LMFDB - Embedded newform 169.4.b.e.168.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,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, 4, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$9.97132279097$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.2
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.4.b.e.168.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.438447i q^{2} -3.68466 q^{3} +7.80776 q^{4} -17.8078i q^{5} +1.61553i q^{6} -5.43845i q^{7} -6.93087i q^{8} -13.4233 q^{9} +O(q^{10})$	\(q-0.438447i q^{2} -3.68466 q^{3} +7.80776 q^{4} -17.8078i q^{5} +1.61553i q^{6} -5.43845i q^{7} -6.93087i q^{8} -13.4233 q^{9} -7.80776 q^{10} +22.4233i q^{11} -28.7689 q^{12} -2.38447 q^{14} +65.6155i q^{15} +59.4233 q^{16} -67.9848 q^{17} +5.88540i q^{18} -80.8078i q^{19} -139.039i q^{20} +20.0388i q^{21} +9.83143 q^{22} -140.531 q^{23} +25.5379i q^{24} -192.116 q^{25} +148.946 q^{27} -42.4621i q^{28} -106.693 q^{29} +28.7689 q^{30} -276.155i q^{31} -81.5009i q^{32} -82.6222i q^{33} +29.8078i q^{34} -96.8466 q^{35} -104.806 q^{36} +4.29168i q^{37} -35.4299 q^{38} -123.423 q^{40} +227.769i q^{41} +8.78596 q^{42} -27.5294 q^{43} +175.076i q^{44} +239.039i q^{45} +61.6155i q^{46} -318.617i q^{47} -218.955 q^{48} +313.423 q^{49} +84.2329i q^{50} +250.501 q^{51} -67.6562 q^{53} -65.3050i q^{54} +399.309 q^{55} -37.6932 q^{56} +297.749i q^{57} +46.7793i q^{58} +291.115i q^{59} +512.311i q^{60} +663.311 q^{61} -121.080 q^{62} +73.0019i q^{63} +439.652 q^{64} -36.2255 q^{66} -425.101i q^{67} -530.810 q^{68} +517.810 q^{69} +42.4621i q^{70} -152.963i q^{71} +93.0351i q^{72} -117.268i q^{73} +1.88167 q^{74} +707.884 q^{75} -630.928i q^{76} +121.948 q^{77} +202.462 q^{79} -1058.20i q^{80} -186.386 q^{81} +99.8647 q^{82} +336.155i q^{83} +156.458i q^{84} +1210.66i q^{85} +12.0702i q^{86} +393.128 q^{87} +155.413 q^{88} -718.194i q^{89} +104.806 q^{90} -1097.23 q^{92} +1017.54i q^{93} -139.697 q^{94} -1439.01 q^{95} +300.303i q^{96} +759.368i q^{97} -137.420i q^{98} -300.994i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9}+O(q^{10}) $ 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9	$ 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9} + 10 q^{10} - 280 q^{12} - 92 q^{14} + 114 q^{16} - 140 q^{17} - 340 q^{22} - 290 q^{23} - 150 q^{25} + 670 q^{27} + 68 q^{29} + 280 q^{30} - 140 q^{35} - 1450 q^{36} - 620 q^{38} - 370 q^{40} - 740 q^{42} - 910 q^{43} - 480 q^{48} + 1130 q^{49} + 466 q^{51} + 1090 q^{53} + 1020 q^{55} + 344 q^{56} + 1004 q^{61} + 1000 q^{62} + 2542 q^{64} - 3196 q^{66} - 1010 q^{68} + 958 q^{69} + 1698 q^{74} + 3450 q^{75} - 510 q^{77} + 480 q^{79} + 244 q^{81} + 3030 q^{82} + 3230 q^{87} + 2040 q^{88} + 1450 q^{90} - 2080 q^{92} + 2080 q^{94} - 2540 q^{95}+O(q^{100}) $ 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9 + 10 * q^10 - 280 * q^12 - 92 * q^14 + 114 * q^16 - 140 * q^17 - 340 * q^22 - 290 * q^23 - 150 * q^25 + 670 * q^27 + 68 * q^29 + 280 * q^30 - 140 * q^35 - 1450 * q^36 - 620 * q^38 - 370 * q^40 - 740 * q^42 - 910 * q^43 - 480 * q^48 + 1130 * q^49 + 466 * q^51 + 1090 * q^53 + 1020 * q^55 + 344 * q^56 + 1004 * q^61 + 1000 * q^62 + 2542 * q^64 - 3196 * q^66 - 1010 * q^68 + 958 * q^69 + 1698 * q^74 + 3450 * q^75 - 510 * q^77 + 480 * q^79 + 244 * q^81 + 3030 * q^82 + 3230 * q^87 + 2040 * q^88 + 1450 * q^90 - 2080 * q^92 + 2080 * q^94 - 2540 * 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 $

$2$	− 0.438447i	− 0.155014i	−0.996992	−	0.0775072i	$-0.975304\pi$
$2$	− 0.438447i	− 0.155014i	0.996992	−	0.0775072i	$-0.0246961\pi$
$3$	−3.68466	−0.709113	−0.354556	−	0.935035i	$-0.615368\pi$
$3$	−3.68466	−0.709113	−0.354556	+	0.935035i	$0.615368\pi$
$4$	7.80776	0.975971
$5$	− 17.8078i	− 1.59277i	−0.604787	−	0.796387i	$-0.706742\pi$
$5$	− 17.8078i	− 1.59277i	0.604787	−	0.796387i	$-0.293258\pi$
$6$	1.61553i	0.109923i
$7$	− 5.43845i	− 0.293649i	−0.989163	−	0.146824i	$-0.953095\pi$
$7$	− 5.43845i	− 0.293649i	0.989163	−	0.146824i	$-0.0469052\pi$
$8$	− 6.93087i	− 0.306304i
$9$	−13.4233	−0.497159
$10$	−7.80776	−0.246903
$11$	22.4233i	0.614625i	0.951609	+	0.307313i	$0.0994297\pi$
$11$	22.4233i	0.614625i	−0.951609	+	0.307313i	$0.900570\pi$
$12$	−28.7689	−0.692073
$13$	0	0
$13$	0	0
$14$	−2.38447	−0.0455198
$15$	65.6155i	1.12946i
$16$	59.4233	0.928489
$17$	−67.9848	−0.969926	−0.484963	−	0.874535i	$-0.661167\pi$
$17$	−67.9848	−0.969926	−0.484963	+	0.874535i	$0.661167\pi$
$18$	5.88540i	0.0770668i
$19$	− 80.8078i	− 0.975714i	−0.872923	−	0.487857i	$-0.837779\pi$
$19$	− 80.8078i	− 0.975714i	0.872923	−	0.487857i	$-0.162221\pi$
$20$	− 139.039i	− 1.55450i
$21$	20.0388i	0.208230i
$22$	9.83143	0.0952758
$23$	−140.531	−1.27403	−0.637017	−	0.770850i	$-0.719832\pi$
$23$	−140.531	−1.27403	−0.637017	+	0.770850i	$0.719832\pi$
$24$	25.5379i	0.217204i
$25$	−192.116	−1.53693
$26$	0	0
$27$	148.946	1.06165
$28$	− 42.4621i	− 0.286592i
$29$	−106.693	−0.683187	−0.341594	−	0.939848i	$-0.610967\pi$
$29$	−106.693	−0.683187	−0.341594	+	0.939848i	$0.610967\pi$
$30$	28.7689	0.175082
$31$	− 276.155i	− 1.59997i	−0.600023	−	0.799983i	$-0.704842\pi$
$31$	− 276.155i	− 1.59997i	0.600023	−	0.799983i	$-0.295158\pi$
$32$	− 81.5009i	− 0.450233i
$33$	− 82.6222i	− 0.435839i
$34$	29.8078i	0.150353i
$35$	−96.8466	−0.467716
$36$	−104.806	−0.485212
$37$	4.29168i	0.0190688i	0.999955	+	0.00953442i	$0.00303495\pi$
$37$	4.29168i	0.0190688i	−0.999955	+	0.00953442i	$0.996965\pi$
$38$	−35.4299	−0.151250
$39$	0	0
$40$	−123.423	−0.487873
$41$	227.769i	0.867598i	0.901010	+	0.433799i	$0.142827\pi$
$41$	227.769i	0.867598i	−0.901010	+	0.433799i	$0.857173\pi$
$42$	8.78596	0.0322787
$43$	−27.5294	−0.0976323	−0.0488162	−	0.998808i	$-0.515545\pi$
$43$	−27.5294	−0.0976323	−0.0488162	+	0.998808i	$0.515545\pi$
$44$	175.076i	0.599856i
$45$	239.039i	0.791862i
$46$	61.6155i	0.197494i
$47$	− 318.617i	− 0.988832i	−0.869225	−	0.494416i	$-0.835382\pi$
$47$	− 318.617i	− 0.988832i	0.869225	−	0.494416i	$-0.164618\pi$
$48$	−218.955	−0.658403
$49$	313.423	0.913771
$50$	84.2329i	0.238247i
$51$	250.501	0.687787
$52$	0	0
$53$	−67.6562	−0.175345	−0.0876726	−	0.996149i	$-0.527943\pi$
$53$	−67.6562	−0.175345	−0.0876726	+	0.996149i	$0.527943\pi$
$54$	− 65.3050i	− 0.164572i
$55$	399.309	0.978960
$56$	−37.6932	−0.0899457
$57$	297.749i	0.691892i
$58$	46.7793i	0.105904i
$59$	291.115i	0.642371i	0.947016	+	0.321186i	$0.104081\pi$
$59$	291.115i	0.642371i	−0.947016	+	0.321186i	$0.895919\pi$
$60$	512.311i	1.10232i
$61$	663.311	1.39227	0.696133	−	0.717913i	$-0.254902\pi$
$61$	663.311	1.39227	0.696133	+	0.717913i	$0.254902\pi$
$62$	−121.080	−0.248018
$63$	73.0019i	0.145990i
$64$	439.652	0.858696
$65$	0	0
$66$	−36.2255	−0.0675613
$67$	− 425.101i	− 0.775140i	−0.921840	−	0.387570i	$-0.873315\pi$
$67$	− 425.101i	− 0.775140i	0.921840	−	0.387570i	$-0.126685\pi$
$68$	−530.810	−0.946619
$69$	517.810	0.903434
$70$	42.4621i	0.0725028i
$71$	− 152.963i	− 0.255681i	−0.991795	−	0.127841i	$-0.959195\pi$
$71$	− 152.963i	− 0.255681i	0.991795	−	0.127841i	$-0.0408046\pi$
$72$	93.0351i	0.152282i
$73$	− 117.268i	− 0.188016i	−0.995571	−	0.0940081i	$-0.970032\pi$
$73$	− 117.268i	− 0.188016i	0.995571	−	0.0940081i	$-0.0299680\pi$
$74$	1.88167	0.00295595
$75$	707.884	1.08986
$76$	− 630.928i	− 0.952268i
$77$	121.948	0.180484
$78$	0	0
$79$	202.462	0.288339	0.144169	−	0.989553i	$-0.453949\pi$
$79$	202.462	0.288339	0.144169	+	0.989553i	$0.453949\pi$
$80$	− 1058.20i	− 1.47887i
$81$	−186.386	−0.255674
$82$	99.8647	0.134490
$83$	336.155i	0.444552i	0.974984	+	0.222276i	$0.0713486\pi$
$83$	336.155i	0.444552i	−0.974984	+	0.222276i	$0.928651\pi$
$84$	156.458i	0.203226i
$85$	1210.66i	1.54487i
$86$	12.0702i	0.0151344i
$87$	393.128	0.484457
$88$	155.413	0.188262
$89$	− 718.194i	− 0.855376i	−0.903927	−	0.427688i	$-0.859328\pi$
$89$	− 718.194i	− 0.855376i	0.903927	−	0.427688i	$-0.140672\pi$
$90$	104.806	0.122750
$91$	0	0
$92$	−1097.23	−1.24342
$93$	1017.54i	1.13456i
$94$	−139.697	−0.153283
$95$	−1439.01	−1.55409
$96$	300.303i	0.319266i
$97$	759.368i	0.794868i	0.917631	+	0.397434i	$0.130099\pi$
$97$	759.368i	0.794868i	−0.917631	+	0.397434i	$0.869901\pi$
$98$	− 137.420i	− 0.141648i
$99$	− 300.994i	− 0.305566i
$100$	−1500.00	−1.50000
$101$	348.697	0.343531	0.171766	−	0.985138i	$-0.445053\pi$
$101$	348.697	0.343531	0.171766	+	0.985138i	$0.445053\pi$
$102$	− 109.831i	− 0.106617i
$103$	580.303	0.555136	0.277568	−	0.960706i	$-0.410472\pi$
$103$	580.303	0.555136	0.277568	+	0.960706i	$0.410472\pi$
$104$	0	0
$105$	356.847	0.331663
$106$	29.6637i	0.0271810i
$107$	571.493	0.516340	0.258170	−	0.966100i	$-0.416881\pi$
$107$	571.493	0.516340	0.258170	+	0.966100i	$0.416881\pi$
$108$	1162.94	1.03614
$109$	176.004i	0.154661i	0.997005	+	0.0773307i	$0.0246397\pi$
$109$	176.004i	0.154661i	−0.997005	+	0.0773307i	$0.975360\pi$
$110$	− 175.076i	− 0.151753i
$111$	− 15.8134i	− 0.0135220i
$112$	− 323.170i	− 0.272649i
$113$	1264.88	1.05301	0.526505	−	0.850172i	$-0.323502\pi$
$113$	1264.88	1.05301	0.526505	+	0.850172i	$0.323502\pi$
$114$	130.547	0.107253
$115$	2502.55i	2.02925i
$116$	−833.035	−0.666770
$117$	0	0
$118$	127.638	0.0995768
$119$	369.732i	0.284817i
$120$	454.773	0.345957
$121$	828.196	0.622236
$122$	− 290.827i	− 0.215821i
$123$	− 839.251i	− 0.615225i
$124$	− 2156.16i	− 1.56152i
$125$	1195.19i	0.855211i
$126$	32.0075	0.0226306
$127$	−2604.11	−1.81950	−0.909752	−	0.415151i	$-0.863729\pi$
$127$	−2604.11	−1.81950	−0.909752	+	0.415151i	$0.863729\pi$
$128$	− 844.772i	− 0.583344i
$129$	101.436	0.0692323
$130$	0	0
$131$	2131.70	1.42174	0.710870	−	0.703324i	$-0.248302\pi$
$131$	2131.70	1.42174	0.710870	+	0.703324i	$0.248302\pi$
$132$	− 645.094i	− 0.425366i
$133$	−439.469	−0.286517
$134$	−186.384	−0.120158
$135$	− 2652.40i	− 1.69098i
$136$	471.194i	0.297092i
$137$	− 687.985i	− 0.429040i	−0.976720	−	0.214520i	$-0.931181\pi$
$137$	− 687.985i	− 0.429040i	0.976720	−	0.214520i	$-0.0688188\pi$
$138$	− 227.032i	− 0.140045i
$139$	−679.580	−0.414685	−0.207343	−	0.978268i	$-0.566482\pi$
$139$	−679.580	−0.414685	−0.207343	+	0.978268i	$0.566482\pi$
$140$	−756.155	−0.456477
$141$	1174.00i	0.701194i
$142$	−67.0662	−0.0396343
$143$	0	0
$144$	−797.656	−0.461607
$145$	1899.97i	1.08816i
$146$	−51.4158	−0.0291452
$147$	−1154.86	−0.647966
$148$	33.5084i	0.0186106i
$149$	− 1975.46i	− 1.08615i	−0.839685	−	0.543074i	$-0.817260\pi$
$149$	− 1975.46i	− 1.08615i	0.839685	−	0.543074i	$-0.182740\pi$
$150$	− 310.370i	− 0.168944i
$151$	− 1803.24i	− 0.971824i	−0.874008	−	0.485912i	$-0.838487\pi$
$151$	− 1803.24i	− 0.971824i	0.874008	−	0.485912i	$-0.161513\pi$
$152$	−560.068	−0.298865
$153$	912.580	0.482208
$154$	− 53.4677i	− 0.0279776i
$155$	−4917.71	−2.54839
$156$	0	0
$157$	−397.168	−0.201894	−0.100947	−	0.994892i	$-0.532187\pi$
$157$	−397.168	−0.201894	−0.100947	+	0.994892i	$0.532187\pi$
$158$	− 88.7689i	− 0.0446967i
$159$	249.290	0.124340
$160$	−1451.35	−0.717120
$161$	764.272i	0.374118i
$162$	81.7206i	0.0396332i
$163$	− 941.393i	− 0.452365i	−0.974085	−	0.226183i	$-0.927375\pi$
$163$	− 941.393i	− 0.452365i	0.974085	−	0.226183i	$-0.0726246\pi$
$164$	1778.37i	0.846750i
$165$	−1471.32	−0.694193
$166$	147.386	0.0689120
$167$	− 3680.43i	− 1.70539i	−0.522408	−	0.852696i	$-0.674966\pi$
$167$	− 3680.43i	− 1.70539i	0.522408	−	0.852696i	$-0.325034\pi$
$168$	138.886	0.0637817
$169$	0	0
$170$	530.810	0.239478
$171$	1084.71i	0.485085i
$172$	−214.943	−0.0952863
$173$	−1422.77	−0.625269	−0.312634	−	0.949874i	$-0.601211\pi$
$173$	−1422.77	−0.625269	−0.312634	+	0.949874i	$0.601211\pi$
$174$	− 172.366i	− 0.0750978i
$175$	1044.82i	0.451318i
$176$	1332.47i	0.570673i
$177$	− 1072.66i	− 0.455514i
$178$	−314.890	−0.132596
$179$	−1167.89	−0.487666	−0.243833	−	0.969817i	$-0.578405\pi$
$179$	−1167.89	−0.487666	−0.243833	+	0.969817i	$0.578405\pi$
$180$	1866.36i	0.772834i
$181$	1133.96	0.465673	0.232836	−	0.972516i	$-0.425199\pi$
$181$	1133.96	0.465673	0.232836	+	0.972516i	$0.425199\pi$
$182$	0	0
$183$	−2444.07	−0.987274
$184$	974.004i	0.390242i
$185$	76.4252	0.0303724
$186$	446.137	0.175873
$187$	− 1524.44i	− 0.596141i
$188$	− 2487.69i	− 0.965071i
$189$	− 810.035i	− 0.311753i
$190$	630.928i	0.240907i
$191$	2682.12	1.01608	0.508040	−	0.861333i	$-0.330370\pi$
$191$	2682.12	1.01608	0.508040	+	0.861333i	$0.330370\pi$
$192$	−1619.97	−0.608913
$193$	1970.67i	0.734983i	0.930027	+	0.367491i	$0.119783\pi$
$193$	1970.67i	0.734983i	−0.930027	+	0.367491i	$0.880217\pi$
$194$	332.943	0.123216
$195$	0	0
$196$	2447.14	0.891813
$197$	− 4016.05i	− 1.45244i	−0.687460	−	0.726222i	$-0.741274\pi$
$197$	− 4016.05i	− 1.45244i	0.687460	−	0.726222i	$-0.258726\pi$
$198$	−131.970	−0.0473672
$199$	4226.06	1.50541	0.752707	−	0.658356i	$-0.228748\pi$
$199$	4226.06	1.50541	0.752707	+	0.658356i	$0.228748\pi$
$200$	1331.53i	0.470768i
$201$	1566.35i	0.549662i
$202$	− 152.885i	− 0.0532523i
$203$	580.245i	0.200617i
$204$	1955.85	0.671260
$205$	4056.06	1.38189
$206$	− 254.432i	− 0.0860541i
$207$	1886.39	0.633398
$208$	0	0
$209$	1811.98	0.599699
$210$	− 156.458i	− 0.0514126i
$211$	1364.67	0.445249	0.222625	−	0.974904i	$-0.428538\pi$
$211$	1364.67	0.445249	0.222625	+	0.974904i	$0.428538\pi$
$212$	−528.244	−0.171132
$213$	563.617i	0.181307i
$214$	− 250.570i	− 0.0800401i
$215$	490.237i	0.155506i
$216$	− 1032.33i	− 0.325189i
$217$	−1501.86	−0.469828
$218$	77.1683	0.0239748
$219$	432.093i	0.133325i
$220$	3117.71	0.955436
$221$	0	0
$222$	−6.93332	−0.00209610
$223$	− 1059.47i	− 0.318149i	−0.987267	−	0.159075i	$-0.949149\pi$
$223$	− 1059.47i	− 0.318149i	0.987267	−	0.159075i	$-0.0508510\pi$
$224$	−443.239	−0.132210
$225$	2578.84	0.764099
$226$	− 554.584i	− 0.163232i
$227$	3464.19i	1.01289i	0.862272	+	0.506446i	$0.169041\pi$
$227$	3464.19i	1.01289i	−0.862272	+	0.506446i	$0.830959\pi$
$228$	2324.75i	0.675266i
$229$	2324.64i	0.670815i	0.942073	+	0.335407i	$0.108874\pi$
$229$	2324.64i	0.670815i	−0.942073	+	0.335407i	$0.891126\pi$
$230$	1097.23	0.314563
$231$	−449.336	−0.127983
$232$	739.476i	0.209263i
$233$	3731.01	1.04904	0.524521	−	0.851398i	$-0.324245\pi$
$233$	3731.01	1.04904	0.524521	+	0.851398i	$0.324245\pi$
$234$	0	0
$235$	−5673.86	−1.57499
$236$	2272.95i	0.626935i
$237$	−746.004	−0.204465
$238$	162.108	0.0441508
$239$	6044.47i	1.63592i	0.575278	+	0.817958i	$0.304894\pi$
$239$	6044.47i	1.63592i	−0.575278	+	0.817958i	$0.695106\pi$
$240$	3899.09i	1.04869i
$241$	5173.96i	1.38292i	0.722414	+	0.691461i	$0.243033\pi$
$241$	5173.96i	1.38292i	−0.722414	+	0.691461i	$0.756967\pi$
$242$	− 363.120i	− 0.0964556i
$243$	−3334.77	−0.880353
$244$	5178.97	1.35881
$245$	− 5581.37i	− 1.45543i
$246$	−367.967	−0.0953688
$247$	0	0
$248$	−1914.00	−0.490076
$249$	− 1238.62i	− 0.315238i
$250$	524.029	0.132570
$251$	−5620.73	−1.41346	−0.706728	−	0.707486i	$-0.749829\pi$
$251$	−5620.73	−1.41346	−0.706728	+	0.707486i	$0.749829\pi$
$252$	569.981i	0.142482i
$253$	− 3151.17i	− 0.783054i
$254$	1141.76i	0.282050i
$255$	− 4460.86i	− 1.09549i
$256$	3146.83	0.768270
$257$	1674.14	0.406342	0.203171	−	0.979143i	$-0.434875\pi$
$257$	1674.14	0.406342	0.203171	+	0.979143i	$0.434875\pi$
$258$	− 44.4745i	− 0.0107320i
$259$	23.3401	0.00559954
$260$	0	0
$261$	1432.17	0.339653
$262$	− 934.640i	− 0.220390i
$263$	−6309.18	−1.47924	−0.739622	−	0.673023i	$-0.764996\pi$
$263$	−6309.18	−1.47924	−0.739622	+	0.673023i	$0.764996\pi$
$264$	−572.644	−0.133499
$265$	1204.81i	0.279285i
$266$	192.684i	0.0444143i
$267$	2646.30i	0.606558i
$268$	− 3319.09i	− 0.756514i
$269$	−2482.73	−0.562731	−0.281366	−	0.959601i	$-0.590787\pi$
$269$	−2482.73	−0.562731	−0.281366	+	0.959601i	$0.590787\pi$
$270$	−1162.94	−0.262126
$271$	− 2835.72i	− 0.635638i	−0.948151	−	0.317819i	$-0.897050\pi$
$271$	− 2835.72i	− 0.635638i	0.948151	−	0.317819i	$-0.102950\pi$
$272$	−4039.88	−0.900566
$273$	0	0
$274$	−301.645	−0.0665075
$275$	− 4307.88i	− 0.944637i
$276$	4042.94	0.881725
$277$	3837.51	0.832396	0.416198	−	0.909274i	$-0.363362\pi$
$277$	3837.51	0.832396	0.416198	+	0.909274i	$0.363362\pi$
$278$	297.960i	0.0642822i
$279$	3706.91i	0.795438i
$280$	671.231i	0.143263i
$281$	9122.13i	1.93659i	0.249819	+	0.968293i	$0.419629\pi$
$281$	9122.13i	1.93659i	−0.249819	+	0.968293i	$0.580371\pi$
$282$	514.735	0.108695
$283$	−2127.85	−0.446952	−0.223476	−	0.974709i	$-0.571740\pi$
$283$	−2127.85	−0.446952	−0.223476	+	0.974709i	$0.571740\pi$
$284$	− 1194.30i	− 0.249537i
$285$	5302.24	1.10203
$286$	0	0
$287$	1238.71	0.254769
$288$	1094.01i	0.223838i
$289$	−291.061	−0.0592430
$290$	833.035	0.168681
$291$	− 2798.01i	− 0.563651i
$292$	− 915.601i	− 0.183498i
$293$	− 8274.77i	− 1.64989i	−0.565215	−	0.824944i	$-0.691207\pi$
$293$	− 8274.77i	− 1.64989i	0.565215	−	0.824944i	$-0.308793\pi$
$294$	506.344i	0.100444i
$295$	5184.10	1.02315
$296$	29.7450	0.00584086
$297$	3339.86i	0.652520i
$298$	−866.136	−0.168369
$299$	0	0
$300$	5526.99	1.06367
$301$	149.717i	0.0286696i
$302$	−790.625	−0.150647
$303$	−1284.83	−0.243602
$304$	− 4801.86i	− 0.905940i
$305$	− 11812.1i	− 2.21757i
$306$	− 400.118i	− 0.0747492i
$307$	3610.49i	0.671211i	0.942003	+	0.335605i	$0.108941\pi$
$307$	3610.49i	0.671211i	−0.942003	+	0.335605i	$0.891059\pi$
$308$	952.140	0.176147
$309$	−2138.22	−0.393654
$310$	2156.16i	0.395037i
$311$	−3331.06	−0.607354	−0.303677	−	0.952775i	$-0.598214\pi$
$311$	−3331.06	−0.607354	−0.303677	+	0.952775i	$0.598214\pi$
$312$	0	0
$313$	−358.125	−0.0646724	−0.0323362	−	0.999477i	$-0.510295\pi$
$313$	−358.125	−0.0646724	−0.0323362	+	0.999477i	$0.510295\pi$
$314$	174.137i	0.0312966i
$315$	1300.00	0.232529
$316$	1580.78	0.281410
$317$	3047.46i	0.539944i	0.962868	+	0.269972i	$0.0870144\pi$
$317$	3047.46i	0.539944i	−0.962868	+	0.269972i	$0.912986\pi$
$318$	− 109.301i	− 0.0192744i
$319$	− 2392.41i	− 0.419904i
$320$	− 7829.23i	− 1.36771i
$321$	−2105.76	−0.366143
$322$	335.093	0.0579938
$323$	5493.70i	0.946371i
$324$	−1455.26	−0.249530
$325$	0	0
$326$	−412.751	−0.0701232
$327$	− 648.514i	− 0.109672i
$328$	1578.64	0.265749
$329$	−1732.78	−0.290369
$330$	645.094i	0.107610i
$331$	7694.77i	1.27777i	0.769301	+	0.638887i	$0.220605\pi$
$331$	7694.77i	1.27777i	−0.769301	+	0.638887i	$0.779395\pi$
$332$	2624.62i	0.433870i
$333$	− 57.6084i	− 0.00948025i
$334$	−1613.68	−0.264360
$335$	−7570.10	−1.23462
$336$	1190.77i	0.193339i
$337$	−4712.21	−0.761693	−0.380846	−	0.924638i	$-0.624367\pi$
$337$	−4712.21	−0.761693	−0.380846	+	0.924638i	$0.624367\pi$
$338$	0	0
$339$	−4660.66	−0.746703
$340$	9452.53i	1.50775i
$341$	6192.31	0.983380
$342$	475.586	0.0751952
$343$	− 3569.92i	− 0.561976i
$344$	190.803i	0.0299052i
$345$	− 9221.03i	− 1.43897i
$346$	623.811i	0.0969257i
$347$	−5261.98	−0.814058	−0.407029	−	0.913415i	$-0.633435\pi$
$347$	−5261.98	−0.814058	−0.407029	+	0.913415i	$0.633435\pi$
$348$	3069.45	0.472815
$349$	− 50.3345i	− 0.00772018i	−0.999993	−	0.00386009i	$-0.998771\pi$
$349$	− 50.3345i	− 0.00772018i	0.999993	−	0.00386009i	$-0.00122871\pi$
$350$	458.096	0.0699608
$351$	0	0
$352$	1827.52	0.276725
$353$	9057.64i	1.36569i	0.730562	+	0.682846i	$0.239258\pi$
$353$	9057.64i	1.36569i	−0.730562	+	0.682846i	$0.760742\pi$
$354$	−470.304	−0.0706112
$355$	−2723.93	−0.407243
$356$	− 5607.49i	− 0.834821i
$357$	− 1362.34i	− 0.201968i
$358$	512.059i	0.0755953i
$359$	− 7177.86i	− 1.05525i	−0.849479	−	0.527623i	$-0.823083\pi$
$359$	− 7177.86i	− 1.05525i	0.849479	−	0.527623i	$-0.176917\pi$
$360$	1656.75	0.242551
$361$	329.105	0.0479815
$362$	− 497.183i	− 0.0721861i
$363$	−3051.62	−0.441235
$364$	0	0
$365$	−2088.28	−0.299467
$366$	1071.60i	0.153042i
$367$	4004.14	0.569522	0.284761	−	0.958599i	$-0.408086\pi$
$367$	4004.14	0.569522	0.284761	+	0.958599i	$0.408086\pi$
$368$	−8350.83	−1.18293
$369$	− 3057.41i	− 0.431334i
$370$	− 33.5084i	− 0.00470816i
$371$	367.945i	0.0514899i
$372$	7944.70i	1.10729i
$373$	−10014.2	−1.39012	−0.695060	−	0.718952i	$-0.744622\pi$
$373$	−10014.2	−1.39012	−0.695060	+	0.718952i	$0.744622\pi$
$374$	−668.388	−0.0924105
$375$	− 4403.88i	− 0.606441i
$376$	−2208.30	−0.302883
$377$	0	0
$378$	−355.158	−0.0483263
$379$	− 8169.12i	− 1.10717i	−0.832791	−	0.553587i	$-0.813258\pi$
$379$	− 8169.12i	− 1.10717i	0.832791	−	0.553587i	$-0.186742\pi$
$380$	−11235.4	−1.51675
$381$	9595.24	1.29023
$382$	− 1175.97i	− 0.157507i
$383$	− 7310.25i	− 0.975290i	−0.873042	−	0.487645i	$-0.837856\pi$
$383$	− 7310.25i	− 0.975290i	0.873042	−	0.487645i	$-0.162144\pi$
$384$	3112.70i	0.413656i
$385$	− 2171.62i	− 0.287470i
$386$	864.033	0.113933
$387$	369.535	0.0485388
$388$	5928.97i	0.775767i
$389$	−8785.47	−1.14509	−0.572546	−	0.819872i	$-0.694044\pi$
$389$	−8785.47	−1.14509	−0.572546	+	0.819872i	$0.694044\pi$
$390$	0	0
$391$	9553.99	1.23572
$392$	− 2172.30i	− 0.279892i
$393$	−7854.60	−1.00817
$394$	−1760.82	−0.225150
$395$	− 3605.40i	− 0.459259i
$396$	− 2350.09i	− 0.298224i
$397$	− 11266.8i	− 1.42434i	−0.702006	−	0.712171i	$-0.747712\pi$
$397$	− 11266.8i	− 1.42434i	0.702006	−	0.712171i	$-0.252288\pi$
$398$	− 1852.90i	− 0.233361i
$399$	1619.29	0.203173
$400$	−11416.2	−1.42702
$401$	− 1576.23i	− 0.196293i	−0.995172	−	0.0981464i	$-0.968709\pi$
$401$	− 1576.23i	− 0.196293i	0.995172	−	0.0981464i	$-0.0312913\pi$
$402$	686.763	0.0852055
$403$	0	0
$404$	2722.54	0.335276
$405$	3319.12i	0.407231i
$406$	254.407	0.0310985
$407$	−96.2335	−0.0117202
$408$	− 1736.19i	− 0.210672i
$409$	6755.78i	0.816753i	0.912814	+	0.408377i	$0.133905\pi$
$409$	6755.78i	0.816753i	−0.912814	+	0.408377i	$0.866095\pi$
$410$	− 1778.37i	− 0.214213i
$411$	2534.99i	0.304238i
$412$	4530.87	0.541796
$413$	1583.21	0.188631
$414$	− 827.083i	− 0.0981858i
$415$	5986.17	0.708072
$416$	0	0
$417$	2504.02	0.294059
$418$	− 794.456i	− 0.0929620i
$419$	10756.2	1.25411	0.627057	−	0.778973i	$-0.284259\pi$
$419$	10756.2	1.25411	0.627057	+	0.778973i	$0.284259\pi$
$420$	2786.17	0.323694
$421$	7886.03i	0.912925i	0.889743	+	0.456463i	$0.150884\pi$
$421$	7886.03i	0.912925i	−0.889743	+	0.456463i	$0.849116\pi$
$422$	− 598.335i	− 0.0690201i
$423$	4276.89i	0.491607i
$424$	468.916i	0.0537089i
$425$	13061.0	1.49071
$426$	247.116	0.0281052
$427$	− 3607.38i	− 0.408837i
$428$	4462.08	0.503932
$429$	0	0
$430$	214.943	0.0241057
$431$	− 14084.6i	− 1.57409i	−0.616897	−	0.787044i	$-0.711610\pi$
$431$	− 14084.6i	− 1.57409i	0.616897	−	0.787044i	$-0.288390\pi$
$432$	8850.86	0.985735
$433$	−1864.14	−0.206894	−0.103447	−	0.994635i	$-0.532987\pi$
$433$	−1864.14	−0.206894	−0.103447	+	0.994635i	$0.532987\pi$
$434$	658.485i	0.0728301i
$435$	− 7000.73i	− 0.771630i
$436$	1374.20i	0.150945i
$437$	11356.0i	1.24309i
$438$	189.450	0.0206673
$439$	−6154.49	−0.669106	−0.334553	−	0.942377i	$-0.608585\pi$
$439$	−6154.49	−0.669106	−0.334553	+	0.942377i	$0.608585\pi$
$440$	− 2767.56i	− 0.299859i
$441$	−4207.17	−0.454289
$442$	0	0
$443$	−14539.3	−1.55933	−0.779663	−	0.626200i	$-0.784609\pi$
$443$	−14539.3	−1.55933	−0.779663	+	0.626200i	$0.784609\pi$
$444$	− 123.467i	− 0.0131970i
$445$	−12789.4	−1.36242
$446$	−464.521	−0.0493177
$447$	7278.90i	0.770202i
$448$	− 2391.03i	− 0.252155i
$449$	7043.87i	0.740358i	0.928960	+	0.370179i	$0.120704\pi$
$449$	7043.87i	0.740358i	−0.928960	+	0.370179i	$0.879296\pi$
$450$	− 1130.68i	− 0.118446i
$451$	−5107.33	−0.533248
$452$	9875.90	1.02771
$453$	6644.32i	0.689133i
$454$	1518.87	0.157013
$455$	0	0
$456$	2063.66	0.211929
$457$	14098.9i	1.44314i	0.692340	+	0.721572i	$0.256580\pi$
$457$	14098.9i	1.44314i	−0.692340	+	0.721572i	$0.743420\pi$
$458$	1019.23	0.103986
$459$	−10126.1	−1.02973
$460$	19539.3i	1.98049i
$461$	14449.7i	1.45985i	0.683529	+	0.729924i	$0.260444\pi$
$461$	14449.7i	1.45985i	−0.683529	+	0.729924i	$0.739556\pi$
$462$	197.010i	0.0198393i
$463$	− 15806.5i	− 1.58659i	−0.608840	−	0.793293i	$-0.708365\pi$
$463$	− 15806.5i	− 1.58659i	0.608840	−	0.793293i	$-0.291635\pi$
$464$	−6340.06	−0.634332
$465$	18120.1	1.80709
$466$	− 1635.85i	− 0.162617i
$467$	15071.3	1.49340	0.746699	−	0.665162i	$-0.231638\pi$
$467$	15071.3	1.49340	0.746699	+	0.665162i	$0.231638\pi$
$468$	0	0
$469$	−2311.89	−0.227619
$470$	2487.69i	0.244146i
$471$	1463.43	0.143166
$472$	2017.68	0.196761
$473$	− 617.299i	− 0.0600073i
$474$	327.083i	0.0316950i
$475$	15524.5i	1.49961i
$476$	2886.78i	0.277973i
$477$	908.169	0.0871744
$478$	2650.18	0.253591
$479$	392.545i	0.0374443i	0.999825	+	0.0187222i	$0.00595980\pi$
$479$	392.545i	0.0374443i	−0.999825	+	0.0187222i	$0.994040\pi$
$480$	5347.73	0.508519
$481$	0	0
$482$	2268.51	0.214373
$483$	− 2816.08i	− 0.265292i
$484$	6466.36	0.607284
$485$	13522.7	1.26605
$486$	1462.12i	0.136467i
$487$	9497.89i	0.883758i	0.897075	+	0.441879i	$0.145688\pi$
$487$	9497.89i	0.883758i	−0.897075	+	0.441879i	$0.854312\pi$
$488$	− 4597.32i	− 0.426457i
$489$	3468.71i	0.320778i
$490$	−2447.14	−0.225613
$491$	1893.82	0.174067	0.0870337	−	0.996205i	$-0.472261\pi$
$491$	1893.82	0.174067	0.0870337	+	0.996205i	$0.472261\pi$
$492$	− 6552.67i	− 0.600442i
$493$	7253.52	0.662641
$494$	0	0
$495$	−5360.04	−0.486699
$496$	− 16410.1i	− 1.48555i
$497$	−831.881	−0.0750804
$498$	−543.068	−0.0488664
$499$	− 13370.1i	− 1.19945i	−0.800205	−	0.599727i	$-0.795276\pi$
$499$	− 13370.1i	− 1.19945i	0.800205	−	0.599727i	$-0.204724\pi$
$500$	9331.79i	0.834661i
$501$	13561.1i	1.20932i
$502$	2464.39i	0.219106i
$503$	5554.71	0.492391	0.246195	−	0.969220i	$-0.420820\pi$
$503$	5554.71	0.492391	0.246195	+	0.969220i	$0.420820\pi$
$504$	505.966	0.0447173
$505$	− 6209.51i	− 0.547168i
$506$	−1381.62	−0.121385
$507$	0	0
$508$	−20332.3	−1.77578
$509$	2197.55i	0.191365i	0.995412	+	0.0956824i	$0.0305033\pi$
$509$	2197.55i	0.191365i	−0.995412	+	0.0956824i	$0.969497\pi$
$510$	−1955.85	−0.169817
$511$	−637.756	−0.0552107
$512$	− 8137.89i	− 0.702437i
$513$	− 12036.0i	− 1.03587i
$514$	− 734.022i	− 0.0629890i
$515$	− 10333.9i	− 0.884206i
$516$	791.991	0.0675687
$517$	7144.45	0.607761
$518$	− 10.2334i	0 0.000868010i
$519$	5242.44	0.443386
$520$	0	0
$521$	17005.2	1.42997	0.714983	−	0.699142i	$-0.246435\pi$
$521$	17005.2	1.42997	0.714983	+	0.699142i	$0.246435\pi$
$522$	− 627.932i	− 0.0526511i
$523$	−14486.2	−1.21116	−0.605581	−	0.795783i	$-0.707059\pi$
$523$	−14486.2	−1.21116	−0.605581	+	0.795783i	$0.707059\pi$
$524$	16643.8	1.38758
$525$	− 3849.79i	− 0.320035i
$526$	2766.24i	0.229304i
$527$	18774.4i	1.55185i
$528$	− 4909.68i	− 0.404671i
$529$	7582.03	0.623163
$530$	528.244	0.0432933
$531$	− 3907.72i	− 0.319361i
$532$	−3431.27	−0.279632
$533$	0	0
$534$	1160.26	0.0940252
$535$	− 10177.0i	− 0.822413i
$536$	−2946.32	−0.237429
$537$	4303.28	0.345810
$538$	1088.55i	0.0872315i
$539$	7027.98i	0.561626i
$540$	− 20709.3i	− 1.65034i
$541$	15266.7i	1.21325i	0.794990	+	0.606623i	$0.207476\pi$
$541$	15266.7i	1.21325i	−0.794990	+	0.606623i	$0.792524\pi$
$542$	−1243.31	−0.0985330
$543$	−4178.27	−0.330215
$544$	5540.83i	0.436693i
$545$	3134.23	0.246341
$546$	0	0
$547$	15260.5	1.19286	0.596430	−	0.802665i	$-0.296586\pi$
$547$	15260.5	1.19286	0.596430	+	0.802665i	$0.296586\pi$
$548$	− 5371.62i	− 0.418731i
$549$	−8903.81	−0.692177
$550$	−1888.78	−0.146432
$551$	8621.64i	0.666595i
$552$	− 3588.87i	− 0.276726i
$553$	− 1101.08i	− 0.0846703i
$554$	− 1682.55i	− 0.129033i
$555$	−281.601	−0.0215374
$556$	−5306.00	−0.404721
$557$	10442.1i	0.794337i	0.917746	+	0.397169i	$0.130007\pi$
$557$	10442.1i	0.794337i	−0.917746	+	0.397169i	$0.869993\pi$
$558$	1625.29	0.123304
$559$	0	0
$560$	−5754.94	−0.434269
$561$	5617.06i	0.422731i
$562$	3999.57	0.300199
$563$	−7145.26	−0.534879	−0.267440	−	0.963575i	$-0.586178\pi$
$563$	−7145.26	−0.534879	−0.267440	+	0.963575i	$0.586178\pi$
$564$	9166.29i	0.684344i
$565$	− 22524.7i	− 1.67721i
$566$	932.950i	0.0692841i
$567$	1013.65i	0.0750783i
$568$	−1060.17	−0.0783162
$569$	−4438.86	−0.327042	−0.163521	−	0.986540i	$-0.552285\pi$
$569$	−4438.86	−0.327042	−0.163521	+	0.986540i	$0.552285\pi$
$570$	− 2324.75i	− 0.170830i
$571$	−10117.3	−0.741497	−0.370748	−	0.928733i	$-0.620899\pi$
$571$	−10117.3	−0.741497	−0.370748	+	0.928733i	$0.620899\pi$
$572$	0	0
$573$	−9882.70	−0.720516
$574$	− 543.109i	− 0.0394929i
$575$	26998.4	1.95810
$576$	−5901.58	−0.426909
$577$	3105.60i	0.224069i	0.993704	+	0.112035i	$0.0357368\pi$
$577$	3105.60i	0.224069i	−0.993704	+	0.112035i	$0.964263\pi$
$578$	127.615i	0.00918352i
$579$	− 7261.23i	− 0.521186i
$580$	14834.5i	1.06202i
$581$	1828.16	0.130542
$582$	−1226.78	−0.0873741
$583$	− 1517.08i	− 0.107772i
$584$	−812.769	−0.0575901
$585$	0	0
$586$	−3628.05	−0.255757
$587$	19662.3i	1.38254i	0.722597	+	0.691270i	$0.242948\pi$
$587$	19662.3i	1.38254i	−0.722597	+	0.691270i	$0.757052\pi$
$588$	−9016.86	−0.632396
$589$	−22315.5	−1.56111
$590$	− 2272.95i	− 0.158603i
$591$	14797.8i	1.02995i
$592$	255.026i	0.0177052i
$593$	− 6395.51i	− 0.442888i	−0.975173	−	0.221444i	$-0.928923\pi$
$593$	− 6395.51i	− 0.442888i	0.975173	−	0.221444i	$-0.0710769\pi$
$594$	1464.35	0.101150
$595$	6584.10	0.453650
$596$	− 15423.9i	− 1.06005i
$597$	−15571.6	−1.06751
$598$	0	0
$599$	8878.48	0.605618	0.302809	−	0.953051i	$-0.402076\pi$
$599$	8878.48	0.605618	0.302809	+	0.953051i	$0.402076\pi$
$600$	− 4906.25i	− 0.333828i
$601$	19100.6	1.29639	0.648194	−	0.761475i	$-0.275525\pi$
$601$	19100.6	1.29639	0.648194	+	0.761475i	$0.275525\pi$
$602$	65.6430	0.00444420
$603$	5706.26i	0.385368i
$604$	− 14079.3i	− 0.948472i
$605$	− 14748.3i	− 0.991082i
$606$	563.330i	0.0377619i
$607$	16595.8	1.10972	0.554861	−	0.831943i	$-0.312771\pi$
$607$	16595.8	1.10972	0.554861	+	0.831943i	$0.312771\pi$
$608$	−6585.91	−0.439299
$609$	− 2138.01i	− 0.142260i
$610$	−5178.97	−0.343755
$611$	0	0
$612$	7125.21	0.470620
$613$	16469.2i	1.08513i	0.840015	+	0.542564i	$0.182546\pi$
$613$	16469.2i	1.08513i	−0.840015	+	0.542564i	$0.817454\pi$
$614$	1583.01	0.104047
$615$	−14945.2	−0.979915
$616$	− 845.205i	− 0.0552829i
$617$	10116.0i	0.660055i	0.943971	+	0.330027i	$0.107058\pi$
$617$	10116.0i	0.660055i	−0.943971	+	0.330027i	$0.892942\pi$
$618$	937.496i	0.0610220i
$619$	− 18854.8i	− 1.22430i	−0.790743	−	0.612148i	$-0.790306\pi$
$619$	− 18854.8i	− 1.22430i	0.790743	−	0.612148i	$-0.209694\pi$
$620$	−38396.3	−2.48715
$621$	−20931.6	−1.35258
$622$	1460.49i	0.0941487i
$623$	−3905.86	−0.251180
$624$	0	0
$625$	−2730.82	−0.174773
$626$	157.019i	0.0100252i
$627$	−6676.51	−0.425254
$628$	−3100.99	−0.197043
$629$	− 291.769i	− 0.0184954i
$630$	− 569.981i	− 0.0360454i
$631$	− 18946.2i	− 1.19531i	−0.801755	−	0.597653i	$-0.796100\pi$
$631$	− 18946.2i	− 1.19531i	0.801755	−	0.597653i	$-0.203900\pi$
$632$	− 1403.24i	− 0.0883194i
$633$	−5028.33	−0.315732
$634$	1336.15	0.0836991
$635$	46373.3i	2.89806i
$636$	1946.40	0.121352
$637$	0	0
$638$	−1048.95	−0.0650912
$639$	2053.27i	0.127114i
$640$	−15043.5	−0.929135
$641$	23586.9	1.45340	0.726698	−	0.686957i	$-0.241054\pi$
$641$	23586.9	1.45340	0.726698	+	0.686957i	$0.241054\pi$
$642$	923.263i	0.0567575i
$643$	− 27153.0i	− 1.66534i	−0.553772	−	0.832669i	$-0.686812\pi$
$643$	− 27153.0i	− 1.66534i	0.553772	−	0.832669i	$-0.313188\pi$
$644$	5967.25i	0.365128i
$645$	− 1806.35i	− 0.110272i
$646$	2408.70	0.146701
$647$	−6856.72	−0.416639	−0.208319	−	0.978061i	$-0.566799\pi$
$647$	−6856.72	−0.416639	−0.208319	+	0.978061i	$0.566799\pi$
$648$	1291.82i	0.0783140i
$649$	−6527.75	−0.394817
$650$	0	0
$651$	5533.83	0.333161
$652$	− 7350.17i	− 0.441495i
$653$	−8073.89	−0.483853	−0.241926	−	0.970295i	$-0.577779\pi$
$653$	−8073.89	−0.483853	−0.241926	+	0.970295i	$0.577779\pi$
$654$	−284.339	−0.0170008
$655$	− 37960.9i	− 2.26451i
$656$	13534.8i	0.805555i
$657$	1574.12i	0.0934739i
$658$	759.734i	0.0450114i
$659$	5305.73	0.313629	0.156815	−	0.987628i	$-0.449877\pi$
$659$	5305.73	0.313629	0.156815	+	0.987628i	$0.449877\pi$
$660$	−11487.7	−0.677512
$661$	− 25848.3i	− 1.52100i	−0.649336	−	0.760502i	$-0.724953\pi$
$661$	− 25848.3i	− 1.52100i	0.649336	−	0.760502i	$-0.275047\pi$
$662$	3373.75	0.198073
$663$	0	0
$664$	2329.85	0.136168
$665$	7825.96i	0.456357i
$666$	−25.2583	−0.00146958
$667$	14993.7	0.870404
$668$	− 28735.9i	− 1.66441i
$669$	3903.78i	0.225604i
$670$	3319.09i	0.191385i
$671$	14873.6i	0.855722i
$672$	1633.18	0.0937521
$673$	14529.1	0.832177	0.416089	−	0.909324i	$-0.363401\pi$
$673$	14529.1	0.832177	0.416089	+	0.909324i	$0.363401\pi$
$674$	2066.06i	0.118073i
$675$	−28615.0	−1.63169
$676$	0	0
$677$	12058.1	0.684535	0.342267	−	0.939603i	$-0.388805\pi$
$677$	12058.1	0.684535	0.342267	+	0.939603i	$0.388805\pi$
$678$	2043.45i	0.115750i
$679$	4129.78	0.233412
$680$	8390.91	0.473201
$681$	− 12764.4i	− 0.718255i
$682$	− 2715.00i	− 0.152438i
$683$	− 30028.8i	− 1.68231i	−0.540792	−	0.841156i	$-0.681875\pi$
$683$	− 30028.8i	− 1.68231i	0.540792	−	0.841156i	$-0.318125\pi$
$684$	8469.13i	0.473429i
$685$	−12251.5	−0.683364
$686$	−1565.22	−0.0871144
$687$	− 8565.50i	− 0.475683i
$688$	−1635.89	−0.0906505
$689$	0	0
$690$	−4042.94	−0.223061
$691$	449.696i	0.0247572i	0.999923	+	0.0123786i	$0.00394034\pi$
$691$	449.696i	0.0247572i	−0.999923	+	0.0123786i	$0.996060\pi$
$692$	−11108.7	−0.610244
$693$	−1636.94	−0.0897291
$694$	2307.10i	0.126191i
$695$	12101.8i	0.660500i
$696$	− 2724.72i	− 0.148391i
$697$	− 15484.8i	− 0.841506i
$698$	−22.0690	−0.00119674
$699$	−13747.5	−0.743889
$700$	8157.67i	0.440473i
$701$	26986.0	1.45399	0.726994	−	0.686644i	$-0.240917\pi$
$701$	26986.0	1.45399	0.726994	+	0.686644i	$0.240917\pi$
$702$	0	0
$703$	346.801	0.0186057
$704$	9858.46i	0.527776i
$705$	20906.2	1.11684
$706$	3971.29	0.211702
$707$	− 1896.37i	− 0.100877i
$708$	− 8375.06i	− 0.444568i
$709$	9098.87i	0.481968i	0.970529	+	0.240984i	$0.0774701\pi$
$709$	9098.87i	0.481968i	−0.970529	+	0.240984i	$0.922530\pi$
$710$	1194.30i	0.0631285i
$711$	−2717.71	−0.143350
$712$	−4977.71	−0.262005
$713$	38808.4i	2.03841i
$714$	−597.312	−0.0313079
$715$	0	0
$716$	−9118.62	−0.475948
$717$	− 22271.8i	− 1.16005i
$718$	−3147.11	−0.163578
$719$	6293.55	0.326439	0.163220	−	0.986590i	$-0.447812\pi$
$719$	6293.55	0.326439	0.163220	+	0.986590i	$0.447812\pi$
$720$	14204.5i	0.735235i
$721$	− 3155.95i	− 0.163015i
$722$	− 144.295i	− 0.00743783i
$723$	− 19064.3i	− 0.980648i
$724$	8853.72	0.454483
$725$	20497.5	1.05001
$726$	1337.97i	0.0683979i
$727$	−18070.7	−0.921878	−0.460939	−	0.887432i	$-0.652487\pi$
$727$	−18070.7	−0.921878	−0.460939	+	0.887432i	$0.652487\pi$
$728$	0	0
$729$	17319.9	0.879944
$730$	915.601i	0.0464218i
$731$	1871.58	0.0946962
$732$	−19082.7	−0.963550
$733$	34771.5i	1.75214i	0.482188	+	0.876068i	$0.339842\pi$
$733$	34771.5i	1.75214i	−0.482188	+	0.876068i	$0.660158\pi$
$734$	− 1755.61i	− 0.0882842i
$735$	20565.4i	1.03206i
$736$	11453.4i	0.573613i
$737$	9532.17	0.476421
$738$	−1340.51	−0.0668631
$739$	− 23631.5i	− 1.17632i	−0.808746	−	0.588158i	$-0.799853\pi$
$739$	− 23631.5i	− 1.17632i	0.808746	−	0.588158i	$-0.200147\pi$
$740$	596.710	0.0296425
$741$	0	0
$742$	161.324	0.00798167
$743$	32502.8i	1.60486i	0.596745	+	0.802431i	$0.296460\pi$
$743$	32502.8i	1.60486i	−0.596745	+	0.802431i	$0.703540\pi$
$744$	7052.42	0.347519
$745$	−35178.6	−1.72999
$746$	4390.69i	0.215489i
$747$	− 4512.31i	− 0.221013i
$748$	− 11902.5i	− 0.581816i
$749$	− 3108.04i	− 0.151622i
$750$	−1930.87	−0.0940072
$751$	−2020.86	−0.0981920	−0.0490960	−	0.998794i	$-0.515634\pi$
$751$	−2020.86	−0.0981920	−0.0490960	+	0.998794i	$0.515634\pi$
$752$	− 18933.3i	− 0.918120i
$753$	20710.5	1.00230
$754$	0	0
$755$	−32111.6	−1.54790
$756$	− 6324.56i	− 0.304262i
$757$	12568.2	0.603434	0.301717	−	0.953398i	$-0.402440\pi$
$757$	12568.2	0.603434	0.301717	+	0.953398i	$0.402440\pi$
$758$	−3581.73	−0.171628
$759$	11611.0i	0.555273i
$760$	9973.56i	0.476025i
$761$	8704.81i	0.414651i	0.978272	+	0.207325i	$0.0664758\pi$
$761$	8704.81i	0.414651i	−0.978272	+	0.207325i	$0.933524\pi$
$762$	− 4207.01i	− 0.200005i
$763$	957.187	0.0454161
$764$	20941.4	0.991665
$765$	− 16251.0i	− 0.768048i
$766$	−3205.16	−0.151184
$767$	0	0
$768$	−11595.0	−0.544790
$769$	− 21915.9i	− 1.02771i	−0.857878	−	0.513853i	$-0.828218\pi$
$769$	− 21915.9i	− 1.02771i	0.857878	−	0.513853i	$-0.171782\pi$
$770$	−952.140	−0.0445620
$771$	−6168.63	−0.288143
$772$	15386.5i	0.717322i
$773$	− 23077.5i	− 1.07379i	−0.843649	−	0.536896i	$-0.819597\pi$
$773$	− 23077.5i	− 1.07379i	0.843649	−	0.536896i	$-0.180403\pi$
$774$	− 162.022i	− 0.00752422i
$775$	53054.0i	2.45904i
$776$	5263.08	0.243471
$777$	−86.0001	−0.00397070
$778$	3851.96i	0.177506i
$779$	18405.5	0.846528
$780$	0	0
$781$	3429.94	0.157148
$782$	− 4188.92i	− 0.191554i
$783$	−15891.5	−0.725309
$784$	18624.6	0.848426
$785$	7072.67i	0.321572i
$786$	3443.83i	0.156282i
$787$	16522.4i	0.748362i	0.927356	+	0.374181i	$0.122076\pi$
$787$	16522.4i	0.748362i	−0.927356	+	0.374181i	$0.877924\pi$
$788$	− 31356.4i	− 1.41754i
$789$	23247.2	1.04895
$790$	−1580.78	−0.0711918
$791$	− 6879.00i	− 0.309215i
$792$	−2086.15	−0.0935962
$793$	0	0
$794$	−4939.89	−0.220794
$795$	− 4439.30i	− 0.198045i
$796$	32996.1	1.46924
$797$	11719.4	0.520855	0.260427	−	0.965493i	$-0.416137\pi$
$797$	11719.4	0.520855	0.260427	+	0.965493i	$0.416137\pi$
$798$	− 709.974i	− 0.0314948i
$799$	21661.2i	0.959095i
$800$	15657.7i	0.691978i
$801$	9640.53i	0.425258i
$802$	−691.096	−0.0304282
$803$	2629.53	0.115559
$804$	12229.7i	0.536454i
$805$	13610.0	0.595886
$806$	0	0
$807$	9148.01	0.399040
$808$	− 2416.77i	− 0.105225i
$809$	−24096.0	−1.04718	−0.523592	−	0.851969i	$-0.675408\pi$
$809$	−24096.0	−1.04718	−0.523592	+	0.851969i	$0.675408\pi$
$810$	1455.26	0.0631267
$811$	16622.6i	0.719729i	0.933005	+	0.359864i	$0.117177\pi$
$811$	16622.6i	0.719729i	−0.933005	+	0.359864i	$0.882823\pi$
$812$	4530.42i	0.195796i
$813$	10448.7i	0.450739i
$814$	42.1933i	0.00181680i
$815$	−16764.1	−0.720516
$816$	14885.6	0.638603
$817$	2224.59i	0.0952613i
$818$	2962.05	0.126609
$819$	0	0
$820$	31668.7	1.34868
$821$	− 38005.5i	− 1.61559i	−0.589461	−	0.807797i	$-0.700660\pi$
$821$	− 38005.5i	− 1.61559i	0.589461	−	0.807797i	$-0.299340\pi$
$822$	1111.46	0.0471613
$823$	15859.5	0.671722	0.335861	−	0.941912i	$-0.390973\pi$
$823$	15859.5	0.671722	0.335861	+	0.941912i	$0.390973\pi$
$824$	− 4022.01i	− 0.170040i
$825$	15873.1i	0.669854i
$826$	− 694.155i	− 0.0292406i
$827$	− 12201.0i	− 0.513023i	−0.966541	−	0.256512i	$-0.917427\pi$
$827$	− 12201.0i	− 0.513023i	0.966541	−	0.256512i	$-0.0825732\pi$
$828$	14728.5	0.618177
$829$	5431.41	0.227552	0.113776	−	0.993506i	$-0.463705\pi$
$829$	5431.41	0.227552	0.113776	+	0.993506i	$0.463705\pi$
$830$	− 2624.62i	− 0.109761i
$831$	−14139.9	−0.590263
$832$	0	0
$833$	−21308.0	−0.886290
$834$	− 1097.88i	− 0.0455834i
$835$	−65540.3	−2.71630
$836$	14147.5	0.585288
$837$	− 41132.2i	− 1.69861i
$838$	− 4716.02i	− 0.194406i
$839$	7960.90i	0.327582i	0.986495	+	0.163791i	$0.0523722\pi$
$839$	7960.90i	0.327582i	−0.986495	+	0.163791i	$0.947628\pi$
$840$	− 2473.26i	− 0.101590i
$841$	−13005.6	−0.533255
$842$	3457.61	0.141517
$843$	− 33611.9i	− 1.37326i
$844$	10655.0	0.434550
$845$	0	0
$846$	1875.19	0.0762062
$847$	− 4504.10i	− 0.182719i
$848$	−4020.35	−0.162806
$849$	7840.40	0.316940
$850$	− 5726.56i	− 0.231082i
$851$	− 603.115i	− 0.0242944i
$852$	4400.59i	0.176950i
$853$	− 13576.7i	− 0.544969i	−0.962160	−	0.272485i	$-0.912155\pi$
$853$	− 13576.7i	− 0.544969i	0.962160	−	0.272485i	$-0.0878454\pi$
$854$	−1581.65	−0.0633756
$855$	19316.2	0.772631
$856$	− 3960.95i	− 0.158157i
$857$	31223.9	1.24456	0.622281	−	0.782794i	$-0.286206\pi$
$857$	31223.9	1.24456	0.622281	+	0.782794i	$0.286206\pi$
$858$	0	0
$859$	−11815.8	−0.469323	−0.234661	−	0.972077i	$-0.575398\pi$
$859$	−11815.8	−0.469323	−0.234661	+	0.972077i	$0.575398\pi$
$860$	3827.65i	0.151770i
$861$	−4564.22	−0.180660
$862$	−6175.36	−0.244006
$863$	− 1790.84i	− 0.0706384i	−0.999376	−	0.0353192i	$-0.988755\pi$
$863$	− 1790.84i	− 0.0706384i	0.999376	−	0.0353192i	$-0.0112448\pi$
$864$	− 12139.2i	− 0.477992i
$865$	25336.4i	0.995912i
$866$	817.328i	0.0320715i
$867$	1072.46	0.0420100
$868$	−11726.1	−0.458538
$869$	4539.87i	0.177220i
$870$	−3069.45	−0.119614
$871$	0	0
$872$	1219.86	0.0473734
$873$	− 10193.2i	− 0.395176i
$874$	4979.01	0.192698
$875$	6500.00	0.251132
$876$	3373.68i	0.130121i
$877$	43542.5i	1.67654i	0.545255	+	0.838270i	$0.316433\pi$
$877$	43542.5i	1.67654i	−0.545255	+	0.838270i	$0.683567\pi$
$878$	2698.42i	0.103721i
$879$	30489.7i	1.16996i
$880$	23728.2	0.908953
$881$	1020.04	0.0390080	0.0195040	−	0.999810i	$-0.493791\pi$
$881$	1020.04	0.0390080	0.0195040	+	0.999810i	$0.493791\pi$
$882$	1844.62i	0.0704214i
$883$	34781.9	1.32560	0.662800	−	0.748797i	$-0.269368\pi$
$883$	34781.9	1.32560	0.662800	+	0.748797i	$0.269368\pi$
$884$	0	0
$885$	−19101.6	−0.725531
$886$	6374.70i	0.241718i
$887$	49785.1	1.88458	0.942288	−	0.334802i	$-0.108670\pi$
$887$	49785.1	1.88458	0.942288	+	0.334802i	$0.108670\pi$
$888$	−109.600	−0.00414183
$889$	14162.3i	0.534295i
$890$	5607.49i	0.211195i
$891$	− 4179.40i	− 0.157144i
$892$	− 8272.08i	− 0.310504i
$893$	−25746.8	−0.964818
$894$	3191.41	0.119392
$895$	20797.5i	0.776743i
$896$	−4594.25	−0.171298
$897$	0	0
$898$	3088.36	0.114766
$899$	29463.9i	1.09308i
$900$	20134.9	0.745738
$901$	4599.60	0.170072
$902$	2239.29i	0.0826611i
$903$	− 551.656i	− 0.0203300i
$904$	− 8766.74i	− 0.322541i
$905$	− 20193.3i	− 0.741712i
$906$	2913.18	0.106826
$907$	−17389.9	−0.636627	−0.318314	−	0.947985i	$-0.603116\pi$
$907$	−17389.9	−0.636627	−0.318314	+	0.947985i	$0.603116\pi$
$908$	27047.6i	0.988553i
$909$	−4680.66	−0.170790
$910$	0	0
$911$	20419.5	0.742621	0.371311	−	0.928509i	$-0.378909\pi$
$911$	20419.5	0.742621	0.371311	+	0.928509i	$0.378909\pi$
$912$	17693.2i	0.642414i
$913$	−7537.71	−0.273233
$914$	6181.60	0.223708
$915$	43523.5i	1.57250i
$916$	18150.2i	0.654695i
$917$	− 11593.2i	− 0.417492i
$918$	4439.75i	0.159623i
$919$	−33231.8	−1.19283	−0.596417	−	0.802674i	$-0.703410\pi$
$919$	−33231.8	−1.19283	−0.596417	+	0.802674i	$0.703410\pi$
$920$	17344.8	0.621567
$921$	− 13303.4i	− 0.475964i
$922$	6335.43	0.226297
$923$	0	0
$924$	−3508.31	−0.124908
$925$	− 824.502i	− 0.0293075i
$926$	−6930.31	−0.245944
$927$	−7789.58	−0.275991
$928$	8695.59i	0.307594i
$929$	− 25222.8i	− 0.890780i	−0.895337	−	0.445390i	$-0.853065\pi$
$929$	− 25222.8i	− 0.890780i	0.895337	−	0.445390i	$-0.146935\pi$
$930$	− 7944.70i	− 0.280126i
$931$	− 25327.0i	− 0.891579i
$932$	29130.9	1.02383
$933$	12273.8	0.430683
$934$	− 6607.97i	− 0.231498i
$935$	−27146.9	−0.949519
$936$	0	0
$937$	−26979.4	−0.940639	−0.470319	−	0.882496i	$-0.655861\pi$
$937$	−26979.4	−0.940639	−0.470319	+	0.882496i	$0.655861\pi$
$938$	1013.64i	0.0352842i
$939$	1319.57	0.0458600
$940$	−44300.2	−1.53714
$941$	7641.67i	0.264730i	0.991201	+	0.132365i	$0.0422572\pi$
$941$	7641.67i	0.264730i	−0.991201	+	0.132365i	$0.957743\pi$
$942$	− 641.635i	− 0.0221928i
$943$	− 32008.7i	− 1.10535i
$944$	17299.0i	0.596434i
$945$	−14424.9	−0.496553
$946$	−270.653	−0.00930200
$947$	2869.32i	0.0984587i	0.998788	+	0.0492293i	$0.0156765\pi$
$947$	2869.32i	0.0984587i	−0.998788	+	0.0492293i	$0.984323\pi$
$948$	−5824.62	−0.199552
$949$	0	0
$950$	6806.67	0.232461
$951$	− 11228.8i	− 0.382881i
$952$	2562.56	0.0872407
$953$	−12313.6	−0.418548	−0.209274	−	0.977857i	$-0.567110\pi$
$953$	−12313.6	−0.418548	−0.209274	+	0.977857i	$0.567110\pi$
$954$	− 398.184i	− 0.0135133i
$955$	− 47762.6i	− 1.61839i
$956$	47193.8i	1.59661i
$957$	8815.22i	0.297759i
$958$	172.110	0.00580442
$959$	−3741.57	−0.125987
$960$	28848.0i	0.969861i
$961$	−46470.7	−1.55989
$962$	0	0
$963$	−7671.32	−0.256703
$964$	40397.1i	1.34969i
$965$	35093.2	1.17066
$966$	−1234.70	−0.0411241
$967$	− 17838.0i	− 0.593207i	−0.955001	−	0.296603i	$-0.904146\pi$
$967$	− 17838.0i	− 0.593207i	0.955001	−	0.296603i	$-0.0958540\pi$
$968$	− 5740.12i	− 0.190593i
$969$	− 20242.4i	− 0.671084i
$970$	− 5928.97i	− 0.196255i
$971$	41525.3	1.37241	0.686206	−	0.727408i	$-0.259275\pi$
$971$	41525.3	1.37241	0.686206	+	0.727408i	$0.259275\pi$
$972$	−26037.1	−0.859199
$973$	3695.86i	0.121772i
$974$	4164.32	0.136995
$975$	0	0
$976$	39416.1	1.29270
$977$	− 31654.4i	− 1.03655i	−0.855213	−	0.518277i	$-0.826574\pi$
$977$	− 31654.4i	− 1.03655i	0.855213	−	0.518277i	$-0.173426\pi$
$978$	1520.85	0.0497253
$979$	16104.3	0.525735
$980$	− 43578.0i	− 1.42046i
$981$	− 2362.55i	− 0.0768913i
$982$	− 830.342i	− 0.0269830i
$983$	39913.2i	1.29505i	0.762045	+	0.647525i	$0.224196\pi$
$983$	39913.2i	1.29505i	−0.762045	+	0.647525i	$0.775804\pi$
$984$	−5816.74	−0.188446
$985$	−71516.8	−2.31342
$986$	− 3180.28i	− 0.102719i
$987$	6384.72	0.205905
$988$	0	0
$989$	3868.74	0.124387
$990$	2350.09i	0.0754453i
$991$	−2700.94	−0.0865773	−0.0432887	−	0.999063i	$-0.513784\pi$
$991$	−2700.94	−0.0865773	−0.0432887	+	0.999063i	$0.513784\pi$
$992$	−22506.9	−0.720358
$993$	− 28352.6i	− 0.906085i
$994$	364.736i	0.0116386i
$995$	− 75256.6i	− 2.39778i
$996$	− 9670.83i	− 0.307663i
$997$	−9729.08	−0.309050	−0.154525	−	0.987989i	$-0.549385\pi$
$997$	−9729.08	−0.309050	−0.154525	+	0.987989i	$0.549385\pi$
$998$	−5862.08	−0.185933
$999$	639.228i	0.0202445i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.b.e.168.2		4
13.2	odd	12		13.4.c.b.9.1	yes	4
13.3	even	3		169.4.e.g.147.3		8
13.4	even	6		169.4.e.g.23.3		8
13.5	odd	4		169.4.a.f.1.2		2
13.6	odd	12		13.4.c.b.3.1	✓	4
13.7	odd	12		169.4.c.f.146.2		4
13.8	odd	4		169.4.a.j.1.1		2
13.9	even	3		169.4.e.g.23.2		8
13.10	even	6		169.4.e.g.147.2		8
13.11	odd	12		169.4.c.f.22.2		4
13.12	even	2	inner	169.4.b.e.168.3		4
39.2	even	12		117.4.g.d.100.2		4
39.5	even	4		1521.4.a.t.1.1		2
39.8	even	4		1521.4.a.l.1.2		2
39.32	even	12		117.4.g.d.55.2		4
52.15	even	12		208.4.i.e.113.1		4
52.19	even	12		208.4.i.e.81.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.b.3.1	✓	4	13.6	odd	12
13.4.c.b.9.1	yes	4	13.2	odd	12
117.4.g.d.55.2		4	39.32	even	12
117.4.g.d.100.2		4	39.2	even	12
169.4.a.f.1.2		2	13.5	odd	4
169.4.a.j.1.1		2	13.8	odd	4
169.4.b.e.168.2		4	1.1	even	1	trivial
169.4.b.e.168.3		4	13.12	even	2	inner
169.4.c.f.22.2		4	13.11	odd	12
169.4.c.f.146.2		4	13.7	odd	12
169.4.e.g.23.2		8	13.9	even	3
169.4.e.g.23.3		8	13.4	even	6
169.4.e.g.147.2		8	13.10	even	6
169.4.e.g.147.3		8	13.3	even	3
208.4.i.e.81.1		4	52.19	even	12
208.4.i.e.113.1		4	52.15	even	12
1521.4.a.l.1.2		2	39.8	even	4
1521.4.a.t.1.1		2	39.5	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.438447i q^{2} -3.68466 q^{3} +7.80776 q^{4} -17.8078i q^{5} +1.61553i q^{6} -5.43845i q^{7} -6.93087i q^{8} -13.4233 q^{9} +O(q^{10})\)	\(q-0.438447i q^{2} -3.68466 q^{3} +7.80776 q^{4} -17.8078i q^{5} +1.61553i q^{6} -5.43845i q^{7} -6.93087i q^{8} -13.4233 q^{9} -7.80776 q^{10} +22.4233i q^{11} -28.7689 q^{12} -2.38447 q^{14} +65.6155i q^{15} +59.4233 q^{16} -67.9848 q^{17} +5.88540i q^{18} -80.8078i q^{19} -139.039i q^{20} +20.0388i q^{21} +9.83143 q^{22} -140.531 q^{23} +25.5379i q^{24} -192.116 q^{25} +148.946 q^{27} -42.4621i q^{28} -106.693 q^{29} +28.7689 q^{30} -276.155i q^{31} -81.5009i q^{32} -82.6222i q^{33} +29.8078i q^{34} -96.8466 q^{35} -104.806 q^{36} +4.29168i q^{37} -35.4299 q^{38} -123.423 q^{40} +227.769i q^{41} +8.78596 q^{42} -27.5294 q^{43} +175.076i q^{44} +239.039i q^{45} +61.6155i q^{46} -318.617i q^{47} -218.955 q^{48} +313.423 q^{49} +84.2329i q^{50} +250.501 q^{51} -67.6562 q^{53} -65.3050i q^{54} +399.309 q^{55} -37.6932 q^{56} +297.749i q^{57} +46.7793i q^{58} +291.115i q^{59} +512.311i q^{60} +663.311 q^{61} -121.080 q^{62} +73.0019i q^{63} +439.652 q^{64} -36.2255 q^{66} -425.101i q^{67} -530.810 q^{68} +517.810 q^{69} +42.4621i q^{70} -152.963i q^{71} +93.0351i q^{72} -117.268i q^{73} +1.88167 q^{74} +707.884 q^{75} -630.928i q^{76} +121.948 q^{77} +202.462 q^{79} -1058.20i q^{80} -186.386 q^{81} +99.8647 q^{82} +336.155i q^{83} +156.458i q^{84} +1210.66i q^{85} +12.0702i q^{86} +393.128 q^{87} +155.413 q^{88} -718.194i q^{89} +104.806 q^{90} -1097.23 q^{92} +1017.54i q^{93} -139.697 q^{94} -1439.01 q^{95} +300.303i q^{96} +759.368i q^{97} -137.420i q^{98} -300.994i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9}+O(q^{10}) \) 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9	\( 4 q + 10 q^{3} - 10 q^{4} + 70 q^{9} + 10 q^{10} - 280 q^{12} - 92 q^{14} + 114 q^{16} - 140 q^{17} - 340 q^{22} - 290 q^{23} - 150 q^{25} + 670 q^{27} + 68 q^{29} + 280 q^{30} - 140 q^{35} - 1450 q^{36} - 620 q^{38} - 370 q^{40} - 740 q^{42} - 910 q^{43} - 480 q^{48} + 1130 q^{49} + 466 q^{51} + 1090 q^{53} + 1020 q^{55} + 344 q^{56} + 1004 q^{61} + 1000 q^{62} + 2542 q^{64} - 3196 q^{66} - 1010 q^{68} + 958 q^{69} + 1698 q^{74} + 3450 q^{75} - 510 q^{77} + 480 q^{79} + 244 q^{81} + 3030 q^{82} + 3230 q^{87} + 2040 q^{88} + 1450 q^{90} - 2080 q^{92} + 2080 q^{94} - 2540 q^{95}+O(q^{100}) \) 4 * q + 10 * q^3 - 10 * q^4 + 70 * q^9 + 10 * q^10 - 280 * q^12 - 92 * q^14 + 114 * q^16 - 140 * q^17 - 340 * q^22 - 290 * q^23 - 150 * q^25 + 670 * q^27 + 68 * q^29 + 280 * q^30 - 140 * q^35 - 1450 * q^36 - 620 * q^38 - 370 * q^40 - 740 * q^42 - 910 * q^43 - 480 * q^48 + 1130 * q^49 + 466 * q^51 + 1090 * q^53 + 1020 * q^55 + 344 * q^56 + 1004 * q^61 + 1000 * q^62 + 2542 * q^64 - 3196 * q^66 - 1010 * q^68 + 958 * q^69 + 1698 * q^74 + 3450 * q^75 - 510 * q^77 + 480 * q^79 + 244 * q^81 + 3030 * q^82 + 3230 * q^87 + 2040 * q^88 + 1450 * q^90 - 2080 * q^92 + 2080 * q^94 - 2540 * q^95

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