LMFDB - Embedded newform 208.10.a.g.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:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 $ x^4 - x^3 - 1602x^2 + 1544x + 342272
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-36.8028$ 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)

$f(q)$	$=$	$q+204.594 q^{3} -258.914 q^{5} +8862.28 q^{7} +22175.6 q^{9} +O(q^{10})$	$q+204.594 q^{3} -258.914 q^{5} +8862.28 q^{7} +22175.6 q^{9} -36087.9 q^{11} -28561.0 q^{13} -52972.2 q^{15} +327405. q^{17} -265525. q^{19} +1.81317e6 q^{21} +2.42458e6 q^{23} -1.88609e6 q^{25} +509973. q^{27} +3.99178e6 q^{29} +6.45220e6 q^{31} -7.38336e6 q^{33} -2.29457e6 q^{35} -8.15498e6 q^{37} -5.84340e6 q^{39} -720241. q^{41} +4.13245e7 q^{43} -5.74158e6 q^{45} -3.67369e7 q^{47} +3.81864e7 q^{49} +6.69850e7 q^{51} +1.67334e7 q^{53} +9.34367e6 q^{55} -5.43248e7 q^{57} +5.89865e7 q^{59} +1.28785e8 q^{61} +1.96527e8 q^{63} +7.39484e6 q^{65} +1.91470e8 q^{67} +4.96054e8 q^{69} -3.40865e8 q^{71} +3.19890e8 q^{73} -3.85882e8 q^{75} -3.19821e8 q^{77} -2.44328e8 q^{79} -3.32145e8 q^{81} +4.38630e8 q^{83} -8.47697e7 q^{85} +8.16693e8 q^{87} -7.90342e8 q^{89} -2.53116e8 q^{91} +1.32008e9 q^{93} +6.87481e7 q^{95} -1.65696e8 q^{97} -8.00272e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9}+O(q^{10}) $ 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9	$ 4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9} + 40140 q^{11} - 114244 q^{13} - 83307 q^{15} + 78717 q^{17} - 209664 q^{19} + 1138431 q^{21} + 4257444 q^{23} - 2900157 q^{25} + 2077801 q^{27} - 1647936 q^{29} + 11366002 q^{31} - 14413222 q^{33} + 13789797 q^{35} + 4636891 q^{37} - 4655443 q^{39} + 13859538 q^{41} + 33368081 q^{43} - 17423928 q^{45} + 3943005 q^{47} + 23294923 q^{49} + 19664471 q^{51} - 171019326 q^{53} + 121160538 q^{55} - 47829030 q^{57} + 63389388 q^{59} + 77050190 q^{61} + 155695476 q^{63} - 13452231 q^{65} + 41174072 q^{67} + 546642556 q^{69} - 252460989 q^{71} + 594415068 q^{73} - 533318748 q^{75} + 561950454 q^{77} - 115998984 q^{79} + 437803700 q^{81} + 79577862 q^{83} + 549463469 q^{85} + 1087526510 q^{87} - 1152240276 q^{89} - 321054201 q^{91} + 1618266556 q^{93} + 1273705170 q^{95} + 1049098084 q^{97} - 2132181050 q^{99}+O(q^{100}) $ 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9 + 40140 * q^11 - 114244 * q^13 - 83307 * q^15 + 78717 * q^17 - 209664 * q^19 + 1138431 * q^21 + 4257444 * q^23 - 2900157 * q^25 + 2077801 * q^27 - 1647936 * q^29 + 11366002 * q^31 - 14413222 * q^33 + 13789797 * q^35 + 4636891 * q^37 - 4655443 * q^39 + 13859538 * q^41 + 33368081 * q^43 - 17423928 * q^45 + 3943005 * q^47 + 23294923 * q^49 + 19664471 * q^51 - 171019326 * q^53 + 121160538 * q^55 - 47829030 * q^57 + 63389388 * q^59 + 77050190 * q^61 + 155695476 * q^63 - 13452231 * q^65 + 41174072 * q^67 + 546642556 * q^69 - 252460989 * q^71 + 594415068 * q^73 - 533318748 * q^75 + 561950454 * q^77 - 115998984 * q^79 + 437803700 * q^81 + 79577862 * q^83 + 549463469 * q^85 + 1087526510 * q^87 - 1152240276 * q^89 - 321054201 * q^91 + 1618266556 * q^93 + 1273705170 * q^95 + 1049098084 * q^97 - 2132181050 * q^99

Display 100 coefficients 10 coefficients

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
$3$	204.594	1.45830	0.729150	−	0.684354i	$-0.239916\pi$
$3$	204.594	1.45830	0.729150	+	0.684354i	$0.239916\pi$
$4$	0	0
$5$	−258.914	−0.185264	−0.0926319	−	0.995700i	$-0.529528\pi$
$5$	−258.914	−0.185264	−0.0926319	+	0.995700i	$0.529528\pi$
$6$	0	0
$7$	8862.28	1.39510	0.697548	−	0.716538i	$-0.254274\pi$
$7$	8862.28	1.39510	0.697548	+	0.716538i	$0.254274\pi$
$8$	0	0
$9$	22175.6	1.12664
$10$	0	0
$11$	−36087.9	−0.743181	−0.371591	−	0.928397i	$-0.621188\pi$
$11$	−36087.9	−0.743181	−0.371591	+	0.928397i	$0.621188\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	−52972.2	−0.270170
$16$	0	0
$17$	327405.	0.950747	0.475373	−	0.879784i	$-0.342313\pi$
$17$	327405.	0.950747	0.475373	+	0.879784i	$0.342313\pi$
$18$	0	0
$19$	−265525.	−0.467428	−0.233714	−	0.972305i	$-0.575088\pi$
$19$	−265525.	−0.467428	−0.233714	+	0.972305i	$0.575088\pi$
$20$	0	0
$21$	1.81317e6	2.03447
$22$	0	0
$23$	2.42458e6	1.80660	0.903299	−	0.429012i	$-0.141138\pi$
$23$	2.42458e6	1.80660	0.903299	+	0.429012i	$0.141138\pi$
$24$	0	0
$25$	−1.88609e6	−0.965677
$26$	0	0
$27$	509973.	0.184676
$28$	0	0
$29$	3.99178e6	1.04803	0.524017	−	0.851708i	$-0.324433\pi$
$29$	3.99178e6	1.04803	0.524017	+	0.851708i	$0.324433\pi$
$30$	0	0
$31$	6.45220e6	1.25482	0.627408	−	0.778691i	$-0.284116\pi$
$31$	6.45220e6	1.25482	0.627408	+	0.778691i	$0.284116\pi$
$32$	0	0
$33$	−7.38336e6	−1.08378
$34$	0	0
$35$	−2.29457e6	−0.258461
$36$	0	0
$37$	−8.15498e6	−0.715344	−0.357672	−	0.933847i	$-0.616429\pi$
$37$	−8.15498e6	−0.715344	−0.357672	+	0.933847i	$0.616429\pi$
$38$	0	0
$39$	−5.84340e6	−0.404460
$40$	0	0
$41$	−720241.	−0.0398062	−0.0199031	−	0.999802i	$-0.506336\pi$
$41$	−720241.	−0.0398062	−0.0199031	+	0.999802i	$0.506336\pi$
$42$	0	0
$43$	4.13245e7	1.84332	0.921658	−	0.388004i	$-0.126835\pi$
$43$	4.13245e7	1.84332	0.921658	+	0.388004i	$0.126835\pi$
$44$	0	0
$45$	−5.74158e6	−0.208725
$46$	0	0
$47$	−3.67369e7	−1.09815	−0.549076	−	0.835772i	$-0.685020\pi$
$47$	−3.67369e7	−1.09815	−0.549076	+	0.835772i	$0.685020\pi$
$48$	0	0
$49$	3.81864e7	0.946295
$50$	0	0
$51$	6.69850e7	1.38647
$52$	0	0
$53$	1.67334e7	0.291302	0.145651	−	0.989336i	$-0.453472\pi$
$53$	1.67334e7	0.291302	0.145651	+	0.989336i	$0.453472\pi$
$54$	0	0
$55$	9.34367e6	0.137685
$56$	0	0
$57$	−5.43248e7	−0.681649
$58$	0	0
$59$	5.89865e7	0.633751	0.316875	−	0.948467i	$-0.397366\pi$
$59$	5.89865e7	0.633751	0.316875	+	0.948467i	$0.397366\pi$
$60$	0	0
$61$	1.28785e8	1.19091	0.595456	−	0.803388i	$-0.296971\pi$
$61$	1.28785e8	1.19091	0.595456	+	0.803388i	$0.296971\pi$
$62$	0	0
$63$	1.96527e8	1.57177
$64$	0	0
$65$	7.39484e6	0.0513829
$66$	0	0
$67$	1.91470e8	1.16082	0.580410	−	0.814324i	$-0.302892\pi$
$67$	1.91470e8	1.16082	0.580410	+	0.814324i	$0.302892\pi$
$68$	0	0
$69$	4.96054e8	2.63456
$70$	0	0
$71$	−3.40865e8	−1.59192	−0.795958	−	0.605351i	$-0.793033\pi$
$71$	−3.40865e8	−1.59192	−0.795958	+	0.605351i	$0.793033\pi$
$72$	0	0
$73$	3.19890e8	1.31840	0.659200	−	0.751968i	$-0.270895\pi$
$73$	3.19890e8	1.31840	0.659200	+	0.751968i	$0.270895\pi$
$74$	0	0
$75$	−3.85882e8	−1.40825
$76$	0	0
$77$	−3.19821e8	−1.03681
$78$	0	0
$79$	−2.44328e8	−0.705751	−0.352876	−	0.935670i	$-0.614796\pi$
$79$	−2.44328e8	−0.705751	−0.352876	+	0.935670i	$0.614796\pi$
$80$	0	0
$81$	−3.32145e8	−0.857325
$82$	0	0
$83$	4.38630e8	1.01449	0.507244	−	0.861802i	$-0.330664\pi$
$83$	4.38630e8	1.01449	0.507244	+	0.861802i	$0.330664\pi$
$84$	0	0
$85$	−8.47697e7	−0.176139
$86$	0	0
$87$	8.16693e8	1.52835
$88$	0	0
$89$	−7.90342e8	−1.33524	−0.667621	−	0.744501i	$-0.732688\pi$
$89$	−7.90342e8	−1.33524	−0.667621	+	0.744501i	$0.732688\pi$
$90$	0	0
$91$	−2.53116e8	−0.386930
$92$	0	0
$93$	1.32008e9	1.82990
$94$	0	0
$95$	6.87481e7	0.0865974
$96$	0	0
$97$	−1.65696e8	−0.190037	−0.0950185	−	0.995476i	$-0.530291\pi$
$97$	−1.65696e8	−0.190037	−0.0950185	+	0.995476i	$0.530291\pi$
$98$	0	0
$99$	−8.00272e8	−0.837296
$100$	0	0
$101$	−7.41636e8	−0.709161	−0.354580	−	0.935026i	$-0.615376\pi$
$101$	−7.41636e8	−0.709161	−0.354580	+	0.935026i	$0.615376\pi$
$102$	0	0
$103$	−1.42175e9	−1.24467	−0.622336	−	0.782750i	$-0.713816\pi$
$103$	−1.42175e9	−1.24467	−0.622336	+	0.782750i	$0.713816\pi$
$104$	0	0
$105$	−4.69454e8	−0.376913
$106$	0	0
$107$	1.44196e9	1.06347	0.531736	−	0.846910i	$-0.321540\pi$
$107$	1.44196e9	1.06347	0.531736	+	0.846910i	$0.321540\pi$
$108$	0	0
$109$	−8.42758e7	−0.0571852	−0.0285926	−	0.999591i	$-0.509103\pi$
$109$	−8.42758e7	−0.0571852	−0.0285926	+	0.999591i	$0.509103\pi$
$110$	0	0
$111$	−1.66846e9	−1.04319
$112$	0	0
$113$	8.60241e8	0.496326	0.248163	−	0.968718i	$-0.420173\pi$
$113$	8.60241e8	0.496326	0.248163	+	0.968718i	$0.420173\pi$
$114$	0	0
$115$	−6.27758e8	−0.334697
$116$	0	0
$117$	−6.33358e8	−0.312473
$118$	0	0
$119$	2.90155e9	1.32638
$120$	0	0
$121$	−1.05561e9	−0.447681
$122$	0	0
$123$	−1.47357e8	−0.0580494
$124$	0	0
$125$	9.94026e8	0.364169
$126$	0	0
$127$	1.44120e9	0.491596	0.245798	−	0.969321i	$-0.420950\pi$
$127$	1.44120e9	0.491596	0.245798	+	0.969321i	$0.420950\pi$
$128$	0	0
$129$	8.45474e9	2.68811
$130$	0	0
$131$	4.54928e9	1.34965	0.674827	−	0.737976i	$-0.264218\pi$
$131$	4.54928e9	1.34965	0.674827	+	0.737976i	$0.264218\pi$
$132$	0	0
$133$	−2.35316e9	−0.652107
$134$	0	0
$135$	−1.32039e8	−0.0342138
$136$	0	0
$137$	−1.99742e9	−0.484426	−0.242213	−	0.970223i	$-0.577873\pi$
$137$	−1.99742e9	−0.484426	−0.242213	+	0.970223i	$0.577873\pi$
$138$	0	0
$139$	−3.55325e8	−0.0807345	−0.0403672	−	0.999185i	$-0.512853\pi$
$139$	−3.55325e8	−0.0807345	−0.0403672	+	0.999185i	$0.512853\pi$
$140$	0	0
$141$	−7.51615e9	−1.60143
$142$	0	0
$143$	1.03071e9	0.206121
$144$	0	0
$145$	−1.03353e9	−0.194163
$146$	0	0
$147$	7.81270e9	1.37998
$148$	0	0
$149$	5.92546e9	0.984881	0.492440	−	0.870346i	$-0.336105\pi$
$149$	5.92546e9	0.984881	0.492440	+	0.870346i	$0.336105\pi$
$150$	0	0
$151$	1.07568e10	1.68378	0.841891	−	0.539647i	$-0.181442\pi$
$151$	1.07568e10	1.68378	0.841891	+	0.539647i	$0.181442\pi$
$152$	0	0
$153$	7.26040e9	1.07115
$154$	0	0
$155$	−1.67056e9	−0.232472
$156$	0	0
$157$	8.87634e9	1.16597	0.582983	−	0.812485i	$-0.301886\pi$
$157$	8.87634e9	1.16597	0.582983	+	0.812485i	$0.301886\pi$
$158$	0	0
$159$	3.42355e9	0.424805
$160$	0	0
$161$	2.14873e10	2.52038
$162$	0	0
$163$	−2.75052e9	−0.305190	−0.152595	−	0.988289i	$-0.548763\pi$
$163$	−2.75052e9	−0.305190	−0.152595	+	0.988289i	$0.548763\pi$
$164$	0	0
$165$	1.91166e9	0.200785
$166$	0	0
$167$	9.17849e9	0.913160	0.456580	−	0.889682i	$-0.349074\pi$
$167$	9.17849e9	0.913160	0.456580	+	0.889682i	$0.349074\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−5.88818e9	−0.526622
$172$	0	0
$173$	9.75049e8	0.0827598	0.0413799	−	0.999143i	$-0.486825\pi$
$173$	9.75049e8	0.0827598	0.0413799	+	0.999143i	$0.486825\pi$
$174$	0	0
$175$	−1.67150e10	−1.34721
$176$	0	0
$177$	1.20683e10	0.924198
$178$	0	0
$179$	8.88356e9	0.646768	0.323384	−	0.946268i	$-0.395180\pi$
$179$	8.88356e9	0.646768	0.323384	+	0.946268i	$0.395180\pi$
$180$	0	0
$181$	−5.22791e9	−0.362055	−0.181028	−	0.983478i	$-0.557942\pi$
$181$	−5.22791e9	−0.362055	−0.181028	+	0.983478i	$0.557942\pi$
$182$	0	0
$183$	2.63485e10	1.73671
$184$	0	0
$185$	2.11144e9	0.132527
$186$	0	0
$187$	−1.18154e10	−0.706577
$188$	0	0
$189$	4.51952e9	0.257641
$190$	0	0
$191$	−9.23471e9	−0.502080	−0.251040	−	0.967977i	$-0.580773\pi$
$191$	−9.23471e9	−0.502080	−0.251040	+	0.967977i	$0.580773\pi$
$192$	0	0
$193$	1.07065e10	0.555441	0.277720	−	0.960662i	$-0.410421\pi$
$193$	1.07065e10	0.555441	0.277720	+	0.960662i	$0.410421\pi$
$194$	0	0
$195$	1.51294e9	0.0749317
$196$	0	0
$197$	−2.77001e10	−1.31034	−0.655169	−	0.755482i	$-0.727403\pi$
$197$	−2.77001e10	−1.31034	−0.655169	+	0.755482i	$0.727403\pi$
$198$	0	0
$199$	−3.32299e10	−1.50207	−0.751035	−	0.660262i	$-0.770445\pi$
$199$	−3.32299e10	−1.50207	−0.751035	+	0.660262i	$0.770445\pi$
$200$	0	0
$201$	3.91737e10	1.69282
$202$	0	0
$203$	3.53763e10	1.46211
$204$	0	0
$205$	1.86481e8	0.00737465
$206$	0	0
$207$	5.37666e10	2.03538
$208$	0	0
$209$	9.58225e9	0.347383
$210$	0	0
$211$	1.49261e10	0.518411	0.259205	−	0.965822i	$-0.416539\pi$
$211$	1.49261e10	0.518411	0.259205	+	0.965822i	$0.416539\pi$
$212$	0	0
$213$	−6.97389e10	−2.32149
$214$	0	0
$215$	−1.06995e10	−0.341500
$216$	0	0
$217$	5.71812e10	1.75059
$218$	0	0
$219$	6.54474e10	1.92262
$220$	0	0
$221$	−9.35101e9	−0.263690
$222$	0	0
$223$	−2.23373e10	−0.604865	−0.302433	−	0.953171i	$-0.597799\pi$
$223$	−2.23373e10	−0.604865	−0.302433	+	0.953171i	$0.597799\pi$
$224$	0	0
$225$	−4.18252e10	−1.08797
$226$	0	0
$227$	−5.20726e9	−0.130165	−0.0650824	−	0.997880i	$-0.520731\pi$
$227$	−5.20726e9	−0.130165	−0.0650824	+	0.997880i	$0.520731\pi$
$228$	0	0
$229$	1.35573e10	0.325773	0.162887	−	0.986645i	$-0.447920\pi$
$229$	1.35573e10	0.325773	0.162887	+	0.986645i	$0.447920\pi$
$230$	0	0
$231$	−6.54335e10	−1.51198
$232$	0	0
$233$	2.06224e8	0.00458393	0.00229196	−	0.999997i	$-0.499270\pi$
$233$	2.06224e8	0.00458393	0.00229196	+	0.999997i	$0.499270\pi$
$234$	0	0
$235$	9.51170e9	0.203448
$236$	0	0
$237$	−4.99880e10	−1.02920
$238$	0	0
$239$	−1.89887e10	−0.376448	−0.188224	−	0.982126i	$-0.560273\pi$
$239$	−1.89887e10	−0.376448	−0.188224	+	0.982126i	$0.560273\pi$
$240$	0	0
$241$	−6.11533e10	−1.16773	−0.583866	−	0.811850i	$-0.698461\pi$
$241$	−6.11533e10	−1.16773	−0.583866	+	0.811850i	$0.698461\pi$
$242$	0	0
$243$	−7.79927e10	−1.43491
$244$	0	0
$245$	−9.88700e9	−0.175314
$246$	0	0
$247$	7.58366e9	0.129641
$248$	0	0
$249$	8.97410e10	1.47943
$250$	0	0
$251$	3.20685e9	0.0509973	0.0254986	−	0.999675i	$-0.491883\pi$
$251$	3.20685e9	0.0509973	0.0254986	+	0.999675i	$0.491883\pi$
$252$	0	0
$253$	−8.74981e10	−1.34263
$254$	0	0
$255$	−1.73433e10	−0.256863
$256$	0	0
$257$	−2.80963e10	−0.401744	−0.200872	−	0.979617i	$-0.564378\pi$
$257$	−2.80963e10	−0.401744	−0.200872	+	0.979617i	$0.564378\pi$
$258$	0	0
$259$	−7.22717e10	−0.997975
$260$	0	0
$261$	8.85201e10	1.18076
$262$	0	0
$263$	1.25524e11	1.61780	0.808902	−	0.587944i	$-0.200062\pi$
$263$	1.25524e11	1.61780	0.808902	+	0.587944i	$0.200062\pi$
$264$	0	0
$265$	−4.33252e9	−0.0539677
$266$	0	0
$267$	−1.61699e11	−1.94718
$268$	0	0
$269$	−1.78558e10	−0.207918	−0.103959	−	0.994582i	$-0.533151\pi$
$269$	−1.78558e10	−0.207918	−0.103959	+	0.994582i	$0.533151\pi$
$270$	0	0
$271$	−1.25204e11	−1.41013	−0.705063	−	0.709144i	$-0.749081\pi$
$271$	−1.25204e11	−1.41013	−0.705063	+	0.709144i	$0.749081\pi$
$272$	0	0
$273$	−5.17859e10	−0.564260
$274$	0	0
$275$	6.80650e10	0.717673
$276$	0	0
$277$	−5.55594e10	−0.567020	−0.283510	−	0.958969i	$-0.591499\pi$
$277$	−5.55594e10	−0.567020	−0.283510	+	0.958969i	$0.591499\pi$
$278$	0	0
$279$	1.43081e11	1.41372
$280$	0	0
$281$	1.01237e11	0.968635	0.484317	−	0.874892i	$-0.339068\pi$
$281$	1.01237e11	0.968635	0.484317	+	0.874892i	$0.339068\pi$
$282$	0	0
$283$	1.08041e11	1.00127	0.500635	−	0.865658i	$-0.333100\pi$
$283$	1.08041e11	1.00127	0.500635	+	0.865658i	$0.333100\pi$
$284$	0	0
$285$	1.40654e10	0.126285
$286$	0	0
$287$	−6.38298e9	−0.0555335
$288$	0	0
$289$	−1.13940e10	−0.0960810
$290$	0	0
$291$	−3.39003e10	−0.277131
$292$	0	0
$293$	3.53224e9	0.0279992	0.0139996	−	0.999902i	$-0.495544\pi$
$293$	3.53224e9	0.0279992	0.0139996	+	0.999902i	$0.495544\pi$
$294$	0	0
$295$	−1.52724e10	−0.117411
$296$	0	0
$297$	−1.84039e10	−0.137248
$298$	0	0
$299$	−6.92485e10	−0.501060
$300$	0	0
$301$	3.66229e11	2.57160
$302$	0	0
$303$	−1.51734e11	−1.03417
$304$	0	0
$305$	−3.33441e10	−0.220633
$306$	0	0
$307$	1.82031e11	1.16956	0.584781	−	0.811191i	$-0.301180\pi$
$307$	1.82031e11	1.16956	0.584781	+	0.811191i	$0.301180\pi$
$308$	0	0
$309$	−2.90881e11	−1.81510
$310$	0	0
$311$	1.51907e11	0.920781	0.460390	−	0.887717i	$-0.347709\pi$
$311$	1.51907e11	0.920781	0.460390	+	0.887717i	$0.347709\pi$
$312$	0	0
$313$	6.10194e10	0.359351	0.179675	−	0.983726i	$-0.442495\pi$
$313$	6.10194e10	0.359351	0.179675	+	0.983726i	$0.442495\pi$
$314$	0	0
$315$	−5.08835e10	−0.291192
$316$	0	0
$317$	6.89227e10	0.383350	0.191675	−	0.981458i	$-0.438608\pi$
$317$	6.89227e10	0.383350	0.191675	+	0.981458i	$0.438608\pi$
$318$	0	0
$319$	−1.44055e11	−0.778880
$320$	0	0
$321$	2.95016e11	1.55086
$322$	0	0
$323$	−8.69341e10	−0.444405
$324$	0	0
$325$	5.38686e10	0.267831
$326$	0	0
$327$	−1.72423e10	−0.0833931
$328$	0	0
$329$	−3.25573e11	−1.53203
$330$	0	0
$331$	−4.70422e10	−0.215408	−0.107704	−	0.994183i	$-0.534350\pi$
$331$	−4.70422e10	−0.215408	−0.107704	+	0.994183i	$0.534350\pi$
$332$	0	0
$333$	−1.80842e11	−0.805934
$334$	0	0
$335$	−4.95744e10	−0.215058
$336$	0	0
$337$	−3.76711e11	−1.59101	−0.795507	−	0.605944i	$-0.792795\pi$
$337$	−3.76711e11	−1.59101	−0.795507	+	0.605944i	$0.792795\pi$
$338$	0	0
$339$	1.76000e11	0.723792
$340$	0	0
$341$	−2.32846e11	−0.932556
$342$	0	0
$343$	−1.92062e10	−0.0749235
$344$	0	0
$345$	−1.28435e11	−0.488089
$346$	0	0
$347$	−1.68331e11	−0.623278	−0.311639	−	0.950201i	$-0.600878\pi$
$347$	−1.68331e11	−0.623278	−0.311639	+	0.950201i	$0.600878\pi$
$348$	0	0
$349$	−9.36126e10	−0.337769	−0.168884	−	0.985636i	$-0.554016\pi$
$349$	−9.36126e10	−0.337769	−0.168884	+	0.985636i	$0.554016\pi$
$350$	0	0
$351$	−1.45653e10	−0.0512199
$352$	0	0
$353$	−3.49043e11	−1.19644	−0.598222	−	0.801330i	$-0.704126\pi$
$353$	−3.49043e11	−1.19644	−0.598222	+	0.801330i	$0.704126\pi$
$354$	0	0
$355$	8.82548e10	0.294924
$356$	0	0
$357$	5.93640e11	1.93426
$358$	0	0
$359$	−5.83173e11	−1.85299	−0.926493	−	0.376312i	$-0.877192\pi$
$359$	−5.83173e11	−1.85299	−0.926493	+	0.376312i	$0.877192\pi$
$360$	0	0
$361$	−2.52184e11	−0.781512
$362$	0	0
$363$	−2.15971e11	−0.652854
$364$	0	0
$365$	−8.28239e10	−0.244252
$366$	0	0
$367$	−5.28604e11	−1.52101	−0.760507	−	0.649329i	$-0.775050\pi$
$367$	−5.28604e11	−1.52101	−0.760507	+	0.649329i	$0.775050\pi$
$368$	0	0
$369$	−1.59718e10	−0.0448472
$370$	0	0
$371$	1.48296e11	0.406394
$372$	0	0
$373$	−4.19405e11	−1.12187	−0.560937	−	0.827858i	$-0.689559\pi$
$373$	−4.19405e11	−1.12187	−0.560937	+	0.827858i	$0.689559\pi$
$374$	0	0
$375$	2.03372e11	0.531067
$376$	0	0
$377$	−1.14009e11	−0.290672
$378$	0	0
$379$	−4.24128e11	−1.05590	−0.527948	−	0.849277i	$-0.677038\pi$
$379$	−4.24128e11	−1.05590	−0.527948	+	0.849277i	$0.677038\pi$
$380$	0	0
$381$	2.94861e11	0.716894
$382$	0	0
$383$	−6.52696e10	−0.154995	−0.0774973	−	0.996993i	$-0.524693\pi$
$383$	−6.52696e10	−0.154995	−0.0774973	+	0.996993i	$0.524693\pi$
$384$	0	0
$385$	8.28062e10	0.192083
$386$	0	0
$387$	9.16396e11	2.07675
$388$	0	0
$389$	−5.91803e11	−1.31040	−0.655200	−	0.755455i	$-0.727416\pi$
$389$	−5.91803e11	−1.31040	−0.655200	+	0.755455i	$0.727416\pi$
$390$	0	0
$391$	7.93819e11	1.71762
$392$	0	0
$393$	9.30755e11	1.96820
$394$	0	0
$395$	6.32600e10	0.130750
$396$	0	0
$397$	−3.05131e11	−0.616495	−0.308248	−	0.951306i	$-0.599743\pi$
$397$	−3.05131e11	−0.616495	−0.308248	+	0.951306i	$0.599743\pi$
$398$	0	0
$399$	−4.81441e11	−0.950967
$400$	0	0
$401$	2.56722e11	0.495808	0.247904	−	0.968785i	$-0.420258\pi$
$401$	2.56722e11	0.495808	0.247904	+	0.968785i	$0.420258\pi$
$402$	0	0
$403$	−1.84281e11	−0.348023
$404$	0	0
$405$	8.59971e10	0.158831
$406$	0	0
$407$	2.94296e11	0.531631
$408$	0	0
$409$	−4.57003e11	−0.807540	−0.403770	−	0.914860i	$-0.632300\pi$
$409$	−4.57003e11	−0.807540	−0.403770	+	0.914860i	$0.632300\pi$
$410$	0	0
$411$	−4.08660e11	−0.706438
$412$	0	0
$413$	5.22755e11	0.884144
$414$	0	0
$415$	−1.13567e11	−0.187948
$416$	0	0
$417$	−7.26973e10	−0.117735
$418$	0	0
$419$	1.65068e11	0.261637	0.130818	−	0.991406i	$-0.458240\pi$
$419$	1.65068e11	0.261637	0.130818	+	0.991406i	$0.458240\pi$
$420$	0	0
$421$	−9.10255e10	−0.141219	−0.0706096	−	0.997504i	$-0.522494\pi$
$421$	−9.10255e10	−0.141219	−0.0706096	+	0.997504i	$0.522494\pi$
$422$	0	0
$423$	−8.14664e11	−1.23722
$424$	0	0
$425$	−6.17514e11	−0.918114
$426$	0	0
$427$	1.14133e12	1.66144
$428$	0	0
$429$	2.10876e11	0.300587
$430$	0	0
$431$	−7.94605e11	−1.10918	−0.554592	−	0.832122i	$-0.687126\pi$
$431$	−7.94605e11	−1.10918	−0.554592	+	0.832122i	$0.687126\pi$
$432$	0	0
$433$	−8.26325e11	−1.12968	−0.564840	−	0.825200i	$-0.691062\pi$
$433$	−8.26325e11	−1.12968	−0.564840	+	0.825200i	$0.691062\pi$
$434$	0	0
$435$	−2.11453e11	−0.283148
$436$	0	0
$437$	−6.43787e11	−0.844453
$438$	0	0
$439$	−1.21562e12	−1.56210	−0.781050	−	0.624469i	$-0.785316\pi$
$439$	−1.21562e12	−1.56210	−0.781050	+	0.624469i	$0.785316\pi$
$440$	0	0
$441$	8.46807e11	1.06613
$442$	0	0
$443$	8.73136e11	1.07712	0.538561	−	0.842586i	$-0.318968\pi$
$443$	8.73136e11	1.07712	0.538561	+	0.842586i	$0.318968\pi$
$444$	0	0
$445$	2.04631e11	0.247372
$446$	0	0
$447$	1.21231e12	1.43625
$448$	0	0
$449$	1.22107e12	1.41785	0.708927	−	0.705282i	$-0.249179\pi$
$449$	1.22107e12	1.41785	0.708927	+	0.705282i	$0.249179\pi$
$450$	0	0
$451$	2.59920e10	0.0295832
$452$	0	0
$453$	2.20077e12	2.45546
$454$	0	0
$455$	6.55352e10	0.0716842
$456$	0	0
$457$	6.32643e11	0.678478	0.339239	−	0.940700i	$-0.389830\pi$
$457$	6.32643e11	0.678478	0.339239	+	0.940700i	$0.389830\pi$
$458$	0	0
$459$	1.66968e11	0.175580
$460$	0	0
$461$	8.09117e11	0.834367	0.417184	−	0.908822i	$-0.363017\pi$
$461$	8.09117e11	0.834367	0.417184	+	0.908822i	$0.363017\pi$
$462$	0	0
$463$	−7.69685e11	−0.778393	−0.389196	−	0.921155i	$-0.627247\pi$
$463$	−7.69685e11	−0.778393	−0.389196	+	0.921155i	$0.627247\pi$
$464$	0	0
$465$	−3.41787e11	−0.339014
$466$	0	0
$467$	−1.59708e12	−1.55382	−0.776909	−	0.629613i	$-0.783213\pi$
$467$	−1.59708e12	−1.55382	−0.776909	+	0.629613i	$0.783213\pi$
$468$	0	0
$469$	1.69686e12	1.61946
$470$	0	0
$471$	1.81604e12	1.70033
$472$	0	0
$473$	−1.49132e12	−1.36992
$474$	0	0
$475$	5.00804e11	0.451384
$476$	0	0
$477$	3.71074e11	0.328192
$478$	0	0
$479$	−1.61308e12	−1.40006	−0.700028	−	0.714115i	$-0.746829\pi$
$479$	−1.61308e12	−1.40006	−0.700028	+	0.714115i	$0.746829\pi$
$480$	0	0
$481$	2.32914e11	0.198401
$482$	0	0
$483$	4.39617e12	3.67547
$484$	0	0
$485$	4.29009e10	0.0352070
$486$	0	0
$487$	−9.29058e11	−0.748449	−0.374225	−	0.927338i	$-0.622091\pi$
$487$	−9.29058e11	−0.748449	−0.374225	+	0.927338i	$0.622091\pi$
$488$	0	0
$489$	−5.62739e11	−0.445058
$490$	0	0
$491$	2.30662e12	1.79106	0.895528	−	0.445005i	$-0.146798\pi$
$491$	2.30662e12	1.79106	0.895528	+	0.445005i	$0.146798\pi$
$492$	0	0
$493$	1.30693e12	0.996415
$494$	0	0
$495$	2.07202e11	0.155121
$496$	0	0
$497$	−3.02084e12	−2.22088
$498$	0	0
$499$	1.31410e11	0.0948800	0.0474400	−	0.998874i	$-0.484894\pi$
$499$	1.31410e11	0.0948800	0.0474400	+	0.998874i	$0.484894\pi$
$500$	0	0
$501$	1.87786e12	1.33166
$502$	0	0
$503$	1.88419e12	1.31241	0.656203	−	0.754584i	$-0.272162\pi$
$503$	1.88419e12	1.31241	0.656203	+	0.754584i	$0.272162\pi$
$504$	0	0
$505$	1.92020e11	0.131382
$506$	0	0
$507$	1.66893e11	0.112177
$508$	0	0
$509$	−1.00390e11	−0.0662919	−0.0331459	−	0.999451i	$-0.510553\pi$
$509$	−1.00390e11	−0.0662919	−0.0331459	+	0.999451i	$0.510553\pi$
$510$	0	0
$511$	2.83495e12	1.83930
$512$	0	0
$513$	−1.35411e11	−0.0863226
$514$	0	0
$515$	3.68110e11	0.230593
$516$	0	0
$517$	1.32576e12	0.816126
$518$	0	0
$519$	1.99489e11	0.120689
$520$	0	0
$521$	−1.04516e11	−0.0621462	−0.0310731	−	0.999517i	$-0.509892\pi$
$521$	−1.04516e11	−0.0621462	−0.0310731	+	0.999517i	$0.509892\pi$
$522$	0	0
$523$	−1.17694e12	−0.687855	−0.343927	−	0.938996i	$-0.611757\pi$
$523$	−1.17694e12	−0.687855	−0.343927	+	0.938996i	$0.611757\pi$
$524$	0	0
$525$	−3.41979e12	−1.96464
$526$	0	0
$527$	2.11248e12	1.19301
$528$	0	0
$529$	4.07744e12	2.26379
$530$	0	0
$531$	1.30806e12	0.714008
$532$	0	0
$533$	2.05708e10	0.0110403
$534$	0	0
$535$	−3.73343e11	−0.197023
$536$	0	0
$537$	1.81752e12	0.943181
$538$	0	0
$539$	−1.37807e12	−0.703269
$540$	0	0
$541$	−2.98502e12	−1.49817	−0.749083	−	0.662476i	$-0.769506\pi$
$541$	−2.98502e12	−1.49817	−0.749083	+	0.662476i	$0.769506\pi$
$542$	0	0
$543$	−1.06960e12	−0.527985
$544$	0	0
$545$	2.18202e10	0.0105943
$546$	0	0
$547$	−3.78628e12	−1.80830	−0.904148	−	0.427219i	$-0.859493\pi$
$547$	−3.78628e12	−1.80830	−0.904148	+	0.427219i	$0.859493\pi$
$548$	0	0
$549$	2.85588e12	1.34173
$550$	0	0
$551$	−1.05992e12	−0.489880
$552$	0	0
$553$	−2.16530e12	−0.984591
$554$	0	0
$555$	4.31987e11	0.193265
$556$	0	0
$557$	1.00551e12	0.442627	0.221313	−	0.975203i	$-0.428966\pi$
$557$	1.00551e12	0.442627	0.221313	+	0.975203i	$0.428966\pi$
$558$	0	0
$559$	−1.18027e12	−0.511244
$560$	0	0
$561$	−2.41735e12	−1.03040
$562$	0	0
$563$	−2.76515e11	−0.115993	−0.0579964	−	0.998317i	$-0.518471\pi$
$563$	−2.76515e11	−0.115993	−0.0579964	+	0.998317i	$0.518471\pi$
$564$	0	0
$565$	−2.22728e11	−0.0919512
$566$	0	0
$567$	−2.94356e12	−1.19605
$568$	0	0
$569$	8.39220e11	0.335638	0.167819	−	0.985818i	$-0.446328\pi$
$569$	8.39220e11	0.335638	0.167819	+	0.985818i	$0.446328\pi$
$570$	0	0
$571$	1.02285e11	0.0402672	0.0201336	−	0.999797i	$-0.493591\pi$
$571$	1.02285e11	0.0402672	0.0201336	+	0.999797i	$0.493591\pi$
$572$	0	0
$573$	−1.88936e12	−0.732184
$574$	0	0
$575$	−4.57297e12	−1.74459
$576$	0	0
$577$	−1.44021e12	−0.540921	−0.270461	−	0.962731i	$-0.587176\pi$
$577$	−1.44021e12	−0.540921	−0.270461	+	0.962731i	$0.587176\pi$
$578$	0	0
$579$	2.19047e12	0.809999
$580$	0	0
$581$	3.88726e12	1.41531
$582$	0	0
$583$	−6.03874e11	−0.216490
$584$	0	0
$585$	1.63985e11	0.0578899
$586$	0	0
$587$	2.13185e11	0.0741116	0.0370558	−	0.999313i	$-0.488202\pi$
$587$	2.13185e11	0.0741116	0.0370558	+	0.999313i	$0.488202\pi$
$588$	0	0
$589$	−1.71322e12	−0.586536
$590$	0	0
$591$	−5.66727e12	−1.91087
$592$	0	0
$593$	−1.76049e12	−0.584640	−0.292320	−	0.956321i	$-0.594427\pi$
$593$	−1.76049e12	−0.584640	−0.292320	+	0.956321i	$0.594427\pi$
$594$	0	0
$595$	−7.51253e11	−0.245731
$596$	0	0
$597$	−6.79863e12	−2.19047
$598$	0	0
$599$	2.06753e12	0.656191	0.328096	−	0.944644i	$-0.393593\pi$
$599$	2.06753e12	0.656191	0.328096	+	0.944644i	$0.393593\pi$
$600$	0	0
$601$	−1.24403e12	−0.388951	−0.194476	−	0.980907i	$-0.562301\pi$
$601$	−1.24403e12	−0.388951	−0.194476	+	0.980907i	$0.562301\pi$
$602$	0	0
$603$	4.24597e12	1.30782
$604$	0	0
$605$	2.73312e11	0.0829391
$606$	0	0
$607$	1.31511e12	0.393200	0.196600	−	0.980484i	$-0.437010\pi$
$607$	1.31511e12	0.393200	0.196600	+	0.980484i	$0.437010\pi$
$608$	0	0
$609$	7.23776e12	2.13219
$610$	0	0
$611$	1.04924e12	0.304573
$612$	0	0
$613$	−3.08252e12	−0.881726	−0.440863	−	0.897574i	$-0.645328\pi$
$613$	−3.08252e12	−0.881726	−0.440863	+	0.897574i	$0.645328\pi$
$614$	0	0
$615$	3.81528e10	0.0107544
$616$	0	0
$617$	−2.29075e10	−0.00636348	−0.00318174	−	0.999995i	$-0.501013\pi$
$617$	−2.29075e10	−0.00636348	−0.00318174	+	0.999995i	$0.501013\pi$
$618$	0	0
$619$	4.45350e12	1.21925	0.609626	−	0.792689i	$-0.291320\pi$
$619$	4.45350e12	1.21925	0.609626	+	0.792689i	$0.291320\pi$
$620$	0	0
$621$	1.23647e12	0.333635
$622$	0	0
$623$	−7.00424e12	−1.86279
$624$	0	0
$625$	3.42640e12	0.898210
$626$	0	0
$627$	1.96047e12	0.506589
$628$	0	0
$629$	−2.66998e12	−0.680111
$630$	0	0
$631$	−3.83831e12	−0.963847	−0.481923	−	0.876213i	$-0.660062\pi$
$631$	−3.83831e12	−0.963847	−0.481923	+	0.876213i	$0.660062\pi$
$632$	0	0
$633$	3.05378e12	0.755998
$634$	0	0
$635$	−3.73148e11	−0.0910749
$636$	0	0
$637$	−1.09064e12	−0.262455
$638$	0	0
$639$	−7.55890e12	−1.79351
$640$	0	0
$641$	9.09238e11	0.212724	0.106362	−	0.994327i	$-0.466080\pi$
$641$	9.09238e11	0.212724	0.106362	+	0.994327i	$0.466080\pi$
$642$	0	0
$643$	−8.23839e12	−1.90061	−0.950305	−	0.311321i	$-0.899229\pi$
$643$	−8.23839e12	−1.90061	−0.950305	+	0.311321i	$0.899229\pi$
$644$	0	0
$645$	−2.18905e12	−0.498009
$646$	0	0
$647$	8.73559e12	1.95985	0.979925	−	0.199365i	$-0.0638880\pi$
$647$	8.73559e12	1.95985	0.979925	+	0.199365i	$0.0638880\pi$
$648$	0	0
$649$	−2.12870e12	−0.470992
$650$	0	0
$651$	1.16989e13	2.55289
$652$	0	0
$653$	−6.13232e11	−0.131982	−0.0659911	−	0.997820i	$-0.521021\pi$
$653$	−6.13232e11	−0.131982	−0.0659911	+	0.997820i	$0.521021\pi$
$654$	0	0
$655$	−1.17787e12	−0.250042
$656$	0	0
$657$	7.09375e12	1.48536
$658$	0	0
$659$	8.83233e12	1.82428	0.912139	−	0.409882i	$-0.134430\pi$
$659$	8.83233e12	1.82428	0.912139	+	0.409882i	$0.134430\pi$
$660$	0	0
$661$	−5.61801e12	−1.14466	−0.572329	−	0.820024i	$-0.693960\pi$
$661$	−5.61801e12	−1.14466	−0.572329	+	0.820024i	$0.693960\pi$
$662$	0	0
$663$	−1.91316e12	−0.384539
$664$	0	0
$665$	6.09265e11	0.120812
$666$	0	0
$667$	9.67839e12	1.89338
$668$	0	0
$669$	−4.57007e12	−0.882075
$670$	0	0
$671$	−4.64757e12	−0.885064
$672$	0	0
$673$	4.50151e12	0.845844	0.422922	−	0.906166i	$-0.361004\pi$
$673$	4.50151e12	0.845844	0.422922	+	0.906166i	$0.361004\pi$
$674$	0	0
$675$	−9.61854e11	−0.178337
$676$	0	0
$677$	4.33098e12	0.792387	0.396193	−	0.918167i	$-0.370331\pi$
$677$	4.33098e12	0.792387	0.396193	+	0.918167i	$0.370331\pi$
$678$	0	0
$679$	−1.46844e12	−0.265120
$680$	0	0
$681$	−1.06537e12	−0.189819
$682$	0	0
$683$	−6.25107e12	−1.09916	−0.549580	−	0.835441i	$-0.685212\pi$
$683$	−6.25107e12	−1.09916	−0.549580	+	0.835441i	$0.685212\pi$
$684$	0	0
$685$	5.17161e11	0.0897466
$686$	0	0
$687$	2.77375e12	0.475075
$688$	0	0
$689$	−4.77923e11	−0.0807926
$690$	0	0
$691$	−7.34082e12	−1.22488	−0.612440	−	0.790517i	$-0.709812\pi$
$691$	−7.34082e12	−1.22488	−0.612440	+	0.790517i	$0.709812\pi$
$692$	0	0
$693$	−7.09223e12	−1.16811
$694$	0	0
$695$	9.19986e10	0.0149572
$696$	0	0
$697$	−2.35810e11	−0.0378456
$698$	0	0
$699$	4.21922e10	0.00668474
$700$	0	0
$701$	−1.71509e11	−0.0268259	−0.0134130	−	0.999910i	$-0.504270\pi$
$701$	−1.71509e11	−0.0268259	−0.0134130	+	0.999910i	$0.504270\pi$
$702$	0	0
$703$	2.16535e12	0.334372
$704$	0	0
$705$	1.94604e12	0.296688
$706$	0	0
$707$	−6.57259e12	−0.989348
$708$	0	0
$709$	−1.00896e13	−1.49956	−0.749780	−	0.661687i	$-0.769841\pi$
$709$	−1.00896e13	−1.49956	−0.749780	+	0.661687i	$0.769841\pi$
$710$	0	0
$711$	−5.41813e12	−0.795126
$712$	0	0
$713$	1.56439e13	2.26695
$714$	0	0
$715$	−2.66865e11	−0.0381868
$716$	0	0
$717$	−3.88497e12	−0.548974
$718$	0	0
$719$	−8.18068e12	−1.14159	−0.570794	−	0.821093i	$-0.693365\pi$
$719$	−8.18068e12	−1.14159	−0.570794	+	0.821093i	$0.693365\pi$
$720$	0	0
$721$	−1.25999e13	−1.73644
$722$	0	0
$723$	−1.25116e13	−1.70290
$724$	0	0
$725$	−7.52885e12	−1.01206
$726$	0	0
$727$	9.98421e11	0.132559	0.0662795	−	0.997801i	$-0.478887\pi$
$727$	9.98421e11	0.132559	0.0662795	+	0.997801i	$0.478887\pi$
$728$	0	0
$729$	−9.41920e12	−1.23521
$730$	0	0
$731$	1.35298e13	1.75253
$732$	0	0
$733$	−2.10457e12	−0.269274	−0.134637	−	0.990895i	$-0.542987\pi$
$733$	−2.10457e12	−0.269274	−0.134637	+	0.990895i	$0.542987\pi$
$734$	0	0
$735$	−2.02282e12	−0.255661
$736$	0	0
$737$	−6.90977e12	−0.862700
$738$	0	0
$739$	1.28931e13	1.59023	0.795113	−	0.606461i	$-0.207412\pi$
$739$	1.28931e13	1.59023	0.795113	+	0.606461i	$0.207412\pi$
$740$	0	0
$741$	1.55157e12	0.189056
$742$	0	0
$743$	−2.72357e11	−0.0327861	−0.0163930	−	0.999866i	$-0.505218\pi$
$743$	−2.72357e11	−0.0327861	−0.0163930	+	0.999866i	$0.505218\pi$
$744$	0	0
$745$	−1.53418e12	−0.182463
$746$	0	0
$747$	9.72689e12	1.14296
$748$	0	0
$749$	1.27790e13	1.48365
$750$	0	0
$751$	1.40530e13	1.61209	0.806045	−	0.591855i	$-0.201604\pi$
$751$	1.40530e13	1.61209	0.806045	+	0.591855i	$0.201604\pi$
$752$	0	0
$753$	6.56102e11	0.0743693
$754$	0	0
$755$	−2.78508e12	−0.311944
$756$	0	0
$757$	8.53220e12	0.944342	0.472171	−	0.881507i	$-0.343470\pi$
$757$	8.53220e12	0.944342	0.472171	+	0.881507i	$0.343470\pi$
$758$	0	0
$759$	−1.79016e13	−1.95796
$760$	0	0
$761$	1.13835e13	1.23039	0.615197	−	0.788373i	$-0.289076\pi$
$761$	1.13835e13	1.23039	0.615197	+	0.788373i	$0.289076\pi$
$762$	0	0
$763$	−7.46876e11	−0.0797789
$764$	0	0
$765$	−1.87982e12	−0.198445
$766$	0	0
$767$	−1.68471e12	−0.175771
$768$	0	0
$769$	1.94858e12	0.200933	0.100466	−	0.994940i	$-0.467967\pi$
$769$	1.94858e12	0.200933	0.100466	+	0.994940i	$0.467967\pi$
$770$	0	0
$771$	−5.74832e12	−0.585864
$772$	0	0
$773$	−1.09355e13	−1.10161	−0.550807	−	0.834633i	$-0.685680\pi$
$773$	−1.09355e13	−1.10161	−0.550807	+	0.834633i	$0.685680\pi$
$774$	0	0
$775$	−1.21694e13	−1.21175
$776$	0	0
$777$	−1.47863e13	−1.45535
$778$	0	0
$779$	1.91242e11	0.0186065
$780$	0	0
$781$	1.23011e13	1.18308
$782$	0	0
$783$	2.03570e12	0.193547
$784$	0	0
$785$	−2.29821e12	−0.216011
$786$	0	0
$787$	−7.50366e12	−0.697247	−0.348623	−	0.937263i	$-0.613351\pi$
$787$	−7.50366e12	−0.697247	−0.348623	+	0.937263i	$0.613351\pi$
$788$	0	0
$789$	2.56814e13	2.35924
$790$	0	0
$791$	7.62369e12	0.692423
$792$	0	0
$793$	−3.67822e12	−0.330300
$794$	0	0
$795$	−8.86406e11	−0.0787011
$796$	0	0
$797$	1.65844e12	0.145592	0.0727962	−	0.997347i	$-0.476808\pi$
$797$	1.65844e12	0.145592	0.0727962	+	0.997347i	$0.476808\pi$
$798$	0	0
$799$	−1.20278e13	−1.04406
$800$	0	0
$801$	−1.75263e13	−1.50434
$802$	0	0
$803$	−1.15441e13	−0.979810
$804$	0	0
$805$	−5.56337e12	−0.466935
$806$	0	0
$807$	−3.65318e12	−0.303207
$808$	0	0
$809$	−7.07921e12	−0.581054	−0.290527	−	0.956867i	$-0.593831\pi$
$809$	−7.07921e12	−0.581054	−0.290527	+	0.956867i	$0.593831\pi$
$810$	0	0
$811$	1.84596e13	1.49840	0.749201	−	0.662343i	$-0.230438\pi$
$811$	1.84596e13	1.49840	0.749201	+	0.662343i	$0.230438\pi$
$812$	0	0
$813$	−2.56161e13	−2.05639
$814$	0	0
$815$	7.12147e11	0.0565406
$816$	0	0
$817$	−1.09727e13	−0.861616
$818$	0	0
$819$	−5.61299e12	−0.435930
$820$	0	0
$821$	2.40377e13	1.84650	0.923250	−	0.384200i	$-0.125523\pi$
$821$	2.40377e13	1.84650	0.923250	+	0.384200i	$0.125523\pi$
$822$	0	0
$823$	1.26494e12	0.0961104	0.0480552	−	0.998845i	$-0.484698\pi$
$823$	1.26494e12	0.0961104	0.0480552	+	0.998845i	$0.484698\pi$
$824$	0	0
$825$	1.39257e13	1.04658
$826$	0	0
$827$	9.81229e12	0.729450	0.364725	−	0.931115i	$-0.381163\pi$
$827$	9.81229e12	0.729450	0.364725	+	0.931115i	$0.381163\pi$
$828$	0	0
$829$	1.26128e13	0.927504	0.463752	−	0.885965i	$-0.346503\pi$
$829$	1.26128e13	0.927504	0.463752	+	0.885965i	$0.346503\pi$
$830$	0	0
$831$	−1.13671e13	−0.826886
$832$	0	0
$833$	1.25024e13	0.899687
$834$	0	0
$835$	−2.37644e12	−0.169175
$836$	0	0
$837$	3.29045e12	0.231734
$838$	0	0
$839$	8.53632e12	0.594760	0.297380	−	0.954759i	$-0.403887\pi$
$839$	8.53632e12	0.594760	0.297380	+	0.954759i	$0.403887\pi$
$840$	0	0
$841$	1.42715e12	0.0983759
$842$	0	0
$843$	2.07124e13	1.41256
$844$	0	0
$845$	−2.11204e11	−0.0142511
$846$	0	0
$847$	−9.35511e12	−0.624559
$848$	0	0
$849$	2.21046e13	1.46015
$850$	0	0
$851$	−1.97724e13	−1.29234
$852$	0	0
$853$	−1.12025e13	−0.724507	−0.362254	−	0.932080i	$-0.617993\pi$
$853$	−1.12025e13	−0.724507	−0.362254	+	0.932080i	$0.617993\pi$
$854$	0	0
$855$	1.52453e12	0.0975639
$856$	0	0
$857$	−2.23241e13	−1.41371	−0.706856	−	0.707357i	$-0.749887\pi$
$857$	−2.23241e13	−1.41371	−0.706856	+	0.707357i	$0.749887\pi$
$858$	0	0
$859$	2.11323e13	1.32428	0.662138	−	0.749382i	$-0.269649\pi$
$859$	2.11323e13	1.32428	0.662138	+	0.749382i	$0.269649\pi$
$860$	0	0
$861$	−1.30592e12	−0.0809845
$862$	0	0
$863$	−1.17061e13	−0.718395	−0.359197	−	0.933262i	$-0.616950\pi$
$863$	−1.17061e13	−0.718395	−0.359197	+	0.933262i	$0.616950\pi$
$864$	0	0
$865$	−2.52454e11	−0.0153324
$866$	0	0
$867$	−2.33115e12	−0.140115
$868$	0	0
$869$	8.81730e12	0.524501
$870$	0	0
$871$	−5.46859e12	−0.321954
$872$	0	0
$873$	−3.67440e12	−0.214103
$874$	0	0
$875$	8.80934e12	0.508051
$876$	0	0
$877$	1.17836e13	0.672638	0.336319	−	0.941748i	$-0.390818\pi$
$877$	1.17836e13	0.672638	0.336319	+	0.941748i	$0.390818\pi$
$878$	0	0
$879$	7.22674e11	0.0408312
$880$	0	0
$881$	−1.93741e13	−1.08350	−0.541751	−	0.840539i	$-0.682238\pi$
$881$	−1.93741e13	−1.08350	−0.541751	+	0.840539i	$0.682238\pi$
$882$	0	0
$883$	1.50199e13	0.831466	0.415733	−	0.909487i	$-0.363525\pi$
$883$	1.50199e13	0.831466	0.415733	+	0.909487i	$0.363525\pi$
$884$	0	0
$885$	−3.12464e12	−0.171220
$886$	0	0
$887$	−1.23661e13	−0.670776	−0.335388	−	0.942080i	$-0.608867\pi$
$887$	−1.23661e13	−0.670776	−0.335388	+	0.942080i	$0.608867\pi$
$888$	0	0
$889$	1.27723e13	0.685824
$890$	0	0
$891$	1.19864e13	0.637148
$892$	0	0
$893$	9.75457e12	0.513306
$894$	0	0
$895$	−2.30008e12	−0.119823
$896$	0	0
$897$	−1.41678e13	−0.730696
$898$	0	0
$899$	2.57558e13	1.31509
$900$	0	0
$901$	5.47860e12	0.276954
$902$	0	0
$903$	7.49283e13	3.75017
$904$	0	0
$905$	1.35358e12	0.0670757
$906$	0	0
$907$	−9.76708e12	−0.479217	−0.239608	−	0.970870i	$-0.577019\pi$
$907$	−9.76708e12	−0.479217	−0.239608	+	0.970870i	$0.577019\pi$
$908$	0	0
$909$	−1.64462e13	−0.798967
$910$	0	0
$911$	1.91556e13	0.921430	0.460715	−	0.887548i	$-0.347593\pi$
$911$	1.91556e13	0.921430	0.460715	+	0.887548i	$0.347593\pi$
$912$	0	0
$913$	−1.58293e13	−0.753949
$914$	0	0
$915$	−6.82200e12	−0.321749
$916$	0	0
$917$	4.03170e13	1.88290
$918$	0	0
$919$	2.82464e13	1.30630	0.653150	−	0.757229i	$-0.273447\pi$
$919$	2.82464e13	1.30630	0.653150	+	0.757229i	$0.273447\pi$
$920$	0	0
$921$	3.72425e13	1.70557
$922$	0	0
$923$	9.73546e12	0.441518
$924$	0	0
$925$	1.53810e13	0.690792
$926$	0	0
$927$	−3.15281e13	−1.40229
$928$	0	0
$929$	1.81017e13	0.797348	0.398674	−	0.917093i	$-0.369471\pi$
$929$	1.81017e13	0.797348	0.398674	+	0.917093i	$0.369471\pi$
$930$	0	0
$931$	−1.01395e13	−0.442324
$932$	0	0
$933$	3.10792e13	1.34277
$934$	0	0
$935$	3.05916e12	0.130903
$936$	0	0
$937$	2.51202e13	1.06462	0.532310	−	0.846550i	$-0.321324\pi$
$937$	2.51202e13	1.06462	0.532310	+	0.846550i	$0.321324\pi$
$938$	0	0
$939$	1.24842e13	0.524041
$940$	0	0
$941$	8.09156e11	0.0336418	0.0168209	−	0.999859i	$-0.494645\pi$
$941$	8.09156e11	0.0336418	0.0168209	+	0.999859i	$0.494645\pi$
$942$	0	0
$943$	−1.74628e12	−0.0719138
$944$	0	0
$945$	−1.17017e12	−0.0477315
$946$	0	0
$947$	1.25569e13	0.507349	0.253675	−	0.967290i	$-0.418361\pi$
$947$	1.25569e13	0.507349	0.253675	+	0.967290i	$0.418361\pi$
$948$	0	0
$949$	−9.13636e12	−0.365658
$950$	0	0
$951$	1.41012e13	0.559040
$952$	0	0
$953$	−3.64201e13	−1.43029	−0.715144	−	0.698977i	$-0.753639\pi$
$953$	−3.64201e13	−1.43029	−0.715144	+	0.698977i	$0.753639\pi$
$954$	0	0
$955$	2.39100e12	0.0930173
$956$	0	0
$957$	−2.94728e13	−1.13584
$958$	0	0
$959$	−1.77017e13	−0.675821
$960$	0	0
$961$	1.51913e13	0.574564
$962$	0	0
$963$	3.19763e13	1.19815
$964$	0	0
$965$	−2.77205e12	−0.102903
$966$	0	0
$967$	−2.96691e13	−1.09115	−0.545575	−	0.838062i	$-0.683689\pi$
$967$	−2.96691e13	−1.09115	−0.545575	+	0.838062i	$0.683689\pi$
$968$	0	0
$969$	−1.77862e13	−0.648076
$970$	0	0
$971$	1.05417e13	0.380563	0.190281	−	0.981730i	$-0.439060\pi$
$971$	1.05417e13	0.380563	0.190281	+	0.981730i	$0.439060\pi$
$972$	0	0
$973$	−3.14899e12	−0.112632
$974$	0	0
$975$	1.10212e13	0.390577
$976$	0	0
$977$	−2.40729e13	−0.845285	−0.422643	−	0.906296i	$-0.638897\pi$
$977$	−2.40729e13	−0.845285	−0.422643	+	0.906296i	$0.638897\pi$
$978$	0	0
$979$	2.85218e13	0.992328
$980$	0	0
$981$	−1.86887e12	−0.0644270
$982$	0	0
$983$	3.57135e13	1.21995	0.609974	−	0.792421i	$-0.291180\pi$
$983$	3.57135e13	1.21995	0.609974	+	0.792421i	$0.291180\pi$
$984$	0	0
$985$	7.17195e12	0.242758
$986$	0	0
$987$	−6.66102e13	−2.23416
$988$	0	0
$989$	1.00195e14	3.33013
$990$	0	0
$991$	−2.48780e13	−0.819376	−0.409688	−	0.912226i	$-0.634362\pi$
$991$	−2.48780e13	−0.819376	−0.409688	+	0.912226i	$0.634362\pi$
$992$	0	0
$993$	−9.62454e12	−0.314129
$994$	0	0
$995$	8.60369e12	0.278279
$996$	0	0
$997$	−3.31717e13	−1.06326	−0.531630	−	0.846977i	$-0.678420\pi$
$997$	−3.31717e13	−1.06326	−0.531630	+	0.846977i	$0.678420\pi$
$998$	0	0
$999$	−4.15882e12	−0.132107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.g.1.4		4
4.3	odd	2		13.10.a.a.1.4	✓	4
12.11	even	2		117.10.a.c.1.1		4
20.19	odd	2		325.10.a.a.1.1		4
52.51	odd	2		169.10.a.a.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.a.a.1.4	✓	4	4.3	odd	2
117.10.a.c.1.1		4	12.11	even	2
169.10.a.a.1.1		4	52.51	odd	2
208.10.a.g.1.4		4	1.1	even	1	trivial
325.10.a.a.1.1		4	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)

\(f(q)\)	\(=\)	\(q+204.594 q^{3} -258.914 q^{5} +8862.28 q^{7} +22175.6 q^{9} +O(q^{10})\)	\(q+204.594 q^{3} -258.914 q^{5} +8862.28 q^{7} +22175.6 q^{9} -36087.9 q^{11} -28561.0 q^{13} -52972.2 q^{15} +327405. q^{17} -265525. q^{19} +1.81317e6 q^{21} +2.42458e6 q^{23} -1.88609e6 q^{25} +509973. q^{27} +3.99178e6 q^{29} +6.45220e6 q^{31} -7.38336e6 q^{33} -2.29457e6 q^{35} -8.15498e6 q^{37} -5.84340e6 q^{39} -720241. q^{41} +4.13245e7 q^{43} -5.74158e6 q^{45} -3.67369e7 q^{47} +3.81864e7 q^{49} +6.69850e7 q^{51} +1.67334e7 q^{53} +9.34367e6 q^{55} -5.43248e7 q^{57} +5.89865e7 q^{59} +1.28785e8 q^{61} +1.96527e8 q^{63} +7.39484e6 q^{65} +1.91470e8 q^{67} +4.96054e8 q^{69} -3.40865e8 q^{71} +3.19890e8 q^{73} -3.85882e8 q^{75} -3.19821e8 q^{77} -2.44328e8 q^{79} -3.32145e8 q^{81} +4.38630e8 q^{83} -8.47697e7 q^{85} +8.16693e8 q^{87} -7.90342e8 q^{89} -2.53116e8 q^{91} +1.32008e9 q^{93} +6.87481e7 q^{95} -1.65696e8 q^{97} -8.00272e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9}+O(q^{10}) \) 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9	\( 4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9} + 40140 q^{11} - 114244 q^{13} - 83307 q^{15} + 78717 q^{17} - 209664 q^{19} + 1138431 q^{21} + 4257444 q^{23} - 2900157 q^{25} + 2077801 q^{27} - 1647936 q^{29} + 11366002 q^{31} - 14413222 q^{33} + 13789797 q^{35} + 4636891 q^{37} - 4655443 q^{39} + 13859538 q^{41} + 33368081 q^{43} - 17423928 q^{45} + 3943005 q^{47} + 23294923 q^{49} + 19664471 q^{51} - 171019326 q^{53} + 121160538 q^{55} - 47829030 q^{57} + 63389388 q^{59} + 77050190 q^{61} + 155695476 q^{63} - 13452231 q^{65} + 41174072 q^{67} + 546642556 q^{69} - 252460989 q^{71} + 594415068 q^{73} - 533318748 q^{75} + 561950454 q^{77} - 115998984 q^{79} + 437803700 q^{81} + 79577862 q^{83} + 549463469 q^{85} + 1087526510 q^{87} - 1152240276 q^{89} - 321054201 q^{91} + 1618266556 q^{93} + 1273705170 q^{95} + 1049098084 q^{97} - 2132181050 q^{99}+O(q^{100}) \) 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9 + 40140 * q^11 - 114244 * q^13 - 83307 * q^15 + 78717 * q^17 - 209664 * q^19 + 1138431 * q^21 + 4257444 * q^23 - 2900157 * q^25 + 2077801 * q^27 - 1647936 * q^29 + 11366002 * q^31 - 14413222 * q^33 + 13789797 * q^35 + 4636891 * q^37 - 4655443 * q^39 + 13859538 * q^41 + 33368081 * q^43 - 17423928 * q^45 + 3943005 * q^47 + 23294923 * q^49 + 19664471 * q^51 - 171019326 * q^53 + 121160538 * q^55 - 47829030 * q^57 + 63389388 * q^59 + 77050190 * q^61 + 155695476 * q^63 - 13452231 * q^65 + 41174072 * q^67 + 546642556 * q^69 - 252460989 * q^71 + 594415068 * q^73 - 533318748 * q^75 + 561950454 * q^77 - 115998984 * q^79 + 437803700 * q^81 + 79577862 * q^83 + 549463469 * q^85 + 1087526510 * q^87 - 1152240276 * q^89 - 321054201 * q^91 + 1618266556 * q^93 + 1273705170 * q^95 + 1049098084 * q^97 - 2132181050 * q^99

Introduction

L-functions

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