LMFDB - Embedded newform 15.12.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,12,Mod(1,15)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("15.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	15.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:	$11.5251477084$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5450x - 7248 $ x^3 - x^2 - 5450*x - 7248
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$74.9776$ of defining polynomial
Character	$\chi$	$=$	15.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-74.9776 q^{2} +243.000 q^{3} +3573.65 q^{4} +3125.00 q^{5} -18219.6 q^{6} -63061.5 q^{7} -114389. q^{8} +59049.0 q^{9} +O(q^{10})$	\(q-74.9776 q^{2} +243.000 q^{3} +3573.65 q^{4} +3125.00 q^{5} -18219.6 q^{6} -63061.5 q^{7} -114389. q^{8} +59049.0 q^{9} -234305. q^{10} +256618. q^{11} +868396. q^{12} -465623. q^{13} +4.72820e6 q^{14} +759375. q^{15} +1.25782e6 q^{16} +1.08463e7 q^{17} -4.42735e6 q^{18} +9.38335e6 q^{19} +1.11676e7 q^{20} -1.53239e7 q^{21} -1.92406e7 q^{22} -2.41377e7 q^{23} -2.77966e7 q^{24} +9.76562e6 q^{25} +3.49113e7 q^{26} +1.43489e7 q^{27} -2.25359e8 q^{28} +2.07639e8 q^{29} -5.69361e7 q^{30} +2.06141e8 q^{31} +1.39961e8 q^{32} +6.23583e7 q^{33} -8.13230e8 q^{34} -1.97067e8 q^{35} +2.11020e8 q^{36} +2.18529e8 q^{37} -7.03542e8 q^{38} -1.13146e8 q^{39} -3.57467e8 q^{40} +8.54426e8 q^{41} +1.14895e9 q^{42} -2.70952e8 q^{43} +9.17063e8 q^{44} +1.84528e8 q^{45} +1.80979e9 q^{46} -2.09625e9 q^{47} +3.05650e8 q^{48} +1.99942e9 q^{49} -7.32204e8 q^{50} +2.63565e9 q^{51} -1.66397e9 q^{52} -1.20106e9 q^{53} -1.07585e9 q^{54} +8.01933e8 q^{55} +7.21356e9 q^{56} +2.28015e9 q^{57} -1.55683e10 q^{58} +6.04665e9 q^{59} +2.71374e9 q^{60} -1.28841e10 q^{61} -1.54560e10 q^{62} -3.72372e9 q^{63} -1.30700e10 q^{64} -1.45507e9 q^{65} -4.67548e9 q^{66} +6.01919e9 q^{67} +3.87608e10 q^{68} -5.86547e9 q^{69} +1.47756e10 q^{70} +7.99639e9 q^{71} -6.75458e9 q^{72} +3.23383e10 q^{73} -1.63848e10 q^{74} +2.37305e9 q^{75} +3.35328e10 q^{76} -1.61827e10 q^{77} +8.48344e9 q^{78} +3.84151e9 q^{79} +3.93068e9 q^{80} +3.48678e9 q^{81} -6.40628e10 q^{82} -1.29073e9 q^{83} -5.47623e10 q^{84} +3.38947e10 q^{85} +2.03154e10 q^{86} +5.04563e10 q^{87} -2.93544e10 q^{88} -6.96454e10 q^{89} -1.38355e10 q^{90} +2.93628e10 q^{91} -8.62597e10 q^{92} +5.00923e10 q^{93} +1.57172e11 q^{94} +2.93230e10 q^{95} +3.40106e10 q^{96} +2.33370e10 q^{97} -1.49912e11 q^{98} +1.51531e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 729 q^{3} + 4757 q^{4} + 9375 q^{5} - 243 q^{6} - 14608 q^{7} - 33999 q^{8} + 177147 q^{9}+O(q^{10}) $ 3 * q - q^2 + 729 * q^3 + 4757 * q^4 + 9375 * q^5 - 243 * q^6 - 14608 * q^7 - 33999 * q^8 + 177147 * q^9	\( 3 q - q^{2} + 729 q^{3} + 4757 q^{4} + 9375 q^{5} - 243 q^{6} - 14608 q^{7} - 33999 q^{8} + 177147 q^{9} - 3125 q^{10} + 540620 q^{11} + 1155951 q^{12} + 840970 q^{13} + 5432712 q^{14} + 2278125 q^{15} + 5062961 q^{16} + 15165038 q^{17} - 59049 q^{18} + 17743756 q^{19} + 14865625 q^{20} - 3549744 q^{21} + 11176076 q^{22} - 28140816 q^{23} - 8261757 q^{24} + 29296875 q^{25} - 52021894 q^{26} + 43046721 q^{27} - 277157944 q^{28} - 67382798 q^{29} - 759375 q^{30} - 206919496 q^{31} - 46592663 q^{32} + 131370660 q^{33} - 1230469666 q^{34} - 45650000 q^{35} + 280896093 q^{36} - 318337278 q^{37} + 653190692 q^{38} + 204355710 q^{39} - 106246875 q^{40} + 2110085854 q^{41} + 1320149016 q^{42} + 418259692 q^{43} + 2558131108 q^{44} + 553584375 q^{45} - 137169096 q^{46} - 1599668584 q^{47} + 1230299523 q^{48} - 316107077 q^{49} - 9765625 q^{50} + 3685104234 q^{51} - 10897289202 q^{52} + 4489142234 q^{53} - 14348907 q^{54} + 1689437500 q^{55} + 7768845960 q^{56} + 4311732708 q^{57} - 24168830726 q^{58} + 11102167484 q^{59} + 3612346875 q^{60} - 3568120958 q^{61} - 35509109136 q^{62} - 862587792 q^{63} - 35608208271 q^{64} + 2628031250 q^{65} + 2715786468 q^{66} + 2229942788 q^{67} - 1367872838 q^{68} - 6838218288 q^{69} + 16977225000 q^{70} + 49842766696 q^{71} - 2007606951 q^{72} + 40752219934 q^{73} - 37519971278 q^{74} + 7119140625 q^{75} + 115970329116 q^{76} - 17819224896 q^{77} - 12641320242 q^{78} + 113159960 q^{79} + 15821753125 q^{80} + 10460353203 q^{81} - 30171431066 q^{82} + 6259660308 q^{83} - 67349380392 q^{84} + 47390743750 q^{85} + 114296127740 q^{86} - 16374019914 q^{87} + 7548672276 q^{88} - 59972401554 q^{89} - 184528125 q^{90} + 118873361824 q^{91} - 221705928648 q^{92} - 50281437528 q^{93} + 92816682800 q^{94} + 55449237500 q^{95} - 11322017109 q^{96} - 207831285882 q^{97} - 288264739625 q^{98} + 31923070380 q^{99}+O(q^{100}) \) 3 * q - q^2 + 729 * q^3 + 4757 * q^4 + 9375 * q^5 - 243 * q^6 - 14608 * q^7 - 33999 * q^8 + 177147 * q^9 - 3125 * q^10 + 540620 * q^11 + 1155951 * q^12 + 840970 * q^13 + 5432712 * q^14 + 2278125 * q^15 + 5062961 * q^16 + 15165038 * q^17 - 59049 * q^18 + 17743756 * q^19 + 14865625 * q^20 - 3549744 * q^21 + 11176076 * q^22 - 28140816 * q^23 - 8261757 * q^24 + 29296875 * q^25 - 52021894 * q^26 + 43046721 * q^27 - 277157944 * q^28 - 67382798 * q^29 - 759375 * q^30 - 206919496 * q^31 - 46592663 * q^32 + 131370660 * q^33 - 1230469666 * q^34 - 45650000 * q^35 + 280896093 * q^36 - 318337278 * q^37 + 653190692 * q^38 + 204355710 * q^39 - 106246875 * q^40 + 2110085854 * q^41 + 1320149016 * q^42 + 418259692 * q^43 + 2558131108 * q^44 + 553584375 * q^45 - 137169096 * q^46 - 1599668584 * q^47 + 1230299523 * q^48 - 316107077 * q^49 - 9765625 * q^50 + 3685104234 * q^51 - 10897289202 * q^52 + 4489142234 * q^53 - 14348907 * q^54 + 1689437500 * q^55 + 7768845960 * q^56 + 4311732708 * q^57 - 24168830726 * q^58 + 11102167484 * q^59 + 3612346875 * q^60 - 3568120958 * q^61 - 35509109136 * q^62 - 862587792 * q^63 - 35608208271 * q^64 + 2628031250 * q^65 + 2715786468 * q^66 + 2229942788 * q^67 - 1367872838 * q^68 - 6838218288 * q^69 + 16977225000 * q^70 + 49842766696 * q^71 - 2007606951 * q^72 + 40752219934 * q^73 - 37519971278 * q^74 + 7119140625 * q^75 + 115970329116 * q^76 - 17819224896 * q^77 - 12641320242 * q^78 + 113159960 * q^79 + 15821753125 * q^80 + 10460353203 * q^81 - 30171431066 * q^82 + 6259660308 * q^83 - 67349380392 * q^84 + 47390743750 * q^85 + 114296127740 * q^86 - 16374019914 * q^87 + 7548672276 * q^88 - 59972401554 * q^89 - 184528125 * q^90 + 118873361824 * q^91 - 221705928648 * q^92 - 50281437528 * q^93 + 92816682800 * q^94 + 55449237500 * q^95 - 11322017109 * q^96 - 207831285882 * q^97 - 288264739625 * q^98 + 31923070380 * 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 $

$2$	−74.9776	−1.65679	−0.828394	−	0.560146i	$-0.810745\pi$
$2$	−74.9776	−1.65679	−0.828394	+	0.560146i	$0.810745\pi$
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	3573.65	1.74494
$5$	3125.00	0.447214
$5$	3125.00	0.447214
$6$	−18219.6	−0.956547
$7$	−63061.5	−1.41816	−0.709079	−	0.705129i	$-0.750889\pi$
$7$	−63061.5	−1.41816	−0.709079	+	0.705129i	$0.750889\pi$
$8$	−114389.	−1.23421
$9$	59049.0	0.333333
$10$	−234305.	−0.740938
$11$	256618.	0.480428	0.240214	−	0.970720i	$-0.422782\pi$
$11$	256618.	0.480428	0.240214	+	0.970720i	$0.422782\pi$
$12$	868396.	1.00744
$13$	−465623.	−0.347813	−0.173906	−	0.984762i	$-0.555639\pi$
$13$	−465623.	−0.347813	−0.173906	+	0.984762i	$0.555639\pi$
$14$	4.72820e6	2.34959
$15$	759375.	0.258199
$16$	1.25782e6	0.299887
$17$	1.08463e7	1.85273	0.926366	−	0.376626i	$-0.122916\pi$
$17$	1.08463e7	1.85273	0.926366	+	0.376626i	$0.122916\pi$
$18$	−4.42735e6	−0.552262
$19$	9.38335e6	0.869387	0.434694	−	0.900578i	$-0.356857\pi$
$19$	9.38335e6	0.869387	0.434694	+	0.900578i	$0.356857\pi$
$20$	1.11676e7	0.780363
$21$	−1.53239e7	−0.818774
$22$	−1.92406e7	−0.795967
$23$	−2.41377e7	−0.781976	−0.390988	−	0.920396i	$-0.627867\pi$
$23$	−2.41377e7	−0.781976	−0.390988	+	0.920396i	$0.627867\pi$
$24$	−2.77966e7	−0.712574
$25$	9.76562e6	0.200000
$26$	3.49113e7	0.576252
$27$	1.43489e7	0.192450
$28$	−2.25359e8	−2.47461
$29$	2.07639e8	1.87984	0.939918	−	0.341402i	$-0.110902\pi$
$29$	2.07639e8	1.87984	0.939918	+	0.341402i	$0.110902\pi$
$30$	−5.69361e7	−0.427781
$31$	2.06141e8	1.29323	0.646615	−	0.762817i	$-0.276184\pi$
$31$	2.06141e8	1.29323	0.646615	+	0.762817i	$0.276184\pi$
$32$	1.39961e8	0.737366
$33$	6.23583e7	0.277375
$34$	−8.13230e8	−3.06958
$35$	−1.97067e8	−0.634220
$36$	2.11020e8	0.581648
$37$	2.18529e8	0.518082	0.259041	−	0.965866i	$-0.416593\pi$
$37$	2.18529e8	0.518082	0.259041	+	0.965866i	$0.416593\pi$
$38$	−7.03542e8	−1.44039
$39$	−1.13146e8	−0.200810
$40$	−3.57467e8	−0.551958
$41$	8.54426e8	1.15176	0.575882	−	0.817533i	$-0.304659\pi$
$41$	8.54426e8	1.15176	0.575882	+	0.817533i	$0.304659\pi$
$42$	1.14895e9	1.35654
$43$	−2.70952e8	−0.281071	−0.140536	−	0.990076i	$-0.544882\pi$
$43$	−2.70952e8	−0.281071	−0.140536	+	0.990076i	$0.544882\pi$
$44$	9.17063e8	0.838320
$45$	1.84528e8	0.149071
$46$	1.80979e9	1.29557
$47$	−2.09625e9	−1.33323	−0.666616	−	0.745401i	$-0.732258\pi$
$47$	−2.09625e9	−1.33323	−0.666616	+	0.745401i	$0.732258\pi$
$48$	3.05650e8	0.173140
$49$	1.99942e9	1.01117
$50$	−7.32204e8	−0.331357
$51$	2.63565e9	1.06967
$52$	−1.66397e9	−0.606914
$53$	−1.20106e9	−0.394501	−0.197251	−	0.980353i	$-0.563201\pi$
$53$	−1.20106e9	−0.394501	−0.197251	+	0.980353i	$0.563201\pi$
$54$	−1.07585e9	−0.318849
$55$	8.01933e8	0.214854
$56$	7.21356e9	1.75031
$57$	2.28015e9	0.501941
$58$	−1.55683e10	−3.11449
$59$	6.04665e9	1.10111	0.550553	−	0.834800i	$-0.314417\pi$
$59$	6.04665e9	1.10111	0.550553	+	0.834800i	$0.314417\pi$
$60$	2.71374e9	0.450543
$61$	−1.28841e10	−1.95316	−0.976581	−	0.215148i	$-0.930977\pi$
$61$	−1.28841e10	−1.95316	−0.976581	+	0.215148i	$0.930977\pi$
$62$	−1.54560e10	−2.14261
$63$	−3.72372e9	−0.472720
$64$	−1.30700e10	−1.52155
$65$	−1.45507e9	−0.155547
$66$	−4.67548e9	−0.459552
$67$	6.01919e9	0.544661	0.272331	−	0.962204i	$-0.412206\pi$
$67$	6.01919e9	0.544661	0.272331	+	0.962204i	$0.412206\pi$
$68$	3.87608e10	3.23291
$69$	−5.86547e9	−0.451474
$70$	1.47756e10	1.05077
$71$	7.99639e9	0.525984	0.262992	−	0.964798i	$-0.415291\pi$
$71$	7.99639e9	0.525984	0.262992	+	0.964798i	$0.415291\pi$
$72$	−6.75458e9	−0.411405
$73$	3.23383e10	1.82575	0.912876	−	0.408236i	$-0.133856\pi$
$73$	3.23383e10	1.82575	0.912876	+	0.408236i	$0.133856\pi$
$74$	−1.63848e10	−0.858352
$75$	2.37305e9	0.115470
$76$	3.35328e10	1.51703
$77$	−1.61827e10	−0.681323
$78$	8.48344e9	0.332699
$79$	3.84151e9	0.140460	0.0702300	−	0.997531i	$-0.477627\pi$
$79$	3.84151e9	0.140460	0.0702300	+	0.997531i	$0.477627\pi$
$80$	3.93068e9	0.134114
$81$	3.48678e9	0.111111
$82$	−6.40628e10	−1.90823
$83$	−1.29073e9	−0.0359671	−0.0179836	−	0.999838i	$-0.505725\pi$
$83$	−1.29073e9	−0.0359671	−0.0179836	+	0.999838i	$0.505725\pi$
$84$	−5.47623e10	−1.42872
$85$	3.38947e10	0.828567
$86$	2.03154e10	0.465675
$87$	5.04563e10	1.08532
$88$	−2.93544e10	−0.592951
$89$	−6.96454e10	−1.32205	−0.661024	−	0.750365i	$-0.729878\pi$
$89$	−6.96454e10	−1.32205	−0.661024	+	0.750365i	$0.729878\pi$
$90$	−1.38355e10	−0.246979
$91$	2.93628e10	0.493254
$92$	−8.62597e10	−1.36450
$93$	5.00923e10	0.746646
$94$	1.57172e11	2.20888
$95$	2.93230e10	0.388802
$96$	3.40106e10	0.425718
$97$	2.33370e10	0.275931	0.137965	−	0.990437i	$-0.455944\pi$
$97$	2.33370e10	0.275931	0.137965	+	0.990437i	$0.455944\pi$
$98$	−1.49912e11	−1.67530
$99$	1.51531e10	0.160143
$100$	3.48989e10	0.348989
$101$	−6.21623e10	−0.588518	−0.294259	−	0.955726i	$-0.595073\pi$
$101$	−6.21623e10	−0.588518	−0.294259	+	0.955726i	$0.595073\pi$
$102$	−1.97615e11	−1.77222
$103$	−1.26451e11	−1.07478	−0.537389	−	0.843334i	$-0.680589\pi$
$103$	−1.26451e11	−1.07478	−0.537389	+	0.843334i	$0.680589\pi$
$104$	5.32623e10	0.429276
$105$	−4.78873e10	−0.366167
$106$	9.00529e10	0.653605
$107$	6.79291e10	0.468215	0.234107	−	0.972211i	$-0.424783\pi$
$107$	6.79291e10	0.468215	0.234107	+	0.972211i	$0.424783\pi$
$108$	5.12779e10	0.335815
$109$	1.15778e11	0.720746	0.360373	−	0.932808i	$-0.382649\pi$
$109$	1.15778e11	0.720746	0.360373	+	0.932808i	$0.382649\pi$
$110$	−6.01270e10	−0.355967
$111$	5.31024e10	0.299115
$112$	−7.93198e10	−0.425287
$113$	−6.09916e10	−0.311414	−0.155707	−	0.987803i	$-0.549766\pi$
$113$	−6.09916e10	−0.311414	−0.155707	+	0.987803i	$0.549766\pi$
$114$	−1.70961e11	−0.831609
$115$	−7.54304e10	−0.349710
$116$	7.42028e11	3.28021
$117$	−2.74945e10	−0.115938
$118$	−4.53364e11	−1.82430
$119$	−6.83983e11	−2.62747
$120$	−8.68644e10	−0.318673
$121$	−2.19459e11	−0.769189
$122$	9.66016e11	3.23598
$123$	2.07626e11	0.664971
$124$	7.36676e11	2.25661
$125$	3.05176e10	0.0894427
$126$	2.79196e11	0.783196
$127$	−1.77862e11	−0.477707	−0.238853	−	0.971056i	$-0.576772\pi$
$127$	−1.77862e11	−0.477707	−0.238853	+	0.971056i	$0.576772\pi$
$128$	6.93315e11	1.78351
$129$	−6.58414e10	−0.162277
$130$	1.09098e11	0.257708
$131$	4.59970e10	0.104169	0.0520843	−	0.998643i	$-0.483414\pi$
$131$	4.59970e10	0.104169	0.0520843	+	0.998643i	$0.483414\pi$
$132$	2.22846e11	0.484004
$133$	−5.91728e11	−1.23293
$134$	−4.51304e11	−0.902388
$135$	4.48403e10	0.0860663
$136$	−1.24070e12	−2.28667
$137$	−4.98200e10	−0.0881943	−0.0440972	−	0.999027i	$-0.514041\pi$
$137$	−4.98200e10	−0.0881943	−0.0440972	+	0.999027i	$0.514041\pi$
$138$	4.39779e11	0.747997
$139$	1.80909e11	0.295718	0.147859	−	0.989008i	$-0.452762\pi$
$139$	1.80909e11	0.295718	0.147859	+	0.989008i	$0.452762\pi$
$140$	−7.04248e11	−1.10668
$141$	−5.09390e11	−0.769742
$142$	−5.99550e11	−0.871444
$143$	−1.19487e11	−0.167099
$144$	7.42728e10	0.0999623
$145$	6.48872e11	0.840688
$146$	−2.42465e12	−3.02488
$147$	4.85860e11	0.583802
$148$	7.80944e11	0.904025
$149$	5.43337e11	0.606101	0.303051	−	0.952974i	$-0.401995\pi$
$149$	5.43337e11	0.606101	0.303051	+	0.952974i	$0.401995\pi$
$150$	−1.77925e11	−0.191309
$151$	−1.19411e12	−1.23786	−0.618930	−	0.785446i	$-0.712434\pi$
$151$	−1.19411e12	−1.23786	−0.618930	+	0.785446i	$0.712434\pi$
$152$	−1.07336e12	−1.07301
$153$	6.40463e11	0.617577
$154$	1.21334e12	1.12881
$155$	6.44191e11	0.578350
$156$	−4.04345e11	−0.350402
$157$	2.21075e12	1.84965	0.924827	−	0.380387i	$-0.124209\pi$
$157$	2.21075e12	1.84965	0.924827	+	0.380387i	$0.124209\pi$
$158$	−2.88027e11	−0.232712
$159$	−2.91858e11	−0.227765
$160$	4.37379e11	0.329760
$161$	1.52216e12	1.10897
$162$	−2.61431e11	−0.184087
$163$	−1.38866e12	−0.945291	−0.472645	−	0.881253i	$-0.656701\pi$
$163$	−1.38866e12	−0.945291	−0.472645	+	0.881253i	$0.656701\pi$
$164$	3.05342e12	2.00976
$165$	1.94870e11	0.124046
$166$	9.67758e10	0.0595899
$167$	7.93403e11	0.472665	0.236332	−	0.971672i	$-0.424055\pi$
$167$	7.93403e11	0.472665	0.236332	+	0.971672i	$0.424055\pi$
$168$	1.75290e12	1.01054
$169$	−1.57536e12	−0.879026
$170$	−2.54134e12	−1.37276
$171$	5.54078e11	0.289796
$172$	−9.68288e11	−0.490454
$173$	2.86863e12	1.40741	0.703706	−	0.710491i	$-0.251527\pi$
$173$	2.86863e12	1.40741	0.703706	+	0.710491i	$0.251527\pi$
$174$	−3.78309e12	−1.79815
$175$	−6.15835e11	−0.283632
$176$	3.22779e11	0.144074
$177$	1.46934e12	0.635724
$178$	5.22185e12	2.19035
$179$	6.99004e10	0.0284307	0.0142154	−	0.999899i	$-0.495475\pi$
$179$	6.99004e10	0.0284307	0.0142154	+	0.999899i	$0.495475\pi$
$180$	6.59438e11	0.260121
$181$	−2.96087e12	−1.13289	−0.566445	−	0.824100i	$-0.691682\pi$
$181$	−2.96087e12	−1.13289	−0.566445	+	0.824100i	$0.691682\pi$
$182$	−2.20156e12	−0.817217
$183$	−3.13083e12	−1.12766
$184$	2.76110e12	0.965126
$185$	6.82902e11	0.231693
$186$	−3.75580e12	−1.23703
$187$	2.78336e12	0.890103
$188$	−7.49127e12	−2.32642
$189$	−9.04863e11	−0.272925
$190$	−2.19857e12	−0.644162
$191$	4.52689e12	1.28860	0.644298	−	0.764775i	$-0.277150\pi$
$191$	4.52689e12	1.28860	0.644298	+	0.764775i	$0.277150\pi$
$192$	−3.17600e12	−0.878465
$193$	−1.04014e12	−0.279593	−0.139796	−	0.990180i	$-0.544645\pi$
$193$	−1.04014e12	−0.279593	−0.139796	+	0.990180i	$0.544645\pi$
$194$	−1.74975e12	−0.457159
$195$	−3.53582e11	−0.0898049
$196$	7.14523e12	1.76444
$197$	5.42576e12	1.30285	0.651427	−	0.758711i	$-0.274170\pi$
$197$	5.42576e12	1.30285	0.651427	+	0.758711i	$0.274170\pi$
$198$	−1.13614e12	−0.265322
$199$	−7.72708e12	−1.75519	−0.877593	−	0.479406i	$-0.840852\pi$
$199$	−7.72708e12	−1.75519	−0.877593	+	0.479406i	$0.840852\pi$
$200$	−1.11708e12	−0.246843
$201$	1.46266e12	0.314460
$202$	4.66078e12	0.975049
$203$	−1.30940e13	−2.66590
$204$	9.41888e12	1.86652
$205$	2.67008e12	0.515084
$206$	9.48103e12	1.78068
$207$	−1.42531e12	−0.260659
$208$	−5.85668e11	−0.104305
$209$	2.40794e12	0.417678
$210$	3.59048e12	0.606661
$211$	2.66403e12	0.438517	0.219258	−	0.975667i	$-0.429636\pi$
$211$	2.66403e12	0.438517	0.219258	+	0.975667i	$0.429636\pi$
$212$	−4.29217e12	−0.688383
$213$	1.94312e12	0.303677
$214$	−5.09317e12	−0.775733
$215$	−8.46726e11	−0.125699
$216$	−1.64136e12	−0.237525
$217$	−1.29996e13	−1.83400
$218$	−8.68080e12	−1.19412
$219$	7.85822e12	1.05410
$220$	2.86582e12	0.374908
$221$	−5.05028e12	−0.644404
$222$	−3.98149e12	−0.495570
$223$	1.73148e12	0.210252	0.105126	−	0.994459i	$-0.466475\pi$
$223$	1.73148e12	0.210252	0.105126	+	0.994459i	$0.466475\pi$
$224$	−8.82616e12	−1.04570
$225$	5.76650e11	0.0666667
$226$	4.57301e12	0.515948
$227$	8.49255e12	0.935181	0.467590	−	0.883945i	$-0.345122\pi$
$227$	8.49255e12	0.935181	0.467590	+	0.883945i	$0.345122\pi$
$228$	8.14847e12	0.875859
$229$	7.60395e12	0.797892	0.398946	−	0.916974i	$-0.369376\pi$
$229$	7.60395e12	0.797892	0.398946	+	0.916974i	$0.369376\pi$
$230$	5.65560e12	0.579396
$231$	−3.93240e12	−0.393362
$232$	−2.37517e13	−2.32012
$233$	1.09237e13	1.04210	0.521051	−	0.853525i	$-0.325540\pi$
$233$	1.09237e13	1.04210	0.521051	+	0.853525i	$0.325540\pi$
$234$	2.06148e12	0.192084
$235$	−6.55079e12	−0.596240
$236$	2.16086e13	1.92137
$237$	9.33486e11	0.0810946
$238$	5.12835e13	4.35315
$239$	5.91005e12	0.490233	0.245117	−	0.969494i	$-0.421174\pi$
$239$	5.91005e12	0.490233	0.245117	+	0.969494i	$0.421174\pi$
$240$	9.55155e11	0.0774305
$241$	2.58394e12	0.204734	0.102367	−	0.994747i	$-0.467358\pi$
$241$	2.58394e12	0.204734	0.102367	+	0.994747i	$0.467358\pi$
$242$	1.64545e13	1.27438
$243$	8.47289e11	0.0641500
$244$	−4.60431e13	−3.40816
$245$	6.24820e12	0.452211
$246$	−1.55673e13	−1.10172
$247$	−4.36910e12	−0.302384
$248$	−2.35804e13	−1.59612
$249$	−3.13647e11	−0.0207656
$250$	−2.28814e12	−0.148188
$251$	−1.89161e13	−1.19847	−0.599234	−	0.800574i	$-0.704528\pi$
$251$	−1.89161e13	−1.19847	−0.599234	+	0.800574i	$0.704528\pi$
$252$	−1.33072e13	−0.824870
$253$	−6.19419e12	−0.375683
$254$	1.33356e13	0.791459
$255$	8.23641e12	0.478373
$256$	−2.52158e13	−1.43335
$257$	−1.55673e13	−0.866125	−0.433062	−	0.901364i	$-0.642567\pi$
$257$	−1.55673e13	−0.866125	−0.433062	+	0.901364i	$0.642567\pi$
$258$	4.93664e12	0.268858
$259$	−1.37807e13	−0.734723
$260$	−5.19991e12	−0.271420
$261$	1.22609e13	0.626612
$262$	−3.44874e12	−0.172585
$263$	−2.33801e13	−1.14575	−0.572874	−	0.819643i	$-0.694172\pi$
$263$	−2.33801e13	−1.14575	−0.572874	+	0.819643i	$0.694172\pi$
$264$	−7.13312e12	−0.342340
$265$	−3.75332e12	−0.176426
$266$	4.43664e13	2.04270
$267$	−1.69238e13	−0.763285
$268$	2.15104e13	0.950403
$269$	1.52689e13	0.660953	0.330476	−	0.943814i	$-0.392791\pi$
$269$	1.52689e13	0.660953	0.330476	+	0.943814i	$0.392791\pi$
$270$	−3.36202e12	−0.142594
$271$	−2.59439e13	−1.07821	−0.539105	−	0.842239i	$-0.681237\pi$
$271$	−2.59439e13	−1.07821	−0.539105	+	0.842239i	$0.681237\pi$
$272$	1.36427e13	0.555610
$273$	7.13517e12	0.284780
$274$	3.73539e12	0.146119
$275$	2.50604e12	0.0960855
$276$	−2.09611e13	−0.787797
$277$	2.84140e13	1.04687	0.523436	−	0.852065i	$-0.324650\pi$
$277$	2.84140e13	1.04687	0.523436	+	0.852065i	$0.324650\pi$
$278$	−1.35641e13	−0.489942
$279$	1.21724e13	0.431076
$280$	2.25424e13	0.782764
$281$	1.47578e12	0.0502500	0.0251250	−	0.999684i	$-0.492002\pi$
$281$	1.47578e12	0.0502500	0.0251250	+	0.999684i	$0.492002\pi$
$282$	3.81928e13	1.27530
$283$	−1.67811e13	−0.549533	−0.274767	−	0.961511i	$-0.588601\pi$
$283$	−1.67811e13	−0.549533	−0.274767	+	0.961511i	$0.588601\pi$
$284$	2.85763e13	0.917814
$285$	7.12548e12	0.224475
$286$	8.95888e12	0.276847
$287$	−5.38814e13	−1.63338
$288$	8.26457e12	0.245789
$289$	8.33702e13	2.43261
$290$	−4.86509e13	−1.39284
$291$	5.67089e12	0.159309
$292$	1.15566e14	3.18584
$293$	−2.95564e13	−0.799612	−0.399806	−	0.916600i	$-0.630923\pi$
$293$	−2.95564e13	−0.799612	−0.399806	+	0.916600i	$0.630923\pi$
$294$	−3.64286e13	−0.967236
$295$	1.88958e13	0.492429
$296$	−2.49973e13	−0.639425
$297$	3.68219e12	0.0924584
$298$	−4.07382e13	−1.00418
$299$	1.12391e13	0.271981
$300$	8.48043e12	0.201489
$301$	1.70867e13	0.398604
$302$	8.95317e13	2.05087
$303$	−1.51054e13	−0.339781
$304$	1.18025e13	0.260718
$305$	−4.02627e13	−0.873481
$306$	−4.80204e13	−1.02319
$307$	−2.76815e13	−0.579333	−0.289666	−	0.957128i	$-0.593544\pi$
$307$	−2.76815e13	−0.579333	−0.289666	+	0.957128i	$0.593544\pi$
$308$	−5.78314e13	−1.18887
$309$	−3.07277e13	−0.620523
$310$	−4.82999e13	−0.958202
$311$	−2.42772e12	−0.0473169	−0.0236585	−	0.999720i	$-0.507531\pi$
$311$	−2.42772e12	−0.0473169	−0.0236585	+	0.999720i	$0.507531\pi$
$312$	1.29427e13	0.247842
$313$	2.50845e13	0.471967	0.235984	−	0.971757i	$-0.424169\pi$
$313$	2.50845e13	0.471967	0.235984	+	0.971757i	$0.424169\pi$
$314$	−1.65756e14	−3.06448
$315$	−1.16366e13	−0.211407
$316$	1.37282e13	0.245095
$317$	1.42145e11	0.00249406	0.00124703	−	0.999999i	$-0.499603\pi$
$317$	1.42145e11	0.00249406	0.00124703	+	0.999999i	$0.499603\pi$
$318$	2.18828e13	0.377359
$319$	5.32840e13	0.903125
$320$	−4.08437e13	−0.680456
$321$	1.65068e13	0.270324
$322$	−1.14128e14	−1.83732
$323$	1.01775e14	1.61074
$324$	1.24605e13	0.193883
$325$	−4.54709e12	−0.0695626
$326$	1.04119e14	1.56615
$327$	2.81342e13	0.416123
$328$	−9.77373e13	−1.42152
$329$	1.32193e14	1.89073
$330$	−1.46109e13	−0.205518
$331$	−9.45872e13	−1.30851	−0.654257	−	0.756272i	$-0.727018\pi$
$331$	−9.45872e13	−1.30851	−0.654257	+	0.756272i	$0.727018\pi$
$332$	−4.61261e12	−0.0627606
$333$	1.29039e13	0.172694
$334$	−5.94875e13	−0.783105
$335$	1.88100e13	0.243580
$336$	−1.92747e13	−0.245540
$337$	8.93839e12	0.112020	0.0560099	−	0.998430i	$-0.482162\pi$
$337$	8.93839e12	0.112020	0.0560099	+	0.998430i	$0.482162\pi$
$338$	1.18116e14	1.45636
$339$	−1.48210e13	−0.179795
$340$	1.21128e14	1.44580
$341$	5.28996e13	0.621303
$342$	−4.15434e13	−0.480130
$343$	−1.39339e12	−0.0158473
$344$	3.09941e13	0.346902
$345$	−1.83296e13	−0.201905
$346$	−2.15083e14	−2.33178
$347$	−1.83665e13	−0.195981	−0.0979907	−	0.995187i	$-0.531242\pi$
$347$	−1.83665e13	−0.195981	−0.0979907	+	0.995187i	$0.531242\pi$
$348$	1.80313e14	1.89383
$349$	1.20668e14	1.24754	0.623769	−	0.781609i	$-0.285601\pi$
$349$	1.20668e14	1.24754	0.623769	+	0.781609i	$0.285601\pi$
$350$	4.61738e13	0.469918
$351$	−6.68117e12	−0.0669366
$352$	3.59166e13	0.354251
$353$	−2.62988e13	−0.255373	−0.127687	−	0.991815i	$-0.540755\pi$
$353$	−2.62988e13	−0.255373	−0.127687	+	0.991815i	$0.540755\pi$
$354$	−1.10167e14	−1.05326
$355$	2.49887e13	0.235227
$356$	−2.48888e14	−2.30690
$357$	−1.66208e14	−1.51697
$358$	−5.24097e12	−0.0471037
$359$	−7.23499e13	−0.640352	−0.320176	−	0.947358i	$-0.603742\pi$
$359$	−7.23499e13	−0.640352	−0.320176	+	0.947358i	$0.603742\pi$
$360$	−2.11081e13	−0.183986
$361$	−2.84430e13	−0.244166
$362$	2.21999e14	1.87696
$363$	−5.33285e13	−0.444092
$364$	1.04932e14	0.860700
$365$	1.01057e14	0.816501
$366$	2.34742e14	1.86829
$367$	−6.60414e13	−0.517789	−0.258894	−	0.965906i	$-0.583358\pi$
$367$	−6.60414e13	−0.517789	−0.258894	+	0.965906i	$0.583358\pi$
$368$	−3.03609e13	−0.234504
$369$	5.04530e13	0.383921
$370$	−5.12023e13	−0.383867
$371$	7.57408e13	0.559466
$372$	1.79012e14	1.30286
$373$	−3.72124e13	−0.266864	−0.133432	−	0.991058i	$-0.542600\pi$
$373$	−3.72124e13	−0.266864	−0.133432	+	0.991058i	$0.542600\pi$
$374$	−2.08690e14	−1.47471
$375$	7.41577e12	0.0516398
$376$	2.39789e14	1.64549
$377$	−9.66813e13	−0.653831
$378$	6.78445e13	0.452178
$379$	1.91105e14	1.25533	0.627664	−	0.778485i	$-0.284011\pi$
$379$	1.91105e14	1.25533	0.627664	+	0.778485i	$0.284011\pi$
$380$	1.04790e14	0.678437
$381$	−4.32203e13	−0.275804
$382$	−3.39416e14	−2.13493
$383$	−2.38924e14	−1.48138	−0.740689	−	0.671848i	$-0.765501\pi$
$383$	−2.38924e14	−1.48138	−0.740689	+	0.671848i	$0.765501\pi$
$384$	1.68476e14	1.02971
$385$	−5.05710e13	−0.304697
$386$	7.79871e13	0.463226
$387$	−1.59995e13	−0.0936904
$388$	8.33981e13	0.481484
$389$	−9.94459e13	−0.566062	−0.283031	−	0.959111i	$-0.591340\pi$
$389$	−9.94459e13	−0.566062	−0.283031	+	0.959111i	$0.591340\pi$
$390$	2.65108e13	0.148788
$391$	−2.61805e14	−1.44879
$392$	−2.28713e14	−1.24801
$393$	1.11773e13	0.0601418
$394$	−4.06810e14	−2.15855
$395$	1.20047e13	0.0628156
$396$	5.41517e13	0.279440
$397$	2.55979e14	1.30274	0.651369	−	0.758762i	$-0.274195\pi$
$397$	2.55979e14	1.30274	0.651369	+	0.758762i	$0.274195\pi$
$398$	5.79358e14	2.90797
$399$	−1.43790e14	−0.711832
$400$	1.22834e13	0.0599774
$401$	3.75203e13	0.180706	0.0903529	−	0.995910i	$-0.471201\pi$
$401$	3.75203e13	0.180706	0.0903529	+	0.995910i	$0.471201\pi$
$402$	−1.09667e14	−0.520994
$403$	−9.59840e13	−0.449802
$404$	−2.22146e14	−1.02693
$405$	1.08962e13	0.0496904
$406$	9.81758e14	4.41684
$407$	5.60784e13	0.248901
$408$	−3.01490e14	−1.32021
$409$	−1.93512e14	−0.836044	−0.418022	−	0.908437i	$-0.637276\pi$
$409$	−1.93512e14	−0.836044	−0.418022	+	0.908437i	$0.637276\pi$
$410$	−2.00196e14	−0.853385
$411$	−1.21063e13	−0.0509190
$412$	−4.51892e14	−1.87543
$413$	−3.81311e14	−1.56154
$414$	1.06866e14	0.431856
$415$	−4.03353e12	−0.0160850
$416$	−6.51691e13	−0.256465
$417$	4.39608e13	0.170733
$418$	−1.80542e14	−0.692003
$419$	−1.29993e14	−0.491748	−0.245874	−	0.969302i	$-0.579075\pi$
$419$	−1.29993e14	−0.491748	−0.245874	+	0.969302i	$0.579075\pi$
$420$	−1.71132e14	−0.638941
$421$	−8.72362e13	−0.321474	−0.160737	−	0.986997i	$-0.551387\pi$
$421$	−8.72362e13	−0.321474	−0.160737	+	0.986997i	$0.551387\pi$
$422$	−1.99743e14	−0.726529
$423$	−1.23782e14	−0.444411
$424$	1.37389e14	0.486899
$425$	1.05921e14	0.370546
$426$	−1.45691e14	−0.503129
$427$	8.12487e14	2.76990
$428$	2.42755e14	0.817009
$429$	−2.90354e13	−0.0964746
$430$	6.34855e13	0.208256
$431$	2.21393e14	0.717033	0.358516	−	0.933523i	$-0.383283\pi$
$431$	2.21393e14	0.717033	0.358516	+	0.933523i	$0.383283\pi$
$432$	1.80483e13	0.0577133
$433$	−4.37191e14	−1.38035	−0.690173	−	0.723644i	$-0.742466\pi$
$433$	−4.37191e14	−1.38035	−0.690173	+	0.723644i	$0.742466\pi$
$434$	9.74677e14	3.03856
$435$	1.57676e14	0.485371
$436$	4.13751e14	1.25766
$437$	−2.26493e14	−0.679840
$438$	−5.89191e14	−1.74642
$439$	1.77794e14	0.520430	0.260215	−	0.965551i	$-0.416207\pi$
$439$	1.77794e14	0.520430	0.260215	+	0.965551i	$0.416207\pi$
$440$	−9.17326e13	−0.265176
$441$	1.18064e14	0.337058
$442$	3.78658e14	1.06764
$443$	5.57996e14	1.55386	0.776928	−	0.629590i	$-0.216777\pi$
$443$	5.57996e14	1.55386	0.776928	+	0.629590i	$0.216777\pi$
$444$	1.89769e14	0.521939
$445$	−2.17642e14	−0.591238
$446$	−1.29822e14	−0.348343
$447$	1.32031e14	0.349933
$448$	8.24212e14	2.15779
$449$	5.52221e14	1.42810	0.714049	−	0.700096i	$-0.246860\pi$
$449$	5.52221e14	1.42810	0.714049	+	0.700096i	$0.246860\pi$
$450$	−4.32359e13	−0.110452
$451$	2.19261e14	0.553339
$452$	−2.17963e14	−0.543401
$453$	−2.90169e14	−0.714679
$454$	−6.36751e14	−1.54940
$455$	9.17589e13	0.220590
$456$	−2.60825e14	−0.619503
$457$	−2.57493e13	−0.0604263	−0.0302132	−	0.999543i	$-0.509619\pi$
$457$	−2.57493e13	−0.0604263	−0.0302132	+	0.999543i	$0.509619\pi$
$458$	−5.70126e14	−1.32194
$459$	1.55632e14	0.356558
$460$	−2.69562e14	−0.610225
$461$	−4.26715e14	−0.954515	−0.477257	−	0.878764i	$-0.658369\pi$
$461$	−4.26715e14	−0.954515	−0.477257	+	0.878764i	$0.658369\pi$
$462$	2.94842e14	0.651717
$463$	1.13760e14	0.248481	0.124241	−	0.992252i	$-0.460351\pi$
$463$	1.13760e14	0.248481	0.124241	+	0.992252i	$0.460351\pi$
$464$	2.61172e14	0.563738
$465$	1.56538e14	0.333910
$466$	−8.19030e14	−1.72654
$467$	−1.60119e14	−0.333580	−0.166790	−	0.985992i	$-0.553340\pi$
$467$	−1.60119e14	−0.333580	−0.166790	+	0.985992i	$0.553340\pi$
$468$	−9.82558e13	−0.202305
$469$	−3.79579e14	−0.772416
$470$	4.91163e14	0.987842
$471$	5.37211e14	1.06790
$472$	−6.91673e14	−1.35900
$473$	−6.95314e13	−0.135034
$474$	−6.99906e13	−0.134357
$475$	9.16343e13	0.173877
$476$	−2.44431e15	−4.58478
$477$	−7.09216e13	−0.131500
$478$	−4.43122e14	−0.812212
$479$	−7.66095e14	−1.38815	−0.694077	−	0.719901i	$-0.744187\pi$
$479$	−7.66095e14	−1.38815	−0.694077	+	0.719901i	$0.744187\pi$
$480$	1.06283e14	0.190387
$481$	−1.01752e14	−0.180196
$482$	−1.93738e14	−0.339200
$483$	3.69885e14	0.640262
$484$	−7.84268e14	−1.34219
$485$	7.29281e13	0.123400
$486$	−6.35277e13	−0.106283
$487$	9.61903e14	1.59119	0.795595	−	0.605829i	$-0.207158\pi$
$487$	9.61903e14	1.59119	0.795595	+	0.605829i	$0.207158\pi$
$488$	1.47380e15	2.41062
$489$	−3.37445e14	−0.545764
$490$	−4.68475e14	−0.749218
$491$	2.32694e14	0.367991	0.183996	−	0.982927i	$-0.441097\pi$
$491$	2.32694e14	0.367991	0.183996	+	0.982927i	$0.441097\pi$
$492$	7.41980e14	1.16034
$493$	2.25211e15	3.48283
$494$	3.27585e14	0.500986
$495$	4.73533e13	0.0716179
$496$	2.59288e14	0.387823
$497$	−5.04264e14	−0.745929
$498$	2.35165e13	0.0344042
$499$	−3.26636e14	−0.472619	−0.236309	−	0.971678i	$-0.575938\pi$
$499$	−3.26636e14	−0.472619	−0.236309	+	0.971678i	$0.575938\pi$
$500$	1.09059e14	0.156073
$501$	1.92797e14	0.272893
$502$	1.41829e15	1.98561
$503$	−1.47765e14	−0.204620	−0.102310	−	0.994753i	$-0.532623\pi$
$503$	−1.47765e14	−0.204620	−0.102310	+	0.994753i	$0.532623\pi$
$504$	4.25954e14	0.583438
$505$	−1.94257e14	−0.263193
$506$	4.64426e14	0.622427
$507$	−3.82812e14	−0.507506
$508$	−6.35614e14	−0.833572
$509$	−1.76808e14	−0.229379	−0.114689	−	0.993401i	$-0.536587\pi$
$509$	−1.76808e14	−0.229379	−0.114689	+	0.993401i	$0.536587\pi$
$510$	−6.17546e14	−0.792563
$511$	−2.03930e15	−2.58921
$512$	4.70714e14	0.591251
$513$	1.34641e14	0.167314
$514$	1.16720e15	1.43498
$515$	−3.95161e14	−0.480655
$516$	−2.35294e14	−0.283164
$517$	−5.37937e14	−0.640522
$518$	1.03325e15	1.21728
$519$	6.97078e14	0.812570
$520$	1.66445e14	0.191978
$521$	−4.91557e14	−0.561004	−0.280502	−	0.959853i	$-0.590501\pi$
$521$	−4.91557e14	−0.561004	−0.280502	+	0.959853i	$0.590501\pi$
$522$	−9.19291e14	−1.03816
$523$	−1.63875e15	−1.83127	−0.915637	−	0.402007i	$-0.868313\pi$
$523$	−1.63875e15	−1.83127	−0.915637	+	0.402007i	$0.868313\pi$
$524$	1.64377e14	0.181769
$525$	−1.49648e14	−0.163755
$526$	1.75298e15	1.89826
$527$	2.23587e15	2.39601
$528$	7.84353e13	0.0831812
$529$	−3.70179e14	−0.388514
$530$	2.81415e14	0.292301
$531$	3.57049e14	0.367035
$532$	−2.11463e15	−2.15139
$533$	−3.97840e14	−0.400598
$534$	1.26891e15	1.26460
$535$	2.12279e14	0.209392
$536$	−6.88531e14	−0.672229
$537$	1.69858e13	0.0164145
$538$	−1.14483e15	−1.09506
$539$	5.13089e14	0.485796
$540$	1.60244e14	0.150181
$541$	−2.49179e14	−0.231168	−0.115584	−	0.993298i	$-0.536874\pi$
$541$	−2.49179e14	−0.231168	−0.115584	+	0.993298i	$0.536874\pi$
$542$	1.94521e15	1.78636
$543$	−7.19492e14	−0.654074
$544$	1.51806e15	1.36614
$545$	3.61808e14	0.322327
$546$	−5.34978e14	−0.471820
$547$	−1.63969e15	−1.43163	−0.715817	−	0.698287i	$-0.753946\pi$
$547$	−1.63969e15	−1.43163	−0.715817	+	0.698287i	$0.753946\pi$
$548$	−1.78039e14	−0.153894
$549$	−7.60791e14	−0.651054
$550$	−1.87897e14	−0.159193
$551$	1.94835e15	1.63430
$552$	6.70947e14	0.557216
$553$	−2.42251e14	−0.199195
$554$	−2.13042e15	−1.73445
$555$	1.65945e14	0.133768
$556$	6.46503e14	0.516011
$557$	−1.62476e15	−1.28406	−0.642032	−	0.766678i	$-0.721909\pi$
$557$	−1.62476e15	−1.28406	−0.642032	+	0.766678i	$0.721909\pi$
$558$	−9.12660e14	−0.714202
$559$	1.26162e14	0.0977602
$560$	−2.47874e14	−0.190194
$561$	6.76356e14	0.513901
$562$	−1.10650e14	−0.0832536
$563$	1.06403e15	0.792791	0.396395	−	0.918080i	$-0.370261\pi$
$563$	1.06403e15	0.792791	0.396395	+	0.918080i	$0.370261\pi$
$564$	−1.82038e15	−1.34316
$565$	−1.90599e14	−0.139269
$566$	1.25820e15	0.910459
$567$	−2.19882e14	−0.157573
$568$	−9.14702e14	−0.649178
$569$	−2.37736e14	−0.167100	−0.0835501	−	0.996504i	$-0.526626\pi$
$569$	−2.37736e14	−0.167100	−0.0835501	+	0.996504i	$0.526626\pi$
$570$	−5.34252e14	−0.371907
$571$	8.27986e14	0.570853	0.285427	−	0.958401i	$-0.407865\pi$
$571$	8.27986e14	0.570853	0.285427	+	0.958401i	$0.407865\pi$
$572$	−4.27005e14	−0.291578
$573$	1.10003e15	0.743971
$574$	4.03990e15	2.70617
$575$	−2.35720e14	−0.156395
$576$	−7.71769e14	−0.507182
$577$	2.25265e15	1.46632	0.733158	−	0.680058i	$-0.238045\pi$
$577$	2.25265e15	1.46632	0.733158	+	0.680058i	$0.238045\pi$
$578$	−6.25090e15	−4.03032
$579$	−2.52753e14	−0.161423
$580$	2.31884e15	1.46695
$581$	8.13953e13	0.0510071
$582$	−4.25190e14	−0.263941
$583$	−3.08215e14	−0.189529
$584$	−3.69916e15	−2.25337
$585$	−8.59204e13	−0.0518489
$586$	2.21607e15	1.32479
$587$	−1.18033e15	−0.699029	−0.349514	−	0.936931i	$-0.613653\pi$
$587$	−1.18033e15	−0.699029	−0.349514	+	0.936931i	$0.613653\pi$
$588$	1.73629e15	1.01870
$589$	1.93430e15	1.12432
$590$	−1.41676e15	−0.815851
$591$	1.31846e15	0.752204
$592$	2.74869e14	0.155366
$593$	2.29778e15	1.28679	0.643394	−	0.765535i	$-0.277526\pi$
$593$	2.29778e15	1.28679	0.643394	+	0.765535i	$0.277526\pi$
$594$	−2.76082e14	−0.153184
$595$	−2.13745e15	−1.17504
$596$	1.94170e15	1.05761
$597$	−1.87768e15	−1.01336
$598$	−8.42679e14	−0.450615
$599$	−9.39685e14	−0.497891	−0.248946	−	0.968517i	$-0.580084\pi$
$599$	−9.39685e14	−0.497891	−0.248946	+	0.968517i	$0.580084\pi$
$600$	−2.71451e14	−0.142515
$601$	−3.30569e15	−1.71970	−0.859850	−	0.510547i	$-0.829443\pi$
$601$	−3.30569e15	−1.71970	−0.859850	+	0.510547i	$0.829443\pi$
$602$	−1.28112e15	−0.660402
$603$	3.55427e14	0.181554
$604$	−4.26733e15	−2.16000
$605$	−6.85808e14	−0.343992
$606$	1.13257e15	0.562945
$607$	1.51796e15	0.747690	0.373845	−	0.927491i	$-0.378039\pi$
$607$	1.51796e15	0.747690	0.373845	+	0.927491i	$0.378039\pi$
$608$	1.31331e15	0.641056
$609$	−3.18185e15	−1.53916
$610$	3.01880e15	1.44717
$611$	9.76063e14	0.463715
$612$	2.28879e15	1.07764
$613$	−1.78314e15	−0.832055	−0.416028	−	0.909352i	$-0.636578\pi$
$613$	−1.78314e15	−0.832055	−0.416028	+	0.909352i	$0.636578\pi$
$614$	2.07549e15	0.959831
$615$	6.48830e14	0.297384
$616$	1.85113e15	0.840899
$617$	−2.06098e15	−0.927911	−0.463955	−	0.885859i	$-0.653570\pi$
$617$	−2.06098e15	−0.927911	−0.463955	+	0.885859i	$0.653570\pi$
$618$	2.30389e15	1.02808
$619$	−1.05386e15	−0.466105	−0.233052	−	0.972464i	$-0.574871\pi$
$619$	−1.05386e15	−0.466105	−0.233052	+	0.972464i	$0.574871\pi$
$620$	2.30211e15	1.00919
$621$	−3.46350e14	−0.150491
$622$	1.82025e14	0.0783941
$623$	4.39194e15	1.87487
$624$	−1.42317e14	−0.0602202
$625$	9.53674e13	0.0400000
$626$	−1.88078e15	−0.781950
$627$	5.85130e14	0.241146
$628$	7.90042e15	3.22755
$629$	2.37022e15	0.959867
$630$	8.72486e14	0.350256
$631$	−1.51057e15	−0.601146	−0.300573	−	0.953759i	$-0.597178\pi$
$631$	−1.51057e15	−0.601146	−0.300573	+	0.953759i	$0.597178\pi$
$632$	−4.39428e14	−0.173358
$633$	6.47360e14	0.253178
$634$	−1.06577e13	−0.00413212
$635$	−5.55817e14	−0.213637
$636$	−1.04300e15	−0.397438
$637$	−9.30976e14	−0.351699
$638$	−3.99511e15	−1.49629
$639$	4.72179e14	0.175328
$640$	2.16661e15	0.797611
$641$	4.65958e15	1.70070	0.850350	−	0.526218i	$-0.176390\pi$
$641$	4.65958e15	1.70070	0.850350	+	0.526218i	$0.176390\pi$
$642$	−1.23764e15	−0.447869
$643$	2.11472e15	0.758739	0.379369	−	0.925245i	$-0.376141\pi$
$643$	2.11472e15	0.758739	0.379369	+	0.925245i	$0.376141\pi$
$644$	5.43967e15	1.93508
$645$	−2.05755e14	−0.0725723
$646$	−7.63082e15	−2.66865
$647$	−4.70348e15	−1.63097	−0.815485	−	0.578778i	$-0.803530\pi$
$647$	−4.70348e15	−1.63097	−0.815485	+	0.578778i	$0.803530\pi$
$648$	−3.98851e14	−0.137135
$649$	1.55168e15	0.529002
$650$	3.40930e14	0.115250
$651$	−3.15889e15	−1.05886
$652$	−4.96260e15	−1.64948
$653$	−3.62919e15	−1.19615	−0.598077	−	0.801439i	$-0.704068\pi$
$653$	−3.62919e15	−1.19615	−0.598077	+	0.801439i	$0.704068\pi$
$654$	−2.10943e15	−0.689427
$655$	1.43741e14	0.0465856
$656$	1.07471e15	0.345399
$657$	1.90955e15	0.608584
$658$	−9.91151e15	−3.13255
$659$	−1.26034e14	−0.0395020	−0.0197510	−	0.999805i	$-0.506287\pi$
$659$	−1.26034e14	−0.0395020	−0.0197510	+	0.999805i	$0.506287\pi$
$660$	6.96395e14	0.216453
$661$	4.45748e15	1.37398	0.686991	−	0.726666i	$-0.258931\pi$
$661$	4.45748e15	1.37398	0.686991	+	0.726666i	$0.258931\pi$
$662$	7.09192e15	2.16793
$663$	−1.22722e15	−0.372047
$664$	1.47646e14	0.0443912
$665$	−1.84915e15	−0.551383
$666$	−9.67503e14	−0.286117
$667$	−5.01193e15	−1.46999
$668$	2.83534e15	0.824774
$669$	4.20750e14	0.121389
$670$	−1.41033e15	−0.403560
$671$	−3.30629e15	−0.938353
$672$	−2.14476e15	−0.603736
$673$	4.32745e15	1.20823	0.604115	−	0.796897i	$-0.293527\pi$
$673$	4.32745e15	1.20823	0.604115	+	0.796897i	$0.293527\pi$
$674$	−6.70180e14	−0.185593
$675$	1.40126e14	0.0384900
$676$	−5.62977e15	−1.53385
$677$	−1.61874e15	−0.437460	−0.218730	−	0.975785i	$-0.570191\pi$
$677$	−1.61874e15	−0.437460	−0.218730	+	0.975785i	$0.570191\pi$
$678$	1.11124e15	0.297882
$679$	−1.47166e15	−0.391314
$680$	−3.87719e15	−1.02263
$681$	2.06369e15	0.539927
$682$	−3.96629e15	−1.02937
$683$	2.93595e15	0.755849	0.377924	−	0.925836i	$-0.376638\pi$
$683$	2.93595e15	0.755849	0.377924	+	0.925836i	$0.376638\pi$
$684$	1.98008e15	0.505677
$685$	−1.55688e14	−0.0394417
$686$	1.04473e14	0.0262556
$687$	1.84776e15	0.460663
$688$	−3.40809e14	−0.0842896
$689$	5.59242e14	0.137213
$690$	1.37431e15	0.334514
$691$	−4.57286e14	−0.110423	−0.0552114	−	0.998475i	$-0.517583\pi$
$691$	−4.57286e14	−0.110423	−0.0552114	+	0.998475i	$0.517583\pi$
$692$	1.02515e16	2.45586
$693$	−9.55574e14	−0.227108
$694$	1.37708e15	0.324699
$695$	5.65339e14	0.132249
$696$	−5.77166e15	−1.33952
$697$	9.26736e15	2.13391
$698$	−9.04743e15	−2.06690
$699$	2.65445e15	0.601658
$700$	−2.20078e15	−0.494922
$701$	6.08159e15	1.35696	0.678481	−	0.734618i	$-0.262639\pi$
$701$	6.08159e15	1.35696	0.678481	+	0.734618i	$0.262639\pi$
$702$	5.00939e14	0.110900
$703$	2.05053e15	0.450414
$704$	−3.35400e15	−0.730993
$705$	−1.59184e15	−0.344239
$706$	1.97182e15	0.423099
$707$	3.92005e15	0.834612
$708$	5.25089e15	1.10930
$709$	−5.51751e15	−1.15661	−0.578307	−	0.815819i	$-0.696286\pi$
$709$	−5.51751e15	−1.15661	−0.578307	+	0.815819i	$0.696286\pi$
$710$	−1.87359e15	−0.389722
$711$	2.26837e14	0.0468200
$712$	7.96669e15	1.63169
$713$	−4.97578e15	−1.01127
$714$	1.24619e16	2.51329
$715$	−3.73398e14	−0.0747289
$716$	2.49799e14	0.0496100
$717$	1.43614e15	0.283036
$718$	5.42463e15	1.06093
$719$	−5.30420e15	−1.02946	−0.514732	−	0.857351i	$-0.672109\pi$
$719$	−5.30420e15	−1.02946	−0.514732	+	0.857351i	$0.672109\pi$
$720$	2.32103e14	0.0447045
$721$	7.97421e15	1.52421
$722$	2.13259e15	0.404531
$723$	6.27898e14	0.118203
$724$	−1.05811e16	−1.97683
$725$	2.02772e15	0.375967
$726$	3.99844e15	0.735765
$727$	7.44972e15	1.36051	0.680254	−	0.732977i	$-0.261870\pi$
$727$	7.44972e15	1.36051	0.680254	+	0.732977i	$0.261870\pi$
$728$	−3.35880e15	−0.608781
$729$	2.05891e14	0.0370370
$730$	−7.57704e15	−1.35277
$731$	−2.93883e15	−0.520750
$732$	−1.11885e16	−1.96770
$733$	5.45148e15	0.951574	0.475787	−	0.879561i	$-0.342163\pi$
$733$	5.45148e15	0.951574	0.475787	+	0.879561i	$0.342163\pi$
$734$	4.95162e15	0.857866
$735$	1.51831e15	0.261084
$736$	−3.37835e15	−0.576602
$737$	1.54463e15	0.261670
$738$	−3.78285e15	−0.636076
$739$	1.63560e15	0.272980	0.136490	−	0.990641i	$-0.456418\pi$
$739$	1.63560e15	0.272980	0.136490	+	0.990641i	$0.456418\pi$
$740$	2.44045e15	0.404292
$741$	−1.06169e15	−0.174581
$742$	−5.67887e15	−0.926916
$743$	−9.47083e15	−1.53444	−0.767219	−	0.641385i	$-0.778360\pi$
$743$	−9.47083e15	−1.53444	−0.767219	+	0.641385i	$0.778360\pi$
$744$	−5.73003e15	−0.921522
$745$	1.69793e15	0.271057
$746$	2.79010e15	0.442136
$747$	−7.62163e13	−0.0119890
$748$	9.94674e15	1.55318
$749$	−4.28371e15	−0.664003
$750$	−5.56017e14	−0.0855561
$751$	−9.72786e15	−1.48593	−0.742964	−	0.669331i	$-0.766581\pi$
$751$	−9.72786e15	−1.48593	−0.742964	+	0.669331i	$0.766581\pi$
$752$	−2.63670e15	−0.399819
$753$	−4.59661e15	−0.691936
$754$	7.24894e15	1.08326
$755$	−3.73160e15	−0.553588
$756$	−3.23366e15	−0.476239
$757$	1.91854e15	0.280507	0.140253	−	0.990116i	$-0.455208\pi$
$757$	1.91854e15	0.280507	0.140253	+	0.990116i	$0.455208\pi$
$758$	−1.43286e16	−2.07981
$759$	−1.50519e15	−0.216901
$760$	−3.35424e15	−0.479865
$761$	−1.17913e16	−1.67474	−0.837369	−	0.546638i	$-0.815907\pi$
$761$	−1.17913e16	−1.67474	−0.837369	+	0.546638i	$0.815907\pi$
$762$	3.24056e15	0.456949
$763$	−7.30116e15	−1.02213
$764$	1.61775e16	2.24853
$765$	2.00145e15	0.276189
$766$	1.79139e16	2.45433
$767$	−2.81546e15	−0.382979
$768$	−6.12745e15	−0.827547
$769$	8.15564e15	1.09361	0.546806	−	0.837259i	$-0.315844\pi$
$769$	8.15564e15	1.09361	0.546806	+	0.837259i	$0.315844\pi$
$770$	3.79170e15	0.504818
$771$	−3.78285e15	−0.500057
$772$	−3.71708e15	−0.487874
$773$	−5.26080e15	−0.685590	−0.342795	−	0.939410i	$-0.611374\pi$
$773$	−5.26080e15	−0.685590	−0.342795	+	0.939410i	$0.611374\pi$
$774$	1.19960e15	0.155225
$775$	2.01310e15	0.258646
$776$	−2.66950e15	−0.340558
$777$	−3.34872e15	−0.424192
$778$	7.45622e15	0.937845
$779$	8.01738e15	1.00133
$780$	−1.26358e15	−0.156705
$781$	2.05202e15	0.252697
$782$	1.96295e16	2.40034
$783$	2.97939e15	0.361774
$784$	2.51491e15	0.303238
$785$	6.90858e15	0.827191
$786$	−8.38045e14	−0.0996422
$787$	−9.41032e15	−1.11108	−0.555538	−	0.831491i	$-0.687488\pi$
$787$	−9.41032e15	−1.11108	−0.555538	+	0.831491i	$0.687488\pi$
$788$	1.93897e16	2.27341
$789$	−5.68136e15	−0.661498
$790$	−9.00085e14	−0.104072
$791$	3.84622e15	0.441635
$792$	−1.73335e15	−0.197650
$793$	5.99911e15	0.679335
$794$	−1.91927e16	−2.15836
$795$	−9.12057e14	−0.101860
$796$	−2.76138e16	−3.06270
$797$	3.36292e15	0.370421	0.185210	−	0.982699i	$-0.440703\pi$
$797$	3.36292e15	0.370421	0.185210	+	0.982699i	$0.440703\pi$
$798$	1.07810e16	1.17935
$799$	−2.27366e16	−2.47012
$800$	1.36681e15	0.147473
$801$	−4.11249e15	−0.440683
$802$	−2.81318e15	−0.299391
$803$	8.29861e15	0.877142
$804$	5.22704e15	0.548716
$805$	4.75675e15	0.495945
$806$	7.19665e15	0.745226
$807$	3.71035e15	0.381601
$808$	7.11071e15	0.726358
$809$	8.02144e15	0.813834	0.406917	−	0.913465i	$-0.366604\pi$
$809$	8.02144e15	0.813834	0.406917	+	0.913465i	$0.366604\pi$
$810$	−8.16971e14	−0.0823264
$811$	−3.16415e15	−0.316695	−0.158348	−	0.987383i	$-0.550617\pi$
$811$	−3.16415e15	−0.316695	−0.158348	+	0.987383i	$0.550617\pi$
$812$	−4.67934e16	−4.65186
$813$	−6.30436e15	−0.622505
$814$	−4.20463e15	−0.412376
$815$	−4.33958e15	−0.422747
$816$	3.31517e15	0.320782
$817$	−2.54244e15	−0.244360
$818$	1.45091e16	1.38515
$819$	1.73385e15	0.164418
$820$	9.54193e15	0.898793
$821$	−1.47257e16	−1.37781	−0.688903	−	0.724853i	$-0.741908\pi$
$821$	−1.47257e16	−1.37781	−0.688903	+	0.724853i	$0.741908\pi$
$822$	9.07699e14	0.0843620
$823$	−1.36058e16	−1.25610	−0.628052	−	0.778171i	$-0.716148\pi$
$823$	−1.36058e16	−1.25610	−0.628052	+	0.778171i	$0.716148\pi$
$824$	1.44647e16	1.32651
$825$	6.08968e14	0.0554750
$826$	2.85898e16	2.58714
$827$	−1.63615e16	−1.47077	−0.735384	−	0.677651i	$-0.762998\pi$
$827$	−1.63615e16	−1.47077	−0.735384	+	0.677651i	$0.762998\pi$
$828$	−5.09355e15	−0.454835
$829$	−1.63816e16	−1.45313	−0.726567	−	0.687096i	$-0.758885\pi$
$829$	−1.63816e16	−1.45313	−0.726567	+	0.687096i	$0.758885\pi$
$830$	3.02425e14	0.0266494
$831$	6.90461e15	0.604412
$832$	6.08567e15	0.529213
$833$	2.16863e16	1.87343
$834$	−3.29607e15	−0.282868
$835$	2.47939e15	0.211382
$836$	8.60513e15	0.728824
$837$	2.95790e15	0.248882
$838$	9.74656e15	0.814722
$839$	1.30757e16	1.08586	0.542929	−	0.839779i	$-0.317315\pi$
$839$	1.30757e16	1.08586	0.542929	+	0.839779i	$0.317315\pi$
$840$	5.47780e15	0.451929
$841$	3.09134e16	2.53378
$842$	6.54076e15	0.532613
$843$	3.58614e14	0.0290119
$844$	9.52031e15	0.765187
$845$	−4.92299e15	−0.393113
$846$	9.28086e15	0.736294
$847$	1.38394e16	1.09083
$848$	−1.51072e15	−0.118306
$849$	−4.07780e15	−0.317273
$850$	−7.94170e15	−0.613916
$851$	−5.27478e15	−0.405128
$852$	6.94403e15	0.529900
$853$	−1.24432e16	−0.943433	−0.471716	−	0.881750i	$-0.656365\pi$
$853$	−1.24432e16	−0.943433	−0.471716	+	0.881750i	$0.656365\pi$
$854$	−6.09184e16	−4.58913
$855$	1.73149e15	0.129601
$856$	−7.77037e15	−0.577878
$857$	−3.66106e15	−0.270528	−0.135264	−	0.990810i	$-0.543188\pi$
$857$	−3.66106e15	−0.270528	−0.135264	+	0.990810i	$0.543188\pi$
$858$	2.17701e15	0.159838
$859$	1.38206e16	1.00824	0.504122	−	0.863632i	$-0.331816\pi$
$859$	1.38206e16	1.00824	0.504122	+	0.863632i	$0.331816\pi$
$860$	−3.02590e15	−0.219338
$861$	−1.30932e16	−0.943034
$862$	−1.65995e16	−1.18797
$863$	1.52984e16	1.08789	0.543946	−	0.839120i	$-0.316929\pi$
$863$	1.52984e16	1.08789	0.543946	+	0.839120i	$0.316929\pi$
$864$	2.00829e15	0.141906
$865$	8.96448e15	0.629414
$866$	3.27796e16	2.28694
$867$	2.02590e16	1.40447
$868$	−4.64559e16	−3.20024
$869$	9.85801e14	0.0674809
$870$	−1.18222e16	−0.804157
$871$	−2.80267e15	−0.189440
$872$	−1.32438e16	−0.889555
$873$	1.37803e15	0.0919769
$874$	1.69819e16	1.12635
$875$	−1.92448e15	−0.126844
$876$	2.80825e16	1.83934
$877$	1.90870e14	0.0124234	0.00621169	−	0.999981i	$-0.498023\pi$
$877$	1.90870e14	0.0124234	0.00621169	+	0.999981i	$0.498023\pi$
$878$	−1.33306e16	−0.862241
$879$	−7.18220e15	−0.461656
$880$	1.00868e15	0.0644319
$881$	7.60210e15	0.482577	0.241288	−	0.970453i	$-0.422430\pi$
$881$	7.60210e15	0.482577	0.241288	+	0.970453i	$0.422430\pi$
$882$	−8.85215e15	−0.558434
$883$	−2.08677e16	−1.30825	−0.654125	−	0.756386i	$-0.726963\pi$
$883$	−2.08677e16	−1.30825	−0.654125	+	0.756386i	$0.726963\pi$
$884$	−1.80479e16	−1.12445
$885$	4.59168e15	0.284304
$886$	−4.18372e16	−2.57441
$887$	3.05601e16	1.86885	0.934427	−	0.356154i	$-0.115912\pi$
$887$	3.05601e16	1.86885	0.934427	+	0.356154i	$0.115912\pi$
$888$	−6.07435e15	−0.369172
$889$	1.12162e16	0.677464
$890$	1.63183e16	0.979555
$891$	8.94773e14	0.0533809
$892$	6.18770e15	0.366879
$893$	−1.96699e16	−1.15909
$894$	−9.89937e15	−0.579764
$895$	2.18439e14	0.0127146
$896$	−4.37215e16	−2.52930
$897$	2.73109e15	0.157028
$898$	−4.14042e16	−2.36605
$899$	4.28029e16	2.43106
$900$	2.06074e15	0.116330
$901$	−1.30271e16	−0.730905
$902$	−1.64397e16	−0.916765
$903$	4.15206e15	0.230134
$904$	6.97679e15	0.384352
$905$	−9.25273e15	−0.506644
$906$	2.17562e16	1.18407
$907$	−2.96304e16	−1.60287	−0.801434	−	0.598083i	$-0.795929\pi$
$907$	−2.96304e16	−1.60287	−0.801434	+	0.598083i	$0.795929\pi$
$908$	3.03494e16	1.63184
$909$	−3.67062e15	−0.196173
$910$	−6.87986e15	−0.365470
$911$	1.41726e16	0.748342	0.374171	−	0.927360i	$-0.377927\pi$
$911$	1.41726e16	0.748342	0.374171	+	0.927360i	$0.377927\pi$
$912$	2.86802e15	0.150526
$913$	−3.31225e14	−0.0172796
$914$	1.93062e15	0.100114
$915$	−9.78383e15	−0.504304
$916$	2.71738e16	1.39228
$917$	−2.90064e15	−0.147728
$918$	−1.16690e16	−0.590741
$919$	2.73944e16	1.37856	0.689281	−	0.724494i	$-0.257927\pi$
$919$	2.73944e16	1.37856	0.689281	+	0.724494i	$0.257927\pi$
$920$	8.62844e15	0.431618
$921$	−6.72660e15	−0.334478
$922$	3.19941e16	1.58143
$923$	−3.72330e15	−0.182944
$924$	−1.40530e16	−0.686395
$925$	2.13407e15	0.103616
$926$	−8.52944e15	−0.411681
$927$	−7.46683e15	−0.358259
$928$	2.90614e16	1.38613
$929$	−1.04825e16	−0.497024	−0.248512	−	0.968629i	$-0.579942\pi$
$929$	−1.04825e16	−0.497024	−0.248512	+	0.968629i	$0.579942\pi$
$930$	−1.17369e16	−0.553218
$931$	1.87613e16	0.879102
$932$	3.90373e16	1.81841
$933$	−5.89936e14	−0.0273184
$934$	1.20053e16	0.552671
$935$	8.69800e15	0.398066
$936$	3.14508e15	0.143092
$937$	−2.52576e16	−1.14241	−0.571207	−	0.820806i	$-0.693525\pi$
$937$	−2.52576e16	−1.14241	−0.571207	+	0.820806i	$0.693525\pi$
$938$	2.84599e16	1.27973
$939$	6.09554e15	0.272490
$940$	−2.34102e16	−1.04040
$941$	8.81490e14	0.0389471	0.0194735	−	0.999810i	$-0.493801\pi$
$941$	8.81490e14	0.0389471	0.0194735	+	0.999810i	$0.493801\pi$
$942$	−4.02788e16	−1.76928
$943$	−2.06239e16	−0.900651
$944$	7.60558e15	0.330207
$945$	−2.82770e15	−0.122056
$946$	5.21330e15	0.223723
$947$	7.90204e15	0.337143	0.168572	−	0.985689i	$-0.446085\pi$
$947$	7.90204e15	0.337143	0.168572	+	0.985689i	$0.446085\pi$
$948$	3.33595e15	0.141506
$949$	−1.50575e16	−0.635020
$950$	−6.87052e15	−0.288078
$951$	3.45413e13	0.00143994
$952$	7.82404e16	3.24286
$953$	−3.64310e15	−0.150127	−0.0750637	−	0.997179i	$-0.523916\pi$
$953$	−3.64310e15	−0.150127	−0.0750637	+	0.997179i	$0.523916\pi$
$954$	5.31753e15	0.217868
$955$	1.41465e16	0.576277
$956$	2.11204e16	0.855430
$957$	1.29480e16	0.521419
$958$	5.74400e16	2.29987
$959$	3.14172e15	0.125074
$960$	−9.92501e15	−0.392861
$961$	1.70857e16	0.672441
$962$	7.62911e15	0.298546
$963$	4.01115e15	0.156072
$964$	9.23410e15	0.357249
$965$	−3.25043e15	−0.125038
$966$	−2.77331e16	−1.06078
$967$	−1.26495e16	−0.481092	−0.240546	−	0.970638i	$-0.577326\pi$
$967$	−1.26495e16	−0.481092	−0.240546	+	0.970638i	$0.577326\pi$
$968$	2.51037e16	0.949345
$969$	2.47312e16	0.929961
$970$	−5.46798e15	−0.204448
$971$	−3.31816e16	−1.23365	−0.616824	−	0.787101i	$-0.711581\pi$
$971$	−3.31816e16	−1.23365	−0.616824	+	0.787101i	$0.711581\pi$
$972$	3.02791e15	0.111938
$973$	−1.14084e16	−0.419375
$974$	−7.21212e16	−2.63626
$975$	−1.10494e15	−0.0401620
$976$	−1.62058e16	−0.585728
$977$	−4.86678e16	−1.74913	−0.874565	−	0.484909i	$-0.838853\pi$
$977$	−4.86678e16	−1.74913	−0.874565	+	0.484909i	$0.838853\pi$
$978$	2.53009e16	0.904215
$979$	−1.78723e16	−0.635148
$980$	2.23288e16	0.789083
$981$	6.83660e15	0.240249
$982$	−1.74469e16	−0.609683
$983$	−6.89914e15	−0.239746	−0.119873	−	0.992789i	$-0.538249\pi$
$983$	−6.89914e15	−0.239746	−0.119873	+	0.992789i	$0.538249\pi$
$984$	−2.37502e16	−0.820717
$985$	1.69555e16	0.582654
$986$	−1.68858e17	−5.77031
$987$	3.21229e16	1.09162
$988$	−1.56136e16	−0.527643
$989$	6.54018e15	0.219791
$990$	−3.55044e15	−0.118656
$991$	−4.39787e16	−1.46163	−0.730814	−	0.682577i	$-0.760859\pi$
$991$	−4.39787e16	−1.46163	−0.730814	+	0.682577i	$0.760859\pi$
$992$	2.88518e16	0.953583
$993$	−2.29847e16	−0.755471
$994$	3.78085e16	1.23585
$995$	−2.41471e16	−0.784943
$996$	−1.12086e15	−0.0362349
$997$	−2.80611e16	−0.902157	−0.451078	−	0.892484i	$-0.648961\pi$
$997$	−2.80611e16	−0.902157	−0.451078	+	0.892484i	$0.648961\pi$
$998$	2.44904e16	0.783029
$999$	3.13565e15	0.0997050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.a.d.1.1	✓	3
3.2	odd	2		45.12.a.f.1.3		3
4.3	odd	2		240.12.a.r.1.3		3
5.2	odd	4		75.12.b.e.49.1		6
5.3	odd	4		75.12.b.e.49.6		6
5.4	even	2		75.12.a.g.1.3		3
15.2	even	4		225.12.b.j.199.6		6
15.8	even	4		225.12.b.j.199.1		6
15.14	odd	2		225.12.a.l.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.d.1.1	✓	3	1.1	even	1	trivial
45.12.a.f.1.3		3	3.2	odd	2
75.12.a.g.1.3		3	5.4	even	2
75.12.b.e.49.1		6	5.2	odd	4
75.12.b.e.49.6		6	5.3	odd	4
225.12.a.l.1.1		3	15.14	odd	2
225.12.b.j.199.1		6	15.8	even	4
225.12.b.j.199.6		6	15.2	even	4
240.12.a.r.1.3		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	15.a (trivial)

\(f(q)\)	\(=\)	\(q-74.9776 q^{2} +243.000 q^{3} +3573.65 q^{4} +3125.00 q^{5} -18219.6 q^{6} -63061.5 q^{7} -114389. q^{8} +59049.0 q^{9} +O(q^{10})\)	\(q-74.9776 q^{2} +243.000 q^{3} +3573.65 q^{4} +3125.00 q^{5} -18219.6 q^{6} -63061.5 q^{7} -114389. q^{8} +59049.0 q^{9} -234305. q^{10} +256618. q^{11} +868396. q^{12} -465623. q^{13} +4.72820e6 q^{14} +759375. q^{15} +1.25782e6 q^{16} +1.08463e7 q^{17} -4.42735e6 q^{18} +9.38335e6 q^{19} +1.11676e7 q^{20} -1.53239e7 q^{21} -1.92406e7 q^{22} -2.41377e7 q^{23} -2.77966e7 q^{24} +9.76562e6 q^{25} +3.49113e7 q^{26} +1.43489e7 q^{27} -2.25359e8 q^{28} +2.07639e8 q^{29} -5.69361e7 q^{30} +2.06141e8 q^{31} +1.39961e8 q^{32} +6.23583e7 q^{33} -8.13230e8 q^{34} -1.97067e8 q^{35} +2.11020e8 q^{36} +2.18529e8 q^{37} -7.03542e8 q^{38} -1.13146e8 q^{39} -3.57467e8 q^{40} +8.54426e8 q^{41} +1.14895e9 q^{42} -2.70952e8 q^{43} +9.17063e8 q^{44} +1.84528e8 q^{45} +1.80979e9 q^{46} -2.09625e9 q^{47} +3.05650e8 q^{48} +1.99942e9 q^{49} -7.32204e8 q^{50} +2.63565e9 q^{51} -1.66397e9 q^{52} -1.20106e9 q^{53} -1.07585e9 q^{54} +8.01933e8 q^{55} +7.21356e9 q^{56} +2.28015e9 q^{57} -1.55683e10 q^{58} +6.04665e9 q^{59} +2.71374e9 q^{60} -1.28841e10 q^{61} -1.54560e10 q^{62} -3.72372e9 q^{63} -1.30700e10 q^{64} -1.45507e9 q^{65} -4.67548e9 q^{66} +6.01919e9 q^{67} +3.87608e10 q^{68} -5.86547e9 q^{69} +1.47756e10 q^{70} +7.99639e9 q^{71} -6.75458e9 q^{72} +3.23383e10 q^{73} -1.63848e10 q^{74} +2.37305e9 q^{75} +3.35328e10 q^{76} -1.61827e10 q^{77} +8.48344e9 q^{78} +3.84151e9 q^{79} +3.93068e9 q^{80} +3.48678e9 q^{81} -6.40628e10 q^{82} -1.29073e9 q^{83} -5.47623e10 q^{84} +3.38947e10 q^{85} +2.03154e10 q^{86} +5.04563e10 q^{87} -2.93544e10 q^{88} -6.96454e10 q^{89} -1.38355e10 q^{90} +2.93628e10 q^{91} -8.62597e10 q^{92} +5.00923e10 q^{93} +1.57172e11 q^{94} +2.93230e10 q^{95} +3.40106e10 q^{96} +2.33370e10 q^{97} -1.49912e11 q^{98} +1.51531e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 729 q^{3} + 4757 q^{4} + 9375 q^{5} - 243 q^{6} - 14608 q^{7} - 33999 q^{8} + 177147 q^{9}+O(q^{10}) \) 3 * q - q^2 + 729 * q^3 + 4757 * q^4 + 9375 * q^5 - 243 * q^6 - 14608 * q^7 - 33999 * q^8 + 177147 * q^9	\( 3 q - q^{2} + 729 q^{3} + 4757 q^{4} + 9375 q^{5} - 243 q^{6} - 14608 q^{7} - 33999 q^{8} + 177147 q^{9} - 3125 q^{10} + 540620 q^{11} + 1155951 q^{12} + 840970 q^{13} + 5432712 q^{14} + 2278125 q^{15} + 5062961 q^{16} + 15165038 q^{17} - 59049 q^{18} + 17743756 q^{19} + 14865625 q^{20} - 3549744 q^{21} + 11176076 q^{22} - 28140816 q^{23} - 8261757 q^{24} + 29296875 q^{25} - 52021894 q^{26} + 43046721 q^{27} - 277157944 q^{28} - 67382798 q^{29} - 759375 q^{30} - 206919496 q^{31} - 46592663 q^{32} + 131370660 q^{33} - 1230469666 q^{34} - 45650000 q^{35} + 280896093 q^{36} - 318337278 q^{37} + 653190692 q^{38} + 204355710 q^{39} - 106246875 q^{40} + 2110085854 q^{41} + 1320149016 q^{42} + 418259692 q^{43} + 2558131108 q^{44} + 553584375 q^{45} - 137169096 q^{46} - 1599668584 q^{47} + 1230299523 q^{48} - 316107077 q^{49} - 9765625 q^{50} + 3685104234 q^{51} - 10897289202 q^{52} + 4489142234 q^{53} - 14348907 q^{54} + 1689437500 q^{55} + 7768845960 q^{56} + 4311732708 q^{57} - 24168830726 q^{58} + 11102167484 q^{59} + 3612346875 q^{60} - 3568120958 q^{61} - 35509109136 q^{62} - 862587792 q^{63} - 35608208271 q^{64} + 2628031250 q^{65} + 2715786468 q^{66} + 2229942788 q^{67} - 1367872838 q^{68} - 6838218288 q^{69} + 16977225000 q^{70} + 49842766696 q^{71} - 2007606951 q^{72} + 40752219934 q^{73} - 37519971278 q^{74} + 7119140625 q^{75} + 115970329116 q^{76} - 17819224896 q^{77} - 12641320242 q^{78} + 113159960 q^{79} + 15821753125 q^{80} + 10460353203 q^{81} - 30171431066 q^{82} + 6259660308 q^{83} - 67349380392 q^{84} + 47390743750 q^{85} + 114296127740 q^{86} - 16374019914 q^{87} + 7548672276 q^{88} - 59972401554 q^{89} - 184528125 q^{90} + 118873361824 q^{91} - 221705928648 q^{92} - 50281437528 q^{93} + 92816682800 q^{94} + 55449237500 q^{95} - 11322017109 q^{96} - 207831285882 q^{97} - 288264739625 q^{98} + 31923070380 q^{99}+O(q^{100}) \) 3 * q - q^2 + 729 * q^3 + 4757 * q^4 + 9375 * q^5 - 243 * q^6 - 14608 * q^7 - 33999 * q^8 + 177147 * q^9 - 3125 * q^10 + 540620 * q^11 + 1155951 * q^12 + 840970 * q^13 + 5432712 * q^14 + 2278125 * q^15 + 5062961 * q^16 + 15165038 * q^17 - 59049 * q^18 + 17743756 * q^19 + 14865625 * q^20 - 3549744 * q^21 + 11176076 * q^22 - 28140816 * q^23 - 8261757 * q^24 + 29296875 * q^25 - 52021894 * q^26 + 43046721 * q^27 - 277157944 * q^28 - 67382798 * q^29 - 759375 * q^30 - 206919496 * q^31 - 46592663 * q^32 + 131370660 * q^33 - 1230469666 * q^34 - 45650000 * q^35 + 280896093 * q^36 - 318337278 * q^37 + 653190692 * q^38 + 204355710 * q^39 - 106246875 * q^40 + 2110085854 * q^41 + 1320149016 * q^42 + 418259692 * q^43 + 2558131108 * q^44 + 553584375 * q^45 - 137169096 * q^46 - 1599668584 * q^47 + 1230299523 * q^48 - 316107077 * q^49 - 9765625 * q^50 + 3685104234 * q^51 - 10897289202 * q^52 + 4489142234 * q^53 - 14348907 * q^54 + 1689437500 * q^55 + 7768845960 * q^56 + 4311732708 * q^57 - 24168830726 * q^58 + 11102167484 * q^59 + 3612346875 * q^60 - 3568120958 * q^61 - 35509109136 * q^62 - 862587792 * q^63 - 35608208271 * q^64 + 2628031250 * q^65 + 2715786468 * q^66 + 2229942788 * q^67 - 1367872838 * q^68 - 6838218288 * q^69 + 16977225000 * q^70 + 49842766696 * q^71 - 2007606951 * q^72 + 40752219934 * q^73 - 37519971278 * q^74 + 7119140625 * q^75 + 115970329116 * q^76 - 17819224896 * q^77 - 12641320242 * q^78 + 113159960 * q^79 + 15821753125 * q^80 + 10460353203 * q^81 - 30171431066 * q^82 + 6259660308 * q^83 - 67349380392 * q^84 + 47390743750 * q^85 + 114296127740 * q^86 - 16374019914 * q^87 + 7548672276 * q^88 - 59972401554 * q^89 - 184528125 * q^90 + 118873361824 * q^91 - 221705928648 * q^92 - 50281437528 * q^93 + 92816682800 * q^94 + 55449237500 * q^95 - 11322017109 * q^96 - 207831285882 * q^97 - 288264739625 * q^98 + 31923070380 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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