LMFDB - Embedded newform 169.8.b.e.168.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,8,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, 8, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 8 $
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:	$52.7930693068$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1591 x^{14} + 998837 x^{12} + 319862003 x^{10} + 57017400035 x^{8} + 5819167911653 x^{6} + \cdots + 11\!\cdots\!00 $ x^16 + 1591x^14 + 998837x^12 + 319862003x^10 + 57017400035x^8 + 5819167911653x^6 + 333227942883751x^4 + 9796730453852209*x^2 + 113340278793830400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 13^{8} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.1
Root			$-20.7945i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.8.b.e.168.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-21.7945i q^{2} -67.8320 q^{3} -346.998 q^{4} -157.804i q^{5} +1478.36i q^{6} +842.108i q^{7} +4772.94i q^{8} +2414.17 q^{9} -3439.24 q^{10} -1991.16i q^{11} +23537.6 q^{12} +18353.3 q^{14} +10704.1i q^{15} +59607.9 q^{16} -9620.90 q^{17} -52615.6i q^{18} +8439.25i q^{19} +54757.5i q^{20} -57121.8i q^{21} -43396.3 q^{22} -34132.3 q^{23} -323758. i q^{24} +53223.0 q^{25} -15409.7 q^{27} -292210. i q^{28} -9672.89 q^{29} +233290. q^{30} -205412. i q^{31} -688185. i q^{32} +135065. i q^{33} +209682. i q^{34} +132888. q^{35} -837714. q^{36} +432002. i q^{37} +183929. q^{38} +753187. q^{40} -113586. i q^{41} -1.24494e6 q^{42} +400363. q^{43} +690930. i q^{44} -380965. i q^{45} +743896. i q^{46} +1.32924e6i q^{47} -4.04332e6 q^{48} +114397. q^{49} -1.15997e6i q^{50} +652605. q^{51} +298608. q^{53} +335846. i q^{54} -314213. q^{55} -4.01933e6 q^{56} -572451. i q^{57} +210815. i q^{58} -1.63424e6i q^{59} -3.71431e6i q^{60} +1.66237e6 q^{61} -4.47685e6 q^{62} +2.03300e6i q^{63} -7.36880e6 q^{64} +2.94366e6 q^{66} -1.42305e6i q^{67} +3.33844e6 q^{68} +2.31526e6 q^{69} -2.89621e6i q^{70} +1.03961e6i q^{71} +1.15227e7i q^{72} +4.43080e6i q^{73} +9.41526e6 q^{74} -3.61022e6 q^{75} -2.92840e6i q^{76} +1.67678e6 q^{77} -7.40813e6 q^{79} -9.40634e6i q^{80} -4.23453e6 q^{81} -2.47556e6 q^{82} -753809. i q^{83} +1.98212e7i q^{84} +1.51821e6i q^{85} -8.72569e6i q^{86} +656131. q^{87} +9.50371e6 q^{88} -2.49177e6i q^{89} -8.30293e6 q^{90} +1.18439e7 q^{92} +1.39335e7i q^{93} +2.89701e7 q^{94} +1.33174e6 q^{95} +4.66809e7i q^{96} -6.38799e6i q^{97} -2.49322e6i q^{98} -4.80702e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 56 q^{3} - 1154 q^{4} + 11976 q^{9} - 13626 q^{10} + 15816 q^{12} + 14484 q^{14} + 122754 q^{16} - 45648 q^{17} + 147452 q^{22} - 144936 q^{23} - 58488 q^{25} + 358616 q^{27} + 443544 q^{29} - 516232 q^{30}+ \cdots - 32534160 q^{95}+O(q^{100}) $ 16 * q + 56 * q^3 - 1154 * q^4 + 11976 * q^9 - 13626 * q^10 + 15816 * q^12 + 14484 * q^14 + 122754 * q^16 - 45648 * q^17 + 147452 * q^22 - 144936 * q^23 - 58488 * q^25 + 358616 * q^27 + 443544 * q^29 - 516232 * q^30 - 63120 * q^35 - 2325986 * q^36 - 2364492 * q^38 + 766030 * q^40 - 3121908 * q^42 + 782984 * q^43 - 9171424 * q^48 - 2217560 * q^49 - 8040712 * q^51 + 3329328 * q^53 + 836432 * q^55 - 7218312 * q^56 - 3527096 * q^61 - 25996776 * q^62 - 32227074 * q^64 + 16245108 * q^66 + 20236182 * q^68 - 9236984 * q^69 - 4412442 * q^74 - 43669976 * q^75 - 35263992 * q^77 + 12644416 * q^79 + 18632544 * q^81 - 18695502 * q^82 - 10399976 * q^87 - 73352808 * q^88 - 132263562 * q^90 + 64127424 * q^92 + 12052256 * q^94 - 32534160 * q^95

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 $

$2$	− 21.7945i	− 1.92638i	−0.268830	−	0.963188i	$-0.586637\pi$
$2$	− 21.7945i	− 1.92638i	0.268830	−	0.963188i	$-0.413363\pi$
$3$	−67.8320	−1.45047	−0.725237	−	0.688499i	$-0.758270\pi$
$3$	−67.8320	−1.45047	−0.725237	+	0.688499i	$0.758270\pi$
$4$	−346.998	−2.71092
$5$	− 157.804i	− 0.564575i	−0.959330	−	0.282288i	$-0.908907\pi$
$5$	− 157.804i	− 0.564575i	0.959330	−	0.282288i	$-0.0910932\pi$
$6$	1478.36i	2.79416i
$7$	842.108i	0.927950i	0.885848	+	0.463975i	$0.153577\pi$
$7$	842.108i	0.927950i	−0.885848	+	0.463975i	$0.846423\pi$
$8$	4772.94i	3.29588i
$9$	2414.17	1.10387
$10$	−3439.24	−1.08758
$11$	− 1991.16i	− 0.451058i	−0.974236	−	0.225529i	$-0.927589\pi$
$11$	− 1991.16i	− 0.451058i	0.974236	−	0.225529i	$-0.0724111\pi$
$12$	23537.6	3.93212
$13$	0	0
$13$	0	0
$14$	18353.3	1.78758
$15$	10704.1i	0.818902i
$16$	59607.9	3.63818
$17$	−9620.90	−0.474946	−0.237473	−	0.971394i	$-0.576319\pi$
$17$	−9620.90	−0.474946	−0.237473	+	0.971394i	$0.576319\pi$
$18$	− 52615.6i	− 2.12648i
$19$	8439.25i	0.282271i	0.989990	+	0.141135i	$0.0450753\pi$
$19$	8439.25i	0.282271i	−0.989990	+	0.141135i	$0.954925\pi$
$20$	54757.5i	1.53052i
$21$	− 57121.8i	− 1.34597i
$22$	−43396.3	−0.868908
$23$	−34132.3	−0.584950	−0.292475	−	0.956273i	$-0.594479\pi$
$23$	−34132.3	−0.584950	−0.292475	+	0.956273i	$0.594479\pi$
$24$	− 323758.i	− 4.78059i
$25$	53223.0	0.681255
$26$	0	0
$27$	−15409.7	−0.150668
$28$	− 292210.i	− 2.51560i
$29$	−9672.89	−0.0736484	−0.0368242	−	0.999322i	$-0.511724\pi$
$29$	−9672.89	−0.0736484	−0.0368242	+	0.999322i	$0.511724\pi$
$30$	233290.	1.57751
$31$	− 205412.i	− 1.23840i	−0.785234	−	0.619199i	$-0.787457\pi$
$31$	− 205412.i	− 1.23840i	0.785234	−	0.619199i	$-0.212543\pi$
$32$	− 688185.i	− 3.71262i
$33$	135065.i	0.654248i
$34$	209682.i	0.914925i
$35$	132888.	0.523898
$36$	−837714.	−2.99252
$37$	432002.i	1.40210i	0.713110	+	0.701052i	$0.247286\pi$
$37$	432002.i	1.40210i	−0.713110	+	0.701052i	$0.752714\pi$
$38$	183929.	0.543760
$39$	0	0
$40$	753187.	1.86077
$41$	− 113586.i	− 0.257385i	−0.991685	−	0.128692i	$-0.958922\pi$
$41$	− 113586.i	− 0.257385i	0.991685	−	0.128692i	$-0.0410780\pi$
$42$	−1.24494e6	−2.59284
$43$	400363.	0.767917	0.383958	−	0.923350i	$-0.374561\pi$
$43$	400363.	0.767917	0.383958	+	0.923350i	$0.374561\pi$
$44$	690930.i	1.22278i
$45$	− 380965.i	− 0.623220i
$46$	743896.i	1.12683i
$47$	1.32924e6i	1.86750i	0.357920	+	0.933752i	$0.383486\pi$
$47$	1.32924e6i	1.86750i	−0.357920	+	0.933752i	$0.616514\pi$
$48$	−4.04332e6	−5.27708
$49$	114397.	0.138909
$50$	− 1.15997e6i	− 1.31235i
$51$	652605.	0.688898
$52$	0	0
$53$	298608.	0.275509	0.137755	−	0.990466i	$-0.456011\pi$
$53$	298608.	0.275509	0.137755	+	0.990466i	$0.456011\pi$
$54$	335846.i	0.290243i
$55$	−314213.	−0.254656
$56$	−4.01933e6	−3.05841
$57$	− 572451.i	− 0.409427i
$58$	210815.i	0.141874i
$59$	− 1.63424e6i	− 1.03594i	−0.855399	−	0.517970i	$-0.826688\pi$
$59$	− 1.63424e6i	− 1.03594i	0.855399	−	0.517970i	$-0.173312\pi$
$60$	− 3.71431e6i	− 2.21998i
$61$	1.66237e6	0.937722	0.468861	−	0.883272i	$-0.344665\pi$
$61$	1.66237e6	0.937722	0.468861	+	0.883272i	$0.344665\pi$
$62$	−4.47685e6	−2.38562
$63$	2.03300e6i	1.02434i
$64$	−7.36880e6	−3.51372
$65$	0	0
$66$	2.94366e6	1.26033
$67$	− 1.42305e6i	− 0.578042i	−0.957323	−	0.289021i	$-0.906670\pi$
$67$	− 1.42305e6i	− 0.578042i	0.957323	−	0.289021i	$-0.0933298\pi$
$68$	3.33844e6	1.28754
$69$	2.31526e6	0.848455
$70$	− 2.89621e6i	− 1.00922i
$71$	1.03961e6i	0.344720i	0.985034	+	0.172360i	$0.0551392\pi$
$71$	1.03961e6i	0.344720i	−0.985034	+	0.172360i	$0.944861\pi$
$72$	1.15227e7i	3.63824i
$73$	4.43080e6i	1.33307i	0.745475	+	0.666533i	$0.232222\pi$
$73$	4.43080e6i	1.33307i	−0.745475	+	0.666533i	$0.767778\pi$
$74$	9.41526e6	2.70098
$75$	−3.61022e6	−0.988143
$76$	− 2.92840e6i	− 0.765215i
$77$	1.67678e6	0.418560
$78$	0	0
$79$	−7.40813e6	−1.69049	−0.845247	−	0.534376i	$-0.820547\pi$
$79$	−7.40813e6	−1.69049	−0.845247	+	0.534376i	$0.820547\pi$
$80$	− 9.40634e6i	− 2.05402i
$81$	−4.23453e6	−0.885335
$82$	−2.47556e6	−0.495820
$83$	− 753809.i	− 0.144706i	−0.997379	−	0.0723532i	$-0.976949\pi$
$83$	− 753809.i	− 0.144706i	0.997379	−	0.0723532i	$-0.0230509\pi$
$84$	1.98212e7i	3.64881i
$85$	1.51821e6i	0.268143i
$86$	− 8.72569e6i	− 1.47930i
$87$	656131.	0.106825
$88$	9.50371e6	1.48663
$89$	− 2.49177e6i	− 0.374664i	−0.982297	−	0.187332i	$-0.940016\pi$
$89$	− 2.49177e6i	− 0.374664i	0.982297	−	0.187332i	$-0.0599841\pi$
$90$	−8.30293e6	−1.20056
$91$	0	0
$92$	1.18439e7	1.58575
$93$	1.39335e7i	1.79626i
$94$	2.89701e7	3.59751
$95$	1.33174e6	0.159363
$96$	4.66809e7i	5.38505i
$97$	− 6.38799e6i	− 0.710662i	−0.934741	−	0.355331i	$-0.884368\pi$
$97$	− 6.38799e6i	− 0.710662i	0.934741	−	0.355331i	$-0.115632\pi$
$98$	− 2.49322e6i	− 0.267590i
$99$	− 4.80702e6i	− 0.497912i
$100$	−1.84683e7	−1.84683
$101$	1.37901e7	1.33182	0.665908	−	0.746034i	$-0.268044\pi$
$101$	1.37901e7	1.33182	0.665908	+	0.746034i	$0.268044\pi$
$102$	− 1.42232e7i	− 1.32708i
$103$	−1.00557e7	−0.906740	−0.453370	−	0.891322i	$-0.649778\pi$
$103$	−1.00557e7	−0.906740	−0.453370	+	0.891322i	$0.649778\pi$
$104$	0	0
$105$	−9.01403e6	−0.759900
$106$	− 6.50800e6i	− 0.530734i
$107$	−6.62417e6	−0.522743	−0.261372	−	0.965238i	$-0.584175\pi$
$107$	−6.62417e6	−0.522743	−0.261372	+	0.965238i	$0.584175\pi$
$108$	5.34713e6	0.408449
$109$	2.15505e7i	1.59391i	0.604036	+	0.796957i	$0.293558\pi$
$109$	2.15505e7i	1.59391i	−0.604036	+	0.796957i	$0.706442\pi$
$110$	6.84809e6i	0.490564i
$111$	− 2.93036e7i	− 2.03372i
$112$	5.01963e7i	3.37605i
$113$	−1.30333e7	−0.849725	−0.424863	−	0.905258i	$-0.639678\pi$
$113$	−1.30333e7	−0.849725	−0.424863	+	0.905258i	$0.639678\pi$
$114$	−1.24763e7	−0.788710
$115$	5.38620e6i	0.330248i
$116$	3.35647e6	0.199655
$117$	0	0
$118$	−3.56174e7	−1.99561
$119$	− 8.10184e6i	− 0.440727i
$120$	−5.10902e7	−2.69900
$121$	1.55224e7	0.796546
$122$	− 3.62305e7i	− 1.80640i
$123$	7.70479e6i	0.373330i
$124$	7.12776e7i	3.35720i
$125$	− 2.07272e7i	− 0.949195i
$126$	4.43080e7	1.97327
$127$	−7.43440e6	−0.322057	−0.161029	−	0.986950i	$-0.551481\pi$
$127$	−7.43440e6	−0.322057	−0.161029	+	0.986950i	$0.551481\pi$
$128$	7.25113e7i	3.05612i
$129$	−2.71574e7	−1.11384
$130$	0	0
$131$	1.19177e7	0.463173	0.231586	−	0.972814i	$-0.425608\pi$
$131$	1.19177e7	0.463173	0.231586	+	0.972814i	$0.425608\pi$
$132$	− 4.68671e7i	− 1.77362i
$133$	−7.10676e6	−0.261933
$134$	−3.10147e7	−1.11353
$135$	2.43170e6i	0.0850633i
$136$	− 4.59200e7i	− 1.56537i
$137$	− 4.91202e7i	− 1.63207i	−0.578004	−	0.816034i	$-0.696168\pi$
$137$	− 4.91202e7i	− 1.63207i	0.578004	−	0.816034i	$-0.303832\pi$
$138$	− 5.04599e7i	− 1.63444i
$139$	−1.87624e7	−0.592566	−0.296283	−	0.955100i	$-0.595747\pi$
$139$	−1.87624e7	−0.592566	−0.296283	+	0.955100i	$0.595747\pi$
$140$	−4.61117e7	−1.42025
$141$	− 9.01651e7i	− 2.70877i
$142$	2.26577e7	0.664060
$143$	0	0
$144$	1.43904e8	4.01609
$145$	1.52642e6i	0.0415801i
$146$	9.65668e7	2.56799
$147$	−7.75978e6	−0.201483
$148$	− 1.49904e8i	− 3.80099i
$149$	− 4.55981e7i	− 1.12926i	−0.825344	−	0.564631i	$-0.809019\pi$
$149$	− 4.55981e7i	− 1.12926i	0.825344	−	0.564631i	$-0.190981\pi$
$150$	7.86828e7i	1.90353i
$151$	− 3.04360e7i	− 0.719397i	−0.933069	−	0.359699i	$-0.882880\pi$
$151$	− 3.04360e7i	− 0.719397i	0.933069	−	0.359699i	$-0.117120\pi$
$152$	−4.02801e7	−0.930331
$153$	−2.32265e7	−0.524281
$154$	− 3.65444e7i	− 0.806303i
$155$	−3.24148e7	−0.699169
$156$	0	0
$157$	−7.21167e7	−1.48726	−0.743631	−	0.668591i	$-0.766898\pi$
$157$	−7.21167e7	−1.48726	−0.743631	+	0.668591i	$0.766898\pi$
$158$	1.61456e8i	3.25653i
$159$	−2.02552e7	−0.399619
$160$	−1.08598e8	−2.09605
$161$	− 2.87431e7i	− 0.542804i
$162$	9.22893e7i	1.70549i
$163$	− 2.74591e7i	− 0.496626i	−0.968680	−	0.248313i	$-0.920124\pi$
$163$	− 2.74591e7i	− 0.496626i	0.968680	−	0.248313i	$-0.0798762\pi$
$164$	3.94143e7i	0.697751i
$165$	2.13137e7	0.369372
$166$	−1.64289e7	−0.278759
$167$	5.65534e6i	0.0939618i	0.998896	+	0.0469809i	$0.0149600\pi$
$167$	5.65534e6i	0.0939618i	−0.998896	+	0.0469809i	$0.985040\pi$
$168$	2.72639e8	4.43615
$169$	0	0
$170$	3.30886e7	0.516544
$171$	2.03738e7i	0.311592i
$172$	−1.38925e8	−2.08176
$173$	6.27713e7	0.921722	0.460861	−	0.887472i	$-0.347541\pi$
$173$	6.27713e7	0.921722	0.460861	+	0.887472i	$0.347541\pi$
$174$	− 1.43000e7i	− 0.205785i
$175$	4.48195e7i	0.632171i
$176$	− 1.18689e8i	− 1.64103i
$177$	1.10854e8i	1.50260i
$178$	−5.43067e7	−0.721744
$179$	−1.33940e8	−1.74551	−0.872757	−	0.488156i	$-0.837670\pi$
$179$	−1.33940e8	−1.74551	−0.872757	+	0.488156i	$0.837670\pi$
$180$	1.32194e8i	1.68950i
$181$	7.06325e6	0.0885379	0.0442690	−	0.999020i	$-0.485904\pi$
$181$	7.06325e6	0.0885379	0.0442690	+	0.999020i	$0.485904\pi$
$182$	0	0
$183$	−1.12762e8	−1.36014
$184$	− 1.62912e8i	− 1.92792i
$185$	6.81715e7	0.791593
$186$	3.03673e8	3.46028
$187$	1.91568e7i	0.214229i
$188$	− 4.61245e8i	− 5.06266i
$189$	− 1.29766e7i	− 0.139812i
$190$	− 2.90246e7i	− 0.306993i
$191$	1.64080e8	1.70388	0.851938	−	0.523643i	$-0.175427\pi$
$191$	1.64080e8	1.70388	0.851938	+	0.523643i	$0.175427\pi$
$192$	4.99840e8	5.09656
$193$	7.93644e6i	0.0794649i	0.999210	+	0.0397324i	$0.0126506\pi$
$193$	7.93644e6i	0.0794649i	−0.999210	+	0.0397324i	$0.987349\pi$
$194$	−1.39223e8	−1.36900
$195$	0	0
$196$	−3.96956e7	−0.376570
$197$	− 5.02645e7i	− 0.468414i	−0.972187	−	0.234207i	$-0.924751\pi$
$197$	− 5.02645e7i	− 0.468414i	0.972187	−	0.234207i	$-0.0752493\pi$
$198$	−1.04766e8	−0.959165
$199$	−6.83961e7	−0.615242	−0.307621	−	0.951509i	$-0.599533\pi$
$199$	−6.83961e7	−0.615242	−0.307621	+	0.951509i	$0.599533\pi$
$200$	2.54031e8i	2.24533i
$201$	9.65286e7i	0.838435i
$202$	− 3.00549e8i	− 2.56558i
$203$	− 8.14562e6i	− 0.0683420i
$204$	−2.26453e8	−1.86755
$205$	−1.79243e7	−0.145313
$206$	2.19159e8i	1.74672i
$207$	−8.24014e7	−0.645712
$208$	0	0
$209$	1.68039e7	0.127321
$210$	1.96456e8i	1.46385i
$211$	−2.52116e8	−1.84762	−0.923810	−	0.382852i	$-0.874942\pi$
$211$	−2.52116e8	−1.84762	−0.923810	+	0.382852i	$0.874942\pi$
$212$	−1.03617e8	−0.746884
$213$	− 7.05188e7i	− 0.500007i
$214$	1.44370e8i	1.00700i
$215$	− 6.31787e7i	− 0.433547i
$216$	− 7.35495e7i	− 0.496583i
$217$	1.72979e8	1.14917
$218$	4.69682e8	3.07048
$219$	− 3.00550e8i	− 1.93358i
$220$	1.09031e8	0.690353
$221$	0	0
$222$	−6.38655e8	−3.91770
$223$	− 1.24043e8i	− 0.749041i	−0.927219	−	0.374520i	$-0.877807\pi$
$223$	− 1.24043e8i	− 0.749041i	0.927219	−	0.374520i	$-0.122193\pi$
$224$	5.79526e8	3.44512
$225$	1.28490e8	0.752020
$226$	2.84053e8i	1.63689i
$227$	1.30892e8i	0.742715i	0.928490	+	0.371358i	$0.121108\pi$
$227$	1.30892e8i	0.742715i	−0.928490	+	0.371358i	$0.878892\pi$
$228$	1.98639e8i	1.10992i
$229$	1.06750e8i	0.587414i	0.955895	+	0.293707i	$0.0948890\pi$
$229$	1.06750e8i	0.587414i	−0.955895	+	0.293707i	$0.905111\pi$
$230$	1.17389e8	0.636182
$231$	−1.13739e8	−0.607110
$232$	− 4.61682e7i	− 0.242736i
$233$	2.23039e8	1.15514	0.577571	−	0.816340i	$-0.304001\pi$
$233$	2.23039e8	1.15514	0.577571	+	0.816340i	$0.304001\pi$
$234$	0	0
$235$	2.09759e8	1.05435
$236$	5.67079e8i	2.80835i
$237$	5.02508e8	2.45202
$238$	−1.76575e8	−0.849005
$239$	− 1.68042e8i	− 0.796204i	−0.917341	−	0.398102i	$-0.869669\pi$
$239$	− 1.68042e8i	− 0.796204i	0.917341	−	0.398102i	$-0.130331\pi$
$240$	6.38050e8i	2.97931i
$241$	− 6.22539e7i	− 0.286488i	−0.989687	−	0.143244i	$-0.954247\pi$
$241$	− 6.22539e7i	− 0.286488i	0.989687	−	0.143244i	$-0.0457534\pi$
$242$	− 3.38303e8i	− 1.53445i
$243$	3.20937e8	1.43482
$244$	−5.76840e8	−2.54209
$245$	− 1.80523e7i	− 0.0784243i
$246$	1.67922e8	0.719174
$247$	0	0
$248$	9.80421e8	4.08161
$249$	5.11323e7i	0.209893i
$250$	−4.51738e8	−1.82851
$251$	2.27779e8	0.909192	0.454596	−	0.890698i	$-0.349784\pi$
$251$	2.27779e8	0.909192	0.454596	+	0.890698i	$0.349784\pi$
$252$	− 7.05445e8i	− 2.77691i
$253$	6.79631e7i	0.263846i
$254$	1.62029e8i	0.620403i
$255$	− 1.02983e8i	− 0.388934i
$256$	6.37137e8	2.37352
$257$	1.21757e7	0.0447435	0.0223717	−	0.999750i	$-0.492878\pi$
$257$	1.21757e7	0.0447435	0.0223717	+	0.999750i	$0.492878\pi$
$258$	5.91880e8i	2.14568i
$259$	−3.63793e8	−1.30108
$260$	0	0
$261$	−2.33520e7	−0.0812986
$262$	− 2.59740e8i	− 0.892245i
$263$	3.47616e8	1.17830	0.589149	−	0.808024i	$-0.299463\pi$
$263$	3.47616e8	1.17830	0.589149	+	0.808024i	$0.299463\pi$
$264$	−6.44655e8	−2.15632
$265$	− 4.71214e7i	− 0.155546i
$266$	1.54888e8i	0.504582i
$267$	1.69021e8i	0.543441i
$268$	4.93797e8i	1.56703i
$269$	−4.98930e7	−0.156281	−0.0781406	−	0.996942i	$-0.524898\pi$
$269$	−4.98930e7	−0.156281	−0.0781406	+	0.996942i	$0.524898\pi$
$270$	5.29976e7	0.163864
$271$	− 3.59296e8i	− 1.09663i	−0.836272	−	0.548315i	$-0.815269\pi$
$271$	− 3.59296e8i	− 1.09663i	0.836272	−	0.548315i	$-0.184731\pi$
$272$	−5.73482e8	−1.72794
$273$	0	0
$274$	−1.07055e9	−3.14398
$275$	− 1.05976e8i	− 0.307286i
$276$	−8.03392e8	−2.30009
$277$	1.25432e7	0.0354591	0.0177296	−	0.999843i	$-0.494356\pi$
$277$	1.25432e7	0.0354591	0.0177296	+	0.999843i	$0.494356\pi$
$278$	4.08917e8i	1.14151i
$279$	− 4.95901e8i	− 1.36704i
$280$	6.34265e8i	1.72670i
$281$	− 4.40328e8i	− 1.18387i	−0.805986	−	0.591935i	$-0.798364\pi$
$281$	− 4.40328e8i	− 1.18387i	0.805986	−	0.591935i	$-0.201636\pi$
$282$	−1.96510e9	−5.21810
$283$	3.05223e8	0.800507	0.400254	−	0.916404i	$-0.368922\pi$
$283$	3.05223e8	0.800507	0.400254	+	0.916404i	$0.368922\pi$
$284$	− 3.60743e8i	− 0.934509i
$285$	−9.03348e7	−0.231152
$286$	0	0
$287$	9.56521e7	0.238840
$288$	− 1.66140e9i	− 4.09826i
$289$	−3.17777e8	−0.774426
$290$	3.32674e7	0.0800988
$291$	4.33310e8i	1.03080i
$292$	− 1.53748e9i	− 3.61384i
$293$	− 4.61504e8i	− 1.07186i	−0.844262	−	0.535931i	$-0.819961\pi$
$293$	− 4.61504e8i	− 1.07186i	0.844262	−	0.535931i	$-0.180039\pi$
$294$	1.69120e8i	0.388132i
$295$	−2.57889e8	−0.584865
$296$	−2.06192e9	−4.62116
$297$	3.06832e7i	0.0679600i
$298$	−9.93784e8	−2.17538
$299$	0	0
$300$	1.25274e9	2.67878
$301$	3.37149e8i	0.712588i
$302$	−6.63337e8	−1.38583
$303$	−9.35413e8	−1.93176
$304$	5.03046e8i	1.02695i
$305$	− 2.62328e8i	− 0.529415i
$306$	5.06210e8i	1.00996i
$307$	− 5.05856e7i	− 0.0997797i	−0.998755	−	0.0498899i	$-0.984113\pi$
$307$	− 5.05856e7i	− 0.0997797i	0.998755	−	0.0498899i	$-0.0158870\pi$
$308$	−5.81838e8	−1.13468
$309$	6.82099e8	1.31520
$310$	7.06462e8i	1.34686i
$311$	3.43240e8	0.647049	0.323524	−	0.946220i	$-0.395132\pi$
$311$	3.43240e8	0.647049	0.323524	+	0.946220i	$0.395132\pi$
$312$	0	0
$313$	4.91976e8	0.906856	0.453428	−	0.891293i	$-0.350201\pi$
$313$	4.91976e8	0.906856	0.453428	+	0.891293i	$0.350201\pi$
$314$	1.57174e9i	2.86502i
$315$	3.20814e8	0.578317
$316$	2.57061e9	4.58280
$317$	− 2.73731e8i	− 0.482633i	−0.970447	−	0.241316i	$-0.922421\pi$
$317$	− 2.73731e8i	− 0.482633i	0.970447	−	0.241316i	$-0.0775791\pi$
$318$	4.41451e8i	0.769817i
$319$	1.92603e7i	0.0332197i
$320$	1.16282e9i	1.98376i
$321$	4.49331e8	0.758226
$322$	−6.26440e8	−1.04564
$323$	− 8.11932e7i	− 0.134064i
$324$	1.46937e9	2.40007
$325$	0	0
$326$	−5.98456e8	−0.956689
$327$	− 1.46181e9i	− 2.31193i
$328$	5.42142e8	0.848310
$329$	−1.11937e9	−1.73295
$330$	− 4.64520e8i	− 0.711550i
$331$	9.35569e7i	0.141800i	0.997483	+	0.0709002i	$0.0225872\pi$
$331$	9.35569e7i	0.141800i	−0.997483	+	0.0709002i	$0.977413\pi$
$332$	2.61570e8i	0.392288i
$333$	1.04293e9i	1.54775i
$334$	1.23255e8	0.181006
$335$	−2.24563e8	−0.326348
$336$	− 3.40491e9i	− 4.89687i
$337$	5.83618e8	0.830661	0.415331	−	0.909671i	$-0.363666\pi$
$337$	5.83618e8	0.830661	0.415331	+	0.909671i	$0.363666\pi$
$338$	0	0
$339$	8.84072e8	1.23250
$340$	− 5.26817e8i	− 0.726915i
$341$	−4.09009e8	−0.558590
$342$	4.44036e8	0.600243
$343$	7.89847e8i	1.05685i
$344$	1.91091e9i	2.53096i
$345$	− 3.65357e8i	− 0.479016i
$346$	− 1.36807e9i	− 1.77558i
$347$	−5.57877e8	−0.716779	−0.358389	−	0.933572i	$-0.616674\pi$
$347$	−5.57877e8	−0.716779	−0.358389	+	0.933572i	$0.616674\pi$
$348$	−2.27676e8	−0.289595
$349$	5.47638e8i	0.689611i	0.938674	+	0.344806i	$0.112055\pi$
$349$	5.47638e8i	0.689611i	−0.938674	+	0.344806i	$0.887945\pi$
$350$	9.76817e8	1.21780
$351$	0	0
$352$	−1.37029e9	−1.67461
$353$	− 2.36367e8i	− 0.286006i	−0.989722	−	0.143003i	$-0.954324\pi$
$353$	− 2.36367e8i	− 0.286006i	0.989722	−	0.143003i	$-0.0456759\pi$
$354$	2.41600e9	2.89458
$355$	1.64054e8	0.194620
$356$	8.64638e8i	1.01569i
$357$	5.49564e8i	0.639263i
$358$	2.91914e9i	3.36251i
$359$	9.18618e8i	1.04786i	0.851761	+	0.523931i	$0.175535\pi$
$359$	9.18618e8i	1.04786i	−0.851761	+	0.523931i	$0.824465\pi$
$360$	1.81833e9	2.05406
$361$	8.22651e8	0.920323
$362$	− 1.53940e8i	− 0.170557i
$363$	−1.05292e9	−1.15537
$364$	0	0
$365$	6.99195e8	0.752616
$366$	2.45759e9i	2.62014i
$367$	7.36607e8	0.777866	0.388933	−	0.921266i	$-0.372844\pi$
$367$	7.36607e8	0.777866	0.388933	+	0.921266i	$0.372844\pi$
$368$	−2.03456e9	−2.12815
$369$	− 2.74218e8i	− 0.284121i
$370$	− 1.48576e9i	− 1.52491i
$371$	2.51460e8i	0.255659i
$372$	− 4.83490e9i	− 4.86953i
$373$	1.03083e9	1.02851	0.514254	−	0.857638i	$-0.328069\pi$
$373$	1.03083e9	1.02851	0.514254	+	0.857638i	$0.328069\pi$
$374$	4.17512e8	0.412685
$375$	1.40597e9i	1.37678i
$376$	−6.34440e9	−6.15507
$377$	0	0
$378$	−2.82818e8	−0.269331
$379$	− 1.51511e9i	− 1.42958i	−0.699341	−	0.714788i	$-0.746523\pi$
$379$	− 1.51511e9i	− 1.42958i	0.699341	−	0.714788i	$-0.253477\pi$
$380$	−4.62112e8	−0.432021
$381$	5.04290e8	0.467135
$382$	− 3.57603e9i	− 3.28230i
$383$	3.96579e8i	0.360690i	0.983603	+	0.180345i	$0.0577214\pi$
$383$	3.96579e8i	0.360690i	−0.983603	+	0.180345i	$0.942279\pi$
$384$	− 4.91858e9i	− 4.43282i
$385$	− 2.64601e8i	− 0.236308i
$386$	1.72970e8	0.153079
$387$	9.66546e8	0.847684
$388$	2.21662e9i	1.92655i
$389$	2.09780e9	1.80693	0.903464	−	0.428664i	$-0.141016\pi$
$389$	2.09780e9	1.80693	0.903464	+	0.428664i	$0.141016\pi$
$390$	0	0
$391$	3.28384e8	0.277820
$392$	5.46011e8i	0.457826i
$393$	−8.08401e8	−0.671820
$394$	−1.09549e9	−0.902341
$395$	1.16903e9i	0.954411i
$396$	1.66803e9i	1.34980i
$397$	− 6.19410e8i	− 0.496834i	−0.968653	−	0.248417i	$-0.920090\pi$
$397$	− 6.19410e8i	− 0.496834i	0.968653	−	0.248417i	$-0.0799104\pi$
$398$	1.49066e9i	1.18519i
$399$	4.82065e8	0.379928
$400$	3.17251e9	2.47853
$401$	7.25576e8i	0.561924i	0.959719	+	0.280962i	$0.0906534\pi$
$401$	7.25576e8i	0.561924i	−0.959719	+	0.280962i	$0.909347\pi$
$402$	2.10379e9	1.61514
$403$	0	0
$404$	−4.78515e9	−3.61045
$405$	6.68224e8i	0.499838i
$406$	−1.77529e8	−0.131652
$407$	8.60188e8	0.632431
$408$	3.11484e9i	2.27052i
$409$	− 1.27241e9i	− 0.919595i	−0.888024	−	0.459797i	$-0.847922\pi$
$409$	− 1.27241e9i	− 0.919595i	0.888024	−	0.459797i	$-0.152078\pi$
$410$	3.90651e8i	0.279928i
$411$	3.33192e9i	2.36727i
$412$	3.48932e9	2.45810
$413$	1.37621e9	0.961300
$414$	1.79589e9i	1.24388i
$415$	−1.18954e8	−0.0816977
$416$	0	0
$417$	1.27269e9	0.859502
$418$	− 3.66232e8i	− 0.245267i
$419$	−2.62230e9	−1.74154	−0.870768	−	0.491693i	$-0.836378\pi$
$419$	−2.62230e9	−1.74154	−0.870768	+	0.491693i	$0.836378\pi$
$420$	3.12785e9	2.06003
$421$	1.20869e9i	0.789457i	0.918798	+	0.394729i	$0.129161\pi$
$421$	1.20869e9i	0.789457i	−0.918798	+	0.394729i	$0.870839\pi$
$422$	5.49474e9i	3.55921i
$423$	3.20902e9i	2.06149i
$424$	1.42524e9i	0.908045i
$425$	−5.12054e8	−0.323560
$426$	−1.53692e9	−0.963201
$427$	1.39990e9i	0.870159i
$428$	2.29858e9	1.41712
$429$	0	0
$430$	−1.37694e9	−0.835174
$431$	− 9.63979e8i	− 0.579959i	−0.957033	−	0.289979i	$-0.906352\pi$
$431$	− 9.63979e8i	− 0.579959i	0.957033	−	0.289979i	$-0.0936484\pi$
$432$	−9.18539e8	−0.548156
$433$	−1.85478e9	−1.09796	−0.548978	−	0.835837i	$-0.684983\pi$
$433$	−1.85478e9	−1.09796	−0.548978	+	0.835837i	$0.684983\pi$
$434$	− 3.76999e9i	− 2.21374i
$435$	− 1.03540e8i	− 0.0603108i
$436$	− 7.47799e9i	− 4.32098i
$437$	− 2.88051e8i	− 0.165114i
$438$	−6.55031e9	−3.72480
$439$	−3.20512e9	−1.80808	−0.904041	−	0.427445i	$-0.859414\pi$
$439$	−3.20512e9	−1.80808	−0.904041	+	0.427445i	$0.859414\pi$
$440$	− 1.49972e9i	− 0.839316i
$441$	2.76175e8	0.153338
$442$	0	0
$443$	1.69743e9	0.927639	0.463819	−	0.885930i	$-0.346479\pi$
$443$	1.69743e9	0.927639	0.463819	+	0.885930i	$0.346479\pi$
$444$	1.01683e10i	5.51324i
$445$	−3.93210e8	−0.211526
$446$	−2.70345e9	−1.44293
$447$	3.09301e9i	1.63796i
$448$	− 6.20532e9i	− 3.26055i
$449$	− 3.01390e9i	− 1.57133i	−0.618655	−	0.785663i	$-0.712322\pi$
$449$	− 3.01390e9i	− 1.57133i	0.618655	−	0.785663i	$-0.287678\pi$
$450$	− 2.80036e9i	− 1.44867i
$451$	−2.26169e8	−0.116096
$452$	4.52252e9	2.30354
$453$	2.06454e9i	1.04347i
$454$	2.85272e9	1.43075
$455$	0	0
$456$	2.73227e9	1.34942
$457$	− 2.13694e9i	− 1.04734i	−0.851922	−	0.523668i	$-0.824563\pi$
$457$	− 2.13694e9i	− 1.04734i	0.851922	−	0.523668i	$-0.175437\pi$
$458$	2.32656e9	1.13158
$459$	1.48255e8	0.0715592
$460$	− 1.86900e9i	− 0.895277i
$461$	− 1.75396e9i	− 0.833809i	−0.908950	−	0.416904i	$-0.863115\pi$
$461$	− 1.75396e9i	− 0.833809i	0.908950	−	0.416904i	$-0.136885\pi$
$462$	2.47888e9i	1.16952i
$463$	− 2.44183e9i	− 1.14336i	−0.820477	−	0.571679i	$-0.806292\pi$
$463$	− 2.44183e9i	− 1.14336i	0.820477	−	0.571679i	$-0.193708\pi$
$464$	−5.76581e8	−0.267946
$465$	2.19876e9	1.01413
$466$	− 4.86102e9i	− 2.22524i
$467$	−3.10574e9	−1.41110	−0.705548	−	0.708662i	$-0.749299\pi$
$467$	−3.10574e9	−1.41110	−0.705548	+	0.708662i	$0.749299\pi$
$468$	0	0
$469$	1.19837e9	0.536394
$470$	− 4.57159e9i	− 2.03107i
$471$	4.89182e9	2.15723
$472$	7.80014e9	3.41433
$473$	− 7.97188e8i	− 0.346375i
$474$	− 1.09519e10i	− 4.72351i
$475$	4.49163e8i	0.192299i
$476$	2.81132e9i	1.19478i
$477$	7.20893e8	0.304128
$478$	−3.66237e9	−1.53379
$479$	− 3.89354e9i	− 1.61871i	−0.587317	−	0.809357i	$-0.699816\pi$
$479$	− 3.89354e9i	− 1.61871i	0.587317	−	0.809357i	$-0.300184\pi$
$480$	7.36641e9	3.04027
$481$	0	0
$482$	−1.35679e9	−0.551884
$483$	1.94970e9i	0.787324i
$484$	−5.38626e9	−2.15938
$485$	−1.00805e9	−0.401222
$486$	− 6.99466e9i	− 2.76401i
$487$	1.80816e9i	0.709390i	0.934982	+	0.354695i	$0.115415\pi$
$487$	1.80816e9i	0.709390i	−0.934982	+	0.354695i	$0.884585\pi$
$488$	7.93441e9i	3.09062i
$489$	1.86260e9i	0.720344i
$490$	−3.93439e8	−0.151075
$491$	−7.46799e8	−0.284720	−0.142360	−	0.989815i	$-0.545469\pi$
$491$	−7.46799e8	−0.284720	−0.142360	+	0.989815i	$0.545469\pi$
$492$	− 2.67355e9i	− 1.01207i
$493$	9.30620e7	0.0349791
$494$	0	0
$495$	−7.58564e8	−0.281109
$496$	− 1.22442e10i	− 4.50551i
$497$	−8.75464e8	−0.319883
$498$	1.11440e9	0.404333
$499$	− 8.17396e8i	− 0.294497i	−0.989100	−	0.147248i	$-0.952958\pi$
$499$	− 8.17396e8i	− 0.294497i	0.989100	−	0.147248i	$-0.0470417\pi$
$500$	7.19229e9i	2.57319i
$501$	− 3.83613e8i	− 0.136289i
$502$	− 4.96432e9i	− 1.75144i
$503$	1.31337e9	0.460149	0.230075	−	0.973173i	$-0.426103\pi$
$503$	1.31337e9	0.460149	0.230075	+	0.973173i	$0.426103\pi$
$504$	−9.70337e9	−3.37610
$505$	− 2.17613e9i	− 0.751910i
$506$	1.48122e9	0.508267
$507$	0	0
$508$	2.57972e9	0.873072
$509$	2.35199e8i	0.0790539i	0.999219	+	0.0395270i	$0.0125851\pi$
$509$	2.35199e8i	0.0790539i	−0.999219	+	0.0395270i	$0.987415\pi$
$510$	−2.24447e9	−0.749234
$511$	−3.73121e9	−1.23702
$512$	− 4.60460e9i	− 1.51617i
$513$	− 1.30046e8i	− 0.0425292i
$514$	− 2.65364e8i	− 0.0861927i
$515$	1.58683e9i	0.511923i
$516$	9.42356e9	3.01954
$517$	2.64674e9	0.842353
$518$	7.92866e9i	2.50637i
$519$	−4.25790e9	−1.33693
$520$	0	0
$521$	−3.47965e9	−1.07796	−0.538982	−	0.842318i	$-0.681191\pi$
$521$	−3.47965e9	−1.07796	−0.538982	+	0.842318i	$0.681191\pi$
$522$	5.08945e8i	0.156612i
$523$	3.91498e8	0.119667	0.0598335	−	0.998208i	$-0.480943\pi$
$523$	3.91498e8	0.119667	0.0598335	+	0.998208i	$0.480943\pi$
$524$	−4.13542e9	−1.25563
$525$	− 3.04020e9i	− 0.916947i
$526$	− 7.57611e9i	− 2.26984i
$527$	1.97625e9i	0.588173i
$528$	8.05091e9i	2.38027i
$529$	−2.23981e9	−0.657834
$530$	−1.02699e9	−0.299639
$531$	− 3.94534e9i	− 1.14355i
$532$	2.46603e9	0.710081
$533$	0	0
$534$	3.68373e9	1.04687
$535$	1.04532e9i	0.295128i
$536$	6.79216e9	1.90516
$537$	9.08538e9	2.53182
$538$	1.08739e9i	0.301056i
$539$	− 2.27784e8i	− 0.0626559i
$540$	− 8.43796e8i	− 0.230600i
$541$	− 5.01138e9i	− 1.36071i	−0.732881	−	0.680357i	$-0.761825\pi$
$541$	− 5.01138e9i	− 1.36071i	0.732881	−	0.680357i	$-0.238175\pi$
$542$	−7.83066e9	−2.11252
$543$	−4.79114e8	−0.128422
$544$	6.62096e9i	1.76329i
$545$	3.40075e9	0.899884
$546$	0	0
$547$	1.06296e9	0.277692	0.138846	−	0.990314i	$-0.455661\pi$
$547$	1.06296e9	0.277692	0.138846	+	0.990314i	$0.455661\pi$
$548$	1.70446e10i	4.42441i
$549$	4.01326e9	1.03513
$550$	−2.30968e9	−0.591948
$551$	− 8.16320e7i	− 0.0207888i
$552$	1.10506e10i	2.79640i
$553$	− 6.23844e9i	− 1.56869i
$554$	− 2.73371e8i	− 0.0683076i
$555$	−4.62421e9	−1.14819
$556$	6.51052e9	1.60640
$557$	− 4.27007e9i	− 1.04699i	−0.852029	−	0.523495i	$-0.824628\pi$
$557$	− 4.27007e9i	− 1.04699i	0.852029	−	0.523495i	$-0.175372\pi$
$558$	−1.08079e10	−2.63343
$559$	0	0
$560$	7.92115e9	1.90603
$561$	− 1.29944e9i	− 0.310733i
$562$	−9.59671e9	−2.28058
$563$	−4.92488e9	−1.16310	−0.581549	−	0.813511i	$-0.697553\pi$
$563$	−4.92488e9	−1.16310	−0.581549	+	0.813511i	$0.697553\pi$
$564$	3.12871e10i	7.34326i
$565$	2.05669e9i	0.479734i
$566$	− 6.65217e9i	− 1.54208i
$567$	− 3.56593e9i	− 0.821547i
$568$	−4.96200e9	−1.13615
$569$	−4.91765e9	−1.11909	−0.559545	−	0.828800i	$-0.689024\pi$
$569$	−4.91765e9	−1.11909	−0.559545	+	0.828800i	$0.689024\pi$
$570$	1.96880e9i	0.445286i
$571$	−1.02161e9	−0.229647	−0.114823	−	0.993386i	$-0.536630\pi$
$571$	−1.02161e9	−0.229647	−0.114823	+	0.993386i	$0.536630\pi$
$572$	0	0
$573$	−1.11298e10	−2.47143
$574$	− 2.08468e9i	− 0.460096i
$575$	−1.81663e9	−0.398500
$576$	−1.77896e10	−3.87870
$577$	− 4.24754e9i	− 0.920496i	−0.887790	−	0.460248i	$-0.847761\pi$
$577$	− 4.24754e9i	− 0.920496i	0.887790	−	0.460248i	$-0.152239\pi$
$578$	6.92577e9i	1.49183i
$579$	− 5.38344e8i	− 0.115262i
$580$	− 5.29664e8i	− 0.112720i
$581$	6.34789e8	0.134280
$582$	9.44375e9	1.98570
$583$	− 5.94578e8i	− 0.124271i
$584$	−2.11479e10	−4.39362
$585$	0	0
$586$	−1.00582e10	−2.06481
$587$	5.10152e8i	0.104104i	0.998644	+	0.0520519i	$0.0165761\pi$
$587$	5.10152e8i	0.104104i	−0.998644	+	0.0520519i	$0.983424\pi$
$588$	2.69263e9	0.546205
$589$	1.73353e9	0.349564
$590$	5.62055e9i	1.12667i
$591$	3.40954e9i	0.679422i
$592$	2.57508e10i	5.10110i
$593$	− 5.55218e9i	− 1.09338i	−0.837334	−	0.546691i	$-0.815887\pi$
$593$	− 5.55218e9i	− 1.09338i	0.837334	−	0.546691i	$-0.184113\pi$
$594$	6.68724e8	0.130916
$595$	−1.27850e9	−0.248823
$596$	1.58224e10i	3.06134i
$597$	4.63944e9	0.892392
$598$	0	0
$599$	2.67091e9	0.507769	0.253884	−	0.967235i	$-0.418292\pi$
$599$	2.67091e9	0.507769	0.253884	+	0.967235i	$0.418292\pi$
$600$	− 1.72314e10i	− 3.25680i
$601$	−2.33696e9	−0.439127	−0.219563	−	0.975598i	$-0.570463\pi$
$601$	−2.33696e9	−0.439127	−0.219563	+	0.975598i	$0.570463\pi$
$602$	7.34797e9	1.37271
$603$	− 3.43550e9i	− 0.638086i
$604$	1.05612e10i	1.95023i
$605$	− 2.44950e9i	− 0.449710i
$606$	2.03868e10i	3.72130i
$607$	8.57060e9	1.55543	0.777715	−	0.628617i	$-0.216379\pi$
$607$	8.57060e9	1.55543	0.777715	+	0.628617i	$0.216379\pi$
$608$	5.80776e9	1.04796
$609$	5.52533e8i	0.0991284i
$610$	−5.71730e9	−1.01985
$611$	0	0
$612$	8.05956e9	1.42129
$613$	5.83442e9i	1.02303i	0.859276	+	0.511513i	$0.170915\pi$
$613$	5.83442e9i	1.02303i	−0.859276	+	0.511513i	$0.829085\pi$
$614$	−1.10248e9	−0.192213
$615$	1.21584e9	0.210773
$616$	8.00315e9i	1.37952i
$617$	− 5.65382e9i	− 0.969045i	−0.874779	−	0.484523i	$-0.838993\pi$
$617$	− 5.65382e9i	− 0.969045i	0.874779	−	0.484523i	$-0.161007\pi$
$618$	− 1.48660e10i	− 2.53358i
$619$	2.81131e9i	0.476422i	0.971213	+	0.238211i	$0.0765609\pi$
$619$	2.81131e9i	0.476422i	−0.971213	+	0.238211i	$0.923439\pi$
$620$	1.12479e10	1.89539
$621$	5.25969e8	0.0881331
$622$	− 7.48073e9i	− 1.24646i
$623$	2.09834e9	0.347670
$624$	0	0
$625$	8.87227e8	0.145363
$626$	− 1.07223e10i	− 1.74695i
$627$	−1.13984e9	−0.184675
$628$	2.50244e10	4.03185
$629$	− 4.15625e9i	− 0.665924i
$630$	− 6.99196e9i	− 1.11406i
$631$	1.40442e9i	0.222533i	0.993791	+	0.111267i	$0.0354907\pi$
$631$	1.40442e9i	0.222533i	−0.993791	+	0.111267i	$0.964509\pi$
$632$	− 3.53586e10i	− 5.57166i
$633$	1.71016e10	2.67992
$634$	−5.96582e9	−0.929732
$635$	1.17317e9i	0.181825i
$636$	7.02851e9	1.08334
$637$	0	0
$638$	4.19768e8	0.0639937
$639$	2.50980e9i	0.380528i
$640$	1.14425e10	1.72541
$641$	9.29598e9	1.39409	0.697047	−	0.717025i	$-0.254497\pi$
$641$	9.29598e9	1.39409	0.697047	+	0.717025i	$0.254497\pi$
$642$	− 9.79292e9i	− 1.46063i
$643$	2.01890e9i	0.299487i	0.988725	+	0.149743i	$0.0478447\pi$
$643$	2.01890e9i	0.299487i	−0.988725	+	0.149743i	$0.952155\pi$
$644$	9.97381e9i	1.47150i
$645$	4.28553e9i	0.628848i
$646$	−1.76956e9	−0.258257
$647$	−1.14928e10	−1.66825	−0.834125	−	0.551575i	$-0.814027\pi$
$647$	−1.14928e10	−1.66825	−0.834125	+	0.551575i	$0.814027\pi$
$648$	− 2.02112e10i	− 2.91796i
$649$	−3.25404e9	−0.467269
$650$	0	0
$651$	−1.17335e10	−1.66684
$652$	9.52826e9i	1.34632i
$653$	3.76318e9	0.528882	0.264441	−	0.964402i	$-0.414813\pi$
$653$	3.76318e9	0.528882	0.264441	+	0.964402i	$0.414813\pi$
$654$	−3.18594e10	−4.45365
$655$	− 1.88066e9i	− 0.261496i
$656$	− 6.77065e9i	− 0.936412i
$657$	1.06967e10i	1.47154i
$658$	2.43960e10i	3.33831i
$659$	1.97143e9	0.268338	0.134169	−	0.990958i	$-0.457164\pi$
$659$	1.97143e9	0.268338	0.134169	+	0.990958i	$0.457164\pi$
$660$	−7.39580e9	−1.00134
$661$	9.23391e8i	0.124360i	0.998065	+	0.0621800i	$0.0198053\pi$
$661$	9.23391e8i	0.124360i	−0.998065	+	0.0621800i	$0.980195\pi$
$662$	2.03902e9	0.273161
$663$	0	0
$664$	3.59789e9	0.476935
$665$	1.12147e9i	0.147881i
$666$	2.27301e10	2.98154
$667$	3.30158e8	0.0430806
$668$	− 1.96239e9i	− 0.254723i
$669$	8.41408e9i	1.08646i
$670$	4.89423e9i	0.628669i
$671$	− 3.31006e9i	− 0.422967i
$672$	−3.93104e10	−4.99706
$673$	−6.88575e8	−0.0870760	−0.0435380	−	0.999052i	$-0.513863\pi$
$673$	−6.88575e8	−0.0870760	−0.0435380	+	0.999052i	$0.513863\pi$
$674$	− 1.27196e10i	− 1.60017i
$675$	−8.20150e8	−0.102643
$676$	0	0
$677$	8.91263e9	1.10394	0.551970	−	0.833864i	$-0.313876\pi$
$677$	8.91263e9	1.10394	0.551970	+	0.833864i	$0.313876\pi$
$678$	− 1.92679e10i	− 2.37427i
$679$	5.37938e9	0.659459
$680$	−7.24634e9	−0.883767
$681$	− 8.87865e9i	− 1.07729i
$682$	8.91414e9i	1.07605i
$683$	− 1.01970e10i	− 1.22461i	−0.790621	−	0.612305i	$-0.790242\pi$
$683$	− 1.01970e10i	− 1.22461i	0.790621	−	0.612305i	$-0.209758\pi$
$684$	− 7.06968e9i	− 0.844701i
$685$	−7.75135e9	−0.921425
$686$	1.72143e10	2.03589
$687$	− 7.24107e9i	− 0.852029i
$688$	2.38648e10	2.79382
$689$	0	0
$690$	−7.96275e9	−0.922765
$691$	3.83838e9i	0.442563i	0.975210	+	0.221282i	$0.0710240\pi$
$691$	3.83838e9i	0.442563i	−0.975210	+	0.221282i	$0.928976\pi$
$692$	−2.17815e10	−2.49872
$693$	4.04803e9	0.462037
$694$	1.21586e10i	1.38079i
$695$	2.96078e9i	0.334548i
$696$	3.13168e9i	0.352083i
$697$	1.09280e9i	0.122244i
$698$	1.19355e10	1.32845
$699$	−1.51292e10	−1.67550
$700$	− 1.55523e10i	− 1.71377i
$701$	1.78976e9	0.196237	0.0981185	−	0.995175i	$-0.468718\pi$
$701$	1.78976e9	0.196237	0.0981185	+	0.995175i	$0.468718\pi$
$702$	0	0
$703$	−3.64578e9	−0.395773
$704$	1.46725e10i	1.58489i
$705$	−1.42284e10	−1.52930
$706$	−5.15149e9	−0.550955
$707$	1.16128e10i	1.23586i
$708$	− 3.84661e10i	− 4.07344i
$709$	− 5.29716e9i	− 0.558189i	−0.960264	−	0.279094i	$-0.909966\pi$
$709$	− 5.29716e9i	− 0.558189i	0.960264	−	0.279094i	$-0.0900342\pi$
$710$	− 3.57547e9i	− 0.374912i
$711$	−1.78845e10	−1.86609
$712$	1.18931e10	1.23485
$713$	7.01120e9i	0.724401i
$714$	1.19774e10	1.23146
$715$	0	0
$716$	4.64768e10	4.73195
$717$	1.13986e10i	1.15487i
$718$	2.00208e10	2.01858
$719$	5.91972e9	0.593950	0.296975	−	0.954885i	$-0.404022\pi$
$719$	5.91972e9	0.593950	0.296975	+	0.954885i	$0.404022\pi$
$720$	− 2.27085e10i	− 2.26739i
$721$	− 8.46800e9i	− 0.841410i
$722$	− 1.79292e10i	− 1.77289i
$723$	4.22280e9i	0.415544i
$724$	−2.45093e9	−0.240020
$725$	−5.14821e8	−0.0501733
$726$	2.29478e10i	2.22568i
$727$	6.36378e9	0.614249	0.307125	−	0.951669i	$-0.400633\pi$
$727$	6.36378e9	0.614249	0.307125	+	0.951669i	$0.400633\pi$
$728$	0	0
$729$	−1.25089e10	−1.19584
$730$	− 1.52386e10i	− 1.44982i
$731$	−3.85185e9	−0.364719
$732$	3.91282e10	3.68724
$733$	2.70194e9i	0.253403i	0.991941	+	0.126702i	$0.0404391\pi$
$733$	2.70194e9i	0.253403i	−0.991941	+	0.126702i	$0.959561\pi$
$734$	− 1.60540e10i	− 1.49846i
$735$	1.22452e9i	0.113752i
$736$	2.34894e10i	2.17170i
$737$	−2.83353e9	−0.260731
$738$	−5.97642e9	−0.547323
$739$	8.27826e9i	0.754543i	0.926103	+	0.377271i	$0.123138\pi$
$739$	8.27826e9i	0.754543i	−0.926103	+	0.377271i	$0.876862\pi$
$740$	−2.36554e10	−2.14595
$741$	0	0
$742$	5.48044e9	0.492495
$743$	3.31677e9i	0.296657i	0.988938	+	0.148329i	$0.0473893\pi$
$743$	3.31677e9i	0.296657i	−0.988938	+	0.148329i	$0.952611\pi$
$744$	−6.65039e10	−5.92027
$745$	−7.19553e9	−0.637553
$746$	− 2.24665e10i	− 1.98129i
$747$	− 1.81983e9i	− 0.159738i
$748$	− 6.64737e9i	− 0.580757i
$749$	− 5.57827e9i	− 0.485080i
$750$	3.06422e10	2.65220
$751$	6.94085e9	0.597961	0.298980	−	0.954259i	$-0.403353\pi$
$751$	6.94085e9	0.597961	0.298980	+	0.954259i	$0.403353\pi$
$752$	7.92333e10i	6.79431i
$753$	−1.54507e10	−1.31876
$754$	0	0
$755$	−4.80291e9	−0.406154
$756$	4.50286e9i	0.379020i
$757$	−1.29261e10	−1.08301	−0.541503	−	0.840699i	$-0.682145\pi$
$757$	−1.29261e10	−1.08301	−0.541503	+	0.840699i	$0.682145\pi$
$758$	−3.30210e10	−2.75390
$759$	− 4.61007e9i	− 0.382702i
$760$	6.35633e9i	0.525242i
$761$	2.15262e10i	1.77060i	0.465019	+	0.885301i	$0.346048\pi$
$761$	2.15262e10i	1.77060i	−0.465019	+	0.885301i	$0.653952\pi$
$762$	− 1.09907e10i	− 0.899878i
$763$	−1.81479e10	−1.47907
$764$	−5.69353e10	−4.61907
$765$	3.66523e9i	0.295996i
$766$	8.64322e9	0.694824
$767$	0	0
$768$	−4.32182e10	−3.44273
$769$	− 9.97009e9i	− 0.790600i	−0.918552	−	0.395300i	$-0.870641\pi$
$769$	− 9.97009e9i	− 0.790600i	0.918552	−	0.395300i	$-0.129359\pi$
$770$	−5.76683e9	−0.455219
$771$	−8.25905e8	−0.0648992
$772$	− 2.75393e9i	− 0.215423i
$773$	1.24324e9i	0.0968115i	0.998828	+	0.0484057i	$0.0154141\pi$
$773$	1.24324e9i	0.0968115i	−0.998828	+	0.0484057i	$0.984586\pi$
$774$	− 2.10653e10i	− 1.63296i
$775$	− 1.09327e10i	− 0.843665i
$776$	3.04895e10	2.34226
$777$	2.46768e10	1.88719
$778$	− 4.57204e10i	− 3.48082i
$779$	9.58585e8	0.0726523
$780$	0	0
$781$	2.07003e9	0.155489
$782$	− 7.15695e9i	− 0.535185i
$783$	1.49056e8	0.0110964
$784$	6.81898e9	0.505374
$785$	1.13803e10i	0.839671i
$786$	1.76187e10i	1.29418i
$787$	5.42091e9i	0.396425i	0.980159	+	0.198212i	$0.0635136\pi$
$787$	5.42091e9i	0.396425i	−0.980159	+	0.198212i	$0.936486\pi$
$788$	1.74417e10i	1.26983i
$789$	−2.35795e10	−1.70909
$790$	2.54783e10	1.83855
$791$	− 1.09754e10i	− 0.788503i
$792$	2.29436e10	1.64106
$793$	0	0
$794$	−1.34997e10	−0.957090
$795$	3.19634e9i	0.225615i
$796$	2.37333e10	1.66787
$797$	−1.30870e9	−0.0915667	−0.0457834	−	0.998951i	$-0.514578\pi$
$797$	−1.30870e9	−0.0915667	−0.0457834	+	0.998951i	$0.514578\pi$
$798$	− 1.05064e10i	− 0.731883i
$799$	− 1.27885e10i	− 0.886965i
$800$	− 3.66273e10i	− 2.52924i
$801$	− 6.01556e9i	− 0.413583i
$802$	1.58135e10	1.08248
$803$	8.82244e9	0.601291
$804$	− 3.34952e10i	− 2.27293i
$805$	−4.53577e9	−0.306454
$806$	0	0
$807$	3.38434e9	0.226682
$808$	6.58196e10i	4.38950i
$809$	2.37346e10	1.57602	0.788011	−	0.615661i	$-0.211111\pi$
$809$	2.37346e10	1.57602	0.788011	+	0.615661i	$0.211111\pi$
$810$	1.45636e10	0.962876
$811$	2.50297e10i	1.64772i	0.566795	+	0.823859i	$0.308183\pi$
$811$	2.50297e10i	1.64772i	−0.566795	+	0.823859i	$0.691817\pi$
$812$	2.82651e9i	0.185270i
$813$	2.43718e10i	1.59063i
$814$	− 1.87473e10i	− 1.21830i
$815$	−4.33314e9	−0.280383
$816$	3.89004e10	2.50633
$817$	3.37876e9i	0.216761i
$818$	−2.77315e10	−1.77148
$819$	0	0
$820$	6.21971e9	0.393933
$821$	− 6.17223e9i	− 0.389261i	−0.980877	−	0.194631i	$-0.937649\pi$
$821$	− 6.17223e9i	− 0.389261i	0.980877	−	0.194631i	$-0.0623508\pi$
$822$	7.26174e10	4.56026
$823$	−1.08887e10	−0.680891	−0.340445	−	0.940264i	$-0.610578\pi$
$823$	−1.08887e10	−0.680891	−0.340445	+	0.940264i	$0.610578\pi$
$824$	− 4.79954e10i	− 2.98851i
$825$	7.18855e9i	0.445710i
$826$	− 2.99937e10i	− 1.85182i
$827$	6.98237e9i	0.429273i	0.976694	+	0.214637i	$0.0688567\pi$
$827$	6.98237e9i	0.429273i	−0.976694	+	0.214637i	$0.931143\pi$
$828$	2.85931e10	1.75047
$829$	1.07848e10	0.657464	0.328732	−	0.944423i	$-0.393379\pi$
$829$	1.07848e10	0.657464	0.328732	+	0.944423i	$0.393379\pi$
$830$	2.59253e9i	0.157380i
$831$	−8.50827e8	−0.0514325
$832$	0	0
$833$	−1.10060e9	−0.0659741
$834$	− 2.77376e10i	− 1.65572i
$835$	8.92433e8	0.0530485
$836$	−5.83093e9	−0.345156
$837$	3.16534e9i	0.186587i
$838$	5.71515e10i	3.35485i
$839$	− 1.41196e10i	− 0.825381i	−0.910871	−	0.412691i	$-0.864589\pi$
$839$	− 1.41196e10i	− 0.825381i	0.910871	−	0.412691i	$-0.135411\pi$
$840$	− 4.30234e10i	− 2.50454i
$841$	−1.71563e10	−0.994576
$842$	2.63428e10	1.52079
$843$	2.98683e10i	1.71717i
$844$	8.74839e10	5.00875
$845$	0	0
$846$	6.99389e10	3.97121
$847$	1.30716e10i	0.739155i
$848$	1.77994e10	1.00235
$849$	−2.07039e10	−1.16111
$850$	1.11599e10i	0.623297i
$851$	− 1.47453e10i	− 0.820161i
$852$	2.44699e10i	1.35548i
$853$	− 1.78844e10i	− 0.986628i	−0.869851	−	0.493314i	$-0.835785\pi$
$853$	− 1.78844e10i	− 0.986628i	0.869851	−	0.493314i	$-0.164215\pi$
$854$	3.05100e10	1.67625
$855$	3.21506e9	0.175917
$856$	− 3.16168e10i	− 1.72290i
$857$	−1.24146e10	−0.673754	−0.336877	−	0.941549i	$-0.609371\pi$
$857$	−1.24146e10	−0.673754	−0.336877	+	0.941549i	$0.609371\pi$
$858$	0	0
$859$	4.99039e9	0.268632	0.134316	−	0.990939i	$-0.457116\pi$
$859$	4.99039e9	0.268632	0.134316	+	0.990939i	$0.457116\pi$
$860$	2.19229e10i	1.17531i
$861$	−6.48827e9	−0.346432
$862$	−2.10094e10	−1.11722
$863$	− 2.82593e10i	− 1.49666i	−0.663326	−	0.748331i	$-0.730856\pi$
$863$	− 2.82593e10i	− 1.49666i	0.663326	−	0.748331i	$-0.269144\pi$
$864$	1.06047e10i	0.559372i
$865$	− 9.90554e9i	− 0.520381i
$866$	4.04239e10i	2.11508i
$867$	2.15554e10	1.12328
$868$	−6.00235e10	−3.11532
$869$	1.47508e10i	0.762511i
$870$	−2.25659e9	−0.116181
$871$	0	0
$872$	−1.02859e11	−5.25335
$873$	− 1.54217e10i	− 0.784482i
$874$	−6.27792e9	−0.318072
$875$	1.74545e10	0.880805
$876$	1.04290e11i	5.24178i
$877$	1.28656e10i	0.644069i	0.946728	+	0.322034i	$0.104367\pi$
$877$	1.28656e10i	0.644069i	−0.946728	+	0.322034i	$0.895633\pi$
$878$	6.98538e10i	3.48305i
$879$	3.13047e10i	1.55471i
$880$	−1.87296e10	−0.926485
$881$	−6.97330e9	−0.343576	−0.171788	−	0.985134i	$-0.554954\pi$
$881$	−6.97330e9	−0.343576	−0.171788	+	0.985134i	$0.554954\pi$
$882$	− 6.01908e9i	− 0.295386i
$883$	1.25092e10	0.611458	0.305729	−	0.952119i	$-0.401100\pi$
$883$	1.25092e10	0.611458	0.305729	+	0.952119i	$0.401100\pi$
$884$	0	0
$885$	1.74931e10	0.848332
$886$	− 3.69946e10i	− 1.78698i
$887$	−7.62507e9	−0.366869	−0.183435	−	0.983032i	$-0.558722\pi$
$887$	−7.62507e9	−0.366869	−0.183435	+	0.983032i	$0.558722\pi$
$888$	1.39864e11	6.70288
$889$	− 6.26057e9i	− 0.298853i
$890$	8.56979e9i	0.407479i
$891$	8.43164e9i	0.399338i
$892$	4.30427e10i	2.03059i
$893$	−1.12178e10	−0.527142
$894$	6.74103e10	3.15533
$895$	2.11361e10i	0.985473i
$896$	−6.10623e10	−2.83593
$897$	0	0
$898$	−6.56862e10	−3.02696
$899$	1.98693e9i	0.0912061i
$900$	−4.45857e10	−2.03867
$901$	−2.87288e9	−0.130852
$902$	4.92924e9i	0.223644i
$903$	− 2.28695e10i	− 1.03359i
$904$	− 6.22070e10i	− 2.80059i
$905$	− 1.11461e9i	− 0.0499863i
$906$	4.49954e10	2.01011
$907$	−4.60454e9	−0.204909	−0.102454	−	0.994738i	$-0.532670\pi$
$907$	−4.60454e9	−0.204909	−0.102454	+	0.994738i	$0.532670\pi$
$908$	− 4.54192e10i	− 2.01344i
$909$	3.32918e10	1.47016
$910$	0	0
$911$	3.96389e9	0.173703	0.0868515	−	0.996221i	$-0.472319\pi$
$911$	3.96389e9	0.173703	0.0868515	+	0.996221i	$0.472319\pi$
$912$	− 3.41226e10i	− 1.48957i
$913$	−1.50096e9	−0.0652711
$914$	−4.65735e10	−2.01756
$915$	1.77942e10i	0.767902i
$916$	− 3.70421e10i	− 1.59243i
$917$	1.00360e10i	0.429801i
$918$	− 3.23114e9i	− 0.137850i
$919$	−4.43139e9	−0.188337	−0.0941685	−	0.995556i	$-0.530019\pi$
$919$	−4.43139e9	−0.188337	−0.0941685	+	0.995556i	$0.530019\pi$
$920$	−2.57080e10	−1.08846
$921$	3.43132e9i	0.144728i
$922$	−3.82266e10	−1.60623
$923$	0	0
$924$	3.94672e10	1.64583
$925$	2.29925e10i	0.955190i
$926$	−5.32184e10	−2.20254
$927$	−2.42763e10	−1.00093
$928$	6.65674e9i	0.273428i
$929$	2.49862e10i	1.02246i	0.859445	+	0.511229i	$0.170810\pi$
$929$	2.49862e10i	1.02246i	−0.859445	+	0.511229i	$0.829190\pi$
$930$	− 4.79207e10i	− 1.95359i
$931$	9.65426e8i	0.0392099i
$932$	−7.73942e10	−3.13150
$933$	−2.32827e10	−0.938527
$934$	6.76880e10i	2.71830i
$935$	3.02301e9	0.120948
$936$	0	0
$937$	4.15588e10	1.65034	0.825171	−	0.564883i	$-0.191079\pi$
$937$	4.15588e10	1.65034	0.825171	+	0.564883i	$0.191079\pi$
$938$	− 2.61177e10i	− 1.03330i
$939$	−3.33717e10	−1.31537
$940$	−7.27860e10	−2.85825
$941$	9.79910e9i	0.383374i	0.981456	+	0.191687i	$0.0613958\pi$
$941$	9.79910e9i	0.383374i	−0.981456	+	0.191687i	$0.938604\pi$
$942$	− 1.06615e11i	− 4.15564i
$943$	3.87697e9i	0.150557i
$944$	− 9.74137e10i	− 3.76893i
$945$	−2.04776e9	−0.0789345
$946$	−1.73743e10	−0.667249
$947$	3.36446e10i	1.28733i	0.765307	+	0.643666i	$0.222587\pi$
$947$	3.36446e10i	1.28733i	−0.765307	+	0.643666i	$0.777413\pi$
$948$	−1.74369e11	−6.64723
$949$	0	0
$950$	9.78925e9	0.370439
$951$	1.85677e10i	0.700046i
$952$	3.86696e10	1.45258
$953$	3.32521e10	1.24450	0.622249	−	0.782819i	$-0.286219\pi$
$953$	3.32521e10	1.24450	0.622249	+	0.782819i	$0.286219\pi$
$954$	− 1.57115e10i	− 0.585864i
$955$	− 2.58923e10i	− 0.961966i
$956$	5.83101e10i	2.15845i
$957$	− 1.30646e9i	− 0.0481843i
$958$	−8.48575e10	−3.11825
$959$	4.13645e10	1.51448
$960$	− 7.88765e10i	− 2.87739i
$961$	−1.46816e10	−0.533631
$962$	0	0
$963$	−1.59919e10	−0.577043
$964$	2.16020e10i	0.776648i
$965$	1.25240e9	0.0448639
$966$	4.24927e10	1.51668
$967$	− 4.41653e9i	− 0.157068i	−0.996911	−	0.0785341i	$-0.974976\pi$
$967$	− 4.41653e9i	− 0.157068i	0.996911	−	0.0785341i	$-0.0250239\pi$
$968$	7.40877e10i	2.62532i
$969$	5.50749e9i	0.194456i
$970$	2.19698e10i	0.772904i
$971$	3.91587e10	1.37265	0.686327	−	0.727293i	$-0.259222\pi$
$971$	3.91587e10	1.37265	0.686327	+	0.727293i	$0.259222\pi$
$972$	−1.11365e11	−3.88969
$973$	− 1.58000e10i	− 0.549872i
$974$	3.94078e10	1.36655
$975$	0	0
$976$	9.90906e10	3.41160
$977$	− 4.27799e10i	− 1.46760i	−0.679364	−	0.733801i	$-0.737744\pi$
$977$	− 4.27799e10i	− 1.46760i	0.679364	−	0.733801i	$-0.262256\pi$
$978$	4.05945e10	1.38765
$979$	−4.96152e9	−0.168995
$980$	6.26411e9i	0.212602i
$981$	5.20267e10i	1.75948i
$982$	1.62761e10i	0.548478i
$983$	4.08692e10i	1.37233i	0.727445	+	0.686166i	$0.240708\pi$
$983$	4.08692e10i	1.37233i	−0.727445	+	0.686166i	$0.759292\pi$
$984$	−3.67745e10	−1.23045
$985$	−7.93192e9	−0.264455
$986$	− 2.02823e9i	− 0.0673828i
$987$	7.59288e10	2.51360
$988$	0	0
$989$	−1.36653e10	−0.449193
$990$	1.65325e10i	0.541521i
$991$	4.28687e10	1.39921	0.699604	−	0.714531i	$-0.253360\pi$
$991$	4.28687e10	1.39921	0.699604	+	0.714531i	$0.253360\pi$
$992$	−1.41362e11	−4.59770
$993$	− 6.34614e9i	− 0.205678i
$994$	1.90803e10i	0.616214i
$995$	1.07932e10i	0.347350i
$996$	− 1.77428e10i	− 0.569004i
$997$	−4.86824e9	−0.155575	−0.0777874	−	0.996970i	$-0.524786\pi$
$997$	−4.86824e9	−0.155575	−0.0777874	+	0.996970i	$0.524786\pi$
$998$	−1.78147e10	−0.567312
$999$	− 6.65702e9i	− 0.211252i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.b.e.168.1		16
13.5	odd	4		169.8.a.e.1.1		8
13.7	odd	12		13.8.c.a.3.1	✓	16
13.8	odd	4		169.8.a.f.1.8		8
13.11	odd	12		13.8.c.a.9.1	yes	16
13.12	even	2	inner	169.8.b.e.168.16		16
39.11	even	12		117.8.g.d.100.8		16
39.20	even	12		117.8.g.d.55.8		16
52.7	even	12		208.8.i.d.81.2		16
52.11	even	12		208.8.i.d.113.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.c.a.3.1	✓	16	13.7	odd	12
13.8.c.a.9.1	yes	16	13.11	odd	12
117.8.g.d.55.8		16	39.20	even	12
117.8.g.d.100.8		16	39.11	even	12
169.8.a.e.1.1		8	13.5	odd	4
169.8.a.f.1.8		8	13.8	odd	4
169.8.b.e.168.1		16	1.1	even	1	trivial
169.8.b.e.168.16		16	13.12	even	2	inner
208.8.i.d.81.2		16	52.7	even	12
208.8.i.d.113.2		16	52.11	even	12

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-21.7945i q^{2} -67.8320 q^{3} -346.998 q^{4} -157.804i q^{5} +1478.36i q^{6} +842.108i q^{7} +4772.94i q^{8} +2414.17 q^{9} -3439.24 q^{10} -1991.16i q^{11} +23537.6 q^{12} +18353.3 q^{14} +10704.1i q^{15} +59607.9 q^{16} -9620.90 q^{17} -52615.6i q^{18} +8439.25i q^{19} +54757.5i q^{20} -57121.8i q^{21} -43396.3 q^{22} -34132.3 q^{23} -323758. i q^{24} +53223.0 q^{25} -15409.7 q^{27} -292210. i q^{28} -9672.89 q^{29} +233290. q^{30} -205412. i q^{31} -688185. i q^{32} +135065. i q^{33} +209682. i q^{34} +132888. q^{35} -837714. q^{36} +432002. i q^{37} +183929. q^{38} +753187. q^{40} -113586. i q^{41} -1.24494e6 q^{42} +400363. q^{43} +690930. i q^{44} -380965. i q^{45} +743896. i q^{46} +1.32924e6i q^{47} -4.04332e6 q^{48} +114397. q^{49} -1.15997e6i q^{50} +652605. q^{51} +298608. q^{53} +335846. i q^{54} -314213. q^{55} -4.01933e6 q^{56} -572451. i q^{57} +210815. i q^{58} -1.63424e6i q^{59} -3.71431e6i q^{60} +1.66237e6 q^{61} -4.47685e6 q^{62} +2.03300e6i q^{63} -7.36880e6 q^{64} +2.94366e6 q^{66} -1.42305e6i q^{67} +3.33844e6 q^{68} +2.31526e6 q^{69} -2.89621e6i q^{70} +1.03961e6i q^{71} +1.15227e7i q^{72} +4.43080e6i q^{73} +9.41526e6 q^{74} -3.61022e6 q^{75} -2.92840e6i q^{76} +1.67678e6 q^{77} -7.40813e6 q^{79} -9.40634e6i q^{80} -4.23453e6 q^{81} -2.47556e6 q^{82} -753809. i q^{83} +1.98212e7i q^{84} +1.51821e6i q^{85} -8.72569e6i q^{86} +656131. q^{87} +9.50371e6 q^{88} -2.49177e6i q^{89} -8.30293e6 q^{90} +1.18439e7 q^{92} +1.39335e7i q^{93} +2.89701e7 q^{94} +1.33174e6 q^{95} +4.66809e7i q^{96} -6.38799e6i q^{97} -2.49322e6i q^{98} -4.80702e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 56 q^{3} - 1154 q^{4} + 11976 q^{9} - 13626 q^{10} + 15816 q^{12} + 14484 q^{14} + 122754 q^{16} - 45648 q^{17} + 147452 q^{22} - 144936 q^{23} - 58488 q^{25} + 358616 q^{27} + 443544 q^{29} - 516232 q^{30}+ \cdots - 32534160 q^{95}+O(q^{100}) \) 16 * q + 56 * q^3 - 1154 * q^4 + 11976 * q^9 - 13626 * q^10 + 15816 * q^12 + 14484 * q^14 + 122754 * q^16 - 45648 * q^17 + 147452 * q^22 - 144936 * q^23 - 58488 * q^25 + 358616 * q^27 + 443544 * q^29 - 516232 * q^30 - 63120 * q^35 - 2325986 * q^36 - 2364492 * q^38 + 766030 * q^40 - 3121908 * q^42 + 782984 * q^43 - 9171424 * q^48 - 2217560 * q^49 - 8040712 * q^51 + 3329328 * q^53 + 836432 * q^55 - 7218312 * q^56 - 3527096 * q^61 - 25996776 * q^62 - 32227074 * q^64 + 16245108 * q^66 + 20236182 * q^68 - 9236984 * q^69 - 4412442 * q^74 - 43669976 * q^75 - 35263992 * q^77 + 12644416 * q^79 + 18632544 * q^81 - 18695502 * q^82 - 10399976 * q^87 - 73352808 * q^88 - 132263562 * q^90 + 64127424 * q^92 + 12052256 * q^94 - 32534160 * q^95

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