LMFDB - Embedded newform 531.8.a.d.1.9

Newspace parameters

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	531.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$165.876448532$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
Defining polynomial:	$x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{10}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.39686$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

$f(q)$	$=$	$q-0.396855 q^{2} -127.843 q^{4} -247.233 q^{5} -652.929 q^{7} +101.532 q^{8} +O(q^{10})$	\(q-0.396855 q^{2} -127.843 q^{4} -247.233 q^{5} -652.929 q^{7} +101.532 q^{8} +98.1157 q^{10} +2477.11 q^{11} +9417.61 q^{13} +259.118 q^{14} +16323.5 q^{16} -12577.2 q^{17} -41390.0 q^{19} +31606.9 q^{20} -983.055 q^{22} +59781.4 q^{23} -17000.9 q^{25} -3737.43 q^{26} +83472.1 q^{28} -212559. q^{29} -7793.74 q^{31} -19474.2 q^{32} +4991.34 q^{34} +161426. q^{35} -447343. q^{37} +16425.8 q^{38} -25102.2 q^{40} -14016.1 q^{41} -896369. q^{43} -316680. q^{44} -23724.6 q^{46} -21326.5 q^{47} -397227. q^{49} +6746.89 q^{50} -1.20397e6 q^{52} -910896. q^{53} -612424. q^{55} -66293.5 q^{56} +84355.0 q^{58} +205379. q^{59} +810282. q^{61} +3092.99 q^{62} -2.08169e6 q^{64} -2.32834e6 q^{65} -3.83166e6 q^{67} +1.60791e6 q^{68} -64062.6 q^{70} +3.35674e6 q^{71} +461542. q^{73} +177530. q^{74} +5.29140e6 q^{76} -1.61738e6 q^{77} -1.37060e6 q^{79} -4.03572e6 q^{80} +5562.36 q^{82} +7.63422e6 q^{83} +3.10951e6 q^{85} +355729. q^{86} +251507. q^{88} +5.26057e6 q^{89} -6.14903e6 q^{91} -7.64260e6 q^{92} +8463.51 q^{94} +1.02330e7 q^{95} -9.23173e6 q^{97} +157641. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) $	$ 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.396855	−0.0350774	−0.0175387	−	0.999846i	$-0.505583\pi$
$2$	−0.396855	−0.0350774	−0.0175387	+	0.999846i	$0.505583\pi$
$3$	0	0
$3$	0	0
$4$	−127.843	−0.998770
$5$	−247.233	−0.884527	−0.442264	−	0.896885i	$-0.645824\pi$
$5$	−247.233	−0.884527	−0.442264	+	0.896885i	$0.645824\pi$
$6$	0	0
$7$	−652.929	−0.719487	−0.359743	−	0.933051i	$-0.617136\pi$
$7$	−652.929	−0.719487	−0.359743	+	0.933051i	$0.617136\pi$
$8$	101.532	0.0701116
$9$	0	0
$10$	98.1157	0.0310269
$11$	2477.11	0.561140	0.280570	−	0.959834i	$-0.409476\pi$
$11$	2477.11	0.561140	0.280570	+	0.959834i	$0.409476\pi$
$12$	0	0
$13$	9417.61	1.18888	0.594442	−	0.804139i	$-0.297373\pi$
$13$	9417.61	1.18888	0.594442	+	0.804139i	$0.297373\pi$
$14$	259.118	0.0252377
$15$	0	0
$16$	16323.5	0.996310
$17$	−12577.2	−0.620889	−0.310444	−	0.950592i	$-0.600478\pi$
$17$	−12577.2	−0.620889	−0.310444	+	0.950592i	$0.600478\pi$
$18$	0	0
$19$	−41390.0	−1.38439	−0.692194	−	0.721711i	$-0.743356\pi$
$19$	−41390.0	−1.38439	−0.692194	+	0.721711i	$0.743356\pi$
$20$	31606.9	0.883439
$21$	0	0
$22$	−983.055	−0.0196833
$23$	59781.4	1.02452	0.512258	−	0.858832i	$-0.328809\pi$
$23$	59781.4	1.02452	0.512258	+	0.858832i	$0.328809\pi$
$24$	0	0
$25$	−17000.9	−0.217611
$26$	−3737.43	−0.0417029
$27$	0	0
$28$	83472.1	0.718602
$29$	−212559.	−1.61840	−0.809199	−	0.587534i	$-0.800099\pi$
$29$	−212559.	−1.61840	−0.809199	+	0.587534i	$0.800099\pi$
$30$	0	0
$31$	−7793.74	−0.0469872	−0.0234936	−	0.999724i	$-0.507479\pi$
$31$	−7793.74	−0.0469872	−0.0234936	+	0.999724i	$0.507479\pi$
$32$	−19474.2	−0.105060
$33$	0	0
$34$	4991.34	0.0217792
$35$	161426.	0.636406
$36$	0	0
$37$	−447343.	−1.45189	−0.725946	−	0.687751i	$-0.758598\pi$
$37$	−447343.	−1.45189	−0.725946	+	0.687751i	$0.758598\pi$
$38$	16425.8	0.0485607
$39$	0	0
$40$	−25102.2	−0.0620156
$41$	−14016.1	−0.0317602	−0.0158801	−	0.999874i	$-0.505055\pi$
$41$	−14016.1	−0.0317602	−0.0158801	+	0.999874i	$0.505055\pi$
$42$	0	0
$43$	−896369.	−1.71928	−0.859641	−	0.510899i	$-0.829313\pi$
$43$	−896369.	−1.71928	−0.859641	+	0.510899i	$0.829313\pi$
$44$	−316680.	−0.560450
$45$	0	0
$46$	−23724.6	−0.0359373
$47$	−21326.5	−0.0299624	−0.0149812	−	0.999888i	$-0.504769\pi$
$47$	−21326.5	−0.0299624	−0.0149812	+	0.999888i	$0.504769\pi$
$48$	0	0
$49$	−397227.	−0.482339
$50$	6746.89	0.00763324
$51$	0	0
$52$	−1.20397e6	−1.18742
$53$	−910896.	−0.840433	−0.420217	−	0.907424i	$-0.638046\pi$
$53$	−910896.	−0.840433	−0.420217	+	0.907424i	$0.638046\pi$
$54$	0	0
$55$	−612424.	−0.496344
$56$	−66293.5	−0.0504444
$57$	0	0
$58$	84355.0	0.0567692
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	0	0
$61$	810282.	0.457069	0.228535	−	0.973536i	$-0.426607\pi$
$61$	810282.	0.457069	0.228535	+	0.973536i	$0.426607\pi$
$62$	3092.99	0.00164819
$63$	0	0
$64$	−2.08169e6	−0.992625
$65$	−2.32834e6	−1.05160
$66$	0	0
$67$	−3.83166e6	−1.55641	−0.778207	−	0.628007i	$-0.783871\pi$
$67$	−3.83166e6	−1.55641	−0.778207	+	0.628007i	$0.783871\pi$
$68$	1.60791e6	0.620125
$69$	0	0
$70$	−64062.6	−0.0223234
$71$	3.35674e6	1.11305	0.556524	−	0.830832i	$-0.312135\pi$
$71$	3.35674e6	1.11305	0.556524	+	0.830832i	$0.312135\pi$
$72$	0	0
$73$	461542.	0.138861	0.0694306	−	0.997587i	$-0.477882\pi$
$73$	461542.	0.138861	0.0694306	+	0.997587i	$0.477882\pi$
$74$	177530.	0.0509286
$75$	0	0
$76$	5.29140e6	1.38269
$77$	−1.61738e6	−0.403733
$78$	0	0
$79$	−1.37060e6	−0.312762	−0.156381	−	0.987697i	$-0.549983\pi$
$79$	−1.37060e6	−0.312762	−0.156381	+	0.987697i	$0.549983\pi$
$80$	−4.03572e6	−0.881264
$81$	0	0
$82$	5562.36	0.00111407
$83$	7.63422e6	1.46552	0.732759	−	0.680488i	$-0.238232\pi$
$83$	7.63422e6	1.46552	0.732759	+	0.680488i	$0.238232\pi$
$84$	0	0
$85$	3.10951e6	0.549193
$86$	355729.	0.0603079
$87$	0	0
$88$	251507.	0.0393424
$89$	5.26057e6	0.790984	0.395492	−	0.918469i	$-0.370574\pi$
$89$	5.26057e6	0.790984	0.395492	+	0.918469i	$0.370574\pi$
$90$	0	0
$91$	−6.14903e6	−0.855386
$92$	−7.64260e6	−1.02326
$93$	0	0
$94$	8463.51	0.00105100
$95$	1.02330e7	1.22453
$96$	0	0
$97$	−9.23173e6	−1.02703	−0.513514	−	0.858081i	$-0.671657\pi$
$97$	−9.23173e6	−1.02703	−0.513514	+	0.858081i	$0.671657\pi$
$98$	157641.	0.0169192
$99$	0	0
$100$	2.17344e6	0.217344
$101$	−4.85554e6	−0.468935	−0.234468	−	0.972124i	$-0.575335\pi$
$101$	−4.85554e6	−0.468935	−0.234468	+	0.972124i	$0.575335\pi$
$102$	0	0
$103$	−4.34436e6	−0.391738	−0.195869	−	0.980630i	$-0.562753\pi$
$103$	−4.34436e6	−0.391738	−0.195869	+	0.980630i	$0.562753\pi$
$104$	956193.	0.0833545
$105$	0	0
$106$	361494.	0.0294802
$107$	6.60510e6	0.521238	0.260619	−	0.965442i	$-0.416073\pi$
$107$	6.60510e6	0.521238	0.260619	+	0.965442i	$0.416073\pi$
$108$	0	0
$109$	−2.74784e6	−0.203235	−0.101617	−	0.994824i	$-0.532402\pi$
$109$	−2.74784e6	−0.203235	−0.101617	+	0.994824i	$0.532402\pi$
$110$	243044.	0.0174104
$111$	0	0
$112$	−1.06581e7	−0.716832
$113$	3.08697e6	0.201260	0.100630	−	0.994924i	$-0.467914\pi$
$113$	3.08697e6	0.201260	0.100630	+	0.994924i	$0.467914\pi$
$114$	0	0
$115$	−1.47799e7	−0.906212
$116$	2.71740e7	1.61641
$117$	0	0
$118$	−81505.7	−0.00456669
$119$	8.21204e6	0.446721
$120$	0	0
$121$	−1.33511e7	−0.685122
$122$	−321565.	−0.0160328
$123$	0	0
$124$	996371.	0.0469294
$125$	2.35182e7	1.07701
$126$	0	0
$127$	1.02662e7	0.444731	0.222365	−	0.974963i	$-0.428622\pi$
$127$	1.02662e7	0.444731	0.222365	+	0.974963i	$0.428622\pi$
$128$	3.31883e6	0.139878
$129$	0	0
$130$	924016.	0.0368874
$131$	3.04995e7	1.18534	0.592671	−	0.805445i	$-0.298073\pi$
$131$	3.04995e7	1.18534	0.592671	+	0.805445i	$0.298073\pi$
$132$	0	0
$133$	2.70247e7	0.996049
$134$	1.52062e6	0.0545950
$135$	0	0
$136$	−1.27700e6	−0.0435315
$137$	−2.18105e7	−0.724676	−0.362338	−	0.932047i	$-0.618021\pi$
$137$	−2.18105e7	−0.724676	−0.362338	+	0.932047i	$0.618021\pi$
$138$	0	0
$139$	5.59222e7	1.76617	0.883085	−	0.469213i	$-0.155462\pi$
$139$	5.59222e7	1.76617	0.883085	+	0.469213i	$0.155462\pi$
$140$	−2.06370e7	−0.635623
$141$	0	0
$142$	−1.33214e6	−0.0390428
$143$	2.33285e7	0.667130
$144$	0	0
$145$	5.25515e7	1.43152
$146$	−183165.	−0.00487089
$147$	0	0
$148$	5.71894e7	1.45011
$149$	−4.25096e7	−1.05277	−0.526387	−	0.850245i	$-0.676454\pi$
$149$	−4.25096e7	−1.05277	−0.526387	+	0.850245i	$0.676454\pi$
$150$	0	0
$151$	−2.69136e7	−0.636140	−0.318070	−	0.948067i	$-0.603035\pi$
$151$	−2.69136e7	−0.636140	−0.318070	+	0.948067i	$0.603035\pi$
$152$	−4.20243e6	−0.0970617
$153$	0	0
$154$	641865.	0.0141619
$155$	1.92687e6	0.0415615
$156$	0	0
$157$	−6.09453e7	−1.25687	−0.628436	−	0.777861i	$-0.716305\pi$
$157$	−6.09453e7	−1.25687	−0.628436	+	0.777861i	$0.716305\pi$
$158$	543928.	0.0109709
$159$	0	0
$160$	4.81467e6	0.0929281
$161$	−3.90330e7	−0.737126
$162$	0	0
$163$	2.43192e7	0.439839	0.219919	−	0.975518i	$-0.429421\pi$
$163$	2.43192e7	0.439839	0.219919	+	0.975518i	$0.429421\pi$
$164$	1.79185e6	0.0317212
$165$	0	0
$166$	−3.02968e6	−0.0514066
$167$	−2.18099e7	−0.362366	−0.181183	−	0.983449i	$-0.557993\pi$
$167$	−2.18099e7	−0.362366	−0.181183	+	0.983449i	$0.557993\pi$
$168$	0	0
$169$	2.59429e7	0.413443
$170$	−1.23402e6	−0.0192643
$171$	0	0
$172$	1.14594e8	1.71717
$173$	−7.57905e7	−1.11289	−0.556446	−	0.830884i	$-0.687836\pi$
$173$	−7.57905e7	−1.11289	−0.556446	+	0.830884i	$0.687836\pi$
$174$	0	0
$175$	1.11004e7	0.156568
$176$	4.04353e7	0.559070
$177$	0	0
$178$	−2.08769e6	−0.0277457
$179$	1.14237e6	0.0148874	0.00744372	−	0.999972i	$-0.497631\pi$
$179$	1.14237e6	0.0148874	0.00744372	+	0.999972i	$0.497631\pi$
$180$	0	0
$181$	1.35346e8	1.69656	0.848280	−	0.529548i	$-0.177638\pi$
$181$	1.35346e8	1.69656	0.848280	+	0.529548i	$0.177638\pi$
$182$	2.44028e6	0.0300047
$183$	0	0
$184$	6.06975e6	0.0718304
$185$	1.10598e8	1.28424
$186$	0	0
$187$	−3.11552e7	−0.348406
$188$	2.72643e6	0.0299255
$189$	0	0
$190$	−4.06101e6	−0.0429533
$191$	−1.11443e8	−1.15727	−0.578637	−	0.815585i	$-0.696415\pi$
$191$	−1.11443e8	−1.15727	−0.578637	+	0.815585i	$0.696415\pi$
$192$	0	0
$193$	−4.58821e7	−0.459402	−0.229701	−	0.973261i	$-0.573775\pi$
$193$	−4.58821e7	−0.459402	−0.229701	+	0.973261i	$0.573775\pi$
$194$	3.66366e6	0.0360254
$195$	0	0
$196$	5.07824e7	0.481745
$197$	−1.22654e8	−1.14301	−0.571504	−	0.820600i	$-0.693640\pi$
$197$	−1.22654e8	−1.14301	−0.571504	+	0.820600i	$0.693640\pi$
$198$	0	0
$199$	−1.02506e7	−0.0922072	−0.0461036	−	0.998937i	$-0.514680\pi$
$199$	−1.02506e7	−0.0922072	−0.0461036	+	0.998937i	$0.514680\pi$
$200$	−1.72614e6	−0.0152571
$201$	0	0
$202$	1.92695e6	0.0164490
$203$	1.38786e8	1.16442
$204$	0	0
$205$	3.46524e6	0.0280928
$206$	1.72408e6	0.0137411
$207$	0	0
$208$	1.53729e8	1.18450
$209$	−1.02528e8	−0.776836
$210$	0	0
$211$	−1.63127e8	−1.19547	−0.597734	−	0.801694i	$-0.703932\pi$
$211$	−1.63127e8	−1.19547	−0.597734	+	0.801694i	$0.703932\pi$
$212$	1.16451e8	0.839399
$213$	0	0
$214$	−2.62127e6	−0.0182837
$215$	2.21612e8	1.52075
$216$	0	0
$217$	5.08876e6	0.0338067
$218$	1.09049e6	0.00712895
$219$	0	0
$220$	7.82938e7	0.495733
$221$	−1.18448e8	−0.738164
$222$	0	0
$223$	2.16056e8	1.30466	0.652332	−	0.757933i	$-0.273791\pi$
$223$	2.16056e8	1.30466	0.652332	+	0.757933i	$0.273791\pi$
$224$	1.27153e7	0.0755890
$225$	0	0
$226$	−1.22508e6	−0.00705968
$227$	9.46539e7	0.537091	0.268546	−	0.963267i	$-0.413457\pi$
$227$	9.46539e7	0.537091	0.268546	+	0.963267i	$0.413457\pi$
$228$	0	0
$229$	5.62090e7	0.309301	0.154651	−	0.987969i	$-0.450575\pi$
$229$	5.62090e7	0.309301	0.154651	+	0.987969i	$0.450575\pi$
$230$	5.86549e6	0.0317876
$231$	0	0
$232$	−2.15816e7	−0.113469
$233$	−3.12602e8	−1.61900	−0.809500	−	0.587120i	$-0.800261\pi$
$233$	−3.12602e8	−1.61900	−0.809500	+	0.587120i	$0.800261\pi$
$234$	0	0
$235$	5.27260e6	0.0265025
$236$	−2.62562e7	−0.130029
$237$	0	0
$238$	−3.25899e6	−0.0156698
$239$	−3.24955e8	−1.53968	−0.769840	−	0.638237i	$-0.779664\pi$
$239$	−3.24955e8	−1.53968	−0.769840	+	0.638237i	$0.779664\pi$
$240$	0	0
$241$	2.55841e8	1.17737	0.588683	−	0.808364i	$-0.299647\pi$
$241$	2.55841e8	1.17737	0.588683	+	0.808364i	$0.299647\pi$
$242$	5.29845e6	0.0240323
$243$	0	0
$244$	−1.03589e8	−0.456507
$245$	9.82075e7	0.426642
$246$	0	0
$247$	−3.89795e8	−1.64588
$248$	−791317.	−0.00329435
$249$	0	0
$250$	−9.33334e6	−0.0377787
$251$	−1.47199e8	−0.587552	−0.293776	−	0.955874i	$-0.594912\pi$
$251$	−1.47199e8	−0.587552	−0.293776	+	0.955874i	$0.594912\pi$
$252$	0	0
$253$	1.48085e8	0.574897
$254$	−4.07420e6	−0.0156000
$255$	0	0
$256$	2.65139e8	0.987718
$257$	3.26518e8	1.19989	0.599944	−	0.800042i	$-0.295190\pi$
$257$	3.26518e8	1.19989	0.599944	+	0.800042i	$0.295190\pi$
$258$	0	0
$259$	2.92083e8	1.04462
$260$	2.97661e8	1.05031
$261$	0	0
$262$	−1.21039e7	−0.0415787
$263$	1.12281e8	0.380592	0.190296	−	0.981727i	$-0.439055\pi$
$263$	1.12281e8	0.380592	0.190296	+	0.981727i	$0.439055\pi$
$264$	0	0
$265$	2.25203e8	0.743386
$266$	−1.07249e7	−0.0349388
$267$	0	0
$268$	4.89849e8	1.55450
$269$	5.12589e7	0.160560	0.0802798	−	0.996772i	$-0.474419\pi$
$269$	5.12589e7	0.160560	0.0802798	+	0.996772i	$0.474419\pi$
$270$	0	0
$271$	−5.28326e8	−1.61254	−0.806269	−	0.591549i	$-0.798516\pi$
$271$	−5.28326e8	−1.61254	−0.806269	+	0.591549i	$0.798516\pi$
$272$	−2.05305e8	−0.618598
$273$	0	0
$274$	8.65561e6	0.0254197
$275$	−4.21131e7	−0.122110
$276$	0	0
$277$	−4.48700e7	−0.126846	−0.0634230	−	0.997987i	$-0.520202\pi$
$277$	−4.48700e7	−0.126846	−0.0634230	+	0.997987i	$0.520202\pi$
$278$	−2.21930e7	−0.0619526
$279$	0	0
$280$	1.63899e7	0.0446194
$281$	5.27134e8	1.41726	0.708629	−	0.705581i	$-0.249314\pi$
$281$	5.27134e8	1.41726	0.708629	+	0.705581i	$0.249314\pi$
$282$	0	0
$283$	3.08524e8	0.809163	0.404582	−	0.914502i	$-0.367417\pi$
$283$	3.08524e8	0.809163	0.404582	+	0.914502i	$0.367417\pi$
$284$	−4.29134e8	−1.11168
$285$	0	0
$286$	−9.25804e6	−0.0234012
$287$	9.15152e6	0.0228511
$288$	0	0
$289$	−2.52152e8	−0.614497
$290$	−2.08553e7	−0.0502139
$291$	0	0
$292$	−5.90047e7	−0.138690
$293$	8.22037e8	1.90921	0.954607	−	0.297869i	$-0.0962760\pi$
$293$	8.22037e8	1.90921	0.954607	+	0.297869i	$0.0962760\pi$
$294$	0	0
$295$	−5.07764e7	−0.115156
$296$	−4.54198e7	−0.101795
$297$	0	0
$298$	1.68702e7	0.0369285
$299$	5.62998e8	1.21803
$300$	0	0
$301$	5.85265e8	1.23700
$302$	1.06808e7	0.0223141
$303$	0	0
$304$	−6.75632e8	−1.37928
$305$	−2.00328e8	−0.404290
$306$	0	0
$307$	1.21225e8	0.239116	0.119558	−	0.992827i	$-0.461852\pi$
$307$	1.21225e8	0.239116	0.119558	+	0.992827i	$0.461852\pi$
$308$	2.06770e8	0.403236
$309$	0	0
$310$	−764688.	−0.00145787
$311$	4.99117e8	0.940894	0.470447	−	0.882428i	$-0.344093\pi$
$311$	4.99117e8	0.940894	0.470447	+	0.882428i	$0.344093\pi$
$312$	0	0
$313$	6.71603e8	1.23796	0.618981	−	0.785406i	$-0.287546\pi$
$313$	6.71603e8	1.23796	0.618981	+	0.785406i	$0.287546\pi$
$314$	2.41865e7	0.0440878
$315$	0	0
$316$	1.75220e8	0.312377
$317$	7.55861e8	1.33271	0.666353	−	0.745636i	$-0.267854\pi$
$317$	7.55861e8	1.33271	0.666353	+	0.745636i	$0.267854\pi$
$318$	0	0
$319$	−5.26531e8	−0.908149
$320$	5.14661e8	0.878004
$321$	0	0
$322$	1.54905e7	0.0258564
$323$	5.20572e8	0.859552
$324$	0	0
$325$	−1.60108e8	−0.258714
$326$	−9.65122e6	−0.0154284
$327$	0	0
$328$	−1.42309e6	−0.00222676
$329$	1.39247e7	0.0215575
$330$	0	0
$331$	7.06940e7	0.107148	0.0535740	−	0.998564i	$-0.482939\pi$
$331$	7.06940e7	0.107148	0.0535740	+	0.998564i	$0.482939\pi$
$332$	−9.75978e8	−1.46372
$333$	0	0
$334$	8.65539e6	0.0127108
$335$	9.47313e8	1.37669
$336$	0	0
$337$	−6.96987e8	−0.992020	−0.496010	−	0.868317i	$-0.665202\pi$
$337$	−6.96987e8	−0.992020	−0.496010	+	0.868317i	$0.665202\pi$
$338$	−1.02956e7	−0.0145025
$339$	0	0
$340$	−3.97527e8	−0.548518
$341$	−1.93060e7	−0.0263664
$342$	0	0
$343$	7.97076e8	1.06652
$344$	−9.10105e7	−0.120542
$345$	0	0
$346$	3.00778e7	0.0390374
$347$	3.13443e8	0.402721	0.201361	−	0.979517i	$-0.435464\pi$
$347$	3.13443e8	0.402721	0.201361	+	0.979517i	$0.435464\pi$
$348$	0	0
$349$	6.69924e8	0.843599	0.421800	−	0.906689i	$-0.361399\pi$
$349$	6.69924e8	0.843599	0.421800	+	0.906689i	$0.361399\pi$
$350$	−4.40524e6	−0.00549201
$351$	0	0
$352$	−4.82399e7	−0.0589531
$353$	1.16673e9	1.41175	0.705875	−	0.708336i	$-0.250554\pi$
$353$	1.16673e9	1.41175	0.705875	+	0.708336i	$0.250554\pi$
$354$	0	0
$355$	−8.29897e8	−0.984521
$356$	−6.72525e8	−0.790011
$357$	0	0
$358$	−453354.	−0.000522212 0
$359$	−1.70302e9	−1.94263	−0.971316	−	0.237794i	$-0.923576\pi$
$359$	−1.70302e9	−1.94263	−0.971316	+	0.237794i	$0.923576\pi$
$360$	0	0
$361$	8.19262e8	0.916532
$362$	−5.37126e7	−0.0595109
$363$	0	0
$364$	7.86108e8	0.854333
$365$	−1.14108e8	−0.122827
$366$	0	0
$367$	9.54527e8	1.00799	0.503996	−	0.863706i	$-0.331863\pi$
$367$	9.54527e8	1.00799	0.503996	+	0.863706i	$0.331863\pi$
$368$	9.75845e8	1.02074
$369$	0	0
$370$	−4.38914e7	−0.0450477
$371$	5.94751e8	0.604681
$372$	0	0
$373$	1.42139e9	1.41819	0.709093	−	0.705115i	$-0.249105\pi$
$373$	1.42139e9	1.41819	0.709093	+	0.705115i	$0.249105\pi$
$374$	1.23641e7	0.0122212
$375$	0	0
$376$	−2.16533e6	−0.00210071
$377$	−2.00179e9	−1.92409
$378$	0	0
$379$	2.98909e8	0.282035	0.141017	−	0.990007i	$-0.454963\pi$
$379$	2.98909e8	0.282035	0.141017	+	0.990007i	$0.454963\pi$
$380$	−1.30821e9	−1.22302
$381$	0	0
$382$	4.42268e7	0.0405941
$383$	2.63974e8	0.240086	0.120043	−	0.992769i	$-0.461697\pi$
$383$	2.63974e8	0.240086	0.120043	+	0.992769i	$0.461697\pi$
$384$	0	0
$385$	3.99869e8	0.357113
$386$	1.82086e7	0.0161146
$387$	0	0
$388$	1.18021e9	1.02576
$389$	1.71619e9	1.47823	0.739116	−	0.673578i	$-0.235243\pi$
$389$	1.71619e9	1.47823	0.739116	+	0.673578i	$0.235243\pi$
$390$	0	0
$391$	−7.51885e8	−0.636111
$392$	−4.03314e7	−0.0338175
$393$	0	0
$394$	4.86758e7	0.0400937
$395$	3.38856e8	0.276647
$396$	0	0
$397$	5.74825e8	0.461072	0.230536	−	0.973064i	$-0.425952\pi$
$397$	5.74825e8	0.461072	0.230536	+	0.973064i	$0.425952\pi$
$398$	4.06802e6	0.00323439
$399$	0	0
$400$	−2.77515e8	−0.216808
$401$	1.14785e9	0.888953	0.444476	−	0.895791i	$-0.353390\pi$
$401$	1.14785e9	0.888953	0.444476	+	0.895791i	$0.353390\pi$
$402$	0	0
$403$	−7.33984e7	−0.0558623
$404$	6.20745e8	0.468358
$405$	0	0
$406$	−5.50778e7	−0.0408447
$407$	−1.10812e9	−0.814715
$408$	0	0
$409$	−4.45907e6	−0.00322265	−0.00161132	−	0.999999i	$-0.500513\pi$
$409$	−4.45907e6	−0.00322265	−0.00161132	+	0.999999i	$0.500513\pi$
$410$	−1.37520e6	−0.000985422 0
$411$	0	0
$412$	5.55394e8	0.391256
$413$	−1.34098e8	−0.0936692
$414$	0	0
$415$	−1.88743e9	−1.29629
$416$	−1.83401e8	−0.124904
$417$	0	0
$418$	4.06887e7	0.0272494
$419$	−2.72506e9	−1.80978	−0.904892	−	0.425642i	$-0.860048\pi$
$419$	−2.72506e9	−1.80978	−0.904892	+	0.425642i	$0.860048\pi$
$420$	0	0
$421$	−1.98552e9	−1.29684	−0.648420	−	0.761283i	$-0.724570\pi$
$421$	−1.98552e9	−1.29684	−0.648420	+	0.761283i	$0.724570\pi$
$422$	6.47379e7	0.0419339
$423$	0	0
$424$	−9.24855e7	−0.0589241
$425$	2.13824e8	0.135112
$426$	0	0
$427$	−5.29057e8	−0.328855
$428$	−8.44413e8	−0.520597
$429$	0	0
$430$	−8.79478e7	−0.0533440
$431$	8.94337e8	0.538060	0.269030	−	0.963132i	$-0.413297\pi$
$431$	8.94337e8	0.538060	0.269030	+	0.963132i	$0.413297\pi$
$432$	0	0
$433$	−2.81913e9	−1.66881	−0.834405	−	0.551152i	$-0.814188\pi$
$433$	−2.81913e9	−1.66881	−0.834405	+	0.551152i	$0.814188\pi$
$434$	−2.01950e6	−0.00118585
$435$	0	0
$436$	3.51290e8	0.202985
$437$	−2.47435e9	−1.41833
$438$	0	0
$439$	1.50706e9	0.850166	0.425083	−	0.905154i	$-0.360245\pi$
$439$	1.50706e9	0.850166	0.425083	+	0.905154i	$0.360245\pi$
$440$	−6.21809e7	−0.0347995
$441$	0	0
$442$	4.70065e7	0.0258929
$443$	−1.30737e9	−0.714471	−0.357235	−	0.934014i	$-0.616281\pi$
$443$	−1.30737e9	−0.714471	−0.357235	+	0.934014i	$0.616281\pi$
$444$	0	0
$445$	−1.30059e9	−0.699647
$446$	−8.57428e7	−0.0457642
$447$	0	0
$448$	1.35919e9	0.714181
$449$	2.59024e9	1.35045	0.675223	−	0.737613i	$-0.264047\pi$
$449$	2.59024e9	1.35045	0.675223	+	0.737613i	$0.264047\pi$
$450$	0	0
$451$	−3.47195e7	−0.0178219
$452$	−3.94646e8	−0.201012
$453$	0	0
$454$	−3.75639e7	−0.0188397
$455$	1.52024e9	0.756612
$456$	0	0
$457$	−3.00834e9	−1.47442	−0.737208	−	0.675666i	$-0.763856\pi$
$457$	−3.00834e9	−1.47442	−0.737208	+	0.675666i	$0.763856\pi$
$458$	−2.23068e7	−0.0108495
$459$	0	0
$460$	1.88950e9	0.905097
$461$	−1.29563e9	−0.615926	−0.307963	−	0.951398i	$-0.599647\pi$
$461$	−1.29563e9	−0.615926	−0.307963	+	0.951398i	$0.599647\pi$
$462$	0	0
$463$	7.89028e7	0.0369453	0.0184726	−	0.999829i	$-0.494120\pi$
$463$	7.89028e7	0.0369453	0.0184726	+	0.999829i	$0.494120\pi$
$464$	−3.46971e9	−1.61243
$465$	0	0
$466$	1.24058e8	0.0567903
$467$	1.97263e9	0.896264	0.448132	−	0.893967i	$-0.352089\pi$
$467$	1.97263e9	0.896264	0.448132	+	0.893967i	$0.352089\pi$
$468$	0	0
$469$	2.50180e9	1.11982
$470$	−2.09246e6	−0.000929639 0
$471$	0	0
$472$	2.08526e7	0.00912775
$473$	−2.22041e9	−0.964758
$474$	0	0
$475$	7.03667e8	0.301259
$476$	−1.04985e9	−0.446172
$477$	0	0
$478$	1.28960e8	0.0540080
$479$	−8.20556e7	−0.0341141	−0.0170571	−	0.999855i	$-0.505430\pi$
$479$	−8.20556e7	−0.0341141	−0.0170571	+	0.999855i	$0.505430\pi$
$480$	0	0
$481$	−4.21290e9	−1.72613
$482$	−1.01532e8	−0.0412989
$483$	0	0
$484$	1.70684e9	0.684279
$485$	2.28239e9	0.908434
$486$	0	0
$487$	2.81731e9	1.10531	0.552655	−	0.833410i	$-0.313615\pi$
$487$	2.81731e9	1.10531	0.552655	+	0.833410i	$0.313615\pi$
$488$	8.22699e7	0.0320458
$489$	0	0
$490$	−3.89742e7	−0.0149655
$491$	4.34251e9	1.65560	0.827800	−	0.561023i	$-0.189592\pi$
$491$	4.34251e9	1.65560	0.827800	+	0.561023i	$0.189592\pi$
$492$	0	0
$493$	2.67340e9	1.00485
$494$	1.54692e8	0.0577330
$495$	0	0
$496$	−1.27221e8	−0.0468138
$497$	−2.19171e9	−0.800823
$498$	0	0
$499$	−2.81881e9	−1.01558	−0.507789	−	0.861481i	$-0.669537\pi$
$499$	−2.81881e9	−1.01558	−0.507789	+	0.861481i	$0.669537\pi$
$500$	−3.00663e9	−1.07569
$501$	0	0
$502$	5.84167e7	0.0206098
$503$	−3.80145e9	−1.33187	−0.665933	−	0.746011i	$-0.731967\pi$
$503$	−3.80145e9	−1.33187	−0.665933	+	0.746011i	$0.731967\pi$
$504$	0	0
$505$	1.20045e9	0.414786
$506$	−5.87684e7	−0.0201659
$507$	0	0
$508$	−1.31246e9	−0.444184
$509$	−1.38927e9	−0.466956	−0.233478	−	0.972362i	$-0.575011\pi$
$509$	−1.38927e9	−0.466956	−0.233478	+	0.972362i	$0.575011\pi$
$510$	0	0
$511$	−3.01354e8	−0.0999089
$512$	−5.30032e8	−0.174525
$513$	0	0
$514$	−1.29580e8	−0.0420889
$515$	1.07407e9	0.346503
$516$	0	0
$517$	−5.28280e7	−0.0168131
$518$	−1.15915e8	−0.0366425
$519$	0	0
$520$	−2.36402e8	−0.0737293
$521$	2.67044e9	0.827277	0.413639	−	0.910441i	$-0.364258\pi$
$521$	2.67044e9	0.827277	0.413639	+	0.910441i	$0.364258\pi$
$522$	0	0
$523$	2.60473e8	0.0796173	0.0398086	−	0.999207i	$-0.487325\pi$
$523$	2.60473e8	0.0796173	0.0398086	+	0.999207i	$0.487325\pi$
$524$	−3.89914e9	−1.18388
$525$	0	0
$526$	−4.45592e7	−0.0133502
$527$	9.80236e7	0.0291738
$528$	0	0
$529$	1.68991e8	0.0496327
$530$	−8.93732e7	−0.0260760
$531$	0	0
$532$	−3.45491e9	−0.994824
$533$	−1.31998e8	−0.0377592
$534$	0	0
$535$	−1.63300e9	−0.461049
$536$	−3.89038e8	−0.109123
$537$	0	0
$538$	−2.03424e7	−0.00563201
$539$	−9.83975e8	−0.270660
$540$	0	0
$541$	−9.32970e7	−0.0253324	−0.0126662	−	0.999920i	$-0.504032\pi$
$541$	−9.32970e7	−0.0253324	−0.0126662	+	0.999920i	$0.504032\pi$
$542$	2.09669e8	0.0565636
$543$	0	0
$544$	2.44932e8	0.0652303
$545$	6.79356e8	0.179767
$546$	0	0
$547$	6.23723e9	1.62943	0.814716	−	0.579861i	$-0.196893\pi$
$547$	6.23723e9	1.62943	0.814716	+	0.579861i	$0.196893\pi$
$548$	2.78831e9	0.723784
$549$	0	0
$550$	1.67128e7	0.00428332
$551$	8.79780e9	2.24049
$552$	0	0
$553$	8.94902e8	0.225028
$554$	1.78069e7	0.00444943
$555$	0	0
$556$	−7.14923e9	−1.76400
$557$	−1.77646e9	−0.435574	−0.217787	−	0.975996i	$-0.569884\pi$
$557$	−1.77646e9	−0.435574	−0.217787	+	0.975996i	$0.569884\pi$
$558$	0	0
$559$	−8.44166e9	−2.04403
$560$	2.63504e9	0.634058
$561$	0	0
$562$	−2.09196e8	−0.0497137
$563$	−2.36725e9	−0.559067	−0.279534	−	0.960136i	$-0.590180\pi$
$563$	−2.36725e9	−0.559067	−0.279534	+	0.960136i	$0.590180\pi$
$564$	0	0
$565$	−7.63200e8	−0.178020
$566$	−1.22439e8	−0.0283833
$567$	0	0
$568$	3.40818e8	0.0780375
$569$	4.20570e9	0.957073	0.478536	−	0.878068i	$-0.341167\pi$
$569$	4.20570e9	0.957073	0.478536	+	0.878068i	$0.341167\pi$
$570$	0	0
$571$	5.59664e9	1.25806	0.629029	−	0.777382i	$-0.283453\pi$
$571$	5.59664e9	1.25806	0.629029	+	0.777382i	$0.283453\pi$
$572$	−2.98237e9	−0.666309
$573$	0	0
$574$	−3.63183e6	−0.000801556 0
$575$	−1.01634e9	−0.222946
$576$	0	0
$577$	−3.81070e9	−0.825828	−0.412914	−	0.910770i	$-0.635489\pi$
$577$	−3.81070e9	−0.825828	−0.412914	+	0.910770i	$0.635489\pi$
$578$	1.00068e8	0.0215549
$579$	0	0
$580$	−6.71831e9	−1.42976
$581$	−4.98460e9	−1.05442
$582$	0	0
$583$	−2.25639e9	−0.471601
$584$	4.68615e7	0.00973579
$585$	0	0
$586$	−3.26230e8	−0.0669702
$587$	−1.29560e9	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$587$	−1.29560e9	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$588$	0	0
$589$	3.22583e8	0.0650486
$590$	2.01509e7	0.00403936
$591$	0	0
$592$	−7.30222e9	−1.44654
$593$	−3.30125e9	−0.650111	−0.325055	−	0.945695i	$-0.605383\pi$
$593$	−3.30125e9	−0.650111	−0.325055	+	0.945695i	$0.605383\pi$
$594$	0	0
$595$	−2.03029e9	−0.395137
$596$	5.43453e9	1.05148
$597$	0	0
$598$	−2.23429e8	−0.0427253
$599$	−3.03218e9	−0.576448	−0.288224	−	0.957563i	$-0.593065\pi$
$599$	−3.03218e9	−0.576448	−0.288224	+	0.957563i	$0.593065\pi$
$600$	0	0
$601$	3.17671e9	0.596922	0.298461	−	0.954422i	$-0.403527\pi$
$601$	3.17671e9	0.596922	0.298461	+	0.954422i	$0.403527\pi$
$602$	−2.32266e8	−0.0433908
$603$	0	0
$604$	3.44071e9	0.635358
$605$	3.30083e9	0.606009
$606$	0	0
$607$	2.80510e9	0.509083	0.254541	−	0.967062i	$-0.418075\pi$
$607$	2.80510e9	0.509083	0.254541	+	0.967062i	$0.418075\pi$
$608$	8.06039e8	0.145443
$609$	0	0
$610$	7.95014e7	0.0141814
$611$	−2.00844e8	−0.0356218
$612$	0	0
$613$	−9.45694e9	−1.65821	−0.829104	−	0.559094i	$-0.811149\pi$
$613$	−9.45694e9	−1.65821	−0.829104	+	0.559094i	$0.811149\pi$
$614$	−4.81088e7	−0.00838755
$615$	0	0
$616$	−1.64216e8	−0.0283064
$617$	−5.14249e9	−0.881405	−0.440702	−	0.897653i	$-0.645271\pi$
$617$	−5.14249e9	−0.881405	−0.440702	+	0.897653i	$0.645271\pi$
$618$	0	0
$619$	7.42513e9	1.25831	0.629153	−	0.777281i	$-0.283402\pi$
$619$	7.42513e9	1.25831	0.629153	+	0.777281i	$0.283402\pi$
$620$	−2.46336e8	−0.0415103
$621$	0	0
$622$	−1.98077e8	−0.0330041
$623$	−3.43478e9	−0.569103
$624$	0	0
$625$	−4.48629e9	−0.735034
$626$	−2.66529e8	−0.0434245
$627$	0	0
$628$	7.79140e9	1.25533
$629$	5.62634e9	0.901464
$630$	0	0
$631$	6.01569e8	0.0953197	0.0476598	−	0.998864i	$-0.484824\pi$
$631$	6.01569e8	0.0953197	0.0476598	+	0.998864i	$0.484824\pi$
$632$	−1.39160e8	−0.0219283
$633$	0	0
$634$	−2.99967e8	−0.0467479
$635$	−2.53815e9	−0.393377
$636$	0	0
$637$	−3.74093e9	−0.573444
$638$	2.08957e8	0.0318555
$639$	0	0
$640$	−8.20524e8	−0.123726
$641$	−3.94333e9	−0.591371	−0.295685	−	0.955285i	$-0.595548\pi$
$641$	−3.94333e9	−0.591371	−0.295685	+	0.955285i	$0.595548\pi$
$642$	0	0
$643$	3.67937e9	0.545802	0.272901	−	0.962042i	$-0.412017\pi$
$643$	3.67937e9	0.545802	0.272901	+	0.962042i	$0.412017\pi$
$644$	4.99008e9	0.736219
$645$	0	0
$646$	−2.06592e8	−0.0301508
$647$	−9.80484e9	−1.42323	−0.711616	−	0.702569i	$-0.752036\pi$
$647$	−9.80484e9	−1.42323	−0.711616	+	0.702569i	$0.752036\pi$
$648$	0	0
$649$	5.08747e8	0.0730542
$650$	6.35396e7	0.00907503
$651$	0	0
$652$	−3.10903e9	−0.439297
$653$	4.40696e9	0.619360	0.309680	−	0.950841i	$-0.399778\pi$
$653$	4.40696e9	0.619360	0.309680	+	0.950841i	$0.399778\pi$
$654$	0	0
$655$	−7.54049e9	−1.04847
$656$	−2.28792e8	−0.0316430
$657$	0	0
$658$	−5.52607e6	−0.000756182 0
$659$	1.35935e10	1.85025	0.925127	−	0.379657i	$-0.123958\pi$
$659$	1.35935e10	1.85025	0.925127	+	0.379657i	$0.123958\pi$
$660$	0	0
$661$	−5.30552e9	−0.714534	−0.357267	−	0.934002i	$-0.616291\pi$
$661$	−5.30552e9	−0.714534	−0.357267	+	0.934002i	$0.616291\pi$
$662$	−2.80553e7	−0.00375847
$663$	0	0
$664$	7.75121e8	0.102750
$665$	−6.68141e9	−0.881033
$666$	0	0
$667$	−1.27070e10	−1.65807
$668$	2.78824e9	0.361920
$669$	0	0
$670$	−3.75946e8	−0.0482907
$671$	2.00716e9	0.256480
$672$	0	0
$673$	3.77543e9	0.477434	0.238717	−	0.971089i	$-0.423273\pi$
$673$	3.77543e9	0.477434	0.238717	+	0.971089i	$0.423273\pi$
$674$	2.76603e8	0.0347975
$675$	0	0
$676$	−3.31661e9	−0.412934
$677$	7.29031e9	0.902995	0.451498	−	0.892272i	$-0.350890\pi$
$677$	7.29031e9	0.902995	0.451498	+	0.892272i	$0.350890\pi$
$678$	0	0
$679$	6.02767e9	0.738933
$680$	3.15716e8	0.0385048
$681$	0	0
$682$	7.66167e6	0.000924865 0
$683$	−7.10884e9	−0.853742	−0.426871	−	0.904313i	$-0.640384\pi$
$683$	−7.10884e9	−0.853742	−0.426871	+	0.904313i	$0.640384\pi$
$684$	0	0
$685$	5.39227e9	0.640996
$686$	−3.16324e8	−0.0374108
$687$	0	0
$688$	−1.46319e10	−1.71294
$689$	−8.57847e9	−0.999177
$690$	0	0
$691$	−1.26683e10	−1.46065	−0.730324	−	0.683100i	$-0.760631\pi$
$691$	−1.26683e10	−1.46065	−0.730324	+	0.683100i	$0.760631\pi$
$692$	9.68924e9	1.11152
$693$	0	0
$694$	−1.24391e8	−0.0141264
$695$	−1.38258e10	−1.56223
$696$	0	0
$697$	1.76284e8	0.0197196
$698$	−2.65863e8	−0.0295913
$699$	0	0
$700$	−1.41910e9	−0.156376
$701$	1.78862e10	1.96113	0.980564	−	0.196201i	$-0.0628606\pi$
$701$	1.78862e10	1.96113	0.980564	+	0.196201i	$0.0628606\pi$
$702$	0	0
$703$	1.85155e10	2.00998
$704$	−5.15657e9	−0.557002
$705$	0	0
$706$	−4.63022e8	−0.0495205
$707$	3.17033e9	0.337393
$708$	0	0
$709$	−1.24346e10	−1.31030	−0.655150	−	0.755499i	$-0.727395\pi$
$709$	−1.24346e10	−1.31030	−0.655150	+	0.755499i	$0.727395\pi$
$710$	3.29349e8	0.0345344
$711$	0	0
$712$	5.34119e8	0.0554572
$713$	−4.65920e8	−0.0481392
$714$	0	0
$715$	−5.76757e9	−0.590095
$716$	−1.46043e8	−0.0148691
$717$	0	0
$718$	6.75854e8	0.0681424
$719$	−2.69952e9	−0.270854	−0.135427	−	0.990787i	$-0.543241\pi$
$719$	−2.69952e9	−0.270854	−0.135427	+	0.990787i	$0.543241\pi$
$720$	0	0
$721$	2.83656e9	0.281850
$722$	−3.25128e8	−0.0321495
$723$	0	0
$724$	−1.73029e10	−1.69447
$725$	3.61368e9	0.352182
$726$	0	0
$727$	−4.43778e9	−0.428346	−0.214173	−	0.976796i	$-0.568706\pi$
$727$	−4.43778e9	−0.428346	−0.214173	+	0.976796i	$0.568706\pi$
$728$	−6.24327e8	−0.0599725
$729$	0	0
$730$	4.52845e7	0.00430844
$731$	1.12738e10	1.06748
$732$	0	0
$733$	8.21884e9	0.770808	0.385404	−	0.922748i	$-0.374062\pi$
$733$	8.21884e9	0.770808	0.385404	+	0.922748i	$0.374062\pi$
$734$	−3.78809e8	−0.0353577
$735$	0	0
$736$	−1.16420e9	−0.107635
$737$	−9.49146e9	−0.873367
$738$	0	0
$739$	−8.71653e9	−0.794490	−0.397245	−	0.917713i	$-0.630034\pi$
$739$	−8.71653e9	−0.794490	−0.397245	+	0.917713i	$0.630034\pi$
$740$	−1.41391e10	−1.28266
$741$	0	0
$742$	−2.36030e8	−0.0212106
$743$	1.01228e9	0.0905396	0.0452698	−	0.998975i	$-0.485585\pi$
$743$	1.01228e9	0.0905396	0.0452698	+	0.998975i	$0.485585\pi$
$744$	0	0
$745$	1.05098e10	0.931207
$746$	−5.64087e8	−0.0497462
$747$	0	0
$748$	3.98296e9	0.347977
$749$	−4.31266e9	−0.375024
$750$	0	0
$751$	1.76039e10	1.51659	0.758295	−	0.651911i	$-0.226032\pi$
$751$	1.76039e10	1.51659	0.758295	+	0.651911i	$0.226032\pi$
$752$	−3.48123e8	−0.0298518
$753$	0	0
$754$	7.94422e8	0.0674919
$755$	6.65393e9	0.562684
$756$	0	0
$757$	1.48620e10	1.24520	0.622602	−	0.782539i	$-0.286076\pi$
$757$	1.48620e10	1.24520	0.622602	+	0.782539i	$0.286076\pi$
$758$	−1.18624e8	−0.00989304
$759$	0	0
$760$	1.03898e9	0.0858537
$761$	−1.01144e10	−0.831939	−0.415970	−	0.909378i	$-0.636558\pi$
$761$	−1.01144e10	−0.831939	−0.415970	+	0.909378i	$0.636558\pi$
$762$	0	0
$763$	1.79414e9	0.146225
$764$	1.42472e10	1.15585
$765$	0	0
$766$	−1.04760e8	−0.00842157
$767$	1.93418e9	0.154779
$768$	0	0
$769$	−1.01233e10	−0.802748	−0.401374	−	0.915914i	$-0.631467\pi$
$769$	−1.01233e10	−0.802748	−0.401374	+	0.915914i	$0.631467\pi$
$770$	−1.58690e8	−0.0125266
$771$	0	0
$772$	5.86568e9	0.458836
$773$	−1.87926e10	−1.46339	−0.731693	−	0.681635i	$-0.761269\pi$
$773$	−1.87926e10	−1.46339	−0.731693	+	0.681635i	$0.761269\pi$
$774$	0	0
$775$	1.32500e8	0.0102250
$776$	−9.37320e8	−0.0720066
$777$	0	0
$778$	−6.81081e8	−0.0518525
$779$	5.80127e8	0.0439685
$780$	0	0
$781$	8.31503e9	0.624576
$782$	2.98389e8	0.0223131
$783$	0	0
$784$	−6.48415e9	−0.480559
$785$	1.50677e10	1.11174
$786$	0	0
$787$	1.04611e10	0.765009	0.382505	−	0.923954i	$-0.375062\pi$
$787$	1.04611e10	0.765009	0.382505	+	0.923954i	$0.375062\pi$
$788$	1.56804e10	1.14160
$789$	0	0
$790$	−1.34477e8	−0.00970405
$791$	−2.01557e9	−0.144804
$792$	0	0
$793$	7.63092e9	0.543402
$794$	−2.28122e8	−0.0161732
$795$	0	0
$796$	1.31047e9	0.0920937
$797$	−2.15776e10	−1.50973	−0.754865	−	0.655880i	$-0.772298\pi$
$797$	−2.15776e10	−1.50973	−0.754865	+	0.655880i	$0.772298\pi$
$798$	0	0
$799$	2.68228e8	0.0186033
$800$	3.31079e8	0.0228622
$801$	0	0
$802$	−4.55529e8	−0.0311821
$803$	1.14329e9	0.0779206
$804$	0	0
$805$	9.65025e9	0.652008
$806$	2.91285e7	0.00195950
$807$	0	0
$808$	−4.92995e8	−0.0328778
$809$	−1.89775e10	−1.26014	−0.630070	−	0.776538i	$-0.716974\pi$
$809$	−1.89775e10	−1.26014	−0.630070	+	0.776538i	$0.716974\pi$
$810$	0	0
$811$	−4.70722e9	−0.309879	−0.154939	−	0.987924i	$-0.549518\pi$
$811$	−4.70722e9	−0.309879	−0.154939	+	0.987924i	$0.549518\pi$
$812$	−1.77427e10	−1.16298
$813$	0	0
$814$	4.39763e8	0.0285781
$815$	−6.01252e9	−0.389049
$816$	0	0
$817$	3.71007e10	2.38015
$818$	1.76961e6	0.000113042 0
$819$	0	0
$820$	−4.43005e8	−0.0280582
$821$	2.74939e10	1.73395	0.866973	−	0.498355i	$-0.166062\pi$
$821$	2.74939e10	1.73395	0.866973	+	0.498355i	$0.166062\pi$
$822$	0	0
$823$	−1.47691e9	−0.0923540	−0.0461770	−	0.998933i	$-0.514704\pi$
$823$	−1.47691e9	−0.0923540	−0.0461770	+	0.998933i	$0.514704\pi$
$824$	−4.41093e8	−0.0274654
$825$	0	0
$826$	5.32175e7	0.00328567
$827$	3.11067e10	1.91243	0.956213	−	0.292671i	$-0.0945439\pi$
$827$	3.11067e10	1.91243	0.956213	+	0.292671i	$0.0945439\pi$
$828$	0	0
$829$	2.06946e9	0.126158	0.0630792	−	0.998009i	$-0.479908\pi$
$829$	2.06946e9	0.126158	0.0630792	+	0.998009i	$0.479908\pi$
$830$	7.49037e8	0.0454705
$831$	0	0
$832$	−1.96045e10	−1.18012
$833$	4.99601e9	0.299479
$834$	0	0
$835$	5.39214e9	0.320522
$836$	1.31074e10	0.775880
$837$	0	0
$838$	1.08145e9	0.0634825
$839$	−2.02258e9	−0.118233	−0.0591166	−	0.998251i	$-0.518828\pi$
$839$	−2.02258e9	−0.118233	−0.0591166	+	0.998251i	$0.518828\pi$
$840$	0	0
$841$	2.79312e10	1.61921
$842$	7.87963e8	0.0454898
$843$	0	0
$844$	2.08546e10	1.19400
$845$	−6.41395e9	−0.365702
$846$	0	0
$847$	8.71731e9	0.492936
$848$	−1.48691e10	−0.837332
$849$	0	0
$850$	−8.48572e7	−0.00473939
$851$	−2.67428e10	−1.48749
$852$	0	0
$853$	−3.22240e10	−1.77770	−0.888849	−	0.458201i	$-0.848494\pi$
$853$	−3.22240e10	−1.77770	−0.888849	+	0.458201i	$0.848494\pi$
$854$	2.09959e8	0.0115354
$855$	0	0
$856$	6.70632e8	0.0365448
$857$	−6.94237e9	−0.376769	−0.188384	−	0.982095i	$-0.560325\pi$
$857$	−6.94237e9	−0.376769	−0.188384	+	0.982095i	$0.560325\pi$
$858$	0	0
$859$	−6.91744e9	−0.372365	−0.186183	−	0.982515i	$-0.559612\pi$
$859$	−6.91744e9	−0.372365	−0.186183	+	0.982515i	$0.559612\pi$
$860$	−2.83314e10	−1.51888
$861$	0	0
$862$	−3.54922e8	−0.0188737
$863$	2.83705e10	1.50255	0.751275	−	0.659990i	$-0.229439\pi$
$863$	2.83705e10	1.50255	0.751275	+	0.659990i	$0.229439\pi$
$864$	0	0
$865$	1.87379e10	0.984384
$866$	1.11878e9	0.0585375
$867$	0	0
$868$	−6.50559e8	−0.0337651
$869$	−3.39512e9	−0.175504
$870$	0	0
$871$	−3.60851e10	−1.85040
$872$	−2.78995e8	−0.0142491
$873$	0	0
$874$	9.81960e8	0.0497512
$875$	−1.53557e10	−0.774895
$876$	0	0
$877$	−3.27420e9	−0.163910	−0.0819552	−	0.996636i	$-0.526116\pi$
$877$	−3.27420e9	−0.163910	−0.0819552	+	0.996636i	$0.526116\pi$
$878$	−5.98083e8	−0.0298216
$879$	0	0
$880$	−9.99693e9	−0.494512
$881$	−1.44579e10	−0.712346	−0.356173	−	0.934420i	$-0.615919\pi$
$881$	−1.44579e10	−0.712346	−0.356173	+	0.934420i	$0.615919\pi$
$882$	0	0
$883$	8.75453e9	0.427928	0.213964	−	0.976842i	$-0.431363\pi$
$883$	8.75453e9	0.427928	0.213964	+	0.976842i	$0.431363\pi$
$884$	1.51426e10	0.737256
$885$	0	0
$886$	5.18835e8	0.0250618
$887$	−1.54047e10	−0.741175	−0.370588	−	0.928798i	$-0.620844\pi$
$887$	−1.54047e10	−0.741175	−0.370588	+	0.928798i	$0.620844\pi$
$888$	0	0
$889$	−6.70311e9	−0.319978
$890$	5.16144e8	0.0245418
$891$	0	0
$892$	−2.76211e10	−1.30306
$893$	8.82702e8	0.0414796
$894$	0	0
$895$	−2.82431e8	−0.0131684
$896$	−2.16696e9	−0.100641
$897$	0	0
$898$	−1.02795e9	−0.0473701
$899$	1.65662e9	0.0760441
$900$	0	0
$901$	1.14566e10	0.521816
$902$	1.37786e7	0.000625147 0
$903$	0	0
$904$	3.13427e8	0.0141107
$905$	−3.34619e10	−1.50065
$906$	0	0
$907$	4.05126e10	1.80287	0.901435	−	0.432915i	$-0.142515\pi$
$907$	4.05126e10	1.80287	0.901435	+	0.432915i	$0.142515\pi$
$908$	−1.21008e10	−0.536430
$909$	0	0
$910$	−6.03317e8	−0.0265400
$911$	−9.95690e9	−0.436325	−0.218162	−	0.975912i	$-0.570006\pi$
$911$	−9.95690e9	−0.436325	−0.218162	+	0.975912i	$0.570006\pi$
$912$	0	0
$913$	1.89108e10	0.822361
$914$	1.19387e9	0.0517186
$915$	0	0
$916$	−7.18590e9	−0.308921
$917$	−1.99140e10	−0.852838
$918$	0	0
$919$	1.67638e10	0.712472	0.356236	−	0.934396i	$-0.384060\pi$
$919$	1.67638e10	0.712472	0.356236	+	0.934396i	$0.384060\pi$
$920$	−1.50064e9	−0.0635360
$921$	0	0
$922$	5.14179e8	0.0216051
$923$	3.16125e10	1.32328
$924$	0	0
$925$	7.60523e9	0.315948
$926$	−3.13130e7	−0.00129594
$927$	0	0
$928$	4.13941e9	0.170028
$929$	2.97001e10	1.21536	0.607678	−	0.794184i	$-0.292101\pi$
$929$	2.97001e10	1.21536	0.607678	+	0.794184i	$0.292101\pi$
$930$	0	0
$931$	1.64412e10	0.667744
$932$	3.99639e10	1.61701
$933$	0	0
$934$	−7.82847e8	−0.0314386
$935$	7.70260e9	0.308174
$936$	0	0
$937$	3.54569e10	1.40803	0.704015	−	0.710185i	$-0.251389\pi$
$937$	3.54569e10	1.40803	0.704015	+	0.710185i	$0.251389\pi$
$938$	−9.92854e8	−0.0392804
$939$	0	0
$940$	−6.74063e8	−0.0264699
$941$	−1.78640e10	−0.698900	−0.349450	−	0.936955i	$-0.613631\pi$
$941$	−1.78640e10	−0.698900	−0.349450	+	0.936955i	$0.613631\pi$
$942$	0	0
$943$	−8.37902e8	−0.0325389
$944$	3.35251e9	0.129709
$945$	0	0
$946$	8.81180e8	0.0338412
$947$	4.19426e10	1.60483	0.802417	−	0.596764i	$-0.203547\pi$
$947$	4.19426e10	1.60483	0.802417	+	0.596764i	$0.203547\pi$
$948$	0	0
$949$	4.34662e9	0.165090
$950$	−2.79254e8	−0.0105674
$951$	0	0
$952$	8.33789e8	0.0313204
$953$	2.78513e10	1.04236	0.521182	−	0.853445i	$-0.325491\pi$
$953$	2.78513e10	1.04236	0.521182	+	0.853445i	$0.325491\pi$
$954$	0	0
$955$	2.75524e10	1.02364
$956$	4.15431e10	1.53779
$957$	0	0
$958$	3.25642e7	0.00119663
$959$	1.42407e10	0.521395
$960$	0	0
$961$	−2.74519e10	−0.997792
$962$	1.67191e9	0.0605482
$963$	0	0
$964$	−3.27074e10	−1.17592
$965$	1.13436e10	0.406353
$966$	0	0
$967$	−1.00043e10	−0.355789	−0.177895	−	0.984050i	$-0.556929\pi$
$967$	−1.00043e10	−0.355789	−0.177895	+	0.984050i	$0.556929\pi$
$968$	−1.35557e9	−0.0480350
$969$	0	0
$970$	−9.05778e8	−0.0318655
$971$	2.19879e10	0.770755	0.385377	−	0.922759i	$-0.374071\pi$
$971$	2.19879e10	0.770755	0.385377	+	0.922759i	$0.374071\pi$
$972$	0	0
$973$	−3.65132e10	−1.27074
$974$	−1.11807e9	−0.0387714
$975$	0	0
$976$	1.32267e10	0.455383
$977$	4.29793e10	1.47444	0.737222	−	0.675651i	$-0.236137\pi$
$977$	4.29793e10	1.47444	0.737222	+	0.675651i	$0.236137\pi$
$978$	0	0
$979$	1.30310e10	0.443853
$980$	−1.25551e10	−0.426117
$981$	0	0
$982$	−1.72335e9	−0.0580741
$983$	2.32107e10	0.779383	0.389691	−	0.920946i	$-0.372582\pi$
$983$	2.32107e10	0.779383	0.389691	+	0.920946i	$0.372582\pi$
$984$	0	0
$985$	3.03240e10	1.01102
$986$	−1.06095e9	−0.0352474
$987$	0	0
$988$	4.98324e10	1.64385
$989$	−5.35862e10	−1.76143
$990$	0	0
$991$	−1.22560e10	−0.400030	−0.200015	−	0.979793i	$-0.564099\pi$
$991$	−1.22560e10	−0.400030	−0.200015	+	0.979793i	$0.564099\pi$
$992$	1.51777e8	0.00493646
$993$	0	0
$994$	8.69793e8	0.0280908
$995$	2.53429e9	0.0815598
$996$	0	0
$997$	−1.51059e10	−0.482740	−0.241370	−	0.970433i	$-0.577597\pi$
$997$	−1.51059e10	−0.482740	−0.241370	+	0.970433i	$0.577597\pi$
$998$	1.11866e9	0.0356238
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.d.1.9		17
3.2	odd	2		177.8.a.b.1.9	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.b.1.9	✓	17	3.2	odd	2
531.8.a.d.1.9		17	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-0.396855 q^{2} -127.843 q^{4} -247.233 q^{5} -652.929 q^{7} +101.532 q^{8} +O(q^{10})\)	\(q-0.396855 q^{2} -127.843 q^{4} -247.233 q^{5} -652.929 q^{7} +101.532 q^{8} +98.1157 q^{10} +2477.11 q^{11} +9417.61 q^{13} +259.118 q^{14} +16323.5 q^{16} -12577.2 q^{17} -41390.0 q^{19} +31606.9 q^{20} -983.055 q^{22} +59781.4 q^{23} -17000.9 q^{25} -3737.43 q^{26} +83472.1 q^{28} -212559. q^{29} -7793.74 q^{31} -19474.2 q^{32} +4991.34 q^{34} +161426. q^{35} -447343. q^{37} +16425.8 q^{38} -25102.2 q^{40} -14016.1 q^{41} -896369. q^{43} -316680. q^{44} -23724.6 q^{46} -21326.5 q^{47} -397227. q^{49} +6746.89 q^{50} -1.20397e6 q^{52} -910896. q^{53} -612424. q^{55} -66293.5 q^{56} +84355.0 q^{58} +205379. q^{59} +810282. q^{61} +3092.99 q^{62} -2.08169e6 q^{64} -2.32834e6 q^{65} -3.83166e6 q^{67} +1.60791e6 q^{68} -64062.6 q^{70} +3.35674e6 q^{71} +461542. q^{73} +177530. q^{74} +5.29140e6 q^{76} -1.61738e6 q^{77} -1.37060e6 q^{79} -4.03572e6 q^{80} +5562.36 q^{82} +7.63422e6 q^{83} +3.10951e6 q^{85} +355729. q^{86} +251507. q^{88} +5.26057e6 q^{89} -6.14903e6 q^{91} -7.64260e6 q^{92} +8463.51 q^{94} +1.02330e7 q^{95} -9.23173e6 q^{97} +157641. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \)	\( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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