LMFDB - Embedded newform 208.10.a.k.1.6

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 = [6,0,68] 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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 72131x^{4} + 310880x^{3} + 1323158708x^{2} - 12926602192x - 1707532200192 $ x^6 - 2x^5 - 72131x^4 + 310880x^3 + 1323158708x^2 - 12926602192*x - 1707532200192
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$204.864$ 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)

$f(q)$	$=$	$q+215.864 q^{3} -1620.37 q^{5} +7157.79 q^{7} +26914.2 q^{9} -36853.5 q^{11} +28561.0 q^{13} -349780. q^{15} +404355. q^{17} +203358. q^{19} +1.54511e6 q^{21} +407597. q^{23} +672480. q^{25} +1.56095e6 q^{27} +225155. q^{29} -4.59794e6 q^{31} -7.95533e6 q^{33} -1.15983e7 q^{35} +1.48411e7 q^{37} +6.16529e6 q^{39} -1.99373e6 q^{41} -1.25478e7 q^{43} -4.36110e7 q^{45} +6.51793e7 q^{47} +1.08804e7 q^{49} +8.72855e7 q^{51} +3.92662e7 q^{53} +5.97163e7 q^{55} +4.38976e7 q^{57} +4.88772e7 q^{59} -4.34425e7 q^{61} +1.92646e8 q^{63} -4.62794e7 q^{65} +5.55762e7 q^{67} +8.79854e7 q^{69} +3.91166e8 q^{71} +3.96635e8 q^{73} +1.45164e8 q^{75} -2.63789e8 q^{77} +1.98103e8 q^{79} -1.92799e8 q^{81} -7.52406e8 q^{83} -6.55205e8 q^{85} +4.86028e7 q^{87} +4.01022e8 q^{89} +2.04434e8 q^{91} -9.92529e8 q^{93} -3.29516e8 q^{95} +8.36965e8 q^{97} -9.91880e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 68 q^{3} - 176 q^{5} + 9664 q^{7} + 26938 q^{9} + 15980 q^{11} + 171366 q^{13} + 10756 q^{15} + 215048 q^{17} + 519868 q^{19} + 384084 q^{21} + 691224 q^{23} - 196082 q^{25} + 1592756 q^{27} - 546932 q^{29}+ \cdots + 1940442636 q^{99}+O(q^{100}) $ 6 * q + 68 * q^3 - 176 * q^5 + 9664 * q^7 + 26938 * q^9 + 15980 * q^11 + 171366 * q^13 + 10756 * q^15 + 215048 * q^17 + 519868 * q^19 + 384084 * q^21 + 691224 * q^23 - 196082 * q^25 + 1592756 * q^27 - 546932 * q^29 + 535300 * q^31 - 3961488 * q^33 + 1042020 * q^35 - 18760 * q^37 + 1942148 * q^39 - 8895252 * q^41 + 21376108 * q^43 - 46124116 * q^45 + 60317048 * q^47 - 76450382 * q^49 + 123419812 * q^51 - 95149004 * q^53 + 160208072 * q^55 - 120699544 * q^57 + 181069068 * q^59 - 169024572 * q^61 + 270169572 * q^63 - 5026736 * q^65 + 356012572 * q^67 - 269915784 * q^69 + 661588848 * q^71 - 315676732 * q^73 + 934109440 * q^75 - 794391744 * q^77 + 1268146056 * q^79 - 1045070882 * q^81 + 1026540484 * q^83 - 1051824708 * q^85 + 1753885344 * q^87 - 201055172 * q^89 + 276013504 * q^91 - 1156090488 * q^93 + 1720777248 * q^95 + 566920580 * q^97 + 1940442636 * 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
$3$	215.864	1.53863	0.769315	−	0.638870i	$-0.220598\pi$
$3$	215.864	1.53863	0.769315	+	0.638870i	$0.220598\pi$
$4$	0	0
$5$	−1620.37	−1.15944	−0.579722	−	0.814814i	$-0.696839\pi$
$5$	−1620.37	−1.15944	−0.579722	+	0.814814i	$0.696839\pi$
$6$	0	0
$7$	7157.79	1.12678	0.563388	−	0.826192i	$-0.309497\pi$
$7$	7157.79	1.12678	0.563388	+	0.826192i	$0.309497\pi$
$8$	0	0
$9$	26914.2	1.36738
$10$	0	0
$11$	−36853.5	−0.758946	−0.379473	−	0.925203i	$-0.623895\pi$
$11$	−36853.5	−0.758946	−0.379473	+	0.925203i	$0.623895\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	−349780.	−1.78395
$16$	0	0
$17$	404355.	1.17420	0.587100	−	0.809514i	$-0.300269\pi$
$17$	404355.	1.17420	0.587100	+	0.809514i	$0.300269\pi$
$18$	0	0
$19$	203358.	0.357989	0.178995	−	0.983850i	$-0.442716\pi$
$19$	203358.	0.357989	0.178995	+	0.983850i	$0.442716\pi$
$20$	0	0
$21$	1.54511e6	1.73369
$22$	0	0
$23$	407597.	0.303707	0.151854	−	0.988403i	$-0.451476\pi$
$23$	407597.	0.303707	0.151854	+	0.988403i	$0.451476\pi$
$24$	0	0
$25$	672480.	0.344310
$26$	0	0
$27$	1.56095e6	0.565265
$28$	0	0
$29$	225155.	0.0591141	0.0295570	−	0.999563i	$-0.490590\pi$
$29$	225155.	0.0591141	0.0295570	+	0.999563i	$0.490590\pi$
$30$	0	0
$31$	−4.59794e6	−0.894202	−0.447101	−	0.894483i	$-0.647544\pi$
$31$	−4.59794e6	−0.894202	−0.447101	+	0.894483i	$0.647544\pi$
$32$	0	0
$33$	−7.95533e6	−1.16774
$34$	0	0
$35$	−1.15983e7	−1.30643
$36$	0	0
$37$	1.48411e7	1.30185	0.650923	−	0.759144i	$-0.274382\pi$
$37$	1.48411e7	1.30185	0.650923	+	0.759144i	$0.274382\pi$
$38$	0	0
$39$	6.16529e6	0.426739
$40$	0	0
$41$	−1.99373e6	−0.110189	−0.0550946	−	0.998481i	$-0.517546\pi$
$41$	−1.99373e6	−0.110189	−0.0550946	+	0.998481i	$0.517546\pi$
$42$	0	0
$43$	−1.25478e7	−0.559705	−0.279852	−	0.960043i	$-0.590286\pi$
$43$	−1.25478e7	−0.559705	−0.279852	+	0.960043i	$0.590286\pi$
$44$	0	0
$45$	−4.36110e7	−1.58540
$46$	0	0
$47$	6.51793e7	1.94836	0.974180	−	0.225771i	$-0.0724903\pi$
$47$	6.51793e7	1.94836	0.974180	+	0.225771i	$0.0724903\pi$
$48$	0	0
$49$	1.08804e7	0.269626
$50$	0	0
$51$	8.72855e7	1.80666
$52$	0	0
$53$	3.92662e7	0.683562	0.341781	−	0.939780i	$-0.388970\pi$
$53$	3.92662e7	0.683562	0.341781	+	0.939780i	$0.388970\pi$
$54$	0	0
$55$	5.97163e7	0.879956
$56$	0	0
$57$	4.38976e7	0.550813
$58$	0	0
$59$	4.88772e7	0.525137	0.262568	−	0.964913i	$-0.415431\pi$
$59$	4.88772e7	0.525137	0.262568	+	0.964913i	$0.415431\pi$
$60$	0	0
$61$	−4.34425e7	−0.401727	−0.200863	−	0.979619i	$-0.564375\pi$
$61$	−4.34425e7	−0.401727	−0.200863	+	0.979619i	$0.564375\pi$
$62$	0	0
$63$	1.92646e8	1.54073
$64$	0	0
$65$	−4.62794e7	−0.321572
$66$	0	0
$67$	5.55762e7	0.336940	0.168470	−	0.985707i	$-0.446117\pi$
$67$	5.55762e7	0.336940	0.168470	+	0.985707i	$0.446117\pi$
$68$	0	0
$69$	8.79854e7	0.467293
$70$	0	0
$71$	3.91166e8	1.82683	0.913415	−	0.407029i	$-0.133435\pi$
$71$	3.91166e8	1.82683	0.913415	+	0.407029i	$0.133435\pi$
$72$	0	0
$73$	3.96635e8	1.63470	0.817350	−	0.576141i	$-0.195442\pi$
$73$	3.96635e8	1.63470	0.817350	+	0.576141i	$0.195442\pi$
$74$	0	0
$75$	1.45164e8	0.529765
$76$	0	0
$77$	−2.63789e8	−0.855163
$78$	0	0
$79$	1.98103e8	0.572228	0.286114	−	0.958196i	$-0.407636\pi$
$79$	1.98103e8	0.572228	0.286114	+	0.958196i	$0.407636\pi$
$80$	0	0
$81$	−1.92799e8	−0.497649
$82$	0	0
$83$	−7.52406e8	−1.74021	−0.870103	−	0.492870i	$-0.835948\pi$
$83$	−7.52406e8	−1.74021	−0.870103	+	0.492870i	$0.835948\pi$
$84$	0	0
$85$	−6.55205e8	−1.36142
$86$	0	0
$87$	4.86028e7	0.0909547
$88$	0	0
$89$	4.01022e8	0.677506	0.338753	−	0.940875i	$-0.389995\pi$
$89$	4.01022e8	0.677506	0.338753	+	0.940875i	$0.389995\pi$
$90$	0	0
$91$	2.04434e8	0.312512
$92$	0	0
$93$	−9.92529e8	−1.37585
$94$	0	0
$95$	−3.29516e8	−0.415069
$96$	0	0
$97$	8.36965e8	0.959919	0.479960	−	0.877291i	$-0.340651\pi$
$97$	8.36965e8	0.959919	0.479960	+	0.877291i	$0.340651\pi$
$98$	0	0
$99$	−9.91880e8	−1.03777
$100$	0	0
$101$	3.16002e8	0.302165	0.151082	−	0.988521i	$-0.451724\pi$
$101$	3.16002e8	0.302165	0.151082	+	0.988521i	$0.451724\pi$
$102$	0	0
$103$	−9.30792e8	−0.814864	−0.407432	−	0.913236i	$-0.633576\pi$
$103$	−9.30792e8	−0.814864	−0.407432	+	0.913236i	$0.633576\pi$
$104$	0	0
$105$	−2.50365e9	−2.01012
$106$	0	0
$107$	1.97923e9	1.45972	0.729860	−	0.683596i	$-0.239585\pi$
$107$	1.97923e9	1.45972	0.729860	+	0.683596i	$0.239585\pi$
$108$	0	0
$109$	−1.15201e9	−0.781692	−0.390846	−	0.920456i	$-0.627817\pi$
$109$	−1.15201e9	−0.781692	−0.390846	+	0.920456i	$0.627817\pi$
$110$	0	0
$111$	3.20366e9	2.00306
$112$	0	0
$113$	−1.48755e9	−0.858262	−0.429131	−	0.903242i	$-0.641180\pi$
$113$	−1.48755e9	−0.858262	−0.429131	+	0.903242i	$0.641180\pi$
$114$	0	0
$115$	−6.60458e8	−0.352132
$116$	0	0
$117$	7.68696e8	0.379243
$118$	0	0
$119$	2.89429e9	1.32306
$120$	0	0
$121$	−9.99771e8	−0.424000
$122$	0	0
$123$	−4.30374e8	−0.169540
$124$	0	0
$125$	2.07512e9	0.760236
$126$	0	0
$127$	3.77601e9	1.28800	0.644001	−	0.765025i	$-0.277273\pi$
$127$	3.77601e9	1.28800	0.644001	+	0.765025i	$0.277273\pi$
$128$	0	0
$129$	−2.70861e9	−0.861179
$130$	0	0
$131$	−1.19884e9	−0.355665	−0.177832	−	0.984061i	$-0.556908\pi$
$131$	−1.19884e9	−0.355665	−0.177832	+	0.984061i	$0.556908\pi$
$132$	0	0
$133$	1.45559e9	0.403374
$134$	0	0
$135$	−2.52932e9	−0.655393
$136$	0	0
$137$	6.95362e9	1.68643	0.843215	−	0.537577i	$-0.180660\pi$
$137$	6.95362e9	1.68643	0.843215	+	0.537577i	$0.180660\pi$
$138$	0	0
$139$	−6.14299e9	−1.39577	−0.697883	−	0.716211i	$-0.745875\pi$
$139$	−6.14299e9	−1.39577	−0.697883	+	0.716211i	$0.745875\pi$
$140$	0	0
$141$	1.40699e10	2.99781
$142$	0	0
$143$	−1.05257e9	−0.210494
$144$	0	0
$145$	−3.64835e8	−0.0685394
$146$	0	0
$147$	2.34868e9	0.414855
$148$	0	0
$149$	1.10782e10	1.84133	0.920663	−	0.390358i	$-0.127649\pi$
$149$	1.10782e10	1.84133	0.920663	+	0.390358i	$0.127649\pi$
$150$	0	0
$151$	−3.34483e9	−0.523573	−0.261787	−	0.965126i	$-0.584312\pi$
$151$	−3.34483e9	−0.523573	−0.261787	+	0.965126i	$0.584312\pi$
$152$	0	0
$153$	1.08829e10	1.60558
$154$	0	0
$155$	7.45037e9	1.03678
$156$	0	0
$157$	1.33338e9	0.175148	0.0875741	−	0.996158i	$-0.472089\pi$
$157$	1.33338e9	0.175148	0.0875741	+	0.996158i	$0.472089\pi$
$158$	0	0
$159$	8.47616e9	1.05175
$160$	0	0
$161$	2.91749e9	0.342211
$162$	0	0
$163$	1.44420e10	1.60245	0.801223	−	0.598366i	$-0.204183\pi$
$163$	1.44420e10	1.60245	0.801223	+	0.598366i	$0.204183\pi$
$164$	0	0
$165$	1.28906e10	1.35393
$166$	0	0
$167$	−1.09420e10	−1.08861	−0.544304	−	0.838888i	$-0.683206\pi$
$167$	−1.09420e10	−1.08861	−0.544304	+	0.838888i	$0.683206\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	5.47321e9	0.489508
$172$	0	0
$173$	1.06493e10	0.903890	0.451945	−	0.892046i	$-0.350730\pi$
$173$	1.06493e10	0.903890	0.451945	+	0.892046i	$0.350730\pi$
$174$	0	0
$175$	4.81347e9	0.387960
$176$	0	0
$177$	1.05508e10	0.807991
$178$	0	0
$179$	−4.13126e9	−0.300777	−0.150388	−	0.988627i	$-0.548052\pi$
$179$	−4.13126e9	−0.300777	−0.150388	+	0.988627i	$0.548052\pi$
$180$	0	0
$181$	−2.29037e10	−1.58618	−0.793090	−	0.609105i	$-0.791529\pi$
$181$	−2.29037e10	−1.58618	−0.793090	+	0.609105i	$0.791529\pi$
$182$	0	0
$183$	−9.37767e9	−0.618109
$184$	0	0
$185$	−2.40482e10	−1.50942
$186$	0	0
$187$	−1.49019e10	−0.891155
$188$	0	0
$189$	1.11730e10	0.636927
$190$	0	0
$191$	2.35516e10	1.28047	0.640237	−	0.768177i	$-0.278836\pi$
$191$	2.35516e10	1.28047	0.640237	+	0.768177i	$0.278836\pi$
$192$	0	0
$193$	−3.24440e10	−1.68316	−0.841582	−	0.540129i	$-0.818375\pi$
$193$	−3.24440e10	−1.68316	−0.841582	+	0.540129i	$0.818375\pi$
$194$	0	0
$195$	−9.99006e9	−0.494780
$196$	0	0
$197$	2.88525e10	1.36485	0.682426	−	0.730954i	$-0.260925\pi$
$197$	2.88525e10	1.36485	0.682426	+	0.730954i	$0.260925\pi$
$198$	0	0
$199$	2.08228e10	0.941239	0.470619	−	0.882336i	$-0.344030\pi$
$199$	2.08228e10	0.941239	0.470619	+	0.882336i	$0.344030\pi$
$200$	0	0
$201$	1.19969e10	0.518426
$202$	0	0
$203$	1.61161e9	0.0666084
$204$	0	0
$205$	3.23058e9	0.127758
$206$	0	0
$207$	1.09701e10	0.415284
$208$	0	0
$209$	−7.49445e9	−0.271695
$210$	0	0
$211$	−2.15144e10	−0.747237	−0.373619	−	0.927582i	$-0.621883\pi$
$211$	−2.15144e10	−0.747237	−0.373619	+	0.927582i	$0.621883\pi$
$212$	0	0
$213$	8.44385e10	2.81082
$214$	0	0
$215$	2.03321e10	0.648946
$216$	0	0
$217$	−3.29111e10	−1.00757
$218$	0	0
$219$	8.56191e10	2.51520
$220$	0	0
$221$	1.15488e10	0.325665
$222$	0	0
$223$	3.08256e10	0.834719	0.417359	−	0.908741i	$-0.362956\pi$
$223$	3.08256e10	0.834719	0.417359	+	0.908741i	$0.362956\pi$
$224$	0	0
$225$	1.80992e10	0.470803
$226$	0	0
$227$	−1.14724e10	−0.286772	−0.143386	−	0.989667i	$-0.545799\pi$
$227$	−1.14724e10	−0.286772	−0.143386	+	0.989667i	$0.545799\pi$
$228$	0	0
$229$	−6.87553e10	−1.65214	−0.826069	−	0.563569i	$-0.809428\pi$
$229$	−6.87553e10	−1.65214	−0.826069	+	0.563569i	$0.809428\pi$
$230$	0	0
$231$	−5.69426e10	−1.31578
$232$	0	0
$233$	−7.01353e10	−1.55896	−0.779480	−	0.626428i	$-0.784516\pi$
$233$	−7.01353e10	−1.55896	−0.779480	+	0.626428i	$0.784516\pi$
$234$	0	0
$235$	−1.05615e11	−2.25901
$236$	0	0
$237$	4.27633e10	0.880448
$238$	0	0
$239$	7.46237e10	1.47940	0.739701	−	0.672936i	$-0.234967\pi$
$239$	7.46237e10	1.47940	0.739701	+	0.672936i	$0.234967\pi$
$240$	0	0
$241$	6.10136e9	0.116506	0.0582532	−	0.998302i	$-0.481447\pi$
$241$	6.10136e9	0.116506	0.0582532	+	0.998302i	$0.481447\pi$
$242$	0	0
$243$	−7.23426e10	−1.33096
$244$	0	0
$245$	−1.76303e10	−0.312617
$246$	0	0
$247$	5.80811e9	0.0992884
$248$	0	0
$249$	−1.62417e11	−2.67753
$250$	0	0
$251$	9.05484e10	1.43996	0.719978	−	0.693997i	$-0.244152\pi$
$251$	9.05484e10	1.43996	0.719978	+	0.693997i	$0.244152\pi$
$252$	0	0
$253$	−1.50213e10	−0.230498
$254$	0	0
$255$	−1.41435e11	−2.09472
$256$	0	0
$257$	1.84543e10	0.263876	0.131938	−	0.991258i	$-0.457880\pi$
$257$	1.84543e10	0.263876	0.131938	+	0.991258i	$0.457880\pi$
$258$	0	0
$259$	1.06230e11	1.46689
$260$	0	0
$261$	6.05987e9	0.0808315
$262$	0	0
$263$	−6.18309e10	−0.796901	−0.398451	−	0.917190i	$-0.630452\pi$
$263$	−6.18309e10	−0.796901	−0.398451	+	0.917190i	$0.630452\pi$
$264$	0	0
$265$	−6.36259e10	−0.792552
$266$	0	0
$267$	8.65662e10	1.04243
$268$	0	0
$269$	−1.86476e10	−0.217139	−0.108569	−	0.994089i	$-0.534627\pi$
$269$	−1.86476e10	−0.217139	−0.108569	+	0.994089i	$0.534627\pi$
$270$	0	0
$271$	3.30066e10	0.371740	0.185870	−	0.982574i	$-0.440490\pi$
$271$	3.30066e10	0.371740	0.185870	+	0.982574i	$0.440490\pi$
$272$	0	0
$273$	4.41298e10	0.480840
$274$	0	0
$275$	−2.47832e10	−0.261313
$276$	0	0
$277$	1.13027e11	1.15352	0.576759	−	0.816915i	$-0.304317\pi$
$277$	1.13027e11	1.15352	0.576759	+	0.816915i	$0.304317\pi$
$278$	0	0
$279$	−1.23750e11	−1.22272
$280$	0	0
$281$	−1.14491e11	−1.09545	−0.547726	−	0.836658i	$-0.684506\pi$
$281$	−1.14491e11	−1.09545	−0.547726	+	0.836658i	$0.684506\pi$
$282$	0	0
$283$	6.05468e10	0.561116	0.280558	−	0.959837i	$-0.409481\pi$
$283$	6.05468e10	0.561116	0.280558	+	0.959837i	$0.409481\pi$
$284$	0	0
$285$	−7.11305e10	−0.638637
$286$	0	0
$287$	−1.42707e10	−0.124159
$288$	0	0
$289$	4.49148e10	0.378747
$290$	0	0
$291$	1.80670e11	1.47696
$292$	0	0
$293$	1.48131e9	0.0117420	0.00587100	−	0.999983i	$-0.498131\pi$
$293$	1.48131e9	0.0117420	0.00587100	+	0.999983i	$0.498131\pi$
$294$	0	0
$295$	−7.91992e10	−0.608866
$296$	0	0
$297$	−5.75264e10	−0.429006
$298$	0	0
$299$	1.16414e10	0.0842333
$300$	0	0
$301$	−8.98145e10	−0.630663
$302$	0	0
$303$	6.82134e10	0.464920
$304$	0	0
$305$	7.03931e10	0.465780
$306$	0	0
$307$	7.86742e9	0.0505486	0.0252743	−	0.999681i	$-0.491954\pi$
$307$	7.86742e9	0.0505486	0.0252743	+	0.999681i	$0.491954\pi$
$308$	0	0
$309$	−2.00924e11	−1.25377
$310$	0	0
$311$	3.06616e11	1.85855	0.929273	−	0.369392i	$-0.120434\pi$
$311$	3.06616e11	1.85855	0.929273	+	0.369392i	$0.120434\pi$
$312$	0	0
$313$	−1.81818e10	−0.107075	−0.0535373	−	0.998566i	$-0.517050\pi$
$313$	−1.81818e10	−0.107075	−0.0535373	+	0.998566i	$0.517050\pi$
$314$	0	0
$315$	−3.12158e11	−1.78639
$316$	0	0
$317$	−2.08655e11	−1.16055	−0.580274	−	0.814422i	$-0.697054\pi$
$317$	−2.08655e11	−1.16055	−0.580274	+	0.814422i	$0.697054\pi$
$318$	0	0
$319$	−8.29774e9	−0.0448644
$320$	0	0
$321$	4.27244e11	2.24597
$322$	0	0
$323$	8.22288e10	0.420351
$324$	0	0
$325$	1.92067e10	0.0954943
$326$	0	0
$327$	−2.48676e11	−1.20273
$328$	0	0
$329$	4.66540e11	2.19537
$330$	0	0
$331$	−2.72957e11	−1.24988	−0.624940	−	0.780673i	$-0.714877\pi$
$331$	−2.72957e11	−1.24988	−0.624940	+	0.780673i	$0.714877\pi$
$332$	0	0
$333$	3.99437e11	1.78012
$334$	0	0
$335$	−9.00541e10	−0.390663
$336$	0	0
$337$	−5.94975e10	−0.251283	−0.125642	−	0.992076i	$-0.540099\pi$
$337$	−5.94975e10	−0.251283	−0.125642	+	0.992076i	$0.540099\pi$
$338$	0	0
$339$	−3.21109e11	−1.32055
$340$	0	0
$341$	1.69450e11	0.678652
$342$	0	0
$343$	−2.10963e11	−0.822968
$344$	0	0
$345$	−1.42569e11	−0.541800
$346$	0	0
$347$	−1.95210e11	−0.722802	−0.361401	−	0.932411i	$-0.617701\pi$
$347$	−1.95210e11	−0.722802	−0.361401	+	0.932411i	$0.617701\pi$
$348$	0	0
$349$	−3.30681e11	−1.19315	−0.596574	−	0.802558i	$-0.703472\pi$
$349$	−3.30681e11	−1.19315	−0.596574	+	0.802558i	$0.703472\pi$
$350$	0	0
$351$	4.45823e10	0.156776
$352$	0	0
$353$	−3.51178e11	−1.20376	−0.601882	−	0.798585i	$-0.705582\pi$
$353$	−3.51178e11	−1.20376	−0.601882	+	0.798585i	$0.705582\pi$
$354$	0	0
$355$	−6.33834e11	−2.11811
$356$	0	0
$357$	6.24772e11	2.03570
$358$	0	0
$359$	−8.92940e10	−0.283725	−0.141862	−	0.989886i	$-0.545309\pi$
$359$	−8.92940e10	−0.283725	−0.141862	+	0.989886i	$0.545309\pi$
$360$	0	0
$361$	−2.81333e11	−0.871844
$362$	0	0
$363$	−2.15814e11	−0.652380
$364$	0	0
$365$	−6.42696e11	−1.89534
$366$	0	0
$367$	1.88824e9	0.00543324	0.00271662	−	0.999996i	$-0.499135\pi$
$367$	1.88824e9	0.00543324	0.00271662	+	0.999996i	$0.499135\pi$
$368$	0	0
$369$	−5.36596e10	−0.150671
$370$	0	0
$371$	2.81060e11	0.770222
$372$	0	0
$373$	−3.13470e11	−0.838506	−0.419253	−	0.907869i	$-0.637708\pi$
$373$	−3.13470e11	−0.838506	−0.419253	+	0.907869i	$0.637708\pi$
$374$	0	0
$375$	4.47944e11	1.16972
$376$	0	0
$377$	6.43066e9	0.0163953
$378$	0	0
$379$	−1.88477e11	−0.469226	−0.234613	−	0.972089i	$-0.575382\pi$
$379$	−1.88477e11	−0.469226	−0.234613	+	0.972089i	$0.575382\pi$
$380$	0	0
$381$	8.15104e11	1.98176
$382$	0	0
$383$	−1.42773e11	−0.339039	−0.169520	−	0.985527i	$-0.554222\pi$
$383$	−1.42773e11	−0.339039	−0.169520	+	0.985527i	$0.554222\pi$
$384$	0	0
$385$	4.27437e11	0.991514
$386$	0	0
$387$	−3.37713e11	−0.765330
$388$	0	0
$389$	−6.20809e11	−1.37463	−0.687313	−	0.726361i	$-0.741210\pi$
$389$	−6.20809e11	−1.37463	−0.687313	+	0.726361i	$0.741210\pi$
$390$	0	0
$391$	1.64814e11	0.356613
$392$	0	0
$393$	−2.58786e11	−0.547236
$394$	0	0
$395$	−3.21001e11	−0.663467
$396$	0	0
$397$	−7.62298e11	−1.54017	−0.770083	−	0.637944i	$-0.779785\pi$
$397$	−7.62298e11	−1.54017	−0.770083	+	0.637944i	$0.779785\pi$
$398$	0	0
$399$	3.14210e11	0.620644
$400$	0	0
$401$	2.58145e11	0.498555	0.249278	−	0.968432i	$-0.419807\pi$
$401$	2.58145e11	0.498555	0.249278	+	0.968432i	$0.419807\pi$
$402$	0	0
$403$	−1.31322e11	−0.248007
$404$	0	0
$405$	3.12407e11	0.576996
$406$	0	0
$407$	−5.46947e11	−0.988031
$408$	0	0
$409$	−4.36915e11	−0.772045	−0.386022	−	0.922489i	$-0.626151\pi$
$409$	−4.36915e11	−0.772045	−0.386022	+	0.922489i	$0.626151\pi$
$410$	0	0
$411$	1.50103e12	2.59479
$412$	0	0
$413$	3.49853e11	0.591712
$414$	0	0
$415$	1.21918e12	2.01767
$416$	0	0
$417$	−1.32605e12	−2.14757
$418$	0	0
$419$	−8.89457e11	−1.40981	−0.704907	−	0.709300i	$-0.749011\pi$
$419$	−8.89457e11	−1.40981	−0.704907	+	0.709300i	$0.749011\pi$
$420$	0	0
$421$	−8.42824e11	−1.30758	−0.653788	−	0.756677i	$-0.726821\pi$
$421$	−8.42824e11	−1.30758	−0.653788	+	0.756677i	$0.726821\pi$
$422$	0	0
$423$	1.75425e12	2.66415
$424$	0	0
$425$	2.71920e11	0.404289
$426$	0	0
$427$	−3.10953e11	−0.452657
$428$	0	0
$429$	−2.27212e11	−0.323872
$430$	0	0
$431$	−9.19676e11	−1.28377	−0.641885	−	0.766801i	$-0.721847\pi$
$431$	−9.19676e11	−1.28377	−0.641885	+	0.766801i	$0.721847\pi$
$432$	0	0
$433$	7.71243e11	1.05438	0.527188	−	0.849748i	$-0.323246\pi$
$433$	7.71243e11	1.05438	0.527188	+	0.849748i	$0.323246\pi$
$434$	0	0
$435$	−7.87547e10	−0.105457
$436$	0	0
$437$	8.28881e10	0.108724
$438$	0	0
$439$	−1.05055e12	−1.34998	−0.674989	−	0.737827i	$-0.735852\pi$
$439$	−1.05055e12	−1.34998	−0.674989	+	0.737827i	$0.735852\pi$
$440$	0	0
$441$	2.92837e11	0.368682
$442$	0	0
$443$	−1.29133e12	−1.59302	−0.796511	−	0.604624i	$-0.793323\pi$
$443$	−1.29133e12	−1.59302	−0.796511	+	0.604624i	$0.793323\pi$
$444$	0	0
$445$	−6.49805e11	−0.785530
$446$	0	0
$447$	2.39138e12	2.83312
$448$	0	0
$449$	3.14592e11	0.365291	0.182645	−	0.983179i	$-0.441534\pi$
$449$	3.14592e11	0.365291	0.182645	+	0.983179i	$0.441534\pi$
$450$	0	0
$451$	7.34758e10	0.0836276
$452$	0	0
$453$	−7.22027e11	−0.805586
$454$	0	0
$455$	−3.31259e11	−0.362340
$456$	0	0
$457$	6.24863e11	0.670134	0.335067	−	0.942194i	$-0.391241\pi$
$457$	6.24863e11	0.670134	0.335067	+	0.942194i	$0.391241\pi$
$458$	0	0
$459$	6.31177e11	0.663734
$460$	0	0
$461$	1.12501e12	1.16012	0.580060	−	0.814573i	$-0.303029\pi$
$461$	1.12501e12	1.16012	0.580060	+	0.814573i	$0.303029\pi$
$462$	0	0
$463$	1.70497e12	1.72426	0.862130	−	0.506687i	$-0.169130\pi$
$463$	1.70497e12	1.72426	0.862130	+	0.506687i	$0.169130\pi$
$464$	0	0
$465$	1.60827e12	1.59522
$466$	0	0
$467$	−4.05226e11	−0.394249	−0.197125	−	0.980378i	$-0.563160\pi$
$467$	−4.05226e11	−0.394249	−0.197125	+	0.980378i	$0.563160\pi$
$468$	0	0
$469$	3.97803e11	0.379656
$470$	0	0
$471$	2.87828e11	0.269488
$472$	0	0
$473$	4.62429e11	0.424786
$474$	0	0
$475$	1.36754e11	0.123259
$476$	0	0
$477$	1.05682e12	0.934690
$478$	0	0
$479$	−3.36752e11	−0.292281	−0.146141	−	0.989264i	$-0.546685\pi$
$479$	−3.36752e11	−0.292281	−0.146141	+	0.989264i	$0.546685\pi$
$480$	0	0
$481$	4.23878e11	0.361067
$482$	0	0
$483$	6.29781e11	0.526535
$484$	0	0
$485$	−1.35619e12	−1.11297
$486$	0	0
$487$	8.16038e11	0.657401	0.328700	−	0.944434i	$-0.393389\pi$
$487$	8.16038e11	0.657401	0.328700	+	0.944434i	$0.393389\pi$
$488$	0	0
$489$	3.11751e12	2.46557
$490$	0	0
$491$	1.56401e12	1.21443	0.607216	−	0.794537i	$-0.292286\pi$
$491$	1.56401e12	1.21443	0.607216	+	0.794537i	$0.292286\pi$
$492$	0	0
$493$	9.10425e10	0.0694118
$494$	0	0
$495$	1.60721e12	1.20324
$496$	0	0
$497$	2.79988e12	2.05843
$498$	0	0
$499$	5.13160e11	0.370510	0.185255	−	0.982690i	$-0.440689\pi$
$499$	5.13160e11	0.370510	0.185255	+	0.982690i	$0.440689\pi$
$500$	0	0
$501$	−2.36198e12	−1.67497
$502$	0	0
$503$	−1.60303e11	−0.111657	−0.0558286	−	0.998440i	$-0.517780\pi$
$503$	−1.60303e11	−0.111657	−0.0558286	+	0.998440i	$0.517780\pi$
$504$	0	0
$505$	−5.12041e11	−0.350343
$506$	0	0
$507$	1.76087e11	0.118356
$508$	0	0
$509$	1.09792e12	0.725006	0.362503	−	0.931983i	$-0.381922\pi$
$509$	1.09792e12	0.725006	0.362503	+	0.931983i	$0.381922\pi$
$510$	0	0
$511$	2.83903e12	1.84194
$512$	0	0
$513$	3.17432e11	0.202359
$514$	0	0
$515$	1.50823e12	0.944789
$516$	0	0
$517$	−2.40208e12	−1.47870
$518$	0	0
$519$	2.29881e12	1.39075
$520$	0	0
$521$	−2.96365e12	−1.76221	−0.881103	−	0.472925i	$-0.843198\pi$
$521$	−2.96365e12	−1.76221	−0.881103	+	0.472925i	$0.843198\pi$
$522$	0	0
$523$	1.79812e12	1.05090	0.525449	−	0.850825i	$-0.323897\pi$
$523$	1.79812e12	1.05090	0.525449	+	0.850825i	$0.323897\pi$
$524$	0	0
$525$	1.03905e12	0.596927
$526$	0	0
$527$	−1.85920e12	−1.04997
$528$	0	0
$529$	−1.63502e12	−0.907762
$530$	0	0
$531$	1.31549e12	0.718062
$532$	0	0
$533$	−5.69429e10	−0.0305610
$534$	0	0
$535$	−3.20709e12	−1.69246
$536$	0	0
$537$	−8.91790e11	−0.462784
$538$	0	0
$539$	−4.00980e11	−0.204632
$540$	0	0
$541$	2.96375e12	1.48749	0.743745	−	0.668464i	$-0.233048\pi$
$541$	2.96375e12	1.48749	0.743745	+	0.668464i	$0.233048\pi$
$542$	0	0
$543$	−4.94408e12	−2.44054
$544$	0	0
$545$	1.86668e12	0.906328
$546$	0	0
$547$	2.09308e10	0.00999639	0.00499820	−	0.999988i	$-0.498409\pi$
$547$	2.09308e10	0.00999639	0.00499820	+	0.999988i	$0.498409\pi$
$548$	0	0
$549$	−1.16922e12	−0.549314
$550$	0	0
$551$	4.57871e10	0.0211622
$552$	0	0
$553$	1.41798e12	0.644774
$554$	0	0
$555$	−5.19113e12	−2.32243
$556$	0	0
$557$	−5.61630e11	−0.247231	−0.123615	−	0.992330i	$-0.539449\pi$
$557$	−5.61630e11	−0.247231	−0.123615	+	0.992330i	$0.539449\pi$
$558$	0	0
$559$	−3.58377e11	−0.155234
$560$	0	0
$561$	−3.21677e12	−1.37116
$562$	0	0
$563$	−2.05243e12	−0.860956	−0.430478	−	0.902601i	$-0.641655\pi$
$563$	−2.05243e12	−0.860956	−0.430478	+	0.902601i	$0.641655\pi$
$564$	0	0
$565$	2.41039e12	0.995106
$566$	0	0
$567$	−1.38002e12	−0.560739
$568$	0	0
$569$	−5.17268e11	−0.206876	−0.103438	−	0.994636i	$-0.532984\pi$
$569$	−5.17268e11	−0.206876	−0.103438	+	0.994636i	$0.532984\pi$
$570$	0	0
$571$	1.09701e12	0.431866	0.215933	−	0.976408i	$-0.430721\pi$
$571$	1.09701e12	0.431866	0.215933	+	0.976408i	$0.430721\pi$
$572$	0	0
$573$	5.08395e12	1.97018
$574$	0	0
$575$	2.74101e11	0.104569
$576$	0	0
$577$	2.16240e12	0.812165	0.406083	−	0.913836i	$-0.366894\pi$
$577$	2.16240e12	0.812165	0.406083	+	0.913836i	$0.366894\pi$
$578$	0	0
$579$	−7.00348e12	−2.58977
$580$	0	0
$581$	−5.38557e12	−1.96082
$582$	0	0
$583$	−1.44710e12	−0.518787
$584$	0	0
$585$	−1.24557e12	−0.439711
$586$	0	0
$587$	−1.08800e12	−0.378232	−0.189116	−	0.981955i	$-0.560562\pi$
$587$	−1.08800e12	−0.378232	−0.189116	+	0.981955i	$0.560562\pi$
$588$	0	0
$589$	−9.35028e11	−0.320115
$590$	0	0
$591$	6.22821e12	2.10000
$592$	0	0
$593$	−1.99045e12	−0.661006	−0.330503	−	0.943805i	$-0.607218\pi$
$593$	−1.99045e12	−0.661006	−0.330503	+	0.943805i	$0.607218\pi$
$594$	0	0
$595$	−4.68982e12	−1.53402
$596$	0	0
$597$	4.49488e12	1.44822
$598$	0	0
$599$	−4.13617e12	−1.31274	−0.656369	−	0.754440i	$-0.727909\pi$
$599$	−4.13617e12	−1.31274	−0.656369	+	0.754440i	$0.727909\pi$
$600$	0	0
$601$	−1.36781e11	−0.0427652	−0.0213826	−	0.999771i	$-0.506807\pi$
$601$	−1.36781e11	−0.0427652	−0.0213826	+	0.999771i	$0.506807\pi$
$602$	0	0
$603$	1.49579e12	0.460725
$604$	0	0
$605$	1.62000e12	0.491605
$606$	0	0
$607$	1.15852e12	0.346381	0.173190	−	0.984888i	$-0.444592\pi$
$607$	1.15852e12	0.346381	0.173190	+	0.984888i	$0.444592\pi$
$608$	0	0
$609$	3.47889e11	0.102486
$610$	0	0
$611$	1.86159e12	0.540378
$612$	0	0
$613$	−1.63345e12	−0.467232	−0.233616	−	0.972329i	$-0.575056\pi$
$613$	−1.63345e12	−0.467232	−0.233616	+	0.972329i	$0.575056\pi$
$614$	0	0
$615$	6.97366e11	0.196572
$616$	0	0
$617$	−2.32758e11	−0.0646578	−0.0323289	−	0.999477i	$-0.510292\pi$
$617$	−2.32758e11	−0.0646578	−0.0323289	+	0.999477i	$0.510292\pi$
$618$	0	0
$619$	−5.19753e12	−1.42295	−0.711475	−	0.702712i	$-0.751972\pi$
$619$	−5.19753e12	−1.42295	−0.711475	+	0.702712i	$0.751972\pi$
$620$	0	0
$621$	6.36238e11	0.171675
$622$	0	0
$623$	2.87043e12	0.763398
$624$	0	0
$625$	−4.67591e12	−1.22576
$626$	0	0
$627$	−1.61778e12	−0.418038
$628$	0	0
$629$	6.00108e12	1.52863
$630$	0	0
$631$	−1.04604e12	−0.262674	−0.131337	−	0.991338i	$-0.541927\pi$
$631$	−1.04604e12	−0.262674	−0.131337	+	0.991338i	$0.541927\pi$
$632$	0	0
$633$	−4.64418e12	−1.14972
$634$	0	0
$635$	−6.11854e12	−1.49337
$636$	0	0
$637$	3.10755e11	0.0747809
$638$	0	0
$639$	1.05279e13	2.49797
$640$	0	0
$641$	1.85450e12	0.433877	0.216938	−	0.976185i	$-0.430393\pi$
$641$	1.85450e12	0.433877	0.216938	+	0.976185i	$0.430393\pi$
$642$	0	0
$643$	5.85750e12	1.35134	0.675668	−	0.737206i	$-0.263855\pi$
$643$	5.85750e12	1.35134	0.675668	+	0.737206i	$0.263855\pi$
$644$	0	0
$645$	4.38896e12	0.998488
$646$	0	0
$647$	5.85418e12	1.31340	0.656699	−	0.754153i	$-0.271952\pi$
$647$	5.85418e12	1.31340	0.656699	+	0.754153i	$0.271952\pi$
$648$	0	0
$649$	−1.80129e12	−0.398550
$650$	0	0
$651$	−7.10432e12	−1.55027
$652$	0	0
$653$	1.18302e12	0.254614	0.127307	−	0.991863i	$-0.459367\pi$
$653$	1.18302e12	0.254614	0.127307	+	0.991863i	$0.459367\pi$
$654$	0	0
$655$	1.94257e12	0.412373
$656$	0	0
$657$	1.06751e13	2.23526
$658$	0	0
$659$	3.74377e12	0.773259	0.386629	−	0.922235i	$-0.373639\pi$
$659$	3.74377e12	0.773259	0.386629	+	0.922235i	$0.373639\pi$
$660$	0	0
$661$	−2.30194e12	−0.469016	−0.234508	−	0.972114i	$-0.575348\pi$
$661$	−2.30194e12	−0.469016	−0.234508	+	0.972114i	$0.575348\pi$
$662$	0	0
$663$	2.49296e12	0.501077
$664$	0	0
$665$	−2.35861e12	−0.467690
$666$	0	0
$667$	9.17725e10	0.0179534
$668$	0	0
$669$	6.65414e12	1.28432
$670$	0	0
$671$	1.60101e12	0.304889
$672$	0	0
$673$	−1.04550e13	−1.96452	−0.982260	−	0.187522i	$-0.939954\pi$
$673$	−1.04550e13	−1.96452	−0.982260	+	0.187522i	$0.939954\pi$
$674$	0	0
$675$	1.04971e12	0.194626
$676$	0	0
$677$	−7.44943e11	−0.136293	−0.0681466	−	0.997675i	$-0.521709\pi$
$677$	−7.44943e11	−0.136293	−0.0681466	+	0.997675i	$0.521709\pi$
$678$	0	0
$679$	5.99082e12	1.08161
$680$	0	0
$681$	−2.47647e12	−0.441237
$682$	0	0
$683$	−8.05108e12	−1.41567	−0.707833	−	0.706379i	$-0.750327\pi$
$683$	−8.05108e12	−1.41567	−0.707833	+	0.706379i	$0.750327\pi$
$684$	0	0
$685$	−1.12674e13	−1.95532
$686$	0	0
$687$	−1.48418e13	−2.54203
$688$	0	0
$689$	1.12148e12	0.189586
$690$	0	0
$691$	1.00440e13	1.67593	0.837963	−	0.545727i	$-0.183747\pi$
$691$	1.00440e13	1.67593	0.837963	+	0.545727i	$0.183747\pi$
$692$	0	0
$693$	−7.09967e12	−1.16933
$694$	0	0
$695$	9.95392e12	1.61831
$696$	0	0
$697$	−8.06173e11	−0.129384
$698$	0	0
$699$	−1.51397e13	−2.39866
$700$	0	0
$701$	−6.14523e12	−0.961186	−0.480593	−	0.876944i	$-0.659578\pi$
$701$	−6.14523e12	−0.961186	−0.480593	+	0.876944i	$0.659578\pi$
$702$	0	0
$703$	3.01807e12	0.466047
$704$	0	0
$705$	−2.27984e13	−3.47579
$706$	0	0
$707$	2.26188e12	0.340472
$708$	0	0
$709$	4.98013e12	0.740172	0.370086	−	0.928997i	$-0.379328\pi$
$709$	4.98013e12	0.740172	0.370086	+	0.928997i	$0.379328\pi$
$710$	0	0
$711$	5.33178e12	0.782455
$712$	0	0
$713$	−1.87411e12	−0.271576
$714$	0	0
$715$	1.70556e12	0.244056
$716$	0	0
$717$	1.61086e13	2.27625
$718$	0	0
$719$	4.89030e12	0.682426	0.341213	−	0.939986i	$-0.389162\pi$
$719$	4.89030e12	0.682426	0.341213	+	0.939986i	$0.389162\pi$
$720$	0	0
$721$	−6.66241e12	−0.918170
$722$	0	0
$723$	1.31706e12	0.179260
$724$	0	0
$725$	1.51412e11	0.0203536
$726$	0	0
$727$	−3.19281e12	−0.423905	−0.211952	−	0.977280i	$-0.567982\pi$
$727$	−3.19281e12	−0.423905	−0.211952	+	0.977280i	$0.567982\pi$
$728$	0	0
$729$	−1.18213e13	−1.55021
$730$	0	0
$731$	−5.07376e12	−0.657206
$732$	0	0
$733$	−6.98794e12	−0.894090	−0.447045	−	0.894511i	$-0.647524\pi$
$733$	−6.98794e12	−0.894090	−0.447045	+	0.894511i	$0.647524\pi$
$734$	0	0
$735$	−3.80574e12	−0.481001
$736$	0	0
$737$	−2.04817e12	−0.255719
$738$	0	0
$739$	−1.40477e12	−0.173263	−0.0866313	−	0.996240i	$-0.527610\pi$
$739$	−1.40477e12	−0.173263	−0.0866313	+	0.996240i	$0.527610\pi$
$740$	0	0
$741$	1.25376e12	0.152768
$742$	0	0
$743$	−5.29417e12	−0.637306	−0.318653	−	0.947871i	$-0.603230\pi$
$743$	−5.29417e12	−0.637306	−0.318653	+	0.947871i	$0.603230\pi$
$744$	0	0
$745$	−1.79508e13	−2.13491
$746$	0	0
$747$	−2.02504e13	−2.37953
$748$	0	0
$749$	1.41669e13	1.64478
$750$	0	0
$751$	1.63972e13	1.88100	0.940502	−	0.339787i	$-0.110355\pi$
$751$	1.63972e13	1.88100	0.940502	+	0.339787i	$0.110355\pi$
$752$	0	0
$753$	1.95461e13	2.21556
$754$	0	0
$755$	5.41987e12	0.607054
$756$	0	0
$757$	6.89161e12	0.762762	0.381381	−	0.924418i	$-0.375449\pi$
$757$	6.89161e12	0.762762	0.381381	+	0.924418i	$0.375449\pi$
$758$	0	0
$759$	−3.24256e12	−0.354651
$760$	0	0
$761$	1.79318e13	1.93817	0.969085	−	0.246728i	$-0.0793554\pi$
$761$	1.79318e13	1.93817	0.969085	+	0.246728i	$0.0793554\pi$
$762$	0	0
$763$	−8.24582e12	−0.880792
$764$	0	0
$765$	−1.76343e13	−1.86158
$766$	0	0
$767$	1.39598e12	0.145647
$768$	0	0
$769$	3.65837e12	0.377242	0.188621	−	0.982050i	$-0.439598\pi$
$769$	3.65837e12	0.377242	0.188621	+	0.982050i	$0.439598\pi$
$770$	0	0
$771$	3.98363e12	0.406007
$772$	0	0
$773$	1.05162e12	0.105938	0.0529692	−	0.998596i	$-0.483132\pi$
$773$	1.05162e12	0.105938	0.0529692	+	0.998596i	$0.483132\pi$
$774$	0	0
$775$	−3.09202e12	−0.307883
$776$	0	0
$777$	2.29312e13	2.25700
$778$	0	0
$779$	−4.05441e11	−0.0394465
$780$	0	0
$781$	−1.44158e13	−1.38647
$782$	0	0
$783$	3.51456e11	0.0334151
$784$	0	0
$785$	−2.16057e12	−0.203074
$786$	0	0
$787$	2.00786e12	0.186572	0.0932862	−	0.995639i	$-0.470263\pi$
$787$	2.00786e12	0.186572	0.0932862	+	0.995639i	$0.470263\pi$
$788$	0	0
$789$	−1.33470e13	−1.22614
$790$	0	0
$791$	−1.06476e13	−0.967070
$792$	0	0
$793$	−1.24076e12	−0.111419
$794$	0	0
$795$	−1.37345e13	−1.21944
$796$	0	0
$797$	−1.60085e13	−1.40536	−0.702681	−	0.711505i	$-0.748014\pi$
$797$	−1.60085e13	−1.40536	−0.702681	+	0.711505i	$0.748014\pi$
$798$	0	0
$799$	2.63555e13	2.28777
$800$	0	0
$801$	1.07932e13	0.926410
$802$	0	0
$803$	−1.46174e13	−1.24065
$804$	0	0
$805$	−4.72742e12	−0.396774
$806$	0	0
$807$	−4.02534e12	−0.334096
$808$	0	0
$809$	2.11454e12	0.173559	0.0867795	−	0.996228i	$-0.472342\pi$
$809$	2.11454e12	0.173559	0.0867795	+	0.996228i	$0.472342\pi$
$810$	0	0
$811$	6.76373e12	0.549025	0.274513	−	0.961583i	$-0.411483\pi$
$811$	6.76373e12	0.549025	0.274513	+	0.961583i	$0.411483\pi$
$812$	0	0
$813$	7.12494e12	0.571970
$814$	0	0
$815$	−2.34014e13	−1.85795
$816$	0	0
$817$	−2.55169e12	−0.200368
$818$	0	0
$819$	5.50217e12	0.427323
$820$	0	0
$821$	9.89188e12	0.759862	0.379931	−	0.925015i	$-0.375948\pi$
$821$	9.89188e12	0.759862	0.379931	+	0.925015i	$0.375948\pi$
$822$	0	0
$823$	−4.33526e12	−0.329394	−0.164697	−	0.986344i	$-0.552665\pi$
$823$	−4.33526e12	−0.329394	−0.164697	+	0.986344i	$0.552665\pi$
$824$	0	0
$825$	−5.34980e12	−0.402063
$826$	0	0
$827$	1.50216e12	0.111671	0.0558355	−	0.998440i	$-0.482218\pi$
$827$	1.50216e12	0.111671	0.0558355	+	0.998440i	$0.482218\pi$
$828$	0	0
$829$	2.26692e13	1.66702	0.833509	−	0.552505i	$-0.186328\pi$
$829$	2.26692e13	1.66702	0.833509	+	0.552505i	$0.186328\pi$
$830$	0	0
$831$	2.43985e13	1.77484
$832$	0	0
$833$	4.39954e12	0.316595
$834$	0	0
$835$	1.77301e13	1.26218
$836$	0	0
$837$	−7.17715e12	−0.505461
$838$	0	0
$839$	1.10333e13	0.768732	0.384366	−	0.923181i	$-0.374420\pi$
$839$	1.10333e13	0.768732	0.384366	+	0.923181i	$0.374420\pi$
$840$	0	0
$841$	−1.44565e13	−0.996506
$842$	0	0
$843$	−2.47145e13	−1.68549
$844$	0	0
$845$	−1.32179e12	−0.0891880
$846$	0	0
$847$	−7.15615e12	−0.477754
$848$	0	0
$849$	1.30699e13	0.863350
$850$	0	0
$851$	6.04920e12	0.395380
$852$	0	0
$853$	−9.89097e12	−0.639688	−0.319844	−	0.947470i	$-0.603631\pi$
$853$	−9.89097e12	−0.639688	−0.319844	+	0.947470i	$0.603631\pi$
$854$	0	0
$855$	−8.86864e12	−0.567557
$856$	0	0
$857$	7.61479e12	0.482219	0.241109	−	0.970498i	$-0.422489\pi$
$857$	7.61479e12	0.482219	0.241109	+	0.970498i	$0.422489\pi$
$858$	0	0
$859$	−6.62850e12	−0.415381	−0.207690	−	0.978195i	$-0.566595\pi$
$859$	−6.62850e12	−0.415381	−0.207690	+	0.978195i	$0.566595\pi$
$860$	0	0
$861$	−3.08053e12	−0.191034
$862$	0	0
$863$	2.17146e13	1.33261	0.666305	−	0.745679i	$-0.267875\pi$
$863$	2.17146e13	1.33261	0.666305	+	0.745679i	$0.267875\pi$
$864$	0	0
$865$	−1.72559e13	−1.04801
$866$	0	0
$867$	9.69547e12	0.582751
$868$	0	0
$869$	−7.30078e12	−0.434291
$870$	0	0
$871$	1.58731e12	0.0934503
$872$	0	0
$873$	2.25262e13	1.31258
$874$	0	0
$875$	1.48533e13	0.856616
$876$	0	0
$877$	−1.07781e13	−0.615237	−0.307618	−	0.951510i	$-0.599532\pi$
$877$	−1.07781e13	−0.615237	−0.307618	+	0.951510i	$0.599532\pi$
$878$	0	0
$879$	3.19762e11	0.0180666
$880$	0	0
$881$	2.91671e13	1.63118	0.815591	−	0.578629i	$-0.196412\pi$
$881$	2.91671e13	1.63118	0.815591	+	0.578629i	$0.196412\pi$
$882$	0	0
$883$	−8.57360e12	−0.474613	−0.237307	−	0.971435i	$-0.576265\pi$
$883$	−8.57360e12	−0.474613	−0.237307	+	0.971435i	$0.576265\pi$
$884$	0	0
$885$	−1.70962e13	−0.936820
$886$	0	0
$887$	−1.12966e13	−0.612764	−0.306382	−	0.951909i	$-0.599118\pi$
$887$	−1.12966e13	−0.612764	−0.306382	+	0.951909i	$0.599118\pi$
$888$	0	0
$889$	2.70279e13	1.45129
$890$	0	0
$891$	7.10532e12	0.377689
$892$	0	0
$893$	1.32547e13	0.697492
$894$	0	0
$895$	6.69418e12	0.348734
$896$	0	0
$897$	2.51295e12	0.129604
$898$	0	0
$899$	−1.03525e12	−0.0528599
$900$	0	0
$901$	1.58775e13	0.802639
$902$	0	0
$903$	−1.93877e13	−0.970356
$904$	0	0
$905$	3.71125e13	1.83909
$906$	0	0
$907$	−3.61284e13	−1.77262	−0.886311	−	0.463090i	$-0.846741\pi$
$907$	−3.61284e13	−1.77262	−0.886311	+	0.463090i	$0.846741\pi$
$908$	0	0
$909$	8.50494e12	0.413175
$910$	0	0
$911$	−5.05617e12	−0.243214	−0.121607	−	0.992578i	$-0.538805\pi$
$911$	−5.05617e12	−0.243214	−0.121607	+	0.992578i	$0.538805\pi$
$912$	0	0
$913$	2.77288e13	1.32072
$914$	0	0
$915$	1.51953e13	0.716663
$916$	0	0
$917$	−8.58105e12	−0.400755
$918$	0	0
$919$	3.93434e13	1.81950	0.909749	−	0.415159i	$-0.136274\pi$
$919$	3.93434e13	1.81950	0.909749	+	0.415159i	$0.136274\pi$
$920$	0	0
$921$	1.69829e12	0.0777757
$922$	0	0
$923$	1.11721e13	0.506672
$924$	0	0
$925$	9.98037e12	0.448238
$926$	0	0
$927$	−2.50515e13	−1.11423
$928$	0	0
$929$	1.76046e13	0.775453	0.387726	−	0.921774i	$-0.373261\pi$
$929$	1.76046e13	0.775453	0.387726	+	0.921774i	$0.373261\pi$
$930$	0	0
$931$	2.21262e12	0.0965234
$932$	0	0
$933$	6.61874e13	2.85962
$934$	0	0
$935$	2.41466e13	1.03324
$936$	0	0
$937$	−3.59967e13	−1.52558	−0.762790	−	0.646646i	$-0.776171\pi$
$937$	−3.59967e13	−1.52558	−0.762790	+	0.646646i	$0.776171\pi$
$938$	0	0
$939$	−3.92479e12	−0.164748
$940$	0	0
$941$	−2.62359e13	−1.09079	−0.545397	−	0.838178i	$-0.683621\pi$
$941$	−2.62359e13	−1.09079	−0.545397	+	0.838178i	$0.683621\pi$
$942$	0	0
$943$	−8.12637e11	−0.0334653
$944$	0	0
$945$	−1.81043e13	−0.738481
$946$	0	0
$947$	−2.36296e11	−0.00954733	−0.00477367	−	0.999989i	$-0.501520\pi$
$947$	−2.36296e11	−0.00954733	−0.00477367	+	0.999989i	$0.501520\pi$
$948$	0	0
$949$	1.13283e13	0.453384
$950$	0	0
$951$	−4.50411e13	−1.78565
$952$	0	0
$953$	2.37919e13	0.934352	0.467176	−	0.884164i	$-0.345271\pi$
$953$	2.37919e13	0.934352	0.467176	+	0.884164i	$0.345271\pi$
$954$	0	0
$955$	−3.81624e13	−1.48464
$956$	0	0
$957$	−1.79118e12	−0.0690297
$958$	0	0
$959$	4.97726e13	1.90023
$960$	0	0
$961$	−5.29856e12	−0.200402
$962$	0	0
$963$	5.32694e13	1.99600
$964$	0	0
$965$	5.25713e13	1.95153
$966$	0	0
$967$	3.61190e13	1.32836	0.664182	−	0.747571i	$-0.268780\pi$
$967$	3.61190e13	1.32836	0.664182	+	0.747571i	$0.268780\pi$
$968$	0	0
$969$	1.77502e13	0.646765
$970$	0	0
$971$	1.78316e13	0.643728	0.321864	−	0.946786i	$-0.395691\pi$
$971$	1.78316e13	0.643728	0.321864	+	0.946786i	$0.395691\pi$
$972$	0	0
$973$	−4.39702e13	−1.57272
$974$	0	0
$975$	4.14603e12	0.146930
$976$	0	0
$977$	−1.06952e13	−0.375545	−0.187772	−	0.982213i	$-0.560127\pi$
$977$	−1.06952e13	−0.375545	−0.187772	+	0.982213i	$0.560127\pi$
$978$	0	0
$979$	−1.47790e13	−0.514191
$980$	0	0
$981$	−3.10053e13	−1.06887
$982$	0	0
$983$	−3.88263e13	−1.32628	−0.663139	−	0.748496i	$-0.730776\pi$
$983$	−3.88263e13	−1.32628	−0.663139	+	0.748496i	$0.730776\pi$
$984$	0	0
$985$	−4.67518e13	−1.58247
$986$	0	0
$987$	1.00709e14	3.37786
$988$	0	0
$989$	−5.11444e12	−0.169987
$990$	0	0
$991$	3.56040e12	0.117265	0.0586324	−	0.998280i	$-0.481326\pi$
$991$	3.56040e12	0.117265	0.0586324	+	0.998280i	$0.481326\pi$
$992$	0	0
$993$	−5.89215e13	−1.92310
$994$	0	0
$995$	−3.37406e13	−1.09131
$996$	0	0
$997$	3.34151e13	1.07106	0.535531	−	0.844515i	$-0.320111\pi$
$997$	3.34151e13	1.07106	0.535531	+	0.844515i	$0.320111\pi$
$998$	0	0
$999$	2.31663e13	0.735887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.k.1.6		6
4.3	odd	2		104.10.a.a.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.a.1.1	✓	6	4.3	odd	2
208.10.a.k.1.6		6	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)

\(f(q)\)	\(=\)	\(q+215.864 q^{3} -1620.37 q^{5} +7157.79 q^{7} +26914.2 q^{9} -36853.5 q^{11} +28561.0 q^{13} -349780. q^{15} +404355. q^{17} +203358. q^{19} +1.54511e6 q^{21} +407597. q^{23} +672480. q^{25} +1.56095e6 q^{27} +225155. q^{29} -4.59794e6 q^{31} -7.95533e6 q^{33} -1.15983e7 q^{35} +1.48411e7 q^{37} +6.16529e6 q^{39} -1.99373e6 q^{41} -1.25478e7 q^{43} -4.36110e7 q^{45} +6.51793e7 q^{47} +1.08804e7 q^{49} +8.72855e7 q^{51} +3.92662e7 q^{53} +5.97163e7 q^{55} +4.38976e7 q^{57} +4.88772e7 q^{59} -4.34425e7 q^{61} +1.92646e8 q^{63} -4.62794e7 q^{65} +5.55762e7 q^{67} +8.79854e7 q^{69} +3.91166e8 q^{71} +3.96635e8 q^{73} +1.45164e8 q^{75} -2.63789e8 q^{77} +1.98103e8 q^{79} -1.92799e8 q^{81} -7.52406e8 q^{83} -6.55205e8 q^{85} +4.86028e7 q^{87} +4.01022e8 q^{89} +2.04434e8 q^{91} -9.92529e8 q^{93} -3.29516e8 q^{95} +8.36965e8 q^{97} -9.91880e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 68 q^{3} - 176 q^{5} + 9664 q^{7} + 26938 q^{9} + 15980 q^{11} + 171366 q^{13} + 10756 q^{15} + 215048 q^{17} + 519868 q^{19} + 384084 q^{21} + 691224 q^{23} - 196082 q^{25} + 1592756 q^{27} - 546932 q^{29}+ \cdots + 1940442636 q^{99}+O(q^{100}) \) 6 * q + 68 * q^3 - 176 * q^5 + 9664 * q^7 + 26938 * q^9 + 15980 * q^11 + 171366 * q^13 + 10756 * q^15 + 215048 * q^17 + 519868 * q^19 + 384084 * q^21 + 691224 * q^23 - 196082 * q^25 + 1592756 * q^27 - 546932 * q^29 + 535300 * q^31 - 3961488 * q^33 + 1042020 * q^35 - 18760 * q^37 + 1942148 * q^39 - 8895252 * q^41 + 21376108 * q^43 - 46124116 * q^45 + 60317048 * q^47 - 76450382 * q^49 + 123419812 * q^51 - 95149004 * q^53 + 160208072 * q^55 - 120699544 * q^57 + 181069068 * q^59 - 169024572 * q^61 + 270169572 * q^63 - 5026736 * q^65 + 356012572 * q^67 - 269915784 * q^69 + 661588848 * q^71 - 315676732 * q^73 + 934109440 * q^75 - 794391744 * q^77 + 1268146056 * q^79 - 1045070882 * q^81 + 1026540484 * q^83 - 1051824708 * q^85 + 1753885344 * q^87 - 201055172 * q^89 + 276013504 * q^91 - 1156090488 * q^93 + 1720777248 * q^95 + 566920580 * q^97 + 1940442636 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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