LMFDB - Embedded newform 320.8.c.g.129.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,8,Mod(129,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.129"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	320.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-60,0,0,0,-6788,0,17808] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$99.9632081549$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{31})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 15x^{2} + 64 $ x^4 - 15*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.4
Root			$2.78388 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.129
Dual form			320.8.c.g.129.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+83.6776i q^{3} +(-237.711 - 147.033i) q^{5} -185.744i q^{7} -4814.95 q^{9} +3561.16 q^{11} +6094.86i q^{13} +(12303.4 - 19891.1i) q^{15} -12470.2i q^{17} +50642.8 q^{19} +15542.6 q^{21} +11442.3i q^{23} +(34887.6 + 69902.6i) q^{25} -219900. i q^{27} +101926. q^{29} +29042.3 q^{31} +297989. i q^{33} +(-27310.4 + 44153.2i) q^{35} -149393. i q^{37} -510003. q^{39} -374382. q^{41} +174226. i q^{43} +(1.14456e6 + 707956. i) q^{45} +428201. i q^{47} +789042. q^{49} +1.04347e6 q^{51} +1.71297e6i q^{53} +(-846525. - 523607. i) q^{55} +4.23767e6i q^{57} -134952. q^{59} +1.39437e6 q^{61} +894345. i q^{63} +(896145. - 1.44881e6i) q^{65} -2.60763e6i q^{67} -957466. q^{69} +4.91925e6 q^{71} +119096. i q^{73} +(-5.84928e6 + 2.91932e6i) q^{75} -661462. i q^{77} -4.70584e6 q^{79} +7.87046e6 q^{81} +9.19870e6i q^{83} +(-1.83352e6 + 2.96429e6i) q^{85} +8.52897e6i q^{87} -6.43438e6 q^{89} +1.13208e6 q^{91} +2.43019e6i q^{93} +(-1.20383e7 - 7.44617e6i) q^{95} -1.26986e7i q^{97} -1.71468e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 60 q^{5} - 6788 q^{9} + 17808 q^{11} + 34960 q^{15} + 63600 q^{19} + 63952 q^{21} + 86100 q^{25} - 169560 q^{29} + 394112 q^{31} - 276720 q^{35} - 1163424 q^{39} + 232488 q^{41} + 2879420 q^{45} + 2520108 q^{49}+ \cdots - 19109776 q^{99}+O(q^{100}) $ 4 * q - 60 * q^5 - 6788 * q^9 + 17808 * q^11 + 34960 * q^15 + 63600 * q^19 + 63952 * q^21 + 86100 * q^25 - 169560 * q^29 + 394112 * q^31 - 276720 * q^35 - 1163424 * q^39 + 232488 * q^41 + 2879420 * q^45 + 2520108 * q^49 + 361088 * q^51 + 526480 * q^55 + 2093520 * q^59 + 5251432 * q^61 + 2761440 * q^65 - 6514864 * q^69 + 7832352 * q^71 - 7397600 * q^75 - 7727040 * q^79 + 16428404 * q^81 + 7995520 * q^85 - 33470040 * q^89 + 5340768 * q^91 - 31904400 * q^95 - 19109776 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			83.6776i			1.78931i	0.446760	+	0.894654i	$0.352578\pi$
$3$			83.6776i			1.78931i	−0.446760	+	0.894654i	$0.647422\pi$
$4$		0			0
$5$	−237.711	−	147.033i	−0.850459	−	0.526041i
$5$	−237.711	−	147.033i	−0.850459	−	0.526041i
$6$		0			0
$7$		−	185.744i		−	0.204678i	−0.994750	−	0.102339i	$-0.967367\pi$
$7$		−	185.744i		−	0.204678i	0.994750	−	0.102339i	$-0.0326326\pi$
$8$		0			0
$9$	−4814.95			−2.20162
$10$		0			0
$11$	3561.16			0.806709			0.403354	−	0.915044i	$-0.367844\pi$
$11$	3561.16			0.806709			0.403354	+	0.915044i	$0.367844\pi$
$12$		0			0
$13$			6094.86i			0.769417i	0.923038	+	0.384708i	$0.125698\pi$
$13$			6094.86i			0.769417i	−0.923038	+	0.384708i	$0.874302\pi$
$14$		0			0
$15$	12303.4	−	19891.1i	0.941249	−	1.52173i
$16$		0			0
$17$		−	12470.2i		−	0.615603i	−0.951451	−	0.307801i	$-0.900407\pi$
$17$		−	12470.2i		−	0.615603i	0.951451	−	0.307801i	$-0.0995933\pi$
$18$		0			0
$19$	50642.8			1.69387			0.846936	−	0.531695i	$-0.178445\pi$
$19$	50642.8			1.69387			0.846936	+	0.531695i	$0.178445\pi$
$20$		0			0
$21$	15542.6			0.366231
$22$		0			0
$23$			11442.3i			0.196095i	0.995182	+	0.0980475i	$0.0312597\pi$
$23$			11442.3i			0.196095i	−0.995182	+	0.0980475i	$0.968740\pi$
$24$		0			0
$25$	34887.6	+	69902.6i	0.446562	+	0.894753i
$26$		0			0
$27$		−	219900.i		−	2.15007i
$28$		0			0
$29$	101926.			0.776058			0.388029	−	0.921647i	$-0.373156\pi$
$29$	101926.			0.776058			0.388029	+	0.921647i	$0.373156\pi$
$30$		0			0
$31$	29042.3			0.175092			0.0875458	−	0.996160i	$-0.472098\pi$
$31$	29042.3			0.175092			0.0875458	+	0.996160i	$0.472098\pi$
$32$		0			0
$33$			297989.i			1.44345i
$34$		0			0
$35$	−27310.4	+	44153.2i	−0.107669	+	0.174070i
$36$		0			0
$37$		−	149393.i		−	0.484870i	−0.970168	−	0.242435i	$-0.922054\pi$
$37$		−	149393.i		−	0.484870i	0.970168	−	0.242435i	$-0.0779461\pi$
$38$		0			0
$39$	−510003.			−1.37672
$40$		0			0
$41$	−374382.			−0.848343			−0.424171	−	0.905582i	$-0.639435\pi$
$41$	−374382.			−0.848343			−0.424171	+	0.905582i	$0.639435\pi$
$42$		0			0
$43$			174226.i			0.334174i	0.985942	+	0.167087i	$0.0534360\pi$
$43$			174226.i			0.334174i	−0.985942	+	0.167087i	$0.946564\pi$
$44$		0			0
$45$	1.14456e6	+	707956.i	1.87239	+	1.15814i
$46$		0			0
$47$			428201.i			0.601597i	0.953688	+	0.300798i	$0.0972531\pi$
$47$			428201.i			0.601597i	−0.953688	+	0.300798i	$0.902747\pi$
$48$		0			0
$49$	789042.			0.958107
$50$		0			0
$51$	1.04347e6			1.10150
$52$		0			0
$53$			1.71297e6i			1.58047i	0.612806	+	0.790233i	$0.290041\pi$
$53$			1.71297e6i			1.58047i	−0.612806	+	0.790233i	$0.709959\pi$
$54$		0			0
$55$	−846525.	−	523607.i	−0.686073	−	0.424362i
$56$		0			0
$57$			4.23767e6i			3.03086i
$58$		0			0
$59$	−134952.			−0.0855458			−0.0427729	−	0.999085i	$-0.513619\pi$
$59$	−134952.			−0.0855458			−0.0427729	+	0.999085i	$0.513619\pi$
$60$		0			0
$61$	1.39437e6			0.786545			0.393273	−	0.919422i	$-0.371343\pi$
$61$	1.39437e6			0.786545			0.393273	+	0.919422i	$0.371343\pi$
$62$		0			0
$63$			894345.i			0.450623i
$64$		0			0
$65$	896145.	−	1.44881e6i	0.404745	−	0.654358i
$66$		0			0
$67$		−	2.60763e6i		−	1.05922i	−0.848243	−	0.529608i	$-0.822339\pi$
$67$		−	2.60763e6i		−	1.05922i	0.848243	−	0.529608i	$-0.177661\pi$
$68$		0			0
$69$	−957466.			−0.350874
$70$		0			0
$71$	4.91925e6			1.63115			0.815576	−	0.578650i	$-0.196420\pi$
$71$	4.91925e6			1.63115			0.815576	+	0.578650i	$0.196420\pi$
$72$		0			0
$73$			119096.i			0.0358316i	0.999839	+	0.0179158i	$0.00570309\pi$
$73$			119096.i			0.0358316i	−0.999839	+	0.0179158i	$0.994297\pi$
$74$		0			0
$75$	−5.84928e6	+	2.91932e6i	−1.60099	+	0.799036i
$76$		0			0
$77$		−	661462.i		−	0.165115i
$78$		0			0
$79$	−4.70584e6			−1.07385			−0.536924	−	0.843631i	$-0.680414\pi$
$79$	−4.70584e6			−1.07385			−0.536924	+	0.843631i	$0.680414\pi$
$80$		0			0
$81$	7.87046e6			1.64552
$82$		0			0
$83$			9.19870e6i			1.76585i	0.469517	+	0.882923i	$0.344428\pi$
$83$			9.19870e6i			1.76585i	−0.469517	+	0.882923i	$0.655572\pi$
$84$		0			0
$85$	−1.83352e6	+	2.96429e6i	−0.323832	+	0.523545i
$86$		0			0
$87$			8.52897e6i			1.38861i
$88$		0			0
$89$	−6.43438e6			−0.967480			−0.483740	−	0.875212i	$-0.660722\pi$
$89$	−6.43438e6			−0.967480			−0.483740	+	0.875212i	$0.660722\pi$
$90$		0			0
$91$	1.13208e6			0.157482
$92$		0			0
$93$			2.43019e6i			0.313293i
$94$		0			0
$95$	−1.20383e7	−	7.44617e6i	−1.44057	−	0.891046i
$96$		0			0
$97$		−	1.26986e7i		−	1.41271i	−0.707858	−	0.706355i	$-0.750338\pi$
$97$		−	1.26986e7i		−	1.41271i	0.707858	−	0.706355i	$-0.249662\pi$
$98$		0			0
$99$	−1.71468e7			−1.77607
$100$		0			0
$101$	−6.66850e6			−0.644026			−0.322013	−	0.946735i	$-0.604359\pi$
$101$	−6.66850e6			−0.644026			−0.322013	+	0.946735i	$0.604359\pi$
$102$		0			0
$103$		−	2.10469e6i		−	0.189783i	−0.995488	−	0.0948916i	$-0.969750\pi$
$103$		−	2.10469e6i		−	0.189783i	0.995488	−	0.0948916i	$-0.0302505\pi$
$104$		0			0
$105$	−3.69464e6	−	2.28527e6i	−0.311465	−	0.192653i
$106$		0			0
$107$			7.43589e6i			0.586799i	0.955990	+	0.293400i	$0.0947867\pi$
$107$			7.43589e6i			0.586799i	−0.955990	+	0.293400i	$0.905213\pi$
$108$		0			0
$109$	−9.12137e6			−0.674633			−0.337316	−	0.941391i	$-0.609519\pi$
$109$	−9.12137e6			−0.674633			−0.337316	+	0.941391i	$0.609519\pi$
$110$		0			0
$111$	1.25009e7			0.867582
$112$		0			0
$113$		−	2.93903e6i		−	0.191615i	−0.995400	−	0.0958075i	$-0.969457\pi$
$113$		−	2.93903e6i		−	0.191615i	0.995400	−	0.0958075i	$-0.0305433\pi$
$114$		0			0
$115$	1.68240e6	−	2.71996e6i	0.103154	−	0.166771i
$116$		0			0
$117$		−	2.93464e7i		−	1.69397i
$118$		0			0
$119$	−2.31625e6			−0.126000
$120$		0			0
$121$	−6.80533e6			−0.349221
$122$		0			0
$123$		−	3.13274e7i		−	1.51795i
$124$		0			0
$125$	1.98482e6	−	2.17462e7i	0.0908942	−	0.995861i
$126$		0			0
$127$		−	1.76254e7i		−	0.763529i	−0.924260	−	0.381764i	$-0.875317\pi$
$127$		−	1.76254e7i		−	0.763529i	0.924260	−	0.381764i	$-0.124683\pi$
$128$		0			0
$129$	−1.45788e7			−0.597939
$130$		0			0
$131$	3.95489e6			0.153704			0.0768520	−	0.997043i	$-0.475513\pi$
$131$	3.95489e6			0.153704			0.0768520	+	0.997043i	$0.475513\pi$
$132$		0			0
$133$		−	9.40658e6i		−	0.346698i
$134$		0			0
$135$	−3.23326e7	+	5.22727e7i	−1.13103	+	1.82855i
$136$		0			0
$137$		−	2.95868e7i		−	0.983051i	−0.870863	−	0.491525i	$-0.836440\pi$
$137$		−	2.95868e7i		−	0.983051i	0.870863	−	0.491525i	$-0.163560\pi$
$138$		0			0
$139$	−4.26561e6			−0.134719			−0.0673596	−	0.997729i	$-0.521457\pi$
$139$	−4.26561e6			−0.134719			−0.0673596	+	0.997729i	$0.521457\pi$
$140$		0			0
$141$	−3.58309e7			−1.07644
$142$		0			0
$143$			2.17047e7i			0.620695i
$144$		0			0
$145$	−2.42290e7	−	1.49865e7i	−0.660005	−	0.408238i
$146$		0			0
$147$			6.60252e7i			1.71435i
$148$		0			0
$149$	7.09039e7			1.75597			0.877987	−	0.478685i	$-0.158886\pi$
$149$	7.09039e7			1.75597			0.877987	+	0.478685i	$0.158886\pi$
$150$		0			0
$151$	−3.36796e7			−0.796063			−0.398032	−	0.917372i	$-0.630307\pi$
$151$	−3.36796e7			−0.796063			−0.398032	+	0.917372i	$0.630307\pi$
$152$		0			0
$153$			6.00432e7i			1.35533i
$154$		0			0
$155$	−6.90366e6	−	4.27017e6i	−0.148908	−	0.0921053i
$156$		0			0
$157$			7.51572e7i			1.54997i	0.631982	+	0.774983i	$0.282241\pi$
$157$			7.51572e7i			1.54997i	−0.631982	+	0.774983i	$0.717759\pi$
$158$		0			0
$159$	−1.43338e8			−2.82794
$160$		0			0
$161$	2.12534e6			0.0401363
$162$		0			0
$163$			8.50523e7i			1.53826i	0.639093	+	0.769130i	$0.279310\pi$
$163$			8.50523e7i			1.53826i	−0.639093	+	0.769130i	$0.720690\pi$
$164$		0			0
$165$	4.38142e7	−	7.08352e7i	0.759314	−	1.22760i
$166$		0			0
$167$			5.10526e7i			0.848223i	0.905610	+	0.424111i	$0.139413\pi$
$167$			5.10526e7i			0.848223i	−0.905610	+	0.424111i	$0.860587\pi$
$168$		0			0
$169$	2.56012e7			0.407998
$170$		0			0
$171$	−2.43843e8			−3.72927
$172$		0			0
$173$			7.71282e7i			1.13254i	0.824221	+	0.566268i	$0.191613\pi$
$173$			7.71282e7i			1.13254i	−0.824221	+	0.566268i	$0.808387\pi$
$174$		0			0
$175$	1.29839e7	−	6.48015e6i	0.183136	−	0.0914012i
$176$		0			0
$177$		−	1.12925e7i		−	0.153068i
$178$		0			0
$179$	9.92044e6			0.129284			0.0646421	−	0.997909i	$-0.479409\pi$
$179$	9.92044e6			0.129284			0.0646421	+	0.997909i	$0.479409\pi$
$180$		0			0
$181$	−5.60330e7			−0.702374			−0.351187	−	0.936305i	$-0.614222\pi$
$181$	−5.60330e7			−0.702374			−0.351187	+	0.936305i	$0.614222\pi$
$182$		0			0
$183$			1.16678e8i			1.40737i
$184$		0			0
$185$	−2.19657e7	+	3.55124e7i	−0.255062	+	0.412362i
$186$		0			0
$187$		−	4.44082e7i		−	0.496612i
$188$		0			0
$189$	−4.08451e7			−0.440072
$190$		0			0
$191$	−8.36160e7			−0.868305			−0.434153	−	0.900839i	$-0.642952\pi$
$191$	−8.36160e7			−0.868305			−0.434153	+	0.900839i	$0.642952\pi$
$192$		0			0
$193$			1.84081e8i			1.84314i	0.388216	+	0.921569i	$0.373092\pi$
$193$			1.84081e8i			1.84314i	−0.388216	+	0.921569i	$0.626908\pi$
$194$		0			0
$195$	1.21233e8	+	7.49873e7i	1.17085	+	0.724213i
$196$		0			0
$197$		−	3.32048e7i		−	0.309435i	−0.987959	−	0.154717i	$-0.950553\pi$
$197$		−	3.32048e7i		−	0.309435i	0.987959	−	0.154717i	$-0.0494467\pi$
$198$		0			0
$199$	1.57568e8			1.41737			0.708683	−	0.705527i	$-0.249290\pi$
$199$	1.57568e8			1.41737			0.708683	+	0.705527i	$0.249290\pi$
$200$		0			0
$201$	2.18200e8			1.89526
$202$		0			0
$203$		−	1.89322e7i		−	0.158842i
$204$		0			0
$205$	8.89945e7	+	5.50465e7i	0.721481	+	0.446263i
$206$		0			0
$207$		−	5.50942e7i		−	0.431727i
$208$		0			0
$209$	1.80347e8			1.36646
$210$		0			0
$211$	−1.86392e8			−1.36597			−0.682983	−	0.730435i	$-0.739318\pi$
$211$	−1.86392e8			−1.36597			−0.682983	+	0.730435i	$0.739318\pi$
$212$		0			0
$213$			4.11631e8i			2.91863i
$214$		0			0
$215$	2.56169e7	−	4.14152e7i	0.175789	−	0.284201i
$216$		0			0
$217$		−	5.39442e6i		−	0.0358373i
$218$		0			0
$219$	−9.96566e6			−0.0641138
$220$		0			0
$221$	7.60038e7			0.473655
$222$		0			0
$223$		−	2.57497e8i		−	1.55491i	−0.628938	−	0.777456i	$-0.716510\pi$
$223$		−	2.57497e8i		−	1.55491i	0.628938	−	0.777456i	$-0.283490\pi$
$224$		0			0
$225$	−1.67982e8	−	3.36577e8i	−0.983160	−	1.96991i
$226$		0			0
$227$			1.10040e8i			0.624398i	0.950017	+	0.312199i	$0.101066\pi$
$227$			1.10040e8i			0.624398i	−0.950017	+	0.312199i	$0.898934\pi$
$228$		0			0
$229$	4.69036e7			0.258096			0.129048	−	0.991638i	$-0.458808\pi$
$229$	4.69036e7			0.258096			0.129048	+	0.991638i	$0.458808\pi$
$230$		0			0
$231$	5.53496e7			0.295442
$232$		0			0
$233$			1.60419e8i			0.830824i	0.909633	+	0.415412i	$0.136363\pi$
$233$			1.60419e8i			0.830824i	−0.909633	+	0.415412i	$0.863637\pi$
$234$		0			0
$235$	6.29597e7	−	1.01788e8i	0.316464	−	0.511633i
$236$		0			0
$237$		−	3.93774e8i		−	1.92144i
$238$		0			0
$239$	−2.51647e8			−1.19234			−0.596169	−	0.802859i	$-0.703311\pi$
$239$	−2.51647e8			−1.19234			−0.596169	+	0.802859i	$0.703311\pi$
$240$		0			0
$241$	2.30623e8			1.06131			0.530655	−	0.847588i	$-0.321946\pi$
$241$	2.30623e8			1.06131			0.530655	+	0.847588i	$0.321946\pi$
$242$		0			0
$243$			1.77660e8i			0.794267i
$244$		0			0
$245$	−1.87564e8	−	1.16015e8i	−0.814831	−	0.504004i
$246$		0			0
$247$			3.08661e8i			1.30329i
$248$		0			0
$249$	−7.69725e8			−3.15964
$250$		0			0
$251$	−2.64559e8			−1.05600			−0.528002	−	0.849243i	$-0.677058\pi$
$251$	−2.64559e8			−1.05600			−0.528002	+	0.849243i	$0.677058\pi$
$252$		0			0
$253$			4.07479e7i			0.158192i
$254$		0			0
$255$	−2.48045e8	−	1.53425e8i	−0.936783	−	0.579436i
$256$		0			0
$257$			1.54580e8i			0.568050i	0.958817	+	0.284025i	$0.0916699\pi$
$257$			1.54580e8i			0.568050i	−0.958817	+	0.284025i	$0.908330\pi$
$258$		0			0
$259$	−2.77489e7			−0.0992421
$260$		0			0
$261$	−4.90771e8			−1.70859
$262$		0			0
$263$			4.30058e8i			1.45775i	0.684649	+	0.728873i	$0.259955\pi$
$263$			4.30058e8i			1.45775i	−0.684649	+	0.728873i	$0.740045\pi$
$264$		0			0
$265$	2.51864e8	−	4.07192e8i	0.831390	−	1.34412i
$266$		0			0
$267$		−	5.38414e8i		−	1.73112i
$268$		0			0
$269$	1.73129e6			0.00542297			0.00271149	−	0.999996i	$-0.499137\pi$
$269$	1.73129e6			0.00542297			0.00271149	+	0.999996i	$0.499137\pi$
$270$		0			0
$271$	2.91507e8			0.889727			0.444864	−	0.895598i	$-0.353252\pi$
$271$	2.91507e8			0.889727			0.444864	+	0.895598i	$0.353252\pi$
$272$		0			0
$273$			9.47298e7i			0.281785i
$274$		0			0
$275$	1.24240e8	+	2.48934e8i	0.360245	+	0.721805i
$276$		0			0
$277$		−	2.01058e8i		−	0.568383i	−0.958767	−	0.284192i	$-0.908275\pi$
$277$		−	2.01058e8i		−	0.568383i	0.958767	−	0.284192i	$-0.0917252\pi$
$278$		0			0
$279$	−1.39837e8			−0.385485
$280$		0			0
$281$	−3.53690e8			−0.950935			−0.475468	−	0.879733i	$-0.657721\pi$
$281$	−3.53690e8			−0.950935			−0.475468	+	0.879733i	$0.657721\pi$
$282$		0			0
$283$		−	9.83086e7i		−	0.257833i	−0.991655	−	0.128917i	$-0.958850\pi$
$283$		−	9.83086e7i		−	0.257833i	0.991655	−	0.128917i	$-0.0411500\pi$
$284$		0			0
$285$	6.23078e8	−	1.00734e9i	1.59436	−	2.57762i
$286$		0			0
$287$			6.95390e7i			0.173637i
$288$		0			0
$289$	2.54834e8			0.621033
$290$		0			0
$291$	1.06258e9			2.52777
$292$		0			0
$293$			2.63252e8i			0.611414i	0.952126	+	0.305707i	$0.0988928\pi$
$293$			2.63252e8i			0.611414i	−0.952126	+	0.305707i	$0.901107\pi$
$294$		0			0
$295$	3.20796e7	+	1.98425e7i	0.0727532	+	0.0450006i
$296$		0			0
$297$		−	7.83100e8i		−	1.73448i
$298$		0			0
$299$	−6.97393e7			−0.150879
$300$		0			0
$301$	3.23613e7			0.0683979
$302$		0			0
$303$		−	5.58004e8i		−	1.15236i
$304$		0			0
$305$	−3.31457e8	−	2.05018e8i	−0.668925	−	0.413755i
$306$		0			0
$307$			4.04138e8i			0.797160i	0.917133	+	0.398580i	$0.130497\pi$
$307$			4.04138e8i			0.797160i	−0.917133	+	0.398580i	$0.869503\pi$
$308$		0			0
$309$	1.76115e8			0.339581
$310$		0			0
$311$	5.84260e8			1.10140			0.550700	−	0.834703i	$-0.314361\pi$
$311$	5.84260e8			1.10140			0.550700	+	0.834703i	$0.314361\pi$
$312$		0			0
$313$			8.58721e8i			1.58288i	0.611250	+	0.791438i	$0.290667\pi$
$313$			8.58721e8i			1.58288i	−0.611250	+	0.791438i	$0.709333\pi$
$314$		0			0
$315$	1.31498e8	−	2.12595e8i	0.237046	−	0.383236i
$316$		0			0
$317$			1.01493e8i			0.178950i	0.995989	+	0.0894748i	$0.0285188\pi$
$317$			1.01493e8i			0.178950i	−0.995989	+	0.0894748i	$0.971481\pi$
$318$		0			0
$319$	3.62976e8			0.626052
$320$		0			0
$321$	−6.22218e8			−1.04996
$322$		0			0
$323$		−	6.31524e8i		−	1.04275i
$324$		0			0
$325$	−4.26046e8	+	2.12635e8i	−0.688438	+	0.343592i
$326$		0			0
$327$		−	7.63255e8i		−	1.20713i
$328$		0			0
$329$	7.95356e7			0.123133
$330$		0			0
$331$	5.45528e8			0.826835			0.413418	−	0.910542i	$-0.364335\pi$
$331$	5.45528e8			0.826835			0.413418	+	0.910542i	$0.364335\pi$
$332$		0			0
$333$			7.19321e8i			1.06750i
$334$		0			0
$335$	−3.83408e8	+	6.19861e8i	−0.557191	+	0.900819i
$336$		0			0
$337$		−	6.98565e8i		−	0.994266i	−0.867674	−	0.497133i	$-0.834386\pi$
$337$		−	6.98565e8i		−	0.994266i	0.867674	−	0.497133i	$-0.165614\pi$
$338$		0			0
$339$	2.45931e8			0.342858
$340$		0			0
$341$	1.03424e8			0.141248
$342$		0			0
$343$		−	2.99527e8i		−	0.400781i
$344$		0			0
$345$	2.27600e8	+	1.40779e8i	0.298404	+	0.184574i
$346$		0			0
$347$			5.05448e8i			0.649417i	0.945814	+	0.324708i	$0.105266\pi$
$347$			5.05448e8i			0.649417i	−0.945814	+	0.324708i	$0.894734\pi$
$348$		0			0
$349$	−1.05055e9			−1.32290			−0.661449	−	0.749990i	$-0.730058\pi$
$349$	−1.05055e9			−1.32290			−0.661449	+	0.749990i	$0.730058\pi$
$350$		0			0
$351$	1.34026e9			1.65430
$352$		0			0
$353$			1.35289e9i			1.63701i	0.574501	+	0.818504i	$0.305196\pi$
$353$			1.35289e9i			1.63701i	−0.574501	+	0.818504i	$0.694804\pi$
$354$		0			0
$355$	−1.16936e9	−	7.23291e8i	−1.38723	−	0.858053i
$356$		0			0
$357$		−	1.93818e8i		−	0.225453i
$358$		0			0
$359$	−4.43730e8			−0.506161			−0.253080	−	0.967445i	$-0.581444\pi$
$359$	−4.43730e8			−0.506161			−0.253080	+	0.967445i	$0.581444\pi$
$360$		0			0
$361$	1.67083e9			1.86920
$362$		0			0
$363$		−	5.69454e8i		−	0.624864i
$364$		0			0
$365$	1.75110e7	−	2.83103e7i	0.0188489	−	0.0304733i
$366$		0			0
$367$		−	1.51050e9i		−	1.59510i	−0.603251	−	0.797551i	$-0.706128\pi$
$367$		−	1.51050e9i		−	1.59510i	0.603251	−	0.797551i	$-0.293872\pi$
$368$		0			0
$369$	1.80263e9			1.86773
$370$		0			0
$371$	3.18174e8			0.323486
$372$		0			0
$373$			1.64359e9i			1.63988i	0.572448	+	0.819941i	$0.305994\pi$
$373$			1.64359e9i			1.63988i	−0.572448	+	0.819941i	$0.694006\pi$
$374$		0			0
$375$	1.81967e9	+	1.66085e8i	1.78190	+	0.162638i
$376$		0			0
$377$			6.21227e8i			0.597112i
$378$		0			0
$379$	−3.65881e7			−0.0345225			−0.0172613	−	0.999851i	$-0.505495\pi$
$379$	−3.65881e7			−0.0345225			−0.0172613	+	0.999851i	$0.505495\pi$
$380$		0			0
$381$	1.47485e9			1.36619
$382$		0			0
$383$			5.68889e8i			0.517406i	0.965957	+	0.258703i	$0.0832952\pi$
$383$			5.68889e8i			0.517406i	−0.965957	+	0.258703i	$0.916705\pi$
$384$		0			0
$385$	−9.72567e7	+	1.57236e8i	−0.0868574	+	0.140424i
$386$		0			0
$387$		−	8.38887e8i		−	0.735724i
$388$		0			0
$389$	2.10183e9			1.81040			0.905199	−	0.424988i	$-0.139722\pi$
$389$	2.10183e9			1.81040			0.905199	+	0.424988i	$0.139722\pi$
$390$		0			0
$391$	1.42688e8			0.120717
$392$		0			0
$393$			3.30936e8i			0.275024i
$394$		0			0
$395$	1.11863e9	+	6.91914e8i	0.913263	+	0.564888i
$396$		0			0
$397$			1.62761e9i			1.30552i	0.757564	+	0.652760i	$0.226389\pi$
$397$			1.62761e9i			1.30552i	−0.757564	+	0.652760i	$0.773611\pi$
$398$		0			0
$399$	7.87120e8			0.620349
$400$		0			0
$401$	−6.33359e8			−0.490506			−0.245253	−	0.969459i	$-0.578871\pi$
$401$	−6.33359e8			−0.490506			−0.245253	+	0.969459i	$0.578871\pi$
$402$		0			0
$403$			1.77009e8i			0.134718i
$404$		0			0
$405$	−1.87089e9	−	1.15722e9i	−1.39945	−	0.865610i
$406$		0			0
$407$		−	5.32013e8i		−	0.391149i
$408$		0			0
$409$	−1.41470e9			−1.02243			−0.511213	−	0.859454i	$-0.670804\pi$
$409$	−1.41470e9			−1.02243			−0.511213	+	0.859454i	$0.670804\pi$
$410$		0			0
$411$	2.47575e9			1.75898
$412$		0			0
$413$			2.50665e7i			0.0175093i
$414$		0			0
$415$	1.35251e9	−	2.18663e9i	0.928908	−	1.50178i
$416$		0			0
$417$		−	3.56936e8i		−	0.241054i
$418$		0			0
$419$	8.14907e7			0.0541202			0.0270601	−	0.999634i	$-0.491385\pi$
$419$	8.14907e7			0.0541202			0.0270601	+	0.999634i	$0.491385\pi$
$420$		0			0
$421$	−2.06996e7			−0.0135199			−0.00675996	−	0.999977i	$-0.502152\pi$
$421$	−2.06996e7			−0.0135199			−0.00675996	+	0.999977i	$0.502152\pi$
$422$		0			0
$423$		−	2.06177e9i		−	1.32449i
$424$		0			0
$425$	8.71696e8	−	4.35054e8i	0.550812	−	0.274905i
$426$		0			0
$427$		−	2.58995e8i		−	0.160988i
$428$		0			0
$429$	−1.81620e9			−1.11062
$430$		0			0
$431$	−2.21145e9			−1.33047			−0.665237	−	0.746633i	$-0.731669\pi$
$431$	−2.21145e9			−1.33047			−0.665237	+	0.746633i	$0.731669\pi$
$432$		0			0
$433$		−	1.16437e9i		−	0.689261i	−0.938738	−	0.344630i	$-0.888004\pi$
$433$		−	1.16437e9i		−	0.689261i	0.938738	−	0.344630i	$-0.111996\pi$
$434$		0			0
$435$	1.25404e9	−	2.02743e9i	0.730464	−	1.18095i
$436$		0			0
$437$			5.79472e8i			0.332160i
$438$		0			0
$439$	7.24010e8			0.408431			0.204216	−	0.978926i	$-0.434536\pi$
$439$	7.24010e8			0.408431			0.204216	+	0.978926i	$0.434536\pi$
$440$		0			0
$441$	−3.79920e9			−2.10939
$442$		0			0
$443$			1.52679e9i			0.834385i	0.908818	+	0.417193i	$0.136986\pi$
$443$			1.52679e9i			0.834385i	−0.908818	+	0.417193i	$0.863014\pi$
$444$		0			0
$445$	1.52952e9	+	9.46066e8i	0.822802	+	0.508934i
$446$		0			0
$447$			5.93307e9i			3.14198i
$448$		0			0
$449$	2.71929e9			1.41773			0.708864	−	0.705345i	$-0.249208\pi$
$449$	2.71929e9			1.41773			0.708864	+	0.705345i	$0.249208\pi$
$450$		0			0
$451$	−1.33323e9			−0.684366
$452$		0			0
$453$		−	2.81823e9i		−	1.42440i
$454$		0			0
$455$	−2.69107e8	−	1.66453e8i	−0.133932	−	0.0828422i
$456$		0			0
$457$			2.20279e9i			1.07961i	0.841790	+	0.539805i	$0.181502\pi$
$457$			2.20279e9i			1.07961i	−0.841790	+	0.539805i	$0.818498\pi$
$458$		0			0
$459$	−2.74219e9			−1.32359
$460$		0			0
$461$	2.41318e9			1.14719			0.573597	−	0.819137i	$-0.305547\pi$
$461$	2.41318e9			1.14719			0.573597	+	0.819137i	$0.305547\pi$
$462$		0			0
$463$			3.44274e9i			1.61202i	0.591902	+	0.806010i	$0.298377\pi$
$463$			3.44274e9i			1.61202i	−0.591902	+	0.806010i	$0.701623\pi$
$464$		0			0
$465$	3.57318e8	−	5.77682e8i	0.164805	−	0.266443i
$466$		0			0
$467$		−	2.27849e9i		−	1.03523i	−0.855613	−	0.517616i	$-0.826820\pi$
$467$		−	2.27849e9i		−	1.03523i	0.855613	−	0.517616i	$-0.173180\pi$
$468$		0			0
$469$	−4.84350e8			−0.216798
$470$		0			0
$471$	−6.28898e9			−2.77337
$472$		0			0
$473$			6.20444e8i			0.269581i
$474$		0			0
$475$	1.76681e9	+	3.54007e9i	0.756418	+	1.51560i
$476$		0			0
$477$		−	8.24788e9i		−	3.47959i
$478$		0			0
$479$	−1.54719e8			−0.0643235			−0.0321618	−	0.999483i	$-0.510239\pi$
$479$	−1.54719e8			−0.0643235			−0.0321618	+	0.999483i	$0.510239\pi$
$480$		0			0
$481$	9.10531e8			0.373067
$482$		0			0
$483$			1.77843e8i			0.0718162i
$484$		0			0
$485$	−1.86711e9	+	3.01858e9i	−0.743144	+	1.20145i
$486$		0			0
$487$		−	2.63948e9i		−	1.03554i	−0.855520	−	0.517770i	$-0.826762\pi$
$487$		−	2.63948e9i		−	1.03554i	0.855520	−	0.517770i	$-0.173238\pi$
$488$		0			0
$489$	−7.11698e9			−2.75242
$490$		0			0
$491$	−1.72366e9			−0.657151			−0.328576	−	0.944478i	$-0.606569\pi$
$491$	−1.72366e9			−0.657151			−0.328576	+	0.944478i	$0.606569\pi$
$492$		0			0
$493$		−	1.27104e9i		−	0.477743i
$494$		0			0
$495$	4.07597e9	+	2.52114e9i	1.51047	+	0.934285i
$496$		0			0
$497$		−	9.13718e8i		−	0.333860i
$498$		0			0
$499$	−2.37910e9			−0.857158			−0.428579	−	0.903504i	$-0.640986\pi$
$499$	−2.37910e9			−0.857158			−0.428579	+	0.903504i	$0.640986\pi$
$500$		0			0
$501$	−4.27196e9			−1.51773
$502$		0			0
$503$		−	4.58013e8i		−	0.160468i	−0.996776	−	0.0802342i	$-0.974433\pi$
$503$		−	4.58013e8i		−	0.160468i	0.996776	−	0.0802342i	$-0.0255668\pi$
$504$		0			0
$505$	1.58517e9	+	9.80489e8i	0.547717	+	0.338784i
$506$		0			0
$507$			2.14225e9i			0.730033i
$508$		0			0
$509$	1.88963e9			0.635132			0.317566	−	0.948236i	$-0.397134\pi$
$509$	1.88963e9			0.635132			0.317566	+	0.948236i	$0.397134\pi$
$510$		0			0
$511$	2.21213e7			0.00733394
$512$		0			0
$513$		−	1.11364e10i		−	3.64195i
$514$		0			0
$515$	−3.09459e8	+	5.00307e8i	−0.0998338	+	0.161403i
$516$		0			0
$517$			1.52489e9i			0.485313i
$518$		0			0
$519$	−6.45390e9			−2.02645
$520$		0			0
$521$	−4.36991e9			−1.35376			−0.676878	−	0.736096i	$-0.736667\pi$
$521$	−4.36991e9			−1.35376			−0.676878	+	0.736096i	$0.736667\pi$
$522$		0			0
$523$		−	7.59971e8i		−	0.232296i	−0.993232	−	0.116148i	$-0.962945\pi$
$523$		−	7.59971e8i		−	0.232296i	0.993232	−	0.116148i	$-0.0370547\pi$
$524$		0			0
$525$	5.42244e8	+	1.08647e9i	0.163545	+	0.327687i
$526$		0			0
$527$		−	3.62162e8i		−	0.107787i
$528$		0			0
$529$	3.27390e9			0.961547
$530$		0			0
$531$	6.49789e8			0.188340
$532$		0			0
$533$		−	2.28180e9i		−	0.652729i
$534$		0			0
$535$	1.09332e9	−	1.76759e9i	0.308681	−	0.499049i
$536$		0			0
$537$			8.30119e8i			0.231329i
$538$		0			0
$539$	2.80990e9			0.772913
$540$		0			0
$541$	3.02681e9			0.821854			0.410927	−	0.911668i	$-0.365205\pi$
$541$	3.02681e9			0.821854			0.410927	+	0.911668i	$0.365205\pi$
$542$		0			0
$543$		−	4.68871e9i		−	1.25676i
$544$		0			0
$545$	2.16825e9	+	1.34114e9i	0.573748	+	0.354884i
$546$		0			0
$547$			1.76298e9i			0.460566i	0.973124	+	0.230283i	$0.0739651\pi$
$547$			1.76298e9i			0.460566i	−0.973124	+	0.230283i	$0.926035\pi$
$548$		0			0
$549$	−6.71382e9			−1.73168
$550$		0			0
$551$	5.16185e9			1.31454
$552$		0			0
$553$			8.74080e8i			0.219793i
$554$		0			0
$555$	−2.97159e9	−	1.83804e9i	−0.737843	−	0.456384i
$556$		0			0
$557$			6.96721e9i			1.70831i	0.520021	+	0.854153i	$0.325924\pi$
$557$			6.96721e9i			1.70831i	−0.520021	+	0.854153i	$0.674076\pi$
$558$		0			0
$559$	−1.06188e9			−0.257119
$560$		0			0
$561$	3.71597e9			0.888592
$562$		0			0
$563$		−	2.44613e9i		−	0.577698i	−0.957375	−	0.288849i	$-0.906728\pi$
$563$		−	2.44613e9i		−	0.577698i	0.957375	−	0.288849i	$-0.0932725\pi$
$564$		0			0
$565$	−4.32134e8	+	6.98638e8i	−0.100797	+	0.162961i
$566$		0			0
$567$		−	1.46189e9i		−	0.336801i
$568$		0			0
$569$	4.26784e9			0.971215			0.485607	−	0.874177i	$-0.338599\pi$
$569$	4.26784e9			0.971215			0.485607	+	0.874177i	$0.338599\pi$
$570$		0			0
$571$	3.15003e9			0.708090			0.354045	−	0.935228i	$-0.384806\pi$
$571$	3.15003e9			0.708090			0.354045	+	0.935228i	$0.384806\pi$
$572$		0			0
$573$		−	6.99679e9i		−	1.55367i
$574$		0			0
$575$	−7.99847e8	+	3.99195e8i	−0.175457	+	0.0875685i
$576$		0			0
$577$		−	1.59513e8i		−	0.0345685i	−0.999851	−	0.0172843i	$-0.994498\pi$
$577$		−	1.59513e8i		−	0.0345685i	0.999851	−	0.0172843i	$-0.00550202\pi$
$578$		0			0
$579$	−1.54034e10			−3.29794
$580$		0			0
$581$	1.70860e9			0.361429
$582$		0			0
$583$			6.10017e9i			1.27498i
$584$		0			0
$585$	−4.31489e9	+	6.97595e9i	−0.891095	+	1.44065i
$586$		0			0
$587$		−	3.17099e9i		−	0.647085i	−0.946214	−	0.323543i	$-0.895126\pi$
$587$		−	3.17099e9i		−	0.647085i	0.946214	−	0.323543i	$-0.104874\pi$
$588$		0			0
$589$	1.47078e9			0.296583
$590$		0			0
$591$	2.77850e9			0.553674
$592$		0			0
$593$			5.65837e9i			1.11429i	0.830414	+	0.557147i	$0.188104\pi$
$593$			5.65837e9i			1.11429i	−0.830414	+	0.557147i	$0.811896\pi$
$594$		0			0
$595$	5.50597e8	+	3.40565e8i	0.107158	+	0.0662813i
$596$		0			0
$597$			1.31849e10i			2.53610i
$598$		0			0
$599$	−3.67540e9			−0.698732			−0.349366	−	0.936986i	$-0.613603\pi$
$599$	−3.67540e9			−0.698732			−0.349366	+	0.936986i	$0.613603\pi$
$600$		0			0
$601$	5.94912e9			1.11787			0.558936	−	0.829211i	$-0.311210\pi$
$601$	5.94912e9			1.11787			0.558936	+	0.829211i	$0.311210\pi$
$602$		0			0
$603$			1.25556e10i			2.33199i
$604$		0			0
$605$	1.61770e9	+	1.00061e9i	0.296998	+	0.183704i
$606$		0			0
$607$			5.91556e9i			1.07358i	0.843715	+	0.536792i	$0.180364\pi$
$607$			5.91556e9i			1.07358i	−0.843715	+	0.536792i	$0.819636\pi$
$608$		0			0
$609$	1.58420e9			0.284217
$610$		0			0
$611$	−2.60982e9			−0.462879
$612$		0			0
$613$		−	2.91525e8i		−	0.0511169i	−0.999673	−	0.0255585i	$-0.991864\pi$
$613$		−	2.91525e8i		−	0.0511169i	0.999673	−	0.0255585i	$-0.00813639\pi$
$614$		0			0
$615$	−4.60616e9	+	7.44685e9i	−0.798502	+	1.29095i
$616$		0			0
$617$			6.88662e9i			1.18034i	0.807278	+	0.590171i	$0.200940\pi$
$617$			6.88662e9i			1.18034i	−0.807278	+	0.590171i	$0.799060\pi$
$618$		0			0
$619$	−5.67213e9			−0.961234			−0.480617	−	0.876931i	$-0.659587\pi$
$619$	−5.67213e9			−0.961234			−0.480617	+	0.876931i	$0.659587\pi$
$620$		0			0
$621$	2.51617e9			0.421619
$622$		0			0
$623$			1.19514e9i			0.198021i
$624$		0			0
$625$	−3.66922e9	+	4.87747e9i	−0.601165	+	0.799125i
$626$		0			0
$627$			1.50910e10i			2.44502i
$628$		0			0
$629$	−1.86296e9			−0.298487
$630$		0			0
$631$	1.36737e9			0.216662			0.108331	−	0.994115i	$-0.465449\pi$
$631$	1.36737e9			0.216662			0.108331	+	0.994115i	$0.465449\pi$
$632$		0			0
$633$		−	1.55969e10i		−	2.44413i
$634$		0			0
$635$	−2.59151e9	+	4.18974e9i	−0.401647	+	0.649350i
$636$		0			0
$637$			4.80910e9i			0.737184i
$638$		0			0
$639$	−2.36859e10			−3.59118
$640$		0			0
$641$	6.50671e9			0.975794			0.487897	−	0.872901i	$-0.337764\pi$
$641$	6.50671e9			0.975794			0.487897	+	0.872901i	$0.337764\pi$
$642$		0			0
$643$		−	1.00325e10i		−	1.48824i	−0.668048	−	0.744118i	$-0.732870\pi$
$643$		−	1.00325e10i		−	1.48824i	0.668048	−	0.744118i	$-0.267130\pi$
$644$		0			0
$645$	3.46553e9	+	2.14356e9i	0.508523	+	0.314541i
$646$		0			0
$647$		−	7.56024e9i		−	1.09741i	−0.836015	−	0.548707i	$-0.815120\pi$
$647$		−	7.56024e9i		−	1.09741i	0.836015	−	0.548707i	$-0.184880\pi$
$648$		0			0
$649$	−4.80587e8			−0.0690106
$650$		0			0
$651$	4.51392e8			0.0641240
$652$		0			0
$653$		−	7.10083e9i		−	0.997960i	−0.866613	−	0.498980i	$-0.833708\pi$
$653$		−	7.10083e9i		−	0.997960i	0.866613	−	0.498980i	$-0.166292\pi$
$654$		0			0
$655$	−9.40120e8	−	5.81499e8i	−0.130719	−	0.0808546i
$656$		0			0
$657$		−	5.73440e8i		−	0.0788877i
$658$		0			0
$659$	2.72098e8			0.0370362			0.0185181	−	0.999829i	$-0.494105\pi$
$659$	2.72098e8			0.0370362			0.0185181	+	0.999829i	$0.494105\pi$
$660$		0			0
$661$	−9.94289e8			−0.133908			−0.0669541	−	0.997756i	$-0.521328\pi$
$661$	−9.94289e8			−0.133908			−0.0669541	+	0.997756i	$0.521328\pi$
$662$		0			0
$663$			6.35982e9i			0.847515i
$664$		0			0
$665$	−1.38308e9	+	2.23604e9i	−0.182377	+	0.294852i
$666$		0			0
$667$			1.16628e9i			0.152181i
$668$		0			0
$669$	2.15468e10			2.78222
$670$		0			0
$671$	4.96557e9			0.634513
$672$		0			0
$673$			4.39633e9i			0.555953i	0.960588	+	0.277976i	$0.0896637\pi$
$673$			4.39633e9i			0.555953i	−0.960588	+	0.277976i	$0.910336\pi$
$674$		0			0
$675$	1.53716e10	−	7.67181e9i	1.92378	−	0.960140i
$676$		0			0
$677$		−	5.81617e9i		−	0.720406i	−0.932874	−	0.360203i	$-0.882708\pi$
$677$		−	5.81617e9i		−	0.720406i	0.932874	−	0.360203i	$-0.117292\pi$
$678$		0			0
$679$	−2.35867e9			−0.289150
$680$		0			0
$681$	−9.20792e9			−1.11724
$682$		0			0
$683$		−	1.01860e10i		−	1.22329i	−0.791130	−	0.611647i	$-0.790507\pi$
$683$		−	1.01860e10i		−	1.22329i	0.791130	−	0.611647i	$-0.209493\pi$
$684$		0			0
$685$	−4.35023e9	+	7.03309e9i	−0.517125	+	0.836044i
$686$		0			0
$687$			3.92478e9i			0.461814i
$688$		0			0
$689$	−1.04403e10			−1.21604
$690$		0			0
$691$	1.04954e10			1.21011			0.605056	−	0.796183i	$-0.293151\pi$
$691$	1.04954e10			1.21011			0.605056	+	0.796183i	$0.293151\pi$
$692$		0			0
$693$			3.18490e9i			0.363522i
$694$		0			0
$695$	1.01398e9	+	6.27185e8i	0.114573	+	0.0708678i
$696$		0			0
$697$			4.66860e9i			0.522242i
$698$		0			0
$699$	−1.34235e10			−1.48660
$700$		0			0
$701$	−1.44399e10			−1.58325			−0.791626	−	0.611006i	$-0.790765\pi$
$701$	−1.44399e10			−1.58325			−0.791626	+	0.611006i	$0.790765\pi$
$702$		0			0
$703$		−	7.56571e9i		−	0.821308i
$704$		0			0
$705$	8.51738e9	+	5.26832e9i	0.915469	+	0.566252i
$706$		0			0
$707$			1.23863e9i			0.131818i
$708$		0			0
$709$	−5.52527e8			−0.0582226			−0.0291113	−	0.999576i	$-0.509268\pi$
$709$	−5.52527e8			−0.0582226			−0.0291113	+	0.999576i	$0.509268\pi$
$710$		0			0
$711$	2.26584e10			2.36421
$712$		0			0
$713$			3.32311e8i			0.0343346i
$714$		0			0
$715$	3.19131e9	−	5.15945e9i	0.326511	−	0.527876i
$716$		0			0
$717$		−	2.10572e10i		−	2.13346i
$718$		0			0
$719$	3.15086e8			0.0316139			0.0158069	−	0.999875i	$-0.494968\pi$
$719$	3.15086e8			0.0316139			0.0158069	+	0.999875i	$0.494968\pi$
$720$		0			0
$721$	−3.90932e8			−0.0388444
$722$		0			0
$723$			1.92980e10i			1.89901i
$724$		0			0
$725$	3.55597e9	+	7.12492e9i	0.346558	+	0.694380i
$726$		0			0
$727$			2.96166e9i			0.285867i	0.989732	+	0.142934i	$0.0456535\pi$
$727$			2.96166e9i			0.285867i	−0.989732	+	0.142934i	$0.954346\pi$
$728$		0			0
$729$	2.34658e9			0.224330
$730$		0			0
$731$	2.17262e9			0.205718
$732$		0			0
$733$		−	2.04158e10i		−	1.91471i	−0.288914	−	0.957355i	$-0.593294\pi$
$733$		−	2.04158e10i		−	1.91471i	0.288914	−	0.957355i	$-0.406706\pi$
$734$		0			0
$735$	9.70788e9	−	1.56949e10i	0.901818	−	1.45798i
$736$		0			0
$737$		−	9.28618e9i		−	0.854478i
$738$		0			0
$739$	1.24593e10			1.13563			0.567817	−	0.823155i	$-0.307788\pi$
$739$	1.24593e10			1.13563			0.567817	+	0.823155i	$0.307788\pi$
$740$		0			0
$741$	−2.58280e10			−2.33199
$742$		0			0
$743$			8.88793e9i			0.794950i	0.917613	+	0.397475i	$0.130114\pi$
$743$			8.88793e9i			0.794950i	−0.917613	+	0.397475i	$0.869886\pi$
$744$		0			0
$745$	−1.68546e10	−	1.04252e10i	−1.49338	−	0.923714i
$746$		0			0
$747$		−	4.42912e10i		−	3.88773i
$748$		0			0
$749$	1.38117e9			0.120105
$750$		0			0
$751$	−7.69514e9			−0.662944			−0.331472	−	0.943465i	$-0.607545\pi$
$751$	−7.69514e9			−0.662944			−0.331472	+	0.943465i	$0.607545\pi$
$752$		0			0
$753$		−	2.21377e10i		−	1.88951i
$754$		0			0
$755$	8.00600e9	+	4.95201e9i	0.677019	+	0.418762i
$756$		0			0
$757$			1.31095e9i			0.109838i	0.998491	+	0.0549189i	$0.0174900\pi$
$757$			1.31095e9i			0.109838i	−0.998491	+	0.0549189i	$0.982510\pi$
$758$		0			0
$759$	−3.40969e9			−0.283053
$760$		0			0
$761$	−4.20324e9			−0.345731			−0.172865	−	0.984945i	$-0.555303\pi$
$761$	−4.20324e9			−0.345731			−0.172865	+	0.984945i	$0.555303\pi$
$762$		0			0
$763$			1.69424e9i			0.138082i
$764$		0			0
$765$	8.82832e9	−	1.42729e10i	0.712957	−	1.15265i
$766$		0			0
$767$		−	8.22516e8i		−	0.0658204i
$768$		0			0
$769$	−4.90370e9			−0.388850			−0.194425	−	0.980917i	$-0.562284\pi$
$769$	−4.90370e9			−0.388850			−0.194425	+	0.980917i	$0.562284\pi$
$770$		0			0
$771$	−1.29349e10			−1.01642
$772$		0			0
$773$			8.59620e9i			0.669389i	0.942327	+	0.334694i	$0.108633\pi$
$773$			8.59620e9i			0.669389i	−0.942327	+	0.334694i	$0.891367\pi$
$774$		0			0
$775$	1.01322e9	+	2.03013e9i	0.0781892	+	0.156664i
$776$		0			0
$777$		−	2.32196e9i		−	0.177575i
$778$		0			0
$779$	−1.89598e10			−1.43698
$780$		0			0
$781$	1.75182e10			1.31586
$782$		0			0
$783$		−	2.24137e10i		−	1.66858i
$784$		0			0
$785$	1.10506e10	−	1.78657e10i	0.815345	−	1.31818i
$786$		0			0
$787$			1.11113e10i			0.812555i	0.913750	+	0.406277i	$0.133173\pi$
$787$			1.11113e10i			0.812555i	−0.913750	+	0.406277i	$0.866827\pi$
$788$		0			0
$789$	−3.59862e10			−2.60836
$790$		0			0
$791$	−5.45906e8			−0.0392193
$792$		0			0
$793$			8.49849e9i			0.605181i
$794$		0			0
$795$	3.40729e10	+	2.10753e10i	2.40505	+	1.48761i
$796$		0			0
$797$			6.73976e9i			0.471564i	0.971806	+	0.235782i	$0.0757650\pi$
$797$			6.73976e9i			0.471564i	−0.971806	+	0.235782i	$0.924235\pi$
$798$		0			0
$799$	5.33974e9			0.370345
$800$		0			0
$801$	3.09812e10			2.13002
$802$		0			0
$803$			4.24119e8i			0.0289057i
$804$		0			0
$805$	−5.05215e8	−	3.12494e8i	−0.0341343	−	0.0211133i
$806$		0			0
$807$			1.44870e8i			0.00970336i
$808$		0			0
$809$	−2.01007e9			−0.133472			−0.0667362	−	0.997771i	$-0.521259\pi$
$809$	−2.01007e9			−0.133472			−0.0667362	+	0.997771i	$0.521259\pi$
$810$		0			0
$811$	−1.56045e10			−1.02725			−0.513626	−	0.858014i	$-0.671698\pi$
$811$	−1.56045e10			−1.02725			−0.513626	+	0.858014i	$0.671698\pi$
$812$		0			0
$813$			2.43926e10i			1.59200i
$814$		0			0
$815$	1.25055e10	−	2.02178e10i	0.809188	−	1.30823i
$816$		0			0
$817$			8.82328e9i			0.566047i
$818$		0			0
$819$	−5.45091e9			−0.346717
$820$		0			0
$821$	7.82354e9			0.493403			0.246702	−	0.969091i	$-0.420653\pi$
$821$	7.82354e9			0.493403			0.246702	+	0.969091i	$0.420653\pi$
$822$		0			0
$823$		−	9.42050e9i		−	0.589080i	−0.955639	−	0.294540i	$-0.904834\pi$
$823$		−	9.42050e9i		−	0.589080i	0.955639	−	0.294540i	$-0.0951665\pi$
$824$		0			0
$825$	−2.08302e10	+	1.03961e10i	−1.29153	+	0.644590i
$826$		0			0
$827$			3.29188e9i			0.202383i	0.994867	+	0.101192i	$0.0322655\pi$
$827$			3.29188e9i			0.202383i	−0.994867	+	0.101192i	$0.967734\pi$
$828$		0			0
$829$	1.72231e10			1.04995			0.524976	−	0.851117i	$-0.324074\pi$
$829$	1.72231e10			1.04995			0.524976	+	0.851117i	$0.324074\pi$
$830$		0			0
$831$	1.68240e10			1.01701
$832$		0			0
$833$		−	9.83948e9i		−	0.589813i
$834$		0			0
$835$	7.50641e9	−	1.21357e10i	0.446200	−	0.721379i
$836$		0			0
$837$		−	6.38642e9i		−	0.376459i
$838$		0			0
$839$	−9.34237e9			−0.546123			−0.273061	−	0.961997i	$-0.588036\pi$
$839$	−9.34237e9			−0.546123			−0.273061	+	0.961997i	$0.588036\pi$
$840$		0			0
$841$	−6.86087e9			−0.397735
$842$		0			0
$843$		−	2.95960e10i		−	1.70152i
$844$		0			0
$845$	−6.08569e9	−	3.76423e9i	−0.346985	−	0.214623i
$846$		0			0
$847$			1.26405e9i			0.0714777i
$848$		0			0
$849$	8.22623e9			0.461343
$850$		0			0
$851$	1.70941e9			0.0950806
$852$		0			0
$853$			1.68999e9i			0.0932312i	0.998913	+	0.0466156i	$0.0148436\pi$
$853$			1.68999e9i			0.0932312i	−0.998913	+	0.0466156i	$0.985156\pi$
$854$		0			0
$855$	5.79640e10	+	3.58529e10i	3.17159	+	1.96175i
$856$		0			0
$857$			3.51222e9i			0.190611i	0.995448	+	0.0953057i	$0.0303829\pi$
$857$			3.51222e9i			0.190611i	−0.995448	+	0.0953057i	$0.969617\pi$
$858$		0			0
$859$	2.67283e10			1.43878			0.719392	−	0.694604i	$-0.244421\pi$
$859$	2.67283e10			1.43878			0.719392	+	0.694604i	$0.244421\pi$
$860$		0			0
$861$	−5.81886e9			−0.310690
$862$		0			0
$863$			7.78348e9i			0.412227i	0.978528	+	0.206113i	$0.0660815\pi$
$863$			7.78348e9i			0.412227i	−0.978528	+	0.206113i	$0.933918\pi$
$864$		0			0
$865$	1.13404e10	−	1.83342e10i	0.595760	−	0.963175i
$866$		0			0
$867$			2.13239e10i			1.11122i
$868$		0			0
$869$	−1.67582e10			−0.866282
$870$		0			0
$871$	1.58931e10			0.814978
$872$		0			0
$873$			6.11429e10i			3.11025i
$874$		0			0
$875$	−4.03922e9	−	3.68667e8i	−0.203830	−	0.0186040i
$876$		0			0
$877$		−	9.46355e9i		−	0.473757i	−0.971539	−	0.236878i	$-0.923876\pi$
$877$		−	9.46355e9i		−	0.473757i	0.971539	−	0.236878i	$-0.0761243\pi$
$878$		0			0
$879$	−2.20283e10			−1.09401
$880$		0			0
$881$	2.29088e10			1.12872			0.564360	−	0.825529i	$-0.309123\pi$
$881$	2.29088e10			1.12872			0.564360	+	0.825529i	$0.309123\pi$
$882$		0			0
$883$		−	1.00814e10i		−	0.492784i	−0.969170	−	0.246392i	$-0.920755\pi$
$883$		−	1.00814e10i		−	0.492784i	0.969170	−	0.246392i	$-0.0792451\pi$
$884$		0			0
$885$	−1.66037e9	+	2.68435e9i	−0.0805199	+	0.130178i
$886$		0			0
$887$		−	2.86392e10i		−	1.37793i	−0.724792	−	0.688967i	$-0.758064\pi$
$887$		−	2.86392e10i		−	1.37793i	0.724792	−	0.688967i	$-0.241936\pi$
$888$		0			0
$889$	−3.27380e9			−0.156277
$890$		0			0
$891$	2.80280e10			1.32745
$892$		0			0
$893$			2.16853e10i			1.01903i
$894$		0			0
$895$	−2.35819e9	−	1.45863e9i	−0.109951	−	0.0680088i
$896$		0			0
$897$		−	5.83562e9i		−	0.269969i
$898$		0			0
$899$	2.96018e9			0.135881
$900$		0			0
$901$	2.13611e10			0.972940
$902$		0			0
$903$			2.70791e9i			0.122385i
$904$		0			0
$905$	1.33196e10	+	8.23869e9i	0.597341	+	0.369478i
$906$		0			0
$907$		−	3.43088e10i		−	1.52679i	−0.645930	−	0.763397i	$-0.723530\pi$
$907$		−	3.43088e10i		−	1.52679i	0.645930	−	0.763397i	$-0.276470\pi$
$908$		0			0
$909$	3.21085e10			1.41790
$910$		0			0
$911$	−1.71466e10			−0.751385			−0.375693	−	0.926744i	$-0.622595\pi$
$911$	−1.71466e10			−0.751385			−0.375693	+	0.926744i	$0.622595\pi$
$912$		0			0
$913$			3.27580e10i			1.42452i
$914$		0			0
$915$	1.71554e10	−	2.77355e10i	0.740335	−	1.19691i
$916$		0			0
$917$		−	7.34596e8i		−	0.0314598i
$918$		0			0
$919$	−1.75219e10			−0.744692			−0.372346	−	0.928094i	$-0.621447\pi$
$919$	−1.75219e10			−0.744692			−0.372346	+	0.928094i	$0.621447\pi$
$920$		0			0
$921$	−3.38173e10			−1.42637
$922$		0			0
$923$			2.99821e10i			1.25504i
$924$		0			0
$925$	1.04430e10	−	5.21198e9i	0.433839	−	0.216524i
$926$		0			0
$927$			1.01340e10i			0.417831i
$928$		0			0
$929$	−1.78228e10			−0.729324			−0.364662	−	0.931140i	$-0.618816\pi$
$929$	−1.78228e10			−0.729324			−0.364662	+	0.931140i	$0.618816\pi$
$930$		0			0
$931$	3.99594e10			1.62291
$932$		0			0
$933$			4.88895e10i			1.97074i
$934$		0			0
$935$	−6.52947e9	+	1.05563e10i	−0.261238	+	0.422348i
$936$		0			0
$937$		−	3.74390e10i		−	1.48674i	−0.668879	−	0.743371i	$-0.733226\pi$
$937$		−	3.74390e10i		−	1.48674i	0.668879	−	0.743371i	$-0.266774\pi$
$938$		0			0
$939$	−7.18557e10			−2.83225
$940$		0			0
$941$	5.59371e9			0.218845			0.109422	−	0.993995i	$-0.465100\pi$
$941$	5.59371e9			0.218845			0.109422	+	0.993995i	$0.465100\pi$
$942$		0			0
$943$		−	4.28380e9i		−	0.166356i
$944$		0			0
$945$	9.70931e9	+	6.00557e9i	0.374263	+	0.231496i
$946$		0			0
$947$			4.72461e10i			1.80776i	0.427785	+	0.903880i	$0.359294\pi$
$947$			4.72461e10i			1.80776i	−0.427785	+	0.903880i	$0.640706\pi$
$948$		0			0
$949$	−7.25872e8			−0.0275695
$950$		0			0
$951$	−8.49273e9			−0.320196
$952$		0			0
$953$			1.44402e10i			0.540442i	0.962798	+	0.270221i	$0.0870967\pi$
$953$			1.44402e10i			0.540442i	−0.962798	+	0.270221i	$0.912903\pi$
$954$		0			0
$955$	1.98764e10	+	1.22943e10i	0.738458	+	0.456764i
$956$		0			0
$957$			3.03730e10i			1.12020i
$958$		0			0
$959$	−5.49555e9			−0.201209
$960$		0			0
$961$	−2.66692e10			−0.969343
$962$		0			0
$963$		−	3.58034e10i		−	1.29191i
$964$		0			0
$965$	2.70659e10	−	4.37579e10i	0.969566	−	1.56751i
$966$		0			0
$967$		−	3.52615e10i		−	1.25403i	−0.779008	−	0.627014i	$-0.784277\pi$
$967$		−	3.52615e10i		−	1.25403i	0.779008	−	0.627014i	$-0.215723\pi$
$968$		0			0
$969$	5.28445e10			1.86580
$970$		0			0
$971$	6.30626e9			0.221057			0.110528	−	0.993873i	$-0.464746\pi$
$971$	6.30626e9			0.221057			0.110528	+	0.993873i	$0.464746\pi$
$972$		0			0
$973$			7.92310e8i			0.0275740i
$974$		0			0
$975$	−1.77928e10	−	3.56505e10i	−0.614792	−	1.23183i
$976$		0			0
$977$		−	2.25969e10i		−	0.775206i	−0.921826	−	0.387603i	$-0.873303\pi$
$977$		−	2.25969e10i		−	0.775206i	0.921826	−	0.387603i	$-0.126697\pi$
$978$		0			0
$979$	−2.29138e10			−0.780474
$980$		0			0
$981$	4.39189e10			1.48529
$982$		0			0
$983$			1.48266e10i			0.497855i	0.968522	+	0.248928i	$0.0800781\pi$
$983$			1.48266e10i			0.497855i	−0.968522	+	0.248928i	$0.919922\pi$
$984$		0			0
$985$	−4.88220e9	+	7.89313e9i	−0.162775	+	0.263161i
$986$		0			0
$987$			6.65535e9i			0.220324i
$988$		0			0
$989$	−1.99354e9			−0.0655298
$990$		0			0
$991$	4.13851e10			1.35079			0.675393	−	0.737458i	$-0.263974\pi$
$991$	4.13851e10			1.35079			0.675393	+	0.737458i	$0.263974\pi$
$992$		0			0
$993$			4.56485e10i			1.47946i
$994$		0			0
$995$	−3.74556e10	−	2.31677e10i	−1.20541	−	0.745593i
$996$		0			0
$997$			4.69228e9i			0.149952i	0.997185	+	0.0749758i	$0.0238879\pi$
$997$			4.69228e9i			0.149952i	−0.997185	+	0.0749758i	$0.976112\pi$
$998$		0			0
$999$	−3.28517e10			−1.04251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.8.c.g.129.4		4
4.3	odd	2		320.8.c.f.129.1		4
5.4	even	2	inner	320.8.c.g.129.1		4
8.3	odd	2		10.8.b.a.9.2	✓	4
8.5	even	2		80.8.c.d.49.1		4
20.19	odd	2		320.8.c.f.129.4		4
24.11	even	2		90.8.c.c.19.3		4
40.3	even	4		50.8.a.i.1.1		2
40.13	odd	4		400.8.a.bf.1.2		2
40.19	odd	2		10.8.b.a.9.3	yes	4
40.27	even	4		50.8.a.j.1.2		2
40.29	even	2		80.8.c.d.49.4		4
40.37	odd	4		400.8.a.t.1.1		2
120.59	even	2		90.8.c.c.19.1		4
120.83	odd	4		450.8.a.bi.1.2		2
120.107	odd	4		450.8.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.8.b.a.9.2	✓	4	8.3	odd	2
10.8.b.a.9.3	yes	4	40.19	odd	2
50.8.a.i.1.1		2	40.3	even	4
50.8.a.j.1.2		2	40.27	even	4
80.8.c.d.49.1		4	8.5	even	2
80.8.c.d.49.4		4	40.29	even	2
90.8.c.c.19.1		4	120.59	even	2
90.8.c.c.19.3		4	24.11	even	2
320.8.c.f.129.1		4	4.3	odd	2
320.8.c.f.129.4		4	20.19	odd	2
320.8.c.g.129.1		4	5.4	even	2	inner
320.8.c.g.129.4		4	1.1	even	1	trivial
400.8.a.t.1.1		2	40.37	odd	4
400.8.a.bf.1.2		2	40.13	odd	4
450.8.a.bd.1.1		2	120.107	odd	4
450.8.a.bi.1.2		2	120.83	odd	4

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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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	320.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+83.6776i q^{3} +(-237.711 - 147.033i) q^{5} -185.744i q^{7} -4814.95 q^{9} +3561.16 q^{11} +6094.86i q^{13} +(12303.4 - 19891.1i) q^{15} -12470.2i q^{17} +50642.8 q^{19} +15542.6 q^{21} +11442.3i q^{23} +(34887.6 + 69902.6i) q^{25} -219900. i q^{27} +101926. q^{29} +29042.3 q^{31} +297989. i q^{33} +(-27310.4 + 44153.2i) q^{35} -149393. i q^{37} -510003. q^{39} -374382. q^{41} +174226. i q^{43} +(1.14456e6 + 707956. i) q^{45} +428201. i q^{47} +789042. q^{49} +1.04347e6 q^{51} +1.71297e6i q^{53} +(-846525. - 523607. i) q^{55} +4.23767e6i q^{57} -134952. q^{59} +1.39437e6 q^{61} +894345. i q^{63} +(896145. - 1.44881e6i) q^{65} -2.60763e6i q^{67} -957466. q^{69} +4.91925e6 q^{71} +119096. i q^{73} +(-5.84928e6 + 2.91932e6i) q^{75} -661462. i q^{77} -4.70584e6 q^{79} +7.87046e6 q^{81} +9.19870e6i q^{83} +(-1.83352e6 + 2.96429e6i) q^{85} +8.52897e6i q^{87} -6.43438e6 q^{89} +1.13208e6 q^{91} +2.43019e6i q^{93} +(-1.20383e7 - 7.44617e6i) q^{95} -1.26986e7i q^{97} -1.71468e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 60 q^{5} - 6788 q^{9} + 17808 q^{11} + 34960 q^{15} + 63600 q^{19} + 63952 q^{21} + 86100 q^{25} - 169560 q^{29} + 394112 q^{31} - 276720 q^{35} - 1163424 q^{39} + 232488 q^{41} + 2879420 q^{45} + 2520108 q^{49}+ \cdots - 19109776 q^{99}+O(q^{100}) \) 4 * q - 60 * q^5 - 6788 * q^9 + 17808 * q^11 + 34960 * q^15 + 63600 * q^19 + 63952 * q^21 + 86100 * q^25 - 169560 * q^29 + 394112 * q^31 - 276720 * q^35 - 1163424 * q^39 + 232488 * q^41 + 2879420 * q^45 + 2520108 * q^49 + 361088 * q^51 + 526480 * q^55 + 2093520 * q^59 + 5251432 * q^61 + 2761440 * q^65 - 6514864 * q^69 + 7832352 * q^71 - 7397600 * q^75 - 7727040 * q^79 + 16428404 * q^81 + 7995520 * q^85 - 33470040 * q^89 + 5340768 * q^91 - 31904400 * q^95 - 19109776 * q^99

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