LMFDB - Embedded newform 208.10.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,10,Mod(1,208)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$107.127453922$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 $ x^5 - 1438x^3 - 4164x^2 + 396957*x - 59580
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.150341$ 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)

$f(q)$	$=$	$q+136.532 q^{3} +2554.62 q^{5} -9399.91 q^{7} -1041.89 q^{9} -44094.1 q^{11} +28561.0 q^{13} +348788. q^{15} +28289.4 q^{17} -273836. q^{19} -1.28339e6 q^{21} +1.12921e6 q^{23} +4.57295e6 q^{25} -2.82962e6 q^{27} -1.63691e6 q^{29} -6.65402e6 q^{31} -6.02028e6 q^{33} -2.40132e7 q^{35} -1.71193e7 q^{37} +3.89950e6 q^{39} -5.15179e6 q^{41} +1.97275e7 q^{43} -2.66162e6 q^{45} -4.82947e7 q^{47} +4.80048e7 q^{49} +3.86242e6 q^{51} -3.06731e7 q^{53} -1.12644e8 q^{55} -3.73875e7 q^{57} +1.15154e7 q^{59} -3.62567e7 q^{61} +9.79364e6 q^{63} +7.29624e7 q^{65} +6.48390e7 q^{67} +1.54174e8 q^{69} +1.47071e8 q^{71} -3.37321e8 q^{73} +6.24356e8 q^{75} +4.14481e8 q^{77} +2.04060e8 q^{79} -3.65828e8 q^{81} -7.61700e8 q^{83} +7.22685e7 q^{85} -2.23491e8 q^{87} -8.29058e8 q^{89} -2.68471e8 q^{91} -9.08490e8 q^{93} -6.99547e8 q^{95} +1.00647e9 q^{97} +4.59410e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27}+ \cdots - 3016199848 q^{99}+O(q^{100}) $ 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9 - 121746 * q^11 + 142805 * q^13 - 105973 * q^15 - 495669 * q^17 + 840738 * q^19 - 1599467 * q^21 + 592152 * q^23 + 1670362 * q^25 - 6847883 * q^27 + 10678182 * q^29 - 12885296 * q^31 + 17278298 * q^33 - 8380731 * q^35 + 7171823 * q^37 - 4598321 * q^39 + 9294012 * q^41 - 12831975 * q^43 + 26135198 * q^45 - 43354215 * q^47 + 25249488 * q^49 - 16905901 * q^51 + 93231780 * q^53 - 99448846 * q^55 + 90173382 * q^57 - 246496182 * q^59 - 132232612 * q^61 + 416955202 * q^63 + 51495483 * q^65 + 369388534 * q^67 - 579986760 * q^69 - 212150457 * q^71 - 252729806 * q^73 + 752457788 * q^75 + 449666118 * q^77 + 1247271728 * q^79 - 317713115 * q^81 - 1696894296 * q^83 - 775363765 * q^85 + 614530466 * q^87 - 753854382 * q^89 - 288437539 * q^91 - 892784668 * q^93 - 1442632962 * q^95 + 3824606 * q^97 - 3016199848 * 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
$3$	136.532	0.973174	0.486587	−	0.873632i	$-0.338242\pi$
$3$	136.532	0.973174	0.486587	+	0.873632i	$0.338242\pi$
$4$	0	0
$5$	2554.62	1.82794	0.913968	−	0.405787i	$-0.133002\pi$
$5$	2554.62	1.82794	0.913968	+	0.405787i	$0.133002\pi$
$6$	0	0
$7$	−9399.91	−1.47973	−0.739865	−	0.672755i	$-0.765111\pi$
$7$	−9399.91	−1.47973	−0.739865	+	0.672755i	$0.765111\pi$
$8$	0	0
$9$	−1041.89	−0.0529333
$10$	0	0
$11$	−44094.1	−0.908058	−0.454029	−	0.890987i	$-0.650014\pi$
$11$	−44094.1	−0.908058	−0.454029	+	0.890987i	$0.650014\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	348788.	1.77890
$16$	0	0
$17$	28289.4	0.0821492	0.0410746	−	0.999156i	$-0.486922\pi$
$17$	28289.4	0.0821492	0.0410746	+	0.999156i	$0.486922\pi$
$18$	0	0
$19$	−273836.	−0.482058	−0.241029	−	0.970518i	$-0.577485\pi$
$19$	−273836.	−0.482058	−0.241029	+	0.970518i	$0.577485\pi$
$20$	0	0
$21$	−1.28339e6	−1.44003
$22$	0	0
$23$	1.12921e6	0.841393	0.420697	−	0.907201i	$-0.361786\pi$
$23$	1.12921e6	0.841393	0.420697	+	0.907201i	$0.361786\pi$
$24$	0	0
$25$	4.57295e6	2.34135
$26$	0	0
$27$	−2.82962e6	−1.02469
$28$	0	0
$29$	−1.63691e6	−0.429768	−0.214884	−	0.976640i	$-0.568937\pi$
$29$	−1.63691e6	−0.429768	−0.214884	+	0.976640i	$0.568937\pi$
$30$	0	0
$31$	−6.65402e6	−1.29407	−0.647033	−	0.762462i	$-0.723991\pi$
$31$	−6.65402e6	−1.29407	−0.647033	+	0.762462i	$0.723991\pi$
$32$	0	0
$33$	−6.02028e6	−0.883698
$34$	0	0
$35$	−2.40132e7	−2.70485
$36$	0	0
$37$	−1.71193e7	−1.50169	−0.750843	−	0.660481i	$-0.770352\pi$
$37$	−1.71193e7	−1.50169	−0.750843	+	0.660481i	$0.770352\pi$
$38$	0	0
$39$	3.89950e6	0.269910
$40$	0	0
$41$	−5.15179e6	−0.284728	−0.142364	−	0.989814i	$-0.545470\pi$
$41$	−5.15179e6	−0.284728	−0.142364	+	0.989814i	$0.545470\pi$
$42$	0	0
$43$	1.97275e7	0.879962	0.439981	−	0.898007i	$-0.354985\pi$
$43$	1.97275e7	0.879962	0.439981	+	0.898007i	$0.354985\pi$
$44$	0	0
$45$	−2.66162e6	−0.0967587
$46$	0	0
$47$	−4.82947e7	−1.44364	−0.721821	−	0.692080i	$-0.756695\pi$
$47$	−4.82947e7	−1.44364	−0.721821	+	0.692080i	$0.756695\pi$
$48$	0	0
$49$	4.80048e7	1.18960
$50$	0	0
$51$	3.86242e6	0.0799454
$52$	0	0
$53$	−3.06731e7	−0.533970	−0.266985	−	0.963701i	$-0.586027\pi$
$53$	−3.06731e7	−0.533970	−0.266985	+	0.963701i	$0.586027\pi$
$54$	0	0
$55$	−1.12644e8	−1.65987
$56$	0	0
$57$	−3.73875e7	−0.469126
$58$	0	0
$59$	1.15154e7	0.123722	0.0618609	−	0.998085i	$-0.480296\pi$
$59$	1.15154e7	0.123722	0.0618609	+	0.998085i	$0.480296\pi$
$60$	0	0
$61$	−3.62567e7	−0.335277	−0.167639	−	0.985849i	$-0.553614\pi$
$61$	−3.62567e7	−0.335277	−0.167639	+	0.985849i	$0.553614\pi$
$62$	0	0
$63$	9.79364e6	0.0783271
$64$	0	0
$65$	7.29624e7	0.506978
$66$	0	0
$67$	6.48390e7	0.393097	0.196548	−	0.980494i	$-0.437027\pi$
$67$	6.48390e7	0.393097	0.196548	+	0.980494i	$0.437027\pi$
$68$	0	0
$69$	1.54174e8	0.818822
$70$	0	0
$71$	1.47071e8	0.686853	0.343427	−	0.939180i	$-0.388412\pi$
$71$	1.47071e8	0.686853	0.343427	+	0.939180i	$0.388412\pi$
$72$	0	0
$73$	−3.37321e8	−1.39024	−0.695122	−	0.718892i	$-0.744650\pi$
$73$	−3.37321e8	−1.39024	−0.695122	+	0.718892i	$0.744650\pi$
$74$	0	0
$75$	6.24356e8	2.27854
$76$	0	0
$77$	4.14481e8	1.34368
$78$	0	0
$79$	2.04060e8	0.589436	0.294718	−	0.955584i	$-0.404774\pi$
$79$	2.04060e8	0.589436	0.294718	+	0.955584i	$0.404774\pi$
$80$	0	0
$81$	−3.65828e8	−0.944265
$82$	0	0
$83$	−7.61700e8	−1.76170	−0.880851	−	0.473394i	$-0.843029\pi$
$83$	−7.61700e8	−1.76170	−0.880851	+	0.473394i	$0.843029\pi$
$84$	0	0
$85$	7.22685e7	0.150163
$86$	0	0
$87$	−2.23491e8	−0.418239
$88$	0	0
$89$	−8.29058e8	−1.40065	−0.700326	−	0.713823i	$-0.746962\pi$
$89$	−8.29058e8	−1.40065	−0.700326	+	0.713823i	$0.746962\pi$
$90$	0	0
$91$	−2.68471e8	−0.410403
$92$	0	0
$93$	−9.08490e8	−1.25935
$94$	0	0
$95$	−6.99547e8	−0.881172
$96$	0	0
$97$	1.00647e9	1.15432	0.577161	−	0.816631i	$-0.304161\pi$
$97$	1.00647e9	1.15432	0.577161	+	0.816631i	$0.304161\pi$
$98$	0	0
$99$	4.59410e7	0.0480665
$100$	0	0
$101$	1.59054e9	1.52089	0.760446	−	0.649401i	$-0.224980\pi$
$101$	1.59054e9	1.52089	0.760446	+	0.649401i	$0.224980\pi$
$102$	0	0
$103$	−1.13889e9	−0.997040	−0.498520	−	0.866878i	$-0.666123\pi$
$103$	−1.13889e9	−0.997040	−0.498520	+	0.866878i	$0.666123\pi$
$104$	0	0
$105$	−3.27858e9	−2.63229
$106$	0	0
$107$	7.21432e8	0.532069	0.266035	−	0.963963i	$-0.414286\pi$
$107$	7.21432e8	0.532069	0.266035	+	0.963963i	$0.414286\pi$
$108$	0	0
$109$	−6.86462e8	−0.465798	−0.232899	−	0.972501i	$-0.574821\pi$
$109$	−6.86462e8	−0.465798	−0.232899	+	0.972501i	$0.574821\pi$
$110$	0	0
$111$	−2.33734e9	−1.46140
$112$	0	0
$113$	−8.33795e8	−0.481068	−0.240534	−	0.970641i	$-0.577323\pi$
$113$	−8.33795e8	−0.481068	−0.240534	+	0.970641i	$0.577323\pi$
$114$	0	0
$115$	2.88470e9	1.53801
$116$	0	0
$117$	−2.97573e7	−0.0146811
$118$	0	0
$119$	−2.65918e8	−0.121559
$120$	0	0
$121$	−4.13658e8	−0.175431
$122$	0	0
$123$	−7.03386e8	−0.277090
$124$	0	0
$125$	6.69264e9	2.45190
$126$	0	0
$127$	4.01307e8	0.136886	0.0684431	−	0.997655i	$-0.478197\pi$
$127$	4.01307e8	0.136886	0.0684431	+	0.997655i	$0.478197\pi$
$128$	0	0
$129$	2.69344e9	0.856356
$130$	0	0
$131$	3.78377e9	1.12255	0.561273	−	0.827631i	$-0.310312\pi$
$131$	3.78377e9	1.12255	0.561273	+	0.827631i	$0.310312\pi$
$132$	0	0
$133$	2.57404e9	0.713317
$134$	0	0
$135$	−7.22860e9	−1.87306
$136$	0	0
$137$	1.45518e9	0.352919	0.176459	−	0.984308i	$-0.443536\pi$
$137$	1.45518e9	0.352919	0.176459	+	0.984308i	$0.443536\pi$
$138$	0	0
$139$	−1.50381e9	−0.341685	−0.170842	−	0.985298i	$-0.554649\pi$
$139$	−1.50381e9	−0.341685	−0.170842	+	0.985298i	$0.554649\pi$
$140$	0	0
$141$	−6.59380e9	−1.40491
$142$	0	0
$143$	−1.25937e9	−0.251850
$144$	0	0
$145$	−4.18168e9	−0.785588
$146$	0	0
$147$	6.55421e9	1.15769
$148$	0	0
$149$	−4.28624e9	−0.712423	−0.356212	−	0.934405i	$-0.615932\pi$
$149$	−4.28624e9	−0.712423	−0.356212	+	0.934405i	$0.615932\pi$
$150$	0	0
$151$	−4.79918e8	−0.0751226	−0.0375613	−	0.999294i	$-0.511959\pi$
$151$	−4.79918e8	−0.0751226	−0.0375613	+	0.999294i	$0.511959\pi$
$152$	0	0
$153$	−2.94743e7	−0.00434843
$154$	0	0
$155$	−1.69985e10	−2.36547
$156$	0	0
$157$	−8.24624e9	−1.08320	−0.541598	−	0.840637i	$-0.682181\pi$
$157$	−8.24624e9	−1.08320	−0.541598	+	0.840637i	$0.682181\pi$
$158$	0	0
$159$	−4.18788e9	−0.519646
$160$	0	0
$161$	−1.06145e10	−1.24504
$162$	0	0
$163$	5.93537e9	0.658573	0.329286	−	0.944230i	$-0.393192\pi$
$163$	5.93537e9	0.658573	0.329286	+	0.944230i	$0.393192\pi$
$164$	0	0
$165$	−1.53795e10	−1.61534
$166$	0	0
$167$	7.41172e9	0.737386	0.368693	−	0.929551i	$-0.379805\pi$
$167$	7.41172e9	0.737386	0.368693	+	0.929551i	$0.379805\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	2.85306e8	0.0255169
$172$	0	0
$173$	5.63923e9	0.478643	0.239322	−	0.970940i	$-0.423075\pi$
$173$	5.63923e9	0.478643	0.239322	+	0.970940i	$0.423075\pi$
$174$	0	0
$175$	−4.29853e10	−3.46457
$176$	0	0
$177$	1.57223e9	0.120403
$178$	0	0
$179$	−1.23881e10	−0.901916	−0.450958	−	0.892545i	$-0.648918\pi$
$179$	−1.23881e10	−0.901916	−0.450958	+	0.892545i	$0.648918\pi$
$180$	0	0
$181$	−2.45852e10	−1.70263	−0.851314	−	0.524657i	$-0.824194\pi$
$181$	−2.45852e10	−1.70263	−0.851314	+	0.524657i	$0.824194\pi$
$182$	0	0
$183$	−4.95022e9	−0.326283
$184$	0	0
$185$	−4.37334e10	−2.74499
$186$	0	0
$187$	−1.24739e9	−0.0745962
$188$	0	0
$189$	2.65982e10	1.51626
$190$	0	0
$191$	1.06604e10	0.579592	0.289796	−	0.957088i	$-0.406413\pi$
$191$	1.06604e10	0.579592	0.289796	+	0.957088i	$0.406413\pi$
$192$	0	0
$193$	−2.09640e10	−1.08759	−0.543797	−	0.839217i	$-0.683014\pi$
$193$	−2.09640e10	−1.08759	−0.543797	+	0.839217i	$0.683014\pi$
$194$	0	0
$195$	9.96174e9	0.493378
$196$	0	0
$197$	1.27051e10	0.601009	0.300504	−	0.953780i	$-0.402845\pi$
$197$	1.27051e10	0.601009	0.300504	+	0.953780i	$0.402845\pi$
$198$	0	0
$199$	−2.57825e9	−0.116543	−0.0582715	−	0.998301i	$-0.518559\pi$
$199$	−2.57825e9	−0.116543	−0.0582715	+	0.998301i	$0.518559\pi$
$200$	0	0
$201$	8.85262e9	0.382551
$202$	0	0
$203$	1.53868e10	0.635941
$204$	0	0
$205$	−1.31608e10	−0.520465
$206$	0	0
$207$	−1.17651e9	−0.0445377
$208$	0	0
$209$	1.20746e10	0.437737
$210$	0	0
$211$	−1.94301e10	−0.674846	−0.337423	−	0.941353i	$-0.609555\pi$
$211$	−1.94301e10	−0.674846	−0.337423	+	0.941353i	$0.609555\pi$
$212$	0	0
$213$	2.00799e10	0.668427
$214$	0	0
$215$	5.03962e10	1.60851
$216$	0	0
$217$	6.25472e10	1.91487
$218$	0	0
$219$	−4.60553e10	−1.35295
$220$	0	0
$221$	8.07973e8	0.0227841
$222$	0	0
$223$	−9.56077e9	−0.258893	−0.129447	−	0.991586i	$-0.541320\pi$
$223$	−9.56077e9	−0.258893	−0.129447	+	0.991586i	$0.541320\pi$
$224$	0	0
$225$	−4.76449e9	−0.123935
$226$	0	0
$227$	2.02032e10	0.505015	0.252507	−	0.967595i	$-0.418745\pi$
$227$	2.02032e10	0.505015	0.252507	+	0.967595i	$0.418745\pi$
$228$	0	0
$229$	−2.45816e10	−0.590678	−0.295339	−	0.955392i	$-0.595433\pi$
$229$	−2.45816e10	−0.590678	−0.295339	+	0.955392i	$0.595433\pi$
$230$	0	0
$231$	5.65901e10	1.30763
$232$	0	0
$233$	7.93260e10	1.76325	0.881625	−	0.471951i	$-0.156450\pi$
$233$	7.93260e10	1.76325	0.881625	+	0.471951i	$0.156450\pi$
$234$	0	0
$235$	−1.23375e11	−2.63888
$236$	0	0
$237$	2.78609e10	0.573623
$238$	0	0
$239$	2.62515e10	0.520431	0.260215	−	0.965551i	$-0.416206\pi$
$239$	2.62515e10	0.520431	0.260215	+	0.965551i	$0.416206\pi$
$240$	0	0
$241$	−1.00766e11	−1.92415	−0.962074	−	0.272787i	$-0.912055\pi$
$241$	−1.00766e11	−1.92415	−0.962074	+	0.272787i	$0.912055\pi$
$242$	0	0
$243$	5.74808e9	0.105753
$244$	0	0
$245$	1.22634e11	2.17452
$246$	0	0
$247$	−7.82103e9	−0.133699
$248$	0	0
$249$	−1.03997e11	−1.71444
$250$	0	0
$251$	8.72780e10	1.38795	0.693973	−	0.720001i	$-0.255858\pi$
$251$	8.72780e10	1.38795	0.693973	+	0.720001i	$0.255858\pi$
$252$	0	0
$253$	−4.97914e10	−0.764034
$254$	0	0
$255$	9.86700e9	0.146135
$256$	0	0
$257$	4.84205e10	0.692358	0.346179	−	0.938169i	$-0.387479\pi$
$257$	4.84205e10	0.692358	0.346179	+	0.938169i	$0.387479\pi$
$258$	0	0
$259$	1.60920e11	2.22209
$260$	0	0
$261$	1.70547e9	0.0227490
$262$	0	0
$263$	4.40656e9	0.0567935	0.0283968	−	0.999597i	$-0.490960\pi$
$263$	4.40656e9	0.0567935	0.0283968	+	0.999597i	$0.490960\pi$
$264$	0	0
$265$	−7.83582e10	−0.976063
$266$	0	0
$267$	−1.13193e11	−1.36308
$268$	0	0
$269$	1.41879e11	1.65209	0.826045	−	0.563604i	$-0.190586\pi$
$269$	1.41879e11	1.65209	0.826045	+	0.563604i	$0.190586\pi$
$270$	0	0
$271$	7.09707e10	0.799313	0.399657	−	0.916665i	$-0.369129\pi$
$271$	7.09707e10	0.799313	0.399657	+	0.916665i	$0.369129\pi$
$272$	0	0
$273$	−3.66550e10	−0.399394
$274$	0	0
$275$	−2.01640e11	−2.12608
$276$	0	0
$277$	1.22293e11	1.24808	0.624039	−	0.781393i	$-0.285491\pi$
$277$	1.22293e11	1.24808	0.624039	+	0.781393i	$0.285491\pi$
$278$	0	0
$279$	6.93274e9	0.0684992
$280$	0	0
$281$	−9.91936e10	−0.949085	−0.474543	−	0.880233i	$-0.657387\pi$
$281$	−9.91936e10	−0.949085	−0.474543	+	0.880233i	$0.657387\pi$
$282$	0	0
$283$	−1.89673e11	−1.75779	−0.878893	−	0.477019i	$-0.841717\pi$
$283$	−1.89673e11	−1.75779	−0.878893	+	0.477019i	$0.841717\pi$
$284$	0	0
$285$	−9.55108e10	−0.857533
$286$	0	0
$287$	4.84264e10	0.421321
$288$	0	0
$289$	−1.17788e11	−0.993252
$290$	0	0
$291$	1.37415e11	1.12336
$292$	0	0
$293$	1.95772e11	1.55183	0.775917	−	0.630835i	$-0.217287\pi$
$293$	1.95772e11	1.55183	0.775917	+	0.630835i	$0.217287\pi$
$294$	0	0
$295$	2.94175e10	0.226155
$296$	0	0
$297$	1.24770e11	0.930475
$298$	0	0
$299$	3.22513e10	0.233360
$300$	0	0
$301$	−1.85437e11	−1.30211
$302$	0	0
$303$	2.17160e11	1.48009
$304$	0	0
$305$	−9.26220e10	−0.612865
$306$	0	0
$307$	7.17504e10	0.461001	0.230500	−	0.973072i	$-0.425964\pi$
$307$	7.17504e10	0.461001	0.230500	+	0.973072i	$0.425964\pi$
$308$	0	0
$309$	−1.55495e11	−0.970293
$310$	0	0
$311$	−2.11023e10	−0.127911	−0.0639554	−	0.997953i	$-0.520372\pi$
$311$	−2.11023e10	−0.127911	−0.0639554	+	0.997953i	$0.520372\pi$
$312$	0	0
$313$	−1.44667e11	−0.851963	−0.425982	−	0.904732i	$-0.640071\pi$
$313$	−1.44667e11	−0.851963	−0.425982	+	0.904732i	$0.640071\pi$
$314$	0	0
$315$	2.50190e10	0.143177
$316$	0	0
$317$	5.78709e10	0.321880	0.160940	−	0.986964i	$-0.448547\pi$
$317$	5.78709e10	0.321880	0.160940	+	0.986964i	$0.448547\pi$
$318$	0	0
$319$	7.21781e10	0.390254
$320$	0	0
$321$	9.84988e10	0.517796
$322$	0	0
$323$	−7.74665e9	−0.0396007
$324$	0	0
$325$	1.30608e11	0.649373
$326$	0	0
$327$	−9.37244e10	−0.453302
$328$	0	0
$329$	4.53966e11	2.13620
$330$	0	0
$331$	1.00283e11	0.459199	0.229600	−	0.973285i	$-0.426258\pi$
$331$	1.00283e11	0.459199	0.229600	+	0.973285i	$0.426258\pi$
$332$	0	0
$333$	1.78364e10	0.0794892
$334$	0	0
$335$	1.65639e11	0.718555
$336$	0	0
$337$	−5.73297e10	−0.242128	−0.121064	−	0.992645i	$-0.538631\pi$
$337$	−5.73297e10	−0.242128	−0.121064	+	0.992645i	$0.538631\pi$
$338$	0	0
$339$	−1.13840e11	−0.468163
$340$	0	0
$341$	2.93403e11	1.17509
$342$	0	0
$343$	−7.19204e10	−0.280562
$344$	0	0
$345$	3.93855e11	1.49675
$346$	0	0
$347$	2.34072e10	0.0866695	0.0433347	−	0.999061i	$-0.486202\pi$
$347$	2.34072e10	0.0866695	0.0433347	+	0.999061i	$0.486202\pi$
$348$	0	0
$349$	3.92804e11	1.41730	0.708649	−	0.705562i	$-0.249305\pi$
$349$	3.92804e11	1.41730	0.708649	+	0.705562i	$0.249305\pi$
$350$	0	0
$351$	−8.08168e10	−0.284197
$352$	0	0
$353$	2.16422e11	0.741849	0.370925	−	0.928663i	$-0.379041\pi$
$353$	2.16422e11	0.741849	0.370925	+	0.928663i	$0.379041\pi$
$354$	0	0
$355$	3.75710e11	1.25552
$356$	0	0
$357$	−3.63064e10	−0.118298
$358$	0	0
$359$	−3.49576e11	−1.11075	−0.555376	−	0.831600i	$-0.687426\pi$
$359$	−3.49576e11	−1.11075	−0.555376	+	0.831600i	$0.687426\pi$
$360$	0	0
$361$	−2.47701e11	−0.767620
$362$	0	0
$363$	−5.64778e10	−0.170725
$364$	0	0
$365$	−8.61727e11	−2.54128
$366$	0	0
$367$	1.83237e9	0.00527248	0.00263624	−	0.999997i	$-0.499161\pi$
$367$	1.83237e9	0.00527248	0.00263624	+	0.999997i	$0.499161\pi$
$368$	0	0
$369$	5.36758e9	0.0150716
$370$	0	0
$371$	2.88325e11	0.790132
$372$	0	0
$373$	5.52030e10	0.147663	0.0738317	−	0.997271i	$-0.476477\pi$
$373$	5.52030e10	0.147663	0.0738317	+	0.997271i	$0.476477\pi$
$374$	0	0
$375$	9.13763e11	2.38612
$376$	0	0
$377$	−4.67518e10	−0.119196
$378$	0	0
$379$	−2.25258e11	−0.560795	−0.280397	−	0.959884i	$-0.590466\pi$
$379$	−2.25258e11	−0.560795	−0.280397	+	0.959884i	$0.590466\pi$
$380$	0	0
$381$	5.47914e10	0.133214
$382$	0	0
$383$	−4.64079e11	−1.10204	−0.551020	−	0.834492i	$-0.685761\pi$
$383$	−4.64079e11	−1.10204	−0.551020	+	0.834492i	$0.685761\pi$
$384$	0	0
$385$	1.05884e12	2.45616
$386$	0	0
$387$	−2.05538e10	−0.0465793
$388$	0	0
$389$	−2.63061e11	−0.582482	−0.291241	−	0.956650i	$-0.594068\pi$
$389$	−2.63061e11	−0.582482	−0.291241	+	0.956650i	$0.594068\pi$
$390$	0	0
$391$	3.19446e10	0.0691198
$392$	0	0
$393$	5.16608e11	1.09243
$394$	0	0
$395$	5.21296e11	1.07745
$396$	0	0
$397$	−1.80382e10	−0.0364449	−0.0182224	−	0.999834i	$-0.505801\pi$
$397$	−1.80382e10	−0.0364449	−0.0182224	+	0.999834i	$0.505801\pi$
$398$	0	0
$399$	3.51440e11	0.694181
$400$	0	0
$401$	7.18792e10	0.138821	0.0694103	−	0.997588i	$-0.477888\pi$
$401$	7.18792e10	0.138821	0.0694103	+	0.997588i	$0.477888\pi$
$402$	0	0
$403$	−1.90046e11	−0.358910
$404$	0	0
$405$	−9.34549e11	−1.72606
$406$	0	0
$407$	7.54862e11	1.36362
$408$	0	0
$409$	5.31862e11	0.939819	0.469910	−	0.882715i	$-0.344287\pi$
$409$	5.31862e11	0.939819	0.469910	+	0.882715i	$0.344287\pi$
$410$	0	0
$411$	1.98680e11	0.343451
$412$	0	0
$413$	−1.08244e11	−0.183075
$414$	0	0
$415$	−1.94585e12	−3.22028
$416$	0	0
$417$	−2.05319e11	−0.332519
$418$	0	0
$419$	−1.45942e11	−0.231322	−0.115661	−	0.993289i	$-0.536899\pi$
$419$	−1.45942e11	−0.231322	−0.115661	+	0.993289i	$0.536899\pi$
$420$	0	0
$421$	3.66247e11	0.568204	0.284102	−	0.958794i	$-0.408305\pi$
$421$	3.66247e11	0.568204	0.284102	+	0.958794i	$0.408305\pi$
$422$	0	0
$423$	5.03176e10	0.0764168
$424$	0	0
$425$	1.29366e11	0.192340
$426$	0	0
$427$	3.40810e11	0.496120
$428$	0	0
$429$	−1.71945e11	−0.245094
$430$	0	0
$431$	−7.66389e11	−1.06980	−0.534899	−	0.844916i	$-0.679650\pi$
$431$	−7.66389e11	−1.06980	−0.534899	+	0.844916i	$0.679650\pi$
$432$	0	0
$433$	−1.24176e12	−1.69763	−0.848813	−	0.528694i	$-0.822682\pi$
$433$	−1.24176e12	−1.69763	−0.848813	+	0.528694i	$0.822682\pi$
$434$	0	0
$435$	−5.70935e11	−0.764513
$436$	0	0
$437$	−3.09218e11	−0.405601
$438$	0	0
$439$	1.48429e12	1.90734	0.953671	−	0.300852i	$-0.0972710\pi$
$439$	1.48429e12	1.90734	0.953671	+	0.300852i	$0.0972710\pi$
$440$	0	0
$441$	−5.00155e10	−0.0629696
$442$	0	0
$443$	−1.03343e12	−1.27487	−0.637433	−	0.770506i	$-0.720004\pi$
$443$	−1.03343e12	−1.27487	−0.637433	+	0.770506i	$0.720004\pi$
$444$	0	0
$445$	−2.11793e12	−2.56030
$446$	0	0
$447$	−5.85211e11	−0.693312
$448$	0	0
$449$	−5.83072e11	−0.677039	−0.338519	−	0.940959i	$-0.609926\pi$
$449$	−5.83072e11	−0.677039	−0.338519	+	0.940959i	$0.609926\pi$
$450$	0	0
$451$	2.27163e11	0.258550
$452$	0	0
$453$	−6.55244e10	−0.0731073
$454$	0	0
$455$	−6.85841e11	−0.750191
$456$	0	0
$457$	7.39681e11	0.793271	0.396635	−	0.917976i	$-0.370178\pi$
$457$	7.39681e11	0.793271	0.396635	+	0.917976i	$0.370178\pi$
$458$	0	0
$459$	−8.00482e10	−0.0841772
$460$	0	0
$461$	1.45843e12	1.50394	0.751971	−	0.659197i	$-0.229104\pi$
$461$	1.45843e12	1.50394	0.751971	+	0.659197i	$0.229104\pi$
$462$	0	0
$463$	1.63194e12	1.65040	0.825200	−	0.564841i	$-0.191063\pi$
$463$	1.63194e12	1.65040	0.825200	+	0.564841i	$0.191063\pi$
$464$	0	0
$465$	−2.32084e12	−2.30201
$466$	0	0
$467$	8.24440e11	0.802109	0.401054	−	0.916054i	$-0.368644\pi$
$467$	8.24440e11	0.802109	0.401054	+	0.916054i	$0.368644\pi$
$468$	0	0
$469$	−6.09481e11	−0.581677
$470$	0	0
$471$	−1.12588e12	−1.05414
$472$	0	0
$473$	−8.69866e11	−0.799056
$474$	0	0
$475$	−1.25224e12	−1.12867
$476$	0	0
$477$	3.19579e10	0.0282648
$478$	0	0
$479$	1.80073e11	0.156293	0.0781463	−	0.996942i	$-0.475100\pi$
$479$	1.80073e11	0.156293	0.0781463	+	0.996942i	$0.475100\pi$
$480$	0	0
$481$	−4.88945e11	−0.416493
$482$	0	0
$483$	−1.44922e12	−1.21164
$484$	0	0
$485$	2.57114e12	2.11003
$486$	0	0
$487$	7.22045e11	0.581680	0.290840	−	0.956772i	$-0.406065\pi$
$487$	7.22045e11	0.581680	0.290840	+	0.956772i	$0.406065\pi$
$488$	0	0
$489$	8.10371e11	0.640905
$490$	0	0
$491$	−1.98313e12	−1.53987	−0.769937	−	0.638119i	$-0.779713\pi$
$491$	−1.98313e12	−1.53987	−0.769937	+	0.638119i	$0.779713\pi$
$492$	0	0
$493$	−4.63072e10	−0.0353051
$494$	0	0
$495$	1.17362e11	0.0878625
$496$	0	0
$497$	−1.38245e12	−1.01636
$498$	0	0
$499$	8.27327e11	0.597344	0.298672	−	0.954356i	$-0.403456\pi$
$499$	8.27327e11	0.597344	0.298672	+	0.954356i	$0.403456\pi$
$500$	0	0
$501$	1.01194e12	0.717605
$502$	0	0
$503$	−4.94554e11	−0.344475	−0.172238	−	0.985055i	$-0.555100\pi$
$503$	−4.94554e11	−0.344475	−0.172238	+	0.985055i	$0.555100\pi$
$504$	0	0
$505$	4.06322e12	2.78009
$506$	0	0
$507$	1.11374e11	0.0748595
$508$	0	0
$509$	−2.31118e12	−1.52617	−0.763087	−	0.646296i	$-0.776317\pi$
$509$	−2.31118e12	−1.52617	−0.763087	+	0.646296i	$0.776317\pi$
$510$	0	0
$511$	3.17079e12	2.05719
$512$	0	0
$513$	7.74852e11	0.493959
$514$	0	0
$515$	−2.90942e12	−1.82252
$516$	0	0
$517$	2.12951e12	1.31091
$518$	0	0
$519$	7.69937e11	0.465803
$520$	0	0
$521$	2.20997e12	1.31406	0.657031	−	0.753864i	$-0.271812\pi$
$521$	2.20997e12	1.31406	0.657031	+	0.753864i	$0.271812\pi$
$522$	0	0
$523$	−2.43867e12	−1.42527	−0.712633	−	0.701537i	$-0.752497\pi$
$523$	−2.43867e12	−1.42527	−0.712633	+	0.701537i	$0.752497\pi$
$524$	0	0
$525$	−5.86889e12	−3.37162
$526$	0	0
$527$	−1.88238e11	−0.106307
$528$	0	0
$529$	−5.26040e11	−0.292057
$530$	0	0
$531$	−1.19978e10	−0.00654900
$532$	0	0
$533$	−1.47140e11	−0.0789694
$534$	0	0
$535$	1.84298e12	0.972588
$536$	0	0
$537$	−1.69138e12	−0.877720
$538$	0	0
$539$	−2.11673e12	−1.08023
$540$	0	0
$541$	−1.84418e12	−0.925585	−0.462792	−	0.886467i	$-0.653152\pi$
$541$	−1.84418e12	−0.925585	−0.462792	+	0.886467i	$0.653152\pi$
$542$	0	0
$543$	−3.35667e12	−1.65695
$544$	0	0
$545$	−1.75365e12	−0.851448
$546$	0	0
$547$	2.62022e12	1.25140	0.625698	−	0.780065i	$-0.284814\pi$
$547$	2.62022e12	1.25140	0.625698	+	0.780065i	$0.284814\pi$
$548$	0	0
$549$	3.77754e10	0.0177473
$550$	0	0
$551$	4.48245e11	0.207173
$552$	0	0
$553$	−1.91815e12	−0.872206
$554$	0	0
$555$	−5.97102e12	−2.67135
$556$	0	0
$557$	2.45908e12	1.08249	0.541245	−	0.840865i	$-0.317953\pi$
$557$	2.45908e12	1.08249	0.541245	+	0.840865i	$0.317953\pi$
$558$	0	0
$559$	5.63437e11	0.244058
$560$	0	0
$561$	−1.70310e11	−0.0725950
$562$	0	0
$563$	1.30659e12	0.548091	0.274046	−	0.961717i	$-0.411638\pi$
$563$	1.30659e12	0.548091	0.274046	+	0.961717i	$0.411638\pi$
$564$	0	0
$565$	−2.13003e12	−0.879361
$566$	0	0
$567$	3.43875e12	1.39726
$568$	0	0
$569$	−2.68810e12	−1.07508	−0.537539	−	0.843239i	$-0.680646\pi$
$569$	−2.68810e12	−1.07508	−0.537539	+	0.843239i	$0.680646\pi$
$570$	0	0
$571$	−2.99602e12	−1.17946	−0.589729	−	0.807601i	$-0.700765\pi$
$571$	−2.99602e12	−1.17946	−0.589729	+	0.807601i	$0.700765\pi$
$572$	0	0
$573$	1.45549e12	0.564044
$574$	0	0
$575$	5.16381e12	1.97000
$576$	0	0
$577$	6.04343e11	0.226982	0.113491	−	0.993539i	$-0.463797\pi$
$577$	6.04343e11	0.226982	0.113491	+	0.993539i	$0.463797\pi$
$578$	0	0
$579$	−2.86227e12	−1.05842
$580$	0	0
$581$	7.15991e12	2.60684
$582$	0	0
$583$	1.35250e12	0.484876
$584$	0	0
$585$	−7.60186e10	−0.0268360
$586$	0	0
$587$	7.41276e11	0.257697	0.128848	−	0.991664i	$-0.458872\pi$
$587$	7.41276e11	0.257697	0.128848	+	0.991664i	$0.458872\pi$
$588$	0	0
$589$	1.82211e12	0.623816
$590$	0	0
$591$	1.73466e12	0.584886
$592$	0	0
$593$	−5.55536e12	−1.84487	−0.922435	−	0.386152i	$-0.873804\pi$
$593$	−5.55536e12	−1.84487	−0.922435	+	0.386152i	$0.873804\pi$
$594$	0	0
$595$	−6.79318e11	−0.222201
$596$	0	0
$597$	−3.52015e11	−0.113417
$598$	0	0
$599$	3.35363e12	1.06437	0.532187	−	0.846627i	$-0.321370\pi$
$599$	3.35363e12	1.06437	0.532187	+	0.846627i	$0.321370\pi$
$600$	0	0
$601$	1.88646e12	0.589809	0.294905	−	0.955527i	$-0.404712\pi$
$601$	1.88646e12	0.589809	0.294905	+	0.955527i	$0.404712\pi$
$602$	0	0
$603$	−6.75548e10	−0.0208079
$604$	0	0
$605$	−1.05674e12	−0.320677
$606$	0	0
$607$	−1.01740e12	−0.304188	−0.152094	−	0.988366i	$-0.548602\pi$
$607$	−1.01740e12	−0.304188	−0.152094	+	0.988366i	$0.548602\pi$
$608$	0	0
$609$	2.10080e12	0.618881
$610$	0	0
$611$	−1.37935e12	−0.400394
$612$	0	0
$613$	−5.21446e11	−0.149155	−0.0745774	−	0.997215i	$-0.523761\pi$
$613$	−5.21446e11	−0.149155	−0.0745774	+	0.997215i	$0.523761\pi$
$614$	0	0
$615$	−1.79688e12	−0.506503
$616$	0	0
$617$	1.14541e12	0.318183	0.159092	−	0.987264i	$-0.449143\pi$
$617$	1.14541e12	0.318183	0.159092	+	0.987264i	$0.449143\pi$
$618$	0	0
$619$	1.10612e12	0.302827	0.151414	−	0.988470i	$-0.451617\pi$
$619$	1.10612e12	0.302827	0.151414	+	0.988470i	$0.451617\pi$
$620$	0	0
$621$	−3.19523e12	−0.862165
$622$	0	0
$623$	7.79308e12	2.07259
$624$	0	0
$625$	8.16561e12	2.14057
$626$	0	0
$627$	1.64857e12	0.425994
$628$	0	0
$629$	−4.84295e11	−0.123362
$630$	0	0
$631$	−3.20443e12	−0.804671	−0.402336	−	0.915492i	$-0.631802\pi$
$631$	−3.20443e12	−0.804671	−0.402336	+	0.915492i	$0.631802\pi$
$632$	0	0
$633$	−2.65284e12	−0.656742
$634$	0	0
$635$	1.02518e12	0.250219
$636$	0	0
$637$	1.37106e12	0.329937
$638$	0	0
$639$	−1.53231e11	−0.0363574
$640$	0	0
$641$	−4.90467e12	−1.14749	−0.573745	−	0.819034i	$-0.694510\pi$
$641$	−4.90467e12	−1.14749	−0.573745	+	0.819034i	$0.694510\pi$
$642$	0	0
$643$	−1.53529e12	−0.354194	−0.177097	−	0.984193i	$-0.556671\pi$
$643$	−1.53529e12	−0.354194	−0.177097	+	0.984193i	$0.556671\pi$
$644$	0	0
$645$	6.88072e12	1.56536
$646$	0	0
$647$	6.61326e11	0.148370	0.0741850	−	0.997244i	$-0.476364\pi$
$647$	6.61326e11	0.148370	0.0741850	+	0.997244i	$0.476364\pi$
$648$	0	0
$649$	−5.07763e11	−0.112347
$650$	0	0
$651$	8.53973e12	1.86350
$652$	0	0
$653$	4.61469e12	0.993192	0.496596	−	0.867982i	$-0.334583\pi$
$653$	4.61469e12	0.993192	0.496596	+	0.867982i	$0.334583\pi$
$654$	0	0
$655$	9.66610e12	2.05194
$656$	0	0
$657$	3.51450e11	0.0735902
$658$	0	0
$659$	9.04687e12	1.86859	0.934294	−	0.356502i	$-0.116031\pi$
$659$	9.04687e12	1.86859	0.934294	+	0.356502i	$0.116031\pi$
$660$	0	0
$661$	3.59295e12	0.732057	0.366028	−	0.930604i	$-0.380717\pi$
$661$	3.59295e12	0.732057	0.366028	+	0.930604i	$0.380717\pi$
$662$	0	0
$663$	1.10315e11	0.0221729
$664$	0	0
$665$	6.57568e12	1.30390
$666$	0	0
$667$	−1.84841e12	−0.361604
$668$	0	0
$669$	−1.30536e12	−0.251948
$670$	0	0
$671$	1.59871e12	0.304451
$672$	0	0
$673$	2.13714e12	0.401573	0.200786	−	0.979635i	$-0.435650\pi$
$673$	2.13714e12	0.401573	0.200786	+	0.979635i	$0.435650\pi$
$674$	0	0
$675$	−1.29397e13	−2.39915
$676$	0	0
$677$	−2.55166e12	−0.466847	−0.233423	−	0.972375i	$-0.574993\pi$
$677$	−2.55166e12	−0.466847	−0.233423	+	0.972375i	$0.574993\pi$
$678$	0	0
$679$	−9.46070e12	−1.70809
$680$	0	0
$681$	2.75840e12	0.491467
$682$	0	0
$683$	−7.63188e12	−1.34196	−0.670978	−	0.741477i	$-0.734126\pi$
$683$	−7.63188e12	−1.34196	−0.670978	+	0.741477i	$0.734126\pi$
$684$	0	0
$685$	3.71743e12	0.645113
$686$	0	0
$687$	−3.35619e12	−0.574832
$688$	0	0
$689$	−8.76056e11	−0.148097
$690$	0	0
$691$	2.36970e12	0.395405	0.197703	−	0.980262i	$-0.436652\pi$
$691$	2.36970e12	0.395405	0.197703	+	0.980262i	$0.436652\pi$
$692$	0	0
$693$	−4.31842e11	−0.0711255
$694$	0	0
$695$	−3.84165e12	−0.624578
$696$	0	0
$697$	−1.45741e11	−0.0233902
$698$	0	0
$699$	1.08306e13	1.71595
$700$	0	0
$701$	−1.72803e12	−0.270284	−0.135142	−	0.990826i	$-0.543149\pi$
$701$	−1.72803e12	−0.270284	−0.135142	+	0.990826i	$0.543149\pi$
$702$	0	0
$703$	4.68789e12	0.723900
$704$	0	0
$705$	−1.68446e13	−2.56809
$706$	0	0
$707$	−1.49509e13	−2.25051
$708$	0	0
$709$	−3.26440e12	−0.485172	−0.242586	−	0.970130i	$-0.577996\pi$
$709$	−3.26440e12	−0.485172	−0.242586	+	0.970130i	$0.577996\pi$
$710$	0	0
$711$	−2.12608e11	−0.0312008
$712$	0	0
$713$	−7.51378e12	−1.08882
$714$	0	0
$715$	−3.21721e12	−0.460365
$716$	0	0
$717$	3.58418e12	0.506469
$718$	0	0
$719$	6.17850e11	0.0862191	0.0431095	−	0.999070i	$-0.486274\pi$
$719$	6.17850e11	0.0862191	0.0431095	+	0.999070i	$0.486274\pi$
$720$	0	0
$721$	1.07054e13	1.47535
$722$	0	0
$723$	−1.37579e13	−1.87253
$724$	0	0
$725$	−7.48550e12	−1.00624
$726$	0	0
$727$	−5.84000e12	−0.775368	−0.387684	−	0.921792i	$-0.626725\pi$
$727$	−5.84000e12	−0.775368	−0.387684	+	0.921792i	$0.626725\pi$
$728$	0	0
$729$	7.98538e12	1.04718
$730$	0	0
$731$	5.58079e11	0.0722882
$732$	0	0
$733$	−7.18908e10	−0.00919826	−0.00459913	−	0.999989i	$-0.501464\pi$
$733$	−7.18908e10	−0.00919826	−0.00459913	+	0.999989i	$0.501464\pi$
$734$	0	0
$735$	1.67435e13	2.11618
$736$	0	0
$737$	−2.85902e12	−0.356954
$738$	0	0
$739$	1.01033e13	1.24613	0.623067	−	0.782168i	$-0.285886\pi$
$739$	1.01033e13	1.24613	0.623067	+	0.782168i	$0.285886\pi$
$740$	0	0
$741$	−1.06783e12	−0.130112
$742$	0	0
$743$	6.09193e12	0.733340	0.366670	−	0.930351i	$-0.380498\pi$
$743$	6.09193e12	0.733340	0.366670	+	0.930351i	$0.380498\pi$
$744$	0	0
$745$	−1.09497e13	−1.30226
$746$	0	0
$747$	7.93604e11	0.0932527
$748$	0	0
$749$	−6.78139e12	−0.787319
$750$	0	0
$751$	1.43572e13	1.64698	0.823491	−	0.567330i	$-0.192024\pi$
$751$	1.43572e13	1.64698	0.823491	+	0.567330i	$0.192024\pi$
$752$	0	0
$753$	1.19163e13	1.35071
$754$	0	0
$755$	−1.22601e12	−0.137319
$756$	0	0
$757$	1.33786e13	1.48074	0.740370	−	0.672200i	$-0.234651\pi$
$757$	1.33786e13	1.48074	0.740370	+	0.672200i	$0.234651\pi$
$758$	0	0
$759$	−6.79815e12	−0.743537
$760$	0	0
$761$	1.00649e13	1.08788	0.543938	−	0.839126i	$-0.316933\pi$
$761$	1.00649e13	1.08788	0.543938	+	0.839126i	$0.316933\pi$
$762$	0	0
$763$	6.45269e12	0.689255
$764$	0	0
$765$	−7.52956e10	−0.00794865
$766$	0	0
$767$	3.28892e11	0.0343142
$768$	0	0
$769$	4.34623e12	0.448172	0.224086	−	0.974569i	$-0.428060\pi$
$769$	4.34623e12	0.448172	0.224086	+	0.974569i	$0.428060\pi$
$770$	0	0
$771$	6.61097e12	0.673784
$772$	0	0
$773$	2.50017e12	0.251862	0.125931	−	0.992039i	$-0.459808\pi$
$773$	2.50017e12	0.251862	0.125931	+	0.992039i	$0.459808\pi$
$774$	0	0
$775$	−3.04285e13	−3.02986
$776$	0	0
$777$	2.19708e13	2.16248
$778$	0	0
$779$	1.41075e12	0.137256
$780$	0	0
$781$	−6.48495e12	−0.623702
$782$	0	0
$783$	4.63183e12	0.440377
$784$	0	0
$785$	−2.10660e13	−1.98001
$786$	0	0
$787$	1.47010e13	1.36603	0.683013	−	0.730406i	$-0.260669\pi$
$787$	1.47010e13	1.36603	0.683013	+	0.730406i	$0.260669\pi$
$788$	0	0
$789$	6.01639e11	0.0552700
$790$	0	0
$791$	7.83761e12	0.711851
$792$	0	0
$793$	−1.03553e12	−0.0929892
$794$	0	0
$795$	−1.06984e13	−0.949879
$796$	0	0
$797$	8.96898e12	0.787373	0.393687	−	0.919245i	$-0.371199\pi$
$797$	8.96898e12	0.787373	0.393687	+	0.919245i	$0.371199\pi$
$798$	0	0
$799$	−1.36623e12	−0.118594
$800$	0	0
$801$	8.63785e11	0.0741411
$802$	0	0
$803$	1.48739e13	1.26242
$804$	0	0
$805$	−2.71159e13	−2.27584
$806$	0	0
$807$	1.93711e13	1.60777
$808$	0	0
$809$	1.74450e13	1.43187	0.715934	−	0.698168i	$-0.246001\pi$
$809$	1.74450e13	1.43187	0.715934	+	0.698168i	$0.246001\pi$
$810$	0	0
$811$	8.34611e12	0.677470	0.338735	−	0.940882i	$-0.390001\pi$
$811$	8.34611e12	0.677470	0.338735	+	0.940882i	$0.390001\pi$
$812$	0	0
$813$	9.68980e12	0.777871
$814$	0	0
$815$	1.51626e13	1.20383
$816$	0	0
$817$	−5.40210e12	−0.424193
$818$	0	0
$819$	2.79716e11	0.0217240
$820$	0	0
$821$	1.88288e13	1.44637	0.723183	−	0.690657i	$-0.242678\pi$
$821$	1.88288e13	1.44637	0.723183	+	0.690657i	$0.242678\pi$
$822$	0	0
$823$	−4.02178e12	−0.305576	−0.152788	−	0.988259i	$-0.548825\pi$
$823$	−4.02178e12	−0.305576	−0.152788	+	0.988259i	$0.548825\pi$
$824$	0	0
$825$	−2.75304e13	−2.06904
$826$	0	0
$827$	−3.84013e12	−0.285477	−0.142738	−	0.989760i	$-0.545591\pi$
$827$	−3.84013e12	−0.285477	−0.142738	+	0.989760i	$0.545591\pi$
$828$	0	0
$829$	1.81041e13	1.33132	0.665660	−	0.746255i	$-0.268150\pi$
$829$	1.81041e13	1.33132	0.665660	+	0.746255i	$0.268150\pi$
$830$	0	0
$831$	1.66969e13	1.21460
$832$	0	0
$833$	1.35803e12	0.0977249
$834$	0	0
$835$	1.89341e13	1.34789
$836$	0	0
$837$	1.88284e13	1.32601
$838$	0	0
$839$	−1.83543e13	−1.27882	−0.639410	−	0.768866i	$-0.720821\pi$
$839$	−1.83543e13	−1.27882	−0.639410	+	0.768866i	$0.720821\pi$
$840$	0	0
$841$	−1.18277e13	−0.815300
$842$	0	0
$843$	−1.35431e13	−0.923625
$844$	0	0
$845$	2.08388e12	0.140610
$846$	0	0
$847$	3.88835e12	0.259591
$848$	0	0
$849$	−2.58965e13	−1.71063
$850$	0	0
$851$	−1.93313e13	−1.26351
$852$	0	0
$853$	−1.20116e13	−0.776835	−0.388418	−	0.921483i	$-0.626978\pi$
$853$	−1.20116e13	−0.776835	−0.388418	+	0.921483i	$0.626978\pi$
$854$	0	0
$855$	7.28848e11	0.0466433
$856$	0	0
$857$	−6.17723e12	−0.391183	−0.195592	−	0.980685i	$-0.562663\pi$
$857$	−6.17723e12	−0.391183	−0.195592	+	0.980685i	$0.562663\pi$
$858$	0	0
$859$	−2.76160e13	−1.73058	−0.865291	−	0.501269i	$-0.832867\pi$
$859$	−2.76160e13	−1.73058	−0.865291	+	0.501269i	$0.832867\pi$
$860$	0	0
$861$	6.61177e12	0.410019
$862$	0	0
$863$	−9.27582e12	−0.569251	−0.284626	−	0.958639i	$-0.591869\pi$
$863$	−9.27582e12	−0.569251	−0.284626	+	0.958639i	$0.591869\pi$
$864$	0	0
$865$	1.44061e13	0.874929
$866$	0	0
$867$	−1.60818e13	−0.966606
$868$	0	0
$869$	−8.99785e12	−0.535242
$870$	0	0
$871$	1.85187e12	0.109025
$872$	0	0
$873$	−1.04862e12	−0.0611021
$874$	0	0
$875$	−6.29103e13	−3.62815
$876$	0	0
$877$	−2.19078e13	−1.25055	−0.625274	−	0.780405i	$-0.715013\pi$
$877$	−2.19078e13	−1.25055	−0.625274	+	0.780405i	$0.715013\pi$
$878$	0	0
$879$	2.67292e13	1.51020
$880$	0	0
$881$	2.92416e12	0.163534	0.0817672	−	0.996651i	$-0.473944\pi$
$881$	2.92416e12	0.163534	0.0817672	+	0.996651i	$0.473944\pi$
$882$	0	0
$883$	−3.35851e13	−1.85919	−0.929594	−	0.368585i	$-0.879842\pi$
$883$	−3.35851e13	−1.85919	−0.929594	+	0.368585i	$0.879842\pi$
$884$	0	0
$885$	4.01645e12	0.220089
$886$	0	0
$887$	3.31699e13	1.79924	0.899619	−	0.436676i	$-0.143844\pi$
$887$	3.31699e13	1.79924	0.899619	+	0.436676i	$0.143844\pi$
$888$	0	0
$889$	−3.77225e12	−0.202555
$890$	0	0
$891$	1.61308e13	0.857447
$892$	0	0
$893$	1.32248e13	0.695920
$894$	0	0
$895$	−3.16469e13	−1.64864
$896$	0	0
$897$	4.40335e12	0.227100
$898$	0	0
$899$	1.08920e13	0.556148
$900$	0	0
$901$	−8.67724e11	−0.0438652
$902$	0	0
$903$	−2.53181e13	−1.26718
$904$	0	0
$905$	−6.28057e13	−3.11229
$906$	0	0
$907$	−2.27086e13	−1.11418	−0.557092	−	0.830451i	$-0.688083\pi$
$907$	−2.27086e13	−1.11418	−0.557092	+	0.830451i	$0.688083\pi$
$908$	0	0
$909$	−1.65716e12	−0.0805059
$910$	0	0
$911$	−4.92090e12	−0.236707	−0.118354	−	0.992972i	$-0.537762\pi$
$911$	−4.92090e12	−0.236707	−0.118354	+	0.992972i	$0.537762\pi$
$912$	0	0
$913$	3.35865e13	1.59973
$914$	0	0
$915$	−1.26459e13	−0.596424
$916$	0	0
$917$	−3.55671e13	−1.66107
$918$	0	0
$919$	9.41621e12	0.435468	0.217734	−	0.976008i	$-0.430133\pi$
$919$	9.41621e12	0.435468	0.217734	+	0.976008i	$0.430133\pi$
$920$	0	0
$921$	9.79626e12	0.448634
$922$	0	0
$923$	4.20049e12	0.190499
$924$	0	0
$925$	−7.82858e13	−3.51597
$926$	0	0
$927$	1.18659e12	0.0527766
$928$	0	0
$929$	−3.87518e13	−1.70695	−0.853476	−	0.521131i	$-0.825510\pi$
$929$	−3.87518e13	−1.70695	−0.853476	+	0.521131i	$0.825510\pi$
$930$	0	0
$931$	−1.31454e13	−0.573458
$932$	0	0
$933$	−2.88114e12	−0.124479
$934$	0	0
$935$	−3.18662e12	−0.136357
$936$	0	0
$937$	1.41425e13	0.599373	0.299687	−	0.954038i	$-0.403118\pi$
$937$	1.41425e13	0.599373	0.299687	+	0.954038i	$0.403118\pi$
$938$	0	0
$939$	−1.97518e13	−0.829108
$940$	0	0
$941$	−2.29704e13	−0.955027	−0.477514	−	0.878624i	$-0.658462\pi$
$941$	−2.29704e13	−0.955027	−0.477514	+	0.878624i	$0.658462\pi$
$942$	0	0
$943$	−5.81744e12	−0.239568
$944$	0	0
$945$	6.79482e13	2.77163
$946$	0	0
$947$	1.50510e12	0.0608120	0.0304060	−	0.999538i	$-0.490320\pi$
$947$	1.50510e12	0.0608120	0.0304060	+	0.999538i	$0.490320\pi$
$948$	0	0
$949$	−9.63423e12	−0.385584
$950$	0	0
$951$	7.90126e12	0.313245
$952$	0	0
$953$	2.42253e13	0.951373	0.475686	−	0.879615i	$-0.342200\pi$
$953$	2.42253e13	0.951373	0.475686	+	0.879615i	$0.342200\pi$
$954$	0	0
$955$	2.72332e13	1.05946
$956$	0	0
$957$	9.85465e12	0.379785
$958$	0	0
$959$	−1.36786e13	−0.522225
$960$	0	0
$961$	1.78364e13	0.674608
$962$	0	0
$963$	−7.51650e11	−0.0281642
$964$	0	0
$965$	−5.35551e13	−1.98805
$966$	0	0
$967$	−2.41172e13	−0.886967	−0.443484	−	0.896282i	$-0.646258\pi$
$967$	−2.41172e13	−0.886967	−0.443484	+	0.896282i	$0.646258\pi$
$968$	0	0
$969$	−1.05767e12	−0.0385383
$970$	0	0
$971$	−3.13584e13	−1.13206	−0.566028	−	0.824386i	$-0.691521\pi$
$971$	−3.13584e13	−1.13206	−0.566028	+	0.824386i	$0.691521\pi$
$972$	0	0
$973$	1.41357e13	0.505602
$974$	0	0
$975$	1.78322e13	0.631953
$976$	0	0
$977$	4.44366e13	1.56033	0.780163	−	0.625576i	$-0.215136\pi$
$977$	4.44366e13	1.56033	0.780163	+	0.625576i	$0.215136\pi$
$978$	0	0
$979$	3.65566e13	1.27187
$980$	0	0
$981$	7.15216e11	0.0246562
$982$	0	0
$983$	−1.53216e13	−0.523376	−0.261688	−	0.965153i	$-0.584279\pi$
$983$	−1.53216e13	−0.523376	−0.261688	+	0.965153i	$0.584279\pi$
$984$	0	0
$985$	3.24567e13	1.09861
$986$	0	0
$987$	6.19812e13	2.07889
$988$	0	0
$989$	2.22765e13	0.740394
$990$	0	0
$991$	−2.32887e13	−0.767034	−0.383517	−	0.923534i	$-0.625287\pi$
$991$	−2.32887e13	−0.767034	−0.383517	+	0.923534i	$0.625287\pi$
$992$	0	0
$993$	1.36919e13	0.446881
$994$	0	0
$995$	−6.58644e12	−0.213033
$996$	0	0
$997$	−1.12247e13	−0.359788	−0.179894	−	0.983686i	$-0.557575\pi$
$997$	−1.12247e13	−0.359788	−0.179894	+	0.983686i	$0.557575\pi$
$998$	0	0
$999$	4.84412e13	1.53876

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.h.1.4		5
4.3	odd	2		13.10.a.b.1.3	✓	5
12.11	even	2		117.10.a.e.1.3		5
20.19	odd	2		325.10.a.b.1.3		5
52.51	odd	2		169.10.a.b.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.a.b.1.3	✓	5	4.3	odd	2
117.10.a.e.1.3		5	12.11	even	2
169.10.a.b.1.3		5	52.51	odd	2
208.10.a.h.1.4		5	1.1	even	1	trivial
325.10.a.b.1.3		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+136.532 q^{3} +2554.62 q^{5} -9399.91 q^{7} -1041.89 q^{9} -44094.1 q^{11} +28561.0 q^{13} +348788. q^{15} +28289.4 q^{17} -273836. q^{19} -1.28339e6 q^{21} +1.12921e6 q^{23} +4.57295e6 q^{25} -2.82962e6 q^{27} -1.63691e6 q^{29} -6.65402e6 q^{31} -6.02028e6 q^{33} -2.40132e7 q^{35} -1.71193e7 q^{37} +3.89950e6 q^{39} -5.15179e6 q^{41} +1.97275e7 q^{43} -2.66162e6 q^{45} -4.82947e7 q^{47} +4.80048e7 q^{49} +3.86242e6 q^{51} -3.06731e7 q^{53} -1.12644e8 q^{55} -3.73875e7 q^{57} +1.15154e7 q^{59} -3.62567e7 q^{61} +9.79364e6 q^{63} +7.29624e7 q^{65} +6.48390e7 q^{67} +1.54174e8 q^{69} +1.47071e8 q^{71} -3.37321e8 q^{73} +6.24356e8 q^{75} +4.14481e8 q^{77} +2.04060e8 q^{79} -3.65828e8 q^{81} -7.61700e8 q^{83} +7.22685e7 q^{85} -2.23491e8 q^{87} -8.29058e8 q^{89} -2.68471e8 q^{91} -9.08490e8 q^{93} -6.99547e8 q^{95} +1.00647e9 q^{97} +4.59410e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27}+ \cdots - 3016199848 q^{99}+O(q^{100}) \) 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9 - 121746 * q^11 + 142805 * q^13 - 105973 * q^15 - 495669 * q^17 + 840738 * q^19 - 1599467 * q^21 + 592152 * q^23 + 1670362 * q^25 - 6847883 * q^27 + 10678182 * q^29 - 12885296 * q^31 + 17278298 * q^33 - 8380731 * q^35 + 7171823 * q^37 - 4598321 * q^39 + 9294012 * q^41 - 12831975 * q^43 + 26135198 * q^45 - 43354215 * q^47 + 25249488 * q^49 - 16905901 * q^51 + 93231780 * q^53 - 99448846 * q^55 + 90173382 * q^57 - 246496182 * q^59 - 132232612 * q^61 + 416955202 * q^63 + 51495483 * q^65 + 369388534 * q^67 - 579986760 * q^69 - 212150457 * q^71 - 252729806 * q^73 + 752457788 * q^75 + 449666118 * q^77 + 1247271728 * q^79 - 317713115 * q^81 - 1696894296 * q^83 - 775363765 * q^85 + 614530466 * q^87 - 753854382 * q^89 - 288437539 * q^91 - 892784668 * q^93 - 1442632962 * q^95 + 3824606 * q^97 - 3016199848 * q^99

Introduction

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