LMFDB - Embedded newform 69.8.a.d.1.5

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 757x^{6} - 1170x^{5} + 170343x^{4} + 424132x^{3} - 9973075x^{2} - 5161010x + 130545120 $ x^8 - 757x^6 - 1170x^5 + 170343x^4 + 424132x^3 - 9973075x^2 - 5161010x + 130545120
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-4.11469$ 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)

$f(q)$	$=$	$q+7.11469 q^{2} +27.0000 q^{3} -77.3812 q^{4} -244.007 q^{5} +192.097 q^{6} +549.357 q^{7} -1461.22 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+7.11469 q^{2} +27.0000 q^{3} -77.3812 q^{4} -244.007 q^{5} +192.097 q^{6} +549.357 q^{7} -1461.22 q^{8} +729.000 q^{9} -1736.03 q^{10} +6497.85 q^{11} -2089.29 q^{12} +6328.07 q^{13} +3908.51 q^{14} -6588.18 q^{15} -491.361 q^{16} +35620.9 q^{17} +5186.61 q^{18} +10650.8 q^{19} +18881.5 q^{20} +14832.7 q^{21} +46230.2 q^{22} +12167.0 q^{23} -39453.0 q^{24} -18585.7 q^{25} +45022.3 q^{26} +19683.0 q^{27} -42509.9 q^{28} -253060. q^{29} -46872.9 q^{30} +138444. q^{31} +183541. q^{32} +175442. q^{33} +253432. q^{34} -134047. q^{35} -56410.9 q^{36} +86831.8 q^{37} +75777.1 q^{38} +170858. q^{39} +356548. q^{40} +299642. q^{41} +105530. q^{42} +283844. q^{43} -502811. q^{44} -177881. q^{45} +86564.4 q^{46} +312235. q^{47} -13266.7 q^{48} -521749. q^{49} -132232. q^{50} +961765. q^{51} -489674. q^{52} +1.01632e6 q^{53} +140038. q^{54} -1.58552e6 q^{55} -802734. q^{56} +287572. q^{57} -1.80044e6 q^{58} +677210. q^{59} +509801. q^{60} +2.34144e6 q^{61} +984986. q^{62} +400482. q^{63} +1.36873e6 q^{64} -1.54409e6 q^{65} +1.24821e6 q^{66} -3.15388e6 q^{67} -2.75639e6 q^{68} +328509. q^{69} -953702. q^{70} -4.36034e6 q^{71} -1.06523e6 q^{72} +2.79812e6 q^{73} +617781. q^{74} -501814. q^{75} -824171. q^{76} +3.56964e6 q^{77} +1.21560e6 q^{78} -6.51090e6 q^{79} +119895. q^{80} +531441. q^{81} +2.13186e6 q^{82} -401787. q^{83} -1.14777e6 q^{84} -8.69174e6 q^{85} +2.01946e6 q^{86} -6.83262e6 q^{87} -9.49480e6 q^{88} +1.16894e7 q^{89} -1.26557e6 q^{90} +3.47637e6 q^{91} -941497. q^{92} +3.73799e6 q^{93} +2.22145e6 q^{94} -2.59887e6 q^{95} +4.95560e6 q^{96} -1.24976e7 q^{97} -3.71208e6 q^{98} +4.73693e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9}+O(q^{10}) $ 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9} + 11720 q^{10} + 6932 q^{11} + 15174 q^{12} + 12404 q^{13} + 30222 q^{14} + 10206 q^{15} + 27058 q^{16} + 24434 q^{17} + 17496 q^{18} - 14682 q^{19} - 3760 q^{20} + 3402 q^{21} + 36294 q^{22} + 97336 q^{23} + 113076 q^{24} + 144644 q^{25} + 325840 q^{26} + 157464 q^{27} - 21566 q^{28} + 255356 q^{29} + 316440 q^{30} + 450764 q^{31} + 647588 q^{32} + 187164 q^{33} + 191822 q^{34} + 1022616 q^{35} + 409698 q^{36} + 206240 q^{37} + 737372 q^{38} + 334908 q^{39} + 590028 q^{40} + 1053344 q^{41} + 815994 q^{42} + 1587806 q^{43} + 589366 q^{44} + 275562 q^{45} + 292008 q^{46} + 443336 q^{47} + 730566 q^{48} + 1944828 q^{49} - 1556112 q^{50} + 659718 q^{51} - 614236 q^{52} - 375530 q^{53} + 472392 q^{54} + 407792 q^{55} - 1316922 q^{56} - 396414 q^{57} - 1413384 q^{58} + 624008 q^{59} - 101520 q^{60} - 2005568 q^{61} - 3908272 q^{62} + 91854 q^{63} - 5082310 q^{64} + 646124 q^{65} + 979938 q^{66} - 2712286 q^{67} - 2289698 q^{68} + 2628072 q^{69} - 16499468 q^{70} - 6287176 q^{71} + 3053052 q^{72} - 10358312 q^{73} - 2000150 q^{74} + 3905388 q^{75} - 25107464 q^{76} - 2156840 q^{77} + 8797680 q^{78} - 8800574 q^{79} + 2384344 q^{80} + 4251528 q^{81} - 31799800 q^{82} + 384948 q^{83} - 582282 q^{84} - 17826684 q^{85} - 11563928 q^{86} + 6894612 q^{87} - 25202782 q^{88} - 3445530 q^{89} + 8543880 q^{90} - 16316740 q^{91} + 6837854 q^{92} + 12170628 q^{93} - 24237616 q^{94} + 26164288 q^{95} + 17484876 q^{96} - 28043764 q^{97} - 9998012 q^{98} + 5053428 q^{99}+O(q^{100}) \) 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9 + 11720 * q^10 + 6932 * q^11 + 15174 * q^12 + 12404 * q^13 + 30222 * q^14 + 10206 * q^15 + 27058 * q^16 + 24434 * q^17 + 17496 * q^18 - 14682 * q^19 - 3760 * q^20 + 3402 * q^21 + 36294 * q^22 + 97336 * q^23 + 113076 * q^24 + 144644 * q^25 + 325840 * q^26 + 157464 * q^27 - 21566 * q^28 + 255356 * q^29 + 316440 * q^30 + 450764 * q^31 + 647588 * q^32 + 187164 * q^33 + 191822 * q^34 + 1022616 * q^35 + 409698 * q^36 + 206240 * q^37 + 737372 * q^38 + 334908 * q^39 + 590028 * q^40 + 1053344 * q^41 + 815994 * q^42 + 1587806 * q^43 + 589366 * q^44 + 275562 * q^45 + 292008 * q^46 + 443336 * q^47 + 730566 * q^48 + 1944828 * q^49 - 1556112 * q^50 + 659718 * q^51 - 614236 * q^52 - 375530 * q^53 + 472392 * q^54 + 407792 * q^55 - 1316922 * q^56 - 396414 * q^57 - 1413384 * q^58 + 624008 * q^59 - 101520 * q^60 - 2005568 * q^61 - 3908272 * q^62 + 91854 * q^63 - 5082310 * q^64 + 646124 * q^65 + 979938 * q^66 - 2712286 * q^67 - 2289698 * q^68 + 2628072 * q^69 - 16499468 * q^70 - 6287176 * q^71 + 3053052 * q^72 - 10358312 * q^73 - 2000150 * q^74 + 3905388 * q^75 - 25107464 * q^76 - 2156840 * q^77 + 8797680 * q^78 - 8800574 * q^79 + 2384344 * q^80 + 4251528 * q^81 - 31799800 * q^82 + 384948 * q^83 - 582282 * q^84 - 17826684 * q^85 - 11563928 * q^86 + 6894612 * q^87 - 25202782 * q^88 - 3445530 * q^89 + 8543880 * q^90 - 16316740 * q^91 + 6837854 * q^92 + 12170628 * q^93 - 24237616 * q^94 + 26164288 * q^95 + 17484876 * q^96 - 28043764 * q^97 - 9998012 * q^98 + 5053428 * 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 $

$2$	7.11469	0.628856	0.314428	−	0.949281i	$-0.398187\pi$
$2$	7.11469	0.628856	0.314428	+	0.949281i	$0.398187\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−77.3812	−0.604540
$5$	−244.007	−0.872985	−0.436493	−	0.899708i	$-0.643779\pi$
$5$	−244.007	−0.872985	−0.436493	+	0.899708i	$0.643779\pi$
$6$	192.097	0.363070
$7$	549.357	0.605357	0.302679	−	0.953093i	$-0.402119\pi$
$7$	549.357	0.605357	0.302679	+	0.953093i	$0.402119\pi$
$8$	−1461.22	−1.00902
$9$	729.000	0.333333
$10$	−1736.03	−0.548982
$11$	6497.85	1.47196	0.735978	−	0.677005i	$-0.236723\pi$
$11$	6497.85	1.47196	0.735978	+	0.677005i	$0.236723\pi$
$12$	−2089.29	−0.349032
$13$	6328.07	0.798858	0.399429	−	0.916764i	$-0.369208\pi$
$13$	6328.07	0.798858	0.399429	+	0.916764i	$0.369208\pi$
$14$	3908.51	0.380683
$15$	−6588.18	−0.504018
$16$	−491.361	−0.0299903
$17$	35620.9	1.75847	0.879233	−	0.476392i	$-0.158056\pi$
$17$	35620.9	1.75847	0.879233	+	0.476392i	$0.158056\pi$
$18$	5186.61	0.209619
$19$	10650.8	0.356242	0.178121	−	0.984009i	$-0.442998\pi$
$19$	10650.8	0.356242	0.178121	+	0.984009i	$0.442998\pi$
$20$	18881.5	0.527755
$21$	14832.7	0.349503
$22$	46230.2	0.925648
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	−39453.0	−0.582561
$25$	−18585.7	−0.237897
$26$	45022.3	0.502367
$27$	19683.0	0.192450
$28$	−42509.9	−0.365963
$29$	−253060.	−1.92677	−0.963386	−	0.268119i	$-0.913598\pi$
$29$	−253060.	−1.92677	−0.963386	+	0.268119i	$0.913598\pi$
$30$	−46872.9	−0.316955
$31$	138444.	0.834657	0.417329	−	0.908756i	$-0.362966\pi$
$31$	138444.	0.834657	0.417329	+	0.908756i	$0.362966\pi$
$32$	183541.	0.990165
$33$	175442.	0.849834
$34$	253432.	1.10582
$35$	−134047.	−0.528468
$36$	−56410.9	−0.201513
$37$	86831.8	0.281821	0.140910	−	0.990022i	$-0.454997\pi$
$37$	86831.8	0.281821	0.140910	+	0.990022i	$0.454997\pi$
$38$	75777.1	0.224025
$39$	170858.	0.461221
$40$	356548.	0.880863
$41$	299642.	0.678984	0.339492	−	0.940609i	$-0.389745\pi$
$41$	299642.	0.678984	0.339492	+	0.940609i	$0.389745\pi$
$42$	105530.	0.219787
$43$	283844.	0.544428	0.272214	−	0.962237i	$-0.412244\pi$
$43$	283844.	0.544428	0.272214	+	0.962237i	$0.412244\pi$
$44$	−502811.	−0.889857
$45$	−177881.	−0.290995
$46$	86564.4	0.131125
$47$	312235.	0.438671	0.219335	−	0.975650i	$-0.429611\pi$
$47$	312235.	0.438671	0.219335	+	0.975650i	$0.429611\pi$
$48$	−13266.7	−0.0173149
$49$	−521749.	−0.633542
$50$	−132232.	−0.149603
$51$	961765.	1.01525
$52$	−489674.	−0.482942
$53$	1.01632e6	0.937698	0.468849	−	0.883278i	$-0.344669\pi$
$53$	1.01632e6	0.937698	0.468849	+	0.883278i	$0.344669\pi$
$54$	140038.	0.121023
$55$	−1.58552e6	−1.28500
$56$	−802734.	−0.610821
$57$	287572.	0.205676
$58$	−1.80044e6	−1.21166
$59$	677210.	0.429281	0.214640	−	0.976693i	$-0.431142\pi$
$59$	677210.	0.429281	0.214640	+	0.976693i	$0.431142\pi$
$60$	509801.	0.304699
$61$	2.34144e6	1.32077	0.660386	−	0.750926i	$-0.270392\pi$
$61$	2.34144e6	1.32077	0.660386	+	0.750926i	$0.270392\pi$
$62$	984986.	0.524879
$63$	400482.	0.201786
$64$	1.36873e6	0.652661
$65$	−1.54409e6	−0.697391
$66$	1.24821e6	0.534423
$67$	−3.15388e6	−1.28110	−0.640551	−	0.767916i	$-0.721294\pi$
$67$	−3.15388e6	−1.28110	−0.640551	+	0.767916i	$0.721294\pi$
$68$	−2.75639e6	−1.06306
$69$	328509.	0.120386
$70$	−953702.	−0.332330
$71$	−4.36034e6	−1.44583	−0.722914	−	0.690939i	$-0.757198\pi$
$71$	−4.36034e6	−1.44583	−0.722914	+	0.690939i	$0.757198\pi$
$72$	−1.06523e6	−0.336341
$73$	2.79812e6	0.841854	0.420927	−	0.907094i	$-0.361705\pi$
$73$	2.79812e6	0.841854	0.420927	+	0.907094i	$0.361705\pi$
$74$	617781.	0.177225
$75$	−501814.	−0.137350
$76$	−824171.	−0.215362
$77$	3.56964e6	0.891060
$78$	1.21560e6	0.290041
$79$	−6.51090e6	−1.48575	−0.742875	−	0.669430i	$-0.766539\pi$
$79$	−6.51090e6	−1.48575	−0.742875	+	0.669430i	$0.766539\pi$
$80$	119895.	0.0261811
$81$	531441.	0.111111
$82$	2.13186e6	0.426983
$83$	−401787.	−0.0771298	−0.0385649	−	0.999256i	$-0.512279\pi$
$83$	−401787.	−0.0771298	−0.0385649	+	0.999256i	$0.512279\pi$
$84$	−1.14777e6	−0.211289
$85$	−8.69174e6	−1.53511
$86$	2.01946e6	0.342367
$87$	−6.83262e6	−1.11242
$88$	−9.49480e6	−1.48524
$89$	1.16894e7	1.75762	0.878811	−	0.477170i	$-0.158338\pi$
$89$	1.16894e7	1.75762	0.878811	+	0.477170i	$0.158338\pi$
$90$	−1.26557e6	−0.182994
$91$	3.47637e6	0.483595
$92$	−941497.	−0.126055
$93$	3.73799e6	0.481890
$94$	2.22145e6	0.275861
$95$	−2.59887e6	−0.310994
$96$	4.95560e6	0.571672
$97$	−1.24976e7	−1.39036	−0.695179	−	0.718836i	$-0.744675\pi$
$97$	−1.24976e7	−1.39036	−0.695179	+	0.718836i	$0.744675\pi$
$98$	−3.71208e6	−0.398407
$99$	4.73693e6	0.490652
$100$	1.43818e6	0.143818
$101$	1.27743e6	0.123371	0.0616856	−	0.998096i	$-0.480352\pi$
$101$	1.27743e6	0.123371	0.0616856	+	0.998096i	$0.480352\pi$
$102$	6.84266e6	0.638446
$103$	−9.61010e6	−0.866558	−0.433279	−	0.901260i	$-0.642643\pi$
$103$	−9.61010e6	−0.866558	−0.433279	+	0.901260i	$0.642643\pi$
$104$	−9.24673e6	−0.806067
$105$	−3.61927e6	−0.305111
$106$	7.23077e6	0.589677
$107$	1.26586e7	0.998948	0.499474	−	0.866329i	$-0.333527\pi$
$107$	1.26586e7	0.998948	0.499474	+	0.866329i	$0.333527\pi$
$108$	−1.52309e6	−0.116344
$109$	−8.26222e6	−0.611088	−0.305544	−	0.952178i	$-0.598838\pi$
$109$	−8.26222e6	−0.611088	−0.305544	+	0.952178i	$0.598838\pi$
$110$	−1.12805e7	−0.808077
$111$	2.34446e6	0.162709
$112$	−269933.	−0.0181549
$113$	−3.78609e6	−0.246841	−0.123420	−	0.992354i	$-0.539386\pi$
$113$	−3.78609e6	−0.246841	−0.123420	+	0.992354i	$0.539386\pi$
$114$	2.04598e6	0.129341
$115$	−2.96883e6	−0.182030
$116$	1.95821e7	1.16481
$117$	4.61316e6	0.266286
$118$	4.81814e6	0.269956
$119$	1.95686e7	1.06450
$120$	9.62681e6	0.508567
$121$	2.27348e7	1.16666
$122$	1.66586e7	0.830576
$123$	8.09035e6	0.392012
$124$	−1.07130e7	−0.504584
$125$	2.35981e7	1.08067
$126$	2.84930e6	0.126894
$127$	2.79370e7	1.21023	0.605114	−	0.796139i	$-0.293128\pi$
$127$	2.79370e7	1.21023	0.605114	+	0.796139i	$0.293128\pi$
$128$	−1.37551e7	−0.579735
$129$	7.66379e6	0.314326
$130$	−1.09857e7	−0.438559
$131$	2.68577e7	1.04380	0.521902	−	0.853006i	$-0.325223\pi$
$131$	2.68577e7	1.04380	0.521902	+	0.853006i	$0.325223\pi$
$132$	−1.35759e7	−0.513759
$133$	5.85110e6	0.215653
$134$	−2.24389e7	−0.805628
$135$	−4.80278e6	−0.168006
$136$	−5.20501e7	−1.77433
$137$	5.57360e7	1.85189	0.925943	−	0.377664i	$-0.123273\pi$
$137$	5.57360e7	1.85189	0.925943	+	0.377664i	$0.123273\pi$
$138$	2.33724e6	0.0757053
$139$	−4.66115e7	−1.47211	−0.736057	−	0.676919i	$-0.763315\pi$
$139$	−4.66115e7	−1.47211	−0.736057	+	0.676919i	$0.763315\pi$
$140$	1.03727e7	0.319480
$141$	8.43034e6	0.253267
$142$	−3.10225e7	−0.909217
$143$	4.11188e7	1.17588
$144$	−358202.	−0.00999677
$145$	6.17483e7	1.68204
$146$	1.99078e7	0.529405
$147$	−1.40872e7	−0.365776
$148$	−6.71915e6	−0.170372
$149$	−1.38043e7	−0.341870	−0.170935	−	0.985282i	$-0.554679\pi$
$149$	−1.38043e7	−0.341870	−0.170935	+	0.985282i	$0.554679\pi$
$150$	−3.57025e6	−0.0863733
$151$	4.84217e6	0.114451	0.0572257	−	0.998361i	$-0.481775\pi$
$151$	4.84217e6	0.114451	0.0572257	+	0.998361i	$0.481775\pi$
$152$	−1.55632e7	−0.359456
$153$	2.59676e7	0.586155
$154$	2.53969e7	0.560348
$155$	−3.37813e7	−0.728643
$156$	−1.32212e7	−0.278827
$157$	−6.95253e7	−1.43382	−0.716909	−	0.697167i	$-0.754444\pi$
$157$	−6.95253e7	−1.43382	−0.716909	+	0.697167i	$0.754444\pi$
$158$	−4.63230e7	−0.934323
$159$	2.74405e7	0.541380
$160$	−4.47852e7	−0.864399
$161$	6.68403e6	0.126226
$162$	3.78104e6	0.0698729
$163$	1.11737e7	0.202087	0.101044	−	0.994882i	$-0.467782\pi$
$163$	1.11737e7	0.202087	0.101044	+	0.994882i	$0.467782\pi$
$164$	−2.31867e7	−0.410474
$165$	−4.28090e7	−0.741893
$166$	−2.85859e6	−0.0485035
$167$	−1.09499e8	−1.81930	−0.909648	−	0.415379i	$-0.863649\pi$
$167$	−1.09499e8	−1.81930	−0.909648	+	0.415379i	$0.863649\pi$
$168$	−2.16738e7	−0.352657
$169$	−2.27040e7	−0.361826
$170$	−6.18391e7	−0.965365
$171$	7.76443e6	0.118747
$172$	−2.19642e7	−0.329129
$173$	4.92182e7	0.722710	0.361355	−	0.932428i	$-0.382314\pi$
$173$	4.92182e7	0.722710	0.361355	+	0.932428i	$0.382314\pi$
$174$	−4.86119e7	−0.699553
$175$	−1.02102e7	−0.144013
$176$	−3.19279e6	−0.0441444
$177$	1.82847e7	0.247845
$178$	8.31661e7	1.10529
$179$	−2.98184e7	−0.388596	−0.194298	−	0.980943i	$-0.562243\pi$
$179$	−2.98184e7	−0.388596	−0.194298	+	0.980943i	$0.562243\pi$
$180$	1.37646e7	0.175918
$181$	−4.59690e7	−0.576222	−0.288111	−	0.957597i	$-0.593027\pi$
$181$	−4.59690e7	−0.576222	−0.288111	+	0.957597i	$0.593027\pi$
$182$	2.47333e7	0.304111
$183$	6.32188e7	0.762549
$184$	−1.77787e7	−0.210396
$185$	−2.11875e7	−0.246025
$186$	2.65946e7	0.303039
$187$	2.31459e8	2.58838
$188$	−2.41611e7	−0.265194
$189$	1.08130e7	0.116501
$190$	−1.84901e7	−0.195570
$191$	−1.04506e8	−1.08524	−0.542621	−	0.839978i	$-0.682568\pi$
$191$	−1.04506e8	−1.08524	−0.542621	+	0.839978i	$0.682568\pi$
$192$	3.69557e7	0.376814
$193$	1.63523e8	1.63730	0.818652	−	0.574289i	$-0.194722\pi$
$193$	1.63523e8	1.63730	0.818652	+	0.574289i	$0.194722\pi$
$194$	−8.89168e7	−0.874335
$195$	−4.16905e7	−0.402639
$196$	4.03736e7	0.383002
$197$	1.04340e8	0.972342	0.486171	−	0.873864i	$-0.338393\pi$
$197$	1.04340e8	0.972342	0.486171	+	0.873864i	$0.338393\pi$
$198$	3.37018e7	0.308549
$199$	−6.79329e7	−0.611075	−0.305538	−	0.952180i	$-0.598836\pi$
$199$	−6.79329e7	−0.611075	−0.305538	+	0.952180i	$0.598836\pi$
$200$	2.71579e7	0.240044
$201$	−8.51548e7	−0.739644
$202$	9.08855e6	0.0775827
$203$	−1.39020e8	−1.16639
$204$	−7.44225e7	−0.613760
$205$	−7.31148e7	−0.592743
$206$	−6.83729e7	−0.544940
$207$	8.86974e6	0.0695048
$208$	−3.10937e6	−0.0239580
$209$	6.92072e7	0.524372
$210$	−2.57500e7	−0.191871
$211$	−1.07654e8	−0.788935	−0.394468	−	0.918910i	$-0.629071\pi$
$211$	−1.07654e8	−0.788935	−0.394468	+	0.918910i	$0.629071\pi$
$212$	−7.86437e7	−0.566877
$213$	−1.17729e8	−0.834749
$214$	9.00621e7	0.628194
$215$	−6.92599e7	−0.475277
$216$	−2.87613e7	−0.194187
$217$	7.60553e7	0.505266
$218$	−5.87831e7	−0.384286
$219$	7.55493e7	0.486045
$220$	1.22689e8	0.776832
$221$	2.25412e8	1.40476
$222$	1.66801e7	0.102321
$223$	1.24220e8	0.750108	0.375054	−	0.927003i	$-0.377624\pi$
$223$	1.24220e8	0.750108	0.375054	+	0.927003i	$0.377624\pi$
$224$	1.00829e8	0.599404
$225$	−1.35490e7	−0.0792990
$226$	−2.69369e7	−0.155227
$227$	2.29102e8	1.29998	0.649992	−	0.759941i	$-0.274772\pi$
$227$	2.29102e8	1.29998	0.649992	+	0.759941i	$0.274772\pi$
$228$	−2.22526e7	−0.124340
$229$	−4.81263e7	−0.264824	−0.132412	−	0.991195i	$-0.542272\pi$
$229$	−4.81263e7	−0.264824	−0.132412	+	0.991195i	$0.542272\pi$
$230$	−2.11223e7	−0.114471
$231$	9.63803e7	0.514454
$232$	3.69777e8	1.94416
$233$	−3.32165e8	−1.72031	−0.860157	−	0.510030i	$-0.829634\pi$
$233$	−3.32165e8	−1.72031	−0.860157	+	0.510030i	$0.829634\pi$
$234$	3.28212e7	0.167456
$235$	−7.61874e7	−0.382953
$236$	−5.24033e7	−0.259518
$237$	−1.75794e8	−0.857799
$238$	1.39225e8	0.669417
$239$	1.03958e8	0.492568	0.246284	−	0.969198i	$-0.420790\pi$
$239$	1.03958e8	0.492568	0.246284	+	0.969198i	$0.420790\pi$
$240$	3.23718e6	0.0151157
$241$	−1.94440e8	−0.894802	−0.447401	−	0.894333i	$-0.647650\pi$
$241$	−1.94440e8	−0.894802	−0.447401	+	0.894333i	$0.647650\pi$
$242$	1.61751e8	0.733658
$243$	1.43489e7	0.0641500
$244$	−1.81183e8	−0.798461
$245$	1.27310e8	0.553073
$246$	5.75603e7	0.246519
$247$	6.73990e7	0.284586
$248$	−2.02298e8	−0.842190
$249$	−1.08482e7	−0.0445309
$250$	1.67893e8	0.679583
$251$	1.51886e8	0.606261	0.303130	−	0.952949i	$-0.401968\pi$
$251$	1.51886e8	0.606261	0.303130	+	0.952949i	$0.401968\pi$
$252$	−3.09897e7	−0.121988
$253$	7.90593e7	0.306924
$254$	1.98763e8	0.761059
$255$	−2.34677e8	−0.886299
$256$	−2.73061e8	−1.01723
$257$	1.66995e8	0.613675	0.306837	−	0.951762i	$-0.400729\pi$
$257$	1.66995e8	0.613675	0.306837	+	0.951762i	$0.400729\pi$
$258$	5.45255e7	0.197665
$259$	4.77017e7	0.170602
$260$	1.19484e8	0.421601
$261$	−1.84481e8	−0.642257
$262$	1.91084e8	0.656402
$263$	−2.50981e7	−0.0850737	−0.0425369	−	0.999095i	$-0.513544\pi$
$263$	−2.50981e7	−0.0850737	−0.0425369	+	0.999095i	$0.513544\pi$
$264$	−2.56360e8	−0.857504
$265$	−2.47988e8	−0.818597
$266$	4.16287e7	0.135615
$267$	3.15612e8	1.01476
$268$	2.44051e8	0.774477
$269$	1.92309e7	0.0602375	0.0301188	−	0.999546i	$-0.490411\pi$
$269$	1.92309e7	0.0602375	0.0301188	+	0.999546i	$0.490411\pi$
$270$	−3.41703e7	−0.105652
$271$	8.71110e7	0.265877	0.132938	−	0.991124i	$-0.457559\pi$
$271$	8.71110e7	0.265877	0.132938	+	0.991124i	$0.457559\pi$
$272$	−1.75027e7	−0.0527369
$273$	9.38621e7	0.279204
$274$	3.96545e8	1.16457
$275$	−1.20767e8	−0.350174
$276$	−2.54204e7	−0.0727781
$277$	−5.71631e8	−1.61598	−0.807991	−	0.589195i	$-0.799445\pi$
$277$	−5.71631e8	−1.61598	−0.807991	+	0.589195i	$0.799445\pi$
$278$	−3.31627e8	−0.925748
$279$	1.00926e8	0.278219
$280$	1.95873e8	0.533237
$281$	−6.95246e8	−1.86925	−0.934623	−	0.355639i	$-0.884263\pi$
$281$	−6.95246e8	−1.86925	−0.934623	+	0.355639i	$0.884263\pi$
$282$	5.99792e7	0.159268
$283$	−2.27178e8	−0.595817	−0.297909	−	0.954594i	$-0.596289\pi$
$283$	−2.27178e8	−0.595817	−0.297909	+	0.954594i	$0.596289\pi$
$284$	3.37408e8	0.874061
$285$	−7.01694e7	−0.179552
$286$	2.92548e8	0.739462
$287$	1.64611e8	0.411028
$288$	1.33801e8	0.330055
$289$	8.58511e8	2.09220
$290$	4.39320e8	1.05776
$291$	−3.37436e8	−0.802724
$292$	−2.16522e8	−0.508935
$293$	−8.01165e7	−0.186074	−0.0930369	−	0.995663i	$-0.529657\pi$
$293$	−8.01165e7	−0.186074	−0.0930369	+	0.995663i	$0.529657\pi$
$294$	−1.00226e8	−0.230020
$295$	−1.65244e8	−0.374756
$296$	−1.26881e8	−0.284364
$297$	1.27897e8	0.283278
$298$	−9.82131e7	−0.214987
$299$	7.69937e7	0.166573
$300$	3.88310e7	0.0830336
$301$	1.55932e8	0.329573
$302$	3.44506e7	0.0719734
$303$	3.44907e7	0.0712284
$304$	−5.23339e6	−0.0106838
$305$	−5.71327e8	−1.15302
$306$	1.84752e8	0.368607
$307$	5.88460e8	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$307$	5.88460e8	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$308$	−2.76223e8	−0.538682
$309$	−2.59473e8	−0.500307
$310$	−2.40343e8	−0.458212
$311$	1.30155e8	0.245358	0.122679	−	0.992446i	$-0.460851\pi$
$311$	1.30155e8	0.245358	0.122679	+	0.992446i	$0.460851\pi$
$312$	−2.49662e8	−0.465383
$313$	−3.53675e8	−0.651927	−0.325964	−	0.945382i	$-0.605689\pi$
$313$	−3.53675e8	−0.651927	−0.325964	+	0.945382i	$0.605689\pi$
$314$	−4.94651e8	−0.901665
$315$	−9.77202e7	−0.176156
$316$	5.03821e8	0.898197
$317$	3.97105e7	0.0700161	0.0350080	−	0.999387i	$-0.488854\pi$
$317$	3.97105e7	0.0700161	0.0350080	+	0.999387i	$0.488854\pi$
$318$	1.95231e8	0.340450
$319$	−1.64434e9	−2.83612
$320$	−3.33979e8	−0.569763
$321$	3.41783e8	0.576743
$322$	4.75548e7	0.0793778
$323$	3.79391e8	0.626438
$324$	−4.11235e7	−0.0671712
$325$	−1.17612e8	−0.190046
$326$	7.94971e7	0.127084
$327$	−2.23080e8	−0.352812
$328$	−4.37845e8	−0.685112
$329$	1.71528e8	0.265553
$330$	−3.04573e8	−0.466544
$331$	−1.13474e9	−1.71988	−0.859938	−	0.510398i	$-0.829498\pi$
$331$	−1.13474e9	−1.71988	−0.859938	+	0.510398i	$0.829498\pi$
$332$	3.10907e7	0.0466281
$333$	6.33004e7	0.0939402
$334$	−7.79053e8	−1.14408
$335$	7.69568e8	1.11838
$336$	−7.28819e6	−0.0104817
$337$	−3.16562e8	−0.450561	−0.225281	−	0.974294i	$-0.572330\pi$
$337$	−3.16562e8	−0.450561	−0.225281	+	0.974294i	$0.572330\pi$
$338$	−1.61532e8	−0.227536
$339$	−1.02225e8	−0.142514
$340$	6.72577e8	0.928039
$341$	8.99588e8	1.22858
$342$	5.52415e7	0.0746748
$343$	−7.39046e8	−0.988877
$344$	−4.14760e8	−0.549341
$345$	−8.01584e7	−0.105095
$346$	3.50172e8	0.454480
$347$	−8.24599e8	−1.05947	−0.529736	−	0.848162i	$-0.677709\pi$
$347$	−8.24599e8	−1.05947	−0.529736	+	0.848162i	$0.677709\pi$
$348$	5.28716e8	0.672504
$349$	−9.77269e8	−1.23062	−0.615312	−	0.788284i	$-0.710970\pi$
$349$	−9.77269e8	−1.23062	−0.615312	+	0.788284i	$0.710970\pi$
$350$	−7.26424e7	−0.0905632
$351$	1.24555e8	0.153740
$352$	1.19262e9	1.45748
$353$	4.67249e8	0.565376	0.282688	−	0.959212i	$-0.408774\pi$
$353$	4.67249e8	0.565376	0.282688	+	0.959212i	$0.408774\pi$
$354$	1.30090e8	0.155859
$355$	1.06395e9	1.26219
$356$	−9.04536e8	−1.06255
$357$	5.28353e8	0.614590
$358$	−2.12149e8	−0.244371
$359$	−1.08943e8	−0.124271	−0.0621356	−	0.998068i	$-0.519791\pi$
$359$	−1.08943e8	−0.124271	−0.0621356	+	0.998068i	$0.519791\pi$
$360$	2.59924e8	0.293621
$361$	−7.80432e8	−0.873092
$362$	−3.27055e8	−0.362361
$363$	6.13840e8	0.673569
$364$	−2.69006e8	−0.292353
$365$	−6.82761e8	−0.734926
$366$	4.49782e8	0.479533
$367$	2.72618e8	0.287888	0.143944	−	0.989586i	$-0.454022\pi$
$367$	2.72618e8	0.287888	0.143944	+	0.989586i	$0.454022\pi$
$368$	−5.97839e6	−0.00625341
$369$	2.18439e8	0.226328
$370$	−1.50743e8	−0.154714
$371$	5.58321e8	0.567643
$372$	−2.89250e8	−0.291322
$373$	8.25069e8	0.823208	0.411604	−	0.911363i	$-0.364969\pi$
$373$	8.25069e8	0.823208	0.411604	+	0.911363i	$0.364969\pi$
$374$	1.64676e9	1.62772
$375$	6.37148e8	0.623923
$376$	−4.56245e8	−0.442629
$377$	−1.60138e9	−1.53922
$378$	7.69312e7	0.0732624
$379$	−5.98480e8	−0.564693	−0.282347	−	0.959312i	$-0.591113\pi$
$379$	−5.98480e8	−0.564693	−0.282347	+	0.959312i	$0.591113\pi$
$380$	2.01103e8	0.188008
$381$	7.54300e8	0.698725
$382$	−7.43531e8	−0.682460
$383$	1.50205e9	1.36612	0.683062	−	0.730361i	$-0.260648\pi$
$383$	1.50205e9	1.36612	0.683062	+	0.730361i	$0.260648\pi$
$384$	−3.71388e8	−0.334710
$385$	−8.71016e8	−0.777882
$386$	1.16342e9	1.02963
$387$	2.06922e8	0.181476
$388$	9.67082e8	0.840528
$389$	4.18026e8	0.360064	0.180032	−	0.983661i	$-0.442380\pi$
$389$	4.18026e8	0.360064	0.180032	+	0.983661i	$0.442380\pi$
$390$	−2.96615e8	−0.253202
$391$	4.33400e8	0.366665
$392$	7.62392e8	0.639260
$393$	7.25157e8	0.602640
$394$	7.42347e8	0.611463
$395$	1.58870e9	1.29704
$396$	−3.66549e8	−0.296619
$397$	−4.57774e8	−0.367184	−0.183592	−	0.983002i	$-0.558773\pi$
$397$	−4.57774e8	−0.367184	−0.183592	+	0.983002i	$0.558773\pi$
$398$	−4.83322e8	−0.384278
$399$	1.57980e8	0.124508
$400$	9.13229e6	0.00713460
$401$	4.25900e8	0.329839	0.164920	−	0.986307i	$-0.447264\pi$
$401$	4.25900e8	0.329839	0.164920	+	0.986307i	$0.447264\pi$
$402$	−6.05850e8	−0.465129
$403$	8.76084e8	0.666773
$404$	−9.88494e7	−0.0745829
$405$	−1.29675e8	−0.0969983
$406$	−9.89087e8	−0.733488
$407$	5.64220e8	0.414828
$408$	−1.40535e9	−1.02441
$409$	−1.12075e8	−0.0809982	−0.0404991	−	0.999180i	$-0.512895\pi$
$409$	−1.12075e8	−0.0809982	−0.0404991	+	0.999180i	$0.512895\pi$
$410$	−5.20189e8	−0.372750
$411$	1.50487e9	1.06919
$412$	7.43641e8	0.523869
$413$	3.72031e8	0.259868
$414$	6.31055e7	0.0437085
$415$	9.80387e7	0.0673332
$416$	1.16146e9	0.791001
$417$	−1.25851e9	−0.849926
$418$	4.92388e8	0.329754
$419$	−1.93151e9	−1.28277	−0.641384	−	0.767220i	$-0.721639\pi$
$419$	−1.93151e9	−1.28277	−0.641384	+	0.767220i	$0.721639\pi$
$420$	2.80063e8	0.184452
$421$	6.61902e7	0.0432321	0.0216161	−	0.999766i	$-0.493119\pi$
$421$	6.61902e7	0.0432321	0.0216161	+	0.999766i	$0.493119\pi$
$422$	−7.65925e8	−0.496126
$423$	2.27619e8	0.146224
$424$	−1.48506e9	−0.946160
$425$	−6.62040e8	−0.418334
$426$	−8.37607e8	−0.524936
$427$	1.28629e9	0.799540
$428$	−9.79539e8	−0.603905
$429$	1.11021e9	0.678897
$430$	−4.92763e8	−0.298881
$431$	−2.55963e9	−1.53995	−0.769975	−	0.638075i	$-0.779731\pi$
$431$	−2.55963e9	−1.53995	−0.769975	+	0.638075i	$0.779731\pi$
$432$	−9.67146e6	−0.00577164
$433$	1.29063e9	0.764001	0.382000	−	0.924162i	$-0.375235\pi$
$433$	1.29063e9	0.764001	0.382000	+	0.924162i	$0.375235\pi$
$434$	5.41110e8	0.317739
$435$	1.66720e9	0.971128
$436$	6.39340e8	0.369428
$437$	1.29588e8	0.0742815
$438$	5.37510e8	0.305652
$439$	−1.48653e8	−0.0838588	−0.0419294	−	0.999121i	$-0.513350\pi$
$439$	−1.48653e8	−0.0838588	−0.0419294	+	0.999121i	$0.513350\pi$
$440$	2.31680e9	1.29659
$441$	−3.80355e8	−0.211181
$442$	1.60373e9	0.883394
$443$	6.67331e8	0.364693	0.182347	−	0.983234i	$-0.441631\pi$
$443$	6.67331e8	0.364693	0.182347	+	0.983234i	$0.441631\pi$
$444$	−1.81417e8	−0.0983643
$445$	−2.85228e9	−1.53438
$446$	8.83785e8	0.471710
$447$	−3.72715e8	−0.197379
$448$	7.51922e8	0.395093
$449$	−1.40061e9	−0.730223	−0.365112	−	0.930964i	$-0.618969\pi$
$449$	−1.40061e9	−0.730223	−0.365112	+	0.930964i	$0.618969\pi$
$450$	−9.63968e7	−0.0498676
$451$	1.94703e9	0.999435
$452$	2.92972e8	0.149225
$453$	1.30739e8	0.0660785
$454$	1.62999e9	0.817502
$455$	−8.48259e8	−0.422171
$456$	−4.20206e8	−0.207532
$457$	−3.43054e9	−1.68134	−0.840671	−	0.541547i	$-0.817839\pi$
$457$	−3.43054e9	−1.68134	−0.840671	+	0.541547i	$0.817839\pi$
$458$	−3.42403e8	−0.166536
$459$	7.01127e8	0.338417
$460$	2.29732e8	0.110044
$461$	1.46443e9	0.696171	0.348085	−	0.937463i	$-0.386832\pi$
$461$	1.46443e9	0.696171	0.348085	+	0.937463i	$0.386832\pi$
$462$	6.85716e8	0.323517
$463$	−4.82051e8	−0.225714	−0.112857	−	0.993611i	$-0.536000\pi$
$463$	−4.82051e8	−0.225714	−0.112857	+	0.993611i	$0.536000\pi$
$464$	1.24344e8	0.0577845
$465$	−9.12094e8	−0.420682
$466$	−2.36325e9	−1.08183
$467$	−4.21545e8	−0.191529	−0.0957645	−	0.995404i	$-0.530530\pi$
$467$	−4.21545e8	−0.191529	−0.0957645	+	0.995404i	$0.530530\pi$
$468$	−3.56972e8	−0.160981
$469$	−1.73261e9	−0.775524
$470$	−5.42050e8	−0.240822
$471$	−1.87718e9	−0.827815
$472$	−9.89556e8	−0.433155
$473$	1.84438e9	0.801374
$474$	−1.25072e9	−0.539432
$475$	−1.97953e8	−0.0847488
$476$	−1.51424e9	−0.643534
$477$	7.40894e8	0.312566
$478$	7.39631e8	0.309754
$479$	−5.85826e8	−0.243553	−0.121777	−	0.992558i	$-0.538859\pi$
$479$	−5.85826e8	−0.243553	−0.121777	+	0.992558i	$0.538859\pi$
$480$	−1.20920e9	−0.499061
$481$	5.49478e8	0.225135
$482$	−1.38338e9	−0.562702
$483$	1.80469e8	0.0728765
$484$	−1.75925e9	−0.705291
$485$	3.04951e9	1.21376
$486$	1.02088e8	0.0403411
$487$	3.35256e9	1.31530	0.657651	−	0.753323i	$-0.271550\pi$
$487$	3.35256e9	1.31530	0.657651	+	0.753323i	$0.271550\pi$
$488$	−3.42136e9	−1.33269
$489$	3.01689e8	0.116675
$490$	9.05774e8	0.347803
$491$	3.71490e9	1.41632	0.708161	−	0.706051i	$-0.249525\pi$
$491$	3.71490e9	1.41632	0.708161	+	0.706051i	$0.249525\pi$
$492$	−6.26040e8	−0.236987
$493$	−9.01422e9	−3.38816
$494$	4.79523e8	0.178964
$495$	−1.15584e9	−0.428332
$496$	−6.80260e7	−0.0250316
$497$	−2.39539e9	−0.875242
$498$	−7.71819e7	−0.0280035
$499$	3.12811e9	1.12702	0.563509	−	0.826110i	$-0.309451\pi$
$499$	3.12811e9	1.12702	0.563509	+	0.826110i	$0.309451\pi$
$500$	−1.82605e9	−0.653306
$501$	−2.95648e9	−1.05037
$502$	1.08062e9	0.381250
$503$	−1.14943e9	−0.402713	−0.201356	−	0.979518i	$-0.564535\pi$
$503$	−1.14943e9	−0.402713	−0.201356	+	0.979518i	$0.564535\pi$
$504$	−5.85193e8	−0.203607
$505$	−3.11703e8	−0.107701
$506$	5.62482e8	0.193011
$507$	−6.13008e8	−0.208900
$508$	−2.16180e9	−0.731632
$509$	−3.96039e9	−1.33115	−0.665573	−	0.746333i	$-0.731813\pi$
$509$	−3.96039e9	−1.33115	−0.665573	+	0.746333i	$0.731813\pi$
$510$	−1.66965e9	−0.557354
$511$	1.53717e9	0.509623
$512$	−1.82087e8	−0.0599563
$513$	2.09640e8	0.0685587
$514$	1.18812e9	0.385913
$515$	2.34493e9	0.756492
$516$	−5.93033e8	−0.190023
$517$	2.02885e9	0.645704
$518$	3.39383e8	0.107284
$519$	1.32889e9	0.417257
$520$	2.25626e9	0.703685
$521$	1.89506e8	0.0587071	0.0293535	−	0.999569i	$-0.490655\pi$
$521$	1.89506e8	0.0587071	0.0293535	+	0.999569i	$0.490655\pi$
$522$	−1.31252e9	−0.403887
$523$	−3.32527e8	−0.101642	−0.0508208	−	0.998708i	$-0.516184\pi$
$523$	−3.32527e8	−0.101642	−0.0508208	+	0.998708i	$0.516184\pi$
$524$	−2.07828e9	−0.631021
$525$	−2.75675e8	−0.0831458
$526$	−1.78565e8	−0.0534991
$527$	4.93150e9	1.46772
$528$	−8.62053e7	−0.0254868
$529$	1.48036e8	0.0434783
$530$	−1.76436e9	−0.514779
$531$	4.93686e8	0.143094
$532$	−4.52765e8	−0.130371
$533$	1.89616e9	0.542412
$534$	2.24549e9	0.638140
$535$	−3.08879e9	−0.872067
$536$	4.60852e9	1.29266
$537$	−8.05097e8	−0.224356
$538$	1.36822e8	0.0378807
$539$	−3.39025e9	−0.932547
$540$	3.71645e8	0.101566
$541$	4.30263e9	1.16827	0.584135	−	0.811656i	$-0.301434\pi$
$541$	4.30263e9	1.16827	0.584135	+	0.811656i	$0.301434\pi$
$542$	6.19768e8	0.167198
$543$	−1.24116e9	−0.332682
$544$	6.53789e9	1.74117
$545$	2.01604e9	0.533471
$546$	6.67800e8	0.175579
$547$	7.00466e9	1.82992	0.914958	−	0.403549i	$-0.132223\pi$
$547$	7.00466e9	1.82992	0.914958	+	0.403549i	$0.132223\pi$
$548$	−4.31292e9	−1.11954
$549$	1.70691e9	0.440258
$550$	−8.59220e8	−0.220209
$551$	−2.69529e9	−0.686396
$552$	−4.80025e8	−0.121472
$553$	−3.57681e9	−0.899410
$554$	−4.06697e9	−1.01622
$555$	−5.72064e8	−0.142043
$556$	3.60686e9	0.889953
$557$	2.99900e9	0.735331	0.367666	−	0.929958i	$-0.380157\pi$
$557$	2.99900e9	0.735331	0.367666	+	0.929958i	$0.380157\pi$
$558$	7.18055e8	0.174960
$559$	1.79619e9	0.434921
$560$	6.58654e7	0.0158489
$561$	6.24940e9	1.49440
$562$	−4.94646e9	−1.17549
$563$	1.11595e9	0.263551	0.131775	−	0.991280i	$-0.457932\pi$
$563$	1.11595e9	0.263551	0.131775	+	0.991280i	$0.457932\pi$
$564$	−6.52349e8	−0.153110
$565$	9.23832e8	0.215488
$566$	−1.61630e9	−0.374683
$567$	2.91951e8	0.0672619
$568$	6.37143e9	1.45887
$569$	−8.66803e8	−0.197255	−0.0986273	−	0.995124i	$-0.531445\pi$
$569$	−8.66803e8	−0.197255	−0.0986273	+	0.995124i	$0.531445\pi$
$570$	−4.99234e8	−0.112912
$571$	2.93241e9	0.659172	0.329586	−	0.944126i	$-0.393091\pi$
$571$	2.93241e9	0.659172	0.329586	+	0.944126i	$0.393091\pi$
$572$	−3.18182e9	−0.710870
$573$	−2.82167e9	−0.626564
$574$	1.17115e9	0.258477
$575$	−2.26132e8	−0.0496050
$576$	9.97804e8	0.217554
$577$	4.78530e9	1.03704	0.518518	−	0.855067i	$-0.326484\pi$
$577$	4.78530e9	1.03704	0.518518	+	0.855067i	$0.326484\pi$
$578$	6.10804e9	1.31569
$579$	4.41513e9	0.945298
$580$	−4.77816e9	−1.01686
$581$	−2.20725e8	−0.0466911
$582$	−2.40075e9	−0.504798
$583$	6.60386e9	1.38025
$584$	−4.08868e9	−0.849451
$585$	−1.12564e9	−0.232464
$586$	−5.70004e8	−0.117014
$587$	2.22146e8	0.0453320	0.0226660	−	0.999743i	$-0.492785\pi$
$587$	2.22146e8	0.0453320	0.0226660	+	0.999743i	$0.492785\pi$
$588$	1.09009e9	0.221126
$589$	1.47454e9	0.297340
$590$	−1.17566e9	−0.235667
$591$	2.81718e9	0.561382
$592$	−4.26658e7	−0.00845189
$593$	−6.63984e9	−1.30757	−0.653787	−	0.756679i	$-0.726821\pi$
$593$	−6.63984e9	−1.30757	−0.653787	+	0.756679i	$0.726821\pi$
$594$	9.09948e8	0.178141
$595$	−4.77487e9	−0.929293
$596$	1.06819e9	0.206674
$597$	−1.83419e9	−0.352804
$598$	5.47786e8	0.104751
$599$	−6.78244e9	−1.28941	−0.644706	−	0.764430i	$-0.723020\pi$
$599$	−6.78244e9	−1.28941	−0.644706	+	0.764430i	$0.723020\pi$
$600$	7.33262e8	0.138589
$601$	6.88273e9	1.29330	0.646651	−	0.762786i	$-0.276169\pi$
$601$	6.88273e9	1.29330	0.646651	+	0.762786i	$0.276169\pi$
$602$	1.10941e9	0.207254
$603$	−2.29918e9	−0.427034
$604$	−3.74693e8	−0.0691905
$605$	−5.54745e9	−1.01847
$606$	2.45391e8	0.0447924
$607$	−6.20980e9	−1.12698	−0.563491	−	0.826122i	$-0.690542\pi$
$607$	−6.20980e9	−1.12698	−0.563491	+	0.826122i	$0.690542\pi$
$608$	1.95486e9	0.352738
$609$	−3.75355e9	−0.673413
$610$	−4.06481e9	−0.725080
$611$	1.97584e9	0.350436
$612$	−2.00941e9	−0.354355
$613$	−5.45404e8	−0.0956327	−0.0478164	−	0.998856i	$-0.515226\pi$
$613$	−5.45404e8	−0.0956327	−0.0478164	+	0.998856i	$0.515226\pi$
$614$	4.18671e9	0.729934
$615$	−1.97410e9	−0.342220
$616$	−5.21604e9	−0.899101
$617$	4.84884e9	0.831073	0.415537	−	0.909576i	$-0.363594\pi$
$617$	4.84884e9	0.831073	0.415537	+	0.909576i	$0.363594\pi$
$618$	−1.84607e9	−0.314621
$619$	8.95916e9	1.51827	0.759137	−	0.650931i	$-0.225621\pi$
$619$	8.95916e9	1.51827	0.759137	+	0.650931i	$0.225621\pi$
$620$	2.61403e9	0.440494
$621$	2.39483e8	0.0401286
$622$	9.26014e8	0.154295
$623$	6.42163e9	1.06399
$624$	−8.39529e7	−0.0138322
$625$	−4.30608e9	−0.705508
$626$	−2.51629e9	−0.409968
$627$	1.86860e9	0.302746
$628$	5.37995e9	0.866801
$629$	3.09303e9	0.495572
$630$	−6.95249e8	−0.110777
$631$	4.66301e9	0.738862	0.369431	−	0.929258i	$-0.379553\pi$
$631$	4.66301e9	0.738862	0.369431	+	0.929258i	$0.379553\pi$
$632$	9.51388e9	1.49916
$633$	−2.90666e9	−0.455492
$634$	2.82528e8	0.0440300
$635$	−6.81682e9	−1.05651
$636$	−2.12338e9	−0.327286
$637$	−3.30167e9	−0.506110
$638$	−1.16990e10	−1.78351
$639$	−3.17869e9	−0.481942
$640$	3.35634e9	0.506100
$641$	9.05541e9	1.35802	0.679008	−	0.734131i	$-0.262410\pi$
$641$	9.05541e9	1.35802	0.679008	+	0.734131i	$0.262410\pi$
$642$	2.43168e9	0.362688
$643$	−6.14386e9	−0.911387	−0.455694	−	0.890137i	$-0.650609\pi$
$643$	−6.14386e9	−0.911387	−0.455694	+	0.890137i	$0.650609\pi$
$644$	−5.17218e8	−0.0763086
$645$	−1.87002e9	−0.274402
$646$	2.69925e9	0.393939
$647$	1.34542e10	1.95295	0.976476	−	0.215626i	$-0.0691791\pi$
$647$	1.34542e10	1.95295	0.976476	+	0.215626i	$0.0691791\pi$
$648$	−7.76554e8	−0.112114
$649$	4.40041e9	0.631883
$650$	−8.36771e8	−0.119511
$651$	2.05349e9	0.291716
$652$	−8.64630e8	−0.122170
$653$	−8.77081e9	−1.23266	−0.616330	−	0.787488i	$-0.711381\pi$
$653$	−8.77081e9	−1.23266	−0.616330	+	0.787488i	$0.711381\pi$
$654$	−1.58714e9	−0.221868
$655$	−6.55345e9	−0.911225
$656$	−1.47233e8	−0.0203629
$657$	2.03983e9	0.280618
$658$	1.22037e9	0.166994
$659$	−1.25720e10	−1.71122	−0.855612	−	0.517618i	$-0.826819\pi$
$659$	−1.25720e10	−1.71122	−0.855612	+	0.517618i	$0.826819\pi$
$660$	3.31261e9	0.448504
$661$	−1.17733e10	−1.58560	−0.792798	−	0.609485i	$-0.791376\pi$
$661$	−1.17733e10	−1.58560	−0.792798	+	0.609485i	$0.791376\pi$
$662$	−8.07330e9	−1.08155
$663$	6.08612e9	0.811041
$664$	5.87100e8	0.0778259
$665$	−1.42771e9	−0.188262
$666$	4.50363e8	0.0590748
$667$	−3.07898e9	−0.401760
$668$	8.47318e9	1.09984
$669$	3.35393e9	0.433075
$670$	5.47524e9	0.703301
$671$	1.52143e10	1.94412
$672$	2.72240e9	0.346066
$673$	−1.27645e10	−1.61418	−0.807088	−	0.590432i	$-0.798958\pi$
$673$	−1.27645e10	−1.61418	−0.807088	+	0.590432i	$0.798958\pi$
$674$	−2.25224e9	−0.283338
$675$	−3.65822e8	−0.0457833
$676$	1.75686e9	0.218738
$677$	−1.31090e10	−1.62371	−0.811854	−	0.583861i	$-0.801541\pi$
$677$	−1.31090e10	−1.62371	−0.811854	+	0.583861i	$0.801541\pi$
$678$	−7.27296e8	−0.0896205
$679$	−6.86567e9	−0.841664
$680$	1.27006e10	1.54897
$681$	6.18575e9	0.750546
$682$	6.40029e9	0.772599
$683$	−5.12267e9	−0.615210	−0.307605	−	0.951514i	$-0.599528\pi$
$683$	−5.12267e9	−0.615210	−0.307605	+	0.951514i	$0.599528\pi$
$684$	−6.00821e8	−0.0717875
$685$	−1.36000e10	−1.61667
$686$	−5.25809e9	−0.621861
$687$	−1.29941e9	−0.152896
$688$	−1.39470e8	−0.0163276
$689$	6.43132e9	0.749088
$690$	−5.70302e8	−0.0660896
$691$	3.65593e9	0.421527	0.210763	−	0.977537i	$-0.432405\pi$
$691$	3.65593e9	0.421527	0.210763	+	0.977537i	$0.432405\pi$
$692$	−3.80856e9	−0.436908
$693$	2.60227e9	0.297020
$694$	−5.86677e9	−0.666256
$695$	1.13735e10	1.28513
$696$	9.98398e9	1.12246
$697$	1.06735e10	1.19397
$698$	−6.95297e9	−0.773885
$699$	−8.96845e9	−0.993224
$700$	7.90077e8	0.0870615
$701$	−1.62181e10	−1.77822	−0.889112	−	0.457691i	$-0.848677\pi$
$701$	−1.62181e10	−1.77822	−0.889112	+	0.457691i	$0.848677\pi$
$702$	8.86173e8	0.0966805
$703$	9.24828e8	0.100396
$704$	8.89379e9	0.960689
$705$	−2.05706e9	−0.221098
$706$	3.32433e9	0.355540
$707$	7.01768e8	0.0746837
$708$	−1.41489e9	−0.149833
$709$	−3.99511e9	−0.420986	−0.210493	−	0.977595i	$-0.567507\pi$
$709$	−3.99511e9	−0.420986	−0.210493	+	0.977595i	$0.567507\pi$
$710$	7.56970e9	0.793733
$711$	−4.74644e9	−0.495250
$712$	−1.70808e10	−1.77348
$713$	1.68445e9	0.174038
$714$	3.75907e9	0.386488
$715$	−1.00333e10	−1.02653
$716$	2.30738e9	0.234922
$717$	2.80687e9	0.284384
$718$	−7.75098e8	−0.0781486
$719$	5.04375e9	0.506060	0.253030	−	0.967458i	$-0.418573\pi$
$719$	5.04375e9	0.506060	0.253030	+	0.967458i	$0.418573\pi$
$720$	8.74038e7	0.00872703
$721$	−5.27938e9	−0.524577
$722$	−5.55253e9	−0.549049
$723$	−5.24989e9	−0.516614
$724$	3.55713e9	0.348350
$725$	4.70330e9	0.458373
$726$	4.36728e9	0.423578
$727$	−8.20105e9	−0.791588	−0.395794	−	0.918339i	$-0.629530\pi$
$727$	−8.20105e9	−0.791588	−0.395794	+	0.918339i	$0.629530\pi$
$728$	−5.07976e9	−0.487959
$729$	3.87420e8	0.0370370
$730$	−4.85763e9	−0.462162
$731$	1.01108e10	0.957358
$732$	−4.89195e9	−0.460991
$733$	−3.11182e9	−0.291844	−0.145922	−	0.989296i	$-0.546615\pi$
$733$	−3.11182e9	−0.291844	−0.145922	+	0.989296i	$0.546615\pi$
$734$	1.93959e9	0.181040
$735$	3.43738e9	0.319317
$736$	2.23314e9	0.206464
$737$	−2.04934e10	−1.88572
$738$	1.55413e9	0.142328
$739$	2.45017e9	0.223326	0.111663	−	0.993746i	$-0.464382\pi$
$739$	2.45017e9	0.223326	0.111663	+	0.993746i	$0.464382\pi$
$740$	1.63952e9	0.148732
$741$	1.81977e9	0.164306
$742$	3.97228e9	0.356965
$743$	1.47214e10	1.31670	0.658350	−	0.752712i	$-0.271255\pi$
$743$	1.47214e10	1.31670	0.658350	+	0.752712i	$0.271255\pi$
$744$	−5.46204e9	−0.486238
$745$	3.36833e9	0.298448
$746$	5.87011e9	0.517679
$747$	−2.92903e8	−0.0257099
$748$	−1.79106e10	−1.56478
$749$	6.95411e9	0.604721
$750$	4.53311e9	0.392357
$751$	−1.60115e10	−1.37941	−0.689703	−	0.724092i	$-0.742259\pi$
$751$	−1.60115e10	−1.37941	−0.689703	+	0.724092i	$0.742259\pi$
$752$	−1.53420e8	−0.0131559
$753$	4.10092e9	0.350025
$754$	−1.13933e10	−0.967946
$755$	−1.18152e9	−0.0999144
$756$	−8.36723e8	−0.0704296
$757$	1.69878e10	1.42332	0.711660	−	0.702524i	$-0.247944\pi$
$757$	1.69878e10	1.42332	0.711660	+	0.702524i	$0.247944\pi$
$758$	−4.25800e9	−0.355111
$759$	2.13460e9	0.177203
$760$	3.79753e9	0.313800
$761$	−2.21862e10	−1.82489	−0.912447	−	0.409195i	$-0.865809\pi$
$761$	−2.21862e10	−1.82489	−0.912447	+	0.409195i	$0.865809\pi$
$762$	5.36661e9	0.439397
$763$	−4.53891e9	−0.369927
$764$	8.08683e9	0.656072
$765$	−6.33628e9	−0.511705
$766$	1.06867e10	0.859095
$767$	4.28544e9	0.342935
$768$	−7.37264e9	−0.587299
$769$	−4.74577e9	−0.376326	−0.188163	−	0.982138i	$-0.560253\pi$
$769$	−4.74577e9	−0.376326	−0.188163	+	0.982138i	$0.560253\pi$
$770$	−6.19701e9	−0.489176
$771$	4.50887e9	0.354305
$772$	−1.26536e10	−0.989817
$773$	−9.43919e9	−0.735032	−0.367516	−	0.930017i	$-0.619792\pi$
$773$	−9.43919e9	−0.735032	−0.367516	+	0.930017i	$0.619792\pi$
$774$	1.47219e9	0.114122
$775$	−2.57308e9	−0.198562
$776$	1.82618e10	1.40291
$777$	1.28795e9	0.0984972
$778$	2.97412e9	0.226428
$779$	3.19143e9	0.241882
$780$	3.22606e9	0.243412
$781$	−2.83328e10	−2.12819
$782$	3.08350e9	0.230580
$783$	−4.98098e9	−0.370807
$784$	2.56367e8	0.0190001
$785$	1.69646e10	1.25170
$786$	5.15927e9	0.378974
$787$	2.24683e10	1.64308	0.821538	−	0.570154i	$-0.193116\pi$
$787$	2.24683e10	1.64308	0.821538	+	0.570154i	$0.193116\pi$
$788$	−8.07396e9	−0.587820
$789$	−6.77648e8	−0.0491173
$790$	1.13031e10	0.815650
$791$	−2.07992e9	−0.149427
$792$	−6.92171e9	−0.495080
$793$	1.48168e10	1.05511
$794$	−3.25692e9	−0.230906
$795$	−6.69567e9	−0.472617
$796$	5.25673e9	0.369420
$797$	1.09382e9	0.0765314	0.0382657	−	0.999268i	$-0.487817\pi$
$797$	1.09382e9	0.0765314	0.0382657	+	0.999268i	$0.487817\pi$
$798$	1.12398e9	0.0782973
$799$	1.11221e10	0.771387
$800$	−3.41123e9	−0.235557
$801$	8.52154e9	0.585874
$802$	3.03015e9	0.207421
$803$	1.81818e10	1.23917
$804$	6.58938e9	0.447145
$805$	−1.63095e9	−0.110193
$806$	6.23306e9	0.419304
$807$	5.19235e8	0.0347781
$808$	−1.86662e9	−0.124485
$809$	1.21344e10	0.805748	0.402874	−	0.915255i	$-0.368011\pi$
$809$	1.21344e10	0.805748	0.402874	+	0.915255i	$0.368011\pi$
$810$	−9.22599e8	−0.0609980
$811$	1.02343e10	0.673730	0.336865	−	0.941553i	$-0.390633\pi$
$811$	1.02343e10	0.673730	0.336865	+	0.941553i	$0.390633\pi$
$812$	1.07576e10	0.705127
$813$	2.35200e9	0.153504
$814$	4.01425e9	0.260867
$815$	−2.72645e9	−0.176419
$816$	−4.72574e8	−0.0304477
$817$	3.02317e9	0.193948
$818$	−7.97376e8	−0.0509362
$819$	2.53428e9	0.161198
$820$	5.65771e9	0.358337
$821$	2.57619e10	1.62472	0.812358	−	0.583159i	$-0.198184\pi$
$821$	2.57619e10	1.62472	0.812358	+	0.583159i	$0.198184\pi$
$822$	1.07067e10	0.672364
$823$	−1.12632e10	−0.704307	−0.352154	−	0.935942i	$-0.614551\pi$
$823$	−1.12632e10	−0.704307	−0.352154	+	0.935942i	$0.614551\pi$
$824$	1.40425e10	0.874378
$825$	−3.26071e9	−0.202173
$826$	2.64688e9	0.163420
$827$	−1.88055e10	−1.15615	−0.578077	−	0.815982i	$-0.696197\pi$
$827$	−1.88055e10	−1.15615	−0.578077	+	0.815982i	$0.696197\pi$
$828$	−6.86351e8	−0.0420185
$829$	−1.31492e9	−0.0801600	−0.0400800	−	0.999196i	$-0.512761\pi$
$829$	−1.31492e9	−0.0801600	−0.0400800	+	0.999196i	$0.512761\pi$
$830$	6.97515e8	0.0423429
$831$	−1.54340e10	−0.932987
$832$	8.66142e9	0.521384
$833$	−1.85852e10	−1.11406
$834$	−8.95392e9	−0.534481
$835$	2.67186e10	1.58822
$836$	−5.35534e9	−0.317004
$837$	2.72499e9	0.160630
$838$	−1.37421e10	−0.806675
$839$	−5.54224e9	−0.323980	−0.161990	−	0.986792i	$-0.551791\pi$
$839$	−5.54224e9	−0.323980	−0.161990	+	0.986792i	$0.551791\pi$
$840$	5.28856e9	0.307865
$841$	4.67894e10	2.71245
$842$	4.70923e8	0.0271868
$843$	−1.87717e10	−1.07921
$844$	8.33039e9	0.476943
$845$	5.53993e9	0.315868
$846$	1.61944e9	0.0919535
$847$	1.24895e10	0.706244
$848$	−4.99378e8	−0.0281219
$849$	−6.13379e9	−0.343995
$850$	−4.71021e9	−0.263072
$851$	1.05648e9	0.0587637
$852$	9.11003e9	0.504639
$853$	−1.06408e10	−0.587021	−0.293511	−	0.955956i	$-0.594824\pi$
$853$	−1.06408e10	−0.587021	−0.293511	+	0.955956i	$0.594824\pi$
$854$	9.15153e9	0.502795
$855$	−1.89457e9	−0.103665
$856$	−1.84971e10	−1.00796
$857$	−4.28265e9	−0.232423	−0.116212	−	0.993224i	$-0.537075\pi$
$857$	−4.28265e9	−0.232423	−0.116212	+	0.993224i	$0.537075\pi$
$858$	7.89879e9	0.426928
$859$	−3.28556e10	−1.76862	−0.884309	−	0.466903i	$-0.845370\pi$
$859$	−3.28556e10	−1.76862	−0.884309	+	0.466903i	$0.845370\pi$
$860$	5.35941e9	0.287324
$861$	4.44449e9	0.237307
$862$	−1.82110e10	−0.968406
$863$	−2.96741e10	−1.57159	−0.785796	−	0.618486i	$-0.787746\pi$
$863$	−2.96741e10	−1.57159	−0.785796	+	0.618486i	$0.787746\pi$
$864$	3.61263e9	0.190557
$865$	−1.20096e10	−0.630915
$866$	9.18242e9	0.480446
$867$	2.31798e10	1.20793
$868$	−5.88525e9	−0.305454
$869$	−4.23068e10	−2.18696
$870$	1.18616e10	0.610699
$871$	−1.99580e10	−1.02342
$872$	1.20729e10	0.616603
$873$	−9.11078e9	−0.463453
$874$	9.21980e8	0.0467123
$875$	1.29638e10	0.654189
$876$	−5.84610e9	−0.293834
$877$	1.68956e10	0.845814	0.422907	−	0.906173i	$-0.361010\pi$
$877$	1.68956e10	0.845814	0.422907	+	0.906173i	$0.361010\pi$
$878$	−1.05762e9	−0.0527351
$879$	−2.16315e9	−0.107430
$880$	7.79062e8	0.0385374
$881$	8.75880e9	0.431548	0.215774	−	0.976443i	$-0.430773\pi$
$881$	8.75880e9	0.431548	0.215774	+	0.976443i	$0.430773\pi$
$882$	−2.70611e9	−0.132802
$883$	2.43303e10	1.18928	0.594641	−	0.803991i	$-0.297294\pi$
$883$	2.43303e10	1.18928	0.594641	+	0.803991i	$0.297294\pi$
$884$	−1.74426e10	−0.849237
$885$	−4.46159e9	−0.216365
$886$	4.74785e9	0.229340
$887$	−9.17722e9	−0.441548	−0.220774	−	0.975325i	$-0.570858\pi$
$887$	−9.17722e9	−0.441548	−0.220774	+	0.975325i	$0.570858\pi$
$888$	−3.42578e9	−0.164178
$889$	1.53474e10	0.732620
$890$	−2.02931e10	−0.964902
$891$	3.45322e9	0.163551
$892$	−9.61227e9	−0.453471
$893$	3.32555e9	0.156273
$894$	−2.65175e9	−0.124123
$895$	7.27589e9	0.339239
$896$	−7.55648e9	−0.350947
$897$	2.07883e9	0.0961712
$898$	−9.96492e9	−0.459205
$899$	−3.50346e10	−1.60819
$900$	1.04844e9	0.0479395
$901$	3.62021e10	1.64891
$902$	1.38525e10	0.628501
$903$	4.21016e9	0.190279
$904$	5.53233e9	0.249068
$905$	1.12167e10	0.503033
$906$	9.30165e8	0.0415539
$907$	1.69074e10	0.752405	0.376202	−	0.926538i	$-0.377230\pi$
$907$	1.69074e10	0.752405	0.376202	+	0.926538i	$0.377230\pi$
$908$	−1.77282e10	−0.785893
$909$	9.31250e8	0.0411237
$910$	−6.03510e9	−0.265485
$911$	7.71107e9	0.337910	0.168955	−	0.985624i	$-0.445961\pi$
$911$	7.71107e9	0.337910	0.168955	+	0.985624i	$0.445961\pi$
$912$	−1.41301e8	−0.00616829
$913$	−2.61075e9	−0.113532
$914$	−2.44072e10	−1.05732
$915$	−1.54258e10	−0.665694
$916$	3.72407e9	0.160097
$917$	1.47545e10	0.631874
$918$	4.98830e9	0.212815
$919$	−2.07829e10	−0.883287	−0.441644	−	0.897191i	$-0.645604\pi$
$919$	−2.07829e10	−0.883287	−0.441644	+	0.897191i	$0.645604\pi$
$920$	4.33812e9	0.183673
$921$	1.58884e10	0.670150
$922$	1.04190e10	0.437791
$923$	−2.75926e10	−1.15501
$924$	−7.45802e9	−0.311008
$925$	−1.61383e9	−0.0670443
$926$	−3.42964e9	−0.141942
$927$	−7.00576e9	−0.288853
$928$	−4.64468e10	−1.90782
$929$	2.73426e10	1.11888	0.559442	−	0.828870i	$-0.311015\pi$
$929$	2.73426e10	1.11888	0.559442	+	0.828870i	$0.311015\pi$
$930$	−6.48927e9	−0.264549
$931$	−5.55705e9	−0.225694
$932$	2.57033e10	1.04000
$933$	3.51419e9	0.141658
$934$	−2.99916e9	−0.120444
$935$	−5.64776e10	−2.25962
$936$	−6.74086e9	−0.268689
$937$	−2.65430e10	−1.05405	−0.527026	−	0.849849i	$-0.676693\pi$
$937$	−2.65430e10	−1.05405	−0.527026	+	0.849849i	$0.676693\pi$
$938$	−1.23270e10	−0.487693
$939$	−9.54923e9	−0.376390
$940$	5.89547e9	0.231511
$941$	−2.60897e10	−1.02072	−0.510358	−	0.859962i	$-0.670487\pi$
$941$	−2.60897e10	−1.02072	−0.510358	+	0.859962i	$0.670487\pi$
$942$	−1.33556e10	−0.520576
$943$	3.64575e9	0.141578
$944$	−3.32755e8	−0.0128743
$945$	−2.63845e9	−0.101704
$946$	1.31222e10	0.503949
$947$	2.11552e10	0.809454	0.404727	−	0.914438i	$-0.367367\pi$
$947$	2.11552e10	0.809454	0.404727	+	0.914438i	$0.367367\pi$
$948$	1.36032e10	0.518574
$949$	1.77067e10	0.672522
$950$	−1.40837e9	−0.0532948
$951$	1.07218e9	0.0404238
$952$	−2.85941e10	−1.07411
$953$	−2.33959e10	−0.875619	−0.437810	−	0.899068i	$-0.644246\pi$
$953$	−2.33959e10	−0.875619	−0.437810	+	0.899068i	$0.644246\pi$
$954$	5.27123e9	0.196559
$955$	2.55003e10	0.947399
$956$	−8.04441e9	−0.297777
$957$	−4.43973e10	−1.63744
$958$	−4.16797e9	−0.153160
$959$	3.06190e10	1.12105
$960$	−9.01744e9	−0.328953
$961$	−8.34587e9	−0.303347
$962$	3.90937e9	0.141577
$963$	9.22813e9	0.332983
$964$	1.50460e10	0.540944
$965$	−3.99008e10	−1.42934
$966$	1.28398e9	0.0458288
$967$	2.79465e10	0.993882	0.496941	−	0.867784i	$-0.334456\pi$
$967$	2.79465e10	0.993882	0.496941	+	0.867784i	$0.334456\pi$
$968$	−3.32207e10	−1.17718
$969$	1.02436e10	0.361674
$970$	2.16963e10	0.763281
$971$	2.96781e10	1.04033	0.520163	−	0.854067i	$-0.325871\pi$
$971$	2.96781e10	1.04033	0.520163	+	0.854067i	$0.325871\pi$
$972$	−1.11034e9	−0.0387813
$973$	−2.56064e10	−0.891156
$974$	2.38524e10	0.827135
$975$	−3.17552e9	−0.109723
$976$	−1.15049e9	−0.0396104
$977$	3.47537e10	1.19226	0.596128	−	0.802889i	$-0.296705\pi$
$977$	3.47537e10	1.19226	0.596128	+	0.802889i	$0.296705\pi$
$978$	2.14642e9	0.0733718
$979$	7.59556e10	2.58714
$980$	−9.85143e9	−0.334355
$981$	−6.02316e9	−0.203696
$982$	2.64304e10	0.890662
$983$	−3.56697e10	−1.19774	−0.598870	−	0.800846i	$-0.704383\pi$
$983$	−3.56697e10	−1.19774	−0.598870	+	0.800846i	$0.704383\pi$
$984$	−1.18218e10	−0.395550
$985$	−2.54597e10	−0.848840
$986$	−6.41334e10	−2.13066
$987$	4.63127e9	0.153317
$988$	−5.21542e9	−0.172044
$989$	3.45353e9	0.113521
$990$	−8.22346e9	−0.269359
$991$	2.52813e10	0.825166	0.412583	−	0.910920i	$-0.364627\pi$
$991$	2.52813e10	0.825166	0.412583	+	0.910920i	$0.364627\pi$
$992$	2.54101e10	0.826448
$993$	−3.06379e10	−0.992971
$994$	−1.70424e10	−0.550401
$995$	1.65761e10	0.533460
$996$	8.39450e8	0.0269207
$997$	−1.68122e10	−0.537268	−0.268634	−	0.963242i	$-0.586572\pi$
$997$	−1.68122e10	−0.537268	−0.268634	+	0.963242i	$0.586572\pi$
$998$	2.22556e10	0.708731
$999$	1.70911e9	0.0542364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.d.1.5	✓	8
3.2	odd	2		207.8.a.e.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.d.1.5	✓	8	1.1	even	1	trivial
207.8.a.e.1.4		8	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)

\(f(q)\)	\(=\)	\(q+7.11469 q^{2} +27.0000 q^{3} -77.3812 q^{4} -244.007 q^{5} +192.097 q^{6} +549.357 q^{7} -1461.22 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+7.11469 q^{2} +27.0000 q^{3} -77.3812 q^{4} -244.007 q^{5} +192.097 q^{6} +549.357 q^{7} -1461.22 q^{8} +729.000 q^{9} -1736.03 q^{10} +6497.85 q^{11} -2089.29 q^{12} +6328.07 q^{13} +3908.51 q^{14} -6588.18 q^{15} -491.361 q^{16} +35620.9 q^{17} +5186.61 q^{18} +10650.8 q^{19} +18881.5 q^{20} +14832.7 q^{21} +46230.2 q^{22} +12167.0 q^{23} -39453.0 q^{24} -18585.7 q^{25} +45022.3 q^{26} +19683.0 q^{27} -42509.9 q^{28} -253060. q^{29} -46872.9 q^{30} +138444. q^{31} +183541. q^{32} +175442. q^{33} +253432. q^{34} -134047. q^{35} -56410.9 q^{36} +86831.8 q^{37} +75777.1 q^{38} +170858. q^{39} +356548. q^{40} +299642. q^{41} +105530. q^{42} +283844. q^{43} -502811. q^{44} -177881. q^{45} +86564.4 q^{46} +312235. q^{47} -13266.7 q^{48} -521749. q^{49} -132232. q^{50} +961765. q^{51} -489674. q^{52} +1.01632e6 q^{53} +140038. q^{54} -1.58552e6 q^{55} -802734. q^{56} +287572. q^{57} -1.80044e6 q^{58} +677210. q^{59} +509801. q^{60} +2.34144e6 q^{61} +984986. q^{62} +400482. q^{63} +1.36873e6 q^{64} -1.54409e6 q^{65} +1.24821e6 q^{66} -3.15388e6 q^{67} -2.75639e6 q^{68} +328509. q^{69} -953702. q^{70} -4.36034e6 q^{71} -1.06523e6 q^{72} +2.79812e6 q^{73} +617781. q^{74} -501814. q^{75} -824171. q^{76} +3.56964e6 q^{77} +1.21560e6 q^{78} -6.51090e6 q^{79} +119895. q^{80} +531441. q^{81} +2.13186e6 q^{82} -401787. q^{83} -1.14777e6 q^{84} -8.69174e6 q^{85} +2.01946e6 q^{86} -6.83262e6 q^{87} -9.49480e6 q^{88} +1.16894e7 q^{89} -1.26557e6 q^{90} +3.47637e6 q^{91} -941497. q^{92} +3.73799e6 q^{93} +2.22145e6 q^{94} -2.59887e6 q^{95} +4.95560e6 q^{96} -1.24976e7 q^{97} -3.71208e6 q^{98} +4.73693e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9}+O(q^{10}) \) 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9} + 11720 q^{10} + 6932 q^{11} + 15174 q^{12} + 12404 q^{13} + 30222 q^{14} + 10206 q^{15} + 27058 q^{16} + 24434 q^{17} + 17496 q^{18} - 14682 q^{19} - 3760 q^{20} + 3402 q^{21} + 36294 q^{22} + 97336 q^{23} + 113076 q^{24} + 144644 q^{25} + 325840 q^{26} + 157464 q^{27} - 21566 q^{28} + 255356 q^{29} + 316440 q^{30} + 450764 q^{31} + 647588 q^{32} + 187164 q^{33} + 191822 q^{34} + 1022616 q^{35} + 409698 q^{36} + 206240 q^{37} + 737372 q^{38} + 334908 q^{39} + 590028 q^{40} + 1053344 q^{41} + 815994 q^{42} + 1587806 q^{43} + 589366 q^{44} + 275562 q^{45} + 292008 q^{46} + 443336 q^{47} + 730566 q^{48} + 1944828 q^{49} - 1556112 q^{50} + 659718 q^{51} - 614236 q^{52} - 375530 q^{53} + 472392 q^{54} + 407792 q^{55} - 1316922 q^{56} - 396414 q^{57} - 1413384 q^{58} + 624008 q^{59} - 101520 q^{60} - 2005568 q^{61} - 3908272 q^{62} + 91854 q^{63} - 5082310 q^{64} + 646124 q^{65} + 979938 q^{66} - 2712286 q^{67} - 2289698 q^{68} + 2628072 q^{69} - 16499468 q^{70} - 6287176 q^{71} + 3053052 q^{72} - 10358312 q^{73} - 2000150 q^{74} + 3905388 q^{75} - 25107464 q^{76} - 2156840 q^{77} + 8797680 q^{78} - 8800574 q^{79} + 2384344 q^{80} + 4251528 q^{81} - 31799800 q^{82} + 384948 q^{83} - 582282 q^{84} - 17826684 q^{85} - 11563928 q^{86} + 6894612 q^{87} - 25202782 q^{88} - 3445530 q^{89} + 8543880 q^{90} - 16316740 q^{91} + 6837854 q^{92} + 12170628 q^{93} - 24237616 q^{94} + 26164288 q^{95} + 17484876 q^{96} - 28043764 q^{97} - 9998012 q^{98} + 5053428 q^{99}+O(q^{100}) \) 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9 + 11720 * q^10 + 6932 * q^11 + 15174 * q^12 + 12404 * q^13 + 30222 * q^14 + 10206 * q^15 + 27058 * q^16 + 24434 * q^17 + 17496 * q^18 - 14682 * q^19 - 3760 * q^20 + 3402 * q^21 + 36294 * q^22 + 97336 * q^23 + 113076 * q^24 + 144644 * q^25 + 325840 * q^26 + 157464 * q^27 - 21566 * q^28 + 255356 * q^29 + 316440 * q^30 + 450764 * q^31 + 647588 * q^32 + 187164 * q^33 + 191822 * q^34 + 1022616 * q^35 + 409698 * q^36 + 206240 * q^37 + 737372 * q^38 + 334908 * q^39 + 590028 * q^40 + 1053344 * q^41 + 815994 * q^42 + 1587806 * q^43 + 589366 * q^44 + 275562 * q^45 + 292008 * q^46 + 443336 * q^47 + 730566 * q^48 + 1944828 * q^49 - 1556112 * q^50 + 659718 * q^51 - 614236 * q^52 - 375530 * q^53 + 472392 * q^54 + 407792 * q^55 - 1316922 * q^56 - 396414 * q^57 - 1413384 * q^58 + 624008 * q^59 - 101520 * q^60 - 2005568 * q^61 - 3908272 * q^62 + 91854 * q^63 - 5082310 * q^64 + 646124 * q^65 + 979938 * q^66 - 2712286 * q^67 - 2289698 * q^68 + 2628072 * q^69 - 16499468 * q^70 - 6287176 * q^71 + 3053052 * q^72 - 10358312 * q^73 - 2000150 * q^74 + 3905388 * q^75 - 25107464 * q^76 - 2156840 * q^77 + 8797680 * q^78 - 8800574 * q^79 + 2384344 * q^80 + 4251528 * q^81 - 31799800 * q^82 + 384948 * q^83 - 582282 * q^84 - 17826684 * q^85 - 11563928 * q^86 + 6894612 * q^87 - 25202782 * q^88 - 3445530 * q^89 + 8543880 * q^90 - 16316740 * q^91 + 6837854 * q^92 + 12170628 * q^93 - 24237616 * q^94 + 26164288 * q^95 + 17484876 * q^96 - 28043764 * q^97 - 9998012 * q^98 + 5053428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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