LMFDB - Embedded newform 208.10.a.l.1.3

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 112810 x^{5} + 1645934 x^{4} + 3493976849 x^{3} - 83049726457 x^{2} + \cdots + 864293655586764 $ x^7 - x^6 - 112810x^5 + 1645934x^4 + 3493976849x^3 - 83049726457x^2 - 21841140807080*x + 864293655586764
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-98.0566$ 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)

$f(q)$	$=$	$q-100.057 q^{3} -2398.21 q^{5} +1546.37 q^{7} -9671.67 q^{9} +12896.0 q^{11} -28561.0 q^{13} +239957. q^{15} -34811.6 q^{17} -29588.5 q^{19} -154724. q^{21} +87242.5 q^{23} +3.79831e6 q^{25} +2.93713e6 q^{27} +4.69746e6 q^{29} -5.76563e6 q^{31} -1.29033e6 q^{33} -3.70853e6 q^{35} +1.47970e7 q^{37} +2.85772e6 q^{39} +9.93166e6 q^{41} +2.28205e7 q^{43} +2.31947e7 q^{45} +4.36131e7 q^{47} -3.79623e7 q^{49} +3.48313e6 q^{51} -2.13528e7 q^{53} -3.09273e7 q^{55} +2.96053e6 q^{57} -1.23658e8 q^{59} +5.30891e7 q^{61} -1.49560e7 q^{63} +6.84954e7 q^{65} -1.85043e7 q^{67} -8.72919e6 q^{69} -3.09090e8 q^{71} +3.60772e8 q^{73} -3.80046e8 q^{75} +1.99420e7 q^{77} -4.18302e8 q^{79} -1.03512e8 q^{81} +4.97861e8 q^{83} +8.34856e7 q^{85} -4.70012e8 q^{87} +3.35112e8 q^{89} -4.41659e7 q^{91} +5.76890e8 q^{93} +7.09596e7 q^{95} -5.37441e8 q^{97} -1.24726e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 13 q^{3} - 801 q^{5} + 1091 q^{7} + 87864 q^{9} - 32218 q^{11} - 199927 q^{13} - 58323 q^{15} - 870531 q^{17} + 128950 q^{19} - 719663 q^{21} - 2198844 q^{23} + 992010 q^{25} - 5441383 q^{27} + 6327710 q^{29}+ \cdots - 5204667600 q^{99}+O(q^{100}) $ 7 * q - 13 * q^3 - 801 * q^5 + 1091 * q^7 + 87864 * q^9 - 32218 * q^11 - 199927 * q^13 - 58323 * q^15 - 870531 * q^17 + 128950 * q^19 - 719663 * q^21 - 2198844 * q^23 + 992010 * q^25 - 5441383 * q^27 + 6327710 * q^29 + 1660376 * q^31 + 14199598 * q^33 + 19231769 * q^35 + 5058787 * q^37 + 371293 * q^39 + 14779748 * q^41 - 9405075 * q^43 + 56644022 * q^45 - 133415721 * q^47 + 44544988 * q^49 - 132066169 * q^51 + 43962424 * q^53 - 191344746 * q^55 + 138963078 * q^57 - 22370274 * q^59 + 23851748 * q^61 - 780720706 * q^63 + 22877361 * q^65 - 12058786 * q^67 + 1036453420 * q^69 - 864303791 * q^71 + 670491202 * q^73 - 693324044 * q^75 + 841052238 * q^77 - 1019547572 * q^79 + 900194079 * q^81 - 1345041700 * q^83 + 824066375 * q^85 - 2589172074 * q^87 + 1409158546 * q^89 - 31160051 * q^91 - 445200892 * q^93 - 3285213058 * q^95 + 320206906 * q^97 - 5204667600 * 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
$3$	−100.057	−0.713182	−0.356591	−	0.934261i	$-0.616061\pi$
$3$	−100.057	−0.713182	−0.356591	+	0.934261i	$0.616061\pi$
$4$	0	0
$5$	−2398.21	−1.71602	−0.858011	−	0.513631i	$-0.828300\pi$
$5$	−2398.21	−1.71602	−0.858011	+	0.513631i	$0.828300\pi$
$6$	0	0
$7$	1546.37	0.243429	0.121714	−	0.992565i	$-0.461161\pi$
$7$	1546.37	0.243429	0.121714	+	0.992565i	$0.461161\pi$
$8$	0	0
$9$	−9671.67	−0.491372
$10$	0	0
$11$	12896.0	0.265575	0.132788	−	0.991145i	$-0.457607\pi$
$11$	12896.0	0.265575	0.132788	+	0.991145i	$0.457607\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	239957.	1.22384
$16$	0	0
$17$	−34811.6	−0.101089	−0.0505445	−	0.998722i	$-0.516096\pi$
$17$	−34811.6	−0.101089	−0.0505445	+	0.998722i	$0.516096\pi$
$18$	0	0
$19$	−29588.5	−0.0520873	−0.0260437	−	0.999661i	$-0.508291\pi$
$19$	−29588.5	−0.0520873	−0.0260437	+	0.999661i	$0.508291\pi$
$20$	0	0
$21$	−154724.	−0.173609
$22$	0	0
$23$	87242.5	0.0650059	0.0325029	−	0.999472i	$-0.489652\pi$
$23$	87242.5	0.0650059	0.0325029	+	0.999472i	$0.489652\pi$
$24$	0	0
$25$	3.79831e6	1.94473
$26$	0	0
$27$	2.93713e6	1.06362
$28$	0	0
$29$	4.69746e6	1.23331	0.616655	−	0.787234i	$-0.288487\pi$
$29$	4.69746e6	1.23331	0.616655	+	0.787234i	$0.288487\pi$
$30$	0	0
$31$	−5.76563e6	−1.12129	−0.560647	−	0.828055i	$-0.689448\pi$
$31$	−5.76563e6	−1.12129	−0.560647	+	0.828055i	$0.689448\pi$
$32$	0	0
$33$	−1.29033e6	−0.189403
$34$	0	0
$35$	−3.70853e6	−0.417729
$36$	0	0
$37$	1.47970e7	1.29797	0.648985	−	0.760801i	$-0.275194\pi$
$37$	1.47970e7	1.29797	0.648985	+	0.760801i	$0.275194\pi$
$38$	0	0
$39$	2.85772e6	0.197801
$40$	0	0
$41$	9.93166e6	0.548901	0.274451	−	0.961601i	$-0.411504\pi$
$41$	9.93166e6	0.548901	0.274451	+	0.961601i	$0.411504\pi$
$42$	0	0
$43$	2.28205e7	1.01793	0.508963	−	0.860788i	$-0.330029\pi$
$43$	2.28205e7	1.01793	0.508963	+	0.860788i	$0.330029\pi$
$44$	0	0
$45$	2.31947e7	0.843205
$46$	0	0
$47$	4.36131e7	1.30370	0.651849	−	0.758349i	$-0.273994\pi$
$47$	4.36131e7	1.30370	0.651849	+	0.758349i	$0.273994\pi$
$48$	0	0
$49$	−3.79623e7	−0.940742
$50$	0	0
$51$	3.48313e6	0.0720948
$52$	0	0
$53$	−2.13528e7	−0.371718	−0.185859	−	0.982576i	$-0.559507\pi$
$53$	−2.13528e7	−0.371718	−0.185859	+	0.982576i	$0.559507\pi$
$54$	0	0
$55$	−3.09273e7	−0.455733
$56$	0	0
$57$	2.96053e6	0.0371477
$58$	0	0
$59$	−1.23658e8	−1.32858	−0.664290	−	0.747475i	$-0.731266\pi$
$59$	−1.23658e8	−1.32858	−0.664290	+	0.747475i	$0.731266\pi$
$60$	0	0
$61$	5.30891e7	0.490932	0.245466	−	0.969405i	$-0.421059\pi$
$61$	5.30891e7	0.490932	0.245466	+	0.969405i	$0.421059\pi$
$62$	0	0
$63$	−1.49560e7	−0.119614
$64$	0	0
$65$	6.84954e7	0.475939
$66$	0	0
$67$	−1.85043e7	−0.112185	−0.0560927	−	0.998426i	$-0.517864\pi$
$67$	−1.85043e7	−0.112185	−0.0560927	+	0.998426i	$0.517864\pi$
$68$	0	0
$69$	−8.72919e6	−0.0463610
$70$	0	0
$71$	−3.09090e8	−1.44352	−0.721759	−	0.692145i	$-0.756666\pi$
$71$	−3.09090e8	−1.44352	−0.721759	+	0.692145i	$0.756666\pi$
$72$	0	0
$73$	3.60772e8	1.48690	0.743448	−	0.668794i	$-0.233189\pi$
$73$	3.60772e8	1.48690	0.743448	+	0.668794i	$0.233189\pi$
$74$	0	0
$75$	−3.80046e8	−1.38695
$76$	0	0
$77$	1.99420e7	0.0646486
$78$	0	0
$79$	−4.18302e8	−1.20828	−0.604140	−	0.796878i	$-0.706483\pi$
$79$	−4.18302e8	−1.20828	−0.604140	+	0.796878i	$0.706483\pi$
$80$	0	0
$81$	−1.03512e8	−0.267182
$82$	0	0
$83$	4.97861e8	1.15148	0.575740	−	0.817633i	$-0.304714\pi$
$83$	4.97861e8	1.15148	0.575740	+	0.817633i	$0.304714\pi$
$84$	0	0
$85$	8.34856e7	0.173471
$86$	0	0
$87$	−4.70012e8	−0.879574
$88$	0	0
$89$	3.35112e8	0.566154	0.283077	−	0.959097i	$-0.408645\pi$
$89$	3.35112e8	0.566154	0.283077	+	0.959097i	$0.408645\pi$
$90$	0	0
$91$	−4.41659e7	−0.0675150
$92$	0	0
$93$	5.76890e8	0.799686
$94$	0	0
$95$	7.09596e7	0.0893830
$96$	0	0
$97$	−5.37441e8	−0.616394	−0.308197	−	0.951323i	$-0.599726\pi$
$97$	−5.37441e8	−0.616394	−0.308197	+	0.951323i	$0.599726\pi$
$98$	0	0
$99$	−1.24726e8	−0.130496
$100$	0	0
$101$	−5.76680e8	−0.551428	−0.275714	−	0.961240i	$-0.588914\pi$
$101$	−5.76680e8	−0.551428	−0.275714	+	0.961240i	$0.588914\pi$
$102$	0	0
$103$	−4.70346e8	−0.411766	−0.205883	−	0.978577i	$-0.566007\pi$
$103$	−4.70346e8	−0.411766	−0.205883	+	0.978577i	$0.566007\pi$
$104$	0	0
$105$	3.71063e8	0.297917
$106$	0	0
$107$	1.43636e8	0.105935	0.0529673	−	0.998596i	$-0.483132\pi$
$107$	1.43636e8	0.105935	0.0529673	+	0.998596i	$0.483132\pi$
$108$	0	0
$109$	1.92524e9	1.30637	0.653186	−	0.757198i	$-0.273432\pi$
$109$	1.92524e9	1.30637	0.653186	+	0.757198i	$0.273432\pi$
$110$	0	0
$111$	−1.48053e9	−0.925688
$112$	0	0
$113$	−6.17526e8	−0.356289	−0.178145	−	0.984004i	$-0.557009\pi$
$113$	−6.17526e8	−0.356289	−0.178145	+	0.984004i	$0.557009\pi$
$114$	0	0
$115$	−2.09226e8	−0.111552
$116$	0	0
$117$	2.76233e8	0.136282
$118$	0	0
$119$	−5.38316e7	−0.0246080
$120$	0	0
$121$	−2.19164e9	−0.929470
$122$	0	0
$123$	−9.93728e8	−0.391466
$124$	0	0
$125$	−4.42514e9	−1.62118
$126$	0	0
$127$	−3.20743e9	−1.09406	−0.547029	−	0.837114i	$-0.684241\pi$
$127$	−3.20743e9	−1.09406	−0.547029	+	0.837114i	$0.684241\pi$
$128$	0	0
$129$	−2.28334e9	−0.725966
$130$	0	0
$131$	−4.61577e8	−0.136938	−0.0684688	−	0.997653i	$-0.521811\pi$
$131$	−4.61577e8	−0.136938	−0.0684688	+	0.997653i	$0.521811\pi$
$132$	0	0
$133$	−4.57548e7	−0.0126796
$134$	0	0
$135$	−7.04387e9	−1.82519
$136$	0	0
$137$	−2.72394e9	−0.660626	−0.330313	−	0.943871i	$-0.607154\pi$
$137$	−2.72394e9	−0.660626	−0.330313	+	0.943871i	$0.607154\pi$
$138$	0	0
$139$	5.18624e9	1.17838	0.589191	−	0.807994i	$-0.299447\pi$
$139$	5.18624e9	1.17838	0.589191	+	0.807994i	$0.299447\pi$
$140$	0	0
$141$	−4.36378e9	−0.929773
$142$	0	0
$143$	−3.68322e8	−0.0736573
$144$	0	0
$145$	−1.12655e10	−2.11639
$146$	0	0
$147$	3.79838e9	0.670920
$148$	0	0
$149$	−4.06533e9	−0.675706	−0.337853	−	0.941199i	$-0.609701\pi$
$149$	−4.06533e9	−0.675706	−0.337853	+	0.941199i	$0.609701\pi$
$150$	0	0
$151$	−9.37428e9	−1.46738	−0.733688	−	0.679486i	$-0.762203\pi$
$151$	−9.37428e9	−1.46738	−0.733688	+	0.679486i	$0.762203\pi$
$152$	0	0
$153$	3.36686e8	0.0496723
$154$	0	0
$155$	1.38272e10	1.92417
$156$	0	0
$157$	8.39317e9	1.10250	0.551249	−	0.834341i	$-0.314152\pi$
$157$	8.39317e9	1.10250	0.551249	+	0.834341i	$0.314152\pi$
$158$	0	0
$159$	2.13649e9	0.265103
$160$	0	0
$161$	1.34909e8	0.0158243
$162$	0	0
$163$	−3.92613e9	−0.435633	−0.217817	−	0.975990i	$-0.569893\pi$
$163$	−3.92613e9	−0.435633	−0.217817	+	0.975990i	$0.569893\pi$
$164$	0	0
$165$	3.09448e9	0.325020
$166$	0	0
$167$	−1.04162e9	−0.103630	−0.0518150	−	0.998657i	$-0.516501\pi$
$167$	−1.04162e9	−0.103630	−0.0518150	+	0.998657i	$0.516501\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	2.86170e8	0.0255942
$172$	0	0
$173$	−9.91250e9	−0.841348	−0.420674	−	0.907212i	$-0.638206\pi$
$173$	−9.91250e9	−0.841348	−0.420674	+	0.907212i	$0.638206\pi$
$174$	0	0
$175$	5.87359e9	0.473404
$176$	0	0
$177$	1.23728e10	0.947519
$178$	0	0
$179$	−4.10530e9	−0.298887	−0.149443	−	0.988770i	$-0.547748\pi$
$179$	−4.10530e9	−0.298887	−0.149443	+	0.988770i	$0.547748\pi$
$180$	0	0
$181$	−2.62132e8	−0.0181538	−0.00907689	−	0.999959i	$-0.502889\pi$
$181$	−2.62132e8	−0.0181538	−0.00907689	+	0.999959i	$0.502889\pi$
$182$	0	0
$183$	−5.31192e9	−0.350124
$184$	0	0
$185$	−3.54863e10	−2.22735
$186$	0	0
$187$	−4.48930e8	−0.0268467
$188$	0	0
$189$	4.54189e9	0.258916
$190$	0	0
$191$	4.98229e9	0.270881	0.135441	−	0.990785i	$-0.456755\pi$
$191$	4.98229e9	0.270881	0.135441	+	0.990785i	$0.456755\pi$
$192$	0	0
$193$	4.89338e9	0.253864	0.126932	−	0.991911i	$-0.459487\pi$
$193$	4.89338e9	0.253864	0.126932	+	0.991911i	$0.459487\pi$
$194$	0	0
$195$	−6.85342e9	−0.339431
$196$	0	0
$197$	−3.92552e10	−1.85695	−0.928473	−	0.371401i	$-0.878877\pi$
$197$	−3.92552e10	−1.85695	−0.928473	+	0.371401i	$0.878877\pi$
$198$	0	0
$199$	2.90641e10	1.31376	0.656882	−	0.753993i	$-0.271875\pi$
$199$	2.90641e10	1.31376	0.656882	+	0.753993i	$0.271875\pi$
$200$	0	0
$201$	1.85148e9	0.0800086
$202$	0	0
$203$	7.26401e9	0.300223
$204$	0	0
$205$	−2.38182e10	−0.941927
$206$	0	0
$207$	−8.43780e8	−0.0319421
$208$	0	0
$209$	−3.81573e8	−0.0138331
$210$	0	0
$211$	−1.12125e9	−0.0389432	−0.0194716	−	0.999810i	$-0.506198\pi$
$211$	−1.12125e9	−0.0389432	−0.0194716	+	0.999810i	$0.506198\pi$
$212$	0	0
$213$	3.09265e10	1.02949
$214$	0	0
$215$	−5.47284e10	−1.74678
$216$	0	0
$217$	−8.91580e9	−0.272955
$218$	0	0
$219$	−3.60977e10	−1.06043
$220$	0	0
$221$	9.94254e8	0.0280370
$222$	0	0
$223$	3.12395e10	0.845926	0.422963	−	0.906147i	$-0.360990\pi$
$223$	3.12395e10	0.845926	0.422963	+	0.906147i	$0.360990\pi$
$224$	0	0
$225$	−3.67360e10	−0.955587
$226$	0	0
$227$	−4.59052e10	−1.14748	−0.573740	−	0.819037i	$-0.694508\pi$
$227$	−4.59052e10	−1.14748	−0.573740	+	0.819037i	$0.694508\pi$
$228$	0	0
$229$	5.14797e10	1.23702	0.618510	−	0.785777i	$-0.287737\pi$
$229$	5.14797e10	1.23702	0.618510	+	0.785777i	$0.287737\pi$
$230$	0	0
$231$	−1.99532e9	−0.0461062
$232$	0	0
$233$	−5.21911e10	−1.16010	−0.580049	−	0.814582i	$-0.696967\pi$
$233$	−5.21911e10	−1.16010	−0.580049	+	0.814582i	$0.696967\pi$
$234$	0	0
$235$	−1.04594e11	−2.23717
$236$	0	0
$237$	4.18539e10	0.861723
$238$	0	0
$239$	−9.76347e10	−1.93559	−0.967795	−	0.251739i	$-0.918998\pi$
$239$	−9.76347e10	−1.93559	−0.967795	+	0.251739i	$0.918998\pi$
$240$	0	0
$241$	4.66786e10	0.891335	0.445667	−	0.895199i	$-0.352966\pi$
$241$	4.66786e10	0.891335	0.445667	+	0.895199i	$0.352966\pi$
$242$	0	0
$243$	−4.74545e10	−0.873070
$244$	0	0
$245$	9.10419e10	1.61434
$246$	0	0
$247$	8.45078e8	0.0144464
$248$	0	0
$249$	−4.98142e10	−0.821214
$250$	0	0
$251$	4.78831e10	0.761466	0.380733	−	0.924685i	$-0.375672\pi$
$251$	4.78831e10	0.761466	0.380733	+	0.924685i	$0.375672\pi$
$252$	0	0
$253$	1.12508e9	0.0172639
$254$	0	0
$255$	−8.35329e9	−0.123716
$256$	0	0
$257$	1.95101e10	0.278972	0.139486	−	0.990224i	$-0.455455\pi$
$257$	1.95101e10	0.278972	0.139486	+	0.990224i	$0.455455\pi$
$258$	0	0
$259$	2.28816e10	0.315963
$260$	0	0
$261$	−4.54323e10	−0.606014
$262$	0	0
$263$	6.50217e10	0.838025	0.419013	−	0.907980i	$-0.362376\pi$
$263$	6.50217e10	0.838025	0.419013	+	0.907980i	$0.362376\pi$
$264$	0	0
$265$	5.12087e10	0.637877
$266$	0	0
$267$	−3.35302e10	−0.403771
$268$	0	0
$269$	−2.04324e9	−0.0237922	−0.0118961	−	0.999929i	$-0.503787\pi$
$269$	−2.04324e9	−0.0237922	−0.0118961	+	0.999929i	$0.503787\pi$
$270$	0	0
$271$	1.24125e11	1.39797	0.698987	−	0.715135i	$-0.253635\pi$
$271$	1.24125e11	1.39797	0.698987	+	0.715135i	$0.253635\pi$
$272$	0	0
$273$	4.41909e9	0.0481505
$274$	0	0
$275$	4.89829e10	0.516473
$276$	0	0
$277$	7.05115e10	0.719617	0.359808	−	0.933026i	$-0.382842\pi$
$277$	7.05115e10	0.719617	0.359808	+	0.933026i	$0.382842\pi$
$278$	0	0
$279$	5.57633e10	0.550972
$280$	0	0
$281$	1.60133e11	1.53215	0.766076	−	0.642750i	$-0.222206\pi$
$281$	1.60133e11	1.53215	0.766076	+	0.642750i	$0.222206\pi$
$282$	0	0
$283$	−1.61246e10	−0.149434	−0.0747169	−	0.997205i	$-0.523805\pi$
$283$	−1.61246e10	−0.149434	−0.0747169	+	0.997205i	$0.523805\pi$
$284$	0	0
$285$	−7.09998e9	−0.0637463
$286$	0	0
$287$	1.53580e10	0.133618
$288$	0	0
$289$	−1.17376e11	−0.989781
$290$	0	0
$291$	5.37746e10	0.439601
$292$	0	0
$293$	1.61823e11	1.28273	0.641364	−	0.767237i	$-0.278369\pi$
$293$	1.61823e11	1.28273	0.641364	+	0.767237i	$0.278369\pi$
$294$	0	0
$295$	2.96558e11	2.27987
$296$	0	0
$297$	3.78772e10	0.282471
$298$	0	0
$299$	−2.49173e9	−0.0180294
$300$	0	0
$301$	3.52889e10	0.247793
$302$	0	0
$303$	5.77007e10	0.393269
$304$	0	0
$305$	−1.27319e11	−0.842450
$306$	0	0
$307$	2.09189e11	1.34405	0.672025	−	0.740528i	$-0.265425\pi$
$307$	2.09189e11	1.34405	0.672025	+	0.740528i	$0.265425\pi$
$308$	0	0
$309$	4.70612e10	0.293664
$310$	0	0
$311$	−2.58404e11	−1.56631	−0.783153	−	0.621829i	$-0.786390\pi$
$311$	−2.58404e11	−1.56631	−0.783153	+	0.621829i	$0.786390\pi$
$312$	0	0
$313$	2.73691e11	1.61180	0.805901	−	0.592051i	$-0.201681\pi$
$313$	2.73691e11	1.61180	0.805901	+	0.592051i	$0.201681\pi$
$314$	0	0
$315$	3.58676e10	0.205260
$316$	0	0
$317$	−3.02680e10	−0.168352	−0.0841759	−	0.996451i	$-0.526826\pi$
$317$	−3.02680e10	−0.168352	−0.0841759	+	0.996451i	$0.526826\pi$
$318$	0	0
$319$	6.05784e10	0.327536
$320$	0	0
$321$	−1.43718e10	−0.0755506
$322$	0	0
$323$	1.03002e9	0.00526545
$324$	0	0
$325$	−1.08483e11	−0.539372
$326$	0	0
$327$	−1.92633e11	−0.931680
$328$	0	0
$329$	6.74420e10	0.317358
$330$	0	0
$331$	3.04175e11	1.39283	0.696414	−	0.717640i	$-0.254778\pi$
$331$	3.04175e11	1.39283	0.696414	+	0.717640i	$0.254778\pi$
$332$	0	0
$333$	−1.43111e11	−0.637786
$334$	0	0
$335$	4.43773e10	0.192513
$336$	0	0
$337$	2.17056e11	0.916719	0.458360	−	0.888767i	$-0.348437\pi$
$337$	2.17056e11	0.916719	0.458360	+	0.888767i	$0.348437\pi$
$338$	0	0
$339$	6.17876e10	0.254099
$340$	0	0
$341$	−7.43535e10	−0.297788
$342$	0	0
$343$	−1.21105e11	−0.472433
$344$	0	0
$345$	2.09345e10	0.0795565
$346$	0	0
$347$	−6.29807e10	−0.233198	−0.116599	−	0.993179i	$-0.537199\pi$
$347$	−6.29807e10	−0.233198	−0.116599	+	0.993179i	$0.537199\pi$
$348$	0	0
$349$	2.13921e11	0.771860	0.385930	−	0.922528i	$-0.373881\pi$
$349$	2.13921e11	0.771860	0.385930	+	0.922528i	$0.373881\pi$
$350$	0	0
$351$	−8.38874e10	−0.294995
$352$	0	0
$353$	−3.21280e11	−1.10128	−0.550639	−	0.834744i	$-0.685616\pi$
$353$	−3.21280e11	−1.10128	−0.550639	+	0.834744i	$0.685616\pi$
$354$	0	0
$355$	7.41263e11	2.47711
$356$	0	0
$357$	5.38620e9	0.0175499
$358$	0	0
$359$	1.30411e11	0.414371	0.207186	−	0.978302i	$-0.433570\pi$
$359$	1.30411e11	0.414371	0.207186	+	0.978302i	$0.433570\pi$
$360$	0	0
$361$	−3.21812e11	−0.997287
$362$	0	0
$363$	2.19288e11	0.662881
$364$	0	0
$365$	−8.65209e11	−2.55155
$366$	0	0
$367$	−3.75855e11	−1.08149	−0.540746	−	0.841186i	$-0.681858\pi$
$367$	−3.75855e11	−1.08149	−0.540746	+	0.841186i	$0.681858\pi$
$368$	0	0
$369$	−9.60557e10	−0.269715
$370$	0	0
$371$	−3.30194e10	−0.0904870
$372$	0	0
$373$	4.46005e10	0.119303	0.0596513	−	0.998219i	$-0.481001\pi$
$373$	4.46005e10	0.119303	0.0596513	+	0.998219i	$0.481001\pi$
$374$	0	0
$375$	4.42765e11	1.15620
$376$	0	0
$377$	−1.34164e11	−0.342059
$378$	0	0
$379$	−7.91117e11	−1.96954	−0.984768	−	0.173872i	$-0.944372\pi$
$379$	−7.91117e11	−1.96954	−0.984768	+	0.173872i	$0.944372\pi$
$380$	0	0
$381$	3.20925e11	0.780262
$382$	0	0
$383$	3.98733e10	0.0946863	0.0473432	−	0.998879i	$-0.484925\pi$
$383$	3.98733e10	0.0946863	0.0473432	+	0.998879i	$0.484925\pi$
$384$	0	0
$385$	−4.78251e10	−0.110939
$386$	0	0
$387$	−2.20712e11	−0.500180
$388$	0	0
$389$	4.45502e11	0.986453	0.493227	−	0.869901i	$-0.335817\pi$
$389$	4.45502e11	0.986453	0.493227	+	0.869901i	$0.335817\pi$
$390$	0	0
$391$	−3.03705e9	−0.00657137
$392$	0	0
$393$	4.61838e10	0.0976614
$394$	0	0
$395$	1.00318e12	2.07344
$396$	0	0
$397$	−1.41918e11	−0.286735	−0.143367	−	0.989670i	$-0.545793\pi$
$397$	−1.41918e11	−0.286735	−0.143367	+	0.989670i	$0.545793\pi$
$398$	0	0
$399$	4.57807e9	0.00904283
$400$	0	0
$401$	7.91193e11	1.52803	0.764016	−	0.645197i	$-0.223225\pi$
$401$	7.91193e11	1.52803	0.764016	+	0.645197i	$0.223225\pi$
$402$	0	0
$403$	1.64672e11	0.310991
$404$	0	0
$405$	2.48243e11	0.458490
$406$	0	0
$407$	1.90821e11	0.344708
$408$	0	0
$409$	7.35512e11	1.29968	0.649838	−	0.760073i	$-0.274837\pi$
$409$	7.35512e11	1.29968	0.649838	+	0.760073i	$0.274837\pi$
$410$	0	0
$411$	2.72549e11	0.471146
$412$	0	0
$413$	−1.91221e11	−0.323415
$414$	0	0
$415$	−1.19398e12	−1.97597
$416$	0	0
$417$	−5.18918e11	−0.840400
$418$	0	0
$419$	6.46595e10	0.102487	0.0512436	−	0.998686i	$-0.483682\pi$
$419$	6.46595e10	0.102487	0.0512436	+	0.998686i	$0.483682\pi$
$420$	0	0
$421$	−3.44906e11	−0.535095	−0.267548	−	0.963545i	$-0.586213\pi$
$421$	−3.44906e11	−0.535095	−0.267548	+	0.963545i	$0.586213\pi$
$422$	0	0
$423$	−4.21812e11	−0.640600
$424$	0	0
$425$	−1.32225e11	−0.196591
$426$	0	0
$427$	8.20954e10	0.119507
$428$	0	0
$429$	3.68531e10	0.0525310
$430$	0	0
$431$	−1.18138e12	−1.64908	−0.824539	−	0.565806i	$-0.808565\pi$
$431$	−1.18138e12	−1.64908	−0.824539	+	0.565806i	$0.808565\pi$
$432$	0	0
$433$	8.62695e11	1.17940	0.589701	−	0.807622i	$-0.299246\pi$
$433$	8.62695e11	1.17940	0.589701	+	0.807622i	$0.299246\pi$
$434$	0	0
$435$	1.12719e12	1.50937
$436$	0	0
$437$	−2.58137e9	−0.00338598
$438$	0	0
$439$	−1.42545e12	−1.83174	−0.915868	−	0.401480i	$-0.868496\pi$
$439$	−1.42545e12	−1.83174	−0.915868	+	0.401480i	$0.868496\pi$
$440$	0	0
$441$	3.67159e11	0.462254
$442$	0	0
$443$	1.30736e11	0.161279	0.0806396	−	0.996743i	$-0.474304\pi$
$443$	1.30736e11	0.161279	0.0806396	+	0.996743i	$0.474304\pi$
$444$	0	0
$445$	−8.03670e11	−0.971533
$446$	0	0
$447$	4.06764e11	0.481901
$448$	0	0
$449$	−3.46674e11	−0.402543	−0.201272	−	0.979535i	$-0.564507\pi$
$449$	−3.46674e11	−0.402543	−0.201272	+	0.979535i	$0.564507\pi$
$450$	0	0
$451$	1.28078e11	0.145775
$452$	0	0
$453$	9.37959e11	1.04651
$454$	0	0
$455$	1.05919e11	0.115857
$456$	0	0
$457$	1.21675e12	1.30491	0.652454	−	0.757829i	$-0.273740\pi$
$457$	1.21675e12	1.30491	0.652454	+	0.757829i	$0.273740\pi$
$458$	0	0
$459$	−1.02246e11	−0.107520
$460$	0	0
$461$	−1.71920e12	−1.77285	−0.886425	−	0.462871i	$-0.846819\pi$
$461$	−1.71920e12	−1.77285	−0.886425	+	0.462871i	$0.846819\pi$
$462$	0	0
$463$	−1.11675e12	−1.12938	−0.564691	−	0.825302i	$-0.691005\pi$
$463$	−1.11675e12	−1.12938	−0.564691	+	0.825302i	$0.691005\pi$
$464$	0	0
$465$	−1.38351e12	−1.37228
$466$	0	0
$467$	−1.49994e12	−1.45931	−0.729654	−	0.683817i	$-0.760319\pi$
$467$	−1.49994e12	−1.45931	−0.729654	+	0.683817i	$0.760319\pi$
$468$	0	0
$469$	−2.86145e10	−0.0273092
$470$	0	0
$471$	−8.39792e11	−0.786281
$472$	0	0
$473$	2.94292e11	0.270336
$474$	0	0
$475$	−1.12386e11	−0.101296
$476$	0	0
$477$	2.06518e11	0.182652
$478$	0	0
$479$	1.02224e12	0.887244	0.443622	−	0.896214i	$-0.353693\pi$
$479$	1.02224e12	0.887244	0.443622	+	0.896214i	$0.353693\pi$
$480$	0	0
$481$	−4.22616e11	−0.359992
$482$	0	0
$483$	−1.34985e10	−0.0112856
$484$	0	0
$485$	1.28890e12	1.05775
$486$	0	0
$487$	−2.00167e12	−1.61254	−0.806272	−	0.591545i	$-0.798518\pi$
$487$	−2.00167e12	−1.61254	−0.806272	+	0.591545i	$0.798518\pi$
$488$	0	0
$489$	3.92836e11	0.310686
$490$	0	0
$491$	−9.02846e11	−0.701047	−0.350523	−	0.936554i	$-0.613996\pi$
$491$	−9.02846e11	−0.701047	−0.350523	+	0.936554i	$0.613996\pi$
$492$	0	0
$493$	−1.63526e11	−0.124674
$494$	0	0
$495$	2.99119e11	0.223934
$496$	0	0
$497$	−4.77967e11	−0.351394
$498$	0	0
$499$	−1.80705e12	−1.30472	−0.652360	−	0.757909i	$-0.726221\pi$
$499$	−1.80705e12	−1.30472	−0.652360	+	0.757909i	$0.726221\pi$
$500$	0	0
$501$	1.04221e11	0.0739070
$502$	0	0
$503$	−4.95765e11	−0.345319	−0.172659	−	0.984982i	$-0.555236\pi$
$503$	−4.95765e11	−0.345319	−0.172659	+	0.984982i	$0.555236\pi$
$504$	0	0
$505$	1.38300e12	0.946263
$506$	0	0
$507$	−8.16193e10	−0.0548601
$508$	0	0
$509$	−2.41843e12	−1.59700	−0.798499	−	0.601996i	$-0.794372\pi$
$509$	−2.41843e12	−1.59700	−0.798499	+	0.601996i	$0.794372\pi$
$510$	0	0
$511$	5.57887e11	0.361953
$512$	0	0
$513$	−8.69053e10	−0.0554011
$514$	0	0
$515$	1.12799e12	0.706599
$516$	0	0
$517$	5.62434e11	0.346230
$518$	0	0
$519$	9.91811e11	0.600034
$520$	0	0
$521$	−7.87743e10	−0.0468398	−0.0234199	−	0.999726i	$-0.507455\pi$
$521$	−7.87743e10	−0.0468398	−0.0234199	+	0.999726i	$0.507455\pi$
$522$	0	0
$523$	−8.05258e11	−0.470628	−0.235314	−	0.971919i	$-0.575612\pi$
$523$	−8.05258e11	−0.470628	−0.235314	+	0.971919i	$0.575612\pi$
$524$	0	0
$525$	−5.87691e11	−0.337623
$526$	0	0
$527$	2.00711e11	0.113350
$528$	0	0
$529$	−1.79354e12	−0.995774
$530$	0	0
$531$	1.19598e12	0.652827
$532$	0	0
$533$	−2.83658e11	−0.152238
$534$	0	0
$535$	−3.44471e11	−0.181786
$536$	0	0
$537$	4.10763e11	0.213161
$538$	0	0
$539$	−4.89562e11	−0.249838
$540$	0	0
$541$	−4.67507e11	−0.234639	−0.117320	−	0.993094i	$-0.537430\pi$
$541$	−4.67507e11	−0.234639	−0.117320	+	0.993094i	$0.537430\pi$
$542$	0	0
$543$	2.62281e10	0.0129469
$544$	0	0
$545$	−4.61715e12	−2.24176
$546$	0	0
$547$	1.59182e12	0.760242	0.380121	−	0.924937i	$-0.375882\pi$
$547$	1.59182e12	0.760242	0.380121	+	0.924937i	$0.375882\pi$
$548$	0	0
$549$	−5.13461e11	−0.241230
$550$	0	0
$551$	−1.38991e11	−0.0642398
$552$	0	0
$553$	−6.46849e11	−0.294130
$554$	0	0
$555$	3.55064e12	1.58850
$556$	0	0
$557$	−4.55275e10	−0.0200413	−0.0100207	−	0.999950i	$-0.503190\pi$
$557$	−4.55275e10	−0.0200413	−0.0100207	+	0.999950i	$0.503190\pi$
$558$	0	0
$559$	−6.51775e11	−0.282322
$560$	0	0
$561$	4.49184e10	0.0191466
$562$	0	0
$563$	1.75827e12	0.737562	0.368781	−	0.929516i	$-0.379775\pi$
$563$	1.75827e12	0.737562	0.368781	+	0.929516i	$0.379775\pi$
$564$	0	0
$565$	1.48096e12	0.611400
$566$	0	0
$567$	−1.60067e11	−0.0650398
$568$	0	0
$569$	1.69143e12	0.676469	0.338235	−	0.941062i	$-0.390170\pi$
$569$	1.69143e12	0.676469	0.338235	+	0.941062i	$0.390170\pi$
$570$	0	0
$571$	1.78493e12	0.702682	0.351341	−	0.936247i	$-0.385726\pi$
$571$	1.78493e12	0.702682	0.351341	+	0.936247i	$0.385726\pi$
$572$	0	0
$573$	−4.98511e11	−0.193187
$574$	0	0
$575$	3.31374e11	0.126419
$576$	0	0
$577$	−3.75906e12	−1.41185	−0.705924	−	0.708288i	$-0.749468\pi$
$577$	−3.75906e12	−1.41185	−0.705924	+	0.708288i	$0.749468\pi$
$578$	0	0
$579$	−4.89616e11	−0.181051
$580$	0	0
$581$	7.69876e11	0.280303
$582$	0	0
$583$	−2.75366e11	−0.0987192
$584$	0	0
$585$	−6.62465e11	−0.233863
$586$	0	0
$587$	−2.10553e11	−0.0731965	−0.0365982	−	0.999330i	$-0.511652\pi$
$587$	−2.10553e11	−0.0731965	−0.0365982	+	0.999330i	$0.511652\pi$
$588$	0	0
$589$	1.70597e11	0.0584052
$590$	0	0
$591$	3.92774e12	1.32434
$592$	0	0
$593$	2.33945e12	0.776906	0.388453	−	0.921469i	$-0.373010\pi$
$593$	2.33945e12	0.776906	0.388453	+	0.921469i	$0.373010\pi$
$594$	0	0
$595$	1.29100e11	0.0422278
$596$	0	0
$597$	−2.90805e12	−0.936953
$598$	0	0
$599$	−2.98402e12	−0.947068	−0.473534	−	0.880775i	$-0.657022\pi$
$599$	−2.98402e12	−0.947068	−0.473534	+	0.880775i	$0.657022\pi$
$600$	0	0
$601$	−1.88352e12	−0.588891	−0.294445	−	0.955668i	$-0.595135\pi$
$601$	−1.88352e12	−0.588891	−0.294445	+	0.955668i	$0.595135\pi$
$602$	0	0
$603$	1.78968e11	0.0551248
$604$	0	0
$605$	5.25603e12	1.59499
$606$	0	0
$607$	−4.77594e12	−1.42794	−0.713969	−	0.700177i	$-0.753104\pi$
$607$	−4.77594e12	−1.42794	−0.713969	+	0.700177i	$0.753104\pi$
$608$	0	0
$609$	−7.26812e11	−0.214114
$610$	0	0
$611$	−1.24563e12	−0.361581
$612$	0	0
$613$	−4.25987e12	−1.21850	−0.609249	−	0.792979i	$-0.708529\pi$
$613$	−4.25987e12	−1.21850	−0.609249	+	0.792979i	$0.708529\pi$
$614$	0	0
$615$	2.38317e12	0.671765
$616$	0	0
$617$	4.63557e12	1.28771	0.643857	−	0.765145i	$-0.277333\pi$
$617$	4.63557e12	1.28771	0.643857	+	0.765145i	$0.277333\pi$
$618$	0	0
$619$	−5.83308e12	−1.59694	−0.798472	−	0.602032i	$-0.794358\pi$
$619$	−5.83308e12	−1.59694	−0.798472	+	0.602032i	$0.794358\pi$
$620$	0	0
$621$	2.56242e11	0.0691415
$622$	0	0
$623$	5.18207e11	0.137818
$624$	0	0
$625$	3.19387e12	0.837255
$626$	0	0
$627$	3.81789e10	0.00986551
$628$	0	0
$629$	−5.15105e11	−0.131210
$630$	0	0
$631$	−5.50540e12	−1.38247	−0.691236	−	0.722629i	$-0.742934\pi$
$631$	−5.50540e12	−1.38247	−0.691236	+	0.722629i	$0.742934\pi$
$632$	0	0
$633$	1.12189e11	0.0277736
$634$	0	0
$635$	7.69210e12	1.87743
$636$	0	0
$637$	1.08424e12	0.260915
$638$	0	0
$639$	2.98941e12	0.709304
$640$	0	0
$641$	−6.97731e12	−1.63240	−0.816201	−	0.577768i	$-0.803924\pi$
$641$	−6.97731e12	−1.63240	−0.816201	+	0.577768i	$0.803924\pi$
$642$	0	0
$643$	−1.83777e12	−0.423976	−0.211988	−	0.977272i	$-0.567994\pi$
$643$	−1.83777e12	−0.423976	−0.211988	+	0.977272i	$0.567994\pi$
$644$	0	0
$645$	5.47593e12	1.24577
$646$	0	0
$647$	−7.60981e11	−0.170728	−0.0853640	−	0.996350i	$-0.527205\pi$
$647$	−7.60981e11	−0.170728	−0.0853640	+	0.996350i	$0.527205\pi$
$648$	0	0
$649$	−1.59469e12	−0.352838
$650$	0	0
$651$	8.92085e11	0.194667
$652$	0	0
$653$	−2.16319e12	−0.465570	−0.232785	−	0.972528i	$-0.574784\pi$
$653$	−2.16319e12	−0.465570	−0.232785	+	0.972528i	$0.574784\pi$
$654$	0	0
$655$	1.10696e12	0.234988
$656$	0	0
$657$	−3.48927e12	−0.730618
$658$	0	0
$659$	6.02579e12	1.24460	0.622300	−	0.782779i	$-0.286199\pi$
$659$	6.02579e12	1.24460	0.622300	+	0.782779i	$0.286199\pi$
$660$	0	0
$661$	1.26118e12	0.256963	0.128481	−	0.991712i	$-0.458990\pi$
$661$	1.26118e12	0.256963	0.128481	+	0.991712i	$0.458990\pi$
$662$	0	0
$663$	−9.94817e10	−0.0199955
$664$	0	0
$665$	1.09730e11	0.0217584
$666$	0	0
$667$	4.09818e11	0.0801724
$668$	0	0
$669$	−3.12572e12	−0.603299
$670$	0	0
$671$	6.84637e11	0.130379
$672$	0	0
$673$	7.37827e12	1.38640	0.693198	−	0.720748i	$-0.256201\pi$
$673$	7.37827e12	1.38640	0.693198	+	0.720748i	$0.256201\pi$
$674$	0	0
$675$	1.11561e13	2.06846
$676$	0	0
$677$	5.88382e12	1.07649	0.538245	−	0.842788i	$-0.319087\pi$
$677$	5.88382e12	1.07649	0.538245	+	0.842788i	$0.319087\pi$
$678$	0	0
$679$	−8.31083e11	−0.150048
$680$	0	0
$681$	4.59312e12	0.818362
$682$	0	0
$683$	4.88451e12	0.858871	0.429436	−	0.903098i	$-0.358713\pi$
$683$	4.88451e12	0.858871	0.429436	+	0.903098i	$0.358713\pi$
$684$	0	0
$685$	6.53260e12	1.13365
$686$	0	0
$687$	−5.15089e12	−0.882220
$688$	0	0
$689$	6.09858e11	0.103096
$690$	0	0
$691$	9.77308e12	1.63072	0.815361	−	0.578953i	$-0.196538\pi$
$691$	9.77308e12	1.63072	0.815361	+	0.578953i	$0.196538\pi$
$692$	0	0
$693$	−1.92872e11	−0.0317665
$694$	0	0
$695$	−1.24377e13	−2.02213
$696$	0	0
$697$	−3.45737e11	−0.0554879
$698$	0	0
$699$	5.22206e12	0.827361
$700$	0	0
$701$	−1.06947e13	−1.67278	−0.836391	−	0.548134i	$-0.815339\pi$
$701$	−1.06947e13	−1.67278	−0.836391	+	0.548134i	$0.815339\pi$
$702$	0	0
$703$	−4.37820e11	−0.0676078
$704$	0	0
$705$	1.04653e13	1.59551
$706$	0	0
$707$	−8.91761e11	−0.134234
$708$	0	0
$709$	6.42483e12	0.954891	0.477445	−	0.878661i	$-0.341563\pi$
$709$	6.42483e12	0.954891	0.477445	+	0.878661i	$0.341563\pi$
$710$	0	0
$711$	4.04568e12	0.593715
$712$	0	0
$713$	−5.03008e11	−0.0728907
$714$	0	0
$715$	8.83316e11	0.126398
$716$	0	0
$717$	9.76899e12	1.38043
$718$	0	0
$719$	−9.15374e12	−1.27738	−0.638688	−	0.769466i	$-0.720523\pi$
$719$	−9.15374e12	−1.27738	−0.638688	+	0.769466i	$0.720523\pi$
$720$	0	0
$721$	−7.27329e11	−0.100236
$722$	0	0
$723$	−4.67050e12	−0.635684
$724$	0	0
$725$	1.78424e13	2.39846
$726$	0	0
$727$	3.78498e12	0.502526	0.251263	−	0.967919i	$-0.419154\pi$
$727$	3.78498e12	0.502526	0.251263	+	0.967919i	$0.419154\pi$
$728$	0	0
$729$	6.78556e12	0.889839
$730$	0	0
$731$	−7.94416e11	−0.102901
$732$	0	0
$733$	3.08194e12	0.394327	0.197164	−	0.980371i	$-0.436827\pi$
$733$	3.08194e12	0.394327	0.197164	+	0.980371i	$0.436827\pi$
$734$	0	0
$735$	−9.10934e12	−1.15131
$736$	0	0
$737$	−2.38631e11	−0.0297937
$738$	0	0
$739$	1.13283e13	1.39721	0.698607	−	0.715505i	$-0.253803\pi$
$739$	1.13283e13	1.39721	0.698607	+	0.715505i	$0.253803\pi$
$740$	0	0
$741$	−8.45556e10	−0.0103029
$742$	0	0
$743$	−1.98944e12	−0.239486	−0.119743	−	0.992805i	$-0.538207\pi$
$743$	−1.98944e12	−0.239486	−0.119743	+	0.992805i	$0.538207\pi$
$744$	0	0
$745$	9.74954e12	1.15953
$746$	0	0
$747$	−4.81514e12	−0.565805
$748$	0	0
$749$	2.22115e11	0.0257875
$750$	0	0
$751$	−1.64086e13	−1.88231	−0.941157	−	0.337971i	$-0.890259\pi$
$751$	−1.64086e13	−1.88231	−0.941157	+	0.337971i	$0.890259\pi$
$752$	0	0
$753$	−4.79102e12	−0.543064
$754$	0	0
$755$	2.24815e13	2.51805
$756$	0	0
$757$	1.47125e13	1.62838	0.814190	−	0.580598i	$-0.197181\pi$
$757$	1.47125e13	1.62838	0.814190	+	0.580598i	$0.197181\pi$
$758$	0	0
$759$	−1.12571e11	−0.0123123
$760$	0	0
$761$	−7.63318e12	−0.825040	−0.412520	−	0.910949i	$-0.635351\pi$
$761$	−7.63318e12	−0.825040	−0.412520	+	0.910949i	$0.635351\pi$
$762$	0	0
$763$	2.97714e12	0.318008
$764$	0	0
$765$	−8.07446e11	−0.0852387
$766$	0	0
$767$	3.53179e12	0.368482
$768$	0	0
$769$	8.30840e12	0.856740	0.428370	−	0.903603i	$-0.359088\pi$
$769$	8.30840e12	0.856740	0.428370	+	0.903603i	$0.359088\pi$
$770$	0	0
$771$	−1.95212e12	−0.198958
$772$	0	0
$773$	−8.08709e12	−0.814675	−0.407338	−	0.913278i	$-0.633543\pi$
$773$	−8.08709e12	−0.814675	−0.407338	+	0.913278i	$0.633543\pi$
$774$	0	0
$775$	−2.18997e13	−2.18062
$776$	0	0
$777$	−2.28945e12	−0.225339
$778$	0	0
$779$	−2.93863e11	−0.0285908
$780$	0	0
$781$	−3.98602e12	−0.383362
$782$	0	0
$783$	1.37970e13	1.31177
$784$	0	0
$785$	−2.01286e13	−1.89191
$786$	0	0
$787$	−7.00768e12	−0.651161	−0.325580	−	0.945514i	$-0.605560\pi$
$787$	−7.00768e12	−0.651161	−0.325580	+	0.945514i	$0.605560\pi$
$788$	0	0
$789$	−6.50585e12	−0.597664
$790$	0	0
$791$	−9.54924e11	−0.0867310
$792$	0	0
$793$	−1.51628e12	−0.136160
$794$	0	0
$795$	−5.12377e12	−0.454922
$796$	0	0
$797$	−2.13611e13	−1.87526	−0.937631	−	0.347632i	$-0.886986\pi$
$797$	−2.13611e13	−1.87526	−0.937631	+	0.347632i	$0.886986\pi$
$798$	0	0
$799$	−1.51824e12	−0.131789
$800$	0	0
$801$	−3.24109e12	−0.278192
$802$	0	0
$803$	4.65251e12	0.394882
$804$	0	0
$805$	−3.23541e11	−0.0271549
$806$	0	0
$807$	2.04440e11	0.0169681
$808$	0	0
$809$	−1.54103e13	−1.26486	−0.632432	−	0.774616i	$-0.717943\pi$
$809$	−1.54103e13	−1.26486	−0.632432	+	0.774616i	$0.717943\pi$
$810$	0	0
$811$	1.10423e13	0.896327	0.448164	−	0.893951i	$-0.352078\pi$
$811$	1.10423e13	0.896327	0.448164	+	0.893951i	$0.352078\pi$
$812$	0	0
$813$	−1.24196e13	−0.997009
$814$	0	0
$815$	9.41571e12	0.747556
$816$	0	0
$817$	−6.75223e11	−0.0530210
$818$	0	0
$819$	4.27158e11	0.0331750
$820$	0	0
$821$	−2.07575e12	−0.159453	−0.0797263	−	0.996817i	$-0.525405\pi$
$821$	−2.07575e12	−0.159453	−0.0797263	+	0.996817i	$0.525405\pi$
$822$	0	0
$823$	−2.47289e13	−1.87890	−0.939452	−	0.342680i	$-0.888665\pi$
$823$	−2.47289e13	−1.87890	−0.939452	+	0.342680i	$0.888665\pi$
$824$	0	0
$825$	−4.90106e12	−0.368339
$826$	0	0
$827$	−1.19341e13	−0.887186	−0.443593	−	0.896228i	$-0.646296\pi$
$827$	−1.19341e13	−0.887186	−0.443593	+	0.896228i	$0.646296\pi$
$828$	0	0
$829$	9.59657e12	0.705701	0.352851	−	0.935680i	$-0.385212\pi$
$829$	9.59657e12	0.705701	0.352851	+	0.935680i	$0.385212\pi$
$830$	0	0
$831$	−7.05515e12	−0.513217
$832$	0	0
$833$	1.32153e12	0.0950986
$834$	0	0
$835$	2.49803e12	0.177831
$836$	0	0
$837$	−1.69344e13	−1.19263
$838$	0	0
$839$	9.04228e12	0.630012	0.315006	−	0.949090i	$-0.397993\pi$
$839$	9.04228e12	0.630012	0.315006	+	0.949090i	$0.397993\pi$
$840$	0	0
$841$	7.55898e12	0.521052
$842$	0	0
$843$	−1.60224e13	−1.09270
$844$	0	0
$845$	−1.95630e12	−0.132002
$846$	0	0
$847$	−3.38909e12	−0.226260
$848$	0	0
$849$	1.61337e12	0.106573
$850$	0	0
$851$	1.29092e12	0.0843757
$852$	0	0
$853$	−2.18412e13	−1.41256	−0.706278	−	0.707935i	$-0.749627\pi$
$853$	−2.18412e13	−1.41256	−0.706278	+	0.707935i	$0.749627\pi$
$854$	0	0
$855$	−6.86298e11	−0.0439203
$856$	0	0
$857$	−1.25968e13	−0.797710	−0.398855	−	0.917014i	$-0.630592\pi$
$857$	−1.25968e13	−0.797710	−0.398855	+	0.917014i	$0.630592\pi$
$858$	0	0
$859$	1.82770e13	1.14534	0.572671	−	0.819785i	$-0.305907\pi$
$859$	1.82770e13	1.14534	0.572671	+	0.819785i	$0.305907\pi$
$860$	0	0
$861$	−1.53667e12	−0.0952942
$862$	0	0
$863$	2.96892e13	1.82201	0.911004	−	0.412398i	$-0.135309\pi$
$863$	2.96892e13	1.82201	0.911004	+	0.412398i	$0.135309\pi$
$864$	0	0
$865$	2.37723e13	1.44377
$866$	0	0
$867$	1.17442e13	0.705894
$868$	0	0
$869$	−5.39441e12	−0.320889
$870$	0	0
$871$	5.28502e11	0.0311146
$872$	0	0
$873$	5.19796e12	0.302878
$874$	0	0
$875$	−6.84291e12	−0.394643
$876$	0	0
$877$	9.21910e12	0.526248	0.263124	−	0.964762i	$-0.415247\pi$
$877$	9.21910e12	0.526248	0.263124	+	0.964762i	$0.415247\pi$
$878$	0	0
$879$	−1.61914e13	−0.914818
$880$	0	0
$881$	8.97398e12	0.501873	0.250936	−	0.968004i	$-0.419262\pi$
$881$	8.97398e12	0.501873	0.250936	+	0.968004i	$0.419262\pi$
$882$	0	0
$883$	1.18978e13	0.658631	0.329316	−	0.944220i	$-0.393182\pi$
$883$	1.18978e13	0.658631	0.329316	+	0.944220i	$0.393182\pi$
$884$	0	0
$885$	−2.96726e13	−1.62596
$886$	0	0
$887$	−5.73960e12	−0.311333	−0.155667	−	0.987810i	$-0.549753\pi$
$887$	−5.73960e12	−0.311333	−0.155667	+	0.987810i	$0.549753\pi$
$888$	0	0
$889$	−4.95987e12	−0.266325
$890$	0	0
$891$	−1.33489e12	−0.0709569
$892$	0	0
$893$	−1.29045e12	−0.0679061
$894$	0	0
$895$	9.84540e12	0.512897
$896$	0	0
$897$	2.49314e11	0.0128582
$898$	0	0
$899$	−2.70838e13	−1.38290
$900$	0	0
$901$	7.43326e11	0.0375766
$902$	0	0
$903$	−3.53088e12	−0.176721
$904$	0	0
$905$	6.28650e11	0.0311523
$906$	0	0
$907$	8.33483e12	0.408944	0.204472	−	0.978872i	$-0.434452\pi$
$907$	8.33483e12	0.408944	0.204472	+	0.978872i	$0.434452\pi$
$908$	0	0
$909$	5.57746e12	0.270956
$910$	0	0
$911$	3.19355e13	1.53618	0.768089	−	0.640343i	$-0.221208\pi$
$911$	3.19355e13	1.53618	0.768089	+	0.640343i	$0.221208\pi$
$912$	0	0
$913$	6.42040e12	0.305804
$914$	0	0
$915$	1.27391e13	0.600820
$916$	0	0
$917$	−7.13768e11	−0.0333346
$918$	0	0
$919$	1.23491e13	0.571106	0.285553	−	0.958363i	$-0.407823\pi$
$919$	1.23491e13	0.571106	0.285553	+	0.958363i	$0.407823\pi$
$920$	0	0
$921$	−2.09307e13	−0.958552
$922$	0	0
$923$	8.82791e12	0.400360
$924$	0	0
$925$	5.62034e13	2.52421
$926$	0	0
$927$	4.54903e12	0.202330
$928$	0	0
$929$	4.29297e12	0.189098	0.0945490	−	0.995520i	$-0.469859\pi$
$929$	4.29297e12	0.189098	0.0945490	+	0.995520i	$0.469859\pi$
$930$	0	0
$931$	1.12325e12	0.0490007
$932$	0	0
$933$	2.58550e13	1.11706
$934$	0	0
$935$	1.07663e12	0.0460695
$936$	0	0
$937$	−2.84490e13	−1.20570	−0.602850	−	0.797855i	$-0.705968\pi$
$937$	−2.84490e13	−1.20570	−0.602850	+	0.797855i	$0.705968\pi$
$938$	0	0
$939$	−2.73846e13	−1.14951
$940$	0	0
$941$	−3.33306e13	−1.38576	−0.692882	−	0.721051i	$-0.743659\pi$
$941$	−3.33306e13	−1.38576	−0.692882	+	0.721051i	$0.743659\pi$
$942$	0	0
$943$	8.66462e11	0.0356818
$944$	0	0
$945$	−1.08924e13	−0.444305
$946$	0	0
$947$	−4.47061e13	−1.80631	−0.903153	−	0.429318i	$-0.858754\pi$
$947$	−4.47061e13	−1.80631	−0.903153	+	0.429318i	$0.858754\pi$
$948$	0	0
$949$	−1.03040e13	−0.412391
$950$	0	0
$951$	3.02852e12	0.120065
$952$	0	0
$953$	−2.37142e13	−0.931303	−0.465651	−	0.884968i	$-0.654180\pi$
$953$	−2.37142e13	−0.931303	−0.465651	+	0.884968i	$0.654180\pi$
$954$	0	0
$955$	−1.19486e13	−0.464838
$956$	0	0
$957$	−6.06127e12	−0.233593
$958$	0	0
$959$	−4.21222e12	−0.160815
$960$	0	0
$961$	6.80292e12	0.257300
$962$	0	0
$963$	−1.38920e12	−0.0520532
$964$	0	0
$965$	−1.17354e13	−0.435637
$966$	0	0
$967$	−8.31095e12	−0.305655	−0.152828	−	0.988253i	$-0.548838\pi$
$967$	−8.31095e12	−0.305655	−0.152828	+	0.988253i	$0.548838\pi$
$968$	0	0
$969$	−1.03061e11	−0.00375522
$970$	0	0
$971$	−2.71942e13	−0.981725	−0.490863	−	0.871237i	$-0.663318\pi$
$971$	−2.71942e13	−0.981725	−0.490863	+	0.871237i	$0.663318\pi$
$972$	0	0
$973$	8.01984e12	0.286852
$974$	0	0
$975$	1.08545e13	0.384670
$976$	0	0
$977$	1.22678e13	0.430767	0.215384	−	0.976530i	$-0.430900\pi$
$977$	1.22678e13	0.430767	0.215384	+	0.976530i	$0.430900\pi$
$978$	0	0
$979$	4.32160e12	0.150356
$980$	0	0
$981$	−1.86203e13	−0.641914
$982$	0	0
$983$	−3.84479e13	−1.31336	−0.656678	−	0.754171i	$-0.728039\pi$
$983$	−3.84479e13	−1.31336	−0.656678	+	0.754171i	$0.728039\pi$
$984$	0	0
$985$	9.41424e13	3.18656
$986$	0	0
$987$	−6.74802e12	−0.226334
$988$	0	0
$989$	1.99091e12	0.0661712
$990$	0	0
$991$	−1.42869e12	−0.0470550	−0.0235275	−	0.999723i	$-0.507490\pi$
$991$	−1.42869e12	−0.0470550	−0.0235275	+	0.999723i	$0.507490\pi$
$992$	0	0
$993$	−3.04347e13	−0.993340
$994$	0	0
$995$	−6.97019e13	−2.25445
$996$	0	0
$997$	−5.20929e13	−1.66974	−0.834872	−	0.550443i	$-0.814459\pi$
$997$	−5.20929e13	−1.66974	−0.834872	+	0.550443i	$0.814459\pi$
$998$	0	0
$999$	4.34606e13	1.38055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.l.1.3		7
4.3	odd	2		104.10.a.c.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.c.1.5	✓	7	4.3	odd	2
208.10.a.l.1.3		7	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-100.057 q^{3} -2398.21 q^{5} +1546.37 q^{7} -9671.67 q^{9} +12896.0 q^{11} -28561.0 q^{13} +239957. q^{15} -34811.6 q^{17} -29588.5 q^{19} -154724. q^{21} +87242.5 q^{23} +3.79831e6 q^{25} +2.93713e6 q^{27} +4.69746e6 q^{29} -5.76563e6 q^{31} -1.29033e6 q^{33} -3.70853e6 q^{35} +1.47970e7 q^{37} +2.85772e6 q^{39} +9.93166e6 q^{41} +2.28205e7 q^{43} +2.31947e7 q^{45} +4.36131e7 q^{47} -3.79623e7 q^{49} +3.48313e6 q^{51} -2.13528e7 q^{53} -3.09273e7 q^{55} +2.96053e6 q^{57} -1.23658e8 q^{59} +5.30891e7 q^{61} -1.49560e7 q^{63} +6.84954e7 q^{65} -1.85043e7 q^{67} -8.72919e6 q^{69} -3.09090e8 q^{71} +3.60772e8 q^{73} -3.80046e8 q^{75} +1.99420e7 q^{77} -4.18302e8 q^{79} -1.03512e8 q^{81} +4.97861e8 q^{83} +8.34856e7 q^{85} -4.70012e8 q^{87} +3.35112e8 q^{89} -4.41659e7 q^{91} +5.76890e8 q^{93} +7.09596e7 q^{95} -5.37441e8 q^{97} -1.24726e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 13 q^{3} - 801 q^{5} + 1091 q^{7} + 87864 q^{9} - 32218 q^{11} - 199927 q^{13} - 58323 q^{15} - 870531 q^{17} + 128950 q^{19} - 719663 q^{21} - 2198844 q^{23} + 992010 q^{25} - 5441383 q^{27} + 6327710 q^{29}+ \cdots - 5204667600 q^{99}+O(q^{100}) \) 7 * q - 13 * q^3 - 801 * q^5 + 1091 * q^7 + 87864 * q^9 - 32218 * q^11 - 199927 * q^13 - 58323 * q^15 - 870531 * q^17 + 128950 * q^19 - 719663 * q^21 - 2198844 * q^23 + 992010 * q^25 - 5441383 * q^27 + 6327710 * q^29 + 1660376 * q^31 + 14199598 * q^33 + 19231769 * q^35 + 5058787 * q^37 + 371293 * q^39 + 14779748 * q^41 - 9405075 * q^43 + 56644022 * q^45 - 133415721 * q^47 + 44544988 * q^49 - 132066169 * q^51 + 43962424 * q^53 - 191344746 * q^55 + 138963078 * q^57 - 22370274 * q^59 + 23851748 * q^61 - 780720706 * q^63 + 22877361 * q^65 - 12058786 * q^67 + 1036453420 * q^69 - 864303791 * q^71 + 670491202 * q^73 - 693324044 * q^75 + 841052238 * q^77 - 1019547572 * q^79 + 900194079 * q^81 - 1345041700 * q^83 + 824066375 * q^85 - 2589172074 * q^87 + 1409158546 * q^89 - 31160051 * q^91 - 445200892 * q^93 - 3285213058 * q^95 + 320206906 * q^97 - 5204667600 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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