LMFDB - Embedded newform 21.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,8,Mod(1,21)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("21.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	21.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$6.56008553517$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1065}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 266 $ x^2 - x - 266
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$16.8172$ of defining polynomial
Character	$\chi$	$=$	21.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.8172 q^{2} -27.0000 q^{3} +305.355 q^{4} +146.343 q^{5} +562.064 q^{6} +343.000 q^{7} -3692.02 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-20.8172 q^{2} -27.0000 q^{3} +305.355 q^{4} +146.343 q^{5} +562.064 q^{6} +343.000 q^{7} -3692.02 q^{8} +729.000 q^{9} -3046.45 q^{10} -7491.69 q^{11} -8244.57 q^{12} +8553.34 q^{13} -7140.29 q^{14} -3951.27 q^{15} +37772.0 q^{16} -5219.65 q^{17} -15175.7 q^{18} -47136.9 q^{19} +44686.6 q^{20} -9261.00 q^{21} +155956. q^{22} +23833.8 q^{23} +99684.5 q^{24} -56708.6 q^{25} -178056. q^{26} -19683.0 q^{27} +104737. q^{28} -183928. q^{29} +82254.3 q^{30} -180272. q^{31} -313728. q^{32} +202276. q^{33} +108658. q^{34} +50195.8 q^{35} +222603. q^{36} -404121. q^{37} +981256. q^{38} -230940. q^{39} -540303. q^{40} +632653. q^{41} +192788. q^{42} +443131. q^{43} -2.28762e6 q^{44} +106684. q^{45} -496152. q^{46} -524989. q^{47} -1.01984e6 q^{48} +117649. q^{49} +1.18051e6 q^{50} +140931. q^{51} +2.61180e6 q^{52} +520667. q^{53} +409744. q^{54} -1.09636e6 q^{55} -1.26636e6 q^{56} +1.27270e6 q^{57} +3.82886e6 q^{58} -1.42159e6 q^{59} -1.20654e6 q^{60} -616389. q^{61} +3.75275e6 q^{62} +250047. q^{63} +1.69611e6 q^{64} +1.25173e6 q^{65} -4.21080e6 q^{66} -1.63770e6 q^{67} -1.59384e6 q^{68} -643513. q^{69} -1.04493e6 q^{70} -595589. q^{71} -2.69148e6 q^{72} -4.41186e6 q^{73} +8.41266e6 q^{74} +1.53113e6 q^{75} -1.43935e7 q^{76} -2.56965e6 q^{77} +4.80752e6 q^{78} +2.03908e6 q^{79} +5.52768e6 q^{80} +531441. q^{81} -1.31700e7 q^{82} -1.08171e6 q^{83} -2.82789e6 q^{84} -763862. q^{85} -9.22473e6 q^{86} +4.96605e6 q^{87} +2.76595e7 q^{88} +8.80625e6 q^{89} -2.22087e6 q^{90} +2.93380e6 q^{91} +7.27776e6 q^{92} +4.86734e6 q^{93} +1.09288e7 q^{94} -6.89817e6 q^{95} +8.47065e6 q^{96} +6.13951e6 q^{97} -2.44912e6 q^{98} -5.46144e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{2} - 54 q^{3} + 317 q^{4} - 360 q^{5} + 243 q^{6} + 686 q^{7} - 5067 q^{8} + 1458 q^{9}+O(q^{10}) $ 2 * q - 9 * q^2 - 54 * q^3 + 317 * q^4 - 360 * q^5 + 243 * q^6 + 686 * q^7 - 5067 * q^8 + 1458 * q^9	\( 2 q - 9 q^{2} - 54 q^{3} + 317 q^{4} - 360 q^{5} + 243 q^{6} + 686 q^{7} - 5067 q^{8} + 1458 q^{9} - 9030 q^{10} - 4932 q^{11} - 8559 q^{12} + 7708 q^{13} - 3087 q^{14} + 9720 q^{15} + 20033 q^{16} - 28584 q^{17} - 6561 q^{18} - 63728 q^{19} + 38790 q^{20} - 18522 q^{21} + 186204 q^{22} + 82260 q^{23} + 136809 q^{24} + 121550 q^{25} - 188046 q^{26} - 39366 q^{27} + 108731 q^{28} - 435996 q^{29} + 243810 q^{30} - 29240 q^{31} - 347355 q^{32} + 133164 q^{33} - 167442 q^{34} - 123480 q^{35} + 231093 q^{36} - 709556 q^{37} + 785196 q^{38} - 208116 q^{39} + 155910 q^{40} - 25056 q^{41} + 83349 q^{42} + 496216 q^{43} - 2257812 q^{44} - 262440 q^{45} + 194280 q^{46} - 1575000 q^{47} - 540891 q^{48} + 235298 q^{49} + 3287025 q^{50} + 771768 q^{51} + 2601958 q^{52} + 2057436 q^{53} + 177147 q^{54} - 2392440 q^{55} - 1737981 q^{56} + 1720656 q^{57} + 850122 q^{58} - 1101024 q^{59} - 1047330 q^{60} + 28996 q^{61} + 5537520 q^{62} + 500094 q^{63} + 3569321 q^{64} + 1679760 q^{65} - 5027508 q^{66} - 4480784 q^{67} - 1865934 q^{68} - 2221020 q^{69} - 3097290 q^{70} + 54540 q^{71} - 3693843 q^{72} + 666604 q^{73} + 4803282 q^{74} - 3281850 q^{75} - 14586668 q^{76} - 1691676 q^{77} + 5077242 q^{78} + 2322952 q^{79} + 14509710 q^{80} + 1062882 q^{81} - 20942298 q^{82} - 7384392 q^{83} - 2935737 q^{84} + 11066520 q^{85} - 8597412 q^{86} + 11771892 q^{87} + 24139932 q^{88} + 1784448 q^{89} - 6582870 q^{90} + 2643844 q^{91} + 7958160 q^{92} + 789480 q^{93} - 1479360 q^{94} + 1502640 q^{95} + 9378585 q^{96} + 16266412 q^{97} - 1058841 q^{98} - 3595428 q^{99}+O(q^{100}) \) 2 * q - 9 * q^2 - 54 * q^3 + 317 * q^4 - 360 * q^5 + 243 * q^6 + 686 * q^7 - 5067 * q^8 + 1458 * q^9 - 9030 * q^10 - 4932 * q^11 - 8559 * q^12 + 7708 * q^13 - 3087 * q^14 + 9720 * q^15 + 20033 * q^16 - 28584 * q^17 - 6561 * q^18 - 63728 * q^19 + 38790 * q^20 - 18522 * q^21 + 186204 * q^22 + 82260 * q^23 + 136809 * q^24 + 121550 * q^25 - 188046 * q^26 - 39366 * q^27 + 108731 * q^28 - 435996 * q^29 + 243810 * q^30 - 29240 * q^31 - 347355 * q^32 + 133164 * q^33 - 167442 * q^34 - 123480 * q^35 + 231093 * q^36 - 709556 * q^37 + 785196 * q^38 - 208116 * q^39 + 155910 * q^40 - 25056 * q^41 + 83349 * q^42 + 496216 * q^43 - 2257812 * q^44 - 262440 * q^45 + 194280 * q^46 - 1575000 * q^47 - 540891 * q^48 + 235298 * q^49 + 3287025 * q^50 + 771768 * q^51 + 2601958 * q^52 + 2057436 * q^53 + 177147 * q^54 - 2392440 * q^55 - 1737981 * q^56 + 1720656 * q^57 + 850122 * q^58 - 1101024 * q^59 - 1047330 * q^60 + 28996 * q^61 + 5537520 * q^62 + 500094 * q^63 + 3569321 * q^64 + 1679760 * q^65 - 5027508 * q^66 - 4480784 * q^67 - 1865934 * q^68 - 2221020 * q^69 - 3097290 * q^70 + 54540 * q^71 - 3693843 * q^72 + 666604 * q^73 + 4803282 * q^74 - 3281850 * q^75 - 14586668 * q^76 - 1691676 * q^77 + 5077242 * q^78 + 2322952 * q^79 + 14509710 * q^80 + 1062882 * q^81 - 20942298 * q^82 - 7384392 * q^83 - 2935737 * q^84 + 11066520 * q^85 - 8597412 * q^86 + 11771892 * q^87 + 24139932 * q^88 + 1784448 * q^89 - 6582870 * q^90 + 2643844 * q^91 + 7958160 * q^92 + 789480 * q^93 - 1479360 * q^94 + 1502640 * q^95 + 9378585 * q^96 + 16266412 * q^97 - 1058841 * q^98 - 3595428 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−20.8172	−1.84000	−0.919998	−	0.391924i	$-0.871810\pi$
$2$	−20.8172	−1.84000	−0.919998	+	0.391924i	$0.871810\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	305.355	2.38558
$5$	146.343	0.523574	0.261787	−	0.965126i	$-0.415688\pi$
$5$	146.343	0.523574	0.261787	+	0.965126i	$0.415688\pi$
$6$	562.064	1.06232
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−3692.02	−2.54946
$9$	729.000	0.333333
$10$	−3046.45	−0.963374
$11$	−7491.69	−1.69709	−0.848546	−	0.529122i	$-0.822521\pi$
$11$	−7491.69	−1.69709	−0.848546	+	0.529122i	$0.822521\pi$
$12$	−8244.57	−1.37732
$13$	8553.34	1.07978	0.539889	−	0.841736i	$-0.318466\pi$
$13$	8553.34	1.07978	0.539889	+	0.841736i	$0.318466\pi$
$14$	−7140.29	−0.695453
$15$	−3951.27	−0.302286
$16$	37772.0	2.30542
$17$	−5219.65	−0.257674	−0.128837	−	0.991666i	$-0.541124\pi$
$17$	−5219.65	−0.257674	−0.128837	+	0.991666i	$0.541124\pi$
$18$	−15175.7	−0.613332
$19$	−47136.9	−1.57661	−0.788303	−	0.615287i	$-0.789040\pi$
$19$	−47136.9	−1.57661	−0.788303	+	0.615287i	$0.789040\pi$
$20$	44686.6	1.24903
$21$	−9261.00	−0.218218
$22$	155956.	3.12264
$23$	23833.8	0.408457	0.204228	−	0.978923i	$-0.434532\pi$
$23$	23833.8	0.408457	0.204228	+	0.978923i	$0.434532\pi$
$24$	99684.5	1.47193
$25$	−56708.6	−0.725870
$26$	−178056.	−1.98679
$27$	−19683.0	−0.192450
$28$	104737.	0.901665
$29$	−183928.	−1.40041	−0.700203	−	0.713943i	$-0.746907\pi$
$29$	−183928.	−1.40041	−0.700203	+	0.713943i	$0.746907\pi$
$30$	82254.3	0.556204
$31$	−180272.	−1.08683	−0.543416	−	0.839464i	$-0.682869\pi$
$31$	−180272.	−1.08683	−0.543416	+	0.839464i	$0.682869\pi$
$32$	−313728.	−1.69250
$33$	202276.	0.979816
$34$	108658.	0.474119
$35$	50195.8	0.197892
$36$	222603.	0.795194
$37$	−404121.	−1.31161	−0.655806	−	0.754929i	$-0.727671\pi$
$37$	−404121.	−1.31161	−0.655806	+	0.754929i	$0.727671\pi$
$38$	981256.	2.90095
$39$	−230940.	−0.623410
$40$	−540303.	−1.33483
$41$	632653.	1.43358	0.716790	−	0.697289i	$-0.245611\pi$
$41$	632653.	1.43358	0.716790	+	0.697289i	$0.245611\pi$
$42$	192788.	0.401520
$43$	443131.	0.849948	0.424974	−	0.905206i	$-0.360283\pi$
$43$	443131.	0.849948	0.424974	+	0.905206i	$0.360283\pi$
$44$	−2.28762e6	−4.04855
$45$	106684.	0.174525
$46$	−496152.	−0.751558
$47$	−524989.	−0.737578	−0.368789	−	0.929513i	$-0.620228\pi$
$47$	−524989.	−0.737578	−0.368789	+	0.929513i	$0.620228\pi$
$48$	−1.01984e6	−1.33103
$49$	117649.	0.142857
$50$	1.18051e6	1.33560
$51$	140931.	0.148768
$52$	2.61180e6	2.57590
$53$	520667.	0.480390	0.240195	−	0.970725i	$-0.422789\pi$
$53$	520667.	0.480390	0.240195	+	0.970725i	$0.422789\pi$
$54$	409744.	0.354107
$55$	−1.09636e6	−0.888553
$56$	−1.26636e6	−0.963607
$57$	1.27270e6	0.910254
$58$	3.82886e6	2.57674
$59$	−1.42159e6	−0.901139	−0.450569	−	0.892741i	$-0.648779\pi$
$59$	−1.42159e6	−0.901139	−0.450569	+	0.892741i	$0.648779\pi$
$60$	−1.20654e6	−0.721127
$61$	−616389.	−0.347697	−0.173848	−	0.984772i	$-0.555620\pi$
$61$	−616389.	−0.347697	−0.173848	+	0.984772i	$0.555620\pi$
$62$	3.75275e6	1.99976
$63$	250047.	0.125988
$64$	1.69611e6	0.808767
$65$	1.25173e6	0.565343
$66$	−4.21080e6	−1.80286
$67$	−1.63770e6	−0.665232	−0.332616	−	0.943062i	$-0.607931\pi$
$67$	−1.63770e6	−0.665232	−0.332616	+	0.943062i	$0.607931\pi$
$68$	−1.59384e6	−0.614702
$69$	−643513.	−0.235823
$70$	−1.04493e6	−0.364121
$71$	−595589.	−0.197489	−0.0987444	−	0.995113i	$-0.531483\pi$
$71$	−595589.	−0.197489	−0.0987444	+	0.995113i	$0.531483\pi$
$72$	−2.69148e6	−0.849822
$73$	−4.41186e6	−1.32737	−0.663685	−	0.748012i	$-0.731008\pi$
$73$	−4.41186e6	−1.32737	−0.663685	+	0.748012i	$0.731008\pi$
$74$	8.41266e6	2.41336
$75$	1.53113e6	0.419081
$76$	−1.43935e7	−3.76112
$77$	−2.56965e6	−0.641440
$78$	4.80752e6	1.14707
$79$	2.03908e6	0.465307	0.232653	−	0.972560i	$-0.425259\pi$
$79$	2.03908e6	0.465307	0.232653	+	0.972560i	$0.425259\pi$
$80$	5.52768e6	1.20706
$81$	531441.	0.111111
$82$	−1.31700e7	−2.63778
$83$	−1.08171e6	−0.207653	−0.103826	−	0.994595i	$-0.533109\pi$
$83$	−1.08171e6	−0.207653	−0.103826	+	0.994595i	$0.533109\pi$
$84$	−2.82789e6	−0.520577
$85$	−763862.	−0.134911
$86$	−9.22473e6	−1.56390
$87$	4.96605e6	0.808525
$88$	2.76595e7	4.32667
$89$	8.80625e6	1.32412	0.662058	−	0.749453i	$-0.269683\pi$
$89$	8.80625e6	1.32412	0.662058	+	0.749453i	$0.269683\pi$
$90$	−2.22087e6	−0.321125
$91$	2.93380e6	0.408118
$92$	7.27776e6	0.974407
$93$	4.86734e6	0.627482
$94$	1.09288e7	1.35714
$95$	−6.89817e6	−0.825470
$96$	8.47065e6	0.977164
$97$	6.13951e6	0.683019	0.341509	−	0.939878i	$-0.389062\pi$
$97$	6.13951e6	0.683019	0.341509	+	0.939878i	$0.389062\pi$
$98$	−2.44912e6	−0.262856
$99$	−5.46144e6	−0.565697
$100$	−1.73162e7	−1.73162
$101$	−4.52457e6	−0.436970	−0.218485	−	0.975840i	$-0.570112\pi$
$101$	−4.52457e6	−0.436970	−0.218485	+	0.975840i	$0.570112\pi$
$102$	−2.93378e6	−0.273733
$103$	−1.07231e7	−0.966923	−0.483461	−	0.875366i	$-0.660621\pi$
$103$	−1.07231e7	−0.966923	−0.483461	+	0.875366i	$0.660621\pi$
$104$	−3.15791e7	−2.75285
$105$	−1.35529e6	−0.114253
$106$	−1.08388e7	−0.883916
$107$	1.51045e7	1.19196	0.595982	−	0.802998i	$-0.296763\pi$
$107$	1.51045e7	1.19196	0.595982	+	0.802998i	$0.296763\pi$
$108$	−6.01029e6	−0.459106
$109$	1.82337e7	1.34860	0.674298	−	0.738460i	$-0.264446\pi$
$109$	1.82337e7	1.34860	0.674298	+	0.738460i	$0.264446\pi$
$110$	2.28231e7	1.63493
$111$	1.09113e7	0.757260
$112$	1.29558e7	0.871367
$113$	6.46569e6	0.421541	0.210771	−	0.977536i	$-0.432403\pi$
$113$	6.46569e6	0.421541	0.210771	+	0.977536i	$0.432403\pi$
$114$	−2.64939e7	−1.67486
$115$	3.48792e6	0.213857
$116$	−5.61632e7	−3.34079
$117$	6.23539e6	0.359926
$118$	2.95934e7	1.65809
$119$	−1.79034e6	−0.0973916
$120$	1.45882e7	0.770666
$121$	3.66382e7	1.88012
$122$	1.28315e7	0.639760
$123$	−1.70816e7	−0.827678
$124$	−5.50468e7	−2.59273
$125$	−1.97320e7	−0.903621
$126$	−5.20527e6	−0.231818
$127$	1.78338e7	0.772557	0.386278	−	0.922382i	$-0.373760\pi$
$127$	1.78338e7	0.772557	0.386278	+	0.922382i	$0.373760\pi$
$128$	4.84900e6	0.204370
$129$	−1.19645e7	−0.490718
$130$	−2.60574e7	−1.04023
$131$	−4.12029e7	−1.60132	−0.800660	−	0.599118i	$-0.795518\pi$
$131$	−4.12029e7	−1.60132	−0.800660	+	0.599118i	$0.795518\pi$
$132$	6.17658e7	2.33743
$133$	−1.61679e7	−0.595901
$134$	3.40923e7	1.22402
$135$	−2.88048e6	−0.100762
$136$	1.92711e7	0.656931
$137$	−2.40754e7	−0.799929	−0.399964	−	0.916531i	$-0.630977\pi$
$137$	−2.40754e7	−0.799929	−0.399964	+	0.916531i	$0.630977\pi$
$138$	1.33961e7	0.433912
$139$	1.19648e7	0.377880	0.188940	−	0.981989i	$-0.439495\pi$
$139$	1.19648e7	0.377880	0.188940	+	0.981989i	$0.439495\pi$
$140$	1.53275e7	0.472089
$141$	1.41747e7	0.425841
$142$	1.23985e7	0.363378
$143$	−6.40790e7	−1.83248
$144$	2.75358e7	0.768473
$145$	−2.69166e7	−0.733217
$146$	9.18425e7	2.44235
$147$	−3.17652e6	−0.0824786
$148$	−1.23400e8	−3.12896
$149$	1.14635e7	0.283900	0.141950	−	0.989874i	$-0.454663\pi$
$149$	1.14635e7	0.283900	0.141950	+	0.989874i	$0.454663\pi$
$150$	−3.18738e7	−0.771108
$151$	−4.04161e7	−0.955291	−0.477645	−	0.878553i	$-0.658510\pi$
$151$	−4.04161e7	−0.955291	−0.477645	+	0.878553i	$0.658510\pi$
$152$	1.74030e8	4.01950
$153$	−3.80513e6	−0.0858913
$154$	5.34928e7	1.18025
$155$	−2.63816e7	−0.569037
$156$	−7.05187e7	−1.48720
$157$	3.24253e7	0.668705	0.334353	−	0.942448i	$-0.391482\pi$
$157$	3.24253e7	0.668705	0.334353	+	0.942448i	$0.391482\pi$
$158$	−4.24478e7	−0.856162
$159$	−1.40580e7	−0.277353
$160$	−4.59120e7	−0.886148
$161$	8.17499e6	0.154382
$162$	−1.10631e7	−0.204444
$163$	3.42478e7	0.619407	0.309703	−	0.950833i	$-0.399770\pi$
$163$	3.42478e7	0.619407	0.309703	+	0.950833i	$0.399770\pi$
$164$	1.93183e8	3.41992
$165$	2.96017e7	0.513006
$166$	2.25181e7	0.382080
$167$	−2.37416e7	−0.394459	−0.197229	−	0.980357i	$-0.563194\pi$
$167$	−2.37416e7	−0.394459	−0.197229	+	0.980357i	$0.563194\pi$
$168$	3.41918e7	0.556339
$169$	1.04112e7	0.165919
$170$	1.59014e7	0.248236
$171$	−3.43628e7	−0.525535
$172$	1.35312e8	2.02762
$173$	−6.21517e6	−0.0912624	−0.0456312	−	0.998958i	$-0.514530\pi$
$173$	−6.21517e6	−0.0912624	−0.0456312	+	0.998958i	$0.514530\pi$
$174$	−1.03379e8	−1.48768
$175$	−1.94511e7	−0.274353
$176$	−2.82976e8	−3.91251
$177$	3.83829e7	0.520273
$178$	−1.83321e8	−2.43637
$179$	5.94712e7	0.775034	0.387517	−	0.921863i	$-0.373333\pi$
$179$	5.94712e7	0.775034	0.387517	+	0.921863i	$0.373333\pi$
$180$	3.25765e7	0.416343
$181$	1.40709e7	0.176379	0.0881893	−	0.996104i	$-0.471892\pi$
$181$	1.40709e7	0.176379	0.0881893	+	0.996104i	$0.471892\pi$
$182$	−6.10734e7	−0.750934
$183$	1.66425e7	0.200743
$184$	−8.79948e7	−1.04135
$185$	−5.91404e7	−0.686726
$186$	−1.01324e8	−1.15456
$187$	3.91040e7	0.437296
$188$	−1.60308e8	−1.75955
$189$	−6.75127e6	−0.0727393
$190$	1.43600e8	1.51886
$191$	4.07460e6	0.0423124	0.0211562	−	0.999776i	$-0.493265\pi$
$191$	4.07460e6	0.0423124	0.0211562	+	0.999776i	$0.493265\pi$
$192$	−4.57949e7	−0.466942
$193$	1.52420e8	1.52613	0.763063	−	0.646325i	$-0.223695\pi$
$193$	1.52420e8	1.52613	0.763063	+	0.646325i	$0.223695\pi$
$194$	−1.27807e8	−1.25675
$195$	−3.37966e7	−0.326401
$196$	3.59247e7	0.340797
$197$	2.27551e7	0.212054	0.106027	−	0.994363i	$-0.466187\pi$
$197$	2.27551e7	0.212054	0.106027	+	0.994363i	$0.466187\pi$
$198$	1.13692e8	1.04088
$199$	−1.86666e8	−1.67911	−0.839557	−	0.543272i	$-0.817185\pi$
$199$	−1.86666e8	−1.67911	−0.839557	+	0.543272i	$0.817185\pi$
$200$	2.09369e8	1.85058
$201$	4.42179e7	0.384072
$202$	9.41886e7	0.804023
$203$	−6.30872e7	−0.529304
$204$	4.30338e7	0.354899
$205$	9.25846e7	0.750585
$206$	2.23225e8	1.77913
$207$	1.73748e7	0.136152
$208$	3.23077e8	2.48934
$209$	3.53135e8	2.67564
$210$	2.82132e7	0.210225
$211$	−7.42684e7	−0.544271	−0.272136	−	0.962259i	$-0.587730\pi$
$211$	−7.42684e7	−0.544271	−0.272136	+	0.962259i	$0.587730\pi$
$212$	1.58988e8	1.14601
$213$	1.60809e7	0.114020
$214$	−3.14433e8	−2.19321
$215$	6.48493e7	0.445011
$216$	7.26700e7	0.490645
$217$	−6.18333e7	−0.410784
$218$	−3.79574e8	−2.48141
$219$	1.19120e8	0.766357
$220$	−3.34778e8	−2.11972
$221$	−4.46455e7	−0.278231
$222$	−2.27142e8	−1.39335
$223$	−7.64759e7	−0.461804	−0.230902	−	0.972977i	$-0.574168\pi$
$223$	−7.64759e7	−0.461804	−0.230902	+	0.972977i	$0.574168\pi$
$224$	−1.07609e8	−0.639704
$225$	−4.13406e7	−0.241957
$226$	−1.34597e8	−0.775634
$227$	−2.38116e7	−0.135113	−0.0675565	−	0.997715i	$-0.521520\pi$
$227$	−2.38116e7	−0.135113	−0.0675565	+	0.997715i	$0.521520\pi$
$228$	3.88623e8	2.17149
$229$	−1.46235e8	−0.804687	−0.402344	−	0.915489i	$-0.631804\pi$
$229$	−1.46235e8	−0.804687	−0.402344	+	0.915489i	$0.631804\pi$
$230$	−7.26086e7	−0.393496
$231$	6.93805e7	0.370336
$232$	6.79065e8	3.57029
$233$	5.59677e7	0.289863	0.144931	−	0.989442i	$-0.453704\pi$
$233$	5.59677e7	0.289863	0.144931	+	0.989442i	$0.453704\pi$
$234$	−1.29803e8	−0.662262
$235$	−7.68287e7	−0.386177
$236$	−4.34088e8	−2.14974
$237$	−5.50551e7	−0.268645
$238$	3.72698e7	0.179200
$239$	1.98698e8	0.941455	0.470728	−	0.882279i	$-0.343991\pi$
$239$	1.98698e8	0.941455	0.470728	+	0.882279i	$0.343991\pi$
$240$	−1.49247e8	−0.696895
$241$	2.81922e8	1.29739	0.648693	−	0.761051i	$-0.275316\pi$
$241$	2.81922e8	1.29739	0.648693	+	0.761051i	$0.275316\pi$
$242$	−7.62704e8	−3.45941
$243$	−1.43489e7	−0.0641500
$244$	−1.88217e8	−0.829459
$245$	1.72172e7	0.0747963
$246$	3.55591e8	1.52292
$247$	−4.03178e8	−1.70238
$248$	6.65567e8	2.77084
$249$	2.92062e7	0.119888
$250$	4.10765e8	1.66266
$251$	−4.91728e8	−1.96276	−0.981379	−	0.192082i	$-0.938476\pi$
$251$	−4.91728e8	−1.96276	−0.981379	+	0.192082i	$0.938476\pi$
$252$	7.63530e7	0.300555
$253$	−1.78555e8	−0.693188
$254$	−3.71249e8	−1.42150
$255$	2.06243e7	0.0778911
$256$	−3.18044e8	−1.18481
$257$	8.50033e7	0.312371	0.156185	−	0.987728i	$-0.450080\pi$
$257$	8.50033e7	0.312371	0.156185	+	0.987728i	$0.450080\pi$
$258$	2.49068e8	0.902918
$259$	−1.38614e8	−0.495743
$260$	3.82220e8	1.34867
$261$	−1.34083e8	−0.466802
$262$	8.57728e8	2.94642
$263$	4.97177e8	1.68526	0.842628	−	0.538496i	$-0.181007\pi$
$263$	4.97177e8	1.68526	0.842628	+	0.538496i	$0.181007\pi$
$264$	−7.46805e8	−2.49801
$265$	7.61961e7	0.251520
$266$	3.36571e8	1.09645
$267$	−2.37769e8	−0.764478
$268$	−5.00079e8	−1.58696
$269$	5.65458e7	0.177120	0.0885600	−	0.996071i	$-0.471773\pi$
$269$	5.65458e7	0.177120	0.0885600	+	0.996071i	$0.471773\pi$
$270$	5.99634e7	0.185401
$271$	4.08584e8	1.24706	0.623532	−	0.781798i	$-0.285697\pi$
$271$	4.08584e8	1.24706	0.623532	+	0.781798i	$0.285697\pi$
$272$	−1.97157e8	−0.594047
$273$	−7.92125e7	−0.235627
$274$	5.01181e8	1.47186
$275$	4.24843e8	1.23187
$276$	−1.96499e8	−0.562574
$277$	9.87316e7	0.279111	0.139556	−	0.990214i	$-0.455433\pi$
$277$	9.87316e7	0.279111	0.139556	+	0.990214i	$0.455433\pi$
$278$	−2.49073e8	−0.695297
$279$	−1.31418e8	−0.362277
$280$	−1.85324e8	−0.504520
$281$	−4.15884e8	−1.11815	−0.559076	−	0.829117i	$-0.688844\pi$
$281$	−4.15884e8	−1.11815	−0.559076	+	0.829117i	$0.688844\pi$
$282$	−2.95077e8	−0.783545
$283$	−2.51481e8	−0.659557	−0.329779	−	0.944058i	$-0.606974\pi$
$283$	−2.51481e8	−0.659557	−0.329779	+	0.944058i	$0.606974\pi$
$284$	−1.81866e8	−0.471126
$285$	1.86251e8	0.476585
$286$	1.33394e9	3.37176
$287$	2.17000e8	0.541842
$288$	−2.28708e8	−0.564166
$289$	−3.83094e8	−0.933604
$290$	5.60328e8	1.34912
$291$	−1.65767e8	−0.394341
$292$	−1.34718e9	−3.16655
$293$	−6.72144e8	−1.56108	−0.780541	−	0.625105i	$-0.785056\pi$
$293$	−6.72144e8	−1.56108	−0.780541	+	0.625105i	$0.785056\pi$
$294$	6.61262e7	0.151760
$295$	−2.08040e8	−0.471813
$296$	1.49202e9	3.34391
$297$	1.47459e8	0.326605
$298$	−2.38638e8	−0.522375
$299$	2.03859e8	0.441042
$300$	4.67538e8	0.999753
$301$	1.51994e8	0.321250
$302$	8.41350e8	1.75773
$303$	1.22163e8	0.252285
$304$	−1.78045e9	−3.63474
$305$	−9.02045e7	−0.182045
$306$	7.92120e7	0.158040
$307$	5.56719e8	1.09812	0.549062	−	0.835781i	$-0.314985\pi$
$307$	5.56719e8	1.09812	0.549062	+	0.835781i	$0.314985\pi$
$308$	−7.84654e8	−1.53021
$309$	2.89525e8	0.558253
$310$	5.49190e8	1.04702
$311$	1.00770e8	0.189963	0.0949816	−	0.995479i	$-0.469721\pi$
$311$	1.00770e8	0.189963	0.0949816	+	0.995479i	$0.469721\pi$
$312$	8.52636e8	1.58936
$313$	−3.63865e6	−0.00670711	−0.00335356	−	0.999994i	$-0.501067\pi$
$313$	−3.63865e6	−0.00670711	−0.00335356	+	0.999994i	$0.501067\pi$
$314$	−6.75002e8	−1.23041
$315$	3.65927e7	0.0659641
$316$	6.22642e8	1.11003
$317$	−1.98937e8	−0.350758	−0.175379	−	0.984501i	$-0.556115\pi$
$317$	−1.98937e8	−0.350758	−0.175379	+	0.984501i	$0.556115\pi$
$318$	2.92648e8	0.510329
$319$	1.37793e9	2.37662
$320$	2.48214e8	0.423449
$321$	−4.07821e8	−0.688180
$322$	−1.70180e8	−0.284062
$323$	2.46038e8	0.406250
$324$	1.62278e8	0.265065
$325$	−4.85048e8	−0.783778
$326$	−7.12942e8	−1.13971
$327$	−4.92310e8	−0.778612
$328$	−2.33577e9	−3.65486
$329$	−1.80071e8	−0.278778
$330$	−6.16223e8	−0.943929
$331$	−5.64752e8	−0.855973	−0.427986	−	0.903785i	$-0.640777\pi$
$331$	−5.64752e8	−0.855973	−0.427986	+	0.903785i	$0.640777\pi$
$332$	−3.30305e8	−0.495373
$333$	−2.94604e8	−0.437204
$334$	4.94232e8	0.725802
$335$	−2.39667e8	−0.348298
$336$	−3.49807e8	−0.503084
$337$	−6.45094e8	−0.918160	−0.459080	−	0.888395i	$-0.651821\pi$
$337$	−6.45094e8	−0.918160	−0.459080	+	0.888395i	$0.651821\pi$
$338$	−2.16731e8	−0.305291
$339$	−1.74574e8	−0.243377
$340$	−2.33249e8	−0.321842
$341$	1.35054e9	1.84445
$342$	7.15336e8	0.966982
$343$	4.03536e7	0.0539949
$344$	−1.63605e9	−2.16691
$345$	−9.41738e7	−0.123471
$346$	1.29382e8	0.167922
$347$	6.99374e8	0.898579	0.449290	−	0.893386i	$-0.351677\pi$
$347$	6.99374e8	0.898579	0.449290	+	0.893386i	$0.351677\pi$
$348$	1.51641e9	1.92880
$349$	−3.15916e8	−0.397816	−0.198908	−	0.980018i	$-0.563740\pi$
$349$	−3.15916e8	−0.397816	−0.198908	+	0.980018i	$0.563740\pi$
$350$	4.04916e8	0.504809
$351$	−1.68355e8	−0.207803
$352$	2.35035e9	2.87232
$353$	−7.33037e8	−0.886982	−0.443491	−	0.896279i	$-0.646260\pi$
$353$	−7.33037e8	−0.886982	−0.443491	+	0.896279i	$0.646260\pi$
$354$	−7.99023e8	−0.957299
$355$	−8.71605e7	−0.103400
$356$	2.68903e9	3.15879
$357$	4.83392e7	0.0562291
$358$	−1.23802e9	−1.42606
$359$	−1.31011e9	−1.49443	−0.747215	−	0.664582i	$-0.768610\pi$
$359$	−1.31011e9	−1.49443	−0.747215	+	0.664582i	$0.768610\pi$
$360$	−3.93881e8	−0.444944
$361$	1.32801e9	1.48569
$362$	−2.92916e8	−0.324536
$363$	−9.89232e8	−1.08549
$364$	8.95848e8	0.973598
$365$	−6.45647e8	−0.694976
$366$	−3.46450e8	−0.369366
$367$	1.28960e9	1.36183	0.680915	−	0.732362i	$-0.261582\pi$
$367$	1.28960e9	1.36183	0.680915	+	0.732362i	$0.261582\pi$
$368$	9.00250e8	0.941664
$369$	4.61204e8	0.477860
$370$	1.23114e9	1.26357
$371$	1.78589e8	0.181570
$372$	1.48626e9	1.49691
$373$	5.26040e8	0.524853	0.262427	−	0.964952i	$-0.415477\pi$
$373$	5.26040e8	0.524853	0.262427	+	0.964952i	$0.415477\pi$
$374$	−8.14035e8	−0.804623
$375$	5.32764e8	0.521706
$376$	1.93827e9	1.88043
$377$	−1.57320e9	−1.51213
$378$	1.40542e8	0.133840
$379$	1.56111e8	0.147298	0.0736491	−	0.997284i	$-0.476536\pi$
$379$	1.56111e8	0.147298	0.0736491	+	0.997284i	$0.476536\pi$
$380$	−2.10639e9	−1.96923
$381$	−4.81512e8	−0.446036
$382$	−8.48216e7	−0.0778547
$383$	6.35727e8	0.578196	0.289098	−	0.957300i	$-0.406645\pi$
$383$	6.35727e8	0.578196	0.289098	+	0.957300i	$0.406645\pi$
$384$	−1.30923e8	−0.117993
$385$	−3.76051e8	−0.335841
$386$	−3.17294e9	−2.80806
$387$	3.23042e8	0.283316
$388$	1.87473e9	1.62940
$389$	3.95833e8	0.340949	0.170474	−	0.985362i	$-0.445470\pi$
$389$	3.95833e8	0.340949	0.170474	+	0.985362i	$0.445470\pi$
$390$	7.03549e8	0.600577
$391$	−1.24404e8	−0.105249
$392$	−4.34362e8	−0.364209
$393$	1.11248e9	0.924523
$394$	−4.73696e8	−0.390178
$395$	2.98406e8	0.243622
$396$	−1.66768e9	−1.34952
$397$	−1.38960e9	−1.11461	−0.557307	−	0.830307i	$-0.688165\pi$
$397$	−1.38960e9	−1.11461	−0.557307	+	0.830307i	$0.688165\pi$
$398$	3.88586e9	3.08956
$399$	4.36535e8	0.344044
$400$	−2.14200e9	−1.67344
$401$	−5.01255e8	−0.388198	−0.194099	−	0.980982i	$-0.562178\pi$
$401$	−5.01255e8	−0.388198	−0.194099	+	0.980982i	$0.562178\pi$
$402$	−9.20492e8	−0.706690
$403$	−1.54193e9	−1.17354
$404$	−1.38160e9	−1.04243
$405$	7.77729e7	0.0581749
$406$	1.31330e9	0.973917
$407$	3.02755e9	2.22593
$408$	−5.20319e8	−0.379279
$409$	2.64451e9	1.91123	0.955617	−	0.294610i	$-0.0951898\pi$
$409$	2.64451e9	1.91123	0.955617	+	0.294610i	$0.0951898\pi$
$410$	−1.92735e9	−1.38107
$411$	6.50035e8	0.461839
$412$	−3.27436e9	−2.30667
$413$	−4.87605e8	−0.340598
$414$	−3.61695e8	−0.250519
$415$	−1.58301e8	−0.108722
$416$	−2.68342e9	−1.82752
$417$	−3.23050e8	−0.218169
$418$	−7.35127e9	−4.92317
$419$	1.83624e9	1.21950	0.609748	−	0.792595i	$-0.291270\pi$
$419$	1.83624e9	1.21950	0.609748	+	0.792595i	$0.291270\pi$
$420$	−4.13843e8	−0.272560
$421$	−1.94343e9	−1.26935	−0.634676	−	0.772778i	$-0.718867\pi$
$421$	−1.94343e9	−1.26935	−0.634676	+	0.772778i	$0.718867\pi$
$422$	1.54606e9	1.00146
$423$	−3.82717e8	−0.245859
$424$	−1.92231e9	−1.22474
$425$	2.95999e8	0.187038
$426$	−3.34759e8	−0.209797
$427$	−2.11421e8	−0.131417
$428$	4.61222e9	2.84353
$429$	1.73013e9	1.05798
$430$	−1.34998e9	−0.818817
$431$	−1.39632e9	−0.840069	−0.420035	−	0.907508i	$-0.637982\pi$
$431$	−1.39632e9	−0.840069	−0.420035	+	0.907508i	$0.637982\pi$
$432$	−7.43466e8	−0.443678
$433$	2.08448e9	1.23393	0.616965	−	0.786991i	$-0.288362\pi$
$433$	2.08448e9	1.23393	0.616965	+	0.786991i	$0.288362\pi$
$434$	1.28719e9	0.755840
$435$	7.26748e8	0.423323
$436$	5.56774e9	3.21719
$437$	−1.12345e9	−0.643975
$438$	−2.47975e9	−1.41009
$439$	2.38118e9	1.34328	0.671638	−	0.740879i	$-0.265591\pi$
$439$	2.38118e9	1.34328	0.671638	+	0.740879i	$0.265591\pi$
$440$	4.04778e9	2.26533
$441$	8.57661e7	0.0476190
$442$	9.29393e8	0.511943
$443$	2.62916e9	1.43683	0.718413	−	0.695616i	$-0.244869\pi$
$443$	2.62916e9	1.43683	0.718413	+	0.695616i	$0.244869\pi$
$444$	3.33181e9	1.80651
$445$	1.28874e9	0.693272
$446$	1.59201e9	0.849717
$447$	−3.09515e8	−0.163910
$448$	5.81765e8	0.305685
$449$	−9.88252e8	−0.515235	−0.257618	−	0.966247i	$-0.582938\pi$
$449$	−9.88252e8	−0.515235	−0.257618	+	0.966247i	$0.582938\pi$
$450$	8.60594e8	0.445199
$451$	−4.73964e9	−2.43292
$452$	1.97433e9	1.00562
$453$	1.09124e9	0.551537
$454$	4.95689e8	0.248607
$455$	4.29342e8	0.213680
$456$	−4.69882e9	−2.32066
$457$	−1.12476e9	−0.551254	−0.275627	−	0.961265i	$-0.588885\pi$
$457$	−1.12476e9	−0.551254	−0.275627	+	0.961265i	$0.588885\pi$
$458$	3.04420e9	1.48062
$459$	1.02738e8	0.0495894
$460$	1.06505e9	0.510174
$461$	−3.37528e9	−1.60456	−0.802281	−	0.596947i	$-0.796380\pi$
$461$	−3.37528e9	−1.60456	−0.802281	+	0.596947i	$0.796380\pi$
$462$	−1.44431e9	−0.681416
$463$	−3.27730e9	−1.53456	−0.767278	−	0.641314i	$-0.778390\pi$
$463$	−3.27730e9	−1.53456	−0.767278	+	0.641314i	$0.778390\pi$
$464$	−6.94732e9	−3.22853
$465$	7.12303e8	0.328533
$466$	−1.16509e9	−0.533346
$467$	−2.42750e9	−1.10293	−0.551467	−	0.834197i	$-0.685932\pi$
$467$	−2.42750e9	−1.10293	−0.551467	+	0.834197i	$0.685932\pi$
$468$	1.90400e9	0.858633
$469$	−5.61731e8	−0.251434
$470$	1.59936e9	0.710563
$471$	−8.75482e8	−0.386077
$472$	5.24853e9	2.29742
$473$	−3.31980e9	−1.44244
$474$	1.14609e9	0.494305
$475$	2.67307e9	1.14441
$476$	−5.46689e8	−0.232336
$477$	3.79566e8	0.160130
$478$	−4.13632e9	−1.73227
$479$	1.88150e9	0.782222	0.391111	−	0.920343i	$-0.372091\pi$
$479$	1.88150e9	0.782222	0.391111	+	0.920343i	$0.372091\pi$
$480$	1.23962e9	0.511618
$481$	−3.45659e9	−1.41625
$482$	−5.86881e9	−2.38718
$483$	−2.20725e8	−0.0891325
$484$	1.11876e10	4.48518
$485$	8.98476e8	0.357611
$486$	2.98704e8	0.118036
$487$	3.47380e9	1.36287	0.681433	−	0.731880i	$-0.261357\pi$
$487$	3.47380e9	1.36287	0.681433	+	0.731880i	$0.261357\pi$
$488$	2.27572e9	0.886440
$489$	−9.24690e8	−0.357615
$490$	−3.58412e8	−0.137625
$491$	−2.04528e9	−0.779772	−0.389886	−	0.920863i	$-0.627486\pi$
$491$	−2.04528e9	−0.779772	−0.389886	+	0.920863i	$0.627486\pi$
$492$	−5.21595e9	−1.97449
$493$	9.60039e8	0.360848
$494$	8.39302e9	3.13238
$495$	−7.99246e8	−0.296184
$496$	−6.80923e9	−2.50560
$497$	−2.04287e8	−0.0746437
$498$	−6.07990e8	−0.220594
$499$	−2.42936e9	−0.875266	−0.437633	−	0.899154i	$-0.644183\pi$
$499$	−2.42936e9	−0.875266	−0.437633	+	0.899154i	$0.644183\pi$
$500$	−6.02526e9	−2.15566
$501$	6.41022e8	0.227741
$502$	1.02364e10	3.61146
$503$	−5.95102e7	−0.0208499	−0.0104249	−	0.999946i	$-0.503318\pi$
$503$	−5.95102e7	−0.0208499	−0.0104249	+	0.999946i	$0.503318\pi$
$504$	−9.23178e8	−0.321202
$505$	−6.62140e8	−0.228786
$506$	3.71702e9	1.27546
$507$	−2.81102e8	−0.0957935
$508$	5.44563e9	1.84300
$509$	−1.41531e9	−0.475708	−0.237854	−	0.971301i	$-0.576444\pi$
$509$	−1.41531e9	−0.475708	−0.237854	+	0.971301i	$0.576444\pi$
$510$	−4.29339e8	−0.143319
$511$	−1.51327e9	−0.501699
$512$	6.00011e9	1.97567
$513$	9.27795e8	0.303418
$514$	−1.76953e9	−0.574760
$515$	−1.56926e9	−0.506256
$516$	−3.65342e9	−1.17065
$517$	3.93306e9	1.25174
$518$	2.88554e9	0.912165
$519$	1.67810e8	0.0526904
$520$	−4.62139e9	−1.44132
$521$	−4.15804e9	−1.28812	−0.644061	−	0.764974i	$-0.722752\pi$
$521$	−4.15804e9	−1.28812	−0.644061	+	0.764974i	$0.722752\pi$
$522$	2.79124e9	0.858914
$523$	−4.80852e9	−1.46979	−0.734895	−	0.678181i	$-0.762768\pi$
$523$	−4.80852e9	−1.46979	−0.734895	+	0.678181i	$0.762768\pi$
$524$	−1.25815e10	−3.82008
$525$	5.25178e8	0.158398
$526$	−1.03498e10	−3.10086
$527$	9.40957e8	0.280048
$528$	7.64035e9	2.25889
$529$	−2.83678e9	−0.833163
$530$	−1.58619e9	−0.462795
$531$	−1.03634e9	−0.300380
$532$	−4.93696e9	−1.42157
$533$	5.41130e9	1.54795
$534$	4.94967e9	1.40664
$535$	2.21044e9	0.624081
$536$	6.04642e9	1.69598
$537$	−1.60572e9	−0.447466
$538$	−1.17712e9	−0.325900
$539$	−8.81390e8	−0.242442
$540$	−8.79567e8	−0.240376
$541$	4.85221e9	1.31749	0.658747	−	0.752364i	$-0.271087\pi$
$541$	4.85221e9	1.31749	0.658747	+	0.752364i	$0.271087\pi$
$542$	−8.50556e9	−2.29459
$543$	−3.79913e8	−0.101832
$544$	1.63755e9	0.436112
$545$	2.66838e9	0.706090
$546$	1.64898e9	0.433552
$547$	2.62737e9	0.686383	0.343191	−	0.939265i	$-0.388492\pi$
$547$	2.62737e9	0.686383	0.343191	+	0.939265i	$0.388492\pi$
$548$	−7.35153e9	−1.90830
$549$	−4.49348e8	−0.115899
$550$	−8.84403e9	−2.26663
$551$	8.66978e9	2.20789
$552$	2.37586e9	0.601221
$553$	6.99404e8	0.175869
$554$	−2.05531e9	−0.513563
$555$	1.59679e9	0.396482
$556$	3.65351e9	0.901464
$557$	−5.10989e9	−1.25291	−0.626453	−	0.779459i	$-0.715494\pi$
$557$	−5.10989e9	−1.25291	−0.626453	+	0.779459i	$0.715494\pi$
$558$	2.73576e9	0.666588
$559$	3.79025e9	0.917755
$560$	1.89600e9	0.456225
$561$	−1.05581e9	−0.252473
$562$	8.65753e9	2.05739
$563$	−4.99752e9	−1.18025	−0.590127	−	0.807311i	$-0.700922\pi$
$563$	−4.99752e9	−1.18025	−0.590127	+	0.807311i	$0.700922\pi$
$564$	4.32831e9	1.01588
$565$	9.46210e8	0.220708
$566$	5.23512e9	1.21358
$567$	1.82284e8	0.0419961
$568$	2.19893e9	0.503491
$569$	7.09719e9	1.61508	0.807539	−	0.589814i	$-0.200799\pi$
$569$	7.09719e9	1.61508	0.807539	+	0.589814i	$0.200799\pi$
$570$	−3.87721e9	−0.876914
$571$	5.32103e9	1.19610	0.598052	−	0.801457i	$-0.295942\pi$
$571$	5.32103e9	1.19610	0.598052	+	0.801457i	$0.295942\pi$
$572$	−1.95668e10	−4.37153
$573$	−1.10014e8	−0.0244291
$574$	−4.51732e9	−0.996987
$575$	−1.35158e9	−0.296486
$576$	1.23646e9	0.269589
$577$	7.62952e8	0.165341	0.0826707	−	0.996577i	$-0.473655\pi$
$577$	7.62952e8	0.165341	0.0826707	+	0.996577i	$0.473655\pi$
$578$	7.97493e9	1.71783
$579$	−4.11533e9	−0.881109
$580$	−8.21911e9	−1.74915
$581$	−3.71027e8	−0.0784853
$582$	3.45079e9	0.725586
$583$	−3.90067e9	−0.815266
$584$	1.62887e10	3.38408
$585$	9.12508e8	0.188448
$586$	1.39921e10	2.87238
$587$	6.02648e7	0.0122979	0.00614895	−	0.999981i	$-0.498043\pi$
$587$	6.02648e7	0.0122979	0.00614895	+	0.999981i	$0.498043\pi$
$588$	−9.69966e8	−0.196760
$589$	8.49745e9	1.71350
$590$	4.33080e9	0.868133
$591$	−6.14387e8	−0.122429
$592$	−1.52645e10	−3.02382
$593$	−4.24574e9	−0.836107	−0.418053	−	0.908422i	$-0.637287\pi$
$593$	−4.24574e9	−0.836107	−0.418053	+	0.908422i	$0.637287\pi$
$594$	−3.06968e9	−0.600952
$595$	−2.62005e8	−0.0509917
$596$	3.50043e9	0.677267
$597$	5.03999e9	0.969437
$598$	−4.24376e9	−0.811515
$599$	3.53794e9	0.672600	0.336300	−	0.941755i	$-0.390824\pi$
$599$	3.53794e9	0.672600	0.336300	+	0.941755i	$0.390824\pi$
$600$	−5.65297e9	−1.06843
$601$	9.73539e9	1.82933	0.914666	−	0.404211i	$-0.132454\pi$
$601$	9.73539e9	1.82933	0.914666	+	0.404211i	$0.132454\pi$
$602$	−3.16408e9	−0.591099
$603$	−1.19388e9	−0.221744
$604$	−1.23413e10	−2.27892
$605$	5.36176e9	0.984382
$606$	−2.54309e9	−0.464203
$607$	−8.59678e9	−1.56018	−0.780091	−	0.625666i	$-0.784827\pi$
$607$	−8.59678e9	−1.56018	−0.780091	+	0.625666i	$0.784827\pi$
$608$	1.47881e10	2.66840
$609$	1.70335e9	0.305594
$610$	1.87780e9	0.334962
$611$	−4.49042e9	−0.796420
$612$	−1.16191e9	−0.204901
$613$	−5.18668e9	−0.909448	−0.454724	−	0.890632i	$-0.650262\pi$
$613$	−5.18668e9	−0.909448	−0.454724	+	0.890632i	$0.650262\pi$
$614$	−1.15893e10	−2.02054
$615$	−2.49978e9	−0.433351
$616$	9.48719e9	1.63533
$617$	1.72270e9	0.295265	0.147632	−	0.989042i	$-0.452835\pi$
$617$	1.72270e9	0.295265	0.147632	+	0.989042i	$0.452835\pi$
$618$	−6.02709e9	−1.02718
$619$	−3.21297e9	−0.544489	−0.272244	−	0.962228i	$-0.587766\pi$
$619$	−3.21297e9	−0.544489	−0.272244	+	0.962228i	$0.587766\pi$
$620$	−8.05574e9	−1.35748
$621$	−4.69121e8	−0.0786075
$622$	−2.09775e9	−0.349532
$623$	3.02054e9	0.500469
$624$	−8.72308e9	−1.43722
$625$	1.54271e9	0.252758
$626$	7.57465e7	0.0123411
$627$	−9.53464e9	−1.54478
$628$	9.90120e9	1.59525
$629$	2.10937e9	0.337968
$630$	−7.61757e8	−0.121374
$631$	−4.60771e9	−0.730100	−0.365050	−	0.930988i	$-0.618948\pi$
$631$	−4.60771e9	−0.730100	−0.365050	+	0.930988i	$0.618948\pi$
$632$	−7.52832e9	−1.18628
$633$	2.00525e9	0.314235
$634$	4.14130e9	0.645392
$635$	2.60986e9	0.404491
$636$	−4.29267e9	−0.661649
$637$	1.00629e9	0.154254
$638$	−2.86846e10	−4.37297
$639$	−4.34184e8	−0.0658296
$640$	7.09619e8	0.107003
$641$	−5.64878e9	−0.847133	−0.423567	−	0.905865i	$-0.639222\pi$
$641$	−5.64878e9	−0.847133	−0.423567	+	0.905865i	$0.639222\pi$
$642$	8.48968e9	1.26625
$643$	−5.31882e9	−0.788999	−0.394500	−	0.918896i	$-0.629082\pi$
$643$	−5.31882e9	−0.788999	−0.394500	+	0.918896i	$0.629082\pi$
$644$	2.49627e9	0.368291
$645$	−1.75093e9	−0.256927
$646$	−5.12182e9	−0.747498
$647$	6.45473e9	0.936942	0.468471	−	0.883479i	$-0.344805\pi$
$647$	6.45473e9	0.936942	0.468471	+	0.883479i	$0.344805\pi$
$648$	−1.96209e9	−0.283274
$649$	1.06501e10	1.52931
$650$	1.00973e10	1.44215
$651$	1.66950e9	0.237166
$652$	1.04577e10	1.47765
$653$	−1.29225e10	−1.81615	−0.908074	−	0.418810i	$-0.862447\pi$
$653$	−1.29225e10	−1.81615	−0.908074	+	0.418810i	$0.862447\pi$
$654$	1.02485e10	1.43264
$655$	−6.02977e9	−0.838410
$656$	2.38966e10	3.30500
$657$	−3.21625e9	−0.442457
$658$	3.74858e9	0.512951
$659$	3.17755e9	0.432507	0.216254	−	0.976337i	$-0.430616\pi$
$659$	3.17755e9	0.432507	0.216254	+	0.976337i	$0.430616\pi$
$660$	9.03901e9	1.22382
$661$	−2.00592e9	−0.270152	−0.135076	−	0.990835i	$-0.543128\pi$
$661$	−2.00592e9	−0.270152	−0.135076	+	0.990835i	$0.543128\pi$
$662$	1.17565e10	1.57499
$663$	1.20543e9	0.160636
$664$	3.99369e9	0.529403
$665$	−2.36607e9	−0.311998
$666$	6.13283e9	0.804453
$667$	−4.38370e9	−0.572005
$668$	−7.24959e9	−0.941014
$669$	2.06485e9	0.266623
$670$	4.98918e9	0.640866
$671$	4.61779e9	0.590073
$672$	2.90543e9	0.369333
$673$	−3.21378e8	−0.0406409	−0.0203205	−	0.999794i	$-0.506469\pi$
$673$	−3.21378e8	−0.0406409	−0.0203205	+	0.999794i	$0.506469\pi$
$674$	1.34290e10	1.68941
$675$	1.11620e9	0.139694
$676$	3.17910e9	0.395814
$677$	2.18777e9	0.270982	0.135491	−	0.990779i	$-0.456739\pi$
$677$	2.18777e9	0.270982	0.135491	+	0.990779i	$0.456739\pi$
$678$	3.63413e9	0.447812
$679$	2.10585e9	0.258157
$680$	2.82019e9	0.343952
$681$	6.42912e8	0.0780075
$682$	−2.81144e10	−3.39378
$683$	−5.01121e9	−0.601825	−0.300912	−	0.953652i	$-0.597291\pi$
$683$	−5.01121e9	−0.601825	−0.300912	+	0.953652i	$0.597291\pi$
$684$	−1.04928e10	−1.25371
$685$	−3.52327e9	−0.418822
$686$	−8.40048e8	−0.0993504
$687$	3.94834e9	0.464586
$688$	1.67379e10	1.95949
$689$	4.45344e9	0.518715
$690$	1.96043e9	0.227185
$691$	−1.19761e10	−1.38083	−0.690417	−	0.723411i	$-0.742573\pi$
$691$	−1.19761e10	−1.38083	−0.690417	+	0.723411i	$0.742573\pi$
$692$	−1.89783e9	−0.217714
$693$	−1.87327e9	−0.213813
$694$	−1.45590e10	−1.65338
$695$	1.75097e9	0.197848
$696$	−1.83347e10	−2.06131
$697$	−3.30223e9	−0.369396
$698$	6.57648e9	0.731980
$699$	−1.51113e9	−0.167352
$700$	−5.93947e9	−0.654492
$701$	8.48978e8	0.0930857	0.0465429	−	0.998916i	$-0.485180\pi$
$701$	8.48978e8	0.0930857	0.0465429	+	0.998916i	$0.485180\pi$
$702$	3.50468e9	0.382357
$703$	1.90490e10	2.06790
$704$	−1.27067e10	−1.37255
$705$	2.07438e9	0.222959
$706$	1.52598e10	1.63204
$707$	−1.55193e9	−0.165159
$708$	1.17204e10	1.24115
$709$	−7.27996e9	−0.767127	−0.383564	−	0.923514i	$-0.625303\pi$
$709$	−7.27996e9	−0.767127	−0.383564	+	0.923514i	$0.625303\pi$
$710$	1.81443e9	0.190255
$711$	1.48649e9	0.155102
$712$	−3.25128e10	−3.37579
$713$	−4.29656e9	−0.443923
$714$	−1.00629e9	−0.103461
$715$	−9.37754e9	−0.959439
$716$	1.81598e10	1.84891
$717$	−5.36483e9	−0.543550
$718$	2.72727e10	2.74975
$719$	8.88359e9	0.891327	0.445664	−	0.895200i	$-0.352968\pi$
$719$	8.88359e9	0.891327	0.445664	+	0.895200i	$0.352968\pi$
$720$	4.02968e9	0.402353
$721$	−3.67804e9	−0.365462
$722$	−2.76455e10	−2.73366
$723$	−7.61189e9	−0.749046
$724$	4.29660e9	0.420766
$725$	1.04303e10	1.01651
$726$	2.05930e10	1.99729
$727$	−1.01358e10	−0.978334	−0.489167	−	0.872190i	$-0.662699\pi$
$727$	−1.01358e10	−0.978334	−0.489167	+	0.872190i	$0.662699\pi$
$728$	−1.08316e10	−1.04048
$729$	3.87420e8	0.0370370
$730$	1.34405e10	1.27875
$731$	−2.31299e9	−0.219009
$732$	5.08186e9	0.478888
$733$	−7.14836e9	−0.670414	−0.335207	−	0.942145i	$-0.608806\pi$
$733$	−7.14836e9	−0.670414	−0.335207	+	0.942145i	$0.608806\pi$
$734$	−2.68458e10	−2.50576
$735$	−4.64863e8	−0.0431837
$736$	−7.47733e9	−0.691312
$737$	1.22691e10	1.12896
$738$	−9.60096e9	−0.879260
$739$	−3.77834e9	−0.344386	−0.172193	−	0.985063i	$-0.555085\pi$
$739$	−3.77834e9	−0.344386	−0.172193	+	0.985063i	$0.555085\pi$
$740$	−1.80588e10	−1.63824
$741$	1.08858e10	0.982872
$742$	−3.71771e9	−0.334089
$743$	1.57256e10	1.40652	0.703261	−	0.710932i	$-0.251727\pi$
$743$	1.57256e10	1.40652	0.703261	+	0.710932i	$0.251727\pi$
$744$	−1.79703e10	−1.59974
$745$	1.67761e9	0.148643
$746$	−1.09507e10	−0.965727
$747$	−7.88567e8	−0.0692176
$748$	1.19406e10	1.04321
$749$	5.18084e9	0.450520
$750$	−1.10906e10	−0.959936
$751$	6.17020e9	0.531569	0.265784	−	0.964033i	$-0.414369\pi$
$751$	6.17020e9	0.531569	0.265784	+	0.964033i	$0.414369\pi$
$752$	−1.98299e10	−1.70043
$753$	1.32766e10	1.13320
$754$	3.27495e10	2.78231
$755$	−5.91464e9	−0.500165
$756$	−2.06153e9	−0.173526
$757$	−9.17672e9	−0.768868	−0.384434	−	0.923152i	$-0.625603\pi$
$757$	−9.17672e9	−0.768868	−0.384434	+	0.923152i	$0.625603\pi$
$758$	−3.24980e9	−0.271028
$759$	4.82100e9	0.400212
$760$	2.54682e10	2.10451
$761$	2.97825e9	0.244971	0.122485	−	0.992470i	$-0.460914\pi$
$761$	2.97825e9	0.244971	0.122485	+	0.992470i	$0.460914\pi$
$762$	1.00237e10	0.820704
$763$	6.25415e9	0.509721
$764$	1.24420e9	0.100940
$765$	−5.56855e8	−0.0449705
$766$	−1.32340e10	−1.06388
$767$	−1.21593e10	−0.973029
$768$	8.58719e9	0.684049
$769$	−8.03668e9	−0.637286	−0.318643	−	0.947875i	$-0.603227\pi$
$769$	−8.03668e9	−0.637286	−0.318643	+	0.947875i	$0.603227\pi$
$770$	7.82832e9	0.617947
$771$	−2.29509e9	−0.180347
$772$	4.65420e10	3.64070
$773$	−1.62540e10	−1.26570	−0.632852	−	0.774273i	$-0.718116\pi$
$773$	−1.62540e10	−1.26570	−0.632852	+	0.774273i	$0.718116\pi$
$774$	−6.72483e9	−0.521300
$775$	1.02230e10	0.788898
$776$	−2.26672e10	−1.74133
$777$	3.74257e9	0.286217
$778$	−8.24013e9	−0.627344
$779$	−2.98213e10	−2.26019
$780$	−1.03199e10	−0.778657
$781$	4.46197e9	0.335157
$782$	2.58974e9	0.193657
$783$	3.62025e9	0.269508
$784$	4.44384e9	0.329346
$785$	4.74522e9	0.350117
$786$	−2.31587e10	−1.70112
$787$	2.09632e10	1.53301	0.766507	−	0.642237i	$-0.221993\pi$
$787$	2.09632e10	1.53301	0.766507	+	0.642237i	$0.221993\pi$
$788$	6.94837e9	0.505872
$789$	−1.34238e10	−0.972983
$790$	−6.21196e9	−0.448264
$791$	2.21773e9	0.159328
$792$	2.01637e10	1.44222
$793$	−5.27219e9	−0.375435
$794$	2.89276e10	2.05088
$795$	−2.05730e9	−0.145215
$796$	−5.69994e10	−4.00566
$797$	1.54193e10	1.07885	0.539425	−	0.842033i	$-0.318642\pi$
$797$	1.54193e10	1.07885	0.539425	+	0.842033i	$0.318642\pi$
$798$	−9.08741e9	−0.633039
$799$	2.74026e9	0.190055
$800$	1.77911e10	1.22853
$801$	6.41975e9	0.441372
$802$	1.04347e10	0.714283
$803$	3.30523e10	2.25267
$804$	1.35021e10	0.916234
$805$	1.19636e9	0.0808304
$806$	3.20986e10	2.15930
$807$	−1.52674e9	−0.102260
$808$	1.67048e10	1.11404
$809$	−2.03191e10	−1.34922	−0.674612	−	0.738172i	$-0.735689\pi$
$809$	−2.03191e10	−1.34922	−0.674612	+	0.738172i	$0.735689\pi$
$810$	−1.61901e9	−0.107042
$811$	−2.24501e10	−1.47790	−0.738951	−	0.673759i	$-0.764679\pi$
$811$	−2.24501e10	−1.47790	−0.738951	+	0.673759i	$0.764679\pi$
$812$	−1.92640e10	−1.26270
$813$	−1.10318e10	−0.719993
$814$	−6.30250e10	−4.09569
$815$	5.01194e9	0.324305
$816$	5.32323e9	0.342973
$817$	−2.08878e10	−1.34003
$818$	−5.50513e10	−3.51666
$819$	2.13874e9	0.136039
$820$	2.82711e10	1.79058
$821$	−2.04023e10	−1.28670	−0.643352	−	0.765570i	$-0.722457\pi$
$821$	−2.04023e10	−1.28670	−0.643352	+	0.765570i	$0.722457\pi$
$822$	−1.35319e10	−0.849781
$823$	2.58877e10	1.61881	0.809403	−	0.587254i	$-0.199791\pi$
$823$	2.58877e10	1.61881	0.809403	+	0.587254i	$0.199791\pi$
$824$	3.95900e10	2.46514
$825$	−1.14708e10	−0.711220
$826$	1.01505e10	0.626699
$827$	−7.49901e9	−0.461036	−0.230518	−	0.973068i	$-0.574042\pi$
$827$	−7.49901e9	−0.461036	−0.230518	+	0.973068i	$0.574042\pi$
$828$	5.30549e9	0.324802
$829$	−1.45731e10	−0.888404	−0.444202	−	0.895927i	$-0.646513\pi$
$829$	−1.45731e10	−0.888404	−0.444202	+	0.895927i	$0.646513\pi$
$830$	3.29538e9	0.200047
$831$	−2.66575e9	−0.161145
$832$	1.45074e10	0.873288
$833$	−6.14087e8	−0.0368106
$834$	6.72498e9	0.401430
$835$	−3.47442e9	−0.206528
$836$	1.07831e11	6.38297
$837$	3.54829e9	0.209161
$838$	−3.82253e10	−2.24387
$839$	2.05684e10	1.20236	0.601179	−	0.799114i	$-0.294698\pi$
$839$	2.05684e10	1.20236	0.601179	+	0.799114i	$0.294698\pi$
$840$	5.00374e9	0.291285
$841$	1.65795e10	0.961140
$842$	4.04568e10	2.33560
$843$	1.12289e10	0.645565
$844$	−2.26782e10	−1.29840
$845$	1.52361e9	0.0868710
$846$	7.96709e9	0.452380
$847$	1.25669e10	0.710619
$848$	1.96666e10	1.10750
$849$	6.78998e9	0.380796
$850$	−6.16187e9	−0.344149
$851$	−9.63174e9	−0.535737
$852$	4.91038e9	0.272005
$853$	1.53676e10	0.847781	0.423890	−	0.905713i	$-0.360664\pi$
$853$	1.53676e10	0.847781	0.423890	+	0.905713i	$0.360664\pi$
$854$	4.40120e9	0.241807
$855$	−5.02877e9	−0.275157
$856$	−5.57661e10	−3.03887
$857$	−1.88779e10	−1.02452	−0.512260	−	0.858831i	$-0.671191\pi$
$857$	−1.88779e10	−1.02452	−0.512260	+	0.858831i	$0.671191\pi$
$858$	−3.60165e10	−1.94668
$859$	2.68718e10	1.44651	0.723255	−	0.690582i	$-0.242645\pi$
$859$	2.68718e10	1.44651	0.723255	+	0.690582i	$0.242645\pi$
$860$	1.98020e10	1.06161
$861$	−5.85900e9	−0.312833
$862$	2.90675e10	1.54572
$863$	−5.42902e9	−0.287530	−0.143765	−	0.989612i	$-0.545921\pi$
$863$	−5.42902e9	−0.287530	−0.143765	+	0.989612i	$0.545921\pi$
$864$	6.17510e9	0.325721
$865$	−9.09550e8	−0.0477826
$866$	−4.33930e10	−2.27042
$867$	1.03435e10	0.539017
$868$	−1.88811e10	−0.979958
$869$	−1.52761e10	−0.789668
$870$	−1.51288e10	−0.778912
$871$	−1.40078e10	−0.718302
$872$	−6.73191e10	−3.43820
$873$	4.47570e9	0.227673
$874$	2.33871e10	1.18491
$875$	−6.76808e9	−0.341537
$876$	3.63739e10	1.82821
$877$	3.41739e10	1.71079	0.855394	−	0.517978i	$-0.173315\pi$
$877$	3.41739e10	1.71079	0.855394	+	0.517978i	$0.173315\pi$
$878$	−4.95693e10	−2.47162
$879$	1.81479e10	0.901291
$880$	−4.14117e10	−2.04849
$881$	−9.26839e9	−0.456655	−0.228328	−	0.973584i	$-0.573326\pi$
$881$	−9.26839e9	−0.456655	−0.228328	+	0.973584i	$0.573326\pi$
$882$	−1.78541e9	−0.0876188
$883$	−1.41218e10	−0.690282	−0.345141	−	0.938551i	$-0.612169\pi$
$883$	−1.41218e10	−0.690282	−0.345141	+	0.938551i	$0.612169\pi$
$884$	−1.36327e10	−0.663742
$885$	5.61708e9	0.272401
$886$	−5.47317e10	−2.64375
$887$	−2.85070e10	−1.37157	−0.685786	−	0.727803i	$-0.740541\pi$
$887$	−2.85070e10	−1.37157	−0.685786	+	0.727803i	$0.740541\pi$
$888$	−4.02846e10	−1.93061
$889$	6.11699e9	0.291999
$890$	−2.68278e10	−1.27562
$891$	−3.98139e9	−0.188566
$892$	−2.33523e10	−1.10167
$893$	2.47464e10	1.16287
$894$	6.44322e9	0.301593
$895$	8.70321e9	0.405788
$896$	1.66321e9	0.0772447
$897$	−5.50419e9	−0.254636
$898$	2.05726e10	0.948031
$899$	3.31570e10	1.52201
$900$	−1.26235e10	−0.577208
$901$	−2.71770e9	−0.123784
$902$	9.86658e10	4.47656
$903$	−4.10383e9	−0.185474
$904$	−2.38714e10	−1.07470
$905$	2.05918e9	0.0923472
$906$	−2.27164e10	−1.01483
$907$	6.85174e9	0.304913	0.152456	−	0.988310i	$-0.451282\pi$
$907$	6.85174e9	0.304913	0.152456	+	0.988310i	$0.451282\pi$
$908$	−7.27097e9	−0.322323
$909$	−3.29841e9	−0.145657
$910$	−8.93768e9	−0.393170
$911$	5.99735e8	0.0262812	0.0131406	−	0.999914i	$-0.495817\pi$
$911$	5.99735e8	0.0262812	0.0131406	+	0.999914i	$0.495817\pi$
$912$	4.80723e10	2.09852
$913$	8.10383e9	0.352406
$914$	2.34142e10	1.01430
$915$	2.43552e9	0.105104
$916$	−4.46535e10	−1.91965
$917$	−1.41326e10	−0.605242
$918$	−2.13872e9	−0.0912442
$919$	2.69384e10	1.14490	0.572449	−	0.819940i	$-0.305993\pi$
$919$	2.69384e10	1.14490	0.572449	+	0.819940i	$0.305993\pi$
$920$	−1.28775e10	−0.545221
$921$	−1.50314e10	−0.634003
$922$	7.02638e10	2.95239
$923$	−5.09428e9	−0.213244
$924$	2.11857e10	0.883466
$925$	2.29171e10	0.952060
$926$	6.82241e10	2.82358
$927$	−7.81717e9	−0.322308
$928$	5.77032e10	2.37019
$929$	−2.01987e10	−0.826549	−0.413274	−	0.910607i	$-0.635615\pi$
$929$	−2.01987e10	−0.826549	−0.413274	+	0.910607i	$0.635615\pi$
$930$	−1.48281e10	−0.604500
$931$	−5.54561e9	−0.225229
$932$	1.70900e10	0.691491
$933$	−2.72079e9	−0.109675
$934$	5.05336e10	2.02939
$935$	5.72261e9	0.228957
$936$	−2.30212e10	−0.917618
$937$	1.26215e10	0.501215	0.250608	−	0.968089i	$-0.419370\pi$
$937$	1.26215e10	0.501215	0.250608	+	0.968089i	$0.419370\pi$
$938$	1.16937e10	0.462637
$939$	9.82436e7	0.00387235
$940$	−2.34600e10	−0.921256
$941$	−1.42405e10	−0.557136	−0.278568	−	0.960416i	$-0.589860\pi$
$941$	−1.42405e10	−0.557136	−0.278568	+	0.960416i	$0.589860\pi$
$942$	1.82251e10	0.710380
$943$	1.50785e10	0.585555
$944$	−5.36962e10	−2.07750
$945$	−9.88004e8	−0.0380844
$946$	6.91088e10	2.65408
$947$	−3.69639e10	−1.41434	−0.707169	−	0.707044i	$-0.750028\pi$
$947$	−3.69639e10	−1.41434	−0.707169	+	0.707044i	$0.750028\pi$
$948$	−1.68113e10	−0.640874
$949$	−3.77362e10	−1.43326
$950$	−5.56457e10	−2.10571
$951$	5.37129e9	0.202510
$952$	6.60997e9	0.248296
$953$	9.76882e9	0.365609	0.182805	−	0.983149i	$-0.441482\pi$
$953$	9.76882e9	0.365609	0.182805	+	0.983149i	$0.441482\pi$
$954$	−7.90149e9	−0.294639
$955$	5.96291e8	0.0221537
$956$	6.06732e10	2.24592
$957$	−3.72041e10	−1.37214
$958$	−3.91675e10	−1.43929
$959$	−8.25786e9	−0.302345
$960$	−6.70178e9	−0.244479
$961$	4.98534e9	0.181202
$962$	7.19564e10	2.60589
$963$	1.10112e10	0.397321
$964$	8.60861e10	3.09502
$965$	2.23056e10	0.799039
$966$	4.59487e9	0.164003
$967$	1.90460e10	0.677346	0.338673	−	0.940904i	$-0.390022\pi$
$967$	1.90460e10	0.677346	0.338673	+	0.940904i	$0.390022\pi$
$968$	−1.35269e11	−4.79330
$969$	−6.64303e9	−0.234549
$970$	−1.87037e10	−0.658002
$971$	2.48451e10	0.870912	0.435456	−	0.900210i	$-0.356587\pi$
$971$	2.48451e10	0.870912	0.435456	+	0.900210i	$0.356587\pi$
$972$	−4.38150e9	−0.153035
$973$	4.10393e9	0.142825
$974$	−7.23146e10	−2.50767
$975$	1.30963e10	0.452515
$976$	−2.32822e10	−0.801587
$977$	4.00843e9	0.137513	0.0687564	−	0.997633i	$-0.478097\pi$
$977$	4.00843e9	0.137513	0.0687564	+	0.997633i	$0.478097\pi$
$978$	1.92494e10	0.658009
$979$	−6.59737e10	−2.24715
$980$	5.25734e9	0.178433
$981$	1.32924e10	0.449532
$982$	4.25769e10	1.43478
$983$	−1.57398e8	−0.00528522	−0.00264261	−	0.999997i	$-0.500841\pi$
$983$	−1.57398e8	−0.00528522	−0.00264261	+	0.999997i	$0.500841\pi$
$984$	6.30657e10	2.11014
$985$	3.33005e9	0.111026
$986$	−1.99853e10	−0.663959
$987$	4.86193e9	0.160953
$988$	−1.23112e11	−4.06118
$989$	1.05615e10	0.347167
$990$	1.66380e10	0.544978
$991$	1.52685e10	0.498355	0.249177	−	0.968458i	$-0.419840\pi$
$991$	1.52685e10	0.498355	0.249177	+	0.968458i	$0.419840\pi$
$992$	5.65563e10	1.83946
$993$	1.52483e10	0.494196
$994$	4.25268e9	0.137344
$995$	−2.73174e10	−0.879140
$996$	8.91824e9	0.286003
$997$	−4.93236e10	−1.57624	−0.788120	−	0.615522i	$-0.788945\pi$
$997$	−4.93236e10	−1.57624	−0.788120	+	0.615522i	$0.788945\pi$
$998$	5.05724e10	1.61049
$999$	7.95432e9	0.252420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.8.a.b.1.1	✓	2
3.2	odd	2		63.8.a.f.1.2		2
4.3	odd	2		336.8.a.n.1.2		2
5.4	even	2		525.8.a.e.1.2		2
7.2	even	3		147.8.e.h.67.2		4
7.3	odd	6		147.8.e.g.79.2		4
7.4	even	3		147.8.e.h.79.2		4
7.5	odd	6		147.8.e.g.67.2		4
7.6	odd	2		147.8.a.c.1.1		2
21.20	even	2		441.8.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.8.a.b.1.1	✓	2	1.1	even	1	trivial
63.8.a.f.1.2		2	3.2	odd	2
147.8.a.c.1.1		2	7.6	odd	2
147.8.e.g.67.2		4	7.5	odd	6
147.8.e.g.79.2		4	7.3	odd	6
147.8.e.h.67.2		4	7.2	even	3
147.8.e.h.79.2		4	7.4	even	3
336.8.a.n.1.2		2	4.3	odd	2
441.8.a.m.1.2		2	21.20	even	2
525.8.a.e.1.2		2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)

\(f(q)\)	\(=\)	\(q-20.8172 q^{2} -27.0000 q^{3} +305.355 q^{4} +146.343 q^{5} +562.064 q^{6} +343.000 q^{7} -3692.02 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-20.8172 q^{2} -27.0000 q^{3} +305.355 q^{4} +146.343 q^{5} +562.064 q^{6} +343.000 q^{7} -3692.02 q^{8} +729.000 q^{9} -3046.45 q^{10} -7491.69 q^{11} -8244.57 q^{12} +8553.34 q^{13} -7140.29 q^{14} -3951.27 q^{15} +37772.0 q^{16} -5219.65 q^{17} -15175.7 q^{18} -47136.9 q^{19} +44686.6 q^{20} -9261.00 q^{21} +155956. q^{22} +23833.8 q^{23} +99684.5 q^{24} -56708.6 q^{25} -178056. q^{26} -19683.0 q^{27} +104737. q^{28} -183928. q^{29} +82254.3 q^{30} -180272. q^{31} -313728. q^{32} +202276. q^{33} +108658. q^{34} +50195.8 q^{35} +222603. q^{36} -404121. q^{37} +981256. q^{38} -230940. q^{39} -540303. q^{40} +632653. q^{41} +192788. q^{42} +443131. q^{43} -2.28762e6 q^{44} +106684. q^{45} -496152. q^{46} -524989. q^{47} -1.01984e6 q^{48} +117649. q^{49} +1.18051e6 q^{50} +140931. q^{51} +2.61180e6 q^{52} +520667. q^{53} +409744. q^{54} -1.09636e6 q^{55} -1.26636e6 q^{56} +1.27270e6 q^{57} +3.82886e6 q^{58} -1.42159e6 q^{59} -1.20654e6 q^{60} -616389. q^{61} +3.75275e6 q^{62} +250047. q^{63} +1.69611e6 q^{64} +1.25173e6 q^{65} -4.21080e6 q^{66} -1.63770e6 q^{67} -1.59384e6 q^{68} -643513. q^{69} -1.04493e6 q^{70} -595589. q^{71} -2.69148e6 q^{72} -4.41186e6 q^{73} +8.41266e6 q^{74} +1.53113e6 q^{75} -1.43935e7 q^{76} -2.56965e6 q^{77} +4.80752e6 q^{78} +2.03908e6 q^{79} +5.52768e6 q^{80} +531441. q^{81} -1.31700e7 q^{82} -1.08171e6 q^{83} -2.82789e6 q^{84} -763862. q^{85} -9.22473e6 q^{86} +4.96605e6 q^{87} +2.76595e7 q^{88} +8.80625e6 q^{89} -2.22087e6 q^{90} +2.93380e6 q^{91} +7.27776e6 q^{92} +4.86734e6 q^{93} +1.09288e7 q^{94} -6.89817e6 q^{95} +8.47065e6 q^{96} +6.13951e6 q^{97} -2.44912e6 q^{98} -5.46144e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} - 54 q^{3} + 317 q^{4} - 360 q^{5} + 243 q^{6} + 686 q^{7} - 5067 q^{8} + 1458 q^{9}+O(q^{10}) \) 2 * q - 9 * q^2 - 54 * q^3 + 317 * q^4 - 360 * q^5 + 243 * q^6 + 686 * q^7 - 5067 * q^8 + 1458 * q^9	\( 2 q - 9 q^{2} - 54 q^{3} + 317 q^{4} - 360 q^{5} + 243 q^{6} + 686 q^{7} - 5067 q^{8} + 1458 q^{9} - 9030 q^{10} - 4932 q^{11} - 8559 q^{12} + 7708 q^{13} - 3087 q^{14} + 9720 q^{15} + 20033 q^{16} - 28584 q^{17} - 6561 q^{18} - 63728 q^{19} + 38790 q^{20} - 18522 q^{21} + 186204 q^{22} + 82260 q^{23} + 136809 q^{24} + 121550 q^{25} - 188046 q^{26} - 39366 q^{27} + 108731 q^{28} - 435996 q^{29} + 243810 q^{30} - 29240 q^{31} - 347355 q^{32} + 133164 q^{33} - 167442 q^{34} - 123480 q^{35} + 231093 q^{36} - 709556 q^{37} + 785196 q^{38} - 208116 q^{39} + 155910 q^{40} - 25056 q^{41} + 83349 q^{42} + 496216 q^{43} - 2257812 q^{44} - 262440 q^{45} + 194280 q^{46} - 1575000 q^{47} - 540891 q^{48} + 235298 q^{49} + 3287025 q^{50} + 771768 q^{51} + 2601958 q^{52} + 2057436 q^{53} + 177147 q^{54} - 2392440 q^{55} - 1737981 q^{56} + 1720656 q^{57} + 850122 q^{58} - 1101024 q^{59} - 1047330 q^{60} + 28996 q^{61} + 5537520 q^{62} + 500094 q^{63} + 3569321 q^{64} + 1679760 q^{65} - 5027508 q^{66} - 4480784 q^{67} - 1865934 q^{68} - 2221020 q^{69} - 3097290 q^{70} + 54540 q^{71} - 3693843 q^{72} + 666604 q^{73} + 4803282 q^{74} - 3281850 q^{75} - 14586668 q^{76} - 1691676 q^{77} + 5077242 q^{78} + 2322952 q^{79} + 14509710 q^{80} + 1062882 q^{81} - 20942298 q^{82} - 7384392 q^{83} - 2935737 q^{84} + 11066520 q^{85} - 8597412 q^{86} + 11771892 q^{87} + 24139932 q^{88} + 1784448 q^{89} - 6582870 q^{90} + 2643844 q^{91} + 7958160 q^{92} + 789480 q^{93} - 1479360 q^{94} + 1502640 q^{95} + 9378585 q^{96} + 16266412 q^{97} - 1058841 q^{98} - 3595428 q^{99}+O(q^{100}) \) 2 * q - 9 * q^2 - 54 * q^3 + 317 * q^4 - 360 * q^5 + 243 * q^6 + 686 * q^7 - 5067 * q^8 + 1458 * q^9 - 9030 * q^10 - 4932 * q^11 - 8559 * q^12 + 7708 * q^13 - 3087 * q^14 + 9720 * q^15 + 20033 * q^16 - 28584 * q^17 - 6561 * q^18 - 63728 * q^19 + 38790 * q^20 - 18522 * q^21 + 186204 * q^22 + 82260 * q^23 + 136809 * q^24 + 121550 * q^25 - 188046 * q^26 - 39366 * q^27 + 108731 * q^28 - 435996 * q^29 + 243810 * q^30 - 29240 * q^31 - 347355 * q^32 + 133164 * q^33 - 167442 * q^34 - 123480 * q^35 + 231093 * q^36 - 709556 * q^37 + 785196 * q^38 - 208116 * q^39 + 155910 * q^40 - 25056 * q^41 + 83349 * q^42 + 496216 * q^43 - 2257812 * q^44 - 262440 * q^45 + 194280 * q^46 - 1575000 * q^47 - 540891 * q^48 + 235298 * q^49 + 3287025 * q^50 + 771768 * q^51 + 2601958 * q^52 + 2057436 * q^53 + 177147 * q^54 - 2392440 * q^55 - 1737981 * q^56 + 1720656 * q^57 + 850122 * q^58 - 1101024 * q^59 - 1047330 * q^60 + 28996 * q^61 + 5537520 * q^62 + 500094 * q^63 + 3569321 * q^64 + 1679760 * q^65 - 5027508 * q^66 - 4480784 * q^67 - 1865934 * q^68 - 2221020 * q^69 - 3097290 * q^70 + 54540 * q^71 - 3693843 * q^72 + 666604 * q^73 + 4803282 * q^74 - 3281850 * q^75 - 14586668 * q^76 - 1691676 * q^77 + 5077242 * q^78 + 2322952 * q^79 + 14509710 * q^80 + 1062882 * q^81 - 20942298 * q^82 - 7384392 * q^83 - 2935737 * q^84 + 11066520 * q^85 - 8597412 * q^86 + 11771892 * q^87 + 24139932 * q^88 + 1784448 * q^89 - 6582870 * q^90 + 2643844 * q^91 + 7958160 * q^92 + 789480 * q^93 - 1479360 * q^94 + 1502640 * q^95 + 9378585 * q^96 + 16266412 * q^97 - 1058841 * q^98 - 3595428 * q^99

Introduction

L-functions

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