LMFDB - Embedded newform 169.6.b.b.168.1

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 201x^{4} + 10512x^{2} + 65536 $ x^6 + 201x^4 + 10512x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.1
Root			$-8.96778i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.6.b.b.168.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.9678i q^{2} -15.7989 q^{3} -88.2923 q^{4} +51.6056i q^{5} +173.279i q^{6} -75.3967i q^{7} +617.402i q^{8} +6.60562 q^{9} +O(q^{10})$	\(q-10.9678i q^{2} -15.7989 q^{3} -88.2923 q^{4} +51.6056i q^{5} +173.279i q^{6} -75.3967i q^{7} +617.402i q^{8} +6.60562 q^{9} +565.999 q^{10} +255.771i q^{11} +1394.92 q^{12} -826.935 q^{14} -815.313i q^{15} +3946.18 q^{16} -53.2245 q^{17} -72.4490i q^{18} -268.895i q^{19} -4556.38i q^{20} +1191.19i q^{21} +2805.25 q^{22} -2080.65 q^{23} -9754.28i q^{24} +461.860 q^{25} +3734.77 q^{27} +6656.95i q^{28} -8177.18 q^{29} -8942.17 q^{30} +4781.97i q^{31} -23523.9i q^{32} -4040.91i q^{33} +583.755i q^{34} +3890.90 q^{35} -583.225 q^{36} -6652.23i q^{37} -2949.18 q^{38} -31861.4 q^{40} -1284.58i q^{41} +13064.7 q^{42} -3808.18 q^{43} -22582.6i q^{44} +340.887i q^{45} +22820.1i q^{46} +3771.22i q^{47} -62345.3 q^{48} +11122.3 q^{49} -5065.58i q^{50} +840.890 q^{51} +24083.8 q^{53} -40962.2i q^{54} -13199.2 q^{55} +46550.1 q^{56} +4248.24i q^{57} +89685.6i q^{58} +24565.5i q^{59} +71985.8i q^{60} +35147.0 q^{61} +52447.7 q^{62} -498.042i q^{63} -131728. q^{64} -44319.8 q^{66} -15093.7i q^{67} +4699.32 q^{68} +32872.0 q^{69} -42674.5i q^{70} -51136.3i q^{71} +4078.32i q^{72} -63005.0i q^{73} -72960.2 q^{74} -7296.89 q^{75} +23741.3i q^{76} +19284.3 q^{77} -80440.1 q^{79} +203645. i q^{80} -60610.5 q^{81} -14089.0 q^{82} -113160. i q^{83} -105173. i q^{84} -2746.69i q^{85} +41767.3i q^{86} +129191. q^{87} -157914. q^{88} -69551.5i q^{89} +3738.78 q^{90} +183705. q^{92} -75550.0i q^{93} +41361.9 q^{94} +13876.5 q^{95} +371653. i q^{96} -18306.5i q^{97} -121987. i q^{98} +1689.53i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9}+O(q^{10}) $ 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9	$ 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9} + 2582 q^{10} + 2182 q^{12} - 1586 q^{14} + 5570 q^{16} - 1816 q^{17} - 6820 q^{22} - 7248 q^{23} - 4046 q^{25} - 8552 q^{27} - 17516 q^{29} - 3434 q^{30} + 8696 q^{35} + 9356 q^{36} - 67572 q^{38} - 87018 q^{40} + 27206 q^{42} - 4064 q^{43} - 152446 q^{48} + 89142 q^{49} - 26936 q^{51} - 25140 q^{53} + 70624 q^{55} + 98574 q^{56} - 25508 q^{61} + 80680 q^{62} - 234786 q^{64} - 224708 q^{66} - 98350 q^{68} + 40224 q^{69} - 263038 q^{74} - 47096 q^{75} + 42104 q^{77} - 123744 q^{79} - 216442 q^{81} - 284520 q^{82} + 278144 q^{87} - 316020 q^{88} - 202516 q^{90} + 601728 q^{92} + 145686 q^{94} + 393120 q^{95}+O(q^{100}) $ 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9 + 2582 * q^10 + 2182 * q^12 - 1586 * q^14 + 5570 * q^16 - 1816 * q^17 - 6820 * q^22 - 7248 * q^23 - 4046 * q^25 - 8552 * q^27 - 17516 * q^29 - 3434 * q^30 + 8696 * q^35 + 9356 * q^36 - 67572 * q^38 - 87018 * q^40 + 27206 * q^42 - 4064 * q^43 - 152446 * q^48 + 89142 * q^49 - 26936 * q^51 - 25140 * q^53 + 70624 * q^55 + 98574 * q^56 - 25508 * q^61 + 80680 * q^62 - 234786 * q^64 - 224708 * q^66 - 98350 * q^68 + 40224 * q^69 - 263038 * q^74 - 47096 * q^75 + 42104 * q^77 - 123744 * q^79 - 216442 * q^81 - 284520 * q^82 + 278144 * q^87 - 316020 * q^88 - 202516 * q^90 + 601728 * q^92 + 145686 * q^94 + 393120 * 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 $

$2$	− 10.9678i	− 1.93885i	−0.245390	−	0.969424i	$-0.578916\pi$
$2$	− 10.9678i	− 1.93885i	0.245390	−	0.969424i	$-0.421084\pi$
$3$	−15.7989	−1.01350	−0.506750	−	0.862093i	$-0.669153\pi$
$3$	−15.7989	−1.01350	−0.506750	+	0.862093i	$0.669153\pi$
$4$	−88.2923	−2.75913
$5$	51.6056i	0.923149i	0.887101	+	0.461575i	$0.152715\pi$
$5$	51.6056i	0.923149i	−0.887101	+	0.461575i	$0.847285\pi$
$6$	173.279i	1.96502i
$7$	− 75.3967i	− 0.581577i	−0.956787	−	0.290789i	$-0.906082\pi$
$7$	− 75.3967i	− 0.581577i	0.956787	−	0.290789i	$-0.0939177\pi$
$8$	617.402i	3.41070i
$9$	6.60562	0.0271836
$10$	565.999	1.78985
$11$	255.771i	0.637339i	0.947866	+	0.318669i	$0.103236\pi$
$11$	255.771i	0.637339i	−0.947866	+	0.318669i	$0.896764\pi$
$12$	1394.92	2.79638
$13$	0	0
$13$	0	0
$14$	−826.935	−1.12759
$15$	− 815.313i	− 0.935613i
$16$	3946.18	3.85369
$17$	−53.2245	−0.0446673	−0.0223336	−	0.999751i	$-0.507110\pi$
$17$	−53.2245	−0.0446673	−0.0223336	+	0.999751i	$0.507110\pi$
$18$	− 72.4490i	− 0.0527049i
$19$	− 268.895i	− 0.170883i	−0.996343	−	0.0854413i	$-0.972770\pi$
$19$	− 268.895i	− 0.170883i	0.996343	−	0.0854413i	$-0.0272300\pi$
$20$	− 4556.38i	− 2.54709i
$21$	1191.19i	0.589429i
$22$	2805.25	1.23570
$23$	−2080.65	−0.820124	−0.410062	−	0.912058i	$-0.634493\pi$
$23$	−2080.65	−0.820124	−0.410062	+	0.912058i	$0.634493\pi$
$24$	− 9754.28i	− 3.45674i
$25$	461.860	0.147795
$26$	0	0
$27$	3734.77	0.985950
$28$	6656.95i	1.60465i
$29$	−8177.18	−1.80555	−0.902773	−	0.430117i	$-0.858472\pi$
$29$	−8177.18	−1.80555	−0.902773	+	0.430117i	$0.858472\pi$
$30$	−8942.17	−1.81401
$31$	4781.97i	0.893723i	0.894603	+	0.446862i	$0.147458\pi$
$31$	4781.97i	0.893723i	−0.894603	+	0.446862i	$0.852542\pi$
$32$	− 23523.9i	− 4.06102i
$33$	− 4040.91i	− 0.645943i
$34$	583.755i	0.0866031i
$35$	3890.90	0.536883
$36$	−583.225	−0.0750032
$37$	− 6652.23i	− 0.798846i	−0.916767	−	0.399423i	$-0.869211\pi$
$37$	− 6652.23i	− 0.798846i	0.916767	−	0.399423i	$-0.130789\pi$
$38$	−2949.18	−0.331316
$39$	0	0
$40$	−31861.4	−3.14858
$41$	− 1284.58i	− 0.119344i	−0.998218	−	0.0596720i	$-0.980995\pi$
$41$	− 1284.58i	− 0.119344i	0.998218	−	0.0596720i	$-0.0190055\pi$
$42$	13064.7	1.14281
$43$	−3808.18	−0.314084	−0.157042	−	0.987592i	$-0.550196\pi$
$43$	−3808.18	−0.314084	−0.157042	+	0.987592i	$0.550196\pi$
$44$	− 22582.6i	− 1.75850i
$45$	340.887i	0.0250945i
$46$	22820.1i	1.59010i
$47$	3771.22i	0.249022i	0.992218	+	0.124511i	$0.0397361\pi$
$47$	3771.22i	0.249022i	−0.992218	+	0.124511i	$0.960264\pi$
$48$	−62345.3	−3.90571
$49$	11122.3	0.661768
$50$	− 5065.58i	− 0.286553i
$51$	840.890	0.0452703
$52$	0	0
$53$	24083.8	1.17770	0.588850	−	0.808242i	$-0.299581\pi$
$53$	24083.8	1.17770	0.588850	+	0.808242i	$0.299581\pi$
$54$	− 40962.2i	− 1.91161i
$55$	−13199.2	−0.588359
$56$	46550.1	1.98358
$57$	4248.24i	0.173190i
$58$	89685.6i	3.50068i
$59$	24565.5i	0.918745i	0.888244	+	0.459372i	$0.151926\pi$
$59$	24565.5i	0.918745i	−0.888244	+	0.459372i	$0.848074\pi$
$60$	71985.8i	2.58148i
$61$	35147.0	1.20938	0.604691	−	0.796460i	$-0.293297\pi$
$61$	35147.0	1.20938	0.604691	+	0.796460i	$0.293297\pi$
$62$	52447.7	1.73279
$63$	− 498.042i	− 0.0158094i
$64$	−131728.	−4.02002
$65$	0	0
$66$	−44319.8	−1.25239
$67$	− 15093.7i	− 0.410780i	−0.978680	−	0.205390i	$-0.934154\pi$
$67$	− 15093.7i	− 0.410780i	0.978680	−	0.205390i	$-0.0658463\pi$
$68$	4699.32	0.123243
$69$	32872.0	0.831196
$70$	− 42674.5i	− 1.04093i
$71$	− 51136.3i	− 1.20388i	−0.798541	−	0.601941i	$-0.794394\pi$
$71$	− 51136.3i	− 1.20388i	0.798541	−	0.601941i	$-0.205606\pi$
$72$	4078.32i	0.0927150i
$73$	− 63005.0i	− 1.38378i	−0.722002	−	0.691891i	$-0.756778\pi$
$73$	− 63005.0i	− 1.38378i	0.722002	−	0.691891i	$-0.243222\pi$
$74$	−72960.2	−1.54884
$75$	−7296.89	−0.149791
$76$	23741.3i	0.471488i
$77$	19284.3	0.370662
$78$	0	0
$79$	−80440.1	−1.45012	−0.725062	−	0.688684i	$-0.758189\pi$
$79$	−80440.1	−1.45012	−0.725062	+	0.688684i	$0.758189\pi$
$80$	203645.i	3.55753i
$81$	−60610.5	−1.02644
$82$	−14089.0	−0.231390
$83$	− 113160.i	− 1.80301i	−0.432771	−	0.901504i	$-0.642464\pi$
$83$	− 113160.i	− 1.80301i	0.432771	−	0.901504i	$-0.357536\pi$
$84$	− 105173.i	− 1.62631i
$85$	− 2746.69i	− 0.0412346i
$86$	41767.3i	0.608962i
$87$	129191.	1.82992
$88$	−157914.	−2.17377
$89$	− 69551.5i	− 0.930746i	−0.885115	−	0.465373i	$-0.845920\pi$
$89$	− 69551.5i	− 0.930746i	0.885115	−	0.465373i	$-0.154080\pi$
$90$	3738.78	0.0486545
$91$	0	0
$92$	183705.	2.26283
$93$	− 75550.0i	− 0.905789i
$94$	41361.9	0.482815
$95$	13876.5	0.157750
$96$	371653.i	4.11585i
$97$	− 18306.5i	− 0.197550i	−0.995110	−	0.0987750i	$-0.968508\pi$
$97$	− 18306.5i	− 0.197550i	0.995110	−	0.0987750i	$-0.0314924\pi$
$98$	− 121987.i	− 1.28307i
$99$	1689.53i	0.0173252i
$100$	−40778.7	−0.407787
$101$	19188.2	0.187168	0.0935839	−	0.995611i	$-0.470168\pi$
$101$	19188.2	0.187168	0.0935839	+	0.995611i	$0.470168\pi$
$102$	− 9222.70i	− 0.0877723i
$103$	78909.2	0.732883	0.366441	−	0.930441i	$-0.380576\pi$
$103$	78909.2	0.732883	0.366441	+	0.930441i	$0.380576\pi$
$104$	0	0
$105$	−61471.9	−0.544131
$106$	− 264146.i	− 2.28338i
$107$	−71123.9	−0.600560	−0.300280	−	0.953851i	$-0.597080\pi$
$107$	−71123.9	−0.600560	−0.300280	+	0.953851i	$0.597080\pi$
$108$	−329752.	−2.72037
$109$	10116.9i	0.0815609i	0.999168	+	0.0407804i	$0.0129844\pi$
$109$	10116.9i	0.0815609i	−0.999168	+	0.0407804i	$0.987016\pi$
$110$	144766.i	1.14074i
$111$	105098.i	0.809631i
$112$	− 297529.i	− 2.24122i
$113$	131480.	0.968639	0.484320	−	0.874891i	$-0.339067\pi$
$113$	131480.	0.968639	0.484320	+	0.874891i	$0.339067\pi$
$114$	46593.8	0.335789
$115$	− 107373.i	− 0.757097i
$116$	721982.	4.98175
$117$	0	0
$118$	269429.	1.78131
$119$	4012.96i	0.0259775i
$120$	503375.	3.19109
$121$	95632.0	0.593799
$122$	− 385485.i	− 2.34481i
$123$	20294.9i	0.120955i
$124$	− 422211.i	− 2.46590i
$125$	185102.i	1.05959i
$126$	−5462.42	−0.0306520
$127$	324455.	1.78503	0.892515	−	0.451018i	$-0.148939\pi$
$127$	324455.	1.78503	0.892515	+	0.451018i	$0.148939\pi$
$128$	691998.i	3.73319i
$129$	60165.1	0.318325
$130$	0	0
$131$	−145557.	−0.741063	−0.370531	−	0.928820i	$-0.620824\pi$
$131$	−145557.	−0.741063	−0.370531	+	0.928820i	$0.620824\pi$
$132$	356781.i	1.78224i
$133$	−20273.8	−0.0993815
$134$	−165545.	−0.796441
$135$	192735.i	0.910179i
$136$	− 32860.9i	− 0.152346i
$137$	253802.i	1.15530i	0.816285	+	0.577649i	$0.196030\pi$
$137$	253802.i	1.15530i	−0.816285	+	0.577649i	$0.803970\pi$
$138$	− 360533.i	− 1.61156i
$139$	138046.	0.606021	0.303011	−	0.952987i	$-0.402008\pi$
$139$	138046.	0.606021	0.303011	+	0.952987i	$0.402008\pi$
$140$	−343536.	−1.48133
$141$	− 59581.1i	− 0.252383i
$142$	−560852.	−2.33414
$143$	0	0
$144$	26066.9	0.104757
$145$	− 421989.i	− 1.66679i
$146$	−691025.	−2.68294
$147$	−175721.	−0.670702
$148$	587341.i	2.20412i
$149$	− 88489.9i	− 0.326534i	−0.986582	−	0.163267i	$-0.947797\pi$
$149$	− 88489.9i	− 0.326534i	0.986582	−	0.163267i	$-0.0522031\pi$
$150$	80030.7i	0.290421i
$151$	243676.i	0.869701i	0.900503	+	0.434850i	$0.143199\pi$
$151$	243676.i	0.869701i	−0.900503	+	0.434850i	$0.856801\pi$
$152$	166016.	0.582829
$153$	−351.581	−0.00121422
$154$	− 211506.i	− 0.718657i
$155$	−246777.	−0.825040
$156$	0	0
$157$	511560.	1.65633	0.828166	−	0.560484i	$-0.189385\pi$
$157$	511560.	1.65633	0.828166	+	0.560484i	$0.189385\pi$
$158$	882250.i	2.81157i
$159$	−380497.	−1.19360
$160$	1.21397e6	3.74893
$161$	156874.i	0.476966i
$162$	664763.i	1.99012i
$163$	− 56212.1i	− 0.165714i	−0.996561	−	0.0828572i	$-0.973595\pi$
$163$	− 56212.1i	− 0.165714i	0.996561	−	0.0828572i	$-0.0264046\pi$
$164$	113418.i	0.329286i
$165$	208534.	0.596302
$166$	−1.24111e6	−3.49576
$167$	51515.8i	0.142938i	0.997443	+	0.0714692i	$0.0227688\pi$
$167$	51515.8i	0.142938i	−0.997443	+	0.0714692i	$0.977231\pi$
$168$	−735441.	−2.01036
$169$	0	0
$170$	−30125.1	−0.0799476
$171$	− 1776.21i	− 0.00464521i
$172$	336233.	0.866601
$173$	550530.	1.39851	0.699255	−	0.714872i	$-0.253515\pi$
$173$	550530.	1.39851	0.699255	+	0.714872i	$0.253515\pi$
$174$	− 1.41693e6i	− 3.54794i
$175$	− 34822.7i	− 0.0859544i
$176$	1.00932e6i	2.45610i
$177$	− 388108.i	− 0.931148i
$178$	−762826.	−1.80458
$179$	472286.	1.10172	0.550862	−	0.834596i	$-0.314299\pi$
$179$	472286.	1.10172	0.550862	+	0.834596i	$0.314299\pi$
$180$	− 30097.7i	− 0.0692392i
$181$	375193.	0.851253	0.425626	−	0.904899i	$-0.360054\pi$
$181$	375193.	0.851253	0.425626	+	0.904899i	$0.360054\pi$
$182$	0	0
$183$	−555284.	−1.22571
$184$	− 1.28460e6i	− 2.79719i
$185$	343292.	0.737454
$186$	−828616.	−1.75619
$187$	− 13613.3i	− 0.0284682i
$188$	− 332969.i	− 0.687084i
$189$	− 281590.i	− 0.573406i
$190$	− 152194.i	− 0.305854i
$191$	−555860.	−1.10251	−0.551255	−	0.834337i	$-0.685851\pi$
$191$	−555860.	−1.10251	−0.551255	+	0.834337i	$0.685851\pi$
$192$	2.08116e6	4.07429
$193$	827884.i	1.59984i	0.600107	+	0.799920i	$0.295125\pi$
$193$	827884.i	1.59984i	−0.600107	+	0.799920i	$0.704875\pi$
$194$	−200782.	−0.383019
$195$	0	0
$196$	−982016.	−1.82591
$197$	− 61091.3i	− 0.112154i	−0.998426	−	0.0560769i	$-0.982141\pi$
$197$	− 61091.3i	− 0.112154i	0.998426	−	0.0560769i	$-0.0178592\pi$
$198$	18530.4	0.0335909
$199$	456275.	0.816760	0.408380	−	0.912812i	$-0.366094\pi$
$199$	456275.	0.816760	0.408380	+	0.912812i	$0.366094\pi$
$200$	285153.i	0.504085i
$201$	238464.i	0.416326i
$202$	− 210452.i	− 0.362890i
$203$	616533.i	1.05006i
$204$	−74244.1	−0.124907
$205$	66291.4	0.110172
$206$	− 865459.i	− 1.42095i
$207$	−13744.0	−0.0222939
$208$	0	0
$209$	68775.5	0.108910
$210$	674211.i	1.05499i
$211$	616069.	0.952628	0.476314	−	0.879275i	$-0.341973\pi$
$211$	616069.	0.952628	0.476314	+	0.879275i	$0.341973\pi$
$212$	−2.12641e6	−3.24943
$213$	807898.i	1.22013i
$214$	780071.i	1.16439i
$215$	− 196524.i	− 0.289947i
$216$	2.30586e6i	3.36277i
$217$	360545.	0.519769
$218$	110960.	0.158134
$219$	995410.i	1.40246i
$220$	1.16539e6	1.62336
$221$	0	0
$222$	1.15269e6	1.56975
$223$	459448.i	0.618692i	0.950950	+	0.309346i	$0.100110\pi$
$223$	459448.i	0.618692i	−0.950950	+	0.309346i	$0.899890\pi$
$224$	−1.77363e6	−2.36180
$225$	3050.87	0.00401761
$226$	− 1.44204e6i	− 1.87805i
$227$	− 1.41499e6i	− 1.82259i	−0.411751	−	0.911297i	$-0.635083\pi$
$227$	− 1.41499e6i	− 1.82259i	0.411751	−	0.911297i	$-0.364917\pi$
$228$	− 375087.i	− 0.477854i
$229$	− 1.42312e6i	− 1.79330i	−0.442745	−	0.896648i	$-0.645995\pi$
$229$	− 1.42312e6i	− 1.79330i	0.442745	−	0.896648i	$-0.354005\pi$
$230$	−1.17765e6	−1.46790
$231$	−304671.	−0.375666
$232$	− 5.04861e6i	− 6.15817i
$233$	−342601.	−0.413427	−0.206713	−	0.978402i	$-0.566277\pi$
$233$	−342601.	−0.413427	−0.206713	+	0.978402i	$0.566277\pi$
$234$	0	0
$235$	−194616.	−0.229884
$236$	− 2.16894e6i	− 2.53494i
$237$	1.27087e6	1.46970
$238$	44013.2	0.0503664
$239$	739257.i	0.837145i	0.908183	+	0.418572i	$0.137469\pi$
$239$	739257.i	0.837145i	−0.908183	+	0.418572i	$0.862531\pi$
$240$	− 3.21737e6i	− 3.60556i
$241$	671299.i	0.744514i	0.928130	+	0.372257i	$0.121416\pi$
$241$	671299.i	0.744514i	−0.928130	+	0.372257i	$0.878584\pi$
$242$	− 1.04887e6i	− 1.15129i
$243$	50030.4	0.0543523
$244$	−3.10321e6	−3.33685
$245$	573975.i	0.610911i
$246$	222590.	0.234514
$247$	0	0
$248$	−2.95240e6	−3.04822
$249$	1.78780e6i	1.82735i
$250$	2.03016e6	2.05438
$251$	251195.	0.251667	0.125833	−	0.992051i	$-0.459840\pi$
$251$	251195.	0.251667	0.125833	+	0.992051i	$0.459840\pi$
$252$	43973.3i	0.0436202i
$253$	− 532171.i	− 0.522697i
$254$	− 3.55855e6i	− 3.46090i
$255$	43394.6i	0.0417913i
$256$	3.37439e6	3.21807
$257$	−250552.	−0.236628	−0.118314	−	0.992976i	$-0.537749\pi$
$257$	−250552.	−0.236628	−0.118314	+	0.992976i	$0.537749\pi$
$258$	− 659878.i	− 0.617184i
$259$	−501556.	−0.464590
$260$	0	0
$261$	−54015.3	−0.0490813
$262$	1.59644e6i	1.43681i
$263$	54253.7	0.0483660	0.0241830	−	0.999708i	$-0.492302\pi$
$263$	54253.7	0.0483660	0.0241830	+	0.999708i	$0.492302\pi$
$264$	2.49487e6	2.20312
$265$	1.24286e6i	1.08719i
$266$	222358.i	0.192686i
$267$	1.09884e6i	0.943312i
$268$	1.33266e6i	1.13340i
$269$	−438210.	−0.369233	−0.184617	−	0.982811i	$-0.559104\pi$
$269$	−438210.	−0.369233	−0.184617	+	0.982811i	$0.559104\pi$
$270$	2.11388e6	1.76470
$271$	− 759175.i	− 0.627941i	−0.949433	−	0.313970i	$-0.898341\pi$
$271$	− 759175.i	− 0.627941i	0.949433	−	0.313970i	$-0.101659\pi$
$272$	−210033.	−0.172134
$273$	0	0
$274$	2.78365e6	2.23995
$275$	118131.i	0.0941956i
$276$	−2.90235e6	−2.29338
$277$	58064.8	0.0454688	0.0227344	−	0.999742i	$-0.492763\pi$
$277$	58064.8	0.0454688	0.0227344	+	0.999742i	$0.492763\pi$
$278$	− 1.51406e6i	− 1.17498i
$279$	31587.9i	0.0242946i
$280$	2.40225e6i	1.83114i
$281$	158274.i	0.119576i	0.998211	+	0.0597880i	$0.0190425\pi$
$281$	158274.i	0.119576i	−0.998211	+	0.0597880i	$0.980958\pi$
$282$	−653473.	−0.489333
$283$	−902391.	−0.669775	−0.334887	−	0.942258i	$-0.608698\pi$
$283$	−902391.	−0.669775	−0.334887	+	0.942258i	$0.608698\pi$
$284$	4.51494e6i	3.32167i
$285$	−219233.	−0.159880
$286$	0	0
$287$	−96852.9	−0.0694078
$288$	− 155390.i	− 0.110393i
$289$	−1.41702e6	−0.998005
$290$	−4.62828e6	−3.23165
$291$	289223.i	0.200217i
$292$	5.56285e6i	3.81804i
$293$	− 840136.i	− 0.571716i	−0.958272	−	0.285858i	$-0.907721\pi$
$293$	− 840136.i	− 0.571716i	0.958272	−	0.285858i	$-0.0922786\pi$
$294$	1.92727e6i	1.30039i
$295$	−1.26772e6	−0.848139
$296$	4.10710e6	2.72462
$297$	955248.i	0.628384i
$298$	−970538.	−0.633099
$299$	0	0
$300$	644259.	0.413292
$301$	287124.i	0.182664i
$302$	2.67258e6	1.68622
$303$	−303153.	−0.189695
$304$	− 1.06111e6i	− 0.658528i
$305$	1.81378e6i	1.11644i
$306$	3856.06i	0.00235419i
$307$	800024.i	0.484459i	0.970219	+	0.242230i	$0.0778787\pi$
$307$	800024.i	0.484459i	−0.970219	+	0.242230i	$0.922121\pi$
$308$	−1.70266e6	−1.02271
$309$	−1.24668e6	−0.742777
$310$	2.70659e6i	1.59963i
$311$	−2.53409e6	−1.48567	−0.742834	−	0.669476i	$-0.766519\pi$
$311$	−2.53409e6	−1.48567	−0.742834	+	0.669476i	$0.766519\pi$
$312$	0	0
$313$	651307.	0.375772	0.187886	−	0.982191i	$-0.439836\pi$
$313$	651307.	0.375772	0.187886	+	0.982191i	$0.439836\pi$
$314$	− 5.61068e6i	− 3.21138i
$315$	25701.8	0.0145944
$316$	7.10224e6	4.00108
$317$	2.01697e6i	1.12733i	0.826003	+	0.563665i	$0.190609\pi$
$317$	2.01697e6i	1.12733i	−0.826003	+	0.563665i	$0.809391\pi$
$318$	4.17321e6i	2.31421i
$319$	− 2.09149e6i	− 1.15074i
$320$	− 6.79790e6i	− 3.71108i
$321$	1.12368e6	0.608667
$322$	1.72056e6	0.924764
$323$	14311.8i	0.00763286i
$324$	5.35144e6	2.83210
$325$	0	0
$326$	−616522.	−0.321295
$327$	− 159836.i	− 0.0826620i
$328$	793100.	0.407046
$329$	284337.	0.144825
$330$	− 2.28715e6i	− 1.15614i
$331$	− 2.91581e6i	− 1.46281i	−0.681942	−	0.731406i	$-0.738864\pi$
$331$	− 2.91581e6i	− 1.46281i	0.681942	−	0.731406i	$-0.261136\pi$
$332$	9.99115e6i	4.97474i
$333$	− 43942.1i	− 0.0217155i
$334$	565014.	0.277136
$335$	778921.	0.379211
$336$	4.70063e6i	2.27148i
$337$	604088.	0.289751	0.144876	−	0.989450i	$-0.453722\pi$
$337$	604088.	0.289751	0.144876	+	0.989450i	$0.453722\pi$
$338$	0	0
$339$	−2.07723e6	−0.981717
$340$	242511.i	0.113772i
$341$	−1.22309e6	−0.569604
$342$	−19481.1	−0.00900636
$343$	− 2.10578e6i	− 0.966446i
$344$	− 2.35118e6i	− 1.07125i
$345$	1.69638e6i	0.767318i
$346$	− 6.03810e6i	− 2.71150i
$347$	−574381.	−0.256080	−0.128040	−	0.991769i	$-0.540869\pi$
$347$	−574381.	−0.256080	−0.128040	+	0.991769i	$0.540869\pi$
$348$	−1.14065e7	−5.04900
$349$	− 855909.i	− 0.376153i	−0.982154	−	0.188076i	$-0.939775\pi$
$349$	− 855909.i	− 0.376153i	0.982154	−	0.188076i	$-0.0602253\pi$
$350$	−381928.	−0.166653
$351$	0	0
$352$	6.01675e6	2.58825
$353$	1.50594e6i	0.643237i	0.946869	+	0.321619i	$0.104227\pi$
$353$	1.50594e6i	0.643237i	−0.946869	+	0.321619i	$0.895773\pi$
$354$	−4.25668e6	−1.80536
$355$	2.63892e6	1.11136
$356$	6.14086e6i	2.56805i
$357$	− 63400.3i	− 0.0263282i
$358$	− 5.17993e6i	− 2.13608i
$359$	974379.i	0.399017i	0.979896	+	0.199509i	$0.0639346\pi$
$359$	974379.i	0.399017i	−0.979896	+	0.199509i	$0.936065\pi$
$360$	−210464.	−0.0855898
$361$	2.40379e6	0.970799
$362$	− 4.11504e6i	− 1.65045i
$363$	−1.51088e6	−0.601816
$364$	0	0
$365$	3.25141e6	1.27744
$366$	6.09024e6i	2.37647i
$367$	−1.62578e6	−0.630083	−0.315041	−	0.949078i	$-0.602018\pi$
$367$	−1.62578e6	−0.630083	−0.315041	+	0.949078i	$0.602018\pi$
$368$	−8.21061e6	−3.16050
$369$	− 8485.43i	− 0.00324420i
$370$	− 3.76516e6i	− 1.42981i
$371$	− 1.81584e6i	− 0.684924i
$372$	6.67048e6i	2.49919i
$373$	1.39838e6	0.520419	0.260210	−	0.965552i	$-0.416208\pi$
$373$	1.39838e6	0.520419	0.260210	+	0.965552i	$0.416208\pi$
$374$	−149308.	−0.0551955
$375$	− 2.92441e6i	− 1.07389i
$376$	−2.32836e6	−0.849336
$377$	0	0
$378$	−3.08842e6	−1.11175
$379$	− 3.24570e6i	− 1.16067i	−0.814377	−	0.580336i	$-0.802921\pi$
$379$	− 3.24570e6i	− 1.16067i	0.814377	−	0.580336i	$-0.197079\pi$
$380$	−1.22519e6	−0.435254
$381$	−5.12604e6	−1.80913
$382$	6.09656e6i	2.13760i
$383$	1.19115e6i	0.414926i	0.978243	+	0.207463i	$0.0665207\pi$
$383$	1.19115e6i	0.414926i	−0.978243	+	0.207463i	$0.933479\pi$
$384$	− 1.09328e7i	− 3.78359i
$385$	995180.i	0.342176i
$386$	9.08006e6	3.10185
$387$	−25155.4	−0.00853795
$388$	1.61633e6i	0.545067i
$389$	2.99954e6	1.00503	0.502517	−	0.864567i	$-0.332407\pi$
$389$	2.99954e6	1.00503	0.502517	+	0.864567i	$0.332407\pi$
$390$	0	0
$391$	110742.	0.0366327
$392$	6.86695e6i	2.25709i
$393$	2.29964e6	0.751067
$394$	−670036.	−0.217449
$395$	− 4.15116e6i	− 1.33868i
$396$	− 149172.i	− 0.0478025i
$397$	3.31310e6i	1.05501i	0.849551	+	0.527507i	$0.176873\pi$
$397$	3.31310e6i	1.05501i	−0.849551	+	0.527507i	$0.823127\pi$
$398$	− 5.00433e6i	− 1.58357i
$399$	320304.	0.100723
$400$	1.82258e6	0.569557
$401$	1.01479e6i	0.315148i	0.987507	+	0.157574i	$0.0503672\pi$
$401$	1.01479e6i	0.315148i	−0.987507	+	0.157574i	$0.949633\pi$
$402$	2.61543e6	0.807193
$403$	0	0
$404$	−1.69417e6	−0.516421
$405$	− 3.12784e6i	− 0.947562i
$406$	6.76200e6	2.03592
$407$	1.70145e6	0.509135
$408$	519167.i	0.154403i
$409$	− 3.24892e6i	− 0.960352i	−0.877172	−	0.480176i	$-0.840573\pi$
$409$	− 3.24892e6i	− 0.960352i	0.877172	−	0.480176i	$-0.159427\pi$
$410$	− 727070.i	− 0.213607i
$411$	− 4.00980e6i	− 1.17090i
$412$	−6.96707e6	−2.02212
$413$	1.85216e6	0.534321
$414$	150741.i	0.0432246i
$415$	5.83969e6	1.66445
$416$	0	0
$417$	−2.18098e6	−0.614203
$418$	− 754315.i	− 0.211160i
$419$	1.45447e6	0.404735	0.202368	−	0.979310i	$-0.435136\pi$
$419$	1.45447e6	0.404735	0.202368	+	0.979310i	$0.435136\pi$
$420$	5.42750e6	1.50133
$421$	− 6.39896e6i	− 1.75956i	−0.475380	−	0.879780i	$-0.657690\pi$
$421$	− 6.39896e6i	− 1.75956i	0.475380	−	0.879780i	$-0.342310\pi$
$422$	− 6.75692e6i	− 1.84700i
$423$	24911.2i	0.00676930i
$424$	1.48694e7i	4.01678i
$425$	−24582.3	−0.00660161
$426$	8.86085e6	2.36566
$427$	− 2.64997e6i	− 0.703349i
$428$	6.27969e6	1.65702
$429$	0	0
$430$	−2.15543e6	−0.562163
$431$	− 2.37413e6i	− 0.615619i	−0.951448	−	0.307809i	$-0.900404\pi$
$431$	− 2.37413e6i	− 0.615619i	0.951448	−	0.307809i	$-0.0995959\pi$
$432$	1.47381e7	3.79954
$433$	−1.10652e6	−0.283622	−0.141811	−	0.989894i	$-0.545293\pi$
$433$	−1.10652e6	−0.283622	−0.141811	+	0.989894i	$0.545293\pi$
$434$	− 3.95438e6i	− 1.00775i
$435$	6.66696e6i	1.68929i
$436$	− 893246.i	− 0.225037i
$437$	559476.i	0.140145i
$438$	1.09174e7	2.71917
$439$	−553247.	−0.137012	−0.0685058	−	0.997651i	$-0.521823\pi$
$439$	−553247.	−0.137012	−0.0685058	+	0.997651i	$0.521823\pi$
$440$	− 8.14924e6i	− 2.00671i
$441$	73469.9	0.0179892
$442$	0	0
$443$	−41818.4	−0.0101241	−0.00506207	−	0.999987i	$-0.501611\pi$
$443$	−41818.4	−0.0101241	−0.00506207	+	0.999987i	$0.501611\pi$
$444$	− 9.27934e6i	− 2.23388i
$445$	3.58925e6	0.859218
$446$	5.03913e6	1.19955
$447$	1.39804e6i	0.330942i
$448$	9.93186e6i	2.33795i
$449$	− 1.67995e6i	− 0.393261i	−0.980478	−	0.196630i	$-0.937000\pi$
$449$	− 1.67995e6i	− 0.393261i	0.980478	−	0.196630i	$-0.0629999\pi$
$450$	− 33461.3i	− 0.00778954i
$451$	328558.	0.0760625
$452$	−1.16086e7	−2.67261
$453$	− 3.84981e6i	− 0.881442i
$454$	−1.55193e7	−3.53373
$455$	0	0
$456$	−2.62287e6	−0.590697
$457$	− 1.59798e6i	− 0.357917i	−0.983857	−	0.178958i	$-0.942727\pi$
$457$	− 1.59798e6i	− 0.357917i	0.983857	−	0.178958i	$-0.0572728\pi$
$458$	−1.56084e7	−3.47693
$459$	−198782.	−0.0440397
$460$	9.48023e6i	2.08893i
$461$	5.88109e6i	1.28886i	0.764664	+	0.644429i	$0.222905\pi$
$461$	5.88109e6i	1.28886i	−0.764664	+	0.644429i	$0.777095\pi$
$462$	3.34157e6i	0.728359i
$463$	778080.i	0.168683i	0.996437	+	0.0843416i	$0.0268787\pi$
$463$	778080.i	0.168683i	−0.996437	+	0.0843416i	$0.973121\pi$
$464$	−3.22686e7	−6.95801
$465$	3.89880e6	0.836179
$466$	3.75757e6i	0.801572i
$467$	6.43008e6	1.36434	0.682172	−	0.731192i	$-0.261035\pi$
$467$	6.43008e6	1.36434	0.682172	+	0.731192i	$0.261035\pi$
$468$	0	0
$469$	−1.13802e6	−0.238900
$470$	2.13451e6i	0.445710i
$471$	−8.08209e6	−1.67869
$472$	−1.51668e7	−3.13356
$473$	− 974024.i	− 0.200178i
$474$	− 1.39386e7i	− 2.84953i
$475$	− 124192.i	− 0.0252556i
$476$	− 354313.i	− 0.0716754i
$477$	159088.	0.0320141
$478$	8.10801e6	1.62310
$479$	2.81814e6i	0.561209i	0.959824	+	0.280604i	$0.0905349\pi$
$479$	2.81814e6i	0.561209i	−0.959824	+	0.280604i	$0.909465\pi$
$480$	−1.91794e7	−3.79954
$481$	0	0
$482$	7.36266e6	1.44350
$483$	− 2.47844e6i	− 0.483405i
$484$	−8.44357e6	−1.63837
$485$	944720.	0.182368
$486$	− 548722.i	− 0.105381i
$487$	2.98105e6i	0.569569i	0.958592	+	0.284785i	$0.0919221\pi$
$487$	2.98105e6i	0.569569i	−0.958592	+	0.284785i	$0.908078\pi$
$488$	2.16998e7i	4.12483i
$489$	888089.i	0.167952i
$490$	6.29523e6	1.18446
$491$	−8.42319e6	−1.57679	−0.788393	−	0.615171i	$-0.789087\pi$
$491$	−8.42319e6	−1.57679	−0.788393	+	0.615171i	$0.789087\pi$
$492$	− 1.79189e6i	− 0.333732i
$493$	435227.	0.0806489
$494$	0	0
$495$	−87189.2	−0.0159937
$496$	1.88705e7i	3.44413i
$497$	−3.85551e6	−0.700150
$498$	1.96083e7	3.54295
$499$	− 7.83744e6i	− 1.40904i	−0.709685	−	0.704519i	$-0.751163\pi$
$499$	− 7.83744e6i	− 1.40904i	0.709685	−	0.704519i	$-0.248837\pi$
$500$	− 1.63431e7i	− 2.92354i
$501$	− 813893.i	− 0.144868i
$502$	− 2.75505e6i	− 0.487944i
$503$	8.98386e6	1.58323	0.791613	−	0.611023i	$-0.209242\pi$
$503$	8.98386e6	1.58323	0.791613	+	0.611023i	$0.209242\pi$
$504$	307492.	0.0539209
$505$	990219.i	0.172784i
$506$	−5.83674e6	−1.01343
$507$	0	0
$508$	−2.86469e7	−4.92514
$509$	4.14912e6i	0.709842i	0.934896	+	0.354921i	$0.115492\pi$
$509$	4.14912e6i	0.709842i	−0.934896	+	0.354921i	$0.884508\pi$
$510$	475943.	0.0810270
$511$	−4.75037e6	−0.804776
$512$	− 1.48656e7i	− 2.50616i
$513$	− 1.00426e6i	− 0.168482i
$514$	2.74800e6i	0.458785i
$515$	4.07216e6i	0.676560i
$516$	−5.31212e6	−0.878301
$517$	−964570.	−0.158711
$518$	5.50096e6i	0.900771i
$519$	−8.69778e6	−1.41739
$520$	0	0
$521$	7.23197e6	1.16725	0.583623	−	0.812025i	$-0.301635\pi$
$521$	7.23197e6	1.16725	0.583623	+	0.812025i	$0.301635\pi$
$522$	592429.i	0.0951612i
$523$	−1.78094e6	−0.284706	−0.142353	−	0.989816i	$-0.545467\pi$
$523$	−1.78094e6	−0.284706	−0.142353	+	0.989816i	$0.545467\pi$
$524$	1.28516e7	2.04469
$525$	550162.i	0.0871148i
$526$	− 595043.i	− 0.0937743i
$527$	− 254518.i	− 0.0399202i
$528$	− 1.59461e7i	− 2.48926i
$529$	−2.10723e6	−0.327396
$530$	1.36314e7	2.10790
$531$	162270.i	0.0249748i
$532$	1.79002e6	0.274207
$533$	0	0
$534$	1.20518e7	1.82894
$535$	− 3.67039e6i	− 0.554406i
$536$	9.31889e6	1.40105
$537$	−7.46161e6	−1.11660
$538$	4.80619e6i	0.715888i
$539$	2.84477e6i	0.421770i
$540$	− 1.70170e7i	− 2.51131i
$541$	− 4.10104e6i	− 0.602423i	−0.953557	−	0.301211i	$-0.902609\pi$
$541$	− 4.10104e6i	− 0.602423i	0.953557	−	0.301211i	$-0.0973909\pi$
$542$	−8.32647e6	−1.21748
$543$	−5.92765e6	−0.862745
$544$	1.25205e6i	0.181395i
$545$	−522090.	−0.0752929
$546$	0	0
$547$	−7.62953e6	−1.09026	−0.545129	−	0.838352i	$-0.683519\pi$
$547$	−7.62953e6	−1.09026	−0.545129	+	0.838352i	$0.683519\pi$
$548$	− 2.24088e7i	− 3.18762i
$549$	232168.	0.0328754
$550$	1.29563e6	0.182631
$551$	2.19880e6i	0.308537i
$552$	2.02952e7i	2.83496i
$553$	6.06492e6i	0.843359i
$554$	− 636842.i	− 0.0881571i
$555$	−5.42365e6	−0.747410
$556$	−1.21884e7	−1.67209
$557$	− 1.40146e7i	− 1.91400i	−0.290093	−	0.956998i	$-0.593686\pi$
$557$	− 1.40146e7i	− 1.91400i	0.290093	−	0.956998i	$-0.406314\pi$
$558$	346449.	0.0471036
$559$	0	0
$560$	1.53542e7	2.06898
$561$	215076.i	0.0288525i
$562$	1.73591e6	0.231840
$563$	−5.27865e6	−0.701863	−0.350931	−	0.936401i	$-0.614135\pi$
$563$	−5.27865e6	−0.701863	−0.350931	+	0.936401i	$0.614135\pi$
$564$	5.26055e6i	0.696360i
$565$	6.78508e6i	0.894199i
$566$	9.89723e6i	1.29859i
$567$	4.56984e6i	0.596957i
$568$	3.15717e7	4.10607
$569$	9.94285e6	1.28745	0.643725	−	0.765257i	$-0.277388\pi$
$569$	9.94285e6	1.28745	0.643725	+	0.765257i	$0.277388\pi$
$570$	2.40450e6i	0.309983i
$571$	4.01361e6	0.515163	0.257581	−	0.966257i	$-0.417074\pi$
$571$	4.01361e6	0.515163	0.257581	+	0.966257i	$0.417074\pi$
$572$	0	0
$573$	8.78199e6	1.11739
$574$	1.06226e6i	0.134571i
$575$	−960970.	−0.121210
$576$	−870145.	−0.109279
$577$	1.14751e6i	0.143489i	0.997423	+	0.0717445i	$0.0228566\pi$
$577$	1.14751e6i	0.143489i	−0.997423	+	0.0717445i	$0.977143\pi$
$578$	1.55416e7i	1.93498i
$579$	− 1.30797e7i	− 1.62144i
$580$	3.72583e7i	4.59890i
$581$	−8.53189e6	−1.04859
$582$	3.17214e6	0.388190
$583$	6.15994e6i	0.750594i
$584$	3.88994e7	4.71966
$585$	0	0
$586$	−9.21443e6	−1.10847
$587$	7.24558e6i	0.867917i	0.900933	+	0.433958i	$0.142884\pi$
$587$	7.24558e6i	0.867917i	−0.900933	+	0.433958i	$0.857116\pi$
$588$	1.55148e7	1.85056
$589$	1.28585e6	0.152722
$590$	1.39040e7i	1.64441i
$591$	965176.i	0.113668i
$592$	− 2.62509e7i	− 3.07850i
$593$	2.17012e6i	0.253424i	0.991940	+	0.126712i	$0.0404424\pi$
$593$	2.17012e6i	0.253424i	−0.991940	+	0.126712i	$0.959558\pi$
$594$	1.04770e7	1.21834
$595$	−207091.	−0.0239811
$596$	7.81297e6i	0.900950i
$597$	−7.20865e6	−0.827786
$598$	0	0
$599$	1.18445e7	1.34881	0.674403	−	0.738363i	$-0.264401\pi$
$599$	1.18445e7	1.34881	0.674403	+	0.738363i	$0.264401\pi$
$600$	− 4.50511e6i	− 0.510890i
$601$	1.25297e7	1.41499	0.707495	−	0.706718i	$-0.249825\pi$
$601$	1.25297e7	1.41499	0.707495	+	0.706718i	$0.249825\pi$
$602$	3.14912e6	0.354159
$603$	− 99703.4i	− 0.0111665i
$604$	− 2.15147e7i	− 2.39962i
$605$	4.93515e6i	0.548165i
$606$	3.32491e6i	0.367789i
$607$	−6.86090e6	−0.755805	−0.377902	−	0.925845i	$-0.623354\pi$
$607$	−6.86090e6	−0.755805	−0.377902	+	0.925845i	$0.623354\pi$
$608$	−6.32546e6	−0.693958
$609$	− 9.74055e6i	− 1.06424i
$610$	1.98932e7	2.16461
$611$	0	0
$612$	31041.9	0.00335019
$613$	1.63240e7i	1.75459i	0.479956	+	0.877293i	$0.340653\pi$
$613$	1.63240e7i	1.75459i	−0.479956	+	0.877293i	$0.659347\pi$
$614$	8.77449e6	0.939293
$615$	−1.04733e6	−0.111660
$616$	1.19062e7i	1.26421i
$617$	− 1.16644e7i	− 1.23353i	−0.787149	−	0.616763i	$-0.788444\pi$
$617$	− 1.16644e7i	− 1.23353i	0.787149	−	0.616763i	$-0.211556\pi$
$618$	1.36733e7i	1.44013i
$619$	− 1.87660e7i	− 1.96855i	−0.176650	−	0.984274i	$-0.556526\pi$
$619$	− 1.87660e7i	− 1.96855i	0.176650	−	0.984274i	$-0.443474\pi$
$620$	2.17885e7	2.27640
$621$	−7.77076e6	−0.808602
$622$	2.77934e7i	2.88048i
$623$	−5.24396e6	−0.541301
$624$	0	0
$625$	−8.10900e6	−0.830361
$626$	− 7.14339e6i	− 0.728566i
$627$	−1.08658e6	−0.110380
$628$	−4.51668e7	−4.57004
$629$	354062.i	0.0356823i
$630$	− 281891.i	− 0.0282964i
$631$	− 7.70194e6i	− 0.770064i	−0.922903	−	0.385032i	$-0.874190\pi$
$631$	− 7.70194e6i	− 0.770064i	0.922903	−	0.385032i	$-0.125810\pi$
$632$	− 4.96639e7i	− 4.94593i
$633$	−9.73322e6	−0.965489
$634$	2.21217e7	2.18572
$635$	1.67437e7i	1.64785i
$636$	3.35950e7	3.29330
$637$	0	0
$638$	−2.29390e7	−2.23112
$639$	− 337787.i	− 0.0327258i
$640$	−3.57110e7	−3.44629
$641$	5.38428e6	0.517586	0.258793	−	0.965933i	$-0.416675\pi$
$641$	5.38428e6	0.517586	0.258793	+	0.965933i	$0.416675\pi$
$642$	− 1.23243e7i	− 1.18011i
$643$	− 1.66697e7i	− 1.59001i	−0.606602	−	0.795005i	$-0.707468\pi$
$643$	− 1.66697e7i	− 1.59001i	0.606602	−	0.795005i	$-0.292532\pi$
$644$	− 1.38508e7i	− 1.31601i
$645$	3.10486e6i	0.293861i
$646$	156969.	0.0147990
$647$	−1.83778e7	−1.72597	−0.862983	−	0.505233i	$-0.831407\pi$
$647$	−1.83778e7	−1.72597	−0.862983	+	0.505233i	$0.831407\pi$
$648$	− 3.74210e7i	− 3.50089i
$649$	−6.28314e6	−0.585552
$650$	0	0
$651$	−5.69622e6	−0.526786
$652$	4.96309e6i	0.457228i
$653$	1.06610e7	0.978395	0.489198	−	0.872173i	$-0.337290\pi$
$653$	1.06610e7	0.978395	0.489198	+	0.872173i	$0.337290\pi$
$654$	−1.75305e6	−0.160269
$655$	− 7.51156e6i	− 0.684111i
$656$	− 5.06917e6i	− 0.459914i
$657$	− 416187.i	− 0.0376162i
$658$	− 3.11855e6i	− 0.280794i
$659$	1.98297e6	0.177870	0.0889352	−	0.996037i	$-0.471654\pi$
$659$	1.98297e6	0.177870	0.0889352	+	0.996037i	$0.471654\pi$
$660$	−1.84119e7	−1.64528
$661$	− 8.53199e6i	− 0.759533i	−0.925082	−	0.379767i	$-0.876004\pi$
$661$	− 8.53199e6i	− 0.759533i	0.925082	−	0.379767i	$-0.123996\pi$
$662$	−3.19799e7	−2.83617
$663$	0	0
$664$	6.98652e7	6.14951
$665$	− 1.04624e6i	− 0.0917439i
$666$	−481947.	−0.0421031
$667$	1.70139e7	1.48077
$668$	− 4.54845e6i	− 0.394386i
$669$	− 7.25878e6i	− 0.627044i
$670$	− 8.54304e6i	− 0.735234i
$671$	8.98960e6i	0.770786i
$672$	2.80214e7	2.39368
$673$	391667.	0.0333333	0.0166667	−	0.999861i	$-0.494695\pi$
$673$	391667.	0.0333333	0.0166667	+	0.999861i	$0.494695\pi$
$674$	− 6.62550e6i	− 0.561784i
$675$	1.72494e6	0.145719
$676$	0	0
$677$	−9.84024e6	−0.825152	−0.412576	−	0.910923i	$-0.635371\pi$
$677$	−9.84024e6	−0.825152	−0.412576	+	0.910923i	$0.635371\pi$
$678$	2.27827e7i	1.90340i
$679$	−1.38025e6	−0.114891
$680$	1.69581e6	0.140639
$681$	2.23554e7i	1.84720i
$682$	1.34146e7i	1.10438i
$683$	− 1.19119e7i	− 0.977075i	−0.872543	−	0.488538i	$-0.837530\pi$
$683$	− 1.19119e7i	− 0.977075i	0.872543	−	0.488538i	$-0.162470\pi$
$684$	156826.i	0.0128168i
$685$	−1.30976e7	−1.06651
$686$	−2.30957e7	−1.87379
$687$	2.24837e7i	1.81751i
$688$	−1.50278e7	−1.21038
$689$	0	0
$690$	1.86055e7	1.48771
$691$	1.02411e7i	0.815930i	0.912998	+	0.407965i	$0.133761\pi$
$691$	1.02411e7i	0.815930i	−0.912998	+	0.407965i	$0.866239\pi$
$692$	−4.86076e7	−3.85868
$693$	127385.	0.0100759
$694$	6.29969e6i	0.496501i
$695$	7.12397e6i	0.559448i
$696$	7.97625e7i	6.24131i
$697$	68371.0i	0.00533077i
$698$	−9.38743e6	−0.729303
$699$	5.41272e6	0.419008
$700$	3.07458e6i	0.237160i
$701$	−1.21924e7	−0.937118	−0.468559	−	0.883432i	$-0.655227\pi$
$701$	−1.21924e7	−0.937118	−0.468559	+	0.883432i	$0.655227\pi$
$702$	0	0
$703$	−1.78875e6	−0.136509
$704$	− 3.36923e7i	− 2.56211i
$705$	3.07472e6	0.232988
$706$	1.65168e7	1.24714
$707$	− 1.44673e6i	− 0.108852i
$708$	3.42669e7i	2.56916i
$709$	7.47906e6i	0.558768i	0.960180	+	0.279384i	$0.0901302\pi$
$709$	7.47906e6i	0.558768i	−0.960180	+	0.279384i	$0.909870\pi$
$710$	− 2.89431e7i	− 2.15476i
$711$	−531357.	−0.0394196
$712$	4.29412e7	3.17449
$713$	− 9.94962e6i	− 0.732964i
$714$	−695361.	−0.0510464
$715$	0	0
$716$	−4.16992e7	−3.03980
$717$	− 1.16795e7i	− 0.848447i
$718$	1.06868e7	0.773634
$719$	−1.24974e7	−0.901564	−0.450782	−	0.892634i	$-0.648855\pi$
$719$	−1.24974e7	−0.901564	−0.450782	+	0.892634i	$0.648855\pi$
$720$	1.34520e6i	0.0967065i
$721$	− 5.94949e6i	− 0.426228i
$722$	− 2.63643e7i	− 1.88223i
$723$	− 1.06058e7i	− 0.754566i
$724$	−3.31267e7	−2.34872
$725$	−3.77671e6	−0.266851
$726$	1.65710e7i	1.16683i
$727$	1.77386e7	1.24475	0.622376	−	0.782719i	$-0.286168\pi$
$727$	1.77386e7	1.24475	0.622376	+	0.782719i	$0.286168\pi$
$728$	0	0
$729$	1.39379e7	0.971359
$730$	− 3.56608e7i	− 2.47676i
$731$	202689.	0.0140293
$732$	4.90273e7	3.38190
$733$	− 1.63154e7i	− 1.12160i	−0.827951	−	0.560800i	$-0.810494\pi$
$733$	− 1.63154e7i	− 1.12160i	0.827951	−	0.560800i	$-0.189506\pi$
$734$	1.78313e7i	1.22164i
$735$	− 9.06818e6i	− 0.619158i
$736$	4.89451e7i	3.33054i
$737$	3.86054e6	0.261806
$738$	−93066.3	−0.00629001
$739$	2.88582e7i	1.94383i	0.235330	+	0.971915i	$0.424383\pi$
$739$	2.88582e7i	1.94383i	−0.235330	+	0.971915i	$0.575617\pi$
$740$	−3.03101e7	−2.03473
$741$	0	0
$742$	−1.99157e7	−1.32796
$743$	5.35871e6i	0.356113i	0.984020	+	0.178057i	$0.0569810\pi$
$743$	5.35871e6i	0.356113i	−0.984020	+	0.178057i	$0.943019\pi$
$744$	4.66447e7	3.08937
$745$	4.56657e6	0.301439
$746$	− 1.53371e7i	− 1.00901i
$747$	− 747492.i	− 0.0490123i
$748$	1.20195e6i	0.0785476i
$749$	5.36251e6i	0.349272i
$750$	−3.20743e7	−2.08211
$751$	−3.13945e6	−0.203121	−0.101560	−	0.994829i	$-0.532383\pi$
$751$	−3.13945e6	−0.203121	−0.101560	+	0.994829i	$0.532383\pi$
$752$	1.48819e7i	0.959651i
$753$	−3.96860e6	−0.255065
$754$	0	0
$755$	−1.25750e7	−0.802864
$756$	2.48622e7i	1.58210i
$757$	2.58865e7	1.64185	0.820926	−	0.571034i	$-0.193458\pi$
$757$	2.58865e7	1.64185	0.820926	+	0.571034i	$0.193458\pi$
$758$	−3.55981e7	−2.25037
$759$	8.40772e6i	0.529754i
$760$	8.56736e6i	0.538038i
$761$	2.51354e7i	1.57335i	0.617370	+	0.786673i	$0.288198\pi$
$761$	2.51354e7i	1.57335i	−0.617370	+	0.786673i	$0.711802\pi$
$762$	5.62213e7i	3.50763i
$763$	762782.	0.0474340
$764$	4.90782e7	3.04197
$765$	− 18143.6i	− 0.00112090i
$766$	1.30643e7	0.804479
$767$	0	0
$768$	−5.33116e7	−3.26151
$769$	2.20737e7i	1.34604i	0.739624	+	0.673021i	$0.235004\pi$
$769$	2.20737e7i	1.34604i	−0.739624	+	0.673021i	$0.764996\pi$
$770$	1.09149e7	0.663428
$771$	3.95845e6	0.239822
$772$	− 7.30958e7i	− 4.41417i
$773$	2.88402e7i	1.73600i	0.496567	+	0.867998i	$0.334594\pi$
$773$	2.88402e7i	1.73600i	−0.496567	+	0.867998i	$0.665406\pi$
$774$	275899.i	0.0165538i
$775$	2.20860e6i	0.132088i
$776$	1.13025e7	0.673783
$777$	7.92404e6	0.470863
$778$	− 3.28983e7i	− 1.94861i
$779$	−345416.	−0.0203938
$780$	0	0
$781$	1.30792e7	0.767280
$782$	− 1.21459e6i	− 0.0710253i
$783$	−3.05399e7	−1.78018
$784$	4.38907e7	2.55025
$785$	2.63994e7i	1.52904i
$786$	− 2.52220e7i	− 1.45621i
$787$	786841.i	0.0452846i	0.999744	+	0.0226423i	$0.00720788\pi$
$787$	786841.i	0.0452846i	−0.999744	+	0.0226423i	$0.992792\pi$
$788$	5.39389e6i	0.309447i
$789$	−857149.	−0.0490190
$790$	−4.55290e7	−2.59550
$791$	− 9.91313e6i	− 0.563339i
$792$	−1.04312e6	−0.0590909
$793$	0	0
$794$	3.63374e7	2.04551
$795$	− 1.96358e7i	− 1.10187i
$796$	−4.02856e7	−2.25355
$797$	−2.04655e7	−1.14124	−0.570620	−	0.821214i	$-0.693297\pi$
$797$	−2.04655e7	−1.14124	−0.570620	+	0.821214i	$0.693297\pi$
$798$	− 3.51302e6i	− 0.195287i
$799$	− 200721.i	− 0.0111231i
$800$	− 1.08648e7i	− 0.600200i
$801$	− 459431.i	− 0.0253011i
$802$	1.11300e7	0.611024
$803$	1.61149e7	0.881938
$804$	− 2.10546e7i	− 1.14870i
$805$	−8.09559e6	−0.440311
$806$	0	0
$807$	6.92323e6	0.374218
$808$	1.18468e7i	0.638372i
$809$	369128.	0.0198292	0.00991460	−	0.999951i	$-0.496844\pi$
$809$	369128.	0.0198292	0.00991460	+	0.999951i	$0.496844\pi$
$810$	−3.43055e7	−1.83718
$811$	1.61942e7i	0.864586i	0.901733	+	0.432293i	$0.142295\pi$
$811$	1.61942e7i	0.864586i	−0.901733	+	0.432293i	$0.857705\pi$
$812$	− 5.44351e7i	− 2.89727i
$813$	1.19941e7i	0.636419i
$814$	− 1.86611e7i	− 0.987136i
$815$	2.90086e6	0.152979
$816$	3.31830e6	0.174458
$817$	1.02400e6i	0.0536716i
$818$	−3.56334e7	−1.86198
$819$	0	0
$820$	−5.85302e6	−0.303980
$821$	− 6.31911e6i	− 0.327189i	−0.986528	−	0.163594i	$-0.947691\pi$
$821$	− 6.31911e6i	− 0.327189i	0.986528	−	0.163594i	$-0.0523088\pi$
$822$	−4.39786e7	−2.27019
$823$	1.20246e7	0.618829	0.309414	−	0.950927i	$-0.399867\pi$
$823$	1.20246e7	0.618829	0.309414	+	0.950927i	$0.399867\pi$
$824$	4.87187e7i	2.49964i
$825$	− 1.86634e6i	− 0.0954673i
$826$	− 2.03140e7i	− 1.03597i
$827$	1.49387e7i	0.759538i	0.925081	+	0.379769i	$0.123997\pi$
$827$	1.49387e7i	0.759538i	−0.925081	+	0.379769i	$0.876003\pi$
$828$	1.21349e6	0.0615120
$829$	1.32004e7	0.667113	0.333557	−	0.942730i	$-0.391751\pi$
$829$	1.32004e7	0.667113	0.333557	+	0.942730i	$0.391751\pi$
$830$	− 6.40485e7i	− 3.22711i
$831$	−917360.	−0.0460826
$832$	0	0
$833$	−591981.	−0.0295594
$834$	2.39205e7i	1.19085i
$835$	−2.65850e6	−0.131954
$836$	−6.07235e6	−0.300498
$837$	1.78596e7i	0.881166i
$838$	− 1.59524e7i	− 0.784720i
$839$	− 1.71730e7i	− 0.842251i	−0.907002	−	0.421126i	$-0.861635\pi$
$839$	− 1.71730e7i	− 0.842251i	0.907002	−	0.421126i	$-0.138365\pi$
$840$	− 3.79529e7i	− 1.85586i
$841$	4.63552e7	2.26000
$842$	−7.01824e7	−3.41152
$843$	− 2.50056e6i	− 0.121190i
$844$	−5.43942e7	−2.62843
$845$	0	0
$846$	273221.	0.0131247
$847$	− 7.21034e6i	− 0.345340i
$848$	9.50388e7	4.53849
$849$	1.42568e7	0.678817
$850$	269613.i	0.0127995i
$851$	1.38410e7i	0.655153i
$852$	− 7.13312e7i	− 3.36651i
$853$	− 2.83579e7i	− 1.33445i	−0.744858	−	0.667223i	$-0.767483\pi$
$853$	− 2.83579e7i	− 1.33445i	0.744858	−	0.667223i	$-0.232517\pi$
$854$	−2.90643e7	−1.36369
$855$	91662.7	0.00428822
$856$	− 4.39120e7i	− 2.04833i
$857$	2.05630e7	0.956391	0.478195	−	0.878253i	$-0.341291\pi$
$857$	2.05630e7	0.956391	0.478195	+	0.878253i	$0.341291\pi$
$858$	0	0
$859$	−2.10953e7	−0.975444	−0.487722	−	0.872999i	$-0.662172\pi$
$859$	−2.10953e7	−0.975444	−0.487722	+	0.872999i	$0.662172\pi$
$860$	1.73515e7i	0.800002i
$861$	1.53017e6	0.0703448
$862$	−2.60390e7	−1.19359
$863$	6.35208e6i	0.290328i	0.989408	+	0.145164i	$0.0463710\pi$
$863$	6.35208e6i	0.290328i	−0.989408	+	0.145164i	$0.953629\pi$
$864$	− 8.78566e7i	− 4.00396i
$865$	2.84104e7i	1.29103i
$866$	1.21361e7i	0.549901i
$867$	2.23874e7	1.01148
$868$	−3.18334e7	−1.43411
$869$	− 2.05743e7i	− 0.924220i
$870$	7.31218e7	3.27528
$871$	0	0
$872$	−6.24620e6	−0.278179
$873$	− 120926.i	− 0.00537012i
$874$	6.13621e6	0.271720
$875$	1.39561e7	0.616231
$876$	− 8.78870e7i	− 3.86959i
$877$	− 1.73593e7i	− 0.762139i	−0.924546	−	0.381069i	$-0.875556\pi$
$877$	− 1.73593e7i	− 0.762139i	0.924546	−	0.381069i	$-0.124444\pi$
$878$	6.06789e6i	0.265645i
$879$	1.32732e7i	0.579435i
$880$	−5.20865e7	−2.26735
$881$	4.39444e7	1.90749	0.953747	−	0.300609i	$-0.0971899\pi$
$881$	4.39444e7	1.90749	0.953747	+	0.300609i	$0.0971899\pi$
$882$	− 805802.i	− 0.0348784i
$883$	1.96964e7	0.850131	0.425065	−	0.905163i	$-0.360251\pi$
$883$	1.96964e7	0.850131	0.425065	+	0.905163i	$0.360251\pi$
$884$	0	0
$885$	2.00285e7	0.859589
$886$	458656.i	0.0196292i
$887$	−3.45823e7	−1.47586	−0.737930	−	0.674878i	$-0.764196\pi$
$887$	−3.45823e7	−1.47586	−0.737930	+	0.674878i	$0.764196\pi$
$888$	−6.48877e7	−2.76140
$889$	− 2.44629e7i	− 1.03813i
$890$	− 3.93661e7i	− 1.66589i
$891$	− 1.55024e7i	− 0.654193i
$892$	− 4.05657e7i	− 1.70705i
$893$	1.01406e6	0.0425535
$894$	1.53334e7	0.641646
$895$	2.43726e7i	1.01706i
$896$	5.21744e7	2.17114
$897$	0	0
$898$	−1.84253e7	−0.762473
$899$	− 3.91031e7i	− 1.61366i
$900$	−269368.	−0.0110851
$901$	−1.28185e6	−0.0526047
$902$	− 3.60355e6i	− 0.147474i
$903$	− 4.53625e6i	− 0.185130i
$904$	8.11757e7i	3.30373i
$905$	1.93621e7i	0.785834i
$906$	−4.22239e7	−1.70898
$907$	−4.48409e7	−1.80991	−0.904953	−	0.425511i	$-0.860094\pi$
$907$	−4.48409e7	−1.80991	−0.904953	+	0.425511i	$0.860094\pi$
$908$	1.24933e8i	5.02878i
$909$	126750.	0.00508789
$910$	0	0
$911$	−6.56107e6	−0.261926	−0.130963	−	0.991387i	$-0.541807\pi$
$911$	−6.56107e6	−0.261926	−0.130963	+	0.991387i	$0.541807\pi$
$912$	1.67643e7i	0.667419i
$913$	2.89431e7	1.14913
$914$	−1.75264e7	−0.693947
$915$	− 2.86558e7i	− 1.13151i
$916$	1.25650e8i	4.94794i
$917$	1.09745e7i	0.430985i
$918$	2.18019e6i	0.0853863i
$919$	2.38079e7	0.929891	0.464946	−	0.885339i	$-0.346074\pi$
$919$	2.38079e7	0.929891	0.464946	+	0.885339i	$0.346074\pi$
$920$	6.62924e7	2.58223
$921$	− 1.26395e7i	− 0.491000i
$922$	6.45025e7	2.49890
$923$	0	0
$924$	2.69001e7	1.03651
$925$	− 3.07240e6i	− 0.118066i
$926$	8.53381e6	0.327051
$927$	521244.	0.0199224
$928$	1.92360e8i	7.33236i
$929$	− 1.06185e7i	− 0.403666i	−0.979420	−	0.201833i	$-0.935310\pi$
$929$	− 1.06185e7i	− 0.403666i	0.979420	−	0.201833i	$-0.0646898\pi$
$930$	− 4.27612e7i	− 1.62122i
$931$	− 2.99073e6i	− 0.113085i
$932$	3.02490e7	1.14070
$933$	4.00359e7	1.50573
$934$	− 7.05237e7i	− 2.64526i
$935$	702524.	0.0262804
$936$	0	0
$937$	−4.34917e7	−1.61829	−0.809146	−	0.587607i	$-0.800070\pi$
$937$	−4.34917e7	−1.61829	−0.809146	+	0.587607i	$0.800070\pi$
$938$	1.24815e7i	0.463192i
$939$	−1.02899e7	−0.380846
$940$	1.71831e7	0.634281
$941$	− 2.00195e7i	− 0.737021i	−0.929624	−	0.368510i	$-0.879868\pi$
$941$	− 2.00195e7i	− 0.737021i	0.929624	−	0.368510i	$-0.120132\pi$
$942$	8.86426e7i	3.25473i
$943$	2.67276e6i	0.0978769i
$944$	9.69396e7i	3.54055i
$945$	1.45316e7	0.529340
$946$	−1.06829e7	−0.388115
$947$	2.19250e7i	0.794447i	0.917722	+	0.397224i	$0.130026\pi$
$947$	2.19250e7i	0.794447i	−0.917722	+	0.397224i	$0.869974\pi$
$948$	−1.12208e8	−4.05510
$949$	0	0
$950$	−1.36211e6	−0.0489669
$951$	− 3.18659e7i	− 1.14255i
$952$	−2.47761e6	−0.0886013
$953$	1.91850e7	0.684274	0.342137	−	0.939650i	$-0.388849\pi$
$953$	1.91850e7	0.684274	0.342137	+	0.939650i	$0.388849\pi$
$954$	− 1.74484e6i	− 0.0620706i
$955$	− 2.86855e7i	− 1.01778i
$956$	− 6.52707e7i	− 2.30979i
$957$	3.30433e7i	1.16628i
$958$	3.09088e7	1.08810
$959$	1.91359e7	0.671895
$960$	1.07399e8i	3.76118i
$961$	5.76187e6	0.201259
$962$	0	0
$963$	−469817.	−0.0163254
$964$	− 5.92705e7i	− 2.05422i
$965$	−4.27235e7	−1.47689
$966$	−2.71830e7	−0.937249
$967$	− 1.40733e7i	− 0.483982i	−0.970279	−	0.241991i	$-0.922200\pi$
$967$	− 1.40733e7i	− 0.483982i	0.970279	−	0.241991i	$-0.0778004\pi$
$968$	5.90433e7i	2.02527i
$969$	− 226111.i	− 0.00773591i
$970$	− 1.03615e7i	− 0.353584i
$971$	−2.57562e6	−0.0876664	−0.0438332	−	0.999039i	$-0.513957\pi$
$971$	−2.57562e6	−0.0876664	−0.0438332	+	0.999039i	$0.513957\pi$
$972$	−4.41729e6	−0.149965
$973$	− 1.04082e7i	− 0.352448i
$974$	3.26955e7	1.10431
$975$	0	0
$976$	1.38696e8	4.66058
$977$	− 1.69297e7i	− 0.567430i	−0.958909	−	0.283715i	$-0.908433\pi$
$977$	− 1.69297e7i	− 0.567430i	0.958909	−	0.283715i	$-0.0915670\pi$
$978$	9.74037e6	0.325633
$979$	1.77893e7	0.593201
$980$	− 5.06776e7i	− 1.68558i
$981$	66828.5i	0.00221712i
$982$	9.23837e7i	3.05715i
$983$	− 1.00192e7i	− 0.330711i	−0.986234	−	0.165355i	$-0.947123\pi$
$983$	− 1.00192e7i	− 0.330711i	0.986234	−	0.165355i	$-0.0528771\pi$
$984$	−1.25301e7	−0.412541
$985$	3.15265e6	0.103535
$986$	− 4.77347e6i	− 0.156366i
$987$	−4.49222e6	−0.146780
$988$	0	0
$989$	7.92349e6	0.257588
$990$	956272.i	0.0310094i
$991$	1.80842e7	0.584944	0.292472	−	0.956274i	$-0.405522\pi$
$991$	1.80842e7	0.584944	0.292472	+	0.956274i	$0.405522\pi$
$992$	1.12491e8	3.62943
$993$	4.60666e7i	1.48256i
$994$	4.22864e7i	1.35748i
$995$	2.35464e7i	0.753991i
$996$	− 1.57849e8i	− 5.04190i
$997$	−4.95896e7	−1.57998	−0.789992	−	0.613117i	$-0.789915\pi$
$997$	−4.95896e7	−1.57998	−0.789992	+	0.613117i	$0.789915\pi$
$998$	−8.59594e7	−2.73191
$999$	− 2.48446e7i	− 0.787622i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.6.b.b.168.1		6
13.5	odd	4		169.6.a.b.1.1		3
13.8	odd	4		13.6.a.b.1.3	✓	3
13.12	even	2	inner	169.6.b.b.168.6		6
39.8	even	4		117.6.a.d.1.1		3
52.47	even	4		208.6.a.j.1.3		3
65.8	even	4		325.6.b.c.274.1		6
65.34	odd	4		325.6.a.c.1.1		3
65.47	even	4		325.6.b.c.274.6		6
91.34	even	4		637.6.a.b.1.3		3
104.21	odd	4		832.6.a.s.1.3		3
104.99	even	4		832.6.a.t.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.b.1.3	✓	3	13.8	odd	4
117.6.a.d.1.1		3	39.8	even	4
169.6.a.b.1.1		3	13.5	odd	4
169.6.b.b.168.1		6	1.1	even	1	trivial
169.6.b.b.168.6		6	13.12	even	2	inner
208.6.a.j.1.3		3	52.47	even	4
325.6.a.c.1.1		3	65.34	odd	4
325.6.b.c.274.1		6	65.8	even	4
325.6.b.c.274.6		6	65.47	even	4
637.6.a.b.1.3		3	91.34	even	4
832.6.a.s.1.3		3	104.21	odd	4
832.6.a.t.1.1		3	104.99	even	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)

\(f(q)\)	\(=\)	\(q-10.9678i q^{2} -15.7989 q^{3} -88.2923 q^{4} +51.6056i q^{5} +173.279i q^{6} -75.3967i q^{7} +617.402i q^{8} +6.60562 q^{9} +O(q^{10})\)	\(q-10.9678i q^{2} -15.7989 q^{3} -88.2923 q^{4} +51.6056i q^{5} +173.279i q^{6} -75.3967i q^{7} +617.402i q^{8} +6.60562 q^{9} +565.999 q^{10} +255.771i q^{11} +1394.92 q^{12} -826.935 q^{14} -815.313i q^{15} +3946.18 q^{16} -53.2245 q^{17} -72.4490i q^{18} -268.895i q^{19} -4556.38i q^{20} +1191.19i q^{21} +2805.25 q^{22} -2080.65 q^{23} -9754.28i q^{24} +461.860 q^{25} +3734.77 q^{27} +6656.95i q^{28} -8177.18 q^{29} -8942.17 q^{30} +4781.97i q^{31} -23523.9i q^{32} -4040.91i q^{33} +583.755i q^{34} +3890.90 q^{35} -583.225 q^{36} -6652.23i q^{37} -2949.18 q^{38} -31861.4 q^{40} -1284.58i q^{41} +13064.7 q^{42} -3808.18 q^{43} -22582.6i q^{44} +340.887i q^{45} +22820.1i q^{46} +3771.22i q^{47} -62345.3 q^{48} +11122.3 q^{49} -5065.58i q^{50} +840.890 q^{51} +24083.8 q^{53} -40962.2i q^{54} -13199.2 q^{55} +46550.1 q^{56} +4248.24i q^{57} +89685.6i q^{58} +24565.5i q^{59} +71985.8i q^{60} +35147.0 q^{61} +52447.7 q^{62} -498.042i q^{63} -131728. q^{64} -44319.8 q^{66} -15093.7i q^{67} +4699.32 q^{68} +32872.0 q^{69} -42674.5i q^{70} -51136.3i q^{71} +4078.32i q^{72} -63005.0i q^{73} -72960.2 q^{74} -7296.89 q^{75} +23741.3i q^{76} +19284.3 q^{77} -80440.1 q^{79} +203645. i q^{80} -60610.5 q^{81} -14089.0 q^{82} -113160. i q^{83} -105173. i q^{84} -2746.69i q^{85} +41767.3i q^{86} +129191. q^{87} -157914. q^{88} -69551.5i q^{89} +3738.78 q^{90} +183705. q^{92} -75550.0i q^{93} +41361.9 q^{94} +13876.5 q^{95} +371653. i q^{96} -18306.5i q^{97} -121987. i q^{98} +1689.53i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9}+O(q^{10}) \) 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9	\( 6 q + 16 q^{3} - 242 q^{4} - 382 q^{9} + 2582 q^{10} + 2182 q^{12} - 1586 q^{14} + 5570 q^{16} - 1816 q^{17} - 6820 q^{22} - 7248 q^{23} - 4046 q^{25} - 8552 q^{27} - 17516 q^{29} - 3434 q^{30} + 8696 q^{35} + 9356 q^{36} - 67572 q^{38} - 87018 q^{40} + 27206 q^{42} - 4064 q^{43} - 152446 q^{48} + 89142 q^{49} - 26936 q^{51} - 25140 q^{53} + 70624 q^{55} + 98574 q^{56} - 25508 q^{61} + 80680 q^{62} - 234786 q^{64} - 224708 q^{66} - 98350 q^{68} + 40224 q^{69} - 263038 q^{74} - 47096 q^{75} + 42104 q^{77} - 123744 q^{79} - 216442 q^{81} - 284520 q^{82} + 278144 q^{87} - 316020 q^{88} - 202516 q^{90} + 601728 q^{92} + 145686 q^{94} + 393120 q^{95}+O(q^{100}) \) 6 * q + 16 * q^3 - 242 * q^4 - 382 * q^9 + 2582 * q^10 + 2182 * q^12 - 1586 * q^14 + 5570 * q^16 - 1816 * q^17 - 6820 * q^22 - 7248 * q^23 - 4046 * q^25 - 8552 * q^27 - 17516 * q^29 - 3434 * q^30 + 8696 * q^35 + 9356 * q^36 - 67572 * q^38 - 87018 * q^40 + 27206 * q^42 - 4064 * q^43 - 152446 * q^48 + 89142 * q^49 - 26936 * q^51 - 25140 * q^53 + 70624 * q^55 + 98574 * q^56 - 25508 * q^61 + 80680 * q^62 - 234786 * q^64 - 224708 * q^66 - 98350 * q^68 + 40224 * q^69 - 263038 * q^74 - 47096 * q^75 + 42104 * q^77 - 123744 * q^79 - 216442 * q^81 - 284520 * q^82 + 278144 * q^87 - 316020 * q^88 - 202516 * q^90 + 601728 * q^92 + 145686 * q^94 + 393120 * q^95

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