LMFDB - Embedded newform 69.8.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.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:	$21.5545667584$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 466x^{4} + 540x^{3} + 48973x^{2} - 77282x - 1061812 $ x^6 - 2x^5 - 466x^4 + 540x^3 + 48973x^2 - 77282*x - 1061812
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$7.53998$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.53998 q^{2} +27.0000 q^{3} -55.0688 q^{4} -446.195 q^{5} -230.579 q^{6} +1770.31 q^{7} +1563.40 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.53998 q^{2} +27.0000 q^{3} -55.0688 q^{4} -446.195 q^{5} -230.579 q^{6} +1770.31 q^{7} +1563.40 q^{8} +729.000 q^{9} +3810.49 q^{10} -5076.40 q^{11} -1486.86 q^{12} +10427.8 q^{13} -15118.4 q^{14} -12047.3 q^{15} -6302.62 q^{16} -976.453 q^{17} -6225.64 q^{18} -47906.2 q^{19} +24571.4 q^{20} +47798.5 q^{21} +43352.3 q^{22} -12167.0 q^{23} +42211.9 q^{24} +120965. q^{25} -89053.1 q^{26} +19683.0 q^{27} -97489.1 q^{28} -70779.7 q^{29} +102883. q^{30} -107045. q^{31} -146291. q^{32} -137063. q^{33} +8338.88 q^{34} -789905. q^{35} -40145.2 q^{36} -400445. q^{37} +409117. q^{38} +281550. q^{39} -697582. q^{40} +351711. q^{41} -408198. q^{42} -561644. q^{43} +279551. q^{44} -325276. q^{45} +103906. q^{46} -414887. q^{47} -170171. q^{48} +2.31047e6 q^{49} -1.03304e6 q^{50} -26364.2 q^{51} -574246. q^{52} -1.16421e6 q^{53} -168092. q^{54} +2.26506e6 q^{55} +2.76772e6 q^{56} -1.29347e6 q^{57} +604457. q^{58} -1.73195e6 q^{59} +663428. q^{60} -2.22856e6 q^{61} +914161. q^{62} +1.29056e6 q^{63} +2.05606e6 q^{64} -4.65283e6 q^{65} +1.17051e6 q^{66} +2.16992e6 q^{67} +53772.1 q^{68} -328509. q^{69} +6.74577e6 q^{70} +2.37410e6 q^{71} +1.13972e6 q^{72} +3.18298e6 q^{73} +3.41979e6 q^{74} +3.26605e6 q^{75} +2.63814e6 q^{76} -8.98682e6 q^{77} -2.40443e6 q^{78} -6.55235e6 q^{79} +2.81220e6 q^{80} +531441. q^{81} -3.00361e6 q^{82} -4.67692e6 q^{83} -2.63221e6 q^{84} +435688. q^{85} +4.79643e6 q^{86} -1.91105e6 q^{87} -7.93646e6 q^{88} -1.99651e6 q^{89} +2.77785e6 q^{90} +1.84605e7 q^{91} +670022. q^{92} -2.89021e6 q^{93} +3.54312e6 q^{94} +2.13755e7 q^{95} -3.94987e6 q^{96} -2.86447e6 q^{97} -1.97314e7 q^{98} -3.70069e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9}+O(q^{10}) $ 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9	\( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9} - 13042 q^{10} - 14824 q^{11} + 4806 q^{12} - 756 q^{13} - 3926 q^{14} - 10044 q^{15} - 13022 q^{16} - 69484 q^{17} - 5832 q^{18} - 43864 q^{19} + 78886 q^{20} - 29808 q^{21} + 98204 q^{22} - 73002 q^{23} - 52812 q^{24} + 228018 q^{25} - 311956 q^{26} + 118098 q^{27} - 545442 q^{28} - 311100 q^{29} - 352134 q^{30} - 245248 q^{31} - 390156 q^{32} - 400248 q^{33} + 235834 q^{34} - 1331256 q^{35} + 129762 q^{36} - 630044 q^{37} + 80910 q^{38} - 20412 q^{39} - 2153982 q^{40} - 969204 q^{41} - 106002 q^{42} - 1770208 q^{43} - 1749140 q^{44} - 271188 q^{45} + 97336 q^{46} - 1400024 q^{47} - 351594 q^{48} + 1985598 q^{49} - 956660 q^{50} - 1876068 q^{51} + 3217272 q^{52} - 1573516 q^{53} - 157464 q^{54} - 431296 q^{55} + 7740702 q^{56} - 1184328 q^{57} + 5987188 q^{58} - 1410320 q^{59} + 2129922 q^{60} - 942172 q^{61} + 3334412 q^{62} - 804816 q^{63} + 1996866 q^{64} - 420944 q^{65} + 2651508 q^{66} - 452072 q^{67} - 9258254 q^{68} - 1971054 q^{69} + 21981136 q^{70} + 122928 q^{71} - 1425924 q^{72} + 16490716 q^{73} - 600104 q^{74} + 6156486 q^{75} + 7428658 q^{76} + 7239696 q^{77} - 8422812 q^{78} + 2458408 q^{79} + 19440230 q^{80} + 3188646 q^{81} + 20510784 q^{82} - 7566456 q^{83} - 14726934 q^{84} + 5817744 q^{85} - 669666 q^{86} - 8399700 q^{87} + 14775668 q^{88} - 20368036 q^{89} - 9507618 q^{90} + 8815576 q^{91} - 2165726 q^{92} - 6621696 q^{93} + 16952576 q^{94} + 5143832 q^{95} - 10534212 q^{96} + 12586972 q^{97} - 39164812 q^{98} - 10806696 q^{99}+O(q^{100}) \) 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9 - 13042 * q^10 - 14824 * q^11 + 4806 * q^12 - 756 * q^13 - 3926 * q^14 - 10044 * q^15 - 13022 * q^16 - 69484 * q^17 - 5832 * q^18 - 43864 * q^19 + 78886 * q^20 - 29808 * q^21 + 98204 * q^22 - 73002 * q^23 - 52812 * q^24 + 228018 * q^25 - 311956 * q^26 + 118098 * q^27 - 545442 * q^28 - 311100 * q^29 - 352134 * q^30 - 245248 * q^31 - 390156 * q^32 - 400248 * q^33 + 235834 * q^34 - 1331256 * q^35 + 129762 * q^36 - 630044 * q^37 + 80910 * q^38 - 20412 * q^39 - 2153982 * q^40 - 969204 * q^41 - 106002 * q^42 - 1770208 * q^43 - 1749140 * q^44 - 271188 * q^45 + 97336 * q^46 - 1400024 * q^47 - 351594 * q^48 + 1985598 * q^49 - 956660 * q^50 - 1876068 * q^51 + 3217272 * q^52 - 1573516 * q^53 - 157464 * q^54 - 431296 * q^55 + 7740702 * q^56 - 1184328 * q^57 + 5987188 * q^58 - 1410320 * q^59 + 2129922 * q^60 - 942172 * q^61 + 3334412 * q^62 - 804816 * q^63 + 1996866 * q^64 - 420944 * q^65 + 2651508 * q^66 - 452072 * q^67 - 9258254 * q^68 - 1971054 * q^69 + 21981136 * q^70 + 122928 * q^71 - 1425924 * q^72 + 16490716 * q^73 - 600104 * q^74 + 6156486 * q^75 + 7428658 * q^76 + 7239696 * q^77 - 8422812 * q^78 + 2458408 * q^79 + 19440230 * q^80 + 3188646 * q^81 + 20510784 * q^82 - 7566456 * q^83 - 14726934 * q^84 + 5817744 * q^85 - 669666 * q^86 - 8399700 * q^87 + 14775668 * q^88 - 20368036 * q^89 - 9507618 * q^90 + 8815576 * q^91 - 2165726 * q^92 - 6621696 * q^93 + 16952576 * q^94 + 5143832 * q^95 - 10534212 * q^96 + 12586972 * q^97 - 39164812 * q^98 - 10806696 * 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 $

$2$	−8.53998	−0.754834	−0.377417	−	0.926043i	$-0.623188\pi$
$2$	−8.53998	−0.754834	−0.377417	+	0.926043i	$0.623188\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−55.0688	−0.430225
$5$	−446.195	−1.59636	−0.798178	−	0.602422i	$-0.794202\pi$
$5$	−446.195	−1.59636	−0.798178	+	0.602422i	$0.794202\pi$
$6$	−230.579	−0.435804
$7$	1770.31	1.95078	0.975388	−	0.220496i	$-0.0707675\pi$
$7$	1770.31	1.95078	0.975388	+	0.220496i	$0.0707675\pi$
$8$	1563.40	1.07958
$9$	729.000	0.333333
$10$	3810.49	1.20498
$11$	−5076.40	−1.14996	−0.574978	−	0.818169i	$-0.694989\pi$
$11$	−5076.40	−1.14996	−0.574978	+	0.818169i	$0.694989\pi$
$12$	−1486.86	−0.248391
$13$	10427.8	1.31641	0.658204	−	0.752839i	$-0.271316\pi$
$13$	10427.8	1.31641	0.658204	+	0.752839i	$0.271316\pi$
$14$	−15118.4	−1.47251
$15$	−12047.3	−0.921656
$16$	−6302.62	−0.384681
$17$	−976.453	−0.0482036	−0.0241018	−	0.999710i	$-0.507673\pi$
$17$	−976.453	−0.0482036	−0.0241018	+	0.999710i	$0.507673\pi$
$18$	−6225.64	−0.251611
$19$	−47906.2	−1.60234	−0.801168	−	0.598439i	$-0.795788\pi$
$19$	−47906.2	−1.60234	−0.801168	+	0.598439i	$0.795788\pi$
$20$	24571.4	0.686792
$21$	47798.5	1.12628
$22$	43352.3	0.868027
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	42211.9	0.623298
$25$	120965.	1.54835
$26$	−89053.1	−0.993670
$27$	19683.0	0.192450
$28$	−97489.1	−0.839273
$29$	−70779.7	−0.538909	−0.269455	−	0.963013i	$-0.586843\pi$
$29$	−70779.7	−0.538909	−0.269455	+	0.963013i	$0.586843\pi$
$30$	102883.	0.695698
$31$	−107045.	−0.645357	−0.322678	−	0.946509i	$-0.604583\pi$
$31$	−107045.	−0.645357	−0.322678	+	0.946509i	$0.604583\pi$
$32$	−146291.	−0.789212
$33$	−137063.	−0.663928
$34$	8338.88	0.0363858
$35$	−789905.	−3.11413
$36$	−40145.2	−0.143408
$37$	−400445.	−1.29968	−0.649841	−	0.760070i	$-0.725164\pi$
$37$	−400445.	−1.29968	−0.649841	+	0.760070i	$0.725164\pi$
$38$	409117.	1.20950
$39$	281550.	0.760029
$40$	−697582.	−1.72340
$41$	351711.	0.796971	0.398486	−	0.917175i	$-0.369536\pi$
$41$	351711.	0.796971	0.398486	+	0.917175i	$0.369536\pi$
$42$	−408198.	−0.850156
$43$	−561644.	−1.07726	−0.538632	−	0.842541i	$-0.681058\pi$
$43$	−561644.	−1.07726	−0.538632	+	0.842541i	$0.681058\pi$
$44$	279551.	0.494740
$45$	−325276.	−0.532118
$46$	103906.	0.157394
$47$	−414887.	−0.582891	−0.291445	−	0.956587i	$-0.594136\pi$
$47$	−414887.	−0.582891	−0.291445	+	0.956587i	$0.594136\pi$
$48$	−170171.	−0.222096
$49$	2.31047e6	2.80553
$50$	−1.03304e6	−1.16875
$51$	−26364.2	−0.0278304
$52$	−574246.	−0.566352
$53$	−1.16421e6	−1.07416	−0.537078	−	0.843533i	$-0.680472\pi$
$53$	−1.16421e6	−1.07416	−0.537078	+	0.843533i	$0.680472\pi$
$54$	−168092.	−0.145268
$55$	2.26506e6	1.83574
$56$	2.76772e6	2.10602
$57$	−1.29347e6	−0.925110
$58$	604457.	0.406787
$59$	−1.73195e6	−1.09788	−0.548940	−	0.835862i	$-0.684968\pi$
$59$	−1.73195e6	−1.09788	−0.548940	+	0.835862i	$0.684968\pi$
$60$	663428.	0.396520
$61$	−2.22856e6	−1.25710	−0.628551	−	0.777768i	$-0.716352\pi$
$61$	−2.22856e6	−1.25710	−0.628551	+	0.777768i	$0.716352\pi$
$62$	914161.	0.487138
$63$	1.29056e6	0.650259
$64$	2.05606e6	0.980406
$65$	−4.65283e6	−2.10146
$66$	1.17051e6	0.501155
$67$	2.16992e6	0.881416	0.440708	−	0.897651i	$-0.354727\pi$
$67$	2.16992e6	0.881416	0.440708	+	0.897651i	$0.354727\pi$
$68$	53772.1	0.0207384
$69$	−328509.	−0.120386
$70$	6.74577e6	2.35065
$71$	2.37410e6	0.787219	0.393609	−	0.919278i	$-0.371226\pi$
$71$	2.37410e6	0.787219	0.393609	+	0.919278i	$0.371226\pi$
$72$	1.13972e6	0.359861
$73$	3.18298e6	0.957642	0.478821	−	0.877912i	$-0.341064\pi$
$73$	3.18298e6	0.957642	0.478821	+	0.877912i	$0.341064\pi$
$74$	3.41979e6	0.981044
$75$	3.26605e6	0.893940
$76$	2.63814e6	0.689365
$77$	−8.98682e6	−2.24331
$78$	−2.40443e6	−0.573696
$79$	−6.55235e6	−1.49521	−0.747605	−	0.664144i	$-0.768796\pi$
$79$	−6.55235e6	−1.49521	−0.747605	+	0.664144i	$0.768796\pi$
$80$	2.81220e6	0.614088
$81$	531441.	0.111111
$82$	−3.00361e6	−0.601581
$83$	−4.67692e6	−0.897815	−0.448907	−	0.893578i	$-0.648187\pi$
$83$	−4.67692e6	−0.897815	−0.448907	+	0.893578i	$0.648187\pi$
$84$	−2.63221e6	−0.484554
$85$	435688.	0.0769501
$86$	4.79643e6	0.813155
$87$	−1.91105e6	−0.311139
$88$	−7.93646e6	−1.24147
$89$	−1.99651e6	−0.300198	−0.150099	−	0.988671i	$-0.547959\pi$
$89$	−1.99651e6	−0.300198	−0.150099	+	0.988671i	$0.547959\pi$
$90$	2.77785e6	0.401661
$91$	1.84605e7	2.56802
$92$	670022.	0.0897081
$93$	−2.89021e6	−0.372597
$94$	3.54312e6	0.439986
$95$	2.13755e7	2.55790
$96$	−3.94987e6	−0.455652
$97$	−2.86447e6	−0.318672	−0.159336	−	0.987224i	$-0.550935\pi$
$97$	−2.86447e6	−0.318672	−0.159336	+	0.987224i	$0.550935\pi$
$98$	−1.97314e7	−2.11771
$99$	−3.70069e6	−0.383319
$100$	−6.66139e6	−0.666139
$101$	−5.82736e6	−0.562791	−0.281396	−	0.959592i	$-0.590797\pi$
$101$	−5.82736e6	−0.562791	−0.281396	+	0.959592i	$0.590797\pi$
$102$	225150.	0.0210073
$103$	3.19790e6	0.288360	0.144180	−	0.989552i	$-0.453946\pi$
$103$	3.19790e6	0.288360	0.144180	+	0.989552i	$0.453946\pi$
$104$	1.63028e7	1.42117
$105$	−2.13274e7	−1.79794
$106$	9.94236e6	0.810810
$107$	1.74875e7	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$107$	1.74875e7	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$108$	−1.08392e6	−0.0827968
$109$	7.57159e6	0.560008	0.280004	−	0.959999i	$-0.409664\pi$
$109$	7.57159e6	0.560008	0.280004	+	0.959999i	$0.409664\pi$
$110$	−1.93436e7	−1.38568
$111$	−1.08120e7	−0.750371
$112$	−1.11576e7	−0.750427
$113$	−1.77479e7	−1.15710	−0.578551	−	0.815646i	$-0.696382\pi$
$113$	−1.77479e7	−1.15710	−0.578551	+	0.815646i	$0.696382\pi$
$114$	1.10462e7	0.698304
$115$	5.42885e6	0.332863
$116$	3.89775e6	0.231852
$117$	7.60186e6	0.438803
$118$	1.47909e7	0.828717
$119$	−1.72863e6	−0.0940345
$120$	−1.88347e7	−0.995004
$121$	6.28265e6	0.322399
$122$	1.90319e7	0.948904
$123$	9.49620e6	0.460132
$124$	5.89483e6	0.277649
$125$	−1.91149e7	−0.875361
$126$	−1.10213e7	−0.490838
$127$	−1.07937e6	−0.0467580	−0.0233790	−	0.999727i	$-0.507442\pi$
$127$	−1.07937e6	−0.0467580	−0.0233790	+	0.999727i	$0.507442\pi$
$128$	1.16659e6	0.0491681
$129$	−1.51644e7	−0.621958
$130$	3.97350e7	1.58625
$131$	−1.94684e7	−0.756625	−0.378312	−	0.925678i	$-0.623495\pi$
$131$	−1.94684e7	−0.756625	−0.378312	+	0.925678i	$0.623495\pi$
$132$	7.54788e6	0.285638
$133$	−8.48090e7	−3.12580
$134$	−1.85310e7	−0.665323
$135$	−8.78245e6	−0.307219
$136$	−1.52659e6	−0.0520398
$137$	−58145.1	−0.00193193	−0.000965965	−	1.00000i	$-0.500307\pi$
$137$	−58145.1	−0.00193193	−0.000965965		1.00000i	$0.500307\pi$
$138$	2.80546e6	0.0908714
$139$	−4.27729e6	−0.135088	−0.0675440	−	0.997716i	$-0.521516\pi$
$139$	−4.27729e6	−0.135088	−0.0675440	+	0.997716i	$0.521516\pi$
$140$	4.34991e7	1.33978
$141$	−1.12019e7	−0.336532
$142$	−2.02748e7	−0.594220
$143$	−5.29356e7	−1.51381
$144$	−4.59461e6	−0.128227
$145$	3.15815e7	0.860291
$146$	−2.71825e7	−0.722861
$147$	6.23827e7	1.61977
$148$	2.20520e7	0.559155
$149$	4.15365e7	1.02867	0.514337	−	0.857588i	$-0.328038\pi$
$149$	4.15365e7	1.02867	0.514337	+	0.857588i	$0.328038\pi$
$150$	−2.78920e7	−0.674777
$151$	−1.83879e7	−0.434623	−0.217311	−	0.976102i	$-0.569729\pi$
$151$	−1.83879e7	−0.434623	−0.217311	+	0.976102i	$0.569729\pi$
$152$	−7.48966e7	−1.72986
$153$	−711834.	−0.0160679
$154$	7.67473e7	1.69333
$155$	4.77629e7	1.03022
$156$	−1.55046e7	−0.326983
$157$	−8.16093e7	−1.68303	−0.841513	−	0.540237i	$-0.818335\pi$
$157$	−8.16093e7	−1.68303	−0.841513	+	0.540237i	$0.818335\pi$
$158$	5.59569e7	1.12864
$159$	−3.14338e7	−0.620164
$160$	6.52745e7	1.25986
$161$	−2.15394e7	−0.406765
$162$	−4.53849e6	−0.0838705
$163$	−7.94412e7	−1.43678	−0.718389	−	0.695642i	$-0.755120\pi$
$163$	−7.94412e7	−1.43678	−0.718389	+	0.695642i	$0.755120\pi$
$164$	−1.93683e7	−0.342877
$165$	6.11567e7	1.05986
$166$	3.99408e7	0.677701
$167$	−6.28625e7	−1.04444	−0.522220	−	0.852811i	$-0.674896\pi$
$167$	−6.28625e7	−1.04444	−0.522220	+	0.852811i	$0.674896\pi$
$168$	7.47283e7	1.21591
$169$	4.59903e7	0.732931
$170$	−3.72077e6	−0.0580846
$171$	−3.49236e7	−0.534112
$172$	3.09291e7	0.463466
$173$	5.08283e7	0.746353	0.373176	−	0.927760i	$-0.378269\pi$
$173$	5.08283e7	0.746353	0.373176	+	0.927760i	$0.378269\pi$
$174$	1.63203e7	0.234859
$175$	2.14146e8	3.02048
$176$	3.19946e7	0.442367
$177$	−4.67628e7	−0.633861
$178$	1.70502e7	0.226600
$179$	−5.40648e7	−0.704578	−0.352289	−	0.935891i	$-0.614597\pi$
$179$	−5.40648e7	−0.704578	−0.352289	+	0.935891i	$0.614597\pi$
$180$	1.79126e7	0.228931
$181$	−3.62140e7	−0.453943	−0.226971	−	0.973901i	$-0.572882\pi$
$181$	−3.62140e7	−0.453943	−0.226971	+	0.973901i	$0.572882\pi$
$182$	−1.57652e8	−1.93843
$183$	−6.01712e7	−0.725788
$184$	−1.90219e7	−0.225109
$185$	1.78676e8	2.07475
$186$	2.46823e7	0.281249
$187$	4.95686e6	0.0554321
$188$	2.28473e7	0.250774
$189$	3.48451e7	0.375427
$190$	−1.82546e8	−1.93079
$191$	2.62687e7	0.272786	0.136393	−	0.990655i	$-0.456449\pi$
$191$	2.62687e7	0.272786	0.136393	+	0.990655i	$0.456449\pi$
$192$	5.55136e7	0.566038
$193$	1.07208e8	1.07344	0.536720	−	0.843760i	$-0.319663\pi$
$193$	1.07208e8	1.07344	0.536720	+	0.843760i	$0.319663\pi$
$194$	2.44625e7	0.240544
$195$	−1.25626e8	−1.21328
$196$	−1.27235e8	−1.20701
$197$	4.98161e7	0.464235	0.232117	−	0.972688i	$-0.425435\pi$
$197$	4.98161e7	0.464235	0.232117	+	0.972688i	$0.425435\pi$
$198$	3.16038e7	0.289342
$199$	−8.59901e7	−0.773505	−0.386752	−	0.922184i	$-0.626403\pi$
$199$	−8.59901e7	−0.773505	−0.386752	+	0.922184i	$0.626403\pi$
$200$	1.89117e8	1.67157
$201$	5.85877e7	0.508886
$202$	4.97656e7	0.424814
$203$	−1.25302e8	−1.05129
$204$	1.45185e6	0.0119733
$205$	−1.56932e8	−1.27225
$206$	−2.73100e7	−0.217664
$207$	−8.86974e6	−0.0695048
$208$	−6.57224e7	−0.506398
$209$	2.43191e8	1.84262
$210$	1.82136e8	1.35715
$211$	1.43600e8	1.05237	0.526183	−	0.850371i	$-0.323623\pi$
$211$	1.43600e8	1.05237	0.526183	+	0.850371i	$0.323623\pi$
$212$	6.41119e7	0.462129
$213$	6.41008e7	0.454501
$214$	−1.49343e8	−1.04169
$215$	2.50603e8	1.71969
$216$	3.07725e7	0.207766
$217$	−1.89503e8	−1.25895
$218$	−6.46612e7	−0.422714
$219$	8.59403e7	0.552895
$220$	−1.24734e8	−0.789781
$221$	−1.01822e7	−0.0634557
$222$	9.23344e7	0.566406
$223$	−2.72368e7	−0.164471	−0.0822356	−	0.996613i	$-0.526206\pi$
$223$	−2.72368e7	−0.164471	−0.0822356	+	0.996613i	$0.526206\pi$
$224$	−2.58982e8	−1.53958
$225$	8.81833e7	0.516117
$226$	1.51566e8	0.873421
$227$	3.16198e7	0.179419	0.0897095	−	0.995968i	$-0.471406\pi$
$227$	3.16198e7	0.179419	0.0897095	+	0.995968i	$0.471406\pi$
$228$	7.12296e7	0.398005
$229$	6.22632e7	0.342616	0.171308	−	0.985218i	$-0.445201\pi$
$229$	6.22632e7	0.342616	0.171308	+	0.985218i	$0.445201\pi$
$230$	−4.63623e7	−0.251256
$231$	−2.42644e8	−1.29517
$232$	−1.10657e8	−0.581797
$233$	−1.76973e8	−0.916559	−0.458279	−	0.888808i	$-0.651534\pi$
$233$	−1.76973e8	−0.916559	−0.458279	+	0.888808i	$0.651534\pi$
$234$	−6.49197e7	−0.331223
$235$	1.85120e8	0.930501
$236$	9.53767e7	0.472335
$237$	−1.76913e8	−0.863260
$238$	1.47624e7	0.0709805
$239$	−2.96990e8	−1.40718	−0.703590	−	0.710606i	$-0.748421\pi$
$239$	−2.96990e8	−1.40718	−0.703590	+	0.710606i	$0.748421\pi$
$240$	7.59293e7	0.354544
$241$	−4.49359e7	−0.206792	−0.103396	−	0.994640i	$-0.532971\pi$
$241$	−4.49359e7	−0.206792	−0.103396	+	0.994640i	$0.532971\pi$
$242$	−5.36537e7	−0.243358
$243$	1.43489e7	0.0641500
$244$	1.22724e8	0.540837
$245$	−1.03092e9	−4.47862
$246$	−8.10973e7	−0.347323
$247$	−4.99556e8	−2.10933
$248$	−1.67354e8	−0.696716
$249$	−1.26277e8	−0.518353
$250$	1.63241e8	0.660752
$251$	2.47284e8	0.987046	0.493523	−	0.869733i	$-0.335709\pi$
$251$	2.47284e8	0.987046	0.493523	+	0.869733i	$0.335709\pi$
$252$	−7.10696e7	−0.279758
$253$	6.17645e7	0.239782
$254$	9.21777e6	0.0352946
$255$	1.17636e7	0.0444272
$256$	−2.73138e8	−1.01752
$257$	5.82176e7	0.213938	0.106969	−	0.994262i	$-0.465885\pi$
$257$	5.82176e7	0.213938	0.106969	+	0.994262i	$0.465885\pi$
$258$	1.29504e8	0.469475
$259$	−7.08914e8	−2.53539
$260$	2.56226e8	0.904099
$261$	−5.15984e7	−0.179636
$262$	1.66260e8	0.571126
$263$	5.61623e8	1.90370	0.951852	−	0.306557i	$-0.0991770\pi$
$263$	5.61623e8	1.90370	0.951852	+	0.306557i	$0.0991770\pi$
$264$	−2.14284e8	−0.716765
$265$	5.19466e8	1.71473
$266$	7.24267e8	2.35946
$267$	−5.39059e7	−0.173319
$268$	−1.19495e8	−0.379207
$269$	3.87212e8	1.21287	0.606437	−	0.795131i	$-0.292598\pi$
$269$	3.87212e8	1.21287	0.606437	+	0.795131i	$0.292598\pi$
$270$	7.50019e7	0.231899
$271$	1.33234e8	0.406651	0.203325	−	0.979111i	$-0.434825\pi$
$271$	1.33234e8	0.406651	0.203325	+	0.979111i	$0.434825\pi$
$272$	6.15421e6	0.0185430
$273$	4.98433e8	1.48265
$274$	496558.	0.00145829
$275$	−6.14066e8	−1.78053
$276$	1.80906e7	0.0517930
$277$	−1.04015e8	−0.294046	−0.147023	−	0.989133i	$-0.546969\pi$
$277$	−1.04015e8	−0.294046	−0.147023	+	0.989133i	$0.546969\pi$
$278$	3.65279e7	0.101969
$279$	−7.80357e7	−0.215119
$280$	−1.23494e9	−3.36196
$281$	2.34656e8	0.630898	0.315449	−	0.948943i	$-0.397845\pi$
$281$	2.34656e8	0.630898	0.315449	+	0.948943i	$0.397845\pi$
$282$	9.56644e7	0.254026
$283$	4.94281e8	1.29635	0.648174	−	0.761492i	$-0.275533\pi$
$283$	4.94281e8	1.29635	0.648174	+	0.761492i	$0.275533\pi$
$284$	−1.30739e8	−0.338681
$285$	5.77138e8	1.47680
$286$	4.52069e8	1.14268
$287$	6.22640e8	1.55471
$288$	−1.06646e8	−0.263071
$289$	−4.09385e8	−0.997676
$290$	−2.69706e8	−0.649377
$291$	−7.73407e7	−0.183985
$292$	−1.75283e8	−0.412002
$293$	3.38248e8	0.785594	0.392797	−	0.919625i	$-0.371507\pi$
$293$	3.38248e8	0.785594	0.392797	+	0.919625i	$0.371507\pi$
$294$	−5.32747e8	−1.22266
$295$	7.72789e8	1.75261
$296$	−6.26057e8	−1.40311
$297$	−9.99188e7	−0.221309
$298$	−3.54721e8	−0.776479
$299$	−1.26875e8	−0.274490
$300$	−1.79857e8	−0.384595
$301$	−9.94287e8	−2.10150
$302$	1.57032e8	0.328068
$303$	−1.57339e8	−0.324928
$304$	3.01934e8	0.616389
$305$	9.94373e8	2.00678
$306$	6.07904e6	0.0121286
$307$	1.70238e8	0.335794	0.167897	−	0.985805i	$-0.446302\pi$
$307$	1.70238e8	0.335794	0.167897	+	0.985805i	$0.446302\pi$
$308$	4.94894e8	0.965127
$309$	8.63433e7	0.166485
$310$	−4.07894e8	−0.777645
$311$	7.57357e8	1.42771	0.713854	−	0.700295i	$-0.246948\pi$
$311$	7.57357e8	1.42771	0.713854	+	0.700295i	$0.246948\pi$
$312$	4.40177e8	0.820514
$313$	7.92575e7	0.146095	0.0730475	−	0.997328i	$-0.476728\pi$
$313$	7.92575e7	0.146095	0.0730475	+	0.997328i	$0.476728\pi$
$314$	6.96941e8	1.27041
$315$	−5.75841e8	−1.03804
$316$	3.60830e8	0.643277
$317$	−4.32284e7	−0.0762187	−0.0381093	−	0.999274i	$-0.512134\pi$
$317$	−4.32284e7	−0.0762187	−0.0381093	+	0.999274i	$0.512134\pi$
$318$	2.68444e8	0.468121
$319$	3.59306e8	0.619722
$320$	−9.17403e8	−1.56508
$321$	4.72163e8	0.796754
$322$	1.83946e8	0.307040
$323$	4.67781e7	0.0772385
$324$	−2.92658e7	−0.0478028
$325$	1.26140e9	2.03826
$326$	6.78426e8	1.08453
$327$	2.04433e8	0.323321
$328$	5.49866e8	0.860397
$329$	−7.34480e8	−1.13709
$330$	−5.22277e8	−0.800022
$331$	−9.69374e7	−0.146924	−0.0734621	−	0.997298i	$-0.523405\pi$
$331$	−9.69374e7	−0.146924	−0.0734621	+	0.997298i	$0.523405\pi$
$332$	2.57552e8	0.386262
$333$	−2.91924e8	−0.433227
$334$	5.36844e8	0.788380
$335$	−9.68205e8	−1.40705
$336$	−3.01256e8	−0.433259
$337$	−6.23180e8	−0.886970	−0.443485	−	0.896282i	$-0.646258\pi$
$337$	−6.23180e8	−0.886970	−0.443485	+	0.896282i	$0.646258\pi$
$338$	−3.92756e8	−0.553242
$339$	−4.79193e8	−0.668054
$340$	−2.39928e7	−0.0331059
$341$	5.43402e8	0.742132
$342$	2.98247e8	0.403166
$343$	2.63233e9	3.52218
$344$	−8.78077e8	−1.16299
$345$	1.46579e8	0.192179
$346$	−4.34072e8	−0.563373
$347$	−7.32460e8	−0.941089	−0.470545	−	0.882376i	$-0.655942\pi$
$347$	−7.32460e8	−0.941089	−0.470545	+	0.882376i	$0.655942\pi$
$348$	1.05239e8	0.133860
$349$	6.88937e8	0.867542	0.433771	−	0.901023i	$-0.357183\pi$
$349$	6.88937e8	0.867542	0.433771	+	0.901023i	$0.357183\pi$
$350$	−1.82880e9	−2.27996
$351$	2.05250e8	0.253343
$352$	7.42633e8	0.907560
$353$	−9.54856e8	−1.15538	−0.577692	−	0.816255i	$-0.696047\pi$
$353$	−9.54856e8	−1.15538	−0.577692	+	0.816255i	$0.696047\pi$
$354$	3.99353e8	0.478460
$355$	−1.05931e9	−1.25668
$356$	1.09946e8	0.129153
$357$	−4.66730e7	−0.0542909
$358$	4.61712e8	0.531840
$359$	3.60052e8	0.410709	0.205355	−	0.978688i	$-0.434165\pi$
$359$	3.60052e8	0.410709	0.205355	+	0.978688i	$0.434165\pi$
$360$	−5.08538e8	−0.574466
$361$	1.40113e9	1.56748
$362$	3.09266e8	0.342652
$363$	1.69632e8	0.186137
$364$	−1.01660e9	−1.10483
$365$	−1.42023e9	−1.52874
$366$	5.13861e8	0.547850
$367$	−1.56087e9	−1.64830	−0.824148	−	0.566375i	$-0.808345\pi$
$367$	−1.56087e9	−1.64830	−0.824148	+	0.566375i	$0.808345\pi$
$368$	7.66840e7	0.0802116
$369$	2.56397e8	0.265657
$370$	−1.52589e9	−1.56609
$371$	−2.06102e9	−2.09544
$372$	1.59160e8	0.160301
$373$	1.07087e9	1.06846	0.534228	−	0.845340i	$-0.320602\pi$
$373$	1.07087e9	1.06846	0.534228	+	0.845340i	$0.320602\pi$
$374$	−4.23315e7	−0.0418420
$375$	−5.16102e8	−0.505390
$376$	−6.48635e8	−0.629279
$377$	−7.38076e8	−0.709425
$378$	−2.97576e8	−0.283385
$379$	1.47354e8	0.139036	0.0695178	−	0.997581i	$-0.477854\pi$
$379$	1.47354e8	0.139036	0.0695178	+	0.997581i	$0.477854\pi$
$380$	−1.17712e9	−1.10047
$381$	−2.91429e7	−0.0269958
$382$	−2.24334e8	−0.205908
$383$	−2.59587e8	−0.236095	−0.118048	−	0.993008i	$-0.537664\pi$
$383$	−2.59587e8	−0.236095	−0.118048	+	0.993008i	$0.537664\pi$
$384$	3.14980e7	0.0283872
$385$	4.00987e9	3.58111
$386$	−9.15557e8	−0.810270
$387$	−4.09439e8	−0.359088
$388$	1.57743e8	0.137101
$389$	−1.59303e9	−1.37215	−0.686075	−	0.727531i	$-0.740668\pi$
$389$	−1.59303e9	−1.37215	−0.686075	+	0.727531i	$0.740668\pi$
$390$	1.07285e9	0.915822
$391$	1.18805e7	0.0100512
$392$	3.61220e9	3.02880
$393$	−5.25646e8	−0.436837
$394$	−4.25428e8	−0.350420
$395$	2.92362e9	2.38689
$396$	2.03793e8	0.164913
$397$	−1.75396e9	−1.40686	−0.703431	−	0.710763i	$-0.748350\pi$
$397$	−1.75396e9	−1.40686	−0.703431	+	0.710763i	$0.748350\pi$
$398$	7.34354e8	0.583868
$399$	−2.28984e9	−1.80468
$400$	−7.62395e8	−0.595621
$401$	1.86247e9	1.44240	0.721199	−	0.692728i	$-0.243591\pi$
$401$	1.86247e9	1.44240	0.721199	+	0.692728i	$0.243591\pi$
$402$	−5.00338e8	−0.384125
$403$	−1.11624e9	−0.849553
$404$	3.20906e8	0.242127
$405$	−2.37126e8	−0.177373
$406$	1.07008e9	0.793551
$407$	2.03282e9	1.49458
$408$	−4.12179e7	−0.0300452
$409$	1.42499e9	1.02987	0.514934	−	0.857230i	$-0.327817\pi$
$409$	1.42499e9	1.02987	0.514934	+	0.857230i	$0.327817\pi$
$410$	1.34019e9	0.960338
$411$	−1.56992e6	−0.00111540
$412$	−1.76104e8	−0.124060
$413$	−3.06611e9	−2.14172
$414$	7.57474e7	0.0524646
$415$	2.08682e9	1.43323
$416$	−1.52550e9	−1.03893
$417$	−1.15487e8	−0.0779930
$418$	−2.07684e9	−1.39087
$419$	7.11501e8	0.472527	0.236263	−	0.971689i	$-0.424077\pi$
$419$	7.11501e8	0.472527	0.236263	+	0.971689i	$0.424077\pi$
$420$	1.17448e9	0.773521
$421$	2.11656e9	1.38243	0.691216	−	0.722649i	$-0.257075\pi$
$421$	2.11656e9	1.38243	0.691216	+	0.722649i	$0.257075\pi$
$422$	−1.22634e9	−0.794362
$423$	−3.02453e8	−0.194297
$424$	−1.82014e9	−1.15964
$425$	−1.18116e8	−0.0746361
$426$	−5.47419e8	−0.343073
$427$	−3.94526e9	−2.45232
$428$	−9.63016e8	−0.593718
$429$	−1.42926e9	−0.874000
$430$	−2.14014e9	−1.29808
$431$	−1.16699e9	−0.702097	−0.351048	−	0.936357i	$-0.614175\pi$
$431$	−1.16699e9	−0.702097	−0.351048	+	0.936357i	$0.614175\pi$
$432$	−1.24054e8	−0.0740320
$433$	1.65689e9	0.980814	0.490407	−	0.871493i	$-0.336848\pi$
$433$	1.65689e9	0.980814	0.490407	+	0.871493i	$0.336848\pi$
$434$	1.61835e9	0.950296
$435$	8.52701e8	0.496689
$436$	−4.16959e8	−0.240930
$437$	5.82874e8	0.334110
$438$	−7.33928e8	−0.417344
$439$	6.46153e8	0.364510	0.182255	−	0.983251i	$-0.441660\pi$
$439$	6.46153e8	0.364510	0.182255	+	0.983251i	$0.441660\pi$
$440$	3.54121e9	1.98183
$441$	1.68433e9	0.935175
$442$	8.69561e7	0.0478985
$443$	2.31866e9	1.26714	0.633569	−	0.773687i	$-0.281589\pi$
$443$	2.31866e9	1.26714	0.633569	+	0.773687i	$0.281589\pi$
$444$	5.95405e8	0.322829
$445$	8.90834e8	0.479222
$446$	2.32602e8	0.124148
$447$	1.12149e9	0.593905
$448$	3.63987e9	1.91255
$449$	−3.13825e9	−1.63616	−0.818080	−	0.575104i	$-0.804961\pi$
$449$	−3.13825e9	−1.63616	−0.818080	+	0.575104i	$0.804961\pi$
$450$	−7.53084e8	−0.389583
$451$	−1.78543e9	−0.916482
$452$	9.77355e8	0.497815
$453$	−4.96473e8	−0.250930
$454$	−2.70032e8	−0.135432
$455$	−8.23697e9	−4.09947
$456$	−2.02221e9	−0.998733
$457$	−1.88096e9	−0.921875	−0.460937	−	0.887433i	$-0.652487\pi$
$457$	−1.88096e9	−0.921875	−0.460937	+	0.887433i	$0.652487\pi$
$458$	−5.31726e8	−0.258618
$459$	−1.92195e7	−0.00927680
$460$	−2.98960e8	−0.143206
$461$	1.48345e9	0.705213	0.352607	−	0.935772i	$-0.385295\pi$
$461$	1.48345e9	0.705213	0.352607	+	0.935772i	$0.385295\pi$
$462$	2.07218e9	0.977642
$463$	3.16239e9	1.48075	0.740375	−	0.672194i	$-0.234648\pi$
$463$	3.16239e9	1.48075	0.740375	+	0.672194i	$0.234648\pi$
$464$	4.46098e8	0.207308
$465$	1.28960e9	0.594797
$466$	1.51134e9	0.691850
$467$	−2.36318e9	−1.07371	−0.536855	−	0.843674i	$-0.680388\pi$
$467$	−2.36318e9	−1.07371	−0.536855	+	0.843674i	$0.680388\pi$
$468$	−4.18625e8	−0.188784
$469$	3.84143e9	1.71945
$470$	−1.58092e9	−0.702374
$471$	−2.20345e9	−0.971695
$472$	−2.70774e9	−1.18525
$473$	2.85113e9	1.23881
$474$	1.51084e9	0.651618
$475$	−5.79496e9	−2.48098
$476$	9.51935e7	0.0404560
$477$	−8.48712e8	−0.358052
$478$	2.53629e9	1.06219
$479$	1.46316e9	0.608301	0.304151	−	0.952624i	$-0.401627\pi$
$479$	1.46316e9	0.608301	0.304151	+	0.952624i	$0.401627\pi$
$480$	1.76241e9	0.727382
$481$	−4.17576e9	−1.71091
$482$	3.83751e8	0.156094
$483$	−5.81564e8	−0.234846
$484$	−3.45978e8	−0.138704
$485$	1.27811e9	0.508713
$486$	−1.22539e8	−0.0484226
$487$	−3.34899e9	−1.31390	−0.656949	−	0.753935i	$-0.728153\pi$
$487$	−3.34899e9	−1.31390	−0.656949	+	0.753935i	$0.728153\pi$
$488$	−3.48414e9	−1.35715
$489$	−2.14491e9	−0.829524
$490$	8.80404e9	3.38061
$491$	2.03782e9	0.776929	0.388465	−	0.921464i	$-0.373006\pi$
$491$	2.03782e9	0.776929	0.388465	+	0.921464i	$0.373006\pi$
$492$	−5.22945e8	−0.197960
$493$	6.91130e7	0.0259774
$494$	4.26619e9	1.59219
$495$	1.65123e9	0.611913
$496$	6.74663e8	0.248257
$497$	4.20291e9	1.53569
$498$	1.07840e9	0.391271
$499$	5.37908e8	0.193801	0.0969006	−	0.995294i	$-0.469107\pi$
$499$	5.37908e8	0.193801	0.0969006	+	0.995294i	$0.469107\pi$
$500$	1.05263e9	0.376602
$501$	−1.69729e9	−0.603008
$502$	−2.11180e9	−0.745056
$503$	5.15747e9	1.80696	0.903481	−	0.428627i	$-0.141003\pi$
$503$	5.15747e9	1.80696	0.903481	+	0.428627i	$0.141003\pi$
$504$	2.01766e9	0.702008
$505$	2.60014e9	0.898415
$506$	−5.27468e8	−0.180996
$507$	1.24174e9	0.423158
$508$	5.94394e7	0.0201165
$509$	−1.44045e9	−0.484156	−0.242078	−	0.970257i	$-0.577829\pi$
$509$	−1.44045e9	−0.484156	−0.242078	+	0.970257i	$0.577829\pi$
$510$	−1.00461e8	−0.0335352
$511$	5.63487e9	1.86815
$512$	2.18327e9	0.718891
$513$	−9.42937e8	−0.308370
$514$	−4.97177e8	−0.161488
$515$	−1.42689e9	−0.460324
$516$	8.35085e8	0.267582
$517$	2.10613e9	0.670299
$518$	6.05411e9	1.91380
$519$	1.37236e9	0.430907
$520$	−7.27424e9	−2.26870
$521$	2.46278e8	0.0762944	0.0381472	−	0.999272i	$-0.487854\pi$
$521$	2.46278e8	0.0762944	0.0381472	+	0.999272i	$0.487854\pi$
$522$	4.40649e8	0.135596
$523$	−5.79562e9	−1.77151	−0.885756	−	0.464152i	$-0.846359\pi$
$523$	−5.79562e9	−1.77151	−0.885756	+	0.464152i	$0.846359\pi$
$524$	1.07210e9	0.325519
$525$	5.78194e9	1.74388
$526$	−4.79624e9	−1.43698
$527$	1.04524e8	0.0311086
$528$	8.63855e8	0.255401
$529$	1.48036e8	0.0434783
$530$	−4.43623e9	−1.29434
$531$	−1.26260e9	−0.365960
$532$	4.67033e9	1.34480
$533$	3.66757e9	1.04914
$534$	4.60355e8	0.130827
$535$	−7.80284e9	−2.20300
$536$	3.39245e9	0.951562
$537$	−1.45975e9	−0.406788
$538$	−3.30678e9	−0.915519
$539$	−1.17289e10	−3.22623
$540$	4.83639e8	0.132173
$541$	1.47643e9	0.400887	0.200444	−	0.979705i	$-0.435762\pi$
$541$	1.47643e9	0.400887	0.200444	+	0.979705i	$0.435762\pi$
$542$	−1.13781e9	−0.306954
$543$	−9.77777e8	−0.262084
$544$	1.42847e8	0.0380429
$545$	−3.37841e9	−0.893972
$546$	−4.25660e9	−1.11915
$547$	−2.90538e8	−0.0759011	−0.0379505	−	0.999280i	$-0.512083\pi$
$547$	−2.90538e8	−0.0759011	−0.0379505	+	0.999280i	$0.512083\pi$
$548$	3.20198e6	0.000831165 0
$549$	−1.62462e9	−0.419034
$550$	5.24411e9	1.34401
$551$	3.39078e9	0.863514
$552$	−5.13592e8	−0.129967
$553$	−1.15997e10	−2.91682
$554$	8.88282e8	0.221956
$555$	4.82427e9	1.19786
$556$	2.35545e8	0.0581182
$557$	7.34166e9	1.80012	0.900059	−	0.435768i	$-0.143523\pi$
$557$	7.34166e9	1.80012	0.900059	+	0.435768i	$0.143523\pi$
$558$	6.66423e8	0.162379
$559$	−5.85671e9	−1.41812
$560$	4.97847e9	1.19795
$561$	1.33835e8	0.0320037
$562$	−2.00395e9	−0.476223
$563$	−2.75015e9	−0.649498	−0.324749	−	0.945800i	$-0.605280\pi$
$563$	−2.75015e9	−0.649498	−0.324749	+	0.945800i	$0.605280\pi$
$564$	6.16878e8	0.144785
$565$	7.91901e9	1.84715
$566$	−4.22115e9	−0.978528
$567$	9.40818e8	0.216753
$568$	3.71168e9	0.849868
$569$	−7.85027e9	−1.78645	−0.893227	−	0.449606i	$-0.851564\pi$
$569$	−7.85027e9	−1.78645	−0.893227	+	0.449606i	$0.851564\pi$
$570$	−4.92874e9	−1.11474
$571$	5.66092e9	1.27251	0.636254	−	0.771479i	$-0.280483\pi$
$571$	5.66092e9	1.27251	0.636254	+	0.771479i	$0.280483\pi$
$572$	2.91510e9	0.651280
$573$	7.09256e8	0.157493
$574$	−5.31733e9	−1.17355
$575$	−1.47178e9	−0.322853
$576$	1.49887e9	0.326802
$577$	−1.42425e9	−0.308653	−0.154326	−	0.988020i	$-0.549321\pi$
$577$	−1.42425e9	−0.308653	−0.154326	+	0.988020i	$0.549321\pi$
$578$	3.49614e9	0.753080
$579$	2.89463e9	0.619751
$580$	−1.73916e9	−0.370119
$581$	−8.27962e9	−1.75143
$582$	6.60488e8	0.138878
$583$	5.91001e9	1.23523
$584$	4.97627e9	1.03385
$585$	−3.39191e9	−0.700485
$586$	−2.88863e9	−0.592994
$587$	−8.50855e8	−0.173629	−0.0868145	−	0.996224i	$-0.527669\pi$
$587$	−8.50855e8	−0.173629	−0.0868145	+	0.996224i	$0.527669\pi$
$588$	−3.43534e9	−0.696866
$589$	5.12811e9	1.03408
$590$	−6.59960e9	−1.32293
$591$	1.34503e9	0.268026
$592$	2.52385e9	0.499963
$593$	−9.13607e8	−0.179915	−0.0899577	−	0.995946i	$-0.528673\pi$
$593$	−9.13607e8	−0.179915	−0.0899577	+	0.995946i	$0.528673\pi$
$594$	8.53304e8	0.167052
$595$	7.71305e8	0.150112
$596$	−2.28736e9	−0.442561
$597$	−2.32173e9	−0.446583
$598$	1.08351e9	0.207195
$599$	−5.87273e9	−1.11647	−0.558234	−	0.829684i	$-0.688521\pi$
$599$	−5.87273e9	−1.11647	−0.558234	+	0.829684i	$0.688521\pi$
$600$	5.10615e9	0.965083
$601$	−3.30873e9	−0.621728	−0.310864	−	0.950454i	$-0.600618\pi$
$601$	−3.30873e9	−0.621728	−0.310864	+	0.950454i	$0.600618\pi$
$602$	8.49119e9	1.58628
$603$	1.58187e9	0.293805
$604$	1.01260e9	0.186986
$605$	−2.80329e9	−0.514664
$606$	1.34367e9	0.245267
$607$	2.06006e8	0.0373869	0.0186934	−	0.999825i	$-0.494049\pi$
$607$	2.06006e8	0.0373869	0.0186934	+	0.999825i	$0.494049\pi$
$608$	7.00826e9	1.26458
$609$	−3.38316e9	−0.606963
$610$	−8.49192e9	−1.51479
$611$	−4.32635e9	−0.767322
$612$	3.91998e7	0.00691281
$613$	−3.16299e8	−0.0554608	−0.0277304	−	0.999615i	$-0.508828\pi$
$613$	−3.16299e8	−0.0554608	−0.0277304	+	0.999615i	$0.508828\pi$
$614$	−1.45383e9	−0.253469
$615$	−4.23716e9	−0.734533
$616$	−1.40500e10	−2.42184
$617$	−4.70958e8	−0.0807205	−0.0403602	−	0.999185i	$-0.512851\pi$
$617$	−4.70958e8	−0.0807205	−0.0403602	+	0.999185i	$0.512851\pi$
$618$	−7.37370e8	−0.125668
$619$	−1.04937e10	−1.77832	−0.889162	−	0.457592i	$-0.848712\pi$
$619$	−1.04937e10	−1.77832	−0.889162	+	0.457592i	$0.848712\pi$
$620$	−2.63024e9	−0.443226
$621$	−2.39483e8	−0.0401286
$622$	−6.46781e9	−1.07768
$623$	−3.53446e9	−0.585618
$624$	−1.77451e9	−0.292369
$625$	−9.21405e8	−0.150963
$626$	−6.76857e8	−0.110277
$627$	6.56615e9	1.06384
$628$	4.49413e9	0.724080
$629$	3.91016e8	0.0626494
$630$	4.91767e9	0.783551
$631$	−7.42264e9	−1.17613	−0.588066	−	0.808813i	$-0.700110\pi$
$631$	−7.42264e9	−1.17613	−0.588066	+	0.808813i	$0.700110\pi$
$632$	−1.02440e10	−1.61420
$633$	3.87721e9	0.607584
$634$	3.69169e8	0.0575325
$635$	4.81608e8	0.0746424
$636$	1.73102e9	0.266810
$637$	2.40931e10	3.69322
$638$	−3.06846e9	−0.467788
$639$	1.73072e9	0.262406
$640$	−5.20527e8	−0.0784898
$641$	1.25757e8	0.0188594	0.00942970	−	0.999956i	$-0.496998\pi$
$641$	1.25757e8	0.0188594	0.00942970	+	0.999956i	$0.496998\pi$
$642$	−4.03226e9	−0.601417
$643$	2.25200e9	0.334064	0.167032	−	0.985951i	$-0.446582\pi$
$643$	2.25200e9	0.334064	0.167032	+	0.985951i	$0.446582\pi$
$644$	1.18615e9	0.175000
$645$	6.76627e9	0.992866
$646$	−3.99484e8	−0.0583023
$647$	2.00995e9	0.291756	0.145878	−	0.989303i	$-0.453399\pi$
$647$	2.00995e9	0.291756	0.145878	+	0.989303i	$0.453399\pi$
$648$	8.30857e8	0.119954
$649$	8.79209e9	1.26251
$650$	−1.07723e10	−1.53855
$651$	−5.11658e9	−0.726853
$652$	4.37473e9	0.618138
$653$	1.29964e10	1.82653	0.913263	−	0.407369i	$-0.133554\pi$
$653$	1.29964e10	1.82653	0.913263	+	0.407369i	$0.133554\pi$
$654$	−1.74585e9	−0.244054
$655$	8.68669e9	1.20784
$656$	−2.21670e9	−0.306580
$657$	2.32039e9	0.319214
$658$	6.27245e9	0.858314
$659$	−5.69361e9	−0.774977	−0.387488	−	0.921875i	$-0.626657\pi$
$659$	−5.69361e9	−0.774977	−0.387488	+	0.921875i	$0.626657\pi$
$660$	−3.36783e9	−0.455980
$661$	2.01432e9	0.271284	0.135642	−	0.990758i	$-0.456690\pi$
$661$	2.01432e9	0.271284	0.135642	+	0.990758i	$0.456690\pi$
$662$	8.27843e8	0.110903
$663$	−2.74921e8	−0.0366362
$664$	−7.31191e9	−0.969265
$665$	3.78413e10	4.98989
$666$	2.49303e9	0.327015
$667$	8.61177e8	0.112370
$668$	3.46176e9	0.449344
$669$	−7.35395e8	−0.0949574
$670$	8.26845e9	1.06209
$671$	1.13131e10	1.44561
$672$	−6.99251e9	−0.888875
$673$	4.37424e8	0.0553159	0.0276580	−	0.999617i	$-0.491195\pi$
$673$	4.37424e8	0.0553159	0.0276580	+	0.999617i	$0.491195\pi$
$674$	5.32194e9	0.669516
$675$	2.38095e9	0.297980
$676$	−2.53263e9	−0.315325
$677$	−1.09393e10	−1.35497	−0.677484	−	0.735538i	$-0.736930\pi$
$677$	−1.09393e10	−1.35497	−0.677484	+	0.735538i	$0.736930\pi$
$678$	4.09230e9	0.504270
$679$	−5.07102e9	−0.621657
$680$	6.81156e8	0.0830741
$681$	8.53734e8	0.103588
$682$	−4.64064e9	−0.560187
$683$	4.05528e9	0.487022	0.243511	−	0.969898i	$-0.421701\pi$
$683$	4.05528e9	0.487022	0.243511	+	0.969898i	$0.421701\pi$
$684$	1.92320e9	0.229788
$685$	2.59441e7	0.00308405
$686$	−2.24801e10	−2.65866
$687$	1.68111e9	0.197809
$688$	3.53983e9	0.414403
$689$	−1.21402e10	−1.41403
$690$	−1.25178e9	−0.145063
$691$	−2.68706e9	−0.309816	−0.154908	−	0.987929i	$-0.549508\pi$
$691$	−2.68706e9	−0.309816	−0.154908	+	0.987929i	$0.549508\pi$
$692$	−2.79905e9	−0.321100
$693$	−6.55139e9	−0.747769
$694$	6.25519e9	0.710366
$695$	1.90850e9	0.215648
$696$	−2.98774e9	−0.335901
$697$	−3.43429e8	−0.0384169
$698$	−5.88351e9	−0.654851
$699$	−4.77826e9	−0.529175
$700$	−1.17928e10	−1.29949
$701$	1.57328e10	1.72502	0.862510	−	0.506041i	$-0.168891\pi$
$701$	1.57328e10	1.72502	0.862510	+	0.506041i	$0.168891\pi$
$702$	−1.75283e9	−0.191232
$703$	1.91838e10	2.08253
$704$	−1.04374e10	−1.12742
$705$	4.99825e9	0.537225
$706$	8.15445e9	0.872124
$707$	−1.03163e10	−1.09788
$708$	2.57517e9	0.272703
$709$	2.79822e8	0.0294862	0.0147431	−	0.999891i	$-0.495307\pi$
$709$	2.79822e8	0.0294862	0.0147431	+	0.999891i	$0.495307\pi$
$710$	9.04651e9	0.948586
$711$	−4.77666e9	−0.498403
$712$	−3.12136e9	−0.324088
$713$	1.30241e9	0.134566
$714$	3.98586e8	0.0409806
$715$	2.36196e10	2.41658
$716$	2.97728e9	0.303127
$717$	−8.01874e9	−0.812435
$718$	−3.07484e9	−0.310018
$719$	1.28407e10	1.28836	0.644181	−	0.764873i	$-0.277198\pi$
$719$	1.28407e10	1.28836	0.644181	+	0.764873i	$0.277198\pi$
$720$	2.05009e9	0.204696
$721$	5.66129e9	0.562525
$722$	−1.19656e10	−1.18319
$723$	−1.21327e9	−0.119391
$724$	1.99426e9	0.195298
$725$	−8.56185e9	−0.834420
$726$	−1.44865e9	−0.140503
$727$	−1.11727e10	−1.07842	−0.539210	−	0.842171i	$-0.681277\pi$
$727$	−1.11727e10	−1.07842	−0.539210	+	0.842171i	$0.681277\pi$
$728$	2.88612e10	2.77239
$729$	3.87420e8	0.0370370
$730$	1.21287e10	1.15394
$731$	5.48419e8	0.0519280
$732$	3.31356e9	0.312252
$733$	−2.11647e9	−0.198494	−0.0992472	−	0.995063i	$-0.531643\pi$
$733$	−2.11647e9	−0.198494	−0.0992472	+	0.995063i	$0.531643\pi$
$734$	1.33298e10	1.24419
$735$	−2.78349e10	−2.58573
$736$	1.77993e9	0.164562
$737$	−1.10154e10	−1.01359
$738$	−2.18963e9	−0.200527
$739$	−1.72711e9	−0.157422	−0.0787110	−	0.996897i	$-0.525080\pi$
$739$	−1.72711e9	−0.157422	−0.0787110	+	0.996897i	$0.525080\pi$
$740$	−9.83950e9	−0.892611
$741$	−1.34880e10	−1.21782
$742$	1.76011e10	1.58171
$743$	1.26059e10	1.12749	0.563746	−	0.825948i	$-0.309360\pi$
$743$	1.26059e10	1.12749	0.563746	+	0.825948i	$0.309360\pi$
$744$	−4.51857e9	−0.402249
$745$	−1.85334e10	−1.64213
$746$	−9.14523e9	−0.806508
$747$	−3.40947e9	−0.299272
$748$	−2.72968e8	−0.0238483
$749$	3.09584e10	2.69211
$750$	4.40750e9	0.381486
$751$	−4.91964e9	−0.423832	−0.211916	−	0.977288i	$-0.567970\pi$
$751$	−4.91964e9	−0.423832	−0.211916	+	0.977288i	$0.567970\pi$
$752$	2.61487e9	0.224227
$753$	6.67666e9	0.569871
$754$	6.30315e9	0.535498
$755$	8.20458e9	0.693812
$756$	−1.91888e9	−0.161518
$757$	−1.19687e10	−1.00280	−0.501398	−	0.865217i	$-0.667181\pi$
$757$	−1.19687e10	−1.00280	−0.501398	+	0.865217i	$0.667181\pi$
$758$	−1.25840e9	−0.104949
$759$	1.66764e9	0.138438
$760$	3.34185e10	2.76146
$761$	−1.72178e9	−0.141622	−0.0708112	−	0.997490i	$-0.522559\pi$
$761$	−1.72178e9	−0.141622	−0.0708112	+	0.997490i	$0.522559\pi$
$762$	2.48880e8	0.0203773
$763$	1.34041e10	1.09245
$764$	−1.44659e9	−0.117359
$765$	3.17617e8	0.0256500
$766$	2.21687e9	0.178213
$767$	−1.80605e10	−1.44526
$768$	−7.37474e9	−0.587465
$769$	−9.97780e9	−0.791211	−0.395606	−	0.918420i	$-0.629465\pi$
$769$	−9.97780e9	−0.791211	−0.395606	+	0.918420i	$0.629465\pi$
$770$	−3.42442e10	−2.70315
$771$	1.57187e9	0.123517
$772$	−5.90384e9	−0.461821
$773$	1.29350e10	1.00725	0.503626	−	0.863922i	$-0.331999\pi$
$773$	1.29350e10	1.00725	0.503626	+	0.863922i	$0.331999\pi$
$774$	3.49660e9	0.271052
$775$	−1.29487e10	−0.999238
$776$	−4.47833e9	−0.344033
$777$	−1.91407e10	−1.46381
$778$	1.36045e10	1.03575
$779$	−1.68491e10	−1.27702
$780$	6.91809e9	0.521982
$781$	−1.20519e10	−0.905267
$782$	−1.01459e8	−0.00758696
$783$	−1.39316e9	−0.103713
$784$	−1.45620e10	−1.07923
$785$	3.64136e10	2.68671
$786$	4.48901e9	0.329740
$787$	−1.86675e10	−1.36513	−0.682565	−	0.730824i	$-0.739136\pi$
$787$	−1.86675e10	−1.36513	−0.682565	+	0.730824i	$0.739136\pi$
$788$	−2.74331e9	−0.199725
$789$	1.51638e10	1.09910
$790$	−2.49677e10	−1.80170
$791$	−3.14193e10	−2.25725
$792$	−5.78568e9	−0.413824
$793$	−2.32390e10	−1.65486
$794$	1.49787e10	1.06195
$795$	1.40256e10	0.990002
$796$	4.73537e9	0.332781
$797$	−1.49176e10	−1.04375	−0.521873	−	0.853023i	$-0.674767\pi$
$797$	−1.49176e10	−1.04375	−0.521873	+	0.853023i	$0.674767\pi$
$798$	1.95552e10	1.36224
$799$	4.05117e8	0.0280975
$800$	−1.76961e10	−1.22198
$801$	−1.45546e9	−0.100066
$802$	−1.59055e10	−1.08877
$803$	−1.61581e10	−1.10125
$804$	−3.22636e9	−0.218935
$805$	9.61078e9	0.649341
$806$	9.53268e9	0.641272
$807$	1.04547e10	0.700253
$808$	−9.11052e9	−0.607580
$809$	−2.05281e10	−1.36310	−0.681551	−	0.731771i	$-0.738694\pi$
$809$	−2.05281e10	−1.36310	−0.681551	+	0.731771i	$0.738694\pi$
$810$	2.02505e9	0.133887
$811$	−3.04612e9	−0.200528	−0.100264	−	0.994961i	$-0.531969\pi$
$811$	−3.04612e9	−0.200528	−0.100264	+	0.994961i	$0.531969\pi$
$812$	6.90025e9	0.452292
$813$	3.59731e9	0.234780
$814$	−1.73602e10	−1.12816
$815$	3.54463e10	2.29361
$816$	1.66164e8	0.0107058
$817$	2.69062e10	1.72614
$818$	−1.21694e10	−0.777379
$819$	1.34577e10	0.856006
$820$	8.64204e9	0.547353
$821$	−2.08127e10	−1.31258	−0.656291	−	0.754508i	$-0.727876\pi$
$821$	−2.08127e10	−1.31258	−0.656291	+	0.754508i	$0.727876\pi$
$822$	1.34071e7	0.000841942 0
$823$	−2.47998e10	−1.55078	−0.775388	−	0.631485i	$-0.782446\pi$
$823$	−2.47998e10	−1.55078	−0.775388	+	0.631485i	$0.782446\pi$
$824$	4.99961e9	0.311308
$825$	−1.65798e10	−1.02799
$826$	2.61845e10	1.61664
$827$	2.75563e10	1.69415	0.847075	−	0.531474i	$-0.178362\pi$
$827$	2.75563e10	1.69415	0.847075	+	0.531474i	$0.178362\pi$
$828$	4.88446e8	0.0299027
$829$	1.17130e10	0.714049	0.357024	−	0.934095i	$-0.383791\pi$
$829$	1.17130e10	0.714049	0.357024	+	0.934095i	$0.383791\pi$
$830$	−1.78214e10	−1.08185
$831$	−2.80839e9	−0.169768
$832$	2.14402e10	1.29061
$833$	−2.25607e9	−0.135237
$834$	9.86254e8	0.0588718
$835$	2.80489e10	1.66730
$836$	−1.33922e10	−0.792740
$837$	−2.10696e9	−0.124199
$838$	−6.07620e9	−0.356679
$839$	−1.70864e10	−0.998811	−0.499406	−	0.866368i	$-0.666448\pi$
$839$	−1.70864e10	−0.998811	−0.499406	+	0.866368i	$0.666448\pi$
$840$	−3.33434e10	−1.94103
$841$	−1.22401e10	−0.709577
$842$	−1.80754e10	−1.04351
$843$	6.33571e9	0.364249
$844$	−7.90790e9	−0.452754
$845$	−2.05207e10	−1.17002
$846$	2.58294e9	0.146662
$847$	1.11223e10	0.628929
$848$	7.33760e9	0.413208
$849$	1.33456e10	0.748447
$850$	1.00871e9	0.0563379
$851$	4.87221e9	0.271002
$852$	−3.52995e9	−0.195538
$853$	−2.01863e10	−1.11362	−0.556809	−	0.830641i	$-0.687974\pi$
$853$	−2.01863e10	−1.11362	−0.556809	+	0.830641i	$0.687974\pi$
$854$	3.36924e10	1.85110
$855$	1.55827e10	0.852633
$856$	2.73400e10	1.48984
$857$	2.54077e9	0.137890	0.0689450	−	0.997620i	$-0.478037\pi$
$857$	2.54077e9	0.137890	0.0689450	+	0.997620i	$0.478037\pi$
$858$	1.22059e10	0.659725
$859$	1.37655e10	0.740996	0.370498	−	0.928833i	$-0.379187\pi$
$859$	1.37655e10	0.740996	0.370498	+	0.928833i	$0.379187\pi$
$860$	−1.38004e10	−0.739856
$861$	1.68113e10	0.897614
$862$	9.96607e9	0.529967
$863$	1.42455e9	0.0754467	0.0377233	−	0.999288i	$-0.487989\pi$
$863$	1.42455e9	0.0754467	0.0377233	+	0.999288i	$0.487989\pi$
$864$	−2.87945e9	−0.151884
$865$	−2.26793e10	−1.19144
$866$	−1.41498e10	−0.740352
$867$	−1.10534e10	−0.576009
$868$	1.04357e10	0.541630
$869$	3.32623e10	1.71943
$870$	−7.28205e9	−0.374918
$871$	2.26274e10	1.16030
$872$	1.18375e10	0.604576
$873$	−2.08820e9	−0.106224
$874$	−4.97773e9	−0.252198
$875$	−3.38394e10	−1.70763
$876$	−4.73263e9	−0.237869
$877$	2.31617e9	0.115950	0.0579751	−	0.998318i	$-0.481536\pi$
$877$	2.31617e9	0.115950	0.0579751	+	0.998318i	$0.481536\pi$
$878$	−5.51813e9	−0.275145
$879$	9.13269e9	0.453563
$880$	−1.42758e10	−0.706174
$881$	1.52561e10	0.751673	0.375836	−	0.926686i	$-0.377356\pi$
$881$	1.52561e10	0.751673	0.375836	+	0.926686i	$0.377356\pi$
$882$	−1.43842e10	−0.705903
$883$	−1.60881e10	−0.786398	−0.393199	−	0.919453i	$-0.628632\pi$
$883$	−1.60881e10	−0.786398	−0.393199	+	0.919453i	$0.628632\pi$
$884$	5.60724e8	0.0273002
$885$	2.08653e10	1.01187
$886$	−1.98013e10	−0.956479
$887$	2.09613e10	1.00852	0.504261	−	0.863551i	$-0.331765\pi$
$887$	2.09613e10	1.00852	0.504261	+	0.863551i	$0.331765\pi$
$888$	−1.69035e10	−0.810088
$889$	−1.91082e9	−0.0912144
$890$	−7.60770e9	−0.361733
$891$	−2.69781e9	−0.127773
$892$	1.49990e9	0.0707596
$893$	1.98756e10	0.933987
$894$	−9.57746e9	−0.448300
$895$	2.41234e10	1.12476
$896$	2.06523e9	0.0959160
$897$	−3.42562e9	−0.158477
$898$	2.68006e10	1.23503
$899$	7.57660e9	0.347789
$900$	−4.85615e9	−0.222046
$901$	1.13680e9	0.0517782
$902$	1.52475e10	0.691792
$903$	−2.68458e10	−1.21330
$904$	−2.77471e10	−1.24919
$905$	1.61585e10	0.724654
$906$	4.23987e9	0.189410
$907$	−1.02188e10	−0.454752	−0.227376	−	0.973807i	$-0.573015\pi$
$907$	−1.02188e10	−0.454752	−0.227376	+	0.973807i	$0.573015\pi$
$908$	−1.74126e9	−0.0771906
$909$	−4.24815e9	−0.187597
$910$	7.03435e10	3.09442
$911$	−2.41894e10	−1.06001	−0.530006	−	0.847994i	$-0.677810\pi$
$911$	−2.41894e10	−1.06001	−0.530006	+	0.847994i	$0.677810\pi$
$912$	8.15223e9	0.355872
$913$	2.37419e10	1.03245
$914$	1.60633e10	0.695863
$915$	2.68481e10	1.15862
$916$	−3.42876e9	−0.147402
$917$	−3.44652e10	−1.47601
$918$	1.64134e8	0.00700244
$919$	−4.68936e9	−0.199301	−0.0996505	−	0.995022i	$-0.531772\pi$
$919$	−4.68936e9	−0.199301	−0.0996505	+	0.995022i	$0.531772\pi$
$920$	8.48749e9	0.359353
$921$	4.59643e9	0.193871
$922$	−1.26686e10	−0.532319
$923$	2.47567e10	1.03630
$924$	1.33621e10	0.557216
$925$	−4.84398e10	−2.01236
$926$	−2.70067e10	−1.11772
$927$	2.33127e9	0.0961199
$928$	1.03545e10	0.425314
$929$	1.87209e10	0.766077	0.383038	−	0.923732i	$-0.374878\pi$
$929$	1.87209e10	0.766077	0.383038	+	0.923732i	$0.374878\pi$
$930$	−1.10131e10	−0.448973
$931$	−1.10686e11	−4.49540
$932$	9.74567e9	0.394326
$933$	2.04486e10	0.824287
$934$	2.01815e10	0.810474
$935$	−2.21173e9	−0.0884893
$936$	1.18848e10	0.473724
$937$	−2.33873e10	−0.928732	−0.464366	−	0.885643i	$-0.653718\pi$
$937$	−2.33873e10	−0.928732	−0.464366	+	0.885643i	$0.653718\pi$
$938$	−3.28057e10	−1.29790
$939$	2.13995e9	0.0843479
$940$	−1.01944e10	−0.400325
$941$	1.44471e10	0.565220	0.282610	−	0.959235i	$-0.408800\pi$
$941$	1.44471e10	0.565220	0.282610	+	0.959235i	$0.408800\pi$
$942$	1.88174e10	0.733469
$943$	−4.27927e9	−0.166180
$944$	1.09159e10	0.422334
$945$	−1.55477e10	−0.599315
$946$	−2.43486e10	−0.935093
$947$	2.81864e10	1.07849	0.539243	−	0.842151i	$-0.318711\pi$
$947$	2.81864e10	1.07849	0.539243	+	0.842151i	$0.318711\pi$
$948$	9.74241e9	0.371396
$949$	3.31914e10	1.26065
$950$	4.94888e10	1.87273
$951$	−1.16717e9	−0.0440049
$952$	−2.70254e9	−0.101518
$953$	−1.35072e10	−0.505521	−0.252761	−	0.967529i	$-0.581339\pi$
$953$	−1.35072e10	−0.505521	−0.252761	+	0.967529i	$0.581339\pi$
$954$	7.24798e9	0.270270
$955$	−1.17210e10	−0.435464
$956$	1.63549e10	0.605404
$957$	9.70126e9	0.357797
$958$	−1.24954e10	−0.459167
$959$	−1.02935e8	−0.00376876
$960$	−2.47699e10	−0.903597
$961$	−1.60540e10	−0.583515
$962$	3.56609e10	1.29145
$963$	1.27484e10	0.460006
$964$	2.47457e9	0.0889671
$965$	−4.78358e10	−1.71359
$966$	4.96655e9	0.177270
$967$	−2.32638e10	−0.827347	−0.413674	−	0.910425i	$-0.635755\pi$
$967$	−2.32638e10	−0.827347	−0.413674	+	0.910425i	$0.635755\pi$
$968$	9.82232e9	0.348057
$969$	1.26301e9	0.0445937
$970$	−1.09151e10	−0.383994
$971$	−2.06909e9	−0.0725291	−0.0362645	−	0.999342i	$-0.511546\pi$
$971$	−2.06909e9	−0.0725291	−0.0362645	+	0.999342i	$0.511546\pi$
$972$	−7.90177e8	−0.0275989
$973$	−7.57214e9	−0.263526
$974$	2.86003e10	0.991776
$975$	3.40577e10	1.17679
$976$	1.40458e10	0.483584
$977$	−1.06684e10	−0.365991	−0.182995	−	0.983114i	$-0.558579\pi$
$977$	−1.06684e10	−0.365991	−0.182995	+	0.983114i	$0.558579\pi$
$978$	1.83175e10	0.626153
$979$	1.01351e10	0.345214
$980$	5.67716e10	1.92681
$981$	5.51969e9	0.186669
$982$	−1.74030e10	−0.586453
$983$	−5.76416e10	−1.93552	−0.967762	−	0.251867i	$-0.918955\pi$
$983$	−5.76416e10	−1.93552	−0.967762	+	0.251867i	$0.918955\pi$
$984$	1.48464e10	0.496750
$985$	−2.22277e10	−0.741084
$986$	−5.90224e8	−0.0196086
$987$	−1.98310e10	−0.656499
$988$	2.75099e10	0.907486
$989$	6.83353e9	0.224625
$990$	−1.41015e10	−0.461893
$991$	5.63203e10	1.83826	0.919131	−	0.393952i	$-0.128893\pi$
$991$	5.63203e10	1.83826	0.919131	+	0.393952i	$0.128893\pi$
$992$	1.56597e10	0.509324
$993$	−2.61731e9	−0.0848267
$994$	−3.58928e10	−1.15919
$995$	3.83684e10	1.23479
$996$	6.95391e9	0.223009
$997$	4.90237e10	1.56665	0.783327	−	0.621610i	$-0.213521\pi$
$997$	4.90237e10	1.56665	0.783327	+	0.621610i	$0.213521\pi$
$998$	−4.59372e9	−0.146288
$999$	−7.88196e9	−0.250124

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.b.1.3	✓	6
3.2	odd	2		207.8.a.c.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.b.1.3	✓	6	1.1	even	1	trivial
207.8.a.c.1.4		6	3.2	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q-8.53998 q^{2} +27.0000 q^{3} -55.0688 q^{4} -446.195 q^{5} -230.579 q^{6} +1770.31 q^{7} +1563.40 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.53998 q^{2} +27.0000 q^{3} -55.0688 q^{4} -446.195 q^{5} -230.579 q^{6} +1770.31 q^{7} +1563.40 q^{8} +729.000 q^{9} +3810.49 q^{10} -5076.40 q^{11} -1486.86 q^{12} +10427.8 q^{13} -15118.4 q^{14} -12047.3 q^{15} -6302.62 q^{16} -976.453 q^{17} -6225.64 q^{18} -47906.2 q^{19} +24571.4 q^{20} +47798.5 q^{21} +43352.3 q^{22} -12167.0 q^{23} +42211.9 q^{24} +120965. q^{25} -89053.1 q^{26} +19683.0 q^{27} -97489.1 q^{28} -70779.7 q^{29} +102883. q^{30} -107045. q^{31} -146291. q^{32} -137063. q^{33} +8338.88 q^{34} -789905. q^{35} -40145.2 q^{36} -400445. q^{37} +409117. q^{38} +281550. q^{39} -697582. q^{40} +351711. q^{41} -408198. q^{42} -561644. q^{43} +279551. q^{44} -325276. q^{45} +103906. q^{46} -414887. q^{47} -170171. q^{48} +2.31047e6 q^{49} -1.03304e6 q^{50} -26364.2 q^{51} -574246. q^{52} -1.16421e6 q^{53} -168092. q^{54} +2.26506e6 q^{55} +2.76772e6 q^{56} -1.29347e6 q^{57} +604457. q^{58} -1.73195e6 q^{59} +663428. q^{60} -2.22856e6 q^{61} +914161. q^{62} +1.29056e6 q^{63} +2.05606e6 q^{64} -4.65283e6 q^{65} +1.17051e6 q^{66} +2.16992e6 q^{67} +53772.1 q^{68} -328509. q^{69} +6.74577e6 q^{70} +2.37410e6 q^{71} +1.13972e6 q^{72} +3.18298e6 q^{73} +3.41979e6 q^{74} +3.26605e6 q^{75} +2.63814e6 q^{76} -8.98682e6 q^{77} -2.40443e6 q^{78} -6.55235e6 q^{79} +2.81220e6 q^{80} +531441. q^{81} -3.00361e6 q^{82} -4.67692e6 q^{83} -2.63221e6 q^{84} +435688. q^{85} +4.79643e6 q^{86} -1.91105e6 q^{87} -7.93646e6 q^{88} -1.99651e6 q^{89} +2.77785e6 q^{90} +1.84605e7 q^{91} +670022. q^{92} -2.89021e6 q^{93} +3.54312e6 q^{94} +2.13755e7 q^{95} -3.94987e6 q^{96} -2.86447e6 q^{97} -1.97314e7 q^{98} -3.70069e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9}+O(q^{10}) \) 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9	\( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9} - 13042 q^{10} - 14824 q^{11} + 4806 q^{12} - 756 q^{13} - 3926 q^{14} - 10044 q^{15} - 13022 q^{16} - 69484 q^{17} - 5832 q^{18} - 43864 q^{19} + 78886 q^{20} - 29808 q^{21} + 98204 q^{22} - 73002 q^{23} - 52812 q^{24} + 228018 q^{25} - 311956 q^{26} + 118098 q^{27} - 545442 q^{28} - 311100 q^{29} - 352134 q^{30} - 245248 q^{31} - 390156 q^{32} - 400248 q^{33} + 235834 q^{34} - 1331256 q^{35} + 129762 q^{36} - 630044 q^{37} + 80910 q^{38} - 20412 q^{39} - 2153982 q^{40} - 969204 q^{41} - 106002 q^{42} - 1770208 q^{43} - 1749140 q^{44} - 271188 q^{45} + 97336 q^{46} - 1400024 q^{47} - 351594 q^{48} + 1985598 q^{49} - 956660 q^{50} - 1876068 q^{51} + 3217272 q^{52} - 1573516 q^{53} - 157464 q^{54} - 431296 q^{55} + 7740702 q^{56} - 1184328 q^{57} + 5987188 q^{58} - 1410320 q^{59} + 2129922 q^{60} - 942172 q^{61} + 3334412 q^{62} - 804816 q^{63} + 1996866 q^{64} - 420944 q^{65} + 2651508 q^{66} - 452072 q^{67} - 9258254 q^{68} - 1971054 q^{69} + 21981136 q^{70} + 122928 q^{71} - 1425924 q^{72} + 16490716 q^{73} - 600104 q^{74} + 6156486 q^{75} + 7428658 q^{76} + 7239696 q^{77} - 8422812 q^{78} + 2458408 q^{79} + 19440230 q^{80} + 3188646 q^{81} + 20510784 q^{82} - 7566456 q^{83} - 14726934 q^{84} + 5817744 q^{85} - 669666 q^{86} - 8399700 q^{87} + 14775668 q^{88} - 20368036 q^{89} - 9507618 q^{90} + 8815576 q^{91} - 2165726 q^{92} - 6621696 q^{93} + 16952576 q^{94} + 5143832 q^{95} - 10534212 q^{96} + 12586972 q^{97} - 39164812 q^{98} - 10806696 q^{99}+O(q^{100}) \) 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9 - 13042 * q^10 - 14824 * q^11 + 4806 * q^12 - 756 * q^13 - 3926 * q^14 - 10044 * q^15 - 13022 * q^16 - 69484 * q^17 - 5832 * q^18 - 43864 * q^19 + 78886 * q^20 - 29808 * q^21 + 98204 * q^22 - 73002 * q^23 - 52812 * q^24 + 228018 * q^25 - 311956 * q^26 + 118098 * q^27 - 545442 * q^28 - 311100 * q^29 - 352134 * q^30 - 245248 * q^31 - 390156 * q^32 - 400248 * q^33 + 235834 * q^34 - 1331256 * q^35 + 129762 * q^36 - 630044 * q^37 + 80910 * q^38 - 20412 * q^39 - 2153982 * q^40 - 969204 * q^41 - 106002 * q^42 - 1770208 * q^43 - 1749140 * q^44 - 271188 * q^45 + 97336 * q^46 - 1400024 * q^47 - 351594 * q^48 + 1985598 * q^49 - 956660 * q^50 - 1876068 * q^51 + 3217272 * q^52 - 1573516 * q^53 - 157464 * q^54 - 431296 * q^55 + 7740702 * q^56 - 1184328 * q^57 + 5987188 * q^58 - 1410320 * q^59 + 2129922 * q^60 - 942172 * q^61 + 3334412 * q^62 - 804816 * q^63 + 1996866 * q^64 - 420944 * q^65 + 2651508 * q^66 - 452072 * q^67 - 9258254 * q^68 - 1971054 * q^69 + 21981136 * q^70 + 122928 * q^71 - 1425924 * q^72 + 16490716 * q^73 - 600104 * q^74 + 6156486 * q^75 + 7428658 * q^76 + 7239696 * q^77 - 8422812 * q^78 + 2458408 * q^79 + 19440230 * q^80 + 3188646 * q^81 + 20510784 * q^82 - 7566456 * q^83 - 14726934 * q^84 + 5817744 * q^85 - 669666 * q^86 - 8399700 * q^87 + 14775668 * q^88 - 20368036 * q^89 - 9507618 * q^90 + 8815576 * q^91 - 2165726 * q^92 - 6621696 * q^93 + 16952576 * q^94 + 5143832 * q^95 - 10534212 * q^96 + 12586972 * q^97 - 39164812 * q^98 - 10806696 * q^99

Introduction

L-functions

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