LMFDB - Embedded newform 13.8.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [13,8,Mod(1,13)] 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("13.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	13.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.06100533129$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 354x^{2} - 640x + 20912 $ x^4 - x^3 - 354x^2 - 640x + 20912
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.26058$ of defining polynomial
Character	$\chi$	$=$	13.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.26058 q^{2} +66.7551 q^{3} -117.369 q^{4} +259.973 q^{5} -217.660 q^{6} +1453.99 q^{7} +800.043 q^{8} +2269.24 q^{9} -847.662 q^{10} -4450.41 q^{11} -7834.95 q^{12} -2197.00 q^{13} -4740.86 q^{14} +17354.5 q^{15} +12414.6 q^{16} -19258.2 q^{17} -7399.03 q^{18} -39326.5 q^{19} -30512.7 q^{20} +97061.4 q^{21} +14510.9 q^{22} +28432.1 q^{23} +53406.9 q^{24} -10539.1 q^{25} +7163.49 q^{26} +5489.83 q^{27} -170653. q^{28} +246229. q^{29} -56585.7 q^{30} -100837. q^{31} -142884. q^{32} -297088. q^{33} +62792.8 q^{34} +377999. q^{35} -266337. q^{36} -100757. q^{37} +128227. q^{38} -146661. q^{39} +207990. q^{40} +49018.9 q^{41} -316476. q^{42} -55968.9 q^{43} +522339. q^{44} +589941. q^{45} -92705.2 q^{46} -777743. q^{47} +828736. q^{48} +1.29055e6 q^{49} +34363.5 q^{50} -1.28558e6 q^{51} +257859. q^{52} +763116. q^{53} -17900.0 q^{54} -1.15699e6 q^{55} +1.16326e6 q^{56} -2.62524e6 q^{57} -802850. q^{58} +1.18041e6 q^{59} -2.03688e6 q^{60} -2.69377e6 q^{61} +328787. q^{62} +3.29946e6 q^{63} -1.12318e6 q^{64} -571161. q^{65} +968677. q^{66} +977111. q^{67} +2.26031e6 q^{68} +1.89799e6 q^{69} -1.23249e6 q^{70} +877610. q^{71} +1.81549e6 q^{72} -3.74799e6 q^{73} +328527. q^{74} -703537. q^{75} +4.61569e6 q^{76} -6.47087e6 q^{77} +478199. q^{78} +6.82797e6 q^{79} +3.22746e6 q^{80} -4.59635e6 q^{81} -159830. q^{82} +5.67139e6 q^{83} -1.13920e7 q^{84} -5.00661e6 q^{85} +182491. q^{86} +1.64371e7 q^{87} -3.56052e6 q^{88} +6.04479e6 q^{89} -1.92355e6 q^{90} -3.19442e6 q^{91} -3.33704e6 q^{92} -6.73138e6 q^{93} +2.53589e6 q^{94} -1.02238e7 q^{95} -9.53825e6 q^{96} -5.51192e6 q^{97} -4.20795e6 q^{98} -1.00990e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 15 q^{2} + 80 q^{3} + 253 q^{4} + 258 q^{5} + 1579 q^{6} + 1692 q^{7} + 1893 q^{8} + 3494 q^{9} - 4495 q^{10} + 1836 q^{11} - 3655 q^{12} - 8788 q^{13} - 18285 q^{14} - 29736 q^{15} - 36159 q^{16}+ \cdots + 6200852 q^{99}+O(q^{100}) $ 4 * q + 15 * q^2 + 80 * q^3 + 253 * q^4 + 258 * q^5 + 1579 * q^6 + 1692 * q^7 + 1893 * q^8 + 3494 * q^9 - 4495 * q^10 + 1836 * q^11 - 3655 * q^12 - 8788 * q^13 - 18285 * q^14 - 29736 * q^15 - 36159 * q^16 + 11814 * q^17 - 2738 * q^18 + 27660 * q^19 - 30369 * q^20 + 69930 * q^21 + 59930 * q^22 + 172920 * q^23 + 26553 * q^24 + 219414 * q^25 - 32955 * q^26 + 174800 * q^27 - 170903 * q^28 + 133344 * q^29 - 624695 * q^30 - 231748 * q^31 - 459291 * q^32 - 33868 * q^33 - 480249 * q^34 + 322296 * q^35 - 157930 * q^36 + 248026 * q^37 - 85326 * q^38 - 175760 * q^39 + 379895 * q^40 + 588108 * q^41 + 132275 * q^42 + 309304 * q^43 + 1270806 * q^44 - 1773144 * q^45 + 574948 * q^46 + 557916 * q^47 + 837429 * q^48 - 767378 * q^49 + 5737920 * q^50 - 1816496 * q^51 - 555841 * q^52 + 2022348 * q^53 - 479 * q^54 - 3770208 * q^55 + 1391625 * q^56 - 5609844 * q^57 - 3027194 * q^58 + 1162668 * q^59 - 8100077 * q^60 - 1340572 * q^61 + 2715564 * q^62 + 4213476 * q^63 - 6734935 * q^64 - 566826 * q^65 + 8124226 * q^66 - 598484 * q^67 + 4515993 * q^68 - 108848 * q^69 - 4329415 * q^70 + 697860 * q^71 + 1607430 * q^72 - 13725816 * q^73 + 14884053 * q^74 + 16604592 * q^75 + 12055070 * q^76 - 5326932 * q^77 - 3469063 * q^78 + 20079576 * q^79 + 3710667 * q^80 - 8368396 * q^81 + 857160 * q^82 - 2024724 * q^83 - 13603483 * q^84 - 16146514 * q^85 - 22922499 * q^86 + 22170952 * q^87 - 2100618 * q^88 + 17646240 * q^89 - 42372890 * q^90 - 3717324 * q^91 + 14504892 * q^92 - 17910680 * q^93 - 3729785 * q^94 + 12508152 * q^95 - 37637751 * q^96 - 6329096 * q^97 - 22166262 * q^98 + 6200852 * 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 $

$2$	−3.26058	−0.288197	−0.144099	−	0.989563i	$-0.546028\pi$
$2$	−3.26058	−0.288197	−0.144099	+	0.989563i	$0.546028\pi$
$3$	66.7551	1.42745	0.713723	−	0.700428i	$-0.247007\pi$
$3$	66.7551	1.42745	0.713723	+	0.700428i	$0.247007\pi$
$4$	−117.369	−0.916942
$5$	259.973	0.930107	0.465054	−	0.885282i	$-0.346035\pi$
$5$	259.973	0.930107	0.465054	+	0.885282i	$0.346035\pi$
$6$	−217.660	−0.411386
$7$	1453.99	1.60221	0.801105	−	0.598524i	$-0.204246\pi$
$7$	1453.99	1.60221	0.801105	+	0.598524i	$0.204246\pi$
$8$	800.043	0.552457
$9$	2269.24	1.03760
$10$	−847.662	−0.268054
$11$	−4450.41	−1.00815	−0.504076	−	0.863659i	$-0.668167\pi$
$11$	−4450.41	−1.00815	−0.504076	+	0.863659i	$0.668167\pi$
$12$	−7834.95	−1.30889
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	−4740.86	−0.461752
$15$	17354.5	1.32768
$16$	12414.6	0.757726
$17$	−19258.2	−0.950701	−0.475351	−	0.879796i	$-0.657679\pi$
$17$	−19258.2	−0.950701	−0.475351	+	0.879796i	$0.657679\pi$
$18$	−7399.03	−0.299034
$19$	−39326.5	−1.31537	−0.657684	−	0.753294i	$-0.728464\pi$
$19$	−39326.5	−1.31537	−0.657684	+	0.753294i	$0.728464\pi$
$20$	−30512.7	−0.852855
$21$	97061.4	2.28707
$22$	14510.9	0.290546
$23$	28432.1	0.487261	0.243631	−	0.969868i	$-0.421662\pi$
$23$	28432.1	0.487261	0.243631	+	0.969868i	$0.421662\pi$
$24$	53406.9	0.788603
$25$	−10539.1	−0.134900
$26$	7163.49	0.0799315
$27$	5489.83	0.0536767
$28$	−170653.	−1.46913
$29$	246229.	1.87476	0.937382	−	0.348302i	$-0.113242\pi$
$29$	246229.	1.87476	0.937382	+	0.348302i	$0.113242\pi$
$30$	−56585.7	−0.382633
$31$	−100837.	−0.607931	−0.303965	−	0.952683i	$-0.598311\pi$
$31$	−100837.	−0.607931	−0.303965	+	0.952683i	$0.598311\pi$
$32$	−142884.	−0.770831
$33$	−297088.	−1.43908
$34$	62792.8	0.273989
$35$	377999.	1.49023
$36$	−266337.	−0.951423
$37$	−100757.	−0.327017	−0.163509	−	0.986542i	$-0.552281\pi$
$37$	−100757.	−0.327017	−0.163509	+	0.986542i	$0.552281\pi$
$38$	128227.	0.379085
$39$	−146661.	−0.395902
$40$	207990.	0.513844
$41$	49018.9	0.111076	0.0555379	−	0.998457i	$-0.482313\pi$
$41$	49018.9	0.111076	0.0555379	+	0.998457i	$0.482313\pi$
$42$	−316476.	−0.659126
$43$	−55968.9	−0.107351	−0.0536757	−	0.998558i	$-0.517094\pi$
$43$	−55968.9	−0.107351	−0.0536757	+	0.998558i	$0.517094\pi$
$44$	522339.	0.924417
$45$	589941.	0.965083
$46$	−92705.2	−0.140427
$47$	−777743.	−1.09268	−0.546341	−	0.837563i	$-0.683980\pi$
$47$	−777743.	−1.09268	−0.546341	+	0.837563i	$0.683980\pi$
$48$	828736.	1.08161
$49$	1.29055e6	1.56708
$50$	34363.5	0.0388778
$51$	−1.28558e6	−1.35707
$52$	257859.	0.254314
$53$	763116.	0.704085	0.352042	−	0.935984i	$-0.385487\pi$
$53$	763116.	0.704085	0.352042	+	0.935984i	$0.385487\pi$
$54$	−17900.0	−0.0154695
$55$	−1.15699e6	−0.937689
$56$	1.16326e6	0.885152
$57$	−2.62524e6	−1.87762
$58$	−802850.	−0.540302
$59$	1.18041e6	0.748257	0.374128	−	0.927377i	$-0.377942\pi$
$59$	1.18041e6	0.748257	0.374128	+	0.927377i	$0.377942\pi$
$60$	−2.03688e6	−1.21740
$61$	−2.69377e6	−1.51952	−0.759761	−	0.650203i	$-0.774684\pi$
$61$	−2.69377e6	−1.51952	−0.759761	+	0.650203i	$0.774684\pi$
$62$	328787.	0.175204
$63$	3.29946e6	1.66246
$64$	−1.12318e6	−0.535575
$65$	−571161.	−0.257965
$66$	968677.	0.414739
$67$	977111.	0.396901	0.198450	−	0.980111i	$-0.436409\pi$
$67$	977111.	0.396901	0.198450	+	0.980111i	$0.436409\pi$
$68$	2.26031e6	0.871738
$69$	1.89799e6	0.695540
$70$	−1.23249e6	−0.429479
$71$	877610.	0.291003	0.145502	−	0.989358i	$-0.453520\pi$
$71$	877610.	0.291003	0.145502	+	0.989358i	$0.453520\pi$
$72$	1.81549e6	0.573231
$73$	−3.74799e6	−1.12763	−0.563817	−	0.825899i	$-0.690668\pi$
$73$	−3.74799e6	−1.12763	−0.563817	+	0.825899i	$0.690668\pi$
$74$	328527.	0.0942454
$75$	−703537.	−0.192563
$76$	4.61569e6	1.20612
$77$	−6.47087e6	−1.61527
$78$	478199.	0.114098
$79$	6.82797e6	1.55811	0.779053	−	0.626958i	$-0.215700\pi$
$79$	6.82797e6	1.55811	0.779053	+	0.626958i	$0.215700\pi$
$80$	3.22746e6	0.704767
$81$	−4.59635e6	−0.960983
$82$	−159830.	−0.0320117
$83$	5.67139e6	1.08872	0.544360	−	0.838852i	$-0.316772\pi$
$83$	5.67139e6	1.08872	0.544360	+	0.838852i	$0.316772\pi$
$84$	−1.13920e7	−2.09711
$85$	−5.00661e6	−0.884254
$86$	182491.	0.0309383
$87$	1.64371e7	2.67613
$88$	−3.56052e6	−0.556961
$89$	6.04479e6	0.908900	0.454450	−	0.890772i	$-0.349836\pi$
$89$	6.04479e6	0.908900	0.454450	+	0.890772i	$0.349836\pi$
$90$	−1.92355e6	−0.278134
$91$	−3.19442e6	−0.444373
$92$	−3.33704e6	−0.446791
$93$	−6.73138e6	−0.867789
$94$	2.53589e6	0.314908
$95$	−1.02238e7	−1.22343
$96$	−9.53825e6	−1.10032
$97$	−5.51192e6	−0.613200	−0.306600	−	0.951838i	$-0.599191\pi$
$97$	−5.51192e6	−0.613200	−0.306600	+	0.951838i	$0.599191\pi$
$98$	−4.20795e6	−0.451627
$99$	−1.00990e7	−1.04606
$100$	1.23696e6	0.123696
$101$	1.35091e7	1.30467	0.652337	−	0.757929i	$-0.273789\pi$
$101$	1.35091e7	1.30467	0.652337	+	0.757929i	$0.273789\pi$
$102$	4.19174e6	0.391105
$103$	1.81950e7	1.64067	0.820335	−	0.571883i	$-0.193787\pi$
$103$	1.81950e7	1.64067	0.820335	+	0.571883i	$0.193787\pi$
$104$	−1.75770e6	−0.153224
$105$	2.52333e7	2.12722
$106$	−2.48820e6	−0.202915
$107$	5.88068e6	0.464071	0.232035	−	0.972707i	$-0.425462\pi$
$107$	5.88068e6	0.464071	0.232035	+	0.972707i	$0.425462\pi$
$108$	−644334.	−0.0492184
$109$	−1.55224e7	−1.14806	−0.574031	−	0.818834i	$-0.694621\pi$
$109$	−1.55224e7	−1.14806	−0.574031	+	0.818834i	$0.694621\pi$
$110$	3.77245e6	0.270239
$111$	−6.72606e6	−0.466799
$112$	1.80507e7	1.21404
$113$	7.99095e6	0.520983	0.260492	−	0.965476i	$-0.416115\pi$
$113$	7.99095e6	0.520983	0.260492	+	0.965476i	$0.416115\pi$
$114$	8.55980e6	0.541124
$115$	7.39159e6	0.453205
$116$	−2.88996e7	−1.71905
$117$	−4.98552e6	−0.287779
$118$	−3.84882e6	−0.215645
$119$	−2.80013e7	−1.52322
$120$	1.38844e7	0.733485
$121$	319007.	0.0163701
$122$	8.78326e6	0.437921
$123$	3.27226e6	0.158555
$124$	1.18351e7	0.557438
$125$	−2.30503e7	−1.05558
$126$	−1.07581e7	−0.479115
$127$	1.73773e7	0.752780	0.376390	−	0.926461i	$-0.377165\pi$
$127$	1.73773e7	0.752780	0.376390	+	0.926461i	$0.377165\pi$
$128$	2.19514e7	0.925182
$129$	−3.73621e6	−0.153238
$130$	1.86231e6	0.0743449
$131$	−3.23735e7	−1.25817	−0.629086	−	0.777336i	$-0.716571\pi$
$131$	−3.23735e7	−1.25817	−0.629086	+	0.777336i	$0.716571\pi$
$132$	3.48688e7	1.31956
$133$	−5.71804e7	−2.10750
$134$	−3.18595e6	−0.114386
$135$	1.42721e6	0.0499251
$136$	−1.54074e7	−0.525222
$137$	9.16741e6	0.304596	0.152298	−	0.988335i	$-0.451333\pi$
$137$	9.16741e6	0.304596	0.152298	+	0.988335i	$0.451333\pi$
$138$	−6.18854e6	−0.200452
$139$	9.03804e6	0.285445	0.142722	−	0.989763i	$-0.454414\pi$
$139$	9.03804e6	0.285445	0.142722	+	0.989763i	$0.454414\pi$
$140$	−4.43652e7	−1.36645
$141$	−5.19183e7	−1.55975
$142$	−2.86152e6	−0.0838662
$143$	9.77756e6	0.279611
$144$	2.81716e7	0.786219
$145$	6.40130e7	1.74373
$146$	1.22206e7	0.324981
$147$	8.61510e7	2.23692
$148$	1.18257e7	0.299856
$149$	4.06925e7	1.00777	0.503887	−	0.863770i	$-0.331903\pi$
$149$	4.06925e7	1.00777	0.503887	+	0.863770i	$0.331903\pi$
$150$	2.29394e6	0.0554960
$151$	−6.97424e6	−0.164846	−0.0824228	−	0.996597i	$-0.526266\pi$
$151$	−6.97424e6	−0.164846	−0.0824228	+	0.996597i	$0.526266\pi$
$152$	−3.14629e7	−0.726685
$153$	−4.37014e7	−0.986451
$154$	2.10988e7	0.465516
$155$	−2.62149e7	−0.565441
$156$	1.72134e7	0.363020
$157$	−8.41804e6	−0.173605	−0.0868025	−	0.996226i	$-0.527665\pi$
$157$	−8.41804e6	−0.173605	−0.0868025	+	0.996226i	$0.527665\pi$
$158$	−2.22631e7	−0.449041
$159$	5.09419e7	1.00504
$160$	−3.71460e7	−0.716956
$161$	4.13401e7	0.780695
$162$	1.49868e7	0.276952
$163$	3.03063e7	0.548121	0.274060	−	0.961712i	$-0.411633\pi$
$163$	3.03063e7	0.548121	0.274060	+	0.961712i	$0.411633\pi$
$164$	−5.75328e6	−0.101850
$165$	−7.72347e7	−1.33850
$166$	−1.84920e7	−0.313766
$167$	−5.08939e7	−0.845586	−0.422793	−	0.906226i	$-0.638950\pi$
$167$	−5.08939e7	−0.845586	−0.422793	+	0.906226i	$0.638950\pi$
$168$	7.76534e7	1.26351
$169$	4.82681e6	0.0769231
$170$	1.63244e7	0.254839
$171$	−8.92411e7	−1.36483
$172$	6.56900e6	0.0984350
$173$	−4.67458e7	−0.686407	−0.343203	−	0.939261i	$-0.611512\pi$
$173$	−4.67458e7	−0.686407	−0.343203	+	0.939261i	$0.611512\pi$
$174$	−5.35943e7	−0.771252
$175$	−1.53238e7	−0.216138
$176$	−5.52500e7	−0.763903
$177$	7.87983e7	1.06810
$178$	−1.97095e7	−0.261942
$179$	1.23980e8	1.61572	0.807859	−	0.589376i	$-0.200626\pi$
$179$	1.23980e8	1.61572	0.807859	+	0.589376i	$0.200626\pi$
$180$	−6.92405e7	−0.884925
$181$	7.81974e7	0.980205	0.490103	−	0.871665i	$-0.336959\pi$
$181$	7.81974e7	0.980205	0.490103	+	0.871665i	$0.336959\pi$
$182$	1.04157e7	0.128067
$183$	−1.79823e8	−2.16904
$184$	2.27469e7	0.269191
$185$	−2.61942e7	−0.304161
$186$	2.19482e7	0.250094
$187$	8.57069e7	0.958451
$188$	9.12827e7	1.00193
$189$	7.98217e6	0.0860013
$190$	3.33355e7	0.352590
$191$	3.55311e7	0.368971	0.184486	−	0.982835i	$-0.440938\pi$
$191$	3.55311e7	0.368971	0.184486	+	0.982835i	$0.440938\pi$
$192$	−7.49780e7	−0.764504
$193$	−6.71166e7	−0.672016	−0.336008	−	0.941859i	$-0.609077\pi$
$193$	−6.71166e7	−0.672016	−0.336008	+	0.941859i	$0.609077\pi$
$194$	1.79720e7	0.176722
$195$	−3.81279e7	−0.368232
$196$	−1.51471e8	−1.43692
$197$	3.27770e7	0.305448	0.152724	−	0.988269i	$-0.451195\pi$
$197$	3.27770e7	0.305448	0.152724	+	0.988269i	$0.451195\pi$
$198$	3.29287e7	0.301472
$199$	−9.03328e7	−0.812568	−0.406284	−	0.913747i	$-0.633176\pi$
$199$	−9.03328e7	−0.812568	−0.406284	+	0.913747i	$0.633176\pi$
$200$	−8.43172e6	−0.0745266
$201$	6.52271e7	0.566555
$202$	−4.40475e7	−0.376003
$203$	3.58016e8	3.00377
$204$	1.50887e8	1.24436
$205$	1.27436e7	0.103312
$206$	−5.93262e7	−0.472836
$207$	6.45193e7	0.505584
$208$	−2.72748e7	−0.210155
$209$	1.75019e8	1.32609
$210$	−8.22753e7	−0.613058
$211$	−1.30945e7	−0.0959625	−0.0479812	−	0.998848i	$-0.515279\pi$
$211$	−1.30945e7	−0.0959625	−0.0479812	+	0.998848i	$0.515279\pi$
$212$	−8.95659e7	−0.645605
$213$	5.85849e7	0.415391
$214$	−1.91744e7	−0.133744
$215$	−1.45504e7	−0.0998483
$216$	4.39210e6	0.0296541
$217$	−1.46616e8	−0.974033
$218$	5.06119e7	0.330868
$219$	−2.50197e8	−1.60964
$220$	1.35794e8	0.859807
$221$	4.23102e7	0.263677
$222$	2.19308e7	0.134530
$223$	2.84081e7	0.171544	0.0857720	−	0.996315i	$-0.472664\pi$
$223$	2.84081e7	0.171544	0.0857720	+	0.996315i	$0.472664\pi$
$224$	−2.07753e8	−1.23503
$225$	−2.39157e7	−0.139973
$226$	−2.60551e7	−0.150146
$227$	−2.74035e8	−1.55495	−0.777474	−	0.628915i	$-0.783499\pi$
$227$	−2.74035e8	−1.55495	−0.777474	+	0.628915i	$0.783499\pi$
$228$	3.08121e8	1.72167
$229$	−2.74411e8	−1.51000	−0.755000	−	0.655725i	$-0.772363\pi$
$229$	−2.74411e8	−1.51000	−0.755000	+	0.655725i	$0.772363\pi$
$230$	−2.41008e7	−0.130612
$231$	−4.31963e8	−2.30571
$232$	1.96994e8	1.03573
$233$	−2.87263e7	−0.148776	−0.0743881	−	0.997229i	$-0.523700\pi$
$233$	−2.87263e7	−0.148776	−0.0743881	+	0.997229i	$0.523700\pi$
$234$	1.62557e7	0.0829372
$235$	−2.02192e8	−1.01631
$236$	−1.38543e8	−0.686109
$237$	4.55802e8	2.22411
$238$	9.13003e7	0.438988
$239$	2.46037e8	1.16575	0.582877	−	0.812560i	$-0.301927\pi$
$239$	2.46037e8	1.16575	0.582877	+	0.812560i	$0.301927\pi$
$240$	2.15449e8	1.00602
$241$	−3.12621e8	−1.43866	−0.719331	−	0.694667i	$-0.755552\pi$
$241$	−3.12621e8	−1.43866	−0.719331	+	0.694667i	$0.755552\pi$
$242$	−1.04015e6	−0.00471781
$243$	−3.18836e8	−1.42543
$244$	3.16165e8	1.39331
$245$	3.35509e8	1.45755
$246$	−1.06694e7	−0.0456950
$247$	8.64003e7	0.364818
$248$	−8.06740e7	−0.335856
$249$	3.78594e8	1.55409
$250$	7.51571e7	0.304215
$251$	3.36400e8	1.34276	0.671380	−	0.741113i	$-0.265702\pi$
$251$	3.36400e8	1.34276	0.671380	+	0.741113i	$0.265702\pi$
$252$	−3.87253e8	−1.52438
$253$	−1.26535e8	−0.491234
$254$	−5.66599e7	−0.216949
$255$	−3.34216e8	−1.26223
$256$	7.21930e7	0.268940
$257$	−3.58158e7	−0.131616	−0.0658081	−	0.997832i	$-0.520963\pi$
$257$	−3.58158e7	−0.131616	−0.0658081	+	0.997832i	$0.520963\pi$
$258$	1.21822e7	0.0441628
$259$	−1.46500e8	−0.523950
$260$	6.70363e7	0.236539
$261$	5.58753e8	1.94526
$262$	1.05556e8	0.362601
$263$	−5.64174e8	−1.91235	−0.956176	−	0.292792i	$-0.905416\pi$
$263$	−5.64174e8	−1.91235	−0.956176	+	0.292792i	$0.905416\pi$
$264$	−2.37683e8	−0.795031
$265$	1.98389e8	0.654875
$266$	1.86441e8	0.607374
$267$	4.03520e8	1.29741
$268$	−1.14682e8	−0.363935
$269$	3.96388e7	0.124162	0.0620809	−	0.998071i	$-0.480226\pi$
$269$	3.96388e7	0.124162	0.0620809	+	0.998071i	$0.480226\pi$
$270$	−4.65352e6	−0.0143883
$271$	1.83544e8	0.560205	0.280102	−	0.959970i	$-0.409632\pi$
$271$	1.83544e8	0.560205	0.280102	+	0.959970i	$0.409632\pi$
$272$	−2.39082e8	−0.720371
$273$	−2.13244e8	−0.634319
$274$	−2.98910e7	−0.0877838
$275$	4.69032e7	0.136000
$276$	−2.22764e8	−0.637770
$277$	2.59168e8	0.732660	0.366330	−	0.930485i	$-0.380614\pi$
$277$	2.59168e8	0.732660	0.366330	+	0.930485i	$0.380614\pi$
$278$	−2.94692e7	−0.0822644
$279$	−2.28823e8	−0.630791
$280$	3.02416e8	0.823287
$281$	−5.83114e6	−0.0156777	−0.00783883	−	0.999969i	$-0.502495\pi$
$281$	−5.83114e6	−0.0156777	−0.00783883	+	0.999969i	$0.502495\pi$
$282$	1.69284e8	0.449514
$283$	9.19774e7	0.241228	0.120614	−	0.992699i	$-0.461514\pi$
$283$	9.19774e7	0.241228	0.120614	+	0.992699i	$0.461514\pi$
$284$	−1.03004e8	−0.266833
$285$	−6.82492e8	−1.74639
$286$	−3.18805e7	−0.0805831
$287$	7.12731e7	0.177967
$288$	−3.24238e8	−0.799817
$289$	−3.94612e7	−0.0961674
$290$	−2.08719e8	−0.502539
$291$	−3.67949e8	−0.875310
$292$	4.39897e8	1.03398
$293$	−4.48184e8	−1.04092	−0.520462	−	0.853885i	$-0.674240\pi$
$293$	−4.48184e8	−1.04092	−0.520462	+	0.853885i	$0.674240\pi$
$294$	−2.80902e8	−0.644673
$295$	3.06875e8	0.695959
$296$	−8.06102e7	−0.180663
$297$	−2.44320e7	−0.0541142
$298$	−1.32681e8	−0.290437
$299$	−6.24654e7	−0.135142
$300$	8.25731e7	0.176569
$301$	−8.13785e7	−0.171999
$302$	2.27400e7	0.0475080
$303$	9.01802e8	1.86235
$304$	−4.88222e8	−0.996689
$305$	−7.00308e8	−1.41332
$306$	1.42492e8	0.284292
$307$	2.89460e8	0.570957	0.285479	−	0.958385i	$-0.407847\pi$
$307$	2.89460e8	0.570957	0.285479	+	0.958385i	$0.407847\pi$
$308$	7.59477e8	1.48111
$309$	1.21461e9	2.34197
$310$	8.54757e7	0.162958
$311$	4.13068e8	0.778683	0.389341	−	0.921094i	$-0.372703\pi$
$311$	4.13068e8	0.778683	0.389341	+	0.921094i	$0.372703\pi$
$312$	−1.17335e8	−0.218719
$313$	5.30900e8	0.978606	0.489303	−	0.872114i	$-0.337251\pi$
$313$	5.30900e8	0.978606	0.489303	+	0.872114i	$0.337251\pi$
$314$	2.74477e7	0.0500324
$315$	8.57770e8	1.54626
$316$	−8.01390e8	−1.42869
$317$	−1.95230e8	−0.344223	−0.172112	−	0.985077i	$-0.555059\pi$
$317$	−1.95230e8	−0.344223	−0.172112	+	0.985077i	$0.555059\pi$
$318$	−1.66100e8	−0.289650
$319$	−1.09582e9	−1.89005
$320$	−2.91997e8	−0.498142
$321$	3.92565e8	0.662436
$322$	−1.34793e8	−0.224994
$323$	7.57356e8	1.25052
$324$	5.39467e8	0.881166
$325$	2.31544e7	0.0374146
$326$	−9.88160e7	−0.157967
$327$	−1.03620e9	−1.63880
$328$	3.92172e7	0.0613646
$329$	−1.13083e9	−1.75071
$330$	2.51830e8	0.385752
$331$	1.98439e8	0.300766	0.150383	−	0.988628i	$-0.451949\pi$
$331$	1.98439e8	0.300766	0.150383	+	0.988628i	$0.451949\pi$
$332$	−6.65644e8	−0.998294
$333$	−2.28642e8	−0.339314
$334$	1.65943e8	0.243695
$335$	2.54022e8	0.369160
$336$	1.20498e9	1.73297
$337$	−9.24024e8	−1.31516	−0.657580	−	0.753385i	$-0.728420\pi$
$337$	−9.24024e8	−1.31516	−0.657580	+	0.753385i	$0.728420\pi$
$338$	−1.57382e7	−0.0221690
$339$	5.33436e8	0.743676
$340$	5.87618e8	0.810810
$341$	4.48767e8	0.612887
$342$	2.90978e8	0.393340
$343$	6.79032e8	0.908574
$344$	−4.47776e7	−0.0593070
$345$	4.93426e8	0.646927
$346$	1.52418e8	0.197820
$347$	−7.88057e8	−1.01252	−0.506261	−	0.862380i	$-0.668973\pi$
$347$	−7.88057e8	−1.01252	−0.506261	+	0.862380i	$0.668973\pi$
$348$	−1.92919e9	−2.45385
$349$	−6.05575e8	−0.762569	−0.381284	−	0.924458i	$-0.624518\pi$
$349$	−6.05575e8	−0.762569	−0.381284	+	0.924458i	$0.624518\pi$
$350$	4.99643e7	0.0622904
$351$	−1.20612e7	−0.0148872
$352$	6.35894e8	0.777115
$353$	−3.09363e8	−0.374333	−0.187166	−	0.982328i	$-0.559930\pi$
$353$	−3.09363e8	−0.374333	−0.187166	+	0.982328i	$0.559930\pi$
$354$	−2.56928e8	−0.307822
$355$	2.28155e8	0.270664
$356$	−7.09468e8	−0.833409
$357$	−1.86923e9	−2.17432
$358$	−4.04246e8	−0.465645
$359$	2.65762e8	0.303153	0.151576	−	0.988446i	$-0.451565\pi$
$359$	2.65762e8	0.303153	0.151576	+	0.988446i	$0.451565\pi$
$360$	4.71978e8	0.533167
$361$	6.52700e8	0.730194
$362$	−2.54969e8	−0.282492
$363$	2.12953e7	0.0233674
$364$	3.74925e8	0.407464
$365$	−9.74376e8	−1.04882
$366$	5.86327e8	0.625109
$367$	−8.08965e8	−0.854276	−0.427138	−	0.904186i	$-0.640478\pi$
$367$	−8.08965e8	−0.854276	−0.427138	+	0.904186i	$0.640478\pi$
$368$	3.52973e8	0.369211
$369$	1.11235e8	0.115253
$370$	8.54081e7	0.0876583
$371$	1.10957e9	1.12809
$372$	7.90053e8	0.795712
$373$	6.57479e8	0.655996	0.327998	−	0.944678i	$-0.393626\pi$
$373$	6.57479e8	0.655996	0.327998	+	0.944678i	$0.393626\pi$
$374$	−2.79454e8	−0.276223
$375$	−1.53872e9	−1.50678
$376$	−6.22228e8	−0.603660
$377$	−5.40966e8	−0.519966
$378$	−2.60265e7	−0.0247853
$379$	1.18091e9	1.11424	0.557119	−	0.830433i	$-0.311907\pi$
$379$	1.18091e9	1.11424	0.557119	+	0.830433i	$0.311907\pi$
$380$	1.19996e9	1.12182
$381$	1.16002e9	1.07455
$382$	−1.15852e8	−0.106336
$383$	1.25293e9	1.13954	0.569772	−	0.821803i	$-0.307032\pi$
$383$	1.25293e9	1.13954	0.569772	+	0.821803i	$0.307032\pi$
$384$	1.46537e9	1.32065
$385$	−1.68225e9	−1.50238
$386$	2.18839e8	0.193673
$387$	−1.27007e8	−0.111388
$388$	6.46927e8	0.562269
$389$	8.65648e8	0.745620	0.372810	−	0.927908i	$-0.378394\pi$
$389$	8.65648e8	0.745620	0.372810	+	0.927908i	$0.378394\pi$
$390$	1.24319e8	0.106123
$391$	−5.47551e8	−0.463240
$392$	1.03250e9	0.865742
$393$	−2.16109e9	−1.79597
$394$	−1.06872e8	−0.0880292
$395$	1.77509e9	1.44921
$396$	1.18531e9	0.959178
$397$	9.76000e8	0.782858	0.391429	−	0.920208i	$-0.371981\pi$
$397$	9.76000e8	0.782858	0.391429	+	0.920208i	$0.371981\pi$
$398$	2.94537e8	0.234180
$399$	−3.81708e9	−3.00834
$400$	−1.30838e8	−0.102217
$401$	1.48866e9	1.15289	0.576447	−	0.817134i	$-0.304439\pi$
$401$	1.48866e9	1.15289	0.576447	+	0.817134i	$0.304439\pi$
$402$	−2.12678e8	−0.163279
$403$	2.21539e8	0.168610
$404$	−1.58555e9	−1.19631
$405$	−1.19493e9	−0.893817
$406$	−1.16734e9	−0.865676
$407$	4.48412e8	0.329683
$408$	−1.02852e9	−0.749726
$409$	3.66344e8	0.264763	0.132381	−	0.991199i	$-0.457738\pi$
$409$	3.66344e8	0.264763	0.132381	+	0.991199i	$0.457738\pi$
$410$	−4.15514e7	−0.0297743
$411$	6.11971e8	0.434795
$412$	−2.13552e9	−1.50440
$413$	1.71631e9	1.19886
$414$	−2.10370e8	−0.145708
$415$	1.47441e9	1.01263
$416$	3.13917e8	0.213790
$417$	6.03335e8	0.407457
$418$	−5.70663e8	−0.382175
$419$	8.82815e8	0.586301	0.293150	−	0.956066i	$-0.405296\pi$
$419$	8.82815e8	0.586301	0.293150	+	0.956066i	$0.405296\pi$
$420$	−2.96160e9	−1.95054
$421$	−5.84465e8	−0.381743	−0.190872	−	0.981615i	$-0.561131\pi$
$421$	−5.84465e8	−0.381743	−0.190872	+	0.981615i	$0.561131\pi$
$422$	4.26957e7	0.0276561
$423$	−1.76489e9	−1.13377
$424$	6.10526e8	0.388977
$425$	2.02963e8	0.128250
$426$	−1.91021e8	−0.119715
$427$	−3.91673e9	−2.43459
$428$	−6.90207e8	−0.425526
$429$	6.52701e8	0.399130
$430$	4.74427e7	0.0287760
$431$	−9.46875e8	−0.569668	−0.284834	−	0.958577i	$-0.591939\pi$
$431$	−9.46875e8	−0.569668	−0.284834	+	0.958577i	$0.591939\pi$
$432$	6.81539e7	0.0406722
$433$	1.63557e9	0.968190	0.484095	−	0.875016i	$-0.339149\pi$
$433$	1.63557e9	0.968190	0.484095	+	0.875016i	$0.339149\pi$
$434$	4.78054e8	0.280713
$435$	4.27319e9	2.48908
$436$	1.82184e9	1.05271
$437$	−1.11814e9	−0.640928
$438$	8.15788e8	0.463893
$439$	−2.41064e8	−0.135990	−0.0679950	−	0.997686i	$-0.521660\pi$
$439$	−2.41064e8	−0.135990	−0.0679950	+	0.997686i	$0.521660\pi$
$440$	−9.25640e8	−0.518033
$441$	2.92858e9	1.62600
$442$	−1.37956e8	−0.0759909
$443$	−1.42230e9	−0.777282	−0.388641	−	0.921389i	$-0.627055\pi$
$443$	−1.42230e9	−0.777282	−0.388641	+	0.921389i	$0.627055\pi$
$444$	7.89429e8	0.428028
$445$	1.57148e9	0.845374
$446$	−9.26269e7	−0.0494385
$447$	2.71643e9	1.43854
$448$	−1.63310e9	−0.858103
$449$	−2.97627e9	−1.55171	−0.775855	−	0.630912i	$-0.782681\pi$
$449$	−2.97627e9	−1.55171	−0.775855	+	0.630912i	$0.782681\pi$
$450$	7.79789e7	0.0403398
$451$	−2.18154e8	−0.111981
$452$	−9.37887e8	−0.477712
$453$	−4.65566e8	−0.235308
$454$	8.93513e8	0.448131
$455$	−8.30464e8	−0.413315
$456$	−2.10031e9	−1.03730
$457$	−1.87138e9	−0.917183	−0.458591	−	0.888647i	$-0.651646\pi$
$457$	−1.87138e9	−0.917183	−0.458591	+	0.888647i	$0.651646\pi$
$458$	8.94737e8	0.435177
$459$	−1.05724e8	−0.0510305
$460$	−8.67540e8	−0.415563
$461$	−1.95250e9	−0.928190	−0.464095	−	0.885785i	$-0.653620\pi$
$461$	−1.95250e9	−0.928190	−0.464095	+	0.885785i	$0.653620\pi$
$462$	1.40845e9	0.664499
$463$	3.23616e9	1.51529	0.757645	−	0.652667i	$-0.226350\pi$
$463$	3.23616e9	1.51529	0.757645	+	0.652667i	$0.226350\pi$
$464$	3.05683e9	1.42056
$465$	−1.74998e9	−0.807137
$466$	9.36642e7	0.0428769
$467$	3.81990e9	1.73557	0.867786	−	0.496939i	$-0.165543\pi$
$467$	3.81990e9	1.73557	0.867786	+	0.496939i	$0.165543\pi$
$468$	5.85143e8	0.263877
$469$	1.42071e9	0.635918
$470$	6.59263e8	0.292898
$471$	−5.61947e8	−0.247812
$472$	9.44379e8	0.413380
$473$	2.49085e8	0.108226
$474$	−1.48618e9	−0.640983
$475$	4.14465e8	0.177443
$476$	3.28647e9	1.39671
$477$	1.73169e9	0.730561
$478$	−8.02222e8	−0.335967
$479$	−4.00597e9	−1.66546	−0.832728	−	0.553682i	$-0.813222\pi$
$479$	−4.00597e9	−1.66546	−0.832728	+	0.553682i	$0.813222\pi$
$480$	−2.47969e9	−1.02342
$481$	2.21364e8	0.0906982
$482$	1.01933e9	0.414618
$483$	2.75966e9	1.11440
$484$	−3.74414e7	−0.0150104
$485$	−1.43295e9	−0.570342
$486$	1.03959e9	0.410804
$487$	−5.88400e8	−0.230845	−0.115423	−	0.993316i	$-0.536822\pi$
$487$	−5.88400e8	−0.230845	−0.115423	+	0.993316i	$0.536822\pi$
$488$	−2.15514e9	−0.839470
$489$	2.02310e9	0.782413
$490$	−1.09395e9	−0.420061
$491$	−3.55177e9	−1.35413	−0.677064	−	0.735924i	$-0.736748\pi$
$491$	−3.55177e9	−1.35413	−0.677064	+	0.735924i	$0.736748\pi$
$492$	−3.84060e8	−0.145386
$493$	−4.74193e9	−1.78234
$494$	−2.81715e8	−0.105139
$495$	−2.62548e9	−0.972950
$496$	−1.25185e9	−0.460645
$497$	1.27604e9	0.466248
$498$	−1.23444e9	−0.447884
$499$	2.12974e9	0.767318	0.383659	−	0.923475i	$-0.374664\pi$
$499$	2.12974e9	0.767318	0.383659	+	0.923475i	$0.374664\pi$
$500$	2.70538e9	0.967905
$501$	−3.39742e9	−1.20703
$502$	−1.09686e9	−0.386979
$503$	−1.46630e9	−0.513730	−0.256865	−	0.966447i	$-0.582690\pi$
$503$	−1.46630e9	−0.513730	−0.256865	+	0.966447i	$0.582690\pi$
$504$	2.63971e9	0.918437
$505$	3.51200e9	1.21349
$506$	4.12576e8	0.141572
$507$	3.22214e8	0.109804
$508$	−2.03954e9	−0.690256
$509$	−5.86398e8	−0.197097	−0.0985485	−	0.995132i	$-0.531420\pi$
$509$	−5.86398e8	−0.197097	−0.0985485	+	0.995132i	$0.531420\pi$
$510$	1.08974e9	0.363770
$511$	−5.44955e9	−1.80671
$512$	−3.04517e9	−1.00269
$513$	−2.15896e8	−0.0706046
$514$	1.16780e8	0.0379314
$515$	4.73020e9	1.52600
$516$	4.38514e8	0.140511
$517$	3.46128e9	1.10159
$518$	4.77676e8	0.151001
$519$	−3.12052e9	−0.979809
$520$	−4.56953e8	−0.142515
$521$	1.66374e9	0.515410	0.257705	−	0.966224i	$-0.417034\pi$
$521$	1.66374e9	0.515410	0.257705	+	0.966224i	$0.417034\pi$
$522$	−1.82186e9	−0.560619
$523$	2.30166e9	0.703534	0.351767	−	0.936088i	$-0.385581\pi$
$523$	2.30166e9	0.703534	0.351767	+	0.936088i	$0.385581\pi$
$524$	3.79963e9	1.15367
$525$	−1.02294e9	−0.308526
$526$	1.83953e9	0.551134
$527$	1.94194e9	0.577961
$528$	−3.68822e9	−1.09043
$529$	−2.59644e9	−0.762576
$530$	−6.46864e8	−0.188733
$531$	2.67863e9	0.776394
$532$	6.71119e9	1.93245
$533$	−1.07694e8	−0.0308069
$534$	−1.31571e9	−0.373908
$535$	1.52882e9	0.431635
$536$	7.81731e8	0.219271
$537$	8.27628e9	2.30635
$538$	−1.29245e8	−0.0357831
$539$	−5.74350e9	−1.57985
$540$	−1.67509e8	−0.0457784
$541$	−6.78529e9	−1.84238	−0.921188	−	0.389118i	$-0.872780\pi$
$541$	−6.78529e9	−1.84238	−0.921188	+	0.389118i	$0.872780\pi$
$542$	−5.98458e8	−0.161449
$543$	5.22007e9	1.39919
$544$	2.75169e9	0.732830
$545$	−4.03540e9	−1.06782
$546$	6.95298e8	0.182809
$547$	3.81983e9	0.997904	0.498952	−	0.866630i	$-0.333718\pi$
$547$	3.81983e9	0.997904	0.498952	+	0.866630i	$0.333718\pi$
$548$	−1.07597e9	−0.279297
$549$	−6.11282e9	−1.57666
$550$	−1.52932e8	−0.0391948
$551$	−9.68333e9	−2.46601
$552$	1.51847e9	0.384256
$553$	9.92783e9	2.49641
$554$	−8.45038e8	−0.211150
$555$	−1.74859e9	−0.434174
$556$	−1.06078e9	−0.261737
$557$	−1.45371e9	−0.356439	−0.178220	−	0.983991i	$-0.557034\pi$
$557$	−1.45371e9	−0.356439	−0.178220	+	0.983991i	$0.557034\pi$
$558$	7.46096e8	0.181792
$559$	1.22964e8	0.0297739
$560$	4.69270e9	1.12918
$561$	5.72137e9	1.36814
$562$	1.90129e7	0.00451826
$563$	−2.63864e9	−0.623161	−0.311580	−	0.950220i	$-0.600858\pi$
$563$	−2.63864e9	−0.623161	−0.311580	+	0.950220i	$0.600858\pi$
$564$	6.09358e9	1.43020
$565$	2.07743e9	0.484570
$566$	−2.99899e8	−0.0695213
$567$	−6.68306e9	−1.53970
$568$	7.02126e8	0.160767
$569$	−1.38350e9	−0.314837	−0.157419	−	0.987532i	$-0.550317\pi$
$569$	−1.38350e9	−0.314837	−0.157419	+	0.987532i	$0.550317\pi$
$570$	2.22532e9	0.503303
$571$	−1.97521e9	−0.444004	−0.222002	−	0.975046i	$-0.571259\pi$
$571$	−1.97521e9	−0.444004	−0.222002	+	0.975046i	$0.571259\pi$
$572$	−1.14758e9	−0.256387
$573$	2.37188e9	0.526686
$574$	−2.32391e8	−0.0512895
$575$	−2.99648e8	−0.0657317
$576$	−2.54877e9	−0.555714
$577$	3.38318e9	0.733178	0.366589	−	0.930383i	$-0.380525\pi$
$577$	3.38318e9	0.733178	0.366589	+	0.930383i	$0.380525\pi$
$578$	1.28666e8	0.0277152
$579$	−4.48037e9	−0.959267
$580$	−7.51311e9	−1.59890
$581$	8.24617e9	1.74436
$582$	1.19972e9	0.252262
$583$	−3.39618e9	−0.709824
$584$	−2.99856e9	−0.622970
$585$	−1.29610e9	−0.267666
$586$	1.46134e9	0.299991
$587$	−6.59389e9	−1.34558	−0.672789	−	0.739835i	$-0.734904\pi$
$587$	−6.59389e9	−1.34558	−0.672789	+	0.739835i	$0.734904\pi$
$588$	−1.01114e10	−2.05112
$589$	3.96557e9	0.799653
$590$	−1.00059e9	−0.200573
$591$	2.18803e9	0.436011
$592$	−1.25086e9	−0.247789
$593$	2.97850e9	0.586552	0.293276	−	0.956028i	$-0.405254\pi$
$593$	2.97850e9	0.586552	0.293276	+	0.956028i	$0.405254\pi$
$594$	7.96624e7	0.0155956
$595$	−7.27957e9	−1.41676
$596$	−4.77603e9	−0.924070
$597$	−6.03017e9	−1.15990
$598$	2.03673e8	0.0389475
$599$	1.71578e9	0.326187	0.163094	−	0.986611i	$-0.447853\pi$
$599$	1.71578e9	0.326187	0.163094	+	0.986611i	$0.447853\pi$
$600$	−5.62860e8	−0.106383
$601$	−4.20852e9	−0.790804	−0.395402	−	0.918508i	$-0.629395\pi$
$601$	−4.20852e9	−0.790804	−0.395402	+	0.918508i	$0.629395\pi$
$602$	2.65341e8	0.0495697
$603$	2.21730e9	0.411826
$604$	8.18557e8	0.151154
$605$	8.29331e7	0.0152259
$606$	−2.94039e9	−0.536724
$607$	1.61275e9	0.292690	0.146345	−	0.989234i	$-0.453249\pi$
$607$	1.61275e9	0.292690	0.146345	+	0.989234i	$0.453249\pi$
$608$	5.61913e9	1.01393
$609$	2.38994e10	4.28772
$610$	2.28341e9	0.407314
$611$	1.70870e9	0.303056
$612$	5.12917e9	0.904518
$613$	8.17364e9	1.43319	0.716595	−	0.697489i	$-0.245699\pi$
$613$	8.17364e9	1.43319	0.716595	+	0.697489i	$0.245699\pi$
$614$	−9.43806e8	−0.164548
$615$	8.50698e8	0.147473
$616$	−5.17698e9	−0.892368
$617$	−9.17589e9	−1.57272	−0.786358	−	0.617771i	$-0.788036\pi$
$617$	−9.17589e9	−1.57272	−0.786358	+	0.617771i	$0.788036\pi$
$618$	−3.96032e9	−0.674949
$619$	−2.25520e9	−0.382180	−0.191090	−	0.981573i	$-0.561202\pi$
$619$	−2.25520e9	−0.382180	−0.191090	+	0.981573i	$0.561202\pi$
$620$	3.07681e9	0.518477
$621$	1.56088e8	0.0261546
$622$	−1.34684e9	−0.224414
$623$	8.78908e9	1.45625
$624$	−1.82073e9	−0.299986
$625$	−5.16908e9	−0.846902
$626$	−1.73104e9	−0.282031
$627$	1.16834e10	1.89292
$628$	9.88014e8	0.159186
$629$	1.94040e9	0.310896
$630$	−2.79682e9	−0.445629
$631$	8.57121e9	1.35812	0.679062	−	0.734081i	$-0.262387\pi$
$631$	8.57121e9	1.35812	0.679062	+	0.734081i	$0.262387\pi$
$632$	5.46267e9	0.860787
$633$	−8.74126e8	−0.136981
$634$	6.36564e8	0.0992041
$635$	4.51762e9	0.700166
$636$	−5.97898e9	−0.921567
$637$	−2.83535e9	−0.434629
$638$	3.57301e9	0.544706
$639$	1.99151e9	0.301946
$640$	5.70677e9	0.860519
$641$	−1.25720e10	−1.88538	−0.942691	−	0.333666i	$-0.891714\pi$
$641$	−1.25720e10	−1.88538	−0.942691	+	0.333666i	$0.891714\pi$
$642$	−1.27999e9	−0.190912
$643$	−9.15051e9	−1.35740	−0.678699	−	0.734417i	$-0.737456\pi$
$643$	−9.15051e9	−1.35740	−0.678699	+	0.734417i	$0.737456\pi$
$644$	−4.85204e9	−0.715852
$645$	−9.71314e8	−0.142528
$646$	−2.46942e9	−0.360397
$647$	1.12770e9	0.163693	0.0818463	−	0.996645i	$-0.473918\pi$
$647$	1.12770e9	0.163693	0.0818463	+	0.996645i	$0.473918\pi$
$648$	−3.67728e9	−0.530902
$649$	−5.25331e9	−0.754357
$650$	−7.54965e7	−0.0107828
$651$	−9.78739e9	−1.39038
$652$	−3.55701e9	−0.502595
$653$	1.30242e10	1.83044	0.915220	−	0.402955i	$-0.132017\pi$
$653$	1.30242e10	1.83044	0.915220	+	0.402955i	$0.132017\pi$
$654$	3.37860e9	0.472296
$655$	−8.41623e9	−1.17023
$656$	6.08549e8	0.0841651
$657$	−8.50508e9	−1.17004
$658$	3.68717e9	0.504548
$659$	1.16522e10	1.58601	0.793007	−	0.609212i	$-0.208514\pi$
$659$	1.16522e10	1.58601	0.793007	+	0.609212i	$0.208514\pi$
$660$	9.06494e9	1.22733
$661$	9.54083e8	0.128493	0.0642467	−	0.997934i	$-0.479536\pi$
$661$	9.54083e8	0.128493	0.0642467	+	0.997934i	$0.479536\pi$
$662$	−6.47024e8	−0.0866798
$663$	2.82442e9	0.376385
$664$	4.53736e9	0.601471
$665$	−1.48654e10	−1.96020
$666$	7.45506e8	0.0977893
$667$	7.00083e9	0.913501
$668$	5.97334e9	0.775354
$669$	1.89639e9	0.244870
$670$	−8.28259e8	−0.106391
$671$	1.19884e10	1.53191
$672$	−1.38686e10	−1.76294
$673$	−7.36144e9	−0.930915	−0.465457	−	0.885070i	$-0.654110\pi$
$673$	−7.36144e9	−0.930915	−0.465457	+	0.885070i	$0.654110\pi$
$674$	3.01285e9	0.379025
$675$	−5.78577e7	−0.00724099
$676$	−5.66516e8	−0.0705340
$677$	9.55774e9	1.18384	0.591922	−	0.805995i	$-0.298369\pi$
$677$	9.55774e9	1.18384	0.591922	+	0.805995i	$0.298369\pi$
$678$	−1.73931e9	−0.214325
$679$	−8.01430e9	−0.982475
$680$	−4.00550e9	−0.488512
$681$	−1.82932e10	−2.21960
$682$	−1.46324e9	−0.176632
$683$	−2.52339e9	−0.303048	−0.151524	−	0.988454i	$-0.548418\pi$
$683$	−2.52339e9	−0.303048	−0.151524	+	0.988454i	$0.548418\pi$
$684$	1.04741e10	1.25147
$685$	2.38328e9	0.283307
$686$	−2.21403e9	−0.261848
$687$	−1.83183e10	−2.15544
$688$	−6.94831e8	−0.0813429
$689$	−1.67657e9	−0.195278
$690$	−1.60885e9	−0.186442
$691$	8.78987e9	1.01347	0.506733	−	0.862103i	$-0.330853\pi$
$691$	8.78987e9	1.01347	0.506733	+	0.862103i	$0.330853\pi$
$692$	5.48650e9	0.629396
$693$	−1.46840e10	−1.67601
$694$	2.56952e9	0.291806
$695$	2.34965e9	0.265494
$696$	1.31504e10	1.47844
$697$	−9.44014e8	−0.105600
$698$	1.97452e9	0.219770
$699$	−1.91762e9	−0.212370
$700$	1.79853e9	0.198186
$701$	−2.87525e7	−0.00315255	−0.00157627	−	0.999999i	$-0.500502\pi$
$701$	−2.87525e7	−0.00315255	−0.00157627	+	0.999999i	$0.500502\pi$
$702$	3.93263e7	0.00429045
$703$	3.96243e9	0.430148
$704$	4.99862e9	0.539941
$705$	−1.34974e10	−1.45073
$706$	1.00870e9	0.107882
$707$	1.96422e10	2.09036
$708$	−9.24845e9	−0.979383
$709$	−1.37312e10	−1.44693	−0.723464	−	0.690362i	$-0.757451\pi$
$709$	−1.37312e10	−1.44693	−0.723464	+	0.690362i	$0.757451\pi$
$710$	−7.43917e8	−0.0780046
$711$	1.54943e10	1.61670
$712$	4.83609e9	0.502128
$713$	−2.86701e9	−0.296221
$714$	6.09476e9	0.626632
$715$	2.54190e9	0.260068
$716$	−1.45513e10	−1.48152
$717$	1.64242e10	1.66405
$718$	−8.66536e8	−0.0873677
$719$	−2.72927e8	−0.0273839	−0.0136919	−	0.999906i	$-0.504358\pi$
$719$	−2.72927e8	−0.0273839	−0.0136919	+	0.999906i	$0.504358\pi$
$720$	7.32387e9	0.731268
$721$	2.64554e10	2.62870
$722$	−2.12818e9	−0.210440
$723$	−2.08691e10	−2.05361
$724$	−9.17792e9	−0.898792
$725$	−2.59503e9	−0.252906
$726$	−6.94350e7	−0.00673442
$727$	−7.05933e8	−0.0681386	−0.0340693	−	0.999419i	$-0.510847\pi$
$727$	−7.05933e8	−0.0681386	−0.0340693	+	0.999419i	$0.510847\pi$
$728$	−2.55568e9	−0.245497
$729$	−1.12317e10	−1.07374
$730$	3.17703e9	0.302267
$731$	1.07786e9	0.102059
$732$	2.11056e10	1.98888
$733$	6.15204e9	0.576973	0.288486	−	0.957484i	$-0.406848\pi$
$733$	6.15204e9	0.576973	0.288486	+	0.957484i	$0.406848\pi$
$734$	2.63769e9	0.246200
$735$	2.23969e10	2.08057
$736$	−4.06250e9	−0.375596
$737$	−4.34855e9	−0.400136
$738$	−3.62692e8	−0.0332155
$739$	−6.51163e9	−0.593519	−0.296759	−	0.954952i	$-0.595906\pi$
$739$	−6.51163e9	−0.593519	−0.296759	+	0.954952i	$0.595906\pi$
$740$	3.07437e9	0.278898
$741$	5.76765e9	0.520757
$742$	−3.61782e9	−0.325113
$743$	−1.93477e10	−1.73049	−0.865245	−	0.501349i	$-0.832837\pi$
$743$	−1.93477e10	−1.73049	−0.865245	+	0.501349i	$0.832837\pi$
$744$	−5.38540e9	−0.479416
$745$	1.05790e10	0.937337
$746$	−2.14376e9	−0.189056
$747$	1.28697e10	1.12966
$748$	−1.00593e10	−0.878844
$749$	8.55046e9	0.743538
$750$	5.01712e9	0.434250
$751$	−2.60822e9	−0.224700	−0.112350	−	0.993669i	$-0.535838\pi$
$751$	−2.60822e9	−0.224700	−0.112350	+	0.993669i	$0.535838\pi$
$752$	−9.65536e9	−0.827954
$753$	2.24564e10	1.91672
$754$	1.76386e9	0.149853
$755$	−1.81311e9	−0.153324
$756$	−9.36857e8	−0.0788582
$757$	−9.81364e9	−0.822233	−0.411116	−	0.911583i	$-0.634861\pi$
$757$	−9.81364e9	−0.822233	−0.411116	+	0.911583i	$0.634861\pi$
$758$	−3.85043e9	−0.321120
$759$	−8.44684e9	−0.701210
$760$	−8.17950e9	−0.675895
$761$	−5.79005e9	−0.476251	−0.238126	−	0.971234i	$-0.576533\pi$
$761$	−5.79005e9	−0.476251	−0.238126	+	0.971234i	$0.576533\pi$
$762$	−3.78233e9	−0.309683
$763$	−2.25694e10	−1.83944
$764$	−4.17024e9	−0.338325
$765$	−1.13612e10	−0.917505
$766$	−4.08527e9	−0.328413
$767$	−2.59336e9	−0.207529
$768$	4.81925e9	0.383897
$769$	−1.82078e10	−1.44383	−0.721913	−	0.691984i	$-0.756737\pi$
$769$	−1.82078e10	−1.44383	−0.721913	+	0.691984i	$0.756737\pi$
$770$	5.48511e9	0.432980
$771$	−2.39089e9	−0.187875
$772$	7.87739e9	0.616200
$773$	1.68623e10	1.31307	0.656537	−	0.754294i	$-0.272021\pi$
$773$	1.68623e10	1.31307	0.656537	+	0.754294i	$0.272021\pi$
$774$	4.14116e8	0.0321017
$775$	1.06273e9	0.0820100
$776$	−4.40978e9	−0.338767
$777$	−9.77965e9	−0.747911
$778$	−2.82251e9	−0.214886
$779$	−1.92774e9	−0.146106
$780$	4.47501e9	0.337647
$781$	−3.90573e9	−0.293375
$782$	1.78533e9	0.133504
$783$	1.35176e9	0.100631
$784$	1.60217e10	1.18741
$785$	−2.18846e9	−0.161471
$786$	7.04641e9	0.517594
$787$	2.34354e10	1.71380	0.856900	−	0.515482i	$-0.172387\pi$
$787$	2.34354e10	1.71380	0.856900	+	0.515482i	$0.172387\pi$
$788$	−3.84699e9	−0.280078
$789$	−3.76615e10	−2.72978
$790$	−5.78781e9	−0.417657
$791$	1.16188e10	0.834724
$792$	−8.07968e9	−0.577904
$793$	5.91822e9	0.421439
$794$	−3.18232e9	−0.225617
$795$	1.32435e10	0.934798
$796$	1.06022e10	0.745078
$797$	−2.70750e10	−1.89437	−0.947185	−	0.320686i	$-0.896086\pi$
$797$	−2.70750e10	−1.89437	−0.947185	+	0.320686i	$0.896086\pi$
$798$	1.24459e10	0.866994
$799$	1.49779e10	1.03881
$800$	1.50587e9	0.103985
$801$	1.37171e10	0.943077
$802$	−4.85388e9	−0.332261
$803$	1.66801e10	1.13683
$804$	−7.65562e9	−0.519498
$805$	1.07473e10	0.726130
$806$	−7.22345e8	−0.0485928
$807$	2.64609e9	0.177234
$808$	1.08079e10	0.720777
$809$	3.46307e9	0.229954	0.114977	−	0.993368i	$-0.463321\pi$
$809$	3.46307e9	0.229954	0.114977	+	0.993368i	$0.463321\pi$
$810$	3.89615e9	0.257595
$811$	−2.19790e10	−1.44689	−0.723445	−	0.690382i	$-0.757442\pi$
$811$	−2.19790e10	−1.44689	−0.723445	+	0.690382i	$0.757442\pi$
$812$	−4.20198e10	−2.75428
$813$	1.22525e10	0.799662
$814$	−1.46208e9	−0.0950136
$815$	7.87882e9	0.509811
$816$	−1.59600e10	−1.02829
$817$	2.20106e9	0.141207
$818$	−1.19449e9	−0.0763039
$819$	−7.24891e9	−0.461083
$820$	−1.49570e9	−0.0947316
$821$	1.70104e10	1.07278	0.536392	−	0.843969i	$-0.319787\pi$
$821$	1.70104e10	1.07278	0.536392	+	0.843969i	$0.319787\pi$
$822$	−1.99538e9	−0.125307
$823$	7.93771e9	0.496359	0.248179	−	0.968714i	$-0.420168\pi$
$823$	7.93771e9	0.496359	0.248179	+	0.968714i	$0.420168\pi$
$824$	1.45568e10	0.906400
$825$	3.13103e9	0.194133
$826$	−5.59616e9	−0.345509
$827$	2.72776e10	1.67702	0.838508	−	0.544890i	$-0.183428\pi$
$827$	2.72776e10	1.67702	0.838508	+	0.544890i	$0.183428\pi$
$828$	−7.57254e9	−0.463592
$829$	1.73055e10	1.05498	0.527488	−	0.849562i	$-0.323134\pi$
$829$	1.73055e10	1.05498	0.527488	+	0.849562i	$0.323134\pi$
$830$	−4.80742e9	−0.291836
$831$	1.73008e10	1.04583
$832$	2.46763e9	0.148542
$833$	−2.48537e10	−1.48982
$834$	−1.96722e9	−0.117428
$835$	−1.32310e10	−0.786486
$836$	−2.05417e10	−1.21595
$837$	−5.53578e8	−0.0326317
$838$	−2.87848e9	−0.168970
$839$	−1.68504e10	−0.985014	−0.492507	−	0.870309i	$-0.663919\pi$
$839$	−1.68504e10	−0.985014	−0.492507	+	0.870309i	$0.663919\pi$
$840$	2.01878e10	1.17520
$841$	4.33790e10	2.51474
$842$	1.90569e9	0.110017
$843$	−3.89258e8	−0.0223790
$844$	1.53689e9	0.0879920
$845$	1.25484e9	0.0715467
$846$	5.75454e9	0.326749
$847$	4.63833e8	0.0262283
$848$	9.47377e9	0.533503
$849$	6.13996e9	0.344341
$850$	−6.61778e8	−0.0369612
$851$	−2.86475e9	−0.159343
$852$	−6.87603e9	−0.380890
$853$	1.48578e10	0.819661	0.409830	−	0.912162i	$-0.365588\pi$
$853$	1.48578e10	0.819661	0.409830	+	0.912162i	$0.365588\pi$
$854$	1.27708e10	0.701642
$855$	−2.32003e10	−1.26944
$856$	4.70480e9	0.256379
$857$	−6.63563e9	−0.360122	−0.180061	−	0.983655i	$-0.557629\pi$
$857$	−6.63563e9	−0.360122	−0.180061	+	0.983655i	$0.557629\pi$
$858$	−2.12818e9	−0.115028
$859$	−1.48045e10	−0.796925	−0.398462	−	0.917185i	$-0.630456\pi$
$859$	−1.48045e10	−0.796925	−0.398462	+	0.917185i	$0.630456\pi$
$860$	1.70776e9	0.0915551
$861$	4.75784e9	0.254038
$862$	3.08736e9	0.164177
$863$	−1.21462e10	−0.643284	−0.321642	−	0.946861i	$-0.604235\pi$
$863$	−1.21462e10	−0.643284	−0.321642	+	0.946861i	$0.604235\pi$
$864$	−7.84410e8	−0.0413757
$865$	−1.21527e10	−0.638432
$866$	−5.33289e9	−0.279029
$867$	−2.63424e9	−0.137274
$868$	1.72082e10	0.893132
$869$	−3.03873e10	−1.57081
$870$	−1.39331e10	−0.717347
$871$	−2.14671e9	−0.110080
$872$	−1.24186e10	−0.634255
$873$	−1.25079e10	−0.636258
$874$	3.64577e9	0.184714
$875$	−3.35149e10	−1.69126
$876$	2.93653e10	1.47595
$877$	−2.24896e10	−1.12586	−0.562929	−	0.826505i	$-0.690326\pi$
$877$	−2.24896e10	−1.12586	−0.562929	+	0.826505i	$0.690326\pi$
$878$	7.86008e8	0.0391919
$879$	−2.99185e10	−1.48586
$880$	−1.43635e10	−0.710512
$881$	−2.25206e9	−0.110960	−0.0554798	−	0.998460i	$-0.517669\pi$
$881$	−2.25206e9	−0.110960	−0.0554798	+	0.998460i	$0.517669\pi$
$882$	−9.54885e9	−0.468609
$883$	−3.28243e10	−1.60448	−0.802239	−	0.597004i	$-0.796358\pi$
$883$	−3.28243e10	−1.60448	−0.802239	+	0.597004i	$0.796358\pi$
$884$	−4.96589e9	−0.241777
$885$	2.04854e10	0.993445
$886$	4.63752e9	0.224010
$887$	1.94755e10	0.937034	0.468517	−	0.883454i	$-0.344788\pi$
$887$	1.94755e10	0.937034	0.468517	+	0.883454i	$0.344788\pi$
$888$	−5.38114e9	−0.257887
$889$	2.52664e10	1.20611
$890$	−5.12393e9	−0.243634
$891$	2.04557e10	0.968816
$892$	−3.33422e9	−0.157296
$893$	3.05859e10	1.43728
$894$	−8.85714e9	−0.414584
$895$	3.22314e10	1.50279
$896$	3.19172e10	1.48234
$897$	−4.16988e9	−0.192908
$898$	9.70436e9	0.447198
$899$	−2.48290e10	−1.13973
$900$	2.80695e9	0.128347
$901$	−1.46962e10	−0.669374
$902$	7.11308e8	0.0322727
$903$	−5.43243e9	−0.245520
$904$	6.39310e9	0.287821
$905$	2.03292e10	0.911696
$906$	1.51801e9	0.0678152
$907$	−1.31624e10	−0.585746	−0.292873	−	0.956151i	$-0.594611\pi$
$907$	−1.31624e10	−0.585746	−0.292873	+	0.956151i	$0.594611\pi$
$908$	3.21631e10	1.42580
$909$	3.06554e10	1.35373
$910$	2.70779e9	0.119116
$911$	3.27282e10	1.43419	0.717097	−	0.696973i	$-0.245470\pi$
$911$	3.27282e10	1.43419	0.717097	+	0.696973i	$0.245470\pi$
$912$	−3.25913e10	−1.42272
$913$	−2.52400e10	−1.09760
$914$	6.10179e9	0.264329
$915$	−4.67491e10	−2.01744
$916$	3.22072e10	1.38458
$917$	−4.70708e10	−2.01585
$918$	3.44721e8	0.0147068
$919$	−1.31775e10	−0.560052	−0.280026	−	0.959992i	$-0.590343\pi$
$919$	−1.31775e10	−0.560052	−0.280026	+	0.959992i	$0.590343\pi$
$920$	5.91359e9	0.250377
$921$	1.93229e10	0.815011
$922$	6.36626e9	0.267502
$923$	−1.92811e9	−0.0807097
$924$	5.06990e10	2.11421
$925$	1.06189e9	0.0441147
$926$	−1.05517e10	−0.436702
$927$	4.12888e10	1.70237
$928$	−3.51823e10	−1.44513
$929$	3.19728e9	0.130836	0.0654178	−	0.997858i	$-0.479162\pi$
$929$	3.19728e9	0.130836	0.0654178	+	0.997858i	$0.479162\pi$
$930$	5.70594e9	0.232614
$931$	−5.07529e10	−2.06128
$932$	3.37156e9	0.136419
$933$	2.75744e10	1.11153
$934$	−1.24551e10	−0.500186
$935$	2.22815e10	0.891462
$936$	−3.98863e9	−0.158986
$937$	−2.69430e10	−1.06993	−0.534967	−	0.844873i	$-0.679676\pi$
$937$	−2.69430e10	−1.06993	−0.534967	+	0.844873i	$0.679676\pi$
$938$	−4.63234e9	−0.183270
$939$	3.54403e10	1.39691
$940$	2.37310e10	0.931899
$941$	−7.15449e9	−0.279908	−0.139954	−	0.990158i	$-0.544695\pi$
$941$	−7.15449e9	−0.279908	−0.139954	+	0.990158i	$0.544695\pi$
$942$	1.83227e9	0.0714186
$943$	1.39371e9	0.0541230
$944$	1.46543e10	0.566974
$945$	2.07515e9	0.0799904
$946$	−8.12161e8	−0.0311905
$947$	2.89770e10	1.10874	0.554368	−	0.832271i	$-0.312960\pi$
$947$	2.89770e10	1.10874	0.554368	+	0.832271i	$0.312960\pi$
$948$	−5.34968e10	−2.03938
$949$	8.23434e9	0.312750
$950$	−1.35139e9	−0.0511387
$951$	−1.30326e10	−0.491360
$952$	−2.24022e10	−0.841515
$953$	1.57016e10	0.587651	0.293826	−	0.955859i	$-0.405072\pi$
$953$	1.57016e10	0.587651	0.293826	+	0.955859i	$0.405072\pi$
$954$	−5.64632e9	−0.210545
$955$	9.23713e9	0.343183
$956$	−2.88770e10	−1.06893
$957$	−7.31517e10	−2.69794
$958$	1.30618e10	0.479979
$959$	1.33294e10	0.488027
$960$	−1.94923e10	−0.711071
$961$	−1.73445e10	−0.630420
$962$	−7.21774e8	−0.0261390
$963$	1.33447e10	0.481521
$964$	3.66919e10	1.31917
$965$	−1.74485e10	−0.625047
$966$	−8.99810e9	−0.321167
$967$	4.38024e10	1.55778	0.778888	−	0.627162i	$-0.215784\pi$
$967$	4.38024e10	1.55778	0.778888	+	0.627162i	$0.215784\pi$
$968$	2.55219e8	0.00904377
$969$	5.05574e10	1.78505
$970$	4.67225e9	0.164371
$971$	2.65581e10	0.930957	0.465479	−	0.885059i	$-0.345882\pi$
$971$	2.65581e10	0.930957	0.465479	+	0.885059i	$0.345882\pi$
$972$	3.74213e10	1.30704
$973$	1.31412e10	0.457343
$974$	1.91852e9	0.0665289
$975$	1.54567e9	0.0534073
$976$	−3.34421e10	−1.15138
$977$	4.86322e10	1.66837	0.834186	−	0.551484i	$-0.185938\pi$
$977$	4.86322e10	1.66837	0.834186	+	0.551484i	$0.185938\pi$
$978$	−6.59647e9	−0.225489
$979$	−2.69018e10	−0.916309
$980$	−3.93783e10	−1.33649
$981$	−3.52240e10	−1.19123
$982$	1.15808e10	0.390256
$983$	5.65601e9	0.189921	0.0949605	−	0.995481i	$-0.469728\pi$
$983$	5.65601e9	0.189921	0.0949605	+	0.995481i	$0.469728\pi$
$984$	2.61795e9	0.0875947
$985$	8.52113e9	0.284100
$986$	1.54614e10	0.513665
$987$	−7.54889e10	−2.49904
$988$	−1.01407e10	−0.334517
$989$	−1.59132e9	−0.0523082
$990$	8.56058e9	0.280401
$991$	−1.64542e10	−0.537054	−0.268527	−	0.963272i	$-0.586537\pi$
$991$	−1.64542e10	−0.537054	−0.268527	+	0.963272i	$0.586537\pi$
$992$	1.44080e10	0.468612
$993$	1.32468e10	0.429327
$994$	−4.16063e9	−0.134371
$995$	−2.34841e10	−0.755775
$996$	−4.44351e10	−1.42501
$997$	−5.55883e10	−1.77644	−0.888219	−	0.459420i	$-0.848057\pi$
$997$	−5.55883e10	−1.77644	−0.888219	+	0.459420i	$0.848057\pi$
$998$	−6.94419e9	−0.221139
$999$	−5.53140e8	−0.0175532

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.8.a.c.1.2	✓	4
3.2	odd	2		117.8.a.e.1.3		4
4.3	odd	2		208.8.a.k.1.2		4
5.4	even	2		325.8.a.c.1.3		4
13.5	odd	4		169.8.b.c.168.5		8
13.8	odd	4		169.8.b.c.168.4		8
13.12	even	2		169.8.a.c.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.a.c.1.2	✓	4	1.1	even	1	trivial
117.8.a.e.1.3		4	3.2	odd	2
169.8.a.c.1.3		4	13.12	even	2
169.8.b.c.168.4		8	13.8	odd	4
169.8.b.c.168.5		8	13.5	odd	4
208.8.a.k.1.2		4	4.3	odd	2
325.8.a.c.1.3		4	5.4	even	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:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	13.a (trivial)

\(f(q)\)	\(=\)	\(q-3.26058 q^{2} +66.7551 q^{3} -117.369 q^{4} +259.973 q^{5} -217.660 q^{6} +1453.99 q^{7} +800.043 q^{8} +2269.24 q^{9} -847.662 q^{10} -4450.41 q^{11} -7834.95 q^{12} -2197.00 q^{13} -4740.86 q^{14} +17354.5 q^{15} +12414.6 q^{16} -19258.2 q^{17} -7399.03 q^{18} -39326.5 q^{19} -30512.7 q^{20} +97061.4 q^{21} +14510.9 q^{22} +28432.1 q^{23} +53406.9 q^{24} -10539.1 q^{25} +7163.49 q^{26} +5489.83 q^{27} -170653. q^{28} +246229. q^{29} -56585.7 q^{30} -100837. q^{31} -142884. q^{32} -297088. q^{33} +62792.8 q^{34} +377999. q^{35} -266337. q^{36} -100757. q^{37} +128227. q^{38} -146661. q^{39} +207990. q^{40} +49018.9 q^{41} -316476. q^{42} -55968.9 q^{43} +522339. q^{44} +589941. q^{45} -92705.2 q^{46} -777743. q^{47} +828736. q^{48} +1.29055e6 q^{49} +34363.5 q^{50} -1.28558e6 q^{51} +257859. q^{52} +763116. q^{53} -17900.0 q^{54} -1.15699e6 q^{55} +1.16326e6 q^{56} -2.62524e6 q^{57} -802850. q^{58} +1.18041e6 q^{59} -2.03688e6 q^{60} -2.69377e6 q^{61} +328787. q^{62} +3.29946e6 q^{63} -1.12318e6 q^{64} -571161. q^{65} +968677. q^{66} +977111. q^{67} +2.26031e6 q^{68} +1.89799e6 q^{69} -1.23249e6 q^{70} +877610. q^{71} +1.81549e6 q^{72} -3.74799e6 q^{73} +328527. q^{74} -703537. q^{75} +4.61569e6 q^{76} -6.47087e6 q^{77} +478199. q^{78} +6.82797e6 q^{79} +3.22746e6 q^{80} -4.59635e6 q^{81} -159830. q^{82} +5.67139e6 q^{83} -1.13920e7 q^{84} -5.00661e6 q^{85} +182491. q^{86} +1.64371e7 q^{87} -3.56052e6 q^{88} +6.04479e6 q^{89} -1.92355e6 q^{90} -3.19442e6 q^{91} -3.33704e6 q^{92} -6.73138e6 q^{93} +2.53589e6 q^{94} -1.02238e7 q^{95} -9.53825e6 q^{96} -5.51192e6 q^{97} -4.20795e6 q^{98} -1.00990e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 15 q^{2} + 80 q^{3} + 253 q^{4} + 258 q^{5} + 1579 q^{6} + 1692 q^{7} + 1893 q^{8} + 3494 q^{9} - 4495 q^{10} + 1836 q^{11} - 3655 q^{12} - 8788 q^{13} - 18285 q^{14} - 29736 q^{15} - 36159 q^{16}+ \cdots + 6200852 q^{99}+O(q^{100}) \) 4 * q + 15 * q^2 + 80 * q^3 + 253 * q^4 + 258 * q^5 + 1579 * q^6 + 1692 * q^7 + 1893 * q^8 + 3494 * q^9 - 4495 * q^10 + 1836 * q^11 - 3655 * q^12 - 8788 * q^13 - 18285 * q^14 - 29736 * q^15 - 36159 * q^16 + 11814 * q^17 - 2738 * q^18 + 27660 * q^19 - 30369 * q^20 + 69930 * q^21 + 59930 * q^22 + 172920 * q^23 + 26553 * q^24 + 219414 * q^25 - 32955 * q^26 + 174800 * q^27 - 170903 * q^28 + 133344 * q^29 - 624695 * q^30 - 231748 * q^31 - 459291 * q^32 - 33868 * q^33 - 480249 * q^34 + 322296 * q^35 - 157930 * q^36 + 248026 * q^37 - 85326 * q^38 - 175760 * q^39 + 379895 * q^40 + 588108 * q^41 + 132275 * q^42 + 309304 * q^43 + 1270806 * q^44 - 1773144 * q^45 + 574948 * q^46 + 557916 * q^47 + 837429 * q^48 - 767378 * q^49 + 5737920 * q^50 - 1816496 * q^51 - 555841 * q^52 + 2022348 * q^53 - 479 * q^54 - 3770208 * q^55 + 1391625 * q^56 - 5609844 * q^57 - 3027194 * q^58 + 1162668 * q^59 - 8100077 * q^60 - 1340572 * q^61 + 2715564 * q^62 + 4213476 * q^63 - 6734935 * q^64 - 566826 * q^65 + 8124226 * q^66 - 598484 * q^67 + 4515993 * q^68 - 108848 * q^69 - 4329415 * q^70 + 697860 * q^71 + 1607430 * q^72 - 13725816 * q^73 + 14884053 * q^74 + 16604592 * q^75 + 12055070 * q^76 - 5326932 * q^77 - 3469063 * q^78 + 20079576 * q^79 + 3710667 * q^80 - 8368396 * q^81 + 857160 * q^82 - 2024724 * q^83 - 13603483 * q^84 - 16146514 * q^85 - 22922499 * q^86 + 22170952 * q^87 - 2100618 * q^88 + 17646240 * q^89 - 42372890 * q^90 - 3717324 * q^91 + 14504892 * q^92 - 17910680 * q^93 - 3729785 * q^94 + 12508152 * q^95 - 37637751 * q^96 - 6329096 * q^97 - 22166262 * q^98 + 6200852 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more