LMFDB - Embedded newform 208.10.a.i.1.2

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 = [5,0,147] 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:	$5$
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} - x^{4} - 5049x^{3} - 87027x^{2} + 2867800x + 48215700 $ x^5 - x^4 - 5049x^3 - 87027x^2 + 2867800*x + 48215700
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$24.5519$ 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)

$f(q)$	$=$	$q-43.6557 q^{3} -1118.78 q^{5} +2400.89 q^{7} -17777.2 q^{9} -89423.9 q^{11} -28561.0 q^{13} +48841.2 q^{15} +6922.38 q^{17} -953407. q^{19} -104812. q^{21} +219400. q^{23} -701452. q^{25} +1.63535e6 q^{27} +3.05756e6 q^{29} -7.36697e6 q^{31} +3.90386e6 q^{33} -2.68607e6 q^{35} +2.13197e7 q^{37} +1.24685e6 q^{39} -2.82208e7 q^{41} -3.92754e7 q^{43} +1.98888e7 q^{45} -6.24103e7 q^{47} -3.45893e7 q^{49} -302201. q^{51} +2.98002e7 q^{53} +1.00046e8 q^{55} +4.16216e7 q^{57} -4.75092e7 q^{59} +6.17261e7 q^{61} -4.26810e7 q^{63} +3.19535e7 q^{65} +2.57132e8 q^{67} -9.57806e6 q^{69} +1.59261e8 q^{71} -2.29436e8 q^{73} +3.06223e7 q^{75} -2.14697e8 q^{77} +3.23364e8 q^{79} +2.78516e8 q^{81} -2.81009e8 q^{83} -7.74463e6 q^{85} -1.33480e8 q^{87} -8.03235e8 q^{89} -6.85718e7 q^{91} +3.21610e8 q^{93} +1.06665e9 q^{95} -6.00909e8 q^{97} +1.58970e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 147 q^{3} + 1721 q^{5} - 3317 q^{7} - 3204 q^{9} - 6562 q^{11} - 142805 q^{13} - 80283 q^{15} - 146493 q^{17} - 110250 q^{19} + 2250015 q^{21} - 1675652 q^{23} + 6048610 q^{25} - 4937895 q^{27} + 7019714 q^{29}+ \cdots - 302611248 q^{99}+O(q^{100}) $ 5 * q + 147 * q^3 + 1721 * q^5 - 3317 * q^7 - 3204 * q^9 - 6562 * q^11 - 142805 * q^13 - 80283 * q^15 - 146493 * q^17 - 110250 * q^19 + 2250015 * q^21 - 1675652 * q^23 + 6048610 * q^25 - 4937895 * q^27 + 7019714 * q^29 - 14112296 * q^31 + 21400890 * q^33 - 19283879 * q^35 + 28285653 * q^37 - 4198467 * q^39 + 55512492 * q^41 - 61336547 * q^43 + 90036738 * q^45 - 78843745 * q^47 + 88081704 * q^49 - 10216377 * q^51 + 105957576 * q^53 - 71100802 * q^55 + 21427866 * q^57 + 59686630 * q^59 - 41439220 * q^61 + 49300830 * q^63 - 49153481 * q^65 + 214570942 * q^67 - 432347172 * q^69 + 477669785 * q^71 - 201698114 * q^73 + 727786788 * q^75 - 521247014 * q^77 + 582737180 * q^79 - 855021987 * q^81 + 1134244092 * q^83 - 552245895 * q^85 + 2198696622 * q^87 - 476858778 * q^89 + 94736837 * q^91 + 1038474204 * q^93 + 866887902 * q^95 + 1473121030 * q^97 - 302611248 * 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
$3$	−43.6557	−0.311168	−0.155584	−	0.987823i	$-0.549726\pi$
$3$	−43.6557	−0.311168	−0.155584	+	0.987823i	$0.549726\pi$
$4$	0	0
$5$	−1118.78	−0.800535	−0.400268	−	0.916398i	$-0.631083\pi$
$5$	−1118.78	−0.800535	−0.400268	+	0.916398i	$0.631083\pi$
$6$	0	0
$7$	2400.89	0.377947	0.188973	−	0.981982i	$-0.439484\pi$
$7$	2400.89	0.377947	0.188973	+	0.981982i	$0.439484\pi$
$8$	0	0
$9$	−17777.2	−0.903174
$10$	0	0
$11$	−89423.9	−1.84156	−0.920781	−	0.390079i	$-0.872448\pi$
$11$	−89423.9	−1.84156	−0.920781	+	0.390079i	$0.872448\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	48841.2	0.249101
$16$	0	0
$17$	6922.38	0.0201018	0.0100509	−	0.999949i	$-0.496801\pi$
$17$	6922.38	0.0201018	0.0100509	+	0.999949i	$0.496801\pi$
$18$	0	0
$19$	−953407.	−1.67837	−0.839184	−	0.543848i	$-0.816967\pi$
$19$	−953407.	−1.67837	−0.839184	+	0.543848i	$0.816967\pi$
$20$	0	0
$21$	−104812.	−0.117605
$22$	0	0
$23$	219400.	0.163479	0.0817394	−	0.996654i	$-0.473952\pi$
$23$	219400.	0.163479	0.0817394	+	0.996654i	$0.473952\pi$
$24$	0	0
$25$	−701452.	−0.359143
$26$	0	0
$27$	1.63535e6	0.592207
$28$	0	0
$29$	3.05756e6	0.802756	0.401378	−	0.915912i	$-0.368531\pi$
$29$	3.05756e6	0.802756	0.401378	+	0.915912i	$0.368531\pi$
$30$	0	0
$31$	−7.36697e6	−1.43272	−0.716360	−	0.697731i	$-0.754193\pi$
$31$	−7.36697e6	−1.43272	−0.716360	+	0.697731i	$0.754193\pi$
$32$	0	0
$33$	3.90386e6	0.573035
$34$	0	0
$35$	−2.68607e6	−0.302560
$36$	0	0
$37$	2.13197e7	1.87014	0.935068	−	0.354469i	$-0.115338\pi$
$37$	2.13197e7	1.87014	0.935068	+	0.354469i	$0.115338\pi$
$38$	0	0
$39$	1.24685e6	0.0863025
$40$	0	0
$41$	−2.82208e7	−1.55970	−0.779852	−	0.625964i	$-0.784706\pi$
$41$	−2.82208e7	−1.55970	−0.779852	+	0.625964i	$0.784706\pi$
$42$	0	0
$43$	−3.92754e7	−1.75192	−0.875958	−	0.482388i	$-0.839770\pi$
$43$	−3.92754e7	−1.75192	−0.875958	+	0.482388i	$0.839770\pi$
$44$	0	0
$45$	1.98888e7	0.723023
$46$	0	0
$47$	−6.24103e7	−1.86559	−0.932794	−	0.360409i	$-0.882637\pi$
$47$	−6.24103e7	−1.86559	−0.932794	+	0.360409i	$0.882637\pi$
$48$	0	0
$49$	−3.45893e7	−0.857156
$50$	0	0
$51$	−302201.	−0.00625504
$52$	0	0
$53$	2.98002e7	0.518773	0.259386	−	0.965774i	$-0.416480\pi$
$53$	2.98002e7	0.518773	0.259386	+	0.965774i	$0.416480\pi$
$54$	0	0
$55$	1.00046e8	1.47424
$56$	0	0
$57$	4.16216e7	0.522254
$58$	0	0
$59$	−4.75092e7	−0.510439	−0.255219	−	0.966883i	$-0.582148\pi$
$59$	−4.75092e7	−0.510439	−0.255219	+	0.966883i	$0.582148\pi$
$60$	0	0
$61$	6.17261e7	0.570801	0.285400	−	0.958408i	$-0.407873\pi$
$61$	6.17261e7	0.570801	0.285400	+	0.958408i	$0.407873\pi$
$62$	0	0
$63$	−4.26810e7	−0.341352
$64$	0	0
$65$	3.19535e7	0.222029
$66$	0	0
$67$	2.57132e8	1.55891	0.779453	−	0.626461i	$-0.215497\pi$
$67$	2.57132e8	1.55891	0.779453	+	0.626461i	$0.215497\pi$
$68$	0	0
$69$	−9.57806e6	−0.0508694
$70$	0	0
$71$	1.59261e8	0.743786	0.371893	−	0.928276i	$-0.378709\pi$
$71$	1.59261e8	0.743786	0.371893	+	0.928276i	$0.378709\pi$
$72$	0	0
$73$	−2.29436e8	−0.945604	−0.472802	−	0.881169i	$-0.656757\pi$
$73$	−2.29436e8	−0.945604	−0.472802	+	0.881169i	$0.656757\pi$
$74$	0	0
$75$	3.06223e7	0.111754
$76$	0	0
$77$	−2.14697e8	−0.696013
$78$	0	0
$79$	3.23364e8	0.934048	0.467024	−	0.884245i	$-0.345326\pi$
$79$	3.23364e8	0.934048	0.467024	+	0.884245i	$0.345326\pi$
$80$	0	0
$81$	2.78516e8	0.718899
$82$	0	0
$83$	−2.81009e8	−0.649933	−0.324966	−	0.945726i	$-0.605353\pi$
$83$	−2.81009e8	−0.649933	−0.324966	+	0.945726i	$0.605353\pi$
$84$	0	0
$85$	−7.74463e6	−0.0160922
$86$	0	0
$87$	−1.33480e8	−0.249792
$88$	0	0
$89$	−8.03235e8	−1.35702	−0.678512	−	0.734589i	$-0.737375\pi$
$89$	−8.03235e8	−1.35702	−0.678512	+	0.734589i	$0.737375\pi$
$90$	0	0
$91$	−6.85718e7	−0.104824
$92$	0	0
$93$	3.21610e8	0.445817
$94$	0	0
$95$	1.06665e9	1.34359
$96$	0	0
$97$	−6.00909e8	−0.689185	−0.344593	−	0.938752i	$-0.611983\pi$
$97$	−6.00909e8	−0.689185	−0.344593	+	0.938752i	$0.611983\pi$
$98$	0	0
$99$	1.58970e9	1.66325
$100$	0	0
$101$	1.50762e9	1.44160	0.720801	−	0.693143i	$-0.243774\pi$
$101$	1.50762e9	1.44160	0.720801	+	0.693143i	$0.243774\pi$
$102$	0	0
$103$	−1.41394e9	−1.23784	−0.618918	−	0.785456i	$-0.712429\pi$
$103$	−1.41394e9	−1.23784	−0.618918	+	0.785456i	$0.712429\pi$
$104$	0	0
$105$	1.17262e8	0.0941469
$106$	0	0
$107$	−4.88063e8	−0.359955	−0.179978	−	0.983671i	$-0.557603\pi$
$107$	−4.88063e8	−0.359955	−0.179978	+	0.983671i	$0.557603\pi$
$108$	0	0
$109$	−1.57823e9	−1.07090	−0.535451	−	0.844566i	$-0.679858\pi$
$109$	−1.57823e9	−1.07090	−0.535451	+	0.844566i	$0.679858\pi$
$110$	0	0
$111$	−9.30725e8	−0.581926
$112$	0	0
$113$	1.93883e9	1.11863	0.559315	−	0.828955i	$-0.311064\pi$
$113$	1.93883e9	1.11863	0.559315	+	0.828955i	$0.311064\pi$
$114$	0	0
$115$	−2.45461e8	−0.130871
$116$	0	0
$117$	5.07734e8	0.250496
$118$	0	0
$119$	1.66199e7	0.00759741
$120$	0	0
$121$	5.63868e9	2.39135
$122$	0	0
$123$	1.23200e9	0.485330
$124$	0	0
$125$	2.96989e9	1.08804
$126$	0	0
$127$	−5.49035e8	−0.187276	−0.0936382	−	0.995606i	$-0.529850\pi$
$127$	−5.49035e8	−0.187276	−0.0936382	+	0.995606i	$0.529850\pi$
$128$	0	0
$129$	1.71460e9	0.545140
$130$	0	0
$131$	4.67425e9	1.38673	0.693364	−	0.720587i	$-0.256128\pi$
$131$	4.67425e9	1.38673	0.693364	+	0.720587i	$0.256128\pi$
$132$	0	0
$133$	−2.28902e9	−0.634334
$134$	0	0
$135$	−1.82960e9	−0.474083
$136$	0	0
$137$	1.31544e9	0.319028	0.159514	−	0.987196i	$-0.449007\pi$
$137$	1.31544e9	0.319028	0.159514	+	0.987196i	$0.449007\pi$
$138$	0	0
$139$	−1.67356e8	−0.0380255	−0.0190128	−	0.999819i	$-0.506052\pi$
$139$	−1.67356e8	−0.0380255	−0.0190128	+	0.999819i	$0.506052\pi$
$140$	0	0
$141$	2.72456e9	0.580512
$142$	0	0
$143$	2.55404e9	0.510758
$144$	0	0
$145$	−3.42074e9	−0.642635
$146$	0	0
$147$	1.51002e9	0.266720
$148$	0	0
$149$	6.81714e9	1.13309	0.566544	−	0.824031i	$-0.308280\pi$
$149$	6.81714e9	1.13309	0.566544	+	0.824031i	$0.308280\pi$
$150$	0	0
$151$	1.96107e9	0.306970	0.153485	−	0.988151i	$-0.450950\pi$
$151$	1.96107e9	0.306970	0.153485	+	0.988151i	$0.450950\pi$
$152$	0	0
$153$	−1.23060e8	−0.0181554
$154$	0	0
$155$	8.24203e9	1.14694
$156$	0	0
$157$	5.66617e8	0.0744288	0.0372144	−	0.999307i	$-0.488152\pi$
$157$	5.66617e8	0.0744288	0.0372144	+	0.999307i	$0.488152\pi$
$158$	0	0
$159$	−1.30095e9	−0.161426
$160$	0	0
$161$	5.26755e8	0.0617863
$162$	0	0
$163$	2.63411e9	0.292274	0.146137	−	0.989264i	$-0.453316\pi$
$163$	2.63411e9	0.292274	0.146137	+	0.989264i	$0.453316\pi$
$164$	0	0
$165$	−4.36757e9	−0.458735
$166$	0	0
$167$	1.46991e9	0.146240	0.0731202	−	0.997323i	$-0.476704\pi$
$167$	1.46991e9	0.146240	0.0731202	+	0.997323i	$0.476704\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	1.69489e10	1.51586
$172$	0	0
$173$	−3.61967e9	−0.307229	−0.153614	−	0.988131i	$-0.549091\pi$
$173$	−3.61967e9	−0.307229	−0.153614	+	0.988131i	$0.549091\pi$
$174$	0	0
$175$	−1.68411e9	−0.135737
$176$	0	0
$177$	2.07405e9	0.158832
$178$	0	0
$179$	1.76991e9	0.128858	0.0644291	−	0.997922i	$-0.479477\pi$
$179$	1.76991e9	0.128858	0.0644291	+	0.997922i	$0.479477\pi$
$180$	0	0
$181$	3.82763e9	0.265079	0.132540	−	0.991178i	$-0.457687\pi$
$181$	3.82763e9	0.265079	0.132540	+	0.991178i	$0.457687\pi$
$182$	0	0
$183$	−2.69469e9	−0.177615
$184$	0	0
$185$	−2.38521e10	−1.49711
$186$	0	0
$187$	−6.19026e8	−0.0370187
$188$	0	0
$189$	3.92629e9	0.223823
$190$	0	0
$191$	6.46855e9	0.351687	0.175844	−	0.984418i	$-0.443735\pi$
$191$	6.46855e9	0.351687	0.175844	+	0.984418i	$0.443735\pi$
$192$	0	0
$193$	−2.13263e10	−1.10639	−0.553194	−	0.833053i	$-0.686591\pi$
$193$	−2.13263e10	−1.10639	−0.553194	+	0.833053i	$0.686591\pi$
$194$	0	0
$195$	−1.39495e9	−0.0690882
$196$	0	0
$197$	3.34250e10	1.58115	0.790575	−	0.612365i	$-0.209782\pi$
$197$	3.34250e10	1.58115	0.790575	+	0.612365i	$0.209782\pi$
$198$	0	0
$199$	2.74180e10	1.23936	0.619678	−	0.784856i	$-0.287263\pi$
$199$	2.74180e10	1.23936	0.619678	+	0.784856i	$0.287263\pi$
$200$	0	0
$201$	−1.12253e10	−0.485082
$202$	0	0
$203$	7.34085e9	0.303399
$204$	0	0
$205$	3.15729e10	1.24860
$206$	0	0
$207$	−3.90032e9	−0.147650
$208$	0	0
$209$	8.52574e10	3.09082
$210$	0	0
$211$	−3.67676e10	−1.27701	−0.638505	−	0.769618i	$-0.720447\pi$
$211$	−3.67676e10	−1.27701	−0.638505	+	0.769618i	$0.720447\pi$
$212$	0	0
$213$	−6.95267e9	−0.231443
$214$	0	0
$215$	4.39407e10	1.40247
$216$	0	0
$217$	−1.76873e10	−0.541492
$218$	0	0
$219$	1.00162e10	0.294242
$220$	0	0
$221$	−1.97710e8	−0.00557524
$222$	0	0
$223$	−3.79284e9	−0.102705	−0.0513526	−	0.998681i	$-0.516353\pi$
$223$	−3.79284e9	−0.102705	−0.0513526	+	0.998681i	$0.516353\pi$
$224$	0	0
$225$	1.24698e10	0.324369
$226$	0	0
$227$	−6.16898e10	−1.54205	−0.771023	−	0.636808i	$-0.780254\pi$
$227$	−6.16898e10	−1.54205	−0.771023	+	0.636808i	$0.780254\pi$
$228$	0	0
$229$	−1.14559e10	−0.275276	−0.137638	−	0.990483i	$-0.543951\pi$
$229$	−1.14559e10	−0.275276	−0.137638	+	0.990483i	$0.543951\pi$
$230$	0	0
$231$	9.37273e9	0.216577
$232$	0	0
$233$	5.80159e10	1.28957	0.644786	−	0.764363i	$-0.276946\pi$
$233$	5.80159e10	1.28957	0.644786	+	0.764363i	$0.276946\pi$
$234$	0	0
$235$	6.98235e10	1.49347
$236$	0	0
$237$	−1.41167e10	−0.290646
$238$	0	0
$239$	−6.92877e10	−1.37362	−0.686808	−	0.726839i	$-0.740989\pi$
$239$	−6.92877e10	−1.37362	−0.686808	+	0.726839i	$0.740989\pi$
$240$	0	0
$241$	2.01694e10	0.385138	0.192569	−	0.981283i	$-0.438318\pi$
$241$	2.01694e10	0.385138	0.192569	+	0.981283i	$0.438318\pi$
$242$	0	0
$243$	−4.43474e10	−0.815905
$244$	0	0
$245$	3.86979e10	0.686184
$246$	0	0
$247$	2.72303e10	0.465495
$248$	0	0
$249$	1.22676e10	0.202238
$250$	0	0
$251$	−7.40441e10	−1.17749	−0.588747	−	0.808317i	$-0.700379\pi$
$251$	−7.40441e10	−1.17749	−0.588747	+	0.808317i	$0.700379\pi$
$252$	0	0
$253$	−1.96196e10	−0.301057
$254$	0	0
$255$	3.38097e8	0.00500738
$256$	0	0
$257$	−7.66758e10	−1.09638	−0.548188	−	0.836355i	$-0.684682\pi$
$257$	−7.66758e10	−1.09638	−0.548188	+	0.836355i	$0.684682\pi$
$258$	0	0
$259$	5.11862e10	0.706812
$260$	0	0
$261$	−5.43548e10	−0.725029
$262$	0	0
$263$	−2.35219e10	−0.303160	−0.151580	−	0.988445i	$-0.548436\pi$
$263$	−2.35219e10	−0.303160	−0.151580	+	0.988445i	$0.548436\pi$
$264$	0	0
$265$	−3.33399e10	−0.415296
$266$	0	0
$267$	3.50658e10	0.422263
$268$	0	0
$269$	−9.12998e10	−1.06313	−0.531563	−	0.847019i	$-0.678395\pi$
$269$	−9.12998e10	−1.06313	−0.531563	+	0.847019i	$0.678395\pi$
$270$	0	0
$271$	−2.92307e10	−0.329214	−0.164607	−	0.986359i	$-0.552636\pi$
$271$	−2.92307e10	−0.329214	−0.164607	+	0.986359i	$0.552636\pi$
$272$	0	0
$273$	2.99355e9	0.0326177
$274$	0	0
$275$	6.27265e10	0.661385
$276$	0	0
$277$	8.96158e10	0.914588	0.457294	−	0.889316i	$-0.348819\pi$
$277$	8.96158e10	0.914588	0.457294	+	0.889316i	$0.348819\pi$
$278$	0	0
$279$	1.30964e11	1.29400
$280$	0	0
$281$	5.24439e9	0.0501783	0.0250892	−	0.999685i	$-0.492013\pi$
$281$	5.24439e9	0.0501783	0.0250892	+	0.999685i	$0.492013\pi$
$282$	0	0
$283$	−7.37444e10	−0.683424	−0.341712	−	0.939805i	$-0.611007\pi$
$283$	−7.37444e10	−0.683424	−0.341712	+	0.939805i	$0.611007\pi$
$284$	0	0
$285$	−4.65655e10	−0.418083
$286$	0	0
$287$	−6.77550e10	−0.589485
$288$	0	0
$289$	−1.18540e11	−0.999596
$290$	0	0
$291$	2.62331e10	0.214452
$292$	0	0
$293$	−1.05352e11	−0.835097	−0.417548	−	0.908655i	$-0.637111\pi$
$293$	−1.05352e11	−0.835097	−0.417548	+	0.908655i	$0.637111\pi$
$294$	0	0
$295$	5.31524e10	0.408624
$296$	0	0
$297$	−1.46239e11	−1.09059
$298$	0	0
$299$	−6.26629e9	−0.0453409
$300$	0	0
$301$	−9.42959e10	−0.662131
$302$	0	0
$303$	−6.58161e10	−0.448580
$304$	0	0
$305$	−6.90581e10	−0.456946
$306$	0	0
$307$	−1.92855e11	−1.23911	−0.619553	−	0.784955i	$-0.712686\pi$
$307$	−1.92855e11	−1.23911	−0.619553	+	0.784955i	$0.712686\pi$
$308$	0	0
$309$	6.17264e10	0.385175
$310$	0	0
$311$	−9.36526e10	−0.567673	−0.283836	−	0.958873i	$-0.591607\pi$
$311$	−9.36526e10	−0.567673	−0.283836	+	0.958873i	$0.591607\pi$
$312$	0	0
$313$	−9.45470e10	−0.556799	−0.278399	−	0.960465i	$-0.589804\pi$
$313$	−9.45470e10	−0.556799	−0.278399	+	0.960465i	$0.589804\pi$
$314$	0	0
$315$	4.77508e10	0.273264
$316$	0	0
$317$	2.31003e11	1.28484	0.642422	−	0.766351i	$-0.277930\pi$
$317$	2.31003e11	1.28484	0.642422	+	0.766351i	$0.277930\pi$
$318$	0	0
$319$	−2.73419e11	−1.47833
$320$	0	0
$321$	2.13067e10	0.112007
$322$	0	0
$323$	−6.59984e9	−0.0337382
$324$	0	0
$325$	2.00342e10	0.0996084
$326$	0	0
$327$	6.88985e10	0.333231
$328$	0	0
$329$	−1.49840e11	−0.705093
$330$	0	0
$331$	3.69233e11	1.69073	0.845364	−	0.534190i	$-0.179383\pi$
$331$	3.69233e11	1.69073	0.845364	+	0.534190i	$0.179383\pi$
$332$	0	0
$333$	−3.79004e11	−1.68906
$334$	0	0
$335$	−2.87675e11	−1.24796
$336$	0	0
$337$	−4.51363e11	−1.90630	−0.953151	−	0.302495i	$-0.902181\pi$
$337$	−4.51363e11	−1.90630	−0.953151	+	0.302495i	$0.902181\pi$
$338$	0	0
$339$	−8.46408e10	−0.348082
$340$	0	0
$341$	6.58783e11	2.63844
$342$	0	0
$343$	−1.79930e11	−0.701906
$344$	0	0
$345$	1.07158e10	0.0407228
$346$	0	0
$347$	−5.25600e10	−0.194613	−0.0973066	−	0.995254i	$-0.531023\pi$
$347$	−5.25600e10	−0.194613	−0.0973066	+	0.995254i	$0.531023\pi$
$348$	0	0
$349$	−3.00136e11	−1.08294	−0.541469	−	0.840721i	$-0.682132\pi$
$349$	−3.00136e11	−1.08294	−0.541469	+	0.840721i	$0.682132\pi$
$350$	0	0
$351$	−4.67072e10	−0.164249
$352$	0	0
$353$	3.39093e11	1.16234	0.581170	−	0.813782i	$-0.302595\pi$
$353$	3.39093e11	1.16234	0.581170	+	0.813782i	$0.302595\pi$
$354$	0	0
$355$	−1.78179e11	−0.595427
$356$	0	0
$357$	−7.25551e8	−0.00236407
$358$	0	0
$359$	−1.12495e11	−0.357443	−0.178721	−	0.983900i	$-0.557196\pi$
$359$	−1.12495e11	−0.357443	−0.178721	+	0.983900i	$0.557196\pi$
$360$	0	0
$361$	5.86297e11	1.81692
$362$	0	0
$363$	−2.46161e11	−0.744113
$364$	0	0
$365$	2.56689e11	0.756990
$366$	0	0
$367$	1.17132e11	0.337038	0.168519	−	0.985698i	$-0.446102\pi$
$367$	1.17132e11	0.337038	0.168519	+	0.985698i	$0.446102\pi$
$368$	0	0
$369$	5.01686e11	1.40868
$370$	0	0
$371$	7.15469e10	0.196069
$372$	0	0
$373$	−5.87942e11	−1.57270	−0.786348	−	0.617784i	$-0.788031\pi$
$373$	−5.87942e11	−1.57270	−0.786348	+	0.617784i	$0.788031\pi$
$374$	0	0
$375$	−1.29653e11	−0.338564
$376$	0	0
$377$	−8.73269e10	−0.222644
$378$	0	0
$379$	−6.33112e10	−0.157617	−0.0788087	−	0.996890i	$-0.525112\pi$
$379$	−6.33112e10	−0.157617	−0.0788087	+	0.996890i	$0.525112\pi$
$380$	0	0
$381$	2.39685e10	0.0582744
$382$	0	0
$383$	2.72035e11	0.645997	0.322999	−	0.946399i	$-0.395309\pi$
$383$	2.72035e11	0.645997	0.322999	+	0.946399i	$0.395309\pi$
$384$	0	0
$385$	2.40199e11	0.557183
$386$	0	0
$387$	6.98207e11	1.58229
$388$	0	0
$389$	2.31998e11	0.513702	0.256851	−	0.966451i	$-0.417315\pi$
$389$	2.31998e11	0.513702	0.256851	+	0.966451i	$0.417315\pi$
$390$	0	0
$391$	1.51877e9	0.00328622
$392$	0	0
$393$	−2.04058e11	−0.431506
$394$	0	0
$395$	−3.61773e11	−0.747738
$396$	0	0
$397$	−5.68473e11	−1.14856	−0.574278	−	0.818660i	$-0.694717\pi$
$397$	−5.68473e11	−1.14856	−0.574278	+	0.818660i	$0.694717\pi$
$398$	0	0
$399$	9.99288e10	0.197384
$400$	0	0
$401$	−1.97098e11	−0.380656	−0.190328	−	0.981721i	$-0.560955\pi$
$401$	−1.97098e11	−0.380656	−0.190328	+	0.981721i	$0.560955\pi$
$402$	0	0
$403$	2.10408e11	0.397365
$404$	0	0
$405$	−3.11599e11	−0.575504
$406$	0	0
$407$	−1.90649e12	−3.44397
$408$	0	0
$409$	−5.80163e11	−1.02517	−0.512584	−	0.858637i	$-0.671312\pi$
$409$	−5.80163e11	−1.02517	−0.512584	+	0.858637i	$0.671312\pi$
$410$	0	0
$411$	−5.74264e10	−0.0992712
$412$	0	0
$413$	−1.14064e11	−0.192919
$414$	0	0
$415$	3.14388e11	0.520294
$416$	0	0
$417$	7.30605e9	0.0118323
$418$	0	0
$419$	7.39387e11	1.17195	0.585974	−	0.810330i	$-0.300712\pi$
$419$	7.39387e11	1.17195	0.585974	+	0.810330i	$0.300712\pi$
$420$	0	0
$421$	−3.41920e11	−0.530462	−0.265231	−	0.964185i	$-0.585448\pi$
$421$	−3.41920e11	−0.530462	−0.265231	+	0.964185i	$0.585448\pi$
$422$	0	0
$423$	1.10948e12	1.68495
$424$	0	0
$425$	−4.85571e9	−0.00721943
$426$	0	0
$427$	1.48197e11	0.215732
$428$	0	0
$429$	−1.11498e11	−0.158931
$430$	0	0
$431$	4.31243e11	0.601970	0.300985	−	0.953629i	$-0.402685\pi$
$431$	4.31243e11	0.601970	0.300985	+	0.953629i	$0.402685\pi$
$432$	0	0
$433$	−9.97252e10	−0.136336	−0.0681678	−	0.997674i	$-0.521715\pi$
$433$	−9.97252e10	−0.136336	−0.0681678	+	0.997674i	$0.521715\pi$
$434$	0	0
$435$	1.49335e11	0.199967
$436$	0	0
$437$	−2.09178e11	−0.274378
$438$	0	0
$439$	−5.36800e11	−0.689798	−0.344899	−	0.938640i	$-0.612087\pi$
$439$	−5.36800e11	−0.689798	−0.344899	+	0.938640i	$0.612087\pi$
$440$	0	0
$441$	6.14901e11	0.774162
$442$	0	0
$443$	−2.73927e11	−0.337923	−0.168961	−	0.985623i	$-0.554041\pi$
$443$	−2.73927e11	−0.337923	−0.168961	+	0.985623i	$0.554041\pi$
$444$	0	0
$445$	8.98645e11	1.08635
$446$	0	0
$447$	−2.97607e11	−0.352581
$448$	0	0
$449$	−2.47519e11	−0.287409	−0.143704	−	0.989621i	$-0.545901\pi$
$449$	−2.47519e11	−0.287409	−0.143704	+	0.989621i	$0.545901\pi$
$450$	0	0
$451$	2.52361e12	2.87229
$452$	0	0
$453$	−8.56117e10	−0.0955194
$454$	0	0
$455$	7.67169e10	0.0839150
$456$	0	0
$457$	6.48193e11	0.695155	0.347577	−	0.937651i	$-0.387004\pi$
$457$	6.48193e11	0.695155	0.347577	+	0.937651i	$0.387004\pi$
$458$	0	0
$459$	1.13205e10	0.0119044
$460$	0	0
$461$	−3.43960e11	−0.354694	−0.177347	−	0.984148i	$-0.556751\pi$
$461$	−3.43960e11	−0.354694	−0.177347	+	0.984148i	$0.556751\pi$
$462$	0	0
$463$	2.31795e10	0.0234418	0.0117209	−	0.999931i	$-0.496269\pi$
$463$	2.31795e10	0.0234418	0.0117209	+	0.999931i	$0.496269\pi$
$464$	0	0
$465$	−3.59811e11	−0.356892
$466$	0	0
$467$	8.96996e11	0.872699	0.436350	−	0.899777i	$-0.356271\pi$
$467$	8.96996e11	0.872699	0.436350	+	0.899777i	$0.356271\pi$
$468$	0	0
$469$	6.17345e11	0.589183
$470$	0	0
$471$	−2.47360e10	−0.0231599
$472$	0	0
$473$	3.51216e12	3.22626
$474$	0	0
$475$	6.68769e11	0.602774
$476$	0	0
$477$	−5.29763e11	−0.468542
$478$	0	0
$479$	1.05580e12	0.916369	0.458185	−	0.888857i	$-0.348500\pi$
$479$	1.05580e12	0.916369	0.458185	+	0.888857i	$0.348500\pi$
$480$	0	0
$481$	−6.08912e11	−0.518682
$482$	0	0
$483$	−2.29958e10	−0.0192259
$484$	0	0
$485$	6.72286e11	0.551717
$486$	0	0
$487$	−5.13456e11	−0.413640	−0.206820	−	0.978379i	$-0.566312\pi$
$487$	−5.13456e11	−0.413640	−0.206820	+	0.978379i	$0.566312\pi$
$488$	0	0
$489$	−1.14994e11	−0.0909462
$490$	0	0
$491$	−1.55403e12	−1.20668	−0.603342	−	0.797482i	$-0.706165\pi$
$491$	−1.55403e12	−1.20668	−0.603342	+	0.797482i	$0.706165\pi$
$492$	0	0
$493$	2.11656e10	0.0161368
$494$	0	0
$495$	−1.77853e12	−1.33149
$496$	0	0
$497$	3.82369e11	0.281112
$498$	0	0
$499$	−8.75125e11	−0.631855	−0.315927	−	0.948783i	$-0.602316\pi$
$499$	−8.75125e11	−0.631855	−0.315927	+	0.948783i	$0.602316\pi$
$500$	0	0
$501$	−6.41700e10	−0.0455053
$502$	0	0
$503$	−8.42711e11	−0.586979	−0.293490	−	0.955962i	$-0.594817\pi$
$503$	−8.42711e11	−0.586979	−0.293490	+	0.955962i	$0.594817\pi$
$504$	0	0
$505$	−1.68670e12	−1.15405
$506$	0	0
$507$	−3.56113e10	−0.0239360
$508$	0	0
$509$	−4.14489e11	−0.273705	−0.136853	−	0.990591i	$-0.543699\pi$
$509$	−4.14489e11	−0.273705	−0.136853	+	0.990591i	$0.543699\pi$
$510$	0	0
$511$	−5.50851e11	−0.357388
$512$	0	0
$513$	−1.55915e12	−0.993941
$514$	0	0
$515$	1.58189e12	0.990931
$516$	0	0
$517$	5.58097e12	3.43560
$518$	0	0
$519$	1.58019e11	0.0955998
$520$	0	0
$521$	1.85381e12	1.10229	0.551146	−	0.834409i	$-0.314191\pi$
$521$	1.85381e12	1.10229	0.551146	+	0.834409i	$0.314191\pi$
$522$	0	0
$523$	−2.31831e10	−0.0135492	−0.00677461	−	0.999977i	$-0.502156\pi$
$523$	−2.31831e10	−0.0135492	−0.00677461	+	0.999977i	$0.502156\pi$
$524$	0	0
$525$	7.35208e10	0.0422370
$526$	0	0
$527$	−5.09969e10	−0.0288003
$528$	0	0
$529$	−1.75302e12	−0.973275
$530$	0	0
$531$	8.44579e11	0.461015
$532$	0	0
$533$	8.06014e11	0.432584
$534$	0	0
$535$	5.46036e11	0.288157
$536$	0	0
$537$	−7.72665e10	−0.0400965
$538$	0	0
$539$	3.09311e12	1.57851
$540$	0	0
$541$	3.16247e12	1.58723	0.793613	−	0.608423i	$-0.208197\pi$
$541$	3.16247e12	1.58723	0.793613	+	0.608423i	$0.208197\pi$
$542$	0	0
$543$	−1.67098e11	−0.0824842
$544$	0	0
$545$	1.76569e12	0.857295
$546$	0	0
$547$	6.34691e11	0.303123	0.151562	−	0.988448i	$-0.451570\pi$
$547$	6.34691e11	0.303123	0.151562	+	0.988448i	$0.451570\pi$
$548$	0	0
$549$	−1.09732e12	−0.515533
$550$	0	0
$551$	−2.91510e12	−1.34732
$552$	0	0
$553$	7.76360e11	0.353020
$554$	0	0
$555$	1.04128e12	0.465853
$556$	0	0
$557$	−3.52826e12	−1.55315	−0.776573	−	0.630027i	$-0.783044\pi$
$557$	−3.52826e12	−1.55315	−0.776573	+	0.630027i	$0.783044\pi$
$558$	0	0
$559$	1.12175e12	0.485894
$560$	0	0
$561$	2.70240e10	0.0115190
$562$	0	0
$563$	−3.38867e12	−1.42148	−0.710741	−	0.703454i	$-0.751640\pi$
$563$	−3.38867e12	−1.42148	−0.710741	+	0.703454i	$0.751640\pi$
$564$	0	0
$565$	−2.16913e12	−0.895503
$566$	0	0
$567$	6.68686e11	0.271705
$568$	0	0
$569$	8.45275e10	0.0338059	0.0169030	−	0.999857i	$-0.494619\pi$
$569$	8.45275e10	0.0338059	0.0169030	+	0.999857i	$0.494619\pi$
$570$	0	0
$571$	2.46141e12	0.968995	0.484497	−	0.874793i	$-0.339003\pi$
$571$	2.46141e12	0.968995	0.484497	+	0.874793i	$0.339003\pi$
$572$	0	0
$573$	−2.82389e11	−0.109434
$574$	0	0
$575$	−1.53899e11	−0.0587123
$576$	0	0
$577$	1.68775e11	0.0633895	0.0316948	−	0.999498i	$-0.489910\pi$
$577$	1.68775e11	0.0633895	0.0316948	+	0.999498i	$0.489910\pi$
$578$	0	0
$579$	9.31013e11	0.344272
$580$	0	0
$581$	−6.74671e11	−0.245640
$582$	0	0
$583$	−2.66485e12	−0.955353
$584$	0	0
$585$	−5.68044e11	−0.200531
$586$	0	0
$587$	3.89965e12	1.35567	0.677835	−	0.735214i	$-0.262919\pi$
$587$	3.89965e12	1.35567	0.677835	+	0.735214i	$0.262919\pi$
$588$	0	0
$589$	7.02372e12	2.40463
$590$	0	0
$591$	−1.45919e12	−0.492004
$592$	0	0
$593$	−2.26068e12	−0.750746	−0.375373	−	0.926874i	$-0.622485\pi$
$593$	−2.26068e12	−0.750746	−0.375373	+	0.926874i	$0.622485\pi$
$594$	0	0
$595$	−1.85940e10	−0.00608200
$596$	0	0
$597$	−1.19695e12	−0.385648
$598$	0	0
$599$	4.43544e12	1.40772	0.703859	−	0.710339i	$-0.251459\pi$
$599$	4.43544e12	1.40772	0.703859	+	0.710339i	$0.251459\pi$
$600$	0	0
$601$	−1.84414e12	−0.576578	−0.288289	−	0.957543i	$-0.593086\pi$
$601$	−1.84414e12	−0.576578	−0.288289	+	0.957543i	$0.593086\pi$
$602$	0	0
$603$	−4.57109e12	−1.40796
$604$	0	0
$605$	−6.30846e12	−1.91436
$606$	0	0
$607$	4.38462e12	1.31094	0.655470	−	0.755221i	$-0.272470\pi$
$607$	4.38462e12	1.31094	0.655470	+	0.755221i	$0.272470\pi$
$608$	0	0
$609$	−3.20470e11	−0.0944081
$610$	0	0
$611$	1.78250e12	0.517421
$612$	0	0
$613$	−4.10734e12	−1.17487	−0.587433	−	0.809273i	$-0.699861\pi$
$613$	−4.10734e12	−1.17487	−0.587433	+	0.809273i	$0.699861\pi$
$614$	0	0
$615$	−1.37834e12	−0.388524
$616$	0	0
$617$	−4.11936e12	−1.14432	−0.572158	−	0.820143i	$-0.693894\pi$
$617$	−4.11936e12	−1.14432	−0.572158	+	0.820143i	$0.693894\pi$
$618$	0	0
$619$	2.10039e12	0.575032	0.287516	−	0.957776i	$-0.407170\pi$
$619$	2.10039e12	0.575032	0.287516	+	0.957776i	$0.407170\pi$
$620$	0	0
$621$	3.58796e11	0.0968134
$622$	0	0
$623$	−1.92848e12	−0.512883
$624$	0	0
$625$	−1.95264e12	−0.511873
$626$	0	0
$627$	−3.72197e12	−0.961764
$628$	0	0
$629$	1.47583e11	0.0375931
$630$	0	0
$631$	−8.64436e11	−0.217071	−0.108535	−	0.994093i	$-0.534616\pi$
$631$	−8.64436e11	−0.217071	−0.108535	+	0.994093i	$0.534616\pi$
$632$	0	0
$633$	1.60511e12	0.397364
$634$	0	0
$635$	6.14250e11	0.149921
$636$	0	0
$637$	9.87906e11	0.237732
$638$	0	0
$639$	−2.83122e12	−0.671769
$640$	0	0
$641$	3.11744e12	0.729352	0.364676	−	0.931135i	$-0.381180\pi$
$641$	3.11744e12	0.729352	0.364676	+	0.931135i	$0.381180\pi$
$642$	0	0
$643$	5.32183e12	1.22776	0.613878	−	0.789401i	$-0.289609\pi$
$643$	5.32183e12	1.22776	0.613878	+	0.789401i	$0.289609\pi$
$644$	0	0
$645$	−1.91826e12	−0.436404
$646$	0	0
$647$	−5.28571e12	−1.18586	−0.592931	−	0.805253i	$-0.702029\pi$
$647$	−5.28571e12	−1.18586	−0.592931	+	0.805253i	$0.702029\pi$
$648$	0	0
$649$	4.24846e12	0.940005
$650$	0	0
$651$	7.72149e11	0.168495
$652$	0	0
$653$	6.83060e11	0.147011	0.0735055	−	0.997295i	$-0.476581\pi$
$653$	6.83060e11	0.147011	0.0735055	+	0.997295i	$0.476581\pi$
$654$	0	0
$655$	−5.22947e12	−1.11012
$656$	0	0
$657$	4.07873e12	0.854046
$658$	0	0
$659$	1.49780e12	0.309364	0.154682	−	0.987964i	$-0.450565\pi$
$659$	1.49780e12	0.309364	0.154682	+	0.987964i	$0.450565\pi$
$660$	0	0
$661$	−5.09414e12	−1.03792	−0.518960	−	0.854798i	$-0.673681\pi$
$661$	−5.09414e12	−1.03792	−0.518960	+	0.854798i	$0.673681\pi$
$662$	0	0
$663$	8.63116e9	0.00173484
$664$	0	0
$665$	2.56092e12	0.507806
$666$	0	0
$667$	6.70829e11	0.131234
$668$	0	0
$669$	1.65579e11	0.0319586
$670$	0	0
$671$	−5.51979e12	−1.05117
$672$	0	0
$673$	−1.29497e12	−0.243327	−0.121664	−	0.992571i	$-0.538823\pi$
$673$	−1.29497e12	−0.243327	−0.121664	+	0.992571i	$0.538823\pi$
$674$	0	0
$675$	−1.14712e12	−0.212687
$676$	0	0
$677$	5.89837e12	1.07915	0.539577	−	0.841936i	$-0.318584\pi$
$677$	5.89837e12	1.07915	0.539577	+	0.841936i	$0.318584\pi$
$678$	0	0
$679$	−1.44272e12	−0.260475
$680$	0	0
$681$	2.69311e12	0.479835
$682$	0	0
$683$	6.57112e12	1.15544	0.577718	−	0.816236i	$-0.303943\pi$
$683$	6.57112e12	1.15544	0.577718	+	0.816236i	$0.303943\pi$
$684$	0	0
$685$	−1.47169e12	−0.255393
$686$	0	0
$687$	5.00113e11	0.0856570
$688$	0	0
$689$	−8.51123e11	−0.143882
$690$	0	0
$691$	−4.95643e12	−0.827023	−0.413512	−	0.910499i	$-0.635698\pi$
$691$	−4.95643e12	−0.827023	−0.413512	+	0.910499i	$0.635698\pi$
$692$	0	0
$693$	3.81670e12	0.628621
$694$	0	0
$695$	1.87235e11	0.0304408
$696$	0	0
$697$	−1.95355e11	−0.0313529
$698$	0	0
$699$	−2.53272e12	−0.401274
$700$	0	0
$701$	−1.24680e13	−1.95014	−0.975068	−	0.221907i	$-0.928772\pi$
$701$	−1.24680e13	−1.95014	−0.975068	+	0.221907i	$0.928772\pi$
$702$	0	0
$703$	−2.03263e13	−3.13877
$704$	0	0
$705$	−3.04819e12	−0.464720
$706$	0	0
$707$	3.61962e12	0.544848
$708$	0	0
$709$	−3.24995e12	−0.483024	−0.241512	−	0.970398i	$-0.577643\pi$
$709$	−3.24995e12	−0.483024	−0.241512	+	0.970398i	$0.577643\pi$
$710$	0	0
$711$	−5.74849e12	−0.843608
$712$	0	0
$713$	−1.61631e12	−0.234219
$714$	0	0
$715$	−2.85741e12	−0.408879
$716$	0	0
$717$	3.02480e12	0.427426
$718$	0	0
$719$	5.14539e12	0.718022	0.359011	−	0.933333i	$-0.383114\pi$
$719$	5.14539e12	0.718022	0.359011	+	0.933333i	$0.383114\pi$
$720$	0	0
$721$	−3.39471e12	−0.467836
$722$	0	0
$723$	−8.80508e11	−0.119843
$724$	0	0
$725$	−2.14473e12	−0.288304
$726$	0	0
$727$	−1.03289e13	−1.37135	−0.685673	−	0.727909i	$-0.740492\pi$
$727$	−1.03289e13	−1.37135	−0.685673	+	0.727909i	$0.740492\pi$
$728$	0	0
$729$	−3.54602e12	−0.465015
$730$	0	0
$731$	−2.71879e11	−0.0352167
$732$	0	0
$733$	1.21857e13	1.55913	0.779564	−	0.626323i	$-0.215441\pi$
$733$	1.21857e13	1.55913	0.779564	+	0.626323i	$0.215441\pi$
$734$	0	0
$735$	−1.68938e12	−0.213518
$736$	0	0
$737$	−2.29938e13	−2.87082
$738$	0	0
$739$	2.06724e12	0.254971	0.127485	−	0.991840i	$-0.459309\pi$
$739$	2.06724e12	0.254971	0.127485	+	0.991840i	$0.459309\pi$
$740$	0	0
$741$	−1.18875e12	−0.144847
$742$	0	0
$743$	−3.20368e12	−0.385655	−0.192827	−	0.981233i	$-0.561766\pi$
$743$	−3.20368e12	−0.385655	−0.192827	+	0.981233i	$0.561766\pi$
$744$	0	0
$745$	−7.62689e12	−0.907078
$746$	0	0
$747$	4.99554e12	0.587003
$748$	0	0
$749$	−1.17178e12	−0.136044
$750$	0	0
$751$	−8.23634e12	−0.944832	−0.472416	−	0.881376i	$-0.656618\pi$
$751$	−8.23634e12	−0.944832	−0.472416	+	0.881376i	$0.656618\pi$
$752$	0	0
$753$	3.23245e12	0.366399
$754$	0	0
$755$	−2.19401e12	−0.245741
$756$	0	0
$757$	1.29938e13	1.43815	0.719076	−	0.694931i	$-0.244565\pi$
$757$	1.29938e13	1.43815	0.719076	+	0.694931i	$0.244565\pi$
$758$	0	0
$759$	8.56507e11	0.0936792
$760$	0	0
$761$	7.50191e12	0.810851	0.405426	−	0.914128i	$-0.367123\pi$
$761$	7.50191e12	0.810851	0.405426	+	0.914128i	$0.367123\pi$
$762$	0	0
$763$	−3.78914e12	−0.404744
$764$	0	0
$765$	1.37678e11	0.0145341
$766$	0	0
$767$	1.35691e12	0.141570
$768$	0	0
$769$	−1.28652e13	−1.32663	−0.663313	−	0.748342i	$-0.730850\pi$
$769$	−1.28652e13	−1.32663	−0.663313	+	0.748342i	$0.730850\pi$
$770$	0	0
$771$	3.34733e12	0.341157
$772$	0	0
$773$	1.51851e13	1.52971	0.764857	−	0.644200i	$-0.222810\pi$
$773$	1.51851e13	1.52971	0.764857	+	0.644200i	$0.222810\pi$
$774$	0	0
$775$	5.16757e12	0.514551
$776$	0	0
$777$	−2.23457e12	−0.219937
$778$	0	0
$779$	2.69059e13	2.61776
$780$	0	0
$781$	−1.42418e13	−1.36973
$782$	0	0
$783$	5.00017e12	0.475398
$784$	0	0
$785$	−6.33921e11	−0.0595829
$786$	0	0
$787$	1.30742e13	1.21486	0.607432	−	0.794372i	$-0.292200\pi$
$787$	1.30742e13	1.21486	0.607432	+	0.794372i	$0.292200\pi$
$788$	0	0
$789$	1.02687e12	0.0943338
$790$	0	0
$791$	4.65491e12	0.422782
$792$	0	0
$793$	−1.76296e12	−0.158312
$794$	0	0
$795$	1.45548e12	0.129227
$796$	0	0
$797$	−1.24204e11	−0.0109037	−0.00545183	−	0.999985i	$-0.501735\pi$
$797$	−1.24204e11	−0.0109037	−0.00545183	+	0.999985i	$0.501735\pi$
$798$	0	0
$799$	−4.32028e11	−0.0375017
$800$	0	0
$801$	1.42793e13	1.22563
$802$	0	0
$803$	2.05171e13	1.74139
$804$	0	0
$805$	−5.89324e11	−0.0494621
$806$	0	0
$807$	3.98576e12	0.330811
$808$	0	0
$809$	−1.52070e13	−1.24817	−0.624087	−	0.781355i	$-0.714529\pi$
$809$	−1.52070e13	−1.24817	−0.624087	+	0.781355i	$0.714529\pi$
$810$	0	0
$811$	9.69769e12	0.787181	0.393590	−	0.919286i	$-0.371233\pi$
$811$	9.69769e12	0.787181	0.393590	+	0.919286i	$0.371233\pi$
$812$	0	0
$813$	1.27609e12	0.102441
$814$	0	0
$815$	−2.94699e12	−0.233975
$816$	0	0
$817$	3.74455e13	2.94036
$818$	0	0
$819$	1.21901e12	0.0946740
$820$	0	0
$821$	1.25402e13	0.963296	0.481648	−	0.876365i	$-0.340038\pi$
$821$	1.25402e13	0.963296	0.481648	+	0.876365i	$0.340038\pi$
$822$	0	0
$823$	−1.72639e13	−1.31171	−0.655856	−	0.754886i	$-0.727692\pi$
$823$	−1.72639e13	−1.31171	−0.655856	+	0.754886i	$0.727692\pi$
$824$	0	0
$825$	−2.73837e12	−0.205802
$826$	0	0
$827$	−1.37562e13	−1.02264	−0.511322	−	0.859389i	$-0.670844\pi$
$827$	−1.37562e13	−1.02264	−0.511322	+	0.859389i	$0.670844\pi$
$828$	0	0
$829$	−2.31796e13	−1.70455	−0.852277	−	0.523091i	$-0.824779\pi$
$829$	−2.31796e13	−1.70455	−0.852277	+	0.523091i	$0.824779\pi$
$830$	0	0
$831$	−3.91224e12	−0.284591
$832$	0	0
$833$	−2.39441e11	−0.0172304
$834$	0	0
$835$	−1.64451e12	−0.117071
$836$	0	0
$837$	−1.20476e13	−0.848467
$838$	0	0
$839$	1.26182e12	0.0879160	0.0439580	−	0.999033i	$-0.486003\pi$
$839$	1.26182e12	0.0879160	0.0439580	+	0.999033i	$0.486003\pi$
$840$	0	0
$841$	−5.15849e12	−0.355583
$842$	0	0
$843$	−2.28947e11	−0.0156139
$844$	0	0
$845$	−9.12625e11	−0.0615796
$846$	0	0
$847$	1.35378e13	0.903804
$848$	0	0
$849$	3.21936e12	0.212660
$850$	0	0
$851$	4.67754e12	0.305728
$852$	0	0
$853$	−9.96982e12	−0.644788	−0.322394	−	0.946606i	$-0.604488\pi$
$853$	−9.96982e12	−0.644788	−0.322394	+	0.946606i	$0.604488\pi$
$854$	0	0
$855$	−1.89621e13	−1.21350
$856$	0	0
$857$	2.26970e13	1.43733	0.718664	−	0.695358i	$-0.244754\pi$
$857$	2.26970e13	1.43733	0.718664	+	0.695358i	$0.244754\pi$
$858$	0	0
$859$	2.47133e12	0.154868	0.0774339	−	0.996997i	$-0.475327\pi$
$859$	2.47133e12	0.154868	0.0774339	+	0.996997i	$0.475327\pi$
$860$	0	0
$861$	2.95789e12	0.183429
$862$	0	0
$863$	1.59942e13	0.981553	0.490777	−	0.871285i	$-0.336713\pi$
$863$	1.59942e13	0.981553	0.490777	+	0.871285i	$0.336713\pi$
$864$	0	0
$865$	4.04963e12	0.245948
$866$	0	0
$867$	5.17494e12	0.311042
$868$	0	0
$869$	−2.89164e13	−1.72011
$870$	0	0
$871$	−7.34395e12	−0.432363
$872$	0	0
$873$	1.06825e13	0.622454
$874$	0	0
$875$	7.13038e12	0.411222
$876$	0	0
$877$	1.72939e13	0.987175	0.493587	−	0.869696i	$-0.335685\pi$
$877$	1.72939e13	0.987175	0.493587	+	0.869696i	$0.335685\pi$
$878$	0	0
$879$	4.59920e12	0.259856
$880$	0	0
$881$	1.28736e13	0.719958	0.359979	−	0.932960i	$-0.382784\pi$
$881$	1.28736e13	0.719958	0.359979	+	0.932960i	$0.382784\pi$
$882$	0	0
$883$	−2.30937e13	−1.27841	−0.639206	−	0.769036i	$-0.720737\pi$
$883$	−2.30937e13	−1.27841	−0.639206	+	0.769036i	$0.720737\pi$
$884$	0	0
$885$	−2.32040e12	−0.127151
$886$	0	0
$887$	9.90454e12	0.537252	0.268626	−	0.963245i	$-0.413430\pi$
$887$	9.90454e12	0.537252	0.268626	+	0.963245i	$0.413430\pi$
$888$	0	0
$889$	−1.31817e12	−0.0707805
$890$	0	0
$891$	−2.49060e13	−1.32390
$892$	0	0
$893$	5.95024e13	3.13114
$894$	0	0
$895$	−1.98014e12	−0.103155
$896$	0	0
$897$	2.73559e11	0.0141086
$898$	0	0
$899$	−2.25249e13	−1.15012
$900$	0	0
$901$	2.06288e11	0.0104283
$902$	0	0
$903$	4.11655e12	0.206034
$904$	0	0
$905$	−4.28228e12	−0.212205
$906$	0	0
$907$	5.97105e10	0.00292966	0.00146483	−	0.999999i	$-0.499534\pi$
$907$	5.97105e10	0.00292966	0.00146483	+	0.999999i	$0.499534\pi$
$908$	0	0
$909$	−2.68012e13	−1.30202
$910$	0	0
$911$	1.79389e13	0.862903	0.431452	−	0.902136i	$-0.358002\pi$
$911$	1.79389e13	0.862903	0.431452	+	0.902136i	$0.358002\pi$
$912$	0	0
$913$	2.51289e13	1.19689
$914$	0	0
$915$	3.01478e12	0.142187
$916$	0	0
$917$	1.12224e13	0.524109
$918$	0	0
$919$	−2.25524e13	−1.04297	−0.521486	−	0.853260i	$-0.674622\pi$
$919$	−2.25524e13	−1.04297	−0.521486	+	0.853260i	$0.674622\pi$
$920$	0	0
$921$	8.41922e12	0.385570
$922$	0	0
$923$	−4.54867e12	−0.206289
$924$	0	0
$925$	−1.49547e13	−0.671646
$926$	0	0
$927$	2.51358e13	1.11798
$928$	0	0
$929$	8.49147e12	0.374035	0.187018	−	0.982357i	$-0.440118\pi$
$929$	8.49147e12	0.374035	0.187018	+	0.982357i	$0.440118\pi$
$930$	0	0
$931$	3.29777e13	1.43862
$932$	0	0
$933$	4.08847e12	0.176642
$934$	0	0
$935$	6.92555e11	0.0296348
$936$	0	0
$937$	2.45074e12	0.103865	0.0519324	−	0.998651i	$-0.483462\pi$
$937$	2.45074e12	0.103865	0.0519324	+	0.998651i	$0.483462\pi$
$938$	0	0
$939$	4.12751e12	0.173258
$940$	0	0
$941$	−3.84726e11	−0.0159955	−0.00799775	−	0.999968i	$-0.502546\pi$
$941$	−3.84726e11	−0.0159955	−0.00799775	+	0.999968i	$0.502546\pi$
$942$	0	0
$943$	−6.19165e12	−0.254979
$944$	0	0
$945$	−4.39266e12	−0.179178
$946$	0	0
$947$	−2.89622e13	−1.17019	−0.585096	−	0.810964i	$-0.698943\pi$
$947$	−2.89622e13	−1.17019	−0.585096	+	0.810964i	$0.698943\pi$
$948$	0	0
$949$	6.55293e12	0.262263
$950$	0	0
$951$	−1.00846e13	−0.399802
$952$	0	0
$953$	−2.95719e13	−1.16134	−0.580672	−	0.814137i	$-0.697210\pi$
$953$	−2.95719e13	−1.16134	−0.580672	+	0.814137i	$0.697210\pi$
$954$	0	0
$955$	−7.23689e12	−0.281538
$956$	0	0
$957$	1.19363e13	0.460008
$958$	0	0
$959$	3.15822e12	0.120575
$960$	0	0
$961$	2.78326e13	1.05269
$962$	0	0
$963$	8.67638e12	0.325103
$964$	0	0
$965$	2.38595e13	0.885702
$966$	0	0
$967$	2.82548e13	1.03914	0.519568	−	0.854429i	$-0.326093\pi$
$967$	2.82548e13	1.03914	0.519568	+	0.854429i	$0.326093\pi$
$968$	0	0
$969$	2.88121e11	0.0104983
$970$	0	0
$971$	−7.35883e12	−0.265657	−0.132829	−	0.991139i	$-0.542406\pi$
$971$	−7.35883e12	−0.265657	−0.132829	+	0.991139i	$0.542406\pi$
$972$	0	0
$973$	−4.01804e11	−0.0143716
$974$	0	0
$975$	−8.74605e11	−0.0309950
$976$	0	0
$977$	−4.47263e13	−1.57050	−0.785248	−	0.619181i	$-0.787465\pi$
$977$	−4.47263e13	−1.57050	−0.785248	+	0.619181i	$0.787465\pi$
$978$	0	0
$979$	7.18284e13	2.49905
$980$	0	0
$981$	2.80564e13	0.967212
$982$	0	0
$983$	2.50576e13	0.855951	0.427975	−	0.903790i	$-0.359227\pi$
$983$	2.50576e13	0.855951	0.427975	+	0.903790i	$0.359227\pi$
$984$	0	0
$985$	−3.73953e13	−1.26577
$986$	0	0
$987$	6.54137e12	0.219402
$988$	0	0
$989$	−8.61704e12	−0.286401
$990$	0	0
$991$	2.32454e13	0.765607	0.382803	−	0.923830i	$-0.374959\pi$
$991$	2.32454e13	0.765607	0.382803	+	0.923830i	$0.374959\pi$
$992$	0	0
$993$	−1.61191e13	−0.526101
$994$	0	0
$995$	−3.06747e13	−0.992149
$996$	0	0
$997$	−4.59157e12	−0.147175	−0.0735873	−	0.997289i	$-0.523445\pi$
$997$	−4.59157e12	−0.147175	−0.0735873	+	0.997289i	$0.523445\pi$
$998$	0	0
$999$	3.48651e13	1.10751

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.i.1.2		5
4.3	odd	2		52.10.a.b.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.10.a.b.1.4	✓	5	4.3	odd	2
208.10.a.i.1.2		5	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)

\(f(q)\)	\(=\)	\(q-43.6557 q^{3} -1118.78 q^{5} +2400.89 q^{7} -17777.2 q^{9} -89423.9 q^{11} -28561.0 q^{13} +48841.2 q^{15} +6922.38 q^{17} -953407. q^{19} -104812. q^{21} +219400. q^{23} -701452. q^{25} +1.63535e6 q^{27} +3.05756e6 q^{29} -7.36697e6 q^{31} +3.90386e6 q^{33} -2.68607e6 q^{35} +2.13197e7 q^{37} +1.24685e6 q^{39} -2.82208e7 q^{41} -3.92754e7 q^{43} +1.98888e7 q^{45} -6.24103e7 q^{47} -3.45893e7 q^{49} -302201. q^{51} +2.98002e7 q^{53} +1.00046e8 q^{55} +4.16216e7 q^{57} -4.75092e7 q^{59} +6.17261e7 q^{61} -4.26810e7 q^{63} +3.19535e7 q^{65} +2.57132e8 q^{67} -9.57806e6 q^{69} +1.59261e8 q^{71} -2.29436e8 q^{73} +3.06223e7 q^{75} -2.14697e8 q^{77} +3.23364e8 q^{79} +2.78516e8 q^{81} -2.81009e8 q^{83} -7.74463e6 q^{85} -1.33480e8 q^{87} -8.03235e8 q^{89} -6.85718e7 q^{91} +3.21610e8 q^{93} +1.06665e9 q^{95} -6.00909e8 q^{97} +1.58970e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 147 q^{3} + 1721 q^{5} - 3317 q^{7} - 3204 q^{9} - 6562 q^{11} - 142805 q^{13} - 80283 q^{15} - 146493 q^{17} - 110250 q^{19} + 2250015 q^{21} - 1675652 q^{23} + 6048610 q^{25} - 4937895 q^{27} + 7019714 q^{29}+ \cdots - 302611248 q^{99}+O(q^{100}) \) 5 * q + 147 * q^3 + 1721 * q^5 - 3317 * q^7 - 3204 * q^9 - 6562 * q^11 - 142805 * q^13 - 80283 * q^15 - 146493 * q^17 - 110250 * q^19 + 2250015 * q^21 - 1675652 * q^23 + 6048610 * q^25 - 4937895 * q^27 + 7019714 * q^29 - 14112296 * q^31 + 21400890 * q^33 - 19283879 * q^35 + 28285653 * q^37 - 4198467 * q^39 + 55512492 * q^41 - 61336547 * q^43 + 90036738 * q^45 - 78843745 * q^47 + 88081704 * q^49 - 10216377 * q^51 + 105957576 * q^53 - 71100802 * q^55 + 21427866 * q^57 + 59686630 * q^59 - 41439220 * q^61 + 49300830 * q^63 - 49153481 * q^65 + 214570942 * q^67 - 432347172 * q^69 + 477669785 * q^71 - 201698114 * q^73 + 727786788 * q^75 - 521247014 * q^77 + 582737180 * q^79 - 855021987 * q^81 + 1134244092 * q^83 - 552245895 * q^85 + 2198696622 * q^87 - 476858778 * q^89 + 94736837 * q^91 + 1038474204 * q^93 + 866887902 * q^95 + 1473121030 * q^97 - 302611248 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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