LMFDB - Embedded newform 1680.3.s.a.1441.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(1441,1680)] 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, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1441"); 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.s (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$45.7766844125$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 7x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1441.7
Root			$1.72286 + 0.178197i$ of defining polynomial
Character	$\chi$	$=$	1680.1441
Dual form			1680.3.s.a.1441.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +2.23607i q^{5} +(-6.70141 - 2.02265i) q^{7} -3.00000 q^{9} -11.9625 q^{11} -9.67294i q^{13} -3.87298 q^{15} +7.09185i q^{17} +17.5508i q^{19} +(3.50333 - 11.6072i) q^{21} +2.84923 q^{23} -5.00000 q^{25} -5.19615i q^{27} -13.6929 q^{29} -19.9468i q^{31} -20.7197i q^{33} +(4.52277 - 14.9848i) q^{35} -11.0672 q^{37} +16.7540 q^{39} +28.2902i q^{41} +72.9666 q^{43} -6.70820i q^{45} -28.3312i q^{47} +(40.8178 + 27.1092i) q^{49} -12.2834 q^{51} -11.7110 q^{53} -26.7490i q^{55} -30.3989 q^{57} -101.160i q^{59} -76.7055i q^{61} +(20.1042 + 6.06794i) q^{63} +21.6294 q^{65} +76.2407 q^{67} +4.93501i q^{69} +95.4493 q^{71} -120.684i q^{73} -8.66025i q^{75} +(80.1657 + 24.1959i) q^{77} +14.8096 q^{79} +9.00000 q^{81} +60.9570i q^{83} -15.8579 q^{85} -23.7169i q^{87} +88.6302i q^{89} +(-19.5649 + 64.8223i) q^{91} +34.5489 q^{93} -39.2448 q^{95} +18.8547i q^{97} +35.8875 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9} + 16 q^{11} - 40 q^{25} - 144 q^{29} + 80 q^{35} - 48 q^{37} + 48 q^{39} + 64 q^{43} - 24 q^{49} + 128 q^{53} + 144 q^{57} + 80 q^{65} + 192 q^{67} - 176 q^{71} + 192 q^{77} + 288 q^{79}+ \cdots - 48 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 + 16 * q^11 - 40 * q^25 - 144 * q^29 + 80 * q^35 - 48 * q^37 + 48 * q^39 + 64 * q^43 - 24 * q^49 + 128 * q^53 + 144 * q^57 + 80 * q^65 + 192 * q^67 - 176 * q^71 + 192 * q^77 + 288 * q^79 + 72 * q^81 - 240 * q^85 - 64 * q^91 + 336 * q^93 - 80 * q^95 - 48 * 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
$3$			1.73205i			0.577350i
$3$			1.73205i			0.577350i
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$	−6.70141	−	2.02265i	−0.957344	−	0.288949i
$7$	−6.70141	−	2.02265i	−0.957344	−	0.288949i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	−11.9625			−1.08750			−0.543750	−	0.839247i	$-0.682996\pi$
$11$	−11.9625			−1.08750			−0.543750	+	0.839247i	$0.682996\pi$
$12$		0			0
$13$		−	9.67294i		−	0.744072i	−0.928218	−	0.372036i	$-0.878660\pi$
$13$		−	9.67294i		−	0.744072i	0.928218	−	0.372036i	$-0.121340\pi$
$14$		0			0
$15$	−3.87298			−0.258199
$16$		0			0
$17$			7.09185i			0.417168i	0.978004	+	0.208584i	$0.0668854\pi$
$17$			7.09185i			0.417168i	−0.978004	+	0.208584i	$0.933115\pi$
$18$		0			0
$19$			17.5508i			0.923727i	0.886951	+	0.461864i	$0.152819\pi$
$19$			17.5508i			0.923727i	−0.886951	+	0.461864i	$0.847181\pi$
$20$		0			0
$21$	3.50333	−	11.6072i	0.166825	−	0.552723i
$22$		0			0
$23$	2.84923			0.123879			0.0619397	−	0.998080i	$-0.480271\pi$
$23$	2.84923			0.123879			0.0619397	+	0.998080i	$0.480271\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		−	5.19615i		−	0.192450i
$28$		0			0
$29$	−13.6929			−0.472170			−0.236085	−	0.971732i	$-0.575864\pi$
$29$	−13.6929			−0.472170			−0.236085	+	0.971732i	$0.575864\pi$
$30$		0			0
$31$		−	19.9468i		−	0.643445i	−0.946834	−	0.321723i	$-0.895738\pi$
$31$		−	19.9468i		−	0.643445i	0.946834	−	0.321723i	$-0.104262\pi$
$32$		0			0
$33$		−	20.7197i		−	0.627869i
$34$		0			0
$35$	4.52277	−	14.9848i	0.129222	−	0.428137i
$36$		0			0
$37$	−11.0672			−0.299113			−0.149556	−	0.988753i	$-0.547785\pi$
$37$	−11.0672			−0.299113			−0.149556	+	0.988753i	$0.547785\pi$
$38$		0			0
$39$	16.7540			0.429590
$40$		0			0
$41$			28.2902i			0.690004i	0.938602	+	0.345002i	$0.112122\pi$
$41$			28.2902i			0.690004i	−0.938602	+	0.345002i	$0.887878\pi$
$42$		0			0
$43$	72.9666			1.69690			0.848449	−	0.529278i	$-0.177537\pi$
$43$	72.9666			1.69690			0.848449	+	0.529278i	$0.177537\pi$
$44$		0			0
$45$		−	6.70820i		−	0.149071i
$46$		0			0
$47$		−	28.3312i		−	0.602792i	−0.953499	−	0.301396i	$-0.902547\pi$
$47$		−	28.3312i		−	0.602792i	0.953499	−	0.301396i	$-0.0974526\pi$
$48$		0			0
$49$	40.8178	+	27.1092i	0.833016	+	0.553248i
$50$		0			0
$51$	−12.2834			−0.240852
$52$		0			0
$53$	−11.7110			−0.220962			−0.110481	−	0.993878i	$-0.535239\pi$
$53$	−11.7110			−0.220962			−0.110481	+	0.993878i	$0.535239\pi$
$54$		0			0
$55$		−	26.7490i		−	0.486345i
$56$		0			0
$57$	−30.3989			−0.533314
$58$		0			0
$59$		−	101.160i		−	1.71458i	−0.514833	−	0.857291i	$-0.672146\pi$
$59$		−	101.160i		−	1.71458i	0.514833	−	0.857291i	$-0.327854\pi$
$60$		0			0
$61$		−	76.7055i		−	1.25747i	−0.777621	−	0.628733i	$-0.783574\pi$
$61$		−	76.7055i		−	1.25747i	0.777621	−	0.628733i	$-0.216426\pi$
$62$		0			0
$63$	20.1042	+	6.06794i	0.319115	+	0.0963165i
$64$		0			0
$65$	21.6294			0.332759
$66$		0			0
$67$	76.2407			1.13792			0.568960	−	0.822365i	$-0.307346\pi$
$67$	76.2407			1.13792			0.568960	+	0.822365i	$0.307346\pi$
$68$		0			0
$69$			4.93501i			0.0715218i
$70$		0			0
$71$	95.4493			1.34436			0.672178	−	0.740389i	$-0.265359\pi$
$71$	95.4493			1.34436			0.672178	+	0.740389i	$0.265359\pi$
$72$		0			0
$73$		−	120.684i		−	1.65321i	−0.562782	−	0.826605i	$-0.690269\pi$
$73$		−	120.684i		−	1.65321i	0.562782	−	0.826605i	$-0.309731\pi$
$74$		0			0
$75$		−	8.66025i		−	0.115470i
$76$		0			0
$77$	80.1657	+	24.1959i	1.04111	+	0.314233i
$78$		0			0
$79$	14.8096			0.187463			0.0937317	−	0.995597i	$-0.470120\pi$
$79$	14.8096			0.187463			0.0937317	+	0.995597i	$0.470120\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			60.9570i			0.734422i	0.930138	+	0.367211i	$0.119687\pi$
$83$			60.9570i			0.734422i	−0.930138	+	0.367211i	$0.880313\pi$
$84$		0			0
$85$	−15.8579			−0.186563
$86$		0			0
$87$		−	23.7169i		−	0.272608i
$88$		0			0
$89$			88.6302i			0.995845i	0.867222	+	0.497922i	$0.165904\pi$
$89$			88.6302i			0.995845i	−0.867222	+	0.497922i	$0.834096\pi$
$90$		0			0
$91$	−19.5649	+	64.8223i	−0.214999	+	0.712333i
$92$		0			0
$93$	34.5489			0.371493
$94$		0			0
$95$	−39.2448			−0.413103
$96$		0			0
$97$			18.8547i			0.194378i	0.995266	+	0.0971891i	$0.0309852\pi$
$97$			18.8547i			0.194378i	−0.995266	+	0.0971891i	$0.969015\pi$
$98$		0			0
$99$	35.8875			0.362500
$100$		0			0
$101$		−	30.1594i		−	0.298608i	−0.988791	−	0.149304i	$-0.952297\pi$
$101$		−	30.1594i		−	0.298608i	0.988791	−	0.149304i	$-0.0477033\pi$
$102$		0			0
$103$			150.869i			1.46475i	0.680903	+	0.732374i	$0.261588\pi$
$103$			150.869i			1.46475i	−0.680903	+	0.732374i	$0.738412\pi$
$104$		0			0
$105$	25.9545	+	7.83368i	0.247185	+	0.0746064i
$106$		0			0
$107$	−88.2689			−0.824943			−0.412472	−	0.910970i	$-0.635334\pi$
$107$	−88.2689			−0.824943			−0.412472	+	0.910970i	$0.635334\pi$
$108$		0			0
$109$	92.7216			0.850657			0.425328	−	0.905039i	$-0.360159\pi$
$109$	92.7216			0.850657			0.425328	+	0.905039i	$0.360159\pi$
$110$		0			0
$111$		−	19.1689i		−	0.172693i
$112$		0			0
$113$	−63.4942			−0.561896			−0.280948	−	0.959723i	$-0.590649\pi$
$113$	−63.4942			−0.561896			−0.280948	+	0.959723i	$0.590649\pi$
$114$		0			0
$115$			6.37107i			0.0554006i
$116$		0			0
$117$			29.0188i			0.248024i
$118$		0			0
$119$	14.3443	−	47.5254i	0.120540	−	0.399373i
$120$		0			0
$121$	22.1016			0.182658
$122$		0			0
$123$	−49.0000			−0.398374
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	252.418			1.98755			0.993773	−	0.111423i	$-0.0355408\pi$
$127$	252.418			1.98755			0.993773	+	0.111423i	$0.0355408\pi$
$128$		0			0
$129$			126.382i			0.979704i
$130$		0			0
$131$		−	206.558i		−	1.57678i	−0.615174	−	0.788391i	$-0.710914\pi$
$131$		−	206.558i		−	1.57678i	0.615174	−	0.788391i	$-0.289086\pi$
$132$		0			0
$133$	35.4991	−	117.615i	0.266910	−	0.884325i
$134$		0			0
$135$	11.6190			0.0860663
$136$		0			0
$137$	179.062			1.30702			0.653512	−	0.756916i	$-0.273295\pi$
$137$	179.062			1.30702			0.653512	+	0.756916i	$0.273295\pi$
$138$		0			0
$139$		−	29.8412i		−	0.214685i	−0.994222	−	0.107342i	$-0.965766\pi$
$139$		−	29.8412i		−	0.214685i	0.994222	−	0.107342i	$-0.0342341\pi$
$140$		0			0
$141$	49.0711			0.348022
$142$		0			0
$143$			115.713i			0.809179i
$144$		0			0
$145$		−	30.6184i		−	0.211161i
$146$		0			0
$147$	−46.9545	+	70.6985i	−0.319418	+	0.480942i
$148$		0			0
$149$	233.638			1.56804			0.784020	−	0.620736i	$-0.213166\pi$
$149$	233.638			1.56804			0.784020	+	0.620736i	$0.213166\pi$
$150$		0			0
$151$	−280.637			−1.85852			−0.929262	−	0.369421i	$-0.879556\pi$
$151$	−280.637			−1.85852			−0.929262	+	0.369421i	$0.879556\pi$
$152$		0			0
$153$		−	21.2756i		−	0.139056i
$154$		0			0
$155$	44.6024			0.287757
$156$		0			0
$157$		−	54.3182i		−	0.345976i	−0.984924	−	0.172988i	$-0.944658\pi$
$157$		−	54.3182i		−	0.345976i	0.984924	−	0.172988i	$-0.0553422\pi$
$158$		0			0
$159$		−	20.2840i		−	0.127572i
$160$		0			0
$161$	−19.0938	−	5.76298i	−0.118595	−	0.0357949i
$162$		0			0
$163$	−200.523			−1.23020			−0.615101	−	0.788448i	$-0.710885\pi$
$163$	−200.523			−1.23020			−0.615101	+	0.788448i	$0.710885\pi$
$164$		0			0
$165$	46.3306			0.280791
$166$		0			0
$167$			282.815i			1.69351i	0.531987	+	0.846753i	$0.321446\pi$
$167$			282.815i			1.69351i	−0.531987	+	0.846753i	$0.678554\pi$
$168$		0			0
$169$	75.4342			0.446356
$170$		0			0
$171$		−	52.6524i		−	0.307909i
$172$		0			0
$173$		−	127.279i		−	0.735714i	−0.929882	−	0.367857i	$-0.880092\pi$
$173$		−	127.279i		−	0.735714i	0.929882	−	0.367857i	$-0.119908\pi$
$174$		0			0
$175$	33.5071	+	10.1132i	0.191469	+	0.0577899i
$176$		0			0
$177$	175.215			0.989914
$178$		0			0
$179$	−122.342			−0.683472			−0.341736	−	0.939796i	$-0.611015\pi$
$179$	−122.342			−0.683472			−0.341736	+	0.939796i	$0.611015\pi$
$180$		0			0
$181$			97.3605i			0.537903i	0.963154	+	0.268952i	$0.0866772\pi$
$181$			97.3605i			0.537903i	−0.963154	+	0.268952i	$0.913323\pi$
$182$		0			0
$183$	132.858			0.725999
$184$		0			0
$185$		−	24.7469i		−	0.133767i
$186$		0			0
$187$		−	84.8363i		−	0.453670i
$188$		0			0
$189$	−10.5100	+	34.8216i	−0.0556083	+	0.184241i
$190$		0			0
$191$	290.175			1.51924			0.759621	−	0.650367i	$-0.225385\pi$
$191$	290.175			1.51924			0.759621	+	0.650367i	$0.225385\pi$
$192$		0			0
$193$	90.7789			0.470357			0.235179	−	0.971952i	$-0.424433\pi$
$193$	90.7789			0.470357			0.235179	+	0.971952i	$0.424433\pi$
$194$		0			0
$195$			37.4631i			0.192119i
$196$		0			0
$197$	−53.3711			−0.270919			−0.135460	−	0.990783i	$-0.543251\pi$
$197$	−53.3711			−0.270919			−0.135460	+	0.990783i	$0.543251\pi$
$198$		0			0
$199$		−	301.563i		−	1.51539i	−0.652607	−	0.757696i	$-0.726325\pi$
$199$		−	301.563i		−	1.51539i	0.652607	−	0.757696i	$-0.273675\pi$
$200$		0			0
$201$			132.053i			0.656979i
$202$		0			0
$203$	91.7620	+	27.6960i	0.452030	+	0.136433i
$204$		0			0
$205$	−63.2587			−0.308579
$206$		0			0
$207$	−8.54768			−0.0412931
$208$		0			0
$209$		−	209.952i		−	1.00455i
$210$		0			0
$211$	74.7911			0.354460			0.177230	−	0.984169i	$-0.443286\pi$
$211$	74.7911			0.354460			0.177230	+	0.984169i	$0.443286\pi$
$212$		0			0
$213$			165.323i			0.776165i
$214$		0			0
$215$			163.158i			0.758876i
$216$		0			0
$217$	−40.3453	+	133.672i	−0.185923	+	0.615999i
$218$		0			0
$219$	209.032			0.954482
$220$		0			0
$221$	68.5991			0.310403
$222$		0			0
$223$		−	413.129i		−	1.85260i	−0.376789	−	0.926299i	$-0.622972\pi$
$223$		−	413.129i		−	1.85260i	0.376789	−	0.926299i	$-0.377028\pi$
$224$		0			0
$225$	15.0000			0.0666667
$226$		0			0
$227$			298.691i			1.31582i	0.753097	+	0.657910i	$0.228559\pi$
$227$			298.691i			1.31582i	−0.753097	+	0.657910i	$0.771441\pi$
$228$		0			0
$229$		−	323.323i		−	1.41189i	−0.708265	−	0.705946i	$-0.750522\pi$
$229$		−	323.323i		−	1.41189i	0.708265	−	0.705946i	$-0.249478\pi$
$230$		0			0
$231$	−41.9086	+	138.851i	−0.181422	+	0.601087i
$232$		0			0
$233$	−323.951			−1.39035			−0.695174	−	0.718841i	$-0.744673\pi$
$233$	−323.951			−1.39035			−0.695174	+	0.718841i	$0.744673\pi$
$234$		0			0
$235$	63.3505			0.269577
$236$		0			0
$237$			25.6510i			0.108232i
$238$		0			0
$239$	362.459			1.51657			0.758283	−	0.651926i	$-0.226039\pi$
$239$	362.459			1.51657			0.758283	+	0.651926i	$0.226039\pi$
$240$		0			0
$241$			124.863i			0.518102i	0.965864	+	0.259051i	$0.0834098\pi$
$241$			124.863i			0.518102i	−0.965864	+	0.259051i	$0.916590\pi$
$242$		0			0
$243$			15.5885i			0.0641500i
$244$		0			0
$245$	−60.6179	+	91.2714i	−0.247420	+	0.372536i
$246$		0			0
$247$	169.768			0.687320
$248$		0			0
$249$	−105.581			−0.424018
$250$		0			0
$251$		−	18.4515i		−	0.0735121i	−0.999324	−	0.0367561i	$-0.988298\pi$
$251$		−	18.4515i		−	0.0735121i	0.999324	−	0.0367561i	$-0.0117025\pi$
$252$		0			0
$253$	−34.0839			−0.134719
$254$		0			0
$255$		−	27.4666i		−	0.107712i
$256$		0			0
$257$			428.659i			1.66793i	0.551816	+	0.833966i	$0.313935\pi$
$257$			428.659i			1.66793i	−0.551816	+	0.833966i	$0.686065\pi$
$258$		0			0
$259$	74.1657	+	22.3850i	0.286354	+	0.0864285i
$260$		0			0
$261$	41.0788			0.157390
$262$		0			0
$263$	133.575			0.507888			0.253944	−	0.967219i	$-0.418272\pi$
$263$	133.575			0.507888			0.253944	+	0.967219i	$0.418272\pi$
$264$		0			0
$265$		−	26.1865i		−	0.0988170i
$266$		0			0
$267$	−153.512			−0.574951
$268$		0			0
$269$		−	493.257i		−	1.83367i	−0.399269	−	0.916834i	$-0.630736\pi$
$269$		−	493.257i		−	1.83367i	0.399269	−	0.916834i	$-0.369264\pi$
$270$		0			0
$271$		−	432.545i		−	1.59611i	−0.602586	−	0.798054i	$-0.705863\pi$
$271$		−	432.545i		−	1.59611i	0.602586	−	0.798054i	$-0.294137\pi$
$272$		0			0
$273$	−112.276	−	33.8875i	−0.411266	−	0.124130i
$274$		0			0
$275$	59.8125			0.217500
$276$		0			0
$277$	−171.941			−0.620724			−0.310362	−	0.950618i	$-0.600450\pi$
$277$	−171.941			−0.620724			−0.310362	+	0.950618i	$0.600450\pi$
$278$		0			0
$279$			59.8404i			0.214482i
$280$		0			0
$281$	74.7499			0.266014			0.133007	−	0.991115i	$-0.457537\pi$
$281$	74.7499			0.266014			0.133007	+	0.991115i	$0.457537\pi$
$282$		0			0
$283$			165.904i			0.586232i	0.956077	+	0.293116i	$0.0946922\pi$
$283$			165.904i			0.586232i	−0.956077	+	0.293116i	$0.905308\pi$
$284$		0			0
$285$		−	67.9740i		−	0.238505i
$286$		0			0
$287$	57.2210	−	189.584i	0.199376	−	0.660571i
$288$		0			0
$289$	238.706			0.825971
$290$		0			0
$291$	−32.6573			−0.112224
$292$		0			0
$293$			495.925i			1.69258i	0.532725	+	0.846289i	$0.321168\pi$
$293$			495.925i			1.69258i	−0.532725	+	0.846289i	$0.678832\pi$
$294$		0			0
$295$	226.201			0.766784
$296$		0			0
$297$			62.1590i			0.209290i
$298$		0			0
$299$		−	27.5604i		−	0.0921753i
$300$		0			0
$301$	−488.979	−	147.586i	−1.62452	−	0.490318i
$302$		0			0
$303$	52.2376			0.172401
$304$		0			0
$305$	171.519			0.562356
$306$		0			0
$307$		−	316.942i		−	1.03238i	−0.856473	−	0.516192i	$-0.827349\pi$
$307$		−	316.942i		−	1.03238i	0.856473	−	0.516192i	$-0.172651\pi$
$308$		0			0
$309$	−261.313			−0.845673
$310$		0			0
$311$			27.1325i			0.0872428i	0.999048	+	0.0436214i	$0.0138895\pi$
$311$			27.1325i			0.0872428i	−0.999048	+	0.0436214i	$0.986110\pi$
$312$		0			0
$313$		−	517.236i		−	1.65251i	−0.563296	−	0.826255i	$-0.690467\pi$
$313$		−	517.236i		−	1.65251i	0.563296	−	0.826255i	$-0.309533\pi$
$314$		0			0
$315$	−13.5683	+	44.9544i	−0.0430740	+	0.142712i
$316$		0			0
$317$	213.041			0.672054			0.336027	−	0.941852i	$-0.390917\pi$
$317$	213.041			0.672054			0.336027	+	0.941852i	$0.390917\pi$
$318$		0			0
$319$	163.802			0.513486
$320$		0			0
$321$		−	152.886i		−	0.476281i
$322$		0			0
$323$	−124.468			−0.385349
$324$		0			0
$325$			48.3647i			0.148814i
$326$		0			0
$327$			160.599i			0.491127i
$328$		0			0
$329$	−57.3040	+	189.859i	−0.174176	+	0.577079i
$330$		0			0
$331$	−292.516			−0.883735			−0.441867	−	0.897080i	$-0.645684\pi$
$331$	−292.516			−0.883735			−0.441867	+	0.897080i	$0.645684\pi$
$332$		0			0
$333$	33.2015			0.0997042
$334$		0			0
$335$			170.479i			0.508894i
$336$		0			0
$337$	−122.881			−0.364633			−0.182316	−	0.983240i	$-0.558360\pi$
$337$	−122.881			−0.364633			−0.182316	+	0.983240i	$0.558360\pi$
$338$		0			0
$339$		−	109.975i		−	0.324411i
$340$		0			0
$341$			238.614i			0.699747i
$342$		0			0
$343$	−218.705	−	264.230i	−0.637623	−	0.770349i
$344$		0			0
$345$	−11.0350			−0.0319855
$346$		0			0
$347$	−207.406			−0.597713			−0.298856	−	0.954298i	$-0.596605\pi$
$347$	−207.406			−0.597713			−0.298856	+	0.954298i	$0.596605\pi$
$348$		0			0
$349$		−	490.264i		−	1.40477i	−0.711798	−	0.702384i	$-0.752119\pi$
$349$		−	490.264i		−	1.40477i	0.711798	−	0.702384i	$-0.247881\pi$
$350$		0			0
$351$	−50.2621			−0.143197
$352$		0			0
$353$		−	454.199i		−	1.28668i	−0.765579	−	0.643342i	$-0.777547\pi$
$353$		−	454.199i		−	1.28668i	0.765579	−	0.643342i	$-0.222453\pi$
$354$		0			0
$355$			213.431i			0.601214i
$356$		0			0
$357$	82.3164	+	24.8451i	0.230578	+	0.0695940i
$358$		0			0
$359$	−378.429			−1.05412			−0.527060	−	0.849828i	$-0.676706\pi$
$359$	−378.429			−1.05412			−0.527060	+	0.849828i	$0.676706\pi$
$360$		0			0
$361$	52.9689			0.146728
$362$		0			0
$363$			38.2810i			0.105457i
$364$		0			0
$365$	269.859			0.739338
$366$		0			0
$367$		−	181.636i		−	0.494920i	−0.968898	−	0.247460i	$-0.920404\pi$
$367$		−	181.636i		−	0.494920i	0.968898	−	0.247460i	$-0.0795959\pi$
$368$		0			0
$369$		−	84.8705i		−	0.230001i
$370$		0			0
$371$	78.4800	+	23.6871i	0.211536	+	0.0638467i
$372$		0			0
$373$	302.597			0.811253			0.405626	−	0.914039i	$-0.367053\pi$
$373$	302.597			0.811253			0.405626	+	0.914039i	$0.367053\pi$
$374$		0			0
$375$	19.3649			0.0516398
$376$		0			0
$377$			132.451i			0.351329i
$378$		0			0
$379$	−706.225			−1.86339			−0.931695	−	0.363243i	$-0.881670\pi$
$379$	−706.225			−1.86339			−0.931695	+	0.363243i	$0.881670\pi$
$380$		0			0
$381$			437.201i			1.14751i
$382$		0			0
$383$		−	238.607i		−	0.622995i	−0.950247	−	0.311498i	$-0.899169\pi$
$383$		−	238.607i		−	0.622995i	0.950247	−	0.311498i	$-0.100831\pi$
$384$		0			0
$385$	−54.1037	+	179.256i	−0.140529	+	0.465600i
$386$		0			0
$387$	−218.900			−0.565632
$388$		0			0
$389$	−523.592			−1.34599			−0.672997	−	0.739645i	$-0.734993\pi$
$389$	−523.592			−1.34599			−0.672997	+	0.739645i	$0.734993\pi$
$390$		0			0
$391$			20.2063i			0.0516785i
$392$		0			0
$393$	357.770			0.910356
$394$		0			0
$395$			33.1153i			0.0838362i
$396$		0			0
$397$			219.940i			0.554004i	0.960869	+	0.277002i	$0.0893408\pi$
$397$			219.940i			0.554004i	−0.960869	+	0.277002i	$0.910659\pi$
$398$		0			0
$399$	203.716	+	61.4862i	0.510565	+	0.154101i
$400$		0			0
$401$	559.885			1.39622			0.698111	−	0.715989i	$-0.254024\pi$
$401$	559.885			1.39622			0.698111	+	0.715989i	$0.254024\pi$
$402$		0			0
$403$	−192.944			−0.478770
$404$		0			0
$405$			20.1246i			0.0496904i
$406$		0			0
$407$	132.391			0.325285
$408$		0			0
$409$		−	477.443i		−	1.16734i	−0.811990	−	0.583672i	$-0.801616\pi$
$409$		−	477.443i		−	1.16734i	0.811990	−	0.583672i	$-0.198384\pi$
$410$		0			0
$411$			310.145i			0.754611i
$412$		0			0
$413$	−204.611	+	677.917i	−0.495427	+	1.64144i
$414$		0			0
$415$	−136.304			−0.328443
$416$		0			0
$417$	51.6865			0.123948
$418$		0			0
$419$			618.800i			1.47685i	0.674335	+	0.738425i	$0.264430\pi$
$419$			618.800i			1.47685i	−0.674335	+	0.738425i	$0.735570\pi$
$420$		0			0
$421$	23.7665			0.0564524			0.0282262	−	0.999602i	$-0.491014\pi$
$421$	23.7665			0.0564524			0.0282262	+	0.999602i	$0.491014\pi$
$422$		0			0
$423$			84.9936i			0.200931i
$424$		0			0
$425$		−	35.4593i		−	0.0834336i
$426$		0			0
$427$	−155.148	+	514.035i	−0.363344	+	1.20383i
$428$		0			0
$429$	−200.420			−0.467180
$430$		0			0
$431$	244.991			0.568424			0.284212	−	0.958762i	$-0.408268\pi$
$431$	244.991			0.568424			0.284212	+	0.958762i	$0.408268\pi$
$432$		0			0
$433$			630.500i			1.45612i	0.685514	+	0.728060i	$0.259578\pi$
$433$			630.500i			1.45612i	−0.685514	+	0.728060i	$0.740422\pi$
$434$		0			0
$435$	53.0325			0.121914
$436$		0			0
$437$			50.0063i			0.114431i
$438$		0			0
$439$		−	79.2321i		−	0.180483i	−0.995920	−	0.0902416i	$-0.971236\pi$
$439$		−	79.2321i		−	0.180483i	0.995920	−	0.0902416i	$-0.0287639\pi$
$440$		0			0
$441$	−122.453	−	81.3275i	−0.277672	−	0.184416i
$442$		0			0
$443$	−494.991			−1.11736			−0.558680	−	0.829383i	$-0.688692\pi$
$443$	−494.991			−1.11736			−0.558680	+	0.829383i	$0.688692\pi$
$444$		0			0
$445$	−198.183			−0.445355
$446$		0			0
$447$			404.673i			0.905308i
$448$		0			0
$449$	539.070			1.20060			0.600301	−	0.799774i	$-0.295048\pi$
$449$	539.070			1.20060			0.600301	+	0.799774i	$0.295048\pi$
$450$		0			0
$451$		−	338.421i		−	0.750379i
$452$		0			0
$453$		−	486.078i		−	1.07302i
$454$		0			0
$455$	−144.947	−	43.7485i	−0.318565	−	0.0961506i
$456$		0			0
$457$	−265.432			−0.580814			−0.290407	−	0.956903i	$-0.593791\pi$
$457$	−265.432			−0.580814			−0.290407	+	0.956903i	$0.593791\pi$
$458$		0			0
$459$	36.8503			0.0802840
$460$		0			0
$461$		−	358.881i		−	0.778484i	−0.921136	−	0.389242i	$-0.872737\pi$
$461$		−	358.881i		−	0.778484i	0.921136	−	0.389242i	$-0.127263\pi$
$462$		0			0
$463$	−112.645			−0.243294			−0.121647	−	0.992573i	$-0.538818\pi$
$463$	−112.645			−0.243294			−0.121647	+	0.992573i	$0.538818\pi$
$464$		0			0
$465$			77.2536i			0.166137i
$466$		0			0
$467$		−	409.478i		−	0.876826i	−0.898774	−	0.438413i	$-0.855541\pi$
$467$		−	409.478i		−	0.876826i	0.898774	−	0.438413i	$-0.144459\pi$
$468$		0			0
$469$	−510.920	−	154.208i	−1.08938	−	0.328802i
$470$		0			0
$471$	94.0819			0.199749
$472$		0			0
$473$	−872.863			−1.84538
$474$		0			0
$475$		−	87.7541i		−	0.184745i
$476$		0			0
$477$	35.1329			0.0736539
$478$		0			0
$479$		−	99.5292i		−	0.207785i	−0.994589	−	0.103893i	$-0.966870\pi$
$479$		−	99.5292i		−	0.207785i	0.994589	−	0.103893i	$-0.0331299\pi$
$480$		0			0
$481$			107.052i			0.222561i
$482$		0			0
$483$	9.98177	−	33.0715i	0.0206662	−	0.0684710i
$484$		0			0
$485$	−42.1603			−0.0869286
$486$		0			0
$487$	123.746			0.254098			0.127049	−	0.991896i	$-0.459449\pi$
$487$	123.746			0.254098			0.127049	+	0.991896i	$0.459449\pi$
$488$		0			0
$489$		−	347.316i		−	0.710258i
$490$		0			0
$491$	−42.2815			−0.0861131			−0.0430565	−	0.999073i	$-0.513710\pi$
$491$	−42.2815			−0.0861131			−0.0430565	+	0.999073i	$0.513710\pi$
$492$		0			0
$493$		−	97.1083i		−	0.196974i
$494$		0			0
$495$			80.2469i			0.162115i
$496$		0			0
$497$	−639.645	−	193.060i	−1.28701	−	0.388451i
$498$		0			0
$499$	−87.3026			−0.174955			−0.0874775	−	0.996166i	$-0.527881\pi$
$499$	−87.3026			−0.174955			−0.0874775	+	0.996166i	$0.527881\pi$
$500$		0			0
$501$	−489.851			−0.977746
$502$		0			0
$503$		−	452.879i		−	0.900356i	−0.892939	−	0.450178i	$-0.851361\pi$
$503$		−	452.879i		−	0.900356i	0.892939	−	0.450178i	$-0.148639\pi$
$504$		0			0
$505$	67.4385			0.133542
$506$		0			0
$507$			130.656i			0.257704i
$508$		0			0
$509$			236.371i			0.464382i	0.972670	+	0.232191i	$0.0745895\pi$
$509$			236.371i			0.464382i	−0.972670	+	0.232191i	$0.925410\pi$
$510$		0			0
$511$	−244.102	+	808.756i	−0.477694	+	1.58269i
$512$		0			0
$513$	91.1967			0.177771
$514$		0			0
$515$	−337.353			−0.655055
$516$		0			0
$517$			338.912i			0.655536i
$518$		0			0
$519$	220.453			0.424765
$520$		0			0
$521$			446.261i			0.856548i	0.903649	+	0.428274i	$0.140878\pi$
$521$			446.261i			0.856548i	−0.903649	+	0.428274i	$0.859122\pi$
$522$		0			0
$523$		−	271.135i		−	0.518423i	−0.965821	−	0.259211i	$-0.916537\pi$
$523$		−	271.135i		−	0.518423i	0.965821	−	0.259211i	$-0.0834626\pi$
$524$		0			0
$525$	−17.5166	+	58.0359i	−0.0333650	+	0.110545i
$526$		0			0
$527$	141.460			0.268425
$528$		0			0
$529$	−520.882			−0.984654
$530$		0			0
$531$			303.481i			0.571527i
$532$		0			0
$533$	273.649			0.513413
$534$		0			0
$535$		−	197.375i		−	0.368926i
$536$		0			0
$537$		−	211.902i		−	0.394603i
$538$		0			0
$539$	−488.283	−	324.294i	−0.905906	−	0.601658i
$540$		0			0
$541$	224.274			0.414555			0.207277	−	0.978282i	$-0.433540\pi$
$541$	224.274			0.414555			0.207277	+	0.978282i	$0.433540\pi$
$542$		0			0
$543$	−168.633			−0.310559
$544$		0			0
$545$			207.332i			0.380425i
$546$		0			0
$547$	−88.4533			−0.161706			−0.0808531	−	0.996726i	$-0.525764\pi$
$547$	−88.4533			−0.161706			−0.0808531	+	0.996726i	$0.525764\pi$
$548$		0			0
$549$			230.116i			0.419155i
$550$		0			0
$551$		−	240.322i		−	0.436157i
$552$		0			0
$553$	−99.2453	−	29.9546i	−0.179467	−	0.0541675i
$554$		0			0
$555$	42.8630			0.0772306
$556$		0			0
$557$	−54.4040			−0.0976733			−0.0488367	−	0.998807i	$-0.515551\pi$
$557$	−54.4040			−0.0976733			−0.0488367	+	0.998807i	$0.515551\pi$
$558$		0			0
$559$		−	705.801i		−	1.26261i
$560$		0			0
$561$	146.941			0.261927
$562$		0			0
$563$			484.965i			0.861394i	0.902497	+	0.430697i	$0.141732\pi$
$563$			484.965i			0.861394i	−0.902497	+	0.430697i	$0.858268\pi$
$564$		0			0
$565$		−	141.977i		−	0.251287i
$566$		0			0
$567$	−60.3127	−	18.2038i	−0.106372	−	0.0321055i
$568$		0			0
$569$	−306.635			−0.538902			−0.269451	−	0.963014i	$-0.586842\pi$
$569$	−306.635			−0.538902			−0.269451	+	0.963014i	$0.586842\pi$
$570$		0			0
$571$	−471.839			−0.826339			−0.413169	−	0.910654i	$-0.635578\pi$
$571$	−471.839			−0.826339			−0.413169	+	0.910654i	$0.635578\pi$
$572$		0			0
$573$			502.598i			0.877134i
$574$		0			0
$575$	−14.2461			−0.0247759
$576$		0			0
$577$		−	81.1961i		−	0.140721i	−0.997522	−	0.0703606i	$-0.977585\pi$
$577$		−	81.1961i		−	0.140721i	0.997522	−	0.0703606i	$-0.0224150\pi$
$578$		0			0
$579$			157.234i			0.271561i
$580$		0			0
$581$	123.294	−	408.498i	0.212211	−	0.703094i
$582$		0			0
$583$	140.093			0.240296
$584$		0			0
$585$	−64.8881			−0.110920
$586$		0			0
$587$			905.925i			1.54331i	0.636039	+	0.771657i	$0.280572\pi$
$587$			905.925i			1.54331i	−0.636039	+	0.771657i	$0.719428\pi$
$588$		0			0
$589$	350.083			0.594368
$590$		0			0
$591$		−	92.4415i		−	0.156415i
$592$		0			0
$593$		−	455.850i		−	0.768719i	−0.923184	−	0.384359i	$-0.874422\pi$
$593$		−	455.850i		−	0.768719i	0.923184	−	0.384359i	$-0.125578\pi$
$594$		0			0
$595$	106.270	+	32.0748i	0.178605	+	0.0539073i
$596$		0			0
$597$	522.323			0.874912
$598$		0			0
$599$	−72.2734			−0.120657			−0.0603284	−	0.998179i	$-0.519215\pi$
$599$	−72.2734			−0.120657			−0.0603284	+	0.998179i	$0.519215\pi$
$600$		0			0
$601$		−	396.088i		−	0.659048i	−0.944147	−	0.329524i	$-0.893112\pi$
$601$		−	396.088i		−	0.659048i	0.944147	−	0.329524i	$-0.106888\pi$
$602$		0			0
$603$	−228.722			−0.379307
$604$		0			0
$605$			49.4206i			0.0816870i
$606$		0			0
$607$		−	6.14590i		−	0.0101250i	−0.999987	−	0.00506252i	$-0.998389\pi$
$607$		−	6.14590i		−	0.0101250i	0.999987	−	0.00506252i	$-0.00161146\pi$
$608$		0			0
$609$	−47.9708	+	158.937i	−0.0787699	+	0.260979i
$610$		0			0
$611$	−274.046			−0.448521
$612$		0			0
$613$	−197.286			−0.321837			−0.160918	−	0.986968i	$-0.551446\pi$
$613$	−197.286			−0.321837			−0.160918	+	0.986968i	$0.551446\pi$
$614$		0			0
$615$		−	109.567i		−	0.178158i
$616$		0			0
$617$	152.122			0.246551			0.123276	−	0.992372i	$-0.460660\pi$
$617$	152.122			0.246551			0.123276	+	0.992372i	$0.460660\pi$
$618$		0			0
$619$		−	700.262i		−	1.13128i	−0.824653	−	0.565639i	$-0.808629\pi$
$619$		−	700.262i		−	1.13128i	0.824653	−	0.565639i	$-0.191371\pi$
$620$		0			0
$621$		−	14.8050i		−	0.0238406i
$622$		0			0
$623$	179.267	−	593.947i	0.287749	−	0.953366i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	363.647			0.579979
$628$		0			0
$629$		−	78.4867i		−	0.124780i
$630$		0			0
$631$	133.714			0.211908			0.105954	−	0.994371i	$-0.466210\pi$
$631$	133.714			0.211908			0.105954	+	0.994371i	$0.466210\pi$
$632$		0			0
$633$			129.542i			0.204648i
$634$		0			0
$635$			564.425i			0.888858i
$636$		0			0
$637$	262.225	−	394.828i	0.411657	−	0.619824i
$638$		0			0
$639$	−286.348			−0.448119
$640$		0			0
$641$	−781.577			−1.21931			−0.609654	−	0.792667i	$-0.708692\pi$
$641$	−781.577			−1.21931			−0.609654	+	0.792667i	$0.708692\pi$
$642$		0			0
$643$		−	346.920i		−	0.539533i	−0.962926	−	0.269766i	$-0.913054\pi$
$643$		−	346.920i		−	0.539533i	0.962926	−	0.269766i	$-0.0869465\pi$
$644$		0			0
$645$	−282.598			−0.438137
$646$		0			0
$647$			971.001i			1.50077i	0.660998	+	0.750387i	$0.270133\pi$
$647$			971.001i			1.50077i	−0.660998	+	0.750387i	$0.729867\pi$
$648$		0			0
$649$			1210.13i			1.86461i
$650$		0			0
$651$	−231.526	−	69.8802i	−0.355647	−	0.107343i
$652$		0			0
$653$	−326.428			−0.499890			−0.249945	−	0.968260i	$-0.580413\pi$
$653$	−326.428			−0.499890			−0.249945	+	0.968260i	$0.580413\pi$
$654$		0			0
$655$	461.879			0.705158
$656$		0			0
$657$			362.053i			0.551070i
$658$		0			0
$659$	−305.606			−0.463743			−0.231871	−	0.972746i	$-0.574485\pi$
$659$	−305.606			−0.463743			−0.231871	+	0.972746i	$0.574485\pi$
$660$		0			0
$661$		−	232.398i		−	0.351586i	−0.984427	−	0.175793i	$-0.943751\pi$
$661$		−	232.398i		−	0.351586i	0.984427	−	0.175793i	$-0.0562489\pi$
$662$		0			0
$663$			118.817i			0.179211i
$664$		0			0
$665$	262.996	+	79.3784i	0.395482	+	0.119366i
$666$		0			0
$667$	−39.0143			−0.0584922
$668$		0			0
$669$	715.561			1.06960
$670$		0			0
$671$			917.589i			1.36750i
$672$		0			0
$673$	283.273			0.420911			0.210455	−	0.977603i	$-0.432505\pi$
$673$	283.273			0.420911			0.210455	+	0.977603i	$0.432505\pi$
$674$		0			0
$675$			25.9808i			0.0384900i
$676$		0			0
$677$			425.511i			0.628524i	0.949336	+	0.314262i	$0.101757\pi$
$677$			425.511i			0.628524i	−0.949336	+	0.314262i	$0.898243\pi$
$678$		0			0
$679$	38.1364	−	126.353i	0.0561655	−	0.186087i
$680$		0			0
$681$	−517.348			−0.759689
$682$		0			0
$683$	126.763			0.185597			0.0927986	−	0.995685i	$-0.470419\pi$
$683$	126.763			0.185597			0.0927986	+	0.995685i	$0.470419\pi$
$684$		0			0
$685$			400.395i			0.584519i
$686$		0			0
$687$	560.013			0.815157
$688$		0			0
$689$			113.279i			0.164411i
$690$		0			0
$691$		−	124.435i		−	0.180080i	−0.995938	−	0.0900401i	$-0.971300\pi$
$691$		−	124.435i		−	0.180080i	0.995938	−	0.0900401i	$-0.0286995\pi$
$692$		0			0
$693$	−240.497	−	72.5878i	−0.347038	−	0.104744i
$694$		0			0
$695$	66.7270			0.0960100
$696$		0			0
$697$	−200.630			−0.287847
$698$		0			0
$699$		−	561.100i		−	0.802718i
$700$		0			0
$701$	273.794			0.390576			0.195288	−	0.980746i	$-0.437436\pi$
$701$	273.794			0.390576			0.195288	+	0.980746i	$0.437436\pi$
$702$		0			0
$703$		−	194.238i		−	0.276299i
$704$		0			0
$705$			109.726i			0.155640i
$706$		0			0
$707$	−61.0018	+	202.111i	−0.0862826	+	0.285871i
$708$		0			0
$709$	−462.551			−0.652399			−0.326200	−	0.945301i	$-0.605768\pi$
$709$	−462.551			−0.652399			−0.326200	+	0.945301i	$0.605768\pi$
$710$		0			0
$711$	−44.4288			−0.0624878
$712$		0			0
$713$		−	56.8330i		−	0.0797096i
$714$		0			0
$715$	−258.741			−0.361876
$716$		0			0
$717$			627.798i			0.875589i
$718$		0			0
$719$		−	773.673i		−	1.07604i	−0.842932	−	0.538020i	$-0.819172\pi$
$719$		−	773.673i		−	1.07604i	0.842932	−	0.538020i	$-0.180828\pi$
$720$		0			0
$721$	305.155	−	1011.04i	0.423238	−	1.40227i
$722$		0			0
$723$	−216.268			−0.299127
$724$		0			0
$725$	68.4647			0.0944341
$726$		0			0
$727$			1233.31i			1.69644i	0.529645	+	0.848219i	$0.322325\pi$
$727$			1233.31i			1.69644i	−0.529645	+	0.848219i	$0.677675\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$			517.468i			0.707891i
$732$		0			0
$733$		−	1421.58i		−	1.93939i	−0.244309	−	0.969697i	$-0.578561\pi$
$733$		−	1421.58i		−	1.93939i	0.244309	−	0.969697i	$-0.421439\pi$
$734$		0			0
$735$	−158.087	−	104.993i	−0.215084	−	0.142848i
$736$		0			0
$737$	−912.030			−1.23749
$738$		0			0
$739$	−518.845			−0.702091			−0.351045	−	0.936358i	$-0.614174\pi$
$739$	−518.845			−0.702091			−0.351045	+	0.936358i	$0.614174\pi$
$740$		0			0
$741$			294.047i			0.396824i
$742$		0			0
$743$	1012.05			1.36212			0.681058	−	0.732230i	$-0.261520\pi$
$743$	1012.05			1.36212			0.681058	+	0.732230i	$0.261520\pi$
$744$		0			0
$745$			522.430i			0.701248i
$746$		0			0
$747$		−	182.871i		−	0.244807i
$748$		0			0
$749$	591.526	+	178.537i	0.789755	+	0.238367i
$750$		0			0
$751$	1274.40			1.69694			0.848470	−	0.529243i	$-0.177524\pi$
$751$	1274.40			1.69694			0.848470	+	0.529243i	$0.177524\pi$
$752$		0			0
$753$	31.9590			0.0424423
$754$		0			0
$755$		−	627.524i		−	0.831157i
$756$		0			0
$757$	1145.83			1.51364			0.756822	−	0.653621i	$-0.226751\pi$
$757$	1145.83			1.51364			0.756822	+	0.653621i	$0.226751\pi$
$758$		0			0
$759$		−	59.0350i		−	0.0777800i
$760$		0			0
$761$		−	580.236i		−	0.762465i	−0.924479	−	0.381233i	$-0.875500\pi$
$761$		−	580.236i		−	0.762465i	0.924479	−	0.381233i	$-0.124500\pi$
$762$		0			0
$763$	−621.365	−	187.543i	−0.814372	−	0.245797i
$764$		0			0
$765$	47.5736			0.0621877
$766$		0			0
$767$	−978.517			−1.27577
$768$		0			0
$769$		−	58.9626i		−	0.0766744i	−0.999265	−	0.0383372i	$-0.987794\pi$
$769$		−	58.9626i		−	0.0766744i	0.999265	−	0.0383372i	$-0.0122061\pi$
$770$		0			0
$771$	−742.459			−0.962981
$772$		0			0
$773$			368.237i			0.476374i	0.971219	+	0.238187i	$0.0765531\pi$
$773$			368.237i			0.476374i	−0.971219	+	0.238187i	$0.923447\pi$
$774$		0			0
$775$			99.7340i			0.128689i
$776$		0			0
$777$	−38.7719	+	128.459i	−0.0498995	+	0.165326i
$778$		0			0
$779$	−496.515			−0.637375
$780$		0			0
$781$	−1141.81			−1.46199
$782$		0			0
$783$			71.1506i			0.0908692i
$784$		0			0
$785$	121.459			0.154725
$786$		0			0
$787$			1063.41i			1.35122i	0.737261	+	0.675608i	$0.236119\pi$
$787$			1063.41i			1.35122i	−0.737261	+	0.675608i	$0.763881\pi$
$788$		0			0
$789$			231.358i			0.293229i
$790$		0			0
$791$	425.501	+	128.426i	0.537928	+	0.162359i
$792$		0			0
$793$	−741.967			−0.935646
$794$		0			0
$795$	45.3564			0.0570520
$796$		0			0
$797$			274.031i			0.343828i	0.985112	+	0.171914i	$0.0549951\pi$
$797$			274.031i			0.343828i	−0.985112	+	0.171914i	$0.945005\pi$
$798$		0			0
$799$	200.921			0.251465
$800$		0			0
$801$		−	265.890i		−	0.331948i
$802$		0			0
$803$			1443.69i			1.79787i
$804$		0			0
$805$	12.8864	−	42.6951i	0.0160080	−	0.0530374i
$806$		0			0
$807$	854.345			1.05867
$808$		0			0
$809$	125.869			0.155586			0.0777930	−	0.996970i	$-0.475213\pi$
$809$	125.869			0.155586			0.0777930	+	0.996970i	$0.475213\pi$
$810$		0			0
$811$		−	566.987i		−	0.699121i	−0.936914	−	0.349560i	$-0.886331\pi$
$811$		−	566.987i		−	0.699121i	0.936914	−	0.349560i	$-0.113669\pi$
$812$		0			0
$813$	749.191			0.921514
$814$		0			0
$815$		−	448.383i		−	0.550163i
$816$		0			0
$817$			1280.62i			1.56747i
$818$		0			0
$819$	58.6948	−	194.467i	0.0716664	−	0.237444i
$820$		0			0
$821$	−444.471			−0.541377			−0.270689	−	0.962667i	$-0.587251\pi$
$821$	−444.471			−0.541377			−0.270689	+	0.962667i	$0.587251\pi$
$822$		0			0
$823$	−1028.74			−1.24999			−0.624995	−	0.780629i	$-0.714899\pi$
$823$	−1028.74			−1.24999			−0.624995	+	0.780629i	$0.714899\pi$
$824$		0			0
$825$			103.598i			0.125574i
$826$		0			0
$827$	515.113			0.622870			0.311435	−	0.950267i	$-0.399190\pi$
$827$	515.113			0.622870			0.311435	+	0.950267i	$0.399190\pi$
$828$		0			0
$829$		−	1334.96i		−	1.61032i	−0.593055	−	0.805162i	$-0.702078\pi$
$829$		−	1334.96i		−	1.61032i	0.593055	−	0.805162i	$-0.297922\pi$
$830$		0			0
$831$		−	297.810i		−	0.358375i
$832$		0			0
$833$	−192.254	+	289.474i	−0.230797	+	0.347508i
$834$		0			0
$835$	−632.394			−0.757359
$836$		0			0
$837$	−103.647			−0.123831
$838$		0			0
$839$			2.13640i			0.00254637i	0.999999	+	0.00127318i	$0.000405267\pi$
$839$			2.13640i			0.00254637i	−0.999999	+	0.00127318i	$0.999595\pi$
$840$		0			0
$841$	−653.503			−0.777055
$842$		0			0
$843$			129.471i			0.153583i
$844$		0			0
$845$			168.676i			0.199617i
$846$		0			0
$847$	−148.112	−	44.7037i	−0.174866	−	0.0527788i
$848$		0			0
$849$	−287.354			−0.338461
$850$		0			0
$851$	−31.5329			−0.0370539
$852$		0			0
$853$			89.0278i			0.104370i	0.998637	+	0.0521851i	$0.0166186\pi$
$853$			89.0278i			0.104370i	−0.998637	+	0.0521851i	$0.983381\pi$
$854$		0			0
$855$	117.734			0.137701
$856$		0			0
$857$			820.566i			0.957486i	0.877955	+	0.478743i	$0.158907\pi$
$857$			820.566i			0.957486i	−0.877955	+	0.478743i	$0.841093\pi$
$858$		0			0
$859$		−	534.947i		−	0.622756i	−0.950286	−	0.311378i	$-0.899210\pi$
$859$		−	534.947i		−	0.622756i	0.950286	−	0.311378i	$-0.100790\pi$
$860$		0			0
$861$	328.369	+	99.1096i	0.381381	+	0.115110i
$862$		0			0
$863$	276.972			0.320941			0.160471	−	0.987041i	$-0.448699\pi$
$863$	276.972			0.320941			0.160471	+	0.987041i	$0.448699\pi$
$864$		0			0
$865$	284.603			0.329021
$866$		0			0
$867$			413.450i			0.476875i
$868$		0			0
$869$	−177.160			−0.203867
$870$		0			0
$871$		−	737.472i		−	0.846695i
$872$		0			0
$873$		−	56.5640i		−	0.0647927i
$874$		0			0
$875$	−22.6139	+	74.9240i	−0.0258444	+	0.0856275i
$876$		0			0
$877$	420.192			0.479125			0.239562	−	0.970881i	$-0.422996\pi$
$877$	420.192			0.479125			0.239562	+	0.970881i	$0.422996\pi$
$878$		0			0
$879$	−858.967			−0.977210
$880$		0			0
$881$		−	897.207i		−	1.01840i	−0.860649	−	0.509198i	$-0.829942\pi$
$881$		−	897.207i		−	1.01840i	0.860649	−	0.509198i	$-0.170058\pi$
$882$		0			0
$883$	1515.08			1.71584			0.857918	−	0.513786i	$-0.171757\pi$
$883$	1515.08			1.71584			0.857918	+	0.513786i	$0.171757\pi$
$884$		0			0
$885$			391.792i			0.442703i
$886$		0			0
$887$			976.074i			1.10042i	0.835026	+	0.550211i	$0.185453\pi$
$887$			976.074i			1.10042i	−0.835026	+	0.550211i	$0.814547\pi$
$888$		0			0
$889$	−1691.56	−	510.553i	−1.90277	−	0.574300i
$890$		0			0
$891$	−107.663			−0.120833
$892$		0			0
$893$	497.236			0.556815
$894$		0			0
$895$		−	273.564i		−	0.305658i
$896$		0			0
$897$	47.7360			0.0532174
$898$		0			0
$899$			273.130i			0.303816i
$900$		0			0
$901$		−	83.0524i		−	0.0921781i
$902$		0			0
$903$	255.626	−	846.937i	0.283085	−	0.937914i
$904$		0			0
$905$	−217.705			−0.240558
$906$		0			0
$907$	−1329.48			−1.46580			−0.732901	−	0.680335i	$-0.761834\pi$
$907$	−1329.48			−1.46580			−0.732901	+	0.680335i	$0.761834\pi$
$908$		0			0
$909$			90.4782i			0.0995360i
$910$		0			0
$911$	1270.75			1.39489			0.697447	−	0.716637i	$-0.254319\pi$
$911$	1270.75			1.39489			0.697447	+	0.716637i	$0.254319\pi$
$912$		0			0
$913$		−	729.198i		−	0.798684i
$914$		0			0
$915$			297.079i			0.324676i
$916$		0			0
$917$	−417.795	+	1384.23i	−0.455610	+	1.50952i
$918$		0			0
$919$	1641.49			1.78617			0.893083	−	0.449893i	$-0.148538\pi$
$919$	1641.49			1.78617			0.893083	+	0.449893i	$0.148538\pi$
$920$		0			0
$921$	548.959			0.596047
$922$		0			0
$923$		−	923.275i		−	1.00030i
$924$		0			0
$925$	55.3359			0.0598225
$926$		0			0
$927$		−	452.607i		−	0.488249i
$928$		0			0
$929$		−	1386.92i		−	1.49292i	−0.665433	−	0.746458i	$-0.731753\pi$
$929$		−	1386.92i		−	1.49292i	0.665433	−	0.746458i	$-0.268247\pi$
$930$		0			0
$931$	−475.788	+	716.386i	−0.511050	+	0.769480i
$932$		0			0
$933$	−46.9949			−0.0503697
$934$		0			0
$935$	189.700			0.202888
$936$		0			0
$937$		−	359.625i		−	0.383805i	−0.981414	−	0.191903i	$-0.938534\pi$
$937$		−	359.625i		−	0.383805i	0.981414	−	0.191903i	$-0.0614658\pi$
$938$		0			0
$939$	895.878			0.954077
$940$		0			0
$941$			1575.46i			1.67424i	0.547020	+	0.837119i	$0.315762\pi$
$941$			1575.46i			1.67424i	−0.547020	+	0.837119i	$0.684238\pi$
$942$		0			0
$943$			80.6051i			0.0854773i
$944$		0			0
$945$	−77.8634	−	23.5010i	−0.0823951	−	0.0248688i
$946$		0			0
$947$	1680.38			1.77442			0.887212	−	0.461361i	$-0.152639\pi$
$947$	1680.38			1.77442			0.887212	+	0.461361i	$0.152639\pi$
$948$		0			0
$949$	−1167.37			−1.23011
$950$		0			0
$951$			368.998i			0.388010i
$952$		0			0
$953$	1419.84			1.48986			0.744931	−	0.667142i	$-0.232483\pi$
$953$	1419.84			1.48986			0.744931	+	0.667142i	$0.232483\pi$
$954$		0			0
$955$			648.851i			0.679425i
$956$		0			0
$957$			283.713i			0.296461i
$958$		0			0
$959$	−1199.97	−	362.180i	−1.25127	−	0.377664i
$960$		0			0
$961$	563.125			0.585978
$962$		0			0
$963$	264.807			0.274981
$964$		0			0
$965$			202.988i			0.210350i
$966$		0			0
$967$	1061.44			1.09766			0.548829	−	0.835934i	$-0.315074\pi$
$967$	1061.44			1.09766			0.548829	+	0.835934i	$0.315074\pi$
$968$		0			0
$969$		−	215.585i		−	0.222481i
$970$		0			0
$971$		−	1375.99i		−	1.41709i	−0.705668	−	0.708543i	$-0.749353\pi$
$971$		−	1375.99i		−	1.41709i	0.705668	−	0.708543i	$-0.250647\pi$
$972$		0			0
$973$	−60.3582	+	199.978i	−0.0620331	+	0.205527i
$974$		0			0
$975$	−83.7701			−0.0859181
$976$		0			0
$977$	541.316			0.554059			0.277030	−	0.960861i	$-0.410650\pi$
$977$	541.316			0.554059			0.277030	+	0.960861i	$0.410650\pi$
$978$		0			0
$979$		−	1060.24i		−	1.08298i
$980$		0			0
$981$	−278.165			−0.283552
$982$		0			0
$983$		−	975.992i		−	0.992871i	−0.868074	−	0.496435i	$-0.834642\pi$
$983$		−	975.992i		−	0.992871i	0.868074	−	0.496435i	$-0.165358\pi$
$984$		0			0
$985$		−	119.341i		−	0.121159i
$986$		0			0
$987$	−328.846	−	99.2535i	−0.333177	−	0.100561i
$988$		0			0
$989$	207.898			0.210211
$990$		0			0
$991$	−459.246			−0.463417			−0.231708	−	0.972785i	$-0.574432\pi$
$991$	−459.246			−0.463417			−0.231708	+	0.972785i	$0.574432\pi$
$992$		0			0
$993$		−	506.653i		−	0.510224i
$994$		0			0
$995$	674.316			0.677704
$996$		0			0
$997$		−	745.998i		−	0.748242i	−0.927380	−	0.374121i	$-0.877944\pi$
$997$		−	745.998i		−	0.748242i	0.927380	−	0.374121i	$-0.122056\pi$
$998$		0			0
$999$			57.5067i			0.0575643i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.s.a.1441.7		8
4.3	odd	2		210.3.f.a.181.6	✓	8
7.6	odd	2	inner	1680.3.s.a.1441.1		8
12.11	even	2		630.3.f.c.181.2		8
20.3	even	4		1050.3.h.b.349.7		16
20.7	even	4		1050.3.h.b.349.10		16
20.19	odd	2		1050.3.f.b.601.3		8
28.27	even	2		210.3.f.a.181.7	yes	8
84.83	odd	2		630.3.f.c.181.4		8
140.27	odd	4		1050.3.h.b.349.15		16
140.83	odd	4		1050.3.h.b.349.2		16
140.139	even	2		1050.3.f.b.601.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.f.a.181.6	✓	8	4.3	odd	2
210.3.f.a.181.7	yes	8	28.27	even	2
630.3.f.c.181.2		8	12.11	even	2
630.3.f.c.181.4		8	84.83	odd	2
1050.3.f.b.601.1		8	140.139	even	2
1050.3.f.b.601.3		8	20.19	odd	2
1050.3.h.b.349.2		16	140.83	odd	4
1050.3.h.b.349.7		16	20.3	even	4
1050.3.h.b.349.10		16	20.7	even	4
1050.3.h.b.349.15		16	140.27	odd	4
1680.3.s.a.1441.1		8	7.6	odd	2	inner
1680.3.s.a.1441.7		8	1.1	even	1	trivial

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.s (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +2.23607i q^{5} +(-6.70141 - 2.02265i) q^{7} -3.00000 q^{9} -11.9625 q^{11} -9.67294i q^{13} -3.87298 q^{15} +7.09185i q^{17} +17.5508i q^{19} +(3.50333 - 11.6072i) q^{21} +2.84923 q^{23} -5.00000 q^{25} -5.19615i q^{27} -13.6929 q^{29} -19.9468i q^{31} -20.7197i q^{33} +(4.52277 - 14.9848i) q^{35} -11.0672 q^{37} +16.7540 q^{39} +28.2902i q^{41} +72.9666 q^{43} -6.70820i q^{45} -28.3312i q^{47} +(40.8178 + 27.1092i) q^{49} -12.2834 q^{51} -11.7110 q^{53} -26.7490i q^{55} -30.3989 q^{57} -101.160i q^{59} -76.7055i q^{61} +(20.1042 + 6.06794i) q^{63} +21.6294 q^{65} +76.2407 q^{67} +4.93501i q^{69} +95.4493 q^{71} -120.684i q^{73} -8.66025i q^{75} +(80.1657 + 24.1959i) q^{77} +14.8096 q^{79} +9.00000 q^{81} +60.9570i q^{83} -15.8579 q^{85} -23.7169i q^{87} +88.6302i q^{89} +(-19.5649 + 64.8223i) q^{91} +34.5489 q^{93} -39.2448 q^{95} +18.8547i q^{97} +35.8875 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} + 16 q^{11} - 40 q^{25} - 144 q^{29} + 80 q^{35} - 48 q^{37} + 48 q^{39} + 64 q^{43} - 24 q^{49} + 128 q^{53} + 144 q^{57} + 80 q^{65} + 192 q^{67} - 176 q^{71} + 192 q^{77} + 288 q^{79}+ \cdots - 48 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 + 16 * q^11 - 40 * q^25 - 144 * q^29 + 80 * q^35 - 48 * q^37 + 48 * q^39 + 64 * q^43 - 24 * q^49 + 128 * q^53 + 144 * q^57 + 80 * q^65 + 192 * q^67 - 176 * q^71 + 192 * q^77 + 288 * q^79 + 72 * q^81 - 240 * q^85 - 64 * q^91 + 336 * q^93 - 80 * q^95 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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