LMFDB - Embedded newform 208.10.a.h.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,10,Mod(1,208)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$107.127453922$
Analytic rank:	$1$
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} - 1438x^{3} - 4164x^{2} + 396957x - 59580 $ x^5 - 1438x^3 - 4164x^2 + 396957*x - 59580
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-24.3176$ 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)

$f(q)$	$=$	$q+195.094 q^{3} -1277.14 q^{5} +2277.98 q^{7} +18378.5 q^{9} +O(q^{10})$	$q+195.094 q^{3} -1277.14 q^{5} +2277.98 q^{7} +18378.5 q^{9} +7177.94 q^{11} +28561.0 q^{13} -249162. q^{15} -447890. q^{17} +528333. q^{19} +444419. q^{21} -2.24354e6 q^{23} -322041. q^{25} -254496. q^{27} +5.98542e6 q^{29} -169630. q^{31} +1.40037e6 q^{33} -2.90930e6 q^{35} +1.26336e7 q^{37} +5.57207e6 q^{39} -2.76549e7 q^{41} +2.27606e7 q^{43} -2.34719e7 q^{45} -5.32103e7 q^{47} -3.51644e7 q^{49} -8.73804e7 q^{51} +3.18756e7 q^{53} -9.16722e6 q^{55} +1.03074e8 q^{57} -1.14800e8 q^{59} -7.80352e7 q^{61} +4.18659e7 q^{63} -3.64764e7 q^{65} -8.40538e7 q^{67} -4.37700e8 q^{69} -1.25752e8 q^{71} -1.88250e8 q^{73} -6.28282e7 q^{75} +1.63512e7 q^{77} +4.28673e8 q^{79} -4.11395e8 q^{81} -2.43067e8 q^{83} +5.72018e8 q^{85} +1.16772e9 q^{87} +2.92716e8 q^{89} +6.50614e7 q^{91} -3.30937e7 q^{93} -6.74754e8 q^{95} +1.14275e9 q^{97} +1.31920e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9}+O(q^{10}) $ 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9	$ 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27} + 10678182 q^{29} - 12885296 q^{31} + 17278298 q^{33} - 8380731 q^{35} + 7171823 q^{37} - 4598321 q^{39} + 9294012 q^{41} - 12831975 q^{43} + 26135198 q^{45} - 43354215 q^{47} + 25249488 q^{49} - 16905901 q^{51} + 93231780 q^{53} - 99448846 q^{55} + 90173382 q^{57} - 246496182 q^{59} - 132232612 q^{61} + 416955202 q^{63} + 51495483 q^{65} + 369388534 q^{67} - 579986760 q^{69} - 212150457 q^{71} - 252729806 q^{73} + 752457788 q^{75} + 449666118 q^{77} + 1247271728 q^{79} - 317713115 q^{81} - 1696894296 q^{83} - 775363765 q^{85} + 614530466 q^{87} - 753854382 q^{89} - 288437539 q^{91} - 892784668 q^{93} - 1442632962 q^{95} + 3824606 q^{97} - 3016199848 q^{99}+O(q^{100}) $ 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9 - 121746 * q^11 + 142805 * q^13 - 105973 * q^15 - 495669 * q^17 + 840738 * q^19 - 1599467 * q^21 + 592152 * q^23 + 1670362 * q^25 - 6847883 * q^27 + 10678182 * q^29 - 12885296 * q^31 + 17278298 * q^33 - 8380731 * q^35 + 7171823 * q^37 - 4598321 * q^39 + 9294012 * q^41 - 12831975 * q^43 + 26135198 * q^45 - 43354215 * q^47 + 25249488 * q^49 - 16905901 * q^51 + 93231780 * q^53 - 99448846 * q^55 + 90173382 * q^57 - 246496182 * q^59 - 132232612 * q^61 + 416955202 * q^63 + 51495483 * q^65 + 369388534 * q^67 - 579986760 * q^69 - 212150457 * q^71 - 252729806 * q^73 + 752457788 * q^75 + 449666118 * q^77 + 1247271728 * q^79 - 317713115 * q^81 - 1696894296 * q^83 - 775363765 * q^85 + 614530466 * q^87 - 753854382 * q^89 - 288437539 * q^91 - 892784668 * q^93 - 1442632962 * q^95 + 3824606 * q^97 - 3016199848 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	195.094	1.39058	0.695292	−	0.718727i	$-0.255275\pi$
$3$	195.094	1.39058	0.695292	+	0.718727i	$0.255275\pi$
$4$	0	0
$5$	−1277.14	−0.913846	−0.456923	−	0.889506i	$-0.651049\pi$
$5$	−1277.14	−0.913846	−0.456923	+	0.889506i	$0.651049\pi$
$6$	0	0
$7$	2277.98	0.358599	0.179299	−	0.983795i	$-0.442617\pi$
$7$	2277.98	0.358599	0.179299	+	0.983795i	$0.442617\pi$
$8$	0	0
$9$	18378.5	0.933725
$10$	0	0
$11$	7177.94	0.147820	0.0739099	−	0.997265i	$-0.476452\pi$
$11$	7177.94	0.147820	0.0739099	+	0.997265i	$0.476452\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	−249162.	−1.27078
$16$	0	0
$17$	−447890.	−1.30062	−0.650311	−	0.759668i	$-0.725361\pi$
$17$	−447890.	−1.30062	−0.650311	+	0.759668i	$0.725361\pi$
$18$	0	0
$19$	528333.	0.930071	0.465036	−	0.885292i	$-0.346041\pi$
$19$	528333.	0.930071	0.465036	+	0.885292i	$0.346041\pi$
$20$	0	0
$21$	444419.	0.498662
$22$	0	0
$23$	−2.24354e6	−1.67170	−0.835851	−	0.548957i	$-0.815025\pi$
$23$	−2.24354e6	−1.67170	−0.835851	+	0.548957i	$0.815025\pi$
$24$	0	0
$25$	−322041.	−0.164885
$26$	0	0
$27$	−254496.	−0.0921603
$28$	0	0
$29$	5.98542e6	1.57146	0.785730	−	0.618569i	$-0.212287\pi$
$29$	5.98542e6	1.57146	0.785730	+	0.618569i	$0.212287\pi$
$30$	0	0
$31$	−169630.	−0.0329894	−0.0164947	−	0.999864i	$-0.505251\pi$
$31$	−169630.	−0.0329894	−0.0164947	+	0.999864i	$0.505251\pi$
$32$	0	0
$33$	1.40037e6	0.205556
$34$	0	0
$35$	−2.90930e6	−0.327704
$36$	0	0
$37$	1.26336e7	1.10820	0.554100	−	0.832450i	$-0.313062\pi$
$37$	1.26336e7	1.10820	0.554100	+	0.832450i	$0.313062\pi$
$38$	0	0
$39$	5.57207e6	0.385679
$40$	0	0
$41$	−2.76549e7	−1.52843	−0.764213	−	0.644964i	$-0.776872\pi$
$41$	−2.76549e7	−1.52843	−0.764213	+	0.644964i	$0.776872\pi$
$42$	0	0
$43$	2.27606e7	1.01526	0.507628	−	0.861576i	$-0.330522\pi$
$43$	2.27606e7	1.01526	0.507628	+	0.861576i	$0.330522\pi$
$44$	0	0
$45$	−2.34719e7	−0.853281
$46$	0	0
$47$	−5.32103e7	−1.59058	−0.795289	−	0.606230i	$-0.792681\pi$
$47$	−5.32103e7	−1.59058	−0.795289	+	0.606230i	$0.792681\pi$
$48$	0	0
$49$	−3.51644e7	−0.871407
$50$	0	0
$51$	−8.73804e7	−1.80862
$52$	0	0
$53$	3.18756e7	0.554904	0.277452	−	0.960740i	$-0.410510\pi$
$53$	3.18756e7	0.554904	0.277452	+	0.960740i	$0.410510\pi$
$54$	0	0
$55$	−9.16722e6	−0.135085
$56$	0	0
$57$	1.03074e8	1.29334
$58$	0	0
$59$	−1.14800e8	−1.23341	−0.616705	−	0.787194i	$-0.711533\pi$
$59$	−1.14800e8	−1.23341	−0.616705	+	0.787194i	$0.711533\pi$
$60$	0	0
$61$	−7.80352e7	−0.721616	−0.360808	−	0.932640i	$-0.617499\pi$
$61$	−7.80352e7	−0.721616	−0.360808	+	0.932640i	$0.617499\pi$
$62$	0	0
$63$	4.18659e7	0.334833
$64$	0	0
$65$	−3.64764e7	−0.253455
$66$	0	0
$67$	−8.40538e7	−0.509590	−0.254795	−	0.966995i	$-0.582008\pi$
$67$	−8.40538e7	−0.509590	−0.254795	+	0.966995i	$0.582008\pi$
$68$	0	0
$69$	−4.37700e8	−2.32464
$70$	0	0
$71$	−1.25752e8	−0.587288	−0.293644	−	0.955915i	$-0.594868\pi$
$71$	−1.25752e8	−0.587288	−0.293644	+	0.955915i	$0.594868\pi$
$72$	0	0
$73$	−1.88250e8	−0.775859	−0.387929	−	0.921689i	$-0.626810\pi$
$73$	−1.88250e8	−0.775859	−0.387929	+	0.921689i	$0.626810\pi$
$74$	0	0
$75$	−6.28282e7	−0.229287
$76$	0	0
$77$	1.63512e7	0.0530080
$78$	0	0
$79$	4.28673e8	1.23824	0.619120	−	0.785297i	$-0.287490\pi$
$79$	4.28673e8	1.23824	0.619120	+	0.785297i	$0.287490\pi$
$80$	0	0
$81$	−4.11395e8	−1.06188
$82$	0	0
$83$	−2.43067e8	−0.562180	−0.281090	−	0.959681i	$-0.590696\pi$
$83$	−2.43067e8	−0.562180	−0.281090	+	0.959681i	$0.590696\pi$
$84$	0	0
$85$	5.72018e8	1.18857
$86$	0	0
$87$	1.16772e9	2.18525
$88$	0	0
$89$	2.92716e8	0.494529	0.247265	−	0.968948i	$-0.420468\pi$
$89$	2.92716e8	0.494529	0.247265	+	0.968948i	$0.420468\pi$
$90$	0	0
$91$	6.50614e7	0.0994573
$92$	0	0
$93$	−3.30937e7	−0.0458746
$94$	0	0
$95$	−6.74754e8	−0.849942
$96$	0	0
$97$	1.14275e9	1.31063	0.655313	−	0.755358i	$-0.272537\pi$
$97$	1.14275e9	1.31063	0.655313	+	0.755358i	$0.272537\pi$
$98$	0	0
$99$	1.31920e8	0.138023
$100$	0	0
$101$	−8.98629e8	−0.859279	−0.429640	−	0.903000i	$-0.641359\pi$
$101$	−8.98629e8	−0.859279	−0.429640	+	0.903000i	$0.641359\pi$
$102$	0	0
$103$	−6.13518e8	−0.537106	−0.268553	−	0.963265i	$-0.586545\pi$
$103$	−6.13518e8	−0.537106	−0.268553	+	0.963265i	$0.586545\pi$
$104$	0	0
$105$	−5.67585e8	−0.455700
$106$	0	0
$107$	−1.46257e9	−1.07868	−0.539338	−	0.842090i	$-0.681325\pi$
$107$	−1.46257e9	−1.07868	−0.539338	+	0.842090i	$0.681325\pi$
$108$	0	0
$109$	7.17715e8	0.487005	0.243502	−	0.969900i	$-0.421704\pi$
$109$	7.17715e8	0.487005	0.243502	+	0.969900i	$0.421704\pi$
$110$	0	0
$111$	2.46473e9	1.54105
$112$	0	0
$113$	9.27444e7	0.0535099	0.0267550	−	0.999642i	$-0.491483\pi$
$113$	9.27444e7	0.0535099	0.0267550	+	0.999642i	$0.491483\pi$
$114$	0	0
$115$	2.86531e9	1.52768
$116$	0	0
$117$	5.24909e8	0.258969
$118$	0	0
$119$	−1.02028e9	−0.466401
$120$	0	0
$121$	−2.30642e9	−0.978149
$122$	0	0
$123$	−5.39529e9	−2.12540
$124$	0	0
$125$	2.90570e9	1.06453
$126$	0	0
$127$	−4.40060e9	−1.50105	−0.750524	−	0.660843i	$-0.770199\pi$
$127$	−4.40060e9	−1.50105	−0.750524	+	0.660843i	$0.770199\pi$
$128$	0	0
$129$	4.44045e9	1.41180
$130$	0	0
$131$	1.21951e9	0.361798	0.180899	−	0.983502i	$-0.442099\pi$
$131$	1.21951e9	0.361798	0.180899	+	0.983502i	$0.442099\pi$
$132$	0	0
$133$	1.20353e9	0.333522
$134$	0	0
$135$	3.25027e8	0.0842204
$136$	0	0
$137$	−6.73258e9	−1.63282	−0.816411	−	0.577472i	$-0.804039\pi$
$137$	−6.73258e9	−1.63282	−0.816411	+	0.577472i	$0.804039\pi$
$138$	0	0
$139$	−4.12694e8	−0.0937695	−0.0468847	−	0.998900i	$-0.514929\pi$
$139$	−4.12694e8	−0.0937695	−0.0468847	+	0.998900i	$0.514929\pi$
$140$	0	0
$141$	−1.03810e10	−2.21183
$142$	0	0
$143$	2.05009e8	0.0409978
$144$	0	0
$145$	−7.64421e9	−1.43607
$146$	0	0
$147$	−6.86035e9	−1.21177
$148$	0	0
$149$	7.17339e9	1.19230	0.596151	−	0.802872i	$-0.296696\pi$
$149$	7.17339e9	1.19230	0.596151	+	0.802872i	$0.296696\pi$
$150$	0	0
$151$	−9.30445e9	−1.45645	−0.728223	−	0.685340i	$-0.759654\pi$
$151$	−9.30445e9	−1.45645	−0.728223	+	0.685340i	$0.759654\pi$
$152$	0	0
$153$	−8.23155e9	−1.21442
$154$	0	0
$155$	2.16641e8	0.0301472
$156$	0	0
$157$	−5.86131e9	−0.769921	−0.384961	−	0.922933i	$-0.625785\pi$
$157$	−5.86131e9	−0.769921	−0.384961	+	0.922933i	$0.625785\pi$
$158$	0	0
$159$	6.21873e9	0.771640
$160$	0	0
$161$	−5.11074e9	−0.599470
$162$	0	0
$163$	−1.60676e10	−1.78282	−0.891409	−	0.453199i	$-0.850283\pi$
$163$	−1.60676e10	−1.78282	−0.891409	+	0.453199i	$0.850283\pi$
$164$	0	0
$165$	−1.78847e9	−0.187847
$166$	0	0
$167$	1.49042e9	0.148281	0.0741404	−	0.997248i	$-0.476379\pi$
$167$	1.49042e9	0.148281	0.0741404	+	0.997248i	$0.476379\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	9.70997e9	0.868431
$172$	0	0
$173$	−6.62250e9	−0.562101	−0.281050	−	0.959693i	$-0.590683\pi$
$173$	−6.62250e9	−0.562101	−0.281050	+	0.959693i	$0.590683\pi$
$174$	0	0
$175$	−7.33604e8	−0.0591276
$176$	0	0
$177$	−2.23967e10	−1.71516
$178$	0	0
$179$	−1.41019e10	−1.02669	−0.513344	−	0.858183i	$-0.671594\pi$
$179$	−1.41019e10	−1.02669	−0.513344	+	0.858183i	$0.671594\pi$
$180$	0	0
$181$	2.38898e10	1.65447	0.827234	−	0.561857i	$-0.189913\pi$
$181$	2.38898e10	1.65447	0.827234	+	0.561857i	$0.189913\pi$
$182$	0	0
$183$	−1.52242e10	−1.00347
$184$	0	0
$185$	−1.61348e10	−1.01272
$186$	0	0
$187$	−3.21493e9	−0.192258
$188$	0	0
$189$	−5.79737e8	−0.0330486
$190$	0	0
$191$	2.32674e10	1.26502	0.632509	−	0.774553i	$-0.282025\pi$
$191$	2.32674e10	1.26502	0.632509	+	0.774553i	$0.282025\pi$
$192$	0	0
$193$	8.87539e9	0.460447	0.230224	−	0.973138i	$-0.426054\pi$
$193$	8.87539e9	0.460447	0.230224	+	0.973138i	$0.426054\pi$
$194$	0	0
$195$	−7.11631e9	−0.352451
$196$	0	0
$197$	−9.96936e9	−0.471595	−0.235798	−	0.971802i	$-0.575770\pi$
$197$	−9.96936e9	−0.471595	−0.235798	+	0.971802i	$0.575770\pi$
$198$	0	0
$199$	5.92044e9	0.267618	0.133809	−	0.991007i	$-0.457279\pi$
$199$	5.92044e9	0.267618	0.133809	+	0.991007i	$0.457279\pi$
$200$	0	0
$201$	−1.63984e10	−0.708628
$202$	0	0
$203$	1.36347e10	0.563524
$204$	0	0
$205$	3.53191e10	1.39675
$206$	0	0
$207$	−4.12330e10	−1.56091
$208$	0	0
$209$	3.79234e9	0.137483
$210$	0	0
$211$	3.18290e10	1.10548	0.552741	−	0.833353i	$-0.313582\pi$
$211$	3.18290e10	1.10548	0.552741	+	0.833353i	$0.313582\pi$
$212$	0	0
$213$	−2.45334e10	−0.816674
$214$	0	0
$215$	−2.90685e10	−0.927788
$216$	0	0
$217$	−3.86413e8	−0.0118300
$218$	0	0
$219$	−3.67264e10	−1.07890
$220$	0	0
$221$	−1.27922e10	−0.360728
$222$	0	0
$223$	2.33567e10	0.632470	0.316235	−	0.948681i	$-0.397581\pi$
$223$	2.33567e10	0.632470	0.316235	+	0.948681i	$0.397581\pi$
$224$	0	0
$225$	−5.91865e9	−0.153958
$226$	0	0
$227$	−2.96928e10	−0.742223	−0.371112	−	0.928588i	$-0.621023\pi$
$227$	−2.96928e10	−0.742223	−0.371112	+	0.928588i	$0.621023\pi$
$228$	0	0
$229$	4.24937e10	1.02109	0.510546	−	0.859850i	$-0.329443\pi$
$229$	4.24937e10	1.02109	0.510546	+	0.859850i	$0.329443\pi$
$230$	0	0
$231$	3.19001e9	0.0737121
$232$	0	0
$233$	−3.84693e10	−0.855092	−0.427546	−	0.903994i	$-0.640622\pi$
$233$	−3.84693e10	−0.855092	−0.427546	+	0.903994i	$0.640622\pi$
$234$	0	0
$235$	6.79569e10	1.45354
$236$	0	0
$237$	8.36314e10	1.72188
$238$	0	0
$239$	1.98946e10	0.394407	0.197203	−	0.980363i	$-0.436814\pi$
$239$	1.98946e10	0.394407	0.197203	+	0.980363i	$0.436814\pi$
$240$	0	0
$241$	−5.31240e10	−1.01441	−0.507206	−	0.861825i	$-0.669322\pi$
$241$	−5.31240e10	−1.01441	−0.507206	+	0.861825i	$0.669322\pi$
$242$	0	0
$243$	−7.52513e10	−1.38448
$244$	0	0
$245$	4.49098e10	0.796332
$246$	0	0
$247$	1.50897e10	0.257955
$248$	0	0
$249$	−4.74209e10	−0.781758
$250$	0	0
$251$	−1.26269e10	−0.200801	−0.100400	−	0.994947i	$-0.532012\pi$
$251$	−1.26269e10	−0.200801	−0.100400	+	0.994947i	$0.532012\pi$
$252$	0	0
$253$	−1.61040e10	−0.247111
$254$	0	0
$255$	1.11597e11	1.65280
$256$	0	0
$257$	1.11000e11	1.58717	0.793583	−	0.608461i	$-0.208213\pi$
$257$	1.11000e11	1.58717	0.793583	+	0.608461i	$0.208213\pi$
$258$	0	0
$259$	2.87790e10	0.397399
$260$	0	0
$261$	1.10003e11	1.46731
$262$	0	0
$263$	−7.01020e10	−0.903503	−0.451752	−	0.892144i	$-0.649201\pi$
$263$	−7.01020e10	−0.903503	−0.451752	+	0.892144i	$0.649201\pi$
$264$	0	0
$265$	−4.07096e10	−0.507097
$266$	0	0
$267$	5.71071e10	0.687685
$268$	0	0
$269$	7.41364e10	0.863269	0.431635	−	0.902049i	$-0.357937\pi$
$269$	7.41364e10	0.863269	0.431635	+	0.902049i	$0.357937\pi$
$270$	0	0
$271$	−8.92477e10	−1.00516	−0.502580	−	0.864531i	$-0.667616\pi$
$271$	−8.92477e10	−1.00516	−0.502580	+	0.864531i	$0.667616\pi$
$272$	0	0
$273$	1.26931e10	0.138304
$274$	0	0
$275$	−2.31159e9	−0.0243733
$276$	0	0
$277$	−6.68313e10	−0.682057	−0.341028	−	0.940053i	$-0.610775\pi$
$277$	−6.68313e10	−0.682057	−0.341028	+	0.940053i	$0.610775\pi$
$278$	0	0
$279$	−3.11754e9	−0.0308031
$280$	0	0
$281$	−8.13670e10	−0.778520	−0.389260	−	0.921128i	$-0.627269\pi$
$281$	−8.13670e10	−0.778520	−0.389260	+	0.921128i	$0.627269\pi$
$282$	0	0
$283$	1.19236e11	1.10502	0.552508	−	0.833507i	$-0.313671\pi$
$283$	1.19236e11	1.10502	0.552508	+	0.833507i	$0.313671\pi$
$284$	0	0
$285$	−1.31640e11	−1.18192
$286$	0	0
$287$	−6.29972e10	−0.548091
$288$	0	0
$289$	8.20174e10	0.691617
$290$	0	0
$291$	2.22943e11	1.82254
$292$	0	0
$293$	−6.95594e10	−0.551381	−0.275690	−	0.961246i	$-0.588906\pi$
$293$	−6.95594e10	−0.551381	−0.275690	+	0.961246i	$0.588906\pi$
$294$	0	0
$295$	1.46615e11	1.12715
$296$	0	0
$297$	−1.82676e9	−0.0136231
$298$	0	0
$299$	−6.40778e10	−0.463647
$300$	0	0
$301$	5.18482e10	0.364069
$302$	0	0
$303$	−1.75317e11	−1.19490
$304$	0	0
$305$	9.96618e10	0.659446
$306$	0	0
$307$	4.02660e10	0.258711	0.129356	−	0.991598i	$-0.458709\pi$
$307$	4.02660e10	0.258711	0.129356	+	0.991598i	$0.458709\pi$
$308$	0	0
$309$	−1.19693e11	−0.746891
$310$	0	0
$311$	1.78556e11	1.08231	0.541157	−	0.840921i	$-0.317986\pi$
$311$	1.78556e11	1.08231	0.541157	+	0.840921i	$0.317986\pi$
$312$	0	0
$313$	9.44550e9	0.0556257	0.0278128	−	0.999613i	$-0.491146\pi$
$313$	9.44550e9	0.0556257	0.0278128	+	0.999613i	$0.491146\pi$
$314$	0	0
$315$	−5.34685e10	−0.305985
$316$	0	0
$317$	5.42593e10	0.301792	0.150896	−	0.988550i	$-0.451784\pi$
$317$	5.42593e10	0.301792	0.150896	+	0.988550i	$0.451784\pi$
$318$	0	0
$319$	4.29630e10	0.232293
$320$	0	0
$321$	−2.85339e11	−1.49999
$322$	0	0
$323$	−2.36635e11	−1.20967
$324$	0	0
$325$	−9.19783e9	−0.0457309
$326$	0	0
$327$	1.40022e11	0.677221
$328$	0	0
$329$	−1.21212e11	−0.570379
$330$	0	0
$331$	−6.66790e10	−0.305325	−0.152663	−	0.988278i	$-0.548785\pi$
$331$	−6.66790e10	−0.305325	−0.152663	+	0.988278i	$0.548785\pi$
$332$	0	0
$333$	2.32186e11	1.03476
$334$	0	0
$335$	1.07348e11	0.465687
$336$	0	0
$337$	2.13457e11	0.901522	0.450761	−	0.892645i	$-0.351153\pi$
$337$	2.13457e11	0.901522	0.450761	+	0.892645i	$0.351153\pi$
$338$	0	0
$339$	1.80938e10	0.0744101
$340$	0	0
$341$	−1.21759e9	−0.00487649
$342$	0	0
$343$	−1.72028e11	−0.671084
$344$	0	0
$345$	5.59004e11	2.12437
$346$	0	0
$347$	9.49978e10	0.351748	0.175874	−	0.984413i	$-0.443725\pi$
$347$	9.49978e10	0.351748	0.175874	+	0.984413i	$0.443725\pi$
$348$	0	0
$349$	2.52647e11	0.911590	0.455795	−	0.890085i	$-0.349355\pi$
$349$	2.52647e11	0.911590	0.455795	+	0.890085i	$0.349355\pi$
$350$	0	0
$351$	−7.26866e9	−0.0255607
$352$	0	0
$353$	4.50376e11	1.54379	0.771896	−	0.635749i	$-0.219309\pi$
$353$	4.50376e11	1.54379	0.771896	+	0.635749i	$0.219309\pi$
$354$	0	0
$355$	1.60602e11	0.536691
$356$	0	0
$357$	−1.99051e11	−0.648570
$358$	0	0
$359$	−3.93740e11	−1.25108	−0.625539	−	0.780193i	$-0.715121\pi$
$359$	−3.93740e11	−1.25108	−0.625539	+	0.780193i	$0.715121\pi$
$360$	0	0
$361$	−4.35522e10	−0.134967
$362$	0	0
$363$	−4.49969e11	−1.36020
$364$	0	0
$365$	2.40422e11	0.709015
$366$	0	0
$367$	2.26305e11	0.651173	0.325586	−	0.945512i	$-0.394438\pi$
$367$	2.26305e11	0.651173	0.325586	+	0.945512i	$0.394438\pi$
$368$	0	0
$369$	−5.08255e11	−1.42713
$370$	0	0
$371$	7.26120e10	0.198988
$372$	0	0
$373$	4.75023e11	1.27065	0.635323	−	0.772246i	$-0.280867\pi$
$373$	4.75023e11	1.27065	0.635323	+	0.772246i	$0.280867\pi$
$374$	0	0
$375$	5.66884e11	1.48031
$376$	0	0
$377$	1.70950e11	0.435845
$378$	0	0
$379$	1.67293e11	0.416486	0.208243	−	0.978077i	$-0.433225\pi$
$379$	1.67293e11	0.416486	0.208243	+	0.978077i	$0.433225\pi$
$380$	0	0
$381$	−8.58528e11	−2.08733
$382$	0	0
$383$	5.36589e11	1.27423	0.637114	−	0.770769i	$-0.280128\pi$
$383$	5.36589e11	1.27423	0.637114	+	0.770769i	$0.280128\pi$
$384$	0	0
$385$	−2.08827e10	−0.0484411
$386$	0	0
$387$	4.18306e11	0.947971
$388$	0	0
$389$	5.95613e11	1.31884	0.659418	−	0.751776i	$-0.270803\pi$
$389$	5.95613e11	1.31884	0.659418	+	0.751776i	$0.270803\pi$
$390$	0	0
$391$	1.00486e12	2.17425
$392$	0	0
$393$	2.37919e11	0.503110
$394$	0	0
$395$	−5.47475e11	−1.13156
$396$	0	0
$397$	−6.68797e11	−1.35125	−0.675627	−	0.737243i	$-0.736127\pi$
$397$	−6.68797e11	−1.35125	−0.675627	+	0.737243i	$0.736127\pi$
$398$	0	0
$399$	2.34801e11	0.463791
$400$	0	0
$401$	4.27579e11	0.825785	0.412892	−	0.910780i	$-0.364519\pi$
$401$	4.27579e11	0.825785	0.412892	+	0.910780i	$0.364519\pi$
$402$	0	0
$403$	−4.84480e9	−0.00914962
$404$	0	0
$405$	5.25408e11	0.970397
$406$	0	0
$407$	9.06830e10	0.163814
$408$	0	0
$409$	−4.37893e10	−0.0773772	−0.0386886	−	0.999251i	$-0.512318\pi$
$409$	−4.37893e10	−0.0773772	−0.0386886	+	0.999251i	$0.512318\pi$
$410$	0	0
$411$	−1.31348e12	−2.27058
$412$	0	0
$413$	−2.61512e11	−0.442299
$414$	0	0
$415$	3.10431e11	0.513746
$416$	0	0
$417$	−8.05139e10	−0.130394
$418$	0	0
$419$	1.14243e11	0.181079	0.0905395	−	0.995893i	$-0.471141\pi$
$419$	1.14243e11	0.181079	0.0905395	+	0.995893i	$0.471141\pi$
$420$	0	0
$421$	−5.06520e11	−0.785828	−0.392914	−	0.919575i	$-0.628533\pi$
$421$	−5.06520e11	−0.785828	−0.392914	+	0.919575i	$0.628533\pi$
$422$	0	0
$423$	−9.77926e11	−1.48516
$424$	0	0
$425$	1.44239e11	0.214453
$426$	0	0
$427$	−1.77763e11	−0.258770
$428$	0	0
$429$	3.99960e10	0.0570110
$430$	0	0
$431$	1.32605e11	0.185103	0.0925513	−	0.995708i	$-0.470498\pi$
$431$	1.32605e11	0.185103	0.0925513	+	0.995708i	$0.470498\pi$
$432$	0	0
$433$	1.21346e12	1.65894	0.829469	−	0.558553i	$-0.188643\pi$
$433$	1.21346e12	1.65894	0.829469	+	0.558553i	$0.188643\pi$
$434$	0	0
$435$	−1.49134e12	−1.99698
$436$	0	0
$437$	−1.18534e12	−1.55480
$438$	0	0
$439$	7.26961e11	0.934159	0.467079	−	0.884215i	$-0.345306\pi$
$439$	7.26961e11	0.934159	0.467079	+	0.884215i	$0.345306\pi$
$440$	0	0
$441$	−6.46270e11	−0.813655
$442$	0	0
$443$	1.02161e12	1.26028	0.630142	−	0.776480i	$-0.282997\pi$
$443$	1.02161e12	1.26028	0.630142	+	0.776480i	$0.282997\pi$
$444$	0	0
$445$	−3.73840e11	−0.451924
$446$	0	0
$447$	1.39948e12	1.65800
$448$	0	0
$449$	5.91722e11	0.687083	0.343542	−	0.939137i	$-0.388373\pi$
$449$	5.91722e11	0.687083	0.343542	+	0.939137i	$0.388373\pi$
$450$	0	0
$451$	−1.98505e11	−0.225932
$452$	0	0
$453$	−1.81524e12	−2.02531
$454$	0	0
$455$	−8.30924e10	−0.0908887
$456$	0	0
$457$	−8.30664e11	−0.890845	−0.445423	−	0.895320i	$-0.646947\pi$
$457$	−8.30664e11	−0.890845	−0.445423	+	0.895320i	$0.646947\pi$
$458$	0	0
$459$	1.13986e11	0.119866
$460$	0	0
$461$	8.73246e11	0.900497	0.450249	−	0.892903i	$-0.351335\pi$
$461$	8.73246e11	0.900497	0.450249	+	0.892903i	$0.351335\pi$
$462$	0	0
$463$	−7.70415e11	−0.779130	−0.389565	−	0.920999i	$-0.627375\pi$
$463$	−7.70415e11	−0.779130	−0.389565	+	0.920999i	$0.627375\pi$
$464$	0	0
$465$	4.22652e10	0.0419223
$466$	0	0
$467$	−4.50995e11	−0.438779	−0.219390	−	0.975637i	$-0.570407\pi$
$467$	−4.50995e11	−0.438779	−0.219390	+	0.975637i	$0.570407\pi$
$468$	0	0
$469$	−1.91473e11	−0.182738
$470$	0	0
$471$	−1.14350e12	−1.07064
$472$	0	0
$473$	1.63374e11	0.150075
$474$	0	0
$475$	−1.70145e11	−0.153355
$476$	0	0
$477$	5.85827e11	0.518128
$478$	0	0
$479$	5.26966e11	0.457376	0.228688	−	0.973500i	$-0.426556\pi$
$479$	5.26966e11	0.457376	0.228688	+	0.973500i	$0.426556\pi$
$480$	0	0
$481$	3.60827e11	0.307360
$482$	0	0
$483$	−9.97072e11	−0.833613
$484$	0	0
$485$	−1.45945e12	−1.19771
$486$	0	0
$487$	−1.88076e12	−1.51514	−0.757570	−	0.652754i	$-0.773614\pi$
$487$	−1.88076e12	−1.51514	−0.757570	+	0.652754i	$0.773614\pi$
$488$	0	0
$489$	−3.13469e12	−2.47916
$490$	0	0
$491$	−1.68889e12	−1.31140	−0.655699	−	0.755022i	$-0.727626\pi$
$491$	−1.68889e12	−1.31140	−0.655699	+	0.755022i	$0.727626\pi$
$492$	0	0
$493$	−2.68081e12	−2.04388
$494$	0	0
$495$	−1.68480e11	−0.126132
$496$	0	0
$497$	−2.86460e11	−0.210601
$498$	0	0
$499$	−1.60699e12	−1.16027	−0.580137	−	0.814519i	$-0.697001\pi$
$499$	−1.60699e12	−1.16027	−0.580137	+	0.814519i	$0.697001\pi$
$500$	0	0
$501$	2.90772e11	0.206197
$502$	0	0
$503$	−2.73565e12	−1.90548	−0.952740	−	0.303786i	$-0.901749\pi$
$503$	−2.73565e12	−1.90548	−0.952740	+	0.303786i	$0.901749\pi$
$504$	0	0
$505$	1.14767e12	0.785249
$506$	0	0
$507$	1.59144e11	0.106968
$508$	0	0
$509$	7.49256e11	0.494766	0.247383	−	0.968918i	$-0.420429\pi$
$509$	7.49256e11	0.494766	0.247383	+	0.968918i	$0.420429\pi$
$510$	0	0
$511$	−4.28830e11	−0.278222
$512$	0	0
$513$	−1.34459e11	−0.0857157
$514$	0	0
$515$	7.83548e11	0.490832
$516$	0	0
$517$	−3.81940e11	−0.235119
$518$	0	0
$519$	−1.29201e12	−0.781649
$520$	0	0
$521$	1.07862e12	0.641355	0.320678	−	0.947188i	$-0.396089\pi$
$521$	1.07862e12	0.641355	0.320678	+	0.947188i	$0.396089\pi$
$522$	0	0
$523$	2.69236e12	1.57353	0.786765	−	0.617253i	$-0.211754\pi$
$523$	2.69236e12	1.57353	0.786765	+	0.617253i	$0.211754\pi$
$524$	0	0
$525$	−1.43121e11	−0.0822219
$526$	0	0
$527$	7.59755e10	0.0429067
$528$	0	0
$529$	3.23232e12	1.79459
$530$	0	0
$531$	−2.10985e12	−1.15167
$532$	0	0
$533$	−7.89851e11	−0.423909
$534$	0	0
$535$	1.86791e12	0.985743
$536$	0	0
$537$	−2.75119e12	−1.42770
$538$	0	0
$539$	−2.52408e11	−0.128811
$540$	0	0
$541$	1.94584e12	0.976604	0.488302	−	0.872675i	$-0.337616\pi$
$541$	1.94584e12	0.976604	0.488302	+	0.872675i	$0.337616\pi$
$542$	0	0
$543$	4.66074e12	2.30068
$544$	0	0
$545$	−9.16622e11	−0.445047
$546$	0	0
$547$	7.49274e11	0.357847	0.178924	−	0.983863i	$-0.442739\pi$
$547$	7.49274e11	0.357847	0.178924	+	0.983863i	$0.442739\pi$
$548$	0	0
$549$	−1.43417e12	−0.673791
$550$	0	0
$551$	3.16229e12	1.46157
$552$	0	0
$553$	9.76509e11	0.444031
$554$	0	0
$555$	−3.14780e12	−1.40828
$556$	0	0
$557$	2.43252e12	1.07080	0.535399	−	0.844599i	$-0.320161\pi$
$557$	2.43252e12	1.07080	0.535399	+	0.844599i	$0.320161\pi$
$558$	0	0
$559$	6.50066e11	0.281581
$560$	0	0
$561$	−6.27212e11	−0.267351
$562$	0	0
$563$	2.26566e12	0.950401	0.475200	−	0.879878i	$-0.342376\pi$
$563$	2.26566e12	0.950401	0.475200	+	0.879878i	$0.342376\pi$
$564$	0	0
$565$	−1.18447e11	−0.0488999
$566$	0	0
$567$	−9.37149e11	−0.380789
$568$	0	0
$569$	4.39567e12	1.75800	0.879002	−	0.476817i	$-0.158210\pi$
$569$	4.39567e12	1.75800	0.879002	+	0.476817i	$0.158210\pi$
$570$	0	0
$571$	−2.81010e12	−1.10626	−0.553132	−	0.833094i	$-0.686567\pi$
$571$	−2.81010e12	−1.10626	−0.553132	+	0.833094i	$0.686567\pi$
$572$	0	0
$573$	4.53931e12	1.75912
$574$	0	0
$575$	7.22513e11	0.275639
$576$	0	0
$577$	−3.06576e12	−1.15146	−0.575728	−	0.817642i	$-0.695281\pi$
$577$	−3.06576e12	−1.15146	−0.575728	+	0.817642i	$0.695281\pi$
$578$	0	0
$579$	1.73153e12	0.640291
$580$	0	0
$581$	−5.53702e11	−0.201597
$582$	0	0
$583$	2.28801e11	0.0820257
$584$	0	0
$585$	−6.70381e11	−0.236658
$586$	0	0
$587$	−1.53677e12	−0.534240	−0.267120	−	0.963663i	$-0.586072\pi$
$587$	−1.53677e12	−0.534240	−0.267120	+	0.963663i	$0.586072\pi$
$588$	0	0
$589$	−8.96210e10	−0.0306825
$590$	0	0
$591$	−1.94496e12	−0.655793
$592$	0	0
$593$	−3.34336e12	−1.11029	−0.555146	−	0.831753i	$-0.687338\pi$
$593$	−3.34336e12	−1.11029	−0.555146	+	0.831753i	$0.687338\pi$
$594$	0	0
$595$	1.30304e12	0.426219
$596$	0	0
$597$	1.15504e12	0.372145
$598$	0	0
$599$	9.97101e11	0.316460	0.158230	−	0.987402i	$-0.449421\pi$
$599$	9.97101e11	0.316460	0.158230	+	0.987402i	$0.449421\pi$
$600$	0	0
$601$	−2.24332e12	−0.701386	−0.350693	−	0.936491i	$-0.614054\pi$
$601$	−2.24332e12	−0.701386	−0.350693	+	0.936491i	$0.614054\pi$
$602$	0	0
$603$	−1.54478e12	−0.475817
$604$	0	0
$605$	2.94562e12	0.893878
$606$	0	0
$607$	5.75690e12	1.72123	0.860617	−	0.509253i	$-0.170078\pi$
$607$	5.75690e12	1.72123	0.860617	+	0.509253i	$0.170078\pi$
$608$	0	0
$609$	2.66003e12	0.783627
$610$	0	0
$611$	−1.51974e12	−0.441147
$612$	0	0
$613$	3.44873e12	0.986479	0.493239	−	0.869894i	$-0.335813\pi$
$613$	3.44873e12	0.986479	0.493239	+	0.869894i	$0.335813\pi$
$614$	0	0
$615$	6.89053e12	1.94229
$616$	0	0
$617$	−5.13402e12	−1.42618	−0.713089	−	0.701073i	$-0.752705\pi$
$617$	−5.13402e12	−1.42618	−0.713089	+	0.701073i	$0.752705\pi$
$618$	0	0
$619$	−1.83851e12	−0.503336	−0.251668	−	0.967814i	$-0.580979\pi$
$619$	−1.83851e12	−0.503336	−0.251668	+	0.967814i	$0.580979\pi$
$620$	0	0
$621$	5.70972e11	0.154065
$622$	0	0
$623$	6.66802e11	0.177338
$624$	0	0
$625$	−3.08200e12	−0.807928
$626$	0	0
$627$	7.39861e11	0.191182
$628$	0	0
$629$	−5.65845e12	−1.44135
$630$	0	0
$631$	−3.01780e12	−0.757805	−0.378903	−	0.925437i	$-0.623699\pi$
$631$	−3.01780e12	−0.757805	−0.378903	+	0.925437i	$0.623699\pi$
$632$	0	0
$633$	6.20963e12	1.53727
$634$	0	0
$635$	5.62017e12	1.37173
$636$	0	0
$637$	−1.00433e12	−0.241685
$638$	0	0
$639$	−2.31113e12	−0.548366
$640$	0	0
$641$	2.47728e12	0.579580	0.289790	−	0.957090i	$-0.406415\pi$
$641$	2.47728e12	0.579580	0.289790	+	0.957090i	$0.406415\pi$
$642$	0	0
$643$	−5.96725e12	−1.37665	−0.688327	−	0.725401i	$-0.741654\pi$
$643$	−5.96725e12	−1.37665	−0.688327	+	0.725401i	$0.741654\pi$
$644$	0	0
$645$	−5.67107e12	−1.29017
$646$	0	0
$647$	4.85153e12	1.08845	0.544227	−	0.838938i	$-0.316823\pi$
$647$	4.85153e12	1.08845	0.544227	+	0.838938i	$0.316823\pi$
$648$	0	0
$649$	−8.24027e11	−0.182322
$650$	0	0
$651$	−7.53867e10	−0.0164506
$652$	0	0
$653$	−1.50523e12	−0.323961	−0.161980	−	0.986794i	$-0.551788\pi$
$653$	−1.50523e12	−0.323961	−0.161980	+	0.986794i	$0.551788\pi$
$654$	0	0
$655$	−1.55749e12	−0.330628
$656$	0	0
$657$	−3.45976e12	−0.724439
$658$	0	0
$659$	3.29009e12	0.679552	0.339776	−	0.940506i	$-0.389649\pi$
$659$	3.29009e12	0.679552	0.339776	+	0.940506i	$0.389649\pi$
$660$	0	0
$661$	7.79224e12	1.58765	0.793827	−	0.608144i	$-0.208086\pi$
$661$	7.79224e12	1.58765	0.793827	+	0.608144i	$0.208086\pi$
$662$	0	0
$663$	−2.49567e12	−0.501622
$664$	0	0
$665$	−1.53708e12	−0.304788
$666$	0	0
$667$	−1.34285e13	−2.62701
$668$	0	0
$669$	4.55675e12	0.879503
$670$	0	0
$671$	−5.60132e11	−0.106669
$672$	0	0
$673$	−6.76161e12	−1.27052	−0.635261	−	0.772297i	$-0.719108\pi$
$673$	−6.76161e12	−1.27052	−0.635261	+	0.772297i	$0.719108\pi$
$674$	0	0
$675$	8.19583e10	0.0151959
$676$	0	0
$677$	6.04459e12	1.10590	0.552952	−	0.833213i	$-0.313501\pi$
$677$	6.04459e12	1.10590	0.552952	+	0.833213i	$0.313501\pi$
$678$	0	0
$679$	2.60316e12	0.469988
$680$	0	0
$681$	−5.79287e12	−1.03212
$682$	0	0
$683$	−3.24587e12	−0.570740	−0.285370	−	0.958417i	$-0.592116\pi$
$683$	−3.24587e12	−0.570740	−0.285370	+	0.958417i	$0.592116\pi$
$684$	0	0
$685$	8.59843e12	1.49215
$686$	0	0
$687$	8.29025e12	1.41992
$688$	0	0
$689$	9.10400e11	0.153903
$690$	0	0
$691$	4.14584e12	0.691769	0.345885	−	0.938277i	$-0.387579\pi$
$691$	4.14584e12	0.691769	0.345885	+	0.938277i	$0.387579\pi$
$692$	0	0
$693$	3.00511e11	0.0494949
$694$	0	0
$695$	5.27067e11	0.0856908
$696$	0	0
$697$	1.23863e13	1.98790
$698$	0	0
$699$	−7.50512e12	−1.18908
$700$	0	0
$701$	−5.47240e12	−0.855947	−0.427974	−	0.903791i	$-0.640772\pi$
$701$	−5.47240e12	−0.855947	−0.427974	+	0.903791i	$0.640772\pi$
$702$	0	0
$703$	6.67473e12	1.03071
$704$	0	0
$705$	1.32580e13	2.02128
$706$	0	0
$707$	−2.04706e12	−0.308136
$708$	0	0
$709$	1.68810e12	0.250895	0.125447	−	0.992100i	$-0.459963\pi$
$709$	1.68810e12	0.250895	0.125447	+	0.992100i	$0.459963\pi$
$710$	0	0
$711$	7.87838e12	1.15618
$712$	0	0
$713$	3.80571e11	0.0551484
$714$	0	0
$715$	−2.61825e11	−0.0374657
$716$	0	0
$717$	3.88131e12	0.548456
$718$	0	0
$719$	3.99409e12	0.557363	0.278681	−	0.960384i	$-0.410103\pi$
$719$	3.99409e12	0.557363	0.278681	+	0.960384i	$0.410103\pi$
$720$	0	0
$721$	−1.39758e12	−0.192605
$722$	0	0
$723$	−1.03642e13	−1.41062
$724$	0	0
$725$	−1.92755e12	−0.259111
$726$	0	0
$727$	1.96928e12	0.261459	0.130730	−	0.991418i	$-0.458268\pi$
$727$	1.96928e12	0.261459	0.130730	+	0.991418i	$0.458268\pi$
$728$	0	0
$729$	−6.58356e12	−0.863350
$730$	0	0
$731$	−1.01942e13	−1.32046
$732$	0	0
$733$	2.48265e12	0.317649	0.158825	−	0.987307i	$-0.449230\pi$
$733$	2.48265e12	0.317649	0.158825	+	0.987307i	$0.449230\pi$
$734$	0	0
$735$	8.76162e12	1.10737
$736$	0	0
$737$	−6.03333e11	−0.0753275
$738$	0	0
$739$	8.83198e12	1.08933	0.544664	−	0.838655i	$-0.316657\pi$
$739$	8.83198e12	1.08933	0.544664	+	0.838655i	$0.316657\pi$
$740$	0	0
$741$	2.94391e12	0.358709
$742$	0	0
$743$	−5.88065e12	−0.707906	−0.353953	−	0.935263i	$-0.615163\pi$
$743$	−5.88065e12	−0.707906	−0.353953	+	0.935263i	$0.615163\pi$
$744$	0	0
$745$	−9.16141e12	−1.08958
$746$	0	0
$747$	−4.46722e12	−0.524921
$748$	0	0
$749$	−3.33171e12	−0.386811
$750$	0	0
$751$	9.39265e12	1.07748	0.538739	−	0.842473i	$-0.318901\pi$
$751$	9.39265e12	1.07748	0.538739	+	0.842473i	$0.318901\pi$
$752$	0	0
$753$	−2.46343e12	−0.279230
$754$	0	0
$755$	1.18831e13	1.33097
$756$	0	0
$757$	1.20690e13	1.33579	0.667897	−	0.744254i	$-0.267195\pi$
$757$	1.20690e13	1.33579	0.667897	+	0.744254i	$0.267195\pi$
$758$	0	0
$759$	−3.14179e12	−0.343628
$760$	0	0
$761$	1.73414e12	0.187436	0.0937182	−	0.995599i	$-0.470125\pi$
$761$	1.73414e12	0.187436	0.0937182	+	0.995599i	$0.470125\pi$
$762$	0	0
$763$	1.63494e12	0.174639
$764$	0	0
$765$	1.05128e13	1.10980
$766$	0	0
$767$	−3.27880e12	−0.342086
$768$	0	0
$769$	−1.87490e13	−1.93335	−0.966675	−	0.256006i	$-0.917593\pi$
$769$	−1.87490e13	−1.93335	−0.966675	+	0.256006i	$0.917593\pi$
$770$	0	0
$771$	2.16553e13	2.20709
$772$	0	0
$773$	−2.23308e12	−0.224956	−0.112478	−	0.993654i	$-0.535879\pi$
$773$	−2.23308e12	−0.224956	−0.112478	+	0.993654i	$0.535879\pi$
$774$	0	0
$775$	5.46278e10	0.00543947
$776$	0	0
$777$	5.61460e12	0.552617
$778$	0	0
$779$	−1.46110e13	−1.42154
$780$	0	0
$781$	−9.02638e11	−0.0868129
$782$	0	0
$783$	−1.52327e12	−0.144826
$784$	0	0
$785$	7.48571e12	0.703590
$786$	0	0
$787$	−7.05497e12	−0.655555	−0.327777	−	0.944755i	$-0.606300\pi$
$787$	−7.05497e12	−0.655555	−0.327777	+	0.944755i	$0.606300\pi$
$788$	0	0
$789$	−1.36765e13	−1.25640
$790$	0	0
$791$	2.11270e11	0.0191886
$792$	0	0
$793$	−2.22876e12	−0.200140
$794$	0	0
$795$	−7.94219e12	−0.705161
$796$	0	0
$797$	1.16654e13	1.02409	0.512045	−	0.858959i	$-0.328888\pi$
$797$	1.16654e13	1.02409	0.512045	+	0.858959i	$0.328888\pi$
$798$	0	0
$799$	2.38323e13	2.06874
$800$	0	0
$801$	5.37969e12	0.461755
$802$	0	0
$803$	−1.35125e12	−0.114687
$804$	0	0
$805$	6.52712e12	0.547823
$806$	0	0
$807$	1.44635e13	1.20045
$808$	0	0
$809$	−1.37043e13	−1.12483	−0.562415	−	0.826855i	$-0.690128\pi$
$809$	−1.37043e13	−1.12483	−0.562415	+	0.826855i	$0.690128\pi$
$810$	0	0
$811$	−2.07440e12	−0.168383	−0.0841917	−	0.996450i	$-0.526831\pi$
$811$	−2.07440e12	−0.168383	−0.0841917	+	0.996450i	$0.526831\pi$
$812$	0	0
$813$	−1.74117e13	−1.39776
$814$	0	0
$815$	2.05206e13	1.62922
$816$	0	0
$817$	1.20252e13	0.944261
$818$	0	0
$819$	1.19573e12	0.0928658
$820$	0	0
$821$	−1.51952e13	−1.16724	−0.583622	−	0.812026i	$-0.698365\pi$
$821$	−1.51952e13	−1.16724	−0.583622	+	0.812026i	$0.698365\pi$
$822$	0	0
$823$	−8.32881e12	−0.632825	−0.316412	−	0.948622i	$-0.602478\pi$
$823$	−8.32881e12	−0.632825	−0.316412	+	0.948622i	$0.602478\pi$
$824$	0	0
$825$	−4.50977e11	−0.0338931
$826$	0	0
$827$	1.50997e13	1.12252	0.561258	−	0.827641i	$-0.310317\pi$
$827$	1.50997e13	1.12252	0.561258	+	0.827641i	$0.310317\pi$
$828$	0	0
$829$	−9.65383e12	−0.709912	−0.354956	−	0.934883i	$-0.615504\pi$
$829$	−9.65383e12	−0.709912	−0.354956	+	0.934883i	$0.615504\pi$
$830$	0	0
$831$	−1.30384e13	−0.948458
$832$	0	0
$833$	1.57498e13	1.13337
$834$	0	0
$835$	−1.90347e12	−0.135506
$836$	0	0
$837$	4.31701e10	0.00304032
$838$	0	0
$839$	−1.96235e13	−1.36725	−0.683623	−	0.729835i	$-0.739597\pi$
$839$	−1.96235e13	−1.36725	−0.683623	+	0.729835i	$0.739597\pi$
$840$	0	0
$841$	2.13181e13	1.46949
$842$	0	0
$843$	−1.58742e13	−1.08260
$844$	0	0
$845$	−1.04180e12	−0.0702959
$846$	0	0
$847$	−5.25399e12	−0.350763
$848$	0	0
$849$	2.32622e13	1.53662
$850$	0	0
$851$	−2.83439e13	−1.85258
$852$	0	0
$853$	−1.66671e13	−1.07793	−0.538963	−	0.842329i	$-0.681184\pi$
$853$	−1.66671e13	−1.07793	−0.538963	+	0.842329i	$0.681184\pi$
$854$	0	0
$855$	−1.24010e13	−0.793613
$856$	0	0
$857$	−8.56434e12	−0.542351	−0.271175	−	0.962530i	$-0.587412\pi$
$857$	−8.56434e12	−0.542351	−0.271175	+	0.962530i	$0.587412\pi$
$858$	0	0
$859$	−1.87649e12	−0.117592	−0.0587958	−	0.998270i	$-0.518726\pi$
$859$	−1.87649e12	−0.117592	−0.0587958	+	0.998270i	$0.518726\pi$
$860$	0	0
$861$	−1.22904e13	−0.762167
$862$	0	0
$863$	−2.54184e11	−0.0155991	−0.00779956	−	0.999970i	$-0.502483\pi$
$863$	−2.54184e11	−0.0155991	−0.00779956	+	0.999970i	$0.502483\pi$
$864$	0	0
$865$	8.45785e12	0.513674
$866$	0	0
$867$	1.60011e13	0.961753
$868$	0	0
$869$	3.07699e12	0.183036
$870$	0	0
$871$	−2.40066e12	−0.141335
$872$	0	0
$873$	2.10021e13	1.22376
$874$	0	0
$875$	6.61913e12	0.381737
$876$	0	0
$877$	−2.88730e13	−1.64814	−0.824069	−	0.566490i	$-0.808301\pi$
$877$	−2.88730e13	−1.64814	−0.824069	+	0.566490i	$0.808301\pi$
$878$	0	0
$879$	−1.35706e13	−0.766742
$880$	0	0
$881$	−1.43284e13	−0.801317	−0.400659	−	0.916227i	$-0.631219\pi$
$881$	−1.43284e13	−0.801317	−0.400659	+	0.916227i	$0.631219\pi$
$882$	0	0
$883$	−3.54065e12	−0.196001	−0.0980007	−	0.995186i	$-0.531245\pi$
$883$	−3.54065e12	−0.196001	−0.0980007	+	0.995186i	$0.531245\pi$
$884$	0	0
$885$	2.86037e13	1.56739
$886$	0	0
$887$	−1.07018e13	−0.580497	−0.290249	−	0.956951i	$-0.593738\pi$
$887$	−1.07018e13	−0.580497	−0.290249	+	0.956951i	$0.593738\pi$
$888$	0	0
$889$	−1.00245e13	−0.538274
$890$	0	0
$891$	−2.95297e12	−0.156967
$892$	0	0
$893$	−2.81127e13	−1.47935
$894$	0	0
$895$	1.80101e13	0.938235
$896$	0	0
$897$	−1.25012e13	−0.644740
$898$	0	0
$899$	−1.01531e12	−0.0518416
$900$	0	0
$901$	−1.42768e13	−0.721720
$902$	0	0
$903$	1.01152e13	0.506269
$904$	0	0
$905$	−3.05106e13	−1.51193
$906$	0	0
$907$	−2.74516e13	−1.34690	−0.673448	−	0.739234i	$-0.735188\pi$
$907$	−2.74516e13	−1.34690	−0.673448	+	0.739234i	$0.735188\pi$
$908$	0	0
$909$	−1.65155e13	−0.802331
$910$	0	0
$911$	−1.57903e13	−0.759550	−0.379775	−	0.925079i	$-0.623999\pi$
$911$	−1.57903e13	−0.759550	−0.379775	+	0.925079i	$0.623999\pi$
$912$	0	0
$913$	−1.74472e12	−0.0831013
$914$	0	0
$915$	1.94434e13	0.917016
$916$	0	0
$917$	2.77803e12	0.129740
$918$	0	0
$919$	−3.33575e13	−1.54267	−0.771335	−	0.636429i	$-0.780411\pi$
$919$	−3.33575e13	−1.54267	−0.771335	+	0.636429i	$0.780411\pi$
$920$	0	0
$921$	7.85564e12	0.359760
$922$	0	0
$923$	−3.59160e12	−0.162885
$924$	0	0
$925$	−4.06853e12	−0.182726
$926$	0	0
$927$	−1.12756e13	−0.501509
$928$	0	0
$929$	−5.90953e12	−0.260305	−0.130152	−	0.991494i	$-0.541547\pi$
$929$	−5.90953e12	−0.260305	−0.130152	+	0.991494i	$0.541547\pi$
$930$	0	0
$931$	−1.85785e13	−0.810471
$932$	0	0
$933$	3.48352e13	1.50505
$934$	0	0
$935$	4.10591e12	0.175694
$936$	0	0
$937$	2.94135e13	1.24657	0.623287	−	0.781993i	$-0.285797\pi$
$937$	2.94135e13	1.24657	0.623287	+	0.781993i	$0.285797\pi$
$938$	0	0
$939$	1.84276e12	0.0773522
$940$	0	0
$941$	−1.78277e13	−0.741212	−0.370606	−	0.928790i	$-0.620850\pi$
$941$	−1.78277e13	−0.741212	−0.370606	+	0.928790i	$0.620850\pi$
$942$	0	0
$943$	6.20448e13	2.55507
$944$	0	0
$945$	7.40404e11	0.0302013
$946$	0	0
$947$	2.20380e13	0.890427	0.445213	−	0.895425i	$-0.353128\pi$
$947$	2.20380e13	0.890427	0.445213	+	0.895425i	$0.353128\pi$
$948$	0	0
$949$	−5.37661e12	−0.215184
$950$	0	0
$951$	1.05856e13	0.419667
$952$	0	0
$953$	3.10318e13	1.21868	0.609340	−	0.792909i	$-0.291435\pi$
$953$	3.10318e13	1.21868	0.609340	+	0.792909i	$0.291435\pi$
$954$	0	0
$955$	−2.97156e13	−1.15603
$956$	0	0
$957$	8.38180e12	0.323023
$958$	0	0
$959$	−1.53367e13	−0.585527
$960$	0	0
$961$	−2.64108e13	−0.998912
$962$	0	0
$963$	−2.68799e13	−1.00719
$964$	0	0
$965$	−1.13351e13	−0.420778
$966$	0	0
$967$	−3.86641e13	−1.42196	−0.710982	−	0.703210i	$-0.751749\pi$
$967$	−3.86641e13	−1.42196	−0.710982	+	0.703210i	$0.751749\pi$
$968$	0	0
$969$	−4.61660e13	−1.68215
$970$	0	0
$971$	−2.91204e13	−1.05126	−0.525631	−	0.850713i	$-0.676171\pi$
$971$	−2.91204e13	−1.05126	−0.525631	+	0.850713i	$0.676171\pi$
$972$	0	0
$973$	−9.40108e11	−0.0336256
$974$	0	0
$975$	−1.79444e12	−0.0635927
$976$	0	0
$977$	−2.43957e13	−0.856618	−0.428309	−	0.903632i	$-0.640891\pi$
$977$	−2.43957e13	−0.856618	−0.428309	+	0.903632i	$0.640891\pi$
$978$	0	0
$979$	2.10110e12	0.0731012
$980$	0	0
$981$	1.31905e13	0.454729
$982$	0	0
$983$	−1.65480e13	−0.565268	−0.282634	−	0.959228i	$-0.591208\pi$
$983$	−1.65480e13	−0.565268	−0.282634	+	0.959228i	$0.591208\pi$
$984$	0	0
$985$	1.27323e13	0.430965
$986$	0	0
$987$	−2.36477e13	−0.793160
$988$	0	0
$989$	−5.10643e13	−1.69721
$990$	0	0
$991$	6.37548e12	0.209982	0.104991	−	0.994473i	$-0.466519\pi$
$991$	6.37548e12	0.209982	0.104991	+	0.994473i	$0.466519\pi$
$992$	0	0
$993$	−1.30086e13	−0.424581
$994$	0	0
$995$	−7.56122e12	−0.244561
$996$	0	0
$997$	1.52049e13	0.487366	0.243683	−	0.969855i	$-0.421644\pi$
$997$	1.52049e13	0.487366	0.243683	+	0.969855i	$0.421644\pi$
$998$	0	0
$999$	−3.21519e12	−0.102132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.h.1.5		5
4.3	odd	2		13.10.a.b.1.2	✓	5
12.11	even	2		117.10.a.e.1.4		5
20.19	odd	2		325.10.a.b.1.4		5
52.51	odd	2		169.10.a.b.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.a.b.1.2	✓	5	4.3	odd	2
117.10.a.e.1.4		5	12.11	even	2
169.10.a.b.1.4		5	52.51	odd	2
208.10.a.h.1.5		5	1.1	even	1	trivial
325.10.a.b.1.4		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q+195.094 q^{3} -1277.14 q^{5} +2277.98 q^{7} +18378.5 q^{9} +O(q^{10})\)	\(q+195.094 q^{3} -1277.14 q^{5} +2277.98 q^{7} +18378.5 q^{9} +7177.94 q^{11} +28561.0 q^{13} -249162. q^{15} -447890. q^{17} +528333. q^{19} +444419. q^{21} -2.24354e6 q^{23} -322041. q^{25} -254496. q^{27} +5.98542e6 q^{29} -169630. q^{31} +1.40037e6 q^{33} -2.90930e6 q^{35} +1.26336e7 q^{37} +5.57207e6 q^{39} -2.76549e7 q^{41} +2.27606e7 q^{43} -2.34719e7 q^{45} -5.32103e7 q^{47} -3.51644e7 q^{49} -8.73804e7 q^{51} +3.18756e7 q^{53} -9.16722e6 q^{55} +1.03074e8 q^{57} -1.14800e8 q^{59} -7.80352e7 q^{61} +4.18659e7 q^{63} -3.64764e7 q^{65} -8.40538e7 q^{67} -4.37700e8 q^{69} -1.25752e8 q^{71} -1.88250e8 q^{73} -6.28282e7 q^{75} +1.63512e7 q^{77} +4.28673e8 q^{79} -4.11395e8 q^{81} -2.43067e8 q^{83} +5.72018e8 q^{85} +1.16772e9 q^{87} +2.92716e8 q^{89} +6.50614e7 q^{91} -3.30937e7 q^{93} -6.74754e8 q^{95} +1.14275e9 q^{97} +1.31920e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9}+O(q^{10}) \) 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9	\( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27} + 10678182 q^{29} - 12885296 q^{31} + 17278298 q^{33} - 8380731 q^{35} + 7171823 q^{37} - 4598321 q^{39} + 9294012 q^{41} - 12831975 q^{43} + 26135198 q^{45} - 43354215 q^{47} + 25249488 q^{49} - 16905901 q^{51} + 93231780 q^{53} - 99448846 q^{55} + 90173382 q^{57} - 246496182 q^{59} - 132232612 q^{61} + 416955202 q^{63} + 51495483 q^{65} + 369388534 q^{67} - 579986760 q^{69} - 212150457 q^{71} - 252729806 q^{73} + 752457788 q^{75} + 449666118 q^{77} + 1247271728 q^{79} - 317713115 q^{81} - 1696894296 q^{83} - 775363765 q^{85} + 614530466 q^{87} - 753854382 q^{89} - 288437539 q^{91} - 892784668 q^{93} - 1442632962 q^{95} + 3824606 q^{97} - 3016199848 q^{99}+O(q^{100}) \) 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9 - 121746 * q^11 + 142805 * q^13 - 105973 * q^15 - 495669 * q^17 + 840738 * q^19 - 1599467 * q^21 + 592152 * q^23 + 1670362 * q^25 - 6847883 * q^27 + 10678182 * q^29 - 12885296 * q^31 + 17278298 * q^33 - 8380731 * q^35 + 7171823 * q^37 - 4598321 * q^39 + 9294012 * q^41 - 12831975 * q^43 + 26135198 * q^45 - 43354215 * q^47 + 25249488 * q^49 - 16905901 * q^51 + 93231780 * q^53 - 99448846 * q^55 + 90173382 * q^57 - 246496182 * q^59 - 132232612 * q^61 + 416955202 * q^63 + 51495483 * q^65 + 369388534 * q^67 - 579986760 * q^69 - 212150457 * q^71 - 252729806 * q^73 + 752457788 * q^75 + 449666118 * q^77 + 1247271728 * q^79 - 317713115 * q^81 - 1696894296 * q^83 - 775363765 * q^85 + 614530466 * q^87 - 753854382 * q^89 - 288437539 * q^91 - 892784668 * q^93 - 1442632962 * q^95 + 3824606 * q^97 - 3016199848 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more