LMFDB - Embedded newform 1680.3.l.b.1121.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(1121,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1121"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,8,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$45.7766844125$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 37 x^{14} - 116 x^{13} + 298 x^{12} - 548 x^{11} + 267 x^{10} + 3264 x^{9} + \cdots + 43046721 $ x^16 - 8x^15 + 37x^14 - 116x^13 + 298x^12 - 548x^11 + 267x^10 + 3264x^9 - 13590x^8 + 29376x^7 + 21627x^6 - 399492x^5 + 1955178x^4 - 6849684x^3 + 19663317x^2 - 38263752*x + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.8
Root			$0.421042 + 2.97031i$ of defining polynomial
Character	$\chi$	$=$	1680.1121
Dual form			1680.3.l.b.1121.7

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.421042 + 2.97031i) q^{3} -2.23607i q^{5} -2.64575 q^{7} +(-8.64545 + 2.50125i) q^{9} +8.16468i q^{11} -5.77792 q^{13} +(6.64181 - 0.941480i) q^{15} -28.0655i q^{17} +4.50420 q^{19} +(-1.11397 - 7.85869i) q^{21} -13.6408i q^{23} -5.00000 q^{25} +(-11.0696 - 24.6265i) q^{27} +15.1126i q^{29} +57.3810 q^{31} +(-24.2516 + 3.43768i) q^{33} +5.91608i q^{35} -9.23344 q^{37} +(-2.43275 - 17.1622i) q^{39} -23.0823i q^{41} +33.1640 q^{43} +(5.59297 + 19.3318i) q^{45} +4.35488i q^{47} +7.00000 q^{49} +(83.3630 - 11.8167i) q^{51} +22.9301i q^{53} +18.2568 q^{55} +(1.89646 + 13.3789i) q^{57} +36.3623i q^{59} -22.8074 q^{61} +(22.8737 - 6.61769i) q^{63} +12.9198i q^{65} +6.78336 q^{67} +(40.5173 - 5.74334i) q^{69} +80.7397i q^{71} +120.569 q^{73} +(-2.10521 - 14.8515i) q^{75} -21.6017i q^{77} +54.4972 q^{79} +(68.4875 - 43.2489i) q^{81} -57.2963i q^{83} -62.7563 q^{85} +(-44.8892 + 6.36306i) q^{87} -150.774i q^{89} +15.2869 q^{91} +(24.1598 + 170.439i) q^{93} -10.0717i q^{95} +117.942 q^{97} +(-20.4219 - 70.5873i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{3} - 10 q^{9} - 10 q^{15} - 48 q^{19} - 14 q^{21} - 80 q^{25} - 28 q^{27} - 24 q^{31} + 92 q^{33} - 40 q^{37} + 54 q^{39} + 168 q^{43} + 40 q^{45} + 112 q^{49} + 186 q^{51} + 80 q^{55} + 252 q^{57}+ \cdots - 26 q^{99}+O(q^{100}) $ 16 * q + 8 * q^3 - 10 * q^9 - 10 * q^15 - 48 * q^19 - 14 * q^21 - 80 * q^25 - 28 * q^27 - 24 * q^31 + 92 * q^33 - 40 * q^37 + 54 * q^39 + 168 * q^43 + 40 * q^45 + 112 * q^49 + 186 * q^51 + 80 * q^55 + 252 * q^57 - 152 * q^61 + 56 * q^63 - 264 * q^67 - 196 * q^69 + 240 * q^73 - 40 * q^75 - 212 * q^79 - 118 * q^81 - 60 * q^85 - 60 * q^87 - 196 * q^91 - 244 * q^93 - 360 * q^97 - 26 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	0.421042	+	2.97031i	0.140347	+	0.990102i
$3$	0.421042	+	2.97031i	0.140347	+	0.990102i
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	−2.64575			−0.377964
$7$	−2.64575			−0.377964
$8$		0			0
$9$	−8.64545	+	2.50125i	−0.960605	+	0.277917i
$10$		0			0
$11$			8.16468i			0.742243i	0.928584	+	0.371122i	$0.121027\pi$
$11$			8.16468i			0.742243i	−0.928584	+	0.371122i	$0.878973\pi$
$12$		0			0
$13$	−5.77792			−0.444455			−0.222228	−	0.974995i	$-0.571333\pi$
$13$	−5.77792			−0.444455			−0.222228	+	0.974995i	$0.571333\pi$
$14$		0			0
$15$	6.64181	−	0.941480i	0.442787	−	0.0627653i
$16$		0			0
$17$		−	28.0655i		−	1.65091i	−0.564469	−	0.825454i	$-0.690919\pi$
$17$		−	28.0655i		−	1.65091i	0.564469	−	0.825454i	$-0.309081\pi$
$18$		0			0
$19$	4.50420			0.237063			0.118532	−	0.992950i	$-0.462181\pi$
$19$	4.50420			0.237063			0.118532	+	0.992950i	$0.462181\pi$
$20$		0			0
$21$	−1.11397	−	7.85869i	−0.0530464	−	0.374223i
$22$		0			0
$23$		−	13.6408i		−	0.593077i	−0.955021	−	0.296538i	$-0.904168\pi$
$23$		−	13.6408i		−	0.593077i	0.955021	−	0.296538i	$-0.0958323\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−11.0696	−	24.6265i	−0.409985	−	0.912092i
$28$		0			0
$29$			15.1126i			0.521125i	0.965457	+	0.260563i	$0.0839080\pi$
$29$			15.1126i			0.521125i	−0.965457	+	0.260563i	$0.916092\pi$
$30$		0			0
$31$	57.3810			1.85100			0.925500	−	0.378748i	$-0.123645\pi$
$31$	57.3810			1.85100			0.925500	+	0.378748i	$0.123645\pi$
$32$		0			0
$33$	−24.2516	+	3.43768i	−0.734897	+	0.104172i
$34$		0			0
$35$			5.91608i			0.169031i
$36$		0			0
$37$	−9.23344			−0.249552			−0.124776	−	0.992185i	$-0.539821\pi$
$37$	−9.23344			−0.249552			−0.124776	+	0.992185i	$0.539821\pi$
$38$		0			0
$39$	−2.43275	−	17.1622i	−0.0623782	−	0.440056i
$40$		0			0
$41$		−	23.0823i		−	0.562983i	−0.959564	−	0.281492i	$-0.909171\pi$
$41$		−	23.0823i		−	0.562983i	0.959564	−	0.281492i	$-0.0908292\pi$
$42$		0			0
$43$	33.1640			0.771255			0.385628	−	0.922654i	$-0.373985\pi$
$43$	33.1640			0.771255			0.385628	+	0.922654i	$0.373985\pi$
$44$		0			0
$45$	5.59297	+	19.3318i	0.124288	+	0.429596i
$46$		0			0
$47$			4.35488i			0.0926571i	0.998926	+	0.0463286i	$0.0147521\pi$
$47$			4.35488i			0.0926571i	−0.998926	+	0.0463286i	$0.985248\pi$
$48$		0			0
$49$	7.00000			0.142857
$50$		0			0
$51$	83.3630	−	11.8167i	1.63457	−	0.231701i
$52$		0			0
$53$			22.9301i			0.432643i	0.976322	+	0.216322i	$0.0694060\pi$
$53$			22.9301i			0.432643i	−0.976322	+	0.216322i	$0.930594\pi$
$54$		0			0
$55$	18.2568			0.331941
$56$		0			0
$57$	1.89646	+	13.3789i	0.0332712	+	0.234717i
$58$		0			0
$59$			36.3623i			0.616311i	0.951336	+	0.308155i	$0.0997117\pi$
$59$			36.3623i			0.616311i	−0.951336	+	0.308155i	$0.900288\pi$
$60$		0			0
$61$	−22.8074			−0.373892			−0.186946	−	0.982370i	$-0.559859\pi$
$61$	−22.8074			−0.373892			−0.186946	+	0.982370i	$0.559859\pi$
$62$		0			0
$63$	22.8737	−	6.61769i	0.363075	−	0.105043i
$64$		0			0
$65$			12.9198i			0.198766i
$66$		0			0
$67$	6.78336			0.101244			0.0506221	−	0.998718i	$-0.483880\pi$
$67$	6.78336			0.101244			0.0506221	+	0.998718i	$0.483880\pi$
$68$		0			0
$69$	40.5173	−	5.74334i	0.587207	−	0.0832368i
$70$		0			0
$71$			80.7397i			1.13718i	0.822622	+	0.568589i	$0.192511\pi$
$71$			80.7397i			1.13718i	−0.822622	+	0.568589i	$0.807489\pi$
$72$		0			0
$73$	120.569			1.65163			0.825813	−	0.563944i	$-0.190717\pi$
$73$	120.569			1.65163			0.825813	+	0.563944i	$0.190717\pi$
$74$		0			0
$75$	−2.10521	−	14.8515i	−0.0280695	−	0.198020i
$76$		0			0
$77$		−	21.6017i		−	0.280542i
$78$		0			0
$79$	54.4972			0.689838			0.344919	−	0.938633i	$-0.387906\pi$
$79$	54.4972			0.689838			0.344919	+	0.938633i	$0.387906\pi$
$80$		0			0
$81$	68.4875	−	43.2489i	0.845525	−	0.533937i
$82$		0			0
$83$		−	57.2963i		−	0.690316i	−0.938545	−	0.345158i	$-0.887825\pi$
$83$		−	57.2963i		−	0.690316i	0.938545	−	0.345158i	$-0.112175\pi$
$84$		0			0
$85$	−62.7563			−0.738309
$86$		0			0
$87$	−44.8892	+	6.36306i	−0.515967	+	0.0731386i
$88$		0			0
$89$		−	150.774i		−	1.69409i	−0.531523	−	0.847044i	$-0.678380\pi$
$89$		−	150.774i		−	1.69409i	0.531523	−	0.847044i	$-0.321620\pi$
$90$		0			0
$91$	15.2869			0.167988
$92$		0			0
$93$	24.1598	+	170.439i	0.259783	+	1.83268i
$94$		0			0
$95$		−	10.0717i		−	0.106018i
$96$		0			0
$97$	117.942			1.21590			0.607949	−	0.793976i	$-0.291993\pi$
$97$	117.942			1.21590			0.607949	+	0.793976i	$0.291993\pi$
$98$		0			0
$99$	−20.4219	−	70.5873i	−0.206282	−	0.713003i
$100$		0			0
$101$		−	140.433i		−	1.39043i	−0.718804	−	0.695213i	$-0.755310\pi$
$101$		−	140.433i		−	1.39043i	0.718804	−	0.695213i	$-0.244690\pi$
$102$		0			0
$103$	−10.1147			−0.0982014			−0.0491007	−	0.998794i	$-0.515636\pi$
$103$	−10.1147			−0.0982014			−0.0491007	+	0.998794i	$0.515636\pi$
$104$		0			0
$105$	−17.5726	+	2.49092i	−0.167358	+	0.0237231i
$106$		0			0
$107$		−	55.1068i		−	0.515017i	−0.966276	−	0.257509i	$-0.917098\pi$
$107$		−	55.1068i		−	0.515017i	0.966276	−	0.257509i	$-0.0829016\pi$
$108$		0			0
$109$	63.5989			0.583477			0.291738	−	0.956498i	$-0.405766\pi$
$109$	63.5989			0.583477			0.291738	+	0.956498i	$0.405766\pi$
$110$		0			0
$111$	−3.88767	−	27.4261i	−0.0350241	−	0.247082i
$112$		0			0
$113$		−	25.6257i		−	0.226776i	−0.993551	−	0.113388i	$-0.963830\pi$
$113$		−	25.6257i		−	0.226776i	0.993551	−	0.113388i	$-0.0361704\pi$
$114$		0			0
$115$	−30.5017			−0.265232
$116$		0			0
$117$	49.9527	−	14.4520i	0.426946	−	0.123522i
$118$		0			0
$119$			74.2542i			0.623985i
$120$		0			0
$121$	54.3380			0.449075
$122$		0			0
$123$	68.5616	−	9.71863i	0.557411	−	0.0790133i
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	69.5279			0.547464			0.273732	−	0.961806i	$-0.411742\pi$
$127$	69.5279			0.547464			0.273732	+	0.961806i	$0.411742\pi$
$128$		0			0
$129$	13.9634	+	98.5072i	0.108244	+	0.763622i
$130$		0			0
$131$			114.353i			0.872927i	0.899722	+	0.436463i	$0.143769\pi$
$131$			114.353i			0.872927i	−0.899722	+	0.436463i	$0.856231\pi$
$132$		0			0
$133$	−11.9170			−0.0896014
$134$		0			0
$135$	−55.0665	+	24.7523i	−0.407900	+	0.183351i
$136$		0			0
$137$		−	165.307i		−	1.20662i	−0.797507	−	0.603310i	$-0.793848\pi$
$137$		−	165.307i		−	1.20662i	0.797507	−	0.603310i	$-0.206152\pi$
$138$		0			0
$139$	−207.941			−1.49598			−0.747989	−	0.663712i	$-0.768980\pi$
$139$	−207.941			−1.49598			−0.747989	+	0.663712i	$0.768980\pi$
$140$		0			0
$141$	−12.9353	+	1.83359i	−0.0917400	+	0.0130042i
$142$		0			0
$143$		−	47.1749i		−	0.329894i
$144$		0			0
$145$	33.7929			0.233054
$146$		0			0
$147$	2.94730	+	20.7921i	0.0200496	+	0.141443i
$148$		0			0
$149$			127.831i			0.857927i	0.903322	+	0.428963i	$0.141121\pi$
$149$			127.831i			0.857927i	−0.903322	+	0.428963i	$0.858879\pi$
$150$		0			0
$151$	−218.450			−1.44669			−0.723343	−	0.690489i	$-0.757396\pi$
$151$	−218.450			−1.44669			−0.723343	+	0.690489i	$0.757396\pi$
$152$		0			0
$153$	70.1987	+	242.638i	0.458815	+	1.58587i
$154$		0			0
$155$		−	128.308i		−	0.827792i
$156$		0			0
$157$	274.083			1.74575			0.872876	−	0.487942i	$-0.162252\pi$
$157$	274.083			1.74575			0.872876	+	0.487942i	$0.162252\pi$
$158$		0			0
$159$	−68.1094	+	9.65454i	−0.428361	+	0.0607204i
$160$		0			0
$161$			36.0901i			0.224162i
$162$		0			0
$163$	283.820			1.74123			0.870613	−	0.491968i	$-0.163722\pi$
$163$	283.820			1.74123			0.870613	+	0.491968i	$0.163722\pi$
$164$		0			0
$165$	7.68688	+	54.2282i	0.0465871	+	0.328656i
$166$		0			0
$167$			101.223i			0.606127i	0.952970	+	0.303064i	$0.0980095\pi$
$167$			101.223i			0.606127i	−0.952970	+	0.303064i	$0.901991\pi$
$168$		0			0
$169$	−135.616			−0.802459
$170$		0			0
$171$	−38.9408	+	11.2661i	−0.227724	+	0.0658838i
$172$		0			0
$173$			332.624i			1.92268i	0.275363	+	0.961340i	$0.411202\pi$
$173$			332.624i			1.92268i	−0.275363	+	0.961340i	$0.588798\pi$
$174$		0			0
$175$	13.2288			0.0755929
$176$		0			0
$177$	−108.007	+	15.3101i	−0.610211	+	0.0864977i
$178$		0			0
$179$		−	15.1419i		−	0.0845918i	−0.999105	−	0.0422959i	$-0.986533\pi$
$179$		−	15.1419i		−	0.0845918i	0.999105	−	0.0422959i	$-0.0134672\pi$
$180$		0			0
$181$	244.178			1.34905			0.674525	−	0.738252i	$-0.264349\pi$
$181$	244.178			1.34905			0.674525	+	0.738252i	$0.264349\pi$
$182$		0			0
$183$	−9.60288	−	67.7450i	−0.0524748	−	0.370191i
$184$		0			0
$185$			20.6466i			0.111603i
$186$		0			0
$187$	229.145			1.22538
$188$		0			0
$189$	29.2874	+	65.1556i	0.154960	+	0.344739i
$190$		0			0
$191$			328.924i			1.72211i	0.508508	+	0.861057i	$0.330197\pi$
$191$			328.924i			1.72211i	−0.508508	+	0.861057i	$0.669803\pi$
$192$		0			0
$193$	304.626			1.57837			0.789186	−	0.614154i	$-0.210502\pi$
$193$	304.626			1.57837			0.789186	+	0.614154i	$0.210502\pi$
$194$		0			0
$195$	−38.3758	+	5.43979i	−0.196799	+	0.0278964i
$196$		0			0
$197$		−	235.048i		−	1.19314i	−0.802563	−	0.596568i	$-0.796531\pi$
$197$		−	235.048i		−	1.19314i	0.802563	−	0.596568i	$-0.203469\pi$
$198$		0			0
$199$	298.075			1.49786			0.748931	−	0.662648i	$-0.230567\pi$
$199$	298.075			1.49786			0.748931	+	0.662648i	$0.230567\pi$
$200$		0			0
$201$	2.85608	+	20.1487i	0.0142094	+	0.100242i
$202$		0			0
$203$		−	39.9843i		−	0.196967i
$204$		0			0
$205$	−51.6136			−0.251774
$206$		0			0
$207$	34.1190	+	117.931i	0.164826	+	0.569713i
$208$		0			0
$209$			36.7753i			0.175959i
$210$		0			0
$211$	113.494			0.537885			0.268942	−	0.963156i	$-0.413326\pi$
$211$	113.494			0.537885			0.268942	+	0.963156i	$0.413326\pi$
$212$		0			0
$213$	−239.822	+	33.9948i	−1.12592	+	0.159600i
$214$		0			0
$215$		−	74.1569i		−	0.344916i
$216$		0			0
$217$	−151.816			−0.699612
$218$		0			0
$219$	50.7646	+	358.126i	0.231802	+	1.63528i
$220$		0			0
$221$			162.160i			0.733755i
$222$		0			0
$223$	−337.861			−1.51507			−0.757536	−	0.652794i	$-0.773597\pi$
$223$	−337.861			−1.51507			−0.757536	+	0.652794i	$0.773597\pi$
$224$		0			0
$225$	43.2272	−	12.5063i	0.192121	−	0.0555834i
$226$		0			0
$227$		−	41.4984i		−	0.182812i	−0.995814	−	0.0914062i	$-0.970864\pi$
$227$		−	41.4984i		−	0.182812i	0.995814	−	0.0914062i	$-0.0291362\pi$
$228$		0			0
$229$	−341.169			−1.48982			−0.744911	−	0.667164i	$-0.767508\pi$
$229$	−341.169			−1.48982			−0.744911	+	0.667164i	$0.767508\pi$
$230$		0			0
$231$	64.1637	−	9.09524i	0.277765	−	0.0393733i
$232$		0			0
$233$		−	113.530i		−	0.487255i	−0.969869	−	0.243627i	$-0.921663\pi$
$233$		−	113.530i		−	0.487255i	0.969869	−	0.243627i	$-0.0783374\pi$
$234$		0			0
$235$	9.73782			0.0414375
$236$		0			0
$237$	22.9456	+	161.873i	0.0968170	+	0.683010i
$238$		0			0
$239$		−	157.758i		−	0.660074i	−0.943968	−	0.330037i	$-0.892939\pi$
$239$		−	157.758i		−	0.660074i	0.943968	−	0.330037i	$-0.107061\pi$
$240$		0			0
$241$	247.947			1.02883			0.514413	−	0.857542i	$-0.328010\pi$
$241$	247.947			1.02883			0.514413	+	0.857542i	$0.328010\pi$
$242$		0			0
$243$	157.299	+	185.219i	0.647319	+	0.762219i
$244$		0			0
$245$		−	15.6525i		−	0.0638877i
$246$		0			0
$247$	−26.0249			−0.105364
$248$		0			0
$249$	170.187	−	24.1242i	0.683484	−	0.0968842i
$250$		0			0
$251$			276.737i			1.10254i	0.834328	+	0.551268i	$0.185856\pi$
$251$			276.737i			1.10254i	−0.834328	+	0.551268i	$0.814144\pi$
$252$		0			0
$253$	111.372			0.440207
$254$		0			0
$255$	−26.4230	−	186.405i	−0.103620	−	0.731001i
$256$		0			0
$257$		−	157.228i		−	0.611782i	−0.952067	−	0.305891i	$-0.901046\pi$
$257$		−	157.228i		−	0.611782i	0.952067	−	0.305891i	$-0.0989543\pi$
$258$		0			0
$259$	24.4294			0.0943219
$260$		0			0
$261$	−37.8005	−	130.655i	−0.144829	−	0.500596i
$262$		0			0
$263$		−	306.093i		−	1.16385i	−0.813242	−	0.581926i	$-0.802299\pi$
$263$		−	306.093i		−	1.16385i	0.813242	−	0.581926i	$-0.197701\pi$
$264$		0			0
$265$	51.2732			0.193484
$266$		0			0
$267$	447.844	−	63.4822i	1.67732	−	0.237761i
$268$		0			0
$269$		−	240.912i		−	0.895584i	−0.894138	−	0.447792i	$-0.852211\pi$
$269$		−	240.912i		−	0.895584i	0.894138	−	0.447792i	$-0.147789\pi$
$270$		0			0
$271$	−0.693234			−0.00255806			−0.00127903	−	0.999999i	$-0.500407\pi$
$271$	−0.693234			−0.00255806			−0.00127903	+	0.999999i	$0.500407\pi$
$272$		0			0
$273$	6.43645	+	45.4069i	0.0235767	+	0.166326i
$274$		0			0
$275$		−	40.8234i		−	0.148449i
$276$		0			0
$277$	−195.345			−0.705215			−0.352608	−	0.935771i	$-0.614705\pi$
$277$	−195.345			−0.705215			−0.352608	+	0.935771i	$0.614705\pi$
$278$		0			0
$279$	−496.084	+	143.524i	−1.77808	+	0.514424i
$280$		0			0
$281$		−	407.985i		−	1.45191i	−0.687745	−	0.725953i	$-0.741399\pi$
$281$		−	407.985i		−	1.45191i	0.687745	−	0.725953i	$-0.258601\pi$
$282$		0			0
$283$	387.819			1.37039			0.685193	−	0.728362i	$-0.259718\pi$
$283$	387.819			1.37039			0.685193	+	0.728362i	$0.259718\pi$
$284$		0			0
$285$	29.9160	−	4.24061i	0.104969	−	0.0148793i
$286$		0			0
$287$			61.0701i			0.212788i
$288$		0			0
$289$	−498.670			−1.72550
$290$		0			0
$291$	49.6586	+	350.324i	0.170648	+	1.20386i
$292$		0			0
$293$			98.4013i			0.335840i	0.985801	+	0.167920i	$0.0537051\pi$
$293$			98.4013i			0.335840i	−0.985801	+	0.167920i	$0.946295\pi$
$294$		0			0
$295$	81.3086			0.275623
$296$		0			0
$297$	201.067	−	90.3796i	0.676995	−	0.304308i
$298$		0			0
$299$			78.8153i			0.263596i
$300$		0			0
$301$	−87.7437			−0.291507
$302$		0			0
$303$	417.129	−	59.1283i	1.37666	−	0.195143i
$304$		0			0
$305$			50.9989i			0.167209i
$306$		0			0
$307$	172.156			0.560769			0.280384	−	0.959888i	$-0.409538\pi$
$307$	172.156			0.560769			0.280384	+	0.959888i	$0.409538\pi$
$308$		0			0
$309$	−4.25874	−	30.0439i	−0.0137823	−	0.0972294i
$310$		0			0
$311$		−	463.864i		−	1.49152i	−0.666212	−	0.745762i	$-0.732086\pi$
$311$		−	463.864i		−	1.49152i	0.666212	−	0.745762i	$-0.267914\pi$
$312$		0			0
$313$	−354.701			−1.13323			−0.566615	−	0.823983i	$-0.691747\pi$
$313$	−354.701			−1.13323			−0.566615	+	0.823983i	$0.691747\pi$
$314$		0			0
$315$	−14.7976	−	51.1472i	−0.0469765	−	0.162372i
$316$		0			0
$317$			44.9323i			0.141742i	0.997485	+	0.0708711i	$0.0225779\pi$
$317$			44.9323i			0.141742i	−0.997485	+	0.0708711i	$0.977422\pi$
$318$		0			0
$319$	−123.390			−0.386802
$320$		0			0
$321$	163.684	−	23.2023i	0.509920	−	0.0722814i
$322$		0			0
$323$		−	126.412i		−	0.391370i
$324$		0			0
$325$	28.8896			0.0888911
$326$		0			0
$327$	26.7779	+	188.908i	0.0818895	+	0.577701i
$328$		0			0
$329$		−	11.5219i		−	0.0350211i
$330$		0			0
$331$	31.4599			0.0950451			0.0475226	−	0.998870i	$-0.484867\pi$
$331$	31.4599			0.0950451			0.0475226	+	0.998870i	$0.484867\pi$
$332$		0			0
$333$	79.8272	−	23.0951i	0.239721	−	0.0693548i
$334$		0			0
$335$		−	15.1681i		−	0.0452778i
$336$		0			0
$337$	−392.031			−1.16330			−0.581648	−	0.813441i	$-0.697592\pi$
$337$	−392.031			−1.16330			−0.581648	+	0.813441i	$0.697592\pi$
$338$		0			0
$339$	76.1163	−	10.7895i	0.224532	−	0.0318275i
$340$		0			0
$341$			468.497i			1.37389i
$342$		0			0
$343$	−18.5203			−0.0539949
$344$		0			0
$345$	−12.8425	−	90.5994i	−0.0372246	−	0.262607i
$346$		0			0
$347$		−	649.989i		−	1.87317i	−0.350444	−	0.936584i	$-0.613969\pi$
$347$		−	649.989i		−	1.87317i	0.350444	−	0.936584i	$-0.386031\pi$
$348$		0			0
$349$	−176.534			−0.505827			−0.252913	−	0.967489i	$-0.581389\pi$
$349$	−176.534			−0.505827			−0.252913	+	0.967489i	$0.581389\pi$
$350$		0			0
$351$	63.9592	+	142.290i	0.182220	+	0.405384i
$352$		0			0
$353$			84.4749i			0.239306i	0.992816	+	0.119653i	$0.0381782\pi$
$353$			84.4749i			0.239306i	−0.992816	+	0.119653i	$0.961822\pi$
$354$		0			0
$355$	180.539			0.508562
$356$		0			0
$357$	−220.558	+	31.2642i	−0.617809	+	0.0875747i
$358$		0			0
$359$		−	175.365i		−	0.488481i	−0.969715	−	0.244241i	$-0.921461\pi$
$359$		−	175.365i		−	0.488481i	0.969715	−	0.244241i	$-0.0785387\pi$
$360$		0			0
$361$	−340.712			−0.943801
$362$		0			0
$363$	22.8786	+	161.401i	0.0630265	+	0.444630i
$364$		0			0
$365$		−	269.600i		−	0.738630i
$366$		0			0
$367$	−643.483			−1.75336			−0.876680	−	0.481075i	$-0.840247\pi$
$367$	−643.483			−1.75336			−0.876680	+	0.481075i	$0.840247\pi$
$368$		0			0
$369$	57.7347	+	199.557i	0.156462	+	0.540805i
$370$		0			0
$371$		−	60.6673i		−	0.163524i
$372$		0			0
$373$	312.351			0.837402			0.418701	−	0.908124i	$-0.362486\pi$
$373$	312.351			0.837402			0.418701	+	0.908124i	$0.362486\pi$
$374$		0			0
$375$	−33.2090	+	4.70740i	−0.0885574	+	0.0125531i
$376$		0			0
$377$		−	87.3196i		−	0.231617i
$378$		0			0
$379$	−129.174			−0.340830			−0.170415	−	0.985372i	$-0.554511\pi$
$379$	−129.174			−0.340830			−0.170415	+	0.985372i	$0.554511\pi$
$380$		0			0
$381$	29.2742	+	206.519i	0.0768352	+	0.542045i
$382$		0			0
$383$		−	371.018i		−	0.968716i	−0.874870	−	0.484358i	$-0.839053\pi$
$383$		−	371.018i		−	0.968716i	0.874870	−	0.484358i	$-0.160947\pi$
$384$		0			0
$385$	−48.3029			−0.125462
$386$		0			0
$387$	−286.717	+	82.9514i	−0.740872	+	0.214345i
$388$		0			0
$389$			239.364i			0.615332i	0.951494	+	0.307666i	$0.0995479\pi$
$389$			239.364i			0.615332i	−0.951494	+	0.307666i	$0.900452\pi$
$390$		0			0
$391$	−382.834			−0.979116
$392$		0			0
$393$	−339.665	+	48.1476i	−0.864287	+	0.122513i
$394$		0			0
$395$		−	121.859i		−	0.308505i
$396$		0			0
$397$	745.866			1.87875			0.939377	−	0.342885i	$-0.111404\pi$
$397$	745.866			1.87875			0.939377	+	0.342885i	$0.111404\pi$
$398$		0			0
$399$	−5.01756	−	35.3971i	−0.0125753	−	0.0887146i
$400$		0			0
$401$		−	338.979i		−	0.845335i	−0.906285	−	0.422667i	$-0.861094\pi$
$401$		−	338.979i		−	0.845335i	0.906285	−	0.422667i	$-0.138906\pi$
$402$		0			0
$403$	−331.543			−0.822687
$404$		0			0
$405$	−96.7074	−	153.143i	−0.238784	−	0.378130i
$406$		0			0
$407$		−	75.3881i		−	0.185229i
$408$		0			0
$409$	−242.273			−0.592355			−0.296177	−	0.955133i	$-0.595712\pi$
$409$	−242.273			−0.592355			−0.296177	+	0.955133i	$0.595712\pi$
$410$		0			0
$411$	491.012	−	69.6012i	1.19468	−	0.169346i
$412$		0			0
$413$		−	96.2057i		−	0.232944i
$414$		0			0
$415$	−128.118			−0.308719
$416$		0			0
$417$	−87.5519	−	617.648i	−0.209957	−	1.48117i
$418$		0			0
$419$		−	116.740i		−	0.278615i	−0.990249	−	0.139308i	$-0.955512\pi$
$419$		−	116.740i		−	0.278615i	0.990249	−	0.139308i	$-0.0444877\pi$
$420$		0			0
$421$	−57.9552			−0.137661			−0.0688304	−	0.997628i	$-0.521927\pi$
$421$	−57.9552			−0.137661			−0.0688304	+	0.997628i	$0.521927\pi$
$422$		0			0
$423$	−10.8927	−	37.6499i	−0.0257510	−	0.0890069i
$424$		0			0
$425$			140.327i			0.330182i
$426$		0			0
$427$	60.3427			0.141318
$428$		0			0
$429$	140.124	−	19.8626i	0.326629	−	0.0462998i
$430$		0			0
$431$		−	683.443i		−	1.58571i	−0.609408	−	0.792857i	$-0.708593\pi$
$431$		−	683.443i		−	1.58571i	0.609408	−	0.792857i	$-0.291407\pi$
$432$		0			0
$433$	710.818			1.64161			0.820806	−	0.571207i	$-0.193525\pi$
$433$	710.818			1.64161			0.820806	+	0.571207i	$0.193525\pi$
$434$		0			0
$435$	14.2282	+	100.375i	0.0327086	+	0.230748i
$436$		0			0
$437$		−	61.4407i		−	0.140597i
$438$		0			0
$439$	113.873			0.259392			0.129696	−	0.991554i	$-0.458600\pi$
$439$	113.873			0.259392			0.129696	+	0.991554i	$0.458600\pi$
$440$		0			0
$441$	−60.5181	+	17.5088i	−0.137229	+	0.0397024i
$442$		0			0
$443$			285.321i			0.644066i	0.946728	+	0.322033i	$0.104366\pi$
$443$			285.321i			0.644066i	−0.946728	+	0.322033i	$0.895634\pi$
$444$		0			0
$445$	−337.140			−0.757619
$446$		0			0
$447$	−379.698	+	53.8223i	−0.849435	+	0.120408i
$448$		0			0
$449$		−	248.688i		−	0.553871i	−0.960888	−	0.276936i	$-0.910681\pi$
$449$		−	248.688i		−	0.553871i	0.960888	−	0.276936i	$-0.0893189\pi$
$450$		0			0
$451$	188.460			0.417871
$452$		0			0
$453$	−91.9766	−	648.863i	−0.203039	−	1.43237i
$454$		0			0
$455$		−	34.1826i		−	0.0751267i
$456$		0			0
$457$	179.703			0.393224			0.196612	−	0.980481i	$-0.437006\pi$
$457$	179.703			0.393224			0.196612	+	0.980481i	$0.437006\pi$
$458$		0			0
$459$	−691.154	+	310.673i	−1.50578	+	0.676847i
$460$		0			0
$461$		−	496.992i		−	1.07807i	−0.842282	−	0.539037i	$-0.818788\pi$
$461$		−	496.992i		−	1.07807i	0.842282	−	0.539037i	$-0.181212\pi$
$462$		0			0
$463$	−289.473			−0.625212			−0.312606	−	0.949883i	$-0.601202\pi$
$463$	−289.473			−0.625212			−0.312606	+	0.949883i	$0.601202\pi$
$464$		0			0
$465$	381.113	−	54.0230i	0.819599	−	0.116179i
$466$		0			0
$467$		−	200.191i		−	0.428674i	−0.976760	−	0.214337i	$-0.931241\pi$
$467$		−	200.191i		−	0.428674i	0.976760	−	0.214337i	$-0.0687590\pi$
$468$		0			0
$469$	−17.9471			−0.0382667
$470$		0			0
$471$	115.401	+	814.111i	0.245012	+	1.72847i
$472$		0			0
$473$			270.773i			0.572459i
$474$		0			0
$475$	−22.5210			−0.0474126
$476$		0			0
$477$	−57.3539	−	198.241i	−0.120239	−	0.415599i
$478$		0			0
$479$			616.585i			1.28723i	0.765348	+	0.643617i	$0.222567\pi$
$479$			616.585i			1.28723i	−0.765348	+	0.643617i	$0.777433\pi$
$480$		0			0
$481$	53.3501			0.110915
$482$		0			0
$483$	−107.199	+	15.1955i	−0.221943	+	0.0314606i
$484$		0			0
$485$		−	263.726i		−	0.543766i
$486$		0			0
$487$	−393.608			−0.808229			−0.404115	−	0.914708i	$-0.632420\pi$
$487$	−393.608			−0.808229			−0.404115	+	0.914708i	$0.632420\pi$
$488$		0			0
$489$	119.500	+	843.032i	0.244377	+	1.72399i
$490$		0			0
$491$		−	602.139i		−	1.22635i	−0.789946	−	0.613176i	$-0.789892\pi$
$491$		−	602.139i		−	1.22635i	0.789946	−	0.613176i	$-0.210108\pi$
$492$		0			0
$493$	424.143			0.860330
$494$		0			0
$495$	−157.838	+	45.6648i	−0.318865	+	0.0922521i
$496$		0			0
$497$		−	213.617i		−	0.429813i
$498$		0			0
$499$	384.486			0.770513			0.385257	−	0.922810i	$-0.374113\pi$
$499$	384.486			0.770513			0.385257	+	0.922810i	$0.374113\pi$
$500$		0			0
$501$	−300.664	+	42.6193i	−0.600128	+	0.0850684i
$502$		0			0
$503$			458.304i			0.911141i	0.890200	+	0.455571i	$0.150565\pi$
$503$			458.304i			0.911141i	−0.890200	+	0.455571i	$0.849435\pi$
$504$		0			0
$505$	−314.018			−0.621817
$506$		0			0
$507$	−57.0999	−	402.820i	−0.112623	−	0.794517i
$508$		0			0
$509$			715.661i			1.40601i	0.711183	+	0.703007i	$0.248160\pi$
$509$			715.661i			1.40601i	−0.711183	+	0.703007i	$0.751840\pi$
$510$		0			0
$511$	−318.995			−0.624256
$512$		0			0
$513$	−49.8596	−	110.923i	−0.0971922	−	0.216223i
$514$		0			0
$515$			22.6172i			0.0439170i
$516$		0			0
$517$	−35.5562			−0.0687741
$518$		0			0
$519$	−987.995	+	140.049i	−1.90365	+	0.269843i
$520$		0			0
$521$			58.2889i			0.111879i	0.998434	+	0.0559394i	$0.0178154\pi$
$521$			58.2889i			0.111879i	−0.998434	+	0.0559394i	$0.982185\pi$
$522$		0			0
$523$	−676.567			−1.29363			−0.646814	−	0.762648i	$-0.723899\pi$
$523$	−676.567			−1.29363			−0.646814	+	0.762648i	$0.723899\pi$
$524$		0			0
$525$	5.56987	+	39.2935i	0.0106093	+	0.0748447i
$526$		0			0
$527$		−	1610.42i		−	3.05583i
$528$		0			0
$529$	342.929			0.648260
$530$		0			0
$531$	−90.9513	−	314.369i	−0.171283	−	0.592031i
$532$		0			0
$533$			133.368i			0.250221i
$534$		0			0
$535$	−123.223			−0.230323
$536$		0			0
$537$	44.9762	−	6.37540i	0.0837546	−	0.0118722i
$538$		0			0
$539$			57.1527i			0.106035i
$540$		0			0
$541$	−430.418			−0.795596			−0.397798	−	0.917473i	$-0.630226\pi$
$541$	−430.418			−0.795596			−0.397798	+	0.917473i	$0.630226\pi$
$542$		0			0
$543$	102.809	+	725.284i	0.189336	+	1.33570i
$544$		0			0
$545$		−	142.212i		−	0.260939i
$546$		0			0
$547$	110.245			0.201546			0.100773	−	0.994909i	$-0.467868\pi$
$547$	110.245			0.201546			0.100773	+	0.994909i	$0.467868\pi$
$548$		0			0
$549$	197.180	−	57.0470i	0.359162	−	0.103911i
$550$		0			0
$551$			68.0703i			0.123540i
$552$		0			0
$553$	−144.186			−0.260734
$554$		0			0
$555$	−61.3267	+	8.69309i	−0.110499	+	0.0156632i
$556$		0			0
$557$		−	463.317i		−	0.831809i	−0.909408	−	0.415904i	$-0.863465\pi$
$557$		−	463.317i		−	0.831809i	0.909408	−	0.415904i	$-0.136535\pi$
$558$		0			0
$559$	−191.619			−0.342789
$560$		0			0
$561$	96.4799	+	680.632i	0.171978	+	1.21325i
$562$		0			0
$563$			878.072i			1.55963i	0.626010	+	0.779815i	$0.284687\pi$
$563$			878.072i			1.55963i	−0.626010	+	0.779815i	$0.715313\pi$
$564$		0			0
$565$	−57.3009			−0.101417
$566$		0			0
$567$	−181.201	+	114.426i	−0.319578	+	0.201809i
$568$		0			0
$569$			579.224i			1.01797i	0.860776	+	0.508984i	$0.169979\pi$
$569$			579.224i			1.01797i	−0.860776	+	0.508984i	$0.830021\pi$
$570$		0			0
$571$	−471.669			−0.826040			−0.413020	−	0.910722i	$-0.635526\pi$
$571$	−471.669			−0.826040			−0.413020	+	0.910722i	$0.635526\pi$
$572$		0			0
$573$	−977.005	+	138.491i	−1.70507	+	0.241694i
$574$		0			0
$575$			68.2038i			0.118615i
$576$		0			0
$577$	291.140			0.504576			0.252288	−	0.967652i	$-0.418817\pi$
$577$	291.140			0.504576			0.252288	+	0.967652i	$0.418817\pi$
$578$		0			0
$579$	128.260	+	904.833i	0.221521	+	1.56275i
$580$		0			0
$581$			151.592i			0.260915i
$582$		0			0
$583$	−187.217			−0.321127
$584$		0			0
$585$	−32.3157	−	111.698i	−0.0552405	−	0.190936i
$586$		0			0
$587$		−	731.515i		−	1.24619i	−0.782145	−	0.623096i	$-0.785874\pi$
$587$		−	731.515i		−	1.24619i	0.782145	−	0.623096i	$-0.214126\pi$
$588$		0			0
$589$	258.455			0.438804
$590$		0			0
$591$	698.164	−	98.9651i	1.18133	−	0.167454i
$592$		0			0
$593$		−	683.187i		−	1.15209i	−0.817419	−	0.576043i	$-0.804596\pi$
$593$		−	683.187i		−	1.15209i	0.817419	−	0.576043i	$-0.195404\pi$
$594$		0			0
$595$	166.037			0.279055
$596$		0			0
$597$	125.502	+	885.373i	0.210221	+	1.48304i
$598$		0			0
$599$		−	464.649i		−	0.775708i	−0.921721	−	0.387854i	$-0.873217\pi$
$599$		−	464.649i		−	0.775708i	0.921721	−	0.387854i	$-0.126783\pi$
$600$		0			0
$601$	−242.259			−0.403093			−0.201547	−	0.979479i	$-0.564597\pi$
$601$	−242.259			−0.403093			−0.201547	+	0.979479i	$0.564597\pi$
$602$		0			0
$603$	−58.6452	+	16.9669i	−0.0972557	+	0.0281374i
$604$		0			0
$605$		−	121.504i		−	0.200832i
$606$		0			0
$607$	143.456			0.236337			0.118168	−	0.992994i	$-0.462298\pi$
$607$	143.456			0.236337			0.118168	+	0.992994i	$0.462298\pi$
$608$		0			0
$609$	118.766	−	16.8351i	0.195017	−	0.0276438i
$610$		0			0
$611$		−	25.1622i		−	0.0411820i
$612$		0			0
$613$	−274.405			−0.447643			−0.223821	−	0.974630i	$-0.571853\pi$
$613$	−274.405			−0.447643			−0.223821	+	0.974630i	$0.571853\pi$
$614$		0			0
$615$	−21.7315	−	153.308i	−0.0353358	−	0.249282i
$616$		0			0
$617$			232.732i			0.377199i	0.982054	+	0.188600i	$0.0603948\pi$
$617$			232.732i			0.377199i	−0.982054	+	0.188600i	$0.939605\pi$
$618$		0			0
$619$	1209.75			1.95437			0.977185	−	0.212392i	$-0.0681253\pi$
$619$	1209.75			1.95437			0.977185	+	0.212392i	$0.0681253\pi$
$620$		0			0
$621$	−335.924	+	150.998i	−0.540941	+	0.243152i
$622$		0			0
$623$			398.910i			0.640305i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−109.234	+	15.4840i	−0.174217	+	0.0246953i
$628$		0			0
$629$			259.141i			0.411988i
$630$		0			0
$631$	−581.467			−0.921500			−0.460750	−	0.887530i	$-0.652420\pi$
$631$	−581.467			−0.921500			−0.460750	+	0.887530i	$0.652420\pi$
$632$		0			0
$633$	47.7856	+	337.111i	0.0754907	+	0.532561i
$634$		0			0
$635$		−	155.469i		−	0.244833i
$636$		0			0
$637$	−40.4454			−0.0634936
$638$		0			0
$639$	−201.950	−	698.031i	−0.316041	−	1.09238i
$640$		0			0
$641$			1052.94i			1.64265i	0.570461	+	0.821325i	$0.306765\pi$
$641$			1052.94i			1.64265i	−0.570461	+	0.821325i	$0.693235\pi$
$642$		0			0
$643$	−307.328			−0.477959			−0.238980	−	0.971025i	$-0.576813\pi$
$643$	−307.328			−0.477959			−0.238980	+	0.971025i	$0.576813\pi$
$644$		0			0
$645$	220.269	−	31.2232i	0.341502	−	0.0484081i
$646$		0			0
$647$			765.017i			1.18241i	0.806523	+	0.591203i	$0.201347\pi$
$647$			765.017i			1.18241i	−0.806523	+	0.591203i	$0.798653\pi$
$648$		0			0
$649$	−296.887			−0.457453
$650$		0			0
$651$	−63.9209	−	450.940i	−0.0981888	−	0.692688i
$652$		0			0
$653$			455.317i			0.697270i	0.937259	+	0.348635i	$0.113355\pi$
$653$			455.317i			0.697270i	−0.937259	+	0.348635i	$0.886645\pi$
$654$		0			0
$655$	255.702			0.390385
$656$		0			0
$657$	−1042.37	+	301.573i	−1.58656	+	0.459015i
$658$		0			0
$659$			345.201i			0.523825i	0.965092	+	0.261913i	$0.0843532\pi$
$659$			345.201i			0.523825i	−0.965092	+	0.261913i	$0.915647\pi$
$660$		0			0
$661$	−487.944			−0.738191			−0.369096	−	0.929391i	$-0.620333\pi$
$661$	−487.944			−0.738191			−0.369096	+	0.929391i	$0.620333\pi$
$662$		0			0
$663$	−481.665	+	68.2762i	−0.726493	+	0.102981i
$664$		0			0
$665$			26.6472i			0.0400710i
$666$		0			0
$667$	206.148			0.309067
$668$		0			0
$669$	−142.254	−	1003.55i	−0.212636	−	1.50008i
$670$		0			0
$671$		−	186.215i		−	0.277519i
$672$		0			0
$673$	−562.562			−0.835902			−0.417951	−	0.908470i	$-0.637252\pi$
$673$	−562.562			−0.835902			−0.417951	+	0.908470i	$0.637252\pi$
$674$		0			0
$675$	55.3479	+	123.132i	0.0819969	+	0.182418i
$676$		0			0
$677$		−	944.244i		−	1.39475i	−0.716708	−	0.697373i	$-0.754352\pi$
$677$		−	944.244i		−	1.39475i	0.716708	−	0.697373i	$-0.245648\pi$
$678$		0			0
$679$	−312.045			−0.459566
$680$		0			0
$681$	123.263	−	17.4726i	0.181003	−	0.0256573i
$682$		0			0
$683$			801.351i			1.17328i	0.809847	+	0.586641i	$0.199550\pi$
$683$			801.351i			1.17328i	−0.809847	+	0.586641i	$0.800450\pi$
$684$		0			0
$685$	−369.637			−0.539616
$686$		0			0
$687$	−143.647	−	1013.38i	−0.209093	−	1.47508i
$688$		0			0
$689$		−	132.488i		−	0.192291i
$690$		0			0
$691$	−360.516			−0.521731			−0.260866	−	0.965375i	$-0.584008\pi$
$691$	−360.516			−0.521731			−0.260866	+	0.965375i	$0.584008\pi$
$692$		0			0
$693$	54.0313	+	186.756i	0.0779672	+	0.269490i
$694$		0			0
$695$			464.970i			0.669021i
$696$		0			0
$697$	−647.816			−0.929434
$698$		0			0
$699$	337.220	−	47.8011i	0.482432	−	0.0683850i
$700$		0			0
$701$		−	818.208i		−	1.16720i	−0.812041	−	0.583601i	$-0.801643\pi$
$701$		−	818.208i		−	1.16720i	0.812041	−	0.583601i	$-0.198357\pi$
$702$		0			0
$703$	−41.5892			−0.0591597
$704$		0			0
$705$	4.10004	+	28.9243i	0.00581565	+	0.0410274i
$706$		0			0
$707$			371.551i			0.525531i
$708$		0			0
$709$	−437.371			−0.616884			−0.308442	−	0.951243i	$-0.599808\pi$
$709$	−437.371			−0.616884			−0.308442	+	0.951243i	$0.599808\pi$
$710$		0			0
$711$	−471.152	+	136.311i	−0.662662	+	0.191717i
$712$		0			0
$713$		−	782.721i		−	1.09778i
$714$		0			0
$715$	−105.486			−0.147533
$716$		0			0
$717$	468.589	−	66.4227i	0.653541	−	0.0926397i
$718$		0			0
$719$			527.093i			0.733091i	0.930400	+	0.366546i	$0.119460\pi$
$719$			527.093i			0.733091i	−0.930400	+	0.366546i	$0.880540\pi$
$720$		0			0
$721$	26.7611			0.0371166
$722$		0			0
$723$	104.396	+	736.479i	0.144393	+	1.01864i
$724$		0			0
$725$		−	75.5632i		−	0.104225i
$726$		0			0
$727$	809.719			1.11378			0.556891	−	0.830586i	$-0.311994\pi$
$727$	809.719			1.11378			0.556891	+	0.830586i	$0.311994\pi$
$728$		0			0
$729$	−483.929	+	545.210i	−0.663825	+	0.747888i
$730$		0			0
$731$		−	930.762i		−	1.27327i
$732$		0			0
$733$	215.991			0.294667			0.147333	−	0.989087i	$-0.452931\pi$
$733$	215.991			0.294667			0.147333	+	0.989087i	$0.452931\pi$
$734$		0			0
$735$	46.4927	−	6.59036i	0.0632553	−	0.00896647i
$736$		0			0
$737$			55.3839i			0.0751478i
$738$		0			0
$739$	1025.08			1.38712			0.693560	−	0.720398i	$-0.256041\pi$
$739$	1025.08			1.38712			0.693560	+	0.720398i	$0.256041\pi$
$740$		0			0
$741$	−10.9576	−	77.3020i	−0.0147876	−	0.104321i
$742$		0			0
$743$			168.132i			0.226287i	0.993579	+	0.113144i	$0.0360921\pi$
$743$			168.132i			0.226287i	−0.993579	+	0.113144i	$0.963908\pi$
$744$		0			0
$745$	285.839			0.383676
$746$		0			0
$747$	143.312	+	495.352i	0.191850	+	0.663121i
$748$		0			0
$749$			145.799i			0.194658i
$750$		0			0
$751$	−998.130			−1.32907			−0.664534	−	0.747258i	$-0.731370\pi$
$751$	−998.130			−1.32907			−0.664534	+	0.747258i	$0.731370\pi$
$752$		0			0
$753$	−821.993	+	116.518i	−1.09162	+	0.154738i
$754$		0			0
$755$			488.468i			0.646978i
$756$		0			0
$757$	−310.157			−0.409719			−0.204860	−	0.978791i	$-0.565674\pi$
$757$	−310.157			−0.409719			−0.204860	+	0.978791i	$0.565674\pi$
$758$		0			0
$759$	46.8925	+	330.810i	0.0617820	+	0.435850i
$760$		0			0
$761$			1386.50i			1.82194i	0.412469	+	0.910972i	$0.364667\pi$
$761$			1386.50i			1.82194i	−0.412469	+	0.910972i	$0.635333\pi$
$762$		0			0
$763$	−168.267			−0.220533
$764$		0			0
$765$	542.556	−	156.969i	0.709223	−	0.205188i
$766$		0			0
$767$		−	210.099i		−	0.273923i
$768$		0			0
$769$	−950.327			−1.23580			−0.617898	−	0.786258i	$-0.712015\pi$
$769$	−950.327			−1.23580			−0.617898	+	0.786258i	$0.712015\pi$
$770$		0			0
$771$	467.015	−	66.1996i	0.605727	−	0.0858621i
$772$		0			0
$773$		−	898.658i		−	1.16256i	−0.813704	−	0.581279i	$-0.802552\pi$
$773$		−	898.658i		−	1.16256i	0.813704	−	0.581279i	$-0.197448\pi$
$774$		0			0
$775$	−286.905			−0.370200
$776$		0			0
$777$	10.2858	+	72.5628i	0.0132378	+	0.0933884i
$778$		0			0
$779$		−	103.967i		−	0.133463i
$780$		0			0
$781$	−659.214			−0.844063
$782$		0			0
$783$	372.171	−	167.291i	0.475314	−	0.213653i
$784$		0			0
$785$		−	612.868i		−	0.780724i
$786$		0			0
$787$	409.324			0.520107			0.260054	−	0.965594i	$-0.416260\pi$
$787$	409.324			0.520107			0.260054	+	0.965594i	$0.416260\pi$
$788$		0			0
$789$	909.190	−	128.878i	1.15233	−	0.163344i
$790$		0			0
$791$			67.7993i			0.0857134i
$792$		0			0
$793$	131.779			0.166178
$794$		0			0
$795$	21.5882	+	152.297i	0.0271550	+	0.191569i
$796$		0			0
$797$		−	163.126i		−	0.204675i	−0.994750	−	0.102337i	$-0.967368\pi$
$797$		−	163.126i		−	0.204675i	0.994750	−	0.102337i	$-0.0326321\pi$
$798$		0			0
$799$	122.222			0.152968
$800$		0			0
$801$	377.123	+	1303.51i	0.470815	+	1.62735i
$802$		0			0
$803$			984.405i			1.22591i
$804$		0			0
$805$	80.6999			0.100248
$806$		0			0
$807$	715.583	−	101.434i	0.886719	−	0.125693i
$808$		0			0
$809$			226.377i			0.279823i	0.990164	+	0.139912i	$0.0446818\pi$
$809$			226.377i			0.279823i	−0.990164	+	0.139912i	$0.955318\pi$
$810$		0			0
$811$	1232.29			1.51948			0.759738	−	0.650230i	$-0.225327\pi$
$811$	1232.29			1.51948			0.759738	+	0.650230i	$0.225327\pi$
$812$		0			0
$813$	−0.291881	−	2.05912i	−0.000359017	−	0.00253274i
$814$		0			0
$815$		−	634.641i		−	0.778700i
$816$		0			0
$817$	149.377			0.182836
$818$		0			0
$819$	−132.162	+	38.2365i	−0.161370	+	0.0466868i
$820$		0			0
$821$		−	499.367i		−	0.608242i	−0.952633	−	0.304121i	$-0.901637\pi$
$821$		−	499.367i		−	0.608242i	0.952633	−	0.304121i	$-0.0983627\pi$
$822$		0			0
$823$	−468.938			−0.569792			−0.284896	−	0.958558i	$-0.591959\pi$
$823$	−468.938			−0.569792			−0.284896	+	0.958558i	$0.591959\pi$
$824$		0			0
$825$	121.258	−	17.1884i	0.146979	−	0.0208344i
$826$		0			0
$827$		−	2.39906i		−	0.00290092i	−0.999999	−	0.00145046i	$-0.999538\pi$
$827$		−	2.39906i		−	0.00290092i	0.999999	−	0.00145046i	$-0.000461695\pi$
$828$		0			0
$829$	487.181			0.587674			0.293837	−	0.955856i	$-0.405068\pi$
$829$	487.181			0.587674			0.293837	+	0.955856i	$0.405068\pi$
$830$		0			0
$831$	−82.2484	−	580.234i	−0.0989752	−	0.698235i
$832$		0			0
$833$		−	196.458i		−	0.235844i
$834$		0			0
$835$	226.342			0.271068
$836$		0			0
$837$	−635.184	−	1413.09i	−0.758881	−	1.68828i
$838$		0			0
$839$			883.573i			1.05313i	0.850136	+	0.526563i	$0.176520\pi$
$839$			883.573i			1.05313i	−0.850136	+	0.526563i	$0.823480\pi$
$840$		0			0
$841$	612.608			0.728429
$842$		0			0
$843$	1211.84	−	171.779i	1.43753	−	0.203771i
$844$		0			0
$845$			303.246i			0.358871i
$846$		0			0
$847$	−143.765			−0.169734
$848$		0			0
$849$	163.288	+	1151.94i	0.192330	+	1.35682i
$850$		0			0
$851$			125.951i			0.148004i
$852$		0			0
$853$	−835.699			−0.979717			−0.489859	−	0.871802i	$-0.662952\pi$
$853$	−835.699			−0.979717			−0.489859	+	0.871802i	$0.662952\pi$
$854$		0			0
$855$	25.1918	+	87.0743i	0.0294641	+	0.101841i
$856$		0			0
$857$		−	1339.06i		−	1.56249i	−0.624222	−	0.781247i	$-0.714584\pi$
$857$		−	1339.06i		−	1.56249i	0.624222	−	0.781247i	$-0.285416\pi$
$858$		0			0
$859$	334.667			0.389600			0.194800	−	0.980843i	$-0.437594\pi$
$859$	334.667			0.389600			0.194800	+	0.980843i	$0.437594\pi$
$860$		0			0
$861$	−181.397	+	25.7131i	−0.210682	+	0.0298642i
$862$		0			0
$863$		−	995.095i		−	1.15306i	−0.817074	−	0.576532i	$-0.804406\pi$
$863$		−	995.095i		−	1.15306i	0.817074	−	0.576532i	$-0.195594\pi$
$864$		0			0
$865$	743.769			0.859849
$866$		0			0
$867$	−209.961	−	1481.20i	−0.242170	−	1.70842i
$868$		0			0
$869$			444.952i			0.512027i
$870$		0			0
$871$	−39.1937			−0.0449985
$872$		0			0
$873$	−1019.66	+	295.003i	−1.16800	+	0.337918i
$874$		0			0
$875$		−	29.5804i		−	0.0338062i
$876$		0			0
$877$	1095.18			1.24878			0.624391	−	0.781112i	$-0.285347\pi$
$877$	1095.18			1.24878			0.624391	+	0.781112i	$0.285347\pi$
$878$		0			0
$879$	−292.282	+	41.4311i	−0.332516	+	0.0471344i
$880$		0			0
$881$		−	1219.05i		−	1.38371i	−0.722036	−	0.691855i	$-0.756794\pi$
$881$		−	1219.05i		−	1.38371i	0.722036	−	0.691855i	$-0.243206\pi$
$882$		0			0
$883$	−1339.04			−1.51646			−0.758232	−	0.651985i	$-0.773936\pi$
$883$	−1339.04			−1.51646			−0.758232	+	0.651985i	$0.773936\pi$
$884$		0			0
$885$	34.2344	+	241.512i	0.0386829	+	0.272894i
$886$		0			0
$887$		−	938.587i		−	1.05816i	−0.848572	−	0.529079i	$-0.822537\pi$
$887$		−	938.587i		−	1.05816i	0.848572	−	0.529079i	$-0.177463\pi$
$888$		0			0
$889$	−183.954			−0.206922
$890$		0			0
$891$	353.113	+	559.178i	0.396311	+	0.627585i
$892$		0			0
$893$			19.6153i			0.0219656i
$894$		0			0
$895$	−33.8584			−0.0378306
$896$		0			0
$897$	−234.106	+	33.1846i	−0.260987	+	0.0369951i
$898$		0			0
$899$			867.178i			0.964602i
$900$		0			0
$901$	643.543			0.714255
$902$		0			0
$903$	−36.9438	−	260.626i	−0.0409123	−	0.288622i
$904$		0			0
$905$		−	545.999i		−	0.603313i
$906$		0			0
$907$	1575.14			1.73665			0.868326	−	0.495994i	$-0.165196\pi$
$907$	1575.14			1.73665			0.868326	+	0.495994i	$0.165196\pi$
$908$		0			0
$909$	351.258	+	1214.11i	0.386423	+	1.33565i
$910$		0			0
$911$			566.329i			0.621656i	0.950466	+	0.310828i	$0.100606\pi$
$911$			566.329i			0.621656i	−0.950466	+	0.310828i	$0.899394\pi$
$912$		0			0
$913$	467.805			0.512383
$914$		0			0
$915$	−151.482	+	21.4727i	−0.165554	+	0.0234674i
$916$		0			0
$917$		−	302.551i		−	0.329935i
$918$		0			0
$919$	−276.281			−0.300632			−0.150316	−	0.988638i	$-0.548029\pi$
$919$	−276.281			−0.300632			−0.150316	+	0.988638i	$0.548029\pi$
$920$		0			0
$921$	72.4850	+	511.356i	0.0787025	+	0.555219i
$922$		0			0
$923$		−	466.507i		−	0.505425i
$924$		0			0
$925$	46.1672			0.0499105
$926$		0			0
$927$	87.4464	−	25.2995i	0.0943327	−	0.0272918i
$928$		0			0
$929$		−	160.080i		−	0.172314i	−0.996282	−	0.0861571i	$-0.972541\pi$
$929$		−	160.080i		−	0.172314i	0.996282	−	0.0861571i	$-0.0274587\pi$
$930$		0			0
$931$	31.5294			0.0338662
$932$		0			0
$933$	1377.82	−	195.306i	1.47676	−	0.209332i
$934$		0			0
$935$		−	512.385i		−	0.548005i
$936$		0			0
$937$	−1084.01			−1.15689			−0.578446	−	0.815720i	$-0.696341\pi$
$937$	−1084.01			−1.15689			−0.578446	+	0.815720i	$0.696341\pi$
$938$		0			0
$939$	−149.344	−	1053.57i	−0.159046	−	1.12201i
$940$		0			0
$941$			692.493i			0.735912i	0.929843	+	0.367956i	$0.119942\pi$
$941$			692.493i			0.735912i	−0.929843	+	0.367956i	$0.880058\pi$
$942$		0			0
$943$	−314.860			−0.333892
$944$		0			0
$945$	145.692	−	65.4885i	0.154172	−	0.0693000i
$946$		0			0
$947$		−	1795.10i		−	1.89556i	−0.318923	−	0.947781i	$-0.603321\pi$
$947$		−	1795.10i		−	1.89556i	0.318923	−	0.947781i	$-0.396679\pi$
$948$		0			0
$949$	−696.636			−0.734074
$950$		0			0
$951$	−133.463	+	18.9184i	−0.140339	+	0.0198932i
$952$		0			0
$953$		−	789.199i		−	0.828121i	−0.910249	−	0.414060i	$-0.864110\pi$
$953$		−	789.199i		−	0.828121i	0.910249	−	0.414060i	$-0.135890\pi$
$954$		0			0
$955$	735.496			0.770153
$956$		0			0
$957$	−51.9523	−	366.505i	−0.0542867	−	0.382973i
$958$		0			0
$959$			437.361i			0.456059i
$960$		0			0
$961$	2331.58			2.42620
$962$		0			0
$963$	137.836	+	476.423i	0.143132	+	0.494728i
$964$		0			0
$965$		−	681.164i		−	0.705870i
$966$		0			0
$967$	−1464.86			−1.51485			−0.757426	−	0.652921i	$-0.773543\pi$
$967$	−1464.86			−1.51485			−0.757426	+	0.652921i	$0.773543\pi$
$968$		0			0
$969$	375.484	−	53.2250i	0.387496	−	0.0549277i
$970$		0			0
$971$		−	1278.47i		−	1.31665i	−0.752733	−	0.658326i	$-0.771265\pi$
$971$		−	1278.47i		−	1.31665i	0.752733	−	0.658326i	$-0.228735\pi$
$972$		0			0
$973$	550.160			0.565426
$974$		0			0
$975$	12.1637	+	85.8110i	0.0124756	+	0.0880113i
$976$		0			0
$977$			403.422i			0.412919i	0.978455	+	0.206460i	$0.0661942\pi$
$977$			403.422i			0.412919i	−0.978455	+	0.206460i	$0.933806\pi$
$978$		0			0
$979$	1231.02			1.25743
$980$		0			0
$981$	−549.841	+	159.077i	−0.560491	+	0.162158i
$982$		0			0
$983$			1086.01i			1.10479i	0.833583	+	0.552394i	$0.186286\pi$
$983$			1086.01i			1.10479i	−0.833583	+	0.552394i	$0.813714\pi$
$984$		0			0
$985$	−525.583			−0.533586
$986$		0			0
$987$	34.2237	−	4.85123i	0.0346745	−	0.00491512i
$988$		0			0
$989$		−	452.382i		−	0.457414i
$990$		0			0
$991$	−70.0427			−0.0706788			−0.0353394	−	0.999375i	$-0.511251\pi$
$991$	−70.0427			−0.0706788			−0.0353394	+	0.999375i	$0.511251\pi$
$992$		0			0
$993$	13.2460	+	93.4457i	0.0133393	+	0.0941044i
$994$		0			0
$995$		−	666.515i		−	0.669865i
$996$		0			0
$997$	719.365			0.721529			0.360765	−	0.932657i	$-0.382516\pi$
$997$	719.365			0.721529			0.360765	+	0.932657i	$0.382516\pi$
$998$		0			0
$999$	102.210	+	227.387i	0.102313	+	0.227615i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.l.b.1121.8		16
3.2	odd	2	inner	1680.3.l.b.1121.7		16
4.3	odd	2		420.3.g.a.281.9	✓	16
12.11	even	2		420.3.g.a.281.10	yes	16
20.3	even	4		2100.3.e.c.449.31		32
20.7	even	4		2100.3.e.c.449.2		32
20.19	odd	2		2100.3.g.d.701.8		16
60.23	odd	4		2100.3.e.c.449.1		32
60.47	odd	4		2100.3.e.c.449.32		32
60.59	even	2		2100.3.g.d.701.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.g.a.281.9	✓	16	4.3	odd	2
420.3.g.a.281.10	yes	16	12.11	even	2
1680.3.l.b.1121.7		16	3.2	odd	2	inner
1680.3.l.b.1121.8		16	1.1	even	1	trivial
2100.3.e.c.449.1		32	60.23	odd	4
2100.3.e.c.449.2		32	20.7	even	4
2100.3.e.c.449.31		32	20.3	even	4
2100.3.e.c.449.32		32	60.47	odd	4
2100.3.g.d.701.7		16	60.59	even	2
2100.3.g.d.701.8		16	20.19	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.421042 + 2.97031i) q^{3} -2.23607i q^{5} -2.64575 q^{7} +(-8.64545 + 2.50125i) q^{9} +8.16468i q^{11} -5.77792 q^{13} +(6.64181 - 0.941480i) q^{15} -28.0655i q^{17} +4.50420 q^{19} +(-1.11397 - 7.85869i) q^{21} -13.6408i q^{23} -5.00000 q^{25} +(-11.0696 - 24.6265i) q^{27} +15.1126i q^{29} +57.3810 q^{31} +(-24.2516 + 3.43768i) q^{33} +5.91608i q^{35} -9.23344 q^{37} +(-2.43275 - 17.1622i) q^{39} -23.0823i q^{41} +33.1640 q^{43} +(5.59297 + 19.3318i) q^{45} +4.35488i q^{47} +7.00000 q^{49} +(83.3630 - 11.8167i) q^{51} +22.9301i q^{53} +18.2568 q^{55} +(1.89646 + 13.3789i) q^{57} +36.3623i q^{59} -22.8074 q^{61} +(22.8737 - 6.61769i) q^{63} +12.9198i q^{65} +6.78336 q^{67} +(40.5173 - 5.74334i) q^{69} +80.7397i q^{71} +120.569 q^{73} +(-2.10521 - 14.8515i) q^{75} -21.6017i q^{77} +54.4972 q^{79} +(68.4875 - 43.2489i) q^{81} -57.2963i q^{83} -62.7563 q^{85} +(-44.8892 + 6.36306i) q^{87} -150.774i q^{89} +15.2869 q^{91} +(24.1598 + 170.439i) q^{93} -10.0717i q^{95} +117.942 q^{97} +(-20.4219 - 70.5873i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} - 10 q^{9} - 10 q^{15} - 48 q^{19} - 14 q^{21} - 80 q^{25} - 28 q^{27} - 24 q^{31} + 92 q^{33} - 40 q^{37} + 54 q^{39} + 168 q^{43} + 40 q^{45} + 112 q^{49} + 186 q^{51} + 80 q^{55} + 252 q^{57}+ \cdots - 26 q^{99}+O(q^{100}) \) 16 * q + 8 * q^3 - 10 * q^9 - 10 * q^15 - 48 * q^19 - 14 * q^21 - 80 * q^25 - 28 * q^27 - 24 * q^31 + 92 * q^33 - 40 * q^37 + 54 * q^39 + 168 * q^43 + 40 * q^45 + 112 * q^49 + 186 * q^51 + 80 * q^55 + 252 * q^57 - 152 * q^61 + 56 * q^63 - 264 * q^67 - 196 * q^69 + 240 * q^73 - 40 * q^75 - 212 * q^79 - 118 * q^81 - 60 * q^85 - 60 * q^87 - 196 * q^91 - 244 * q^93 - 360 * q^97 - 26 * q^99

Introduction

L-functions

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