LMFDB - Embedded newform 169.2.b.b.168.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,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, 2, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$1.34947179416$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.3
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.2.b.b.168.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.554958i q^{2} +0.801938 q^{3} +1.69202 q^{4} -2.80194i q^{5} -0.445042i q^{6} +2.69202i q^{7} -2.04892i q^{8} -2.35690 q^{9} +O(q^{10})$	\(q-0.554958i q^{2} +0.801938 q^{3} +1.69202 q^{4} -2.80194i q^{5} -0.445042i q^{6} +2.69202i q^{7} -2.04892i q^{8} -2.35690 q^{9} -1.55496 q^{10} +1.19806i q^{11} +1.35690 q^{12} +1.49396 q^{14} -2.24698i q^{15} +2.24698 q^{16} -1.13706 q^{17} +1.30798i q^{18} +1.93900i q^{19} -4.74094i q^{20} +2.15883i q^{21} +0.664874 q^{22} +4.60388 q^{23} -1.64310i q^{24} -2.85086 q^{25} -4.29590 q^{27} +4.55496i q^{28} -7.89977 q^{29} -1.24698 q^{30} +5.89977i q^{31} -5.34481i q^{32} +0.960771i q^{33} +0.631023i q^{34} +7.54288 q^{35} -3.98792 q^{36} -0.951083i q^{37} +1.07606 q^{38} -5.74094 q^{40} +3.31767i q^{41} +1.19806 q^{42} -7.15883 q^{43} +2.02715i q^{44} +6.60388i q^{45} -2.55496i q^{46} +7.69202i q^{47} +1.80194 q^{48} -0.246980 q^{49} +1.58211i q^{50} -0.911854 q^{51} +5.87263 q^{53} +2.38404i q^{54} +3.35690 q^{55} +5.51573 q^{56} +1.55496i q^{57} +4.38404i q^{58} +0.0120816i q^{59} -3.80194i q^{60} -8.03684 q^{61} +3.27413 q^{62} -6.34481i q^{63} +1.52781 q^{64} +0.533188 q^{66} -9.25667i q^{67} -1.92394 q^{68} +3.69202 q^{69} -4.18598i q^{70} -13.7409i q^{71} +4.82908i q^{72} -12.8170i q^{73} -0.527811 q^{74} -2.28621 q^{75} +3.28083i q^{76} -3.22521 q^{77} +0.807315 q^{79} -6.29590i q^{80} +3.62565 q^{81} +1.84117 q^{82} -16.3327i q^{83} +3.65279i q^{84} +3.18598i q^{85} +3.97285i q^{86} -6.33513 q^{87} +2.45473 q^{88} +14.7289i q^{89} +3.66487 q^{90} +7.78986 q^{92} +4.73125i q^{93} +4.26875 q^{94} +5.43296 q^{95} -4.28621i q^{96} +3.13169i q^{97} +0.137063i q^{98} -2.82371i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 6 q^{9}+O(q^{10}) $ 6 * q - 4 * q^3 - 6 * q^9	$ 6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100}) $ 6 * q - 4 * q^3 - 6 * q^9 - 10 * q^10 - 10 * q^14 + 4 * q^16 + 4 * q^17 + 6 * q^22 + 10 * q^23 + 10 * q^25 + 2 * q^27 - 2 * q^29 + 2 * q^30 + 8 * q^35 + 14 * q^36 - 24 * q^38 - 6 * q^40 + 16 * q^42 - 26 * q^43 + 2 * q^48 + 8 * q^49 + 2 * q^51 + 2 * q^53 + 12 * q^55 + 8 * q^56 + 8 * q^61 - 2 * q^62 + 22 * q^64 + 10 * q^66 - 42 * q^68 + 12 * q^69 - 16 * q^74 - 30 * q^75 - 16 * q^77 - 10 * q^79 - 2 * q^81 + 28 * q^82 - 36 * q^87 - 30 * q^88 + 24 * q^90 + 10 * q^94 - 6 * 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 $

$2$	− 0.554958i	− 0.392415i	−0.980562	−	0.196207i	$-0.937137\pi$
$2$	− 0.554958i	− 0.392415i	0.980562	−	0.196207i	$-0.0628626\pi$
$3$	0.801938	0.462999	0.231499	−	0.972835i	$-0.425637\pi$
$3$	0.801938	0.462999	0.231499	+	0.972835i	$0.425637\pi$
$4$	1.69202	0.846011
$5$	− 2.80194i	− 1.25306i	−0.779395	−	0.626532i	$-0.784474\pi$
$5$	− 2.80194i	− 1.25306i	0.779395	−	0.626532i	$-0.215526\pi$
$6$	− 0.445042i	− 0.181688i
$7$	2.69202i	1.01749i	0.860918	+	0.508744i	$0.169890\pi$
$7$	2.69202i	1.01749i	−0.860918	+	0.508744i	$0.830110\pi$
$8$	− 2.04892i	− 0.724402i
$9$	−2.35690	−0.785632
$10$	−1.55496	−0.491721
$11$	1.19806i	0.361229i	0.983554	+	0.180615i	$0.0578087\pi$
$11$	1.19806i	0.361229i	−0.983554	+	0.180615i	$0.942191\pi$
$12$	1.35690	0.391702
$13$	0	0
$13$	0	0
$14$	1.49396	0.399277
$15$	− 2.24698i	− 0.580168i
$16$	2.24698	0.561745
$17$	−1.13706	−0.275778	−0.137889	−	0.990448i	$-0.544032\pi$
$17$	−1.13706	−0.275778	−0.137889	+	0.990448i	$0.544032\pi$
$18$	1.30798i	0.308293i
$19$	1.93900i	0.444837i	0.974951	+	0.222419i	$0.0713952\pi$
$19$	1.93900i	0.444837i	−0.974951	+	0.222419i	$0.928605\pi$
$20$	− 4.74094i	− 1.06011i
$21$	2.15883i	0.471096i
$22$	0.664874	0.141752
$23$	4.60388	0.959974	0.479987	−	0.877275i	$-0.340641\pi$
$23$	4.60388	0.959974	0.479987	+	0.877275i	$0.340641\pi$
$24$	− 1.64310i	− 0.335397i
$25$	−2.85086	−0.570171
$26$	0	0
$27$	−4.29590	−0.826746
$28$	4.55496i	0.860806i
$29$	−7.89977	−1.46695	−0.733475	−	0.679716i	$-0.762103\pi$
$29$	−7.89977	−1.46695	−0.733475	+	0.679716i	$0.762103\pi$
$30$	−1.24698	−0.227666
$31$	5.89977i	1.05963i	0.848113	+	0.529815i	$0.177739\pi$
$31$	5.89977i	1.05963i	−0.848113	+	0.529815i	$0.822261\pi$
$32$	− 5.34481i	− 0.944839i
$33$	0.960771i	0.167249i
$34$	0.631023i	0.108219i
$35$	7.54288	1.27498
$36$	−3.98792	−0.664653
$37$	− 0.951083i	− 0.156357i	−0.996939	−	0.0781785i	$-0.975090\pi$
$37$	− 0.951083i	− 0.156357i	0.996939	−	0.0781785i	$-0.0249104\pi$
$38$	1.07606	0.174561
$39$	0	0
$40$	−5.74094	−0.907722
$41$	3.31767i	0.518133i	0.965860	+	0.259066i	$0.0834148\pi$
$41$	3.31767i	0.518133i	−0.965860	+	0.259066i	$0.916585\pi$
$42$	1.19806	0.184865
$43$	−7.15883	−1.09171	−0.545856	−	0.837879i	$-0.683795\pi$
$43$	−7.15883	−1.09171	−0.545856	+	0.837879i	$0.683795\pi$
$44$	2.02715i	0.305604i
$45$	6.60388i	0.984448i
$46$	− 2.55496i	− 0.376708i
$47$	7.69202i	1.12200i	0.827817	+	0.560998i	$0.189583\pi$
$47$	7.69202i	1.12200i	−0.827817	+	0.560998i	$0.810417\pi$
$48$	1.80194	0.260087
$49$	−0.246980	−0.0352828
$50$	1.58211i	0.223743i
$51$	−0.911854	−0.127685
$52$	0	0
$53$	5.87263	0.806667	0.403334	−	0.915053i	$-0.367851\pi$
$53$	5.87263	0.806667	0.403334	+	0.915053i	$0.367851\pi$
$54$	2.38404i	0.324427i
$55$	3.35690	0.452644
$56$	5.51573	0.737070
$57$	1.55496i	0.205959i
$58$	4.38404i	0.575653i
$59$	0.0120816i	0.00157289i	1.00000		0.000786444i	$0.000250333\pi$
$59$	0.0120816i	0.00157289i	−1.00000		0.000786444i	$0.999750\pi$
$60$	− 3.80194i	− 0.490828i
$61$	−8.03684	−1.02901	−0.514506	−	0.857487i	$-0.672025\pi$
$61$	−8.03684	−1.02901	−0.514506	+	0.857487i	$0.672025\pi$
$62$	3.27413	0.415815
$63$	− 6.34481i	− 0.799371i
$64$	1.52781	0.190976
$65$	0	0
$66$	0.533188	0.0656309
$67$	− 9.25667i	− 1.13088i	−0.824789	−	0.565441i	$-0.808706\pi$
$67$	− 9.25667i	− 1.13088i	0.824789	−	0.565441i	$-0.191294\pi$
$68$	−1.92394	−0.233311
$69$	3.69202	0.444467
$70$	− 4.18598i	− 0.500320i
$71$	− 13.7409i	− 1.63075i	−0.578934	−	0.815375i	$-0.696531\pi$
$71$	− 13.7409i	− 1.63075i	0.578934	−	0.815375i	$-0.303469\pi$
$72$	4.82908i	0.569113i
$73$	− 12.8170i	− 1.50012i	−0.661372	−	0.750058i	$-0.730025\pi$
$73$	− 12.8170i	− 1.50012i	0.661372	−	0.750058i	$-0.269975\pi$
$74$	−0.527811	−0.0613568
$75$	−2.28621	−0.263989
$76$	3.28083i	0.376337i
$77$	−3.22521	−0.367547
$78$	0	0
$79$	0.807315	0.0908300	0.0454150	−	0.998968i	$-0.485539\pi$
$79$	0.807315	0.0908300	0.0454150	+	0.998968i	$0.485539\pi$
$80$	− 6.29590i	− 0.703903i
$81$	3.62565	0.402850
$82$	1.84117	0.203323
$83$	− 16.3327i	− 1.79275i	−0.443296	−	0.896375i	$-0.646191\pi$
$83$	− 16.3327i	− 1.79275i	0.443296	−	0.896375i	$-0.353809\pi$
$84$	3.65279i	0.398552i
$85$	3.18598i	0.345568i
$86$	3.97285i	0.428404i
$87$	−6.33513	−0.679197
$88$	2.45473	0.261675
$89$	14.7289i	1.56126i	0.624996	+	0.780628i	$0.285101\pi$
$89$	14.7289i	1.56126i	−0.624996	+	0.780628i	$0.714899\pi$
$90$	3.66487	0.386312
$91$	0	0
$92$	7.78986	0.812149
$93$	4.73125i	0.490608i
$94$	4.26875	0.440288
$95$	5.43296	0.557410
$96$	− 4.28621i	− 0.437459i
$97$	3.13169i	0.317975i	0.987281	+	0.158987i	$0.0508229\pi$
$97$	3.13169i	0.317975i	−0.987281	+	0.158987i	$0.949177\pi$
$98$	0.137063i	0.0138455i
$99$	− 2.82371i	− 0.283793i
$100$	−4.82371	−0.482371
$101$	−5.29052	−0.526426	−0.263213	−	0.964738i	$-0.584782\pi$
$101$	−5.29052	−0.526426	−0.263213	+	0.964738i	$0.584782\pi$
$102$	0.506041i	0.0501055i
$103$	13.5308	1.33323	0.666614	−	0.745403i	$-0.267743\pi$
$103$	13.5308	1.33323	0.666614	+	0.745403i	$0.267743\pi$
$104$	0	0
$105$	6.04892	0.590314
$106$	− 3.25906i	− 0.316548i
$107$	5.63102	0.544371	0.272186	−	0.962245i	$-0.412253\pi$
$107$	5.63102	0.544371	0.272186	+	0.962245i	$0.412253\pi$
$108$	−7.26875	−0.699436
$109$	4.17629i	0.400016i	0.979794	+	0.200008i	$0.0640969\pi$
$109$	4.17629i	0.400016i	−0.979794	+	0.200008i	$0.935903\pi$
$110$	− 1.86294i	− 0.177624i
$111$	− 0.762709i	− 0.0723931i
$112$	6.04892i	0.571569i
$113$	7.64310	0.719003	0.359501	−	0.933145i	$-0.382947\pi$
$113$	7.64310	0.719003	0.359501	+	0.933145i	$0.382947\pi$
$114$	0.862937	0.0808214
$115$	− 12.8998i	− 1.20291i
$116$	−13.3666	−1.24106
$117$	0	0
$118$	0.00670477	0.000617224 0
$119$	− 3.06100i	− 0.280601i
$120$	−4.60388	−0.420274
$121$	9.56465	0.869513
$122$	4.46011i	0.403799i
$123$	2.66056i	0.239895i
$124$	9.98254i	0.896459i
$125$	− 6.02177i	− 0.538604i
$126$	−3.52111	−0.313685
$127$	−6.77777	−0.601430	−0.300715	−	0.953714i	$-0.597225\pi$
$127$	−6.77777	−0.601430	−0.300715	+	0.953714i	$0.597225\pi$
$128$	− 11.5375i	− 1.01978i
$129$	−5.74094	−0.505461
$130$	0	0
$131$	−13.6799	−1.19522	−0.597611	−	0.801786i	$-0.703883\pi$
$131$	−13.6799	−1.19522	−0.597611	+	0.801786i	$0.703883\pi$
$132$	1.62565i	0.141494i
$133$	−5.21983	−0.452617
$134$	−5.13706	−0.443775
$135$	12.0368i	1.03597i
$136$	2.32975i	0.199774i
$137$	− 12.9879i	− 1.10963i	−0.831973	−	0.554816i	$-0.812789\pi$
$137$	− 12.9879i	− 1.10963i	0.831973	−	0.554816i	$-0.187211\pi$
$138$	− 2.04892i	− 0.174415i
$139$	12.0465	1.02177	0.510886	−	0.859648i	$-0.329317\pi$
$139$	12.0465	1.02177	0.510886	+	0.859648i	$0.329317\pi$
$140$	12.7627	1.07865
$141$	6.16852i	0.519483i
$142$	−7.62565	−0.639930
$143$	0	0
$144$	−5.29590	−0.441325
$145$	22.1347i	1.83818i
$146$	−7.11290	−0.588668
$147$	−0.198062	−0.0163359
$148$	− 1.60925i	− 0.132280i
$149$	− 0.740939i	− 0.0607001i	−0.999539	−	0.0303500i	$-0.990338\pi$
$149$	− 0.740939i	− 0.0607001i	0.999539	−	0.0303500i	$-0.00966220\pi$
$150$	1.26875i	0.103593i
$151$	19.0737i	1.55219i	0.630614	+	0.776097i	$0.282803\pi$
$151$	19.0737i	1.55219i	−0.630614	+	0.776097i	$0.717197\pi$
$152$	3.97285	0.322241
$153$	2.67994	0.216660
$154$	1.78986i	0.144231i
$155$	16.5308	1.32779
$156$	0	0
$157$	−4.02177	−0.320972	−0.160486	−	0.987038i	$-0.551306\pi$
$157$	−4.02177	−0.320972	−0.160486	+	0.987038i	$0.551306\pi$
$158$	− 0.448026i	− 0.0356430i
$159$	4.70948	0.373486
$160$	−14.9758	−1.18394
$161$	12.3937i	0.976763i
$162$	− 2.01208i	− 0.158084i
$163$	− 15.1371i	− 1.18563i	−0.805340	−	0.592813i	$-0.798017\pi$
$163$	− 15.1371i	− 1.18563i	0.805340	−	0.592813i	$-0.201983\pi$
$164$	5.61356i	0.438346i
$165$	2.69202	0.209574
$166$	−9.06398	−0.703502
$167$	6.26337i	0.484674i	0.970192	+	0.242337i	$0.0779140\pi$
$167$	6.26337i	0.484674i	−0.970192	+	0.242337i	$0.922086\pi$
$168$	4.42327	0.341263
$169$	0	0
$170$	1.76809	0.135606
$171$	− 4.57002i	− 0.349478i
$172$	−12.1129	−0.923600
$173$	−16.3913	−1.24621	−0.623105	−	0.782138i	$-0.714129\pi$
$173$	−16.3913	−1.24621	−0.623105	+	0.782138i	$0.714129\pi$
$174$	3.51573i	0.266527i
$175$	− 7.67456i	− 0.580142i
$176$	2.69202i	0.202919i
$177$	0.00968868i	0 0.000728246i
$178$	8.17390	0.612660
$179$	2.45473	0.183475	0.0917376	−	0.995783i	$-0.470758\pi$
$179$	2.45473	0.183475	0.0917376	+	0.995783i	$0.470758\pi$
$180$	11.1739i	0.832853i
$181$	−11.8073	−0.877631	−0.438815	−	0.898577i	$-0.644602\pi$
$181$	−11.8073	−0.877631	−0.438815	+	0.898577i	$0.644602\pi$
$182$	0	0
$183$	−6.44504	−0.476431
$184$	− 9.43296i	− 0.695407i
$185$	−2.66487	−0.195925
$186$	2.62565	0.192522
$187$	− 1.36227i	− 0.0996192i
$188$	13.0151i	0.949221i
$189$	− 11.5646i	− 0.841204i
$190$	− 3.01507i	− 0.218736i
$191$	−8.99330	−0.650732	−0.325366	−	0.945588i	$-0.605488\pi$
$191$	−8.99330	−0.650732	−0.325366	+	0.945588i	$0.605488\pi$
$192$	1.22521	0.0884219
$193$	− 13.5254i	− 0.973581i	−0.873519	−	0.486790i	$-0.838168\pi$
$193$	− 13.5254i	− 0.973581i	0.873519	−	0.486790i	$-0.161832\pi$
$194$	1.73795	0.124778
$195$	0	0
$196$	−0.417895	−0.0298496
$197$	12.9758i	0.924490i	0.886752	+	0.462245i	$0.152956\pi$
$197$	12.9758i	0.924490i	−0.886752	+	0.462245i	$0.847044\pi$
$198$	−1.56704	−0.111365
$199$	13.5864	0.963116	0.481558	−	0.876414i	$-0.340071\pi$
$199$	13.5864	0.963116	0.481558	+	0.876414i	$0.340071\pi$
$200$	5.84117i	0.413033i
$201$	− 7.42327i	− 0.523597i
$202$	2.93602i	0.206577i
$203$	− 21.2664i	− 1.49261i
$204$	−1.54288	−0.108023
$205$	9.29590	0.649254
$206$	− 7.50902i	− 0.523179i
$207$	−10.8509	−0.754187
$208$	0	0
$209$	−2.32304	−0.160688
$210$	− 3.35690i	− 0.231648i
$211$	10.4601	0.720103	0.360052	−	0.932932i	$-0.382759\pi$
$211$	10.4601	0.720103	0.360052	+	0.932932i	$0.382759\pi$
$212$	9.93661	0.682449
$213$	− 11.0194i	− 0.755035i
$214$	− 3.12498i	− 0.213619i
$215$	20.0586i	1.36799i
$216$	8.80194i	0.598896i
$217$	−15.8823	−1.07816
$218$	2.31767	0.156972
$219$	− 10.2784i	− 0.694553i
$220$	5.67994	0.382941
$221$	0	0
$222$	−0.423272	−0.0284081
$223$	− 11.4058i	− 0.763790i	−0.924206	−	0.381895i	$-0.875272\pi$
$223$	− 11.4058i	− 0.763790i	0.924206	−	0.381895i	$-0.124728\pi$
$224$	14.3884	0.961362
$225$	6.71917	0.447945
$226$	− 4.24160i	− 0.282147i
$227$	10.6407i	0.706249i	0.935576	+	0.353124i	$0.114881\pi$
$227$	10.6407i	0.706249i	−0.935576	+	0.353124i	$0.885119\pi$
$228$	2.63102i	0.174244i
$229$	1.13946i	0.0752974i	0.999291	+	0.0376487i	$0.0119868\pi$
$229$	1.13946i	0.0752974i	−0.999291	+	0.0376487i	$0.988013\pi$
$230$	−7.15883	−0.472040
$231$	−2.58642	−0.170174
$232$	16.1860i	1.06266i
$233$	10.8509	0.710863	0.355432	−	0.934702i	$-0.384334\pi$
$233$	10.8509	0.710863	0.355432	+	0.934702i	$0.384334\pi$
$234$	0	0
$235$	21.5526	1.40593
$236$	0.0204423i	0.00133068i
$237$	0.647416	0.0420542
$238$	−1.69873	−0.110112
$239$	− 11.9293i	− 0.771643i	−0.922573	−	0.385822i	$-0.873918\pi$
$239$	− 11.9293i	− 0.771643i	0.922573	−	0.385822i	$-0.126082\pi$
$240$	− 5.04892i	− 0.325906i
$241$	3.64848i	0.235019i	0.993072	+	0.117510i	$0.0374911\pi$
$241$	3.64848i	0.235019i	−0.993072	+	0.117510i	$0.962509\pi$
$242$	− 5.30798i	− 0.341210i
$243$	15.7952	1.01326
$244$	−13.5985	−0.870555
$245$	0.692021i	0.0442116i
$246$	1.47650	0.0941383
$247$	0	0
$248$	12.0881	0.767598
$249$	− 13.0978i	− 0.830042i
$250$	−3.34183	−0.211356
$251$	−1.37329	−0.0866813	−0.0433406	−	0.999060i	$-0.513800\pi$
$251$	−1.37329	−0.0866813	−0.0433406	+	0.999060i	$0.513800\pi$
$252$	− 10.7356i	− 0.676277i
$253$	5.51573i	0.346771i
$254$	3.76138i	0.236010i
$255$	2.55496i	0.159998i
$256$	−3.34721	−0.209200
$257$	−29.4359	−1.83616	−0.918082	−	0.396391i	$-0.870263\pi$
$257$	−29.4359	−1.83616	−0.918082	+	0.396391i	$0.870263\pi$
$258$	3.18598i	0.198350i
$259$	2.56033	0.159091
$260$	0	0
$261$	18.6189	1.15248
$262$	7.59179i	0.469023i
$263$	10.6963	0.659564	0.329782	−	0.944057i	$-0.393025\pi$
$263$	10.6963	0.659564	0.329782	+	0.944057i	$0.393025\pi$
$264$	1.96854	0.121155
$265$	− 16.4547i	− 1.01081i
$266$	2.89679i	0.177613i
$267$	11.8116i	0.722860i
$268$	− 15.6625i	− 0.956738i
$269$	−10.1860	−0.621050	−0.310525	−	0.950565i	$-0.600505\pi$
$269$	−10.1860	−0.621050	−0.310525	+	0.950565i	$0.600505\pi$
$270$	6.67994	0.406528
$271$	29.4523i	1.78910i	0.446966	+	0.894551i	$0.352505\pi$
$271$	29.4523i	1.78910i	−0.446966	+	0.894551i	$0.647495\pi$
$272$	−2.55496	−0.154917
$273$	0	0
$274$	−7.20775	−0.435436
$275$	− 3.41550i	− 0.205963i
$276$	6.24698	0.376024
$277$	10.2446	0.615538	0.307769	−	0.951461i	$-0.400418\pi$
$277$	10.2446	0.615538	0.307769	+	0.951461i	$0.400418\pi$
$278$	− 6.68532i	− 0.400959i
$279$	− 13.9051i	− 0.832480i
$280$	− 15.4547i	− 0.923597i
$281$	11.5646i	0.689889i	0.938623	+	0.344944i	$0.112102\pi$
$281$	11.5646i	0.689889i	−0.938623	+	0.344944i	$0.887898\pi$
$282$	3.42327	0.203853
$283$	30.7090	1.82546	0.912730	−	0.408562i	$-0.133970\pi$
$283$	30.7090	1.82546	0.912730	+	0.408562i	$0.133970\pi$
$284$	− 23.2500i	− 1.37963i
$285$	4.35690	0.258080
$286$	0	0
$287$	−8.93123	−0.527194
$288$	12.5972i	0.742295i
$289$	−15.7071	−0.923946
$290$	12.2838	0.721330
$291$	2.51142i	0.147222i
$292$	− 21.6866i	− 1.26911i
$293$	18.6082i	1.08710i	0.839376	+	0.543551i	$0.182921\pi$
$293$	18.6082i	1.08710i	−0.839376	+	0.543551i	$0.817079\pi$
$294$	0.109916i	0.00641045i
$295$	0.0338518	0.00197093
$296$	−1.94869	−0.113265
$297$	− 5.14675i	− 0.298645i
$298$	−0.411190	−0.0238196
$299$	0	0
$300$	−3.86831	−0.223337
$301$	− 19.2717i	− 1.11080i
$302$	10.5851	0.609103
$303$	−4.24267	−0.243735
$304$	4.35690i	0.249885i
$305$	22.5187i	1.28942i
$306$	− 1.48725i	− 0.0850207i
$307$	− 8.94438i	− 0.510483i	−0.966877	−	0.255241i	$-0.917845\pi$
$307$	− 8.94438i	− 0.510483i	0.966877	−	0.255241i	$-0.0821549\pi$
$308$	−5.45712	−0.310948
$309$	10.8509	0.617284
$310$	− 9.17390i	− 0.521042i
$311$	−21.0398	−1.19306	−0.596529	−	0.802591i	$-0.703454\pi$
$311$	−21.0398	−1.19306	−0.596529	+	0.802591i	$0.703454\pi$
$312$	0	0
$313$	−7.12737	−0.402863	−0.201432	−	0.979503i	$-0.564559\pi$
$313$	−7.12737	−0.402863	−0.201432	+	0.979503i	$0.564559\pi$
$314$	2.23191i	0.125954i
$315$	−17.7778	−1.00166
$316$	1.36599	0.0768431
$317$	23.9651i	1.34601i	0.739636	+	0.673007i	$0.234997\pi$
$317$	23.9651i	1.34601i	−0.739636	+	0.673007i	$0.765003\pi$
$318$	− 2.61356i	− 0.146561i
$319$	− 9.46442i	− 0.529906i
$320$	− 4.28083i	− 0.239306i
$321$	4.51573	0.252043
$322$	6.87800	0.383296
$323$	− 2.20477i	− 0.122677i
$324$	6.13467	0.340815
$325$	0	0
$326$	−8.40044	−0.465257
$327$	3.34913i	0.185207i
$328$	6.79763	0.375336
$329$	−20.7071	−1.14162
$330$	− 1.49396i	− 0.0822397i
$331$	2.89546i	0.159149i	0.996829	+	0.0795745i	$0.0253561\pi$
$331$	2.89546i	0.159149i	−0.996829	+	0.0795745i	$0.974644\pi$
$332$	− 27.6353i	− 1.51669i
$333$	2.24160i	0.122839i
$334$	3.47591	0.190193
$335$	−25.9366	−1.41707
$336$	4.85086i	0.264636i
$337$	3.10560	0.169173	0.0845865	−	0.996416i	$-0.473043\pi$
$337$	3.10560	0.169173	0.0845865	+	0.996416i	$0.473043\pi$
$338$	0	0
$339$	6.12929	0.332898
$340$	5.39075i	0.292354i
$341$	−7.06829	−0.382770
$342$	−2.53617	−0.137140
$343$	18.1793i	0.981589i
$344$	14.6679i	0.790838i
$345$	− 10.3448i	− 0.556946i
$346$	9.09651i	0.489031i
$347$	−11.3787	−0.610839	−0.305419	−	0.952218i	$-0.598797\pi$
$347$	−11.3787	−0.610839	−0.305419	+	0.952218i	$0.598797\pi$
$348$	−10.7192	−0.574608
$349$	3.34721i	0.179172i	0.995979	+	0.0895859i	$0.0285544\pi$
$349$	3.34721i	0.179172i	−0.995979	+	0.0895859i	$0.971446\pi$
$350$	−4.25906	−0.227656
$351$	0	0
$352$	6.40342	0.341303
$353$	0.637727i	0.0339428i	0.999856	+	0.0169714i	$0.00540242\pi$
$353$	0.637727i	0.0339428i	−0.999856	+	0.0169714i	$0.994598\pi$
$354$	0.00537681	0.000285774 0
$355$	−38.5013	−2.04343
$356$	24.9215i	1.32084i
$357$	− 2.45473i	− 0.129918i
$358$	− 1.36227i	− 0.0719983i
$359$	21.4590i	1.13256i	0.824211	+	0.566282i	$0.191619\pi$
$359$	21.4590i	1.13256i	−0.824211	+	0.566282i	$0.808381\pi$
$360$	13.5308	0.713136
$361$	15.2403	0.802120
$362$	6.55257i	0.344395i
$363$	7.67025	0.402584
$364$	0	0
$365$	−35.9124	−1.87974
$366$	3.57673i	0.186959i
$367$	−9.38703	−0.489999	−0.244999	−	0.969523i	$-0.578788\pi$
$367$	−9.38703	−0.489999	−0.244999	+	0.969523i	$0.578788\pi$
$368$	10.3448	0.539261
$369$	− 7.81940i	− 0.407062i
$370$	1.47889i	0.0768840i
$371$	15.8092i	0.820775i
$372$	8.00538i	0.415059i
$373$	27.7265	1.43562	0.717811	−	0.696238i	$-0.245144\pi$
$373$	27.7265	1.43562	0.717811	+	0.696238i	$0.245144\pi$
$374$	−0.756004	−0.0390921
$375$	− 4.82908i	− 0.249373i
$376$	15.7603	0.812776
$377$	0	0
$378$	−6.41789	−0.330101
$379$	35.8702i	1.84253i	0.388935	+	0.921265i	$0.372843\pi$
$379$	35.8702i	1.84253i	−0.388935	+	0.921265i	$0.627157\pi$
$380$	9.19269	0.471575
$381$	−5.43535	−0.278462
$382$	4.99090i	0.255357i
$383$	4.85517i	0.248087i	0.992277	+	0.124044i	$0.0395863\pi$
$383$	4.85517i	0.248087i	−0.992277	+	0.124044i	$0.960414\pi$
$384$	− 9.25236i	− 0.472157i
$385$	9.03684i	0.460560i
$386$	−7.50604	−0.382047
$387$	16.8726	0.857684
$388$	5.29888i	0.269010i
$389$	−2.38537	−0.120943	−0.0604716	−	0.998170i	$-0.519260\pi$
$389$	−2.38537	−0.120943	−0.0604716	+	0.998170i	$0.519260\pi$
$390$	0	0
$391$	−5.23490	−0.264740
$392$	0.506041i	0.0255589i
$393$	−10.9705	−0.553387
$394$	7.20105	0.362783
$395$	− 2.26205i	− 0.113816i
$396$	− 4.77777i	− 0.240092i
$397$	15.2664i	0.766196i	0.923708	+	0.383098i	$0.125143\pi$
$397$	15.2664i	0.766196i	−0.923708	+	0.383098i	$0.874857\pi$
$398$	− 7.53989i	− 0.377941i
$399$	−4.18598	−0.209561
$400$	−6.40581	−0.320291
$401$	− 12.7584i	− 0.637124i	−0.947902	−	0.318562i	$-0.896800\pi$
$401$	− 12.7584i	− 0.637124i	0.947902	−	0.318562i	$-0.103200\pi$
$402$	−4.11960	−0.205467
$403$	0	0
$404$	−8.95167	−0.445362
$405$	− 10.1588i	− 0.504797i
$406$	−11.8019	−0.585720
$407$	1.13946	0.0564807
$408$	1.86831i	0.0924953i
$409$	− 25.3588i	− 1.25391i	−0.779054	−	0.626956i	$-0.784300\pi$
$409$	− 25.3588i	− 1.25391i	0.779054	−	0.626956i	$-0.215700\pi$
$410$	− 5.15883i	− 0.254777i
$411$	− 10.4155i	− 0.513759i
$412$	22.8944	1.12793
$413$	−0.0325239	−0.00160040
$414$	6.02177i	0.295954i
$415$	−45.7633	−2.24643
$416$	0	0
$417$	9.66056	0.473080
$418$	1.28919i	0.0630565i
$419$	−11.6673	−0.569983	−0.284992	−	0.958530i	$-0.591991\pi$
$419$	−11.6673	−0.569983	−0.284992	+	0.958530i	$0.591991\pi$
$420$	10.2349	0.499412
$421$	− 8.29291i	− 0.404172i	−0.979368	−	0.202086i	$-0.935228\pi$
$421$	− 8.29291i	− 0.404172i	0.979368	−	0.202086i	$-0.0647720\pi$
$422$	− 5.80492i	− 0.282579i
$423$	− 18.1293i	− 0.881476i
$424$	− 12.0325i	− 0.584351i
$425$	3.24160	0.157241
$426$	−6.11529	−0.296287
$427$	− 21.6353i	− 1.04701i
$428$	9.52781	0.460544
$429$	0	0
$430$	11.1317	0.536818
$431$	0.932296i	0.0449071i	0.999748	+	0.0224536i	$0.00714779\pi$
$431$	0.932296i	0.0449071i	−0.999748	+	0.0224536i	$0.992852\pi$
$432$	−9.65279	−0.464420
$433$	13.3502	0.641569	0.320785	−	0.947152i	$-0.396053\pi$
$433$	13.3502	0.641569	0.320785	+	0.947152i	$0.396053\pi$
$434$	8.81402i	0.423086i
$435$	17.7506i	0.851077i
$436$	7.06638i	0.338418i
$437$	8.92692i	0.427032i
$438$	−5.70410	−0.272553
$439$	−13.9922	−0.667813	−0.333906	−	0.942606i	$-0.608367\pi$
$439$	−13.9922	−0.667813	−0.333906	+	0.942606i	$0.608367\pi$
$440$	− 6.87800i	− 0.327896i
$441$	0.582105	0.0277193
$442$	0	0
$443$	23.7017	1.12610	0.563051	−	0.826422i	$-0.309627\pi$
$443$	23.7017	1.12610	0.563051	+	0.826422i	$0.309627\pi$
$444$	− 1.29052i	− 0.0612454i
$445$	41.2693	1.95635
$446$	−6.32975	−0.299722
$447$	− 0.594187i	− 0.0281041i
$448$	4.11290i	0.194316i
$449$	12.5864i	0.593990i	0.954879	+	0.296995i	$0.0959844\pi$
$449$	12.5864i	0.593990i	−0.954879	+	0.296995i	$0.904016\pi$
$450$	− 3.72886i	− 0.175780i
$451$	−3.97477	−0.187165
$452$	12.9323	0.608284
$453$	15.2959i	0.718664i
$454$	5.90515	0.277142
$455$	0	0
$456$	3.18598	0.149197
$457$	− 33.6383i	− 1.57353i	−0.617250	−	0.786767i	$-0.711753\pi$
$457$	− 33.6383i	− 1.57353i	0.617250	−	0.786767i	$-0.288247\pi$
$458$	0.632351	0.0295478
$459$	4.88471	0.227999
$460$	− 21.8267i	− 1.01767i
$461$	− 1.40283i	− 0.0653363i	−0.999466	−	0.0326681i	$-0.989600\pi$
$461$	− 1.40283i	− 0.0653363i	0.999466	−	0.0326681i	$-0.0104004\pi$
$462$	1.43535i	0.0667787i
$463$	15.2010i	0.706453i	0.935538	+	0.353226i	$0.114915\pi$
$463$	15.2010i	0.706453i	−0.935538	+	0.353226i	$0.885085\pi$
$464$	−17.7506	−0.824052
$465$	13.2567	0.614763
$466$	− 6.02177i	− 0.278953i
$467$	39.3414	1.82050	0.910250	−	0.414058i	$-0.135889\pi$
$467$	39.3414	1.82050	0.910250	+	0.414058i	$0.135889\pi$
$468$	0	0
$469$	24.9191	1.15066
$470$	− 11.9608i	− 0.551709i
$471$	−3.22521	−0.148610
$472$	0.0247542	0.00113940
$473$	− 8.57673i	− 0.394358i
$474$	− 0.359289i	− 0.0165027i
$475$	− 5.52781i	− 0.253633i
$476$	− 5.17928i	− 0.237392i
$477$	−13.8412	−0.633743
$478$	−6.62027	−0.302804
$479$	22.3690i	1.02206i	0.859561	+	0.511032i	$0.170737\pi$
$479$	22.3690i	1.02206i	−0.859561	+	0.511032i	$0.829263\pi$
$480$	−12.0097	−0.548165
$481$	0	0
$482$	2.02475	0.0922250
$483$	9.93900i	0.452240i
$484$	16.1836	0.735618
$485$	8.77479	0.398443
$486$	− 8.76569i	− 0.397620i
$487$	22.9205i	1.03863i	0.854584	+	0.519313i	$0.173812\pi$
$487$	22.9205i	1.03863i	−0.854584	+	0.519313i	$0.826188\pi$
$488$	16.4668i	0.745418i
$489$	− 12.1390i	− 0.548944i
$490$	0.384043	0.0173493
$491$	−1.84356	−0.0831987	−0.0415993	−	0.999134i	$-0.513245\pi$
$491$	−1.84356	−0.0831987	−0.0415993	+	0.999134i	$0.513245\pi$
$492$	4.50173i	0.202954i
$493$	8.98254	0.404553
$494$	0	0
$495$	−7.91185	−0.355611
$496$	13.2567i	0.595242i
$497$	36.9909	1.65927
$498$	−7.26875	−0.325720
$499$	− 12.0344i	− 0.538736i	−0.963037	−	0.269368i	$-0.913185\pi$
$499$	− 12.0344i	− 0.538736i	0.963037	−	0.269368i	$-0.0868147\pi$
$500$	− 10.1890i	− 0.455664i
$501$	5.02284i	0.224404i
$502$	0.762118i	0.0340150i
$503$	−30.5056	−1.36018	−0.680088	−	0.733130i	$-0.738058\pi$
$503$	−30.5056	−1.36018	−0.680088	+	0.733130i	$0.738058\pi$
$504$	−13.0000	−0.579066
$505$	14.8237i	0.659646i
$506$	3.06100	0.136078
$507$	0	0
$508$	−11.4681	−0.508816
$509$	− 1.51142i	− 0.0669924i	−0.999439	−	0.0334962i	$-0.989336\pi$
$509$	− 1.51142i	− 0.0669924i	0.999439	−	0.0334962i	$-0.0106642\pi$
$510$	1.41789	0.0627854
$511$	34.5036	1.52635
$512$	− 21.2174i	− 0.937687i
$513$	− 8.32975i	− 0.367767i
$514$	16.3357i	0.720538i
$515$	− 37.9124i	− 1.67062i
$516$	−9.71379	−0.427626
$517$	−9.21552	−0.405298
$518$	− 1.42088i	− 0.0624298i
$519$	−13.1448	−0.576994
$520$	0	0
$521$	−5.64012	−0.247098	−0.123549	−	0.992338i	$-0.539428\pi$
$521$	−5.64012	−0.247098	−0.123549	+	0.992338i	$0.539428\pi$
$522$	− 10.3327i	− 0.452251i
$523$	−31.7506	−1.38836	−0.694179	−	0.719802i	$-0.744232\pi$
$523$	−31.7506	−1.38836	−0.694179	+	0.719802i	$0.744232\pi$
$524$	−23.1468	−1.01117
$525$	− 6.15452i	− 0.268605i
$526$	− 5.93602i	− 0.258823i
$527$	− 6.70841i	− 0.292223i
$528$	2.15883i	0.0939512i
$529$	−1.80433	−0.0784492
$530$	−9.13169	−0.396655
$531$	− 0.0284750i	− 0.00123571i
$532$	−8.83207	−0.382919
$533$	0	0
$534$	6.55496	0.283661
$535$	− 15.7778i	− 0.682133i
$536$	−18.9661	−0.819213
$537$	1.96854	0.0849488
$538$	5.65279i	0.243709i
$539$	− 0.295897i	− 0.0127452i
$540$	20.3666i	0.876438i
$541$	− 24.3297i	− 1.04602i	−0.852327	−	0.523009i	$-0.824809\pi$
$541$	− 24.3297i	− 1.04602i	0.852327	−	0.523009i	$-0.175191\pi$
$542$	16.3448	0.702070
$543$	−9.46873	−0.406342
$544$	6.07739i	0.260566i
$545$	11.7017	0.501246
$546$	0	0
$547$	−8.18896	−0.350135	−0.175067	−	0.984556i	$-0.556014\pi$
$547$	−8.18896	−0.350135	−0.175067	+	0.984556i	$0.556014\pi$
$548$	− 21.9758i	− 0.938761i
$549$	18.9420	0.808424
$550$	−1.89546	−0.0808227
$551$	− 15.3177i	− 0.652555i
$552$	− 7.56465i	− 0.321973i
$553$	2.17331i	0.0924185i
$554$	− 5.68532i	− 0.241546i
$555$	−2.13706	−0.0907133
$556$	20.3830	0.864431
$557$	− 25.3327i	− 1.07338i	−0.843779	−	0.536691i	$-0.819674\pi$
$557$	− 25.3327i	− 1.07338i	0.843779	−	0.536691i	$-0.180326\pi$
$558$	−7.71678	−0.326677
$559$	0	0
$560$	16.9487	0.716213
$561$	− 1.09246i	− 0.0461236i
$562$	6.41789	0.270723
$563$	25.3937	1.07022	0.535109	−	0.844783i	$-0.320270\pi$
$563$	25.3937	1.07022	0.535109	+	0.844783i	$0.320270\pi$
$564$	10.4373i	0.439488i
$565$	− 21.4155i	− 0.900957i
$566$	− 17.0422i	− 0.716338i
$567$	9.76032i	0.409895i
$568$	−28.1540	−1.18132
$569$	31.1347	1.30523	0.652617	−	0.757688i	$-0.273671\pi$
$569$	31.1347	1.30523	0.652617	+	0.757688i	$0.273671\pi$
$570$	− 2.41789i	− 0.101274i
$571$	20.5090	0.858276	0.429138	−	0.903239i	$-0.358817\pi$
$571$	20.5090	0.858276	0.429138	+	0.903239i	$0.358817\pi$
$572$	0	0
$573$	−7.21206	−0.301288
$574$	4.95646i	0.206879i
$575$	−13.1250	−0.547350
$576$	−3.60089	−0.150037
$577$	15.6890i	0.653143i	0.945172	+	0.326572i	$0.105893\pi$
$577$	15.6890i	0.653143i	−0.945172	+	0.326572i	$0.894107\pi$
$578$	8.71678i	0.362570i
$579$	− 10.8465i	− 0.450767i
$580$	37.4523i	1.55512i
$581$	43.9681	1.82410
$582$	1.39373	0.0577720
$583$	7.03577i	0.291392i
$584$	−26.2610	−1.08669
$585$	0	0
$586$	10.3268	0.426595
$587$	30.5687i	1.26171i	0.775903	+	0.630853i	$0.217295\pi$
$587$	30.5687i	1.26171i	−0.775903	+	0.630853i	$0.782705\pi$
$588$	−0.335126	−0.0138203
$589$	−11.4397	−0.471363
$590$	− 0.0187864i	0 0.000773422i
$591$	10.4058i	0.428038i
$592$	− 2.13706i	− 0.0878328i
$593$	29.6883i	1.21915i	0.792727	+	0.609576i	$0.208660\pi$
$593$	29.6883i	1.21915i	−0.792727	+	0.609576i	$0.791340\pi$
$594$	−2.85623	−0.117193
$595$	−8.57673	−0.351612
$596$	− 1.25368i	− 0.0513529i
$597$	10.8955	0.445922
$598$	0	0
$599$	24.2325	0.990113	0.495057	−	0.868861i	$-0.335147\pi$
$599$	24.2325	0.990113	0.495057	+	0.868861i	$0.335147\pi$
$600$	4.68425i	0.191234i
$601$	16.4819	0.672310	0.336155	−	0.941807i	$-0.390873\pi$
$601$	16.4819	0.672310	0.336155	+	0.941807i	$0.390873\pi$
$602$	−10.6950	−0.435896
$603$	21.8170i	0.888457i
$604$	32.2731i	1.31317i
$605$	− 26.7995i	− 1.08956i
$606$	2.35450i	0.0956451i
$607$	1.43190	0.0581188	0.0290594	−	0.999578i	$-0.490749\pi$
$607$	1.43190	0.0581188	0.0290594	+	0.999578i	$0.490749\pi$
$608$	10.3636	0.420300
$609$	− 17.0543i	− 0.691075i
$610$	12.4969	0.505986
$611$	0	0
$612$	4.53452	0.183297
$613$	3.84846i	0.155438i	0.996975	+	0.0777190i	$0.0247637\pi$
$613$	3.84846i	0.155438i	−0.996975	+	0.0777190i	$0.975236\pi$
$614$	−4.96376	−0.200321
$615$	7.45473	0.300604
$616$	6.60819i	0.266251i
$617$	− 15.0388i	− 0.605437i	−0.953080	−	0.302719i	$-0.902106\pi$
$617$	− 15.0388i	− 0.605437i	0.953080	−	0.302719i	$-0.0978943\pi$
$618$	− 6.02177i	− 0.242231i
$619$	− 12.8170i	− 0.515159i	−0.966257	−	0.257579i	$-0.917075\pi$
$619$	− 12.8170i	− 0.515159i	0.966257	−	0.257579i	$-0.0829249\pi$
$620$	27.9705	1.12332
$621$	−19.7778	−0.793655
$622$	11.6762i	0.468174i
$623$	−39.6504	−1.58856
$624$	0	0
$625$	−31.1269	−1.24508
$626$	3.95539i	0.158089i
$627$	−1.86294	−0.0743985
$628$	−6.80492	−0.271546
$629$	1.08144i	0.0431199i
$630$	9.86592i	0.393068i
$631$	− 25.7517i	− 1.02516i	−0.858640	−	0.512579i	$-0.828690\pi$
$631$	− 25.7517i	− 1.02516i	0.858640	−	0.512579i	$-0.171310\pi$
$632$	− 1.65412i	− 0.0657974i
$633$	8.38835	0.333407
$634$	13.2996	0.528195
$635$	18.9909i	0.753631i
$636$	7.96854	0.315973
$637$	0	0
$638$	−5.25236	−0.207943
$639$	32.3860i	1.28117i
$640$	−32.3274	−1.27785
$641$	−24.4571	−0.965998	−0.482999	−	0.875621i	$-0.660453\pi$
$641$	−24.4571	−0.965998	−0.482999	+	0.875621i	$0.660453\pi$
$642$	− 2.50604i	− 0.0989055i
$643$	− 9.97344i	− 0.393314i	−0.980472	−	0.196657i	$-0.936991\pi$
$643$	− 9.97344i	− 0.393314i	0.980472	−	0.196657i	$-0.0630086\pi$
$644$	20.9705i	0.826352i
$645$	16.0858i	0.633376i
$646$	−1.22355	−0.0481401
$647$	−11.8431	−0.465600	−0.232800	−	0.972525i	$-0.574789\pi$
$647$	−11.8431	−0.465600	−0.232800	+	0.972525i	$0.574789\pi$
$648$	− 7.42865i	− 0.291825i
$649$	−0.0144745	−0.000568173 0
$650$	0	0
$651$	−12.7366	−0.499188
$652$	− 25.6122i	− 1.00305i
$653$	−7.47411	−0.292484	−0.146242	−	0.989249i	$-0.546718\pi$
$653$	−7.47411	−0.292484	−0.146242	+	0.989249i	$0.546718\pi$
$654$	1.85862	0.0726780
$655$	38.3303i	1.49769i
$656$	7.45473i	0.291058i
$657$	30.2083i	1.17854i
$658$	11.4916i	0.447988i
$659$	34.1739	1.33123	0.665613	−	0.746297i	$-0.268170\pi$
$659$	34.1739	1.33123	0.665613	+	0.746297i	$0.268170\pi$
$660$	4.55496	0.177302
$661$	− 33.6088i	− 1.30723i	−0.756827	−	0.653615i	$-0.773252\pi$
$661$	− 33.6088i	− 1.30723i	0.756827	−	0.653615i	$-0.226748\pi$
$662$	1.60686	0.0624524
$663$	0	0
$664$	−33.4644	−1.29867
$665$	14.6256i	0.567158i
$666$	1.24400	0.0482039
$667$	−36.3696	−1.40824
$668$	10.5978i	0.410040i
$669$	− 9.14675i	− 0.353634i
$670$	14.3937i	0.556078i
$671$	− 9.62863i	− 0.371709i
$672$	11.5386	0.445110
$673$	−48.0320	−1.85150	−0.925750	−	0.378137i	$-0.876565\pi$
$673$	−48.0320	−1.85150	−0.925750	+	0.378137i	$0.876565\pi$
$674$	− 1.72348i	− 0.0663860i
$675$	12.2470	0.471386
$676$	0	0
$677$	−33.6582	−1.29359	−0.646794	−	0.762665i	$-0.723891\pi$
$677$	−33.6582	−1.29359	−0.646794	+	0.762665i	$0.723891\pi$
$678$	− 3.40150i	− 0.130634i
$679$	−8.43057	−0.323535
$680$	6.52781	0.250330
$681$	8.53319i	0.326992i
$682$	3.92261i	0.150204i
$683$	− 15.9041i	− 0.608553i	−0.952584	−	0.304276i	$-0.901585\pi$
$683$	− 15.9041i	− 0.608553i	0.952584	−	0.304276i	$-0.0984147\pi$
$684$	− 7.73258i	− 0.295663i
$685$	−36.3913	−1.39044
$686$	10.0887	0.385190
$687$	0.913773i	0.0348626i
$688$	−16.0858	−0.613264
$689$	0	0
$690$	−5.74094	−0.218554
$691$	33.1903i	1.26262i	0.775531	+	0.631309i	$0.217482\pi$
$691$	33.1903i	1.26262i	−0.775531	+	0.631309i	$0.782518\pi$
$692$	−27.7345	−1.05431
$693$	7.60148	0.288756
$694$	6.31468i	0.239702i
$695$	− 33.7536i	− 1.28035i
$696$	12.9801i	0.492011i
$697$	− 3.77240i	− 0.142890i
$698$	1.85756	0.0703097
$699$	8.70171	0.329129
$700$	− 12.9855i	− 0.490807i
$701$	14.9129	0.563253	0.281627	−	0.959524i	$-0.409126\pi$
$701$	14.9129	0.563253	0.281627	+	0.959524i	$0.409126\pi$
$702$	0	0
$703$	1.84415	0.0695534
$704$	1.83041i	0.0689863i
$705$	17.2838	0.650946
$706$	0.353912	0.0133197
$707$	− 14.2422i	− 0.535633i
$708$	0.0163935i	0 0.000616104i
$709$	− 38.4312i	− 1.44331i	−0.692252	−	0.721656i	$-0.743381\pi$
$709$	− 38.4312i	− 1.44331i	0.692252	−	0.721656i	$-0.256619\pi$
$710$	21.3666i	0.801874i
$711$	−1.90276	−0.0713589
$712$	30.1782	1.13098
$713$	27.1618i	1.01722i
$714$	−1.36227	−0.0509818
$715$	0	0
$716$	4.15346	0.155222
$717$	− 9.56657i	− 0.357270i
$718$	11.9089	0.444435
$719$	11.4373	0.426538	0.213269	−	0.976993i	$-0.431589\pi$
$719$	11.4373	0.426538	0.213269	+	0.976993i	$0.431589\pi$
$720$	14.8388i	0.553008i
$721$	36.4252i	1.35654i
$722$	− 8.45771i	− 0.314764i
$723$	2.92585i	0.108814i
$724$	−19.9782	−0.742485
$725$	22.5211	0.836413
$726$	− 4.25667i	− 0.157980i
$727$	−3.63640	−0.134867	−0.0674333	−	0.997724i	$-0.521481\pi$
$727$	−3.63640	−0.134867	−0.0674333	+	0.997724i	$0.521481\pi$
$728$	0	0
$729$	1.78986	0.0662910
$730$	19.9299i	0.737639i
$731$	8.14005	0.301071
$732$	−10.9051	−0.403066
$733$	− 3.52217i	− 0.130094i	−0.997882	−	0.0650472i	$-0.979280\pi$
$733$	− 3.52217i	− 0.130094i	0.997882	−	0.0650472i	$-0.0207198\pi$
$734$	5.20941i	0.192283i
$735$	0.554958i	0.0204699i
$736$	− 24.6069i	− 0.907021i
$737$	11.0901	0.408508
$738$	−4.33944	−0.159737
$739$	− 0.420288i	− 0.0154605i	−0.999970	−	0.00773027i	$-0.997539\pi$
$739$	− 0.420288i	− 0.0154605i	0.999970	−	0.00773027i	$-0.00246064\pi$
$740$	−4.50902	−0.165755
$741$	0	0
$742$	8.77346	0.322084
$743$	− 25.3623i	− 0.930452i	−0.885192	−	0.465226i	$-0.845973\pi$
$743$	− 25.3623i	− 0.930452i	0.885192	−	0.465226i	$-0.154027\pi$
$744$	9.69394	0.355397
$745$	−2.07606	−0.0760611
$746$	− 15.3870i	− 0.563359i
$747$	38.4946i	1.40844i
$748$	− 2.30499i	− 0.0842790i
$749$	15.1588i	0.553892i
$750$	−2.67994	−0.0978576
$751$	−0.650874	−0.0237507	−0.0118754	−	0.999929i	$-0.503780\pi$
$751$	−0.650874	−0.0237507	−0.0118754	+	0.999929i	$0.503780\pi$
$752$	17.2838i	0.630276i
$753$	−1.10129	−0.0401333
$754$	0	0
$755$	53.4432	1.94500
$756$	− 19.5676i	− 0.711668i
$757$	−16.7909	−0.610276	−0.305138	−	0.952308i	$-0.598703\pi$
$757$	−16.7909	−0.610276	−0.305138	+	0.952308i	$0.598703\pi$
$758$	19.9065	0.723036
$759$	4.42327i	0.160555i
$760$	− 11.1317i	− 0.403789i
$761$	− 30.9221i	− 1.12093i	−0.828179	−	0.560463i	$-0.810623\pi$
$761$	− 30.9221i	− 1.12093i	0.828179	−	0.560463i	$-0.189377\pi$
$762$	3.01639i	0.109272i
$763$	−11.2427	−0.407012
$764$	−15.2168	−0.550526
$765$	− 7.50902i	− 0.271489i
$766$	2.69441	0.0973531
$767$	0	0
$768$	−2.68425	−0.0968596
$769$	− 43.7689i	− 1.57835i	−0.614169	−	0.789174i	$-0.710509\pi$
$769$	− 43.7689i	− 1.57835i	0.614169	−	0.789174i	$-0.289491\pi$
$770$	5.01507	0.180730
$771$	−23.6058	−0.850142
$772$	− 22.8853i	− 0.823660i
$773$	− 42.4209i	− 1.52577i	−0.646532	−	0.762886i	$-0.723782\pi$
$773$	− 42.4209i	− 1.52577i	0.646532	−	0.762886i	$-0.276218\pi$
$774$	− 9.36360i	− 0.336568i
$775$	− 16.8194i	− 0.604171i
$776$	6.41657	0.230341
$777$	2.05323	0.0736592
$778$	1.32378i	0.0474598i
$779$	−6.43296	−0.230485
$780$	0	0
$781$	16.4625	0.589075
$782$	2.90515i	0.103888i
$783$	33.9366	1.21280
$784$	−0.554958	−0.0198199
$785$	11.2687i	0.402199i
$786$	6.08815i	0.217157i
$787$	36.0116i	1.28368i	0.766841	+	0.641838i	$0.221828\pi$
$787$	36.0116i	1.28368i	−0.766841	+	0.641838i	$0.778172\pi$
$788$	21.9554i	0.782129i
$789$	8.57779	0.305378
$790$	−1.25534	−0.0446630
$791$	20.5754i	0.731577i
$792$	−5.78554	−0.205580
$793$	0	0
$794$	8.47219	0.300667
$795$	− 13.1957i	− 0.468002i
$796$	22.9885	0.814806
$797$	31.7101	1.12323	0.561614	−	0.827399i	$-0.310181\pi$
$797$	31.7101	1.12323	0.561614	+	0.827399i	$0.310181\pi$
$798$	2.32304i	0.0822349i
$799$	− 8.74632i	− 0.309422i
$800$	15.2373i	0.538720i
$801$	− 34.7144i	− 1.22657i
$802$	−7.08038	−0.250017
$803$	15.3556	0.541886
$804$	− 12.5603i	− 0.442969i
$805$	34.7265	1.22395
$806$	0	0
$807$	−8.16852	−0.287546
$808$	10.8398i	0.381344i
$809$	−45.2814	−1.59201	−0.796005	−	0.605290i	$-0.793057\pi$
$809$	−45.2814	−1.59201	−0.796005	+	0.605290i	$0.793057\pi$
$810$	−5.63773	−0.198090
$811$	− 42.8635i	− 1.50514i	−0.658511	−	0.752571i	$-0.728813\pi$
$811$	− 42.8635i	− 1.50514i	0.658511	−	0.752571i	$-0.271187\pi$
$812$	− 35.9831i	− 1.26276i
$813$	23.6189i	0.828352i
$814$	− 0.632351i	− 0.0221639i
$815$	−42.4131	−1.48567
$816$	−2.04892	−0.0717265
$817$	− 13.8810i	− 0.485634i
$818$	−14.0731	−0.492054
$819$	0	0
$820$	15.7289	0.549276
$821$	− 7.82776i	− 0.273191i	−0.990627	−	0.136595i	$-0.956384\pi$
$821$	− 7.82776i	− 0.273191i	0.990627	−	0.136595i	$-0.0436160\pi$
$822$	−5.78017	−0.201606
$823$	36.7754	1.28191	0.640955	−	0.767579i	$-0.278539\pi$
$823$	36.7754	1.28191	0.640955	+	0.767579i	$0.278539\pi$
$824$	− 27.7235i	− 0.965793i
$825$	− 2.73902i	− 0.0953604i
$826$	0.0180494i	0 0.000628019i
$827$	− 47.3293i	− 1.64580i	−0.568186	−	0.822900i	$-0.692355\pi$
$827$	− 47.3293i	− 1.64580i	0.568186	−	0.822900i	$-0.307645\pi$
$828$	−18.3599	−0.638050
$829$	−25.2687	−0.877620	−0.438810	−	0.898580i	$-0.644600\pi$
$829$	−25.2687	−0.877620	−0.438810	+	0.898580i	$0.644600\pi$
$830$	25.3967i	0.881533i
$831$	8.21552	0.284993
$832$	0	0
$833$	0.280831	0.00973023
$834$	− 5.36121i	− 0.185643i
$835$	17.5496	0.607328
$836$	−3.93064	−0.135944
$837$	− 25.3448i	− 0.876045i
$838$	6.47484i	0.223670i
$839$	37.6883i	1.30114i	0.759444	+	0.650572i	$0.225471\pi$
$839$	37.6883i	1.30114i	−0.759444	+	0.650572i	$0.774529\pi$
$840$	− 12.3937i	− 0.427624i
$841$	33.4064	1.15194
$842$	−4.60222	−0.158603
$843$	9.27413i	0.319418i
$844$	17.6987	0.609215
$845$	0	0
$846$	−10.0610	−0.345904
$847$	25.7482i	0.884720i
$848$	13.1957	0.453141
$849$	24.6267	0.845187
$850$	− 1.79895i	− 0.0617036i
$851$	− 4.37867i	− 0.150099i
$852$	− 18.6450i	− 0.638768i
$853$	31.0121i	1.06183i	0.847424	+	0.530917i	$0.178152\pi$
$853$	31.0121i	1.06183i	−0.847424	+	0.530917i	$0.821848\pi$
$854$	−12.0067	−0.410861
$855$	−12.8049	−0.437919
$856$	− 11.5375i	− 0.394344i
$857$	−12.4692	−0.425940	−0.212970	−	0.977059i	$-0.568314\pi$
$857$	−12.4692	−0.425940	−0.212970	+	0.977059i	$0.568314\pi$
$858$	0	0
$859$	−17.3163	−0.590826	−0.295413	−	0.955370i	$-0.595457\pi$
$859$	−17.3163	−0.590826	−0.295413	+	0.955370i	$0.595457\pi$
$860$	33.9396i	1.15733i
$861$	−7.16229	−0.244090
$862$	0.517385	0.0176222
$863$	− 3.46383i	− 0.117910i	−0.998261	−	0.0589550i	$-0.981223\pi$
$863$	− 3.46383i	− 0.117910i	0.998261	−	0.0589550i	$-0.0187769\pi$
$864$	22.9608i	0.781141i
$865$	45.9275i	1.56158i
$866$	− 7.40880i	− 0.251761i
$867$	−12.5961	−0.427786
$868$	−26.8732	−0.912136
$869$	0.967213i	0.0328105i
$870$	9.85086	0.333975
$871$	0	0
$872$	8.55688	0.289772
$873$	− 7.38106i	− 0.249811i
$874$	4.95407	0.167574
$875$	16.2107	0.548023
$876$	− 17.3913i	− 0.587599i
$877$	57.2549i	1.93336i	0.255989	+	0.966680i	$0.417599\pi$
$877$	57.2549i	1.93336i	−0.255989	+	0.966680i	$0.582401\pi$
$878$	7.76510i	0.262059i
$879$	14.9226i	0.503327i
$880$	7.54288	0.254270
$881$	43.1782	1.45471	0.727355	−	0.686261i	$-0.240749\pi$
$881$	43.1782	1.45471	0.727355	+	0.686261i	$0.240749\pi$
$882$	− 0.323044i	− 0.0108775i
$883$	−49.9560	−1.68115	−0.840576	−	0.541693i	$-0.817784\pi$
$883$	−49.9560	−1.68115	−0.840576	+	0.541693i	$0.817784\pi$
$884$	0	0
$885$	0.0271471	0.000912539 0
$886$	− 13.1535i	− 0.441899i
$887$	17.6746	0.593454	0.296727	−	0.954962i	$-0.404105\pi$
$887$	17.6746	0.593454	0.296727	+	0.954962i	$0.404105\pi$
$888$	−1.56273	−0.0524417
$889$	− 18.2459i	− 0.611948i
$890$	− 22.9028i	− 0.767702i
$891$	4.34375i	0.145521i
$892$	− 19.2989i	− 0.646174i
$893$	−14.9148	−0.499106
$894$	−0.329749	−0.0110284
$895$	− 6.87800i	− 0.229906i
$896$	31.0592	1.03761
$897$	0	0
$898$	6.98493	0.233090
$899$	− 46.6069i	− 1.55443i
$900$	11.3690	0.378966
$901$	−6.67755	−0.222461
$902$	2.20583i	0.0734462i
$903$	− 15.4547i	− 0.514301i
$904$	− 15.6601i	− 0.520847i
$905$	33.0834i	1.09973i
$906$	8.48858	0.282014
$907$	−7.73423	−0.256811	−0.128406	−	0.991722i	$-0.540986\pi$
$907$	−7.73423	−0.256811	−0.128406	+	0.991722i	$0.540986\pi$
$908$	18.0043i	0.597494i
$909$	12.4692	0.413577
$910$	0	0
$911$	39.6179	1.31260	0.656299	−	0.754501i	$-0.272121\pi$
$911$	39.6179	1.31260	0.656299	+	0.754501i	$0.272121\pi$
$912$	3.49396i	0.115697i
$913$	19.5676	0.647594
$914$	−18.6679	−0.617478
$915$	18.0586i	0.596999i
$916$	1.92798i	0.0637024i
$917$	− 36.8267i	− 1.21612i
$918$	− 2.71081i	− 0.0894700i
$919$	14.6213	0.482313	0.241157	−	0.970486i	$-0.422473\pi$
$919$	14.6213	0.482313	0.241157	+	0.970486i	$0.422473\pi$
$920$	−26.4306	−0.871390
$921$	− 7.17283i	− 0.236353i
$922$	−0.778512	−0.0256389
$923$	0	0
$924$	−4.37627	−0.143969
$925$	2.71140i	0.0891502i
$926$	8.43594	0.277222
$927$	−31.8907	−1.04743
$928$	42.2228i	1.38603i
$929$	3.55735i	0.116713i	0.998296	+	0.0583565i	$0.0185860\pi$
$929$	3.55735i	0.116713i	−0.998296	+	0.0583565i	$0.981414\pi$
$930$	− 7.35690i	− 0.241242i
$931$	− 0.478894i	− 0.0156951i
$932$	18.3599	0.601398
$933$	−16.8726	−0.552385
$934$	− 21.8328i	− 0.714391i
$935$	−3.81700	−0.124829
$936$	0	0
$937$	34.5526	1.12878	0.564392	−	0.825507i	$-0.309111\pi$
$937$	34.5526	1.12878	0.564392	+	0.825507i	$0.309111\pi$
$938$	− 13.8291i	− 0.451536i
$939$	−5.71571	−0.186525
$940$	36.4674	1.18944
$941$	20.6233i	0.672299i	0.941809	+	0.336149i	$0.109125\pi$
$941$	20.6233i	0.672299i	−0.941809	+	0.336149i	$0.890875\pi$
$942$	1.78986i	0.0583167i
$943$	15.2741i	0.497394i
$944$	0.0271471i	0 0.000883562i
$945$	−32.4034	−1.05408
$946$	−4.75973	−0.154752
$947$	− 29.4999i	− 0.958619i	−0.877646	−	0.479309i	$-0.840887\pi$
$947$	− 29.4999i	− 0.958619i	0.877646	−	0.479309i	$-0.159113\pi$
$948$	1.09544	0.0355783
$949$	0	0
$950$	−3.06770	−0.0995295
$951$	19.2185i	0.623203i
$952$	−6.27173	−0.203268
$953$	−26.2389	−0.849963	−0.424981	−	0.905202i	$-0.639719\pi$
$953$	−26.2389	−0.849963	−0.424981	+	0.905202i	$0.639719\pi$
$954$	7.68127i	0.248690i
$955$	25.1987i	0.815409i
$956$	− 20.1847i	− 0.652818i
$957$	− 7.58987i	− 0.245346i
$958$	12.4138	0.401073
$959$	34.9638	1.12904
$960$	− 3.43296i	− 0.110798i
$961$	−3.80731	−0.122817
$962$	0	0
$963$	−13.2717	−0.427676
$964$	6.17331i	0.198829i
$965$	−37.8974	−1.21996
$966$	5.51573	0.177466
$967$	17.5176i	0.563330i	0.959513	+	0.281665i	$0.0908866\pi$
$967$	17.5176i	0.563330i	−0.959513	+	0.281665i	$0.909113\pi$
$968$	− 19.5972i	− 0.629877i
$969$	− 1.76809i	− 0.0567991i
$970$	− 4.86964i	− 0.156355i
$971$	20.5120	0.658262	0.329131	−	0.944284i	$-0.393244\pi$
$971$	20.5120	0.658262	0.329131	+	0.944284i	$0.393244\pi$
$972$	26.7259	0.857233
$973$	32.4295i	1.03964i
$974$	12.7199	0.407572
$975$	0	0
$976$	−18.0586	−0.578042
$977$	25.4450i	0.814059i	0.913415	+	0.407030i	$0.133435\pi$
$977$	25.4450i	0.814059i	−0.913415	+	0.407030i	$0.866565\pi$
$978$	−6.73663	−0.215414
$979$	−17.6461	−0.563971
$980$	1.17092i	0.0374035i
$981$	− 9.84309i	− 0.314266i
$982$	1.02310i	0.0326484i
$983$	39.5244i	1.26063i	0.776339	+	0.630316i	$0.217074\pi$
$983$	39.5244i	1.26063i	−0.776339	+	0.630316i	$0.782926\pi$
$984$	5.45127	0.173780
$985$	36.3575	1.15845
$986$	− 4.98493i	− 0.158753i
$987$	−16.6058	−0.528568
$988$	0	0
$989$	−32.9584	−1.04802
$990$	4.39075i	0.139547i
$991$	−29.8377	−0.947826	−0.473913	−	0.880572i	$-0.657159\pi$
$991$	−29.8377	−0.947826	−0.473913	+	0.880572i	$0.657159\pi$
$992$	31.5332	1.00118
$993$	2.32198i	0.0736858i
$994$	− 20.5284i	− 0.651121i
$995$	− 38.0683i	− 1.20685i
$996$	− 22.1618i	− 0.702224i
$997$	−4.93123	−0.156174	−0.0780868	−	0.996947i	$-0.524881\pi$
$997$	−4.93123	−0.156174	−0.0780868	+	0.996947i	$0.524881\pi$
$998$	−6.67861	−0.211408
$999$	4.08575i	0.129268i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.b.b.168.3		6
3.2	odd	2		1521.2.b.l.1351.4		6
4.3	odd	2		2704.2.f.o.337.1		6
13.2	odd	12		169.2.c.c.22.2		6
13.3	even	3		169.2.e.b.147.4		12
13.4	even	6		169.2.e.b.23.4		12
13.5	odd	4		169.2.a.b.1.2	✓	3
13.6	odd	12		169.2.c.c.146.2		6
13.7	odd	12		169.2.c.b.146.2		6
13.8	odd	4		169.2.a.c.1.2	yes	3
13.9	even	3		169.2.e.b.23.3		12
13.10	even	6		169.2.e.b.147.3		12
13.11	odd	12		169.2.c.b.22.2		6
13.12	even	2	inner	169.2.b.b.168.4		6
39.5	even	4		1521.2.a.r.1.2		3
39.8	even	4		1521.2.a.o.1.2		3
39.38	odd	2		1521.2.b.l.1351.3		6
52.31	even	4		2704.2.a.z.1.1		3
52.47	even	4		2704.2.a.ba.1.1		3
52.51	odd	2		2704.2.f.o.337.2		6
65.34	odd	4		4225.2.a.bb.1.2		3
65.44	odd	4		4225.2.a.bg.1.2		3
91.34	even	4		8281.2.a.bj.1.2		3
91.83	even	4		8281.2.a.bf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.2	✓	3	13.5	odd	4
169.2.a.c.1.2	yes	3	13.8	odd	4
169.2.b.b.168.3		6	1.1	even	1	trivial
169.2.b.b.168.4		6	13.12	even	2	inner
169.2.c.b.22.2		6	13.11	odd	12
169.2.c.b.146.2		6	13.7	odd	12
169.2.c.c.22.2		6	13.2	odd	12
169.2.c.c.146.2		6	13.6	odd	12
169.2.e.b.23.3		12	13.9	even	3
169.2.e.b.23.4		12	13.4	even	6
169.2.e.b.147.3		12	13.10	even	6
169.2.e.b.147.4		12	13.3	even	3
1521.2.a.o.1.2		3	39.8	even	4
1521.2.a.r.1.2		3	39.5	even	4
1521.2.b.l.1351.3		6	39.38	odd	2
1521.2.b.l.1351.4		6	3.2	odd	2
2704.2.a.z.1.1		3	52.31	even	4
2704.2.a.ba.1.1		3	52.47	even	4
2704.2.f.o.337.1		6	4.3	odd	2
2704.2.f.o.337.2		6	52.51	odd	2
4225.2.a.bb.1.2		3	65.34	odd	4
4225.2.a.bg.1.2		3	65.44	odd	4
8281.2.a.bf.1.2		3	91.83	even	4
8281.2.a.bj.1.2		3	91.34	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.554958i q^{2} +0.801938 q^{3} +1.69202 q^{4} -2.80194i q^{5} -0.445042i q^{6} +2.69202i q^{7} -2.04892i q^{8} -2.35690 q^{9} +O(q^{10})\)	\(q-0.554958i q^{2} +0.801938 q^{3} +1.69202 q^{4} -2.80194i q^{5} -0.445042i q^{6} +2.69202i q^{7} -2.04892i q^{8} -2.35690 q^{9} -1.55496 q^{10} +1.19806i q^{11} +1.35690 q^{12} +1.49396 q^{14} -2.24698i q^{15} +2.24698 q^{16} -1.13706 q^{17} +1.30798i q^{18} +1.93900i q^{19} -4.74094i q^{20} +2.15883i q^{21} +0.664874 q^{22} +4.60388 q^{23} -1.64310i q^{24} -2.85086 q^{25} -4.29590 q^{27} +4.55496i q^{28} -7.89977 q^{29} -1.24698 q^{30} +5.89977i q^{31} -5.34481i q^{32} +0.960771i q^{33} +0.631023i q^{34} +7.54288 q^{35} -3.98792 q^{36} -0.951083i q^{37} +1.07606 q^{38} -5.74094 q^{40} +3.31767i q^{41} +1.19806 q^{42} -7.15883 q^{43} +2.02715i q^{44} +6.60388i q^{45} -2.55496i q^{46} +7.69202i q^{47} +1.80194 q^{48} -0.246980 q^{49} +1.58211i q^{50} -0.911854 q^{51} +5.87263 q^{53} +2.38404i q^{54} +3.35690 q^{55} +5.51573 q^{56} +1.55496i q^{57} +4.38404i q^{58} +0.0120816i q^{59} -3.80194i q^{60} -8.03684 q^{61} +3.27413 q^{62} -6.34481i q^{63} +1.52781 q^{64} +0.533188 q^{66} -9.25667i q^{67} -1.92394 q^{68} +3.69202 q^{69} -4.18598i q^{70} -13.7409i q^{71} +4.82908i q^{72} -12.8170i q^{73} -0.527811 q^{74} -2.28621 q^{75} +3.28083i q^{76} -3.22521 q^{77} +0.807315 q^{79} -6.29590i q^{80} +3.62565 q^{81} +1.84117 q^{82} -16.3327i q^{83} +3.65279i q^{84} +3.18598i q^{85} +3.97285i q^{86} -6.33513 q^{87} +2.45473 q^{88} +14.7289i q^{89} +3.66487 q^{90} +7.78986 q^{92} +4.73125i q^{93} +4.26875 q^{94} +5.43296 q^{95} -4.28621i q^{96} +3.13169i q^{97} +0.137063i q^{98} -2.82371i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 6 q^{9}+O(q^{10}) \) 6 * q - 4 * q^3 - 6 * q^9	\( 6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100}) \) 6 * q - 4 * q^3 - 6 * q^9 - 10 * q^10 - 10 * q^14 + 4 * q^16 + 4 * q^17 + 6 * q^22 + 10 * q^23 + 10 * q^25 + 2 * q^27 - 2 * q^29 + 2 * q^30 + 8 * q^35 + 14 * q^36 - 24 * q^38 - 6 * q^40 + 16 * q^42 - 26 * q^43 + 2 * q^48 + 8 * q^49 + 2 * q^51 + 2 * q^53 + 12 * q^55 + 8 * q^56 + 8 * q^61 - 2 * q^62 + 22 * q^64 + 10 * q^66 - 42 * q^68 + 12 * q^69 - 16 * q^74 - 30 * q^75 - 16 * q^77 - 10 * q^79 - 2 * q^81 + 28 * q^82 - 36 * q^87 - 30 * q^88 + 24 * q^90 + 10 * q^94 - 6 * q^95

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