LMFDB - Embedded newform 169.6.b.a.168.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,6,Mod(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	169.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$27.1048655484$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.4
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.6.b.a.168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.56155i q^{2} -1.63068 q^{3} +11.1922 q^{4} +103.462i q^{5} -7.43845i q^{6} +126.309i q^{7} +197.024i q^{8} -240.341 q^{9} +O(q^{10})$	\(q+4.56155i q^{2} -1.63068 q^{3} +11.1922 q^{4} +103.462i q^{5} -7.43845i q^{6} +126.309i q^{7} +197.024i q^{8} -240.341 q^{9} -471.948 q^{10} -14.8296i q^{11} -18.2510 q^{12} -576.164 q^{14} -168.714i q^{15} -540.582 q^{16} +1051.12 q^{17} -1096.33i q^{18} +213.723i q^{19} +1157.97i q^{20} -205.969i q^{21} +67.6458 q^{22} +4231.16 q^{23} -321.283i q^{24} -7579.41 q^{25} +788.176 q^{27} +1413.68i q^{28} -504.955 q^{29} +769.597 q^{30} -4783.58i q^{31} +3838.86i q^{32} +24.1823i q^{33} +4794.75i q^{34} -13068.2 q^{35} -2689.95 q^{36} -4635.74i q^{37} -974.911 q^{38} -20384.5 q^{40} -7944.15i q^{41} +939.541 q^{42} +8516.41 q^{43} -165.976i q^{44} -24866.2i q^{45} +19300.7i q^{46} +24921.2i q^{47} +881.518 q^{48} +853.113 q^{49} -34573.9i q^{50} -1714.05 q^{51} -7808.46 q^{53} +3595.31i q^{54} +1534.30 q^{55} -24885.8 q^{56} -348.515i q^{57} -2303.38i q^{58} -37337.5i q^{59} -1888.29i q^{60} -18172.2 q^{61} +21820.5 q^{62} -30357.1i q^{63} -34809.8 q^{64} -110.309 q^{66} +34559.9i q^{67} +11764.4 q^{68} -6899.68 q^{69} -59611.1i q^{70} -41255.7i q^{71} -47352.8i q^{72} -1056.42i q^{73} +21146.2 q^{74} +12359.6 q^{75} +2392.04i q^{76} +1873.10 q^{77} -47719.3 q^{79} -55929.8i q^{80} +57117.6 q^{81} +36237.7 q^{82} +74799.0i q^{83} -2305.26i q^{84} +108751. i q^{85} +38848.0i q^{86} +823.421 q^{87} +2921.78 q^{88} +9799.26i q^{89} +113428. q^{90} +47356.1 q^{92} +7800.50i q^{93} -113679. q^{94} -22112.3 q^{95} -6259.97i q^{96} +138432. i q^{97} +3891.52i q^{98} +3564.15i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 56 q^{3} + 86 q^{4} + 424 q^{9}+O(q^{10}) $ 4 * q - 56 * q^3 + 86 * q^4 + 424 * q^9	$ 4 q - 56 q^{3} + 86 q^{4} + 424 q^{9} - 890 q^{10} - 1714 q^{12} - 1010 q^{14} + 930 q^{16} + 5260 q^{17} + 452 q^{22} + 5248 q^{23} - 16464 q^{25} - 9464 q^{27} - 1624 q^{29} + 118 q^{30} - 46088 q^{35} + 23396 q^{36} - 2036 q^{38} - 37330 q^{40} - 1874 q^{42} - 4840 q^{43} - 51270 q^{48} - 17368 q^{49} - 86696 q^{51} - 87440 q^{53} - 41328 q^{55} - 40690 q^{56} + 3968 q^{61} + 46216 q^{62} - 6434 q^{64} - 8572 q^{66} + 123970 q^{68} + 70960 q^{69} + 53010 q^{74} + 59136 q^{75} - 113496 q^{77} - 110592 q^{79} + 185524 q^{81} + 72384 q^{82} + 17840 q^{87} + 26052 q^{88} + 251236 q^{90} - 7528 q^{92} - 214250 q^{94} - 32144 q^{95}+O(q^{100}) $ 4 * q - 56 * q^3 + 86 * q^4 + 424 * q^9 - 890 * q^10 - 1714 * q^12 - 1010 * q^14 + 930 * q^16 + 5260 * q^17 + 452 * q^22 + 5248 * q^23 - 16464 * q^25 - 9464 * q^27 - 1624 * q^29 + 118 * q^30 - 46088 * q^35 + 23396 * q^36 - 2036 * q^38 - 37330 * q^40 - 1874 * q^42 - 4840 * q^43 - 51270 * q^48 - 17368 * q^49 - 86696 * q^51 - 87440 * q^53 - 41328 * q^55 - 40690 * q^56 + 3968 * q^61 + 46216 * q^62 - 6434 * q^64 - 8572 * q^66 + 123970 * q^68 + 70960 * q^69 + 53010 * q^74 + 59136 * q^75 - 113496 * q^77 - 110592 * q^79 + 185524 * q^81 + 72384 * q^82 + 17840 * q^87 + 26052 * q^88 + 251236 * q^90 - 7528 * q^92 - 214250 * q^94 - 32144 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	4.56155i	0.806376i	0.915117	+	0.403188i	$0.132098\pi$
$2$	4.56155i	0.806376i	−0.915117	+	0.403188i	$0.867902\pi$
$3$	−1.63068	−0.104608	−0.0523042	−	0.998631i	$-0.516657\pi$
$3$	−1.63068	−0.104608	−0.0523042	+	0.998631i	$0.516657\pi$
$4$	11.1922	0.349757
$5$	103.462i	1.85079i	0.379008	+	0.925393i	$0.376265\pi$
$5$	103.462i	1.85079i	−0.379008	+	0.925393i	$0.623735\pi$
$6$	− 7.43845i	− 0.0843537i
$7$	126.309i	0.974290i	0.873321	+	0.487145i	$0.161962\pi$
$7$	126.309i	0.974290i	−0.873321	+	0.487145i	$0.838038\pi$
$8$	197.024i	1.08841i
$9$	−240.341	−0.989057
$10$	−471.948	−1.49243
$11$	− 14.8296i	− 0.0369527i	−0.999829	−	0.0184764i	$-0.994118\pi$
$11$	− 14.8296i	− 0.0369527i	0.999829	−	0.0184764i	$-0.00588155\pi$
$12$	−18.2510	−0.0365875
$13$	0	0
$13$	0	0
$14$	−576.164	−0.785644
$15$	− 168.714i	− 0.193608i
$16$	−540.582	−0.527912
$17$	1051.12	0.882126	0.441063	−	0.897476i	$-0.354602\pi$
$17$	1051.12	0.882126	0.441063	+	0.897476i	$0.354602\pi$
$18$	− 1096.33i	− 0.797552i
$19$	213.723i	0.135821i	0.997691	+	0.0679107i	$0.0216333\pi$
$19$	213.723i	0.135821i	−0.997691	+	0.0679107i	$0.978367\pi$
$20$	1157.97i	0.647326i
$21$	− 205.969i	− 0.101919i
$22$	67.6458	0.0297978
$23$	4231.16	1.66778	0.833892	−	0.551928i	$-0.186108\pi$
$23$	4231.16	1.66778	0.833892	+	0.551928i	$0.186108\pi$
$24$	− 321.283i	− 0.113857i
$25$	−7579.41	−2.42541
$26$	0	0
$27$	788.176	0.208072
$28$	1413.68i	0.340765i
$29$	−504.955	−0.111495	−0.0557477	−	0.998445i	$-0.517754\pi$
$29$	−504.955	−0.111495	−0.0557477	+	0.998445i	$0.517754\pi$
$30$	769.597	0.156121
$31$	− 4783.58i	− 0.894022i	−0.894528	−	0.447011i	$-0.852488\pi$
$31$	− 4783.58i	− 0.894022i	0.894528	−	0.447011i	$-0.147512\pi$
$32$	3838.86i	0.662716i
$33$	24.1823i	0.00386557i
$34$	4794.75i	0.711325i
$35$	−13068.2	−1.80320
$36$	−2689.95	−0.345930
$37$	− 4635.74i	− 0.556692i	−0.960481	−	0.278346i	$-0.910214\pi$
$37$	− 4635.74i	− 0.556692i	0.960481	−	0.278346i	$-0.0897862\pi$
$38$	−974.911	−0.109523
$39$	0	0
$40$	−20384.5	−2.01442
$41$	− 7944.15i	− 0.738054i	−0.929419	−	0.369027i	$-0.879691\pi$
$41$	− 7944.15i	− 0.738054i	0.929419	−	0.369027i	$-0.120309\pi$
$42$	939.541	0.0821850
$43$	8516.41	0.702401	0.351201	−	0.936300i	$-0.385774\pi$
$43$	8516.41	0.702401	0.351201	+	0.936300i	$0.385774\pi$
$44$	− 165.976i	− 0.0129245i
$45$	− 24866.2i	− 1.83053i
$46$	19300.7i	1.34486i
$47$	24921.2i	1.64560i	0.568330	+	0.822801i	$0.307590\pi$
$47$	24921.2i	1.64560i	−0.568330	+	0.822801i	$0.692410\pi$
$48$	881.518	0.0552241
$49$	853.113	0.0507594
$50$	− 34573.9i	− 1.95579i
$51$	−1714.05	−0.0922777
$52$	0	0
$53$	−7808.46	−0.381835	−0.190917	−	0.981606i	$-0.561146\pi$
$53$	−7808.46	−0.381835	−0.190917	+	0.981606i	$0.561146\pi$
$54$	3595.31i	0.167784i
$55$	1534.30	0.0683916
$56$	−24885.8	−1.06043
$57$	− 348.515i	− 0.0142081i
$58$	− 2303.38i	− 0.0899073i
$59$	− 37337.5i	− 1.39642i	−0.715894	−	0.698209i	$-0.753980\pi$
$59$	− 37337.5i	− 1.39642i	0.715894	−	0.698209i	$-0.246020\pi$
$60$	− 1888.29i	− 0.0677157i
$61$	−18172.2	−0.625292	−0.312646	−	0.949870i	$-0.601215\pi$
$61$	−18172.2	−0.625292	−0.312646	+	0.949870i	$0.601215\pi$
$62$	21820.5	0.720918
$63$	− 30357.1i	− 0.963628i
$64$	−34809.8	−1.06231
$65$	0	0
$66$	−110.309	−0.00311710
$67$	34559.9i	0.940559i	0.882518	+	0.470279i	$0.155847\pi$
$67$	34559.9i	0.940559i	−0.882518	+	0.470279i	$0.844153\pi$
$68$	11764.4	0.308530
$69$	−6899.68	−0.174464
$70$	− 59611.1i	− 1.45406i
$71$	− 41255.7i	− 0.971265i	−0.874163	−	0.485632i	$-0.838589\pi$
$71$	− 41255.7i	− 0.971265i	0.874163	−	0.485632i	$-0.161411\pi$
$72$	− 47352.8i	− 1.07650i
$73$	− 1056.42i	− 0.0232022i	−0.999933	−	0.0116011i	$-0.996307\pi$
$73$	− 1056.42i	− 0.0232022i	0.999933	−	0.0116011i	$-0.00369283\pi$
$74$	21146.2	0.448903
$75$	12359.6	0.253718
$76$	2392.04i	0.0475045i
$77$	1873.10	0.0360027
$78$	0	0
$79$	−47719.3	−0.860253	−0.430126	−	0.902769i	$-0.641531\pi$
$79$	−47719.3	−0.860253	−0.430126	+	0.902769i	$0.641531\pi$
$80$	− 55929.8i	− 0.977053i
$81$	57117.6	0.967291
$82$	36237.7	0.595149
$83$	74799.0i	1.19179i	0.803061	+	0.595896i	$0.203203\pi$
$83$	74799.0i	1.19179i	−0.803061	+	0.595896i	$0.796797\pi$
$84$	− 2305.26i	− 0.0356469i
$85$	108751.i	1.63263i
$86$	38848.0i	0.566400i
$87$	823.421	0.0116634
$88$	2921.78	0.0402198
$89$	9799.26i	0.131135i	0.997848	+	0.0655675i	$0.0208858\pi$
$89$	9799.26i	0.131135i	−0.997848	+	0.0655675i	$0.979114\pi$
$90$	113428.	1.47610
$91$	0	0
$92$	47356.1	0.583320
$93$	7800.50i	0.0935222i
$94$	−113679.	−1.32697
$95$	−22112.3	−0.251376
$96$	− 6259.97i	− 0.0693257i
$97$	138432.i	1.49385i	0.664906	+	0.746927i	$0.268472\pi$
$97$	138432.i	1.49385i	−0.664906	+	0.746927i	$0.731528\pi$
$98$	3891.52i	0.0409312i
$99$	3564.15i	0.0365484i
$100$	−84830.5	−0.848305
$101$	−139151.	−1.35733	−0.678663	−	0.734450i	$-0.737440\pi$
$101$	−139151.	−1.35733	−0.678663	+	0.734450i	$0.737440\pi$
$102$	− 7818.71i	− 0.0744106i
$103$	−98512.2	−0.914950	−0.457475	−	0.889223i	$-0.651246\pi$
$103$	−98512.2	−0.914950	−0.457475	+	0.889223i	$0.651246\pi$
$104$	0	0
$105$	21310.0	0.188630
$106$	− 35618.7i	− 0.307902i
$107$	−26848.4	−0.226704	−0.113352	−	0.993555i	$-0.536159\pi$
$107$	−26848.4	−0.226704	−0.113352	+	0.993555i	$0.536159\pi$
$108$	8821.45	0.0727747
$109$	63220.9i	0.509676i	0.966984	+	0.254838i	$0.0820222\pi$
$109$	63220.9i	0.509676i	−0.966984	+	0.254838i	$0.917978\pi$
$110$	6998.78i	0.0551494i
$111$	7559.43i	0.0582346i
$112$	− 68280.2i	− 0.514340i
$113$	114434.	0.843058	0.421529	−	0.906815i	$-0.361494\pi$
$113$	114434.	0.843058	0.421529	+	0.906815i	$0.361494\pi$
$114$	1589.77	0.0114570
$115$	437765.i	3.08671i
$116$	−5651.57	−0.0389964
$117$	0	0
$118$	170317.	1.12604
$119$	132766.i	0.859446i
$120$	33240.6	0.210725
$121$	160831.	0.998634
$122$	− 82893.4i	− 0.504221i
$123$	12954.4i	0.0772066i
$124$	− 53538.9i	− 0.312691i
$125$	− 460863.i	− 2.63813i
$126$	138476.	0.777047
$127$	248871.	1.36919	0.684596	−	0.728922i	$-0.259978\pi$
$127$	248871.	1.36919	0.684596	+	0.728922i	$0.259978\pi$
$128$	− 35943.2i	− 0.193906i
$129$	−13887.6	−0.0734770
$130$	0	0
$131$	102963.	0.524205	0.262102	−	0.965040i	$-0.415584\pi$
$131$	102963.	0.524205	0.262102	+	0.965040i	$0.415584\pi$
$132$	270.654i	0.00135201i
$133$	−26995.1	−0.132329
$134$	−157647.	−0.758444
$135$	81546.3i	0.385097i
$136$	207096.i	0.960117i
$137$	− 36037.4i	− 0.164041i	−0.996631	−	0.0820204i	$-0.973863\pi$
$137$	− 36037.4i	− 0.164041i	0.996631	−	0.0820204i	$-0.0261373\pi$
$138$	− 31473.3i	− 0.140684i
$139$	152655.	0.670151	0.335076	−	0.942191i	$-0.391238\pi$
$139$	152655.	0.670151	0.335076	+	0.942191i	$0.391238\pi$
$140$	−146262.	−0.630683
$141$	− 40638.6i	− 0.172144i
$142$	188190.	0.783205
$143$	0	0
$144$	129924.	0.522136
$145$	− 52243.7i	− 0.206354i
$146$	4818.92	0.0187097
$147$	−1391.16	−0.00530986
$148$	− 51884.3i	− 0.194707i
$149$	− 72547.1i	− 0.267704i	−0.991001	−	0.133852i	$-0.957265\pi$
$149$	− 72547.1i	− 0.267704i	0.991001	−	0.133852i	$-0.0427346\pi$
$150$	56379.0i	0.204592i
$151$	489021.i	1.74536i	0.488291	+	0.872681i	$0.337621\pi$
$151$	489021.i	1.74536i	−0.488291	+	0.872681i	$0.662379\pi$
$152$	−42108.6	−0.147830
$153$	−252627.	−0.872473
$154$	8544.26i	0.0290317i
$155$	494919.	1.65464
$156$	0	0
$157$	89467.9	0.289680	0.144840	−	0.989455i	$-0.453733\pi$
$157$	89467.9	0.289680	0.144840	+	0.989455i	$0.453733\pi$
$158$	− 217674.i	− 0.693687i
$159$	12733.1	0.0399431
$160$	−397177.	−1.22655
$161$	534432.i	1.62490i
$162$	260545.i	0.780000i
$163$	− 225668.i	− 0.665275i	−0.943055	−	0.332637i	$-0.892061\pi$
$163$	− 225668.i	− 0.665275i	0.943055	−	0.332637i	$-0.107939\pi$
$164$	− 88912.8i	− 0.258140i
$165$	−2501.95	−0.00715434
$166$	−341200.	−0.961033
$167$	209528.i	0.581367i	0.956819	+	0.290683i	$0.0938827\pi$
$167$	209528.i	0.581367i	−0.956819	+	0.290683i	$0.906117\pi$
$168$	40580.9	0.110930
$169$	0	0
$170$	−496074.	−1.31651
$171$	− 51366.5i	− 0.134335i
$172$	95317.6	0.245670
$173$	465184.	1.18171	0.590853	−	0.806779i	$-0.298791\pi$
$173$	465184.	1.18171	0.590853	+	0.806779i	$0.298791\pi$
$174$	3756.08i	0.00940506i
$175$	− 957345.i	− 2.36305i
$176$	8016.60i	0.0195078i
$177$	60885.7i	0.146077i
$178$	−44699.9	−0.105744
$179$	472573.	1.10239	0.551197	−	0.834375i	$-0.314171\pi$
$179$	472573.	1.10239	0.551197	+	0.834375i	$0.314171\pi$
$180$	− 278308.i	− 0.640243i
$181$	74099.4	0.168120	0.0840598	−	0.996461i	$-0.473211\pi$
$181$	74099.4	0.168120	0.0840598	+	0.996461i	$0.473211\pi$
$182$	0	0
$183$	29633.1	0.0654108
$184$	833638.i	1.81524i
$185$	479624.	1.03032
$186$	−35582.4	−0.0754141
$187$	− 15587.7i	− 0.0325970i
$188$	278924.i	0.575561i
$189$	99553.5i	0.202722i
$190$	− 100866.i	− 0.202704i
$191$	−224128.	−0.444543	−0.222271	−	0.974985i	$-0.571347\pi$
$191$	−224128.	−0.444543	−0.222271	+	0.974985i	$0.571347\pi$
$192$	56763.8	0.111127
$193$	− 662265.i	− 1.27979i	−0.768463	−	0.639895i	$-0.778978\pi$
$193$	− 662265.i	− 1.27979i	0.768463	−	0.639895i	$-0.221022\pi$
$194$	−631467.	−1.20461
$195$	0	0
$196$	9548.24	0.0177535
$197$	− 666464.i	− 1.22352i	−0.791043	−	0.611760i	$-0.790462\pi$
$197$	− 666464.i	− 1.22352i	0.791043	−	0.611760i	$-0.209538\pi$
$198$	−16258.1	−0.0294717
$199$	−645720.	−1.15588	−0.577938	−	0.816081i	$-0.696142\pi$
$199$	−645720.	−1.15588	−0.577938	+	0.816081i	$0.696142\pi$
$200$	− 1.49332e6i	− 2.63985i
$201$	− 56356.3i	− 0.0983903i
$202$	− 634746.i	− 1.09451i
$203$	− 63780.1i	− 0.108629i
$204$	−19184.0	−0.0322748
$205$	821919.	1.36598
$206$	− 449369.i	− 0.737794i
$207$	−1.01692e6	−1.64953
$208$	0	0
$209$	3169.43	0.00501897
$210$	97206.9i	0.152107i
$211$	−868021.	−1.34222	−0.671110	−	0.741357i	$-0.734182\pi$
$211$	−868021.	−1.34222	−0.671110	+	0.741357i	$0.734182\pi$
$212$	−87394.1	−0.133550
$213$	67274.9i	0.101602i
$214$	− 122470.i	− 0.182808i
$215$	881125.i	1.29999i
$216$	155289.i	0.226468i
$217$	604207.	0.871037
$218$	−288385.	−0.410991
$219$	1722.69i	0.00242715i
$220$	17172.2	0.0239205
$221$	0	0
$222$	−34482.7	−0.0469590
$223$	58342.9i	0.0785644i	0.999228	+	0.0392822i	$0.0125071\pi$
$223$	58342.9i	0.0785644i	−0.999228	+	0.0392822i	$0.987493\pi$
$224$	−484882.	−0.645678
$225$	1.82164e6	2.39887
$226$	521995.i	0.679822i
$227$	− 111768.i	− 0.143964i	−0.997406	−	0.0719820i	$-0.977068\pi$
$227$	− 111768.i	− 0.143964i	0.997406	−	0.0719820i	$-0.0229324\pi$
$228$	− 3900.67i	− 0.00496937i
$229$	1.14984e6i	1.44893i	0.689309	+	0.724467i	$0.257914\pi$
$229$	1.14984e6i	1.44893i	−0.689309	+	0.724467i	$0.742086\pi$
$230$	−1.99689e6	−2.48905
$231$	−3054.44	−0.00376618
$232$	− 99488.0i	− 0.121353i
$233$	−630470.	−0.760807	−0.380404	−	0.924821i	$-0.624215\pi$
$233$	−630470.	−0.760807	−0.380404	+	0.924821i	$0.624215\pi$
$234$	0	0
$235$	−2.57840e6	−3.04566
$236$	− 417891.i	− 0.488408i
$237$	77815.0	0.0899897
$238$	−605618.	−0.693037
$239$	165628.i	0.187559i	0.995593	+	0.0937797i	$0.0298949\pi$
$239$	165628.i	0.187559i	−0.995593	+	0.0937797i	$0.970105\pi$
$240$	91203.8i	0.102208i
$241$	690968.i	0.766329i	0.923680	+	0.383165i	$0.125166\pi$
$241$	690968.i	0.766329i	−0.923680	+	0.383165i	$0.874834\pi$
$242$	733639.i	0.805275i
$243$	−284667.	−0.309259
$244$	−203387.	−0.218700
$245$	88264.9i	0.0939448i
$246$	−59092.1	−0.0622575
$247$	0	0
$248$	942478.	0.973065
$249$	− 121974.i	− 0.124671i
$250$	2.10225e6	2.12733
$251$	386887.	0.387615	0.193807	−	0.981040i	$-0.437916\pi$
$251$	386887.	0.387615	0.193807	+	0.981040i	$0.437916\pi$
$252$	− 339764.i	− 0.337036i
$253$	− 62746.2i	− 0.0616292i
$254$	1.13524e6i	1.10408i
$255$	− 177339.i	− 0.170786i
$256$	−949957.	−0.905950
$257$	−258260.	−0.243907	−0.121953	−	0.992536i	$-0.538916\pi$
$257$	−258260.	−0.243907	−0.121953	+	0.992536i	$0.538916\pi$
$258$	− 63348.8i	− 0.0592501i
$259$	585535.	0.542379
$260$	0	0
$261$	121361.	0.110275
$262$	469669.i	0.422706i
$263$	−1.19053e6	−1.06133	−0.530665	−	0.847582i	$-0.678058\pi$
$263$	−1.19053e6	−1.06133	−0.530665	+	0.847582i	$0.678058\pi$
$264$	−4764.49	−0.00420733
$265$	− 807879.i	− 0.706695i
$266$	− 123140.i	− 0.106707i
$267$	− 15979.5i	− 0.0137178i
$268$	386803.i	0.328967i
$269$	850968.	0.717022	0.358511	−	0.933525i	$-0.383285\pi$
$269$	850968.	0.717022	0.358511	+	0.933525i	$0.383285\pi$
$270$	−371978.	−0.310533
$271$	− 40926.1i	− 0.0338514i	−0.999857	−	0.0169257i	$-0.994612\pi$
$271$	− 40926.1i	− 0.0338514i	0.999857	−	0.0169257i	$-0.00538788\pi$
$272$	−568218.	−0.465685
$273$	0	0
$274$	164387.	0.132279
$275$	112399.i	0.0896256i
$276$	−77222.8	−0.0610201
$277$	−1.00054e6	−0.783490	−0.391745	−	0.920074i	$-0.628128\pi$
$277$	−1.00054e6	−0.783490	−0.391745	+	0.920074i	$0.628128\pi$
$278$	696342.i	0.540394i
$279$	1.14969e6i	0.884239i
$280$	− 2.57474e6i	− 1.96263i
$281$	− 1.73596e6i	− 1.31151i	−0.754972	−	0.655757i	$-0.772350\pi$
$281$	− 1.73596e6i	− 1.31151i	0.754972	−	0.655757i	$-0.227650\pi$
$282$	185375.	0.138813
$283$	1.27363e6	0.945319	0.472660	−	0.881245i	$-0.343294\pi$
$283$	1.27363e6	0.945319	0.472660	+	0.881245i	$0.343294\pi$
$284$	− 461743.i	− 0.339707i
$285$	36058.1	0.0262961
$286$	0	0
$287$	1.00342e6	0.719078
$288$	− 922636.i	− 0.655464i
$289$	−315001.	−0.221854
$290$	238312.	0.166399
$291$	− 225739.i	− 0.156270i
$292$	− 11823.7i	− 0.00811515i
$293$	− 2043.70i	− 0.00139075i	−1.00000		0.000695374i	$-0.999779\pi$
$293$	− 2043.70i	− 0.00139075i	1.00000		0.000695374i	$-0.000221344\pi$
$294$	− 6345.84i	− 0.00428174i
$295$	3.86302e6	2.58447
$296$	913351.	0.605910
$297$	− 11688.3i	− 0.00768883i
$298$	330927.	0.215870
$299$	0	0
$300$	138332.	0.0887398
$301$	1.07570e6i	0.684342i
$302$	−2.23070e6	−1.40742
$303$	226912.	0.141988
$304$	− 115535.i	− 0.0717018i
$305$	− 1.88013e6i	− 1.15728i
$306$	− 1.15237e6i	− 0.703541i
$307$	− 401308.i	− 0.243014i	−0.992591	−	0.121507i	$-0.961227\pi$
$307$	− 401308.i	− 0.243014i	0.992591	−	0.121507i	$-0.0387728\pi$
$308$	20964.2	0.0125922
$309$	160642.	0.0957114
$310$	2.25760e6i	1.33427i
$311$	1.92628e6	1.12933	0.564663	−	0.825322i	$-0.309006\pi$
$311$	1.92628e6	1.12933	0.564663	+	0.825322i	$0.309006\pi$
$312$	0	0
$313$	1.64519e6	0.949196	0.474598	−	0.880203i	$-0.342593\pi$
$313$	1.64519e6	0.949196	0.474598	+	0.880203i	$0.342593\pi$
$314$	408113.i	0.233591i
$315$	3.14081e6	1.78347
$316$	−534085.	−0.300880
$317$	1.99476e6i	1.11492i	0.830205	+	0.557459i	$0.188223\pi$
$317$	1.99476e6i	1.11492i	−0.830205	+	0.557459i	$0.811777\pi$
$318$	58082.8i	0.0322092i
$319$	7488.26i	0.00412006i
$320$	− 3.60150e6i	− 1.96611i
$321$	43781.2	0.0237151
$322$	−2.43784e6	−1.31028
$323$	224649.i	0.119812i
$324$	639273.	0.338317
$325$	0	0
$326$	1.02940e6	0.536462
$327$	− 103093.i	− 0.0533164i
$328$	1.56519e6	0.803306
$329$	−3.14777e6	−1.60329
$330$	− 11412.8i	− 0.00576909i
$331$	675924.i	0.339100i	0.985522	+	0.169550i	$0.0542314\pi$
$331$	675924.i	0.339100i	−0.985522	+	0.169550i	$0.945769\pi$
$332$	837168.i	0.416838i
$333$	1.11416e6i	0.550600i
$334$	−955772.	−0.468800
$335$	−3.57564e6	−1.74077
$336$	111343.i	0.0538042i
$337$	2.13552e6	1.02430	0.512152	−	0.858895i	$-0.328848\pi$
$337$	2.13552e6	1.02430	0.512152	+	0.858895i	$0.328848\pi$
$338$	0	0
$339$	−186605.	−0.0881909
$340$	1.21717e6i	0.571023i
$341$	−70938.3	−0.0330366
$342$	234311.	0.108325
$343$	2.23063e6i	1.02374i
$344$	1.67793e6i	0.764502i
$345$	− 713855.i	− 0.322896i
$346$	2.12196e6i	0.952899i
$347$	−2.57257e6	−1.14695	−0.573473	−	0.819225i	$-0.694404\pi$
$347$	−2.57257e6	−1.14695	−0.573473	+	0.819225i	$0.694404\pi$
$348$	9215.92	0.00407935
$349$	2.02363e6i	0.889339i	0.895695	+	0.444670i	$0.146679\pi$
$349$	2.02363e6i	0.889339i	−0.895695	+	0.444670i	$0.853321\pi$
$350$	4.36698e6	1.90551
$351$	0	0
$352$	56928.7	0.0244892
$353$	− 2.04810e6i	− 0.874810i	−0.899265	−	0.437405i	$-0.855898\pi$
$353$	− 2.04810e6i	− 0.874810i	0.899265	−	0.437405i	$-0.144102\pi$
$354$	−277733.	−0.117793
$355$	4.26840e6	1.79760
$356$	109676.i	0.0458654i
$357$	− 216499.i	− 0.0899053i
$358$	2.15567e6i	0.888944i
$359$	− 1.59901e6i	− 0.654808i	−0.944885	−	0.327404i	$-0.893826\pi$
$359$	− 1.59901e6i	− 0.654808i	0.944885	−	0.327404i	$-0.106174\pi$
$360$	4.89922e6	1.99238
$361$	2.43042e6	0.981553
$362$	338008.i	0.135568i
$363$	−262265.	−0.104466
$364$	0	0
$365$	109299.	0.0429424
$366$	135173.i	0.0527457i
$367$	−3.86389e6	−1.49747	−0.748737	−	0.662867i	$-0.769339\pi$
$367$	−3.86389e6	−1.49747	−0.748737	+	0.662867i	$0.769339\pi$
$368$	−2.28729e6	−0.880444
$369$	1.90930e6i	0.729977i
$370$	2.18783e6i	0.830824i
$371$	− 986276.i	− 0.372018i
$372$	87305.0i	0.0327101i
$373$	−1.56702e6	−0.583179	−0.291589	−	0.956544i	$-0.594184\pi$
$373$	−1.56702e6	−0.583179	−0.291589	+	0.956544i	$0.594184\pi$
$374$	71104.0	0.0262854
$375$	751521.i	0.275971i
$376$	−4.91007e6	−1.79109
$377$	0	0
$378$	−454118.	−0.163471
$379$	− 3.19239e6i	− 1.14161i	−0.821086	−	0.570805i	$-0.806631\pi$
$379$	− 3.19239e6i	− 1.14161i	0.821086	−	0.570805i	$-0.193369\pi$
$380$	−247486.	−0.0879208
$381$	−405829.	−0.143229
$382$	− 1.02237e6i	− 0.358469i
$383$	− 400432.i	− 0.139486i	−0.997565	−	0.0697432i	$-0.977782\pi$
$383$	− 400432.i	− 0.139486i	0.997565	−	0.0697432i	$-0.0222180\pi$
$384$	58611.9i	0.0202842i
$385$	193795.i	0.0666333i
$386$	3.02096e6	1.03199
$387$	−2.04684e6	−0.694715
$388$	1.54937e6i	0.522487i
$389$	−413440.	−0.138528	−0.0692642	−	0.997598i	$-0.522065\pi$
$389$	−413440.	−0.138528	−0.0692642	+	0.997598i	$0.522065\pi$
$390$	0	0
$391$	4.44746e6	1.47120
$392$	168083.i	0.0552471i
$393$	−167899.	−0.0548362
$394$	3.04011e6	0.986618
$395$	− 4.93714e6i	− 1.59214i
$396$	39890.8i	0.0127831i
$397$	− 102926.i	− 0.0327753i	−0.999866	−	0.0163877i	$-0.994783\pi$
$397$	− 102926.i	− 0.0327753i	0.999866	−	0.0163877i	$-0.00521659\pi$
$398$	− 2.94548e6i	− 0.932071i
$399$	44020.5	0.0138428
$400$	4.09729e6	1.28040
$401$	2.23365e6i	0.693671i	0.937926	+	0.346836i	$0.112744\pi$
$401$	2.23365e6i	0.693671i	−0.937926	+	0.346836i	$0.887256\pi$
$402$	257072.	0.0793396
$403$	0	0
$404$	−1.55741e6	−0.474734
$405$	5.90950e6i	1.79025i
$406$	290937.	0.0875958
$407$	−68746.0	−0.0205713
$408$	− 337708.i	− 0.100436i
$409$	4.46150e6i	1.31878i	0.751801	+	0.659390i	$0.229185\pi$
$409$	4.46150e6i	1.31878i	−0.751801	+	0.659390i	$0.770815\pi$
$410$	3.74923e6i	1.10149i
$411$	58765.6i	0.0171600i
$412$	−1.10257e6	−0.320010
$413$	4.71606e6	1.36052
$414$	− 4.63874e6i	− 1.33014i
$415$	−7.73887e6	−2.20575
$416$	0	0
$417$	−248931.	−0.0701034
$418$	14457.5i	0.00404718i
$419$	4.22792e6	1.17650	0.588250	−	0.808679i	$-0.299817\pi$
$419$	4.22792e6	1.17650	0.588250	+	0.808679i	$0.299817\pi$
$420$	238507.	0.0659748
$421$	− 4.11791e6i	− 1.13233i	−0.824293	−	0.566163i	$-0.808427\pi$
$421$	− 4.11791e6i	− 1.13233i	0.824293	−	0.566163i	$-0.191573\pi$
$422$	− 3.95952e6i	− 1.08233i
$423$	− 5.98959e6i	− 1.62759i
$424$	− 1.53845e6i	− 0.415594i
$425$	−7.96688e6	−2.13952
$426$	−306878.	−0.0819298
$427$	− 2.29531e6i	− 0.609216i
$428$	−300493.	−0.0792912
$429$	0	0
$430$	−4.01930e6	−1.04828
$431$	− 1.15324e6i	− 0.299038i	−0.988759	−	0.149519i	$-0.952227\pi$
$431$	− 1.15324e6i	− 0.299038i	0.988759	−	0.149519i	$-0.0477725\pi$
$432$	−426074.	−0.109844
$433$	33734.3	0.00864673	0.00432337	−	0.999991i	$-0.498624\pi$
$433$	33734.3	0.00864673	0.00432337	+	0.999991i	$0.498624\pi$
$434$	2.75612e6i	0.702383i
$435$	85192.9i	0.0215864i
$436$	707583.i	0.178263i
$437$	904298.i	0.226521i
$438$	−7858.13	−0.00195719
$439$	−7.48363e6	−1.85332	−0.926661	−	0.375898i	$-0.877334\pi$
$439$	−7.48363e6	−1.85332	−0.926661	+	0.375898i	$0.877334\pi$
$440$	302293.i	0.0744383i
$441$	−205038.	−0.0502039
$442$	0	0
$443$	−3.28028e6	−0.794148	−0.397074	−	0.917786i	$-0.629974\pi$
$443$	−3.28028e6	−0.794148	−0.397074	+	0.917786i	$0.629974\pi$
$444$	84606.9i	0.0203680i
$445$	−1.01385e6	−0.242703
$446$	−266134.	−0.0633525
$447$	118301.i	0.0280040i
$448$	− 4.39678e6i	− 1.03500i
$449$	7.95356e6i	1.86185i	0.365205	+	0.930927i	$0.380999\pi$
$449$	7.95356e6i	1.86185i	−0.365205	+	0.930927i	$0.619001\pi$
$450$	8.30951e6i	1.93439i
$451$	−117808.	−0.0272731
$452$	1.28077e6	0.294866
$453$	− 797439.i	− 0.182579i
$454$	509837.	0.116089
$455$	0	0
$456$	68665.8	0.0154642
$457$	3.35187e6i	0.750753i	0.926872	+	0.375377i	$0.122487\pi$
$457$	3.35187e6i	0.750753i	−0.926872	+	0.375377i	$0.877513\pi$
$458$	−5.24506e6	−1.16839
$459$	828468.	0.183546
$460$	4.89956e6i	1.07960i
$461$	− 4.68627e6i	− 1.02701i	−0.858086	−	0.513505i	$-0.828347\pi$
$461$	− 4.68627e6i	− 1.02701i	0.858086	−	0.513505i	$-0.171653\pi$
$462$	− 13933.0i	− 0.00303696i
$463$	6.64697e6i	1.44102i	0.693442	+	0.720512i	$0.256093\pi$
$463$	6.64697e6i	1.44102i	−0.693442	+	0.720512i	$0.743907\pi$
$464$	272969.	0.0588599
$465$	−807056.	−0.173090
$466$	− 2.87592e6i	− 0.613497i
$467$	3.14141e6	0.666549	0.333275	−	0.942830i	$-0.391846\pi$
$467$	3.14141e6	0.666549	0.333275	+	0.942830i	$0.391846\pi$
$468$	0	0
$469$	−4.36522e6	−0.916377
$470$	− 1.17615e7i	− 2.45594i
$471$	−145894.	−0.0303029
$472$	7.35638e6	1.51988
$473$	− 126295.i	− 0.0259556i
$474$	354957.i	0.0725655i
$475$	− 1.61990e6i	− 0.329423i
$476$	1.48595e6i	0.300598i
$477$	1.87669e6	0.377656
$478$	−755521.	−0.151243
$479$	− 6.68286e6i	− 1.33083i	−0.746473	−	0.665416i	$-0.768254\pi$
$479$	− 6.68286e6i	− 1.33083i	0.746473	−	0.665416i	$-0.231746\pi$
$480$	647670.	0.128307
$481$	0	0
$482$	−3.15189e6	−0.617950
$483$	− 871489.i	− 0.169979i
$484$	1.80006e6	0.349280
$485$	−1.43225e7	−2.76481
$486$	− 1.29853e6i	− 0.249379i
$487$	4.06478e6i	0.776631i	0.921526	+	0.388316i	$0.126943\pi$
$487$	4.06478e6i	0.776631i	−0.921526	+	0.388316i	$0.873057\pi$
$488$	− 3.58035e6i	− 0.680575i
$489$	367993.i	0.0695933i
$490$	−402625.	−0.0757549
$491$	2.10434e6	0.393923	0.196962	−	0.980411i	$-0.436893\pi$
$491$	2.10434e6	0.393923	0.196962	+	0.980411i	$0.436893\pi$
$492$	144989.i	0.0270036i
$493$	−530768.	−0.0983530
$494$	0	0
$495$	−368755.	−0.0676432
$496$	2.58592e6i	0.471966i
$497$	5.21095e6	0.946293
$498$	556389.	0.100532
$499$	− 5.96715e6i	− 1.07279i	−0.843966	−	0.536396i	$-0.819785\pi$
$499$	− 5.96715e6i	− 1.07279i	0.843966	−	0.536396i	$-0.180215\pi$
$500$	− 5.15808e6i	− 0.922706i
$501$	− 341673.i	− 0.0608158i
$502$	1.76481e6i	0.312563i
$503$	−1.00144e7	−1.76483	−0.882417	−	0.470467i	$-0.844085\pi$
$503$	−1.00144e7	−1.76483	−0.882417	+	0.470467i	$0.844085\pi$
$504$	5.98108e6	1.04882
$505$	− 1.43969e7i	− 2.51212i
$506$	286220.	0.0496963
$507$	0	0
$508$	2.78542e6	0.478885
$509$	8.47321e6i	1.44962i	0.688950	+	0.724809i	$0.258072\pi$
$509$	8.47321e6i	1.44962i	−0.688950	+	0.724809i	$0.741928\pi$
$510$	808940.	0.137718
$511$	133435.	0.0226057
$512$	− 5.48346e6i	− 0.924442i
$513$	168452.i	0.0282606i
$514$	− 1.17807e6i	− 0.196681i
$515$	− 1.01923e7i	− 1.69338i
$516$	−155433.	−0.0256991
$517$	369571.	0.0608095
$518$	2.67095e6i	0.437362i
$519$	−758567.	−0.123616
$520$	0	0
$521$	−197614.	−0.0318951	−0.0159476	−	0.999873i	$-0.505076\pi$
$521$	−197614.	−0.0318951	−0.0159476	+	0.999873i	$0.505076\pi$
$522$	553596.i	0.0889235i
$523$	8.27263e6	1.32248	0.661240	−	0.750174i	$-0.270030\pi$
$523$	8.27263e6	1.32248	0.661240	+	0.750174i	$0.270030\pi$
$524$	1.15238e6	0.183345
$525$	1.56113e6i	0.247195i
$526$	− 5.43066e6i	− 0.855831i
$527$	− 5.02812e6i	− 0.788640i
$528$	− 13072.5i	− 0.00204068i
$529$	1.14664e7	1.78150
$530$	3.68518e6	0.569862
$531$	8.97374e6i	1.38114i
$532$	−302136.	−0.0462832
$533$	0	0
$534$	72891.3	0.0110617
$535$	− 2.77779e6i	− 0.419580i
$536$	−6.80913e6	−1.02372
$537$	−770618.	−0.115320
$538$	3.88174e6i	0.578190i
$539$	− 12651.3i	− 0.00187570i
$540$	912686.i	0.134690i
$541$	363216.i	0.0533546i	0.999644	+	0.0266773i	$0.00849266\pi$
$541$	363216.i	0.0533546i	−0.999644	+	0.0266773i	$0.991507\pi$
$542$	186686.	0.0272970
$543$	−120833.	−0.0175867
$544$	4.03511e6i	0.584599i
$545$	−6.54097e6	−0.943302
$546$	0	0
$547$	−620452.	−0.0886624	−0.0443312	−	0.999017i	$-0.514116\pi$
$547$	−620452.	−0.0886624	−0.0443312	+	0.999017i	$0.514116\pi$
$548$	− 403339.i	− 0.0573745i
$549$	4.36752e6	0.618449
$550$	−512715.	−0.0722719
$551$	− 107921.i	− 0.0151435i
$552$	− 1.35940e6i	− 0.189889i
$553$	− 6.02736e6i	− 0.838136i
$554$	− 4.56400e6i	− 0.631788i
$555$	−782114.	−0.107780
$556$	1.70855e6	0.234390
$557$	3.89737e6i	0.532272i	0.963935	+	0.266136i	$0.0857471\pi$
$557$	3.89737e6i	0.532272i	−0.963935	+	0.266136i	$0.914253\pi$
$558$	−5.24437e6	−0.713029
$559$	0	0
$560$	7.06442e6	0.951933
$561$	25418.5i	0.00340992i
$562$	7.91865e6	1.05757
$563$	−510725.	−0.0679073	−0.0339536	−	0.999423i	$-0.510810\pi$
$563$	−510725.	−0.0679073	−0.0339536	+	0.999423i	$0.510810\pi$
$564$	− 454837.i	− 0.0602085i
$565$	1.18395e7i	1.56032i
$566$	5.80975e6i	0.762283i
$567$	7.21445e6i	0.942422i
$568$	8.12834e6	1.05714
$569$	−9.75625e6	−1.26329	−0.631644	−	0.775259i	$-0.717619\pi$
$569$	−9.75625e6	−1.26329	−0.631644	+	0.775259i	$0.717619\pi$
$570$	164481.i	0.0212045i
$571$	1.41952e7	1.82201	0.911006	−	0.412393i	$-0.135307\pi$
$571$	1.41952e7	1.82201	0.911006	+	0.412393i	$0.135307\pi$
$572$	0	0
$573$	365482.	0.0465029
$574$	4.57713e6i	0.579847i
$575$	−3.20697e7	−4.04506
$576$	8.36622e6	1.05069
$577$	1.16423e6i	0.145579i	0.997347	+	0.0727896i	$0.0231902\pi$
$577$	1.16423e6i	0.145579i	−0.997347	+	0.0727896i	$0.976810\pi$
$578$	− 1.43689e6i	− 0.178898i
$579$	1.07994e6i	0.133877i
$580$	− 584723.i	− 0.0721740i
$581$	−9.44777e6	−1.16115
$582$	1.02972e6	0.126012
$583$	115796.i	0.0141098i
$584$	208140.	0.0252536
$585$	0	0
$586$	9322.45	0.00112147
$587$	− 6.58038e6i	− 0.788234i	−0.919060	−	0.394117i	$-0.871050\pi$
$587$	− 6.58038e6i	− 0.788234i	0.919060	−	0.394117i	$-0.128950\pi$
$588$	−15570.2	−0.00185716
$589$	1.02236e6	0.121427
$590$	1.76214e7i	2.08406i
$591$	1.08679e6i	0.127991i
$592$	2.50600e6i	0.293885i
$593$	− 1.91423e6i	− 0.223541i	−0.993734	−	0.111771i	$-0.964348\pi$
$593$	− 1.91423e6i	− 0.223541i	0.993734	−	0.111771i	$-0.0356522\pi$
$594$	53316.8	0.00620009
$595$	−1.37362e7	−1.59065
$596$	− 811964.i	− 0.0936313i
$597$	1.05296e6	0.120914
$598$	0	0
$599$	−2.33678e6	−0.266104	−0.133052	−	0.991109i	$-0.542478\pi$
$599$	−2.33678e6	−0.266104	−0.133052	+	0.991109i	$0.542478\pi$
$600$	2.43514e6i	0.276150i
$601$	1.04273e7	1.17757	0.588786	−	0.808289i	$-0.299606\pi$
$601$	1.04273e7	1.17757	0.588786	+	0.808289i	$0.299606\pi$
$602$	−4.90684e6	−0.551837
$603$	− 8.30617e6i	− 0.930266i
$604$	5.47324e6i	0.610453i
$605$	1.66399e7i	1.84826i
$606$	1.03507e6i	0.114495i
$607$	−120274.	−0.0132495	−0.00662474	−	0.999978i	$-0.502109\pi$
$607$	−120274.	−0.0132495	−0.00662474	+	0.999978i	$0.502109\pi$
$608$	−820455.	−0.0900111
$609$	104005.i	0.0113635i
$610$	8.57633e6	0.933205
$611$	0	0
$612$	−2.82747e6	−0.305154
$613$	− 1.34576e7i	− 1.44649i	−0.690592	−	0.723245i	$-0.742650\pi$
$613$	− 1.34576e7i	− 1.44649i	0.690592	−	0.723245i	$-0.257350\pi$
$614$	1.83059e6	0.195961
$615$	−1.34029e6	−0.142893
$616$	369046.i	0.0391858i
$617$	6.84879e6i	0.724270i	0.932126	+	0.362135i	$0.117952\pi$
$617$	6.84879e6i	0.724270i	−0.932126	+	0.362135i	$0.882048\pi$
$618$	732778.i	0.0771794i
$619$	− 5.40663e6i	− 0.567153i	−0.958950	−	0.283577i	$-0.908479\pi$
$619$	− 5.40663e6i	− 0.567153i	0.958950	−	0.283577i	$-0.0915210\pi$
$620$	5.53925e6	0.578724
$621$	3.33490e6	0.347019
$622$	8.78684e6i	0.910661i
$623$	−1.23773e6	−0.127763
$624$	0	0
$625$	2.39962e7	2.45721
$626$	7.50463e6i	0.765409i
$627$	−5168.33	−0.000525027 0
$628$	1.00135e6	0.101318
$629$	− 4.87273e6i	− 0.491072i
$630$	1.43270e7i	1.43815i
$631$	− 9.62552e6i	− 0.962390i	−0.876614	−	0.481195i	$-0.840203\pi$
$631$	− 9.62552e6i	− 0.962390i	0.876614	−	0.481195i	$-0.159797\pi$
$632$	− 9.40183e6i	− 0.936310i
$633$	1.41547e6	0.140408
$634$	−9.09920e6	−0.899043
$635$	2.57487e7i	2.53408i
$636$	142512.	0.0139704
$637$	0	0
$638$	−34158.1	−0.00332232
$639$	9.91542e6i	0.960636i
$640$	3.71876e6	0.358879
$641$	1.58752e7	1.52607	0.763037	−	0.646355i	$-0.223707\pi$
$641$	1.58752e7	1.52607	0.763037	+	0.646355i	$0.223707\pi$
$642$	199710.i	0.0191233i
$643$	− 1.57235e7i	− 1.49976i	−0.661574	−	0.749880i	$-0.730111\pi$
$643$	− 1.57235e7i	− 1.49976i	0.661574	−	0.749880i	$-0.269889\pi$
$644$	5.98149e6i	0.568322i
$645$	− 1.43684e6i	− 0.135990i
$646$	−1.02475e6	−0.0966132
$647$	1.58173e7	1.48549	0.742747	−	0.669572i	$-0.233523\pi$
$647$	1.58173e7	1.48549	0.742747	+	0.669572i	$0.233523\pi$
$648$	1.12535e7i	1.05281i
$649$	−553699.	−0.0516015
$650$	0	0
$651$	−985270.	−0.0911178
$652$	− 2.52573e6i	− 0.232685i
$653$	5.31229e6	0.487527	0.243763	−	0.969835i	$-0.421618\pi$
$653$	5.31229e6	0.487527	0.243763	+	0.969835i	$0.421618\pi$
$654$	470265.	0.0429931
$655$	1.06527e7i	0.970192i
$656$	4.29447e6i	0.389628i
$657$	253901.i	0.0229483i
$658$	− 1.43587e7i	− 1.29286i
$659$	9.90554e6	0.888514	0.444257	−	0.895899i	$-0.353468\pi$
$659$	9.90554e6	0.888514	0.444257	+	0.895899i	$0.353468\pi$
$660$	−28002.5	−0.00250228
$661$	− 1.29988e7i	− 1.15717i	−0.815621	−	0.578587i	$-0.803604\pi$
$661$	− 1.29988e7i	− 1.15717i	0.815621	−	0.578587i	$-0.196396\pi$
$662$	−3.08326e6	−0.273442
$663$	0	0
$664$	−1.47372e7	−1.29716
$665$	− 2.79297e6i	− 0.244913i
$666$	−5.08229e6	−0.443991
$667$	−2.13654e6	−0.185950
$668$	2.34508e6i	0.203337i
$669$	− 95138.8i	− 0.00821850i
$670$	− 1.63105e7i	− 1.40372i
$671$	269486.i	0.0231062i
$672$	790688.	0.0675433
$673$	1.32503e7	1.12769	0.563844	−	0.825881i	$-0.309322\pi$
$673$	1.32503e7	1.12769	0.563844	+	0.825881i	$0.309322\pi$
$674$	9.74129e6i	0.825975i
$675$	−5.97391e6	−0.504660
$676$	0	0
$677$	2.23310e7	1.87257	0.936284	−	0.351245i	$-0.114242\pi$
$677$	2.23310e7	1.87257	0.936284	+	0.351245i	$0.114242\pi$
$678$	− 851208.i	− 0.0711151i
$679$	−1.74852e7	−1.45545
$680$	−2.14266e7	−1.77697
$681$	182259.i	0.0150598i
$682$	− 323589.i	− 0.0266399i
$683$	− 1.40049e7i	− 1.14876i	−0.818588	−	0.574380i	$-0.805243\pi$
$683$	− 1.40049e7i	− 1.14876i	0.818588	−	0.574380i	$-0.194757\pi$
$684$	− 574906.i	− 0.0469847i
$685$	3.72851e6	0.303605
$686$	−1.01751e7	−0.825523
$687$	− 1.87502e6i	− 0.151571i
$688$	−4.60382e6	−0.370806
$689$	0	0
$690$	3.25629e6	0.260376
$691$	5.24817e6i	0.418132i	0.977902	+	0.209066i	$0.0670423\pi$
$691$	5.24817e6i	0.418132i	−0.977902	+	0.209066i	$0.932958\pi$
$692$	5.20645e6	0.413310
$693$	−450183.	−0.0356087
$694$	− 1.17349e7i	− 0.924870i
$695$	1.57940e7i	1.24031i
$696$	162233.i	0.0126945i
$697$	− 8.35027e6i	− 0.651056i
$698$	−9.23090e6	−0.717142
$699$	1.02810e6	0.0795868
$700$	− 1.07148e7i	− 0.826495i
$701$	2.13994e7	1.64477	0.822386	−	0.568930i	$-0.192642\pi$
$701$	2.13994e7	1.64477	0.822386	+	0.568930i	$0.192642\pi$
$702$	0	0
$703$	990767.	0.0756107
$704$	516214.i	0.0392553i
$705$	4.20456e6	0.318601
$706$	9.34250e6	0.705426
$707$	− 1.75760e7i	− 1.32243i
$708$	681447.i	0.0510915i
$709$	4.35333e6i	0.325242i	0.986689	+	0.162621i	$0.0519947\pi$
$709$	4.35333e6i	0.325242i	−0.986689	+	0.162621i	$0.948005\pi$
$710$	1.94705e7i	1.44954i
$711$	1.14689e7	0.850839
$712$	−1.93069e6	−0.142729
$713$	− 2.02401e7i	− 1.49104i
$714$	987571.	0.0724975
$715$	0	0
$716$	5.28915e6	0.385570
$717$	− 270087.i	− 0.0196203i
$718$	7.29395e6	0.528021
$719$	−2.21389e7	−1.59710	−0.798552	−	0.601926i	$-0.794400\pi$
$719$	−2.21389e7	−1.59710	−0.798552	+	0.601926i	$0.794400\pi$
$720$	1.34422e7i	0.966361i
$721$	− 1.24430e7i	− 0.891426i
$722$	1.10865e7i	0.791501i
$723$	− 1.12675e6i	− 0.0801645i
$724$	829338.	0.0588010
$725$	3.82726e6	0.270422
$726$	− 1.19633e6i	− 0.0842385i
$727$	4.83218e6	0.339084	0.169542	−	0.985523i	$-0.445771\pi$
$727$	4.83218e6	0.339084	0.169542	+	0.985523i	$0.445771\pi$
$728$	0	0
$729$	−1.34154e7	−0.934940
$730$	498575.i	0.0346277i
$731$	8.95178e6	0.619606
$732$	331661.	0.0228779
$733$	2.47827e7i	1.70368i	0.523803	+	0.851840i	$0.324513\pi$
$733$	2.47827e7i	1.70368i	−0.523803	+	0.851840i	$0.675487\pi$
$734$	− 1.76253e7i	− 1.20753i
$735$	− 143932.i	− 0.00982741i
$736$	1.62428e7i	1.10527i
$737$	512509.	0.0347562
$738$	−8.70939e6	−0.588636
$739$	7.09289e6i	0.477762i	0.971049	+	0.238881i	$0.0767807\pi$
$739$	7.09289e6i	0.477762i	−0.971049	+	0.238881i	$0.923219\pi$
$740$	5.36806e6	0.360361
$741$	0	0
$742$	4.49895e6	0.299986
$743$	− 1.95117e7i	− 1.29665i	−0.761362	−	0.648327i	$-0.775469\pi$
$743$	− 1.95117e7i	− 1.29665i	0.761362	−	0.648327i	$-0.224531\pi$
$744$	−1.53688e6	−0.101791
$745$	7.50588e6	0.495462
$746$	− 7.14803e6i	− 0.470262i
$747$	− 1.79773e7i	− 1.17875i
$748$	− 174461.i	− 0.0114010i
$749$	− 3.39118e6i	− 0.220875i
$750$	−3.42810e6	−0.222536
$751$	−1.66103e7	−1.07468	−0.537339	−	0.843366i	$-0.680570\pi$
$751$	−1.66103e7	−1.07468	−0.537339	+	0.843366i	$0.680570\pi$
$752$	− 1.34720e7i	− 0.868733i
$753$	−630891.	−0.0405477
$754$	0	0
$755$	−5.05952e7	−3.23029
$756$	1.11423e6i	0.0709037i
$757$	1.22902e6	0.0779508	0.0389754	−	0.999240i	$-0.487591\pi$
$757$	1.22902e6	0.0779508	0.0389754	+	0.999240i	$0.487591\pi$
$758$	1.45622e7	0.920567
$759$	102319.i	0.00644693i
$760$	− 4.35664e6i	− 0.273601i
$761$	1.37482e7i	0.860569i	0.902693	+	0.430284i	$0.141587\pi$
$761$	1.37482e7i	0.860569i	−0.902693	+	0.430284i	$0.858413\pi$
$762$	− 1.85121e6i	− 0.115496i
$763$	−7.98535e6	−0.496572
$764$	−2.50850e6	−0.155482
$765$	− 2.61374e7i	− 1.61476i
$766$	1.82659e6	0.112478
$767$	0	0
$768$	1.54908e6	0.0947699
$769$	1.01549e7i	0.619240i	0.950860	+	0.309620i	$0.100202\pi$
$769$	1.01549e7i	0.619240i	−0.950860	+	0.309620i	$0.899798\pi$
$770$	−884007.	−0.0537315
$771$	421140.	0.0255147
$772$	− 7.41222e6i	− 0.447616i
$773$	− 1.41511e7i	− 0.851807i	−0.904769	−	0.425903i	$-0.859956\pi$
$773$	− 1.41511e7i	− 0.851807i	0.904769	−	0.425903i	$-0.140044\pi$
$774$	− 9.33677e6i	− 0.560202i
$775$	3.62567e7i	2.16837i
$776$	−2.72745e7	−1.62593
$777$	−954821.	−0.0567374
$778$	− 1.88593e6i	− 0.111706i
$779$	1.69785e6	0.100243
$780$	0	0
$781$	−611803.	−0.0358909
$782$	2.02873e7i	1.18634i
$783$	−397993.	−0.0231991
$784$	−461178.	−0.0267965
$785$	9.25654e6i	0.536136i
$786$	− 765882.i	− 0.0442186i
$787$	− 1.03095e6i	− 0.0593338i	−0.999560	−	0.0296669i	$-0.990555\pi$
$787$	− 1.03095e6i	− 0.0593338i	0.999560	−	0.0296669i	$-0.00944465\pi$
$788$	− 7.45923e6i	− 0.427935i
$789$	1.94137e6	0.111024
$790$	2.25210e7	1.28387
$791$	1.44540e7i	0.821383i
$792$	−702222.	−0.0397797
$793$	0	0
$794$	469500.	0.0264292
$795$	1.31740e6i	0.0739262i
$796$	−7.22705e6	−0.404276
$797$	1.43337e7	0.799303	0.399651	−	0.916667i	$-0.369131\pi$
$797$	1.43337e7	0.799303	0.399651	+	0.916667i	$0.369131\pi$
$798$	200802.i	0.0111625i
$799$	2.61952e7i	1.45163i
$800$	− 2.90963e7i	− 1.60736i
$801$	− 2.35516e6i	− 0.129700i
$802$	−1.01889e7	−0.559360
$803$	−15666.3	−0.000857386 0
$804$	− 630753.i	− 0.0344127i
$805$	−5.52935e7	−3.00735
$806$	0	0
$807$	−1.38766e6	−0.0750065
$808$	− 2.74161e7i	− 1.47733i
$809$	1.55020e7	0.832751	0.416376	−	0.909193i	$-0.363300\pi$
$809$	1.55020e7	0.832751	0.416376	+	0.909193i	$0.363300\pi$
$810$	−2.69565e7	−1.44361
$811$	− 2.45861e7i	− 1.31261i	−0.754494	−	0.656307i	$-0.772118\pi$
$811$	− 2.45861e7i	− 1.31261i	0.754494	−	0.656307i	$-0.227882\pi$
$812$	− 713842.i	− 0.0379938i
$813$	66737.5i	0.00354114i
$814$	− 313589.i	− 0.0165882i
$815$	2.33481e7	1.23128
$816$	926583.	0.0487146
$817$	1.82016e6i	0.0954011i
$818$	−2.03513e7	−1.06343
$819$	0	0
$820$	9.19911e6	0.477761
$821$	3.33396e6i	0.172624i	0.996268	+	0.0863122i	$0.0275083\pi$
$821$	3.33396e6i	0.172624i	−0.996268	+	0.0863122i	$0.972492\pi$
$822$	−268062.	−0.0138375
$823$	−3.08787e6	−0.158913	−0.0794564	−	0.996838i	$-0.525318\pi$
$823$	−3.08787e6	−0.158913	−0.0794564	+	0.996838i	$0.525318\pi$
$824$	− 1.94092e7i	− 0.995842i
$825$	− 183288.i	− 0.00937559i
$826$	2.15125e7i	1.09709i
$827$	6.89555e6i	0.350595i	0.984516	+	0.175297i	$0.0560886\pi$
$827$	6.89555e6i	0.350595i	−0.984516	+	0.175297i	$0.943911\pi$
$828$	−1.13816e7	−0.576936
$829$	2.00007e7	1.01078	0.505391	−	0.862890i	$-0.331348\pi$
$829$	2.00007e7	1.01078	0.505391	+	0.862890i	$0.331348\pi$
$830$	− 3.53012e7i	− 1.77867i
$831$	1.63156e6	0.0819596
$832$	0	0
$833$	896725.	0.0447762
$834$	− 1.13551e6i	− 0.0565297i
$835$	−2.16782e7	−1.07599
$836$	35473.0	0.00175542
$837$	− 3.77030e6i	− 0.186021i
$838$	1.92859e7i	0.948701i
$839$	5.23988e6i	0.256990i	0.991710	+	0.128495i	$0.0410146\pi$
$839$	5.23988e6i	0.256990i	−0.991710	+	0.128495i	$0.958985\pi$
$840$	4.19858e6i	0.205307i
$841$	−2.02562e7	−0.987569
$842$	1.87841e7	0.913081
$843$	2.83079e6i	0.137195i
$844$	−9.71509e6	−0.469452
$845$	0	0
$846$	2.73218e7	1.31245
$847$	2.03144e7i	0.972959i
$848$	4.22111e6	0.201575
$849$	−2.07689e6	−0.0988883
$850$	− 3.63413e7i	− 1.72526i
$851$	− 1.96146e7i	− 0.928442i
$852$	752957.i	0.0355362i
$853$	− 1.32853e7i	− 0.625170i	−0.949890	−	0.312585i	$-0.898805\pi$
$853$	− 1.32853e7i	− 0.625170i	0.949890	−	0.312585i	$-0.101195\pi$
$854$	1.04702e7	0.491257
$855$	5.31449e6	0.248626
$856$	− 5.28976e6i	− 0.246747i
$857$	8.34473e6	0.388115	0.194057	−	0.980990i	$-0.437835\pi$
$857$	8.34473e6	0.388115	0.194057	+	0.980990i	$0.437835\pi$
$858$	0	0
$859$	−4.07521e7	−1.88437	−0.942187	−	0.335088i	$-0.891234\pi$
$859$	−4.07521e7	−1.88437	−0.942187	+	0.335088i	$0.891234\pi$
$860$	9.86176e6i	0.454683i
$861$	−1.63625e6	−0.0752216
$862$	5.26056e6	0.241137
$863$	1.73853e7i	0.794614i	0.917686	+	0.397307i	$0.130055\pi$
$863$	1.73853e7i	0.794614i	−0.917686	+	0.397307i	$0.869945\pi$
$864$	3.02570e6i	0.137893i
$865$	4.81289e7i	2.18708i
$866$	153881.i	0.00697252i
$867$	513667.	0.0232078
$868$	6.76243e6	0.304652
$869$	707656.i	0.0317887i
$870$	−388612.	−0.0174068
$871$	0	0
$872$	−1.24560e7	−0.554738
$873$	− 3.32710e7i	− 1.47751i
$874$	−4.12500e6	−0.182661
$875$	5.82109e7	2.57030
$876$	19280.7i	0 0.000848912i
$877$	− 2.58279e6i	− 0.113394i	−0.998391	−	0.0566971i	$-0.981943\pi$
$877$	− 2.58279e6i	− 0.113394i	0.998391	−	0.0566971i	$-0.0180569\pi$
$878$	− 3.41370e7i	− 1.49447i
$879$	3332.63i	0 0.000145484i
$880$	−829414.	−0.0361048
$881$	1.66814e7	0.724090	0.362045	−	0.932161i	$-0.382079\pi$
$881$	1.66814e7	0.724090	0.362045	+	0.932161i	$0.382079\pi$
$882$	− 935291.i	− 0.0404833i
$883$	−2.36384e7	−1.02027	−0.510137	−	0.860093i	$-0.670405\pi$
$883$	−2.36384e7	−1.02027	−0.510137	+	0.860093i	$0.670405\pi$
$884$	0	0
$885$	−6.29936e6	−0.270358
$886$	− 1.49632e7i	− 0.640382i
$887$	−5.25660e6	−0.224334	−0.112167	−	0.993689i	$-0.535779\pi$
$887$	−5.25660e6	−0.224334	−0.112167	+	0.993689i	$0.535779\pi$
$888$	−1.48939e6	−0.0633833
$889$	3.14345e7i	1.33399i
$890$	− 4.62474e6i	− 0.195710i
$891$	− 847029.i	− 0.0357441i
$892$	652988.i	0.0274785i
$893$	−5.32625e6	−0.223508
$894$	−539638.	−0.0225818
$895$	4.88935e7i	2.04030i
$896$	4.53994e6	0.188921
$897$	0	0
$898$	−3.62806e7	−1.50135
$899$	2.41549e6i	0.0996795i
$900$	2.03882e7	0.839022
$901$	−8.20763e6	−0.336826
$902$	− 537389.i	− 0.0219924i
$903$	− 1.75412e6i	− 0.0715879i
$904$	2.25461e7i	0.917595i
$905$	7.66648e6i	0.311153i
$906$	3.63756e6	0.147228
$907$	−3.22789e7	−1.30287	−0.651435	−	0.758705i	$-0.725833\pi$
$907$	−3.22789e7	−1.30287	−0.651435	+	0.758705i	$0.725833\pi$
$908$	− 1.25094e6i	− 0.0503525i
$909$	3.34438e7	1.34247
$910$	0	0
$911$	4.20975e7	1.68058	0.840292	−	0.542134i	$-0.182383\pi$
$911$	4.20975e7	1.68058	0.840292	+	0.542134i	$0.182383\pi$
$912$	188401.i	0.00750061i
$913$	1.10924e6	0.0440400
$914$	−1.52897e7	−0.605389
$915$	3.06590e6i	0.121061i
$916$	1.28693e7i	0.506775i
$917$	1.30051e7i	0.510728i
$918$	3.77910e6i	0.148007i
$919$	−2.19460e7	−0.857168	−0.428584	−	0.903502i	$-0.640987\pi$
$919$	−2.19460e7	−0.857168	−0.428584	+	0.903502i	$0.640987\pi$
$920$	−8.62500e7	−3.35961
$921$	654407.i	0.0254213i
$922$	2.13767e7	0.828157
$923$	0	0
$924$	−34186.0	−0.00131725
$925$	3.51362e7i	1.35021i
$926$	−3.03205e7	−1.16201
$927$	2.36765e7	0.904937
$928$	− 1.93845e6i	− 0.0738899i
$929$	1.35921e7i	0.516710i	0.966050	+	0.258355i	$0.0831804\pi$
$929$	1.35921e7i	0.516710i	−0.966050	+	0.258355i	$0.916820\pi$
$930$	− 3.68143e6i	− 0.139575i
$931$	182330.i	0.00689421i
$932$	−7.05637e6	−0.266098
$933$	−3.14116e6	−0.118137
$934$	1.43297e7i	0.537489i
$935$	1.61273e6	0.0603300
$936$	0	0
$937$	3.01018e7	1.12007	0.560033	−	0.828470i	$-0.310788\pi$
$937$	3.01018e7	1.12007	0.560033	+	0.828470i	$0.310788\pi$
$938$	− 1.99122e7i	− 0.738944i
$939$	−2.68279e6	−0.0992938
$940$	−2.88581e7	−1.06524
$941$	1.06275e7i	0.391252i	0.980679	+	0.195626i	$0.0626739\pi$
$941$	1.06275e7i	0.391252i	−0.980679	+	0.195626i	$0.937326\pi$
$942$	− 665503.i	− 0.0244356i
$943$	− 3.36130e7i	− 1.23091i
$944$	2.01840e7i	0.737187i
$945$	−1.03000e7	−0.375196
$946$	576099.	0.0209300
$947$	− 2.41709e7i	− 0.875826i	−0.899017	−	0.437913i	$-0.855718\pi$
$947$	− 2.41709e7i	− 0.875826i	0.899017	−	0.437913i	$-0.144282\pi$
$948$	870924.	0.0314745
$949$	0	0
$950$	7.38925e6	0.265639
$951$	− 3.25282e6i	− 0.116630i
$952$	−2.61580e7	−0.935432
$953$	−4.27043e7	−1.52314	−0.761569	−	0.648084i	$-0.775570\pi$
$953$	−4.27043e7	−1.52314	−0.761569	+	0.648084i	$0.775570\pi$
$954$	8.56063e6i	0.304533i
$955$	− 2.31888e7i	− 0.822754i
$956$	1.85375e6i	0.0656003i
$957$	− 12211.0i	0 0.000430993i
$958$	3.04842e7	1.07315
$959$	4.55184e6	0.159823
$960$	5.87290e6i	0.205672i
$961$	5.74655e6	0.200724
$962$	0	0
$963$	6.45276e6	0.224223
$964$	7.73348e6i	0.268029i
$965$	6.85193e7	2.36862
$966$	3.97535e6	0.137067
$967$	4.42692e7i	1.52242i	0.648504	+	0.761211i	$0.275395\pi$
$967$	4.42692e7i	1.52242i	−0.648504	+	0.761211i	$0.724605\pi$
$968$	3.16875e7i	1.08693i
$969$	− 366332.i	− 0.0125333i
$970$	− 6.53329e7i	− 2.22947i
$971$	3.88962e7	1.32391	0.661956	−	0.749542i	$-0.269726\pi$
$971$	3.88962e7	1.32391	0.661956	+	0.749542i	$0.269726\pi$
$972$	−3.18606e6	−0.108166
$973$	1.92816e7i	0.652922i
$974$	−1.85417e7	−0.626257
$975$	0	0
$976$	9.82357e6	0.330099
$977$	2.71611e7i	0.910354i	0.890401	+	0.455177i	$0.150424\pi$
$977$	2.71611e7i	0.910354i	−0.890401	+	0.455177i	$0.849576\pi$
$978$	−1.67862e6	−0.0561184
$979$	145319.	0.00484580
$980$	987881.i	0.0328579i
$981$	− 1.51946e7i	− 0.504099i
$982$	9.59904e6i	0.317650i
$983$	− 1.98048e7i	− 0.653714i	−0.945074	−	0.326857i	$-0.894011\pi$
$983$	− 1.98048e7i	− 0.653714i	0.945074	−	0.326857i	$-0.105989\pi$
$984$	−2.55232e6	−0.0840326
$985$	6.89538e7	2.26448
$986$	− 2.42113e6i	− 0.0793096i
$987$	5.13301e6	0.167718
$988$	0	0
$989$	3.60343e7	1.17145
$990$	− 1.68209e6i	− 0.0545459i
$991$	−1.44104e7	−0.466115	−0.233057	−	0.972463i	$-0.574873\pi$
$991$	−1.44104e7	−0.466115	−0.233057	+	0.972463i	$0.574873\pi$
$992$	1.83635e7	0.592483
$993$	− 1.10222e6i	− 0.0354727i
$994$	2.37700e7i	0.763068i
$995$	− 6.68075e7i	− 2.13928i
$996$	− 1.36516e6i	− 0.0436048i
$997$	3.16635e7	1.00884	0.504418	−	0.863459i	$-0.331707\pi$
$997$	3.16635e7	1.00884	0.504418	+	0.863459i	$0.331707\pi$
$998$	2.72195e7	0.865074
$999$	− 3.65378e6i	− 0.115832i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.6.b.a.168.4		4
13.5	odd	4		169.6.a.a.1.2		2
13.8	odd	4		13.6.a.a.1.1	✓	2
13.12	even	2	inner	169.6.b.a.168.1		4
39.8	even	4		117.6.a.c.1.2		2
52.47	even	4		208.6.a.h.1.1		2
65.8	even	4		325.6.b.b.274.4		4
65.34	odd	4		325.6.a.b.1.2		2
65.47	even	4		325.6.b.b.274.1		4
91.34	even	4		637.6.a.a.1.1		2
104.21	odd	4		832.6.a.p.1.1		2
104.99	even	4		832.6.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.a.1.1	✓	2	13.8	odd	4
117.6.a.c.1.2		2	39.8	even	4
169.6.a.a.1.2		2	13.5	odd	4
169.6.b.a.168.1		4	13.12	even	2	inner
169.6.b.a.168.4		4	1.1	even	1	trivial
208.6.a.h.1.1		2	52.47	even	4
325.6.a.b.1.2		2	65.34	odd	4
325.6.b.b.274.1		4	65.47	even	4
325.6.b.b.274.4		4	65.8	even	4
637.6.a.a.1.1		2	91.34	even	4
832.6.a.i.1.2		2	104.99	even	4
832.6.a.p.1.1		2	104.21	odd	4

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.56155i q^{2} -1.63068 q^{3} +11.1922 q^{4} +103.462i q^{5} -7.43845i q^{6} +126.309i q^{7} +197.024i q^{8} -240.341 q^{9} +O(q^{10})\)	\(q+4.56155i q^{2} -1.63068 q^{3} +11.1922 q^{4} +103.462i q^{5} -7.43845i q^{6} +126.309i q^{7} +197.024i q^{8} -240.341 q^{9} -471.948 q^{10} -14.8296i q^{11} -18.2510 q^{12} -576.164 q^{14} -168.714i q^{15} -540.582 q^{16} +1051.12 q^{17} -1096.33i q^{18} +213.723i q^{19} +1157.97i q^{20} -205.969i q^{21} +67.6458 q^{22} +4231.16 q^{23} -321.283i q^{24} -7579.41 q^{25} +788.176 q^{27} +1413.68i q^{28} -504.955 q^{29} +769.597 q^{30} -4783.58i q^{31} +3838.86i q^{32} +24.1823i q^{33} +4794.75i q^{34} -13068.2 q^{35} -2689.95 q^{36} -4635.74i q^{37} -974.911 q^{38} -20384.5 q^{40} -7944.15i q^{41} +939.541 q^{42} +8516.41 q^{43} -165.976i q^{44} -24866.2i q^{45} +19300.7i q^{46} +24921.2i q^{47} +881.518 q^{48} +853.113 q^{49} -34573.9i q^{50} -1714.05 q^{51} -7808.46 q^{53} +3595.31i q^{54} +1534.30 q^{55} -24885.8 q^{56} -348.515i q^{57} -2303.38i q^{58} -37337.5i q^{59} -1888.29i q^{60} -18172.2 q^{61} +21820.5 q^{62} -30357.1i q^{63} -34809.8 q^{64} -110.309 q^{66} +34559.9i q^{67} +11764.4 q^{68} -6899.68 q^{69} -59611.1i q^{70} -41255.7i q^{71} -47352.8i q^{72} -1056.42i q^{73} +21146.2 q^{74} +12359.6 q^{75} +2392.04i q^{76} +1873.10 q^{77} -47719.3 q^{79} -55929.8i q^{80} +57117.6 q^{81} +36237.7 q^{82} +74799.0i q^{83} -2305.26i q^{84} +108751. i q^{85} +38848.0i q^{86} +823.421 q^{87} +2921.78 q^{88} +9799.26i q^{89} +113428. q^{90} +47356.1 q^{92} +7800.50i q^{93} -113679. q^{94} -22112.3 q^{95} -6259.97i q^{96} +138432. i q^{97} +3891.52i q^{98} +3564.15i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 56 q^{3} + 86 q^{4} + 424 q^{9}+O(q^{10}) \) 4 * q - 56 * q^3 + 86 * q^4 + 424 * q^9	\( 4 q - 56 q^{3} + 86 q^{4} + 424 q^{9} - 890 q^{10} - 1714 q^{12} - 1010 q^{14} + 930 q^{16} + 5260 q^{17} + 452 q^{22} + 5248 q^{23} - 16464 q^{25} - 9464 q^{27} - 1624 q^{29} + 118 q^{30} - 46088 q^{35} + 23396 q^{36} - 2036 q^{38} - 37330 q^{40} - 1874 q^{42} - 4840 q^{43} - 51270 q^{48} - 17368 q^{49} - 86696 q^{51} - 87440 q^{53} - 41328 q^{55} - 40690 q^{56} + 3968 q^{61} + 46216 q^{62} - 6434 q^{64} - 8572 q^{66} + 123970 q^{68} + 70960 q^{69} + 53010 q^{74} + 59136 q^{75} - 113496 q^{77} - 110592 q^{79} + 185524 q^{81} + 72384 q^{82} + 17840 q^{87} + 26052 q^{88} + 251236 q^{90} - 7528 q^{92} - 214250 q^{94} - 32144 q^{95}+O(q^{100}) \) 4 * q - 56 * q^3 + 86 * q^4 + 424 * q^9 - 890 * q^10 - 1714 * q^12 - 1010 * q^14 + 930 * q^16 + 5260 * q^17 + 452 * q^22 + 5248 * q^23 - 16464 * q^25 - 9464 * q^27 - 1624 * q^29 + 118 * q^30 - 46088 * q^35 + 23396 * q^36 - 2036 * q^38 - 37330 * q^40 - 1874 * q^42 - 4840 * q^43 - 51270 * q^48 - 17368 * q^49 - 86696 * q^51 - 87440 * q^53 - 41328 * q^55 - 40690 * q^56 + 3968 * q^61 + 46216 * q^62 - 6434 * q^64 - 8572 * q^66 + 123970 * q^68 + 70960 * q^69 + 53010 * q^74 + 59136 * q^75 - 113496 * q^77 - 110592 * q^79 + 185524 * q^81 + 72384 * q^82 + 17840 * q^87 + 26052 * q^88 + 251236 * q^90 - 7528 * q^92 - 214250 * q^94 - 32144 * q^95

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