LMFDB - Embedded newform 208.10.a.m.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,10,Mod(1,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-141] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$107.127453922$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3 x^{7} - 135356 x^{6} - 24398 x^{5} + 5213582205 x^{4} + 598076469 x^{3} + \cdots + 15\!\cdots\!64 $ x^8 - 3x^7 - 135356x^6 - 24398x^5 + 5213582205x^4 + 598076469x^3 - 44131402568826x^2 + 60107523849756*x + 1558753465556664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$270.514$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+252.514 q^{3} +1395.15 q^{5} +11230.7 q^{7} +44080.4 q^{9} +11945.0 q^{11} +28561.0 q^{13} +352296. q^{15} -357016. q^{17} +812674. q^{19} +2.83590e6 q^{21} +875590. q^{23} -6671.05 q^{25} +6.16067e6 q^{27} -3.81480e6 q^{29} -1.70447e6 q^{31} +3.01628e6 q^{33} +1.56685e7 q^{35} -2.01332e7 q^{37} +7.21205e6 q^{39} +7.51492e6 q^{41} +8.77900e6 q^{43} +6.14989e7 q^{45} -5.56456e7 q^{47} +8.57741e7 q^{49} -9.01516e7 q^{51} +1.84201e7 q^{53} +1.66651e7 q^{55} +2.05212e8 q^{57} -9.16484e7 q^{59} -4.97163e7 q^{61} +4.95051e8 q^{63} +3.98470e7 q^{65} -1.84401e8 q^{67} +2.21099e8 q^{69} +1.75076e8 q^{71} -3.00349e8 q^{73} -1.68453e6 q^{75} +1.34150e8 q^{77} -5.38224e8 q^{79} +6.88023e8 q^{81} -3.07653e8 q^{83} -4.98092e8 q^{85} -9.63291e8 q^{87} +4.56635e8 q^{89} +3.20759e8 q^{91} -4.30403e8 q^{93} +1.13381e9 q^{95} -1.45156e9 q^{97} +5.26539e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 141 q^{3} + 2051 q^{5} + 2417 q^{7} + 115741 q^{9} + 53118 q^{11} + 228488 q^{13} + 464555 q^{15} + 433095 q^{17} + 434954 q^{19} + 906875 q^{21} + 1124296 q^{23} + 5966065 q^{25} - 7820643 q^{27}+ \cdots + 641626736 q^{99}+O(q^{100}) $ 8 * q - 141 * q^3 + 2051 * q^5 + 2417 * q^7 + 115741 * q^9 + 53118 * q^11 + 228488 * q^13 + 464555 * q^15 + 433095 * q^17 + 434954 * q^19 + 906875 * q^21 + 1124296 * q^23 + 5966065 * q^25 - 7820643 * q^27 - 1913432 * q^29 - 5220148 * q^31 - 1626882 * q^33 - 950707 * q^35 - 419409 * q^37 - 4027101 * q^39 + 12148046 * q^41 - 32143187 * q^43 + 18893544 * q^45 - 57898411 * q^47 + 104302637 * q^49 - 198521477 * q^51 + 273106866 * q^53 - 141214662 * q^55 + 253032058 * q^57 - 50734502 * q^59 + 139520018 * q^61 + 20344158 * q^63 + 58578611 * q^65 - 528144786 * q^67 + 483506576 * q^69 - 334814973 * q^71 + 630864344 * q^73 - 2265796264 * q^75 + 939664146 * q^77 - 865037608 * q^79 + 3194281040 * q^81 - 1520229952 * q^83 + 2130158385 * q^85 - 1167867006 * q^87 + 894700240 * q^89 + 69031937 * q^91 + 3803209548 * q^93 + 2695926542 * q^95 + 980232164 * q^97 + 641626736 * q^99

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
$3$	252.514	1.79986	0.899932	−	0.436029i	$-0.143616\pi$
$3$	252.514	1.79986	0.899932	+	0.436029i	$0.143616\pi$
$4$	0	0
$5$	1395.15	0.998291	0.499145	−	0.866518i	$-0.333647\pi$
$5$	1395.15	0.998291	0.499145	+	0.866518i	$0.333647\pi$
$6$	0	0
$7$	11230.7	1.76793	0.883963	−	0.467557i	$-0.154866\pi$
$7$	11230.7	1.76793	0.883963	+	0.467557i	$0.154866\pi$
$8$	0	0
$9$	44080.4	2.23951
$10$	0	0
$11$	11945.0	0.245991	0.122995	−	0.992407i	$-0.460750\pi$
$11$	11945.0	0.245991	0.122995	+	0.992407i	$0.460750\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	352296.	1.79679
$16$	0	0
$17$	−357016.	−1.03673	−0.518367	−	0.855158i	$-0.673460\pi$
$17$	−357016.	−1.03673	−0.518367	+	0.855158i	$0.673460\pi$
$18$	0	0
$19$	812674.	1.43062	0.715312	−	0.698806i	$-0.246285\pi$
$19$	812674.	1.43062	0.715312	+	0.698806i	$0.246285\pi$
$20$	0	0
$21$	2.83590e6	3.18203
$22$	0	0
$23$	875590.	0.652417	0.326209	−	0.945298i	$-0.394229\pi$
$23$	875590.	0.652417	0.326209	+	0.945298i	$0.394229\pi$
$24$	0	0
$25$	−6671.05	−0.00341558
$26$	0	0
$27$	6.16067e6	2.23096
$28$	0	0
$29$	−3.81480e6	−1.00157	−0.500785	−	0.865572i	$-0.666955\pi$
$29$	−3.81480e6	−1.00157	−0.500785	+	0.865572i	$0.666955\pi$
$30$	0	0
$31$	−1.70447e6	−0.331484	−0.165742	−	0.986169i	$-0.553002\pi$
$31$	−1.70447e6	−0.331484	−0.165742	+	0.986169i	$0.553002\pi$
$32$	0	0
$33$	3.01628e6	0.442750
$34$	0	0
$35$	1.56685e7	1.76490
$36$	0	0
$37$	−2.01332e7	−1.76606	−0.883030	−	0.469317i	$-0.844500\pi$
$37$	−2.01332e7	−1.76606	−0.883030	+	0.469317i	$0.844500\pi$
$38$	0	0
$39$	7.21205e6	0.499193
$40$	0	0
$41$	7.51492e6	0.415333	0.207667	−	0.978200i	$-0.433413\pi$
$41$	7.51492e6	0.415333	0.207667	+	0.978200i	$0.433413\pi$
$42$	0	0
$43$	8.77900e6	0.391595	0.195798	−	0.980644i	$-0.437270\pi$
$43$	8.77900e6	0.391595	0.195798	+	0.980644i	$0.437270\pi$
$44$	0	0
$45$	6.14989e7	2.23569
$46$	0	0
$47$	−5.56456e7	−1.66338	−0.831689	−	0.555242i	$-0.812625\pi$
$47$	−5.56456e7	−1.66338	−0.831689	+	0.555242i	$0.812625\pi$
$48$	0	0
$49$	8.57741e7	2.12556
$50$	0	0
$51$	−9.01516e7	−1.86598
$52$	0	0
$53$	1.84201e7	0.320664	0.160332	−	0.987063i	$-0.448744\pi$
$53$	1.84201e7	0.320664	0.160332	+	0.987063i	$0.448744\pi$
$54$	0	0
$55$	1.66651e7	0.245570
$56$	0	0
$57$	2.05212e8	2.57493
$58$	0	0
$59$	−9.16484e7	−0.984670	−0.492335	−	0.870406i	$-0.663857\pi$
$59$	−9.16484e7	−0.984670	−0.492335	+	0.870406i	$0.663857\pi$
$60$	0	0
$61$	−4.97163e7	−0.459742	−0.229871	−	0.973221i	$-0.573830\pi$
$61$	−4.97163e7	−0.459742	−0.229871	+	0.973221i	$0.573830\pi$
$62$	0	0
$63$	4.95051e8	3.95929
$64$	0	0
$65$	3.98470e7	0.276876
$66$	0	0
$67$	−1.84401e8	−1.11796	−0.558980	−	0.829181i	$-0.688807\pi$
$67$	−1.84401e8	−1.11796	−0.558980	+	0.829181i	$0.688807\pi$
$68$	0	0
$69$	2.21099e8	1.17426
$70$	0	0
$71$	1.75076e8	0.817644	0.408822	−	0.912614i	$-0.365940\pi$
$71$	1.75076e8	0.817644	0.408822	+	0.912614i	$0.365940\pi$
$72$	0	0
$73$	−3.00349e8	−1.23787	−0.618933	−	0.785444i	$-0.712435\pi$
$73$	−3.00349e8	−1.23787	−0.618933	+	0.785444i	$0.712435\pi$
$74$	0	0
$75$	−1.68453e6	−0.00614758
$76$	0	0
$77$	1.34150e8	0.434893
$78$	0	0
$79$	−5.38224e8	−1.55468	−0.777341	−	0.629080i	$-0.783432\pi$
$79$	−5.38224e8	−1.55468	−0.777341	+	0.629080i	$0.783432\pi$
$80$	0	0
$81$	6.88023e8	1.77591
$82$	0	0
$83$	−3.07653e8	−0.711558	−0.355779	−	0.934570i	$-0.615784\pi$
$83$	−3.07653e8	−0.711558	−0.355779	+	0.934570i	$0.615784\pi$
$84$	0	0
$85$	−4.98092e8	−1.03496
$86$	0	0
$87$	−9.63291e8	−1.80269
$88$	0	0
$89$	4.56635e8	0.771462	0.385731	−	0.922611i	$-0.373949\pi$
$89$	4.56635e8	0.771462	0.385731	+	0.922611i	$0.373949\pi$
$90$	0	0
$91$	3.20759e8	0.490334
$92$	0	0
$93$	−4.30403e8	−0.596626
$94$	0	0
$95$	1.13381e9	1.42818
$96$	0	0
$97$	−1.45156e9	−1.66480	−0.832402	−	0.554172i	$-0.813035\pi$
$97$	−1.45156e9	−1.66480	−0.832402	+	0.554172i	$0.813035\pi$
$98$	0	0
$99$	5.26539e8	0.550899
$100$	0	0
$101$	2.02443e9	1.93578	0.967890	−	0.251372i	$-0.0808818\pi$
$101$	2.02443e9	1.93578	0.967890	+	0.251372i	$0.0808818\pi$
$102$	0	0
$103$	1.17562e9	1.02920	0.514598	−	0.857431i	$-0.327941\pi$
$103$	1.17562e9	1.02920	0.514598	+	0.857431i	$0.327941\pi$
$104$	0	0
$105$	3.95652e9	3.17659
$106$	0	0
$107$	−1.79278e9	−1.32221	−0.661105	−	0.750294i	$-0.729912\pi$
$107$	−1.79278e9	−1.32221	−0.661105	+	0.750294i	$0.729912\pi$
$108$	0	0
$109$	−1.47094e9	−0.998100	−0.499050	−	0.866573i	$-0.666318\pi$
$109$	−1.47094e9	−0.998100	−0.499050	+	0.866573i	$0.666318\pi$
$110$	0	0
$111$	−5.08392e9	−3.17867
$112$	0	0
$113$	−1.43367e9	−0.827175	−0.413587	−	0.910464i	$-0.635724\pi$
$113$	−1.43367e9	−0.827175	−0.413587	+	0.910464i	$0.635724\pi$
$114$	0	0
$115$	1.22158e9	0.651302
$116$	0	0
$117$	1.25898e9	0.621129
$118$	0	0
$119$	−4.00953e9	−1.83287
$120$	0	0
$121$	−2.21527e9	−0.939489
$122$	0	0
$123$	1.89762e9	0.747544
$124$	0	0
$125$	−2.73422e9	−1.00170
$126$	0	0
$127$	6.66749e8	0.227429	0.113714	−	0.993513i	$-0.463725\pi$
$127$	6.66749e8	0.227429	0.113714	+	0.993513i	$0.463725\pi$
$128$	0	0
$129$	2.21682e9	0.704818
$130$	0	0
$131$	−6.41886e7	−0.0190431	−0.00952153	−	0.999955i	$-0.503031\pi$
$131$	−6.41886e7	−0.0190431	−0.00952153	+	0.999955i	$0.503031\pi$
$132$	0	0
$133$	9.12687e9	2.52924
$134$	0	0
$135$	8.59509e9	2.22714
$136$	0	0
$137$	6.53226e9	1.58424	0.792119	−	0.610366i	$-0.208978\pi$
$137$	6.53226e9	1.58424	0.792119	+	0.610366i	$0.208978\pi$
$138$	0	0
$139$	3.61791e9	0.822036	0.411018	−	0.911627i	$-0.365173\pi$
$139$	3.61791e9	0.822036	0.411018	+	0.911627i	$0.365173\pi$
$140$	0	0
$141$	−1.40513e10	−2.99385
$142$	0	0
$143$	3.41161e8	0.0682255
$144$	0	0
$145$	−5.32224e9	−0.999857
$146$	0	0
$147$	2.16592e10	3.82572
$148$	0	0
$149$	8.85288e9	1.47145	0.735726	−	0.677279i	$-0.236841\pi$
$149$	8.85288e9	1.47145	0.735726	+	0.677279i	$0.236841\pi$
$150$	0	0
$151$	−5.05046e9	−0.790560	−0.395280	−	0.918561i	$-0.629352\pi$
$151$	−5.05046e9	−0.790560	−0.395280	+	0.918561i	$0.629352\pi$
$152$	0	0
$153$	−1.57374e10	−2.32178
$154$	0	0
$155$	−2.37800e9	−0.330917
$156$	0	0
$157$	6.78519e9	0.891278	0.445639	−	0.895213i	$-0.352976\pi$
$157$	6.78519e9	0.891278	0.445639	+	0.895213i	$0.352976\pi$
$158$	0	0
$159$	4.65132e9	0.577151
$160$	0	0
$161$	9.83345e9	1.15343
$162$	0	0
$163$	4.06629e9	0.451185	0.225592	−	0.974222i	$-0.427568\pi$
$163$	4.06629e9	0.451185	0.225592	+	0.974222i	$0.427568\pi$
$164$	0	0
$165$	4.20817e9	0.441993
$166$	0	0
$167$	1.48051e10	1.47294	0.736472	−	0.676468i	$-0.236490\pi$
$167$	1.48051e10	1.47294	0.736472	+	0.676468i	$0.236490\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	3.58230e10	3.20390
$172$	0	0
$173$	1.97114e10	1.67306	0.836528	−	0.547924i	$-0.184582\pi$
$173$	1.97114e10	1.67306	0.836528	+	0.547924i	$0.184582\pi$
$174$	0	0
$175$	−7.49203e7	−0.00603849
$176$	0	0
$177$	−2.31425e10	−1.77227
$178$	0	0
$179$	5.20046e9	0.378620	0.189310	−	0.981917i	$-0.439375\pi$
$179$	5.20046e9	0.378620	0.189310	+	0.981917i	$0.439375\pi$
$180$	0	0
$181$	1.03831e10	0.719073	0.359536	−	0.933131i	$-0.382935\pi$
$181$	1.03831e10	0.719073	0.359536	+	0.933131i	$0.382935\pi$
$182$	0	0
$183$	−1.25541e10	−0.827474
$184$	0	0
$185$	−2.80889e10	−1.76304
$186$	0	0
$187$	−4.26455e9	−0.255027
$188$	0	0
$189$	6.91884e10	3.94417
$190$	0	0
$191$	8.63545e9	0.469499	0.234750	−	0.972056i	$-0.424573\pi$
$191$	8.63545e9	0.469499	0.234750	+	0.972056i	$0.424573\pi$
$192$	0	0
$193$	1.67569e10	0.869331	0.434666	−	0.900592i	$-0.356867\pi$
$193$	1.67569e10	0.869331	0.434666	+	0.900592i	$0.356867\pi$
$194$	0	0
$195$	1.00619e10	0.498339
$196$	0	0
$197$	1.87783e10	0.888297	0.444149	−	0.895953i	$-0.353506\pi$
$197$	1.87783e10	0.888297	0.444149	+	0.895953i	$0.353506\pi$
$198$	0	0
$199$	−3.52425e9	−0.159304	−0.0796521	−	0.996823i	$-0.525381\pi$
$199$	−3.52425e9	−0.159304	−0.0796521	+	0.996823i	$0.525381\pi$
$200$	0	0
$201$	−4.65638e10	−2.01218
$202$	0	0
$203$	−4.28427e10	−1.77070
$204$	0	0
$205$	1.04845e10	0.414624
$206$	0	0
$207$	3.85963e10	1.46110
$208$	0	0
$209$	9.70738e9	0.351920
$210$	0	0
$211$	−4.12902e10	−1.43409	−0.717045	−	0.697027i	$-0.754506\pi$
$211$	−4.12902e10	−1.43409	−0.717045	+	0.697027i	$0.754506\pi$
$212$	0	0
$213$	4.42092e10	1.47165
$214$	0	0
$215$	1.22481e10	0.390926
$216$	0	0
$217$	−1.91423e10	−0.586038
$218$	0	0
$219$	−7.58424e10	−2.22799
$220$	0	0
$221$	−1.01967e10	−0.287538
$222$	0	0
$223$	4.75372e10	1.28725	0.643623	−	0.765343i	$-0.277431\pi$
$223$	4.75372e10	1.28725	0.643623	+	0.765343i	$0.277431\pi$
$224$	0	0
$225$	−2.94062e8	−0.00764924
$226$	0	0
$227$	2.67975e10	0.669851	0.334925	−	0.942245i	$-0.391289\pi$
$227$	2.67975e10	0.669851	0.334925	+	0.942245i	$0.391289\pi$
$228$	0	0
$229$	−1.48840e9	−0.0357650	−0.0178825	−	0.999840i	$-0.505692\pi$
$229$	−1.48840e9	−0.0357650	−0.0178825	+	0.999840i	$0.505692\pi$
$230$	0	0
$231$	3.38748e10	0.782749
$232$	0	0
$233$	−7.20283e10	−1.60104	−0.800519	−	0.599307i	$-0.795443\pi$
$233$	−7.20283e10	−1.60104	−0.800519	+	0.599307i	$0.795443\pi$
$234$	0	0
$235$	−7.76342e10	−1.66053
$236$	0	0
$237$	−1.35909e11	−2.79822
$238$	0	0
$239$	−4.32647e10	−0.857715	−0.428858	−	0.903372i	$-0.641084\pi$
$239$	−4.32647e10	−0.857715	−0.428858	+	0.903372i	$0.641084\pi$
$240$	0	0
$241$	6.59730e10	1.25977	0.629883	−	0.776690i	$-0.283103\pi$
$241$	6.59730e10	1.25977	0.629883	+	0.776690i	$0.283103\pi$
$242$	0	0
$243$	5.24750e10	0.965438
$244$	0	0
$245$	1.19668e11	2.12193
$246$	0	0
$247$	2.32108e10	0.396784
$248$	0	0
$249$	−7.76868e10	−1.28071
$250$	0	0
$251$	−1.34901e9	−0.0214528	−0.0107264	−	0.999942i	$-0.503414\pi$
$251$	−1.34901e9	−0.0214528	−0.0107264	+	0.999942i	$0.503414\pi$
$252$	0	0
$253$	1.04589e10	0.160488
$254$	0	0
$255$	−1.25775e11	−1.86279
$256$	0	0
$257$	2.04936e10	0.293034	0.146517	−	0.989208i	$-0.453194\pi$
$257$	2.04936e10	0.293034	0.146517	+	0.989208i	$0.453194\pi$
$258$	0	0
$259$	−2.26109e11	−3.12226
$260$	0	0
$261$	−1.68158e11	−2.24303
$262$	0	0
$263$	2.32322e10	0.299426	0.149713	−	0.988730i	$-0.452165\pi$
$263$	2.32322e10	0.299426	0.149713	+	0.988730i	$0.452165\pi$
$264$	0	0
$265$	2.56988e10	0.320115
$266$	0	0
$267$	1.15307e11	1.38853
$268$	0	0
$269$	−4.76939e10	−0.555363	−0.277682	−	0.960673i	$-0.589566\pi$
$269$	−4.76939e10	−0.555363	−0.277682	+	0.960673i	$0.589566\pi$
$270$	0	0
$271$	−1.02000e11	−1.14878	−0.574391	−	0.818581i	$-0.694761\pi$
$271$	−1.02000e11	−1.14878	−0.574391	+	0.818581i	$0.694761\pi$
$272$	0	0
$273$	8.09961e10	0.882536
$274$	0	0
$275$	−7.96856e7	−0.000840200 0
$276$	0	0
$277$	−4.31461e10	−0.440335	−0.220167	−	0.975462i	$-0.570660\pi$
$277$	−4.31461e10	−0.440335	−0.220167	+	0.975462i	$0.570660\pi$
$278$	0	0
$279$	−7.51337e10	−0.742362
$280$	0	0
$281$	1.21086e11	1.15855	0.579274	−	0.815133i	$-0.303336\pi$
$281$	1.21086e11	1.15855	0.579274	+	0.815133i	$0.303336\pi$
$282$	0	0
$283$	1.93259e11	1.79102	0.895512	−	0.445038i	$-0.146810\pi$
$283$	1.93259e11	1.79102	0.895512	+	0.445038i	$0.146810\pi$
$284$	0	0
$285$	2.86302e11	2.57053
$286$	0	0
$287$	8.43975e10	0.734279
$288$	0	0
$289$	8.87263e9	0.0748190
$290$	0	0
$291$	−3.66540e11	−2.99642
$292$	0	0
$293$	2.21207e11	1.75345	0.876727	−	0.480988i	$-0.159722\pi$
$293$	2.21207e11	1.75345	0.876727	+	0.480988i	$0.159722\pi$
$294$	0	0
$295$	−1.27864e11	−0.982987
$296$	0	0
$297$	7.35891e10	0.548795
$298$	0	0
$299$	2.50077e10	0.180948
$300$	0	0
$301$	9.85940e10	0.692311
$302$	0	0
$303$	5.11197e11	3.48414
$304$	0	0
$305$	−6.93618e10	−0.458956
$306$	0	0
$307$	5.56848e10	0.357778	0.178889	−	0.983869i	$-0.442750\pi$
$307$	5.56848e10	0.357778	0.178889	+	0.983869i	$0.442750\pi$
$308$	0	0
$309$	2.96860e11	1.85242
$310$	0	0
$311$	−5.64253e9	−0.0342021	−0.0171010	−	0.999854i	$-0.505444\pi$
$311$	−5.64253e9	−0.0342021	−0.0171010	+	0.999854i	$0.505444\pi$
$312$	0	0
$313$	−1.20514e11	−0.709723	−0.354862	−	0.934919i	$-0.615472\pi$
$313$	−1.20514e11	−0.709723	−0.354862	+	0.934919i	$0.615472\pi$
$314$	0	0
$315$	6.90673e11	3.95253
$316$	0	0
$317$	−1.29791e11	−0.721901	−0.360950	−	0.932585i	$-0.617548\pi$
$317$	−1.29791e11	−0.721901	−0.360950	+	0.932585i	$0.617548\pi$
$318$	0	0
$319$	−4.55677e10	−0.246377
$320$	0	0
$321$	−4.52702e11	−2.37980
$322$	0	0
$323$	−2.90138e11	−1.48318
$324$	0	0
$325$	−1.90532e8	−0.000947311 0
$326$	0	0
$327$	−3.71432e11	−1.79645
$328$	0	0
$329$	−6.24937e11	−2.94073
$330$	0	0
$331$	3.06818e11	1.40493	0.702464	−	0.711719i	$-0.252083\pi$
$331$	3.06818e11	1.40493	0.702464	+	0.711719i	$0.252083\pi$
$332$	0	0
$333$	−8.87479e11	−3.95511
$334$	0	0
$335$	−2.57268e11	−1.11605
$336$	0	0
$337$	1.02414e11	0.432539	0.216270	−	0.976334i	$-0.430611\pi$
$337$	1.02414e11	0.432539	0.216270	+	0.976334i	$0.430611\pi$
$338$	0	0
$339$	−3.62023e11	−1.48880
$340$	0	0
$341$	−2.03599e10	−0.0815418
$342$	0	0
$343$	5.10102e11	1.98991
$344$	0	0
$345$	3.08467e11	1.17226
$346$	0	0
$347$	−3.10216e11	−1.14863	−0.574317	−	0.818633i	$-0.694732\pi$
$347$	−3.10216e11	−1.14863	−0.574317	+	0.818633i	$0.694732\pi$
$348$	0	0
$349$	−3.98911e11	−1.43933	−0.719667	−	0.694319i	$-0.755706\pi$
$349$	−3.98911e11	−1.43933	−0.719667	+	0.694319i	$0.755706\pi$
$350$	0	0
$351$	1.75955e11	0.618756
$352$	0	0
$353$	−1.11096e11	−0.380815	−0.190407	−	0.981705i	$-0.560981\pi$
$353$	−1.11096e11	−0.380815	−0.190407	+	0.981705i	$0.560981\pi$
$354$	0	0
$355$	2.44258e11	0.816246
$356$	0	0
$357$	−1.01246e12	−3.29892
$358$	0	0
$359$	−2.78159e11	−0.883829	−0.441914	−	0.897057i	$-0.645700\pi$
$359$	−2.78159e11	−0.883829	−0.441914	+	0.897057i	$0.645700\pi$
$360$	0	0
$361$	3.37752e11	1.04668
$362$	0	0
$363$	−5.59386e11	−1.69095
$364$	0	0
$365$	−4.19033e11	−1.23575
$366$	0	0
$367$	−2.28585e11	−0.657733	−0.328867	−	0.944376i	$-0.606667\pi$
$367$	−2.28585e11	−0.657733	−0.328867	+	0.944376i	$0.606667\pi$
$368$	0	0
$369$	3.31260e11	0.930145
$370$	0	0
$371$	2.06869e11	0.566909
$372$	0	0
$373$	5.79749e11	1.55078	0.775390	−	0.631482i	$-0.217553\pi$
$373$	5.79749e11	1.55078	0.775390	+	0.631482i	$0.217553\pi$
$374$	0	0
$375$	−6.90428e11	−1.80293
$376$	0	0
$377$	−1.08955e11	−0.277785
$378$	0	0
$379$	−6.30593e11	−1.56990	−0.784952	−	0.619557i	$-0.787312\pi$
$379$	−6.30593e11	−1.56990	−0.784952	+	0.619557i	$0.787312\pi$
$380$	0	0
$381$	1.68363e11	0.409341
$382$	0	0
$383$	5.07531e11	1.20523	0.602613	−	0.798034i	$-0.294126\pi$
$383$	5.07531e11	1.20523	0.602613	+	0.798034i	$0.294126\pi$
$384$	0	0
$385$	1.87160e11	0.434150
$386$	0	0
$387$	3.86982e11	0.876983
$388$	0	0
$389$	3.19286e11	0.706979	0.353490	−	0.935438i	$-0.384995\pi$
$389$	3.19286e11	0.706979	0.353490	+	0.935438i	$0.384995\pi$
$390$	0	0
$391$	−3.12600e11	−0.676384
$392$	0	0
$393$	−1.62085e10	−0.0342749
$394$	0	0
$395$	−7.50906e11	−1.55202
$396$	0	0
$397$	−2.65511e11	−0.536445	−0.268222	−	0.963357i	$-0.586436\pi$
$397$	−2.65511e11	−0.536445	−0.268222	+	0.963357i	$0.586436\pi$
$398$	0	0
$399$	2.30466e12	4.55228
$400$	0	0
$401$	−3.57381e11	−0.690210	−0.345105	−	0.938564i	$-0.612157\pi$
$401$	−3.57381e11	−0.690210	−0.345105	+	0.938564i	$0.612157\pi$
$402$	0	0
$403$	−4.86814e10	−0.0919370
$404$	0	0
$405$	9.59898e11	1.77287
$406$	0	0
$407$	−2.40491e11	−0.434434
$408$	0	0
$409$	−6.53884e11	−1.15544	−0.577718	−	0.816236i	$-0.696057\pi$
$409$	−6.53884e11	−1.15544	−0.577718	+	0.816236i	$0.696057\pi$
$410$	0	0
$411$	1.64949e12	2.85142
$412$	0	0
$413$	−1.02927e12	−1.74082
$414$	0	0
$415$	−4.29224e11	−0.710342
$416$	0	0
$417$	9.13572e11	1.47955
$418$	0	0
$419$	−5.63993e11	−0.893945	−0.446972	−	0.894548i	$-0.647498\pi$
$419$	−5.63993e11	−0.893945	−0.446972	+	0.894548i	$0.647498\pi$
$420$	0	0
$421$	1.02848e12	1.59561	0.797806	−	0.602914i	$-0.205994\pi$
$421$	1.02848e12	1.59561	0.797806	+	0.602914i	$0.205994\pi$
$422$	0	0
$423$	−2.45288e12	−3.72516
$424$	0	0
$425$	2.38167e9	0.00354105
$426$	0	0
$427$	−5.58346e11	−0.812790
$428$	0	0
$429$	8.61479e10	0.122797
$430$	0	0
$431$	8.40771e11	1.17363	0.586813	−	0.809722i	$-0.300382\pi$
$431$	8.40771e11	1.17363	0.586813	+	0.809722i	$0.300382\pi$
$432$	0	0
$433$	−7.90716e11	−1.08100	−0.540499	−	0.841344i	$-0.681765\pi$
$433$	−7.90716e11	−1.08100	−0.540499	+	0.841344i	$0.681765\pi$
$434$	0	0
$435$	−1.34394e12	−1.79961
$436$	0	0
$437$	7.11569e11	0.933363
$438$	0	0
$439$	4.11309e11	0.528540	0.264270	−	0.964449i	$-0.414869\pi$
$439$	4.11309e11	0.528540	0.264270	+	0.964449i	$0.414869\pi$
$440$	0	0
$441$	3.78095e12	4.76023
$442$	0	0
$443$	2.28836e11	0.282298	0.141149	−	0.989988i	$-0.454920\pi$
$443$	2.28836e11	0.282298	0.141149	+	0.989988i	$0.454920\pi$
$444$	0	0
$445$	6.37077e11	0.770143
$446$	0	0
$447$	2.23548e12	2.64842
$448$	0	0
$449$	−9.73693e10	−0.113061	−0.0565306	−	0.998401i	$-0.518004\pi$
$449$	−9.73693e10	−0.113061	−0.0565306	+	0.998401i	$0.518004\pi$
$450$	0	0
$451$	8.97656e10	0.102168
$452$	0	0
$453$	−1.27531e12	−1.42290
$454$	0	0
$455$	4.47508e11	0.489496
$456$	0	0
$457$	−5.00855e10	−0.0537142	−0.0268571	−	0.999639i	$-0.508550\pi$
$457$	−5.00855e10	−0.0537142	−0.0268571	+	0.999639i	$0.508550\pi$
$458$	0	0
$459$	−2.19946e12	−2.31291
$460$	0	0
$461$	−4.96407e11	−0.511899	−0.255949	−	0.966690i	$-0.582388\pi$
$461$	−4.96407e11	−0.511899	−0.255949	+	0.966690i	$0.582388\pi$
$462$	0	0
$463$	−9.05832e11	−0.916080	−0.458040	−	0.888932i	$-0.651448\pi$
$463$	−9.05832e11	−0.916080	−0.458040	+	0.888932i	$0.651448\pi$
$464$	0	0
$465$	−6.00478e11	−0.595606
$466$	0	0
$467$	−3.32492e11	−0.323486	−0.161743	−	0.986833i	$-0.551712\pi$
$467$	−3.32492e11	−0.323486	−0.161743	+	0.986833i	$0.551712\pi$
$468$	0	0
$469$	−2.07094e12	−1.97647
$470$	0	0
$471$	1.71336e12	1.60418
$472$	0	0
$473$	1.04865e11	0.0963287
$474$	0	0
$475$	−5.42139e9	−0.00488641
$476$	0	0
$477$	8.11963e11	0.718130
$478$	0	0
$479$	−4.39783e11	−0.381706	−0.190853	−	0.981619i	$-0.561125\pi$
$479$	−4.39783e11	−0.381706	−0.190853	+	0.981619i	$0.561125\pi$
$480$	0	0
$481$	−5.75025e11	−0.489817
$482$	0	0
$483$	2.48308e12	2.07601
$484$	0	0
$485$	−2.02515e12	−1.66196
$486$	0	0
$487$	1.54739e12	1.24657	0.623287	−	0.781993i	$-0.285797\pi$
$487$	1.54739e12	1.24657	0.623287	+	0.781993i	$0.285797\pi$
$488$	0	0
$489$	1.02680e12	0.812072
$490$	0	0
$491$	1.17406e12	0.911638	0.455819	−	0.890072i	$-0.349346\pi$
$491$	1.17406e12	0.911638	0.455819	+	0.890072i	$0.349346\pi$
$492$	0	0
$493$	1.36195e12	1.03836
$494$	0	0
$495$	7.34603e11	0.549958
$496$	0	0
$497$	1.96622e12	1.44553
$498$	0	0
$499$	1.52477e12	1.10091	0.550455	−	0.834865i	$-0.314454\pi$
$499$	1.52477e12	1.10091	0.550455	+	0.834865i	$0.314454\pi$
$500$	0	0
$501$	3.73849e12	2.65110
$502$	0	0
$503$	−1.45642e12	−1.01445	−0.507224	−	0.861814i	$-0.669328\pi$
$503$	−1.45642e12	−1.01445	−0.507224	+	0.861814i	$0.669328\pi$
$504$	0	0
$505$	2.82439e12	1.93247
$506$	0	0
$507$	2.05983e11	0.138451
$508$	0	0
$509$	−1.34729e12	−0.889677	−0.444838	−	0.895611i	$-0.646739\pi$
$509$	−1.34729e12	−0.889677	−0.444838	+	0.895611i	$0.646739\pi$
$510$	0	0
$511$	−3.37312e12	−2.18846
$512$	0	0
$513$	5.00662e12	3.19166
$514$	0	0
$515$	1.64017e12	1.02744
$516$	0	0
$517$	−6.64686e11	−0.409175
$518$	0	0
$519$	4.97741e12	3.01128
$520$	0	0
$521$	1.69421e12	1.00739	0.503694	−	0.863882i	$-0.331974\pi$
$521$	1.69421e12	1.00739	0.503694	+	0.863882i	$0.331974\pi$
$522$	0	0
$523$	1.40302e12	0.819983	0.409992	−	0.912089i	$-0.365532\pi$
$523$	1.40302e12	0.819983	0.409992	+	0.912089i	$0.365532\pi$
$524$	0	0
$525$	−1.89184e10	−0.0108685
$526$	0	0
$527$	6.08524e11	0.343661
$528$	0	0
$529$	−1.03450e12	−0.574352
$530$	0	0
$531$	−4.03989e12	−2.20518
$532$	0	0
$533$	2.14634e11	0.115193
$534$	0	0
$535$	−2.50120e12	−1.31995
$536$	0	0
$537$	1.31319e12	0.681465
$538$	0	0
$539$	1.02457e12	0.522868
$540$	0	0
$541$	2.57026e12	1.29000	0.645000	−	0.764183i	$-0.276857\pi$
$541$	2.57026e12	1.29000	0.645000	+	0.764183i	$0.276857\pi$
$542$	0	0
$543$	2.62187e12	1.29423
$544$	0	0
$545$	−2.05218e12	−0.996394
$546$	0	0
$547$	−2.34041e12	−1.11776	−0.558882	−	0.829247i	$-0.688769\pi$
$547$	−2.34041e12	−1.11776	−0.558882	+	0.829247i	$0.688769\pi$
$548$	0	0
$549$	−2.19151e12	−1.02960
$550$	0	0
$551$	−3.10019e12	−1.43287
$552$	0	0
$553$	−6.04461e12	−2.74856
$554$	0	0
$555$	−7.09285e12	−3.17324
$556$	0	0
$557$	−3.29174e12	−1.44903	−0.724514	−	0.689260i	$-0.757936\pi$
$557$	−3.29174e12	−1.44903	−0.724514	+	0.689260i	$0.757936\pi$
$558$	0	0
$559$	2.50737e11	0.108609
$560$	0	0
$561$	−1.07686e12	−0.459014
$562$	0	0
$563$	1.31650e12	0.552248	0.276124	−	0.961122i	$-0.410950\pi$
$563$	1.31650e12	0.552248	0.276124	+	0.961122i	$0.410950\pi$
$564$	0	0
$565$	−2.00020e12	−0.825761
$566$	0	0
$567$	7.72696e12	3.13968
$568$	0	0
$569$	−3.81990e12	−1.52773	−0.763866	−	0.645375i	$-0.776701\pi$
$569$	−3.81990e12	−1.52773	−0.763866	+	0.645375i	$0.776701\pi$
$570$	0	0
$571$	−1.53477e12	−0.604199	−0.302100	−	0.953276i	$-0.597687\pi$
$571$	−1.53477e12	−0.604199	−0.302100	+	0.953276i	$0.597687\pi$
$572$	0	0
$573$	2.18057e12	0.845035
$574$	0	0
$575$	−5.84110e9	−0.00222838
$576$	0	0
$577$	−3.14093e12	−1.17969	−0.589844	−	0.807517i	$-0.700811\pi$
$577$	−3.14093e12	−1.17969	−0.589844	+	0.807517i	$0.700811\pi$
$578$	0	0
$579$	4.23135e12	1.56468
$580$	0	0
$581$	−3.45515e12	−1.25798
$582$	0	0
$583$	2.20027e11	0.0788802
$584$	0	0
$585$	1.75647e12	0.620068
$586$	0	0
$587$	1.66414e12	0.578521	0.289261	−	0.957250i	$-0.406591\pi$
$587$	1.66414e12	0.578521	0.289261	+	0.957250i	$0.406591\pi$
$588$	0	0
$589$	−1.38518e12	−0.474228
$590$	0	0
$591$	4.74179e12	1.59882
$592$	0	0
$593$	1.27000e12	0.421752	0.210876	−	0.977513i	$-0.432368\pi$
$593$	1.27000e12	0.421752	0.210876	+	0.977513i	$0.432368\pi$
$594$	0	0
$595$	−5.59391e12	−1.82974
$596$	0	0
$597$	−8.89922e11	−0.286726
$598$	0	0
$599$	2.20905e12	0.701107	0.350553	−	0.936543i	$-0.385994\pi$
$599$	2.20905e12	0.701107	0.350553	+	0.936543i	$0.385994\pi$
$600$	0	0
$601$	2.43469e12	0.761218	0.380609	−	0.924736i	$-0.375714\pi$
$601$	2.43469e12	0.761218	0.380609	+	0.924736i	$0.375714\pi$
$602$	0	0
$603$	−8.12846e12	−2.50369
$604$	0	0
$605$	−3.09064e12	−0.937883
$606$	0	0
$607$	−2.89583e11	−0.0865814	−0.0432907	−	0.999063i	$-0.513784\pi$
$607$	−2.89583e11	−0.0865814	−0.0432907	+	0.999063i	$0.513784\pi$
$608$	0	0
$609$	−1.08184e13	−3.18702
$610$	0	0
$611$	−1.58929e12	−0.461338
$612$	0	0
$613$	1.65691e12	0.473944	0.236972	−	0.971517i	$-0.423845\pi$
$613$	1.65691e12	0.473944	0.236972	+	0.971517i	$0.423845\pi$
$614$	0	0
$615$	2.64748e12	0.746266
$616$	0	0
$617$	4.15316e12	1.15371	0.576854	−	0.816847i	$-0.304280\pi$
$617$	4.15316e12	1.15371	0.576854	+	0.816847i	$0.304280\pi$
$618$	0	0
$619$	−2.15564e12	−0.590157	−0.295079	−	0.955473i	$-0.595346\pi$
$619$	−2.15564e12	−0.590157	−0.295079	+	0.955473i	$0.595346\pi$
$620$	0	0
$621$	5.39422e12	1.45552
$622$	0	0
$623$	5.12832e12	1.36389
$624$	0	0
$625$	−3.80162e12	−0.996573
$626$	0	0
$627$	2.45125e12	0.633408
$628$	0	0
$629$	7.18788e12	1.83094
$630$	0	0
$631$	1.01169e12	0.254049	0.127024	−	0.991900i	$-0.459457\pi$
$631$	1.01169e12	0.254049	0.127024	+	0.991900i	$0.459457\pi$
$632$	0	0
$633$	−1.04264e13	−2.58117
$634$	0	0
$635$	9.30217e11	0.227040
$636$	0	0
$637$	2.44979e12	0.589525
$638$	0	0
$639$	7.71741e12	1.83112
$640$	0	0
$641$	−1.91146e12	−0.447201	−0.223601	−	0.974681i	$-0.571781\pi$
$641$	−1.91146e12	−0.447201	−0.223601	+	0.974681i	$0.571781\pi$
$642$	0	0
$643$	−1.76778e12	−0.407831	−0.203915	−	0.978989i	$-0.565367\pi$
$643$	−1.76778e12	−0.407831	−0.203915	+	0.978989i	$0.565367\pi$
$644$	0	0
$645$	3.09281e12	0.703614
$646$	0	0
$647$	1.06872e12	0.239769	0.119885	−	0.992788i	$-0.461748\pi$
$647$	1.06872e12	0.239769	0.119885	+	0.992788i	$0.461748\pi$
$648$	0	0
$649$	−1.09474e12	−0.242220
$650$	0	0
$651$	−4.83371e12	−1.05479
$652$	0	0
$653$	3.97859e12	0.856287	0.428144	−	0.903711i	$-0.359168\pi$
$653$	3.97859e12	0.856287	0.428144	+	0.903711i	$0.359168\pi$
$654$	0	0
$655$	−8.95529e10	−0.0190105
$656$	0	0
$657$	−1.32395e13	−2.77222
$658$	0	0
$659$	−4.17558e12	−0.862447	−0.431224	−	0.902245i	$-0.641918\pi$
$659$	−4.17558e12	−0.862447	−0.431224	+	0.902245i	$0.641918\pi$
$660$	0	0
$661$	−1.40927e12	−0.287137	−0.143568	−	0.989640i	$-0.545858\pi$
$661$	−1.40927e12	−0.287137	−0.143568	+	0.989640i	$0.545858\pi$
$662$	0	0
$663$	−2.57482e12	−0.517530
$664$	0	0
$665$	1.27334e13	2.52491
$666$	0	0
$667$	−3.34020e12	−0.653441
$668$	0	0
$669$	1.20038e13	2.31687
$670$	0	0
$671$	−5.93860e11	−0.113092
$672$	0	0
$673$	1.88918e12	0.354982	0.177491	−	0.984122i	$-0.443202\pi$
$673$	1.88918e12	0.354982	0.177491	+	0.984122i	$0.443202\pi$
$674$	0	0
$675$	−4.10982e10	−0.00762001
$676$	0	0
$677$	−1.84519e12	−0.337591	−0.168796	−	0.985651i	$-0.553988\pi$
$677$	−1.84519e12	−0.337591	−0.168796	+	0.985651i	$0.553988\pi$
$678$	0	0
$679$	−1.63020e13	−2.94325
$680$	0	0
$681$	6.76675e12	1.20564
$682$	0	0
$683$	−1.42232e12	−0.250094	−0.125047	−	0.992151i	$-0.539908\pi$
$683$	−1.42232e12	−0.250094	−0.125047	+	0.992151i	$0.539908\pi$
$684$	0	0
$685$	9.11350e12	1.58153
$686$	0	0
$687$	−3.75841e11	−0.0643722
$688$	0	0
$689$	5.26095e11	0.0889361
$690$	0	0
$691$	−4.24814e11	−0.0708838	−0.0354419	−	0.999372i	$-0.511284\pi$
$691$	−4.24814e11	−0.0708838	−0.0354419	+	0.999372i	$0.511284\pi$
$692$	0	0
$693$	5.91338e12	0.973949
$694$	0	0
$695$	5.04753e12	0.820631
$696$	0	0
$697$	−2.68295e12	−0.430591
$698$	0	0
$699$	−1.81882e13	−2.88165
$700$	0	0
$701$	−3.72982e12	−0.583387	−0.291694	−	0.956512i	$-0.594219\pi$
$701$	−3.72982e12	−0.583387	−0.291694	+	0.956512i	$0.594219\pi$
$702$	0	0
$703$	−1.63617e13	−2.52657
$704$	0	0
$705$	−1.96037e13	−2.98874
$706$	0	0
$707$	2.27357e13	3.42232
$708$	0	0
$709$	4.10258e12	0.609746	0.304873	−	0.952393i	$-0.401386\pi$
$709$	4.10258e12	0.609746	0.304873	+	0.952393i	$0.401386\pi$
$710$	0	0
$711$	−2.37251e13	−3.48173
$712$	0	0
$713$	−1.49242e12	−0.216266
$714$	0	0
$715$	4.75972e11	0.0681089
$716$	0	0
$717$	−1.09249e13	−1.54377
$718$	0	0
$719$	2.13034e12	0.297282	0.148641	−	0.988891i	$-0.452510\pi$
$719$	2.13034e12	0.297282	0.148641	+	0.988891i	$0.452510\pi$
$720$	0	0
$721$	1.32030e13	1.81954
$722$	0	0
$723$	1.66591e13	2.26741
$724$	0	0
$725$	2.54487e10	0.00342094
$726$	0	0
$727$	1.17072e13	1.55435	0.777175	−	0.629284i	$-0.216652\pi$
$727$	1.17072e13	1.55435	0.777175	+	0.629284i	$0.216652\pi$
$728$	0	0
$729$	−2.91681e11	−0.0382503
$730$	0	0
$731$	−3.13425e12	−0.405980
$732$	0	0
$733$	1.06511e13	1.36278	0.681391	−	0.731919i	$-0.261375\pi$
$733$	1.06511e13	1.36278	0.681391	+	0.731919i	$0.261375\pi$
$734$	0	0
$735$	3.02179e13	3.81919
$736$	0	0
$737$	−2.20267e12	−0.275008
$738$	0	0
$739$	−9.91154e12	−1.22248	−0.611239	−	0.791446i	$-0.709329\pi$
$739$	−9.91154e12	−1.22248	−0.611239	+	0.791446i	$0.709329\pi$
$740$	0	0
$741$	5.86105e12	0.714157
$742$	0	0
$743$	5.37923e12	0.647546	0.323773	−	0.946135i	$-0.395049\pi$
$743$	5.37923e12	0.647546	0.323773	+	0.946135i	$0.395049\pi$
$744$	0	0
$745$	1.23511e13	1.46894
$746$	0	0
$747$	−1.35615e13	−1.59354
$748$	0	0
$749$	−2.01341e13	−2.33757
$750$	0	0
$751$	1.28935e13	1.47908	0.739542	−	0.673111i	$-0.235042\pi$
$751$	1.28935e13	1.47908	0.739542	+	0.673111i	$0.235042\pi$
$752$	0	0
$753$	−3.40645e11	−0.0386122
$754$	0	0
$755$	−7.04617e12	−0.789208
$756$	0	0
$757$	−1.15892e12	−0.128269	−0.0641344	−	0.997941i	$-0.520429\pi$
$757$	−1.15892e12	−0.128269	−0.0641344	+	0.997941i	$0.520429\pi$
$758$	0	0
$759$	2.64102e12	0.288858
$760$	0	0
$761$	1.08521e13	1.17296	0.586481	−	0.809963i	$-0.300513\pi$
$761$	1.08521e13	1.17296	0.586481	+	0.809963i	$0.300513\pi$
$762$	0	0
$763$	−1.65196e13	−1.76457
$764$	0	0
$765$	−2.19561e13	−2.31781
$766$	0	0
$767$	−2.61757e12	−0.273098
$768$	0	0
$769$	1.44473e12	0.148977	0.0744885	−	0.997222i	$-0.476268\pi$
$769$	1.44473e12	0.148977	0.0744885	+	0.997222i	$0.476268\pi$
$770$	0	0
$771$	5.17492e12	0.527422
$772$	0	0
$773$	−3.75524e11	−0.0378295	−0.0189147	−	0.999821i	$-0.506021\pi$
$773$	−3.75524e11	−0.0378295	−0.0189147	+	0.999821i	$0.506021\pi$
$774$	0	0
$775$	1.13706e10	0.00113221
$776$	0	0
$777$	−5.70958e13	−5.61965
$778$	0	0
$779$	6.10718e12	0.594186
$780$	0	0
$781$	2.09128e12	0.201133
$782$	0	0
$783$	−2.35018e13	−2.23446
$784$	0	0
$785$	9.46638e12	0.889755
$786$	0	0
$787$	−8.84811e12	−0.822175	−0.411088	−	0.911596i	$-0.634851\pi$
$787$	−8.84811e12	−0.822175	−0.411088	+	0.911596i	$0.634851\pi$
$788$	0	0
$789$	5.86645e12	0.538926
$790$	0	0
$791$	−1.61011e13	−1.46238
$792$	0	0
$793$	−1.41995e12	−0.127510
$794$	0	0
$795$	6.48931e12	0.576165
$796$	0	0
$797$	3.16529e12	0.277876	0.138938	−	0.990301i	$-0.455631\pi$
$797$	3.16529e12	0.277876	0.138938	+	0.990301i	$0.455631\pi$
$798$	0	0
$799$	1.98664e13	1.72448
$800$	0	0
$801$	2.01286e13	1.72770
$802$	0	0
$803$	−3.58767e12	−0.304503
$804$	0	0
$805$	1.37192e13	1.15145
$806$	0	0
$807$	−1.20434e13	−0.999579
$808$	0	0
$809$	1.83262e13	1.50420	0.752099	−	0.659050i	$-0.229041\pi$
$809$	1.83262e13	1.50420	0.752099	+	0.659050i	$0.229041\pi$
$810$	0	0
$811$	−1.29703e12	−0.105283	−0.0526414	−	0.998613i	$-0.516764\pi$
$811$	−1.29703e12	−0.105283	−0.0526414	+	0.998613i	$0.516764\pi$
$812$	0	0
$813$	−2.57564e13	−2.06765
$814$	0	0
$815$	5.67310e12	0.450413
$816$	0	0
$817$	7.13447e12	0.560225
$818$	0	0
$819$	1.41392e13	1.09811
$820$	0	0
$821$	−7.29992e12	−0.560756	−0.280378	−	0.959890i	$-0.590460\pi$
$821$	−7.29992e12	−0.560756	−0.280378	+	0.959890i	$0.590460\pi$
$822$	0	0
$823$	−2.12216e13	−1.61243	−0.806213	−	0.591625i	$-0.798487\pi$
$823$	−2.12216e13	−1.61243	−0.806213	+	0.591625i	$0.798487\pi$
$824$	0	0
$825$	−2.01217e10	−0.00151225
$826$	0	0
$827$	−2.55292e13	−1.89785	−0.948925	−	0.315501i	$-0.897828\pi$
$827$	−2.55292e13	−1.89785	−0.948925	+	0.315501i	$0.897828\pi$
$828$	0	0
$829$	−1.95149e13	−1.43506	−0.717531	−	0.696526i	$-0.754728\pi$
$829$	−1.95149e13	−1.43506	−0.717531	+	0.696526i	$0.754728\pi$
$830$	0	0
$831$	−1.08950e13	−0.792543
$832$	0	0
$833$	−3.06227e13	−2.20364
$834$	0	0
$835$	2.06554e13	1.47043
$836$	0	0
$837$	−1.05007e13	−0.739526
$838$	0	0
$839$	1.43041e13	0.996626	0.498313	−	0.866997i	$-0.333953\pi$
$839$	1.43041e13	0.996626	0.498313	+	0.866997i	$0.333953\pi$
$840$	0	0
$841$	4.55685e10	0.00314111
$842$	0	0
$843$	3.05758e13	2.08523
$844$	0	0
$845$	1.13807e12	0.0767916
$846$	0	0
$847$	−2.48789e13	−1.66095
$848$	0	0
$849$	4.88007e13	3.22360
$850$	0	0
$851$	−1.76284e13	−1.15221
$852$	0	0
$853$	2.57693e11	0.0166660	0.00833300	−	0.999965i	$-0.497347\pi$
$853$	2.57693e11	0.0166660	0.00833300	+	0.999965i	$0.497347\pi$
$854$	0	0
$855$	4.99786e13	3.19842
$856$	0	0
$857$	2.11761e13	1.34101	0.670507	−	0.741904i	$-0.266077\pi$
$857$	2.11761e13	1.34101	0.670507	+	0.741904i	$0.266077\pi$
$858$	0	0
$859$	−9.07036e12	−0.568401	−0.284201	−	0.958765i	$-0.591728\pi$
$859$	−9.07036e12	−0.568401	−0.284201	+	0.958765i	$0.591728\pi$
$860$	0	0
$861$	2.13116e13	1.32160
$862$	0	0
$863$	−4.14464e12	−0.254354	−0.127177	−	0.991880i	$-0.540592\pi$
$863$	−4.14464e12	−0.254354	−0.127177	+	0.991880i	$0.540592\pi$
$864$	0	0
$865$	2.75005e13	1.67020
$866$	0	0
$867$	2.24046e12	0.134664
$868$	0	0
$869$	−6.42908e12	−0.382437
$870$	0	0
$871$	−5.26667e12	−0.310066
$872$	0	0
$873$	−6.39854e13	−3.72835
$874$	0	0
$875$	−3.07071e13	−1.77093
$876$	0	0
$877$	4.96197e11	0.0283241	0.0141620	−	0.999900i	$-0.495492\pi$
$877$	4.96197e11	0.0283241	0.0141620	+	0.999900i	$0.495492\pi$
$878$	0	0
$879$	5.58579e13	3.15598
$880$	0	0
$881$	1.32510e13	0.741067	0.370533	−	0.928819i	$-0.379175\pi$
$881$	1.32510e13	0.741067	0.370533	+	0.928819i	$0.379175\pi$
$882$	0	0
$883$	3.05486e13	1.69110	0.845548	−	0.533899i	$-0.179274\pi$
$883$	3.05486e13	1.69110	0.845548	+	0.533899i	$0.179274\pi$
$884$	0	0
$885$	−3.22874e13	−1.76924
$886$	0	0
$887$	2.55124e13	1.38387	0.691933	−	0.721961i	$-0.256759\pi$
$887$	2.55124e13	1.38387	0.691933	+	0.721961i	$0.256759\pi$
$888$	0	0
$889$	7.48803e12	0.402077
$890$	0	0
$891$	8.21843e12	0.436857
$892$	0	0
$893$	−4.52218e13	−2.37967
$894$	0	0
$895$	7.25545e12	0.377973
$896$	0	0
$897$	6.31480e12	0.325682
$898$	0	0
$899$	6.50222e12	0.332004
$900$	0	0
$901$	−6.57626e12	−0.332443
$902$	0	0
$903$	2.48964e13	1.24607
$904$	0	0
$905$	1.44860e13	0.717844
$906$	0	0
$907$	−2.27135e13	−1.11443	−0.557213	−	0.830370i	$-0.688129\pi$
$907$	−2.27135e13	−1.11443	−0.557213	+	0.830370i	$0.688129\pi$
$908$	0	0
$909$	8.92375e13	4.33521
$910$	0	0
$911$	1.11760e12	0.0537595	0.0268798	−	0.999639i	$-0.491443\pi$
$911$	1.11760e12	0.0537595	0.0268798	+	0.999639i	$0.491443\pi$
$912$	0	0
$913$	−3.67491e12	−0.175037
$914$	0	0
$915$	−1.75148e13	−0.826059
$916$	0	0
$917$	−7.20880e11	−0.0336667
$918$	0	0
$919$	2.65728e13	1.22890	0.614452	−	0.788954i	$-0.289377\pi$
$919$	2.65728e13	1.22890	0.614452	+	0.788954i	$0.289377\pi$
$920$	0	0
$921$	1.40612e13	0.643953
$922$	0	0
$923$	5.00035e12	0.226774
$924$	0	0
$925$	1.34310e11	0.00603211
$926$	0	0
$927$	5.18216e13	2.30490
$928$	0	0
$929$	−6.66310e11	−0.0293498	−0.0146749	−	0.999892i	$-0.504671\pi$
$929$	−6.66310e11	−0.0293498	−0.0146749	+	0.999892i	$0.504671\pi$
$930$	0	0
$931$	6.97064e13	3.04088
$932$	0	0
$933$	−1.42482e12	−0.0615591
$934$	0	0
$935$	−5.94970e12	−0.254591
$936$	0	0
$937$	2.81754e13	1.19410	0.597051	−	0.802203i	$-0.296339\pi$
$937$	2.81754e13	1.19410	0.597051	+	0.802203i	$0.296339\pi$
$938$	0	0
$939$	−3.04315e13	−1.27741
$940$	0	0
$941$	−3.37670e12	−0.140391	−0.0701956	−	0.997533i	$-0.522362\pi$
$941$	−3.37670e12	−0.140391	−0.0701956	+	0.997533i	$0.522362\pi$
$942$	0	0
$943$	6.57998e12	0.270971
$944$	0	0
$945$	9.65285e13	3.93743
$946$	0	0
$947$	3.50923e13	1.41787	0.708937	−	0.705272i	$-0.249175\pi$
$947$	3.50923e13	1.41787	0.708937	+	0.705272i	$0.249175\pi$
$948$	0	0
$949$	−8.57827e12	−0.343322
$950$	0	0
$951$	−3.27740e13	−1.29932
$952$	0	0
$953$	−8.80454e12	−0.345771	−0.172886	−	0.984942i	$-0.555309\pi$
$953$	−8.80454e12	−0.345771	−0.172886	+	0.984942i	$0.555309\pi$
$954$	0	0
$955$	1.20478e13	0.468697
$956$	0	0
$957$	−1.15065e13	−0.443445
$958$	0	0
$959$	7.33615e13	2.80082
$960$	0	0
$961$	−2.35344e13	−0.890119
$962$	0	0
$963$	−7.90264e13	−2.96111
$964$	0	0
$965$	2.33784e13	0.867845
$966$	0	0
$967$	−7.25751e12	−0.266912	−0.133456	−	0.991055i	$-0.542608\pi$
$967$	−7.25751e12	−0.266912	−0.133456	+	0.991055i	$0.542608\pi$
$968$	0	0
$969$	−7.32639e13	−2.66952
$970$	0	0
$971$	−7.54291e12	−0.272303	−0.136151	−	0.990688i	$-0.543473\pi$
$971$	−7.54291e12	−0.272303	−0.136151	+	0.990688i	$0.543473\pi$
$972$	0	0
$973$	4.06315e13	1.45330
$974$	0	0
$975$	−4.81120e10	−0.00170503
$976$	0	0
$977$	6.09261e12	0.213933	0.106966	−	0.994263i	$-0.465886\pi$
$977$	6.09261e12	0.213933	0.106966	+	0.994263i	$0.465886\pi$
$978$	0	0
$979$	5.45450e12	0.189772
$980$	0	0
$981$	−6.48393e13	−2.23526
$982$	0	0
$983$	4.00886e13	1.36940	0.684700	−	0.728825i	$-0.259933\pi$
$983$	4.00886e13	1.36940	0.684700	+	0.728825i	$0.259933\pi$
$984$	0	0
$985$	2.61986e13	0.886779
$986$	0	0
$987$	−1.57805e14	−5.29291
$988$	0	0
$989$	7.68681e12	0.255483
$990$	0	0
$991$	1.09696e13	0.361294	0.180647	−	0.983548i	$-0.442181\pi$
$991$	1.09696e13	0.361294	0.180647	+	0.983548i	$0.442181\pi$
$992$	0	0
$993$	7.74758e13	2.52868
$994$	0	0
$995$	−4.91686e12	−0.159032
$996$	0	0
$997$	2.19450e13	0.703407	0.351703	−	0.936111i	$-0.385603\pi$
$997$	2.19450e13	0.703407	0.351703	+	0.936111i	$0.385603\pi$
$998$	0	0
$999$	−1.24034e14	−3.94000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.m.1.8		8
4.3	odd	2		104.10.a.d.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.d.1.1	✓	8	4.3	odd	2
208.10.a.m.1.8		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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q+252.514 q^{3} +1395.15 q^{5} +11230.7 q^{7} +44080.4 q^{9} +11945.0 q^{11} +28561.0 q^{13} +352296. q^{15} -357016. q^{17} +812674. q^{19} +2.83590e6 q^{21} +875590. q^{23} -6671.05 q^{25} +6.16067e6 q^{27} -3.81480e6 q^{29} -1.70447e6 q^{31} +3.01628e6 q^{33} +1.56685e7 q^{35} -2.01332e7 q^{37} +7.21205e6 q^{39} +7.51492e6 q^{41} +8.77900e6 q^{43} +6.14989e7 q^{45} -5.56456e7 q^{47} +8.57741e7 q^{49} -9.01516e7 q^{51} +1.84201e7 q^{53} +1.66651e7 q^{55} +2.05212e8 q^{57} -9.16484e7 q^{59} -4.97163e7 q^{61} +4.95051e8 q^{63} +3.98470e7 q^{65} -1.84401e8 q^{67} +2.21099e8 q^{69} +1.75076e8 q^{71} -3.00349e8 q^{73} -1.68453e6 q^{75} +1.34150e8 q^{77} -5.38224e8 q^{79} +6.88023e8 q^{81} -3.07653e8 q^{83} -4.98092e8 q^{85} -9.63291e8 q^{87} +4.56635e8 q^{89} +3.20759e8 q^{91} -4.30403e8 q^{93} +1.13381e9 q^{95} -1.45156e9 q^{97} +5.26539e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 141 q^{3} + 2051 q^{5} + 2417 q^{7} + 115741 q^{9} + 53118 q^{11} + 228488 q^{13} + 464555 q^{15} + 433095 q^{17} + 434954 q^{19} + 906875 q^{21} + 1124296 q^{23} + 5966065 q^{25} - 7820643 q^{27}+ \cdots + 641626736 q^{99}+O(q^{100}) \) 8 * q - 141 * q^3 + 2051 * q^5 + 2417 * q^7 + 115741 * q^9 + 53118 * q^11 + 228488 * q^13 + 464555 * q^15 + 433095 * q^17 + 434954 * q^19 + 906875 * q^21 + 1124296 * q^23 + 5966065 * q^25 - 7820643 * q^27 - 1913432 * q^29 - 5220148 * q^31 - 1626882 * q^33 - 950707 * q^35 - 419409 * q^37 - 4027101 * q^39 + 12148046 * q^41 - 32143187 * q^43 + 18893544 * q^45 - 57898411 * q^47 + 104302637 * q^49 - 198521477 * q^51 + 273106866 * q^53 - 141214662 * q^55 + 253032058 * q^57 - 50734502 * q^59 + 139520018 * q^61 + 20344158 * q^63 + 58578611 * q^65 - 528144786 * q^67 + 483506576 * q^69 - 334814973 * q^71 + 630864344 * q^73 - 2265796264 * q^75 + 939664146 * q^77 - 865037608 * q^79 + 3194281040 * q^81 - 1520229952 * q^83 + 2130158385 * q^85 - 1167867006 * q^87 + 894700240 * q^89 + 69031937 * q^91 + 3803209548 * q^93 + 2695926542 * q^95 + 980232164 * q^97 + 641626736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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