i	$0.414749\pi$
$59$			113.793i			1.92870i	−0.264634	+	0.964349i	$0.585251\pi$
$60$		0			0
$61$	−106.204			−1.74105			−0.870525	−	0.492125i	$-0.836220\pi$
$61$	−106.204			−1.74105			−0.870525	+	0.492125i	$0.836220\pi$
$62$	36.5569	+	12.8517i	0.589628	+	0.207286i
$63$		0			0
$64$	−28.2724	+	57.4166i	−0.441757	+	0.897135i
$65$	14.2480			0.219199
$66$		0			0
$67$	−25.0449			−0.373805			−0.186903	−	0.982378i	$-0.559845\pi$
$67$	−25.0449			−0.373805			−0.186903	+	0.982378i	$0.559845\pi$
$68$	−47.4191	+	59.1069i	−0.697340	+	0.869219i
$69$		0			0
$70$	−5.65462	−	1.98791i	−0.0807803	−	0.0283987i
$71$		−	67.2579i		−	0.947295i	−0.880715	−	0.473647i	$-0.842937\pi$
$71$		−	67.2579i		−	0.947295i	0.880715	−	0.473647i	$-0.157063\pi$
$72$		0			0
$73$	99.5142			1.36321			0.681604	−	0.731721i	$-0.261283\pi$
$73$	99.5142			1.36321			0.681604	+	0.731721i	$0.261283\pi$
$74$	20.7747	−	59.0940i	0.280740	−	0.798567i
$75$		0			0
$76$	−41.8487	+	63.4404i	−0.550641	+	0.834742i
$77$			68.9549i			0.895518i
$78$		0			0
$79$	−84.4233			−1.06865			−0.534324	−	0.845279i	$-0.679434\pi$
$79$	−84.4233			−1.06865			−0.534324	+	0.845279i	$0.679434\pi$
$80$	−10.1077	+	2.24501i	−0.126346	+	0.0280626i
$81$		0			0
$82$	72.0199	+	25.3189i	0.878291	+	0.308767i
$83$	−92.6388			−1.11613			−0.558065	−	0.829797i	$-0.688456\pi$
$83$	−92.6388			−1.11613			−0.558065	+	0.829797i	$0.688456\pi$
$84$		0			0
$85$	−12.2593			−0.144227
$86$	21.3189	−	60.6420i	0.247895	−	0.705139i
$87$		0			0
$88$	62.9306	+	101.134i	0.715120	+	1.14925i
$89$	120.417			1.35300			0.676499	−	0.736444i	$-0.263496\pi$
$89$	120.417			1.35300			0.676499	+	0.736444i	$0.263496\pi$
$90$		0			0
$91$	−101.966			−1.12051
$92$	42.8003	+	34.3370i	0.465221	+	0.373228i
$93$		0			0
$94$	142.288	+	50.0219i	1.51370	+	0.532147i
$95$	−12.2422	+	1.14164i	−0.128866	+	0.0120173i
$96$		0			0
$97$			140.014i			1.44345i	0.692182	+	0.721723i	$0.256650\pi$
$97$			140.014i			1.44345i	−0.692182	+	0.721723i	$0.743350\pi$
$98$	−51.9858	−	18.2758i	−0.530467	−	0.186488i
$99$		0			0
$100$	76.6942	+	61.5287i	0.766942	+	0.615287i
$101$			34.2779i			0.339385i	0.985497	+	0.169692i	$0.0542774\pi$
$101$			34.2779i			0.339385i	−0.985497	+	0.169692i	$0.945723\pi$
$102$		0			0
$103$	−185.830			−1.80418			−0.902088	−	0.431553i	$-0.857966\pi$
$103$	−185.830			−1.80418			−0.902088	+	0.431553i	$0.857966\pi$
$104$	−149.550	+	93.0576i	−1.43798	+	0.894785i
$105$		0			0
$106$	−114.632	−	40.2994i	−1.08143	−	0.380183i
$107$			161.771i			1.51188i	0.654643	+	0.755938i	$0.272819\pi$
$107$			161.771i			1.51188i	−0.654643	+	0.755938i	$0.727181\pi$
$108$		0			0
$109$			100.731i			0.924138i	0.886844	+	0.462069i	$0.152893\pi$
$109$			100.731i			0.924138i	−0.886844	+	0.462069i	$0.847107\pi$
$110$	−6.39118	+	18.1798i	−0.0581016	+	0.165271i
$111$		0			0
$112$	72.3358	−	16.0665i	0.645856	−	0.143451i
$113$	−5.03853			−0.0445887			−0.0222944	−	0.999751i	$-0.507097\pi$
$113$	−5.03853			−0.0445887			−0.0222944	+	0.999751i	$0.507097\pi$
$114$		0			0
$115$			8.87719i			0.0771929i
$116$	−68.1324	−	54.6599i	−0.587348	−	0.471206i
$117$		0			0
$118$	75.4804	−	214.705i	0.639665	−	1.81953i
$119$	87.7343			0.737263
$120$		0			0
$121$	100.692			0.832165
$122$	200.386	+	70.4465i	1.64251	+	0.577430i
$123$		0			0
$124$	−60.4509	−	48.4973i	−0.487507	−	0.391107i
$125$			32.0852i			0.256682i
$126$		0			0
$127$	31.2889			0.246369			0.123185	−	0.992384i	$-0.460689\pi$
$127$	31.2889			0.246369			0.123185	+	0.992384i	$0.460689\pi$
$128$	91.4296	−	89.5803i	0.714294	−	0.699846i
$129$		0			0
$130$	−26.8831	−	9.45085i	−0.206793	−	0.0726989i
$131$	18.9095			0.144347			0.0721736	−	0.997392i	$-0.477006\pi$
$131$	18.9095			0.144347			0.0721736	+	0.997392i	$0.477006\pi$
$132$		0			0
$133$	87.6120	−	8.17021i	0.658737	−	0.0614301i
$134$	47.2548	+	16.6126i	0.352648	+	0.123975i
$135$		0			0
$136$	128.677	−	80.0693i	0.946153	−	0.588745i
$137$			140.158i			1.02305i	0.859269	+	0.511524i	$0.170919\pi$
$137$			140.158i			1.02305i	−0.859269	+	0.511524i	$0.829081\pi$
$138$		0			0
$139$			155.210i			1.11662i	0.829633	+	0.558309i	$0.188550\pi$
$139$			155.210i			1.11662i	−0.829633	+	0.558309i	$0.811450\pi$
$140$	9.35054	+	7.50156i	0.0667896	+	0.0535826i
$141$		0			0
$142$	−44.6130	+	126.902i	−0.314176	+	0.893678i
$143$			327.824i			2.29247i
$144$		0			0
$145$		−	14.1313i		−	0.0974572i
$146$	−187.764	−	66.0091i	−1.28605	−	0.452117i
$147$		0			0
$148$	−78.3956	+	97.7184i	−0.529700	+	0.660260i
$149$		−	204.301i		−	1.37115i	−0.728002	−	0.685575i	$-0.759551\pi$
$149$		−	204.301i		−	1.37115i	0.728002	−	0.685575i	$-0.240449\pi$
$150$		0			0
$151$	−194.202			−1.28610			−0.643052	−	0.765823i	$-0.722332\pi$
$151$	−194.202			−1.28610			−0.643052	+	0.765823i	$0.722332\pi$
$152$	121.041	−	91.9406i	0.796322	−	0.604872i
$153$		0			0
$154$	45.7386	−	130.104i	0.297004	−	0.844832i
$155$		−	12.5381i		−	0.0808909i
$156$		0			0
$157$	−124.358			−0.792087			−0.396043	−	0.918232i	$-0.629617\pi$
$157$	−124.358			−0.792087			−0.396043	+	0.918232i	$0.629617\pi$
$158$	159.290	+	55.9990i	1.00816	+	0.354424i
$159$		0			0
$160$	20.5603	+	2.46866i	0.128502	+	0.0154291i
$161$		−	63.5299i		−	0.394595i
$162$		0			0
$163$		−	194.930i		−	1.19589i	−0.801538	−	0.597944i	$-0.795984\pi$
$163$		−	194.930i		−	1.19589i	0.801538	−	0.597944i	$-0.204016\pi$
$164$	−119.093	−	95.5434i	−0.726176	−	0.582582i
$165$		0			0
$166$	174.791	+	61.4485i	1.05296	+	0.370171i
$167$		−	6.41401i		−	0.0384073i	−0.999816	−	0.0192036i	$-0.993887\pi$
$167$		−	6.41401i		−	0.0384073i	0.999816	−	0.0192036i	$-0.00611308\pi$
$168$		0			0
$169$	−315.764			−1.86843
$170$	23.1309	+	8.13177i	0.136064	+	0.0478339i
$171$		0			0
$172$	−80.4492	+	100.278i	−0.467728	+	0.583013i
$173$	−99.2301			−0.573585			−0.286792	−	0.957993i	$-0.592589\pi$
$173$	−99.2301			−0.573585			−0.286792	+	0.957993i	$0.592589\pi$
$174$		0			0
$175$		−	113.840i		−	0.650512i
$176$	−51.6542	−	232.562i	−0.293490	−	1.32137i
$177$		0			0
$178$	−227.202	−	79.8740i	−1.27642	−	0.448730i
$179$		−	127.149i		−	0.710330i	−0.934804	−	0.355165i	$-0.884425\pi$
$179$		−	127.149i		−	0.710330i	0.934804	−	0.355165i	$-0.115575\pi$
$180$		0			0
$181$			3.33163i			0.0184068i	0.999958	+	0.00920341i	$0.00292958\pi$
$181$			3.33163i			0.0184068i	−0.999958	+	0.00920341i	$0.997070\pi$
$182$	192.389	+	67.6353i	1.05709	+	0.371623i
$183$		0			0
$184$	−57.9795	−	93.1770i	−0.315106	−	0.506397i
$185$	−20.2677			−0.109555
$186$		0			0
$187$		−	282.068i		−	1.50839i
$188$	−235.288	−	188.763i	−1.25153	−	1.00406i
$189$		0			0
$190$	23.8559	+	5.96638i	0.125558	+	0.0314020i
$191$	99.7437			0.522219			0.261109	−	0.965309i	$-0.415912\pi$
$191$	99.7437			0.522219			0.261109	+	0.965309i	$0.415912\pi$
$192$		0			0
$193$		−	55.1904i		−	0.285961i	−0.989726	−	0.142980i	$-0.954331\pi$
$193$		−	55.1904i		−	0.285961i	0.989726	−	0.142980i	$-0.0456686\pi$
$194$	92.8733	−	264.179i	0.478728	−	1.36175i
$195$		0			0
$196$	85.9642	+	68.9656i	0.438593	+	0.351865i
$197$		−	148.744i		−	0.755043i	−0.926001	−	0.377522i	$-0.876776\pi$
$197$		−	148.744i		−	0.755043i	0.926001	−	0.377522i	$-0.123224\pi$
$198$		0			0
$199$			119.642i			0.601216i	0.953748	+	0.300608i	$0.0971897\pi$
$199$			119.642i			0.601216i	−0.953748	+	0.300608i	$0.902810\pi$
$200$	−103.894	−	166.965i	−0.519470	−	0.834823i
$201$		0			0
$202$	22.7369	−	64.6755i	0.112559	−	0.320176i
$203$			101.131i			0.498182i
$204$		0			0
$205$		−	24.7010i		−	0.120493i
$206$	350.624	+	123.263i	1.70206	+	0.598366i
$207$		0			0
$208$	343.897	−	76.3828i	1.65335	−	0.367225i
$209$	−26.2674	−	281.675i	−0.125682	−	1.34773i
$210$		0			0
$211$	−273.915			−1.29817			−0.649087	−	0.760714i	$-0.724849\pi$
$211$	−273.915			−1.29817			−0.649087	+	0.760714i	$0.724849\pi$
$212$	189.557	+	152.074i	0.894135	+	0.717329i
$213$		0			0
$214$	107.305	−	305.229i	0.501423	−	1.42630i
$215$	−20.7986			−0.0967378
$216$		0			0
$217$			89.7292i			0.413498i
$218$	66.8162	−	190.059i	0.306496	−	0.871832i
$219$		0			0
$220$	24.1177	−	30.0623i	0.109626	−	0.136647i
$221$	417.104			1.88735
$222$		0			0
$223$	208.529			0.935108			0.467554	−	0.883965i	$-0.345135\pi$
$223$	208.529			0.935108			0.467554	+	0.883965i	$0.345135\pi$
$224$	−147.140	−	17.6670i	−0.656877	−	0.0788707i
$225$		0			0
$226$	9.50669	+	3.34212i	0.0420650	+	0.0147881i
$227$		−	218.093i		−	0.960762i	−0.877060	−	0.480381i	$-0.840498\pi$
$227$		−	218.093i		−	0.960762i	0.877060	−	0.480381i	$-0.159502\pi$
$228$		0			0
$229$	104.878			0.457983			0.228991	−	0.973428i	$-0.426457\pi$
$229$	104.878			0.457983			0.228991	+	0.973428i	$0.426457\pi$
$230$	5.88835	−	16.7495i	0.0256015	−	0.0728238i
$231$		0			0
$232$	92.2956	+	148.325i	0.397826	+	0.639333i
$233$			85.0991i			0.365232i	0.983184	+	0.182616i	$0.0584566\pi$
$233$			85.0991i			0.365232i	−0.983184	+	0.182616i	$0.941543\pi$
$234$		0			0
$235$		−	48.8010i		−	0.207664i
$236$	−284.833	+	355.038i	−1.20692	+	1.50440i
$237$		0			0
$238$	−165.537	−	58.1952i	−0.695534	−	0.244518i
$239$	208.984			0.874411			0.437205	−	0.899362i	$-0.355968\pi$
$239$	208.984			0.874411			0.437205	+	0.899362i	$0.355968\pi$
$240$		0			0
$241$		−	291.886i		−	1.21115i	−0.795790	−	0.605573i	$-0.792944\pi$
$241$		−	291.886i		−	1.21115i	0.795790	−	0.605573i	$-0.207056\pi$
$242$	−189.986	−	66.7902i	−0.785064	−	0.275993i
$243$		0			0
$244$	−331.360	−	265.837i	−1.35803	−	1.08950i
$245$			17.8298i			0.0727746i
$246$		0			0
$247$	416.523	−	38.8426i	1.68633	−	0.157257i
$248$	81.8899	+	131.603i	0.330201	+	0.530656i
$249$		0			0
$250$	21.2825	−	60.5384i	0.0851301	−	0.242154i
$251$	234.938			0.936006			0.468003	−	0.883727i	$-0.344974\pi$
$251$	234.938			0.936006			0.468003	+	0.883727i	$0.344974\pi$
$252$		0			0
$253$	−204.250			−0.807313
$254$	−59.0359	−	20.7543i	−0.232425	−	0.0817099i
$255$		0			0
$256$	−231.929	+	108.374i	−0.905973	+	0.423335i
$257$	54.0528			0.210322			0.105161	−	0.994455i	$-0.466464\pi$
$257$	54.0528			0.210322			0.105161	+	0.994455i	$0.466464\pi$
$258$		0			0
$259$	145.047			0.560025
$260$	44.4541	+	35.6638i	0.170977	+	0.137168i
$261$		0			0
$262$	−35.6784	−	12.5429i	−0.136177	−	0.0478736i
$263$	482.093			1.83305			0.916526	−	0.399975i	$-0.130981\pi$
$263$	482.093			1.83305			0.916526	+	0.399975i	$0.130981\pi$
$264$		0			0
$265$			39.3158i			0.148362i
$266$	−170.726	−	42.6986i	−0.641826	−	0.160521i
$267$		0			0
$268$	−78.1411	−	62.6894i	−0.291571	−	0.233916i
$269$	−99.6712			−0.370525			−0.185262	−	0.982689i	$-0.559314\pi$
$269$	−99.6712			−0.370525			−0.185262	+	0.982689i	$0.559314\pi$
$270$		0			0
$271$		−	78.1924i		−	0.288533i	−0.989539	−	0.144266i	$-0.953918\pi$
$271$		−	78.1924i		−	0.288533i	0.989539	−	0.144266i	$-0.0460822\pi$
$272$	−295.898	+	65.7218i	−1.08786	+	0.241624i
$273$		0			0
$274$	92.9684	−	264.450i	0.339301	−	0.965145i
$275$	−365.998			−1.33090
$276$		0			0
$277$	87.8522			0.317156			0.158578	−	0.987346i	$-0.449309\pi$
$277$	87.8522			0.317156			0.158578	+	0.987346i	$0.449309\pi$
$278$	102.953	−	292.850i	0.370333	−	1.05342i
$279$		0			0
$280$	−12.6667	−	20.3563i	−0.0452383	−	0.0727010i
$281$	−223.682			−0.796023			−0.398011	−	0.917380i	$-0.630300\pi$
$281$	−223.682			−0.796023			−0.398011	+	0.917380i	$0.630300\pi$
$282$		0			0
$283$			427.937i			1.51214i	0.654488	+	0.756072i	$0.272884\pi$
$283$			427.937i			1.51214i	−0.654488	+	0.756072i	$0.727116\pi$
$284$	168.352	−	209.847i	0.592788	−	0.738898i
$285$		0			0
$286$	217.450	−	618.538i	0.760313	−	2.16272i
$287$			176.773i			0.615935i
$288$		0			0
$289$	−69.8875			−0.241825
$290$	−9.37346	+	26.6629i	−0.0323223	+	0.0919411i
$291$		0			0
$292$	310.488	+	249.092i	1.06331	+	0.853054i
$293$	−264.532			−0.902841			−0.451420	−	0.892311i	$-0.649082\pi$
$293$	−264.532			−0.902841			−0.451420	+	0.892311i	$0.649082\pi$
$294$		0			0
$295$	−73.6383			−0.249621
$296$	212.735	−	132.374i	0.718698	−	0.447211i
$297$		0			0
$298$	−135.516	+	385.476i	−0.454751	+	1.29354i
$299$		−	302.032i		−	1.01014i
$300$		0			0
$301$	148.846			0.494505
$302$	366.420	+	128.816i	1.21331	+	0.426544i
$303$		0			0
$304$	−289.366	+	93.1856i	−0.951860	+	0.306532i
$305$		−	68.7272i		−	0.225335i
$306$		0			0
$307$	−567.515			−1.84858			−0.924291	−	0.381689i	$-0.875343\pi$
$307$	−567.515			−1.84858			−0.924291	+	0.381689i	$0.875343\pi$
$308$	−172.599	+	215.141i	−0.560388	+	0.698511i
$309$		0			0
$310$	−8.31667	+	23.6569i	−0.0268280	+	0.0763125i
$311$	121.048			0.389222			0.194611	−	0.980880i	$-0.437655\pi$
$311$	121.048			0.389222			0.194611	+	0.980880i	$0.437655\pi$
$312$		0			0
$313$	109.296			0.349190			0.174595	−	0.984640i	$-0.444138\pi$
$313$	109.296			0.349190			0.174595	+	0.984640i	$0.444138\pi$
$314$	234.638	+	82.4880i	0.747255	+	0.262701i
$315$		0			0
$316$	−263.403	−	211.318i	−0.833555	−	0.668728i
$317$	−494.489			−1.55990			−0.779952	−	0.625840i	$-0.784756\pi$
$317$	−494.489			−1.55990			−0.779952	+	0.625840i	$0.784756\pi$
$318$		0			0
$319$	325.139			1.01924
$320$	−37.1557	−	18.2958i	−0.116112	−	0.0571743i
$321$		0			0
$322$	−42.1402	+	119.868i	−0.130870	+	0.372261i
$323$	−358.387	+	33.4212i	−1.10956	+	0.103471i
$324$		0			0
$325$		−	541.214i		−	1.66527i
$326$	−129.299	+	367.794i	−0.396624	+	1.12820i
$327$		0			0
$328$	161.329	+	259.267i	0.491858	+	0.790449i
$329$			349.246i			1.06154i
$330$		0			0
$331$	488.208			1.47495			0.737475	−	0.675375i	$-0.236018\pi$
$331$	488.208			1.47495			0.737475	+	0.675375i	$0.236018\pi$
$332$	−289.036	−	231.882i	−0.870590	−	0.698440i
$333$		0			0
$334$	−4.25450	+	12.1020i	−0.0127380	+	0.0362334i
$335$		−	16.2072i		−	0.0483797i
$336$		0			0
$337$			322.191i			0.956056i	0.878345	+	0.478028i	$0.158648\pi$
$337$			322.191i			0.956056i	−0.878345	+	0.478028i	$0.841352\pi$
$338$	595.784	+	209.450i	1.76267	+	0.619676i
$339$		0			0
$340$	−38.2495	−	30.6860i	−0.112499	−	0.0902531i
$341$	288.482			0.845988
$342$		0			0
$343$		−	354.526i		−	1.03360i
$344$	218.307	−	135.842i	0.634614	−	0.394889i
$345$		0			0
$346$	187.227	+	65.8206i	0.541120	+	0.190233i
$347$	−269.767			−0.777425			−0.388713	−	0.921359i	$-0.627080\pi$
$347$	−269.767			−0.777425			−0.388713	+	0.921359i	$0.627080\pi$
$348$		0			0
$349$	248.416			0.711795			0.355897	−	0.934525i	$-0.384175\pi$
$349$	248.416			0.711795			0.355897	+	0.934525i	$0.384175\pi$
$350$	−75.5113	+	214.793i	−0.215747	+	0.613694i
$351$		0			0
$352$	−56.8000	+	473.061i	−0.161364	+	1.34392i
$353$		−	318.157i		−	0.901294i	−0.892702	−	0.450647i	$-0.851193\pi$
$353$		−	318.157i		−	0.901294i	0.892702	−	0.450647i	$-0.148807\pi$
$354$		0			0
$355$	43.5242			0.122603
$356$	375.704	+	301.412i	1.05535	+	0.846664i
$357$		0			0
$358$	−84.3395	+	239.905i	−0.235585	+	0.670125i
$359$	−203.903			−0.567976			−0.283988	−	0.958828i	$-0.591658\pi$
$359$	−203.903			−0.567976			−0.283988	+	0.958828i	$0.591658\pi$
$360$		0			0
$361$	−354.775	+	66.7492i	−0.982757	+	0.184901i
$362$	2.20991	−	6.28613i	0.00610474	−	0.0173650i
$363$		0			0
$364$	−318.137	−	255.229i	−0.874003	−	0.701178i
$365$			64.3981i			0.176433i
$366$		0			0
$367$		−	411.595i		−	1.12151i	−0.827981	−	0.560756i	$-0.810511\pi$
$367$		−	411.595i		−	1.12151i	0.827981	−	0.560756i	$-0.189489\pi$
$368$	47.5903	+	214.265i	0.129321	+	0.582242i
$369$		0			0
$370$	38.2411	+	13.4438i	0.103354	+	0.0363347i
$371$		−	281.365i		−	0.758396i
$372$		0			0
$373$			455.722i			1.22177i	0.791718	+	0.610887i	$0.209187\pi$
$373$			455.722i			1.22177i	−0.791718	+	0.610887i	$0.790813\pi$
$374$	−187.099	+	532.206i	−0.500266	+	1.42301i
$375$		0			0
$376$	318.734	+	512.227i	0.847697	+	1.36231i
$377$			480.795i			1.27532i
$378$		0			0
$379$	326.021			0.860213			0.430106	−	0.902778i	$-0.358476\pi$
$379$	326.021			0.860213			0.430106	+	0.902778i	$0.358476\pi$
$380$	−41.0538	−	27.0813i	−0.108036	−	0.0712666i
$381$		0			0
$382$	−188.197	−	66.1613i	−0.492661	−	0.173197i
$383$		−	239.458i		−	0.625217i	−0.949882	−	0.312609i	$-0.898797\pi$
$383$		−	239.458i		−	0.625217i	0.949882	−	0.312609i	$-0.101203\pi$
$384$		0			0
$385$	−44.6224			−0.115902
$386$	−36.6085	+	104.133i	−0.0948407	+	0.269776i
$387$		0			0
$388$	−350.467	+	436.849i	−0.903265	+	1.12590i
$389$		−	381.887i		−	0.981714i	−0.871240	−	0.490857i	$-0.836684\pi$
$389$		−	381.887i		−	0.981714i	0.871240	−	0.490857i	$-0.163316\pi$
$390$		0			0
$391$			259.876i			0.664646i
$392$	−116.451	−	187.145i	−0.297070	−	0.477412i
$393$		0			0
$394$	−98.6635	+	280.649i	−0.250415	+	0.712308i
$395$		−	54.6323i		−	0.138310i
$396$		0			0
$397$	67.7120			0.170559			0.0852796	−	0.996357i	$-0.472822\pi$
$397$	67.7120			0.170559			0.0852796	+	0.996357i	$0.472822\pi$
$398$	79.3601	−	225.741i	0.199397	−	0.567188i
$399$		0			0
$400$	85.2774	+	383.943i	0.213194	+	0.959858i
$401$	393.401			0.981050			0.490525	−	0.871427i	$-0.336805\pi$
$401$	393.401			0.981050			0.490525	+	0.871427i	$0.336805\pi$
$402$		0			0
$403$			426.588i			1.05853i
$404$	−85.8001	+	106.948i	−0.212377	+	0.264723i
$405$		0			0
$406$	67.0815	−	190.814i	0.165225	−	0.469985i
$407$		−	466.329i		−	1.14577i
$408$		0			0
$409$		−	497.707i		−	1.21689i	−0.793597	−	0.608443i	$-0.791794\pi$
$409$		−	497.707i		−	1.21689i	0.793597	−	0.608443i	$-0.208206\pi$
$410$	−16.3845	+	46.6058i	−0.0399621	+	0.113673i
$411$		0			0
$412$	−579.796	−	465.147i	−1.40727	−	1.12900i
$413$	526.995			1.27602
$414$		0			0
$415$		−	59.9488i		−	0.144455i
$416$	−699.531	−	83.9922i	−1.68157	−	0.201904i
$417$		0			0
$418$	−137.277	+	548.888i	−0.328414	+	1.31313i
$419$	328.054			0.782946			0.391473	−	0.920190i	$-0.371966\pi$
$419$	328.054			0.782946			0.391473	+	0.920190i	$0.371966\pi$
$420$		0			0
$421$			194.786i			0.462675i	0.972874	+	0.231338i	$0.0743102\pi$
$421$			194.786i			0.462675i	−0.972874	+	0.231338i	$0.925690\pi$
$422$	516.822	+	181.691i	1.22470	+	0.430547i
$423$		0			0
$424$	−256.783	−	412.668i	−0.605621	−	0.973274i
$425$			465.675i			1.09571i
$426$		0			0
$427$			491.848i			1.15187i
$428$	−404.925	+	504.730i	−0.946085	+	1.17928i
$429$		0			0
$430$	39.2429	+	13.7960i	0.0912625	+	0.0320837i
$431$			314.574i			0.729870i	0.931033	+	0.364935i	$0.118909\pi$
$431$			314.574i			0.729870i	−0.931033	+	0.364935i	$0.881091\pi$
$432$		0			0
$433$		−	234.253i		−	0.540999i	−0.962720	−	0.270500i	$-0.912811\pi$
$433$		−	234.253i		−	0.540999i	0.962720	−	0.270500i	$-0.0871889\pi$
$434$	59.5185	−	169.301i	0.137139	−	0.390095i
$435$		0			0
$436$	−252.138	+	314.284i	−0.578297	+	0.720836i
$437$	24.2009	+	259.514i	0.0553795	+	0.593854i
$438$		0			0
$439$	−239.334			−0.545181			−0.272590	−	0.962130i	$-0.587880\pi$
$439$	−239.334			−0.545181			−0.272590	+	0.962130i	$0.587880\pi$
$440$	−65.4461	+	40.7239i	−0.148741	+	0.0925543i
$441$		0			0
$442$	−786.992	−	276.670i	−1.78053	−	0.625951i
$443$	−253.331			−0.571854			−0.285927	−	0.958251i	$-0.592301\pi$
$443$	−253.331			−0.571854			−0.285927	+	0.958251i	$0.592301\pi$
$444$		0			0
$445$			77.9246i			0.175111i
$446$	−393.453	−	138.320i	−0.882181	−	0.310134i
$447$		0			0
$448$	265.906	+	130.934i	0.593540	+	0.292264i
$449$	−352.503			−0.785085			−0.392543	−	0.919734i	$-0.628404\pi$
$449$	−352.503			−0.785085			−0.392543	+	0.919734i	$0.628404\pi$
$450$		0			0
$451$	568.331			1.26016
$452$	−15.7204	−	12.6118i	−0.0347796	−	0.0279023i
$453$		0			0
$454$	−144.664	+	411.498i	−0.318643	+	0.906383i
$455$		−	65.9846i		−	0.145021i
$456$		0			0
$457$	−33.8092			−0.0739807			−0.0369903	−	0.999316i	$-0.511777\pi$
$457$	−33.8092			−0.0739807			−0.0369903	+	0.999316i	$0.511777\pi$
$458$	−197.884	−	69.5670i	−0.432061	−	0.151893i
$459$		0			0
$460$	−22.2203	+	27.6971i	−0.0483050	+	0.0602111i
$461$		−	159.193i		−	0.345321i	−0.984981	−	0.172660i	$-0.944764\pi$
$461$		−	159.193i		−	0.345321i	0.984981	−	0.172660i	$-0.0552363\pi$
$462$		0			0
$463$			383.183i			0.827610i	0.910366	+	0.413805i	$0.135800\pi$
$463$			383.183i			0.827610i	−0.910366	+	0.413805i	$0.864200\pi$
$464$	−75.7573	−	341.081i	−0.163270	−	0.735089i
$465$		0			0
$466$	56.4473	−	160.565i	0.121132	−	0.344560i
$467$	136.387			0.292049			0.146025	−	0.989281i	$-0.453352\pi$
$467$	136.387			0.292049			0.146025	+	0.989281i	$0.453352\pi$
$468$		0			0
$469$			115.987i			0.247308i
$470$	−32.3703	+	92.0778i	−0.0688731	+	0.195910i
$471$		0			0
$472$	772.925	−	480.953i	1.63755	−	1.01897i
$473$		−	478.544i		−	1.01172i
$474$		0			0
$475$	43.3657	+	465.026i	0.0912962	+	0.979002i
$476$	273.734	+	219.606i	0.575071	+	0.461356i
$477$		0			0
$478$	−394.312	−	138.622i	−0.824920	−	0.290004i
$479$	−31.4695			−0.0656982			−0.0328491	−	0.999460i	$-0.510458\pi$
$479$	−31.4695			−0.0656982			−0.0328491	+	0.999460i	$0.510458\pi$
$480$		0			0
$481$	689.577			1.43363
$482$	−193.612	+	550.731i	−0.401684	+	1.14260i
$483$		0			0
$484$	314.162	+	252.040i	0.649095	+	0.520743i
$485$	−90.6067			−0.186818
$486$		0			0
$487$	707.449			1.45267			0.726333	−	0.687343i	$-0.241223\pi$
$487$	707.449			1.45267			0.726333	+	0.687343i	$0.241223\pi$
$488$	448.877	+	721.376i	0.919830	+	1.47823i
$489$		0			0
$490$	11.8267	−	33.6412i	0.0241362	−	0.0686556i
$491$	−549.851			−1.11986			−0.559930	−	0.828540i	$-0.689172\pi$
$491$	−549.851			−1.11986			−0.559930	+	0.828540i	$0.689172\pi$
$492$		0			0
$493$		−	413.689i		−	0.839125i
$494$	−811.660	−	202.997i	−1.64304	−	0.410924i
$495$		0			0
$496$	−67.2162	−	302.627i	−0.135517	−	0.610134i
$497$	−311.482			−0.626725
$498$		0			0
$499$		−	92.5348i		−	0.185440i	−0.995692	−	0.0927202i	$-0.970444\pi$
$499$		−	92.5348i		−	0.185440i	0.995692	−	0.0927202i	$-0.0295562\pi$
$500$	−80.3117	+	100.107i	−0.160623	+	0.200214i
$501$		0			0
$502$	−443.280	−	155.837i	−0.883029	−	0.310432i
$503$	−29.7026			−0.0590510			−0.0295255	−	0.999564i	$-0.509400\pi$
$503$	−29.7026			−0.0590510			−0.0295255	+	0.999564i	$0.509400\pi$
$504$		0			0
$505$	−22.1820			−0.0439248
$506$	385.380	+	135.482i	0.761620	+	0.267751i
$507$		0			0
$508$	97.6223	+	78.3185i	0.192170	+	0.154170i
$509$	−573.572			−1.12686			−0.563431	−	0.826163i	$-0.690519\pi$
$509$	−573.572			−1.12686			−0.563431	+	0.826163i	$0.690519\pi$
$510$		0			0
$511$		−	460.867i		−	0.901892i
$512$	509.490	−	50.6380i	0.995097	−	0.0989023i
$513$		0			0
$514$	−101.987	−	35.8539i	−0.198418	−	0.0697547i
$515$		−	120.255i		−	0.233505i
$516$		0			0
$517$	1122.84			2.17183
$518$	−273.674	−	96.2112i	−0.528328	−	0.185736i
$519$		0			0
$520$	−60.2198	−	96.7774i	−0.115807	−	0.186110i
$521$	546.984			1.04987			0.524937	−	0.851141i	$-0.324089\pi$
$521$	546.984			1.04987			0.524937	+	0.851141i	$0.324089\pi$
$522$		0			0
$523$	144.876			0.277009			0.138504	−	0.990362i	$-0.455771\pi$
$523$	144.876			0.277009			0.138504	+	0.990362i	$0.455771\pi$
$524$	58.9982	+	47.3319i	0.112592	+	0.0903280i
$525$		0			0
$526$	−909.613	−	319.778i	−1.72930	−	0.607943i
$527$		−	367.048i		−	0.696486i
$528$		0			0
$529$	−340.819			−0.644271
$530$	26.0787	−	74.1811i	0.0492051	−	0.139964i
$531$		0			0
$532$	293.803	+	193.808i	0.552261	+	0.364301i
$533$			840.411i			1.57676i
$534$		0			0
$535$	−104.686			−0.195674
$536$	105.854	+	170.114i	0.197489	+	0.317378i
$537$		0			0
$538$	188.060	+	66.1132i	0.349553	+	0.122887i
$539$	−410.236			−0.761105
$540$		0			0
$541$	−933.960			−1.72636			−0.863179	−	0.504898i	$-0.831530\pi$
$541$	−933.960			−1.72636			−0.863179	+	0.504898i	$0.831530\pi$
$542$	−51.8660	+	147.534i	−0.0956938	+	0.272202i
$543$		0			0
$544$	601.895	+	72.2691i	1.10643	+	0.132848i
$545$	−65.1855			−0.119606
$546$		0			0
$547$	670.814			1.22635			0.613176	−	0.789947i	$-0.289892\pi$
$547$	670.814			1.22635			0.613176	+	0.789947i	$0.289892\pi$
$548$	−350.826	+	437.297i	−0.640193	+	0.797987i
$549$		0			0
$550$	690.565	+	242.771i	1.25557	+	0.441402i
$551$	−38.5245	−	413.112i	−0.0699175	−	0.749749i
$552$		0			0
$553$			390.978i			0.707013i
$554$	−165.760	−	58.2735i	−0.299205	−	0.105187i
$555$		0			0
$556$	−388.502	+	484.260i	−0.698745	+	0.870971i
$557$			29.7179i			0.0533535i	0.999644	+	0.0266767i	$0.00849248\pi$
$557$			29.7179i			0.0533535i	−0.999644	+	0.0266767i	$0.991508\pi$
$558$		0			0
$559$	707.640			1.26590
$560$	10.3970	+	46.8103i	0.0185661	+	0.0835897i
$561$		0			0
$562$	422.044	+	148.371i	0.750968	+	0.264006i
$563$		−	909.252i		−	1.61501i	−0.589859	−	0.807506i	$-0.700817\pi$
$563$		−	909.252i		−	1.61501i	0.589859	−	0.807506i	$-0.299183\pi$
$564$		0			0
$565$		−	3.26055i		−	0.00577089i
$566$	283.856	−	807.432i	0.501512	−	1.42656i
$567$		0			0
$568$	−456.840	+	284.269i	−0.804296	+	0.500474i
$569$	−808.539			−1.42098			−0.710491	−	0.703706i	$-0.751527\pi$
$569$	−808.539			−1.42098			−0.710491	+	0.703706i	$0.751527\pi$
$570$		0			0
$571$			384.360i			0.673135i	0.941659	+	0.336567i	$0.109266\pi$
$571$			384.360i			0.673135i	−0.941659	+	0.336567i	$0.890734\pi$
$572$	−820.568	+	1022.82i	−1.43456	+	1.78815i
$573$		0			0
$574$	117.256	−	333.536i	0.204279	−	0.581073i
$575$	337.203			0.586440
$576$		0			0
$577$	−197.780			−0.342773			−0.171387	−	0.985204i	$-0.554825\pi$
$577$	−197.780			−0.342773			−0.171387	+	0.985204i	$0.554825\pi$
$578$	131.864	+	46.3573i	0.228138	+	0.0802029i
$579$		0			0
$580$	35.3717	−	44.0901i	0.0609857	−	0.0760174i
$581$			429.025i			0.738426i
$582$		0			0
$583$	−904.596			−1.55162
$584$	−420.603	−	675.937i	−0.720210	−	1.15743i
$585$		0			0
$586$	499.120	+	175.468i	0.851740	+	0.299433i
$587$	−779.727			−1.32833			−0.664163	−	0.747588i	$-0.731212\pi$
$587$	−779.727			−1.32833			−0.664163	+	0.747588i	$0.731212\pi$
$588$		0			0
$589$	−34.1811	−	366.536i	−0.0580325	−	0.622303i
$590$	138.941	+	48.8452i	0.235493	+	0.0827885i
$591$		0			0
$592$	−489.193	+	108.655i	−0.826340	+	0.183538i
$593$			118.684i			0.200142i	0.994980	+	0.100071i	$0.0319069\pi$
$593$			118.684i			0.200142i	−0.994980	+	0.100071i	$0.968093\pi$
$594$		0			0
$595$			56.7750i			0.0954201i
$596$	511.382	−	637.427i	0.858024	−	1.06951i
$597$		0			0
$598$	−200.342	+	569.874i	−0.335020	+	0.952967i
$599$		−	278.133i		−	0.464329i	−0.972677	−	0.232164i	$-0.925419\pi$
$599$		−	278.133i		−	0.464329i	0.972677	−	0.232164i	$-0.0745808\pi$
$600$		0			0
$601$			769.770i			1.28082i	0.768035	+	0.640408i	$0.221235\pi$
$601$			769.770i			1.28082i	−0.768035	+	0.640408i	$0.778765\pi$
$602$	−280.843	−	98.7315i	−0.466516	−	0.164006i
$603$		0			0
$604$	−605.916	−	486.102i	−1.00317	−	0.804804i
$605$			65.1602i			0.107703i
$606$		0			0
$607$	341.395			0.562430			0.281215	−	0.959645i	$-0.409263\pi$
$607$	341.395			0.562430			0.281215	+	0.959645i	$0.409263\pi$
$608$	607.786	+	16.1172i	0.999649	+	0.0265086i
$609$		0			0
$610$	−45.5876	+	129.674i	−0.0747338	+	0.212581i
$611$			1660.38i			2.71748i
$612$		0			0
$613$	192.021			0.313247			0.156624	−	0.987658i	$-0.449939\pi$
$613$	192.021			0.313247			0.156624	+	0.987658i	$0.449939\pi$
$614$	1070.79	+	376.440i	1.74395	+	0.613094i
$615$		0			0
$616$	468.367	−	291.442i	0.760336	−	0.473120i
$617$		−	607.759i		−	0.985023i	−0.870306	−	0.492511i	$-0.836079\pi$
$617$		−	607.759i		−	0.985023i	0.870306	−	0.492511i	$-0.163921\pi$
$618$		0			0
$619$			118.381i			0.191246i	0.995418	+	0.0956231i	$0.0304844\pi$
$619$			118.381i			0.191246i	−0.995418	+	0.0956231i	$0.969516\pi$
$620$	31.3838	−	39.1192i	0.0506190	−	0.0630955i
$621$		0			0
$622$	−228.394	−	80.2928i	−0.367193	−	0.129088i
$623$		−	557.670i		−	0.895136i
$624$		0			0
$625$	593.768			0.950028
$626$	−206.221	−	72.4977i	−0.329426	−	0.115811i
$627$		0			0
$628$	−388.000	−	311.277i	−0.617834	−	0.495664i
$629$	−593.330			−0.943291
$630$		0			0
$631$		−	1215.14i		−	1.92574i	−0.269960	−	0.962871i	$-0.587011\pi$
$631$		−	1215.14i		−	1.92574i	0.269960	−	0.962871i	$-0.412989\pi$
$632$	356.820	+	573.434i	0.564588	+	0.907332i
$633$		0			0
$634$	933.003	+	328.001i	1.47161	+	0.517352i
$635$			20.2478i			0.0318863i
$636$		0			0
$637$		−	606.630i		−	0.952323i
$638$	−613.473	−	215.669i	−0.961556	−	0.338039i
$639$		0			0
$640$	57.9695	+	59.1663i	0.0905774	+	0.0924473i
$641$	507.836			0.792255			0.396128	−	0.918195i	$-0.370354\pi$
$641$	507.836			0.792255			0.396128	+	0.918195i	$0.370354\pi$
$642$		0			0
$643$			517.738i			0.805191i	0.915378	+	0.402596i	$0.131892\pi$
$643$			517.738i			0.805191i	−0.915378	+	0.402596i	$0.868108\pi$
$644$	159.020	−	198.215i	0.246926	−	0.307788i
$645$		0			0
$646$	698.374	+	174.664i	1.08107	+	0.270377i
$647$	174.389			0.269535			0.134768	−	0.990877i	$-0.456971\pi$
$647$	174.389			0.269535			0.134768	+	0.990877i	$0.456971\pi$
$648$		0			0
$649$		−	1694.30i		−	2.61064i
$650$	−358.994	+	1021.16i	−0.552298	+	1.57102i
$651$		0			0
$652$	487.924	−	608.188i	0.748350	−	0.932803i
$653$			1222.16i			1.87161i	0.352513	+	0.935807i	$0.385327\pi$
$653$			1222.16i			1.87161i	−0.352513	+	0.935807i	$0.614673\pi$
$654$		0			0
$655$			12.2368i			0.0186821i
$656$	−132.421	−	596.197i	−0.201861	−	0.908837i
$657$		0			0
$658$	231.659	−	658.958i	0.352066	−	1.00146i
$659$		−	289.924i		−	0.439945i	−0.975506	−	0.219973i	$-0.929403\pi$
$659$		−	289.924i		−	0.439945i	0.975506	−	0.219973i	$-0.0705968\pi$
$660$		0			0
$661$		−	116.537i		−	0.176304i	−0.996107	−	0.0881520i	$-0.971904\pi$
$661$		−	116.537i		−	0.176304i	0.996107	−	0.0881520i	$-0.0280961\pi$
$662$	−921.152	−	323.835i	−1.39147	−	0.489176i
$663$		0			0
$664$	391.543	+	629.236i	0.589673	+	0.947645i
$665$	5.28714	+	56.6958i	0.00795058	+	0.0852569i
$666$		0			0
$667$	−299.559			−0.449114
$668$	16.0548	−	20.0119i	0.0240341	−	0.0299580i
$669$		0			0
$670$	−10.7504	+	30.5797i	−0.0160454	+	0.0456414i
$671$	1581.31			2.35664
$672$		0			0
$673$		−	436.189i		−	0.648126i	−0.946035	−	0.324063i	$-0.894951\pi$
$673$		−	436.189i		−	0.648126i	0.946035	−	0.324063i	$-0.105049\pi$
$674$	213.713	−	607.910i	0.317082	−	0.901944i
$675$		0			0
$676$	−985.195	−	790.382i	−1.45739	−	1.16920i
$677$	384.501			0.567949			0.283974	−	0.958832i	$-0.408347\pi$
$677$	384.501			0.567949			0.283974	+	0.958832i	$0.408347\pi$
$678$		0			0
$679$	648.429			0.954977
$680$	51.8148	+	83.2698i	0.0761982	+	0.122456i
$681$		0			0
$682$	−544.308	−	191.354i	−0.798105	−	0.280577i
$683$			670.786i			0.982116i	0.871127	+	0.491058i	$0.163390\pi$
$683$			670.786i			0.982116i	−0.871127	+	0.491058i	$0.836610\pi$
$684$		0			0
$685$	−90.6994			−0.132408
$686$	−235.162	+	668.920i	−0.342801	+	0.975103i
$687$		0			0
$688$	−502.008	+	111.501i	−0.729663	+	0.162065i
$689$		−	1337.66i		−	1.94145i
$690$		0			0
$691$		−	710.048i		−	1.02757i	−0.857920	−	0.513783i	$-0.828244\pi$
$691$		−	710.048i		−	1.02757i	0.857920	−	0.513783i	$-0.171756\pi$
$692$	−309.601	−	248.381i	−0.447401	−	0.358932i
$693$		0			0
$694$	508.996	+	178.940i	0.733423	+	0.257838i
$695$	−100.440			−0.144518
$696$		0			0
$697$		−	723.112i		−	1.03746i
$698$	−468.712	−	164.778i	−0.671508	−	0.236071i
$699$		0			0
$700$	284.950	−	355.184i	0.407071	−	0.507405i
$701$		−	1318.94i		−	1.88152i	−0.339078	−	0.940758i	$-0.610115\pi$
$701$		−	1318.94i		−	1.88152i	0.339078	−	0.940758i	$-0.389885\pi$
$702$		0			0
$703$	−592.503	+	55.2535i	−0.842821	+	0.0785968i
$704$	420.957	−	854.895i	0.597951	−	1.21434i
$705$		0			0
$706$	−211.037	+	600.298i	−0.298920	+	0.850281i
$707$	158.746			0.224535
$708$		0			0
$709$	352.865			0.497694			0.248847	−	0.968543i	$-0.419948\pi$
$709$	352.865			0.497694			0.248847	+	0.968543i	$0.419948\pi$
$710$	−82.1215	−	28.8702i	−0.115664	−	0.0406622i
$711$		0			0
$712$	−508.948	−	817.914i	−0.714815	−	1.14876i
$713$	−265.786			−0.372771
$714$		0			0
$715$	−212.143			−0.296703
$716$	318.264	−	396.709i	0.444502	−	0.554063i
$717$		0			0
$718$	384.725	+	135.252i	0.535829	+	0.188373i
$719$	123.090			0.171196			0.0855979	−	0.996330i	$-0.472720\pi$
$719$	123.090			0.171196			0.0855979	+	0.996330i	$0.472720\pi$
$720$		0			0
$721$			860.609i			1.19363i
$722$	713.666	+	109.384i	0.988457	+	0.151502i
$723$		0			0
$724$	−8.33934	+	10.3948i	−0.0115184	+	0.0143575i
$725$	−536.782			−0.740389
$726$		0			0
$727$			668.993i			0.920211i	0.887864	+	0.460105i	$0.152188\pi$
$727$			668.993i			0.920211i	−0.887864	+	0.460105i	$0.847812\pi$
$728$	430.965	+	692.590i	0.591985	+	0.951360i
$729$		0			0
$730$	42.7160	−	121.506i	0.0585151	−	0.166447i
$731$	−608.873			−0.832931
$732$		0			0
$733$	−954.382			−1.30202			−0.651011	−	0.759068i	$-0.725655\pi$
$733$	−954.382			−1.30202			−0.651011	+	0.759068i	$0.725655\pi$
$734$	−273.016	+	776.597i	−0.371956	+	1.05803i
$735$		0			0
$736$	52.3313	−	435.842i	0.0711023	−	0.592177i
$737$	372.902			0.505973
$738$		0			0
$739$			387.814i			0.524782i	0.964962	+	0.262391i	$0.0845110\pi$
$739$			387.814i			0.524782i	−0.964962	+	0.262391i	$0.915489\pi$
$740$	−63.2359	−	50.7317i	−0.0854540	−	0.0685563i
$741$		0			0
$742$	−186.633	+	530.880i	−0.251527	+	0.715471i
$743$			478.165i			0.643560i	0.946815	+	0.321780i	$0.104281\pi$
$743$			478.165i			0.643560i	−0.946815	+	0.321780i	$0.895719\pi$
$744$		0			0
$745$	132.208			0.177461
$746$	302.286	−	859.856i	0.405209	−	1.15262i
$747$		0			0
$748$	706.038	−	880.062i	0.943902	−	1.17655i
$749$	749.186			1.00025
$750$		0			0
$751$	−938.092			−1.24912			−0.624562	−	0.780975i	$-0.714723\pi$
$751$	−938.092			−1.24912			−0.624562	+	0.780975i	$0.714723\pi$
$752$	−261.621	−	1177.89i	−0.347900	−	1.56634i
$753$		0			0
$754$	318.917	−	907.163i	0.422967	−	1.20313i
$755$		−	125.673i		−	0.166454i
$756$		0			0
$757$	−710.155			−0.938117			−0.469059	−	0.883167i	$-0.655407\pi$
$757$	−710.155			−0.938117			−0.469059	+	0.883167i	$0.655407\pi$
$758$	−615.136	−	216.254i	−0.811525	−	0.285295i
$759$		0			0
$760$	59.4970	+	78.3285i	0.0782855	+	0.103064i
$761$		−	1234.32i		−	1.62197i	−0.585065	−	0.810986i	$-0.698931\pi$
$761$		−	1234.32i		−	1.62197i	0.585065	−	0.810986i	$-0.301069\pi$
$762$		0			0
$763$	466.502			0.611405
$764$	311.204	+	249.666i	0.407335	+	0.326788i
$765$		0			0
$766$	−158.836	+	451.810i	−0.207357	+	0.589830i
$767$	2505.43			3.26653
$768$		0			0
$769$	−343.803			−0.447078			−0.223539	−	0.974695i	$-0.571761\pi$
$769$	−343.803			−0.447078			−0.223539	+	0.974695i	$0.571761\pi$
$770$	84.1935	+	29.5986i	0.109342	+	0.0384397i
$771$		0			0
$772$	138.146	−	172.196i	0.178945	−	0.223052i
$773$	−172.010			−0.222522			−0.111261	−	0.993791i	$-0.535489\pi$
$773$	−172.010			−0.222522			−0.111261	+	0.993791i	$0.535489\pi$
$774$		0			0
$775$	−476.263			−0.614533
$776$	951.029	−	591.779i	1.22555	−	0.762601i
$777$		0			0
$778$	−253.310	+	720.544i	−0.325592	+	0.926149i
$779$	−67.3394	−	722.104i	−0.0864434	−	0.926963i
$780$		0			0
$781$			1001.43i			1.28223i
$782$	172.379	−	490.335i	0.220434	−	0.627027i
$783$		0			0
$784$	95.5848	+	430.350i	0.121919	+	0.548916i
$785$		−	80.4748i		−	0.102516i
$786$		0			0
$787$	−783.490			−0.995540			−0.497770	−	0.867309i	$-0.665848\pi$
$787$	−783.490			−0.995540			−0.497770	+	0.867309i	$0.665848\pi$
$788$	372.317	−	464.085i	0.472483	−	0.588940i
$789$		0			0
$790$	−36.2383	+	103.080i	−0.0458713	+	0.130481i
$791$			23.3342i			0.0294997i
$792$		0			0
$793$			2338.33i			2.94872i

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.88680 - 0.663313i) q^{2} +(3.12003 + 2.50308i) q^{4} +0.647124i q^{5} -4.63116i q^{7} +(-4.22656 - 6.79236i) q^{8} +O(q^{10})\)	\(q+(-1.88680 - 0.663313i) q^{2} +(3.12003 + 2.50308i) q^{4} +0.647124i q^{5} -4.63116i q^{7} +(-4.22656 - 6.79236i) q^{8} +(0.429246 - 1.22099i) q^{10} -14.8893 q^{11} -22.0174i q^{13} +(-3.07191 + 8.73808i) q^{14} +(3.46921 + 15.6194i) q^{16} +18.9443i q^{17} +(1.76418 + 18.9179i) q^{19} +(-1.61980 + 2.01905i) q^{20} +(28.0932 + 9.87628i) q^{22} +13.7179 q^{23} +24.5812 q^{25} +(-14.6044 + 41.5424i) q^{26} +(11.5922 - 14.4494i) q^{28} -21.8371 q^{29} -19.3751 q^{31} +(3.81482 - 31.7718i) q^{32} +(12.5660 - 35.7442i) q^{34} +2.99694 q^{35} +31.3197i q^{37} +(9.21984 - 36.8645i) q^{38} +(4.39550 - 2.73511i) q^{40} -38.1704 q^{41} +32.1401i q^{43} +(-46.4552 - 37.2691i) q^{44} +(-25.8830 - 9.09926i) q^{46} -75.4122 q^{47} +27.5523 q^{49} +(-46.3799 - 16.3050i) q^{50} +(55.1112 - 68.6949i) q^{52} +60.7547 q^{53} -9.63524i q^{55} +(-31.4565 + 19.5739i) q^{56} +(41.2022 + 14.4848i) q^{58} +113.793i q^{59} -106.204 q^{61} +(36.5569 + 12.8517i) q^{62} +(-28.2724 + 57.4166i) q^{64} +14.2480 q^{65} -25.0449 q^{67} +(-47.4191 + 59.1069i) q^{68} +(-5.65462 - 1.98791i) q^{70} -67.2579i q^{71} +99.5142 q^{73} +(20.7747 - 59.0940i) q^{74} +(-41.8487 + 63.4404i) q^{76} +68.9549i q^{77} -84.4233 q^{79} +(-10.1077 + 2.24501i) q^{80} +(72.0199 + 25.3189i) q^{82} -92.6388 q^{83} -12.2593 q^{85} +(21.3189 - 60.6420i) q^{86} +(62.9306 + 101.134i) q^{88} +120.417 q^{89} -101.966 q^{91} +(42.8003 + 34.3370i) q^{92} +(142.288 + 50.0219i) q^{94} +(-12.2422 + 1.14164i) q^{95} +140.014i q^{97} +(-51.9858 - 18.2758i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 8 q^{4}+O(q^{10}) \) 80 * q - 8 * q^4	\( 80 q - 8 q^{4} - 56 q^{16} - 400 q^{25} - 464 q^{49} - 272 q^{58} - 352 q^{61} - 200 q^{64} + 480 q^{73} + 152 q^{76} + 32 q^{82} + 704 q^{85}+O(q^{100}) \) 80 * q - 8 * q^4 - 56 * q^16 - 400 * q^25 - 464 * q^49 - 272 * q^58 - 352 * q^61 - 200 * q^64 + 480 * q^73 + 152 * q^76 + 32 * q^82 + 704 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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