LMFDB - Embedded newform 69.8.a.d.1.7

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			1.7
Root			$-14.7586$ 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)

$f(q)$	$=$	$q+17.7586 q^{2} +27.0000 q^{3} +187.368 q^{4} +32.5148 q^{5} +479.482 q^{6} +1672.82 q^{7} +1054.29 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+17.7586 q^{2} +27.0000 q^{3} +187.368 q^{4} +32.5148 q^{5} +479.482 q^{6} +1672.82 q^{7} +1054.29 q^{8} +729.000 q^{9} +577.418 q^{10} -3105.35 q^{11} +5058.93 q^{12} -7087.75 q^{13} +29707.0 q^{14} +877.901 q^{15} -5260.35 q^{16} +18708.2 q^{17} +12946.0 q^{18} +34552.7 q^{19} +6092.24 q^{20} +45166.2 q^{21} -55146.7 q^{22} +12167.0 q^{23} +28465.9 q^{24} -77067.8 q^{25} -125869. q^{26} +19683.0 q^{27} +313434. q^{28} +250815. q^{29} +15590.3 q^{30} -81911.3 q^{31} -228366. q^{32} -83844.5 q^{33} +332232. q^{34} +54391.6 q^{35} +136591. q^{36} -413414. q^{37} +613608. q^{38} -191369. q^{39} +34280.1 q^{40} -538737. q^{41} +802089. q^{42} -430259. q^{43} -581843. q^{44} +23703.3 q^{45} +216069. q^{46} -533167. q^{47} -142029. q^{48} +1.97480e6 q^{49} -1.36862e6 q^{50} +505122. q^{51} -1.32802e6 q^{52} -1.03569e6 q^{53} +349543. q^{54} -100970. q^{55} +1.76364e6 q^{56} +932923. q^{57} +4.45412e6 q^{58} +458111. q^{59} +164490. q^{60} +1.16409e6 q^{61} -1.45463e6 q^{62} +1.21949e6 q^{63} -3.38213e6 q^{64} -230457. q^{65} -1.48896e6 q^{66} -4.02241e6 q^{67} +3.50532e6 q^{68} +328509. q^{69} +965919. q^{70} +2.96942e6 q^{71} +768579. q^{72} +12116.2 q^{73} -7.34166e6 q^{74} -2.08083e6 q^{75} +6.47407e6 q^{76} -5.19470e6 q^{77} -3.39845e6 q^{78} -5.33176e6 q^{79} -171039. q^{80} +531441. q^{81} -9.56721e6 q^{82} -3.28547e6 q^{83} +8.46271e6 q^{84} +608295. q^{85} -7.64081e6 q^{86} +6.77199e6 q^{87} -3.27395e6 q^{88} -401235. q^{89} +420938. q^{90} -1.18566e7 q^{91} +2.27971e6 q^{92} -2.21161e6 q^{93} -9.46830e6 q^{94} +1.12348e6 q^{95} -6.16588e6 q^{96} +1.54716e6 q^{97} +3.50696e7 q^{98} -2.26380e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	17.7586	1.56965	0.784827	−	0.619715i	$-0.212752\pi$
$2$	17.7586	1.56965	0.784827	+	0.619715i	$0.212752\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	187.368	1.46381
$5$	32.5148	0.116329	0.0581643	−	0.998307i	$-0.481475\pi$
$5$	32.5148	0.116329	0.0581643	+	0.998307i	$0.481475\pi$
$6$	479.482	0.906240
$7$	1672.82	1.84335	0.921674	−	0.387966i	$-0.126822\pi$
$7$	1672.82	1.84335	0.921674	+	0.387966i	$0.126822\pi$
$8$	1054.29	0.728024
$9$	729.000	0.333333
$10$	577.418	0.182596
$11$	−3105.35	−0.703455	−0.351727	−	0.936102i	$-0.614406\pi$
$11$	−3105.35	−0.703455	−0.351727	+	0.936102i	$0.614406\pi$
$12$	5058.93	0.845132
$13$	−7087.75	−0.894760	−0.447380	−	0.894344i	$-0.647643\pi$
$13$	−7087.75	−0.894760	−0.447380	+	0.894344i	$0.647643\pi$
$14$	29707.0	2.89342
$15$	877.901	0.0671624
$16$	−5260.35	−0.321066
$17$	18708.2	0.923552	0.461776	−	0.886997i	$-0.347212\pi$
$17$	18708.2	0.923552	0.461776	+	0.886997i	$0.347212\pi$
$18$	12946.0	0.523218
$19$	34552.7	1.15570	0.577849	−	0.816143i	$-0.303892\pi$
$19$	34552.7	1.15570	0.577849	+	0.816143i	$0.303892\pi$
$20$	6092.24	0.170283
$21$	45166.2	1.06426
$22$	−55146.7	−1.10418
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	28465.9	0.420325
$25$	−77067.8	−0.986468
$26$	−125869.	−1.40446
$27$	19683.0	0.192450
$28$	313434.	2.69831
$29$	250815.	1.90968	0.954838	−	0.297127i	$-0.0960283\pi$
$29$	250815.	1.90968	0.954838	+	0.297127i	$0.0960283\pi$
$30$	15590.3	0.105422
$31$	−81911.3	−0.493831	−0.246915	−	0.969037i	$-0.579417\pi$
$31$	−81911.3	−0.493831	−0.246915	+	0.969037i	$0.579417\pi$
$32$	−228366.	−1.23199
$33$	−83844.5	−0.406140
$34$	332232.	1.44966
$35$	54391.6	0.214434
$36$	136591.	0.487937
$37$	−413414.	−1.34177	−0.670887	−	0.741560i	$-0.734086\pi$
$37$	−413414.	−1.34177	−0.670887	+	0.741560i	$0.734086\pi$
$38$	613608.	1.81405
$39$	−191369.	−0.516590
$40$	34280.1	0.0846901
$41$	−538737.	−1.22077	−0.610384	−	0.792106i	$-0.708985\pi$
$41$	−538737.	−1.22077	−0.610384	+	0.792106i	$0.708985\pi$
$42$	802089.	1.67051
$43$	−430259.	−0.825260	−0.412630	−	0.910899i	$-0.635390\pi$
$43$	−430259.	−0.825260	−0.412630	+	0.910899i	$0.635390\pi$
$44$	−581843.	−1.02973
$45$	23703.3	0.0387762
$46$	216069.	0.327295
$47$	−533167.	−0.749067	−0.374534	−	0.927213i	$-0.622197\pi$
$47$	−533167.	−0.749067	−0.374534	+	0.927213i	$0.622197\pi$
$48$	−142029.	−0.185368
$49$	1.97480e6	2.39793
$50$	−1.36862e6	−1.54841
$51$	505122.	0.533213
$52$	−1.32802e6	−1.30976
$53$	−1.03569e6	−0.955576	−0.477788	−	0.878475i	$-0.658561\pi$
$53$	−1.03569e6	−0.955576	−0.477788	+	0.878475i	$0.658561\pi$
$54$	349543.	0.302080
$55$	−100970.	−0.0818320
$56$	1.76364e6	1.34200
$57$	932923.	0.667243
$58$	4.45412e6	2.99753
$59$	458111.	0.290395	0.145197	−	0.989403i	$-0.453618\pi$
$59$	458111.	0.290395	0.145197	+	0.989403i	$0.453618\pi$
$60$	164490.	0.0983131
$61$	1.16409e6	0.656647	0.328323	−	0.944565i	$-0.393516\pi$
$61$	1.16409e6	0.656647	0.328323	+	0.944565i	$0.393516\pi$
$62$	−1.45463e6	−0.775143
$63$	1.21949e6	0.614449
$64$	−3.38213e6	−1.61273
$65$	−230457.	−0.104086
$66$	−1.48896e6	−0.637499
$67$	−4.02241e6	−1.63390	−0.816949	−	0.576710i	$-0.804336\pi$
$67$	−4.02241e6	−1.63390	−0.816949	+	0.576710i	$0.804336\pi$
$68$	3.50532e6	1.35191
$69$	328509.	0.120386
$70$	965919.	0.336587
$71$	2.96942e6	0.984616	0.492308	−	0.870421i	$-0.336153\pi$
$71$	2.96942e6	0.984616	0.492308	+	0.870421i	$0.336153\pi$
$72$	768579.	0.242675
$73$	12116.2	0.00364532	0.00182266	−	0.999998i	$-0.499420\pi$
$73$	12116.2	0.00364532	0.00182266	+	0.999998i	$0.499420\pi$
$74$	−7.34166e6	−2.10612
$75$	−2.08083e6	−0.569537
$76$	6.47407e6	1.69173
$77$	−5.19470e6	−1.29671
$78$	−3.39845e6	−0.810867
$79$	−5.33176e6	−1.21668	−0.608339	−	0.793677i	$-0.708164\pi$
$79$	−5.33176e6	−1.21668	−0.608339	+	0.793677i	$0.708164\pi$
$80$	−171039.	−0.0373492
$81$	531441.	0.111111
$82$	−9.56721e6	−1.91618
$83$	−3.28547e6	−0.630702	−0.315351	−	0.948975i	$-0.602122\pi$
$83$	−3.28547e6	−0.630702	−0.315351	+	0.948975i	$0.602122\pi$
$84$	8.46271e6	1.55787
$85$	608295.	0.107436
$86$	−7.64081e6	−1.29537
$87$	6.77199e6	1.10255
$88$	−3.27395e6	−0.512132
$89$	−401235.	−0.0603301	−0.0301650	−	0.999545i	$-0.509603\pi$
$89$	−401235.	−0.0603301	−0.0301650	+	0.999545i	$0.509603\pi$
$90$	420938.	0.0608652
$91$	−1.18566e7	−1.64935
$92$	2.27971e6	0.305226
$93$	−2.21161e6	−0.285113
$94$	−9.46830e6	−1.17578
$95$	1.12348e6	0.134441
$96$	−6.16588e6	−0.711288
$97$	1.54716e6	0.172121	0.0860606	−	0.996290i	$-0.472572\pi$
$97$	1.54716e6	0.172121	0.0860606	+	0.996290i	$0.472572\pi$
$98$	3.50696e7	3.76392
$99$	−2.26380e6	−0.234485
$100$	−1.44400e7	−1.44400
$101$	1.09935e7	1.06172	0.530862	−	0.847458i	$-0.321868\pi$
$101$	1.09935e7	1.06172	0.530862	+	0.847458i	$0.321868\pi$
$102$	8.97026e6	0.836959
$103$	−1.08340e7	−0.976916	−0.488458	−	0.872588i	$-0.662440\pi$
$103$	−1.08340e7	−0.976916	−0.488458	+	0.872588i	$0.662440\pi$
$104$	−7.47256e6	−0.651407
$105$	1.46857e6	0.123804
$106$	−1.83925e7	−1.49992
$107$	1.48870e6	0.117480	0.0587399	−	0.998273i	$-0.481292\pi$
$107$	1.48870e6	0.117480	0.0587399	+	0.998273i	$0.481292\pi$
$108$	3.68796e6	0.281711
$109$	1.41429e7	1.04603	0.523017	−	0.852322i	$-0.324807\pi$
$109$	1.41429e7	1.04603	0.523017	+	0.852322i	$0.324807\pi$
$110$	−1.79309e6	−0.128448
$111$	−1.11622e7	−0.774673
$112$	−8.79964e6	−0.591837
$113$	1.47726e7	0.963122	0.481561	−	0.876413i	$-0.340070\pi$
$113$	1.47726e7	0.963122	0.481561	+	0.876413i	$0.340070\pi$
$114$	1.65674e7	1.04734
$115$	395608.	0.0242562
$116$	4.69946e7	2.79541
$117$	−5.16697e6	−0.298253
$118$	8.13542e6	0.455819
$119$	3.12956e7	1.70243
$120$	925564.	0.0488958
$121$	−9.84397e6	−0.505151
$122$	2.06726e7	1.03071
$123$	−1.45459e7	−0.704811
$124$	−1.53476e7	−0.722875
$125$	−5.04607e6	−0.231083
$126$	2.16564e7	0.964472
$127$	1.42674e7	0.618061	0.309031	−	0.951052i	$-0.399995\pi$
$127$	1.42674e7	0.618061	0.309031	+	0.951052i	$0.399995\pi$
$128$	−3.08311e7	−1.29943
$129$	−1.16170e7	−0.476464
$130$	−4.09260e6	−0.163379
$131$	2.80585e7	1.09047	0.545237	−	0.838282i	$-0.316440\pi$
$131$	2.80585e7	1.09047	0.545237	+	0.838282i	$0.316440\pi$
$132$	−1.57098e7	−0.594512
$133$	5.78006e7	2.13035
$134$	−7.14324e7	−2.56465
$135$	639990.	0.0223875
$136$	1.97239e7	0.672368
$137$	2.90657e7	0.965736	0.482868	−	0.875693i	$-0.339595\pi$
$137$	2.90657e7	0.965736	0.482868	+	0.875693i	$0.339595\pi$
$138$	5.83386e6	0.188964
$139$	4.89962e7	1.54743	0.773714	−	0.633535i	$-0.218397\pi$
$139$	4.89962e7	1.54743	0.773714	+	0.633535i	$0.218397\pi$
$140$	1.01912e7	0.313891
$141$	−1.43955e7	−0.432474
$142$	5.27327e7	1.54551
$143$	2.20099e7	0.629423
$144$	−3.83480e6	−0.107022
$145$	8.15520e6	0.222150
$146$	215166.	0.00572189
$147$	5.33195e7	1.38444
$148$	−7.74606e7	−1.96410
$149$	1.19196e7	0.295194	0.147597	−	0.989048i	$-0.452846\pi$
$149$	1.19196e7	0.295194	0.147597	+	0.989048i	$0.452846\pi$
$150$	−3.69526e7	−0.893976
$151$	−4.94565e7	−1.16897	−0.584486	−	0.811404i	$-0.698704\pi$
$151$	−4.94565e7	−1.16897	−0.584486	+	0.811404i	$0.698704\pi$
$152$	3.64286e7	0.841377
$153$	1.36383e7	0.307851
$154$	−9.22507e7	−2.03539
$155$	−2.66333e6	−0.0574467
$156$	−3.58565e7	−0.756191
$157$	1.24765e7	0.257302	0.128651	−	0.991690i	$-0.458935\pi$
$157$	1.24765e7	0.257302	0.128651	+	0.991690i	$0.458935\pi$
$158$	−9.46846e7	−1.90976
$159$	−2.79637e7	−0.551702
$160$	−7.42528e6	−0.143315
$161$	2.03533e7	0.384364
$162$	9.43765e6	0.174406
$163$	2.14224e7	0.387446	0.193723	−	0.981056i	$-0.437944\pi$
$163$	2.14224e7	0.387446	0.193723	+	0.981056i	$0.437944\pi$
$164$	−1.00942e8	−1.78697
$165$	−2.72619e6	−0.0472457
$166$	−5.83454e7	−0.989984
$167$	−7.32594e7	−1.21718	−0.608591	−	0.793484i	$-0.708265\pi$
$167$	−7.32594e7	−1.21718	−0.608591	+	0.793484i	$0.708265\pi$
$168$	4.76184e7	0.774805
$169$	−1.25123e7	−0.199404
$170$	1.08025e7	0.168637
$171$	2.51889e7	0.385233
$172$	−8.06168e7	−1.20803
$173$	−7.87905e7	−1.15694	−0.578472	−	0.815702i	$-0.696351\pi$
$173$	−7.87905e7	−1.15694	−0.578472	+	0.815702i	$0.696351\pi$
$174$	1.20261e8	1.73062
$175$	−1.28921e8	−1.81840
$176$	1.63352e7	0.225856
$177$	1.23690e7	0.167660
$178$	−7.12538e6	−0.0946974
$179$	1.25066e8	1.62988	0.814938	−	0.579548i	$-0.196771\pi$
$179$	1.25066e8	1.62988	0.814938	+	0.579548i	$0.196771\pi$
$180$	4.44124e6	0.0567611
$181$	9.63252e7	1.20744	0.603719	−	0.797197i	$-0.293685\pi$
$181$	9.63252e7	1.20744	0.603719	+	0.797197i	$0.293685\pi$
$182$	−2.10556e8	−2.58891
$183$	3.14304e7	0.379115
$184$	1.28276e7	0.151804
$185$	−1.34421e7	−0.156087
$186$	−3.92750e7	−0.447529
$187$	−5.80956e7	−0.649677
$188$	−9.98984e7	−1.09649
$189$	3.29262e7	0.354752
$190$	1.99514e7	0.211026
$191$	5.43389e7	0.564279	0.282140	−	0.959373i	$-0.408956\pi$
$191$	5.43389e7	0.564279	0.282140	+	0.959373i	$0.408956\pi$
$192$	−9.13176e7	−0.931108
$193$	−7.87494e7	−0.788491	−0.394245	−	0.919005i	$-0.628994\pi$
$193$	−7.87494e7	−0.788491	−0.394245	+	0.919005i	$0.628994\pi$
$194$	2.74754e7	0.270171
$195$	−6.22234e6	−0.0600942
$196$	3.70014e8	3.51012
$197$	1.24009e8	1.15564	0.577819	−	0.816165i	$-0.303904\pi$
$197$	1.24009e8	1.15564	0.577819	+	0.816165i	$0.303904\pi$
$198$	−4.02019e7	−0.368060
$199$	1.93717e8	1.74254	0.871268	−	0.490807i	$-0.163298\pi$
$199$	1.93717e8	1.74254	0.871268	+	0.490807i	$0.163298\pi$
$200$	−8.12519e7	−0.718172
$201$	−1.08605e8	−0.943331
$202$	1.95230e8	1.66654
$203$	4.19569e8	3.52020
$204$	9.46437e7	0.780523
$205$	−1.75169e7	−0.142010
$206$	−1.92396e8	−1.53342
$207$	8.86974e6	0.0695048
$208$	3.72840e7	0.287277
$209$	−1.07298e8	−0.812982
$210$	2.60798e7	0.194329
$211$	3.81020e7	0.279228	0.139614	−	0.990206i	$-0.455414\pi$
$211$	3.81020e7	0.279228	0.139614	+	0.990206i	$0.455414\pi$
$212$	−1.94056e8	−1.39878
$213$	8.01742e7	0.568468
$214$	2.64372e7	0.184403
$215$	−1.39898e7	−0.0960014
$216$	2.07516e7	0.140108
$217$	−1.37023e8	−0.910301
$218$	2.51158e8	1.64191
$219$	327137.	0.00210463
$220$	−1.89185e7	−0.119787
$221$	−1.32599e8	−0.826357
$222$	−1.98225e8	−1.21597
$223$	−2.83677e8	−1.71300	−0.856499	−	0.516149i	$-0.827365\pi$
$223$	−2.83677e8	−1.71300	−0.856499	+	0.516149i	$0.827365\pi$
$224$	−3.82016e8	−2.27098
$225$	−5.61824e7	−0.328823
$226$	2.62340e8	1.51177
$227$	−3.20198e8	−1.81689	−0.908445	−	0.418004i	$-0.862730\pi$
$227$	−3.20198e8	−1.81689	−0.908445	+	0.418004i	$0.862730\pi$
$228$	1.74800e8	0.976718
$229$	2.11267e8	1.16254	0.581269	−	0.813712i	$-0.302556\pi$
$229$	2.11267e8	1.16254	0.581269	+	0.813712i	$0.302556\pi$
$230$	7.02545e6	0.0380738
$231$	−1.40257e8	−0.748657
$232$	2.64432e8	1.39029
$233$	1.21921e8	0.631441	0.315721	−	0.948852i	$-0.397754\pi$
$233$	1.21921e8	0.631441	0.315721	+	0.948852i	$0.397754\pi$
$234$	−9.17582e7	−0.468154
$235$	−1.73359e7	−0.0871380
$236$	8.58354e7	0.425084
$237$	−1.43958e8	−0.702450
$238$	5.55765e8	2.67222
$239$	2.27248e8	1.07673	0.538367	−	0.842711i	$-0.319042\pi$
$239$	2.27248e8	1.07673	0.538367	+	0.842711i	$0.319042\pi$
$240$	−4.61807e6	−0.0215636
$241$	−2.85965e8	−1.31599	−0.657996	−	0.753022i	$-0.728596\pi$
$241$	−2.85965e8	−1.31599	−0.657996	+	0.753022i	$0.728596\pi$
$242$	−1.74815e8	−0.792913
$243$	1.43489e7	0.0641500
$244$	2.18113e8	0.961207
$245$	6.42102e7	0.278948
$246$	−2.58315e8	−1.10631
$247$	−2.44901e8	−1.03407
$248$	−8.63584e7	−0.359521
$249$	−8.87077e7	−0.364136
$250$	−8.96111e7	−0.362720
$251$	−2.37970e8	−0.949868	−0.474934	−	0.880021i	$-0.657528\pi$
$251$	−2.37970e8	−0.949868	−0.474934	+	0.880021i	$0.657528\pi$
$252$	2.28493e8	0.899438
$253$	−3.77828e7	−0.146680
$254$	2.53369e8	0.970142
$255$	1.64240e7	0.0620279
$256$	−1.14605e8	−0.426936
$257$	2.33952e8	0.859726	0.429863	−	0.902894i	$-0.358562\pi$
$257$	2.33952e8	0.859726	0.429863	+	0.902894i	$0.358562\pi$
$258$	−2.06302e8	−0.747884
$259$	−6.91569e8	−2.47335
$260$	−4.31803e7	−0.152363
$261$	1.82844e8	0.636559
$262$	4.98281e8	1.71167
$263$	2.83534e7	0.0961082	0.0480541	−	0.998845i	$-0.484698\pi$
$263$	2.83534e7	0.0961082	0.0480541	+	0.998845i	$0.484698\pi$
$264$	−8.83965e7	−0.295680
$265$	−3.36754e7	−0.111161
$266$	1.02646e9	3.34392
$267$	−1.08334e7	−0.0348316
$268$	−7.53671e8	−2.39172
$269$	2.33693e8	0.732002	0.366001	−	0.930615i	$-0.380727\pi$
$269$	2.33693e8	0.732002	0.366001	+	0.930615i	$0.380727\pi$
$270$	1.13653e7	0.0351406
$271$	3.27606e8	0.999906	0.499953	−	0.866053i	$-0.333351\pi$
$271$	3.27606e8	0.999906	0.499953	+	0.866053i	$0.333351\pi$
$272$	−9.84118e7	−0.296521
$273$	−3.20127e8	−0.952255
$274$	5.16166e8	1.51587
$275$	2.39322e8	0.693935
$276$	6.15521e7	0.176222
$277$	4.83558e8	1.36700	0.683502	−	0.729949i	$-0.260456\pi$
$277$	4.83558e8	1.36700	0.683502	+	0.729949i	$0.260456\pi$
$278$	8.70104e8	2.42893
$279$	−5.97134e7	−0.164610
$280$	5.73446e7	0.156113
$281$	−4.37393e8	−1.17598	−0.587990	−	0.808868i	$-0.700081\pi$
$281$	−4.37393e8	−1.17598	−0.587990	+	0.808868i	$0.700081\pi$
$282$	−2.55644e8	−0.678835
$283$	−2.22456e8	−0.583434	−0.291717	−	0.956505i	$-0.594227\pi$
$283$	−2.22456e8	−0.583434	−0.291717	+	0.956505i	$0.594227\pi$
$284$	5.56373e8	1.44129
$285$	3.03339e7	0.0776195
$286$	3.90866e8	0.987977
$287$	−9.01212e8	−2.25030
$288$	−1.66479e8	−0.410662
$289$	−6.03412e7	−0.147052
$290$	1.44825e8	0.348699
$291$	4.17733e7	0.0993742
$292$	2.27018e6	0.00533606
$293$	1.81014e8	0.420413	0.210207	−	0.977657i	$-0.432586\pi$
$293$	1.81014e8	0.420413	0.210207	+	0.977657i	$0.432586\pi$
$294$	9.46880e8	2.17310
$295$	1.48954e7	0.0337812
$296$	−4.35859e8	−0.976844
$297$	−6.11226e7	−0.135380
$298$	2.11675e8	0.463353
$299$	−8.62367e7	−0.186570
$300$	−3.89881e8	−0.833696
$301$	−7.19748e8	−1.52124
$302$	−8.78278e8	−1.83488
$303$	2.96825e8	0.612987
$304$	−1.81759e8	−0.371056
$305$	3.78502e7	0.0763868
$306$	2.42197e8	0.483219
$307$	−3.98750e8	−0.786533	−0.393266	−	0.919425i	$-0.628655\pi$
$307$	−3.98750e8	−0.786533	−0.393266	+	0.919425i	$0.628655\pi$
$308$	−9.73321e8	−1.89814
$309$	−2.92517e8	−0.564022
$310$	−4.72971e7	−0.0901713
$311$	5.26090e8	0.991743	0.495871	−	0.868396i	$-0.334849\pi$
$311$	5.26090e8	0.991743	0.495871	+	0.868396i	$0.334849\pi$
$312$	−2.01759e8	−0.376090
$313$	2.70320e8	0.498279	0.249140	−	0.968468i	$-0.419852\pi$
$313$	2.70320e8	0.498279	0.249140	+	0.968468i	$0.419852\pi$
$314$	2.21565e8	0.403876
$315$	3.96515e7	0.0714780
$316$	−9.99001e8	−1.78099
$317$	7.35356e8	1.29655	0.648276	−	0.761405i	$-0.275490\pi$
$317$	7.35356e8	1.29655	0.648276	+	0.761405i	$0.275490\pi$
$318$	−4.96596e8	−0.865981
$319$	−7.78867e8	−1.34337
$320$	−1.09970e8	−0.187606
$321$	4.01949e7	0.0678270
$322$	3.61445e8	0.603319
$323$	6.46420e8	1.06735
$324$	9.95750e7	0.162646
$325$	5.46237e8	0.882652
$326$	3.80432e8	0.608156
$327$	3.81858e8	0.603928
$328$	−5.67986e8	−0.888748
$329$	−8.91895e8	−1.38079
$330$	−4.84133e7	−0.0741594
$331$	−8.38584e8	−1.27101	−0.635505	−	0.772097i	$-0.719208\pi$
$331$	−8.38584e8	−1.27101	−0.635505	+	0.772097i	$0.719208\pi$
$332$	−6.15592e8	−0.923230
$333$	−3.01379e8	−0.447258
$334$	−1.30098e9	−1.91055
$335$	−1.30788e8	−0.190069
$336$	−2.37590e8	−0.341697
$337$	1.13260e8	0.161203	0.0806015	−	0.996746i	$-0.474316\pi$
$337$	1.13260e8	0.161203	0.0806015	+	0.996746i	$0.474316\pi$
$338$	−2.22201e8	−0.312995
$339$	3.98859e8	0.556058
$340$	1.13975e8	0.157265
$341$	2.54363e8	0.347388
$342$	4.47320e8	0.604682
$343$	1.92585e9	2.57687
$344$	−4.53619e8	−0.600809
$345$	1.06814e7	0.0140043
$346$	−1.39921e9	−1.81600
$347$	1.44981e9	1.86276	0.931381	−	0.364046i	$-0.118605\pi$
$347$	1.44981e9	1.86276	0.931381	+	0.364046i	$0.118605\pi$
$348$	1.26885e9	1.61393
$349$	−3.64693e8	−0.459239	−0.229620	−	0.973280i	$-0.573748\pi$
$349$	−3.64693e8	−0.459239	−0.229620	+	0.973280i	$0.573748\pi$
$350$	−2.28945e9	−2.85426
$351$	−1.39508e8	−0.172197
$352$	7.09156e8	0.866647
$353$	−1.99838e8	−0.241805	−0.120903	−	0.992664i	$-0.538579\pi$
$353$	−1.99838e8	−0.241805	−0.120903	+	0.992664i	$0.538579\pi$
$354$	2.19656e8	0.263167
$355$	9.65501e7	0.114539
$356$	−7.51786e7	−0.0883119
$357$	8.44980e8	0.982896
$358$	2.22100e9	2.55834
$359$	7.71291e6	0.00879808	0.00439904	−	0.999990i	$-0.498600\pi$
$359$	7.71291e6	0.00879808	0.00439904	+	0.999990i	$0.498600\pi$
$360$	2.49902e7	0.0282300
$361$	3.00018e8	0.335639
$362$	1.71060e9	1.89526
$363$	−2.65787e8	−0.291649
$364$	−2.22154e9	−2.41434
$365$	393956.	0.000424055 0
$366$	5.58160e8	0.595079
$367$	5.29899e8	0.559580	0.279790	−	0.960061i	$-0.409735\pi$
$367$	5.29899e8	0.559580	0.279790	+	0.960061i	$0.409735\pi$
$368$	−6.40027e7	−0.0669470
$369$	−3.92739e8	−0.406923
$370$	−2.38713e8	−0.245002
$371$	−1.73253e9	−1.76146
$372$	−4.14384e8	−0.417352
$373$	−1.47849e9	−1.47515	−0.737577	−	0.675263i	$-0.764030\pi$
$373$	−1.47849e9	−1.47515	−0.737577	+	0.675263i	$0.764030\pi$
$374$	−1.03170e9	−1.01977
$375$	−1.36244e8	−0.133416
$376$	−5.62114e8	−0.545339
$377$	−1.77771e9	−1.70870
$378$	5.84723e8	0.556838
$379$	4.26713e8	0.402623	0.201312	−	0.979527i	$-0.435480\pi$
$379$	4.26713e8	0.402623	0.201312	+	0.979527i	$0.435480\pi$
$380$	2.10503e8	0.196796
$381$	3.85219e8	0.356838
$382$	9.64983e8	0.885723
$383$	1.13179e9	1.02937	0.514683	−	0.857381i	$-0.327910\pi$
$383$	1.13179e9	1.02937	0.514683	+	0.857381i	$0.327910\pi$
$384$	−8.32440e8	−0.750229
$385$	−1.68905e8	−0.150845
$386$	−1.39848e9	−1.23766
$387$	−3.13659e8	−0.275087
$388$	2.89888e8	0.251953
$389$	−1.02164e9	−0.879981	−0.439991	−	0.898002i	$-0.645018\pi$
$389$	−1.02164e9	−0.879981	−0.439991	+	0.898002i	$0.645018\pi$
$390$	−1.10500e8	−0.0943271
$391$	2.27623e8	0.192574
$392$	2.08201e9	1.74575
$393$	7.57581e8	0.629586
$394$	2.20223e9	1.81395
$395$	−1.73361e8	−0.141535
$396$	−4.24164e8	−0.343242
$397$	−2.88404e7	−0.0231331	−0.0115666	−	0.999933i	$-0.503682\pi$
$397$	−2.88404e7	−0.0231331	−0.0115666	+	0.999933i	$0.503682\pi$
$398$	3.44014e9	2.73518
$399$	1.56062e9	1.22996
$400$	4.05404e8	0.316722
$401$	−1.62456e9	−1.25815	−0.629073	−	0.777346i	$-0.716565\pi$
$401$	−1.62456e9	−1.25815	−0.629073	+	0.777346i	$0.716565\pi$
$402$	−1.92868e9	−1.48070
$403$	5.80567e8	0.441860
$404$	2.05983e9	1.55417
$405$	1.72797e7	0.0129254
$406$	7.45095e9	5.52549
$407$	1.28380e9	0.943877
$408$	5.32546e8	0.388192
$409$	−2.35821e9	−1.70432	−0.852159	−	0.523284i	$-0.824707\pi$
$409$	−2.35821e9	−1.70432	−0.852159	+	0.523284i	$0.824707\pi$
$410$	−3.11076e8	−0.222907
$411$	7.84773e8	0.557568
$412$	−2.02994e9	−1.43002
$413$	7.66340e8	0.535299
$414$	1.57514e8	0.109098
$415$	−1.06827e8	−0.0733688
$416$	1.61860e9	1.10233
$417$	1.32290e9	0.893408
$418$	−1.90547e9	−1.27610
$419$	1.67574e9	1.11290	0.556451	−	0.830881i	$-0.312163\pi$
$419$	1.67574e9	1.11290	0.556451	+	0.830881i	$0.312163\pi$
$420$	2.75164e8	0.181225
$421$	−1.67013e9	−1.09085	−0.545423	−	0.838161i	$-0.683631\pi$
$421$	−1.67013e9	−1.09085	−0.545423	+	0.838161i	$0.683631\pi$
$422$	6.76639e8	0.438292
$423$	−3.88679e8	−0.249689
$424$	−1.09192e9	−0.695683
$425$	−1.44180e9	−0.911054
$426$	1.42378e9	0.892298
$427$	1.94732e9	1.21043
$428$	2.78934e8	0.171968
$429$	5.94269e8	0.363398
$430$	−2.48440e8	−0.150689
$431$	1.45319e9	0.874285	0.437142	−	0.899392i	$-0.355991\pi$
$431$	1.45319e9	0.874285	0.437142	+	0.899392i	$0.355991\pi$
$432$	−1.03539e8	−0.0617892
$433$	−1.74251e9	−1.03150	−0.515748	−	0.856740i	$-0.672486\pi$
$433$	−1.74251e9	−1.03150	−0.515748	+	0.856740i	$0.672486\pi$
$434$	−2.43334e9	−1.42886
$435$	2.20190e8	0.128258
$436$	2.64993e9	1.53120
$437$	4.20403e8	0.240980
$438$	5.80949e6	0.00330353
$439$	1.03334e9	0.582932	0.291466	−	0.956581i	$-0.405857\pi$
$439$	1.03334e9	0.582932	0.291466	+	0.956581i	$0.405857\pi$
$440$	−1.06452e8	−0.0595756
$441$	1.43963e9	0.799310
$442$	−2.35478e9	−1.29709
$443$	2.81649e8	0.153920	0.0769601	−	0.997034i	$-0.475479\pi$
$443$	2.81649e8	0.153920	0.0769601	+	0.997034i	$0.475479\pi$
$444$	−2.09143e9	−1.13398
$445$	−1.30461e7	−0.00701812
$446$	−5.03770e9	−2.68881
$447$	3.21828e8	0.170430
$448$	−5.65771e9	−2.97281
$449$	4.17577e8	0.217708	0.108854	−	0.994058i	$-0.465282\pi$
$449$	4.17577e8	0.217708	0.108854	+	0.994058i	$0.465282\pi$
$450$	−9.97721e8	−0.516137
$451$	1.67297e9	0.858755
$452$	2.76790e9	1.40983
$453$	−1.33533e9	−0.674906
$454$	−5.68628e9	−2.85189
$455$	−3.85514e8	−0.191867
$456$	9.83573e8	0.485769
$457$	−1.24124e9	−0.608341	−0.304171	−	0.952618i	$-0.598379\pi$
$457$	−1.24124e9	−0.608341	−0.304171	+	0.952618i	$0.598379\pi$
$458$	3.75180e9	1.82478
$459$	3.68234e8	0.177738
$460$	7.41243e7	0.0355065
$461$	−1.87145e9	−0.889663	−0.444832	−	0.895614i	$-0.646736\pi$
$461$	−1.87145e9	−0.889663	−0.444832	+	0.895614i	$0.646736\pi$
$462$	−2.49077e9	−1.17513
$463$	−3.16121e9	−1.48020	−0.740100	−	0.672497i	$-0.765222\pi$
$463$	−3.16121e9	−1.48020	−0.740100	+	0.672497i	$0.765222\pi$
$464$	−1.31937e9	−0.613133
$465$	−7.19100e7	−0.0331668
$466$	2.16515e9	0.991144
$467$	3.42208e9	1.55482	0.777412	−	0.628992i	$-0.216532\pi$
$467$	3.42208e9	1.55482	0.777412	+	0.628992i	$0.216532\pi$
$468$	−9.68125e8	−0.436587
$469$	−6.72879e9	−3.01184
$470$	−3.07860e8	−0.136776
$471$	3.36865e8	0.148554
$472$	4.82983e8	0.211415
$473$	1.33611e9	0.580533
$474$	−2.55649e9	−1.10260
$475$	−2.66290e9	−1.14006
$476$	5.86378e9	2.49203
$477$	−7.55020e8	−0.318525
$478$	4.03561e9	1.69010
$479$	−2.03897e9	−0.847691	−0.423845	−	0.905735i	$-0.639320\pi$
$479$	−2.03897e9	−0.847691	−0.423845	+	0.905735i	$0.639320\pi$
$480$	−2.00483e8	−0.0827432
$481$	2.93018e9	1.20057
$482$	−5.07834e9	−2.06565
$483$	5.49538e8	0.221913
$484$	−1.84444e9	−0.739447
$485$	5.03057e7	0.0200226
$486$	2.54817e8	0.100693
$487$	1.92494e9	0.755208	0.377604	−	0.925967i	$-0.376748\pi$
$487$	1.92494e9	0.755208	0.377604	+	0.925967i	$0.376748\pi$
$488$	1.22729e9	0.478055
$489$	5.78405e8	0.223692
$490$	1.14028e9	0.437851
$491$	4.53370e8	0.172849	0.0864246	−	0.996258i	$-0.472456\pi$
$491$	4.53370e8	0.172849	0.0864246	+	0.996258i	$0.472456\pi$
$492$	−2.72543e9	−1.03171
$493$	4.69229e9	1.76368
$494$	−4.34910e9	−1.62314
$495$	−7.36071e7	−0.0272773
$496$	4.30882e8	0.158552
$497$	4.96731e9	1.81499
$498$	−1.57533e9	−0.571568
$499$	−1.01960e9	−0.367348	−0.183674	−	0.982987i	$-0.558799\pi$
$499$	−1.01960e9	−0.367348	−0.183674	+	0.982987i	$0.558799\pi$
$500$	−9.45472e8	−0.338262
$501$	−1.97800e9	−0.702740
$502$	−4.22601e9	−1.49096
$503$	−5.25300e9	−1.84043	−0.920215	−	0.391413i	$-0.871986\pi$
$503$	−5.25300e9	−1.84043	−0.920215	+	0.391413i	$0.871986\pi$
$504$	1.28570e9	0.447334
$505$	3.57453e8	0.123509
$506$	−6.70970e8	−0.230238
$507$	−3.37833e8	−0.115126
$508$	2.67325e9	0.904725
$509$	−3.47786e9	−1.16896	−0.584481	−	0.811408i	$-0.698702\pi$
$509$	−3.47786e9	−1.16896	−0.584481	+	0.811408i	$0.698702\pi$
$510$	2.91667e8	0.0973624
$511$	2.02682e7	0.00671959
$512$	1.91116e9	0.629294
$513$	6.80101e8	0.222414
$514$	4.15465e9	1.34947
$515$	−3.52265e8	−0.113643
$516$	−2.17665e9	−0.697454
$517$	1.65567e9	0.526935
$518$	−1.22813e10	−3.88231
$519$	−2.12734e9	−0.667962
$520$	−2.42969e8	−0.0757773
$521$	1.14363e9	0.354287	0.177143	−	0.984185i	$-0.443314\pi$
$521$	1.14363e9	0.354287	0.177143	+	0.984185i	$0.443314\pi$
$522$	3.24705e9	0.999177
$523$	3.78149e9	1.15587	0.577933	−	0.816084i	$-0.303860\pi$
$523$	3.78149e9	1.15587	0.577933	+	0.816084i	$0.303860\pi$
$524$	5.25727e9	1.59625
$525$	−3.48086e9	−1.04986
$526$	5.03517e8	0.150857
$527$	−1.53242e9	−0.456078
$528$	4.41051e8	0.130398
$529$	1.48036e8	0.0434783
$530$	−5.98028e8	−0.174484
$531$	3.33963e8	0.0967983
$532$	1.08300e10	3.11844
$533$	3.81843e9	1.09229
$534$	−1.92385e8	−0.0546735
$535$	4.84048e7	0.0136663
$536$	−4.24080e9	−1.18952
$537$	3.37679e9	0.941009
$538$	4.15005e9	1.14899
$539$	−6.13244e9	−1.68683
$540$	1.19914e8	0.0327710
$541$	3.46679e9	0.941320	0.470660	−	0.882315i	$-0.344016\pi$
$541$	3.46679e9	0.941320	0.470660	+	0.882315i	$0.344016\pi$
$542$	5.81782e9	1.56951
$543$	2.60078e9	0.697115
$544$	−4.27232e9	−1.13780
$545$	4.59854e8	0.121684
$546$	−5.68501e9	−1.49471
$547$	3.72645e9	0.973507	0.486754	−	0.873539i	$-0.338181\pi$
$547$	3.72645e9	0.973507	0.486754	+	0.873539i	$0.338181\pi$
$548$	5.44598e9	1.41366
$549$	8.48621e8	0.218882
$550$	4.25003e9	1.08924
$551$	8.66632e9	2.20701
$552$	3.46344e8	0.0876438
$553$	−8.91910e9	−2.24276
$554$	8.58732e9	2.14572
$555$	−3.62937e8	−0.0901167
$556$	9.18032e9	2.26514
$557$	6.64263e8	0.162872	0.0814360	−	0.996679i	$-0.474049\pi$
$557$	6.64263e8	0.162872	0.0814360	+	0.996679i	$0.474049\pi$
$558$	−1.06043e9	−0.258381
$559$	3.04957e9	0.738410
$560$	−2.86119e8	−0.0688476
$561$	−1.56858e9	−0.375091
$562$	−7.76749e9	−1.84588
$563$	−6.14488e9	−1.45122	−0.725612	−	0.688104i	$-0.758443\pi$
$563$	−6.14488e9	−1.45122	−0.725612	+	0.688104i	$0.758443\pi$
$564$	−2.69726e9	−0.633061
$565$	4.80327e8	0.112039
$566$	−3.95051e9	−0.915789
$567$	8.89007e8	0.204816
$568$	3.13063e9	0.716824
$569$	−7.57121e9	−1.72295	−0.861474	−	0.507801i	$-0.830458\pi$
$569$	−7.57121e9	−1.72295	−0.861474	+	0.507801i	$0.830458\pi$
$570$	5.38687e8	0.121836
$571$	6.99906e9	1.57331	0.786654	−	0.617395i	$-0.211812\pi$
$571$	6.99906e9	1.57331	0.786654	+	0.617395i	$0.211812\pi$
$572$	4.12396e9	0.921358
$573$	1.46715e9	0.325787
$574$	−1.60043e10	−3.53219
$575$	−9.37684e8	−0.205693
$576$	−2.46557e9	−0.537575
$577$	−4.05834e9	−0.879495	−0.439747	−	0.898121i	$-0.644932\pi$
$577$	−4.05834e9	−0.879495	−0.439747	+	0.898121i	$0.644932\pi$
$578$	−1.07157e9	−0.230821
$579$	−2.12623e9	−0.455235
$580$	1.52802e9	0.325186
$581$	−5.49602e9	−1.16260
$582$	7.41836e8	0.155983
$583$	3.21619e9	0.672205
$584$	1.27740e7	0.00265388
$585$	−1.68003e8	−0.0346954
$586$	3.21456e9	0.659903
$587$	4.47493e9	0.913172	0.456586	−	0.889679i	$-0.349072\pi$
$587$	4.47493e9	0.913172	0.456586	+	0.889679i	$0.349072\pi$
$588$	9.99037e9	2.02657
$589$	−2.83026e9	−0.570719
$590$	2.64522e8	0.0530249
$591$	3.34825e9	0.667208
$592$	2.17470e9	0.430798
$593$	−2.72369e9	−0.536373	−0.268186	−	0.963367i	$-0.586424\pi$
$593$	−2.72369e9	−0.536373	−0.268186	+	0.963367i	$0.586424\pi$
$594$	−1.08545e9	−0.212500
$595$	1.01757e9	0.198041
$596$	2.23334e9	0.432109
$597$	5.23036e9	1.00605
$598$	−1.53144e9	−0.292851
$599$	5.09808e9	0.969199	0.484600	−	0.874736i	$-0.338965\pi$
$599$	5.09808e9	0.969199	0.484600	+	0.874736i	$0.338965\pi$
$600$	−2.19380e9	−0.414637
$601$	−1.25245e9	−0.235343	−0.117671	−	0.993053i	$-0.537543\pi$
$601$	−1.25245e9	−0.235343	−0.117671	+	0.993053i	$0.537543\pi$
$602$	−1.27817e10	−2.38782
$603$	−2.93234e9	−0.544633
$604$	−9.26656e9	−1.71116
$605$	−3.20075e8	−0.0587636
$606$	5.27120e9	0.962177
$607$	−4.63329e9	−0.840870	−0.420435	−	0.907323i	$-0.638123\pi$
$607$	−4.63329e9	−0.840870	−0.420435	+	0.907323i	$0.638123\pi$
$608$	−7.89066e9	−1.42381
$609$	1.13284e10	2.03239
$610$	6.72166e8	0.119901
$611$	3.77896e9	0.670236
$612$	2.55538e9	0.450635
$613$	2.89197e9	0.507086	0.253543	−	0.967324i	$-0.418404\pi$
$613$	2.89197e9	0.507086	0.253543	+	0.967324i	$0.418404\pi$
$614$	−7.08125e9	−1.23458
$615$	−4.72957e8	−0.0819897
$616$	−5.47673e9	−0.944037
$617$	6.68525e9	1.14583	0.572914	−	0.819615i	$-0.305813\pi$
$617$	6.68525e9	1.14583	0.572914	+	0.819615i	$0.305813\pi$
$618$	−5.19469e9	−0.885320
$619$	6.61297e9	1.12067	0.560337	−	0.828265i	$-0.310672\pi$
$619$	6.61297e9	1.12067	0.560337	+	0.828265i	$0.310672\pi$
$620$	−4.99023e8	−0.0840911
$621$	2.39483e8	0.0401286
$622$	9.34263e9	1.55669
$623$	−6.71196e8	−0.111209
$624$	1.00667e9	0.165860
$625$	5.85685e9	0.959586
$626$	4.80050e9	0.782126
$627$	−2.89705e9	−0.469375
$628$	2.33769e9	0.376642
$629$	−7.73424e9	−1.23920
$630$	7.04155e8	0.112196
$631$	4.18438e9	0.663023	0.331511	−	0.943451i	$-0.392441\pi$
$631$	4.18438e9	0.663023	0.331511	+	0.943451i	$0.392441\pi$
$632$	−5.62123e9	−0.885772
$633$	1.02875e9	0.161213
$634$	1.30589e10	2.03514
$635$	4.63902e8	0.0718982
$636$	−5.23950e9	−0.807588
$637$	−1.39969e10	−2.14557
$638$	−1.38316e10	−2.10863
$639$	2.16470e9	0.328205
$640$	−1.00247e9	−0.151162
$641$	4.58753e9	0.687980	0.343990	−	0.938973i	$-0.388221\pi$
$641$	4.58753e9	0.687980	0.343990	+	0.938973i	$0.388221\pi$
$642$	7.13804e8	0.106465
$643$	−1.08232e10	−1.60552	−0.802762	−	0.596300i	$-0.796637\pi$
$643$	−1.08232e10	−1.60552	−0.802762	+	0.596300i	$0.796637\pi$
$644$	3.81355e9	0.562637
$645$	−3.77725e8	−0.0554264
$646$	1.14795e10	1.67537
$647$	2.87142e9	0.416804	0.208402	−	0.978043i	$-0.433174\pi$
$647$	2.87142e9	0.416804	0.208402	+	0.978043i	$0.433174\pi$
$648$	5.60294e8	0.0808916
$649$	−1.42260e9	−0.204280
$650$	9.70041e9	1.38546
$651$	−3.69963e9	−0.525563
$652$	4.01387e9	0.567148
$653$	1.04014e10	1.46183	0.730914	−	0.682470i	$-0.239094\pi$
$653$	1.04014e10	1.46183	0.730914	+	0.682470i	$0.239094\pi$
$654$	6.78127e9	0.947957
$655$	9.12319e8	0.126853
$656$	2.83394e9	0.391947
$657$	8.83269e6	0.00121511
$658$	−1.58388e10	−2.16736
$659$	−1.11771e9	−0.152135	−0.0760677	−	0.997103i	$-0.524237\pi$
$659$	−1.11771e9	−0.152135	−0.0760677	+	0.997103i	$0.524237\pi$
$660$	−5.10801e8	−0.0691588
$661$	−8.43061e9	−1.13541	−0.567706	−	0.823231i	$-0.692169\pi$
$661$	−8.43061e9	−1.13541	−0.567706	+	0.823231i	$0.692169\pi$
$662$	−1.48921e10	−1.99504
$663$	−3.58018e9	−0.477098
$664$	−3.46385e9	−0.459167
$665$	1.87938e9	0.247821
$666$	−5.35207e9	−0.702040
$667$	3.05166e9	0.398195
$668$	−1.37265e10	−1.78173
$669$	−7.65927e9	−0.988999
$670$	−2.32262e9	−0.298343
$671$	−3.61490e9	−0.461921
$672$	−1.03144e10	−1.31115
$673$	8.81023e9	1.11413	0.557063	−	0.830470i	$-0.311928\pi$
$673$	8.81023e9	1.11413	0.557063	+	0.830470i	$0.311928\pi$
$674$	2.01135e9	0.253033
$675$	−1.51693e9	−0.189846
$676$	−2.34441e9	−0.291890
$677$	−6.28975e9	−0.779064	−0.389532	−	0.921013i	$-0.627363\pi$
$677$	−6.28975e9	−0.779064	−0.389532	+	0.921013i	$0.627363\pi$
$678$	7.08318e9	0.872819
$679$	2.58813e9	0.317279
$680$	6.41320e8	0.0782157
$681$	−8.64536e9	−1.04898
$682$	4.51714e9	0.545278
$683$	9.76407e9	1.17262	0.586311	−	0.810086i	$-0.300580\pi$
$683$	9.76407e9	1.17262	0.586311	+	0.810086i	$0.300580\pi$
$684$	4.71960e9	0.563909
$685$	9.45066e8	0.112343
$686$	3.42003e10	4.04479
$687$	5.70420e9	0.671192
$688$	2.26332e9	0.264963
$689$	7.34073e9	0.855012
$690$	1.89687e8	0.0219819
$691$	−7.16059e9	−0.825611	−0.412806	−	0.910819i	$-0.635451\pi$
$691$	−7.16059e9	−0.825611	−0.412806	+	0.910819i	$0.635451\pi$
$692$	−1.47628e10	−1.69355
$693$	−3.78694e9	−0.432237
$694$	2.57466e10	2.92389
$695$	1.59310e9	0.180010
$696$	7.13966e9	0.802685
$697$	−1.00788e10	−1.12744
$698$	−6.47644e9	−0.720846
$699$	3.29187e9	0.364563
$700$	−2.41556e10	−2.66180
$701$	−9.03880e7	−0.00991055	−0.00495527	−	0.999988i	$-0.501577\pi$
$701$	−9.03880e7	−0.00991055	−0.00495527	+	0.999988i	$0.501577\pi$
$702$	−2.47747e9	−0.270289
$703$	−1.42846e10	−1.55069
$704$	1.05027e10	1.13448
$705$	−4.68068e8	−0.0503092
$706$	−3.54884e9	−0.379551
$707$	1.83902e10	1.95713
$708$	2.31756e9	0.245422
$709$	1.66026e10	1.74950	0.874751	−	0.484573i	$-0.161025\pi$
$709$	1.66026e10	1.74950	0.874751	+	0.484573i	$0.161025\pi$
$710$	1.71460e9	0.179787
$711$	−3.88685e9	−0.405560
$712$	−4.23019e8	−0.0439218
$713$	−9.96615e8	−0.102971
$714$	1.50057e10	1.54281
$715$	7.15650e8	0.0732200
$716$	2.34334e10	2.38583
$717$	6.13571e9	0.621652
$718$	1.36971e8	0.0138099
$719$	−1.35703e10	−1.36156	−0.680782	−	0.732486i	$-0.738360\pi$
$719$	−1.35703e10	−1.36156	−0.680782	+	0.732486i	$0.738360\pi$
$720$	−1.24688e8	−0.0124497
$721$	−1.81233e10	−1.80079
$722$	5.32791e9	0.526837
$723$	−7.72105e9	−0.759788
$724$	1.80483e10	1.76746
$725$	−1.93297e10	−1.88383
$726$	−4.72001e9	−0.457788
$727$	−1.71708e10	−1.65737	−0.828685	−	0.559715i	$-0.810911\pi$
$727$	−1.71708e10	−1.65737	−0.828685	+	0.559715i	$0.810911\pi$
$728$	−1.25003e10	−1.20077
$729$	3.87420e8	0.0370370
$730$	6.99610e6	0.000665619 0
$731$	−8.04939e9	−0.762170
$732$	5.88905e9	0.554953
$733$	−1.40815e10	−1.32064	−0.660322	−	0.750983i	$-0.729580\pi$
$733$	−1.40815e10	−1.32064	−0.660322	+	0.750983i	$0.729580\pi$
$734$	9.41026e9	0.878346
$735$	1.73368e9	0.161051
$736$	−2.77853e9	−0.256887
$737$	1.24910e10	1.14937
$738$	−6.97450e9	−0.638727
$739$	1.44525e10	1.31731	0.658656	−	0.752444i	$-0.271125\pi$
$739$	1.44525e10	1.31731	0.658656	+	0.752444i	$0.271125\pi$
$740$	−2.51862e9	−0.228482
$741$	−6.61233e9	−0.597022
$742$	−3.07673e10	−2.76488
$743$	1.88767e10	1.68836	0.844181	−	0.536059i	$-0.180088\pi$
$743$	1.88767e10	1.68836	0.844181	+	0.536059i	$0.180088\pi$
$744$	−2.33168e9	−0.207569
$745$	3.87562e8	0.0343396
$746$	−2.62559e10	−2.31548
$747$	−2.39511e9	−0.210234
$748$	−1.08852e10	−0.951005
$749$	2.49033e9	0.216556
$750$	−2.41950e9	−0.209417
$751$	−9.16087e9	−0.789218	−0.394609	−	0.918849i	$-0.629120\pi$
$751$	−9.16087e9	−0.789218	−0.394609	+	0.918849i	$0.629120\pi$
$752$	2.80465e9	0.240500
$753$	−6.42518e9	−0.548407
$754$	−3.15697e10	−2.68207
$755$	−1.60807e9	−0.135985
$756$	6.16931e9	0.519291
$757$	−1.77258e10	−1.48515	−0.742573	−	0.669765i	$-0.766395\pi$
$757$	−1.77258e10	−1.48515	−0.742573	+	0.669765i	$0.766395\pi$
$758$	7.57783e9	0.631979
$759$	−1.02014e9	−0.0846860
$760$	1.18447e9	0.0978762
$761$	8.48556e9	0.697966	0.348983	−	0.937129i	$-0.386527\pi$
$761$	8.48556e9	0.697966	0.348983	+	0.937129i	$0.386527\pi$
$762$	6.84096e9	0.560112
$763$	2.36586e10	1.92820
$764$	1.01814e10	0.825999
$765$	4.43447e8	0.0358118
$766$	2.00990e10	1.61575
$767$	−3.24698e9	−0.259834
$768$	−3.09433e9	−0.246491
$769$	1.80190e10	1.42886	0.714430	−	0.699707i	$-0.246686\pi$
$769$	1.80190e10	1.42886	0.714430	+	0.699707i	$0.246686\pi$
$770$	−2.99952e9	−0.236774
$771$	6.31669e9	0.496363
$772$	−1.47551e10	−1.15420
$773$	−1.25368e10	−0.976246	−0.488123	−	0.872775i	$-0.662318\pi$
$773$	−1.25368e10	−0.976246	−0.488123	+	0.872775i	$0.662318\pi$
$774$	−5.57015e9	−0.431791
$775$	6.31272e9	0.487148
$776$	1.63116e9	0.125308
$777$	−1.86724e10	−1.42799
$778$	−1.81429e10	−1.38127
$779$	−1.86148e10	−1.41084
$780$	−1.16587e9	−0.0879667
$781$	−9.22108e9	−0.692633
$782$	4.04226e9	0.302274
$783$	4.93678e9	0.367517
$784$	−1.03881e10	−0.769894
$785$	4.05671e8	0.0299316
$786$	1.34536e10	0.988232
$787$	1.06633e10	0.779797	0.389899	−	0.920858i	$-0.372510\pi$
$787$	1.06633e10	0.779797	0.389899	+	0.920858i	$0.372510\pi$
$788$	2.32353e10	1.69164
$789$	7.65543e8	0.0554881
$790$	−3.07866e9	−0.222160
$791$	2.47119e10	1.77537
$792$	−2.38671e9	−0.170711
$793$	−8.25077e9	−0.587541
$794$	−5.12165e8	−0.0363110
$795$	−9.09236e8	−0.0641788
$796$	3.62963e10	2.55075
$797$	−1.99461e10	−1.39558	−0.697789	−	0.716303i	$-0.745833\pi$
$797$	−1.99461e10	−1.39558	−0.697789	+	0.716303i	$0.745833\pi$
$798$	2.77144e10	1.93061
$799$	−9.97461e9	−0.691803
$800$	1.75996e10	1.21532
$801$	−2.92500e8	−0.0201100
$802$	−2.88499e10	−1.97485
$803$	−3.76250e7	−0.00256432
$804$	−2.03491e10	−1.38086
$805$	6.61783e8	0.0447126
$806$	1.03101e10	0.693567
$807$	6.30970e9	0.422621
$808$	1.15904e10	0.772961
$809$	−2.22638e10	−1.47836	−0.739179	−	0.673509i	$-0.764786\pi$
$809$	−2.22638e10	−1.47836	−0.739179	+	0.673509i	$0.764786\pi$
$810$	3.06864e8	0.0202884
$811$	8.40970e9	0.553615	0.276807	−	0.960925i	$-0.410724\pi$
$811$	8.40970e9	0.553615	0.276807	+	0.960925i	$0.410724\pi$
$812$	7.86137e10	5.15291
$813$	8.84536e9	0.577296
$814$	2.27984e10	1.48156
$815$	6.96546e8	0.0450711
$816$	−2.65712e9	−0.171197
$817$	−1.48666e10	−0.953752
$818$	−4.18785e10	−2.67519
$819$	−8.64343e9	−0.549785
$820$	−3.28211e9	−0.207876
$821$	−1.15961e10	−0.731324	−0.365662	−	0.930748i	$-0.619157\pi$
$821$	−1.15961e10	−0.731324	−0.365662	+	0.930748i	$0.619157\pi$
$822$	1.39365e10	0.875189
$823$	1.52874e10	0.955951	0.477975	−	0.878373i	$-0.341371\pi$
$823$	1.52874e10	0.955951	0.477975	+	0.878373i	$0.341371\pi$
$824$	−1.14222e10	−0.711218
$825$	6.46171e9	0.400644
$826$	1.36091e10	0.840233
$827$	3.33339e9	0.204935	0.102468	−	0.994736i	$-0.467326\pi$
$827$	3.33339e9	0.204935	0.102468	+	0.994736i	$0.467326\pi$
$828$	1.66191e9	0.101742
$829$	2.92842e10	1.78522	0.892612	−	0.450825i	$-0.148870\pi$
$829$	2.92842e10	1.78522	0.892612	+	0.450825i	$0.148870\pi$
$830$	−1.89709e9	−0.115164
$831$	1.30561e10	0.789240
$832$	2.39717e10	1.44300
$833$	3.69449e10	2.21461
$834$	2.34928e10	1.40234
$835$	−2.38202e9	−0.141593
$836$	−2.01043e10	−1.19005
$837$	−1.61226e9	−0.0950377
$838$	2.97587e10	1.74687
$839$	−4.45381e9	−0.260355	−0.130177	−	0.991491i	$-0.541555\pi$
$839$	−4.45381e9	−0.260355	−0.130177	+	0.991491i	$0.541555\pi$
$840$	1.54831e9	0.0901320
$841$	4.56581e10	2.64686
$842$	−2.96592e10	−1.71225
$843$	−1.18096e10	−0.678952
$844$	7.13910e9	0.408738
$845$	−4.06836e8	−0.0231964
$846$	−6.90239e9	−0.391925
$847$	−1.64672e10	−0.931169
$848$	5.44811e9	0.306803
$849$	−6.00631e9	−0.336846
$850$	−2.56044e10	−1.43004
$851$	−5.03001e9	−0.279779
$852$	1.50221e10	0.832131
$853$	−2.71758e10	−1.49921	−0.749603	−	0.661887i	$-0.769756\pi$
$853$	−2.71758e10	−1.49921	−0.749603	+	0.661887i	$0.769756\pi$
$854$	3.45816e10	1.89995
$855$	8.19014e8	0.0448136
$856$	1.56952e9	0.0855282
$857$	1.59445e10	0.865323	0.432662	−	0.901556i	$-0.357574\pi$
$857$	1.59445e10	0.865323	0.432662	+	0.901556i	$0.357574\pi$
$858$	1.05534e10	0.570409
$859$	−3.19563e10	−1.72020	−0.860102	−	0.510122i	$-0.829600\pi$
$859$	−3.19563e10	−1.72020	−0.860102	+	0.510122i	$0.829600\pi$
$860$	−2.62124e9	−0.140528
$861$	−2.43327e10	−1.29921
$862$	2.58067e10	1.37232
$863$	−1.24014e10	−0.656800	−0.328400	−	0.944539i	$-0.606509\pi$
$863$	−1.24014e10	−0.656800	−0.328400	+	0.944539i	$0.606509\pi$
$864$	−4.49492e9	−0.237096
$865$	−2.56186e9	−0.134586
$866$	−3.09446e10	−1.61909
$867$	−1.62921e9	−0.0849006
$868$	−2.56738e10	−1.33251
$869$	1.65570e10	0.855879
$870$	3.91027e9	0.201321
$871$	2.85099e10	1.46195
$872$	1.49107e10	0.761538
$873$	1.12788e9	0.0573737
$874$	7.46577e9	0.378255
$875$	−8.44119e9	−0.425966
$876$	6.12949e7	0.00308078
$877$	−9.66888e9	−0.484036	−0.242018	−	0.970272i	$-0.577809\pi$
$877$	−9.66888e9	−0.484036	−0.242018	+	0.970272i	$0.577809\pi$
$878$	1.83507e10	0.915001
$879$	4.88739e9	0.242726
$880$	5.31138e8	0.0262735
$881$	−1.13975e10	−0.561556	−0.280778	−	0.959773i	$-0.590592\pi$
$881$	−1.13975e10	−0.561556	−0.280778	+	0.959773i	$0.590592\pi$
$882$	2.55658e10	1.25464
$883$	−6.25195e9	−0.305600	−0.152800	−	0.988257i	$-0.548829\pi$
$883$	−6.25195e9	−0.305600	−0.152800	+	0.988257i	$0.548829\pi$
$884$	−2.48448e10	−1.20963
$885$	4.02176e8	0.0195036
$886$	5.00170e9	0.241601
$887$	−4.13242e9	−0.198826	−0.0994128	−	0.995046i	$-0.531696\pi$
$887$	−4.13242e9	−0.198826	−0.0994128	+	0.995046i	$0.531696\pi$
$888$	−1.17682e10	−0.563981
$889$	2.38668e10	1.13930
$890$	−2.31681e8	−0.0110160
$891$	−1.65031e9	−0.0781616
$892$	−5.31519e10	−2.50751
$893$	−1.84224e10	−0.865696
$894$	5.71521e9	0.267517
$895$	4.06651e9	0.189601
$896$	−5.15750e10	−2.39531
$897$	−2.32839e9	−0.107716
$898$	7.41559e9	0.341726
$899$	−2.05446e10	−0.943057
$900$	−1.05268e10	−0.481334
$901$	−1.93760e10	−0.882524
$902$	2.97095e10	1.34795
$903$	−1.94332e10	−0.878289
$904$	1.55746e10	0.701176
$905$	3.13200e9	0.140460
$906$	−2.37135e10	−1.05937
$907$	1.64191e10	0.730674	0.365337	−	0.930875i	$-0.380954\pi$
$907$	1.64191e10	0.730674	0.365337	+	0.930875i	$0.380954\pi$
$908$	−5.99949e10	−2.65959
$909$	8.01427e9	0.353908
$910$	−6.84619e9	−0.301165
$911$	−2.35385e10	−1.03149	−0.515745	−	0.856742i	$-0.672485\pi$
$911$	−2.35385e10	−1.03149	−0.515745	+	0.856742i	$0.672485\pi$
$912$	−4.90750e9	−0.214229
$913$	1.02025e10	0.443671
$914$	−2.20426e10	−0.954885
$915$	1.02195e9	0.0441020
$916$	3.95846e10	1.70174
$917$	4.69370e10	2.01012
$918$	6.53932e9	0.278986
$919$	−2.89668e10	−1.23111	−0.615554	−	0.788095i	$-0.711068\pi$
$919$	−2.89668e10	−1.23111	−0.615554	+	0.788095i	$0.711068\pi$
$920$	4.17086e8	0.0176591
$921$	−1.07663e10	−0.454105
$922$	−3.32344e10	−1.39646
$923$	−2.10465e10	−0.880995
$924$	−2.62797e10	−1.09589
$925$	3.18609e10	1.32362
$926$	−5.61387e10	−2.32340
$927$	−7.89796e9	−0.325639
$928$	−5.72775e10	−2.35270
$929$	−2.22692e10	−0.911277	−0.455638	−	0.890165i	$-0.650589\pi$
$929$	−2.22692e10	−0.911277	−0.455638	+	0.890165i	$0.650589\pi$
$930$	−1.27702e9	−0.0520604
$931$	6.82346e10	2.77128
$932$	2.28441e10	0.924312
$933$	1.42044e10	0.572583
$934$	6.07713e10	2.44053
$935$	−1.88897e9	−0.0755760
$936$	−5.44749e9	−0.217136
$937$	3.15430e10	1.25261	0.626303	−	0.779580i	$-0.284567\pi$
$937$	3.15430e10	1.25261	0.626303	+	0.779580i	$0.284567\pi$
$938$	−1.19494e11	−4.72755
$939$	7.29864e9	0.287682
$940$	−3.24818e9	−0.127554
$941$	−1.49779e10	−0.585987	−0.292993	−	0.956114i	$-0.594651\pi$
$941$	−1.49779e10	−0.585987	−0.292993	+	0.956114i	$0.594651\pi$
$942$	5.98226e9	0.233178
$943$	−6.55481e9	−0.254548
$944$	−2.40983e9	−0.0932360
$945$	1.07059e9	0.0412679
$946$	2.37274e10	0.911236
$947$	2.53390e10	0.969539	0.484769	−	0.874642i	$-0.338904\pi$
$947$	2.53390e10	0.969539	0.484769	+	0.874642i	$0.338904\pi$
$948$	−2.69730e10	−1.02825
$949$	−8.58764e7	−0.00326169
$950$	−4.72894e10	−1.78950
$951$	1.98546e10	0.748565
$952$	3.29946e10	1.23941
$953$	7.27301e9	0.272201	0.136100	−	0.990695i	$-0.456543\pi$
$953$	7.27301e9	0.272201	0.136100	+	0.990695i	$0.456543\pi$
$954$	−1.34081e10	−0.499975
$955$	1.76682e9	0.0656419
$956$	4.25791e10	1.57614
$957$	−2.10294e10	−0.775596
$958$	−3.62093e10	−1.33058
$959$	4.86218e10	1.78019
$960$	−2.96918e9	−0.108315
$961$	−2.08031e10	−0.756131
$962$	5.20358e10	1.88447
$963$	1.08526e9	0.0391600
$964$	−5.35807e10	−1.92636
$965$	−2.56052e9	−0.0917241
$966$	9.75902e9	0.348326
$967$	−4.43467e10	−1.57714	−0.788568	−	0.614948i	$-0.789177\pi$
$967$	−4.43467e10	−1.57714	−0.788568	+	0.614948i	$0.789177\pi$
$968$	−1.03784e10	−0.367762
$969$	1.74533e10	0.616233
$970$	8.93359e8	0.0314286
$971$	−4.92204e9	−0.172535	−0.0862675	−	0.996272i	$-0.527494\pi$
$971$	−4.92204e9	−0.172535	−0.0862675	+	0.996272i	$0.527494\pi$
$972$	2.68853e9	0.0939036
$973$	8.19620e10	2.85245
$974$	3.41843e10	1.18542
$975$	1.47484e10	0.509599
$976$	−6.12352e9	−0.210827
$977$	−3.86252e10	−1.32507	−0.662537	−	0.749029i	$-0.730520\pi$
$977$	−3.86252e10	−1.32507	−0.662537	+	0.749029i	$0.730520\pi$
$978$	1.02717e10	0.351119
$979$	1.24598e9	0.0424395
$980$	1.20309e10	0.408327
$981$	1.03102e10	0.348678
$982$	8.05121e9	0.271313
$983$	1.46424e10	0.491673	0.245836	−	0.969311i	$-0.420937\pi$
$983$	1.46424e10	0.491673	0.245836	+	0.969311i	$0.420937\pi$
$984$	−1.53356e10	−0.513119
$985$	4.03214e9	0.134434
$986$	8.33286e10	2.76837
$987$	−2.40812e10	−0.797200
$988$	−4.58866e10	−1.51369
$989$	−5.23497e9	−0.172079
$990$	−1.30716e9	−0.0428159
$991$	8.70242e9	0.284042	0.142021	−	0.989864i	$-0.454640\pi$
$991$	8.70242e9	0.284042	0.142021	+	0.989864i	$0.454640\pi$
$992$	1.87057e10	0.608393
$993$	−2.26418e10	−0.733818
$994$	8.82125e10	2.84890
$995$	6.29868e9	0.202707
$996$	−1.66210e10	−0.533027
$997$	−2.46012e10	−0.786182	−0.393091	−	0.919500i	$-0.628594\pi$
$997$	−2.46012e10	−0.786182	−0.393091	+	0.919500i	$0.628594\pi$
$998$	−1.81067e10	−0.576609
$999$	−8.13723e9	−0.258224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.d.1.7	✓	8
3.2	odd	2		207.8.a.e.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.d.1.7	✓	8	1.1	even	1	trivial
207.8.a.e.1.2		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)

\(f(q)\)	\(=\)	\(q+17.7586 q^{2} +27.0000 q^{3} +187.368 q^{4} +32.5148 q^{5} +479.482 q^{6} +1672.82 q^{7} +1054.29 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+17.7586 q^{2} +27.0000 q^{3} +187.368 q^{4} +32.5148 q^{5} +479.482 q^{6} +1672.82 q^{7} +1054.29 q^{8} +729.000 q^{9} +577.418 q^{10} -3105.35 q^{11} +5058.93 q^{12} -7087.75 q^{13} +29707.0 q^{14} +877.901 q^{15} -5260.35 q^{16} +18708.2 q^{17} +12946.0 q^{18} +34552.7 q^{19} +6092.24 q^{20} +45166.2 q^{21} -55146.7 q^{22} +12167.0 q^{23} +28465.9 q^{24} -77067.8 q^{25} -125869. q^{26} +19683.0 q^{27} +313434. q^{28} +250815. q^{29} +15590.3 q^{30} -81911.3 q^{31} -228366. q^{32} -83844.5 q^{33} +332232. q^{34} +54391.6 q^{35} +136591. q^{36} -413414. q^{37} +613608. q^{38} -191369. q^{39} +34280.1 q^{40} -538737. q^{41} +802089. q^{42} -430259. q^{43} -581843. q^{44} +23703.3 q^{45} +216069. q^{46} -533167. q^{47} -142029. q^{48} +1.97480e6 q^{49} -1.36862e6 q^{50} +505122. q^{51} -1.32802e6 q^{52} -1.03569e6 q^{53} +349543. q^{54} -100970. q^{55} +1.76364e6 q^{56} +932923. q^{57} +4.45412e6 q^{58} +458111. q^{59} +164490. q^{60} +1.16409e6 q^{61} -1.45463e6 q^{62} +1.21949e6 q^{63} -3.38213e6 q^{64} -230457. q^{65} -1.48896e6 q^{66} -4.02241e6 q^{67} +3.50532e6 q^{68} +328509. q^{69} +965919. q^{70} +2.96942e6 q^{71} +768579. q^{72} +12116.2 q^{73} -7.34166e6 q^{74} -2.08083e6 q^{75} +6.47407e6 q^{76} -5.19470e6 q^{77} -3.39845e6 q^{78} -5.33176e6 q^{79} -171039. q^{80} +531441. q^{81} -9.56721e6 q^{82} -3.28547e6 q^{83} +8.46271e6 q^{84} +608295. q^{85} -7.64081e6 q^{86} +6.77199e6 q^{87} -3.27395e6 q^{88} -401235. q^{89} +420938. q^{90} -1.18566e7 q^{91} +2.27971e6 q^{92} -2.21161e6 q^{93} -9.46830e6 q^{94} +1.12348e6 q^{95} -6.16588e6 q^{96} +1.54716e6 q^{97} +3.50696e7 q^{98} -2.26380e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9}+O(q^{10}) \) 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

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