LMFDB - Embedded newform 208.10.a.d.1.3

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 = [3,0,-156] 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:	$3$
Coefficient field:	3.3.2119705.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 376x + 1820 $ x^3 - x^2 - 376*x + 1820
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$16.8848$ 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)

$f(q)$	$=$	$q+178.768 q^{3} -2343.01 q^{5} -10882.2 q^{7} +12274.9 q^{9} -80220.9 q^{11} -28561.0 q^{13} -418855. q^{15} +193066. q^{17} +290332. q^{19} -1.94538e6 q^{21} -1.11158e6 q^{23} +3.53659e6 q^{25} -1.32434e6 q^{27} -2.97253e6 q^{29} -7.98567e6 q^{31} -1.43409e7 q^{33} +2.54971e7 q^{35} -762787. q^{37} -5.10578e6 q^{39} -4.07656e6 q^{41} +4.14147e6 q^{43} -2.87601e7 q^{45} +2.04292e7 q^{47} +7.80686e7 q^{49} +3.45140e7 q^{51} +5.05420e7 q^{53} +1.87959e8 q^{55} +5.19020e7 q^{57} +6.13428e7 q^{59} +1.31190e8 q^{61} -1.33577e8 q^{63} +6.69188e7 q^{65} -3.22558e8 q^{67} -1.98714e8 q^{69} +5.50167e7 q^{71} +1.03200e8 q^{73} +6.32227e8 q^{75} +8.72979e8 q^{77} -6.04613e8 q^{79} -4.78354e8 q^{81} +5.67238e8 q^{83} -4.52357e8 q^{85} -5.31391e8 q^{87} +1.37799e8 q^{89} +3.10806e8 q^{91} -1.42758e9 q^{93} -6.80252e8 q^{95} -1.19935e7 q^{97} -9.84699e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 156 q^{3} - 1272 q^{5} - 17058 q^{7} + 42273 q^{9} - 73974 q^{11} - 85683 q^{13} - 393756 q^{15} + 374976 q^{17} - 418338 q^{19} - 284694 q^{21} - 1026168 q^{23} + 3337287 q^{25} - 4218588 q^{27}+ \cdots + 2480087466 q^{99}+O(q^{100}) $ 3 * q - 156 * q^3 - 1272 * q^5 - 17058 * q^7 + 42273 * q^9 - 73974 * q^11 - 85683 * q^13 - 393756 * q^15 + 374976 * q^17 - 418338 * q^19 - 284694 * q^21 - 1026168 * q^23 + 3337287 * q^25 - 4218588 * q^27 - 3075834 * q^29 - 9286482 * q^31 - 25456452 * q^33 + 31802592 * q^35 - 17647776 * q^37 + 4455516 * q^39 - 47257110 * q^41 + 60023760 * q^43 - 81112122 * q^45 + 40824726 * q^47 + 45921177 * q^49 + 27559188 * q^51 + 8072046 * q^53 + 36910236 * q^55 + 245224476 * q^57 - 138035310 * q^59 + 264203886 * q^61 - 436094838 * q^63 + 36329592 * q^65 - 203167074 * q^67 - 397637544 * q^69 - 123067110 * q^71 - 433013250 * q^73 + 884469216 * q^75 + 380043264 * q^77 - 406418748 * q^79 + 100121427 * q^81 - 105365610 * q^83 + 5268186 * q^85 + 12753768 * q^87 + 1365375798 * q^89 + 487193538 * q^91 - 95909856 * q^93 + 85600692 * q^95 + 669691662 * q^97 + 2480087466 * 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
$3$	178.768	1.27422	0.637108	−	0.770774i	$-0.280130\pi$
$3$	178.768	1.27422	0.637108	+	0.770774i	$0.280130\pi$
$4$	0	0
$5$	−2343.01	−1.67652	−0.838262	−	0.545268i	$-0.816428\pi$
$5$	−2343.01	−1.67652	−0.838262	+	0.545268i	$0.816428\pi$
$6$	0	0
$7$	−10882.2	−1.71307	−0.856536	−	0.516088i	$-0.827388\pi$
$7$	−10882.2	−1.71307	−0.856536	+	0.516088i	$0.827388\pi$
$8$	0	0
$9$	12274.9	0.623627
$10$	0	0
$11$	−80220.9	−1.65204	−0.826019	−	0.563642i	$-0.809400\pi$
$11$	−80220.9	−1.65204	−0.826019	+	0.563642i	$0.809400\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	−418855.	−2.13625
$16$	0	0
$17$	193066.	0.560643	0.280322	−	0.959906i	$-0.409559\pi$
$17$	193066.	0.560643	0.280322	+	0.959906i	$0.409559\pi$
$18$	0	0
$19$	290332.	0.511097	0.255549	−	0.966796i	$-0.417744\pi$
$19$	290332.	0.511097	0.255549	+	0.966796i	$0.417744\pi$
$20$	0	0
$21$	−1.94538e6	−2.18282
$22$	0	0
$23$	−1.11158e6	−0.828258	−0.414129	−	0.910218i	$-0.635914\pi$
$23$	−1.11158e6	−0.828258	−0.414129	+	0.910218i	$0.635914\pi$
$24$	0	0
$25$	3.53659e6	1.81073
$26$	0	0
$27$	−1.32434e6	−0.479581
$28$	0	0
$29$	−2.97253e6	−0.780431	−0.390216	−	0.920724i	$-0.627600\pi$
$29$	−2.97253e6	−0.780431	−0.390216	+	0.920724i	$0.627600\pi$
$30$	0	0
$31$	−7.98567e6	−1.55304	−0.776522	−	0.630090i	$-0.783018\pi$
$31$	−7.98567e6	−1.55304	−0.776522	+	0.630090i	$0.783018\pi$
$32$	0	0
$33$	−1.43409e7	−2.10505
$34$	0	0
$35$	2.54971e7	2.87201
$36$	0	0
$37$	−762787.	−0.0669107	−0.0334554	−	0.999440i	$-0.510651\pi$
$37$	−762787.	−0.0669107	−0.0334554	+	0.999440i	$0.510651\pi$
$38$	0	0
$39$	−5.10578e6	−0.353404
$40$	0	0
$41$	−4.07656e6	−0.225303	−0.112651	−	0.993635i	$-0.535934\pi$
$41$	−4.07656e6	−0.225303	−0.112651	+	0.993635i	$0.535934\pi$
$42$	0	0
$43$	4.14147e6	0.184734	0.0923669	−	0.995725i	$-0.470557\pi$
$43$	4.14147e6	0.184734	0.0923669	+	0.995725i	$0.470557\pi$
$44$	0	0
$45$	−2.87601e7	−1.04553
$46$	0	0
$47$	2.04292e7	0.610677	0.305339	−	0.952244i	$-0.401230\pi$
$47$	2.04292e7	0.610677	0.305339	+	0.952244i	$0.401230\pi$
$48$	0	0
$49$	7.80686e7	1.93461
$50$	0	0
$51$	3.45140e7	0.714381
$52$	0	0
$53$	5.05420e7	0.879855	0.439927	−	0.898033i	$-0.355004\pi$
$53$	5.05420e7	0.879855	0.439927	+	0.898033i	$0.355004\pi$
$54$	0	0
$55$	1.87959e8	2.76968
$56$	0	0
$57$	5.19020e7	0.651249
$58$	0	0
$59$	6.13428e7	0.659067	0.329533	−	0.944144i	$-0.393109\pi$
$59$	6.13428e7	0.659067	0.329533	+	0.944144i	$0.393109\pi$
$60$	0	0
$61$	1.31190e8	1.21316	0.606578	−	0.795024i	$-0.292542\pi$
$61$	1.31190e8	1.21316	0.606578	+	0.795024i	$0.292542\pi$
$62$	0	0
$63$	−1.33577e8	−1.06832
$64$	0	0
$65$	6.69188e7	0.464984
$66$	0	0
$67$	−3.22558e8	−1.95556	−0.977782	−	0.209626i	$-0.932775\pi$
$67$	−3.22558e8	−1.95556	−0.977782	+	0.209626i	$0.932775\pi$
$68$	0	0
$69$	−1.98714e8	−1.05538
$70$	0	0
$71$	5.50167e7	0.256940	0.128470	−	0.991713i	$-0.458993\pi$
$71$	5.50167e7	0.256940	0.128470	+	0.991713i	$0.458993\pi$
$72$	0	0
$73$	1.03200e8	0.425330	0.212665	−	0.977125i	$-0.431786\pi$
$73$	1.03200e8	0.425330	0.212665	+	0.977125i	$0.431786\pi$
$74$	0	0
$75$	6.32227e8	2.30727
$76$	0	0
$77$	8.72979e8	2.83006
$78$	0	0
$79$	−6.04613e8	−1.74645	−0.873223	−	0.487320i	$-0.837975\pi$
$79$	−6.04613e8	−1.74645	−0.873223	+	0.487320i	$0.837975\pi$
$80$	0	0
$81$	−4.78354e8	−1.23472
$82$	0	0
$83$	5.67238e8	1.31194	0.655970	−	0.754787i	$-0.272260\pi$
$83$	5.67238e8	1.31194	0.655970	+	0.754787i	$0.272260\pi$
$84$	0	0
$85$	−4.52357e8	−0.939932
$86$	0	0
$87$	−5.31391e8	−0.994438
$88$	0	0
$89$	1.37799e8	0.232805	0.116402	−	0.993202i	$-0.462864\pi$
$89$	1.37799e8	0.232805	0.116402	+	0.993202i	$0.462864\pi$
$90$	0	0
$91$	3.10806e8	0.475120
$92$	0	0
$93$	−1.42758e9	−1.97891
$94$	0	0
$95$	−6.80252e8	−0.856867
$96$	0	0
$97$	−1.19935e7	−0.0137554	−0.00687768	−	0.999976i	$-0.502189\pi$
$97$	−1.19935e7	−0.0137554	−0.00687768	+	0.999976i	$0.502189\pi$
$98$	0	0
$99$	−9.84699e8	−1.03026
$100$	0	0
$101$	−4.21284e8	−0.402836	−0.201418	−	0.979505i	$-0.564555\pi$
$101$	−4.21284e8	−0.402836	−0.201418	+	0.979505i	$0.564555\pi$
$102$	0	0
$103$	1.32027e9	1.15584	0.577918	−	0.816095i	$-0.303865\pi$
$103$	1.32027e9	1.15584	0.577918	+	0.816095i	$0.303865\pi$
$104$	0	0
$105$	4.55806e9	3.65956
$106$	0	0
$107$	2.86202e8	0.211079	0.105540	−	0.994415i	$-0.466343\pi$
$107$	2.86202e8	0.211079	0.105540	+	0.994415i	$0.466343\pi$
$108$	0	0
$109$	−1.50243e9	−1.01947	−0.509737	−	0.860330i	$-0.670257\pi$
$109$	−1.50243e9	−1.01947	−0.509737	+	0.860330i	$0.670257\pi$
$110$	0	0
$111$	−1.36362e8	−0.0852587
$112$	0	0
$113$	−1.46902e9	−0.847569	−0.423784	−	0.905763i	$-0.639299\pi$
$113$	−1.46902e9	−0.847569	−0.423784	+	0.905763i	$0.639299\pi$
$114$	0	0
$115$	2.60445e9	1.38859
$116$	0	0
$117$	−3.50582e8	−0.172963
$118$	0	0
$119$	−2.10099e9	−0.960422
$120$	0	0
$121$	4.07744e9	1.72923
$122$	0	0
$123$	−7.28756e8	−0.287084
$124$	0	0
$125$	−3.71007e9	−1.35921
$126$	0	0
$127$	−3.62878e9	−1.23778	−0.618890	−	0.785478i	$-0.712417\pi$
$127$	−3.62878e9	−1.23778	−0.618890	+	0.785478i	$0.712417\pi$
$128$	0	0
$129$	7.40361e8	0.235391
$130$	0	0
$131$	2.56547e9	0.761108	0.380554	−	0.924759i	$-0.375733\pi$
$131$	2.56547e9	0.761108	0.380554	+	0.924759i	$0.375733\pi$
$132$	0	0
$133$	−3.15945e9	−0.875546
$134$	0	0
$135$	3.10294e9	0.804028
$136$	0	0
$137$	−2.71592e9	−0.658680	−0.329340	−	0.944211i	$-0.606826\pi$
$137$	−2.71592e9	−0.658680	−0.329340	+	0.944211i	$0.606826\pi$
$138$	0	0
$139$	−1.38690e9	−0.315122	−0.157561	−	0.987509i	$-0.550363\pi$
$139$	−1.38690e9	−0.315122	−0.157561	+	0.987509i	$0.550363\pi$
$140$	0	0
$141$	3.65209e9	0.778135
$142$	0	0
$143$	2.29119e9	0.458193
$144$	0	0
$145$	6.96467e9	1.30841
$146$	0	0
$147$	1.39561e10	2.46512
$148$	0	0
$149$	2.31719e9	0.385143	0.192572	−	0.981283i	$-0.438317\pi$
$149$	2.31719e9	0.385143	0.192572	+	0.981283i	$0.438317\pi$
$150$	0	0
$151$	8.24464e9	1.29055	0.645276	−	0.763950i	$-0.276742\pi$
$151$	8.24464e9	1.29055	0.645276	+	0.763950i	$0.276742\pi$
$152$	0	0
$153$	2.36986e9	0.349632
$154$	0	0
$155$	1.87105e10	2.60372
$156$	0	0
$157$	6.19786e8	0.0814130	0.0407065	−	0.999171i	$-0.487039\pi$
$157$	6.19786e8	0.0814130	0.0407065	+	0.999171i	$0.487039\pi$
$158$	0	0
$159$	9.03527e9	1.12113
$160$	0	0
$161$	1.20964e10	1.41886
$162$	0	0
$163$	−4.01230e9	−0.445194	−0.222597	−	0.974911i	$-0.571453\pi$
$163$	−4.01230e9	−0.445194	−0.222597	+	0.974911i	$0.571453\pi$
$164$	0	0
$165$	3.36009e10	3.52917
$166$	0	0
$167$	−8.70691e9	−0.866244	−0.433122	−	0.901335i	$-0.642588\pi$
$167$	−8.70691e9	−0.866244	−0.433122	+	0.901335i	$0.642588\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	3.56378e9	0.318734
$172$	0	0
$173$	−8.79323e9	−0.746347	−0.373174	−	0.927762i	$-0.621730\pi$
$173$	−8.79323e9	−0.746347	−0.373174	+	0.927762i	$0.621730\pi$
$174$	0	0
$175$	−3.84858e10	−3.10191
$176$	0	0
$177$	1.09661e10	0.839793
$178$	0	0
$179$	−1.75285e10	−1.27616	−0.638080	−	0.769970i	$-0.720271\pi$
$179$	−1.75285e10	−1.27616	−0.638080	+	0.769970i	$0.720271\pi$
$180$	0	0
$181$	8.13160e9	0.563148	0.281574	−	0.959540i	$-0.409144\pi$
$181$	8.13160e9	0.563148	0.281574	+	0.959540i	$0.409144\pi$
$182$	0	0
$183$	2.34525e10	1.54582
$184$	0	0
$185$	1.78722e9	0.112177
$186$	0	0
$187$	−1.54879e10	−0.926204
$188$	0	0
$189$	1.44117e10	0.821556
$190$	0	0
$191$	2.60980e10	1.41892	0.709458	−	0.704748i	$-0.248940\pi$
$191$	2.60980e10	1.41892	0.709458	+	0.704748i	$0.248940\pi$
$192$	0	0
$193$	−2.60832e10	−1.35317	−0.676586	−	0.736364i	$-0.736541\pi$
$193$	−2.60832e10	−1.35317	−0.676586	+	0.736364i	$0.736541\pi$
$194$	0	0
$195$	1.19629e10	0.592490
$196$	0	0
$197$	−1.78237e10	−0.843142	−0.421571	−	0.906795i	$-0.638521\pi$
$197$	−1.78237e10	−0.843142	−0.421571	+	0.906795i	$0.638521\pi$
$198$	0	0
$199$	−1.37597e10	−0.621969	−0.310985	−	0.950415i	$-0.600659\pi$
$199$	−1.37597e10	−0.621969	−0.310985	+	0.950415i	$0.600659\pi$
$200$	0	0
$201$	−5.76630e10	−2.49181
$202$	0	0
$203$	3.23476e10	1.33693
$204$	0	0
$205$	9.55143e9	0.377725
$206$	0	0
$207$	−1.36445e10	−0.516524
$208$	0	0
$209$	−2.32907e10	−0.844353
$210$	0	0
$211$	−2.34073e9	−0.0812980	−0.0406490	−	0.999173i	$-0.512943\pi$
$211$	−2.34073e9	−0.0812980	−0.0406490	+	0.999173i	$0.512943\pi$
$212$	0	0
$213$	9.83520e9	0.327397
$214$	0	0
$215$	−9.70352e9	−0.309711
$216$	0	0
$217$	8.69017e10	2.66048
$218$	0	0
$219$	1.84488e10	0.541962
$220$	0	0
$221$	−5.51417e9	−0.155494
$222$	0	0
$223$	2.60853e10	0.706356	0.353178	−	0.935556i	$-0.385101\pi$
$223$	2.60853e10	0.706356	0.353178	+	0.935556i	$0.385101\pi$
$224$	0	0
$225$	4.34111e10	1.12922
$226$	0	0
$227$	1.32079e10	0.330156	0.165078	−	0.986281i	$-0.447212\pi$
$227$	1.32079e10	0.330156	0.165078	+	0.986281i	$0.447212\pi$
$228$	0	0
$229$	−3.78030e10	−0.908379	−0.454190	−	0.890905i	$-0.650071\pi$
$229$	−3.78030e10	−0.908379	−0.454190	+	0.890905i	$0.650071\pi$
$230$	0	0
$231$	1.56060e11	3.60611
$232$	0	0
$233$	2.12787e10	0.472982	0.236491	−	0.971634i	$-0.424003\pi$
$233$	2.12787e10	0.472982	0.236491	+	0.971634i	$0.424003\pi$
$234$	0	0
$235$	−4.78660e10	−1.02382
$236$	0	0
$237$	−1.08085e11	−2.22535
$238$	0	0
$239$	−8.35483e10	−1.65633	−0.828165	−	0.560484i	$-0.810615\pi$
$239$	−8.35483e10	−1.65633	−0.828165	+	0.560484i	$0.810615\pi$
$240$	0	0
$241$	9.98011e10	1.90572	0.952859	−	0.303414i	$-0.0981264\pi$
$241$	9.98011e10	1.90572	0.952859	+	0.303414i	$0.0981264\pi$
$242$	0	0
$243$	−5.94473e10	−1.09372
$244$	0	0
$245$	−1.82916e11	−3.24342
$246$	0	0
$247$	−8.29217e9	−0.141753
$248$	0	0
$249$	1.01404e11	1.67169
$250$	0	0
$251$	3.96659e9	0.0630792	0.0315396	−	0.999503i	$-0.489959\pi$
$251$	3.96659e9	0.0630792	0.0315396	+	0.999503i	$0.489959\pi$
$252$	0	0
$253$	8.91719e10	1.36831
$254$	0	0
$255$	−8.08668e10	−1.19768
$256$	0	0
$257$	1.94473e10	0.278074	0.139037	−	0.990287i	$-0.455599\pi$
$257$	1.94473e10	0.278074	0.139037	+	0.990287i	$0.455599\pi$
$258$	0	0
$259$	8.30080e9	0.114623
$260$	0	0
$261$	−3.64873e10	−0.486698
$262$	0	0
$263$	8.88949e10	1.14571	0.572857	−	0.819656i	$-0.305835\pi$
$263$	8.88949e10	1.14571	0.572857	+	0.819656i	$0.305835\pi$
$264$	0	0
$265$	−1.18421e11	−1.47510
$266$	0	0
$267$	2.46340e10	0.296644
$268$	0	0
$269$	4.82616e10	0.561974	0.280987	−	0.959712i	$-0.409338\pi$
$269$	4.82616e10	0.561974	0.280987	+	0.959712i	$0.409338\pi$
$270$	0	0
$271$	6.58608e10	0.741763	0.370881	−	0.928680i	$-0.379056\pi$
$271$	6.58608e10	0.741763	0.370881	+	0.928680i	$0.379056\pi$
$272$	0	0
$273$	5.55621e10	0.605406
$274$	0	0
$275$	−2.83708e11	−2.99140
$276$	0	0
$277$	−1.32558e11	−1.35284	−0.676418	−	0.736518i	$-0.736469\pi$
$277$	−1.32558e11	−1.35284	−0.676418	+	0.736518i	$0.736469\pi$
$278$	0	0
$279$	−9.80229e10	−0.968521
$280$	0	0
$281$	1.77381e11	1.69719	0.848593	−	0.529046i	$-0.177450\pi$
$281$	1.77381e11	1.69719	0.848593	+	0.529046i	$0.177450\pi$
$282$	0	0
$283$	3.52850e10	0.327003	0.163502	−	0.986543i	$-0.447721\pi$
$283$	3.52850e10	0.327003	0.163502	+	0.986543i	$0.447721\pi$
$284$	0	0
$285$	−1.21607e11	−1.09183
$286$	0	0
$287$	4.43619e10	0.385959
$288$	0	0
$289$	−8.13133e10	−0.685679
$290$	0	0
$291$	−2.14404e9	−0.0175273
$292$	0	0
$293$	−7.69061e10	−0.609616	−0.304808	−	0.952414i	$-0.598592\pi$
$293$	−7.69061e10	−0.609616	−0.304808	+	0.952414i	$0.598592\pi$
$294$	0	0
$295$	−1.43727e11	−1.10494
$296$	0	0
$297$	1.06239e11	0.792286
$298$	0	0
$299$	3.17478e10	0.229717
$300$	0	0
$301$	−4.50683e10	−0.316462
$302$	0	0
$303$	−7.53118e10	−0.513300
$304$	0	0
$305$	−3.07380e11	−2.03389
$306$	0	0
$307$	−1.38440e11	−0.889484	−0.444742	−	0.895659i	$-0.646705\pi$
$307$	−1.38440e11	−0.889484	−0.444742	+	0.895659i	$0.646705\pi$
$308$	0	0
$309$	2.36022e11	1.47279
$310$	0	0
$311$	−6.65986e10	−0.403686	−0.201843	−	0.979418i	$-0.564693\pi$
$311$	−6.65986e10	−0.403686	−0.201843	+	0.979418i	$0.564693\pi$
$312$	0	0
$313$	−1.16628e11	−0.686838	−0.343419	−	0.939182i	$-0.611585\pi$
$313$	−1.16628e11	−0.686838	−0.343419	+	0.939182i	$0.611585\pi$
$314$	0	0
$315$	3.12974e11	1.79106
$316$	0	0
$317$	1.15396e11	0.641835	0.320918	−	0.947107i	$-0.396009\pi$
$317$	1.15396e11	0.641835	0.320918	+	0.947107i	$0.396009\pi$
$318$	0	0
$319$	2.38459e11	1.28930
$320$	0	0
$321$	5.11637e10	0.268961
$322$	0	0
$323$	5.60533e10	0.286543
$324$	0	0
$325$	−1.01008e11	−0.502207
$326$	0	0
$327$	−2.68587e11	−1.29903
$328$	0	0
$329$	−2.22315e11	−1.04613
$330$	0	0
$331$	−3.35395e11	−1.53578	−0.767892	−	0.640580i	$-0.778694\pi$
$331$	−3.35395e11	−1.53578	−0.767892	+	0.640580i	$0.778694\pi$
$332$	0	0
$333$	−9.36310e9	−0.0417273
$334$	0	0
$335$	7.55759e11	3.27855
$336$	0	0
$337$	−2.66081e11	−1.12377	−0.561887	−	0.827214i	$-0.689924\pi$
$337$	−2.66081e11	−1.12377	−0.561887	+	0.827214i	$0.689924\pi$
$338$	0	0
$339$	−2.62613e11	−1.07999
$340$	0	0
$341$	6.40618e11	2.56569
$342$	0	0
$343$	−4.10422e11	−1.60106
$344$	0	0
$345$	4.65591e11	1.76937
$346$	0	0
$347$	4.29253e11	1.58939	0.794696	−	0.607008i	$-0.207630\pi$
$347$	4.29253e11	1.58939	0.794696	+	0.607008i	$0.207630\pi$
$348$	0	0
$349$	−2.94419e10	−0.106231	−0.0531155	−	0.998588i	$-0.516915\pi$
$349$	−2.94419e10	−0.106231	−0.0531155	+	0.998588i	$0.516915\pi$
$350$	0	0
$351$	3.78244e10	0.133012
$352$	0	0
$353$	1.59952e11	0.548280	0.274140	−	0.961690i	$-0.411607\pi$
$353$	1.59952e11	0.548280	0.274140	+	0.961690i	$0.411607\pi$
$354$	0	0
$355$	−1.28905e11	−0.430766
$356$	0	0
$357$	−3.75588e11	−1.22378
$358$	0	0
$359$	−5.47928e11	−1.74100	−0.870499	−	0.492170i	$-0.836204\pi$
$359$	−5.47928e11	−1.74100	−0.870499	+	0.492170i	$0.836204\pi$
$360$	0	0
$361$	−2.38395e11	−0.738779
$362$	0	0
$363$	7.28914e11	2.20342
$364$	0	0
$365$	−2.41798e11	−0.713075
$366$	0	0
$367$	2.04029e11	0.587075	0.293538	−	0.955948i	$-0.405167\pi$
$367$	2.04029e11	0.587075	0.293538	+	0.955948i	$0.405167\pi$
$368$	0	0
$369$	−5.00391e10	−0.140505
$370$	0	0
$371$	−5.50008e11	−1.50725
$372$	0	0
$373$	4.85466e11	1.29858	0.649290	−	0.760541i	$-0.275066\pi$
$373$	4.85466e11	1.29858	0.649290	+	0.760541i	$0.275066\pi$
$374$	0	0
$375$	−6.63241e11	−1.73193
$376$	0	0
$377$	8.48983e10	0.216453
$378$	0	0
$379$	3.51410e11	0.874859	0.437430	−	0.899253i	$-0.355889\pi$
$379$	3.51410e11	0.874859	0.437430	+	0.899253i	$0.355889\pi$
$380$	0	0
$381$	−6.48708e11	−1.57720
$382$	0	0
$383$	−7.81319e11	−1.85539	−0.927693	−	0.373345i	$-0.878211\pi$
$383$	−7.81319e11	−1.85539	−0.927693	+	0.373345i	$0.878211\pi$
$384$	0	0
$385$	−2.04540e12	−4.74466
$386$	0	0
$387$	5.08359e10	0.115205
$388$	0	0
$389$	−3.32630e11	−0.736525	−0.368263	−	0.929722i	$-0.620047\pi$
$389$	−3.32630e11	−0.736525	−0.368263	+	0.929722i	$0.620047\pi$
$390$	0	0
$391$	−2.14609e11	−0.464357
$392$	0	0
$393$	4.58623e11	0.969816
$394$	0	0
$395$	1.41662e12	2.92796
$396$	0	0
$397$	1.97334e11	0.398698	0.199349	−	0.979929i	$-0.436117\pi$
$397$	1.97334e11	0.398698	0.199349	+	0.979929i	$0.436117\pi$
$398$	0	0
$399$	−5.64807e11	−1.11564
$400$	0	0
$401$	−4.44050e11	−0.857595	−0.428798	−	0.903401i	$-0.641063\pi$
$401$	−4.44050e11	−0.857595	−0.428798	+	0.903401i	$0.641063\pi$
$402$	0	0
$403$	2.28079e11	0.430737
$404$	0	0
$405$	1.12079e12	2.07003
$406$	0	0
$407$	6.11915e10	0.110539
$408$	0	0
$409$	−1.98941e11	−0.351536	−0.175768	−	0.984432i	$-0.556241\pi$
$409$	−1.98941e11	−0.351536	−0.175768	+	0.984432i	$0.556241\pi$
$410$	0	0
$411$	−4.85519e11	−0.839301
$412$	0	0
$413$	−6.67544e11	−1.12903
$414$	0	0
$415$	−1.32905e12	−2.19950
$416$	0	0
$417$	−2.47933e11	−0.401534
$418$	0	0
$419$	−5.54213e11	−0.878444	−0.439222	−	0.898379i	$-0.644746\pi$
$419$	−5.54213e11	−0.878444	−0.439222	+	0.898379i	$0.644746\pi$
$420$	0	0
$421$	3.86989e11	0.600384	0.300192	−	0.953879i	$-0.402949\pi$
$421$	3.86989e11	0.600384	0.300192	+	0.953879i	$0.402949\pi$
$422$	0	0
$423$	2.50766e11	0.380835
$424$	0	0
$425$	6.82796e11	1.01517
$426$	0	0
$427$	−1.42764e12	−2.07822
$428$	0	0
$429$	4.09590e11	0.583837
$430$	0	0
$431$	9.78340e11	1.36566	0.682829	−	0.730578i	$-0.260749\pi$
$431$	9.78340e11	1.36566	0.682829	+	0.730578i	$0.260749\pi$
$432$	0	0
$433$	−9.07692e10	−0.124092	−0.0620459	−	0.998073i	$-0.519763\pi$
$433$	−9.07692e10	−0.124092	−0.0620459	+	0.998073i	$0.519763\pi$
$434$	0	0
$435$	1.24506e12	1.66720
$436$	0	0
$437$	−3.22727e11	−0.423320
$438$	0	0
$439$	−1.86949e11	−0.240233	−0.120117	−	0.992760i	$-0.538327\pi$
$439$	−1.86949e11	−0.240233	−0.120117	+	0.992760i	$0.538327\pi$
$440$	0	0
$441$	9.58281e11	1.20648
$442$	0	0
$443$	1.60031e12	1.97418	0.987090	−	0.160165i	$-0.0512028\pi$
$443$	1.60031e12	1.97418	0.987090	+	0.160165i	$0.0512028\pi$
$444$	0	0
$445$	−3.22865e11	−0.390303
$446$	0	0
$447$	4.14238e11	0.490756
$448$	0	0
$449$	−4.68942e11	−0.544516	−0.272258	−	0.962224i	$-0.587770\pi$
$449$	−4.68942e11	−0.544516	−0.272258	+	0.962224i	$0.587770\pi$
$450$	0	0
$451$	3.27025e11	0.372209
$452$	0	0
$453$	1.47387e12	1.64444
$454$	0	0
$455$	−7.28224e11	−0.796551
$456$	0	0
$457$	−1.32147e12	−1.41721	−0.708603	−	0.705607i	$-0.750674\pi$
$457$	−1.32147e12	−1.41721	−0.708603	+	0.705607i	$0.750674\pi$
$458$	0	0
$459$	−2.55685e11	−0.268874
$460$	0	0
$461$	1.51257e12	1.55977	0.779885	−	0.625923i	$-0.215278\pi$
$461$	1.51257e12	1.55977	0.779885	+	0.625923i	$0.215278\pi$
$462$	0	0
$463$	−7.55476e11	−0.764022	−0.382011	−	0.924158i	$-0.624768\pi$
$463$	−7.55476e11	−0.764022	−0.382011	+	0.924158i	$0.624768\pi$
$464$	0	0
$465$	3.34484e12	3.31770
$466$	0	0
$467$	6.15598e11	0.598923	0.299462	−	0.954108i	$-0.403193\pi$
$467$	6.15598e11	0.598923	0.299462	+	0.954108i	$0.403193\pi$
$468$	0	0
$469$	3.51014e12	3.35002
$470$	0	0
$471$	1.10798e11	0.103738
$472$	0	0
$473$	−3.32232e11	−0.305187
$474$	0	0
$475$	1.02678e12	0.925461
$476$	0	0
$477$	6.20396e11	0.548701
$478$	0	0
$479$	1.26054e12	1.09408	0.547039	−	0.837107i	$-0.315755\pi$
$479$	1.26054e12	1.09408	0.547039	+	0.837107i	$0.315755\pi$
$480$	0	0
$481$	2.17860e10	0.0185577
$482$	0	0
$483$	2.16245e12	1.80794
$484$	0	0
$485$	2.81009e10	0.0230612
$486$	0	0
$487$	7.25359e11	0.584350	0.292175	−	0.956365i	$-0.405621\pi$
$487$	7.25359e11	0.584350	0.292175	+	0.956365i	$0.405621\pi$
$488$	0	0
$489$	−7.17270e11	−0.567274
$490$	0	0
$491$	2.15372e11	0.167233	0.0836166	−	0.996498i	$-0.473353\pi$
$491$	2.15372e11	0.167233	0.0836166	+	0.996498i	$0.473353\pi$
$492$	0	0
$493$	−5.73895e11	−0.437543
$494$	0	0
$495$	2.30716e12	1.72725
$496$	0	0
$497$	−5.98703e11	−0.440157
$498$	0	0
$499$	1.51150e12	1.09133	0.545665	−	0.838004i	$-0.316277\pi$
$499$	1.51150e12	1.09133	0.545665	+	0.838004i	$0.316277\pi$
$500$	0	0
$501$	−1.55651e12	−1.10378
$502$	0	0
$503$	1.93478e12	1.34765	0.673823	−	0.738893i	$-0.264651\pi$
$503$	1.93478e12	1.34765	0.673823	+	0.738893i	$0.264651\pi$
$504$	0	0
$505$	9.87073e11	0.675364
$506$	0	0
$507$	1.45826e11	0.0980166
$508$	0	0
$509$	7.92174e11	0.523107	0.261553	−	0.965189i	$-0.415765\pi$
$509$	7.92174e11	0.523107	0.261553	+	0.965189i	$0.415765\pi$
$510$	0	0
$511$	−1.12304e12	−0.728620
$512$	0	0
$513$	−3.84497e11	−0.245112
$514$	0	0
$515$	−3.09342e12	−1.93779
$516$	0	0
$517$	−1.63885e12	−1.00886
$518$	0	0
$519$	−1.57194e12	−0.951008
$520$	0	0
$521$	−4.77706e11	−0.284047	−0.142024	−	0.989863i	$-0.545361\pi$
$521$	−4.77706e11	−0.284047	−0.142024	+	0.989863i	$0.545361\pi$
$522$	0	0
$523$	2.40926e12	1.40808	0.704039	−	0.710162i	$-0.251378\pi$
$523$	2.40926e12	1.40808	0.704039	+	0.710162i	$0.251378\pi$
$524$	0	0
$525$	−6.88002e12	−3.95251
$526$	0	0
$527$	−1.54176e12	−0.870704
$528$	0	0
$529$	−5.65543e11	−0.313989
$530$	0	0
$531$	7.52973e11	0.411012
$532$	0	0
$533$	1.16431e11	0.0624877
$534$	0	0
$535$	−6.70576e11	−0.353880
$536$	0	0
$537$	−3.13352e12	−1.62610
$538$	0	0
$539$	−6.26273e12	−3.19606
$540$	0	0
$541$	2.74866e12	1.37954	0.689768	−	0.724030i	$-0.257712\pi$
$541$	2.74866e12	1.37954	0.689768	+	0.724030i	$0.257712\pi$
$542$	0	0
$543$	1.45367e12	0.717572
$544$	0	0
$545$	3.52022e12	1.70917
$546$	0	0
$547$	2.92795e12	1.39836	0.699182	−	0.714944i	$-0.253548\pi$
$547$	2.92795e12	1.39836	0.699182	+	0.714944i	$0.253548\pi$
$548$	0	0
$549$	1.61034e12	0.756557
$550$	0	0
$551$	−8.63019e11	−0.398876
$552$	0	0
$553$	6.57952e12	2.99179
$554$	0	0
$555$	3.19497e11	0.142938
$556$	0	0
$557$	−1.25060e12	−0.550516	−0.275258	−	0.961370i	$-0.588763\pi$
$557$	−1.25060e12	−0.550516	−0.275258	+	0.961370i	$0.588763\pi$
$558$	0	0
$559$	−1.18285e11	−0.0512359
$560$	0	0
$561$	−2.76874e12	−1.18018
$562$	0	0
$563$	1.44758e12	0.607233	0.303617	−	0.952794i	$-0.401806\pi$
$563$	1.44758e12	0.607233	0.303617	+	0.952794i	$0.401806\pi$
$564$	0	0
$565$	3.44193e12	1.42097
$566$	0	0
$567$	5.20555e12	2.11516
$568$	0	0
$569$	2.86081e12	1.14415	0.572077	−	0.820200i	$-0.306138\pi$
$569$	2.86081e12	1.14415	0.572077	+	0.820200i	$0.306138\pi$
$570$	0	0
$571$	−3.97308e12	−1.56410	−0.782051	−	0.623214i	$-0.785827\pi$
$571$	−3.97308e12	−1.56410	−0.782051	+	0.623214i	$0.785827\pi$
$572$	0	0
$573$	4.66547e12	1.80801
$574$	0	0
$575$	−3.93120e12	−1.49975
$576$	0	0
$577$	−4.01486e12	−1.50792	−0.753962	−	0.656918i	$-0.771860\pi$
$577$	−4.01486e12	−1.50792	−0.753962	+	0.656918i	$0.771860\pi$
$578$	0	0
$579$	−4.66283e12	−1.72423
$580$	0	0
$581$	−6.17279e12	−2.24745
$582$	0	0
$583$	−4.05452e12	−1.45355
$584$	0	0
$585$	8.21418e11	0.289977
$586$	0	0
$587$	5.58149e12	1.94034	0.970172	−	0.242418i	$-0.0779405\pi$
$587$	5.58149e12	1.94034	0.970172	+	0.242418i	$0.0779405\pi$
$588$	0	0
$589$	−2.31850e12	−0.793757
$590$	0	0
$591$	−3.18631e12	−1.07435
$592$	0	0
$593$	−1.67772e12	−0.557150	−0.278575	−	0.960414i	$-0.589862\pi$
$593$	−1.67772e12	−0.557150	−0.278575	+	0.960414i	$0.589862\pi$
$594$	0	0
$595$	4.92264e12	1.61017
$596$	0	0
$597$	−2.45978e12	−0.792524
$598$	0	0
$599$	−1.29126e12	−0.409818	−0.204909	−	0.978781i	$-0.565690\pi$
$599$	−1.29126e12	−0.409818	−0.204909	+	0.978781i	$0.565690\pi$
$600$	0	0
$601$	3.19814e12	0.999914	0.499957	−	0.866050i	$-0.333349\pi$
$601$	3.19814e12	0.999914	0.499957	+	0.866050i	$0.333349\pi$
$602$	0	0
$603$	−3.95936e12	−1.21954
$604$	0	0
$605$	−9.55349e12	−2.89910
$606$	0	0
$607$	−2.45377e12	−0.733643	−0.366822	−	0.930291i	$-0.619554\pi$
$607$	−2.45377e12	−0.733643	−0.366822	+	0.930291i	$0.619554\pi$
$608$	0	0
$609$	5.78270e12	1.70354
$610$	0	0
$611$	−5.83479e11	−0.169371
$612$	0	0
$613$	6.03421e12	1.72603	0.863015	−	0.505179i	$-0.168573\pi$
$613$	6.03421e12	1.72603	0.863015	+	0.505179i	$0.168573\pi$
$614$	0	0
$615$	1.70749e12	0.481304
$616$	0	0
$617$	−2.05037e12	−0.569572	−0.284786	−	0.958591i	$-0.591923\pi$
$617$	−2.05037e12	−0.569572	−0.284786	+	0.958591i	$0.591923\pi$
$618$	0	0
$619$	1.38918e12	0.380322	0.190161	−	0.981753i	$-0.439099\pi$
$619$	1.38918e12	0.380322	0.190161	+	0.981753i	$0.439099\pi$
$620$	0	0
$621$	1.47211e12	0.397216
$622$	0	0
$623$	−1.49956e12	−0.398811
$624$	0	0
$625$	1.78536e12	0.468021
$626$	0	0
$627$	−4.16362e12	−1.07589
$628$	0	0
$629$	−1.47269e11	−0.0375130
$630$	0	0
$631$	6.68443e12	1.67854	0.839272	−	0.543712i	$-0.182982\pi$
$631$	6.68443e12	1.67854	0.839272	+	0.543712i	$0.182982\pi$
$632$	0	0
$633$	−4.18446e11	−0.103591
$634$	0	0
$635$	8.50227e12	2.07517
$636$	0	0
$637$	−2.22972e12	−0.536565
$638$	0	0
$639$	6.75322e11	0.160235
$640$	0	0
$641$	−5.15458e12	−1.20596	−0.602978	−	0.797757i	$-0.706020\pi$
$641$	−5.15458e12	−1.20596	−0.602978	+	0.797757i	$0.706020\pi$
$642$	0	0
$643$	−6.56700e11	−0.151502	−0.0757509	−	0.997127i	$-0.524135\pi$
$643$	−6.56700e11	−0.151502	−0.0757509	+	0.997127i	$0.524135\pi$
$644$	0	0
$645$	−1.73467e12	−0.394638
$646$	0	0
$647$	−6.44064e12	−1.44497	−0.722487	−	0.691385i	$-0.757001\pi$
$647$	−6.44064e12	−1.44497	−0.722487	+	0.691385i	$0.757001\pi$
$648$	0	0
$649$	−4.92097e12	−1.08880
$650$	0	0
$651$	1.55352e13	3.39002
$652$	0	0
$653$	−4.37326e12	−0.941231	−0.470615	−	0.882338i	$-0.655968\pi$
$653$	−4.37326e12	−0.941231	−0.470615	+	0.882338i	$0.655968\pi$
$654$	0	0
$655$	−6.01093e12	−1.27602
$656$	0	0
$657$	1.26676e12	0.265247
$658$	0	0
$659$	−1.68924e12	−0.348904	−0.174452	−	0.984666i	$-0.555815\pi$
$659$	−1.68924e12	−0.348904	−0.174452	+	0.984666i	$0.555815\pi$
$660$	0	0
$661$	3.67415e12	0.748600	0.374300	−	0.927308i	$-0.377883\pi$
$661$	3.67415e12	0.748600	0.374300	+	0.927308i	$0.377883\pi$
$662$	0	0
$663$	−9.85755e11	−0.198134
$664$	0	0
$665$	7.40263e12	1.46787
$666$	0	0
$667$	3.30420e12	0.646398
$668$	0	0
$669$	4.66320e12	0.900050
$670$	0	0
$671$	−1.05242e13	−2.00418
$672$	0	0
$673$	−9.07116e12	−1.70449	−0.852247	−	0.523140i	$-0.824760\pi$
$673$	−9.07116e12	−1.70449	−0.852247	+	0.523140i	$0.824760\pi$
$674$	0	0
$675$	−4.68363e12	−0.868392
$676$	0	0
$677$	3.34601e12	0.612179	0.306089	−	0.952003i	$-0.400979\pi$
$677$	3.34601e12	0.612179	0.306089	+	0.952003i	$0.400979\pi$
$678$	0	0
$679$	1.30515e11	0.0235639
$680$	0	0
$681$	2.36115e12	0.420690
$682$	0	0
$683$	3.30378e12	0.580923	0.290461	−	0.956887i	$-0.406191\pi$
$683$	3.30378e12	0.580923	0.290461	+	0.956887i	$0.406191\pi$
$684$	0	0
$685$	6.36344e12	1.10429
$686$	0	0
$687$	−6.75796e12	−1.15747
$688$	0	0
$689$	−1.44353e12	−0.244028
$690$	0	0
$691$	−4.90706e12	−0.818785	−0.409392	−	0.912358i	$-0.634259\pi$
$691$	−4.90706e12	−0.818785	−0.409392	+	0.912358i	$0.634259\pi$
$692$	0	0
$693$	1.07157e13	1.76490
$694$	0	0
$695$	3.24953e12	0.528310
$696$	0	0
$697$	−7.87046e11	−0.126314
$698$	0	0
$699$	3.80395e12	0.602681
$700$	0	0
$701$	−7.43425e12	−1.16280	−0.581402	−	0.813617i	$-0.697496\pi$
$701$	−7.43425e12	−1.16280	−0.581402	+	0.813617i	$0.697496\pi$
$702$	0	0
$703$	−2.21462e11	−0.0341979
$704$	0	0
$705$	−8.55689e12	−1.30456
$706$	0	0
$707$	4.58449e12	0.690087
$708$	0	0
$709$	−4.41374e12	−0.655992	−0.327996	−	0.944679i	$-0.606373\pi$
$709$	−4.41374e12	−0.655992	−0.327996	+	0.944679i	$0.606373\pi$
$710$	0	0
$711$	−7.42153e12	−1.08913
$712$	0	0
$713$	8.87671e12	1.28632
$714$	0	0
$715$	−5.36828e12	−0.768172
$716$	0	0
$717$	−1.49357e13	−2.11052
$718$	0	0
$719$	−4.26760e12	−0.595530	−0.297765	−	0.954639i	$-0.596241\pi$
$719$	−4.26760e12	−0.595530	−0.297765	+	0.954639i	$0.596241\pi$
$720$	0	0
$721$	−1.43675e13	−1.98003
$722$	0	0
$723$	1.78412e13	2.42830
$724$	0	0
$725$	−1.05126e13	−1.41315
$726$	0	0
$727$	1.33040e13	1.76635	0.883177	−	0.469040i	$-0.155400\pi$
$727$	1.33040e13	1.76635	0.883177	+	0.469040i	$0.155400\pi$
$728$	0	0
$729$	−1.21181e12	−0.158913
$730$	0	0
$731$	7.99579e11	0.103570
$732$	0	0
$733$	−3.02373e12	−0.386879	−0.193439	−	0.981112i	$-0.561964\pi$
$733$	−3.02373e12	−0.386879	−0.193439	+	0.981112i	$0.561964\pi$
$734$	0	0
$735$	−3.26994e13	−4.13282
$736$	0	0
$737$	2.58759e13	3.23067
$738$	0	0
$739$	8.35198e12	1.03012	0.515062	−	0.857153i	$-0.327769\pi$
$739$	8.35198e12	1.03012	0.515062	+	0.857153i	$0.327769\pi$
$740$	0	0
$741$	−1.48237e12	−0.180624
$742$	0	0
$743$	−9.08574e12	−1.09373	−0.546866	−	0.837220i	$-0.684179\pi$
$743$	−9.08574e12	−1.09373	−0.546866	+	0.837220i	$0.684179\pi$
$744$	0	0
$745$	−5.42920e12	−0.645702
$746$	0	0
$747$	6.96276e12	0.818161
$748$	0	0
$749$	−3.11451e12	−0.361594
$750$	0	0
$751$	7.46214e11	0.0856019	0.0428010	−	0.999084i	$-0.486372\pi$
$751$	7.46214e11	0.0856019	0.0428010	+	0.999084i	$0.486372\pi$
$752$	0	0
$753$	7.09098e11	0.0803765
$754$	0	0
$755$	−1.93173e13	−2.16364
$756$	0	0
$757$	−8.87740e12	−0.982549	−0.491274	−	0.871005i	$-0.663469\pi$
$757$	−8.87740e12	−0.982549	−0.491274	+	0.871005i	$0.663469\pi$
$758$	0	0
$759$	1.59410e13	1.74353
$760$	0	0
$761$	1.73096e12	0.187092	0.0935462	−	0.995615i	$-0.470180\pi$
$761$	1.73096e12	0.187092	0.0935462	+	0.995615i	$0.470180\pi$
$762$	0	0
$763$	1.63498e13	1.74643
$764$	0	0
$765$	−5.55262e12	−0.586167
$766$	0	0
$767$	−1.75201e12	−0.182792
$768$	0	0
$769$	1.57031e12	0.161926	0.0809628	−	0.996717i	$-0.474201\pi$
$769$	1.57031e12	0.161926	0.0809628	+	0.996717i	$0.474201\pi$
$770$	0	0
$771$	3.47654e12	0.354326
$772$	0	0
$773$	1.11736e13	1.12560	0.562800	−	0.826593i	$-0.309724\pi$
$773$	1.11736e13	1.12560	0.562800	+	0.826593i	$0.309724\pi$
$774$	0	0
$775$	−2.82420e13	−2.81215
$776$	0	0
$777$	1.48391e12	0.146054
$778$	0	0
$779$	−1.18356e12	−0.115152
$780$	0	0
$781$	−4.41349e12	−0.424475
$782$	0	0
$783$	3.93663e12	0.374280
$784$	0	0
$785$	−1.45217e12	−0.136491
$786$	0	0
$787$	−3.15643e12	−0.293298	−0.146649	−	0.989189i	$-0.546849\pi$
$787$	−3.15643e12	−0.293298	−0.146649	+	0.989189i	$0.546849\pi$
$788$	0	0
$789$	1.58915e13	1.45989
$790$	0	0
$791$	1.59862e13	1.45195
$792$	0	0
$793$	−3.74692e12	−0.336469
$794$	0	0
$795$	−2.11698e13	−1.87959
$796$	0	0
$797$	1.96411e13	1.72426	0.862131	−	0.506686i	$-0.169130\pi$
$797$	1.96411e13	1.72426	0.862131	+	0.506686i	$0.169130\pi$
$798$	0	0
$799$	3.94420e12	0.342372
$800$	0	0
$801$	1.69146e12	0.145183
$802$	0	0
$803$	−8.27877e12	−0.702661
$804$	0	0
$805$	−2.83421e13	−2.37876
$806$	0	0
$807$	8.62761e12	0.716077
$808$	0	0
$809$	2.00366e13	1.64458	0.822292	−	0.569066i	$-0.192695\pi$
$809$	2.00366e13	1.64458	0.822292	+	0.569066i	$0.192695\pi$
$810$	0	0
$811$	−1.11737e13	−0.906994	−0.453497	−	0.891258i	$-0.649824\pi$
$811$	−1.11737e13	−0.906994	−0.453497	+	0.891258i	$0.649824\pi$
$812$	0	0
$813$	1.17738e13	0.945166
$814$	0	0
$815$	9.40088e12	0.746379
$816$	0	0
$817$	1.20240e12	0.0944170
$818$	0	0
$819$	3.81510e12	0.296298
$820$	0	0
$821$	−1.01042e13	−0.776175	−0.388087	−	0.921623i	$-0.626864\pi$
$821$	−1.01042e13	−0.776175	−0.388087	+	0.921623i	$0.626864\pi$
$822$	0	0
$823$	−2.46694e12	−0.187438	−0.0937192	−	0.995599i	$-0.529876\pi$
$823$	−2.46694e12	−0.187438	−0.0937192	+	0.995599i	$0.529876\pi$
$824$	0	0
$825$	−5.07178e13	−3.81169
$826$	0	0
$827$	−1.01053e13	−0.751229	−0.375615	−	0.926776i	$-0.622568\pi$
$827$	−1.01053e13	−0.751229	−0.375615	+	0.926776i	$0.622568\pi$
$828$	0	0
$829$	−2.96694e12	−0.218179	−0.109090	−	0.994032i	$-0.534794\pi$
$829$	−2.96694e12	−0.218179	−0.109090	+	0.994032i	$0.534794\pi$
$830$	0	0
$831$	−2.36970e13	−1.72381
$832$	0	0
$833$	1.50724e13	1.08463
$834$	0	0
$835$	2.04004e13	1.45228
$836$	0	0
$837$	1.05757e13	0.744810
$838$	0	0
$839$	2.10306e13	1.46529	0.732644	−	0.680612i	$-0.238286\pi$
$839$	2.10306e13	1.46529	0.732644	+	0.680612i	$0.238286\pi$
$840$	0	0
$841$	−5.67124e12	−0.390927
$842$	0	0
$843$	3.17100e13	2.16258
$844$	0	0
$845$	−1.91127e12	−0.128963
$846$	0	0
$847$	−4.43715e13	−2.96230
$848$	0	0
$849$	6.30782e12	0.416673
$850$	0	0
$851$	8.47899e11	0.0554193
$852$	0	0
$853$	−1.45181e13	−0.938944	−0.469472	−	0.882947i	$-0.655556\pi$
$853$	−1.45181e13	−0.938944	−0.469472	+	0.882947i	$0.655556\pi$
$854$	0	0
$855$	−8.34999e12	−0.534366
$856$	0	0
$857$	6.94178e12	0.439600	0.219800	−	0.975545i	$-0.429460\pi$
$857$	6.94178e12	0.439600	0.219800	+	0.975545i	$0.429460\pi$
$858$	0	0
$859$	2.82565e12	0.177072	0.0885358	−	0.996073i	$-0.471781\pi$
$859$	2.82565e12	0.177072	0.0885358	+	0.996073i	$0.471781\pi$
$860$	0	0
$861$	7.93047e12	0.491796
$862$	0	0
$863$	−1.81122e12	−0.111153	−0.0555766	−	0.998454i	$-0.517700\pi$
$863$	−1.81122e12	−0.111153	−0.0555766	+	0.998454i	$0.517700\pi$
$864$	0	0
$865$	2.06027e13	1.25127
$866$	0	0
$867$	−1.45362e13	−0.873704
$868$	0	0
$869$	4.85026e13	2.88520
$870$	0	0
$871$	9.21259e12	0.542376
$872$	0	0
$873$	−1.47218e11	−0.00857822
$874$	0	0
$875$	4.03738e13	2.32843
$876$	0	0
$877$	3.34755e13	1.91086	0.955430	−	0.295217i	$-0.0953919\pi$
$877$	3.34755e13	1.91086	0.955430	+	0.295217i	$0.0953919\pi$
$878$	0	0
$879$	−1.37483e13	−0.776783
$880$	0	0
$881$	−1.66782e12	−0.0932731	−0.0466366	−	0.998912i	$-0.514850\pi$
$881$	−1.66782e12	−0.0932731	−0.0466366	+	0.998912i	$0.514850\pi$
$882$	0	0
$883$	1.05950e12	0.0586512	0.0293256	−	0.999570i	$-0.490664\pi$
$883$	1.05950e12	0.0586512	0.0293256	+	0.999570i	$0.490664\pi$
$884$	0	0
$885$	−2.56937e13	−1.40793
$886$	0	0
$887$	2.50470e13	1.35862	0.679312	−	0.733849i	$-0.262278\pi$
$887$	2.50470e13	1.35862	0.679312	+	0.733849i	$0.262278\pi$
$888$	0	0
$889$	3.94891e13	2.12041
$890$	0	0
$891$	3.83740e13	2.03980
$892$	0	0
$893$	5.93126e12	0.312116
$894$	0	0
$895$	4.10694e13	2.13951
$896$	0	0
$897$	5.67548e12	0.292710
$898$	0	0
$899$	2.37376e13	1.21204
$900$	0	0
$901$	9.75796e12	0.493285
$902$	0	0
$903$	−8.05675e12	−0.403241
$904$	0	0
$905$	−1.90525e13	−0.944131
$906$	0	0
$907$	1.28009e13	0.628070	0.314035	−	0.949411i	$-0.398319\pi$
$907$	1.28009e13	0.628070	0.314035	+	0.949411i	$0.398319\pi$
$908$	0	0
$909$	−5.17119e12	−0.251219
$910$	0	0
$911$	−1.98066e12	−0.0952747	−0.0476373	−	0.998865i	$-0.515169\pi$
$911$	−1.98066e12	−0.0952747	−0.0476373	+	0.998865i	$0.515169\pi$
$912$	0	0
$913$	−4.55043e13	−2.16737
$914$	0	0
$915$	−5.49496e13	−2.59161
$916$	0	0
$917$	−2.79180e13	−1.30383
$918$	0	0
$919$	3.44433e12	0.159289	0.0796444	−	0.996823i	$-0.474622\pi$
$919$	3.44433e12	0.159289	0.0796444	+	0.996823i	$0.474622\pi$
$920$	0	0
$921$	−2.47485e13	−1.13339
$922$	0	0
$923$	−1.57133e12	−0.0712624
$924$	0	0
$925$	−2.69766e12	−0.121157
$926$	0	0
$927$	1.62062e13	0.720811
$928$	0	0
$929$	3.95599e13	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$929$	3.95599e13	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$930$	0	0
$931$	2.26658e13	0.988776
$932$	0	0
$933$	−1.19057e13	−0.514383
$934$	0	0
$935$	3.62885e13	1.55280
$936$	0	0
$937$	−2.52511e13	−1.07017	−0.535084	−	0.844799i	$-0.679720\pi$
$937$	−2.52511e13	−1.07017	−0.535084	+	0.844799i	$0.679720\pi$
$938$	0	0
$939$	−2.08493e13	−0.875180
$940$	0	0
$941$	−8.69444e12	−0.361483	−0.180742	−	0.983531i	$-0.557850\pi$
$941$	−8.69444e12	−0.361483	−0.180742	+	0.983531i	$0.557850\pi$
$942$	0	0
$943$	4.53142e12	0.186609
$944$	0	0
$945$	−3.37668e13	−1.37736
$946$	0	0
$947$	−2.11484e13	−0.854481	−0.427241	−	0.904138i	$-0.640514\pi$
$947$	−2.11484e13	−0.854481	−0.427241	+	0.904138i	$0.640514\pi$
$948$	0	0
$949$	−2.94749e12	−0.117965
$950$	0	0
$951$	2.06290e13	0.817837
$952$	0	0
$953$	−1.01534e13	−0.398744	−0.199372	−	0.979924i	$-0.563890\pi$
$953$	−1.01534e13	−0.398744	−0.199372	+	0.979924i	$0.563890\pi$
$954$	0	0
$955$	−6.11479e13	−2.37885
$956$	0	0
$957$	4.26287e13	1.64285
$958$	0	0
$959$	2.95552e13	1.12837
$960$	0	0
$961$	3.73314e13	1.41195
$962$	0	0
$963$	3.51309e12	0.131635
$964$	0	0
$965$	6.11133e13	2.26862
$966$	0	0
$967$	−6.43077e12	−0.236507	−0.118254	−	0.992983i	$-0.537730\pi$
$967$	−6.43077e12	−0.236507	−0.118254	+	0.992983i	$0.537730\pi$
$968$	0	0
$969$	1.00205e13	0.365118
$970$	0	0
$971$	−1.74913e12	−0.0631446	−0.0315723	−	0.999501i	$-0.510051\pi$
$971$	−1.74913e12	−0.0631446	−0.0315723	+	0.999501i	$0.510051\pi$
$972$	0	0
$973$	1.50925e13	0.539827
$974$	0	0
$975$	−1.80570e13	−0.639920
$976$	0	0
$977$	1.50942e13	0.530012	0.265006	−	0.964247i	$-0.414626\pi$
$977$	1.50942e13	0.530012	0.265006	+	0.964247i	$0.414626\pi$
$978$	0	0
$979$	−1.10544e13	−0.384602
$980$	0	0
$981$	−1.84422e13	−0.635772
$982$	0	0
$983$	4.41201e13	1.50711	0.753557	−	0.657383i	$-0.228336\pi$
$983$	4.41201e13	1.50711	0.753557	+	0.657383i	$0.228336\pi$
$984$	0	0
$985$	4.17613e13	1.41355
$986$	0	0
$987$	−3.97427e13	−1.33300
$988$	0	0
$989$	−4.60357e12	−0.153007
$990$	0	0
$991$	−3.75113e13	−1.23547	−0.617733	−	0.786388i	$-0.711949\pi$
$991$	−3.75113e13	−1.23547	−0.617733	+	0.786388i	$0.711949\pi$
$992$	0	0
$993$	−5.99577e13	−1.95692
$994$	0	0
$995$	3.22391e13	1.04275
$996$	0	0
$997$	5.56595e12	0.178407	0.0892034	−	0.996013i	$-0.471568\pi$
$997$	5.56595e12	0.178407	0.0892034	+	0.996013i	$0.471568\pi$
$998$	0	0
$999$	1.01019e12	0.0320891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.d.1.3		3
4.3	odd	2		26.10.a.e.1.1	✓	3
12.11	even	2		234.10.a.k.1.3		3
52.51	odd	2		338.10.a.e.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.10.a.e.1.1	✓	3	4.3	odd	2
208.10.a.d.1.3		3	1.1	even	1	trivial
234.10.a.k.1.3		3	12.11	even	2
338.10.a.e.1.1		3	52.51	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+178.768 q^{3} -2343.01 q^{5} -10882.2 q^{7} +12274.9 q^{9} -80220.9 q^{11} -28561.0 q^{13} -418855. q^{15} +193066. q^{17} +290332. q^{19} -1.94538e6 q^{21} -1.11158e6 q^{23} +3.53659e6 q^{25} -1.32434e6 q^{27} -2.97253e6 q^{29} -7.98567e6 q^{31} -1.43409e7 q^{33} +2.54971e7 q^{35} -762787. q^{37} -5.10578e6 q^{39} -4.07656e6 q^{41} +4.14147e6 q^{43} -2.87601e7 q^{45} +2.04292e7 q^{47} +7.80686e7 q^{49} +3.45140e7 q^{51} +5.05420e7 q^{53} +1.87959e8 q^{55} +5.19020e7 q^{57} +6.13428e7 q^{59} +1.31190e8 q^{61} -1.33577e8 q^{63} +6.69188e7 q^{65} -3.22558e8 q^{67} -1.98714e8 q^{69} +5.50167e7 q^{71} +1.03200e8 q^{73} +6.32227e8 q^{75} +8.72979e8 q^{77} -6.04613e8 q^{79} -4.78354e8 q^{81} +5.67238e8 q^{83} -4.52357e8 q^{85} -5.31391e8 q^{87} +1.37799e8 q^{89} +3.10806e8 q^{91} -1.42758e9 q^{93} -6.80252e8 q^{95} -1.19935e7 q^{97} -9.84699e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 156 q^{3} - 1272 q^{5} - 17058 q^{7} + 42273 q^{9} - 73974 q^{11} - 85683 q^{13} - 393756 q^{15} + 374976 q^{17} - 418338 q^{19} - 284694 q^{21} - 1026168 q^{23} + 3337287 q^{25} - 4218588 q^{27}+ \cdots + 2480087466 q^{99}+O(q^{100}) \) 3 * q - 156 * q^3 - 1272 * q^5 - 17058 * q^7 + 42273 * q^9 - 73974 * q^11 - 85683 * q^13 - 393756 * q^15 + 374976 * q^17 - 418338 * q^19 - 284694 * q^21 - 1026168 * q^23 + 3337287 * q^25 - 4218588 * q^27 - 3075834 * q^29 - 9286482 * q^31 - 25456452 * q^33 + 31802592 * q^35 - 17647776 * q^37 + 4455516 * q^39 - 47257110 * q^41 + 60023760 * q^43 - 81112122 * q^45 + 40824726 * q^47 + 45921177 * q^49 + 27559188 * q^51 + 8072046 * q^53 + 36910236 * q^55 + 245224476 * q^57 - 138035310 * q^59 + 264203886 * q^61 - 436094838 * q^63 + 36329592 * q^65 - 203167074 * q^67 - 397637544 * q^69 - 123067110 * q^71 - 433013250 * q^73 + 884469216 * q^75 + 380043264 * q^77 - 406418748 * q^79 + 100121427 * q^81 - 105365610 * q^83 + 5268186 * q^85 + 12753768 * q^87 + 1365375798 * q^89 + 487193538 * q^91 - 95909856 * q^93 + 85600692 * q^95 + 669691662 * q^97 + 2480087466 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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