LMFDB - Embedded newform 13.8.a.c.1.3

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			1.3
Root			$-12.1994$ 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)

$f(q)$	$=$	\(q+16.1994 q^{2} +71.3782 q^{3} +134.421 q^{4} -532.467 q^{5} +1156.29 q^{6} -6.33667 q^{7} +104.021 q^{8} +2907.85 q^{9} -8625.66 q^{10} +4955.31 q^{11} +9594.75 q^{12} -2197.00 q^{13} -102.650 q^{14} -38006.6 q^{15} -15520.8 q^{16} +15165.4 q^{17} +47105.4 q^{18} -8043.90 q^{19} -71574.9 q^{20} -452.300 q^{21} +80273.2 q^{22} +18427.3 q^{23} +7424.84 q^{24} +205396. q^{25} -35590.1 q^{26} +51452.8 q^{27} -851.783 q^{28} +69407.9 q^{29} -615684. q^{30} -236034. q^{31} -264743. q^{32} +353701. q^{33} +245670. q^{34} +3374.07 q^{35} +390877. q^{36} +35408.8 q^{37} -130307. q^{38} -156818. q^{39} -55387.8 q^{40} +72574.1 q^{41} -7327.00 q^{42} +258384. q^{43} +666100. q^{44} -1.54833e6 q^{45} +298511. q^{46} +379959. q^{47} -1.10785e6 q^{48} -823503. q^{49} +3.32730e6 q^{50} +1.08248e6 q^{51} -295324. q^{52} -84188.0 q^{53} +833505. q^{54} -2.63854e6 q^{55} -659.147 q^{56} -574159. q^{57} +1.12437e6 q^{58} +1.65041e6 q^{59} -5.10889e6 q^{60} +1.68652e6 q^{61} -3.82361e6 q^{62} -18426.1 q^{63} -2.30202e6 q^{64} +1.16983e6 q^{65} +5.72976e6 q^{66} -2.29784e6 q^{67} +2.03855e6 q^{68} +1.31530e6 q^{69} +54658.0 q^{70} -1.65536e6 q^{71} +302477. q^{72} -6.62342e6 q^{73} +573602. q^{74} +1.46608e7 q^{75} -1.08127e6 q^{76} -31400.2 q^{77} -2.54036e6 q^{78} +877934. q^{79} +8.26434e6 q^{80} -2.68686e6 q^{81} +1.17566e6 q^{82} -5.80198e6 q^{83} -60798.7 q^{84} -8.07506e6 q^{85} +4.18567e6 q^{86} +4.95421e6 q^{87} +515457. q^{88} +4.14974e6 q^{89} -2.50821e7 q^{90} +13921.7 q^{91} +2.47702e6 q^{92} -1.68476e7 q^{93} +6.15511e6 q^{94} +4.28312e6 q^{95} -1.88969e7 q^{96} +1.51387e7 q^{97} -1.33403e7 q^{98} +1.44093e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	16.1994	1.43184	0.715920	−	0.698182i	$-0.246007\pi$
$2$	16.1994	1.43184	0.715920	+	0.698182i	$0.246007\pi$
$3$	71.3782	1.52630	0.763152	−	0.646219i	$-0.223651\pi$
$3$	71.3782	1.52630	0.763152	+	0.646219i	$0.223651\pi$
$4$	134.421	1.05017
$5$	−532.467	−1.90501	−0.952507	−	0.304518i	$-0.901505\pi$
$5$	−532.467	−1.90501	−0.952507	+	0.304518i	$0.901505\pi$
$6$	1156.29	2.18542
$7$	−6.33667	−0.00698261	−0.00349131	−	0.999994i	$-0.501111\pi$
$7$	−6.33667	−0.00698261	−0.00349131	+	0.999994i	$0.501111\pi$
$8$	104.021	0.0718301
$9$	2907.85	1.32961
$10$	−8625.66	−2.72767
$11$	4955.31	1.12253	0.561263	−	0.827637i	$-0.310315\pi$
$11$	4955.31	1.12253	0.561263	+	0.827637i	$0.310315\pi$
$12$	9594.75	1.60287
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	−102.650	−0.00999798
$15$	−38006.6	−2.90763
$16$	−15520.8	−0.947317
$17$	15165.4	0.748655	0.374328	−	0.927297i	$-0.377874\pi$
$17$	15165.4	0.748655	0.374328	+	0.927297i	$0.377874\pi$
$18$	47105.4	1.90378
$19$	−8043.90	−0.269048	−0.134524	−	0.990910i	$-0.542950\pi$
$19$	−8043.90	−0.269048	−0.134524	+	0.990910i	$0.542950\pi$
$20$	−71574.9	−2.00058
$21$	−452.300	−0.0106576
$22$	80273.2	1.60728
$23$	18427.3	0.315801	0.157900	−	0.987455i	$-0.449527\pi$
$23$	18427.3	0.315801	0.157900	+	0.987455i	$0.449527\pi$
$24$	7424.84	0.109635
$25$	205396.	2.62908
$26$	−35590.1	−0.397121
$27$	51452.8	0.503078
$28$	−851.783	−0.00733290
$29$	69407.9	0.528465	0.264232	−	0.964459i	$-0.414881\pi$
$29$	69407.9	0.528465	0.264232	+	0.964459i	$0.414881\pi$
$30$	−615684.	−4.16326
$31$	−236034.	−1.42301	−0.711505	−	0.702681i	$-0.751986\pi$
$31$	−236034.	−1.42301	−0.711505	+	0.702681i	$0.751986\pi$
$32$	−264743.	−1.42824
$33$	353701.	1.71332
$34$	245670.	1.07195
$35$	3374.07	0.0133020
$36$	390877.	1.39631
$37$	35408.8	0.114923	0.0574613	−	0.998348i	$-0.481699\pi$
$37$	35408.8	0.114923	0.0574613	+	0.998348i	$0.481699\pi$
$38$	−130307.	−0.385233
$39$	−156818.	−0.423321
$40$	−55387.8	−0.136837
$41$	72574.1	0.164452	0.0822259	−	0.996614i	$-0.473797\pi$
$41$	72574.1	0.164452	0.0822259	+	0.996614i	$0.473797\pi$
$42$	−7327.00	−0.0152600
$43$	258384.	0.495594	0.247797	−	0.968812i	$-0.420293\pi$
$43$	258384.	0.495594	0.247797	+	0.968812i	$0.420293\pi$
$44$	666100.	1.17884
$45$	−1.54833e6	−2.53292
$46$	298511.	0.452176
$47$	379959.	0.533819	0.266910	−	0.963722i	$-0.413997\pi$
$47$	379959.	0.533819	0.266910	+	0.963722i	$0.413997\pi$
$48$	−1.10785e6	−1.44589
$49$	−823503.	−0.999951
$50$	3.32730e6	3.76442
$51$	1.08248e6	1.14268
$52$	−295324.	−0.291264
$53$	−84188.0	−0.0776756	−0.0388378	−	0.999246i	$-0.512366\pi$
$53$	−84188.0	−0.0776756	−0.0388378	+	0.999246i	$0.512366\pi$
$54$	833505.	0.720328
$55$	−2.63854e6	−2.13843
$56$	−659.147	−0.000501561 0
$57$	−574159.	−0.410649
$58$	1.12437e6	0.756677
$59$	1.65041e6	1.04619	0.523095	−	0.852274i	$-0.324777\pi$
$59$	1.65041e6	1.04619	0.523095	+	0.852274i	$0.324777\pi$
$60$	−5.10889e6	−3.05350
$61$	1.68652e6	0.951342	0.475671	−	0.879623i	$-0.342205\pi$
$61$	1.68652e6	0.951342	0.475671	+	0.879623i	$0.342205\pi$
$62$	−3.82361e6	−2.03752
$63$	−18426.1	−0.00928412
$64$	−2.30202e6	−1.09769
$65$	1.16983e6	0.528356
$66$	5.72976e6	2.45320
$67$	−2.29784e6	−0.933380	−0.466690	−	0.884421i	$-0.654554\pi$
$67$	−2.29784e6	−0.933380	−0.466690	+	0.884421i	$0.654554\pi$
$68$	2.03855e6	0.786212
$69$	1.31530e6	0.482008
$70$	54658.0	0.0190463
$71$	−1.65536e6	−0.548893	−0.274447	−	0.961602i	$-0.588495\pi$
$71$	−1.65536e6	−0.548893	−0.274447	+	0.961602i	$0.588495\pi$
$72$	302477.	0.0955057
$73$	−6.62342e6	−1.99275	−0.996374	−	0.0850869i	$-0.972883\pi$
$73$	−6.62342e6	−1.99275	−0.996374	+	0.0850869i	$0.972883\pi$
$74$	573602.	0.164551
$75$	1.46608e7	4.01277
$76$	−1.08127e6	−0.282545
$77$	−31400.2	−0.00783817
$78$	−2.54036e6	−0.606128
$79$	877934.	0.200340	0.100170	−	0.994970i	$-0.468061\pi$
$79$	877934.	0.200340	0.100170	+	0.994970i	$0.468061\pi$
$80$	8.26434e6	1.80465
$81$	−2.68686e6	−0.561755
$82$	1.17566e6	0.235469
$83$	−5.80198e6	−1.11379	−0.556895	−	0.830583i	$-0.688007\pi$
$83$	−5.80198e6	−1.11379	−0.556895	+	0.830583i	$0.688007\pi$
$84$	−60798.7	−0.0111922
$85$	−8.07506e6	−1.42620
$86$	4.18567e6	0.709611
$87$	4.95421e6	0.806598
$88$	515457.	0.0806312
$89$	4.14974e6	0.623959	0.311980	−	0.950089i	$-0.399008\pi$
$89$	4.14974e6	0.623959	0.311980	+	0.950089i	$0.399008\pi$
$90$	−2.50821e7	−3.62673
$91$	13921.7	0.00193663
$92$	2.47702e6	0.331643
$93$	−1.68476e7	−2.17195
$94$	6.15511e6	0.764344
$95$	4.28312e6	0.512539
$96$	−1.88969e7	−2.17992
$97$	1.51387e7	1.68418	0.842088	−	0.539340i	$-0.181326\pi$
$97$	1.51387e7	1.68418	0.842088	+	0.539340i	$0.181326\pi$
$98$	−1.33403e7	−1.43177
$99$	1.44093e7	1.49252
$100$	2.76097e7	2.76097
$101$	−7.98328e6	−0.771004	−0.385502	−	0.922707i	$-0.625972\pi$
$101$	−7.98328e6	−0.771004	−0.385502	+	0.922707i	$0.625972\pi$
$102$	1.75355e7	1.63613
$103$	−8.43494e6	−0.760592	−0.380296	−	0.924865i	$-0.624178\pi$
$103$	−8.43494e6	−0.760592	−0.380296	+	0.924865i	$0.624178\pi$
$104$	−228534.	−0.0199221
$105$	240835.	0.0203028
$106$	−1.36380e6	−0.111219
$107$	2.09587e7	1.65394	0.826971	−	0.562245i	$-0.190062\pi$
$107$	2.09587e7	1.65394	0.826971	+	0.562245i	$0.190062\pi$
$108$	6.91635e6	0.528316
$109$	8.90985e6	0.658989	0.329494	−	0.944158i	$-0.393122\pi$
$109$	8.90985e6	0.658989	0.329494	+	0.944158i	$0.393122\pi$
$110$	−4.27429e7	−3.06189
$111$	2.52742e6	0.175407
$112$	98350.4	0.00661475
$113$	460673.	0.0300343	0.0150172	−	0.999887i	$-0.495220\pi$
$113$	460673.	0.0300343	0.0150172	+	0.999887i	$0.495220\pi$
$114$	−9.30105e6	−0.587983
$115$	−9.81191e6	−0.601605
$116$	9.32990e6	0.554976
$117$	−6.38854e6	−0.368766
$118$	2.67357e7	1.49798
$119$	−96097.9	−0.00522757
$120$	−3.95348e6	−0.208855
$121$	5.06796e6	0.260066
$122$	2.73206e7	1.36217
$123$	5.18021e6	0.251003
$124$	−3.17279e7	−1.49440
$125$	−6.77679e7	−3.10341
$126$	−298492.	−0.0132934
$127$	1.45973e7	0.632355	0.316178	−	0.948700i	$-0.397600\pi$
$127$	1.45973e7	0.632355	0.316178	+	0.948700i	$0.397600\pi$
$128$	−3.40427e6	−0.143479
$129$	1.84430e7	0.756427
$130$	1.89506e7	0.756521
$131$	1.38092e7	0.536686	0.268343	−	0.963323i	$-0.413524\pi$
$131$	1.38092e7	0.536686	0.268343	+	0.963323i	$0.413524\pi$
$132$	4.75450e7	1.79927
$133$	50971.5	0.00187866
$134$	−3.72237e7	−1.33645
$135$	−2.73969e7	−0.958371
$136$	1.57752e6	0.0537759
$137$	9.14763e6	0.303939	0.151970	−	0.988385i	$-0.451438\pi$
$137$	9.14763e6	0.303939	0.151970	+	0.988385i	$0.451438\pi$
$138$	2.13072e7	0.690159
$139$	−4.36263e7	−1.37783	−0.688917	−	0.724841i	$-0.741913\pi$
$139$	−4.36263e7	−1.37783	−0.688917	+	0.724841i	$0.741913\pi$
$140$	453547.	0.0139693
$141$	2.71208e7	0.814771
$142$	−2.68159e7	−0.785928
$143$	−1.08868e7	−0.311333
$144$	−4.51322e7	−1.25956
$145$	−3.69575e7	−1.00673
$146$	−1.07296e8	−2.85330
$147$	−5.87801e7	−1.52623
$148$	4.75970e6	0.120688
$149$	5.93143e7	1.46895	0.734476	−	0.678635i	$-0.237428\pi$
$149$	5.93143e7	1.46895	0.734476	+	0.678635i	$0.237428\pi$
$150$	2.37497e8	5.74564
$151$	6.80038e6	0.160736	0.0803681	−	0.996765i	$-0.474390\pi$
$151$	6.80038e6	0.160736	0.0803681	+	0.996765i	$0.474390\pi$
$152$	−836735.	−0.0193257
$153$	4.40986e7	0.995416
$154$	−508665.	−0.0112230
$155$	1.25680e8	2.71085
$156$	−2.10797e7	−0.444557
$157$	−1.54043e7	−0.317682	−0.158841	−	0.987304i	$-0.550776\pi$
$157$	−1.54043e7	−0.317682	−0.158841	+	0.987304i	$0.550776\pi$
$158$	1.42220e7	0.286854
$159$	−6.00919e6	−0.118557
$160$	1.40967e8	2.72081
$161$	−116767.	−0.00220511
$162$	−4.35255e7	−0.804343
$163$	−8.64806e7	−1.56409	−0.782045	−	0.623222i	$-0.785823\pi$
$163$	−8.64806e7	−1.56409	−0.782045	+	0.623222i	$0.785823\pi$
$164$	9.75551e6	0.172702
$165$	−1.88334e8	−3.26389
$166$	−9.39888e7	−1.59477
$167$	3.31038e7	0.550010	0.275005	−	0.961443i	$-0.411321\pi$
$167$	3.31038e7	0.550010	0.275005	+	0.961443i	$0.411321\pi$
$168$	−47048.7	−0.000765535 0
$169$	4.82681e6	0.0769231
$170$	−1.30811e8	−2.04209
$171$	−2.33904e7	−0.357727
$172$	3.47323e7	0.520456
$173$	3.03784e7	0.446071	0.223036	−	0.974810i	$-0.428403\pi$
$173$	3.03784e7	0.446071	0.223036	+	0.974810i	$0.428403\pi$
$174$	8.02554e7	1.15492
$175$	−1.30153e6	−0.0183578
$176$	−7.69106e7	−1.06339
$177$	1.17804e8	1.59681
$178$	6.72234e7	0.893410
$179$	4.63505e7	0.604044	0.302022	−	0.953301i	$-0.402338\pi$
$179$	4.63505e7	0.604044	0.302022	+	0.953301i	$0.402338\pi$
$180$	−2.08129e8	−2.65998
$181$	−5.98710e7	−0.750484	−0.375242	−	0.926927i	$-0.622440\pi$
$181$	−5.98710e7	−0.750484	−0.375242	+	0.926927i	$0.622440\pi$
$182$	225523.	0.00277294
$183$	1.20381e8	1.45204
$184$	1.91682e6	0.0226840
$185$	−1.88540e7	−0.218929
$186$	−2.72922e8	−3.10988
$187$	7.51491e7	0.840385
$188$	5.10746e7	0.560599
$189$	−326039.	−0.00351280
$190$	6.93840e7	0.733874
$191$	8.28798e7	0.860660	0.430330	−	0.902672i	$-0.358397\pi$
$191$	8.28798e7	0.860660	0.430330	+	0.902672i	$0.358397\pi$
$192$	−1.64314e8	−1.67541
$193$	9.24163e7	0.925333	0.462666	−	0.886532i	$-0.346893\pi$
$193$	9.24163e7	0.925333	0.462666	+	0.886532i	$0.346893\pi$
$194$	2.45238e8	2.41147
$195$	8.35004e7	0.806432
$196$	−1.10696e8	−1.05012
$197$	6.38854e7	0.595347	0.297673	−	0.954668i	$-0.403789\pi$
$197$	6.38854e7	0.595347	0.297673	+	0.954668i	$0.403789\pi$
$198$	2.33422e8	2.13705
$199$	−4.29220e7	−0.386095	−0.193047	−	0.981189i	$-0.561837\pi$
$199$	−4.29220e7	−0.386095	−0.193047	+	0.981189i	$0.561837\pi$
$200$	2.13656e7	0.188847
$201$	−1.64016e8	−1.42462
$202$	−1.29324e8	−1.10395
$203$	−439815.	−0.00369006
$204$	1.45508e8	1.20000
$205$	−3.86434e7	−0.313283
$206$	−1.36641e8	−1.08905
$207$	5.35836e7	0.419891
$208$	3.40993e7	0.262738
$209$	−3.98601e7	−0.302013
$210$	3.90139e6	0.0290704
$211$	1.45303e8	1.06484	0.532422	−	0.846479i	$-0.321282\pi$
$211$	1.45303e8	1.06484	0.532422	+	0.846479i	$0.321282\pi$
$212$	−1.13167e7	−0.0815723
$213$	−1.18157e8	−0.837778
$214$	3.39518e8	2.36818
$215$	−1.37581e8	−0.944112
$216$	5.35217e6	0.0361361
$217$	1.49567e6	0.00993632
$218$	1.44334e8	0.943566
$219$	−4.72768e8	−3.04154
$220$	−3.54676e8	−2.24571
$221$	−3.33183e7	−0.207640
$222$	4.09427e7	0.251154
$223$	−2.34165e8	−1.41402	−0.707008	−	0.707205i	$-0.749956\pi$
$223$	−2.34165e8	−1.41402	−0.707008	+	0.707205i	$0.749956\pi$
$224$	1.67759e6	0.00997282
$225$	5.97262e8	3.49563
$226$	7.46263e6	0.0430044
$227$	−1.93730e8	−1.09928	−0.549638	−	0.835403i	$-0.685235\pi$
$227$	−1.93730e8	−1.09928	−0.549638	+	0.835403i	$0.685235\pi$
$228$	−7.71792e7	−0.431249
$229$	3.22985e8	1.77729	0.888643	−	0.458599i	$-0.151649\pi$
$229$	3.22985e8	1.77729	0.888643	+	0.458599i	$0.151649\pi$
$230$	−1.58947e8	−0.861402
$231$	−2.24129e6	−0.0119634
$232$	7.21988e6	0.0379597
$233$	−2.21714e8	−1.14828	−0.574139	−	0.818758i	$-0.694663\pi$
$233$	−2.21714e8	−1.14828	−0.574139	+	0.818758i	$0.694663\pi$
$234$	−1.03491e8	−0.528014
$235$	−2.02316e8	−1.01693
$236$	2.21851e8	1.09867
$237$	6.26653e7	0.305779
$238$	−1.55673e6	−0.00748504
$239$	−9.88382e7	−0.468309	−0.234154	−	0.972199i	$-0.575232\pi$
$239$	−9.88382e7	−0.468309	−0.234154	+	0.972199i	$0.575232\pi$
$240$	5.89894e8	2.75445
$241$	1.44036e8	0.662843	0.331421	−	0.943483i	$-0.392472\pi$
$241$	1.44036e8	0.662843	0.331421	+	0.943483i	$0.392472\pi$
$242$	8.20980e7	0.372373
$243$	−3.04310e8	−1.36049
$244$	2.26704e8	0.999067
$245$	4.38488e8	1.90492
$246$	8.39164e7	0.359397
$247$	1.76725e7	0.0746204
$248$	−2.45525e7	−0.102215
$249$	−4.14135e8	−1.69998
$250$	−1.09780e9	−4.44359
$251$	−1.29025e8	−0.515011	−0.257506	−	0.966277i	$-0.582901\pi$
$251$	−1.29025e8	−0.515011	−0.257506	+	0.966277i	$0.582901\pi$
$252$	−2.47685e6	−0.00974987
$253$	9.13128e7	0.354495
$254$	2.36469e8	0.905431
$255$	−5.76384e8	−2.17681
$256$	2.39512e8	0.892250
$257$	1.38838e8	0.510202	0.255101	−	0.966914i	$-0.417891\pi$
$257$	1.38838e8	0.510202	0.255101	+	0.966914i	$0.417891\pi$
$258$	2.98765e8	1.08308
$259$	−224374.	−0.000802459 0
$260$	1.57250e8	0.554861
$261$	2.01828e8	0.702650
$262$	2.23702e8	0.768448
$263$	5.75906e8	1.95212	0.976060	−	0.217501i	$-0.0697905\pi$
$263$	5.75906e8	1.95212	0.976060	+	0.217501i	$0.0697905\pi$
$264$	3.67924e7	0.123068
$265$	4.48273e7	0.147973
$266$	825710.	0.00268993
$267$	2.96201e8	0.952352
$268$	−3.08879e8	−0.980204
$269$	7.31135e7	0.229015	0.114508	−	0.993422i	$-0.463471\pi$
$269$	7.31135e7	0.229015	0.114508	+	0.993422i	$0.463471\pi$
$270$	−4.43814e8	−1.37223
$271$	−2.89317e7	−0.0883041	−0.0441521	−	0.999025i	$-0.514059\pi$
$271$	−2.89317e7	−0.0883041	−0.0441521	+	0.999025i	$0.514059\pi$
$272$	−2.35379e8	−0.709214
$273$	993703.	0.00295588
$274$	1.48186e8	0.435192
$275$	1.01780e9	2.95121
$276$	1.76805e8	0.506189
$277$	1.31878e8	0.372816	0.186408	−	0.982472i	$-0.440315\pi$
$277$	1.31878e8	0.372816	0.186408	+	0.982472i	$0.440315\pi$
$278$	−7.06721e8	−1.97284
$279$	−6.86349e8	−1.89204
$280$	350974.	0.000955481 0
$281$	−4.01620e8	−1.07980	−0.539900	−	0.841729i	$-0.681538\pi$
$281$	−4.01620e8	−1.07980	−0.539900	+	0.841729i	$0.681538\pi$
$282$	4.39341e8	1.16662
$283$	1.88671e8	0.494826	0.247413	−	0.968910i	$-0.420420\pi$
$283$	1.88671e8	0.494826	0.247413	+	0.968910i	$0.420420\pi$
$284$	−2.22515e8	−0.576429
$285$	3.05721e8	0.782291
$286$	−1.76360e8	−0.445779
$287$	−459878.	−0.00114830
$288$	−7.69833e8	−1.89899
$289$	−1.80350e8	−0.439516
$290$	−5.98689e8	−1.44148
$291$	1.08057e9	2.57057
$292$	−8.90328e8	−2.09272
$293$	−6.51697e8	−1.51359	−0.756796	−	0.653651i	$-0.773236\pi$
$293$	−6.51697e8	−1.51359	−0.756796	+	0.653651i	$0.773236\pi$
$294$	−9.52204e8	−2.18532
$295$	−8.78791e8	−1.99301
$296$	3.68326e6	0.00825489
$297$	2.54965e8	0.564719
$298$	9.60858e8	2.10330
$299$	−4.04847e7	−0.0875874
$300$	1.97073e9	4.21407
$301$	−1.63729e6	−0.00346054
$302$	1.10162e8	0.230149
$303$	−5.69832e8	−1.17679
$304$	1.24848e8	0.254873
$305$	−8.98016e8	−1.81232
$306$	7.14371e8	1.42528
$307$	7.24498e8	1.42907	0.714534	−	0.699601i	$-0.246639\pi$
$307$	7.24498e8	1.42907	0.714534	+	0.699601i	$0.246639\pi$
$308$	−4.22085e6	−0.00823138
$309$	−6.02071e8	−1.16090
$310$	2.03595e9	3.88151
$311$	−6.51271e8	−1.22772	−0.613861	−	0.789414i	$-0.710385\pi$
$311$	−6.51271e8	−1.22772	−0.613861	+	0.789414i	$0.710385\pi$
$312$	−1.63124e7	−0.0304072
$313$	−5.50326e8	−1.01441	−0.507207	−	0.861824i	$-0.669322\pi$
$313$	−5.50326e8	−1.01441	−0.507207	+	0.861824i	$0.669322\pi$
$314$	−2.49541e8	−0.454870
$315$	9.81128e6	0.0176864
$316$	1.18013e8	0.210390
$317$	8.55988e8	1.50925	0.754624	−	0.656158i	$-0.227819\pi$
$317$	8.55988e8	1.50925	0.754624	+	0.656158i	$0.227819\pi$
$318$	−9.73453e7	−0.169754
$319$	3.43938e8	0.593216
$320$	1.22575e9	2.09111
$321$	1.49599e9	2.52442
$322$	−1.89156e6	−0.00315737
$323$	−1.21989e8	−0.201424
$324$	−3.61171e8	−0.589936
$325$	−4.51256e8	−0.729174
$326$	−1.40094e9	−2.23953
$327$	6.35969e8	1.00582
$328$	7.54924e6	0.0118126
$329$	−2.40767e6	−0.00372745
$330$	−3.05091e9	−4.67337
$331$	9.42127e8	1.42795	0.713973	−	0.700173i	$-0.246894\pi$
$331$	9.42127e8	1.42795	0.713973	+	0.700173i	$0.246894\pi$
$332$	−7.79910e8	−1.16966
$333$	1.02963e8	0.152802
$334$	5.36262e8	0.787526
$335$	1.22353e9	1.77810
$336$	7.02008e6	0.0100961
$337$	−6.21108e8	−0.884020	−0.442010	−	0.897010i	$-0.645734\pi$
$337$	−6.21108e8	−0.884020	−0.442010	+	0.897010i	$0.645734\pi$
$338$	7.81915e7	0.110142
$339$	3.28820e7	0.0458415
$340$	−1.08546e9	−1.49774
$341$	−1.16962e9	−1.59737
$342$	−3.78912e8	−0.512208
$343$	1.04368e7	0.0139649
$344$	2.68774e7	0.0355985
$345$	−7.00357e8	−0.918232
$346$	4.92113e8	0.638703
$347$	−9.33642e7	−0.119958	−0.0599788	−	0.998200i	$-0.519103\pi$
$347$	−9.33642e7	−0.119958	−0.0599788	+	0.998200i	$0.519103\pi$
$348$	6.65952e8	0.847062
$349$	9.52954e8	1.20000	0.600002	−	0.799998i	$-0.295166\pi$
$349$	9.52954e8	1.20000	0.600002	+	0.799998i	$0.295166\pi$
$350$	−2.10840e7	−0.0262854
$351$	−1.13042e8	−0.139529
$352$	−1.31189e9	−1.60323
$353$	−1.30072e9	−1.57389	−0.786943	−	0.617025i	$-0.788338\pi$
$353$	−1.30072e9	−1.57389	−0.786943	+	0.617025i	$0.788338\pi$
$354$	1.90835e9	2.28637
$355$	8.81425e8	1.04565
$356$	5.57814e8	0.655261
$357$	−6.85930e6	−0.00797886
$358$	7.50851e8	0.864895
$359$	−4.65446e8	−0.530932	−0.265466	−	0.964120i	$-0.585526\pi$
$359$	−4.65446e8	−0.530932	−0.265466	+	0.964120i	$0.585526\pi$
$360$	−1.61059e8	−0.181940
$361$	−8.29167e8	−0.927613
$362$	−9.69876e8	−1.07457
$363$	3.61742e8	0.396940
$364$	1.87137e6	0.00203378
$365$	3.52675e9	3.79621
$366$	1.95010e9	2.07909
$367$	−1.21650e9	−1.28464	−0.642320	−	0.766436i	$-0.722028\pi$
$367$	−1.21650e9	−1.28464	−0.642320	+	0.766436i	$0.722028\pi$
$368$	−2.86007e8	−0.299164
$369$	2.11035e8	0.218656
$370$	−3.05424e8	−0.313471
$371$	533471.	0.000542378 0
$372$	−2.26468e9	−2.28090
$373$	−1.73611e9	−1.73219	−0.866097	−	0.499876i	$-0.833379\pi$
$373$	−1.73611e9	−1.73219	−0.866097	+	0.499876i	$0.833379\pi$
$374$	1.21737e9	1.20330
$375$	−4.83715e9	−4.73675
$376$	3.95237e7	0.0383443
$377$	−1.52489e8	−0.146570
$378$	−5.28165e6	−0.00502977
$379$	−5.52595e7	−0.0521398	−0.0260699	−	0.999660i	$-0.508299\pi$
$379$	−5.52595e7	−0.0521398	−0.0260699	+	0.999660i	$0.508299\pi$
$380$	5.75742e8	0.538252
$381$	1.04193e9	0.965166
$382$	1.34260e9	1.23233
$383$	1.84846e9	1.68118	0.840592	−	0.541669i	$-0.182208\pi$
$383$	1.84846e9	1.68118	0.840592	+	0.541669i	$0.182208\pi$
$384$	−2.42991e8	−0.218993
$385$	1.67196e7	0.0149318
$386$	1.49709e9	1.32493
$387$	7.51341e8	0.658944
$388$	2.03496e9	1.76867
$389$	−1.04622e9	−0.901156	−0.450578	−	0.892737i	$-0.648782\pi$
$389$	−1.04622e9	−0.901156	−0.450578	+	0.892737i	$0.648782\pi$
$390$	1.35266e9	1.15468
$391$	2.79456e8	0.236426
$392$	−8.56616e7	−0.0718266
$393$	9.85678e8	0.819146
$394$	1.03491e9	0.852441
$395$	−4.67471e8	−0.381650
$396$	1.93692e9	1.56739
$397$	−1.61599e9	−1.29620	−0.648100	−	0.761555i	$-0.724436\pi$
$397$	−1.61599e9	−1.29620	−0.648100	+	0.761555i	$0.724436\pi$
$398$	−6.95311e8	−0.552826
$399$	3.63826e6	0.00286740
$400$	−3.18793e9	−2.49057
$401$	−1.13909e9	−0.882170	−0.441085	−	0.897465i	$-0.645406\pi$
$401$	−1.13909e9	−0.882170	−0.441085	+	0.897465i	$0.645406\pi$
$402$	−2.65696e9	−2.03983
$403$	5.18566e8	0.394672
$404$	−1.07312e9	−0.809682
$405$	1.43066e9	1.07015
$406$	−7.12475e6	−0.00528358
$407$	1.75462e8	0.129004
$408$	1.12600e8	0.0820785
$409$	−7.19496e8	−0.519992	−0.259996	−	0.965610i	$-0.583721\pi$
$409$	−7.19496e8	−0.519992	−0.259996	+	0.965610i	$0.583721\pi$
$410$	−6.26000e8	−0.448571
$411$	6.52941e8	0.463904
$412$	−1.13384e9	−0.798748
$413$	−1.04581e7	−0.00730514
$414$	8.68024e8	0.601216
$415$	3.08937e9	2.12178
$416$	5.81641e8	0.396122
$417$	−3.11397e9	−2.10299
$418$	−6.45710e8	−0.432435
$419$	−5.04960e8	−0.335357	−0.167679	−	0.985842i	$-0.553627\pi$
$419$	−5.04960e8	−0.335357	−0.167679	+	0.985842i	$0.553627\pi$
$420$	3.23733e7	0.0213214
$421$	−8.92548e8	−0.582967	−0.291484	−	0.956576i	$-0.594149\pi$
$421$	−8.92548e8	−0.582967	−0.291484	+	0.956576i	$0.594149\pi$
$422$	2.35383e9	1.52469
$423$	1.10486e9	0.709769
$424$	−8.75732e6	−0.00557944
$425$	3.11491e9	1.96827
$426$	−1.91407e9	−1.19956
$427$	−1.06869e7	−0.00664285
$428$	2.81729e9	1.73691
$429$	−7.77082e8	−0.475189
$430$	−2.22873e9	−1.35182
$431$	9.66453e7	0.0581447	0.0290724	−	0.999577i	$-0.490745\pi$
$431$	9.66453e7	0.0581447	0.0290724	+	0.999577i	$0.490745\pi$
$432$	−7.98590e8	−0.476575
$433$	3.23967e9	1.91775	0.958877	−	0.283824i	$-0.0916030\pi$
$433$	3.23967e9	1.91775	0.958877	+	0.283824i	$0.0916030\pi$
$434$	2.42289e7	0.0142272
$435$	−2.63796e9	−1.53658
$436$	1.19767e9	0.692048
$437$	−1.48227e8	−0.0849655
$438$	−7.65856e9	−4.35500
$439$	−2.07987e9	−1.17330	−0.586650	−	0.809840i	$-0.699554\pi$
$439$	−2.07987e9	−1.17330	−0.586650	+	0.809840i	$0.699554\pi$
$440$	−2.74464e8	−0.153603
$441$	−2.39462e9	−1.32954
$442$	−5.39737e8	−0.297307
$443$	7.81890e8	0.427300	0.213650	−	0.976910i	$-0.431465\pi$
$443$	7.81890e8	0.427300	0.213650	+	0.976910i	$0.431465\pi$
$444$	3.39739e8	0.184206
$445$	−2.20960e9	−1.18865
$446$	−3.79333e9	−2.02465
$447$	4.23375e9	2.24207
$448$	1.45871e7	0.00766474
$449$	2.20537e9	1.14979	0.574897	−	0.818226i	$-0.305042\pi$
$449$	2.20537e9	1.14979	0.574897	+	0.818226i	$0.305042\pi$
$450$	9.67529e9	5.00519
$451$	3.59628e8	0.184601
$452$	6.19242e7	0.0315410
$453$	4.85399e8	0.245333
$454$	−3.13832e9	−1.57399
$455$	−7.41283e6	−0.00368930
$456$	−5.97247e7	−0.0294969
$457$	−2.16001e8	−0.105864	−0.0529320	−	0.998598i	$-0.516857\pi$
$457$	−2.16001e8	−0.105864	−0.0529320	+	0.998598i	$0.516857\pi$
$458$	5.23216e9	2.54479
$459$	7.80300e8	0.376632
$460$	−1.31893e9	−0.631785
$461$	−1.09842e9	−0.522176	−0.261088	−	0.965315i	$-0.584081\pi$
$461$	−1.09842e9	−0.522176	−0.261088	+	0.965315i	$0.584081\pi$
$462$	−3.63076e7	−0.0171297
$463$	1.71237e9	0.801798	0.400899	−	0.916122i	$-0.368698\pi$
$463$	1.71237e9	0.801798	0.400899	+	0.916122i	$0.368698\pi$
$464$	−1.07727e9	−0.500624
$465$	8.97082e9	4.13758
$466$	−3.59163e9	−1.64415
$467$	2.39543e9	1.08836	0.544181	−	0.838968i	$-0.316840\pi$
$467$	2.39543e9	1.08836	0.544181	+	0.838968i	$0.316840\pi$
$468$	−8.58756e8	−0.387266
$469$	1.45607e7	0.00651743
$470$	−3.27740e9	−1.45608
$471$	−1.09953e9	−0.484880
$472$	1.71678e8	0.0751479
$473$	1.28037e9	0.556317
$474$	1.01514e9	0.437827
$475$	−1.65219e9	−0.707347
$476$	−1.29176e7	−0.00548981
$477$	−2.44806e8	−0.103278
$478$	−1.60112e9	−0.670543
$479$	−2.89543e9	−1.20376	−0.601878	−	0.798588i	$-0.705581\pi$
$479$	−2.89543e9	−1.20376	−0.601878	+	0.798588i	$0.705581\pi$
$480$	1.00620e10	4.15278
$481$	−7.77931e7	−0.0318738
$482$	2.33330e9	0.949085
$483$	−8.33465e6	−0.00336568
$484$	6.81241e8	0.273113
$485$	−8.06086e9	−3.20838
$486$	−4.92965e9	−1.94800
$487$	2.23314e9	0.876121	0.438061	−	0.898945i	$-0.355665\pi$
$487$	2.23314e9	0.876121	0.438061	+	0.898945i	$0.355665\pi$
$488$	1.75433e8	0.0683350
$489$	−6.17283e9	−2.38728
$490$	7.10326e9	2.72754
$491$	−3.98109e9	−1.51781	−0.758904	−	0.651202i	$-0.774265\pi$
$491$	−3.98109e9	−1.51781	−0.758904	+	0.651202i	$0.774265\pi$
$492$	6.96331e8	0.263595
$493$	1.05260e9	0.395638
$494$	2.86284e8	0.106844
$495$	−7.67248e9	−2.84327
$496$	3.66344e9	1.34804
$497$	1.04895e7	0.00383271
$498$	−6.70875e9	−2.43410
$499$	1.87040e9	0.673879	0.336939	−	0.941526i	$-0.390608\pi$
$499$	1.87040e9	0.673879	0.336939	+	0.941526i	$0.390608\pi$
$500$	−9.10945e9	−3.25910
$501$	2.36289e9	0.839482
$502$	−2.09013e9	−0.737414
$503$	−2.70239e9	−0.946803	−0.473402	−	0.880847i	$-0.656974\pi$
$503$	−2.70239e9	−0.946803	−0.473402	+	0.880847i	$0.656974\pi$
$504$	−1.91670e6	−0.000666879 0
$505$	4.25083e9	1.46877
$506$	1.47922e9	0.507580
$507$	3.44529e8	0.117408
$508$	1.96219e9	0.664078
$509$	−4.37033e8	−0.146893	−0.0734466	−	0.997299i	$-0.523400\pi$
$509$	−4.37033e8	−0.146893	−0.0734466	+	0.997299i	$0.523400\pi$
$510$	−9.33708e9	−3.11685
$511$	4.19704e7	0.0139146
$512$	4.31570e9	1.42104
$513$	−4.13881e8	−0.135352
$514$	2.24909e9	0.730527
$515$	4.49133e9	1.44894
$516$	2.47913e9	0.794374
$517$	1.88282e9	0.599226
$518$	−3.63473e6	−0.00114899
$519$	2.16836e9	0.680841
$520$	1.21687e8	0.0379518
$521$	2.58147e9	0.799714	0.399857	−	0.916578i	$-0.369060\pi$
$521$	2.58147e9	0.799714	0.399857	+	0.916578i	$0.369060\pi$
$522$	3.26949e9	1.00608
$523$	4.76055e9	1.45513	0.727564	−	0.686040i	$-0.240652\pi$
$523$	4.76055e9	1.45513	0.727564	+	0.686040i	$0.240652\pi$
$524$	1.85626e9	0.563609
$525$	−9.29008e7	−0.0280196
$526$	9.32935e9	2.79512
$527$	−3.57954e9	−1.06534
$528$	−5.48974e9	−1.62306
$529$	−3.06526e9	−0.900270
$530$	7.26177e8	0.211874
$531$	4.79915e9	1.39102
$532$	6.85166e6	0.00197290
$533$	−1.59445e8	−0.0456107
$534$	4.79829e9	1.36362
$535$	−1.11598e10	−3.15078
$536$	−2.39024e8	−0.0670448
$537$	3.30841e9	0.921955
$538$	1.18440e9	0.327913
$539$	−4.08071e9	−1.12247
$540$	−3.68273e9	−1.00645
$541$	−2.81854e9	−0.765303	−0.382652	−	0.923893i	$-0.624989\pi$
$541$	−2.81854e9	−0.765303	−0.382652	+	0.923893i	$0.624989\pi$
$542$	−4.68676e8	−0.126437
$543$	−4.27348e9	−1.14547
$544$	−4.01493e9	−1.06926
$545$	−4.74421e9	−1.25538
$546$	1.60974e7	0.00423235
$547$	1.97278e9	0.515375	0.257688	−	0.966228i	$-0.417039\pi$
$547$	1.97278e9	0.515375	0.257688	+	0.966228i	$0.417039\pi$
$548$	1.22964e9	0.319187
$549$	4.90414e9	1.26491
$550$	1.64878e10	4.22566
$551$	−5.58311e8	−0.142182
$552$	1.36819e8	0.0346227
$553$	−5.56318e6	−0.00139889
$554$	2.13635e9	0.533813
$555$	−1.34577e9	−0.334152
$556$	−5.86430e9	−1.44695
$557$	2.20358e9	0.540301	0.270151	−	0.962818i	$-0.412926\pi$
$557$	2.20358e9	0.540301	0.270151	+	0.962818i	$0.412926\pi$
$558$	−1.11185e10	−2.70910
$559$	−5.67669e8	−0.137453
$560$	−5.23684e7	−0.0126012
$561$	5.36401e9	1.28268
$562$	−6.50602e9	−1.54610
$563$	4.39412e9	1.03775	0.518875	−	0.854850i	$-0.326351\pi$
$563$	4.39412e9	1.03775	0.518875	+	0.854850i	$0.326351\pi$
$564$	3.64561e9	0.855645
$565$	−2.45293e8	−0.0572158
$566$	3.05636e9	0.708511
$567$	1.70257e7	0.00392251
$568$	−1.72192e8	−0.0394271
$569$	−4.86249e9	−1.10654	−0.553269	−	0.833003i	$-0.686620\pi$
$569$	−4.86249e9	−1.10654	−0.553269	+	0.833003i	$0.686620\pi$
$570$	4.95251e9	1.12012
$571$	−6.48375e9	−1.45747	−0.728736	−	0.684795i	$-0.759892\pi$
$571$	−6.48375e9	−1.45747	−0.728736	+	0.684795i	$0.759892\pi$
$572$	−1.46342e9	−0.326951
$573$	5.91581e9	1.31363
$574$	−7.44976e6	−0.00164419
$575$	3.78489e9	0.830264
$576$	−6.69393e9	−1.45949
$577$	−6.82993e9	−1.48013	−0.740067	−	0.672533i	$-0.765206\pi$
$577$	−6.82993e9	−1.48013	−0.740067	+	0.672533i	$0.765206\pi$
$578$	−2.92157e9	−0.629316
$579$	6.59651e9	1.41234
$580$	−4.96787e9	−1.05724
$581$	3.67652e7	0.00777716
$582$	1.75047e10	3.68064
$583$	−4.17178e8	−0.0871929
$584$	−6.88975e8	−0.143139
$585$	3.40169e9	0.702504
$586$	−1.05571e10	−2.16722
$587$	6.26198e9	1.27784	0.638922	−	0.769271i	$-0.279380\pi$
$587$	6.26198e9	1.27784	0.638922	+	0.769271i	$0.279380\pi$
$588$	−7.90130e9	−1.60280
$589$	1.89863e9	0.382857
$590$	−1.42359e10	−2.85367
$591$	4.56003e9	0.908681
$592$	−5.49574e8	−0.108868
$593$	1.25498e9	0.247141	0.123571	−	0.992336i	$-0.460565\pi$
$593$	1.25498e9	0.247141	0.123571	+	0.992336i	$0.460565\pi$
$594$	4.13028e9	0.808587
$595$	5.11690e7	0.00995858
$596$	7.97311e9	1.54264
$597$	−3.06369e9	−0.589298
$598$	−6.55829e8	−0.125411
$599$	−6.75930e9	−1.28501	−0.642507	−	0.766280i	$-0.722106\pi$
$599$	−6.75930e9	−1.28501	−0.642507	+	0.766280i	$0.722106\pi$
$600$	1.52504e9	0.288238
$601$	6.01563e9	1.13037	0.565185	−	0.824964i	$-0.308805\pi$
$601$	6.01563e9	1.13037	0.565185	+	0.824964i	$0.308805\pi$
$602$	−2.65232e7	−0.00495494
$603$	−6.68178e9	−1.24103
$604$	9.14116e8	0.168800
$605$	−2.69852e9	−0.495430
$606$	−9.23095e9	−1.68497
$607$	−6.19648e9	−1.12457	−0.562283	−	0.826945i	$-0.690077\pi$
$607$	−6.19648e9	−1.12457	−0.562283	+	0.826945i	$0.690077\pi$
$608$	2.12957e9	0.384264
$609$	−3.13932e7	−0.00563216
$610$	−1.45473e10	−2.59495
$611$	−8.34770e8	−0.148055
$612$	5.92779e9	1.04535
$613$	7.63839e9	1.33934	0.669669	−	0.742660i	$-0.266436\pi$
$613$	7.63839e9	1.33934	0.669669	+	0.742660i	$0.266436\pi$
$614$	1.17365e10	2.04620
$615$	−2.75829e9	−0.478165
$616$	−3.26628e6	−0.000563016 0
$617$	−1.68888e8	−0.0289469	−0.0144734	−	0.999895i	$-0.504607\pi$
$617$	−1.68888e8	−0.0289469	−0.0144734	+	0.999895i	$0.504607\pi$
$618$	−9.75320e9	−1.66222
$619$	8.29914e9	1.40642	0.703212	−	0.710981i	$-0.251749\pi$
$619$	8.29914e9	1.40642	0.703212	+	0.710981i	$0.251749\pi$
$620$	1.68941e10	2.84684
$621$	9.48133e8	0.158873
$622$	−1.05502e10	−1.75790
$623$	−2.62955e7	−0.00435686
$624$	2.43395e9	0.401019
$625$	2.00376e10	3.28296
$626$	−8.91497e9	−1.45248
$627$	−2.84514e9	−0.460964
$628$	−2.07066e9	−0.333619
$629$	5.36987e8	0.0860373
$630$	1.58937e8	0.0253240
$631$	−5.89418e9	−0.933943	−0.466972	−	0.884272i	$-0.654655\pi$
$631$	−5.89418e9	−0.933943	−0.466972	+	0.884272i	$0.654655\pi$
$632$	9.13236e7	0.0143904
$633$	1.03715e10	1.62528
$634$	1.38665e10	2.16100
$635$	−7.77261e9	−1.20464
$636$	−8.07762e8	−0.124504
$637$	1.80924e9	0.277337
$638$	5.57160e9	0.849390
$639$	−4.81353e9	−0.729812
$640$	1.81266e9	0.273330
$641$	−9.17050e9	−1.37528	−0.687638	−	0.726053i	$-0.741352\pi$
$641$	−9.17050e9	−1.37528	−0.687638	+	0.726053i	$0.741352\pi$
$642$	2.42342e10	3.61456
$643$	4.28119e9	0.635076	0.317538	−	0.948246i	$-0.397144\pi$
$643$	4.28119e9	0.635076	0.317538	+	0.948246i	$0.397144\pi$
$644$	−1.56960e7	−0.00231574
$645$	−9.82028e9	−1.44100
$646$	−1.97615e9	−0.288407
$647$	6.12043e9	0.888417	0.444208	−	0.895923i	$-0.353485\pi$
$647$	6.12043e9	0.888417	0.444208	+	0.895923i	$0.353485\pi$
$648$	−2.79490e8	−0.0403509
$649$	8.17832e9	1.17438
$650$	−7.31009e9	−1.04406
$651$	1.06758e8	0.0151658
$652$	−1.16248e10	−1.64256
$653$	−5.55840e9	−0.781185	−0.390592	−	0.920564i	$-0.627730\pi$
$653$	−5.55840e9	−0.781185	−0.390592	+	0.920564i	$0.627730\pi$
$654$	1.03023e10	1.44017
$655$	−7.35297e9	−1.02239
$656$	−1.12641e9	−0.155788
$657$	−1.92599e10	−2.64957
$658$	−3.90029e7	−0.00533711
$659$	−3.85307e9	−0.524454	−0.262227	−	0.965006i	$-0.584457\pi$
$659$	−3.85307e9	−0.524454	−0.262227	+	0.965006i	$0.584457\pi$
$660$	−2.53162e10	−3.42763
$661$	1.35130e10	1.81990	0.909949	−	0.414721i	$-0.136121\pi$
$661$	1.35130e10	1.81990	0.909949	+	0.414721i	$0.136121\pi$
$662$	1.52619e10	2.04459
$663$	−2.37820e9	−0.316921
$664$	−6.03528e8	−0.0800036
$665$	−2.71407e7	−0.00357886
$666$	1.66795e9	0.218787
$667$	1.27900e9	0.166890
$668$	4.44986e9	0.577601
$669$	−1.67143e10	−2.15822
$670$	1.98204e10	2.54596
$671$	8.35723e9	1.06791
$672$	1.19743e8	0.0152216
$673$	−2.71258e9	−0.343028	−0.171514	−	0.985182i	$-0.554866\pi$
$673$	−2.71258e9	−0.343028	−0.171514	+	0.985182i	$0.554866\pi$
$674$	−1.00616e10	−1.26578
$675$	1.05682e10	1.32263
$676$	6.48826e8	0.0807820
$677$	−1.09739e10	−1.35926	−0.679629	−	0.733556i	$-0.737859\pi$
$677$	−1.09739e10	−1.35926	−0.679629	+	0.733556i	$0.737859\pi$
$678$	5.32669e8	0.0656378
$679$	−9.59289e7	−0.0117599
$680$	−8.39977e8	−0.102444
$681$	−1.38281e10	−1.67783
$682$	−1.89472e10	−2.28717
$683$	4.35591e8	0.0523126	0.0261563	−	0.999658i	$-0.491673\pi$
$683$	4.35591e8	0.0523126	0.0261563	+	0.999658i	$0.491673\pi$
$684$	−3.14417e9	−0.375673
$685$	−4.87081e9	−0.579008
$686$	1.69070e8	0.0199955
$687$	2.30541e10	2.71268
$688$	−4.01033e9	−0.469484
$689$	1.84961e8	0.0215433
$690$	−1.13454e10	−1.31476
$691$	−5.75145e9	−0.663139	−0.331569	−	0.943431i	$-0.607578\pi$
$691$	−5.75145e9	−0.663139	−0.331569	+	0.943431i	$0.607578\pi$
$692$	4.08351e9	0.468449
$693$	−9.13069e7	−0.0104217
$694$	−1.51245e9	−0.171760
$695$	2.32296e10	2.62479
$696$	5.15342e8	0.0579380
$697$	1.10061e9	0.123118
$698$	1.54373e10	1.71821
$699$	−1.58255e10	−1.75262
$700$	−1.74953e8	−0.0192788
$701$	1.25087e10	1.37151	0.685756	−	0.727832i	$-0.259472\pi$
$701$	1.25087e10	1.37151	0.685756	+	0.727832i	$0.259472\pi$
$702$	−1.83121e9	−0.199783
$703$	−2.84825e8	−0.0309196
$704$	−1.14072e10	−1.23219
$705$	−1.44409e10	−1.55215
$706$	−2.10710e10	−2.25355
$707$	5.05874e7	0.00538362
$708$	1.58353e10	1.67691
$709$	9.96620e9	1.05019	0.525095	−	0.851044i	$-0.324030\pi$
$709$	9.96620e9	1.05019	0.525095	+	0.851044i	$0.324030\pi$
$710$	1.42786e10	1.49720
$711$	2.55290e9	0.266373
$712$	4.31661e8	0.0448190
$713$	−4.34945e9	−0.449388
$714$	−1.11117e8	−0.0114244
$715$	5.79688e9	0.593093
$716$	6.23049e9	0.634347
$717$	−7.05489e9	−0.714782
$718$	−7.53996e9	−0.760209
$719$	3.13607e9	0.314655	0.157327	−	0.987546i	$-0.449712\pi$
$719$	3.13607e9	0.314655	0.157327	+	0.987546i	$0.449712\pi$
$720$	2.40314e10	2.39947
$721$	5.34494e7	0.00531092
$722$	−1.34320e10	−1.32819
$723$	1.02810e10	1.01170
$724$	−8.04794e9	−0.788133
$725$	1.42561e10	1.38937
$726$	5.86000e9	0.568355
$727$	−7.77184e9	−0.750159	−0.375079	−	0.926993i	$-0.622385\pi$
$727$	−7.77184e9	−0.750159	−0.375079	+	0.926993i	$0.622385\pi$
$728$	1.44815e6	0.000139108 0
$729$	−1.58450e10	−1.51476
$730$	5.71314e10	5.43557
$731$	3.91849e9	0.371029
$732$	1.61817e10	1.52488
$733$	1.14653e10	1.07528	0.537638	−	0.843176i	$-0.319317\pi$
$733$	1.14653e10	1.07528	0.537638	+	0.843176i	$0.319317\pi$
$734$	−1.97066e10	−1.83940
$735$	3.12985e10	2.90749
$736$	−4.87849e9	−0.451038
$737$	−1.13865e10	−1.04774
$738$	3.41864e9	0.313080
$739$	−1.77009e9	−0.161339	−0.0806697	−	0.996741i	$-0.525706\pi$
$739$	−1.77009e9	−0.161339	−0.0806697	+	0.996741i	$0.525706\pi$
$740$	−2.53438e9	−0.229912
$741$	1.26143e9	0.113893
$742$	8.64193e6	0.000776599 0
$743$	2.70827e9	0.242232	0.121116	−	0.992638i	$-0.461353\pi$
$743$	2.70827e9	0.242232	0.121116	+	0.992638i	$0.461353\pi$
$744$	−1.75251e9	−0.156011
$745$	−3.15829e10	−2.79837
$746$	−2.81240e10	−2.48023
$747$	−1.68713e10	−1.48090
$748$	1.01016e10	0.882544
$749$	−1.32808e8	−0.0115488
$750$	−7.83591e10	−6.78227
$751$	2.13875e9	0.184256	0.0921279	−	0.995747i	$-0.470633\pi$
$751$	2.13875e9	0.184256	0.0921279	+	0.995747i	$0.470633\pi$
$752$	−5.89728e9	−0.505696
$753$	−9.20959e9	−0.786064
$754$	−2.47024e9	−0.209864
$755$	−3.62098e9	−0.306205
$756$	−4.38266e7	−0.00368902
$757$	4.36664e9	0.365857	0.182929	−	0.983126i	$-0.441442\pi$
$757$	4.36664e9	0.365857	0.182929	+	0.983126i	$0.441442\pi$
$758$	−8.95171e8	−0.0746559
$759$	6.51775e9	0.541067
$760$	4.45534e8	0.0368157
$761$	8.78007e9	0.722190	0.361095	−	0.932529i	$-0.382403\pi$
$761$	8.78007e9	0.722190	0.361095	+	0.932529i	$0.382403\pi$
$762$	1.68787e10	1.38196
$763$	−5.64588e7	−0.00460146
$764$	1.11408e10	0.903836
$765$	−2.34811e10	−1.89628
$766$	2.99440e10	2.40719
$767$	−3.62596e9	−0.290161
$768$	1.70959e10	1.36185
$769$	−5.08927e9	−0.403565	−0.201782	−	0.979430i	$-0.564673\pi$
$769$	−5.08927e9	−0.403565	−0.201782	+	0.979430i	$0.564673\pi$
$770$	2.70847e8	0.0213800
$771$	9.91000e9	0.778723
$772$	1.24227e10	0.971753
$773$	8.47655e9	0.660071	0.330036	−	0.943968i	$-0.392939\pi$
$773$	8.47655e9	0.660071	0.330036	+	0.943968i	$0.392939\pi$
$774$	1.21713e10	0.943503
$775$	−4.84805e10	−3.74120
$776$	1.57474e9	0.120974
$777$	−1.60154e7	−0.00122480
$778$	−1.69482e10	−1.29031
$779$	−5.83779e8	−0.0442454
$780$	1.12242e10	0.846887
$781$	−8.20282e9	−0.616148
$782$	4.52703e9	0.338524
$783$	3.57123e9	0.265859
$784$	1.27815e10	0.947271
$785$	8.20228e9	0.605189
$786$	1.59674e10	1.17289
$787$	1.34947e10	0.986854	0.493427	−	0.869787i	$-0.335744\pi$
$787$	1.34947e10	0.986854	0.493427	+	0.869787i	$0.335744\pi$
$788$	8.58756e9	0.625213
$789$	4.11071e10	2.97953
$790$	−7.57276e9	−0.546462
$791$	−2.91913e6	−0.000209718 0
$792$	1.49887e9	0.107208
$793$	−3.70528e9	−0.263855
$794$	−2.61781e10	−1.85595
$795$	3.19970e9	0.225852
$796$	−5.76963e9	−0.405464
$797$	5.29553e9	0.370515	0.185257	−	0.982690i	$-0.440688\pi$
$797$	5.29553e9	0.370515	0.185257	+	0.982690i	$0.440688\pi$
$798$	5.89377e7	0.00410566
$799$	5.76222e9	0.399646
$800$	−5.43774e10	−3.75494
$801$	1.20668e10	0.829620
$802$	−1.84526e10	−1.26313
$803$	−3.28211e10	−2.23691
$804$	−2.20472e10	−1.49609
$805$	6.21748e7	0.00420077
$806$	8.40046e9	0.565107
$807$	5.21871e9	0.349547
$808$	−8.30429e8	−0.0553812
$809$	4.26471e9	0.283185	0.141592	−	0.989925i	$-0.454778\pi$
$809$	4.26471e9	0.283185	0.141592	+	0.989925i	$0.454778\pi$
$810$	2.31759e10	1.53228
$811$	−1.82946e10	−1.20434	−0.602172	−	0.798366i	$-0.705698\pi$
$811$	−1.82946e10	−1.20434	−0.602172	+	0.798366i	$0.705698\pi$
$812$	−5.91205e7	−0.00387518
$813$	−2.06509e9	−0.134779
$814$	2.84238e9	0.184713
$815$	4.60481e10	2.97961
$816$	−1.68010e10	−1.08248
$817$	−2.07841e9	−0.133338
$818$	−1.16554e10	−0.744546
$819$	4.04821e7	0.00257495
$820$	−5.19449e9	−0.328999
$821$	−1.76114e10	−1.11069	−0.555344	−	0.831620i	$-0.687414\pi$
$821$	−1.76114e10	−1.11069	−0.555344	+	0.831620i	$0.687414\pi$
$822$	1.05773e10	0.664236
$823$	−1.14592e10	−0.716567	−0.358283	−	0.933613i	$-0.616638\pi$
$823$	−1.14592e10	−0.716567	−0.358283	+	0.933613i	$0.616638\pi$
$824$	−8.77412e8	−0.0546334
$825$	7.26490e10	4.50444
$826$	−1.69416e8	−0.0104598
$827$	6.35186e9	0.390509	0.195255	−	0.980753i	$-0.437447\pi$
$827$	6.35186e9	0.390509	0.195255	+	0.980753i	$0.437447\pi$
$828$	7.20278e9	0.440955
$829$	2.43061e10	1.48175	0.740873	−	0.671645i	$-0.234412\pi$
$829$	2.43061e10	1.48175	0.740873	+	0.671645i	$0.234412\pi$
$830$	5.00460e10	3.03806
$831$	9.41324e9	0.569031
$832$	5.05754e9	0.304444
$833$	−1.24887e10	−0.748619
$834$	−5.04445e10	−3.01115
$835$	−1.76267e10	−1.04778
$836$	−5.35804e9	−0.317164
$837$	−1.21446e10	−0.715885
$838$	−8.18006e9	−0.480178
$839$	−1.07498e10	−0.628394	−0.314197	−	0.949358i	$-0.601735\pi$
$839$	−1.07498e10	−0.628394	−0.314197	+	0.949358i	$0.601735\pi$
$840$	2.50519e7	0.00145836
$841$	−1.24324e10	−0.720725
$842$	−1.44588e10	−0.834716
$843$	−2.86669e10	−1.64810
$844$	1.95318e10	1.11826
$845$	−2.57012e9	−0.146539
$846$	1.78981e10	1.01628
$847$	−3.21140e7	−0.00181594
$848$	1.30667e9	0.0735834
$849$	1.34670e10	0.755255
$850$	5.04598e10	2.81825
$851$	6.52487e8	0.0362926
$852$	−1.58828e10	−0.879807
$853$	2.15567e10	1.18922	0.594609	−	0.804015i	$-0.297307\pi$
$853$	2.15567e10	1.18922	0.594609	+	0.804015i	$0.297307\pi$
$854$	−1.73122e8	−0.00951150
$855$	1.24546e10	0.681475
$856$	2.18014e9	0.118803
$857$	−6.98089e9	−0.378859	−0.189430	−	0.981894i	$-0.560664\pi$
$857$	−6.98089e9	−0.378859	−0.189430	+	0.981894i	$0.560664\pi$
$858$	−1.25883e10	−0.680394
$859$	1.93619e10	1.04225	0.521125	−	0.853481i	$-0.325513\pi$
$859$	1.93619e10	1.04225	0.521125	+	0.853481i	$0.325513\pi$
$860$	−1.84938e10	−0.991475
$861$	−3.28253e7	−0.00175266
$862$	1.56560e9	0.0832539
$863$	2.93466e10	1.55425	0.777123	−	0.629349i	$-0.216678\pi$
$863$	2.93466e10	1.55425	0.777123	+	0.629349i	$0.216678\pi$
$864$	−1.36218e10	−0.718515
$865$	−1.61755e10	−0.849772
$866$	5.24807e10	2.74592
$867$	−1.28731e10	−0.670835
$868$	2.01049e8	0.0104348
$869$	4.35044e9	0.224887
$870$	−4.27334e10	−2.20014
$871$	5.04836e9	0.258873
$872$	9.26812e8	0.0473352
$873$	4.40210e10	2.23929
$874$	−2.40119e9	−0.121657
$875$	4.29423e8	0.0216699
$876$	−6.35500e10	−3.19412
$877$	−2.27937e10	−1.14108	−0.570541	−	0.821269i	$-0.693267\pi$
$877$	−2.27937e10	−1.14108	−0.570541	+	0.821269i	$0.693267\pi$
$878$	−3.36926e10	−1.67998
$879$	−4.65169e10	−2.31020
$880$	4.09524e10	2.02577
$881$	−2.33000e10	−1.14800	−0.573998	−	0.818857i	$-0.694608\pi$
$881$	−2.33000e10	−1.14800	−0.573998	+	0.818857i	$0.694608\pi$
$882$	−3.87915e10	−1.90369
$883$	9.72562e9	0.475395	0.237698	−	0.971339i	$-0.423607\pi$
$883$	9.72562e9	0.475395	0.237698	+	0.971339i	$0.423607\pi$
$884$	−4.47869e9	−0.218056
$885$	−6.27265e10	−3.04194
$886$	1.26662e10	0.611825
$887$	2.34686e10	1.12916	0.564579	−	0.825379i	$-0.309038\pi$
$887$	2.34686e10	1.12916	0.564579	+	0.825379i	$0.309038\pi$
$888$	2.62905e8	0.0125995
$889$	−9.24986e7	−0.00441549
$890$	−3.57943e10	−1.70196
$891$	−1.33142e10	−0.630585
$892$	−3.14767e10	−1.48495
$893$	−3.05635e9	−0.143623
$894$	6.85843e10	3.21028
$895$	−2.46801e10	−1.15071
$896$	2.15717e7	0.00100186
$897$	−2.88972e9	−0.133685
$898$	3.57258e10	1.64632
$899$	−1.63826e10	−0.752010
$900$	8.02847e10	3.67100
$901$	−1.27674e9	−0.0581522
$902$	5.82576e9	0.264320
$903$	−1.16867e8	−0.00528183
$904$	4.79197e7	0.00215737
$905$	3.18794e10	1.42968
$906$	7.86318e9	0.351277
$907$	3.38151e10	1.50482	0.752410	−	0.658695i	$-0.228891\pi$
$907$	3.38151e10	1.50482	0.752410	+	0.658695i	$0.228891\pi$
$908$	−2.60415e10	−1.15442
$909$	−2.32141e10	−1.02513
$910$	−1.20084e8	−0.00528249
$911$	3.83265e10	1.67952	0.839760	−	0.542957i	$-0.182695\pi$
$911$	3.83265e10	1.67952	0.839760	+	0.542957i	$0.182695\pi$
$912$	8.91144e9	0.389015
$913$	−2.87506e10	−1.25026
$914$	−3.49908e9	−0.151580
$915$	−6.40988e10	−2.76615
$916$	4.34160e10	1.86645
$917$	−8.75045e7	−0.00374747
$918$	1.26404e10	0.539277
$919$	2.20149e10	0.935648	0.467824	−	0.883822i	$-0.345038\pi$
$919$	2.20149e10	0.935648	0.467824	+	0.883822i	$0.345038\pi$
$920$	−1.02065e9	−0.0432133
$921$	5.17134e10	2.18119
$922$	−1.77938e10	−0.747673
$923$	3.63682e9	0.152236
$924$	−3.01277e8	−0.0125636
$925$	7.27284e9	0.302140
$926$	2.77395e10	1.14805
$927$	−2.45275e10	−1.01129
$928$	−1.83753e10	−0.754773
$929$	−3.29628e10	−1.34887	−0.674434	−	0.738335i	$-0.735612\pi$
$929$	−3.29628e10	−1.34887	−0.674434	+	0.738335i	$0.735612\pi$
$930$	1.45322e11	5.92436
$931$	6.62418e9	0.269035
$932$	−2.98030e10	−1.20588
$933$	−4.64865e10	−1.87388
$934$	3.88045e10	1.55836
$935$	−4.00145e10	−1.60095
$936$	−6.64543e8	−0.0264885
$937$	7.93079e9	0.314940	0.157470	−	0.987524i	$-0.449666\pi$
$937$	7.93079e9	0.314940	0.157470	+	0.987524i	$0.449666\pi$
$938$	2.35874e8	0.00933192
$939$	−3.92813e10	−1.54830
$940$	−2.71955e10	−1.06795
$941$	3.02026e10	1.18163	0.590813	−	0.806808i	$-0.298807\pi$
$941$	3.02026e10	1.18163	0.590813	+	0.806808i	$0.298807\pi$
$942$	−1.78118e10	−0.694270
$943$	1.33734e9	0.0519340
$944$	−2.56158e10	−0.991074
$945$	1.73605e8	0.00669193
$946$	2.07413e10	0.796557
$947$	1.94863e10	0.745598	0.372799	−	0.927912i	$-0.378398\pi$
$947$	1.94863e10	0.745598	0.372799	+	0.927912i	$0.378398\pi$
$948$	8.42356e9	0.321119
$949$	1.45517e10	0.552689
$950$	−2.67645e10	−1.01281
$951$	6.10989e10	2.30357
$952$	−9.99621e6	−0.000375496 0
$953$	−2.96980e10	−1.11148	−0.555741	−	0.831355i	$-0.687565\pi$
$953$	−2.96980e10	−1.11148	−0.555741	+	0.831355i	$0.687565\pi$
$954$	−3.96571e9	−0.147877
$955$	−4.41308e10	−1.63957
$956$	−1.32860e10	−0.491802
$957$	2.45497e10	0.905428
$958$	−4.69042e10	−1.72359
$959$	−5.79655e7	−0.00212229
$960$	8.74919e10	3.19168
$961$	2.81992e10	1.02496
$962$	−1.26020e9	−0.0456381
$963$	6.09446e10	2.19909
$964$	1.93615e10	0.696095
$965$	−4.92087e10	−1.76277
$966$	−1.35016e8	−0.00481911
$967$	−9.53037e9	−0.338936	−0.169468	−	0.985536i	$-0.554205\pi$
$967$	−9.53037e9	−0.338936	−0.169468	+	0.985536i	$0.554205\pi$
$968$	5.27174e8	0.0186806
$969$	−8.70734e9	−0.307434
$970$	−1.30581e11	−4.59388
$971$	3.07862e10	1.07917	0.539583	−	0.841932i	$-0.318582\pi$
$971$	3.07862e10	1.07917	0.539583	+	0.841932i	$0.318582\pi$
$972$	−4.09058e10	−1.42874
$973$	2.76445e8	0.00962087
$974$	3.61756e10	1.25447
$975$	−3.22098e10	−1.11294
$976$	−2.61762e10	−0.901222
$977$	−2.24313e9	−0.0769528	−0.0384764	−	0.999260i	$-0.512250\pi$
$977$	−2.24313e9	−0.0769528	−0.0384764	+	0.999260i	$0.512250\pi$
$978$	−9.99962e10	−3.41820
$979$	2.05633e10	0.700411
$980$	5.89422e10	2.00048
$981$	2.59085e10	0.876195
$982$	−6.44914e10	−2.17326
$983$	1.76690e10	0.593301	0.296650	−	0.954986i	$-0.404130\pi$
$983$	1.76690e10	0.593301	0.296650	+	0.954986i	$0.404130\pi$
$984$	5.38851e8	0.0180296
$985$	−3.40169e10	−1.13414
$986$	1.70515e10	0.566490
$987$	−1.71855e8	−0.00568923
$988$	2.37555e9	0.0783638
$989$	4.76131e9	0.156509
$990$	−1.24290e11	−4.07110
$991$	2.39182e10	0.780675	0.390338	−	0.920672i	$-0.372358\pi$
$991$	2.39182e10	0.780675	0.390338	+	0.920672i	$0.372358\pi$
$992$	6.24883e10	2.03239
$993$	6.72474e10	2.17948
$994$	1.69923e8	0.00548783
$995$	2.28546e10	0.735516
$996$	−5.56686e10	−1.78526
$997$	−3.75834e10	−1.20106	−0.600528	−	0.799604i	$-0.705043\pi$
$997$	−3.75834e10	−1.20106	−0.600528	+	0.799604i	$0.705043\pi$
$998$	3.02993e10	0.964886
$999$	1.82188e9	0.0578150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.8.a.c.1.3	✓	4
3.2	odd	2		117.8.a.e.1.2		4
4.3	odd	2		208.8.a.k.1.1		4
5.4	even	2		325.8.a.c.1.2		4
13.5	odd	4		169.8.b.c.168.2		8
13.8	odd	4		169.8.b.c.168.7		8
13.12	even	2		169.8.a.c.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.a.c.1.3	✓	4	1.1	even	1	trivial
117.8.a.e.1.2		4	3.2	odd	2
169.8.a.c.1.2		4	13.12	even	2
169.8.b.c.168.2		8	13.5	odd	4
169.8.b.c.168.7		8	13.8	odd	4
208.8.a.k.1.1		4	4.3	odd	2
325.8.a.c.1.2		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)

\(f(q)\)	\(=\)	\(q+16.1994 q^{2} +71.3782 q^{3} +134.421 q^{4} -532.467 q^{5} +1156.29 q^{6} -6.33667 q^{7} +104.021 q^{8} +2907.85 q^{9} -8625.66 q^{10} +4955.31 q^{11} +9594.75 q^{12} -2197.00 q^{13} -102.650 q^{14} -38006.6 q^{15} -15520.8 q^{16} +15165.4 q^{17} +47105.4 q^{18} -8043.90 q^{19} -71574.9 q^{20} -452.300 q^{21} +80273.2 q^{22} +18427.3 q^{23} +7424.84 q^{24} +205396. q^{25} -35590.1 q^{26} +51452.8 q^{27} -851.783 q^{28} +69407.9 q^{29} -615684. q^{30} -236034. q^{31} -264743. q^{32} +353701. q^{33} +245670. q^{34} +3374.07 q^{35} +390877. q^{36} +35408.8 q^{37} -130307. q^{38} -156818. q^{39} -55387.8 q^{40} +72574.1 q^{41} -7327.00 q^{42} +258384. q^{43} +666100. q^{44} -1.54833e6 q^{45} +298511. q^{46} +379959. q^{47} -1.10785e6 q^{48} -823503. q^{49} +3.32730e6 q^{50} +1.08248e6 q^{51} -295324. q^{52} -84188.0 q^{53} +833505. q^{54} -2.63854e6 q^{55} -659.147 q^{56} -574159. q^{57} +1.12437e6 q^{58} +1.65041e6 q^{59} -5.10889e6 q^{60} +1.68652e6 q^{61} -3.82361e6 q^{62} -18426.1 q^{63} -2.30202e6 q^{64} +1.16983e6 q^{65} +5.72976e6 q^{66} -2.29784e6 q^{67} +2.03855e6 q^{68} +1.31530e6 q^{69} +54658.0 q^{70} -1.65536e6 q^{71} +302477. q^{72} -6.62342e6 q^{73} +573602. q^{74} +1.46608e7 q^{75} -1.08127e6 q^{76} -31400.2 q^{77} -2.54036e6 q^{78} +877934. q^{79} +8.26434e6 q^{80} -2.68686e6 q^{81} +1.17566e6 q^{82} -5.80198e6 q^{83} -60798.7 q^{84} -8.07506e6 q^{85} +4.18567e6 q^{86} +4.95421e6 q^{87} +515457. q^{88} +4.14974e6 q^{89} -2.50821e7 q^{90} +13921.7 q^{91} +2.47702e6 q^{92} -1.68476e7 q^{93} +6.15511e6 q^{94} +4.28312e6 q^{95} -1.88969e7 q^{96} +1.51387e7 q^{97} -1.33403e7 q^{98} +1.44093e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 15 q^{2} + 80 q^{3} + 253 q^{4} + 258 q^{5} + 1579 q^{6} + 1692 q^{7} + 1893 q^{8} + 3494 q^{9} - 4495 q^{10} + 1836 q^{11} - 3655 q^{12} - 8788 q^{13} - 18285 q^{14} - 29736 q^{15} - 36159 q^{16}+ \cdots + 6200852 q^{99}+O(q^{100}) \) 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

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