LMFDB - Embedded newform 126.8.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,8,Mod(1,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-16,0,128,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.3605132110$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{499}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 499 $ x^2 - 499
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$22.3383$ of defining polynomial
Character	$\chi$	$=$	126.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000 q^{2} +64.0000 q^{4} +162.030 q^{5} -343.000 q^{7} -512.000 q^{8} -1296.24 q^{10} -1306.63 q^{11} +9652.63 q^{13} +2744.00 q^{14} +4096.00 q^{16} +29490.0 q^{17} -40827.4 q^{19} +10369.9 q^{20} +10453.0 q^{22} -27081.9 q^{23} -51871.3 q^{25} -77221.0 q^{26} -21952.0 q^{28} +201460. q^{29} -195315. q^{31} -32768.0 q^{32} -235920. q^{34} -55576.2 q^{35} +291580. q^{37} +326619. q^{38} -82959.3 q^{40} -183410. q^{41} -255395. q^{43} -83624.1 q^{44} +216655. q^{46} +980650. q^{47} +117649. q^{49} +414971. q^{50} +617768. q^{52} +1.94956e6 q^{53} -211713. q^{55} +175616. q^{56} -1.61168e6 q^{58} +312867. q^{59} +1.44334e6 q^{61} +1.56252e6 q^{62} +262144. q^{64} +1.56401e6 q^{65} +2.24319e6 q^{67} +1.88736e6 q^{68} +444610. q^{70} +1.13349e6 q^{71} -1.82440e6 q^{73} -2.33264e6 q^{74} -2.61295e6 q^{76} +448173. q^{77} +6.31863e6 q^{79} +663674. q^{80} +1.46728e6 q^{82} +7.56653e6 q^{83} +4.77826e6 q^{85} +2.04316e6 q^{86} +668993. q^{88} +5.98398e6 q^{89} -3.31085e6 q^{91} -1.73324e6 q^{92} -7.84520e6 q^{94} -6.61526e6 q^{95} +3.97074e6 q^{97} -941192. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{2} + 128 q^{4} + 56 q^{5} - 686 q^{7} - 1024 q^{8} - 448 q^{10} + 3016 q^{11} - 4284 q^{13} + 5488 q^{14} + 8192 q^{16} + 22792 q^{17} - 15176 q^{19} + 3584 q^{20} - 24128 q^{22} + 52792 q^{23}+ \cdots - 1882384 q^{98}+O(q^{100}) $ 2 * q - 16 * q^2 + 128 * q^4 + 56 * q^5 - 686 * q^7 - 1024 * q^8 - 448 * q^10 + 3016 * q^11 - 4284 * q^13 + 5488 * q^14 + 8192 * q^16 + 22792 * q^17 - 15176 * q^19 + 3584 * q^20 - 24128 * q^22 + 52792 * q^23 - 118754 * q^25 + 34272 * q^26 - 43904 * q^28 + 5120 * q^29 - 255528 * q^31 - 65536 * q^32 - 182336 * q^34 - 19208 * q^35 - 152396 * q^37 + 121408 * q^38 - 28672 * q^40 + 603288 * q^41 + 269800 * q^43 + 193024 * q^44 - 422336 * q^46 + 1693776 * q^47 + 235298 * q^49 + 950032 * q^50 - 274176 * q^52 + 2094000 * q^53 - 670040 * q^55 + 351232 * q^56 - 40960 * q^58 + 2357936 * q^59 + 2978892 * q^61 + 2044224 * q^62 + 524288 * q^64 + 3041712 * q^65 + 2895176 * q^67 + 1458688 * q^68 + 153664 * q^70 + 6490792 * q^71 + 3097724 * q^73 + 1219168 * q^74 - 971264 * q^76 - 1034488 * q^77 + 5641968 * q^79 + 229376 * q^80 - 4826304 * q^82 + 5489344 * q^83 + 5488456 * q^85 - 2158400 * q^86 - 1544192 * q^88 + 14285880 * q^89 + 1469412 * q^91 + 3378688 * q^92 - 13550208 * q^94 - 9335072 * q^95 + 9198140 * q^97 - 1882384 * q^98

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	162.030	0.579696	0.289848	−	0.957073i	$-0.406395\pi$
$5$	162.030	0.579696	0.289848	+	0.957073i	$0.406395\pi$
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−512.000	−0.353553
$9$	0	0
$10$	−1296.24	−0.409907
$11$	−1306.63	−0.295990	−0.147995	−	0.988988i	$-0.547282\pi$
$11$	−1306.63	−0.295990	−0.147995	+	0.988988i	$0.547282\pi$
$12$	0	0
$13$	9652.63	1.21855	0.609276	−	0.792959i	$-0.291460\pi$
$13$	9652.63	1.21855	0.609276	+	0.792959i	$0.291460\pi$
$14$	2744.00	0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	29490.0	1.45581	0.727904	−	0.685679i	$-0.240495\pi$
$17$	29490.0	1.45581	0.727904	+	0.685679i	$0.240495\pi$
$18$	0	0
$19$	−40827.4	−1.36557	−0.682785	−	0.730619i	$-0.739232\pi$
$19$	−40827.4	−1.36557	−0.682785	+	0.730619i	$0.739232\pi$
$20$	10369.9	0.289848
$21$	0	0
$22$	10453.0	0.209297
$23$	−27081.9	−0.464122	−0.232061	−	0.972701i	$-0.574547\pi$
$23$	−27081.9	−0.464122	−0.232061	+	0.972701i	$0.574547\pi$
$24$	0	0
$25$	−51871.3	−0.663953
$26$	−77221.0	−0.861646
$27$	0	0
$28$	−21952.0	−0.188982
$29$	201460.	1.53390	0.766949	−	0.641708i	$-0.221774\pi$
$29$	201460.	1.53390	0.766949	+	0.641708i	$0.221774\pi$
$30$	0	0
$31$	−195315.	−1.17752	−0.588762	−	0.808306i	$-0.700384\pi$
$31$	−195315.	−1.17752	−0.588762	+	0.808306i	$0.700384\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	−235920.	−1.02941
$35$	−55576.2	−0.219104
$36$	0	0
$37$	291580.	0.946349	0.473175	−	0.880969i	$-0.343108\pi$
$37$	291580.	0.946349	0.473175	+	0.880969i	$0.343108\pi$
$38$	326619.	0.965604
$39$	0	0
$40$	−82959.3	−0.204953
$41$	−183410.	−0.415604	−0.207802	−	0.978171i	$-0.566631\pi$
$41$	−183410.	−0.415604	−0.207802	+	0.978171i	$0.566631\pi$
$42$	0	0
$43$	−255395.	−0.489861	−0.244930	−	0.969541i	$-0.578765\pi$
$43$	−255395.	−0.489861	−0.244930	+	0.969541i	$0.578765\pi$
$44$	−83624.1	−0.147995
$45$	0	0
$46$	216655.	0.328184
$47$	980650.	1.37775	0.688877	−	0.724879i	$-0.258104\pi$
$47$	980650.	1.37775	0.688877	+	0.724879i	$0.258104\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	414971.	0.469486
$51$	0	0
$52$	617768.	0.609276
$53$	1.94956e6	1.79875	0.899374	−	0.437180i	$-0.144023\pi$
$53$	1.94956e6	1.79875	0.899374	+	0.437180i	$0.144023\pi$
$54$	0	0
$55$	−211713.	−0.171584
$56$	175616.	0.133631
$57$	0	0
$58$	−1.61168e6	−1.08463
$59$	312867.	0.198325	0.0991626	−	0.995071i	$-0.468384\pi$
$59$	312867.	0.198325	0.0991626	+	0.995071i	$0.468384\pi$
$60$	0	0
$61$	1.44334e6	0.814168	0.407084	−	0.913391i	$-0.366546\pi$
$61$	1.44334e6	0.814168	0.407084	+	0.913391i	$0.366546\pi$
$62$	1.56252e6	0.832635
$63$	0	0
$64$	262144.	0.125000
$65$	1.56401e6	0.706389
$66$	0	0
$67$	2.24319e6	0.911180	0.455590	−	0.890190i	$-0.349428\pi$
$67$	2.24319e6	0.911180	0.455590	+	0.890190i	$0.349428\pi$
$68$	1.88736e6	0.727904
$69$	0	0
$70$	444610.	0.154930
$71$	1.13349e6	0.375848	0.187924	−	0.982184i	$-0.439824\pi$
$71$	1.13349e6	0.375848	0.187924	+	0.982184i	$0.439824\pi$
$72$	0	0
$73$	−1.82440e6	−0.548896	−0.274448	−	0.961602i	$-0.588495\pi$
$73$	−1.82440e6	−0.548896	−0.274448	+	0.961602i	$0.588495\pi$
$74$	−2.33264e6	−0.669170
$75$	0	0
$76$	−2.61295e6	−0.682785
$77$	448173.	0.111874
$78$	0	0
$79$	6.31863e6	1.44188	0.720938	−	0.693000i	$-0.243711\pi$
$79$	6.31863e6	1.44188	0.720938	+	0.693000i	$0.243711\pi$
$80$	663674.	0.144924
$81$	0	0
$82$	1.46728e6	0.293876
$83$	7.56653e6	1.45252	0.726262	−	0.687418i	$-0.241256\pi$
$83$	7.56653e6	1.45252	0.726262	+	0.687418i	$0.241256\pi$
$84$	0	0
$85$	4.77826e6	0.843925
$86$	2.04316e6	0.346384
$87$	0	0
$88$	668993.	0.104648
$89$	5.98398e6	0.899757	0.449879	−	0.893090i	$-0.351467\pi$
$89$	5.98398e6	0.899757	0.449879	+	0.893090i	$0.351467\pi$
$90$	0	0
$91$	−3.31085e6	−0.460569
$92$	−1.73324e6	−0.232061
$93$	0	0
$94$	−7.84520e6	−0.974219
$95$	−6.61526e6	−0.791615
$96$	0	0
$97$	3.97074e6	0.441744	0.220872	−	0.975303i	$-0.429110\pi$
$97$	3.97074e6	0.441744	0.220872	+	0.975303i	$0.429110\pi$
$98$	−941192.	−0.101015
$99$	0	0
$100$	−3.31977e6	−0.331977
$101$	−1.11442e6	−0.107627	−0.0538137	−	0.998551i	$-0.517138\pi$
$101$	−1.11442e6	−0.107627	−0.0538137	+	0.998551i	$0.517138\pi$
$102$	0	0
$103$	−1.40649e6	−0.126826	−0.0634129	−	0.997987i	$-0.520199\pi$
$103$	−1.40649e6	−0.126826	−0.0634129	+	0.997987i	$0.520199\pi$
$104$	−4.94214e6	−0.430823
$105$	0	0
$106$	−1.55965e7	−1.27191
$107$	−9.78235e6	−0.771969	−0.385985	−	0.922505i	$-0.626138\pi$
$107$	−9.78235e6	−0.771969	−0.385985	+	0.922505i	$0.626138\pi$
$108$	0	0
$109$	1.24723e7	0.922472	0.461236	−	0.887278i	$-0.347406\pi$
$109$	1.24723e7	0.922472	0.461236	+	0.887278i	$0.347406\pi$
$110$	1.69370e6	0.121328
$111$	0	0
$112$	−1.40493e6	−0.0944911
$113$	1.75625e7	1.14502	0.572510	−	0.819898i	$-0.305970\pi$
$113$	1.75625e7	1.14502	0.572510	+	0.819898i	$0.305970\pi$
$114$	0	0
$115$	−4.38808e6	−0.269049
$116$	1.28935e7	0.766949
$117$	0	0
$118$	−2.50294e6	−0.140237
$119$	−1.01151e7	−0.550244
$120$	0	0
$121$	−1.77799e7	−0.912390
$122$	−1.15467e7	−0.575704
$123$	0	0
$124$	−1.25002e7	−0.588762
$125$	−2.10633e7	−0.964586
$126$	0	0
$127$	1.09866e7	0.475940	0.237970	−	0.971273i	$-0.423518\pi$
$127$	1.09866e7	0.475940	0.237970	+	0.971273i	$0.423518\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	−1.25121e7	−0.499492
$131$	−2.50735e7	−0.974462	−0.487231	−	0.873273i	$-0.661993\pi$
$131$	−2.50735e7	−0.974462	−0.487231	+	0.873273i	$0.661993\pi$
$132$	0	0
$133$	1.40038e7	0.516137
$134$	−1.79455e7	−0.644301
$135$	0	0
$136$	−1.50989e7	−0.514706
$137$	−2.29312e7	−0.761913	−0.380957	−	0.924593i	$-0.624405\pi$
$137$	−2.29312e7	−0.761913	−0.380957	+	0.924593i	$0.624405\pi$
$138$	0	0
$139$	2.68924e6	0.0849333	0.0424666	−	0.999098i	$-0.486478\pi$
$139$	2.68924e6	0.0849333	0.0424666	+	0.999098i	$0.486478\pi$
$140$	−3.55688e6	−0.109552
$141$	0	0
$142$	−9.06790e6	−0.265765
$143$	−1.26124e7	−0.360679
$144$	0	0
$145$	3.26426e7	0.889194
$146$	1.45952e7	0.388128
$147$	0	0
$148$	1.86611e7	0.473175
$149$	−6.11013e7	−1.51321	−0.756603	−	0.653874i	$-0.773143\pi$
$149$	−6.11013e7	−1.51321	−0.756603	+	0.653874i	$0.773143\pi$
$150$	0	0
$151$	7.26479e7	1.71713	0.858567	−	0.512702i	$-0.171355\pi$
$151$	7.26479e7	1.71713	0.858567	+	0.512702i	$0.171355\pi$
$152$	2.09036e7	0.482802
$153$	0	0
$154$	−3.58538e6	−0.0791067
$155$	−3.16469e7	−0.682605
$156$	0	0
$157$	6.77448e6	0.139710	0.0698549	−	0.997557i	$-0.477746\pi$
$157$	6.77448e6	0.139710	0.0698549	+	0.997557i	$0.477746\pi$
$158$	−5.05490e7	−1.01956
$159$	0	0
$160$	−5.30939e6	−0.102477
$161$	9.28909e6	0.175422
$162$	0	0
$163$	3.54014e6	0.0640271	0.0320136	−	0.999487i	$-0.489808\pi$
$163$	3.54014e6	0.0640271	0.0320136	+	0.999487i	$0.489808\pi$
$164$	−1.17382e7	−0.207802
$165$	0	0
$166$	−6.05322e7	−1.02709
$167$	1.72103e7	0.285944	0.142972	−	0.989727i	$-0.454334\pi$
$167$	1.72103e7	0.285944	0.142972	+	0.989727i	$0.454334\pi$
$168$	0	0
$169$	3.04247e7	0.484867
$170$	−3.82261e7	−0.596745
$171$	0	0
$172$	−1.63453e7	−0.244930
$173$	2.13356e7	0.313288	0.156644	−	0.987655i	$-0.449933\pi$
$173$	2.13356e7	0.313288	0.156644	+	0.987655i	$0.449933\pi$
$174$	0	0
$175$	1.77919e7	0.250951
$176$	−5.35194e6	−0.0739975
$177$	0	0
$178$	−4.78719e7	−0.636225
$179$	4.31704e7	0.562602	0.281301	−	0.959620i	$-0.409234\pi$
$179$	4.31704e7	0.562602	0.281301	+	0.959620i	$0.409234\pi$
$180$	0	0
$181$	1.39688e8	1.75099	0.875496	−	0.483226i	$-0.160535\pi$
$181$	1.39688e8	1.75099	0.875496	+	0.483226i	$0.160535\pi$
$182$	2.64868e7	0.325671
$183$	0	0
$184$	1.38659e7	0.164092
$185$	4.72446e7	0.548595
$186$	0	0
$187$	−3.85325e7	−0.430905
$188$	6.27616e7	0.688877
$189$	0	0
$190$	5.29221e7	0.559757
$191$	9.77447e7	1.01502	0.507512	−	0.861645i	$-0.330565\pi$
$191$	9.77447e7	1.01502	0.507512	+	0.861645i	$0.330565\pi$
$192$	0	0
$193$	−1.54728e8	−1.54924	−0.774619	−	0.632428i	$-0.782059\pi$
$193$	−1.54728e8	−1.54924	−0.774619	+	0.632428i	$0.782059\pi$
$194$	−3.17659e7	−0.312360
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	−4.12582e7	−0.384484	−0.192242	−	0.981348i	$-0.561576\pi$
$197$	−4.12582e7	−0.384484	−0.192242	+	0.981348i	$0.561576\pi$
$198$	0	0
$199$	−1.38189e8	−1.24305	−0.621524	−	0.783395i	$-0.713486\pi$
$199$	−1.38189e8	−1.24305	−0.621524	+	0.783395i	$0.713486\pi$
$200$	2.65581e7	0.234743
$201$	0	0
$202$	8.91534e6	0.0761041
$203$	−6.91009e7	−0.579759
$204$	0	0
$205$	−2.97179e7	−0.240924
$206$	1.12520e7	0.0896794
$207$	0	0
$208$	3.95372e7	0.304638
$209$	5.33462e7	0.404195
$210$	0	0
$211$	1.82690e8	1.33884	0.669418	−	0.742886i	$-0.266544\pi$
$211$	1.82690e8	1.33884	0.669418	+	0.742886i	$0.266544\pi$
$212$	1.24772e8	0.899374
$213$	0	0
$214$	7.82588e7	0.545865
$215$	−4.13816e7	−0.283970
$216$	0	0
$217$	6.69931e7	0.445062
$218$	−9.97783e7	−0.652286
$219$	0	0
$220$	−1.35496e7	−0.0857921
$221$	2.84656e8	1.77398
$222$	0	0
$223$	−1.72714e8	−1.04295	−0.521473	−	0.853268i	$-0.674617\pi$
$223$	−1.72714e8	−1.04295	−0.521473	+	0.853268i	$0.674617\pi$
$224$	1.12394e7	0.0668153
$225$	0	0
$226$	−1.40500e8	−0.809651
$227$	8.47439e7	0.480859	0.240430	−	0.970667i	$-0.422712\pi$
$227$	8.47439e7	0.480859	0.240430	+	0.970667i	$0.422712\pi$
$228$	0	0
$229$	−1.21817e8	−0.670324	−0.335162	−	0.942161i	$-0.608791\pi$
$229$	−1.21817e8	−0.670324	−0.335162	+	0.942161i	$0.608791\pi$
$230$	3.51046e7	0.190247
$231$	0	0
$232$	−1.03148e8	−0.542315
$233$	−1.48834e8	−0.770827	−0.385414	−	0.922744i	$-0.625941\pi$
$233$	−1.48834e8	−0.770827	−0.385414	+	0.922744i	$0.625941\pi$
$234$	0	0
$235$	1.58895e8	0.798677
$236$	2.00235e7	0.0991626
$237$	0	0
$238$	8.09206e7	0.389081
$239$	2.98943e8	1.41643	0.708216	−	0.705996i	$-0.249500\pi$
$239$	2.98943e8	1.41643	0.708216	+	0.705996i	$0.249500\pi$
$240$	0	0
$241$	−2.69575e8	−1.24057	−0.620283	−	0.784378i	$-0.712982\pi$
$241$	−2.69575e8	−1.24057	−0.620283	+	0.784378i	$0.712982\pi$
$242$	1.42239e8	0.645157
$243$	0	0
$244$	9.23737e7	0.407084
$245$	1.90626e7	0.0828137
$246$	0	0
$247$	−3.94092e8	−1.66402
$248$	1.00001e8	0.416318
$249$	0	0
$250$	1.68506e8	0.682065
$251$	3.22653e8	1.28789	0.643944	−	0.765072i	$-0.277297\pi$
$251$	3.22653e8	1.28789	0.643944	+	0.765072i	$0.277297\pi$
$252$	0	0
$253$	3.53859e7	0.137375
$254$	−8.78931e7	−0.336540
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−4.00216e8	−1.47071	−0.735357	−	0.677679i	$-0.762986\pi$
$257$	−4.00216e8	−1.47071	−0.735357	+	0.677679i	$0.762986\pi$
$258$	0	0
$259$	−1.00012e8	−0.357686
$260$	1.00097e8	0.353194
$261$	0	0
$262$	2.00588e8	0.689048
$263$	−1.13867e8	−0.385969	−0.192985	−	0.981202i	$-0.561817\pi$
$263$	−1.13867e8	−0.385969	−0.192985	+	0.981202i	$0.561817\pi$
$264$	0	0
$265$	3.15886e8	1.04273
$266$	−1.12030e8	−0.364964
$267$	0	0
$268$	1.43564e8	0.455590
$269$	4.53156e8	1.41943	0.709717	−	0.704487i	$-0.248823\pi$
$269$	4.53156e8	1.41943	0.709717	+	0.704487i	$0.248823\pi$
$270$	0	0
$271$	−3.00573e8	−0.917397	−0.458699	−	0.888592i	$-0.651684\pi$
$271$	−3.00573e8	−0.917397	−0.458699	+	0.888592i	$0.651684\pi$
$272$	1.20791e8	0.363952
$273$	0	0
$274$	1.83450e8	0.538754
$275$	6.77765e7	0.196524
$276$	0	0
$277$	2.22674e8	0.629491	0.314746	−	0.949176i	$-0.398081\pi$
$277$	2.22674e8	0.629491	0.314746	+	0.949176i	$0.398081\pi$
$278$	−2.15139e7	−0.0600569
$279$	0	0
$280$	2.84550e7	0.0774651
$281$	−5.40173e8	−1.45232	−0.726158	−	0.687528i	$-0.758696\pi$
$281$	−5.40173e8	−1.45232	−0.726158	+	0.687528i	$0.758696\pi$
$282$	0	0
$283$	−3.96552e8	−1.04003	−0.520017	−	0.854156i	$-0.674074\pi$
$283$	−3.96552e8	−1.04003	−0.520017	+	0.854156i	$0.674074\pi$
$284$	7.25432e7	0.187924
$285$	0	0
$286$	1.00899e8	0.255039
$287$	6.29096e7	0.157083
$288$	0	0
$289$	4.59323e8	1.11938
$290$	−2.61141e8	−0.628755
$291$	0	0
$292$	−1.16762e8	−0.274448
$293$	4.61303e8	1.07140	0.535698	−	0.844410i	$-0.320049\pi$
$293$	4.61303e8	1.07140	0.535698	+	0.844410i	$0.320049\pi$
$294$	0	0
$295$	5.06938e7	0.114968
$296$	−1.49289e8	−0.334585
$297$	0	0
$298$	4.88810e8	1.07000
$299$	−2.61412e8	−0.565556
$300$	0	0
$301$	8.76005e7	0.185150
$302$	−5.81184e8	−1.21420
$303$	0	0
$304$	−1.67229e8	−0.341393
$305$	2.33864e8	0.471970
$306$	0	0
$307$	−1.59543e8	−0.314698	−0.157349	−	0.987543i	$-0.550295\pi$
$307$	−1.59543e8	−0.314698	−0.157349	+	0.987543i	$0.550295\pi$
$308$	2.86831e7	0.0559369
$309$	0	0
$310$	2.53175e8	0.482675
$311$	−6.05166e8	−1.14081	−0.570405	−	0.821363i	$-0.693214\pi$
$311$	−6.05166e8	−1.14081	−0.570405	+	0.821363i	$0.693214\pi$
$312$	0	0
$313$	4.32355e8	0.796958	0.398479	−	0.917177i	$-0.369538\pi$
$313$	4.32355e8	0.796958	0.398479	+	0.917177i	$0.369538\pi$
$314$	−5.41958e7	−0.0987898
$315$	0	0
$316$	4.04392e8	0.720938
$317$	−9.05398e8	−1.59636	−0.798182	−	0.602416i	$-0.794205\pi$
$317$	−9.05398e8	−1.59636	−0.798182	+	0.602416i	$0.794205\pi$
$318$	0	0
$319$	−2.63233e8	−0.454019
$320$	4.24752e7	0.0724620
$321$	0	0
$322$	−7.43128e7	−0.124042
$323$	−1.20400e9	−1.98801
$324$	0	0
$325$	−5.00695e8	−0.809061
$326$	−2.83211e7	−0.0452740
$327$	0	0
$328$	9.39059e7	0.146938
$329$	−3.36363e8	−0.520742
$330$	0	0
$331$	−7.73135e8	−1.17181	−0.585905	−	0.810379i	$-0.699261\pi$
$331$	−7.73135e8	−1.17181	−0.585905	+	0.810379i	$0.699261\pi$
$332$	4.84258e8	0.726262
$333$	0	0
$334$	−1.37682e8	−0.202193
$335$	3.63464e8	0.528207
$336$	0	0
$337$	−7.30546e8	−1.03978	−0.519892	−	0.854232i	$-0.674028\pi$
$337$	−7.30546e8	−1.03978	−0.519892	+	0.854232i	$0.674028\pi$
$338$	−2.43397e8	−0.342853
$339$	0	0
$340$	3.05809e8	0.421963
$341$	2.55204e8	0.348535
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	1.30762e8	0.173192
$345$	0	0
$346$	−1.70685e8	−0.221528
$347$	5.80259e8	0.745537	0.372768	−	0.927924i	$-0.378409\pi$
$347$	5.80259e8	0.745537	0.372768	+	0.927924i	$0.378409\pi$
$348$	0	0
$349$	−3.69718e8	−0.465566	−0.232783	−	0.972529i	$-0.574783\pi$
$349$	−3.69718e8	−0.465566	−0.232783	+	0.972529i	$0.574783\pi$
$350$	−1.42335e8	−0.177449
$351$	0	0
$352$	4.28155e7	0.0523241
$353$	−2.92874e8	−0.354380	−0.177190	−	0.984177i	$-0.556701\pi$
$353$	−2.92874e8	−0.354380	−0.177190	+	0.984177i	$0.556701\pi$
$354$	0	0
$355$	1.83659e8	0.217878
$356$	3.82975e8	0.449879
$357$	0	0
$358$	−3.45364e8	−0.397819
$359$	−1.51190e9	−1.72462	−0.862309	−	0.506383i	$-0.830982\pi$
$359$	−1.51190e9	−1.72462	−0.862309	+	0.506383i	$0.830982\pi$
$360$	0	0
$361$	7.73005e8	0.864783
$362$	−1.11750e9	−1.23814
$363$	0	0
$364$	−2.11894e8	−0.230285
$365$	−2.95607e8	−0.318193
$366$	0	0
$367$	6.30746e8	0.666075	0.333037	−	0.942914i	$-0.391926\pi$
$367$	6.30746e8	0.666075	0.333037	+	0.942914i	$0.391926\pi$
$368$	−1.10927e8	−0.116030
$369$	0	0
$370$	−3.77957e8	−0.387915
$371$	−6.68698e8	−0.679863
$372$	0	0
$373$	−7.84566e8	−0.782796	−0.391398	−	0.920222i	$-0.628008\pi$
$373$	−7.84566e8	−0.782796	−0.391398	+	0.920222i	$0.628008\pi$
$374$	3.08260e8	0.304696
$375$	0	0
$376$	−5.02093e8	−0.487109
$377$	1.94462e9	1.86913
$378$	0	0
$379$	2.87718e8	0.271475	0.135737	−	0.990745i	$-0.456660\pi$
$379$	2.87718e8	0.271475	0.135737	+	0.990745i	$0.456660\pi$
$380$	−4.23376e8	−0.395808
$381$	0	0
$382$	−7.81958e8	−0.717731
$383$	−8.22441e7	−0.0748013	−0.0374006	−	0.999300i	$-0.511908\pi$
$383$	−8.22441e7	−0.0748013	−0.0374006	+	0.999300i	$0.511908\pi$
$384$	0	0
$385$	7.26174e7	0.0648527
$386$	1.23782e9	1.09548
$387$	0	0
$388$	2.54127e8	0.220872
$389$	−4.81980e8	−0.415150	−0.207575	−	0.978219i	$-0.566557\pi$
$389$	−4.81980e8	−0.415150	−0.207575	+	0.978219i	$0.566557\pi$
$390$	0	0
$391$	−7.98646e8	−0.675672
$392$	−6.02363e7	−0.0505076
$393$	0	0
$394$	3.30065e8	0.271871
$395$	1.02381e9	0.835849
$396$	0	0
$397$	1.82447e9	1.46342	0.731712	−	0.681614i	$-0.238721\pi$
$397$	1.82447e9	1.46342	0.731712	+	0.681614i	$0.238721\pi$
$398$	1.10551e9	0.878967
$399$	0	0
$400$	−2.12465e8	−0.165988
$401$	7.61935e8	0.590082	0.295041	−	0.955485i	$-0.404667\pi$
$401$	7.61935e8	0.590082	0.295041	+	0.955485i	$0.404667\pi$
$402$	0	0
$403$	−1.88530e9	−1.43487
$404$	−7.13227e7	−0.0538137
$405$	0	0
$406$	5.52807e8	0.409951
$407$	−3.80986e8	−0.280110
$408$	0	0
$409$	1.20258e9	0.869127	0.434563	−	0.900641i	$-0.356903\pi$
$409$	1.20258e9	0.869127	0.434563	+	0.900641i	$0.356903\pi$
$410$	2.37743e8	0.170359
$411$	0	0
$412$	−9.00156e7	−0.0634129
$413$	−1.07313e8	−0.0749599
$414$	0	0
$415$	1.22600e9	0.842022
$416$	−3.16297e8	−0.215411
$417$	0	0
$418$	−4.26769e8	−0.285809
$419$	2.80914e9	1.86563	0.932814	−	0.360359i	$-0.117346\pi$
$419$	2.80914e9	1.86563	0.932814	+	0.360359i	$0.117346\pi$
$420$	0	0
$421$	1.68081e9	1.09782	0.548910	−	0.835881i	$-0.315043\pi$
$421$	1.68081e9	1.09782	0.548910	+	0.835881i	$0.315043\pi$
$422$	−1.46152e9	−0.946699
$423$	0	0
$424$	−9.98173e8	−0.635954
$425$	−1.52969e9	−0.966588
$426$	0	0
$427$	−4.95066e8	−0.307727
$428$	−6.26071e8	−0.385985
$429$	0	0
$430$	3.31053e8	0.200797
$431$	3.06658e9	1.84495	0.922474	−	0.386060i	$-0.126164\pi$
$431$	3.06658e9	1.84495	0.922474	+	0.386060i	$0.126164\pi$
$432$	0	0
$433$	1.72660e9	1.02208	0.511039	−	0.859557i	$-0.329261\pi$
$433$	1.72660e9	1.02208	0.511039	+	0.859557i	$0.329261\pi$
$434$	−5.35944e8	−0.314707
$435$	0	0
$436$	7.98226e8	0.461236
$437$	1.10568e9	0.633791
$438$	0	0
$439$	−2.00732e9	−1.13238	−0.566189	−	0.824275i	$-0.691583\pi$
$439$	−2.00732e9	−1.13238	−0.566189	+	0.824275i	$0.691583\pi$
$440$	1.08397e8	0.0606642
$441$	0	0
$442$	−2.27725e9	−1.25439
$443$	−2.34022e9	−1.27892	−0.639461	−	0.768824i	$-0.720842\pi$
$443$	−2.34022e9	−1.27892	−0.639461	+	0.768824i	$0.720842\pi$
$444$	0	0
$445$	9.69584e8	0.521585
$446$	1.38172e9	0.737474
$447$	0	0
$448$	−8.99154e7	−0.0472456
$449$	−7.62271e8	−0.397418	−0.198709	−	0.980059i	$-0.563675\pi$
$449$	−7.62271e8	−0.397418	−0.198709	+	0.980059i	$0.563675\pi$
$450$	0	0
$451$	2.39648e8	0.123015
$452$	1.12400e9	0.572510
$453$	0	0
$454$	−6.77951e8	−0.340019
$455$	−5.36457e8	−0.266990
$456$	0	0
$457$	3.43888e9	1.68543	0.842715	−	0.538360i	$-0.180956\pi$
$457$	3.43888e9	1.68543	0.842715	+	0.538360i	$0.180956\pi$
$458$	9.74538e8	0.473990
$459$	0	0
$460$	−2.80837e8	−0.134525
$461$	3.74754e9	1.78153	0.890766	−	0.454462i	$-0.150169\pi$
$461$	3.74754e9	1.78153	0.890766	+	0.454462i	$0.150169\pi$
$462$	0	0
$463$	−7.54718e7	−0.0353388	−0.0176694	−	0.999844i	$-0.505625\pi$
$463$	−7.54718e7	−0.0353388	−0.0176694	+	0.999844i	$0.505625\pi$
$464$	8.25181e8	0.383474
$465$	0	0
$466$	1.19067e9	0.545057
$467$	3.19925e9	1.45358	0.726791	−	0.686859i	$-0.241011\pi$
$467$	3.19925e9	1.45358	0.726791	+	0.686859i	$0.241011\pi$
$468$	0	0
$469$	−7.69414e8	−0.344394
$470$	−1.27116e9	−0.564750
$471$	0	0
$472$	−1.60188e8	−0.0701185
$473$	3.33706e8	0.144994
$474$	0	0
$475$	2.11777e9	0.906675
$476$	−6.47365e8	−0.275122
$477$	0	0
$478$	−2.39154e9	−1.00157
$479$	1.32434e9	0.550586	0.275293	−	0.961360i	$-0.411225\pi$
$479$	1.32434e9	0.550586	0.275293	+	0.961360i	$0.411225\pi$
$480$	0	0
$481$	2.81451e9	1.15318
$482$	2.15660e9	0.877212
$483$	0	0
$484$	−1.13791e9	−0.456195
$485$	6.43378e8	0.256077
$486$	0	0
$487$	−4.26092e9	−1.67168	−0.835838	−	0.548976i	$-0.815018\pi$
$487$	−4.26092e9	−1.67168	−0.835838	+	0.548976i	$0.815018\pi$
$488$	−7.38990e8	−0.287852
$489$	0	0
$490$	−1.52501e8	−0.0585581
$491$	−4.33474e9	−1.65264	−0.826319	−	0.563202i	$-0.809569\pi$
$491$	−4.33474e9	−1.65264	−0.826319	+	0.563202i	$0.809569\pi$
$492$	0	0
$493$	5.94107e9	2.23306
$494$	3.15273e9	1.17664
$495$	0	0
$496$	−8.00010e8	−0.294381
$497$	−3.88786e8	−0.142057
$498$	0	0
$499$	−2.15621e9	−0.776852	−0.388426	−	0.921480i	$-0.626981\pi$
$499$	−2.15621e9	−0.776852	−0.388426	+	0.921480i	$0.626981\pi$
$500$	−1.34805e9	−0.482293
$501$	0	0
$502$	−2.58123e9	−0.910675
$503$	5.08504e9	1.78158	0.890792	−	0.454412i	$-0.150151\pi$
$503$	5.08504e9	1.78158	0.890792	+	0.454412i	$0.150151\pi$
$504$	0	0
$505$	−1.80569e8	−0.0623912
$506$	−2.83088e8	−0.0971391
$507$	0	0
$508$	7.03145e8	0.237970
$509$	−2.54536e9	−0.855532	−0.427766	−	0.903889i	$-0.640699\pi$
$509$	−2.54536e9	−0.855532	−0.427766	+	0.903889i	$0.640699\pi$
$510$	0	0
$511$	6.25770e8	0.207463
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	3.20173e9	1.03995
$515$	−2.27894e8	−0.0735204
$516$	0	0
$517$	−1.28134e9	−0.407801
$518$	8.00095e8	0.252923
$519$	0	0
$520$	−8.00775e8	−0.249746
$521$	−2.93447e8	−0.0909070	−0.0454535	−	0.998966i	$-0.514473\pi$
$521$	−2.93447e8	−0.0909070	−0.0454535	+	0.998966i	$0.514473\pi$
$522$	0	0
$523$	2.18899e9	0.669096	0.334548	−	0.942379i	$-0.391416\pi$
$523$	2.18899e9	0.669096	0.334548	+	0.942379i	$0.391416\pi$
$524$	−1.60470e9	−0.487231
$525$	0	0
$526$	9.10936e8	0.272921
$527$	−5.75985e9	−1.71425
$528$	0	0
$529$	−2.67140e9	−0.784591
$530$	−2.52709e9	−0.737319
$531$	0	0
$532$	8.96243e8	0.258069
$533$	−1.77039e9	−0.506435
$534$	0	0
$535$	−1.58503e9	−0.447507
$536$	−1.14851e9	−0.322151
$537$	0	0
$538$	−3.62525e9	−1.00369
$539$	−1.53723e8	−0.0422843
$540$	0	0
$541$	−2.76115e8	−0.0749720	−0.0374860	−	0.999297i	$-0.511935\pi$
$541$	−2.76115e8	−0.0749720	−0.0374860	+	0.999297i	$0.511935\pi$
$542$	2.40458e9	0.648698
$543$	0	0
$544$	−9.66329e8	−0.257353
$545$	2.02088e9	0.534753
$546$	0	0
$547$	−7.24010e9	−1.89142	−0.945712	−	0.325005i	$-0.894634\pi$
$547$	−7.24010e9	−1.89142	−0.945712	+	0.325005i	$0.894634\pi$
$548$	−1.46760e9	−0.380957
$549$	0	0
$550$	−5.42212e8	−0.138963
$551$	−8.22510e9	−2.09465
$552$	0	0
$553$	−2.16729e9	−0.544978
$554$	−1.78139e9	−0.445118
$555$	0	0
$556$	1.72111e8	0.0424666
$557$	2.64234e9	0.647881	0.323941	−	0.946077i	$-0.394992\pi$
$557$	2.64234e9	0.647881	0.323941	+	0.946077i	$0.394992\pi$
$558$	0	0
$559$	−2.46523e9	−0.596920
$560$	−2.27640e8	−0.0547761
$561$	0	0
$562$	4.32139e9	1.02694
$563$	3.12718e9	0.738539	0.369269	−	0.929322i	$-0.379608\pi$
$563$	3.12718e9	0.738539	0.369269	+	0.929322i	$0.379608\pi$
$564$	0	0
$565$	2.84566e9	0.663763
$566$	3.17241e9	0.735415
$567$	0	0
$568$	−5.80346e8	−0.132882
$569$	−7.94358e9	−1.80769	−0.903844	−	0.427863i	$-0.859267\pi$
$569$	−7.94358e9	−1.80769	−0.903844	+	0.427863i	$0.859267\pi$
$570$	0	0
$571$	1.29883e9	0.291961	0.145981	−	0.989287i	$-0.453366\pi$
$571$	1.29883e9	0.291961	0.145981	+	0.989287i	$0.453366\pi$
$572$	−8.07192e8	−0.180340
$573$	0	0
$574$	−5.03277e8	−0.111075
$575$	1.40477e9	0.308155
$576$	0	0
$577$	−3.63635e9	−0.788044	−0.394022	−	0.919101i	$-0.628917\pi$
$577$	−3.63635e9	−0.788044	−0.394022	+	0.919101i	$0.628917\pi$
$578$	−3.67459e9	−0.791518
$579$	0	0
$580$	2.08913e9	0.444597
$581$	−2.59532e9	−0.549003
$582$	0	0
$583$	−2.54734e9	−0.532412
$584$	9.34093e8	0.194064
$585$	0	0
$586$	−3.69043e9	−0.757591
$587$	−6.61703e9	−1.35030	−0.675149	−	0.737681i	$-0.735921\pi$
$587$	−6.61703e9	−1.35030	−0.675149	+	0.737681i	$0.735921\pi$
$588$	0	0
$589$	7.97421e9	1.60799
$590$	−4.05551e8	−0.0812948
$591$	0	0
$592$	1.19431e9	0.236587
$593$	−6.54428e9	−1.28876	−0.644378	−	0.764707i	$-0.722884\pi$
$593$	−6.54428e9	−1.28876	−0.644378	+	0.764707i	$0.722884\pi$
$594$	0	0
$595$	−1.63894e9	−0.318974
$596$	−3.91048e9	−0.756603
$597$	0	0
$598$	2.09129e9	0.399908
$599$	8.93561e9	1.69875	0.849376	−	0.527788i	$-0.176978\pi$
$599$	8.93561e9	1.69875	0.849376	+	0.527788i	$0.176978\pi$
$600$	0	0
$601$	4.09443e9	0.769365	0.384682	−	0.923049i	$-0.374311\pi$
$601$	4.09443e9	0.769365	0.384682	+	0.923049i	$0.374311\pi$
$602$	−7.00804e8	−0.130921
$603$	0	0
$604$	4.64947e9	0.858567
$605$	−2.88087e9	−0.528908
$606$	0	0
$607$	5.73788e9	1.04134	0.520668	−	0.853759i	$-0.325683\pi$
$607$	5.73788e9	1.04134	0.520668	+	0.853759i	$0.325683\pi$
$608$	1.33783e9	0.241401
$609$	0	0
$610$	−1.87091e9	−0.333733
$611$	9.46585e9	1.67886
$612$	0	0
$613$	1.13990e9	0.199874	0.0999370	−	0.994994i	$-0.468136\pi$
$613$	1.13990e9	0.199874	0.0999370	+	0.994994i	$0.468136\pi$
$614$	1.27635e9	0.222525
$615$	0	0
$616$	−2.29465e8	−0.0395533
$617$	−3.12068e9	−0.534874	−0.267437	−	0.963575i	$-0.586177\pi$
$617$	−3.12068e9	−0.534874	−0.267437	+	0.963575i	$0.586177\pi$
$618$	0	0
$619$	−3.90534e9	−0.661823	−0.330912	−	0.943662i	$-0.607356\pi$
$619$	−3.90534e9	−0.661823	−0.330912	+	0.943662i	$0.607356\pi$
$620$	−2.02540e9	−0.341303
$621$	0	0
$622$	4.84133e9	0.806675
$623$	−2.05251e9	−0.340076
$624$	0	0
$625$	6.39567e8	0.104787
$626$	−3.45884e9	−0.563535
$627$	0	0
$628$	4.33567e8	0.0698549
$629$	8.59870e9	1.37770
$630$	0	0
$631$	−5.31681e9	−0.842458	−0.421229	−	0.906954i	$-0.638401\pi$
$631$	−5.31681e9	−0.842458	−0.421229	+	0.906954i	$0.638401\pi$
$632$	−3.23514e9	−0.509780
$633$	0	0
$634$	7.24318e9	1.12880
$635$	1.78016e9	0.275900
$636$	0	0
$637$	1.13562e9	0.174079
$638$	2.10587e9	0.321040
$639$	0	0
$640$	−3.39801e8	−0.0512383
$641$	−8.68550e8	−0.130254	−0.0651271	−	0.997877i	$-0.520745\pi$
$641$	−8.68550e8	−0.130254	−0.0651271	+	0.997877i	$0.520745\pi$
$642$	0	0
$643$	−1.04976e10	−1.55723	−0.778614	−	0.627503i	$-0.784077\pi$
$643$	−1.04976e10	−1.55723	−0.778614	+	0.627503i	$0.784077\pi$
$644$	5.94502e8	0.0877108
$645$	0	0
$646$	9.63201e9	1.40573
$647$	−5.79034e9	−0.840503	−0.420251	−	0.907408i	$-0.638058\pi$
$647$	−5.79034e9	−0.840503	−0.420251	+	0.907408i	$0.638058\pi$
$648$	0	0
$649$	−4.08801e8	−0.0587023
$650$	4.00556e9	0.572092
$651$	0	0
$652$	2.26569e8	0.0320136
$653$	−4.33544e9	−0.609308	−0.304654	−	0.952463i	$-0.598541\pi$
$653$	−4.33544e9	−0.609308	−0.304654	+	0.952463i	$0.598541\pi$
$654$	0	0
$655$	−4.06265e9	−0.564891
$656$	−7.51247e8	−0.103901
$657$	0	0
$658$	2.69090e9	0.368220
$659$	−9.78432e9	−1.33178	−0.665889	−	0.746051i	$-0.731947\pi$
$659$	−9.78432e9	−1.33178	−0.665889	+	0.746051i	$0.731947\pi$
$660$	0	0
$661$	5.12312e9	0.689968	0.344984	−	0.938608i	$-0.387884\pi$
$661$	5.12312e9	0.689968	0.344984	+	0.938608i	$0.387884\pi$
$662$	6.18508e9	0.828595
$663$	0	0
$664$	−3.87406e9	−0.513545
$665$	2.26903e9	0.299202
$666$	0	0
$667$	−5.45593e9	−0.711915
$668$	1.10146e9	0.142972
$669$	0	0
$670$	−2.90771e9	−0.373499
$671$	−1.88591e9	−0.240986
$672$	0	0
$673$	1.33592e10	1.68938	0.844692	−	0.535253i	$-0.179784\pi$
$673$	1.33592e10	1.68938	0.844692	+	0.535253i	$0.179784\pi$
$674$	5.84437e9	0.735238
$675$	0	0
$676$	1.94718e9	0.242433
$677$	4.62265e9	0.572573	0.286286	−	0.958144i	$-0.407579\pi$
$677$	4.62265e9	0.572573	0.286286	+	0.958144i	$0.407579\pi$
$678$	0	0
$679$	−1.36196e9	−0.166963
$680$	−2.44647e9	−0.298373
$681$	0	0
$682$	−2.04163e9	−0.246452
$683$	−2.42230e9	−0.290908	−0.145454	−	0.989365i	$-0.546464\pi$
$683$	−2.42230e9	−0.290908	−0.145454	+	0.989365i	$0.546464\pi$
$684$	0	0
$685$	−3.71554e9	−0.441678
$686$	3.22829e8	0.0381802
$687$	0	0
$688$	−1.04610e9	−0.122465
$689$	1.88183e10	2.19187
$690$	0	0
$691$	4.95179e9	0.570938	0.285469	−	0.958388i	$-0.407851\pi$
$691$	4.95179e9	0.570938	0.285469	+	0.958388i	$0.407851\pi$
$692$	1.36548e9	0.156644
$693$	0	0
$694$	−4.64207e9	−0.527174
$695$	4.35737e8	0.0492354
$696$	0	0
$697$	−5.40877e9	−0.605039
$698$	2.95774e9	0.329205
$699$	0	0
$700$	1.13868e9	0.125475
$701$	1.15124e10	1.26227	0.631137	−	0.775672i	$-0.282589\pi$
$701$	1.15124e10	1.26227	0.631137	+	0.775672i	$0.282589\pi$
$702$	0	0
$703$	−1.19044e10	−1.29231
$704$	−3.42524e8	−0.0369988
$705$	0	0
$706$	2.34299e9	0.250584
$707$	3.82245e8	0.0406794
$708$	0	0
$709$	−1.79274e10	−1.88910	−0.944549	−	0.328372i	$-0.893500\pi$
$709$	−1.79274e10	−1.88910	−0.944549	+	0.328372i	$0.893500\pi$
$710$	−1.46927e9	−0.154063
$711$	0	0
$712$	−3.06380e9	−0.318112
$713$	5.28950e9	0.546514
$714$	0	0
$715$	−2.04358e9	−0.209084
$716$	2.76291e9	0.281301
$717$	0	0
$718$	1.20952e10	1.21949
$719$	3.93062e9	0.394376	0.197188	−	0.980366i	$-0.436819\pi$
$719$	3.93062e9	0.394376	0.197188	+	0.980366i	$0.436819\pi$
$720$	0	0
$721$	4.82428e8	0.0479357
$722$	−6.18404e9	−0.611494
$723$	0	0
$724$	8.94003e9	0.875496
$725$	−1.04500e10	−1.01844
$726$	0	0
$727$	1.05669e10	1.01994	0.509971	−	0.860191i	$-0.329656\pi$
$727$	1.05669e10	1.01994	0.509971	+	0.860191i	$0.329656\pi$
$728$	1.69516e9	0.162836
$729$	0	0
$730$	2.36486e9	0.224996
$731$	−7.53160e9	−0.713143
$732$	0	0
$733$	−1.71976e9	−0.161288	−0.0806442	−	0.996743i	$-0.525698\pi$
$733$	−1.71976e9	−0.161288	−0.0806442	+	0.996743i	$0.525698\pi$
$734$	−5.04597e9	−0.470986
$735$	0	0
$736$	8.87420e8	0.0820459
$737$	−2.93101e9	−0.269700
$738$	0	0
$739$	1.93328e9	0.176213	0.0881066	−	0.996111i	$-0.471918\pi$
$739$	1.93328e9	0.176213	0.0881066	+	0.996111i	$0.471918\pi$
$740$	3.02366e9	0.274297
$741$	0	0
$742$	5.34958e9	0.480736
$743$	2.07767e10	1.85830	0.929150	−	0.369702i	$-0.120540\pi$
$743$	2.07767e10	1.85830	0.929150	+	0.369702i	$0.120540\pi$
$744$	0	0
$745$	−9.90023e9	−0.877199
$746$	6.27652e9	0.553520
$747$	0	0
$748$	−2.46608e9	−0.215452
$749$	3.35535e9	0.291777
$750$	0	0
$751$	−6.08859e9	−0.524538	−0.262269	−	0.964995i	$-0.584471\pi$
$751$	−6.08859e9	−0.524538	−0.262269	+	0.964995i	$0.584471\pi$
$752$	4.01674e9	0.344438
$753$	0	0
$754$	−1.55570e10	−1.32168
$755$	1.17711e10	0.995415
$756$	0	0
$757$	−8.77776e9	−0.735442	−0.367721	−	0.929936i	$-0.619862\pi$
$757$	−8.77776e9	−0.735442	−0.367721	+	0.929936i	$0.619862\pi$
$758$	−2.30174e9	−0.191962
$759$	0	0
$760$	3.38701e9	0.279878
$761$	1.03563e10	0.851843	0.425922	−	0.904760i	$-0.359950\pi$
$761$	1.03563e10	0.851843	0.425922	+	0.904760i	$0.359950\pi$
$762$	0	0
$763$	−4.27799e9	−0.348662
$764$	6.25566e9	0.507512
$765$	0	0
$766$	6.57953e8	0.0528925
$767$	3.01999e9	0.241669
$768$	0	0
$769$	1.65740e10	1.31427	0.657137	−	0.753771i	$-0.271767\pi$
$769$	1.65740e10	1.31427	0.657137	+	0.753771i	$0.271767\pi$
$770$	−5.80939e8	−0.0458578
$771$	0	0
$772$	−9.90259e9	−0.774619
$773$	5.15312e9	0.401275	0.200637	−	0.979666i	$-0.435699\pi$
$773$	5.15312e9	0.401275	0.200637	+	0.979666i	$0.435699\pi$
$774$	0	0
$775$	1.01313e10	0.781821
$776$	−2.03302e9	−0.156180
$777$	0	0
$778$	3.85584e9	0.293555
$779$	7.48815e9	0.567536
$780$	0	0
$781$	−1.48105e9	−0.111247
$782$	6.38917e9	0.477772
$783$	0	0
$784$	4.81890e8	0.0357143
$785$	1.09767e9	0.0809892
$786$	0	0
$787$	−2.59807e10	−1.89994	−0.949969	−	0.312344i	$-0.898886\pi$
$787$	−2.59807e10	−1.89994	−0.949969	+	0.312344i	$0.898886\pi$
$788$	−2.64052e9	−0.192242
$789$	0	0
$790$	−8.19045e9	−0.591035
$791$	−6.02395e9	−0.432777
$792$	0	0
$793$	1.39320e10	0.992106
$794$	−1.45958e10	−1.03480
$795$	0	0
$796$	−8.84410e9	−0.621524
$797$	−2.32372e10	−1.62585	−0.812924	−	0.582369i	$-0.802126\pi$
$797$	−2.32372e10	−1.62585	−0.812924	+	0.582369i	$0.802126\pi$
$798$	0	0
$799$	2.89194e10	2.00574
$800$	1.69972e9	0.117371
$801$	0	0
$802$	−6.09548e9	−0.417251
$803$	2.38381e9	0.162468
$804$	0	0
$805$	1.50511e9	0.101691
$806$	1.50824e10	1.01461
$807$	0	0
$808$	5.70582e8	0.0380521
$809$	3.64347e9	0.241933	0.120967	−	0.992657i	$-0.461401\pi$
$809$	3.64347e9	0.241933	0.120967	+	0.992657i	$0.461401\pi$
$810$	0	0
$811$	1.79425e10	1.18117	0.590583	−	0.806977i	$-0.298898\pi$
$811$	1.79425e10	1.18117	0.590583	+	0.806977i	$0.298898\pi$
$812$	−4.42246e9	−0.289879
$813$	0	0
$814$	3.04789e9	0.198068
$815$	5.73609e8	0.0371162
$816$	0	0
$817$	1.04271e10	0.668939
$818$	−9.62066e9	−0.614565
$819$	0	0
$820$	−1.90195e9	−0.120462
$821$	3.74203e9	0.235997	0.117998	−	0.993014i	$-0.462352\pi$
$821$	3.74203e9	0.235997	0.117998	+	0.993014i	$0.462352\pi$
$822$	0	0
$823$	−2.72013e10	−1.70095	−0.850473	−	0.526019i	$-0.823684\pi$
$823$	−2.72013e10	−1.70095	−0.850473	+	0.526019i	$0.823684\pi$
$824$	7.20125e8	0.0448397
$825$	0	0
$826$	8.58507e8	0.0530046
$827$	−4.47569e9	−0.275163	−0.137582	−	0.990490i	$-0.543933\pi$
$827$	−4.47569e9	−0.275163	−0.137582	+	0.990490i	$0.543933\pi$
$828$	0	0
$829$	−6.87198e9	−0.418930	−0.209465	−	0.977816i	$-0.567172\pi$
$829$	−6.87198e9	−0.418930	−0.209465	+	0.977816i	$0.567172\pi$
$830$	−9.80803e9	−0.595399
$831$	0	0
$832$	2.53038e9	0.152319
$833$	3.46947e9	0.207973
$834$	0	0
$835$	2.78858e9	0.165760
$836$	3.41416e9	0.202098
$837$	0	0
$838$	−2.24731e10	−1.31920
$839$	−3.28739e10	−1.92170	−0.960848	−	0.277077i	$-0.910634\pi$
$839$	−3.28739e10	−1.92170	−0.960848	+	0.277077i	$0.910634\pi$
$840$	0	0
$841$	2.33364e10	1.35284
$842$	−1.34465e10	−0.776276
$843$	0	0
$844$	1.16922e10	0.669418
$845$	4.92971e9	0.281075
$846$	0	0
$847$	6.09850e9	0.344851
$848$	7.98539e9	0.449687
$849$	0	0
$850$	1.22375e10	0.683481
$851$	−7.89654e9	−0.439221
$852$	0	0
$853$	9.30525e9	0.513342	0.256671	−	0.966499i	$-0.417374\pi$
$853$	9.30525e9	0.513342	0.256671	+	0.966499i	$0.417374\pi$
$854$	3.96052e9	0.217596
$855$	0	0
$856$	5.00856e9	0.272932
$857$	−2.05780e10	−1.11679	−0.558394	−	0.829576i	$-0.688582\pi$
$857$	−2.05780e10	−1.11679	−0.558394	+	0.829576i	$0.688582\pi$
$858$	0	0
$859$	−4.50298e9	−0.242395	−0.121197	−	0.992628i	$-0.538673\pi$
$859$	−4.50298e9	−0.242395	−0.121197	+	0.992628i	$0.538673\pi$
$860$	−2.64842e9	−0.141985
$861$	0	0
$862$	−2.45326e10	−1.30457
$863$	7.07888e9	0.374910	0.187455	−	0.982273i	$-0.439976\pi$
$863$	7.07888e9	0.374910	0.187455	+	0.982273i	$0.439976\pi$
$864$	0	0
$865$	3.45700e9	0.181611
$866$	−1.38128e10	−0.722719
$867$	0	0
$868$	4.28756e9	0.222531
$869$	−8.25609e9	−0.426781
$870$	0	0
$871$	2.16527e10	1.11032
$872$	−6.38581e9	−0.326143
$873$	0	0
$874$	−8.84547e9	−0.448158
$875$	7.22471e9	0.364579
$876$	0	0
$877$	−3.13680e10	−1.57032	−0.785160	−	0.619292i	$-0.787420\pi$
$877$	−3.13680e10	−1.57032	−0.785160	+	0.619292i	$0.787420\pi$
$878$	1.60586e10	0.800712
$879$	0	0
$880$	−8.67175e8	−0.0428960
$881$	−2.70407e10	−1.33230	−0.666150	−	0.745818i	$-0.732059\pi$
$881$	−2.70407e10	−1.33230	−0.666150	+	0.745818i	$0.732059\pi$
$882$	0	0
$883$	2.13360e10	1.04292	0.521460	−	0.853275i	$-0.325387\pi$
$883$	2.13360e10	1.04292	0.521460	+	0.853275i	$0.325387\pi$
$884$	1.82180e10	0.886988
$885$	0	0
$886$	1.87218e10	0.904334
$887$	−4.80221e9	−0.231051	−0.115526	−	0.993304i	$-0.536855\pi$
$887$	−4.80221e9	−0.231051	−0.115526	+	0.993304i	$0.536855\pi$
$888$	0	0
$889$	−3.76842e9	−0.179888
$890$	−7.75667e9	−0.368817
$891$	0	0
$892$	−1.10537e10	−0.521473
$893$	−4.00374e10	−1.88142
$894$	0	0
$895$	6.99490e9	0.326138
$896$	7.19323e8	0.0334077
$897$	0	0
$898$	6.09817e9	0.281017
$899$	−3.93482e10	−1.80620
$900$	0	0
$901$	5.74925e10	2.61863
$902$	−1.91719e9	−0.0869845
$903$	0	0
$904$	−8.99202e9	−0.404825
$905$	2.26336e10	1.01504
$906$	0	0
$907$	3.43972e10	1.53072	0.765362	−	0.643600i	$-0.222560\pi$
$907$	3.43972e10	1.53072	0.765362	+	0.643600i	$0.222560\pi$
$908$	5.42361e9	0.240430
$909$	0	0
$910$	4.29165e9	0.188790
$911$	−1.64680e9	−0.0721652	−0.0360826	−	0.999349i	$-0.511488\pi$
$911$	−1.64680e9	−0.0721652	−0.0360826	+	0.999349i	$0.511488\pi$
$912$	0	0
$913$	−9.88663e9	−0.429933
$914$	−2.75111e10	−1.19178
$915$	0	0
$916$	−7.79630e9	−0.335162
$917$	8.60020e9	0.368312
$918$	0	0
$919$	−2.93595e10	−1.24780	−0.623899	−	0.781505i	$-0.714453\pi$
$919$	−2.93595e10	−1.24780	−0.623899	+	0.781505i	$0.714453\pi$
$920$	2.24670e9	0.0951233
$921$	0	0
$922$	−2.99804e10	−1.25973
$923$	1.09411e10	0.457990
$924$	0	0
$925$	−1.51246e10	−0.628332
$926$	6.03774e8	0.0249883
$927$	0	0
$928$	−6.60145e9	−0.271157
$929$	1.83459e10	0.750731	0.375365	−	0.926877i	$-0.377517\pi$
$929$	1.83459e10	0.750731	0.375365	+	0.926877i	$0.377517\pi$
$930$	0	0
$931$	−4.80330e9	−0.195082
$932$	−9.52539e9	−0.385414
$933$	0	0
$934$	−2.55940e10	−1.02784
$935$	−6.24341e9	−0.249794
$936$	0	0
$937$	−2.16246e10	−0.858736	−0.429368	−	0.903130i	$-0.641264\pi$
$937$	−2.16246e10	−0.858736	−0.429368	+	0.903130i	$0.641264\pi$
$938$	6.15531e9	0.243523
$939$	0	0
$940$	1.01693e10	0.399339
$941$	−2.48532e10	−0.972341	−0.486170	−	0.873864i	$-0.661607\pi$
$941$	−2.48532e10	−0.972341	−0.486170	+	0.873864i	$0.661607\pi$
$942$	0	0
$943$	4.96709e9	0.192891
$944$	1.28150e9	0.0495813
$945$	0	0
$946$	−2.66965e9	−0.102526
$947$	1.75432e10	0.671248	0.335624	−	0.941996i	$-0.391053\pi$
$947$	1.75432e10	0.671248	0.335624	+	0.941996i	$0.391053\pi$
$948$	0	0
$949$	−1.76103e10	−0.668858
$950$	−1.69422e10	−0.641116
$951$	0	0
$952$	5.17892e9	0.194540
$953$	3.02964e10	1.13388	0.566938	−	0.823760i	$-0.308128\pi$
$953$	3.02964e10	1.13388	0.566938	+	0.823760i	$0.308128\pi$
$954$	0	0
$955$	1.58376e10	0.588405
$956$	1.91323e10	0.708216
$957$	0	0
$958$	−1.05947e10	−0.389323
$959$	7.86541e9	0.287976
$960$	0	0
$961$	1.06354e10	0.386563
$962$	−2.25161e10	−0.815418
$963$	0	0
$964$	−1.72528e10	−0.620283
$965$	−2.50706e10	−0.898087
$966$	0	0
$967$	2.38289e10	0.847443	0.423722	−	0.905792i	$-0.360724\pi$
$967$	2.38289e10	0.847443	0.423722	+	0.905792i	$0.360724\pi$
$968$	9.10331e9	0.322579
$969$	0	0
$970$	−5.14702e9	−0.181074
$971$	2.38158e10	0.834831	0.417416	−	0.908716i	$-0.362936\pi$
$971$	2.38158e10	0.834831	0.417416	+	0.908716i	$0.362936\pi$
$972$	0	0
$973$	−9.22410e8	−0.0321018
$974$	3.40874e10	1.18205
$975$	0	0
$976$	5.91192e9	0.203542
$977$	−2.20743e10	−0.757279	−0.378640	−	0.925544i	$-0.623608\pi$
$977$	−2.20743e10	−0.757279	−0.378640	+	0.925544i	$0.623608\pi$
$978$	0	0
$979$	−7.81883e9	−0.266319
$980$	1.22001e9	0.0414068
$981$	0	0
$982$	3.46779e10	1.16859
$983$	4.47350e10	1.50214	0.751070	−	0.660223i	$-0.229538\pi$
$983$	4.47350e10	1.50214	0.751070	+	0.660223i	$0.229538\pi$
$984$	0	0
$985$	−6.68505e9	−0.222884
$986$	−4.75286e10	−1.57901
$987$	0	0
$988$	−2.52219e10	−0.832009
$989$	6.91658e9	0.227355
$990$	0	0
$991$	−2.13803e10	−0.697842	−0.348921	−	0.937152i	$-0.613452\pi$
$991$	−2.13803e10	−0.697842	−0.348921	+	0.937152i	$0.613452\pi$
$992$	6.40008e9	0.208159
$993$	0	0
$994$	3.11029e9	0.100450
$995$	−2.23907e10	−0.720589
$996$	0	0
$997$	−4.58361e10	−1.46479	−0.732394	−	0.680881i	$-0.761597\pi$
$997$	−4.58361e10	−1.46479	−0.732394	+	0.680881i	$0.761597\pi$
$998$	1.72497e10	0.549318
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.8.a.j.1.2	✓	2
3.2	odd	2		126.8.a.m.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.8.a.j.1.2	✓	2	1.1	even	1	trivial
126.8.a.m.1.1	yes	2	3.2	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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +64.0000 q^{4} +162.030 q^{5} -343.000 q^{7} -512.000 q^{8} -1296.24 q^{10} -1306.63 q^{11} +9652.63 q^{13} +2744.00 q^{14} +4096.00 q^{16} +29490.0 q^{17} -40827.4 q^{19} +10369.9 q^{20} +10453.0 q^{22} -27081.9 q^{23} -51871.3 q^{25} -77221.0 q^{26} -21952.0 q^{28} +201460. q^{29} -195315. q^{31} -32768.0 q^{32} -235920. q^{34} -55576.2 q^{35} +291580. q^{37} +326619. q^{38} -82959.3 q^{40} -183410. q^{41} -255395. q^{43} -83624.1 q^{44} +216655. q^{46} +980650. q^{47} +117649. q^{49} +414971. q^{50} +617768. q^{52} +1.94956e6 q^{53} -211713. q^{55} +175616. q^{56} -1.61168e6 q^{58} +312867. q^{59} +1.44334e6 q^{61} +1.56252e6 q^{62} +262144. q^{64} +1.56401e6 q^{65} +2.24319e6 q^{67} +1.88736e6 q^{68} +444610. q^{70} +1.13349e6 q^{71} -1.82440e6 q^{73} -2.33264e6 q^{74} -2.61295e6 q^{76} +448173. q^{77} +6.31863e6 q^{79} +663674. q^{80} +1.46728e6 q^{82} +7.56653e6 q^{83} +4.77826e6 q^{85} +2.04316e6 q^{86} +668993. q^{88} +5.98398e6 q^{89} -3.31085e6 q^{91} -1.73324e6 q^{92} -7.84520e6 q^{94} -6.61526e6 q^{95} +3.97074e6 q^{97} -941192. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} + 128 q^{4} + 56 q^{5} - 686 q^{7} - 1024 q^{8} - 448 q^{10} + 3016 q^{11} - 4284 q^{13} + 5488 q^{14} + 8192 q^{16} + 22792 q^{17} - 15176 q^{19} + 3584 q^{20} - 24128 q^{22} + 52792 q^{23}+ \cdots - 1882384 q^{98}+O(q^{100}) \) 2 * q - 16 * q^2 + 128 * q^4 + 56 * q^5 - 686 * q^7 - 1024 * q^8 - 448 * q^10 + 3016 * q^11 - 4284 * q^13 + 5488 * q^14 + 8192 * q^16 + 22792 * q^17 - 15176 * q^19 + 3584 * q^20 - 24128 * q^22 + 52792 * q^23 - 118754 * q^25 + 34272 * q^26 - 43904 * q^28 + 5120 * q^29 - 255528 * q^31 - 65536 * q^32 - 182336 * q^34 - 19208 * q^35 - 152396 * q^37 + 121408 * q^38 - 28672 * q^40 + 603288 * q^41 + 269800 * q^43 + 193024 * q^44 - 422336 * q^46 + 1693776 * q^47 + 235298 * q^49 + 950032 * q^50 - 274176 * q^52 + 2094000 * q^53 - 670040 * q^55 + 351232 * q^56 - 40960 * q^58 + 2357936 * q^59 + 2978892 * q^61 + 2044224 * q^62 + 524288 * q^64 + 3041712 * q^65 + 2895176 * q^67 + 1458688 * q^68 + 153664 * q^70 + 6490792 * q^71 + 3097724 * q^73 + 1219168 * q^74 - 971264 * q^76 - 1034488 * q^77 + 5641968 * q^79 + 229376 * q^80 - 4826304 * q^82 + 5489344 * q^83 + 5488456 * q^85 - 2158400 * q^86 - 1544192 * q^88 + 14285880 * q^89 + 1469412 * q^91 + 3378688 * q^92 - 13550208 * q^94 - 9335072 * q^95 + 9198140 * q^97 - 1882384 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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