LMFDB - Embedded newform 208.10.a.i.1.5

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			1.5
Root			$-49.9322$ 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)

$f(q)$	$=$	$q+179.797 q^{3} +2632.50 q^{5} +5669.07 q^{7} +12643.9 q^{9} -8955.44 q^{11} -28561.0 q^{13} +473315. q^{15} -213770. q^{17} +192892. q^{19} +1.01928e6 q^{21} -927462. q^{23} +4.97693e6 q^{25} -1.26562e6 q^{27} +6.36060e6 q^{29} +3.37329e6 q^{31} -1.61016e6 q^{33} +1.49238e7 q^{35} +1.95683e7 q^{37} -5.13517e6 q^{39} +1.82849e6 q^{41} -2.75317e7 q^{43} +3.32850e7 q^{45} +1.35834e7 q^{47} -8.21523e6 q^{49} -3.84351e7 q^{51} +4.04014e7 q^{53} -2.35752e7 q^{55} +3.46813e7 q^{57} +1.46844e8 q^{59} +1.35392e8 q^{61} +7.16789e7 q^{63} -7.51868e7 q^{65} -2.89686e8 q^{67} -1.66755e8 q^{69} +7.81904e7 q^{71} -5.03612e7 q^{73} +8.94836e8 q^{75} -5.07691e7 q^{77} -4.72679e8 q^{79} -4.76422e8 q^{81} -2.16097e8 q^{83} -5.62749e8 q^{85} +1.14361e9 q^{87} +1.11485e8 q^{89} -1.61914e8 q^{91} +6.06506e8 q^{93} +5.07788e8 q^{95} +1.48535e9 q^{97} -1.13231e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	179.797	1.28155	0.640776	−	0.767728i	$-0.278613\pi$
$3$	179.797	1.28155	0.640776	+	0.767728i	$0.278613\pi$
$4$	0	0
$5$	2632.50	1.88366	0.941832	−	0.336084i	$-0.109103\pi$
$5$	2632.50	1.88366	0.941832	+	0.336084i	$0.109103\pi$
$6$	0	0
$7$	5669.07	0.892423	0.446212	−	0.894928i	$-0.352773\pi$
$7$	5669.07	0.892423	0.446212	+	0.894928i	$0.352773\pi$
$8$	0	0
$9$	12643.9	0.642374
$10$	0	0
$11$	−8955.44	−0.184425	−0.0922126	−	0.995739i	$-0.529394\pi$
$11$	−8955.44	−0.184425	−0.0922126	+	0.995739i	$0.529394\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	473315.	2.41401
$16$	0	0
$17$	−213770.	−0.620763	−0.310382	−	0.950612i	$-0.600457\pi$
$17$	−213770.	−0.620763	−0.310382	+	0.950612i	$0.600457\pi$
$18$	0	0
$19$	192892.	0.339565	0.169782	−	0.985482i	$-0.445694\pi$
$19$	192892.	0.339565	0.169782	+	0.985482i	$0.445694\pi$
$20$	0	0
$21$	1.01928e6	1.14369
$22$	0	0
$23$	−927462.	−0.691068	−0.345534	−	0.938406i	$-0.612302\pi$
$23$	−927462.	−0.691068	−0.345534	+	0.938406i	$0.612302\pi$
$24$	0	0
$25$	4.97693e6	2.54819
$26$	0	0
$27$	−1.26562e6	−0.458316
$28$	0	0
$29$	6.36060e6	1.66996	0.834981	−	0.550278i	$-0.185478\pi$
$29$	6.36060e6	1.66996	0.834981	+	0.550278i	$0.185478\pi$
$30$	0	0
$31$	3.37329e6	0.656033	0.328016	−	0.944672i	$-0.393620\pi$
$31$	3.37329e6	0.656033	0.328016	+	0.944672i	$0.393620\pi$
$32$	0	0
$33$	−1.61016e6	−0.236350
$34$	0	0
$35$	1.49238e7	1.68103
$36$	0	0
$37$	1.95683e7	1.71651	0.858254	−	0.513226i	$-0.171550\pi$
$37$	1.95683e7	1.71651	0.858254	+	0.513226i	$0.171550\pi$
$38$	0	0
$39$	−5.13517e6	−0.355438
$40$	0	0
$41$	1.82849e6	0.101057	0.0505283	−	0.998723i	$-0.483909\pi$
$41$	1.82849e6	0.101057	0.0505283	+	0.998723i	$0.483909\pi$
$42$	0	0
$43$	−2.75317e7	−1.22808	−0.614038	−	0.789276i	$-0.710456\pi$
$43$	−2.75317e7	−1.22808	−0.614038	+	0.789276i	$0.710456\pi$
$44$	0	0
$45$	3.32850e7	1.21002
$46$	0	0
$47$	1.35834e7	0.406039	0.203019	−	0.979175i	$-0.434925\pi$
$47$	1.35834e7	0.406039	0.203019	+	0.979175i	$0.434925\pi$
$48$	0	0
$49$	−8.21523e6	−0.203581
$50$	0	0
$51$	−3.84351e7	−0.795540
$52$	0	0
$53$	4.04014e7	0.703323	0.351662	−	0.936127i	$-0.385617\pi$
$53$	4.04014e7	0.703323	0.351662	+	0.936127i	$0.385617\pi$
$54$	0	0
$55$	−2.35752e7	−0.347395
$56$	0	0
$57$	3.46813e7	0.435170
$58$	0	0
$59$	1.46844e8	1.57769	0.788844	−	0.614594i	$-0.210680\pi$
$59$	1.46844e8	1.57769	0.788844	+	0.614594i	$0.210680\pi$
$60$	0	0
$61$	1.35392e8	1.25201	0.626004	−	0.779820i	$-0.284689\pi$
$61$	1.35392e8	1.25201	0.626004	+	0.779820i	$0.284689\pi$
$62$	0	0
$63$	7.16789e7	0.573270
$64$	0	0
$65$	−7.51868e7	−0.522434
$66$	0	0
$67$	−2.89686e8	−1.75627	−0.878135	−	0.478413i	$-0.841212\pi$
$67$	−2.89686e8	−1.75627	−0.878135	+	0.478413i	$0.841212\pi$
$68$	0	0
$69$	−1.66755e8	−0.885639
$70$	0	0
$71$	7.81904e7	0.365166	0.182583	−	0.983190i	$-0.441554\pi$
$71$	7.81904e7	0.365166	0.182583	+	0.983190i	$0.441554\pi$
$72$	0	0
$73$	−5.03612e7	−0.207560	−0.103780	−	0.994600i	$-0.533094\pi$
$73$	−5.03612e7	−0.207560	−0.103780	+	0.994600i	$0.533094\pi$
$74$	0	0
$75$	8.94836e8	3.26564
$76$	0	0
$77$	−5.07691e7	−0.164585
$78$	0	0
$79$	−4.72679e8	−1.36535	−0.682676	−	0.730721i	$-0.739184\pi$
$79$	−4.72679e8	−1.36535	−0.682676	+	0.730721i	$0.739184\pi$
$80$	0	0
$81$	−4.76422e8	−1.22973
$82$	0	0
$83$	−2.16097e8	−0.499801	−0.249900	−	0.968272i	$-0.580398\pi$
$83$	−2.16097e8	−0.499801	−0.249900	+	0.968272i	$0.580398\pi$
$84$	0	0
$85$	−5.62749e8	−1.16931
$86$	0	0
$87$	1.14361e9	2.14014
$88$	0	0
$89$	1.11485e8	0.188349	0.0941744	−	0.995556i	$-0.469979\pi$
$89$	1.11485e8	0.188349	0.0941744	+	0.995556i	$0.469979\pi$
$90$	0	0
$91$	−1.61914e8	−0.247514
$92$	0	0
$93$	6.06506e8	0.840740
$94$	0	0
$95$	5.07788e8	0.639626
$96$	0	0
$97$	1.48535e9	1.70356	0.851779	−	0.523902i	$-0.175524\pi$
$97$	1.48535e9	1.70356	0.851779	+	0.523902i	$0.175524\pi$
$98$	0	0
$99$	−1.13231e8	−0.118470
$100$	0	0
$101$	−2.45128e8	−0.234394	−0.117197	−	0.993109i	$-0.537391\pi$
$101$	−2.45128e8	−0.234394	−0.117197	+	0.993109i	$0.537391\pi$
$102$	0	0
$103$	−2.38832e8	−0.209086	−0.104543	−	0.994520i	$-0.533338\pi$
$103$	−2.38832e8	−0.209086	−0.104543	+	0.994520i	$0.533338\pi$
$104$	0	0
$105$	2.68326e9	2.15432
$106$	0	0
$107$	−2.49128e9	−1.83737	−0.918683	−	0.394995i	$-0.870746\pi$
$107$	−2.49128e9	−1.83737	−0.918683	+	0.394995i	$0.870746\pi$
$108$	0	0
$109$	−6.58467e8	−0.446801	−0.223401	−	0.974727i	$-0.571716\pi$
$109$	−6.58467e8	−0.446801	−0.223401	+	0.974727i	$0.571716\pi$
$110$	0	0
$111$	3.51832e9	2.19979
$112$	0	0
$113$	−3.08304e9	−1.77879	−0.889397	−	0.457135i	$-0.848876\pi$
$113$	−3.08304e9	−1.77879	−0.889397	+	0.457135i	$0.848876\pi$
$114$	0	0
$115$	−2.44154e9	−1.30174
$116$	0	0
$117$	−3.61121e8	−0.178163
$118$	0	0
$119$	−1.21188e9	−0.553984
$120$	0	0
$121$	−2.27775e9	−0.965987
$122$	0	0
$123$	3.28756e8	0.129509
$124$	0	0
$125$	7.96018e9	2.91627
$126$	0	0
$127$	4.74367e9	1.61807	0.809036	−	0.587760i	$-0.199990\pi$
$127$	4.74367e9	1.61807	0.809036	+	0.587760i	$0.199990\pi$
$128$	0	0
$129$	−4.95011e9	−1.57384
$130$	0	0
$131$	−4.82156e9	−1.43043	−0.715215	−	0.698904i	$-0.753671\pi$
$131$	−4.82156e9	−1.43043	−0.715215	+	0.698904i	$0.753671\pi$
$132$	0	0
$133$	1.09352e9	0.303035
$134$	0	0
$135$	−3.33173e9	−0.863313
$136$	0	0
$137$	6.84563e9	1.66024	0.830120	−	0.557584i	$-0.188272\pi$
$137$	6.84563e9	1.66024	0.830120	+	0.557584i	$0.188272\pi$
$138$	0	0
$139$	1.40576e9	0.319407	0.159704	−	0.987165i	$-0.448946\pi$
$139$	1.40576e9	0.319407	0.159704	+	0.987165i	$0.448946\pi$
$140$	0	0
$141$	2.44225e9	0.520360
$142$	0	0
$143$	2.55776e8	0.0511503
$144$	0	0
$145$	1.67443e10	3.14565
$146$	0	0
$147$	−1.47707e9	−0.260900
$148$	0	0
$149$	−5.00961e9	−0.832657	−0.416328	−	0.909214i	$-0.636683\pi$
$149$	−5.00961e9	−0.832657	−0.416328	+	0.909214i	$0.636683\pi$
$150$	0	0
$151$	5.20038e9	0.814028	0.407014	−	0.913422i	$-0.366570\pi$
$151$	5.20038e9	0.814028	0.407014	+	0.913422i	$0.366570\pi$
$152$	0	0
$153$	−2.70287e9	−0.398763
$154$	0	0
$155$	8.88018e9	1.23575
$156$	0	0
$157$	−1.10243e10	−1.44811	−0.724055	−	0.689743i	$-0.757724\pi$
$157$	−1.10243e10	−1.44811	−0.724055	+	0.689743i	$0.757724\pi$
$158$	0	0
$159$	7.26404e9	0.901345
$160$	0	0
$161$	−5.25785e9	−0.616725
$162$	0	0
$163$	3.80298e9	0.421968	0.210984	−	0.977489i	$-0.432333\pi$
$163$	3.80298e9	0.421968	0.210984	+	0.977489i	$0.432333\pi$
$164$	0	0
$165$	−4.23875e9	−0.445205
$166$	0	0
$167$	9.08007e9	0.903369	0.451685	−	0.892178i	$-0.350823\pi$
$167$	9.08007e9	0.903369	0.451685	+	0.892178i	$0.350823\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	2.43890e9	0.218128
$172$	0	0
$173$	6.99066e9	0.593349	0.296675	−	0.954979i	$-0.404122\pi$
$173$	6.99066e9	0.593349	0.296675	+	0.954979i	$0.404122\pi$
$174$	0	0
$175$	2.82146e10	2.27406
$176$	0	0
$177$	2.64020e10	2.02189
$178$	0	0
$179$	9.27045e9	0.674935	0.337468	−	0.941337i	$-0.390430\pi$
$179$	9.27045e9	0.674935	0.337468	+	0.941337i	$0.390430\pi$
$180$	0	0
$181$	−8.93118e9	−0.618522	−0.309261	−	0.950977i	$-0.600082\pi$
$181$	−8.93118e9	−0.618522	−0.309261	+	0.950977i	$0.600082\pi$
$182$	0	0
$183$	2.43430e10	1.60451
$184$	0	0
$185$	5.15136e10	3.23332
$186$	0	0
$187$	1.91440e9	0.114484
$188$	0	0
$189$	−7.17486e9	−0.409011
$190$	0	0
$191$	1.76817e9	0.0961332	0.0480666	−	0.998844i	$-0.484694\pi$
$191$	1.76817e9	0.0961332	0.0480666	+	0.998844i	$0.484694\pi$
$192$	0	0
$193$	3.18546e10	1.65259	0.826294	−	0.563239i	$-0.190445\pi$
$193$	3.18546e10	1.65259	0.826294	+	0.563239i	$0.190445\pi$
$194$	0	0
$195$	−1.35183e10	−0.669527
$196$	0	0
$197$	−1.86231e10	−0.880955	−0.440478	−	0.897764i	$-0.645191\pi$
$197$	−1.86231e10	−0.880955	−0.440478	+	0.897764i	$0.645191\pi$
$198$	0	0
$199$	−7.85919e9	−0.355254	−0.177627	−	0.984098i	$-0.556842\pi$
$199$	−7.85919e9	−0.355254	−0.177627	+	0.984098i	$0.556842\pi$
$200$	0	0
$201$	−5.20846e10	−2.25075
$202$	0	0
$203$	3.60587e10	1.49031
$204$	0	0
$205$	4.81349e9	0.190357
$206$	0	0
$207$	−1.17267e10	−0.443924
$208$	0	0
$209$	−1.72743e9	−0.0626242
$210$	0	0
$211$	−2.89592e10	−1.00581	−0.502904	−	0.864342i	$-0.667735\pi$
$211$	−2.89592e10	−1.00581	−0.502904	+	0.864342i	$0.667735\pi$
$212$	0	0
$213$	1.40584e10	0.467979
$214$	0	0
$215$	−7.24773e10	−2.31328
$216$	0	0
$217$	1.91234e10	0.585459
$218$	0	0
$219$	−9.05478e9	−0.265999
$220$	0	0
$221$	6.10548e9	0.172169
$222$	0	0
$223$	−2.62550e10	−0.710951	−0.355475	−	0.934686i	$-0.615681\pi$
$223$	−2.62550e10	−0.710951	−0.355475	+	0.934686i	$0.615681\pi$
$224$	0	0
$225$	6.29276e10	1.63689
$226$	0	0
$227$	5.10050e10	1.27496	0.637479	−	0.770467i	$-0.279977\pi$
$227$	5.10050e10	1.27496	0.637479	+	0.770467i	$0.279977\pi$
$228$	0	0
$229$	−2.73225e10	−0.656539	−0.328269	−	0.944584i	$-0.606465\pi$
$229$	−2.73225e10	−0.656539	−0.328269	+	0.944584i	$0.606465\pi$
$230$	0	0
$231$	−9.12811e9	−0.210924
$232$	0	0
$233$	5.31510e9	0.118144	0.0590718	−	0.998254i	$-0.481186\pi$
$233$	5.31510e9	0.118144	0.0590718	+	0.998254i	$0.481186\pi$
$234$	0	0
$235$	3.57583e10	0.764841
$236$	0	0
$237$	−8.49862e10	−1.74977
$238$	0	0
$239$	−8.67256e10	−1.71932	−0.859660	−	0.510866i	$-0.829325\pi$
$239$	−8.67256e10	−1.71932	−0.859660	+	0.510866i	$0.829325\pi$
$240$	0	0
$241$	−3.11273e10	−0.594381	−0.297190	−	0.954818i	$-0.596050\pi$
$241$	−3.11273e10	−0.594381	−0.297190	+	0.954818i	$0.596050\pi$
$242$	0	0
$243$	−6.07481e10	−1.11765
$244$	0	0
$245$	−2.16266e10	−0.383478
$246$	0	0
$247$	−5.50918e9	−0.0941783
$248$	0	0
$249$	−3.88535e10	−0.640520
$250$	0	0
$251$	1.39275e10	0.221484	0.110742	−	0.993849i	$-0.464677\pi$
$251$	1.39275e10	0.221484	0.110742	+	0.993849i	$0.464677\pi$
$252$	0	0
$253$	8.30583e9	0.127450
$254$	0	0
$255$	−1.01180e11	−1.49853
$256$	0	0
$257$	−9.46548e10	−1.35345	−0.676727	−	0.736234i	$-0.736602\pi$
$257$	−9.46548e10	−1.35345	−0.676727	+	0.736234i	$0.736602\pi$
$258$	0	0
$259$	1.10934e11	1.53185
$260$	0	0
$261$	8.04224e10	1.07274
$262$	0	0
$263$	−9.11688e10	−1.17502	−0.587510	−	0.809217i	$-0.699892\pi$
$263$	−9.11688e10	−1.17502	−0.587510	+	0.809217i	$0.699892\pi$
$264$	0	0
$265$	1.06357e11	1.32482
$266$	0	0
$267$	2.00447e10	0.241379
$268$	0	0
$269$	3.28436e10	0.382442	0.191221	−	0.981547i	$-0.438755\pi$
$269$	3.28436e10	0.382442	0.191221	+	0.981547i	$0.438755\pi$
$270$	0	0
$271$	1.09848e11	1.23717	0.618586	−	0.785717i	$-0.287706\pi$
$271$	1.09848e11	1.23717	0.618586	+	0.785717i	$0.287706\pi$
$272$	0	0
$273$	−2.91117e10	−0.317201
$274$	0	0
$275$	−4.45707e10	−0.469950
$276$	0	0
$277$	−5.21653e10	−0.532381	−0.266190	−	0.963920i	$-0.585765\pi$
$277$	−5.21653e10	−0.532381	−0.266190	+	0.963920i	$0.585765\pi$
$278$	0	0
$279$	4.26513e10	0.421419
$280$	0	0
$281$	−9.83370e9	−0.0940889	−0.0470444	−	0.998893i	$-0.514980\pi$
$281$	−9.83370e9	−0.0940889	−0.0470444	+	0.998893i	$0.514980\pi$
$282$	0	0
$283$	−1.63436e11	−1.51464	−0.757319	−	0.653045i	$-0.773491\pi$
$283$	−1.63436e11	−1.51464	−0.757319	+	0.653045i	$0.773491\pi$
$284$	0	0
$285$	9.12985e10	0.819713
$286$	0	0
$287$	1.03658e10	0.0901852
$288$	0	0
$289$	−7.28904e10	−0.614653
$290$	0	0
$291$	2.67061e11	2.18320
$292$	0	0
$293$	−2.07425e11	−1.64421	−0.822104	−	0.569338i	$-0.807200\pi$
$293$	−2.07425e11	−1.64421	−0.822104	+	0.569338i	$0.807200\pi$
$294$	0	0
$295$	3.86566e11	2.97183
$296$	0	0
$297$	1.13341e10	0.0845249
$298$	0	0
$299$	2.64892e10	0.191668
$300$	0	0
$301$	−1.56079e11	−1.09596
$302$	0	0
$303$	−4.40732e10	−0.300388
$304$	0	0
$305$	3.56418e11	2.35836
$306$	0	0
$307$	2.39297e11	1.53750	0.768748	−	0.639552i	$-0.220880\pi$
$307$	2.39297e11	1.53750	0.768748	+	0.639552i	$0.220880\pi$
$308$	0	0
$309$	−4.29413e10	−0.267955
$310$	0	0
$311$	−1.68255e11	−1.01987	−0.509936	−	0.860213i	$-0.670331\pi$
$311$	−1.68255e11	−1.01987	−0.509936	+	0.860213i	$0.670331\pi$
$312$	0	0
$313$	1.61730e11	0.952449	0.476225	−	0.879324i	$-0.342005\pi$
$313$	1.61730e11	0.952449	0.476225	+	0.879324i	$0.342005\pi$
$314$	0	0
$315$	1.88695e11	1.07985
$316$	0	0
$317$	4.04406e10	0.224932	0.112466	−	0.993656i	$-0.464125\pi$
$317$	4.04406e10	0.224932	0.112466	+	0.993656i	$0.464125\pi$
$318$	0	0
$319$	−5.69620e10	−0.307983
$320$	0	0
$321$	−4.47924e11	−2.35468
$322$	0	0
$323$	−4.12344e10	−0.210789
$324$	0	0
$325$	−1.42146e11	−0.706741
$326$	0	0
$327$	−1.18390e11	−0.572599
$328$	0	0
$329$	7.70052e10	0.362358
$330$	0	0
$331$	−2.68780e11	−1.23076	−0.615378	−	0.788232i	$-0.710996\pi$
$331$	−2.68780e11	−1.23076	−0.615378	+	0.788232i	$0.710996\pi$
$332$	0	0
$333$	2.47419e11	1.10264
$334$	0	0
$335$	−7.62599e11	−3.30822
$336$	0	0
$337$	−9.14118e10	−0.386072	−0.193036	−	0.981192i	$-0.561833\pi$
$337$	−9.14118e10	−0.386072	−0.193036	+	0.981192i	$0.561833\pi$
$338$	0	0
$339$	−5.54320e11	−2.27962
$340$	0	0
$341$	−3.02093e10	−0.120989
$342$	0	0
$343$	−2.75340e11	−1.07410
$344$	0	0
$345$	−4.38982e11	−1.66825
$346$	0	0
$347$	−3.75389e11	−1.38995	−0.694975	−	0.719033i	$-0.744585\pi$
$347$	−3.75389e11	−1.38995	−0.694975	+	0.719033i	$0.744585\pi$
$348$	0	0
$349$	−5.23032e10	−0.188718	−0.0943592	−	0.995538i	$-0.530080\pi$
$349$	−5.23032e10	−0.188718	−0.0943592	+	0.995538i	$0.530080\pi$
$350$	0	0
$351$	3.61472e10	0.127114
$352$	0	0
$353$	2.20823e11	0.756933	0.378466	−	0.925615i	$-0.376452\pi$
$353$	2.20823e11	0.756933	0.378466	+	0.925615i	$0.376452\pi$
$354$	0	0
$355$	2.05836e11	0.687851
$356$	0	0
$357$	−2.17891e11	−0.709959
$358$	0	0
$359$	−7.53809e10	−0.239517	−0.119758	−	0.992803i	$-0.538212\pi$
$359$	−7.53809e10	−0.239517	−0.119758	+	0.992803i	$0.538212\pi$
$360$	0	0
$361$	−2.85480e11	−0.884696
$362$	0	0
$363$	−4.09532e11	−1.23796
$364$	0	0
$365$	−1.32576e11	−0.390973
$366$	0	0
$367$	−2.75198e10	−0.0791859	−0.0395929	−	0.999216i	$-0.512606\pi$
$367$	−2.75198e10	−0.0791859	−0.0395929	+	0.999216i	$0.512606\pi$
$368$	0	0
$369$	2.31191e10	0.0649161
$370$	0	0
$371$	2.29038e11	0.627662
$372$	0	0
$373$	3.11639e11	0.833610	0.416805	−	0.908996i	$-0.363150\pi$
$373$	3.11639e11	0.833610	0.416805	+	0.908996i	$0.363150\pi$
$374$	0	0
$375$	1.43121e12	3.73735
$376$	0	0
$377$	−1.81665e11	−0.463164
$378$	0	0
$379$	1.28321e11	0.319464	0.159732	−	0.987160i	$-0.448937\pi$
$379$	1.28321e11	0.319464	0.159732	+	0.987160i	$0.448937\pi$
$380$	0	0
$381$	8.52896e11	2.07364
$382$	0	0
$383$	−3.34432e11	−0.794169	−0.397085	−	0.917782i	$-0.629978\pi$
$383$	−3.34432e11	−0.794169	−0.397085	+	0.917782i	$0.629978\pi$
$384$	0	0
$385$	−1.33650e11	−0.310023
$386$	0	0
$387$	−3.48107e11	−0.788885
$388$	0	0
$389$	−3.73715e11	−0.827499	−0.413749	−	0.910391i	$-0.635781\pi$
$389$	−3.73715e11	−0.827499	−0.413749	+	0.910391i	$0.635781\pi$
$390$	0	0
$391$	1.98263e11	0.428990
$392$	0	0
$393$	−8.66901e11	−1.83317
$394$	0	0
$395$	−1.24433e12	−2.57187
$396$	0	0
$397$	6.40261e11	1.29360	0.646799	−	0.762660i	$-0.276107\pi$
$397$	6.40261e11	1.29360	0.646799	+	0.762660i	$0.276107\pi$
$398$	0	0
$399$	1.96611e11	0.388355
$400$	0	0
$401$	7.61577e11	1.47084	0.735418	−	0.677614i	$-0.236986\pi$
$401$	7.61577e11	1.47084	0.735418	+	0.677614i	$0.236986\pi$
$402$	0	0
$403$	−9.63444e10	−0.181951
$404$	0	0
$405$	−1.25418e12	−2.31640
$406$	0	0
$407$	−1.75243e11	−0.316567
$408$	0	0
$409$	4.53304e10	0.0801004	0.0400502	−	0.999198i	$-0.487248\pi$
$409$	4.53304e10	0.0801004	0.0400502	+	0.999198i	$0.487248\pi$
$410$	0	0
$411$	1.23082e12	2.12768
$412$	0	0
$413$	8.32467e11	1.40796
$414$	0	0
$415$	−5.68875e11	−0.941456
$416$	0	0
$417$	2.52751e11	0.409337
$418$	0	0
$419$	−4.33209e11	−0.686649	−0.343324	−	0.939217i	$-0.611553\pi$
$419$	−4.33209e11	−0.686649	−0.343324	+	0.939217i	$0.611553\pi$
$420$	0	0
$421$	2.32634e11	0.360914	0.180457	−	0.983583i	$-0.442242\pi$
$421$	2.32634e11	0.360914	0.180457	+	0.983583i	$0.442242\pi$
$422$	0	0
$423$	1.71746e11	0.260829
$424$	0	0
$425$	−1.06392e12	−1.58182
$426$	0	0
$427$	7.67545e11	1.11732
$428$	0	0
$429$	4.59878e10	0.0655518
$430$	0	0
$431$	8.52061e11	1.18939	0.594693	−	0.803953i	$-0.297274\pi$
$431$	8.52061e11	1.18939	0.594693	+	0.803953i	$0.297274\pi$
$432$	0	0
$433$	9.97062e11	1.36310	0.681549	−	0.731773i	$-0.261307\pi$
$433$	9.97062e11	1.36310	0.681549	+	0.731773i	$0.261307\pi$
$434$	0	0
$435$	3.01056e12	4.03131
$436$	0	0
$437$	−1.78900e11	−0.234662
$438$	0	0
$439$	4.94971e10	0.0636047	0.0318023	−	0.999494i	$-0.489875\pi$
$439$	4.94971e10	0.0636047	0.0318023	+	0.999494i	$0.489875\pi$
$440$	0	0
$441$	−1.03872e11	−0.130775
$442$	0	0
$443$	1.69415e11	0.208994	0.104497	−	0.994525i	$-0.466677\pi$
$443$	1.69415e11	0.208994	0.104497	+	0.994525i	$0.466677\pi$
$444$	0	0
$445$	2.93485e11	0.354786
$446$	0	0
$447$	−9.00712e11	−1.06709
$448$	0	0
$449$	−6.66846e11	−0.774313	−0.387157	−	0.922014i	$-0.626543\pi$
$449$	−6.66846e11	−0.774313	−0.387157	+	0.922014i	$0.626543\pi$
$450$	0	0
$451$	−1.63749e10	−0.0186374
$452$	0	0
$453$	9.35012e11	1.04322
$454$	0	0
$455$	−4.26240e11	−0.466233
$456$	0	0
$457$	−8.94784e11	−0.959611	−0.479806	−	0.877375i	$-0.659293\pi$
$457$	−8.94784e11	−0.959611	−0.479806	+	0.877375i	$0.659293\pi$
$458$	0	0
$459$	2.70550e11	0.284506
$460$	0	0
$461$	−1.29373e12	−1.33410	−0.667052	−	0.745011i	$-0.732444\pi$
$461$	−1.29373e12	−1.33410	−0.667052	+	0.745011i	$0.732444\pi$
$462$	0	0
$463$	−4.54513e11	−0.459655	−0.229827	−	0.973231i	$-0.573816\pi$
$463$	−4.54513e11	−0.459655	−0.229827	+	0.973231i	$0.573816\pi$
$464$	0	0
$465$	1.59663e12	1.58367
$466$	0	0
$467$	−6.41780e10	−0.0624396	−0.0312198	−	0.999513i	$-0.509939\pi$
$467$	−6.41780e10	−0.0624396	−0.0312198	+	0.999513i	$0.509939\pi$
$468$	0	0
$469$	−1.64225e12	−1.56734
$470$	0	0
$471$	−1.98213e12	−1.85583
$472$	0	0
$473$	2.46559e11	0.226488
$474$	0	0
$475$	9.60010e11	0.865275
$476$	0	0
$477$	5.10829e11	0.451797
$478$	0	0
$479$	1.28336e12	1.11388	0.556942	−	0.830551i	$-0.311975\pi$
$479$	1.28336e12	1.11388	0.556942	+	0.830551i	$0.311975\pi$
$480$	0	0
$481$	−5.58891e11	−0.476073
$482$	0	0
$483$	−9.45344e11	−0.790365
$484$	0	0
$485$	3.91019e12	3.20893
$486$	0	0
$487$	2.14921e12	1.73140	0.865701	−	0.500561i	$-0.166873\pi$
$487$	2.14921e12	1.73140	0.865701	+	0.500561i	$0.166873\pi$
$488$	0	0
$489$	6.83763e11	0.540774
$490$	0	0
$491$	7.79359e10	0.0605161	0.0302580	−	0.999542i	$-0.490367\pi$
$491$	7.79359e10	0.0605161	0.0302580	+	0.999542i	$0.490367\pi$
$492$	0	0
$493$	−1.35970e12	−1.03665
$494$	0	0
$495$	−2.98082e11	−0.223158
$496$	0	0
$497$	4.43267e11	0.325883
$498$	0	0
$499$	−1.15785e12	−0.835985	−0.417992	−	0.908450i	$-0.637266\pi$
$499$	−1.15785e12	−0.835985	−0.417992	+	0.908450i	$0.637266\pi$
$500$	0	0
$501$	1.63257e12	1.15771
$502$	0	0
$503$	7.69460e11	0.535958	0.267979	−	0.963425i	$-0.413644\pi$
$503$	7.69460e11	0.535958	0.267979	+	0.963425i	$0.413644\pi$
$504$	0	0
$505$	−6.45300e11	−0.441520
$506$	0	0
$507$	1.46666e11	0.0985809
$508$	0	0
$509$	1.25208e12	0.826801	0.413401	−	0.910549i	$-0.364341\pi$
$509$	1.25208e12	0.826801	0.413401	+	0.910549i	$0.364341\pi$
$510$	0	0
$511$	−2.85501e11	−0.185231
$512$	0	0
$513$	−2.44127e11	−0.155628
$514$	0	0
$515$	−6.28726e11	−0.393848
$516$	0	0
$517$	−1.21645e11	−0.0748838
$518$	0	0
$519$	1.25690e12	0.760408
$520$	0	0
$521$	2.68068e12	1.59395	0.796977	−	0.604010i	$-0.206431\pi$
$521$	2.68068e12	1.59395	0.796977	+	0.604010i	$0.206431\pi$
$522$	0	0
$523$	1.64731e12	0.962759	0.481380	−	0.876512i	$-0.340136\pi$
$523$	1.64731e12	0.962759	0.481380	+	0.876512i	$0.340136\pi$
$524$	0	0
$525$	5.07289e12	2.91433
$526$	0	0
$527$	−7.21106e11	−0.407241
$528$	0	0
$529$	−9.40967e11	−0.522425
$530$	0	0
$531$	1.85667e12	1.01347
$532$	0	0
$533$	−5.22234e10	−0.0280280
$534$	0	0
$535$	−6.55830e12	−3.46098
$536$	0	0
$537$	1.66680e12	0.864964
$538$	0	0
$539$	7.35710e10	0.0375455
$540$	0	0
$541$	9.59735e11	0.481685	0.240843	−	0.970564i	$-0.422576\pi$
$541$	9.59735e11	0.481685	0.240843	+	0.970564i	$0.422576\pi$
$542$	0	0
$543$	−1.60580e12	−0.792668
$544$	0	0
$545$	−1.73341e12	−0.841624
$546$	0	0
$547$	3.82447e11	0.182653	0.0913267	−	0.995821i	$-0.470889\pi$
$547$	3.82447e11	0.182653	0.0913267	+	0.995821i	$0.470889\pi$
$548$	0	0
$549$	1.71187e12	0.804258
$550$	0	0
$551$	1.22691e12	0.567060
$552$	0	0
$553$	−2.67965e12	−1.21847
$554$	0	0
$555$	9.26197e12	4.14367
$556$	0	0
$557$	1.03867e12	0.457224	0.228612	−	0.973518i	$-0.426581\pi$
$557$	1.03867e12	0.457224	0.228612	+	0.973518i	$0.426581\pi$
$558$	0	0
$559$	7.86334e11	0.340607
$560$	0	0
$561$	3.44203e11	0.146718
$562$	0	0
$563$	9.49393e11	0.398252	0.199126	−	0.979974i	$-0.436190\pi$
$563$	9.49393e11	0.398252	0.199126	+	0.979974i	$0.436190\pi$
$564$	0	0
$565$	−8.11610e12	−3.35065
$566$	0	0
$567$	−2.70087e12	−1.09744
$568$	0	0
$569$	−4.62973e12	−1.85161	−0.925807	−	0.377996i	$-0.876613\pi$
$569$	−4.62973e12	−1.85161	−0.925807	+	0.377996i	$0.876613\pi$
$570$	0	0
$571$	−3.03010e11	−0.119288	−0.0596438	−	0.998220i	$-0.518996\pi$
$571$	−3.03010e11	−0.119288	−0.0596438	+	0.998220i	$0.518996\pi$
$572$	0	0
$573$	3.17911e11	0.123200
$574$	0	0
$575$	−4.61592e12	−1.76097
$576$	0	0
$577$	1.10138e12	0.413663	0.206832	−	0.978377i	$-0.433685\pi$
$577$	1.10138e12	0.413663	0.206832	+	0.978377i	$0.433685\pi$
$578$	0	0
$579$	5.72735e12	2.11788
$580$	0	0
$581$	−1.22507e12	−0.446034
$582$	0	0
$583$	−3.61812e11	−0.129710
$584$	0	0
$585$	−9.50652e11	−0.335598
$586$	0	0
$587$	3.99914e12	1.39026	0.695128	−	0.718886i	$-0.255347\pi$
$587$	3.99914e12	1.39026	0.695128	+	0.718886i	$0.255347\pi$
$588$	0	0
$589$	6.50679e11	0.222765
$590$	0	0
$591$	−3.34837e12	−1.12899
$592$	0	0
$593$	8.78271e11	0.291664	0.145832	−	0.989309i	$-0.453414\pi$
$593$	8.78271e11	0.291664	0.145832	+	0.989309i	$0.453414\pi$
$594$	0	0
$595$	−3.19026e12	−1.04352
$596$	0	0
$597$	−1.41306e12	−0.455276
$598$	0	0
$599$	−5.17796e12	−1.64338	−0.821690	−	0.569935i	$-0.806968\pi$
$599$	−5.17796e12	−1.64338	−0.821690	+	0.569935i	$0.806968\pi$
$600$	0	0
$601$	−2.53790e12	−0.793486	−0.396743	−	0.917930i	$-0.629860\pi$
$601$	−2.53790e12	−0.793486	−0.396743	+	0.917930i	$0.629860\pi$
$602$	0	0
$603$	−3.66275e12	−1.12818
$604$	0	0
$605$	−5.99617e12	−1.81960
$606$	0	0
$607$	2.92326e12	0.874013	0.437006	−	0.899458i	$-0.356039\pi$
$607$	2.92326e12	0.874013	0.437006	+	0.899458i	$0.356039\pi$
$608$	0	0
$609$	6.48323e12	1.90991
$610$	0	0
$611$	−3.87955e11	−0.112615
$612$	0	0
$613$	−1.89601e12	−0.542335	−0.271168	−	0.962532i	$-0.587410\pi$
$613$	−1.89601e12	−0.542335	−0.271168	+	0.962532i	$0.587410\pi$
$614$	0	0
$615$	8.65450e11	0.243952
$616$	0	0
$617$	−7.27725e11	−0.202155	−0.101077	−	0.994879i	$-0.532229\pi$
$617$	−7.27725e11	−0.202155	−0.101077	+	0.994879i	$0.532229\pi$
$618$	0	0
$619$	−4.35291e12	−1.19171	−0.595857	−	0.803090i	$-0.703188\pi$
$619$	−4.35291e12	−1.19171	−0.595857	+	0.803090i	$0.703188\pi$
$620$	0	0
$621$	1.17381e12	0.316727
$622$	0	0
$623$	6.32019e11	0.168087
$624$	0	0
$625$	1.12346e13	2.94508
$626$	0	0
$627$	−3.10586e11	−0.0802562
$628$	0	0
$629$	−4.18311e12	−1.06554
$630$	0	0
$631$	−1.83736e12	−0.461383	−0.230692	−	0.973027i	$-0.574099\pi$
$631$	−1.83736e12	−0.461383	−0.230692	+	0.973027i	$0.574099\pi$
$632$	0	0
$633$	−5.20677e12	−1.28900
$634$	0	0
$635$	1.24877e13	3.04790
$636$	0	0
$637$	2.34635e11	0.0564632
$638$	0	0
$639$	9.88628e11	0.234573
$640$	0	0
$641$	4.54945e12	1.06438	0.532191	−	0.846624i	$-0.321369\pi$
$641$	4.54945e12	1.06438	0.532191	+	0.846624i	$0.321369\pi$
$642$	0	0
$643$	−4.75234e12	−1.09637	−0.548186	−	0.836356i	$-0.684681\pi$
$643$	−4.75234e12	−1.09637	−0.548186	+	0.836356i	$0.684681\pi$
$644$	0	0
$645$	−1.30312e13	−2.96459
$646$	0	0
$647$	5.58258e11	0.125247	0.0626233	−	0.998037i	$-0.480053\pi$
$647$	5.58258e11	0.125247	0.0626233	+	0.998037i	$0.480053\pi$
$648$	0	0
$649$	−1.31505e12	−0.290965
$650$	0	0
$651$	3.43832e12	0.750296
$652$	0	0
$653$	−8.16374e11	−0.175703	−0.0878516	−	0.996134i	$-0.528000\pi$
$653$	−8.16374e11	−0.175703	−0.0878516	+	0.996134i	$0.528000\pi$
$654$	0	0
$655$	−1.26928e13	−2.69445
$656$	0	0
$657$	−6.36760e11	−0.133331
$658$	0	0
$659$	5.10551e12	1.05452	0.527260	−	0.849704i	$-0.323219\pi$
$659$	5.10551e12	1.05452	0.527260	+	0.849704i	$0.323219\pi$
$660$	0	0
$661$	−4.50499e12	−0.917884	−0.458942	−	0.888466i	$-0.651771\pi$
$661$	−4.50499e12	−0.917884	−0.458942	+	0.888466i	$0.651771\pi$
$662$	0	0
$663$	1.09774e12	0.220643
$664$	0	0
$665$	2.87868e12	0.570817
$666$	0	0
$667$	−5.89921e12	−1.15406
$668$	0	0
$669$	−4.72055e12	−0.911120
$670$	0	0
$671$	−1.21249e12	−0.230902
$672$	0	0
$673$	−5.92192e11	−0.111274	−0.0556372	−	0.998451i	$-0.517719\pi$
$673$	−5.92192e11	−0.111274	−0.0556372	+	0.998451i	$0.517719\pi$
$674$	0	0
$675$	−6.29888e12	−1.16788
$676$	0	0
$677$	−3.80993e12	−0.697056	−0.348528	−	0.937298i	$-0.613318\pi$
$677$	−3.80993e12	−0.697056	−0.348528	+	0.937298i	$0.613318\pi$
$678$	0	0
$679$	8.42057e12	1.52029
$680$	0	0
$681$	9.17053e12	1.63393
$682$	0	0
$683$	9.05115e11	0.159151	0.0795757	−	0.996829i	$-0.474643\pi$
$683$	9.05115e11	0.159151	0.0795757	+	0.996829i	$0.474643\pi$
$684$	0	0
$685$	1.80211e13	3.12734
$686$	0	0
$687$	−4.91249e12	−0.841388
$688$	0	0
$689$	−1.15390e12	−0.195067
$690$	0	0
$691$	−6.55695e12	−1.09408	−0.547042	−	0.837105i	$-0.684246\pi$
$691$	−6.55695e12	−1.09408	−0.547042	+	0.837105i	$0.684246\pi$
$692$	0	0
$693$	−6.41917e11	−0.105725
$694$	0	0
$695$	3.70067e12	0.601656
$696$	0	0
$697$	−3.90875e11	−0.0627322
$698$	0	0
$699$	9.55638e11	0.151407
$700$	0	0
$701$	3.06995e12	0.480176	0.240088	−	0.970751i	$-0.422824\pi$
$701$	3.06995e12	0.480176	0.240088	+	0.970751i	$0.422824\pi$
$702$	0	0
$703$	3.77457e12	0.582865
$704$	0	0
$705$	6.42922e12	0.980183
$706$	0	0
$707$	−1.38965e12	−0.209179
$708$	0	0
$709$	7.61524e12	1.13182	0.565908	−	0.824469i	$-0.308526\pi$
$709$	7.61524e12	1.13182	0.565908	+	0.824469i	$0.308526\pi$
$710$	0	0
$711$	−5.97649e12	−0.877067
$712$	0	0
$713$	−3.12859e12	−0.453363
$714$	0	0
$715$	6.73332e11	0.0963500
$716$	0	0
$717$	−1.55930e13	−2.20340
$718$	0	0
$719$	−9.51130e12	−1.32727	−0.663636	−	0.748056i	$-0.730988\pi$
$719$	−9.51130e12	−1.32727	−0.663636	+	0.748056i	$0.730988\pi$
$720$	0	0
$721$	−1.35396e12	−0.186593
$722$	0	0
$723$	−5.59659e12	−0.761729
$724$	0	0
$725$	3.16563e13	4.25538
$726$	0	0
$727$	−9.21476e12	−1.22343	−0.611715	−	0.791078i	$-0.709520\pi$
$727$	−9.21476e12	−1.22343	−0.611715	+	0.791078i	$0.709520\pi$
$728$	0	0
$729$	−1.54488e12	−0.202592
$730$	0	0
$731$	5.88545e12	0.762345
$732$	0	0
$733$	−5.85946e12	−0.749704	−0.374852	−	0.927085i	$-0.622306\pi$
$733$	−5.85946e12	−0.749704	−0.374852	+	0.927085i	$0.622306\pi$
$734$	0	0
$735$	−3.88839e12	−0.491447
$736$	0	0
$737$	2.59427e12	0.323900
$738$	0	0
$739$	1.07791e13	1.32948	0.664740	−	0.747075i	$-0.268543\pi$
$739$	1.07791e13	1.32948	0.664740	+	0.747075i	$0.268543\pi$
$740$	0	0
$741$	−9.90533e11	−0.120694
$742$	0	0
$743$	1.46255e13	1.76061	0.880303	−	0.474411i	$-0.157339\pi$
$743$	1.46255e13	1.76061	0.880303	+	0.474411i	$0.157339\pi$
$744$	0	0
$745$	−1.31878e13	−1.56845
$746$	0	0
$747$	−2.73230e12	−0.321059
$748$	0	0
$749$	−1.41233e13	−1.63971
$750$	0	0
$751$	4.52434e12	0.519010	0.259505	−	0.965742i	$-0.416441\pi$
$751$	4.52434e12	0.519010	0.259505	+	0.965742i	$0.416441\pi$
$752$	0	0
$753$	2.50412e12	0.283843
$754$	0	0
$755$	1.36900e13	1.53335
$756$	0	0
$757$	−4.43997e12	−0.491415	−0.245708	−	0.969344i	$-0.579020\pi$
$757$	−4.43997e12	−0.491415	−0.245708	+	0.969344i	$0.579020\pi$
$758$	0	0
$759$	1.49336e12	0.163334
$760$	0	0
$761$	−1.21097e13	−1.30888	−0.654442	−	0.756112i	$-0.727096\pi$
$761$	−1.21097e13	−1.30888	−0.654442	+	0.756112i	$0.727096\pi$
$762$	0	0
$763$	−3.73290e12	−0.398736
$764$	0	0
$765$	−7.11532e12	−0.751135
$766$	0	0
$767$	−4.19400e12	−0.437572
$768$	0	0
$769$	−2.20914e12	−0.227800	−0.113900	−	0.993492i	$-0.536334\pi$
$769$	−2.20914e12	−0.227800	−0.113900	+	0.993492i	$0.536334\pi$
$770$	0	0
$771$	−1.70186e13	−1.73452
$772$	0	0
$773$	2.35745e12	0.237484	0.118742	−	0.992925i	$-0.462114\pi$
$773$	2.35745e12	0.237484	0.118742	+	0.992925i	$0.462114\pi$
$774$	0	0
$775$	1.67886e13	1.67170
$776$	0	0
$777$	1.99456e13	1.96315
$778$	0	0
$779$	3.52700e11	0.0343152
$780$	0	0
$781$	−7.00229e11	−0.0673458
$782$	0	0
$783$	−8.05007e12	−0.765370
$784$	0	0
$785$	−2.90214e13	−2.72775
$786$	0	0
$787$	−1.12444e13	−1.04484	−0.522418	−	0.852689i	$-0.674970\pi$
$787$	−1.12444e13	−1.04484	−0.522418	+	0.852689i	$0.674970\pi$
$788$	0	0
$789$	−1.63918e13	−1.50585
$790$	0	0
$791$	−1.74780e13	−1.58744
$792$	0	0
$793$	−3.86692e12	−0.347245
$794$	0	0
$795$	1.91226e13	1.69783
$796$	0	0
$797$	2.64697e12	0.232374	0.116187	−	0.993227i	$-0.462933\pi$
$797$	2.64697e12	0.232374	0.116187	+	0.993227i	$0.462933\pi$
$798$	0	0
$799$	−2.90372e12	−0.252054
$800$	0	0
$801$	1.40961e12	0.120990
$802$	0	0
$803$	4.51007e11	0.0382792
$804$	0	0
$805$	−1.38413e13	−1.16170
$806$	0	0
$807$	5.90518e12	0.490120
$808$	0	0
$809$	−2.90663e10	−0.00238573	−0.00119286	−	0.999999i	$-0.500380\pi$
$809$	−2.90663e10	−0.00238573	−0.00119286	+	0.999999i	$0.500380\pi$
$810$	0	0
$811$	−3.57442e12	−0.290143	−0.145071	−	0.989421i	$-0.546341\pi$
$811$	−3.57442e12	−0.290143	−0.145071	+	0.989421i	$0.546341\pi$
$812$	0	0
$813$	1.97503e13	1.58550
$814$	0	0
$815$	1.00113e13	0.794846
$816$	0	0
$817$	−5.31064e12	−0.417011
$818$	0	0
$819$	−2.04722e12	−0.158996
$820$	0	0
$821$	6.57405e12	0.504997	0.252498	−	0.967597i	$-0.418748\pi$
$821$	6.57405e12	0.504997	0.252498	+	0.967597i	$0.418748\pi$
$822$	0	0
$823$	−9.79180e12	−0.743983	−0.371992	−	0.928236i	$-0.621325\pi$
$823$	−9.79180e12	−0.743983	−0.371992	+	0.928236i	$0.621325\pi$
$824$	0	0
$825$	−8.01366e12	−0.602266
$826$	0	0
$827$	−9.38817e12	−0.697921	−0.348961	−	0.937137i	$-0.613465\pi$
$827$	−9.38817e12	−0.697921	−0.348961	+	0.937137i	$0.613465\pi$
$828$	0	0
$829$	−7.04824e12	−0.518305	−0.259152	−	0.965836i	$-0.583443\pi$
$829$	−7.04824e12	−0.518305	−0.259152	+	0.965836i	$0.583443\pi$
$830$	0	0
$831$	−9.37914e12	−0.682274
$832$	0	0
$833$	1.75617e12	0.126376
$834$	0	0
$835$	2.39033e13	1.70164
$836$	0	0
$837$	−4.26928e12	−0.300670
$838$	0	0
$839$	1.83602e13	1.27923	0.639617	−	0.768694i	$-0.279093\pi$
$839$	1.83602e13	1.27923	0.639617	+	0.768694i	$0.279093\pi$
$840$	0	0
$841$	2.59500e13	1.78878
$842$	0	0
$843$	−1.76807e12	−0.120580
$844$	0	0
$845$	2.14741e12	0.144897
$846$	0	0
$847$	−1.29127e13	−0.862069
$848$	0	0
$849$	−2.93853e13	−1.94109
$850$	0	0
$851$	−1.81489e13	−1.18622
$852$	0	0
$853$	1.28564e13	0.831472	0.415736	−	0.909485i	$-0.363524\pi$
$853$	1.28564e13	0.831472	0.415736	+	0.909485i	$0.363524\pi$
$854$	0	0
$855$	6.42039e12	0.410879
$856$	0	0
$857$	−1.31309e13	−0.831538	−0.415769	−	0.909470i	$-0.636487\pi$
$857$	−1.31309e13	−0.831538	−0.415769	+	0.909470i	$0.636487\pi$
$858$	0	0
$859$	2.20323e13	1.38067	0.690337	−	0.723488i	$-0.257462\pi$
$859$	2.20323e13	1.38067	0.690337	+	0.723488i	$0.257462\pi$
$860$	0	0
$861$	1.86374e12	0.115577
$862$	0	0
$863$	−2.44095e13	−1.49799	−0.748997	−	0.662573i	$-0.769464\pi$
$863$	−2.44095e13	−1.49799	−0.748997	+	0.662573i	$0.769464\pi$
$864$	0	0
$865$	1.84029e13	1.11767
$866$	0	0
$867$	−1.31054e13	−0.787709
$868$	0	0
$869$	4.23305e12	0.251805
$870$	0	0
$871$	8.27373e12	0.487102
$872$	0	0
$873$	1.87806e13	1.09432
$874$	0	0
$875$	4.51268e13	2.60255
$876$	0	0
$877$	9.70479e12	0.553972	0.276986	−	0.960874i	$-0.410664\pi$
$877$	9.70479e12	0.553972	0.276986	+	0.960874i	$0.410664\pi$
$878$	0	0
$879$	−3.72943e13	−2.10714
$880$	0	0
$881$	−1.44632e13	−0.808859	−0.404430	−	0.914569i	$-0.632530\pi$
$881$	−1.44632e13	−0.808859	−0.404430	+	0.914569i	$0.632530\pi$
$882$	0	0
$883$	−1.02994e13	−0.570152	−0.285076	−	0.958505i	$-0.592019\pi$
$883$	−1.02994e13	−0.570152	−0.285076	+	0.958505i	$0.592019\pi$
$884$	0	0
$885$	6.95033e13	3.80856
$886$	0	0
$887$	−1.16087e13	−0.629688	−0.314844	−	0.949143i	$-0.601952\pi$
$887$	−1.16087e13	−0.629688	−0.314844	+	0.949143i	$0.601952\pi$
$888$	0	0
$889$	2.68922e13	1.44400
$890$	0	0
$891$	4.26657e12	0.226793
$892$	0	0
$893$	2.62012e12	0.137876
$894$	0	0
$895$	2.44045e13	1.27135
$896$	0	0
$897$	4.76268e12	0.245632
$898$	0	0
$899$	2.14561e13	1.09555
$900$	0	0
$901$	−8.63660e12	−0.436597
$902$	0	0
$903$	−2.80626e13	−1.40453
$904$	0	0
$905$	−2.35113e13	−1.16509
$906$	0	0
$907$	3.89275e13	1.90996	0.954979	−	0.296675i	$-0.0958777\pi$
$907$	3.89275e13	1.90996	0.954979	+	0.296675i	$0.0958777\pi$
$908$	0	0
$909$	−3.09936e12	−0.150569
$910$	0	0
$911$	1.89482e13	0.911453	0.455726	−	0.890120i	$-0.349380\pi$
$911$	1.89482e13	0.911453	0.455726	+	0.890120i	$0.349380\pi$
$912$	0	0
$913$	1.93524e12	0.0921758
$914$	0	0
$915$	6.40829e13	3.02236
$916$	0	0
$917$	−2.73338e13	−1.27655
$918$	0	0
$919$	1.56774e12	0.0725029	0.0362514	−	0.999343i	$-0.488458\pi$
$919$	1.56774e12	0.0725029	0.0362514	+	0.999343i	$0.488458\pi$
$920$	0	0
$921$	4.30247e13	1.97038
$922$	0	0
$923$	−2.23320e12	−0.101279
$924$	0	0
$925$	9.73902e13	4.37399
$926$	0	0
$927$	−3.01976e12	−0.134312
$928$	0	0
$929$	1.43481e13	0.632010	0.316005	−	0.948758i	$-0.397658\pi$
$929$	1.43481e13	0.632010	0.316005	+	0.948758i	$0.397658\pi$
$930$	0	0
$931$	−1.58465e12	−0.0691289
$932$	0	0
$933$	−3.02516e13	−1.30702
$934$	0	0
$935$	5.03967e12	0.215650
$936$	0	0
$937$	1.16894e13	0.495410	0.247705	−	0.968836i	$-0.420324\pi$
$937$	1.16894e13	0.495410	0.247705	+	0.968836i	$0.420324\pi$
$938$	0	0
$939$	2.90786e13	1.22061
$940$	0	0
$941$	−1.54364e13	−0.641792	−0.320896	−	0.947115i	$-0.603984\pi$
$941$	−1.54364e13	−0.641792	−0.320896	+	0.947115i	$0.603984\pi$
$942$	0	0
$943$	−1.69585e12	−0.0698370
$944$	0	0
$945$	−1.88878e13	−0.770440
$946$	0	0
$947$	−4.57619e13	−1.84897	−0.924483	−	0.381224i	$-0.875503\pi$
$947$	−4.57619e13	−1.84897	−0.924483	+	0.381224i	$0.875503\pi$
$948$	0	0
$949$	1.43837e12	0.0575667
$950$	0	0
$951$	7.27108e12	0.288261
$952$	0	0
$953$	4.10090e13	1.61050	0.805250	−	0.592935i	$-0.202031\pi$
$953$	4.10090e13	1.61050	0.805250	+	0.592935i	$0.202031\pi$
$954$	0	0
$955$	4.65470e12	0.181083
$956$	0	0
$957$	−1.02416e13	−0.394696
$958$	0	0
$959$	3.88084e13	1.48164
$960$	0	0
$961$	−1.50606e13	−0.569621
$962$	0	0
$963$	−3.14994e13	−1.18028
$964$	0	0
$965$	8.38573e13	3.11292
$966$	0	0
$967$	−9.77642e12	−0.359551	−0.179776	−	0.983708i	$-0.557537\pi$
$967$	−9.77642e12	−0.359551	−0.179776	+	0.983708i	$0.557537\pi$
$968$	0	0
$969$	−7.41381e12	−0.270137
$970$	0	0
$971$	3.19450e13	1.15323	0.576616	−	0.817015i	$-0.304373\pi$
$971$	3.19450e13	1.15323	0.576616	+	0.817015i	$0.304373\pi$
$972$	0	0
$973$	7.96936e12	0.285046
$974$	0	0
$975$	−2.55574e13	−0.905725
$976$	0	0
$977$	−2.10587e13	−0.739444	−0.369722	−	0.929142i	$-0.620547\pi$
$977$	−2.10587e13	−0.739444	−0.369722	+	0.929142i	$0.620547\pi$
$978$	0	0
$979$	−9.98401e11	−0.0347363
$980$	0	0
$981$	−8.32556e12	−0.287014
$982$	0	0
$983$	−3.15106e13	−1.07638	−0.538189	−	0.842824i	$-0.680892\pi$
$983$	−3.15106e13	−1.07638	−0.538189	+	0.842824i	$0.680892\pi$
$984$	0	0
$985$	−4.90253e13	−1.65942
$986$	0	0
$987$	1.38453e13	0.464381
$988$	0	0
$989$	2.55346e13	0.848685
$990$	0	0
$991$	−2.57481e13	−0.848037	−0.424018	−	0.905654i	$-0.639381\pi$
$991$	−2.57481e13	−0.848037	−0.424018	+	0.905654i	$0.639381\pi$
$992$	0	0
$993$	−4.83258e13	−1.57728
$994$	0	0
$995$	−2.06893e13	−0.669179
$996$	0	0
$997$	9.81382e12	0.314565	0.157282	−	0.987554i	$-0.449727\pi$
$997$	9.81382e12	0.314565	0.157282	+	0.987554i	$0.449727\pi$
$998$	0	0
$999$	−2.47660e13	−0.786702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.i.1.5		5
4.3	odd	2		52.10.a.b.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.10.a.b.1.1	✓	5	4.3	odd	2
208.10.a.i.1.5		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)

\(f(q)\)	\(=\)	\(q+179.797 q^{3} +2632.50 q^{5} +5669.07 q^{7} +12643.9 q^{9} -8955.44 q^{11} -28561.0 q^{13} +473315. q^{15} -213770. q^{17} +192892. q^{19} +1.01928e6 q^{21} -927462. q^{23} +4.97693e6 q^{25} -1.26562e6 q^{27} +6.36060e6 q^{29} +3.37329e6 q^{31} -1.61016e6 q^{33} +1.49238e7 q^{35} +1.95683e7 q^{37} -5.13517e6 q^{39} +1.82849e6 q^{41} -2.75317e7 q^{43} +3.32850e7 q^{45} +1.35834e7 q^{47} -8.21523e6 q^{49} -3.84351e7 q^{51} +4.04014e7 q^{53} -2.35752e7 q^{55} +3.46813e7 q^{57} +1.46844e8 q^{59} +1.35392e8 q^{61} +7.16789e7 q^{63} -7.51868e7 q^{65} -2.89686e8 q^{67} -1.66755e8 q^{69} +7.81904e7 q^{71} -5.03612e7 q^{73} +8.94836e8 q^{75} -5.07691e7 q^{77} -4.72679e8 q^{79} -4.76422e8 q^{81} -2.16097e8 q^{83} -5.62749e8 q^{85} +1.14361e9 q^{87} +1.11485e8 q^{89} -1.61914e8 q^{91} +6.06506e8 q^{93} +5.07788e8 q^{95} +1.48535e9 q^{97} -1.13231e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 147 q^{3} + 1721 q^{5} - 3317 q^{7} - 3204 q^{9} - 6562 q^{11} - 142805 q^{13} - 80283 q^{15} - 146493 q^{17} - 110250 q^{19} + 2250015 q^{21} - 1675652 q^{23} + 6048610 q^{25} - 4937895 q^{27} + 7019714 q^{29}+ \cdots - 302611248 q^{99}+O(q^{100}) \) 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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