LMFDB - Embedded newform 208.10.a.j.1.4

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,-60] 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:	$1$
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} - 69754x^{4} - 2752492x^{3} + 1089377733x^{2} + 50183965132x - 2195812679340 $ x^6 - 69754x^4 - 2752492x^3 + 1089377733x^2 + 50183965132x - 2195812679340
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$28.3984$ 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)

$f(q)$	$=$	$q+18.3984 q^{3} +2458.39 q^{5} -6435.39 q^{7} -19344.5 q^{9} +75667.5 q^{11} -28561.0 q^{13} +45230.4 q^{15} -574152. q^{17} +839418. q^{19} -118401. q^{21} -2.18346e6 q^{23} +4.09055e6 q^{25} -718044. q^{27} -4.54258e6 q^{29} +3.24725e6 q^{31} +1.39216e6 q^{33} -1.58207e7 q^{35} -1.47111e7 q^{37} -525477. q^{39} -5.26815e6 q^{41} -1.06362e7 q^{43} -4.75563e7 q^{45} -2.85092e6 q^{47} +1.06069e6 q^{49} -1.05635e7 q^{51} -4.50292e7 q^{53} +1.86020e8 q^{55} +1.54440e7 q^{57} -2.49085e7 q^{59} +1.72663e8 q^{61} +1.24489e8 q^{63} -7.02140e7 q^{65} +2.69226e8 q^{67} -4.01722e7 q^{69} -2.82052e8 q^{71} +1.32314e8 q^{73} +7.52595e7 q^{75} -4.86950e8 q^{77} +2.75182e8 q^{79} +3.67547e8 q^{81} -4.46977e8 q^{83} -1.41149e9 q^{85} -8.35762e7 q^{87} -1.12425e8 q^{89} +1.83801e8 q^{91} +5.97442e7 q^{93} +2.06362e9 q^{95} -1.44020e9 q^{97} -1.46375e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 60 q^{3} + 176 q^{5} - 3416 q^{7} + 22010 q^{9} + 101316 q^{11} - 171366 q^{13} - 257460 q^{15} - 170304 q^{17} + 92084 q^{19} - 424004 q^{21} - 2369944 q^{23} + 2043598 q^{25} + 6428196 q^{27} - 9021404 q^{29}+ \cdots - 282095980 q^{99}+O(q^{100}) $ 6 * q - 60 * q^3 + 176 * q^5 - 3416 * q^7 + 22010 * q^9 + 101316 * q^11 - 171366 * q^13 - 257460 * q^15 - 170304 * q^17 + 92084 * q^19 - 424004 * q^21 - 2369944 * q^23 + 2043598 * q^25 + 6428196 * q^27 - 9021404 * q^29 + 9894388 * q^31 + 6717200 * q^33 + 1910916 * q^35 + 2974376 * q^37 + 1713660 * q^39 - 2977668 * q^41 + 31585340 * q^43 - 99427116 * q^45 + 1766688 * q^47 - 25397750 * q^49 + 68102932 * q^51 - 137836444 * q^53 + 64702520 * q^55 - 253569848 * q^57 + 128523972 * q^59 + 33822404 * q^61 + 359085076 * q^63 - 5026736 * q^65 + 618333476 * q^67 - 837582264 * q^69 + 52534456 * q^71 - 667586460 * q^73 + 940557360 * q^75 - 944797672 * q^77 + 399417496 * q^79 - 821518498 * q^81 + 467205356 * q^83 - 2583200620 * q^85 + 219585328 * q^87 - 1630231268 * q^89 + 97564376 * q^91 - 585888632 * q^93 + 3079232768 * q^95 - 3266492188 * q^97 - 282095980 * 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
$3$	18.3984	0.131140	0.0655699	−	0.997848i	$-0.479113\pi$
$3$	18.3984	0.131140	0.0655699	+	0.997848i	$0.479113\pi$
$4$	0	0
$5$	2458.39	1.75908	0.879540	−	0.475826i	$-0.157851\pi$
$5$	2458.39	1.75908	0.879540	+	0.475826i	$0.157851\pi$
$6$	0	0
$7$	−6435.39	−1.01306	−0.506529	−	0.862223i	$-0.669072\pi$
$7$	−6435.39	−1.01306	−0.506529	+	0.862223i	$0.669072\pi$
$8$	0	0
$9$	−19344.5	−0.982802
$10$	0	0
$11$	75667.5	1.55827	0.779134	−	0.626857i	$-0.215659\pi$
$11$	75667.5	1.55827	0.779134	+	0.626857i	$0.215659\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	45230.4	0.230685
$16$	0	0
$17$	−574152.	−1.66727	−0.833637	−	0.552313i	$-0.813745\pi$
$17$	−574152.	−1.66727	−0.833637	+	0.552313i	$0.813745\pi$
$18$	0	0
$19$	839418.	1.47770	0.738851	−	0.673868i	$-0.235369\pi$
$19$	839418.	1.47770	0.738851	+	0.673868i	$0.235369\pi$
$20$	0	0
$21$	−118401.	−0.132852
$22$	0	0
$23$	−2.18346e6	−1.62694	−0.813468	−	0.581610i	$-0.802423\pi$
$23$	−2.18346e6	−1.62694	−0.813468	+	0.581610i	$0.802423\pi$
$24$	0	0
$25$	4.09055e6	2.09436
$26$	0	0
$27$	−718044.	−0.260024
$28$	0	0
$29$	−4.54258e6	−1.19265	−0.596323	−	0.802745i	$-0.703372\pi$
$29$	−4.54258e6	−1.19265	−0.596323	+	0.802745i	$0.703372\pi$
$30$	0	0
$31$	3.24725e6	0.631521	0.315761	−	0.948839i	$-0.397740\pi$
$31$	3.24725e6	0.631521	0.315761	+	0.948839i	$0.397740\pi$
$32$	0	0
$33$	1.39216e6	0.204351
$34$	0	0
$35$	−1.58207e7	−1.78205
$36$	0	0
$37$	−1.47111e7	−1.29044	−0.645221	−	0.763996i	$-0.723235\pi$
$37$	−1.47111e7	−1.29044	−0.645221	+	0.763996i	$0.723235\pi$
$38$	0	0
$39$	−525477.	−0.0363716
$40$	0	0
$41$	−5.26815e6	−0.291159	−0.145580	−	0.989347i	$-0.546505\pi$
$41$	−5.26815e6	−0.291159	−0.145580	+	0.989347i	$0.546505\pi$
$42$	0	0
$43$	−1.06362e7	−0.474439	−0.237219	−	0.971456i	$-0.576236\pi$
$43$	−1.06362e7	−0.474439	−0.237219	+	0.971456i	$0.576236\pi$
$44$	0	0
$45$	−4.75563e7	−1.72883
$46$	0	0
$47$	−2.85092e6	−0.0852208	−0.0426104	−	0.999092i	$-0.513567\pi$
$47$	−2.85092e6	−0.0852208	−0.0426104	+	0.999092i	$0.513567\pi$
$48$	0	0
$49$	1.06069e6	0.0262849
$50$	0	0
$51$	−1.05635e7	−0.218646
$52$	0	0
$53$	−4.50292e7	−0.783885	−0.391943	−	0.919990i	$-0.628197\pi$
$53$	−4.50292e7	−0.783885	−0.391943	+	0.919990i	$0.628197\pi$
$54$	0	0
$55$	1.86020e8	2.74112
$56$	0	0
$57$	1.54440e7	0.193786
$58$	0	0
$59$	−2.49085e7	−0.267617	−0.133809	−	0.991007i	$-0.542721\pi$
$59$	−2.49085e7	−0.267617	−0.133809	+	0.991007i	$0.542721\pi$
$60$	0	0
$61$	1.72663e8	1.59667	0.798334	−	0.602214i	$-0.205715\pi$
$61$	1.72663e8	1.59667	0.798334	+	0.602214i	$0.205715\pi$
$62$	0	0
$63$	1.24489e8	0.995635
$64$	0	0
$65$	−7.02140e7	−0.487881
$66$	0	0
$67$	2.69226e8	1.63223	0.816115	−	0.577890i	$-0.196124\pi$
$67$	2.69226e8	1.63223	0.816115	+	0.577890i	$0.196124\pi$
$68$	0	0
$69$	−4.01722e7	−0.213356
$70$	0	0
$71$	−2.82052e8	−1.31724	−0.658622	−	0.752474i	$-0.728860\pi$
$71$	−2.82052e8	−1.31724	−0.658622	+	0.752474i	$0.728860\pi$
$72$	0	0
$73$	1.32314e8	0.545322	0.272661	−	0.962110i	$-0.412096\pi$
$73$	1.32314e8	0.545322	0.272661	+	0.962110i	$0.412096\pi$
$74$	0	0
$75$	7.52595e7	0.274654
$76$	0	0
$77$	−4.86950e8	−1.57861
$78$	0	0
$79$	2.75182e8	0.794872	0.397436	−	0.917630i	$-0.369900\pi$
$79$	2.75182e8	0.794872	0.397436	+	0.917630i	$0.369900\pi$
$80$	0	0
$81$	3.67547e8	0.948703
$82$	0	0
$83$	−4.46977e8	−1.03379	−0.516897	−	0.856048i	$-0.672913\pi$
$83$	−4.46977e8	−1.03379	−0.516897	+	0.856048i	$0.672913\pi$
$84$	0	0
$85$	−1.41149e9	−2.93287
$86$	0	0
$87$	−8.35762e7	−0.156403
$88$	0	0
$89$	−1.12425e8	−0.189937	−0.0949683	−	0.995480i	$-0.530275\pi$
$89$	−1.12425e8	−0.189937	−0.0949683	+	0.995480i	$0.530275\pi$
$90$	0	0
$91$	1.83801e8	0.280972
$92$	0	0
$93$	5.97442e7	0.0828176
$94$	0	0
$95$	2.06362e9	2.59940
$96$	0	0
$97$	−1.44020e9	−1.65177	−0.825885	−	0.563839i	$-0.809324\pi$
$97$	−1.44020e9	−1.65177	−0.825885	+	0.563839i	$0.809324\pi$
$98$	0	0
$99$	−1.46375e9	−1.53147
$100$	0	0
$101$	−1.02501e9	−0.980129	−0.490065	−	0.871686i	$-0.663027\pi$
$101$	−1.02501e9	−0.980129	−0.490065	+	0.871686i	$0.663027\pi$
$102$	0	0
$103$	−3.46579e8	−0.303413	−0.151707	−	0.988426i	$-0.548477\pi$
$103$	−3.46579e8	−0.303413	−0.151707	+	0.988426i	$0.548477\pi$
$104$	0	0
$105$	−2.91076e8	−0.233697
$106$	0	0
$107$	1.69911e8	0.125312	0.0626562	−	0.998035i	$-0.480043\pi$
$107$	1.69911e8	0.125312	0.0626562	+	0.998035i	$0.480043\pi$
$108$	0	0
$109$	−3.11815e8	−0.211582	−0.105791	−	0.994388i	$-0.533737\pi$
$109$	−3.11815e8	−0.211582	−0.105791	+	0.994388i	$0.533737\pi$
$110$	0	0
$111$	−2.70661e8	−0.169228
$112$	0	0
$113$	−1.57007e9	−0.905872	−0.452936	−	0.891543i	$-0.649623\pi$
$113$	−1.57007e9	−0.905872	−0.452936	+	0.891543i	$0.649623\pi$
$114$	0	0
$115$	−5.36780e9	−2.86191
$116$	0	0
$117$	5.52498e8	0.272580
$118$	0	0
$119$	3.69489e9	1.68904
$120$	0	0
$121$	3.36762e9	1.42820
$122$	0	0
$123$	−9.69255e7	−0.0381826
$124$	0	0
$125$	5.25461e9	1.92506
$126$	0	0
$127$	−7.73812e8	−0.263948	−0.131974	−	0.991253i	$-0.542132\pi$
$127$	−7.73812e8	−0.263948	−0.131974	+	0.991253i	$0.542132\pi$
$128$	0	0
$129$	−1.95690e8	−0.0622178
$130$	0	0
$131$	1.31015e9	0.388687	0.194343	−	0.980934i	$-0.437742\pi$
$131$	1.31015e9	0.388687	0.194343	+	0.980934i	$0.437742\pi$
$132$	0	0
$133$	−5.40199e9	−1.49700
$134$	0	0
$135$	−1.76523e9	−0.457403
$136$	0	0
$137$	−4.78493e9	−1.16047	−0.580234	−	0.814450i	$-0.697039\pi$
$137$	−4.78493e9	−1.16047	−0.580234	+	0.814450i	$0.697039\pi$
$138$	0	0
$139$	−4.08752e9	−0.928739	−0.464369	−	0.885642i	$-0.653719\pi$
$139$	−4.08752e9	−0.928739	−0.464369	+	0.885642i	$0.653719\pi$
$140$	0	0
$141$	−5.24525e7	−0.0111758
$142$	0	0
$143$	−2.16114e9	−0.432186
$144$	0	0
$145$	−1.11674e10	−2.09796
$146$	0	0
$147$	1.95150e7	0.00344700
$148$	0	0
$149$	1.60352e9	0.266525	0.133262	−	0.991081i	$-0.457455\pi$
$149$	1.60352e9	0.266525	0.133262	+	0.991081i	$0.457455\pi$
$150$	0	0
$151$	−7.79529e8	−0.122021	−0.0610107	−	0.998137i	$-0.519432\pi$
$151$	−7.79529e8	−0.122021	−0.0610107	+	0.998137i	$0.519432\pi$
$152$	0	0
$153$	1.11067e10	1.63860
$154$	0	0
$155$	7.98299e9	1.11090
$156$	0	0
$157$	7.27110e9	0.955107	0.477553	−	0.878603i	$-0.341524\pi$
$157$	7.27110e9	0.955107	0.477553	+	0.878603i	$0.341524\pi$
$158$	0	0
$159$	−8.28465e8	−0.102799
$160$	0	0
$161$	1.40514e10	1.64818
$162$	0	0
$163$	−1.56583e10	−1.73741	−0.868703	−	0.495333i	$-0.835046\pi$
$163$	−1.56583e10	−1.73741	−0.868703	+	0.495333i	$0.835046\pi$
$164$	0	0
$165$	3.42247e9	0.359470
$166$	0	0
$167$	−1.24905e10	−1.24267	−0.621333	−	0.783546i	$-0.713409\pi$
$167$	−1.24905e10	−1.24267	−0.621333	+	0.783546i	$0.713409\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−1.62381e10	−1.45229
$172$	0	0
$173$	−2.31060e10	−1.96118	−0.980588	−	0.196079i	$-0.937179\pi$
$173$	−2.31060e10	−1.96118	−0.980588	+	0.196079i	$0.937179\pi$
$174$	0	0
$175$	−2.63243e10	−2.12171
$176$	0	0
$177$	−4.58277e8	−0.0350953
$178$	0	0
$179$	−4.31517e9	−0.314166	−0.157083	−	0.987585i	$-0.550209\pi$
$179$	−4.31517e9	−0.314166	−0.157083	+	0.987585i	$0.550209\pi$
$180$	0	0
$181$	−1.46244e10	−1.01280	−0.506401	−	0.862298i	$-0.669024\pi$
$181$	−1.46244e10	−1.01280	−0.506401	+	0.862298i	$0.669024\pi$
$182$	0	0
$183$	3.17672e9	0.209387
$184$	0	0
$185$	−3.61657e10	−2.26999
$186$	0	0
$187$	−4.34446e10	−2.59806
$188$	0	0
$189$	4.62089e9	0.263420
$190$	0	0
$191$	1.75992e10	0.956845	0.478423	−	0.878130i	$-0.341209\pi$
$191$	1.75992e10	0.956845	0.478423	+	0.878130i	$0.341209\pi$
$192$	0	0
$193$	−1.38011e10	−0.715986	−0.357993	−	0.933724i	$-0.616539\pi$
$193$	−1.38011e10	−0.715986	−0.357993	+	0.933724i	$0.616539\pi$
$194$	0	0
$195$	−1.29183e9	−0.0639806
$196$	0	0
$197$	−5.16948e9	−0.244539	−0.122270	−	0.992497i	$-0.539017\pi$
$197$	−5.16948e9	−0.244539	−0.122270	+	0.992497i	$0.539017\pi$
$198$	0	0
$199$	−1.40212e10	−0.633790	−0.316895	−	0.948461i	$-0.602640\pi$
$199$	−1.40212e10	−0.633790	−0.316895	+	0.948461i	$0.602640\pi$
$200$	0	0
$201$	4.95334e9	0.214050
$202$	0	0
$203$	2.92333e10	1.20822
$204$	0	0
$205$	−1.29512e10	−0.512172
$206$	0	0
$207$	4.22380e10	1.59896
$208$	0	0
$209$	6.35167e10	2.30266
$210$	0	0
$211$	2.51083e10	0.872061	0.436030	−	0.899932i	$-0.356384\pi$
$211$	2.51083e10	0.872061	0.436030	+	0.899932i	$0.356384\pi$
$212$	0	0
$213$	−5.18930e9	−0.172743
$214$	0	0
$215$	−2.61480e10	−0.834576
$216$	0	0
$217$	−2.08973e10	−0.639767
$218$	0	0
$219$	2.43437e9	0.0715134
$220$	0	0
$221$	1.63984e10	0.462418
$222$	0	0
$223$	5.54861e10	1.50249	0.751246	−	0.660022i	$-0.229453\pi$
$223$	5.54861e10	1.50249	0.751246	+	0.660022i	$0.229453\pi$
$224$	0	0
$225$	−7.91295e10	−2.05834
$226$	0	0
$227$	2.94375e10	0.735841	0.367921	−	0.929857i	$-0.380070\pi$
$227$	2.94375e10	0.735841	0.367921	+	0.929857i	$0.380070\pi$
$228$	0	0
$229$	6.74856e10	1.62163	0.810814	−	0.585304i	$-0.199025\pi$
$229$	6.74856e10	1.62163	0.810814	+	0.585304i	$0.199025\pi$
$230$	0	0
$231$	−8.95911e9	−0.207019
$232$	0	0
$233$	−4.22644e9	−0.0939449	−0.0469725	−	0.998896i	$-0.514957\pi$
$233$	−4.22644e9	−0.0939449	−0.0469725	+	0.998896i	$0.514957\pi$
$234$	0	0
$235$	−7.00868e9	−0.149910
$236$	0	0
$237$	5.06290e9	0.104239
$238$	0	0
$239$	−4.94236e10	−0.979815	−0.489908	−	0.871774i	$-0.662969\pi$
$239$	−4.94236e10	−0.979815	−0.489908	+	0.871774i	$0.662969\pi$
$240$	0	0
$241$	4.07735e10	0.778577	0.389289	−	0.921116i	$-0.372721\pi$
$241$	4.07735e10	0.778577	0.389289	+	0.921116i	$0.372721\pi$
$242$	0	0
$243$	2.08955e10	0.384437
$244$	0	0
$245$	2.60759e9	0.0462373
$246$	0	0
$247$	−2.39746e10	−0.409841
$248$	0	0
$249$	−8.22367e9	−0.135572
$250$	0	0
$251$	9.94646e9	0.158175	0.0790873	−	0.996868i	$-0.474799\pi$
$251$	9.94646e9	0.158175	0.0790873	+	0.996868i	$0.474799\pi$
$252$	0	0
$253$	−1.65217e11	−2.53520
$254$	0	0
$255$	−2.59691e10	−0.384615
$256$	0	0
$257$	−1.00123e10	−0.143165	−0.0715824	−	0.997435i	$-0.522805\pi$
$257$	−1.00123e10	−0.143165	−0.0715824	+	0.997435i	$0.522805\pi$
$258$	0	0
$259$	9.46720e10	1.30729
$260$	0	0
$261$	8.78739e10	1.17213
$262$	0	0
$263$	−9.20736e10	−1.18668	−0.593341	−	0.804952i	$-0.702191\pi$
$263$	−9.20736e10	−1.18668	−0.593341	+	0.804952i	$0.702191\pi$
$264$	0	0
$265$	−1.10699e11	−1.37892
$266$	0	0
$267$	−2.06844e9	−0.0249082
$268$	0	0
$269$	3.08802e10	0.359580	0.179790	−	0.983705i	$-0.442458\pi$
$269$	3.08802e10	0.359580	0.179790	+	0.983705i	$0.442458\pi$
$270$	0	0
$271$	1.18521e11	1.33486	0.667429	−	0.744674i	$-0.267395\pi$
$271$	1.18521e11	1.33486	0.667429	+	0.744674i	$0.267395\pi$
$272$	0	0
$273$	3.38165e9	0.0368466
$274$	0	0
$275$	3.09521e11	3.26357
$276$	0	0
$277$	−1.63843e11	−1.67212	−0.836062	−	0.548635i	$-0.815148\pi$
$277$	−1.63843e11	−1.67212	−0.836062	+	0.548635i	$0.815148\pi$
$278$	0	0
$279$	−6.28164e10	−0.620660
$280$	0	0
$281$	−3.80342e10	−0.363911	−0.181956	−	0.983307i	$-0.558243\pi$
$281$	−3.80342e10	−0.363911	−0.181956	+	0.983307i	$0.558243\pi$
$282$	0	0
$283$	−7.53549e10	−0.698349	−0.349174	−	0.937058i	$-0.613538\pi$
$283$	−7.53549e10	−0.698349	−0.349174	+	0.937058i	$0.613538\pi$
$284$	0	0
$285$	3.79672e10	0.340884
$286$	0	0
$287$	3.39026e10	0.294961
$288$	0	0
$289$	2.11063e11	1.77980
$290$	0	0
$291$	−2.64974e10	−0.216613
$292$	0	0
$293$	−8.75188e10	−0.693740	−0.346870	−	0.937913i	$-0.612756\pi$
$293$	−8.75188e10	−0.693740	−0.346870	+	0.937913i	$0.612756\pi$
$294$	0	0
$295$	−6.12348e10	−0.470760
$296$	0	0
$297$	−5.43326e10	−0.405188
$298$	0	0
$299$	6.23618e10	0.451231
$300$	0	0
$301$	6.84484e10	0.480634
$302$	0	0
$303$	−1.88586e10	−0.128534
$304$	0	0
$305$	4.24472e11	2.80867
$306$	0	0
$307$	6.03418e10	0.387700	0.193850	−	0.981031i	$-0.437903\pi$
$307$	6.03418e10	0.387700	0.193850	+	0.981031i	$0.437903\pi$
$308$	0	0
$309$	−6.37650e9	−0.0397895
$310$	0	0
$311$	−2.10937e11	−1.27859	−0.639294	−	0.768963i	$-0.720773\pi$
$311$	−2.10937e11	−1.27859	−0.639294	+	0.768963i	$0.720773\pi$
$312$	0	0
$313$	3.16216e10	0.186223	0.0931117	−	0.995656i	$-0.470319\pi$
$313$	3.16216e10	0.186223	0.0931117	+	0.995656i	$0.470319\pi$
$314$	0	0
$315$	3.06043e11	1.75140
$316$	0	0
$317$	2.40969e11	1.34028	0.670139	−	0.742235i	$-0.266234\pi$
$317$	2.40969e11	1.34028	0.670139	+	0.742235i	$0.266234\pi$
$318$	0	0
$319$	−3.43725e11	−1.85846
$320$	0	0
$321$	3.12609e9	0.0164334
$322$	0	0
$323$	−4.81954e11	−2.46373
$324$	0	0
$325$	−1.16830e11	−0.580871
$326$	0	0
$327$	−5.73691e9	−0.0277468
$328$	0	0
$329$	1.83468e10	0.0863335
$330$	0	0
$331$	4.00948e11	1.83595	0.917977	−	0.396633i	$-0.129822\pi$
$331$	4.00948e11	1.83595	0.917977	+	0.396633i	$0.129822\pi$
$332$	0	0
$333$	2.84580e11	1.26825
$334$	0	0
$335$	6.61863e11	2.87122
$336$	0	0
$337$	1.55183e11	0.655403	0.327702	−	0.944781i	$-0.393726\pi$
$337$	1.55183e11	0.655403	0.327702	+	0.944781i	$0.393726\pi$
$338$	0	0
$339$	−2.88868e10	−0.118796
$340$	0	0
$341$	2.45711e11	0.984079
$342$	0	0
$343$	2.52865e11	0.986429
$344$	0	0
$345$	−9.87589e10	−0.375310
$346$	0	0
$347$	−6.98583e10	−0.258664	−0.129332	−	0.991601i	$-0.541283\pi$
$347$	−6.98583e10	−0.258664	−0.129332	+	0.991601i	$0.541283\pi$
$348$	0	0
$349$	−5.24407e11	−1.89214	−0.946072	−	0.323955i	$-0.894987\pi$
$349$	−5.24407e11	−1.89214	−0.946072	+	0.323955i	$0.894987\pi$
$350$	0	0
$351$	2.05080e10	0.0721178
$352$	0	0
$353$	2.63624e11	0.903646	0.451823	−	0.892108i	$-0.350774\pi$
$353$	2.63624e11	0.903646	0.451823	+	0.892108i	$0.350774\pi$
$354$	0	0
$355$	−6.93393e11	−2.31714
$356$	0	0
$357$	6.79802e10	0.221501
$358$	0	0
$359$	−4.18402e11	−1.32944	−0.664720	−	0.747093i	$-0.731449\pi$
$359$	−4.18402e11	−1.32944	−0.664720	+	0.747093i	$0.731449\pi$
$360$	0	0
$361$	3.81935e11	1.18361
$362$	0	0
$363$	6.19588e10	0.187294
$364$	0	0
$365$	3.25279e11	0.959265
$366$	0	0
$367$	3.13025e11	0.900703	0.450352	−	0.892851i	$-0.351299\pi$
$367$	3.13025e11	0.900703	0.450352	+	0.892851i	$0.351299\pi$
$368$	0	0
$369$	1.01910e11	0.286152
$370$	0	0
$371$	2.89781e11	0.794121
$372$	0	0
$373$	4.18923e11	1.12058	0.560292	−	0.828295i	$-0.310689\pi$
$373$	4.18923e11	1.12058	0.560292	+	0.828295i	$0.310689\pi$
$374$	0	0
$375$	9.66764e10	0.252453
$376$	0	0
$377$	1.29741e11	0.330780
$378$	0	0
$379$	−5.47158e11	−1.36219	−0.681093	−	0.732197i	$-0.738495\pi$
$379$	−5.47158e11	−1.36219	−0.681093	+	0.732197i	$0.738495\pi$
$380$	0	0
$381$	−1.42369e10	−0.0346141
$382$	0	0
$383$	−1.08005e11	−0.256477	−0.128239	−	0.991743i	$-0.540932\pi$
$383$	−1.08005e11	−0.256477	−0.128239	+	0.991743i	$0.540932\pi$
$384$	0	0
$385$	−1.19711e12	−2.77691
$386$	0	0
$387$	2.05753e11	0.466280
$388$	0	0
$389$	2.46852e10	0.0546592	0.0273296	−	0.999626i	$-0.491300\pi$
$389$	2.46852e10	0.0546592	0.0273296	+	0.999626i	$0.491300\pi$
$390$	0	0
$391$	1.25364e12	2.71255
$392$	0	0
$393$	2.41046e10	0.0509723
$394$	0	0
$395$	6.76503e11	1.39824
$396$	0	0
$397$	8.16247e10	0.164917	0.0824583	−	0.996595i	$-0.473723\pi$
$397$	8.16247e10	0.164917	0.0824583	+	0.996595i	$0.473723\pi$
$398$	0	0
$399$	−9.93879e10	−0.196316
$400$	0	0
$401$	4.97530e11	0.960881	0.480441	−	0.877027i	$-0.340477\pi$
$401$	4.97530e11	0.960881	0.480441	+	0.877027i	$0.340477\pi$
$402$	0	0
$403$	−9.27447e10	−0.175152
$404$	0	0
$405$	9.03573e11	1.66884
$406$	0	0
$407$	−1.11315e12	−2.01085
$408$	0	0
$409$	−7.11698e11	−1.25760	−0.628798	−	0.777569i	$-0.716453\pi$
$409$	−7.11698e11	−1.25760	−0.628798	+	0.777569i	$0.716453\pi$
$410$	0	0
$411$	−8.80351e10	−0.152184
$412$	0	0
$413$	1.60296e11	0.271112
$414$	0	0
$415$	−1.09884e12	−1.81853
$416$	0	0
$417$	−7.52039e10	−0.121795
$418$	0	0
$419$	−2.51824e11	−0.399148	−0.199574	−	0.979883i	$-0.563956\pi$
$419$	−2.51824e11	−0.399148	−0.199574	+	0.979883i	$0.563956\pi$
$420$	0	0
$421$	7.79431e9	0.0120923	0.00604614	−	0.999982i	$-0.498075\pi$
$421$	7.79431e9	0.0120923	0.00604614	+	0.999982i	$0.498075\pi$
$422$	0	0
$423$	5.51497e10	0.0837552
$424$	0	0
$425$	−2.34859e12	−3.49187
$426$	0	0
$427$	−1.11115e12	−1.61752
$428$	0	0
$429$	−3.97615e10	−0.0566768
$430$	0	0
$431$	7.28020e11	1.01624	0.508119	−	0.861287i	$-0.330341\pi$
$431$	7.28020e11	1.01624	0.508119	+	0.861287i	$0.330341\pi$
$432$	0	0
$433$	3.00825e11	0.411262	0.205631	−	0.978630i	$-0.434075\pi$
$433$	3.00825e11	0.411262	0.205631	+	0.978630i	$0.434075\pi$
$434$	0	0
$435$	−2.05463e11	−0.275126
$436$	0	0
$437$	−1.83284e12	−2.40413
$438$	0	0
$439$	2.09332e11	0.268996	0.134498	−	0.990914i	$-0.457058\pi$
$439$	2.09332e11	0.268996	0.134498	+	0.990914i	$0.457058\pi$
$440$	0	0
$441$	−2.05186e10	−0.0258329
$442$	0	0
$443$	1.18518e12	1.46207	0.731036	−	0.682339i	$-0.239037\pi$
$443$	1.18518e12	1.46207	0.731036	+	0.682339i	$0.239037\pi$
$444$	0	0
$445$	−2.76385e11	−0.334113
$446$	0	0
$447$	2.95023e10	0.0349520
$448$	0	0
$449$	−4.69649e11	−0.545336	−0.272668	−	0.962108i	$-0.587906\pi$
$449$	−4.69649e11	−0.545336	−0.272668	+	0.962108i	$0.587906\pi$
$450$	0	0
$451$	−3.98628e11	−0.453704
$452$	0	0
$453$	−1.43421e10	−0.0160019
$454$	0	0
$455$	4.51855e11	0.494251
$456$	0	0
$457$	1.47311e12	1.57984	0.789919	−	0.613212i	$-0.210123\pi$
$457$	1.47311e12	1.57984	0.789919	+	0.613212i	$0.210123\pi$
$458$	0	0
$459$	4.12266e11	0.433532
$460$	0	0
$461$	6.46731e10	0.0666913	0.0333457	−	0.999444i	$-0.489384\pi$
$461$	6.46731e10	0.0666913	0.0333457	+	0.999444i	$0.489384\pi$
$462$	0	0
$463$	1.89611e11	0.191756	0.0958780	−	0.995393i	$-0.469434\pi$
$463$	1.89611e11	0.191756	0.0958780	+	0.995393i	$0.469434\pi$
$464$	0	0
$465$	1.46874e11	0.145683
$466$	0	0
$467$	1.27878e11	0.124414	0.0622069	−	0.998063i	$-0.480186\pi$
$467$	1.27878e11	0.124414	0.0622069	+	0.998063i	$0.480186\pi$
$468$	0	0
$469$	−1.73258e12	−1.65354
$470$	0	0
$471$	1.33777e11	0.125252
$472$	0	0
$473$	−8.04818e11	−0.739303
$474$	0	0
$475$	3.43368e12	3.09484
$476$	0	0
$477$	8.71067e11	0.770404
$478$	0	0
$479$	2.46378e10	0.0213842	0.0106921	−	0.999943i	$-0.496597\pi$
$479$	2.46378e10	0.0213842	0.0106921	+	0.999943i	$0.496597\pi$
$480$	0	0
$481$	4.20165e11	0.357904
$482$	0	0
$483$	2.58524e11	0.216142
$484$	0	0
$485$	−3.54057e12	−2.90559
$486$	0	0
$487$	−7.93791e11	−0.639479	−0.319739	−	0.947506i	$-0.603595\pi$
$487$	−7.93791e11	−0.639479	−0.319739	+	0.947506i	$0.603595\pi$
$488$	0	0
$489$	−2.88088e11	−0.227843
$490$	0	0
$491$	−9.39198e11	−0.729274	−0.364637	−	0.931150i	$-0.618807\pi$
$491$	−9.39198e11	−0.729274	−0.364637	+	0.931150i	$0.618807\pi$
$492$	0	0
$493$	2.60813e12	1.98847
$494$	0	0
$495$	−3.59846e12	−2.69398
$496$	0	0
$497$	1.81511e12	1.33444
$498$	0	0
$499$	−5.14451e11	−0.371443	−0.185721	−	0.982602i	$-0.559462\pi$
$499$	−5.14451e11	−0.371443	−0.185721	+	0.982602i	$0.559462\pi$
$500$	0	0
$501$	−2.29805e11	−0.162963
$502$	0	0
$503$	−7.42503e11	−0.517181	−0.258590	−	0.965987i	$-0.583258\pi$
$503$	−7.42503e11	−0.517181	−0.258590	+	0.965987i	$0.583258\pi$
$504$	0	0
$505$	−2.51988e12	−1.72412
$506$	0	0
$507$	1.50081e10	0.0100877
$508$	0	0
$509$	7.70649e11	0.508893	0.254447	−	0.967087i	$-0.418107\pi$
$509$	7.70649e11	0.508893	0.254447	+	0.967087i	$0.418107\pi$
$510$	0	0
$511$	−8.51493e11	−0.552443
$512$	0	0
$513$	−6.02739e11	−0.384239
$514$	0	0
$515$	−8.52025e11	−0.533728
$516$	0	0
$517$	−2.15722e11	−0.132797
$518$	0	0
$519$	−4.25113e11	−0.257188
$520$	0	0
$521$	1.46940e12	0.873715	0.436858	−	0.899531i	$-0.356091\pi$
$521$	1.46940e12	0.873715	0.436858	+	0.899531i	$0.356091\pi$
$522$	0	0
$523$	2.94075e12	1.71870	0.859352	−	0.511385i	$-0.170867\pi$
$523$	2.94075e12	1.71870	0.859352	+	0.511385i	$0.170867\pi$
$524$	0	0
$525$	−4.84325e11	−0.278240
$526$	0	0
$527$	−1.86441e12	−1.05292
$528$	0	0
$529$	2.96635e12	1.64692
$530$	0	0
$531$	4.81843e11	0.263015
$532$	0	0
$533$	1.50464e11	0.0807531
$534$	0	0
$535$	4.17706e11	0.220434
$536$	0	0
$537$	−7.93922e10	−0.0411997
$538$	0	0
$539$	8.02599e10	0.0409590
$540$	0	0
$541$	6.51159e11	0.326813	0.163406	−	0.986559i	$-0.447752\pi$
$541$	6.51159e11	0.326813	0.163406	+	0.986559i	$0.447752\pi$
$542$	0	0
$543$	−2.69066e11	−0.132819
$544$	0	0
$545$	−7.66563e11	−0.372189
$546$	0	0
$547$	1.65143e12	0.788711	0.394355	−	0.918958i	$-0.370968\pi$
$547$	1.65143e12	0.788711	0.394355	+	0.918958i	$0.370968\pi$
$548$	0	0
$549$	−3.34008e12	−1.56921
$550$	0	0
$551$	−3.81312e12	−1.76238
$552$	0	0
$553$	−1.77090e12	−0.805251
$554$	0	0
$555$	−6.65391e11	−0.297686
$556$	0	0
$557$	−1.10492e12	−0.486387	−0.243194	−	0.969978i	$-0.578195\pi$
$557$	−1.10492e12	−0.486387	−0.243194	+	0.969978i	$0.578195\pi$
$558$	0	0
$559$	3.03782e11	0.131586
$560$	0	0
$561$	−7.99312e11	−0.340709
$562$	0	0
$563$	3.85170e11	0.161572	0.0807858	−	0.996731i	$-0.474257\pi$
$563$	3.85170e11	0.161572	0.0807858	+	0.996731i	$0.474257\pi$
$564$	0	0
$565$	−3.85985e12	−1.59350
$566$	0	0
$567$	−2.36531e12	−0.961090
$568$	0	0
$569$	5.78791e11	0.231482	0.115741	−	0.993279i	$-0.463076\pi$
$569$	5.78791e11	0.231482	0.115741	+	0.993279i	$0.463076\pi$
$570$	0	0
$571$	2.70011e12	1.06297	0.531483	−	0.847069i	$-0.321635\pi$
$571$	2.70011e12	1.06297	0.531483	+	0.847069i	$0.321635\pi$
$572$	0	0
$573$	3.23796e11	0.125481
$574$	0	0
$575$	−8.93155e12	−3.40739
$576$	0	0
$577$	−1.44536e12	−0.542856	−0.271428	−	0.962459i	$-0.587496\pi$
$577$	−1.44536e12	−0.542856	−0.271428	+	0.962459i	$0.587496\pi$
$578$	0	0
$579$	−2.53918e11	−0.0938943
$580$	0	0
$581$	2.87648e12	1.04729
$582$	0	0
$583$	−3.40724e12	−1.22150
$584$	0	0
$585$	1.35825e12	0.479490
$586$	0	0
$587$	1.07907e12	0.375125	0.187563	−	0.982253i	$-0.439941\pi$
$587$	1.07907e12	0.375125	0.187563	+	0.982253i	$0.439941\pi$
$588$	0	0
$589$	2.72580e12	0.933200
$590$	0	0
$591$	−9.51101e10	−0.0320688
$592$	0	0
$593$	1.03467e12	0.343604	0.171802	−	0.985132i	$-0.445041\pi$
$593$	1.03467e12	0.343604	0.171802	+	0.985132i	$0.445041\pi$
$594$	0	0
$595$	9.08348e12	2.97116
$596$	0	0
$597$	−2.57967e11	−0.0831151
$598$	0	0
$599$	−2.51684e12	−0.798793	−0.399396	−	0.916778i	$-0.630780\pi$
$599$	−2.51684e12	−0.798793	−0.399396	+	0.916778i	$0.630780\pi$
$600$	0	0
$601$	1.01442e12	0.317164	0.158582	−	0.987346i	$-0.449308\pi$
$601$	1.01442e12	0.317164	0.158582	+	0.987346i	$0.449308\pi$
$602$	0	0
$603$	−5.20805e12	−1.60416
$604$	0	0
$605$	8.27892e12	2.51232
$606$	0	0
$607$	1.84602e12	0.551935	0.275967	−	0.961167i	$-0.411002\pi$
$607$	1.84602e12	0.551935	0.275967	+	0.961167i	$0.411002\pi$
$608$	0	0
$609$	5.37846e11	0.158445
$610$	0	0
$611$	8.14253e10	0.0236360
$612$	0	0
$613$	−5.10934e12	−1.46148	−0.730740	−	0.682656i	$-0.760825\pi$
$613$	−5.10934e12	−1.46148	−0.730740	+	0.682656i	$0.760825\pi$
$614$	0	0
$615$	−2.38281e11	−0.0671662
$616$	0	0
$617$	4.21621e12	1.17122	0.585611	−	0.810593i	$-0.300855\pi$
$617$	4.21621e12	1.17122	0.585611	+	0.810593i	$0.300855\pi$
$618$	0	0
$619$	7.24019e11	0.198218	0.0991088	−	0.995077i	$-0.468401\pi$
$619$	7.24019e11	0.198218	0.0991088	+	0.995077i	$0.468401\pi$
$620$	0	0
$621$	1.56782e12	0.423043
$622$	0	0
$623$	7.23500e11	0.192417
$624$	0	0
$625$	4.92852e12	1.29198
$626$	0	0
$627$	1.16861e12	0.301970
$628$	0	0
$629$	8.44643e12	2.15152
$630$	0	0
$631$	2.90019e11	0.0728273	0.0364136	−	0.999337i	$-0.488407\pi$
$631$	2.90019e11	0.0728273	0.0364136	+	0.999337i	$0.488407\pi$
$632$	0	0
$633$	4.61953e11	0.114362
$634$	0	0
$635$	−1.90233e12	−0.464306
$636$	0	0
$637$	−3.02944e10	−0.00729013
$638$	0	0
$639$	5.45615e12	1.29459
$640$	0	0
$641$	−5.41213e11	−0.126621	−0.0633107	−	0.997994i	$-0.520166\pi$
$641$	−5.41213e11	−0.126621	−0.0633107	+	0.997994i	$0.520166\pi$
$642$	0	0
$643$	−1.10572e12	−0.255091	−0.127546	−	0.991833i	$-0.540710\pi$
$643$	−1.10572e12	−0.255091	−0.127546	+	0.991833i	$0.540710\pi$
$644$	0	0
$645$	−4.81082e11	−0.109446
$646$	0	0
$647$	3.04797e12	0.683820	0.341910	−	0.939733i	$-0.388926\pi$
$647$	3.04797e12	0.683820	0.341910	+	0.939733i	$0.388926\pi$
$648$	0	0
$649$	−1.88477e12	−0.417019
$650$	0	0
$651$	−3.84477e11	−0.0838989
$652$	0	0
$653$	5.03816e11	0.108433	0.0542166	−	0.998529i	$-0.482734\pi$
$653$	5.03816e11	0.108433	0.0542166	+	0.998529i	$0.482734\pi$
$654$	0	0
$655$	3.22085e12	0.683730
$656$	0	0
$657$	−2.55955e12	−0.535944
$658$	0	0
$659$	7.03410e12	1.45286	0.726431	−	0.687240i	$-0.241178\pi$
$659$	7.03410e12	1.45286	0.726431	+	0.687240i	$0.241178\pi$
$660$	0	0
$661$	1.37076e12	0.279290	0.139645	−	0.990202i	$-0.455404\pi$
$661$	1.37076e12	0.279290	0.139645	+	0.990202i	$0.455404\pi$
$662$	0	0
$663$	3.01704e11	0.0606415
$664$	0	0
$665$	−1.32802e13	−2.63334
$666$	0	0
$667$	9.91854e12	1.94036
$668$	0	0
$669$	1.02086e12	0.197037
$670$	0	0
$671$	1.30650e13	2.48804
$672$	0	0
$673$	−1.63464e12	−0.307153	−0.153576	−	0.988137i	$-0.549079\pi$
$673$	−1.63464e12	−0.307153	−0.153576	+	0.988137i	$0.549079\pi$
$674$	0	0
$675$	−2.93719e12	−0.544584
$676$	0	0
$677$	8.41927e12	1.54037	0.770186	−	0.637820i	$-0.220163\pi$
$677$	8.41927e12	1.54037	0.770186	+	0.637820i	$0.220163\pi$
$678$	0	0
$679$	9.26825e12	1.67334
$680$	0	0
$681$	5.41603e11	0.0964981
$682$	0	0
$683$	−4.06625e12	−0.714992	−0.357496	−	0.933915i	$-0.616369\pi$
$683$	−4.06625e12	−0.714992	−0.357496	+	0.933915i	$0.616369\pi$
$684$	0	0
$685$	−1.17632e13	−2.04135
$686$	0	0
$687$	1.24163e12	0.212660
$688$	0	0
$689$	1.28608e12	0.217411
$690$	0	0
$691$	−4.31792e11	−0.0720482	−0.0360241	−	0.999351i	$-0.511469\pi$
$691$	−4.31792e11	−0.0720482	−0.0360241	+	0.999351i	$0.511469\pi$
$692$	0	0
$693$	9.41981e12	1.55147
$694$	0	0
$695$	−1.00487e13	−1.63372
$696$	0	0
$697$	3.02472e12	0.485442
$698$	0	0
$699$	−7.77598e10	−0.0123199
$700$	0	0
$701$	−3.07495e12	−0.480957	−0.240479	−	0.970654i	$-0.577304\pi$
$701$	−3.07495e12	−0.480957	−0.240479	+	0.970654i	$0.577304\pi$
$702$	0	0
$703$	−1.23488e13	−1.90689
$704$	0	0
$705$	−1.28949e11	−0.0196592
$706$	0	0
$707$	6.59636e12	0.992927
$708$	0	0
$709$	−6.01531e12	−0.894026	−0.447013	−	0.894528i	$-0.647512\pi$
$709$	−6.01531e12	−0.894026	−0.447013	+	0.894528i	$0.647512\pi$
$710$	0	0
$711$	−5.32325e12	−0.781202
$712$	0	0
$713$	−7.09024e12	−1.02744
$714$	0	0
$715$	−5.31292e12	−0.760249
$716$	0	0
$717$	−9.09316e11	−0.128493
$718$	0	0
$719$	−8.51552e12	−1.18831	−0.594157	−	0.804349i	$-0.702514\pi$
$719$	−8.51552e12	−1.18831	−0.594157	+	0.804349i	$0.702514\pi$
$720$	0	0
$721$	2.23037e12	0.307375
$722$	0	0
$723$	7.50168e11	0.102102
$724$	0	0
$725$	−1.85816e13	−2.49783
$726$	0	0
$727$	3.46201e12	0.459646	0.229823	−	0.973232i	$-0.426185\pi$
$727$	3.46201e12	0.459646	0.229823	+	0.973232i	$0.426185\pi$
$728$	0	0
$729$	−6.84998e12	−0.898288
$730$	0	0
$731$	6.10682e12	0.791019
$732$	0	0
$733$	7.11267e12	0.910049	0.455025	−	0.890479i	$-0.349630\pi$
$733$	7.11267e12	0.910049	0.455025	+	0.890479i	$0.349630\pi$
$734$	0	0
$735$	4.79755e10	0.00606355
$736$	0	0
$737$	2.03717e13	2.54345
$738$	0	0
$739$	1.31328e13	1.61978	0.809891	−	0.586581i	$-0.199526\pi$
$739$	1.31328e13	1.61978	0.809891	+	0.586581i	$0.199526\pi$
$740$	0	0
$741$	−4.41095e11	−0.0537465
$742$	0	0
$743$	−1.28936e12	−0.155212	−0.0776058	−	0.996984i	$-0.524728\pi$
$743$	−1.28936e12	−0.155212	−0.0776058	+	0.996984i	$0.524728\pi$
$744$	0	0
$745$	3.94208e12	0.468838
$746$	0	0
$747$	8.64655e12	1.01602
$748$	0	0
$749$	−1.09344e12	−0.126949
$750$	0	0
$751$	−2.31793e12	−0.265901	−0.132951	−	0.991123i	$-0.542445\pi$
$751$	−2.31793e12	−0.265901	−0.132951	+	0.991123i	$0.542445\pi$
$752$	0	0
$753$	1.82999e11	0.0207430
$754$	0	0
$755$	−1.91638e12	−0.214645
$756$	0	0
$757$	−6.40292e12	−0.708674	−0.354337	−	0.935118i	$-0.615293\pi$
$757$	−6.40292e12	−0.708674	−0.354337	+	0.935118i	$0.615293\pi$
$758$	0	0
$759$	−3.03973e12	−0.332466
$760$	0	0
$761$	−5.79503e12	−0.626362	−0.313181	−	0.949694i	$-0.601395\pi$
$761$	−5.79503e12	−0.626362	−0.313181	+	0.949694i	$0.601395\pi$
$762$	0	0
$763$	2.00666e12	0.214345
$764$	0	0
$765$	2.73045e13	2.88243
$766$	0	0
$767$	7.11413e11	0.0742237
$768$	0	0
$769$	1.83778e12	0.189506	0.0947532	−	0.995501i	$-0.469794\pi$
$769$	1.83778e12	0.189506	0.0947532	+	0.995501i	$0.469794\pi$
$770$	0	0
$771$	−1.84211e11	−0.0187746
$772$	0	0
$773$	−1.45836e13	−1.46912	−0.734560	−	0.678543i	$-0.762612\pi$
$773$	−1.45836e13	−1.46912	−0.734560	+	0.678543i	$0.762612\pi$
$774$	0	0
$775$	1.32830e13	1.32263
$776$	0	0
$777$	1.74181e12	0.171438
$778$	0	0
$779$	−4.42218e12	−0.430247
$780$	0	0
$781$	−2.13421e13	−2.05262
$782$	0	0
$783$	3.26177e12	0.310117
$784$	0	0
$785$	1.78752e13	1.68011
$786$	0	0
$787$	1.05513e13	0.980441	0.490220	−	0.871599i	$-0.336916\pi$
$787$	1.05513e13	0.980441	0.490220	+	0.871599i	$0.336916\pi$
$788$	0	0
$789$	−1.69401e12	−0.155621
$790$	0	0
$791$	1.01040e13	0.917700
$792$	0	0
$793$	−4.93143e12	−0.442836
$794$	0	0
$795$	−2.03669e12	−0.180831
$796$	0	0
$797$	−1.55803e13	−1.36778	−0.683888	−	0.729587i	$-0.739712\pi$
$797$	−1.55803e13	−1.36778	−0.683888	+	0.729587i	$0.739712\pi$
$798$	0	0
$799$	1.63686e12	0.142086
$800$	0	0
$801$	2.17481e12	0.186670
$802$	0	0
$803$	1.00119e13	0.849758
$804$	0	0
$805$	3.45439e13	2.89928
$806$	0	0
$807$	5.68147e11	0.0471552
$808$	0	0
$809$	1.89112e13	1.55221	0.776106	−	0.630602i	$-0.217192\pi$
$809$	1.89112e13	1.55221	0.776106	+	0.630602i	$0.217192\pi$
$810$	0	0
$811$	−5.34506e12	−0.433869	−0.216934	−	0.976186i	$-0.569606\pi$
$811$	−5.34506e12	−0.433869	−0.216934	+	0.976186i	$0.569606\pi$
$812$	0	0
$813$	2.18061e12	0.175053
$814$	0	0
$815$	−3.84943e13	−3.05624
$816$	0	0
$817$	−8.92826e12	−0.701080
$818$	0	0
$819$	−3.55554e12	−0.276139
$820$	0	0
$821$	8.66499e12	0.665616	0.332808	−	0.942995i	$-0.392004\pi$
$821$	8.66499e12	0.665616	0.332808	+	0.942995i	$0.392004\pi$
$822$	0	0
$823$	2.49997e13	1.89948	0.949741	−	0.313038i	$-0.101347\pi$
$823$	2.49997e13	1.89948	0.949741	+	0.313038i	$0.101347\pi$
$824$	0	0
$825$	5.69470e12	0.427984
$826$	0	0
$827$	1.12047e13	0.832960	0.416480	−	0.909145i	$-0.363264\pi$
$827$	1.12047e13	0.832960	0.416480	+	0.909145i	$0.363264\pi$
$828$	0	0
$829$	−1.40565e13	−1.03367	−0.516835	−	0.856085i	$-0.672890\pi$
$829$	−1.40565e13	−1.03367	−0.516835	+	0.856085i	$0.672890\pi$
$830$	0	0
$831$	−3.01445e12	−0.219282
$832$	0	0
$833$	−6.08998e11	−0.0438242
$834$	0	0
$835$	−3.07064e13	−2.18595
$836$	0	0
$837$	−2.33167e12	−0.164211
$838$	0	0
$839$	8.22132e12	0.572813	0.286406	−	0.958108i	$-0.407539\pi$
$839$	8.22132e12	0.572813	0.286406	+	0.958108i	$0.407539\pi$
$840$	0	0
$841$	6.12786e12	0.422403
$842$	0	0
$843$	−6.99768e11	−0.0477233
$844$	0	0
$845$	2.00538e12	0.135314
$846$	0	0
$847$	−2.16720e13	−1.44685
$848$	0	0
$849$	−1.38641e12	−0.0915813
$850$	0	0
$851$	3.21212e13	2.09947
$852$	0	0
$853$	−8.63701e12	−0.558589	−0.279295	−	0.960205i	$-0.590101\pi$
$853$	−8.63701e12	−0.558589	−0.279295	+	0.960205i	$0.590101\pi$
$854$	0	0
$855$	−3.99196e13	−2.55469
$856$	0	0
$857$	5.69915e11	0.0360908	0.0180454	−	0.999837i	$-0.494256\pi$
$857$	5.69915e11	0.0360908	0.0180454	+	0.999837i	$0.494256\pi$
$858$	0	0
$859$	−2.40033e13	−1.50419	−0.752094	−	0.659056i	$-0.770956\pi$
$859$	−2.40033e13	−1.50419	−0.752094	+	0.659056i	$0.770956\pi$
$860$	0	0
$861$	6.23754e11	0.0386811
$862$	0	0
$863$	−1.23684e13	−0.759041	−0.379521	−	0.925183i	$-0.623911\pi$
$863$	−1.23684e13	−0.759041	−0.379521	+	0.925183i	$0.623911\pi$
$864$	0	0
$865$	−5.68034e13	−3.44986
$866$	0	0
$867$	3.88322e12	0.233403
$868$	0	0
$869$	2.08223e13	1.23862
$870$	0	0
$871$	−7.68938e12	−0.452699
$872$	0	0
$873$	2.78599e13	1.62336
$874$	0	0
$875$	−3.38155e13	−1.95020
$876$	0	0
$877$	2.72862e13	1.55756	0.778780	−	0.627298i	$-0.215839\pi$
$877$	2.72862e13	1.55756	0.778780	+	0.627298i	$0.215839\pi$
$878$	0	0
$879$	−1.61021e12	−0.0909770
$880$	0	0
$881$	−2.76782e13	−1.54791	−0.773955	−	0.633241i	$-0.781724\pi$
$881$	−2.76782e13	−1.54791	−0.773955	+	0.633241i	$0.781724\pi$
$882$	0	0
$883$	−1.80925e13	−1.00156	−0.500778	−	0.865576i	$-0.666953\pi$
$883$	−1.80925e13	−1.00156	−0.500778	+	0.865576i	$0.666953\pi$
$884$	0	0
$885$	−1.12662e12	−0.0617354
$886$	0	0
$887$	−2.13275e13	−1.15687	−0.578433	−	0.815730i	$-0.696335\pi$
$887$	−2.13275e13	−1.15687	−0.578433	+	0.815730i	$0.696335\pi$
$888$	0	0
$889$	4.97979e12	0.267395
$890$	0	0
$891$	2.78113e13	1.47833
$892$	0	0
$893$	−2.39312e12	−0.125931
$894$	0	0
$895$	−1.06084e13	−0.552643
$896$	0	0
$897$	1.14736e12	0.0591743
$898$	0	0
$899$	−1.47509e13	−0.753181
$900$	0	0
$901$	2.58536e13	1.30695
$902$	0	0
$903$	1.25934e12	0.0630302
$904$	0	0
$905$	−3.59524e13	−1.78160
$906$	0	0
$907$	2.18772e13	1.07339	0.536696	−	0.843776i	$-0.319672\pi$
$907$	2.18772e13	1.07339	0.536696	+	0.843776i	$0.319672\pi$
$908$	0	0
$909$	1.98284e13	0.963273
$910$	0	0
$911$	7.21991e12	0.347295	0.173648	−	0.984808i	$-0.444445\pi$
$911$	7.21991e12	0.347295	0.173648	+	0.984808i	$0.444445\pi$
$912$	0	0
$913$	−3.38216e13	−1.61093
$914$	0	0
$915$	7.80962e12	0.368328
$916$	0	0
$917$	−8.43132e12	−0.393762
$918$	0	0
$919$	1.68962e13	0.781394	0.390697	−	0.920519i	$-0.372234\pi$
$919$	1.68962e13	0.781394	0.390697	+	0.920519i	$0.372234\pi$
$920$	0	0
$921$	1.11019e12	0.0508429
$922$	0	0
$923$	8.05568e12	0.365338
$924$	0	0
$925$	−6.01766e13	−2.70265
$926$	0	0
$927$	6.70439e12	0.298195
$928$	0	0
$929$	2.94634e13	1.29781	0.648906	−	0.760869i	$-0.275227\pi$
$929$	2.94634e13	1.29781	0.648906	+	0.760869i	$0.275227\pi$
$930$	0	0
$931$	8.90364e11	0.0388413
$932$	0	0
$933$	−3.88090e12	−0.167674
$934$	0	0
$935$	−1.06804e14	−4.57019
$936$	0	0
$937$	3.67745e12	0.155854	0.0779271	−	0.996959i	$-0.475170\pi$
$937$	3.67745e12	0.155854	0.0779271	+	0.996959i	$0.475170\pi$
$938$	0	0
$939$	5.81787e11	0.0244213
$940$	0	0
$941$	−4.72700e13	−1.96532	−0.982659	−	0.185424i	$-0.940634\pi$
$941$	−4.72700e13	−1.96532	−0.982659	+	0.185424i	$0.940634\pi$
$942$	0	0
$943$	1.15028e13	0.473697
$944$	0	0
$945$	1.13600e13	0.463376
$946$	0	0
$947$	2.25228e13	0.910014	0.455007	−	0.890488i	$-0.349637\pi$
$947$	2.25228e13	0.910014	0.455007	+	0.890488i	$0.349637\pi$
$948$	0	0
$949$	−3.77902e12	−0.151245
$950$	0	0
$951$	4.43345e12	0.175764
$952$	0	0
$953$	3.02491e13	1.18794	0.593970	−	0.804487i	$-0.297560\pi$
$953$	3.02491e13	1.18794	0.593970	+	0.804487i	$0.297560\pi$
$954$	0	0
$955$	4.32656e13	1.68317
$956$	0	0
$957$	−6.32400e12	−0.243718
$958$	0	0
$959$	3.07929e13	1.17562
$960$	0	0
$961$	−1.58950e13	−0.601181
$962$	0	0
$963$	−3.28684e12	−0.123157
$964$	0	0
$965$	−3.39284e13	−1.25948
$966$	0	0
$967$	−2.87362e13	−1.05684	−0.528421	−	0.848983i	$-0.677216\pi$
$967$	−2.87362e13	−1.05684	−0.528421	+	0.848983i	$0.677216\pi$
$968$	0	0
$969$	−8.86718e12	−0.323094
$970$	0	0
$971$	−4.90160e13	−1.76950	−0.884751	−	0.466064i	$-0.845672\pi$
$971$	−4.90160e13	−1.76950	−0.884751	+	0.466064i	$0.845672\pi$
$972$	0	0
$973$	2.63048e13	0.940865
$974$	0	0
$975$	−2.14949e12	−0.0761753
$976$	0	0
$977$	1.70416e13	0.598392	0.299196	−	0.954192i	$-0.403282\pi$
$977$	1.70416e13	0.598392	0.299196	+	0.954192i	$0.403282\pi$
$978$	0	0
$979$	−8.50693e12	−0.295972
$980$	0	0
$981$	6.03191e12	0.207943
$982$	0	0
$983$	4.37056e13	1.49295	0.746476	−	0.665412i	$-0.231744\pi$
$983$	4.37056e13	1.49295	0.746476	+	0.665412i	$0.231744\pi$
$984$	0	0
$985$	−1.27086e13	−0.430164
$986$	0	0
$987$	3.37552e11	0.0113218
$988$	0	0
$989$	2.32238e13	0.771881
$990$	0	0
$991$	2.74983e12	0.0905680	0.0452840	−	0.998974i	$-0.485581\pi$
$991$	2.74983e12	0.0905680	0.0452840	+	0.998974i	$0.485581\pi$
$992$	0	0
$993$	7.37680e12	0.240767
$994$	0	0
$995$	−3.44695e13	−1.11489
$996$	0	0
$997$	−1.61703e13	−0.518310	−0.259155	−	0.965836i	$-0.583444\pi$
$997$	−1.61703e13	−0.518310	−0.259155	+	0.965836i	$0.583444\pi$
$998$	0	0
$999$	1.05632e13	0.335546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.j.1.4		6
4.3	odd	2		104.10.a.b.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.b.1.3	✓	6	4.3	odd	2
208.10.a.j.1.4		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)

\(f(q)\)	\(=\)	\(q+18.3984 q^{3} +2458.39 q^{5} -6435.39 q^{7} -19344.5 q^{9} +75667.5 q^{11} -28561.0 q^{13} +45230.4 q^{15} -574152. q^{17} +839418. q^{19} -118401. q^{21} -2.18346e6 q^{23} +4.09055e6 q^{25} -718044. q^{27} -4.54258e6 q^{29} +3.24725e6 q^{31} +1.39216e6 q^{33} -1.58207e7 q^{35} -1.47111e7 q^{37} -525477. q^{39} -5.26815e6 q^{41} -1.06362e7 q^{43} -4.75563e7 q^{45} -2.85092e6 q^{47} +1.06069e6 q^{49} -1.05635e7 q^{51} -4.50292e7 q^{53} +1.86020e8 q^{55} +1.54440e7 q^{57} -2.49085e7 q^{59} +1.72663e8 q^{61} +1.24489e8 q^{63} -7.02140e7 q^{65} +2.69226e8 q^{67} -4.01722e7 q^{69} -2.82052e8 q^{71} +1.32314e8 q^{73} +7.52595e7 q^{75} -4.86950e8 q^{77} +2.75182e8 q^{79} +3.67547e8 q^{81} -4.46977e8 q^{83} -1.41149e9 q^{85} -8.35762e7 q^{87} -1.12425e8 q^{89} +1.83801e8 q^{91} +5.97442e7 q^{93} +2.06362e9 q^{95} -1.44020e9 q^{97} -1.46375e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 60 q^{3} + 176 q^{5} - 3416 q^{7} + 22010 q^{9} + 101316 q^{11} - 171366 q^{13} - 257460 q^{15} - 170304 q^{17} + 92084 q^{19} - 424004 q^{21} - 2369944 q^{23} + 2043598 q^{25} + 6428196 q^{27} - 9021404 q^{29}+ \cdots - 282095980 q^{99}+O(q^{100}) \) 6 * q - 60 * q^3 + 176 * q^5 - 3416 * q^7 + 22010 * q^9 + 101316 * q^11 - 171366 * q^13 - 257460 * q^15 - 170304 * q^17 + 92084 * q^19 - 424004 * q^21 - 2369944 * q^23 + 2043598 * q^25 + 6428196 * q^27 - 9021404 * q^29 + 9894388 * q^31 + 6717200 * q^33 + 1910916 * q^35 + 2974376 * q^37 + 1713660 * q^39 - 2977668 * q^41 + 31585340 * q^43 - 99427116 * q^45 + 1766688 * q^47 - 25397750 * q^49 + 68102932 * q^51 - 137836444 * q^53 + 64702520 * q^55 - 253569848 * q^57 + 128523972 * q^59 + 33822404 * q^61 + 359085076 * q^63 - 5026736 * q^65 + 618333476 * q^67 - 837582264 * q^69 + 52534456 * q^71 - 667586460 * q^73 + 940557360 * q^75 - 944797672 * q^77 + 399417496 * q^79 - 821518498 * q^81 + 467205356 * q^83 - 2583200620 * q^85 + 219585328 * q^87 - 1630231268 * q^89 + 97564376 * q^91 - 585888632 * q^93 + 3079232768 * q^95 - 3266492188 * q^97 - 282095980 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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