LMFDB - Embedded newform 15.12.a.c.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:	$2$
Coefficient field:	$\Q(\sqrt{1801}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 450 $ x^2 - x - 450
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			$21.7191$ 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)

$f(q)$	$=$	$q-27.7191 q^{2} -243.000 q^{3} -1279.65 q^{4} -3125.00 q^{5} +6735.74 q^{6} -72157.2 q^{7} +92239.5 q^{8} +59049.0 q^{9} +O(q^{10})$	\(q-27.7191 q^{2} -243.000 q^{3} -1279.65 q^{4} -3125.00 q^{5} +6735.74 q^{6} -72157.2 q^{7} +92239.5 q^{8} +59049.0 q^{9} +86622.2 q^{10} -509499. q^{11} +310955. q^{12} +1.85516e6 q^{13} +2.00013e6 q^{14} +759375. q^{15} +63931.4 q^{16} +5.94676e6 q^{17} -1.63678e6 q^{18} +6.02189e6 q^{19} +3.99891e6 q^{20} +1.75342e7 q^{21} +1.41228e7 q^{22} -4.82627e7 q^{23} -2.24142e7 q^{24} +9.76562e6 q^{25} -5.14234e7 q^{26} -1.43489e7 q^{27} +9.23361e7 q^{28} -1.13550e7 q^{29} -2.10492e7 q^{30} -1.72184e8 q^{31} -1.90679e8 q^{32} +1.23808e8 q^{33} -1.64839e8 q^{34} +2.25491e8 q^{35} -7.55622e7 q^{36} +6.25191e8 q^{37} -1.66921e8 q^{38} -4.50805e8 q^{39} -2.88248e8 q^{40} -5.53971e8 q^{41} -4.86032e8 q^{42} +1.52106e9 q^{43} +6.51981e8 q^{44} -1.84528e8 q^{45} +1.33780e9 q^{46} +1.19456e9 q^{47} -1.55353e7 q^{48} +3.22934e9 q^{49} -2.70694e8 q^{50} -1.44506e9 q^{51} -2.37396e9 q^{52} +1.22320e9 q^{53} +3.97739e8 q^{54} +1.59218e9 q^{55} -6.65575e9 q^{56} -1.46332e9 q^{57} +3.14750e8 q^{58} -5.83637e9 q^{59} -9.71736e8 q^{60} -6.61097e9 q^{61} +4.77277e9 q^{62} -4.26081e9 q^{63} +5.15451e9 q^{64} -5.79738e9 q^{65} -3.43185e9 q^{66} +1.66261e10 q^{67} -7.60978e9 q^{68} +1.17278e10 q^{69} -6.25042e9 q^{70} +7.36200e9 q^{71} +5.44665e9 q^{72} -6.35726e9 q^{73} -1.73297e10 q^{74} -2.37305e9 q^{75} -7.70593e9 q^{76} +3.67640e10 q^{77} +1.24959e10 q^{78} +2.47565e10 q^{79} -1.99786e8 q^{80} +3.48678e9 q^{81} +1.53556e10 q^{82} +3.59416e10 q^{83} -2.24377e10 q^{84} -1.85836e10 q^{85} -4.21625e10 q^{86} +2.75926e9 q^{87} -4.69959e10 q^{88} +7.47690e10 q^{89} +5.11495e9 q^{90} -1.33863e11 q^{91} +6.17594e10 q^{92} +4.18406e10 q^{93} -3.31121e10 q^{94} -1.88184e10 q^{95} +4.63349e10 q^{96} -1.66300e10 q^{97} -8.95144e10 q^{98} -3.00854e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 13 q^{2} - 486 q^{3} - 3111 q^{4} - 6250 q^{5} + 3159 q^{6} + 7784 q^{7} + 35139 q^{8} + 118098 q^{9}+O(q^{10}) $ 2 * q - 13 * q^2 - 486 * q^3 - 3111 * q^4 - 6250 * q^5 + 3159 * q^6 + 7784 * q^7 + 35139 * q^8 + 118098 * q^9	\( 2 q - 13 q^{2} - 486 q^{3} - 3111 q^{4} - 6250 q^{5} + 3159 q^{6} + 7784 q^{7} + 35139 q^{8} + 118098 q^{9} + 40625 q^{10} + 295568 q^{11} + 755973 q^{12} + 657492 q^{13} + 3176796 q^{14} + 1518750 q^{15} + 2974065 q^{16} + 8579948 q^{17} - 767637 q^{18} + 17627976 q^{19} + 9721875 q^{20} - 1891512 q^{21} + 25972696 q^{22} - 29841072 q^{23} - 8538777 q^{24} + 19531250 q^{25} - 69052066 q^{26} - 28697814 q^{27} - 54064108 q^{28} - 201881948 q^{29} - 9871875 q^{30} - 71057008 q^{31} - 30902261 q^{32} - 71823024 q^{33} - 126080702 q^{34} - 24325000 q^{35} - 183701439 q^{36} + 705858484 q^{37} + 3909548 q^{38} - 159770556 q^{39} - 109809375 q^{40} - 327655148 q^{41} - 771961428 q^{42} + 3192552120 q^{43} - 822376568 q^{44} - 369056250 q^{45} + 1608947832 q^{46} + 2053064720 q^{47} - 722697795 q^{48} + 7642614770 q^{49} - 126953125 q^{50} - 2084927364 q^{51} - 180610022 q^{52} - 2304299452 q^{53} + 186535791 q^{54} - 923650000 q^{55} - 11220431460 q^{56} - 4283598168 q^{57} - 2489635114 q^{58} - 1478770576 q^{59} - 2362415625 q^{60} - 8264891460 q^{61} + 6261263448 q^{62} + 459637416 q^{63} + 1546316681 q^{64} - 2054662500 q^{65} - 6311365128 q^{66} + 24212177528 q^{67} - 12432065594 q^{68} + 7251380496 q^{69} - 9927487500 q^{70} - 20218888256 q^{71} + 2074922811 q^{72} + 25879583268 q^{73} - 16142374034 q^{74} - 4746093750 q^{75} - 28960704764 q^{76} + 101122045248 q^{77} + 16779652038 q^{78} + 22324995440 q^{79} - 9293953125 q^{80} + 6973568802 q^{81} + 18686740478 q^{82} + 48014508984 q^{83} + 13137578244 q^{84} - 26812337500 q^{85} - 17559698092 q^{86} + 49057313364 q^{87} - 92965609896 q^{88} + 79209683076 q^{89} + 2398865625 q^{90} - 229606712048 q^{91} + 28023036264 q^{92} + 17266852944 q^{93} - 20475612776 q^{94} - 55087425000 q^{95} + 7509249423 q^{96} - 37075227452 q^{97} - 24554976677 q^{98} + 17452994832 q^{99}+O(q^{100}) \) 2 * q - 13 * q^2 - 486 * q^3 - 3111 * q^4 - 6250 * q^5 + 3159 * q^6 + 7784 * q^7 + 35139 * q^8 + 118098 * q^9 + 40625 * q^10 + 295568 * q^11 + 755973 * q^12 + 657492 * q^13 + 3176796 * q^14 + 1518750 * q^15 + 2974065 * q^16 + 8579948 * q^17 - 767637 * q^18 + 17627976 * q^19 + 9721875 * q^20 - 1891512 * q^21 + 25972696 * q^22 - 29841072 * q^23 - 8538777 * q^24 + 19531250 * q^25 - 69052066 * q^26 - 28697814 * q^27 - 54064108 * q^28 - 201881948 * q^29 - 9871875 * q^30 - 71057008 * q^31 - 30902261 * q^32 - 71823024 * q^33 - 126080702 * q^34 - 24325000 * q^35 - 183701439 * q^36 + 705858484 * q^37 + 3909548 * q^38 - 159770556 * q^39 - 109809375 * q^40 - 327655148 * q^41 - 771961428 * q^42 + 3192552120 * q^43 - 822376568 * q^44 - 369056250 * q^45 + 1608947832 * q^46 + 2053064720 * q^47 - 722697795 * q^48 + 7642614770 * q^49 - 126953125 * q^50 - 2084927364 * q^51 - 180610022 * q^52 - 2304299452 * q^53 + 186535791 * q^54 - 923650000 * q^55 - 11220431460 * q^56 - 4283598168 * q^57 - 2489635114 * q^58 - 1478770576 * q^59 - 2362415625 * q^60 - 8264891460 * q^61 + 6261263448 * q^62 + 459637416 * q^63 + 1546316681 * q^64 - 2054662500 * q^65 - 6311365128 * q^66 + 24212177528 * q^67 - 12432065594 * q^68 + 7251380496 * q^69 - 9927487500 * q^70 - 20218888256 * q^71 + 2074922811 * q^72 + 25879583268 * q^73 - 16142374034 * q^74 - 4746093750 * q^75 - 28960704764 * q^76 + 101122045248 * q^77 + 16779652038 * q^78 + 22324995440 * q^79 - 9293953125 * q^80 + 6973568802 * q^81 + 18686740478 * q^82 + 48014508984 * q^83 + 13137578244 * q^84 - 26812337500 * q^85 - 17559698092 * q^86 + 49057313364 * q^87 - 92965609896 * q^88 + 79209683076 * q^89 + 2398865625 * q^90 - 229606712048 * q^91 + 28023036264 * q^92 + 17266852944 * q^93 - 20475612776 * q^94 - 55087425000 * q^95 + 7509249423 * q^96 - 37075227452 * q^97 - 24554976677 * q^98 + 17452994832 * 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 $

$2$	−27.7191	−0.612511	−0.306256	−	0.951949i	$-0.599076\pi$
$2$	−27.7191	−0.612511	−0.306256	+	0.951949i	$0.599076\pi$
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	−1279.65	−0.624830
$5$	−3125.00	−0.447214
$5$	−3125.00	−0.447214
$6$	6735.74	0.353634
$7$	−72157.2	−1.62271	−0.811355	−	0.584554i	$-0.801269\pi$
$7$	−72157.2	−1.62271	−0.811355	+	0.584554i	$0.801269\pi$
$8$	92239.5	0.995227
$9$	59049.0	0.333333
$10$	86622.2	0.273923
$11$	−509499.	−0.953857	−0.476929	−	0.878942i	$-0.658250\pi$
$11$	−509499.	−0.953857	−0.476929	+	0.878942i	$0.658250\pi$
$12$	310955.	0.360746
$13$	1.85516e6	1.38578	0.692889	−	0.721044i	$-0.256338\pi$
$13$	1.85516e6	1.38578	0.692889	+	0.721044i	$0.256338\pi$
$14$	2.00013e6	0.993928
$15$	759375.	0.258199
$16$	63931.4	0.0152424
$17$	5.94676e6	1.01581	0.507904	−	0.861414i	$-0.330421\pi$
$17$	5.94676e6	1.01581	0.507904	+	0.861414i	$0.330421\pi$
$18$	−1.63678e6	−0.204170
$19$	6.02189e6	0.557941	0.278971	−	0.960300i	$-0.410007\pi$
$19$	6.02189e6	0.557941	0.278971	+	0.960300i	$0.410007\pi$
$20$	3.99891e6	0.279432
$21$	1.75342e7	0.936872
$22$	1.41228e7	0.584248
$23$	−4.82627e7	−1.56354	−0.781769	−	0.623568i	$-0.785682\pi$
$23$	−4.82627e7	−1.56354	−0.781769	+	0.623568i	$0.785682\pi$
$24$	−2.24142e7	−0.574594
$25$	9.76562e6	0.200000
$26$	−5.14234e7	−0.848804
$27$	−1.43489e7	−0.192450
$28$	9.23361e7	1.01392
$29$	−1.13550e7	−0.102801	−0.0514005	−	0.998678i	$-0.516368\pi$
$29$	−1.13550e7	−0.102801	−0.0514005	+	0.998678i	$0.516368\pi$
$30$	−2.10492e7	−0.158150
$31$	−1.72184e8	−1.08020	−0.540098	−	0.841602i	$-0.681613\pi$
$31$	−1.72184e8	−1.08020	−0.540098	+	0.841602i	$0.681613\pi$
$32$	−1.90679e8	−1.00456
$33$	1.23808e8	0.550710
$34$	−1.64839e8	−0.622194
$35$	2.25491e8	0.725698
$36$	−7.55622e7	−0.208277
$37$	6.25191e8	1.48219	0.741094	−	0.671401i	$-0.234307\pi$
$37$	6.25191e8	1.48219	0.741094	+	0.671401i	$0.234307\pi$
$38$	−1.66921e8	−0.341745
$39$	−4.50805e8	−0.800079
$40$	−2.88248e8	−0.445079
$41$	−5.53971e8	−0.746751	−0.373376	−	0.927680i	$-0.621800\pi$
$41$	−5.53971e8	−0.746751	−0.373376	+	0.927680i	$0.621800\pi$
$42$	−4.86032e8	−0.573844
$43$	1.52106e9	1.57787	0.788934	−	0.614478i	$-0.210633\pi$
$43$	1.52106e9	1.57787	0.788934	+	0.614478i	$0.210633\pi$
$44$	6.51981e8	0.595998
$45$	−1.84528e8	−0.149071
$46$	1.33780e9	0.957685
$47$	1.19456e9	0.759747	0.379873	−	0.925038i	$-0.375968\pi$
$47$	1.19456e9	0.759747	0.379873	+	0.925038i	$0.375968\pi$
$48$	−1.55353e7	−0.00880023
$49$	3.22934e9	1.63318
$50$	−2.70694e8	−0.122502
$51$	−1.44506e9	−0.586477
$52$	−2.37396e9	−0.865875
$53$	1.22320e9	0.401774	0.200887	−	0.979614i	$-0.435618\pi$
$53$	1.22320e9	0.401774	0.200887	+	0.979614i	$0.435618\pi$
$54$	3.97739e8	0.117878
$55$	1.59218e9	0.426578
$56$	−6.65575e9	−1.61496
$57$	−1.46332e9	−0.322127
$58$	3.14750e8	0.0629667
$59$	−5.83637e9	−1.06281	−0.531407	−	0.847117i	$-0.678336\pi$
$59$	−5.83637e9	−1.06281	−0.531407	+	0.847117i	$0.678336\pi$
$60$	−9.71736e8	−0.161330
$61$	−6.61097e9	−1.00219	−0.501096	−	0.865392i	$-0.667070\pi$
$61$	−6.61097e9	−1.00219	−0.501096	+	0.865392i	$0.667070\pi$
$62$	4.77277e9	0.661632
$63$	−4.26081e9	−0.540903
$64$	5.15451e9	0.600064
$65$	−5.79738e9	−0.619739
$66$	−3.43185e9	−0.337316
$67$	1.66261e10	1.50445	0.752226	−	0.658905i	$-0.228980\pi$
$67$	1.66261e10	1.50445	0.752226	+	0.658905i	$0.228980\pi$
$68$	−7.60978e9	−0.634707
$69$	1.17278e10	0.902709
$70$	−6.25042e9	−0.444498
$71$	7.36200e9	0.484256	0.242128	−	0.970244i	$-0.422155\pi$
$71$	7.36200e9	0.484256	0.242128	+	0.970244i	$0.422155\pi$
$72$	5.44665e9	0.331742
$73$	−6.35726e9	−0.358917	−0.179458	−	0.983766i	$-0.557435\pi$
$73$	−6.35726e9	−0.358917	−0.179458	+	0.983766i	$0.557435\pi$
$74$	−1.73297e10	−0.907857
$75$	−2.37305e9	−0.115470
$76$	−7.70593e9	−0.348618
$77$	3.67640e10	1.54783
$78$	1.24959e10	0.490058
$79$	2.47565e10	0.905191	0.452595	−	0.891716i	$-0.350498\pi$
$79$	2.47565e10	0.905191	0.452595	+	0.891716i	$0.350498\pi$
$80$	−1.99786e8	−0.00681663
$81$	3.48678e9	0.111111
$82$	1.53556e10	0.457393
$83$	3.59416e10	1.00154	0.500769	−	0.865581i	$-0.333050\pi$
$83$	3.59416e10	1.00154	0.500769	+	0.865581i	$0.333050\pi$
$84$	−2.24377e10	−0.585385
$85$	−1.85836e10	−0.454283
$86$	−4.21625e10	−0.966462
$87$	2.75926e9	0.0593522
$88$	−4.69959e10	−0.949304
$89$	7.47690e10	1.41931	0.709653	−	0.704551i	$-0.248852\pi$
$89$	7.47690e10	1.41931	0.709653	+	0.704551i	$0.248852\pi$
$90$	5.11495e9	0.0913078
$91$	−1.33863e11	−2.24871
$92$	6.17594e10	0.976945
$93$	4.18406e10	0.623651
$94$	−3.31121e10	−0.465354
$95$	−1.88184e10	−0.249519
$96$	4.63349e10	0.579985
$97$	−1.66300e10	−0.196629	−0.0983144	−	0.995155i	$-0.531345\pi$
$97$	−1.66300e10	−0.196629	−0.0983144	+	0.995155i	$0.531345\pi$
$98$	−8.95144e10	−1.00034
$99$	−3.00854e10	−0.317952
$100$	−1.24966e10	−0.124966
$101$	−5.30247e10	−0.502008	−0.251004	−	0.967986i	$-0.580761\pi$
$101$	−5.30247e10	−0.502008	−0.251004	+	0.967986i	$0.580761\pi$
$102$	4.00558e10	0.359224
$103$	1.00437e9	0.00853666	0.00426833	−	0.999991i	$-0.498641\pi$
$103$	1.00437e9	0.00853666	0.00426833	+	0.999991i	$0.498641\pi$
$104$	1.71119e11	1.37916
$105$	−5.47944e10	−0.418982
$106$	−3.39061e10	−0.246091
$107$	−1.65238e11	−1.13893	−0.569466	−	0.822015i	$-0.692850\pi$
$107$	−1.65238e11	−1.13893	−0.569466	+	0.822015i	$0.692850\pi$
$108$	1.83616e10	0.120249
$109$	7.04515e10	0.438576	0.219288	−	0.975660i	$-0.429627\pi$
$109$	7.04515e10	0.438576	0.219288	+	0.975660i	$0.429627\pi$
$110$	−4.41339e10	−0.261284
$111$	−1.51921e11	−0.855741
$112$	−4.61312e9	−0.0247341
$113$	2.07466e11	1.05929	0.529646	−	0.848219i	$-0.322325\pi$
$113$	2.07466e11	1.05929	0.529646	+	0.848219i	$0.322325\pi$
$114$	4.05619e10	0.197307
$115$	1.50821e11	0.699235
$116$	1.45304e10	0.0642331
$117$	1.09546e11	0.461926
$118$	1.61779e11	0.650985
$119$	−4.29102e11	−1.64836
$120$	7.00444e10	0.256966
$121$	−2.57228e10	−0.0901567
$122$	1.83250e11	0.613854
$123$	1.34615e11	0.431137
$124$	2.20335e11	0.674938
$125$	−3.05176e10	−0.0894427
$126$	1.18106e11	0.331309
$127$	1.17811e11	0.316422	0.158211	−	0.987405i	$-0.449427\pi$
$127$	1.17811e11	0.316422	0.158211	+	0.987405i	$0.449427\pi$
$128$	2.47632e11	0.637017
$129$	−3.69618e11	−0.910983
$130$	1.60698e11	0.379597
$131$	4.60288e11	1.04241	0.521204	−	0.853432i	$-0.325483\pi$
$131$	4.60288e11	1.04241	0.521204	+	0.853432i	$0.325483\pi$
$132$	−1.58431e11	−0.344100
$133$	−4.34523e11	−0.905376
$134$	−4.60860e11	−0.921494
$135$	4.48403e10	0.0860663
$136$	5.48526e11	1.01096
$137$	−5.80435e11	−1.02752	−0.513760	−	0.857934i	$-0.671748\pi$
$137$	−5.80435e11	−1.02752	−0.513760	+	0.857934i	$0.671748\pi$
$138$	−3.25085e11	−0.552920
$139$	6.94847e11	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$139$	6.94847e11	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$140$	−2.88550e11	−0.453438
$141$	−2.90278e11	−0.438640
$142$	−2.04068e11	−0.296612
$143$	−9.45203e11	−1.32183
$144$	3.77509e9	0.00508081
$145$	3.54843e10	0.0459740
$146$	1.76217e11	0.219841
$147$	−7.84730e11	−0.942920
$148$	−8.00027e11	−0.926115
$149$	−1.12749e11	−0.125774	−0.0628869	−	0.998021i	$-0.520031\pi$
$149$	−1.12749e11	−0.125774	−0.0628869	+	0.998021i	$0.520031\pi$
$150$	6.57787e10	0.0707267
$151$	2.45742e11	0.254746	0.127373	−	0.991855i	$-0.459346\pi$
$151$	2.45742e11	0.254746	0.127373	+	0.991855i	$0.459346\pi$
$152$	5.55457e11	0.555278
$153$	3.51150e11	0.338602
$154$	−1.01907e12	−0.948065
$155$	5.38074e11	0.483078
$156$	5.76873e11	0.499913
$157$	−5.83317e11	−0.488042	−0.244021	−	0.969770i	$-0.578467\pi$
$157$	−5.83317e11	−0.488042	−0.244021	+	0.969770i	$0.578467\pi$
$158$	−6.86227e11	−0.554439
$159$	−2.97239e11	−0.231964
$160$	5.95871e11	0.449254
$161$	3.48250e12	2.53717
$162$	−9.66505e10	−0.0680568
$163$	2.65801e12	1.80936	0.904681	−	0.426089i	$-0.140109\pi$
$163$	2.65801e12	1.80936	0.904681	+	0.426089i	$0.140109\pi$
$164$	7.08890e11	0.466592
$165$	−3.86901e11	−0.246285
$166$	−9.96269e11	−0.613454
$167$	−1.68070e12	−1.00126	−0.500632	−	0.865660i	$-0.666899\pi$
$167$	−1.68070e12	−1.00126	−0.500632	+	0.865660i	$0.666899\pi$
$168$	1.61735e12	0.932400
$169$	1.64947e12	0.920380
$170$	5.15121e11	0.278253
$171$	3.55587e11	0.185980
$172$	−1.94643e12	−0.985900
$173$	−2.11009e12	−1.03525	−0.517627	−	0.855606i	$-0.673185\pi$
$173$	−2.11009e12	−1.03525	−0.517627	+	0.855606i	$0.673185\pi$
$174$	−7.64841e10	−0.0363539
$175$	−7.04661e11	−0.324542
$176$	−3.25730e10	−0.0145391
$177$	1.41824e12	0.613616
$178$	−2.07253e12	−0.869341
$179$	8.73323e10	0.0355208	0.0177604	−	0.999842i	$-0.494346\pi$
$179$	8.73323e10	0.0355208	0.0177604	+	0.999842i	$0.494346\pi$
$180$	2.36132e11	0.0931442
$181$	2.60066e12	0.995066	0.497533	−	0.867445i	$-0.334239\pi$
$181$	2.60066e12	0.995066	0.497533	+	0.867445i	$0.334239\pi$
$182$	3.71057e12	1.37736
$183$	1.60646e12	0.578616
$184$	−4.45173e12	−1.55607
$185$	−1.95372e12	−0.662854
$186$	−1.15978e12	−0.381993
$187$	−3.02987e12	−0.968935
$188$	−1.52862e12	−0.474713
$189$	1.03538e12	0.312291
$190$	5.21630e11	0.152833
$191$	−2.28556e12	−0.650593	−0.325296	−	0.945612i	$-0.605464\pi$
$191$	−2.28556e12	−0.650593	−0.325296	+	0.945612i	$0.605464\pi$
$192$	−1.25255e12	−0.346447
$193$	3.11630e12	0.837671	0.418836	−	0.908062i	$-0.362438\pi$
$193$	3.11630e12	0.837671	0.418836	+	0.908062i	$0.362438\pi$
$194$	4.60968e11	0.120437
$195$	1.40876e12	0.357806
$196$	−4.13243e12	−1.02046
$197$	3.68099e12	0.883894	0.441947	−	0.897041i	$-0.354288\pi$
$197$	3.68099e12	0.883894	0.441947	+	0.897041i	$0.354288\pi$
$198$	8.33940e11	0.194749
$199$	−2.20305e12	−0.500417	−0.250208	−	0.968192i	$-0.580499\pi$
$199$	−2.20305e12	−0.500417	−0.250208	+	0.968192i	$0.580499\pi$
$200$	9.00776e11	0.199045
$201$	−4.04014e12	−0.868596
$202$	1.46980e12	0.307485
$203$	8.19343e11	0.166816
$204$	1.84918e12	0.366448
$205$	1.73116e12	0.333957
$206$	−2.78402e10	−0.00522880
$207$	−2.84986e12	−0.521179
$208$	1.18603e11	0.0211226
$209$	−3.06815e12	−0.532196
$210$	1.51885e12	0.256631
$211$	−1.08697e13	−1.78921	−0.894607	−	0.446853i	$-0.852545\pi$
$211$	−1.08697e13	−1.78921	−0.894607	+	0.446853i	$0.852545\pi$
$212$	−1.56528e12	−0.251040
$213$	−1.78897e12	−0.279585
$214$	4.58024e12	0.697609
$215$	−4.75332e12	−0.705644
$216$	−1.32354e12	−0.191531
$217$	1.24243e13	1.75284
$218$	−1.95285e12	−0.268633
$219$	1.54481e12	0.207221
$220$	−2.03744e12	−0.266539
$221$	1.10322e13	1.40768
$222$	4.21112e12	0.524151
$223$	9.89456e12	1.20149	0.600744	−	0.799441i	$-0.294871\pi$
$223$	9.89456e12	1.20149	0.600744	+	0.799441i	$0.294871\pi$
$224$	1.37588e13	1.63011
$225$	5.76650e11	0.0666667
$226$	−5.75078e12	−0.648829
$227$	5.42930e12	0.597862	0.298931	−	0.954275i	$-0.403370\pi$
$227$	5.42930e12	0.597862	0.298931	+	0.954275i	$0.403370\pi$
$228$	1.87254e12	0.201275
$229$	2.17366e12	0.228085	0.114042	−	0.993476i	$-0.463620\pi$
$229$	2.17366e12	0.228085	0.114042	+	0.993476i	$0.463620\pi$
$230$	−4.18062e12	−0.428290
$231$	−8.93366e12	−0.893642
$232$	−1.04738e12	−0.102310
$233$	−1.98130e13	−1.89013	−0.945067	−	0.326876i	$-0.894004\pi$
$233$	−1.98130e13	−1.89013	−0.945067	+	0.326876i	$0.894004\pi$
$234$	−3.03650e12	−0.282935
$235$	−3.73299e12	−0.339769
$236$	7.46853e12	0.664078
$237$	−6.01583e12	−0.522612
$238$	1.18943e13	1.00964
$239$	1.03457e12	0.0858169	0.0429085	−	0.999079i	$-0.486338\pi$
$239$	1.03457e12	0.0858169	0.0429085	+	0.999079i	$0.486338\pi$
$240$	4.85479e10	0.00393558
$241$	1.36081e13	1.07821	0.539106	−	0.842238i	$-0.318762\pi$
$241$	1.36081e13	1.07821	0.539106	+	0.842238i	$0.318762\pi$
$242$	7.13011e11	0.0552220
$243$	−8.47289e11	−0.0641500
$244$	8.45973e12	0.626199
$245$	−1.00917e13	−0.730382
$246$	−3.73140e12	−0.264076
$247$	1.11716e13	0.773182
$248$	−1.58821e13	−1.07504
$249$	−8.73381e12	−0.578239
$250$	8.45920e11	0.0547847
$251$	−2.28023e13	−1.44469	−0.722344	−	0.691534i	$-0.756935\pi$
$251$	−2.28023e13	−1.44469	−0.722344	+	0.691534i	$0.756935\pi$
$252$	5.45236e12	0.337972
$253$	2.45898e13	1.49139
$254$	−3.26562e12	−0.193812
$255$	4.51582e12	0.262280
$256$	−1.74206e13	−0.990244
$257$	6.13077e12	0.341101	0.170550	−	0.985349i	$-0.445445\pi$
$257$	6.13077e12	0.341101	0.170550	+	0.985349i	$0.445445\pi$
$258$	1.02455e13	0.557987
$259$	−4.51121e13	−2.40516
$260$	7.41863e12	0.387231
$261$	−6.70500e11	−0.0342670
$262$	−1.27588e13	−0.638486
$263$	−3.25384e13	−1.59456	−0.797278	−	0.603613i	$-0.793727\pi$
$263$	−3.25384e13	−1.59456	−0.797278	+	0.603613i	$0.793727\pi$
$264$	1.14200e13	0.548081
$265$	−3.82251e12	−0.179679
$266$	1.20446e13	0.554553
$267$	−1.81689e13	−0.819437
$268$	−2.12756e13	−0.940027
$269$	3.66970e13	1.58852	0.794261	−	0.607577i	$-0.207858\pi$
$269$	3.66970e13	1.58852	0.794261	+	0.607577i	$0.207858\pi$
$270$	−1.24293e12	−0.0527166
$271$	1.87361e13	0.778659	0.389329	−	0.921099i	$-0.372707\pi$
$271$	1.87361e13	0.778659	0.389329	+	0.921099i	$0.372707\pi$
$272$	3.80185e11	0.0154834
$273$	3.25288e13	1.29830
$274$	1.60891e13	0.629368
$275$	−4.97557e12	−0.190771
$276$	−1.50075e13	−0.564040
$277$	−3.03044e13	−1.11652	−0.558260	−	0.829666i	$-0.688531\pi$
$277$	−3.03044e13	−1.11652	−0.558260	+	0.829666i	$0.688531\pi$
$278$	−1.92605e13	−0.695700
$279$	−1.01673e13	−0.360065
$280$	2.07992e13	0.722234
$281$	4.93501e13	1.68037	0.840183	−	0.542304i	$-0.182448\pi$
$281$	4.93501e13	1.68037	0.840183	+	0.542304i	$0.182448\pi$
$282$	8.04623e12	0.268672
$283$	−5.45122e12	−0.178512	−0.0892562	−	0.996009i	$-0.528449\pi$
$283$	−5.45122e12	−0.178512	−0.0892562	+	0.996009i	$0.528449\pi$
$284$	−9.42079e12	−0.302578
$285$	4.57288e12	0.144060
$286$	2.62002e13	0.809638
$287$	3.99730e13	1.21176
$288$	−1.12594e13	−0.334854
$289$	1.09207e12	0.0318649
$290$	−9.83592e11	−0.0281596
$291$	4.04108e12	0.113524
$292$	8.13507e12	0.224262
$293$	4.85807e13	1.31429	0.657146	−	0.753763i	$-0.271763\pi$
$293$	4.85807e13	1.31429	0.657146	+	0.753763i	$0.271763\pi$
$294$	2.17520e13	0.577549
$295$	1.82387e13	0.475305
$296$	5.76673e13	1.47511
$297$	7.31075e12	0.183570
$298$	3.12531e12	0.0770378
$299$	−8.95352e13	−2.16672
$300$	3.03667e12	0.0721491
$301$	−1.09756e14	−2.56042
$302$	−6.81176e12	−0.156035
$303$	1.28850e13	0.289834
$304$	3.84988e11	0.00850439
$305$	2.06593e13	0.448194
$306$	−9.73357e12	−0.207398
$307$	−6.68455e13	−1.39898	−0.699489	−	0.714643i	$-0.746589\pi$
$307$	−6.68455e13	−1.39898	−0.699489	+	0.714643i	$0.746589\pi$
$308$	−4.70451e13	−0.967132
$309$	−2.44061e11	−0.00492864
$310$	−1.49149e13	−0.295891
$311$	7.26443e13	1.41586	0.707928	−	0.706284i	$-0.249630\pi$
$311$	7.26443e13	1.41586	0.707928	+	0.706284i	$0.249630\pi$
$312$	−4.15820e13	−0.796260
$313$	6.27938e13	1.18147	0.590735	−	0.806866i	$-0.298838\pi$
$313$	6.27938e13	1.18147	0.590735	+	0.806866i	$0.298838\pi$
$314$	1.61690e13	0.298931
$315$	1.33150e13	0.241899
$316$	−3.16797e13	−0.565590
$317$	−4.20135e13	−0.737161	−0.368581	−	0.929596i	$-0.620156\pi$
$317$	−4.20135e13	−0.737161	−0.368581	+	0.929596i	$0.620156\pi$
$318$	8.23919e12	0.142081
$319$	5.78534e12	0.0980574
$320$	−1.61078e13	−0.268357
$321$	4.01527e13	0.657563
$322$	−9.65318e13	−1.55404
$323$	3.58108e13	0.566761
$324$	−4.46187e12	−0.0694256
$325$	1.81168e13	0.277156
$326$	−7.36778e13	−1.10825
$327$	−1.71197e13	−0.253212
$328$	−5.10980e13	−0.743187
$329$	−8.61960e13	−1.23285
$330$	1.07245e13	0.150852
$331$	1.36334e14	1.88604	0.943019	−	0.332740i	$-0.107973\pi$
$331$	1.36334e14	1.88604	0.943019	+	0.332740i	$0.107973\pi$
$332$	−4.59927e13	−0.625792
$333$	3.69169e13	0.494063
$334$	4.65874e13	0.613286
$335$	−5.19565e13	−0.672811
$336$	1.12099e12	0.0142802
$337$	9.94099e13	1.24585	0.622924	−	0.782282i	$-0.285944\pi$
$337$	9.94099e13	1.24585	0.622924	+	0.782282i	$0.285944\pi$
$338$	−4.57218e13	−0.563743
$339$	−5.04143e13	−0.611583
$340$	2.37806e13	0.283850
$341$	8.77273e13	1.03035
$342$	−9.85655e12	−0.113915
$343$	−9.03418e13	−1.02747
$344$	1.40302e14	1.57034
$345$	−3.66495e13	−0.403704
$346$	5.84897e13	0.634105
$347$	7.96482e13	0.849892	0.424946	−	0.905219i	$-0.360293\pi$
$347$	7.96482e13	0.849892	0.424946	+	0.905219i	$0.360293\pi$
$348$	−3.53089e12	−0.0370850
$349$	−2.33239e13	−0.241135	−0.120568	−	0.992705i	$-0.538471\pi$
$349$	−2.33239e13	−0.241135	−0.120568	+	0.992705i	$0.538471\pi$
$350$	1.95326e13	0.198786
$351$	−2.66196e13	−0.266693
$352$	9.71505e13	0.958209
$353$	−7.20065e13	−0.699215	−0.349608	−	0.936896i	$-0.613685\pi$
$353$	−7.20065e13	−0.699215	−0.349608	+	0.936896i	$0.613685\pi$
$354$	−3.93123e13	−0.375846
$355$	−2.30062e13	−0.216566
$356$	−9.56782e13	−0.886825
$357$	1.04272e14	0.951681
$358$	−2.42077e12	−0.0217569
$359$	1.02696e13	0.0908941	0.0454470	−	0.998967i	$-0.485529\pi$
$359$	1.02696e13	0.0908941	0.0454470	+	0.998967i	$0.485529\pi$
$360$	−1.70208e13	−0.148360
$361$	−8.02270e13	−0.688702
$362$	−7.20880e13	−0.609489
$363$	6.25063e12	0.0520520
$364$	1.71299e14	1.40506
$365$	1.98664e13	0.160513
$366$	−4.45298e13	−0.354409
$367$	1.03116e14	0.808470	0.404235	−	0.914655i	$-0.367538\pi$
$367$	1.03116e14	0.808470	0.404235	+	0.914655i	$0.367538\pi$
$368$	−3.08550e12	−0.0238321
$369$	−3.27114e13	−0.248917
$370$	5.41554e13	0.406006
$371$	−8.82631e13	−0.651962
$372$	−5.35414e13	−0.389676
$373$	1.30172e14	0.933509	0.466755	−	0.884387i	$-0.345423\pi$
$373$	1.30172e14	0.933509	0.466755	+	0.884387i	$0.345423\pi$
$374$	8.39852e13	0.593484
$375$	7.41577e12	0.0516398
$376$	1.10185e14	0.756120
$377$	−2.10653e13	−0.142459
$378$	−2.86997e13	−0.191281
$379$	−6.26815e13	−0.411740	−0.205870	−	0.978579i	$-0.566002\pi$
$379$	−6.26815e13	−0.411740	−0.205870	+	0.978579i	$0.566002\pi$
$380$	2.40810e13	0.155907
$381$	−2.86282e13	−0.182686
$382$	6.33537e13	0.398495
$383$	1.06310e14	0.659145	0.329572	−	0.944130i	$-0.393095\pi$
$383$	1.06310e14	0.659145	0.329572	+	0.944130i	$0.393095\pi$
$384$	−6.01745e13	−0.367782
$385$	−1.14888e14	−0.692212
$386$	−8.63809e13	−0.513083
$387$	8.98173e13	0.525956
$388$	2.12806e13	0.122860
$389$	−7.42336e13	−0.422549	−0.211275	−	0.977427i	$-0.567761\pi$
$389$	−7.42336e13	−0.422549	−0.211275	+	0.977427i	$0.567761\pi$
$390$	−3.90497e13	−0.219160
$391$	−2.87007e14	−1.58825
$392$	2.97873e14	1.62539
$393$	−1.11850e14	−0.601834
$394$	−1.02034e14	−0.541395
$395$	−7.73640e13	−0.404814
$396$	3.84988e13	0.198666
$397$	−1.48552e14	−0.756018	−0.378009	−	0.925802i	$-0.623391\pi$
$397$	−1.48552e14	−0.756018	−0.378009	+	0.925802i	$0.623391\pi$
$398$	6.10665e13	0.306511
$399$	1.05589e14	0.522719
$400$	6.24331e11	0.00304849
$401$	−9.20974e13	−0.443561	−0.221781	−	0.975097i	$-0.571187\pi$
$401$	−9.20974e13	−0.443561	−0.221781	+	0.975097i	$0.571187\pi$
$402$	1.11989e14	0.532025
$403$	−3.19429e14	−1.49691
$404$	6.78531e13	0.313669
$405$	−1.08962e13	−0.0496904
$406$	−2.27115e13	−0.102177
$407$	−3.18534e14	−1.41380
$408$	−1.33292e14	−0.583677
$409$	−3.23531e13	−0.139778	−0.0698889	−	0.997555i	$-0.522264\pi$
$409$	−3.23531e13	−0.139778	−0.0698889	+	0.997555i	$0.522264\pi$
$410$	−4.79862e13	−0.204553
$411$	1.41046e14	0.593239
$412$	−1.28524e12	−0.00533396
$413$	4.21137e14	1.72464
$414$	7.89956e13	0.319228
$415$	−1.12317e14	−0.447902
$416$	−3.53740e14	−1.39210
$417$	−1.68848e14	−0.655764
$418$	8.50463e13	0.325976
$419$	−1.61755e14	−0.611900	−0.305950	−	0.952048i	$-0.598974\pi$
$419$	−1.61755e14	−0.611900	−0.305950	+	0.952048i	$0.598974\pi$
$420$	7.01178e13	0.261792
$421$	−1.14462e14	−0.421803	−0.210902	−	0.977507i	$-0.567640\pi$
$421$	−1.14462e14	−0.421803	−0.210902	+	0.977507i	$0.567640\pi$
$422$	3.01297e14	1.09591
$423$	7.05374e13	0.253249
$424$	1.12828e14	0.399856
$425$	5.80738e13	0.203161
$426$	4.95885e13	0.171249
$427$	4.77029e14	1.62627
$428$	2.11447e14	0.711639
$429$	2.29684e14	0.763161
$430$	1.31758e14	0.432215
$431$	1.01052e14	0.327279	0.163639	−	0.986520i	$-0.447677\pi$
$431$	1.01052e14	0.327279	0.163639	+	0.986520i	$0.447677\pi$
$432$	−9.17346e11	−0.00293341
$433$	1.21838e14	0.384680	0.192340	−	0.981328i	$-0.438392\pi$
$433$	1.21838e14	0.384680	0.192340	+	0.981328i	$0.438392\pi$
$434$	−3.44390e14	−1.07364
$435$	−8.62268e12	−0.0265431
$436$	−9.01534e13	−0.274035
$437$	−2.90633e14	−0.872362
$438$	−4.28208e13	−0.126925
$439$	2.85736e14	0.836393	0.418197	−	0.908357i	$-0.362662\pi$
$439$	2.85736e14	0.836393	0.418197	+	0.908357i	$0.362662\pi$
$440$	1.46862e14	0.424542
$441$	1.90689e14	0.544395
$442$	−3.05803e14	−0.862222
$443$	4.82498e14	1.34362	0.671808	−	0.740725i	$-0.265518\pi$
$443$	4.82498e14	1.34362	0.671808	+	0.740725i	$0.265518\pi$
$444$	1.94406e14	0.534693
$445$	−2.33653e14	−0.634733
$446$	−2.74268e14	−0.735925
$447$	2.73981e13	0.0726155
$448$	−3.71935e14	−0.973729
$449$	6.86974e14	1.77658	0.888291	−	0.459281i	$-0.151893\pi$
$449$	6.86974e14	1.77658	0.888291	+	0.459281i	$0.151893\pi$
$450$	−1.59842e13	−0.0408341
$451$	2.82248e14	0.712294
$452$	−2.65485e14	−0.661878
$453$	−5.97154e13	−0.147078
$454$	−1.50495e14	−0.366197
$455$	4.18323e14	1.00566
$456$	−1.34976e14	−0.320590
$457$	−2.14744e14	−0.503944	−0.251972	−	0.967734i	$-0.581079\pi$
$457$	−2.14744e14	−0.503944	−0.251972	+	0.967734i	$0.581079\pi$
$458$	−6.02519e13	−0.139705
$459$	−8.53295e13	−0.195492
$460$	−1.92998e14	−0.436903
$461$	−3.22127e14	−0.720563	−0.360281	−	0.932844i	$-0.617319\pi$
$461$	−3.22127e14	−0.720563	−0.360281	+	0.932844i	$0.617319\pi$
$462$	2.47633e14	0.547366
$463$	8.81227e13	0.192483	0.0962415	−	0.995358i	$-0.469318\pi$
$463$	8.81227e13	0.192483	0.0962415	+	0.995358i	$0.469318\pi$
$464$	−7.25940e11	−0.00156694
$465$	−1.30752e14	−0.278905
$466$	5.49199e14	1.15773
$467$	3.86746e14	0.805718	0.402859	−	0.915262i	$-0.368016\pi$
$467$	3.86746e14	0.805718	0.402859	+	0.915262i	$0.368016\pi$
$468$	−1.40180e14	−0.288625
$469$	−1.19969e15	−2.44129
$470$	1.03475e14	0.208112
$471$	1.41746e14	0.281771
$472$	−5.38344e14	−1.05774
$473$	−7.74980e14	−1.50506
$474$	1.66753e14	0.320106
$475$	5.88076e13	0.111588
$476$	5.49101e14	1.02994
$477$	7.22290e13	0.133925
$478$	−2.86774e13	−0.0525638
$479$	−5.88123e14	−1.06567	−0.532835	−	0.846219i	$-0.678873\pi$
$479$	−5.88123e14	−1.06567	−0.532835	+	0.846219i	$0.678873\pi$
$480$	−1.44797e14	−0.259377
$481$	1.15983e15	2.05398
$482$	−3.77205e14	−0.660417
$483$	−8.46248e14	−1.46483
$484$	3.29162e13	0.0563326
$485$	5.19687e13	0.0879350
$486$	2.34861e13	0.0392926
$487$	3.77139e14	0.623867	0.311933	−	0.950104i	$-0.399023\pi$
$487$	3.77139e14	0.623867	0.311933	+	0.950104i	$0.399023\pi$
$488$	−6.09792e14	−0.997408
$489$	−6.45898e14	−1.04464
$490$	2.79732e14	0.447367
$491$	−3.71374e13	−0.0587305	−0.0293652	−	0.999569i	$-0.509349\pi$
$491$	−3.71374e13	−0.0587305	−0.0293652	+	0.999569i	$0.509349\pi$
$492$	−1.72260e14	−0.269387
$493$	−6.75253e13	−0.104426
$494$	−3.09667e14	−0.473583
$495$	9.40168e13	0.142193
$496$	−1.10079e13	−0.0164648
$497$	−5.31222e14	−0.785806
$498$	2.42093e14	0.354178
$499$	2.72085e14	0.393688	0.196844	−	0.980435i	$-0.436931\pi$
$499$	2.72085e14	0.393688	0.196844	+	0.980435i	$0.436931\pi$
$500$	3.90519e13	0.0558865
$501$	4.08410e14	0.578081
$502$	6.32060e14	0.884888
$503$	1.30904e15	1.81272	0.906358	−	0.422511i	$-0.138851\pi$
$503$	1.30904e15	1.81272	0.906358	+	0.422511i	$0.138851\pi$
$504$	−3.93015e14	−0.538321
$505$	1.65702e14	0.224505
$506$	−6.81606e14	−0.913494
$507$	−4.00821e14	−0.531382
$508$	−1.50757e14	−0.197710
$509$	1.41934e15	1.84135	0.920677	−	0.390326i	$-0.127638\pi$
$509$	1.41934e15	1.84135	0.920677	+	0.390326i	$0.127638\pi$
$510$	−1.25174e14	−0.160650
$511$	4.58722e14	0.582418
$512$	−2.42674e13	−0.0304817
$513$	−8.64076e13	−0.107376
$514$	−1.69939e14	−0.208928
$515$	−3.13865e12	−0.00381771
$516$	4.72983e14	0.569209
$517$	−6.08626e14	−0.724690
$518$	1.25047e15	1.47319
$519$	5.12751e14	0.597704
$520$	−5.34748e14	−0.616780
$521$	−5.15525e14	−0.588359	−0.294180	−	0.955750i	$-0.595046\pi$
$521$	−5.15525e14	−0.588359	−0.294180	+	0.955750i	$0.595046\pi$
$522$	1.85856e13	0.0209889
$523$	1.15496e15	1.29064	0.645322	−	0.763911i	$-0.276723\pi$
$523$	1.15496e15	1.29064	0.645322	+	0.763911i	$0.276723\pi$
$524$	−5.89008e14	−0.651328
$525$	1.71233e14	0.187374
$526$	9.01935e14	0.976683
$527$	−1.02393e15	−1.09727
$528$	7.91524e12	0.00839416
$529$	1.37648e15	1.44465
$530$	1.05957e14	0.110055
$531$	−3.44632e14	−0.354271
$532$	5.56038e14	0.565706
$533$	−1.02771e15	−1.03483
$534$	5.03624e14	0.501914
$535$	5.16367e14	0.509346
$536$	1.53358e15	1.49727
$537$	−2.12217e13	−0.0205080
$538$	−1.01721e15	−0.972988
$539$	−1.64534e15	−1.55782
$540$	−5.73800e13	−0.0537768
$541$	−1.08911e15	−1.01038	−0.505190	−	0.863008i	$-0.668578\pi$
$541$	−1.08911e15	−1.01038	−0.505190	+	0.863008i	$0.668578\pi$
$542$	−5.19347e14	−0.476937
$543$	−6.31961e14	−0.574502
$544$	−1.13392e15	−1.02044
$545$	−2.20161e14	−0.196137
$546$	−9.01669e14	−0.795221
$547$	9.36391e14	0.817574	0.408787	−	0.912630i	$-0.365952\pi$
$547$	9.36391e14	0.817574	0.408787	+	0.912630i	$0.365952\pi$
$548$	7.42754e14	0.642025
$549$	−3.90371e14	−0.334064
$550$	1.37918e14	0.116850
$551$	−6.83784e13	−0.0573569
$552$	1.08177e15	0.898400
$553$	−1.78636e15	−1.46886
$554$	8.40010e14	0.683881
$555$	4.74754e14	0.382699
$556$	−8.89163e14	−0.709692
$557$	−3.48191e14	−0.275178	−0.137589	−	0.990489i	$-0.543935\pi$
$557$	−3.48191e14	−0.275178	−0.137589	+	0.990489i	$0.543935\pi$
$558$	2.81827e14	0.220544
$559$	2.82182e15	2.18658
$560$	1.44160e13	0.0110614
$561$	7.36258e14	0.559415
$562$	−1.36794e15	−1.02924
$563$	8.78985e14	0.654916	0.327458	−	0.944866i	$-0.393808\pi$
$563$	8.78985e14	0.654916	0.327458	+	0.944866i	$0.393808\pi$
$564$	3.71454e14	0.274075
$565$	−6.48332e14	−0.473730
$566$	1.51103e14	0.109341
$567$	−2.51597e14	−0.180301
$568$	6.79067e14	0.481944
$569$	−1.45202e14	−0.102060	−0.0510299	−	0.998697i	$-0.516250\pi$
$569$	−1.45202e14	−0.102060	−0.0510299	+	0.998697i	$0.516250\pi$
$570$	−1.26756e14	−0.0882382
$571$	−1.99915e15	−1.37831	−0.689156	−	0.724613i	$-0.742019\pi$
$571$	−1.99915e15	−1.37831	−0.689156	+	0.724613i	$0.742019\pi$
$572$	1.20953e15	0.825921
$573$	5.55391e14	0.375620
$574$	−1.10802e15	−0.742216
$575$	−4.71315e14	−0.312708
$576$	3.04368e14	0.200021
$577$	−7.69862e13	−0.0501125	−0.0250562	−	0.999686i	$-0.507976\pi$
$577$	−7.69862e13	−0.0501125	−0.0250562	+	0.999686i	$0.507976\pi$
$578$	−3.02712e13	−0.0195176
$579$	−7.57260e14	−0.483630
$580$	−4.54075e13	−0.0287259
$581$	−2.59345e15	−1.62521
$582$	−1.12015e14	−0.0695345
$583$	−6.23221e14	−0.383235
$584$	−5.86390e14	−0.357204
$585$	−3.42330e14	−0.206580
$586$	−1.34661e15	−0.805019
$587$	8.86073e14	0.524759	0.262379	−	0.964965i	$-0.415493\pi$
$587$	8.86073e14	0.524759	0.262379	+	0.964965i	$0.415493\pi$
$588$	1.00418e15	0.589164
$589$	−1.03687e15	−0.602685
$590$	−5.05559e14	−0.291129
$591$	−8.94480e14	−0.510316
$592$	3.99694e13	0.0225922
$593$	9.32426e14	0.522172	0.261086	−	0.965316i	$-0.415919\pi$
$593$	9.32426e14	0.522172	0.261086	+	0.965316i	$0.415919\pi$
$594$	−2.02647e14	−0.112439
$595$	1.34094e15	0.737169
$596$	1.44280e14	0.0785872
$597$	5.35341e14	0.288916
$598$	2.48183e15	1.32714
$599$	1.16299e15	0.616208	0.308104	−	0.951353i	$-0.400305\pi$
$599$	1.16299e15	0.616208	0.308104	+	0.951353i	$0.400305\pi$
$600$	−2.18889e14	−0.114919
$601$	−2.63081e15	−1.36861	−0.684306	−	0.729195i	$-0.739895\pi$
$601$	−2.63081e15	−1.36861	−0.684306	+	0.729195i	$0.739895\pi$
$602$	3.04233e15	1.56829
$603$	9.81753e14	0.501484
$604$	−3.14465e14	−0.159173
$605$	8.03836e13	0.0403193
$606$	−3.57160e14	−0.177527
$607$	1.56499e14	0.0770857	0.0385429	−	0.999257i	$-0.487728\pi$
$607$	1.56499e14	0.0770857	0.0385429	+	0.999257i	$0.487728\pi$
$608$	−1.14825e15	−0.560487
$609$	−1.99100e14	−0.0963113
$610$	−5.72656e14	−0.274524
$611$	2.21610e15	1.05284
$612$	−4.49350e14	−0.211569
$613$	−1.63056e15	−0.760859	−0.380429	−	0.924810i	$-0.624224\pi$
$613$	−1.63056e15	−0.760859	−0.380429	+	0.924810i	$0.624224\pi$
$614$	1.85290e15	0.856890
$615$	−4.20672e14	−0.192810
$616$	3.39109e15	1.54044
$617$	3.55060e15	1.59858	0.799289	−	0.600947i	$-0.205210\pi$
$617$	3.55060e15	1.59858	0.799289	+	0.600947i	$0.205210\pi$
$618$	6.76516e12	0.00301885
$619$	−7.99938e14	−0.353800	−0.176900	−	0.984229i	$-0.556607\pi$
$619$	−7.99938e14	−0.353800	−0.176900	+	0.984229i	$0.556607\pi$
$620$	−6.88547e14	−0.301842
$621$	6.92517e14	0.300903
$622$	−2.01363e15	−0.867228
$623$	−5.39512e15	−2.30312
$624$	−2.88206e13	−0.0121952
$625$	9.53674e13	0.0400000
$626$	−1.74059e15	−0.723664
$627$	7.45560e14	0.307264
$628$	7.46443e14	0.304943
$629$	3.71786e15	1.50562
$630$	−3.69081e14	−0.148166
$631$	3.38323e14	0.134639	0.0673194	−	0.997731i	$-0.478555\pi$
$631$	3.38323e14	0.134639	0.0673194	+	0.997731i	$0.478555\pi$
$632$	2.28353e15	0.900870
$633$	2.64133e15	1.03300
$634$	1.16457e15	0.451520
$635$	−3.68160e14	−0.141508
$636$	3.80362e14	0.144938
$637$	5.99095e15	2.26323
$638$	−1.60364e14	−0.0600613
$639$	4.34719e14	0.161419
$640$	−7.73849e14	−0.284883
$641$	2.78171e15	1.01529	0.507647	−	0.861565i	$-0.330515\pi$
$641$	2.78171e15	1.01529	0.507647	+	0.861565i	$0.330515\pi$
$642$	−1.11300e15	−0.402765
$643$	−1.37901e15	−0.494774	−0.247387	−	0.968917i	$-0.579572\pi$
$643$	−1.37901e15	−0.494774	−0.247387	+	0.968917i	$0.579572\pi$
$644$	−4.45639e15	−1.58530
$645$	1.15506e15	0.407404
$646$	−9.92642e14	−0.347147
$647$	−4.51705e15	−1.56632	−0.783162	−	0.621818i	$-0.786394\pi$
$647$	−4.51705e15	−1.56632	−0.783162	+	0.621818i	$0.786394\pi$
$648$	3.21619e14	0.110581
$649$	2.97362e15	1.01377
$650$	−5.02182e14	−0.169761
$651$	−3.01910e15	−1.01200
$652$	−3.40133e15	−1.13054
$653$	−4.67633e15	−1.54128	−0.770642	−	0.637268i	$-0.780064\pi$
$653$	−4.67633e15	−1.54128	−0.770642	+	0.637268i	$0.780064\pi$
$654$	4.74543e14	0.155095
$655$	−1.43840e15	−0.466179
$656$	−3.54162e13	−0.0113823
$657$	−3.75390e14	−0.119639
$658$	2.38927e15	0.755133
$659$	−2.44697e15	−0.766935	−0.383468	−	0.923554i	$-0.625270\pi$
$659$	−2.44697e15	−0.766935	−0.383468	+	0.923554i	$0.625270\pi$
$660$	4.95098e14	0.153886
$661$	−1.00863e15	−0.310902	−0.155451	−	0.987844i	$-0.549683\pi$
$661$	−1.00863e15	−0.310902	−0.155451	+	0.987844i	$0.549683\pi$
$662$	−3.77906e15	−1.15522
$663$	−2.68083e15	−0.812726
$664$	3.31523e15	0.996758
$665$	1.35789e15	0.404896
$666$	−1.02330e15	−0.302619
$667$	5.48022e14	0.160733
$668$	2.15071e15	0.625620
$669$	−2.40438e15	−0.693680
$670$	1.44019e15	0.412105
$671$	3.36828e15	0.955948
$672$	−3.34340e15	−0.941146
$673$	3.14110e15	0.877000	0.438500	−	0.898731i	$-0.355510\pi$
$673$	3.14110e15	0.877000	0.438500	+	0.898731i	$0.355510\pi$
$674$	−2.75555e15	−0.763096
$675$	−1.40126e14	−0.0384900
$676$	−2.11075e15	−0.575081
$677$	−1.77722e15	−0.480291	−0.240145	−	0.970737i	$-0.577195\pi$
$677$	−1.77722e15	−0.480291	−0.240145	+	0.970737i	$0.577195\pi$
$678$	1.39744e15	0.374601
$679$	1.19997e15	0.319071
$680$	−1.71414e15	−0.452114
$681$	−1.31932e15	−0.345176
$682$	−2.43172e15	−0.631102
$683$	−4.74129e15	−1.22063	−0.610313	−	0.792160i	$-0.708956\pi$
$683$	−4.74129e15	−1.22063	−0.610313	+	0.792160i	$0.708956\pi$
$684$	−4.55027e14	−0.116206
$685$	1.81386e15	0.459521
$686$	2.50419e15	0.629340
$687$	−5.28200e14	−0.131685
$688$	9.72438e13	0.0240506
$689$	2.26924e15	0.556770
$690$	1.01589e15	0.247273
$691$	2.44203e15	0.589688	0.294844	−	0.955545i	$-0.404732\pi$
$691$	2.44203e15	0.589688	0.294844	+	0.955545i	$0.404732\pi$
$692$	2.70018e15	0.646858
$693$	2.17088e15	0.515944
$694$	−2.20777e15	−0.520568
$695$	−2.17140e15	−0.507952
$696$	2.54513e14	0.0590688
$697$	−3.29433e15	−0.758555
$698$	6.46517e14	0.147698
$699$	4.81456e15	1.09127
$700$	9.01720e14	0.202783
$701$	3.91146e15	0.872749	0.436375	−	0.899765i	$-0.356262\pi$
$701$	3.91146e15	0.872749	0.436375	+	0.899765i	$0.356262\pi$
$702$	7.37870e14	0.163353
$703$	3.76483e15	0.826973
$704$	−2.62621e15	−0.572375
$705$	9.07117e14	0.196166
$706$	1.99596e15	0.428277
$707$	3.82611e15	0.814613
$708$	−1.81485e15	−0.383405
$709$	−7.52555e15	−1.57755	−0.788776	−	0.614681i	$-0.789285\pi$
$709$	−7.52555e15	−1.57755	−0.788776	+	0.614681i	$0.789285\pi$
$710$	6.37712e14	0.132649
$711$	1.46185e15	0.301730
$712$	6.89665e15	1.41253
$713$	8.31004e15	1.68893
$714$	−2.89032e15	−0.582915
$715$	2.95376e15	0.591142
$716$	−1.11755e14	−0.0221945
$717$	−2.51401e14	−0.0495464
$718$	−2.84665e14	−0.0556737
$719$	3.82053e15	0.741507	0.370753	−	0.928731i	$-0.379099\pi$
$719$	3.82053e15	0.741507	0.370753	+	0.928731i	$0.379099\pi$
$720$	−1.17971e13	−0.00227221
$721$	−7.24724e13	−0.0138525
$722$	2.22382e15	0.421838
$723$	−3.30677e15	−0.622506
$724$	−3.32794e15	−0.621747
$725$	−1.10888e14	−0.0205602
$726$	−1.73262e14	−0.0318824
$727$	−7.15305e15	−1.30633	−0.653163	−	0.757217i	$-0.726558\pi$
$727$	−7.15305e15	−1.30633	−0.653163	+	0.757217i	$0.726558\pi$
$728$	−1.23475e16	−2.23798
$729$	2.05891e14	0.0370370
$730$	−5.50679e14	−0.0983158
$731$	9.04540e15	1.60281
$732$	−2.05572e15	−0.361536
$733$	3.23209e15	0.564172	0.282086	−	0.959389i	$-0.408974\pi$
$733$	3.23209e15	0.564172	0.282086	+	0.959389i	$0.408974\pi$
$734$	−2.85829e15	−0.495197
$735$	2.45228e15	0.421687
$736$	9.20266e15	1.57067
$737$	−8.47096e15	−1.43503
$738$	9.06731e14	0.152464
$739$	−7.04037e15	−1.17504	−0.587518	−	0.809211i	$-0.699895\pi$
$739$	−7.04037e15	−1.17504	−0.587518	+	0.809211i	$0.699895\pi$
$740$	2.50008e15	0.414171
$741$	−2.71470e15	−0.446397
$742$	2.44657e15	0.399334
$743$	1.76769e15	0.286396	0.143198	−	0.989694i	$-0.454261\pi$
$743$	1.76769e15	0.286396	0.143198	+	0.989694i	$0.454261\pi$
$744$	3.85936e15	0.620674
$745$	3.52342e14	0.0562477
$746$	−3.60825e15	−0.571785
$747$	2.12232e15	0.333846
$748$	3.87717e15	0.605420
$749$	1.19231e16	1.84816
$750$	−2.05558e14	−0.0316299
$751$	−6.97015e15	−1.06469	−0.532344	−	0.846528i	$-0.678689\pi$
$751$	−6.97015e15	−1.06469	−0.532344	+	0.846528i	$0.678689\pi$
$752$	7.63698e13	0.0115804
$753$	5.54097e15	0.834091
$754$	5.83912e14	0.0872579
$755$	−7.67945e14	−0.113926
$756$	−1.32492e15	−0.195128
$757$	−1.10779e16	−1.61968	−0.809841	−	0.586650i	$-0.800447\pi$
$757$	−1.10779e16	−1.61968	−0.809841	+	0.586650i	$0.800447\pi$
$758$	1.73747e15	0.252196
$759$	−5.97532e15	−0.861055
$760$	−1.73580e15	−0.248328
$761$	1.03991e15	0.147700	0.0738500	−	0.997269i	$-0.476471\pi$
$761$	1.03991e15	0.147700	0.0738500	+	0.997269i	$0.476471\pi$
$762$	7.93546e14	0.111897
$763$	−5.08359e15	−0.711681
$764$	2.92472e15	0.406510
$765$	−1.09734e15	−0.151428
$766$	−2.94682e15	−0.403734
$767$	−1.08274e16	−1.47282
$768$	4.23319e15	0.571717
$769$	−5.06484e15	−0.679158	−0.339579	−	0.940578i	$-0.610285\pi$
$769$	−5.06484e15	−0.679158	−0.339579	+	0.940578i	$0.610285\pi$
$770$	3.18458e15	0.423987
$771$	−1.48978e15	−0.196935
$772$	−3.98777e15	−0.523402
$773$	−9.32127e15	−1.21475	−0.607376	−	0.794414i	$-0.707778\pi$
$773$	−9.32127e15	−1.21475	−0.607376	+	0.794414i	$0.707778\pi$
$774$	−2.48965e15	−0.322154
$775$	−1.68148e15	−0.216039
$776$	−1.53394e15	−0.195690
$777$	1.09622e16	1.38862
$778$	2.05769e15	0.258816
$779$	−3.33596e15	−0.416643
$780$	−1.80273e15	−0.223568
$781$	−3.75093e15	−0.461911
$782$	7.95557e15	0.972823
$783$	1.62931e14	0.0197841
$784$	2.06456e14	0.0248937
$785$	1.82287e15	0.218259
$786$	3.10038e15	0.368630
$787$	8.41800e15	0.993912	0.496956	−	0.867776i	$-0.334451\pi$
$787$	8.41800e15	0.993912	0.496956	+	0.867776i	$0.334451\pi$
$788$	−4.71038e15	−0.552283
$789$	7.90683e15	0.920617
$790$	2.14446e15	0.247953
$791$	−1.49702e16	−1.71892
$792$	−2.77506e15	−0.316435
$793$	−1.22644e16	−1.38882
$794$	4.11774e15	0.463069
$795$	9.28871e14	0.103738
$796$	2.81913e15	0.312676
$797$	1.32426e16	1.45866	0.729328	−	0.684164i	$-0.239833\pi$
$797$	1.32426e16	1.45866	0.729328	+	0.684164i	$0.239833\pi$
$798$	−2.92684e15	−0.320171
$799$	7.10375e15	0.771757
$800$	−1.86210e15	−0.200913
$801$	4.41503e15	0.473102
$802$	2.55286e15	0.271686
$803$	3.23901e15	0.342356
$804$	5.16997e15	0.542725
$805$	−1.08828e16	−1.13466
$806$	8.85427e15	0.916875
$807$	−8.91738e15	−0.917134
$808$	−4.89097e15	−0.499611
$809$	−2.72166e15	−0.276132	−0.138066	−	0.990423i	$-0.544089\pi$
$809$	−2.72166e15	−0.276132	−0.138066	+	0.990423i	$0.544089\pi$
$810$	3.02033e14	0.0304359
$811$	1.27431e16	1.27544	0.637722	−	0.770267i	$-0.279877\pi$
$811$	1.27431e16	1.27544	0.637722	+	0.770267i	$0.279877\pi$
$812$	−1.04847e15	−0.104232
$813$	−4.55286e15	−0.449559
$814$	8.82947e15	0.865965
$815$	−8.30630e15	−0.809171
$816$	−9.23850e13	−0.00893934
$817$	9.15968e15	0.880358
$818$	8.96800e14	0.0856155
$819$	−7.90450e15	−0.749571
$820$	−2.21528e15	−0.208666
$821$	2.99243e15	0.279986	0.139993	−	0.990152i	$-0.455292\pi$
$821$	2.99243e15	0.279986	0.139993	+	0.990152i	$0.455292\pi$
$822$	−3.90966e15	−0.363366
$823$	1.31983e16	1.21848	0.609239	−	0.792987i	$-0.291475\pi$
$823$	1.31983e16	1.21848	0.609239	+	0.792987i	$0.291475\pi$
$824$	9.26423e13	0.00849591
$825$	1.20906e15	0.110142
$826$	−1.16735e16	−1.05636
$827$	−8.07855e15	−0.726195	−0.363097	−	0.931751i	$-0.618281\pi$
$827$	−8.07855e15	−0.726195	−0.363097	+	0.931751i	$0.618281\pi$
$828$	3.64683e15	0.325648
$829$	−9.24683e15	−0.820244	−0.410122	−	0.912031i	$-0.634514\pi$
$829$	−9.24683e15	−0.820244	−0.410122	+	0.912031i	$0.634514\pi$
$830$	3.11334e15	0.274345
$831$	7.36397e15	0.644623
$832$	9.56245e15	0.831555
$833$	1.92041e16	1.65900
$834$	4.68031e15	0.401663
$835$	5.25218e15	0.447779
$836$	3.92616e15	0.332532
$837$	2.47065e15	0.207884
$838$	4.48370e15	0.374796
$839$	−7.02444e14	−0.0583339	−0.0291669	−	0.999575i	$-0.509285\pi$
$839$	−7.02444e14	−0.0583339	−0.0291669	+	0.999575i	$0.509285\pi$
$840$	−5.05421e15	−0.416982
$841$	−1.20716e16	−0.989432
$842$	3.17278e15	0.258359
$843$	−1.19921e16	−0.970159
$844$	1.39094e16	1.11795
$845$	−5.15459e15	−0.411606
$846$	−1.95523e15	−0.155118
$847$	1.85608e15	0.146298
$848$	7.82012e13	0.00612402
$849$	1.32465e15	0.103064
$850$	−1.60975e15	−0.124439
$851$	−3.01734e16	−2.31746
$852$	2.28925e15	0.174693
$853$	4.32915e15	0.328234	0.164117	−	0.986441i	$-0.447523\pi$
$853$	4.32915e15	0.328234	0.164117	+	0.986441i	$0.447523\pi$
$854$	−1.32228e16	−0.996106
$855$	−1.11121e15	−0.0831729
$856$	−1.52414e16	−1.13350
$857$	2.43435e16	1.79882	0.899412	−	0.437103i	$-0.143995\pi$
$857$	2.43435e16	1.79882	0.899412	+	0.437103i	$0.143995\pi$
$858$	−6.36664e15	−0.467445
$859$	1.46780e16	1.07079	0.535394	−	0.844603i	$-0.320163\pi$
$859$	1.46780e16	1.07079	0.535394	+	0.844603i	$0.320163\pi$
$860$	6.08260e15	0.440908
$861$	−9.71344e15	−0.699610
$862$	−2.80106e15	−0.200462
$863$	2.68085e16	1.90640	0.953199	−	0.302342i	$-0.0977685\pi$
$863$	2.68085e16	1.90640	0.953199	+	0.302342i	$0.0977685\pi$
$864$	2.73603e15	0.193328
$865$	6.59403e15	0.462980
$866$	−3.37724e15	−0.235621
$867$	−2.65373e14	−0.0183972
$868$	−1.58988e16	−1.09523
$869$	−1.26134e16	−0.863422
$870$	2.39013e14	0.0162579
$871$	3.08441e16	2.08484
$872$	6.49841e15	0.436482
$873$	−9.81983e14	−0.0655429
$874$	8.05608e15	0.534332
$875$	2.20206e15	0.145140
$876$	−1.97682e15	−0.129478
$877$	1.33110e16	0.866386	0.433193	−	0.901301i	$-0.357387\pi$
$877$	1.33110e16	0.866386	0.433193	+	0.901301i	$0.357387\pi$
$878$	−7.92035e15	−0.512300
$879$	−1.18051e16	−0.758807
$880$	1.01791e14	0.00650209
$881$	−1.89533e16	−1.20314	−0.601570	−	0.798820i	$-0.705458\pi$
$881$	−1.89533e16	−1.20314	−0.601570	+	0.798820i	$0.705458\pi$
$882$	−5.28573e15	−0.333448
$883$	−3.10076e15	−0.194395	−0.0971975	−	0.995265i	$-0.530988\pi$
$883$	−3.10076e15	−0.194395	−0.0971975	+	0.995265i	$0.530988\pi$
$884$	−1.41174e16	−0.879563
$885$	−4.43200e15	−0.274417
$886$	−1.33744e16	−0.822980
$887$	−2.17069e16	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$887$	−2.17069e16	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$888$	−1.40132e16	−0.851657
$889$	−8.50094e15	−0.513461
$890$	6.47665e15	0.388781
$891$	−1.77651e15	−0.105984
$892$	−1.26616e16	−0.750726
$893$	7.19350e15	0.423894
$894$	−7.59451e14	−0.0444778
$895$	−2.72913e14	−0.0158854
$896$	−1.78684e16	−1.03369
$897$	2.17570e16	1.25095
$898$	−1.90423e16	−1.08818
$899$	1.95514e15	0.111045
$900$	−7.37912e14	−0.0416553
$901$	7.27411e15	0.408125
$902$	−7.82365e15	−0.436288
$903$	2.66706e16	1.47826
$904$	1.91366e16	1.05424
$905$	−8.12707e15	−0.445007
$906$	1.65526e15	0.0900866
$907$	−7.15520e15	−0.387063	−0.193532	−	0.981094i	$-0.561994\pi$
$907$	−7.15520e15	−0.387063	−0.193532	+	0.981094i	$0.561994\pi$
$908$	−6.94761e15	−0.373562
$909$	−3.13105e15	−0.167336
$910$	−1.15955e16	−0.615975
$911$	2.19413e16	1.15854	0.579270	−	0.815135i	$-0.303338\pi$
$911$	2.19413e16	1.15854	0.579270	+	0.815135i	$0.303338\pi$
$912$	−9.35522e13	−0.00491001
$913$	−1.83122e16	−0.955325
$914$	5.95252e15	0.308672
$915$	−5.02020e15	−0.258765
$916$	−2.78153e15	−0.142514
$917$	−3.32131e16	−1.69152
$918$	2.36526e15	0.119741
$919$	−3.69170e15	−0.185777	−0.0928884	−	0.995677i	$-0.529610\pi$
$919$	−3.69170e15	−0.185777	−0.0928884	+	0.995677i	$0.529610\pi$
$920$	1.39116e16	0.695898
$921$	1.62435e16	0.807700
$922$	8.92906e15	0.441353
$923$	1.36577e16	0.671071
$924$	1.14320e16	0.558374
$925$	6.10538e15	0.296438
$926$	−2.44268e15	−0.117898
$927$	5.93069e13	0.00284555
$928$	2.16515e15	0.103270
$929$	2.85838e16	1.35530	0.677648	−	0.735387i	$-0.263000\pi$
$929$	2.85838e16	1.35530	0.677648	+	0.735387i	$0.263000\pi$
$930$	3.62432e15	0.170833
$931$	1.94467e16	0.911221
$932$	2.53537e16	1.18101
$933$	−1.76526e16	−0.817445
$934$	−1.07203e16	−0.493512
$935$	9.46833e15	0.433321
$936$	1.01044e16	0.459721
$937$	1.92375e16	0.870125	0.435062	−	0.900400i	$-0.356726\pi$
$937$	1.92375e16	0.870125	0.435062	+	0.900400i	$0.356726\pi$
$938$	3.32544e16	1.49532
$939$	−1.52589e16	−0.682122
$940$	4.77693e15	0.212298
$941$	−2.64255e16	−1.16756	−0.583781	−	0.811911i	$-0.698427\pi$
$941$	−2.64255e16	−1.16756	−0.583781	+	0.811911i	$0.698427\pi$
$942$	−3.92907e15	−0.172588
$943$	2.67361e16	1.16757
$944$	−3.73128e14	−0.0161999
$945$	−3.23555e15	−0.139661
$946$	2.14817e16	0.921867
$947$	−1.92888e16	−0.822961	−0.411481	−	0.911418i	$-0.634988\pi$
$947$	−1.92888e16	−0.822961	−0.411481	+	0.911418i	$0.634988\pi$
$948$	7.69816e15	0.326544
$949$	−1.17937e16	−0.497379
$950$	−1.63009e15	−0.0683490
$951$	1.02093e16	0.425600
$952$	−3.95801e16	−1.64049
$953$	1.47724e16	0.608750	0.304375	−	0.952552i	$-0.401552\pi$
$953$	1.47724e16	0.608750	0.304375	+	0.952552i	$0.401552\pi$
$954$	−2.00212e15	−0.0820304
$955$	7.14238e15	0.290954
$956$	−1.32389e15	−0.0536210
$957$	−1.40584e15	−0.0566135
$958$	1.63022e16	0.652735
$959$	4.18826e16	1.66737
$960$	3.91420e15	0.154936
$961$	4.23870e15	0.166822
$962$	−3.21495e16	−1.25809
$963$	−9.75711e15	−0.379644
$964$	−1.74137e16	−0.673699
$965$	−9.73843e15	−0.374618
$966$	2.34572e16	0.897228
$967$	−6.09293e15	−0.231729	−0.115865	−	0.993265i	$-0.536964\pi$
$967$	−6.09293e15	−0.231729	−0.115865	+	0.993265i	$0.536964\pi$
$968$	−2.37265e15	−0.0897263
$969$	−8.70202e15	−0.327219
$970$	−1.44052e15	−0.0538612
$971$	2.81975e16	1.04835	0.524174	−	0.851611i	$-0.324374\pi$
$971$	2.81975e16	1.04835	0.524174	+	0.851611i	$0.324374\pi$
$972$	1.08423e15	0.0400829
$973$	−5.01383e16	−1.84310
$974$	−1.04539e16	−0.382125
$975$	−4.40239e15	−0.160016
$976$	−4.22649e14	−0.0152759
$977$	−1.84959e16	−0.664745	−0.332372	−	0.943148i	$-0.607849\pi$
$977$	−1.84959e16	−0.664745	−0.332372	+	0.943148i	$0.607849\pi$
$978$	1.79037e16	0.639851
$979$	−3.80947e16	−1.35382
$980$	1.29138e16	0.456365
$981$	4.16009e15	0.146192
$982$	1.02942e15	0.0359731
$983$	−2.21617e16	−0.770121	−0.385060	−	0.922891i	$-0.625819\pi$
$983$	−2.21617e16	−0.770121	−0.385060	+	0.922891i	$0.625819\pi$
$984$	1.24168e16	0.429079
$985$	−1.15031e16	−0.395289
$986$	1.87174e15	0.0639621
$987$	2.09456e16	0.711785
$988$	−1.42958e16	−0.483108
$989$	−7.34106e16	−2.46706
$990$	−2.60606e15	−0.0870946
$991$	7.87374e15	0.261683	0.130842	−	0.991403i	$-0.458232\pi$
$991$	7.87374e15	0.261683	0.130842	+	0.991403i	$0.458232\pi$
$992$	3.28317e16	1.08512
$993$	−3.31292e16	−1.08890
$994$	1.47250e16	0.481315
$995$	6.88453e15	0.223793
$996$	1.11762e16	0.361301
$997$	−6.16649e15	−0.198251	−0.0991254	−	0.995075i	$-0.531604\pi$
$997$	−6.16649e15	−0.198251	−0.0991254	+	0.995075i	$0.531604\pi$
$998$	−7.54196e15	−0.241138
$999$	−8.97081e15	−0.285247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.12.a.c.1.1	✓	2
3.2	odd	2		45.12.a.c.1.2		2
4.3	odd	2		240.12.a.m.1.2		2
5.2	odd	4		75.12.b.d.49.1		4
5.3	odd	4		75.12.b.d.49.4		4
5.4	even	2		75.12.a.c.1.2		2
15.2	even	4		225.12.b.i.199.4		4
15.8	even	4		225.12.b.i.199.1		4
15.14	odd	2		225.12.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.a.c.1.1	✓	2	1.1	even	1	trivial
45.12.a.c.1.2		2	3.2	odd	2
75.12.a.c.1.2		2	5.4	even	2
75.12.b.d.49.1		4	5.2	odd	4
75.12.b.d.49.4		4	5.3	odd	4
225.12.a.i.1.1		2	15.14	odd	2
225.12.b.i.199.1		4	15.8	even	4
225.12.b.i.199.4		4	15.2	even	4
240.12.a.m.1.2		2	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)

\(f(q)\)	\(=\)	\(q-27.7191 q^{2} -243.000 q^{3} -1279.65 q^{4} -3125.00 q^{5} +6735.74 q^{6} -72157.2 q^{7} +92239.5 q^{8} +59049.0 q^{9} +O(q^{10})\)	\(q-27.7191 q^{2} -243.000 q^{3} -1279.65 q^{4} -3125.00 q^{5} +6735.74 q^{6} -72157.2 q^{7} +92239.5 q^{8} +59049.0 q^{9} +86622.2 q^{10} -509499. q^{11} +310955. q^{12} +1.85516e6 q^{13} +2.00013e6 q^{14} +759375. q^{15} +63931.4 q^{16} +5.94676e6 q^{17} -1.63678e6 q^{18} +6.02189e6 q^{19} +3.99891e6 q^{20} +1.75342e7 q^{21} +1.41228e7 q^{22} -4.82627e7 q^{23} -2.24142e7 q^{24} +9.76562e6 q^{25} -5.14234e7 q^{26} -1.43489e7 q^{27} +9.23361e7 q^{28} -1.13550e7 q^{29} -2.10492e7 q^{30} -1.72184e8 q^{31} -1.90679e8 q^{32} +1.23808e8 q^{33} -1.64839e8 q^{34} +2.25491e8 q^{35} -7.55622e7 q^{36} +6.25191e8 q^{37} -1.66921e8 q^{38} -4.50805e8 q^{39} -2.88248e8 q^{40} -5.53971e8 q^{41} -4.86032e8 q^{42} +1.52106e9 q^{43} +6.51981e8 q^{44} -1.84528e8 q^{45} +1.33780e9 q^{46} +1.19456e9 q^{47} -1.55353e7 q^{48} +3.22934e9 q^{49} -2.70694e8 q^{50} -1.44506e9 q^{51} -2.37396e9 q^{52} +1.22320e9 q^{53} +3.97739e8 q^{54} +1.59218e9 q^{55} -6.65575e9 q^{56} -1.46332e9 q^{57} +3.14750e8 q^{58} -5.83637e9 q^{59} -9.71736e8 q^{60} -6.61097e9 q^{61} +4.77277e9 q^{62} -4.26081e9 q^{63} +5.15451e9 q^{64} -5.79738e9 q^{65} -3.43185e9 q^{66} +1.66261e10 q^{67} -7.60978e9 q^{68} +1.17278e10 q^{69} -6.25042e9 q^{70} +7.36200e9 q^{71} +5.44665e9 q^{72} -6.35726e9 q^{73} -1.73297e10 q^{74} -2.37305e9 q^{75} -7.70593e9 q^{76} +3.67640e10 q^{77} +1.24959e10 q^{78} +2.47565e10 q^{79} -1.99786e8 q^{80} +3.48678e9 q^{81} +1.53556e10 q^{82} +3.59416e10 q^{83} -2.24377e10 q^{84} -1.85836e10 q^{85} -4.21625e10 q^{86} +2.75926e9 q^{87} -4.69959e10 q^{88} +7.47690e10 q^{89} +5.11495e9 q^{90} -1.33863e11 q^{91} +6.17594e10 q^{92} +4.18406e10 q^{93} -3.31121e10 q^{94} -1.88184e10 q^{95} +4.63349e10 q^{96} -1.66300e10 q^{97} -8.95144e10 q^{98} -3.00854e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 13 q^{2} - 486 q^{3} - 3111 q^{4} - 6250 q^{5} + 3159 q^{6} + 7784 q^{7} + 35139 q^{8} + 118098 q^{9}+O(q^{10}) \) 2 * q - 13 * q^2 - 486 * q^3 - 3111 * q^4 - 6250 * q^5 + 3159 * q^6 + 7784 * q^7 + 35139 * q^8 + 118098 * q^9	\( 2 q - 13 q^{2} - 486 q^{3} - 3111 q^{4} - 6250 q^{5} + 3159 q^{6} + 7784 q^{7} + 35139 q^{8} + 118098 q^{9} + 40625 q^{10} + 295568 q^{11} + 755973 q^{12} + 657492 q^{13} + 3176796 q^{14} + 1518750 q^{15} + 2974065 q^{16} + 8579948 q^{17} - 767637 q^{18} + 17627976 q^{19} + 9721875 q^{20} - 1891512 q^{21} + 25972696 q^{22} - 29841072 q^{23} - 8538777 q^{24} + 19531250 q^{25} - 69052066 q^{26} - 28697814 q^{27} - 54064108 q^{28} - 201881948 q^{29} - 9871875 q^{30} - 71057008 q^{31} - 30902261 q^{32} - 71823024 q^{33} - 126080702 q^{34} - 24325000 q^{35} - 183701439 q^{36} + 705858484 q^{37} + 3909548 q^{38} - 159770556 q^{39} - 109809375 q^{40} - 327655148 q^{41} - 771961428 q^{42} + 3192552120 q^{43} - 822376568 q^{44} - 369056250 q^{45} + 1608947832 q^{46} + 2053064720 q^{47} - 722697795 q^{48} + 7642614770 q^{49} - 126953125 q^{50} - 2084927364 q^{51} - 180610022 q^{52} - 2304299452 q^{53} + 186535791 q^{54} - 923650000 q^{55} - 11220431460 q^{56} - 4283598168 q^{57} - 2489635114 q^{58} - 1478770576 q^{59} - 2362415625 q^{60} - 8264891460 q^{61} + 6261263448 q^{62} + 459637416 q^{63} + 1546316681 q^{64} - 2054662500 q^{65} - 6311365128 q^{66} + 24212177528 q^{67} - 12432065594 q^{68} + 7251380496 q^{69} - 9927487500 q^{70} - 20218888256 q^{71} + 2074922811 q^{72} + 25879583268 q^{73} - 16142374034 q^{74} - 4746093750 q^{75} - 28960704764 q^{76} + 101122045248 q^{77} + 16779652038 q^{78} + 22324995440 q^{79} - 9293953125 q^{80} + 6973568802 q^{81} + 18686740478 q^{82} + 48014508984 q^{83} + 13137578244 q^{84} - 26812337500 q^{85} - 17559698092 q^{86} + 49057313364 q^{87} - 92965609896 q^{88} + 79209683076 q^{89} + 2398865625 q^{90} - 229606712048 q^{91} + 28023036264 q^{92} + 17266852944 q^{93} - 20475612776 q^{94} - 55087425000 q^{95} + 7509249423 q^{96} - 37075227452 q^{97} - 24554976677 q^{98} + 17452994832 q^{99}+O(q^{100}) \) 2 * q - 13 * q^2 - 486 * q^3 - 3111 * q^4 - 6250 * q^5 + 3159 * q^6 + 7784 * q^7 + 35139 * q^8 + 118098 * q^9 + 40625 * q^10 + 295568 * q^11 + 755973 * q^12 + 657492 * q^13 + 3176796 * q^14 + 1518750 * q^15 + 2974065 * q^16 + 8579948 * q^17 - 767637 * q^18 + 17627976 * q^19 + 9721875 * q^20 - 1891512 * q^21 + 25972696 * q^22 - 29841072 * q^23 - 8538777 * q^24 + 19531250 * q^25 - 69052066 * q^26 - 28697814 * q^27 - 54064108 * q^28 - 201881948 * q^29 - 9871875 * q^30 - 71057008 * q^31 - 30902261 * q^32 - 71823024 * q^33 - 126080702 * q^34 - 24325000 * q^35 - 183701439 * q^36 + 705858484 * q^37 + 3909548 * q^38 - 159770556 * q^39 - 109809375 * q^40 - 327655148 * q^41 - 771961428 * q^42 + 3192552120 * q^43 - 822376568 * q^44 - 369056250 * q^45 + 1608947832 * q^46 + 2053064720 * q^47 - 722697795 * q^48 + 7642614770 * q^49 - 126953125 * q^50 - 2084927364 * q^51 - 180610022 * q^52 - 2304299452 * q^53 + 186535791 * q^54 - 923650000 * q^55 - 11220431460 * q^56 - 4283598168 * q^57 - 2489635114 * q^58 - 1478770576 * q^59 - 2362415625 * q^60 - 8264891460 * q^61 + 6261263448 * q^62 + 459637416 * q^63 + 1546316681 * q^64 - 2054662500 * q^65 - 6311365128 * q^66 + 24212177528 * q^67 - 12432065594 * q^68 + 7251380496 * q^69 - 9927487500 * q^70 - 20218888256 * q^71 + 2074922811 * q^72 + 25879583268 * q^73 - 16142374034 * q^74 - 4746093750 * q^75 - 28960704764 * q^76 + 101122045248 * q^77 + 16779652038 * q^78 + 22324995440 * q^79 - 9293953125 * q^80 + 6973568802 * q^81 + 18686740478 * q^82 + 48014508984 * q^83 + 13137578244 * q^84 - 26812337500 * q^85 - 17559698092 * q^86 + 49057313364 * q^87 - 92965609896 * q^88 + 79209683076 * q^89 + 2398865625 * q^90 - 229606712048 * q^91 + 28023036264 * q^92 + 17266852944 * q^93 - 20475612776 * q^94 - 55087425000 * q^95 + 7509249423 * q^96 - 37075227452 * q^97 - 24554976677 * q^98 + 17452994832 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more